## 315 Members Subjected To Torsion

Forces or moments that tend to twist a member are called torisonal loads. In shafts, the stresses and corresponding strains induced by these loads depend on both the shape and size of the cross section.

Suppose that a circular shaft is fixed at one end and a twisting couple, or torque, is applied at the other end (Fig. 3.21a). When the angle of twist is small, the circular cross section remains circular during twist. Also, the distance between any two sections remains the same, indicating that there is no longitudinal stress along the length of the member.

Figure 3.21b shows a cylindrical section with length dx isolated from the shaft. The lower cross section has rotated with respect to its top section through an angle d6, where 6 is the

total rotation of the shaft with respect to the fixed end. With no stress normal to the cross section, the section is in a state of pure shear (Art. 3.9). The shear stresses act normal to the radii of the section. The magnitude of the shear strain y at a given radius r is given by y = AA = rd6 = 6 (3 52)

A1A2 dx L

where L = total length of the shaft dd/dx = 6/L = angle of twist per unit length of shaft

Incorporation of Hooke's law (v = Gy) into Eq. (3.52) gives the shear stress at a given radius r:

where G is the shear modulus of elasticity. This equation indicates that the shear stress in a circular shaft varies directly with distance r from the axis of the shaft (Fig. 3.21c). The maximum shear stress occurs at the surface of the shaft.

From conditions of equilibrium, the twisting moment T and the shear stress v are related by v = j (3.54)

where J = fr2 dA = tt4/2 = polar moment of inertia dA = differential area of the circular section

By Eqs. (3.53) and (3.54), the applied torque T is related to the relative rotation of one end of the member to the other end by

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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