Shaking table testing and numerical simulation of the seismic response of a typical Mexican colonial...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS

Shaking table testing and numerical simulation of the seismicresponse of a typical Mexican colonial temple

Marcos Chávez and Roberto Meli∗,†

Institute of Engineering, National University of Mexico (UNAM), Ciudad Universitaria,Coyoacán 04510, D.F., México

SUMMARY

A shaking table testing program was undertaken with the main objective of providing basic informationfor the calibration of analytical models, and procedures for determining seismic response of typical stonemasonry temples of the 16–18th centuries stone masonry construction in Mexico. A typical colonialtemple was chosen as a prototype. A model at a 1:8 geometric scale was built with the same materialsand techniques as the prototype, and was subjected to horizontal and vertical motions of increasingintensities. The maximum applied intensity corresponded to a base shear force of about 58% of the totalbuilding weight. Vertical component of the base motion significantly affected the response and increasedthe damage of the model. Damage patterns were similar to those observed in actual temples. Dampingcoefficients of the response ranged from 7% for undamaged state, reached about 14% for severe damage.

The main features of the measured response were compared with those computed using a nonlinear,finite element model; for the latter, a constitutive law developed for plain concrete was adopted forreproducing cracking and crushing of the irregular stone masonry. Observed damage patterns as well asmeasured response could be reproduced with reasonable accuracy by the analytical simulation, except forsome local vibrations, as those at the top of the bell towers. It can be concluded that the simple constitutivelaw adopted for the simulation was able to reproduce the experimental response with reasonable level ofaccuracy. Copyright � 2011 John Wiley & Sons, Ltd.

Received 9 August 2010; Revised 21 December 2010; Accepted 26 January 2011

KEY WORDS: shaking table test; seismic response; stone masonry; nonlinear analyses; damping

1. INTRODUCTION

In the recent years, great advances in the development of analytical tools for nonlinear analysisof large and complex historical masonry structures have been made; nevertheless, a thoroughvalidation is still lacking regarding the reliability of their results in terms of capability for predictingthe seismic intensities corresponding to different levels of performance, including collapse. Maindoubts concern the validity of constitutive laws of the materials and accuracy in the representationof effects of structural damage. With the main objective of acquiring basic information for thecalibration of analytical models, a shaking table testing program was undertaken at the Instituteof Engineering of UNAM, pursuing the main objective of providing basic information for thecalibration of analytical models for determining the seismic response of ancient stone masonrystructures. Additionally, it is aimed at a better understanding of the seismic response and of the

∗Correspondence to: Roberto Meli, Institute of Engineering, National University of Mexico (UNAM), CiudadUniversitaria, Coyoacán 04510, D.F., México.

†E-mail: [email protected]

Copyright � 2011 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2012; 41:233–253Published online 23 May 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1127

M. CHÁVEZ AND R. MELI

modes of failure of these buildings, as well as at evaluating the efficacy of some rehabilitationtechniques, commonly used for reducing their vulnerability to severe earthquakes.

A typical colonial temple was chosen as a prototype. Owing to the size of the shaking table,and to the maximum weight it can carry, the model ought to be scaled at a 1:8 geometric scale,for which the same constituting materials and construction techniques of the prototype could stillbe used.

Summarized in this paper are the results of the first stage of this program, which consisted ofthe design of the experiment, the construction of the model, and its testing under increasing levelsof seismic intensities. Maximum intensity applied in this stage corresponded to the onset of thecritical modes of failure, and to a level of damage that was feasible to be repaired with traditionalrehabilitation techniques.

From the experimental results, main dynamic properties of the model were derived, and theirvariation with the level of damage was studied. The response of the model temple was thensimulated, first in its undamaged state through linear elastic finite element models, and then fordifferent levels of damage, through nonlinear models.

2. DESCRIPTION OF THE TEST SPECIMEN

The temple selected as a prototype is typical of central Mexico. This being a region where moderateseismic activity is rather frequent, but where very strong earthquakes had not been experiencedin colonial times, religious architecture is not as robust as in areas that are repeatedly struck bysevere earthquakes. The temple is tall and has light buttressing (Figure 1(a)).

In order to limit the scale factor of the model and to ease its construction, the prototype wassimplified by eliminating two bays, including the transept and its dome, but preserving all othercharacteristics (Figure 1(b)). The model to be tested in the shaking table was reduced at a scale1:8 (Figure 2) to fit the 4×4m shaking table of the Institute of Engineering at UNAM. The modelwas built with the same materials as the prototype, in order to reproduce as close as possible itsmechanical properties and the expected modes of failure.

In such case, to meet the requirements of the laws of similarity [1], the materials should have amass density eight times greater than the prototype. This condition cannot be met without alteringthe mechanical properties. One possible solution is the addition of properly distributed masses, toincrease the total mass of the model in the desired ratio; however, in this case the additional massrequired could not be placed without affecting the dynamic properties of the model; moreover, thetotal weight of the model would exceed the maximum allowed for operating the shaking table.

By reducing all dimensions by the scale factor and using the same materials as in the prototype,the dynamic behavior and stresses in the model could be reproduced, but stresses due to gravityloads would be smaller by one scale factor. It must be taken into account that in this kind ofstructures, the vertical compressive forces produced by their weight govern the resistance to out-of-plane bending, and also significantly increase the plane shear strength. In order to increase verticalstresses on the walls, even if marginally on the vault additional masses were placed on the vault

Figure 1. Prototype of the temple: (a) Prototype and (b) simplified prototype.

