IMPLICIT AND EXPLICIT TRANSIENT STRUCTURAL ANALYSIS OF ... · time-history analysis) were performed...

SAHC2014 – 9th International Conference on Structural Analysis of Historical Constructions

F. Peña & M. Chávez (eds.) Mexico City, Mexico, 14–17 October 2014

IMPLICIT AND EXPLICIT TRANSIENT STRUCTURAL ANALYSIS OF ANCIENT MASONRY CHURCHES IN SOUTHERN MEXICO

UNDER EARTHQUAKE LOADING

Marcos M. Chávez1, Hugo C. Gomez2, Tim Milot2, and Nikolay Doumbalsky2

1 Instituto de Ingeniería, Universidad Nacional Autónoma de México Circuito Escolar, Ciudad Universitaria, 04510 México City, México

[email protected]

2 MMI Engineering, Inc 1111 Broadway, Oakland, CA [email protected]

Keywords: Structural Analysis, Finite Element, Ancient Church, Earthquake Loading.

Abstract. The earthquake response of cultural heritage masonry churches located in the south of Mexico has been object of ongoing research during the last two decades. This paper aims to complement the research efforts made in order to understand and prevent the collapse mechanisms of ancient churches during seismic events. The paper presents a case study of the assessment of the seismic performance of a small and a large scale masonry church. The main objectives of the paper are (1) to develop computer models that represent the actual structural response with reasonable accuracy, (2) to verify the computer models by means of a comparison against their experimental and empirical counterpart, and (3) to define a fre-quency versus damage state relationship for condition monitoring. In order to achieve such objectives, an approach based on the formulation of computer models of the ancient churches is presented. Assessment of the seismic response is conducted by means of computer-aided techniques in combination with experimental findings from a small scale church model. The computer models are defined using the Finite Element software ANSYS® and LS-DYNA®. In the models the failure of the structure is initiated by tensile cracking of the walls perpendicu-lar to the shear walls. In conjunction, experimental results obtained from a shaking table test of a 1:8 scale model are used to verify the results from the structural analysis. Additionally, empirical findings on a full scale structure damaged during the 7.1 magnitude 1999 Tehuacan earthquake are used to further assess the accuracy of the computer models. The results are then used to describe empirical relationships between vibration frequencies and damage state. These relationships can be used for future condition assessment of the structure based on sen-sor data. Based on the results, the development of constitutive models that accurately repre-sent the masonry failure requires further research.

Marcos M. Chavez, Hugo C. Gomez, Tim Milot, and Nikolay Doumbalsky

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1 INTRODUCTION Finite Element (FE) models have been widely used to study the dynamic behavior and

seismic response of ancient masonry structures. With the advent of sophisticated FE software and more efficient personal computers, this approach has become the preferred approach to study the dynamic response of historical constructions such as the typical ancient churches located in the south of Mexico. These monuments are masonry structures that were built be-tween the 16th and 18th centuries. Given that this type of structures consists of massive thick elements, a common approach is to define a computer model that idealizes the structure as a finite number of solid elements connected at node locations. The connected elements define a mesh that shows how the elements are connected to form the structural system. Earthquake loads are applied to the model as input base accelerations. Then, the equation of motion is solved for the system and the response is computed at every node location in the mesh. In solving the equation of motion, different algorithms are available in the literature and imple-mented in software packages, both implicit and explicit methods have been applied to esti-mate the dynamic response of historical constructions [1-3]. Results have been proved to approximate the observed behavior of existing structures. This paper aims to complement this body of literature focused on providing insight on the seismic behavior of ancient churches by means of FE models. The main objectives of the paper are (1) to develop computer models that represent the actual structural response with reasonable accuracy, (2) to verify the com-puter models by means of a comparison against their experimental and empirical counterpart, and (3) to define a frequency versus damage state relationship for condition monitoring. The followed approach is to estimate the seismic response of a 1:8 scale laboratory test specimen using FE models that consider material and geometric nonlinearities. A comparison between implicit and explicit dynamic analysis is presented. Damage patterns are idealized as cracks that propagate through the finite elements. Experimental results from a shaking table test are compared against their analytical counterpart. In the shaking table test program the 1:8 scale church models was subjected to an earthquake with increasing intensity until it was consid-ered safe for the laboratory facilities [4]. To further evaluate the FE modeling techniques, damage patterns that were observed in an existing ancient church after the 7.1 magnitude 1999 Tehuacan earthquake are estimated using computer models. Structural frequencies are estimated using the acceleration response obtained from the FE analysis of the scaled model and relationships are defined that relates the natural frequency with the observed damage.

