Temperature Dependent Deformation

Disconnecting one support converts a statically indeterminate problem into a determinate one, allowing the rod to experience free thermal expansion or contraction. This simplification enables calculation of deformations in each material section before applying equilibrium constraints to find the unknown reaction force.

Apply an unknown force at the free end and calculate resulting deformations in each material section. Sum the individual deformations, then assert that total deformation must equal zero due to the fixed boundary constraints. This constraint equation solves for the unknown force.

Although the total deformation of a composite rod restrained at both ends equals zero, individual deformations in each material section are non-zero. The steel and brass portions deform in opposite directions, with their deformations summing to zero while each section experiences measurable strain independently.

In a composite rod subjected to normal strain under axial loading, the same force transmits through both the steel and brass sections due to force equilibrium and continuity at their interface. Equal forces in each section allow calculation of corresponding stress values, which differ because each material has distinct cross-sectional area and elastic properties.

When a restrained composite rod experiences temperature change, thermal expansion or contraction is prevented by support reactions, generating internal stresses. The magnitude and direction of stress depend on the thermal expansion coefficients of each material and the constraint forces required to maintain zero total deformation.

After determining the stress in each material section using the equilibrium force, calculate strain by dividing stress by the modulus of elasticity for that material. Express individual deformations by multiplying strain by the original length of each section, confirming their sum equals zero.

Static indeterminacy means equilibrium equations alone cannot determine all reaction forces. Temperature-dependent deformation problems require compatibility conditions—specifically that total deformation must be zero—combined with material properties and deformation of member under multiple loadings principles to solve for unknown forces and resulting stresses in each section.