An objective seismic damage index to

evaluate RC structures

J.C. Vielma a A.H. Barbat a,b S. Oller a

aPolitechnical University of Catalonia (UPC), Edificio C1, Campus Nord, C/Gran Capitan s/n, 08034, Barcelona, Spain

bCorresponding author

Abstract

In modern earthquake-resistant design codes is considered that the structural el-ements (columns and beams) have a nonlinear behaviour during the action of anearthquake similar to the considered in design process. This implies that these ele-ments are damaged and it is very interesting for the designer to be able to estimatethe expected global damage in the structure and to relate it to the design ductility,and also with the ductility demand. The damage index calculated applying finiteelements method, have values that do not reflect the deterioration in the case ofbuildings designed for low ductility, this feature is contrary with the damage in-dex calculated for ductile buildings. Therefore, in this work an objective damageindex is proposed, based on the ductility and the values of the elastic and ultimatestiffness, that is independent of the selected structural typology. The procedure isillustrated by means of the assessment of the index from damage to three buildings,two of which have been designed for low ductility (building with wae slabs andframed building with wide beams) and a third one that is framed building withdepth beams, designed for high ductility. For the three buildings the static nonlin-ear response has been determined by means of a force-based procedure, and alsothe performance point corresponding to the three buildings are calculated applyingthe N2 method. The results obtained demonstrate that the objective damage in-dex proposed provides values that characterize suitably the damage suffered by thethree buildings, at the instant of collapse.

Key words: Damage Index, Seismic Damage, Pushover Analysis, Limit States, RCbuildingsPACS:

Preprint submitted to Elsevier 11 January 2008

1 Introduction

In current earthquake-resistant design procedures elastic procedures are us-ing, applying Response reduction factors to reduce elastic response to convertit in equivalent elasto-plastic response.This approach implicitly accepts thatstructures has a plastic deformation capacity without loss of stability. But theconcept of ductility also impply that the structure reachs a state of damagewhen are subjected to earthquakes. It is ussefull for the structural designerthe assessment of the magnitude of this damage, and coorrelate this with thestructural ductility and the dcutility demands Vielma et al.[1].

Damage Indexes have received special attention during past two decades,mainly based in the possibility of correlate this Damage Indexes with the LimitStates of the Performace-based design. For Kunnath [2] in the performance-based design procedures, the process to transform calculated demands intodemands that suitably quantify the behaviour of the buildings, is a question-able part of the global procedure. For this reason, it is necessary to considerindex that accounts in objective way the seismic damage in buildings.

Global seismic damage indexes provides a measure of the structural deteriora-tion. They are computed from numarical simulations of structures due lateralstatic or dynamic forces, that represent seismic forces. Deppending on the loadtype, various damage indexes have been formulated. These damage indexesinclude some of the main characteristics of the non-linear response (static ordynamic) of the structure. For RC structures, damage indexes can be clasifiedaccording to the parameters considerated in their formulation. This parame-ters are related to: maximum lateral displacements, plastic dissipated energyand a combination of both.

Some indices measure the overall seismic damage of a structure from its localdamage, ie, the contribution of cumulative damage in the structural elementsin a given instant to the structure being subjected to a seismic demand. Amongthe indices which have served as the baseline for many researches, it can beciting the proposed by Park and Ang [3] that can determine the damage in anelement, based on the non-linear dynamic response by the following expression:

DIe =mu

+

uPy

dEh (1)

Where m is the maximum displacement of the element, uis the last move-ment, is a parameter that is adjusted depending on the materials and thestructural type, Py is the yield strength and

dEh is the dissipated hyteretic

energy. This damage index is at local level, at an element, however it is possi-ble to apply this index in the computation of values for an specific structural

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level, or for the whole structure.

