Beam Calculator: Reactions, Bending Moment, Stress and Deflection
Work out the reactions, maximum bending moment, bending stress and deflection of a cantilever or simply supported beam, with the formulas shown step by step.
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Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
A beam carries load by bending. The load makes a bending moment M along the beam, and the moment makes a stress in the material that is largest at the outer fibre: σ = M / Z, where Z is the elastic section modulus of the cross-section. The beam also sags, and the maximum deflection depends on the load, the span cubed or to the fourth power, the stiffness of the material E and the stiffness of the section I.
This calculator covers the four standard textbook cases: a cantilever with a point load at the free end, a cantilever with a uniform load, a simply supported beam with a point load at mid-span, and a simply supported beam with a uniform load. Enter the span, the load, E, I and Z. If you add the yield strength it also gives the factor of safety and the span, load or section that would reach your target.
Self-weight of the beam is not included, so add it to the load if it matters. The formulas are the elastic ones, so they hold only while the stress stays below yield.
Formulas used
- \sigma_{max} = M_{max} / Z
- \delta_{max} = \frac{P L^3}{3 E I}\ \text{(cantilever, end point load)}
- \delta_{max} = \frac{5 w L^4}{384 E I}\ \text{(simply supported, uniform load)}
Worked example
With the default numbers in the calculator above:
- Reaction at the fixed end: 10 kN
- Maximum bending moment (at the fixed end): 30 kN m
- Maximum bending stress: 150 MPa
- Maximum deflection: 22.5 mm
- Span to deflection ratio: 133
- Factor of safety against first yield: 1.67
- Smallest section modulus that reaches FoS 1.5: 180 ×10³ mm³
- Largest point load at FoS 1.5: 11.1 kN
- Longest span at FoS 1.5: 3.33 m
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
Where do I find I and Z for a steel section?
They are listed as Ix and Zx in the steel section tables from the supplier. For a simple shape you can work them out with the section properties calculator.
What span-to-deflection ratio is acceptable?
It depends on what the beam carries and what the deflection could damage. Limits are commonly written as span divided by 200 to 500. Take the limit from the project specification or AS 1170.0.
Does this include self-weight?
No. Add the beam weight per metre to the uniform load, or add the weight as part of a point load.
Related calculators
Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.
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