A simply supported beam of span L is subjected to a temperature gradient which increases linearly from zero at the left-hand support to $t_{0}$ at the right-hand support. If the temperature gradient also varies linearly through the depth, h, of the beam and is zero on its undersurface, calculate the deflection of the beam at its mid-span point. The coefficient of linear expansion of the material of the beam is α.

Ans. $-\alpha t_{0} L^{2} / 48 h$ .

Question

A simply supported beam of span L is subjected to a temperature gradient which increases linearly from zero at the left-hand support to $t_{0}$ at the right-hand support. If the temperature gradient also varies linearly through the depth, h, of the beam and is zero on its undersurface, calculate the deflection of the beam at its mid-span point. The coefficient of linear expansion of the material of the beam is α.

Ans. $-\alpha t_{0} L^{2} / 48 h$ .

Ans. [latex]-\alpha t_{0} L^{2} / 48 h[/latex] .

Accepted Answer

The temperature at any section a distance x from the left hand support is given by [latex]t=(x / L) t_{0}[/latex]. With unit load applied at the mid-span point of the beam Eq. (15.43)

[latex]\Delta_{ Te , B }=-\int_{0}^{L} M_{1} \frac{\alpha t}{h} d x[/latex] (15.43)

becomes

[latex]\delta=-\int_{0}^{L / 2}\left[\frac{\left(\frac{1}{2}\right) x \alpha\left(\frac{x}{L}\right) t_{0}}{h}\right] d x[/latex]

[latex]\delta=\frac{-\alpha L^{2} t_{0}}{48 h}[/latex] .

Question 15.P.18: A simply supported beam of span L is subjected to a temperat...

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