INFLUENCE OF THE GEOMETRIC EMPIRICAL RULES IN THE … · ALEJO Leslie Edith, ZAVALA Marisela,...

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

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

INFLUENCE OF THE GEOMETRIC EMPIRICAL RULES IN THE

SEISMIC FRAGILITY OF RELIGIOUS HISTORICAL BUILDINGS OF

MORELIA CITY, MEXICO

ALEJO Leslie Edith1, ZAVALA Marisela

1, MARTINEZ Guillermo

2, JARA José Manuel

2

1 Master Student, Universidad Michoacana de San Nicolás de Hidalgo Ciudad Universitaria Francisco J. Mújica S/N, Col. Molino de Parras, 58040 Morelia, Mich. Telephone, (+52 443) 3041002

[email protected], [email protected]

2 Professor and Researcher, Universidad Michoacana de San Nicolás de Hidalgo Ciudad Universitaria Francisco J. Mújica S/N, Col. Molino de Parras, 58040 Morelia, Mich. Telephone, (+52 443) 3041002

[email protected]; [email protected]

Keywords: Macroelement, Gothic rules, capacity spectrum, fragility curves.

Abstract. This article describes the structural behavior under lateral loads for thirteen naves of heritage buildings of the historical downtown of Morelia city, Mexico. The transversal macroelements has been defined for each structure with the actual geometrical characteristics, which were modified according to the ancient first and second design rules Gothic. The capacity under lateral loading of the different structures in the transverse direction of the buildings was analyzed by performing nonlinear static incremental analysis, from which capacity spectrum were generated for each macroelement. Probabilistic damage scenarios were established by generating fragility curves thus it was possible to identify the buildings with the major seismic risk for both macroelement sections, those with actual geometry and the modified ones with the ancient rules of the Gothic. The results show us in most cases, a better behavior for the sections with actual geometries unlike with the macroelements with modified geometries where the vulnerability was increased. However, there were cases where the ancient rules of the Gothic confer a better performance thanks to the presence of structural elements with bigger dimensions.

mailto:[email protected]

ALEJO Leslie Edith, ZAVALA Marisela, MARTINEZ Guillermo, JARA José Manuel

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INTRODUCTION

The architecture of the towns is usually studied in view of its further development. It is said that if a colony becomes a great nation its origins cause great curiosity, so it becomes a center of attention for the scientists. However, in Mexico, the architecture from 16th century does not require such justification because the mastery achieved by indigenous builders and the originality of many Mexican structural and ornamental forms in that first period are powerful enough to justify its study without reference to subsequent events of its national history [1].

Michoacan state has about 6000 of the 85000 buildings listed as heritage buildings within Mexican territory. Only in the historic downtown of Morelia city, there are 1100 structures dating from the 16th-19th centuries, of which just over 95% are civil and the other are religious type. Despite this, within the multidisciplinary context which involves the proper study and conservation of this kind of buildings in Mexico, the corresponding part to earthquake impact and structural engineering knowledge is currently abandoned, so that when the diagnostic stage to establish some kind of intervention is defined, in most cases these interventions are unfortunate and with low scientific rigor.

Morelia city is located approximately 340 kilometers from the Mexican subduction zone, and it suffer local and normal earthquakes more rarely, but these type of events historically have demonstrated a highly destructive nature for the heritage structures, a proof is the earthquake of June 19th, 1858, where several buildings within Michoacan and surrounding states collapsed. For this earthquake, an estimated intensity IX level has been reported for the epicentral zone of Patzcuaro city, located at 55km from Morelia city and severe damages to religious structures were reported, such as the Cathedral of Morelia [2], a condition that has not been repeated during recent earthquakes.

The present research is motivated by the high seismic hazard and the large number of heritage structures in this city distinguished by UNESCO since 1991 as a World Heritage, and is part of a series of studies aimed to the better understanding of the expected structural behavior of these buildings to future seismic actions, in order to achieve a better understanding of the Mexican Heritage. Additionally, the experience of the ancient geometrical dimensioning techniques is analyzed as an attempt to find if some of the old geometrical rules were also effective to resist seismic actions.

