Moment Distribution method

The Moment Distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross, in 1930 .The moment distribution method falls into the category of displacement method of structural analysis.

The method only accounts for flexural effects and ignores axial and shear effects.

In the moment distribution method, every joint of the structure to be analysed is fixed so as to develop the fixed-end moments. Then each fixed joint is sequentially released and the fixed-end moments (which by the time of release are not in equilibrium) are distributed to adjacent members until equilibrium is achieved.

The moment distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration.

Distribution factors

When a joint is released and begins to rotate under the unbalanced moment, resisting forces develop at each member framed together at the joint. Although the total resistance is equal to the unbalanced moment, the magnitudes of resisting forces developed at each member differ by the members' flexural stiffness. Distribution factors can be defined as the proportions of the unbalanced moments carried by each of the members.

In mathematical terms, distribution factor of member $k$ framed at joint $j$ is given as:

$D_{jk} = \frac{\frac{E_k I_k}{L_k}}{\sum_{i=1}^{i=n} \frac{E_i I_i}{L_i}}$

where n is the number of members framed at the joint.

The sum of all distribution factors at joint will always equal to 'one'

Carryover factors

When a joint is released, balancing moment occurs to counterbalance the unbalanced moment which is initially the same as the fixed-end moment. This balancing moment is then carried over to the member's other end. The ratio of the carried-over moment at the other end to the fixed-end moment of the initial end is the carryover factor. It is equal to 0.5
Method of Analysis

Calculate stiffness factors for each member
Calculate distribution factors at both ends of each member
Determine carryover factors at both ends of each member
Assume all joints are fixed and calculate fixed-end moments for each member
Balance pinned (to zero) and cantilevered ends and distribute half the moment to the opposite end.
Distribute the unbalanced moments at all other joints to each adjacent member based on the distribution factor.
Carryover the distributed moments to the opposite ends of the each member using the carryover factors.
Iterate steps 6 and 7 until moment imbalance at each joint approaches zero

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