Copyright � 2011 John Wiley & Sons, Ltd.DOI: 10.1002/eqe

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Earthquake Engng Struct. Dyn. 2012; 41:233–253

SHAKING TABLE TESTING AND NUMERICAL SIMULATION

Figure 2. Model temple at 1:8 scale (m).

Figure 3. Basic construction materials of the model temple: (a) Stone masonry and (b) brick masonry.

to reach the maximum load capacity of the shaking table (2 t more). With this additional mass, themagnitude of vertical forces would equal those produced by a model built to a 1:5 scale. Giventhe above-mentioned considerations, the extrapolation to the prototype of the results of the scaleddown model should be limited to those parameters of the response that are not significantly affectedby the vertical compression due to gravity loads. Therefore, the model was designed and testedas a full-scale model of a miniature temple. The time scale of the seismic input was reduced bythe geometric scale factor and the acceleration was increased by this same factor. It is consideredthat the study is useful for the basic purpose of the investigation, which is to check the validityof analytical models for predicting response of actual structures of the same type and, also, toascertain some basic dynamic properties and features of their behavior under seismic actions.

3. CONSTRUCTION OF THE MODEL

For the walls, the basic construction material of these historical buildings is a masonry conglomeratecomposed of stones of different sizes, agglutinated by a lime–sand mortar (Figure 3(a)). Thisheterogeneous masonry can be considered as a kind of low-strength concrete. The vault and thebell towers were built with hand-made clay roof tiles, which were cut to the required size to scaledown a solid brick (Figure 3(b)).

The compressive strength and Young’s modulus of stone and brick masonry were obtainedfrom tests on small walls having a height-to-thickness ratio of 2 and 4, respectively. Diagonalcompression tests were performed on walls with a height-to-thickness ratio of 1.5 and 3, for the


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Figure 4. Flexural test.

stone and brick, respectively. Since most of the main failure modes of the masonry walls aregoverned by out-of-plane bending, the tensile strength of the masonry was obtained indirectly by abeam-bending test for each type of material. A schematic view of this test is shown in Figure 4. Thetensile strength obtained for stone masonry was found to be between 2 and 5% of the compressivestrength. The great variation in the ratio may be due to difficulties in obtaining perfectly flatand horizontal surfaces in the contact areas of the supports of the testing device, causing someuncontrolled restriction to the rotation at the beam. For brick masonry, the results were much moreuniform and the tensile strength was, on an average, 2% the compressive capacity. As it will beseen, masonry tensile strength plays an important role in determining the lateral load capacity ofthe model, according to the constitutive law adopted in this study.

Mechanical properties obtained from small walls are given in Table V [2].

4. MODEL INSTRUMENTATION AND TESTING PROGRAM

A profuse instrumentation array was installed on the model for monitoring. Twenty-three accelerom-eters were placed on the table, on the basis and on the roof of the model, as well as at differentheights of the towers. Vertical acceleration was measured at the vault center. Six displacementtransducers were placed on stiff frames inside the nave, to measure opening and closing of the vault.The total lateral displacements of the model were measured by attaching displacement transducersto two stiff steel frames placed outside the shaking table (Figure 5). The total weight of the modelwas 117 kN (11.96 t).

4.1. Seismic input

Acceleration time histories, called ‘Cal,’ applied to the table in the horizontal and vertical directionswere derived from a strong ground motion record obtained in 1985 near the epicenter of thegreat Michoacán earthquake (Ms 8.1). The acceleration scale was increased to reach almost thecapacity of the table. The time scale was adjusted to produce its highest spectral amplitudes inthe neighborhood of the fundamental period of vibration of the structure. The horizontal motionwas applied in the transverse direction of the model. Some modifications were made to the initialpart of the vertical record in order that its maximum acceleration occurs at the same time thatof the horizontal record. The resulting time histories and their acceleration response spectra for5% damping are shown in Figure 6. As it can be appreciated, the combination of accelerationhistories is very severe for the structure under study. The maximum acceleration is very highin both directions (near to 0.6 g). The duration of the strong motion is very large (about 50 s);the frequency content of both records gives rise to large spectral amplitudes, not only for thefundamental period of the structure corresponding to its undamaged state, but also for the longerperiods that could be expected once the structure is damaged by the earthquake.

The response to ambient vibration and to a small amplitude white-noise shaking was measuredbefore starting the tests with the selected ground motions. Initial natural periods of vibrationderived from these tests were consistently around 0.088 s.


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Figure 5. Model instrumentation.

Figure 6. Basic seismic input: (a) Acceleration time histories and (b) acceleration response spectra.

When initiating the seismic tests, a malfunctioning of the software controlling the actuators gaverise to a strong, abrupt shaking of the table that produced significant damage to parts of the modeltemple: diagonal cracking of the façade, flexural cracking on longitudinal wall and buttresses of theright side, and some dislocation of the vault. The model was carefully repaired, first restituting itsinitial geometry by tensioning external steel tie rods, which were later removed, and then injectingcracks with a lime grout. Fundamental period measured at the end of the process was close to thatof the undamaged model (0.08 vs 0.088 s).

The repair was intended to restore the original conditions of the model. The great similarityin the fundamental frequencies of the original and the repaired models shows that the purposewas nearly achieved. On the other hand, the slightly greater vibration frequency and some lack ofsymmetry in the response and in the patterns of damage showed that the strength and stiffness of theleft side of the temple, which was subject to more intense grout injection, were greater than thoseon the right side. This condition is not expected to have significantly altered the model response.

