Shafts A and B are made of the same material and have the...

Shafts $A$ and $B$ are made of the same material and have the same crosssectional area, but $A$ has a circular cross section and $B$ has a square cross section. Determine the ratio of the maximum shearing stresses occurring in $A$ and $B$, respectively, when the two shafts are subjected to the same torque $\left(T_{A}=T_{B}\right)$. Assume both deformations to be elastic.

Key Concepts

Torsion

This concept deals with how a shaft or beam twists when subjected to a torque. In torsion problems one uses the relationship between the applied torque, the geometry of the cross?section, and the resulting shear stress or angular twist. In the elastic regime the shear stress is related to the applied torque through geometric properties and material constants, and understanding torsion is fundamental when analyzing shafts subjected to twisting loads.

Polar Moment of Inertia

For solid circular cross?sections the polar moment of inertia (often denoted J) is a geometric property that indicates resistance to twisting. It appears directly in the formula ? = T·c/J, where c is the distance from the center to the outer fiber. Because a circular section has a simple, linear shear stress distribution with radius, its polar moment of inertia can be expressed exactly and is a key element in comparing torsional performance of shafts with other cross?section shapes.

Torsion Constant for Non?Circular Sections

Unlike circular sections, noncircular cross?sections such as a square do not have a simple linear distribution of shear stress, and the concept of a polar moment of inertia does not fully describe their torsional behavior. Instead, a torsion constant (sometimes also denoted J or Jt) is defined – usually computed through more complicated methods or numerical solutions – that accounts for the warping and nonuniform stress distribution. This constant is lower in noncircular sections than in circular ones of equal area, meaning that under the same torque the maximum shear stress is higher.

Shear Stress Distribution in Torsion

When a shaft is twisted, the shear stresses vary over the cross?section. In a circular shaft the stress distribution is linear with the radial coordinate and the maximum occurs at the boundary. In contrast, for noncircular shapes the stress distribution is not linear and its maximum does not simply occur at a geometric ‘radius’ but must be determined from solving the torsion differential equations with appropriate boundary conditions. This difference in stress distributions is key for understanding why, for a given area and torque, different shapes (for example circular versus square) will develop different maximum shear stresses.

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