Continuous Beam Calculator
Moments, shears, support reactions and deflections for a prismatic continuous beam of one to five spans under uniform dead and live load. Live load is placed on every combination of spans, so the results are true envelopes, not a single full-load case. Each end can be pinned or fixed.
| Span | +M max | at x | |V| max |
|---|---|---|---|
| 1 | — | — | — |
| 2 | — | — | — |
| 3 | — | — | — |
| 4 | — | — | — |
| 5 | — | — | — |
| Support | −M max | R max | R min |
| 1 (left end) | — | — | — |
| 2 | — | — | — |
| 3 | — | — | — |
| 4 | — | — | — |
| 5 | — | — | — |
| 6 | — | — | — |
Moments in kip-ft, shears and reactions in kip, x in ft from the left support of that span. Negative R min means the support must resist uplift under some live-load pattern.
How the calculator works
The unknowns are the rotations at each support. A fixed end has no rotation unknown. Each span contributes the slope-deflection stiffness 4EI/L and 2EI/L plus the fixed-end moments ∓wL²/12 of its uniform load. The page then enforces moment equilibrium at every free joint and solves the linear system. Once the end moments are known, it gets moment, shear and deflection anywhere in a span by statics. Deflection is the fixed-end uniform-load shape wx²(L−x)²/24EI plus the Hermite cubic driven by the end rotations.
Dead load acts on all spans. Live load runs through all 2n on/off patterns, and the page samples 200 points per span for each pattern and keeps the maximum and minimum. That captures the positive moment with alternate spans loaded, the negative moment with the adjacent spans loaded, and uplift at end supports of short spans next to long ones. The moment and shear envelopes use the load factors. Deflections are recomputed with unfactored loads.
Check values (equal spans, uniform load on all spans)
| Beam | Interior support M | Max +M | End reaction | 1st interior reaction |
|---|---|---|---|---|
| Simple span | — | 0.1250 | 0.500 | — |
| 2 equal spans | −0.1250 | 0.0703 | 0.375 | 1.250 |
| 3 equal spans | −0.1000 | 0.0800 (end span) | 0.400 | 1.100 |
| Fixed–fixed single span | −0.0833 (ends) | 0.0417 | 0.500 | — |
| Propped cantilever (fixed–pinned) | −0.1250 (fixed end) | 0.0703 | 0.625 / 0.375 | — |
Worked example — two spans, live load pattern
Two equal 40 ft spans, wD = 1.0 klf, wL = 0.8 klf, service loads
Limits
- Uniform loads only. The page has no point loads, moving trucks or influence lines, so for HL-93 it covers the 0.64 klf lane load but not the design truck or tandem.
- Prismatic beam: one EI for all spans, and no shear deformation.
- No support settlement, no cantilever overhangs, and no internal hinges.
- This is a line-beam analysis. It does not include distribution factors, composite-section staging, creep or restraint moments.
References: Hibbeler, R.C., Structural Analysis, slope-deflection and stiffness methods (Ch. 11–15). AISC Steel Construction Manual, 16th ed., Table 3-23 (continuous-beam cases 28–36), used as the check values above. AASHTO LRFD Bridge Design Specifications, 9th ed., Art. 3.6.1.2.4 (design lane load 0.64 klf), Table 3.4.1-1 (Strength I), Art. 2.5.2.6.2 (deflection criteria).
Related tools
- Beam deflection — single simply supported or cantilever beam, point and uniform load
- Bridge scour (HEC-18) — contraction, pier and abutment scour at the substructure
- Pile capacity — size pier and abutment piles for the reactions found here
- Bearing capacity — spread footings under piers and abutments