Modeling and Stress Analysis of a Pre-Shaped Curved Beam: Influence of High Modes of Buckling

Abstract

In this paper, we investigate the effect of high modes of buckling on the mechanical behavior of a pre-shaped curved beam. In a first stage, the presented modeling develops further the snapping forces solution and bistability conditions in order to include high modes of buckling. In a second stage, we develop the analytical solution of the stresses inside the beam during deflection between the two sides of buckling. The buckling with or without mechanical conditions on antisymmetric modes, the force characteristics, bistability conditions and stresses are described in this paper based on mathematical approach in order to provide a clear physical understanding of the curved beam behavior and its design parameters. The accurate knowledge of the design parameters is important in order to achieve the best integration of the curved beam in a complete microstructure. The analytical results are compared with and without considering high modes of buckling and have shown to be in excellent agreement with FEM simulations. The results show the importance of the high modes in calculating stresses and snapping forces.

Analytical Study of the Snap-Through and Bistability of Beams With Arbitrarily Initial Shape

“…The same conclusion can be extracted by minimizing the variation of the total energy of the system as in [44,55]. Let u b , u c , u f and u t are the bending, compression, actuation and total energy and let U (·) = u (·) l 3 EIh 2 is the normalization for each of them.…”

“…This work showed a bifurcation in the snap-through behavior of the curved beam where the second or third mode of buckling becomes activated at a certain level of the axial stress. Using the same derivation, Hussein et al [55] developed explicit analytical expressions of the snapping forces and internal stresses that account for all modes of buckling. A generalized model for arbitrarily initial shape and loading is developed in [56] and solved using two different iterative algorithms.…”

“…The snap-through governing expressions contain infinite sums depending on the modes of buckling and Fourier series coefficients. Some of these sums are already evaluated in [55], while the other sums are dependent on the initial shape. The infinite sums are evaluated for two initial shape cases, curved and inclined, and explicit analytical snap-through expressions are obtained.…”

“…Mathematically, N j is calculated to have a non-trivial solution after introducing the boundary conditions in the homogeneous problem (similar to the problem of a straight beam). For clampedclamped boundary conditions, N j is the j th positive solution of the following periodic equation [44,55]:…”

“…Equation (9) is the same equation governing a straight beam with an axial load. Its solution is an infinite superposition of the modes of buckling as follows [44,55]:…”