The paper presents a strategy for the non-linear dynamical analysis of ancient
masonry structures through two case studies ("Maniace Castle" in Syracuse, Italy and the Qutb
Minar in Delhi, India).(undefined

Casolo, Siro

Lourenço, Paulo B.

Peña, Fernando

Universidade do Minho: RepositoriUM

IX Internatinal Conference on Computational PlasticityCOMPLAS IXE. On˜ate and D.R.J. Owen (Eds)c©CIMNE, Barcelona, 2007SEISMIC ANALYSIS OF MASONRY MONUMENTS BY ANINTEGRATED APPROACH THAT COMBINES THEFINITE ELEMENT MODELS WITH A SPECIFICMECHANISTIC MODELFernando Pen˜a∗, Siro Casolo† and Paulo B. Lourenc¸o↑∗Institute of Engineering (IdeI)Universidad Nacional Auto´noma de Me´xicoCiudad Universitaria, 04510 Mexico City, Mexicoe-mail: fpem@pumas.iingen.unam.mx, web page: http://www.iingen.unam.mx†Department of Structural Engineering (DIS)Politecnico di MilanoPiazza Leonardo da Vinci 32, 20144 Milan, Italye-mail: Siro.Casolo@polimi.it - Web page: http://www.stru.polimi.it↑Department of Civil Engineering (DEC)University of MinhoCampus Azurem, 4800-058 Guimara˜es, Portugale-mail: pbl@civil.uminho.pt - Web page: http:// www.civil.uminho.pt/masonryKey words: Rigid Element Method, Non-linear Dynamic Analysis, Ancient Masonry Struc-tures.Summary. The paper presents a strategy for the non-linear dynamical analysis of ancientmasonry structures through two case studies (”Maniace Castle” in Syracuse, Italy and the QutbMinar in Delhi, India).1 INTRODUCTIONThe evaluation of the seismic behaviour of ancient masonry structures requires peculiar pro-cedures, since their response to dynamical loads often differ substantially from those of ordinarybuildings. In order to obtain a reliable estimation of the seismic risk, it is desirable to performfull dynamical analyses that describe the effective transmission and dissipation of the energycoming from the ground motion into the structure. In general, modelling the non-linear me-chanical behaviour of ancient masonry structures by means of three-dimensional models requirea great amount of computational resources that are not commonly available. Luckily, whenperforming complete dynamical analyses it is often effective to adopt a two-dimensional modelrather than a three-dimensional, even if the definition of a simpler model often requires a processof tuning in order to approximate as well as possible the features of the specific kinematics thatis of interest.1Fernando Pen˜a, Siro Casolo and Paulo B. Lourenc¸oIn this context, the use of two or more analysis tools allows to overcome the complexity ofthe study of the seismic behaviour of ancient masonry structures. The results obtained fromone analysis tool are used as the basis for a better conception of another model which uses adifferent analysis tool. Moreover, by combining the results of the different models it is possibleto obtain a better and more comprehensive interpretation of the seismic behaviour.The two case studies presented in this paper (the “Maniace” Castle, in Syracuse, Italy andthe Qutb Minar, in Delhi, India) show the application of this strategy of analysis combiningthe well-known Finite Element Method with the specific mechanistic rigid body and springmodel (RBSM) [1], here named as Rigid Element Method. The first one allows a good three-dimensional discretization of the geometry of the structure while the second one allows fullynon-linear dynamical analyses with few computational resources.2 THE RIGID ELEMENT METHODThe rigid element method (REM) [1, 3] idealises the masonry structure as a mechanismmade of rigid elements and springs. The elements are quadrilateral and have the kinematics ofrigid bodies with two linear displacements and one rotation. Three springs-devices connect thecommon side between two rigid elements or the restrained sides.These connections are two axialsprings separated by a distance 2b, and one shear spring. A volume of pertinence is assignedto each connection point. Out-of the linear elastic field, the main macroscopic constitutiveaspects are: the very low tensile strength; the significant post-elastic orthotropy combined withthe texture effects; the dependence of the shear strength on the vertical compression stress;the progressive mechanical degradation during repeated loading; and the energy dissipationcapability. To do this, a simplified heuristic approach is adopted, based on the phenomenologicalconsideration of the main