4.2. Behavior of the model during the seismic tests

Intensity of the selected seismic input was scaled down to different fractions of the referencerecords, starting with 5% and increasing it up to 60%. For each level of intensity, first, onlythe horizontal component was applied, and then horizontal–vertical motions were simultaneously


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Table I. Steps of the test program and observed damage.

Maximum base

Step Acceleration (g) Disp. (mm)Applied

# % components H V H V Damage description

1 5 H 0.05 0.01 0.01 0.99 None

2 5 H and V 0.05 0.05 0.05 0.98 None

3 10 H 0.07 0.01 0.01 2.04 None

4 10 H and V 0.09 0.08 0.08 2.04 None

5 20 H 0.15 0.02 0.02 4.15 Small longitudinal cracks in the vault

6 40 H 0.35 0.05 8.43 Failure at bottom of the piers of theupper body of the bell tower

7 40 H and V 0.35 0.28 8.41 4.96 Cracking of the left longitudinal wall,especially near the façade. Increase ofcracking of the vault

8 60 H 0.47 0.14 12.8 Collapse of upper part of both belltowers; upper part of the left towerseparates from the façade; increase ofwall cracking, especially in the leftside; descent (5mm) of the left frontalpart of de vault, the lateral drift ofthe nave, produced flexural cracks atthe base of the longitudinal wall andtheir buttresses, as well as longitudinalcracking of the right side of vault at itsgroin

9 60 H and V 0.44 0.36 13 7.49 Collapse of the lower part of the left belltower, as a result of its lateral vibrationand diagonal cracking of the façade,favored by the openings for windowsand door

H: Horizontal; V: Vertical.

Figure 7. Points where the relative displacements were measured.

induced. The description of tests performed and the maximum acceleration applied at the shakingtable for each intensity level are shown in Table I.

The main damages at the end of the test program are showed in Figure 19. The damage wasgreater on the left side of the temple, presumably because the right side had been more intenselygrouted after the previously mentioned accidental damage suffered by the model.

4.3. Measured response

Maximum displacements measured at some relevant points (see Figure 7) are shown in TableII for different intensities of the imposed motion. In all cases displacements shown are relative


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Table II. Maximum measured displacements for different intensities (mm).

Intensity (%) Applied component TD-1 TD-2 TD-3 BI-1 BC-1 BD-1 BI-2 BC-2 BD-2

5 H 0.090 0.022 0.086 0.092 0.026 0.084HV 0.094 0.026 0.084 0.102 0.031 0.068

10 H 0.167 0.038 0.130 0.165 0.039 0.123HV 0.94 0.63 1.12 0.169 0.031 0.149 0.165 0.036 0.135

20 H 1.18 1.21 2.18 0.411 0.069 0.296 0.420 0.079 0.28840 H 3.34 3.70 8.50 1.481 0.289 0.955 1.506 0.322 0.970

HV 7.00 7.68 7.70 1.745 0.367 1.090 1.755 0.401 1.09760 H 3.99 6.09 19.40 4.809 0.866 2.890 4.796 0.961 2.890

HV 9.68 5.890 1.201 3.132 5.761 1.292 3.143

See Figure 7 for location of measurement points.

to the base of the table. As it can be appreciated, the displacements of the main body of thetemple exceeded 1mm only for the maximum applied intensity, for which the damage was rathersevere; they were significantly amplified at the top of the towers, though never exceeding a fewmillimeters. It is important to notice that the horizontal displacement at the center of the navewas about twice than those measured at each end of the nave, thus showing that the roof bendshorizontally producing greater displacements in its center part, which has less lateral stiffness thanthe two ends of the nave.

When the horizontal and vertical components of acceleration were simultaneously applied, themaximum lateral displacements at the top of the walls were almost 20% greater than those measuredwhen only the horizontal component was applied, principally on the keystone. The tower bell topreached a maximum displacement of 19mm before its collapse.

The lateral displacements recorded in the left side of vault (BI-1, BI-2) were, on the average,10% greater than those on the right side (points, BD-1, BD-2) for low intensities; this differenceincreased for greater intensity of the applied motion, reaching 84%. This may be a consequenceof the right wall having greater lateral stiffness, because it had received a greater amount than theleft wall of mortar injection to repair the damage caused by the unintentional shaking applied priorto the test.

Maximum accelerations recorded at the base and at the vault and towers tops, for differentintensities of the applied motion, are shown in Table III. As it can be seen, for 10% of the seismicinput the acceleration at the table level is amplified only 15% at the top of the vault, thus indicatingthat the temple as a whole vibrates almost as a rigid body. On the other hand, the amplificationreached 300% at the top of the tower, due to an appendix-like response of these members. Forincreasing intensities, the structural damage gave rise to a decreasing stiffness of the structureand to an increase in its energy dissipation. The two factors have counteracting effects on theamplification; thus, in some points of measurement, response increases as amplification increases,while other points show an opposite trend.

The effect of the vertical component of the motion on the keystone of the vault is shown inFigure 8. As it can be appreciated, the maximum vertical displacement at the keystone of the vaultis significantly greater when the vertical component of the base motion is added. This shows thatthe vertical motion applied at the base activates the mode of vertical vibration of the vault.

