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

If the assembly fits snugly between the two 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 $\alpha$. The flexible supports at $A$ and $C$ each have a stiffness $k$.

Key Concepts

Compatibility Conditions

Compatibility conditions ensure that the deformations or displacements of different parts of an assembly are consistent with each other and with the boundary conditions. In the context of thermal loading, these conditions serve to relate the free thermal expansion of a material to the deformations permitted by the supports, ultimately leading to internal force and stress development. This concept is key when linking the material’s thermal behavior with the mechanical constraints imposed by its supports.

Modulus of Elasticity

The modulus of elasticity, also known as Young's modulus, is a fundamental material property that measures a material's stiffness or resistance to deformation under load. It relates the stress generated within a material to the strain it experiences. In problems involving thermal stress, the modulus of elasticity is used to determine how much stress is generated for a given deformation that is induced by temperature changes.

Support Stiffness

Support stiffness refers to the resistance provided by the supports or constraints in an assembly against any displacement or deformation. When supports have finite stiffness, they allow some degree of movement under load, which influences the distribution and magnitude of induced stresses. This concept is critical to understanding how the flexible nature of supports can moderate or develop internal forces and stresses when thermal expansion occurs.

Thermal Expansion

Thermal expansion is the tendency of materials to change their dimensions in response to changes in temperature. This phenomenon is characterized by the coefficient of thermal expansion, which quantifies how much a material expands per unit temperature change. Understanding thermal expansion is crucial when analyzing assemblies subjected to temperature variations, as unconstrained expansion can lead to displacements or forces when movement is restricted.

Thermal Stress

Thermal stress arises when a material or an assembly is prevented from expanding or contracting freely due to temperature changes. When constraints are applied, such as fixed supports or boundary conditions, the thermal expansion or contraction induces internal stresses within the material. This concept is essential in assessing how temperature variations can lead to potentially damaging stress levels in structural elements or assemblies.

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