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Love-Kirchhoff hypotheses are accurately achieved when the thickness ratio t¡i of the plate satisfies, say, t ¡i< 1/10. In addition, hypothesis 2 also implies that the middle surface of the plate is free from any "in-plane" stress, oxx |z=0 = oyy |z=0 =

This not necessarily implies that the flexure is small compared to the thickness. For instance, the case of a cylindrically bent plate treated hereafter may lead to a large flexure with low stress level.

1.13.7 Bending of Thin Plates and Developable Surfaces

From Love-Kirchhoff hypotheses, the components u, v, and w of the displacement take a greatly simplified form if we consider the basic case of a thin plate. Let us also consider a plate geometry and loads such as the shape of the flexure is a developable surface. The simplest of such surfaces is a cylinder. A cylindrical flexure may be obtained by opposite bending moments uniformly distributed on the long edges in the y-direction of a rectangular plate (Fig. 1.51).

We assume that the long edges allow considering a plate strip element of width dy as taking a strict cylindrical deformation when bent, so the shear strain exy can be neglected, £xy = 0. This could be achieved by maintaining the long edges as straight lines, so the curvature in any section x = constant is null (1/Ry = 0). If no "in-plane" force is applied in the x-direction on the long edges, there is no stretching of the middle surface during bending; the stresses axx and oyy in the strip element are only generated by the curvature 1/Rx due to the moments. Denoting t the thickness of the plate, we will show that for thin plate i.e. t/\Rx \ C 1, these stresses are null at the middle surface or neutral surface of the plate and linearly increase up to maximum opposite values on the faces where z = ±t/2.

Restricting hereafter to the case of a thin plate, the stress component ozz in the volume element of a strip-plate can be neglected when no external load is applied, thus ozz = 0. Assuming that the long y-length is constant during bending entails that the component £yy is null. The stress-strain linear relations (1.123b) provide

1 (Oxx - VOyy) , £yy = E (Oyy - V 0XX )= 0 , (1.139a)

from where

For a thin plate, the displacement or flexure component w = uz is the same at any level z of the plate and does not vary along y, thus w = w(x). Denoting 1/R = d2w/dx2 the curvature, the strain exx at a distance z from the middle surface is £xx = -z/R. Hence from (1.139b),

Depending on the boundary conditions at the long edges, the plate may also be submitted to the action of tensile or compression forces acting in the x-direction. The corresponding induced stress must be added to the above stress.

From the expression of oxx, the bending moment in the strip-plate element per unit length in the y-direction is rt/2 Et3 d2w

The quantity linking the bending moment M to the curvature d2w/dx2 is usually denoted

and called the flexural rigidity or rigidity.

The representation of the z-displacement or flexure curve w of the strip-plate element is a solution of d2 w M

Compared to the flexure of rods or beams such as represented by (1.95), the flexural rigidity D is similar to the quantity EI where now the dimension is decreased of a length because M in (1.143) is a moment per unit length.

If the origin of the coordinates is taken at the center of the neutral surface of the plate which is only submitted to the bending moments ±M0 at the edges x = ±1/2 of the plate, the two integration constants are null. This leads to the flexure w = Md x2, (1.144)

which is a parabola. In fact the true flexure is a circular arc and the parabola must be considered as the result of a first-order approximation for a constant curvature. Sign Convention: We have adopted a positive sign convention for the bending moment when the curvature of the deformation is also positive. Since M0 > 0 for x > 0, and D > 0, then w(±i/2) > 0.14

In the general case of cylindrical bending of a long plate, we may consider the following various loads q: uniform load per unit length applied to the element strip, F: force applied to the positive edge in the x-direction, M0: bending moment applied to the positive edge, whose resulting effect at any cross section of the strip is represented by the bending moment

0 0