Mechanical Engineering

Young's Modulus Calculator

Calculate Young's modulus, stress, strain, axial force, area, original length, or length change using E = σ / ε and E = F L0 / (A ΔL).

Formulas E = σ / ε · E = F L0 / (A ΔL)Reviewed Sep 8, 2026

Young's modulus, also called the elastic modulus or modulus of elasticity in axial loading, measures how strongly a material resists elastic stretching or compression. In the linear elastic range, Hooke's law gives the constitutive relation σ = E ε, so E = σ / ε. This calculator also supports the equivalent direct axial-deformation form E = F L0 / (A ΔL). Use it for small-strain elastic checks, material-test calculations, and quick consistency checks between stress, strain, force, and measured elongation.

Calculation Bench

Method

Solve for

01Stress

σ · Average normal stress used with Hooke's law in the elastic range.

02Engineering Strain

ε · Engineering strain expressed as a decimal ratio, percent, or microstrain.

Stress / Strain mode assumes one-dimensional linear elastic behavior. Solved strain is reported in percent, with microstrain shown as the alternate conversion. If the material is beyond proportional loading, a single constant Young's modulus is no longer a faithful model.

Solution

Enter the required values to calculate young's modulus.

E = σ / ε

Formula Sheet

E = \dfrac{\sigma}{\varepsilon}

\sigma = E\varepsilon

\varepsilon = \dfrac{\sigma}{E}

EYoung's Modulus
σStress
εEngineering Strain
FAxial Force
AArea
L0Original Length
ΔLLength Change

Variables & Units

Symbol	Variable	Description	Common Units
E	Young's Modulus	Elastic modulus relating normal stress to engineering strain in one-dimensional linear elasticity.	MPa, GPa, psi, ksi
σ	Stress	Average normal stress magnitude in the linear elastic range.	Pa, MPa, GPa, psi, ksi
ε	Engineering Strain	Dimensionless normal strain based on change in length divided by original length.	1, %, µε
F	Axial Force	Direct axial load magnitude used in the uniform-member deformation relation.	N, kN, lbf
A	Area	Uniform cross-sectional area carrying the axial force in the elastic member relation.	mm², cm², m², in²
L0	Original Length	Initial undeformed length of the prismatic member or gauge section.	mm, cm, m, in, ft
ΔL	Length Change	Elastic extension magnitude used in the axial deformation relation.	mm, cm, m, in

How to Use This Calculator

01Choose the method first. Use Stress / Strain when you already know normal stress and engineering strain. Use Axial Deformation when you know force, area, original length, and measured length change for a prismatic member or coupon.
02Choose which variable to solve for. The calculator will show only the inputs needed for that equation.
03Enter stress or modulus in any supported pressure units, and enter engineering strain as a decimal ratio, percent, or microstrain.
04When estimating E from test data, use the slope of the initial straight-line stress-strain region. Do not use yield strength, ultimate strength, or a point after permanent deformation starts.
05Axial Deformation mode uses positive magnitudes for force and length change. It is intended for direct elastic elongation-style calculations, not large-deformation plastic behavior.
06For lab-style axial measurements, use the original gauge length and measured elastic extension for the same load step; subtract machine or grip compliance if your test setup reports it separately.
07This page assumes the material remains in its linear elastic range. If the stress-strain response is nonlinear, the reported result is not a valid constant Young's modulus.

How the Formula Works

In one-dimensional linear elasticity, normal stress and engineering strain are proportional through Hooke's law: σ = E ε. Rearranging gives E = σ / ε, so Young's modulus is simply the slope of the linear stress-strain relation. A larger modulus means a stiffer material that strains less under the same stress.

For a uniform axial member in linear elasticity, the elongation relation is ΔL = F L0 / (A E). Rearranging gives E = F L0 / (A ΔL). This form is especially useful when you measure load and extension directly during a tensile or compression test, or when you want to estimate the force, area, or deformation of a prismatic member in the elastic range.

Worked Example 01

Young's modulus from stress and strain

Known

Stress (σ): 200 MPa
Engineering Strain (ε): 0.1%

Formula

E = σ / ε

Substitution

E = 200 MPa / 0.001

Result

E = 200 GPa

A material carrying 200 MPa of stress at 0.1 percent engineering strain has a Young's modulus of 200 GPa in the linear range.

