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396 W. Prager In terms of the deflexion w of the plate, the curvatures Ky and twist A at general point x,y are given by d2w d2w . d2w jB Kx = ~da?’ Ky = ~dy*’ <10 Elimination of w between these equations furnishes the following equations o compatibihty: dx fay 0, dx dy (11 fax dX dy dx The field equations listed above must be supplemented by appropriate boundary conditions and by stress-strain laws, which characterize the mechanical behaviour of the disk or plate. The stress-strain law of a disk relates the stress resultants Sx, Sy, T to the strains ex, ey, y; the stress-strain law of a plate relates the bending and] twisting moments Mx, My, N to the curvatures kx, Ky, and the twist A.
On the analogy between plates and disks 397 _ Gy and du/dy — y — 0)are also known along this boundary. Now, since we ake 4> = 0, dv = ey->Mx, , 1 (16) d2u dyd o) dN d\Jr _ , ~dy*~ dy + dy^ dy+ dy ~ qx\ khere q'x = qx + dN/dy is Kirchhoff’s equivalent shear force, the plus sign corre- iponding to the sign conventions implicit in (6) and (9).