2 CONFIGURATION OF A TYPICAL MASONRY ANCIENT CHURCH IN SOUTHERN MEXICO

Typical ancient churches located in the south of Mexico are masonry structures that were built between the 16th and 18th centuries. A considerable number of these structures are locat-ed within an earthquake prone region in the states of Puebla and Oaxaca in the south of Mexi-co. In the state of Puebla, a common geometric configuration exhibits a cross shape in plan view (Figure 1a & b). Common dimensions for this configuration are a width of 20 m, length of 40 m and height of 18 m measured to the top of vaulted roofs. At the intersection of vaulted roofs, a hemispherical dome rests on top of an octagonal drum. In the front, either one or two bell towers are attached to the façade. The towers are commonly 20 to 30 m in height. A hem-ispherical dome is placed over the transept bay and it is supported by the drum. Small but-tresses are attached to the lateral walls to provide lateral resistance. On the other hand, in the state of Oaxaca earthquakes caused recurrent damage of cross shaped churches, which result-ed in more recent constructions of shorter heights, heavily buttressed walls, and limited deco-rations (Figure 1c).

Implicit and Explicit Transient Analysis of Ancient Churches in Southern Mexico Under Earthquake Loading

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Figure 1: Typical ancient masonry church construction in Southern Mexico: (a) Ajalpan, Puebla;

(b) aerial view, Ajalpan, Puebla; (c) San Miguel Ejutla, Oaxaca

3 OBSERVED SEISMIC BEHAVIOR For decades, numerous seismic events have induced damage patterns to the church struc-

tures located in Southern Mexico questioning their stability. One of the most significant seis-mic events occurred in 1999, when the 7.1 magnitude Tehuacan earthquake stroke the states of Puebla and Oaxaca damaging more than 1500 of the churches located within a radius of 100 km from the epicenter [5].

3.1 Earthquake damage in existing ancient church structures Earthquake damage patterns have been reported for existing ancient churches in the states

of Puebla and Oaxaca. Common patterns include tensile cracks developed through the con-stituent materials of the masonry. This is due to the low tensile resistance of the stone mason-ry with lime mortar joints. Another reason for the presence of multiple cracks is that these structures were conceived as compression only because reinforcing rebar was not available at the time they were built. Cracks concentrate vertically along the walls, across lower bodies of the towers, along meridians of the domes, and along the vaulted roofs. Due to excessive weight, the vaulted roofs tend to open at the end supports developing cracks in the interior of the vault at the middle span, and in the exterior at one and two thirds of the span. Another damage pattern is the partial collapse of the bell towers. This is because the towers behave as an appendix attached to the main structure of the church. Other damage patterns have been well documented in a vast literature collection [1-5].

3.2 Shaking table test The recurrent damage of ancient churches motivated a research program at the Engineering

Institute of the Universidad Nacional Autónoma de México to study the seismic response of a typical ancient church. A shaking table test was conducted where the test specimen was sub-jected to earthquakes of increasing intensity. The test specimen and its basic dimensions are shown in Figure 2. The specimen was built using constituent materials similar to those found in existing ancient churches. The material properties are listed in Table 1.

Table 1: Material properties of the shaking table test specimen [4].

