Beam Deflection Calculator — Free Tool with AI Insights & Diagram
This beam deflection calculator uses standard Euler-Bernoulli formulas, checks your result against the common L/360 serviceability limit, and explains what it means for your design.
Beam & load details
Choose your support condition, load type, beam length, material, and section.
What is Beam Deflection?
Beam deflection is the displacement of a point on a beam, measured perpendicular to its original axis, caused by an applied load. It is usually expressed in millimeters and is calculated separately from — but alongside — checks on bending stress and shear strength, because a beam can be strong enough not to break while still bending more than is acceptable for serviceability.
The amount a beam deflects depends on four things: how it is supported (its boundary conditions), how the load is applied (concentrated at a point or spread out evenly), the span length, and the beam's stiffness — a combination of the material's modulus of elasticity (E) and the cross-section's moment of inertia (I). The four most common textbook scenarios, and the formulas this calculator uses, are summarized below.
| Scenario | Formula (max deflection, δ) | Description |
|---|---|---|
| Cantilever, point load at free end | δ = P·L³ / (3·E·I) | Maximum deflection occurs at the free (unsupported) tip. |
| Cantilever, UDL over full length | δ = w·L⁴ / (8·E·I) | Load spread evenly along the beam; maximum deflection still at the free tip. |
| Simply supported, point load at center | δ = P·L³ / (48·E·I) | Maximum deflection occurs at mid-span, directly under the load. |
| Simply supported, UDL over full length | δ = 5·w·L⁴ / (384·E·I) | Load spread evenly along the beam; maximum deflection at mid-span. |
How the Beam Deflection Calculator Works — The Formula
This calculator picks one of four standard Euler-Bernoulli beam deflection formulas based on your support condition and load type:
Cantilever, point load at free end: δ = P·L³ / (3·E·I)
Cantilever, UDL: δ = w·L⁴ / (8·E·I)
Simply supported, point load at center: δ = P·L³ / (48·E·I)
Simply supported, UDL: δ = 5·w·L⁴ / (384·E·I)
- P
- Point load applied to the beam, in newtons (N).
- w
- Distributed load per unit length, in N/m. If you know the total UDL load, w = total load (N) / L (m).
- L
- Span (or cantilever length), in meters (m).
- E
- Modulus of elasticity of the material, in pascals (Pa) — convert from GPa by multiplying by 10⁹.
- I
- Second moment of area (moment of inertia) of the cross-section, in m⁴ — convert from cm⁴ by multiplying by 10⁻⁸.
Worked example
Take a simply supported steel beam, 4m long, with a 5kN point load at mid-span and an IPE 200 section (I = 1,940 cm⁴), E = 200 GPa.
P = 5,000 N L = 4 m E = 200 × 10⁹ Pa I = 1,940 × 10⁻⁸ m⁴ = 1.94 × 10⁻⁵ m⁴
δ = P·L³ / (48·E·I)
δ = (5,000 × 4³) / (48 × 200×10⁹ × 1.94×10⁻⁵)
δ = 320,000 / (1.8624 × 10⁸)
δ ≈ 1.718 × 10⁻³ m = 1.72 mm
The L/x ratio is 4000mm / 1.72mm ≈ L/2326, which is far below (better than) the L/360 limit of 4000/360 ≈ 11.1mm — this beam passes comfortably.
Unit conversion notes
- E: entered in GPa, convert to Pa by multiplying by 10⁹ (1 GPa = 10⁹ Pa).
- I: entered in cm⁴, convert to m⁴ by multiplying by 10⁻⁸ (1 cm⁴ = 10⁻⁸ m⁴).
- Result: the formula returns δ in meters — multiply by 1,000 to get millimeters.
The most common error is forgetting to convert I from cm⁴ to m⁴. Since 1 cm⁴ = 10⁻⁸ m⁴, leaving I in cm⁴ while E is in Pa makes the denominator (E·I) about 10⁸ times too small — which makes the calculated deflection roughly 10⁸ times too large. This calculator handles the conversion for you automatically.
