---
author: Akbar Rahman
date: \today
title: MMME2044 // Shaft Design
tags: [ mmme2044, shafts, uni ]
uuid: 8e0928a6-c20c-4f80-9691-beb2defa4022
---

# Shaft Design Considerations

- Function and loading
- Size and connection to components
- Material selection and treatments
- Deflection and rigidity
- Stress and strength
- Critical speed
- Manufacturing constraints

# Shaft-Hub Connections

![](./images/vimscrot-2022-10-31T15:09:11,091932569+00:00.png)

![](./images/vimscrot-2022-10-31T15:10:02,106108248+00:00.png)

# Shaft-Shaft Connections

Shaft-shaft connections can use either rigid or flexible couplings.

![](./images/vimscrot-2022-10-31T15:11:05,058803741+00:00.png)

# Location of Bearing on Shaft

![](./images/vimscrot-2022-10-31T15:18:36,571125403+00:00.png)

# Shaft Loading

- Axial stresses

    - cause by self-weight in vertical shafts
    - due to axial restraint at bearings and associated axial load

- bending stress

    - due to bending moment in belt drivers, gear forces, mounted component weights
    - dynamic forces which can load to fatigue and resonance

- shear stresses

    - due to torque load/direct shear

# Shaft Diameter (ASME Design Code)

$$d = \left[
        \frac{32n_s}{\pi}
        \sqrt{
                \left(\frac{M}{\sigma_e}\right)^2 
                + \frac34 \left(\frac{T}{\sigma_y}\right)^2
        }
      \right]^{\frac13}
$$

where:

- $n_s$ is [safety factor](#safety-factor)
- $M$ is max bending moment (Nm)
- $T$ is max torque (Nm)
- $\sigma_e$ is endurance limit stress (Pa)
- $\sigma_y$ is yield strength of shaft (Pa)


<details>

<summary>

#### ASME Design Code Derivation

</summary>

- Bending moment creates alternating tensile/compressive stresses ($\sigma_a$):

  \begin{align}
  \sigma &= \frac{My}{I} \\
  I &= \frac{\pi d^4}{64}
  \end{align}

  Combine to get:

  $$\sigma_a = \frac{32M}{\pi d^3}$$

- Torque normally generates constant shear stress ($\tau_m$):

  \begin{align}
  \tau &= \frac{TR}{J} \\
  J &= \frac{\pi d^4}{32}
  \end{align}

  Combine to get:

  $$\rightarrow \tau_m = \frac{16T}{\pi d^3}$$

- von Mises stress in plane stress condition:

  $$\sigma_\text{von Mises} = (\sigma^2 + 3\tau^2)^{\frac12}$$

- ASME Fatigue Failure Criterion

  $$\left(\frac{n_s\sigma_a}{\sigma_e}\right)^2 + \left(\frac{n_s\sigma_m}{\sigma_Y}\right)^2 = 1$$

  Finally substitute $\sigma_a$ and $\tau_m$ to make $d$ the subject.

</details>

## Endurance Limit Stress, $\sigma_e$

$\sigma_e$ is the level of stress with which fatigue failure wouldn't occur in cycling or
alternating load conditions:

![](./images/vimscrot-2022-11-04T10:13:44,527741950+00:00.png)

- $\sigma$ at failure decreases with the number of cycles up until a certain point ($\sigma_e$ after
  around $10^6$ cycles)
- if you never exceed this point then the material will last for "infinite" cycles
- most steels have this fatigue behaviour so they are often used for shafts

Time for a silly equation:

$$\sigma_e = k_ak_bk_ck_dk_ek_fk_g\sigma_e'$$

where:

- $k_a$ --- surface factor
- $k_b$ --- size factor
- $k_c$ --- reliability factor
- $k_d$ --- temperature factor
- $k_e$ --- duty cycle factor
- $k_f$ --- fatigue stress concentration factor
- $k_g$ --- miscellaneous effects factor
- $\sigma_e'$ --- endurance limit of test specimen

this is a joke of an equation

## Reducing Stress Concentrations

![](./images/vimscrot-2022-11-04T10:31:41,364654063+00:00.png)

![](./images/vimscrot-2022-11-04T10:31:50,672814681+00:00.png)

## Critical Speed of Shaft (Natural Frequency)

- operational speed of shat should be half the critical speed
- Centre of Mass should be on the Centre of Rotation
- in practice this is not the case
- this imbalance causes a deflection (due to centrifugal force, $mr\omega^2$
- the critical speed (or natural frequency) is the speed at which the shaft is unstable

  when this is the case it may cause damage to the shaft, bearings, and other destructive
  vibrations

## Critical Speed Equation

$$\omega_c = \sqrt\frac{g}{\delta_\text{st}}$$

where $g$ is acceleration due to gravity and $\delta_\text{st}$ is 
[static deflection](#shaft-deflection) of the shaft.

## Rayleigh Ritz Equation

When you have multiple masses the Rayleigh-Ritz equation may be more suitable:

$$\omega_c = \sqrt{g\frac{\Sigma w_i\delta_i}{\Sigma w_i\delta_i^2}} $$

where $w_i$ is the weight of node $i$ and $\delta_i$ is the static deflection at node $i$.

# Shaft Deflection

- Shaft deflection is required to determine the critical speed.
- Macaulay's method for the deflection in beam bending:

  $$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \frac{M}{EI}$$
  $$y = \int^x_0\int^x_0\left(\frac{M}{EI}\right) \mathrm{d}x + C_1x + C_2$$

## Shaft Deflection Equations

![](./images/shaft-design-064.jpg)

- Maximum deflection:

  $$\delta_\text{max} = \frac{PL^3}{3EI}$$

- Deflection at any point $x$

  $$\delta = \frac{Px^3}{6EI}(3L-x)$$

![](./images/shaft-design-065.jpg)

- Maximum deflection:

  $$\delta_\text{max} = \frac{PL^3}{48EI}$$

- Deflection at any point:

  $$\delta = \frac{Px}{12EI} \left(\frac{3L^2}{4} - x^2 \right)$$

![](./images/shaft-design-066.jpg)


- Maximum deflection

  $$\delta_\text{max} = \frac{Pb^2L}{3EI}$$

- Deflection at any point

  - For $0 \le x \le a$:

    $$\delta = \frac{Pbx}{6aEI}(x^2-a^2)$$

  - For $0 \le z \le b$:

    $$\delta = \frac{Pbx}{6aEI}\left(z^3 - b(2L+b) + 2b^2L\right)$$

# Safety Factor

Safety Factor (also known as reserve factor) is a simple way to accommodate for uncertainties in
design.

Simply multiply the maximum stresses and loads you expect by the safety factor, $n_s$, and assume
that as your maximum stress and load.

$n_s$ | Operational conditions and use of materials
----- | -------------------------------------------
1.25-1.50 | Reliable materials under controlled conditions, known stresses with certainty
1.50-2.00 | Well-known materials under reasonably constant environmental condition, known stresses
2.00-2.50 | Average materials subjected to known loads and stresses and environment (LSE)
2.50-3.00 | Lesser well-known materials under average conditions LSE
3.00-3.40 | Untried materials under average conditions of stresses and environment, or well known materials under uncertain LSE