Material Density (Kg/m3)

Young’s Modulus (KPa)

Compression capacity (KPa)

Tension Capacity (KPa)

Brick masonry 1.58 1.2 ×106 4 000 320 Stone masonry 1.83 4.5 ×105 1 300 104 Concrete 1.45 2 ×107 - -


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Figure 2: Shaking table test specimen: (a) 1:8 scale church; (b) front view; (c) elevation view; (d) plan view; (e) cross section. Dimensions in meters

The test program consisted of a total of 35 tests including ambient, white noise and earth-quake loadings. The program was divided into three stages. The first stage is object of study in this paper. In the second and third stages, the specimen was retrofitted and subjected to higher earthquake intensities. These stages were used to assess the effectiveness of the retrofit techniques. These are beyond the scope of this paper and are not presented herein. A detailed description of the shaking table test results can be found elsewhere [3, 4].

In the first stage of the test, the specimen was subjected to a maximum base acceleration of 0.44g in the transverse direction and 0.36g in the vertical direction. Figure 3 shows the earth-quake records applied to the test specimen. These were defined such that the test specimen frequencies fell within the interval of maximum acceleration of the response spectrum.

Figure 3: Shaking table test input base acceleration: (a) transverse direction; (b) transverse response spectrum

with 5% damping; (c) vertical direction; (d) vertical response spectrum with 5% damping


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Table 2: Shaking table summary of results

Intensity level

Base acceleration (g) Roof mid-span displacement (mm)

Top of right tower displacement (mm)

Identified structure frequency (Hz)

Transverse Vertical

White noise 0.03 0.03 - - Very low 0.05 0.05 0.03 - 11.65 Low 0.09 0.08 0.04 1.12 11.50 Moderate 0.35 0.28 0.40 7.7 9.03 Strong 0.44 0.36 1.29 - 7.09 Remarks: the symbol - denotes measurement is not available

4 TRANSIENT ANALYSIS The motions recorded by the control system of the shaking table were used to estimate the

seismic response of the test specimen using FE models. Transient analyses (also known as time-history analysis) were performed using both implicit and explicit methods. The software ANSYS® was used to compute the seismic response of the scaled church by means of an im-plicit method whereas LS-DYNA® was used to estimate the response by an explicit method.

4.1 FE models of the test specimen The finite elements of the generated computer model were defined as solid elements of ei-

ther eight or six nodes. Tetrahedral elements were avoided as the selected constitutive models are not recommended for this shape. The mesh was defined such that the elements have an aspect ratio less than 20 (see Figure 4). The model has a total of 29,201 elements. The model was generated in both ANSYS and LS-DYNA software for implicit and explicit analysis, re-spectively. The material properties assigned to the model are those listed in Table 1.

Figure 4: Finite element model of the church specimen

4.2 Dynamic characteristics Prior to the transient analysis, a modal analysis was performed to estimate the dynamic

characteristics of the model. Figure 5 shows the first three natural frequencies estimated with both ANSYS and LS-DYNA. Although the frequency values are consistent, there are small differences due to the different algorithms implemented in each software package.


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For the test specimen, acceleration data was processed to identify the structural natural fre-quencies. The identified frequency of the first mode was 11.65 Hz. The difference between the analytical and experimental results is attributed to the fact that the stone masonry possess-es variations in its properties across the entire structure, which is not represented in the model.

Figure 5: Natural frequencies of the church specimen FE model

4.3 Implicit analysis The FE model defined in ANSYS was used to estimate the seismic response by means of

an implicit method. The solid finite elements in this model were defined as nonlinear elements with a nonlocal smeared cracking constitutive model (SOLID65), which was originally devel-oped for the analysis of concrete structures [6]. In this model, the elements can develop crack-ing in tension (in three orthogonal directions), crushing in compression, and plastic deformation. The presence of a crack is represented through a modification of the stress-strain relations by introducing a plane of weakness or softening in a direction normal to the crack face (Figure 6). The stress normal to the crack direction is softened until it reaches a zero val-ue. Once cracks form, they can close and reopen but their directions remain fixed. Also, the shear strength is reduced for subsequent loads which induce sliding across the crack face.

