response-based damage assessment of structures (p 79-104)

* Correspondence to: A. Ghobarah, Department of Civil Engineering, McMaster University, Hamilton, Canada L8S4L7. E-mail: ghobara@mcmaster. ca

CCC 0098—8847/99/010079—26$17)50 Received 28 October 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 16 April 1998

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS

Earthquake Engng. Struct. Dyn. 28, 79—104 (1999)

RESPONSE-BASED DAMAGE ASSESSMENT OFSTRUCTURES

A. GHOBARAH1,*, H. ABOU-ELFATH1, AND ASHRAF BIDDAH2

1 Department of Civil Engineering, McMaster University, Hamilton, Canada L8S 4L72 Structural Engineering Department, Ain Shams University, Cairo, Egypt

SUMMARY

The structure’s ability to survive an earthquake may be measured in terms of the expected state of damage ofthe structure after the earthquake. Damage may be quantified by using any of several damage indices definedas functions whose values can be related to particular structural damage states. A number of availableresponse-based damage indices are discussed and critically evaluated for their applicability in seismicdamage evaluation. A new rational approach for damage assessment is presented which provides a measureof the physical response characteristics of the structure and is better suited for non-linear structural analysis.A practical method based on the static pushover analysis is proposed to estimate the expected damage tostructures when subjected to earthquakes of different intensities. Results of the analysis of ductile andnon-ductile reinforced concrete buildings show that the proposed procedure for damage assessment givesa simple, consistent and rational damage indicator for structures. Copyright ( 1999 John Wiley & Sons, Ltd.

KEY WORDS: damage; assessment; structures; frames; index; pushover analysis

1. INTRODUCTION

There is renewed interest in damage analysis for application in the seismic assessment andrehabilitation of existing buildings as well as performance-based engineering approaches. Theidea of describing the state of damage of the structure by one number on a defined scale in theform of a damage index is attractive because of its simplicity. However, its development iscomplex since the index should apply to various structural systems at advanced stages of inelasticdeformation and up to collapse. The damage state of a structure can be defined in several ways: (a)a binary damage state (failure/no failure); and (b) a discrete valued damage state using qualitativeindicators such as none, minor, reparable, severe and failure. Empirical and theoretical ap-proaches have been applied to yield various estimates of structural damage.1

The empirical damage models are based on statistics of observed structural damage followingearthquakes.2 Although, these damage observations may be subjective, they provide usefulqualitative information on the overall seismic performance of structural systems. However, the

empirical evaluation does not lend itself well to rationally predicting the strength reserve andresponse characteristics of a structure with a specified degree of damage because: (a) it completelydisregards the mechanics of materials that undergo large inelastic cyclic deformation; (b) futureearthquakes may have different intensities, duration, and frequency content; (c) buildings in otherlocations and recently built structures that are designed to current codes may differ significantlyfrom the structures used to develop the damage statistics; and (d) the dynamic characteristics ofthe population of structures included in the statistical analysis may have altered due to repairsand damage accumulation from previous earthquakes.

The analytical damage models may involve various degrees of complexity as they account forthe characteristics of the structure and its seismic response. They can be broadly divided into twoclasses1 which are: (a) strength-based damage indices; and (b) response-based damage indices.Strength-based damage indices are simple and do not require response analysis. However, theindex must be calibrated against observed damage using a large database. If field observations ofdamaged structures due to seismic loads are not available, the damage index may be calibratedbased on damage prediction using non-linear dynamic analysis. Strength-based damage indiceswere first proposed in 1968 by Shiga et al.3 and later applied by Yang and Yang.4 These indicesdepend on the geometry of structural elements such as the column and wall area and their generalmaterial properties. A strength-based damage index is proposed by the Japan Building DisasterPrevention Association Standard for the evaluation of the seismic capacity of existing buildings.5Damage evaluation based on the non-linear response analysis is relatively complex but mayrequire less data for calibration. Detailed information of structural and material models anddescription of ground motion consistent with the site of structure are needed.

