Temperature and Support Settlement in the Force Method

In a statically indeterminate structure, thermal deformation and support settlement cannot generally occur freely because redundant restraints limit deformation and therefore generate additional reactions and internal forces. In the force method these effects enter the compatibility equations directly.

Compatibility with imposed actions

A general equation in redundant direction $i$ may be written as

$$\sum_{j=1}^{n}\delta_{ij}X_j+\Delta_{iP}+\Delta_{iT}=\Delta_i^{\mathrm{given}},$$

where $\Delta_{iT}$ is the free displacement of the primary structure caused by temperature and $\Delta_i^{\mathrm{given}}$ is a prescribed displacement such as support settlement. Exact signs depend on the adopted positive directions.

Uniform temperature change

A free member has thermal elongation $\alpha\Delta T L$. If redundant restraints prevent that elongation, the unknown forces $X_j$ must generate compensating elastic deformation.

Temperature gradient

Nonuniform temperature through the section depth creates free curvature. In frames and continuous beams, restraint of this curvature can generate bending moments even when no external mechanical forces are applied.

Support settlement

If the original structure has a prescribed nonzero displacement in a released direction, the corresponding compatibility equation must reproduce that displacement. Settlement should therefore not automatically be modeled as an additional external force.

Superposition and verification

For linear material behavior, mechanical, thermal, and prescribed-movement states may be evaluated separately and superposed. Check both equilibrium and that the restored structure reproduces the specified support and restraint displacements.