Torsion of Circular Shaft Problem

In this worked example, a stepped solid circular shaft made of A284 Grade C steel transmits 7000 Nm torque at 900 rpm from a turbine to a generator. Using the torsion relation τ = T·c / J and standard yield criteria for ductile materials in plane stress, we determine the maximum shear stress, principal stresses, and check whether the shaft design is safe.

The stepped shaft shown in the figure is to rotate at 900 rpm as it transmits 7000 Nm torque from a turbine to a generator and this is the only loading case on the shaft. The material specified in the design is A284 Steel (grade C) and the design factor is 2. Determine/evaluate the following for the shaft:

a) Maximum shear stress on the shaft

b) Principal stresses on the shaft

c) Material yield criteria for the selected material and the resulting stresses.

Solution of the Torsion Problem for Circular Shaft

Step 1 – Input parameters and material properties

Write down the input parameters (including material properties) which are defined in the sample example.

Step 2 – Maximum shear stress and angle of twist

Go to the " Torsion of Solid and Hollow Shafts Calculator " page to calculate the maximum shear stress on the shaft. Larger shear stresses occur on the smaller diameter section of the shaft, so analysis of the smaller diameter section is sufficient for this example.

Step 3 – Stress concentration at shaft shoulder fillet

There is a shoulder fillet in the shaft design and this geometry will raise the local stress. The stress concentration factor and the maximum shear stress at the shoulder fillet are calculated for torsional loading. Go to the " Shoulder fillet in stepped circular shaft " page for calculations.

The maximum shear stress of 357 MPa occurs at the outer radius of the shoulder fillet. This is the answer for clause a) of the sample example.

Step 4 – Principal stresses at the shaft surface

To calculate the principal stresses occurring on the shaft surface, go to the " Principal/Maximum Shear Stress Calculator For Plane Stress " page. Note that the torsional loading of the shaft results in a plane stress state on the surface of the shaft, so this calculator can be used.

The principal stresses are calculated as 357 MPa and -357 MPa. This is the answer for clause b) of the sample example.

Step 5 – Yield criteria for ductile material under plane stress

The selected material (A284 Steel) is ductile since the elongation at break is greater than 5%. For the evaluation of yield criteria for a ductile material with a plane stress state, we can use the " Yield Criteria For Ductile Materials Under Plane Stress (Static Loading) " page.

Summary

According to the results, the design is not safe for the given parameters and conditions. The shaft diameter and/or material must be changed to satisfy the required design criteria. The steps listed above shall be repeated to find dimensions or materials that satisfy the required conditions.

Note: In this example, the loading case is static and the shaft material is ductile. According to Shigley's Mechanical Engineering Design Chapter 3, for ductile materials in static loading, the stress-concentration factor is not usually applied to predict the critical stress, because plastic strain in the region of the stress is localized and has a strengthening effect.
According to Peterson's Stress Concentration Factors Chapter 1, the notch sensitivity q usually lies in the range of 0 to 0.1 for ductile materials. If a statically loaded member is also subjected to shock loading or subjected to high and low temperature, or if the part contains sharp discontinuities, a ductile material may behave like a brittle material. These are special cases and if there is a doubt, K_t (q = 1) shall be applied.
In this example, since there is no information about temperature and shock loading conditions of the shaft, the notch sensitivity factor q is taken as 1 and K_t is applied.

Related calculators used in this example

The problem is fully solved with the following calculators:

Frequently Asked Questions

Is the shaft design safe for the given loading?

No. For the given torque, geometry, and A284 Grade C steel with a design factor of 2, both the Maximum Shear Stress (Tresca) and Distortion Energy (von Mises) criteria are not satisfied, so the design is not safe.

What causes the highest stress in this shaft?

The highest stress occurs at the shoulder fillet of the stepped shaft, where the geometry change introduces a stress concentration under torsional loading.

How can I make the shaft design safe?

You can increase the shaft diameter, select a stronger material with a higher yield strength, or reduce the applied torque. After each change, repeat the calculations to verify the design against the selected yield criteria.