The Am 1004-T 61 magnesium tube is bonded to the A-36 steel...

The $\mathrm{Am} 1004-\mathrm{T} 61$ magnesium tube is bonded to the $A-36$ steel rod. If the allowable shear stresses for the magnesium and steel are $\left(\tau_{\text {allow }}\right)_{\mathrm{mg}}=45 \quad \mathrm{MPa}$ and $\left(\tau_{\text {allow }}\right)_{\mathrm{st}}=75 \mathrm{MPa}$, respectively, determine the maximum allowable torque that can be applied at $A .$ Also, find the corresponding angle of twist of end $A$

Key Concepts

Torsion in Circular Shafts

This concept involves understanding how an applied torque causes shear stress in circular members. The basic torsion equation, ? = T·r/J, relates the applied torque (T) to the shear stress (?) at a given radius (r) using the polar moment of inertia (J). It is fundamental in determining how much torque a component can withstand before reaching its allowable stress limits.

Polar Moment of Inertia

The polar moment of inertia is a geometrical property that measures a cross?section’s resistance to torsion. For circular and hollow sections, the calculation of J is different and crucial, as it influences stress distribution. It essentially quantifies how effectively a cross section can resist twisting forces, and it directly appears in the torsion formula.

Composite Torsion Analysis

When dealing with bonded members of distinct materials under torsion, each component must satisfy its shear stress limits while undergoing the same angle of twist. This analysis involves partitioning the applied torque between the components based on their individual geometries and material properties. The overall design is governed by the material with the lower allowable stress, making it crucial to compare both behaviors in the composite system.

Allowable Shear Stress

Allowable shear stress is the maximum shear stress a material can sustain without failure. It is used as a safety limit in design calculations, ensuring that neither component in a composite system is overstressed when subjected to an applied torque. This concept is critical because it directly governs the design and maximum load that can be applied to the member.

Angle of Twist

The angle of twist in a torsional scenario is a measure of the rotational deformation of a member under an applied torque. It is calculated using the relationship ? = T·L/(J·G), where L is the length of the member, J is the polar moment of inertia, and G is the shear modulus. In composite systems, the angle of twist is the same for all bonded materials, and this compatibility condition is key in the analysis.

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