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Special Publication No. I.99.195

Response Spectrum Design of a RC Frame Protected with

Energy Dissipation Devices

Fabio Taucer, Francesco Marazzi and Georges Magonette

October 1999

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Abstract

A procedure for the design of building frame structures protected with energy dissipation devices is

presented, based on the use of displacement response spectra and equivalent stiffness and damping

properties. The methodology is simple and is aimed towards its use by design engineers by means of

iterative analysis of a single degree of freedom system until the desired target drifts of the structure arereached. The procedure envisions the design of regular and irregular structures, forcing the protected system

to respond as a regular structure. An analytical procedure is then presented for linear and nonlinear analysis,

with emphasis on the calculation of the modal equivalent properties of the structure, and on a simple bilinear

model representing the hysteretic behaviour of high damping rubber energy dissipation devices. Two

examples are presented to demonstrate the applicability of the proposed methodologies.

Acknowledgements

The present work was developed as part of the REEDS Brite EuRam Project dedicated to the

optimisation of energy dissipation devices, rolling-systems and hydraulic couplers for reducing seismic risk

to structures and industrial facilities. The authors would like to thank Javier Molina, Guido Verzeletti and

all the staff of the ELSA Laboratory for the experimental tests performed on a two-storey reinforced

concrete frame protected with energy dissipation devices made of high damping rubber fabricated by

TARRC*.

The authors would also like to thank Mike Griffith from the University of Adelaide, Australia, and Artur

Pinto and Umberto Varum from the ELSA Laboratory, for their collaboration in adopting the proposed

design methodology towards one of the retrofit options suggested for the seismic retrofit of a reinforced

concrete frame building with masonry infill walls [Griffith, 1999] currently tested at the ELSA Laboratory

under the ICONS TMR Network Programme.

* TARRC (Tun Abdul Razak Research Centre, Brickendonbury, Hertford SG13 8NL, England)

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Table of Contents

1. Introduction..........................................................................................................................4

2. Scope....................................................................................................................................4

3. Design Procedure.................................................................................................................4

Stiffness Procedure...........................................................................................................7

Damping Procedure ........................................................................................................10

4. Analytical Procedure..........................................................................................................11

Linear Analysis............................................................................................................... 12

Nonlinear Analysis .........................................................................................................13

Step-by-step Analysis .....................................................................................................15

5. Conclusions........................................................................................................................17

6. Recommendations..............................................................................................................17

7. References..........................................................................................................................17

Appendix:

A1. Design and Analysis of a 4-storey RC Frame.................................................................... 19

Example 1....................................................................................................................... 22

Example 2....................................................................................................................... 28

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1. Introduction

The work presented herein is part of the REEDS Brite EuRam Project dedicated to the optimisation of

energy dissipation devices, rolling-systems and hydraulic couplers for reducing seismic risk to structures

and industrial facilities, and corresponds to Task 12 for the incorporation of devices into design practices.

During the last decade it has been recognised the need for the rehabilitation of structures to reduce theirseismic risk. In addition, lifeline structures such as hospitals, police and fire stations, etc. are required to

withstand earthquakes with very little levels of damage. To this aim, the use of energy dissipation devices

(ED) has been recognised by the engineering community as a viable way to reduce the seismic risk of

structures. Although sophisticated analytical tools are currently available to predict the dynamic response of

structures protected with ED devices with sufficient accuracy, there is a strong need for simple design and

analytical procedures. The work that is presented here is directed towards filling this gap, by exploiting the

results obtained from the REEDS Project and from other researchers in the field.

A Design Procedure for the dimensioning of ED devices working in parallel with a building frame structure

is first presented. Two design methodologies, namely the Stiffness and the Damping approaches, based on

the use of displacement response spectra, are proposed for the design of regular and irregular building

structures that can be described with equivalent secant stiffness and damping properties, forced to respond

with constant drift at low levels of damage. An Analytical Procedure is then presented for the Linear and

Nonlinear analysis of structures once the ED devices have been chosen from the Design Procedure.

Two examples are presented in the Appendix for the retrofit design and analysis of a regular and an irregular

building frame by means of the aforementioned methodologies.

2. Scope

The proposed procedure is applicable to building frames that satisfy the following conditions:

The retrofit design must prevent the formation of any sequential plastic hinges or storey mechanisms.This condition may be relaxed if plastic hinges are formed in all storeys at approximately the same

earthquake intensity (i.e., the presence of a soft storey mechanism is prevented).

The infill masonry walls must not contribute to the stiffness of the building structure.

The secant stiffness and equivalent damping properties of the structure are known for the range ofdisplacements of the retrofit design.

The retrofit system works in parallel with the structure, adding stiffness and equivalent damping.

3. Design Procedure

The design procedure is aimed towards limiting the drift of a building at acceptable levels of damage

through the addition of ED devices. The procedure is based on the use of displacement response spectra,

adopted with success in the numerical analysis of structure mock-ups and components in Task 10 of the

REEDS Project [Taucer, 1999b] for predicting the response to Pseudodynamic (PsD) tests performed on a

two-storey Reinforced Concrete (RC) building frame mock-up. The results of these tests, corresponding to

the experimental analysis of structure mock-ups and components of Task 8 are reported in [Taucer, 1999a

and Renda 2000].

The variables used throughout this presentation are introduced in Table 1, and correspond to the barestructure (represented by a RC frame), the retrofit system (represented by ED energy dissipation devices in

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series with a steel bracing system), and the retrofitted structure (represented by the system in parallel of the

RC frame and the ED devices). The schematic representation of the system is shown in Figure 1.

Description RC Frame ED Device RC Frame +

ED Devices

Global secant stiffness Kc Kd Ks

Storey i secant stiffness Kci - -

Period of vibration Tc - Ts

Angular frequency c - s

Loss factor c d -

Equivalent damping ratio c d s

Total mass Mc - -

Interstorey i shear Vci Vdi Vsi

Base Shear - - Vbs

Table 1: Nomenclature

+ =

Reinforced Concrete Frame Energy Dissipation Devices

(+ Bracing System)

Retrofitted Structure

Frame Flexural

Stiffness = 0

Figure 1. Retrofit System Schematic Representation

The geometry of the building structure is described by the following properties:

H: frame total height

Hi: height from ground to storey i

hi: interstorey i height

In addition, the following variables are defined:

Tyear: period of return of the earthquake

targetyears: target drift corresponding to Tyear

dtarget years: target spectral displacement corresponding to Tyear

di target years: interstorey i target displacement corresponding to Tyear

dtop: top storey displacement of the frame

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The information needed to start the step-by-step procedure is:

RC frame properties: geometry, mass, stiffness and damping properties as a function of drift .

