Exact two-dimensional piezoelasticity solution for buckling of hybrid beams and cross-ply panels using transfer matrices

Abstract

Two-dimensional exact piezoelasticity solution is presented for buckling of simply supported symmetrically laminated hybrid beam and cross-ply panel with elastic substrate and piezoelectric layers. Buckling is considered under axial strain and actuation potentials for movable inplane end conditions and under actuation potential alone for immovable inplane end conditions. The governing equations for buckling mode are formulated in terms of six variables: displacements u, w, potential φ, stresses σ_z, τ_zx and electric displacement D_z. These entities are expanded in Fourier series that satisfy the end conditions. The governing equations reduce to differential equations in z with coefficients dependent on the axial load and actuation potentials. The solution has six constants for each layer. The transfer matrix is derived relating the six primary variables at the top and bottom of a layer. The six conditions σ_z=τ_zx=0, φ=0 at the top and bottom of the beam are used to set up three homogeneous equations for u, w, Dz at the bottom. The determinant of their coefficient matrix is set to zero to obtain the buckling axial strain/potential. The present benchmark solution would help assess one-dimensional theories for buckling of hybrid beams.