An Exact finite element using hyperbolic shear deformation beam theory

Abstract

A hyperbolic shear deformation theory for thick isotropic beams is developed. This theory satisfies shear stress free boundary condition at top and bottom of the beam and doesn’t need shear correction factor. A displacement based finite element model of this theory is formulated using the variational principle. Displacements are approximated using the homogeneous solutions of the governing differential equations that describe the deformations of the cross-section according to the high order theory, which includes cubic variation of the axial displacements over the cross-section of the beam. Also, this model gives the exact stiffness coefficients for the high order isotropic beam element. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two components of end moments. Results obtained for displacements using the present beam element for static flexure of uniform isotropic cantilever carrying an end load are compared with the solutions derived using the present beam theory, other beam theories and the exact two-dimensional elasticity solution to validate the present beam element. A continuous thick beam problem is also solved for displacements and compared with the results obtained using ‘ABAQUS’ software (version 6.14) with different type of beam elements and a two-dimensional model.