damping characteristics of reinforced … · prestressed normal- and high-strength concrete beams...

DAMPING CHARACTERISTICS OF REINFORCED AND

PRESTRESSED NORMAL- AND HIGH-STRENGTH

CONCRETE BEAMS

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy

by

Angela Salzmann BEng (Hons1), J.P.

from

School of Engineering

Faculty of Engineering and Information Technology

GRIFFITH UNIVERSITY

GOLD COAST CAMPUS

November 2002

To My Parents

Declaration i

Declaration

This work has not been previously submitted for a degree or diploma in any university.

To the best of my knowledge and belief, the thesis contains no material previously

published or written by another person except where due reference is made in the thesis

itself.

________________________

Angela Salzmann

November 2002

Damping Characteristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams

Acknowledgements ii

Acknowledgements

The research from which this thesis has been composed was undertaken at the School of

Engineering, Griffith University Gold Coast Campus under the supervision of Dr. Sam

Fragomeni and Professor Yew-Chaye Loo. The author is greatly indebted to Dr.

Fragomeni and Professor Loo, whose continued support, encouragement, inspiration

and technical contributions helped to guide and shape the research effort. In particular,

the provision of significant guidance regarding the scope of the experimental work

provided invaluable assistance.

A special thankyou is given to all the technical staff and in particular to Charles Allport

without whose help, the experimental work could not have been possible. Grateful

thanks are also extended to the administrative staff of the School who provided constant

encouragement and to many final year students who enthusiastically assisted in the

laboratory testing tasks.

The author also wishes to thank the Australian Government Australian Postgraduate

Award (APA) Scholarship Scheme and the School of Engineering for providing the

financial assistance which allowed the research to be completed in this form.

Finally, her deep heartfelt thanks goes to her partner, Peter, and her parents, Gus and

Fiona and her sister Monique for their constant encouragement, understanding, financial

assistance, and belief. The completion of this research is but a small gift for their

efforts and great expectations.


List of Publications iii

List of Publications

Salzmann, A., Fragomeni, S. and Loo, Y.C. (2002a) Damping behaviour of reinforced

concrete beams – Review and new developments, 17th Australasian Conference

on the Mechanics of Structures and Materials, 12-14 June 2002, Gold Coast,

Australia.

Salzmann, A., Fragomeni, S. and Loo, Y.C. (2002b) The damping analysis of

experimental concrete beams under free-vibration, Advances in Structural

Engineering – An International Journal, Accepted for Publication.

Salzmann, A., Fragomeni, S. and Loo, Y.C. (2002c) Estimation of the free-vibration

damping characteristics of untested reinforced concrete beams, Electronic Journal

of Structural Engineering, Submitted for Publication.

Salzmann, A., Fragomeni, S. and Loo, Y.C. (2001a) Verification of damping formulae

using experimental results from full-scale concrete beams reinforced with 500

MPA steel, The Australian Structural Engineering Conference, Surfers Paradise

Marriott Resort, Gold Coast, Australia, 29 April – 2 May 2001, pp. 95-102.

Salzmann, A., Fragomeni, S. and Loo, Y.C. (2001b) Investigation of damping in high-

strength prestressed concrete beams, The Eighth East Asia-Pacific Conference on

Structural Engineering & Construction, 5-7 December 2001, Singapore.

Salzmann, A. and Fragomeni, S. (2000) Experimental determination of damping from

full-scale reinforced and prestressed concrete beams, Civil Engineering

Challenges in the 21st Century, Queensland Civil Engineering Postgraduate

Conference, December 12-13, 2000, Physical Infrastructure Centre Queensland

University of Technology, pp 75-84.


Synopsis iv

Synopsis

In the last few decades there has been a significant increase in the design strength and

performance of different building materials. In particular, new methods, materials

and admixtures for the production of concrete have allowed for strengths as high as

100 MPa to be readily available. In addition, the standard manufactured yield

strength of reinforcing steel in Australia has increased from 400 MPa to 500 MPa.

A perceived design advantage of higher-strength materials is that structural elements

can have longer spans and be more slender than previously possible. An emerging

problem with slender concrete members is that they can be more vulnerable to loading

induced vibration. The damping capacity is an inherent fundamental quantity of all

structural concrete members that affects their vibrational response. It is defined as the

rate at which a structural member can dissipate the vibrational energy imparted to it.

Generally damping capacity measurements, to indicate the integrity of structural

members, are taken once the structure is in service. This type of non-destructive testing

has been the subject of much research. The published non-destructive testing research

on damping capacity is conflicting and a unified method to describe the effect of

damage on damping capacity has not yet been proposed.

Significantly, there is not one method in the published literature or national design

codes, including the Australian Standard AS 3600-2001, available to predict the

damping capacity of concrete beam members at the design stage. Further, little

research has implemented full-scale testing with a view to developing damping

capacity design equations, which is the primary focus of this thesis.

To examine the full-range damping behaviour of concrete beams, two categories of

testing were proposed. The categories are the ‘untested’ and ‘tested’ beam states.

These beam states have not been separately investigated in previous work and are

considered a major shortcoming of previous research on the damping behaviour of

concrete beams.


Synopsis v An extensive experimental programme was undertaken to obtain residual deflection and

damping capacity data for thirty-one reinforced and ten prestressed concrete beams.

The concrete beams had compressive strengths ranging between 23.1 MPa and 90.7

MPa, reinforcement with yield strengths of 400 MPa or 500 MPa, and tensile

reinforcement ratios between 0.76% and 2.90%. The full- and half-scale beams tested

had lengths of 6.0 m and 2.4 m, respectively. The testing regime consisted of a series of

on-off load increments, increasing until failure, designed to induce residual deflections

with increasing amounts of internal damage at which damping capacity (logarithmic

decrement) was measured.

The inconsistencies that were found between the experimental damping capacity of the

beams and previous research prompted an initial investigation into the data obtained.

It was found that the discrepancies were due to the various interpretations of the

method used to extract damping capacity from the free-vibration decay curve.

Therefore, a logarithmic decrement calculation method was proposed to ensure

consistency and accuracy of the extracted damping capacity data to be used in the

subsequent analytical research phase.

The experimental test data confirmed that the ‘untested’ damping capacity of reinforced

concrete beams is dependent upon the beam reinforcement ratio and distribution. This

quantity was termed the total longitudinal reinforcement distribution. For the

prestressed concrete beams, the ‘untested’ damping capacity was shown to be

proportional to the product of the prestressing force and prestressing eccentricity.

Separate ‘untested’ damping capacity equations for reinforced and prestressed concrete

beams were developed to reflect these quantities.

To account for the variation in damping capacity due to damage in ‘tested’ beams, a

residual deflection mechanism was utilised. The proposed residual deflection

mechanism estimates the magnitude of permanent deformation in the beam and

attempts to overcome traditional difficulties in calculating the damping capacity during

low loading levels. Residual deflection equations, based on the instantaneous

deflection data for the current experimental programme, were proposed for both the

reinforced and prestressed concrete beams, which in turn were utilised with the

proposed ‘untested’ damping equation to calculate the total damping capacity.


Synopsis vi

The proposed ‘untested’ damping, residual deflection and total damping capacity

equations were compared to published test data and an additional series of test beams.

These verification investigations have shown that the proposed equations are reliable

and applicable for a range of beam designs, test setups, constituent materials and

loading regimes.


Table of Contents vii

Table of Contents

Declaration ....................................................................................................... i Acknowledgements .......................................................................................... ii List of Publications .......................................................................................... iii Synopsis ............................................................................................................ iv Table of Contents ............................................................................................ vii List of Figures .................................................................................................. xi List of Tables ................................................................................................... xv List of Plates .................................................................................................... xvi Notation ............................................................................................................ xvii

CHAPTER

1. Introduction ..................................................................................................... 1-1 1.1 General Remarks ................................................................................... 1-1 1.2 Research Objectives .............................................................................. 1-2 1.3 Research Methodology ......................................................................... 1-2 1.4 Layout of the Thesis .............................................................................. 1-3 1.5 Summary ............................................................................................... 1-4

2. Damping in Concrete ....................................................................................... 2-1 2.1 General Remarks ................................................................................... 2-1 2.2 Undamped Systems ............................................................................... 2-2 2.2.1 Single-DOF and multi-DOF structures .................................. 2-2 2.3 Damped Systems ................................................................................... 2-4 2.3.1 The idealized MDOF system ................................................. 2-4 2.3.2 Viscous damping .................................................................... 2-5 2.3.3 Coulomb damping .................................................................. 2-11 2.3.4 Hysteretic damping ................................................................ 2-12 2.3.5 Equivalent viscous damping .................................................. 2-14 2.4 Experimental Determination of Damping ............................................. 2-15 2.4.1 Free-vibration damping .......................................................... 2-15 2.4.2 Forced excitation damping ..................................................... 2-16 2.4.2.1 Half-power (bandwidth) method ............................... 2-17 2.4.2.2 Resonant amplification ............................................. 2-17 2.4.2.3 Energy loss per cycle ................................................ 2-18 2.5 Literature Review of Damping in Concrete ............................................ 2-21 2.5.1 Material damping ..................................................................... 2-21 2.5.2 Member damping ..................................................................... 2-24 2.5.3 Structural damping................................................................... 2-32 2.6 Summary ............................................................................................... 2-34

3. Theoretical Consideration ............................................................................. 3-1 3.1 General Remarks ................................................................................... 3-1 3.2 Pilot Study............................................................................................... 3-1 3.2.1 Verifying the accuracy of logdec ........................................... 3-2


Table of Contents viii

3.2.2 Logdec versus stage of testing ............................................... 3-2 3.2.3 Damage mechanisms in concrete beams ................................ 3-3 3.2.4 Residual deflection ................................................................. 3-4 3.3 The Total Damping Capacity Equation .................................................. 3-4 3.3.1 ‘Untested’ beams .................................................................... 3-5 3.3.2 ‘Tested’ beams ....................................................................... 3-6 3.4 Summary ............................................................................................... 3-7 4. Experimental Programme ............................................................................. 4-1 4.1 General Remarks ................................................................................... 4-1 4.2 Design of Beam Test Specimens ........................................................... 4-1 4.2.1 Geometrical and mechanical details ........................................ 4-2 4.2.2 Primary test variables .............................................................. 4-2 4.3 Materials.................................................................................................. 4-2 4.3.1 Concrete .................................................................................. 4-2 4.3.2 Reinforcement ......................................................................... 4-9 4.4 Fabrication .............................................................................................. 4-13 4.4.1 Reinforced concrete beams .................................................... 4-13 4.4.2 Prestressed concrete beams ..................................................... 4-13 4.4.3 Curing ...................................................................................... 4-15 4.5 Test Set-Up ........................................................................................... 4-15 4.5.1 Beam support system ............................................................... 4-15 4.5.2 Loading beam width (LBW) ..................................................... 4-15 4.5.3 Hammer excitation position (HEP).......................................... 4-15 4.5.4 Hammer weight (HW) .............................................................. 4-18 4.6 Test Procedures ..................................................................................... 4-18 4.7 Instrumentation ..................................................................................... 4-18 4.7.1 Damping ................................................................................. 4-19 4.7.2 Crack width ............................................................................ 4-21 4.7.3 Crack Patterns ......................................................................... 4-22 4.7.4 Deflection ............................................................................... 4-22 4.8 Summary ............................................................................................... 4-23

5. A Method for Extracting Damping Capacity ............................................... 5-1 5.1 General Remarks ................................................................................... 5-1 5.2 Experimental Techniques ...................................................................... 5-1 5.3 Analytical Technique for Calculating Logdec ...................................... 5-3 5.4 Applying the TLT and DCM Techniques .............................................. 5-4 5.4.1 Differences between the TLT and DCM output ...................... 5-6 5.4.2 Proposed rules for calculating logdec .................................... 5-7 5.5 Effect of Experimental Test Variables .................................................. 5-10 5.5.1 Hammer weight (HW) ............................................................ 5-10 5.5.2 Hammer excitation position (HEP) ........................................ 5-10 5.6 Summary ............................................................................................... 5-11

6. Damping Prediction in ‘Untested’ Concrete Beams .................................... 6-1 6.1 General Remarks ................................................................................... 6-1 6.2 Damping in ‘Untested’ Reinforced Concrete Beams ............................ 6-1 6.2.1 Historical review .................................................................... 6-1 6.2.2 Experimental effect of fcm and fsy ........................................... 6-3 6.2.3 Proposed damping equation ................................................... 6-3


Table of Contents ix

6.3 Damping in ‘Untested’ Prestressed Concrete Beams ............................. 6-7 6.3.1 Historical review .................................................................... 6-7 6.3.2 Experimental observations ..................................................... 6-8 6.3.3 Hop’s prestressed equation .................................................... 6-8 6.3.4 Proposed damping equation ..................................................... 6-10 6.4 Verification ............................................................................................ 6-11 6.4.1 Original beam data .................................................................. 6-11 6.4.2 F-Series Beams ....................................................................... 6-11 6.4.3 Neild’s Beam ........................................................................... 6-13 6.5 Summary ............................................................................................... 6-14

7. Residual Deflection Mechanisms in Concrete Beams ................................. 7-1 7.1 General Remarks ................................................................................... 7-1 7.2 Residual Deflection in Reinforced Concrete Beams .............................. 7-1 7.2.1 Effect of fsy .............................................................................. 7-1 7.2.2 Effect of fcm .............................................................................. 7-2 7.2.3 Effect of ρt................................................................................ 7-2 7.2.4 Effect of loading conditions ..................................................... 7-2 7.2.5 Summary of effects ................................................................. 7-2 7.2.6 The proposed equation ............................................................ 7-7 7.3 Residual Deflection in Prestressed Concrete Beams ............................. 7-7 7.3.1 Effect of fcm and e ..................................................................... 7-9 7.3.2 Effect of H ............................................................................... 7-9 7.3.3 Effect of H and e ...................................................................... 7-10 7.3.4 Summary of effects ................................................................. 7-10 7.3.5 The proposed equation ............................................................. 7-10 7.4 Verification ............................................................................................ 7-14 7.4.1 Original beam data .................................................................. 7-14 7.4.2 F-Series Beams ....................................................................... 7-16 7.4.3 James’ Beams........................................................................... 7-17 7.5 Summary ............................................................................................... 7-20

8. Damping in ‘Tested’ Concrete Beams............................................................. 8-1 8.1 General Remarks ................................................................................... 8-1 8.2 Development of Total Damping Equations ........................................... 8-1 8.3 Verification ........................................................................................... 8-7 8.3.1 F-Series Beams ....................................................................... 8-7 8.3.2 Chowdhury’s beams ............................................................... 8-8 8.4 Advantages of Proposed Residual Deflection Equations ...................... 8-10 8.5 Summary ............................................................................................... 8-12

9. Conclusions and Recommendations .............................................................. 9-1 9.1 General Remarks ................................................................................... 9-1 9.2 Research Objectives and Outcomes ...................................................... 9-1 9.2.1 Summary of test results ......................................................... 9-2 9.2.2 Verification of the proposed methods ................................... 9-4 9.3 Recommendations and Scope for Future Research ............................... 9-4 9.4 Closure ................................................................................................ 9-5

References .................................................................................................................Re-1


Table of Contents x

Bibliography .............................................................................................................Bi-1 Appendix A Literature Review Summary Tabulations ........................................... A-1 Appendix B RC and PSC Beam Calculations ......................................................... B-1 Appendix C Beam Crack Pattern Photographs ....................................................... C-1 Appendix D Logdec Comparative Graphs .............................................................. D-1 Appendix E Damping Tabulations ......................................................................... E-1 Appendix F Serviceability Curves ......................................................................... F-1


List of Figures xi

List of Figures

Page

Figure 2.1 The undamped free-vibration response (Clough and Penzien, 1975)

2-3

Figure 2.2 Freely vibrating body in the x,y,z direction (Fertis, 1995) 2-3Figure 2.3 Example of a structure modelled as a SDOF system (Fertis,

1995) 2-4

Figure 2.4 Beam system (Fertis, 1995): (a) MDOF beam (b) Idealized SDOF mass-spring system

2-3

Figure 2.5 Decay of free-vibration under the assumption of viscous damping (Newland, 1989): (a) Mechanical model (b) Decay curve characteristics

2-6

Figure 2.6 Free-vibration response of critical and overdamped systems (Clough and Penzien, 1975)

2-8

Figure 2.7 Free-vibration response of an underdamped system (Fertis, 1995)

2-9

Figure 2.8 Decay of free-vibration under the assumption of Coulomb damping (Newland, 1989): (a) Mechanical model (b) Decay curve characteristics

2-11

Figure 2.9 Decay of vibration under the assumption of hystetic damping (Newland, 1989): (a) Mechanical model (b) Decay curve characteristics

2-13

Figure 2.10 Force-displacement Hysteresis loop (Fertis, 1995) 2-13Figure 2.11 Frequency response curve for moderately damped system

(Clough and Penzien, 1975) 2-18

Figure 2.12 Actual and equivalent damping energy per cycle (Clough and Penzien, 1975)

2-19

Figure 2.13 Elastic stiffness and strain energy (Clough and Penzien, 1975)

2-20

Figure 2.14 Damping for the (Dieterle and Bachman, 1981): (a) Uncracked (b) Cracked beam sections

2-27

Figure 2.15 Damping ratio as a function of the relative steel stress (Dieterle and Bachman, 1981)

2-27

Figure 2.16 SDOF RC cantilever beam element during one loading cycle (Flesch, 1981)

2-28

Figure 2.17 Damping in beams without load as a function of the maximum load (Wang et al., 1998)

2-30

Figure 3.1 Schematic residual load-deflection curve for concrete beams 3-5


List of Figures xii Figure 4.1 Geometric detailing for B-, CS, and F-Series test beams 4-3Figure 4.2 Geometric detailing for PS-Series test beams 4-3Figure 4.3 B-Series – Primary test variables 4-5Figure 4.4 PS-Series – Primary test variables 4-6Figure 4.5 CS-Series– Primary test variables 4-7Figure 4.6 F-Series – Primary test variables 4-8Figure 4.7 Stress-strain curve for 400 MPa reinforcing steel (BHP

Laboratory, Brisbane, Australia) 4-11

Figure 4.8 Stress-strain curve for 500 MPa reinforcing steel (BHP Laboratory, Brisbane, Australia)

4-12

Figure 4.9 Stress-strain curve for prestressing tendons (BHP Laboratory, Brisbane, Australia)

4-12

Figure 4.10 Rig used for prestressing the tendons 4-14Figure 4.11 Diagram of beam test set-up 4-16Figure 4.12 Location of beam testing equipment 4-19Figure 4.13 Impact energy frequency spectrum induced by hammer

excitation (Døssing, 1988b) 4-20

Figure 5.1 Experimental beams and HEP test variables 5-2Figure 5.2 Analytical algorithm used by the DCM 5-4Figure 5.3 Definition of data lengths using the TLT and DCM 5-5Figure 5.4 Calculation of logdec (TLT) using cycle number (n) 5-6Figure 5.5 Calculation of logdec (DCM) using NDP 5-6Figure 5.6 Variation of logdec (TLT) with n 5-8Figure 5.7 Plot of natural logarithm of An versus cycle number 5-8Figure 5.8 Example calculation of:

(a) Peak ratio (b) “Optimal peak ratio” curves

5-9

Figure 5.9 Effect of HW on logdec (TLT) 5-10Figure 5.10 Effect of HEP on logdec (TLT) 5-11 Figure 6.1 Classification of historical ‘untested’ damping research 6-2Figure 6.2 Variation of logdec with steel yield strength – B-Series

Beams 6-4

Figure 6.3 Variation of logdec with concrete compressive strength (a) B-Series (b) CS-Series beams

6-5

Figure 6.4 Dependence of ‘untested’ logdec on LRD in RC beams: (a) Separate trendlines (b) Unified prediction equation

6-6

Figure 6.5 Prestressing force versus ‘untested’ logdec for PS-Series beams

6-9

Figure 6.6 Prestressing eccentricity versus ‘untested’ logdec for PS-Series beams

6-9

Figure 6.7 ‘Untested’ logdec versus logdec prediction using Hop (1991) 6-10Figure 6.8 ‘Untested’ logdec versus initial prestress in beam 6-11Figure 6.9 Comparison between experimental logdec and logdec

calculated by Equation 6.2 and 6.3 6-12

Figure 6.10 Observed versus predicted ‘untested’ logdec for the F-Series beams

6-12


List of Figures xiii Figure 6.11 Details of test beams and testing arrangement of Neild

(2001) 6-13

Figure 6.12 ‘Optimal peak ratio’ analysis of Neild’s (2001) free-decay curve

6-14

Figure 7.1 Effect of reinforcement yield strength for B-Series beams on

residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment

7-3

Figure 7.2 Effect of concrete compressive strength for B-Series beams on residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment

7-4

Figure 7.3 Effect of tensile reinforcement ratio for B-Series beams on residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment

7-5

Figure 7.4 Effect of LBW for CS-Series beams on residual deflection versus: (a) Instantaneous deflection (b) Normalised bending moment

7-6

Figure 7.5 Effect of reinforcement ratio on the instantaneous versus residual deflection relationship: (a) B-Series (b) CS-Series

7-8

Figure 7.6 Selection of curve coefficient, αrc for the calculation of residual deflection

7-9

Figure 7.7 Residual deflection for PS3, PS6, PS8 and PS9 versus: (a) and (b) Concrete compressive strength and prestress eccentricity (c) Normalised mid-span bending moment

7-11

Figure 7.8 Residual deflection for PS9 and PS10 versus: (a) Prestressing force (b) Normalised mid-span bending moment

7-12

Figure 7.9 Residual deflection for PS5 and PS6 versus: (a) Prestressing force and prestressing eccentricity (b) Normalised mid-span bending moment

7-13

Figure 7.10 Correlation between instantaneous and residual deflection for PSC beams

7-14

Figure 7.11 Experimental versus calculated residual deflection for: (a) B-Series (b) CS-Series (c) PS-Series test beams

7-15

Figure 7.12 Experimental versus calculated residual deflection for F-Series test beams

7-16

Figure 7.13 Experimental versus calculated residual deflection for James’ beams

7-19

Figure 8.1 Logdec versus residual deflection for B-Series:

(a) BI-, BI-3 and BII-4 (b) BI-5 to BII-8

8-2


List of Figures xiv

(c) BI-9 to BII-12 Figure 8.2 Logdec versus residual deflection for CS-Series:

(a) CS1 to CS3 (b) CS4 to CS6 (c) CS7 to CS9

8-3

Figure 8.3 Logdec versus residual deflection for PS-Series: (a) PS3 to PS6 (b) PS7 to PS10

8-4

Figure 8.4 Dependence of D-R slope on concrete compressive strength: (a) B- and PS-Series beams (b) CS-Series beams

8-6

Figure 8.5 Calculated versus experimental logdec – F-Series beams 8-7Figure 8.6 Details of test beams and testing arrangement of Chowdhury

(1999) 8-8

Figure 8.7 Chowdhury’s experimental versus calculated logdec (Equation 8.2)

8-11


List of Tables xv

List of Tables

Page

Table 4.1 Geometrical and reinforcement details – RC beams 4-4Table 4.2 Geometrical and reinforcement details – PSC beams 4-4Table 4.3 Concrete technical data – Materials (CSR Construction

Materials) 4-9

Table 4.4 Concrete technical data – Mix design (CSR Construction Materials)

4-9

Table 4.5 Details of test beams - Concrete 4-10Table 4.6 Technical design data for 400 and 500 MPa reinforcing steel

(Patrick, 1999) 4-11

Table 4.7 Details of test beams – Reinforcing bars 4-13Table 4.8 Loading beam width specifications 4-17Table 4.9 Specifications for ICP® impulse-force hammer (PCB ®

Piezotronics, 1992) 4-20

Table 6.1 Experimental ‘untested’ damping data – RC beams 6-4Table 6.2 Experimental prestressed ‘untested’ damping data 6-8 Table 7.1 Details of James’ (1997) reinforced concrete box beams 7-17Table 7.2 Deflection data for beam 5 (James, 1997) 7-17Table 7.3 Deflection data for beam 7 (James, 1997) 7-18Table 7.4 Deflection data for beam 16 (James, 1997) 7-18Table 7.5 Deflection data for beam 17 (James, 1997) 7-18Table 7.6 Deflection data for beam 18 (James, 1997) 7-19 Table 8.1 Details of Chowdhury’s (1997) reinforced concrete box beams 8-8Table 8.2 Deflection versus damping data for beam 5 (James, 1997) 8-9Table 8.3 Deflection versus damping data for beam 7 (James, 1997) 8-9Table 8.4 Deflection versus damping data for beam 16 (James, 1997) 8-9Table 8.5 Deflection versus damping data for beam 17 (James, 1997) 8-10Table 8.6 Deflection versus damping data for beam 18 (James, 1997) 8-10


List of Plates xvi

List of Plates

Page

Plate 4.1 Photograph of beam test set-up 4-16Plate 4.2 Beam support system:

(a) Roller (b) Knife supports

4-17

Plate 4.3 Test set-up of oscilloscope during experimentation 4-22Plate 4.4 Crack width microscope 4-23


Notation xvii

Notation

a = Shear span during testing (mm)

A0 = Initial displacement amplitude, or zero-frequency displacement

A1,2,3 = Displacement amplitude at the first, second and third cycle, respectively

A = Displacement amplitude, or displacement amplitude under forced-

vibration, or Proportionality coefficient (Dieterle and Bachmann, 1981)

Ac = Area of concrete in the tensile zone (mm2)

Agt = Uniform elongation of reinforcement sample (elongation at max. stress)

Amax = Maximum amplitude on the harmonic-response curve

An = Displacement amplitude after n number of cycles

Ap = Area of steel in the tensile zone (mm2)

Ast = Area of tension reinforcement (mm2)

Asc = Area of compression reinforcement (mm2)

b = Width of rectangular beam section (mm)

C = Viscous damping matrix in linear equation of motion

c = Viscous damping coefficient of a system, or concrete cover (mm)

ccr = Critical damping coefficient

co = Dimensionless material constant for hysteresis damping

ce = Equivalent viscous damping coefficient (considering hysteresis damping)

ceq = Equivalent viscous damping coefficient

d = Effective depth to the centroid of the tensile steel of a rectangular beam

section (mm), or specific damping coefficient (MPa) (Dieterle and

Bachmann, 1981)

D = Total depth of the rectangular beam section (mm)

Dm = Dynamic magnification factor evaluated at maximum amplitude (Amax)

e = Eccentricity of the prestressing tendons (mm)

Ec = Young’s modulus of elasticity of concrete (MPa)

ED = Dynamic modulus of elasticity of concrete (MPa)

Em = Young’s modulus of elasticity of mortar (MPa)

Ep = Dynamic modulus of elasticity (MPa) (Cole and Spooner, 1968)

Es = Young’s modulus of elasticity of steel (GPa)

f = Vibration frequency defined by the number of cycles per time unit, in Hz

(= 1/τ)


Notation xviii f1 = Fundamental natural frequency of vibration (Hz)

f’c = Characteristic compressive strength of concrete at 28 days (MPa)

f’cf = Flexural stress in tension (MPa)

fcm = Concrete compressive strength on the day of testing (MPa)

fD = Damping force developed during energy loss per cycle test

fD,max = Maximum damping force developed during energy loss per cycle test

fs = Stress in the steel (MPa), or static force applied during energy loss per

cycle forced vibration test

fs,max = Maximum static force applied during energy loss per cycle forced

vibration test

fsy = Nominal reinforcement yield strength (MPa)

F = Resisting force developed in a body due to hysteresis damping

Ff = Frictional damping force produced between a vibrating mass and surface

during Coulomb damping

H = Prestressing force in the member (kN)

I = Moment of inertia of a section (mm4)

Icr = Fully-cracked moment of inertia of the section (mm4)

Ief = Effective moment of inertia of a section (mm4)

Ig = Gross moment of inertia of the uncracked section (mm4)

k = Stiffness (also termed the spring force)

ku = Compressive stress block parameter defining the effective depth of the

member

K = Stiffness matrix in linear equation of motion, stiffness (Flesch, 1981)

L = Total beam length (mm)

m = Lumped mass of structure in an idealized system

M = Mass matrix in linear equation of motion, or age of beam specimen

(Cole, 1966)

Mcr = Bending moment at which first cracking is observed (kNm)

Ms = Service bending moment (kNm)

Mu = Ultimate bending moment capacity of the beam (kNm)

n = Cycle number from a free-vibration decay record

P = General load applied to the structure

0P = Maximum force amplitude at resonant frequency (Flesch, 1981)

Pcr = Load causing first cracking in the member (kN)


Notation xix Ps = Service load (kN)

Py = Load causing yielding of the tensile steel (kN)

Re = Yield strength of reinforcing steel (MPa) (Patrick, 1999)

Rm = Tensile strength of reinforcing steel (MPa) (Patrick, 1999)

s = Spacing of the reinforcing bars (mm)

st = Spacing of the tension reinforcing bars (mm)

sc = Spacing of the compression reinforcing bars (mm)

S = Represents a surface

t = Time (sec)

U = Steel peak stress on stress-strain curve (MPa)

vo = Initial velocity given to a mass

vmax = Maximum displacement during forced vibration

v& max = Maximum velocity during forced vibration

V = Absolute volume fraction of coarse aggregates (Swamy and Rigby, 1971)

wD = Area under the force-displacement curve during energy loss per cycle

forced vibration test

wi = Average instantaneous crack width (mm)

ws = Area under the static-force-displacement during energy loss per cycle

forced vibration test

Wr = Average residual crack width (mm)

x,y,z = Generalised response co-ordinates defining displacement

X = Evaporable water content (Cole, 1966)

0y = Initial displacement given to a body

0y& = Initial velocity given to a body

0y&& = Initial acceleration given to a body

yt = Distance between the neutral axis and the extreme fibres in tension of the

uncracked section (mm)

Y = Steel off-yield stress on stress-strain curve (MPa)

y& = Velocity of motion as a function of time

y&& = Acceleration of motion as a function of time

∆ = Generalised deflection (mm)

∆i = Mid-span instantaneous deflection (mm)

∆r = Mid-span residual deflection (mm)


Notation xx x = Mean of the data set

∆U = Area within a hysteresis loop representing the amount of energy

dissipated

Φ = Diameter of reinforcing bar (mm)

α = Compressive stress block parameter defining the effective width of the

member

αrc = Residual deflection curve coefficient for RC beams

αp = Loading constant for deflection calculation

αps = Residual deflection curve coefficient for PSC beams

αw = Loading constant for deflection calculation

β = Frequency response during a forced vibration experiment

β1, β2 = Frequencies at which the harmonic-response curve has reduced to 1/√2

βfl = Flexural damage function for the calculation of the ‘tested’ logdec

δ = Logarithmic decrement (logdec)

δcr = ‘Cracked’ damping capacity

δm = Damping capacity of the mortar

δtest = Damping capacity of the ‘tested’ concrete beam

δtotal = Total damping capacity for RC and PSC beams (in terms of logdec)

δuncr = ‘Uncracked’ damping capacity

δuntest = Damping capacity of the ‘untested’ concrete beam

εc = Ultimate compressive strain in the concrete at failure

φ = Phase angle of motion

γ = Effective depth of the compressive stress block

η = Effective prestressing coefficient

ϕ = Initial camber (upwards deflection) in a section due to prestressing (mm)

ρ = Density of concrete

ρt = Tensile reinforcement ratio (=Ast/bd) sometimes defined using, ρ

ρc = Compression reinforcement ratio (=Asc/bd)

σn-1 = Standard deviation of the data set

σe,max = Steel stress (Dieterle and Bachmann, 1981)

σp = Stress in the prestressing tendon (daN/cm2) (Hop, 1991)

σpi = Stress in the prestressing tendon immediately after transfer (MPa)


Notation xxi σpu = Ultimate stress in the prestressing tendon (MPa)

σbp = Flexural stress provided by prestressing (MPa)

τ = Period of time required to complete one cycle of motion

τd = Damped period of motion

ω = Undamped natural circular frequency or angular velocity of motion

(=2π f )

ωd = Damped circular frequency of motion

ξ = Damping ratio

ξVD = Viscous damping ratio (Dieterle and Bachmann, 1981)

ξFD = Friction (Coulomb) damping ratio (Dieterle and Bachmann, 1981)

ξ cr = Damping ratio for a cracked beam (Dieterle and Bachmann, 1981)

ξ un = Damping ratio for an uncracked beam (Dieterle and Bachmann, 1981)

ξe = Equivalent damping ratio for a hysteretically damped system

ξs = Material damping of concrete (Flesch, 1981)

ξv = Slip damping (Flesch,1981)

ξtotal = Total damping of an uncracked beam (Flesch, 1981)

ζ = Equivalent viscous damping ratio


Chapter 1: Introduction 1-1

CHAPTER 1

Introduction

1.1 General Remarks

“All structures exhibit vibration damping, but despite a large literature

on the subject damping remains one of the least well-understood

aspects of general vibration analysis” Woodhouse (1998).

Damping is a broad topic that has been the subject of a wide variety of research efforts

over the years. Pioneering work was undertaken by Coulomb in 1784, who speculated

on the micro-structural mechanisms of damping in his ‘Memoir on Torsion’. Between

the period of 1784 and 1968 well over 2500 articles had been published with a scientific

or engineering interest in the damping of polymers, elastomers and other non-metallic

materials, with approximately four papers specifically devoted to concrete (Lazan,

1968). Since 1968, there has been a significant amount of concrete specific damping

research examining a very wide range of damping topics.

The introduction of high-strength concrete (currently in AS3600-2001 the compressive

strength is > 65MPa, but is constantly being revised with each new publication) and

steel reinforcement (a nominal yield strength of 500 MPa was introduced in Australia in

2000) has resulted in the design of more slender concrete members. In concrete column

design for example, significant savings in cross-sectional dimensions have been

achieved using high strength concrete whilst maintaining the same load capacity. Even

though reduced sections are seen as more cost effective some related side effects occur.

High-strength concrete beams with reduced cross-sections may be more susceptibility to

vibration and serviceability problems. This particular vibration response has not been

studied adequately by the literature and is therefore a focus of this thesis.


Chapter 1: Introduction 1-2 It is well known that damping capacity is, in some way, dependent on the level of

damage that exists in a concrete beam (Van Den Abeele and De Visscher, 2000).

Therefore, it has been common for researchers to adopt an expression relating damping

capacity to the cracking in a concrete beam. Dieterle and Bachman (1981) based

damping capacity on the level of stress in the reinforcing bars (derived from cracking

theory), whilst Chowdhury (1999) utilised a direct measure of residual crack width to

calculate damping. These previous studies are not, however, able to fully explain all the

experimentally observed damping characteristics. They do not indicate how damping

should be calculated prior to the beam cracking. This is particularly important for

prestressed beams that commence cracking at 70% of their ultimate capacity. Another

omission in previous work is the calculation of the basic inherent damping capacity of a

beam when it is first cast. This has been termed the ‘untested’ damping capacity and

along with ‘tested’ damping, forms the major focus of this research.

1.2 Research Objectives

There are two interrelated core objectives within this thesis, the investigation of the (1)

damping; and (2) residual deflection characteristics of reinforced and prestressed

concrete beams. To achieve both of these research objectives the following four items

will be addressed:

(a) The importance of technique for accurately extracting the logarithmic decrement

(a measure of damping capacity) from the experimental data;

(b) The identification and quantification of the experimental variables for the

development of a damping prediction formula for ‘untested’ reinforced and

prestressed concrete beams;

(c) The examination of the residual deflection characteristics of reinforced and

prestressed concrete beams in order to calculate ‘tested’ damping capacity. Even

though residual deflection implies the consideration of long-term effects, crack

patterns and stress relaxation etc., for the purposes of this thesis it relates

specifically to the experimental regime implemented; and

(d) Using residual deflection along with ‘untested’ damping predictions as a predictor

of total damping capacity for beams in-service.

1.3 Research Methodology

To achieve the aims and objectives, multiple stages of research are required. The initial


Chapter 1: Introduction 1-3 stage is comprised of an extensive literature review, used primarily to highlight the

major gaps and omissions in previous damping research so that the framework for the

research could be established.

The experimental programme stage developed in response to the desired outcomes of

the research, which includes the development of the design equations in which a

substantial experimental database is required.

Following the experimental programme, an extensive analytical investigation is

undertaken. In this phase, a review of the difference between the damping capacities of

the various tested beam types is made and the proposed damping models incorporating

residual deflection developed and verified.

1.4 Layout of the Thesis

Following the introduction presented here, Chapter 2 presents a brief overview of

vibration and damping theory and also a literature review of the published concrete

damping research. Chapter 3 presents the development of the theoretical framework for

the prediction of the total damping capacity of concrete beams, that includes the

‘untested’ and ‘tested’ damping components.

Chapter 4 presents details of the extensive experimental programme carried out to

investigate the damping and deflection characteristics of reinforced and

prestressed concrete beams. A total of forty-one beams were tested over a period of

three years. The complete data on the geometrical and mechanical details of these test

beams, supports and loading systems, and test set-up are also given.

Chapter 5 examines how the logarithmic decrement is calculated from the experimental

free-vibration decay records. This Chapter arose because of the initial difficulties found

in calculating logarithmic decrement consistently and accurately.

Utilising the findings of Chapter 5, Chapter 6 is concerned with the determination of

the damping capacity of ‘untested’ concrete beams. The resulting ‘untested’ damping

capacity equation defines the starting point from which to calculate the total damping

capacity of a beam at any stage of its service life. This Chapter examines the dependent

variables for both reinforced and prestressed beams and presents proposed damping


Chapter 1: Introduction 1-4 prediction equations.

Chapter 7 evaluates the calculation of the residual deflection characteristics of

reinforced and prestressed concrete beams. This is necessary because the prediction of

residual deflection is not well understood. Initial verification of the proposed equation

for the calculation of residual deflection is made using the experimental test results of

James (1997).

Chapter 8 presents the proposed equation for the calculation of the total damping

capacity for beams in-service using residual deflection along with ‘untested’ damping

capacity predictions. Verification of the equation, using test data and the published

experimental damping test data of Chowdhury (1999), is conducted.

The final Chapter, Chapter 9, summarises the main findings of the research, draws

conclusions and identifies the shortcomings. Recommendations are then made as to the

anticipated applications of the current findings and suggestions are made for further

research studies.

1.5 Summary

As initially highlighted by Woodhouse (1998) at the beginning of this Chapter, the

quantification of damping is perhaps one of the most vexing problems in structural

engineering. Unlike the unique physical properties of a structural system, such as

inertial and stiffness properties, which can be related to deflection and cracking,

damping is dependent upon a variety of features, such as component materials and

external influences. Damping has traditionally been treated as a relatively unknown

quantity, simply because it has been difficult to define and quantify.

This thesis provides essential experimental data and analytical proposals that act as the

foundation for further knowledge that will contribute to the meagre existing database on

the damping characteristics of reinforced and prestressed concrete beams. Such

proposals will benefit many areas of damping knowledge, including:

The development of a set of rules that will allow researchers to fully compare

their research to existing published research. Up to this point, this has not been

feasible because the style in which damping data has been reported is inconsistent;


Chapter 1: Introduction 1-5 The development of a unique technique to simply and easily determine the

damping capacity of a structural concrete member. This is currently not available

to structural and civil engineers (Wilyman and Ranzi, 2001).


Chapter 2: Damping in Concrete 2-1

CHAPTER 2

Damping in Concrete

2.1 General Remarks

All materials, members and structures, regardless of shape or function, are governed by

the same basic fundamental laws of motion. For vibration damping, these fundamental

theories have been well researched. Nevertheless, despite the extensive amount of

literature available on the subject, damping remains one of the least understood aspects

of general vibration analysis. The major reason proposed for this is “the absence of a

universal mathematical model to represent damping forces” (Woodhouse, 1998).

A discretely damped vibrating system with N degrees of freedom, executing small

vibrations about a stable equilibrium position, obeys the governing linear equation of

motion:

xyKyCyM =++ &&& (2.1)

where M, C, and K are the mass, damping and stiffness matrices respectively, y is the

vector of generalised response co-ordinates, and x is the vector of generalised forces

driving the vibration. Also, y and x are functions of time.

The modelling of the mass and stiffness matrices in Equation 2.1 is well established in

the literature, where, for example, the mass matrix for a highrise building would

conprise of the weights of each floor (Béliveau, 1976). However, as discussed by

Woodhouse (1998), in mathematically constructing the linear damping matrix required

by Equation 2.1, it is still not clear which variables the damping forces will depend on

i.e. viscous or Coulomb damping which will be discussed presently (Mo, 1994; Kareem

and Gurley, 1996). Furthermore, Kana (1981) discussed the difficulty in assigning

damping values for individual components in different parts of the structural system.



Nonlinear behaviour is generally considered in the analysis of structures subject to

motions well into the inelastic range, such as during earthquakes, and as such is not

considered here (further information may be found in Tilley, 1986; Economou et al.,

1993; Fajfar et al., 1993; Jeary, 1996; Kunnath et al., 1997; Xiao and Ma, 1998).

Initially, vibration concepts will be introduced, as an understanding of these

fundamental laws is important to the development of the thesis. A discussion of the

methods of experimental determination of fundamental vibration and damping

quantities, is also presented. The Chapter concludes with a review of selected available

published concrete damping research. The bibliography provides a complete listing.

2.2 Undamped Systems

In an undamped single-degree-of-freedom (SDOF) vibrating system, the displacements

of motion about equilibrium are time-dependent and in the absence of attenuating

forces, will continue on forever. If these vibration repetitions are at regular time

intervals, the motion is called periodic, where a period, τ, is the amount of time required

to complete one cycle of motion. The frequency of vibration, f, is the number of cycles

per time unit (f = 1/τ). The frequency of vibration may also be discussed in terms of

number of radians per unit time, called the circular frequency or angular velocity of

motion, ω (where ω = 2π f). These various definitions may be seen in Figure 2.1. In

many structural engineering cases, motion is harmonic. Harmonic motions are naturally

periodic and may be described in terms of sine and cosine functions which allows for

mathematical simplicity.

2.2.1 Single-DOF and Multi-DOF Structures

The degree-of-freedoms (DOF’s) of a freely vibrating body are defined as the number

of independent co-ordinates that are required to identify it’s displacement configuration

during vibration. Each possibility of free-vibration is defined as a mode of vibration.

For example, the rigid block in Figure 2.2 may vibrate in six different ways: three

displacement translations (x, y and z coordinates) and three angular displacement

rotations (x, y and z axes). If the rigid block now moves only in the vertical direction

(SDOF), a single vertical coordinate y(t) is sufficient to completely define the position

of the mass m, with stiffness k, during vertical vibration.

Most structures, regardless of complexity, may be reduced to a form that readily allows

computations. Without this simplification, it would be practically impossible for an



engineer to conduct a dynamic analysis. Therefore, for an elastic body, such as the

simply supported beam modelled as a SDOF structure, shown in Figure 2.3, there are an

infinite number of modes of vibration and, subsequently, an infinite number of degrees

of freedom. That is, there are an infinite number of particles requiring an infinite

number of coordinates to determine the position of each particle during vibration. Thus,

in general, even the simplest of structures, such as the simply-supported beam, are in

reality multi-degree-of-freedom (MDOF) systems with an infinite number of DOF’s.

An important assumption involved in any SDOF approach is that one mode of vibration

dominates. This would be inappropriate for structures with high modal density, but for

the present beam problem it is satisfactory (Fahey and Pratt, 1998a).

τ = 2π/ω = 1/f

A

DisplacementResponse

y(t)

Time (t)

Motion described by:y(t)= A cos (ω t - φ )

φ/ω

φ =Phase Angle

φ

Figure 2.1: The Undamped Free-Vibration Response (Clough and Penzien, 1975)

kz

xm

y(t)y

kkz

xm

y(t)y

Figure 2.2: Freely Vibrating Body in the X, Y, Z Direction (Fertis, 1995)



Deflected Shape = Fundamental Mode Shape= Fundamental Mode of Vibration

Figure 2.3: Example of a Structure Modelled as a SDOF System (Fertis, 1995)

The natural frequencies of a freely vibrating body are equal in number to its DOF’s and

there is a mode shape associated with each frequency. One important mode of vibration

is at the lowest frequency and is called the fundamental frequency of vibration. This

correlates to the fundamental mode of vibration, as shown in Figure 2.3. The

fundamental mode of vibration has the same shape as it’s deflected shape. In many

structural problems, the fundamental mode of vibration is of particular importance

because the amplitudes of vibration are the largest, and are often used to define the

rigidity of the structure. In a study of the response of tall buildings, with a structural

system consisting of frames, it has been shown that the fundamental mode contributes

about 80% of the total response (Penelis and Kappos, 1997). Since the rigidity of a

structure is a function of its free frequency of vibration, as stiffness (k) increases,

frequency (f) of vibration also increases.

2.3 Damped Systems

Damping is a collective term describing the non-conservative forces that act upon

bodies to resist motion. The amplitudes of a freely vibrating damped body are reduced

by the resisting force that is developed during the period of free-vibration. This

resisting force dissipates energy and in time the vibrations die out. Damping is present

to some degree in all structural systems, but its nature and magnitude are generally not

well understood (Irvine, 1986). If damping were not present, vibrations would never die

out, but continue on forever.

2.3.1 The Idealized MDOF System

When undertaking a dynamic analysis of beam and frame MDOF systems with

continuous mass and elasticity, such as that shown in Figure 2.4a, sufficient accuracy

may be obtained by using a simpler SDOF model, such as that shown in Figure 2.4b.

This is commonly termed the idealized system.



P = Applied Force

EI

L/2

m

P = Applied Force

kc

a) b)L/2

m

∆

Figure 2.4: Beam System: a) MDOF Beam, b) Idealized SDOF Mass-Spring System

(Fertis, 1995)

In Figure 2.4b, the spring constant k represents the stiffness of the simply-supported

beam at its centre and c represents the damping force. The deflection, ∆ of the beam is

given by:

EIPL

48

3

=∆ (2.2)

where E is the Young’s modulus for the beam material and I is the moment of inertia of

the beam’s cross-sectional area about the neutral axis.

The equivalent stiffness, k, of the beam is the ratio of the applied load to the deflection

at the point of application of the load, giving:

3

48LEIkP

==∆

(2.3)

where P is the vertical load which produces a vertical displacement ∆ equal to unity.

The damping of the system c, in Figure 2.4b, may be mathematically modelled by one,

or a combination of the three primary types of damping: viscous, Coulomb and

hysteretic. Generally, one form dominates, thus allowing a reasonable analysis to be

undertaken (Tedesco et al., 1999). Each different type of damping is detailed below.

2.3.2 Viscous Damping

Viscous damping is considered to be proportional to the velocity of the oscillatory

motion. The equation of motion can be written by summing the forces shown in Figure



2.5a in the x-direction. Figure 2.5b is representative of the decaying vibratory

oscillations under viscous damping action. The resulting summation produces the

differential equation that describes the motion of this viscously damped system as:

0=++ kyycym &&& (2.4)

where and y are the acceleration, velocity and displacement of the body,

respectively.

yy &&& ,

The solution y(t) to Equation 2.4 takes the form of an exponential function, given by:

pteAty =)( (2.5)

where A and p are constants.

By substituting Equation 2.5 into Equation 2.4, followed by the cancellation of common

factors, the characteristic equation that describes the system is arrived at, namely:

02 =++ kcpmp (2.6)

The solution to this quadratic equation is given by the following two roots:

mk

mc

mcp −⎟

⎠⎞

⎜⎝⎛±−=

2

2,1 22 (2.7)

k m

a) Mechanical Models b) Decay Curve Characteristics

Exponential rate of decayAmplitude

ofVibration

Time

Viscous Damping

c

Displacement output, measuredfrom neutral position of spring

y

Force input= F

Figure 2.5: Decay of Free-Vibration Under the Assumption of Viscous Damping: a)

Mechanical Model; and b) Decay Curve Characteristics (Newland, 1989)



The general solution to Equation 2.5 is, therefore, given by the superposition of the two

possible solutions given by Equation 2.8, namely:

tptp eAeAty 2121)( += (2.8)

where A1 and A2 are constants determined from the initial conditions of the vibratory

motion.

For example, solving for constants A1 and A2, at time t=0, with an initial displacement

of A0 and initial velocity of , gives the following solution: 0y&

)()( /)2/(2

/)2/(

1)2/(

22 tmkmctmkmctmc eAeAety ⎥⎦⎤

⎢⎣⎡ −−⎥⎦

⎤⎢⎣⎡ −− += (2.9)

where the factor e-(c/2m)t is an exponentially decaying function of time, which shows that

the damped vibratory motion has an exponentially decaying amplitude with time.

The final form of Equation 2.6 is dependent upon the sign of the expression under the

radical sign in Equation 2.7. It may either be zero, positive or negative. Where it is

zero, the case is called critical damping. This case will be considered first.

Critical damping is the value of the damping coefficient for which the system will not

oscillate when disturbed initially, but will simply return to the equilibrium position. A

useful definition is that it is the smallest amount of damping for which no oscillation

occurs in the free response (Clough and Penzien, 1975). This condition does not usually

occur in practice (Fertis, 1995).

The critical damping value, ccr is defined as the value of c in Equation 2.7 that makes

the algebraic sum of the terms under the radical equal to zero. Thus:

ωω kmkmccr

222 === (2.10)

where ω is the undamped natural frequency of vibration of the spring-mass system

(ω = km ), and the damping of the spring-mass system may now be specified in terms

of c and the damping ratio, ξ, given by (Buchholdt, 1997):



crcc

=ξ (2.11)

which may be further expressed by the following expressions (Fertis, 1995):

ωξ=mc

2 (2.12)

and

( 222

12

ξω −=⎟⎠⎞

⎜⎝⎛−

mc

mk ) (2.13)

In an overdamped system, the damping coefficient (c) is greater than the critical

damping (ccr), i.e. c>ccr. For this case, the term under the radical sign in Equation 2.7 is

positive and the solution may be determined directly from Equation 2.7. The motion of

critically damped and overdamped systems is not oscillatory. Figure 2.6 depicts,

graphically, the motion for both the critically and overdamped system. In the case of

the critically system, the curve would return to the neutral position more quickly

(Meirovitch, 1975).

y

Time, t

y(t)y&

Overdamped andCriticallyDamped SystemResponse

Figure 2.6: Free-Vibration Response of Critical and Overdamped Systems (Clough and

Penzien, 1975)

A graphical record of the response of an underdamped system, with initial displacement

A0 and zero initial velocity ( =0) and phase angle, φ, is shown in Figure 2.7. In an 0y&



underdamped system, motion is oscillatory, but not periodic, and the amplitude of

vibration is not constant, but decreases exponentially for each successive cycle. The

oscillations occur at equal intervals of time. This is termed the damped period of

vibration (τd) and is defined by:

21

22

ξω

πω

πτ−

==d

d (2.14)

where ωd is the damped circular frequency.

t = time

Amplitude ofVibration

τd

A0 e -ξ ω t

A2

A1

t1 t2+τd

A0

A0 sin φ

φ

Figure 2.7: Free-Vibration Response of an Underdamped System (Fertis, 1995)

In an underdamped system, the damping coefficient (c) is less than the critical damping

(ccr), i.e. c<ccr. For this case, the term under the radical sign in Equation 2.7 is negative

and the solution is given by complex conjugates, so that:

2

2,1 22⎟⎠⎞

⎜⎝⎛−±−=

mc

mki

mcp (2.15)

where i = √-1 is the imaginary root.

Substituting the roots from Equation 2.15 into Equation 2.8 will give the general

solution of the displacement of the underdamped system, given by:



)sincos()( )2/( tBtAety ddtmc ωω += − (2.16)

where A and B are constants of integration, and the damped frequency of the system

(ωd) is given by:

2

2⎟⎠⎞

⎜⎝⎛−=

mc

mk

d ωω (2.17)

Substituting the expression for the undamped natural frequency (Equation 2.13) gives:

21 ξωω −=d (2.18)

In real structures, the damping coefficient (c) is generally much less than the critical

damping coefficient (ccr), thus indicating an underdamped condition, usually of the

order of 2 to 10% of critical damping. At 10% of critical (i.e. ξ = 0.10), Equation 2.18,

gives the following:

ωd = 0.995ω (2.19)

Thus Equation 2.19 indicates that the frequency of vibration for a system with as much

as 10% of critical damping is essentially equal to the undamped natural frequency.

Therefore in practice, the natural frequency of a damped system, ωd is taken as being

equal to the undamped natural frequency, ω (Paz, 1997).

In practice, the effect of damping on the natural frequency is ignored, and the

underdamped motion is described by the following (Irvine, 1986):

⎟⎠⎞

⎜⎝⎛=

δπξ 1ln

21n

(2.20)

where δ is the logarithmic decrement (an alternative measure of damping discussed in

Section 2.4.1) and n is the number of free vibration cycles.

For example, if after 10 cycles of free vibration (n = 10), the effect of damping has

reduced the peak displacement to 50% of its original height, the percentage of critical



damping is:

( ) %1011.02ln20

1≈==

πξ (2.21)

2.3.3 Coulomb Damping

Coulomb damping, also known as dry friction damping, is the result of rubbing and

sliding between vibrating dry surfaces. It assumes a frictional damping force, Ff is

produced between the mass m and the surface S (see Figure 2.8a). It is constant in

magnitude but changes sign according to the sign of the vibrational velocity.

A freely vibrating system subject to pure Coulomb damping may be diagrammatically

represented by Figure 2.8, which shows the linearly decaying amplitude of a Coulomb

damped system. The ‘linear-decay’ property of vibrating systems exhibiting pure

Coulomb damping was first found by Lorenz (1924), and Malushte and Singh (1987)

discussed the mechanism as found in a single storey frame building.

Coulomb / Friction Damping

k m

a) Mechanical Model b) Decay Curve Characteristics

Linear rate of decay

Amplitudeof

Vibration

Time, t

t=2π/ω

4Ff /k

Ff /k

y

Ao

Ff

S

A0

Figure 2.8: Decay of Free-Vibration under the Assumption of Coulomb Damping: a)


In Coulomb damping, motion ceases when the amplitude is less than Ff/k and the spring

force k is no longer able to overcome the static friction force. Coulomb damping is

harmonic, and the frequency of oscillation (f) is the same as the free undamped



frequency of the spring-mass system. The natural period of vibration, τ is unchanged by

this form of damping. The equation of motion of the free body diagram in Figure 2.8a

is given by:

00 >−=+<=+ yFykymandyFykym ff &&&&&& (2.22)

The mathematical solution to Equation 2.22, that describes the displacement y at time

t=2π /ω, is given by (Figure 2.8b):

kF

Ay ft

40/2 −== ωπ (2.23)

where A0 is the initial displacement given to the mass m, k is the spring stiffness, and Ff

is the resisting force which is produced from the friction between the mass m, and the

surface S (assumed to remain constant).

