# Torsion Spring Rate and Bending Stress Calculator

> Source: https://netray.co/tools/torsion-spring-rate-and-bending-stress-calculator
> Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems.
> Section: Free Tools

## Short answer

Torsion spring rate, moment, inner-fiber bending stress, arm force and the loaded inside diameter that sets your arbor size.

## Category

engineering-plm

## Industry

Springs and Fasteners

## Audience

- Design engineer
- Spring estimator
- Quality engineer

## Tool

**Kind:** calculator

**Inputs:**

#### Wire diameter

**Key:** d

**Type:** number

**Default:** 0.072

**Min:** 0.005

**Max:** 0.5

**Step:** 0.001

**Unit:** in

#### Mean coil diameter (free)

**Key:** dm

**Type:** number

**Default:** 0.6

**Min:** 0.05

**Max:** 6

**Step:** 0.005

**Unit:** in

#### Body coils

**Key:** nb

**Type:** number

**Default:** 6

**Min:** 1

**Max:** 80

**Step:** 0.25

**Unit:** coils

#### Arm length 1

**Key:** l1

**Type:** number

**Default:** 1

**Min:** 0

**Max:** 20

**Step:** 0.05

**Unit:** in

**Help:** Measured from the coil axis to the point of load.

#### Arm length 2

**Key:** l2

**Type:** number

**Default:** 1

**Min:** 0

**Max:** 20

**Step:** 0.05

**Unit:** in

#### Deflection angle

**Key:** theta

**Type:** number

**Default:** 90

**Min:** 0

**Max:** 720

**Step:** 5

**Unit:** deg

**Help:** Winding-up direction, in degrees from the free position.

#### Elastic modulus E

**Key:** e

**Type:** select

**Default:** 29500000

**Options:**

##### Music wire / chrome alloys (29.5 Mpsi)

**Value:** 29500000

##### Hard drawn / oil tempered (28.5 Mpsi)

**Value:** 28500000

##### Stainless 302 / 304 (28.0 Mpsi)

**Value:** 28000000

##### Phosphor bronze (15.0 Mpsi)

**Value:** 15000000

##### Beryllium copper (17.0 Mpsi)

**Value:** 17000000

#### Wire tensile strength

**Key:** sut

**Type:** number

**Default:** 294000

**Min:** 20000

**Max:** 400000

**Step:** 1000

**Unit:** psi

#### Allowable bending stress

**Key:** pct

**Type:** slider

**Default:** 70

**Min:** 50

**Max:** 90

**Step:** 1

**Unit:** % of tensile

**Help:** Torsion springs load the wire in bending. About 70% to 80% is common for music wire without set removal.

**Outputs:**

#### Spring index C

**Key:** c

**Expr:** dm / d

**Format:** decimal

#### Active coils incl. arm allowance

**Key:** na

**Expr:** nb + (l1 + l2) / (3 * pi * dm)

**Format:** decimal

**Description:** Na = Nb + (L1 + L2) / (3 pi D), Shigley's allowance for arm deflection.

#### Spring rate per turn

**Key:** rate

**Expr:** e * pow(d, 4) / (10.8 * dm * na)

**Format:** decimal

**Primary:** true

**Description:** k' = E d^4 / (10.8 D Na), lbf-in per revolution. 10.8 includes a friction allowance; 10.19 is the frictionless value.

#### Moment at deflection

**Key:** moment

**Expr:** rate * theta / 360

**Format:** decimal

**Primary:** true

**Description:** Torque in lbf-in at the working angle.

#### Bending stress (inside fiber)

**Key:** sigma

**Expr:** ((4 * c * c - c - 1) / (4 * c * (c - 1))) * 32 * moment / (pi * pow(d, 3))

**Format:** integer

**Description:** Wire bending stress with the inner-fiber curvature factor Ki, psi.

#### Safety factor

**Key:** sf

**Expr:** pct / 100 * sut / sigma

**Format:** decimal

**Description:** Allowable bending stress divided by working stress.

#### Inside diameter at deflection

**Key:** id_loaded

**Expr:** nb * dm / (nb + theta / 360) - d

**Format:** decimal

**Description:** Coils tighten as the spring is wound. Arbor must be smaller than this, inches.

#### Force at arm 1 tip

**Key:** arm_force

**Expr:** moment / l1

**Format:** decimal

**Description:** Load at the end of arm 1, assuming perpendicular loading, lbf.

**Chart:**

#### Moment and bending stress vs deflection angle

**Type:** line

**X:** theta

**From:** 0

**To:** 180

**Steps:** 36

**Series:**

- moment
- sigma

**Disclaimer:** Round wire, static loading, torque applied in the winding-up direction. Arm loading angle, friction on the arbor and fatigue life are not modelled. Confirm with your spring maker.