4.4. Variation of the fundamental period of vibration

Ratios of the spectral amplitudes of the motion measured at the roof, related to the one measuredat the shaking table allow an accurate determination of the fundamental modal frequency of thestructure and, with less accuracy, also of the frequencies for higher modes. Shown in Figure 9are Fourier spectral ratios between the motions at the roof and at the base, for different motionintensities; their peaks reveal the fundamental period of vibration. For increasing intensities of theapplied motion, the structure becomes more flexible because of the increased cracking; therefore,its natural frequencies decrease.


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Table III. Maximum measured accelerations for different intensities of applied horizontal motion (g).

Bell tower Amplification relative to the base

Vault Base Top Vault Bell tower base Bell tower topIntensity Maximum% acceleration H HV H HV H HV H HV H HV H HV

5 0.05 0.06 0.07 0.06 0.05 0.17 0.22 1.15 0.93 1.17 1.27 3.25 4.0310 0.07 0.13 0.11 0.12 0.10 0.35 0.23 1.78 1.15 1.60 1.28 4.90 2.6320 0.15 0.24 0.20 0.39 1.56 1.27 2.5440 0.35 0.54 0.55 0.46 0.54 1.12 1.36 1.53 1.53 1.30 1.54 3.20 3.8460 0.47 0.89 0.82 0.89 0.84 3.66 1.88 1.90 1.88 1.86 7.72

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5

Dis

pla

cem

ent

[mm

]

Maximum base acceleration [g]

BC-2-H BC-2-HV

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

Acc

eler

atio

n [

g]


BC-2-H-Hor BC-2-HV-HorBC-2-H-Ver BC-2-HV-Ver

H: Base input horizontal; Hor: horizontal direction; HV: Base input vertical; Vert: vertical direction

Figure 8. Effect of vertical component of table motion on the displacements and accelerations at the vault.

Figure 9. Ratios of Fourier’s spectral amplitudes measured at the top of the vault to those at the base.

Fundamental periods of vibration determined from the records of different tests are shownin Table IV; both the fundamental period related to the lateral drift of the nave and the periodcorresponding to the vertical vibration mode of the vault are reported in the table. During the test,the initial period remained constant until the base acceleration reached 0.05 g. For the maximumapplied intensity, the horizontal period increased 37.5% and the vertical one 75%.

4.5. Evaluation of the damping coefficient

Damping coefficients involved in the measured response of the structure were derived from theshape of the ratio of spectral accelerations recorded at the roof and at the base, near the peak atthe fundamental frequency of vibration of the structure. The procedure proposed by Rinawi andClough [3] was adopted, by which the theoretical transfer function of a single degree of freedom


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Table IV. Vibration periods derived from response to different intensities of the motion.

Period (s)

Maximum baseacceleration (g) Lateral Vertical

Condition ofTest the model H V H HV V

∗Computed 0.098 0.0223†White noise Initial 0.09Ambient vibration Damaged 0.14Ambient vibration Repaired 0.08 0.04White noise Repaired 0.08 — 0.045% 0.05 0.05 0.08 0.09 0.04

0.07

0.09

0.11

0.13

0.15

0 0.1 0.2 0.3 0.4 0.5

Per

iod

[s]


Lateral vibration

H

HV

10% 0.07 0.08 0.08 0.09 0.04

0.03

0.05

0.07

0.09

0 0.1 0.2 0.3 0.4 0.5

Per

iod

[s]

Maximun base acceleration [g]

Vertical vibration

Variation of vibration period with the intensity of motion

20% Repaired 0.15 0.09 —40% 0.35 0.28 0.10 0.11 0.0660% 0.47 0.36 0.13 0.14 0.08White noise Repaired 0.11 0.07

∗From linear analysis of FEM model.†Measured before accidental damage caused by failure of the control system.

system is fitted to the experimental shape. Computed damping coefficients are shown in Figure 10for different motion intensities. They start at about 7% for the response to a low-intensity motion,and reach about 14% for the highest applied motion, which caused significant structural damageto the model. The high damping ratios for low-intensity motions can be attributed to the plasticdeformation of the lime mortar. The extensive cracking that masonry members had suffered at thehighest intensities of the shaking was the cause of additional increase in damping. These resultscorrespond to those obtained by the strong motion network installed in the Mexico City Cathedral,where, from records of the response of different points of the structures to moderate earthquakes,damping coefficients between 6 and 12% were derived. Damping was found to depend on theintensity of the motion and on the severity of the cracking that the corresponding part of thebuilding had suffered due to severe differential settlements of its foundation [4].

The slight decrease in damping between the first and second level of intensity could be attributedto the consolidation of the mortar that produced an increase in the stiffness and a reduction inenergy dissipation of the material.

5. NONLINEAR SIMULATION OF THE RESPONSE OF THE EXPERIMENTAL MODEL

Numerical simulation of the model test was performed through a widely used finite elementprogram ANSYS and specifically its SOLID65 element [5], which allows reproducing crackingand crushing of the material, thus making it a commonly used simulation for studying rocks andunreinforced concrete.

This element is initially considered as isotropic with linear elastic behavior, and then it canundergo tensile cracking or crushing according to the following equation

F

fm−S�0 (1)

where F is a function of the principal stresses state (�xp,�yp,�zp), S is the failure surface expressedin terms of principal stresses, and five input parameters of the material, fm is the uniaxial compres-sive strength, �xp,�yp,�zp are the principal stresses.

