rehabilitation of existing structures by optimal placement of viscous dampers

7/29/2019 Rehabilitation of Existing Structures by Optimal Placement of Viscous Dampers

1/8

1 INTRODUCTIONThe current approach to seismic design of build-

ings supposes three different levels of performance.The structure needs to behave differently for differ-ent earthquake intensities. In the event of a minor

earthquake the structure must not sustain damage,while in the event of a moderate earthquake thestructure can sustain damage in certain parts of thestructure. In the event of a severe earthquake, build-ings must not collapse, however considerable dam-age is allowed. This principle requires buildings to

be repaired after each moderate or severe earth-quake. A modern approach to rehabilitation is to in-crease the ability of the structure to absorb energythrough passive devices.

The viscous damper is a passive device, whichhas been shown to significantly reduce the responseof a structure during an earthquake. References to

these devices have been implemented in designcodes like the FEMA 356 (2000). It is shown that, ingeneral, this type of device improves the seismic be-havior of structures.

While there are some studies on the effect of pas-sive damping devices in the structure, their optimal

placement is not as well researched. Most studies onthe optimal distribution use iterative nonlinear trialand error analyses to obtain an optimal damper dis-tribution. In their study, Martinez and Romero(2003) distribute the highest damping capacity in thestories with the maximum relative velocity. The

evaluation of this parameter is, however, dependenton a certain seismic action, thus a more theoreticalapproach needs to be studied. Such an approach has

been developed by Takewaki (2009) using optimaldesign theory, and will be used in the followingstudy.

In addition, the study aims to address optimaldamper placement for the particular seismic condi-tions of pulse like, long period earthquakes produced

by the Vrancea source.Currently, design codes in Romania want to in-

crease the level of seismic hazard, to a more suitablemean return period of 475 years. It is argued that forthese types of earthquakes the forces in the viscousdampers need to be exceedingly high, introducingadditional efforts into the adjacent member.

The main objectives of the study are the follow-ing:

1. Test the Optimal Damper Placement methoddeveloped by Takewaki (2009).

2. Determine the opportunity of using viscousdampers for the rehabilitation of structuresunder the particularities of Vrancea earth-quakes and the new design provisions.

2 VISCOUS DAMPERSViscous dampers are passive dissipation devices.

This type of damper is very robust and has been

Rehabilitation of Existing Structures by Optimal Placement of ViscousDampers

A.Gh. Pricopie & D. CretuTechnical University of Civil Engineering Bucharest, Bucharest, Romania

ABSTRACT: The aim of this paper is to study the optimal placement of viscous dampers in existing struc-tures, in order to minimize the displacement response, for pulse like seismic conditions of Romania. The op-timal behavior is determined using the dynamic equation of motion, written in the frequency domain. The op-timization process is based on minimizing the sum of the mean square response of interstorey displacement.Using the sensitivities of the transfer function, the optimal position of the viscous dampers will be evaluatedfor a structural configuration and increased in an iterative process. A case study on an existing concrete frame

structure is presented. The structure is modeled with finite elements, and the optimal viscous damper place-ment is determined. The seismic responses of the existing structure and of the structure rehabilitated with op-timal viscous dampers are compared. Also the influence of the dampers is assessed on the adjacent connectingelements.


2/8

Figure 1. Elevation of originalstructure

used in both new and existing projects. The viscousdamper is built like a piston with two chambers, oneof which is filled with viscous fluid. When the pis-ton moves, it forces the liquid through an orificegenerating a resisting force. The value of the force

developed in the damper Fvs is:

vs

v= sign(v)CF (1)

where v is the relative speed between the ends of thedamper, C is the damper constant and is a powerexponent of relative speed, between 0.3-1.5. The ar-ticle will refer only to the linear viscous damper, forwhich =1.

3 NUMERICAL PROCEDURESThe study aims to present and apply an optimal

damper placement strategy and test it to the specific,pulse like, Vrancea earthquake.

First of all, the dynamic analysis of the structureis computed without dampers. The code require-ments are assessed using the performance levels ex-

pressed in FEMA 356(2000), which will also beadapted to the Romanian code P100(2006).