Energy dissipation devices: d* (See Equation 29 for a definition of the loss factor).

Displacement response spectra for a Tyear earthquake for different levels of damping.

targetyears.

* For the design procedure presented here d is taken as a constant. Characterisation tests performed on EDdevices made of High Damping Rubber (HDR) used for the retrofit of a RC frame tested with the PsD

method at the ELSA Laboratory [Dumoulin, 1998; Molina, 2000 and Taucer, 1999a] showed that d remainsfairly constant for the range of deformations and frequencies expected for civil structures. However, in its

general form d is a function ofs and of the deformation amplitude (i.e., drift) of the structure.

The expected output at the end of the step-by-step procedure is:

Vdi at each storey i

s

ED devices will be designed with a loss factor d to develop at each storey i a force equal to Vdi at adisplacement di target years and angular frequency s.

The stiffness properties of the RC frame are computed as a function of drift, based on the effective moment

of inertia Ieff of members relative to their gross moment of inertia Igross. The interstorey displacements (and

stiffness) of the RC frame are then computed by applying to a Finite Element Model (FEM) of the frame a

triangular inverted distribution of external forces, as shown on Figure 2, compatible with a first mode

dominated triangular distribution of displacements.

di

dtop

Vbc

Ftop

Figure 2. FEM: Distribution of External Forces, Deformed Shape.

The loss factor proportioned by the RC frame is also computed as a function of drift, with typical values

ranging to about 10% (c=5%). The derivation of relationships for expressing the hysteretic damping ratio ofRC members in terms of displacements has been the subject of recent research, as shown by a series of

experimental cyclic tests performed at the ELSA Laboratory [Verzeletti, 1991] on RC column specimens.

The step-by-step procedures that are proposed in the following are iterative and are based on the additional

stiffness and equivalent damping provided by the ED devices.

The first procedure, from now on called Stiffness Procedure, is based on assumptions made on theadditional stiffness provided by the ED devices. For this, the definition of the stiffness ratio is introduced,to represent the stiffness participation of the device relative to the total stiffness of the structure:

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s

d

K

K= ; 0 < 1

(1)

For each value of the correspondent period and equivalent damping of the structure are computed, andspectral displacement dyears is obtained from the displacement response spectrum. The designer increases or

decreases the value of until the desired target spectral displacement dtarget years matches the computedspectral displacement. The procedure assumes that the mode shape of the structure remains unchanged, only

the stiffness (or the frequency) and the equivalent damping change through the iterations. Therefore, to

warrant a uniform drift of the retrofitted structure, the RC frame must be regular and the distribution of

stiffness of the ED devices along the height of the structure must follow that of the RC frame.

The second procedure, calledDamping Procedure, is based on assumptions made on the additional damping

provided by the ED devices. For this, the procedure iterates on the expected damping s of the retrofittedstructure to achieve the desired target spectral displacement dtargetyears. The correspondent period of vibration

of the structure is computed from the displacement response spectrum, and the base shear is calculated and

distributed through the height of the structure in correspondence with a triangular inverted distribution of

external forces. The correspondent interstorey displacements of the RC frame are computed, and compared

with the target interstorey displacements. At this stage the necessary force that must be shared by the EDdevices to keep the interstorey displacements equal to the target displacement is computed. The equivalent

damping provided by the ED devices is then calculated and compared with the equivalent damping assumed

at the beginning of the iteration. The procedure converges when the assumed and computed equivalent

damping at the beginning and at the end of the iteration coincide.

The Damping Procedure can be applied to non-regular structures (i.e. soft-storey structures) since it makes

no reference to the mode shape of the bare RC frame. It is based on the principle that the resulting mode

shape of the retrofitted structure must have a constant drift. As a result, the stiffness distribution of the ED

devices will not necessarily follow that of the RC frame.

Since the structure is retrofitted to achieve a constant drift throughout its height, all variables used in

response spectrum analysis correspond to, or approximate to first modal properties, with the spectral

displacement located at 2/3 of the total height of the structure.

Stiffness Procedure

Step 1. Define targetyears corresponding to Tyears of the earthquake.

Step 2. Calculate dtarget years and interstorey di target years:

dtarget years = targetyears (2/3) H (2)

di target years = hitargetyears (3)

Step 3. Compute Kc as a function oftargetyears:

Apply an inverted triangular distribution of forces to a FEM of the RC frame and obtain top

displacement dtop for corresponding base shear Vbc, as shown in Figure 2. Members in the FEM

are modelled with an effective moment of inertia Ieff set as a function of drift target years. Thestiffness of the system is then obtained as:

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top

bc

c

d3

2

VK =

(4)

Step 4. Compute Tc and c:

c

c

cK

M2T =

c

cT

2=

(5, 6)

Step 5. Compute c as a function oftargetyears.

Step 6. Choose (start with low values and move up to higher ones until convergence is achieved).

Step 7. Compute s and Ts:

=

1

1cs

s

s

2T

=(7, 8)

Step 8. Compute device stiffness Kd:

=1

KK cd(9)

Step 9. Compute the damping contributions d and c as a function of the stiffness participation ratios

and (1-) of ED devices and RC members in the structure (See Equations 24 and 25, Section 4):

=2

d

d (if needed d may be updated as a function ofs)(10)

( )

= 12

c

c

(11)

Step 10. Compute s:

s = d + c (12)

Step 11. Obtain spectral displacement dyears corresponding to Ts and s for an earthquake of period of returnTyears:

enter with Ts intersect displacement spectra for s obtain dyears

Step 12. Compare dyears with dtarget years:

If dyears > dtarget years increase and go to step 7

If dyears < dtarget years decrease and go to step 7

If dyears ~ dtarget years and go to step 13.

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Step 13. Compute Vbs:

Vbs = Mc dtarget yearss2 (13)

Step 14. Compute Vsi at each storey i:

=

=

=n

1i

i

1i

0i

i

bssi

H

H

1VV

(14)

Step 15. Compute Vdi at each storey i:

Vdi = Vsi (15)

Step 16. Compute di at each storey i:

In order to be consistent with the procedure di should be computed by applying (1-)Vbsdistributed triangular inverted through the height of the structure (Figure 3) to the FEM of the

frame with Ieffas derived in step 3. Since the procedure assumes constant drift through the height

of the structure, di can be safely approximated to di target years:

di = di target years (16)

Step 17. ED devices will be designed at each storey i with the following information:

Develop force Vdi at displacement di at a frequency s with a loss factor d.

