Deflections of Timoshenko beam with varying cross-section. (English) Zbl 0869.73035

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.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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