to my parents, brother, sister, my wife and ll a my...
TRANSCRIPT
1
EFFECT OF SHORT-DURATION-HIGH-IMPULSE VARIABLE AXIAL AND TRANSVERSE LOADS
ON REINFORCED CONCRETE COLUMN
By
THIEN PHUOC TRAN
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2009
2
© 2009 Thien Phuoc Tran
3
To my parents, brother, sister, my wife and all my children
4
ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. Theodor Krauthammer, for all his valuable advices
and guidance. I would also like to extend my appreciation to Dr. Serdar Astarlioglu for all his
constructive ideas and his attentiveness to facilitate the completion of this thesis paper. I am
grateful to the Canadian Armed Forces, particularly the Defence Research and Development
Canada in Suffield for providing the opportunity for me to complete this postgraduate program.
I would like to thank all my friends and colleagues in Canada and at the Center for Infrastructure
Protection and Physical Security, University of Florida for all the supports during the last two
years.
Lastly, I am indebted to my parents, my brother, my sister, my wife and all my children
for all the sacrifices that they have made to provide the opportunity for my achievement.
5
TABLE OF CONTENTS page
ACKNOWLEDGMENTS.................................................................................................................... 4
LIST OF TABLES................................................................................................................................ 7
LIST OF FIGURES.............................................................................................................................. 8
LIST OF SYMBOLS AND ABBREVIATIONS ............................................................................. 12
ABSTRACT........................................................................................................................................ 15
CHAPTER
1 INTRODUCTION....................................................................................................................... 16
1.1 Problem Statement ............................................................................................................ 16 1.2 Objectives and Scope........................................................................................................ 16 1.3 Research Significance ....................................................................................................... 17
2 LITERATURE REVIEW ........................................................................................................... 18
2.1 Introduction ....................................................................................................................... 18 2.2 Blast Loading on Structure ............................................................................................... 18 2.3 Structure and Its Equivalent System ................................................................................ 23
2.3.1 Introduction ........................................................................................................... 23 2.3.2 Equivalent Mass.................................................................................................... 24 2.3.3 Equivalent Load.................................................................................................... 25 2.3.4 Shape Functions.................................................................................................... 27 2.3.5 Resistance Functions ............................................................................................ 27
2.4 Static Analysis ................................................................................................................... 28 2.4.1 Flexural Behavior ................................................................................................ . 28 2.4.2 Influence of Shear on Flexural Response............................................................ 30 2.4.3 Influence of Axial Force on Shear and Flexural Responses .............................. 32 2.4.4 Direct Shear Mode of Response .......................................................................... 34
2.5 Dynamic Analysis ............................................................................................................. 36 2.5.1 Newmark-Beta Method ........................................................................................ 36 2.5.2 Dynamic Resistance Functions............................................................................ 37 2.5.3 Modified Equivalent Parameters for SDOF System........................................... 40
2.5.3.1 Mass factor................................................................................................ 40 2.5.3.2 Load factor ................................................................................................ 41
2.5.4 Dynamic Reactions............................................................................................... 41 2.6 Pressure-Impulse (P-I) Diagrams..................................................................................... 43
2.6.1 Characteristics of P-I Diagrams........................................................................... 43 2.6.2 Derivation of P-I Diagrams.................................................................................. 44
2.7 Summary............................................................................................................................ 46
6
3 METHODOLOGY ...................................................................................................................... 47
3.1 Introduction ....................................................................................................................... 47 3.2 Load Determination .......................................................................................................... 47
3.2.1 Overview of Structure .......................................................................................... 47 3.2.2 Effects of Transverse Loads................................................................................. 50 3.2.3 Axial Loads ........................................................................................................... 50
3.3 Load Deformation Analysis.............................................................................................. 51 3.4 Computations of Dynamic Reactions, Shear and Flexural Responses .......................... 54
3.4.1 Computation of Dynamic Reactions for the Supported Mass ........................... 54 3.4.2 Computation of Shear and Flexural Responses on Columns............................. 55
3.5 Summary............................................................................................................................ 58
4 ANALYSIS ................................................................................................................................. 59
4.1 Introduction ....................................................................................................................... 59 4.2 Description of DSAS ........................................................................................................ 59 4.3 Validations with Experimental Data ................................................................................ 60
4.3.1 Experimental Data ................................................................................................ 60 4.3.1.1 Material and physical properties.............................................................. 60 4.3.1.2 Loading functions..................................................................................... 61
4.3.2 ABAQUS Validations .......................................................................................... 64 4.3.3 DSAS Validations................................................................................................ . 73 4.3.4 Comparison of Results from ABAQUS and DSAS to Experimental Data....... 74
4.4 Validation for Beam Subject to Uniform Load Using ABAQUS and DSAS ............... 77 4.5 Validation for Column Using ABAQUS and DSAS ...................................................... 79 4.6 Summary............................................................................................................................ 82
5 PARAMETRIC STUDY ............................................................................................................ 83
5.1 Description of Columns .................................................................................................... 83 5.2 Columns Subject to Transverse and Constant Axial Load ............................................. 84 5.3 .........................Columns Subject to Transverse, Constant and Variable Axial Loads 101 5.4 Summary.......................................................................................................................... 106
6 CONCLUSIONS AND RECOMMENDATIONS ................................................................. 107
6.1 Summary.......................................................................................................................... 107 6.2 Conclusions ..................................................................................................................... 108 6.3 Recommendations ........................................................................................................... 108
APPENDIX SAMPLE ABAQUS INPUT FILE – BEAM 1-C .................................................... 110
LIST OF REFERENCES ................................................................................................................. 118
BIOGRAPHICAL SKETCH ........................................................................................................... 121
7
LIST OF TABLES
Table
page
2-1 Definition of direct shear-slip δ relationships ...................................................................... 35
4-1 Concrete material properties................................................................................................ .. 60
4-2 Steel reinforcements material properties .............................................................................. 60
4-3 Material model properties for beams in ABAQUS.............................................................. 67
4-4 Strain rate hardening and material enhancement factors ..................................................... 67
4-5 Comparison on ABAQUS, DSAS and experiment results.................................................. 74
5-1 Summary of columns physical and material properties....................................................... 83
5-2 Constant axial load cases ....................................................................................................... 87
5-3 .........Comparisons on displacements resulted from ABAQUS and DSAS for P1 and P2 88
5-4 .........Comparisons on displacements resulted from ABAQUS and DSAS for P3 and P4 88
5-5 ................Comparisons on displacements induced by constant and variable axial loads 101
8
LIST OF FIGURES
Figure
page
2-1 Air-burst explosion................................................................................................................. 19
2-2 Ground-burst explosion ......................................................................................................... 19
2-3 Typical blast wave pressure – time-history graph................................................................ 20
2-4 Partial and simplied time-history graph................................................................................ 22
2-5 Real and typical equivalent SDOF system ........................................................................... 23
2-6 Mass diagram.......................................................................................................................... 24
2-7 Load diagram.......................................................................................................................... 25
2-8 Resistance functions............................................................................................................... 28
2-9 Stress and strain diagram of a cross section ......................................................................... 29
2-10 Typical shear failure of a column.......................................................................................... 30
2-11 Flexure-shear interaction model – “valley of diagonal failure”. ......................................... 31
2-12 Flexure-shear interaction model............................................................................................ 32
2-13 Direct shear-slip relationship................................................................................................ . 34
2-14 Degrading stiffness method ................................................................................................ ... 38
2-15 Idealized hysteresis loops for reinforced concrete ............................................................... 38
2-16 Typical response of a SDOF system ..................................................................................... 39
2-17 Dynamic reactions for beam with arbitrary boundary conditions....................................... 42
2-18 Typical pressure-impulse diagram. ....................................................................................... 44
2-19 Search algorithm in developing P-I diagram........................................................................ 45
2-20 Typical P-I diagram for multi failure modes. ....................................................................... 46
3-1 Blast loads on structure.......................................................................................................... 48
3-2 Load diagram for supported mass ......................................................................................... 49
3-3 Load diagram for column ...................................................................................................... 49
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3-4 Newton-Rhapson method ...................................................................................................... 51
3-5 Typical load-deformation diagram of a strucutre................................................................ . 52
3-6 Spherical constant arc length criterion for SDOF system ................................................... 53
3-7 ........................................................................................................Supported mass diagram 55
3-8 Column with axial load.......................................................................................................... 55
3-9 Flow chart for determining dynamic reactions based on failure mode............................... 56
3-10 Newmark-Beta method for computing displacement .......................................................... 57
4-1 Beam 1-C layout..................................................................................................................... 60
4-2 Load function for beam 1-C ................................................................................................ .. 62
4-3 Load function for beam 1-G ................................................................................................ .. 62
4-4 Load function for beam 1-H ................................................................................................ .. 63
4-5 Load function for beam 1-I.................................................................................................... 63
4-6 Load function for beam 1-J ................................................................................................ ... 64
4-7 Typical modeling of experimental beam using ABAQUS.................................................. 65
4-8 ..........Graphical presentation of parameters in the Modified Drucker-Prager/Cap Model 66
4-9 Stress-strain relationship – tension reinforcements – beam 1-C ......................................... 68
4-10 Stress-strain relationship – compression reinforcement – beam 1-C................................ .. 68
4-11 Stress-strain relationship – tension reinforcements – beam 1-G......................................... 69
4-12 Stress-strain relationship – compression reinforcement – beam 1-G ................................ . 69
4-13 Stress-strain relationship – tension reinforcements – beam 1-H......................................... 70
4-14 Stress-strain relationship – compression reinforcement – beam 1-H ................................ . 70
4-15 Stress-strain relationship – tension reinforcements – beam 1-I .......................................... 71
4-16 Stress-strain relationship – compression reinforcement – beam 1-I ................................ ... 71
4-17 Stress-strain relationship – tension reinforcements – beam 1-J .......................................... 72
4-18 Stress-strain relationship – compression reinforcement – beam 1-J................................ ... 72
10
4-19 Typical DSAS data entry screen ........................................................................................... 73
4-20 Comparison of displacement-time history for beam 1-C..................................................... 75
4-21 Comparison of displacement-time history for beam 1-G .................................................... 75
4-22 Comparison of displacement-time history for beam 1-H .................................................... 76
4-23 Comparison of displacement-time history for beam 1-I ...................................................... 76
4-24 Comparison of displacement-time history for beam 1-J...................................................... 77
4-25 Loading function for 500 pounds of Trinitrotoluene (TNT) at 20 ft ................................ .. 78
4-26 Displacement-time history of beam 1-C ............................................................................... 78
4-27 Typical column layout in ABAQUS..................................................................................... 80
4-28 Loads and boundary conditions for column in ABAQUS................................................... 80
4-29 Longitudinal and transverse steel reinforcement layout in ABAQUS................................ 81
4-30 Displacement-time history – column subject to blast load.................................................. 81
5-1 Axial-moment interaction diagram – confined conrete – DSAS versus CRSI .................. 84
5-2 Stress-strain relationship – tension and compression reinforcements – column 1............. 85
5-3 Stress-strain relationship – tension and compression reinforcements – column 2............. 86
5-4 Stress-strain relationship – tension and compression reinforcements – column 3............. 86
5-5 Stress-strain relationship – tension and compression reinforcements – column 4............. 87
5-6 Axial-moment interaction diagram – 8 No. 7 RC – confined ............................................. 90
5-7 Displacement-time history diagram – 8 No. 7 RC – confined ............................................ 91
5-8 Flexure-resistance diagram – 8 No. 7 RC – confined ……................................................. 91
5-9 Moment-curvature diagram – 8 No. 7 RC – confined ......................................................... 92
5-10 Pressure-impuluse diagram – 8 No. 7 RC – confined.......................................................... 92
5-11 Axial-moment interaction diagram – 8 No. 10 RC – confined ........................................... 93
5-12 Displacement-time history diagram – 8 No. 10 RC – confined .......................................... 93
5-13 Flexure-resistance diagram – 8 No. 10 RC – confined........................................................ 94
11
5-14 Moment-curvature diagram – 8 No. 10 RC – confined ....................................................... 94
5-15 Pressure-impuluse diagram – 8 No. 10 RC – confined........................................................ 95
5-16 Axial-moment interaction diagram – 12 No. 11 RC – confined ......................................... 95
5-17 Displacement-time history diagram – 12 No. 11 RC, P = 0 to 570 kips – confined.......... 96
5-18 Displacement-time history diagram – 12 No. 11 RC, P > 570 kips – confined................. 96
5-19 Flexure-resistance diagram – 12 No. 11 RC – confined...................................................... 97
5-20 Moment-curvature diagram – 12 No. 11 RC – confined ..................................................... 97
5-21 Pressure-impuluse diagram – 12 No. 11 RC – confined...................................................... 98
5-22 Axial-moment interaction diagram – 4 No. 14 RC – confined ........................................... 98
5-23 ..........................................Displacement-time history diagram – 4 No. 14 RC – confined 99
5-24 Flexure-resistance diagram – 4 No. 14 RC – confined ........................................................ 99
5-25 Moment-curvature diagram – 4 No. 14 RC – confined ..................................................... 100
5-26 Pressure-impuluse diagram – 4 No. 14 RC – confined...................................................... 100
5-27 Variable axial load profile ................................................................................................... 102
5-28 Displacement-time history – 8 No. 7 RC – P ≤ Pbal + Pvar ............................................. 102
5-29 Displacement-time history – 8 No. 7 RC – P > Pbal + Pvar ............................................. 103
5-30 Displacement-time history – 8 No. 10 RC – P ≤ Pbal + Pvar ........................................... 103
5-31 Displacement-time history – 8 No. 10 RC – P > Pbal + Pvar ........................................... 104
5-32 Displacement-time history – 12 No. 11 RC – P ≤ Pbal + Pvar ......................................... 104
5-33 Displacement-time history – 12 No. 11 RC – P > Pbal + Pvar ......................................... 105
5-34 Displacement-time history – 4 No. 14 RC – P ≤ Pbal + Pvar ........................................... 105
5-35 Displacement-time history – 4 No. 14 RC – P > Pbal + Pvar ........................................... 106
12
LIST OF SYMBOLS AND ABBREVIATIONS
a Distance from support to the load
a Speed of sound 0
A Cross-section gross area g
ATM Atmospheric pressure
d Depth from the top of concrete to the layer of reinforced bars
f’ Specified compressive strength c
f Yield strength y
F(t) Load time function
h Depth of concrete cross-section
I Positive incident impulse s
I Positive normal reflected impulse r
K Tangent stiffness matrix
K Coefficient matrix τ
K Equivalent stiffness e
K Load factor L
K Mass factor M
L Wavelength of positive phase w
m Mass
M Moment
M Equivalent mass e
M Ultimate moment due to pure flexure fl
M Nominal flexural strength n
13
M Ultimate moment due to shear and flexure u
M Total mass t
N Factored axial force normal to cross section u
P Equivalent load e
Pd Downward load (t)
P Reflected pressure r
P Incident pressure so
P Actual total load t
Pu Upward load (t)
q Dynamic pressure s
R Stand-off distance G
R Load vector
R Maximum plastic-limit load m
R(t) Resistance function
SRF Shear reduction factor
t Time of arrival of blast wave A
t Positive duration of positive phase pos
U Shock front velocity
V Nominal shear strength c
w Width of concrete beam cross-section
W Trinitrotoluene (TNT) equivalent charge weight
α Angle of incident
ε Maximum compression strain cm
14
Δ Displacement
Δu Incremental displacement
ΔR Incremental load
γ Inertia and load proportionality factors
λ Load multiplier
φ Curvature
φ(x) Assume shape function
ρ Reinforcement ratio
ρ Air density behind the shock front s
ρ Density of air beyond the blast wave at atmospheric pressure 0
Ψ(x) Shape function
σ Stress
τ Maximum shear stress m
15
Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
EFFECT OF SHORT-DURATION-HIGH-IMPULSE VARIABLE AXIAL AND TRANSVERSE LOADS
ON REINFORCED CONCRETE COLUMN
By
Thien Phuoc Tran
May 2009
Chair: Theodor Krauthammer Major: Civil Engineering
Previous studies were conducted on the deformations of reinforced concrete columns
induced by blast load that combined both axial and transverse loading components. Most of
those studies assumed that the response of the mass supported by the column in its axial direction
developed much slower compared to that in the lateral movement. Thus, the load transferred
from the supported mass to the column in its axial direction could be treated as a static load.