The procedure outlined above is summarised in Figure 3:

Ts

target years

Tyears

= 20 %

= 15%

= 10%

= 5%

Period

Spectraldisplacement

Tc

Increase Ks,

Increase s

Solution

Figure 3. Graphic Representation of the Step-by-step Stiffness Procedure.

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Damping Procedure

Step 1. Define targetyears corresponding to Tyears of the earthquake.

Step 2. Calculate dtarget years and interstorey di target years from Equations 2 and 3.

Step 3. Compute the effective moment of inertia Ieffof RC members as a function oftargetyears.

Step 4. Compute c as a function oftargetyears.

Step 5. Choose the expected equivalent damping s*

of the retrofitted structure, a value of 15% is

recommended.

Step 6. Compute Ts as a function of dtarget years and s*

for the displacement spectra corresponding to Tyearsof the earthquake:

enter with dtarget years intersect displacement spectra for s* obtain Ts

Step 7. Compute s:

s

sT

2=

(17)

Step 8. Compute Vbs from Equation 13.

Step 9. Compute Vsi at each storey i from Equation 14.

Step 10. Compute di by applying Vbs distributed triangular inverted through the height of the structure tothe FEM of the RC bare frame with Ieffas derived in step 3.

Step 11. Compute Vdi necessary at each storey i to reduce di to di target years:

0d

d1VV

i

yearsetargti

sidi

=

(18)

If Vdi < 0, then the procedure is not applicable, since it is not possible to achieve a uniform drift

distribution for the corresponding Tyears earthquake.

Step 12. Compute d and c as the ratio between the energy absorbed by the ED devices or RC members,and the energy absorbed by the retrofitted structure (See Equations 24 and 25, Section 4):

=

=

=n

1i

isi

n

1i

idi

dd

dV

dV

2(if needed d may be updated as a function ofs) (19)

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( )

=

=

=n

1i

isi

n

1i

idisi

cc

dV

dVV

2(20)

Step 13. Compute s from Equation 12.

Step 14. Compare s computed in step 13 with expected s*

from step 5:

Ifss*

then make s*

= s and go to step 6

Ifs ~ s*

then go to step 15.

Step 15. ED devices will be designed at each storey i with the following information:

Develop force Vdi at displacement di target years at a frequency s with a loss factor d.

4. Analytical Procedure

Once the energy dissipation devices have been chosen to protect the RC frame at the desired target drift

for the design earthquake, more refined analyses must be performed to verify the validity of the design. In

addition, it may be required to analyse the retrofitted structure for other earthquakes of longer periods of

return in order to check the levels of demand imposed on the structure and to derive fragility curves.

Two approaches are proposed, the first and more simple one is based on linear analysis, while the second is

based on nonlinear time history analysis. Both procedures assume that the properties of the ED devices are

known, based on data provided by the device manufacturers, where for a displacement of the device d u,

working at frequency s, a resisting force Fu and loss factor d are given to the designer. This data could bepresented as closed form equations, or expressed in charts as shown in Figure 4.

du

s F

u,

d

Figure 4. Device Design Chart

When analysing the structure for the same Tyear earthquake used in the design procedure outlined in Section

3, the interstorey displacements di resulting from the analysis are compared with those assumed in the

design. If the resulting values are different, the analysis is performed again with updated values of Ieff, c, duand Fu as a function of the new drift . Convergence is achieved when the resulting interstoreydisplacements are the same as those assumed at the beginning of the iteration. The designer should then

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evaluate how different the resulting values are from those assumed in the design and if they are within

acceptable limits.

When the analysis is performed for a different Tyear earthquake, the same procedure outlined above is

followed, starting with parameters corresponding to a converged state of the closest Tyear earthquake. The

results of the analysis are evaluated in terms of the drift demands imposed on RC members and ED devices

for the new period of return Tyears of the earthquake.

Linear Analysis

Response spectrum and time history analysis are performed based on secant stiffness properties of RC

members and ED devices, and on modal equivalent damping ratios of the structure.

The secant stiffness properties of RC members is taken into account by I eff, while for ED devices the secant

stiffness is obtained at each storey by dividing the resisting force Fu by the corresponding maximum

deformation du expected for the device.

The modal equivalent damping ratios s n of the structure are obtained as the sum of the modal dampingratios d n and c n of ED devices and RC members, computed as a function of loss factors d and c, modalmass Mc n of the RC frame, eigenfrequencies s n of the structure, and modal stiffness Kd n and Kc n of thedevices and the RC frame, based on eigenvectors s n obtained from modal analysis of the structureinclusive of ED devices:

n

TnncM scs M= (21)

n

TnndK sds K= (22)

n

TnncK scs K= (23)

The stiffness matrix Kd represents the stiffness contribution of the ED devices in the structure, and is

assembled in the FEM by assigning zero lateral stiffness to the RC frame (i.e., zero flexural and shear

stiffness of RC members). The stiffness matrix Kc corresponds to the stiffness contribution of the RC frame

in the structure.

The expression used for computing d n and c n is derived in [Taucer, 1999b] by means of the modal strainenergy method, as presented in [Chang, 1991; Chang, 1995 and Johnson, 1992]:

2nsnc

ndd

nd M2

K

=

(24)

2

nsnc

ncc

ncM2

K

=

(25)

The expressions presented in Equations 10 and 11 for computing d and c are directly derived fromEquations 1, 24 and 25 for n=1 (first mode) by substituting Ms

2with Ks. The same relation is maintained

for the expressions shown in Equations 19 and 20, by multiplying the numerator and denominator of

Equations 24 and 25 by the square of modal displacement d s. The total energy absorbed by the structure, ED

devices or RC members is then computed as the sum of the individual energies absorbed at each storey,

equal to the product of the interstorey shear times the interstorey displacement.

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Nonlinear Analysis

Nonlinear time history analysis is performed on the structure by modelling the ED devices with a bilinear

model and the RC members of the frame with equivalent linear properties Ieffand loss factor c, derived as afunction of the expected drift. The damping contribution of the RC members is given explicitly by c n, whilethe damping contribution of the ED devices is inherent in the bilinear model.

The bilinear model of the ED devices is shown in Figure 5, as described by the following parameters:

K0 : initial stiffness

Kh : hardening tangent stiffness

Fy : yield force of the bilinear model

du

Fu

W

Kh

K0

2Fy

Figure 5. ED Device Bilinear Model.

derived based on the following values:

Fu : maximum force

du : maximum displacement

: yield force to ultimate force ratio

: strain hardening stiffness to initial stiffness ratio

The properties of the bilinear model K0, Kh and Fy are computed as:

( )

+=

11

d

FK

u

u0

(26)

0h KK = (27)

uy FF = (28)

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The loss factor d is computed in terms of the area W inside of the hysteresis loop and in terms of workW.The expressions shown in Equations 29 through 31 are derived from experimental studies performed for

viscoelastic materials [Uehara, 1991].

d = tan (29)

=

W

W2sin 1(30)

W = 2 Fu du (31)

The value of W is obtained from geometric relations as a function of and . For simplicity the Fu, du andK0 terms are introduced in the equations, however, they do not determine the value of d. Similarexpressions for computing W have been proposed by other researchers for different types of yielding

oscillators showing hysteretic behaviour [DebChaudhury, 1985].

