Considering torsional stresses only, compute the maximum...

Considering torsional stresses only, compute the maximum shear stress in a "slinky" toy shown in Fig. P. 14.27 when it is hanging vertically under its own weight. A typical slinky weighs about $1 / 2 \mathrm{lb}$, has about 90 coils, and has a diameter $D$ of about 2.7 in . The rectangular cross section of the wire itself is $b=.01 \mathrm{in}$. and $a=0.025 \mathrm{in}$. For a rectangular cross-section, we will learn in Section 14.7 that $\tau_{\text {max }}=$ $\mathrm{M}_k / \mathrm{kba}^2$. Here, $\kappa \propto .280$.

Key Concepts

Torsional Stress

Torsional stress refers to the internal shear stress generated within a material when it is subjected to a twisting moment. This concept is fundamental in understanding how elements such as rods, shafts, or springs react when their ends are rotated in opposite directions, which results in internal forces trying to oppose the applied twist.

Shear Stress Distribution in Torsion

In torsion, the shear stress varies across the section of a member, reaching a maximum at specific points. For most cross sections, the maximum shear stress occurs at the surface and is crucial in design to ensure the material does not yield or fail. Understanding the distribution is essential for predicting where cracks or deformations might begin in real-world applications.

Torsional Constant for Non-Circular Sections

For non-circular sections, such as a rectangular cross-section, the torsional rigidity is not as straightforward to calculate as in circular sections due to geometric irregularities. The torsional constant (often influenced by a correction factor) reflects the section’s ability to resist twisting. It appears in the formula for maximum shear stress and is essential for accurately assessing the material’s performance under torsional loading.

Self-Weight Induced Loading

Self-weight induced loading refers to the phenomenon where the weight of a structure or component, such as a slinky toy, generates internal forces or moments. In the case of a hanging structure, the weight creates a distributed load that results in a varying internal moment along its length. This concept is crucial for analyzing how elements deform or stress under their own weight in vertical configurations.

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