The seismic performance of structures is commonly related to the capacity to undergo inelasticdeformations, defined as the ratio of peak inelastic response to the corresponding yield responseor ductility. Experimental studies show that ductility as well as alternative measures of seismicstructural performance based solely on the low-cycle fatigue theory do not seem to providea satisfactory index for seismic damage.6 These test results are consistent with the notion thatfailure of brittle systems is caused by excessive deformation while failure of ideal ductile systems isinitiated by repeated inelastic deformations. Damage indices for structures that are neither idealbrittle nor ideal ductile, should account for the damage effects of both excessive and repeatedinelastic deformations.7 There is a need for more general and reliable indices to characterize theperformance of structures.

The objective of this study is to review and evaluate some of the available response-baseddamage models and to develop a new rational approach for damage evaluation. The proposeddamage assessment is based on the determined response of the structure and the performancecharacteristics of its members. The concept is general and applies to both steel and concretestructures of different lateral load resisting systems. A ductile and non-ductile three-storeyreinforced concrete frame office buildings are used to illustrate the application of the damageevaluation procedure and to compare the results to a number of response-based indices.

2. RESPONSE-BASED DAMAGE MODELS

The response-based damage indices can be divided into three groups according to what the indexaccounts for: (a) maximum deformation; (b) cumulative damage; and( c) maximum deformationand cumulative damage.

80 A. GHOBARAH, H. ABOU-ELFATH AND A. BIDDAH

Copyright ( 1999 John Wiley & Sons, Ltd. Earthquake Engng. Struct. Dyn. 28, 79—104 (1999)

2.1. Damage indices based on maximum deformation

2.1.1. Ductility ratio (DR). The ductility ratio is defined as the ratio of the maximum deforma-tion to the yield deformation. It has been used extensively in seismic analysis to evaluate thecapacity of structures undergoing inelastic deformation and develop inelastic response spectra.8As a damage index, the ductility ratio may be unsatisfactory,9 especially when shear distortion injoints and beam bottom bars pullout are anticipated. As demonstrated by experimental studies,the ductility ratio does not account for the effect of the duration and frequency content of theground motion. It is normally assumed that failure occurs when the ductility demand (response)exceeds the structural ductility (capacity) that is equal to the ratio of the ultimate deformationunder monotonic static load to the yield deformation.

2.1.2. Interstorey drift (ID). The interstorey drift is the maximum relative displacement betweentwo storeys normalized to the storey height. According to Sozen10 the percentage of damage tothe structure is given by

% of damage"50 (maximum interstorey drift in percentage)!25 (1)

From the analysis of test data on components and small-scale structures, it was found that an IDvalue smaller than 1 per cent corresponds to damage of non-structural components while valuesof ID larger than 4 per cent may result in irreparable structural damage or collapse. Collapse isconsidered to occur when ID exceeds 6 per cent.11 Similar to the damage index based on theductility ratio, the interstorey drift does not account for effects of cumulative damage due torepeated inelastic deformation. In addition, the relationship between damage and interstorey driftvaries depending on the maximum deformation at collapse which depends on the ductility class ofthe structure.

2.1.3. Slope ratio (SR). The slope ratio is a measure of damage due to stiffness degradationduring seismic loading. It is defined as the ratio of the slope of the loading branch of theforce—displacement diagram to the slope of the unloading branch. From tests on small-scalestructural systems, it has been determined that SR with values of 1)0 and 0)2 correspond to safestructural behaviour and critically damaged structures,12 respectively.

2.1.4. Flexural damage ratio (FDR). Roufaiel and Meyer11 suggested that the ratio of initialstiffness to the reduced secant stiffness at the maximum displacement can be used as a measure ofdamage. Damage indices based on extreme inelastic deformations seem to be strongly correlated6

so that their predictions are usually similar. For example, the correlation coefficient betweenthe two ratios DR and FDR have been found to be 0)95. The FDR index does not account foreffects of cumulative damage caused by repeated load reversals. Critical values of the DR, SR andFDR damage indices are determined from laboratory tests and field observations. Therefore,their use in the prediction of seismic damage for structures with characteristics significantlydifferent from those used in the calibration process requires caution. Additional difficulties in theuse of these damage indices relate to the differences between the characteristics of the expectedearthquake and the earthquakes used in the calibration such as intensity, duration and frequencycontent.

