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15 Sophie Germain [68] established a biharmonic equation in memoirs to the French Academy, the last in 1815, as attempts to explain the vibration modes of an elastic plate. This problem was brought to her attention by the fascinating experiments by Chladni [28] showing various patterns formed by the sand which moved on a vibrating plate, when excited by acoustic strings of different tones, until reaching the two-dimensional positions of the nodes. After seeing Chladni's patterns -or nodal curves - in 1810, Napoleon funded the Institute of France for an extraordinary prize to be awarded for a theory providing a mathematical explanation. At the issue of the contest announcement, the examining committee noticed some flaws in Germain's last memoir but deduced that her approach was correct, and the prize was awarded to her in 1816. In fact a full analysis of Germain's problem would have required first knowing a boundary condition involving the twisting moment Mrt at the plate edge. This condition for non-axisymmetrical deformations was found much later by Kirchhoff, in 1850 (see hereafter).

Turning to the case of the vibrations of an elastic membrane, only involving second derivatives, the nodal curves of a circular membrane - like drum heads - were first derived by Clebsch [34], in 1862, where the radial part of the solutions are Bessel functions.

Germain's problem for a circular plate was finally solved by Rayleigh [124, 125], in 1873, by using the Kirchhoff boundary condition of the thin plate theory. He also solved this problem for the nodal curves of a square plate. Ritz [133] later generalized Rayleigh's extremals method of the total energy (potential and kinetic), classically known as the Rayleigh-Ritz method.

1 fd2Mx 2 d2Mxy d2M, D \ dx2 dxdy dy f d Mx 2d2Mxy d2My\ ^2^2

Hence, from (1.164d), this second derivative sum of the moments must be zero if there is no transverse load (q = 0).

The shearing forces Qx and Qy per unit length act on the sections of the element in the z-direction. The equilibrium equations of the statics, obtained from the moments around the y- and x-axes, and from the forces in the z-directions are respectively dMx dMxy dMy dMyx

From (1.167), we obtain the expression of the shearing forces d o d o

Substituting them into (1.168) entails il V2w + V2w - q = 0, dx2 dy2 W D '

which is the biharmonic Poisson equation.

Finally Kirchhoff [88, 89] defined, in 1850, the net shearing forces Vx and Vy per unit length as

Mxy Myx

which are the resulting axial forces acting into the plate. Theses forces enter in the formulation of a boundary condition, known as Kirchhoff's condition at the edge.16 For a plate with a free edge, the net shearing forces become null at the contour.

16 The Kirchhoff definition of the net shearing force is an important formulation for a boundary condition at a plate edge. This attracted the attention of Rayleigh and permitted him to elaborate the theory of the vibrations of plates.

Its physical significance entailed a clarified statement of the boundaries as was explained by Kelvin and Tait [83]. Independently the same question was explained by Boussinesq [18]. A detailed account on Kirchhoff's condition is given by Love [97], p. 460, and by Timoshenko and Woinowsky-Krieger [155], p. 84.

1.13.9 Bending of Rectangular Plates of Constant Thickness

• Plate distorted by bending moments: Let us consider a flat rectangular plate of lengths a, b in the x,y directions only submitted to opposite bending moments Ma and Mb, i.e. where no transverse load is applied to its surface (q = 0). We assume that the bending moments are in the principal directions of the frame, positive for x and y, and that the frame origin remains at the center of the middle surface when at rest and when bent (Fig. 1.54).

Setting the origin at the center of the mid-surface, and for symmetry reasons, dw = 0, ^

The Kirchhoff conditions for a plate with free edges is that the net shearing forces Vx and Vy are zeroed on this contour,

Since the x- and y-axis are in the principal directions of the bending moments, the twisting moments Mxy and Myx vanish whatever x and y which entails that Qx = Qy = 0 and Vx = Vy = 0. The bending moments Mx and My are constant moments whatever x and y. Equating them to the moments Ma and Mb applied to the edges, we obtain from (1.165a) to (1.165b), d2w Ma - vMb d2w Mb - vMa dx2 D(1 - v2)' dy2 D(1 - v2)'

From the origin and boundary conditions, all the integration constants are equal to zero. The solution is

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