The approximate elastic critical loads of rigidly jointed plane frameworks having redundant members

Abstract

Several methods are already available for determining the approximate elastic critical loads of rigidly jointed plane braced frameworks which if pin jointed would be statically determinate. The present paper breaks new ground by describing a method enabling the critical loads to be determined approximately when any number of redundant members are present. In the method described the load pattern in a redundant framework is expressed as a function of the unknown forces in one non-redundant member and the redundant members. It is shown that the magnitude of the load pattern must acquire a maximum value subject to a relation which exists between the forces when the framework is in a critical condition. This relation arises because at the critical condition any joint has zero stiffness. To obtain maximum accuracy attention is here directed to the ‘least stiff’ joint. The relation between the forces to give zero stiffness of this particular joint is approximately represented by an elliptical expression and, by using the method of Lagrange, the magnitude of the load pattern in terms of the forces is maximized subject to the elliptical approximation, giving an estimate of the elastic critical load of the redundant framework. The method is applied to a singly redundant frame, where the critical load can be determined indirectly by trial and error, the agreement being to within about 1 %. The method is also applied to a more complex framework having five redundant members, where the critical load cannot otherwise be determined. Application of a similar method to the determination of the critical loads of space frameworks is being considered.