GENERATION OF MACROELEMENTS

A two-dimensional modeling has been employed for the analysis of selected macroelements, which correspond to a transversal cross section of thirteen Religious heritage buildings located on firm soil of the historical downtown of the city of Morelia, Michoacan. For the definition of these macroelements it was considered the most vulnerable areas of each case in study, using the criterion that where there is greater stiffness, greater shear stresses will be presented.

In Figure 1, the studied sections are presented, in chronological order. In these images it is showed the actual dimensions of each macroelement.

Influence of the geometric empirical rules in the seismic behavior of religious historical buildings

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Church of Saint Agustin

(1550) Franciscan Church of Saint

Buenaventura (1585) Church of el Carmen

(1596)

Church of Saint Francisco Javier

(1660) Church of la Cruz

(1680) Church of Saint Juan Bautista

(1696)

Church of Guadalupe (1708) Church of las Monjas (1732) Church of Capuchinas (1734)

Church of la Merced

(1736) Church of Saint Rosa de Lima

(1757) Church of Saint Jose

(1760)

Transfiguration Cathedral (1660)

Figure 1: Actual transversal Macroelements.

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MODIFICATION OF THE ACTUAL SECTIONS

The modification of the sections was carried out according to two rules of Gothic, named First geometric gothic rule and Second geometric gothic rule [4].

The first geometric gothic rule was printed in a Derand treatise in 1643, but then it was discovered that also was cited in earlier sources dating from the 16th century, such as the letter of Martínez de Aranda. Later, around 1675, Blondel includes it in his Traité d'architecture, due to this it is also known as the "Blondel’s Rule" [4] for the widths of walls or abutments of arches, which establish that the arch ABCD is divided into three equal parts by the points B and C, CD distance is extended so that CD = DE and Then E defines the outer edge of the abutment or wall (Figure 2).

Figure 2: First geometric gothic rule. Figure 3: Second geometric gothic rule.

The Second geometric rule appears in the treatise of Hernan Ruiz the Younger around 1560

where addresses the problem of how to find the thickness of the piers or buttresses for any kind of arch [3].

The Second rule establish following steps for a semi-arch (Figure 3) : 1. The line of extrados is divided in two equal parts. 2. From the point mentioned in 1 a tangent line is extended. 3. The point in which the tangent cuts the projection of the impost line, give us the thickness

of the vertical supports or buttresses. The thickness of the buttress varies depending on the thickness and shape of the arch. As seen

in Figure 3, for a pointed arc a lower buttress thickness will be obtained. So, following these geometrical rules, we proceeded to verify the existence of those mentioned

ancient rules in the transversal sections presented in Figure 1. For those macroelements that did not have any kind of ancient rule in their actual geometric features, a hypothetic modification following these rules was proposed in order to make a comparative between the lateral capacity between the actual and modified structural systems.

PA

B C

D

E


5

(A) (B) (C)

Figure 4: Application of the geometrical gothic rules to the transversal macroelement from the Church of Saint Francisco; A) Actual proportions of the section, B) Application of the first geometrical rule and C) Application of the

second geometrical rule.

For the particular case of Figure 4 it can be observe that the actual geometrical proportion doesn't have any of the features that correspond to some Gothic geometric rule proposed for this study, in other words the width of the support a ≠ b ≠ c.

Actually, each of the actual sections was subjected to a formal study in order to know if the geometry has the corresponding rules of the gothic proportions. Of all the analyzed building group was found that the most approached to submit any of these proportions is the cross section of the Church of La Merced (1736), which presents the first rule, for the remaining sections two models were proposed respecting both proportions.

It is noteworthy that the rules of Gothic apply to the thickness of the supports, so we set out to work these changes only in the transversal cross sections of the thirteen buildings.

