Beam input form
The page uses one vertical workflow, while the input grid expands to three columns on large screens, two on smaller screens, and one on mobile.
Example data table
| Case | Span | Load | E | I | Approx. max deflection |
|---|---|---|---|---|---|
| Simply supported, center point load | 6 m | 12 kN | 200 GPa | 8,500,000 mm⁴ | 31.76 mm |
| Cantilever, end point load | 2.5 m | 3 kN | 69 GPa | 4,200,000 mm⁴ | 53.91 mm |
| Simply supported, full UDL | 5 m | 5 kN/m | 200 GPa | 12,000,000 mm⁴ | 16.95 mm |
| Fixed-fixed, full UDL | 7 m | 8 kN/m | 200 GPa | 18,000,000 mm⁴ | 13.89 mm |
Formula used
This tool uses classic Euler-Bernoulli elastic beam relations. Symbols: L = span, E = Young’s modulus, I = second moment of area, P = point load, w = uniform load, M = applied moment, a = load position from the left or fixed support.
Simply supported, center point load: δmax = P L³ / (48 E I), Mmax = P L / 4.
Simply supported, point load at any position: reactions are RL = P(L-a)/L and RR = Pa/L. The plotted curve uses the standard piecewise elastic equations.
Simply supported, full UDL: δmax = 5 w L⁴ / (384 E I), Mmax = w L² / 8.
Cantilever, end point load: δmax = P L³ / (3 E I), fixed-end moment = P L.
Cantilever, point load at any position: free-end deflection = P a²(3L-a) / (6 E I), fixed-end moment = P a.
Cantilever, full UDL: δmax = w L⁴ / (8 E I), fixed-end moment = w L² / 2.
Cantilever, end moment: δtip = M L² / (2 E I), with constant bending moment along the beam.
Fixed-fixed, center point load: δmax = P L³ / (192 E I), support moments = −P L / 8, center moment = +P L / 8.
Fixed-fixed, full UDL: δmax = w L⁴ / (384 E I), support moments = −w L² / 12, center moment = +w L² / 24.
Serviceability check: allowable deflection = L / selected ratio. The calculator compares the absolute maximum deflection against that limit.
How to use this calculator
- Select the beam case that matches your support and loading condition.
- Enter span length, material stiffness, and second moment of area.
- Choose consistent input units for force, moment, and section properties.
- Enter the point load, UDL, load position, or end moment required by the selected case.
- Set a serviceability ratio such as L/240, L/360, or L/480.
- Press the calculate button to view the result summary, support actions, and deflection graph above the form.
FAQs
1) What does this calculator compute?
It estimates elastic beam deflection, slope, reactions, and bending moments for common loading cases. The chart plots the deflected shape along the span for quick visual review.
2) When is Euler-Bernoulli beam theory appropriate?
It works best for slender beams, small deflections, linear elastic materials, and sections that remain plane during bending. Deep beams or large rotations may need more advanced methods.
3) Why do E and I have such a strong effect?
E measures material stiffness, while I measures section stiffness about the bending axis. Their product, EI, controls curvature resistance, so larger EI values reduce deflection and slope.
4) Can I use metric and imperial units?
Yes. The tool converts supported units internally before solving the beam equations. Keep the chosen units consistent with the actual bending axis and section property direction.
5) What does the L/δ ratio mean?
It compares span to maximum deflection and is widely used for serviceability screening. Larger ratios indicate a stiffer response. Final limits should always follow your project standard.
6) Why are fixed-fixed cases usually stiffer?
End restraints reduce rotation and lower midspan displacement. The same load usually causes less deflection than a simply supported beam, but support moments become more important.
7) Does this version support partial UDLs or multiple loads?
No. This single-file version covers major standard cases cleanly. For several loads or partial distributions, extend the equations, superpose results, or use matrix stiffness analysis.
8) Does the pass or review status replace design checks?
No. The serviceability check is only a quick screen based on your selected deflection limit. Final design still needs strength, stability, code, vibration, and connection review.