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Stress and Strain Calculator

Find the stress, strain and stiffness (Young's modulus) of a bar from a pull test, or how far a bar stretches under a load.

Level: Grades 9–10

  • Common Core standard HSA.CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  • Common Core standard HSN.Q.A.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Printable worksheet

Stress and Strain Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Find
Cross-section

How much the piece stretched (or shortened) when the force was on it.

Result

Young's modulus (E)

199.8806 GPa

Stress 127.324 MPa, strain 0.000637, E = 199.8806 GPa.

Stress (σ)
127.324 MPa
Strain (ε)
0.000637
Strain as a percentage
0.0637%
Microstrain
637 µε
Cross-sectional area
78.5398 mm²
Young's modulus in psi
28,990,233 psi

See the step-by-step solution ↓

Step-by-step solution

  1. Change the force to N, the standard unit the formula uses (1 kN = 1,000 N)

    10 kN×1,000=10,000 N

  2. Change the diameter to m, the standard unit the formula uses (1 mm = 0.001 m)

    10 mm×0.001=0.01 m

  3. Find the area of the bar's round end (A) from its diameter (d), because stress spreads over that area

    3.14159265×0.0124=0.0000785398 m²

  4. Divide the force by the area to get the stress, the load each square meter of the bar carries

    10,0000.0000785398=127,323,954.473516 Pa

  5. Change the change in length to m, the standard unit the formula uses (1 mm = 0.001 m)

    0.637 mm×0.001=0.000637 m

  6. Divide the stretch by the original length to get the strain, the stretch per meter of bar

    0.0006371=0.000637

  7. Divide the stress by the strain to get Young's modulus, a measure of how stiff the material is

    127,323,954.4735160.000637=199,880,619,267.68643 Pa

  8. Convert to GPa, the unit you chose to show

    199,880,619,267.686431,000,000,000=199.880619 GPa

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how hard a bar is being pulled, how much it stretches, and how stiff its material is. Pull on a steel rod and it stretches a tiny amount. Engineers describe that with two ratios. Stress is the force spread over the bar's cross-section (the area of its cut end). That way it doesn't matter whether the rod is thick or thin. Strain is the stretch divided by the original length. That way it doesn't matter whether the rod is long or short. While the bar still springs back fully (its elastic range), the two rise together in step. The number linking them is Young's modulus, E, a measure of how stiff the material is.

This stress and strain calculator works both ways. Give it a test result: a force, the bar's size, its length and the measured stretch. It finds the stress, strain and Young's modulus. Or give it a material's modulus, and it predicts how far a bar will stretch under a load. It's a learning and estimating tool for straight bars that are pulled (tension) or squeezed (compression) along their length. Designing a structural member or a load-bearing part needs an engineer.

Where it's used

Why this math matters outside the classroom:

  • Materials lab. Students pull a 10 mm rod with 10 kN, measure 0.637 mm of stretch over 1 m and find E ≈ 199.88 GPa.
  • Guitar strings. A builder estimates how much a steel string stretches as it is tightened to pitch, from F × L ÷ (A × E).
  • Hanging loads. A rigger checks that a 1 m, 10 mm steel rod holding 10 kN stretches only about 0.64 mm.
  • Choosing materials. An engineer sees an aluminum rod stretches 200 ÷ 70, about 2.9 times, as much as an identical steel rod.
  • Strain gauges. A technician reads 637 microstrain on a steel bar and converts it to about 127 MPa of stress with σ = E × ε.

Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSA.CED.A.4, HSN.Q.A.1. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose what to find: Young's modulus from a test, or the stretch from a known modulus.
  2. Enter the axial force (the pull or push along the bar's length). Then enter the cross-section: a round bar's diameter, or the area itself.
  3. Enter the original length. Then enter either the measured change in length or the material.
  4. Pick the units you want to see. Read the stress, strain and the modulus or stretch.

