CONSTRUCTION PROCESS NUMERICAL SIMULATION AND SEISMIC ASSESSMENT OF MALLORCA CATHEDRAL

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1 CONSTRUCTION PROCESS NUMERICAL SIMULATION AND SEISMIC ASSESSMENT OF MALLORCA CATHEDRAL Roca, Pere 1 ; Pelà, Luca 2 ; Cerera, Miguel 3 ; Clemente, Roberto 4 1 PhD, Professor, Technical Uniersity of Catalonia (UPC), EC Department, pere.roca.fabregat@upc.edu 2 PhD, Lecturer, Technical Uniersity of Catalonia (UPC), EC Department, luca.pela@upc.edu 3 PhD, Professor, Technical Uniersity of Catalonia (UPC), RMEE Department, miguel.cerera@upc.edu 4 PhD, Researcher, Technical Uniersity of Catalonia (UPC), CIMNE, clemente@cimne.upc.edu This paper presents a numerical study of Mallorca Cathedral carried out by means of a FE approach deised for the study of this complex historical construction. Preious studies, including inspection and historical research, hae shown that part of the existing damage and deformation might hae been experienced during the construction process itself, while later historical processes causing long-term deformation, may also hae contributed significantly to the final deformation. In order to analyse the possible influence of the construction process and long term deformation on the deformation of the structure, a numerical tool has been deeloped to carry out sequential-eolutionary analyses, inoling the superposition of consecutie construction stages. A constitutie model has been implemented accounting for both iscoelasticity and mechanical damage by means of an enhanced continuum damage model. This tool has been used to carry out the sequential FE analysis of a typical bay structure of the main nae of the building. The proposed numerical tool has been also used to assess the seismic performance of the typical bay, in the transerse direction, through a nonlinear static analysis. The proposed numerical strategy seems effectie to describe deformation and damage and could be applied to other similar historical masonry constructions. Keywords: Historical Construction, Continuum Model, Long-term Effects, Creep, Seismic Analysis, Localized Damage. INTRODUCTION The Cathedral of Santa Maria in Palma, Mallorca, Spain, is one of the most imposing Gothic constructions of the Mediterranean area. The structure combines extraordinary dimensions and ery slender members, see Figure 1a. The piers slenderness, reaching a ratio of 14.2 between height and circumscribed diameter, constitutes one of the more audacious aspects of the structure. A detailed historical inestigation was carried out in order to understand the construction process of the cathedral (Domenge, 1997). It was possible, at least for one of the bays (the 4 th one starting from the East end), to identify the process leading to its complete construction, see Figure 1b-e. According to this inestigation, the lateral chapels were erected firstly, followed by the pillars, then one lateral ault, then the other and finally the central one. It is

2 worth noticing that during a period of about 5 years, the lateral aults were pushing against the pillars while the central ault was not yet there to counteract their thrust. Figure 1: Mallorca cathedral: internal iew (a) and construction stages (b-e). One of the more noticeable structural anomalies detected in the construction are the significant lateral deformation affecting the piers. In some of the piers, the lateral deformation reaches up to 26 cm, i.e. almost 1/90 of their height at the springing of the lateral aults. Howeer, this lateral deformation is ery ariable both in magnitude and direction among the piers, making it difficult to identify a common trend. Mallorca Cathedral has been monitored since 2003 (Roca & Gonzales, 2008). Monitoring has shown that this deformation is still increasing at present at a slow rate. A ariation ratio of the distance of the piers across the transerse span of the bays of about 0.1 mm per year has been measured by means of baseline extensometers. The study presented herein is aimed at characterizing the performance of the typical bay of Mallorca Cathedral under graity and seismic actions. A particular aim is found in the inestigation of the possible influence of the construction process and later long-term deformation on the deformed condition of the building. For this purpose, a sequential numerical analysis is carried out, in which the changes experienced by the construction are subsequently simulated and superposed. The time-dependent analysis includes the simulation of joint influence of geometric nonlinearity and long term deformation. The FE analysis of the typical bay subjected to horizontal earthquake static equialent forces in the direction transerse to the nae is also presented.. It is worth noticing that an enhanced FE tool has been deised specifically for the present study.. It includes a iscoelasticity and mechanical damage model, a FE actiation strategy for sequential analysis and a crack-tracking algorithm for localize damage simulation. The computational model is summarized in the following sections and then the results of the FE analyses are presented and discussed. VISCOELASTICITY MODEL The rheological model can be schematized through the Maxwell s chain shown in Figure 2a. The first chain element is composed by a spring with elastic stiffness E, whereas the second element is composed by a spring with elastic stiffness E, arranged in series with a dashpot with a iscosity parameter. The springs response is linear elastic whereas the iscous stress in the dashpot is proportional to the iscous strain rate.