Figure 6: SOLID65 stress-strain relation

In this analysis the response of the structure is governed by the equation of motion,

ft: uniaxial tensile cracking stress Tc: multiplier for tensile stress relaxation E: Young’s modulus Rt: secant Young’s modulus εck: strain at crack


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[M]�̈�𝑖 + [C]�̇�𝑖 + [K]𝑖𝑢𝑖 = −[M]𝜄�̈�𝑖𝑔 (1)

The solution requires computation of the tangent stiffness matrix at each time step 𝑖

�K��𝑖

= [K]𝑖 +2∆𝑡

[C] +4

(∆𝑡)2[M] (2)

A Newton-Raphson method is applied to solve for the response 𝑢𝑖+1 at time step 𝑖 + 1,

𝑢𝑖+1 = 𝑢𝑖 + ∆𝑢𝑖 (3)

Since the response at time 𝑖 + 1 is determined from the equilibrium condition at time 𝑖 + 1, this method is called an implicit method.

Estimation of the damping matrix [C] is also required. In this study, the procedure used to estimate the damping matrix was the so-called Rayleigh damping given by,

[C] = 𝛼[M] + 𝛽[K] (4)

In a previous study, the coefficients 𝛼 and 𝛽 were estimated based on equivalent damping ratios obtained from the shaking table test measurements [3, 4]. It was found the damping ra-tios for the scaled model oscillates between 8 and 14% of the critical value. These values de-pend on earthquake intensity and damage level. Rayleigh damping coefficients initially assigned to the ANSYS model were 𝛼=9.88 and 𝛽=0.0005. Since the frequency of the model decreases as cracks progressively increase, the corresponding damping ratio increases expo-nentially below 10 Hz (under the mass proportionally damping curve). In this area the mass proportional damping effectively damps a rigid body motion. This can result in the underes-timation of the structural response. For this reason, alternative 𝛼 and 𝛽 values were assigned to the model such that the assigned damping ratios were in the order of 8 to 14% for frequen-cies below 10 Hz. The results are compared for both damping cases later in the paper.

4.4 Explicit analysis The suitability of an explicit analysis to solve for the seismic response of the scaled model

was studied using the FE model defined in LS-DYNA. This is a general purpose finite ele-ment program for the analysis of large deformation static and dynamic response of structures. In this approach, the solid finite elements were defined as nonlinear elements with a basic plasticity constitutive model (Winfrith concrete model) [7]. The model includes strain soften-ing in tension. Tensile cracking with up to three orthogonal crack planes per element is avail-able. Once a crack is formed it can be closed and reopened. As the opening size increases, the length also increases and the work done in crack propagation is utilized to create a new crack surface.

In this analysis, the central difference method is used to solve Equation (1). Here, the struc-tural response is given by,

�K��𝑢𝑖+1 = �P��𝑖 (5)

Where

�K�� =[M]

(Δ𝑡)2−

[C]2Δ𝑡

(6)

�P��𝑖

= −[M]𝜄�̈�𝑖𝑔 − �[M]

(Δ𝑡)2−

[C]2Δ𝑡

� 𝑢𝑖−1 − [K]𝑖𝑢𝑖 + �2[M](Δ𝑡)2

� 𝑢𝑖 (7)


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The resisting force [K]𝑖𝑢𝑖 appears explicitly and it depends only on the response at time 𝑖,

not on the unknown response at time 𝑖 + 1. The implementation of Rayleigh damping in LS-DYNA is not well defined and other op-

tions are available to define damping. In this study, an approximately constant damping ratio was assumed through a frequency interval 1 Hz < 𝑓 < 70 Hz. Since damping ratios greater than 5% are not recommended in the selected algorithm, a conservative damping ratio of 5% was assigned to the LS-DYNA model.

