Solve a two-span indeterminate continuous beam by the slope-deflection (displacement) method. Adjust the span lengths, flexural rigidity and distributed loads to see the joint rotations, interior support moment, reactions and bending-moment diagram update in real time — and build an intuition for how indeterminate structures are solved.
Parameters
Span 1 (AB) length L1
m
Span 2 (BC) length L2
m
Flexural rigidity EI
kN·m²
Product of Young's modulus E and second moment of area I
UDL on span 1, w1
kN/m
UDL on span 2, w2
kN/m
Results
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Interior support moment M_B (kN·m)
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Rotation at A, θ_A (rad)
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Rotation at B, θ_B (rad)
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Max sagging moment, span 1 (kN·m)
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Interior reaction R_B (kN)
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Analysis status
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Continuous-beam model — loads & deflected shape
A two-span continuous beam with three supports: a pin at A, an interior support at B and a roller at C. The uniformly distributed loads and the deflected shape, including joint rotation, are shown.
The slope-deflection equation. M_ij: end moment at joint i of member ij, θ: joint rotation, M^F: fixed-end moment. For a uniformly distributed load w on a span of length L, the fixed-end moment is wL²/12.
$$M_{BA}+M_{BC}=0$$
Moment equilibrium at the interior joint B. Because the simple end supports do not restrain rotation, they impose the boundary conditions M_AB = M_CB = 0; together these three equations solve for the three unknown rotations θ_A, θ_B and θ_C.
What is the Slope-Deflection Method?
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The "slope-deflection method" is a way to solve indeterminate beams, right? But first — why is an "indeterminate" beam hard to solve at all?
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Good question. For a statically determinate beam, the equilibrium equations alone — vertical force and moment — fix all the reactions. But once you have three or more supports, like this continuous beam, there are more unknown reactions than equilibrium equations. In other words, you simply do not have enough equations. The missing information has to come from how the beam actually deforms — its deflections and rotations.
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I see. So how does the slope-deflection method bring that "deformation" into the picture?
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Roughly speaking, it flips the problem around. You would normally want to make the forces (the moments) the unknowns. The slope-deflection method instead takes how much each joint rotates — the rotation θ — as the unknown. Each member's end moment is written as a linear function of θ: M = (2EI/L)(2θ_i+θ_j) + the fixed-end moment. Then you just enforce "the sum of moments is zero" at each joint, which gives a set of simultaneous equations in θ. Solve those, and the rotations are known — and the moments and reactions all fall out from there.
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What is the "fixed-end moment" that appears in the equation? Changing the load on the left moves the results, so it must be doing something.
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The fixed-end moment (FEM) is the end moment the load produces if you first imagine both ends clamped rigidly. For a uniformly distributed load its magnitude is wL²/12. The slope-deflection procedure has two stages: first you picture a fully-restrained state where every joint is locked, then you release the locks and let the joints rotate. The FEM is the "locked-state" share, and the (2EI/L)(2θ_i+θ_j) term is the "release correction". So changing the load w changes the FEM, and the results change with it.
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With the default settings, the rotation at B comes out almost zero. Is that a bug, or does it actually mean something?
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That is a correct result. Both spans have the same length and carry the same uniformly distributed load — the beam is symmetric. When a symmetric structure carries a symmetric load, the central joint B has the "tendency to rotate left" and the "tendency to rotate right" cancel exactly, so it does not rotate at all. Hence θ_B = 0. In that case the interior support moment becomes M_B = wL²/8, the familiar textbook value. Make the span lengths or loads asymmetric and B will start to rotate.
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Is this method actually used in practice, or is it an old technique that computer analysis has replaced?
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As a hand calculation it is no longer used for large structures, but the idea itself is very much alive. The slope-deflection method follows the flow "take displacements (rotations) as unknowns, write stiffness relations, and solve with equilibrium." That is exactly what today's structural-analysis software does — the matrix stiffness method, which is the finite-element method. The slope-deflection method is, in effect, the hand-calculation version and the direct ancestor of FEM. Understanding it makes it click what the software is doing inside.
Frequently Asked Questions
The slope-deflection method is a displacement (stiffness) method that solves an indeterminate structure by taking the joint rotations as the unknowns. Each member's end moment is written as a linear function of the joint rotations plus the fixed-end moment, M_ij = (2EI/L)(2θ_i+θ_j) + M^F_ij, and moment equilibrium is enforced at every joint to solve for the rotations. Once the rotations are known, all end moments, reactions and the bending-moment diagram follow. It is the direct conceptual ancestor of the modern finite-element (matrix stiffness) method.
A fixed-end moment is the moment that appears at a member end when both ends are fully clamped and the member carries its load. The slope-deflection method first uses the FEMs as the end moments of an imagined fully-restrained state, then relaxes the joints and lets them rotate. For a uniformly distributed load w on a span of length L, the magnitude of the FEM is wL²/12. This tool takes clockwise member-end moments as positive, with the left-side end negative and the right-side end positive.
A simple support (pin or roller) does not restrain rotation, so the member cannot carry a bending moment there. When both ends A and C of a continuous beam are simply supported, the boundary conditions are M_AB = 0 and M_CB = 0. These two equations, together with the moment equilibrium at the interior joint B (M_BA + M_BC = 0), give three equations for the three unknown rotations θ_A, θ_B and θ_C. This tool solves that system internally.
