How to Calculate Combined Torsion and Bending in Shafts

In mining and heavy industrial machinery, shafts are rarely subjected to just one type of load. Instead, they typically experience both torsional (twisting) and bending (flexural) stresses simultaneously — especially in components like drive shafts, winches, pulleys, or conveyor rollers.

Properly calculating combined torsion and bending is critical to prevent fatigue failure, excessive deflection, or even catastrophic shaft breakage.

1. Why Combined Loading Happens in Mining Applications

Let’s consider a conveyor system:

• The motor torque applies torsion (T)

• The weight of the belt and material applies a bending moment (M)

• Additional misalignment or dynamic loading increases complexity

Neglecting either stress component results in underestimating the true danger to the shaft.

2. Step-by-Step Calculation of Combined Stress

The shaft is analyzed for equivalent stress, using either the Von Mises or Maximum Shear Stress Theory.

👉 Von Mises Equivalent Stress:

σeq=σb2+3τ2σ_{eq} = \sqrt{σ_b^2 + 3τ^2}σeq​=σb2​+3τ2​

Where:

• σbσ_bσb​ = Bending stress = M⋅cI\frac{M \cdot c}{I}IM⋅c​

• $\tau$ = Torsional shear stress = T⋅cJ\frac{T \cdot c}{J}JT⋅c​

This gives a single value to compare against the material’s yield strength.

3. Moment and Torsion Formulas Refresher

• Bending stress (outer fiber):

  σb=32Mπd3σ_b = \frac{32M}{πd^3}σb​=πd332M​

• Torsional shear stress:

  τ=16Tπd3τ = \frac{16T}{πd^3}τ=πd316T​

Combine both into the Von Mises equation to evaluate total stress on the shaft’s surface.

4. Real-World Example

A mining shaft:

• $M=1500\text{Nm}$

• $T=900\text{Nm}$

• Shaft diameter $d=60\text{mm}$

1. Calculate bending stress:
   σb=32×1500π×603≈7.08 MPaσ_b = \frac{32 × 1500}{π × 60^3} ≈ 7.08 \, \text{MPa}σb​=π×60332×1500​≈7.08MPa

2. Calculate torsional stress:
   τ=16×900π×603≈2.83 MPaτ = \frac{16 × 900}{π × 60^3} ≈ 2.83 \, \text{MPa}τ=π×60316×900​≈2.83MPa

3. Use Von Mises:
   σeq=7.082+3×2.832≈9.17 MPaσ_{eq} = \sqrt{7.08^2 + 3 × 2.83^2} ≈ 9.17 \, \text{MPa}σeq​=7.082+3×2.832​≈9.17MPa

Compare with material yield strength to verify design safety.

5. Fatigue Check: Repeated Combined Loads

Shafts in mining operate continuously and are cyclically loaded. Use a modified Goodman diagram or S-N curve with equivalent stress to evaluate fatigue life.

✅ Always apply dynamic safety factors (2.0–3.0)
✅ Choose materials with high fatigue strength like 42CrMo4 or alloy steels

6. Best Practices for Shaft Design Under Combined Load

✔️ Use fillets at shoulder transitions
✔️ Avoid sharp keyways or grooves near high-stress zones
✔️ Perform FEM analysis for non-circular or stepped shafts
✔️ Increase diameter slightly to reduce stress significantly (d³ relation)

Combined torsion and bending loads are the reality in industrial shaft design — especially in the harsh, repetitive environments of mining operations. By correctly calculating and checking these stresses using robust methods, engineers can ensure shaft performance, safety, and durability under all working conditions.