Flexure of narrow rectangular deep beams with built-in ends

Abstract

The paper presents the exact analytical solutions for built-in uniform, isotropic deep beams using sinusoidal refined shear deformation theory under transverse bending. The theory is built upon the classical beam theory including sinusoidal function in terms of thickness coordinate to include the shear deformation effects. The kinematics of the theory enforces transverse shear stress to satisfy the shear stress-free conditions on the top and bottom planes of the beam. The shear stress distribution through the thickness is realistic and requires no shear correction factor. Using the principle of virtual work, the equilibrium equations and boundary conditions have been obtained based on kinematics of the theory. To demonstrate the efficacy of the theory, the exact analytical solutions for beams, with narrow rectangular cross sections, subjected to parabolic and cosine loads are obtained to examine the complete flexural response. Results for these specialized loading cases are obtained for the first time and are discussed critically with those of other theories. The solutions obtained can be served as a benchmark for comparison of results by other refined theories.