Beam Inputs
Example Data Table
| Case | L (m) | E (Pa) | I (m⁴) | P (N) | a (m) | w (N/m) | Notes |
|---|---|---|---|---|---|---|---|
| A | 6 | 2.00e11 | 8.00e-6 | 1000 | 3 | 200 | Combined load, midspan point load. |
| B | 4 | 7.00e10 | 3.50e-6 | 1500 | 1 | 0 | Single point load near the left. |
| C | 8 | 2.00e11 | 1.20e-5 | 0 | 0 | 500 | Uniform load only across the span. |
Formula Used
Support Reactions
- Point load at a: R1 = P(L−a)/L, R2 = Pa/L
- Full-span UDL: R1 = R2 = wL/2
- Combined loads use superposition: add the reactions.
Bending Moment
- Moment function: M(x)=R1x − (w x²)/2 − P(x−a)H(x−a)
- H(·) is the step function, active after the point load.
- The peak moment occurs where shear changes sign.
Deflection
- UDL: y(x)= w x (L³ − 2Lx² + x³)/(24EI)
- Point load: piecewise closed-form deflection for x≤a and x≥a
- Total deflection: y = y_udl + y_point
Assumptions
- Linear-elastic material behavior.
- Small deflections and constant E and I.
- Loads are vertical and the beam is prismatic.
How to Use This Calculator
- Enter the span length L between supports.
- Provide material stiffness using E.
- Enter section stiffness using I.
- Set point load P and its position a, or use 0.
- Add uniform load w across the span, or use 0.
- Press Submit to see results above the form.
- Export CSV or print to save a PDF report.
Technical Notes
Load modeling and superposition
A simply supported span is commonly checked with a point load and a full-span uniform load. The calculator adds their effects using superposition, so reactions, shear, moment, and deflection are summed from each case. This approach fits many beams in buildings, platforms, and machinery frames. It is valid when response is linear and deflections remain small compared with span.
Reactions and equilibrium checks
Support reactions follow static equilibrium. For a point load P at position a, the left reaction equals P(L−a)/L and the right reaction equals Pa/L. For a uniform load w over length L, each reaction equals wL/2. A quick quality check is R1+R2 ≈ P+wL. If your units are N and m, the reaction output stays in N.
Shear diagram behavior
Shear starts at +R1 at the left support, decreases linearly with slope −w, and drops by P at the point load. The largest absolute shear usually occurs at a support or just to either side of the point load. Knowing the peak shear helps size webs, check bearing, and select fasteners. For steel beams, shear stress is often compared to 0.4Fy, while timber checks use code-specific factors.
Bending moment and peak location
The bending moment integrates shear, producing a quadratic curve under uniform load and a kink at the point load. The maximum moment typically occurs where shear crosses zero, so its location shifts as P, a, and w change. The tool evaluates candidate locations to report the peak magnitude and its x-position, supporting stress checks with σ = M·c/I. Use consistent geometry: c is the distance from neutral axis to extreme fiber.
Deflection limits and serviceability
Deflection is computed using classic closed-form beam functions and a fine scan along the span. Typical service limits are L/360 for floors, L/240 for roofs, and tighter criteria for brittle finishes. Increasing I, shortening L, or reducing w often improves serviceability more efficiently than increasing strength alone. If you enter E in Pa and I in m⁴, deflection returns in meters; multiply by 1000 for millimeters. For vibration-sensitive spans, also review mass, damping, and dynamic load effects, since static deflection alone may not govern.
FAQs
1) What load cases does this beam tool cover?
It combines one point load at position a and one uniform load applied over the full span. Results are summed using superposition for reactions, shear, moment, and deflection.
2) Which unit system should I use?
Any consistent system works. For SI, use meters, newtons, and pascals with I in m⁴. Outputs follow those inputs: reactions in N, moments in N·m, and deflection in m.
3) Why is my deflection extremely large?
Usually E or I is too small, or units are mixed, such as mm entered as m. Confirm I is in m⁴, not mm⁴, and confirm E is in Pa, not GPa without conversion.
4) Where does the maximum moment occur?
The peak moment typically occurs where shear crosses zero. With only a uniform load, it is at midspan. With a point load, the location depends on a and the load magnitude.
5) Does this include self-weight or partial UDL regions?
Self-weight is not automatic. Add it by increasing w. Partial-span UDLs and multiple point loads are not modeled here; split the problem into cases or use a more general beam solver.
6) Can I save the results as a PDF report?
Yes. After calculating, use the Print / Save PDF button in your browser to produce a PDF. Use the Results CSV button to export the numeric table for spreadsheets.