If, in the beam section of Example 15.15, the temperature change in the upper flange is 2T0 but in the web varies linearly from 2T0 at its junction with the upper flange to zero at its junction with the lower flange, determine the values of the stress resultants; the temperature change in the lower

Question

If, in the beam section of Example 15.15, the temperature change in the upper flange is [latex]2 T_{0}[/latex] but in the web varies linearly from [latex]2 T_{0}[/latex] at its junction with the upper flange to zero at its junction with the lower flange, determine the values of the stress resultants; the temperature change in the lower flange remains zero.

Accepted Answer

The temperature change at any point in the web is given by

[latex]T_{ w }=2 T_{0}(a+y) / 2 a=\frac{T_{0}}{a}(a+y)[/latex]

Then, from Eqs. (15.48) and (15.51),

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

[latex]N_{T}=\int_{A} E \alpha \Delta T(x, y) t d s[/latex] (15.51)

[latex]N_{T}=E \alpha 2 T_{0} a t+\int_{-a}^{a} E \alpha \frac{T_{0}}{a}(a+y) t d s[/latex]

that is,

[latex]N_{T}=E \alpha T_{0}\left\{2 a t+\frac{1}{a}\left[a y+\frac{y^{2}}{2}\right]_{-a}^{a}\right\}[/latex]

which gives

[latex]N_{T}=4 E \alpha T_{0} a t[/latex]

Note that, in this case, the answer is identical to that in Example 15.15, which is to be expected, since the average temperature change in the web is [latex]\left(2 T_{0}+0\right) / 2=T_{0}[/latex], which is equal to the constant temperature change in the web in Example 15.15. From Eqs. (15.49) and (15.52),

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

[latex]M_{x T}=\int_{A} E \alpha \Delta T(x, y) t y d s[/latex] (15.52)

[latex]M_{x T}=E \alpha 2 T_{0} a t(a)+\int_{-a}^{a} E \alpha \frac{T_{0}}{a}(a+y) y t d s[/latex]

that is,

[latex]M_{x T}=E \alpha T_{0}\left\{2 a^{2} t+\frac{1}{a}\left[\frac{a y^{2}}{2}+\frac{y^{3}}{3}\right]_{-a}^{a}\right\}[/latex]

from which

[latex]M_{x T}=\frac{8 E \alpha a^{2} t T_{0}}{3}[/latex]

Alternatively, the average temperature change [latex]T_{0}[/latex] in the web may be considered to act at the centroid of the temperature change distribution. Then,

[latex]M_{x T}=E \alpha 2 T_{0} a t(a)+E \alpha T_{0} 2 a t\left(\frac{a}{3}\right)[/latex]

that is,

[latex]M_{x T}=\frac{8 E \alpha a^{2} t T_{0}}{3}[/latex]

as before. The contribution of the temperature change in the web to [latex]M_{y T}[/latex] remains zero, since the section centroid is in the web; the value of [latex]M_{y T}[/latex] is therefore [latex]-E \alpha a^{2} t T_{0}[/latex] as in Example 15.14.

Question 15.16: If, in the beam section of Example 15.15, the temperature ch...

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