For a symmetric continuous beam — two equal spans carrying equal uniformly distributed loads — the interior joint B does not rotate, so θ_B = 0. The interior support moment is then M_B = wL²/8, the end rotations are θ_A = wL³/(48EI), the maximum sagging moment near mid-span is 9wL²/128, and the interior reaction is R_B = 1.25wL. Entering the default values (L1 = L2 = 6 m, w1 = w2 = 10 kN/m) reproduces these textbook results.
Real-World Applications
Design of bridges with continuous girders: The girders of road bridges, footbridges and viaducts are often designed as continuous beams spanning several piers. A continuous beam has a smaller mid-span deflection than a chain of simple spans, allowing a shallower girder. In exchange, a large negative (hogging) bending moment develops over the supports, so the layout of reinforcement or steel at that section becomes a critical point of the design. The slope-deflection method is the classical hand tool for estimating those support moments.
Building floor and foundation beams: In reinforced-concrete and steel buildings, the main floor beams supported by a row of columns act as continuous beams. Because tension develops on the top fibre over a support, RC beams need extra top reinforcement near the supports. Changing the span ratio and loads in a tool like this to see how the support moment grows or shrinks builds a feel for reinforcement planning.
An educational gateway to the matrix stiffness method and FEM: The slope-deflection method shows the skeleton of every displacement method — take displacements (rotations) as unknowns, link forces and displacements with stiffness coefficients, and solve with equilibrium — in its simplest form. This is exactly what the matrix stiffness method inside modern structural-analysis software does. Solving slope-deflection problems by hand before learning FEM makes stiffness matrices and boundary-condition handling intuitive.
Sanity-checking CAE results: When you build a continuous-beam model in beam-element structural-analysis software, you can verify whether the support moments and reactions are reasonable by comparing them with a slope-deflection estimate. A symmetric model should match the known results θ_B ≈ 0 and M_B ≈ wL²/8; if the output is wildly different, it is a clue to suspect a mistake in the support conditions or load input.
Common Misconceptions and Pitfalls
The most common mistake is mixing up the sign conventions. The slope-deflection method usually adopts the convention "clockwise member-end moments are positive", which is a different convention from "sagging (concave up) is positive" used for the bending-moment diagram in mechanics of materials. The sign of the fixed-end moments, the way the equilibrium equations are written and the conversion when finally drawing the BMD — confuse the convention at any one of these points and the sign flips, reversing the positive/negative of the support moment. This tool keeps a consistent convention: clockwise positive, left end negative, right end positive. When you cross-check by hand, always fix your convention first.
Next is confusing the "degree of indeterminacy" with the "number of unknowns". This two-span continuous beam has four reactions and two equilibrium equations, so its degree of indeterminacy is two — but the number of unknowns the slope-deflection method actually solves is three rotations (θ_A, θ_B, θ_C). In a displacement method the number of unknowns is set by the degrees of freedom (the number of joints that can rotate), not by the degree of indeterminacy itself. Conversely, the force method (the dual of the slope-deflection method) introduces as many unknown forces as the degree of indeterminacy. For the same problem, the number and type of unknowns change with the method you choose.
Finally, the range of validity of this model. This tool solves under the assumptions of no support settlement, straight members with uniform flexural rigidity EI, and small, linear-elastic deformations. In real bridges and buildings, differential settlement of the supports can change the support moments substantially, and temperature change, creep and axial-force effects (buckling, P-δ effects) cannot always be ignored. The slope-deflection method can be extended to treat support settlement as an extra term in θ, but this tool is limited to the basic no-settlement form. Use the results understanding that they are values for an idealised model.
How to Use
Enter span lengths L1 and L2 (meters) for your continuous beam—typical range 3–8 m for building frames.
Input flexural rigidity EI (kN·m²): for steel I-beams use 8000–25000; for reinforced concrete use 4000–12000 depending on section.
Specify distributed load w (kN/m) applied to span 1; the simulator solves joint rotations θ_A and θ_B, support moment M_B, and max sagging moment using slope-deflection equilibrium equations.
Review outputs: interior reaction R_B (kN) and moment distribution to verify structural behavior.
Worked Example
A two-span continuous beam: L1 = 5 m, L2 = 6 m, EI = 12000 kN·m² (RC beam), w = 8 kN/m on span 1. The simulator applies slope-deflection equations M_AB = (2EI/L)(2θ_A + θ_B − 3Δ/L) + FEM to compute: M_B ≈ 34.2 kN·m (negative, indicating tension in top fiber), θ_B ≈ −0.0018 rad, max sagging moment in span 1 ≈ 18.6 kN·m, R_B ≈ 28.5 kN. These values reflect typical mid-story floor beam design in reinforced concrete structures.
Practical Notes
Unequal spans (L1 ≠ L2) create asymmetric moment distribution; span 1 typically governs sagging design moment when it carries load.
For cantilever-like overhang effects, ensure EI accurately reflects actual section properties; undercounting stiffness inflates deflections by 15–30%.
Interior support moment M_B changes sign if load shifts to span 2; use this sensitivity to identify critical load cases for reinforcement placement.
Check that rotation magnitudes (θ in radians) remain under 0.01 rad for serviceability; higher rotations suggest oversized spans or undersized sections.