The wires A B and A C are made of steel, and wire A D is...

The wires $A B$ and $A C$ are made of steel, and wire $A D$ is made of copper. Before the 150 -lb force is applied, $A B$ and $A C$ are each 60 in. long and $A D$ is 40 in. long. If the temperature is increased by $80^{\circ} \mathrm{F}$, determine the force in each wire needed to support the load. Each wire has a crosssectional area of 0.0123 in $^{2} .$ Take $E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}$ \[ E_{\mathrm{cu}}=17\left(10^{3}\right) \mathrm{ksi}, \alpha_{\mathrm{st}}=8\left(10^{-6}\right) /^{\circ} \mathrm{F}, \alpha_{\mathrm{cu}}=9.60\left(10^{-6}\right) /^{\circ} \mathrm{F} \]

The wires $A B$ and $A C$ are made of steel, and wire $A D$ is made of copper. Before the 150 -lb force is applied, $A B$ and $A C$ are each 60 in. long and $A D$ is 40 in. long. If the temperature is increased by $80^{\circ} \mathrm{F}$, determine the force in each wire needed to support the load. Each wire has a crosssectional area of 0.0123 in $^{2} .$ Take $E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}$
\[
E_{\mathrm{cu}}=17\left(10^{3}\right) \mathrm{ksi}, \alpha_{\mathrm{st}}=8\left(10^{-6}\right) /^{\circ} \mathrm{F}, \alpha_{\mathrm{cu}}=9.60\left(10^{-6}\right) /^{\circ} \mathrm{F}
\]

Key Concepts

Thermal Expansion

Thermal expansion is the change in a material's length or volume in response to a change in temperature. In the context of this problem, the increase in temperature causes the wires to want to expand by an amount proportional to their initial length, the coefficient of thermal expansion, and the temperature change. This free expansion, however, is constrained by the load and the connection of the wires, leading to additional stresses.

Thermal Stress

Thermal stress arises when a material's desired thermal expansion (or contraction) is restricted by external constraints. In this scenario, the wires are subject to both a temperature increase and a load, so the free expansion is partially or entirely restrained, creating internal forces (stresses) that counteract the thermal expansion. These stresses are determined by combining the thermal strain with the elastic strain imposed by the load.

Young's Modulus and Stress-Strain Relationship

Young's modulus is a measure of a material's stiffness and is used to relate stress (force per unit area) to strain (the fractional change in length) in the elastic regime through Hooke's law. In this problem, each wire’s response to the applied force and temperature change is governed by its material's Young's modulus, which enables the calculation of the elastic force arising from the deformation.

Compatibility of Deformations

Compatibility of deformations ensures that all parts of a structure or system, which are connected, must undergo a consistent overall displacement. For the wires supporting a common load, the combination of thermal expansion and elastic extension must be compatible with the fixed geometric constraints. This concept is critical in setting up equations that reflect the fact that the wires, though with different properties, share a common displacement at the junction where the load is applied.

Equilibrium Load Sharing

Equilibrium load sharing involves distributing an applied external load among various connected structural members such that the net force is balanced. In the problem at hand, multiple wires with different material properties and lengths must together support the overall load, and their individual forces are determined by solving the equilibrium equations that account for both the mechanical forces and the effects of temperature-induced expansion.

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