Beam bending stress - Study Guide | Wiki

Core idea

Overview

The flexure formula defines the relationship between the internal bending moment and the resulting normal stress distributed across a beam's cross-section. It assumes the beam is composed of a linear-elastic material where plane sections remain plane and perpendicular to the neutral axis after deformation.

When to use: This formula is applied to straight beams subjected to pure bending within their elastic range, where the material is considered homogeneous. It is used to determine the stress at any point located at a vertical distance y from the neutral axis, specifically for symmetric cross-sections.

Why it matters: This equation is the foundation for designing structural members like floor joists and bridge girders, ensuring they can withstand loads without permanent deformation. By calculating stress peaks, engineers can select appropriate materials and cross-sectional shapes, such as I-beams, to optimize weight and safety.

Remember it

Memory Aid

Phrase: Stress Makes You Intense.

Visual Analogy: Bending a thick rubber eraser: the outer edges stretch and compress the most (high y), while the center line remains unchanged (neutral axis).

Exam Tip: Always measure 'y' from the neutral axis, not the base. To find maximum stress, use the distance to the furthest outer fiber.

Why it makes sense

Intuition

Imagine a straight beam bending like a flexible ruler under a load. The top surface shortens (compresses), the bottom surface lengthens (stretches), and somewhere in between, there's an imaginary line (the neutral axis)

Symbols

Variables

M = Bending Moment, y = Distance from Neutral Axis, I = Second Moment of Area, \sigma = Bending Stress

M

Bending Moment

N \cdot m

y

Distance from Neutral Axis

m

I

Second Moment of Area

m^{4}

σ

Bending Stress

P a

Walkthrough

Derivation

Derivation of Beam Bending Stress

The flexure formula gives the normal stress in a beam under a bending moment and shows stress varies linearly with distance from the neutral axis.

The beam is initially straight and has a uniform cross-section.
The material is homogeneous, isotropic, and linearly elastic (Hooke's Law).
Plane sections remain plane after bending (Euler–Bernoulli hypothesis).
Deflections and slopes are small (geometric linearity).

1

Relate strain to curvature:

From beam geometry, fibres a distance y from the neutral axis experience longitudinal strain proportional to curvature 1/ $ρ$ .

ε = - \frac{y}{ρ}

2

Apply Hooke's Law:

In linear elasticity, normal stress is proportional to normal strain with Young’s modulus E.

σ = E ε = - \frac{E y}{ρ}

3

Relate internal bending moment to the stress distribution:

The internal moment is the resultant of stress times lever arm y over the cross-sectional area A (sign depends on convention).

M = \int_{A} σ y d A = \int_{A} (- \frac{E y}{ρ}) y d A

4

Introduce the second moment of area:

The integral $\int_{A}$ $y^{2}$ dA is the second moment of area I about the neutral axis.

I = \int_{A} y^{2} d A \Rightarrow M = - \frac{E}{ρ} I

5

Eliminate curvature to obtain the flexure formula:

Combine $σ$ = -Ey/ $ρ$ with M = -(E/ $ρ$ )I to remove $ρ$ and relate stress directly to bending moment.

σ = - \frac{M y}{I}

Note: Magnitude: | $σ$ | = $\frac{∣ M ∣ ∣ y ∣}{I}$ . Maximum stress occurs at the outermost fibre where |y| is largest.

Result

σ = - \frac{M y}{I}

Source: Mechanics of Materials — Hibbeler, Chapter 6

Where it shows up

Real-World Context

Analyzing stress in a bridge girder.

Avoid these traps

Common Mistakes

Using diameter instead of radius for y (if circular).
Wrong units for I ( $m^{4}$ ).

Study smarter

Tips

Identify the neutral axis where stress is zero before measuring the distance y.
The maximum stress occurs at the outermost fibers, where y is at its maximum value.
Ensure all units for moment, distance, and inertia are consistent, typically using meters and Newtons.

Common questions

Frequently Asked Questions

The flexure formula gives the normal stress in a beam under a bending moment and shows stress varies linearly with distance from the neutral axis.

This formula is applied to straight beams subjected to pure bending within their elastic range, where the material is considered homogeneous. It is used to determine the stress at any point located at a vertical distance y from the neutral axis, specifically for symmetric cross-sections.

This equation is the foundation for designing structural members like floor joists and bridge girders, ensuring they can withstand loads without permanent deformation. By calculating stress peaks, engineers can select appropriate materials and cross-sectional shapes, such as I-beams, to optimize weight and safety.

Using diameter instead of radius for y (if circular). Wrong units for I (m^4).

Analyzing stress in a bridge girder.

Identify the neutral axis where stress is zero before measuring the distance y. The maximum stress occurs at the outermost fibers, where y is at its maximum value. Ensure all units for moment, distance, and inertia are consistent, typically using meters and Newtons.