The net shearing force Vn which is the axial reaction at the edge, is given by

dMn ds

The previous equations representing Mn and Vn at the contour C completely determine the boundary conditions for the elliptical plate. The net shearing forces can be readily determined in the x and y directions. Considering Vn on y-axis (x = 0) for y = ay at the contour, the second term in (5.40) vanishes and Vy{0, ±ay} = Qn{0, ±ay} at the ends of the minor axis ay, that is

ax2 ay3

Similarly, on the x-axis and at the contour C, the reaction at the ends of the major axis ax is Vx{±ax, 0} = Qn{±ax, 0}, and


Its absolute value gives a slightly lower intensity than for |Vy{0, ±ay}\. Degenerating the ellipse into a circle with ax = ay = a, the two latter relations allow recovering the well-known result for circular plates, Vr{a} = Qr{a} = -qa/2.

While the above analysis applies to a built-in contour, let us consider now a flexure z composed of a co-added quadratic mode such as

1 x2 y2 ax2 ay2

1 x2 y2 ax2 ay2

where k is a free parameter. The new sag is z{0,0} = (1 + k) z0 and this flexure also satisfies V4z = q/D with z0 as given by (5.28). k = 0 is corresponding to the previous case of null slope at edge. So, with k = 0, the principal curvatures are changed which implies modification of the bending moments along the plate edge.

At the extremities of the x, y axes, the bending moments are respectively

ax2 ay2


0 0

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