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P.5.5 An elliptic plate of major and minor axes 2a and 2b and of small thickness t is clamped along its boundary and is subjected to a uniform pressure difference p between the two faces. Show that the usual differential equation for normal displacements of a thin flat plate subject to lateral loading is satisfied by the solution

where w0 is the deflection at the centre which is taken as the origin.

Determine w0 in terms of p and the relevant material properties of the plate and hence expressions for the greatest stresses due to bending at the centre and at the ends of the minor axis.

Ans.

2 Et3

Centre, a.

Ends of minor axis

±6paAb2

P.5.6 Use the energy method to determine the deflected shape of a rectangular plate ax b, simply supported along each edge and carrying a concentrated load W at a position (£,77) referred to axes through a corner of the plate. The deflected shape of the plate can be represented by the series w m=1»=1

Ans.

P.5.7 If, in addition to the point load W, the plate of problem P.5.6 supports an in-plane compressive load of Nx per unit length on the edges x = 0 and x = a, calculate the resulting deflected shape.

Ans. Am

### Ans. Am

P.5.8 A square plate of side a is simply supported along all four sides and is subjected to a transverse uniformly distributed load of intensity q0. It is proposed to determine the deflected shape of the plate by the Rayleigh-Ritz method employing a 'guessed' form for the deflection of in which the origin is taken at the centre of the plate.

Comment on the degree to which the boundary conditions are satisfied and find the central deflection assuming f = 0.3.

P.5.9 A rectangular plate ax b, simply supported along each edge, possesses a small initial curvature in its unloaded state given by

Determine, using the energy method, its final deflected shape when it is subjected to a compressive load Nx per unit length along the edges x = 0, x = a.

A large proportion of an aircraft's structure comprises thin webs stiffened by slender longerons or stringers. Both are susceptible to failure by buckling at a buckling stress or critical stress, which is frequently below the limit of proportionality and seldom appreciably above the yield stress of the material. Clearly, for this type of structure, buckling is the most critical mode of failure so that the prediction of buckling loads of columns, thin plates and stiffened panels is extremely important in aircraft design. In this chapter we consider the buckling failure of all these structural elements and also the flexuraMorsional failure of thin-walled open tubes of low torsional rigidity.

Two types of structural instability arise: primary and secondary. The former involves the complete element, there being no change in cross-sectional area while the wavelength of the buckle is of the same order as the length of the element. Generally, solid and thick-walled columns experience this type of failure. In the latter mode, changes in cross-sectional area occur and the wavelength of the buckle is of the order of the cross-sectional dimensions of the element. Thin-walled columns and stiffened plates may fail in this manner.

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