4.5 Comparison of results The response of the scaled model computed by means of implicit and explicit analyses was

compared. To validate the accuracy of these, measured response from the shaking table test was also compared. Figure 7 shows a comparison of the acceleration time-history response at the top of the left tower and at the middle span of the vaulted roof. Rayleigh damping defined from measurements was considered in the implicit analysis whereas approximately 5% con-stant damping was used in the explicit analysis. Although the amplitude of the acceleration is similar between the two methods, the response from the implicit analysis shows narrower cy-cles. Conversely, the response from the explicit analysis shows wider cycles. The frequency of the church during the applied earthquake was estimated for each analysis using an Eigen-system Realization Algorithm (ERA) for system identification [8]. The frequency identified from the implicit analysis is approximately 7.6 Hz. As expected, this is greater than that iden-tified from the explicit analysis, which is approximately 2.7 Hz (Figure 8).

Figure 7: Acceleration response of the church specimen: (a) top of right tower;

(b) middle span of vaulted roof

Figure 8: Identified frequencies using ERA: (a) implicit analysis; (b) explicit analysis. The used symbols are: ‘ ’ for a stable pole; ‘.v’ for a pole with stable frequency and vector; ‘.d’ for a pole with stable frequency and damp-

ing; ‘.f’ for a pole with stable frequency, ‘.’ for a new pole and ‘- -’ for FRF function.

(a) (b)


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Figure 9 shows a comparison of the displacement response at the top of the left tower and at the middle span of the vaulted roof. Here, the response from the implicit analysis is given for two damping cases. A first case with high damping where Rayleigh damping was defined from measurements, and a second case with low damping where Rayleigh damping was as-sumed such that the damping ratios remained below 10% for structural frequencies in the in-terval 4 Hz < 𝑓 < 48 Hz. As it was expected, the low damping case of the implicit analysis approximates the explicit response because damping values are similar. However, with low damping the response from the implicit analysis “blows up” around 15 seconds. This is be-cause large deformations are experienced at the top body of the left tower and the analysis is terminated as it does not achieve convergence.

Figure 9: Displacement response of the church specimen: (a) top of right tower;

(b) middle span of vaulted roof

Figure 10 shows the hysteretic loops at the mid-span of the vaulted roof. The wider loops from the explicit analysis show the displacement of the structure can be overestimated when damping is assumed to be constant and approximately 5% of the critical.

Figure 10: Hysteretic loops at the mid-span roof

Figure 11 shows a comparison of crack plots generated in each model nearly the peak re-sponse at 18 seconds. The implicit analysis using high damping ratios (Figure 11a) shows most of the cracks concentrated in the façade and the lower body of the towers. Here, the first crack for an element at an integration point is shown in red color, the second crack is shown in green, and the third crack in blue. As expected, the assumption of low damping results in a considerably greater amount of cracks (Figure 11b). For low damping the deformed shape shows large displacements and cracks concentrated in the upper body of the left tower. This analysis was terminated at time 15 s due to excessive deformation on the left tower. In the


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case of the explicit analysis, a considerable amount of cracks is shown (Figure 11c). This is attributed to the conservative 5% damping assigned to the model.

Figure 11: Crack plots: (a) implicit analysis with high damping; (b) implicit analysis with low damping;

(c) explicit analysis approximately 5% damping


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Figure 12 shows a comparison between the response at the top of the tower and at the mid-dle span of the vaulted roof obtained with the high damping implicit analysis and that meas-ured during the shaking table test. The closeness between the time-histories suggests the assumption of high damping gives a better approximation to the actual response.

