A Hyperbolic Shear Deformation Theory for Flexure and Vibration of Thick Isotropic Beams

Abstract

A Hyperbolic Shear Deformation Theory taking into account transverse shear deformation effects, is presented for the static flexure and free flexural vibration analysis of thick isotropic beams. The displacement field of the theory contains two variables and does not require shear correction factor. The hyperbolic sine function is used in the displacement field in terms of thickness coordinate to represent shear deformation. The most important feature of the theory is that the transverse shear stress can be obtained directly from the use of constitutive relation, satisfying the stress free boundary conditions at top and bottom of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions of the theory are obtained using the principle of virtual work. Results obtained for flexure and free vibration of simply supported uniform, isotropic beams are compared with those of elementary, refined, and exact beam theories to validate the accuracy of the theory.

Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory

“…Ghugal and Shimpi [6] developed an equivalent single layer trigonometric theory without considering effect of transverse normal strain for isotropic beams which is further extended by Arya et al [34] for laminated composite beams. The displacement field of the theory is…”

“…These theories are developed by assuming higher order variation of axial displacement. Many displacement based equivalent single layer shear deformation theories are available in the literature for the bending, buckling and free vibration analysis of laminated composite beams such as: Levinson [2,3], Reddy [4], Krishna Murty [5], Ghugal and Shimpi [6], Soldatos [7], Karama et al [8] and many more. All these theories are represented by single unified shear deformation theories by Sayyad [9] and Sayyad et al [10].…”

A refined shear deformation theory for flexure of thick beams

“…However, with these theories shear stress free boundary conditions are not satisfied at top and bottom surfaces of the beam. This discrepancy is removed by Ghugal and Shimpi [7] and developed a variationally consistent refined trigonometric shear deformation theory for flexure and free vibration of thick isotropic beams. Ghugal and Nakhate [5] obtained the general bending solutions for thick beams using variationally consistent refined trigonometric shear deformation theory.…”

“…Ghugal and Nakhate [5] obtained the general bending solutions for thick beams using variationally consistent refined trigonometric shear deformation theory. Ghugal and Sharma [6] developed the variationally consistent hyperbolic shear deformation theory for flexural analysis of thick beams and obtained the displacements, stresses and fundamental frequencies of flexural mode and thickness shear modes from free vibration of simply supported beams.…”

“…In this paper, a variationally consistent hyperbolic shear deformation theory previously developed by Ghugal and Sharma [6] for thick beams is used to obtain the general bending solutions for thick isotropic beams. The theory is applied to uniform isotropic solid beams of rectangular cross-section for static flexure with various boundary and loading conditions.…”