The beam section shown in Fig. 15.37 is subjected to a temperature rise of 2T0 in its upper flange, a temperature rise of T0 in its web, and no temperature change in its lower flange. Determine the normal force on the beam section and the moments about the centroidal x and y axes.The beam section

Question

The beam section shown in Fig. 15.37 is subjected to a temperature rise of [latex]2 T_{0}[/latex] in its upper flange, a temperature rise of [latex]2 T_{0}[/latex] in its web, and no temperature change in its lower flange. Determine the normal force on the beam section and the moments about the centroidal x and y axes. The beam section has a Young’s modulus E and the coefficient of linear expansion of the material of the beam is [latex]\alpha .[/latex]

Accepted Answer

From Eq. (15.48),

[latex]N_{T}=\sum E \alpha \Delta T A_{i}[/latex] (15.48)

[latex]N_{T}=E \alpha\left(2 T_{0} a t+T_{0} 2 a t\right)=4 E \alpha a t T_{0}[/latex]

From Eq. (15.49),

[latex]M_{x T}=\Sigma E \alpha \Delta T \bar{y}_{i} A_{i}[/latex] (15.49)

[latex]M_{x T}=E \alpha\left[2 T_{0} a t(a)+T_{0} 2 a t(0)\right]=2 E \alpha a^{2} t T_{0}[/latex]

and from Eq. (15.50),

[latex]M_{y T}=\Sigma E \alpha \Delta T \bar{x}_{i} A_{i}[/latex] (15.50)

[latex]M_{y T}=E \alpha\left[2 T_{0} a t(-a / 2)+T_{0} 2 a t(0)\right]=-E \alpha a^{2} t T_{0}[/latex]

Note that [latex]M_{y T}[/latex] is negative, which means that the upper flange tends to rotate out of the paper about the web, which agrees with a temperature rise for this part of the section. The stresses corresponding to these stress resultants are calculated in the normal way and are added to those produced by any applied loads.

Question 15.15: The beam section shown in Fig. 15.37 is subjected to a tempe...

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