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EasyBeam

Continuous Beam Calculator

Analyse a multi-span continuous beam with the three-moment equation. Set span lengths, per-span distributed loads, point loads and end conditions, and get reactions, shear, bending moment, deflection and stress across every span.

Spans (2 spans, total 10.0 m)
Distributed loads (N/m)
Point loads (N)
Cross-section & material
Project (for reports)
Beam layout
Beam layout diagramA continuous beam, total length 10.00 m, with 2 supports.

Support reactions

SupportLocation (m)Force (N)Moment (N·m)
R10.0018.75k—
R25.0062.50k—
R310.0018.75k—
Max shear
31.25k N
Max moment
17.58k N·m
Max deflection
0.38 mm
Max stress
5.86 MPa

Vertical equilibrium residual: 0.0e+0 — ΣR vs ΣP + ΣwL. A sound solve keeps this at zero.

Shear force
Shear force along the beam length in m. Values in N. Use left and right arrow keys to read values at a point.-30.00k-20.00k-10.00k010.00k20.00k30.00k0.001.002.003.004.005.006.007.008.009.0010.00
Bending moment
Bending moment along the beam length in m. Values in N·m. Use left and right arrow keys to read values at a point.-30.00k-20.00k-10.00k010.00k0.001.002.003.004.005.006.007.008.009.0010.00
Deflection
Deflection along the beam length in m. Values in mm. Use left and right arrow keys to read values at a point.-3.50e-1-3.00e-1-2.50e-1-2.00e-1-1.50e-1-1.00e-1-5.00e-20.00e+00.001.002.003.004.005.006.007.008.009.0010.00
Bending stress
Bending stress along the beam length in m. Values in MPa. Use left and right arrow keys to read values at a point.-10.00-8.00-6.00-4.00-2.0002.004.000.001.002.003.004.005.006.007.008.009.0010.00
Data table (accessibility fallback)
x (m)Shear (N)Moment (N·m)Deflection (mm)Stress (MPa)
0.0018.75k00.00e+00
0.909.75k12.82k-2.38e-14.27
1.80750.0017.55k-3.67e-15.85
2.70-8.25k14.17k-3.44e-14.72
3.60-17.25k2.70k-1.99e-10.90
4.50-26.25k-16.88k-3.65e-2-5.63
5.3827.50k-20.23k-2.14e-2-6.74
6.2818.50k465.63-1.75e-10.16
7.189.50k13.07k-3.29e-14.36
8.07500.0017.57k-3.73e-15.86
8.97-8.50k13.97k-2.64e-14.66
9.88-17.50k2.27k-3.62e-20.76
10.00-18.75k-1.5e-11-5.43e-5-4.9e-15

The three-moment equation

For three consecutive supports A, B and C with spans L₁ and L₂, the three-moment (Clapeyron) equation is:

MAL₁ + 2MB(L₁+L₂) + MCL₂ = −6(A₁ā₁/L₁ + A₂b̄₂/L₂)

where the A terms are the areas of the simply-supported moment diagrams of each span. Writing one equation per interior support and solving the tridiagonal system yields the support moments. The calculator builds and solves that system for up to six spans, then reconstructs the full shear and moment diagrams span by span.

Checking the answer

Every solve reports a vertical equilibrium residual — the difference between the sum of reactions and the total applied load. A correct continuous-beam solution drives this to zero, which is the single fastest sanity check on a hand or computer analysis.

Frequently asked questions

What is a continuous beam?+

A continuous beam spans over three or more supports without a hinge between them. Unlike a simply supported beam it is statically indeterminate: equilibrium alone cannot find the reactions, because there are more unknowns than equations.

How are continuous beams analysed?+

This calculator uses the three-moment equation (Clapeyron's theorem), which relates the bending moments at three consecutive supports to the loading on the two adjacent spans. Solving the resulting system of simultaneous equations gives the support moments, from which reactions, shear, moment and deflection all follow.

Why are the support moments negative for a downward load?+

A downward load on a continuous beam makes it hog over the interior supports — the top fibre is in tension there. By the sagging-positive convention used here, hogging moments are negative, which is why the moment diagram dips below zero over the supports.

What end conditions can I model?+

Pinned–pinned, fixed–pinned, pinned–fixed, fixed–fixed, and a pinned start with an overhang at the far end. Fixed ends develop a end moment; pinned ends carry zero moment.