## Vgh

Still following Timoshenko and using hereafter his convention of negative sign (which does not hold for all previous chapters) for expressing the two bending moments with respect to w and its derivatives (see footnote in Sect. 3.2), we write

where D(x) = Et3(x)/12(1 — v ) is the flexural rigidity. Returning to (10.14) and eliminating dQx/dx by using (10.13), we find d2 Mx 1

After substitution of Nv and Mx, the general differential equation of the flexure of a cylinder is d2 f d2w\ E

This equation was first obtained by S. Timoshenko - probably around 1930 - and was included in the 1940 issue of Theory of Plates and Shells [27]. 4 Introducing the dimensionless variable x = [12(1 — v2)]1/4 a (10.23)

into (10.22) and multiplying it by a/E, we obtain dWfl +Lw=qa (1024)

For clarity it is also useful to define dimensionless thickness T and flexure W as the following

aqa where C is an unknown constant. After substitutions, we obtain a general normalized equation of variable thickness cylinders

4 For a cylinder with constant thickness, the general equation becomes t3 d4w t q ---1--w = —

Although in a somewhat different form, this equation was obtained by Augustus Love [29] around 1915. In the preface of the fourth issue of Mathematical Theory of Elasticity, the author refers to the section where it was published (Sect. 339) as one of the sections with letter-A that were added in the third issue.

If the flexure W(x) to generate is in a polynomial form, at least quadratic in x, the integration allows deriving the radial thickness distribution T(x) from a central thickness T(0).

If the flexure is a constant or a linear form in x, then the first left-hand term vanishes and the thickness distribution is either the reciprocal constant or the reciprocal linear form, respectively.

10.2.3 Radial Thickness Distributions and Parabolic Flexure

We consider hereafter various cases of obtaining a parabolic flexure. These are generated by a uniform load applied all over the surface of the cylinder.

• Uniform load in reaction with simply supported ends: A representation of a purely parabolic flexure can be simply expressed by a quadratic function of the axial ordinate as

where P is the length parameter such that x = ±P at the cylinder edges (or ends), and C2 = C is the constant in (10.26) for the parabolic case. The radial reacting forces at the edges ensure that the radial displacement is zero at x = ±P. The intensity of these two equal forces Fp = F_p, per length unit, is determined from the statics by

where L = 2xmax is the length of the cylinder. With the initial variables, this flexure is represented by

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