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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.

MSC:

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