Elena Mele, Aldo Giordano, Antonello De Luca · PDF file · 2014-05-16Nonlinear analysis of some typical elements of a basilica plan church Elena Mele, Aldo Giordano, Antonello De

Nonlinear analysis of some typical elements of a

basilica plan church

Elena Mele, Aldo Giordano, Antonello De LucaDept. of Structural Analysis and Design, University of Naples "Federicoir, P.le Tecchio 80, 80125 Naples, Italy. Email: [email protected]

Abstract

In this paper the strength and deformation capacity of the main structuralelements of a basilica church are studied via static nonlinear analysis. Inparticular, the nave arcades, the cross section at the triumphal arch and the endwall of chancel are analysed through FEM, using the general purpose softwareprogram ABAQUS. The push-over analyses performed on the main structuralelements of the church provide useful information on the non-linear behaviour,the stress pattern, the collapse mode and the ultimate strength. These informationare particularly valuable since the structural schemes herein analysed are veryrepetitive within basilica type churches, which represent a large portion of theItalian cultural heritage.The results, in terms of ultimate loads, have been compared with the collapseloads obtained through the application of the limit analysis. This comparison hasallowed to demonstrate the capacity of ABAQUS code to provide reliable resultsand to investigate on the effect of material properties on the ultimate capacity.

1 Introduction

The very recent earthquakes which occurred in Italy have highlighted thevulnerability of historical masonry churches and the need to address the problemof the assessment of their seismic capacity. Church buildings deserve attentionfrom research since they represent a large portion of the building patrimonywithin the Italian cultural heritage. Furthermore these buildings are characterisedby quite repetitive structural schemes which therefore allow to extrapolate theresults obtained on a single case study to many other similar cases.

Transactions on the Built Environment vol 38 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509

534 Earthquake Resistant Engineering Structures

The seismic analysis of complex masonry buildings, and in particular ofchurches, presents objective difficulties deriving from the complexity ofgeometrical configuration. In addition the analytical treatment of the masonrymaterial non-linearity, which displays nearly no-tension characteristics, reflectsin several different models (Luciano [1], Shing [2], Lemos [3], Pegon [4],Abruzzese [5] etc.) proposed in the inherent literature for the prediction of thebehaviour under statically applied loads. For these reasons the capacity ofexisting nonlinear programs in providing reliable results has to be verified anddemonstrated.In this paper the general purpose software program ABAQUS is adopted for thepush-over analyses of the main structural elements of a basilica church. Theresults are verified against limit analysis application.

2 Object of the paper

Object of this paper is to assess the non-linear behaviour, the stress pattern, thecollapse mode and the ultimate strength of the typical structural elements of abasilica plan church. For this purpose the 2D models of the single elements areanalysed through FEM, using the general purpose computer code ABAQUS.A secondary object of the paper is to check the potential of the ABAQUS code inthe modelling of masonry structures and to investigate on the effect of materialproperties on the ultimate capacity. For this reason the results, in terms ofultimate loads, have been compared with the collapse loads obtained through theapplication of the limit analysis. In a companion paper (Mele [6]) these collapseloads, representing the horizontal strength capacity of the elements, arecompared to the seismic elastic demands on the same elements computedthrough a complete 3D linear model.

3 The case study

In this paper it is made reference to a case study (Mele [7]): the S. IppolistoMart ire church which seems representative of many other buildings of this typecharacterised by a basilica plan. The church located in Atripalda (Avellino,Italy), was built between 1584 and 1612 on a previous basilica which dates backto the IV century a.c. (Min.BB.CC.AA. [8]). The plan and the elevationsevidence the typical basilica plan with the main nave, the two lateral aisles andthe clerestory, transept and chancel. The nave is 11.6m wide, 28 m long and hasa maximum height of 16.5 m, while the aisle are 5.0 m wide and 8.5 m high. Themasonry walls have thickness varying between 1.0 and 1.2 m, the columns of thenave arcade have rectangular section 1.2 x 1.2 m. The chancel has a rectangularplan shape, 8.8 x 11.6 m, and height variable between 14.8 and 8.0 m. Thestructural elements are made of tuff masonry.The structural system of the roof of the nave is king-post timber roof, while thelateral aisles are covered by quadripartite vaulting systems with four diagonal


Earthquake Resistant Engineering Structures 535

ribs. In figure 1 is provided the schematic layout of the three elementary sub-structures studied in this paper, namely the end wall of the chancel, the triumphalarch and the longitudinal section on the nave arcade.