Failure criteria are a simplification of those proposed by Willam and Warnke [6]. They areshown in Figure 11, in which failure surface is defined only in terms of uniaxial compressive and


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0

5

10

15

20

0% c

riti

cal

dam

pm

ig

Maximum acceleration base [g]

0.1 0.2 0.3 0.4 0.5

Figure 10. Damping derived from response to different intensities of the motion.

Uniaxial tensile strength Uniaxial compressive strength Biaxial compressive strength 1.20 Compressive strength for a state of biaxial compression superimposed on hydrostatic stress state.

1.45

Compressive strength for a state of uniaxial compression superimposed on hydrostatic stress state.

1.725

Figure 11. Concrete material and failure surface in principal stress.

Table V. Material properties used for the analysis [2].

Properties Stone masonry Brick masonry

Mass density (kg/m3) 1830 1580Young’s Modulus (MPa) 441 1177Poisson ratio 0.3 0.3Compression strength (MPa) 1.28 3.92Tension strength (MPa) 0.10 0.31

tensile strengths. A distributed cracking is assumed for the modification of the stiffness matrix.The details on the criterion for simulating cracking can be found in [5]. Additionally, plastificationcan be represented through the Drucker–Prager law.

Considering that the particular type of irregular stone masonry of the temple under study is akind of conglomerate with low anisotropic behavior, it can be expected that its response couldbe approximated by the above-described SOLID65 element, as proposed by Giordano [7]. Thebrick masonry used for the vaults and bell towers has a more definite anisotropic behavior thanthe stone masonry; nevertheless, it was modeled using the same constitutive law, considering thatthe small size of the bricks and the large number of joints would give rise to cracking patterns notsignificantly different from that of an isotropic material.

Material properties required for describing constitutive laws of this specific element are shownin Table V. These properties were obtained from the previously described tests on small masonrywalls, except for the tensile strength (ft) which did not correspond to the one experimentallymeasured, because with that strength, the model showed convergence problems since low-intensityseismic actions, and the cracking patterns neither correspond to those observed in experimentaltests nor to those common in buildings damaged by earthquakes. Similarly, in a study on nonlinearbehavior of vaulted stone masonry structures subjected to gravity loads, a much greater tensilestrength had to be used in order to reproduce observed modes of failure [8]. In this study, it wasfound that the nonlinear response of the model was very sensitive to the tensile strength assumed.Only by making ft =0.08 fm, the response of the model was close to the one measured in theexperimental tests (see Figure 12). The effect of the tensile strength of masonry on the response


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0

10

20

30

40

5 6 7 8 9 10 11

Sti

ffn

ess

[kN

/mm

]

Percentage of fm

Stiffness experimental =16 kN/mm

Figure 12. Calibration of the tension strength of the masonry.

seems to depend mostly on the specific constitutive law assumed in the SOLID65 finite modelof the ANSYS program, probably derived from the sharp decreasing branch of its post-crackingbehavior. More specific studies on this topic are required aiming at devising better constitutive lawsand testing procedure that most closely reproduce the out-of-plane bending of this kind of masonry.

Additional parameters to be defined are those controlling shear transfer along cracks. For opencrack, shear transfer coefficient (�t) has been found to range from 0.15 to 0.5 in several studieson reinforced concrete [9, 10] and on brick masonry [11, 12]. In this case, �t was assumed 0.6,since in stone masonry, cracks have rougher surfaces than in other types of masonry. For closedcracks, shear transfer coefficient (�c) was taken as 0.8, based on studies by Hemmaty [13] onthe modeling of the shear forces between cracks in reinforced concrete. After cracking, a certainamount of stress relaxation can be included through the tensile crack factor (Tc), which in thiscase was taken as 0.7.

5.1. Modal elastic analysis

Eigenvalue analyses of the model show that, even if largest part of the participating mass corre-sponds to the fundamental mode of vibration, participation of very high modes cannot be disre-garded; this can be attributed to the complex geometry and distribution of mass of this structure.Displacements of the modes with greatest participating mass are shown in Figure 13. Fundamentalperiods of vibration are 0.082 and 0.046 s, for transverse and vertical directions, respectively; thesevalues are very close to those derived from the shaking table test for the linear range of response,0.088 and 0.040 s, respectively).

5.2. Time history nonlinear analysis

For a nonlinear time history analysis, a damping matrix must be defined in order to reproducedamping variation as the stiffness matrix changes. ANSYS does it through the Rayleigh damping.

For our purposes, modal damping coefficients were derived from white-noise tests performedbefore initiating the seismic tests (see Figure 14). The two coefficients were computed for thefundamental modes corresponding to lateral drift and to vertical vibration. Nonlinear analyses wereperformed for the acceleration histories recorded at the base of the shaking table for four tests:Cal 40% H, Cal 40% HV, Cal 60% H, and Cal 60% HV.

6. RESULTS OF NUMERICAL ANALYSES

6.1. Damage patterns

Damage patterns obtained from the numerical simulation for the simultaneous application of hori-zontal and vertical motions at 60% of the basic intensity (Cal HV 60%) are shown in Figure 19(a).When compared with the damage observed in the experimental model for the same seismic input,the damage in the numerical model looks much more diffused and distributed in wide strips,instead of a few large cracks, as in the experimental model. It must be taken into account that


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Modal shape 1 lateral Modal shape 4 vertical

P = 0.082 s P = 0.046 s

Figure 13. Main modal shapes.

0

0.1

0.2

0 200 400 600 800

ω (rad/s)

α dampingβ damping

(ξ , )

2

(ξ , )

Figure 14. Parameters for computing Rayleigh damping.