Two sets of constraints on the viscous dampersare imposed. Firstly, using equivalent viscous damp-ing, an overall sum of damping coefficients (Ctot) isdetermined for the structure. Secondly, for technicaland economic reasons a limit value is imposed on

the damping constant, CClim.From the full finite element model of the struc-ture a shear building model is constructed, in orderto simplify the amount of calculation in the optimi-zation process. Using the original structure and theconstraints determined for the dampers, the optimaldamper distribution is determined, considering theMarch 4

th, 1977 Vrancea earthquake.

Once the optimal distribution is obtained, themodel with a uniform damper distribution and theoptimal damper distribution are subjected to an

Figure 2. Elevation of Retrofitted Structure

incremental dynamic analysis. For the incrementaldynamic analysis 4 accelerograms are generated us-ing Vanmarke (1967) method. Each of these accel-erograms PGA is scaled using 4 levels (Sf={0.6, 1,1.5, 2}).

For each one of the cases, a nonlinear dynamicanalysis is run. A total of 3 damper distributions (nodampers, uniform dampers distribution, optimaldamper distribution) are tested against 5 accelero-grams, each with 4 scaling coefficients, resulting in60 nonlinear dynamic analyses. Each dynamic non-linear analysis is performed using S.A.P.v14 soft-ware. The results of the analyses are compared andthe performance levels are assessed for the 3 distri-

butions.

4 THE TEST STRUCTUREFor the numeric experiments, a symmetric con-

crete structure is used. Only one of the centralframes of the structure is studied and its elevation isshown in figure 1. The structure has 6 stories of 3meach and 4 spans of 6m each. The building ischecked and complies with current design specifica-tions corresponding to a 100 year mean return periodPGA. The rebar grade considered is S235 and theconcrete class is C20/25. The design includes deadloads (5 kPa), which comprise slab finishing layers,

partitions, and live loads (2 kPa).The inelastic response of the structure is modeled

using plastic hinges which can form in both ends ofeach bar element. The plastic hinges are assigned aTakeda type hysteretic behavior. For the beam plas-tic hinges only the moment curvature relation is con-sidered and idealized for bilinear behavior. For col-umn plastic hinges, the interaction between axialforce and moment is considered. The relation mo-ment curvature for each level of axial force is con-sidered bilinear.

Each of the moment curvature relations are de-duced using the average strength of the concrete andsteel used. Additionally, for the concrete strength

B35x60

6@3m

C80x80

C60x60

4@6m

Linear viscous damper

B35x60

B35x60

B35x60

B35x60

B35x60


3/8

and strain, the effect of confinement is consideredusing formulas given by the P100 (2006) designstandard. The acceptance criteria for the plastichinge rotation are extracted from FEMA 356 andare:

Table 1. Acceptance Criteria For Plastic RotationAngle (rad)

Performance Level IO LS CP

Beams 0.010 0.020 0.025Columns 0.005 0.015 0.020

All of the elements have adequate transverse rein-forcement. Also, it is considered that the beam col-umn connection is strong enough to avoid any sheardeformation or yielding. The model is consideredfixed to the ground at the base of the bottom storey.In order to determine the optimal distribution of theviscous dampers, the frame is further simplified to ashear building model.

5 DESIGN METHODIn the following chapter, the proposed design

method is developed using theory by Takewaki(2009). Firstly, some theoretical considerations are

presented and explained. In the second part of thechapter the logical steps for programming are pre-sented and in the last part of the chapter the used ac-celerograms are discussed.

5.1 Theoretical ConsiderationsThe problem which needs to be solved can be

formulated in the following manner. Given a struc-ture, its dynamic characteristics and the PSD of theinput accelerogram, the optimal position of the vis-cous dampers needs to be evaluated, so that it mini-mizes the sum of mean squares of interstorey dis-

placements. The problem needs to be solved whileaccounting for two constraints. Firstly, the sum ofthe damper coefficients (Ci) is equal to a set value(Ctot). Secondly, each of the damper coefficients will

be smaller than a certain value (Clim).