Worked Example 02

Stress from modulus and strain

Known

Young's Modulus (E): 70 GPa
Engineering Strain (ε): 500 µε

Formula

σ = E ε

Substitution

σ = 70 GPa × 0.0005

Result

σ = 35 MPa

An aluminum-like modulus of 70 GPa combined with 500 microstrain produces 35 MPa of elastic stress.

Worked Example 03

Young's modulus from axial force and measured elongation

Known

Axial Force (F): 45 kN
Original Length (L0): 2 m
Area (A): 300 mm²
Length Change (ΔL): 1.5 mm

Formula

E = F L0 / (A ΔL)

Substitution

E = 45,000 × 2 / (0.0003 × 0.0015)

Result

E = 200 GPa

A 45 kN load elongating a 2 m specimen with 300 mm² area by 1.5 mm implies a Young's modulus of 200 GPa.

Worked Example 04

Elastic length change from force, stiffness, area, and length

Known

Axial Force (F): 20 kN
Original Length (L0): 1.2 m
Area (A): 500 mm²
Young's Modulus (E): 210 GPa

Formula

ΔL = F L0 / (E A)

Substitution

ΔL = 20,000 × 1.2 / (210,000,000,000 × 0.0005)

Result

ΔL ≈ 0.229 mm

A steel-like member with 500 mm² area and 1.2 m gauge length stretches only about 0.229 mm under a 20 kN elastic axial load.

Applications

01Estimating Young's modulus from linear stress-strain test data
02Checking elastic elongation of rods, bars, tie members, and coupons under axial load
03Relating stress, strain, and stiffness before beam deflection or more detailed structural calculations

Assumptions

01The material remains in the linear elastic range, so Hooke's law is applicable.
02Stress / Strain mode uses one-dimensional normal stress and engineering strain, not shear or true strain.
03Axial mode assumes a prismatic member or gauge section with uniform area and small elastic deformation.

Where This Model Stops

01Not for plastic deformation, nonlinear stress-strain curves, secant/tangent modulus extraction, viscoelasticity, or temperature-dependent modulus changes.
02Axial mode does not model stress concentrations, taper, varying area, multi-material members, or compatibility constraints in indeterminate systems.
03The result is a scalar elastic modulus for the chosen loading direction; anisotropic materials may have different modulus values in different directions.
04Does not validate material identification from the modulus alone. Different alloys, heat treatments, composites, and test conditions can overlap or shift apparent E values.

References

[1]
12.3 Stress, Strain, and Elastic Modulus
OpenStax University Physics Volume 1
Defines Young's modulus as tensile stress divided by tensile strain and shows the equivalent axial form using force, area, original length, and length change.
[2]
Material
MechRef, University of Illinois Urbana-Champaign
Summarizes the linear elastic stress-strain relation σ = Eε and typical modulus values for common materials.
[3]
Axial
MechRef, University of Illinois Urbana-Champaign
Gives the axial deformation relation ΔL = F L / (E A) used for the direct elastic member calculation mode.

Frequently Asked Questions

Is Young's modulus the same as modulus of elasticity?

For simple axial normal stress-strain behavior, yes. Engineers often use the terms interchangeably for E. More broadly, modulus of elasticity can refer to other elastic moduli as well, such as shear modulus or bulk modulus, which are different properties.

Why does this only work in the linear elastic range?

Because Young's modulus is the proportionality constant in Hooke's law, where stress is directly proportional to strain. Once the material response becomes nonlinear or plastic, a single constant E no longer describes the full curve accurately.

How is this different from the Stress Calculator and Strain Calculator?

Those pages calculate stress from force and area, or strain from geometry, independently. This page connects stiffness to those quantities through Hooke's law and the axial deformation relation, so it can solve for the elastic modulus itself or use E to predict elastic response.

Can I use yield stress or ultimate strength to calculate Young's modulus?

No. Young's modulus comes from the elastic slope of the stress-strain curve. Yield stress and ultimate strength describe failure or permanent-deformation behavior, not the initial elastic stiffness.

Where should typical Young's modulus values come from?

Use a material datasheet, recognized engineering handbook, or a linear-region tensile/compression test. This calculator can check consistency, but it should not be the only source used to identify a material.

Related Calculators