Observations on Higher‐Order Beam Theory

A parabolic shear‐deformation beam theory assuming a higher‐order variation for axial displacement has been recently presented. In this theory, the axial displacement variation can be selected so that it results in a suitable admissible transverse shear‐strain variation across the depth of the beam. This paper examines several transverse shear‐strain variations that can go with the aforementioned higher‐order theory. Apart from the usual simple parabolic variation, six other shear‐strain variations are considered: the sinusoidal variation, cubic, quartic, quintic, and sixth‐order polynomials. All these variations for transverse shear‐strain satisfy the requirement that the shear strain be zero at the extreme fibers $(z=\pm h/2)$ and nonzero elsewhere along the depth of the beam. Comparison of the results from this paper with results from others show that the simple parabolic distribution for transverse shear strain gives most accurate results. Also, Timoshenko's theory (with a shear factor of five‐sixths) and the current formulation which uses the parabolic shear‐strain distribution, give identical values for deflections.