Numerical modelling for earthquake engineering: the case of lightly RC structural walls

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2004; 28:857–874 (DOI: 10.1002/nag.363)

Numerical modelling for earthquake engineering: the case oflightly RC structural walls

J. Mazars1,n,y, F. Ragueneau2, G. Casaux2, A. Colombo1 and P. Kotronis1

1Laboratoire Sols, Solides, Structures (L3S), RNVO network, Domaine Universitaire BP 53,

38041 Grenoble cedex 9, France2Laboratoire de M !eecanique et de Technologie (LMT), 61, av. du Pr !eesident Wilson, 94230 Cachan, France

SUMMARY

Different types of numerical models exist to describe the non-linear behaviour of reinforced concretestructures. Based on the level of discretization they are often classified as refined or simplified ones. Theefficiency of two simplified models using beam elements and damage mechanics in describing the global andlocal behaviour of lightly reinforced concrete structural walls subjected to seismic loadings is investigatedin this paper. The first model uses an implicit and the second an explicit numerical scheme. For each case,the results of the CAMUS 2000 experimental programme are used to validate the approaches. Copyright# 2004 John Wiley & Sons, Ltd.

KEY WORDS: reinforced concrete; damage mechanics; beam elements; simplified analysis

INTRODUCTION

Reinforced concrete (R/C) bearing walls with limited reinforcement are often used in Franceand other European countries. Recent experimental programs (e.g. CASSBA, CAMUS) haveshown that this type of structural element exhibits good behaviour under seismic loading,although its ductility might be limited due to the light reinforcement and the existence of largesections in the walls with practically no steel [1]. Some unconventional mechanisms ofearthquake resistance have also been highlighted such as rigid block-type rotations of the walls(at the interface between the foundation and the soil or at the level of construction joints) or theexcitation of high frequency vibrations corresponding to pumping vertical motions (due to theopening and closing of wide horizontal cracks and the conversion of part of the seismic intopotential energy). Furthermore, in case of an out of plane loading additional fluctuation of theaxial force may arise and heavy damage is expected since cracks do not completely close at loadreversal. It is therefore easily understood that the conventional elastic seismic analysis is not

Received 9 May 2003Revised 21 January 2004

Accepted 21 January 2004Copyright # 2004 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: J. Mazars, Laboratoire Sols, Solides, Structures (L3S), RNVO network, Domaine UniversitaireBP 53, 38041 Grenoble cedex 9, France.

adequate to take into account all these effects. More reliable numerical tools are necessary toassist engineers during the design phase.

Various modelling strategies have been proposed up to now for the non-linear analysis of R/Cstructures. Their level of complexity is usually proportional to the dimension of the problem.Detailed 2D or 3D finite element models with local constitutive relationships are typically usedfor predicting the response of structural elements or substructures, whereas simplified global orlocal models are useful for the dynamic response analysis of structures.

In this paper the performance of two simplified models}a fibre model and a beam modelwith multiple integration points}is evaluated using the experimental results of two five-storeylightly R/C walls subjected to dynamic loading. Both models use local constitutive laws basedon damage mechanics and plasticity. The use of an implicit (for the fibre model) and an explicitnumerical scheme (for the beam model) is also investigated.

THE CAMUS 2000 RESEARCH PROGRAMME

The 3-years experimental and numerical research program CAMUS 2000 was launched in 1998with the aim of evaluating the effects of torsion and the behaviour of lightly R/C walls subjectedto bi-directional motions [2]. The design of the mock-ups was made according to the ‘multifuse’concept commonly used in France that privileges diffuse rupture at several stories instead ofconcentration at the base of the building with the creation of one plastic hinge. Low percentagesof reinforcement combined with an appropriate distribution at several levels helps to dissipateenergy via wide crack patterns at different heights of the wall and leads into multiplication of thedissipation zones [3].

