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.