2.3.4 Hysteretic Damping

Hysteretic damping, also known as solid or structural damping, is generally attributed to

internal material friction created during motion. These frictional forces develop

between material matrix planes that slip, relative to one another during oscillatory

motion. This type of damping is considered to be independent of the frequency of

vibration, but approximately proportional to the amplitude of the deformed elastic body.

Figure 2.9 shows the time-history of an oscillating, purely hysteretically damped

system: the rate of reduction of its amplitude, A, depends on the size of the area within

the hysteresis loop (Figure 2.10).

Hysteresis damping causes a reduction in amplitude depending on the size of the area

within the hysteresis loop. Rubber materials have loops containing a much larger area

when compared to metallic materials, like steel. This is the reason why artificial

dampers use rubber-like materials.

Considering one cycle of motion of a vibrating single-degree-of-freedom spring-mass

system (Figure 2.9) and plotting the force-displacement diagram shown in Figure 2.10

(F is the resisting force developed in the body due to hysteresis damping and A is the

amplitude of displacement), the area within the loop represents the amount of energy,



∆U, transformed into heat, per cycle of motion, due to the internal friction in the

material.

Experiments have shown that ∆U (from Figure 2.10) could be obtained from the

approximate expression (Fertis, 1995):

2AckU oπ=∆ (2.24)

where co is the dimensionless constant of the material for solid damping, and k is the

force required to deflect the spring by an amount equal to unity.

k m

F

a) Mechanical Model b) Decay Curve CharacteristicsAmplitude

of Vibration

Time, t

Hysteretic Damping

2π

A1A2

A3

Figure 2.9: Decay of Free-Vibration under the Assumption of Hysteretic Damping: a)


O

A A

F =Force

y(t)

∆U is areawithin hysteresisloop

Figure 2.10: Force-Displacement Hysteresis Loop (Fertis, 1995)



The amount of hysterestis damping in engineering structures is generally very small and

is usually neglected in an analysis (Fertis, 1995). However, if hysterestic damping is to

be considered then the damping, ξ, is termed the equivalent viscous damping ratio for

hysteretically damped systems, ξe. For the SDOF mass-spring system shown in Figure

2.9a, the equivalent damping ratio, ξe is given by the expression:

2o

ec

=ξ (2.25)

where co is the hysteresis damping coefficient, and for small hysteresis damping is

defined by the approximate expression (refer to Figure 2.9b):

ocAA

π+≈ 13

1 (2.26)

The equivalent viscous damping coefficient, ce, may be determined from:

ωξ

kckmccc o

oece === (2.27)

and the time-displacement response y(t) of a hysteretically damped system is:

( )[ ]tAety ete 21sin)( ξωωξ −= − (2.28)

where it is possible to assume, without appreciable loss of accuracy, that the undamped

and damped frequencies are the same (ω = ωd = (k/m)½) (Fertis, 1995).

2.3.5 Equivalent Viscous Damping

The true damping characteristics of typical structural systems are very complex and

difficult to define. The original theory (based on hysteretic damping discussed in

Section 2.3.4) to describe internal damping was first proposed by Kelvin (1865). It was

later shown that the theory was not universally applicable to all solids and, since then,

researchers have been unable to develop a theory to describe solid damping for general

applications (James et al., 1964). This is because, even though structural systems

exhibit a combination of linear (damping independent of amplitude) and non-linear

damping elements, (damping amplitude dependent, i.e. damping increases as motion-

generated stress levels increase) (Irvine, 1986)), it is material type, material history,



environment and test conditions that provide enormous complications to the

development of a ‘general’ theory to describe the dynamic response of vibrating

materials. In other words, all vibrating structures cannot be modelled by a single

mathematical equation.

In general, viscous damping is the most conducive to mathematical formulation (due to

its velocity dependence) and it is generally employed in a vibration analysis. Even

when researchers know that viscous damping is not operating, i.e. hysteresis or

Coulomb damping actually exists, an ‘equivalent’ viscous damping is assumed

(Tedesco et al., 1999). This is when the linear equations for viscous damping are

simply adopted for a vibration analysis, and thus the ‘equivalent viscous damping’

concept was introduced to allow the evaluation of the internal damping (of any kind) of

a member. Under these circumstances, structural damping is defined as the equivalent

viscous damping ratio, ζ (Sun and Lu, 1995).

It has been discussed within the literature that researchers tend to use damping

terminology interchangeably and indiscriminately without specifying whether pure

viscous damping exists, or whether equivalent viscous damping has been assumed. A

significant amount of confusion has thus been created, and has contributed to some of

the general ‘misdirection’ with regards to damping research (Jeary, 1996).

2.4 Experimental Determination of Damping

In undertaking a dynamic response analysis of a SDOF structure it is assumed that the

physical properties of the system (mass, stiffness and damping) are known. In the

majority of cases, structural mass and stiffness can be evaluated rather easily, usually

through simple generalized mathematical equations. On the other hand, the basic

energy-loss mechanisms that exist in practical structures are seldom fully understood.

Consequently it has not been traditionally feasible to determine the damping coefficient

by means of a generalized mathematical damping expression (Clough and Penzien,

1975). For this reason, the damping in most structural systems must be evaluated

directly from experimental tests. As these measures will be discussed within the

following Chapter, a brief survey of the principal procedures for evaluating damping

from experimental measurements follows.

2.4.1 Free-Vibration Damping

Logarithmic decrement, δ (generally shortened to logdec), is a measure of the rate of



decay of free-vibration. It is probably the simplest and most frequently used

experimental damping technique (Clough and Penzien, 1975; Kummer et al., 1981).

The original idea of utilising the free-vibration decay of a viscously damped system was

proposed by Helmholtz (1877), who used it to determine frequency information from

musical tones. It was, however, Rayleigh (1945) who coined the term ‘logarithmic

decrement’ for the estimation of the damping. The equation that analytically describes

the damped free-vibrating system represented by the response in Figure 2.7 is:

( )φωωξ += − teAty dt sin)( (2.29)

where Equation 2.29 represents pseudoharmonic motion with an exponentially decaying

amplitude (Ae-ξωt) and a phase angle (φ). The damped period of vibration (τd) may be

obtained from Equation 2.14. Since the amplitude (Ae-ξωt) depends on the damping

ratio (ξ), the rate of decay of the amplitude therefore depends on the amount of damping

in the system.

Using the resultant of Equation 2.29, logdec (δ) can be calculated from the ratio of

amplitudes several cycles apart (refer to Figure 2.7). If A2 is the amplitude n number of

cycles after the initial amplitude A1, logdec (δ) may be determined experimentally from

the traditional logdec technique (TLT) equation:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1ln1AA

nδ (2.30)

A major advantage of the traditional logdec technique (TLT) is that equipment and

instrumentation requirements are minimal; free-vibrations can be initiated by many

convenient and appropriate methods, i.e. simple vibration hammer or from ambient

excitations (traffic) and only relative displacements need to be measured and recorded

(Ibrahim and Mikulcik, 1977). Furthermore, when the structure is set into free-vibration

by a shock load, the fundamental mode dominates since all the higher modes are

damped out quite quickly. It is not usually possible to excite any mode other than the

fundamental mode using this method (Beards, 1996).

2.4.2 Forced Excitation Damping

Forced vibration techniques rely on observations of the steady-state harmonic response



behaviour of a SDOF structure, and thus require a means of applying harmonic

excitations to the structure at specific frequencies and amplitudes. A frequency-

response curve for the structure can be constructed from the application of a harmonic

load at a specified sequence of frequencies (that span the resonant frequency response

range), whereby the resulting displacement amplitudes can be plotted as a function of

the applied frequencies (as shown in Figure 2.11). The three main methods of harmonic

excitation are described below.

2.4.2.1 Half-Power (Bandwidth) Method

One of the most convenient and utilised methods is the half-power, or bandwidth

method, whereby the damping is determined from the frequencies at which the

maximum response (Amax) of the general harmonic-response curve shown in Figure

2.11, is reduced to (1/√2). These points correlating to β1 and β2 are indicated in Figure

2.11. Provided that the damping is small, and the natural frequencies are sufficiently

separated, and if the excitation frequency is close to a natural frequency, only one mode

will dominate the response (assuming that the point of excitation does not happen to be

close to a node for the mode in question) (Newland, 1989).

The equivalent viscous damping ratio is given by half the difference between the half-

power frequencies:

( 1221 ββζ −= ) (2.32)

This technique helps to avoid the need for determining the static response, but it does

require the response curve to be plotted accurately in the half-power range and at

resonance (Figure 2.11).

2.4.2.2 Resonant Amplification

From Figure 2.11, the equivalent viscous damping ratio (ζ) may also be evaluated from

the expression:

m

o

DAA

21

21

max

=≅ζ (2.33)

where Dm is the dynamic magnification factor at resonance, Amax (Clough and Penzien,

1975).



Ao

2Ao

3Ao

Amax

√ 2

β2 - β1 = 2ζ

Amax= Ao 2ζ

0 1 2β2β1

Frequency ratio, β

Har

mon

ic re

spon

se a

mpl

itude

~

.Ao = Zero-frequency or static displacement

Figure 2.11: Frequency Response Curve for Moderately Damped System (Clough and

Penzien, 1975)

In practice, the equivalent viscous damping ratio, ζ, is determined from the dynamic

magnification factor Dm evaluated at maximum amplitude (Amax), namely:

0

max

AA

Dm = (2.34)

where Amax is the maximum response amplitude, and A0 is the zero-frequency or static

displacement (see Figure 2.11).

This technique has the advantage of requiring only simple instrumentation capable of

vibrating a structure over a range of frequencies that span the resonant frequency, and a

simple transducer capable of measuring relative displacement amplitudes. However,

evaluation of the static displacement can be difficult because many types of loading

systems cannot operate at zero frequency and it can be difficult, in practice, to apply a

static lateral load to a structure (Paz, 1997).

2.4.2.3 Energy Loss per Cycle

If instrumentation is available to measure the phase relationship between the input force

and the resulting displacements, damping can be evaluated from tests run only at

resonance, and there is no need to construct the frequency-response curve as given in

Figure 2.11. The procedure involves establishing resonance by adjusting the input

frequency until the response is 90o out of phase with the applied loading. Then the



damping force (FD) exactly balances the applied load, so that if the relationship between

the applied load and the resulting displacements is plotted for one loading cycle, as

shown in Figure 2.12, the result can be interpreted as the damping-force-displacement

diagram.

DampingForce =fD

Area=wD

Ellipse (viscous damping)(Equivalent area=wD)

Displacement

Amax

Figure 2.12: Actual and Equivalent Damping Energy Per Cycle (Clough and Penzien,

1975)

If the structure has linear viscous damping, the curve will be an ellipse, as shown by the

dashed line in Figure 2.12. In this case, the damping coefficient can be determined

directly from the ratio of the maximum damping force (fD,max) to the maximum velocity

( ) where the maximum velocity is given by the product of the frequency, ω and

displacement amplitude, A:

maxv&

max

max,

vf

c D

&= (2.35)

If the damping is not linear viscous, the shape of the force-displacement diagram will

not be elliptical, such as the solid line as shown in Figure 2.12, and the equivalent

viscous-damping coefficient is then given by:

2maxv

wc D

eq&πω

= (2.36)



In most cases, it is more convenient to define damping in terms of the critical damping

ratio rather than the damping coefficient. For this purpose, it is necessary to define a

measure of the critical damping coefficient of the structure. This is expressed in terms

of stiffness (k) and frequency (f):

ωk

ccr2

= (2.37)

This expression is more convenient because the stiffness of the structure can be

measured by the same instrumentation used to measure the damping energy loss per

cycle, merely by operating the system very slowly at essentially static conditions. The

static-force-displacement diagram, obtained in this way, will be of the form shown in

Figure 2.13, if the structure is linearly elastic, then the slope of the curve represents the

stiffness (Clough and Penzien, 1975).

1k

Area=ws

Displacement

StaticForce

fs

fs,max

Amax

Figure 2.13: Elastic Stiffness and Strain Energy (Clough and Penzien, 1975)

Alternatively, the stiffness may be expressed by the area under the force-displacement

diagram, ws, as follows:

2

2Aw

k s= (2.38)

Thus the equivalent viscous damping ratio, ζ, can be obtained by combining Equations

2.36 to 2.38:

s

D

c ww

cc

πζ

4== (2.39)



2.5 Literature Review of Damping in Concrete

In the past it has been extremely difficult to develop equations to predict the level of

damping in real-life structures, and so designers were generally only able to conduct a

dynamic analysis once the structure had been built using non-destructive testing (NDT).

Despite the wealth of damping research undertaken in a huge variety of research areas,

there remains to be found: (a) a consensus regarding basic damping levels in reinforced

and prestressed concrete beams; (b) clarification of the mechanisms that affect in-

service damping; and (c) an accurate prediction methodology to estimate real-life

damping levels in concrete beams at any stage of their service life. In this review,

concrete damping research has been broadly classified into the three main categories of

material, member and structure damping.

2.5.1 Material Damping

Jones (1957) investigated the effect forced frequency of vibration has on the dynamic

modulus and damping coefficient of concrete cylinders. The results implied that

damping arises from the interfacial boundaries in concrete. Contrary to Kesler and

Higuchi (1953), damping was found not to depend on the frequency of vibration.

Cole and Spooner (1965) also examined the effect of forced vibration at very low

frequencies on the damping capacity of small rectangular cement paste beams and found

that damping capacity is influenced by variations in both the frequency and stress

amplitudes of vibration.

Cole (1966) measured the damping capacity of small rectangular cement paste beams

vibrated at low stress amplitudes where the variation of logdec was attributed primarily

to changes in age and water content. An approximate value for the logarithmic

decrement, δ, was given as

δ = a + bX – (c + dX) loge M (2.40)

where X is the evaporable water content, M the age of the specimen in months, and the

constants a, b, c and d have values of, a = 0.026, b = 0.022, c = 0.0039 and d = 0.0030.

Jones and Welch (1967) described a series of free-vibration decay experiments to

determine the vibrational damping coefficient of three types of plain concrete and

corresponding sand/cement mortars. No appreciable difference in damping could be



detected with change in frequency, size of specimen or method of measurement (i.e.

forced versus free-vibration experimental techniques).

Swamy and Rigby (1971) experimentally investigated the dynamic moduli (Young’s

Modulus) and damping properties of hardened pastes, mortars and concrete prisms.

Equation 2.47 was proposed for the determination of logdec for concrete in the dry

state, undergoing forced flexural vibration.

cmmc

VEE

0000913.0265.010640.01113.00174.0 −+−+= δδ (2.41)

where Ec and Em are the dynamic modulus of concrete and mortar (psi) respectively, δm

is the logdec of the mortar matrix, and Vc is the absolute volume fraction of coarse

aggregates.

Spooner and Dougill (1975) developed an analytical technique to quantitatively describe

the extent of damage experienced by a concrete specimen in compression. It was found

that, at low strains, logdec is approximately 0.09 and that the energy dissipated in

damage correlates to the change in initial elastic modulus during initial loading.

Ashbee et al. (1976) described an experimental method which purported to overcome

the inherent problem of frequency and load control commonly encountered in the

experimental study of damping. The logdec of concrete was found to range from 0.03

to 0.15, with an average value of about 0.05.

Spooner et al. (1976) suggested that damping depends specifically on the strain range

and that it is independent of the degree of damage exhibited by the specimen. This

result supported the findings of Swamy (1970). Thus, for cement paste prisms, the

relative contribution to damping by sliding friction between cracks (degree of damage)

and movement of capillary water was insignificant.

Using forced vibration techniques, Jordan (1980) studied the effects of stress (amplitude

and rate of vibration), frequency of vibration, and curing regime and age, upon the

damping of concrete cylinders. The presence of microcracks was found to be of great

importance in determining the damping capacity.

Sri Ravindrarajah and Tam (1985) investigated the use of recycled concrete as coarse



aggregates in concrete. For both normal and recycled aggregate concrete, damping

increased with a decrease in compressive strength. For all grades of concrete, those

with recycled concrete aggregates had higher levels of damping capacity.

Fu and Chung (1996) undertook experimental investigations into the effect of latex,

methylcellulose, silica fume and short fibres on the damping capacity of cement pastes.

It was found that the damping capacity of cement pastes is increased by the addition of

silica fume, latex, methyl-cellulose or short fibres.

The dependence on frequency of the damping capacity of hardened cement paste in the

temperature region of –900C and forced vibration frequency was studied by Xu and

Setzer (1997). The temperature dependence of damping was attributed to the interaction

between the pore ice and internal solid surface.

Furthering the experimental work of Fu and Chung (1996), Fu et al. (1998) investigated

the use of additives to improve the vibration damping capacity of cement. Silica fume

and latex were found to increase damping capacity by up to 390%. Methylcellulose was

found to only marginally affect the damping capacity. Li and Chung (1998) found that

the damping capacity at all temperatures and frequencies was increased by the surface

treatment of the silica fume. This increase was up to 300% in some cases.

Wang and Chung (1998) studied the effect of the addition of sand and silica fume on the

damping capacity of cement mortars. It was found that the loss modulus was decreased

greatly by the addition of sand, but the addition of silica fume increased the loss

modulus to a level comparable to, or exceeding that, of plain cement paste.

Orak (2000) investigated the damping capacity of polymer concrete. The damping

capacity of polymer concrete was found to be four to seven times larger than a

corresponding cast iron sample (it was not compared to that of normal concrete).

From the preceding discussion, it can be seen that a wide range of very different

damping variables have been identified and explored within the literature. For this

reason a summary table has been developed to attempt to extricate the most important

topics (Table A.1 may be found in Appendix A). The complete summary table is not

presented here due to its complexity, rather a point summary of the most important

conclusions, are outlined below.



(1) State of Material: In the early weeks of a beam’s life, damping is affected

considerably by hydration. Hydration has, within the literature, been termed as

moisture content, degree of hydration, curing conditions and age. Typically, it has

been well established that as the beam dries out, damping is reduced. After 28 days,

damping becomes stable. These findings do not hold significant implications for the

current experiments and indeed for general concrete construction because concrete

specimens are generally subjected to 28 day minimum curing periods. Interestingly,

recent work by Almansa et al. (1993), who investigated the effect of the early

unforming of concrete beams found that damping was not significantly affected by

unforming age (between 2 and 90 days).

(2) Concrete Composition: Generally, with respect to concrete strength (f’c) and the

elastic or dynamic modulus of elasticity (ED and Ec, respectively), an inverse

relationship with damping is suggested, but not conclusively (Kesler and Higuchi,

1953; Jones and Welch, 1967; Swamy and Rigby, 1971; Sri Ravindrarajah and Tam,

1985). For the effect of the various aggregates and additives, previous research has

not been conclusive as to the nature of their effect on damping.

(3) Testing Procedures: An extremely wide variety of testing procedures has been

employed. For this reason, there has been an equally wide variety of testing results

and conclusions. The interdependency between frequency and damping has been

discussed, with tests showing damping as ‘very dependent on frequency’ (Kesler

and Highuchi, 1953; Cole and Spooner, 1965) to ‘not dependent on frequency’

(Jones and Welch, 1967; Jordan, 1980).

(4) Cracking Mechanisms: The effect of cracking on damping capacity has been of

interest to researchers as it gives an indication of the internal structure. Jones and

Welch (1967), Swamy (1970) and Spooner et al. (1976) have stated that from tests,

sliding friction within the solid gel structure (hysteretic damping) is unimportant,

whereas the primary energy loss is due to moisture movement within the solid

structure (viscous damping). Jordan (1980) strongly indicated that the component of

damping from microcracking (Coulomb damping) is by far the most important

damping mechanism. Also the bonding between the different types of aggregate-

cement pastes under load was investigated by Sri Ravindrarajah and Tam (1985)

who examined the suitability and resiliance of recycled-aggregate concretes.

2.5.2 Member Damping

The following discussion reviews damping research on reinforced and prestressed



concrete members. A complete summary of published research on concrete member

damping is presented in Tables A.2 and A.3 (Appendix A).

Bock (1942) undertook forced vibrations experimental tests and found that damping

was independent of forced vibration amplitude and was higher in beams not containing

reinforcement.

James et al. (1964) tested beams that were subjected to forced vibrations at various

stages of cracking in order to examine flexural rigidity and damping properties. They

found that the dynamic response of reinforced concrete beams was significantly affected

by test history. James et al. (1964) also undertook forced vibration experimental tests

on six prestressed concrete beams. It was found that the damping characteristics of

prestressed and reinforced concrete beams were generally the same. Test history or

cracking did not affect the dynamic response characteristics.

In his reinforced concrete experimental investigation, Penzien (1964) found that for

both forced and free vibration, damping increased with an increase in cracking. Penzien

(1964) also undertook free-vibration experiments on prestressed concrete beams and

found that damping was affected primarily by the cracked state of the beam, thus the

degree and type of prestress only indirectly influenced damping by influencing the

cracking of the beam.

In 1974, Wills’ finding that the value of logdec regularly being used for chimney design

only applies to small amplitudes of vibration whilst for large vibration amplitudes,

tensile stresses are induced and the damping increases by a factor of five to ten. From

an experimental investigation, Jordan (1977) found that the material damping did not

increase as tensile stresses were induced.

Dieterle and Bachmann (1981) developed theoretical models to explain the damping

behaviour exhibited by reinforced concrete beams in the uncracked and cracked states.

The theoretical damping models are shown in Figure 2.14. Forced vibration

experimental investigations were undertaken from which the curves given in Figure

2.15 were developed. The curves show that the damping, as a result of cracking, can

tend towards a very small value after an initial growth. For the uncracked beam the

damping due to viscous damping was described by



⎥⎦

⎤⎢⎣

⎡= 2

02 cm

cunVD fC

Edπ

ξ (2.42)

( )c

ect E

EnH

hH

HhnC =

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −−+=

2

0

2

'231 ρρ (2.43)

For a cracked beam element, total damping, ξ cr

was specified as the sum of two

components, viscous damping, ξVD, and friction damping, ξFD

⎥⎥⎦

⎤

⎢⎢⎣

⎡×+⎥

⎦

⎤⎢⎣

⎡=+= 2

max,

max,

2

2

21

62 e

emvtc

cm

ccrFD

unVD

cr VBhCAjEn

fCEd

σσφτρ

πξξξ (2.44)

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+= 3

2

3

2

1''331

Na

Nac

Na

Nat x

hxhnxxhhnC ρρ (2.45)

( )

2

2

3

2'

3 ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−++

−=

Na

Nact

Na

Na

xhhxnn

xhhxC ρρ (2.46)

where fcm is concrete compressive strength (MPa), d is a specific damping coefficient

(MPa) that depends on the material and must be determined experimentally, σe,max is the

steel stress (MPa), Ec is the modulus of elasticity for concrete (MPa), A is a

proportionality coefficient, and h, H, h’, xNa, are defined in Figure 2.14.

The reinforcement ratio is considered in the coefficient Co, but its influence on damping

appears to be inversely proportional to damping capacity. Interestingly, despite the

development of Equation 2.42, the viscous damping component (damping ratio, ) of

normal concrete beams was established as 0.006 (δ = 0.0377) in virtually all test data.

This suggests that the development of a prediction formula was unnecessary.

unVDξ

Equations 2.42 and 2.44 are the only known attempt to theoretically derive a damping

prediction equation for the uncracked beam state. However, they are complicated and

would be difficult for practitioners to use, also their validity was not proven.



a)

UNCRACKED BENDING ELEMENT

h

h’

HM M

Fe’=µ’ b h

Fe=µ b h

s

Zb

D

Ze

Bending Element Model

TensionZone

CompressionZone

k

mz

ViscousDamping

b)

CRACKED BENDING ELEMENT

xNA

h’

h

M M

Fe’=µ’ b h

Fe=µ b h

s

D

Z

Bending Element Model

CompressionZone

k

mz

ViscousDamping

Friction

Damping

)(,

eFNµ

Figure 2.14: Damping for the: a) Uncracked; and b) Cracked Beam Sections (Dieterle

and Bachmann, 1981)

crFD

unVD

cr ξξξ +=

ξ

σe, relTP1: Initial stressing & crackingTP2: After cracking & repeat of test

ξunVD

=0.0

06

TP1=Test Phase 1Viscous Damping (ξun

VD) Friction Damping (ξcr

FD)

TP1TP1

TP1

TP2

TP2

ξπδ 2=

Figure 2.15: Damping Ratio as a Function of the Relative Steel Stress (Dieterle and

Bachmann, 1981)



Flesch (1981) developed a mathematical model to predict the damping capacity of a

reinforced concrete cantilever beam element during one cycle of loading for both the

uncracked and cracked condition. This cantilever element is shown in Figure 2.16. The

total damping of the uncracked beam, ξtotal may be calculated from

ξtotal = ξv + ξs (2.47)

where ξv = slip damping and ξs = material damping of concrete (material damping of

steel neglected below yield point).

In addition to material damping, the hysteretic damping caused by the slip between steel

and concrete was derived using the equivalent viscous damping mechanism for the

uncracked section, it is given by the iterative formula

02

48 20

2

1

6

513

6

32 =⎟⎟

⎠

⎞⎜⎜⎝

⎛+− −−− n

nn

c

n Pclc

hh

hbEhl ξπξ (2.48)

where l, b and h are defined in Figure 2.16 and are in m, Ec is the modulus of elasticity

for concrete (kN/cm2), n = material constant evaluated from tests, Po = maximum force

amplitude at resonant frequency, and h5 , h6 , c1 and c2 are constants described by

additional equations.

my tP ωsin0

h

l

AA

xE E

Cross-Section A-A

Uncracked Cracked

h

εc

b

εc

εs

y

b

Figure 2.16: SDOF RC Cantilever Beam During One Loading Cycle (Flesch, 1981)

The prediction equation for the equivalent damping ratio with cracks is given by

Equation 2.49:



312

2

23

23

/44

vvpvK

pv

pv

Kp

vp

ooooo −⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −−⎟⎠

⎞⎜⎝

⎛ −=ππ

ξ (2.49)

where K = stiffness given by 3EI/l3 and I = bh3/12 and v1, v2 and v3 are constants

described by additional equations.

The equations of Flesch (1981) are similar in concept to Dieterle and Bachmann (1981)

and as such, are complicated and would be difficult for the practitioner to implement.

Furthermore, the equations by Flesch (1981) were not verified using experimental tests,

rather the equations were utilised to undertake a parametric study.

Askegaard and Langsœ (1986) experimentally investigated damping measurements to

determine the extent of damage and deterioration in concrete caused by the freeze-thaw

cycle. Using both free- and forced-vibration, it was found that the formation of cracks

in the test beams approximately doubled the damping capacity of the freeze-thaw

resistant beams from 0.04, to 0.07 in the fully-cracked beam.

Hop (1991) found that an increase in prestressing, σp (daN/cm2), caused a considerable

decrease in the logarithmic decrement, δ, according to the following

325 00109.01316.023.18945010 ppp σσσδ −+−= (2.50)

Almansa et al. (1993) undertook free-vibration experimental tests to determine the

effectiveness of using natural frequency and modal damping factors to determine the

degree of cracking of early stripped reinforced concrete beams. Most significantly, they

found that damping was not influenced by age of stripping or by the level of cracking

found in the beams.

Wang et al. (1998) studied the free-vibration dynamic behaviour of reinforced concrete

beams and found that when initial cracks first develop resonant frequency could be 20 –

25 % greater than the original value and damping (dB/ms) could change by a factor of

four. The empirical formula to estimate the experimentally observed damping

behaviour of the progressively cracked beam in Figure 2.17 was given by

( ) una

crcr M

M δααδβ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−= 122 (2.51)



where δcr is the damping of a cracked beam; α2 = 4; the constant β is between 0.7 to 0.8

but was not described; δun is the decay rate of an uncracked beam; Mcr and Ma are the

cracking and applied bending moments in Nm.

The primary difficulty with Equation 2.51 is that it does not describe how to determine

the uncracked damping capacity (δun) and would therefore be difficult to use for design

purposes.

D

ecay

rate

in d

B/m

sec

0

0.2

0.4

0.6

0.8

0 50 100 150 200Bending moment in Nm

Proposed curve to fitexperimental damping data(δcr)

δcr is the decay rate ofthe beam

Figure 2.17: Damping in Beams Without Load as a Function of the Maximum Load

(Wang et al., 1998)

Chowdhury (1999) undertook free-vibration experiments on full-scale box beams. The

research suggested that the level of cracking that exists in a reinforced or partially

prestressed concrete beam is of primary significance in the damping response of that

beam according to

rW205.010075.0 ×=δ (2.52)

where Wr is the average residual crack width in mm.

According to Chowdhury (1999), at low or no residual crack widths, the measured

damping values varied widely for partially-prestressed beams of similar geometric

properties. He suggests that this phenomena may be due to the variation in internal

cracking due to different levels of prestressing in the beams. Similar to Wang et al.’s

(1998) equation, Equation 2.52 also does not provide a method to calculate the

uncracked logdec (δun), but rather ‘suggests’ a base logdec value of 0.075 for the

uncracked condition.



Ndambi et al. (2000) found from concrete beam tests that damping increased with

excitation amplitude, and that modal damping ratios were highly influenced by non-

linear effects, and are, therefore, highly subjective and difficult to estimate.

Damping capacity experiments of small plain mortar beams were undertaken by Wen

and Chung (2000). It was found that damping increased three-fold with the embedment

of steel reinforcing bars into the mortar. Damping was also increased by at least two

orders of magnitude with the addition of silica fume.

Yan et al. (2000a) found that damping was doubled in the polyolefin fibre-reinforced

concrete as compared to plain reinforced concrete specimens. Further work of Yan et

al. (2000b) found that fibres with a crimped or wavy surface produced a significant

increase in damping and a discernible decrease in response frequency.

Shield (1997) undertook experimental tests on two prestressed concrete beams to assess

the practice of using acoustic emissions as an indicator of the relative ‘health’ of the

beams. The test results clearly showed that the formation and propagation of cracks in

concrete is preceded by a significant increase in acoustic emission activity (AEA) rate

(i.e. a measurement similar to damping).

The primary damping variables identified from the above literature review can be

classed into three groups: beam constituent materials, testing procedures, and the effect

of cracking. Significantly, there is a lack of research into the prediction of damping

capacity prior to the construction of the beam for design purposes. That is, the initial

damping value of the ‘untested’ beam needs to be established. In summary, the

following overall conclusions on member damping can be made:

(1) Beam Constituent Materials: The literature did not fully investigate the effect of

age, concrete compressive strength (fcm), percentage of tensile reinforcement (ρt), or

beam dimensions on damping capacity, which are the fundamental features defining

the basic damping capacity of a member. Of the limited research on the influence of

the amount of steel, Flesch (1981) found that damping may decrease slightly, or

increase strongly, depending on the state of cracking, whereas Wen and Chung

(2000) found that damping decreased by three orders of magnitude with the addition

of reinforcing bars.



(2) Testing Procedures: In terms of testing procedures, test results are also

inconclusive. For example, James et al (1964) found that damping (δ) is

independent of vibration amplitude, whilst Yan et al. (2000a,b) found that damping

increases as the maximum response amplitude increases. Penzien (1964) found that

when undertaking both forced- and free-vibrations on the same beam, the damping

value for free-vibration was much greater than that for forced.

(3) Effects of Cracking: An interrelationship exists between damping and cracking,

however the exact nature of this relationship has still not been confirmed in the

literature. Dieterle and Bachmann (1981) found that damping can sink to a lower

value than the initial damping capacity with cracking present, whilst Wang et al.

(1998) found the opposite.

2.5.3 Structural Damping

The most common full-scale experiments on damping in concrete structures have been

conducted on bridges and buildings. Generally, the intent is to develop damping

databases that assist structural designers in assigning nominal damping values in a

dynamic analysis. Assigning damping values is not easy to achieve because the

fundamental knowledge about essential damping mechanisms of individual structural

components is not yet fully understood.

Jeary (1974) took frequency and damping measurements on full-scale reinforced

concrete multi-flue chimneys using accelerometers. It was found that the value of

logdec = 0.06, commonly adopted in industry for multi-flue chimney design was

excessive and a more appropriate value of δ = 0.03 was suggested.

Leonard and Eyre (1975) took damping and frequency measurements from eight full-

scale box girder bridges. It was concluded that the damping of bridges is significantly

influenced by construction conditions therefore damping values cannot be assigned until

the mechanisms of damping for individual bridge components are fully understood.

Douglas et al. (1981) undertook dynamic field tests on a reinforced concrete bridge,

producing transverse vibrations by quick-release pullback using tractors, vertical

vibrations by truck traffic and by dropping a sand-laden truck onto the bridge. The

modal data was useful for assessing whether the concrete bridge had been overloaded or

damaged at any stage during its lifetime.



From experimental studies of twenty-one full-scale footbridges under pedestrian

excitation, Wheeler (1982) found that logdec was affected primarily by the first mode of

frequency (f1) rather than the construction materials. The variation of damping with

frequency was organised into categories that aid the design of footbridges to remain free

from damaging vibrations induced by pedestrian traffic.

Jeary (1986) proposed two formulae for the calculation of damping for a particular

building, vibrating at a specific amplitude. The equations were divided into two parts:

a) those buildings with low-amplitude damping; and b) those whose damping increases

with amplitude.

Lagomarsino (1993) found that the mechanisms which produce dissipation during

oscillation in a building are varied in nature and include: (a) damping intrinsic to the

structural material; (b) damping due to friction in the structural joints and between

structural and non-structural elements; (c) energy dissipated in the foundation soil; (d)

aerodynamical damping; and (e) passive and active dissipation systems.

Brownjohn (1994) examined published suspension bridge damping databases in order to

determine the primary factors contributing to the damping characteristics exhibited by

suspension bridges. In practice, this is extremely difficult because a suspension bridge

is comprised of friction and bearing joints, construction joints, hysteresis damping in

wire hangers, aero-dynamic damping and foundation damping. Therefore, determining

the overall damping cannot be defined by a small set of certain factors, such as material

or dimensions alone.

Lutes and Sarkani (1995) assessed the effect that building foundations have on the

damping characteristics of an entire structure in a parametric study. It was suggested

that the two separate dynamic characteristics of both the structure and foundation should

be merged to obtain a mathematical expression for the whole system.

Denoon and Kwok (1996) and Glanville et al. (1996), developed and installed

equipment for the measurement of the dynamical response of an 84 m high reinforced

concrete control tower. The focus of the investigation was on the effects that different

wind turbulence regimes, natural frequencies, mode shapes and structural damping,

have on the code specified serviceability criteria for human comfort.



Full-scale experimental data was collected by Suda et al. (1996) on the dynamic

properties of 123 steel structure buildings and 66 reinforced concrete buildings in Japan.

It was found that the damping ratio is dependent, to varying degrees, on building height,

foundation height, building usage, vibration amplitude and type of damping evaluation

method. Despite the large amount of collected data, the authors were unable to develop

any relationship between damping and these variables.

Using data collected from a 78-storey high-rise building in China, Fang et al. (1998)

developed an empirical equation to predict the value of damping at high amplitudes of

vibration in reinforced concrete buildings (using Jeary’s (1986) proposal that damping

is highly amplitude dependent).

Boroschek and Yáñez (2000) obtained strong earthquake motion and ambient vibration

records of 22-storey high Chilean buildings. The data was used to determine damping

ratios for the whole structure and to add to the database used for the dynamic modelling

of a Chilean building’s response to seismic actions.

Pagnini and Solari (2001) found that the availability of experimental data on the

damping of steel poles and tubular towers is extremely rare. In response to this, full-

scale testing was conducted, where it was found that damping (in terms of logdec),

increased considerably with motion amplitude.

From this overview of structural damping, it may be concluded that a wide range of

different concrete structures have been studied by many different researchers. Table A.4

in Appendix A presents a summary of selected structural damping studies. Most

significantly the studies indicate that it is difficult to undertake a meaningful dynamic

analysis without in-depth knowledge about the damping of the individual structural

components.

2.6 Summary

In this Chapter, published research on the damping behaviour of concrete materials,

members and structures has been briefly reviewed. From the discussions it was deduced

that:

• free-vibration excitation is an appropriate experimental technique for the



determination of damping;

• there is no published research focussing on the ‘untested’ damping capacity;

• even though it has been established that damping is affected by the damage in the

‘tested’ beam, a method to describe the relationship has yet to be proposed;

• For both ‘untested’ and ‘tested’ concrete beam damping, variables such as the effect

of constituent materials (i.e. high-strength versus normal-strength concrete and 400

MPa versus 500 MPa reinforcing steel) on the damping capacity has not been

adequately examined; and

• the calculation of member damping needs to be better understood before structure

damping can be examined.

These points form the basis of the pilot study presented in Chapter 3.


Chapter 3: Theoretical Considerations 3-1

CHAPTER 3

Theoretical Considerations

3.1 General Remarks

It is evident that there have been many damping studies undertaken. However, many of

the studies are conflicting and have yet to determine the primary variables responsible

for the observed damping mechanisms in concrete beams. In view of this, a

comprehensive experimental programme designed to examine all of the possible

variables should be conducted. Before attempting the comprehensive experimental

programme, a pilot study aided in identifying the major variables.

3.2 Pilot Study

A pilot test programme, involving six full-scale reinforced concrete test beams (BI-1 to

BII-6 in Table 4.1), was carried out to investigate the literature review findings

presented in Chapter 2. It was found that:

(a) previous studies cannot verify the accuracy of their reported logdec making it

difficult to compare to the current test data;

(b) logdec is dependent upon the stage of testing from which is taken. Previous

methods are not able to calculate the logdec of the concrete beams at any specific

stage of testing; and

(c) all beams experience different damage mechanisms according to their

construction, constituent materials, test set-up and loading regime, which will

affect logdec and cannot be explained by direct measures of cracking.

These findings formed the basis of the remaining experimental and analytical

investigations and are described in more depth in the following sections.



3.2.1 Verifying the Accuracy of Logdec

The traditional logarithmic decrement technique (TLT), may be employed for any type

of vibrational response experiment (Section 2.4.1). The TLT utilises the free-vibration

decay curve for the extraction of the inherent damping capacity, logdec, of that member

(Cole and Spooner, 1968; Leonard and Eyre, 1975). Generally, for systems exhibiting

less than 0.5% of critical damping, which is generally so in concrete structures, this time

domain technique is vastly superior to frequency domain techniques, such as the

Bandwidth method (Fahey and Pratt, 1998a,b).

Despite the use of the TLT method by previous researchers, it was difficult to make

meaningful comparisons between the pilot study test data and various published

damping results. Table 3.1 shows some conflicting basic data within the literature that

describes damping levels of both the ‘uncracked’ and ‘cracked’ conditions for

reinforced and prestressed concrete beams. Note that for the ‘uncracked’ condition

there is up to a ten-fold difference in the experimental damping levels between Hop’s

(1991) and Dieterle & Bachman’s (1981) results. For the ‘cracked’ condition similar

large discrepancies were found. Plunkett (1960), identified a similar problem, by

investigating which damping result would be most correct amongst different results for

the same material. Chapter 5 is devoted to resolving this issue.

Table 3.1. Summary of Historical Experimental Damping Research ‘Uncracked’ ‘Cracked’

Researcher Beam Type (δuncr)* Eqn. ? (δcr)* Eqn. ?

Dieterle & Bachman (1981) RC 0.038 Y 0.06 Y

Hop (1991) RC 0.11-0.33 N ~0.224 N

Wang et al. (1998) RC 0.02-0.04 N ~0.032 Y

Chowdhury (1999) RC 0.075 N ≥ 0.075 Y

* All damping values expressed as logarithmic decrement (logdec) in the uncracked (δuncr) or cracked condition (δcr)

3.2.2 Logdec versus Stage of Testing

Another difficulty identified with previous damping research is that they focussed on

individual stages of testing and did not seem to link all the stages. For instance, Hop

(1991) reported on the effect of prestressing force with respect to damping capacity,

however, it was unclear whether the beams were ‘tested’, ‘untested’, ‘cracked’ or

‘uncracked’.



It is therefore proposed in this thesis that damping falls into one of two categories,

‘untested’ and ‘tested’. The ‘tested’ category further contains the ‘uncracked’ and

‘cracked’ sub-categories.

The ‘untested’ beam classification encompasses beams that have not yet received any

service load. It is essential to ascertain the “start” value of damping in a beam based on

its material properties. This category would utilise material damping research

information on concrete.

The ‘uncracked’ beam sub-category covers those beams that have been in service but

have not yet received significant loading to ‘crack’ them. This sub-category is the most

ambiguous, because beams can be in service for many years without any outward signs

of cracking but contain micro-cracking or some form of internal micro-damage.

The ‘cracked’ beam category is self-explanatory. The presence of cracks can suggest

that the beam has experienced a significantly damaging event in the past or, the beam

could be suffering from corrosion, poor concrete quality, etc.

3.2.3 Damage Mechanisms in Concrete Beams

Although some previous work (Chowdhury, 1999) advocated cracking as being the

most appropriate method by which to calculate logdec, the following highlight why this

is not advocated:

(1) Significant disagreements exist regarding the nature of the crack-dependent

damping mechanism. For instance, Dieterle and Bachman (1981) suggest that,

after cracking, damping can sink to a value lower than that to begin with, whilst

various other researchers suggest the opposite (Askegaard and Langsœ, 1986;

Almansa et al., 1993);

(2) Crack-dependent damping does not fully explain the behaviour of many beams

without obvious cracking, such as prestressed beams that may never exhibit

obvious in-service cracking or damage, yet they increase in damping capacity

(Hop, 1991); and

(3) The selection of an appropriate crack-width equation is difficult practically when

many different methods and types of loading can be used to calculate crack

width.



3.2.4 Residual Deflection

This thesis focuses primarily on the use of residual deflection to predict damping. It is

considered that residual deflection can be used for the ‘tested’ sub-categories of

‘uncracked’ and ‘cracked’ stress stages more adequately than crack width alone. This

mechanism has not previously been proposed as a method for calculating logdec.

In many practical situations, certain structural members will be subjected to repeated,

damage inducing impact or fatigue loadings (Dexter and Fisher, 1997). The

accumulated damage that results from repetitive loads is reflected in a flexural member

in the form of non-recoverable deflection. This condition is termed here as residual

deflection and as briefly discussed in Chapter 1, for the current testing regime the term

residual deflection does not include a study of the effects of long-term loading, cyclic

loading, creep and shrinkage effects or stress relaxation in the prestressed beams.

Furthermore, it does not consider the special case of load-reversal, which can occur

during events such as earthquakes.

For further definition of residual deflection refer to the generalised residual load-

deflection (L-D) curve for a two-point loaded simply-supported rectangular reinforced

concrete beam in Figure 3.1. The solid line traces the load deflection history from

linear to non-linear behaviour where the beam becomes cracked. If the load is released

from point C, the load path will return along the dotted line to point D. A residual

deflection would remain (Distance A-D) due to incomplete crack closure. It has also

been suggested that the amount of recovery of deflection is a direct indication of the

integrity of the bond between concrete and steel (Swamy and Anand, 1974).

It is proposed here that the easiest method by which to estimate the residual deflection is

to correlate it to the instantaneous deflection using the relationship in Equation 3.1.

∆r = f1 (∆i) (3.1)

where the residual deflection ∆r is a function of the instantaneous deflection ∆i for the

experimental programme described in this thesis. This relationship is established

experimentally in Chapter 7.

3.3 The Total Damping Capacity Equation

The proposed equation to calculate the total logdec of a concrete beam at any stage of



its service life will be the sum of the ‘untested’ and ‘tested’ components.

δtotal = δuntest + δtest (3.2)

The presentation of the theory for the δuntest and δtest components are presented below.

.

Py = Yield LoadPs = Service LoadPcr = Cracking Load

Fully Cracked

Load

, PPy

Ps

Pcr

Deflection, ∆

. ..A

B

C

D

Residualdeflection, ∆r

PartiallyCracked

Figure 3.1: Schematic Residual Load-Deflection Curve for Concrete Beams

3.3.1 ‘Untested’ Beams

One of the most elementary studies on material damping is that by Lazan (1968) who

described reinforced concrete as being a composite material and stated that the

properties, number and location of the individual reinforcing bars in the concrete must

be known before the damping properties of the reinforced concrete can be determined.

This particular statement has not been researched further in any subsequent studies.

Lazan (1968) further stated that the material damping of any whole composite may be

determined by summing the damping contribution of each component part or member.

Even though a method for doing so was not provided, the research of Lazan (1968)

does, however, provide a starting point for the development of the theory for the

calculation of the ‘untested’ damping capacity (δuntest) for the present research.

Following the findings of Lazan (1968), the damping capacity, in terms of logdec (δ), of

an ‘untested’ concrete beam, is expected to incorporate

δuntest = f2 (ρ, fcm) (3.3)



where ρ is the reinforcement ratio and fcm mean concrete compressive strength.

The relative contributions of each of these variables, if any, are to be determined by

experimentation in Chapter 6.

3.3.2 ‘Tested’ Beams

Residual deflection is proposed as the theoretical foundation for the calculation of the

damping capacity for the ‘tested’ beam sub-categories of ‘uncracked’ and ‘cracked’.

Residual deflection will reflect the unseen micro-cracking in an ‘uncracked’ beam,

something that cracking models do not.

Crack closure in reinforced concrete is a complex mechanism where, once a crack has

opened up, small amounts of solid particles become dislodged and prevent the crack

from closing (Neild et al., 2002). At a given level of damage, concrete exhibits energy

dissipation due to the frictional sliding between the crack lips (due to crack closure)

(Ragueneau et al., 2000). This mechanism operates even at very low loads, which are

insufficient to cause visible cracking, for instance in the ‘uncracked’ beam (Zhang and

Wu, 1997; Zhang, 1998). This mechanism also helps to explain why damping increases

from the instance it is first loaded. Additionally, this provides justification for the use

of residual deflection to estimate damping capacity, as opposed to explicit

measurements of crack width.

It has also been shown that the presence of cracks increases the flexibility of a structure.

Thus the experimentally observed changes in the flexural stiffness rigidity can be

interpreted as an indicator of damage in the structure (Jerath and Shibani, 1985; Carr

and Tabuchi, 1993; Pandey and Biswas, 1994; Zhao and DeWolf, 1999). This is why

non-destructive testing is used extensively in industry as an on-site monitoring tool to

detect the changes, and thus an extensive amount of research has been conducted in this

area (Johns and Belanger, 1981; Stephens and Yao, 1987; Ahlborn et al., 1997; Ohtsu et

al., 1998; Wang et al., 1998; Wahab and De Roeck, 1999; Kisa and Brandon, 2000;

Maeck et al., 2000; Ravi and Liew, 2000; Van Den Abeele and De Visscher, 2000;

Dems and Mróz, 2001; Ohtsu and Watanabe, 2001; Razak and Choi, 2001; Tan et al.,

2001; Capozucca and Cerri, 2002; Ndambi et al., 2002).

The relationship that would result between logdec δ and residual deflection ∆r, in the



‘tested’ beam is represented by the following equation:

δtest = f3 (∆r) (3.4)

where the function defining the relationship is determined by experimentation in

Chapter 8.

3.4 Summary

Using the pilot study test results, this Chapter has presented a collection of concepts and

general equations used to develop a unified method for the determination of the

damping capacity of both reinforced and prestressed concrete beams.

The first concept proposed is that damping capacity, as reported in the literature and

observed in the pilot study, can very enormously even though they have been obtained

using the same technique. Chapter 5 presents an in-depth investigation to solve this

problem.

It has also been proposed that damping capacity measurements fall within the ‘untested’

and ‘tested’ stages of beam testing. This division is important as it allows for research

efforts to be compared. The divisions have not been previously proposed.

For the ‘untested’ damping capacity, Chapter 6 presents the quantification of the

contribution by concrete and reinforcement to damping capacity. To investigate the

total damping capacity in Chapter 8, the residual deflection is examined and quantified

in Chapter 7.


Chapter 4: Experimental Programme 4-1

CHAPTER 4

Experimental Programme

4.1 General Remarks

An extensive experimental programme was undertaken to investigate the damping and

deflection behaviour of reinforced and prestressed concrete beams containing normal-

and high-strength concrete and/or 400 MPa or 500 MPa reinforcement. This Chapter

describes the geometrical and mechanical details of the beams, the primary test

variables and details of the constituent materials, the testing regime and the

instrumentation employed. A total of forty-one reinforced and prestressed concrete

beams were tested. These beams were divided into four test series as follows:

B-Series Test Beams: 12 reinforced concrete beams, 6-metres in length;

PS-Series Test Beams: 10 prestressed concrete beams, 6-metres in length;

CS-Series Test Beams: 9 reinforced concrete beams, 2.4-metres in length;

F-Series Test Beams: 10 reinforced concrete beams, 2.4-metres in length, beams

with small self-weight induced stresses.

The beams are grouped so that comparisons can be made between each test series. For

instance, comparisons between the B- and CS-Series beams provides valuable

information on the size effect in reinforced concrete beams, whilst comparisons between

the CS- and F-Series highlight the effect of small self-weight induced stresses.

4.2 Design of Beam Test Specimens

The reinforced and prestressed concrete beams were generally under-reinforced and

designed to fail in flexure, a condition preferred by most national codes of practice.

Some beams were designed to fail in shear and were included for comparative purposes



and also to provide a wider range of different test results. The primary test variables

selected for investigation in the reinforced concrete beams (B-Series, CS-Series and F-

Series) were (note that all test variables implemented are in SI Units):

Concrete compressive strength on day of testing (fcm in MPa);

Nominal reinforcement yield strength (fsy in MPa); and

Tensile reinforcement ratio (ρt = Ast/bd), bar spacing (s in mm) and bar diameter

(Φ in mm).

For the prestressed concrete beams (PS-Series), the primary test variables selected were:

Prestressing force (H in kN);

Prestressing eccentricity (e in mm); and

Concrete compressive strength on day of testing (fcm in MPa).

4.2.1 Geometrical and Mechanical Details

The geometric details of the reinforced and prestressed concrete beams are given in

Figures 4.1 and 4.2. Tables 4.1 and 4.2 present the complete tabulated variables for

each reinforced and prestressed concrete beam, respectively. The calculation of the

section moment of inertias (Ig) and the ultimate moment capacity (Mu) for the design of

each beam may be found in Appendix B.

4.2.2 Primary Test Variables

Figures 4.3 to 4.6 give a diagrammatic summary of primary variables and comparisons

to be investigated within each test series.

4.3 Materials

The concrete was supplied by a local ready-mix supply company (CSR Construction

Materials), and the reinforcing steel was supplied by BHP OneSteel. All materials are

currently in widespread use within the Australian construction industry.

4.3.1 Concrete

The normal- and high-strength concrete mixes ordered from the supplier for the test

beams were required to be 32 MPa and 80 MPa, respectively. The technical data for the

ready-mix concrete, as provided by the manufacturer, is presented in Tables 4.3 and 4.4.

Due to variations in slump values and test dates, a range of normal and high-strength



compressive concrete strengths were obtained (20.0 MPa<fcm<90.7 MPa). Excessive

slump variations are attributed to overzealous truck drivers introducing extra water to

the mix on delivery.

CR SR

TR100100

All dimensions are in mm Indicates reinforcing bar

Variables Defined in Table 4.1:Tension Reinforcement (TR) Compression Reinforcement (CR)Shear Reinforcement (SR)Beam Effective Depth (d)TR Spacing (st) and CR Spacing (sc)Concrete Cover (c) = 20 mm

Section A-A

b

D=300 mm for B-SeriesD= 250 mm for CS- and F-Series

c

d

c

A

AL = 6000 mm for B-SeriesL = 2400 mm for CS- and F-Series

st

sc

D

b = 200 mm for B-Seriesb = 150 mm for CS- and F-Series

Figure 4.1: Geometric Detailing for B-, CS- and F-Series Test Beams

SR

TR100100

All dimensions are in mm Indicates prestressing tendon

Additional Variables Defined in Table 4.2:

Prestressing Force from Tendons (H)Prestressing Eccentricity (e)Concrete Cover (c) = 20 mm

Section A-A

b = 200 mm

D =

300

mm

c

c

A

AL = 6000 mm

150

NA

e

Horizontal spacing between tendons = 40mmVertical spacing between tendons = 20mm

Figure 4.2: Geometric Detailing for PS-Series Test Beams



Table 4.1. Geometrical and Reinforcement Details – RC Beams Beam d (mm) TR CR SR TR Spacing

st, (mm) CR Spacing

sc, (mm) BI-1 264 4Y20 2Y12 R6@90mm 22.7 124 BII-2 264 3N20 2N12 R6@90 44.0 124 BI-3 262 3Y24 2Y12 R6@90 38.0 124 BII-4 262 2N24 2N12 R6@90 100.0 124 BII-5 264 4N20 2N12 R6@90 22.7 124 BII-6 264 3N20 2N12 R6@90 44.0 124 BI-7 266 2Y16 2Y12 R6@90 116.0 124 BII-8 266 2N16 2N12 R6@90 116.0 124 BI-9 262 2Y24 2Y12 R6@90 100.0 124 BII-10 262 2N24 2N12 R6@90 100.0 124 BII-11 264 3N20 2N12 R6@90 44.0 124 BII-12 264 4N20 2N12 R6@90 22.7 124 CS1 214 3N20 2N12 R6@90 14 74 CS2 214 3N20 2N12 R6@90 14 74 CS3 210 3N20 2N12 R10@125 13 66 CS4 213 2N24 None None 52 - CS5 213 2N24 None None 52 - CS6 213 2N24 None None 52 - CS7 212 2N24 2N12 R6@90 40 74 CS8 212 2N24 2N12 R6@90 40 74 CS9 208 2N24 2N12 R10@125 32 66 F1 220 3Y20 None None 25 - F2 216 2Y16 1Y6 R6@150 66 98 F3 210 2Y20 2Y10 R10@110 50 70 F4 222 3Y16 None None 31 - F5 220 2N20 None None 35 - F6 208 2N24 None None 31 - F7 220 3N20 None None 25 - F8 208 2N24 2N12 R10@125 42 66 F9 210 2N20 2N12 R10@125 50 66 F10 214 2N20 2N12 R6@110 58 74

Table 4.2. Geometrical and Reinforcement Details – PSC Beams

Beam

Prestressing Force

H (kN)

Prestressing Eccentricity

e (mm)

Initial Camber due to Prestress

ϕ (mm)

d (mm) TR # CR SR

PS1 239 111.7 8 261.7 9HS5 None R10@225mm PS2 293 110.5 9 260.5 11HS5 None R10@225 PS3 346 97.0 9 247.0 13HS5 None R10@225 PS4 585 80.5 14 230.5 22HS5 4Y12 R10@225 PS5 612 60.0 15 210.0 23HS5 None R10@225 PS6 400 99.7 11 249.7 15HS5 None R10@225 PS7 450 97.5 14 247.5 13HS5 None R6@200 PS8 400 80.0 13 230.0 15HS5 None R6@200 PS9 346 90.0 7 240.0 13HS5 None R6@200 PS10 480 90.0 17.5 240.0 18HS5 None R6@200

# The first number refers to the number of tendons, the last number refers to the diameter in mm of the tendon (see Figure 4.4 for tendon layout details).



