Bending Moment Formula
Learn the bending moment formula used in structural engineering to calculate internal forces in beams under load.
The Formula
The bending moment formula, also known as the flexure formula, describes how internal stresses develop inside a beam when it bends under load. This equation is one of the most important relationships in structural and mechanical engineering. It connects the bending moment acting on a beam cross-section to the stress experienced at any point within that cross-section.
When a load is applied to a beam — whether it is the weight of a person on a bridge, books on a shelf, or snow on a roof — the beam curves. The material on the outer edge of the curve gets stretched (tension), while the material on the inner edge gets compressed (compression). Somewhere in between, there is a layer that experiences zero stress. This is called the neutral axis.
The stress at any point depends on three things: how large the bending moment is at that cross-section, how far the point is from the neutral axis, and the beam's moment of inertia (a geometric property that measures how well the cross-section resists bending). A beam with a larger moment of inertia — like an I-beam compared to a flat plate — can handle much greater loads before the stress becomes dangerous.
Structural engineers use this formula every day when designing buildings, bridges, aircraft wings, and machine components. The maximum bending stress always occurs at the point farthest from the neutral axis. If this maximum stress exceeds the material's yield strength, the beam will permanently deform. If it exceeds the ultimate strength, the beam will fracture. This is why engineers always ensure the calculated bending stress stays well below the material's capacity, applying a factor of safety for additional protection.
Variables
| Symbol | Meaning |
|---|---|
| σ (sigma) | Bending stress at the point of interest (Pa or psi) |
| M | Bending moment at the cross-section (N·m or lb·ft) |
| y | Distance from the neutral axis to the point of interest (m or in) |
| I | Second moment of area (moment of inertia) of the cross-section (m⁴ or in⁴) |
Example 1: Simply Supported Beam with Center Load
Problem: A rectangular steel beam (width 50 mm, height 100 mm) has a maximum bending moment of 5,000 N·m. Find the maximum bending stress.
The neutral axis is at the center, so y_max = 100/2 = 50 mm = 0.05 m.
Moment of inertia for a rectangle: I = bh³/12 = (0.05)(0.1)³/12 = 4.167 × 10⁻⁶ m⁴
σ = M × y / I = 5000 × 0.05 / (4.167 × 10⁻⁶)
σ = 60 MPa (well within the yield strength of structural steel at ~250 MPa)
Example 2: Wooden Shelf Under Load
Problem: A pine shelf (width 200 mm, thickness 25 mm) spans 1 m and supports 50 kg of books evenly distributed. Find the maximum bending stress.
Distributed load: w = 50 × 9.81 / 1.0 = 490.5 N/m
Maximum bending moment for a uniformly loaded simply supported beam: M = wL²/8 = 490.5 × 1² / 8 = 61.3 N·m
I = (0.2)(0.025)³/12 = 2.604 × 10⁻⁷ m⁴, y_max = 0.0125 m
σ = 61.3 × 0.0125 / (2.604 × 10⁻⁷) = 2.94 MPa (safe for pine, which yields around 40 MPa)
When to Use It
The bending moment formula is used whenever you need to check if a beam or structural member can safely carry a load.
- Designing beams for buildings, bridges, and other structures
- Checking if shelves, floors, or balconies can support their intended load
- Sizing aircraft wing spars and fuselage frames
- Analyzing machine components like shafts, axles, and cantilever arms
- Selecting the correct beam profile (I-beam, channel, rectangular) for a given application