Moment distribution method
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross.
From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
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.
In order to apply the moment distribution method to analyse a structure, the following things must be considered.
Fixed end moments are the moments produced at member ends by external loads.Spanwise calculation is carried out assuming each support to be fixed and implementing formulas as per the nature of load ,i.e. point load ( mid span or unequal) ,udl,uvl or couple.
What is needed in the moment distribution method is not the specific values but the ratios of bending stiffnesses between all members.
When a joint is being 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' bending stiffness.
Distribution factors can be defined as the proportions of the unbalanced moments carried by each of the members.
In mathematical terms, the distribution factor of member
This balancing moment is then carried over to the member's other end.
developed at end B is found, the carryover factor of this member is given as the ratio of
: In case of a beam of length L with constant cross-section whose flexural rigidity is
, therefore the carryover factor Once a sign convention has been chosen, it has to be maintained for the whole structure.
In the BMD case, the left side moment is clockwise direction and other is anticlockwise direction so the bending is positive and is called sagging.
Framed structure with or without sidesway can be analysed using the moment distribution method.
The statically indeterminate beam shown in the figure is to be analysed.
The beam is considered to be three separate members, AB, BC, and CD, connected by fixed end (moment resisting) joints at B and C. In the following calculations, clockwise moments are positive.
The bending stiffness of members AB, BC and CD are
Therefore, expressing the results in repeating decimal notation: The distribution factors of joints A and D are
Step 2 ends with carry-over of balanced moment
Joint B is released once again to induce moment distribution and to achieve equilibrium.
* Steps 5 - 10: Joints are released and fixed again until every joint has unbalanced moments of size zero or neglectably small in required precision.
For comparison purposes, the following are the results generated using a matrix method.
Note that in the analysis above, the iterative process was carried to >0.01 precision.
The fact that the matrix analysis results and the moment distribution analysis results match to 0.001 precision is mere coincidence.
Developing complete bending moment diagrams require additional calculations using the determined joint moments and internal section equilibrium.
As the Hardy Cross method provides only approximate results, with a margin of error inversely proportionate to the number of iterations, it is important[citation needed] to have an idea of how accurate this method might be.
Replacing the values presented above in the equation and solving it for