W = A B sin c (32)

( ) ( ) 1KK

F12A

2

0

0

u +

=

(33)

1KK

F2B

2

0

0

u +

=

(34)

( )01

0

1KtanKtanc = (35)

The parameters and of the bilinear model could be given by the ED device manufacturer as closed form

solutions or expressed in charts, as shown in Figure 6, as a function of deformation du of the device workingat frequency s.

du

s ,

Figure 6. Device Bilinear Model Parameters Chart

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Step-by-step Analysis

The procedure is common for both linear and nonlinear approaches, with differences highlighted for each

approach when computing the modal damping and the equivalent secant stiffness or bilinear properties of

the devices. The base acceleration time history or displacement response spectra of the earthquake for T yearsis known.

Step 1. Start the analysis with the expected drift *, device forces Fui*

and displacements dui*

at each storey

i of the structure:

* =

dui*

= di Last converged state.

Fui*

= Vdi

Step 2. Calculate Ieff as a function of*

(Ieff may be computed for each storey i as a function interstorey

drift dui*/hi).

Step 3. Calculate c as a function of*.

Step 4. Calculate ED device stiffness properties:

ForLinear and Nonlinear Analysis compute secant stiffness Kdi:

*

ui

*

ui

did

FK =

(36)

ForNonlinear Analysis compute the properties of the bilinear model of the ED devices at eachstorey i:

Compute i and i by entering the Device Bilinear Model Parameters Chart (Figure 6)with dui

*and first mode eigenvalue s 1 from step 5 (For most practical cases it will not be

necessary to iterate over s 1).

Compute K0i, Khi and Fyi as a function of Fui*, dui

*, i and i from Equations 26, 27 and 28.

Step 5. Calculate d n and c n:

ForLinear Analysis compute d n and c n:

By modal decomposition of the n (equal to the number of stories) modes of the structure -inclusive of ED devices, with secant stiffness properties derived in steps 2 and 4 -

compute eigenvalues s n and eigenvectors s n.

Compute Mc n from Equation 21, as a function ofMc and s n.

Compute Kd n and Kc n from Equations 22 and 23, as a function ofKd, Kc and s n.

Compute d n and c n from Equations 24 and 25 as a function ofd, c, Kd n, Kc n, Mc n, sn.

ForNonlinear Analysisd n is made equal to zero for all modes:

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Compute c n as for Linear Analysis, solving the eigenvalue problem using the secantstiffness properties of the structure.

Step 6. Calculate s n from Equation 12 for all modes n of the structure.

Step 7. Perform FEM analysis of the structure and obtain di and dtop:

ForLinear and Nonlinear Analysis perform modal decomposition.

ForLinear Analysis proceed with response spectrum or time history analysis.

ForNonlinear Analysis perform time history analysis.

Step 8. Compare di with expected dui*:

Ifdi dui*

then:

H

d top* =

dui*

= di

Fui*

= Fui computed by entering the Device Design Chart (Figure 4) with di and s 1

Go to step 2.

Else, the solution to the analysis has converged and di = dui*.

For most structures the modal decomposition of step 5 may be too onerous, with no appreciable

improvement in the response for the contribution of higher modes. Alternatively, it may be possible to

compute the participation factor from the frequency ratio (c/s)2

and compute d and c from Equations10 and 11:

Step 5. Calculate d n and c n:

ForLinear and Nonlinear Analysis compute s 1, c 1 and :

By modal decomposition compute eigenvalue s 1 corresponding to the first mode ofvibration of the protected structure.

By modal decomposition compute eigenvalue c 1 corresponding to the first mode ofvibration of the bare frame with Ieffas derived in step 2.

Compute :

2

1s

1c1

=

(37)

ForLinear Analysis compute d n and c n from Equations 10 and 11 using the corresponding

value of computed from Equation 37, equal for all modes n of the structure.

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For Nonlinear Analysis compute c n from Equation 11 using the corresponding value of computed from Equation 37, equal for all modes n of the structure. Damping ratio d n is set tozero for all modes of the structure.

The iterative procedure for nonlinear analysis is needed due to the dependency of the proposed bilinear

model as a function of the maximum expected deformation (and frequency of the system). However, no

iterations are necessary when more sophisticated models are used (i.e., the hysteresis loop changes as a

function of the maximum deformation and frequency of the system) or where this dependency on

deformation and frequency does not exist, such as for steel yielding devices.

5. Conclusions

A step-by-step procedure for the design of structures protected with ED devices by means of

displacement response spectra has been proposed. The procedure is aimed for the design of structures that

can be linearised by equivalent secant stiffness properties and equivalent viscous damping, and can be

applied for both regular and irregular building frames. In addition, a procedure for performing linear and

nonlinear analysis has been presented, once the ED devices have been selected for the structure. Two

examples demonstrate the implementation of the proposed procedures, showing good correlation between

the results of linear and nonlinear analysis with respect to the expected response from the design procedure.

6. Recommendations

The design procedure proposed in this work is aimed towards easing the introduction of the of ED

devices into design practice. For this it is also necessary the collaboration and feedback from the producers

of seismic devices, to provide the necessary information for the designers to dimension the structure in

terms of parameters such as working frequency, deformation, temperature and number of cycles.

Further research is required for the design of buildings protected with ED devices by taking into account theinteraction with masonry infills throughout the height of the structure.

7. References

ACI Committee 318 (1983). Building Code Requirements for Reinforced Concrete (ACI-318-83),

American Concrete Institute, Detroit, USA.

Chang, K.C., Soong, T.T., O, S-T., and Lai, M.L. (1991). Effect of Ambient Temperatures on

Viscoelastically Damped Structures, Journal of Structural Engineering Division, ASCE, 118(7), pp.

1955-1973.

Chang, K.C., Hsu, C.J., Soong, T.T., Lai, M.L. and Nielsen, E.J. (1995). Viscoelastic Damping System for

Mild and Strong Earthquake Applications, 10th European Conference on Earthquake Engineering,

Balkema, pp. 1877-1882.

Clough, R. W. and Penzien, J. (1975). Dynamics of Structures,McGraw-Hill, Inc.,USA, pp. 557-558.