How to Use the Beam Deflection Calculator
- Choose your Support condition — Cantilever or Simply supported.
- Select the Load type that matches your situation — a point load or a uniformly distributed load (UDL).
- Enter the Load magnitude (P) and choose its unit (N or kN).
- Enter the Beam length / span (L) in meters, choose the Modulus of elasticity (E) preset (or Custom), and pick a Steel section for I.
- Click Calculate to see your results instantly.
- Scroll to the AI Insights section to understand what your result means.
How to Interpret Your Beam Deflection Results
What a Good Result Looks Like
A "good" result is a deflection ratio that is better than L/360 — meaning the L/x number is larger than 360 (e.g. L/500 or L/1000), so the beam deflects less than the L/360 serviceability limit allows. L/360 is the most commonly cited limit for floor members supporting brittle finishes, but it is not the only one in use: L/240 is sometimes used for roof members where a little more sag is acceptable, while L/480 (or even stricter) is used for floors supporting sensitive equipment, long-span glazing, or finishes that crack easily.
Warning Signs in Your Results
If your result shows "Fail", your deflection ratio is worse than L/360 — the L/x number is smaller than 360 (e.g. L/200), meaning the beam bends more than the limit allows. This does not necessarily mean the beam will break, but it does mean it may visibly sag, feel "springy" underfoot, or crack rigid finishes like plaster, tile, or drywall over time. A "Fail" here is a signal that the section, span, or load combination needs to be revisited before finalizing the design.
How to Improve Your Result
- Increase the moment of inertia (I) by choosing a larger or deeper section. Deflection is inversely proportional to I, so doubling I (e.g. moving from IPE 200 to IPE 300) cuts deflection roughly in half.
- Reduce the span (L), for example by adding an intermediate support. Deflection scales with L³ (point loads) or L⁴ (UDL), so even a modest reduction in span has an outsized effect.
- Use a stiffer material (higher E) — for example, switching from timber or aluminum to steel significantly increases stiffness for the same section size.
- Reduce the applied load — re-check whether the assumed load (P or w) accurately reflects the actual service load, since deflection scales linearly with load.
Beam Deflection Calculator Examples
Cantilever Beam Deflection Formula Example
A steel bracket acts as a cantilever, fixed to a wall at one end with a point load applied at the free tip.
- Support: Cantilever
- Load: Point load at free end, 5 kN
- Span: L = 4 m
- Material: Steel, E = 200 GPa
- Section: IPE 200, I = 1,940 cm⁴
Using δ = P·L³ / (3·E·I), the result is ≈ 27.5 mm of deflection — about L/145, which is well beyond the L/360 limit (Fail). What this means: a cantilever deflects 16 times more than the same load and span on a simply supported beam, because there is no second support to share the load. What to do next: try a deeper section (e.g. IPE 400) or shorten the cantilever's reach.
Simply Supported Beam with UDL Example
A floor joist spans between two walls and carries a uniformly distributed load along its full length.
- Support: Simply supported
- Load: UDL, total 5 kN over the span
- Span: L = 4 m (so w = 5,000 N / 4 m = 1,250 N/m)
- Material: Steel, E = 200 GPa
- Section: IPE 200, I = 1,940 cm⁴
Using δ = 5·w·L⁴ / (384·E·I), the result is ≈ 1.07 mm, roughly L/3727 — comfortably better than L/360 (Pass). What this means: spreading the same total load evenly along the span produces far less deflection than concentrating it at mid-span. What to do next: this section has plenty of margin — you could consider a smaller, lighter section if strength checks also allow it.
Steel I-Beam Deflection Comparison (IPE 200 vs IPE 300 vs IPE 400)
Same simply supported beam, same 5 kN center point load, same 4 m span and 200 GPa steel — only the section changes.
