# Extension Spring Load and Hook Stress Calculator

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

## Short answer

Rate, load with initial tension, body stress and full-loop hook stresses for round-wire extension springs, with an extension sweep.

## 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:** 0.063

**Min:** 0.005

**Max:** 0.5

**Step:** 0.001

**Unit:** in

#### Mean coil diameter

**Key:** dm

**Type:** number

**Default:** 0.5

**Min:** 0.05

**Max:** 6

**Step:** 0.005

**Unit:** in

**Help:** Outside diameter minus one wire diameter.

#### Body coils

**Key:** nb

**Type:** number

**Default:** 12

**Min:** 3

**Max:** 200

**Step:** 0.25

**Unit:** coils

**Help:** Coils in the wound body, excluding the hook loops.

#### Initial tension

**Key:** fi

**Type:** number

**Default:** 3

**Min:** 0

**Max:** 200

**Step:** 0.1

**Unit:** lbf

**Help:** Force that must be exceeded before the coils begin to separate.

#### Working extension beyond free length

**Key:** y

**Type:** number

**Default:** 0.75

**Min:** 0

**Max:** 20

**Step:** 0.05

**Unit:** in

#### Hook side bend radius

**Key:** r2

**Type:** number

**Default:** 0.1

**Min:** 0.01

**Max:** 2

**Step:** 0.005

**Unit:** in

**Help:** Mean radius of the bend where the hook meets the body.

#### Shear modulus G

**Key:** g

**Type:** select

**Default:** 11500000

**Options:**

##### Music wire / hard drawn / oil tempered (11.5 Mpsi)

**Value:** 11500000

##### Chrome silicon / chrome vanadium (11.2 Mpsi)

**Value:** 11200000

##### Stainless 302 / 304 / 316 (10.0 Mpsi)

**Value:** 10000000

##### Phosphor bronze (6.0 Mpsi)

**Value:** 6000000

#### Wire tensile strength

**Key:** sut

**Type:** number

**Default:** 300000

**Min:** 20000

**Max:** 400000

**Step:** 1000

**Unit:** psi

#### Allowable body shear

**Key:** pct

**Type:** slider

**Default:** 45

**Min:** 30

**Max:** 70

**Step:** 1

**Unit:** % of tensile

**Outputs:**

#### Spring index C

**Key:** c

**Expr:** dm / d

**Format:** decimal

**Description:** Mean diameter over wire diameter.

#### Spring rate

**Key:** k

**Expr:** g * pow(d, 4) / (8 * pow(dm, 3) * (nb + 0.4))

**Format:** decimal

**Primary:** true

**Description:** Uses effective coils Na = Nb + 0.4 for the end-loop flexibility (Shigley's correction of G/E for steel). lbf per inch.

#### Load at working extension

**Key:** force

**Expr:** fi + k * y

**Format:** decimal

**Primary:** true

**Description:** Initial tension plus rate times extension, lbf.

#### Body shear stress

**Key:** tau_body

**Expr:** ((4 * c - 1) / (4 * c - 4) + 0.615 / c) * 8 * force * dm / (pi * pow(d, 3))

**Format:** integer

**Description:** Wahl-corrected torsion in the coils at working load, psi.

#### Stress from initial tension

**Key:** tau_init

**Expr:** ((4 * c - 1) / (4 * c - 4) + 0.615 / c) * 8 * fi * dm / (pi * pow(d, 3))

**Format:** integer

**Description:** Typical as-wound initial stress is in the low-to-mid teens of ksi at moderate index; compare to your spring maker's guide.

#### Hook bending stress (full loop)

**Key:** sigma_hook

**Expr:** force * ((4 * c * c - c - 1) / (4 * c * (c - 1)) * 16 * dm / (pi * pow(d, 3)) + 4 / (pi * d * d))

**Format:** integer

**Description:** Bending plus axial at the loop-body transition, assuming hook radius equal to half the mean coil diameter, psi.

