How to use this calculator
- Choose the load case. Pick the support and loading condition closest to your beam.
- Enter span and load. Use point load P or total uniform load W over the full span.
- Enter material stiffness. Use Young's modulus E for the beam material.
- Enter section properties. Use I for deflection and S for bending stress about the loaded axis.
- Check both outputs. Compare deflection and stress against your design limits.
How it works
Beam deflection depends on load, span, material stiffness and area moment of
inertia. For a simply supported beam with a center point load:
delta = P x L^3 / (48 x E x I)
A cantilever with an end load is much more flexible:
delta = P x L^3 / (3 x E x I).
Uniform-load cases use the total load W over the span:
5W L^3/(384EI) for simply supported, and W L^3/(8EI)
for a cantilever. Maximum bending stress is always calculated from
sigma = M / S.
Worked example
Verified against the live calculator
For a simply supported beam with L = 1000 mm, P = 1000 N,
E = 200000 MPa and I = 1,000,000 mm^4, maximum
deflection is 0.1042 mm.
The maximum moment is P L / 4 = 250 N*m. With
S = 10,000 mm^3, bending stress is 25 MPa and the
span/deflection ratio is about 9600.
Frequently asked questions
What beam deflection formulas does this use?
It uses closed-form Euler-Bernoulli results for four common cases: simply supported center point load, simply supported uniform total load, cantilever end point load, and cantilever uniform total load.
Is the uniform load entered as line load or total load?
Enter the total uniform load over the span. If you know a line load w, multiply by span first: W = w * L.
How is bending stress calculated?
Maximum bending stress is sigma = M/S, where M is the maximum bending moment for the selected load case and S is the section modulus about the bending axis.
Where do I get I and S?
Use section data from a steel manual, CAD section properties, manufacturer data, or the MachineCalcs section modulus calculator for common shapes.
Does this include shear deflection?
No. It uses Euler-Bernoulli bending theory, which ignores shear deflection. Deep beams, short spans and soft materials may need Timoshenko beam theory or FEA.
Can I use imperial units?
Yes. Toggle units to enter span, I and S in imperial display units while the formulas continue to run in unit-safe internal units.
Method & assumptions
- Closed-form small-deflection Euler-Bernoulli beam formulas for prismatic members.
- Uniform-load inputs are total load over the span, not load per unit length.
- Section properties
IandSmust be about the bending axis being loaded. - For wood LVL members where you want to enter plies, span and line loads directly, use the LVL beam calculator.
- Shear deflection, local holes, tapered members, combined loading, buckling, self-weight unless entered, and dynamic loading are not included.