Variables : ρt (reinforcement ratio)φ (bar diameter) s (bar spacing) f’c (concrete strength (E)) fsy (steel strength)

Investigate:f’c: BII-2 vs BII-6 f’c, fsy : BI-1 vs BII-5 BII-2 vs BII-11 BI-1 vs BII-12 BII-4 vs BII-10 BI-4 vs BII-9 BII-5 vs BII-12 fsy : BI-7 vs BII-8 BII-6 vs BII-11 BI-9 vs BII-10

30.0 MPa400 MPa

4Y20= 1257 mm2

2Y12’s

30.0 MPa500 MPa

3N20 = 942 mm2

2N12’s

23.1 MPa400 MPa

3Y24= 1357 mm2

2Y12’s

23.1 MPa500 MPa

2N24 = 905 mm2

2N12’s

41.5 MPa500 MPa

3N20 = 942 mm2

2N12’s

41.5 MPa500 MPa

4N20 = 1257mm2

2N12’s

BI-1 BII-2

BI-3 BII-4

BII-5 BII-6

BII-5Comparefsy, , f’c

BII-6Compare

f’c

BI-9Compare

fsy , f’c

BII-10Compare

f’c

BII-12Compare

f’c

BII-11Compare

f’c

Compare fsy

64.5 MPa400 MPa

2Y16 = 402 mm2

2Y12’s

64.5 MPa500 MPa

2N16 = 402 mm2

2N12’s

53.0 MPa400 MPa

2Y24 = 905 mm2

2Y12’s

53.0 MPa500 MPa

2N24 = 905 mm2

2N12’s

90.7 MPa500 MPa

3N20 = 942 mm2

2N12’s

90.7 MPa500 MPa

4N20 =1257 mm2

2N12’s

BI-7 BII-8

BI-9 BII-10

BII-11 BII-12

Compare fsy BII-4

Compare f’c

BII-5Compare

f’c

BI-1Compare

f’c, fsy

BII-6Compare

f’c

BII-2Compare

f’c

BII-4Compare

f’c, fsy

BII-12Comparefsy, , f’c

BII-11Compare

f’c

BII-1Compare

fsy , f’c

BII-2Compare

f’c

Compare ρt

Compare ρt

Compare ρt

Figure 4.3: B-Series – Primary Test Variables



69.82 MPaH = 612 kNe = 60.0 mm

23 HS5 =452 mm2

69.82 MPaH = 400.0 kNe = 99.7 mm

15 HS5 =295 mm2

PS1 PS2

PS5 PS6

60.63 MPaH = 239 kN

e = 111.7 mm

9 HS5 =177 mm2

60.63 MPaH = 293 kN

e = 110.5 mm

11 HS5 = 216 mm2

PS3 PS4

60.18 MPaH = 585 kNe = 80.5 mm

22 HS5=432 mm2

4Y12’s60.18 MPaH = 346 kNe = 97.0 mm

13 HS5=255 mm2

CompareH

CompareH, e

PS7Compare

H

PS7 PS8

f’c = 52.5 MPaH = 450 kNe = 97.5 mm

13 HS5 =255 mm2

f’c = 52.5 MPaH = 400 kNe =80.0 mm

15 HS5 =295 mm2

PS9 PS10

f’c = 83.5 MPaH = 480 kNe = 90 mm

18 HS5 =353 mm2

f’c = 83.5 MPaH = 346 kNe = 90.0 mm

13 HS5 =255 mm2

PS3Compare

H

CompareH, e, f’c,

Comparef’c ,e

Investigate: H: PS1 vs PS2 H , e: PS5 vs PS6 PS3 vs PS7 PS7 vs PS8

f’c, e : PS3 vs PS9 PS6 vs PS8

Variables: H (prestressing force) e (eccentricity) f’c (concrete strength (E))Secondary Variables: ρt (tensile reinforcement ratio)ρc (compression reinforcement ratio)

PS3Compare

f’c, e

PS9Compare

f’c, e

PS9Compare

f’c, e

CompareH

CompareH , e, ρc

Figure 4.4: PS-Series – Primary Test Variables



Variables : fsy (steel strength)ρ (reinforcement ratio) φ (bar diameter)s (bar spacing) f’c (concrete strength (E))Loading Conditions

22.5 MPa400 MPa

3N20 = 942 mm2

2N12

CS2

32.0 MPa500 MPa

2N24 = 905 mm2

CS4

31.5 MPa500 MPa

2N24 = 905 mm2

2N12

CS7

315 MPa500 MPa

2N24 = 905 mm2

2N12

CS8

31.5 MPa500 MPa

2N24 = 905 mm2

2N12

CS9

22.5 MPa400 MPa

2N12

CS1

3N20 = 942 mm2

22.5 MPa400 MPa

2N12

CS3

3N20 = 942 mm2

32.0 MPa500 MPa

CS5

2N24 = 905 mm2

32.0 MPa500 MPa

CS6

2N24 = 905 mm2

Beam Length

Load

Beam Length

Load400 mm

Beam Length

Load800 mm

Beam Length

Load

Beam Length

Load500 mm

Beam Length

Load700 mm

Beam Length

Load

Beam Length

Load500 mm

Beam Length

Load700 mm

FLEXURALCS1,2,3CompareLoading

Conditions

SHEARCS4,5,6CompareLoading

Conditions

FLEXURALCS7,8,9CompareLoading

Conditions

Figure 4.5: CS-Series – Primary Test Variables



Variables : ρc (compression reinforcement ratio)ρt (tensile reinforcement ratio)s (bar spacing) f’c (concrete strength (E))φ (bar diameter)

SHEAR32.0 MPa500 MPa

1N20 =314 mm2

F1

FLEXURAL 32.0 MPa500 MPa

F2

2N16 =402 mm2

1N6

FLEXURAL32.0 MPa500 MPa

F3

2N20 =628 mm2

2N10


F4

3N16 =603 mm2


2N20 =628 mm2

F5


2N24 =905 mm2

F6


3N20 =942 mm2

F7


2N24 = 905 mm2

2N12

F8


2N20 = 628 mm2

2N12

F9


2N20 = 628 mm2

2N12

F10

F4Compare

ρt

F5,6,7Compare

f’c, ρt

F5,6Compare

ρt

F1,4Compare

f’c, ρt

F6,7Compare

ρt

F1,4Compare

f’c, ρt

F5,7Compare

ρt

F1,4Compare

f’c, ρt

F5,6,7Compare

ρt

F1Compare

f’c, ρt

F3,8,9,10Compare

ρc, ρt

F2Compareρc, ρt

F8Compareρc, ρt, f’c

F9,10Compareρc, f’c

F2,3Compareρc, ρt, f’c

F9,10Compare

ρt


F8Compare

ρc


F8Compare

ρc

Figure 4.6: F-Series – Primary Test Variables



During the casting of each beam specimen, six compression cylinders (200 mm depth

and 100 mm diameter) were cast for each and every beam specimen. The cylinders

were cured under exactly the same conditions as the beams. This involved the cylinders

being continuously kept moist with wet hessian sacking. On the day of testing, these

cylinders were used to determine an average for the concrete compressive strength at

testing, fcm (see Table 4.5). The testing of the concrete cylinders was carried out using

the DMG Denison Type 7640 with automatic printout.

4.3.2 Reinforcement

Table 4.6 presents the manufacturers specifications for 400 MPa and 500 MPa

reinforcing steel used. In determining the actual strength and ductility properties of the

three different types of reinforcing bars/tendons used in the current test beams, stress-

strain tests were undertaken by the BHP Laboratories in Brisbane, Australia. The

resulting stress-strain curves for the 400 and 500 MPa reinforcement bars are given in

Figures 4.7 and 4.8 respectively, and for the prestressing steel, Figure 4.9. Each figure

contains two different curves: a) the entire stress-strain curve where the steel peak stress

U may be observed; and, b) a close up of the yield portion of the curve where the yield

strength is denoted by Y. A minimum of ten samples of each bar type was tested. This

approach was used to obtain averages as detailed in Table 4.7.

Table 4.3. Concrete Technical Data – Materials (CSR Construction Materials)

Material No Supplier Works, Quarry, Pit Specification

Type GP Cement 1 QCL Bulwer Island AS3972 Fly Ash 2 Pozzolanic Calide AS3582.1 Silica Fume 3 10 mm Aggregate 4 CSR Readymix Beenleigh AS2758.1 Coarse Sand 5 CSR Readymix Oxenford AS2758.1 Fine Sand 6 CSR Readymix Oxenford AS2758.1 Water Reducing Agent 10 WR Grace Archerfield AS1478/79 Superplasticiser 11 WR Grace Archerfield AS1478/79

Table 4.4. Concrete Technical Data – Mix Design (CSR Construction Materials)

Mass of Materials (kg/m3) Admixtures

Mix Name Slump (mm)

Water (l/m3)

1 2 3 4 5 6 10

(ml/100kg cement)

11 (ml/m3)

N32/10mm 120 115 235 75 - 990 620 310 400 100

S80/10mm 120 100 550 40 60 1060 540 180 400 9000 Note: Coarse and fine sand moisture contents are at 6% and 8%, respectively.



Table 4.5. Details of Test Beams – Concrete

Beam Age at Testing (days)

Date of Testing

Maximum Aggregate Size (mm)

Concrete Density (kg/m3)

Nominal

Concrete Compressive

Strength at Testing fcm (MPa)

BI-1 28 17/5/00 10 2400 30.0 BII-2 28 17/5/00 10 2400 30.0 BI-3 30 14/7/00 10 2400 23.1 BII-4 30 14/7/00 10 2400 23.1 BII-5 28 24/7/00 10 2400 41.5 BII-6 28 24/7/00 10 2400 41.5 BI-7 63 15/5/01 10 2400 64.5 BII-8 63 15/5/01 10 2400 64.5 BI-9 61 29/5/01 10 2400 53.0 BII-10 61 29/5/01 10 2400 53.0 BII-11 90 9/7/01 10 2500 90.7 BII-12 90 9/7/01 10 2500 90.7

PS1 36 18/9/00 10 2400 60.6 PS2 36 18/9/00 10 2400 60.6 PS3 28 19/9/00 10 2400 60.2 PS4 28 19/9/00 10 2400 60.2 PS5 35 21/9/00 10 2500 69.8 PS6 35 21/9/00 10 2500 69.8 PS7 45 17/8/01 10 2400 52.5 PS8 45 21/8/01 10 2400 52.5 PS9 60 10/10/01 10 2500 83.5 PS10 60 9/10/01 10 2500 83.5

CS1 35 18/9/00 10 2400 22.5 CS2 35 18/9/00 10 2400 22.5 CS3 35 18/9/00 10 2400 22.5 CS4 28 1/10/01 10 2400 32.0 CS5 28 1/10/01 10 2400 32.0 CS6 28 1/10/01 10 2400 32.0 CS7 35 8/10/01 10 2400 31.5 CS8 35 8/10/01 10 2400 31.5 CS9 35 8/10/01 10 2400 31.5

F1 ~ 4 yrs 10/4/02 10 2400 32.0 F2 ~ 4 yrs 10/4/02 10 2400 32.0 F3 ~ 4 yrs 10/4/02 10 2400 32.0 F4 ~ 4 yrs 10/4/02 10 2400 32.0 F5 28 17/4/02 10 2500 72.0 F6 28 17/4/02 10 2500 72.0 F7 28 17/4/02 10 2500 72.0 F8 28 17/4/02 10 2500 72.0 F9 28 17/4/02 10 2500 72.0 F10 28 17/4/02 10 2500 72.0



Table 4.6. Technical Design Data for 400 and 500 MPa Reinforcing Steel (Patrick, 1999)

Ductility Properties Rebar Type Yield Strength

(MPa) #Ratio Rm/Re Elongation (Agt%)

OneSteel TEMPCORE/Microalloy 400Y 400 minimum 1.10 min

16% min on gauge length of 5d at

fracture fsy,L 500 Class L

Low Ductility fsy,U 750 1.03 1.5

fsy,L 500 OneSteel 500PLUS

Class N Normal Ductility fsy,U 650

1.08 6

# fsy,L and fsy,U are the lower and upper characteristics yield strengths.

a)

U

b)

Figure 4.7: Stress-Strain Curve for 400 MPa Reinforcement (BHP Laboratory,

Brisbane, Australia)



a)

b)

Figure 4.8: Stress-Strain Curve for 500 MPa Reinforcement (BHP Laboratory,

Brisbane, Australia)

Figure 4.9: Stress-Strain Curve for Prestressing Tendons (BHP Laboratory, Brisbane,

Australia)



Table 4.7. Details of Test Beams – Reinforcing Bars

Bar Type

Nominal Steel Yield

Strength fsy (MPa)

Steel Peak Stress U (MPa)

Steel Off-Yield Stress

Y (MPa)

Modulus of Elasticity for Steel

Es (GPa)

N 400 526.2 423.5 203.4

Y 500 642.8 569.4 196.8

Tendon 1710 1470.4 1461.5 227.0

4.4 Fabrication

This section contains complete details regarding the construction, casting and curing of

each beam specimen.

4.4.1 Reinforced Concrete Beams

The smooth inner surface of the beam moulds were oiled with laboratory grease to

produce a smooth finish to all sides of the beam specimen. The reinforcement cages

were constructed and carefully placed into the beam moulds (using 20 mm chairs). If

required, small mortar wedges were placed down the sides of the mould to ensure that

the reinforcement cage would stay positioned exactly in the centre of the mould during

pouring.

Concrete was then poured and, using a poker vibrator, vibrated to ensure the concrete

mix was distributed evenly in the formwork. A rubber mallet was pounded along the

exterior mould to ensure that all bubbles were removed from the concrete mix. The top

of the beam was trowelled to a smooth finish. This was to ensure a uniform surface for

the damping hammer impacts.

4.4.2 Prestressed Concrete Beams

In the prestressed beam preparation, the primary steps involved positioning and

prestressing the tendons, fixing the shear reinforcement, pouring the concrete and then

cutting the tendons. The prestressing test rig is diagrammatically shown in Figure 4.10.

The beam casting moulds were first positioned between two prestressing buttresses.

Each prestressing buttress was bolted to a two-metre thick concrete strong floor, 7000

mm apart. Twenty-millimetre thick plywood plates were attached to the ends of the



beams. They contained holes with the number, diameter and spacing corresponding to

the strand arrangements as detailed in Figures 4.2 and 4.4. After positioning the

tendons in the beam moulds, each tendon was anchored to one buttress through male

and female cones at the passive end. Each prestressing buttress that was set-up had a

shearing resistance of 30 tonnes, with the entire prestressing rig able to withstand 60

tonnes of prestressing force. At the active end, the tendons were tensioned by a

hydraulic jack.

Anchor bolts

7.0 metres

1.0 metre thickconcrete strong floor

ButtressSteel reinforced beam mould6.0m length, 300mm deep200mm wide

Male/Femalecones

ElectricHydraulicJackingSystem

Hydraulic jack

PASSIVEEND

ACTIVEEND

Figure 4.10: Rig Used for Prestressing the Tendons

To implement the required extension needed to reach the required prestress force (H), a

Vernier calliper was used. Hook’s Law was used to calculate the required wire

extension, using Young’s Modulus of Elasticity as determined from the steel stress-

strain tests (Table 4.7). Each tendon was prestressed individually in a symmetrical

pattern to ensure that the beam was evenly stressed around its longitudinal axis

eliminating excessive longitudinal eccentricity. The complete calculation details used

for the prestressing extension process are given in Appendix B.

Stirrups intended to limit shear failure were then fixed to the prestressing tendons at the

required spacing and the concrete was cast. Following the minimum concrete curing

period required to achieve minimum concrete compressive strength (see Appendix B for

calculations), the tendons were cut individually using an oxy-acetylene gas flame. This

type of cutting method ensures the gradual transfer of compressive forces to the

concrete. Compression cylinders were tested periodically to monitor when the

minimum compressive strength was reached. Following the tendon cutting, the beams

were ready for testing.



4.4.3 Curing

Immediately following the pouring of the concrete the beams, and their respective

cylinders, were covered with wet hessian sacking. They remained under this condition

for the entirety of the curing period. As indicated in Table 4.5, the concrete beams were

tested between 28 and 90 days. The variable test days were due to the unavailability of

the laboratory facilities or extra curing time to reach the desired strength.

4.5 Test Set-Up

All experiments conducted on the beams used the third point loading test set-up as seen

in Figure 4.11 and Plate 4.1. This kind of loading system is the most common type of

loading arrangement and is favoured for laboratory experiments because it has the

advantage of offering a substantial region of nearly uniform moment coupled with very

small shears (Bungey and Millard, 1996). All of the beams were simply-supported, and

subject to static concentrated point loads of various magnitudes. The load was applied

by a calibrated hydraulic jack with a capacity of 100 tonnes, and transmitted through a

load-cell and thick metal plate to the loading beam, which in turn produced two point

loads on the test beam.

4.5.1 Beam Support System

As shown in Figure 4.11, each concrete beam was simply-supported at each end using

roller bars and angles, with the concrete supports affixed firmly to the floor. This type

of support provides minimum friction and restraining moments, and helps to isolate the

damping measurements to the beam itself. In addition, the steel bearing plates were

grouted to the supports, this was to ensure that the load was uniformly distributed over

the beam, thereby avoiding bearing failure at the supports. Detailed photos of the beam

support system may be seen in Plate 4.2.

4.5.2 Loading Beam Width (LBW)

As shown in Figure 4.11, the LBW’s were designed to vary between each test series so

as to provide information of the effect of bending moment distribution on the

instantaneous and residual deflection. The LBW’s along with shear span length for

every test beam are presented in Table 4.8.

4.5.3 Hammer Excitation Position (HEP)

The test variable of hammer excitation position (HEP) was part of the investigations



because ensuring consistency during experimental testing was always of great concern

in this research. Previous literature has indicated that concrete beam damping should be

relatively constant, regardless of the hammer excitation position (HEP) (Chowdhury,

1999). A portion of the current study however did focus on examining and verifying the

impact of HEP on logdec. Each beam was impacted during testing at various locations

along its length, as indicated in Figure 4.11. Chapter 5 presents the conclusions drawn

on the effect of HEP on damping measurements.

Base plate boltedto floor with 100

tonne capacityeach plate

CL

Hammer Excitation Position(HEP)

Variable

Loading cell = 18.5 kg (All beams)Weight of loading beam= 221 kg for B-and PS-Series Beams

= 70.5 kg for CS- and F-Series Beams

100 tonne capacity cylinder

Loading Beam Width (LBW)

Hammer Weight(HW)

Shear Span (a)

Figure 4.11. Diagram of Beam Test Set-Up

Plate 4.1. Photograph of Beam Test Set-Up



a)

b)

Plate 4.2. Beam Support System a) Roller and b) Knife Supports

Table 4.8. Loading Beam Width Specifications Beam LBW (mm) a# (mm) All B-Series Beams 2000 1900 All PS-Series Beams 2000 1900 CS1 0 1150 CS2 400 950 CS3 800 750 CS4 0 1050 CS5 500 800 CS6 700 700 CS7 0 1050 CS8 500 800 CS9 700 700 All F-Series Beams 700 700

# Represents the shear span – the distance between the support and the loading beam (Figure 4.13).



4.5.4 Hammer Weight (HW)

In the current experimental tests, two different hammer weights were used (HW1=

163.1 g and HW2= 239.5 g). The primary purpose of this was to examine the effect

hammer weight has on the vibration decay curve, and the subsequent calculation of

logdec. According to French (1999) the imparted frequency spectrum and force

duration is affected by hammer tip (i.e. steel versus rubber) and hammer weight.

A discussion on the effect of hammer weight on logdec will also be presented in

Chapter 5.

4.6 Test Procedures

Static load testing of the beams was induced by using the servo-controlled 100 ton

capacity, Enerpac hydraulic jacks. The loading regime followed a set cycle of ‘load on’

and ‘load off’ increments, increasing at each step until beam failure. The ‘load-on’

testing phase was necessary in order to induce various stages of damage to the beam,

which corresponded to in-service loading damage. Secondly, the ‘load-off’ phase

allowed residual deflection and logdec measurements to be taken from the beam.

The loading regime consisted of a series of ‘load-on’ and ‘load-off’ increments. Each

increment was approximately 1/10th of the anticipated failure load. The load was

released at the same rate as it was applied. Between pre-load and first crack the

increments were smaller so that first crack could be accurately identified.

A rest period of approximately one-minute was given prior to the taking of

measurements at each ‘load-on’ and ‘load-off’ position. This was a sufficient amount of

time that would account for the primary phase of creep in the beams immediately

following initial loading (Penzien and Hansen, 1954).

At each ‘load-on’ position, deflection, crack widths by crack microscope and

reinforcement strains were obtained. At each ‘load-off’ position, deflection, crack

widths by crack microscope, reinforcement strains and free-vibration decay curve

signatures were obtained. The regime was repeated at ever increasing intervals until the

beam failed. All of the raw data taken from the beam is presented in Appendix F.

4.7 Instrumentation

The current testing programme involved a number of types of measurements.



Generally, these measurements were of two categories: dynamic and static. The

dynamic measurements were of the free-vibration decay curves for damping calculation,

and the static measurements included deflections, crack widths and reinforcement

strains. Figure 4.12 presents a diagram with the locations of all the relevant testing

equipment.

4.7.1 Damping

One of the most popular excitation techniques used for modal damping analysis is

impact, or hammer excitation. The waveform, produced by an impact, is a transient (or

short duration) energy transfer event. Figure 4.13 shows a common type of energy

frequency spectrum produced by a short duration impact. The duration, and thus the

shape of the spectrum of an impact, is determined by the mass and stiffness of both the

impactor and the structure (Døssing, 1988a,b). As outlined previously, care was taken

to ensure the relative stiffness of the beam testing structure was much greater than that

of the beam itself, helping isolate vibration to within the beam. Also, it is extremely

important that the impact technique is as consistent as possible, so to ensure the

repeatability of the acquired signatures (Bhuvanagiri and Swartz, 2000).

L/3 L/3

Beam Vibration DetectedBy Accelerometer

Model 353A Quartz ShearMode Accelerometer(PCB® Piezotronics)

Vibration signalrecorded onoscilloscope

TDS 460A DigitisingOscilloscope (Tektronix)

Vibration decay curvenow ready for analysis

on PC(i.e. TLT or DCM)

Dial Gauge Deflection

LVDT Mid-Span Deflection

ICP® Impulse-Force Hammer (PCB® Piezotronics)

Crack Width

Microscope Levelof TR

Figure 4.12. Location of Beam Testing Equipment



TDuration

time

F(t)

Figure 4.13. Impact Energy Frequency Spectrum Induced by Hammer Excitation

(Døssing, 1988b)

As discussed in Section 4.5.4, the current experimentation included a research

component that examined the effect of the hammer weight on the quality and accuracy

of the free-vibration decay curves. In general, massive structures, with lower stiffness,

require the use of the extender (added weight) and soft impact tip (also employed) to

adequately excite low frequency resonance (Døssing, 1988b). For the current

experiments, an ICP® Impulse-Force Hammer (PCB® Piezotronics brand) was used to

excite the test beam into free-vibration. The technical specifications, for this hammer,

as provided by the manufacturer, are shown in Table 4.9.

Table 4.9. Specifications for ICP® Impulse-Force Hammer (PCB® Piezotronics, 1992) Model No: 086CO4Hammer Range: 8 kHz (approx)Hammer Range: 4400N (5V output) Hammer Sensitivity: 1.2 mV/N (approx) Resonant Frequency: 31 kHzHammer Mass (excluding counterweight): 163.1 gMass of Hammer Counterweight: 76.4 g

This particular type of hammer is commonly employed in industrial applications

because of its simple implementation and suitability for on-site monitoring. The impact

hammer supplies a short impulse, with a large frequency range, including the first

several resonance frequencies of the beam. Usually, in vibration monitoring, the first

several modes of the beam are most important, since their energy content is

comparatively large when the beam vibrates. A Fast Fourier Transform (FFT) of the



vibration decay curve signature, is used to convert the time signal into its component

frequencies, and confirms whether a single dominant 1st modal frequency response has

been induced in the beam. Details of the procedure for doing this will be presented in

Section 5.3.

Figure 4.13 shows the frequency response of a beam, excited by a hammer impact,

which is detected by a Model 353A Quartz Shear Mode Accelerometer (PCB®

Piezotronics). These accelerometers convert mechanical accelerations into proportional

electrical signals for measurement and analysis. The electrical signal that is generated,

is the result of a force imposed upon the quartz crystal by a mass, which undergoes

acceleration.

As shown previously in Figure 4.12, the TDS 460A Digitizing Oscilloscope (Tektronix,

1995) captured the acquisition of the vibration response. The waveform acquisition

details for the current experiments included:

A sampling rate of 5 kilosamples/second (5000 Hz) and a high-resolution

sampling mode. The sampling rate of the oscilloscope is, according to the

Shannon Sampling Theorem, required to be at least twice that of the fundamental

frequency (McClellan et al., 1999). For the current test beams the fundamental

frequency of the test beams was approximately 800 Hz, therefore the sampling

frequency of 5000 Hz more than satisfied this criterion (>1600 Hz).

The waveform record length selected was 450 data points; and

Edge triggering, was the waveform acquisition method used, and it is through this

method that the oscilloscope knows when to begin acquiring and displaying a

waveform.

Shown in Plate 4.3, is a photograph of the oscilloscope set-up during experimentation.

4.7.2 Crack Width

Both the instantaneous and residual crack widths were measured using a crack

microscope (correlating to the ‘load-on’ and ‘load-off’ testing phases). The crack

widths were measured at the level of the tension steel to maintain consistency in the

readings. For the prestressed beams, crack widths were measured at a similar distance

from the bottom beam edge as for the RC beams. The crack width microscope has an

accuracy of 0.005 mm.



Plate 4.3. Test Set-Up of Oscilloscope during Experimentation

The measurement of crack widths, using the crack microscope, required the identifying

of a minimum of ten cracks per beam face, located within the constant moment region.

Each of these cracks were marked and measured at every ‘load on’ and ‘load off’

position. This approach ensured consistency and accuracy in the test measurements.

The experimental use of the crack microscope is shown in Plate 4.4.

4.7.3 Crack Patterns

A number of photographs were taken of the beams following failure. The primary

reason for this was to demonstrate the exact type of failure they experienced, to provide

information on the cracking pattern and to ensure that both beam sides experienced

symmetrical crack patterns. The photographs may be found in Appendix C.

4.7.4 Deflection

Deflection measurements were taken at the third points of the beam as shown in Figure

4.12. A linear variable differential transducer (LVDT) recorded the mid-span deflection

of the beam. The deflection at the third points under the loading beam nodes, was

monitored with dial gauges. This monitoring was to ensure that beam loading and

deflection was symmetrical about the longitudinal x- and y-axes.



Plate 4.4. Crack Width Microscope

4.8 Summary

This Chapter described the extensive experimental programme for the investigation into

the damping, deflection and cracking behaviour of the reinforced and prestressed

concrete test beams. Geometrical and mechanical details of the experimental beams and

constituent materials were outlined, and the beam test set-up was explained.


Chapter 5: A Method for Extracting Damping Capacity 5-1

CHAPTER 5

A Method for Extracting Damping Capacity

5.1 General Remarks

This Chapter initially presents a detailed review of the experimental implementation of

the logdec technique (TLT) and describes the shortcomings of the use of the TLT by

previous researchers. To investigate these identified shortcomings, another analytical

technique based on the free-vibration decay curve, the “Decay Curve Method” (DCM),

was devised.

The next portion of the Chapter establishes a ‘set of rules’ governing the determination

and reporting of damping capacity to ensure confidence in the ‘accuracy’ of the

presented data.

Finally, the effect of experimental variables such as hammer weight and hammer

excitation position are also explored to determine if they themselves, affect the

calculation of logdec.

5.2 Experimental Techniques

A series of beam vibration tests, using a wide variety of beam types, was undertaken.

Four types of beams, impacted as various HEP, were used for the experiments as shown

in Figure 5.1. The different beam materials, shapes and sizes, were necessary to ensure

that the test findings were consistent and repeatable.

In this verification stage, all of the damping measurements were taken immediately

prior to testing, when the beam was in its final test set-up position.



150250

100 100L=2400

CL Concrete Beam

RC Beam - Series CS

RC Beam - Series BL=6000

200300

100 100CL

Concrete Beam

200

PS Concrete Beam - Series PS

300

100 100L=6000

CL

Concrete Beam

Beam and Experimental Variables

+++ B1 to B640011001800AC B

B/15B/6B/3

+++ B760010001300AC B

B/10B/6B/4.6

+++ B863011001300AC B

B/10B/6B/4.6

+++ B9 & B10100012501500

AC B

B/6B/5B/4

+++ B11 & B12100012501500BD C

B/6B/5B/4

+750A

B/8

+ PS3 to PS61150

A

B/5.22

+++ PS7 to PS1075011501500AC B

B/8B/5.22B/4

+ CS1 to CS3450A

B/ 5

++ CS4450650AB

B/ 5 B/32/3

Steel Beam - Series S

200

100 100L=2000

CL Steel I-Beam130

105

Accelerometer

Hammer Weight(HW)

Hammer ExcitationPosition (HEP)

+ CS5 & CS 6450A

B/ 5

++ CS7450650AB

B/ 5 B/32/3

+ CS8 & CS9450A

B/ 5

SB1+++500600800B/4B/3B/2

HEP (distance in mm) from end of beam

HEP as a fractionof Total BeamLength (FTL)

KEY

+++75011501500

AC B

B/8B/5.22B/4

HEPReferenceNode

End ofBeam

Note: All dimensions are in mm

Figure 5.1: Experimental Beams and HEP Test Variables



It is well known from in-field damping experiments that the support conditions can

contribute greatly to the damping of the system as a whole (Leonard and Eyre, 1975).

Consequently, care was taken to ensure that there was minimal contact between the

testing rig and the concrete beam and this was generally why small-amplitude free-

vibrations were initiated when no static load was on the beams. Furthermore, the

supports were made significantly stiffer than the beam itself and this was achieved by

ensuring that the solid concrete supports were fixed to the floor.

The testing regime for this component of research consisted of a minimum of 10

hammer impacts, for each HEP/HW combination, in order to obtain high-quality free-

vibration decay signatures for one single logdec value. This approach was used

because, up to 10-20% of the hammer ‘hits’ can be spoiled due to the rattling

phenomenon from hard hits, or from ambient influences (Reynolds and Pavic, 2000).

Jones and Welch (1967) undertook a similar regime, taking the mean value of the

lowest set of logdec values because it was thought that these readings would be least

influenced by external influences.

5.3 Analytical Technique for Calculating Logdec

Equation 2.30 and Figure 2.7 presented the TLT experimental method for extracting the

logdec from a free-vibration decay curve. It is based on the assumption that the free-

vibration decay is purely viscous or exponential (Swamy, 1970). The TLT method is

very easy to use because viscous damping introduces a linear term into the equation of

motion, thus making the system very easy to analyse (Kelly, 1993).

It is, however, well known that in real-life one pure type of damping does not exist,

rather it is a combination of viscous, frictional or hysteretic damping (Bachmann et al.,

1995; Jeary, 1997a,b). However, the definition of logdec, as given by Equation 2.30,

holds true for whatever type of damping is present (Newland, 1989). However, some

conditions are required when using Equation 2.30 including:

That only one dominant mode of vibration is operating, meaning that there must

be a single-frequency response; and

That the decay of vibration must be strictly exponential, otherwise logdec, δ,

becomes a function of the absolute time at which it is measured and the systems

damping cannot be described by a single parameter (Newland, 1989).



As mentioned in the introduction, a program called the Decay Curve Method (DCM),

based on free-vibration theory, was devised to compare and verify the results obtained

by the TLT. Complete details of the “Decay Curve Method” (DCM), are given in

Appendix D, but a summary of the DCM analytical algorithm is shown in Figure 5.2. A

flowchart of the output produced by the DCM is shown in Figure D.1.

Decay Curve Method

a) Obtain raw vibration curve data

b) Apply a FFT* to remove ‘noise’ to obtain fundamental frequency, f

c) Plot a line through a plot of the natural log of oscillation peaks. The

slope of this line

= -2π f δ (since ln(A× exp(-δ ωt)) = ln(A)+ -δ ωt)

d) Use equation of Decay Curve Envelope to extract damping ratio ξ

(= A e -ξω t). Then use logdec formula to determine logdec: δ = 2 π ξ

* FFT: The Fast Fourier Transform algorithm computes the frequency spectrum from the

time domain signal and returns the fundamental frequency.

Figure 5.2: Analytical Algorithm used by the DCM

Note that the focus here is on first modal damping, as much of the literature has justified

that first modal bending is of primary importance for dynamic calculations undertaken

during the design process (Jeary, 1974). Furthermore, first modal bending, as calculated

from tests, tends to be on the lower end of the range of damping, and therefore

conservative (Wheeler, 1982).

5.4 Applying the TLT and DCM Techniques

Every recorded free-vibration decay curve obtained from the oscilloscope was of a

standard 450 data points in length. To each free-vibration decay curve both the TLT and

DCM were applied, thus giving two logdec values for each curve. Both the TLT and

DCM involve the calculation of logdec at nominal points on the free-vibration decay

curve. These nominal data lengths (DL) are defined for the TLT in terms of the number

of cycles n that are contained within 50/450, 100/450, 150/450, 200/450, 250/450 and 300/450 data

point windows. Equation 2.30 requires the height of the nth peak, and to make sure that



the TLT and DCM outcomes were comparable, therefore the DL’s of both methods

needed to be roughly equal. Measurements ceased beyond the DL of 300/450 because of

the difficulty associated with reading low amplitude oscillations (Penzien, 1964). For

the DCM, the data length’s (DL) processed by the DCM are called the number of data

points or NDP contained in windows of 50/450, 100/450, 150/450, 200/450, 250/450 and 300/450 data

point lengths. These definitions are shown diagrammatically in Figure 5.3.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Time (s)

Vib

ratio

n Am

plitu

de (

mV

)

50/450

100/450

150/450

200/450

250/450

300/450For the DCM:

Data Length (DL) isdefined in terms of

Number of Data Pointsor NDP

For TLT n=7(in 50/450 NDP)δ = 1/7 ln A1/An

A1

A7 For the TLT: Data Length (DL)is defined in terms

of Number ofCycles or n

For DCMNDP = 50/450

Figure 5.3: Definition of the Data Lengths Used by the TLT and DCM

Examples of the calculated logdec using the TLT and DCM for selected beams are

presented in Figures 5.4 and 5.5. Complete curves for every test beam may be observed

in Appendix D, Section D.2 and D.3, respectively and also Section D.4.

From Figure 5.4 it may be seen that there are significant variations in calculated logdec

depending on the number of cycles used. As the decay curve sample gets longer,

moving along the x-axis, the calculated logdec becomes smaller. Again, Figure 5.5

indicates there are variations in the calculated logdec, depending on the NDP used.



11

1

1 11

77

7

7 7

7

B

B

BB B B

f

f

f

f f f

Cycle Number (n)

Logd

ec(T

LT)

0 25 50 75 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2SB1CS1CS7BII-2PS6

17Bf

Figure 5.4: Calculation of Logdec (TLT) using Cycle Number (n)

1

11

11

17 77

77 7

B

BB

BB B

f

f ff

ff

Number of Data Points (NDP)

Logd

ec(D

CM

)

0 100 200 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24 SB1CS1CS7BII-2PS6

17Bf

Figure 5.5: Calculation of Logdec (DCM) using NDP

5.4.1 Differences Between the TLT and DCM Output

Figures 5.4 and 5.5 indicate that the early portion of the curve produces significantly

elevated logdec values, as compared to the latter portions. This “NDP reduction effect”

appears to be more pronounced for the TLT than the DCM. The most obvious

assumption for the “NDP reduction effect”, is that the curve does not decay in a strictly

exponential manner. This is illustrated for a typical free-vibration decay curve in

Figure 5.6 where using the TLT, logdecs for every successive peak of the free-vibration



decay curve are calculated. It seems that logdec is not strictly exponential in the initial

portion but begins to decay exponentially after an initial period as shown in Figure 5.7.

When a beam is subjected to free vibration, the fundamental (800 Hz) and harmonic

modes (1200 Hz, 1600 Hz etc) are all excited. This has an effect on the initial portion

of Figure 5.7, because the higher harmonic modes override the fundamental mode. The

region where logdec becomes linear is the region where the fundamental frequency is

predominant, which occurs after the ‘non-exponential’ portion (i.e. when n > 10 in

Figure 5.7) of the curve.

In the literature, researchers using the TLT took their single logdec measurement from

somewhere (usually unknown) along the free-vibration decay curve. This location is

occasionally specified, but they vary enormously from researcher to researcher.

Penzien (1964) used n=4, whilst Cole and Spooner (1965) calculated logdec with n very

much greater than 10 (with the total number of oscillations specified as at least 10).

Cole (1966) measured An over ‘at least 40 cycles’. Leonard and Eyre (1975) used n=10

cycles, and despite observing that the plot of loge An versus decay cycle n did not

produce a straight line (i.e. non-viscous damping), they argued that for practical

purposes, an ‘average’ value of logdec is an adequate way of presenting damping

information. From Figure 5.6, a huge range of logdec values could be extracted from

the same curve, the question is: Which one is correct?

5.4.2 Proposed Rules for Calculating Logdec

In Figure 5.8a, the method of calculating the ratio between the initial (A1) and

subsequent peak (An) is shown. In Figure 5.8b, the effect of the peak ratio (An/A1) on the

calculated logdec for various test beams (using the TLT) is shown.

In Figure 5.8b, the last three nodes for each curve indicate where the decay becomes

exponential (according to the fundamental frequency), and it is at this point where

logdec would be most appropriately and consistently measured (see Figure 5.7). This

region, is termed the “optimal peak ratio An/A1” and produces a conservative, stable

estimate of damping capacity.

The “optimal peak ratio An/A1” region for each experimental specimen does vary for each

specimen, but nevertheless, it was found to range between a peak ratio of approximately

10 to 15% for all beam specimens. The “optimal peak ratio An/A1” curves for all the test

beams may be found in Appendix D, Section D.5, Figures D.9 to D.15.



Time (ms)

Vib

ratio

nA

mpl

itude

(mV

)

0 0.01 0.02 0.03

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2 Logdec, δ = 1/n ln A1/An

δ=0.

2284

A1 = 0.222844

δ=0.2

401

δ=0.2

539

δ=0.0

999

δ=0.116

7

δ=0.110

6

δ=0.1

450

δ=0.101

3

δ=0.107

1

δ=0.1

119

δ=0.126

3δ=0.1

064δ=

0.1479δ=

0.1028

δ=0.1

039

δ=0.119

0

Non-Exponential Decay

Figure 5.6: Variation of Logdec (TLT) with n

Cycle Number, n

Nat

ural

Loga

rithm

ofA

mpl

itude

(lnA

n)

0 10 20 30-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Region of stableexponential decay

A1= ln(0.222844)

Non-exponentialdecay

Figure 5.7: Plot of Natural Logarithm of An versus Cycle Number



a)

Time, (ms)V

ibra

tion

Am

plitu

de, A . Vibration

Amplitudeat n=4

A4 = 0.5.

MaximumVibrationAmplitude

A1 = 1

A4

A1Peak Ratio =

= 0.5 1 = 0.5

= 50%

b) Peak Ratio An/A1 (%)

Logd

ec(T

LT)

0102030400

0.025

0.05

0.075

0.1

0.125

0.15

0.175SB1CS6BII-11PS9

DCM 300NDP

DCM 50NDP

DCM 100NDP

DCM 150NDP

DCM 200NDP

DCM 250NDP

Figure 5.8: Example Calculation of: a) Peak Ratio; and b) “Optimal Peak Ratio” Curves

Finally, it should be noted that Figure 5.8b would allow all types of logdec research to

be comparable. Thus, it will help avoid the current identified situation where it is

difficult to effectively utilise published data. The TLT has been employed for this

thesis.



5.5 Effect Of Experimental Test Variables

Two primary variables were examined: a) hammer weight HW, and b) hammer

excitation position HEP. A summary of their effect is presented below.

5.5.1 Hammer Weight (HW)

In the current experimental tests, two different hammer weights were used (HW1=

163.1 g and HW2= 239.5 g). Figure 5.9 shows that logdec is unaffected by hammer

weight. It should be noted that by using different hammer weights, the actual excitation

force and response amplitude varied with each hammer impact. Thus, it can be

concluded that minor variations in impact weights during small-amplitude damping

laboratory experiments do not appear to affect the obtained values of damping.

o

oo o o o

v

v

v

v vv

A

A

A

AA

A

*

*

*

** *

Cycle Number (n)

Logd

ec(T

LT)

0 25 50 75 1000.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18SB1-With CWSB1-No CWCS6-With CWCS6-No CWPS10-With CWPS10-No CW

o

vA*

Figure 5.9: Effect of HW on Logdec (TLT)

5.5.2 Hammer Excitation Position (HEP)

Previous studies suggest that concrete beam damping should be relatively constant,

regardless of the hammer impact position (Chowdhury, 1999). It is difficult to

consistently impact the beam in the exact same positions, therefore researchers need to

be confident that the selection of hammer impact position will not affect the obtained

damping values. In Figure 5.10, it should be noted that the HEP’s are specified in

terms of FTL, these are described in Figure 5.1.



Figure 5.10 shows all data points from selected beams that have been extracted from the

free-vibration decay curve at their “optimal peak ratio An/A1”. For Beams SB1, BII-11

and PS9 the optimal peak ratios are at 14%, 17% and 16%, respectively.

For each beam in Figure 5.10, the average logdec for the three HEP’s given.

Accompanying the average logdec is the range of all three points from the average

logdec. For beams SB1, BII-1 and PS9 all data is contained within 15%, 13 % and 15%

of the average, respectively. This is acceptable.

Hammer Excitation Position, HEP (FTL)

Logd

ec(T

LT)

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1SB1- 200NDPBII-11- 200NDPPS9- 200NDP

L/L/ L/L/L/L/

PS9Average Logdec0.0489 ± 15%Optimal Peak Ratio= 16%

SB1Average Logdec=0.0336 ± 15%Optimal Peak Ratio= 14%

BII-11Average Logdec0.0494 ± 13%Optimal Peak Ratio= 17%

Figure 5.10: Effect of HEP on Logdec (TLT)

5.6 Summary

The experimental test results show that free-vibration decay in concrete beams is not

strictly exponential, as is required by the TLT logdec equation (Equation 2.30). The

literature has acknowledged this peculiarity of damping calculation, however no

meaningful research has been conducted to assist researchers in overcoming these

difficulties.

The non-exponential decay effect has been demonstrated well, herein, whereby logdec

decreases for increasing cycles. However, logdec starts to stabilise and become

consistent at some region within the free-vibration decay curve. This ‘stabilisation’



region is different for every beam, and it is recommended that it be determined from

trial and error solutions. Commonly, stabilisation occurs when the peaks have been

reduced to about 10-15% of their original height and this has been termed herein as the

“optimal peak ratio An/A1”.

Regardless of experimental logdec technique used it is extremely important to report

full and complete details. A diagram such as Figure 5.8b would allow researchers to

both understand the published damping data and be able to make meaningful

comparisons. As mentioned, it is extremely difficult for researchers to make

comparative reviews when presented with the ‘logdec’ of a specimen.

The test results showed that the experimental variables of hammer weight and hammer

excitation position did not produce any detectable impact on calculated damping. It is

recommended to excite the beam at reasonable distances from the support and loading

nodes and that a number of measurements be taken from various HEP locations with

different HW’s to verify that the obtained results are consistent and repeatable.


Chapter 6: Damping Prediction in ‘Untested’ Concrete Beams 6-1

CHAPTER 6

Damping Prediction in ‘Untested’ Concrete Beams

6.1 General Remarks

The purpose of this Chapter is to utilise the experimental ‘untested’ logdec test data to

derive an equation to predict the ‘untested’ damping capacity. Establishing the

‘untested’ damping capacity is important, as it is the initial component in the calculation

of the total damping capacity.

6.2 Damping In ‘Untested’ Reinforced Concrete Beams

In the ‘untested’ phase, the concrete beam has not yet received any service loading and

is considered to be completely undamaged.

6.2.1 Historical Review

A majority of early damping research was concerned with the damping capacity of the

concrete material itself, utilising tests on concrete cylinders. Figure 6.1 outlines

examples of four such research programs. These studies tried to ascertain if damping

could be used to determine concrete compressive strength or other material properties.

Kesler and Higuchi (1953) found that the concrete strength, fcm, could not be estimated

merely by determining the damping capacity of a cylinder.

Cole and Spooner (1968) established that the measurement of damping capacity is not

suitable for the quality control of concrete, particularly in view of Akashi’s (1960)

equation that the compressive strength, fcm, equals:

410346.079.1 −×⎟⎠⎞

⎜⎝⎛ +×=

δpcm Ef ±30% (6.1)



where Ep is the dynamic modulus; δ is the logarithmic decrement, regardless of age,

water content and mix design. Cole and Spooner (1968) state that “assessing the

compressive strength to ±30% is not a particularly practical advantage for a

measurement of this complexity”.

Swamy and Rigby (1971) developed an equation to predict the damping capacity of

concrete cylinders based on the properties of the concrete constituents (see Equation

3.2), but would be difficult to apply to structural elements other than cylinders.

Sri Ravindrarajah and Tam (1985) looked at the variation of logdec with different types

of aggregates. For the same composition, logdec was found to increase with a decrease

in compressive strength, in the case of recycled aggregates, logdec increased by up to

27%. No prediction equation was developed because of insufficient data trends.

Evidently, it is difficult to utilise damping equations developed for concrete cylinders

for full-scale beams as they do not consider the effect of more important variables, such

as steel reinforcement, dimensions or support conditions.

Figure 6.1 also highlights the work of Dieterle & Bachman (1981) and Flesch (1981),

who developed equations to model damping in ‘untested’ beams. As discussed in

Chapter 2, their equations rely on constants that need to be established by experimental

investigations. This therefore defeats the purpose of providing practitioners with a

simple yet accurate means of modelling material damping prior to construction.

CYLINDERSCompare Dampingand fcm or Concrete

Constituents Swamy and Rigby (1971)δc=0.0174-0.1131/Ec-0.06401/Em+0.265 δm-0.0000913Vc

Sri Ravindrarajah and Tam (1985)δ increased for decrease in fcm (Aggregate type)

⎥⎦

⎤⎢⎣

⎡= 2

02 cm

cunVD fC

Edπ

ξ

Dieterle and Bachman (1981)BEAMS

Measure Beam Constituents to

Predict DampingFlesch (1981)

Kesler and Higuchi (1953)Measuringδ alone cannot predict fcm

Cole and Spooner (1968)E cannot be used to calculateδ

Figure 6.1: Classification Of Historical ‘Untested’ Damping Research



6.2.2 Experimental Effect of fcm and fsy

The effect of concrete compressive strength, fcm, and reinforcement yield strength, fsy, on

the ‘untested’ damping capacity are discussed here. Table 6.1 gives the experimentally

determined logdec of each specimen in the ‘untested’ condition along with relevant

beam variables, fcm, fsy and total longitudinal reinforcement distribution, LRD.

The LRD variable gives an indication of not only the amount of total longitudinal

reinforcement, but also of how it is distributed. Also, by creating the primary variable

of LRD, a normalised cross-sectional area is produced, allowing beams with different

cross-sections (b×d) and lengths (L) to be compared such as the B- and CS-Series

beams.

Note that logdec values in Table 6.1 range from 0.0361 to 0.0700, where Appendix E

gives full details of the logdec values obtained according to the optimal peak ratio

method as described in Section 5.4.2.

The effect of steel yield strength on damping capacity appears to be negligible as also

shown in Figure 6.2. Considering the beam pairs of BI-7/BII-8 and BI-9/BII-10, which

contain the same LRD and concrete compressive strength but differ in steel yield

strength, only slight increases in logdec have occurred for the higher strength

reinforcement.

In Figures 6.3a and 6.3b, the ‘untested’ logdec versus the longitudinal reinforcement

distribution (LRD) curve for the B- and CS-Series beams are presented. It may be seen

in both figures, that when the beam is in the ‘untested’ state, the damping capacity

appears not to be affected by concrete compressive strength, fcm, alone. Despite a

natural expected variation in concrete constituents and casting variability, damping

capacity remains relatively constant for each LRD group.

6.2.3 Proposed Damping Equation

Figure 6.4 shows the variation in logdec versus the total longitudinal reinforcement

distribution for all the ‘untested’ RC beams.

In Figure 6.4a, a general power-fit trend for both sets of data shows that damping

capacity increases as LRD increases (i.e. with the bar spacing getting smaller).



Table 6.1. Experimental ‘Untested’ Damping Data – RC Beams

Beam Code

Logdec of ‘Untested’ Beam,

δuntest Appendix E


Strength fcm (MPa)

Reinforcement Yield

Strength fsy (MPa)

Total Longitudinal Reinforcement

LRD #

(ρt /st)+ (ρt /sc)

BI-1 0.0648 30.0 400 0.00108 BII-2 0.0548 30.0 500 0.00044 BI-3 0.0585 23.1 400 0.00072 BII-4 0.0500 23.1 500 0.00021 BII-5 0.0573 41.5 500 0.00108 BII-6 0.0534 41.5 500 0.00044 BI-7 0.0361 64.5 400 0.00010 BII-8 0.0388 64.5 500 0.00010 BI-9 0.0444 53.0 400 0.00021 BII-10 0.0457 53.0 500 0.00021 BII-11 0.0518 90.7 500 0.00044 BII-12 0.0566 90.7 500 0.00108 CS1 0.0700 22.5 400 0.00222 CS2 0.0693 22.5 400 0.00222 CS4 0.0495 32.0 500 0.00054 CS5 0.0501 32.0 500 0.00054 CS6 0.0519 32.0 500 0.00054 CS7 0.0631 31.5 500 0.00082 CS8 0.0591 31.5 500 0.00082 CS9 0.0627 31.5 500 0.00101

# Total longitudinal reinforcement distribution is defined in Section 7.2.3 where ρt = Ast/bd and ρc = Asc/bd

GH

IJ

Longitudinal Reinforcement Distribution (ρt /st + ρc /sc)

'Unt

este

d'Lo

gdec

,δun

test

0 0.0005 0.001 0.0015 0.002 0.00250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)

GHIJ

BI-9fsy = 400 MPa

BII-10fsy = 500 MPa

BI-7fsy = 400 MPa

BII-8fsy = 500 MPa

Figure 6.2: Variation of Logdec with Steel Yield Strength – B-Series Beams



a)

B EFK

L


'Unt

este

d'Lo

gdec

,δun

test

0 0.0005 0.001 0.0015 0.002 0.00250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

BII-2 (B)BII-5 (E)BII-6 (F)BII-11 (K)BII-12 (L)

BEFKL

BII-2fcm = 30.0 MPa

BII-6fcm = 41.5 MPa

BII-11fcm = 90.7 MPa

BII-5fcm = 41.5 MPa

BII-12fcm = 90.7 MPa

b)

12

456

78

9


'Unt

este

d'Lo

gdec

,δun

test

0 0.0005 0.001 0.0015 0.002 0.00250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

CS1CS2CS4CS5CS6CS7CS8CS9

12456789

CS4fcm = 32.0 MPa

CS5fcm = 32.0 MPa

CS6fcm = 32.0 MPa

CS1fcm = 22.5 MPa

CS2fcm = 22.5 MPa

Figure 6.3: Variation of Logdec with Concrete Compressive Strength: a) B-Series; b)

CS-Series Beams



a)

A

AA

A

AA

AA

AA

A

A

BB

BBB

BB

B


'Unt

este

dLo

gdec

',δ u

ntes

t

0 0.0005 0.001 0.0015 0.002 0.00250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08RC BeamsCS Beams

AB

For CS-Series Beams

Y = e-1.2811458 * X 0.22185349

For B-Series Beams

Y = e-1.5140317 * X 0.18673056

b)

A

AA

A

AA

AA

AA

A

A

BB

BBB

BB

B


'Unt

este

d'Lo

gdec

,δun

test

0 0.0005 0.001 0.0015 0.002 0.00250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

RC BeamsCS BeamsEquation 6.2

AB

For RC Beams

δuntest = 0.223 * (ρt /st+ρc /sc)0.19

R2 = 0.90

Figure 6.4: Dependence of ‘Untested’ Logdec on LRD in RC Beams: a) Separate

Trendlines; b) Unified Prediction Equation



In other words, for an equivalent LRD, large amounts of smaller bars give greater

damping capacities than fewer bigger bars (c.f. CS1 versus CS8 or BII-2 versus BII-4).

The R2 values of 0.89 and 0.87 for the B- and CS-Series beams respectively, indicates a

good measure of correlation. The identical nature of the two curves for the B- and CS-

Series in Figure 6.4a suggests that total beam length does not affect damping capacity.

Since the two trendlines in Figures 6.4a are reasonably similar, and independent of

length, a single equation was fitted to both sets of data as shown in Figure 6.4b. The

single equation for estimating the damping capacity of ‘untested’ reinforced concrete

beams is

19.0

223.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛+×=

c

c

t

tuntest ss

ρρδ (6.2)

where st and sc are the tension and compression reinforcement spacings, respectively in

mm and are given in Table 4.1; ρt = Ast/bd and ρc = Asc/bd. The equation is valid for

LRD distributions between 0.0001 and 0.0023 (see Table 6.1).

The relationship of Equation 6.2 to the current data set may be observed in Figure 6.4b.

As discussed previously, comparing this prediction equation to existing published

damping data for verification would be difficult. However, an attempt at doing so is

made later within this Chapter where further verification is undertaken using the F-

Series beam tests and comparisons to published experimental data.

6.3 Damping in ‘Untested’ Prestressed Concrete Beams

The following section will discuss the ‘untested’ damping characteristics of the PS-

Series, 6-metre prestressed concrete beams.

6.3.1 Historical Review

Only three researchers have examined damping in full-prestressed concrete (PSC)

beams. James et al. (1964) and Penzien (1964) did not examine damping prestressed

concrete beams in the ‘untested’ state at all, whilst Hop (1991) found that an increase in

prestressing caused a decrease in the beam’s logdec, δ as illustrated by his Equation 3.9.