Computers and Structures Inc. (1997). SAP 2000 Integrated Finite Element Analysis and Design of

Structures: Analysis Reference Volume 1, Berkeley (California), USA.

DebChaudhury, A. (1985). Periodic Response of Yielding Oscillators, Journal of Enegineering

Mechanics Division, ASCE, Vol. 111, No. 8, pp. 977-994.

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Dumoulin C., Magonette G., Taucer F. F., Fuller K.N.G., Goodchild I.R, Ahmadi H.R. (1998). "Viscoelastic

Energy Dissipaters for Earthquake Protection of Reinforced Concrete Buildings", 11th European

Conference on Earthquake Engineering, Paris.

Eurocode 8 (1993). Part 1-1: General Rules and Rules for Buildings Seismic Actions and General

Requirements for Structures,European Committee for Standarisation, Brussels.

Griffith, M. (1999). Seismic Retrofit of RC Frame Buildings with Masonry Infill Walls LiteratureReview and Preliminary Case Study, Special Publication No. I.99.xx (to be published), ELSA

Laboratory - JRC, Ispra, Italy

Johnson, C.D. and Kienholz, D.A. (1982). Finite Element Prediction of Damping in Structures with

Constrained Viscoelastic Layers,AIAA Journal, Vol. 20, No. 9, pp. 1284-1290.

Molina, F. J., Verzeletti, G., Magonette, G., and Taucer, F. (2000). Dynamic and Pseudodynamic

Responses in a Two Storey Building Retrofitted with Rate-sensitive Rubber Dissipators, 12th World

Conference on Earthquake Engineering, Auckland, New Zeland.

Renda, V., Taucer, F., Magonette, G. and Molina J., G. Verzeletti (2000). Pseudo-dynamic Seismic Testing

of Large Scale Models of Structures at the European Laboratory for Structural Assessment of theEuropean Commission, Ninth International Conference on Pressure Vessel Technology, Sydney,

Australia.

Taucer F., Magonette G. and Marazzi F. (1999a). Seismic Retrofit of a Reinforced Concrete Frame with

Energy Dissipation Devices, Workshop on Seismic Performance of Built Heritage in Small Historic

Centres, Assisi, Italy.

Taucer, F., Magonette, G. and Marazzi, F. (1999b). Numerical Analysis of PsD Tests Performed on a RC

Frame Protected with Energy Dissipation Devices, Special Publication No. I.99.47, ELSA Laboratory -

JRC, Ispra, Italy.

Tolis Stavros V. and Faccioli E. (1999), Displacement Design Spectra, Journal of EarthquakeEngineering, Vol. 3, No. 1, Imperial College Press, pp. 107-125.

Verzeletti, G., Bousias, S. N., Gutierrez, E., Magonette, G. and Tognoli, P. (1991). Experimental Study of

Hysteretic Energy Absorption and Load Capacity of Reinforced Concrete Columns Subjected to

Different Cyclic Loading Rates, Proceedings of International Conference Earthquake Blast and Impact,

Llyods Register, pp. 275-284.

Uehara, K., Katano, Y., Ogino, N., Katoh, T. and Sakao, K. (1991). Experimental Studies on a Vibration

Control Wall with Viscoelastic Material, SmiRT 11 Transactions, Vol. K, Tokyo.

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Appendix

A1. Design and Analysis of a 4-storey RC Frame

Two examples are presented for the implementation of the proposed procedure for the retrofit design and

successive analysis of a four storey RC frame. The Stiffness Procedure is adopted in Example 1, while the Damping Procedure is adopted in Example 2, as shown in Figure A1. Linear time history analysis is

performed for both examples, while nonlinear analysis is performed for Example 1 only.

4 x 3 m

3 x 3 m

5 m

Example 1 Regular Structure Example 2 Irregular Structure

Figure A1: FEM Geometry of the Retrofitted Structure showing the RC Frame and ED Devices

The RC frame is a single-bay four storey building structure with interstorey heights h i of 3 m (5 m for the

first storey of Example 2) and beam span of 6 m. The cross section of columns is 40x30 cm (longer

dimension oriented in the direction of loading) for the first and second storeys, and 30x30 cm for the third

and fourth storeys. Beams are 55 cm deep with a width of 25 cm. The structure is fixed at the ground, withmasses of 10000 Kg at each storey.

The concrete modulus of elasticity is Ec = 28 MPa, based on a compressive strength of fc = 35 MPa,

computed according to the following expression proposed by ACI on section 8.5.1 [ACI, 1983]:

cc f4730E = , where f c and Ec are expressed in MPa.(A1)

The relation between the effective and the gross moment of inertia of RC members I eff/ Igross is computed as

a function of drift, according to the following empirical relation proposed for the two examples for all

members in the structure, as shown in Figure A2:

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4.0

gross

eff 3.0I

I = (= 1 for < 0.0493), where is expressed in %.(A2)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Drift (%)

Ieff/Igross

Figure A2. Effective Moment of Inertia of RC members as a Function of Drift.

Similarly, the loss factor proportioned by RC members is computed as a function of drift, with ordinate

values of c varying between 6% and 16%, as shown on Figure A3 as given by the following empiricalequation:

2.0

c 14 = , where c and are expressed in %. (A3)

0

1

2

3

4

5

6

7

8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Drift (%)

[c

/2](%)

Figure A3. RC Equivalent Damping Ratio as a Function of Drift.

The seismic input corresponds to a reference period of return of the earthquake Tyears equal to 475 year and

is generated based on the acceleration response spectrum proposed in Section 4.2.2 of Part 1-1 of Eurocode

8 [Eurocode 8, 1998] for Type B medium soils, with a peak ground acceleration apeak of 0.3g. The

displacement response spectrum is derived from the acceleration spectrum by dividing the ordinate values

by the square of the angular frequency. The ordinate values for damping ratios higher than 5% are obtained

by multiplying the ordinate values by the expression proposed in Equation 4.5 [Eurocode 8, 1998]:

correction factor =s2

7

+

(A4)

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Note that the lowest limit of the correction factor of 0.7 is not adopted, in order to be able to represent

spectral values for damping ratios higher than 12.3%. Although not adopted in the current example, a

similar expression with different coefficients (15 in the numerator and 10 in the denominator) has been

proposed by [Tolis, 1999] for a better estimate of the correction factor for higher damping ratios.

An artificial time history with a time duration of 15s (Figure A4) was generated by the Instituto Superior

Tecnico (IST) in Lisbon by Prof. Azevedo for Task 1 of the REEDS Project to fit the EC8 elastic response

spectrum.

