A more complex polynomial for the stress function is
\phi=\frac{A x^{3}}{6}+\frac{B x^{2} y}{2}+\frac{C x y^{2}}{2}+\frac{D y^{3}}{6}
Question 2.2: A more complex polynomial for the stress function is Φ= Ax^3...
As before,
\frac{\partial^{4} \phi}{\partial x^{4}}=\frac{\partial^{4} \phi}{\partial x^{2} \partial y^{2}}=\frac{\partial^{4} \phi}{\partial y^{4}}=0
so that the compatibility equation (2.9) is identically satisfied. The stresses are given by
\frac{\partial^{4} \phi}{\partial x^{4}}+2 \frac{\partial^{4} \phi}{\partial x^{2} \partial y^{2}}+\frac{\partial^{4} \phi}{\partial y^{4}}=0 (2.9)
\sigma_{x}=\frac{\partial^{2} \phi}{\partial y^{2}}=C x+D y
\sigma_{y}=\frac{\partial^{2} \phi}{\partial x^{2}}=A x+B y
\tau_{x y}=-\frac{\partial^{2} \phi}{\partial x \partial y}=-B x-C y
We may choose any number of values of the coefficients A, B, C, and D to produce a variety of loading conditions on a rectangular plate. For example, if we assume A = B = C = 0, then \sigma_{x}=D y, \sigma_{y}=0, and \tau_{x y}=0, so that, for axes referred to an origin at the mid-point of a vertical side of the plate, we obtain the state of pure bending shown in Fig. 2.2(a). Alternatively, Fig. 2.2(b) shows the loading conditions corresponding to A = C = D = 0 in which \sigma_{x}=0, \sigma_{y}=B y, \text { and } \tau_{x y}=-B x.
Related Answered Questions
The stress function must satisfy Eq. (2.9). From t...
Equation (2.9) is identically satisfied, since eac...