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Combined Bending and Torsion

Design shafts under simultaneous bending and torsion: combine normal and shear stress, find the critical section, and check strength with von Mises or Tresca.

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Combined bending and torsion is typical of shafts that transmit torque while also carrying transverse forces from gears, pulleys, chain drives, or other machine elements. At a critical section, bending produces normal stress and torsion produces shear stress.

Resultant bending moment

If a shaft bends in two mutually perpendicular planes, determine the components $M_x$ and $M_y$ and construct the corresponding bending-moment diagrams.

If mutually perpendicular bending-moment components $M_x$ and $M_y$ act simultaneously at a section, the magnitude of the resultant bending moment is:

$$M_b=\sqrt{M_x^2+M_y^2}.$$

For a circular section, whose geometric properties are identical about any centroidal axis in the section plane, the maximum bending normal stress can be calculated as:

$$|\sigma_b|=\frac{M_b}{W}.$$

  • $M_x$, $M_y$ — bending-moment components;
  • $M_b$ — resultant bending moment;
  • $W$ — bending section modulus;
  • $\sigma_b$ — bending normal stress.

For noncircular sections, the orientation of the bending moment and the second moments of area about the corresponding principal axes must generally be considered explicitly.

Bending and torsional stresses

For a circular shaft, maximum bending normal stress occurs at the outer surface. Torsional shear stress is also maximum at the outer surface, so outer points of a critical section are typical candidates for multiaxial strength assessment.

At a critical point of a circular shaft subjected to bending and torsion, the bending moment produces normal stress $\sigma_b$ and the torque produces shear stress $\tau_t$:

$$\sigma_b=\frac{M_b}{W},\qquad \tau_t=\frac{T}{W_p}.$$

For this plane stress state, the von Mises equivalent stress is:

$$\sigma_{\mathrm{eq,VM}}=\sqrt{\sigma_b^2+3\tau_t^2}.$$

According to the Tresca criterion:

$$\sigma_{\mathrm{eq,T}}=\sqrt{\sigma_b^2+4\tau_t^2}.$$

  • $M_b$ — resultant bending moment;
  • $T$ — torque;
  • $W$ — bending section modulus;
  • $W_p$ — polar section modulus.

The resulting equivalent stress is compared with an allowable value appropriate to the material and the adopted design method.

The calculator uses the von Mises criterion for the bending-plus-torsion plane stress state.

Failure criteria

For ductile isotropic materials under a multiaxial stress state, the Tresca and von Mises criteria are widely used.

Tresca:

$$\sigma_{\mathrm{eq,T}}=\max\left(|\sigma_1-\sigma_2|,|\sigma_2-\sigma_3|,|\sigma_3-\sigma_1|\right).$$

von Mises:

$$\sigma_{\mathrm{eq,VM}}=\sqrt{\frac{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}{2}}.$$

  • $\sigma_1$, $\sigma_2$, $\sigma_3$ — principal stresses;
  • $\sigma_{\mathrm{eq,T}}$ — Tresca equivalent stress;
  • $\sigma_{\mathrm{eq,VM}}$ — von Mises equivalent stress.

In allowable-stress design, the corresponding equivalent stress is compared with an allowable value consistent with the material properties and the adopted design method.

For ductile isotropic materials, the Tresca or von Mises criterion is commonly used. The criterion and allowable value must be consistent with the material properties and adopted design method.

Solid circular shaft

For diameter $d$:

$$W=\frac{\pi d^3}{32},\qquad W_p=\frac{\pi d^3}{16}=2W.$$

Therefore, for a solid circular shaft the von Mises criterion can also be written in terms of moments:

$$\sigma_{\mathrm{eq,VM}}=\frac{32}{\pi d^3}\sqrt{M_b^2+\frac{3}{4}T^2}.$$

The Tresca equivalent stress becomes:

$$\sigma_{\mathrm{eq,T}}=\frac{32}{\pi d^3}\sqrt{M_b^2+T^2}.$$

Example

Let $M_b=600\ \text{N·m}$, $T=400\ \text{N·m}$, and $d=40\ \text{mm}$ for a solid shaft. Converting moments to N·mm:

$$\sigma_b=\frac{32\cdot600000}{\pi40^3}\approx95.5\ \text{MPa},$$

$$\tau_t=\frac{16\cdot400000}{\pi40^3}\approx31.8\ \text{MPa}.$$

According to von Mises:

$$\sigma_{\mathrm{eq,VM}}\approx\sqrt{95.5^2+3\cdot31.8^2}\approx110.2\ \text{MPa}.$$

Shaft calculation procedure

  1. Determine forces from transmissions and support reactions.
  2. Construct $M_x$, $M_y$, and $T$ diagrams.
  3. Find $M_b$ and identify critical sections.
  4. Calculate $W$ and $W_p$.
  5. Determine $\sigma_b$ and $\tau_t$.
  6. Calculate equivalent stress using the selected failure criterion.
  7. Check strength and, where required, stiffness and fatigue.

Important practical note

Keyways, fillets, shoulders, fits, and other stress concentrators can significantly increase local stresses. Under cyclic shaft loading, a static equivalent-stress check is not sufficient by itself; fatigue must also be assessed.

About this topic

Simultaneous bending and torsion represents the primary loading state for power transmission shafts and gearboxes. This section covers combined moment diagrams, critical section identification, and equivalent bending moment equations evaluated via the maximum shear stress (Tresca) and distortion energy (von Mises) failure criteria.