Two 1/3rd scaled models were tested on the Azal!eee shaking table of Commissariat "aa l’EnergieAtomique (CEA) in the Saclay Nuclear Centre. The specimens were composed of two parallelbraced walls linked with six square slabs. A highly reinforced footing allowed the anchorage tothe shaking table. Two shear walls in one direction and a steel bracing system in the orthogonaldirection provided structural stiffness. Due to similarity laws additional masses of 6:55 ton werepositioned at each storey. The first structure (CAMUS 2000-1) was subjected to a horizontal bi-directional excitation (Figure 1). A set of accelerograms of increasing amplitude (0.15, 0.22,0.25, 0.40, 0.55 and 0:65g) was applied to the specimen. The accelerograms were modified intime according to the ratio 1=

ffiffiffi3

pto account for similarity rules. For the second test (CAMUS

2000-2) only an in plane excitation was applied. The torsional response was caused by thedissymmetry in the horizontal dimensions of the two walls (Figure 2). The distribution ofreinforcement for both specimens is described in Tables I and II.

FIBRE MODEL (IMPLICIT PROCEDURE)

Numerical tools

Non-linear dynamic analysis of civil engineering buildings requires large-scale calculations andthe use of delicate solution strategies. The response of a structure subjected to severe loadingdepends on a strong interaction between ‘material’ (plasticity, cracks), ‘structural’ (geometry,mass distribution, construction joints) and ‘environmental’ effects (boundary conditions,

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2004; 28:857–874

J. MAZARS ET AL.858

soil–structure interaction). The major sources of non-linearity for a R/C structure are often onthe ‘material’ level. In order to reproduce correctly the main physical phenomena at this level(damage, permanent strain, crack-reclosure) one has to use advanced local constitutiverelationships. For the structural level however, a ‘simplified approach’ helps to reduce thecomputational cost and facilitates parametrical studies.

The choice of a multifibre finite element configuration combines the advantage of usingBernoulli beam type finite elements with 1D local constitutive laws. Each finite element isa beam discretized into several fibres where different material properties can be assigned(Figure 3). In case where shear deformations are prevailing the Timoshenko beam theory canalso be adopted [4]. For the general 3D case the cross section behaviour}the relation betweenthe generalized strains e and the generalized stresses s}becomes

s ¼ Kse

s ¼ ðN Sy Sz Mx My MzÞT

ð1Þ

and

e ¼ ðe gy gz yx wy wzÞT

where N is the normal force, Sy and Sx the shear forces, Mx the torque, My and Mz the bendingmoments, e the axial strain, yx the twist, wy and wz the curvatures. The matrix Ks has the

Figure 1. CAMUS 2000-1: layout of the specimen.


NUMERICAL MODELLING FOR EARTHQUAKE ENGINEERING 859

following form [5]:

Ks ¼

Ks11 0 0 0 Ks15 Ks16

Ks22 0 Ks24 0 0

Ks33 Ks34 0 0

Ks44 0 0

Ks55 Ks56

sym Ks66

2666666666664

3777777777775

ð2Þ

Figure 2. CAMUS 2000-2: non-symmetric specimen anchored on the Azal!eeeShaking table (CEA-Saclay, France).


J. MAZARS ET AL.860

where the coefficients are obtained by integrating over the cross section (y- and z-axis):

Ks11 ¼ZS

E dS; Ks15 ¼ZS

Ez dS

Ks16 ¼ �ZS

Ey dS; Ks22 ¼ ky

ZS

G dS

Ks24 ¼ �ky

ZS

Gz dS; Ks33 ¼ kz

ZS

G dS

Ks34 ¼ kz

ZS

Gy dS; Ks44 ¼ZS

Gðkzy2 þ kyz2Þ dS

Ks55 ¼ZS

Ez2 dS; Ks56 ¼ �ZS

Eyz dS; Ks66 ¼ZS

Ey2 dS

ð3Þ

Table II. CAMUS 2000-2: reinforcement areas.

Wall Floor Reinforcement areas ðcm2Þ

1:30 m� 0:06 m 1st 4.152nd 2.863rd 1.574th 0.785th 0.16

2:10 m� 0:06 m 1st 1.642nd 1.483rd 1.004th 0.505th 0.28

Table I. CAMUS 2000-1: reinforcement areas ðcm2Þ:

Floor 1 Floor 2 Floor 3 Floor 4 Floor 5

Reinforcement areas 0.95 0.48 0.16 0.16 0.160.79 0.320.64 0.160.48

Figure 3. Multifibre discretization principle for a 2-node beam element.



