Finite element formulation for the analysis of multilayered beams based on the principle of stationary complementary strain energy

A family of finite elements for the analysis of orthotropic multilayered beams with mono-symmetric cross sections is developed based on the principle of stationary complementary energy. The longitudinal normal stress field is postulated as polynomial and Heaviside step function series and substituted into the infinitesimal equilibrium conditions to develop expressions for the shear and transverse stress fields. The statically admissible stress fields thus derived are then adopted within the complementary energy variational principle framework to develop a family of finite elements. The distinguishing features of the solution are: (i) it captures the nonlinear distribution of the stress fields along the section depth and steep stress gradients typically occurring near bondline ends of multilayer beams, (ii) unlike conventional solutions based on the principle of stationary potential energy which predict jumps in the shear and peeling stresses at interfaces of adjacent layers, the present solution satisfies equilibrium in an exact infinitesimal sense at layer interfaces and thus ensures continuity of the stress fields across the interface, (iii) it naturally captures the effects of transverse shear and transverse normal stresses, and (iv) it consistently converges to the displacements from above, in contrast to conventional finite element solutions where convergence is typically from below. The versatility of the solution is then illustrated in applications involving wood beams and steel beams strengthened with GFRP plates and sandwich beams with soft cores.