If the assembly fits snugly between two rigid supports A and...

If the assembly fits snugly between two rigid supports $A$ and $C$ when the temperature is $T_1$, determine the normal stress developed in both segments when the temperature rises to $T_2$. Both solid segments are made of the same material, having a modulus of elasticity of $E$ and coefficient of thermal expansion of $\alpha$.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of a material to change its dimensions in response to a change in temperature. The coefficient of thermal expansion quantifies this behavior by relating the change in temperature to the relative change in length of the material. In scenarios where the material is unconstrained, the dimensional change is purely thermal; however, when the material is restrained, such as by rigid supports, this expansion can lead to the development of internal stresses.

Modulus of Elasticity

The modulus of elasticity, or Young's modulus, is a measure of a material's stiffness. It defines the relationship between stress (force per unit area) and strain (deformation relative to original dimensions) in the elastic region of the material's behavior. In the context of thermal stresses, it is used to relate the strain induced by thermal expansion (or its restraint) to the resulting stress within the material.

Thermal Stress in Constrained Structures

Thermal stress in constrained structures arises when a material, which would normally expand or contract with temperature change, is prevented from doing so by external restraints. This restraint leads to the development of internal stresses within the material as it attempts to expand or contract against the rigid boundaries. The magnitude of these stresses is directly related to the coefficient of thermal expansion, the temperature change, and the material's modulus of elasticity.

Compatibility and Equilibrium Conditions

Compatibility and equilibrium conditions refer to the requirements that deformations in a structure must be consistent with the constraints imposed by supports or connections, and that the sum of forces and moments within the structure must balance. In problems involving thermal expansion with rigid supports, these conditions help determine the induced stresses by ensuring that the net displacement of the assembly is zero despite the thermal expansion tendency.

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