Formula & Calculator
Bending Stress (Flexure Formula)
Calculates the maximum bending stress in a beam from the applied bending moment, distance to the outer fiber, and the section's moment of inertia.
Interpretation
Bending stress at a point in a beam cross‑section is proportional to the bending moment and the distance from the neutral axis, and inversely proportional to the moment of inertia. The maximum stress occurs at the outermost fibres. σ = M·c / I.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| sigma | Bending stress | MPa |
| M | Bending moment at the section | N.mm |
| c | Distance from the neutral axis to the outer fiber | mm |
| I | Second moment of area (moment of inertia) of the cross-section | mm4 |
What it means
The flexure formula relates bending stress to the internal bending moment in a beam. It states that σ = M·c / I, where M is the bending moment at the section, c is the distance from the neutral axis to the point of interest (usually the extreme fibre), and I is the area moment of inertia about the neutral axis. This formula assumes linear elastic material behaviour and plane sections remain plane. The neutral axis passes through the centroid of the cross‑section. The maximum bending stress is at the top and bottom surfaces (c = c_max). This formula is fundamental for designing beams to prevent failure due to bending. Engineers use it to select cross‑sectional shapes and dimensions that can safely carry loads. The bending stress distribution is linear, with compression on one side and tension on the other. The formula is also used in composite beams and in calculating deflection via the curvature relationship.
Worked example
Bending Stress – Two Examples
Real‑World| Parameter | Value |
|---|---|
| M | 2,000,000 N·mm |
| c | 50 mm |
| I | 4,000,000 mm⁴ |
| Parameter | Value |
|---|---|
| M | 10,000,000 N·mm |
| c | 100 mm |
| I | 20,000,000 mm⁴ |
Common mistakes
- Distance c: c is the distance from the neutral axis to the point of interest, not the full depth of the beam.
- Moment of inertia I: Use the second moment of area about the neutral axis – not the polar moment.
- Unit harmony: M in N·m, c in m, I in m⁴ → Pa; for MPa use N·mm and mm⁴.
- Stress sign: Tension on one side, compression on the other – be explicit about signs.
- Maximum stress: At the outermost fibre, use σmax = M / S where S = I/c (section modulus).
Applications
Bending stress, given by the flexure formula σ = M c / I, describes the stress distribution in a beam cross‑section under bending moments. This is perhaps the most important stress type in structural and mechanical engineering, as beams are ubiquitous in buildings, bridges, vehicles, and machinery. The formula allows engineers to determine the maximum stress at the outermost fibres, which dictates the required section modulus to avoid yielding or fracture. It is used in the design of floor joists, crane girders, aircraft wings, and even bicycle frames. By adjusting the moment of inertia through cross‑sectional shape (e.g., I‑beams), engineers can optimise strength and weight. Bending stress analysis is also essential for fatigue life prediction and the assessment of structural integrity under cyclic loads.
- Design of steel and concrete beams
- Aircraft wing spar and fuselage design
- Bridge girder and deck analysis
- Machine tool spindle and support beam design
- Fatigue analysis of cyclically loaded components
Frequently Asked Questions
The flexure formula is σ = M·c / I. It calculates the normal stress due to bending at any point in a beam's cross‑section. It applies to beams in pure bending (no shear) and assumes linear elastic material behaviour (Hooke's Law) and that plane sections remain plane.
- σ = bending stress (Pa, MPa, psi) – positive for tension, negative for compression.
- M = internal bending moment at the section (N·m, lb·in).
- c = perpendicular distance from the neutral axis to the point of interest (m, in).
- I = second moment of area (moment of inertia) about the neutral axis (m⁴, in⁴).
The neutral axis is the line through the cross‑section where bending stress is zero. It passes through the centroid of the area. For symmetric sections (rectangle, I‑beam), it's the centreline. For unsymmetric sections, locate the centroid using ȳ = Σ(Aᵢ·yᵢ) / ΣAᵢ. The distance c is measured from this axis.
- Using the total depth instead of c – c is from the neutral axis to the extreme fibre, not the full depth (unless the NA is at the centre and you use half the depth).
- Using the wrong I – always use I about the neutral axis, not about some other axis.
- Ignoring sign convention – positive moment causes tension on the bottom fibre (depending on the sign convention).
- Applying it beyond the elastic limit – the formula is valid only while stress is proportional to strain.
Bending stress varies linearly from zero at the neutral axis to a maximum at the top and bottom surfaces. The distribution is σ(y) = (M / I) · y, where y is the signed distance from the neutral axis. This linear variation is a key assumption of the Euler‑Bernoulli beam theory.
For a rectangle of width b and height h, I = b·h³/12 and c = h/2. Thus σ_max = M·(h/2) / (b·h³/12) = 6M / (b·h²). This simplified form is widely used in preliminary design.
The section modulus is defined as S = I / c. Then σ_max = M / S. It is a geometric property that directly indicates a beam's resistance to bending. Larger S means lower stress for a given moment. It is tabulated for standard sections (e.g., W‑beams).
For composite beams (e.g., reinforced concrete, steel‑concrete), the simple formula requires modification. The transformed‑section method converts different materials into an equivalent single material using the modular ratio (n = E₁/E₂). Then the same flexure formula applies to the transformed section.
Elastic bending assumes σ ≤ σ_yield and uses the elastic section modulus S. Plastic bending occurs when the outer fibres yield; the stress distribution becomes non‑linear, and the plastic section modulus Z is used. The plastic moment M_p = Z·σ_yield is higher than the elastic yield moment. Plastic analysis is used for ultimate strength design.
For sharply curved beams (e.g., hooks, rings), the simple straight‑beam formula is inaccurate because the neutral axis shifts from the centroid. The curved‑beam theory uses a different stress distribution: σ = M·(rₙ – r) / (A·e·r), where rₙ is the neutral axis radius and e is the eccentricity. Always check the curvature ratio (r/h) before applying the straight‑beam formula.