Continuous Beam Analysis: Methods, Moment Distribution, and Software Solutions
Understand continuous beam analysis including the moment distribution method, three-moment equation, and how structural analysis software handles multi-span beams. Practical examples for building structures.
Continuous beams — beams that span over three or more supports — are everywhere in structural engineering. Floor beams in multi-storey buildings, bridge decks, transfer beams, and even domestic floor joists often behave as continuous members. Unlike simply supported beams, continuous beams are statically indeterminate, meaning the equations of static equilibrium alone cannot determine all the reactions and internal forces. This guide explains the methods used to analyse continuous beams and how modern software makes the process efficient.
Why Continuous Beams Are Different
A simply supported beam with a UDL has a maximum moment of wL²/8 at midspan. But make that same beam continuous over an intermediate support, and the moment distribution changes dramatically. The hogging (negative) moment at the internal support can be as large as wL²/8 or more, while the sagging (positive) moments in the spans reduce significantly. This redistribution of moments is why continuous beams use steel more efficiently — smaller sections can carry the same loads. The trade-off is that the analysis is more complex, and you need to design for both hogging and sagging moments.
The Three-Moment Equation (Clapeyron's Theorem)
The three-moment equation relates the bending moments at three consecutive supports of a continuous beam. For a beam with equal spans L and uniform EI, the equation simplifies to: M(n-1) + 4Mn + M(n+1) = -(6EI/L²) × (δ terms). By writing one equation for each intermediate support and solving the resulting system of simultaneous equations, you obtain the support moments. This was the classical method used before computers, and it's still taught in university structural analysis courses.
Moment Distribution Method (Hardy Cross)
The moment distribution method, developed by Hardy Cross in 1930, revolutionised structural analysis. It's an iterative technique that avoids solving simultaneous equations directly. The process is: calculate fixed-end moments for each span assuming all joints are locked, release one joint at a time and distribute the out-of-balance moment according to stiffness factors, carry over half the distributed moment to the far end, and repeat until convergence. Most continuous beam problems converge within 3-5 cycles.
Distribution factors at each joint are calculated as: DF = k/(Σk), where k is the stiffness of each member meeting at the joint. For a prismatic beam with far end fixed, k = 4EI/L. For far end pinned, k = 3EI/L. Understanding these distribution factors is essential for grasping how continuous beams work.
Software Solutions: The Modern Approach
In practice, most engineers now use software for continuous beam analysis. The stiffness method (direct stiffness matrix method) is the foundation of all modern structural analysis programs. The beam is discretised into elements, each element's stiffness matrix is assembled, and the global system of equations [K]{d} = {F} is solved for nodal displacements, from which internal forces are recovered.
BeamBuddy handles continuous beams directly within Excel. Define your spans, supports, and loads — the stiffness matrix solver runs behind the scenes and outputs shear force diagrams, bending moment diagrams, and deflection profiles for the entire continuous beam. No need to set up matrices by hand.
Patterned Loading for Continuous Beams
A critical consideration for continuous beam design is patterned loading. Unlike simply supported beams where maximum effects occur with all spans loaded, continuous beams can have worse effects when some spans are loaded and others are not. For example, to find the maximum sagging moment in span 1, you load span 1 and span 3 (alternate spans) while leaving span 2 unloaded. Eurocode requires engineers to consider all adverse loading patterns. This typically means analysing multiple load cases to find the envelope of maximum effects.
Moment Redistribution in Eurocode
Eurocode 3 permits moment redistribution in continuous steel beams, allowing up to 15% redistribution from support moments to span moments for Class 1 sections, and up to 10% for Class 2 sections. This means if the elastic analysis gives a large hogging moment at a support, you can reduce it by up to 15% and increase the corresponding span moments to maintain equilibrium. This often leads to a more economical design by balancing the hogging and sagging moment demands.
Key Takeaways for Continuous Beam Design
- Continuous beams are more efficient than simply supported beams for multi-span structures.
- You must consider patterned loading — not just all spans loaded.
- Moment redistribution can make designs more economical.
- Software (like BeamBuddy) handles the complex analysis automatically.
- Always check both hogging and sagging moment regions — they may require different restraint conditions.
- Design the connections at internal supports for hogging moments and check for lateral torsional buckling of the bottom flange.
Ready to try beam analysis in Excel?
BeamBuddy brings professional structural beam analysis directly into Microsoft Excel. Shear force diagrams, bending moment diagrams, deflection checks, and 397 steel sections — all in your spreadsheet.