E and G are the Young’s and shear modulus that vary in y and z; ky and kz are the shearreduction factors. The chosen modulus can be initial, secant or tangent, depending on theiterative algorithm used to solve the global equilibrium equations. The components of theconstitutive matrix are computed by numerical integration according to (3), with one Gausspoint per fibre. An implicit numerical scheme using the initial Young’s modulus is chosen for thefollowing calculations done with the fibre model.

The non-linear behaviour of the materials is described within the thermodynamic frame-work for irreversible processes [6]. To model the behaviour of reinforcement bars we choosethe classical plasticity theory with a non-linear kinematic hardening in order to reproducecorrectly the observed hysteresis loops [7]. The free energy for this model takes the follow-ing form:

rc ¼ 12ðe� epÞ : H : ðe� epÞ þ 1

2ba : a ð4Þ

in which H is the Hooke’s elasticity tensor, ep is the plastic strain and a the hardening internalvariable. The constitutive equations are obtained by derivation:

s ¼@ðrcÞ@e

¼ H : ðe� epÞ

X ¼@ðrcÞ@a

¼ ba

ð5Þ

where X is the stress like hardening variable. The latter is used to describe a modified form of theplasticity criterion allowing remaining within the associated plasticity theory:

f ¼ J2ðs� XÞ þ 34aX : X � sy40 ð6Þ

where a; b and sy are material parameters. Due to the particular geometric characteristics ofsteel bars only a 1D implementation of the model is carried out. The evolution equation hasbeen modified to account for the particular behaviour of reinforcing steel used in civilengineering. Based on the normality rule in an associated framework we have

’aa ¼ �’llgðemaxp Þ

@f

@X

gðemaxp Þ ¼ 0 if jemax

p j4elimp

ð7Þ

and

gðemaxp Þ ¼ 1 if jemax

p j > elimp

elimp is a material parameter corresponding to the stress–strain plateau length without hardening.A typical stress–strain curve predicted by this model is given in Figure 4.

The constitutive model for concrete in earthquake engineering ought to take into accountsome observed phenomena such as decrease in material stiffness due to cracking, stiffnessrecovery which occurs at crack closure, inelastic strains concomitant to damage and strain rateeffects. To account for such complex phenomena we use a damage model with two scalardamage variables}D1 for damage in tension and D2 for damage in compression (La Borderiemodel [8]). Unilateral effect and stiffness recovery (damage deactivation) are also included.Inelastic strains are taken into account thanks to an isotropic tensor. To account for strain rateeffects in dynamics, the model has been improved by modifying the evolution laws (as it is


J. MAZARS ET AL.862

usually done for visco-plastic models [9]). The Gibbs free energy of the model can be expressed as

w ¼hsiþ : hsiþ2Eð1�D1Þ

þhsi� : hsi�2Eð1�D2Þ

þuEðs : s� Trðs2ÞÞ þ

b1D1

Eð1�D1Þf ðsÞ

þb2D2

Eð1�D2ÞTrðsÞ ð8Þ

where b1 and b2 are material coefficients, whereas h:iþ and h:i� denotes the positive or negativevalues of the given variable. f ðsÞ and sf are the crack closure function and the crack closurestress, respectively. E is the initial Young’s modulus and n the Poisson ratio.

For a 1D implementation of the model, the total strain is

e ¼ ee þ ein

ee ¼sþ

Eð1�D1Þþ

s�Eð1�D2Þ

ein ¼b1D1

Eð1�D1Þf ðsÞ þ

b2D2

Eð1�D2Þ

ð9Þ

where ee is the elastic strain and ein the inelastic strain. The partition of the stress is obtained as

s > 0 ! sþ ¼ s; s� ¼ 0

s50 ! sþ ¼ 0; s� ¼ sð10Þ

Damage criteria are expressed as fi ¼ Yi � Zi � Y0i (i ¼ 1 for tension or 2 for compression, Yi

is the associated force to the damage variable Di and Zi a threshold depending on the hardeningvariables). The evolution laws for the damage variablesDi are obtained thanks to normality rules:

’DDi ¼ ’ll@fi@Yi

ð11Þ

As for visco-plasticity, the plastic multiplier is imposed as function of the threshold criterion,

’ll ¼1

mi

hYi � Zi � Y0iiY0i

� �ni

ð12Þ

0

1×108

2×108

3×108

4×108

5×108

6×108

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

experimentmodel

stre

ss (

Pa)

strain

Figure 4. Uniaxial steel modelling. Local model parameter identification.



where Y0i is the initial elastic threshold and mi and ni are parameters to be identified for tensionand compression to recover the relative effects of strain rate on the peak stress observed onconcrete specimens subject to various dynamic loads.

