# Moment Distribution Method

## Displacement Method of Analysis (Moment Distribution Method)

The moment distribution method is the most suitable manual method for the analysis of continuous beams and plane frames. The method was presented by Prof. Hardy Cross of the USA in 1929.

The method consists in solving indirectly the equations of equilibrium as formulated in the slope deflection method without finding the displacements. This is an iterative procedure. This is also known as a relaxation method.

In this method the analysis begins by assuming each joint in the structure to be fixed. Then, by unlocking and locking each joint in succession, the internal moments at the joints are distributed and balanced until the joints have rotated to the final and nearly positions. The following examples has been given to illustrate the basic concept.

### Sign Convention

The clockwise moment is taken as +ve and the anti-clockwise moment is taken as -ve i.e., the sign convention is the same as that in slope deflection method for member end moment calculation.

## Member Stiffness Factor

Stiffness M/θ = K = 4EI/l and θ = Ml/4EI

Stiffness M/θ = K = 3EI/l and θ = Ml/3EI

## Joint Stiffness Factor

As joint A is rigidly connected, all member at joint A will rotate by the same amount θA. If a moment M is applied at joint A the moment gets distributed in all the connected members. As all connected members rotate by same amount θA at A. We have-

M = (4EI3θA/l3 + 4EI2θA/l2 + 4EI1θA/l1)

M/ θA = K1 + K2 + K3 = Joint Stiffness Factor (K)

## Distribution Factor (D.F.)

If a moment M is applied at joint A due to which the joint rotates by θA then,

Moment Distributed in AC= 4EI2θA/l2 = KAC θA = MDAC

Moment Distributed in AB= 3EI1θA/l1 = KAB θA = MDAB

MDAD : MDAC : MDAB = I3/l3 : I2/l2 : 3I1/4l1

Where,

I3/l3 = Relative stiffness of AD

I2/l2 = Relative stiffness of AC

3I1/4l1 = Relative stiffness of AB

Relative Stiffness when far end is fixed = I/l

Relative Stiffness when far end is hinged = 3I/4l

Distribution Factor of a Member = Stiffness of member/Total Stiffness of all members at the joint

## Carry Over Factor

When M moment is applied at pin, Moment carried over to fixed far end= M/2

Carry Over Factor= Carried over moment/Applied moment

However, In the following case, carry over factor = 0

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