- IPE 200 (I = 1,940 cm⁴): ≈ 1.72 mm, ≈ L/2326 (Pass)
- IPE 300 (I = 8,360 cm⁴): ≈ 0.40 mm, ≈ L/10,000 (Pass)
- IPE 400 (I = 23,130 cm⁴): ≈ 0.14 mm, ≈ L/27,800 (Pass)
What this means: all three sections pass L/360 for this light load, but the deflection drops dramatically as I increases — IPE 300 deflects about 23% as much as IPE 200, and IPE 400 about 8% as much. What to do next: for this load, IPE 200 is likely sufficient on deflection grounds alone — but always verify bending stress and shear capacity too, which this tool does not check.
Frequently Asked Questions about the Beam Deflection Calculator
What is beam deflection and why does it matter in design?
Beam deflection is the distance a beam bends or sags under load, measured perpendicular to its original axis. It matters because excessive deflection — even if the beam is strong enough not to break — can crack plaster, jam doors, cause floors to feel "bouncy," or damage finishes. Building codes therefore limit deflection separately from strength, typically to a fraction of the span such as L/360.
What does the L/360 deflection limit mean and where does it come from?
L/360 means the maximum allowable deflection is the span length divided by 360. For a 4m (4000mm) span, that is 4000 / 360 ≈ 11.1mm. It comes from building codes (such as IBC and AISC guidance) as a long-standing serviceability limit for floor members supporting brittle finishes like plaster or tile, where larger movements would cause visible cracking.
What is the difference between a cantilever and a simply supported beam?
A cantilever beam is fixed (rigidly held) at one end and completely free at the other, like a diving board. A simply supported beam rests on two supports — typically a pin at one end and a roller at the other — and can rotate freely at both ends. For the same load and span, a cantilever deflects far more: a point load at a cantilever tip can deflect 16 times more than the same load at the center of a simply supported beam of equal length.
How does the moment of inertia (I) affect deflection?
Deflection is inversely proportional to the moment of inertia (I) — double I and deflection is cut in half, all else being equal. I depends on the cross-section shape: taller, deeper sections resist bending far more efficiently than shallow ones. For example, going from an IPE 200 (I = 1,940 cm⁴) to an IPE 300 (I = 8,360 cm⁴) reduces deflection to roughly 23% of its original value.
How do I choose between the four deflection formulas?
Pick the formula based on two things: how the beam is supported (cantilever vs. simply supported) and how the load is applied (a single point load vs. a load spread evenly along the span, called a UDL). This calculator selects the correct formula automatically once you choose your support condition and load type — there are four combinations in total, each with its own formula.
How accurate is this beam deflection calculator?
It uses standard linear-elastic beam theory (Euler-Bernoulli), which is accurate for typical small-deflection scenarios with prismatic (constant cross-section) beams and idealized point or uniform loads. It assumes the material stays within its elastic range and does not account for shear deformation, dynamic/impact loads, connection flexibility, or combined loading — real structures often involve these additional factors.
What should I do if my beam fails the deflection check?
A "Fail" result means your deflection ratio is worse (a smaller L/x number) than the L/360 limit — the beam will sag more than the code typically allows. Your main options are: choose a deeper or stiffer section (higher I), shorten the span (e.g. add an intermediate support), switch to a stiffer material (higher E), or reduce the applied load. Increasing I is usually the most efficient fix since deflection scales directly with 1/I.
Does this calculator replace a structural engineer's calculations?
No. This tool is for quick estimates, learning, and preliminary sizing only. It does not check bending stress, shear capacity, buckling, connections, dynamic loading, code-specific load combinations, or local material standards. Final structural designs — especially anything affecting life safety — must be reviewed and stamped by a licensed structural engineer per your local building code.
This calculator is intended for educational use and preliminary, back-of-envelope estimation only. It does not check bending stress, shear, buckling, or code-specific load combinations. Final structural designs must be verified and stamped by a licensed structural engineer in accordance with applicable building codes.