The cylinder C D of the assembly is heated from T1=30^∘ C to T2=180^∘ C using electrical resista

step 1 In this question we have given that a 100 kilogram piano rolls down on a 20 degree inclined plan

The cylinder C D of the assembly is heated from T1=30^∘ C to...

The cylinder $C D$ of the assembly is heated from $T_{1}=30^{\circ} \mathrm{C}$ to $T_{2}=180^{\circ} \mathrm{C}$ using electrical resistance. Also, the two end rods $A B$ and $E F$ are heated from $T_{1}=30^{\circ} \mathrm{C}$ to $T_{2}=50^{\circ} \mathrm{C} . \mathrm{At}$ 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 steel, and each has a cross-sectional area of $125 \mathrm{mm}^{2} . C D$ is made of aluminum and has a crosssectional area of $375 \mathrm{mm}^{2} . E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa}$ \[ \alpha_{\mathrm{st}}=12\left(10^{6}\right) /^{\circ} \mathrm{C}, \text { and } \alpha_{\mathrm{al}}=23\left(10^{6}\right) /^{\circ} \mathrm{C} \]

The cylinder $C D$ of the assembly is heated from $T_{1}=30^{\circ} \mathrm{C}$ to $T_{2}=180^{\circ} \mathrm{C}$ using electrical resistance. Also, the two end rods $A B$ and $E F$ are heated from $T_{1}=30^{\circ} \mathrm{C}$ to $T_{2}=50^{\circ} \mathrm{C} . \mathrm{At}$ 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 steel, and each has a cross-sectional area of $125 \mathrm{mm}^{2} . C D$ is made of aluminum and has a crosssectional area of $375 \mathrm{mm}^{2} . E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa}$
\[
\alpha_{\mathrm{st}}=12\left(10^{6}\right) /^{\circ} \mathrm{C}, \text { and } \alpha_{\mathrm{al}}=23\left(10^{6}\right) /^{\circ} \mathrm{C}
\]

Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon where materials change in size or shape as a result of temperature changes. In solids, this typically manifests as a change in length proportional to the temperature change, and it forms the basic principle behind calculating deformations induced by heating or cooling in structures.

Coefficient of Thermal Expansion

The coefficient of thermal expansion quantifies how much a material expands per unit length per degree change in temperature. This parameter is crucial for predicting the extent of dimensional changes in different materials when they experience a temperature variation, and it is fundamental in designing assemblies that include components made of different materials.

Thermal Stress

Thermal stress arises when the natural expansion or contraction of a material due to temperature changes is restrained by external constraints. When a material is prevented from expanding or contracting freely, internal forces develop, which can lead to significant stresses. These stresses are calculated by relating the induced strain, the material’s modulus of elasticity, and the degree of constraint.

Young's Modulus

Young's modulus, or the modulus of elasticity, is a measure of the stiffness of a material. It relates stress to strain in the elastic region and is a key parameter in determining how much a material will deform under a given load, including internally generated loads such as those from restrained thermal expansion.

Structural Compatibility

Structural compatibility refers to the condition where the deformations or displacements of different parts of an assembly must be consistent with one another under loads or temperature changes. In thermally stressed systems, ensuring compatibility means accounting for the different expansions of components and the constraints imposed by gaps or rigid connections, which ultimately influences the force distribution within the structure.

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