RIGID ELEMENT METHOD

The criterion for the structural analysis chosen was the rigid element method [5, 6] which is an appropriate simplification for linear and non-linear analysis of unreinforced masonry structures, considering that the elements have the kinematics of rigid body with two linear displacements and one rotation. These connections are two axial springs and one shear device connected to the common side between two rigid elements.

The meshing criteria for each macroelement were based on the geometry of each studied section conforming a mesh as regular as possible. The thickness for arches, walls and buttresses were obtained directly from the actual structures and they stayed without any change both for actual geometries as the modified ones with the golden proportion. It is noteworthy that the stiffening effect provided by the vaults, apse and narthex were not included in the macroelements. The mechanical properties used in the analyses are shown in Table 1.

b ca b ca b ca


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Table 1: Mechanical properties

Young’s Modulus (MPa)

Poisson’s ratio

Density (Kg/m3)

Stone quarry 1000 0.2 1800

Fill material 500 0.2 1600

STRUCTURAL CAPACITY

Nonlinear static incremental analysis is a powerful tool for the knowledge of the lateral capacity in structural systems. Using this procedure, capacity curves of each macroelement were obtained plotting the base shear against the maximum drift displacement. In this analysis a horizontal acceleration proportional to the mass was applied up to failure.

In the Rigid Element formulation for incremental accelerations a sine function is desirable rather than a linear, because the method becomes more stable numerically at the final stages of loading [6].

CAPACITY SPECTRUM

The obtained capacity curves are in units of base shear (kN) versus displacement (cm), however the work units used for fragility purposes are spectral acceleration (Sa) against spectral displacement (cm). To perform this conversion, Freeman [7] proposes the use of the dynamic properties of the structure based on the fundamental modal participation factor, Г1, which indicates the participation grade of the fundamental vibration mode on the total response of the structure. So we have

(1)

Where Δ is displacement and Sd is the spectral displacement. Thus the spectral displacement is

(2)

Assuming that the base shear is the sum of the seismic lateral forces acting on the structure

∑ (3)

(4)

(5)

Where m corresponds to the mass of the macroelement and Sa to the spectral acceleration. In equation (5) is called the effective modal mass which can be calculated as

(6)


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It is recommended that the total effective mass should never be less than 0.75 times the total mass of the structure [7].

Finally, spectral acceleration is obtained by

(7)

Applying the equations (2) and (7) to the capacity curves the corresponding capacity spectrum was obtained for each one of the macroelements.

With the aim to facilitate the obtaining of the damage thresholds and the fragility curves, it is recommended simplify the capacity spectrum performing a bilinealization of each spectrum. The point where the elastic and inelastic branches are crossed is called yield point. In the present research the bilinealization criteria was based on equalizing the areas above and below of the continuous capacity spectrum.

Figures 5 to 17 show a comparison between the bilinear capacity spectrums of each of the thirteen sections of this study, this comparison corresponds to the behavior that exhibits each of the buildings with the actual features and their geometric modifications. It can be seen that in most cases the behavior of buildings is better without the geometrical modifications, this is because of the existence of buttresses that contributes in a positive way to the capacity of the macroelements and therefore to the capacity of the whole structure. In some cases as in Figures 6 and 14, the capacity is improved when the rules of the Gothic are applied, because in these particular cases the thickness of the buttresses increases instead of decreasing.

Figure 5: Church of Saint Agustin. Figure 6: Franciscan Church of Saint Buenaventura.

Figure 7: Church of el Carmen.

Figure 8: Church of Saint Francisco Javier.

Figure 9: Church of la Cruz. Figure 10: Church of Saint Juan Bautista.

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Figure 11: Church of Guadalupe. Figure 12: Church of las Monjas. Figure 13: Church of Capuchinas.

Figure 14: -Church of la Merced. Figure 15: Church of Saint Rosa de Lima.

Figure 16: Church of Saint Jose.

Figure 17: Transfiguration Cathedral.

