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22nd CAD-FEM Users Meeting 2004International Congress on FEM Technologywith ANSYS CFX & ICEM CFD Conference

November 10-12, 2004, International Congress Center Dresden, Germany

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1.2.3

Damage Analysis of Masonry Structures underEarthquake Action using ANSYS-Software

Song Ha Nguyen, Wolfram Jger

Chair of Structural Design, Faculty of Architecture, TU Dresden,Mommsenstrasse 13, D-01062 Dresden, Germany

Summary

Under earthquake action, masonry structures may collapse or suffer severe damages. In order toassess the damages, a complex nonlinear computational model is introduced to take intoconsideration both material and structural damage.

The material damage can be simulated by adopting nonlinear material laws and the structural damageoccurs e.g. as a result of crack formation. In this paper, a realistic nonlinear material law for masonryon the macro-level based on the elasto-plastic theory [7] is introduced and implemented with theANSYS-Software. The tensile failure of brick, the compressive failure of masonry and the shear failure

between mortar and brick according to the Mann/Mller theory (see [12]) are realistically described.Softening effects for tension, compression and initial shear strength are included.

For the evaluation of damage, the global damage indicators based on changes of the eigenfrequencies of the structure and eigenvalues of the system stiffness matrix are applied (see [10], [9]).

Damage analysis with the aid of the introduced damage indicators under earthquake action isdemonstrated by means of a numerical example.

Keywords

Masonry, Damage Indicator, Earthquake, Material Law


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1 Introduction

Masonry buildings are the most important construction type around the world. In Germany, most of thehistorical monumental structures and about 83.9% of the domestic buildings are made of masonry.The masonry structures are used even in seismic zones.

Because of the inherent weakness to withstand horizontal loads, the behavior of masonry buildingsunder seismic loads has been a major concern. In the process of deciding whether a structure

damaged during an earthquake can still be safely used in the future or if the structure has to berepaired, assessment of the damage and estimation of the residual load-bearing capacity of thestructure are of primary importance.

Damage to a structure leads to significant changes in its stiffness characteristics and a reduction ofthe load-bearing capacity. For assessment of the damage of a structure under earthquake action, thenonlinear time history analysis is employed. With this nonlinear simulation, the geometrical andphysical nonlinearities can be taken into account. The material and structural damages areaccumulated in the tangential system stiffness matrix and yield to changes of eigen angularfrequencies.

2 Realistic Masonry Analysis under Earthquake Action

2.1 General

For assessment of the seismic effects on structures, different numerical models are developed. Mostbuilding codes suggest three methods: the equivalent static loads analysis (or lateral force method ofanalysis), modal response spectrum analysis and time history analysis. The equivalent static loadsanalysis may be applied to buildings whose response is not significantly affected by contributions frommodes of vibration higher than the fundamental mode in each principal direction. The conditions forthe buildings are given in DIN 4149 (new) [2] or EC 8 [5].

The modal response spectrum analysis has been developed for application in slender constructionslike towers and high-rise buildings. With this method, the maximum responses of the system underseismic action are determined; the time parameter is neglected. This method is more or less restrictedto linear systems. Nonlinear behavior can be considered approximately by introducing a behaviorfactor q. In DIN 4149 (new) [2], this factor is fixed for masonry with 1.5. The fixing is empirical,

unfounded and unsatisfactory.In time history analysis, the nonlinear differential equations of motion are integrated in small stepsover the time. The time-dependent response of the system can be obtained by direct numericalintegration of its differential equations of motion. The dynamic response of the structure behavesconformably in relation to the progressive damage process.

)t(v)d,v,t(M)t(v)d,v,t(K)t(v)d,v,t(D)t(v)d,v,t(M IT =++ &&& (1)

whereM mass matrixD damping matrixKT tangential stiffness matrixt time

d scalar damagev, v& , v&& incremental displacement, velocity, acceleration vectors

Iv&& incremental acceleration vector of the ground motion

For the solution of the differential equations Eq. (1), a variety of explicit and implicit time operatorshave been proposed. For nonlinear analysis, the modified NEWMARK operator (s. [6], pp. 497) isused. The nonlinear equations are then solved with the modified NEWTON-RAPHSON method.

