The beam section shown in Fig. P.20.2 has been idealized into an arrangement of direct stress carrying booms and shear stress only carrying panels. If the beam section is subjected to a vertical shear load of 1495 N through its shear centre, booms 1, 4, 5 and 8 each have an area of 200 mm²

Question

The beam section shown in Fig. P.20.2 has been idealized into an arrangement of direct stress carrying booms and shear stress only carrying panels. If the beam section is subjected to a vertical shear load of 1495 N through its shear centre, booms 1, 4, 5 and 8 each have an area of 200 mm² and booms 2, 3, 6 and 7 each have an area of 250 mm2 determine the shear flow distribution and the position of the shear centre.

Accepted Answer

From Eq. (20.6)

[latex]\begin{aligned}q_s=&-\left(\frac{S_x I_{x x}-S_y I_{x y}}{I_{x x} I_{y y}-I_{x y}^2} ight)\left(\int_0^s t_{\mathrm{D}} x \mathrm{~d} s+\sum_{r=1}^n B_r x_r ight) \&-\left(\frac{S_y I_{y y}-S_x I_{x y}}{I_{x x} I_{y y}-I_{x y}^2} ight)\left(\int_0^s t_{\mathrm{D}} y \mathrm{~d} s+\sum_{r=1}^n B_r y_r ight)\end{aligned}[/latex] (20.6)

[latex]q_s=-\frac{S_y}{I_{x x}} \sum_{r=1}^n B_r y_r[/latex]

where
[latex]I_{x x}=4 imes 2.0 imes 80^2+2 imes 200 imes 50^2+2 imes 200 imes 40^2[/latex]
i.e.
[latex]I_{x x}=8.04 imes 10^6 \mathrm{~mm}^4[/latex]
Then

[latex]q_s=-1.86 imes 10^{-4} \sum_{r=1}^n B_r y_r[/latex]

from which

[latex]\begin{aligned}&q_{12}=-1.86 imes 10^{-4} imes 200 imes(-50)=1.86 \mathrm{~N} / \mathrm{mm} \&q_{43}=-1.86 imes 10^{-4} imes 200 imes(-40)=1.49 \mathrm{~N} / \mathrm{mm} \&q_{32}=1.49-1.86 imes 10^{-4} imes 250 imes(-80)=5.21 \mathrm{~N} / \mathrm{mm} \&q_{27}=1.86+5.21-1.86 imes 10^{-4} imes 250(-80)=10.79 \mathrm{~N} / \mathrm{mm}\end{aligned}[/latex]

The remaining shear flow distribution follows from symmetry; the complete distribution is shown in Fig. S.20.2(b).
Taking moments about the mid-point of web 27

[latex]S_y x_{\mathrm{S}}=2\left(q_{12} imes 150 imes 80-q_{32} imes 200 imes 80-q_{43} imes 150 imes 80-q_{43} imes 40 imes 200 ight)[/latex]

which gives

[latex]x_{\mathrm{S}}=-122 \mathrm{~mm}[/latex] (i.e. to the left of web 27)

Question p.20.2: The beam section shown in Fig. P.20.2 has been idealized int...

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