Large Deflection of Determinate and Indeterminate Bars of Variable Stiffness

Abstract

The research here deals with the development of an analytical method for the computation of large deflections and rotations of members with varying stiffness along their length. The member may be statically determinate, or statically indeterminate, and its loading may be arbitrary. This method involves the use of equivalent nonlinear and equivalent pseudolinear systems of uniform stiffness that replace the original variable‐stiffness member. The large deflections and rotations of the equivalent system are identical to the corresponding ones of the original variable‐stiffness member, and they can be obtained by either: (1) Using the equivalent pseudolinear system and applying elementary linear analysis; or (2) using a simplified equivalent nonlinear system and applying nonlinear analysis. A mathematical proof regarding the existence of equivalent nonlinear and equivalent pseudolinear systems is obtained by using the nonlinear second‐order differential equation. The use of equivalent systems simplifies a great deal the mathematical complexity of the variable‐stiffness problem. The Young's modulus of elasticity is assumed to be constant, but the method applies equally well when this modulus varies along the length of the member.