With this method, geometrical as well as physical nonlinearities can be taken into account. The input

earthquake loads Iv&& (t) are presented as registered acceleration time functions or artificially generated

acceleration time functions.


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For a realistic analysis of masonry, only the time history method is suitable. For this method, the

acceleration time function of the ground motion Iv&& (t) and the realistic description of the nonlinear

material behavior are required. The acceleration time function of the ground motion Iv&& (t) is described

in section 2.2. The realistic description of the nonlinear material behavior based on elasto-plastictheory is presented in section 2.3. The inhomogeneous and anisotropic characteristics are modelledthrough yield conditions and failure laws of a homogeneous comparison material (smeared model).

2.2 Earthquake Action

Acceleration time functions of the ground motion are the most important initial information to describethe earthquake action. In application of the time history method, these functions are used as loadfunctions.

The acceleration time functions in general have an arbitrary time response which doesnt show anyperiodicity; they are transient functions. Beside existent acceleration time functions in variousearthquake libraries, artificially generated acceleration time functions may be used.

In order to generate the acceleration time functions numerically, the algorithm developed in [8] ispresented here (see Fig. 1).

Fig. 1 Generation of the acceleration time function

The algorithm, as outlined in Fig. 1, ischaracterized by the following steps:

- Generation of a random function with constantpower spectral density S0 (uniform randomnoise)

- With the aid of discrete FOURIERtransformation, the random function istransformed into the frequency range andmodified by multiplication with the elastic

response spectrum (see EC 8 or DIN 4149(new)) or with the transfer functions of a high-pass, low-pass and band-pass filter in eachcase (see [8] or [13]).

- After back-transformation into the timedomain, the stationary soil acceleration ismodified by modulation with an intensityfunction. The intensity function may take ondifferent forms. Two possible intensityfunctions are shown in Fig. 2. The factors a,b, c, d1, d2 are determined by evaluating theseismographs for a particular location.

Fig. 2 functions Intensity

random functionx(t)

FFT from time domain

to frequency domainX() Y()

y(t)a(t)

filter functions or

elastic response function

Multiplication with an

enveloped intensity function I(t)

FFT from frequency domainto time domain


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A horizontal accelerationtime function generated byusing this algorithm forearthquake zone 1 (ag = 0.4m/s2) and ground type CT(according to DIN 4149

(new)) is presented in Fig. 3.The elastic responsespectrum is shown in Fig. 4.

Fig. 3 Horizontal ground acceleration time function ZV1 for theearthquake zone 1(ag= 0,4 m/s

2) and the ground type CT(E DIN 4149 2004-05)

Fig. 4 Elastic horizontalspectrum for theearthquake zone1 (ag= 0.4 m/s

2)with the groundtype CT and theresponsespectrum of thegroundacceleration timefunction ZV1 inFig. 3

2.3 Material Law for Masonry

2.3.1 Material Model

For the description of the nonlinear material behavior of masonry, in the context of this work a simpleand robust material law on the macro level, based on the elasto-plastic theory is presented. With thismaterial law, the elasto-plastic behavior of masonry under cyclic loading can be described realistically.At every cross-section point, the material behavior in the inelastic zone and the progressive materialdamage (softening effects) can be included.

Due to the variety of possible failure mechanisms of masonry, the yield surfaces are derived from thefailure criteria. The failure criteria under combination of shear and compression stress occur according

to the failure model of Mann/Mller (see Fig. 5). The first failure mode, the adhesive tensile failure,occurs if the vertical stress is a tensile stress and exceeds the adhesive tensile strength between unitand mortar. The second failure mode gets substantial if the principal tensile stress exceeds the shearstrength between unit and mortar. The third failure mode describes the compressive failure.

