Flexure Formula (Bending Stress)

Core idea

Overview

This formula assumes the beam material is linear-elastic, isotropic, and homogeneous, with a cross-section symmetric about the plane of bending. It relates the internal moment to the stress distribution across the depth of the member, showing that stress varies linearly with the distance from the neutral axis. The negative sign is a convention indicating that a positive moment causes compression on the top fibers of a simply supported beam.

When to use: Use this to determine the internal normal stress in a beam subjected to pure bending or bending combined with other loads.

Why it matters: It is fundamental for structural safety, ensuring that the induced bending stress does not exceed the yield strength or allowable stress of the material.

Symbols

Variables

sigma = Bending Stress, M = Bending Moment, y = Distance from Neutral Axis, I = Moment of Inertia

sigma

Bending Stress

Variable

M

Bending Moment

Variable

y

Distance from Neutral Axis

Variable

I

Moment of Inertia

Variable

Walkthrough

Derivation

Derivation of Flexure Formula (Bending Stress)

This derivation relates the internal bending moment of a beam to the internal normal stress by enforcing geometric compatibility (linear strain) and constitutive behavior (Hooke's Law).

The beam is initially straight and prismatic.
The material is linear-elastic, homogeneous, and isotropic.
Plane sections remain plane and perpendicular to the longitudinal axis after bending (Bernoulli-Euler hypothesis).
The beam is subjected to pure bending.

1

Kinematic Relation (Strain)

Assuming a radius of curvature $ρ$ , the longitudinal strain $ϵ_{x}$ varies linearly with the distance $y$ from the neutral axis.

ϵ_{x} = - \frac{y}{ρ}

Note: The negative sign indicates that for positive bending (concave up), fibers above the neutral axis are in compression.

2

Constitutive Relation (Hooke's Law)

By applying Hooke's Law ( $σ = E ϵ$ ), we express the stress as a function of the elastic modulus $E$ and curvature.

σ_{x} = E ϵ_{x} = - \frac{E y}{ρ}

Note: This assumes the material is within the linear elastic range.

3

Moment Equilibrium

The internal moment $M$ is the integral of the moment generated by the stress distribution over the cross-sectional area $A$ .

M = \int_{A} - y σ_{x} d A = \int_{A} - y (- \frac{E y}{ρ}) d A = \frac{E}{ρ} \int_{A} y^{2} d A

Note: The integral $\int y^{2} d A$ is defined as the area moment of inertia $I$ .

4

Relating Moment and Curvature

We replace the integral with $I$ to solve for the curvature term $1/ ρ$ in terms of the applied moment.

M = \frac{E I}{ρ} ⟹ \frac{1}{ρ} = \frac{M}{E I}

Note: The term $E I$ is known as the flexural rigidity of the beam.

5

Final Flexure Formula

Substitute the curvature expression back into the stress equation to obtain the final formula.

σ = - \frac{E y}{E I} \cdot \frac{M}{1} = - \frac{M y}{I}

Note: Always ensure units are consistent (e.g., N/mm² for MPa).

Result

σ = - \frac{E y}{E I} \cdot \frac{M}{1} = - \frac{M y}{I}

Source: Beer, F. P., Johnston, E. R., DeWolf, J. T., & Mazurek, D. F. (2015). Mechanics of Materials.

Free formulas

Rearrangements

Solve for $σ$

Make $σ$ the subject

σ = - \frac{M y}{I}

The formula is already expressed with $σ$ as the subject.

Difficulty: 1/5

Solve for $M$

Make M the subject

M = - \frac{σ I}{y}

Rearrange the equation to isolate the bending moment M by multiplying both sides by I and dividing by negative y.

Difficulty: 2/5

Solve for $I$

Make I the subject

I = - \frac{M y}{σ}

Rearrange to solve for the moment of inertia I by multiplying by I and dividing by sigma.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Imagine bending a thick rubber eraser. As you bend it, the outer side stretches (tension) and the inner side compresses. The neutral axis (the center plane) remains unstretched. The equation describes this as a linear 'ramp' of stress: the further you move away from the center (y), the more the material must stretch or squash to accommodate the bend, with the slope of this ramp determined by the moment (M) and the resistance of the shape (I).

s

Bending Stress

The internal 'push' or 'pull' force per unit area acting on the material fibers at a specific location.

M

Bending Moment

The 'twisting' force applied to the beam; a larger M creates a more intense internal struggle between tension and compression.

y

Centroidal Distance

The 'leverage arm'; how far you are from the center line where no stress exists.

I

Area Moment of Inertia

The geometric 'stiffness'; it measures how efficiently the shape distributes material away from the center to resist bending.

Signs and relationships

Negative sign (-): This is a sign convention: it ensures that for a positive bending moment (causing curvature concave upward), points above the neutral axis (positive y) result in negative stress (compression), while points below (negative y) result in positive stress (tension).

One free problem

Practice Problem

A beam has a moment of inertia I = 5000 cm^4 and is subjected to a bending moment M = 10 kN-m. Calculate the bending stress at a point 10 cm from the neutral axis.

Bending Moment10000000

Distance from Neutral Axis100

Moment of Inertia50000000

Solve for: sigma

Hint: Convert all units to Newtons and millimeters to maintain consistency (N/mm^2 = MPa).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In the stress at the top and bottom edges of a steel I-beam supporting a bridge deck to ensure the steel does not yield under traffic loads, Flexure Formula (Bending Stress) is used to calculate Bending Stress from Bending Moment and Distance from Neutral Axis. The result matters because it helps estimate likelihood and make a risk or decision statement rather than treating the number as certainty.

Study smarter

Tips

Ensure the distance 'y' is measured from the centroidal neutral axis of the cross-section.
Check that units for M, y, and I are consistent (usually N, mm, and mm^4).
Remember that maximum stress occurs at the outermost fibers (maximum 'y').

Avoid these traps

Common Mistakes

Using the wrong Moment of Inertia (I) for the specific axis of bending.
Confusing the distance from the outer surface with the distance from the neutral axis.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation relates the internal bending moment of a beam to the internal normal stress by enforcing geometric compatibility (linear strain) and constitutive behavior (Hooke's Law).

Use this to determine the internal normal stress in a beam subjected to pure bending or bending combined with other loads.

It is fundamental for structural safety, ensuring that the induced bending stress does not exceed the yield strength or allowable stress of the material.

Using the wrong Moment of Inertia (I) for the specific axis of bending. Confusing the distance from the outer surface with the distance from the neutral axis.

In the stress at the top and bottom edges of a steel I-beam supporting a bridge deck to ensure the steel does not yield under traffic loads, Flexure Formula (Bending Stress) is used to calculate Bending Stress from Bending Moment and Distance from Neutral Axis. The result matters because it helps estimate likelihood and make a risk or decision statement rather than treating the number as certainty.

Ensure the distance 'y' is measured from the centroidal neutral axis of the cross-section. Check that units for M, y, and I are consistent (usually N, mm, and mm^4). Remember that maximum stress occurs at the outermost fibers (maximum 'y').

References

Sources

Hibbeler, R. C. (2017). Mechanics of Materials.
Beer, F. P., Johnston, E. R., DeWolf, J. T., & Mazurek, D. F. (2014). Mechanics of Materials.
Beer, F. P., Johnston, E. R., DeWolf, J. T., & Mazurek, D. F. (2015). Mechanics of Materials.

Overview

Variables

Derivation

Kinematic Relation (Strain)

Constitutive Relation (Hooke's Law)

Moment Equilibrium

Relating Moment and Curvature

Final Flexure Formula

Rearrangements

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Section Modulus

Frequently Asked Questions

Sources