5

1

i tot

i

C C (2)

limiC C (3)

5

2

1id

i

d (4)

This problem has been studied by Takewaki (2009).The method uses optimal design theory, supposingthe structure remains elastic. The article will apply

the method to the present problem and extrapolatethe results for the nonlinear response of the struc-ture. The problem can be formulated using general-ized Lagrange formulation and Lagrange multipliers( , , ):

6 6 62

1 1 1

6

lim

1

( , , , ) ( ) (0 )

( )

ii i i i d

i i i

i i

i

L C C W C

C C

(5)

For the reduced model the equation of motion iswritten in the frequency domain:

2 gK i C M v Mrv (6)where M is the mass matrix, C is the damping ma-trix, K is the stiffness matrix, r is a column vectorwith 1 on every position, v() is the Fourier trans-form of the displacement vector and vg () is theFourier transform of the ground acceleration. In or-der to simplify the statement the following notationsare made:

2A K i C M (7)

The equation of motion is written:

( ) ( )gAv rMv (8)

The relation between displacement and interstoreydisplacement is expressed using a transformationmatrix(T):

1 1

2 2

3 3

4 4

5 5

6 6

1 0 0 0 0 01 1 0 0 0 0

0 1 1 0 0 0

0 0 1 1 0 0

0 0 0 1 1 0

0 0 0 0 1 1

d v

d v

d v

d v

d v

d v

(9)

1( ) ( )i gd TA rMv (10)

or writing 1( )Hd TA rM as the transfer func-tions for each of the interstorey displacements. Us-

ing random vibration theory, the mean square re-sponse of the interstorey displacement di

2can be

expressed:

22

( ) ( )ii

d gdH P d (11)

Where, Pg represents the power spectral densityfunction of the input acceleration. The next step is toassess the first order sensitivity of the mean square


4/8

response of the interstory displacement to eachdamper (Cj):

2

( ) ( )( ) ( ) ( ) ( )i i i

i i

d d dd g d g

j j j

H HH P d H P d

C C C

(12)

And the second order sensitivity:

2( ) ( ) ( ) ( )

( ) ( )

( ) ( )( ) ( ) ( ) ( )

i i i i i

i i

i i

d d d d d g g

j l j l l j

d dd g d g

j l j l

H H H HP d P d

C C C C C C

H HH P d H P d

C C C C

(13)

After determining the first and second order sen-sitivities, the following algorithm is used to solve theoptimal distribution problem.

5.2 Optimal Distribution AlgorithmA routine has been programmed into MATLAB tosolve the optimal linear viscous damper placement.It is shown that the following algorithm solves theLagrange problem.

Step 1. Initialize all damping constants Cj=0;Step 2. Define Dynamic Characteristics (M,K,C)

and constraints (Ctot,Clim), PSD function(Pg) andnumber of steps(n)

Step 3. Find l damper so that / lD C is mini-mum and increase the damping constant of damper

l /lC W n Step 4. Update the objective function using a lin-ear approximation / lD C D C and its sensitivity 2/ /

i i lD C D C C ;

Step 5. If there is another damper m such that/ /m lD C D C , compute the increment mC

Step 6. Update C matrix and continue from step 3for the remaining number of steps.

If in step 3 there are multiple dampers l1lkwith the same sensitivity, all of their damping coef-ficients are increased using the following relation:

2 2

11 11 1

...lk lk

l lkl l li lk lk li i l i l

D D D DC C

C C C C C C

(14)

5.3 Input AccelerogramsThe most important accelerogram to be used is therecord of the March 4

th, 1977 Vrancea Earthquake

(N-S principal component). For this accelerogramthe PSD is plotted in figure 3.

