Power transmission shafts, for example in engines and gearboxes, are subject to torque loads that result in the shaft twisting or twisting around its axis. Similar to structures under tension or compression, two important mechanical properties of shafts under torque loads are shear stress and shear stress.

Stress is the resistance of a material to an applied force, and stress is the deformation resulting from tension. **Shear stress and shear stress** (which are loads caused by torsion) occur when a force is applied parallel or tangential to an area. **Normal stress and normal strain** (caused by tension and compression) occur when a force is applied normally (perpendicular) to an area.

Torque on a shaft causes shear stresses.

The twist, or twist, induced when torque is applied to a shaft causes a distribution of stress over the cross-sectional area of the shaft. (Note that this is different from tensile and compressive loads, which produce a *uniform* stress across the cross section of the object.)

In the elastic range of a material, shear stress is distributed along the radius of an axis, from zero at the center of the axis to a maximum at the outer edge.

*Couple vs. Time:*

*Torque is a force applied over a distance that causes a change in angular momentum. A moment is also a force applied over a distance, but it does not cause a change in angular momentum. In other words, torque causes a body to rotate about an axis, while a momentary load does not cause rotation.*

The shear stress depends on the applied torque, the distance along the axis radius and the polar moment of inertia. (Note that the polar moment of inertia is a function of geometry and does not depend on shaft material.)

*τ = shear stress (N/m ^{2} Pai)*

*T = applied torque (Nm)*

*r = distance along the radius of the tree (m)*

*J = polar moment of inertia (m ^{4} )*

*When measuring the shear stress at the outer edge of the shaft, the letter “c” is sometimes used in place of “r” to indicate that the radius is at its maximum.*

The polar moment of inertia (also known as second polar moment of area) for a solid cylinder is given as:

The amount of shear deformation is determined by the angle of twist, the distance along the radius of the shaft, and the length of the shaft. The shear deformation equation is valid in both the elastic and plastic ranges of the material. It is important to note that shear deformation and shaft length are inversely proportional: the longer the shaft, the smaller the shear deformation.

*γ = shear strain (radians)*

*r = distance along the radius of the tree (m)*

*θ = twist angle (radians)*

*L = shaft length (m)*

Also note that at the center of the shaft (r = 0), there is no shear deformation (γ = 0). In contrast, the shear strain is at its maximum value (γ = γ _{max} ) at the outer surface of the shaft (r = r _{max} ).

Similar to the modulus of elasticity (E) for a body in tension, a shaft in torsion has a property known as shear modulus (also known as shear modulus or modulus of rigidity). Shear modulus (G) is the ratio of shear stress to shear strain. Like the modulus of elasticity, the shear modulus is governed by Hooke's law: the relationship between shear stress and shear strain is proportional up to the proportional limit of the material.

**OR**

*G = shear modulus (Pa)*

Note that the failure process of a shaft in torsion is not as simple as the failure process of a structure in tension. This occurs because bodies under tension experience constant tension throughout their cross-section. Therefore, the failure occurs simultaneously throughout the body.

As described above, for a shaft in torsion, the shear stress varies from zero at the center of the shaft (the shaft) to a maximum at the surface of the shaft. When the surface reaches the elastic limit and begins to fail, the interior will still exhibit elastic behavior for an additional amount of torque. At some point, the applied torque causes the shaft to enter its plastic region, where the deformation increases while the torque is constant. Only when the torque causes fully plastic behavior does the entire cross section fail.