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7 Torsion of Prismatic Bars
Fig. 7.10 Cross-section of a
thin-walled multiple-cell
beam and moment axis
−τ 1 t 1 dl + τ 2 t 2 dl + τ 3 t 3 dl = 0
or
τ 1 t 1 = τ 2 t 2 + τ 3 t 3
i.e.
q 1 = q 2 + q 3
(7.25)
This is again equivalent to a fluid flow dividing itself into two streams. Now,
choose moment axis, such as point O as shown in Fig. 7.10.
The shear flow in the web is considered to be made of q 1 and –q 2 , since q 3 = q 1
− q 2 .
Moment about point O due to q 1 flowing in cell 1 (including web) is
M t1 = 2q 1 A 1
(7.26)
Similarly, the moment about point O due to q 2 flowing in cell 2 (including web)
is
M t2 = 2q 2 (A 2 + A 1 ) − 2q 2 A 1
(7.27)
The second term with the negative sign on the right-hand side is the moment due
to shear flow q 2 in the middle web.
Therefore, the total torque is
M t = M t 1 + M t 2
M t = 2q 1 A 1 + 2q 2 A 2
(7.28)
7 Torsion of Prismatic Bars
Fig. 7.10 Cross-section of a
thin-walled multiple-cell
beam and moment axis
−τ 1 t 1 dl + τ 2 t 2 dl + τ 3 t 3 dl = 0
or
τ 1 t 1 = τ 2 t 2 + τ 3 t 3
i.e.
q 1 = q 2 + q 3
(7.25)
This is again equivalent to a fluid flow dividing itself into two streams. Now,
choose moment axis, such as point O as shown in Fig. 7.10.
The shear flow in the web is considered to be made of q 1 and –q 2 , since q 3 = q 1
− q 2 .
Moment about point O due to q 1 flowing in cell 1 (including web) is
M t1 = 2q 1 A 1
(7.26)
Similarly, the moment about point O due to q 2 flowing in cell 2 (including web)
is
M t2 = 2q 2 (A 2 + A 1 ) − 2q 2 A 1
(7.27)
The second term with the negative sign on the right-hand side is the moment due
to shear flow q 2 in the middle web.
Therefore, the total torque is
M t = M t 1 + M t 2
M t = 2q 1 A 1 + 2q 2 A 2
(7.28)