Figure 15. Fourier’s spectral of motion at vault and bell tower tops (measured vs computed):(a) Vault and (b) bell tower top.

small cracks were difficult to appreciate in the experimental model because most of them appearin the stone–mortar interface, and they close after the shaking when the model could be inspected.Despite this difference it can be said that the main patterns of damage are similar in both cases.


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Table VI. Vibration periods of the nave (s).

Fourier’s spectrum Transfer functions vault/base

Vault Bell tower top Transverse VerticalIntensity Maximum base(%) acceleration (g) Exp. Calculated Exp. Calculated Exp. Calculated Exp. Calculated

Initial state of the models 0.08 0.082 0.04 0.04640 H H=0.33 0.125 0.128 0.123 0.128 0.105 0.11440 HV V=0.26 0.127 0.128 0.127 0.128 0.111 0.11260 H H=0.45 0.136 0.155 0.137 0.157 0.130 0.14160 HV V=0.37 0.157 0.171 0.157∗ 0.205∗ 0.141 0.141

*Bell tower base.

Table VII. Comparison of measured and computed maximum base shear and average stiffness.

Base force shear (kN) Seismic coefficient (g) Average stiffness (kN/mm)Intensity Maximum base(%) acceleration (g) Exp. Calculated Linear Exp. Calculated Linear Exp. Calculated Linear

40 H H=0.33; −43.08 −41.98 0.35 0.34 16.3 21.3 44.2640 HV V=0.26 −51.74 −42.70 −46.5 0.41 0.34 0.36 18.0 23.060 H H=0.45; −68.14 −52.81 0.54 0.42 11.1 13.460 HV V=0.37 −72.10 −52.38 0.58 0.42 8.5 10.5

6.2. Computed modal periods

Measured and computed Fourier spectra are compared in Figure 15 for the center of the vault andfor the top of the bell tower. As it can be appreciated, despite some differences in the spectralamplitude, the frequencies of the peaks corresponding to the natural modes are quite similar. Thevariation of the natural frequencies was studied through the comparison of the spectral ratiosbetween accelerations at the center of the vault and at the base of the table. The results aresummarized in Table VI. Numerical simulations could closely reproduce the highest frequencies,as well as their reduction when horizontal and vertical motions are simultaneously applied.

In most cases, the fundamental vibration period of the experimental model increased whenhorizontal and vertical components of acceleration were simultaneously applied, in comparison towhen only the horizontal component was applied. The horizontal vibration periods obtained forthe analytical and experimental models were very close, 0.114 and 0.105 s, respectively for thetest with 40% of the maximum intensity.

6.3. Base shear force and hysteretic behavior

Base shear force is a good indicator of the total equivalent lateral force induced in a structureduring an earthquake. The hysteretic behavior of the models, the softening of the stiffness, andof the energy dissipation capacity were studied by means of curves relating lateral displacementsat the base of the vaults to base shear forces induced in each time interval. For the experimentalmodel, the base shear force was estimated by dividing the temple into several horizontal stripsand assigning to each of them the acceleration acting at its center of mass, as estimated byinterpolating the readings of accelerometers placed at the closest points. For the analytical model,the base shear was derived from the inertia force acting at each finite element. Maximum valuesof base shear forces obtained in this way for the measured and for the computed accelerationhistories are shown in Table VII, along with the corresponding seismic coefficients obtained bydividing the base shear force by the total mass of the structure. As it can be appreciated, computedmaximum shear force on the model was close to 0.42 times its weight, 28% smaller than theexperimental one. Experimental maximum base shear was greater when the horizontal and verticalcomponents were applied simultaneously, than when only the horizontal component is applied,thus explaining the greater damage of the model when submitted to the combined effects of the two


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Figure 16. Measured and computed base shear force–displacement relationship.

Table VIII. Percentage of critical damping.

Calculated Experimental

Intensity (%) Maximum acceleration base (g) � (%) P (s) � (%) P (s)

10 H 0.07 7.4 0.08220 H 0.15 7.7 0.08940 H 0.35 17 0.11 10.7 0.10240 HV 16 0.1160 H 0.47 18 0.13 13.9 0.11660 HV 19 0.14

components. On the other hand, in the analytical models, the difference between these two cases wasminimal.

Measured and computed base shear versus lateral displacement relationships are shown inFigure 16, for the four tests considered. Measured curves show a greater variability than thecomputed ones, but a reasonable similarity between the two groups can be observed. A straight linewas adjusted by mean squares to the points of each graph, thus defining an average stiffness, whichis reported in Table VII. The analytical model is in general terms stiffer than the experimental one,and in all cases, a progressive reduction of the stiffness can be observed as the intensity of seismicinput increases. The computed stiffness of the model was 30% greater than the experimental one,and 50% smaller than the one obtained with a linear analysis. The loss of stiffness due to nonlinearbehavior of the masonry, and to the structural damage increases with the intensity of the appliedbase motion, as it can be observed in the results of the analytical models.

6.4. Damping coefficient

The percentage of critical damping computed by the previously described Rinawi method is shownin Table VIII. The minimum damping coefficient for 40% H of the reference intensity was 17%, thatis, greater by 58% than the corresponding experimental value. The maximum damping coefficientfor an intensity of 60% H was 18%, that is, greater by 30% than the corresponding experimentalvalue. This indicates that the amount of energy dissipated by nonlinear deformation was greaterin the analytical model than in the experimental model. On the other hand, the area enclosed by


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Figure 17. Comparison between calculated and measured responses.