A typical stress-strain response of the model for a uniaxial cyclic loading (tension,compression) and for two different strain rates is given in Figure 5.

Numerical simulation

The finite element mesh used for the calculations is presented in Figure 6. The additional massesand the weight load of each floor are concentrated at each storey. The stiffness of the springsbelow the shaking table is calibrated to fit the first eigenmodes measured before the applicationof the seismic loads. The following calculations are done with the finite element code CAST3Mdeveloped at CEA.

-50

-40

-30

-20

-10

0

10

-4 -3 -2 -1 0 1

100. s-1

static

stre

ss (

MPa

)

strain (10-3)

Figure 5. La Borderie model: uniaxial constitutive relations for concrete. Strain rateeffects in tension and compression.

Figure 6. CAMUS 2000-1 fibre model: finite element mesh.


J. MAZARS ET AL.864

A first series of calculations (modal analysis) is performed to check the ability of the proposedmodel to reproduce the main characteristics of the CAMUS 2000-1 specimen (boundaryconditions, mass distributions). The shaking table has to be included to the mesh (withorthogonal beams) to insure the good correlation of the calculations. The values of the first foureigen frequencies and mode shapes are presented in Figure 7 (fibre model). Comparison with themeasured ones shows a good agreement.

Despite the lack of physical meaning, damping is generally introduced in the analysis throughviscous forces generated by the means of a damping matrix (the classical viscous Rayleighdamping matrix, derived from the general expression proposed by Caughey [10]). Twoparameters}the Rayleigh damping coefficients}allow calibrating the matrix by imposing thevalue of the damping ratio for two eigenmodes. For the following calculations the Rayleighdamping coefficients have been adjusted to ensure a value of 1% on the first mode and 2% onthe second mode. It is important for concrete structures}where cracking induces loss ofstiffness and shift of the fundamental frequency}to keep these damping values constantthroughout the analysis even during strong non-linear behaviour. Therefore, the damping of thefirst eigenmode has been chosen so as to remain around the minimum (almost constant) range ofthe Rayleigh diagram.

Results for the transient dynamic calculations are presented in terms of horizontal topdisplacements in the plane (X direction) of the walls (Figure 8) and global flexural moment inthe Y direction (Figure 9) for the signal corresponding to 0:55g: Tables III and IV allow acomparison between computation and experimental results (global and local quantities) for theCAMUS 2000-1 and CAMUS 2000-2 mock-ups. The results have been obtained without furthercalibration of the model. Work is in progress on the effects of damping [11] and improvementsof the modelling are carried out to account for torsion and 3D material behaviour using anenhanced beam formulation [12].

BEAM MODEL (EXPLICIT PROCEDURE)

Numerical tools

Explicit procedures combined with simplified modelling strategies are very useful for solvingtransient dynamic problems. Taking advantage of the small number of degrees of freedom

Mode In-planeflexion

Out of planeflexion

Torsion Pumping(verticalmode)

Shape

Experiment 6.0 Hz 5.45 Hz - -ComputationFiber model Beam model

6.0 Hz6.02 Hz

5.5 Hz5.5 Hz

10.5 Hz12.56 Hz

17.2 Hz19.63 Hz

Figure 7. CAMUS 2000-1: natural eigen frequencies and mode shapes.



needed in a simplified modelling strategy, explicit procedures contribute to a significantreduction of the computational cost since they require no iterations and no calculation of thetangent stiffness matrix. Even though they are conditionally stable, this is not always aconstraint in Earthquake Engineering because a much smaller time step should be chosen toobtain results that are accurate.

