# Theorem of three moments

In civil engineering and structural analysis Clapeyron's theorem of three moments is a relationship among the bending moments at three consecutive supports of a horizontal beam.

Let A,B,C be the three consecutive points of support, and denote by l the length of AB by $l'$ the length of BC, by w and $w'$ the weight per unit of length in these segments. Then[1] the bending moments $M_A,\, M_B,\, M_C$ at the three points are related by:

$M_A l + 2 M_B (l+l') +M_C l' = \frac{1}{4} w l^3 + \frac{1}{4} w' (l')^3.$

This equation can also be written as [2]

$M_A l + 2 M_B (l+l') +M_C l' = \frac{6 a_1 x_1}{l} + \frac{6 a_2 x_2}{l'}$

where a1 is the area on the bending moment diagram due to vertical loads on AB, a2 is the area due to loads on BC, x1 is the distance from A to the center of gravity for the b.m. diagram for AB, x2 is the distance from C to the c.g. for the b.m. diagram for BC.

The second equation is more general as it does not require that the weight of each segment be distributed uniformly.

Figure 01-Sample continuous beam section

## Derivation of Three Moments Equations

Mohr's Theorem[3] can be used to derive the Three Moment Theorem[4] (TMT).

### Mohr's First Theorem

The change in slope of a deflection curve between two points of a beam is equal to the area of the M/EI diagram between those two points.(Figure 02)

Figure 02-Mohr's First Theorem

### Mohr's Second Theorem

Consider two points k1 and k2 on a beam. The deflection of k1 and k2 relative to the point of intersection between tangent at k1 and k2 and vertical through k1 is equal to the M/EI diagram between k1 and k2 about k1.(Figure 03)

Figure03-Mohr's Second Theorem

Three Moment Equation expresses the relation between bending moments at three successive supports of a continuous beam, subject to a loading on a two adjacent span with or without settlement of the supports.

### The Sign Convention

According to the Figure 04,

1. The moment M1, M2, and M3 be positive if they cause compression in the upper part of the beam. ( Sagging positive)
2. The deflection downward positive. (Downward settlement positive)
3. Let ABC is a continuous beam with support at A,B, and C. Then moment at A,B, and C are M1, M2, and M3, respectively.
4. Let A' B' and C' be the final positions of the beam ABC due to support settlements.
Figure 04-Deflection Curve of a Continuous Beam Under Settlement

### Derivation of Three Moment Theorem

PB'Q is a tangent drawn at B' for final Elastic Curve A'B'C' of the beam ABC. RB'S is a horizontal line drawn through B'. Consider, Triangles RB'P and QB'S.

$\dfrac{PR}{RB'} = \dfrac{SQ}{B'S},$

$\dfrac{PR}{L1} = \dfrac{SQ}{L2}$

(1)

$PR = \Delta B - \Delta A + PA'$

(2)

$SQ = \Delta C - \Delta B - QC'$

(3)

From (1), (2), and (3),

$\dfrac{\Delta B - \Delta A + PA'}{L1} = \dfrac{\Delta C - \Delta B - QC'}{L2}$

$\dfrac{PA'}{L1} + \dfrac{QC'}{L2} = \dfrac{\Delta A -\Delta B}{L1} + \dfrac{\Delta C -\Delta B}{L2}$

(a)

Draw The M/EI diagram to find the PA' and QC'.

Figure 05 - M / EI Diagram

From Mohr's Second Theorem
PA' = First moment of area of M/EI diagram between A and B about A.

$PA' = \left(\frac{1}{2} \times \frac{M_1}{E_1 I_1} \times L_1\right)\times L_1\times \frac{1}{3} + \left(\frac{1}{2} \times \frac{M_2}{E_2 I_2} \times L_1\right)\times L_1\times\frac{2}{3}+ \frac{A_1 X_1}{E_1 I_1}$

QC' = First moment of area of M/EI diagram between B and C about C.

$QC' = \left(\frac{1}{2} \times \frac{M_3}{E_2 I_2} \times L_2\right)\times L_2\times\frac{1}{3} + \left(\frac{1}{2} \times \frac{M_2}{E_2 I_2} \times L_2\right)\times L_2\times\frac{2}{3}+ \frac{A_2 X_2}{E_2 I_2}$

Substitute in PA' and QC' on equation (a), Three Moment Theorem (TMT)can be obtain.

## Three Moment Equation

$\frac{M_1 L_1}{E_1 I_1}+ 2M_2\left(\frac{L_1}{E_1 I_1} + \frac{L_2}{E_2 I_2}\right)+\frac{M_3 L_2}{E_2 I_2} = 6 [\frac{\Delta A - \Delta B}{L_1} + \frac{\Delta C - \Delta B}{L_2}] - 6 [\frac{A_1 X_1}{E_1 I_1 L_1} + \frac{A_2 X_2}{E_2 I_2 L_2}]$

## Notes

1. ^ J. B. Wheeler: An Elementary Course of Civil Engineering, 1876, Page 118 [1]
2. ^ Srivastava and Gope: Strength of Materials, page 73
3. ^
4. ^