WORLD FOR CIVIL

Topic 3.81: Thermal Stress, Strain & Deformation II

1. Completely Constrained Thermal Deformation
The thermal stress which develops if a structure or member is completely constrained (not allowed to move at all) is the product of the coefficient of linear expansion and the temperature change and Young's modulus for the material, or

Example:
A twelve foot horizontal steel rod is fixed between two concrete walls. The rod is initially at temperature of 0^oF and experiences a temperature increase to a final temperature of 80^oF. If the steel rod was initially unstressed, what is the stress in the steel at 80^oF? [Young's modulus for steel is 30 x 10⁶ lb/in², and the coefficient of linear expansion of steel is 6.5 x 10^-6/^oF.]

Solution:
In a completely constrained problem, where the member can not move at all, the thermal stress which develops is given by: = (6.5 x 10^-6/^oF) (80^oF - 0^oF)(30 x 10⁶ lb/in²) = 15,600 lb/in².

Notice that this is quite a sizable stress. In this case there was no initial stress, so the stress which developed is well within the range of allowable stresses for steel. However, there are many cases where structures and materials are near or at their allowable stresses. In that case, if a thermal stress develops, the total stress may well exceed the allowable stress and cause the structure to fail. This, of course, is the reason bridges are built with expansion joints which allow the structure to expand and contract freely and thus avoid thermal stresses. Additionally, this is why concrete sidewalks are built with spaces separating adjacent slabs, allowing expansions to avoid thermal stresses. Concrete highways used to also have expansion spaces built-in, however modern concrete highways are designed without expansion spaces to withstand thermal stresses which develop. Normally they do withstand these stresses, but occasionally long hot periods will allow stresses to built up until the highway actually exploded in a area, producing a large hole in the concrete.

2. Partially Constrained Thermal Deformation:

A more normal situation in a structure, rather than completely constrained or completely free thermal deformation, is a partially constrained thermal deformation. This means a member may expand (or contract) but not as much as it would if unconstrained. Perhaps the best way to demonstrate this situation is to work slowly through an example(s). Please select the following examples:
Example 1 ; Example 2 ; Example 3