Moreover, when comparing the vertical displacement with the lateral displacement of the
column, it was assumed that the former was much smaller, and therefore was negligible.
Consequently, the failure of the column was assumed to be governed by the flexure caused by
transverse loads. While this may be true, the effect of variable axial loads may still be an
important factor in determining the failure of the column. Thus, the above simplified assumption
should be re-examined to determine the actual effect of variable axial loads on the behavior of a
column.
16
CHAPTER 1 INTRODUCTION
1.1 Problem Statement
Progressive collapse of a building is normally caused by an abrupt failure of one or more
structural bearing members such as beams or columns. Therefore, the endurance of these
members under short duration but highly impulsive loads is crucial for the survivability of the
building. While beams are normally subject to transverse loads, columns are always exposed to
both transverse and axial loads. In practice, it is assumed that failure of a column is normally
caused by transverse rather than axial loads. This may not be accurate, particularly in the case
where a structure is subjected to short duration but highly impulsive loads such as blast loads.
While the failure of the column will most likely be induced by the transverse loads, the effect of
variable axial loads should also be considered as a contributing factor. The column resistance
may be reduced due to the variable axial loads under the same material and physical properties,
and the column may fail sooner. On the other hand, the alterations in directions and the
eccentricity of the variable axial loads over the time period may act as an enhancement factor to
the strength of the column, thus preventing it from failing in the early stage. The discussion on
eccentricity is, however, not a in the scope of this work and therefore is not included in this
study.
1.2 Objective and Scope
The objective of this research is to determine the actual effect of variable axial loads on
the column, allowing them to be properly accounted for during the design stage of structure
subjected to blast loads.
17
This research will:
• Develop a Single Degree of Freedom (SDOF) algorithm for axial and transverse loads on a reinforced concrete column and implement it within the Dynamic Structure Analysis Suite (DSAS) Version 2.0 (Center for Infrastructure and Physical Security, University of Florida (CIPPS, UF)).
• Model the columns in ABAQUS Version 6.8-1 (Dassault Systèmes, 2008) using the same
physical and material properties. • Validate the above software applications with available experimental data. • Validate analytically between the two software applications. • Conduct parametric study using DSAS Version 2.0 based on the above results
1.3 Research Significance
The outcome of this research will verify the assumption stated in the problem statement.
By knowing the significant effect of axial loads, building structures in high threat environment
will have a better chance to resist and endure when subject to blast effects. On the other hand,
for structure that may not be subject to these conditions, excluding the effect of variable axial
loads will reduce the cost and the runtime required for the analysis and design.
18
CHAPTER 2 LITERATURE REVIEW
2.1 Introduction
In the past few decades, there are numbers of books, publications, and technical papers
explaining and discussing the effects of blast load on structures, particularly on the combined
effects of flexure, axial, diagonal and direct shears (Biggs 1964, Murtha and Holland 1982,
Baker et al. 1983, Krauthammer et al. 1988). Properties and behavior of blast when exerted on a
structure were experimentally studied and recorded. Equations for calculations of all necessary
parameters were either derived mathematically or empirically. The results were well explained
and summarized in tables and charts allowing the users to expediently obtain pertinent
information for using in the design of structures (Department of the Army, 1990). Although
fragmentation is normally associated with blast, it is not within the scope of this work.
Therefore, the following sections in this chapter will provide a brief summary of these studies
with respect to blast only as well as the required information to be used in this research.
2.2 Blast loading on Structures
When explosive detonates, it generates a sudden, violent release of energy. After the
arrival time, tA, following the detonation, the pressure peaks at its highest value, Pso above the
atmospheric pressure (1 ATM = 14.7 psi). There are two types of blast bursts with each type
generates different effect . These are air-burst and ground-burst. If the charge is air-burst,
depending of the angle of attack, the Pso is further increased by the reflection off from the ground
as shown in Fig. 2-1. If the charge is ground-burst, then the Pso
is at its maximum as the
reflection of the blast wave occurs immediately as shown in Fig. 2-2.
19
Figure 2-1. Air-burst explosion (TM5-1300, 1990)
Figure 2-2. Ground-burst explosion (TM5-1300, 1990)
20
As the shock wave moves through air, it is followed by an air pressure pocket which
travels at a slower speed. This air pressure, a function of time, is known as the dynamic pressure
q(t). As shown in Fig. 2-3, two stages are formed within a very short duration. An outward
burst of the blast wave immediately raises the pressure over the ambient atmospheric pressure.
This is also known as the positive phase. Shortly after that, this pressure drops below the
ambient atmospheric pressure (negative phase) as the distance increases.
Figure 2-3. Typical blast wave pressure – time-history graph
Where: • Pso• P
Incident pressure, r
• I Reflected pressure,
s/W1/3
• I Scaled unit positive incident impulse,
r/W1/3
• t Scaled unit positive normal reflected impulse,
A/W1/3
• t Scaled time of arrival of blast wave,
pos/W1/3
• U Shock front velocity, Scaled positive duration of positive phase,
• Lw/W1/3
Ps0
P0
P(t)
ttA t0 t0
-
1 ATM
TPOS TNEG
Area iS
Ps0-
Pr
Pr-
Reflected Pressure
Incident Pressure
Scaled wavelength of positive phase.
21
A “scaled-distance”, Z, expressed in terms of stand-off distance, RG
3 WR
Z G=
, and the
Trinitrotoluene (TNT) equivalent charge weight, W, is used as a common factor to determine all
the above parameters. The scaled-distance, Z, is calculated as follow:
(2-1)
Depending on the category of explosion, air-burst (spherical) or ground-burst
(hemispherical), data for the listed parameters can be obtained from the charts provided in the
US Army TM 5-1300 (1990). Calculations for these parameters can be found in numerous
references. Brode (1955) introduced the calculations for the over pressure in unit of bars:
17.63 +=
ZPso bar for 10>soP bars
(2-2)
019.085.5455.1975.032 −++=
ZZZPso bars for 101.0 << soP bars
(2-3)
This formula was later refined by Newmark and Hansen (1961) for ground-burst
explosion:
21
33 936784
⋅+⋅=
RW
RWPso bars
(2-4)
Other parameters such as shock front velocity, U, air density behind the shock front, ρs,
and dynamic pressure, qs
00
0
776
aP
PPU so ⋅
⋅⋅+⋅
=
, were introduced by Rankine and Hugoniot (1870):
(2-5)
00
0
776
ρρ ⋅⋅+⋅+⋅
=PP
PP
so
sos
(2-6)
)7(25
0
2
PPP
qso
sos ⋅+⋅
⋅=
(2-7)
22
Where a0 is the speed of sound and ρ0
( )( )so
sosor PP
PPPP
+⋅⋅+⋅
⋅⋅=0
0
747
2
is the density of air beyond the blast wave at
atmospheric pressure.
In the case where reflection occurs, then:
(2-8)
To compute the value of pressure, P, along the curve, one can use the Modified
Friedlander equation:
( )αt
pos
eT
tPtP −⋅
−⋅= 1)( max
(2-9)
The impulse, the area under the positive portion of the blast wave – time history graph as
shown in Fig. 2-3, can be calculated by integrating Equation 2-9
∫=posT
s dttPi0
)(
(2-10)
However, approximation can be made for Fig. 2-3 by considering only the positive
portion of the curve as a triangle. It can be simplified and re-drawn as shown in Fig. 2-4
A B Figure 2-4. Partial and simplified Time-History graph. A) partial blast wave, B) simplified blast
wave
P(t)
t
P max i S
T pos
P(t)
t T pos
P max i S
23
The relation of impulse with respect to a blast pressure and time can then be
approximated based on Fig. 4B:
poss TPi ⋅⋅= max21
(2-11)
Since is and Pmax can be determined from Equations 2-4 and 2-10, the Tpos
2.3 Structure and Its Equivalent System
can be
computed by manipulating Equation 2-11.
2.3.1 Introduction
For most structure, there will be an infinite number of degrees of freedom. It will be
cumbersome and inefficient to analyze the structure in this form, not to mention the unfeasibility
at times. Therefore, it is often possible to reduce the system to a single degree of freedom
(SDOF) system to simplify the process. To achieve this, equivalent parameters for the SDOF
system such as mass, Me, stiffness, Ke, load, Fe
A
and load time function F(t) need to be setup.
Although the response of the equivalent system, in term of forces and stress, are not the same as
that of the actual system; the deformation and time, however, are the same in both systems. As
such, selection of the equivalent system should be based on the criteria that the deflection of the
concentrated mass is the same as that for a significant point on the actual structure as shown in
Fig. 2-5.
B
Figure 2-5. Real system and typical equivalent SDOF system. A) Actual structural member. B) Equivalent SDOF system
C K e
M e
W
u
L L
w(x)
u
24
2.3.2 Equivalent Mass
As shown in Fig. 2-6, m(x) is the continuous mass of the element, u is the deflection at
mid-span, u(x) represents the deflection along the element, and φ(x) is the assumed-shape
function of the element.
Figure 2-6. Mass diagram
Assuming that the displacement, u, and the velocity, u , can be approximated as:
)()(),( tuxtxu ⋅= φ
)()(),( tuxtxu ⋅= φ
(2-12)
(2-13)
Then the kinetic energy of the distributed mass system can be:
dxtxuxmKEL
∫ ⋅⋅=0
2)),(()(21
dxtuxxmKEL
∫ ⋅⋅⋅=0
22 ))(())(()(21
φ
(2-14)
(2-15)
And the kinetic energy of the equivalent system is
2))((21 tuMKE ee ⋅⋅=
(2-16)
If Me is selected so that the kinetic energy of the real system is the same as that of the
equivalent system, then by equating and simplifying the right-hand sides of Equations. 2-15 and
2-16, Me
m(x)
uu(x)x
LLϕ(x)
for the distributed mass system can be obtained and expressed as below:
or
25
∫ ⋅=L
e dxxxmM0
2 )()( φ
(2-17) Similar process can be applied for the lumped mass system, which will yield:
)]([ 2
1i
n
iie xmM φ⋅= ∑
=
(2-18)
Thus the mass factor KM is the ratio of equivalent load, Me, to actual total load, Mt
t
eM M
MK =
.
(2-19)
2.3.3 Equivalent Load
Using the parameters shown in Fig. 2-7 as well as defining w(x) as the distributed load
and Pi
Figure 2-7. Load diagram
The work done by the external load, WE, on the real system must be equal to the work
done by the external load, WE
as the concentrated load at location i, the equivalent load can also be determined
following similar approach.
e, on the equivalent system. Thus by defining, equating and
simplifying the expressions for WE and WEe, the equivalent load, Pe
P1 P2 Pnw(x)
uu(x)x1x2
xn
LL
ϕ(x)
, for the distributed load
system can be expressed as follow.
26
∫ ⋅=L
dxtxutxwWE0
),(),(
(2-20)
Where u(x, t) is as previously defined, w(x, t) is expressed as below, and φ2(x) represents
the assumed-shape function for the distributed load. Since this is a uniformly distributed-load
case, φ2
)()()(),( 2 twtwxtxw =⋅= φ
(x) = 1. The expression for w(x, t) becomes
(2-21)
Therefore,
∫ ⋅⋅=L
dxtuxtwWE0
)()()( φ
(2-22)
Or
∫⋅⋅=L
dxxtutwWE0
)()()( φ
(2-23)
The work done by the equivalent external load on the equivalent system will be:
)(tuPWE ee ⋅=
(2-24)
Equating and modifying the right-hand sides of the expressions for the work done on the
real system, WE, and the work done on the equivalent system, WEe, Pe
∫⋅=L
e dxxtwP0
)()( φ
for the distributed load
system can be obtained:
(2-25)
The same process can be repeated again for the concentrated load case; of which:
))((1
i
n
iie xPP φ⋅= ∑
=
(2-26)
Thus the load factor KL is the ratio of equivalent load, Pe, to actual total load, Pt
t
eL P
PK =
.
(2-27)
27
2.3.4 Shape Functions
As shown in Section 2.3.4, to obtain the equivalent mass and equivalent load, it is
required to use the assumed-shape functions φ and φ(x). Along with the derivation of the above
factors, Biggs (1964) also provided some assumed-shape functions for simply supported beams
and one-way slabs under various types of loading conditions. The data is, however, only
applicable for either elastic or fully plastic and nothing in between. Employment of these
assumed-shape functions will introduce some small errors and will be limited to the outlined load
cases. Therefore, to correct these errors and eliminate these constraints, modifications to the
employment of these shape functions will be required. Various approaches were done in the past
with the attempt to overcome the above-mentioned limitations. Summary of some of these
approaches will be provided in the later section.
2.3.5 Resistance Functions
In general, when an external load is exerted on a column, the column tends to produce a
resistance force trying to reinstate it to its original position. Biggs (1964) suggested, as shown in
Fig. 2-8A, the three possible shapes corresponding to the three categories of materials: brittle,
ductile, and plain concrete or instable structures. Simplification is made for most structures by
using the bilinear functions in computing the resistant factor as shown in Fig. 2-8B; where Rm is
the maximum plastic-limit load that the beam could support statically. Thus in the linear elastic
range, the resistance factor, KR, is the same as the load factor, KL, due the fact that the deflection
is the same for both systems. Since this does not apply nonlinear situation, further development
will be required and will be discussed in another section.
28
A B
Figure 2-8. Resistance functions. A) Actual. B) Simplify (Biggs, 1964)
The general equation of motion and its applicable form for a linear elastic system can be
expressed as shown in Equations 2-28 and 2-29, repectively.
)(tFukucum =⋅+⋅+⋅ (2-28)
)()( tPtRKuCuM eLe =⋅+⋅+⋅
(2-29)
Where R(t) is the resistance function that replaces the product of the spring stiffness and
the displacement for an elastic beam and is defined as the “restoring force in the spring, and the
maximum resistance is the ultimate load the beam can carry under static conditions” for an
inelastic beam (Krauthammer et al., 1988).