RESPONSE-BASED DAMAGE ASSESSMENT OF STRUCTURES 81


2.1.5. Maximum permanent drift. Permanent drift is closely related to the plastic deformationsin a structural system. Toussi and Yao12 and Stephens and Yao13 introduced a qualitativeclassification of damage, which among other things included the permanent drift experienced bythe building. They defined four levels of structural damage as: (a) Safe with storey drift notexceeding 1 per cent of the storey height and no permanent drift; (b) lightly damaged withpermanent displacement of approximately 0)5 per cent of the storey height; (c) damaged withpermanent displacement of 1 per cent of the storey height; and (d) critically damaged with topstorey displacement showing some aperiodicity at the end of the record with poor correlationbetween base shear and top level displacement.

The shortcoming of the maximum permanent drift as a measure of damage is that light damageimplies a maximum permanent drift of 0)5 per cent or less. However, a permanent drift of 0)5 percent does not necessarily indicate light damage. For example, a damaged non-ductile framestructure may exhibit no permanent drift, even after severe inelastic deformation.

2.2. Damage indices based on cumulative damage

2.2.1. Normalized cumulative rotation (NCR). A simple measure of structural deteriorationduring a seismic event is the sum of all inelastic excursions experienced by the structure. The valueof this measure depends on the duration and intensity of the earthquake. The normalizedcumulative rotation is defined as the ratio of the sum of the inelastic rotations during half cyclesto the yield rotation.6 Statistical analysis of data on beam-column elements subjected to cyclicloads shows that damage indices based only on cumulative inelastic deformation or dissipatedenergy, may be inadequate to characterize the complex process of damage propagation andsubsequent failure in concrete members.9

2.2.2. Low cycle fatigue (LCF). The theory of low-cycle fatigue has been applied to the seismicanalysis of structures subjected to strong ground motion to estimate the state of damage.14 Thedetermination of the damage index is somewhat complex and involves the entire response history.However, the index does not account for the effect of maximum inelastic deformation.

2.3. Damage indices accounting for maximum deformation and cumulative damage

2.3.1. Park and Ang+s local damage index. The index is based on scaled values of ductility anddissipated energy of the local element during the seismic ground shaking.7,15 The ductility definedas the ratio of the maximum experienced deformation d

., to the yield deformation, is scaled by

the ratio of the ultimate deformation d6, under a monotonic static load to the yield deformation.

The dissipated energy is scaled by b%/(Q

:d6) where Q

:is the yield force and b

%is a constant

determined from experimental calibration. The ultimate deformation is determined when theconcrete reaches a specified ultimate strain. Although the value of the index may exceed unity,failure of the structure is assumed to occur when the damage index equals 0.8—1.0. Undermonotonically increasing loads, the dissipated energy is zero and the value of the damage index isd./d

6so that failure is predicted to occur, as expected when d

."d

6.



The assumptions used in the development of the damage index expression are: (a)the contributions to damage of the extreme deformation and dissipated energy can be superim-posed linearly, and (b) the related evolution in time of these components can be disregarded. Theresults obtained by Banon and Veneziano6 do not support these assumptions. In addition, thevalue of the constant b

%is not specified and has to be obtained by calibration using laboratory or

field data. The behaviour of this index is strongly dependent on the hysteretic model of theelements.

2.3.2. Chung, Meyer and Shinozuka+s local damage index. Chung et al.16 proposed a damageindex which contains damage modifiers that reflect the effect of the loading history. This indexconsiders the difference in response of members to positive and negative moments. The effect ofthe loading history is taken into account by a damage modifier which includes the change instiffness and the bending moment sustained up to the calculation cycle. The damage indexdefinition does not explicitly account for the damage caused by the maximum deformationexperienced by the element.

2.3.3. Maximum softening. DiPasquale and Cakmak17 developed a damage model based on theevolution of the natural period of a time-varying linear system equivalent to the actual non-linearsystem for a series of non-overlapping time windows. This global damage index depends ona combined effect of stiffness degradation and plastic deformation. However, to compute themaximum softening it is necessary to have the input ground acceleration and the acceleration atanother location such as at the top of the structure. The maximum softening, index does notexplicitly account for the dissipated hysteretic energy and strength deterioration and does notprovide information concerning the extent of local damage sustained by the members.