3.2

10.3

14.8

11.6 5.0 11.6

18.014.8

30.0

I 9.0

9.15

1 7, J

17.3u

I 4.61.6' 48.8

Thickness = 1.2 m

Figure 1: Geometry of the analysed church structural elements.

4 Non linear analysis of single structural elements of the church

4.1 Premise

In this section the results of the nonlinear structural analyses performed on the2D models representing the major structural elements of the church are provided.The analyses have been carried out using ABAQUS (HKS [9]), a generalpurpose FEM computer code. In order to provide a detailed description of thenonlinear behaviour of the church elements, only the results obtained on threemajor structural elements are reported. In particular these structural elements arethe end wall of the chancel, the transversal section at the triumphal arch and thelongitudinal section on the nave arcade.



The structural elements have been analysed under vertical loads, deriving fromthe own weight and from the supported elements, and horizontal seismic typeactions. The earthquake type loading has been simulated through a set ofhorizontal forces proportional to masses distributed throughout the joints of the2D F.E. models.In the following sub-sections, the modelling strategies, the material propertiesand the calibration of the numerical model are briefly discussed. The results ofthe FEM push-over analysis on the elements are provided in terms of force-displacement curve, stress distribution, collapse mode and ultimate strength.Finally, the collapse mechanism and the values of strength determined via FEMare compared with the results obtained through limit analysis of simplified 2Dmodels.

4.2 Calibration of the ABAQUS model

The ABAQUS computer code provides a constitutive model for brittle materials,namely the "concrete" model, originally intended to simulate plain concrete.Cracking occurs when the stresses reach a failure surface, expressed in terms ofprincipal stress components. The model is a smeared crack model, since it doesnot track individual "macro" cracks, rather cracking phenomena are taken intoaccount by affecting the stress and stiffness of the material at each integrationpoint. The calibration has been carried out through extensive sensitivity analysesdevoted to assess the effect of the mesh size, material model parameters andnonlinear solution strategy on the inelastic response of the simplest structuralscheme: a rectangular wall.

4.3 End wall of chancel

The end wall of the chancel is the structural element of the church characterisedby the simplest geometrical configuration. Though extremely simple, thesimulation of this element through ABAQUS clears out the difficulty inobtaining a complete softening curve. The 2D mesh used for this structuralelement consists of 195 joints connected by means of 168 shell elements.In figure 2 the results of the push-over analysis are shown in terms of theresultant of the horizontal force distribution, F, normalised to the total verticalload W, vs the horizontal displacement of the right top joint evidenced in thesame figure. The analysis stopped at increment 45 due to numerical instability,revealing a certain nonlinear behaviour, but the inability, with the chosenparameters, for the model to progress the analysis toward the descending branch.In the same figure, the collapse multiplier of the horizontal forces, /L resultingfrom the overturning mechanism of the wall is also depicted as the straight lineat F/W=0.78. This multiplier has been obtained adopting the classical limitanalysis (Heyman [10]). The common assumptions were therefore adopted: notensile resisting capability, infinite compression strength, sliding of a part of thestructure upon another cannot occur. The collapse multiplier computed underthese assumptions is ̂ = 0.784.



Limit Analysis

p

1 : 1 : 1 1 1 1 I I I

V---,

w

r--.̂

0 5 10 15 20 25 30 35

Figure 2: Chancel end wall: nonlinear force displacement curve.

The deformed configuration of the wall and the vertical stress pattern due to thevertical plus horizontal loads at the last increment of the analysis are respectivelyshown in figures 3(a) and 3(b). The stress distribution shows good agreementwith the failure mechanism.

Figure 3: Chancel end wall: (a) deformed configuration; (b) vertical stresses.



4.4 Transversal section at the triumphal arch

The triumphal arch, which is the separation element between the chancel and thetransept, has a particular importance in the seismic analysis of basilica typechurches. In the statistical analysis of the damage pattern observed in theelements of churches subjected to earthquakes (Doglioni [11]), it has beenobserved that very often the triumphal arch experiences severe damage resultingfrom in-plane horizontal actions, which leads to partial or total collapse.A fairly fine 2D mesh, consisting of 1102 joints connected by 968 shellelements, has been used to model the transversal section of the church at thetriumphal arch.In figure 4 the results of the push-over analysis are provided in terms ofhorizontal force resultant, F, normalised to the vertical load W, vs the horizontaldisplacement of the arch's top right joint. Three different curves resulting formanalyses carried out with different values of the compressive and tensile strength,are reported in the chart. In this scheme the push over analysis seems to benumerically more stable than in the previous case. This can be possiblyconnected with the scheme geometrical configuration, however furtherinvestigations are in progress by the authors on this topic. Furthermore the effectof the material properties on the nonlinear behaviour can be deduced from thethree curves reported in the chart. In particular it is evident that by increasing thecompression strength, the ultimate strength does not significantly vary, while thea larger deformation capacity can be observed.