Figure 3. PSD of March 4th, 1977 recorded accelerogram and

the Critical PSD

This record, being one of the only strong motionsrecorded in Romania, will constitute the basis for theoptimal design procedure. In order to account for thevariability of the PSD function, a critical approachmethod is used, also proposed by Takewaki(2007). Itis considered that the critical excitation PSD has thesame total power as of the March 4th, 1977 accelero-

gram and the amplitude is equal to the peak value ofthe recorded accelerogram PSD. The critical PSD iscentered on the first angular frequency of the struc-ture. Thus, for the proposed critical excitation, theresponse is almost resonant. The peak value of thePSD is s=756 cm

2/s

3and the area of the PSD

S=2075cm2/s4. The critical excitation PSD used forthe optimal design process has a constant amplitudeof 756 cm

2/s

3on a 2.7 rad/s interval centered on the

first period of the studied frame 1=10.4 rad/s(T1=0.6s). Figure 3 plots the critical excitation PSDand the PSD of the March 4

th, 1977 earthquake. In

order to confirm the obtained results through thenonlinear time history analysis, another series of 4accelerograms are generated. The accelerograms arespectrum compatible and have been generated usingVanmarke (1976) algorithm. In figure 4 the targetspectrum is presented, along with the spectra of thegenerated accelerograms. It must be noted that a

Figure 4. Target Spectrum and Generated AccelerogramSpectra

0

100

200

300

400

500

600

700

800

0 5 10 15 20

PowerSpectrum(cm2/s3)

Angular Frequency(rad/s)

PSDMarch

4th1977

PSDCritical

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4

Acceleration(g)

Period (s)

Generated 1 Generated 2Generated 3 Generated 4Target Spectrum


5/8

series of 50 accelerograms were generated out ofwhich 4 have been chosen to resemble the March 4

th,

1977 accelerogram in terms of PGA and Ariasintensity. All the accelerograms used, have a PGA of0.24g, which corresponds to current Romanian de-sign codes for conditions in Bucharest. This PGAlevel corresponds to a mean return period of 100years. The next generation of Romanian codes aims

to raise the mean return period of the design earth-quake to 475 years. Table 2 presents the levels ofseismic hazard and the acceptance criteria whichneed to be fulfilled for each hazard level.

Table 2. Levels of Seismic Hazard Considered

Mean Return Period (years) 50 100 475 975

Scaling Factor (Sf) 0.6 1 1.5 2Performance Level IO LS LS CP

6 RESULTS OF OPTIMAL DISTRIBUTIONALGORITHM

After the complete frame is modeled using finiteelements, a reduced shear building model is pro-duced.

The characteristics of the reduced shear beammodel are presented in table 3.

Table 3. Characteristics of the Shear BuildingModel______________________________________________--

Setting mi k c ______________________________10

3kg kN /m kN s/m

______________________________________________

Storey 1 111 230367 3128Storey 2 111 110856 1505Storey 3 111 88911 1207Storey 4 122 70183 953Storey 5 122 67628 918Storey 6 122 59768 811

_______________________________________

The mass matrix is determined using the finite el-ement model. The structural damping is assumed to

be Rayleigh proportional to the stiffness of the struc-ture. The damping matrix results considering 5%fraction of critical damping for the first period of thestructure.

A certain level of damping is imposed. Becausethis study aims to prove the opportunity of usingviscous dampers for particular seismic conditions, anaverage level of equivalent viscous damping is cho-sen for structures outfitted with viscous dampers. Bychoosing an equivalent viscous damping level ofd=25% of critical damping the following formulacan be used to compute an uniform distribution of sf

Figure 5. Evolution of damping coefficients with respect to de-

sign step

linear viscous dampers(Cunf):2

2 2

1

4

4000 /

cos

d i i

iunf

i ri ii

m

C kNs m

T m

(15)

Where d is the equivalent viscous damping intro-duced by the dampers mi i ri,,T1 are the storeymass, normalized displacement in the first mode,relative normalized displacement in the first mode,the angle between the damper and horizontal, re-spectively the first period of the structure. Thus, theuniform distribution uses 6 linear viscous dampers,each one with a damping constant(Cunf=4000kNs/m).

For the optimal distribution of the dampers the

following constraints are employed. Firstly, the sumof the damping coefficients for the whole structurewill be the same as in the uniform distribution case(Cs=24000kNs/m). The second constraint of the al-gorithm needs to be chosen considering two aspects.Firstly, a very high damping constant results in largeforces which can cause local structural failure, forthe elements with which the damper is connected.