2.4 Static Analysis
2.4.1 Flexural Behavior
Figure 2-9 shows a typical stress and strain diagram of a cross section in the plastic
deformation state under axial load. The axial force, P, at any instance, t, can be computed from
the blast pressure p(t) as outlined in Equation 2-9.
R
∆
BrittleDuctile
Plain / Instable
R
∆
k
1
∆el
Rm
29
Figure 2-9. Stress and strain diagram of the cross section
Moment-curvature relationship can be derived using the computed values of stresses and
strains at any given point on the curve. Defining ε h/2 as the strain value at the mid-depth of the
cross section and φ is the curvature at the corresponding point, the value of strain at any depth z,
εz
zhhz
−
−=
)2()tan( )2(εε
φ
can be calculated using similar triangle. Thus:
(2-30)
Assuming small angle is used, the above equation can be re-written as:
( ) φεε ⋅
−+= zh
hz 22
(2-31)
From the constitutive relation of stress and strain, stress at any point on the cross section
can be expressed as a function of strain at the corresponding point.
( )zz f εσ =
(2-32)
The associated moment can then be defined as:
dzzhMh
z ⋅
−⋅= ∫0 2
σ
(2-33)
Equations 2-31 and 2-33 define the Moment-Curvature relationship for the cross-section.
h
b
dsb
h/2P
Strain, ε Stress, σ
εcm σc
dst z
φ
εz
εh/2
30
2.4.2 Influence of Shear on Flexural Response
Aside from flexure failure, shear failure is also a factor that needs to be considered. Two
types of shear failures commonly known are diagonal shear and direct shear. Fig. 2-10 shows a
typical shear failure of a column.
Figure 2-10. Typical shear failure of a column (MacGregor and Wight, 2009)
To account for shear effect in the design for members that are subject to shear and flexure
only, ACI 318-05 uses Equation 2-34. However, further knowledge is required to fully
understand the actual behavior and the interaction between shear and flexure.
dbfV wcc ⋅⋅⋅= '2
(2-34)
Where:
• Vc
• f' the nominal shear strength.
c
Numerous studies were conducted to examine the influence of shear on flexural response
(Kani, 1966, Placas and Regan, 1971, Haddadin et al., 1971, Bazant and Kim (1984), Ahamad
the compressive strength.
31
and Lue., 1987, Krauthammer et al., 1988). It was determined that the failure due to shear
mainly depends on the reinforcement ratio, ρ, as well as the shear span-effective depth ratio, a /d;
where a is the distance from the support to the load and d is depth taken from the top of concrete
beam to the first layer of reinforcing bars. Figs. 2-11 and 2-12 show two models proposed by
Kani (1966) and Ahamad et al. (1987) respectively; where Mu/Mfl
Figure 2-11. Flexure-shear interaction model – “valley of diagonal failure” (Kani, 1966)
is the ratio between the
ultimate moment due to shear and flexure and the ultimate moment due to pure flexure. This
ratio is known as the shear reduction factor (SRF).
32
Figure 2-12. Flexure-shear interaction model (Ahmad et al., 1987)
These two models were evaluated and further developed and modified by Krauthammer
et al. (1988) and Russo et al., (1991 and 1997). Detailed discussion on the developments by
these two authors is not part of the scope of this work and can be found in Krauthammer et al.
(2004). Based on the result found, approach by Krauthammer et al. (1988) concluded in the
closer match with the experimental data presented by Kani (1966). Hence, this research paper
will adopt this approach. This means that to take the shear effect into account more correctly,
the moment will be multiplied by the SRF and the curvature will be divided by the SRF
(Krauthammer et al., 2004). This, in turn, produces a more accurate result for both moment and
curvature.
2.4.3 Influence of Axial Force on Shear and Flexural Responses
The presence of axial load enhances the moment capacity of the cross section as well as
delays cracks from occurring (Krauthammer et al., 1988). This is because the compressive load
increases the normal stress and reduces effect of the principal tensile stress. To maintain
equilibrium, the sum of forces must still be zero. Thus:
33
∑ ∫ =−=h
z PdzFx0
0σ
(2-35)
The current code, ACI 318-05, accounts for the axial load effect on flexure and shear by
employing the equations for nominal shear strength, Vc, and nominal flexural strength, Mn
dbfA
NV wc
g
uc ⋅⋅⋅
⋅+⋅= '
200012
.
Equations 2-36 and 2-37 represent the lower and upper bound of the shear strength respectively:
(2-36)
g
uwcc A
NdbfV
⋅+⋅⋅⋅⋅≤
50015.3 '
(2-37)
( )( )
−⋅⋅−=
84 dhNMM uun
(2-38)
Where: • Vc
: nominal shear strength provided by concrete.
• Nu
: factored axial force normal to cross section.
• Ag
: section gross area.
• f’c
: specified concrete compressive strength.
• bw
: width of the section.
• d: is the effective depth which is from the top compression fiber to the centroid of the longitudinal tension reinforcement.
Mattock and Wang’s studies (1984) conducted a study and found that Equations 2-36 and
2-37 were too conservative. The recommended replacements for Equations 2-36 and 2-37,
respectively, are:
⋅⋅
+⋅⋅= '' 3
12cg
ucc fA
NfV
(2-39)
34
g
ucc A
NfV
⋅+⋅≤
3.05.3 '
(2-40)
2.4.4 Direct Shear Mode of Response
There have not been many studies done on the direct shear effect. In addition to the
above mentioned factors, Park and Paulay (1975) also suggested that the transferring of high
shear stress across a weak section where cracks have formed was another factor that indicated
the significant effects of direct shear on flexural members. The failure due to direct shear was
further proven by the dynamic tests that were conducted by Kiger et al. (1984) and Slawson
(1984). Hawkins (1974) proposed an empirical model on the behavior of shear stress versus slip;
which did not include the effect of compression loads. This relationship was later modified by
the Krauthammer et al. (1988) to include the effects of load reversals.
Figure 2-13. Direct shear – slip relationship (Krauthammer et al., 2002)
Summary of the slip range and the associated descriptions for each segment in Fig. 2-13
is shown in Table 2-1.
O
A
B C
D E
τm
τLτe
δ1 δ2 δ3 δ4 δmax
Shear Stress, τ
Slip, δ
Ke
Ku
35
Table 2-1. Definition of direct shear – slip δ relationship (Hawkins, 1974)
Segment Slip
δ x 10 Description -3,
in OA 0 – 4 • Elastic response.
• Positive slope δτ eeK = .
• 2
157.0165 ' mce f
ττ ≤⋅+=
(2-41)
AB 4 – 12 • Slope decreases but remains positive. • Reaches maximum shear stress, τm
• .
'' 35.08.08 cyvtcm fff ⋅≤⋅⋅+⋅= ρτ
(2-42)
• Where ρvt
g
svt A
A=ρ
is the ratio between the area of reinforcement crossing the
shear plane and the gross area,
fy steel yield strength BC 12 – 24 Maximum shear stress, τm remains constant.
CD ≥ 24 • Constant decrease slope. • Independent of reinforcement crossing the shear plane. • '75.02000 cu fK ⋅+= (2-43)
DE > 24 • Shear capacity remains constant. • Deformation at E varies with the level of damage.
Krauthammer et al. (2002) went further to define the upper limits of segments CD and
DE. Equations 2-42 and 2-43 show the upper limits of D and E respectively:
g
ssL A
fA '85.0 ⋅⋅=τ
(2-44)
−=
601
max
xeδ
Where
b
c
df
x'
86.2
900
⋅
=
(2-45)
36
Based on the proven fact that axial load increases the strength of the beam in a sense that
it deters the cracks from extending into the compression block, it is safe to assume that the
maximum stress will also increase. Hence, it is proposed that with the effect of axial load, the
value of τm (Equation 2-42) will increase its value by 1 + (Nu / (2000*Ag)). Thus the equation
of τm
'' 35.08.02000
18 cyvtcg
um fff
AN
⋅≤⋅⋅+⋅
⋅+⋅= ρτ
can be re-written as below. This point still needs to be proven as part of this research.
(2-46)
2.5 Dynamic Analysis
2.5.1 Newmark-Beta Method
For a nonlinear system, a different approach will be required to obtain the solution for the
equation of motion (Equation 2-28). There are numbers of methods that can be used to for
computing the displacements, some of which are the acceleration method (implicit method) and
the central difference method (explicit method). The main difference between these two is the
time at which the equation of motion is satisfied. The former method satisfies the equation of
motion at time ti+1 and the latter one is at time ti
• Knowing y
. For this research, the linear acceleration
(implicit) method will be used. It is basically a special case of Newmark-Beta Method
(Newmark and Rosenblueth, 1971), where γ = 1/2 and β = 1/6. A brief summary for this method
is as follow:
i iy and at time ti
iy and with Δt is the time step, from the equation of motion
(Equation 2-12), compute for . • Estimate 1+iy . • Compute 1+iy , and yi+1
using the two equations below with the values of γ = 1/2, β = 1/6.
[ ]iiii yytyy ⋅−+⋅⋅∆+= ++ )1(11 γγ (2-47)
37
( )
⋅
−+⋅⋅∆+⋅∆+= ++ iiiii yytytyy ββ
21
12
1
(2-48)
Using the equation of motion, Equation 2-28, and the above computed values of 1+iy and
yi+1, 1+iycompute for .
• Check if convergence is satisfied. If so, move to the next time step; if not, use the newly computed value of 1+iy for the next iteration.
To ensure stability and accuracy for the nonlinear system, Δt should be less than πT⋅3 ,
where T is the natural period of the system. A flowchart for this method will be presented in the
next chapter.
2.5.2 Dynamic Resistance Functions
The dynamic resistance responses of a structure depend on a number of factors. These
include the stability nonlinearity, the geometric nonlinearity, and material nonlinearity of the
structure. When an external load that acts on a structure is less than the yield load of the
structure, the response of the structure is still in the elastic range. Thus, the resistance force or
the internal force is the product of structure stiffness and the displacement (Fint = R = k*u).
However, when the external load is greater than the yield load, the resistance force or the internal
force is no longer linear and becomes a function of displacement (Fint = R = fn(u)).
Figs 2-14 and 2-15 represent two models that were proposed by Clough et al. (1966) and
Sozen (1974) that illustrated the nonlinear behaviors of a structure. Both of these models were
portrayed by a number of bilinear resistance functions, of which each segment represented a
loading stage. The order of loading sequence is as shown in alphabetical order.
38
Figure 2-14. Degrading stiffness model (Clough, 1966)
Figure 2-15. Idealized hysteresis loops for reinforced concrete (Sozen, 1974)
Based on these two models, Krauthammer et al. (1988) proposed a piecewise multi-
linear-curve model (Fig. 2-16) that included all the above mentioned nonlinearities and portrayed
both elastic and inelastic response of a beam with the following assumptions:
a bh
/ Yi
g
co
de
f
M/MU
- 1
1
1
a, b, c, ... is the loading sequence
39
a. Beam is symmetrically reinforced. For unsymmetrical reinforced beam, two R-Δ curves will be required for both negative and negative loadings.
b. Maximum displacement is reached during the first positive cycle, which is valid only for
blast or impact loads.
The elastic range exists when the maximum dynamic displacement (Δmax) is less than the
yield displacement (Δy). Within this range, the beam will behave elastically and oscillation will
occur along line A-A’. This oscillation will eventually come to rest once external damping
dissipates all the energy. As Δmax becomes greater than Δy, the beam will be in the inelastic
range, at which point permanent deformation will take place. The order of points A, B, D, E, F,
G, D’, and E’ in the figure below (quadrants I and IV) describes the sequence of loading and
unloading that eventually forms the plastic deformation (Δplastic
) of the beam when all the energy
is completely dissipated. Point C represents the flexure failure of the beam.
Figure 2-16. Typical response of a SDOF system (Krauthammer et al., 1988)
40
2.5.3 Modified Equivalent Parameters for SDOF System.
As mentioned in Section 2.3, the mass factor and the load factor derived by Biggs (1964)
were based on the assumed-shape functions that were either for elastic or fully plastic and
nothing in between. Moreover, these assumed-shape functions were also derived for specific
cases of end supports and loading conditions. With a better insight on the behaviors of a structure
member in both elastic and inelastic range based on the above model proposed by Krauthammer
et al. (1988), these factors can be re-evaluated to account for the behaviors of the member in the
transition stage between elastic and inelastic. The procedures to compute mass factor and load
factor were extracted from the above mentioned reference.
2.5.3.1 Mass factor
The mass factor can now be computed along the load-deflection curve by taking the
integration over the length of the beam for each load step j. Using Fig. 2-6 and all previously
defined variables, the equivalent mass and the mass factor at load step j can be expressed as
below.
( )[ ]∑∫=
⋅+⋅⋅=n
ijiji
L
je xmdxxxmMj
1
2
0
2 )()()( φφ
(2-49)
t
eM M
MK j
j=
(2-50)
Thus the mass factor, KM ,
( )jjj
MMMM
jJ
j
KKKK ∆−∆⋅
∆−∆
−+=
+
+
)1(
)1(
for each time step can be computed:
(2-51)
Where Δj < Δ ≤ Δ
Reference to Fig. 2-16, K
(j+1).
M is computed for every time step until it reaches point B,
which is the maximum inelastic displacement. Beyond that, it will remain constant based on the
41
assumption that the shape functions do not significantly change after the formation of the plastic
hinges.
2.5.3.2 Load factor
The procedure to re-evaluate the load factor for the transition stage between elastic and
plastic is similar to that of the mass factor. Hence, using Fig. 2-7, for each load step j with shape
function φ(x) (j,i)
),()( ijL xKj
φ=
where i is the location to be evaluated, the load factor for each load step can be
expressed as:
(2-52)
Thus the load factor for each time step will be
)()1(
)1(
jjj
LLLL
jJ
j
KKKK ∆−∆⋅
∆−∆
−+=
+
+
(2-53)
Where Δj < Δ ≤ Δ (j+1)
2.5.4 Dynamic Reactions
.
Biggs (1964) provided a series of tables of equations for the dynamic reactions of various
loading scenarios with different boundary conditions. As mentioned earlier, the limitations of
these equations are that they are only applicable to either perfectly elastic or plastic structures
and are bounded by specific load cases. Computations of the dynamic reactions based on these
equations will introduce inaccurate results for the purpose of this paper. Therefore, another
approach is required. Krauthammer et al. (1988) introduced a procedure that could be used to
neutralize the above limitations. This procedure was based on the assumption that the
“distribution of the inertia forces is identical to the deformed shape function of the beam” as
shown in Fig. 2-17.
42
Figure 2-17. Dynamic reactions for beam with arbitrary boundary conditions (Krauthammer et al., 1988)
Summary of the procedure is as follow (Krauthammer et al., 1988):
a. For each load step i, obtaining the reactions at each end of the of the element and compute the corresponding load proportionality factor γ:
iii QQ /11 =γ (2-54)
iii QQ /22 =γ (2-55)
Where:
Q1i and Q2i Q
are the static reactions at load step i of end 1 and 2. i
γ is the load at step i.
