Effects of approximations in analyses of beams of open thin-walled cross-section - part II: 3-D non-linear behaviour

In a companion paper, the e ects of approximations in the 8exural-torsional stability analysis of beams was studied, and it was shown that a second-order rotation matrix was su9ciently accurate for a 8exural-torsional stability analysis. However, the second-order rotation matrix is not necessarily accurate in formulating ;nite element model for a 3-D non-linear analysis of thin-walled beams of open cross-section. The approximations in the second-order rotation matrix may introduce ‘self-straining’ due to superimposed rigid-body motions, which may lead to physically incorrect predictions of the 3-D non-linear behaviour of beams. In a 3-D non-linear elastic–plastic analysis, numerical integration over the cross-section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross-section that is not consistent with the strain distributions may lead to incorrect predictions of the 3-D non-linear elastic–plastic behaviour of beams. This paper investigates the e ects of approximations on the 3-D non-linear analysis of beams. It is found that a ;nite element model for 3-D non-linear analysis based on the second-order rotation matrix leads to over-sti predictions of the 8exural-torsional buckling and postbuckling response and to an overestimate of the maximum load-carrying capacities of beams in some cases. To perform a correct 3-D non-linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross-section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3-D non-linear elastic–plastic behaviour of beams.