18.7:
Thermal expansion and Thermal stress: Problem Solving
A 75 centimeter long metal wire is placed inside a furnace at 450 degrees Celsius. If the wire length changes to 75.77 centimeters, and the room temperature is 20 degrees Celsius, evaluate the coefficient of linear expansion of the wire's material.
After cooling, both ends of the wire are attached to a stand and placed inside the furnace at 200 degrees Celsius. Evaluate the thermal stress generated if Young's modulus is 7.0 x 1010 Pascals, and identify the wire's material.
The coefficient of linear expansion of the wire depends on the initial length, the temperature gradient, and the change in length and is calculated as 2.4 x10-5 per degrees Celsius.
Further, the thermal stress generated can be calculated using Young's modulus, the temperature gradient, and the linear expansion coefficient of the wire's material and is calculated as 3.02 x102 megapascals.
The values of the coefficient of expansion and Young's modulus suggest that the wire is most likely made of aluminum.
18.7:
Thermal expansion and Thermal stress: Problem Solving
San Francisco's Golden Gate Bridge is exposed to temperatures ranging from -15 °C to 40 °C. At its coldest, the main span of the bridge is 1275 m long. Assuming that the bridge is made entirely of steel, what is the change in its length between these temperatures?
To solve the problem, first, identify the known and unknown quantities. The initial length (L) of the bridge is 1275 m, the coefficient of linear expansion (α) for steel is 12 x 10-6/°C, and the change in temperature (ΔT) is 55 °C.
Recall that thermal expansion is the expansion in material with increased temperature. Linear expansion is the change in the length of a substance by applying temperature, and it is proportional to the initial length and change in temperature. The proportionality constant is the coefficient of linear expansion. Using the equation for linear thermal expansion to calculate the change in length, ΔL
Substituting all the known values into the equation to solve for the change in length, ΔL
The change in length of the Golden Gate Bridge is found to be 0.84 m. Although not large compared with the length of the bridge, this change in length is observable. It is generally spread over many expansion joints so that the expansion at each joint is small.
Further, if changing the temperature of an object while preventing it from expanding or contracting, the object is subjected to thermal stress. To calculate the thermal stress in an object where both ends are fixed rigidly, the stress can be thought of as developing in two steps. First, let the ends be free to expand or contract and find the expansion or contraction. Second, find the stress necessary to compress or extend the object to its original length by the methods studied in static equilibrium and elasticity on static equilibrium elasticity. In other words, the ΔL of the thermal expansion equals the ΔL of the elastic distortion, except that the signs are opposite.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; pp 15, 18.