Shaft design is a common task in machines because shafts are everywhere. While common, the amount of thought that goes into it can be surprisingly large. In this article we'll look at the various considerations to create a shaft design.

## Things to Check in a Shaft Design

**Static stress** The shaft must withstand stress that results from applied loads and torque. This could also include the weight of attachments such as gears and flywheels.

**Deflection** Excessive deflection can cause problems such as gear mesh.

**Twist** Excessive twist can cause timing problems.

**Slope** The slope at bearings must be kept within the bearing's specified limits to avoid binding and premature wear.

**Fatigue** Since alternating stresses can be present, fatigue may be an important consideration.

**First critical speed** The first critical speed should be calculated to avoid operating near it.

We'll discuss each of these items in more detail as we go along.

## Static Stress

Shafts can be subjected to several loading conditions such as radial and axial loads, as well as torque. We'll consider a general case of a shaft with all three loads.

A shaft is shown below with a radial load and torque centered between the bearings. There is an opposing torque and axial load at the right end of the shaft. Point A is the worst-case combined loading case. Bending moment is greatest at A, is under torque, and has an axial load in tension.

Point A is at the surface of the shaft where bending stress and shear stress from torque is greatest. We'll assume that this is a plane stress condition, and a stress element at A is shown below.

Since bending stress is tensile and the stress from the axial load is tensile, these stresses are added.

Now let's consider a similar shaft, except that the axial load is compressive. The worst-case point on the shaft is now at B.

The stress element at point B is shown below. B is worst-case since bending stress is in compression as well as axial stress.

As we see from the examples, the axial load determines which location to analyze, MAYBE. This assumes that compressive and tensile yield strengths are the same. This may not be the case, and this could govern which stress state should be used.

#### Combined Static Stresses

In the example shafts, we have combined stresses. Therefore we need to use an appropriate failure theory to determine effective stress and hence a safety factor. We'll consider three failure theories: maximum distortion energy, maximum shear stress, and maximum normal stress.

Maximum normal stress should NOT be used for ductile materials as it is not safe.

*Always do your due diligence when selecting a failure theory based on the situation.*

#### Principal Stress Convention

Since the worst-case points are on the surface of the shaft, the stress normal to the surface is zero. In the formulas above, we calculate σ1 and σ2 which are the in-plane principal stresses. By the definition of plane stress, σz = 0 and σ3 = 0.

To determine maximum shear stress, we need to reorder the principal stresses according to the following convention: σ1 > σ2 > σ3.

## Deflection, Bearing Slope and Twist

A shaft with too much deflection can cause operational problems such as gears not meshing properly, misalignment of shaft components, etc. Deflection should be checked so that each item mounted on the shaft and their interfaces work properly.

For shafts with a length to diameter ratio of less than 10, the deflection due to transverse shear can be significant, and should be included when determining deflection [2].

The shaft slope at each bearing should be checked to ensure that the slope is less than the bearing's slope specification.

For precision machines, shaft twist may be a consideration. If the shaft is supposed to turn through a precise angle, shaft twist may result in the wrong angle. Excessive shaft twist can also affect timing during start-up where the shaft goes from zero torque to operating torque and the shaft twist is changing.

## Fatigue

When radial loads are present we have alternating bending stress and fatigue can be an important consideration. It's possible that torque or axial loading can fluctuate, but we will assume constant torque and axial loading.

Also, for this article, we will only discuss stress-life fatigue analysis for infinite life.

To start, we need to separate mean and alternating stresses. Since we are assuming constant torque and axial load, these result in mean stress as shown below. This is for the first example shaft with a tensile load.

The alternating stress is the result of bending.

Since the mean stress element has combined stress, we need to select an appropriate failure theory to compute effective mean stress. Again, you must do proper due diligence when selecting the failure theory.

#### Fatigue Safety Factors

When mean stress is non-zero, an appropriate mean stress failure theory must be applied to determine a design is safe for infinite life (or a long life for materials without an endurance limit).

Fatigue safety factor can be calculated directly, but first we'll show it graphically. The diagram below shows various failure theories and an operating point where mean stress = 9 and alternating stress = 8. The load line is drawn from the origin through the operating point. In our case, we'll use the Modified Goodman line and the safety factor is the ratio of distance A to B.

The equations for various fatigue failure theories is listed below.

## Stress Concentrations

Thus far we haven't mentioned stress concentrations, but it doesn't mean that they aren't important, because they are. Shafts are rarely a simple round bar. Keyways, grooves, holes, steps, etc. are necessary for a functioning shaft and stress concentrations for static and fatigue stress should be used appropriately.

## First Critical Speed

At the first critical speed (and higher critical speeds), the shaft will become unstable and deflect wildly. Therefore, it's important to calculate first critical speed and, if at all possible, operate well below this speed.

## References

[1] Budynas, Nisbett, *Shigley's Mechanical Engineering Design*, 8th Ed., 2006, McGraw-Hill.

[2] Shigley, J., Mischke, C., *Standard Handbook of Machine Design*, 3rd Ed., 2004, McGraw-Hill, page 17.8.

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