Stresses in Thin Cylindrical Shells – Torsion of Shafts and Springs

Definition

Thin cylindrical shells are pressure vessels where the wall thickness is small compared to the radius (typically $t < d/20$). Torsion refers to the twisting of a structural member (like a shaft or spring) caused by an applied torque, leading to shear stresses throughout the material.

Main Content

1. Stresses in Thin Cylindrical Shells

Hoop (Circumferential) Stress: The stress acting along the circumference of the shell, trying to burst it open longitudinally.
Longitudinal Stress: The stress acting along the axis of the cylinder, trying to pull the ends of the cylinder apart.

      [ Hoop Stress (σh) ]
          _______
        /         \
       |     O     |  <-- Internal Pressure (p)
        \_______/
      [ Hoop Stress (σh) ]

2. Torsion of Circular Shafts

Shear Stress Distribution: When a shaft is twisted, shear stress varies linearly from zero at the center (neutral axis) to a maximum at the outer surface.
Torsional Rigidity: This is the product of the modulus of rigidity ($G$) and the polar moment of inertia ($J$), determining how much a shaft twists under torque.

3. Torsion of Helical Springs

Coil Stress: Springs are essentially rods twisted into a helix. The load applied axially causes torsional shear stress in the wire cross-section.
Deflection: The spring constant ($k$) depends on the material, wire diameter, coil diameter, and the number of active turns.

Working / Process

1. Calculating Stress in Thin Cylinders

Identify the internal pressure ($p$), diameter ($d$), and thickness ($t$).
Calculate Hoop Stress ($\sigma_h = pd/2t$) and Longitudinal Stress ($\sigma_l = pd/4t$). Note that hoop stress is twice the magnitude of longitudinal stress.

2. Analyzing Shaft Torsion

Use the Torsion Formula: $\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$.
Determine the maximum shear stress ($\tau_{max} = \frac{Tr}{J}$) where $T$ is the applied torque, $r$ is the outer radius, and $J$ is the polar moment of inertia.

3. Determining Spring Mechanics

Relate axial force ($F$) to torque ($T = F \times R$, where $R$ is the mean coil radius).
Calculate the wire shear stress: $\tau = \frac{16TR}{\pi d^3}$, considering the Wahl’s correction factor for stress concentration in tight coils.

Advantages / Applications

Pressure Vessels: Storage of gases and liquids in industrial tanks and boilers requires thin-shell analysis to minimize material weight while ensuring safety.
Power Transmission: Circular shafts are universally used in automotive engines and industrial machinery to transmit rotary power via torsion.
Energy Storage/Absorption: Helical springs are critical for suspension systems, valves, and measuring instruments, utilizing the principles of torsional deflection.

Summary

This topic covers the analysis of thin-walled pressure vessels subjected to fluid pressure and the study of structural members subjected to twisting moments. It establishes the mathematical relationship between physical loads and internal stresses in cylindrical geometries, as well as in circular shafts and helical springs.

Important terms to remember: Hoop Stress, Longitudinal Stress, Polar Moment of Inertia, Torsional Rigidity, Shear Stress.