#### Hook torsion stress at side bend

**Key:** tau_hook

**Expr:** force * ((4 * (2 * r2 / d) - 1) / (4 * (2 * r2 / d) - 4)) * 8 * dm / (pi * pow(d, 3))

**Format:** integer

**Description:** Torsion at the bend where the hook leaves the body, psi.

#### Body safety factor

**Key:** sf_body

**Expr:** pct / 100 * sut / tau_body

**Format:** decimal

**Description:** Allowable body shear divided by working stress.

**Chart:**

#### Load and body stress vs extension

**Type:** line

**X:** y

**From:** 0

**To:** 1.6

**Steps:** 32

**Series:**

- force
- tau_body

**Disclaimer:** Round wire, static loading, full round loop ends. Hook failures are usually fatigue or bend-radius driven; use the hook stresses as a screening check and confirm allowables and fatigue life with your spring maker.

## Content

**Intro:** Extension springs fail at the hooks far more often than in the body. This calculator gives rate, load including initial tension, body stress and both hook stresses so you can see the weak point before ordering parts.

**Sections:**

#### How to use it

**Body:** Enter wire and mean diameter, the number of body coils and the initial tension from your drawing. Set the working extension beyond free length (not the overall length). Hook side bend radius is the mean radius where the hook wire leaves the body; a tighter radius raises torsion there sharply. Choose the modulus and tensile strength for your wire. The outputs show load at the working point, stress in the body, stress from initial tension alone, and the hook bending and torsion stresses. If the hook stresses are well above the body stress, widen the bend radius or consider a different end style.

**Bullets:**

- Initial tension is a function of index and winding; ask the maker what range is holdable.
- Extension is measured from the free length of the body plus hooks, as defined on your drawing.
- Hook stress is normally the limit, so check it first.

#### Method and formulas

**Body:** Rate uses the helical spring equation with effective active coils Na = Nb + G/E, which Shigley approximates as Nb + 0.4 for full-loop ends in steel. Load is F = Fi + k y. Body shear uses the Wahl factor. At the hook, bending stress is sigma = F [ Ka 16 D / (pi d^3) + 4 / (pi d^2) ] with Ka = (4 C1^2 - C1 - 1) / (4 C1 (C1 - 1)) and C1 = 2 R1 / d, here with R1 = D/2 for a full loop. Side-bend torsion is tau = Kb 8 F D / (pi d^3) with Kb = (4 C2 - 1) / (4 C2 - 4) and C2 = 2 R2 / d.

#### Worked example

**Body:** With the defaults (0.063 in wire, 0.500 in mean diameter, 12 body coils, 3 lbf initial tension) the rate is about 14.6 lbf per inch. At 0.75 in of extension the load is about 14.0 lbf and body shear is near 84,000 psi, a safety factor of about 1.6 against 45% of tensile. The hook is the real limit: bending is about 161,000 psi and side-bend torsion about 95,700 psi, which is why the same spring with a 0.05 in side radius would be a likely field failure.

#### Common mistakes

**Body:** Extension spring problems are repetitive and mostly avoidable at the design stage.

**Bullets:**

- Forgetting initial tension when computing load at a given extension.
- Specifying an overall length without stating the hook style and the loop opening.
- Designing to the body stress and skipping the hook check.
- Using a hook that carries load off-axis, which adds bending the formulas do not capture.

**Cta:** If hook styles, tolerances and quote notes are scattered across drawings and email, we can help connect your ERP and document store to a private assistant that surfaces them at quote time.

**Faq:**

#### What is initial tension in an extension spring?

**Answer:** It is the internal force that presses the coils together as they are wound. The spring will not start to extend until the applied load exceeds it. It is set by the coiling process and index, and it is toleranced on the drawing as a force at a given length.

#### Why do extension springs usually fail at the hook?

**Answer:** The hook has a sharp bend that concentrates bending and torsion stress, and it is formed by cold working, which leaves residual stress. Cyclic loads then crack the wire at the transition. A larger bend radius or a different end style is the usual fix.

#### What does the 0.4 added to body coils mean?

**Answer:** The loop ends flex slightly, so the spring behaves as if it had a bit more than its body coils. Shigley uses G divided by E, about 0.4 for steel, to add this to the body coil count when computing rate.

#### Is this valid for double-twisted or extended hook ends?

**Answer:** The body calculations apply, but the hook stress formulas are for a full round loop. Machine hooks, extended hooks and double-twisted loops have different geometries, so use the body results and confirm hook stress with your spring maker.

## Search terms

extension spring calculator, extension spring initial tension calculator, extension spring hook stress, extension spring rate formula, tension spring load calculator, extension spring design, spring hook bending stress, extension spring force at extension

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