Moment Distribution
Moment distribution is an analysis technique suited to statically indeterminate beams
and simple frame structures. Although moment distribution is an analysis technique it
is typically used in design to calculate design moments.
The method of moment distribution is a simplified version of the stiffness method of
analysis where all the degrees of freedom are joint rotations. The stiffness method
involves solving a series of equilibrium equations. These equations are typically
written in matrix form as,
S D  FR 
Where S  is the stiffness matrix, FR  is the vector of restraint forces required to
prevent movement of the degrees of freedom when the loads are applied, and D is
the vector of the unknown displacements. Each equation represented by this matrix
expression has the form
si1d1  si 2 d 2    sij d j    sin d n  f R i  0
Where a stiffness coefficient sij is the force at degree of freedom i associated with a
unit displacement at degree of freedom j. If all the degrees of freedom are joint
rotations, then the general term sij d j equals the moment at degree of freedom i
associated with a unit displacement at j.
By solving the system of linear equilibrium equations represented by the matrix
equation the unknown displacements can be calculated. However, prior to the advent
of digital computers the solution of large systems of simultaneous linear equations
was infeasible. Nevertheless iterative techniques were developed that facilitated
identifying the solution to an acceptable degree of accuracy. Such methods are
sometimes referred to as relaxation techniques. The moment distribution procedure is
such a method.
Consider the analysis problem shown below
A
2q kN/m
L
B
q kN/m
L
C
D
L
In the typical stiffness formulation we consider the degrees of freedom to be
restrained initially and then attempt to calculate the magnitude of the final rotations of
each of the nodes. In the moment distribution approach we don’t release all the
restraints simultaneously. Rather, we release and then re-clamp them in turns until we
reach the stage that releasing the restraints does not alter the beam’s displaced shape.
This procedure works because at each stage we are attempting to find only a single
variable. In essence we are freezing the rotation of all nodes except the released node
and then solving for the rotation of the released node that is required for equilibrium.
To make the method easy to manage, and recognising that what is typically required
are the member moments rather than joint rotations, the equilibrium equations are
arranged such that the member end moments are the variables that are sought.
Carry over factor
Consider the stiffness matrix for a simple beam element.
2EI/L
1
 4 EI

S    2LEI

 L
4EI/L
2 EI 
L 
4 EI 

L 
If one end of a beam is rotated by an angle  then the moment in that end of the beam
4 EI
  while the moment in the other end of the beam, where the end is
will be
L
2 EI
  (which is exactly half the moment in the end
prevented from rotating will be
L
which was rotated).
Consider a number of beam elements intersecting at a point, assume that all the beams
are connected at the intersection point such that there can be no relative rotation of
their ends.
If a concentrated moment M is applied to the intersection point then the intersection
point will rotate. If the beams are not all collinear then the intersection point will not
translate.
2E1I 1
L1
4E3I3
L3
4E1I 1
L1
2E3I3
L3
4E2I 2
L2
2E2I 2
L2
The amount of rotation of the intersection point will be such that the sum of the
internal moments developed in the beam ends equals the externally applied moment
M. Furthermore, because all the member ends undergo the same rotation,
M  M R1  M R 2    M R _ No.M   
No . M

i 1
4Ei I i
Li
Thus, if a concentrated moment is applied to the intersection point the moment is
resisted by the members intersecting at the point in direct proportion to their stiffness.
That is, the stiffest members resist most moment.
Note:
If any of the beam elements are pinned then their contribution to resisting the moment
is
1

3EI
L
3EI/L
Overview of the moment distribution procedure
The moment distribution analysis method can be broken into a number of simple
steps.
Step 1
Calculate the stiffness of each of the beam elements of the structure. Use these
stiffnesses to calculate the relative stiffness of all the members meeting at each
intersection.
Step 2
Imagine that all the intersection points have been restrained against rotation. Apply
the loads to the structure and calculate the magnitude of the fixed-end moments for
each beam element. This equates to calculating the magnitude of the external
restraints necessary to prevent rotation of the joints.
Step 3
This is an iterative step and it will be necessary to apply this step successively until
the system has converged, of a sufficiently accurate solution has been obtained.
Release an intersection point, i.e. remove the external restraint. Once the restraint has
been removed the joint would rotate until there is no external moment necessary for
equilibrium. Now the end moments of the beams intersecting at the node equal the
applied moment at that node. However, the change in the end moments of each of the
members intersecting at the released node affects the restraining force required at the
members other ends. The change in the restraining moment at the members’ other
ends equals half the change in the beams end moment at the released node. Once the
node in question has been released, the end moments balanced and the carry over
factors distributed, the node is then restrained and a different node is released.