Twisting moments, or torques, are forces acting through distances (“lever arms") so as to promote rotation. The simple example is that of using a wrench to tighten a nut on a bolt as shown in Fig.: if the bolt, wrench, and force are all perpendicular to one another, the moment is just the force F times the length l of the wrench: T = F.l. This relation will suffice when the geometry of torsional loading is simple as in this case, when the torque is applied “straight".
Often, however, the geometry of the applied moment is a bit more complicated. Consider a not-uncommon case where for instance a spark plugs must be loosened and there just isn't room to put a wrench on it properly. Here a swivelled socket wrench might be needed, which can result in the lever arm not being perpendicular to the spark plug axis, and the applied force (from your hand) not being perpendicular to the lever arm. Vector algebra can make the geometrical calculations easier in such cases. Here the moment vector around a point O is obtained by crossing the vector representation of the lever arm r from O with the force vector F:
T = r ×F
This vector is in a direction given by the right hand rule, and is normal to the plane containing the point O and the force vector. The torque tending to loosen the spark plug is then the component of this moment vector along the plug axis:
T = i . (r × F)
where i is a unit vector along the axis. The result, a torque or twisting moment around an axis, is a scalar quantity.
Torsional Stresses and Displacements
The stresses and deformations induced in a circular shaft by a twisting moment can be found by what is sometimes called the direct method of stress analysis. Here an expression of the geometrical form of displacement in the structure is proposed, after which the kinematic, constitutive, and equilibrium equations are applied sequentially to develop expressions for the strains and stresses. In the case of simple twisting of a circular shaft, the geometric statement is simply that the circular symmetry of the shaft is maintained, which implies in turn that plane cross sections remain plane, without warping. As depicted in Fig., the deformation is like a stack of poker chips that rotate relative to one another while remaining flat. The sequence of direct analysis then takes the following form:
1. Geometrical statement: To quantify the geometry of deformation, consider an increment of length dz from the shaft as seen in Fig. 10, in which the top rotates relative to the bottom by an increment of angle dθ. The relative tangential displacement of the top of a vertical line drawn at a distance r from the center is then:
2. Kinematic or strain-displacement equation: The geometry of deformation exactly our earlier description of shear strain, so we can write:
The subscript indicates a shearing of the z plane (the plane normal to the z axis) in the θ direction. As with the shear stresses, 
, so the order of subscripts is arbitrary.
Constitutive equation: If the material is in its linear elastic regime, the shear stress is given directly from Hooke's Law as:
The sign convention here is that positive twisting moments (moment vector along the +z axis) produce positive shear stresses and strains. However, it is probably easier simply to intuit in which direction the applied moment will tend to slip adjacent horizontal planes.
Here the upper (+z) plane is clearly being twisted to the right relative to the lower (−z) plane, so the upper arrow points to the right. The other three arrows are then determined as well.
1. Equilibrium equation: In order to maintain rotational equilibrium, the sum of the moments contributed by the shear stress acting on each differential area dA on the cross section must balance the applied moment T as shown in Fig.
2. An explicit formula for the stress can be obtained by using this in Eqn.:
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