An advantage of the model is the low number of the initial parameters which can be found in DIN1053-1 [1] or DIN 1053-100 [2]. Since the stresses in direction parallel to the bed joints are low andconsiderably smaller than the experimentally determined strength of masonry, no failure criteria oryield rules are set to capture a possible failure in this direction. The stresses are in the elastic rangeand the strains follow HOOKEs law.


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1. ( )h

HZ1,R +=

2. ( )

h1

1HS2,R

+

+=

3.RZ

RZ3,R 145,0

+=

4. ( )h

R4,R =

Fig. 5 Failure Criteria by Mann/Mller [12]

2.3.2 Implementation with ANSYS

In this research project, the material law presented above is implemented in the FE program packageANSYS with the help of a user defined material routine.

The implementation of the elasto-plastic constitutive relations in a FE program requires theconsideration of two different levels, in the following called the global level and the material levelrespectively.

On the global level, equilibrium must be satisfied in any linear or nonlinear FE computation. The nodedisplacements as well as the strains of the integration points at each cross-section in every particulartime step or time point are calculated by solving the differential equations Eq. (1) with the help of theNEWMARK operator and the modified NEWTON-RAPHSON method. Together with the strains andthe stresses of the previous time step, the state variables of the user routine are determined at every

integration point.

On the material level, the plasticity relations must be satisfied. The resulting stresses as well as thetangent stiffness matrix at each integration point are then estimated by local NEWTON-RAPHSONiterations (return mapping method). The stresses and the tangent stiffness matrix are passed back tothe main program. The softening effects are also included in the model with different softeningparameters for all regions. The problem of the mesh objectivity is tackled by the introduction of anequivalent length related to the element size and fracture energies.

On the global level, the residual forces are calculated at the end of the iteration step. If the fixedtolerance criterions are not fulfilled, a new iteration step is carried out with updated state values fromthe user routine.

2.3.3 Verification of the material model

The material model was verified by comparing the numerical results to the experimental results fromtes [15] (s. Fig. 2).

The comparison of the hysteresis diagrams of the experimental results and the numerical resultsshows that this nonlinear material model can well describe the behaviour of masonry under cyclic loadand the collapse load can be well estimated by this model.

The collapse deformation calculated by this model is considerably smaller than the collapsedeformation in the test. A reason for this is that the discontinuity in the masonry caused by crackformation in the test is not included in this model which is based on the continuum mechanics.Furthermore, the input parameters used for calculating and the parameters of the tested wall might benot quite identical. The parameters of the wall tested by tes have not been measured before the test.

friction failure

Failure due tocrushing

Tensile failurefvk,

Dd

gaping


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The input parameters used in the model were taken from the same material component of ibac(institute of building material research, Aachen university).

-50,0

-40,0

-30,0

-20,0

-10,0

0,0

10,0

20,030,0

40,0

50,0

-6,0 -4,0 -2,0 0,0 2,0 4,0 6,0

Fig. 6 Hysteresis diagrams of the experimental result V6 by tes [15] and the numerical result

3 Damage Assessment

3.1 Damage phenomena

Damage occurs during the loading process; it is therefore a function of time. In general, a supportingstructure initially remains in a damage-free state (undamaged state) for a certain period of time. Evenafter the first signs of local damage zones become evident, the global load-bearing behavior generallyremains unaffected. However, if the damage zones continue to develop over-proportionally in bothspace and time, this progressive nonlinear damage process leads to system failure. A particular

structure is susceptible to specific types of external loading. Especially in the case of alternating orcyclic loading, it is only possible to forecast damage processes by means of suitable realistic models.

