Deflections of Timoshenko beam with varying cross-section (Q1126241)

The author presents closed-form solutions to the problem of a beam of varying cross-section, subject to transverse in-plane loading. The effects of transverse shear deformation are included by making use of the Timoshenko hypothesis: plane sections normal to the beam axis remain plane, but not necessarily normal. For a beam of generally varying cross-section the resulting equation is of fourth order, with non-constant coefficients. Three specific cross-sectional forms are considered: either (i) the depth varies linearly; or (ii) it varies parabolically according to the binomial form \(h(x)= (h_0^{1/2}+ \alpha x)^2\); or (iii) the width varies linearly. The transverse load is assumed to vary parabolically. The author presents results in graphical form for a number of examples. The linearly varying case is considered in the context of a beam with piecewise linearly varying depth, symmetric about the centre. The closed-form results are compared either with numerical results, or with closed form results for the case of the Bernoulli approximation. There are no surprises in the results.