Buckling analysis of pre-cracked beam-columns by Liapunovʹs second method

Interaction between reduced axial and lateral torsional buckling strengths of thin beam-columns of double symmetric sections with an edge crack at the middle is examined via Liapunovʹs second method. This is done in order to show how critical loads for pre-cracked thin-walled beam-columns can be obtained with out the rigour of meticulous solution of a set of differential equations. Having derived an asymptotic expression for the normalized energy release rates for the beam-column, a metric space and a Liapunovʹs Functional for a pre-cracked beam-column are proposed. By using Zubovʹs stability theorems and appropriate eigen value inequalities, expressions for reduced buckling strengths due to edge cracks are obtained. A crack reduction parameter k is proposed which is used to calculate reserve strengths for published lateral buckling strength equations. Results obtained using the proposed parameter are well in agreement with findings of earlier published works in the literature when the parameter is set to zero. The results also show that cracks cause loss of stiffness and can reduce the lateral buckling strengths of beam-columns considerably.