The three-moment equation is a structural analysis tool used to find support moments in continuous beams. It connects the bending moments at three consecutive supports using span lengths, flexural rigidity, loads, and support settlements. This cheat sheet helps engineering students organize the equation, identify the needed load-area terms, and apply consistent signs.
It is especially useful when analyzing indeterminate beams by hand before using matrix methods or software.
The core idea is that beam continuity creates compatibility of rotation between adjacent spans. For constant EI, the standard equation relates M_A, M_B, and M_C to the first moments of the simple-span bending moment diagrams. If EI varies, each span contribution is weighted by L/EI and by the load diagram area divided by EI.
Settlement terms must be included when supports move vertically, because they introduce additional rotation compatibility effects.
Key Facts
- For two adjacent spans AB and BC with constant EI, the three-moment equation is M_A L_1 + 2 M_B(L_1 + L_2) + M_C L_2 = -6(a_1 x_1/L_1 + a_2 x_2/L_2) + 6EI(Delta_1/L_1 + Delta_2/L_2).
- In the constant EI equation, L_1 and L_2 are the lengths of spans AB and BC, and M_A, M_B, and M_C are the support moments at A, B, and C.
- The term a_1 x_1 is the first moment of the simple-span bending moment diagram on span AB about support A, and a_2 x_2 is the first moment of the simple-span bending moment diagram on span BC about support C.
- For a uniformly distributed load w over a simple span L, the area of the simple-span moment diagram is a = wL^3/12, and its centroid is at L/2 from either support.
- For a point load P at distance a from the left support and b from the right support, where L = a + b, the area of the simple-span moment diagram is a_m = Pab/2.
- End supports that are simple pins or rollers usually have zero end moment, so M_A = 0 or M_C = 0 when the end is not fixed.
- If all supports are at the same elevation, the settlement term is zero, so the equation contains only support moment terms and load-area terms.
- A consistent sign convention is required, and hogging support moments are commonly taken as negative while sagging span moments are commonly taken as positive.
Vocabulary
- Continuous beam
- A beam that extends over more than two supports and has internal force continuity across intermediate supports.
- Three-moment equation
- An equation that relates the bending moments at three consecutive supports of a continuous beam using compatibility of rotations.
- Flexural rigidity
- Flexural rigidity, written EI, is the product of the material modulus of elasticity E and the section moment of inertia I.
- Support moment
- A support moment is the bending moment at a beam support caused by continuity, loading, fixity, or settlement.
- First moment of area
- The first moment of a bending moment diagram is the area of the diagram multiplied by the distance from a chosen reference support to its centroid.
- Support settlement
- Support settlement is vertical movement of a support that changes beam compatibility and can create additional bending moments.
Common Mistakes to Avoid
- Using the real continuous-beam moment diagram for a_1 and a_2 is wrong because the three-moment equation requires the simple-span bending moment diagrams caused by loads on each span.
- Forgetting that x_1 and x_2 are measured from different reference supports is wrong because the first moment terms must match the form of the equation being used.
- Dropping the EI factors when flexural rigidity is not constant is wrong because spans with different EI values do not contribute equally to rotation compatibility.
- Assigning nonzero moment at a simple end support is wrong because a pin or roller cannot resist bending moment unless an external couple is applied.
- Mixing sagging-positive and hogging-positive sign conventions is wrong because inconsistent signs can make the computed support moments have the wrong direction.
Practice Questions
- 1 A continuous beam has two equal spans, L_1 = L_2 = 6 m, constant EI, simple end supports at A and C, and a uniformly distributed load w = 12 kN/m on both spans. Using M_A = 0 and M_C = 0 with no settlement, find M_B.
- 2 For a simple span of length 8 m carrying a uniformly distributed load of 5 kN/m, calculate the area of the simple-span bending moment diagram and the location of its centroid.
- 3 A point load P = 20 kN is placed 3 m from the left support and 5 m from the right support of a simple span. Calculate the area of the simple-span bending moment diagram.
- 4 Explain why the three-moment equation is a compatibility equation rather than only an equilibrium equation.
Understanding Three-Moment Equation for Continuous Beams
The equation comes from beam curvature. A bending moment causes curvature, and curvature builds up into slope and deflection along a span. At an interior support, the beam is one physical member, so the two spans cannot choose unrelated rotations there.
The three-moment method turns that condition into a relationship among nearby support moments. The loading part is found from the bending moment diagram of a simply supported version of each span.
This is not the final diagram for the continuous beam. It is a temporary reference diagram that measures how much the loads tend to rotate the ends of that span.
A reliable hand method starts by drawing the whole beam, naming supports from left to right, and listing every span length. Mark fixed ends, pin ends, rollers, internal supports, loads, and any stated settlement. Then write one three-moment equation for each group of three neighboring supports.
A beam with four supports gives two equations, while a beam with five supports gives three. Apply known end moments before solving. The unknown interior support moments can then be found as a set of simultaneous equations.
After that, analyze each individual span using equilibrium. The known end moments and applied loads give the reactions, shear force diagram, and final bending moment diagram.
Sign errors are the main source of wrong answers. Choose one convention before doing any arithmetic and keep it in every diagram and equation. A common choice treats sagging moments as positive and hogging moments as negative.
This makes interior support moments under downward gravity loading negative in many ordinary beams. Check the physical shape of the final diagram. Near an internal support, a continuous beam often bends upward because the neighboring spans restrain it.
A result showing a large sagging moment at every interior support should be checked carefully. Units provide another quick check. Moment has units of force times length, while terms involving diagram areas carry higher powers of length before they are combined in the equation.
Support settlement matters because a beam can be forced to bend even when no external load is added. If one column settles in a building frame, or one pier moves in a bridge, the connected beam must adjust to the new support positions. This adjustment changes reactions and support moments.
The direction of movement matters, so settlement values need a stated positive direction. Students should separate settlement effects from load effects at first. Solve and interpret each part clearly, then combine them.
The three-moment method is most useful for straight continuous beams with small deflections and linear elastic behavior. It becomes less suitable when members have major changes in stiffness, hinges inside spans, large deformation, or material yielding. In those cases, stiffness methods and structural software are usually better, but the same ideas about equilibrium, compatibility, and sign control still apply.