Like slope deflection method, moment distribution is a displacement method of analysis of indeterminate structures. In fact essentially it consists of solving the simultaneous equations of the slope deflection by an iterative technique. It’s not an approximate method but a method of successive approximation where any degree of accuracy can be obtained by repeated iteration. The moment distribution is not always recognized as a deformable variable method because the computations are carried in terms of moment produced by the changes in the deformable variable. This is all the more advantageous because in structural problems we are interested in finding the bending moments rather than slopes and deflections. Before the advent of computers, this method was developed by Hardy cross of the university of Illinois, Urbana, USA, in 1930. Even in this age of computer, the method will be useful in preliminary designs and in checking of computer results.
The moment distribution method starts from the same basic assumption made in the slope deflection method. In the analysis of continuous beams and frames all joints are assumed fixed and the moments are then corrected.
SIGN CONVENTION
Though different types of sign conventions are adopted by different authors in their books, yet the following sign conventions, which are widely used and internationally recognized will be used in this post:
1)All the clockwise moments at the ends are taken as positive
2) All the anticlockwise moments at the ends are taken as negative
CARRY OVER FACTOR
That the moments are applied on all the end joints of a structure, whose effects are evaluated on the joints. The ratio of moment produced at a joint to the moments applied at the other joint, without displacing it, is called CARRY OVER FACTOR
The important steps in moment distribution method are:
1) Lock all joints and determine the fixed-end moments that result.
2)Release the lock on a joint and apply the balancing moment to that joint.
3) Distribute the balancing moment and carry over moments to the (still -locked) adjacent joints. 4) Re -lock the joint. 5) Considering the next joint, repeat steps 2 to 4. 6) Repeat until the balancing and carry over moments are only a few percent of the original moments.
The reason this is an iterative procedure is (as we will see) that carrying over to a previously balanced joint unbalances it again. This can go on ad infinitum and so we stop when the moments being balanced are sufficiently small (about 1 or 2% of the start moments). Also note that some simple structures do not require iterations. Thus we have the following rule: For structures requiring distribution iterations, always finish on a distribution, never on a carry over This leaves all joints balanced (i.e. no unbalancing carry-over moment) at the end.
Stiffness factor
It is the moment required to rotate the end, while acting on it, through a unit angle without translation of the far end.
DISTRIBUTION FACTOR AT A PINNED END AND AT A FIXED END
At a pinned end we can imagine that there is a fictitious member outside the structure exists but it has I=0. Therefore, the distribution factors at a pinned joint are 1 for the real member side and 0 for other side. At a fixed joint we can think of the wall as a member with infinite stiffness, the distribution factors yield a value of Zero for the member and 1 for the wall.
Example 2
