Q4
A deep beam of negligible weight and unit width is simply supported
and carries a uniformly distributed load w per unit length on its lower
edge, i.e. at y = -h/2 as shown in figure Q4. It is thought that the
following stress function ? describes this problem, where A is a
constant and x, y, h and L are as shown in figure Q4.
$\phi = A(-4y^5 + 20x^2y^3 - 15x^2yh^2 - 20y^3L^2 + 2y^3h^2 + 5x^2h^3)$
(a) Show that this stress function represents a valid solution to a problem
in two dimensional elasticity with zero body forces.
(b) Show that the boundary conditions on the upper and lower edges
(y = h/2, y = -h/2) are satisfied, and hence determine the value of the
constant A.
(c) Determine expressions representing the normal stress ($\sigma_x$ and $\sigma_y$)
distributions at the cross section where x = 0 and determine the
expression representing shear stress $\tau_{xy}$ distribution at the cross
section were x = L. Sketch all three distributions.