-6

-4

-2

0

2

4

6

0 5 10 15

Time (s)

Acce

lera

tion

(m/s2)

Figure A4: Synthetic Acceleration Time History

Displacement spectra derived from the EC8 elastic acceleration spectra (i.e., Sd -2

Sa, where Sd = spectral

displacement and Sa = spectral acceleration) for damping ratios of 5, 10, 15 and 20% are shown in Figure

A5 for periods of vibration between 0 and 1s, and are compared with displacement spectra from the artificial

acceleration time history, showing good correlation for periods below 0.6s (equal to TC for type B soils inEurocode 8). Displacement spectra for higher periods of vibration (in the range of values up to 3s) are

available in literature, however, the proposed spectra is considered sufficient as the example structure shows

periods of vibration below TC.

0

25

50

75

100

125

150

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Period (s)

SpectralDisplacem

ent(mm)

EC8 5% damping

EC8 10% damping

EC8 15% damping

EC8 20% damping

TH 5% damping

TH 10% damping

TH 15% dampingTH 20% damping

Figure A5: Elastic Displacement Response Spectra for EC8 and Synthetic Time History

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The RC frame, representative of a strong-beam weak-column structure, is retrofitted in order to limit storey

drifts targetyears to 0.3% for the earthquake of Figure A4. The retrofit system consists of ED devices mountedon K braces of infinite stiffness at each storey (X braces are used for the first storey of Example 2 to

maintain the inclination of the braces close to 45). The ED devices, similar to those used in the REEDSProject and developed by TARRC (Tun Abdul Razak Research Centre) [Dumoulin, 1998], are made of

HDR and deform in shear at the interface between the RC beam and the K brace. The loss factor d of the

devices is made equal to 0.39 and is considered to remain constant for the range of frequencies anddisplacements of the retrofitted structure. The and values used in the bilinear model are equal to 1/3 and0.10, as derived from characterisation tests performed on ED devices at the ELSA Laboratory [Molina, 2000

and Taucer, 1999b].

Note that if the stiffness of the braces is comparable to the stiffness of the ED devices, the procedure would

be applied by replacing the stiffness of the device with the stiffness of the device in series with the stiffness

of the brace. The displacements of the device would then be obtained as the difference between the

interstorey displacements and the brace displacements, the latter computed as the force of the system in

series divided by the stiffness of the brace. The equivalent damping would be computed as the sum of the

damping contributions of the ED device and steel brace in proportion to their relative displacements with

respect to the total displacement of the series system.

In addition, the retrofitted structure is checked for Example 1 for a higher intensity earthquake of maximum

base acceleration equal to 0.45g.

Example 1

The structure is the regular frame shown in Figure A1.

Stiffness Design Procedure:

Step 1. targetyears = 0.3%

Step 2.

dtarget years = 24 mm

di target years = 9 mm

Step 3.

With Ieff/ Igross = 0.49 from Equation A2

Vbc = 1000 KN is applied as shown in Figure 2 to a FEM of the structure, and d top = 0.458 mm is

obtained

Then Kc = 3.27 MN/m.

Step 4.

Tc = 0.698 s

c = 9.01 rad

Step 5. c = 11% from Equation A3.

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Step 6. First iteration starts with = 0.

Steps 7through 12 are summarised on Table A1, where several iterations are performed until convergence is

achieved for = 0.60.

Step6/12

Step8

Styep10

Step11

Iteration s Ts Kd d c s dyearsrad s MN/m % mm

1 0.00 9.01 0.698 0.00 0.0 5.5 5.5 75

2 0.10 9.49 0.662 0.36 2.0 5.0 6.9 66

3 0.20 10.1 0.624 0.81 3.9 4.4 8.3 58

4 0.30 10.8 0.584 1.39 5.9 3.9 9.7 49

5 0.40 11.6 0.540 2.16 7.8 3.3 11.1 40

6 0.50 12.7 0.493 3.24 9.8 2.8 12.5 32

7 0.60 14.2 0.441 4.87 11.7 2.2 13.9 24

Step7

Step9

%

Table A1: Stiffness Design Procedure, iterations steps 7 through 12 Example 1.

Step 13.

Vbs = 195 KN

Step 14.

Vs = [195, 176, 137, 78.1]T

KN for storeys i 1 through 4.

Step 15.

Vd = [117, 105, 82.0, 46.9]T

KN for storeys i 1 through 4.

Step 16.

d = [9, 9, 9, 9]T mm for storeys i 1 through 4.

Step 17. Design ED devices with the following characteristics:

(d, Vd) = [(9, 115); (9, 105); (9, 80); (9, 45)] (mm, KN) for storeys i 1 through 4

with d = 0.39 at a frequency ofs = 14.2 Hz.

It is important to note that on Table A1 only iteration 7 corresponds to a converged state. For a better

understanding of the procedure, the different converged states of the structure are presented on Table A2 for

values comprised between 0 and 1, where the damping ratio and effective moment of inertia of the RCframe are updated for each of the target drifts assumed in step 1.

The graphic representation of Table A2 is shown on Figure A6, along with the EC8 displacement spectra for

damping ratios between 5 and 20%.

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24

Step1

Step2

Step5

Step6/12

Step8

Styep10

Step11

target dtarget Ieff/Igross Kc Tc c c s Ts Kd d c s dyears% mm MN/m s rad % rad s MN/m % mm

1.11 89 0.29 1.92 0.906 6.93 14.3 0.00 6.93 0.906 0.00 0.0 7.1 7.1 89

0.96 77 0.30 2.04 0.880 7.14 13.9 0.10 7.52 0.835 0.23 2.0 6.2 8.2 77

0.84 67 0.32 2.15 0.857 7.33 13.5 0.20 8.20 0.767 0.54 3.9 5.4 9.3 67

0.72 58 0.34 2.29 0.831 7.56 13.1 0.30 9.04 0.695 0.98 5.9 4.6 10.4 580.62 50 0.36 2.43 0.807 7.79 12.7 0.40 10.1 0.625 1.62 7.8 3.8 11.6 50

0.46 37 0.41 2.73 0.760 8.27 12.0 0.50 11.7 0.537 2.73 9.8 3.0 12.7 370.30 24 0.49 3.24 0.698 9.01 11.0 0.60 14.2 0.441 4.87 11.1 2.2 13.9 24

0.17 14 0.61 4.07 0.623 10.1 9.8 0.70 18.4 0.341 9.50 11.7 1.5 15.1 14

0.09 7 0.79 5.25 0.548 11.5 8.6 0.80 25.6 0.245 21.0 13.7 0.9 16.5 7

0.04 3 1.00 6.68 0.486 12.9 7.4 0.90 40.9 0.154 60.1 15.6 0.4 17.9 30.00 0 1.00 6.68 0.486 12.9 0.0 1.00 infinity 0.000 infinity 19.5 0.0 19.5 0

