A steel shaft (Fig. P.14.22) having diameter D1 =6 in. and...

A steel shaft (Fig. P.14.22) having diameter $D_1$ $=6$ in. and an outer sleeve of aluminum having an inner diameter of 12 in . and a thickness $t$ of 2 in . is held by rigid end plates on which a torque $T$ of $5000 \mathrm{ft}-\mathrm{lb}$ is applied. What are the largest torsion strewses in each material? Take $L=10 \mathrm{ft}$, $G_{\mathrm{sa}}=15 \times 10^6$ pri and $G_{\mathrm{AL}}=10 \times 10^6 \mathrm{psi}$. Figure P. 1422

Key Concepts

Torsion in Circular Shafts

Torsion refers to the twisting of an object due to an applied torque, resulting in shear stresses that vary linearly from the center (axis) to the outer surface of the shaft. The analysis involves understanding how the applied moment influences the distribution of stress along the cross-section of a circular shaft.

Polar Moment of Inertia

This is a geometrical property of a cross-section that quantifies its resistance to torsional deformation. For circular and annular (hollow) sections, formulas for the polar moment of inertia are used to determine how the cross-sectional shape contributes to the overall stiffness of the shaft against twisting.

Shear Modulus and Material Properties

The shear modulus, or modulus of rigidity, is a material property that indicates how a material deforms under shear stress. It is crucial in torsional analysis because different materials, such as steel and aluminum, will resist torsion differently due to variations in their shear moduli.

Composite Shaft Analysis

In a composite shaft consisting of more than one material, the distribution of torsional stresses must account for the differences in both geometry and material properties. Techniques such as converting the cross-sectional areas using a modular ratio allow for the combined analysis of the different materials to determine the overall behavior under applied torque.

Torsion Stress Formula

The fundamental relation for torsional analysis in circular members is given by the equation ? = T*r/J, where ? is the shear stress, T is the applied torque, r is the radial distance from the center, and J is the polar moment of inertia. This equation is essential for determining the maximum shear stress within different sections of the shaft.

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