Torsion

124 CHAPTER 5. TORSION
Q 1
i 3
R
i 2
i 3
Circular
cylinder
i 2
R i
R o
i 3
Circular
annulus
Figure 5.1: Circular cylinder with end torques.
written as the projection of this displacement along directions ī2 and ī3, respectively,
u2(x1, r, α) = −rφ1(x1) sin α; u3(x1, r, α) = rφ1(x1) cos α. (5.1)
Since the cross-section does not deform out of its own plane, the axial displacement field must vanish,
i.e. u1(x1, x2, x3) = 0. Finally, the transformation from polar to Cartesian coordinates is
i 1
i 2
Q1
x2 = r cos α; x3 = r sin α. (5.2)
The complete displacement field describing the torsion of circular cylinders expressed in Cartesian coordinates
can now be written by substituting eq. (5.2) in (5.1) to yield
and
u2(x1, x2, x3) = −x3φ1(x1); u3(x1, x2, x3) = x2φ1(x1) (5.3)
u1(x1, x2, x3) = 0. (5.4)
The corresponding strain field is readily obtained using the strain-displacement equations as:
ε1 = ∂u1
= 0; (5.5)
∂x1
ε2 = ∂u2
= 0; ε3 =
∂x2
∂u3
= 0; γ23 =
∂x3
∂u2
+
∂x3
∂u3
= 0; (5.6)
∂x2
γ12 = ∂u1
+
∂x2
∂u2
∂x1
= −x3 κ1(x1); γ13 = ∂u1
+
∂x3
∂u3
∂x1
= x2 κ1(x1), (5.7)
where the sectional twist rate is defined as
κ1(x1) = dφ1
. (5.8)
dx1
The section twist rate, κ1, measures the deformation of the circular cylinder. The twist angle, φ1, simply
measures the rotation of any section with respect to a reference. Note that a constant twist angle implies a
rigid body rotation of the cylinder about its axis, but no deformation.