Effect of Curvature on Stability of Curved Beams

Abstract

This paper intends to derive the nonlinear differential equations of equilibrium for a horizontally curved I‐beam. Based on the principle of virtual displacements, the equilibrium of a bar is established for its deformed or buckled configuration using a Lagrangian approach. Central to the consideration of the effect of curvature is the expression of various quantities in cylindrical coordinates and the incorporation of radial stresses in the virtual work statement. The governing differential equations are obtained for the curved beam as the Euler‐Lagrange equations of the functional using a variational procedure. Rationality of the present theory is demonstrated for some typical examples, where the sources of errors in existing theories are traced. Also illustrated is the inconsistency involved in a conventional finite element analysis in which a curved beam is represented by several straight beam elements.