Material Behaviour

A GBT formulation applicable to members made of materials other than isotropic and linear elastic requires modifications that depend on the specific type of material behaviour.[3] This task has been carried out for (i) orthotropic linear elastic materials (e. g., laminated plate fibre-reinforced plastics – FRP) and (ii) isotropic non-linear elastic-plastic materials (e. g., stainless steel or aluminium). In the former case, first addressed about four years ago (Silvestre and Camotim, 2002), the layer (lamina) plane-stress constitutive law reads


" Q11







Q 22



, (2)



Q 23



where Qij are “transformed reduced stiffness components” depending on the layer (i) fibre and plastic matrix material properties and (ii) fibre orientation (Jones, 1999). Moreover, the mechanical behaviour of a laminated plate member also varies with its layer properties and configuration (Sil – vestre and Camotim, 2002). In the most general case (arbitrary orthotropy, i. e., anisotropy), the GBT system (1) becomes

Cik(Pk, xxxx + Hikfk, xxx Dikfk, xx + Fikfk, x + ВїкФк XjikWj.0 Фк, хх = 0 (3)

and its boundary conditions must be modified accordingly. It is worth noting (i) the additional tensors Hik and Fik, accounting for material coupling effects between torsion and longitudinal/transversal flexure, and (ii) that, due to the layer-variation of the material properties, the various tensor com­ponents are now mechanical properties – material constants and geometrical characteristics fused together. Another aspect that deserves to be mentioned is the need to include in the analysis de­formation modes that take into consideration the non-linearity of the warping displacement variation within the width of each wall and, therefore, are also associated with membrane shear strains, i. e., do not satisfy Vlassov’s assumption (Silvestre and Camotim, 2004c; Silvestre, 2005). The warping
configurations of this set of additional shear deformation modes are shown in Figure 9, for the case of a lipped channel cross-section.

Concerning the buckling analysis of isotropic non-linear elastic-plastic members, one begins by recalling that, if no strain reversal occurs along the fundamental equilibrium path, an elastic – plastic solid and its “hypoelastic comparison solid” have identical critical bifurcation stresses/loads (Hill, 1958). Thus, by (i) monitoring the evolution of the instantaneous moduli (on the fundamental path) and (ii) adopting incremental constitutive relations, it is possible to determine the member plastic bifurcation behaviour using non-linear elastic stability theory. For fundamental states with only longitudinal normal stresses, the plane-stress incremental constitutive relations read


where (i) dij and sij are stress-rate and strain-rate components, (ii) E ij and G are the instantaneous elastic and shear moduli, (iii) ET is the longitudinal uniaxial tangent modulus and (iv) (■)B and (■)M are superscripts identifying bending and membrane terms. The material uniaxial stress-strain law is commonly described by Ramberg-Osgood type expressions (Rasmussen, 2003) and, due to the well-known “plate plastic buckling paradox” (Hutchinson, 1974), both J2-deformation and J2- flow small strain plasticity theories were included in the GBT formulation (Gonsalves and Camotim, 2004,2005). After the incorporation of the instantaneous moduli, the hypoelastic bifurcation analysis leads to the system of incremental GBT equations

СікФк, хххх — Dik<pk, xx + ВікФк + Xik<Pk, xx — 0 (5)

and associated boundary conditions, where (i) functions фк provide the deformation mode amplitude rates and (ii) all tensor components are load-dependent through the instantaneous moduli (Gonqalves and Camotim, 2005). Note that (5) is applicable to rather general (uniaxial-stress) loading conditions – the only restriction is that they must satisfy the basic hypothesis of Hill’s “comparison solid” concept: the material behaviour may be assumed as hypoelastic in the close vicinity of the bifurcation point (Hill, 1958; Hutchinson, 1974).