Beam and elastic curve inputs
Example data table
| Support condition | Load case | Span | E | I | Load | Key output |
|---|---|---|---|---|---|---|
| Simply supported | Interior point load at midspan | 4.0 m | 200 GPa | 8,500,000 mm4 | 12 kN | Midspan deflection and support reactions |
| Cantilever | Full-span uniformly distributed load | 2.5 m | 70 GPa | 4,200,000 mm4 | 3.5 kN/m | Free-end slope and tip deflection |
Formula used
The calculator applies Euler-Bernoulli beam theory, where curvature is proportional to bending moment.
Core relationship: M(x) = E I y''(x)
Curvature: kappa(x) = M(x) / (E I)
Slope: theta(x) = dy/dx
Deflection: y(x) is obtained by integrating the bending-moment function and enforcing support boundary conditions.
Cantilever with end point load: y(x) = P x² (3L − x) / (6 E I)
Cantilever with full-span uniformly distributed load: y(x) = w x² (6L² − 4Lx + x²) / (24 E I)
Simply supported with interior point load: the page uses the standard piecewise elastic-curve equations on both sides of the load position.
Simply supported with full-span uniformly distributed load: y(x) = w x (L³ − 2Lx² + x³) / (24 E I)
Assumptions: linear elastic material, small deflections, constant E and I, slender beam behavior, and static loading.
How to use this calculator
- Select the support condition that matches your beam.
- Choose the load case.
- Enter span, elastic modulus, and second moment of area.
- Fill in the point load or distributed load magnitude.
- Enter point load location when using the interior point-load option.
- Set the evaluation position x and the number of table stations.
- Press the calculate button.
- Review the summary cards, plot, and station-by-station table.
- Use the CSV or PDF buttons to save the results.
Frequently asked questions
1) What does the elastic curve represent?
It shows the deformed shape of a beam under load. The curve describes how the beam bends from one support condition to another across the full span.
2) Why are slope and deflection both reported?
Slope measures rotation at a point, while deflection measures displacement. Engineers often need both because serviceability checks may limit one or both values.
3) When should I use the point-load position field?
Use it for the simply supported interior point-load case. It is not needed for cantilever end loads or full-span uniformly distributed loads.
4) What is the role of E and I?
E is material stiffness and I is section stiffness. Their product, E I, controls how strongly the beam resists bending under the applied load.
5) Are the results suitable for large deflections?
No. The formulas assume small deflection beam theory. Large rotations, geometric nonlinearity, or changing stiffness require more advanced analysis.
6) Why does the plot look exaggerated?
The graph scales deflection vertically so the curve is visible on screen. The numeric table provides the actual displacement values for design work.
7) Can I use imperial units?
Yes. The calculator supports feet, inches, pounds-force, psi, and inch-based area moments. Internally, all values are converted to consistent SI units.
8) What are reaction values used for?
Reactions help verify static equilibrium and support design. They show how much force or fixed-end moment each support must resist.