3.2 Damage Indicators

In order to assess the damage of structures, several damage indicators have been introduced (see[10] and [14]). Whereas the indicators for local damage refer to the hysteretic characteristics ofparticular system details, those for global damage predominantly refer to changes in the stiffnessparameters. In this paper, two damage indicators based on changes of eigen frequencies of thestructure and eigenvalues of stiffness matrix are presented in the following. These damage indicatorsprovide indirect estimates of the reduction of the global bearing capacity as a result of local damage ofarbitrary kind.

3.2.1 Formulation of damage indicator using eigenvalues of the system stiffness matrix

The global damage indicator presented in this section is based on the changes in the system stiffnessmatrix during earthquake (see [10] or [14]). This damage indicator is defined in the form of Eq. (2).

structureundamagedtheofstiffness

structuredamagedtheofstiffnessD = (2)

The stiffness of a structure may be expressed with the aid of the eigenvalues of the system stiffnessmatrix. This leads to a set of fuzzy damage indicators:

)0d,v,0t(

)d,v,t(1)t(D

K

i

K

iK

i==

= (3)

H [kN]

d [mm]


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in which (t,v,d) is the ith eigenvalue of the system stiffness matrix at time point t (in damaged state)and (t=0,v,d=0) is the ith eigenvalue of the system stiffness matrix in the undamaged state.

If a valueK

iD is equal to unity, the system stiffness matrix becomes singular; that is, the

displacements increase in an uncontrolled manner and the structure fails. The maximum value ofK

iD

may be interpreted as a global damage indicator. This determines the minimum separation distancebetween the damage and the failure state:

KD (t) = max { KiD (t)}, i = 1, , N (4)

in which N denotes the number of degrees of freedom of the structure. The damage indicatorKD

according to Eq. (4) describes the evolution of the local damage globally.

3.2.2 Formulation of a damage indicator using eigen angular frequencies

If material behavior has been formulated in a damage-oriented manner, damage alters the globaltangential system stiffness matrix KT(t, v, d). On that basis, the eigen angular frequencies of astructure may be chosen as characteristic parameters for (nonlinear) damage indicators (see [10] or[14]).

)0d,v,0t(

)d,v,t(1)t(D

i

ii

==

= 1 i N (5)

where Di(t) is the damage indicator; i(t, v, d) is the ith eigen angular frequency at time t, dependent on

the displacement state v(t) and the degree of damage d(t); i(t = 0, v, d = 0) is the ith eigen angular

frequency at the beginning of the service life (t = 0) for the undamaged system (d = 0).

Owing to the simplicity of the latter and its high sensitivity, this provides a suitable means of estimatingthe residual load-bearing capacity of the system and interactive control of the numerical simulation.Critical loading states and critical local damage states are sensitively indicated in relation to globalload-bearing behavior.

The eigen angular frequencies i required to evaluate Eq. (5) follow from the eigenvalue problem

without the inclusion of damping:(KT(t, v, d) -

2M)v = 0 (6)

Here KT changes as a result of damage during the loading process. IfKT is known at time t, the eigenangular frequencies i may be computed form Eq. (6).

4 Example

4.1 System and Loads

For demonstration a standard terraced house made of masonry is studied under earthquake action.The geometry of the building was taken from [15]. It is a 1-storied residential house with a woodenroof construction on a ground plan of 10.0 x 6.0 m. The terraced house is stiffened in the longitudinal

direction by the gable walls against horizontal loads. In the transversal direction the interior walls(sideways of the staircase) carry the majority of the horizontal load. The roof construction is a purlintype roof with the roof areas acting as stiff plates due to the integration of wind panicles.

In order to model the terraced house as realistically as possible, a 3D model with SOLID elements isused (see Fig. 7). The roof construction is modeled as a single unit with SHELL elements. Theequivalent stiffness of roof construction is approximately determined.


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-1,00

-0,80

-0,60

-0,40

-0,20

0,00

0,00 2,00 4,00 6,00 8,00 10,00

The earthquake load is applied asacceleration time functions of ground motion.These loads consist of three accelerationtime functions independent of each other,which act simultaneously on the structure inthe three perpendicular directions x, y and z.The acceleration time functions aregenerated according to earthquake zone 1(ag = 0.4 m/s

2) and ground type CT(according to DIN 4149 (new) [2]).