2.3.4. Final softening. DiPasquale and Cakmak17 used the change in the fundamental period ofthe structure as a measure of the change in the stiffness caused by the earthquake. However, theinstantaneous fundamental period includes the effect of the inertia and damping forces. Theadvantage of the final softening is that it can be evaluated from the initial natural period and thefinal period determined from vibration field testing after the earthquake. In effect, it is notnecessary to know the actual structural response. However, the measured change in period couldbe caused by cracking of infill walls while the structural system may remain intact. Thefinal period is affected by the changes in the fundamental mode of the structure due toinelastic response. These changes will cause a corresponding change in the modal mass leadingto final softening index that is no longer representative of the global stiffness deterioration.A shortcoming of damage measurements based on the final softening is that local element andstorey damage as well as the information contained in the response to the earthquake are notavailable. A recognized difficulty in the calculation of the final period is due to the idealizationused in the analytical procedure. The period calculation at the final time step of the earthquakeloading may be affected by the randomness of the instantaneous tangent stiffness at the end of thedynamic load. In the inelastic hysteretic response range of reinforced concrete, the stiffness of theloading direction may be significantly different from the stiffness in the unloading direction. Inaddition, the stiffness at the zero load position may differ from the stiffness at the loadedpositions.



2.4. Global damage indices

Several of the discussed damage indices such as DR, NCR, LCF, Park and Ang and Chunget al. represent the local damage sustained by the individual structural elements. The analysis oflocal damage indices identifies the weak or vulnerable elements that should be rehabilitated.However, it is difficult to get a clear idea of a structural system’s response to a given input groundmotion from a long list of element damage indices. Important decisions concerning the residualstrength and safety of a damaged structure are currently based on a single overall or globaldamage index. Global indices are required for post-earthquake evaluation of structures, reliabil-ity studies and applications in the performance-based engineering approach.

The global damage index is obtained from special combinations of local damage measures. Thesimplest technique for combining local damage indices is to use a weighing scheme. The weighingfactor can reflect the replacement cost and/or the relative importance of the member or substruc-ture in maintaining the integrity of the structure. For example, the lower storey of a buildingmight be assigned more importance than the upper storeys. The weighing factor for any storeycould also depend on the magnitude of the damage index for that storey, so that severelydamaged storeys are weighted more heavily. Due to the integration of detailed damage informa-tion of an entire structure into a single global estimator, much information is lost thus allowingonly a crude estimate of structural performance during seismic events. Park and Ang7 presentedan approach to calculate the global damage index by using weight factors for local indices definedas functions of the hysteretic energy dissipated by each element. Chung et al.16 used the damageindex from each storey to define the global damage index. The storey damage index is obtained asa weighted average of the local damage indices of all elements in the storey, with the energydissipated in the member as the weighing function.

The use of weighted average procedure to calculate global damage index does not properlyaccount for the local concentration of damage, does not distinguish between a column anda beam, and may lead to misleading results. It is possible for a few structural members of thebuilding to have undergone severe damage without being reflected in the global index. Toillustrate the point, consider two identical portal frames A and B. The first frame A is subjected toa lateral force causing one plastic hinge on the right side of the beam with local damage index d

R,

hence, the global damage index is the also dR. The second frame B is subjected to a higher lateral

force causing two plastic hinges to form at both ends of the beam, one to the right and one to theleft. The local damage indices of the plastic hinges are d

R, d

Land the dissipated hysteretic energy

of each plastic hinge are ER

and EL. The global damage index is (d

RER#d

LEL)/(E

R#E

L). If (d

R,

ER) of frame B are slightly larger than (d

R, E

R) of frame A and (d

L, E

L) of frame B are fractions of

(dR, E

R) of frame B the calculation may yield a global damage index of frame B that is lower than

that of frame A. This is incorrect since frame A is at an earlier stage of loading and should haveless damage than frame B.

