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## Q. p.24.2

Calculate the shear flows in the web panels and direct load in the flanges and stiffeners of the beam shown in Fig. P.24.2 if the web panels resist shear stresses only.

## Verified Solution

Referring to Fig. P.24.2 and considering the vertical equilibrium of the stiffener CDF
$8000 \sin 30^{\circ}-q_1 \times 200-q_2 \times 200=0$
from which
$q_1+q_2=20$ (i)
Now considering the horizontal equilibrium of the stiffener ED
$8000 \cos 30^{\circ}-q_1 \times 300+q_2 \times 300=0$

whence
$q_1-q_2=23.1$ (ii)
$2 q_1=43.1$
i.e.
$q_1$ = 21.6 N/mm
so that, from Eq. (i)
$q_2$ = −1.6 N/mm
The vertical shear load at any section in the panel ABEGH is 8000 sin 30° = 4000 N. Hence

400$q_3$ = 4000
i.e.
$q_3$ = 10 N/mm
Now consider the equilibrium of the flange ABC in Fig. S.24.2(a). At any section z between C and B
$P_{\mathrm{CB}}=21.6 z$ (iii)

so that $P_{\mathrm{CB}}$ varies linearly from zero at C to 6480 N (tension) at B. Also at any section z between B and A

$P_{\mathrm{AB}}$ = 21.6 × 300 + 10(z − 300)

i.e.
$P_{\mathrm{AB}}$ = 3480 + 10z (iv)
Thus $P_{\mathrm{BA}}$ varies linearly from 6480 N (tension) at B to 9480 N (tension) at A.
Referring to Fig. S.24.2(b) for the bottom flange HGF, the flange load $P_{\mathrm{FG}}$ at any section z is given by
$P_{\mathrm{FG}}=1.6 z$ (v)

Thus $P_{\mathrm{FG}}$ varies linearly from zero at F to 480 N (tension) at G. Also at any section z between G and H
$P_{\mathrm{GH}}$ + 10(z − 300) − 1.6 × 300 = 0
i.e.
$P_{\mathrm{GH}}$ = 3480 − 10z (vi)

Hence $P_{\mathrm{GH}}$ varies linearly from 480 N (tension) at G to −2520 N (compression) at H.
The forces acting on the stiffener DE are shown in Fig. S.24.2(c). At any section a distance z from D

$P_{\mathrm{DE}}+21.6 z+1.6 z-8000 \cos 30^{\circ}=0$

i.e.
$P_{\mathrm{DE}}$ = −23.2z + 6928.2 (vii)

Therefore $P_{\mathrm{DE}}$ varies linearly from 6928 N (tension) at D to zero at E. (The small value of $P_{\mathrm{DE}}$ at E given by Eq. (vii) is due to rounding off errors in the values of the shear flows.)

The forces in the stiffener CDF are shown in Fig. S.24.2(d). At any section in CD a distance h from C the stiffener load, $P_{\mathrm{CD}}$, is given by

$P_{\mathrm{CD}}$ = 21.6h (viii)

so that $P_{\mathrm{CD}}$ varies linearly from zero at C to 4320 N (tension) at D. In DF

$P_{\mathrm{DF}}+8000 \sin 30^{\circ}+1.6(h-200)-21.6 \times 200=0$

from which
$P_{\mathrm{DF}}=640-1.6 h$ (ix)

Hence $P_{\mathrm{DF}}$ varies linearly from 320 N (tension) at D to zero at F.
The stiffener BEG is shown in Fig. S.24.2(e). In BE at any section a distance h from B
$P_{\mathrm{BE}}+21.6 h-10 h=0$
i.e.
$P_{\mathrm{BE}}$ = −11.6h (x)

$P_{\mathrm{BE}}$ therefore varies linearly from zero at B to −2320 N (compression) at E. In EG
$P_{\mathrm{EG}}$ − 1.6(h − 200) + 21.6 × 200 − 10h = 0
i.e.
$P_{\mathrm{EG}}$ = 11.6h − 4640 (xi)
Thus $P_{\mathrm{EG}}$ varies linearly from −2320 N (compression) at E to zero at G.