Fig. 7 FE model of the row house

4.2 Results

The behavior of the terraced house isstudied for different levels of earthquakeaction through an earthquake load factor k.The factor k is equal to 1 for earthquakeaction in zone 1 with ag = 0.4 m/s

2.

In this example, the damage indicatorbased on the change of first eigen angularfrequency (see Eq. (5) with i = 1) is used.

Damage of the row house is studied withdifferent strong levels of earthquake actionthrough an earthquake load factor k. In thisexample, the damage indicator based on

the change of first eigen angular frequency(see Eq. (5) with i = 1) is used.

Fig. 8 shows the damaged areas at thestiffening walls for the case of theearthquake load factor being equal to 2.5.In the figure, the damaged areas arepresented as equivalent plastic strains.

Fig. 8 Damage areas at the stiff walls with k = 2.5

Fig. 9 Variation of the stress in vertical direction at the point 1 (see Fig. 8 )during the earthquakeprocess with the load factor k = 1

point 1


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-1,50

-1,25

-1,00

-0,75

-0,50

-0,25

0,00

0,25

0,50

0 2 4 6 8 10

linear analysis

nonlinearanalysis

Fig. 9 shows the variation of the stress in z direction (vertical direction) at the point 1 (see Fig. 8)during the earthquake process with an earthquake load factor k = 1 (ag = 0.4 m/s

2). At this load level,major areas of the masonry walls work in the elastic range. The structure remains undamaged. Thestresses calculated by nonlinear and linear models are identical. The damage indicator is equal tozero.

For an earthquake action with the factor k = 2.5 (ag = 1.0 m/s2), the stress in the vertical direction at

the point 1 is represented in Fig. 10. At this load level, major areas in stiff wall work in plastic range.The structure experiences local damage. The stresses from nonlinear and linear models are differentsignificantly. At t = 4.5 s the damage indicator calculated through Eq. (5) with i = 1 is

6044.04673.6

5585.21

)0,,0(

),,5.4(1)(

1

11 ==

==

==

dvt

dvttD

Fig. 10 Variation of the stress in vertical direction at the point 1 (see Fig. 8 ) during the earthquake

process with the load factor k = 2.5

Conclusion

In this paper, the time history method is used to analyze masonry structures under earthquake action.The earthquake loadings are taken as artificially generated acceleration time functions. The nonlinearmaterial law based on elasto-plastic theory for masonry is introduced at the macro level andimplemented in the ANSYS program package. For damage assessment, the different damageindicators are discussed. As an example, damage of a terraced house under earthquake action isstudied.

With the algorithm presented in this paper, the residual load-bearing capacity of masonry structures

after earthquake action can been estimated. The results help to make a decision whether a damagedstructure can still be safely used in the future or if the structure has to be repaired.

The material law based on elasto-plastic theory is not quite correct for the precise description ofdamage. In the future, the new material law based on e.g. endochron theory or damage theory will bedeveloped.

Acknowledgment

The research discussed in this contribution has been financed by the federal office for building andregional planning BBR (Bundesamt fr Bauwesen und Raumordnung) of Germany (project Z 6 10.07.03-03.09 / II 13 80 01 03-9).


