Fourier series method for finding displacements and stress fields in hyperbolic shear deformable thick beams subjected to distributed transverse loads

This paper presents a systematic formulation of the hyperbolic shear deformation theory for bending problems of thick beams; and the Fourier series method for solving the resulting system of coupled differential equations and ultimately finding the displacements and stress fields. Hyperbolic sine and cosine functions are used in formulating the displacement field components such that transverse shear stress free conditions are achieved at the top and bottom surfaces of the beam, thus obviating the shear correction factors of the first order shear deformation theories. The vanishing of the first variation of the total potential energy functional is used to obtain the system of coupled differential equations for the domain and the boundary conditions. The domain equations are solved using Fourier series method for simply supported ends for linearly distributed and uniformly distributed loads. The solutions are found as infinite series with good convergence. Solutions obtained for the axial and transverse displacements, and normal and shear stresses at critical points on the beam agree remarkably well with previous solutions, and for normal stresses, the errors of the present method are less than 0.5% for aspect ratio of 4 and less than 1.9% for aspect ratio of 10.