in-plane damage mechanisms that is described at the meso-scaleby adopting specific separate hysteretic laws for the axial and shear deformation between theelements [3]. This separation reduces the computational effort, even though a Coulomb-like lawis adopted in order to relate the strength of the shear springs to the vertical axial loading.3 THE MANIACE CASTLEAs first, we present the seismic analysis of the large medieval “Maniace Castle” in Siracusa,Italy. Three different numerical models [2] were implemented: a) a global three-dimensionalfinite element model of the whole castle; b) a detailed three-dimensional finite element modelof the architectural part of the building that suffered the main static problems; c) a plane rigidelement model. Only linear-elastic analyses were carried out with the first two models, whilethe third model was formulated with the aim to investigate the non-linear dynamical responseand the seismic damage of the structural system constituted by the pillars, the buttresses andthe arches of the masonry monuments.The global model of Maniace Castle was made by means of a total number of 108625 solidtetrahedron elements. In this model the geometric details were simplified by disregarding theexact three-dimensional shape of the internal vaults, as well as many other structural details.A simple process of tuning allowed to approximate the basic fundamental modes of vibration,as well as the mean values of stress due to the gravity in the most critical parts of the building,the columns and the buttresses, as it resulted from the previously published experimental and2Fernando Pen˜a, Siro Casolo and Paulo B. Lourenc¸ocomputational studies.The numerical analysis made on the whole Castle showed that in the case of seismic excitationthe most critical part of the building is the vaulted room, i.e. the rest of the ancient hypostylechamber named Salone. In particular, the load bearing capacity of the four columns and ofthe two buttresses in the internal courtyard are particularly at risk. Thus, a refined three-dimensional finite element model was also assembled in order to get a deeper knowledge ofthe kinematics and the stress situation of this part of the building in the linear elastic field.In this case, the geometry was modelled with great detailed, and the real shape of the cross-vaults, the ribs, and all the structural details that can be significant for the stress analysis wereimplemented. Moreover, the different construction materials adopted in the real building couldbe assigned with some precision within the limits of the linear elastic field. This detailed modelhas a total number of 48004 8-node brick elements.The three-dimensional models showed that the kinematics of the whole castle is dominated bythe first natural mode of vibration, that can be very well described on the basis of a plane model.Clearly, to do this, it was necessary to perform a fine tuning on the characteristics the differentmaterials that constitute the model with the objective of obtaining a good approximation of thereal three-dimensional behaviour. Six different materials were assigned to this discrete model.The numerical response of this model proved to be satisfactory in the linear elastic field. Thefirst natural mode of vibration was in good agreement with the previous finite element models,both in terms of frequency and of global shape. Moreover, also the estimated values of stresswere correct, in particular for what regards the vertical axial component in the columns and inthe buttress that were the more critical parts of this structure.4 THE QUTB MINARThree different numerical models were considered to evaluate the structural behaviour of theMinaret [4]. The first model uses 3-D solid elements based on Finite Element Method (3-DModel), the second model was performed with 3-D composite beams based on Mindlin-Reissnertheory (Beam-Model) and 2D in-plane elements based on the Rigid Element Method were usedin the third model (Rigid- Model).The Three Dimensional Solid Model (3-D model) was implemented by using the Finite Ele-ment Method. Eight-node isoparametric solid brick elements, based on quadratic interpolationand 3x3x3 Gauss integration, were employed. The foundation, the doors and windows of theminaret were also modelled. The full model involves 65,912 elements with 57,350 nodes, result-ing in about 172,000 degrees of freedom (DOF). As