WORLD FOR CIVIL

Thermal Stress, Strain & Deformation - I

Changes in temperatures causes thermal effects on materials. Some of these thermal effects include thermal stress, strain, and deformation. The first effect we will consider is thermal deformation. Thermal deformation simply means that as the "thermal" energy (and temperature) of a material increases, so does the vibration of its atoms/molecules; and this increased vibration results in what can be considered a stretching of the molecular bonds - which causes the material to expand. Of course, if the thermal energy (and temperature) of a material decreases, the material will shrink or contract. For a long rod the main thermal deformation occurs along the length of the rod, and is given by:

where (alpha) is the linear coefficient of expansion for the material, and is the fractional change in length per degree change in temperature. [Some values of the linear coefficient of expansion are: Steel = 12 x 10^-6/^oC = 6.5 x 10^-6/^oF; Brass = 20 x 10^-6/^oC = 11 x 10^-6/^oF; Aluminum = 23 x 10^-6/^oC = 13 x 10^-6/^oF.] The term is the temperature change the material experiences, which represents (T_f - T_o), the final temperature minus the original temperature. If the change in temperature is positive we have thermal expansion, and if negative, thermal contraction. The term 'L' represents the initial length of the rod.

Example 1
A twelve foot steel rod is initially at a temperature of 0^oF and experiences a temperature increase to a final temperature of 80^oF. What is the resultant change in length of the steel?

Solution: Deformation = = 6.5 x 10^-6/^oF (80^oF- 0^oF) (144 inches) = .075 inches

(The length of the rod was converted into inches in the equation since the deformations are normally quite small.) We see the deformation is indeed quite small, and in many cases the thermal deformation has no significant effect on the structure. However, if the structure or members of the structure are constrained such that the thermal expansion can not occur, then a significant thermal stress may arise which can effect the structure substantially - and which we will address shortly.

In addition to the length, both the area and volume of a material will change with a corresponding change in temperature. The resulting changes in area and volume are given by:

; and

These formulas, as written, are not exact. In the derivations [using (L + L)² for area, and (L + L)³ for Volume] there are cross terms involving the linear coefficient of expansion squared in the area formula, and the coefficient of expansion squared and cubed in the volume formula. These terms are very small and can be ignored, resulting in the two equations above.

While unconstrained thermal expansion is relatively straight forward effect, it still requires a bit of thought, such as in the following question.

A flat round copper plate has a hole in the center. The plate is heated and expands. What happens to the hole in the center of the plate - expands, stays the same, or shrinks?

When I ask this question in my classroom it is not unusual for the majority of the answers to be incorrect. Our first thought often is that since the plate is expanding, the hole is the center must be getting smaller. However, this is not the case. The atoms/molecules all move away from each other with the result that the hole expands just as if it were made of the same material as the plate. This is also true of volume expansion. The inside volume of a glass bottle expands as if it were made of glass.

A somewhat more interesting aspect of thermal expansion is when it "can't" - that is, what happens when we constrain a structure or member so it can not expand. (or contract)? When this happens a force and resulting stress develop in the structure. A simple way to determine the amount of stress is to let the material expand freely due to thermal expansion, and then compress it back to its original length (a mechanical deformation) . See diagram below.

If we equate these two effects (deformations) we have: = FL/EA ;note that we can cancel the length L from each side of the equation, and then cross multiply by E, arriving at: = F/A , however, F/A is stress and we can finally write:

; 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.