Moment Distribution Method
Moment distribution method was developed by Hardy Cross in 1932. It is used for solving statically indeterminate beams and frames. It belongs to the category of displacement method of structural analysis.
This is an iterative method in which all the joints are considered to be fixed at the start of the problem which means that when loads are applied to each span, fixed end moments are developed at each end of the loaded span. After calculating the fixed end moments the joints are released and the unbalanced moment is distributed among the members attached to that joint in proportion to the distribution factors.
The steps involved in moment distribution method are described below
1. Calculate the fixed-end moments for all the spans of the beams by considering all the joints as fixed.
2. Calculate the stiffness coefficients for all the members. The stiffness coefficient (kAB) for a member AB is calculated as follows;
kAB = 4 EI/L if the far end B is fixed
kAB = 3 EI/L if the far end B is pinned or on a roller.
E is modulus of elasticity, I is moment of inertia of section and L is span AB. While calculating the stiffness coefficients all the intermediate joints are taken as fixed but the joints at the end of beam are kept as they are.
3. Calculate the distribution factors based on the stiffness coefficient of the member.
Distribution factor can be defined as the ratio of stiffness coefficient of a member to the sum of the stiffness coefficient of all the members meeting at that joint. If BA, BC and BD are connected at joint B, then the Distribution factor (D) can be easily calculated as follows;
DBA = kBA /(kBA + kBC + kBD )
DBC = kBC /(kBA + kBC + kBD )
DBD = kBD /(kBA + kBC + kBD )
Distribution factor for a pinned support or roller at the end of beam is taken as 1 whereas for a fixed support at the end of beam the distribution factor is taken as zero.
4. Balance all the joints by applying the balancing moments in the proportions of distribution factors.
5. Carry over half of the balancing moments to the opposite ends of the span. If the opposite end is pinned there should be no carry over moment to that end (as in the case of pinned support at the ends of the beam).
6. Continue these cycles of balancing and carry-over till the joints reach equilibrium state when the unbalanced moment is negligible (based on desired accuracy).
7. Take the sum of all the moments (fixed end moment, balancing moment, carry-over moment) at each end to get the Final end moments on the left and right hand side of the supports.
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