Mechanical Engineering Design - Year 2 - DSGN 215 and DSGN 221

Designing for Combined Loading

In many engineering situations components frequently have to withstand more than one type of load. Shafts often have to withstand torque and bending moments. To solve this type of design problem the components are assumed to behave in a linear manner and superposition is used. The stresses due to each type of loading are determined in turn and then combined using appropriate equations or Mohr's circle.

If the design is to be based on the maximum shear stress theory (Tresca) then the maximum shear stress in the component (after combining the contributions due to all the loads) must be found. In some configurations it will not be obvious where the maximum combined shear stress occurs, in these cases it will be necessary to check the combined stresses at a number of locations.

For the example of a uniform solid shaft, diameter d, subject to torsion T and tension P, the maximum shear stress is constant at all points on the surface.

The direct stress due to P is = 4 P/(3.142 d2)

The max. shear stress due to T (at the surface) is = 16 T/(3.142 d3)

The maximum combined shear stress can be found from Mohr's circle to be:

max. shear stress = [(direct stress/2)2 +(shear stress)2]0.5

For a safe design the max. combined shear stress must be less than or = yield shear stress / factor of safety.

If the design is to be assessed against the distortion energy (von Mises) theory, then the von Mises stress needs to be calculated, which means the 3 principal stresses have to be determined.

In many cases of three dimensional solid components, it will often be adequate to carry out a 2D stress analysis. An example of this is in the design of most shafting, where there are no interference fits and consequently the radial stresses are usually insignificant compared to axial stresses caused by bending and / or tension / compression and shear stresses (on axial and circumferential planes) caused by torques transmitted.

It should also be noted when analysing shafts for bending stresses, particularly those in gearboxes, that their length to diameter ratio will often be too low for an accurate stress calculation results to be obtained using simple bending theory.
Additional complications are frequently present in such shafts in the form of section changes (shoulders, fillets) grooves and splines. Stress concentration factors for the individual features may be available, but there may be two or three in close proximity and allowing for their interaction may be very difficult.
In these cases the only way to obtain accurate stress values will be to carry out FEA, taking care to use appropriate loads and restraints.
For heavily stressed shafts in gearboxes, it will be necessary not just to check maximum stress levels, but also deflections to ensure that the required shaft separations are maintained within tolerance. To evaluate this accurately the model will have to include the housing and representations of the bearings as well as the shafts to give an accurate result. Such a complete analysis, including the casing, will also be needed where there is a requirement for low noise (and low vibration) levels.

Example of calculation to design a shaft against failure under combined loading. (200kb file).

This page contains Java Script to rapidly carry out the shaft design calculation shown in the above example.

Further Reading: 'Mechanical Engineering Design', by J E Shigley, chapter 6.

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David J Grieve, modified: 9th June 2009, 21st November 2006, 21st July 2004, original: 24th October 2001.