Exact critical temperatures for snap-through of an active simply supported buckled beam under a longitudinal half sine wave temperature profile

The buckled beam is a fundamental building block used to produce morphing structures. The advancement of responsive materials has enabled these building blocks to actively transform structures between multiple states. Often the driving stimulus is applied to these structural elements in a longitudinally nonuniform fashion. However, most models assume uniform application of the stimulus or rely on numerical results, which obscure connections between the system parameters and the system response. In this work, we examine a thermally responsive, buckled beam and consider the temperature profile to be a half sine wave along its length. This assumption makes the analysis convenient and enables the derivation of an exact expression for the critical temperature rise leading to loss of stability under a range of conditions. In addition, a very simple expression is obtained for predicting the crossover from the symmetric snap-through mode to unsymmetrical transitions between buckled states. The results give insight into system design strategies and also lay the foundation for analyzing more complex systems.