Carry out the integration details to develop the displacements (5.7.4) in Example 5.1.

Equation 5.7.4

Data from example 5.1

As an example of a simple direct integration problem, consider the case of a uniform prismatic bar stretched by its own weight, as shown in Fig. 5.11. The body forces for this problem are F_x = F_y = 0, F_z =–ρg, where ρ is the material mass density and g is the acceleration of gravity. Assuming that on each cross-section we have uniform tension produced by the weight of the lower portion of the bar, the stress field would take the form:

fig 5.11

This equation can be integrated directly, and applying the boundary condition σ_z = 0 at z = 0 gives the result σ_z(z) = ρgz. Next, by using Hooke’s law, the strains are easily calculated as:

The displacements follow from integrating the strain-displacement relations (5.1.1), and for the case with boundary conditions of zero displacement and rotation at point A (x = y = 0, z = l), the final result is:

Equation 5.1.1

Transcribed Image Text:

u= W = vpgxz E V = vpgyz E pg 21 [2 +v(x +y) 1]