Determine the stresses induced in steel bolt and aluminium collar assembly of Example 1.14 if the temperature also increases by 50 °C. Take the coefficient of thermal expansion for steel and aluminium to be αs = 18 x 10^-6 mm/mm/°C and αs = 23 x 10^-6 mm/mm/°C, respectively.

Question

Determine the stresses induced in steel bolt and aluminium collar assembly of Example 1.14 if the temperature also increases by 50 °C. Take the coefficient of thermal expansion for steel and aluminium to be [latex]α_s[/latex] = 18 x [latex]10^{-6}[/latex] mm/mm/°C and [latex]α_s[/latex] = 23 x [latex] 10^{-6}[/latex] mm/mm/°C, respectively.

Accepted Answer

First consider the temperature stresses only.

Equilibrium condition: Since no external force is acting at the final position of the assembly, the equilibrium of internal forces in free body diagram [Fig. 1.25(d)] gives,

[latex]F_s = F_a[/latex] (i)

Compatibility condition: Fig. 1.25(b) shows the free expansion of the steel bolt and aluminium collar. The expansion of aluminium collar will be more than steel bolt since its coefficient of thermal expansion is higher. Thus at equilibrium aluminium collar will shorten by an amount [latex]\Delta L_a[/latex] and steel bolt will extend by an amount [latex]\Delta L_s[/latex], as shown in Fig. 1.25(c) such that,

[latex]\Delta L_s+\Delta L_a=\left(\alpha_a-\alpha_s ight) imes L imes \Delta t [/latex] (ii)

Substituting stress-strain relation in Eq. (ii),
[latex]\frac{F_s imes L_s}{A_s imes E_s}+\frac{F_a imes L_a}{A_a imes E_a}=\left(\alpha_a-\alpha_s ight) imes L imes \Delta t [/latex] (iii)

or, [latex]\frac{F_s}{600 imes 210000}+\frac{F_a}{900 imes 70000} =(23-18) imes 10^{-6} imes 50 [/latex]

or , [latex] F_s=F_a[/latex] = 10500 kN

Stress in steel, [latex]\sigma_s=10500 / 600 =17.5 MPa [/latex] (Tension)
Stress in aluminium, [latex]\sigma_a =10500 / 900 [/latex] = 11.67 MPa (Compression)

Combined stresses: If the external loads are also acting with rise in temperature, the mechanical and thermal stress will superimpose as follows.

Stress in steel, [latex]\sigma _s=\sigma _s {_{,mech}}+\sigma _s {_{,therm}} [/latex] = 175 + 17.5 = 192.5 MPa (Tension)

Stress in aluminium,

[latex]\sigma _a=\sigma _a {_{,mech}}+\sigma _a {_{,therm}}[/latex] = 116.67 + 11.67 = 128.34 MPa (Compression)

Question 1.15: Determine the stresses induced in steel bolt and aluminium c...

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