Post-Buckling Analysis

A previously developed and numerically implemented non-linear GBT formulation to analyse the post-buckling behaviour of initially imperfect isotropic linear elastic folded-plate members (Silvestre and Camotim, 2003) was recently extended to encompass orthotropic FRP composite members (Silvestre and Camotim, 2004e; Silvestre, 2005). This involved major modifications with respect to the conventional GBT, namely (i) to employ the stress-strain relations given in (2) and (ii) to consider the non-linear strain-displacement relations

where the bars identify the terms associated with the initial imperfections. Moreover, one must also include an additional set of transverse extension deformation modes in the cross-section analysis,

Figure 10. Lipped channel eight most relevant transverse extension modes.

which (i) account for the “bowing effect” stemming from the transverse bending of cross-section walls and (ii) do not comply with Vlassov’s assumption – they stem from the sequential imposition of unit transverse and null warping displacements at both the natural and intermediate nodes. The configurations of the eight most relevant transverse extension deformation modes of a lipped channel cross-section are displayed in Figure 10.

After (i) incorporating (10) into the principle of virtual work, (ii) performing several operations, described in detail elsewhere (Silvestre and Camotim, 2003; Silvestre, 2005) and (iii) including the special cross-section analysis, one is led to the member equilibrium equations, written variationally as

SU1 + SU2 + SU3 – SU1 – SU2 – SU3 + Snq + 5nw = 0, (11)

where (i) the strain energy terms SU1,SU2 and SU3 are linear, quadratic and cubic functionals of the mode amplitude functions фі, (ii) their “bar counterparts” contain the imperfection amplitude functions фі and (iii) the Sn terms stand for the virtual work done by distributed or concentrated external loads.