Figure 6. Evolution of Interstorey Displacement Sensitivitywith respect to design step

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 100 200 300 400

DampingCoefficientkNs/m

Algorithm Step Number

C1

C2

C3

C4

C5

C6

-5.00E-10

-4.00E-10

-3.00E-10

-2.00E-10

-1.00E-10

0.00E+00

0 100 200 300 400

InterstoreyDisplacementSe

nsitivity

Algorithm Step Number

C1

C2

C3

C4

C5

C6


6/8

Secondly, technical aspects need to be taken into ac-count as producers usually have a limit force fortheir dampers. In the case of this study the limit onthe damping constant considered is Clim=8000kNs/m). For these constraints, considering the criti-cal PSD, shown in figure 3, and choosing a numberof steps (n=400), the damper distribution presentedin figure 5 is obtained. The optimal damper distribu-tion uses 3 dampers on stories 2,3 and 4 each with a

damping coefficient Clim=8000kNs/m.The results indicate that the 4th

storey damper isthe most useful for the structure. As its value in-creases the sensitivity of the 4

thstorey damper (fig-

ure 6) reaches the sensitivity of the 2nd

storey damp-er and the increase in damping coefficient isdistributed between the two dampers. In the laststeps of the process both the 2nd and the 3rd storeydampers reach the imposed constraint of the algo-rithm and their increase is transferred to the 4

thsto-

rey damper which also reaches the constraint in thelast step of the algorithm.

Using equation (15), the equivalent viscous damp-

ing can be assessed for both distributions. Theequivalent viscous damping for the uniform distribu-tion is equal to 26%, while the equivalent viscousdamping for the optimum distribution is almost 1.5times higher (38%).

In figure 7 the objective function, the sum of in-terstorey displacement is plotted through the algo-rithm steps for the optimal distribution and for theuniform distribution. It is clear that through all thedesign steps, the optimal distribution obtained pro-vides lower interstorey displacement than the uni-form distribution. The gap between the two curves

starts to decrease towards the end as the constraintsin the algorithm limit the values of the most usefuldamping coefficients. The difference between thetwo objective functions (sum of the interstorey dis-

placements) at the end of the algorithm is 13%.

Figure 7. Evolution of objective functions with design step.

Figure 8. I.D.A. Maximum drift for March 4th, 1977 accelero-

gram.

7 RESULTS OF NONLINEAR DYNAMICANALYSIS

The nonlinear dynamic analysis needs to establishthe opportunity of using the linear viscous dampersfor specified seismic conditions. The structure is an-alyzed without viscous dampers and with two distri-

butions of viscous dampers (uniform distributionand optimal distribution).

In figure 8, the results for the nonlinear analysisare displayed for the structure in the conditions ofthe 1977 accelerogram and all of the three damperdistributions. The positive effect of the dampers isevident on the maximum drifts of the structure.From the figure it results that the dampers are evenmore effective in reducing displacements as the level

of the PGA increases. If the decrease from the nodamper distribution to the optimal distribution is22% for Sf=0.6, it increases to 50% for Sf=2. Thedifferences in drift between the uniform distributionand the optimal distribution range between 8% and16%, approximately the same as the elastic structure.

Figure 9. I.D.A. Maximum drift for generated accelerograms.

1E04

0.0001

0.0003

0.0005

0.00070.0009

0.0011

0.0013

0.0015

0 100 200 300 400

ObjectiveFun

ction

AlgorithmStepNumberUniformDistribution OptimalDistribution

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.005 0.01 0.015 0.02

PGA

ScalingFa

ctor

MaximumDrift

NoDampers

Uniform

Distribution

Optimal

Distribution

0

0.5

1

1.5

2

0 0.005 0.01 0.015 0.02

PGA

ScalingFactor

Drift

NoDampers

Uniform

Distribution

Optimum

Distribution


7/8


8/8

Figure 13. I.D.A. Envelope of central column axial force

In the event of an increase of the seismic hazard inRomanian codes (from Sf=1 to Sf=1.5), the structurewithout dampers would be in need of rehabilitation.Both cases of distribution meet the design objectives

which is life safety for Sf=1.5 and collapse preven-tion for Sf=2. The optimal distribution performseven better ensuring the life safety objective.

