so the the differential equation of the flexure (1.144) becomes d2w 4^2 q (I2 2\ Mo dX2 - IT w = - 2D I J - x + D ■

The particular solution wi is of the form wi = A1x2 + A2. After identification of A1 and A2 coefficients, we obtain q£2 , q£4 2 - y2 Mq£2 w1 = - D2 x2 - 32D y4 - 4Dy ■ (1.151)

The solution without the right-hand side of (1.150) is w2 = C1coshy + C2sinh2yx, (1.152)

Since the loading of the plate is symmetrical with respect to the z,y plane, then C2 = 0. The general solution is of the form w = w1 + C1 cosh (2yx/£). Setting the origin of the flexure on the neutral surface and midway between the edges, w(0) = 0 is achieved if q£4 2 - y2 M0£2 4Dy2

which entails w = w1(0) (1 - cosh2^) - x2- (1.154a)

Substituting 1 - cosh 2yx/£ = -2sinh2 yx/£, we can represent the flexure as a function of the force F (or y) and moment M0 by w = (qq^2-2+My) ^ ^ - ^

The most interesting case is when the opposite forces F, -F, are generated by the deformation of the plate itself when the edges are not allowed to move in the x-direction. During the bending, the plate elongates and its curvilinear width - £ when not stressed - is increased by 8£ which represents the difference of the length between the arc along the flexure curve and the length £ of the cord.

We have seen with (1.148b) that the (movable) built-in edge condition entails dw/dx\±£/2 = 0 and M0 = q£2/12. Now the intensity of the forces ±F due to the unmovable built-in edges can be obtained. This was solved by Timoshenko and Woinowsky-Krieger ([155], Chap. 1), who showed that for small flexures, an element strip elongates of

This elongation is only caused by the extension of the plate from the blocked edges. Still assuming that the lateral strain £yy of the strip is unchanged in the long plate the co-added component of constant stress is, from (1.139b), o!(x = -E 8£/(1 - v2)£ = -F/t. Also using (1.149) and (1.142), we obtain

Equalizing the latter two expressions and substituting the flexure represented by (1.154b), we thus obtain for unmovable built-in edges a relation which now contains the unknown p only,

where ¥(p) is derived from (1.155a) and includes finite term polynomials in pn and tanhmp. Any numerical solving in function of the dimensionless ratio (1 - v2)2(q/E)2(i/t)8 allows us to obtain the intensity of force F in (1.149) and the flexure from (1.154b).

1.13.8 Bending of Thin Plates and Non-developable Surfaces

Let us consider hereafter a thin plate of thickness t and in-plane typical dimension t, such as t/t C 1, and assume that the absolute value of the maximum curvature |1/Rmin| occurring in the bending is t/|Rmin| C 1. The latter condition is equivalent to state that the maximum value wmax of the flexure taken over all the surface remains small compared the thickness of the plate. Hence the conditions are respectively t/t C 1, |Wmax|/t C 1. (1.157)

Under these conditions, the bending introduces a negligible strain along the middle surface and the perpendicular stress distribution is null on it; this surface is then called the neutral surface.

One generally distinguishes between plates bent into an synclastic surface and into a anticlastic surface. The principal radii of curvature Rx and Ry of a synclastic surface are RxRy > 0, whilst for an anticlastic surface RxRy < 0 (Fig. 1.53).

Let us assume that the plate is flat when not bent, and set a coordinate frame x, y, z with the origin at the neutral surface and the z-axis normal to the surface. The above hypotheses of thin plate and small deformation entail that the flexure w is a function of x and y only. Since the middle surface is also the neutral surface, the components of the displacement vector at this surface are

Fig. 1.53 Left: Synclastic surface, RxRy > 0. Right: Anticlastic surface, RxRy < 0

For a thin plate and a small deformation, the second of Love-Kirchhoff hypotheses states that the stresses

at the outer surface of the plate as well as at the inner volume of it. The flexure of all points originally located at any distance z from the middle surface can be derived by orthogonal and equal lengths from the flexure of the middle surface. The third equation of the stress-strain relations (1.123a) leads to

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