1i and γ2i
are the load proportionality factors at end 1 and 2.
b. For every load step, compute the Inertial Load Factor, IFL:
dxxL
ILFL
ii ))((10∫⋅= ψ
(2-56)
Where:
IFL is the load factor associated with the distribution of the inertial forces. ψ(x)i is the deflected shape function at load step i. c. Compute the inertia proportionality factors γ’1i and γ’2i
at every load step i. These factors are approximated using the principles of the linear beam theory.
d. Compute the dynamic reaction at each end of the element for each time step:
γ'1i γ'
2i
Q(t)
γ1Q(t) γ2Q(t)
Ψ (x)
M1 M2
43
( ) ( )XMILFtQV tii ⋅⋅⋅+⋅= 111 )( γγ
(2-57)
( ) ( )XMILFtQV tii ⋅⋅⋅+⋅= 222 )( γγ
(2-58)
Where:
Q(t) is the forcing function. Mt
is the mass of the beam.
X is the acceleration. γ1i, γ2i i1γ, and i2γ specific displacement at every time step by using the following linear
interpolation equation:
( )iii
iii ∆−∆⋅
∆−∆
−+=
+
+
1
1 γγγγ for 1+∆<∆<∆ ii
(2-59)
Δ is the dynamic displacement at the specific time step. γ is the generic name for inertia and load proportionality factors.
2.6 Pressure-Impulse (P-I) Diagrams
2.6.1 Characteristics of P-I Diagrams
P-I diagrams are graphical tools used to determine the potential damage of a structure
caused by dynamic loads. Detailed descriptions on P-I diagrams can be found in numerous
references (Krauthammer, 2008). There are three distinguished regions on the P-I curve, as
shown in Fig. 2-18. These are the Impulsive Loading Region, the Quasi-Static Loading Region
and the Dynamic Loading Region. In addition, there are two asymptotes. The Impulse
Asymptote is tangent to the Impulsive Loading Region and the Pressure Asymptote is tangent to
the Quasi-Static Loading Region. In the Impulsive Loading Region, the response time of the
structure is much longer than the duration of the loading. Hence, before the structure can
experience any permanent deformation, the load is already dissipated. In the Dynamic Loading
Region, the duration for both loading and natural period is approximately the same. The
44
response of the structure in this region depends on the loading history. In the Quasi-Static
Loading Region, the loading duration is much longer than the natural period. Therefore, the
structure experiences maximum deformation before the load completely dissolves. (Smith and
Hetherington, 1994)
Figure 2-18. Typical pressure-impulse diagram
2.6.2 Derivation of P-I Diagrams
Although approach to develop a P-I diagram for a complex non-linear structure could be
possible, it would be, however, extremely cumbersome. Hence, numerical approach should be
used. The P-I diagram is derived from the results of numerous single dynamic analyses, where
the computed threshold points are used to plot the P-I curve. Since the process to obtain these
threshold points is intensive in term of computational time, an effective and efficient search
algorithm is required.
Impulse, psi-msec
Pres
sure
, psi
0 2 4 6 8 10 12 14 16 18 20-100
0
100
200
300
400
500
Impulsive Loading Region
Quasi-Static Loading Region
Pressure Asymptote
Impulse Asymptote
Dynamic Loading Region
45
Blasko et al. (2007) developed a good search engine where a single radial search
direction was originated from an arbitrary pivot point that was located in the failure zone of the
P-I diagram as shown in Fig. 2-19. The iteration process continued where another arbitrary point
between the point in the safe zone and the first assumed point was evaluated. The same
procedure was repeated until all the threshold points were successfully acquired.
This process can be completed for structure that may also experience more than one
failure modes such as shear and flexure. The results from both of these failure modes can then
be plotted together for use in evaluating the structure to determine the likelihood of the mode of
failure as shown in Fig. 2-20.
Figure 2-19. Search algorithm in developing P-I diagram (Blasko et al., 2007)
46
Figure 2-20. Typical P-I diagram for multi failure modes (Chee, 2008)
2.7 Summary
In this chapter, the property of explosive blast and the behavior of a beam under the
influence of transverse and axial loads due to blast pressure were briefly reviewed along with the
possible failure modes due to shear and flexure. The effect of axial load on these modes of
failures was also considered. Moreover, the transformation from an actual structure to an
equivalent SDOF system through the use of equivalent parameters such as mass factor, load
factor and the dynamic reactions that were based on the employment of the assumed-shaped
functions was discussed. Since this problem includes nonlinearity, closed-form approach would
be very cumbersome and inefficient. Hence, direct integration techniques – both implicit and
explicit – were considered. A short summary on the Pressure-Impulse diagram and how it can be
used to quickly determine the failure of a structure was provided. The brief discussions on each
topic in this chapter provide adequate source of information to form a basis for the analysis that
will be discussed in the next chapter.
Pressure
Impulse
Mode 1 (Flexure)
Mode 2 (Direct Shear)
Failure in Mode 1 & 2
Failure in Mode 1
Failure in Mode 2
Safe
47
CHAPTER 3 METHODOLOGY
3.1 Introduction
This chapter provides an overview of the structure being considered as well as outlines
the approaches used in determining the effects of variable and constant axial loads on a
reinforced concrete column that is also subjected to transverse loads. Assumptions and
simplifications used in the implementation steps will also be included in the appropriate sections
of this chapter.
3.2 Load Determination
3.2.1 Overview of Structure
In either case of air or ground-blast as shown in Figs. 2-1 or 2-2, once the first wave of
blast strikes the building, it will destroy architectural items such as windows and doors. This
creates openings in the structure allowing subsequent blast waves to act as internal pressures in
the outwards and upwards directions against walls, floor and roof of the building. The structure
considered in this case is as shown in Fig. 3-1, of which only the column is being investigated. It
should be noted that the loading diagram caused by both exterior and interior pressure is much
more complicated than shown. For simplicity purpose, only loads that have significant effects on
the column are being considered. In addition, although the arrival time, tA, of the blast load will
be different for the column and the supported mass; for ease of computations, it is assumed that
tA is the same for the above mentioned structures. Moreover, to reduce the computer runtime,
the end boundary conditions for the column will be reduced to simply support condition rather
than fixed-fixed condition.
48
Figure 3-1. Blast loads on structure
The column of the above structure will be subject to flexure, diagonal shear and direct
shear induced by:
a. Lateral dynamic loads F(t). b. Downward loads, Pdc. Upward loads, P
(t). u
The equivalent system for the above structure will be a multi-degree of freedom (MDOF)
system under the influence of transverse loads and axial loads. Solving for the above structure
using multi-degree of freedom approach may be inept. Thus the system will be “de-coupled”
into two independent members that will eventually be reduced to two equivalent single degree of
freedom systems.
The first member to be considered is the supported mass. As shown in Fig. 3-2, P(t) is the
net load resulting from the differential pressure between upward and downward pressure. R
(t).
1(t)
and R2
F(t) HH
Pd(t)
Pu(t)
LL
(t) are the dynamic reactions, M is the internal moment, and Δ is the deflection due to
load.
49
A Β
Figure 3-2. Load diagram for supported mass. A) Load diagram, B) Free-body diagram
The second member to be considered is the column; which is the main focus of this
research. Loads that act on this member include the transverse load F(t), the internal moment M,
the weight of the structure above, the variable axial loads which are the dynamic reactions
resulting from the first member, R(t), and the self-weight of the column.
It should be noted that in both cases, members are subject to transverse loads that could
induce failure by either shear or flexure. Thus, consideration of the effects of transverse load is
required.
A
B
Figure 3-3. Load diagram for column. A) Initial stage, B) Deformed stage
R(t)
δ
M Plastic Hinge 3 Locations (Typ)
M
H H F(t)
R(t)
H H F(t)
? M M
P(t)
L L L L
P(t)
50
3.2.2 Effect of Transverse Loads.
Previous studies on box-type buried reinforced concrete structure subject to blast load
(Kiger et al., 1984, Slawson et al., 1984, and Ross, T.J. 1983 and 1985) indicated that when
element of the studied structure failed due to shear, the flexural response was negligible. On the
other hand, when element of the studied structure failed due to flexure, the element was able to
withstand the shear forces at the early stage. Based on the result, Krauthammer et al. (1988)
suggested the employment of two separate SDOF systems for evaluating flexural and direct
shear responses of a beam. Verification against the failure criterion of the computed results from
these two SDOFs was conducted at the end of each time step to determine the mode of failure.
As mentioned above, the failure of the two members could also be induced by either flexure or
direct shear. Therefore, this research paper will employ the approach suggested by
Krauthammer et al. (1988). This means that each of the two members described above will have
two SDOF systems. The failure mode of the first member will be used as a governing factor in
the computation for the second member. In other word, the dynamic reactions resulting from the
failure mode in the first member, by direct shear or flexure, will be used as the variable axial
loads acting on the second member.
3.2.3 Axial Loads
As indicated above, two types of axial loads will be used in the modeling and the
required computations. These are constant and variable axial loads. The constant axial loads are
assumed to be induced by the weight of the supported mass over the period of time. Application
of various magnitudes of constant axial loads to the column will be considered for comparison
purpose. Variable axial loads are derived from the computations of the dynamic reactions
caused by the effect of the blast load on the supported mass.
51
3.3 Load Deformation Analysis
Within the elastic range, the stress distribution of a cross section at any point along the
beam will remain linear. As the load increases, the resulting moment also increases until it
passes the yield point; when the stress distribution of the cross section becomes nonlinear and a
plastic hinge is formed. Since the load-deformation relationship is no longer linear, equations
derived for the linear elastic are no longer valid. As such, a different method is required to trace
the nonlinear path of the beam behavior.
One of the most commonly used methods was the Newton-Rhapson method (Fig. 3-4);
where for any load-displacement function, the displacement value at point B could be determined
based on the known value at point A. The aim of each iteration, i, was to reduce the “out-of-
balance” load ΔR i or the displacement Δu i
)1()1( −∆=∆⋅− iiB RuK
i
to a satisfactorily small value. Thus:
(3-1)
Where:
• i the iteration number. • K tangent stiffness matrix. • Δu i incremental displacement at ith
• ΔR iteration.
(i-1) incremental load at (i-1)th
iteration.
Figure 3-4. Newton-Rhapson method (Bathe, 1996)
uA uB Displacement, uu1 u2
KB
1Load
RA
RB
∆u1 ∆u2
RB - FB
RB - FBKB
1A
A 1
1
52
It was, however, found that this method did not converge for zero slopes. Thus for
typical load-deformation diagram as shown in Fig. 3-5, this method would not yield the best
result as it might not be able to pass the post-buckling response point.
Figure 3-5. Typical load-deformation diagram of a structure (Bathe, 1996)
To account for this issue and to allow for the post-buckling response of a structure, the
Cylindrical Arc-Length method developed by Crisfield M.A. (1981) is used. This procedure
allows the tracing to be possible even when the slope of load-deflection curve is negative. The
main difference between this method and the Newton-Rhapson method is the assumption that the
load vector, R, varies proportionally during the response calculation and the use of the load
multiplier; which will need to be determined. In short, Newton-Rhapson is a forced-controlled
method; whereas, Cylindrical Arc-Length is a displacement-controlled method. Using Fig. 3-6,
the algorithm of this procedure can be described as follow with detailed discussion can be found
in the reference (Cook et al., 1974):
Displacement
Load
Large load increments
Small load increments
Loaddecreases
Postbucklingresponse
53
Figure 3-6. Spherical constant arc length criterion for SDOF system (Cook et al., 1974)
a. The governing finite element equations for n equations in (n+1) unknowns
0)()( =−⋅ ∆+∆+ tttt FRλ
(3-2)
Or
[ ]{ } [ ])1()()1()( −∆+−∆+ −⋅∆+=∆⋅
ittiitti FRuK λλτ
(3-3)
Where:
• λ(t+Δt)
is the load multiplier at time (t+Δt) that needs to be determine and can be increased or decreased.
• R is the reference load vector for n DOFs of the FEA model. It can contain any loading on the structure but is constant throughout the response calculation.
• F(t+Δt)
is the vector of n nodal point forces corresponding to the element stresses at time (t+Δt).
• i is the iteration order. • Kτ
is the coefficient matrix.
b. Additional equation requires for determining the unknowns – vector of displacement increments, Δu, and load multiplier increment, Δλ – is a constraint equation between Δu i and Δλ i
.
Displacement, u
Load
λRA
λRB
∆l
54
0),( =∆∆ ii uf λ (3-4)
Where:
[ ] titti uuu −= ∆+ )(
(3-5)
[ ] titti λλλ −= ∆+ )( (3-6)
c. Effective constraint equation is given by the spherical constant arc length criterion
( ) 22)( luu i
Ti
i ∆=⋅
+β
λ
(3-7)
Where
• ui is the total increment of displacement within each load step for the ith
• λ iteration.
i is the total increment in the load multiplier for the ith
• iteration.
l∆ is the arc length for the time step. • β is the normalizing factor.
3.4 Computations of Dynamic Reactions, Shear and Flexural Responses
As mentioned in Sect. 3.2.1, two separate SDOF systems will be used for the supported
mass and the column respectively. Within the SDOF system for the supported mass, a sub-set of
two SDOF systems will be used to determine the mode of failure. The governing dynamic
reactions of the supported mass – produced either by shear or flexure – will be used as the time-
dependant axial loads applied to the column.
3.4.1 Computation of Dynamic Reactions for the Supported Mass
The equivalent SDOF system and the associated free-body diagram, as shown in Fig. 3-7,
can be used to illustrate all the pertinent loads used in the computations. The complete process
in determining the mode of failure of a structure as well as obtaining the required parameters –
such as acceleration, velocity, displacement and resistance – associated with the failure mode is
shown in Fig 3-9. Figure 3-10 presents the Newmark-Beta method, which is a part of the
55
complete process. Both of Figs 3-9 and 3-10 are applicable to Section 3.4.1 and Section 3.4.2.
The results obtained from the complete process, particularly the dynamic reactions, will be used
for the necessary computations in the next section
A B
Figure 3-7. Supported mass diagram. A) SDOF for supported mass. B) Free-body diagram 3.4.2 Computation of Shear and Flexural Responses on Column.
The column is subjected to three different types of loads. Firstly, it is experienced by a
transverse load that could cause failure in either flexure or shear. Secondly, it is exerted by the
dead load of the structure directly above it as well as its own weight. These are considered to be
the static loads. Finally, it is also experienced by the variable axial loads which are the dynamic
reactions resulted from the supported mass being subjected to the blast load.
A
B
Figure 3-8. Column with axial load. A) Equivalent SDOF system. B) Free-body diagram
R F (t)
R A (t)
C*x' M Fe *x" F(t) C
K F
M Fe F(t)
x
P(t)
R A (t) C*y'
M Ae *y" y
C K A
M Ae
P(t)
56
No
yes
Input Parameters, Pertinent Data
Compute Velocity, Displacement, Acceleration, Mass and Load Factors, Resistance
START
Initialize All Variables
Initial TimeIncrement by Δt
Last Time Step?
