MEboost's shaft design tool allows for comprehensive analysis of shaft designs. The shaft design tool allows for the following analysis:

- Static stress.
- Deflection and slope.
- Stress life fatigue analysis for infinite life.
- Static and fatigue stress concentrations.
- Critical speed calculation.

For more general information on shaft design, see shaft design considerations.

## Shaft Design Tool Coordinate System

The coordinate system used by the MEboost shaft design tool assumes the centerline of the shaft is along the x-axis. The coordinate system applied to a shaft is shown below.

## Example Shaft

To illustrate the use of the shaft design tool, we'll analyze the shaft shown below. Dimensions are in inches.

The shaft has two gears mounted on it. Each gear is mounted on the Ø1.25 steps. The radial loads are shown below.

The load shown in Section A-A is assumed to be concentrated at 3.4375". The load shown in Section B-B is assumed to be concentrated at 6.0625". There is also a compressive axial load from 3.875" to 5.625".

A torque is applied to the shaft starting at 3", and a resisting torque is applied ending at 6.5".

## Using the Shaft Design Tool

To start the shaft design tool, click the *Shaft Design* button on the MEboost ribbon.

The shaft design form will appear. It's divided into seven pages. Click the appropriate button on the left side menu to access a page.

### Model Page

A shaft design can be saved as a model on a new worksheet in the current workbook. This allows you to load a model and run new reports without having to re-enter data. You can also load a model, make changes and run a report. The revised design can be saved as a new model by selecting “New model” and saving it. You can revise the original model by making changes and clicking the Save Model button.

NOTE: You must save the workbook to keep any models.

#### Units

Units can be SI or US customary. When entering data, the appropriate units to use will appear throughout the form.

#### X Increment

For most diagrams and calculations, the shaft is divided into increments determined by this setting. A smaller x increment will result in more accurate data but will take longer to process. The default value is .001 for SI units which results in 1000 increments per meter. The default value is .04 for US units which results in 25 increments per inch.

As we'll see later, there will be stress concentrations applied over a .03" interval where diameter changes. Since the default X increment is .04", some of these stress concentrations may not be evaluated since the increment may skip over the shaft length where they are applied. To ensure this doesn't happen, we will use an X increment of .01".

### Report Page

#### Transverse Shear

For shafts of length to diameter ratios of less than 10, deflection due to transverse shear can be significant [1]. To include the deflection from transverse shear, check the box. Deflections due to bending and transverse shear are superposed in the y and z direction prior to calculating resultant deflection.

For bearing slope, the slope due to bending and the slope due to shear are superposed prior to calculating the resultant slope. At diameter changes or shear change locations, the slope prior to and after the change is averaged per the procedure outlined in [1].

Note: Slope due to transverse shear is only calculated at the bearings. The slope diagram will not contain slope due to transverse shear.

Since the example shaft has a minimum diameter of 1" and a length of 9.5", the length to diameter is less than 10. Therefore, we should check transverse shear deflection.

[1] Shigley, J., Mischke, C. *Standard Handbook of Machine Design*, 3^{rd} Ed., 2004, McGraw-Hill, pages 17.8-17.12.

#### Report Diagrams

Numerous diagrams can be generated in the report. The following is a description of each diagram.

**Torque** The applied torque as a function of x.

**Twist** Angular deflection of the shaft. The left end of the shaft starts with zero twist.

**Shear** If radial loads are present, this is the shear diagram. There are two curves, one for the shear in the y direction and one for the z direction.

**Y & Z moments** If radial loads are present, this is the moment diagram. There are two curves, one for moment in the y direction and one for the z direction.

**Resultant moment** This is the resultant of the Y and Z moments. Since this is a resultant, all values will be positive.

**Slope** The resultant slope of the shaft due to bending. Since this is a resultant, all values will be positive.

**Deflection** The resultant deflection of the shaft due to bending, and if selected, also due to transverse shear. Since this is a resultant, all values will be positive. This diagram also shows resultant deflection angle measured counterclockwise from the positive Z axis.

