Displacements due to Temperature and Support Settlement

Structural displacements can occur without applied mechanical forces because of temperature change, a temperature gradient, or prescribed support movement. In a statically determinate structure these actions can often produce free deformation without additional stress resultants; in an indeterminate structure compatibility may generate reactions and internal forces.

Uniform temperature change

For a free straight member of length $L$ subjected to a uniform temperature change $\Delta T$,

$$\Delta L=\alpha\Delta T\,L,$$

where $\alpha$ is the coefficient of linear thermal expansion.

Temperature gradient

If temperature varies across the section depth, one side tends to expand more than the other and a thermal curvature develops. For a linear temperature difference between extreme fibers, the curvature is proportional to $\alpha\Delta T_h/h$, where $h$ is the distance between those fibers.

Support settlement and prescribed movement

Support settlement is a prescribed kinematic action. A determinate structure may accommodate it without additional reactions when its kinematics permit. In an indeterminate structure, prescribed support movements enter the compatibility conditions and can generate additional internal forces.

Superposition

In a linear analysis, displacements caused by mechanical loading, temperature, and prescribed support movements can be evaluated separately and then combined algebraically.

Signs and units

Define the sign of $\Delta T$, the direction of support movement, and the positive direction of the required displacement explicitly. Thermal elongation has units of length, whereas thermal curvature has units of inverse length.