---
title: "ASME code equation for shaft diameter"
subject: "Machine Design I"
chapter: 5
key: "semester_6/machine_design_i:asme-code-shaft-diameter-derivation"
source: https://www.bctengineer.com/q/semester_6/machine_design_i/asme-code-shaft-diameter-derivation
site: BCT Engineering
---

# ASME code equation for shaft diameter

Machine Design I · Chapter 5: Design of shafts

- Asked in: year not given, 8 marks
- Video: [6.8 ASME Code for Design of Shaft _ Machine Design I](https://www.youtube.com/watch?v=jzkWQgu3BIQ) — yogesh shinde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/machine_design_i/asme-code-shaft-diameter-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_6/machine_design_i/chapter-5

## Question

Derive the equation for the diameter of a solid shaft subjected to combined bending moment M and torque T using the maximum shear stress theory with the ASME code shock and fatigue factors. State how the ASME code fixes the allowable shear stress of the shaft material and the effect of a keyway.

## Short answer

By maximum shear stress theory, τ_max = (16/πd³)√(M²+T²). Introducing ASME shock and fatigue factors Kb and Kt, the design equation is d³ = (16/(π τ_allow))√((Kb M)²+(Kt T)²). Allowable shear stress is the smaller of 0.30 Syt and0.18 Sut, decreased by25% for a keyway.

## Answer

The ASME code equation gives the diameter of a shaft under combined bending and torsion by applying the **maximum shear stress theory** and multiplying the moment and torque by factors that allow for shock and fatigue.

#### Derivation

For a solid shaft of diameter $d$, the maximum bending stress and the torsional shear stress at the surface are

$$
\sigma_b = \frac{32M}{\pi d^3}, \qquad \tau = \frac{16T}{\pi d^3}
$$

The maximum shear stress at that point is

$$
\begin{aligned} \tau_{max} &= \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau^2} \\ &= \sqrt{\left(\frac{16M}{\pi d^3}\right)^2 + \left(\frac{16T}{\pi d^3}\right)^2} \\ &= \frac{16}{\pi d^3}\sqrt{M^2 + T^2} \end{aligned}
$$

The quantity $\sqrt{M^2+T^2}$ is the **equivalent twisting moment** $T_e$. In practice the loads are not steady, so the ASME code multiplies the bending moment by $K_b$ (combined shock and fatigue factor for bending) and the torque by $K_t$ (for torsion):

$$
\tau_{max} = \frac{16}{\pi d^3}\sqrt{(K_b M)^2 + (K_t T)^2}
$$

Putting $\tau_{max} = \tau_{allow}$ gives

$$
d^3 = \frac{16}{\pi \tau_{allow}}\sqrt{(K_b M)^2 + (K_t T)^2}
$$

For a hollow shaft with $k = d_i/d_o$, replace $d^3$ by $d_o^3(1-k^4)$.

#### Values of $K_b$ and $K_t$ (rotating shaft)

| Load condition | $K_b$ | $K_t$ |
|---|---|---|
| Gradually applied or steady | 1.5 | 1.0 |
| Suddenly applied, minor shock | 1.5 to 2.0 | 1.0 to 1.5 |
| Suddenly applied, heavy shock | 2.0 to 3.0 | 1.5 to 3.0 |

For a stationary shaft the values are smaller (1.0 for gradually applied loads, 1.5 to 2.0 for suddenly applied loads).

#### Allowable shear stress by the code

- $\tau_{allow} = 0.30\,S_{yt}$ or $0.18\,S_{ut}$, whichever is **smaller**.
- If the shaft has a keyway, reduce the value by **25 percent** (multiply by 0.75) because of stress concentration.

The calculated diameter is rounded up to the nearest standard shaft size.

## Check yourself

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

1. For combined bending and torsion, the maximum shear stress at the surface is
   A. (16/(pi d^3)) (M + T)
   B. (32/(pi d^3)) sqrt(M^2 + T^2)
   C. (16/(pi d^3)) sqrt(M^2 + 3T^2)
   D. (16/(pi d^3)) sqrt(M^2 + T^2)
   Answer: D. tau_max = sqrt((sigma_b/2)^2 + tau^2) with sigma_b = 32M/(pi d^3) and tau = 16T/(pi d^3). (Key point: Maximum shear stress)

2. How are Kb and Kt used in the ASME equation?
   A. Kb multiplies the bending moment and Kt multiplies the torque inside the square root
   B. Both multiply the diameter
   C. They divide the allowable stress
   D. They are added to M and T
   Answer: A. d^3 = 16/(pi tau) sqrt((KbM)^2 + (KtT)^2). (Key point: Shock and fatigue factors)

3. How does the code fix allowable shear stress?
   A. The larger of 0.30 Syt and 0.18 Sut
   B. The smaller of 0.30 Syt and 0.18 Sut
   C. 0.5 Sut always
   D. 0.18 Syt only
   Answer: B. Whichever is smaller governs. (Key point: Allowable stress)

4. What is the effect of a keyway?
   A. Allowable stress is increased by 25 percent
   B. Diameter is reduced by 25 percent
   C. Allowable shear stress is reduced by 25 percent because of stress concentration
   D. Torque is reduced by 25 percent
   Answer: C. The value is multiplied by 0.75. (Key point: Keyway effect)

## Related in Machine Design I

- [Shaft diameter for power and bending](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Astatic-shaft-power-bending-diameter) — Machine Design I, chapter 5
- [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
- [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
- [Diameter of a shaft carrying a spur gear](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Agear-loaded-shaft-diameter) — Machine Design I, chapter 5
- [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
- [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

## Related in other subjects

- [Solid shaft 120 Nm: diameter from limits](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asolid-shaft-120nm-twist-limit) — Strength of Materials, chapter 6, asked 2×
- [Gearbox shaft design with spur gear](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Agearbox-shaft-design-asme) — Machine Design II, chapter 5
- [Principal stresses in a shaft under combined loads](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Ashaft-combined-bending-torsion-stresses) — Strength of Materials, chapter 6
- [Stepped shaft: maximum torque at C](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astepped-shaft-fixed-max-torque-at) — Strength of Materials, chapter 6