3. PROPOSED DAMAGE MODEL

From a series of earthquake simulation tests of concrete elements, Otani et al.18 observed thatwhen loading reversal was repeated at the same newly attained maximum displacement, theloading stiffness in the second cycle was lower than that in the first cycle, although the resistances



Figure 1. Pushover analysis. Change of stiffness for calculating the damage index of: (a) the whole frame;(b) a storey

at peak displacement were almost identical. Otani and Sozen19 observed from testing ofmulti-storey frames that if a reinforced concrete structure, subjected to a base motion of sufficientintensity to develop yield, was tested for a second time with a base motion of similar intensity, itdeveloped essentially the same maximum drift it did in the first test. The stiffness of the structureat the beginning of the second test was lower than the initial stiffness of the structure. From thisobservation, Sozen20 concluded that the maximum drift response appeared to be a function of theinitial properties of the structure and not related to the stiffness of the structure at the beginningof the second test. However, when a reinforced concrete member is subjected to cyclic loading atconstant displacement level, the member stiffness will decrease with the loading cycles. In otherwords, the damage to the structure after it was tested for the second time is more than the damageto the structure after the first test. This is the reason why drift alone may not be the best measureof damage. The stiffness before and after the loading is a more consistent indicator of damage.This is the concept of the final softening damage index.

To overcome the analytical difficulties in the calculation of the change in period as a measure ofdamage, a new approach for determining the change in stiffness of the structure is proposed. Theapproach is to perform pushover analysis for the structure twice; once before subjecting thestructure to the earthquake and once after subjecting the structure to the ground motion, asshown in Figure 1. Before performing the second pushover analysis, the structure is returned tothe unloaded static state. Relating the initial stiffness before and after subjecting the structure toan earthquake, a new global damage index as well as damage indices for each storey level can bedetermined. The damage indices of the storeys are useful in determining the deterioration instiffness and the distribution of damage to the various storeys. The stiffness damage index (DI)

Kof

the whole frame can be calculated as follows:

(DI)K"1!(K

&*/!-/ K

*/*5*!-) (2)

where K*/*5*!-

is the initial slope of the base shear-top deflection relationship resulting from thepushover analysis of the frame before subjecting it to the earthquake ground motion and K

&*/!-is

the initial slope of the same relationship but after subjecting the frame to the earthquake time



history. The stiffness damage index of the ith storey is defined as:

(DI)iK"1!(Ki

&*/!-/Ki

*/*5*!-) (3)

where Ki*/*5*!-

and Ki&*/!-

are the initial slopes of the base shear-storey drift relationships of the ithstorey resulting from the pushover analysis of the frame before and after subjecting it to theearthquake, respectively.

The values of the damage indices range from zero to one depending on the amount of damageexperienced. A value of zero represents no damage while a damage index value of one corres-ponds to collapse. However, in practical terms, collapse may be defined at a lower damage indexcorresponding to a certain percentage loss of stiffness.

An inverted triangular loading representing the code lateral load distribution may be used inthe pushover analyses for structures whose response is dominated by the first mode. The loadingcan be modified to represent the contribution of higher modes. The proposed index is based onrelative stiffness before and after the earthquake. For this reason, the index is not sensitive to thedistribution of the applied load in the pushover analysis. The effect of torsion and buildingirregularities can be represented by the 3D pushover analysis.21 Research programs to widen theapplicability of the pushover analysis are underway by several researchers.

4. ADVANTAGES OF THE PROPOSED PUSHOVER DAMAGE INDEX

The advantages of the proposed damage evaluation approach are numerous. Some of theseadvantages include: (a) The calculation of the damage index is based on rational response analysisprocedure with little need for calibration. (b) The stiffness is calculated after removing the inertiaand damping force effects and bringing the frame to a static state. (c) The damage can beestimated at any stage of loading without the need to guess the maximum displacement ordeformation of the structure near collapse. (d) Two different final stiffness can be calculateddepending on the direction of the load in the pushover analysis (either from right to left or fromleft to right). The smaller stiffness can be used in calculating the stiffness index; (e) A veryimportant advantage of the proposed procedure is that it provides information regarding theelement and storey damage and the sequence of element damage and failure. A stiffness damageindex is calculated for each storey level separately as well as for the whole frame which can beused to identify which storey is contributing to the damage; (f ) The storey and global damageindices for the whole frame are obtained without the need for an averaging or weightingprocedure to integrate the effect of the frame elements. (g) The proposed index is capable ofmodelling damage due to mechanisms other than flexural yielding. In this case, the models used inthe analysis should include all possible failure modes. For example, in case of reinforced concretebuildings, the model should consider shear deformation and reinforcing bar bond slip. Williamset al.22 noted that most of the available damage indices consider flexural yielding only and do notconsider the possibility of shear failure for example. These issues are significant in the damageanalysis of existing non-ductile structures.