3 The initial stiffness of the system is gien by the sum of the stiffnesses of the two springs, being the dashpot of the Maxwell s chain infinitely stiff at the beginning of the deformation process. Thus, the instantaneous elastic modulus E can be defined as follows: E E E (1) The stiffness of the system for t =+ is equal to E, since the dashpot is completely slackened at the end of the deformation process. The total stress sustained by the Maxwell s chain is gien by the sum of the stresses in the two elements: E E (2) in which E E is the participation ratio which denotes the amount of stiffness susceptible to iscosity. The total deformation of the system is denoted by, whereas denotes the iscous strain of the chain which increases with time under a constant stress. The phenomenological behaiour of the model is depicted in Figures 2b-d, which also show the effect of the so-called retardation time / E on the time-dependent increase of strain or decrease of stiffness. Figure 2: Viscoelasticity model: a) schematization through a Maxwell chain and strain (b), stress (c) and stiffness (d) time-dependent laws. The strain rate of the system is defined by the following equation: E (3) Thus, the first order differential equation goerning the eolution of the iscous stress is gien by:

4 E (4) The preious equation can be rewritten for the multidimensional case, using the tensorial counterparts of the scalar terms used for the uniaxial model: Cε σ σ (5) With the aim of assuming the iscous strain in the Maxwell s chain as internal ariable, the relationship σ C ε ε (6) can be included in Equation (5), leading finally to the eolution law for the iscous strain: 1 ε εε (7) The solution of the differential equation for a generic time step t n+1 can be obtained by integrating the preious equation, leading finally to (Cerera, 2003) ε ε t n1 n n1 n ε ε (8) t t t t TENSION-COMPRESSION DAMAGE MODEL The mechanical damage in masonry due to cracking and crushing is described by the Tension- Compression Damage Model deeloped by Cerera et al. (1995). The model is based on a split of the effectie stress tensor into tensile and compressie components: 3 i i i and i1 σ p p σ σ σ (9) where i denotes the i-th principal stress alue from effectie stress tensor σ, p i represents the unit ector associated with its respectie principal direction and the symbols. are the Macaulay brackets x x, if x 0, x 0, if x 0. Two internal damage ariables d and d are defined, each related with the sign of the stress and thus with tension and compression. They are equal to zero when the material is undamaged and equal to one when it is completely damaged. The constitutie equation takes the form: 1d 1d σ σ σ (10)