Table IX. Maximum and minimum accelerations on vault and bell tower top andamplification of the acceleration.

Acceleration (g) Amplification of the acceleration

Vault Bell tower top Vault Bell tower topIntensity(%) Value Exp. Calculated Linear Exp. Calculated Linear Exp. Calculated Linear Exp. Calculated Linear

40 H Max 0.46 0.34 1.00 2.01 1.5 1.3 3.3 6.3Min −0.47 −0.39 −1.09 −1.05

40 HV Max 0.53 0.39 0.40 1.31 2.00 0.69 1.5 1.4 1.7 3.9 6 2.4Min −0.49 −0.35 −0.41−1.06 −1.10 −0.86

60 H Max 0.73 0.41 1.769 2.396 1.9 1.3 7.8 7Min −0.76 −0.49 −3.765 −3.077

60 HV Max 0.8 0.43 1.64 1.75 1.9 1.3 2 3.9Min −0.7 −0.49 −2.98 −0.65

the hysteretic loops in the analytical model was smaller than that obtained from the response ofthe experimental model (see Figure 16). The two results are contradictory; therefore, the topicdeserves further interpretation.

6.5. Accelerations and displacements

Measured and computed acceleration time histories for the four selected tests are shown inFigure 17. Maximum values are reported in Table IX. Despite the reasonable similarity amongthe graphs, accelerations computed at the vault top are consistently smaller than those measured,while at the tower top the opposite trend is found. The differences could be attributed to anover-estimation of the stiffness in the analytical model towers.

When horizontal and vertical components of acceleration were simultaneously applied, theacceleration amplification relative to the base in the vault was greater than when only the horizontalcomponent was applied. This behavior was also observed experimentally. The average amplificationof the acceleration in the vault was 1.9 experimentally and 1.3 analytically. At the bell towers


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Table X. Maximum and minimum displacements on vault and bell tower top.

Vault (mm) Bell tower top (mm)

Intensity (%) Value Exp Calculated Linear Exp Calculated Linear

40 H Max 1.12 1.22 −10.12 3.41Min −1.74 −2.12 −10.12 −2.16

40 HV Max 1.68 1.75 0.88 7.86 2.42 1.53Min −1.84 −1.05 −0.94 −10.09 −3.49 −1.55

60 H Max 3.11 2.11 25.15 12.88Min −5.83 −5.16 −26.17 −12.653

60 HV Max 3.8 2.36 5.03 4.36Min −6.6 −5.68 −5.60 −1.59

Figure 18. Basic seismic input for prototype.

top, the acceleration amplification relative to the base was 3.9 experimentally and 6.0 analyticallyunder small intensities of motion.

Maximum displacements did not exceed a few millimeters, even at the top of the towers ascan be observed in Table X. This is mainly due to the high frequencies of the applied motion.Similarity among measured and computed displacements is poorer than for acceleration timehistories, especially regarding the motion at the tower top.

6.6. Comparison of the computed response of the prototype and of its scaled down models

In addition to the simulation of the experimental models, their prototype was also modeled andanalyzed, with the main objective of comparing its response with that of the experimental and ofthe scaled down analytical model. The same material properties were used in the full scale and inthe scaled down model.

The acceleration histories applied to the scaled down model were modified in their accelerationand time scales to comply with the similitude laws (see Figure 18). An additional adjustment ofthe time scale was made to achieve that the highest spectral ordinates correspond to the respectivefundamental periods of vibration of the full-scale model. Analyses were performed for 60 and100% of the base intensity; both components of acceleration were applied simultaneously.

The damage caused by the seismic input is shown in Figure 20. The façade showed diagonalcracking in an opposite direction to that observed in the experimental model, indicating that thetowers separated from the façade. The differences in the pattern of damage are attributed to thegreater vertical compressive stresses that increased both the shear and the flexural strength of theprototype walls. Cracking of the walls concentrated on the openings, as in the experimental model,although the prototype did not show any horizontal cracks at the base of the walls, as it happenedin the scaled model.

The vibration periods of the prototype, shown in Table XI, were eight times greater than those ofthe scaled model, as expected from the similitude laws. The modal shapes of the prototype and theexperimental model were similar for the modes with more participation of mass in the transverseand vertical directions. The fundamental periods of vibration (Figure 21) and the hysteretic behavior


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Figure 19. Crack patterns in the model: (a) Measured and (b) calculated.

of the full-scale model were similar to those observed in the model. Hysteresis loops did not showthe same pinching effect as those of the scaled down model. The stiffness degradation was smallerthan that observed in the scaled model (Figure 22).

Table XI shows some of the differences observed between the response of the prototype andthat of the scaled model; they can be summarized as follows:

• The final stiffness of the scaled model decreased 54%, while that of the prototype only 12%.• The maximum percentage of critical damping derived from response of the scaled model was19%, and that of the prototype was 8.2% despite the heavy damage suffered.


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Figure 20. Crack patterns in the prototype.

Table XI. Comparison between the response of the prototype and of the model.

Maximumbase

accelera-tion

AveragestiffnesskN/mm � (%) Drift (%)

Periodtransversal

(s)

Amplificationof the

acceleration

ModelCondition ofthe model (%) (g) Exp. Calc. Exp. Calc. Exp. Calc. Exp. Calc. Exp. Calc.