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References

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[2] E DIN 4149: 2004-05: Bauten in deutschen Erdbebengebieten. Auslegung von Hochbautengegen Erdbeben. Entwurf. DIN: Berlin Oktober 2004

[3] DIN 1053-1: 1996-11. Mauerwerk. Teil 1: Berechnung und Ausfhrung. DIN: Berlin 1996

[4] EC 6: Bemessung und Konstruktion von Mauerwerksbauten. Teil 1-1: Allgemeine Regeln. Re-geln fr bewehrtes und unbewehrtes Mauerwerk. Deutsche Fassung ENV 1996-1-1: 1995

[5] EC 8: Auslegung von Bauwerken gegen Erdbeben. Teil 1-1. ENV 1998-1-1. Grundlagen Erdbebeneinwirkung und allgemeine Anforderungen an Bauwerke. Deutsche Fassung ENV1998-1-1: 1994. Teil 1-2: Grundlagen Allgemeine Regeln fr Hochbauten. Deutsche FassungENV 1998-1-2: 1994. Teil 1-3: Grundlagen Baustoffspezifische Regeln fr Hochbauten.Deutsche Fassung ENV 1998-1-3: 1995. Teil 1-4: Grundlagen Verstrkung und Reparatur vonHochbauten. Deutsche Fassung ENV 1998-1-4: 1996

[6] Argyris, J; Mlejnek, H.P.: Die Methode der Finiten Flemente, Band III: Einfhrung in dieDynamik. Vieweg Verlag: Braunschweig 1988

[7] Chen, W.F.; Han, D.J.: Plasticity for Structural Engineers. Springer-Verlag New York, 1988

[8] Hosser, D. Ahorner, L.; Schneider, G.: Realistische seismische Lastannahmen fr Bauwerke.Institut fr Bautechnik, IRB Verlag: Berlin 1986

[9] Jger, W.; Nguyen, S.H.: Erdbebenbeanspruchung von Mauerwerk Numerische Simulationzur Bewertung der Tragwerksschdigung. In: Brameshuber, W.; Schuber, P. (Hrsg.) 30.Aachener Baustofftag "Mauerwerk", Festschrift zum Dank an Dr.-Ing. Peter Schubert, Aachen,2004, S. 175-191

[10] Krtzig, W:B.; Noh, S.Y.: Computersimulation progressiver Schdigungsprozesse vonStahlbetonkonstruktionen. Finite Elemente in der Baupraxis, 1998, S. 123-132

[11] Lourenco, P.B.: An orthotropic continuum model for the analysis of masonry structures. DelftUniversity of Technology. Report No. 95-NM-R0712, 1995

[12] Mann, W.; Mller, H.: Schubtragfhigkeit von Mauerwerk. In: Mauerwerk-Kalender1978, Ernst& Sohn Berlin, 1978

[13] Meskouris, K.: Structural Dynamics, Models, Methods, Examples. Verlag Ernst & Sohn: Berlin2000

[14] Nguyen, S.H.; Mller, B.; Graf, W.: Modeling of time-dependent loading as a fuzzy-process. In:Fifth European Conference on Structural Dynamics (EURODYN 2002), Eds. Grundmann, H.;Schuller, G.I., TU Mnchen, A.A. Balkema Publishers: Rotterdam 2002, Vol. 1, pp. 679-684

[15] tes, A., Lring, S.: Zur Horizontalaussteifung von Reihenhusern. In: Das Mauerwerk 4

(2000), S. 152-158[16] tes, A.; Lring, S.: Tastversuche zur Identifizierung des Verhaltensfaktors von

Mauerwerksbauten fr den Erdbebennachweis. Abschlussbericht der Lehrstuhl frTragkonstruktionen, Universitt Dortmund, 2003

[17] Rautenstrauch, K.; Schlegel, R.: Numerische Modellierung von Mauerwerk. In: Brameshuber,W. & Schubert, P. (Hrsg.): 30. Aachener Baustofftag "Mauerwerk". Festschrift zum Dank an Dr.-Ing. Schubert, P., Aachen 2004, S. 193-217

[18] Zilch, K.; Schermer, D.: Experimentelle und numerische Untersuchungen zumErdbebentragverhalten unbewehrter Mauerwerksbauten. In: Mauerwerk Kalender 2004. Hrsgg.v. P. Schubert, H.-J. Irmschler u. W. Jger. Ernst & Sohn: Berlin 2004, S. 649 664


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