On the negative side, the calculation of the proposed damage index involves more analysis andeffort than what is needed to evaluate some other damage models. In addition, the applicabilityand reliability of the proposed damage assessment are affected by the limitations of the pushoveranalysis technique.



5. BUILDING DESIGN

As examples, the damage analysis is performed for two typical low-rise three storey reinforcedconcrete frame office buildings. One of the buildings is designed according to the 1963 ACI23 codeand the other according to the 1995 NBCC24 code. The current code designed structure and theexisting building designed to earlier code are considered to represent ductile and nonductilemoment resisting frames. The dimensions of the buildings are shown in Figure 2. The design liveload is taken as 2)4 kN/mm2 which is typical for an office building. The steel reinforcement detailsin the existing frame includes: (a) light confinement reinforcement in the columns; (b) the beambottom longitudinal reinforcement is embedded 150 mm into the beam-column joint; (c) widelyspaced transverse reinforcement in beams and columns; (d) column lap splices are 20 times thecolumn longitudinal bar diameter and located just above the floor level; and (e) no transversereinforcement in the beam-column joint. Section dimensions for the columns and beams and thearea of reinforcement as a percentage of the cross-sectional area (reinforcement ratio) are shownin Figure 2. The NBCC24 code designed building is assumed to be located in the city of Victoriaon Canada’s west Coast. The design base shear for the building is calculated to be equal to 0)147times the weight of the building. The design loads on the building are further increased by theeffect of the accidental torsion provisions of the code to 0)177 times the weight of the building. Thesectional dimensions of the beams and columns of the current code designed frame and thereinforcement ratios are shown in Figure 2.

6. THE ELEMENT MODEL

An inelastic single-component element was developed for modelling the beams and columns ofthe frame. All essential characteristics of the hysteretic behaviour of reinforced concrete membersincluding stiffness degradation, pinching and strength deterioration, are explicitly taken intoaccount. The element was implemented in the general-purpose program DRAIN-2DX.25 Thehysteretic rules are shown in Figure 3. The element utilizes a non-symmetric bilinear curve inconjunction with four variable user-specified parameters. The stiffness degradation parameter, c,and the strength deterioration due to low cycle fatigue parameter, b, are similar to the onesproposed by Park et al.26 The pinching effect is introduced in the loops by an input parameter, a,similar to the parameter proposed by Chung et al.15. The main characteristics of the element is inits ability to model the softening behaviour of reinforced concrete members after reachinga specified deformation corresponding to ultimate strength. The model is also capable ofrepresenting reinforcement pullout and takes into account inadequate shear capacity of thecolumn. Details of the model are presented by Ghobarah et al.27.

7. DAMAGE ANALYSIS

The damage analysis was conducted in three steps: (a) a static nonlinear pushover analysis of theframe was performed; (b) the frame response to an applied earthquake ground motion wasdetermined; and( c) a second pushover analysis was performed. The damage index is evaluated forthe whole structure and for each storey using the proposed approach as given by equations (2)and (3), as well as three selected damage analysis approaches for comparison of the results.



Figure 2. Dimensions and reinforcement of the three-storey office building



Figure 3. Hysteretic modelling of the moment—curvature relationship: (a) stiffness degradation; (b) pinching; (c) strengthdeterioration; (d) softening

7.1. Pushover analysis

The first pushover analysis of the interior frame was conducted using an inverted triangularload to determine the initial stiffness of the frame. The second pushover analysis was conductedafter subjecting the frame to the selected earthquake ground motion to determine the finalstiffness. In the second pushover analysis, the selected stiffness is the smaller of: (a) applying theload to the frame from left to right; or (b) applying the load to the frame in the opposite directionfrom right to left. Examples of the pushover analyses before and after the application of 1940 ElCentro earthquake record scaled to PGA of 0)3g, are shown in Figures 4(a) and 4(b) for the ACIand NBCC code designed buildings, respectively. The analysis results shown in Figures 4(a) and(b) do not include strength deterioration in order to enable the comparison between the differentdamage indices on equal bases.