Compute Applied Force(for beam and column)
Convergence?
Compute Plastic Displacement
Direct Shear Analysis?
Compute Shear Force
Compute Shear Velocity, Displacement, Shear Resistance and Acceleration
Convergence?
Shear Failure?
Outputs
STOP
Flexural Failure?
No
yes
No
No
No
yes
Process to compute
listed variables
using Newmark-
Beta Method
Process to compute
listed variables
using Newmark-
Beta Method
Figure 3-9. Flow chart for determining dynamic reactions based on failure mode
57
Yes
Yes
START
Known Initial Conditions:
Select Time Step Δh
Estimate
Output
No
Start Counter
Increment CounterCounter - 1
STOP
Fe, Me, Re
Compute 0umuutFuuu /),,(2 0
200 =++ ωξω
)1( +iu
)1( +← itrial uu
( ) ( )( ) 21)1( 2
1 huuhuuu iiiii ∆⋅⋅+⋅−+∆⋅+= ++ ββ
)2()( )1()1( huuuu iiii ∆⋅++= ++
[ ]{ } ( ) )1(2
)1()1()1()1()1( 2,, ++++++ ⋅−⋅⋅⋅−−= iieeeiiii uuMRMuutFu ωωξ
0)1( ≈−+ triali uu
uuu ,,
00 ,uu
Figure 3-10. Newmark-Beta method for computing displacement
58
3.5 Summary
This chapter provided the layout of the structure as well as outlined the assumptions and
limitations applied to the column being considered. It described the loads exerted on each
member of the structure and the computation process to transfer the applicable loads from one
member to another. The effect of transverse load on the column was also discussed. Approach
on load-deformation analysis was outlined along with the approach on the computations of the
dynamic reactions, shear and flexural response of the equivalent SDOF system.
59
CHAPTER 4 ANALYSIS
4.1 Introduction
Numerous steps were taken in order to confirm the objective. Firstly, validations for the
ABAQUS Version 6.8-1 (Dassault Systèmes, 2008) and the Dynamic Structure Analysis Suite
(DSAS) Version 2.0 (CIPPS, 2008)) were required to ensure that they could produce reliable
results. This was completed by comparing the results obtaining from these two applications
against known data of a series of experiment on reinforced concrete beams. Secondly, the
validation was completed on a reinforced concrete column. However, since there was no
experimental data available for the column, these two software applications were validated
analytically using a standard size column with the same material properties of the experimental
beam. Lastly, a series of columns of the same physical and material properties but different
reinforcements configurations was arbitrarily picked from the Concrete Reinforcing Steel
Institute Design Handbook (CRSI, 2002) for further analysis in the parametric study to determine
the effect of axial loads on the P-I relationships. This was followed by the analysis on the effects
of variable axial loads on the columns. The computer code DSAS was modified to address the
effects of axial force on RC columns, and it was used to derive the corresponding P-I curves.
4.2 Description of DSAS
DSAS (CIPPS 2008) is a comprehensive software suite developed for the analysis and
assessment of structural members subject to severe dynamic loads such as blast and impact. The
primary analysis engine in DSAS is based on an advanced single degree of freedom (SDOF)
formulation and is capable of developing fully non-linear resistance functions for reinforced
concrete, steel, masonry and other members with diverse end conditions using force or
displacement-controlled solution procedures (Krauthammer et al., 1990 and 2003, DSAS User
60
Manual, 2008). The moment curvature relationships for RC that are derived by DSAS are based
on layered section analysis with fully nonlinear material models for steel and confined and/or
unconfined concrete. The resistance function is based on a displacement controlled solution
approach, and the Direct Shear function uses the Hawkins model. The present study enabled the
development of an enhanced version of DSAS that allows for constant gravity loads to be
specified and modifications can be made to account for dynamic variations in axial force.
Moreover, DSAS is also capable to conduct Physics-based P-I analysis and produce the P-I
diagrams.
4.3 Validations with Experimental Data
4.3.1 Experimental Data
4.3.1.1 Material and physical properties
Data of five beams from the experiment conducted by Feldman and Siess (1958) were
used as part of the validation. Detailed layouts of steel reinforcements for all beams are shown
in Fig. 4-2. The main difference in the physical configuration between beam 1-C and the other
four beams was the transverse reinforcements. Beam 1-C used opened stirrups and the other two
beams used closed stirrups. Other properties of the beams are described in Tables 4-1 and 4-2.
Figure 4-1. Beam 1-C layout (Feldman and Siess, 1958)
61
Table 4-1. Concrete material properties b = 6 in, h = 12 in, d = 10 in, d’ = 2.0 in, span =106 in
Beam Compressive Strength, f’c
Modulus of Elasticity, Ec, ksi , ksi
Rupture Strength, fr, ksi
1-C 5.67 4292.1 0.9 1-G 6.21 4491.8 1.0 1-H 6.15 4450 0.935 1-I 6.50 4730 0.85 1-J 6.09 3890 0.935
Table 4-2. Steel reinforcements material properties Tension reinforcement = 2 #7 bars, compression reinforcement = 2 #6 bars, stirrups = 16 #3 bars at 7 in on center Beam Compression reinforcements Tension reinforcements
f’y E’, Ksi s ε', Ksi y ε', in/in sh f, in/in y E, Ksi s ε, Ksi y ε, in/in sh, in/in 1-C 46.70 --- --- --- 46.08 29520 0.0016 0.0144 1-G 48.30 --- --- --- 47.75 --- --- --- 1-H 47.61 32280 0.0015 0.015 47.17 34900 0.0014 0.0125 1-I 47.95 --- --- --- 47.00 32600 0.0014 0.015 1-J 48.86 29560 0.0016 0.012 47.42 --- --- ---
4.3.1.2 Loading functions
Loading functions for all beams were reproduced by extracting the points from the
loading graphs provided in the experimental report. For beam 1-G, due to the failure of test
recording equipment, the actual loads were not properly recorded. Hence, it had to be estimated.
Two main criteria were based on when conducting this process. Firstly, the load profile was
assumed to follow the shape of the sum of the reactions curve. Secondly, it must take into
account the inertia effect of the beam under loading. Loading for beam 1-G was stopped at
0.072 sec when the wooden stop was hit. For beam 1-J, it should be noted that it was subjected
to two separate sets of impact loads. Similar to the loading situation for beam 1-G, the first set
of loads stopped at 0.072 sec and resulted in a displacement of 0.5 in. It was then subjected to
the second set of loads which was stopped at approximately 0.068 sec. For simplicity, these two
sets of loads were combined in one simulation. Figs 4-2 to 4-6 show the loading functions for
the beams.
62
Figure 4-2. Load function for beam 1-C (Feldman et al., 1958)
Figure 4-3. Load function for beam 1-G (Feldman et al., 1958)
Time (msec)
Load
(kip
s)
0 10 20 30 40 50 60 70 80 90 100 110 120-5
0
5
10
15
20
25
30
35
Time (msec)
Load
(kip
s)
0 10 20 30 40 50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
Loads, Kips (Actual)Loads, Kips (Estimated)
63
Figure 4-4. Load function for beam 1-H (Feldman et al., 1958)
Figure 4-5. Load function for beam 1-I (Feldman et al., 1958)
Time (msec)
Load
(kip
s)
0 10 20 30 40 50 60 70 80 90 100 110 120-5
0
5
10
15
20
25
30
35
40
Time (msec)
Load
(kip
s)
0 20 40 60 80 100 1200
10
20
30
40
Stop at 72 msec
64
Figure 4-6. Load function for beam 1-J (Feldman et al., 1958)
4.3.2 ABAQUS Validations
All five beams were modeled with ABAQUS Version 6.8.1 (Dassault Systèmes, 2008).
Although various mesh sizes were evaluated, it was found that a cubical mesh size of 1 inch
yielded the most effective and economical results in terms of computer runtime and the accuracy
of the outcomes. Hence, this mesh size and the solid element type C3D8I were used in the
modeling of all the beams. Various types of material models in ABAQUS were also explored
for the modeling of the concrete beam. It was determined that the Modified Drucker-Prager/Cap
Model was the most suitable one to be used in this case for numerous reasons. Firstly, it could
respond to large stress reversals in the cap region. Secondly, it provided the required inelastic
hardening mechanism to account for the plastic compactions as well as the necessary controlling
of the material expansion when yielding in shear (ABAQUS Analysis User’s Manual). With
Time (msec)
Load
(kip
s)
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
First stop @ 72 msec
First stop @ 68 msec
65
regards to the longitudinal and transverse reinforcements, two different element types were used.
Beam element, type B31, was used to model the compression reinforcements. Surface element,
type SFM3D4R, was used for the modeling of tension and transverse reinforcements. This was
done because the surface-element type required less work to model compared to the beam-
element type while still yielding the same result. Bond slip between concrete and steel
reinforcements was taken into account using the built-in capability of ABAQUS known as the
embedded region constraint. Fig. 4-7 shows a typical ABAQUS (Dassault Systèmes, 2008)
model for all the above-mentioned beams.
Figure 4-7. Typical modeling of experimental beams using ABAQUS
Six parameters were required for the Modified Drucker-Prager/Cap Model. These were
the material cohesion (d), the angle of friction (β), the cap eccentricity (R), the initial yield
surface position (α), the transition surface radius (𝜖𝜖𝑣𝑣𝑣𝑣𝑣𝑣𝑝𝑝𝑣𝑣 ), and the flow stress ratio (K). Fig. 4-8
66
provides the graphical illustration of these parameters. Detailed explanations and derivations of
these parameters can be found in the Section 19.3.2 of the ABAQUS Analysis User’s Manual,
ABAQUS Version 6.8.1.
Figure 4-8. Graphical presentation of parameters in the Modified Drucker-Prager/Cap Model
(ABAQUS analysis user’s manual)
Default values for most of the parameters were used with the exception of two parameters
that had significant effects on the results. These were the material cohesion and the angle of
friction. Computation of the material cohesion was based on the following procedure:
𝑓𝑓𝑐𝑐 = 𝜈𝜈 ∗ 𝑓𝑓𝑐𝑐′ (4-1)
𝜈𝜈 = 0.7 − 𝑓𝑓𝑐𝑐′
200 (4-2)
𝑑𝑑 = 𝑓𝑓𝑐𝑐4
(4-3)
With regards to the angle of friction, for normal concrete, it should be taken about 37
degree. However, it was found that this value of the angle of friction and those values produced
from Equations 4-1 to 4-3 for the material cohesion only provided a good starting point. The
calculated values of these two parameters for each beam still required adjustments in order to
67
obtain the desired results. Numerous trials were made with different values of material cohesion
and angle of friction in attempt to produce the outcomes that were comparable to the
experimental results. Table 4-3 provides the summary of the best values of the parameters used
for all five beams.
Table 4-3. Material model properties for beams in ABAQUS Beam Material
Cohesion, d
Angle of Friction,
β
Cap Eccentricity,
R
Initial Yd Surf Pos, 𝜖𝜖𝑣𝑣𝑣𝑣𝑣𝑣𝑝𝑝𝑣𝑣
Trans Surf Rad, 𝜖𝜖𝑣𝑣𝑣𝑣𝑣𝑣𝑝𝑝𝑣𝑣
Flow Stress Ratio,
K 1-C 0.85 60 0.5 0.003 0.01 0 1-G 0.87 53 0.5 0.003 0.01 0 1-H 0.85 60 0.5 0.003 0.01 0 1-I 1.00 59 0.5 0.003 0.01 0 1-J 1.07 47.7 0.5 0.003 0.01 0
With regards to the material property of steel, Hsu theory (Hsu, 1993) on stress-strain
relationship of mild steel was applied taking into account the strain rate hardening for both
concrete and steel reinforcements. Detailed explanations on the computational procedure can be
found in Chapter 7 of the above mentioned reference. The strain rate hardening and the
enhancement factors for all five beams, which were required for the computations, were obtained
from the previous research (Shanaa, 1991), and were implemented in the calculations of the
stress-strain relationships of concrete and steel reinforcements. Using the data provided in Table
4-4, the stress and strain relationships for all five beams were calculated. The results are
illustrated in Figs. 4-9 to 4-18 respectively.
Table 4-4. Strain rate hardening and material enhancement factors (Shanaa et al., 1991) Beam Strain rate,
in/in/sec Strain rate enhancement factors
Compression - Reinforcements
Tension - Reinforcements
Compression - Concrete
Tension - Concrete
1-C 0.28 1.23 1.23 1.36 2.20 1-G 0.46 1.24 1.24 1.37 2.36 1-H 0.46 1.24 1.24 1.37 2.36 1-I 0.46 1.24 1.24 1.37 2.36 1-J 0.27 1.23 1.23 1.36 2.22
68
Figure 4-9. Stress-strain relationship – tension reinforcements – beam 1-C
Figure 4-10. Stress-strain relationship – compression reinforcements – beam 1-C
Strain (in/in)
Stre
ss (k
si)
0 0.004 0.008 0.012 0.016 0.020
15
30
45
60
75
Strain, (in/in)
Stre
ss, (
ksi)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
69
Figure 4-11. Stress-strain relationship – tension reinforcements – beam 1-G
Figure 4-12. Stress-strain relationship – compression reinforcements – beam 1-G
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
70
Figure 4-13. Stress-strain relationship – tension reinforcements – beam 1-H
Figure 4-14. Stress-strain relationship – compression reinforcements – beam 1-H
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
71
Figure 4-15. Stress-strain relationship – tension reinforcements – beam 1-I
Figure 4-16. Stress-strain relationship – compression reinforcements – beam 1-I
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
72
Figure 4-17. Stress-strain relationship – tension reinforcements – beam 1-J
Figure 4-18. Stress-strain relationship – compression reinforcements – beam 1-J
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
15
30
45
60
75
73
Since ABAQUS Explicit required the use of true stress and logarithmic strain, the above
calculated stress-strain values were then converted using the two equations provided in the
ABAQUS manual (2008) prior to input the data into the ABAQUS program:
𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝜎𝜎𝑛𝑛𝑣𝑣𝑛𝑛 ∗ (1 + 𝜀𝜀𝑛𝑛𝑣𝑣𝑛𝑛 ) (4-4)
𝜀𝜀𝑣𝑣𝑛𝑛𝑝𝑝𝑣𝑣 = ln(1 + 𝜀𝜀𝑛𝑛𝑣𝑣𝑛𝑛 ) −
𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐸𝐸𝑠𝑠
(4-5)
4.3.3 DSAS Validations
Validation using DSAS was a much simpler task. This software application required the
inputs of physical and material properties of the beam as well as load time history. Strain
hardening could either be applied manually to the material property prior to input or through the
built-in switch. For this validation, an average strain rate of 0.3 was used for all beams.
Figure 4-19. Typical DSAS data entry screen
74
4.3.4 Comparison of Results from ABAQUS and DSAS to Experimental Data
Validation results for the displacements at the mid-span of all the beams from ABAQUS
(Dassault Systèmes, 2008) and DSAS (CIPPS, UF) are summarized in Table 4-5 and illustrated
in Figs. 4-20 to 4-24. It should be noted that the unloading portion in the ABAQUS result did
not match up well compared to those of experimental and DSAS results. This was because
hysteresis damping was not captured by the concrete material model in ABAQUS when
modeling the beams.
