Beam Stress Calculator Formula
Understand the math behind the beam stress calculator. Each variable explained with a worked example.
Formulas Used
Maximum Bending Stress
bending_stress = m_nmm * dist_to_na / i_mm4Variables
| Variable | Description | Default |
|---|---|---|
moment | Bending Moment (M)(kN-m) | 50 |
dist_to_na | Distance to Neutral Axis (c)(mm) | 150 |
inertia | Moment of Inertia (I)(cm^4) | 8356 |
m_nmm | Derived value= moment * 1e6 | calculated |
i_mm4 | Derived value= inertia * 1e4 | calculated |
How It Works
Bending Stress in Beams
The flexure formula relates the internal bending moment to the normal stress at any fibre of the cross-section.
Formula
sigma = M c / I
M is the bending moment, c is the perpendicular distance from the neutral axis to the outermost fibre, and I is the second moment of area about the neutral axis. The result is the peak tensile or compressive stress at the extreme fibre.
Worked Example
A steel beam with I = 8356 cm^4, depth 300 mm (c = 150 mm), under a moment of 50 kN-m.
- 01Convert moment: 50 kN-m = 50 x 10^6 N-mm
- 02Convert inertia: 8356 cm^4 = 8356 x 10^4 mm^4 = 8.356 x 10^7 mm^4
- 03sigma = (50 x 10^6 x 150) / (8.356 x 10^7)
- 04sigma = 7.5 x 10^9 / 8.356 x 10^7 = 89.76 MPa
When to Use This Formula
- Verifying that a beam will not yield or fracture under the maximum expected load by comparing the calculated bending stress to the material allowable stress.
- Sizing a beam cross-section — choosing the minimum section modulus (S = I/c) required to keep bending stress below the material strength with an appropriate safety factor.
- Analyzing a failure to determine whether the beam was overstressed — calculating the actual stress at the failure point and comparing it to the material yield strength.
- Comparing rectangular, I-beam, and circular cross-sections to find the most weight-efficient shape for a given bending moment.
- Evaluating whether an existing beam can support a new, heavier load — for example, adding HVAC equipment to a roof beam originally designed for lighter loads.
Common Mistakes to Avoid
- Using the wrong value for c (distance from the neutral axis to the extreme fiber) — for a symmetric cross-section like a rectangle, c is half the depth, but for an asymmetric section like a T-beam, the neutral axis is not at the center and c differs for the top and bottom faces.
- Confusing the moment of inertia (I) with the section modulus (S) — the stress formula can be written as σ = M/S where S = I/c, or as σ = Mc/I. Using I where S belongs (or vice versa) gives a result off by a factor of c.
- Forgetting to find the maximum bending moment (M) before calculating stress — the stress varies along the beam length. You must first determine where M is largest (usually at midspan for uniform loads or at the support for cantilevers) and use that value.
- Neglecting shear stress in short, deep beams — the bending stress formula assumes slender beams where bending dominates. For short spans with heavy loads, shear stress near the neutral axis can be the critical failure mode instead.
Frequently Asked Questions
What happens when stress exceeds yield strength?
The beam begins to yield plastically. For structural steel with yield strength around 250 MPa, the beam can still carry load through plastic redistribution, but permanent deformation occurs.
Is bending stress the same on both sides?
In magnitude yes, but the sign changes: one side is in tension and the other in compression. For symmetric sections, both extreme fibres have the same absolute stress.
How does section modulus relate to this formula?
Section modulus S = I / c, so the formula simplifies to sigma = M / S. This is a convenient shortcut when S is tabulated for standard sections.
Learn More
Guide
Beam Stress Calculation Guide: From Theory to Practice
Learn how to calculate beam stress step by step. Covers bending stress, shear stress, the flexure formula, stress distributions, and practical design checks for structural beams.
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