7.2. Dynamic analysis

The seismic behaviour of the interior frame was studied using scaled ground motion records.The selected ground motions are El Centro record of the 1940 Imperial Valley earthquake (S00E



Figure 4. Pushover analysis of the buildings when subjected to El Centro earthquake scaled to PGA of 0)3 g

component), the 1985 Mexico earthquake (Zihuatenejo, Guerrero Array, S00E component), the1971 San Fernando earthquake (800 W. First St., L.A., N53W component) and the 1985 Nahanniearthquake (Iverson-Site 1, N10E). These earthquake records were selected to represent a widerange of frequency content as evaluated by the A/» ratio (A is the peak ground acceleration PGAin g and » is the peak ground velocity in m/s). The two extreme cases are the far-field Mexicoearthquake record which is rich in low-frequency content and the near-field Nahanni recordwhich is rich in high frequencies. The fundamental period of the two frames are almost the sameand equals approximately 1)0 s. The dynamic analysis was carried out using a time step incrementof 0)005 s taking into account the P!* effect. Three of the discussed damage models werecalculated after the dynamic analysis for comparison with the proposed damage model. The three



Figure 5. Damage index variation with PGA for the existing frame

models are: (a) Park and Ang7;( b) Roufaiel and Meyer11; and (c) DiPasquale and Cakmak17 finalsoftening damage model.

7.3. Results of the damage analysis

Figures 5 and 6 show the damage index versus PGA when the existing 1963 ACI and the 1995NBCC code designed buildings are subjected to various earthquakes scaled to different PGAlevels. The ultimate rotations used in calculating Park and Ang’s index are obtained according to



Figure 5. Continued

the concrete material model given by Park and Paulay.28 The parameter b%used in calculating

Park and Ang’s index is taken 0)1 according to Park et al.26 for nominal strength deterioration.From the figures, there is a correlation between the PGA and the proposed damage level. Asexpected, the current code designed ductile building shows much lower damage index values thanthose of the existing non-ductile building when subjected to the same earthquake at the same levelof PGA.

The final softening index is the closest to the proposed pushover index since they are bothconceptually similar. In most cases, it gives lower values than the proposed index especially athigh PGA levels, as shown in Figure 6. In some cases, the final softening index was found todecrease with increasing PGA level which is not physically correct. However, this is mainly due tothe analytical difficulties in determining the final period as discussed in Section 2.3. The finalfundamental period used in calculating the final softening index is determined after the last



Figure 6. Damage index variation with PGA for the current code designed frame

increment of the earthquake loading with no control on the loading direction which may givedifferent loading or unloading stiffness. The figure shows that there is a degree of randomness inthe calculation of the instantaneous stiffness of the structure at the final step of the earthquake.

In the case of the response of the current code designed frame to the San Fernando earthquake(Figure 6), the building was more damaged at PGA"0)3g in terms of local damage indices,storey drift and the proposed damage index as compared to the damage state at PGA"0)2g.However, the averaging procedure used in calculating Roufaiel and Meyer and Park and Angindices yielded global damage indices for the case of PGA"0)3g that is less than or almost equal



Figure 6. Continued

to the case of PGA"0)2g. These results confirm the earlier discussion that in some cases theaveraging procedure may give incorrect results. The difference between the results may betangible for one specific ground motion, as shown in Figure 6. However, these inconsistentdamage values almost disappear when the response to several earthquakes is averaged.

Figures 7 and 8 show the damage index variation with the maximum interstorey drift for theexisting and current code designed frames, respectively. The maximum interstorey drift occurredat the first storey. From the figures, some correlation between the proposed damage index and themaximum interstorey drift is observed. Comparing Figures 7 and 8, the interstorey drift limits aredependent on the ductility class of the structures. Collapse occurs at maximum interstorey drift of2)25 per cent for the existing nonductile building (Figure 7) and from 4 to 5 per cent for currentcode designed ductile building (Figure 8). Figure 9 shows the proposed pushover damage indexvariation with PGA of the El Centro record for each storey and for the whole existing frame. The



Figure 7. Damage index variation with the maximum interstorey drift for the existing frame

damage indices for the storeys are useful in determining the distribution of damage to the variousstoreys and identifying which storey control the performance of the whole frame. In thisparticular case, the first storey damage controlled the performance of the frame. Once a storeyindex reaches 1)0, the damage index of the whole structure will be also 1)0.

