A shaft under bending and torsion at once

A transmission shaft is almost never in pure torsion. It carries the torque it was put there for, and it carries bending from the gear or pulley loads that deliver that torque — and the two combine into a stress state neither one predicts on its own. This is a solid round shaft in a gearbox: 40 mm diameter, S355, carrying 800 N·m of torque with 500 N·m of bending at the section being checked.

use steel

Four significant figures: a stress is not known to six.

digits 4

d

40 mm

M

500 N \cdot m

T

800 N \cdot m

The two stresses

Bending is largest at the surface, and so is torsional shear, and on a round section they are largest at the same place — which is why the combination has to be taken at a point rather than averaged.

Z

\frac{π \cdot d^{3}}{32}

\frac{π \cdot {(40 mm)}^{3}}{32}

6283 {mm}^{3}

Z_{p}

\frac{π \cdot d^{3}}{16}

\frac{π \cdot {(40 mm)}^{3}}{16}

12570 {mm}^{3}

σ_{b}

\frac{M}{Z}

\frac{500 J}{6283 {mm}^{3}}

79.58 MPa

τ

\frac{T}{Z_{p}}

\frac{800 J}{12570 {mm}^{3}}

63.66 MPa

Two failure theories, and the gap between them

Distortion energy (von Mises) is the usual choice for a ductile steel and matches test data best. Maximum shear (Tresca) is the more conservative, by about 15% in pure shear, and some codes require it.

σ_{vm}

\sqrt{{σ_{b}}^{2} + 3 \cdot τ^{2}}

\sqrt{{(79.58 MPa)}^{2} + 3 \cdot {(63.66 MPa)}^{2}}

136 MPa

σ_{tresca}

\sqrt{{σ_{b}}^{2} + 4 \cdot τ^{2}}

\sqrt{{(79.58 MPa)}^{2} + 4 \cdot {(63.66 MPa)}^{2}}

150.1 MPa

The factor of safety against first yield, on each theory.

n_{vm}

\frac{{Fy}_{S355}}{σ_{vm}}

\frac{355 MPa}{136 MPa}

2.611

n_{tresca}

\frac{{Fy}_{S355}}{σ_{tresca}}

\frac{355 MPa}{150.1 MPa}

2.364

check

n_{tresca} \geq 2

2.364 \geq 2

pass

What diameter the load actually needs

Rearranged rather than iterated: the equivalent stress is proportional to 1/d³, so the diameter for a chosen factor comes out in one line — and the cube root divides the dimension, which is what a length is doing inside a cubed quantity in the first place.

n_{target}

2

d_{required}

nthroot (\frac{32 \cdot n_{target} \cdot \sqrt{M^{2} + 0.75 \cdot T^{2}}}{π \cdot {Fy}_{S355}}, 3)

nthroot (\frac{32 \cdot 2 \cdot \sqrt{{(500 J)}^{2} + 0.75 \cdot {(800 J)}^{2}}}{π \cdot 355 MPa}, 3)

36.6 mm

check

d \geq d_{required}

40 mm \geq 36.6 mm

pass

The same state, as principal stresses

Bending and torsion at a point are a plane stress state, and its principal stresses are the eigenvalues of the tensor that holds it. They are what the two failure theories above are written in terms of — von Mises is a distance between them and Tresca is the largest difference — so computing them says the same thing a third way and shows where the numbers come from.

state

[\begin{matrix} σ_{b} & τ \\ τ & 0 MPa \end{matrix}]

[\begin{matrix} 79.58 MPa & 63.66 MPa \\ 63.66 MPa & 0 MPa \end{matrix}]

=[[79580000 Pa, 63660000 Pa], [63660000 Pa, 0 Pa]]

principal

reverse (eigenvalues (state))

reverse (eigenvalues ([[79580000 Pa, 63660000 Pa], [63660000 Pa, 0 Pa]]))

=[114.9 MPa, -35.28 MPa]

Largest first, which is how a stress table is read: σ₁ is tension, σ₂ is compression, and the third principal stress of a plane state is zero.

σ_{1}

{principal}_{1}

{([114900000 Pa, -35280000 Pa])}_{1}

114.9 MPa

σ_{2}

{principal}_{2}

{([114900000 Pa, -35280000 Pa])}_{2}

-35.28 MPa

Tresca is the largest difference between any two of the three, and the third one here is zero — so it is σ₁ − σ₂, and it agrees with the formula above.

{tresca}_{check}

σ_{1} - σ_{2}

114.9 MPa - -35.28 MPa

150.1 MPa

check

| {tresca}_{check} - σ_{tresca} | \leq 0.001 MPa

| 150.1 MPa - 150.1 MPa | \leq 0.001 MPa

pass

What this calculation does not include

First yield at a plain section, and nothing else. A shaft fails at its shoulders, keyways and cross-holes, where the stress concentration factor multiplies these numbers by two or more, and it fails in fatigue rather than by yielding — the torque is steady but the bending is fully reversed once per revolution. Both belong in a design check and neither is here.

Safety factor against diameter

fn factor defined

plot (factor, 25, 60)