Equal torques are applied to thin-walled tubes of the same...

Equal torques are applied to thin-walled tubes of the same thickness $t$ and same radius $c .$ One of the tubes has been slit lengthwise as shown. Determine the ratio $\tau_{b} / \tau_{a}$ of the maximum shearing stresses in the tubes.

Key Concepts

Influence of Longitudinal Cuts on Stress Distribution

Introducing a longitudinal cut in a tube transforms a closed cross-section into an open one. This change disrupts the typical load-carrying characteristics of a closed section, leading to a distinct distribution of shear stresses and a reduction in torsional rigidity, which is reflected in the increased maximum shear stress for the same applied torque.

Maximum Shear Stress Calculation under Torsion

Calculating the maximum shear stress in a twisted thin-walled member involves relating the applied torque to the moment resulting from the shear flow, which depends on the wall thickness and the geometry (such as the enclosed area and the perimeter). This calculation is essential for ensuring that the structure can safely withstand the applied loads.

Shear Flow

Shear flow in thin-walled sections describes how distributed shear forces, generated by an applied torque, flow along the wall’s contours. It is a crucial concept for determining the internal force distribution and is used to derive the expression for shear stress based on the geometry of the cross-section and the wall thickness.

Torsion of Thin-Walled Structures

This concept involves analyzing the behavior of thin-walled members when subjected to twisting loads. In such analyses, key assumptions include that the wall thickness is small compared to other dimensions, allowing for simplifications in stress distribution and enabling the use of specialized formulas to relate applied torque to shear stresses.

Effect of Cross-Section Geometry on Torsional Rigidity

The geometry of a structure’s cross-section, whether it is closed or open, significantly impacts its ability to resist twisting (torsional rigidity). Closed sections tend to have higher torsional rigidity, meaning they sustain lower shear stresses under a given torque, whereas open sections (e.g., when a tube is slit) exhibit lower rigidity and consequently higher shear stresses.

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