Overcoming stretching and shortening assumptions in Euler–Bernoulli theory using nonlinear Hencky’s beam models: Applicable to partly-shortened and partly-stretched beams

Abstract

This paper addresses the challenges of the Euler–Bernoulli beam theory regarding shortening and stretching assumptions. Certain boundary conditions, such as a cantilever with a horizontal spring attached to its end, result in beams that partly shorten or stretch, depending on the spring stiffness. The traditional Euler–Bernoulli beam model may not accurately capture the geometrical nonlinearity in these cases. To address this, nonlinear Hencky’s beam models are proposed to describe such conditions. The validity of these models is assessed against the nonlinear Euler–Bernoulli model using the Galerkin method, with examples including cantilever and clamped–clamped configurations representing shortened and stretched beams. An analysis of a cantilever with a horizontal spring, where stiffness varies, using the nonlinear Hencky’s model, indicates that increasing horizontal stiffness stiffens the system. This analysis reveals a transition from softening to linear behavior to hardening near the second resonance frequency, suggesting a bifurcation point. Despite the computational demands of nonlinear Hencky’s models, this study highlights their effectiveness in overcoming the inherent assumptions of stretching and shortening in Euler–Bernoulli beam theory. These models enable a comprehensive nonlinear analysis of partly shortened or stretched beams.