In general, most of the differences in the results compared to the experiment were in the
range within ±10%, which was acceptable. With regards to beam 1-G, the material model
parameters for ABAQUS required some minor adjustment. It was found that these parameters,
particularly the material cohesion and the angle of friction, were quite sensitive. A small
adjustment would change the result significantly. For beam 1-I, the recording instrument did not
properly recorded the loads during the experiment. Therefore, the loading function had to be
estimated based on two criteria. It must equal to the sum of the dynamic reactions less the inertia
effect due to the weight of the beam. It must also follow the shape of the experimental loading
function. For beam 1-J, the load was accidentally applied and stopped. The damaged beam was
then re-subjected to another set of loading.
Table 4-5. Comparison of ABAQUS and DSAS to Experiment Results Beam Midspan
Displacement, in
(Experiment)
Midspan Displacement, in (ABAQUS)
% Difference
Midspan Displacement,
in (DSAS)
% Difference
1-C 3.01 2.9 3.65% 3.07 1.99% 1-G 4.14 4.23 2.17% 4.02 2.90% 1-H 8.86 8.77 1.02% 8.2 7.45% 1-I 10.57 9.94 5.96% 9.55 9.65% 1-J - 1st Set 0.95 1.21 27.37% 0.84 11.58% 1-J - 2nd Set 10.17 10.6 4.23% 8.54 16.03%
75
Figure 4-20. Comparison of displacement-time history for beam 1-C
Figure 4-21. Comparison of displacement-time history for beam 1-G
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.1 0.12-1
0
1
2
3
4
ExperimentDSASABAQUS
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.1 0.120
1
2
3
4
5
ExperimentDSASABAQUS
76
Figure 4-22. Comparison of displacement-time history for beam 1-H
Figure 4-23. Comparison of displacement-time history for beam 1-I
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.1 0.120
1
2
3
4
5
6
7
8
9
10ExperimentDSASABAQUS
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.1 0.12-1
0
1
2
3
4
5
6
7
8
9
10
11ExperimentDSASABAQUS
Hit wooden stop at 0.072 sec
77
Figure 4-24. Comparison of displacement-time history for beam 1-J
4.4 Validation for Beam Subject to Uniform Load Using ABAQUS and DSAS
Since data for beam under uniform load was not available, beam 1-C was also used to
further validate for the beam case under uniform loading. In this case, Conventional Weapons
Effects (CONWEP, US Army Engineer Waterways Experiment Station, 1992) was used to
derive the loading function for a blast load of 500 pounds of TNT at a distance of 20 feet. This
loading function, as shown in Fig. 4-25, was used throughout the study in this paper. The results
computed from ABAQUS and DSAS for beam 1-C under uniform pressure load is shown in Fig.
4-26. A difference of 0.03 inch or 2.3% for the peak displacement exists between the two
outcomes, which is acceptable. As noted earlier, there is a more significant difference for the
residual displacement that is due to the inability of the ABAQUS material model to capture the
hysteretic behavior.
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-1
0
1
2
3
4
5
6
7
8
9
10
11ExperimentDSASABAQUS
1st stop @ 0.072 sec
2nd stop @ 0.068 sec
78
Figure 4-25. Loading function for 500 pounds of Trinitrotoluene (TNT) at 20 feet
Figure 4-26. Displacement-time history of beam 1-C under blast load
Time (msec)
Pres
sure
(psi
)
2 4 6 8 10 12 140
200
400
600
800
1000
1200
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.1 0.12-0.25
0
0.25
0.5
0.75
1
1.25
1.5ABAQUSDSAS
79
4.5 Validation for Columns Using ABAQUS and DSAS
A 16 in x 16 in x 144 in reinforced concrete column was generated in ABAQUS (Dassault
Systèmes, 2008) and DSAS (CIPPS, 2008). The column is subject to the above-mentioned
loading function. Similarly to the beam models, a cubical grid of 1 inch was found to be the
most effective and most economical size for the modeling of the column. Material properties
were also taken to be within the range of those of the beams in the previous step. Spacing for
transverse reinforcements was set at 12 inches on center. Longitudinal reinforcements consisted
of 8 No. 7 and were placed into 3 layers with a minimum concrete cover of 1.5 inches all around.
For ABAQUS to work properly, both ends of the column were extended by 6 inches and
were used as the end supports. The span length of the column was, therefore, still at 12 feet.
Typical layout of the column, loads, boundary conditions and reinforcement layouts are shown in
Figs. 4-27 to 4-29, respectively. A summary of the results for the column subjected to transverse
load only is shown in Fig. 4-30. It should be noted that the difference between DSAS and
ABAQUS was 0.75 inch or 35.8% for the maximum displacement at the mid-span of the column.
This was most likely caused by the following factors:
• The material model parameters in ABAQUS required some minor adjustment.
• The current version of DSAS did not include the effect of the shear reduction factor (SRF)
in the computations.
• The column was modeled in ABAQUS using three-dimensional solid element while the
same column was modeled in DSAS using beam element.
80
Figure 4-27. Typical column layout in ABAQUS
Figure 4-28. Loads and boundary conditions for column in ABAQUS
81
Figure 4-29. Longitudinal and transverse steel reinforcement layout in ABAQUS
Figure 4-30. Displacement-time history – column subject to blast load
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.10
1
2
3
P=0 - ABAQUSP=0 - DSAS
82
4.6 Summary
This chapter described the steps taken in the determination of the axial loads effects on a
column. First, in order to establish the proper material model to be used and the accuracy of the
software applications, validations of ABAQUS Version 6.8-1 (Dassault Systèmes, 2008) and the
Dynamic Structure Analysis Suite (DSAS) Version 2.0 (CIPPS, 2008) were completed against
past experimental data. During this step, discussion on the development of the steel stress-strain
relationship using Hsu Theory and how it was incorporated in the required computations were
also included. This was followed by an analytical validation of the above mentioned software
applications on a beam and a column that are subjected to the same blast load. The results from
both experimental and analytical validations indicated that either software application could be
used for the parametric study.
83
CHAPTER 5 PARAMETRIC STUDY
5.1 Description of Columns
The parametric study was based on four confined reinforced concrete columns with the
same dimensions stated in Chapter 4, Section 4.5, but with various configurations of steel
reinforcements. These columns were arbitrarily selected from the CRSI Design Handbook
(2002) along with the sizes and spacing of the longitudinal and transverse reinforcements.
Spacing of the transverse reinforcements was at 12 inches on center for all configurations. A
summary of the column material used is shown in Table 5-1.
Table 5-1. Summary of columns physical and material properties f’c = 4000 psi, fy = 60000 psi, Es = 29000 ksi, 16” x 16” x 144”
Column Bars Stirrups ρ s', in Mmax P, kips-ft max, kips 1 8 #7 #3 1.88 12 323.7 1720 2 8 #10 #3 4.88 12 450.7 2140 3 12 #11 #4 7.31 12 598.8 2820 4 4 #14 #4 3.52 12 474.5 2050 With the outcome from the above validations, both experimentally and analytically,
DSAS and ABAQUS can now be used for the necessary computations. However, since
experimental data on column was not available, one additional step was required to ensure the
consistency of the result.
An unconfined reinforced concrete column of 16 inch x 16 inch x 144 inch consisted of 8
No. 7 grade 60 steel reinforcements was used for this confirmation. The result obtained from
DSAS was compared with the pre-defined data provided in the CRSI Design Handbook (2002).
The compression strength of the concrete, f’c, was 4000 psi. The yield strength of the steel, fy,
was 60000 psi. The layout of steel reinforcement was based on the recommendation outlined in
Table 3-1 of the CRSI. An axial load-moment interaction diagram for the axial loads, P, acting
on this column was generated from the data computed by DSAS. The result, shown in Fig. 4-29,
84
indicated that the axial load and moment capacity values obtained from DSAS closely matched
those outlined in the CRSI (2002).
Figure 5-1. Axial-moment interaction diagram – unconfined concrete – DSAS versus CRSI
5.2 Columns Subject to Transverse and Constant Axial Load
Since using ABAQUS Version 6.8-1 (Dassault Systèmes, 2008) to produce the required
axial load-moment interaction (P-M) diagrams, the flexure-resistance (P-d) diagrams, and the
pressure-impulse (P-I) diagrams would be time consuming and cumbersome, only DSAS
Version 2.0 (CIPPS, 2008) was used to generate these diagrams for the columns listed in Table
5-1. However, to produce time-displacement history diagrams for the above columns, both
DSAS and ABAQUS were used.
Similar to the beam cases, Hsu theory (Hsu, 1993) on stress-strain relationship of mild
steel was applied to the computations of nominal stress and strain values for concrete and steel
Moment (ft-kips)
Axi
al L
oad
(kip
s)
0 30 60 90 120 150 180 210 240 270-500
-250
0
250
500
750
1000
1250
1500
DSASCRSI
85
reinforcements. A value of 0.3 was used as the strain rate hardening for both concrete and steel
reinforcements. Equations 4-4 and 4-5 were then used to convert the computed nominal stress
and strain values to the true stress and logarithm strain values prior to input into ABAQUS.
Figures 5-2 to 5-5 plotted the computed nominal stress-strain relationships of the steel
reinforcements used in the columns. Since the tension and the compression steel reinforcements
were assumed to have the same material properties for the four columns listed in Table 5-1, the
computations were the same for both types of steel and only one plot was used for each type of
steel reinforcements.
Figure 5-2. Stress-strain relationship – tension and compression reinforcements – column 1
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
20
40
60
80
100
86
Figure 5-3. Stress-strain relationship – tension and compression reinforcements – column 2
Figure 5-4. Stress-strain relationship – tension and compression reinforcements – column 3
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
20
40
60
80
100
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
20
40
60
80
100
87
Figure 5-5. Stress-strain relationship – tension and compression reinforcements – column 4
Four different load magnitudes, as shown in Table 5-2, were arbitrary chosen to be used
in the analyses for each column. In all cases, one load magnitude at zero, one below balance
load, one at balance load and one above balance load were used in both DSAS and ABAQUS.
Summary of the results obtained from these two software applications are provided in Tables 5-3
and 5-4. As mentioned earlier, the differences in the results between DSAS and ABAQUS were
due to the fact that the current version of DSAS did not include the SRF in the computations.
Table 5-2. Summary of load cases used in the analyses Column Rebar Pbal, Pkips 1 P, kips 2 P, kips 3 P, kips 4, kips
1 8 No. 7 560 0 250 560 1000 2 8 No. 10 560 0 250 560 1000 3 12 No. 11 570 0 250 570 1500 4 4 No. 14 530 0 250 530 1000
Strain (in/in)
Stre
ss (k
si)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
20
40
60
80
100
88
Table 5-3. Comparisons on displacements resulted from ABAQUS and DSAS for P1 and P2 Load ABAQUS DSAS %
Difference ABAQUS DSAS %
Difference Column P1 P2 1 2.84 2.09 26.37 2.40 1.55 35.30 2 1.89 1.36 28.05 1.73 1.16 32.92 3 2.79 1.07 61.72 2.48 0.99 60.00 4 2.00 1.40 30.16 1.74 1.16 33.59
Table 5-4. Comparisons on displacements resulted from ABAQUS and DSAS for P3 and P4
Load ABAQUS DSAS % Difference
ABAQUS DSAS % Difference Column P3 P4
1 2.28 0.99 56.45 2.66 0.68 74.32 2 1.88 1.12 40.49 1.81 0.85 53.38 3 2.30 0.95 58.87 24.61 1.27 94.85 4 1.59 1.09 31.62 1.78 0.90 49.70
Figures 5-6 to 5-26 illustrate the four different approaches in presenting the outcomes of
the study. The first approach showed the time-displacement history of the columns computed
from both ABAQUS and DSAS. Since this version of DSAS did not include the shear reduction
factor (SRF) in the computations, the results produced by DSAS were, therefore, lower than
those produced by ABAQUS. However, interpretation from these results yielded some common
points on the behavior of the column. When being subjected to the transverse load only, the
column experienced a larger displacement at mid-span compared to those obtained when axial
load were exerted on the column. For all columns, as the axial loads, P, increased, the difference
between the displacements in comparing to zero axial load became larger until P reached the
balance load, Pbal. As P surpassed Pbal, the difference in displacement became smaller. This was
an indication that the applications of axial loads actually enhanced the flexure resistance of the
columns. In the case of Column 2, the displacement at balance load was the same with that at
zero axial load. This was due to the fact that the yielding of steel reinforcements and the
crushing of concrete occurred early and Column 2 reached the plastic stage.
89
Two other approaches used the Moment-Curvature and Flexure Resistance relationships
to demonstrate column behavior from the elastic to plastic range based on the analysis of loads
and deformations. The results showed the changes in column stiffness for various magnitudes of
constant axial loads, P. Within the elastic and elasto-plastic range, the stiffness of the column
decreased slowly as P increased. The plastic range of the column depended on its ductility. In
this range, the lower P were, the larger the displacements the column could undertake. This was
probably due to the formation of the hinge at the column mid-span that allowed the column to
behave in such a way. As P increased, the plastic range of the column became much shorter and
failure took place sharply. Again, the time between the formations of the hinges at the column
mid-span and at the two end supports occurred much faster during this stage, and therefore
caused the column failure. It should also be noted that the area under the curve decreased as P
increased and the rate of change also intensified significantly once P approached closer to Pbal.
This implied that the strain energy, represented by the area under the Flexure-Resistance curve,
dissipated much faster once the first hinge was formed. Moreover, for P much less than Pbal
The Pressure-Impulse (P-I) diagram was the last approach for analyzing the results.
Unlike the other three methods, the P-I diagrams predicted the points of failure of the column
, P
actually enhanced the moment capacity and strengthened the column. Hence the displacement
due to the transverse load was actually less than that of P = 0 kips. Once the balance point was
surpassed, P became the contributing factor to the column failure. From Fig. 5-8, it should also
be noted that the final lateral displacement of the column was approximately 19 inches for P = 0,
while Fig. 5-7 showed the column displacement of 2.1 inches (DSAS) or 2.85 inches (ABAQUS)
for the same load. This was because Fig 5-8 also included the failure of the column in the
tension membrane mode.
90
based on each magnitude of load, P, and impulse, I, that the column experienced over the time
period. The Impulsive Loading region for P < Pbal had higher tolerance than that of P > Pbal.
This implied that the strain energy decreased as P approached Pbal
Figure 5-6. Axial-moment interaction diagram – 8 No. 7 RC – confined
. As P increased, the Dynamic
Loading region became shorter and the columns exhibited longer Quasi-Static Loading Regions;
where the column deformation became larger and permanent damage could take place. Hence,
the columns achieved failure.
