Computation of Different Stresses Acting on Cantilever Beam Using Hyperbolic Shear Deformation Theory

Abstract

The shear deformation effects are more significant in the thick beams. The classical beam theory (ETB) is based on Bernoulli-Euler hypothesis and the classical plate theory (CPT) is based on Kirchhoff’s hypothesis. The shear deformation effects are more pronounced in the thick beams when subjected to transverse loads than in the thin beams under same loading condition. These effects are neglected in elementary theory of beam (ETB). In order to describe the correct bending behavior of thick beams including shear deformation effects and the associated cross sectional warping, shear deformation theories are required. This led to the development shear deformation theories for thick beams and plates. The various methods of development of refined theories based on the reduction of the three-dimensional problems of mechanics of elastic bodies are discussed by Gol Denveizer, Kil Chevskiy, Donnell, Vlasov and Leontev, Sayir and Mitropoulos. In the present paper, a hyperbolic shear deformation theory is developed for computation of different stresses acting on thick isotropic beam. The theory assumes a parabolic variation of transverse shear stress across the thickness of the beams. Thick isotropic cantilever beams subjected to varying load analyzed for the axial displacement, transverse displacement, axial bending stress and transverse shear stress and numerical results are compared with those of Elementary, Timoshenko, Trigonometric and other higher order refined theories.
Keywords: Thick beam, shear deformation, isotropic beam, transverse shear stress, static flexure, hyperbolic shear deformation theory, principle of virtual work