Numerical Methods for the Calculation of Elastic Instability

Abstract

Three numerical methods for the evaluation of buckling loads requiring an operations table similar to that used in Southwell's relaxation procedure are developed and described: the Determinant Method, the Convergence Method, and the EnergyMethod. The Determinant Method makes use of the fact that the determinant made up by the elements of the operations table vanishes at buckling. The Convergence Method states tha t a systematic relaxation process based on the operations table is convergent only if the structure is stable The principle that the second variation of the total potential is zero during buckling forms the basis of the Energy Method. The theoretical background of the three methods is discussed after a review of the setting up of operations tables and of Southwell's method in general. The procedure to be followed in each method is given and illustrated in numerical examples. Experiments on sheet and stringer combinations carried out at the Polytechnic Institute of Brooklyn Aeronautical Laboratories (PIBAL) are described, the results of which are in good agreement with those obtained from each of the three methods of calculation.