Seismic analysis of dry stone masonry structures based on combined finite-discrete element method

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1 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: ; ISBN: Seismic analysis of dry stone masonry structures based on combined finite-discrete element method Željana Nikolić 1, Hrvoje Smoljanović 1, Nikolina Živaljić 1 1 University of Split, Faculty of Civil Engineering, Architecture and Geodesy, Split, Croatia zeljana.nikolic@gradst.hr, hrvoje.smoljanovic@ gradst.hr, nikolina.zivaljic@gradst.hr ABSTRACT: Dry stone masonry building technique is one of the oldest techniques which has survived to the present. Most of ancient dry stone masonry structures were constructed by assembling regular stone blocks without the addition of mortar between bed and head joints, and if used, the mortar was usually of low strength and hence it has experienced mechanical degradation over time which means that its influence on the behaviour of such structures is negligible. This paper presents the performance of a combined finite-discrete element method (FEM/DEM) for seismic analysis of dry stone masonry structures. In the proposed modelling approach each stone block is modelled as a discrete element which is discretized by constant strain triangular finite elements. Material non-linearity including fracture and fragmentation of discrete elements as well as cyclic behaviour during dynamic load are considered through contact elements which are implemented within a finite element mesh. The interaction between discrete elements is considered through the contact interaction algorithm for the normal forces and through the Coulomb-type law for friction. The contact interaction is resolved on the principle of potential contact forces. The method uses an explicit numerical integration of the equation of motion. The numerical analysis based on experimental test data has been carried out to simulate the main features of dry stone structures. The performed analysis shows high accuracy of the numerical results in comparison with the experimental ones and demonstrates the potential of the FEM/DEM method for realistic modelling of the response of dry stone masonry structures. KEY WORDS: Finite-discrete element method (FEM/DEM); Cyclic load; Seismic load; Dry stone masonry structures. 1 INTRODUCTION The dry stone masonry building technique is one of the oldest techniques which have survived to the present. Most of ancient dry stone masonry structures were constructed by assembling regular stone blocks without the addition of mortar between bed and head joints, and if used, the mortar was usually of low strength and hence it has experienced mechanical degradation over time which means that its influence on the behaviour of such structures is negligible. The main characteristic of the stone structure is its composite nature consisting of stone blocks separated by bed and head joints which may or may not be filled with mortar. The composite nature of dry stone masonry structures is the cause of extremely complex behaviour which represents a real challenge for numerical modelling. The main motivation for writing this paper was to present a numerical model suitable for seismic analysis of dry stone masonry structures. Its application enables the evaluation of the seismic resistance of dry stone masonry structures and assists in making right decisions when undertaking certain interventions to increase the seismic resistance. The FEM/DEM was developed mainly for the simulation of fracturing problems considering deformable blocks that may split and separate during the analysis. Within the framework of this method the blocks are discretized by constant strain triangular finite elements. Material non-linearity including fracture and fragmentation of discrete elements as well as cyclic behaviour during dynamic load are considered through contact elements which are implemented within a finite element mesh. The interaction between discrete elements is considered through the contact interaction algorithm for the normal forces and through the Coulomb-type law for friction. The contact interaction is resolved on the principle of potential contact forces. The method uses an explicit numerical integration of the equation of motion. The advantage of the combined finite-discrete element method in the analysis of the response of dry stone masonry structures is the possibility of modelling the fragmentation of each block which is especially interesting when analysing the failure modes of structures under hazardous loads. 2 THE COMBINED FINITE-DISCRETE ELEMENT METHOD The combined finite-discrete element method simulation [1] comprises a large number of particles which are represented by a single discrete element that interacts with discrete elements close to it. Each discrete element has its own finite element mesh which is used to analyze the particle deformability. The material non-linearity including elastic hysteresis, fracture and fragmentation of discrete elements is considered through contact elements which are implemented within the finite element mesh. The main processes included in the FEM/DEM method are contact detection, contact interaction, finite strain elasticity as well as fracture and fragmentation. A contact detection algorithm has to detect couples of discrete elements close to each other and eliminate couples which are too far and cannot possibly be in contact. In the FEM/DEM code a Munjiza NBS (no binary search) contact detection algorithm [2, 3] is implemented. 1777

