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 $T_{1}=60^{\circ} \mathrm{F}$. Each rod has a diameter of 1.25 in. Determine the average normal stress in each rod if $T_{2}=400^{\circ} \mathrm{F},$ and also calculate the new length of the aluminum segment. Take $\alpha_{\mathrm{al}}=13\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}$ \begin{array}{l} \left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}, \alpha_{\mathrm{cu}}=9.4\left(10^{-6}\right) /^{\circ} \mathrm{F},\left(\sigma_{Y}\right)_{\mathrm{cu}}=50 \mathrm{ksi}, \text { and } \\ E_{\mathrm{cu}}=15\left(10^{3}\right) \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 $T_{1}=60^{\circ} \mathrm{F}$. Each rod has a diameter of 1.25 in. Determine the average normal stress in each rod if $T_{2}=400^{\circ} \mathrm{F},$ and also calculate the new length of the aluminum segment. Take $\alpha_{\mathrm{al}}=13\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}$
\begin{array}{l}
\left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}, \alpha_{\mathrm{cu}}=9.4\left(10^{-6}\right) /^{\circ} \mathrm{F},\left(\sigma_{Y}\right)_{\mathrm{cu}}=50 \mathrm{ksi}, \text { and } \\
E_{\mathrm{cu}}=15\left(10^{3}\right) \mathrm{ksi}
\end{array}

Key Concepts

Gap Clearance and Rigid Constraints

In problems where components have an initial gap or clearance, such as the gap between rod segments, the gap can initially accommodate thermal expansion without inducing stress. However, once the gap is closed due to continued expansion, further temperature-induced length changes generate significant stresses as the rigid constraints prevent additional deformation. Understanding this mechanism is crucial in analyzing the transition from free thermal expansion to a state where thermal stresses develop.

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a material property that quantifies the amount by which a material expands per unit length for a one-degree change in temperature. It is essential in predicting dimensional changes due to temperature variations and, when combined with the elastic modulus in restrained conditions, allows for calculation of thermal stresses.

Thermal Stress

Thermal stress develops when a material experiences temperature-induced expansion or contraction but is prevented from freely deforming due to constraints. When such a restraint is applied, the material attempts to expand or contract but the fixed boundary conditions generate internal stresses. These stresses can be estimated using the material's elastic modulus and coefficient of thermal expansion along with the imposed temperature change, playing a critical role in the design and analysis of constrained structures.

Elastic Modulus (Young's Modulus)

The elastic modulus is a measure of a material's stiffness, representing the ratio of stress to strain in the elastic (linear) deformation region of the material. It is used to determine how much a material will deform under a given load or, in the case of thermal expansion, how much stress will develop when the material is restrained.

Thermal Expansion

Thermal expansion refers to the tendency of a material's dimensions to change in response to a change in temperature. In engineering problems, the free expansion of a component is calculated using the product of its coefficient of thermal expansion, the original length, and the change in temperature. This concept is fundamental when analyzing how temperature changes affect the dimensions of materials, especially when they are either free to expand or are restrained by external constraints.

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