Engineering · Structural mechanics

Euler–Bernoulli beam moment–curvature relation

Linear Euler–Bernoulli beam theory relates bending moment to beam curvature through flexural rigidity EI.

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Formula

M (x) = E I κ (x) \approx E I \frac{d ^{2} w}{d x ^{2}}

M in N·m; E in Pa; I in m⁴; κ in 1/m; w in m

Variables & units

Variables & units
Symbol	Meaning	Unit
$M(x)$	Bending moment along the beam	N·m
$E$	Young’s modulus	Pa
$I$	Second moment of area about the bending axis	m⁴
$κ(x)$	Curvature of the deflected beam axis	1/m
$w(x)$	Transverse deflection along coordinate x	m

LaTeX

M(x)=EI\,\kappa(x)\approx EI\frac{d^2w}{dx^2}

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Derivation & background

For small slopes, curvature is approximately the second derivative of deflection w″; the sign depends on the selected moment and deflection convention. EI is flexural rigidity. The theory assumes a slender beam, small deformation, linear elasticity and negligible transverse shear deformation. Short or thick beams, large deflection, composite sections and anisotropic materials may need Timoshenko or more advanced models. Boundary conditions and loading are needed to integrate a deflection curve.

Exam blueprint

Validity range

Applies to slender beams, small deformation, linear elasticity and negligible transverse shear.

Derivation steps

Integrating the linear bending stress distribution over the section yields M=EIκ.

1Strain increases with distance from the neutral axis.
2Hooke’s law relates strain and stress.
3Section integration yields flexural rigidity EI.

Rearrangements

Curvature

kappa=M/(EI)

I is the area moment of inertia about the bending axis.

Task variant

Given E=200 GPa, I=8×10⁻⁶ m⁴, M=1.6 kN·m, compute κ.

κ=1600/(200×10⁹×8×10⁻⁶)=0.001 m⁻¹.

Common mistakes

Confusing I with mass moment of inertia.

I describes section geometry and has units m⁴.

Exam context

Bending moment, curvature and deflection.

Formula cluster

Beam stiffness

E describes material and I the cross-section.

E = σ/ε = FL₀/(A₀ΔL)

Young's modulus

F = k·x

Hooke's Law (Spring Force)

M = F·r·sin(θ)

Torque

Worked example

For E=200 GPa, I=8×10⁻⁶ m⁴ and M=1.6 kN·m, curvature magnitude is M/(EI)=0.001 m⁻¹.

Applications

Model beams and girders under transverse loading and estimate stiffness and deflection.

How do you calculate with Euler–Bernoulli beam moment–curvature relation?

Here is how to work through a typical Euler–Bernoulli beam moment–curvature relation (M = EIκ ≈ EI·w″) task step by step:

1
Task
Given E=200 GPa, I=8×10⁻⁶ m⁴, M=1.6 kN·m, compute κ.
Solution path
κ=1600/(200×10⁹×8×10⁻⁶)=0.001 m⁻¹.

Frequently asked questions about Euler–Bernoulli beam moment–curvature relation

What does EI mean for a beam?+

E is Young’s modulus of the material; I is the area moment of the cross-section about the bending axis. Their product EI is flexural rigidity. A larger E or I produces less curvature for the same moment. Material and section shape therefore affect response together but in different ways. I is the geometric area moment of inertia about the bending axis, not mass moment of inertia. The relation gives local curvature from moment, but not a full deflection without loads and boundary conditions. Small deformation, slender geometry and linear elasticity are key assumptions. Check the sign convention used in the structural problem.

How do I find curvature from a bending moment?+

Rearrange M=EIκ to κ=M/(EI). Use M in N·m, E in Pa and I in m⁴; the result has units 1/m. For M=1.6 kN·m, E=200 GPa and I=8×10⁻⁶ m⁴, κ=0.001 m⁻¹. Sign follows the selected convention. I is the geometric area moment of inertia about the bending axis, not mass moment of inertia. The relation gives local curvature from moment, but not a full deflection without loads and boundary conditions. Small deformation, slender geometry and linear elasticity are key assumptions. Check the sign convention used in the structural problem.

Why is deflection not directly known from M=EIκ?+

The equation relates local moment to local curvature. To obtain deflection w(x), integrate the curvature relation and determine the moment field and integration constants. Loads and supports provide boundary conditions. Without them, many beam shapes can satisfy the same local moment–curvature relation. I is the geometric area moment of inertia about the bending axis, not mass moment of inertia. The relation gives local curvature from moment, but not a full deflection without loads and boundary conditions. Small deformation, slender geometry and linear elasticity are key assumptions. Check the sign convention used in the structural problem.

When is Euler–Bernoulli theory unsuitable?+

In short or deep beams, transverse shear may contribute significantly, although Euler–Bernoulli neglects it. Large deflection, plastic behavior or composite and anisotropic sections can also violate the linear theory. Timoshenko theory can suit shear-flexible beams; nonlinear cases need suitable models. I is the geometric area moment of inertia about the bending axis, not mass moment of inertia. The relation gives local curvature from moment, but not a full deflection without loads and boundary conditions. Small deformation, slender geometry and linear elasticity are key assumptions. Check the sign convention used in the structural problem.

How does cross-section shape affect stiffness?+

I depends on how far section area lies from the neutral axis. Material farther from the axis contributes strongly because distance is squared. A deeper beam can therefore be much stiffer in bending than a shallow one with similar material area. Specify the axis and section geometry when computing I. I is the geometric area moment of inertia about the bending axis, not mass moment of inertia. The relation gives local curvature from moment, but not a full deflection without loads and boundary conditions. Small deformation, slender geometry and linear elasticity are key assumptions. Check the sign convention used in the structural problem.

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]MIT OpenCourseWare, 6.777J Design and Fabrication of Microelectromechanical Devices, Lecture 7: “Curvature in Pure Bending”.

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Quanta Study editorial team

Symbol

Meaning

Unit

M(x)

Bending moment along the beam

N·m

E

Young’s modulus

I

Second moment of area about the bending axis

m⁴

κ(x)

Curvature of the deflected beam axis

1/m

w(x)

Transverse deflection along coordinate x

Derivation & background

Exam blueprint

Validity range

Applies to slender beams, small deformation, linear elasticity and negligible transverse shear.

Derivation steps

Integrating the linear bending stress distribution over the section yields M=EIκ.

1Strain increases with distance from the neutral axis.
2Hooke’s law relates strain and stress.
3Section integration yields flexural rigidity EI.

Frequently asked questions about Euler–Bernoulli beam moment–curvature relation

What does EI mean for a beam?+

How do I find curvature from a bending moment?+

Why is deflection not directly known from M=EIκ?+

When is Euler–Bernoulli theory unsuitable?+

How does cross-section shape affect stiffness?+

Retain Euler–Bernoulli beam moment–curvature relation for exams

Create a curated FSRS exam set for M = EIκ ≈ EI·w″: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential