A cooling tube having the cross section shown is formed from...

A cooling tube having the cross section shown is formed from a sheet of stainless steel of 3 -mm thickness. The radii $c_{1}=150 \mathrm{mm}$ and $c_{2}=$ $100 \mathrm{mm}$ are measured to the center line of the sheet metal. Knowing that a torque of magnitude $T=3 \mathrm{kN} \cdot \mathrm{m}$ is applied to the tube, determine $(a)$ the maximum shearing stress in the tube, (b) the magnitude of the torque carried by the outer circular shell. Neglect the dimension of the small opening where the outer and inner shells are connected.

Key Concepts

Torque Distribution in Composite Sections

In structures composed of different connected segments, such as composite or multi-shell sections, the applied torque is not uniformly distributed but instead is shared among the components based on their geometry and stiffness. Understanding the torque distribution requires evaluating each segment’s contribution to the overall resistance to twisting, which is crucial for ensuring that all parts of the structure are adequately designed to handle the load.

Thin-Walled Tube Theory

Thin-walled tube theory is used to simplify the analysis of hollow structural elements, where the wall thickness is small compared to other dimensions of the cross section. This theory allows for approximations that make it easier to calculate stresses, moments, and deformations by linearizing the stress distribution over the thickness. It is particularly useful in designing and analyzing components such as pipes, tubes, and shells in engineering applications.

Shear Stress under Torsion

Shear stress due to torsion is generated when a twisting moment is applied to a member, producing a distribution of internal shear stresses across the cross section. The maximum shear stress typically occurs at locations farthest from the center of twist. Analyzing shear stress under torsion involves evaluating the material’s response to the applied moment and ensuring that the stresses do not exceed the material’s allowable limits.

Torsional Loading

Torsional loading refers to the application of a twisting moment, or torque, about the longitudinal axis of a structural element. This action induces shear stresses within the material, which must be analyzed to ensure the structural integrity. The ability of a member to resist torsion is largely dependent on its geometry and material properties, making torsion a fundamental topic in mechanics of materials and structural analysis.

Transcript

Please give Ace some feedback