Research Article Quasistatic Seismic Damage …downloads.hindawi.com/journals/mpe/2014/615792.pdfResearch Article Quasistatic Seismic Damage Indicators for RC Structures from Dissipating

Research ArticleQuasistatic Seismic Damage Indicators for RC Structures fromDissipating Energies in Tangential Subspaces

Wilfried B Kraumltzig12 and Yuri S Petryna3

1 Institute for Statics amp Dynamics Ruhr-University Bochum 44780 Bochum Germany2 Kraetzig amp Partners Engineering Consultants Buscheyplatz 9-13 44801 Bochum Germany3 Technical University of Berlin Gustav-Meyer-Allee 25 13355 Berlin Germany

Correspondence should be addressed to Yuri S Petryna yuriypetrynatu-berlinde

Received 4 September 2013 Accepted 29 September 2013 Published 12 January 2014

Academic Editor Anaxagoras Elenas

Copyright copy 2014 W B Kratzig and Y S Petryna This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper applies recent research on structural damage description to earthquake-resistant design concepts Based on the primarydesign aim of life safety this work adopts the necessity of additional protection aims for property installation and equipmentThis requires the definition of damage indicators which are able to quantify the arising structural damage As in present designit applies nonlinear quasistatic (pushover) concepts due to code provisions as simplified dynamic design tools Substituting sononlinear time-history analyses seismic low-cycle fatigue of RC structures is approximated in similar manner The treatment willbe embedded into a finite element environment and the tangential stiffness matrix K

119879in tangential subspaces then is identified as

the most general entry for structural damage information Its spectra of eigenvalues 120582119894or natural frequencies 120596

119894of the structure

serve to derive damage indicators119863119894 applicable to quasistatic evaluation of seismic damage Because detK

119879= 0 denotes structural

failure such damage indicators range from virgin situation 119863119894= 0 to failure 119863

119894= 1 and thus correspond with Fema proposals on

performance-based seismic design Finally the developed concept is checked by reanalyses of two experimentally investigated RCframes

1 Introduction

The remarkable reduction of life losses in seismic events overthe last 75 years is clearly a merit of the enormous increasein research on structural seismic safety Consequently dueto improvements of earthquake-resistant design concepts andof modern seismic codes for example EC 8 [1] earthquakeswith thousands of fatalities until far into the 20th centurydemand nowadays only few ones But while in severalstrong earthquakes (San Fernando 1971 Imperial Valley 1979Loma Prieta 1989 Northridge 1994 Tohoku 2011) life lossesremained limited material losses raised dramatically witheach event This trend cries for new seismic design conceptsbeyond life protection namely a better control of earthquakedamage along the structuresrsquo paths to seismic failure Thisrequires a more intensive emphasis on structural seismicdamage which by experience correlates with material lossesfrom installations and equipment

Structural safety concepts compare actions 119878 on a par-ticular structure with its resistance 119877 generally for code-prescribed limit states Structural failure is then defined as

failure = 119891 = 119877 le 119878 (1)

Code standards transform (1) into limit state conditions 119892(X)for state vectors X the latter as functions of the designvariables X

119889

119892 (X) = (120574119877)minus1119877 (X) minus 120574

119878times 119878 (X) le 0 for failure

gt 0 for survival(2)

in which the partial safety factors 120574119878of the actions 119878 and

120574119877of the resistance 119877 are due to national code regulations

Obviously (2) formulates structural safety by virtue of forcevariables at failure point a customary manner in structuralengineering

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 615792 11 pageshttpdxdoiorg1011552014615792

2 Mathematical Problems in Engineering

Such design concepts are also code prescribed in case ofseismic failure But the explained extension of seismic designaims requires a turn towards structural damage whereinthe seismic damage formationmdashnot force resistance 119877mdashhasto be prosecuted until limited by final damage boundsConsequently such more advanced protection aims likereduction of material losses must be combined with the basicsafety objective of human life protection which can be donewithin techniques of performance-based seismic design [2]

Codified seismic design concepts [1] are generally basedon (linear elastic) response spectra Codes favor the useof quasistatic analyses instead of full time histories corre-sponding well with quasistatic constitutive laws [3] frommonotonic material tests If linear elastic structural modelsare applied the evaluation of damage is impossible sincedesign formats (2) are tailored towards force-limitations [4]It is a usual design practice Inclusion of damage alwaysrequires consideration of material nonlinearity The simplestquasistatic seismic simulation technique then uses nonlinearpushover analyses

On the other hand seismic loading creates low-cyclefatigue processes in the materials involved which after 20to 1000 sufficiently intensive load cycles will lead to seismicfailure Those damage paths to failure can be marked bydamage indicators [5] generally in dynamic settings Butin structural earthquake-resistant design practice time-his-tory computationsmdashespecially with time-dependentmateriallawsmdashare very rare So this paper addresses the questionwhether realistic estimates for such time-dependent struc-tural damage evolutions can be approximated by nonlinearquasistatic damage evaluation

To approach these aims the paper will be based ondamage-controlled limit states as proposed in performance-based seismic engineering concepts [2] It then depictspushover analyses followed by description and time-inva-riant approximation of low-cycle (RC) fatigue processesjustifying a quasistatic treatment The work will close withnecessary steps for quasistatic seismic damage evaluationfollowed by two applications