For non linear analisys due static horizontal loads, it is usefull to consideratedamage indexes that incorporate the stiffness degradation. Skrbk et al.[4]proposes the following damage index:

DIe = 1

KiKi1

(2)

Where DIe is the damage index of the beam or column, Ki is the currenttangent stiffness and Ki1 is the initial tangent stiffness.

Period degradation provides a measure of the stiffness degradation. For thisreason, Hori and Inoue [5] has been formulated an expression to calculate thisperiod degradation based on the design ductility as follows:

T = 2pi

yT0 (3)

where T is te period on the collapse state, is the design ductility, y is astiffness-degradation dependent coefficient and T0 is the elastic period of thestructure. Other damage index based on stiffness degradation, is the proposedfor Gupta et al. [6]. They formulated an expression basen on the relationshipbetween the ultimate and yielding displacements, that is equivalent to ultimateand yielding stiffnesses. Also this formulation includes the design ductilityvalue according to:

DI =xmax/z00 1

1(4)

Among the desirable features that should have a damage index, Catbas andAktan [7] includes:

Must be sensitive to the accumulation of deterioration. There should be sensitive to changes in the properties of the structures orthe acelerogramas applied.

Must remain valid and meaning through the State Boundaries of serviceand collapse.

It should allow the location and quantification of damage to the index cor-related with the integrity of the structure.

These damage indexes, especially those calculated with the stiffnesses rela-tionship, have the shortcoming that produces consistent results in the case of

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structures with ductile behavior. However, in the case of structures designed tolow ductilities, i.e. framed buildings with wide beams or wae slabbs build-ings, this damage indexes does not describe objectively the overall state ofdamage when the response is close to the collapse threshold. To overcome thisdrawback, in this article it is developed an objective seismic damage index, in-dependent of the structural typology, formulated as a function that depends onthe stiffnesses relationship and maximum ductility values, computed directlyfrom the capacity curve of the buildings. Numerical examples of the applica-tion of this index are presented, which consist in three RC buildings that havebeen designed for different values of ductility, typified on the Spanish seismiccode NCSE-02 [8], and characterized by the corresponding performance pointcomputed by the method N2 Fajfar [9].

2 Formulation

The above mentioned indexes, have been developed in order to quantify theglobal damage in ductile structures. However, when the non-linear responseof restricted-ductility structures is studied, it is possible to observe that thedamage index velues corresponding to collapse threshold, are lower than thecorresponding to ductile structures Vielma et al. [10]. This shortcoming doesnot allow that the referenced indexes can be used in order to carry out anobjective characterization of the damage in restricted-ductility buildings.

The following analysis is done starting from the assumed hypothesis that thenon-linear behaviour of the structures follows the principles of the Mechanical-damage Theory Oliver et al. [11]. This theory, based on the continuum me-chanics, fulfills the fundamental thermodynamics principles. Not all the ma-terials used with structural purposes follow a behavior that can be asimilableto the damage (degradation/loss of stiffness) instead their behavior follow thePlasticity Theory (development of irreversible deformations), see Figure 1.Other materials combined both behaviour, and they have loss of stiffness withirreversible deformations, which is the case of the RC structures.

To determine whether damage or plastic behaviour has occurred, it is necessaryto observe the unload branch in Figure 1. It is accepted that damage hasoccurred if the unload branch pass throgh the origin; on the other hand, ifthe unload branch is paralel to the load branch, the behaviour corresponds toplasticity.

Reinforced concrete has a combined behaviour (plasticity and damage) but themain feature corresponds to the degradation Oller [12]. The last affirmationcan be validated by laboratory test or by numerical simulations using theMixing Theory of simple substances Car et al. [13] and [14]

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Fig. 1. Schemes of the damage and plastic behaviour

The following procedure has been proposed with the aim to describe the struc-tural degradation under seismic loads starting from a few non-linear charac-teristics. This feature allows the procedure in a simple way and its applicationis quickly and efficiently in evaluating the seismic behaviour.