To verify the efficiency of an explicit procedure coupled with a simplified approach indescribing the seismic behaviour of R/C wall structures we use a simplified finite element meshto model the CAMUS 2000-1 specimen. The mesh is realized using 3D beam elements withmultiple integration points. It differs from the one presented in the previous paragraphsfrom the fact that all structural members are characterized by homogeneous cross-sections.

-0.02

-0.01

-0.007

0

0.007

0.01

0.02

0 2 4 6 8 10

computationexperiment

In p

lane

dis

plac

emen

t (m

)

time (s)

Figure 8. CAMUS 2000-1 fibre model: in plane top horizontal displacement ð0:55gÞ:

-5×105

-3×105

-2×105

0

2×105

3×105

5×105

0 2 4 6 8 10

computation

experimentOut

pla

ne b

endi

ng m

omen

t (N

. m)

time (s)

Figure 9. CAMUS 2000-1 fibre model: out of plane bending moment ð0:55gÞ:


J. MAZARS ET AL.866

The explicit version of the commercial computer code ABAQUS is chosen for the analyses. Thedescription of the mesh follows (Figure 10):

* The structure is modelled using Timoshenko beams. The transverse shear deformation isconsidered linear elastic, independent of the axial and bending behaviour.

* The sections of the Timoshenko beam elements differ according to the various structuralcomponents. More specifically:

Walls. Beam elements with rectangular section (25 integration points per section) arechosen to model R/C walls. The longitudinal reinforcement is introduced using boxsection elements (16 integration points per section). The area of the reinforcement bars istransformed into an equivalent area of the box section. Concentrated masses simulatethe additional masses placed along the walls.

Table IV. CAMUS 2000-2 fibre model: numerical versus experimental results.

0:80g 1:12g

PGA Comparisons Comp. Exp. Comp. Exp.

Left wall in-plane 34.2 32.4 65.2 57.9Displacement (mm)Right wall in-plane 33.6 31.8 64.8 63.4Displacement (mm)Out of plane displacement (mm) 10.0 6.0 16.4 17.9Out of plane moment, wall 130 (kN.m) 348 469 393 519In-plane moment, wall 210 (kN.m) 458 727 503 632Total axial force, wall 130 (kN) �8.5 +36 �36.2 +64

�301 �296 �321 �444Total axial force, wall 210 (kN) �52.6 �64 �57 +13

�303 �278 �335 �398

Table III. CAMUS 2000-1 fibre model: numerical versus experimental results.

0:15g 0:40g 0:55g 0:65g

PGA Comparisons Comp. Exp. Comp. Exp. Comp. Exp. Comp. Exp.

Left wall 4.92 4.08 12.9 13.2 22.4 18.7 26.9 31.0in-planeDisp. (mm)Right wall 4.40 4.31 15.7 16.1 20.6 18.3 29.5 40.3in-planeDisp. (mm)Out of plane +3.58 +4.93 +9.82 +12.8 +12.4 +27.9 +12.3 +45.7Disp. (mm) �3.99 �4.56 �11.8 �14.3 �18.1 �21.5 �22.4In-plane 293 225 382 362 452 473 392 407Moment ðkN mÞOut of plane moment 305 331 545 492 649 578 621 500Left wall } } } } +46.9 +79 +9 +28vertical load(kN) �484 �467 �470 �401



Slab. Rectangular and box section elements are used to mesh the slabs. The mass blocksadded to the floors to represent the dead load of the structure are modelled using fictivebox sections. The density of the material is chosen according to weight of the blocks,whereas the stiffness of the elements is limited in order to avoid their influence on thedynamic behaviour of the structure.Bracing system. The bracing system is modelled with I section elements (13 integrationpoints per section). The geometry of sections corresponds to the geometry of the steelsections adopted in reality.Basement. A network of elements simulates the stiffness of the basement in thethree directions. Rectangular section elements are used for concrete and box sectionelements for reinforcement. The specimen is connected to the table via four circular

Figure 10. CAMUS 2000-1: simplified beam model (a) Finite element mesh; (b) section shape adopted tomodel the bracing system; (c) rectangular section element for concrete members; and (d) box section usedto model the reinforcement and the mass blocks on the floors. Position and number of the integration

points in each section are also indicated.