2 Extended Seismic Protection Based onDamage Indicators

Losses in strong seismic events human as well as materialones depend on the seismically dismissed energy the arearsquospopulation density the degree of economic development andthe level of earthquake-resistant safety provisions Buildingshave been primarily designed for protection of human livesbut we will also be aware of material losses As a consequenceof this not only one seismic event (of 475 years of returnperiod) should be considered but also the buildingrsquos func-tionality has to be differentiated

Figure 1 elucidates this idea in the well-known conceptof the Federal Emergency Management Agency (Fema) [2]and of the Structural Engineers Association of California(Seaoc) [6] Such performance-based seismic design conceptcenters the attention towards the quantification of seismicdamage in the design assuming the latter to be correlated

Occurrence ofdesign earthquakeFrequently

Occasionally

Seldom

Rare

Performance levels of buildingconstructionplant

Fullyoperational Life safeOperational

Buildings of highest safety levelProtection aim of DIN 4149 and EC8

I II III IV

High occupancyEmergency responseEa

rthq

uake

des

ign

leve

l

Unacceptable performance

Nearcollapse

Tw = 43a

Tw = 72a

Tw = 475a

Tw = 970a

Ordinary buildings

Figure 1 Recommended performance objectives for buildings [2]

Table 1 Limit states and damage levels according to Fema 273

Structural limit states Damage levels Damage indicesND No damage virgin structure 0 000IO Immediate occupancy I 005ID Incipient damage 010LS Life safety threatens II 020025CP Collapse prevention III 040050IC Incipient collapse IV gt090

with material losses of installation and equipmentThis basicassumption of seismic damage correlation avoids modelingof each installationequipment element as attached structuralcomponent in the computational model but it remains anempirical working hypothesis requiring the determination ofthe generated seismic structural damage

Figure 1 proposes in this sense different protection aimsof buildings in the columns and seismic events of increasingintensities in the rows measured in return periods 119879

119908[1]

In the diagonal classes of structures are distinguished withdifferent design damagesvulnerabilities for various seismicevents Table 1 explains the damage states in more detailintroducing damage levels as follows [2]

(i) Level 0 no damage virgin structure(ii) Level I very limited structural damage no risk of life-

threatening injury(iii) Level II significant damage low risk of life-

threatening injury(iv) Level III substantial damage significant risk of injury

from falling hazards(v) Level IV collapse is imminent highest risk of life to

occupants

Fema documents for example [2 6] order numerical valuesto these damage levels they also propose a normalizationfrom 000 (virgin) to 100 (seismic failure) But the interestingfact is that no advice is given by Fema which of the existingdamage indicators is preferred

In the literature there exist a great variety of damageindices as can be seen from the survey in [5] Most indi-cators describe damage in local cross-sections thus theyare unsuited for damage description of complete structures

Mathematical Problems in Engineering 3

But most important All indices are rather empirical andnone of them has a sound foundation from mathematics orstructural mechanics as we see on the hybrid Park-Angindex [7] which combines dissipated energies with inelasticstory drifts

119863 =120575119898

120575119891

+ 120573119864119903

119865119910120575119891

(3)

Herein 120575119898(120575119891) denotes the maximum occurring (bearable)

structural displacement 119864119903is the total dissipated energy

119865119910the structural yield resistance and 120573 a parameter to

be determined by tests [7] The combination of dissipatedenergies and story drifts in (3) is really too empirical to beapplied in a theoretical sound earthquake-resistant designconcepts

In Section 6 we will derive a set of damage indices119863119894 119894 =

1 119899 from inherent mechanical properties of a particularstructurersquos path to seismic failureThese parameters will havethe following main properties serving as an optimal basis forearthquake-resistant designs

(i) They will be related to the entire structure(ii) They will describe the seismic low-cycle process with

a sufficient accuracy from the mechanical viewpoint(iii) They will be based on mechanics for example the

structural state variables of the seismic process(iv) They will range from 000 (undamaged virgin state)

to 100 (failure)

The authors have certain experienceswith indices119863119894for dam-

age problems other than seismic ones The have found thatvalues around 06 separate economically structural repairsfrom reconstruction This holds especially for RC structureson which we will now concentrate

3 Quasistatic Nonlinear Simulation

For application in present anti-seismic design concepts thefurther treatment will be quasistatic Then for monotonicdeformation processes of materially nonlinear respondingstructures tangential subspace simulations up to failure arewell-established tools in present engineering Because ofmaterial degradation of the structural resistance 119877 duringearthquakes seismic damage simulations are nonlinear pro-cesses To derive their governing response equations weimbed the treatment for arbitrary structures into a FE-environment in which the nodal degrees of freedom arearranged in the119898-dimensional vector field

V = 11988111198812

sdot sdot sdot 119881119894

sdot sdot sdot 119881119898 (4)

and the nodal forces in dual order to V in

P = 11987511198752

sdot sdot sdot 119875119894

sdot sdot sdot 119875119898 (5)

In this frame the structural response is governed by thefollowing nonlinear stiffness equation with G as the vectorfunctional of the structural resistance 119877

G (V d) = P (6)

This 119898-dimensional set of nonlinear algebraic equationsdescribes equilibriumcompatibility in terms of the internalresistance Gmdashwith inelastic deformation (V) and damaging(d) material model propertiesmdashdue to an external forceprocess P All nonlinear structural properties anchoredgenerally in the constitutive laws on Gauss point level (seeFigure 2) are assumed to be fixed in required quality in G

