Application of the orthotropic Rankine-type model to masonry panels P. Bilko 1 , L. Małyszko 2 1 Faculty of Technical Sciences, University of Warmia and Mazury in Olsztyn, POLAND ABSTRACT: The second author presented the orthotropic Rankine-type plasticity model for the analysis of structural problems in the plane stress state during the previous local seminar of IASS PC in Warsaw. The model included a maximum principal stress failure criterion of Rankine both for tension and compression regimes by incorporating the second order strength tensor. Within the framework of the finite element method for the elastoplasticity theory of small strains for softening/hardening materials, two yield surfaces resulting from the ortho- tropic principal stress criterion were implemented at the integration point level into the proprietary finite element program by means of user- defined subroutines. This paper demonstrates an application of the model to the analysis of a masonry panel. The numerical tests have been done in order to check the possibility of the model to reproduce an orthotropic behaviour of masonry panels with different tensile and com- pressive strengths along the material axes as well as different inelastic behaviour for each material axis after the own numerical implementa- tion in the finite element code. The ability of the model to reproduce a failure mode of the panels is also of interest. Key words: masonry, orthotropic failure criteria, rate-independent softening plasticity, numerical implementation, finite element method 1. INTRODUCTION A large number of buildings, including these that may correspond with the lightweight structure definition, are constructed with masonry infill walls for architectural needs or the fire rating and sound transmission reasons. The walls can also greatly stiffen a flexible steel or reinforced concrete frame and significantly affect the distribution of lateral loads to various parts of the building. However, unreinforced masonry may be assumed a homogeneous but obviously anisotropic material, which ex- hibits distinct directional properties due to the influence of the mortar joints acting as planes of weakness. A model must reproduce an orthotropic material with different tensile and compressive strengths along the material axes as well as different inelastic behaviour for each material axis. A reduced number of orthotropic material models that may be specific for masonry have been proposed. An attempt of formu- lating one of them is the orthotropic Rankine-type model proposed by Malyszko (Ref 4). The model was presented during previous PC IASS seminar in Warsaw. This paper continues the previous work and illus- trates behaviour of the masonry panels regarded as equivalent orthotropic continuum under a directional loading. Masonry is an exam- ple of a material for which the model applies, having different strengths parallel and perpendicular to the bed joints. The possibility of formulat- ing robust numerical algorithms by means of user-defined subroutines of the proprietary nonlinear finite element program is also of prime impor- tance. The constitutive model for the plane stress is provided with the framework of the mathematical elastoplasticity theory of small strains for softening materials. Here, the model for compression regime is a novel development. The implementation of the model is done within a framework of an incremental-iterative algorithm of finite element method using both the return-mapping algorithm allowing the stresses to be returned to the yield surface and a consistent tangent stiffness opera- tor. The paper is concluded by presenting some numerical results of a masonry panel analysis verifying own implementation. 2. ORTHOTROPIC RANKINE-TYPE PLASTICITY MODEL For the sake of simplicity, the mechanical description of the elastic- plastic model is presented in a state of plane stress parallel to the XZ- plane based on the assumption that the principal axes of orthotropy co- incided with the frame of reference. The constitutive relation between stresses xz z x , , σ and strains xz z x , , ε are given in a ma- terial point. Within the framework of the theory of elastoplasticity, the orthotropic Rankine-type failure criteria (2) and (4) serve as the yield surfaces and have to distinguish between domains of the different ma- terial response by means of the yield condition given by the yield sur- face f. Within the yield surface (f < 0), the material behaves elastically. On the yield surface ( f = 0), the material begins to yield. Thus, in the phenomenological approach that is applied in the present model, the geometrical nonlinearity like cracking is accounted for by the introduc- tion of a loading function f describing the failure criterion. By the as- sumption of small strains, the function f is set up with quantities that re- fer to the undeformed configuration. 2.1. Orthotropic failure criteria In an arbitrary right-handed Cartesian coordinate system {xi} coaxial with the axes of the principal stresses and for a so-called generalized plane state problem when 2 is equal to zero, the failure criterion for the tension regime can be written in the following form 1 sin cos sin cos 3 2 2 3 1 1 2 2 f f f f f f tX tZ tZ tX tZ tX (1) where the angle in the direction cosines measures the rotation be- tween the first axis x1, i.e. the axis of the first principal stress 1 and the first material X-axis. The parameters ftX and ftZ denote the tensile strengths that may be determined from the simple tension tests along the direction of the material axes. In the {xi} frame coaxial with the XZ axes, where X and Z are the principal axes of material orthotropy, the failure criterion can be written as 0 2 xz z tZ x tX f f (2) For the compression regime, the similar form of the failure criterion can be obtained by replacing the tensile strengths by the compressive strengths with positive values, i.e. by the strengths -fcX and -fcZ . The failure condition caused by the compression can be written as 1 sin cos sin cos 3 2 2 3 1 1 2 2 f f f f f f cX cZ cZ cX cZ cX (3) instead of Eqn (1) and as 0 2 xz z cZ x cX f f (4) instead of Eqn (2). As one can see, a representation of an orthotropic failure surface in term of principal stresses only is not possible. For plane stress situation, a