5 Different damage criteria are assumed for tension and compression stress states (Cerera, 2003) in order to describe different failure mechanisms for masonry, i.e. cracking and crushing of the material, see Figure 3. Exponential eolution laws are assumed for the damage indexes d, depending on the material tensile and compressie fracture energies G. f Figure 3: Damage criteria adopted for masonry. TENSILE CRACK LOCALIZATION The classical smeared crack approach, based on standard finite elements and Continuum Damage Mechanics models, proides an approximate representation of the damaging process occurring on the material. This is more eident in case of tensile damage, which is portrayed as a spreading phenomenon inoling large regions of the construction. Conersely, indiidual large cracks are normally experienced by masonry structures in the ultimate condition. Such limitation is oercome in this work adopting the crack-tracking technique proposed by Cerera et al. (2010), which forces the tensile crack to deelop along a single row of finite elements according to the direction of the main tensile stress. The generation of localized cracks, acting as plastic hinges, represents more realistically the behaiour of the structure in the ultimate condition (Pelà, 2009). The proposed method is applied at eery time step during the FE analysis, just before the stress ealuation. The algorithm is able to detect the point of the boundary of the structure where a crack is originated. Making use of a flag system, finite elements are then labelled to delimit the zones where cracks will appear or deelop. The criteria used to define these zones depend on the magnitude and direction of the principal stresses at each element. A minimum distance between two crack root elements, called usion radius, is used to guarantee the creation of separated discrete cracks. The algorithm ensures mesh-bias and element-size objectie FE results and has been implemented for 2D problems using three-noded triangular elements. FE ACTIVATION TECHNIQUE FOR SEQUENTIAL ANALYSIS A finite element actiation procedure has been deeloped to reproduce the addition of different structure portions during the building stages. This strategy classifies the elements of the oerall FE mesh into actie and inactie. At the beginning of the analysis, the elements which define the first portion built are actiated, i.e. computed and assembled into the global matrix, whereas the inactie elements are disregarded in calculations. In the following step, the elements corresponding to the next construction stage are actiated and the calculation proceeds, considering the first portion already deformed. By repeating such procedure until

6 the completion of all building stages, it is possible to obtain a numerical simulation of the whole construction process. An important adantage of the proposed actiation technique is the possibility of defining the computational mesh independently of the construction process. Different hypotheses about the building stages can be considered by simply changing the actiation sequence or the grouping of elements. This is ery useful in case of historical constructions, where comparatie studies are often necessary in order to assess the most critical construction process that might be experienced by the structure. NUMERICAL SIMULATION OF THE CONSTRUCTION PROCESS AND LATER LONG TERM DEFORMATION The iscoelasticity model, the tension-compression damage model, the FE actiation strategy and the crack-tracking techniques discussed preiously hae been implemented into the FE program COMET (Cerera et al., 2002) deeloped at the International Centre for Numerical Methods in Engineering (CIMNE, Barcelona). Pre- and post-processing hae been carried out with GiD (2002), also deeloped at CIMNE. The analysis of a single typical bay has been carried out on a model including piers, buttresses, flying arches and aults of the nae and the aisles. Such macroelement has been considered as the most representatie for the purpose of inestigating the possible link between construction process and existing deformation in the transerse direction A macromodelling approach with a continuum FE model has been considered in computations, see also Roca et al. (2010) and Pelà et al. (2011). Based on preious inspection work, three groups of materials hae been distinguished for different structural members. The first includes buttresses, aults, ribs and clerestory, whose properties were assumed as follows: Young s modulus E=2000 MPa, Poisson s ratio =0.2, tensile and compressie strengths f + =0.1 MPa and f - =2 MPa. The second group includes columns and flying arches, with E=8000 MPa, =0.2, f + =0.4 MPa and f - =8 MPa. The properties of the material of the central ault backing are E=1000 MPa, =0.2, f + =0.05 MPa and f - =1 MPa. Values for the fracture energies hae been assumed for all materials (G f + =100 J/m 2, G f + =40000 J/m 2 ) based on preious experience in similar masonry types. The retardation time is assumed arbitrarily as =50 time units. Its effectie entity is not significant and has to be related only to the total number of time steps in calculations. The time is measured in pseudo-time without fixed quantitatie relationship with real time. In this way, the possible influence of long-term deformation on the structure can be studied een if its real deelopment ratios in the historical time are not known. Concerning the participation ratio, two different alues hae been considered, and, so as to produce different responses to the structure. These assumed alues are great enough to analyse the structure under extremely aderse conditions. Geometric nonlinearity has been considered through a total Lagrangian formulation with the assumption of small-strain/largedisplacement. The numerical simulation of the construction process consists of three subsequent analysis steps, in compliance with the information about the building stages proided by the historical inestigation. In the first step (see Figure 4a), the pier, the aisle ault and the buttress are actiated in the FE model. In the second analysis step (see Figure 4b), the upper part of the buttress, the flying arches, the clerestory, the nae ault are subsequently actiated and the calculus is carried on starting from the stress-strain state obtained at the end of first analysis.