To derive the basis for incremental-iterative solutionstrategies from (6) we decompose the external nodal kine-matics V and loads P of a certain structural state intovariables V P of a known fundamental state and theirunknown increments 120575V 120575P reaching from fundamental toneighboring position

V = V + 120575V P = P + 120575P (7)

Assuming now all increments as infinitesimally small varia-tions we gain from the first variation of (6) with respect tothe fundamental stateVP the119898-dimensional set of algebraicequation for the increments 120575V 120575P namely

120597G120597V

10038161003816100381610038161003816100381610038161003816Vsdot 120575V = 120575P (8)

Substituting herein for the load variation 120575P = P minus P thenfor P the left-hand side of the original stiffness relation (6)we receive the tangential stiffness equation which forms thebasis of finally applied iterative solution techniques

K119879(V d) sdot 120575V = 120575P = P minus F

119868(V d) (9)

with the vector 120575V of the increments of the global degrees offreedom and

K119879=

120597G120597V

10038161003816100381610038161003816100381610038161003816Vthe tangential stiffness matrix (10)

F119868= G (V d) the internal equilibrium force vector

(11)

Since K119879and F

119868depend on the reached deformation V

the tangential stiffness equation requires iterative solutiontechniques for the increments 120575V In (9) P represents thetotal applied load at a certain solution point and d stands forthe structural damage later on detected as function of K

119879

Materially nonlinear structural analyses here applied aspushover analyses can be carried out on rather differentlevels of mechanical exactness and complexity beams withplastic hinges with or without stiffness iterations are theleast complex alternative nonlinear multilevel (multiscale)simulations form the other extreme [8]Thus different modelcomplexity may influence the simulation accuracy of (9) andso the later seismic damage detection

In order to detail (macroscopic) materially nonlinearstructural FE simulations [8] we return to a multilevel(multi-scale) simulation concept as the most complex anal-ysis scheme and distinguish the following four simulationlevels due to Figure 2

(i) The first one is the structural level on which allvariables of the tangential stiffness relation (9) aredefined

4 Mathematical Problems in Engineering

V = V ++V

+V = Kminus1

T

+P = Kminus1

T (P minus FI) P

+120576+120581

+120574

120574ij +120574ij

V+V

Structurallevel

Finite elementlevel

hGauss pointlevel

Material pointlevel

KT(V) FI(V)

120576 = 120572(120572120573)120573(120572120573)120574120572

+120576 =

+120572(120572120573)

+120573(120572120573)

+120574120572

120590 = n120572120573m120572120573q120572

+n

+m

+q

+120590 = 120572120573 120572120573 120572

or eg NMQ

+N

+M

+Q

ij120590ij +120590

120576 120581 120574or eg

g1

g2g3

a1a1

a2a2 a3a3

b

d

RijmnC

mnkl+120574kl = Clowastijkl+120574kl =+120590kl

a1

a2a3

x3

x2

x1

vp vp+

int1205741198961198970

Clowastmnkl(120574op )d120574kl = 120590ij

kpT(vp) fp(vp)

Figure 2 Variables and analysis scheme of multilevel simulation technique

(ii) The second one is the level of classical FEs (beamsplates and shells) connected with the previous oneby kinematic relations in the nodal points

(iii) In the third level with regard to materially nonlinearactions and damage phenomena the applied FEsmdashsee Figure 2mdashare equipped in theirGauss pointswitha layered structure such that each FE-Gauss pointconsists of a chain of integration subpoints one ineach layer

(iv) Finally at these subpoints on material level the con-stitutive relations including their nonlinear propertiesand failure bounds are defined

The material point level is generally the lowest macro-scopic simulation level in design practice It collects allmaterial information usually in a 3D (119909119894 1199091 1199092 1199093) or 2D(119909120572 119909

1 1199092) simulation space Here suitable constitutive

laws of the material components have to be activated [9]for incorporation in K

119879and F

119868 Applying beam plate or

shell elements engineering simplifications for constitutivelaws and failure conditions are often in favor particularly fordesign purposes (see Section 8)

4 Nonlinear Pushover Analyses

We now return to seismic problems that means to problemsof nonlinear dynamics if the structurersquos path until seismic

failure will be investigated Instead of attempting to solve theoriginal nonlinear equation of motion

M sdot V + G (VV d) = P (119905) (12)

we concentrate for a shortmoment on the tangential subspaceto V VVP and thus on the incremental tangential equa-tion of motion

M sdot 120575V + C119879sdot 120575V + K

119879sdot 120575V = P (119905) minus F

119868 (119905) (13)

The derivation of (13) from the original nonlinear equation ofmotion (12) can be found in detail in [10] In (13) the vectors120575V 120575V 120575V represent increments of the nodal equivalentsnamely accelerations velocities and deformation degreesof freedom As a nonlinear differential equation in time 119905(13) would require time-history solution for an (spectrum-compatible) acceleration functions 119886(119905) with stiffness itera-tions in each time step

In the present structural design practice such complexnonlinear time-history verifications are completely unusualNational seismic codes generally require linear dynamicanalyses if time-history simulations are selected mostlylinear quasistatic formats are applied Pushover analysesserve since 30 years as approximate quasistatic but nonlinearsubstitution

