Beam Bending Stress Equation
Calculates the normal bending stress in a straight beam subjected to transverse loading and bending moments.
Primary Mathematical Expression
Design Schematic
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| M | Bending moment | N·m | in·lb | The applied bending moment acting on the cross-section. |
| y | Distance to neutral axis | mm | in | Distance from neutral axis to the point of interest. |
| I | Area moment of inertia | mm⁴ | in⁴ | The geometric stiffness property of the cross-section. |
| \sigma | Bending stress | MPa | psi | The resulting normal stress at distance y. |
Step-by-Step Derivation
- 1
Consider a straight beam subjected to pure bending moments. The bending moment causes deformation where top fibers contract and bottom fibers stretch.
- 2
The axial strain at any fiber is proportional to its distance from the neutral surface: \epsilon = -y / \rho, where \rho is the radius of curvature.
- 3
Assuming elastic behavior, Hooke's Law applies: \sigma = E * \epsilon = -E * y / \rho.
- 4
The internal bending moment must balance the external moment: M = \int -\sigma * y * dA = \int (E * y² / \rho) * dA = (E / \rho) * I.
- 5
Substituting E / \rho = M / I back into the stress equation yields the bending stress formula: \sigma = M * y / I.
Worked Example Calculation
A rectangular steel beam undergoes a bending moment of 1200 N·m. The distance from the neutral axis to the outer fiber is 25 mm, and the area moment of inertia of the cross-section is 1.6 \times 10^5 mm⁴. Calculate the maximum bending stress.
- •Identify input parameters: M = 1200 N·m = 1,200,000 N·mm, y = 25 mm, I = 160,000 mm⁴.
- •Apply the Bending Stress Formula: \sigma = M * y / I.
- •Compute: \sigma = (1,200,000 * 25) / 160,000 = 30,000,000 / 160,000 = 187.5 MPa.
Engineering Assumptions
- •The beam material is homogeneous, isotropic, and obeys Hooke's Law.
- •The beam is straight with a uniform cross-section.
- •Plane sections remain plane after bending (Bernoulli-Euler hypothesis).
- •The loading is applied in the plane of symmetry.
Design Limitations
- •Not applicable for short, deep beams where shear deformation is significant (requires Timoshenko beam theory).
- •Not applicable beyond the material's elastic limit (yield strength).
- •Does not account for stress concentration zones near holes, fillets, or notches.
Academic References & Standards
Standard Specification for Carbon Structural Steel
The standard detailing carbon steel structural shapes.
Shigley's Mechanical Engineering Design, 11th Edition
Textbook covering bending and normal stresses in mechanical components.