Keywords: 3D finite elements, solid finite elements, masonry structures, stress analysis, failure analysis.

The implementation of three-dimensional solid finite elements when performing finite element analysis of historical structures and monuments is discussed. For this category of structures, consisting usually of masonry systems, high standard requirements, contradict modeling assumptions that lead to simplifications implied by the use of shell elements. To overcome lost accuracy, the use of solid elements is proposed.

The same necessity also appears especially when abrupt changes in the building shape occur, for example at the areas of roof-wall intersections, openings, vaults, domes etc. As a result of these geometrical peculiarities, area dimensions of finite elements are necessarily small due to the proper meshing requirements and their thickness, which usually represents the width of the wall, even greater than their other two dimensions. In this case the use of three-dimensional solid elements is imposed in order to overcome lost accuracy.

The solid finite elements used in this paper are an eight-node hexahedron brick and a six-node pentahedron wedge. They are isotropic and they have three translational degrees of freedom for each node. Global Cartesian coordinates are expressed on the basis of local coordinates, by linear polynomial shape functions. They are considered to be isoparametric. The Jacobian matrix and stiffness matrix are evaluated as in [1,4].

The efficiency of the method is demonstrated through application to a Byzantine monastery, situated in Athens. Its support system consists mainly of regular shape stone masonry. Wall width variations, niches, openings and other particular geometrical configurations reveal the efficiency of three-dimensional solid finite elements in simulating the structure in the most realistic way that looks like the physical system.

In the case of three-dimensional elasticity consideration, three principal stresses are produced, each one corresponding to the relevant principal axis. In order to adapt the problem of determination of mechanical failure to considerations made for two-dimensional assumptions, the influence of the third occurring principal stress over the other two remaining principal stresses has to be evaluated. The proposed process is repeated three times. Each time, a principal stress , or is eliminated and the pair consisting of the two remaining principal stresses, , and relatively, is assumed to apply to the finite element. The impact of the eliminated stress on the other two is calculated according to Hookes' law. Equations providing equivalent values of the transformed stresses are extracted.

In order to perform failure analysis the establishment of a failure criterion is crucial. For two-dimensional assumptions, a modified Von Mises criterion, adapted especially for masonry structures, is presented. Failure occurs when a biaxial stress state of a node is represented by a point outside the surface area described in [3]. Failure analysis takes place using "FAILURE", a software program developed by the author's research team (NTUA, Greece), [5]. Graphical outputs including failed areas are produced, for each plane.

The methodology proposed is applied to the structure considered, which is subjected to horizontal seismic loading, according to [2]. Elimination of one principal stress at a time, and subsequent modifications of remaining principal stresses applied to the solid nodes, permit reduction of the three-dimensional state to biaxial. Two-dimensional data are elaborated considering, in turn, all three sets of stresses, , , . In this way, graphical outputs of failure results are obtained. Comparative results concerning failure over the wall surface, for solid and shell elements, reveal that the use of solid elements provides additional, more accurate information related to failure resulting from out of plane stresses.

1

O.C. Zienkiewicz, R.L. Taylor, "The finite Element Method, Basic Formulation and Linear Problems", 153, 1967.

2

Greek Aseismic Code, EAK, 2000.

3

C.A. Syrmakezis, P.G. Asteris, "Masonry Failure Criterion under Biaxial Stress State", Journal of Materials in Civil Engineering, ASCE, 58-64, 2001. doi:10.1061/(ASCE)0899-1561(2001)13:1(58)

4

A.K. Antonopoulos, "Comparative Exploration of Analysis Methods for the Dynamic Behaviour of Structures with Continuously Distributed Mass and Stiffness", MSc thesis, 2005.

5

C.A. Syrmakezis, P.G. Asteris, A.A. Sophocleous, "Earthquake Resistant Design of Masonry Tower Structures", Proceedings, 5th STREMA Conference on Structural studies, Repairs and Maintenance of Historical Buildings, 1, 377-386, St. Sebastian, Spain, 1997.

6

F. Casciati, L. Faravelli, "Stochastic Nonlinear Controllers", Proc. IUTAM Symposium, NonLinearity-Stochastic Structural Dynamics, Madras, Kluwer, 2001.

7

F. Casciati, and H.J. Lagorio, "Urban Renewal Aspects and Technological Devices in Infrastructure Rehabilitation", Proc.1st European Conference on Structural Control, Barcelona, 173-181, 1996.

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