For an structure an static non-linear analysis (pushover analysis) is applied.Calculated roof displacements are plotted vs base shear V . The resultantcurve called Capacity curve, has a initial slope that corresponds to the initialstiffness K0, see Figure 2.

Fig. 2. Determination of the initial stiffness from capacity curve

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If the yield base shear is knonw, and for a specific ductility value postulatedin seismic code, the damage at the point C where the maximun damage isreached, can be evaluated according to the continuum-damage mechanical:

DC = 1KCKo

= 1Vy/uVy/y

= 11

=

1

(5)

According to Equation 5, the maximum damage is developed at the collapsepoint C ; this impply that the damage deppends only on the adopted structuralductility, then it is possible to affirm that:

Ductile structure = 4 DC = 0, 75

Fragile structure = 2 DC = 0, 50

In other words, ductile structures have a damage value greather than thedamage value reached for fragile structures at the collapse point. However,this way to compute damage values possibilitates a misunderstandings: struc-tural designer can interpretate that ductile structures have a greather damagethan the dcorresponding to fragile ones, at the Collapse Limit State. Thisshortcoming impply that the damage index must be reformulated, in order toavoid its deppendence on the structurla fragility. This aim is possible if thedamage index is normalized respect to the maximum damage that can occursin the structure. Thus, the objective damage index 0 DobjP 1 achieved bya structure at any point P is defined as:

DobjP =DPDC

= DP1

=

(1KP/K0)

1(6)

For example, P might be the performance point, resulting from the intersectionbetween inelastic spectrum (demand) and the capacity curve (obtanied frompushover analysis). Under these conditions, Equation 6 provides that maxi-mum damage would reach the structure subjected to earthquake prescribedby the code.

3 Numerical Examples

In this section, the objective damage index is applied on evaluating the non-linear behaviour of three buildings that are designed according differents duc-tility levels. The first building is designed to a ductility value of two, is awae slabs buildings. The second one, is a framed building with wide beams,

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designed to ductility of two, and the third one is a framed moment-resistingbuilding, designed to ductility value of four. In order to compute the non-linearresponse of these buildings a force-bases procedure is applied. The selected pat-tern of the forces corresponds a inverted traingle, this shape is reccomendableonly if the buildings have plan and elevation regularities, see Figure 3. Thismethod has the advantage of that the pattern of forces is suitably to reproducethe seismic forces and produce a damage pattern that is similar to the damagepatern that an earthquake produce. This procedure has a shortcoming that itis stable until a singular point is reached; a singular point is a point in whichthe base shear does not increases with the displacement increases. In order toavoid this shortcoming, a force-based procedure is used; the collapse displace-ment is obtained when a minimun value of finite element-based damage indexis reached.

Fn

F2

F1 hn

h2

h1

Fig. 3. Forces distribution according to inverted triangle pattern

Seismic equivallent forces are computed to all buildings levels, when focesvalues deppends on the level height. Ths procedure requires some iterationsto achieve its convergence. A well criteria to initialize the iterations it is rec-ommendable to start from a base shear that corresponds to the design shearbase. This value is used to compute the initial levels forces, that obviouslydoes not produce the wanted maximum displacement or not fulfill the con-vergence criteria. In next iteration, base shear is incremented, levels forcesare recalculated and the non-linear analysis is performed again. Iterations arerepeated until the covergence criteria is reached. The algoritm of the completeprcedure is shown in the Figure 4.

In the Figure 5 is shonw a typical capacity curve obtained from non-linearanalysis and the associated curve of finite elements-based damage. In thisfigure, it is possible to identify three important points that reflects the mainfeatures of the non linear response: point A corresponding to the formationof first plastic hinges at the ends of the beams, point B corresponding tothe formation of the plastic hinges at the end of the columns and point C,corresponding to the collapse threshold, in which a widespread distribution ofplastic hinges at beams and columns is observed,.

