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Chapter 24

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.

Screenshot 2022-10-11 132926

Step-by-Step

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)
Adding Eqs (i) and (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.

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