Pushover analyses [4 11] are based on standardizeddesign spectra of the respective site [12] generally prescribedin the respective code and on dynamic structural proper-ties namely mode shapes as well as natural frequencies of

Mathematical Problems in Engineering 5

the undamaged structure A pushover analysis consists of thefollowing essential steps [13]

(i) We select a (code-given) design spectrum of seismicaccelerations for the building site

(ii) The considered structure is represented by a suit-able FE discretization and all (nonlinear) materialproperties on suitable structural levels (Figure 2) arespecified

(iii) Execution of an eigenvalue analysis of the undamagedlinear-elastic structural model to receive critical natu-ral frequencies (or periods) and corresponding modeshapes From them follow the modal participationfactors which form the seismic load bases for eachmode shape in each structural point

(iv) With all these information the design spectrum deliv-ers maximum seismic modal excitation loads andmaximummodal pseudodeformations

(v) Both groups of variables are then united by modalsuperposition if different mode shapes participate inthe seismic response

(vi) These superposed seismic nodal loads are applied inincremental manner on the structure for a quasistaticnonlinear simulation process the pushover analysisIt stops after all seismic loads are applied the struc-tural model fails or the deformation limit has beenreachedThe gained stress resultants then are the basisof the seismic design as detailed in [12 14]

In summary pushover verifications are nonlinear qua-sistatic approximations of a highly nonlinear dynamic load-ing process So one can never expect exact correspondenceof their solutions compared to those ones of nonlinear time-history analyses But from comparable simulations in theliterature it seems to hold true that pushover analyses delivermostly some upper bounds of the internal forces not faraway from physical reality In engineering design practicethese strong deficiencies of pushover concepts mainly theincognizance of the true alternating seismic actions towardsfailure [11] and the ignorance of the structural stiffnessweakening are allowed if nonlinearity will be taken intoaccount in a simple manner

All values of dynamic structural stiffness used in thisapproximate analysis stem from the undamaged (virgin)structure The fact that they will decrease during the seismicdamage process and may finally end up with those ones ofthe deadly injured structure may confirm the suspicion thatthe pushover concept delivers upper bounds of the designvariables since generally inelastic mechanisms reduce thedesign forces [4 15]

In spite of all these deficiencies pushover analyses willbe considered as the background of this present work fortime-independent seismic damage evaluation and to estimateseismic vulnerabilities of structures under design Beforedetailing this we will now have a look into low-cycle fatigueprocesses to also find for them a quasistatic approximation

5 Quasistatic Damage Approximation ofLow-Cycle Fatigue

Seismic failure of RC structures or their components followsa low-cycle fatigue process in which after 20ndash100 stress-strain alternations of sufficient intensity cross-sections willfail In modern research such fatigue processes are treated ascomplex phenomena on mesoscale with detailed descriptionof crack-formation crack-closing bond-slip and internalstress redistribution In the design of structures empiricalmacroscopic models for low-cycle fatigue failure are appliedthe most modern ones are based on dissipated energieswhereby close correspondence to monotonic failure of RCspecimen is aspired [16 17]

Some energetic contribution has been modeled also inthe second contribution of the Park-Ang indicator (3) inan extremely simplified and unsatisfying manner Acceptingthe properties of damage indices from Section 2 a muchbetter damage description is the 119863

119876-index introduced in

[17] This indicator consists solely of dissipated energies ofcyclic deformations up to failure119863

119876has a sound theoretical

(macroscopic) basis and has been tested extensively in large-scale experiments [17] on beams columns and beam-columnconnections Bending as well as normal and shear forcefailure has been considered For cyclic failure on one side of aparticular cross-section the119863

119876-index distinguishes primary

half cycles (energies of new deformation amplitudes) andsecondary half cycles (energies of redeformation amplitudes)of the cyclic damage process a distinction discovered by [18]For all further119863

119876-details the reader is referred to [5 17]

In the following we present a brief explanation for across-section with double reinforcement under cyclic bend-ing In the moment-curvature (119872-120581) plane as explained inFigure 3 two damage contributions119863+

119876for positive deforma-

tion and119863minus

119876for negative deformation are defined as

119863+

119876=

119864+

119875+ 119864+

119878

119864+

119872+ 119864+

119878

119863minus

119876=

119864minus

119875+ 119864minus

119878

119864minus

119872+ 119864minus

119878

(14)

with 119864+

119875as complete dissipated energies of all primary half

cycles 119864+119878as complete dissipated energies of all secondary

half cycles and119864+119872as dissipated failure energy formonotonic

deformation all for positive deformation The minus signin (14) holds for negative deformation (reverse deformationcycles) From these constituents the final 119863

119876-indicator is

defined as follows

119863119876= 119863+

119876+ 119863minus

119876+ 119863+

119876lowast 119863minus

119876 (15)

This indicator can be formed successively during numericaltime-step integration or from recorded dynamic test data Asobserved from (14) and (15)119863

119876is scaled such that the virgin

situation starts with 119863119876

= 0 and failure is designated for119863119876

= 1 Figures 4 and 5 demonstrate these properties by arecalculation of an elderly beam experiment from Berkeley[5]

In quasistatic approximations for example pushoveranalyses all time-history integrations have to be avoided So