Moment (ft-kips)
Axi
al F
orc
e (k
ips)
0 50 100 150 200 250 300 350-600
-400
-200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
P4 = 1000 kips > Pb
P3 = Pb = 560 kips
P2 = 250 kips < Pb
P1 = 0
91
Figure 5-7. Displacement-time history diagram – 8 No. 7 RC – confined
Figure 5-8. Flexure-resistance diagram – 8 No. 7 RC – confined
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
2.5
3
P=250 kips - ABAQUS
P=560 kips - ABAQUS
P=1000 kips - ABAQUS
P=0 kips - DSAS
P=250 kips - DSASP=560 kips - DSAS
P=1000 kips - DSAS
P=0 kips - ABAQUS
Displacement (in)
Pres
sure
(psi
)
0 2 4 6 8 10 12 14 16 18 200
20
40
60
80
100
P1 = 0
P2 = 250 kips < Pb
P3 = Pb = 560 kips
P4 = 1000 kips > Pb
92
Figure 5-9. Moment-curvature diagram – 8 No. 7 RC – confined
Figure 5-10. Pressure-impulse diagram – 8 No. 7 RC – confined
Curvature (1/in)
Mom
ent (
ft-ki
ps)
0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.0240
50
100
150
200
250
300
350
P1= 0
P2 = 250 kips < Pb
P3 = Pb = 560 kips
P4 = 1000 kips > Pb
Impulse (psi-sec)
Pres
sure
(psi
)
0 2 4 6 8 10 12 14 16 18 200
100
200
300
400
500
600
700
800
900
1000
Direct Shear
P = 0
P = 250 kips < Pb
P = Pb = 560 kips
P = 1000 kips > Pb
93
Figure 5-11. Axial-moment interaction diagram – 8 No. 10 RC – confined
Figure 5-12. Displacement-time history diagram – 8 No. 10 RC – confined
Moment (ft-kips)
Axi
al F
orce
(kip
s)
0 50 100 150 200 250 300 350 400 450-1200
-800
-400
0
400
800
1200
1600
2000
P = 0
P = 250 kips < Pb
P = Pb = 560 kips
P = 1000 kips > Pb
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
P=250 kips - ABAQUS
P=560 kips - ABAQUSP=1000 kips - ABAQUS
P=0 kips - DSASP=250 kips - DSAS
P=560 kips - DSAS
P=1000 kips - DSAS
P=0 kips - ABAQUS
94
Figure 5-13. Flexure-resistance diagram – 8 No. 10 RC – confined
Figure 5-14. Moment-curvature diagram – 8 No. 10 RC – confined
Displacement (in)
Pres
sure
(psi
)
0 2 4 6 8 10 12 14 16 18 20 22 240
20
40
60
80
100
120
140
160
P = 0
P = 250 kips < Pb
P = Pb = 560 kips
P = 1000 kips > Pb
Curvature (1/in)
Mom
ent (
ft-ki
ps)
0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.0240
50
100
150
200
250
300
350
400
450
500
P = 0
P = 250 kips < Pb
P = Pb = 560 kips
P = 1000 kips > Pb
95
Figure 5-15. Pressure-impulse diagram – 8 No. 10 RC – confined
Figure 5-16. Axial-moment interaction diagram – 12 No. 11 RC – confined
Impulse (psi-sec)
Pres
sure
(psi
)
0 3 6 9 12 15 18 21 24 27 300
100
200
300
400
500
600
700
800
900
1000
Direct Shear
P = 0
P = 250 kips < Pb
P = Pb = 560 kipsP = 1000 kips > Pb
Moment (ft-kips)
Axi
al F
orce
(kip
s)
0 50 100 150 200 250 300 350 400 450 500 550 600-2000
-1600
-1200
-800
-400
0
400
800
1200
1600
2000
2400
2800
P = 0
P = 250 kips < Pb
P = 1500 kips > Pb
96
Figure 5-17. Displacement-time history diagram – 12 No. 11 RC, P = 0 to 570 kips – confined
Figure 5-18. Displacement-time history diagram – 12 No. 11 RC – confined
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.1-0.5
0
0.5
1
1.5
2
2.5
3
P=250 kips - ABAQUSP=570 kips - ABAQUS
P=0 kips - DSASP=250 kips - DSAS
P=570 kips - DSAS
P=0 kips - ABAQUS
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.10
5
10
15
20
25
P=1000 kips - ABAQUS
P=1500 kips - ABAQUS
P=1500 kips - DSAS
P=0 kips - ABAQUS
97
Figure 5-19. Flexure-resistance diagram – 12 No. 11 RC – confined
Figure 5-20. Moment-curvature diagram – 12 No. 11 RC – confined
Displacement (in)
Pres
sure
(psi
)
0 2 4 6 8 10 12 14 16 18 20 22 24 260
20
40
60
80
100
120
140
160
180
200
P = 0P = 250 kips < Pb
P = Pb = 570 kips
P = 1500 kips > Pb
Curvature (1/in)
Mom
ent (
ft-ki
ps)
0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.030
100
200
300
400
500
600
700
800
P = 0
P = 250 kips < Pb
P = Pb = 570 kips
P = 1500 kips > Pb
98
Figure 5-21. Pressure-impulse diagram – 12 No. 11 RC – confined
Figure 5-22. Axial-moment interaction diagram – 4 No. 14 RC – confined
Impulse (psi-sec)
Pres
sure
(psi
)
0 3 6 9 12 15 18 21 24 27 300
100
200
300
400
500
600
700
800
900
1000
Direct ShearP = 0
P = 250 kips < Pb
P = Pb = 570 kipsP = 1500 kips > Pb
Moment (ft-kips)
Axi
al F
orce
(kip
s)
0 50 100 150 200 250 300 350 400 450 500-1000
-600
-200
200
600
1000
1400
1800
2200
P = 0
P = 250 kips < Pb
P = Pb = 530 kips
P = 1000 kips > Pb
99
Figure 5-23. Displacement-time history diagram – 4 No. 14 RC – confined
Figure 5-24. Flexure-resistance diagram – 4 No. 14 RC – confined
Time (sec)
Dis
plac
emen
t (in
)
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
2.5
P=250 kips - ABAQUSP=530 kips - ABAQUS
P=1000 kips - ABAQUS
P=0 kips - DSASP=250 kips - DSAS
P=530 kips - DSAS
P=1000 kips - DSAS
P=0 kips - ABAQUS
Displacement (in)
Pres
sure
(psi
)
0 2 4 6 8 10 12 14 16 18 200
20
40
60
80
100
120
140
160
P = 0
P = 250 kips < Pb
P = Pb = 530 kips
P = 1000 kips > Pb
100
Figure 5-25. Moment-curvature diagram – 4 No. 14 RC – confined
Figure 5-26. Pressure-impulse diagram – 4 No. 14 RC – confined
Curvature (1/in)
Mom
ent (
ft-ki
ps)
0 0.003 0.006 0.009 0.012 0.0150
50
100
150
200
250
300
350
400
450
500
550
600
P = 0
P = 250 kips < Pb
P = Pb = 530 kips
P = 1000 kips > Pb
Impulse (psi-sec)
Pres
sure
(psi
)
0 2 4 6 8 10 12 14 16 18 200
100
200
300
400
500
600
700
800
900
1000
Direct Shear
P = 0
P = 250 kips < Pb
P = Pb = 530 kips
P = 1000 kips > Pb
101
5.3 Columns Subject to Transverse, Constant and Variable Axial Loads
With the known effects of constant axial loads, P, the same columns were then subjected
to the variable axial loads, Pvar, to determine the significance of their effects. As indicated in
Chapter 4, Section 4.4, during the course of the required validations, beam 1-C was subjected to
the same blast load as the column. The dynamic reactions at the two supports generated by this
blast load were then used as the variable axial loads that acted on the columns. To reduce the
computer runtime, it was assumed that the arrival time of the blast load for the beam was the
same as that of the column. Hence, in ABAQUS, the variable axial loads were activated at the
same time as the variable transverse loads. The profile of the variable axial loads and the results
of the applications of the variable axial loads on the columns are shown in Figs. 5-27 to 5-35.
From Table 5-5, it should be noted that for Column 3, at load P4 the column did not completely
fail until it reached 24.61 inches. However, upon the exertion Pvar
Table 5-5. Comparisons on displacements induced by constant and variable axial loads
, the column actually failed at
4.46 inches.
Load P2 P2 + P
% Increase var
P3 P3 + P
% Increase var
P4 P4 + P
% Increase var Column
1 2.40 2.62 8.68 2.28 4.90 53.37 2.66 24.57 89.15 2 1.73 1.82 4.61 1.88 2.19 14.30 1.81 23.08 92.14 3 2.48 2.54 2.28 2.30 3.32 30.77 24.61 4.46 -451.89 4 1.74 1.78 2.54 1.59 2.14 25.62 1.78 18.78 90.50
In all cases, for P ≤ Pbal, by exerting the variable axial loads, Pvar, in addition to the
constant axial loads, P, the effects on the column behavior were reversed. Within this range,
while the applications of P enhanced the flexure resistance of the columns and reduced the peak
displacement at the column mid-span, the applications of Pvar actually reduced the flexural
resistance and increased the displacement at the same location. As P reached Pbal, the effect of
additional Pvar became more significant and the columns failed at sooner.
102
Figure 5-27. Variable axial load profile
Figure 5-28. Displacement-time-history – 8 No. 7 RC – P ≤ Pbal + P
Time (sec)
Dyn
amic
Rea
ctio
n (k
ips)
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-100
0
100
200
300
400
var
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
1
2
3
4
5
P=250 kipsP=560 kips
P=250 kips + Pvar
P=560 kips + Pvar
P=0 kips
103
Figure 5-29. Displacement-time history – 8 No. 7 RC – P > Pbal + P
Figure 5-30. Displacement-time history – 8 No. 10 RC – P ≤ P
var
bal + P
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
5
10
15
20
25
P=1000 kips
P=1000 kips + Pvar
P=0 kips
var
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
2.5
P=250 kips
P=560 kips
P=250 kips + Pvar
P=560 kips + Pvar
P=0 kips
104
Figure 5-31. Displacement-time history – 8 No. 10 RC – P > Pbal + P
Figure 5-32. Displacement-time history – 12 No. 11 RC – P ≤ P
var
bal + P
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
5
10
15
20
25
P=1000 kips
P=1000 kips + Pvar
P=0 kips
var
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
2.5
3
3.5
P=250 kipsP=570 kips
P=250 kips + Pvar P=570 kips + Pvar P=0 kips
105
Figure 5-33. Displacement-time history – 12 No. 11 RC – P > Pbal + P
Figure 5-34. Displacement-time history – 4 No. 14 RC – P ≤ P
var
bal + P
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
5
10
15
20
25
30
P=1000 kips
P=1500 kips
P=1000 kips + Pvar
P=1500 kips + Pvar
P=0 kips
var
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
0.5
1
1.5
2
2.5
P=250 kipsP=530 kips P=250 kips + Pvar
P=530 kips + Pvar P=0 kips
106
Figure 5-35. Displacement-time history – 4 No. 14 RC – P > Pbal + P
5.4 Summary
var
Parametric study on four reinforced concrete columns with different steel configurations
was conducted in this chapter. The study was completed in two stages. In the first stage, only
constant axial loads were applied to the columns that were subjected to variable transverse load.
Four different methods were used to illustrate the outcomes of the study along with the
associated interpretation on the column behavior in each method. These methods were flexure-
resistance diagrams, moment-curvature diagrams, and pressure-impulse diagrams. In the second
stage, variable axial loads were applied to the columns in addition to the constant axial loads.
The displacement-time histories associated with the loads for each column obtained from both
stages were compared to determine the effects of the variable axial loads on the columns.
Time (sec)
Disp
lace
men
t (in
)
0 0.02 0.04 0.06 0.08 0.10
4
8
12
16
20
P=1000 kips
P=1000 + Pvar
P=0 kips
107
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS
6.1 Summary
A study on the effects of short-duration-high-impulsive loads such as blast load on a
reinforced concrete column was presented in this paper.
A summary on the computation of blast load and its effect were provided in Chapter 2. A
review on a structure and its equivalent SDOF system along with the associated shape and
resistance functions were also included in this chapter. This was followed by the discussion of
the behaviors of flexure and shear and approaches to obtain the solutions for the equation of
motion of the equivalent SDOF system. Chapter 3 discussed the load determination and the load
deformation analysis approach using Newmark-Beta method and the computations of the
dynamic reactions. Chapter 4 included the implementation of the above discussion using
ABAQUS Version 6.8-1 (Dassault Systèmes, 2008) and the Dynamic Structure Analysis Suite
(DSAS) Version 2.0 (CIPPS, 2008)). A series of steps were required to validate the two
software applications.
Upon the completion of the validations, a parametric study on column behaviors under
the influence of short-duration-high-impulsive transverse and axial loads was conducted as
shown in Chapter 5. Four approaches were used to present the effects of variable transverse and
axial loads on a column. The first approach used time-displacement history to determine the
effects of these above-mentioned loads on the columns. Two other approaches included the use
of moment–curvature and load–deflection relationships to describe the columns behaviors in the
elastic, elasto-plastic, and fully plastic ranges. Pressure-impulse relationship was the last
approach used in analyzing the results.
108
6.2 Conclusions
Under normal loading conditions, a reinforced concrete column of a structure can be
designed to support a given mass. However, consideration for maximum support capacity for the
column should be taken into account when the structure is likely to be subjected to short-
duration-high-impulse variable axial and transverse loads as this will significantly diminish the
loading capacity of the column.
For a column with low and normal ductility, under the influence of short duration
transverse loads, the column fails as the load resulted from the supported mass is greater than the
balance load of the column. It should be noted that for loads less than the balance load, loads
resulted from the supported mass act as an enhancement factor to the strength of the column.
However, chance of a column failure increases as the loads of the supported mass surpass the
balance load. For column with high ductility, the column lasts longer even when its balance load
capacity is surpassed to a certain load magnitude where failure of the column occurs.
The probability of the shear failure of a column depends on the magnitudes of the
transverse loads. The higher the transverse loads are, the higher the probability the column can
fail in shear failure mode. In this study, the failure of the column was governed by flexure.
Four methods of analyzing the performance of a structure were used in this study. While
the employment of time-displacement history diagram, load deflection diagram and moment
curvature diagram provide a more thorough analysis, P-I diagram proves to be the most
expedient one to in determining the state of the structure.
ABAQUS Version 6.8-1 (Dassault Systèmes, 2008) and the Dynamic Structure Analysis
Suite (DSAS) Version 2.0 (CIPPS, 2008)) were heavily used in the study of this paper.
Although DSAS is a non-commercial software application, its portability, speed and ability to
109
provide accurate results prove that it is a valuable tool to be used for a quick prediction of a
structure behavior. On the other hand, while being a commercial software application,
ABAQUS shows that it is too cumbersome and lacks of flexibility. The properties of the
material model used in ABAQUS are too sensitive. As such, a slight change in these properties
may produce different results. In addition, the level of public technical supports such as the
internet forum for ABAQUS is also found to be limited compared to that of other finite element
software applications.
6.3 Recommendations
The following recommendations can be deduced from the above results and observations:
• The results from this study should be verified by actual experiments using the same boundary conditions.
• Further study should be conducted on the column behaviors in the tension membrane state of the steel reinforcements.