8 CONCLUSIONSThe seismic performance of a six-storey concrete

frame to particular seismic conditions given by theVrancea source is analysed in view of upcomingchange of design codes. A comparative study is per-formed in order to study the opportunity of usinglinear viscous dampers to diminish the seismic re-sponse of the structure. At the same time the methoddeveloped by Takewaki (2009) to assess optimaldamper distribution is tested. The proposed structureis outfitted with two damper distributions, a uniformdistribution and an optimal distribution. The twoconfigurations are studied using the only strong mo-tion recorded earthquake from the Vrancea source(March 4

th, 1977). Moreover a set of 4 spectrumcompatible accelerograms are generated in such amanner as to resemble as much as possible the rec-orded accelerogram. Four degrees of the PGA areconsidered corresponding to important mean return

periods, aiming to prove that an eventual increase ofseismic hazard can be compensated by the introduc-tion of the linear viscous dampers.

The results show that the linear viscous dampersreduce the displacement of the structure in the eventof a pulse like earthquake produced by the Vranceasource. It is shown that the introduction of a usuallevel of dampers reduces displacements from 22%up to 50% depending on the PGA of the earthquake.

The tested optimal distribution algorithm pro-vides a decrease in displacements from the uniformdistributions of about 50%.

Another interesting aspect is the change in effortsfor the members to which the dampers are connect-ed. The results show that although the moment in thecolumn does not decrease, the axial force varies ex-tensively, by up to 60%. Although there is a de-crease in axial force, the nonlinear analyses do notshow the formation of additional plastic hinges onthe columns adjacent to the dampers, which meansthat the decrease in axial force is not in phase with

the maximum moments. One must note that for thecurrent case study the change in efforts does not af-fect the structural response, however the changes,

being extensive, could have a negative effect on adifferent structure.

The eventual increase of the mean return periodfor the design earthquake in Romanian codes couldrender a lot of structures in need of rehabilitation.For the case study the structure does not meet therequired acceptance criteria for the increased hazard,unless outfitted with viscous dampers. The viscousdampers prove their effectiveness in reducing dis-

placements and plastic rotations. Furthermore, with

the use of the optimal distribution algorithm, damperdistributions can be improved. For the case study,the optimal distribution uses half of the number ofdampers the uniform distribution does, limiting bothcosts and extent of intervention.

It is interesting to note that although linear vis-cous dampers provide a better structural response,the nonlinear viscous dampers are generally pre-ferred because they limit the force which they de-velop. This study shows that the linear viscousdampers can successfully be employed, but a studyof the nonlinear viscous dampers is already being

developed and the results are promising.

REFERENCES

Symans M.D., Constantinou M.C. 1999. Semi-active controlfor seismic protection of structures: astate-of-the-art review.Eng Struct; 21(6):469-487;

Takewaki I. 2009, Building Control with Passive Dampers Op-timal Performance-based Design for Earthquakes, Kyoto,John Wiley & Sons (Asia):205-230;

Takewaki I. 2000, Optimal damper placemnt for critical exi-tation, Probabilistic Engineering Mechanics, no. 4, 317-325,Elsvier B.V.

Gasparini D.A., Vanmarke EH 1976, Simulated earthquakemotions compatible with prescribed response spectra. MITcivil engineering reaserch report R76-4.

Martinez-Rodrigo M., Romero M.L. An optimum retrofit strat-egy for moment resisting frames with nonlinear viscousdampers for seismic applications. Engineering Struc-tures25(2003) 913-925

P100/2006- Seismic Design Code for Buildings.(RomanianCode)

A.S.C.E. FEMA 356 2000Prestandard and Commentary for theSeismic Rehabilitation of Buildings

0

10

20

30

40

50

60

NoDampers

Uniform

Optimal

NoDampers

Uniform

Optimal

NoDampers

Uniform

Optimal

NoDampers

Uniform

Optimal

Nu

mberofPlasticHinges

rehabilitation of existing structures by optimal placement of viscous dampers

Documents