Figure 4: Triumphal arch: nonlinear force displacement curve.

0,6

0,5

0,4

0,3

0,2

0,1

F/W

Limit Analysis

3 MPa dt = 0.3 MPc

7MPa dt =0.7MPc

7 MPa dt = 0.3 MPc

10 15 20 25' A (mm)30 35

In figure 4 the horizontal force multiplier, obtained through the application ofthe limit analysis is also reported as the horizontal line at F/W=0.283. Thecorresponding collapse mechanism is schematically depicted in the same figure.The collapse multiplier has been defined through the application of the cinematictheorem, i.e. as the minimum value among the horizontal force multiplierscorresponding to all possible mechanisms. With regard to the geometrical



theorem, i.e. as the minimum value among the horizontal force multiplierscorresponding to all possible mechanisms. With regard to the geometricalcharacteristics of the arch, clearly different kinds of mechanisms are to beconsidered, each involving four hinges. If the arch is slender, as in this case, thefailure will occur following one of the general patterns shown in figure 5.Varying the position of the hinges in all possible configurations, the class of thecinematic multipliers is defined, and their minimum value represents thesearched collapse multiplier.

a) global mechanism b) semi-global mechanism

Figure 5: Possible types of collapse mechanisms of slender arches.

The deformed configuration and the vertical stress pattern due to the vertical plushorizontal loads at the last increment of the analysis are respectively shown infigure 6(a) and 6(b). From these figures the correspondence between the collapsemechanisms determined through the limit analysis and the FEM nonlinearanalysis can be derived quite simply.

a) b)Figure 6: Triumphal arch: (a) deformed configuration; (b) vertical stresses.

It should be noticed that, while in the limit analysis the masonry is thought tohave unlimited compressive strength and no tensile capacity, for the materialmodel used in the FEM analysis finite values of the compression and tensilestrength have been assumed. In order to investigate if higher strength valuescould possibly lead the model to overcome the limit analysis multiplier, asensitivity analysis has been carried out, by varying the compression and the



4.5 Longitudinal section on the nave arcade

A special attention has been paid to the analysis of the section of the churchalong the nave arcade, since it displays a typical architectural/structural solution,given by the assemblage of columns supporting arches and vaults, whichfrequently occurs in the masonry monumental constructions. It is worth tomention that a full-scale test model of a very similar structural layout,representing part of the S. Vincente de Fora Monastery of Lisbon, has beenrecently developed and subjected to a large experimental program in the contextof the Cosismo European research project (Pinto [12]).

0,6

0,5

0,4

0,3

0,2

0,1

F/W

861

F IWarc I U.

0 5 10 15 20 25 30 35

Figure 7: Longitudinal section: nonlinear force displacement curve.

The 2D mesh used for the L2 element consists of 942 joints connected by 753shell elements. The results of the push-over analysis are reported in threedifferent curves, provided in figure 7, in which the horizontal force resultant isplotted vs the displacements of three different joints, which have been monitoredduring the analysis. The joints which the three curves refer to are indicated in theschematic layout of the element provided in figure 7. The curves show very closetrends, in particular characterised by a remarkable deformation capacity.In the analysis a sudden decay of resistance occurs at the load step in which themajor arch fails, with a subsequent redistribution of stresses. The analysis hasshown that since the beginning of the loading process the major arch is subjectedto an almost uniform tensile stress state, which give rise to large lateraldeformation.In order to compare the results of the FE analysis to the outcomes of simplifiedmodels, the structural element has been subdivided into two parts, namely thewall and the arcade. For these single sub-elements the horizontal collapse forceshave been evaluated and reported (normalised to the total vertical load acting onthe complete structural element) in figure 7, along with their sum. The actual



have been evaluated and reported (normalised to the total vertical load acting onthe complete structural element) in figure 7, along with their sum. The actualmultiplier derived through FEM is strictly comprised between those two valuesand fairly lower than the summation.The deformed configuration and the vertical stress pattern due to the vertical plushorizontal loads at the last increment of the FEM analysis are respectively shownin figures 8 a) and 8 b).

