If the attachment between a joist and a column in Ex. 13.25 prevents rotation of the joist calculate the bending moment in the joist at the built-in end.
Ans. -3.98 kNm (hogging).

Question

Accepted Answer

Suppose that the reaction in the column is R. Then, as before, the shortening of the column is [latex]1.061 imes 10^{-5} R mm[/latex]. The bending moment induced in a joist at the column and at the built in end is then

[latex]BM =\frac{6 E I imes 1.06 imes 10^{-5} R}{\left(1.5 imes 10^{3} ight)^{2}}[/latex]

Substituting for E and I gives
BM (due to column shortening) = 45.23 RNmm

Taking the origin for x at the column end of a joist, Eq. (13.10)

[latex]x_{ B }\left(\frac{ d v}{ d x} ight)_{ B }-x_{ A }\left(\frac{ d v}{ d x} ight)_{ A }-\left(v_{ B }-v_{ A } ight)=\int_{ A }^{ B } \frac{M}{E I} x d x[/latex] (13.10)

gives

[latex]1.061 imes 10^{-5} R=\frac{1}{E I} \int_{0}^{1500}\left(R x-\frac{12.05 x^{2}}{2}+45.23 R ight) x d x[/latex],

which simplifies to

[latex]1.061 imes 10^{-5} R=73.689 imes 10^{-5} R-4.78[/latex]

from which R = 6.58 kN
The total bending moment at the built in end is then

Total BM (built inend) [latex]=6.58 imes 1.5-\frac{12.05^{2}}{2}-\frac{45.23 imes 6.58}{10^{3}}[/latex]

= -3.98 kNm (hogging)

Question 13.P.30: If the attachment between a joist and a column in Ex. 13.25 ...

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