J. MAZARS ET AL.868

section beams elements. A preload is applied to these components to reproducethe reality.Shaking table. A rigid body composed by rectangular section beams simulates theshaking table. Circular beam elements simulate the compliance of the table. The mass ofthe table is taken into account through an appropriate material density.

- A lumped mass formulation is adopted.- The cross-section of each beam is integrated numerically to obtain the force–moment/strain–

curvature relations for the section. In that way a complete generality in material response isachieved, since each point of the section is considered independently by the constitutiveroutines.

The elastic–plastic model of ABAQUS is chosen to describe the behaviour of steel members.The PRM model simulates the cyclic behaviour of concrete (PRM for Pontiroli–Rouquand–Mazars: [13–15]). This model distinguishes the behaviour under ‘traction’ and ‘compression’ atthe level of a transition zone defined by ðsft; eftÞ where cracks close (Figure 11 for the cyclicresponse). In these co-ordinates, the main equations of the PRM model for an uniaxial loadingtake the following form:

* Partition of strain and stress tensors:

e ¼ ed þ eft and s ¼ sd þ sft ð13Þ

Figure 11. PRM model: uniaxial stress–strain relationship.



* Constitutive equations (E0 is the initial Young’s modulus):under traction

ðs� sftÞ ¼ E0 � ð1�DtÞ � ðe� eftÞ ð14Þ

under compression

ðs� sftÞ ¼ E0 � ð1�DcÞ � ðe� eftÞ

Dt is evolved in traction and in compression through the variable *ee ¼ffiffiffiffiffiffiffiffiffiffiffihxi2þ

q[13], with x ¼ e

when load is traction and x ¼ ðe� eftÞ when load is compression; the maximum value of bothevolutions is introduced in the calculation.

Dc evolved through the same variable with x ¼ e: Then Di ¼ fct (*ee; ed0; Ai; Bi) with i ¼ t; c:*ee drives the damage evolution after the initial threshold ed0; Ai; Bi are material para-meters. eft ¼ eft0 (material parameter) when Dc ¼ 0 and is directly link to Dc afterwards;sft ¼ f ðeft;DcÞ:

* In order to describe dissipation due to hysteretic loops (Figure 11) a hysteretic stress termis added:

shyst ¼ ðb1 þ b2DiÞE0ð1�DiÞðe� eftÞf ðe� eftÞ ð15Þ

b1; b2 are the ‘Rayleigh’ parameters and f is a function used to calibrate the evolution with thestrain.

This model has been set up in order to describe 3D situations with an explicit formulationuseful for FEM calculation particularly for dynamic loadings. The general 3D formulation ofthe model relating strain and stress tensors (in bold) is reported below:

ðs� sftÞ ¼ L0ð1�DÞðe� eftÞ ¼ ð1�DÞ½l0 traceðe� eftÞ1þ 2m0ðe� eftÞ�: ð16Þ

where sft and eft are the crack closure stress and strain thresholds used to manage permanenteffects; L0 is related to the initial mechanical characteristics. D is issued from a combination ofthe two modes of damage:

D ¼ atDt þ ð1� atÞDc ð17Þ

at evolved in between 0 and 1 and the actual values depends on (e� eft).The PRM model takes into account crack-closure effects, permanent strains, hysteretic loops

and includes in its general form the effects of strain rate. It is formulated in an explicit form,compatible with the use of an explicit algorithm.

Numerical simulations for the CAMUS 2001 mock-up

A modal analysis is first carried out in order to check the performance of the model. As it wasthe case for the fibre model, the material parameters are issued from tests performed on concreteand re-bar samples. The calibration of boundary conditions (footing-support interface) comesfrom the measurement of the two first natural modes (5.5 and 6:00 Hz for the out-of-plane andthe in-plane flexure). Figure 7 shows that the results obtained for the other modes are similar tothe ones calculated with the fibre model.