6 Mathematical Problems in Engineering

Quasi-static M-120581 backbone curve

Secondary half cycles EminusS

Primary half cycles EminusP Secondary half cycles E+

S

Primary half cycles E+P

+M

+120581

Figure 3 Cyclic deformation and dissipated energies

Bend

ing

mom

ent M

(kN

m)

50

100

200

250

150

0

Curvature 120581 middot 103 (1m)

minus150

minus100

minus50

0 50 100minus100 minus50

ExperimentSimulation

Figure 4 Moment-curvature relation of a cyclically loaded testbeam

0000

200100

050

100

Accumulated increments of ductility

0968

Dam

age i

ndic

ator

sDQ

Figure 5119863119876-damage evolution corresponding to Figure 4

still for one single cross-section we set up the question if 119863119876

can be approximated by a monotonic deformation processonly using the backbone curve and its monotonic failure

energy For proof we transform one of the parameters from(14) into an infinite Taylor-series

119863+

119876=

119864+

119875

119864+

119872

+119864+

119878

119864+

119872

(1 minus119864+

119875

119864+

119872

) minus (119864+

119878

119864+

119872

)

2

(1 minus119864+

119875

119864+

119872

)

2

+ sdot sdot sdot

(16)

We observe thereby that the normalized monotonic energies119864+

119875119864+

119872 determinable just from quasistatic analyses form the

first and dominant terms in119863119876 respectively All the following

energy contributions in (16) obviously can be considered ascorrection terms of higher order

We now transform this recognition to complete struc-tures For design purposes we are primarily interested in themaximum seismic damage The upper limit119863+

119876= 1 of (16) is

formed for 119864+119875rArr 119864+

119872 so a quasistatic failure analysis seems

well applicable to determine the upper limits of contributingsectional damage Consequently estimation of structuraldamage from pushover concepts can be transferred as anupper limit estimate to the entire structures

6 Time-Invariant Approximation ofSeismic Damage

During a seismic event of sufficient intensity RC structureswill be weakened by inelastic actions and eventually fail Thisdegradation process can bemonitored as a successive stiffnessreduction of the structural resistance119877 reflected in the vectorfunctional G (6) Since structural damage d depends alsoon the forming inelastic deformation the tangential stiffnessmatrix K

119879(10) is considered as the most suitable entry for

damage description [19 20]The damage measures later defined in (18) had been

derived mainly for deterioration and life-duration problemsof structures in which the time scale is enlarged comparedto physical time 119905 such that quasistatic treatment is justifiedUsing time-independent pushover analyses as approximationof nonlinear structural seismic processes time is completelydeleted such that quasistatic damage description is the onlyalternative

How is K119879connected with structural softening and fail-

ure Figure 6 repeats the basic fact that for integer structures

Mathematical Problems in Engineering 7

Definition structural integrity is defined by stable equilibriumpaths (for simplicity structural instabilities are excluded)

Prerequisites for structural integrity

KT nonsingular det KT ne 0

all eigenvalues 120582i gt 0 1 le i le m

positive definite+VT middot KT middot gt 0

+V

[+V] = [Kminus1

T ] middot ([P] minus [Fi])[KT] middot [+V] = [P] minus [Fi]

Figure 6 Conditions for structural integrity

Damage indicatorsDamage parameters

Measure of structural integrity

120582lowasti eigenvalues 120582i 1 le i le m of KT(V d)

(KT(V d) minus 120582iI) middot i = 0

The corresponding eigenvectors i 1 le i le m map the damage

formation on the external deformations

Di(V d) =120582lowasti ( d = 0) minus 120582lowasti (V d)

120582lowasti ( d = 0)= 1 minus

120582lowasti (V d)120582lowasti ( d = 0)

Φ

Φ

Vo Vo Vo

1205821(Vd) 1205822(Vd) 120582i(V d) 120582m(V d)

Φ1(Vd) Φ2(V d) Φi(V d) Φm(V d)

Figure 7 Definition of damage indicators

K119879is regular and positive definite thus all its (hypothetically)

transformed main diagonal elements are positive In case ofstructural failure K

119879gets singular det K

119879= 0 and thus at

least one main diagonal stiffness element vanishes So K119879

accompanies all quasistatic structural responses also time-independently approximated seismic ones from structuralbirth until final failure [9 19ndash21]

Since K119879(generally) is quadratic and symmetric it can

be transformed into a diagonal form with the eigenvalues 120582119894

119894 = 1 2 119898 as principal stiffness elements

K119879= diag 120582

11205822

sdot sdot sdot 120582119894

sdot sdot sdot 120582119898 (17)

The diagonal matrix (17) can be used to form sets of damageindicators119863

119894 119894 = 1 2 119898 as repeated in detail in Figure 7

119863119894=120582119894(V0 d = 0) minus 120582

119894 (V d)120582119894(V0 d = 0)

= 1 minus120582119894 (V d)

120582119894(V0 d = 0)

(18)

From this definition we observe that the elements of 119863119894take

values between 119863119894= 0 for the undamaged state and 119863

119894=

1 for the failure state Obviously these damage indicatorsare able to denote properly all additional limit states fromTable 1 Further they have been gained as inherent structuralmechanics properties of a considered structure controlled byits nonlinear quasistatic seismic response pathObviously the119863119894have amuchmore sound foundation than all other damage

indicators also that Park-Ang index (3)