• Reinforced concrete columns with different size configurations, strengths and boundary
conditions can be used to further validate the conclusions. • Another finite element software application should be used for the modeling of the
columns and the results obtained can be compared with those from ABAQUS. • Shear reduction factor should be included in the computations in the next version of
DSAS.
110
APPENDIX SAMPLE ABAQUS INPUT FILE – BEAM 1-C
*Heading ** Job name: Beam_C1_R17_9 Model name: Beam_C1_R17_9 ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=BEAM_C1-1 *Node **<included in the electronic version only> *Element, type=C3D8R **<included in the electronic version only> *Element, type=B31 **<included in the electronic version only> *Element, type=SFM3D4R **<included in the electronic version only> *Nset, nset=BEAM_C1, generate 1, 12068, 1 *Elset, elset=BEAM_C1, generate 1, 9144, 1 *Elset, elset=TOP_RIGHT, generate 9145, 9262, 1 *Elset, elset=TOP_LEFT, generate 9263, 9380, 1 *Elset, elset=SF_BARSBOTTOM, generate 9381, 9498, 1 *Elset, elset=SF_STIRRUPS, generate 9499, 12254, 1 *Elset, elset=STEEL_PLATE, generate 12255, 12398, 1 *Nset, nset=MIDSPAN_GAUGE_3 3151, ** Section: Section-1-BEAM_C1 *Solid Section, elset=BEAM_C1, material=CONCRETE 1., ** Section: Section-2-TOP_RIGHT Profile: Profile-1 *Beam Section, elset=TOP_RIGHT, material=STEEL_COMP, temperature=GRADIENTS, section=CIRC 0.375 0.,0.,1. ** Section: Section-3-TOP_LEFT Profile: Profile-2 *Beam Section, elset=TOP_LEFT, material=STEEL_COMP, temperature=GRADIENTS, section=CIRC
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0.375 0.,0.,1. ** Section: Section-4-SF_BARSBOTTOM *Surface Section, elset=SF_BARSBOTTOM *Rebar Layer BARS_BOTTOM, 0.60132, 3.125, , STEEL, 0., 1 ** Section: Section-5-SF_STIRRUPS *Surface Section, elset=SF_STIRRUPS *Rebar Layer STIRRUPS, 0.11045, 6.625, , STEEL_TRANS, 90., 1 ** Section: Section-6-STEEL_PLATE *Solid Section, elset=STEEL_PLATE, material=STEEL_PLATE 1., *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=BEAM_C1-1, part=BEAM_C1-1 *End Instance ** *Elset, elset=__PICKEDSURF8_S4, internal, instance=BEAM_C1-1, generate 7, 504, 7 *Nset, nset=SET-1, instance=BEAM_C1-1 53, *Elset, elset=SF_BARSBOTTOM, instance=BEAM_C1-1, generate 9381, 9498, 1 *Elset, elset=SF_STIRRUPS, instance=BEAM_C1-1, generate 9499, 12254, 1 *Elset, elset=__PICKEDSURF13_S4, internal, instance=BEAM_C1-1, generate 7, 504, 7 *Nset, nset=_PICKEDSET16, internal, instance=BEAM_C1-1 18, 19, 201, 202, 203, 204, 205 *Nset, nset=_PICKEDSET18, internal, instance=BEAM_C1-1 11, 12, 124, 125, 126, 127, 128 *Elset, elset=__PICKEDSURF26_S4, internal, instance=BEAM_C1-1, generate 7, 504, 7 *Elset, elset=_M12, internal, instance=BEAM_C1-1, generate 9381, 9498, 1 *Elset, elset=_M13, internal, instance=BEAM_C1-1, generate 9499, 12254, 1 *Elset, elset=_M14, internal, instance=BEAM_C1-1, generate 9263, 9380, 1 *Elset, elset=_M15, internal, instance=BEAM_C1-1, generate
112
9145, 9262, 1 *Elset, elset=_M16, internal, instance=BEAM_C1-1, generate 9381, 9498, 1 *Elset, elset=_M17, internal, instance=BEAM_C1-1, generate 9499, 12254, 1 *Elset, elset=_M18, internal, instance=BEAM_C1-1, generate 9263, 9380, 1 *Elset, elset=_M19, internal, instance=BEAM_C1-1, generate 9145, 9262, 1 *Elset, elset=_M20, internal, instance=BEAM_C1-1, generate 9381, 9498, 1 *Elset, elset=_M21, internal, instance=BEAM_C1-1, generate 9499, 12254, 1 *Elset, elset=_M22, internal, instance=BEAM_C1-1, generate 9263, 9380, 1 *Elset, elset=_M23, internal, instance=BEAM_C1-1, generate 9145, 9262, 1 *Nset, nset=STRAIN_GAUGE, instance=BEAM_C1-1 2822, 2829, 2843, 2850 *Nset, nset=LEFT_RF, instance=BEAM_C1-1 18, 19, 201, 202, 203, 204, 205 *Nset, nset=RIGHT_RF, instance=BEAM_C1-1 11, 12, 124, 125, 126, 127, 128 *Nset, nset=_PickedSet38, internal, instance=BEAM_C1-1 12023, *Nset, nset=MIDSPAN_GAUGE_3, instance=BEAM_C1-1 3151, ** Constraint: BOTTOMBARS *Embedded Element _M20 ** Constraint: STIRRUPS *Embedded Element _M21 ** Constraint: TOPLEFT *Embedded Element _M22 ** Constraint: TOPRIGHT *Embedded Element _M23 *End Assembly *Amplitude, name=IMPACT2 0., 0., 0.003, 5., 0.004375, 10., 0.00625, 15. 0.006875, 20., 0.01125, 25., 0.01375, 30., 0.014, 31.2 0.0163, 29.2, 0.0175, 33.8, 0.02, 29.2, 0.022, 31.6 0.0225, 29.2, 0.025, 29., 0.027, 30.6, 0.027625, 28.6 0.031, 28.5, 0.032, 28.8, 0.034, 29.5, 0.03625, 28.8
113
0.037, 28., 0.03825, 28.05, 0.0385, 28., 0.04181, 25. 0.04375, 22., 0.045, 20., 0.0475, 17., 0.04875, 15. 0.05, 14.5, 0.0525, 12., 0.05375, 10., 0.055, 7. 0.05625, 5., 0.0569, 4.8, 0.058, 5.5, 0.06, 3.8 0.063, 0., 0.064, -0.3, 0.066, 0., 0.072, 0. 0.075, -0.5, 0.07625, -0.1, 0.07725, -0.15, 0.08, -0.6 0.081, -0.7, 0.0875, 0., 0.09, -0.25, 0.0925, -0.25 0.094, -0.4, 0.0975, 0., 0.099, 0., 0.1025, -0.2 0.105, -0.2, 0.1075, -0.25, 0.11125, -0.1, 0.11375, -0.15 0.11625, 0., 0.11875, 0.05, 0.12, 0. ** ** MATERIALS ** *Material, name=CONCRETE *Cap Plasticity 0.85, 60., 0.5, 0.003, 0.01, 0. *Cap Hardening 6.,0. *Density 2.27e-07, *Elastic 5005.37, 0.2 *Material, name=STEEL *Density 7.33024e-07, *Elastic 36309.6, 0.3 *Plastic 49., 0. 50.6802, 0.000602224 51.6936, 0.00157182 52.709, 0.00254037 53.7253, 0.0035079 54.7445, 0.00447436 55.7656, 0.00543978 56.7887, 0.00640416 57.8137, 0.0073675 58.8406, 0.00832981 59.8694, 0.00929108 60.9001, 0.0102513 61.9328, 0.0112105 62.9664, 0.0121688 64.0029, 0.0131259 65.0413, 0.0140821 66.0816, 0.0150372 67.1239, 0.0159913
114
68.168, 0.0169443 69.2141, 0.0178964 *Material, name=STEEL_COMP *Density 7.33024e-07, *Elastic 36309.6, 0.3 *Plastic 48.5, 0. 50.5028, 0.000607109 51.5933, 0.00157458 52.6869, 0.00254098 53.7826, 0.00350632 54.8803, 0.00447062 55.9801, 0.00543387 57.082, 0.00639608 58.186, 0.00735725 59.2921, 0.00831737 60.4002, 0.00927646 61.5104, 0.0102345 62.6226, 0.0111915 63.737, 0.0121475 64.8534, 0.0131025 65.9719, 0.0140564 67.0925, 0.0150093 68.2152, 0.0159612 69.3399, 0.0169121 70.4667, 0.0178619 *Material, name=STEEL_PLATE *Density 7.33024e-07, *Elastic 36000., 0.3 *Plastic 75.,0. *Material, name=STEEL_TRANS *Density 7.33024e-07, *Elastic 36309.6, 0.3 *Plastic 49., 0. 50.6802, 0.000602224 51.6936, 0.00157182 52.709, 0.00254037 53.7253, 0.0035079
115
54.7445, 0.00447436 55.7656, 0.00543978 56.7887, 0.00640416 57.8137, 0.0073675 58.8406, 0.00832981 59.8694, 0.00929108 60.9001, 0.0102513 61.9328, 0.0112105 62.9664, 0.0121688 64.0029, 0.0131259 65.0413, 0.0140821 66.0816, 0.0150372 67.1239, 0.0159913 68.168, 0.0169443 69.2141, 0.0178964 ** ---------------------------------------------------------------- ** ** STEP: BeamLoad ** *Step, name=BeamLoad *Dynamic, Explicit , 0.12 *Bulk Viscosity 0.06, 0.12 ** ** BOUNDARY CONDITIONS ** ** Name: Disp-BC-1 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 1, 1 ** Name: Disp-BC-2 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 2, 2 ** Name: Disp-BC-3 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 3, 3 ** Name: Disp-BC-4 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 4, 4 ** Name: Disp-BC-5 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 5, 5 ** Name: Disp-BC-6 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 1, 1 ** Name: Disp-BC-7 Type: Displacement/Rotation
116
*Boundary, amplitude=IMPACT2 _PICKEDSET18, 2, 2 ** Name: Disp-BC-8 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 3, 3 ** Name: Disp-BC-9 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 4, 4 ** Name: Disp-BC-10 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 5, 5 ** ** LOADS ** ** Name: GRAVITY-BM Type: Gravity *Dload, amplitude=IMPACT2 BEAM_C1-1.BEAM_C1, GRAV, 386.1, 0., -1., 0. ** Name: GRAVITY-STEELPLATE Type: Gravity *Dload BEAM_C1-1.STEEL_PLATE, GRAV, 386.1, 0., -1., 0. ** Name: Load-7 Type: Concentrated force *Cload, amplitude=IMPACT2 _PickedSet38, 2, -1. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field *Node Output, nset=MIDSPAN_GAUGE_3 A, RF, U, V ** ** FIELD OUTPUT: F-Output-2 ** *Node Output, nset=STRAIN_GAUGE A, U, V ** ** HISTORY OUTPUT: DRF_LEFT ** *Output, history *Node Output, nset=LEFT_RF CF2, CM3, RF1, RF2, RF3, RM1, RM2, RM3 ** ** HISTORY OUTPUT: DEF_MIDSPN_3
117
** *Node Output, nset=MIDSPAN_GAUGE_3 A1, A2, A3, AR, AR1, AR2, AR3, U1 U2, U3, UR, UR1, UR2, UR3, V1, V2 V3, VR, VR1, VR2, VR3 ** ** HISTORY OUTPUT: DRF_RIGHT ** *Node Output, nset=RIGHT_RF CF2, CM3, RF1, RF2, RF3, RM1, RM2, RM3 ** ** HISTORY OUTPUT: STRAIN ** *Node Output, nset=STRAIN_GAUGE A1, A2, A3, AR1, AR2, AR3, U1, U2 U3, UR3, V1, V2, V3, VR1, VR2, VR3 *End Step
118
LIST OF REFERENCES ACI 318-05, 2005. “Building Code Requirements for Structural Concrete (ACI 318-05) and
Commentary (ACI 318R-05)”. American Concrete Institute. Baker, W.E., Cox, P.A., Westine, P.S., Kulesz, J.J. and Strehlow, R.A., 1983. “Explosion
Hazards and Evaluation.” Elsevier Scientific Publishing Company: Amsterdam. Bathe, K.J., 1996. “Finite Element Procedures.” Prentice-Hall, Inc., Englewood Cliffs, New
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elements subjected to dynamic loads.” Technical report PTC-TR-002-2007, Protective Technology Center, The Pennsylvania State University, PA.
Brode, H.L., 1955, “Numerical Solution of Spherical Blast Waves”, Journal of Applied Physics,
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Concrete Box Structures Subjected To Airblast Loads.” Technical report CIPPS-TR-002-2008. Center for Infrastructure Protection and Physical Security, University of Florida.
Clough, R.W., Johnston, S.B., 1966. “Effect of Stiffness Degradation on Earthquake Ductility
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CRSI Design Handbook, 9th Edition, 2002. Concrete Reinforcing Steel Institute, Schaumburg
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Under Axial Slamming.” Journal of Engineering Mechanics. Vol. 125, No. 5, May. pp 513-520.
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TM5-1300. Washington, DC.
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ElMandooh Galal, K., Ghobarah, A., 2003. “Flexural And Shear Hysteretic Behaviour Of Reinforced Concrete Columns With Variable Axial Load.” Engineering Structures. Vol. 25, Issue 11, September. pp 1353 – 1367.
Hawkins, N.M., 1974. “The Strength of Stud Shear Connections.” Civil Engineering
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Shallow-Buried Flat Roof Structures.” U.S. Army Engineer Waterways Experiment Station, Techical Report SL-80-7. Six part, Sept.
Krauthammer, T., Bazeos, N., Holmquist, T.J., 1986 “Modified SDOF Analysis of R. C. Box-
Type Structures.” Journal of Structural Engineering, Vol. 112, No. 4, pgs 726-744 Krauthammer, T., S. Shahriar, 1988. ESL-TR-87-60. “A Computational Method for Evaluating
Modular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects.” Engineering & Services Laboratory Air Force Engineering & Services Center, Tyndall Air Force Base, Florida.
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MacGregor, G.J., Wight K.J., 2009. “Reinforced Concrete – Mechanics and Design.” Pearson
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Ross, T.J., 1983. “Direct Shear Failure in Reinforced Concrete Beams Under Impulsive
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121
BIOGRAPHICAL SKETCH
Thien Phuoc Tran was born in Hue, Viet Nam in 1964. He escaped from the Vietnamese
Communist Regime and arrived in Canada as a political refugee in 1980.
He started his undergraduate studies in the civil engineering program at the University of
Alberta, Canada in 1983. He graduated from this program and obtained his Bachelor of Science
in engineering in 1988.
He worked as a structural and construction engineer until 1997 when he joined the
Canadian Armed Forces as a Combat Engineer. He served two oversea-tours. Upon his
returning from the tour in Afghanistan in 2006, he was awarded a postgraduate scholarship to
pursue a master’s degree in civil engineering at the University of Florida, specializing in the field
of Force Protection. He received his Master of Science from the University of Florida in the
spring of 2009.