MPa

\

MV

b)

\

Figure 8: Longitudinal section (a) deformed configuration; (b) vertical stresses.

5 Conclusive remarks

In this paper the major structural elements identified in a basilica plan churchhave been studied through non linear static FEM analysis. The numerical results,which are in good agreement with the ones derived on the basis of the limitanalysis, have provided indications on the collapse mode, ultimate strength anddeformation capacity of these substructures. The ABAQUS computer codeadopted for the FEM nonlinear analyses has demonstrated to be a reliable tool inanalysing masonry models, though some aspects are still under investigation bythe authors. The effect of material properties on the ultimate capacity and on thenonlinear behaviour have also been assessed.



In a companion paper (Mele [6]) the results of these nonlinear analyses,expressed in terms of nondimensional horizontal strength capacity of theelements, are compared to the seismic elastic demands on the same elementscomputed through a complete 3D linear model.

AcknowledgementsThis research is supported by CNR (Progetto Finalizzato Beni Culturali) and byMURST PRIN 1997 "Protezione sismica dell'edilizia esistente e di nuovaedificazione attraverso sistemi innovativi".

References

1. Luciano, R. & Sacco, E. Un modello di danno per Fanalisi di strutturemurarie. Atti Convegno Nazionale "La Meccanica delle Murature tra Teoriae Progetto". Pitagora Editrice Bologna, Messina, Italy, pp.327-336, 1996.

2. Shing, B. Finite element analysis of masonry structures. Atti ConvegnoNazionale "La Meccanica delle Murature tra Teoria e Progetto ". PitagoraEditrice Bologna, Messina, Italy, pp. 1-16, 1996.

3. Lemos, J.V. Numerical models for seismic analysis of monuments. Proc.MONUMENT '98 Workshop on Seismic Performance of Monuments,Lisbon, Portugal, pp. K19-K36, 1998.

4. Pegon, P. & Pinto A. Numerical modelling in support of experimental modeldefinition-Hie S. Vincente de Fora model. Proc. MONUMENT '98 Workshopon Seismic Performance of Monuments, Lisbon, Portugal, pp. 3-12, 1998.

5. Abruzzese, D. Como, M., Lanni, G. Some results on the strength evaluationof vaulted masonry structures. Structural Studies of Historical Buildings IV- Vol.1: Architectural Studies, Materials & Analysis. Eds. C.A. Brebbia, B.Leftheris. Computational Mechanics Publications, pp.431-440, 1995.

6. Mele, E. & De Luca, A. Behaviour and modelling of masonry churchbuildings in seismic regions of. Proc. T** Int. Symposium on EarthquakeResist. Engrg. Struct. ERES '99, Catania, Italy, 1999.

7. Mele, E., Modano, M. & De Luca, A. The seismic retrofit of historicmasonry buildings through BIS: preliminary analysis for application tochurch typology. Proc. MONUMENT '98 Workshop on SeismicPerformance of Monuments, Lisbon, Portugal, pp. 269-280, 1998.

8. Min. Beni Cult, e Amb. Soprintendenza Generale Interventi Post-Sismici inCampania e Basilicata. Dopo lapolvere. Tomo II - Provincia di Avellino. (inItalian). Istituto Poligrafico e Zecca dello Stato, Rome, pp. 211-212, 1994.

9. HSK, Inc. ABAQUS Theory Manual, version 5.8, 1998.10. Heyman, J., The stone skeleton, Int. J. of Solids and Structures, Vol.2, pp.

249-279, 1966.11. Doglioni, F., Moretti A. & Petrini V., Le chiese e il terremoto, (in Italian),

LINT, Trieste, 1994.12. Pinto, A.V., Verzelleti, G., Molina, F.J., Plumier, C. Seismic tests on the

S.Vincente de Fora model. Proc. MONUMENT '98 Workshop on SeismicPerformance of Monuments, Lisbon, Portugal, pp. 33-46,1998.


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Elena Mele, Aldo Giordano, Antonello De Luca · PDF file · 2014-05-16Nonlinear analysis of some typical elements of a basilica plan church Elena Mele, Aldo Giordano, Antonello De