Three-moment equation for adjacent spans with constant EI and uniform load q:
$$M_{i-1}L_i + 2M_i(L_i+L_{i+1}) + M_{i+1}L_{i+1}= -\frac{q_i L_i^3}{4}- \frac{q_{i+1}L_{i+1}^3}{4}$$Support reaction: $R_{i,L}= \dfrac{q_i L_i}{2}- \dfrac{M_{i+1}-M_i}{L_i}$
Verification: 2 equal spans, $L=6$ m, $q=20$ kN/m → interior support $M_B=-90$ kN·m ($=-qL^2/8$), reaction $R_B=150$ kN ($=5qL/4$).
The Three-Moment (Clapeyron) Theorem
A continuous beam has three or more supports and is statically indeterminate, so it cannot be solved by superposing simple beams. The three-moment (Clapeyron) theorem takes the support moments as unknowns and writes equations from the condition that the slope is continuous across each support (compatibility). For supports $i-1, i, i+1$, spans $L_i, L_{i+1}$, and distributed loads $w_i, w_{i+1}$:
$M_{i-1}L_i + 2M_i(L_i+L_{i+1}) + M_{i+1}L_{i+1} = -\dfrac{w_i L_i^3}{4} - \dfrac{w_{i+1} L_{i+1}^3}{4}$
Writing this at each interior support gives a system of linear equations for the support moments $M_i$. This simulator solves it automatically for 2–5 spans and draws the support moments, reactions, and bending moment diagram.
Continuous Beam Coefficients (equal spans, uniform load w)
Representative values for equal spans $L$ with a uniform load $w$ on all spans. Interior supports develop a negative (hogging) moment, and the midspan moment is smaller than for a simple beam — the advantage of continuity.
| Form | Interior support moment | Reactions | Max positive (span) moment |
|---|---|---|---|
| 2 equal spans | $M_B=-wL^2/8$ | $R_A=R_C=3wL/8$, $R_B=5wL/4$ | $9wL^2/128$ (at $3L/8$) |
| 3 equal spans | $M_B=M_C=-wL^2/10$ | $R_A=R_D=0.4wL$, $R_B=R_C=1.1wL$ | end span $\approx wL^2/12.5$, interior $\approx wL^2/40$ |
For reference, a simple beam (uniform load) has a midspan moment $wL^2/8$ and reactions $wL/2$. Continuity reduces the midspan moment by about 36–45%, transferring it to the negative support moment, so RC beams need top reinforcement over the supports.
Characteristics vs. a Simple Beam
Indeterminate: a continuous beam is statically indeterminate, so reactions and moments depend on member stiffness ($EI$) and support conditions. Support settlement or temperature change also induces stresses — unlike a determinate simple beam.
Negative bending moment: over interior supports the beam deflects concave-up (hogging) with tension on top; the BMD swings to the negative side there and positive at midspan.
Pattern (chequerboard) live load: the maximum positive and support moments occur when live load is placed on alternate spans, which can be more severe than uniform loading on all spans — check this with the "distributed load (staggered)" option.
How to Use
- Enter Young's modulus (E) in GPa—typically 200 GPa for steel, 30 GPa for concrete, 70 GPa for aluminum
- Define second moment of inertia (I) in cm⁴ or select standard section (e.g., IPE 300 = 8356 cm⁴)
- Input distributed load (q) in kN/m across each span
- Specify all span lengths in meters, separated by commas (e.g., 4, 5, 3.5 for three spans)
- Inputs update the three-moment equation solution for internal support moments in real time
- Review M_max+, M_max−, Max Reaction [kN], and Max Deflection [mm] in results
Worked Example
Steel 3-span continuous beam: E=200 GPa, I=23,500 cm⁴, spans 5m + 5m + 5m, uniform load q=15 kN/m. The three-moment equation gives support moments at B and C of about −37.5 kN·m, maximum positive bending moment about +30.0 kN·m, maximum support reaction 82.5 kN, and maximum deflection about 1.37 mm.
Practical Notes
- For precast concrete elements with I=12000 cm⁴ and sustained loads, apply 1.5× creep factor to deflection results
- Check support moments against plastic section modulus—negative moments govern for ductile steels, often requiring top-flange reinforcement in continuous RC beams
- Span ratios exceeding 2:1 (e.g., 8m then 3m) create unbalanced settlements; verify foundation stiffness
- Account for temperature gradients in long multi-span structures; continuous beams develop axial restraint forces increasing support reactions by 5–8% per 20°C