As explained in Figure 7 damage indicators (18) aredefined by using the eigenvalues of K

119879as state variables of a

certain deformed structural situation For evaluation in eachdamage state these eigenvalues have to be solved from theeigenvalue problem

[K119879 (V d) minus Λ] sdotΦ120582 = 0 (19)

Herein Λ forms a (119898lowast119898) diagonal matrix of all eigenvalues120582119894 the actual principal stiffness elements of the structureΦ

120582

assembles the rows of eigenvectors (eigenmode shapes) ofK119879

If we remember orthogonality ofΦ120582 we find from (19) by left

multiplication withΦ119879120582

Φ119879

120582Φ120582= I 997904rArr Λ = K

119879= Φ119879

120582K119879 (V d)Φ120582 (20)

that the matrix Λ describes physically an energy BecauseK119879is defined in the tangential subspace of the increments

120575V 120575P Λ can be interpreted as virtually dissipated energyat the point of K

119879

Classical nonlinear structural dynamics offers a highlyefficient extension of (17) (19) for practical damage detectionin numerical simulation as well as in physical testing There-fore we superpose upon an actual nonlinearly deformed qua-sistatic state of response (PV d) of the considered structurean infinite set of infinitesimally small harmonic vibrations

120575P = 0 (K119879minus 1205962

119894M) sdot 120575V = 0 119894 = 1 119898 (21)

8 Mathematical Problems in Engineering

254 1016 254

1524 1524

1524

1524

3120601127

2120601127

2120601127

All dimensions (mm)

Structure P3 P3 P3

381

381

2120601953

2286 h

Concrete fcm = 4082MPa

Reinforcement ES = 209GPa 120601953 fy = 47238370MPa = 0077

120601127 fy = 45557026MPa = 0110fsu

fsu 120576su

120576su

Dimensions and materials = 54864mm (18ft) h = 18288mm (6ft)

l4 l4l4l4

l

l

Figure 8 Dimensions and material specifications of RC test frame

Studying the (squared) frequency spectrum 1205962

119894as a solution

of the eigenvalue problem (21) the latter delivers for zeros ofK119879the statement 1205962

119894= 0 identical with the quasistatic failure

condition det K119879= 0 So for M distributed evenly over the

structure eigenfrequencies 120596119894are likewise well suited as the

eigenvalues 120582119894in (17) for damage detection

[K119879 (V d) minusΩM] sdotΦ

120596= 0 (22)

HereinΩ forms a (119898lowast119898) diagonal matrix with the (squared)frequency spectrum 120596

2

119894on its main diagonal The (119898 lowast 119898)

square matrix Φ120596collects the rows of all 119898 vibration mode

shapes From (21) we derive by left-hand multiplication withΦ119879

120596

Φ119879

120596MΦ120596= I 997904rArr Ω = diag 1205962

119894 = Φ

119879

120596K119879 (V d)Φ120596 (23)

interpreting 1205962119894as virtually dissipating energies of the modal

forces (K119879Φ120596) along the virtual vibration mode shapes Φ119879

120596

An analogous energy interpretation has just been drawnfrom (20) The use of eigenfrequencies 120596

119894for evaluation of

damage indicators compared to eigenvalues 120582119894is advanta-

geous because of generally less problems in setting up (22)compared to (19) This substitution holds true because bothmatrices Λ andΩ can be related to each other by

Λ = Φ119879

120596MΦ120582ΩΦminus1

120582Φ120596 (24)

In the literature alsomore complex functions of the eigenval-ues 120582119894are recommended for formation of damage indicators

[6] due to the definition on top of Figure 7

Computer simulation

Experiment

Displacement u (mm)20 40 600

12

08

06

04

02

00

Load

fact

or1205822

80 100 120 140

10

uH

Plowast0 3 Plowast

0 3 Plowast0 3

Figure 9 Load-deflection diagram for horizontal loading119867

7 Example 1 Damage Formation inRC Test Frame

We now demonstrate the application of the derived damageconcept by hand of a RC test frame the testing of which hadbeen published in detail 40 years ago [22] The test frameas well as all necessary geometrical and material data of thisexperiment is reprinted in Figure 8

The 2-hinge test frame carries three equal loads 1198753 inthe quarter-points of the girder and one horizontal load 119867

on the girder due to Figure 9 For an anti-seismic designthe evaluation of 119867 from a seismic design spectrum dueto the steps in Chapter 4 is straight forward In one of theexperiments [22] the test frame first was predamaged by atotal vertical loading up to 119875

lowast

0= 265 kN Maintaining this

load then the horizontal (pushover) load 119867 as sketched inFigure 9 has been applied and increased until failure of thestructure at119867 = 142 kN

Mathematical Problems in Engineering 9

0

08

06

04

02

00

10

02 04 06 08 10 02 04 06 08 10

LS

CP

IC

Dam

age i

ndic

ator

sDi

D1 D1

D2

(a)

D2

(b)

Load factor 1205821 Load factor 120582lowast1 P0 Load factor 1205822 Load P0 + 1205822H

Figure 10 Damage indicators for vertical (a) load and additional horizontal (b) load119867

