The composite bar shown in Fig. P.7.16 is initially unstressed. If the temperature of the bar is reduced by an amount T uniformly along its length, find an expression for the tensile stress induced. The coefficients of linear expansion of steel and aluminium are αS and αA per unit temperature

Question

The composite bar shown in Fig. P.7.16 is initially unstressed. If the temperature of the bar is reduced by an amount T uniformly along its length, find an expression for the tensile stress induced. The coefficients of linear expansion of steel and aluminium are [latex] \alpha_{ S } ext { and } \alpha_{ A } [/latex] per unit temperature change, respectively, while the corresponding values of Young’s modulus are [latex] E_{ S } ext { and } E_{ A } [/latex].

Accepted Answer

If the steel portion of the bar were disconnected from the aluminium part the separate parts would take up the positions shown in Fig. S.7.16 where

[latex]\delta_{ S }=\alpha_{ S } T L_{1}, \quad \delta_{ A }=\alpha_{ A } T L_{2}[/latex] (i)

Suppose that the connected parts of the bar take up the position shown so that the junction of the two parts has suffered a displacement δ. Then

[latex]\begin{gathered} ext { extensionofsteel }=\delta_{ S }-\delta \ ext { extensionofaluminium }=\delta_{ A }+\delta\end{gathered}[/latex]

These extensions produce tensile stresses in the steel and aluminium which must be equal since their areas of cross section are the same. Therefore

[latex] \sigma=\frac{E_{S}\left(\delta_{S}-\delta ight)}{L_{1}}=\frac{E_{A}\left(\delta_{A}+\delta ight)}{L_{2}}[/latex] (ii)

Rearranging Eq. (ii) gives

[latex] \delta=\frac{\left(\frac{E_{S} \delta_{S}}{L_{1}} ight)-\left(\frac{E_{A} \delta_{A}}{L_{2}} ight)}{\left(\frac{E_{S}}{L_{1}} ight)+\left(\frac{E_{A}}{L_{2}} ight)} [/latex] (iii)

Now substituting for δ in the first of Eqs (ii) (or the second) and also for [latex] \delta_{ S } ext { and } \delta_{ A } [/latex] from Eq. (i) and rearranging gives

[latex]\sigma=\frac{T\left(\alpha_{S} L_{1}+\alpha_{A} L_{2} ight)}{\left(\frac{L_{1}}{E_{S}}+\frac{L_{2}}{E_{A}} ight)}[/latex]

Question 7.16: The composite bar shown in Fig. P.7.16 is initially unstress...

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