The two cylindrical rod segments are fixed to the rigid...

The two cylindrical rod segments are fixed to the rigid walls such that there is a gap of 0.01 in. between them when $T_{1}=60^{\circ} \mathrm{F} .$ What larger temperature $T_{2}$ is required in order to just close the gap? Each rod has a diameter of 1.25 in. Determine the average normal stress in each rod if \[ \begin{array}{l} T_{2}=300^{\circ} \mathrm{F} . \text { Take } \alpha_{\mathrm{al}}=13\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi} \\ \left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}, \alpha_{\mathrm{cu}}=9.4\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{cu}}=15\left(10^{3}\right) \mathrm{ksi}, \text { and } \\ \left(\sigma_{Y}\right)_{\mathrm{cu}}=50 \mathrm{ksi} \end{array} \]

The two cylindrical rod segments are fixed to the rigid walls such that there is a gap of 0.01 in. between them when $T_{1}=60^{\circ} \mathrm{F} .$ What larger temperature $T_{2}$ is required in order to just close the gap? Each rod has a diameter of 1.25 in. Determine the average normal stress in each rod if
\[
\begin{array}{l}
T_{2}=300^{\circ} \mathrm{F} . \text { Take } \alpha_{\mathrm{al}}=13\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi} \\
\left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}, \alpha_{\mathrm{cu}}=9.4\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{cu}}=15\left(10^{3}\right) \mathrm{ksi}, \text { and } \\
\left(\sigma_{Y}\right)_{\mathrm{cu}}=50 \mathrm{ksi}
\end{array}
\]

Key Concepts

Geometric Properties of Circular Sections

For cylindrical rods, the cross-sectional area is determined by the diameter using the area formula A = ?d²/4. This geometric property is crucial when calculating stress, as the force resulting from thermal expansion is distributed over the rod's area, influencing the average normal stress experienced by the material.

Gap Closure and Compatibility Conditions

In problems involving multiple members with gaps or clearances, such as gap-closing scenarios, compatibility conditions are used to relate the thermal expansions of individual components to the overall displacement requirements. This ensures that the collective expansion, when constrained, leads to specific conditions (like the closing of a gap), thereby linking thermal expansion with structural response.

Material Properties

Material properties such as the coefficient of thermal expansion, modulus of elasticity, and yield strength are essential for analyzing thermal and mechanical behavior of materials. These properties define how a material responds to temperature changes and applied loads, and are fundamental in assessing safety and performance in engineering applications.

Thermal Expansion

Thermal expansion describes the phenomenon by which a material changes in length (or other dimensions) as its temperature changes. It is typically quantified using the coefficient of thermal expansion (?) and the formula ?L = ? L ?T, which is critical for determining how much a material expands or contracts with temperature variations.

Thermal Stress in Constrained Members

When a material is prevented from expanding or contracting freely, thermal stresses develop as a result of temperature changes. These stresses are calculated using the relation ? = E ? ?T, where E is the modulus of elasticity, and play a key role in structural integrity when thermal expansion is restricted by connections or fixed boundaries.

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