Beam Bending Stress Formula (Flexure Equation σ = My/I)
Calculate normal flexural and maximum beam bending stress (σ = My/I = M/S) in beams, shafts, and structural shapes per Euler-Bernoulli beam theory.
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Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| M | Bending moment | N·m | in·lb | The internal bending moment load acting on the cross-section. |
| y | Distance to neutral axis | mm | in | Perpendicular distance from the neutral axis to the stress evaluation fiber. |
| c | Extreme fiber distance | mm | in | Distance from neutral axis to the outermost surface fiber (y_max). |
| I | Second moment of area | mm⁴ | in⁴ | The cross-sectional moment of inertia resisting flexural bending. |
| S | Elastic section modulus | mm³ | in³ | Geometric ratio S = I / c defining peak flexural capacity. |
| \sigma | Flexural bending stress | MPa | psi | The resulting normal tensile or compressive stress at distance y. |
Step-by-Step Derivation
- 1
Under transverse loading, a straight beam deflects into a curve with radius of curvature \rho.
- 2
Based on the Euler-Bernoulli hypothesis, plane cross-sections remain plane and perpendicular to the longitudinal axis after bending.
- 3
Longitudinal normal strain varies linearly with distance y from the neutral axis: \epsilon(y) = -y / \rho.
- 4
For linear elastic, isotropic materials obeying Hooke's Law: \sigma(y) = E * \epsilon(y) = -E * y / \rho.
- 5
Moment equilibrium requires the internal resisting moment to balance the applied moment M: M = \int -\sigma * y * dA = \int (E * y² / \rho) * dA = (E / \rho) * I.
- 6
Solving for curvature (E / \rho = M / I) and substituting back into the stress equation yields the classic flexure formula: \sigma = M * y / I.
- 7
At the extreme outer fiber (y = c), the maximum bending stress simplifies to \sigma_{max} = M * c / I = M / S, where S = I / c is the elastic section modulus.
Worked Example Calculation
A structural rectangular steel beam (width $b = 50\text{ mm}$, height $h = 100\text{ mm}$) supports an applied bending moment $M = 3,500\text{ N}\cdot\text{m}$. Calculate the area moment of inertia, the section modulus, and the maximum flexural bending stress.
- •Identify input dimensions: b = 50 mm, h = 100 mm, M = 3,500 N·m = 3,500,000 N·mm, c = h / 2 = 50 mm.
- •Calculate second moment of area (I): I = (b * h³) / 12 = (50 * 100³) / 12 = 50,000,000 / 12 ≈ 4,166,667 mm⁴.
- •Calculate elastic section modulus (S): S = (b * h²) / 6 = (50 * 100²) / 6 = 500,000 / 6 ≈ 83,333 mm³.
- •Apply the Flexure Formula: \sigma_{max} = M / S = 3,500,000 N·mm / 83,333 mm³ = 42.0 MPa.
- •Stress distribution: Outer top fibers undergo 42.0 MPa compression, bottom fibers undergo 42.0 MPa tension, and stress at the neutral axis (y = 0) is 0 MPa.
Engineering Assumptions
- •The beam is straight, prismatic (constant cross-section), and subjected to pure bending in a principal plane of symmetry.
- •Plane sections before bending remain planar and perpendicular to the neutral axis (Euler-Bernoulli hypothesis).
- •The material is homogeneous, isotropic, and operates strictly within its linear elastic regime (below yield strength $S_y$).
- •Deflections and slopes are small such that geometric nonlinearities are negligible.
Design Limitations
- •Short, deep beams ($L/h < 5$) require Timoshenko beam theory to account for transverse shear deformation.
- •Does not apply to inelastic, plastic hinge formation (use plastic section modulus $Z$ for AISC limit-state design).
- •Local geometric discontinuities (fillets, holes, notches) cause localized stress concentrations requiring fatigue notch factors ($K_t, K_f$).
- •For rotating shafts under combined bending and torque, use the ASME B106.1M combined stress equations.
Academic References & Standards
Shigley's Mechanical Engineering Design, 11th Edition
Flexural stress, beam deflections, and transverse shear equations.
Roark's Formulas for Stress and Strain, 9th Edition
Formulas for beams under transverse bending and point loads.
Specification for Structural Steel Buildings
Allowable bending stress and section properties for structural shapes.
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