**Von Mises stress** Von Mises stress due to combined loading of torque, radial and axial loads.

** Shear stress** Maximum shear stress due to combined loading of torque, radial and axial loads.

** Normal stress** Maximum and minimum principal stresses due to combined loading of torque, radial and axial loads. Maximum and minimum principal stresses may not be at the same point around the circumference of the shaft. For example, max principal stress may be at the point furthest from the neutral axis in bending tension and min principal stress may be at the point furthest from the neutral axis in bending compression.

**Fatigue** A fatigue diagram can have Goodman, Soderberg, Gerber and yield lines. The operating point of alternating stress and mean stress is plotted using stresses according to the effective stress type chosen.

**Alternating & mean stress** Plot of alternating and mean stress using the effective stress chosen.

**Fatigue safety factor plots** For each criterion checked, a plot of x vs. 1/safety factor.

If there are no radial loads, i.e. no bending, the shear, moment, slope and deflection diagrams will not appear on the report even if they are selected. Additionally, fatigue diagrams and data will not appear on the report if they are selected.

#### Effective Stress

Since combined stresses are present, we need to choose an effective stress to use for fatigue analysis. Alternating and mean stress values will depend on the effective stress chosen.

The shaft is a ductile material so we could choose Von Mises, Signed Von Mises, or Tresca criterion. We will use Von Mises.

#### Plot Worst-Case By

On the fatigue diagram, the operating point is plotted to indicate where the worst-case stresses are in relation to the failure criterion selected. For each failure criterion, the alternating and mean stress is calculated at each x increment. Then the fatigue safety factor is calculated for each criterion. Finally, the x location that results in the lowest safety factor is found for each criterion. Usually it’s at the same location for all criteria.

Select the failure criterion to use. The alternating and mean stress at the worst-case location for that criterion will be plotted.

We will choose ASME elliptic.

### Shaft Page

Material properties and dimensions of the shaft are entered on this page.

#### Shaft Dimensions

The shaft can be solid or hollow. Enter the overall shaft length, and if hollow is selected, enter the inside dimension.

#### Material Properties

Enter the shaft material properties.

Fatigue limit is the maximum stress level that the material can be subjected to for a very large number of cycles without failure. The term endurance limit is often used for the stress that a material can withstand for an unlimited number of cycles. Some materials, aluminum for example, do not have an endurance limit, and a fatigue limit is assumed for a large number of cycles, such as 10^{7} or 10^{8} cycles. The fatigue limit should be adjusted for various Marin factors such as surface finish, loading, temperature, etc.

E is the modulus of elasticity and G is the modulus of rigidity.

#### Shaft Steps

Stepped shaft dimensions are entered here. For each “step”, enter the starting x coordinate from the left end of the shaft for a given diameter, and click the *Add* button. The last step entered will be assumed to go to the right end of the shaft.

If the diameter of the shaft is constant, enter the starting x value and the diameter. The diameter will be constant for the entire length.

### Torque, Radial Loads, Bearings Page

Applied torques, radial loads, and bearing locations are specified on this page.

Two bearings are required. Additional bearings are not allowed. Bearing reactions are treated as point loads on the shaft, and the shaft is assumed to be simply supported.

Shaft weight is ignored for static and fatigue analysis. To include shaft weight, add a distributed load for each shaft step. Likewise, the weights of any attached components such as gears, pulleys, etc. are ignored for static and fatigue analysis. To include the weight of these components, add an appropriate concentrated or distributed load for each item.

#### Adding Items to the Model

Select an element type to add to the model. Depending on the selection, parameter boxes will appear to define the element.

**Bearing** Enter the x coordinate of the bearing centerline.

**Torque** Specify the location of the applied torque. Enter a positive value when the torque is clockwise when viewed from the left end of the shaft. Enter a negative value when the torque is counterclockwise.

**Concentrated Load** Enter the location of the load. The force must be entered as y and z component forces. If the force acts in the positive y or z direction, enter a positive number, otherwise enter a negative number.

**Distributed Load** Specify the start and end location of the load. The distributed load value (force/distance) must be entered as y and z component loads.

After all parameters are entered, click the *Add* button. The element will appear in the listbox.

For the example, we will ignore the shaft weight. To enter the radial loads, we need to break each force into its y and z components. The left gear has a 1200 lb load in the negative y direction. Therefore, the y component is -1200 and the z component is 0.

