Postcritical Imperfection-Sensitive Buckling and Optimal Bracing of Large Regular Frames

Periodic interior buckling of regular multistory and multibay rectangular elastic frames with elastic bracing is analyzed. It is shown that there exists a certain critical bracing stiffness for which the critical loads for the nonsway (symmetric) and sway (antisymmetric) buckling modes coincide. Simple formulae for the critical stiffness are given. For the critical and softer bracing, the type of postcritical buckling behavior is the unstable symmetric bifurcation, exhibiting imperfection sensitivity according to Koiter's 2/3-power law. For stiffer bracing, there is no imperfection sensitivity. The critical bracing, however, represents a naive optimal design which should be avoided because the imperfection sensitivity is the strongest. It is recommended that the truly optimal bracing should be significantly stiffer (perhaps 1.1 to 2 times as stiff). The buckling behavior, including the postcritical imperfection sensitivity, is similar to that of a portal frame analyzed before. The solution also provides a demonstration of a simple method for the initial postcritical analysis of frames recently proposed by Bažant and Cedolin, which is based on energy minimization. In this method, the distribution of cross section rotations is assumed to be the same as in the classical linearized theory. The curvatures and deflections are obtained from the rotations by integration with at least a second-order accuracy (in terms of the rotations), and the axial shortening with at least a fourth-order accuracy.