By J Prescott

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63) has already shown, that the axis of the resultant couple does not coincide with the neutral axis except when the neutral axis coincides with one of the principal axes with OY'. of the section. 39. The form of the strained rod. The displacements of points originally on in the expressions by putting y=zo, 2 = the ji'-axis are obtained for the displacements. 65) write 2 for old' jr-axis is w strained. 66) I APPLIED ELASTICITY 38 The radius of curvature of this parabola is R (I +a2x^)^ a ni (-S) = — approximately (3-^7) ci This last approximation is obtained by neglecting ^'- in comparison with x'-, z being the displacement in this equation.

13), -\- = (w — n) = {m n) The and the -\- (c — (m — n)a -\- c -\- a) and c, by means of terms of a alone. 23) SOME PARTICULAR SOLUTIONS OF THE EQUATIONS Also c m—n = a m+n — (l— 2g) _ ' I I+(I— 20) This last case would 31 be useful if a we were with dealing the stretching of thin sheets by a pull in one direction, the length in a perpendicular stresses P^^ 36. A being kept direction and F^ are by some means. constant The parallel to the faces of the sheet. prism (or rod) hanging vertically under its own weight.

10). volume of a small block of natural dimensions is 6x(i -\-a)xdy(i -\- P)x = dxdydz(i dz(i -\-y) 4-« + + /^ J') neglecting products of «, fi and y. Then it follows that the ratio of the increase of volume to the original volume is (a-\- fi -{- y)- Call this A. 20) . ^) and n are constants, n being the modulus of E and In terms of rigidity. 22) j^ du dv dw 25. Relations We between stresses and external with a rectangular block of dimensions . '"3) ^Si + 8y + T. forces. x x dy dx dx having its centre at x, y, 2, and take the means stresses on the faces to be the stresses at the middle of the face concerned.