By James. Harkness

ISBN-10: 1144722683

ISBN-13: 9781144722683

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Vols. 170, 171, Berlin, 1915; see also his book "Elastische Platten," p. 42, Berlin, 1925. 8 We always assume very small deflections or else bending to a developable surface. The case of bending to a nondevelopable surface when the deflections are not small will be discussed later: see p. 47 acting on sections perpendicular to the x axis, it can be concluded from a comparison with the usual beam theory that the effect of the shearing forces is negligible in the case of thin plates, and the equations developed for the case of pure bending can be used with sufficient accuracy for lateral loading.

By using this table, Eq. (21) can be readily solved by the trial-and-error method. For any particular plate we first calculate the left-hand side of the equation and, by using the curves in Figs. 4 and 8, determine the values of the parameter u (1) for simply supported edges and (2) for absolutely built-in edges. Naturally u for elastically built-in edges must have a value intermediate between these two. Assuming one such value for u, we calculate Uo, Ui1 and U2 by using Table 1 and determine the value of the right-hand side of Eq.

In the same manner the slope in the y direction is iv — dw/dy. Taking now any direction an in the xy plane (Fig. 166) making an angle a with the x axis, we find that the difference in the deflections of the two adjacent points a and ax in the an direction is and that the corresponding slope is p IG 16 (a) To find the direction a\ for which the slope is a maximum we equate to zero the derivative with respect to a of expression (a) In this way we obtain , dw /dw ... tana, = - / (b) Substituting the corresponding values of sin ax and cos a\ in (a), we obtain for the maximum slope the expression («) By setting expression (a) equal to zero we obtain the direction for which the slope of the surface is zero.

### A Treatise On the Theory of Functions by James. Harkness

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