# Moment Distribution Method

By Aina Parasher|Updated : June 16th, 2022

Moment Distribution Method: There are two basic methods of structural analysis in which one is the displacement method/ equilibrium method and another one is the force method/ compatibility method. The moment distribution method is one of the displacement methods. The moment distribution method is used for statically indeterminate beams and frames.

The moment distribution method is one of the most widely practised methods. In this article, we will have a brief discussion on the displacement method and force method. We will discuss the moment distribution method in detail.

## What is Moment Distribution Method?

The moment distribution method is a practical method for analysing statically indeterminate beams and rigid frames. Every joint of the structure to be analysed is fixed in the moment distribution method in order to create fixed-end moments. Then, one by one, each fixed joint is released, and the fixed-end moments (which are not in equilibrium at the time of release) are transferred to nearby members until equilibrium is attained. Furthermore, the moment distribution approach may be proved mathematically as the process of iteratively solving a set of simultaneous equations. Let us discuss the displacement method and force method.

### Displacement Method

The displacement method is a method of structural analysis in which displacements are treated as unknowns. The number of unknowns in the displacement method is treated as equal to Kinematic Indeterminacy. Different types of displacement methods are Slope Deflection Method, Moment Distribution Method, Kani’s Method & Stiffness Matrix Method.

### Force Method

The Force Method is a method of structural analysis in which redundant forces are treated as unknowns. The number of redundant forces is treated as equal to static indeterminacy. Different types of force methods are flexibility matrix method, consistent deformation method, column analogy method & elastic centre method.

## Analysis of Beam Using Moment Distribution Method

The moment distribution method was developed by Hardy Cross. In the moment distribution method, displacements are treated as unknowns. The moment distribution method is also known as the equilibrium or displacement method. It is suitable for analyzing indeterminate structures like beams and rigid jointed frames.

In the moment distribution method, the acting axial forces and corresponding axial force deformations are neglected. The moment distribution method is applicable for both prismatic and non-prismatic members.

## Carry Over Factor (COF)

Carry Over Factor is the ratio of the far-end moment and near-end moment. Carry Over Moment (COM) is also called a developed moment or induced moment. It is the moment developed at one end due to the moment at another end.

COF= COM/ Applied Moment

Carry Over Factors for different cases are:

Case 1: When a far end is fixed

COF = ½

Case 2: One end is hinge

COF = 0

Case 3: Cantilever Beam

COF = -1

Case 4: Far-end guided roller

COF = -1

### Stiffness Factor/ Rotational Stiffness

The Stiffness Factor is the moment required to produce a unit rotation.

Case 1: Far end is fixed

M= 4EI/L

Case 2: Far end is hinged

M= 3EI/L

Case 3: Far end is free

M= 0

Case 4: Far end is a guided roller

M= EI/L

## Moment Distribution Method Problems

1. Find the stiffness factor or rotational stiffness at O.

Solution-

δ= 3EI/L+ 4EI/L+0+ EI/L

δ= 8EI/L

2. Find the relative stiffness & distribution factors of members.

Solution –

KAB=3EI/L

KAC=3EI

KAE=0

The distribution factor is the ratio in which the moment sharing capacity of various members meets at a rigid joint.

DF= K/∑K

(DF)AB= KAB/ ∑KA= 4/11

(DF)AC= KAC/ ∑KA= 3/11

(DF)AE= KAE/ ∑KA= 0

Distribution factor at a rigid joint = 1

For fixed support, distribution factor = 0

For hinged or roller beam, distribution factor = 1

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## FAQs on Moment Distribution Method

• Moment Distribution Method was developed by Hardy Cross. It is a structural analysis method to analyze statically indeterminate structures like beams, rigid jointed frames with internal hinges. The moment distribution method only accounts for flexural effects and not for axial & shear effects.

• Moment distribution method is not a force method, it is a displacement method in which the number of unknown displacements is treated equal to kinematic indeterminacy. In moment distribution method, every joint of the structure to be analyzed is fixed so as to develop the fixed-end moments.

• The primary unknown obtained moment distribution method are forces in members of beams, rigid jointed frames etc. Compatibility equations are written for Carry Over Factor, Stiffness Factor and Distribution Factor.

• Carry over factor is the ratio of far-end moment and near-end moment. It is also known as a developed moment or induced moment. The balancing moment is the same as the end moments initially and then gets carried over to other end. This ratio of a carried-over moment to the other end to the fixed end is called carry over factor.

• Stiffness Factor is the moment required to produce a unit rotation. Rotational stiffness is directly proportional to material and geometric stiffness. Beam stiffness is the moment required to produce unit rotation at the simply supported end of a beam, the other end being rigidly fixed. GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 help@byjusexamprep.com