The state of stress of an elastically supported transversely isotropic beam (Q1316703)

The bending stress-strain state of a three-layered beam with stiff outer layers and an elastic soft middle layer is analyzed. The lower layer behaves as a Bernoulli-Euler beam if a load is applied. A uniformly distributed compressing load acts on the upper layer, whose ends are free of any loads. The plane state of stress of the transversely isotropic layer is determined by the equations of linear elasticity. The exact solution of the problem is obtained in terms of Fourier series. Numerical results show that the transversal tensile stresses \(\sigma_ y\) may arise with substantial values at the ends of the layer as the compliance coefficient \(\lambda_ 0\) decreases. The shear stresses near the free surface are also large. Thus, when computing the strength of a multi-layer structure, it is advisable to determine the shear and normal stresses in the stiff layers. The axial stresses \(\sigma_ x\) vary linearly across the thickness of the beam, what provides a qualitative confirmation of the applicability of the fundamental assumptions (e.g. Kirchhoff-Love hypothesis) of the theory of thin shells.