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. Take $E_{\mathrm{w}}=29\left(10^{3}\right) \mathrm{ksi}, E_{\mathrm{cm}}=17\left(10^{3}\right) \mathrm{ksi}, \alpha_{\mathrm{st}}=8\left(10^{-6}\right) /^{\circ} \mathrm{F}$ $\alpha_{\mathrm{cu}}=9.60\left(10^{-6}\right) /^{\circ} \mathrm{F}$. Each wire has a cross-sectional area of 0.0123 in $^{2}$.

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. Take $E_{\mathrm{w}}=29\left(10^{3}\right) \mathrm{ksi}, E_{\mathrm{cm}}=17\left(10^{3}\right) \mathrm{ksi}, \alpha_{\mathrm{st}}=8\left(10^{-6}\right) /^{\circ} \mathrm{F}$
$\alpha_{\mathrm{cu}}=9.60\left(10^{-6}\right) /^{\circ} \mathrm{F}$. Each wire has a cross-sectional area of 0.0123 in $^{2}$.

Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon in which a material changes its dimensions in response to a change in temperature. In problems like this, where temperature increases, the lengths of the wires would increase if free to expand. However, when they are restrained by an external load or connected in a structure, the expansion induces additional stresses that must be accounted for in the analysis.

Modulus of Elasticity

The modulus of elasticity, often denoted as E, is a measure of a material's stiffness and relates stress to strain within the elastic regime. It is essential for determining how much a material deforms under a given load and plays a crucial role when calculating the induced stresses due to both mechanical loads and temperature changes.

Hooke’s Law and Elastic Deformation

Hooke’s law establishes a linear relationship between stress and strain for materials behaving elastically. This principle is used to quantify the deformation (or strain) in each wire due to applied forces as well as thermal changes, by relating the force applied, the cross-sectional area, and the material’s elastic properties.

Equilibrium in Structural Analysis

In structural analysis, equilibrium refers to the state where all forces and moments acting on a structure are balanced. In the context of this problem, the forces in the wires must collectively support the external load, ensuring that the sum of the forces in the constrained members equals the applied load while also accounting for the effects of temperature-induced strains.

Compatibility of Deformations in Composite Systems

When different materials are connected or constrained together, their respective deformations must remain compatible, meaning that the overall deformation of the system is consistent across all its parts. This concept is crucial in multi-material structures where differences in thermal expansion coefficients can cause internal stresses, necessitating a careful analysis of both mechanical and thermal effects to ensure structural integrity.

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