040310570

460

040

180

160

040

040

090 090040 3502

202

00Test frame

030

040

Mechanical model Cross-section2 times 4 120601 20BSt 500

Concrete

Reinforcement

700kN700kN700kN700kN

H H

asympC 2530

Figure 11 2-story RC test frame

The load-deformation diagram P(V d) of the secondloading phase the application of the horizontal force inFigure 9 plots 119867 against the horizontal girder displacement119906 Therein both the experimentally monitored and the com-puted deformations are plotted the latter evaluated by thedescribed multilevel-simulation concept in a complicatednonlinear analysis Both curves show sufficient correspon-dence for encouragement of the damage simulation

Those results are depicted in Figure 10 for both damagevariables 119863

1 1198632 based on the eigenvalues 120582

1 1205822 The

computed damage variables11986311198632are plotted there in the left

part of the Figure as functions of the vertical loading then inthe right part as functions of the horizontal (pushover) load119867 Details of the complicated nonlinear load-deformationprocess can be observed much clearer by hand of the damageevolution on this latter Figure Obviously 119863

1describes the

structurersquos path to final failure although its effect undervertical loading is less than 119863

2 At the point of the extreme

load 119867 structural equilibrium could be iterated for the lasttime terminating the final damage evolution in Figure 10(b)somewhere between 090 and 100 close to the theoreticalunit value

8 Example 2 Damage Formation ofa 2-Story RC Frame

A 2-story RC frame will serve as second example the testingof which had been described in [23] Figure 11 depicts all

necessary dimensions with equal cross-sections of bothbeams and columns In this example the frame modelapplied in the nonlinear simulation is a simple beamcolumnstructure The concrete strength corresponds to C2530 andthe reinforcement 4 bars of 20mmdiameter on each side hasthe European steel quality BSt 500

To demonstrate the general applicability of the deriveddamage concept this example will use an extremely simplestraightforward design analysis without any sophisticatednonlinear simulation techniques It will again show only thenonlinear pushover analysis and omit any seismic evaluationof the applied horizontal load 119867 The analysis is carried outas in simple design practice by the use of a linear elastic frameprogram and successive load application on the structure Ateach structural position reaching the yield moment of thebeam model a plastic hinge is introduced and the furtheranalysis is carried out with this reduced redundant structureThe stiffness reductions of the progressively cracking RCmembers on their way to reinforcement yielding are consid-ered by iterations As a foundation of thismodeling Figure 12shows the 119872-120581 diagrams purely based on the (monotonic)prescriptions in [3]

The gained load-displacement curve is compared inFigure 13 with the global backbone curve of the cyclic swelltest [23] demonstrating excellent agreement The damageevolution on Figure 14 uses eigenfrequencies evaluated foreach different structural systems of each load step Because

10 Mathematical Problems in Engineering

0 50 100 1500

100

200

300

25301312

1950938

18901089

690125

280051

55000

kNm

2

15500kNm

2

15500kNm

2

Columns for N = minus700kNColumns for N = 0

Beams for N = 0

120581 middot 103 (1m)

M (k

Nm

)

Figure 12119872-120581 diagrams for all sections of the test frame

Forc

e (kN

)

0Horizontal displacement (mm)

350

300

250

150

100

50

0

200

20 40 60 80 100 120 140

Pushover analysis

Experiment

Figure 13 Load-deflection diagram of test frame under horizontalload119867

50 75100

00

02

04

06

08

10

0 200 300115 155 195 235 274296 326319325

2 plastic hinges

Plastic hinge

Primary cracking phase

2 plastic hinges

Plastic hingeCrack widening

Failure

Load steps H (kN)

Stru

ctur

al d

amag

eD

Figure 14 Damage indicators (eigenfrequencies) of test frameunder load119867

of the applied plastic hinge model the analysis is straight-forward by standard software Looking at Figure 14 we canclearly distinguish the brief uncracked phase with 119863 equiv 0

from the further primary cracking and crack-wideningphases Close to failure at 119863 = 1 we observe the expected

rapidly successive formation of plastic hinges transformingthe originally resistant structure into a kinematic chain

9 Summary

In the center of present seismic design codes stands thesafe protection of human lives but with growing buildinginvestments installation and equipment also viewpoints ofproperty protection increase As proposed by Fema Stan-dards [2] such extension requires the prosecution of astructurersquos damage path by damage indicators which are ableto scale the arising structural damage

The present manuscript derives such a damage indicatorbased on dissipated energies of virtual mode shapesmodalforces and is connected to the structural response by thetangential stiffness matrix K

119879during the nonlinear solution

process Thereby it adopts the quasistatic seismic safetyconcepts of most design codes and proposes for damageevaluation the seismic load-path evolution of the tangentialstiffnessK

119879as well known from pushover analyses Using the

evolution of its eigenvalues or natural frequencies the paperderives damage indicators which fulfill all requirements from[2] and aremdashas accompanying data fields of design analysesmdashrelatively easy to determine The present examples should befurther extended for instance to new structural types inorder to get more general experience with the applicabilityof the proposed damage measures

Seismic damage is known as a low-cycle fatigue processwhich is approximated in this manuscript in a quasistaticmanner Similar damage indicators have been successfullyapplied to the analysis of longtime structural deterioration[9 19] and also to high-cycle fatigue structural phenomena[24]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] EC 8 Design Provisions for Earthquake Resistance of StructuresCEN European Committee for Standardization Brussels Bel-gium 1994

