Advanced Beam Calculator
Formula Used
The calculator uses Euler-Bernoulli beam theory for small elastic deflection. Flexural rigidity is:
Common formulas include:
- Simply supported center point load: δmax = PL³ / 48EI, θend = PL² / 16EI.
- Simply supported uniform load: δmax = 5wL⁴ / 384EI, θend = wL³ / 24EI.
- Cantilever end point load: δmax = PL³ / 3EI, θfree = PL² / 2EI.
- Cantilever uniform load: δmax = wL⁴ / 8EI, θfree = wL³ / 6EI.
- Fixed ended uniform load: δmax = wL⁴ / 384EI, ideal end slope = 0.
All loads are converted to newtons. Span is measured in meters. Deflection is reported in millimeters. Slope is reported in milliradians.
How to Use This Calculator
- Select the support and load case that best matches your beam.
- Enter the beam span in meters.
- Enter elastic modulus in GPa.
- Enter second moment of area in cm⁴.
- Enter the point load or uniform load value.
- Use point load position for the offset point load case.
- Enter a service load factor if you want factored service load.
- Set the deflection limit ratio, such as L/360.
- Press the calculate button.
- Review the result panel, graph, reactions, and downloads.
Example Data Table
| Case | Span | E | I | Load | Common Use |
|---|---|---|---|---|---|
| Simply supported, center point load | 6 m | 200 GPa | 8,000 cm⁴ | 12 kN | Midspan machine load check |
| Simply supported, uniform load | 5 m | 200 GPa | 6,500 cm⁴ | 4 kN/m | Floor beam estimate |
| Cantilever, end point load | 2.5 m | 70 GPa | 3,000 cm⁴ | 3 kN | Bracket or canopy check |
| Fixed ended, uniform load | 7 m | 200 GPa | 12,000 cm⁴ | 5 kN/m | Restrained beam comparison |
Beam Slope and Deflection Guide
Why Beam Movement Matters
Beam slope and deflection describe how a beam bends under load. Slope is the rotation of the elastic curve. Deflection is the vertical movement from the original straight line. Both values are important in design, repair, and classroom checks.
Serviceability Comes First
A beam may look strong by stress alone. Yet it can still feel weak if movement is high. Floors may bounce. Shelves may sag. Doors may bind near a lintel. Machines may lose alignment. This calculator helps compare common support and load cases before a final detailed design.
What the Calculator Assumes
The tool uses standard elastic beam equations. It assumes small deflection, straight members, linear material behavior, and constant flexural rigidity. You enter span, elastic modulus, second moment of area, and the applied load. The calculator converts engineering units and reports rotation, maximum deflection, stiffness, and a span limit check.
Support Conditions Change Results
The support case matters. A simply supported beam can rotate at its ends. A cantilever is fixed at one end and free at the other. A fixed ended beam has restrained rotation at both supports. Because boundary conditions change the elastic curve, the same load can create very different movement.
Load Location Matters
Load position also matters. A centered point load gives a simple symmetric curve. A uniform load spreads force across the span. A custom point load changes reactions, slope, and maximum deflection location. The graph makes these differences easier to see.
Use Reliable Stiffness Values
Use results as a planning guide. Check units carefully. Use realistic material stiffness. Use the correct section inertia. For steel, a common elastic modulus is near 200 GPa. For concrete, timber, and aluminum, values vary more. Always confirm the value from a trusted specification.
Deflection Limits
Deflection limits depend on the job. Many building checks use ratios such as L/240, L/360, or L/480. Sensitive finishes may need tighter control. Structural safety also needs separate bending, shear, bearing, buckling, and connection checks.
Best Use
This page gives fast estimates and downloadable reports. It is useful for study, early sizing, and comparison. A licensed engineer should review final work where safety, code compliance, or public use is involved. For better accuracy, run several cases. Compare load positions, section sizes, and limit ratios before choosing a final member.
FAQs
What is beam deflection?
Beam deflection is the vertical movement of a beam from its original straight position after loading. It is usually checked for comfort, appearance, alignment, and serviceability.
What is beam slope?
Beam slope is the rotation of the elastic curve at a selected point. It is commonly measured in radians or milliradians.
Which units does this calculator use?
Span uses meters. Elastic modulus uses GPa. Inertia uses cm⁴. Point load uses kN. Uniform load uses kN/m. Results show deflection in millimeters.
What does EI mean?
EI is flexural rigidity. It combines material stiffness and section stiffness. A larger EI usually means lower slope and lower deflection.
Can I use this for steel beams?
Yes, for early elastic checks. Use a realistic steel modulus and correct section inertia. Final design should also check stress, shear, buckling, and code rules.
What is an L/360 limit?
L/360 means allowable deflection equals span divided by 360. For a 6 meter beam, the limit is 16.67 millimeters.
Why does support type change deflection?
Support type changes end rotation and restraint. A cantilever, simply supported beam, and fixed ended beam bend differently under the same load.
Is this a final structural design tool?
No. It is best for estimates, learning, and comparison. A qualified professional should review final structural work and safety decisions.