Moment Distribution Method - Theory (Part -2)

Second principle we need to know is

• Carry over factor

Let us see the ideolized free body diagram of single beam.

Moment required to give rotation q_A  is M_A . we can write this

Therefore

From these two equations we know that  M_B = 1/2 M_A.

And we can say that 1/2 of the moment from pinned end is carried over to the fixed end. Note that this is positive here because of same directional rotation.

• Fixed End Moments 

Fixed end moments of the elements will be needed to proceed the moment distribution method. The formulae are presented in the appendix part of structural analysis books.

Moment Distribution Method - Theory (Part 1)

Dear Friends...

We know the moment distribution method in analysis of structures. How it works, the theoretical explanation given below.

We need to know some basic principles to understand the moment distribution.

• Distribution Factor

Consider the following structure with an externally applied moment M at the joint.

Due to applied moment M the joint undergoes rotation q. And the rotation of the each beam is equivalent to q because of compatibility. That means q₁= q₂= q₃= q . Lets see the free body diagram of the beams to resist the applied moment M.

We know from moment equilibrium that M₁ = M₂ = M₃ = M. Now consider the free bending moment diagram of beam 1 shown below. We can notice that at the right end it can rotate but not translate, ie it can’t move along X or Y direction (neglecting axial deformation).

Since the beam acting as a spring resisting rotation we can say that M₁ = k₁ q₁. Similarly for beam 2 and beam 3 M₂ = k₂ q₂ and M₃ = k₃ q₃. For the whole structure M = k q. Where k is the total rotational spring stiffness of the joint of three intersecting  beams.

We know that

q₁= q₂= q₃= q

Then

We can rewrite them  

Considering the moment equilibrium

k₁ q₁ + k₂ q₂ + k₃ q₃ = k q

Since  q₁= q₂= q₃= q

(k₁ + k₂ + k₃)q = k q

Or      k= k₁ + k₂ + k₃ =∑ k_i                                i=1,2,3

The term

is called as distribution factors. And it is the distribution of the rotation to each beam in that joint due to applied moment. The sum of all distribution factors in a joint is always unity. And individual moment resisting of each beam is proportional to their stiffness.

 

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