When considering thermal expansion or shrinkage of a structure, it can be useful to consider the resulting stresses in terms of primary stresses and secondary stresses
Apply the thermal load to the structure assuming it is fully restrained (i.e. \(\epsilon = 0\)) everywhere. This puts a stresses into the structure determined by:
\[ \sigma_{res} = -E \alpha \Delta T \]
Note that \(\alpha\) and \(\Delta T\) can vary across the cross-section.
The net effect of these stresses over the section are called the Restraint Forces and comprise an axial force and moment.
\[\begin{align} F_{res} &= \int \sigma_{res} \ dA \\ M_{res} &= \int \sigma_{res} y \ dA \end{align}\]
where \(y\) is the distance above the centroid.
The Release Forces are simply the Restraint Forces with the opposite sign
\[\begin{align} F_{rel} &= -F_{res} \\ M_{rel} &= -M_{res} \end{align}\]
The Primary Stresses are those that result from adding the Restraint Stresses to the stress field resulting from applying the Release Forces assuming plane sections remain plane.
\[\begin{align} \sigma_{1} & = \sigma_{res} + \sigma_{rel} \\ \sigma_{1} & = -E \alpha \Delta T + \frac{F_{rel}}{A} + \frac{M_{rel}y}{I} \end{align}\]
Note that these stresses will have a net force of zero and a net moment of zero, but will still cause the structure to deflect.
The Secondary Stresses arise from the global structure’s resistance to deflect due to the primary stresses. These are calculated by applying the release forces everywhere in the structure, running a global analysis, and subtracting off the original release forces at the end.
Suppose we calculated the distribution of release moments shown in Figure 1.
We can apply these moments to an analysis model as shown in Figure 2.
Running for this analysis gives the results in Figure 3.
Then in order to obtain secondary moments, we subtract the moment diagram in Figure 1 from the moment diagram in Figure 3:
