Exact Modelling of Stress Fields In Bimaterial Beams Using Readily Available Mathematical Software

Abstract

A little-used first principles solution was proposed by Hess in 1969 for the stress fields in a bimaterial elastic beam under any arbitrary self equilibrating free end loading. In this paper Hess's solution is implemented using Matlab to calculate axial and normal stresses at any required horizontal or vertical cross-section of the beam. The approach uses numerical methods to develop an eigenvalue solution for any given set of layer thicknesses and material properties. A novel finite element mesh design, originally presented in 1989 by Schiermeier and Szabo, is used to validate the results from the above analysis. The mesh (of p-elements) is strongly graded around singularities, ensuring their effects are isolated. More remote areas of the model, where stresses and gradients are low, are sparsely populated by elements. The rapid changes in interfacial peeling stress and interfacial shear stress close to the free edge are coped with quite effectively by this mesh design. The two methods are used to examine the stress fields in the bimaterial beam. Although both methods can be used to calculate stresses at any required horizontal or vertical cross-section in the beam, the first principles method has the advantage of not requiring FEA software. Instead Excel or Matlab can readily display a plot of the stress distribution in the selected cross section. The method can be applied to axial, shear and peeling (normal) stresses in bimaterial beams. The solution has applications in many varied areas of engineering, from thermal stresses in IC packages to the behaviour of armour plating under mechanical loads