J. MAZARS ET AL.870

Seven seismic signals of increasing amplitude have been applied during the experimentalcampaign. Among them, only those corresponding to an intensity of 0.15, 0.4 and 0:55g areconsidered in the numerical study. In spite of the simplified approach adopted, numerical andexperimental data are in good agreement both at global and at local level. Displacementtime histories obtained for 0.4 and 0:55g in the direction parallel to direction of the walls ofthe CAMUS 2000-1 specimen are shown in Figure 12. A satisfactory agreement betweenexperimental and numerical axial load and moment time histories can also be observed for 0:55gin Figure 13 and Table V (maximum values). These results are also comparable to thoseobtained by the fibre implicit model (Table III) which confirm the good performance of thebeam explicit model.

A complex state of stress inside the walls caused by the combination of bi-directional flexureand shear}similar to that observed during the tests}is also highlighted by the numericalanalyses. PRM model allows studying the variation of damage at different points of the section.For example, the damage concentrated at the point TR of the left wall is plotted in Figure 14ð0:55gÞ: The effects of tension and compression are considered separately. It can be observedthat the damage due to tension is important at the base of the wall and decreases in the upperstories. On the contrary, compression causes limited damage only in the first storey, close to thebasement and to the connection with the slab of the first storey. That is where the maximum

25.00

20.00

15.00

10.00

5.00

0.00

-5.00

-10.00

-15.00

-20.00

20.00

15.00

10.00

5.00

0.00

-5.00

-10.00

-15.00

10 2 3 4 5 6 7

Time [sec]

10 2 3 4 5 6 7

Time [sec]

Dis

p [m

m]

Dis

p [m

m]

(b)

(a)

Figure 12. CAMUS 2000-1 beam model: comparison between calculated (dotted line) and measured (solidline) horizontal top in plane displacements-right wall: (a) 0:4g; and (b) 0:55g:



Mo

men

t [k

Nm

]

300.0

200.0

100.0

0.0

-100.0

-200.0

-300.0

600.0

400.0

200.0

0.0

-200.0

-400.0

-600.0

0 1 2 3 4 5 6 7

Time [sec]

0 1 2 3 4 5 6 7

Time [sec]

Fo

rce

[kN

]

(a)

(b)

Figure 13. CAMUS 2000-1 beam model: comparison between calculated (dotted line) and measured (solidline): (a) axial load in the right wall; and (b) global-moment at the base of the walls for 0:55g:

Table V. CAMUS 2000-1 beam model: numerical versus experimental results.

0:40g 0:55g

PGA Comparisons Comp. Exp. Comp. Exp.

Left Wall 9.2 13.2 16.5 18.7in-planeDisp. (mm)Right wall 8.1 16.1 15.4 18.3in-planeDisp. (mm)Out of plane +8.9 +12.8 +11.6 +27.9Disp. (mm) �7.5 �14.3 �9.9 �21.5In-plane 393 362 445 473Moment ðkN mÞLeft wall +165 } +135 +79vertical load (kN) �318 } �363 �467


J. MAZARS ET AL.872

damage occurred during the tests. We can therefore conclude that the simplified approachadopted using a model based on damage mechanics associated to an explicit scheme ofresolution is adequate to simulate the response of the CAMUS 2000-1 specimen subjected to aseries of severe ground motions.

CONCLUSIONS

The dynamic behaviour of two R/C wall buildings tested on a shaking table during the CAMUS2000 experimental research programme was simulated using two different simplified modellingstrategies (a fibre and a beam model) and two resolution schemes (implicit and explicit,respectively). The constitutive relationships used for the materials were based on damagemechanics for concrete and plasticity for steel.

Damage

5.0

4.0

3.0

2.0

1.0

0.0

H [m

]

5.0

4.0

3.0

2.0

1.0

0.0

H [m

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Damage

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

TL

BL

TR

BR

Y

(a)

(b)

Figure 14. CAMUS 2000-1 beam model: damage evolution at the point TR along the left wall: (a) due tocompression; and (b) due to tension.



Both models were able to reproduce correctly the global response of the structures in terms ofmaximum values and frequency content. Local phenomena (e.g. elongation of reinforcementbars, concentration of damage) were also qualitatively well simulated. However, in order toreproduce quantitatively all the local phenomena a 3D refined finite element model has to beadopted. A simplified modelling strategy can predict only the general trend of the localindicators. This limited accuracy characterizing the results obtained with simplified models isbalanced by their reduced computational cost. This aspect takes a fundamental importancewhen there is a need for parametric studies or vulnerability analyses [16].

