Timoshenko Beam Theory for the Flexural Analysis of Moderately Thick Beams – Variational Formulation, and Closed Form Solution

Abstract

In this study, the Timoshenko first order shear deformation beam theory for the flexural behaviour of moderately thick beams of rectangular cross-section is formulated from vartiational principles, and applied to obtain closed form solutions to the flexural problem of moderately thick rectangular beams. The total potential energy functional for the moderately thick beam flexure problem was formulated by considering the contribution of shearing deformation to the strain energy. Euler-Lagrange conditions were then applied to obtain the system of two coupled ordinary differential equations of equilibrium. The problem of moderately thick beam with simply supported ends subject to uniformly distributed transverse load on the entire span was solved in closed form to obtain the transverse deflection as the sum of the flexural and shear components. Another problem of moderately thick cantilever beam under point load at the free end was solved in closed form to illustrate the solution of the governing equations. The transverse deflection was similarly obtained as the sum of shear and bending components. The bending component of deflection was found to be identical with the Euler-Bernoulli results while the shear component was found to be dependent on the square of the ratio of the beam thickness, t, to the span, l. It was found that as t/l<0.02, the contribution of shear to the overall deflection is insignificant; but becomes significant for t/l>0.10. The findings are in excellent agreement with the technical literature.