[2] Federal Emergency Management Agency ldquoGuidelines for theseismic rehabilitation of buildingsrdquo FEMA 273 Federal Emer-gency Management Agency Washington DC USA 1997

[3] EC 2 Design of Concrete Structures Part 1 General Rules andRules for Buildings CEN European Committee for Standardiza-tion Brussels Belgium 1991

[4] A J Kappos andG Panagopoulos ldquoPerformance-based seismicdesign of 3D RC buildings using inelastic static and dynamicanalysis proceduresrdquo ISET Journal of Earthquake Technologyvol 41 pp 141ndash158 2004

[5] W B Kratzig and K Meskouris ldquoAssessment of seismic struc-tural vulnerability as a low-cycle fatigue processrdquo in Proceedingsof the 11th European Conference on Earthquake Engineering PBisch et al Ed pp 161ndash178 AA Balkema Rotterdam TheNetherlands 1999

Mathematical Problems in Engineering 11

[6] SEAOC ldquoVision 2000mdashperformance based seismic design ofbuildingsrdquo Committee Report Structural Engineers Associa-tion of California Sacramento Calif USA 1995

[7] Y Park and A H-S Ang ldquoMechanistic seismic damage modelfor reinforced concreterdquo Journal of Structural Engineering vol111 no 4 pp 722ndash739 1985

[8] WBKratzig ldquoMulti-levelmodeling technique for elasto-plasticstructural responsesrdquo in Proceedings of the 5th InternationalConference Computational Plasticity D R J Owen et al Edvol 1 pp 457ndash468 CIMNE Barcelona Spain 1997

[9] WBKratzig andY S Petryna ldquoStructural damage and life-timeestimates by nonlinear FE simulationrdquo Engineering Structuresvol 27 no 12 pp 1726ndash1740 2005

[10] W B Kratzig ldquoDamage indicators for estimates of seismicvulnerabilityrdquo in Innovations in Computational Structures Tech-nologies B H V Topping et al Ed pp 111ndash132 Saxe-CoburgPublications Stirlingshire UK 2006

[11] H Krawinkler and G D P K Seneviratna ldquoPros and consof a pushover analysis of seismic performance evaluationrdquoEngineering Structures vol 20 no 4ndash6 pp 452ndash464 1998

[12] B Gupta and S K Kunnath ldquoAdaptive spectra-based pushoverprocedure for seismic evaluation of structuresrdquo EarthquakeSpectra vol 16 no 2 pp 367ndash392 2000

[13] V Kilar and P Fajfar ldquoSimplified push-over analysis of buildingstructuresrdquo in Proceedings of the 11th World Conference onEarthquake Engineering pp 59ndash66 Acapulco Mexico 1996

[14] A K Chopra Dynamics of Structures Theory and Applicationsto Earthquake Engineering Prentice Hall Upper Saddle RiverNJ USA 2001

[15] L Petti L Marino M De Iuliis and G Giannatasio ldquoThe roleof modeling in the push-over analysis of existing framed struc-turesrdquo in Proceedings of the (COMPDYN rsquo07) M Papadrakakiset al Ed Civil Engineering Software Rethymno Greece June2007

[16] E DiPasquale and A S Cakmak ldquoDetection and assessmentof seismic structural damagerdquo National Center for Research onEarthquake Engineering Research Technical Report NCEER-87-0015 State University of New York Buffalo NY USA 1987

[17] I F Meyer W B Kratzig and F Stangenberg ldquoDamageprediction in reinforced concrete frames under seismic actionrdquoEEE-European Earthquake Engineering vol 3 pp 9ndash15 1988

[18] A Otes ldquoZur werkstoffgerechten Berechnung der Erdbeben-beanspruchung in Stahlbetontragwerkenrdquo Tech Rep 25 Insti-tute f Reinforced Concrete TH Darmstadt 1985

[19] W B Kratzig and Y S Petryna ldquoFundamental tools forstructural damage indication and lifetime managementrdquo inProceedings of the 2nd International Conference on Lifetime-Oriented Design Concepts (ICLODC rsquo04) F Stangenberg et alEd SFB 398 Ruhr-University Bochum Germany 2004

[20] Y S Petryna ldquoStructural degradation reliability and lifetimesimulations of civil engineering structuresrdquo in Proceedings of the9th International Conference on Structural Safety and Reliability(ICOSSARrsquo05) Rome Italy G Augusti G I Schueller and MCiampoli Eds Millpress Rotterdam The Netherlands 2005

[21] Y S Petryna and W B Kratzig ldquoCompliance-based structuraldamagemeasure and its sensitivity to uncertaintiesrdquoComputersand Structures vol 83 no 14 pp 1113ndash1133 2005

[22] G C Ernst G M Smith A R Riveland and D N PierceldquoBasic reinforced concrete frame performance under verticaland horizontal loadsrdquo American Concrete Institute Journal vol70 no 4 pp 261ndash269 1973

[23] F J Vecchio and M B Emara ldquoShear deformations in rein-forced concrete framesrdquo ACI Structural Journal vol 89 no 1pp 46ndash56 1992

[24] Y S Petryna D Pfanner F Stangenberg and W B KratzigldquoReliability of reinforced concrete structures under fatiguerdquoReliability Engineering and System Safety vol 77 no 3 pp 253ndash261 2002

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Mathematical Problems in Engineering

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