## Content

**Intro:** Torsion springs store energy by bending the wire, not twisting it. This calculator gives rate per turn, moment at the working angle, bending stress with the inner-fiber factor, and the loaded inside diameter that limits your arbor size.

**Sections:**

#### How to use it

**Body:** Enter wire diameter, free mean diameter, body coils and the two arm lengths measured from the coil axis to the load point. Set the working deflection in degrees from the free position, in the direction that winds the coils tighter. Choose the modulus for your wire and the tensile strength for its diameter. Read the moment at your angle, the bending stress and the safety factor. Then check the loaded inside diameter: the spring shrinks on the arbor as it winds, so the arbor must be smaller than that value. The chart shows how moment and stress rise with angle so you can see the allowable limit.

**Bullets:**

- Arm length is from the axis, not from the body edge.
- Always wind the spring in the direction that tightens the coils.
- Keep arbor clearance of about 10% of the inside diameter.

#### Method and formulas

**Body:** Active coils include an arm allowance, Na = Nb + (L1 + L2) / (3 pi D). Rate per revolution is k' = E d^4 / (10.8 D Na); Shigley uses 10.8 in place of the theoretical 10.19 to cover friction between coils and the arbor. Moment is the rate times the turns. Bending stress is sigma = Ki x 32 M / (pi d^3), where Ki = (4 C^2 - C - 1) / (4 C (C - 1)) is the inner-fiber curvature factor. Loaded diameter is D' = Nb D / (Nb + turns).

#### Worked example

**Body:** With the defaults (0.072 in music wire, 0.600 in mean diameter, 6 body coils, 1 in arms, 90 degrees) active coils are about 6.35, rate is about 19.3 lbf-in per turn, and moment at 90 degrees is about 4.8 lbf-in. Bending stress is about 144,000 psi against 70% of a 294 ksi tensile, a safety factor near 1.4. The coils shrink so the loaded inside diameter is near 0.50 in, which limits the arbor to around 0.45 in.

#### Common mistakes

**Body:** Torsion springs are forgiving on paper and unforgiving in the field when these points are missed.

**Bullets:**

- Using the free inside diameter to size the arbor; the loaded diameter is smaller.
- Winding the spring open, which cuts allowable stress and is not covered by these formulas.
- Quoting torque at one angle without stating the angle and arm radius.
- Using long, unsupported spring bodies; guide them on an arbor and lubricate to cut friction.

**Cta:** If torque specs, arm geometry and test data sit in separate files, we can connect them to your ERP and quality records so engineers and estimators find them in one place.

**Faq:**

#### Why is torsion spring stress a bending stress?

**Answer:** The moment applied at the arms bends the wire along its length, the way a curved beam bends. The wire itself is not twisted about its axis, which is why the formulas use bending stress with a curvature correction rather than torsional shear.

#### What is the difference between 10.8 and 10.19 in the rate equation?

**Answer:** 10.19 (64 divided by 2 pi) is the theoretical frictionless value. Shigley recommends 10.8 because friction between the coils and the arbor makes real springs stiffer than theory. Use 10.8 for prediction and verify with a torque test on samples.

#### How much does the spring shrink when it is wound?

**Answer:** The body length of wire stays about the same, so adding turns reduces the coil diameter in proportion. The loaded diameter is the free diameter times the body coils divided by body coils plus turns. This output gives the inside diameter at your working angle.

#### What deflection direction do the allowable stresses assume?

**Answer:** They assume winding that tightens the coils. Residual stress from coiling then opposes the working stress. Loading the spring in the opening direction cancels that benefit, so the allowable stress drops and you should design more conservatively.

## Search terms

torsion spring calculator, torsion spring rate formula, torsion spring bending stress, torsion spring torque calculator, torsion spring arbor diameter, torsion spring design, spring torque per degree, torsion spring arm force

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