What the inputs mean

Original length (L₀)
Engineers call this the gauge length. It's the original length of the part being measured, before the load went on.
Measured change in length (ΔL)
Enter a positive number for stretch when the bar is pulled. The same formulas work for shortening when it's squeezed, as long as the bar can't buckle (bow sideways). Enter the amount it shortened.
Material (Young's modulus E)
The presets are typical values from a university physics reference: steel 200 GPa, aluminum 70 GPa, glass 65 GPa, concrete 30 GPa, wood 13 GPa and polystyrene 3 GPa. GPa (gigapascals) is the unit of stiffness here. Real values vary by alloy, grade, moisture and the direction of the grain or fibers.

Formula and method

σ = F ÷ A
ε = ΔL ÷ L₀
E = σ ÷ ε = (F × L₀) ÷ (A × ΔL)
ΔL = F × L₀ ÷ (A × E)
A = π × d² ÷ 4   (round bar)

In the formulas, F is the force, A is the cross-sectional area and d is a round bar's diameter. L₀ is the original length and ΔL is the change in length. σ (sigma) is stress, ε (epsilon) is strain and E is Young's modulus.

Stress has the same units as pressure: N/m², called pascals (Pa), or psi (pounds per square inch). Strain is one length divided by another, so it has no units. It's often written as a percentage or in microstrain (millionths).

Hooke's law for materials says stress rises in step with strain while the bar still springs back: σ = E × ε. Rearranging it gives E from a test, or the stretch from a known E. The calculator converts your inputs to metric (SI) units first.

Worked example

A rod 10 mm across and 1 m long stretches 0.637 mm under a 10 kN pull.

  1. Area: π × 10² ÷ 4 = 78.54 mm².
  2. Stress: 10,000 N ÷ 78.54 mm² = 127.32 MPa.
  3. Strain: 0.637 mm ÷ 1,000 mm = 0.000637 (637 microstrain).
  4. Young's modulus: 127.32 MPa ÷ 0.000637 = 199,881 MPa, which is 199.88 GPa.

Result: E ≈ 199.88 GPa, which matches steel's typical 200 GPa.

Understanding the result

A modulus near 200 GPa points to steel, and one near 70 GPa points to aluminum. A result far from what you expected usually has one of three causes. It may be a measuring error, a load past the elastic range, or the bar slipping in the grips.

Stress, not force, decides whether a part yields (bends or stretches for good). Compare the stress with the material's yield strength from its specification. Once stress passes yield, strain no longer rises in step, and these formulas stop applying.

Assumptions and limitations

  • The bar is straight and the same size along its length. It's loaded along its center line, with the stress spread evenly over the cut end.
  • The material behaves in a straight-line, spring-back way (below its proportional limit). Stress and strain are based on the bar's original size, which engineers call engineering stress and strain.
  • It doesn't cover yielding or necking (the bar thinning before it breaks). It also leaves out true stress and strain, temperature effects, and the extra stress around holes and threads.
  • A bar being squeezed can buckle long before it reaches these stresses. That needs a separate check.
  • For learning and estimating only. Structural and load-bearing parts must be designed or approved by a qualified engineer.

Frequently asked questions

What is the difference between stress and strain?

Stress is force per area: how hard the material inside is being loaded. Strain is the change in length as a fraction of the original: how much it deforms as a result. Stress causes strain, and Young's modulus connects them.

How do I calculate Young's modulus?

Divide stress by strain: E = (F ÷ A) ÷ (ΔL ÷ L₀). For example, put a 10 kN load on a 78.54 mm² rod that stretches 0.637 mm over 1 m. That gives 127.32 MPa ÷ 0.000637 = 199.88 GPa.

How much does a steel rod stretch under load?

Use ΔL = F × L ÷ (A × E). Take a steel rod 10 mm across and 1 m long, under 10 kN. It stretches 10,000 × 1 ÷ (0.00007854 × 200,000,000,000) = 0.637 mm.

Why is strain unitless?

It's one length divided by another, so the units cancel out. A strain of 0.001 means 1 mm per meter, or 0.001 in per inch. That's 0.1%, or 1,000 microstrain.

Sources

How this calculator is checked

Method: Definitions of engineering stress (F/A) and strain (ΔL/L₀) and Hooke's law σ = Eε; typical E values from HyperPhysics (Georgia State University). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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