The right gear has a radial load of 900 lb at a 30° angle from the positive z axis. The y component is 900sin(30°) = 450. The z component is 900cos(30°) = 779.42.

We add a torque at the start of the left gear keyseat and an opposing torque at the end of the right gear keyseat.

### Axial Loading Page

Axial loading in the x direction is specified on this page. Axial loads are specified by the state of loading along the shaft instead of applying specific loads. The loading specified is the net loading due to the sum of tensile or compressive loads.

#### Adding Items to the Model

Select either compressive or tensile loading from the drop-down box. Then specify the start and end x coordinates of the loading. Finally, enter the loading value as a positive number.

After all parameters are entered, click the *Add* button. The loading will appear in the listbox.

We have a 335 lb compressive load on the Ø1.5" section of the shaft, so we'll add that loading.

### Stress Concentration Page

Stress concentrations are specified on this page. A stress concentration is applied over an x interval. Any stress calculations for the type (static or fatigue) are applied while x is within the interval.

#### Adding a Stress Concentration

Select the type of stress concentration. For example, if the type is static, the stress concentration will only be applied for static stress calculations. It will not affect fatigue calculations.

Select the type of loading. Each stress concentration is for one loading type and will only be applied when calculating stress for the loading type. For combined stress, enter a separate stress concentration for each load type.

Enter the starting x location of the stress concentration and the ending x location. The stress concentration will be applied for the appropriate stress calculations during this interval. Enter the stress concentration factor in the Factor box.

After all parameters are selected or entered, click the *Add* button. The stress concentration will appear in the listbox.

For the example shaft, since it is a ductile material we will not use stress concentrations for static stress. However, we will use stress concentrations for fatigue. At each diameter change there will be a stress concentration for bending and torsion. We'll also add stress concentrations in bending and torsion for the keyseats.

Axial loading only occurs in the Ø1.5" section, so we will not use axial loading stress concentrations.

### Critical Speed Page

The first critical speed of the shaft can be estimated using the Rayleigh-Ritz and Dunkerley methods. Weights on the shaft are entered. These weights are due to items such as gears, pulleys, flywheels, etc. and not applied loads. All weights are concentrated at the specified location. To include the weight of the shaft, enter a shaft density. Otherwise, enter 0 for density to ignore shaft weight.

When shaft weight is included, the shaft is divided into segments of length = X increment to approximate the shaft’s critical speed. If there is a diameter change within a segment, the average of the starting and ending diameters is used to calculate shaft weight for the segment.

If deflections due to transverse shear was selected on the Report page, these deflections will be accounted for in critical speed calculations.

The estimated critical speed will appear in the report heading.

#### Adding Weights

Enter the x coordinate and weight of the item. Then click the *Add* button. The weight and location will appear in the listbox.

The left side gear weighs 15 lbs and the right side gear weighs 18 lbs so we add the weights at each gear's centerline location.

## Saving the Model

Now that we've entered everything in the design, we can save it for later use. Go to the Model page and click the *Save Model* button. If you modify an existing model, saving the model again will save the changes. Remember that you still need to save the workbook to keep the model worksheet.

A new worksheet will be created that contains the shaft information. Do not modify this worksheet. If it's modified, the shaft design tool may not be able to read it.

## Shaft Design Report

Once all information has been entered, click the Create Report button. The report will appear in a new worksheet in either the current workbook or a new workbook, depending on what was selected.

The shaft design report contains three sections:

- Report heading.
- Diagrams.
- Data for diagrams.

### Report Heading

The report heading contains material properties and calculated results.

### Diagrams

Diagrams are located below the report heading. We checked all diagrams for the report, so there will be 16 charts on the report.

### Data for Diagrams

The data used to generate diagrams is located to the right of the diagrams. This data is useful for finding values at a specific x location. Click on a curve and the data will be highlighted. For example, to find the Von Mises stress at x=4.25 go to the x column for Von Mises stress and locate the row where x = 4.25 (or the nearest x value if not exactly 4.25) and then find the corresponding Von Mises stress.

## Wrapping it Up

We've now run a report for the shaft. There is a lot of information in this report to analyze. If the design is OK, then we're done. Otherwise, we can reload the model, make the appropriate changes and run a new report quickly.