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### Torsion in Thin-Walled Noncircular Shells Calculator

Torsion in Thin-Walled Noncircular Shells Formulas and Calculator

Shear stress due to torsion in a thin-walled, noncircular shell (also known as a closed box) acts around the perimeter of the shell, as shown in Fig. 1. The shear stress, (, is given by Eq. 1. A is the area enclosed by the centerline of the shell.

Eq. 1
τ = T / ( 2 · A · t )

Figure 1 Torsion in Thin-Walled Shells

The shear stress at any point is not proportional to the distance from the centroid of the cross section. Rather, the shear flow, q, around the shell is constant, regardless of whether the wall thickness is constant or variable.

The shear flow is the shear per-unit length of the centerline path.26 At any point where the shell thickness is t,

Eq. 2
q = τ · t = T / ( 2 · A )

When the wall thickness, t, is constant, the angular twist depends on the perimeter, p, of the shell as measured along the centerline of the shell wall.

Eq. 3
γ = T · L · p / ( 4 · A2 · t · G )

Where:

τ = shear stress, lb/in2, N/m2
t = wall thickness, in, m
T = torque, in-lbf, m-N
A = section area, in2, m2
L = length, in, m
p = perimeter as measured from centerline, in, m
G = shear modulus, lb/in2, N/m2
γ = angle of twist, radian
q = shear flow lbf/in, N/m

The concept of shear flow can also be applied to a regular beam in bending, although there is little to be gained by doing so. Removing the dimension b in the general beam shear stress equation, q = VQ/I.

Reference

Civil Engineering Reference Manual, Fifteenth Edition, Michael R. Lindeburg, PE

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