DAMAGE THRESHOLDS

Once the bilinear capacity spectrums were defined, it was possible to evaluate the expected damage degrees from damage thresholds predefined by experts. In this paper the criteria of Lagomarsino [8] was used, whose values are based on the European Macroseismic Scale [11] (see Table 2), thus, the damage thresholds were calculated for each of the studied buildings. In Table 2, Sdy and Sdu are the yield and ultimate lateral displacements respectively.

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Sa (

g)

Sd (cm)

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0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0

Sa (

g)

Sd (cm)

0.00.20.40.60.81.01.21.41.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Sa (

g)

Sd (cm)

0.00.20.40.60.81.01.21.41.61.8

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Sa (

g)

Sd (cm)

0.00.20.40.60.81.01.21.41.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Sa (

g)

Sd (cm)

0.00.20.40.60.81.01.21.41.6

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Sa (

g)

Sd (cm)

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Sa (

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Sd (cm)


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Table 1: Values and description for the damage thresholds [8].

Damage thresholds Spectral displacement Damage description Sd1 0.7Sdy Slight Sd2 Sdy Moderate Sd3 Sdy + 0.25 (Sdu-Sdy) Extensive Sd4 Sdu Collapse

FRAGILITY CURVES

As a closer approximation to the seismic risk it is common to represent the damage probability for a building through fragility curves which express the probability that the expected damage level in the structure equals or exceeds a certain damage degree in terms of a certain seismic demand.

For each one of the studied macroelements that probability was obtained using equation (8), Sd denotes spectral displacement (seismic hazard parameter), dsSd , is the median value of the spectral displacement at which the building reaches a certain threshold of the damage state ds ;

ds is the standard deviation of the natural logarithm of the spectral displacement of the damage state, ds ; and is the standard normal cumulative distribution function. The parameters (

dsdsSd ,, ) proposed by the experts [8] to model fragility were used in this work.

[ | ] *

(

)+ (8)

If an specific spectral displacement as a result of any time-history analysis, performance point or even an arbitrary displacement for each of the macroelements set here is provided, it is now possible to enter these values at the fragility curves and to know the probability of exceedance for each damage degree (slight, moderate, extensive and collapse).

Figures 19 through 30 show the fragility curves of each of the macroelements, both cases, for

its real characteristics as well as its geometric modifications incorporating the Gothic rules. In Figure 18 we have the interpretation of those graphs, in which the probability of exceedance against spectral displacement are plotted, and the four curves that define the regions of light damage, moderate damage, extensive damage and collapse are plotted as well.

Figure 18: Characteristics of the fragility curve.

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Slight damage

Moderate damage

Extensive damage

Collapse


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Actual Proportions 1st Gothic rule 2nd Gothic rule

Figure 19: Fragility curves. Church of Saint Agustin (1550).


Figure 20. Fragility curves. Franciscan Church of Saint Buenaventura (1585).


Figure 21: Fragility curves. Church of el Carmen (1596).


Figure 22: Fragility curves. Church of Saint Francisco Javier (1660).

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Figure 23: Fragility curves. Church of la Cruz (1680).


Figure 24: Fragility curves. Church of Saint Juan Bautista (1696).


Figure 25: Fragility curves. Church of Guadalupe (1708).


Figure 26: Fragility curves. Church of las Monjas (1732).

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Figure 27: Fragility curves. Church of Capuchinas (1734).


Figure 28: Fragility curves. Church of la Merced (1736).


Figure 29: Fragility curves. Church of Saint Rosa de Lima (1757).


Figure 30: Fragility curves. Church of Saint Jose (1760).

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Figure 31: Fragility curves. Transfiguration Cathedral (1660).

CONCLUSIONS

We concluded that most of the studied sections have not geometric features that correspond to some gothic rule, except the macroelement of the Church of la Merced (1736), which exhibits the first Gothic rule in actual geometry.

The empirical ancient design rules do not improve the structural behavior of buildings except those where the proposed geometric rules derived in thicker supports, as in the case of the Franciscan church of San Buenaventura (1585) and the Church of la Merced (1736).

