Monday, 23 January 2012

How To Calculate Moment Distributions

Calculating moment distributions from distribution factors and rotational stiffness.

When a load is applied to any continuous structure it is important to understand how the moments are distributed throughout the members and joints. You achieve this by using the Moment Distribution method.

A stiffness factor relates the stiffness and length of the member. Stiffness is represented by EI (E- Young's Modulus, I - Second moment of area) and length by L. The formula is EI / L.
If a member has fixed ends at both of its extremities then its stiffness = 4EI / L
If the member has one fixed end and another pinned then the equation becomes = 3EI / L

You take stiffness factors and distribution factors (DF) about a joint that connects 2 or more members. In order to find the distribution factors of the member about that joint you need to first sum up the stiffness factors at that joint e.g.
•    If you have two members connected at a joint, one with 2 fixed ends and one with a fixed/pinned combination then the first stage is : 4EI / L + 3EI / L = 7EI / L
The next step is to calculate the distribution factor for each member. To do this you divide the member stiffness factor by the total stiffness at the joint e.g.
•    For the fixed/pinned combination member the distribution factor would be calculated by : (3EI / L) / (7EI / L) = 0.43

This value 0.43 means that for any moment exerted at the joint this member will take 43% of it. There is a simple way check your calculation. Distribution factors around a joint have to be equal to 1. Therefore, to check our 0.43 we need to calculate the DF of the doubly fixed end member.
•    2x fixed end member : (4EI / L) / (7EI / L) = 0.57
•    0.57+0.43=1
Therefore, calculation is correct.

From this point onwards you will be able to identify how a load/moment is distributed amongst its members. This allows you to design the structure in greater detail and with higher accuracy.
The best way to find the moments exerted on each member is to calculate the fixed end moments (FEM) and balance out each joint as you proceed.
•    FEM for 2x fixed end member = Point Load : WL/8
•    UDL : WL/12
•    Fixed/Pinned combination member = WL/8
Once the end moments have been found you can set up a table representing each joint and its corresponding moments. It is important to remember that the moments around a joint must =0 otherwise the structure will not be in equilibrium. e.g
•    If the net moment around a joint is -100kN. Then an additional +100kN is required to maintain equilibrium in the structure. This is added to the joint and distributed to each member using their distribution factors i.e. 43% and 57%.
Carry-over moments are crucial to maintain equilibrium. If a member has fixed ends at both of its ends then you need to use the carry-over method. e.g.
•    Member A has two fixed ends. It has a moment of 10kN exerted on its right hand end. Under the carry over method 5kN is transferred to its left hand end to maintain equilibrium.
By combining stiffness factors, distribution factors, fixed end moments and the carry over moment method you can identify what magnitude of moment will be exerted on any particular member in a structure.


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