Research in progress concerns the validation of the two proposed strategies with a new R/Cwall specimen that is going to be tested at the shaking table facility of LNEC in Portugal.Instead of a steel bracing system, a more realistic wall with openings is used this time to providestiffness in the transversal direction. Finally, research in also undertaken to verify the efficiencyof explicit methods and the PRM model for dynamic problems of different nature as the effectsof impact of blocks on R/C slabs [17].

REFERENCES

1. Mazars J. French advanced research on structural walls: an overview on recent seismic programs. Proceedings of the11th European Conference on Earthquake Engineering, Invited Lectures, Paris, CD-ROM, 1998.

2. Bisch P, Coin A. The ‘CAMUS 2000’ research. Proceedings of the 12th European Conference on EarthquakeEngineering, London, CD-ROM, 2002.

3. PS 92. R "eegles de construction parasismique. DTU R"eegles PS92, AFNOR, 1995.4. Kotronis P, Davenne L, Mazars J. Poutre 3D multifibre Timoshenko pour la mod!eelisation des structures en b!eeton

arm!ee soumises "aa des chargements s!eev"eeres. Revue Fran-caise de G !eenie Civil, 2004, accepted for publication.

5. Guedes J, P!eegon P, Pinto A. A fibre Timoshenko Beam Element in CASTEM 2000. Special publication No. I.94.31,JRC, I-21020 Ispra: Italy, 1994.

6. Lemaitre J, Chaboche JL. Mechanics of Solids Materials. Cambridge University Press: Cambridge, 1990.7. Armstrong PJ, Frederick CO. A mathematical representation of the multiaxial bauschinger effect. G.E.G.B. Report

RD/B/N 731, 1966.8. La Borderie Ch. Ph!eenom"eenes unilat!eeraux dans un mat!eeriau endommageable: mod!eelisation et application "aa l’analyse

de structures en b!eeton. Ph.D. Thesis, University of Paris 6, 1991.9. Dub!ee JF, Pijaudier-Cabot G, La Borderie Ch. Rate dependent damage model for concrete in dynamics. Journal of

Engineering Mechanics 1996; 122(10).10. Caughey T. Classical normal modes in damped linear systems. Journal of Applied Mechanics 1960; 27:269–271.11. Mazars J, Ragueneau F. Ultimate behavior of R/C bearing walls: experiment and modelling. In ASCE Commitee,

Reports, Modeling of Inelastic Behavior of RC Structures Under Seismic Loads, Shing PB, Tanab!ee T (eds). ISBN0-7844-0553-0: 2001; 454–470.

12. Casaux G, Ragueneau F, Mazars J. Multifibre beam element in dynamics: torsional response properties. In VIIInternational Conference on Computational plasticity, COMPLAS 2003, Onate E, Owen JR (eds). CIMNE:Barcelona, 2003.

13. Mazars J. A description of micro- and macroscale damage of concrete structures. Engineering Fracture Mechanics1986; 25(5/6).

14. Rouquand A, Pontiroli C. Some considerations on implicit damage models including crack closure effects andanisotropic behaviour. In Proceedings FRAMCOS-2, Wittmann FH (ed). AEDIFICATIO Publisher: Freiburg,1995.

15. Rouquand A, Mazars J. Mod"eele incluant endommagement et dissipation hysteretique coupl!eee. Internal report CEG-DGA Gramat, France, 2001.

16. Negro P, Colombo A. How reliable are global computer models? Correlation with large-scale tests. EarthquakeSpectra 1998; 14(3):441–467.

17. Mazars J, Berthet-Rambaud Ph, Daudeville L, Nicot Fr. Rockfall protection: Impact effects on structures. Analysisand modeling. In ISRM International Symposium on Rock Engineering for Mountain Regions. Funchal, November25–28th, Dinis da Gama C, Ribeiro e Sousa L (eds). Sociedade Portuguesa de Geotecnia, 2002.


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Numerical modelling for earthquake engineering: the case of lightly RC structural walls