Extension of Boley’s continuum mechanics-based successive approximation method to two-layer rectangular beams

Abstract

An extension of Boley’s continuum mechanics-based successive approximation method is presented for rectangular beams composed of two isotropic linear elastic layers. The solution is cast into the form of tables, in complete analogy to the tables originally presented by Boley and Tolins for single-layer strips. The first column in these tables corresponds to the classical Bernoulli–Euler theory of beams. The further columns represent comparatively fast converging correction terms of an increasing refinement. Our two-layer formulation automatically satisfies the stress continuity conditions at the interface of the two layers. Enforcing displacement continuity at the interface between the layers, we derive results that do satisfy the equilibrium field equations, the stress continuity conditions at the interface and the stress boundary conditions at the upper and lower edges. When converged, the field constitutive relations and the displacement continuity at the interface between the two layers are also satisfied. We present a compact formulation, which allows writing down the results for more than the three successive steps considered by Boley and Tolins. The elasticity solutions presented subsequently can be used as novel analytic benchmarks for comparison with refined structural mechanics beam theories. Interior solutions for beams with a finite axial extent can be obtained by assigning approximate boundary conditions at the lateral ends. Comparisons to finite element computations for a clamped–clamped beam give strong evidence for the correctness of our analytic results.