7 Finally, the structure is subject to constant loading and the time starts elapsing in order to ealuate the deformation accumulation due to creep (see Figure 4c). Figure 4: Tensile damage in Mallorca Cathedral typical bay after the stages of construction (ab), and due to material creep (c). Figure 5: Horizontal displacement increase at pier top due to creep. Although significant monitoring information is aailable at present, it is still difficult to identify and simulate in an accurate way the real long-term deformation trends experienced along the history of the building. The iscolelastic model adopted, howeer, permits an inestigation on the joint influence of long term deformation and geometric nonlinear effects on the stability of the building. Figure 5 shows the maximum horizontal displacements eolution at the pier due to long term deformation. The maximum displacement occurs at the leel of the lateral ault. For the lower alue of the participation ratio,, the pier horizontal time-dependent displacement reaches a stable alue of 12 cm after 3,000 time units. It is worth noticing that a conentional instantaneous analysis of the cathedral bay, i.e. without resorting to the construction process simulation with iscoelasticity model and geometric nonlinearity, leads to a horizontal displacement at the pier top of only 0.76 cm. The assumptions of geometric nonlinearity and lead to the simulation of the structure collapse due to the buckling of the piers as shown by the corresponding cure in Figure 5 at 2,000 time units. The cure shows a significant increase of the deformation

8 elocity for about 40 cm of maximum lateral deformation reached. It can be concluded that the numerical simulation can represent the failure condition only for extremely high alues of the participation ratio. This order of magnitude is comparable to the real displacements recently measured in Mallorca Cathedral bays, showing the possibility that creep phenomena and geometric effects had played a significant role during the life of the structure. SEISMIC LOAD ANALYSIS The typical bay seismic performance has been assessed by means of a pushoer analysis consisting of the gradual application of a system of lateral equialent static forces on the structure, see also Pelà et al. (2009). The analysis has been carried out using the cracktracking technique to simulate the localized tensile damage. Since such numerical tool is implemented for 2D problems, a plane-stress FE model equialent to the 3D model of the bay has been prepared by maintaining the weights of different structural elements. The thickness of different components hae been modified in such a manner that the 2D and the 3D FE models present equialent deformed shapes after a linear elastic analysis. Two loading conditions hae been applied in consecutie phases. The graity load is applied in the first step. In the second step, the lateral forces proportional to mass distribution are applied and increased gradually until reaching failure. Different analyses hae been carried out considering three different alues for the usion radius r in order to understand the influence on results: 1 m, 2 m and 3 m. The usion radius defines the minimum distance imposed between two crack root elements, and it is used to guarantee the creation of separated discrete cracks (Cerera et al., 2010). The lowest alue of r that has been assumed in the analyses corresponds approximately to the dimension of a stone unit. Figure 6 shows the seismic load multiplier (defined as a fraction of graity acceleration) against the horizontal displacement at the top of the piers. The smeared damage model causes failure for a load factor of about The localized damage model produces a higher failure load factor, ranging between 0.1 for r 1 m and 0.12 for r 3 m, as expected due to restrictions that the model imposes to the formation and propagation of damage. It is worth noticing that such seismic load multipliers are similar to the design alues which can be deried from the Spanish seismic proisions NCSE-02 (2002) for Mallorca Island with a return period of 1000 years. Figure 6 Seismic load multiplier s. horizontal displacement.