Figure 12: Comparison between computed and measured acceleration response; (a) top of the tower;

(b) middle span of the vaulted roof

An estimation of the ultimate displacement at the top of the left tower using video images is shown in Figure 13. Three time frames of the video were selected, a first frame was cap-tured at rest to represent the undeformed shape, a second and a third frame were captured be-fore and after the tower collapsed during the shaking table test at t≈13s and t≈21s, respectively. The estimated displacement at collapse, nearly 21 seconds, is approximately 4 cm ± 6 cm. The error was estimated based on the angle of the church in the captured frames. Although this approximation is too crude, the video images suggest the displacement at the top of the tower right before collapse is in the order of 4 to 10 cm. This contradicts the displacements estimated using an implicit analysis with high damping, which are in the order of 2 mm. Con-versely, a displacement of approximately 10cm, obtained from the explicit analysis, suggest the collapse behavior is well represented in the LS-DYNA model even when the response is overestimated.

Figure 13: Approximated ultimate displacement at the top of the towers from video

frame captures. Dimensions in meters


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5 SEISMIC BEHAVIOR OF A FULL-SCALE CHURCH To confirm the applicability of the constitutive models presented above for the estimation

of the response of a full-scale church, an explicit analysis was applied to qualitatively assess the seismic behavior of an existing ancient church. Since the applicability of Rayleigh damp-ing is not suitable in LS-DYNA, the explicit analysis was performed using a constant damp-ing ratio of 5% of the critical. This is the recommended value to solve transient analysis in LS-DYNA. Values greater than 5% are not recommended [7]. For this reason a quantitative assessment of the seismic response of a full-scale church by means of explicit methods was not attainable.

It is expected the assumption of 5% to be conservative. Equivalent damping ratios for an-cient masonry structures has been estimated to be greater than 5% in previous studies, for in-stance, the damping ratio for the Mexico City Cathedral varies from 6 to 12% [9].

The application of an implicit analysis was not be attainable at the time of this study as the required running time considerably escalated in comparison to the explicit analysis.

5.1 Description of the Almolonga church The object of study in this section is the Almolonga church. This church was built during

the 17th century in the town of Almolonga, Puebla. Its structure was built with lime-mortar masonry. It has a cross shape in plan view. A dome is supported on the drum at the top center of the church. There is a single bell tower. The roof system consists of vaulted masonry roofs. Figure 14a shows a view of the church.

5.2 Observed damage and repairs Based on a site visit and a survey report after the 1999 Tehuacan earthquake [10], the ob-

served damage included major cracks on the walls, especially on the transversal walls that support the East and West vaulted roofs. Also, major cracks appeared on the lower body of the tower. A vertical crack was observed along the connection between the tower and the fa-çade. Retrofit techniques have been applied to the church through the years. Unfortunately, these have not been documented rendering difficult to include those in a computer model. Figure 14b-d shows some damage observed during a site visit in 2008.

Figure 14: Almolonga church: (a) view; (b) cracks on interior vaulted roof;

(c) cracks on interior walls; (d) cracks on exterior walls


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5.3 Earthquake records Figure 15 shows the earthquake record used in the analysis. This was recorded during the

1999 Tehuacan earthquake in the state of Puebla. The Almolonga church is located approxi-mately 80 km from the epicenter.

Figure 15: Earthquake records for the FE analysis of Almolonga church: (a) transverse direction; (b) transverse

response spectrum with 5% damping; (c) vertical direction; (d) vertical response spectrum with 5% damping

5.4 Computed response The computed response was estimated by means of an explicit analysis. Figure 16 shows

the cracks developed in the model, cracks are developed mainly in the walls, the rear vaulted roofs, the lower body of the tower and the connection between tower and the left lateral wall. According to the field observations the amount of cracks on the exterior walls and on the tow-er appears to be overestimated, especially the amount of cracks on the tower results unrealistic. Similar to the scaled model, this is attributed to the approach adopted for modeling of the damping. Since the damping is constant and limited to 5% of the critical, the estimated re-sponse (see Figure 17) results in increased displacements. In this regard, experimental studies could lead to a more accurate estimation of damping.

