Stress Estimation in Shafts: Bending, Torsion, and Combined Loading

This cheat sheet covers the fundamental principles of stress estimation in shafts subjected to bending, torsion, and combined loading, including theoretical force analysis, stress superposition, and failure criteria.

Core Principles

Cantilever shafts are fixed at one end and free at the other, with internal forces being statically determinate.
Simply supported shafts rest on two bearings, offering a common configuration for rotating shafts.
Combined bending and torsion in shafts, such as those with gears, require superposition of stresses.
Stress concentration at critical points on the shaft surface is crucial for accurate analysis.
Equivalent stress criteria (Tresca and von Mises) are used to predict yielding under combined stress states.
Factor of safety is calculated to ensure the design can withstand applied loads with a margin of safety.

Action Steps

Identify the type of shaft support (cantilever, simply supported, etc.).
Determine the applied loads (forces, moments, torques) on the shaft.
Calculate the internal bending moments and torques along the shaft.
Identify the critical points on the shaft where stresses are maximum.
Calculate the bending normal stress (σb) and torsional shear stress (τT) at these critical points.
Use equivalent stress criteria (Tresca or von Mises) to combine stresses.
Calculate the factor of safety (n) using the yield strength (Sy) and equivalent stress (σeq).

Formulas

$M_{max} = FL$ (Maximum bending moment in a cantilever shaft with end load)
$M_{max} = \frac{FL}{4}$ (Maximum bending moment in a simply supported shaft with central load)
$\sigma_b = \frac{My}{I}, \quad \sigma_{b,max} = \frac{M}{W}$ (Bending normal stress)
$\tau_T = \frac{T\rho}{J}, \quad \tau_{T,max} = \frac{T}{W_T}$ (Torsional shear stress)
$W = \frac{\pi d^3}{32}$ (Section modulus in bending)
$W_T = \frac{\pi d^3}{16} = 2W$ (Section modulus in torsion)
$\sigma_{1,2} = \frac{\sigma_b}{2} \pm \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau_T^2}$ (Principal stresses)
$\tau_{max} = \frac{\sigma_1 - \sigma_2}{2} = \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau_T^2}$ (Maximum shear stress - Tresca criterion)
$\sigma_{eq, Tresca} = \sqrt{\sigma_b^2 + 4\tau_T^2}$ (Equivalent stress - Tresca criterion)
$\sigma_{eq, von Mises} = \sqrt{\sigma_b^2 + 3\tau_T^2}$ (Equivalent stress - von Mises criterion)
$n = \frac{S_y}{\sigma_{eq}}$ (Factor of safety)

Key Terms

Cantilever Shaft: A shaft fixed at one end and free at the other.
Simply Supported Shaft: A shaft supported at two points, typically bearings.
Section Modulus (W, WT): Geometric property used in stress calculations for bending (W) and torsion (WT).
Equivalent Stress: A single stress value that represents the combined effect of multiple stresses, used in failure criteria.
Tresca Criterion: A failure criterion based on the maximum shear stress.
Von Mises Criterion: A failure criterion based on distortion energy theory.
Factor of Safety (n): The ratio of the material's strength to the actual stress, indicating the margin of safety.

Real World Examples

A simply supported shaft with a gear at mid-span transmitting torque and radial force.: Calculating bending and torsional stresses, then using von Mises criterion to find equivalent stress and factor of safety.