For the stress state in Exercise 3.8, plot contours of the von Mises stress in the region 0≤x ≤L, –c≤ y≤ c, with L = 1, c = 0.1, and P = 1.

Data from exercise 3.8

Exercise 8.2 provides the plane stress (see Exercise 3.5) solution for a cantilever beam of unit thickness, with depth 2c, and carrying an end load of P with stresses given by:

Data from exercise 8.2

Show that the Airy function:

solves the following cantilever beam problem, as shown in the following figure. As usual for such problems, boundary conditions at the ends (x = 0 and L) should be formulated only in terms of the resultant force system, while at y =± c the exact pointwise specification should be used. For the case with N = 0, compare the elasticity stress field with the corresponding results from strength of materials theory.

Data from exercise 3.5

A two-dimensional state of plane stress in the x, y-plane is defined by σ_z = τ_yz =  τ _zx = 0. Using general principal value theory, show that for this case the in-plane principal stresses and maximum shear stress are given by:

Transcribed Image Text:

0x = 3P 2c3y, dy = 0, Txy= 1,25 = Show that the principal stresses are given by 3P 1/ [xy ( - y) + x 403 and the principal directions are 3P 4c 20(1.2) + = [(xy (c y) + xy )e1 + (2 3) e] Note that the principal directions do not depend on the loading P. X constant