Elasticity Problems and Airy Stress Functions Explained

(a) How is the Airy stress function determined in elasticity problems? What are the steps involved?

In elasticity problems, it is often useful to calculate stresses by first determining the Airy stress function, φ. The Airy stress function φ can be calculated by following certain steps. This function is expressed as: $$\\phi = \\frac{1}{2E}(M_x^2+2N_{xy}^2-M_y^2)$$ where M_x, M_y, and N_xy represent the bending moments about the x and y-axes and the twisting moment, respectively.

To determine the Airy stress function, we first take the first and second derivatives of φ with respect to x and y. By setting these derivatives equal to zero, we can solve for the stress components σx, σy, and σxy. The Airy stress function is crucial in understanding the stress distribution in structures under loading conditions.