Engineering · Strength of materials

Bending stress in an Euler–Bernoulli beam

In elastic beam bending, longitudinal normal stress varies linearly with distance from the neutral axis.

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Formula

σ_{x} (y) = - \frac{M y}{I}

σ in Pa; M in N·m; y in m; I in m⁴

Variables & units

Variables & units
Symbol	Meaning	Unit
$σₓ$	Longitudinal stress at distance y from the neutral axis	Pa
$M$	Bending moment at the section	N·m
$y$	Signed distance from the neutral axis	m
$I$	Second moment of area about the bending axis	m⁴

LaTeX

\sigma_x(y)=-\frac{M y}{I}

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For positive bending moment M and upward y, σx=−M·y/I: compression above, zero at the neutral axis, and tension below.

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Derivation & background

The relation assumes linear elasticity, small deformation and classical beam behavior for a suitable cross-section. Sign depends on the chosen moment and y conventions; magnitude grows with distance from the neutral axis. I is the area second moment, not mass moment of inertia. Notch and section-change peaks require local analysis.

Exam blueprint

Validity range

Derivation steps

In elastic beam bending, longitudinal normal stress varies linearly with distance from the neutral axis.

1State the model assumptions and list quantities with units.
2Substitute into σ_x(y) = −My/I; convert quantities to SI first.
3For M=1.2 kN·m, y=0.05 m and I=8×10⁻⁶ m⁴, |σ|=7.5 MPa.

Rearrangements

Base equation for the target quantity

\sigma_x(y)=-\frac{M y}{I}

Preserve signs, boundary conditions and units when rearranging.

Task variant

Work through the numerical example for “Bending stress in an Euler–Bernoulli beam” and check its unit and magnitude.

For M=1.2 kN·m, y=0.05 m and I=8×10⁻⁶ m⁴, |σ|=7.5 MPa.

Common mistakes

Substituting a related quantity that shares the same symbol.

Use variable definitions and check: σ in Pa; M in N·m; y in m; I in m⁴.

Exam context

Estimate stress distribution in slender beams under bending.

Formula cluster

Strength of materials

In elastic beam bending, longitudinal normal stress varies linearly with distance from the neutral axis.

M = EIκ ≈ EI·w″

Euler–Bernoulli beam moment–curvature relation

E = σ/ε = FL₀/(A₀ΔL)

Young's modulus

M = F·r·sin(θ)

Torque

Worked example

For M=1.2 kN·m, y=0.05 m and I=8×10⁻⁶ m⁴, |σ|=7.5 MPa.

Applications

Estimate stress distribution in slender beams under bending.

How do you calculate with Bending stress in an Euler–Bernoulli beam?

Here is how to work through a typical Bending stress in an Euler–Bernoulli beam (σₓ(y) = −My/I) task step by step:

1
Task
Work through the numerical example for “Bending stress in an Euler–Bernoulli beam” and check its unit and magnitude.
Solution path
For M=1.2 kN·m, y=0.05 m and I=8×10⁻⁶ m⁴, |σ|=7.5 MPa.

Frequently asked questions about Bending stress in an Euler–Bernoulli beam

Where is bending stress greatest?+

Because |σ| is proportional to |y|, the ideal beam model has its largest stress at the outermost fibers farthest from the neutral axis. Longitudinal stress is zero at that axis. The outer-fiber location and I must refer to the same bending axis. The linear stress distribution refers to a defined neutral axis and bending axis. Check the section property I and the y-coordinate carefully. Small deformation and elastic material behavior are central assumptions; the formula does not capture notch stress or plastic redistribution. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

Why does I appear in the bending-stress formula?+

I describes how section area is distributed relative to the neutral axis. A larger I places more material away from the axis, reducing stress at a given y for the same moment. I has units m⁴; use area moment, not mass moment with units kg·m². The linear stress distribution refers to a defined neutral axis and bending axis. Check the section property I and the y-coordinate carefully. Small deformation and elastic material behavior are central assumptions; the formula does not capture notch stress or plastic redistribution. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

What changes when the moment reverses?+

Reversing moment sign swaps tension and compression sides under the same sign convention. The stress magnitude stays the same for equal |M|. State the sign convention and y direction before evaluating σ=−My/I; otherwise the magnitude may be right while the tension side is wrong. The linear stress distribution refers to a defined neutral axis and bending axis. Check the section property I and the y-coordinate carefully. Small deformation and elastic material behavior are central assumptions; the formula does not capture notch stress or plastic redistribution. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

How do I check the result and units for Bending stress in an Euler–Bernoulli beam?+

Write the quantity and unit from the variable list next to every value: σ in Pa; M in N·m; y in m; I in m⁴. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For M=1.2 kN·m, y=0.05 m and I=8×10⁻⁶ m⁴, |σ|=7.5 MPa. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The linear stress distribution refers to a defined neutral axis and bending axis. Check the section property I and the y-coordinate carefully. Small deformation and elastic material behavior are central assumptions; the formula does not capture notch stress or plastic redistribution. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

Which model limit matters for Bending stress in an Euler–Bernoulli beam?+

The background states this starting condition: The relation assumes linear elasticity, small deformation and classical beam behavior for a suitable cross-section. Sign depends on. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The linear stress distribution refers to a defined neutral axis and bending axis. Check the section property I and the y-coordinate carefully. Small deformation and elastic material behavior are central assumptions; the formula does not capture notch stress or plastic redistribution. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

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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, Unified Engineering: Materials and Structures, bending stress σ=−My/I.

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\sum_i \vec F_i=\vec 0,\quad \sum_i M_{i,O}=0

Static equilibrium conditions

\sigma_\mathrm{m} = \frac{F}{A}

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Bending stress in an Euler–Bernoulli beam

In elastic beam bending, longitudinal normal stress varies linearly with distance from the neutral axis.

AdvancedUniversity/collegeBachelor foundationsSubject specialization

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Symbol

Meaning

Unit

σₓ

Longitudinal stress at distance y from the neutral axis

M

Bending moment at the section

N·m

y

Signed distance from the neutral axis

I

Second moment of area about the bending axis

m⁴

Derivation & background

Exam blueprint

Validity range

Derivation steps

In elastic beam bending, longitudinal normal stress varies linearly with distance from the neutral axis.

1State the model assumptions and list quantities with units.
2Substitute into σ_x(y) = −My/I; convert quantities to SI first.
3For M=1.2 kN·m, y=0.05 m and I=8×10⁻⁶ m⁴, |σ|=7.5 MPa.

How do you calculate with Bending stress in an Euler–Bernoulli beam?

Here is how to work through a typical Bending stress in an Euler–Bernoulli beam (σₓ(y) = −My/I) task step by step:

Task

Work through the numerical example for “Bending stress in an Euler–Bernoulli beam” and check its unit and magnitude.

Solution path

For M=1.2 kN·m, y=0.05 m and I=8×10⁻⁶ m⁴, |σ|=7.5 MPa.

Frequently asked questions about Bending stress in an Euler–Bernoulli beam

Where is bending stress greatest?+

Why does I appear in the bending-stress formula?+

What changes when the moment reverses?+

How do I check the result and units for Bending stress in an Euler–Bernoulli beam?+

Which model limit matters for Bending stress in an Euler–Bernoulli beam?+

Retain Bending stress in an Euler–Bernoulli beam for exams

Create a curated FSRS exam set for σₓ(y) = −My/I: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

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