The equation did not relate logdec to any particular beam state and it was unclear

whether the equation related to the ‘untested’ or ‘tested’ beam. This equation will be

examined in more depth in Section 6.3.3.



6.3.2 Experimental Observations

In Table 6.2, the experimentally determined logdec of each PSC specimen in the

‘untested’ condition is given along with relevant beam variables, fcm, H and e. Appendix

E gives full details of the logdec values obtained. Note that logdec values range from

0.0412 to 0.0547.

In Figures 6.5 and 6.6, the relationships between logdec and prestressing force, H, and

logdec versus prestressing eccentricity, e, respectively, may be observed. From both

figures it seems that neither prestressing force or eccentricity ‘alone’ provides a

consistent trend to model the ‘untested’ logdec. It is proposed in Section 6.3.4 that the

total initial prestress, He, gives a better prediction of logdec.

6.3.3 Hop’s Prestressed Equation

Figure 6.7 gives a plot of the ‘untested’ logdec values from the current experiments

(using the initial prestress value, H) compared to predicted logdec values using Hop’s

(1991) Equation 3.9. The location of the data points in Figure 6.7 indicates the

following:

Equation 3.9 consistently overestimates the damping values of the current beams at

the pre-test stage; and

Equation 3.9 suggests that beams with a higher prestressing force give lower

amounts of damping. This was not found for the current tests as indicated in Table

6.2 and Figure 6.5.

Table 6.2. Experimental Prestressed ‘Untested’ Damping Data

Beam Code

Logdec of ‘Untested’

Beam, δuntest Appendix E


Strength fcm (MPa)

Table 4.5

Prestressing Force

H (kN)

Prestressing Eccentricity

e (mm)

Total Initial Prestress

He (kNmm)

PS3 0.0420 60.2 346 97.0 33,562.0

PS4 0.0478 60.2 585 80.5 47,092.5

PS5 0.0430 69.8 612 60.0 36,720.0

PS6 0.0467 69.8 400 99.7 39,880.0

PS7 0.0565 52.5 450 97.5 43,875.0

PS8 0.0412 52.5 400 80.0 32,000.0

PS9 0.0424 83.5 346 90.0 31,140.0

PS10 0.0547 83.5 480 90.0 43,200.0



x

x

x

x

x

xx

x

Prestressing Force, H (kN)

'Unt

este

d'Lo

gdec

,δun

test

200 300 400 500 600 700 8000.03

0.04

0.05

0.06

All PS Beamsx

Figure 6.5: Prestressing Force versus ‘Untested’ Logdec for PS-Series Beams

x

x

x

x

x

xx

x

Prestressing Eccentricity, e (mm)

'Unt

este

d'Lo

gdec

,δun

test

50 75 100 1250.03

0.04

0.05

0.06

All PS Beamsx

PS7

PS10

PS4

PS6

PS9PS3

PS8

PS5

Figure 6.6: Prestressing Eccentricity versus ‘Untested’ Logdec for PS-Series Beams



The current evaluation has demonstrated that, contrary to the findings of Hop (1991), a

damping prediction equation for the ‘untested’ beam appears not to be dependent upon

the single variable of prestressing force, H alone. In view of this, the development of a

more appropriate logdec prediction equation incorporating the eccentricity of prestress,

e, along with prestressing force, H, is undertaken in Section 6.3.4.

x

xx

x

x

xx

x

Untested Logdec By Hop (1991), δ

'Unt

este

d'Lo

gdec

,δun

test

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1All PS Series Beamsx

Figure 6.7: ‘Untested’ Logdec versus Logdec Prediction using Hop (1991)

6.3.4 Proposed Damping Equation

For the current prestressed test beams in the ‘untested’ state, the most obvious

distinguishing feature is the overall location and amount of prestressing that exists in

the beam. That is, it is not only the amount of prestressing force, H, significant, but

how it is located within the beam (i.e. its eccentricity, e). Shown in Figure 6.8, this

variable, He, is plotted against the ‘untested’ logdec of each beam. This variable He

gives an excellent correlation to the ‘untested’ logdec (R2 = 0.99), as shown by the

second order polynomial curve shown in Figure 6.8, and given by

δuntest = 1.4(×10-10)He2 – 9.4(×10-6)He + 0.2 (R2 = 1.0) (6.3)

where Equation 6.3 is valid for He between 30,000 and 45,000 (kNmm).



xx

x

x

xx

x

Total Initial Prestress, He (kNmm)

'Unt

este

d'Lo

gdec

,δun

test

30000 35000 40000 45000 500000.03

0.04

0.05

0.06

All PS Beamsx

Figure 6.8: ‘Untested’ Logdec versus Initial Prestress in Beam

6.4 Verification

Three verification checks of the proposed reinforced concrete logdec equation are

presented here using, a) the original beam data, b) the test data from the additional F-

Series beams. The final check, c) uses a published free-decay curve from Nield (2001).

6.4.1 Original Beam Data

The overall performance of both Equations 6.2 and 6.3 may be represented by the

scattergram shown in Figure 6.9. As may be seen, all values fall within the ±20%

envelope, and in fact close to the 45o line, indicating a very good prediction.

6.4.2 F-Series Beams

As the F-Series beams are of reinforced concrete only, Equation 6.2 is investigated here.

Shown in Figure 6.10 are the experimental versus calculated ‘untested’ logdec values

for the F-Series reinforced concrete beams. As seen, the ‘untested’ logdec values

predicted by Equation 6.2 are very good, with all the values within ±20%. A slightly

higher damping is given by Equation 6.2, indicating a conservative prediction.



Calculated 'Untested' Logdec

Expe

rimen

tal'

Unt

este

d'Lo

gdec

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

B-SeriesCS-SeriesPS-Series

-20%

+20%

Figure 6.9: Comparison between Experimental Logdec and Logdec Calculated by

Equation 6.2 and 6.3

Calculated 'Untested' Logdec

Expe

rimen

tal'

Unt

este

d'Lo

gdec

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

All F-Series Beams

-20%

+20%Prediction Equationδuntest=0.223x(ρt/st +ρc/sc)

0.19

Figure 6.10: Observed versus Predicted ‘Untested’ Logdec for the F-Series Beams



6.4.3 Neild’s Beam

The use of non-linear vibration techniques, for the detection of damage in a reinforced

concrete beam, under low-amplitude, cyclic loading was reported by Neild (2001).

Free-vibration experiments were performed at incremental levels of damage loading,

where the impact excitation was applied 70 mm away from mid-span using an

instrumented (5.45 kg) sledge hammer dropped from a height of approximately 50 mm

above the beam. The cross-sectional details of Neild’s (2001) test beam, and test set-up

are given in Figure 6.11. Using Equation 6.2, the ‘untested’ damping capacity may be

established for the single beam as:

19.0

4.1380027.0

2.570161.0223.0 ⎟

⎠⎞

⎜⎝⎛ +×=untestδ

∴ δuntest,calc = 0.0478

In the free-vibration experiments, Neild (2001) did not actually calculate the damping

capacity, but did present a mid-span, free-vibration decay record for the beam prior to

testing. This type of published information is very rare. Using this free-vibration decay

record, the ‘optimal peak ratio’ technique was applied, the results of which are shown

below in Figure 6.12. From the figure, the δuntest,exp, can be established as being

between 0.043 and 0.049 and therefore the ratio between δuntest,calc/δuntest,exp is 1.11 and

0.98, respectively. This is more than acceptable.

200 mm

105 mm 3@12mm fsy = 410 MPa

2@6mm fsy = 240 MPa

Pin Roller

L = 3000 mm

l = 2800 mmst = 57.2 mm

sc = 138.4 mmStirrups =

4.8@80mm

f’c = 30 MPa

Three-point loading, withdamage levels in steps of

10% of failure load

Brüel and Kjær 4382 accelerometersat half and quarter span points

Figure 6.11: Details of Test Beam and Testing Arrangement of Neild (2001)



Peak Ratio An/A1 (%)

Logd

ec,δ

(TLT

)

01020304050600

0.02

0.04

0.06

0.08

0.1

0.12

Neild (2001)

Region ofOptimal Peak Ratio

δ = 0.043

δ = 0.049

Figure 6.12: ‘Optimal Peak Ratio’ Analysis of Neild’s (2001) Free-Decay Curve

6.5 Summary

A critical review of the pertinent existing literature on concrete beam damping capacity

has highlighted a major omission in available ‘untested’ logdec computation methods

for both reinforced and prestressed concrete beams. From an analysis and design point

of view, the initial establishment of the ‘untested’ damping capacity is extremely

important.

From the extensive experimental investigation, an equation has been proposed that

illustrates a strong correlation between the damping capacity of ‘untested’ reinforced

concrete beams and the quantity and distribution of the longitudinal reinforcement of

the beam. The developed equation incorporates both tension and compression

longitudinal reinforcement ratios and respective spacing values. The effects of concrete

compressive strength and reinforcement yield strength were found to exert negligible

impact on the ‘untested’ damping capacity for the current series of reinforced concrete

beam tests.

For the prestressed concrete beams, the prediction of the ‘untested’ logdec is related to



the initial total prestress in the beam, He by a second order polynomial equation.

Although a damping equation was proposed, it is expected that future research will

allow further improvement of the developed equation.

Verification of the reinforced concrete ‘untested’ damping prediction equation using the

additional F-Series experimental test data and published damping data of Nield (2001)

has shown the proposed equation to be reliable.


Chapter 7: Residual Deflection Mechanisms in Concrete Beams 7-1

CHAPTER 7

Residual Deflection Mechanisms in Concrete Beams

7.1 General Remarks

This Chapter develops and verifies residual deflection equations for both the reinforced

and prestressed concrete beams. The prediction of the ‘tested’ damping capacity is

based upon the calculation of the residual deflection as developed for the current

experimental programme. As discussed in Section 3.2.4, the computation of the residual

deflection is to be directly related to the instantaneous deflection. Sections F.1 and F.2

of Appendix F present respectively, the raw instantaneous and residual deflection

experimental data. Figures 7.1 to 7.4 were produced using this raw data.

7.2 Residual Deflection in Reinforced Concrete Beams

This section considers the effect of a) fsy, b) fcm, c) ρt and d) loading conditions on the

residual deflection of concrete beams.

7.2.1 Effect of fsy

Figure 7.1a demonstrates the relationship between the experimental instantaneous

deflection, ∆i,exp, and the experimental residual deflection, ∆r,exp, and highlights the

effects of the two different grades of reinforcing steel. As indicated in Table 4.1, each

of the beam pairs contained identical geometrical and mechanical details. As shown in

Figure 7.1a, the initial linear elastic portion for each beam pair was very similar. In

Figure 7.1b it may be seen that the 500 MPa beams attained greater residual deflection

at failure.

As discussed previously, because 500 MPa reinforcing steel is the primary reinforcing

steel available in the current Australian market, the remaining discussion will focus on

the characteristics of this steel.



7.2.2 Effect of fcm

The curves in Figure 7.2a show similar instantaneous versus residual deflection

characteristics suggesting that fcm alone does not impact significantly on the residual

deflection characteristics of the beam. Examining the normalised1 moment versus

residual deflection curve in Figure 7.2b, it may be seen that for the portion of each

curve up until about 80% of the ultimate load the curves are very similar, again

indicating that the effect of concrete compressive strength alone is minimal.

7.2.3 Effect of ρt

Figure 7.3a, compares beams containing different percentages of tensile reinforcement

(BII-5 versus BII-6 and BII-11 and BII-12). It shows that for a particular instantaneous

deflection, the beam containing lower amounts of reinforcing steel exhibit higher levels

of residual deflection. Interestingly however, the total residual deflection at failure, for

each beam pair was very similar. This can be verified in Figure 7.3b, where at an

equivalent normalised moment level, the curves are remarkably similar.

7.2.4 Effect of Loading Conditions

The CS-Series flexural beams were tested to investigate the variable of loading beam

width, LBW (as given in Table 4.8). The tensile and compressive reinforcement ratios,

and the concrete compressive strengths were very similar, with the tensile reinforcement

yield strengths being either 400 MPa or 500 MPa for these beams.

At first glance, Figure 7.4a seems to indicate that varying LBW does not affect the

instantaneous versus residual deflection characteristics. However, Figure 7.4b clearly

indicates that at an equivalent bending moment below the region of yielding say around

0.5 or 50% of ultimate the loading condition does impact on the residual deflection

characteristics of the CS-Series flexural beams. It can be concluded therefore, that the

calculation of the instantaneous deflection incorporates the effect of loading conditions.

This further demonstrates that the calculation of instantaneous deflection is an excellent

method for the prediction of the residual deflection characteristics.

7.2.5 Summary of Effects

In conclusion, it was found that ρt and LBW affect instantaneous and residual deflection.

1 Normalised moment equals the mid-span service moment divided by the failure moment.



a)

G

GGGGG

G

G

HH

HHHHHHH

H

H

I

I

I

I

I

I

J

JJJJ

JJJ

J

J

J

J

Mid-Span Residual Deflection (mm)

Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 10 20 30 40 50 600

25

50

75

100

125

BI-7 (fsy = 400 MPa)BII-8 (fsy = 500 MPa)BI-9 (fsy = 400 MPa)BII-10 (fsy = 500 MPa)

GHIJ

Variable = fsy

b)

G

G

G

G

G

G

G

G

H

HH

HHH

H

H

H

H

H

I

I

I

I

I

I

J

J

J

J

J

J

J

J

J

JJ J


Mi-S

pan

Ben

ding

Mom

ent(

Nor

mal

ised

)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)

GHIJ

Figure 7.1: Effect of Reinforcement Yield Strength for B-Series Beams On Residual

Deflection Versus: a) Instantaneous Deflection; b) Normalised Bending Moment



a)

B

BB

BBBBB

BB

B

D

D

D

D

D

D

D

D

F

F

F

F

FF

FF

J

JJJJ

JJJ

J

J

J

J

K

K

KK

K

K

K

K

K


Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 10 20 30 40 50 600

25

50

75

100

125

BII-2 (ρ =1.78%, fcm=30.0 MPa)BII-4 (ρ =1.73%, fcm=23.0 MPa)BII-6 (ρ =1.78%, fcm=41.5 MPa)BII-10 (ρ =1.73%, fcm=53.0 MPa)BII-11 (ρ =1.78%, fcm=90.7 MPa)

BDFJK

Variable = fcm

500 MPaSteel Only

b)

B

B

B

B

B

B

B

B

B

B

B

F

F

F

F

F

FF

F

K

K

K

K

K

K

K

K

K


Mid

-Spa

nB

endi

ngM

omen

t(N

orm

alis

ed)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

BII-2 (B)BII-6 (F)BII-11 (K)

BFK

90.7 MPa

30.0 MPa 41.5 MPa

Figure 7.2: Effect of Concrete Compressive Strength for B-Series Beams On Residual




a)

E

E

E

E

E

E

E

F

F

F

F

FF

FF

K

K

KK

K

K

K

K

K

L

LLLLLL

LL

L

L


Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 10 20 30 40 50 600

25

50

75

100

125

BII-5 (ρt = 2.38%)BII-6 (ρt =1.78%)BII-11 (ρt =1.78%)BII-12 (ρt =2.38%)

EFKL

Variable = ρt

2.38%90.7 MPa

1.78%90.7 MPa

2.38%41.5 MPa

1.78%41.5 MPa

500 MPaSteel Only

b)

E

E

E

E

E

EE

F

F

F

F

F

FF

F

K

K

K

K

K

K

K

K

K

L

L

L

L

L

L

L

L

L

LL


Mid

-Spa

nB

endi

ngM

omen

t(N

orm

alis

ed)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

BII-5 (E)BII-6 (F)BII-11 (K)BII-12 (L)

EFKL

Figure 7.3: Effect of Tensile Reinforcement Ratio for B-Series Beams On Residual




a)

1

1

1

2

2

2

3

3

3

7

7

7

7

7

8

8

8

8

8

9

9

9

9

9

9


Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 2 4 6 8 10 120

5

10

15

20

25

30

CS1 (a=1150mm)CS2 (a=950mm)CS3 (a=750mm)CS7 (a=1050mm)CS8 (a=800mm)CS9 (a=700mm)

123789

b)

1

1

2

2

3

3

7

7

7

7

7

8

8

8

8

8

9

9

9

9

9

9


Mid

-Spa

nB

endi

ngM

omen

t(N

orm

alis

ed)

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CS1CS2CS3CS7CS8CS9

123789

Figure 7.4: Effect of LBW for CS-Series Beams On Residual Deflection Versus: a)

Instantaneous Deflection; b) Normalised Bending Moment



7.2.6 The Proposed Equation

As discussed in Section 3.2.4, there is no established method by which to calculate the

residual deflection, ∆r for the current series of test beams.

A plot of the experimental instantaneous ∆i,exp versus residual deflection ∆r,exp is given

in Figures 7.5a and 7.5b, for all flexural B-Series and CS-Series test beams (with 500

MPa reinforcement only). The main variable influencing the slope of the curve is the

tensile reinforcement ratio, ρt. Note that the data is only plotted prior to the yielding

moment, after which it is not considered linear. In Figures 7.5a and 7.5b, each beam’s

instantaneous verus residual deflection curve was fitted to the linear equation x = αrc ⋅ y,

where y and x are the instantaneous and residual deflections, respectively. The value of

αrc is the residual deflection curve coefficient for reinforced concrete beams.

In Figure 7.6 the αrc values from Figure 7.5 are plotted against the tensile

reinforcement ratio. Even though beams of different sizes are shown, i.e. the B-Series

beam are 6.0 m in length and the CS-Series beams are 2.4 m, a reasonably correlated

linear relationship has been fitted to all data points, thereby indicating that beam size

does not significantly affect the calculation of the curve coefficient, αrc.

Therefore prior to yielding, an estimate of the residual deflection of a flexural reinforced

concrete beam having a length between 2.4 m and 6.0 m, reinforcement yield strength

of 500 MPa and variable concrete compressive strength, may be found from the

instantaneous deflection of that beam by the relationship

∆r,calc = αrc ×∆i,exp (7.1)

where αrc is the curve coefficient found from Equation 7.2 and derived from Figure 7.6

and ∆r,calc and ∆i,exp are in mm.

αrc = -0.08ρt + 0.39 (7.2)

where Equation 7.2 is valid for 0.76% < ρt < 3.0% .

7.3 Residual Deflection in Prestressed Concrete Beams

The prestressed beam pair comparisons, shown in Figures 7.7, 7.8 and 7.9, examine the

effect of the beam test variables, fcm, H and e on the residual deflection characteristics.



The figures utilised the raw data presented in Table 4.2 and Sections F.1 and F.2 of

Appendix F.

a)

B2x=0.22y

R2 = 0.95

B4x=0.20y

R2 = 0.98

B5x=0.17y

R2 = 0.99

B6x=0.23y

R2 = 0.95

B8x=0.37y

R2 = 0.90

B10x=0.24y

R2 = 0.89

B11x=0.23y

R2 = 0.87

B12x=0.19y

R2 = 0.95

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0 2 4 6 8 10 12 14 16 18 20

Mid-Span Residual Deflection (x)

Mid

-Spa

n In

stan

tane

ous

Def

lect

ion

(y)

BII-2

BII-4

BII-5

BII-6

BII-8

BII-10

BII-11

BII-12

Linear (BII-2)

Linear (BII-4)

Linear (BII-5)

Linear (BII-6)

Linear (BII-8)

Linear (BII-10)

Linear (BII-11)

Linear (BII-12)

Linear (BII-12)

b)

CS8x=0.19y

R2 = 0.99

CS9x=0.18y

R2 = 0.99

CS7x=0.12y

R2 = 0.93

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8


Mid

-Spa

n In

stan

tane

ous

Def

lect

ion

(y)

CS7

CS8

CS9

Linear (CS8)

Linear (CS9)

Linear (CS7)

Slope of Curve = αrc

Figure 7.5: Effect of Reinforcement Ratio on the Instantaneous versus Residual

Deflection Relationship: a) B-Series; b) CS-Series



BD

E

F

H

JK

L

7

89

Tensile Reinforcement Ratio, ρt

Cur

veC

oeff

icie

nt,α

rc

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

BII-2BII-4BII-5BII-6BII-8BII-10BII-11BII-12CS7CS8CS9αrc = - 0.08ρt + 0.39

BDEFHJKL789

R2 = 0.78

(%)

Figure 7.6: Selection of Curve Coefficient, αrc for the Calculation of Residual

Deflection

7.3.1 Effect of fcm and e

The first examination, made in Figures 7.7a and 7.7b, is of the combined effect of fcm

and prestress eccentricity, e, on the residual versus instantaneous deflection relationship.

Figures 7.7a and 7.7b indicate that the residual deflection trend for prestressed concrete

beams with the same H and varying combined effect of fcm and e is inconclusive.

Figure 7.7c shows similar normalised mid-span bending moment versus residual

deflection trends, prior to the occurrence of yielding, for the four beams shown.

7.3.2 Effect of H

Figure 7.8a shows that the beam with less prestressing force, H, only (PS9) exhibits

marginally less residual deflection after an equivalent instantaneous deflection. It

should be noted here that only that portion of the curve prior to the occurrence of

yielding is shown. This effect is confirmed in Figure 7.8b, where at a normalised



bending moment of say 0.5 (50% of ultimate), beam PS9 exhibits less residual

deflection.

7.3.3 Effect of H and e

Figure 7.9a and 7.9b shows the beam with less overall total prestressing, He (PS5)

exhibits more residual deflection than PS6. This is confirmed by the normalised mid-

span bending moment versus residual deflection plot in Figure 7.9b

7.3.4 Summary of Effects

Clearly, there is insufficient data (or significant trends) to identify if residual deflection

is directly impacted on by any particular prestressing variable. Consequently, all PS-

Series beams will be grouped together for the determination of the curve coefficient that

relates the residual deflection to the instantaneous deflection.

7.3.5 The Proposed Equation

A plot containing the instantaneous versus residual deflection curves for all PSC beams

(except PS4, which contained compression reinforcement and is omitted from these

calculations) is given in Figure 7.10.

The form of the equation to the right of the figure is x = αps y, where y and x are the

instantaneous and residual mid-span deflections, respectively. The value of αps is the

residual deflection curve coefficient for prestressed concrete beams.

Using the coefficients obtained in Figure 7.10, the average of αps is found to be 0.09

with a standard deviation of 0.02. This average value process is used as more

supporting experimental data is required before αps can be conclusively associated to a

particularly beam variable (as was done in Section 7.2.6 for the reinforced concrete

beams).

Thus, for the current PSC beams, the residual deflection may be approximated from the

instantaneous deflection by

∆r = 0.09 ∆i (7.3)

where ∆r and ∆i are the residual and instantaneous deflections in mm, at a given service

loading condition, respectively.



a)

cc

c

c

i ii

i

i

i


Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 10 20 300

25

50

75

100

125

PS3PS9

ci

PS3H=346 kNe=97.0 mmfcm= 60.2 MPa


Compare fcm and e

b)

f

f

f

hhh h

hhhh

hh

h

h

h


Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 10 20 300

25

50

75

100

125

PS6PS8

fh

Compare fcm and ePS6H=400 kNe=99.7 mmfcm=69.8 MPa PS8

H=400 kNe=80.0 mmfcm= 52.5 MPa

c) c

c

c

c

f

f

f

h

h

h

h

h

hhh

hh

hh h

i

i

i

i

i

i


Mid

-Spa

nB

endi

ngM

omen

t(N

orm

alis

ed)

0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PS3 (c)PS6 (f)PS8 (h)PS9 (i)

cfhi

Figure 7.7: Residual Deflection for PS3, PS6, PS8 and PS9 Versus: a) and b) Concrete

Compressive Strength and Prestress Eccentricity; c) Normalised Mid-Span Bending

Moment



a)

i ii

i

i

i

j jjj j

jjjjj

j

j

j


Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 10 20 300

25

50

75

100

125

PS9PS10

ij



Compare H

b)

i

i

i

i

i

i

j

j

j

jj

jjj

jj

j

jj


Mid

-Spa

nB

endi

ngM

omen

t(N

orm

alis

ed)

0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PS9 (i)PS10 (j)

ij

Figure 7.8: Residual Deflection for PS9 and PS10 Versus: a) Prestressing Force; b)

Normalised Mid-Span Bending Moment



a)

e

ee

e

f

f

f


Mid

-Spa

nIn

stan

tane

ous

Def

lect

ion

(mm

)

0 10 20 300

25

50

75

100

125

PS5PS6

ef

PS6He = 39,880fcm= 69.8 MPa

PS5He = 36,720fcm= 69.8 MPa

Compare H and e

b)

e

e

e

e

f

f

f


Mid

-Spa

nB

endi

ngM

omen

t(N

orm

alis

ed)

0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PS5 (e)PS6 (f)

ef

Figure 7.9: Residual Deflection for PS5 and PS6 Versus: a) Prestressing Force and

Prestressing Eccentricity; b) Normalised Mid-Span Bending Moment



PS3x=0.08y

R2 = 0.90

PS5x=0.11y

R2 = 0.99

PS6x=0.06y

R2 = 0.99

PS7x=0.08y

R2 = 0.99

PS8x=0.10y

R2 = 0.98

PS9x=0.09y

R2 = 0.97

PS10x=0.12y

R2 = 0.950

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12


Mid

-Spa

n In

stan

tane

ous

Def

lect

ion

(y)

PS3

PS5

PS6

PS7

PS8

PS9

PS10

Linear (PS3)

Linear (PS5)

Linear (PS6)

Linear (PS7)

Linear (PS8)

Linear (PS9)

Linear (PS10)

Figure 7.10: Correlation between Instantaneous and Residual Deflection for PSC Beams

7.4 Verification

Three verification checks of the reinforced concrete residual deflection prediction

equation are presented using, a) the original beam data to recheck that the developed

equations are satisfactory, b) the test data from the F-Series beams, and c) the published

experimental data of James (1997).

7.4.1 Original Beam Data

The overall performance of Equations 7.1 and 7.3 may be represented by the

scattergram shown in Figure 7.11. The following can be observed:

84% of the RC beam deflection data points in Figures 7.11a and 7.11b fall within

the ±20% envelope indicating satisfactory prediction of the reinforced concrete

residual form;

In Figure 7.11c, 68.4% of the PSC beam data falls within the envelope. This is

reasonable considering that the residual deflection equation (Equation 7.3) was

made on limited data and it was recommended that more measurements were

required.



a)

A

A

AA

AA

AA

A

A

A

A

B

B

B

B

BB

BB

B

C

C

C

C

C

C

C

D

D

D

D

E

E

E

E

F

F

F

F

F

G

G

GG

GG

H

H

H

HH

HH

I

I

I

I

J

J

JJ

J

J

J

K

K

KK

K

K

L

L LL

L

Calculated Residual Deflection (mm)

Expe

rimen

talR

esid

ualD

efle

ctio

n(m

m)

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12

ABCDEFGHIJKL

r,calc = αrc i,exp (7.1)

αrc = -0.08ρt + 0.39 (7.2)

20%

20%

b)

1

1

2

2

3

3

4

4

4

4

5

55

6

6

6

7

7

7

7

8

8

8

8

9

9

9

9


Expe

rimen

talR

esid

ualD

efle

ctio

n(m

m)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

CS1CS2CS3CS4CS5CS6CS7CS8CS9

123456789


αrc = -0.08ρt + 0.39 (7.2)

20%

20%

c)

cc

cc

e

e

e

e

f

f

f

gg

gg

gg

g

g

hh

h

h

h

hh h

h

h

h

h

i

ii

i

i

j

jj

jj

jj

j

j


Expe

rimen

talR

esid

ualD

efle

ctio

n(m

m)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

PS3PS5PS6PS7PS8PS9PS10

cefghij

r,calc = 0.09 i,exp (7.3)

20%

20%

Figure 7.11: Experimental versus Calculated Residual Deflection for a) B-Series; b) CS-

Series; and c) PS-Series Test Beams




In Figure 7.12 the experimental versus calculated mid-span residual deflection

scattergram for the additional F-Series reinforced concrete beams, using Equation 7.1

are presented. It seems that at low residual deflections (< 0.5 mm) the scatter of the

data suggests the measurements may have larger margins of error.

Two general observations can be made regarding Figures 7.11 and 7.12:

The residual deflection around the region of first cracking (i.e. at small residual

deflections) was either over- or under-predicted. In spite of this, and considering

the expected variation in deflection measurements at smaller loads, the linear

relationships proposed appear overall to be more than acceptable; and

The prediction of the residual deflection post-cracking was good, where almost all

data values fall within the ±20% envelope. This was generally after 0.5 mm

residual deflection for all beams.

q

qqqq

qq

qq

rr

rr

r

rr

rr

rr

r r

r

r

s

ss

s

ss s

s

ss

s

s

ttt

tt

tt

tt t

t tt

uuuuuu

uu

u u uu

uu

v

vvv

vv

vv

w

ww w

w ww

x

x

xx

xx

x

x

x

y

y

y

y

y

y

y

z

z

z

z

z

z


Expe

rimen

talR

esid

ualD

efle

ctio

n(m

m)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

F1F2F3F4F5F6F7F8F9F10

qrstuvwxyz


αrc = -0.08ρt + 0.39 (7.2)

20%

20%

Figure 7.12: Experimental versus Calculated Residual Deflection for F-Series Test

Beams



7.4.3 James’ Beams

James (1997) undertook an extensive study of the calculation of the instantaneous

deflection using the AS3600-1994 Concrete Structures Code. Details of James’ (1997)

reinforced concrete box beams tested are presented in Table 7.1.

Tables 7.2 to 7.6 present the instantaneous and residual experimental test results of

James (1997) and details of the calculated residual deflection, at every incremental

residual load level, using proposed Equations 7.1 and 7.2. Obtaining tabulated residual

deflection data conducted under similar testing regimes is very rare.

The average ratios ∆r,calc/∆r,exp are significantly close to unity, with acceptable variations,

indicating that Equations 7.1 and 7.2 give very good predictions. To further emphasise

this fact, Figure 7.13 has been produced, showing that most values fall within the ±20%

envelope.

Table 7.1. Details of James’ (1997) Reinforced Concrete Box Beams

Spacing of Reinforcement, s (mm)

Reinforcement Beam# fcm

(MPa) fsy

(MPa) Tension, st Comp., sc Tension Comp. Shear

Beam Type*

5 37.7 400 80 100 3Y20 2R6 R6@300 SS 7 32.4 400 28 100 6Y20 2R6 R6@125 SS 16 34.1 400 80 100 3Y20 2R10 R6@125 C 17 34.2 400 28 100 6Y20 2Y20 R10@130 C 18 30.6 400 55 100 4Y24 2Y24 R10@120 C * Beam types are SS – Simply Supported (Total length 6 metres); and C – Two–Span Continuous (Total length 12 metres). # All beams had b = 300 mm, D = 300 mm and internal void of b = 180 mm, D = 180 mm. Table 7.2. Deflection Data for Beam 5 (James, 1997)

Experimental In-Service

Instantaneous Deflection ∆i,exp (mm)

Curve Coefficient

αrc

Calculated In-Service Residual

Deflection ∆r,calc (mm)

Experimental In-Service Residual

Deflection ∆r,exp (mm)

Load (kN)

(James, 1997) Equation 7.2 Equation 7.1 (James, 1997)

∆ r,calc

∆ r,exp

0 0 0 0 - 24.5 3.38 0.98 1.22 0.80 34.3 9.81 2.84 2.30 1.23 44.9 14.18 4.11 3.27 1.26 54.0 18.54 5.38 3.93 1.37 64.4 22.61 6.56 4.47 1.47 74.4 26.63

ρt = 1.2% ∴ αrc = 0.29

7.72 4.89 1.58 Mean, x = 1.29

Standard Deviation, σn-1 = 0.27



Table 7.3. Deflection Data for Beam 7 (James, 1997) Experimental

In-Service Instantaneous

Deflection ∆i,exp (mm)

Curve Coefficient

αrc





Load (kN)


∆ r,calc

∆ r,exp

0 0 0 0 - 24.5 3.62 0.76 0.89 0.85 39.2 6.71 1.41 1.83 0.77 54.2 10.05 2.1 2.42 0.87 68.7 13.27 2.8 3.00 0.93 83.4 16.51 3.47 3.36 1.03 117.7 24.46 5.14 4.58 1.12 157.0 33.83

ρt = 2.3% ∴ αrc = 0.21

7.10 5.88 1.21 Mean, x = 0.97





Curve Coefficient

αrc





Load (kN)


∆ r,calc

∆ r,exp

0 0 0 0 - 29.4 2.04 0.59 0.66 0.89 40.2 4.82 1.40 1.25 1.12 49.8 6.70 1.94 1.66 1.17 59.8 8.69 2.52 2.01 1.25 80.0 12.99 3.77 2.98 1.27 90.3 15.12

ρt = 1.2% ∴ αrc = 0.29

4.38 3.33 1.32 Mean, x = 1.17





Curve Coefficient

αrc





Load (kN)


∆ r,calc

∆ r,exp

0 0 0 0 - 21.0 1.82 0.33 0.38 0.87 44.7 4.4 0.79 0.95 0.83 68.9 7.66 1.38 1.52 0.91 118.9 15.23 2.74 2.74 1.0 167.0 22.65 4.08 3.79 1.08 216.4 30.29

ρt = 2.3% ∴ αrc = 0.21

5.45 4.87 1.12 Mean, x = 0.97







Curve Coefficient

αrc





Load (kN)


∆ r,calc

∆ r,exp

0 0 0 0 - 21.8 1.74 0.37 0.42 0.87 50.4 5.09 1.07 1.26 0.85 80.4 9.36 1.97 2.06 0.95 108.7 13.73 2.88 2.84 1.02 138.1 18.88 3.96 3.90 1.02 196.6 28.68

ρt = 2.2% ∴ αrc = 0.21

6.02 5.77 1.04 Mean, x = 0.96


⊕

⊕

⊕

⊕

⊕⊕

⊕

⊕

∞

∞

∞

∞∞

∞∞

•

•

•

•

•

•

•


Jam

es'E

xper

imen

talR

esid

ualD

efle

ctio

n(m

m)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

James 5James 7James 16James 17James 18

⊕∞

•


αrc = -0.08ρt + 0.39 (7.2)

20%

20%

Figure 7.13: Experimental versus Calculated Residual Deflection for James’ Beams



The following general conclusions can be noted from Tables 7.2 to 7.6 and Figure 7.13:

All James’ test beams were constructed as hollow box beams. Furthermore, they

all contained 400 MPa reinforcement. Thus, Equations 7.1 and 7.2 give

remarkably good residual deflection predictions for these beams;

Beams 5 and 7 were simply-supported (single span beams), whilst beams 16, 17

and 18 were two-span continuous beams. Therefore Equations 7.1 and 7.2 seem

to be equally applicable for a range of beam and test setups;

A good range of tensile reinforcement ratios, between 1.2% and 2.3%, as used to

verify Equations 7.1 and 7.2, also the compression and shear reinforcement

arrangements varied, as shown in Table 7.1; and

The average ratios ∆r,calc/∆r,exp were 1.29, 0.97, 1.17, 0.97 and 0.96 for beams 5, 7,

16, 17 and 18, respectively. They are significantly close to unity, with acceptable

variations, indicating that Equations 7.1 and 7.2 give very good predictions.

7.5 Summary

The residual deflection characteristics of the reinforced concrete beams were modelled

using a linear relationship to instantaneous deflection, and selected according to the

tensile reinforcement ratio, ρt. The extensive verifications of the reinforced concrete

residual deflection equation indicate that:

The prediction of residual deflection for shear beams is good, however, Equations

7.1 and 7.2 tend to overpredict values particularly at low load levels as shown in

Figure 7.1;

The equations are considered valid for concrete compressive strengths between

22.5 and 90.7 MPa and reinforcement of either 400 or 500 MPa.

For the PS-Series beams a generic linear relationship for prestressed beams was

suggested. It gives reasonable prediction of residual deflections as shown in Figure

7.11c but as indicated in Section 7.3.5, more supporting data is required so a better

estimation of αps is derived.


Chapter 8: Total Damping in Concrete Beams 8-1

CHAPTER 8

Total Damping in Concrete Beams

8.1 General Remarks

As presented in Section 3.3, the equation for computing the total damping capacity of a

concrete beam at any stage of its service life is the sum of the contributions by the

‘untested’ and ‘tested’ logdec components. To estimate the ‘untested’ logdec

component for reinforced and prestressed concrete beams, Equations 6.2 and 6.3 were

developed and verified in Sections 6.2.3 and 6.3.4, respectively.

To estimate the ‘tested’ logdec component, residual deflection will be used. Sections

3.2.3 and 3.2.4, highlighted reasons why residual deflection is considered superior to

explicit measures of residual crack width.

Section 8.2 makes general observations of the experimental ‘tested’ logdec data versus

the residual deflection, and develops an equation to model the relationship. Chapter 7

focused on the development of residual deflection utilising instantaneous deflection.

Finally, in Section 8.3, verification of the developed total logdec prediction equation is

made using the F-Series beams and the data of Chowdhury (1999).

8.2 Development of Total Damping Equations

Presented in Figures 8.1, 8.2 and 8.3, are the experimental logdec versus mid-span

residual deflections for all the B-, CS- and PS-Series test beams, respectively. It should

be noted here that only the residual deflection prior to yielding is shown. Also, the y-

intercept is the experimental ‘untested’ logdec. In the figures, both flexural and shear

beams are presented, varying in concrete compressive strength and reinforcement yield

strength.



a)

B4 = 0.0008x + 0.05

R2 = 0.74

B3 = 0.001x + 0.0585

R2 = 0.6104

B1 = 0.0011x + 0.0648

R2 = 0.8257

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

0 10 20 30


Logd

ec

B1

B3

B4

Linear (B4)

Linear (B3)

Linear (B1)

δ

δ

δ

δ

b)

B6 = 0.0013x + 0.0534

R2 = 0.3593

B8 = 0.0019x + 0.0388

R2 = 0.5983

B7 = 0.0019x + 0.0361

R2 = 0.5885

B5 = 0.0016x + 0.0573

R2 = 0.8474

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0 5 10 15 20 25 30


Logd

ec

B5B6B7B8Linear (B6)Linear (B8)Linear (B7)Linear (B5)

δ

δ

δ

δ

δ

c)

B10 = 0.0015x + 0.0457

R2 = 0.6378

B12 = 0.0022x + 0.0566

R2 = 0.6238

B11 = 0.0033x + 0.0518

R2 = 0.4417

B9 = 0.0023x + 0.0444

R2 = 0.7741

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

0 5 10 15 20 25 30


Logd

ec

B9

B10B11

B12

Linear (B10)Linear (B12)

Linear (B11)Linear (B9)

δ

δ

δ

δ

δ

Figure 8.1: Logdec versus Residual Deflection for B-Series: a) BI-1, BI-3, BII-4; b) BI-

5 to BII-8; and c) BI-9 to BII-12



a)

CS2 = 0.0098x + 0.0693

R2 = 1

CS1 = 0.0108x + 0.07

R2 = 1

CS3 = 0.0175x + 0.059

R2 = 1

0.045

0.055

0.065

0.075

0.085

0.095

0.105

0.115

0 2 4 6


Logd

ecCS1

CS2

CS3

Linear (CS2)

Linear (CS1)

Linear (CS3)

δ

δ

δ

δ

b)

CS6 = 0.0318x + 0.0519

R2 = 0.8931

CS5 = 0.018x + 0.0501

R2 = 0.9721

CS4 = 0.0292x + 0.0495

R2 = 0.9741

0.045

0.055

0.065

0.075

0.085

0.095

0.105

0.115

0 2 4 6Mid-Span Residual Deflection (mm)

Logd

ec

CS4

CS5

CS6

Linear (CS6)

Linear (CS5)

Linear (CS4)

δ

δ

δ

δ

c)

CS9 = 0.0188x + 0.0627

R2 = 0.8544

CS8 = 0.0191x + 0.0591

R2 = 0.8458

CS7 = 0.0173x + 0.0631

R2 = 0.7841

0.0450.0550.0650.0750.0850.0950.1050.1150.1250.1350.145

0 2 4 6


Logd

ec

CS7

CS8

CS9

Linear (CS9)

Linear (CS8)

Linear (CS7)

δ

δ

δ

δ

Figure 8.2: Logdec versus Residual Deflection for CS-Series: a) CS1 to CS3; b) CS4 to

CS6; and c) CS7 to CS9



a)

PS5 = 0.0018x + 0.043

R2 = 1

PS3 = 0.0019x + 0.042

R2 = 1

PS6 = 0.0029x + 0.0467

R2 = 0.8302

0.040.045

0.050.055

0.060.065

0.070.075

0.080.085

0.09


Logd

ec

PS3

PS5

PS6

Linear (PS5)

Linear (PS3)

Linear (PS6)

δ

δ

δ

δ

b)

PS8 = 0.0027x + 0.0412

R2 = 0.9354

PS7 = 0.0019x + 0.042

R2 = 1

PS9 = 0.005x + 0.0424

R2 = 0.253

PS10 = 0.0027x + 0.0547

R2 = 0.8571

0.04

0.05

0.06

0.07

0.08

0.09

0.1


Logd

ec

PS7PS8PS9PS10Linear (PS8)Linear (PS7)Linear (PS9)Linear (PS10)

δ

δ

δ

δ

δ

Figure 8.3: Logdec versus Residual Deflection for: a) PS3 to PS6; b) PS7 to PS10

For each beam data presented in Figures 8.1, 8.2 and 8.3, the line-of-best-fit has been

used and consists of two components, the last number in each equation shown is the y-

intercept which is in fact the ‘untested’ logdec, δuntest. The first number defines the

slope of the curve, and is termed here the damping-residual deflection (D-R) slope, βfl,

for concrete beams.

The D-R Slopes, βfl, for each test beam have been plotted against the concrete



compressive strength, fcm in Figure 8.4. In Figure 8.4a, both the B- and PS-Series

flexural beams having a length of 6.0 m and the same cross-sectional dimensions have

been plotted together. For these beams a relationship was found as

βfl = 0.0007e0.018fcm (8.1)

where fcm ranges between 22.5 MPa and 90.7 MPa.

In Figure 8.4b, all the CS-Series flexural and shear beams have been plotted. There is

some difference between the flexural and shear beams. In the absence of a significant

number of flexural test beams, the D-R Slope, βfl for beams CS1 to CS3 and CS7 to

CS9 has been averaged to 0.016.

Finally, the equation to predict the total logdec, δtotal, in reinforced and prestressed

flexural beams is given by

δtotal = βfl ∆r + δuntest (8.2)

where δuntest is the relevant ‘untested’ damping capacity as given in Chapter 6; ∆r is the

calculated residual deflection of the beam in mm for any service loading level as

detailed in Chapter 7; and βfl is calculated from Equation 8.1 for full-scale reinforced

and prestressed concrete beams similar to the B- and PS-Series beams, and equals 0.016

for half-scale concrete beams similar to the CS-Series beams.

From the above discussion and formulation, the following points may be noted:

The establishment of the ‘untested’ logdec is very important because the

calculation of the logdec in-service is based on this quantity;

It should be noted here, that even though the PSC beams do not exhibit visual

cracking until they reach 60% of failure, internal damage begins immediately, and

this is reflected by the presence of residual deflection, and thus an increase in

damping capacity;

The D-R Slope, βfl, defining the relationship between damping capacity and

residual deflection appears to be affected by beam size. This is concluded by the

separate plots of B- and CS-Series beams; and

Interestingly, the D-R Slope, βfl, was very similar for both the reinforced and

prestressed concrete beams.



a) Concrete Compressive Strength, fcm (MPa)

D-R

Slop

e-F

lexu

ralB

eam

s(β

fl)

0 20 40 60 80 1000

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

B-SeriesPS-Series(β fl ) = 0.0007e0.018 fcm

(β fl ) = 0.0007e0.018 fcm

R2 = 0.68

b)

xx

x xxx

Concrete Compressive Strength, fcm (MPa)

D-R

Slop

e-F

lexu

ralB

eam

s(β

fl)

0 20 40 60 80 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

CS-Series (Flexural Only)CS-Series (Shear Only)

x

Figure 8.4: Dependence of D-R Slope on Concrete Compressive Strength: a) B- and PS-

Series Beams; b) CS-Series Beams



8.3 Verification

In order to verify the proposed total damping prediction equation, comparisons are

made with the F-Series test beams and Chowdhury’s (1999) test data. As mentioned

through this thesis, it is extremely difficult to obtain published work that is acceptable

for comparison due to the fact that they lack the necessary beam details, material

properties, tabulated loading history, logdec and residual deflection details. This is why

the additional F-Series was developed.


In Figure 8.5, a comparison of the experimentally observed F-Series damping test data

versus that predicted using Equation 8.2 is given. The scattergram shows that:

In general, the proposed equation is satisfactory for both flexural and shear beams

at all residual deflections, as most of the calculated and experimental logdecs are

within 20% of a perfect correlation;

For some flexural beams (F2, F3, F8 and F9), there appears to be a slight

underestimation of damping towards failure. This is most likely due to the fact

that the total damping equation was developed using pre-yield data, where the

data included in Figure 8.5 is for all load levels.

Overall however, the correlation is excellent.

qq qqq qqqq

r rrrr

r rr rrr rrr

r

ss s

ss sss

ssss

t tt ttt t t tttt

t

uu uuu uuuu

vv vv

vv

ww ww

w ww

xx

xxxxx

x

x

y

yy

yy

y

y

zz

zzz

z

z

Experimental Logdec, δtotal

Cal

cula

ted

Logd

ec,δ

tota

l

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.02

0.04

0.06

0.08

0.1

0.12

0.14

F1F2F3F4F5F6F7F8F9F10

qrstuvwxyz

+ 20%

- 20%F-Series Beams

Figure 8.5: Calculated versus Experimental Logdec – F-Series Beams



8.3.2 Chowdhury’s Beams

Chowdhury (1999) performed free-vibration experiments at incremental levels of

damage loading. A typical cross-sectional detail of Chowdhury’s (1999) test beams,

and test set-up are given in Figure 8.6. Table 8.1 presents details for each test beam.

Pin Roller

L varies according to beam typeSS = 6 metresC = 12 metres

l = 2000 mmfor all beams

PCB Piezotronics Model 353AAccelerometer at the mid point

Embeddedpolystyrene

void

21

112

30

12030 120 30

60 60

1446

180

60

180CR

SR

TR

Figure 8.6: Details of Test Beams and Testing Arrangement of Chowdhury (1999)

Table 8.1. Details of Chowdhury’s (1999) Reinforced Concrete Box Beams Spacing of

Reinforcement, s (mm) Reinforcement

Beam# fcm(MPa)

fsy(MPa)

Tension, st Comp., scTension

TR Comp.

CR Shear SR

Beam Type*

5 37.7 400 80 100 3Y20 2R6 R6@300 SS 7 32.4 400 28 100 6Y20 2R6 R6@125 SS 16 34.1 400 80 100 3Y20 2R10 R6@125 C 17 34.2 400 28 100 6Y20 2Y20 R10@130 C 18 30.6 400 55 100 4Y24 2Y24 R10@120 C * Beam types are SS – Simply Supported (Total length 6 metres); and C – Two–Span Continuous (Total length 12 metres). # All beams had b = 300 mm, D = 300 mm and internal void of b = 180 mm, D = 180 mm.

Tables 8.2 to 8.6 presents a comparison of the calculated ‘tested’ damping results (using

Equation 9.2), to the experimental damping test results for the beams of Chowdhury

(1999). The tables show that the ratio between the experimental and calculated total

logdec values of Chowdhury’s (1999) beams indicates a very good correlation, being

0.95, 0.97, 0.94, 1.04 and 1.18 for beams 5, 7, 16, 17 and 18, respectively.

Figure 8.7 shows that an excellent correlation exists between all ‘tested’ calculated and

experimental logdec values for Chowdhury’s (1999) beams where 95% of all points lie

within ±20% of a perfect correlation.



Table 8.2. Deflection versus Damping Data for Beam 5 (Chowdhury, 1999) Experimental

In-Service Logdec δexp


Residual Deflection ∆r,exp (mm)

Calculated In-Service

δtotal(Equation 8.2)

Load (kN) #

(Chowdhury, 1999) (James, 1997) δtotal = βfl ∆r + δuntest

δtotal = 0.0014∆r + 0.073

δtotal,exp

δtotal,calc

0 0.073 0 0.073 (δuntest) 1.00 24.5 ** 0.074 1.22 0.075 0.99 34.3 0.075 2.30 0.076 1.0 44.9 0.071 3.27 0.078 0.91 54.0 0.069 3.93 0.079 0.87 64.4 0.074 4.47 0.080 0.93 74.4 0.074 4.89 0.080 0.93

Mean, x = 0.95 Standard Deviation, σn-1 = 0.05

** This is the load at which the beam first cracked. # The maximum (failure) load was 105.0 kN. Table 8.3. Deflection versus Damping Data for Beam 7 (Chowdhury, 1999)


Logdec δexp





Load (kN) #


δtotal = 0.0013∆r + 0.073

δtotal,exp

δtotal,calc

0 0.073 0 0.073 (δuntest) 1.00 24.5 0.070 0.89 0.074 0.95 39.2 ** 0.075 1.83 0.075 1.00 54.2 0.079 2.42 0.076 1.04 68.7 0.073 3.00 0.077 0.95 83.4 0.074 3.36 0.077 0.96 117.7 0.076 4.58 0.079 0.96 157.0 0.075 5.88 0.081 0.92


** This is the load at which the beam first cracked. # The maximum (failure) load was 230.0 kN. Table 8.4. Deflection versus Damping Data for Beam 16 (Chowdhury, 1999)


Logdec δexp





Load (kN) #


δtotal = 0.0013∆r + 0.082

δtotal,exp

δtotal,calc

0 0.082 0 0.082 (δuntest) 1.00 29.4 ** 0.081 0.66 0.083 0.98 40.2 0.078 1.25 0.084 0.93 49.8 0.078 1.66 0.084 0.93 59.8 0.077 2.01 0.085 0.91 80.0 0.074 2.98 0.086 0.86 90.3 0.083 3.33 0.086 0.97


** This is the load at which the beam first cracked. # The maximum (failure) load was 160.0 kN.









Load (kN) #


δtotal = 0.0013∆r + 0.076

δtotal,exp

δtotal,calc

0 0.076 0 0.076 (δuntest) 1.00 21.0** 0.084 0.38 0.077 1.09 44.7 0.076 0.95 0.077 0.99 68.9 0.088 1.52 0.078 1.13 118.9 0.083 2.74 0.080 1.04 167.0 0.079 3.79 0.081 0.98 216.4 0.086 4.87 0.082 1.05









Load (kN) #


δtotal = 0.0012∆r + 0.071

δtotal,exp

δtotal,calc

0 0.071 0 0.071 (δuntest) 1.00 21.8 ** 0.086 0.42 0.072 1.19 50.4 0.083 1.26 0.073 1.14 80.4 0.098 2.06 0.073 1.34 108.7 0.088 2.84 0.074 1.19 138.1 0.085 3.90 0.076 1.12 196.6 0.099 5.77 0.078 1.27



8.4 Advantages of Proposed Residual Deflection Equations

From the discussions above, the advantages of the proposed total damping model are as

follows:

For prestressed beams: it has been demonstrated that damping will increase

during its service life. It is common for these beams to not exhibit visual cracking

and thus, residual deflection is an excellent way of predicting the in-service

damping levels;

For shear beams: flexural cracking could not be properly measured. However,

similar to the prestressed beams, damping still increased during the experiments



and thus the use of residual deflection was found to be very effective;

For flexural beams: the residual deflection mechanism can be reliably predicted

from a simply equation;

Concrete compressive strength: was found to affect the damping versus residual

deflection relationship for the flexural beams (the D-R Slope, βfl). Therefore,

concrete compressive strength (and associated beam constituents) most likely

affects the internal beam damage, thus affecting damping capacity;

For simply-supported and continuous beams: the calculation of residual

deflection and damping capacity using the proposed equation was excellent for

both types of test beams (using Chowdhury’s, 1999 test beams);

For concrete box beams: the calculation of total logdec in Chowdhury’s (1999)

hollow box beams was excellent; and

Establishing the ‘untested’ damping capacity: is of primary importance before

modelling the in-service, total damping capacity.

PPPPP

P

P

∅

∅

∅

∅

∅∅

∅

Total Logdec, δtotal (Calculated from Equation 8.2)

Tota

lLog

dec,δ t

otal

(Cho

wdh

ury'

sEx

perim

enta

lDat

a)

0.06 0.08 0.1 0.12

0.06

0.08

0.1

0.12

Beam 5Beam 7Beam 16Beam 17Beam 18

P∅

-20%

+20%

Figure 8.7: Chowdhury’s Experimental versus Calculated using Equation 8.2



8.5 Summary

A method by which to calculate the residual deflection of a beam, subject to transient

damage causing loads, was proposed in this Chapter. The experimental data from the

current series of test beams were used to develop a relationship, for the current testing

regime, between the experimental residual deflection and logdec values. Verifications

using the additional F-Series beams and Chowdhury’s (1999) test beams have shown

the proposed equation to be very good.


Chapter 9: Conclusions and Recommendations 9-1

CHAPTER 9

Conclusions and Recommendations

9.1 General Remarks

This thesis focussed on the problems associated with a current world-wide trend

towards optimised structures, in which structural members are increasingly longer,

lighter and more slender. One negative effect of these seemingly cost effective trends is

their vulnerability to vibrations causing damage. Following an extensive review of the

literature on damping, the omission of two broad areas were identified as follows:

There is an extreme paucity of damping data available;

There is no method currently available to predict the damping capacity of

reinforced and prestressed concrete beams at the design stage.

In response to these findings, an extensive experimental programme was devised to

investigate and provide the experimental data necessary to elucidate the necessary

information. These test results were used to evaluate the damping research presented

within the literature and to assist in formulating a new technique to analyse and

calculate the damping capacity of concrete beams over the full loading regime. The

developed equations were constructed using the extensive experimental data and

verified by: a) an additional series of test beams to provide supporting data; and b)

comparative experimental data extracted from the literature.

9.2 Research Objectives and Outcomes

In Section 1.2, a number of main research objectives were identified. Each of the

objectives have been addressed in depth within the thesis and the following outcomes



have been derived:

(a) A damping analysis technique was created for the extraction of the logdec from

the raw free-vibration decay curve. The proposed method is a generic technique

allowing researchers to interpret and compare experimental damping data

satisfactorily;

(b) Identification of damping categories to describe the damping characteristics of a

beam at any stage of its service life (i.e. ‘untested’ versus ‘tested’). Previous work

has used damping terminology inconsistently and interchangeably, making

damping data comparisons exceedingly difficult;

(c) The calculation of the ‘untested’ logdec has not been adequately addressed by the

literature. Equations to determine the ‘untested’ damping capacity of reinforced

and prestressed concrete beams were proposed using experimental results and

verified with supporting data;

(d) A residual deflection calculation method has been proposed and is based on the

instantaneous deflection characteristics of reinforced and prestressed concrete

beams for the current experimental programme. Verifications using additional

experimental data and data from previous research showed the proposed equation

to be very versatile; and

(e) A total damping capacity equation has been developed to predict the full-range

damping behaviour of reinforced and prestressed concrete beams. The proposed

equation has been verified using additional experimental data and data from

previous research.

