Formula & Calculator
Bending Stress in a Beam
Normal stress due to bending moment.
Interpretation
Bending stress in a beam: σ = M·y / I, where M is bending moment, y is distance from neutral axis, and I is second moment of area. Maximum at top and bottom fibres. Example: M=1000 N·m, y=0.05 m, I=4e-6 m⁴ → σ = 12.5 MPa.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| σ | Bending stress | Pa |
| M | Bending moment | N·m |
| y | Distance from neutral axis | m |
| I | Second moment of area | m⁴ |
What it means
The flexure formula σ = M·y / I is the fundamental equation for calculating bending stresses in beams subjected to pure bending. It assumes linear elastic material behaviour and that plane sections remain plane. The stress varies linearly from zero at the neutral axis (centroid) to a maximum at the extreme fibres (top and bottom). The bending moment M is the internal moment at the section, y is the perpendicular distance from the neutral axis to the point of interest, and I is the area moment of inertia of the cross‑section about the neutral axis. This formula is used to design beams against flexural failure, ensuring that the maximum stress does not exceed the material's yield strength. It is also the basis for selecting beam cross‑sections (I‑beams, channels, etc.) to optimise strength and weight. In practice, the section modulus Z = I / c (where c is the extreme fibre distance) is often used, giving σ_max = M / Z. The formula is a cornerstone of structural engineering and is applied in buildings, bridges, and machine components.
Worked example
Bending Stress – Two Examples
Real‑World| Parameter | Value |
|---|---|
| M | 50,000 N·m |
| y | 0.05 m |
| I | 1.25×10⁻⁴ m⁴ |
| Parameter | Value |
|---|---|
| M | 3000 N·m |
| y | 0.03 m |
| I | 0.50×10⁻⁴ m⁴ |
Common mistakes
- Distance y: y is the distance from the neutral axis to the point of interest, not the total beam depth. Maximum stress occurs at the outermost fibre (y = half depth).
- Moment of inertia I: Use the second moment of area about the bending axis – not the polar moment or about the wrong axis.
- Units: M in N·m, y in m, I in m⁴ → stress in Pa (or N/mm² if using mm units consistently).
- Sign convention: Tensile stress is positive, compressive is negative – indicate the sign clearly.
- Section modulus: For maximum stress, use σ_max = M / S where S = I/y_max (section modulus).
Applications
Bending stress in a beam is the internal resistance per unit area induced by an applied bending moment. The flexure formula σ = M·y/I relates the stress at any point to the moment (M), the distance from the neutral axis (y), and the second moment of area (I). This stress is maximum at the extreme fibres (top and bottom) and zero at the neutral axis. In structural engineering, bending stress analysis is essential for sizing beams, girders, and joists in buildings, bridges, and industrial structures. Engineers use it to select cross‑sectional shapes (I‑beams, channels, rectangular sections) that can safely carry the anticipated loads while minimising weight and cost. The formula also guides the design of crane beams, floor systems, and even aircraft wings where bending moments are significant. Proper calculation of bending stress prevents excessive deflection, yielding, and eventual fracture, ensuring the longevity and safety of the structure. By understanding this relationship, engineers can optimise material usage and achieve economical yet robust designs.
- Design of steel and concrete beams in buildings and bridges
- Sizing of crane girders and industrial support beams
- Analysis of floor joists and roof purlins
- Structural optimisation for weight and cost
- Verification of bending capacity in existing structures
Frequently Asked Questions
The bending stress formula σ = M·y / I gives the normal stress at any point in a beam cross‑section due to an internal bending moment M. y is the distance from the neutral axis, and I is the second moment of area (moment of inertia) about the neutral axis. It assumes linear elastic material and pure bending.
- σ = bending stress – Pa, MPa, or psi.
- M = internal bending moment – N·m or lb·in.
- y = distance from neutral axis to point of interest – m or in.
- I = area moment of inertia about the neutral axis – m⁴ or in⁴.
The neutral axis is the line across the cross‑section where bending stress is zero. It passes through the centroid of the cross‑sectional area. For symmetric sections (e.g., rectangle, I‑beam), it is the axis of symmetry. For unsymmetric shapes, compute the centroid using Σ(Aᵢ·yᵢ)/ΣAᵢ.
- Using the wrong y – y is measured from the neutral axis, not from the top or bottom.
- Using the wrong I – I must be about the neutral axis, not any arbitrary axis.
- Ignoring sign convention – positive moment produces tension on the bottom fibre (depending on convention).
- Applying it beyond the elastic limit – the formula is valid only while stress remains proportional to strain.
Maximum stress occurs at the extreme fibre where |y| is largest. If c is the distance from the neutral axis to the extreme fibre, then σ_max = M·c / I = M / S, where S = I/c is the section modulus. For a rectangle of width b and height h, S = bh²/6.
The section modulus S = I/c combines geometry into a single value. Then σ_max = M / S. It allows quick comparison of beam strengths: a larger S means lower stress for the same moment. Standard sections (e.g., W‑beams) have tabulated S values.
For composite beams (e.g., reinforced concrete, steel‑concrete), use the transformed section method. Convert all materials into an equivalent single material using the modular ratio n = E₁/E₂, then apply the same flexure formula to the transformed section.
- Material is homogeneous and linearly elastic (Hooke's Law).
- Plane sections remain plane (no warping).
- Pure bending (no shear force) or shear is negligible.
- Small deflections.
The stress distribution is linear: zero at the neutral axis, tensile on one side (positive y), compressive on the other (negative y). The variation is σ(y) = (M/I)·y. The maximum tensile and compressive stresses are equal in magnitude for symmetric sections.
Elastic bending (σ ≤ σ_y) uses the elastic section modulus S. When the extreme fibre yields, the stress distribution becomes non‑linear; plastic analysis uses the plastic section modulus Z, and the plastic moment M_p = Z·σ_y. Plastic design is used for ultimate strength checks.