The center rod C D of the assembly is heated from T1=30^∘ C to T2=180^∘ C using electrical resis

step 1 In this question we have to find the force in rods AB and EF that caused by the increase in temp

The center rod C D of the assembly is heated from T1=30^∘ C...

The center rod $C D$ of the assembly is heated from $T_{1}=30^{\circ} \mathrm{C}$ to $T_{2}=180^{\circ} \mathrm{C}$ using electrical resistance heating At the lower temperature $T_{1}$ the gap between $C$ and the rigid bar is $0.7 \mathrm{mm}$. Determine the force in rods $A B$ and $E F$ caused by the increase in temperature. Rods $A B$ and $E F$ are made of stecl, and each has a cross-sectional area of $125 \mathrm{mm}^{2}$. CD is made of aluminum and has a cross-sectional area of $375 \mathrm{mm}^{2}$ \[ E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa}, \text { and } \alpha_{\mathrm{al}}=23\left(10^{-6}\right) /^{\mathrm{e}} \mathrm{C} \]

The center rod $C D$ of the assembly is heated from $T_{1}=30^{\circ} \mathrm{C}$ to $T_{2}=180^{\circ} \mathrm{C}$ using electrical resistance heating At the lower temperature $T_{1}$ the gap between $C$ and the rigid bar is $0.7 \mathrm{mm}$. Determine the force in rods $A B$ and $E F$ caused by the increase in temperature. Rods $A B$ and $E F$ are made of stecl, and each has a cross-sectional area of $125 \mathrm{mm}^{2}$. CD is made of aluminum and has a cross-sectional area of $375 \mathrm{mm}^{2}$
\[
E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa}, \text { and } \alpha_{\mathrm{al}}=23\left(10^{-6}\right) /^{\mathrm{e}} \mathrm{C}
\]

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of a material to change its dimensions in response to a change in temperature. This is characterized by the coefficient of thermal expansion, a material-specific parameter that quantifies the change in length (or volume) per unit temperature change. Understanding thermal expansion is crucial in problems where temperature variations lead to dimensional changes, which may interact with other components in a mechanical system.

Thermal Stress

Thermal stress arises when a material that is free to expand or contract is instead restrained, preventing it from achieving its free thermal expansion. This restraint creates internal forces and stresses as the material is forced to accommodate the difference between its free and actual deformed states. Calculating thermal stress is essential in predicting structural behavior and ensuring that components can withstand the additional loads induced by temperature changes.

Hooke's Law

Hooke's Law is a foundational principle in elasticity, stating that the stress in a material is proportional to the strain within the elastic limit of that material. This relationship, expressed mathematically as stress equals the modulus of elasticity multiplied by strain, is crucial for analyzing deformations under load, including those due to thermal effects. Hooke's Law provides a means to compute the force generated in a component when its expansion or compression is constrained.

Compatibility of Deformations

Compatibility of deformations is a concept that ensures that all parts of a connected mechanical system deform in a manner that allows them to remain joined without creating gaps or overlaps. In thermal problems, compatibility conditions link the thermal expansion of one element with the elastic deformation of another, ensuring that the overall system maintains equilibrium. This concept is crucial when different components with varying thermal properties interact.

Material Properties in Thermal Analysis

Material properties such as the coefficient of thermal expansion, modulus of elasticity, and cross-sectional area play a key role in thermal analysis. These properties determine how much a material will expand or contract with temperature change and how much stress or force is generated when this expansion is constrained. Understanding and utilizing these properties is essential for designing structures that can safely accommodate temperature-induced deformations.

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