9.2.1 Summary of Test Results

In order to achieve the research objectives, a large amount of experimental test data was

generated. Using the data, the following was found:

(a) Damping Analysis Technique

Free-vibration decay in reinforced and fully-prestressed concrete beams is not

strictly exponential in the initial portion of decay, as required by the theory, but

becomes exponential as the free-vibration stabilises;

The point from which logdec is calculated on the free-vibration decay curve

cannot be defined by a cycle number as traditionally suggested. It is proposed



here that the ‘optimal peak ratio’ (as defined in Section 5.4.2) be used to define

the point where logdec is extracted from decay curves;

The weight and location of the impact hammer does not affect the resulting

calculation of logdec of concrete beams.

(b) Damping Prediction in ‘Untested’ Concrete Beams

The damping capacity of an ‘untested’ beam was found to be the integral

component of any damping model, as it serves as the initial starting point in

calculating damping, a fact not previously considered or developed;

In an ‘untested’ reinforced concrete beam, the concrete compressive strength and

reinforcement yield strength did not affect damping capacity, but longitudinal

reinforcement ratio (LRD) did;

A damping prediction equation for ‘untested’ reinforced concrete beams, δuntest,

was thus proposed (Equation 6.2) utilising LRD as the primary variable. This

equation normalises the cross-section dimensions by utilising the reinforcement

ratio, and it was also found to be equally applicable for beams of length 2.4 m and

6.0 m. It is reproduced here:

19.0

223.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛+×=

c

c

t

tuntest ss

ρρδ (6.2)

where st and sc are the tension and compression reinforcement spacings,

respectively in mm and are given in Table 4.1; ρt = Ast/bd and ρc = Asc/bd. The

equation is valid for LRD distributions between 0.0001 and 0.0023 (see Table

6.1).

An ‘untested’ damping prediction equation for fully-prestressed beams was also

proposed (Equation 6.3) utilising the overall amount of prestress in a beam (He)

as the primary variable. It is reproduced here:

δuntest = 1.4×10-10He2 - 9.4×10-6He + 0.2 (6.3)

where Equation 6.3 is valid for He between 30,000 and 45,000 (kNmm).

(c) Calculation of Residual Deflection

In reinforced concrete beams the instantaneous versus residual deflection



relationship were affected primarily by the tensile reinforcement ratio, ρt;

A residual deflection equation, based on the instantaneous deflection was

proposed (Equations 7.1 and 7.2). They are reproduced here:

∆r = αrc ×∆i (7.1)

where αrc is the curve coefficient found from Equation 7.2 and derived from

Figure 7.6 and ∆r and ∆i are in mm.

αrc = -0.08ρt + 0.39 (7.2)

where Equation 7.2 is valid for 0.76% < ρt < 3.0% .

For prestressed concrete beams a residual deflection equation was also proposed

(Equation 7.3). It is reproduced here:

∆r = 0.09 ∆i (7.3)

where ∆r and ∆i are the residual and instantaneous deflections in mm, at a given

service loading condition, respectively.

(d) Total Damping in Concrete Beams

The equation to predict the total logdec, δtotal, in reinforced and prestressed

concrete beams (Equation 8.2), is the sum of the contribution of the ‘untested’

damping capacity, δuntest, and the ‘tested’ damping capacity defined by the residual

deflection relationship, βfl ∆r. It is reproduced here:

δtotal = βfl ∆r + δuntest (8.2)

For full-scale reinforced and prestressed concrete beams, βfl is the correlative

function, it may be found from Equation 8.1. It is reproduced here:

βfl = 0.0007e0.018fcm (8.1)

where δuntest is the relevant ‘untested’ damping capacity as given in Chapter 6; ∆r

is the calculated residual deflection of the beam in mm for any service loading

level as detailed in Chapter 7; and βfl is calculated from Equation 8.1 for full-

scale reinforced and prestressed concrete beams similar to the B- and PS-Series



beams, and equals 0.016 for half-scale concrete beams similar to the CS-Series

beams.

For half-scale reinforced concrete beams similar to the CS-Series, βfl is equal to

0.016.

9.2.2 Verification of the Proposed Methods

As mentioned throughout this thesis, the paucity of available and useful damping data

made verification of the proposed methods difficult. Therefore, it was necessary to

devise an additional test series (F-Series), along with the data of James (1997),

Chowdhury (1999) and Neild (2001) for validation purposes. Verification of the

proposed methods was undertaken in the appropriate sections of Chapters 6, 7, and 8.

These comparative investigations have shown that the proposed equations are reliable

and applicable for a range of beam designs, test set-ups, constituent materials and

loading regimes.

9.3 Recommendations and Scope for Future Research

A number of prediction equations have been presented. These equations appear to offer

promising methods of damping prediction for a range of variables. Further research

should be directed towards continually improving the prediction methods by

investigating:

Other damage sources: the effect of damage on the residual deflection/damping

capacity due to corrosion of reinforcement or spalling of concrete;

Other loading regimes: by examining damage caused by other cyclic and impact

loads and also load-reversal effects such as that created during earthquakes ; and

Structure damping: the effect of different types of joints and connections, such as

beam-column or beam-panel connections, on total damping.

9.4 Closure

The need for a user friendly damping calculation method has been addressed within this

thesis. In particular, a number of important damping considerations identified as being

absent from the literature were investigated.

This thesis has addressed the perceived gaps in the literature by presenting details and

results of a series of extensive experiments that will be an invaluable source of reference



not only to damping researchers but structural designers and practitioners. The

subsequent extensive analysis has resulted in a straightforward methodology for

deriving and calculating the total damping capacity of reinforced and fully-prestressed

concrete beams.


References Re-1

REFERENCES

Ahlborn, T.M., Shield, C.K. and French, C.W. (1997) Full-scale testing of prestressed concrete bridge girders, Experimental Techniques, 21(1), 33-35.

Akashi, T. (1960) On the measurement of logarithmic decrement of concrete, General

Meeting Reviews, Cement Association of Japan, May, pp. 103-104. Almansa, F.L., Casas, J.R., Serrà, I. And Canas, J.A. (1993) Experimental study on the

dynamic behaviour of early unformed reinforced concrete beams, Proceedings of the Second European Conference on Structural Dynamics: EURODYN ’93, June 21-23, Trondheim, Norway, pp.911-917.

AS 3600 – 2001, Australian Standard for Concrete Structures, Standards Association of

Australia, North Sydney, NSW, Australia. Ashbee, R.A., Heritage, C.A.R. and Jordan, R.W. (1976) The expanded hysteresis loop

method for measuring the damping properties of concrete, Magazine of Concrete Research, 28(96), 148-156.

Askegaard, V. and Langsœ, H.E. (1986) Correlation between changes in dynamic

properties and remaining carrying capacity: Laboratory tests with reinforced concrete beams, Materials and Structures (RILEM), 19(109), 11-19.

Bachmann, H., Ammann, W.J. and Deischl, F. et al. (1995) Vibration Problems in

Structures: Practical Guidelines, Birkhäuser Verlag Basel, Basel. Beards, C.F. (1996) Structural Vibration: Analysis and Damping, Arnold, London. Béliveau, J.-G. (1976) Identification of viscous damping in structures from modal

information, Journal of Applied Mechanics, Transactions of the American Society of Civil Engineers, June, 335-339.

Bhuvanagiri, V.K. and Swartz, S.E. (2000) Ensuring consistency in impact-vibration

signature tests, Experimental Techniques, 24(1), 24-26. Bishop, R.E.D. (1955) The treatment of damping forces in vibration theory, Journal of

the Royal Aeronautical Society, 59, November, 738-742. Bock, E. (1942) Behaviour of concrete and reinforced concrete subjected to vibrations


References Re-2

causing bending, Zeitschrift des Vereins Deutscher Ingenieure, 86(9/10), 145-147. (In German).

Boroschek, R.L. and Yáñez, F.V. (2000) Experimental verification of basic analytical

assumptions used in the analysis of structural wall buildings, Engineering Structures, 22, 657-669.

Brownjohn, J.M.W. (1994) Estimation of damping in suspension bridges, Proceedings

of the Institution of Civil Engineers: Structures and Buildings, 104, 401-415. Buchholdt, H. (1997) Structural Dynamics for Engineers, Thomas Telford, London. Bungey, J.H. and Millard, S.G. (1996) Testing of Concrete in Structures, Blackie

Academic & Professional, Cambridge. Capozucca, R. and Cerri, R.N. (2002) Static and dynamic behaviour of RC beam model

strengthened by CFRP-sheets, Construction and Buildings, 16, 91-99. Carr, A.J. and Tabuchi, M. (1993) The structural ductility and the damage index for

reinforced concrete structure under seismic excitations, Proceedings of the Second European Conference on Structural Dynamics: EURODYN ’93, June 21-23, Trondheim, Norway, pp.169-176.

Chowdhury, S.H. (1999) Damping Characteristics of Reinforced and Partially

Prestressed Concrete Beams, PhD Thesis, Griffith University, Australia. Clough, R.W. and Penzien, J. (1975) Dynamics of Structures, McGraw-Hill, New York. Cole, D.G. (1966) The damping capacity of hardened cement paste, mortar, and

concrete specimens, Vibration in Civil Engineering: Proceedings of a Symposium Organized by the British National Section of the International Association for Earthquake Engineering, Imperial College of Science and Technology, London, April 1965, (Ed. B.O. Skipp), Butterworths, London, pp. 235-247.

Cole, D.G. and Spooner, D.C. (1965) The damping capacity of hardened cement paste

and mortar in specimens vibrating at very low frequencies, Proceedings, American Society for Testing and Materials, Committee Reports and Technical Papers, 65, 661-667.

Cole, D.G. and Spooner, D.C. (1968) The damping capacity of concrete, The Structure

of Concrete and Its Behaviour Under Load: Proceedings of an International Conference on the Structure of Concrete, London, September, (Eds. A.E. Brooks and K. Newmann) Cement and Concrete Association, London, pp. 217-225.

Dems, K. and Mróz, Z. (2001) Identification of damage in beam and plate structures

using parameter-dependent frequency changes, Engineering Computations, 18(1/2), 96-120.

Denoon, R.O. and Kwok, K.C.S. (1996) Full-scale measurements of wind-induced

response of an 84m high concrete control tower, Journal of Wind Engineering and Industrial Aerodynamics, 60, 155-165.


References Re-3

Dexter, R.J. and Fisher, J.W. (1997) Fatigue and fracture, Chapter in Handbook of Structural Engineering, (W.F. Chen Ed.), CRC Press LLC, Florida, USA.

Dieterle, R. and Bachmann, H. (1981) Experiments and models for the damping

behaviour of vibrating reinforced concrete beams in the uncracked and cracked condition, International Association for Bridge and Structural Engineering, Reports of the Working Commissions, 34, 69-82.

Døssing, O. (1988a) Structural Testing Part 1: Mechanical Mobility Measurements,

Brüel & Kjær, Denmark. Døssing, O. (1988b) Structural Testing Part 2: Modal Analysis and Simulation, Brüel &

Kjær, Denmark. Douglas, B.M., Drown, C.D. and Gordon,M.L. (1981) Experimental dynamics of

highway bridges, Proceedings of the Second Specialty Conference on Dynamic Response of Structures: Experimental Observation, Prediction and Control, January 15-16, Atlanta, Georgia, pp. 698-712.

Economou, S.N., Fardis, M.N. and Harisis, A. (1993) Linear elastic v nonlinear

dynamic seismic response analysis of RC buildings, Proceedings of the Second European Conference on Structural Dynamics: EURODYN ’93, June 21-23, Trondheim, Norway, pp. 63-70.

Fahey, S.O’F. and Pratt, J. (1998a) Frequency domain modal estimation techniques,

Experimental Techniques, 22(5), 33-37. Fahey, S.O’F. and Pratt, J. (1998b) Time domain modal estimation techniques,

Experimental Techniques, 22(6), 45-49. Fajfar, P., Vidic, T. and Fischinger, M. (1993) Influence of damping model on the

seismic response of nonlinear SDOF systems, Proceedings of the Second European Conference on Structural Dynamics: EURODYN ’93, June 21-23, Trondheim, Norway, pp. 77-84.

Fang, J.Q., Li, Q.S., Jeary, A.P. and Wong, C.K. (1998) Full scale measurement of

damping in a 78-storey tall building, Advances in Structural Engineering, 2(1), 41-48.

Fertis, D.G. (1995) Mechanical and Structural Vibrations, John Wiley & Sons, New

York. Flesch, R. (1981) The damping behaviour of R/C cantilever elements, International

Association for Bridge and Structural Engineering, Reports of the Working Commissions, 34, 83-98.

French, M. (1999) What makes a good impact function?, Experimental Techniques,

23(6), 33-35. Fu, X. and Chung, D.D.L. (1996) Vibration damping admixtures for cement, Cement

and Concrete Research, 26(1), 69-75.


References Re-4

Fu, X., Li, X. and Chung, D.D.L. (1998) Improving the vibration damping capacity of cement, Journal of Materials Science, 33, 3601-3605.

Glanville, M.J., Kwok, K.C.S. and Denoon, R.O. (1996) Full-scale damping

measurements of structures in Australia, Journal of Wind Engineering and Industrial Aerodynamics, 59, 349-364.

Helmholtz, H.L.F. (1877) On the sensations of tones as a physical basis for the theory of

music, Dover, New York, 406 pp. Translation by A.J. Ellis of Die Lehre von dem Tonempfindungen, Fourth Edition, 1877; first edition published in 1863.

Hop, T. (1991) The effect of degree of prestressing and age of concrete beams on

frequency and damping of their free vibration, Materials and Structures, 24, 210-220.

Ibrahim, S.R. and Mikulcik, E.C. (1977) A method for the direct identification of

vibration parameters from the free response, The Shock and Vibration Bulletin, 47(4), 183-198.

Irvine, H.M. (1986) Structural Dynamics for the Practising Engineer, E&FN Spon,

London. James, M.I. (1997) Prediction of Deflection in Simply Supported and Continuous

Reinforced Concrete Box Beams, BEng Thesis, Griffith University, Australia. James, M.L., Lutes, L.D. and Smith, G.M. (1964) Dynamic properties of reinforced and

prestressed concrete structural components, Journal of the American Concrete Institute, 61(11), 1359-1381.

Jeary, A. P. (1974) Damping measurements from the dynamic behaviour of several

large multi-flue chimneys, Proceedings, The Institution of Civil Engineers, Part 2: Research and Theory, 57, 321-329.

Jeary, A.P. (1986) Damping in tall buildings – A mechanism and a predictor,

Earthquake Engineering and Structural Dynamics, 14, 733-750. Jeary, A.P. (1996) The description and measurement of nonlinear damping in structures,

Journal of Wind Engineering and Industrial Aerodynamics, 59, 103-114. Jeary, A. P. (1997a) Designer’s Guide to the Dynamic Response of Structures, E & FN

Spon, Cambridge. Jeary, A.P. (1997b) Damping in structures, Journal of Wind Engineering and Industrial

Aerodynamics, 72, 345-355. Jerath, S. and Shibani, M.M. (1985) Dynamic stiffness and vibration of reinforced

concrete beams, ACI Journal, March-April, 196-202. Johns, K.C. and Belanger, M.D. (1981) Dynamic stiffness of concrete beams, ACI

Journal, May-June, 201-205.


References Re-5

Jones, R. (1957) The effect of frequency on the dynamic modulus and damping coefficient of concrete, Magazine of Concrete Research, 9(26), 69-72.

Jones, R. and Welch, G.B. (1967) The damping properties of plain concrete: Effect of

composition and relations with elasticity and strength, Road Research Laboratory Report No. LR111, 16pp.

Jordan, R.W. (1977) Tensile stress effects on damping, Concrete: The Journal of the

Concrete Society, 11(3), 31-33. Jordan, R.W. (1980) The effect of stress, frequency, curing, mix and age upon the

damping of concrete, Magazine of Concrete Research, 32(113), 195-205. Kana, D.D. (1981) Energy methods for damping synthesis in seismic design of complex

structural systems, Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, 107(EM1), 255-259.

Kareem, A. and Gurley, K. (1996) Damping in structures: its evaluation and treatment

of uncertainty, Journal of Wind Engineering and Industrial Aerodynamics, 59, 131-157.

Kelly, S.G. (1993) Fundamentals of Mechanical Vibrations, McGraw-Hill, New York. Kelvin, Lord. (1865) On elasticity and viscosity of metal, Proceedings Royal Society of

London, May 1865. Kesler, C.E. and Higuchi, Y. (1953) Determination of compressive strength of concrete

by using its sonic properties, Proceedings, American Society for Testing and Materials, Committee Reports and Technical Papers, 53, 1044-1051.

Kisa, M. and Brandon, J.A. (2000) Free vibration analysis of multiple open-edge

cracked beams by component mode synthesis, Structural Engineering and Mechanics, 10(1), 81-92.

Kummer, E., Yang, J.C.S. and Dagalakis, N. (1981) Detection of fatigue cracks in

structural members, Proceedings of the Second Specialty Conference on Dynamic Response of Structures: Experimental Observation, Prediction and Control, Janurary 15-16, Atlanta, Georgia, pp. 445-459.

Kunnath, S.K., Mander, J.B. and Fang, L. (1997) Parameter identification for degrading

and pinched hysteretic structural concrete systems, Engineering Structures, 19(3), 224-232.

Lagomarsino, S. (1993) Forecast models for damping and vibration periods of

buildings, Journal of Wind Engineering and Industrial Aerodynamics, 48, 221-239.

Lazan, B.J. (1968) Damping of Materials and Members in Structural Mechanics,

Pergamon Press, Oxford. Leonard, D.R. and Eyre, R. (1975) Damping and frequency measurements on eight box


References Re-6

girder bridges, TRRL Laboratory Report 682, Transport and Road Research Laboratory, Crowthorne, Berkshire.

Li, X. and Chung, D.D.L. (1998) Improving silica fume for concrete by surface

treatment, Cement and Concrete Research, 28(4), 493-498. Lorenz, H. (1924) Lehrbuch der Technischen Physik, Erster Band: Technische

Mechanik starrer Gebilde, Verlag von Julius Springer, Berlin, 1924. Lutes, L.D. and Sarkani, S. (1995) Structural damping for soil-structure interaction

studies, Structural Engineering and Mechanics, 3(2), 107-120. Maeck, J., Wahab, M.A., Peeters, B., De Roeck, G., De Visscher, J., De Wilde, W.P.,

Ndambi, J.-M. and Vantomme, J. (2000) Damage identification in reinforced concrete structures by dynamic stiffness determination, Engineering Structures, 22, 1339-1349.

Malushte, S.R. and Singh, M.P. (1987) Seismic response of simple structures with

Coulomb damping, In Dynamics of Structures, Proceedings of the Sessions at Structures Congress ’87, Orlando, Florida, Aug 17-20, (Ed. Roesset, J.M.), ASCE, New York, pp. 50-65.

McClellan, J.H., Schafer, R.W. and Yoder, M.A. (1999) DSP First: A Multimedia

Approach, Prentice Hall, New Jersey. Meirovitch, L. (1975) Elements of Vibration Analysis, McGraw-Hill, New York. Mo, Y.L. (1994) Dynamic Behaviour of Concrete Structures, Elsevier, Amsterdam. Ndambi, J.M., Peeters, B., Maeck, J., De Visscher, J., Wahab, M.A., Vantomme, J. De

Roeck, G. and De Wilde, W.P. (2000) Comparison of techniques for modal analysis of concrete structures, Engineering Structures, 22, 1159-1166.

Ndambi, J.M., Vantomme, J. and Harri, K. (2002) Damage assessment in reinforced

concrete beams using eigenfrequencies and mode shape derivatives, Engineering Structures, 24, 501-515.

Neild, S.A. (2001) Using Non-Linear Vibration Techniques to Detect Damage in

Concrete Bridges, D.Phil, Department of Engineering Science, University of Oxford.

Neild, S.A., Williams, M.S. and McFadden, P.D. (2002) Non-linear behaviour of

reinforced concrete beams under low- amplitude cyclic and vibration loads, Engineering Structures, 24, 707-718.

Newland, D.E. (1989) Mechanical Vibration Analysis and Computation, John Wiley &

Sons, New York. Ohtsu, M., Okamoto, T. and Yuyama, S. (1998) Moment tensor analysis of acoustic

emission for cracking mechanisms in concrete, ACI Structural Journal, 95(2), 87-95.


References Re-7

Ohtsu, M. and Watanabe, H. (2001) Quantitative damage estimation of concrete by acoustic emission, Construction and Building Materials, 15, 217-224.

Orak, S. (2000) Investigation of vibration damping on polymer concrete with polyester

resin, Cement and Concrete Research, 30(), 171-174. Pagnini, L.C. and Solari, G. (2001) Damping measurements of steel poles and tubular

towers, Engineering Structures, 23(9), 1085-1095. Pandey, A.K. and Biswas, M (1994) Damage detection in structures using changes in

flexibility, Journal of Sound and Vibration, 169(1), 3-17. Patrick, M. (1999) Australian 500 MPa reinforcing steels and new AS3600 ductility

design provisions, AS3600 Amendment 2 (1999): A Seminar presented by the Concrete Institute of Australia, 14th April, Sydney.

Paz, M. (1997) Structural Dynamics: Theory and Computations, Chapman & Hall, New

York, NY. PCB® Piezotronics (1992) Impact Hammer Instrumentation for Structural Behaviour

Testing – Operating Instructions, PCB® Piezotronics Inc., New York. Penelis, G.G and Kappos, A.J. (1997) Earthquake-Resistant Concrete Structures, E &

FN Spon, Great Britain. Penzien, J. (1964) Damping characteristics of prestressed concrete, Journal of the

American Concrete Institute, 61(9), 1125-1148. Penzien, J. and Hansen R.J. (1954) Static and dynamic elastic behaviour of reinforced

concrete beams, Journal of the ACI, Proceedings, 50(7), 545-567. Plunkett, R. (1960) Measurement of Damping, Structural Damping: Papers Presented

at a Colloquium on Structural Damping held at the ASME Annual Meeting in Atlantic City, New Jersey in December 1959, (Ed. J.E. Ruzicka), Pergamon Press, New York, 117-131.

Ragueneau, F., La Borderie, Ch. And Mazars, J. (2000) Damage model for concrete-like

materials coupling cracking and friction, contribution towards structural damping: first uniaxial applications, Mechanics of Cohesive-Frictional Materials, 5, 607-625.

Ravi, D. and Liew, K.M. (2000) A study of the effect of microcrack on the vibration

mode shape, Engineering Structures, 22, 1097-1102. Rayleigh, Lord, (1945) The Theory of Sound, Vol.1, reprinted by Dover, New York,

Originally published in 1877, 46-51. Razak, H.A. and Choi, F.C. (2001) The effect of corrosion on the natural frequency and

modal damping of reinforced concrete beams, Engineering Structures, 1126-1133.

Reynolds, P. and Pavic, A. (2000) Impulse hammer versus shaker excitation for the


References Re-8

modal testing of building floors, Experimental Techniques, 24(3), 39-44. Shield, C.K. (1997) Comparison of acoustic emission activity in reinforced and

prestressed concrete beams under bending, Construction and Building Materials, 11(3), 189-194.

Spooner, D.C. and Dougill, J.W. (1975) A quantitative assessment of damage sustained

in concrete during compressive loading, Magazine of Concrete Research, 27(92), 151-160.

Spooner, D.C., Pomeroy, C.D. and Dougill, J.W. (1976) Damage and energy dissipation

in cement pastes in compression, Magazine of Concrete Research, 28(94), 21-29.

Sri Ravindrarajah, R. and Tam C.T. (1985) Properties of concrete made with crushed

concrete as coarse aggregate, Magazine of Concrete Research, 37(130), 29-38. Stephens, J.E. and Yao, J.T.P. (1986) Damage assessment using response

measurements, Journal of Structural Engineering, 113(4), 787-801. Suda, K., Satake, N., Ono, J. and Saski, A. (1996) Damping properties of buildings in

Japan, Journal of Wind Engineering and Industrial Aerodynamics, 59, 383-392. Sun, C.T. and Lu, Y.P. (1995) Vibration Damping of Structural Elements, Prentice-

Hall, Englewood Cliffs, New Jersey. Swamy, R.N. (1970) Damping mechanisms in cementitious systems, Dynamics Waves

in Civil Engineering, Proceedings of a Conference Organized by the Society for Earthquake and Civil Engineering Dynamics, University College of Swansea on 7-9 July, (Ed’s D.A. Howells, I.P. Haigh & C. Taylor), Wiley-Interscience, London, 521-542.

Swamy, R.N. and Anand, K.L. (1974) Influence of steel stress and concrete strength on

the deflection characteristics of reinforced and prestressed beams, ACI Committee 435, Deflections of Concrete Structures, Publication SP-43, American Concrete Institute, Detroit, Michigan, 443-471.

Swamy, R.N. and Rigby, G. (1971) Dynamic properties of hardened paste, mortar and

concrete, Materials and Structures: Research and Testing, 4(19), 13-40. Tan, H.C., Famiyesin, O.O.R. and Imbabi, M.S.E. (2001) Dynamic deformation

signatures in reinforced concrete slabs for conditioning monitoring, Computers and Structures, 79, 2413-2423.

Tedesco, J.M., McDougal, W.G. and Ross, C.A. (1999) Structural Dynamics: Theory

and Applications, Addison Wesley Longman, Menlo Park, California. Tektronix (1995) TDS 410A, TDS 420A & TDS 460A Digitizing Oscilloscopes – User

Manual, Tektronix Inc., Wilsonville, Oregon, USA. Tilley, G.P (1986) Dynamic Behaviour of Concrete Structures, Elsevier, The

Netherlands.


References Re-9

Van Den Abeele, K. and De Visscher, J. (2000) Damage assessment in reinforced concrete using spectral and temporal nonlinear vibration techniques, Cement and Concrete Research, 30(9), 1453-1464.

Wahab, M.M.A. and De Roeck, G. (1999) Effect of excitation type on dynamic system

parameters of a reinforced concrete bridge, Structural Engineering and Mechanics, 7(4), 387-400.

Wang, Y. and Chung, D.D.L. (1998) Effects of sand and silica fume on the vibration

damping behaviour of cement, Cement and Concrete Research, 28(10), 1353-1356.

Wang, Z., Man, X-T.C., Finch, R.D. and Jansen, B.H. (1998) Dynamic behaviour and

vibration monitoring of reinforced concrete beams, Journal of Testing and Evaluation, 26(5), 405-419.

Wen, S. and Chung, D.D.L. (2000) Enhancing the vibration reduction ability of concrete

by using steel reinforcement and steel surface treatments, Cement and Concrete Research, 30, 327-330.

Wheeler, J.E. (1982) Prediction and control of pedestrian-induced vibration in

footbridges, Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, 108(ST9), 2045-2065.

Wills, L.G. (1974) Letter to the editor, Concrete, 8(3). Wilyman, K.N. and Ranzi, G. (2001) Dynamic behaviour of a piled foundation

supporting a coal crushing mill, Proceedings Australasian Structural Engineering Conference incorporating the Fifth International Kerensky Conference of the Institution of Structural Engineers, 29 April – 2 May 2001 Surfers Paradise Marriott Resort Gold Coast, pp. 81-88.

Woodhouse, J. (1998) Linear damping models for structural vibration, Journal of Sound

and Vibration, 215(3), 547-569. Xiao, Y. and Ma, R. (1998) Seismic behaviour of high strength concrete beams,

Structural Design of Tall Buildings, 7(1), 73-90. Xu, X. and Setzer, M.J. (1997) Damping maximum of hardened cement paste (hcp) in

the region of –90oC: A mechanical relaxation process, Advanced Cement Based Materials, 5, 69-74.

Yan, L., Jenkins, C.H. and Pendleton, R.L. (2000a) Polyolefin fiber-reinforced concrete

composites, Part 1. Damping and frequency measurements, Cement and Concrete Research, 30(3), 391-401.

Yan, L., Jenkins, C.H. and Pendleton, R.L. (2000b) Polyolefin fiber-reinforced concrete

composites, Part 2. Damping and interface debonding, Cement and Concrete Research, 30(3), 403-410.

Zhang, B. (1998) Relationship between pore structure and mechanical properties of

ordinary concrete under bending fatigue, Cement and Concrete Research, 28(5), 699-711.


References Re-10

Zhang, B. and Wu, K. (1997) Residual fatigure strength and stiffness of ordinary concrete under bending, Cement and Concrete Research, 27(1), 115-126.

Zhao, J. and DeWolf, J.T. (1999) Sensitivity study for vibrational parameters used in

damage detection, Journal of Structural Engineering, 125(4), 410-416.


Bibliography Bi-1

BIBLIOGRAPHY

Aalami, B. and Javaherian, H. (1973) Free vibration of rectangular bars, Fourth

Australasian Conference on the Mechanics of Structures and Materials, Proceedings, Department of Civil Engineering, University of Queensland, Brisbane, Australia, August 20-22, pp. 1-8.

Abeles, P.W. and Gill, V.L. (1969) High-strength strand reinforcement for concrete,

Concrete, 3(4), April, 127-130. ACI Committee 318 (1995) Building Code Requirements for Structural Concrete (ACI

318-95) and Commentary (ACI 318R-95), American Concrete Institute, Farmington Hills, MI.

ACI Committee 363 (1984) State-of-the-Art report on high-strength concrete, ACI

Journal, July-August, 364-411. ACI Committee 363 (1987) Research needs for high-strength concrete, ACI Materials

Journal, Nov-Dec, 559-561. ACI Committee 435 (1974) Deflections of Concrete Structures, Publication SP-43,

American Concrete Institute, Detroit, Michigan. Adams, J., Walsh, P., Marsden, W. and Patrick, M. (1999) Factors affecting the ductility

of stiffened rafts, Concrete ’99: Our Concrete Environment, Proceedings of the CIA 19th Biennial Conference, (D. Baweja, S. Bernard, R.Wheen and P. Shea eds.) Sydney Hilton Hotel, May 5-7, pp. 255-263.

Ahmad, S.H. and Barker, R. (1991) Flexural behaviour of reinforced high-strength

lightweight concrete beams, ACI Structural Journal, 88(1), 69-77. Ahmad, S.H. and Shah, S.P. (1985) Structural properties of high strength concrete and

its implications for precast prestressed concrete, PCI Journal, 30(6), 92-119. Aïtcin, P.-C. (1999) The art and science of high-performance concrete, Concrete ’99:

Our Concrete Environment, Proceedings of the CIA 19th Biennial Conference, (D. Baweja, S. Bernard, R.Wheen and P. Shea eds.) May 5-7, Sydney Hilton Hotel, Sydney, Australia, pp. 478-487.

Aïtcin, P.-C. (2000) Cements of yesterday and today: Concrete of tomorrow, Cement


Bibliography Bi-2

and Concrete Research, 30(9), 1349-1359. Albertini, C., Cadoni, E. and Labibes, K. (1999) Study of the mechanical properties of

plain concrete under dynamic loading, Experimental Mechanics, 39(2), 137-141. Alca, N., Alexander, S.D.B. and MacGregor, J.G. (1997) Effect of size on flexural

behaviour of high-strength concrete beams, ACI Structural Journal, 94(1), 59-67.

Allen, A.H. (1988) Reinforced Concrete Design to BS 8110: Simply Explained, E & FN

Spon, London, UK. Argyris, J. and Mlejnek, H.-P. (1991) Dynamics of Structures: Texts on Computational

Mechanics Volume V, North-Holland, Amsterdam, The Netherlands. AS 3600 – 1994, Australian Standard for Concrete Structures, Standards Australia,

Homebush, NSW. AS 3600 Supp1 – 1994, Australian Standard for Concrete Structures – Commentary,

Standards Australia, Homebush, NSW. Ashour, S.A. (2000) Effects of compressive strength and tensile reinforcement ratio on

flexural behaviour of high-strength concrete beams, Engineering Structures, 22, 413-423.

Ashour, S.A., Mahmood, K. and Wafa, F.F. (1997) Influence of steel fibers and

compression reinforcement on deflection of high-strength concrete beams, ACI Structural Journal, 94(6), 611-624.

Ashour, S.A., Wafa, F.F. and Kamal, M.I. (2000) Effect of the concrete compressive

strength and tensile reinforcement ratio on the flexural behaviour of fibrous concrete beams, Engineering Structures, 22, 1145-1158.

Attard, M.M. and Stewart, M.G. (1998) A two parameter stress block for high-strength

concrete, ACI Structural Journal, 95(3), 305-317. Bachmann, H. (1986) Schwingungsverhalten teilweise vorgespannter Konstruktionen

aus Leichtbeton und Normalbeton, Beton- und Stahlbetonbau, 7. Bachmann, H. and Ammann, W. (1987) Vibrations in Structures: Induced by Man and

Machines, Structural Engineering Documents 3e, International Association for Bridge and Structural Engineering, Switzerland.

Balakrishnan, A.V. (1999) Damping performance of strain actuated beams,

Computational Applied Mathematics, 18(1), 31-86. Balan, T.A., Filippou, F.C. and Popov, E.P. (1998) Hysteretic model of ordinary and

high-strength reinforcing steel, Journal of Structural Engineering, 124(3), 288-297.

Barrett, P.R., Foadian, H. and Rashid, Y.R. (1991) Effects of cracking on damping of

concrete dams, Lifeline Earthquake Engineering: Third US Conference in


Bibliography Bi-3

Lifeline Earthquake Engineering, Los Angeles, August 22-23, (M.A.Cassaro Ed.), pp. 985-994.

Base, G.D., Read, J.B., Beeby, A.W. and Taylor, H.P.J. (1966) An investigation of the

crack control characteristics of various types of bar in reinforced concrete beams, Research Report 18: Part 1, Cement and Concrete Association, pp. 44.

Bažant, Z. and Oh, B.H. (1984) Deformation of progressively cracking reinforced

concrete beams, ACI Journal, May-June, 268-278. Beeby, A.W. (1971) The prediction and control of flexural cracking in reinforced

concrete members, In International Symposium on Cracking, Deflection, and Ultimate Load of Concrete Slab Systems, Publication SP-30, ACI, Detroit, Mich., pp. 55-75.

Beeby, A. W. (1974) A note on an aspect of the variability of deflections, Magazine of

Concrete Research, 26(88), 161-168. Beeby, A.W. (1997) Ductility in reinforced concrete: Why is it needed and how is it

achieved?, The Structural Engineer, 75(18), 311-318. Beeby, A.W. (1998) Discussion: Ductility in reinforced concrete: Why is it needed and

how is it achieved?, The Structural Engineer, 76(9), 180-183. Beeby, A.W. and Narayanan, R.S. (1995) Designers’ Handbook to Eurocode 2 – Part

1.1: Design of Concrete Structures, Thomas Telford, London. Bennett, E.W. and Chandrasekhar, C.S. (1972) Calculation of the width of cracks in

Class 3 prestressed beams, Proceedings of the Institution of Engineers, 49, 333-346.

Bert, C.W. (1973) Material damping: An introductory review of mathematical models,

measures and experimental techniques, Journal of Sound and Vibration, 29(2), 129-153.

Bishop, R.E.D. (1955) The treatment of damping forces in vibration theory, Journal of

the Royal Aeronautical Society, 59, November, 738-742. Bishop, R.E.D. and Johnson, D.C. (1960) The Mechanics of Vibration, Cambridge

University Press, London. Bjerkeli, L., Tomaszewicz, A. and Jensen, J.J. (1990) Deformation properties and

ductility of high-strength concrete, High-Strength Concrete SP-121, American Concrete Institute, Detroit, Mich.,pp. 215-238.

Blanchard, J., Davies, B.L. and Smith, J.W. (1977) Design criteria and analysis for

dynamic loading of footbridges, Proceedings Symposium of Dynamic Behaviour of Bridges, Supplementary Report 275, Transport and Road Resesrch Laboratory, Berkshire, England, May, pp. 90-106.

Bode, L., Tailhan, J.L., Pijaudier-Cabot, G., La Borderie, C. and Clement, J.L. (1997)

Failure analysis of initially cracked concrete structures, Journal of Engineering Mechanics, 123(11), 1153-1160.


Bibliography Bi-4

Bolton, A. (1978) Natural frequencies of structures for designers, The Structural Engineer, 9(56A), 245-253.

Bolton, A. (1994) Structural Dynamics in Practice: A Guide for Professional Engineers,

McGraw-Hill, Cambridge. Brebbia, C.A., Tottenham, H., Warburton, G.B., Wilson, J. and Wilson, R. (1976)

Vibrations of Engineering Structures, Computational Mechanics Ltd., Southampton, UK.

Bridge, R., Patrick, M. and Wheeler, A. (2000) Reinforced Concrete Building Series

Design Booklet RCB-1.1(1): Crack Control of Beams Part 1: AS 3600 Design, Construction Technology Research Group, BHP Reinforcing Products and The University of Western Sydney-Nepean.

Broms, B.B. and Lutz, L.A. (1965) Effects of arrangement of reinforcement on crack

width and spacing of reinforced concrete members, Journal of the ACI, Proceedings, 62(11), 1395-1410.

CP 110 (1972) The structural use of concrete: Part 1: Design, materials and

workmanship, British Standards Institution, London, pp. 156. BS 1881 (1970) Methods of testing of concrete: Parts 1 to 5, British Standards

Institution, London. Butt, A.S. and Akl, F.A. (1997) Experimental analysis of impact-damped flexible

beams, Journal of Engineering Mechanics, 123(4), 377-383. Carr, A.J. and Tabuchi, M. (1993) The structural ductility and the damage index for

reinforced concrete structure under seismic excitations, Proceedings of the Second European Conference on Structural Dynamics: EURODYN ’93, June 21-23, Trondheim, Norway, pp.169-176.

Carrasquillo, R.L., Nilson, A.H. and Slate, F.O. (1981) Properties of high strength

concrete subject to short-term loads, ACI Journal, May-June, 171-178. Caughey, T.K. and O’Kelly, M.E.J. (1961) Effect of damping on the natural frequencies

of linear dynamic systems, Journal of the Acoustical Society of America, 33(11), 1458-1461.

Cement and Concrete Association of Australia (1992) High Strength Concrete, Cement

and Concrete Association of Australia and National Ready Mixed Concrete Association of Australasia, Sydney.

Chi, M. and Kirstein, A.F. (1958) Flexural cracks in reinforced concrete, Journal of the

ACI, Proceedings, 54, April, 865-878. Chick, C., Patrick, M. and Wong, K. (1999) Ductility of reinforced-concrete beams and

slabs, and AS 3600 design requirements, Concrete ’99: Our Concrete Environment, Proceedings of the CIA 19th Biennial Conference, (D. Baweja, S. Bernard, R.Wheen and P. Shea eds.) Sydney Hilton Hotel, May 5-7, pp. 570-578.


Bibliography Bi-5

Christiansen, M.B. and Nielsen, M.P. (1997) Modelling Tension Stiffening in Reinforced Concrete Structures: Rods, Beams and Disks, Department of Structural Engineering and Materials, Technical University of Denmark, Series R, No. 22.

Chu, F.H. and Wang, B.P. (1980) Experimental determination of damping in materials

and structures, Damping Applications for Vibration Control, The Winter Annual Meeting of the American Society of Mechanical Engineers, Chicago, Illinois, Nov 16-21, 1980, (Ed. P.J. Torvik), ASME, New York, pp. 113-122.

Chu, K.H., Garg, V.K. and Wang, T.L. (1986) Impact in railway prestressed concrete

bridges, Journal of Structural Engineering, 112(5), 1036-1051. Collins, M.P., Mitchell, D. and MacGregor, J.G. (1993) Structural design considerations

for high-strength concrete, ACI Concrete International, 15(5), 27-34. Clark, A.P. (1956) Cracking in reinforced concrete flexural members, Journal of the

ACI, Proceedings, 52(8), 851-862. Comite Euro-International Du Beton (CEB) (1993) CEB-FIP Model Code 1990,

Thomas Telford, London. Comite Euro-International Du Beton (1996) RC Elements Under Cyclic Loading – State

of the Art Report, Thomas Telford, London. Comite Euro-International Du Beton (1996) RC Frames Under Earthquake Loading –

State of the Art Report, Thomas Telford, London. Crandall, S.H. (1970) The role of damping in vibration theory, Journal of Sound and

Vibration, 11(1), 3-18. Crawley, D.B. and McPherson, I. (1986) Minimum cost design for moderate earthquake

risk in Australia, applied to Aser project structures, Proceedings of the Tenth Australasian Conference on the Mechanics of Structures and Materials, August 20-22, University of Adelaide, Adelaide, Australia, pp. 205-210.

Desayi, P. and Rao, K.B. (1989) Probabilistic analysis of cracking moment of reinforced

concrete beams, ACI Structural Journal, 86(3), 235-241. Dowell, E. H. (1986) Damping in beams and plates due to slipping at the support

boundaries, Journal of Sound and Vibration, 105(2), 243-253. Elices, M. and Planas, J. (1996) Fracture mechanics parameters of concrete, Advanced

Cement Based Materials, 4, 116-127. Esfahani, M.R. and Rangan, B.V. (1999) Evaluation of proposed revisions to AS3600

bond strength provisions, Australian Journal of Structural Engineering, SE2(1), 31-35.

European Committee for Standardization (CEN) (1991) Eurocode 2: Design of Concrete

Structures – Part 1: General rules and rules for buildings, ENV 1992-1-1:1991, CEN, Brussels.


Bibliography Bi-6

European Committee for Standardization (CEN) (1994) Eurocode 8: Design provisions for earthquake resistance of structures – Part 1-1: General rules – Seismic actions and general requirements for structures, CEN, Brussels., CEN, Brussels.

Falati, S. (1999) The Contribution of Non-Structural Components to the Overall

Dynamic Behaviour of Concrete Floor Slabs, D.Phil, University of Oxford. Farag, H.M. and Leach, P. (1997) The elasto-plastic design of reinforced concrete beams

and slabs subject to dynamic loading, Proceedings of the Institution of Civil Engineers: Structures and Buildings, 122, 117-123.

Farrar, C.R., Baker, W.E. and Dove, R.C. (1994) Dynamic parameter similitude for

concrete models, ACI Structural Journal, 91(1), 90-99. Feeny, B.F. and Liang, J.W. (1996) A decrement method for the simultaneous

estimation of Coulomb and viscous friction, Journal of Sound and Vibration, 195(1), 149-154.

Ferry Borges, J. (1966) Cracking and deformability of reinforced concrete beams,

Publications, International Association for Bridge and Structural Engineering, Zurich, 26, 75-95.

FIP-CEB Working Group on High Strength Concrete (1990) High Strength Concrete:

State of the Art Report, Federation Internationale De La Precontrainte (FIP) and Comite Euro-International Du Beton (CEB), London.

Foster, S.J. and Gilbert, R.I. (1996) The design of nonflexural members with normal and

high-strength concretes, ACI Structural Journal, 93(1), 3-10. Foster, S.J. and Gilbert, R.I. (1998) Experimental studies on high-strength concrete deep

beams, ACI Structural Journal, 95(4), 382-390. Ghali, A. and Azarnejad, A. (1999) Deflection prediction of members of any concrete

strength, ACI Structural Journal, 96(5), 807-816. Gilbert, R.I. (1983) Deflection calculations for reinforced concrete beams, Civil

Engineering Transactions, The Institution of Engineers, Australia, 128-134. Gilbert, R.I. (1997a) Anchorage of reinforcement in high strength concrete, Proceedings

of the USA-Australia Workshop on High Performance Concrete (HPC), Sydney, Australia, August 20-23, (Eds. D.V. Reddy and B.V. Rangan), pp. 425-443.

Gilbert, R.I. (1997b) Comments on the serviceability requirements of AS3600-1994,

Seminar Proceedings The Use of High Strength Concrete and High Strength Reinforcement in Concrete Structures – Design Implications, The Munro Centre for Civil and Environmental Engineering, UNSW, 20 November, 1997.

Gilbert, R.I. (1997c) High strength reinforcement in concrete structures: Serviceability

implications, Proceedings of the 15th Australasian Conference on the Mechanics of Structures and Materials, Melbourne, Australia, 8-10 December, pp. 179-185.

Gilbert, R.I. (1998) Deflection calculation and control for reinforced concrete structures,


Bibliography Bi-7

Proceedings of the Australasian Structural Engineering Conference (1998) Auckland, 30 September – 2 October, (Ed. J.W. Butterworth), pp. 269-274.

Gilbert, R.I. (1999a) Flexural crack control for reinforced concrete beams and slabs: An

evaluation of design procedures, Proceedings of the 16th Australasian Conference on the Mechanics of Structures and Materials, Sydney, Australia, 8-10 December, (Eds. M.A. Bradford, R.Q. Bridge and S.J. Foster), pp. 175-180.

Gilbert, R.I. (1999b) Deflection calculation for reinforced concrete structures – Why we

sometimes get it wrong, ACI Structural Journal, 96(6), 1027-1032. Gilbert, R.I., Patrick, M. and Adams, J.C. (1999) Evaluation of crack control design

rules for reinforced concrete beams and slabs, Concrete ’99: Our Concrete Environment, Proceedings of the CIA 19th Biennial Conference, (Eds. D. Baweja, S. Bernard, R.Wheen and P. Shea) Sydney Hilton Hotel, May 5-7, pp. 21-29.

Gravina, R.J. and Warner, R.F. (1999) Modelling of high-moment plastification regions

in concrete structures, Proceedings of the 16th Australasian Conference on the Mechanics of Structures and Materials, Sydney, Australia, 8-10 December, (Eds. M.A. Bradford, R.Q. Bridge and S.J. Foster), pp. 103-108.

Gudmundson, P. (1983) The dynamic behaviour of slender structures with cross-

sectional cracks, Journal Mechanical and Physical Solids, 31(4), 329-345. Cement and Concrete Association of Australia (1993) High-Strength High-Performance

Concrete Seminar, Cement and Concrete Association of Australia and National Readymixed Concrete Association, CSIRO-National Building Technology Centre, North Ryde, Thursday 11 February 1993.

Hagel, W.C. and Clark, J.W. (1957) The specific damping energy of fixed-fixed beam

specimens, Journal of Applied Mechanics, 24(3), 426-430. Hao, H., Ang, T.T. and Shen, J. (2001) Building vibration to traffic-induced ground

motion, Building and Environment, 36, 321-336. Harris, A.J. (1948) Prestressed concrete in highway bridges, Journal of the Institute of

Highway Engineers, 6, 8-44. Hill, T. and Palmer, M. (2001) Predicting the response of slender steel staircases, The

Structural Engineer, 79(7), 16-18. Hoff, G.C. (1997) The Hibernia offshore concrete platform, Concrete ’97: For the

Future, Proceedings of the CIA 18th Biennial Conference, Adelaide Convention Centre, May 14-16, pp. 119-128.

Hognestad, E. (1962) High strength bars as concrete reinforcement, Part 2, Control of

flexural cracking, Journal of the Portland Cement Association, Research and Development Laboratories, 4(1), 46-63.

Huang, C.S. and Lin, H.L. (2001) Modal identification of structures from ambient

vibration, free vibration, and seismic response data via a subspace approach, Earthquake Engineering and Structural Dynamics, 20(12), 1857-1878.


Bibliography Bi-8

Huang, C.S., Yang, Y.B., Lu, L.Y. and Chen, C.H. (1999) Dynamic testing and system identification of a multi-span highway bridge, Earthquake Engineering and Structural Dynamics, 28, 857-878.

Hughes, G. and Beeby, A.W. (1982) Investigation of the the effect of impact loading on

concrete beams, The Structural Engineer, 60B(3), 45-52. Hussain, A.T.M.K., Lowrey, M.J. and Loo, Y.C. (1990) Damping properties of

concrete- An overview, Proceedings of the Twelfth Australasian Conference on the Mechanics of Structures and Materials, 24-26 September, Queensland University of Technology, Queensland, Australia, pp. 151-156.

Ibrahim, H.H.H. and MacGregor, J.G. (1997) Modification of the ACI rectangular stress

block for high-strength concrete, ACI Structural Journal, 94(1), 40-48. Irwin, A.W. (1978) Human response to dynamic motion of structures, The Structural

Engineer, 9(56A), 237-244. Jacobsen, L.S. and Ayre, R.S. (1958) Engineering Vibration, McGraw-Hill, New York. Janjua, M.A. and Welch, G.B. (1972) Magnitude and distribution of cracks in

reinforced concrete flexural members, UNICIV Report No. R.78, July, University of New South Wales, Kensington.

Jensen, J.J. (1994) Structural aspects of high strength concretes, in Concrete

Technology: New Trends, Industrial Applications, Proceedings of the International RILEM Workshop on Technology Transfer of the New Trends in Concrete ConTech ’94, 7-9 November, Barcelona, (Eds. A. Agvado, G. Ravindra and S.P. Shah), E & FN Spon, pp. 197-212.

Jones, D.I.G. (1979) A simple low-cost technique for measuring material damping

behaviour, The Shock and Vibration Bulletin, 42, Part 2, September, 97-103. Kaar, P.H. and Mattock, A.H. (1963) High strength bars as concrete reinforcement,

Journal of the Portland Cement Association Research and Development Laboratories, 5(1), 15-38.

Kankam, C.K. (1997) Relationship of bond stress, steel stress, and slip in reinforced

concrete, Journal of Structural Engineering, 123(1), 79-85. Kareem, A. and Sun, W.-J. (1990) Dynamic response of structures with uncertain

damping, Engineering Structures, 12, 2-8. Kimura, H., Sugano, S., Nagashima, T. and Ichikawa, A. (1993) Seismic loading tests of

reinforced concrete beams using high strength concrete and high strength steel bars, Utilization of High Strength Concrete, Proceedings of a Symposium, Lillehammer, Norway, June 20-23, (Eds. Holand, I. And Sellevold, E.) pp. 377-384.

Kiureghian, A.D. (1980) Structural response to stationary excitation, Journal of the

Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, 106(EM6), 1195-1213.


Bibliography Bi-9

Kmita, A. (2000) A new generation of concrete in civil engineering, Journal of Materials Processing Technology, 106, 80-86.

Kong, P.Y.L. and Rangan, B.V. (1997) Reinforced high strength concrete (HSC) beams

in shear, Australian Civil/Structural Engineering Transactions, CE39(1), 43-50. Kowalsky, M.J., Priestly, M.J.N. and Seible, F. (1999) Shear and flexural behaviour of

lightweight concrete bridge columns in seismic regions, ACI Structural Journal, 96(1), 136-148.

Kowalsky, M.J., Priestly, M.J.N. and Seible, F. (2000) Dynamic behaviour of

lightweight concrete bridges, ACI Structural Journal, 97(4), 602-618. Lazan, B.J. (1960) Energy dissipation mechanisms in structures, with particular

reference to material damping, Structural Damping: Papers Presented at a Colloquium on Structural Damping held at the ASME Annual Meeting in Atlantic City, New Jersey in December 1959, (Ed. J.E. Ruzicka), Pergamon Press, New York, 1-34.

Leslie, K.E., Rajagopalan, K.S. and Everard, N.J. (1976) Flexural behaviour of high-

strength concrete beams, ACI Journal, September, 517-521. Li, Q.S., Liu, D.K., Fang, J.Q., Jeary, A.P. and Wong, C.K. (2000) Damping in

buildings: its neural network model and AR model, Engineering Structures, 22, 1216-1223.

Liang, J.W. and Feeny, B.F. (1998) Identifying coulomb and viscous friction from free-

vibration decrements, Nonlinear Dynamics, 16, 337-347. Lin, C.M. and Restrepo, J.I. (1998) Experimental verification of the concrete structures

standard recommendations for the design of beam-column joints, Proceedings of the Australasian Structural Engineering Conference, Auckland, 30 September – 2 October, (Ed. J.W. Butterworth), pp. 309-316.

Loo, Y.-C. and Santos, A.P. (1986) Impact deflection analysis of concrete beams,

Journal of Structural Engineering, 112(6), 1297-1312. Loo, Y.-C. (1990) Reinforced Concrete Analysis and Design: With Emphasis on the

Application of AS3600-1988, University of Wollongong Press, North Wollongong.

Lorrain, M., Maurel, O., Boukari, S. and Pinto-Barbosa, M. (1999) Numerical and

experimental analysis of high strength concrete beams, reinforced with high yield steel bars submitted to flexure, Materials and Structures, 32(224), 708-718.

Macchi, G., Pinto, P.E. and Sanpaolesi, L. (1996) Ductility requirements for

reinforcement under Eurocodes, Structural Engineering International, 4(96), 249-254.

Magrab, E.B. (1979) Vibrations of Elastic Structural Members, Sijthoff & Noordhoff,

The Netherlands. Mansur, M.A., Chin, M.S. and Wee, T.H. (1997) Flexural behaviour of high-strength


Bibliography Bi-10

concrete beams, ACI Structural Journal, 94(6), 663-674. Malvar, L.J. (1998) Review of static and dynamic properties of steel reinforcing bars,

ACI Materials Journal, 95(5), 609-616. Mathey, R.G. and Watstein, D. (1960) Effect of tensile properties of reinforcement on

the flexural characteristics of beams, Journal of the American Concrete Institute, 56(12), 1253-1273.

Mendis, P.A. and Pendyala, R.S. (1997) Structural design with high-strength/high-

performance concrete – Beams and columns, Concrete in Australia, 23(2), 26-28.

Melby, K., Jordet, E.A. and Hansvold, C. (1996) Long-span bridges in Norway

constructed in high-strength LWA concrete, Engineering Structures, 18(11), 845-849.

Mickleborough, N.C. (1984) Vibrational response of reinforced concrete structural

components, The Ninth Australasian Conference on the Mechanics of Structures and Materials, The University of Sydney, Australia, 29-31 August, pp. 273-278.

Mickleborough, N.C. and Gilbert, R.I. (1986) Control of concrete floor slab vibration by

L/D limits, Proceedings of the Tenth Australasian Conference on the Mechanics of Structures and Materials, August 20-22, University of Adelaide, Adelaide, Australia, pp. 51-56.

Milne, R.J.W. (1997) Structural Engineering: History and Development, E & FN Spon,

Great Britain. Mo, Y.L. and Yang, R.Y. (1996) Response of reinforced/prestressed concrete box

structures to dynamically applied torsion, Nuclear Engineering and Design, 165, 25-41.

