What is Moment Distribution Method?
The moment distribution method is a practical method for analysing statically indeterminate beams and rigid frames. Every joint of the structure to be analysed is fixed in the moment distribution method in order to create fixed-end moments. Then, one by one, each fixed joint is released, and the fixed-end moments (which are not in equilibrium at the time of release) are transferred to nearby members until equilibrium is attained. Furthermore, the moment distribution approach may be proved mathematically as the process of iteratively solving a set of simultaneous equations. Let us discuss the displacement method and force method.
The displacement method is a method of structural analysis in which displacements are treated as unknowns. The number of unknowns in the displacement method is treated as equal to Kinematic Indeterminacy. Different types of displacement methods are Slope Deflection Method, Moment Distribution Method, Kani’s Method & Stiffness Matrix Method.
The Force Method is a method of structural analysis in which redundant forces are treated as unknowns. The number of redundant forces is treated as equal to static indeterminacy. Different types of force methods are flexibility matrix method, consistent deformation method, column analogy method & elastic centre method.
Analysis of Beam Using Moment Distribution Method
The moment distribution method was developed by Hardy Cross. In the moment distribution method, displacements are treated as unknowns. The moment distribution method is also known as the equilibrium or displacement method. It is suitable for analyzing indeterminate structures like beams and rigid jointed frames.
In the moment distribution method, the acting axial forces and corresponding axial force deformations are neglected. The moment distribution method is applicable for both prismatic and non-prismatic members.
Carry Over Factor (COF)
Carry Over Factor is the ratio of the far-end moment and near-end moment. Carry Over Moment (COM) is also called a developed moment or induced moment. It is the moment developed at one end due to the moment at another end.
COF= COM/ Applied Moment
Carry Over Factors for different cases are:
Case 1: When a far end is fixed
COF = ½
Case 2: One end is hinge
COF = 0
Case 3: Cantilever Beam
COF = -1
Case 4: Far-end guided roller
COF = -1
Stiffness Factor/ Rotational Stiffness
The Stiffness Factor is the moment required to produce a unit rotation.
Case 1: Far end is fixed
Case 2: Far end is hinged
Case 3: Far end is free
Case 4: Far end is a guided roller
Moment Distribution Method Problems
1. Find the stiffness factor or rotational stiffness at O.
δ= 3EI/L+ 4EI/L+0+ EI/L
2. Find the relative stiffness & distribution factors of members.
The distribution factor is the ratio in which the moment sharing capacity of various members meets at a rigid joint.
(DF)AB= KAB/ ∑KA= 4/11
(DF)AC= KAC/ ∑KA= 3/11
(DF)AD= KAD/ ∑KA= 4/11
(DF)AE= KAE/ ∑KA= 0
Distribution factor at a rigid joint = 1
For fixed support, distribution factor = 0
For hinged or roller beam, distribution factor = 1