a boundary condition, it was consideredthat the base of the foundation was fully restrained.The three dimensional beam model (Beam model) was performed based on the Finite ElementMethod. Three dimensional beams elements of three nodes based on the Mindlin- Reissnertheory were used. In order to model the different layers of materials, as well as the centre core ofthe minaret, composite beams were used. The model has 41 nodes (120 DOF) and 20 elements.Each composite beam element was defined with four different pipe sections in order to take intoaccount the different layers and one circular section that defines the centre core. In this modelis not possible to model the influence of the staircase as in the three-dimensional solid modeland a perfect connection between the shaft and the core is considered.3Fernando Pen˜a, Siro Casolo and Paulo B. Lourenc¸oA simplified in-plane model of the minaret based on the Rigid Element Method (REM) wasdeveloped. In total, the numerical model has 39 elements and 117 DOF. The rigid elementsused can be only defined with a rectangular cross section. For this reason an equivalent squarecross section and equivalent isotropic material were considered.The advantage of this simplified model is that it is possible to perform non-linear dynamicanalysis with few computational resources and in less time than Finite Element Models (Beamand 3-D models). For example, the Beam model requires 23 hours to perform a non-lineardynamic analysis, while the RE model needs only 20 minutes. Therefore, this simplified modelwas used to have preliminary results about the global dynamic behaviour of the structure.5 CONCLUSIONS1. The use of a simplified model (REM) allowed performing non-linear dynamic analyses withfew computational resources with less time than other more complex models, allowing toobtain valuable results about the global behaviour of the structure.2. The use of two or more analysis tools allows to overcome the complexity on the study ofthe seismic behaviour of ancient masonry structures.3. The results obtained from one analysis tool are used as preliminary results for the betterconception of another model using a different analysis tool. Moreover, by combining theresults of the different models it is possible to obtain a better and more comprehensiveinterpretation of the seismic behaviour.4. The approach followed in the Maniace Castle was to create three-dimensional linear finiteelement models in order to find the most critical part of the structure. The Rigid ElementMethod was then used to perform non-linear dynamic analysis of the critical part of thestructure.5. In the case of the Qutb Minar, the Rigid Element Method was used to obtain preliminaryresults about the global behaviour of the structure, in which the more complex finiteelement models were based.REFERENCES[1] S. Casolo, Modelling in–plane micro–structure of masonry walls by rigid elements. Interna-tional Journal of Solids and Structures, 41, 3625–3641, 2004.[2] S. Casolo, A specific Rigid Element Model for Macro-Scale Dynamics of Monumental Ma-sonry considering Damage and Micro-Structure effects. Proc. Structural Analysis of Histor-ical Constructions, SAHC, Lourenc¸o, Roca, Modena & Agrawal Eds., 939–946, (2006).[3] S. Casolo, F. Pen˜a, Rigid element model for in–plane dynamics of masonry walls consideringhysteretic behaviour and damage. Earthquake Engineering and Structural Dynamics, 36,1029–1048, 2007.[4] ECCP, Benchmarking on the seismic behaviour of the Qutb Minar. EU–India EconomicCross Cultural Programme, (2006).4

Seismic analysis of masonry monuments by an integrated approach that combines the finite element models with a specific mechanistic model

Summary. The paper presents a strategy for the non-linear dynamical analysis of ancient masonry structures through two case studies (”Maniace Castle ” in Syracuse, Italy and the Qutb Minar in Delhi, India). 

D. R. J. Owen (eds

Siro Casolo

Paulo B. Lourenço

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SEISMIC ANALYSIS OF MASONRY MONUMENTS BY AN INTEGRATED APPROACH THAT COMBINES THE FINITE ELEMENT MODELS WITH A SPECIFIC MECHANISTIC MODEL

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Seismic analysis of masonry monuments by an integrated approach that combines the finite element models with a specific mechanistic model

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