Figure 16: Cracks developed in LS-DYNA FE model for the Almolonga church


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Figure 17: Displacement response at the top of the tower

6 STRUCTURAL FREQUENCY DECAY The acceleration response computed for the scaled model was used to estimate the fre-

quency decay during the applied earthquake. The Frequency Response Function (FRF) was calculated for 5-second segments along the time-history (Figure 18). The FRF estimated at the mid-span of the vaulted roof with respect to the base acceleration (Figure 19). In the implicit analysis with high damping the frequency decays from 10.7 Hz in segment 1 to 7.4 Hz in segment 7. In the case of an explicit analysis with 5% damping the frequency decays from 8 Hz in segment 1 to 2.7 Hz in segment 7. Clearly, the LS-DYNA model used in the explicit analysis represents a softer structure. Once the cracks start developing the large displacements introduce weak areas that results in a considerable loss of stiffness.

Figure 18: Segmentation of the mid-span roof acceleration response for structural frequency identification:

(a) implicit analysis with high damping; (b) explicit analysis with 5% damping

Figure 19: Frequency response function at the mid-span roof: (a), (b) from implicit analysis with low damping;

(c), (d) from explicit analysis with 5% damping


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7 CONCLUSIONS The following conclusions are derived from this study:

• Engineering assumptions such as constitutive laws and damping values are crucial for FE modeling of the seismic response of ancient masonry churches

• Special attention should be given to the way damping is assigned to the model as the seismic response is considerably sensitive to damping parameters

• Rayleigh damping is suitable for implicit transient analysis provided the mass and stiff-ness proportional damping coefficients are estimated based on measured response

• Approximately 5% constant damping ratio is suggested only if the assessment of the seismic response is qualitative and an explicit analysis is applied using LS-DYNA

• Instrumentation of full-scale church structures is deemed necessary to assess the accuracy of an FE approach for computation of the seismic response and frequency decay

REFERENCES [1] R. Meli, F. Peña, On elastic models for evaluation of the seismic vulnerability of ma-

sonry churches, SAHC 4th International Conference, Padova, Italy, November 2005.

[2] T.T. Bakeer, Collapse Analysis of Masonry Structures Under Earthquake Actions, Vol-ume 8 of Bauforschung und Baupraxis. Technische Universtat Dresden, 2009.

[3] M. Chavez; R. Meli, Shaking table testing and numerical simulation of the seismic re-sponse of a typical mexican colonial temple, Earthquake Engineering And Structural Dynamics, Vol.41, Pag.233-253, 2012. DOI: 10.1002/EQE.1127

[4] M. Chávez; R. Meli, Shaking table tests of a typical mexican colonial temple: evalua-tion of two retrofitting techniques, Earthquake Spectra, Vol. 29, pp. 1209-1231, 2013. DOI: http://dx.doi.org/10.1193/030512EQS064M

[5] Mexican Institute of Antropology and History (INAH), Rehabilitación de inmuebles históricos dañados por los sismos de junio y septiembre de 1999 en los estados de Gue-rrero, Estado de México, Morelos, Oaxaca, Puebla Tlaxcala y Veracruz, Memoria FONDEN, 2000.

[6] ANSYS® Release 14.0, Reference manual.

[7] LS-DYNA® Keyword user’s manual, Volume 1, Livermore Software Technology Cor-poration, 1997.

[8] J-N. Juang, R.S. Pappa RS, An Eigensystem Realization Algorithm for Model Parame-ter Identification and Model reduction. Journal of Guidance, Control, and Dynamics 8(5):620–627, 1985.

[9] D. Rivera, R. Meli, R. Sanchez and B. Orozco, Evaluation of the measured seismic re-sponse of the Mexico City Cathedral. Earthquake Engineering and Structural Dynamics; 37:1249–1268, 2008.

[10] G. García, Templo de Todos los Santos, Almolonga, Tepexi, Puebla, Proyecto de Inter-vención, Reporte técnico, 2000.

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