Cases studied Cases studied consist on three RC buildings designed according

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Fig. 4. Algoritm of force-based non-linear analysis

to Spanish seismic code NCSE-02, for two different ductility: low ( = 2) andintermediate ( = 4). Two buildings belong to the first goup, they are a waeslabs building and a framed building with wide beams. In the second groupis a moment-resisting frame building. The buildings have 15mx24m in plandimensions, and are 4,5m high.Framed buildings have four and three uniformspaced spans in x and x direction, respectively. Wae slabs building havenon uniform spans because its columns are not aligned according to straightlines, and consecuantly, does not form resistant frames. The plan and elevationviews of the three buildings are sow in Figure 6.

The buildings are modelized as 2D frames and equivalent frame in the caseof the wae slabs building. Discretization was performed according to thedifferent confinment applied, thus it is necessary to define elements for theconfined zone near the nodes and in the middle of beams and columns, inFigure 7 a typical frame discretization is show. Longitudes of the confinementzones deppend on the dimensions of the sections of beams or colums, spanlongitude, interstory high or diameter of the reinforcement steel.

Sectional discretization is also applied. This consist in to split sections in stripsparallels to the main flexure axis. Reinforcement chracteristics are incorpo-rated by means of the application of the Mixing Theory, Mata et al. [15]. Allthe strips have a particular combination of materials (concrete or reinforcesteel) expressed in percentage. The effect of the different confinement pro-vided by the longitudinal and transversal reinforcement is incorporated withthe modiffication of the strength of the concrete according to the Mander etal. [16] procedure.

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Fig. 5. Algoritm of force-based non-linear analysis

Capacity curves are product of the static non-linear analysis of the frames orequivalent frame, deppending of the case studied. These capacity cuves plotsnormalized base shear (V/W ) vs. normalized roof displacement (/H) of thewhole structure modelized as a multiple degree of freedon (MDOF) system.In order to determine its performance point, it is necessary to intersect thecapacity cuve with the demand, tipyfied on the seismic codes by the inelasticspectrum. Therefore it is necessary to convert the non-linear of the MDOF intothe response of the equivalent SDOF by means of the dynamic characteristicsof the first mode. Roof displacements are converted in pseudo-displacementsaccording to the Equation 7:

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Fig. 6. Algoritm of force-based non-linear analysis

Sd =c

FPM(7)

where Sd is the pseudo-displacement, c is the roof displacement and FPM isthe modal participation factor, obtained from dynamic characteristics of theframes:

FPM =

ni=1 mi1,ini=1 mi

21,i

(8)

In Equation 8 n is the story number, mi is the mass of the story i, 1,i is thenormalized amplitude of the first mode at a story i. By the other hand, inelasticspectrum is plotted in Sa vs. Sd format, that requires the transformation ofthe base shear of the capacity curve into pseudo-acceleration. This is achieved

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Fig. 7. Typical frame discretization

through the application of Equation 9:

Sa =V/W

(9)

where V is the base shear, W is the seismic weight and is a dimensionlessparameter computed from:

=(n

i=1 mi1,i)2n

i=1 mi21,i

(10)

Generally, he codes prescribes spectra in Tvs.Sa format. Thus, it is necessaryto convert this format into Sdvs.Sa format. This conversion is achieved byappling the Equation 11

Sd =SagT

2

4pi2(11)

here, g is the gravity and T is the first mode period. Once transformationshave been made, it is necessary to plot together the capacity spectrum andthe elastic and inelastic demand spectra in order to obtain the performancepoint.

The performace point represents the point with maximum lateral displacementof the equivalent SDOF system, produced for seismid demand. In this articlethe performance points are computed through the N2 procedure, that consistin to determine a bilinear idealized shape of the capacity spectrum by followingthese steps:

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The plastic branch of the capacity spectrum must be horizontal, ensuringthe compensation of thr areas above and below of the plastic branche.

The elastic branch is determined by means of a secant from the origin to apoint at the capacity spectrum with 60% of the maximum base shear.

