Multi-span continuous beam analysis via the three-moment equation (Clapeyron). Computes support moments, reactions, bending moment diagram, and deflection in real time for 2–5 span beams.
For adjacent spans $i$ and $i+1$:
$$M_{i-1}L_i + 2M_i(L_i+L_{i+1}) + M_{i+1}L_{i+1}= -\frac{q_i L_i^3}{4}- \frac{q_{i+1}L_{i+1}^3}{4}$$(equal EI, uniform distributed loads $q$)
Support reactions (span i):
$$R_{i,L}= \frac{q_i L_i}{2}- \frac{M_{i+1}-M_i}{L_i}, \quad R_{i,R}= \frac{q_i L_i}{2}+ \frac{M_{i+1}-M_i}{L_i}$$Location of max positive moment: $x^* = R_{i,L}/ q_i$
The fundamental governing equation is the Three-Moment Equation (Clapeyron's Theorem). It enforces slope continuity at the interior support i between two adjacent spans, assuming a constant flexural rigidity $EI$ and a uniformly distributed load.
$$M_{i-1}L_i + 2M_i(L_i+L_{i+1}) + M_{i+1}L_{i+1}= -\frac{q_i L_i^3}{4}- \frac{q_{i+1}L_{i+1}^3}{4}$$Where:
$M_{i-1}, M_i, M_{i+1}$ = Bending moments at supports (unknowns we solve for).
$L_i, L_{i+1}$ = Lengths of the two adjacent spans.
$q_i, q_{i+1}$ = Intensities of the uniformly distributed loads on each span (use the "Load intensity q" and pattern).
The right-hand side represents the fixed-end moments from the loads.
Once support moments are known, we calculate the Reactions. The reaction at support i has contributions from the spans to its left and right.
$$R_{i}= R_{i,L}+ R_{i,R}= \left( \frac{q_i L_i}{2}- \frac{M_{i}-M_{i-1}}{L_i}\right) + \left( \frac{q_{i+1}L_{i+1}}{2}+ \frac{M_{i+1}-M_{i}}{L_{i+1}} \right)$$Physical Meaning: The first term in each parenthesis ($qL/2$) is the simple beam reaction. The second term ($\Delta M / L$) is a shear correction caused by the continuity moments. This is why reactions at middle supports are higher—they help carry the load from adjacent spans.
Bridge Design: Multi-span highway and railway bridges are classic continuous beams. Using continuity reduces the depth of the deck required, leading to more economical and slender designs. Engineers use this exact analysis to determine maximum moments and shear forces for reinforcement.
Building Floor Systems: Continuous reinforced concrete beams supporting floor slabs are analyzed this way. The load pattern parameter lets you model different live load arrangements (like alternate spans loaded) to find the most critical bending envelope for design.
Industrial Gantry Cranes: The runway beams for overhead cranes are often continuous over multiple supports. Accurate reaction force calculation is vital for designing the supporting columns and foundations to handle the moving load.
CAE Model Validation: Before trusting complex Finite Element Analysis (FEA) of a structure, engineers create simple beam models. The results from this calculator serve as an exact benchmark to verify that the FEM software's beam elements and boundary conditions are set up correctly.
When you start using this tool, there are a few common pitfalls. First, are you perhaps assuming that all supports are the same as pin supports? The model in this tool treats internal supports as being free to rotate but restrained in vertical movement, much like rollers. However, actual bridge piers are often rigidly connected to foundations or ground and can deform. Keep in mind that the tool's results are a first approximation assuming "ideal, rigid supports."
Next, when setting parameters, get into the habit of considering the "Second Moment of Area I" and the "Modulus of Elasticity E" together. Their product "EI" is the flexural rigidity, the crucial value that determines deflection. For example, using a section with the same I for steel (E=205 GPa) and concrete (E=30 GPa) results in a deflection difference of about 7 times. If you reduce E by a factor of 10 in the tool, you should see the deflection jump by 10 times. In practice, once you choose a material, understanding the EI is your starting line.
Finally, note that the location of maximum deflection does not coincide with the location of maximum bending moment. For a simply supported beam with a uniformly distributed load, it's at the center, but for a continuous beam, it shifts away from the mid-span. For a two-span beam with a uniform load, the maximum deflection is slightly toward the ends from each span's center. Take a close look at the tool's deflection curve and check where the peak is. In design, you check whether the deflection at this location exceeds regulatory limits.