Mechanics of deformable solids Thin bodies and structures Theorem

Three-moment equation

Builds on Euler–Bernoulli beam equation Bending moment Shear and moment diagram Centroid Statically indeterminate structure Conjugate beam method

Statement

Let a continuous beam with constant bending stiffness $EI$ rest on supports $0, 1, \dots, N$ at the same level, which do not settle. Let span $i$ run from support $i - 1$ to support $i$ and have length $L_i$ , and assume the Euler–Bernoulli beam equation. Let $M_i$ be the bending moment at support $i$ , with sagging taken positive. For each span $i$ , let $A_i$ be the area of the moment diagram of its loads on the span taken as simply supported. Let $a_i$ be the distance of the centroid of that area from support $i - 1$ , and $b_i = L_i - a_i$ its distance from support $i$ . Then for each interior support $n$ , with $1 \le n \le N - 1$ , $M_{n-1} L_n + 2 M_n (L_n + L_{n+1}) + M_{n+1} L_{n+1} = -\frac{6 A_n a_n}{L_n} - \frac{6 A_{n+1} b_{n+1}}{L_{n+1}}.$

Remarks

The proof writes the moment in each span as the free moment plus the linear moment from the support moments. The slope of the beam at support $n$ is then computed from the left span and from the right span, for example by the moment-area theorems. Setting the two slopes equal, since the beam is continuous over the support, gives the equation.

Example: for a uniform load $w_i$ on span $i$ , the free diagram is a parabola with $A_i = w_i L_i^3/12$ and $a_i = b_i = L_i/2$ , so the right side is $-(w_n L_n^3 + w_{n+1} L_{n+1}^3)/4$ . For two equal spans $L$ with load $w$ and simply supported ends, $M_0 = M_2 = 0$ gives $4 M_1 L = -wL^3/2$ , so $M_1 = -wL^2/8$ , a hogging moment over the middle support.

If the supports settle or the stiffness differs between spans, extra terms appear, and the equation above is false as written.

Going deeper

For a student

A continuous beam is a statically indeterminate structure, and the support moments are a natural choice of redundant unknowns. The equations form a tridiagonal linear system, which is easy to solve even for many spans. Once the $M_i$ are known, each span is a simply supported beam with known end moments, and its shear and moment diagram follows by superposition. The same equation is often derived with the conjugate beam method.

Reading path

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Bending moment concept
Centroid concept
Conjugate beam method concept
Euler–Bernoulli beam equation concept
Shear and moment diagram concept
Statically indeterminate structure concept
Three-moment equation statement

Tag	`ZGCR9KA1PT`
URI	`m2://ZGCR9KA1PT`
Kind	statement, theorem
Writing	W0 stub
MSC	74K10, 74G05
Status	This copy of the site has no build data. Run `m2 status ZGCR9KA1PT`.
About	Bending moment
Uses	Euler–Bernoulli beam equation, Bending moment, Shear and moment diagram, Centroid, Statically indeterminate structure, Conjugate beam method
Same as	Wikidata Q7782348 Wikipedia Theorem of three moments
Changed	3 November 2026

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