2 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 The contact interaction between discrete elements including Coulomb-type law for friction is taken into account by a distributed potential contact force [1]. The stress and strain relation, in constant strain triangular finite element, is implemented using Hooke's law according to: E ( E ( T = Ed + Es + μd (1) 1+ υ 1 2υ where T is the Cauchy stress tensor, E is the modulus of elasticity, υ is Poisson ratio, E ( d is the shape changing part and E ( s is the volume changing part of Green-St. Venant strain tensor, μ is the damping coefficient and D is the rate of the deformation tensor [4, 5]. Fracture and fragmentation are realized by a combined single and a smeared crack model [6], where the behaviour of the stone in compression is linear elastic while a stress strain curve for stone in tension is divided into two sections (Fig. 1): strain-hardening prior to reaching the peak stress ft, which is implemented through the constitutive law [6] and strainsoftening which is based on an approximation of the experimental stress-displacement curves. The cracks are assumed to coincide with the finite element edges which are achieved in advance through the topology of adjacent elements being described by different nodes. Separation of these edges induces a bonding stress which is taken to be a function of the size of separation δ (Fig. 1). features related to cyclic tensile behaviour with stiffness degradation between cycles [8]. Discretization of dry stone masonry structure in the framework of FEM/DEM method is shown in Fig. 2. Each stone block is modelled as a discrete element which is discretized by triangular finite elements. Material nonlinearity, fracture and fragmentation are considered through contact elements which are implemented within the finite element mesh. Figure 2. Discretization of dry stone masonry structure. 3 NUMERICAL EXAMPLES 3.1 Shear behaviour of stone masonry joints Shear behaviour between two stone blocks was analyzed in order to validate the relation between normal and shear stresses in dry stone masonry joints. This example was tested by Vasconcelos and Lourenço [9] with two different precompression stresses (0.5 MPa, 0.75 MPa and 1.0 MPa). After applying the pre-compression vertical load the shear tests were carried out under displacement control by imposing reversal displacement history shown in Fig Figure 1. Strain hardening and strain softening curves defined in terms of strains and the strain softening curve defined in terms of displacements. In this paper the experimental stress-displacement curve for softening branch is taken according to [7] where is an existing constitutive interface model extended to capture the main displacement / mm time Figure 3. Displacement history for cyclic shear test on dry masonry joints. 1778

3 The geometry of the blocks and finite element mesh are shown in Fig. 4. In the numerical analyses the penalty term for friction k t was equal to Figure 4. Test specimen: geometry of the blocks and finite element mesh. The results of this model based on the FEM/DEM method were compared with the experimental results in Fig. 5. The shear stress-displacement curves show very good agreement of the results obtained by the code based on the FEM/DEM method with the experimental ones. These results show that the FEM/DEM model is capable of predicting the shear behaviour of dry joints between stone blocks. (c) Figure 5. Shear stress-shear displacement diagrams under cyclic loading for the pre-compression stress: 0.5 MPa; 0.75 MPa and (c) 1.0 MPa. 3.2 Seismic analysis of the Prothyron in Split Croatia The following example shows the application of the FEM/DEM method for the simulation of the dynamic response of the structure of the Prothyron in Split (Fig. 6) to seismic load. Prothyron is a space which connects or separates the north part of Diocletian s Palace, which was used by servants, army etc., from the south part where the emperor s apartment was situated. The Prothyron is located on the south side of the Peristyle (the square in front of the Cathedral of St. Domnius). Four massive granite columns with Dorian capitals are located at the entrance of the Prothyron. The capitals support the broad gable with an arch in the middle. The structure was originally built of stone blocks with mortar of very low strength, so that its influence on the mechanical characteristics and behaviour of the whole structure do not have to be considered. Throughout history deformations of the stone blocks which constitute the broad gable have occurred with the movement of the central columns. Due to the movement of the blocks, restoration of the structure was performed using metal clamps during the period of the Austro-Hungary Empire. 1779

4 Figure 6. The Prothyron entrance at the Peristyle in Split. In order to evaluate the dynamic response of the structure, the incremental dynamic analysis of the original structures before the restoration was applied. The structure was exposed to horizontal ground acceleration (Fig. 7) which was recorded on in Dubrovnik on rock soil during an earthquake with the epicentre in Petrovac (Montenegro). The accelelogram was firstly scaled on peak ground acceleration a g =0.22g which is valid for Split. After that the acceleration was gradually increased to the collapse of the structure. acceleration / m/s time / s Figure 7. Horizontal ground acceleration recorded during the Petrovac earthquake (1979. Montenegro). Fig. 8 and Fig. 9 respectively show the geometry of the structure and finite element mesh. Figure 9. Finite element mesh of structure. Material characteristics used in the numerical analysis are shown in Table 1. Table 1. Material characteristics of the stone. Young s Modulus E (MPa) Static sliding friction µ st Dynamic sliding friction µ din Dumping coefficient Dynamic analysis of the structure showed that for the acceleration of a g =0.22g, the movement of the blocks, i.e. opening of the joints appeared, as shown in Fig 10. μ Figure 8. Geometry of the Prothyron structure. In the numerical analysis it was assumed that a collapse would be caused due to the loss of stability of the structure, so fracture and fragmentation of stone blocks were not considered. Figure 10. The Prothyron structure for acceleration a g =0.22g: whole structure and enlarged central section between the two central columns. Subsequently, the acceleration was gradually increased. For a g =0.35g the displacements of the gable stone blocks were significant. Fig. 11a shows the form of the structure for the 1780

