About Continuous Beam Calculator
The continuous beam calculator analyses a multi-span beam resting on simple supports using the Three-Moment Theorem (Clapeyron's theorem). It returns the bending moment at each support, the vertical reaction at each support, the maximum sagging moment in every span, and a sampled bending-moment diagram for the whole member. The beam may have two, three, or four spans of any length.
Each span can carry a uniformly distributed load and any number of concentrated point loads. The solver assembles the three-moment equations for the interior supports, solves the resulting linear system for the unknown support moments, and then treats each span as a simply supported beam carrying its loads plus the end moments to recover the reactions and span moments by static equilibrium.
How It Works
- Enter the number of spans (2 to 4) and the length of each span.
- Add loads to each span: a uniformly distributed load (UDL) over the whole span and any concentrated point loads at a position measured from the left end of the span.
- The calculator builds the three-moment equation at each interior support, M_{i-1}*L1 + 2*M_i*(L1+L2) + M_{i+1}*L2 = -(load terms), and solves the system for the support moments.
- With the support moments known, each span is analysed as a simply supported beam under its loads plus the end moments to obtain the reactions and the bending-moment diagram.
Worked Example
A two-span continuous beam with two equal spans of L = 10 m carries a uniformly distributed load w = 12 kN/m over both spans. The end supports are simple so their moments are zero. The three-moment equation at the interior support reduces to 4*M*L = -w*L^3/2, giving the interior support moment M = -w*L^2/8 = -12*10^2/8 = -150 kN.m (hogging). The end reactions are 3*w*L/8 = 3*12*10/8 = 45 kN each, and the interior reaction is 10*w*L/8 = 1.25*w*L = 150 kN, which sum to the total load 2*w*L = 240 kN. The maximum sagging moment in each span is 9*w*L^2/128 = 84.375 kN.m, occurring 3*L/8 = 3.75 m from each end support.
Formulas
- Three-Moment Theorem (Clapeyron)
M_{i-1}*L1 + 2*M_i*(L1 + L2) + M_{i+1}*L2 = -( 6*A1*x1bar/L1 + 6*A2*x2bar/L2 )- Load term for a UDL span
6*A*xbar/L = w*L^3 / 4 (equal from both ends)- Load term for a point load (a from left, b = L - a from right)
from left: (P*b/L)*(L^2 - b^2) ; from right: (P*a/L)*(L^2 - a^2)- Two equal spans, full UDL (closed form)
M_interior = -w*L^2/8, R_end = 3*w*L/8, R_interior = 10*w*L/8 = 1.25*w*L
Standards & References
- Three-Moment Theorem (Clapeyron's theorem)
- Hibbeler, Structural Analysis
- Eurocode (EN 1992 / EN 1993 continuous-beam design)
Frequently Asked Questions
What is the Three-Moment Theorem used by the continuous beam calculator?
Clapeyron's Three-Moment Theorem relates the bending moments at three consecutive supports of a continuous beam to the loads on the two spans between them. Writing one equation per interior support gives a linear system whose solution is the set of support moments.
How many spans can the continuous beam calculator handle?
It analyses continuous beams on simple supports with two, three, or four spans of any length. Each span can carry a uniformly distributed load and any number of concentrated point loads.
Why are the support moments shown as negative?
Over an interior support a continuous beam bends with tension on top, which is hogging. The calculator uses the sagging-positive convention, so hogging support moments are reported as negative values.
Does the result depend on EI or the section properties?
For a beam of constant EI the flexural rigidity cancels out of the three-moment equations, so the support moments, reactions, and span moments depend only on the span lengths and the loads. EI is therefore not required as an input.