The differences observed between actual and modified geometries using the gothic ancient geometric rules, are due mainly to the fact that a single arc produces supports with different thicknesses for each of the proportions, however, a case is presented (see Figure 10) where both changes match the width of the support, this is because the arc shape of the Church of Saint Juan Bautista allows both ratios to converge to the same thickness. This arc has a radius ratio in the base of 2.85m to 2.78m height, in other words the radius at the base is about 2.5% greater than the radius in height.

Both Gothic proportions exhibited differences based on the shape of the arc of each macroelment, so for the first Gothic rule while more pointed is an arch, its resulting thickness for the supports will be bigger in relation to the second Gothic rule and vice versa, which explains the change in position for the capacity curves in both cases (Figures 5 to 17).

When analyzing fragility curves we see that most cases, both the first and the second rule of Gothic exhibits a more vulnerable behavior, except for the cases where both geometric rules improved the structural performance, that were the least. In other words, in the majority of the studied macroelements the use of the provided empirical design rules did not provide improvement to the structural behavior of the buildings presented in this article.

In order to make a comparison between the vulnerability of each building with their actual proportions it can be observed that for a proposed 1 cm spectral displacement, the most vulnerable macroelements are the Franciscan church of San Buenaventura (Figure 20), the Church of el Carmen (Figure 21), the church of la Cruz (Figure 23) the Church of Saint Juan Bautista (Figure 24), the Church of la Merced (Figure 28) and finally the Transfiguration Cathedral (Figure 31).

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REFERENCES

[1] George Kubler, Arquitectura mexicana del siglo XVI. Mexico: Fondo de Cultura Económica, 2012.

[2] V. García Acosta and G. Suárez Reynoso, Los sismos en la historia de México, FCE-CIESAS-UNAM, vol. 1, 1996.

[3] Santiago Huerta, Diseño estructural de arcos, bóbedas y cúpulas en España ca.1600 - ca.1800, Escuela Técnica Superior de Arquitectura de Madrid , Madrid , Tésis 1990.

[4] Santiago Huerta, Arcos, bóvedas y cúpulas Geometría y equilibrio en el cálculo tradicional de estructuras de fábrica. Madrid, España, 2004.

[5] Siro Casasolo and Fernando Peña, Rigid element model for in-plane dynamics of masonry walls considering hysteretic behabiour and damage, Earthquake Engineering and Structural Dynamics, vol. 36, 1029-1048, 2007.

[6] Fernando Peña, Programa RIGID v.4.0.1, Manual del usuario, Instituto de Ingeniería, UNAM, 2010.

[7] Freeman, S.A., Development and use of capacity spectrum method, in Proceedn Sixth U.S. National Conference on Earthquake Engineering. Earthquake Engineering Reserch Inst., Oakland California, 1998.

[8] Giovinazzi S., Podestà S., Resemini S. Lagomarsino S., Wp5 –Vulnerability of historical and monumental buildings Handbook. Risk- UE: An advanced approach to earthquake risk scenarios with applications to different European towns, Contract No. EVK4-CT-2000-00014, 2003

[9] A.H. Barbat, L. Pujades, and N. Lantada, Performance of buildings under earthquakes in Barcelona, Spain, in Computer-Aided Civil and Infrastructure Engineering, 2006, 573–593.

[10] Siro Casasolo and Fernando Peña, Modelo de elementos rígidos para el análisis de estructuras de mampostería, Revista Internacional de Métodos Numéricos para el Cálculo y Diseño en Ingeniería, vol. 21, no. 2, 193-211, 2005

[11] EMS-98. European Macroseismic Scale. G. Grünthal, Chairman of the ESC Working Group Macroseismic Scales, GeoForschungsZentrum Potsdam, Germany. http://www.gfz-potsdam.de/pb5/pb53/projekt/ems/.

http://www.gfz-potsdam.de/pb5/pb53/projekt/ems/

http://www.gfz-potsdam.de/pb5/pb53/projekt/ems/

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