5 applied accelelogram, while Fig. 11b shows the enlarged central section between the two central columns. The figures show major displacements of the blocks in the upper part of the central columns with separation of the capitals from the upper beams. Figure 12. The Prothyron structure for acceleration a g =0.50g: whole structure and enlarged central section between the two central columns. Figure 11. The Prothyron structure for acceleration a g =0.35g: whole structure and enlarged central section between the two central columns. The form of the structure for acceleration a g =0.5g is shown in Fig. 12. The collapse of the structure occurred at an acceleration of a g =0.6g. Fig. 13 shows gradual separation of the two central columns, with the ultimate collapse of the central part of the structure after the central columns were sufficiently separated. (c) 1781

6 an analysis for a set of seven accelelograms compatible with the soil type [10]. (d) 4 CONCLUSIONS This paper presents the application of the FEM/DEM method for a 2D analysis of dry stone masonry structures. The model based on a combined finite discrete element method has the capability to describe the monotonic and cyclic behaviour of dry stone masonry walls with the possibility of the fragmentation of each block which can be important in the analyses of structures under high compressive loads. The comparison between numerical results obtained by a model based on a combined finite discrete element method and the experimental results shows that this numerical model simulates the main features of dry stone structures such as shear behaviour in dry joints between stone blocks. The performed incremental dynamic analysis of a real historical structure showed that the model is capable of predicting the collapse mechanism of dry stone masonry structures under seismic loads as well as determining the safety of the structure with regard to the occurrence of the collapse load. ACKNOWLEDGMENTS (e) (f) Figure 13. Collapse mechanism of the Prothyron structure for acceleration a g =0.60g in time: t=0.0 s; t=11.91 s; (c) t=13.27 s; (d) t=16.33 s; (e) t=17.86 s; (f) t=18.54 s. The performed analysis shows that the original structure before its restoration with the metal clamps had low seismic resistance because the significant displacement of the central block occurred at design acceleration a g =0.22g. Large displacement of the blocks arises for peak ground acceleration of a g =0.35g (which is 59% higher in relation to the design acceleration), while the collapse of the structure occurs for a g =0.5g. The purpose of this analysis was to show the possibility of the presented model based on a combined finite discrete element method in seismic analyses of dry stone masonry structures. Relevant conclusions for the seismic resistance of this structure can be carried out on the basis of The partial financial support provided by the Ministry of Science, Education and Sports of the Republic of Croatia for the Non-linear Dynamic Analysis of Three-dimensional Reinforced Concrete Structures project, Grant. No , is gratefully acknowledged. REFERENCES [1] A. Munjiza, The combined finite-discrete element method, UK: John Wiley & Sons, [2] A. Munjiza, K.R.F. Andrews, J.K. White, NBS contact detection algorithm for bodies of similar size, International Journal for Numerical Methods in Engineering, Vol. 43, pp , [3] A. Munjiza, E. Rougier, N.W.M. John, MR linear contact detection algorithm, International Journal for Numerical Methods in Engineering, Vol. 66 (1), pp , [4] A. Munjiza, D.R.J. Owen, N. Bicanic, A combined finite-discrete element method in transient dynamics of fracturing solids, Engineering Computations Vol. 12, pp , [5] A. Munjiza, D.R.J. Owen, A.J.L. Crook, An M(M-1K)m proportional damping in explicit integration of dynamic structural systems, International Journal for Numerical Methods in Engineering Vol. 41 (7), pp , [6] A. Munjiza, K.R.F. Andrews, J.K. White, Combined single and smeared crack model in combined finite-discrete element method, International Journal for Numerical Methods in Engineering, Vol. 44, pp , [7] N. Živaljić, H. Smoljanović, Ž. Nikolić, A combined finite-discrete element model for RC structures under dynamic loading, Engineering computations, Vol. 30, pp , [8] H. Smoljanović, N. Živaljić, Ž. Nikolić, A combined finite-discrete element analysis of dry stone masonry structure, Engineering structures, Vol. 52, pp , [9] G. Vasconcelos, P.B. Lourenço, Experimental characterization of stone masonry in shear and compression, Construction and Building Materials, Vol. 23, pp , [10] Comite European de Normalization (CEN), Eurocode 8: Design of structures for Earthquake resistance,en , Brussels,