9 Figure 7 depicts the deformations and the tensile damage distribution obtained by the analyses with smeared damage model and localized damage model. It is eident how the former approach can only proide an approximate description of the damage experienced by the structure under horizontal loads. On the other hand, the latter model is more accurate and it can show the possible locations of the tensile cracks, allowing us a better understanding of the real collapse mechanism under seismic loading. As expected, the model with r 3 m leads to the representation of a lesser number of cracks. The use of lower usion radii seems more suitable since it leads to a clear representation of the tensile cracks experienced by the structure under ultimate conditions. The more affected portions are the base of columns and buttresses, the aults and the flying arches. It is worth noticing how the presence of the big false windows (modelled as real windows in the analysis) entails the propagation of cracks which weaken the buttresses. Figure 7 Deformed shape and tensile damage obtained by seismic analysis: a) smeared damage model, b) localized damage model with r 1m, c) r 2m and d) r 3m. CONCLUSIONS This paper has presented a numerical study of Mallorca Cathedral, as a final step of a detailed research including also structural inspection and historical inestigation. A special FE tool has been deised specifically for this complex structure to understand the possible reason of large deformation of the structure (and particularly that of the nae piers) and to assess the structural behaiour under seismic horizontal forces. The deeloped computational tool includes a iscoelasticity and mechanical damage model, a FE actiation strategy for sequential analysis and a crack-tracking algorithm for localize damage simulation. The analyses suggest that the current large deformation obsered in the bays are due to deformation attained during the construction process and later time-dependent iscous phenomena experienced by the different masonries of the structure. The analyses hae shown that a significant increase of the maximum lateral deformation attained by the piers might lead

10 to collapse due to geometrical instability. Although the real deformation of the structure is still far from the limits for which such instability is attained, and the increasing deformation ratio is low at present (as shown by the monitoring), a long-term surey of deformation by means of detailed monitoring is recommendable. The FE analysis of the typical bay structure to transerse earthquake equialent forces has proided an understanding of the seismic performance of the typical bay structure at a ery reasonable computer effort. REFERENCES Cerera, M., Olier, J., Faria, R. Seismic ealuation of concrete dams ia continuum damage models Earthquake Engineering & Structural Dynamics, 24, 9, 1995, pp Cerera M., Agelet de Saracibar C., Chiumenti M. COMET: COupled MEchanical and thermal analysis data input manual ersion 5.0. CIMNE, Technical Uniersity of Catalonia, Barcelona, 2002, 182pp. Cerera, M. Viscoelasticity and rate-dependent continuum damage models. CIMNE, Barcelona, 2003, 76pp. Cerera, M., Pelà L., Clemente, R. and Roca, P. "A crack-tracking technique for localized damage in quasi-brittle materials", Engineering Fracture Mechanics, 77, 2010, Domenge J. L obra de la Seu. El procés de construcció de la catedral de Mallorca en el trescents (in Catalan), Palma de Mallorca, NCSE-02. Norma de construcción sismorresistente. Parte general y edificación (in Spanish). Boletín Oficial del Estado, 2002, 70pp. Pelà, L. Continuum Damage Model for Nonlinear Analysis of Masonry Structures. PhD- Thesis, Technical Uniersity of Catalonia, Uniersity of Ferrara, 2009, 276pp. Pelà, L., Aprile, A. and Benedetti A. "Seismic assessment of masonry arch bridges", Engineering Structures, 31, 2009, pp Pelà, L., Cerera, M. and Roca, P. Continuum damage model for orthotropic materials: Application to masonry", Computer Methods in Applied Mechanics and Engineering, 200, 2011, pp Roca, P., González, J.L. Estudio, diagnóstico y peritación y en su caso planteamiento de actuaciones sobre el comportamiento constructio-estructural de la catedral de Santa María, en la ciudad de Palma, isla de Mallorca, Baleares (in Spanish). Technical Uniersity of Catalonia, Barcelona, Roca, P., Cerera, M., Gariup, G. and Pelà, L. "Structural analysis of masonry historical constructions. Classical and adanced approaches". Archies of Computational Methods in Engineering, 17, 2010, pp website of CIMNE, Technical Uniersity of Catalonia, Barcelona.