%

Step7

Step3

Step4

Step9

Table A2: Stiffness Procedure, Converged States for different values of - Example 1

0

25

50

75

100

125

150

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Period (s)

SpectralDisplacement(mm)

Structure Converged States

EC8 5% damping

EC8 10% damping

EC8 15% damping

EC8 20% damping

RC Frame

RC Frame + ED Devices

Figure A6: Stiffness Procedure, Converged States for different values of - Example 1

The two converged states corresponding to = 0 and = 0.60 are highlighted, showing how the drift of thestructure decreases by 73% when ED devices are added: the frequency of the structure increases by a factor

of two (equivalent to an increase in stiffness by a factor of 4) while the equivalent damping ratio of the

structure increases from 7 to 14%, with an increase of the base shear of only 15%. It is interesting to note

that if the target drift were to be achieved by means of additional stiffness only (i.e., devices with d = 0)with

s= 5% (to account for some damping contribution from the steel braces), the required value of

would had gone up to 0.73, equivalent to an increase of the base shear forces of 48% with respect to thedamped solution.

Linear Time History Analysis:

The analysis of the structure is performed based on the Device Design Charts provided by the

manufacturer of the ED devices. Let us assume that these charts are obtained from the following

hypothetical equation, independent of frequency, based on values di and Vdi as obtained from step 17of theDesign Procedure:

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25

diu

i

di V17.0dd

V83.0 +

for dudi

Fu = (KN, mm units)

diu

i

di V25.0dd

V25.1

for du > di

(A5)

Equation A5 is linear and is divided in two parts to reflect the strain hardening effect for deformations d u of

the device larger than di, as shown in Figure A7.

Device Displacement du

Dev

ice

Force

Fu

(di, Vdi)

Design Chart Envelope

Bilinear Model for du = di

Kh

Figure A7: Design Chart Envelope

The parameters have been chosen such that the slope of the proposed equation is larger than slope K h of thebilinear model, in order to simulate the expansion of the hysteresis loops with the increase of displacement

du, as shown on Figure A8 for characterisation tests performed at the ELSA Laboratory (at a frequency of

1Hz and 100% shear strain of the material) on HDR ED devices manufactured by TARRC for the REEDS

Project [Molina, 2000].

Figure A8: Characterisation Tests performed on TARRC ED Devices at the ELSA Laboratory.

(du, Fu) Envelope

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The iterative procedure is presented on Table A3, starting at step 1 with the last converged state from the

design procedure. For the second iteration the Fu*

values are obtained from Equation A5, as a function of

displacements du*

obtained from the previous iteration.

The analysis converges in two iterations with an error tolerance of 6%. The difference between the design

displacements and the analytical displacements derive from the approximation assumed in step 16 of the

Stiffness Procedure where the interstorey displacements are directly computed from the global drift, not

acknowledging the differences in drift that may exist between the different storeys of the structure.

Nonetheless, the difference is acceptable, and for this particular example is conservative. In addition, the

calculation of the modal stiffness to derive the modal damping of the devices in step 5 may not be necessary,

as in most cases it can be safely substituted for all modes of the structure by the device damping derived in

the design procedure.

Step2

Step3

Step4

Iteration

dui* Fui

* Ieff/Igross c Kdi% mm KN % MN/m

1 0.30 [9.0, 9.0, 9.0, 9.0] [115, 105, 80.0, 45.0] 0.49 11.0 [12.8, 11.7, 8.89, 5.00]

2 0.27 [7.5, 8.3, 8.6, 6.5] [99.6, 98.0, 77.2, 34.5] 0.51 10.8 [13.2, 11.8, 8.96, 5.33]

Step1/8

Steps 1 through 4

Iteration s n Kd n Kc n d n c nrad

1 [14.5, 38.6, 61.9, 90.3] [125, 748, 1847, 3442] [93.1, 722, 1945, 4502] [11.6, 9.9, 9.5, 8.4] [2.4, 2.7, 2.8, 3.1]2 [14.7, 39.3, 63.0, 91.6] [123, 763, 1910, 3494] [87.3, 720, 1947, 4506] [11.8, 9.7, 9.5, 8.2] [2.4, 2.6, 2.8, 3.1]

%

Step5

Step 5

Step6

Iteration s n di dtop% mm mm

1 [14.0, 12.5, 12.3, 11.4] [7.5, 8.3, 8.6, 6.5] 322 [14.2, 12.3, 12.3, 11.3] [7.2, 7.9, 8.3, 6.1] 31

Step7

Steps 6 through 7

Table A3: Linear Time History Analysis, Iterative Procedure - Example 1

Nonlinear Time History Analysis:

Nonlinear analysis is performed on the structure with the FEM code SAP2000 Nonlinear [CSI, 1997],

starting with the last converged state obtained from linear analysis (Fu*

values in step 1 are obtained as a

function of du*

using Equation A5). The iterative procedure is shown in Table A4. Step 6is not shown since

the modal damping ratio s n of the structure is made equal to the modal damping ratio c n of RC members.Stiffness Khi from Step 4 is not shown, since it is easily obtained as K0i, with = 0.10.

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Step2

Step3

Iteration

dui*

Fui*

Ieff/Igross c Kdi K0i Fyi% mm KN % MN/m MN/m KN

1 0.26 [7.2, 8.0, 8.3, 6.1] [95.8, 94.8, 74.7, 32.8] 0.51 11.7 [13.3, 11.9, 9.02, 5.41] [93.3, 83.5, 63.3, 37.9] [31.9, 31.6, 24.9, 11.0]

2 0.29 [7.4, 8.8, 9.8, 7.4] [98.3, 103, 89.1, 38.2] 0.49 10.9 [13.2, 11.7, 9.07, 5.19] [92.7, 82.0, 63.5, 36.3] [32.8, 34.4, 29.7, 12.7]

Step1/8

Step4

Steps 1 through 4

Iteration n Kc n c n di dtoprad % mm mm

1 [14.8, 39.4, 63.1, 91.7] [92.8, 719, 1947, 4506] [2.4, 2.6, 2.7, 3.0] [7.4, 8.8, 9.8, 7.4] 35

2 [14.6, 38.8, 62.3, 90.5] [93.0, 722, 1944, 4502] [2.4, 2.6, 2.8, 3.0] [7.0, 8.5, 9.2, 6.9] 33

Step5

Step7

Steps 5 through 7

Table A4: Nonlinear Time History Analysis, Iterative Procedure - Example 1

The hysteresis cycles of the ED devices for storeys 1 through 4 resulting from nonlinear analysis are shownon Figure A9.