MVTEMP 01 Initial 0.082Low damage 40 0.35 18 23 10.7 16 0.115 0.11 0.111 0.112 1.5 1.4Several damage 60 0.47 8.5 10.5 13.9 19 0.405 0.355 0.141 0.141 1.9 1.3

Prototype Initial 0.61Low damage 60 0.054 165 7.6 0.12 0.719 2.5Several damage 100 0.09 147 8.2 0.19 0.833 3.1

• The distortion corresponding to severe damages was 0.35% in the scaled model, while thatof the prototype was 0.19 %.

• The acceleration amplification in the vault of the scaled model was 1.3, while in the prototypeit was 3.1.

The above described results demonstrate that due to the inadequate reproduction of verticalstress in the scaled down model, the behavior of the facade was different from that obtained in theprototype. However, for other modes of failure of walls and vault, the differences were minimal.


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Figure 21. Ratios of Fourier’s spectra and transfer functions of prototype for 60% HV seismic input:(a) Ratios of Fourier spectra and (b) transfer function.

Figure 22. Computed base shear force–displacement relationship of prototype for 60% HV seismic input:(a) MVTEMP-01 and (b) prototype.

The displacements and the accelerations do not correspond to those computed by multiplying thosemeasured experimentally by the corresponding scale factors; therefore, experimental response ofthe scaled down models in the nonlinear range cannot be directly extrapolated to the prototype.

7. CONCLUDING REMARKS

The experimental program presented in this paper provided valuable information about the seismicresponse of scaled model and it was useful to calibrate its analytical nonlinear model.

The motion applied to the shaking table was particularly severe for this type of building, becauseof its high spectral ordinates in a wide range of periods, covering the fundamental period ofvibration of the model in its initial and damaged states. Maximum induced accelerations reached0.89g at the roof and 3.66g at the towers tops, and the maximum base shear coefficient wasalmost 0.58.

The amplification of the acceleration at the roof was rather small, not exceeding 1.53 times ofthat measured at the base. The acceleration at the top of bell towers was almost three times theacceleration at the base for low-intensity base motions.

Damage patterns induced in the model corresponded to those observed in actual temples hitby intense earthquakes. Most vulnerable parts of the structure were: the two bell towers, whichshowed great amplifications due to their great flexibility; the main body of the towers, which isprone to separating from the façade and failing by shear; and the central part of the nave, whoselateral drift produces bending damage in walls and buttresses, as well as dislocation of the vault.It must be taken into account that modes of failure due to the ground motion in the longitudinaldirection of the temple were not reproduced in the test. The effect of the vertical component ofthe motion showed to be rather severe; therefore, buildings of this kind are particularly vulnerableto earthquakes with nearby epicenters, whose ground motion shows high vertical accelerations.


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Lateral displacements of the model did not exceed a few millimeters. It can be concluded that,due to their high stiffness and to their low period of vibration, structures of this kind suffer quitesmall displacements until a near to failure condition is reached. Nevertheless, significant nonlinearresponse appeared since intensities as low as 20% of the maximum applied in these tests.

Damping coefficients were above 5%, even for very low-intensity motions. This shows that thiskind of structure could count on a significant capacity of energy dissipation since early stages ofloading; this significant damping could be attributed to the presence of many small cavities betweenthe stone and the mortar, as well as to the great deformability of the mortar. Damping coefficientsreached 14% for high intensity shaking, thus indicating that damping constitutes a more importantsource of capacity for dissipating energy introduced by great earthquakes than for common modernbuildings, thus partially compensating the rather low ductility of their nonlinear behavior.

The vault did not act as a rigid diaphragm, as it was shown by the significantly differentlateral displacements recorded along the nave. Therefore, the inertia forces applied to each lateralload-resisting member cannot be computed in terms of its relative lateral stiffness. Only a tridi-mensional analysis considering the lateral stiffness of the roof could provide a close prediction ofthe distribution of lateral forces.

Despite the limitations of the selected finite element, nonlinear numerical simulation providedan acceptable prediction of the observed damage patterns, and of the measured response not onlyfor moderate ground shaking, but also for high intensity events producing severe damage. This isespecially true in terms of modal frequencies and of maximum horizontal and vertical accelerations.In terms of maximum lateral and horizontal displacements, differences between computed andmeasured results are more significant, but they could be still considered as acceptable for practicalpurposes. Prediction of observed response was better for the behavior of the nave than for that ofthe top of the bell towers.

The nonlinear response that could be expected in the prototype will show significant differencesfrom the observed in the experimental model, at least for some modes of failure. This is becauseit was not possible to fulfill all the requirements imposed by the laws of similitude, mainly thoseregarding the model mass. The compressive stresses due to gravity loads are much smaller in theexperimental model than in the prototype, thus causing, premature flexural and shear crackingin the former. However, the information obtained experimentally could be successfully used tocalibrate the parameters of the analytical model, which provided a reasonably close prediction ofthe experimental response. Therefore, it can be expected that nonlinear analyses as those performedin this research when applied on models of full-scale temples would give a similar degree ofreliability.

ACKNOWLEDGEMENTS

This paper was based on the research work performed by its first author, for attaining his Ph.D. degree bythe National University of Mexico (UNAM). The experimental work was carried out at the shaking tableof the Institute of Engineering of UNAM. The valuable and generous support of the technical personnelof this Laboratory and especially of its head, Roberto Durán, was decisive for the success of the project.

The many valuable contributions to this research by Roberto Sánchez, Fernando Pena and NataliaGarcia, all of them from the Institute of Engineering, are greatly appreciated.

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