Using the stress-strain diagram shown, determine (a) the...

Using the stress-strain diagram shown, determine $(a)$ the torque that causes a maximum shearing stress of 15 ksi in a 0.8 -in.-diameter solid rod, (b) the corresponding angle of twist in a 20 -in. length of the rod.

Key Concepts

Torsion in Circular Shafts

This concept involves the study of how circular rods respond when subjected to twisting forces. When a torque is applied to a circular shaft, the resulting shear stress is not uniform; it increases with the radius and reaches a maximum at the outer surface. The relationship is given by the formula ? = Tc/J, where ? is the shear stress, T is the applied torque, c is the outer radius, and J is the polar moment of inertia.

Polar Moment of Inertia

The polar moment of inertia is a geometrical property of a cross-section that measures its resistance to torsional deformation. For a solid circular rod, it is calculated with J = (?d^4)/32, where d is the diameter. This concept is critical in determining how the applied torque translates into shear stress across the rod’s cross-section.

Angle of Twist and Torsional Deformation

The angle of twist refers to the rotational displacement between the two ends of a rod under an applied torque. The deformation due to torsion is described by the formula ? = TL/(JG), where ? is the angle of twist, T is the applied torque, L is the length of the rod, J is the polar moment of inertia, and G is the modulus of rigidity. This relationship is important for understanding how structures deform under torsional loads.

Stress-Strain Diagram in Torsion Analysis

A stress-strain diagram represents the relationship between the applied stress and the resulting strain in a material. In torsion, such diagrams help identify key properties like the maximum shear stress and the modulus of rigidity. They are used to ensure that the material remains within its elastic range and to determine the corresponding deformation behavior under applied loads.

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