Nashif, A.D., Jones, D.I.G. and Henderson, J.P. (1985) Vibration Damping, John Wiley

& Sons, New York. Nawy, E.G. (1968) Crack control in reinforced concrete structures, ACI Journal,

Proceedings, 65(10), October, 825-836. Nawy, E.G. and Potyondy, J.G. (1971) Flexural cracking behaviour of pretensioned I-

and T-Beams, ACI Journal, May, 355-360. Nichols, J.M. and Totoev, Y.Z. (1997) Experimental determination of the dynamic

modulus of elasticity of masonry units, Proceedings of the 15th Australasian Conference on the Mechanics of Structures and Materials, Melbourne, Australia, 8-10 December, pp. 639-644.

Oh, B.H. and Jung, B.S. (1998) Structural damage assessment with combined data of

static and modal tests, Journal Structural Engineering, 124(8), 956-965. OneSteel Reinforcing (2000) Designing Concrete Structures with 500PLUS Rebar,

National Seminar Series: Crack Control of Beams and Cross-Section Strength of Columns, August 2000, Centre for Construction Technology Research,


Bibliography Bi-11

University of Western Sydney. Owen, J.S., Eccles, B.J., Choo, B.S. and Woodings, M.A. (2001) The application of

auto-regressive time series modelling for the time-frequency analysis of civil engineering structures, Engineering Structures, 23, 521-536.

Park, R. (1995) Opportunities in New Zealand for high strength reinforcing steel,

Concrete ’95 – The Concrete Future, Proceedings of the New Zealand Concrete Society, Technical Conference and AGM TR17, The Sheraton Hotel, Auckland, 30 August – 1 September, pp. 124-133.

Park, R. (1998) Some current and future aspects of design and construction of structural

concrete for earthquake resistance, Proceedings of the Australasian Structural Engineering Conference, Auckland, 30 September – 2 October, (Ed. J.W. Butterworth), pp. 1-16.

Park, R. and Paulay, T. (1975) Reinforced Concrete Structures, John Wiley & Sons,

New York. Patrick, M., Akbarshahi, E. and Warner, R.F. (1997) Ductility limits for the design of

concrete structures containing high-strength, low-elongation steel reinforcement, Concrete ’97: For the Future, 18th Biennial Conference, Adelaide Convention Centre, May 14-16, pp 509-517.

Pavic, A. and Reynolds, P. (1999) Experimental assessment of vibration serviceability

of existing office floors under human-induced excitation, Experimental Techniques, 23(5), 41-45.

Pavic, A., Reynolds, P., Waldron, P. and Bennett, K.J. (2001) Critical review of

guidelines for checking serviceability of post-tensioned concrete floors, Cement and Concrete Composites, 23, 21-31.

Pendyala, R. S. (1994) Flexural design using high performance concrete, High

Performance Concrete: Technology, Design and Applications, Proceedings of a Seminar held by the Department of Civil and Environmental Engineering, 10th February, Melbourne, Australia (P. Mendis Ed.), Draft Report Ch 2, pp. 55-86.

Pendyala, R.S., Mendis, P.A. and Bajaj, A.S. (1997) Design of high-strength concrete

members, Proceedings of the 15th Australasian Conference on the Mechanics of Structures and Materials, Melbourne, Australia, 8-10 December, pp. 211-216.

Pendyala, R.S., Mendis, P. and Baweja, D. (1997) Towards the development of new

codes and standards to increase the field application of high performance concretes, Concrete ’97: For the Future, Proceedings of the CIA 18th Biennial Conference, Adelaide Convention Centre, May 14-16, pp. 175-185.

Pendyala, R.S., Mendis, P. and Patnaikuni, I. (1996) Full-range behaviour of high-

strength concrete flexural members: Comparison of ductility parameters of high and normal-strength concrete members, ACI Structural Journal, 93(1), 30-35.

Person, N.L. and Lazan, B.J. (1956) The effect of static mean stress on the damping

properties of materials, Proceedings of the American Society for Testing and


Bibliography Bi-12

Materials, 56, 1399-1414. Qu, Z.-Q. and Chang, W. (2000) Dynamic condensation method for viscously damped

vibration systems in engineering, Engineering Structures, 23, 1426-1432. Raggett, J.D. (1975) Estimating damping of real structures, Journal of the Structural

Division, Proceedings of the American Society of Civil Engineers, 101(ST9), 1823-1835.

Rangan, B.V. (1996) Applications of high-strength concrete (HSC), in Large Concrete

Buildings (Eds. B.V. Rangan and R.F. Warner), Longman, England, 158-182. Rangan, B.V. (1997) Design of high performance high strength concrete, Proceedings of

the USA-Australia Workshop on High Performance Concrete (HPC), Sydney, Australia, August 20-23, (Eds. D.V. Reddy and B.V. Rangan), pp. 445-470.

Rangan, B.V. (1998) Suggestions for design of high performance high strength concrete

(HPHSC) structural members, Australian Journal of Structural Engineering, SE1(2), 103-111.

Rasmussen, L.J. and Baker, G. (1995a) Torsion in reinforced normal and high-strength

concrete beams – Part 1: Experimental test series, ACI Structural Journal, 92(1), 56-62.

Rasmussen, L.J. and Baker, G. (1995b) Torsion in reinforced normal and high-strength

concrete beams – Part 2: Theory and design, ACI Structural Journal, 92(2), 149-156.

Ruzzene, M., Fasana, A., Garibaldi, L. and Piombo, B. (1997) Natural frequencies and

dampings identification using wavelet transform: Application to real data, Mechanical Systems and Signal Processing, 11(2), 207-218.

Saliger, I.R. (1936) High-grade steel in reinforced concrete, Proceedings 2nd Congress,

International Association for Bridge and Structural Engineers, Berlin, 293-315. Sanders, P.T. (1999) Advances in fire design for reinforced concrete structures –

Moving to more rational design methods, Concrete ’99: Our Concrete Environment, Proceedings of the CIA 19th Biennial Conference, (D. Baweja, S. Bernard, R.Wheen and P. Shea eds.) Sydney Hilton Hotel, May 5-7, pp. 598-606.

Sanjayan, J.G. and Jeevanayagam, P.W. (1998) Long term deflection of high strength

concrete beams, Australian Journal of Structural Engineering, SE1(2), 93-101. Sarkar, S.L., Adwan, O., and Munday, J.G.L. (1997) High strength concrete: An

investigation of the flexural behaviour of high strength RC beams, The Structural Engineer, 75(7), 115-121.

Scanlan, R.H. and Mendelson, A. (1963) Structural damping, AIAA Journal, 1(4), 938-

939. Scholz, H. (1993) Alternate crack width computation for prestressed members,

Proceedings Institution Civil Engineers, Structures and Buildings, 99, May, 135-140.


Bibliography Bi-13

Senjayan, J.G. and Jeevanayagam, P.W. (1998) Long term deflection of high strength concrete beams, Australian Journal of Structural Engineering, SE1(2), 93-101.

Setunge, S (1994) Engineering properties of high performance concrete, High

Performance Concrete: Technology, Design and Applications, Proceedings of a Seminar held by the Department of Civil and Environmental Engineering, 10th February, Melbourne, Australia (P. Mendis Ed.), Draft Report Ch 1, pp. 1-54.

Severn, R.T. (1999) European experimental research in earthquake engineering for

Eurocode 8, Proceedings of the Institution of Civil Engineers: Structures and Buildings, 134, 205-217.

Severn, R.T., Brownjohn, J.M.W., Dumanoglu, A.A. and Taylor, C.A. (1989) A review

of dynamic testing methods for civil engineering structures, Civil Engineering Dynamics, Proceedings of the Conference Organised by the University of Bristol, in Association with the Society for Earthquake and Civil Engineering Dynamics held at the University of Bristol, 24-25 March 1988, pp. 1-23.

Shah, S.P. (1997) High-performance concrete: Controlled performance concrete,

Magazine of Concrete Research, 49(178), 1-3. Shah, S. And Ouyang, C. (1996) Tensile response of reinforced high strength concrete

members, Worldwide Advances in Structural Concrete and Masonry, Proceedings of the CCMS Symposium held in Conjunction with Structures Congress XIV, 15-18 April, Chicago, Illinois, (Eds. A.E. Schultz and S.L. McCabe), ASCE, New York, New York, pp. 431-442.

Shears, M. (1989) The role of dynamic analysis in engineering design, Civil Engineering

Dynamics, Proceedings of the Conference Organised by the University of Bristol, in Association with the Society for Earthquake and Civil Engineering Dynamics held at the University of Bristol, 24-25 March 1988, pp. 245-269.

Shin, S.-W., Ghosh, S.K. and Moreno, J. (1989) Flexural ductility of ultra-high-strength

concrete members, ACI Structural Journal, 86(4), 394-400. Shkolnik, I.E. (1996) Evaluation of dynamic strength of concrete from results of static

tests, Journal of Engineering Mechanics, 122(12), 1133-1138. Smith, J.W. (1988) Vibration of Structures: Applications in Civil Engineering Design,

Chapman & Hall, Bristol. Soh, C.K., Chiew, S.P. and Dong, Y.X. (1999) Damage model for concrete-steel

interface, Journal of Engineering Mechanics, 125(8), 979-983. Sparrow, C.J. (1989) High strength concrete in the Melbourne Central project, In

Concrete’89, XIV Biennial Conference, 10-13 May, Adelaide Convention Centre, Adelaide.

Srinivasulu, P., Lakshmanan, N., Muthumani, K. and Sivarama Sarma, B. (1987)

Dynamic behaviour of fibre reinforced concrete beams, Proceedings of the International Symposium on Fibre Reinforced Concrete, December 16-19, 1987, Madras, India.


Bibliography Bi-14

Steffens, R.J. (1966) Some aspects of structural vibration, Vibration in Civil Engineering: Proceedings of a Symposium Organized by the British National Section of the International Association for Earthquake Engineering, Imperial College of Science and Technology, London, April 1965, (Ed. B.O. Skipp), Butterworths, London, pp. 1-30.

Steffens, R.J. (1974) Structural Vibration and Damage, Building Research

Establishment Report, Department of the Environment Building Research Establishment, Her Majesty’s Stationary Office, London.

Stephens, R.W.B. (1958) The applications of damping capacity for investigating the

structure of solids, Progress in Non-Destructive Testing, 1, 167-198. Stevens, R.F. (1969) Tests on prestressed reinforced concrete beams, Concrete, 3(11),

November, 457-462. Stevenson, A.C. and Humphreys, A.W. (1998) Quantification of steel floor vibration

ansd its relationship to human comfort, Journal of Constructional Steel Research, 46(1-3), 106-107.

Suzuki, K., Ohno, Y. and Srisompong, S. (1986) Internal cracking characteristics of

prestressed concrete beams, Proceedings of the Tenth Australasian Conference on the Mechanics of Structures and Materials, August 20-22, University of Adelaide, Adelaide, Australia, pp.423-428.

Tamura, Y., Yamada, M. and Yokota, H. (1994) Estimation of structural damping of

buildings, Proceedings of Structures Congress ’94, Atlanta, GA, April 24-28, (Eds. Baker, N.C. and Goodno, B.J.), ASCE, New York, pp. 1012-1017.

Teng, S., Ma, W., Tan, K.H. and Kong, F.K. (1998) Fatigue tests of reinforced concrete

deep beams, The Structural Engineer, 76(18), 347-352. Teng, T.-L. and Hu, N.-K. (2001) Analysis of damping characteristics for viscoelastic

laminated beams, Computer Methods in Applied Mechanics and Engineering, 190, 3881-3892.

Terenzi, G. (1999) Dynamics of SDOF systems with nonlinear viscous damping,

Journal Engineering Mechanics, 125(8), 956-963. Timoshenko, S. (1937) Vibration Problems in Engineering, Van Nostrand Co., New

York. Thorkildsen, E. and Imbsen, R.A. (2000) A realistic approach to seismic evaluation of

concrete bridges, Concrete International, 22(10), 33-35. Trifunac, M.O. (1972) Comparisons between ambient and forced vibration experiments,

Earthquake Engineering and Structural Dynamics, 1(2), 133-150. Turner, M.D. (1999) Introduction of 500 MPa reinforcing steel and it’s effect on AS

3600, Concrete ’99: Our Concrete Environment, Proceedings of the CIA 19th Biennial Conference, (D. Baweja, S. Bernard, R.Wheen and P. Shea eds.) Sydney Hilton Hotel, May 5-7, pp. 579-584.


Bibliography Bi-15

Uchikawa, H., Hanehara, S. and Hirao, H. (1997) Influence of microstructural change under stress on the strength-related properties of hardened cement mortar and paste, Advance Cement Based Materials, 6, 87-98.

Ungar, E.E. (1973) The status of engineering knowledge concerning the damping of

built-up structures, Journal of Sound and Vibration, 26(1), 141-154. Vandiver, J.K., Dunwoody, A.B., Campbell, R.B. and Code, M.F. (1982) A

mathematical basis for the random decrement vibration signature analysis technique, Transactions of the ASME, Journal of Mechanical Design, 104(2), 307-313.

Wakim, G. (1994) High strength prestressed concrete members, High Performance

Concrete: Technology, Design and Applications, Proceedings of a Seminar held by the Department of Civil and Environmental Engineering, 10th February, Melbourne, Australia (P. Mendis Ed.), Draft Report Ch 7, pp. 159-161.

Wang, Z. and Finch, R.D. (1992) The dynamic behaviour of a reinforced concrete beam

with or without cracks, Journal of the Acoustical Society of America, 92(4) Part 2, 2442.

Warner, R.F. and Faulkes, K.A. (1988) Prestressed Concrete, Longman Cheshire,

Melbourne, Australia. Warner, R.F., Rangan, B.V., Hall, A.S. and Faulkes, K.A. (1998) Concrete Structures,

Longman, Malaysia. Watson, A.J., Al-Azawi, T.K.A.L. and Ang, T.H. (1989) Impact testing of model

reinforced concrete structures, Civil Engineering Dynamics, Proceedings of the Conference Organised by the University of Bristol, in Association with the Society for Earthquake and Civil Engineering Dynamics held at the University of Bristol, 24-25 March 1988, pp. 461-478.

Wheen, R.J. (1995) After cracking – What then?, Prestressed Concrete: Current

Developments and Innovative Applications, 4 May, Concrete Institute of Australia, NSW Branch, Sydney, pp. 1-19.

Willford, M. (2001) An investigation into crowd-induced vertical dynamic loads using

available measurements, The Structural Engineer, 79(12), 21-25. Witakowski, P. (1998) Strength development dynamic of cement paste: Testing and

control, Cement and Concrete Research, 28(11), 1629-1638. Wyatt, T.A. (1977) Mechanisms of damping, Symposium on Dynamic Behaviour of

Bridges, Transportation and Road Research Laboratory, Crowthorne, Berkshire, UK.

Yeih, W., Huang, R., Chang, J.J. and Yang, C.C. (1997) A pullout test for determining

interface properties between rebar and concrete, Advanced Cement Based Materials, 5, 57-65.

Yon, J.-H., Hawkins, N.M. and Kobayashi, A.S. (1997) Comparisons of concrete


Bibliography Bi-16

fracture models, Journal Engineering Mechanics, 123(3), 196-203. Young, K., Fenwick, R. and Lawley, D. (1999) Mechanical threaded rebar couplers and

plate anchors in seismic resistant concrete frames, Concrete ’99: Our Concrete Environment, Proceedings of the CIA 19th Biennial Conference, (D. Baweja, S. Bernard, R.Wheen and P. Shea eds.) Sydney Hilton Hotel, May 5-7, pp. 351-360.

Zabegayev, A.V. (1989) Analysis of reinforced concrete structures under impulsive and

impact loadings, Civil Engineering Dynamics, Proceedings of the Conference Organised by the University of Bristol, in Association with the Society for Earthquake and Civil Engineering Dynamics held at the University of Bristol, 24-25 March 1988, pp. 481-492.

Zachar, J.A. and Naik, T.R. (1993) Analysis of the flexural interaction between high-

strength concrete and high-strength reinforcement, Utilization of High Strength Concrete, Proceedings of a Symposium, Lillehammer, Norway, June 20-23, (Eds. Holand, I. and Sellevold, E.) pp. 428-434.

Zhang, W., Chen, Y. and Jin, Y. (2000) A study of dynamic responses of incorporating

damaged materials and structures, Structural Engineering and Mechanics, 10(2), 139-156.

Xiao, Y. and Ma. (1998) Seismic behaviour of high strength concrete beams, The

Structural Design of Tall Buildings, 7, 73-90.


APPENDIX A: Literature Review Summary Tabulations A-1

APPENDIX A

Literature Review Summary Tabulations


Table A.1. Material Damping Literature Review Summary Effect on Concrete Material Damping (Part 1)

Author’s

Moisture or Water

Content (mc) (wc)

Size,Shape, Type

Aggregate (Recycled -

r/a)

Stress/Strain Amplitude

or State of Stress

Degree of Hydration

or Age

Interfacial Damping /

Microcracking

Frequency (F) or

Vibration Amplitude

(A)

Curing Condition

Water/ Cement Ratio (w/c)

Aggregate/ Cement

Ratio (a/c)

Vibration Mode (Flexural-

Longitudinal- Torsional Fl-Lo-To

Concrete Compressive or Flexural Strength f’c

Air Voids or

Additives Added

Dynamic (ED) or

Elastic (Ec) Modulus

Testing/ Measurement Methods or

Errors

Kesler and Higuchi (1953)

δ as the mc of the

specimen δ with

in age δ is dependent on F

δ less dep.on w/c as mc

Measuring δ cannot predict

f’c

Measuring δ

cannot predict Ec

Jones (1957) δ is interfacial

or frictional

No ∆ in δ with F 70-10,000 c/s

δ same for Fl, Lo & To

Cole and Spooner (1965)

δ linearly with max.

strain amplitude

F below 2.5 cps: δ with

F

Cole (1966) δ very

sensitive to mc (Until dry)

δ with

age and mc.

δ in cement paste, is not interfacial

For small specimens, no ∆ in δ with ∆

in A

No difference in δ btw

concrete or paste

For sml specimens, δ

loss in mountings is

v.sml

Jones and Welch (1967)

Influences δ. Is a complex interaction.

‘Q’ value increases

(∴δ ) as specimen dries out

Not signif. Similar to Kesler and

Higuchi (1953)

δ as coarse

aggregate content

δ influenced by max. agg. size. Difficult

to define

Hard to define.

Disagrees with Kesler

and Highuchi (1953)

Influences δ. More

pronounced for higher f’c.

Swamy (1970)

Highly dependent. δ

is related to % of water-filled

pores

agg. size δ.

Angular agg’s δ.

Reactive agg.s infl. δ

For omplex stresses, δ . Infl. of stress

state unknown

δ for age

(&∴ mc)

Microcracks friction type δ

At very low frequency, δ

with a in freq

δ for Lo. vibration is highest and To

lowest

Air voids do not affect δ

Swamy and Rigby (1971)

Important – can

overshadow aggregate effects.

δ with

mc.

δ lge specif. area.

δ for crushed

δ smooth δ for sml δ

for lge

The more complex the state of stress the smaller the

δ

Ageing effect less

than drying effect. δ

significant in first 28

days

δ as aggregate content . Drying has

greater influence on δ

than microcracks

Not completely

established, δ may depend on frequency

Drying δ, but rapidly stabilised with time

in % water filled pores δ

More aggregates

δ. δ occurs in

matrix, less in agg. &

agg./matrix interface

Lo. vibration highest δ capacity, then Fl, then To

Not conclusive but δ gradually with in f’c

Adds little to δ

δ with an in dynamic

modulus. Not conclusive

Estimates accuracy of

experiment δ to be approx. ±

6%. Thought to be satisfactory

Effect on Concrete Material Damping (Part 2)

Author’s

Moisture or Water

Content (mc) (wc)

Size,Shape, Type


r/a)


or State of Stress

Degree of Hydration

or Age


Microcracking

Frequency (F) or

Vibration Amplitude

(A)

Curing Condition


Aggregate/ Cement

Ratio (a/c)




Air Voids or

Additives Added

Dynamic (ED) or



Errors

Spooner and Dougill (1975)

δ by another process both during an

or of strain

Majority of δ from one

mechanism during 1st load

With strain & initial Ec

δ

Ashbee et al. (1976)

Shear stress plays a larger role in δ than

(mc)

δ dependent on mean &

dynamic load histories

δ depend. on age.

Not quantified

or discussed

Non-linear effects from

shrinkage cracks

δ dependent on curing

history

δ cyclic dependent. (ie. δ

with the no. of cycles )

Uses steel specimens to reduce errors and calibrate

Spooner et al. (1976)

Strain range applied to a specimen is

VIP in determining δ

δ of ‘solid’ material

independ. of age

δ is independ. of degree of damage

(cracking) of a specimen

δ of ‘solid’ material

independ. of w/c

For mortar specimens δ is ∝ to quantity

of ‘solid’ material

Jordan (1980)

Signif. in δ

for in dynamic stress

Age is signif. 20%

in δ with age

Micro-cracking is VIP Not signif.

δ of dry greater than

wet: cracking

w/c with a/c ratio,

δ

a/c gave δ

Sri Ravindraraj-ah and Tam (1985)

δ due to a higher wc in

recycled-aggregate (r-a)

concretes.

For all grades of concrete,

r/a had a δ.

δ due to a larger amount of micro-cracks in the r-a concrete.

δ with a in f’c,

possibly due to the in

total porosity

Fu and Chung (1996)

Latex andsilica fume

δ

Xu and Setzer (1997)

At verylow temps. δ is temp. dependent

At very low temps. δ is F

dependent

Fu et al. (1998)

Latex andsilica fume

δ by 390%

Effect on Concrete Material Damping (Part 3)

Author’s

Moisture or Water

Content (mc) (wc)

Size,Shape, Type


r/a)


or State of Stress

Degree of Hydration

or Age


Microcracking

Frequency (F) or

Vibration Amplitude

(A)

Curing Condition


Aggregate/ Cement

Ratio (a/c)




Air Voids or

Additives Added

Dynamic (ED) or



Errors

Li and Chung (1998)

in δ because of high tensile ductility of silica fume

concrete

Latex and silica fume

δ by 300% at all F & Temps.

Wang and Chung (1998)

Aggregatesany shape or

size degrade δ

Addition ofsand δ

Aggregate+ Silica Fume =

large δ

Orak (2000)

ymer Poconcrete δ

is 4-7 times higher than

for steel

l

Table A.2 Member Damping Literature Review Summary

Effect on Plain RC Member Damping (Part 1)

Author’s

Concrete Strength/ Cracking/

Type

Type or Percentage of

Reinforcement

Amplitude (A) or Mode of Vibration

Test History (Repeated

Loading) or Test Method

Steel Stresses or Strains in

Member

Size Effects or Beam

Dimensions (h, b)

Steel Area

Elastic Modulus of Concrete

(Ec)

Material constant, n (Es/Ec)

Beam Displacement

/Support Influence

Crack spacing,

s Other

Bock (1942)

Beams without reinforcement had δ than those with reinforcement

Damping not affected by (A)

Penzien and Hansen (1954)

δ in RCB’s subject to

dynamic forces reduces max. strains signif.

James et al. (1964)

δ rapidly with the mode of vibration. δ

independent of (A)

Loading history has a significant

effect on δ.

δ not viscous for sml amplitudes, but viscous for lge

amplitudes. Unconfirmed that as

temp. δ

Penzien (1964)

Significant increases in δ with

cracking

Jordan (1977)

δ only by 25% when

severe cracking appeared

No evidence to suggest that

material δ as tensile stresses

are induced Un-

Cracked δ for reinf. amount δ till cracks

fully formed Dieterle and Bachman (1981) Cracked δ to nearly

zero at fsy

Small infl. on δ

ratio

Un-Cracked δ slightly δ slightly for

increasing h,b δ slightly δ slightly VIP Es neglected Flesch

(1981) Cracked δ strongly δ strongly for

increasing b,h δ strongly Strong in δ for displ.

Effect on Plain RC Member Damping (Part 2)

Author’s

Concrete Strength/ Cracking/

Type

Type or Percentage of

Reinforcement

Amplitude (A) or Mode of Vibration



Steel Stresses or Strains in

Member

Size Effects or Beam

Dimensions (h, b)

Steel Area Elastic

Modulus of Concrete (Ec)

Material constant, n (Es/Ec)

Beam Displacement

/Support Influence

Crack spacing

s Other

Un-cracked

After frost- exposure δ

30-40%

Askegaard and Langsæ (1986)

Cracked

Well- developed cracking alters δ

Type of testing regimes Effect ons δ can be up

to 100%

Humidity changes may affect

operational δ significantly. Not

quantified

δ is dependent on mc. mc from 0 to

4% δ by 80%. Humidity and

temperature affects δ signif.

Almansa et al. (1993)

δ factors had no influence wrt cracking

levels

δ factors were found to exhibit no influence wrt the age of beam

un-forming

Wang et al. (1998)

On first cracking δ

by a factor of 4

δ is a function of the maximum load

to which it has been subjected

Chowdhury (1999) Effect of

cracking on δ is VIP

Ndambi et al. (2000)

δ with the excitation

amplitude up to 30%.

δ very sensitive to excitation method

(ie. impact vs shaker.

Found non-linear behaviour

introduced into system by impact

hammer excitation.

Weng and Chung (2000)

Addition of silica fume δ by two

or more orders of

magnitude

Sandblasted rebars δ 91% over

plain rebars. Considered to be

signif. for practical use

Concrete with rebars

δ (by 3 orders of

magnitude) than no rebar

mortar. Additional

reo δ more

Considers the high damping capacity of

rebars to be responsible for high damping capacities

of RC

Yan et al. (2000a/b) δ as max. response A

FRC exhibit δ with an in

number of vibration cycles

Concluded that δ is strain dependent

δ significantly with wavy fibres

Table A.3 Prestressed Damping Literature Review Summary

Effect on Prestressed Concrete Member Damping Initial Prestress

Condition Type of Vibration

Author’s

Cracked State of Concrete

Test History (Repeated / Past

Loading)

Magnitude/Degree of Prestressing

Type of Prestress Uniform

Axial Eccentric (triangle)

Steady-State Free-Decay

Strength of Concrete f’c

Age Amplitude of Vibration

Frequency of Vibration

Effect of Support Damping

James et al (1964) Little effect on δ

δ not viscous for sml amplitudes,

but viscous for lge amplitudes

Penzien (1964)

VIP. δ with tension crack

development. δ may be 3-6% of critical for sig.

cracking and 1% for uncracked

δ depends on load/stress level

history and initial amplitude

displacements that may cause cracking

Only indirect influence on δ as it

influences cracking

Only indirect influence on δ

as it influences cracking which ∴ affects δ

δ as degree of prestress is

(0.75-1.5% of critical)

Mag. of prestress

did not sig. influence δ (0.5-1% of

critical)

δ greater for free

vibration than steady-

state

Effect is entirely

masked by effect of cracking

δ with amplitude of

oscillation for all cases

Important, however not

studied or quantified

VIP, effect higher in free vibration than steady-state. Magnitude of influence not

quantified

Hop (1991)

δ is approx. 35% higher after

cracking than prior to the onset

of cracking

of axial & eccentric

prestressing δ signif.

in degree of axial

causes large in δ

in degree of eccentric

causes large in δ

δ with age. At 20

yrs δ 40% - 75% for

beams with high E

Shield (1997)

Prior to formation of cracks, AEA

activity increased

Table A.4 Structural Damping Literature Review Summary

Effect on RC Structural Damping (Part 1)

Author’s

Concrete Strength/ Cracking/ Type

(Crack Spacing/Width)

Type or Percentage (Steel Area) of Reinforcement

(Stresses/ Strains)

Amplitude or Mode or




Structure Interaction

Effects

Structure Size Effects, Type or

Dimensions

Beam Displacement

/Support Influence

Applicability or Usefulness of

Damping Results

Seismic Aspects Type of δ Other

Jeary (1974)

All energy dissipation in

chimney’s in 1st mode of bending

Using acceleration response, signif.

Inaccuracies when δ reaches 0.1

Leonard and Eyre (1975)

δ with higher frequencies and

higher amplitudes of

vibration

RC bridges had δ with

structural movement

Determined that abutment

interaction accounted for

2/3rds of damping value

Roller supports do not contribute to δ.

Therefore overall δ from superstructure and movement of supporting piers

Suggested δ differences btw bridges due to

support conditions

δ results from one bridge can only be

used for predicting δof an identical

bridge. Even then, not with confidence

Deduced that RC bridge

damping was viscous damping

Douglas et al. (1981)

Free-vibration tests very effective in determining if

structure has ever been overloaded

during it's lifetime

Free-Vibration tests very

effective in determining δ

Soil structure interaction VIP

Wheeler (1982)

Single test pedestrian

adequate for footbridge studies

Proposes methods for design of

damping devices in footbridges

Jeary (1986)

δ predictor btw building base

dimension

Shears (1989)

In Oil Platform dynamic

modelling, soil-structure

interaction VIP

Longspan lightweight

floors have low δ, ∴ induce

signif. dynamic response

Lagomars-ino (1993)

δ highly dependent upon the size and

quantity of cracking

δ influenced by the stress state of the

structural components

Damping highly dependent on fundamental

vibration period

Initial damping highly

correlated to joint slippage

Effect on RC Structural Damping (Part 2)

Author’s

Concrete Strength/ Cracking/ Type

(Crack Spacing/Width)

Type or Percentage (Steel Area) of Reinforcement

(Stresses/ Strains)

Amplitude or Mode or




Structure Interaction

Effects

Structure Size Effects, Type or

Dimensions

Beam Displacement

/Support Influence

Applicability or Usefulness of

Damping Results

Seismic Aspects Type of δ Other

Brownjohn (1994)

Suspensionbridge hangers may provide

hysteresis δ if inclined

The bridge in suspension bridges

influences higher mode δ

Deck bearings may provide

significant δ at low amplitudes

Suggests that for design calcs

only structural δ be considered

Farrar et al. (1994)

δ distortion affects will however, be more pronounced when the structure cracks and δ

δ will be in scale models compared to full-size structures. thought not to affect prediction of elastic dynamic response of

RC structures

Scaling δ from prototype tests

to full-size structures

affected by δ mechanism

Lutes and Sarkani (1995)

Using a fixed-base model in

soil-structure (s-s) interaction

studies will give sig. δ than a

free-based model

Wrong choice of fixed or free-

base may attribute

structural δ to s-s interaction, when actually

due to choice of δ

Denoon and Kwok (1996)

δ dependent on

vibration amplitude

δ dependent on type of test

method

δ dependent on building and

foundation height δ dependent on

building usage

Suda et al. (1996)

δ ratio with natural

frequency

δ ratios are in the longer

direction than shorter direction for office blocks

δ ratios of hotels or apartments (non-structural members) are

larger than office buildings

δ ratios as buildings become taller

δ ratios for pile foundation

buildings are than spread

foundations

Fang et al. (1998)

δ is amplitude dependent (buildings)

Li et al. (2000)

δ is amplitude dependent (buildings)

Table A.5. High-Strength Concrete and Reinforcement Specific Research Author’s Location Research Focus and Conclusions Drawn

Sparrow (1989) Connell Group High strength concrete in the Melbourne Central Project HSC 65 and 70 MPa usage described in the Melbourne Central office tower.

Collins, Mitchell and MacGregor (1993)

Canada Structural design considerations for high-strength concrete. - Questions and investigates the applicability of traditional design procedures for use with HSC.

Jensen (1994) Norway

Structural aspects of high strength concretes -Presents results of recent HSC research (overview) is given -Comments on HSC (or lack of it) in concrete design codes, Eurocode EC2 (up to 60 MPa), Norwegian Standard (recently changed from 65 to 105 MPa.

Park (1995) University of Canterbury, NZ

Opportunities in New Zealand for high strength reinforcing steel - Discusses HSC columns in particular. - Discusses NZS 3101 and its provisions for HSC up to 70 and 100 MPa concrete - Discusses NZS 3101 wrt 500 MPa steel.

Rasmussen and Baker (1995)

University of Queensland

Torsion in reinforced normal and high-strength concrete beams - Examines HSC beams subject to pure torsion (30, 50, 70 and 110 MPa). - Uses HS Danish Steel – Grade 550 MPa.

Foster and Gilbert (1996)

University of NSW

The design of nonflexural members with normal and high-strength concretes - Investigates concretes ranging btw 20 to 100 MPa for analysis techniques of nonflexural members (strut and tie model, plastic truss model). - Also looks at the main failure modes of nonflexural members.

Macchi, Pinto and Sanpaolesi (1996)

University of (Italy)

Ductility requirements for reinforcement under Eurocodes - Experimental research is reported in which cyclic tests showed that the currently accepted properties of reinforcing steel do not provide sufficient local ductility for the highest ductility class of structures envisaged by the Eurocodes. New requirements are proposed for reinforcing steel to be used in seismic regions, particularly with reference to uniform elongation at maximum load, and the ratio btw ultimate stress and yield stress.

Pendyala, Mendis and Patnaikuni (1996)

University of Melbourne

Full-range behaviour of high-strength concrete flexural members: Comparison of ductility parameters of high and normal-strength concrete members - Compares HSC flexural members against NSC in the 3 ductility parameters of hinge lengths, softening slopes and hinge rotation capacities. - The implications of designing with HSC are discussed.

Shah and Ouyang (1996)

Northwestern University

Tensile response of reinforced high strength concrete members - Effect of various parameters (reinforcement ratio and distribution of steel bars) were experimentally examined for HSC tensile members (fcm = 99 MPa). - Also uses 500 MPa steel.

Gilbert (1997a) University of NSW

Anchorage of reinforcement in high strength concrete - A comparison is made of the provisions in several international codes of practice for the anchorage of reinforcement bars in concrete with compressive strength up to 100 MPa. - Presents a case study on development lengths with f’c = 25, 32, 50,70 and 100 MPa. Also steel yield stresses fsy of 400 and 500 MPa.

Gilbert (1997b) University of NSW

High strength reinforcement in concrete structures: Serviceability implications - Serviceability of reinforced concrete beams and slabs is investigated for elements designed using Grade 500 reinforcement. - The short and long term behaviour of beams and slabs are considered.

Hoff (1997) Texas, US

The Hibernia offshore concrete platform - Case study of this project which is, to date, the largest single use of HSC (80 MPa). Additionally 500 MPa steel was used, as well as 400 steel.

Kong and Rangan (1997)

Curtin University

Reinforced high strength concrete (HSC) beams in shear - f’c ranged from 60 – 90 MPa. - Proposal to modify the value of minimum shear reinforcement given in AS 3600.

Mansur, Chin and Wee (1997)

University of Singapore

Flexural behaviour of high-strength concrete beams - 11 reinforced HSC beams tested in flexure - Yield strength of bars = 550 MPa - f’c ranged from 50 to 100 MPa.

Pendyala, Mendis and Bajaj (1997)


Design of high-strength concrete members - Ductility of beams and columns - Rectangular stress block for HSC - Shear design for HSC - Bond and anchorage for reinforcement in HSC members.

Pendyala, Mendis and Baweja (1997)


Towards the development of new codes and standards to increase the field application of high performance concretes - Investigates and discusses the broad attribues of HPC. - Reviews AS 3600 and summarises current research.

Attard and Stewart (1998)

Uni NSW – Uni Newcastle

A two parameter stress block for high-strength concrete - Looks at the applicability of the ACI rectangular stress block parameters to high-strength concretes. - Strengths btw 20 and 120 MPa.

Foster and Gilbert (1998)

University of NSW

Experimental studies on high-strength concrete deep beams - Presents the results for 16 HSC deep beams tested to destruction. Variables considered were shear-span to depth ratio, concrete strength (50 to 120 MPa) and the provision of secondary reinforcement.

Lin and Restrepo (1998)

University of Canterbury, NZ

Experimental verification of the concrete structures standard recommendations for the design of beam-column joints - Determining the shear strength of beam-column joints under seismic loading. - Provides experimental verification of the seismic performance of frames built using Grade 500 (threaded) longitudinal reinforcement (concrete 30 Mpa).

Lorrain, Maurel and Seffo (1998)

University of France

Cracking behaviour of reinforced high-strength concrete tension ties - The cracking behaviour of NSC and HSC tension ties under short term load was investigated experimentally. - The mechanical strength of the concrete, the reinforcement ratio, and the yield strength of deformed steel bars were taken as test parameters - f’c = 40 – 100 MPa - fsy = 620 and 830 MPa.

Park (1998) University of Canterbury, NZ

Some current and future aspects of design and construction of structural concrete for earthquake resistance - Investigates HSC (up to 100 MPA) and HSS (500 MPa) wrt ductile performance of a structure during an earthquake (specifically columns, precast floors and frames).

Rangan (1998) Curtin University

Suggestions for design of high performance high strength concrete (HPHSC structural members) - Proposes design rules of HPHSC beams, columns and walls in the range 20 – 100 MPa. - Briefly mentions 500 MPa steel in an example on calculating the max. tensile steel ratio for the flexural strength of HPHSC beams.

Sanjayan and Jeevanayagam (1998)

Monash University

Long term deflection of high strength concrete beams - Compares predicted results against experimental results for long term deflection of HSC beams. - Also develops a relationship btw basic creep factor and

compressive strength of concrete, for strengths up to 100 MPa.

Teng, Ma, Tan and Kong (1998)

Nanyang, Singapore

Fatigue tests of reinforced concrete deep beams - f’c = 50 MPa - fsy = 577 MPa - 3 different types of web reinforcement were investigated, and the tests revealed that web reinforcement has significant influence on the structural response of deep beams under fatigue loading

Adams, Walsh, Marsden, Patrick (1999)

BHP – University of Newcastle (Walsh)

Factors affecting the ductility of stiffened rafts - Theoretical and experimental study to assess the likely impact that steel ductility has on the behaviour of stiffened rafts detailed in accordance with AS 2870. - Uses Grade 500 reinforcing mesh.

Chick, Patrick and Wong (1999)

BHP – University of Adelaide

Ductility of reinforced-concrete beams and slabs, and AS3600 design requirements - Uses 500 MPa steel to study the overload behaviour of concrete beams and slabs. - In particular, the effect that low-ductility reinforcing steel used in welded mesh has on the load-carrying capacity of these types of flexural members is examined. Implications for AS 3600 are discussed.

Esfahani and Rangan (1999)

Curtin University

Evaluation of proposed revisions to AS3600 bond strength provisions - Presents a comparison of the proposed AS3600 bond strength provisions for NSC and HSC.

Gilbert (1999) University of NSW

Flexural crack control for reinforced concrete beams and slabs: An evaluation of design procedures - The advent of high strength reinforcing steels will inevitably lead to higher steel stresses under in-service conditions, thereby exacerbating the problem of crack control. - In the paper, the current flexural crack control provisions of AS3600 are presented and the crack width calculation procedure in several of the major international codes, ACI318, EC2 are assessed.

Gilbert, Patrick and Adams (1999)

University of NSW – BHP

Evaluation of crack control design rules for reinforced concrete beams and slabs - Recommendations are made as to a suitable crack control model for inclusion in an amendment to AS 3600 to allow the design yield strength to increase to 500 MPa.

Gravina and Warner (1999)

University of Adelaide

Modelling of high-moment plastification regions in concrete structures - A review of recent research into the rotation capacity of reinforced concrete flexural members shows renewed interest in the topic has been generated by the introduction, in various countries, of high-strength reinforcing steel with limited uniform elongation. This has caused concern over the possibility of steel fracture in high-moment regions - Grade 500 steel, 10% uniform elongation, - f’c = 30 MPa.

Panagopoulos, Mendis and Portella (1999)


Seismic performance of frame structures with high-strength concrete and 500 MPa steel - f’c = 50 and 100 MPa - grade 400 and 500 MPa

Patrick (1999) BHP Research

Australian 500 MPa reinforcing steels and new AS 3600 ductility design provisions - Discusses experimental and analytical studies of the overload behaviour of concrete beams and slabs. - In particular, the effect that low-ductility reinforcing steel used in welded mesh has on the load-carrying capacity of these types of flexural members is examined.

Sanders (1999) SRIA Advances in fire design for reinforced concrete structures- Moving to more rational design methods

- grade 400 and 500 MPa

Turner (1999) SRIA

Introduction of 500 MPa reinforcing steel and its effect on AS 3600 - Discusses mainly the new AS/NZS xxxx proposed reinforcing standard and its interaction with AS 3600. Reviews major technical changes esp. wrt ductility

Young, Fenwick and Lawley (1999)

University of Auckland, NZ

Mechanical threaded rebar couplers and plate anchors in seismic resistant concrete frames - Assessment of how couplers perform in plastic hinge zones under seismic loading, particularly beam-column connections. - Uses Grade 500 threaded bars, concrete Grade 35 MPa.

APPENDIX B: RC and PSC Beam Calculations B-1

APPENDIX B

RC and PSC Beam Calculations

B.1 General Remarks

If the stress for a reinforced concrete member, subjected to a bending moment causing

deflection, has never exceeded its tensile strength, the member is free from cracks

(Ghali and Favre, 1986). In this case, the reinforcement and concrete undergo similar

strains; this region is defined herein as ‘uncracked’. The analytical equation to calculate

the bending moment that causes first cracking, Mcr in kNm, is based on the flexural

strength of concrete in tension and the beam’s cross-sectional dimensions. It is defined

as Equation B.1 (AS3600-2001)

t

gcfcr y

IfM '= (B.1)

where f’cf is the characteristic flexural tensile strength of the concrete in MPa (f’cf =

0.6√fcm, AS3600-2001); yt is the distance between the neutral axis and extreme fibres in

tension of the uncracked section in mm; Ig is the gross moment of inertia of the

uncracked section in mm4 (determined here using Ig = bD3/12).

The ultimate moment capacity, Mu,calc in kNm of a doubly-reinforced (under-reinforced)

concrete beam is found from

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −= c

u

cscsystcalcu d

dkd

AdfAM2

16002,

αα (B.2)

where Ast is the cross-sectional area of reinforcing steel in mm2; d is the depth to the

centroid of the reinforcing steel in mm; and α and ku are the compressive stress block

parameters.

Damping Characteristics of Reinforced and Prestressed Normal- and High Strength Concrete Beams


The cracking moment, Mcr (kNm) of a prestressed concrete beam may be calculated

from the following formula:

Mcr = (f’cf + σbp) I/yB (B.3)

where f’cf is the characteristic flexural tensile strength of concrete in MPa; and σbp is the

flexural stress provided by the prestress in MPa.

Similarly, the ultimate moment capacity may be calculated from the following:

Mu = [σpu Apt dp + fsy Ast ds - fsy Asc dsc - (0.85 f’c b(γ ku d)2/2)] (B.4)

Where γ is defined in Figure B.1; σpu, Apt and dp are the ultimate stress (MPa), cross-

section area (CSA) in mm2, and depth to the prestressing tendons (mm), respectively; fsy

Ast and ds are the yield stress (MPa), area of tension steel (mm2), and depth to the

tension reinforcement (mm), respectively; and Asc and dsc are the CSA (mm2) and depth

to the compression reinforcement (mm), respectively.

The formula for the calculation of the instantaneous static deflection in mm (∆i) of a 2-

point loaded reinforced concrete and prestressed concrete beams is given by

IElw

IElP

wpi

43

αα +=∆ (B.5)

where P is the applied load in kN, l is the effective span in m, w is the self-weight of the

beam in kN/m, and αp and αw are constants that depend on the loading conditions. The effective moment of inertia Ief (the second moment of area) of the member,

incorporating tension stiffening (see Bažant and Oh, 1984), is calculated using

Branson’s formula:

Ief = Icr + (Ig – Icr)(Mcr/Ms)3 ≤ Ig (B.6)

where Ief is defined by the following limits:

Icr ≤ Ief ≤ Ig (B.7)

and Ig is the second moment of area of the gross concrete cross section in mm4 about the

centroidal axis as discussed previously; Icr is the second moment of area of a cracked



section in mm4 with the reinforcement transformed to an equivalent area of concrete;

Mcr is the cracking moment at the section in kNm; and Ms is the applied bending

moment in kNm at the section for the loading increment being considered.

The Australian Concrete Structures Design Code (AS3600-1994), previously allowed,

as a further simplification (for rectangular RC members only with a width of b in mm

and effective depth to the centroid of the tensile steel in mm), that:

Ief = 0.045bd3 (B.8)

As a simplification (for RC members only), Ief is now calculated using the following

equation (AS3600-2001: Clause 8.5.3.1):

Ief = (0.02+(2.5ρt))bd3 (B.9)

where ρt is the tensile reinforcement ratio.

T

C = 0.85 f’c γ ku bd

0.85 f’c for NSCα f’c for HSC

γ kudkud

NA

γ kud/2

η f’c

C = αηf’cku bd

β kud

For NSC (AS3600-2001): γ = 0.85 for f’c ≤ 28 MPa andγ = 0.85 – 0.007(f’c-28) forf’c ≥ 28 MPa where 0.65 ≤ γ ≤ 0.85 for f’c ≤ 65 MPa.

For HSC (Mendis, 2002):γ = 0.65 – 0.00125(f’c-60)

60 MPa ≤ f’c ≤ 100 MPa

α = 0.85 – 0.0025(f’c-60)

60 MPa ≤ f’c ≤ 100 MPa

Figure B.1: Compressive Block Parameters



B.2 Calculations for B-Series Beams

Beam is under-reinforced

pt = Ast/bd = 1257/200×264 = 0.0238pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0.01933(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy

= 0.0194(pt - pc)lim > pt - pc

∴ Asc will not yield_________________________η = (ptfsy - 600pc) 1.7 γ f’c

= 0.1618υ = 600 pc

0.85 γ f’c

= 0.1174

200

300

20

264

2044

Beam BII-2

30.0 MPa500 MPa

3N20’s = 942 mm2

2N12’s = 226 mm2

dc = 32 mmdt = 264 mm

γ = 0.85 - 0.007(f’c - 28)

= 0.836



= 0.03101(pt - pc)lim > pt - pc


= 0.1477υ = 600 pc

0.85 γ f’c

= 0.1174

200

300

20

262

2038

Beam BI-3

23.1 MPa400 MPa

3Y24’s = 1357 mm2

2Y12’s = 226 mm2

dc = 32 mm

γ = 0.85 - 0.007(f’c - 28)

= 0.85



= 0.01529(pt - pc)lim < pt - pc

∴ Asc will yield_________________________a = (Ast - Asc) fsy

0.85 f’c b = 115.1 mm

ku = η + √ η2 + υ (dc/d) = 0.3628a = γ ku d = 80.1 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 111.8 kNm

Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4


Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4

Mu = Astfsy(d-a/2) + Ascfsy (a/2 - dc) = 112.7 kNm

Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4

200

300

20

dt = 264

20 22.7

Beam BI-1

30.0 MPa400 MPa

4Y20’s = 1257 mm2

2Y12’s = 226 mm2

dc = 32 mmdt = 264 mm

γ = 0.85 - 0.007(f’c - 28)

= 0.836

dc



Beam BII-5

41.5 MPa500 MPa

4N20’s = 1257 mm 2

2N12’s = 226 mm 2

200

300

20

264

22.720

Beam BII-4

23.1 MPa500 MPa

2N24’s = 905 mm2

2N12’s = 226 mm2

200

300

20

262

10020dc = 32 mm

γ = 0.85 - 0.007(f’c - 28)

= 0.85

dc = 32 mm

γ = 0.85 - 0.007(f’c - 28)

= 0.756

Beam BII-6

200

300

20

264

2044

41.54 MPa500 MPa

3N20’s = 942 mm 2

2N12’s = 226 mm 2

dc = 32 mm

γ = 0.85 - 0.007(f’c - 28)

= 0.756



= 0.02446(pt - pc)lim > pt - pc


= 0.1818υ = 600 pc

0.85 γ f’c

= 0.1510



= 0.0388(pt - pc)lim > pt - pc


= 0.1749υ = 600 pc

0.85 γ f’c

= 0.0938



= 0.0388(pt - pc)lim > pt - pc


= 0.1182υ = 600 pc

0.85 γ f’c

= 0.0938


Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4


Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4


Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4



η = (ptfsy - 600pc) (2α) γ f’c

= 0.01791 Normal = 0.01874 HSυ = 600 pc

α γ f’c

= 0.0697 Normal = 0.07297 HSku = η + √ η2 + υ (dc/d) = 0.1112 Normal = 0.1143 HSa = γ ku d = 19.2 Normal = 19.8 HSMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 51.0 kNm Normal = 51.1 HS

a = (Ast - Asc) fsy

0.85 f’c b = 6.6 mm


For Both BI-7 and BII-8Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4

Beam BI-7200

300

20

266

20 116

64.5 MPa400 MPa

2Y16’s = 402 mm 2

2Y12’s

dc = 32 mm

For Normalγ = 0.85 - 0.007(f’c - 28) = 0.5945 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.812pt = Ast/bd = 402/200×266 = 0.00756pc = Asc/bd = 226/200×266 = 0.00425pt - pc = 0.00338(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy

= 0.03215 Normal

For Normal (pt - pc)lim < pt - pc

∴ Asc will yield

γ = 0.65 - 0.00125(f’c - 57)= 0.6406 Pendyala and Mendis (1997)

= 0.574 Rangan (1998)γ = 0.85 - 0.008(f’c - 30)

α = 0.85 - 0.0025(f’c - 57)= 0.8313 Pendyala and Mendis (1997)

= 0.812 Rangan (1998)α = 0.85 - 0.004(f’c - 55)

Beam BII-8200

300

20

266

116

64.5 MPa500 MPa

2N16’s = 402 mm 2

2N12’s = 226 mm 2

For Normalγ = 0.85 - 0.007(f’c - 28) = 0.5945 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.812pt = Ast/bd = 402/200×266 = 0. 00756pc = Asc/bd = 226/200×266 = 0. 00425pt - pc = 0. 00338(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy

= 0.04705 Normal = 0.05144 HSFor Normal (pt - pc)lim > pt - pc

∴ Asc will not yield

Beam BI-9200

300

20

262

20 100

53.0 MPa400 MPa

2Y24’s = 905 mm2

2Y12’s = 226 mm 2

dc = 32 mmγ = 0.85 - 0.007(f’c - 28) = 0.675α = 0.85

For Normalpt = Ast/bd = 905/200×262 = 0.0173pc = Asc/bd = 226/200×262 = 0.00431pt - pc = 0.01298(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy

= 0.04860(pt - pc)lim > pt - pc

∴ Asc will not yieldη = (ptfsy - 600pc) (2α) γ f’c

= 0.07156υ = 600 pc

α γ f’c

= 0.08287


For Both BI-9 and BII-10Ig = bD3/12 = 200×3003/12 = 450 × 106 mm4



Beam BII-10

200

300

20

262

100

53.0 MPa500 MPa

2N24’s = 905 mm2

2N12’s = 226 mm2

Beam BII-11200

300

20

264

2044

90.7 MPa500 MPa

3N20’s = 942 mm2

2N12’s = 226 mm 2

Beam BII-12200

300

20

264

20 22.7

80 MPa500 MPa

4N20’s = 1257 mm2

2N12’s

dc = 32 mmγ = 0.85 - 0.007(f’c - 28) = 0.675α = 0.85

For Normalpt = Ast/bd = 905/200×262 = 0.0173pc = Asc/bd = 226/200×262 = 0.00431pt - pc = 0.01298(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy

= 0.04457(pt - pc)lim > pt - pc

∴ Asc will not yield

η = (ptfsy - 600pc) (2α) γ f’c

= 0.0998

υ = 600 pc

α γ f’c

= 0.0829ku = η + √ η2 + υ (dc/d) = 0.2415a = γ ku d = 42.7 mmMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 107.6 kNm

γ = 0.65 - 0.00125(f’c - 57)= 0.6079 Pendyala and Mendis (1997)

= 0.3644 Rangan (1998)γ = 0.85 - 0.008(f’c - 30)

α = 0.85 - 0.0025(f’c - 57)= 0.7658 Pendyala and Mendis (1997)

= 0.7072 Rangan (1998)α = 0.85 - 0.004(f’c - 55)

For Normalγ = 0.85 - 0.007(f’c - 28) = 0.4111 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.7072pt = Ast/bd = 942/200×264 = 0. 0178pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0. 01344(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy



= 0.07870 Normal = 0.09459 HS

For Normalγ = 0.85 - 0.007(f’c - 28) = 0.4111 ≈ 0.65α = 0.85For HSγ = 0.65α = 0.7072pt = Ast/bd = 1257/200×264 = 0. 0238pc = Asc/bd = 226/200×264 = 0.00428pt - pc = 0. 01931(pt - pc)lim = 510γ f’c (dc/d) (600-fsy) fsy



= 0.1080 Normal = 0.1298 HS

υ = 600 pc

α γ f’c

= 0.04993 Normal = 0.06001 HSku = η + √ η2 + υ (dc/d) = 0.2411 Normal = 0.2851 HSa = γ ku d = 41.4 mm Normal = 48.9 HSMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 150.1 kNm Normal = 150.1 kNm HS

υ = 600 pc

α γ f’c

= 0.04993 Normal = 0.06001 HSku = η + √ η2 + υ (dc/d) = 0.1894 Normal = 0.2220 HSa = γ ku d = 32.5 mm Normal = 38.1 mm HSMu = Astfsy(d-a/2) + 600Asc(1-(dc/kud))(a/2 - dc) = 114.5 kNm Normal = 114.5 kNm HS



B.3 Calculations for PS-Series Beams

Beam PS1

60.6 MPa

200

300235

NA

e=111.67 mm

1. Computation of the Effective Prestress Coefficient, η

Is determined from the sum of:a) Elastic shortening of concrete;b) Shrinkage of concrete, and;c) Relaxation of steel wires.Each of these will be considered in turn.

a) Loss of prestress due to elastic shortening of concrete =

where σcp = Stress in concrete at the time of transfer; Ecp = Young’s Modulus of concrete at the time of transfer = 32093 MPa Ep = Young’s Modulus of steel = 227000 MPa (From BHP tests)

For beam PS1, the total number of tendons was 9, each transferring a force of26344.77N to the concrete.

Thus, the initial stress transferred by the wires to the concrete section = σcp = H / Ac = 9×26344.77/60,000 = 3.95 MPa.

∴ Loss of prestress due to elastic shortening of concrete =

= 28.0 MPa

% Loss of stress due to elastic shortening of concrete = 28.0 / 1432.4 (σpi)×100 = 2.50%

b) Loss of prestress due to shrinkage of concrete = ∆ σshrinkage = Ep εcs

where εcs = k1 εcs.b; εcs.b = Basic shrinkage factor, for high-strength concrete = 750×10-6; k1 = 0.21.

