Numerical solutions are obtained for elastic and elastic-plastic bending of a curved beam via couples at its end sections, under plane stress presupposition. The model proposed is based on the von Mises’ yield criterion, Henky’s deformation theory, and nonlinear strain hardening material behavior. Using formal nondimensional variables, and an appropriate stress function, a single second order nonlinear differential equation describing the deformation behavior of the curved beam is obtained. A shooting technique using Newton iterations with numerically approximated tangents is used for the numerical integration of the governing equation. The computational model is verified both in elastic and partially plastic cases making use of published analytical and numerical FEM solutions.
Stresses and displacement in a partially plastic curved beam.
The distributions of stress and displacement in a partially plastic nonlinearly hardening curved beam at θ= 0° plane is displayed in Fig. 1. Inner radius to outer radius ratio,
, has been taken as 5. 1. The stress variable φ in this figure is computed from von Mises’ yield equation so that φ ≥ 1 in a plastically deformed region. The two vertical lines labeled as
, which correspond to φ = 1, indicate elastic-plastic borders. Other vertical line
points the neutral axis at which σ (
) = 0. The curved beam is composed of an inner plastic region in 1 ≤
≤ 1.081, an elastic region in 1.081 ≤
≤ 1.434, and an outer plastic region in 1.434 ≤