An improved shear deformation theory for bending beams with symmetrically varying mechanical properties in the depth direction

Krzysztof Magnucki,Ewa Magnucka-Blandzi,Jerzy Lewinski

doi:10.1007/s00707-020-02763-y

Abstract

The paper is devoted to simply supported beams under three-point bending. Their mechanical properties symmetrically vary in the depth direction. The individual shear deformation theory for beams of such features is proposed. Based on the principle of stationary total potential energy the differential equations of equilibrium are obtained. The system of the equations is analytically solved, and the shear coefficients and deflections of example beams are calculated. The solution is compared with other analytical results obtained with the use of another deformation function. Moreover, the bending problem of these beams is also numerically studied using the finite element method. Results of analytical and numerical studies are presented in Figures and Tables.

Highlights

The shearing effect occurring in bent constructions was noticed already in the nineteenth century, and studied in detail for homogeneous and layered constructions in the twentieth century
Reddy [1] developed a theoretical model of bending of functionally graded rectangular plates considering the shearing effect
Detailed analysis is made taking into account the first and third order shear deformation theory

Summary

Introduction

The shearing effect occurring in bent constructions was noticed already in the nineteenth century, and studied in detail for homogeneous and layered constructions in the twentieth century. Thai and Vo [7] developed various higher-order shear deformation theories for testing bending and free vibration of functionally graded beams These theories account for higher-order variation of transverse shear strain in the depth direction of the beam, and satisfy the stress-free boundary conditions on the upper and lower surfaces of the beam. Grover et al [10] proposed a new inverse hyperbolic shear deformation theory of laminated composite and sandwich plates This theory is formulated based on the shear strain shape function and validated by numerical studies of the bending and buckling problem of rectangular plates. Mahi et al [14] presented a new hyperbolic shear deformation theory describing bending and free vibration of isotropic, functionally graded, sandwich and laminated composite plates.

Comparative analysis—analytical approach

Findings

Conclusions