Thus, loss of stress due to shrinkage of concrete is = ∆ σshrinkage = 227000×0.21×750×10-6 = 35.8 MPa.% Loss of stress due to shrinkage of concrete = 32.70 / 1432.4 ×100

= 2.47%

c) (%) Loss of prestress due to relaxation of steel wires =

where R = Design relaxation of the tendon and is given by = k4 k5 k6 Rb; Rb = Basic relaxation of tendon = 1% for low relaxation wire; k4 = log[5.4(j)1/6] where j is the number of days after prestressing (36) = 0.992; k5 = 1.5; k6 = 1.00 for an average temperature of 200C.∴ R = 0.992 × 1.5 ×1.00 = 1.488

% Loss of stress due to relaxation of steel wires = 1.488[1-35.8/ 1432.4 ] = 1.45%

Thus, loss of stress due to relaxation of steel wires = (1.45/100) × 1432.4 = 20.8 MPa.______________________________________________________________________________________Total loss of stress at the day of testing the beam (i.e. 36 days after casting)

= Loss due to (Elastic shortening of concrete+Shrinkage of concrete+Creep of concrete+relaxation of tendon)= 28.0 + 35.8 + 0 + 20.8 = 84.6 MPa.Total loss of stress at the day of beam testing = 84.6/ 1432.4 ×100 = 5.91%

∴ The effective prestress coefficient, η to be used for design and analysis of PS1 = (100-6.34)/100 = 0.94.Thus, in summary for all prestressed beams:

cpc

pp E

Eσσ ×=∆

95.332093227000 ×=∆

pσ

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ∆+∆−=∆

pi

creepshrinkagerelaxation R

σσσ

σ 1

∆σp

∆σshrinkage

∆σrelaxation

η

PS191.95%28.0 MPa

2.50%35.8 MPa

1.45%20.8 MPa

0.94

PS2112.4%34.2

2.50%35.8

1.45%20.8

0.94

PS3132.82%40.4

2.50%35.8

1.45%20.8

0.93

PS4224.8%68.3

2.50%35.8

1.45%20.8

0.91

PS5234.98%71.4

2.50%35.8

1.45%20.8

0.91

PS6153.3%46.6

2.50%35.8

1.45%20.8

0.93

PS7132.82%40.4

2.50%35.8

1.45%20.8

0.93

PS8153.3%46.6

2.50%35.8

1.45%20.8

0.93

PS9132.82%40.4

2.50%35.8

1.43%20.8

0.93

PS1018

3.9%55.9

2.50%35.8

1.45%20.8

0.92



2. Flexural and Shear Design of Beam Test Specimens

The design of a fully prestressed concrete member, implies that cracking in not allowed during service. The main method ofensuring the serviceability requirements of structures are met, is by limiting the tensile and compressive stresses in concrete bothat transfer and under full service loads. The way in which these tensile and compressive stresses as controlled is through the safedesign of the prestressing force and prestressing eccentricity.

The main differences of prestressed design to that of reinforced concrete design include:a) Checking load transfer stresses;b) Limit state design at service loading;c) Limit state design at failure loading.These checks help to control for short and long term effects, cracking and deflection at service loads.

Graphical Representation of StressesThe following diagram shows the stresses that occur in the extreme fibers of a section when subject to various stresses that occurin prestressed concrete members.

H/A fct

+ +

+

+

+

+

+-

--

H eB ytI

H eB ytI

Mw ytI

= OR

i) Stresses due to prestress force

ii) Stresses due to moment created by eccentricity of prestressing force

iii) Stresses due to moment created by applied live anddead loads

iv) Combined effects of prestressing forces and applied loads

IyM

IyeH

AH twtB +−

IyM

IyeH

AH twtB −⎟

⎠⎞

⎜⎝⎛ +

IyM

IyeH

AH twtB +⎟

⎠⎞

⎜⎝⎛ −η

IyM

IyeH

AH twtB −⎟

⎠⎞

⎜⎝⎛ +η

General Equations for Bending DesignEquations describing the stresses at the top and bottom fibers under working stressed are defined as follows.

At Transfer: fCF = Equation 1.1

fCB = Equation 1.2

After Prestress Losses: fCF = Equation 1.3

fCB = Equation 1.4

Types of Prestressing CasesOne of two analysis and design methods can be used, depending on the type of moments the beam will be subject to during its life.They are Case A and Case B prestressing.

Case A PrestressingThis case applies if both the minimum moment (M1) and maximum moment (M2) are positive.

Case B PrestressingThis case applies if the minimum moment (M1) is negative and the maximum moment (M2) is positive.

For simply-supported beams, M1 is the bending moment due to beam self-weight, and M2 is the moment caused by the sum of thedead and live loads (it is always positive), an example follows.

Due to g = M1 = g L2 / 8 = 5.86 kNm

Due to Q = M2 = (Q L / 6) + (g L2 / 8)

g = self-weight

Q

gL2/8

QL/6



If we examine the prestress force alone, the stress in the top fibre (fct) is given by:

fct =

But I / A = k2 where k = = the radius of gyration

Therefore, fct = NB

Hence, fCT is negative (in tension) if This is referred to as Case A Prestressing

The positive fCT due to M1 should therefore be counteracted by a negative prestressing force.

Consequently, if fCT is positive (in compression) if This is referred to as Case B Prestressing

The negative fCT due to M1 should therefore be counteracted by a positive prestressing force.

Critical Stress State Equations for Design (Case A Only Exists in Current Investigations)

Case A Prestressing i.e.

When subject to M1, the following conditions must be satisfied:

Top fibre stress (fCT): Equation A1

Bottom fibre stress (fCB): Equation A2

When subject to M1, the following conditions must be satisfied:



:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

AIye

AH

IyeH

AH tBtB 1

tB y

ke2

>

AI

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

t

BtB

yke

AH

kyeH

AH

22 1

tB y

ke2

<

6

2 Dyk

t

=

+

+

+-

-

M1 yTI

⎟⎟⎠

⎞⎜⎜⎝

⎛−

t

B

yke

AH

21

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

t

B

yke

AH

21η

⎟⎟⎠

⎞⎜⎜⎝

⎛+

t

B

yke

AH

21

M1 yBI

NANA

+

-

=

≤ C

≤ Ct or ≥ -Ct

tB y

ke2

>

tTTB C

IyM

IyeH

AH

−≥+− 1

CIyM

IyeH

AH BBB ≤−+ 1

tBBB C

IyM

IyeH

AH

−≥+⎥⎦⎤

⎢⎣⎡ − 2η

CI

yMI

yeHAH TTB ≤−⎥⎦

⎤⎢⎣⎡ + 2η

+

+

+-

-

M2 yT

I

⎟⎟⎠

⎞⎜⎜⎝

⎛−

T

B

yke

AH

21

NANA

+

-

=

≥ -Ct

≤ C

η

η




fct =


Therefore, fct = N






Under M1



Under M2



3. Example Design of Prestressed Beam PS1

B:

a) Calculating maximum service moment allowable for the section:

Combining EquationsA1 and A3, we obtain: Equation A1.1

For PS1 = 102.8×106 Nmm ∴ M2 ≤ 102.8 kNm


For PS1 = 98.3×106 Nmm ∴ M2 ≤ 98.3 kNm

b) Obtain maximum 1/H and maximum eB:

We rearrange to obtain the following equations to obtain max. H and max. eB (whilst ensuring concrete cover ismaintained):

Solving Equations A1 and A4 simultaneously, we obtain a min. H of 714.1 kN and min. eB of 76.57 mm.

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

AIye

AH

IyeH

AH tBtB 1

tB y

ke2

>

AI

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

t

BtB

yke

AH

kyeH

AH

22 1

tB y

ke2

<

6

2 Dyk

t

=

tTTB C

IyM

IyeH

AH

−≥+− 1

CIyM

IyeH

AH BBB ≤−+ 1

tBBB C

IyM

IyeH

AH

−≥+⎥⎦⎤

⎢⎣⎡ − 2η

CI

yMI


⎤⎢⎣⎡ + 2η

tT

T CCMMZ

yI

ηη

+−

≥= 12

tB

B CCMMZ

yI

+−

≥=η

η 12

45.4937.0253.281086.5937.0103

626

×+××−

≥×M

45.4253.28937.01086.5937.0103

626

+×××−

≥×M




fct =


Therefore, fct = N






Under M1



Under M2



3. Example Design of Prestressed Beam PS1

B:

a) Calculating maximum service moment allowable for the section:


For PS1 = 102.8×106 Nmm ∴ M2 ≤ 102.8 kNm


For PS1 = 98.3×106 Nmm ∴ M2 ≤ 98.3 kNm

b) Obtain maximum 1/H and maximum eB:

We rearrange to obtain the following equations to obtain max. H and max. eB (whilst ensuring concrete cover ismaintained):

Solving Equations A1 and A4 simultaneously, we obtain a min. H of 714.1 kN and min. eB of 76.57 mm.

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

AIye

AH

IyeH

AH tBtB 1

tB y

ke2

>

AI

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

t

BtB

yke

AH

kyeH

AH

22 1

tB y

ke2

<

6

2 Dyk

t

=

tTTB C

IyM

IyeH

AH

−≥+− 1

CIyM

IyeH

AH BBB ≤−+ 1

tBBB C

IyM

IyeH

AH

−≥+⎥⎦⎤

⎢⎣⎡ − 2η

CI

yMI


⎤⎢⎣⎡ + 2η

tT

T CCMMZ

yI

ηη

+−

≥= 12

tB

B CCMMZ

yI

+−

≥=η

η 12

45.4937.0253.281086.5937.0103

626

×+××−

≥×M

45.4253.28937.01086.5937.0103

626

+×××−

≥×M



4. Example Computation of Maximum Bending Moment and Shear Capacity of Prestressed Beam PS1

Task 1: Ensure a tension failure by checking that ku≤0.4 (AS 3600-1994: Clause 8.1.3) by using:

(ku d) = [σpy Apt + fsy (Ast - Asc)] / (0.85 f’c γ b)

where σpu = ultimate stress in bonded tendons (Clause 8.1.5) = fp (1 - k1 k2/γ ) = 1710 (1-(0.4×0.0971/0.65)) = 1608.0 MPa

where k1 = 0.4 for fpy/fp < 0.9 = 0.28 for fpy/fp ≥ 0.9 k2 = [fp Apt + fsy (Ast - Asc)] / (bef dp f’c)

and γ = for HSC (Rangan, 1988) For NSC: γ = AS3600-1994 Clause 8.1.2.2(b) = 0.85 - 0.008(f’c-30), where 0.65≤ γ ≤0.85 = 0.85 - 0.007(f’c-28) = 0.6052 = 0.6218∴ γ = 0.65 = 0.65

Finally ku = 0.162 which is ≤ 0.4, ∴ means the beam is under-reinforced and will fail in tension.

Task 2: Calculate Ultimate Moment Capacity of under-reinforced beam PS1:

Mu = [σpu Apt dp + fsy Ast ds - fsy Asc dsc - (0.85 f’c b(γ ku d)2/2)] Conditions satisfied

Mu = [(1608×176.7 ×261.7)+0-0-((0.85 ×60.6 ×200 ×(0.65 ×0.162 ×261.7)2)/2)] = 70.5 kNm

Task 3: Calculate Cracking Moment of under-reinforced beam PS1:

Mcr = (f’cf + σbp) I/yB

where f’cf = 0.6√f’c = 4.7 MPa

where σbp = ηH(1/A + eB yB/I) = 0.94×239000 (1/60000 + (111.7×150/450×106)) = 12.51 MPa

∴ Mcr = (4.7+12.51)(450×10^6/150) = 51.6 MPa.

5. Example Computation of Initial Camber of Prestressed Beam PS1

Initial camber of the beam is given by: ϕc = (H eB L2 )/8EcIg

= 239000 × 111.7 × 60002

8 × 32093 × 450×106

= 8.3 mm (upwards deflection).



Computation of Initial Prestress Force, H (For One Wire)

AS3600-1994 Clause 6.3.1 -b(ii)Yield strength of tendons, fpy = 0.85 fp.

∴ fpy = 0.85 × 1710 = 1453.5 MPa

Allowable tensile strength in steel wires (Clause 19.3.4.6): = 0.80 fp = 0.80 × 1710 = 1368 MPa

Allowable strain in the wire: =Allowable Stress Young’s Modulus (BHP Tests) = 1368 227000 = 0.00603

Elastic Strain = ∆ L LThus the allowable extension in the wire = Elastic Strain × L

= 0.00603 × 7000 = 42.2 mm

Average slip in wire through the gripping cones (2 mm observed during stretching)∴ Net Extension = 44.2 mm

Strain in wire after releasing jack = 44.2/7000 = 0.00631Stress in wire after loosening jack = 0.00631 × 227000

= 1432.4Initial prestress force in each wire just after transfer = fpi × Ap

= 1432.4 ×19.63/1000 = 28118 kN

5. Example Computation of Moment of Inertias

Ief = Icr + [(I - Icr)(Mcr/Ms)3] ≤ Ig (Gross Moment of Inertia)

Icr = bd3 (4k3 + 12 ρ n (1-k)2)

whereModular Ratio, n = Es/ Ec

Es = Young’s Modulus of Steel = 227 Gpa

Ec = Young’s Modulus of Concrete on Day of Testing = 32093 MPa used for all calculations

Steel Ratio, ρ = Apt/bd

k = √ (ρ n)2 + 2ρ n - ρ n

Ms = Maximum Moment = Mu (Measured Ultimate Moment Capacity of Test Beam)

Mcr = Measured Cracking Moment of Test Beam



9 HS 5 = 177 mm2

Density of Concrete 2328 kg/m3

Beam Dead Load wd = 2328×9.81×200×300/1000= 1.37 kN/m

Moment due to Dead Load, M1 = wL2/8 = 1.37×5.82/8 = 5.86 kNm

η = 0.94C = 0.5 fcm = 0.5 × 60.6 = 30.3 MPaCt = 0.6√fcm = 0.6 √60.6 = 4.67 MPa__________________________________________

CI

yMI

yeHAH TTB ≤+⎥⎦

⎤⎢⎣⎡ − 2η

3.3010450150

104501507.111239000

6000023900094.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M

2. Find Ultimate Moment Capacity of Beam

Satisfy Equation A3:

Satisfy Equation A4:kNmM 6.10472 ≤

tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

67.410450150

104501507.111239000

6000023900094.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 3.502 ≤

11 HS 5 = 216 mm2




η = 0.94C = 0.5 f’cp = 0.5 × 60.6 = 30.3 MPaCt = 0.6√f’c = 0.6 √60.6 = 4.67 MPa__________________________________________

tTTB C

IyM

IyeH

AH

−≥+− 1

67.410450

1501086.510450

1505.11029300060000293000

6

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

3.3010450

1501086.510450

1505.11029300060000293000

6

6

6 ≤×××

−×

××+

1. Ensure Strength at Tendon Transfer

For Top Fibre Stress, Satisfy Equation A1:

For Bottom Fibre Stress, Satisfy Equation A2:

67.496.3 −≥−

3.307.13 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

3.3010450150

104501505.110293000

6000029300094.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

67.410450150

104501505.110293000

6000029300094.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 2.582 ≤

σpu = 1608 MPaku = 0.162γ = 0.65d = 261.7 mm

Mu = 70.5 kNmMcr = 51.6 kNmInitial Camber ϕ = 8.3 mm

Icr = 63.9 × 106 mm4

Ief = 285.9 × 106 mm4

___________________________

σpu = 1608 MPaku = 0.133γ = 0.65d = 260.5 mm


Icr = 75.9 × 106 mm4

Ief = 150.1 × 106 mm4

___________________________

Beam PS2

fcm = 60.6 MPaH = 293.0 kN

200

300 NA

eB =110.5 mm

226

Beam PS1

200

300227

NA

eB =111.7 mm

fcm = 60.6 MPaH = 239.0 kN

tTTB C

IyM

IyeH

AH

−≥+− 1

4.6710450

1501086.510450

150111.72390006

6

6−≥

×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

3.3010450

1501086.510450

1507.11123900060000239000

6

6

6 ≤×××

−×

××+




67.496.2 −≥−

3.309.10 ≤

60000239000



Beam PS3

200

fcm = 60.2 MPaH = 346 kN

300222

NA

eB =97.0 mm

tTTB C

IyM

IyeH

AH

−≥+− 1

66.410450

1501086.510450

1509734600060000

3460006

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

2010450

1501086.510450

1509734600060000

3460006

6

6 ≤×××

−×

××+




66.447.3 −≥−

1.300.15 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

1.3010450150

1045015097346000

6000034600093.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

66.410450150

1045015097346000

6000034600093.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 3.612 ≤

tTTB C

IyM

IyeH

AH

−≥+− 1

66.410450

1501086.510450

1505.8058500060000

5850006

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

1.3010450

1501086.510450

1505.8058500060000

5850006

6

6 ≤×××

−×

××+




66.499.3 −≥−

1.305.23 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

1.3010450150

104501505.80585000

6000058500091.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

66.410450150

104501505.80585000

6000058500091.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 5.832 ≤

22 HS 5 = 432 mm2





σpu = 1608 MPaku = 0.304γ = 0.65d = 230.5 mm


Icr = 98.0 × 106 mm4

Ief = 210 × 106 mm4

___________________________

13 HS 5 = 255 mm2





σpu = 1608 MPaku = 0.191γ = 0.65d = 247.0 mm


Icr = 77.7 × 106 mm4

Ief = 193.1 × 106 mm4

___________________________

Beam PS4

fcm = 60.2 MPaH = 585.0 kN

200

300 NAeB = 80.5 mm

172

4Y12’s = 440 mm2



tTTB C

IyM

IyeH

AH

−≥+− 1

01.510450

1501086.510450

1506061200060000

6120006

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

9.3410450

1501086.510450

1506061200060000

6120006

6

6 ≤×××

−×

××+




01.5614.3 −≥−

9.345.20 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

9.3410450150

1045015060612000

6000061200091.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

01.510450150

1045015060612000

6000061200091.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 3.762 ≤

tTTB C

IyM

IyeH

AH

−≥+− 1

01.510450

1501086.510450

1507.9940000060000400000

6

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

9.3410450

1501086.510450

1507.9940000060000400000

6

6

6 ≤×××

−×

××+




01.567.4 −≥−

9.340.18 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

9.3410450150

104501507.99400000

6000040000093.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

01.510450150

104501507.99400000

6000040000093.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 7.702 ≤

Beam PS6

fcm = 69.8 MPaH = 400.0 kN

200

300 NA

eB = 99.7 mm

207

15 HS 5 = 295 mm2





σpu = 1608 MPaku = 0.246γ = 0.65d = 249.7 mm


Icr = 89.7 × 106 mm4

Ief = 245.2 × 106 mm4

___________________________

22 HS 5 = 452 mm2





σpu = 1608 MPaku = 0.447γ = 0.65d = 210.0 mm


Icr = 85.7 × 106 mm4

Ief = 173.1 × 106 mm4

___________________________

Beam PS5

200

fcm = 69.8 MPaH = 612 kN

300160

NAeB = 60.0 mm



Beam PS7

200

fcm = 52.5 MPaH = 450 kN

300223

NA

eB = 97.5 mm

tTTB C

IyM

IyeH

AH

−≥+− 1

35.410450

1501086.510450

1505.9745000060000450000

6

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

3.2610450

1501086.510450

1505.9745000060000450000

6

6

6 ≤×××

−×

××+




35.417.5 −≥−

3.2617.20 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

3.2610450150

104501505.97450000

6000045000093.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

35.410450150

104501505.97450000

6000045000093.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 8.742 ≤

tTTB C

IyM

IyeH

AH

−≥+− 1

35.410450

1501086.510450

1508040000060000400000

6

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

3.2610450

1501086.510450

1508040000060000400000

6

6

6 ≤×××

−×

××+




35.405.2 −≥−

3.2638.15 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

3.2610450150

1045015080400000

6000040000093.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

35.410450150

1045015080400000

6000040000093.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 4.612 ≤

Beam PS8

fcm = 52.5 MPaH = 400.0 kNe = 80.0 mm

200

300 NA

eB = 80 mm

193

15 HS 5 = 295 mm2





σpu = 1608 MPaku = 0.345γ = 0.67d = 236 mm


Icr = 79.3 × 106 mm4

Ief = 109.9 × 106 mm4

___________________________

13 HS 5 = 295 mm2





σpu = 1608 MPaku = 0.277γ = 0.67d = 247.5 mm


Icr = 78.0 × 106 mm4

Ief = 174.4 × 106 mm4

___________________________



Beam PS9

200

fcm = 83.5 MPaH = 346 kN

300215

NA

eB = 90.0 mm

tTTB C

IyM

IyeH

AH

−≥+− 1

48.510450

1501086.510450

1509034600060000

3460006

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

8.4110450

1501086.510450

1509034600060000

3460006

6

6 ≤×××

−×

××+




48.502.3 −≥−

8.4156.14 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

8.4110450150

1045015090346000

6000034600093.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

48.510450150

1045015090346000

6000034600093.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 5.612 ≤

tTTB C

IyM

IyeH

AH

−≥+− 1

48.510450

1501086.510450

1509048000060000480000

6

6

6 −≥×××

+×

××−

CIyM

IyeH

AH BBB ≤−+ 1

8.4110450

1501086.510450

1509048000060000480000

6

6

6 ≤×××

−×

××+




48.545.4 −≥−

8.415.20 ≤

CI

yMI


⎤⎢⎣⎡ − 2η

8.4110450150

1045015090480000

6000048000092.0 6

26 ≤

××

+⎥⎦⎤

⎢⎣⎡

×××

−M




tBBB C

IyM

IyeH

AH

≥−⎥⎦⎤

⎢⎣⎡ + 2η

48.510450150

1045015090480000

6000048000092.0 6

26 −≥

××

−⎥⎦⎤

⎢⎣⎡

×××

+M

kNmM 3.782 ≤

Beam PS10

fcm = 83.5 MPaH = 480 kN

200

300 NAeB = 90.0 mm

203

18 HS 5 = 353 mm2





σpu = 1608 MPaku = 0.256γ = 0.65d = 240.0 mm


Icr = 95.0 × 106 mm4

Ief = 126.2 × 106 mm4

___________________________

13 HS 5 = 295 mm2





σpu = 1608 MPaku = 0.185γ = 0.65d = 240.0 mm


Icr = 83.6 × 106 mm4

Ief = 144.0 × 106 mm4

___________________________



B.4 Calculations for CS-Series Beams

dc = 32 mmdt = 214 mm

γ = 0.85 - 0.007(f’c - 28)

≈ 0.85

20

150

250

20

214

14

2Y12’s=226 mm2

22.5 MPa400 MPa

CS1 and CS2 and CS3

3N20 = 942 mm2



= 0.01823(pt - pc)lim < pt - pc

∴ Asc will yield_________________________a = (Ast - Asc) fsy

0.85 f’c b = 99.0 mm


Ig = bD3/12 = 150×2503/12 = 195.3 × 106 mm4

32.0 MPa500 MPa

2N24 = 905 mm2

CS4 and CS5 and CS6

dc = 32 mmdt = 213 mm

γ = 0.85 - 0.007(f’c - 28)

= 0.822

213

52

Under-Reinforced (Bending)pt = Ast/bd = 905/150×213 = 0.0283pB = 0.85 f’c γ kuB

fsy

= 0.03354pt < pB

∴ Under-Reinforced_________________________Mu = Astfsyd [1-(0.6×Ast/bd×

fsy/f’c) = 60.5 kNm

Ig = bD3/12 = 150×2503/12 = 195.3 × 106 mm4

ku = pt fsy / 0.85 γ f’c

= 0.630

Shear Capacity

Vuc = β1β2β3 bwd0 Ast f’c

bwd0

= 47.1 kN

3

31.5 MPa500 MPa

2N24 = 905 mm2

2Y12’s

CS7 and CS8 and CS9

20

150

250

20

210

40/32



= 0.04044(pt - pc)lim > pt - pc


= 0.2283υ = 600 pc

0.85 γ f’c

= 0.1894

dc = 32 mmdt = 213 mm

γ = 0.85 - 0.007(f’c - 28)= 0.826


Ig = bD3/12 = 150×2503/12 = 195.3 × 106 mm4


APPENDIX C: Beam Crack Pattern Photographs C-1

APPENDIX C

Beam Crack Pattern Photographs


Appendix D: Logdec Comparative Graphs D-1

APPENDIX D

LOGDEC COMPARATIVE GRAPHS

D.1 Analytical Decay Curve Method Implemented Using Matlab

______________________________________________________________________ home; clear; close; fname = 'beam I-1 00 F.txt'; datfle = load(fname); t = datfle(:,1); v = datfle(:,2); plot(t,v); axis([0 max(t) min(v) max(v)]); Stage 1

Replotting Excel file of original vibration decay waveform detected by oscilloscope.

axis([min(t) max(t) -max(v) max(v)]); title('CONCRETE BEAM DAMPING SIGNAL'); xlabel('Time (s)'); ylabel('Amplitude (mV)'); grid on; disp('PRESS A KEY TO CONTINUE'); pause; % Find when time > 0 sec pos = 1; tmp = 1; while (pos==1)&(tmp<length(t)) if (t(tmp)>0) pos = tmp; end tmp = tmp+1; end if (pos>1) tmp = length(t); Stage 2

Replot Stage 1 plot by removing negative portion (of time) from the waveform record.

t = t(pos:tmp); v = v(pos:tmp); end plot(t,v); axis([0 max(t) min(v) max(v)]); axis([min(t) max(t) -max(v) max(v)]); title('CONCRETE BEAM DAMPING SIGNAL'); xlabel('Time (s)'); ylabel('Amplitude (mV)'); grid on; disp('PRESS A KEY TO CONTINUE'); pause;



% Power Spectrum % Get the Frequency data, and calibrate it, and graph it P = abs(fft(v)); dt = t(2)-t(1); % Time between each sample fmax = 1/dt; % Max freq = 1 / ( time between each sample ) freq = t/max(t) * fmax; % Scale freq from 0 to fmax Stage 3

Replotting Excel file of original vibration decay waveform detected by oscilloscope.

bar(freq,P); axis([0 max(freq)/2 0 max(P)]); % Graph is mirrored so only wanthalf grid on; title('SPECTRUM OF COMMUNCATION SIGNAL'); xlabel('Frequency (Hz)'); ylabel('Amplitude (mV)'); pause; % Choose freq between 800 - 1200 Hz tmp1 = 1; while (freq(tmp1)<800) % 800 Hz tmp1 = tmp1 + 1; end; tmp2 = 1; while (freq(tmp2)<1200) % 1200 Hz tmp2 = tmp2 + 1; end; P = P(tmp1:tmp2); freq = freq(tmp1:tmp2); % Show the new Freq range of data bar(freq,P); axis([min(freq) max(freq) 0 max(P)]); % Graph is mirrored so only wanthalf grid on; title('SPECTRUM OF COMMUNCATION SIGNAL'); xlabel('Frequency (Hz)'); ylabel('Amplitude (mV)'); % find the frequency of the peak position = min(find(P==max(P))); % The Position of the first peak of the fft graph

Stage 4 Undertaking the FFT.

Stage 4 As above

fpeak = freq(position); % The frequency of the peak of the fft graph disp(['The frequency peaks at: ' num2str(fpeak) ' Hz']); text(fpeak,max(P)/1.1,['Peak = ' num2str(fpeak) 'Hz']); disp('PRESS A KEY TO CONTINUE'); pause; % Remove negative parts of amplitude - time graph v = abs(v); % Get the peaks of the wave & Take the natural log num = 1; for tmp = 2:size(t) if ((v(tmp-1)<v(tmp))&(v(tmp-1)<v(tmp))&(v(tmp)>0)) tn(num) = t(tmp); vn(num) = log(v(tmp)); num = num + 1; Stage 5

Plotting the natural logarithm of each of the peaks (with respect to amplitude).

end end % Plot Log Graph plot(tn,vn,'o'); %axis([0 0.05 0 max(vn)]); title('LOG GRAPH OF DAMPING'); xlabel('Time (s)'); ylabel('ln(Amplitude)'); hold on; % Fit Line p = polyfit(tn,vn,1); lnV = p(2)+p(1)*tn; plot(tn,lnV,'-r'); text(max(tn)/2,max(lnV)-0.1,['Slope = ' num2str(p(1))]); Stage 6

Calculating slope, and logdec.

% Display Answer disp([ 'Intercept = ' num2str(p(2)) ]); disp([ 'Slope = ' num2str(p(1)) ]); w = 2*pi*fpeak; disp([ 'w = ' num2str(w) ]); s = -p(1)/w; disp([ 'Decay - s = ' num2str(s) ]); disp([ 'Log Dec = ' num2str(s*2*pi) ]);



1 2 3 4 5 6 7 8 9

x 10- 3

-0. 1

-0. 05

0

0. 05

0. 1

CONCRETE BEAM DAMP ING S IGNAL

Time (s )

Am

plitu

de

(mV

)

500 1 000 1 500 2 0000

0.2

0.4

0.6

0.8

1

1.2

SP ECTRUM O F CO MMUNCATION S IGNAL

F re quency (Hz)

Am

plitu

de (m

V)

Pea k = 819. 64 52Hz

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0. 01-3.6

-3.4

-3.2

-3

-2.8

-2.6

-2.4

-2.2

-2

-1.8LOG GRAP H O F DAMP IN G

Time (s )

ln(A

mpl

itude

)

S lope = -98 .5 093

The frequency peaks at: 820 HzIntercept = -2.2Slope = -98.5w = 5150.0Decay - s = 0.019Log Dec = 0.120

0. 005 0.01 0. 015 0.02 0. 025 0.03 0. 035

-0.1

-0.05

0

0.05

0.1

CO NCRETE BEAM DAMPING S IG NAL

Time (s)

Am

plitu

de (m

V)

0 500 1 000 1 500 2 000 2 5000

0.5

1

1.5

2

2.5

S P ECTRUM OF CO MMUNCATION S IG NAL

F re quency (Hz)

Am

plitu

de (m

V)

0 0. 005 0.01 0. 015 0.02 0. 025 0.03 0. 035 0.04-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5LOG GRAP H O F DAMP IN G

Time (s)

ln(A

mpl

itude

)

S lope = -61 .6 302

The frequency peaks at: 780 HzIntercept = -2.4Slope = -61.6w = 4899.1Decay - s = 0.013Log Dec = 0.079

CS6-200-01CS6-50-01

a) e)

b) f)

c)

d) h)

g)

Figure D.1: Diagrammatic Flowchart of the DCM: a) and e) Original Signal for 50 and

200 NDP (see Section 5.4); b) and f) FFT of Time Spectrum; c) and g) Checking logeAn

versus n; d) and h) DCM Algorithm Extracting Logdec



D.2 Interrelationship between Logdec (TLT) and Cycle Number, n

a) Cycle Number (n)

Logd

ec(T

LT)

0 25 50 75 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18SB1SB2

b)

11

1

1 11

2

2

2

2 22

33

3

3 3 3

4

4

4

4 4 4

5

5

5 55 5

66

6

6

66

77

7

7 7

7

8

8

8

8 8 8

9

9

99

9 9

Cycle Number (n)

Logd

ec(T

LT)

0 25 50 75 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22CS1CS2CS3CS4CS5CS6CS7CS8CS9

123456789

c)

A

A

A

AAA

B

B

BB

B B

C

C C

CC C

D

D

DD

D D

E

E

E

E E E

F

F

F

F F F

G

G

G G G G

H

H

H H H H

I

II

I II

J

J

JJ

JJ

K

K

K

KK K

L

L

L

LL L

Cycle Number (n)

Logd

ec(T

LT)

0 25 50 75 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12

ABCDEFGHIJKL

d)

c

c

c

cc

c

d

d

dd

dd

e

e

e

ee e

f

f

f

ff f

g

g

g

g

g g

hh

h

hh h

i i

i

ii

i

j

j

j

j

jj

Cycle Number (n)

Logd

ec(T

LT)

0 25 50 75 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16PS3PS4PS5PS6PS7PS8PS9PS10

cdefghij

Figure D.2: Logdec (TLT) versus Cycle Number (n) for: a) S-Series; b) CS-Series; c) B-

Series; d) PS-Series Beams



D.3 Interrelationship between Logdec (DCM) and Number of Data Points

(NDP)

a)

A

AA

A

AA

B

B

B

BB B

C

CC

CC C

D

D D D

D D

E

EE

EE

E

FF F

F FFG

G

G

GG

G

H

HH

H HH

I

I II I I

J

J

JJ J J

K

K K

K K K

L

L L L L L


Logd

ec(D

CM

)

0 100 200 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


ABCDEFGHIJKL

b)

1

11

11

1

2

22

2 2

2

3

3

33 3

3

4

44

44

4

5

5

5 5 55

66

66

66

7 77

77 7

8

8

8

8 88

9

9

9

99 9

Number of Data Points (NDP)Lo

gdec

(DC

M)

0 100 200 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24 CS1CS2CS3CS4CS5CS6CS7CS8CS9

123456789

c)

A

AA

A

AA

B

B

B

BB B

C

CC

CC C

D

D D D

D D

E

EE

EE

E

FF F

F FFG

G

G

GG

G

H

HH

H HH

I

I II I I

J

J

JJ J J

K

K K

K K K

L

L L L L L


Logd

ec(D

CM

)

0 100 200 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


ABCDEFGHIJKL

d)

c

c

c cc c

d

d d d d d

e

e ee

ee

f

ff

ff

f

g

g

g

g

gg

h

hh

hh h

i

i

i

i

i i

j

j jj j

j


Logd

ec(D

CM

)

0 100 200 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14


cdefghij

Figure D.3: Logdec (DCM) versus Number of Data Points (NDP) for: a) S-Series; b)

CS-Series; c) B-Series; d) PS-Series Beams



a) Number of Data Points (NDP)

Free

-Vib

ratio

nFr

eque

ncy

(Hz)

0 100 200 3000

200

400

600

800

1000

1200

1400

1600

1800

2000SB1SB2

b)

11 1 1 1 1

2 2 2 2 2 23 3 3

3 3 34 4 4 4 4 4

55 5 5 5 5

66 6 6 6 688

8 8 8 89 9 9 9 9 9


Free

-Vib

ratio

nFr

eque

ncy

(Hz)

0 100 200 3000

250

500

750

1000

1250

1500

1750

2000CS1CS2CS3CS4CS5CS6CS8CS9

12345689

c)

A AA A

A A

B B B B B

BC

C C C C C

D

D D D D D

EE

E

E E E

F F F F F F

G

G G

G

G GH HH H H H

I II I I I

J J

J J JJ

K K K K KK

LL L L L L


Free

-Vib

ratio

nFr

eque

ncy

(Hz)

0 100 200 3000

250

500

750

1000

1250

1500

1750

2000

2250

2500

2750

3000BI-1BII-2BI-3BII-4BII-5BII-6BI-7BII-8BI-9BII-10BII-11BII-12

ABCDEFGHIJKL

d)

c

c c c c c

d

dd

d d d

ee

e e e ef

f

f f f f

gg g

g g g

h

h hh h hi

i

i

i i

i

jj j

j jj


Free

-Vib

ratio

nFr

eque

ncy

(Hz)

0 100 200 3000

250

500

750

1000

1250

1500

1750

2000

2250

2500PS3PS4PS5PS6PS7PS8PS9PS10

cdefghij

Figure D.4: Frequency (Hz) versus Number of Data Points (NDP) for: a) S-Series; b)

CS-Series; c) B-Series; d) PS-Series Beams



D.4 Correlation between Logdec (TLT) and Logdec (DCM) Techniques

a) Logdec (DCM)

Logd

ec(T

LT)

0 0.02 0.04 0.06 0.08 0.1 0.120

0.02

0.04

0.06

0.08

0.1

0.12

SB1

b) Logdec (DCM)

Logd

ec(T

LT)

0 0.02 0.04 0.06 0.08 0.1 0.120

0.02

0.04

0.06

0.08

0.1

0.12

SB2

Figure D.5: Logdec (TLT) versus Logdec (DCM) for: a) SB1 and b) SB2

a)

1 11

111

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS11

b)

2

2

2

222

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS22

c)

33

3333

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS33

d)

4

4

4

444

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS44

Figure D.6: Logdec (TLT) versus Logdec (DCM) for: a) CS1, b) CS2, c) CS3, d) CS4,

e) CS5, f) CS6, g) CS7, h) CS8, and i) CS9 (Continued Overleaf)



e)

5

5

5555

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS55

f)

66

6

66

6

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS66

g)

77

7

777

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS77

h)

8

8

8

888

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS88

i)

9

9

99

99

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

CS99

Figure D.6: Logdec (TLT) versus Logdec (DCM) for: a) CS1, b) CS2, c) CS3, d) CS4,

e) CS5, f) CS6, g) CS7, h) CS8, and i) CS9



a)

AA

AA

AA

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BI-1A

b)

B

B

BBBB

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-2B

c)

C

C CC

CC

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BI-3C

d)

D

D

DDDD

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-4D

e)

E

E

EEEE

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-5E

f)

F

F

FFFF

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-6F

Figure D.7: Logdec (TLT) versus Logdec (DCM) for: a) BI-1, b) BII-2, c) BI-3, d) BII-

4, e) BII-5, f) BII-6, g) BI-7, h) BII-8, i) BI-9, j) BII-10, k) BII-11, and l) BII-12

(Continued Overleaf)



g)

G

G

GG GG

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BI-7G

h)

H

H

HHH H

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-8H

i)

I

II

III

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BI-9I

j)

J

J

JJ

JJ

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-10J

k)

K

KK

KKK

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-11K

l)

L

L

LLLL

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

BII-12L

Figure D.7: Logdec (TLT) versus Logdec (DCM) for: a) BI-1, b) BII-2, c) BI-3, d) BII-

4, e) BII-5, f) BII-6, g) BI-7, h) BII-8, i) BI-9, j) BII-10, k) BII-11, and l) BII-12



a)

c

c

cccc

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS3c

b)

d

ddddd

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS4d

c)

e

e

ee

ee

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS5e

d)

ff

f

fff

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS6f

e)

g

g

gg

gg

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS7g

f)

hh

hhhh

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS8h

Figure D.8: Logdec (TLT) versus Logdec (DCM) for: a) PS3, b) PS4, c) PS5, d) PS6, e)

PS7, f) PS8, g) PS9, and h) PS10 (Continued Overleaf)



g)

ii

iiii

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS9i

h)

j

j

jjj

j

Logdec (DCM)

Logd

ec(T

LT)

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

PS10j

Figure D.8: Logdec (TLT) versus Logdec (DCM) for: a) PS3, b) PS4, c) PS5, d) PS6, e)

PS7, f) PS8, g) PS9, and h) PS10



D.5 ‘Optimal Peak Ratio’ Curves for All Beams

Peak Ratio An/A1 (%)

Logd

ec(T

LT)

01020304050600

0.025

0.05

0.075

0.1

0.125

0.15

0.175SB1SB2

DCM 300NDP

DCM 50NDP

DCM 100NDP

DCM 150NDPDCM 200NDP

DCM 250NDP

Figure D.9: Optimal Peak Ratio (An/A1 %) Curves for All S-Series Beams

A

A

A

AA

B

B

B

BB

C

C

C CC

D

D

DD

D

E

E

E

E E

F

F

F

FF

G

G

G G G

H

H

H HH

I

II I I

J

J

J J

J

K

K

KKK

L L

L

LL

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14


ABCDEFGHIJKL

17%> An/A1>10%

Figure D.10: Optimal Peak Ratio (An/A1 %) Curves for All B-Series Beams



a)

A

A

A

AA

B

B

B

BB

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

BI-1BII-2

AB

17%> An/A1>10%

BII-2TLT (150)

BI-1TLT (150)

b)

C

C

C CC

D

D

DD

D

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

BI-3BII-4

CD

17%> An/A1>10%

BI-3TLT (150)

BII-4TLT (200)

c)

E

E

E

E E

F

F

F

FF

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16BII-5BII-6

EF

17%> An/A1>10%

BII-5TLT (150)

BII-6TLT (150)

d)

G

G

G G G

H

H

H HH

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16BI-7BII-8

GH

17%> An/A1>10%BI-7

TLT (200)

BII-8TLT (200)



e)

I

II I I

J

JJ J

J

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

BI-9BII-10

IJ

17%> An/A1>10%

BII-10TLT (150)

BI-9TLT (150)

f)

K

K

KKK

L L

L

LL

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

BII-11BII-12

KL

17%> An/A1>10%BII-12

TLT (150)

BII-11TLT (150)

Figure D.11: Optimal Peak Ratio (An/A1 %) Curves for: a) BI-1 and BII-2, b) BI-3 and

BII-4, c) BII-5 and BII-6, d) BI-7 and BII-8, e) BI-9 and BII-10, f) BII-11 and BII-12

1 11

1 1

2

2

2

22

33

33

3

4

4

4

4 4

5

5

5

5 5

6

6

6

6 6

7

7

7

7

8

8

8

88

9

9

99

9 9

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

05101520253035404550550

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26CS1CS2CS3CS4CS5CS6CS7CS8CS9

123456789

17%> An/A1>8%

Figure D.12: Optimal Peak Ratio (An/A1 %) Curves for All CS-Series Beams



a)

1 11

1 1

2

2

2

22

33

33

3

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

05101520253035404550550

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26CS1CS2CS3

123

17%> An/A1>8%

CS2TLT (200)

CS1TLT (200)

CS3TLT (200)

b)

4

4

4

4 4

5

5

5

5 5

6

6

6

6 6

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

05101520253035404550550

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26CS4CS5CS6

456

17%> An/A1>8%

CS4TLT (200)

CS5TLT (200)

CS6TLT (200)

Figure D.13: Optimal Peak Ratio (An/A1 %) Curves for: a) CS1, CS2 and CS3, b) CS4,

CS5 and CS6, c) CS7, CS8 and CS9 (Continued Overleaf)

c)

7

7

7

7

8

8

8

88

9

9

99

9 9

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

05101520253035404550550

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26CS7CS8CS9

789

17%> An/A1>8%

CS8TLT (200)

CS9TLT (250)

CS7TLT (200)

Figure D.13: Optimal Peak Ratio (An/A1 %) Curves for: a) CS1, CS2 and CS3, b) CS4,

CS5 and CS6, c) CS7, CS8 and CS9



a

a

a

a a

b

b

bb

b

c

c

c

cc

d

d

d

dd

e

e

e

ee

ff

f

f f

g g

g

g g

h

h

h

h

h

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14


abcdefgh

20%> An/A1>10%

Figure D.14: Optimal Peak Ratio (An/A1 %) Curves for All PS-Series Beams

a)

a

a

a

a a

b

b

bb

b

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16PS3PS4

ab

20%> An/A1>10%

PS3TLT (200)

PS4TLT (200)

b)

c

c

c

cc

d

d

d

dd

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16PS5PS6

cd

PS6TLT (200)

PS5TLT (200)

20%> An/A1>10%



c)

e

e

e

ee

ff

f

f f

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16PS7PS8

ef

PS8TLT (200)

PS7TLT (200)

20%> An/A1>10%

d)

g g

g

g g

h

h

h

h

h

An/A1 (%)

'Unt

este

d'Lo

garit

hmic

Dec

rem

ent(

TLT)

051015202530354045500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16PS9PS10

gh

PS9TLT (200)

PS10TLT (200)

20%> An/A1>10%

Figure D.15: Optimal Peak Ratio (An/A1 %) Curves for: a) PS3 and PS4, b) PS5 and PS6,

c) PS7 and PS8, d) PS9 and PS10


APPENDIX E: Damping Tabulations E-1

APPENDIX E

Damping Tabulations


Table E.1. Optimal Peak Ratio Damping Tabulations for Beam BI-1

BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#

Calc. Logdec (TLT) 100#

Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*

Calc. Logdec (TLT) 150*

Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1024 0.0600 0.0953 1012 0.0843 0.0821 975 0.0776 0.0648 981 0.0554 0.0512 714 0.0727 0.044918.6 0.16 2.38 0.005 1120 0.0778 0.0828 975 0.0649 0.0661 28.3 0.27 3.42 0.02 1025 0.0987 0.1812 969 0.0752 0.0923 980 0.0592 0.0677 956 0.0532 0.0515 945 0.0453 0.052437.4 0.37 4.24 0.022 969 0.0702 0.0847 969 0.0605 0.0696 46.6 0.42 5.05 0.025 1045 0.0944 0.1537 968 0.0613 0.0827 946 0.0542 0.0729 961 0.0496 0.0575 949 0.0479 0.053665.3 0.57 6.6 0.03 1047 0.0996 0.1643 962 0.0559 0.0882 941 0.0542 0.0724 937 0.0469 0.0598 949 0.0451 0.051074.6 0.62 7.37 0.03 969 0.0597 0.0893 946 0.0523 0.0714 83.9 0.70 8.51 0.032 1068 0.0731 0.1237 975 0.0575 0.0807 950 0.0508 0.0712 942 0.0477 0.0590 933 0.0452 0.050893.2 1.26 9.61 0.034 961 0.0555 0.0956 940 0.0513 0.0740

102.5 1.93 10.81 0.041 1023 0.0669 0.1398 1097 0.0417 0.0931 915 0.0464 0.0790 905 0.0504 0.0562 904 0.0482 0.0485Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 113.0 kNm. Date beam tested: 17 May 2000 Age of beam at testing: 28 days

Table E.2. Optimal Peak Ratio Damping Tabulations for Beam BII-2

BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1584 0.083 0.0916 1550 0.047 0.0702 1533 0.0339 0.0548 1483 0.0295 0.0466 1477 0.0247 0.042618.6 0.55 2.87 0.016 1432 0.0452 0.0952 1198 0.0519 0.0711 28.1 0.71 4.13 0.019 1038 0.1304 0.1156 969 0.0807 0.0992 945 0.0613 0.0739 931 0.0693 0.0599 935 0.0626 0.057437.2 0.84 5.43 0.025 976 0.0548 0.0946 951 0.0618 0.0685 47.1 0.94 6.44 0.026 1023 0.119 0.0965 962 0.0578 0.0917 941 0.071 0.0748 930 0.0667 0.0665 831 0.0642 0.064656.0 1.02 7.02 0.027 960 0.0543 0.0855 940 0.0584 0.068 65.3 1.07 7.73 0.028 1022 0.1072 0.0928 961 0.0698 0.0842 940 0.0731 0.0752 930 0.0645 0.0669 924 0.0602 0.062974.6 1.13 8.55 0.03 962 0.0637 0.0842 941 0.0695 0.0688 83.9 1.20 9.60 0.031 1023 0.0974 0.1256 961 0.0626 0.1057 941 0.079 0.089 930 0.0647 0.0786 830 0.0607 0.069993.2 1.28 10.95 0.034 961 0.0591 0.1107 941 0.0568 0.0735 98.0 2.70 13.0 0.041 1025 0.0684 0.1315 962 0.0509 0.0798 908 0.058 0.0699 906 0.0505 0.0576 904 0.0519 0.0514

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 149.6 kNm. Date beam tested: 17 May 2000 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations2

Dam

ping Charateristics of Reinforced and Prestressed N

ormal- and H

igh-Strength Concrete Beam

s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -# 0 0 1038 0.0695 0.1563 922 0.054 0.0695 922 0.0631 0.0585 911 0.0471 0.059 894 0.0425 0.051632.6 -# 1.88 0.005 918 0.0449 0.0731 914 0.0502 0.0583 51.3 -# 3.45 0.014 1022 0.0828 0.1247 909 0.0345 0.0726 906 0.0441 0.0623 886 0.0494 0.0518 984 0.0517 0.04969.9 -# 5.43 0.02 1553 0.0845 0.1311 922 0.0541 0.0757 898 0.0565 0.0642 886 0.0604 0.0619 889 0.0546 0.048788.5 -# 7.6 0.026 1075 0.045 0.1312 912 0.0503 0.0807 875 0.0504 0.0627 881 0.0513 0.0527 865 0.0505 0.049197.9 -# 9.44 0.030 919 0.0523 0.0800 879 0.0516 0.0721

107.2 -# 11.91 0.038 1022 0.0724 0.1412 960 0.0578 0.0978 962 0.0454 0.0789 855 0.0491 0.0609 844 0.0481 0.0593116.5 -# 16.65 0.045 1097 0.0575 0.1281 1150 0.0553 0.0923 935 0.0555 0.0693 836 0.056 0.061 833 0.0509 0.0545

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 116.5 kNm. -# Strain gauge was faulty on this beam Date beam tested: 14 July 2000 Age of beam at testing: 30 days


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1178 0.0273 0.1544 912 0.0535 0.0740 908 0.0504 0.0574 906 0.0484 0.0500 905 0.0428 0.045632.6 - 4.87 0.006 908 0.0545 0.0679 891 0.0593 0.062 51.3 0.77 7.01 0.011 1126 0.0686 0.1407 911 0.0625 0.0714 874 0.0634 0.0736 881 0.0622 0.0635 864 0.0572 0.05769.9 0.92 9.35 0.013 995 0.057 0.0772 873 0.057 0.0677 83.9 0.95 11.21 0.018 1125 0.0594 0.1021 1163 0.0563 0.0845 874 0.0615 0.0745 855 0.0588 0.0599 844 0.0519 0.05391.3 2.51 16.8 0.03 938 0.0676 0.0919 872 0.0623 0.0822 848 0.0497 0.0778 837 0.0489 0.0610 827 0.0484 0.047498.4 6.3 22.13 0.046 1047 0.0361 0.1420 973 0.0433 0.0950 949 0.0447 0.0824 811 0.0517 0.0643 939 0.0441 0.059398.8 10.84 28.86 0.075 1241 0.0419 0.1171 901 0.0778 0.0923 807 0.0653 0.0664 805 0.0604 0.0749 784 0.0555 0.0583

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 98.8 kNm. Date beam tested: 14 July 2000 Age of beam at testing: 30 days

Appendix E: Dam

ping Tabulations3

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1149 0.0231 0.1285 1074 0.0499 0.1052 1419 0.0463 0.0573 1062 0.0533 0.0447 1050 0.0432 0.045146.6 0.20 4.33 0.005 1241 0.0313 0.1152 1387 0.0786 0.1018 1369 0.0665 0.0865 1360 0.0566 0.0577 1469 0.0401 0.055574.6 0.21 6.5 0.005 1330 0.0672 0.156 1264 0.0782 0.1075 1175 0.0761 0.0892 1223 0.0584 0.0581 1219 0.044 0.054993.2 0.22 8.05 0.01 1241 0.0591 0.1354 1254 0.0743 0.0886 1183 0.0789 0.0756 1118 0.0605 0.0565 1160 0.0478 0.0503

111.8 0.80 10.41 0.02 936 0.0743 0.1129 1069 0.0589 0.079 1057 0.063 0.0606 848 0.0655 0.0618 839 0.0541 0.0501121.1 0.85 15.70 0.037 1090 0.06 0.1301 867 0.1001 0.1007 844 0.0815 0.0789 997 0.0663 0.0611 991 0.0549 0.0572128.6 - 35.60 0.044 0.1315 0.0943

Failure Mode: Beam failed in compression (crushing of concrete) at a Bending Moment (BM) of 128.6 kNm. Date beam tested: 24 July 2000 Age of beam at testing: 28 days


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1229 0.0488 0.1188 1181 0.0604 0.0929 1154 0.0523 0.0534 1149 0.043 0.0419 1139 0.0402 0.039737.2 0.60 4.85 - 1130 0.0479 0.0809 1120 0.0438 0.0703 56.0 0.60 6.5 0.005 1545 0.0451 0.0877 1137 0.0461 0.0638 1125 0.0423 0.0618 1110 0.0448 0.0462 1101 0.045 0.045674.6 0.60 8.03 0.01 1175 0.059 0.0964 1137 0.053 0.0761 1125 0.0557 0.0658 1102 0.0519 0.0485 987 0.0531 0.046593.2 0.60 10.15 0.01 1159 0.0546 0.0944 1112 0.0595 0.0783 1086 0.0631 0.065 1089 0.0597 0.0541 1199 0.046 0.0536

102.5 1.00 14.36 0.01 1159 0.0674 0.1286 1112 0.0627 0.0861 1086 0.0573 0.0697 1089 0.0538 0.0567 1078 0.0481 0.0551120.4 11.0 36.24 0.02 1045 0.0825 0.1011 1073 0.0506 0.0788 1065 0.0505 0.0746 1061 0.0571 0.0564 909 0.055 0.0498

Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete) at a Bending Moment (BM) of 120.4 kNm. Date beam tested: 24 July 2000 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations4

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1545 0.0277 0.1122 1119 0.0328 0.0490 1113 0.0356 0.0364 1100 0.0227 0.0361 1088 0.0298 0.034615.8 1.55 6.31 0.051 923 0.0509 0.0917 1076 0.0495 0.0505 1050 0.0446 0.0396 813 0.0398 0.055 704 0.0439 0.047220.5 1.74 8.3 0.053 1067 0.0472 0.0401 1044 0.0373 0.0324 25.2 1.89 9.05 0.055 1139 0.0842 0.1402 1012 0.0525 0.0603 1008 0.0457 0.0458 1009 0.0371 0.0541 1007 0.0367 0.045528.0 1.94 9.65 0.057 1035 0.0919 0.104 1017 0.0703 0.0537 1011 0.0527 0.0512 1009 0.0407 0.0465 987 0.039 0.043332.6 2.05 10.30 0.063 1024 0.0846 0.1152 1011 0.0584 0.0638 1008 0.0516 0.0598 991 0.0415 0.0503 985 0.0406 0.048635.4 4.40 11.48 0.065 937 0.0629 0.1153 1011 0.075 0.0579 974 0.0655 0.0501 925 0.046 0.0652 867 0.05 0.0543



BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1128 0.0475 0.1043 1170 0.0288 0.0593 1091 0.0337 0.0401 1119 0.0316 0.0388 1106 0.0311 0.035615.8 1.50 6.43 0.050 1068 0.0768 0.0765 1062 0.0796 0.0691 0.0589 18.8 1.60 7.36 0.053 1140 0.0686 0.1357 1069 0.0415 0.0637 1046 0.038 0.0625 1035 0.0366 0.0489 1028 0.0366 0.043221.4 1.72 8.26 0.056 1069 0.0426 0.0651 1024 0.0412 0.0536 0.0603 24.4 1.80 9.01 0.060 1140 0.045 0.1411 1069 0.0414 0.0632 912 0.0399 0.0533 1010 0.0373 0.0489 1008 0.0368 0.046328.0 1.86 9.73 0.069 1125 0.0547 0.1451 910 0.0436 0.0577 1008 0.0358 0.0554 1006 0.0382 0.0550 1005 0.037 0.045832.6 1.94 10.4 0.069 1126 0.0667 0.1391 1011 0.0441 0.0624 1008 0.0344 0.0567 1006 0.0343 0.0577 1005 0.0359 0.048937.3 2.05 10.9 0.070 1140 0.0738 0.1435 1019 0.0436 0.0569 1013 0.0414 0.0548 1010 0.0378 0.0635 994 0.0378 0.0498