Figures 10(a) and 11(a) show the correlation between the proposed damage model and Parkand Ang’s model calculated for the existing and current code designed frames when subjected tothe selected earthquakes scaled to various PGA levels. The figures indicate that the value of theproposed damage index is about 0)66 of the value of the Park and Ang’s index in case of the



Figure 7. Continued

non-ductile building design and is about 0)85 in case of the current code designed ductile building.It is also interesting to note that for the existing building, failure of Park and Ang’s indexcorresponds to 0)66 of the proposed index. Figures 10(b) and 11(b) show the relationship betweenthe proposed model and the weighted average of Roufaiel and Meyer’s index for the existing andcurrent code designed buildings, respectively. The weight factors are defined as functions of thehysteretic energy dissipated by each element. The correlation with the proposed model is good incase of the existing nonductile building. The value of the proposed damage index is about 0)8 thevalue of the weighted average of Roufaiel and Meyer’s model in case of the existing building and is1)06 in case of the current code designed ductile building. Figures 10(c) and 11(c) show that thecorrelation between the final softening damage index and the proposed index is reasonable. Ingeneral, there is more scatter in the results in case of the ductile frame (Figure 11) than the values



Figure 8. Damage index variation with the maximum interstorey drift for the current code designed frame



Figure 8. Continued



Figure 9. Proposed damage index of the whole frame and of the different storeys versus the peak ground acceleration forthe existing frame

in the case of non-ductile frame (Figure 10). The reason for this behaviour is that the accuracy ofthe indices in estimating the ultimate behaviour decrease with increased ductility. The compari-sons shown in Figures 10 and 11 indicate that the consistency and agreement between severalresponse-based damage indices is good in the case of the non-ductile building. These goodcorrelations between the indices is remarkable considering that they are arrived at usingcompletely different definitions. However, these results are for the simple case of three-storeyframes and the correlation may not be as good in other cases of more complex response.

According to the damage states defined by Gunturi29 and the average relationship between theproposed index and Park and Ang’s index, damage states for structures can be defined as listedin Table I. Figures 12 and 13 show the results of comparing the relationship betweenthe proposed damage model and the maximum interstorey drift for the non-ductile andductile frames, respectively. The definition of damage to the three-storey frame for a giveninterstorey drift presented by Sozen10 and the proposed damage states are included. It is observedthat the definition of the percentage of structural damage by Sozen10 is realistic for existing(non-ductile) buildings and is overestimating the damage for the current code designed (ductile)buildings. The existing building is expected to fail at interstorey drift of 2)5%. However, this limitis lower than the expected interstorey drift at failure for code designed building which isapproximately 4—5%.



Figure 10. Correlation between the damage indices for the existing frame



Figure 11. Correlation between the damage indices for the current code designed frame



Table I. Proposed damage index for different damagestates

Damage stateRange of proposed

damage index

Minor 0)0—0)15Moderate (reparable) 0)15—0.3Severe (irreparable) 0)3—0)8

Collapse '0)8

Figure 12. Proposed damage index variation with the maximum interstorey drift for the non-ductile frame

8. CONCLUSIONS

1. The proposed approach to evaluate the damage by performing two pushover analyses,before and after subjecting the structure to an earthquake results in a simple and rationaldamage indicator that is based solely on structural analysis. The damage model appliesequally to ductile and non-ductile structures.

2. For the studied cases, the proposed damage model at the storey level or for the wholestructure is a consistent and robust damage index that avoids the use of weighing functionsand other analytical difficulties. Local members information is available and can be easilyextracted from the pushover analysis.

3. There is good correlation between the response-based damage evaluation procedures thataccount for maximum deformation and cumulative damage. This was demonstrated for thesimple case of three-storey ductile and non-ductile reinforced concrete frames.



Figure 13. Proposed damage index variation with the maximum interstorey drift for the ductile frame

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