---
title: "Goodman and Soderberg lines for shafts"
subject: "Machine Design I"
chapter: 5
key: "semester_6/machine_design_i:goodman-soderberg-diagram"
source: https://www.bctengineer.com/q/semester_6/machine_design_i/goodman-soderberg-diagram
site: BCT Engineering
---

# Goodman and Soderberg lines for shafts

Machine Design I · Chapter 5: Design of shafts

- Asked in: year not given, 6 marks
- Video: [Modified Goodman Diagram Explained | Design for Fatigue load | GATE Machine Design](https://www.youtube.com/watch?v=xNij5v7X5z8) — Ekeeda GATE & ESE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/machine_design_i/goodman-soderberg-diagram
- Whole chapter: https://www.bctengineer.com/md/semester_6/machine_design_i/chapter-5

## Question

Draw a fatigue diagram (mean stress against alternating stress) showing the Soderberg and modified Goodman lines. Write the equations of both lines and the factor of safety expressions, and compare their use in shaft design.

## Short answer

Soderberg line: σm/Sy + σa/Se =1/n, using yield strength; Goodman line: σm/Sut + σa/Se =1/n, using ultimate strength. Soderberg is more conservative and protects against yielding; Goodman is less conservative and closer to test data. For shaft design under combined loads, equivalent stresses from distortion energy are substituted. Stress concentration factor Kf is applied to alternating stress.

## Answer

In a fatigue diagram the **mean stress** $\sigma_m$ is plotted on the horizontal axis and the **alternating stress** $\sigma_a$ on the vertical axis. A line is drawn to show the combinations of stress that the material can bear for infinite life.

```text
 sigma_a
    ^
 Se *
    |\ ' .
    | \     ' .      Goodman line
    |  \        ' .
    |   \ Soderberg   ' .
    |    \  line          ' .
    +-----*-------------------*----> sigma_m
          Sy                  Sut
```

(Soderberg line: from $S_e$ on the vertical axis to $S_y$ on the horizontal axis. Goodman line: from $S_e$ to $S_{ut}$. The Soderberg line is the lower one.)

#### Equations

**Soderberg line** (based on yield strength):

$$
\frac{\sigma_m}{S_y} + \frac{\sigma_a}{S_e} = \frac{1}{n}
$$

**Modified Goodman line** (based on ultimate strength):

$$
\frac{\sigma_m}{S_{ut}} + \frac{\sigma_a}{S_e} = \frac{1}{n}
$$

where $n$ is the factor of safety and $S_e$ is the corrected endurance limit. The stress concentration factor $K_f$ is applied to the alternating component (and, for brittle materials, also to the mean component).

#### Comparison

| Point | Soderberg | Goodman |
|---|---|---|
| Static limit used | Yield strength $S_y$ | Ultimate strength $S_{ut}$ |
| Position | Lower, more conservative | Higher, closer to test data |
| Protects against yielding | Yes (the line lies inside the yield limit) | Not by itself; a yield check is also needed |
| Use | Shafts and ductile parts where yielding is not allowed | Fatigue design of ductile or brittle parts when somewhat less conservatism is acceptable |

For a shaft with bending and torsion, the stresses are combined using the distortion energy theory: the equivalent mean and alternating stresses are found and substituted in the line equation.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the fatigue diagram, what is plotted on each axis?
   A. Alternating stress horizontal, mean stress vertical
   B. Yield strength horizontal, Sut vertical
   C. Mean stress horizontal, alternating stress vertical
   D. Time horizontal, stress vertical
   Answer: C. The line gives combinations allowing infinite life. (Key point: Axes)

2. The Soderberg line is
   A. sigma_m/Sut + sigma_a/Se = 1/n
   B. sigma_m/Se + sigma_a/Sy = 1/n
   C. sigma_m + sigma_a = Se/n
   D. sigma_m/Sy + sigma_a/Se = 1/n
   Answer: D. It uses yield strength on the mean stress axis. (Key point: Soderberg equation)

3. The modified Goodman line uses which static strength?
   A. Ultimate strength Sut
   B. Yield strength Sy
   C. Endurance limit only
   D. Shear strength only
   Answer: A. sigma_m/Sut + sigma_a/Se = 1/n. (Key point: Goodman equation)

4. Which statement compares them correctly?
   A. Goodman is more conservative than Soderberg
   B. Soderberg is lower and more conservative and protects against yielding
   C. Both lines coincide
   D. Soderberg is based on ultimate strength
   Answer: B. The Soderberg line lies inside the Goodman line. (Key point: Comparison)

## Related in Machine Design I

- [Soderberg design with reversed bending and steady torque](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Asoderberg-bending-torsion-shaft) — Machine Design I, chapter 5
- [Bending stress and safety factor of a spur pinion](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aagma-bending-stress-safety-factor) — Machine Design I, chapter 9
- [Hollow shaft against solid shaft](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Ahollow-versus-solid-shaft) — Machine Design I, chapter 5
- [Roller chain drive: length, speed and factor of safety](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aroller-chain-drive-selection) — Machine Design I, chapter 8
- [AGMA bending stress equation and its factors](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aagma-factors-stress-concentration-geometry) — Machine Design I, chapter 9
- [ASME code equation for shaft diameter](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aasme-code-shaft-diameter-derivation) — Machine Design I, chapter 5

## Related in other subjects

- [Normal, shear, bearing stress and factor of safety](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Astress-types-factor-of-safety) — Strength of Materials, chapter 1
- [Gumbel design flood, factor of safety 1.3, 45-year life](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-25yr-life45-fos1-3) — Engineering Hydrology, chapter 6
- [Gumbel design flood, safety factor, confidence limits](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-20yr-life35-rel87) — Engineering Hydrology, chapter 6
- [Factor of safety: WSM and LSM](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Afactor-of-safety-wsm-and-lsm) — Design of RCC Structure, chapter 1