Finite element analysis of layered wooden shells under application of an orthotropic single-surface plasticity model

The analysis of layered wooden shells requires a suitable constitutive model for multi-axially loaded wood. In this contribution a brief overview of the development of an orthotropic material model for the simulation of spruce wood under simultaneous biaxial in-plane stresses and transverse shear stresses is given. The model considers an initially linear elastic domain as well as hardening and softening behaviour at higher states of stress and strain, respectively. Combining the advantage of a smooth single-surface plasticity model with the identification of several modes of failure is the key to the proposed mathematical formulation. The applicability of the constitutive model will be demonstrated by means of a nonlinear finite element analysis of a layered cylindrical shell with an opening and surrounding stiffeners. Figure 1. Shell geometry and material coordinate system for orthotropic layered wooden shells. Since a typical wooden shell is constructed with initially straight boards, it is assumed that the wood fibres are always parallel to the middle surface of the shell. In general, this cannot be assumed for the Rand T-direction. The stress-strain behaviour in these directions, however, is reasonable similar to justify a simplified approach such that one characteristic stress-strain function may be used and scaled by the respective uniaxial stiffness and strength parameters. Throughout this paper it will be assumed that the tangential direction of the stem is aligned with the normal to the shell surface. This yields an orientation of the material as illustrated in Figure 1. This also defines the general stress state in the material, where the thickness stress σT is assumed to vanish for thin shells. 3 CONSTITUTIVE MATERIAL MODEL 3.1 Material model for in-plane stress states The properties and the concept of the used singlesurface plasticity model are summarised in (Müllner et al. 2004a). The formulation of evolution laws requires control variables. These so-called primary variables are collected in a vector α. The determination of α is subject to a non-associated hardening and softening rule. Its formulation is required because of the brittle tensile and the ductile compressive behaviour of wood. The rule considers different modes of failure which were identified by (Mackenzie-Helnwein et al. 2003) as: − brittle tensile failure in fibre direction, − compressive failure in fibre direction, − brittle tensile failure perpendicular to grain, and − ductile compressive behaviour perpendicular to grain. 3.2 Material model for layered wooden shells Concerning stress states in layered wooden shells, a fifth failure mode may become relevant. It is controlled by transverse shear stresses and cannot be observed in the tests by (Eberhardsteiner 2002). The extension of the material model for transverse shear stresses was done by (MackenzieHelnwein et al. 2005). Experimental observations by (Lucena-Simon et al. 2000) show typical shear failure as cracking parallel to the fibres. Hence failure due to transverse shear has to be controlled by τLR, τRT and τTL. In order to keep the model as simple as possible, shear failure in given planes perpendicular to the Rand T-direction shall be characterized by the respective effective shear stresses τR and τT. These effective stresses are shown in Figure 1. The appropriate definitions can be found in Figure 2. Figure 2. Types of material models – model for in-plane stress states and model for transverse shear stress states. Figure 3. Yield surface of single-surface plasticity model in the orthotropic stress space and evolution laws for the strength values depending on the strain-like primary variables αi (i ∈ {(t,L), (c,L), (t,R), (c,R), (ref), (shr,R), (shr,T)}) by (Müllner et al. 2004b). The latter shows the chosen approach of the material modelling. The extension of the material model leads to an increase of the number of stress invariants σ and the vector of the primary variables α. The used elliptical yield surface and the appropriate evolution laws for the consideration of transverse shear stress states are shown in Figure 3.