The intersection of the projection of the elastic branch of the capacity spec-trum with the elastic demand spectrum, provides the displacement of theperformance point. An alternative procedure is to determine the performancepoint by means of the intersection of the inelastic branch of the idealizes capac-ity spectrum with the inelastic demand spectrum, computed from the elasticdemand spectrum reduced by a response reduction factor R, defined as:

R =

( 1)

T

TC+ 1 when T TC

when T > TC

(12)

In Equation 12 T is the first mode period, is the design ductility and TC is thecorner period of the elastic design spectrum, that limit the branch of constantacceleration to the decreasing one.

Sd (mm)

Sa (g)

0 40 80 120 160 200 240 280 320 360 4000

0,4

0,8

1,2

Performance point

Capacity curveIdealized capacity curveElastic spectrumDemand spectrum

Fig. 8. Determination of the performance point of the wae slabs building

Computed values of the performance points displacements are show in Table 1.It is necessary to point out that tese displacements are computed by applyingthe Equation 7, in order to transform the displacements of the equivalentSDOF system to the MDOF one.

Capacity curves of case studied are show in Figures 11, 12 and 13. In thesefigures, the base shear V is normalized respect to the siesmic wieght W . Oth-

12

Sd (mm)

Sa (g)

0 40 80 120 160 200 240 280 320 360 4000

0,4

0,8

1,2

Performance point

Capacity curveIdealized capacity curveElastic spectrumDemand spectrum

Fig. 9. Determination of the performance point of the framed building with widebeams

Sd (mm)

Sa (g)

0 40 80 120 160 200 240 280 320 360 4000

0,4

0,8

1,2

Performance point

Capacity curveIdealized capacity curveElastic SpectrumDemand spectrum

Fig. 10. Determination of the performance point of the moment-resisting framedbuilding

erwise, it can be seen the stiffnesses evolution, from the elastic behavior tothe collapse threshold. Note that for restricted ductility buildings, the perfor-mance points are nearer to the collapse threshold than the in the case of themoment-resisting framed building.

The objective damage index is computed by applying the Equation 6 and isplotted vs the normalized roof drift. In Figure shows the evolution of the dam-

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Table 1Performance points displacements of the of the studied buildings

Building Performance point displacement (mm)

Wae slabs building 222.07

Framed building with wide beams 170.22

Moment-resisting framed building 120.18

Roof drift (mm)

Ba

se

sh

ea

r co

eff

icie

nt (V

/W)

0 40 80 120 160 200 240 280 320 360 4000

0,2

0,4

0,6

Capacity curveOriginal stiffnessPerformance point stiffnessUltimate stiffness

Fig. 11. Performance point displacement and stiffness degradation of the wae slabsbuilding

age index of the buildings studied, a special chracteristic of these curves is thesmooth aproximation of the ductile building curve to the collapse threshold,;by the other hand the curves of the two restricted-ductility buildings have apronunciated slope near to the collapse threshold. This feature highligth thatrestricted-ductility have a abrupt collapse in contrast to the ductile behaviorof the moment-resisting framed buildings.

In Table 2 are show the values of damage index computed for the performancepoint displacements.

Table 2Damage index (Dobj) computed for the performance points of cases studied

Building Damage index (Dobj)

Wae slabs building 0.79

Framed building with wide beams 0.80

Moment-resisting framed building 0.69

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Roof drift (mm)

Ba

se

sh

ea

r co

eff

icie

nt (V

/W)

0 25 50 75 100 125 150 175 200 225 2500

0,2

0,4

0,6

Capacity curveOriginal stiffnessPerformance point stiffnessUltimate stiffness

Fig. 12. Performance point displacement and stiffness degradation of the framedbuilding with wide beams

Roof drift (mm)

Ba

se

sh

ea

r co

eff

icie

nt (V

/W)

0 50 100 150 200 250 300 350 400 450 5000

0,2

0,4

0,6

Capacity curveOriginal stiffnessPerformance point stiffnessUltimate stiffness

Fig. 13. Performance point displacement and stiffness degradation of the momen-t-resisting framed building

According to the computed values of the objective damage index, is possibleto identify that the damage that occur in the restricted ductility buildingsnear to the performance point is greather than the damage obtained for theductile moment-resisting framed building. Also this last typology have an ade-quate ductility value that exceed the design ductility prescribed in the Spanishseismic code.