Appendix E: Dam

ping Tabulations5

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1294 0.0973 0.1195 1255 0.0354 0.0502 1111 0.0426 0.0444 1107 0.0389 0.0439 1095 0.0370 0.042128.0 0.91 6.13 0.040 1278 0.0972 0.141 1239 0.0639 0.0622 1041 0.0561 0.0571 1220 0.041 0.0522 1216 0.0373 0.048442.0 1.07 7.5 0.045 1279 0.0841 0.1323 1239 0.0607 0.0628 1209 0.0528 0.0480 1207 0.0493 0.0563 1196 0.0437 0.05 56.0 1.20 8.59 0.050 1370 0.0812 0.1188 983 0.0749 0.0931 1267 0.0461 0.0609 942 0.0634 0.0609 940 0.0608 0.052369.9 1.27 9.34 0.060 1715 0.0817 0.1607 1136 0.0848 0.0923 1135 0.0616 0.0697 1127 0.0544 0.068 1128 0.0485 0.055976.4 13.15 15.13 0.070 922 0.1397 0.1889 1011 0.1042 0.0876 873 0.0832 0.0869 679 0.0919 0.0809 677 0.0741 0.0757



BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1410 0.0879 0.1031 1469 0.0261 0.0536 1240 0.0337 0.0457 1245 0.0285 0.0442 1256 0.0281 0.034718.6 0.79 5.14 0.052 948 0.0733 0.0817 928 0.0629 0.0654 27.9 1.00 7.33 0.058 1199 0.0656 0.0742 1031 0.0618 0.0577 37.2 1.12 7.97 0.058 1330 0.0236 0.109 1483 0.0461 0.0773 1231 0.0454 0.0532 1458 0.0343 0.0454 1219 0.0367 0.041946.6 1.19 8.57 0.067 1487 0.0368 0.0731 1234 0.0435 0.0534 56.0 1.27 9.65 0.071 1328 0.0407 0.0906 1484 0.0424 0.0771 1232 0.0473 0.0591 1223 0.044 0.0474 1446 0.0315 0.041765.3 1.34 10.61 0.079 1585 0.0224 0.0865 1295 0.0467 0.0789 1261 0.0426 0.0617 1169 0.0418 0.0512 1156 0.0379 0.044174.6 1.42 11.35 0.088 1585 0.0254 0.1061 1025 0.0666 0.0808 1158 0.0487 0.0633 1509 0.0301 0.0511 1156 0.0407 0.046783.9 1.62 13.69 0.091 1532 0.0281 0.1194 1146 0.0606 0.0841 1197 0.0519 0.0628 1144 0.0429 0.0517 1486 0.0282 0.039993.2 2.35 17.55 0.108 1026 0.0668 0.1449 1022 0.0809 0.0865 853 0.0686 0.0761 956 0.051 0.0579 965 0.0424 0.049999.8 2.76 25.90 0.196 1024 0.0776 0.1368 840 0.0696 0.0853 819 0.0518 0.0786 939 0.0496 0.0725 830 0.0477 0.0597


Appendix E: Dam

ping Tabulations6

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1021 0.0702 0.1182 1049 0.0481 0.0931 1025 0.0521 0.0518 1012 0.037 0.0489 999 0.0343 0.044223.3 0.74 4.83 0.027 990 0.0664 0.0943 993 0.0584 0.0794 37.2 1.16 6.19 0.03 885 0.1931 0.1165 943 0.0795 0.1232 928 0.063 0.0795 804 0.0709 0.0604 904 0.0505 0.050351.3 1.26 6.94 0.037 987 0.0748 0.1214 891 0.0758 0.081 65.3 1.37 7.93 0.047 1342 0.0865 0.1261 905 0.093 0.0993 811 0.0927 0.0798 793 0.075 0.0654 560 0.0842 0.057179.2 1.41 8.99 0.050 872 0.1371 0.1286 905 0.0974 0.1134 1164 0.0751 0.0808 1027 0.066 0.062 1022 0.0586 0.058193.2 1.51 9.73 0.055 1163 0.0947 0.1016 1142 0.0771 0.0851

107.2 2.41 13.29 0.085 952 0.1281 0.1427 792 0.1247 0.1037 761 0.0868 0.0816 997 0.0616 0.0603 744 0.0673 0.061115.4 12.38 23.94 0.126 836 0.0864 0.1198 987 0.0618 0.0934 970 0.062 0.0891 951 0.0571 0.0625 948 0.0518 0.0541



BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 1126 0.0578 0.0949 1037 0.0329 0.0954 1025 0.0338 0.0566 1018 0.0322 0.0434 1015 0.0292 0.037337.2 0.91 4.18 0.018 1103 0.0431 0.0904 1002 0.042 0.0686 56.0 1.11 5.29 0.024 1257 0.0551 0.1336 1011 0.0537 0.0746 999 0.0374 0.0628 687 0.0468 0.051 1044 0.0326 0.046569.9 1.20 6.13 0.025 1008 0.0455 0.0824 1265 0.0302 0.0635 83.9 1.26 6.44 0.028 1237 0.0238 0.1376 1004 0.0448 0.0722 1061 0.0404 0.0646 1035 0.0384 0.0434 1148 0.0297 0.04197.9 1.31 7.00 0.031 1090 0.0449 0.1111 940 0.0455 0.0879 1027 0.0341 0.0747 1031 0.0366 0.0538 1018 0.0351 0.0489

111.8 1.35 7.52 0.032 1107 0.096 0.1101 935 0.0473 0.0775 901 0.0454 0.0759 901 0.038 0.0557 887 0.0355 0.0491125.8 1.38 6.94 0.035 971 0.0613 0.1467 935 0.0554 0.0703 901 0.0419 0.0732 901 0.0386 0.0561 767 0.0383 0.0494139.8 1.64 7.46 0.038 970 0.0378 0.1689 904 0.0414 0.0881 869 0.0362 0.0773 901 0.0339 0.0581 767 0.0361 0.0565177.7 - 12.35 0.175 937 0.038 0.1786 841 0.0409 0.0828 743 0.0661 0.0804 817 0.0584 0.0676 814 0.0525 0.064


Appendix E: Dam

ping Tabulations7

Dam


ormal- and H


s

Table E.13. Optimal Peak Ratio Damping Tabulations for Beam CS1

BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 819 0.0765 0.0979 759 0.1136 0.0931 740 0.1068 0.0815 730 0.0846 0.0700 724 0.0734 0.068075.9 -% 4.87 0.038 720 0.2233 0.1985 660 0.1554 0.1896 640 0.1091 0.1283 630 0.0876 0.0915 644 0.0828 0.0910

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 75.9 kNm. % No strain gauges attached. Date beam tested: 18 September 2000 Age of beam at testing: 35 days


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 822 0.1548 0.2122 811 0.0829 0.1234 774 0.0890 0.0848 780 0.0651 0.0693 784 0.0678 0.061172.2 -% 5.23 0.042 1329 0.0960 0.1590 608 0.1317 0.1282 605 0.1241 0.1226 604 0.1169 0.0820 583 0.0909 0.0718



BM(kNm)

εres (mV) ∆res

(mm) Wres

(mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 823 0.1115 0.0981 811 0.0551 0.0727 808 0.0360 0.0590 755 0.0248 0.0478 744 0.0204 0.036759.0 -% 3.89 0.028 633 0.2858 0.1212 616 0.1131 0.1168 588 0.1192 0.0976 583 0.0982 0.0883 587 0.0842 0.0753


Appendix E: Dam

ping Tabulations8

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 819 0.1202 0.1716 810 0.0864 0.1160 806 0.0720 0.0875 780 0.0790 0.0495 784 0.0724 0.047325.8 -% 0.63 0 818 0.1563 0.1506 817 0.1292 0.1343 812 0.0863 0.1018 779 0.0698 0.0691 783 0.0668 0.065936.1 -% 0.86 0 820 0.0994 0.1828 817 0.1565 0.1449 778 0.1095 0.1165 780 0.0593 0.0769 744 0.0515 0.065143.8 -% 1.18 0.01 866 0.1002 0.2267 780 0.1805 0.1702 781 0.1307 0.1263 791 0.0704 0.0880 753 0.0506 0.074352.0 -% 1.89 - 922 0.2093 0.1675 870 0.1722 0.2279 846 0.1352 0.1537 830 0.0968 0.1007 -& -& -&

Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 52.0 kNm. % No shear reinforcement (NSR) in beam. Because beam was designed to fail in shear, therefore there were no bending cracks or strain gauges attached. & Logdec unable to be calculated because of increased decay due to damage. Date beam tested: 1 October 2001 Age of beam at testing: 28 days


BM(kNm)

εres (mV) ∆res

(mm) Wres

(mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 923 0.1253 0.1891 861 0.0582 0.1009 841 0.0349 0.0701 830 0.0491 0.0501 824 0.0457 0.049724.2 -% 0.87 0 943 0.1083 0.1920 815 0.0837 0.1308 810 0.0567 0.0813 811 0.0478 0.0639 808 0.0475 0.069735.5 -% 1.03 0 943 0.1199 0.1851 815 0.0940 0.1459 810 0.0544 0.0878 810 0.0460 0.0702 808 0.0485 0.0733

Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 37.4 kNm. % No shear reinforcement (NSR) in beam Because beam was designed to fail in shear, therefore there were no bending cracks or strain gauges attached. Date beam tested: 1 October 2001 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations9

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


-% No 0 934 0.1034 0.1630 866 0.1109 0.1139 844 0.0726 0.0863 833 0.0653 0.0502 804 0.0653 0.0468Pre-Test -% With 0 930 0.0976 0.1528 867 0.1096 0.1398 845 0.0780 0.0977 832 0.0690 0.0519 821 0.0628 0.051017.0 -% 0.52 0 968 0.1254 0.1716 822 0.1358 0.1654 815 0.1169 0.1142 817 0.0603 0.0735 813 0.0492 0.068625.6 -% 0.76 0 843 0.1125 0.1675 815 0.1499 0.1627 802 0.1209 0.1225 785 0.0695 0.0811 809 0.0515 0.067329.5 -% 1.28 0.01 1757 0.1231 0.1606 1408 0.0925 0.1619 677 0.1448 0.1303 661 0.1048 0.0875 669 0.0808 0.0665

Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 33.7 kNm. # No shear reinforcement (NSR) in beam. * Residual measurements not taken due to catastrophic shear failure. % Because beam was designed to fail in shear there were no bending cracks or strain gauges attached. Date beam tested: 1 October 2001 Age of beam at testing: 28 days


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


0 No 0 1581 0.1147 0.1715 1557 0.0880 0.1103 1722 0.0455 0.0930 1541 0.0491 0.0620 -& -& -&Pre-T

0 With 0 1745 0.0681 0.2533 1722 0.0649 0.1518 1716 0.0466 0.0940 1696 0.0320 0.0631 -& -& -&

30.7 0.12 0.23 0.005 923 0.1860 0.2472 819 0.1665 0.2189 807 0.1148 0.1548 805 0.0789 0.0839 804 0.0628 0.077850.5 0.24 0.58 0.023 892 0.2435 0.2161 834 0.1459 0.2219 817 0.1247 0.1597 810 0.0789 0.0782 808 0.0644 0.087162.2 0.35 1.06 0.03 1635 0.0985 0.1503 1222 0.0984 0.1477 1198 0.0848 0.1007 1533 0.0404 0.0786 1546 0.0330 0.070188.0 0.47 3.33 0.04 667 0.2021 0.1784 1124 0.1563 0.1798 1110 0.1127 0.1150 604 0.1002 0.0886 -& -& 0.0730

Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete under point load) at a Bending Moment (BM) of 88.0 kNm. & Logdec unable to be calculated because of increased decay due to damage (See ……..). Date beam tested: 8 October 2001 Age of beam at testing: 35 days

Appendix E: Dam

ping Tabulations10

Dam


ormal- and H


s

Table D.20. Optimal Peak Ratio Damping Tabulations for Beam CS8

BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


0 No 0 819 0.1638 0.1490 759 0.0902 0.1480 806 0.0715 0.0889 805 0.0387 0.0550 804 0.0419 0.0564Pre-T 0 With 0 818 0.2127 0.1964 763 0.0856 0.1643 809 0.0663 0.0884 804 0.0378 0.0591 804 0.0405 0.0571

23.9 0.39 0.56 0.005 819 0.1468 0.2063 809 0.1091 0.1367 781 0.0742 0.1068 780 0.0790 0.0814 784 0.0555 0.072639.2 0.50 0.95 0.015 819 0.1477 0.1807 810 0.1398 0.1696 773 0.0941 0.1131 780 0.0700 0.0854 764 0.0603 0.075951.4 0.60 1.60 0.02 1143 0.1124 0.1502 715 0.1350 0.1265 685 0.1317 0.1255 685 0.1093 0.0998 678 0.1089 0.099878.1 0.90 3.87 0.03 641 0.1937 0.2110 564 0.1340 0.1958 543 0.1049 0.1567 535 0.0979 0.1251 538 0.0908 0.1144

Failure Mode: Beam failed in tension and compression (yielding of steel plus crushing of concrete) at a Bending Moment (BM) of 78.1 kNm. Date beam tested: 8 October 2001 Age of beam at testing: 28 days


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T 0 0 0 842 0.2242 0.1982 814 0.0980 0.1108 809 0.0707 0.0865 785 0.0519 0.0789 788 0.0424 0.0627 22.2 - 0.74 0 842 0.2111 0.2400 821 0.0847 0.1421 780 0.0678 0.1337 760 0.0577 0.0782 769 0.0620 0.0824 34.5 0.56 1.16 0.015 843 0.2068 0.2117 765 0.0867 0.1502 743 0.0808 0.1178 761 0.0627 0.0729 749 0.0594 0.0886 48.1 0.70 1.83 0.027 720 0.1713 0.2185 665 0.1317 0.1841 676 0.0898 0.1348 655 0.0893 0.0948 644 0.0864 0.0921 61.9 0.71 2.41 0.04 720 0.2372 0.2714 659 0.1699 0.2415 606 0.1691 0.1692 605 0.1207 0.1147 604 0.0902 0.0861 73.9 0.85 10.38 0.07 617 0.3201 0.3637 507 0.1562 0.2108 505 0.1498 0.1716 479 0.1137 0.1121 483 0.0951 0.0954

Failure Mode: Beam failed in tension and shear (yielding of steel with minor shear cracking) at a Bending Moment (BM) of 73.9 kNm. Date beam tested: 8 October 2001 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations11

Dam


ormal- and H


s

Table E.22. Optimal Peak Ratio Damping Tabulations for Beam PS3

BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 1276 0.0496 0.0983 1120 0.0305 0.0612 1114 0.0404 0.0519 1085 0.0385 0.0420 1088 0.0325 0.039882.9 -% 8.87 0.042 1127 0.0963 0.1049 1063 0.0606 0.0815 1075 0.0500 0.0641 1056 0.0455 0.0587 1052 0.0429 0.0518

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 82.9 kNm. Date beam tested: 19 September 2000 Age of beam at testing: 28 days % No strain gauges attached. 1st Crack = 56.1 kNm = 62.9% Table E.23. Optimal Peak Ratio Damping Tabulations for Beam PS4

BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 1276 0.0536 0.0877 1121 0.0390 0.0761 1214 0.0365 0.0540 1085 0.0371 0.0478 1088 0.0348 0.042638.8 -% 0.92 0 1124 0.0441 0.0964 1112 0.0177 0.0571 1074 0.0348 0.0522 1081 0.0390 0.0500 1085 0.0399 0.0413

129.4 -% 6 0.031 1500 0.0838 0.1453 1450 0.0657 0.1139 885 0.0812 0.0842 814 0.0732 0.0663 724 0.0730 0.0606Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 129.4 kNm. Date beam tested: 19 September 2000 Age of beam at testing: 28 days 1st Crack = 78.3 kNm = 60.5% Table E.24. Optimal Peak Ratio Damping Tabulations for Beam PS5

BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 1375 0.0574 0.0935 1304 0.0367 0.0667 1046 0.0374 0.0518 1035 0.0331 0.0430 1028 0.0270 0.0359135.0 -% 11.1 0.049 1025 0.1228 0.1329 962 0.0678 0.1222 941 0.0660 0.0841 931 0.0549 0.0627 925 0.0517 0.0571

Failure Mode: Beam failed in compression (under load support) at a Bending Moment (BM) of 135.0 kNm. Date beam tested: 21 September 2000 Age of beam at testing: 35 days 1st Crack = 83.9 kNm = 62.2%

Appendix E: Dam

ping Tabulations12

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 1125 0.0612 0.0764 1297 0.0348 0.0635 1041 0.0392 0.0467 1030 0.0291 0.0321 1024 0.0215 0.028247.7 -% 0.76 0 1616 0.0789 0.1371 1775 0.0310 0.0710 1046 0.0533 0.0599 1035 0.0485 0.0527 1028 0.0385 0.044979.2 -% 2.54 0.01 1849 0.0839 0.1017 1774 0.0508 0.0872 1749 0.0419 0.0845 1737 0.0352 0.0505 1727 0.0281 0.044895.1 -% 12.9 0.048 1039 0.0970 0.1558 969 0.0802 0.0935 979 0.0624 0.0839 960 0.0585 0.0600 947 0.0416 0.0527

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 95.1 kNm. Date beam tested: 21 September 2000 Age of beam at testing: 35 days 1st Crack =71.6 kNm = 75.3%


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 806 0.0886 0.1326 874 0.0719 0.0982 841 0.0801 0.0743 739 0.0663 0.0565 746 0.0570 0.051637.2 -% 1.0 0 820 0.0664 0.1518 713 0.1014 0.0993 675 0.0979 0.0835 772 0.0627 0.0609 771 0.0501 0.055246.6 -% 1.36 0 733 0.0864 0.1737 822 0.0764 0.0971 675 0.0858 0.0864 658 0.0709 0.0660 580 0.0703 0.061256.0 -% 1.78 0 734 0.0938 0.1721 715 0.1039 0.2926 676 0.0841 0.0913 658 0.0644 0.0769 493 0.0757 0.062465.3 -% 2.74 0.01 820 0.1228 0.1165 772 0.0970 0.1131 672 0.0785 0.0924 754 0.0530 0.0689 744 0.0496 0.059073.1 -% 3.69 0.013 734 0.2126 0.1518 665 0.1182 0.1036 676 0.0782 0.0962 658 0.0655 0.0687 647 0.0602 0.0602

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 73.1 kNm. Date beam tested: 17 August 2001 Age of beam at testing: 43 days 1st Crack = 46.6 kNm = 63.7%

Appendix E: Dam

ping Tabulations13

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 0 897 0.1087 0.0829 1167 0.0429 0.0791 1161 0.0454 0.0536 1226 0.0325 0.0412 1216 0.0279 0.039027.9 -% 0.63 0 1152 0.0648 0.0960 1214 0.0437 0.1005 1200 0.0362 0.0569 956 0.0420 0.0421 1131 0.0349 0.039337.2 -% 1.5 0 1023 0.0706 0.1190 1327 0.0493 0.0925 1142 0.0460 0.0646 1156 0.0376 0.0480 1158 0.0362 0.041055.9 -% 2.65 0.01 1022 0.0857 0.1205 779 0.1175 0.1057 772 0.0719 0.0728 704 0.0632 0.0585 864 0.0504 0.050765.3 -% 2.86 0.013 1357 0.0935 0.1491 1015 0.0742 0.0699 836 0.0553 0.0624 712 0.0647 0.0517 710 0.0519 0.042583.9 -% 7.72 0.059 1057 0.1019 0.1395 746 0.0811 0.1024 781 0.0698 0.0740 947 0.0498 0.0610 944 0.0446 0.051085.8 -% 18.10 0.120 717 0.0796 0.1396 659 0.0945 0.1053 740 0.0757 0.0778 704 0.0571 0.0674 697 0.0584 0.0568

Failure Mode: Beam failed in compression (under load support) at a Bending Moment (BM) of 85.8 kNm. Date beam tested: 17 August 2001 (Tension crack in left front face of beam prior to testing). Age of beam at testing: 47 days 1st Crack = 37.3 kNm = 43.5%


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 No 1391 0.0807 0.0989 1289 0.0609 0.0828 1452 0.0306 0.0605 1260 0.0291 0.0414 - - -Pre-T -% 0 With 1200 0.1071 0.0964 910 0.0802 0.0947 1359 0.0358 0.0612 1187 0.0448 0.0424 1146 0.0360 0.042118.6 -% 0.52 0 1105 0.0931 0.0938 928 0.0648 0.0904 1020 0.0439 0.0668 909 0.0512 0.0523 914 0.0451 0.050537.2 -% 0.82 0 1023 0.0681 0.1354 898 0.0640 0.0943 899 0.0508 0.0693 874 0.0543 0.0626 874 0.0493 0.056055.4 -% 1.52 0 1022 0.0381 0.1103 898 0.0540 0.0901 899 0.0456 0.0720 874 0.0455 0.0612 874 0.0466 0.056769.9 -% 2.52 0.012 921 0.0750 0.1285 901 0.0341 0.0946 893 0.0392 0.0718 780 0.0424 0.0634 874 0.0353 0.051779.2 -% 5.65 0.028 933 0.0594 0.1394 901 0.0467 0.0884 867 0.0397 0.0697 871 0.0363 0.0641 856 0.0325 0.048285.0 -% 54.35 0.048 804 0.1188 0.1809 790 0.0921 0.1298 760 0.0744 0.0997 751 0.0572 0.0824 826 0.0464 0.0687

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 85.0 kNm. Date beam tested: 10 October 2001 Age of beam at testing: 60 days 1st Crack = 46.6 kNm = 54.8%

Appendix E: Dam

ping Tabulations14

Dam


ormal- and H


s


BM(kNm)

εres (mV)

∆res (mm)

Wres (mm)

Freq. DCM (Hz) 50#

Meas. Logdec (DCM)

50#

Calc. Logdec (TLT)

50#

Freq. DCM (Hz) 100#

Meas. Logdec (DCM)

100#


Freq. DCM (Hz) 150*

Meas. Logdec (DCM)

150*


Freq. DCM (Hz) 200*

Meas. Logdec (DCM)

200*


Freq. DCM (Hz) 250*

Meas. Logdec (DCM)

250*


Pre-T -% 0 No 914 0.0803 0.1250 983 0.0452 0.0873 984 0.0381 0.0623 844 0.0357 0.0471 971 0.0251 0.0370Pre-T -% 0 With 825 0.0827 0.1177 939 0.0515 0.0934 901 0.0480 0.0685 844 0.0399 0.0547 829 0.0426 0.046037.3 -% 1.35 0 908 0.1281 0.1857 825 0.0815 0.1132 754 0.0528 0.0608 833 0.0484 0.0564 831 0.0407 0.049555.9 -% 2.01 0 823 0.1306 0.1694 760 0.0950 0.0982 740 0.0469 0.0711 1082 0.0412 0.0662 744 0.0465 0.044665.3 -% 2.33 0 890 0.1424 0.1667 939 0.0654 0.1038 909 0.0404 0.0689 1013 0.0370 0.0605 750 0.0503 0.049174.6 -% 2.74 0 907 0.1250 0.1396 894 0.0726 0.1112 867 0.0448 0.0773 839 0.0525 0.0625 831 0.0508 0.057583.9 -% 3.21 0 907 0.1285 0.1287 900 0.0833 0.1003 829 0.0431 0.0723 908 0.0436 0.0653 826 0.0480 0.0537

102.5 -% 4.39 0.012 919 0.1192 0.1582 778 0.1130 0.1058 794 0.0597 0.0809 748 0.0582 0.0667 914 0.0458 0.0600121.1 -% 8.61 0.016 897 0.1252 0.1576 762 0.0966 0.1175 833 0.0652 0.0843 817 0.0569 0.0765 809 0.0562 0.0691125.8 -% 20.01 0.066 820 0.1250 0.1427 667 0.1108 0.0878 706 0.0672 0.0855 612 0.0681 0.0784 597 0.0682 0.0596

Failure Mode: Beam failed in tension (yielding of steel) at a Bending Moment (BM) of 125.8 kNm. Date beam tested: 9 October 2000 (Tension crack in centre upper top of beam prior to testing). Age of beam at testing: 59 days 1st Crack = 55.9 kNm = 44.4%

Appendix E: Dam

ping Tabulations15

Dam


ormal- and H


s

Table E.30. Optimal Peak Ratio Damping Tabulations for Beam F1 NDP=50# NDP=100# NDP=150* NDP=200* NDP=250* NDP=300*BM

(kNm) Load

Level %∆res

(mm) Wres

(mm) A1/An * δ A1/An δ A1/An δ A1/An δ A1/An δ A1/An δ

1 0 0 37 0.1435 18 0.0820 15 0.0634 10 0.0571 8 0.0552 5 0.0604Standard Deviation 4.1 0.016 1.3 0.004 1.2 0.003 1.5 0.003 0.5 0.002 1.1 0.006


Pre-T

OVERALL AVERAGE LOGDEC 36 0.1496 18 0.0836 15 0.0686 12 0.0581 10 0.0531 7 0.0549

2.7 0.39 0 17 0.0601 12 0.06403.8 & 0.46 0 14 0.0664 10 0.06975.7 0.53 0 13 0.0652 8 0.06857.9 0.55 0 15 0.0615 10 0.062911.8 0.72 0 12 0.0816 8 0.084114.8 0.88 0 15 0.0716 10 0.077922.5 1.01 0 14 0.0671 6 0.073625.0

-

1.10 0

-

15 0.0661 7 0.0700

-

% The load levels are described in Figure E.1. * The determination of the ‘Optimal Peak Ratio’ A1/An is described in Section 5.4.2. # The average logdec is calculated at various NDP (see Section 5.4). & This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 28.8 kNm. Date beam tested: 10 April 2002 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations16

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres



2 0 0 30 0.1533 23 0.0981 14 0.0768 14 0.0531 9 0.0556 9 0.0557Standard Deviation 2.0 0.043 2.5 0.015 2.5 0.008 1.2 0.005 2.1 0.010 - -


Pre-T

OVERALL AVERAGE LOGDEG 31 0.1632 22 0.1004 16 0.0705 14 0.0583 10 0.0545 9 0.0491

1.9 0.15 - 16 0.0654 8 0.06544.1& 0.28 - 14 0.0668 9 0.06196.3 0.39 - 12 0.0706 9 0.06188.2 0.55 - 12 0.0765 9 0.063111.2 0.81 - 14 0.0706 10 0.061613.3 0.95 - 15 0.0764 10 0.071215.3 1.08 - 14 0.0721 10 0.068718.3 1.21 - 14 0.0777 9 0.073221.4 1.36 - 14 0.0721 9 0.073724.8 1.52 - 18 0.0692 8 0.072528.0 1.69 - 14 0.0805 9 0.074931.9 1.69 - 14 0.0745 10 0.074534.7 2.03 - 12 0.0706 10 0.062237.5 3.41 - 12 0.0859 9 0.080142.4

-

20.98 -

-

10 0.0770 8 0.0707

-

& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 42.4 kNm. Date beam tested: 10 April 2002 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations17

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres


1 0 0 33 0.1142 24 0.0676 22 0.0494 18 0.0446 14 0.0463 12 0.0377Standard Deviation 5.0 0.0 1.7 0.006 1.5 0.003 2.0 0.002 5.7 - - 0.000



Pre-T


2.9 0.31 - 10 0.0734 12 0.05644.9 & 0.42 - 13 0.0716 9 0.066310.2 0.78 - 18 0.0648 14 0.065619.4 1.16 - 16 0.0806 11 0.079324.0 1.28 - 10 0.0961 10 0.085626.5 1.29 - 15 0.0694 10 0.072633.1 1.41 - - - 8 0.075340.4 1.73 - - - 8 0.075843.0 1.87 - 10 0.0835 9 0.073249.1 2.03 - 10 0.0794 9 0.072655.0 2.32 - - - 12 0.063757.0 4.49 - - - 11 0.066460.4

-

14.12 -

-

- - 12 0.0650

-

& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 8.7 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 42.4 kNm. Date beam tested: 11 April 2002 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations18

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres



2 0 0 33 0.1188 25 0.0666 17 0.0620 14 0.0551 11 0.0495 11 0.0448Standard Deviation 12.5 0.028 4.4 0.003 2.5 0.003 - - 2.1 0.000 1.5 0.004


Pre-T


5.6 0.11 - 14 0.0540 10 0.04887.7 0.31 - 11 0.0572 10 0.0487

9.3 & 0.42 - 9 0.0712 7 0.064911.8 0.55 - 11 0.0634 7 0.069914.5 0.71 - 11 0.0587 7 0.063017.7 0.86 - 10 0.0676 7 0.063721.2 0.96 - 11 0.0745 9 0.073724.6 1.11 - 10 0.0872 7 0.076227.3 1.16 - 11 0.0883 11 0.072630.8 1.29 - 12 0.0843 9 0.070434.6 1.38 - 12 0.0768 9 0.070337.9

-

1.60 -

-

15 0.0755 13 0.0678

-

& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 49.4 kNm. Date beam tested: 11 April 2002 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations19

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres



2 0 0 34 0.1373 18 0.0974 15 0.0799 12 0.0654 11 0.0539 9 0.0533Standard Deviation 4.4 0.017 5.4 0.007 3.7 0.014 - 0.015 1.0 0.004 1.9 0.006


Pre-T


2.6 - - 11 0.0689 8 0.06993.8 0.13 - 11 0.0698 10 0.06745.7 0.35 - 11 0.0656 6 0.069613.3 0.58 - 10 0.0695 8 0.0637

15.3 & 0.68 - 10 0.0681 7 0.065118.2 0.83 - 11 0.0738 8 0.072520.9 0.92 - 11 0.0772 8 0.069022.9 0.96 - 11 0.0754 8 0.061327.4 1.06 - 11 0.0856 8 0.064931.7 1.17 - 12 0.0910 10 0.077838.0

-

1.32 -

-

13 0.0878 9 0.0745& This is the mid-span bending moment where 1st cracking appeared. - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in shear at a mid-span Bending Moment (BM) of 43.4 kNm. Date beam tested: 17 April 2002 Age of beam at testing: 28 days

Appendix E: Dam

ping Tabulations20

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres




3 0 0 36 0.1354 23 0.1078 17 0.0856 15 0.0667 10 0.0675 8 0.0606Standard Deviation 3.6 0.008 2.1 0.006 2.6 0.007 - - 1.0 0.005 1.4 0.009

Pre-T


5.9 0.29 - 9 0.0787 6 0.07049.9 0.33 - 10 0.0873 7 0.0751

16.3 & 0.48 - 10 0.0811 7 0.075128.6 0.72 - 10 0.0767 8 0.064041.2

-

0.95 -

-


Appendix E: Dam

ping Tabulations21

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres





Pre-T


11.9 0.48 - 11 0.0612 9 0.056117.0 & 0.59 - 10 0.0692 8 0.056422.6 0.68 - 11 0.0680 8 0.057229.7 0.83 - 11 0.0569 10 0.051334.9 0.91 - 9 0.0651 7 0.056242.7

-

1.01 -

-


Appendix E: Dam

ping Tabulations22

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres





Pre-T


11.8 0.35 0 11 0.0504 9 0.047120.0& 0.59 0 9 0.0538 8 0.049430.2 0.77 0.02 10 0.0537 9 0.049340.9 0.95 0.04 10 0.0542 8 0.051952.9 1.15 0.05 9 0.0547 9 0.051165.3 1.35 0.06 10 0.0553 8 0.052077.5 1.63 0.07 9 0.0585 7 0.054494.0

-

2.39 0.09

-

8 0.0645 7 0.0603& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 18.1 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 94.0 kNm. Date beam tested: 18 April 2002 Age of beam at testing: 29 days

Appendix E: Dam

ping Tabulations23

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres



2 0 0 55 0.0991 35 0.0815 22 0.0718 16 0.0579 12 0.0561 7 0.0528Standard Deviation 9.1 0.041 5.1 0.030 3.0 0.017 2.6 0.003 - - 0.7 0.001


Pre-T


9.6 & 0.49 0 12 0.0550 9 0.050519.2 0.86 0.01 10 0.0549 9 0.052229.3 1.14 0.03 10 0.0591 10 0.048242.5 1.39 0.04 10 0.0600 9 0.052655.7 1.68 0.08 10 0.0584 10 0.050168.5 2.94 0.12 12 0.0751 8 0.073571.9

-

17.77 -

-

13 0.0659 12 0.0546& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 13.2 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure at a mid-span Bending Moment (BM) of 71.9 kNm. Date beam tested: 18 April 2002 Age of beam at testing: 29 days

Appendix E: Dam

ping Tabulations24

Dam


ormal- and H


s


(kNm) Load

Level %∆res

(mm) Wres


1 0 0 59 0.0832 37 0.0663 28 0.0514 20 0.0425 12 0.0400 10 0.0378Standard Deviation 2.0 0.022 - - - - 4.5 0.007 1.7 0.006 0.7 0.004



Pre-T


11.8 & 0.45 0 14 0.0498 11 0.045522.1 0.81 0.03 15 0.0538 12 0.048033.8 1.06 0.06 13 0.0560 8 0.051545.4 1.32 0.07 13 0.0550 11 0.052456.3 1.68 0.09 10 0.0765 9 0.062067.3

-

2.51 0.12

-

12 0.0829 10 0.0681& This is the mid-span bending moment where 1st cracking appeared (It actually occurred at 12.8 kNm). - This indicates that a reading was not taken, or was not applicable. Failure Mode: Beam failed in flexure and shear at a mid-span Bending Moment (BM) of 73.6 kNm. Date beam tested: 18 April 2002 Age of beam at testing: 29 days

Appendix E: Dam

ping Tabulations25

Dam


ormal- and H


s

E.1 F-Series Beams Load Levels Load Level 1 – Totally Unloaded

This phase induced free-vibration to the beam whilst it had no loading whatsoever on it. This provided

the ‘benchmark’ for comparison with the following stages.

Load Level 2 – Loading Beam Only

At this stage, the loading beam and loading cell were placed into position, at which point more free-

vibration damping testes were conducted. It should be noted here, that the ‘untested’ damping

experiments for the B-, CS- and PS-Series beams were conducted at this Load Level.

Load Level 3 – Additional Weights Hung

Two large concrete blocks were hung from the pin and roller supports of the loading beam. These third

points were selected so that the suspension slings did not come into contact with the test beam, and to

ensure minimal interference with the free-vibration damping tests. Each concrete block weighed 136.5

kg. The tests were limited to two of these blocks due to the size restrictions of the blocks. Nevertheless,

they were sufficiently large enough to induce a small observable level of stress in the beam, which was

monitored via the deflection gauges.

100 tonne capacitycylinder


Loading CellWeight of loading beam (221 kg)+ Weight of cylinder loading discs

(40 kg)

2000mm 2000mm2000mm


Loading CellWeight of loading beam (221 kg)+ Weight of cylinder loading discs

(40 kg)

2000mm 2000mm2000mm

136.5 kg 136.5 kg

LoadLevel

1

LoadLevel

2

LoadLevel

3

Figure E.1: Load Levels for F-Series Beam Tests

Appendix E: Damping Tabulations26

Damping Charateristics of Reinforced and Prestressed Normal- and High-Strength Concrete Beams

APPENDIX F: Serviceability Curves F-1

APPENDIX F

Serviceability Curves



F.1 Bending Moment versus Instantaneous Deflection

a) AA

AAA

AA

AA

AA

AA

AA

BBB

BBB

BB

BB

BB

BBB

CCC

C

C

C

CC

CC

DD

D

D

D

D

DD

DD

E

EE

E

E

E

EE

E

FF

FF

F

F

F

FF

F

F

GGGGGG

GG

GG G

G GGGGG G

HHHHH H HH H H H

HH

H HH HHHH H H

II I

I

I

I

I

III I I I

JJ

JJ

JJ

JJ

JJ

JJJ J J

KKK

KK

K

K

K

K

K

KK K

LLL

LL

L

L

L

L

L

L

L

L

LL

LL

LL L

Instantaneous Deflection (mm)

Ben

ding

Mom

ent(

kNm

)

0 25 50 75 100 1250

25

50

75

100

125

150

175


ABCDEFGHIJKL

b) 1

11

11

11

11

11

11

11

11

11

11

11

1

22

2222 2

22 2 22

22

22

22

22

22

22

33

3333

33

33 3

33

33

33

33

33

3 3

444

444

44

444

44

44

44

44

4

555

555

555555

5555555 5

66666

66666

66666

66

77777

7777

777

777

77

77 7

77

77

77

77

77 7

77

7

888

88888

888888888

8888 88 8 8 88 8 88

8 88 8 8 8 8 8 8 8 8

9999999999999999999 9 99 9 99 99 9 9

999 9 9 9 9 9 9 9 9 9


Ben

ding

Mom

ent(

kNm

)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

70

80

90


123456789

c) a a

aaaaaa

a a a a a a a a a a a a

bbbbbbbbbbb

bb

bb

bb

bbb

ccccccccccc

cc

cc

cc

c

dddddddddddddddddddd

dd

dd

dd d d

dd

eeeee

eeeeeeeeeeeee

ee

e

ee

ee

e

fff

f

f

f

fff

ff

ff

ff f f

g

g

g

g

g

g

gg g

h

h

h

h

h

hh

hh

hh

h h

i

i

i

i

i

ii

j

j

j

j

j

j

j

j

j

j

j

jj


Ben

ding

Mom

ent(

kNm

)

-20 0 20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90

100

110

120

130

PS1 (a)PS2 (b)PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)

abcdefghij

Figure F.1: BM (kNm) versus Instantaneous Deflection for: a) BI-1 to BII-12; b) CS1 to

CS9; c) PS1 to PS10



a) AA

AA

A

A

A

A

A

A

A

A

A

A

A

BB

BB

B

B

B

B

B

B

B

B

BBB

C

CC

C

C

C

C

C

C

C

D

DD

D

D

D

D

DDD

E

EE

E

E

E

E

EE

F

FF

F

F

F

F

F

FF

F

GGGGGG

G

G

G

G

G

G

GGGGG G

HH

HH

HH

HH

HH

H

H

H

H

HH HH

HH H H

I

II

I

I

I

I

I

I

I II

I

J

J

J

J

J

J

J

J

J

J

JJ J

J J

KKK

K

K

K

K

K

K

K

KK

K

LLL

LL

L

L

L

L

L

L

L

L

LLL L

LL L


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)

ABCDEFGHIJKL

b) 1

11

11

11

11

11

1

11

11

11

11

11

11

2

222

22

22

22

22

22

22

22

22

22

22

3

33

33

33

33

33

33

33

33

33

33

33

444444

44

444

4

44

44

44

44

5

55

555

555555

555555

55

666

66

666

66

66

666

6

6

77777

777

77

77

777

77

77

77

77

7 77 7

77

77

77

7

88888

888

888

888

88888

888

8 88 8

8 8 88

8 888

8 8 88

88 8

999

9999

999

9999999

999 99

9 9999 9 9

9 999 9

9 99 9 9 9

9 9


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 2 4 6 8 10 12 14 16 18 20 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


123456789

c) a

a

a

aa

a

a

a

aa

aa

aa

aa

aa

aa

bbbbbbbbbbb

bb

bb

bb

bbb

c

c

ccccc

c

c

ccc

cc

cc

cc

dddddddddddddddddddd

dd

dd

dd d d

dd

eeeee

eeeeeeeeeeeee

ee

e

ee

ee

e

f

ff

f

f

f

ff

f

f

f f

f f

f f f

g

g

g

g

g

g

g

gg

h

h

h

h

h

hh

hh

hh

h h

i

i

i

i

i

i

i

j

j

j

j

j

j

j

j

j

j

j

jj


Ben

ding

Mom

ent(

Nor

mal

ised

)

-20 0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PS1 (a)PS2 (b)PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)

abcdefghij

Figure F.2: BM (normalised) versus Instantaneous Deflection for: a) BI-1 to BII-12; b)

CS1 to CS9; c) PS1 to PS10



F.2 Bending Moment versus Residual Deflection

a) A

AAAAAAA

AA

AA

B

BB

BBBBB

BB

B

C

C

C

C

CC

CC

D

D

D

D

DD

D D

E

E

E

E

EE

E

F

F

F

F

FF

F

F

G

GGGG

GG G

H

H HHHHHHH

HH

I

I

I

I

II

J

JJJJ

JJJ

JJ

J J

K

K

K

K

K

K

K

KK

L

L

L

L

L

L

L

L

L

LL

Residual Deflection (mm)

Ben

ding

Mom

ent(

kNm

)

0 20 40 600

20

40

60

80

100

120

140

160

180


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

4

4

4

4

5

5

5

6

6

6

7

7

7

7

7

8

8

8

8

8

9

9

9

9

9

9


Ben

ding

Mom

ent(

kNm

)

0 2 4 6 8 10 120

10

20

30

40

50

60

70

80

90


123456789

c) a

a

a

a

b

b

b

b

c

c

c

c

d

d

d

e

e

e

e

e

e

eee

f

f

f

f

f

ffff

ff

f f

g

g

g

g

g

gg

h

h

h

h

h

h

h

h

h

h

h

hh


Ben

ding

Mom

ent(

kNm

)

0 20 40 600

20

40

60

80

100

120

140

PS1 (a)PS2 (b)PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)

abcdefgh

Figure F.3: BM (kNm) versus Residual Deflection for: a) BI-1 to BII-12; b) CS1 to

CS9; c) PS3 to PS10



a) A

A

A

A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

C

C

C

C

C

C

C

C

D

D

D

D

D

DD D

E

E

E

E

E

EE

F

F

F

F

F

FF

F

G

G

G

G

G

G

G

G

H

HH

HHH

H

H

H

H

H

I

I

I

I

I

I

J

J

J

J

J

J

J

J

J

JJ J

K

K

K

K

K

K

K

K

K

L

L

L

L

L

L

L

L

L

LL


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 20 40 600


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

4

4

4

4

5

5

5

6

6

6

7

7

7

7

7

8

8

8

8

8

9

9

9

9

9

9


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 1 2 3 4 5 6 7 8 9 10 110

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


123456789

c) c

c

c

c

d

d

d

d

e

e

e

e

f

f

f

g

g

g

g

g

g

gg g

h

h

h

h

h

hhh

hh

hh h

i

i

i

i

i

i

j

j

j

j

j

j

j

j

j

j

j

jj


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 2 4 6 8 10 12 14 16 18 20 220

20

40

60

80

100

120

140

PS3 (c)PS4 (d)PS5 (e)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)

cdefghij

Figure F.4: BM (Normalised) versus Residual Deflection for: a) BI-1 to BII-12; b) CS1

to CS9; c) PS3 to PS10



F.3 Bending Moment versus Average Instantaneous Crack Width

a) A

AAAAAAAA

AA

B

BBB

BBBB

BBB

B

C

C

C

C

C

CCC

C

D

D

D

D

DD

DD

E

E

E

E

EE

E

F

F

F

F

F

FF

F

G

GG

GGGGG

H

H HH HHHHH

HH

I

I

I

I

II

JJ

JJ

JJJ

JJ

JJ

J

K

K

K

K

K

K

K

K

L

L

L

L

L

L

L

L

L

L

Average Instantaneous Crack Width (mm)

Ben

ding

Mom

ent(

kNm

)

0 0.5 1 1.50

25

50

75

100

125

150

175


ABCDEFGHIJKL

b) c

c

c

d

d

d

f

f

f

g

ggg

h

h

h

h

h h

i

i

ii

j

j

jj


Ben

ding

Mom

ent(

kNm

)

0 0.02 0.04 0.06 0.08 0.1 0.120

20

40

60

80

100

120

140

PS3 (c)PS4 (d)PS6 (f)PS7 (g)PS8 (h)PS9 (i)PS10 (j)

cdfghij

Figure F.5: BM (kNm) versus Average Instantaneous Crack Width for: a) BI-1 to BII-

12; b) PS3 to PS10



a) A

AA

A

A

A

A

A

A

A

A

B

BB

B

B

B

B

B

B

B

BB

C

C

C

C

C

C

C

C

C

D

D

D

D

D

DDD

E

E

E

E

E

EE

F

F

F

F

F

F

FF

G

G

G

G

G

G

G

G

H

HH

HH

H

H

H

H

H

H

I

I

I

I

I

I

J

J

J

J

J

J

J

J

J

J

JJ

K

K

K

K

K

K

K

K

L

L

L

L

L

L

L

L

L

L


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.25 0.5 0.750

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


ABCDEFGHIJKL

b) c

ccc

cc

cc

d

dd

d

dd

ddd

ddd

f

f

f

f

g

g

g

gg

h

h

h

h

h

h

h

i

i

i

i

j

j

j

j

j

j

j

j


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


cdfghij

Figure F.6: BM (Normalised) versus Average Instantaneous Crack Width for: a) BI-1 to

BII-12; b) PS3 to PS10



F.4 Bending Moment versus Average Residual Crack Width

a) A

AAA

AA

AA

AA

B

BB

BBBB

BB

BB

C

C

C

C

CC

C

D

D

D

D

DD

D

E

E

E

E

EE

E

F

F

F

FF

F

G

GGGG

GG

H

H HHHH HHHH

H

I

I

I

I

II

J

JJJ

JJ

JJ

JJ

J

K

K

K

K

K

K

K

KK

L

L

L

L

L

L

L

L

L

L

Average Residual Crack Width (mm)

Ben

ding

Mom

ent(

kNm

)

0 0.05 0.1 0.15 0.20

25

50

75

100

125

150

175


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

7

7

7

7

7

8

8

8

8

8

9

9

9

9

9


Ben

ding

Mom

ent(

kNm

)

0 0.02 0.04 0.06 0.080

25

50

75

CS1CS2CS3CS7CS8CS9

123789

c) c

c

c

d

d

d

f

f

f

g

ggg

h

h

h

h

h h

i

i

ii

j

j

jj


Ben

ding

Mom

ent(

kNm

)

0 0.02 0.04 0.06 0.08 0.1 0.120

20

40

60

80

100

120

140


cdfghij

Figure F.7: BM (kNm) versus Average Residual Crack Width for: a) BI-1 to BII-12; b)

CS1 to CS3 and CS7 to CS9; c) PS3 to PS10



a) A

A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

BB

C

C

C

C

C

C

C

D

D

D

D

D

DD

E

E

E

E

E

EE

F

F

F

F

FF

G

G

G

G

G

G

G

H

HH

HH

H

H

H

H

H

H

I

I

I

I

I

I

J

J

J

J

J

J

J

J

J

JJ

K

K

K

K

K

K

K

K

K

L

L

L

L

L

L

L

L

L

L


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

7

7

7

7

7

8

8

8

8

8

9

9

9

9

9


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.02 0.04 0.06 0.080

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CS1CS2CS3CS7CS8CS9

123789

c) c

c

c

d

d

d

f

f

f

g

g

gg

h

h

h

h

h h

i

i

i

i

j

j

jj


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1


cdfghij

Figure F.8: BM (Normalised) versus Average Residual Crack Width for: a) BI-1 to BII-

12; b) CS1 to CS3 and CS7 to CS9; c) PS3 to PS10



F.5 Bending Moment versus Instantaneous Steel Strain

a) A

AA

A

A

A

A

A

A

A

A

BB

BB

B

B

B

B

B

C

CC

C

C

C

C

D

DD

D

D

E

E

E

E

E

F

F

F

F

F

GGGGG G G

GG

GG

HHH

H HH

HH

HH

H

II

II

I

I

J

J

J

J

J

J

K

KK

K

K

K

K

L

LL

LL

L

L

L

L

Instantaneous Steel Strain (mm/mm)

Ben

ding

Mom

ent(

kNm

)

0 0.0005 0.001 0.0015 0.0020

10

20

30

40

50

60

70

80

90

100BI-1 (A)BII-2 (B)BI-3 (C)BII-4 (D)BII-5 (E)BII-6 (F)BI-7 (G)BII-8 (H)BI-9 (I)BII-10 (J)BII-11 (K)BII-12 (L)

ABCDEFGHIJKL

b) 1

11

11

11

11

11

11

111

11

11

11

11

22

22222

222 22

22

22

22

22

22

2

33

3333

33

33 3

33

33

33

33

33

3 3

777

777

77

77

77

777

77

77 7

77

77

77

77

77 7

77

7

888

88888 8888

88 8888 88 88 888 8888 8

8 88 8 8 8 8 8 8 8

999

99999999999 999999 9999 99999

999 999 99 9999 99


Ben

ding

Mom

ent(

kNm

)

0 0.001 0.002 0.003 0.004 0.0050

10

20

30

40

50

60

70

80

90

CS1CS2CS3CS7CS8CS9

123789

Figure F.9: BM (kNm) versus Instantaneous Steel Strain for: a) BI-1 to BII-12; b) CS1

to CS3 and CS7 to CS9



a) A

AA

A

A

A

A

A

A

A

A

BB

BB

B

B

B

B

B

C

CC

C

C

C

C

D

DD

D

D

E

E

E

E

E

F

F

F

F

F

GGG

GG

GG

G

G

G

G

HH

HH

HH

HH

HH

H

I

II

I

I

I

J

J

J

J

J

J

KK

KK

K

K

K

LLLL L

L

L

L

L


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.0005 0.001 0.0015 0.0020

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


ABCDEFGHIJKL

b) 1

11

11

11

11

11

11

11

11

11

11

11

1

2

222

22

22

22

22

22

22

22

22

22

2

3

33

33

33

33

33

33

33

33

33

33

33

777

777

777

77

77

777

77

77

77

77 7

777

77

77

77

88888

888

88 8

88 8

8888 8

8 88

888 8

88 88

8 8 88

8 8 88

88

999

99 99

999

9 99 9999

999 99

99 999 99

99 9

99 999

999999


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.001 0.002 0.003 0.0040

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CS1CS2CS3CS7CS8CS9

123789

Figure F.10: BM (Normalised) versus Instantaneous Steel Strain for: a) BI-1 to BII-12;

b) CS1 to CS3 and CS7 to CS9



F.6 Bending Moment versus Instantaneous Steel Stress

a) A

AA

A

A

A

A

A

A

A

A

BB

BB

B

B

B

B

B

C

CC

C

C

C

C

D

DD

D

D

E

E

E

E

E

F

F

F

F

F

GGGGG G G

GG

GG

HHH

H HH

HH

HH

H

II

II

I

I

J

J

J

J

J

J

K

KK

K

K

K

K

L

LL

LL

L

L

L

L

Instantaneous Steel Stress (MPa)

Ben

ding

Mom

ent(

kNm

)

0 100 200 300 4000

10

20

30

40

50

60

70

80

90


ABCDEFGHIJKL

b) 1

11

11

11

11

11

11

11

1

2

22

22

22

22

22

22

2

22

2

2

3

33

33 3

33

33

33

33

33

33

77

77

77

77

7

77

77

77

77

77

77

88

88

88

88

88

88

88

88 8

88

88

88

88

99

99

99

99 9

99

99

99

99

99

99

99

99

9


Ben

ding

Mom

ent(

kNm

)

0 100 200 3000

10

20

30

40

50

CS1CS2CS3CS7CS8CS9

123789

Figure F.11: BM (kNm) versus Instantaneous Steel Stress for: a) BI-1 to BII-12; b) CS1

to CS3 and CS7 to CS9



a) 1

1

1

2

2

2

3

3

3

7

7

7

8

8

8

9

9

9


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CS1CS2CS3CS7CS8CS9

123789

b) 1

11

11

11

11

11

11

11

1

2

22 2

22

22

22

22

22

22

22

3

33

33

33

33

33

33

33

33

3

77

77 7

77 7

77

77

77 7

77

77

77

88

8 8 88

8 88

8 88

8 88 8 8

8 88 8

88 8

8

99 9

99 9 9

9 99

9 9 9 99 9 9

9 99 9 9

9 9 99


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CS1CS2CS3CS7CS8CS9

123789

Figure F.12: BM (Normalised) versus Instantaneous Steel Stress for: a) BI-1 to BII-12;

b) CS1 to CS3 and CS7 CS9



F.7 Bending Moment versus Residual Steel Strain

a) A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

C

C

C

C

C

D

D

D

E

E

E

E

F

F

F

F

G

GG

GG

H

HH

HH

HH

I

I

I

J

J

J

J

J

K

K

K

K

L

L

L

L

L

Residual Steel Strain (mm/mm)

Ben

ding

Mom

ent(

kNm

)

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.00080

10

20

30

40

50

60

70

80

90


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

7

7

7

8

8

8

9

9

9


Ben

ding

Mom

ent(

kNm

)

0 5E-05 0.0001 0.00015 0.0002 0.000250

5

10

15

20

25

30

35

40

45

50

55

CS1CS2CS3CS7CS8CS9

123789

Figure F.13: BM (kNm) versus Residual Steel Strain for: a) BI-1 to BII-12; b) CS1 to

CS3 and CS7 to CS9



a) A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

C

C

C

C

C

D

D

D

E

E

E

E

F

F

F

F

G

G

G

G

G

H

HH

HHH

H

I

I

I

J

J

J

J

J

K

K

K

K

L

L

L

L

L


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 0.0005 0.001 0.0015 0.0020

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

7

7

7

8

8

8

9

9

9


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 5E-05 0.0001 0.00015 0.0002 0.000250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CS1CS2CS3CS7CS8CS9

123789

Figure F.14: BM (Normalised) versus Residual Steel Strain for: a) BI-1 to BII-12; b)

CS1 to CS3 and CS7 to CS9



F.8 Bending Moment versus Residual Steel Stress

a) A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

C

C

C

C

C

D

D

D

E

E

E

E

F

F

F

F

G

GG

GG

H

HH

HH

HH

I

I

I

J

J

J

J

J

K

K

K

K

L

L

L

L

L

Residual Steel Stress (MPa)

Ben

ding

Mom

ent(

kNm

)

0 40 80 120 1600

10

20

30

40

50

60

70

80

90


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

7

7

7

8

8

8

9

9

9


Ben

ding

Mom

ent(

kNm

)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

55

CS1CS2CS3CS7CS8CS9

123789

Figure F.15: BM (kNm) versus Residual Steel Stress for: a) BI-1 to BII-12; b) CS1 to

CS3 and CS7 to CS9



a) A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

C

C

C

C

C

D

D

D

E

E

E

E

F

F

F

F

G

G

G

G

G

H

HH

HH

H

H

I

I

I

J

J

J

J

J

K

K

K

K

L

L

L

L

L


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 40 80 120 160 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


ABCDEFGHIJKL

b) 1

1

1

2

2

2

3

3

3

7

7

7

8

8

8

9

9

9


Ben

ding

Mom

ent(

Nor

mal

ised

)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CS1CS2CS3CS7CS8CS9

123789

Figure F.16: BM (Normalised) versus Residual Steel Stress for: a) BI-1 to BII-12; b)

CS1 to CS3 and CS7 to CS9
