Thermal Stresses in a Rectangular Plate Clamped Along an Edge

An approximate solution has been obtained for the stresses induced by a uniform change in temperature of a thin rectangular plate, clamped along an edge. The solution has been carried to completion for plates whose clamped edge is long, i.e., more than 5 times the length of the perpendicular free edge. The solution for smaller ratios of clamped to perpendicular lengths is expressed in terms of six determined functions whose coefficients are to be evaluated by satisfying two boundary conditions. The thermal-stress problem is first converted to one of specified boundary tractions. The normal stress, σx, parallel to the clamped edge is assumed of the form σx = f1 (x) + y f2(x) + y2f3(x), where fi(x) are as yet undetermined functions, and where y is the co-ordinate at right angles to the clamped edge. Using the equations of equilibrium and the boundary conditions, τxy and σy are expressed in terms of powers of y and the derivatives of fi(x). The integral representing the strain energy is then expressed in terms of the expressions for σx, σy, and τxy. In accordance with the principle of least work, the integral representing the strain energy is minimized, using the calculus of variations. The resulting simultaneous differential equations for fi(x) are solved as a linear combination of twelve functions (six of which drop out, by symmetry). Given f1(x), then f2(x) and f3(x) are determinate by virtue of the simultaneous equations. The six coefficients in the expression for f1 are evaluated by satisfying the boundary conditions along the free edges. The maximum normal stress concentration, over 10, occurs at the junction of the free and clamped edges.