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Normalized displacement (%)

Ob

jective

da

ma

ge

in

de

x (

DI o

bj)

0 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25 3,50

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Moment-resisting framed buildingFramed building with wide beamsWaffle slabs building

Fig. 14. Damage index curves and damage at performance point displacement ofthe cases studied

4 Conclusions

According to computed nonlinear response of reinforced concrete buildings,conventional damage indexes values deppend on the structural typology. Thus,for RC restricted ductility buildings, damage conventional indexes do not pro-vide results comparable to those calculated by applying the finite elementmethod.

The structural analysis previously performed allows objective assessment ofstructural damage in a simple manner. Specifically, the use of the equation(6) allows to obtain indexes values very close to those that result from moreexpensive computational procedures. Thus, it is possible to know the level ofglobal structural damage, in an specific point, for example the performancepoint obtained by means of the intersection of the demand curve, or demandspectrum, with the capacity curve of the structure.

The objective damage index, which incorporates the stiffness degradation andthe maximum value of the structural ductility enables appropriate values of theglobal structural damage, regardless of the typology of the analyzed structure.

The moment-resisting framed buildings has an acceptable value of damageat the performance point and their behaviour remain ductile, this nonlinearresponse feature exceeded the expected design values.

Among the three cases studied, it is possible to affirm that the framed building

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with wide beams and the wae slabs building, it is possible to anticipate ahigh value of the damage index corresponding to the performance point. Also,these buildings has an insufficient structural ductility compared with Spanishseismic code requirements.

A new procedure for calculating the response nonlinear static-controlled forcesis proposed. This solves the problem of singularity at the threshold of collapseby implementing an iterative process of calculation which considers obtaininga certain damage index as a convergence criterion.

References

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[2] S. Kunnath, Performance-based seismic design and evaluation of buildingstructures. Earthquake engineering for structural design, Boca Raton: CRCPress, 1th ed., 2006.

[3] Y. J. Park and Ang, A. H.-S., Mechanistic seismic damage model for reinforcedconcrete, Journal of Structural Engineering, vol. 111, pp. 722739, 1985.

[4] P. S. Skrbk, S. R. Nielsen, P. H. Kierkegaard and A. S. Cakmak, Damagelocalization and quantification earthquake excited RC-frames, EarthquakeEngineering and Structural Dynamics, vol. 27, pp. 903916, 1998.

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[11] X. Oliver, M. Cervera, S. Oller and J. Lubliner, Isotropic damage models andsmeared cracks analysis of concrete, Computer aided analysis and design ofconcrete structures, vol. 2, pp. 945958, 1990.

[12] S. Oller, Modelizacion numerica de materiales friccionales, Barcelona:Monografa N 3. Centro Internacional de Metodos Numericos en Ingeniera,1991.

[13] E. Car, S. Oller, and E. Onate, An Anisotropic Elasto Plastic ConstitutiveModel for Large Strain Analysis of Fiber Reinforced Composite Materials,Computer Methods in Applied Mechanics and Engineering, vol. 185, pp. 245277, 2000.

[14] E. Car, S. Oller, and E. Onate, A Large Strain Plasticity for AnisotropicMaterials: Composite Material Application, International Journal ofPlasticity, vol. 17, pp. 14371463, 2001.

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[16] J. B. Mander, M. J. N. Priestley, and R. Park, Observed stress-strain behaviourof confined concrete, Journal of Structural Engineering, vol. 114, pp. 18271849, 1988.

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