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A 4-m -long steel member has a W 310 ×60 cross section....

A $4-\mathrm{m}$ -long steel member has a $\mathrm{W} 310 \times 60$ cross section. Knowing that $G=77.2 \mathrm{GPa}$ and that the allowable shearing stress is $40 \mathrm{MPa},$ determine $(a)$ the largest torque $\mathrm{T}$ that can be applied, (b) the corresponding angle of twist. Refer to Appendix E for the dimensions of the cross section and neglect the effect of stress concentrations. (Hint: Consider the web and flanges separately and obtain a relation between the torques exerted on the web and a flange, respectively, by expressing that the resulting angles of twist are equal.)

Key Concepts

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross section that represents its ability to resist torsion. It is a central term in formulas relating the applied torque to the resulting shear stress and the angle of twist. In complex or non-circular sections, the correct estimation or computation of the polar moment of inertia is necessary for accurate torsional analysis.

Composite Section Analysis

For members with built-up or non-homogeneous cross sections, such as those comprising separate flanges and webs, composite section analysis is used. This method involves decomposing the cross section into simpler parts, analyzing the torsion in each component, and ensuring that the resulting angles of twist are compatible. This approach is crucial when the geometry does not readily yield a straightforward polar moment of inertia.

Torsion in Structural Members

Torsion is the phenomenon where a structural member undergoes twisting when subjected to an applied torque. The analysis involves understanding how the torque leads to shear stresses distributed across the member’s cross section and produces an angle of twist along its length. Key equations in torsion relate the applied torque to the shear stress and the twist per unit length using parameters like the shear modulus and the polar moment of inertia.

Shear Stress and Allowable Stress

In the context of torsional loading, the induced shear stress is a critical parameter, determined by the relationship between the applied torque, the distance from the center of the cross section, and the polar moment of inertia. The concept of allowable shear stress ensures that the design remains within material limits to prevent failure. This safety check is essential in structural design to guarantee that the applied loads do not exceed the capacity of the member.

Angle of Twist

The angle of twist is the measure of rotational displacement experienced by a member under applied torque. It is determined by the relationship ? = TL/(GJ), where T is the applied torque, L is the length of the member, G is the shear modulus, and J is the polar moment of inertia. Evaluating the angle of twist is important for understanding how much a member deforms under load, which is critical for serviceability in design.

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