-200

-100

0

100

200

-10 -5 0 5 10

Displacement (mm)

Force

(KN)

First Storey

Second Storey

-100

0

100

-10 -5 0 5 10

Displacement (mm)

Force

(KN)

Third Storey

Fourth Storey

First and Second Storeys Third and Fourth Storeys

Figure A9: ED Devices Force - Displacement Hysteretic Cycles from Nonlinear Analysis Example 1

-40

-20

0

20

40

0 5 10 15

Time (s)

Disp

lacemen

t(mm

)

Linear

Nonlinear

-300

-200

-100

0

100

200

300

0 5 10 15

Time (s)

Force

(KN)

Linear

Nonlinear

Top Storey Displacement Base Shear

Figure A10: Comparison Between Linear and Nonlinear Time History Analysis Example 1

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The nonlinear analysis converges in 2 iterations with an error tolerance in the interstorey displacements of

7%. A comparison between linear and nonlinear analysis for the base shear and top storey displacement time

histories is shown on Figure A10, confirming that the methodology presented for linear analysis constitutes

a valid alternative for performing nonlinear analysis (ED device bilinear model + secant properties of RC

members).

It is important to note that during the last 7 seconds of the time history, linear analysis overestimates the top

storey displacements; this is explained from the higher secant stiffness of the bilinear model at small

displacements, with no relevant influence on the peak response.

The iterations for the nonlinear analysis of the structure corresponding to the 0.45g peak ground

acceleration is shown on Table A5. The alternative method for computing c based on the participationfactor is adopted, and is shown on step 5. Step 6is not shown since s is made equal to c (i.e., d = 0) forall modes of the structure. The iterations start with the last converged state from the nonlinear analysis

corresponding to 0.3g.

Step2

Step3

Iteration

dui*

Fui*

Ieff/Igross c% mm KN %

1 0.28 [7.0, 8.5, 9.2, 6.9] [94.1, 99.7, 83.3, 36.3] 0.50 10.8

2 0.40 [10.2, 12.4, 13.6, 10.0] [123, 155, 131, 51.3] 0.43 11.7

3 0.43 [11.4, 13.2, 14.5, 10.8] [140, 166, 141, 56.3] 0.42 11.8

Step1/8

Steps 1 through 3

Iteration Kdi K0i Fyi s 1 c 1 c di dtopMN/m KN % mm mm

1 [13.4, 11.8, 83.3, 36.3] [93.7, 82.6, 63.0, 36.8] [31.4, 33.2, 27.8, 12.1] 14.7 9.43 0.59 2.2 [13.4, 16.2, 17.9, 13.0] 63

2 [12.0, 12.5, 9.64, 5.13] [84.1, 87.3, 67.5, 35.8] [40.8, 51.5, 43.7, 17.0] 14.3 8.74 0.63 2.2 [12.5, 13.9, 15.4, 11.5] 553 [12.3, 12.6, 9.73, 5.21] [86.0, 88.0, 68.1, 36.5] [46.7, 55.3, 47.0, 18.8] 14.3 8.64 0.64 2.2 [11.8, 13.1, 14.5, 10.9] 52

Step7

Step4

rad

Step

5

Steps 4 through 7

Table A5: Nonlinear Time History Analysis, Iterative Procedure with apeak = 0.45g Example 1

A converged solution is obtained after 3 iterations, with an increase of interstorey drifts of 58%. To obtain a

faster convergence the target displacements dui*

for iteration i were obtained as the average of the converged

displacements di (from step 7) and the target displacements dui*

(from step 1) from iteration i-1.

Example 2

The structure is the irregular frame shown in Figure A1.

Damping Design Procedure:

Step 1. targetyears = 0.3%

Step 2.

dtarget years = 28 mm

di target years = [15, 9, 9, 9]T

mm

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29

Step 3. Ieff/ Igross = 0.49 from Equation A2.

Step 4. c = 11% from Equation A3.

Step 5. s*

= 15% is chosen to start the iterations.

The iterative procedure for steps 6 through 14 is shown on Table A6, with convergence achieved after 2iterations:

Step5/14

Step6

Step7

Step8

Step9

Step10

Step11

Step13

Iteration s*

Ts n Vbs Vsi di(RC bare frame) Vdi d c s% sec rad KN KN mm KN %

1 15.0 0.484 13.0 189 [189, 164, 124, 69.5] [62.1, 26.3, 27.8, 15.4] [143, 108, 84.0, 28.9] 13.8 1.6 15.2

2 15.2 0.487 12.9 186 [186, 162, 123, 68.7] [61.3, 26.0, 27.5, 15.2] [141, 106, 82.5, 28.0] 13.7 1.6 15.3

%

Step12

Table A6: Damping Design Procedure, iterations steps 6 through 14 Example 2

Step 15. Design ED devices with the following characteristics:

(d, Vd) = [(15, 140); (9, 105); (9, 80); (9, 30)] (mm, KN) for storeys i 1 through 4

with d = 0.39 at a frequency ofs = 12.9 Hz.

Linear Time History Analysis:

Time history analysis is performed on the structure with the envelope of the ED devices described by

Equation A5 and design values di

and Vdi derived from the Damping Design Procedure. The iterative

procedure is shown on Table A7 for a peak ground acceleration of the earthquake equal to 0.3g. The

alternative based on the derivation of participation factor is used in step 5 for computing the equivalentdamping ratio contributions of the ED devices and the RC frame.

Step2

Step3

Step4

Iteration

dui* Fui

* Ieff/Igross c Kdi% mm KN % MN/m

1 0.30 [15, 9.0, 9.0, 9.0] [140, 105, 80, 30] 0.49 11.0 [9.33, 12.2, 9.44, 3.33]

2 0.30 [15, 8.0, 8.5, 7.5] [140, 95.5, 76.6, 26.0] 0.49 11.0 [9.34, 11.91, 8.97, 3.44]

Step1/8

Steps 1 through 4

Step6

Iteration s 1 c 1 d c s di dtop% mm mm

1 12.6 6.95 0.70 13.6 1.7 15.2 [15.0, 8.0, 8.5, 7.5] 40

2 12.6 6.95 0.70 13.6 1.7 15.2 [14.9, 8.1, 8.8, 7.5] 42

rad %

Step5

Step7

Steps 5 through 7Table A7: Linear Time History Analysis, Iterative Procedure - Example 2

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The analysis converges in two iterations, showing the effectiveness of the Damping Design Procedure.

Nonlinear analysis is not performed for this example, in view of the good results obtained for the linear time

history analysis of Example 1 shown in Figure A10.