## TOS ~ Moment Distribution Method

## TOS

Moment Distribution Method

### Introduction of Moment Distribution Method

- Suggested by Prof. Hardy Cross in the early 1930s.
- The method is widely used for the analysis of indeterminate structures.
- Ideally suited for a fairly higher degree of indeterminate structures

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### Indeterminate structures of Moment Distribution Method

- These are the structures which cannot be fully analysed by the condition of equilibrium.
- The equations of statical equilibrium which are based on Sir Isaac Newtonâ€™s law governing the motion of bodies 1687 which are

- The summation of all forces in any axis is zero.
- The summation of all moment in any axis is zero.

If summation is taken about x, y and z axes, symbolically this becomes.

- âˆ‘Fx = 0; âˆ‘Fy = 0; and âˆ‘Fz = 0;
- âˆ‘Mx = 0; âˆ‘My = 0; and âˆ‘Mz = 0;

For Planar structures and forces acting in the same plane, the equation reduces to

- âˆ‘Fx = 0, âˆ‘Fy = 0, and âˆ‘Mz = 0,

### Types of Supports in Moment Distribution Method

- Simple support
- Roller support
- Hinge or Pin support
- Fixed support

**Calculation of Indeterminacy by Formulae**

- Degree of Indeterminacy of Frame: i=(3m +r) â€“ (3j +c)
- Or statically determinate if 3j = 3m +r

- Degree of Indeterminacy of Plane Truss: i=(2j -3) â€“ m
- Or statically determinate if m=2j -3

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### Terminology on Moment Distribution Method

**Carryover moment**

- When a moment is applied at one end of a member allowing rotation of that end and fixing the far end, some moment develops at the far end also, this moment is called carryover moment

**Carryover factor**

- The ratio of carryover moment to applied moment is called carryover factor.
- Carryover factor = Mâ€™/M

**Stiffness**

- The moment required to rotate an end by the unit angle 1 radian, when rotation is permitted at that end, is called stiffness of the beam. Thus in the beam above, if Î¸A is the rotation at end A,
- Stiffness of the beam AB = k (M/ Î¸A)

*OR*- When a structural member of uniform section is subjected to a moment at one end only, then the moment required so as to rotate that end to produce a unit slope, it is called the stiffness of the member.

**Distribution factor of Moment Distribution Method**

- If a moment is connected to a rigid joint where a number of members are meeting, the connected moment is shared with the branches meeting about that joint.
- The ratio of the moment shared by a member to the applied moment at the joint is called the distribution factor of that member.
- Thus, if MoA is the moment shared by member OA when moment M is applied at joint O, then the distribution factor for member OA is dOA = MOA/M

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**Application of Moment Distribution Method to Continuous Beams with Fixed ends**

- Assuming all ends are fixed, find the fixed end moment s developed.
- Calculate distribution factors for all members meeting at a joint.
- Balance a joint by distributing balance moment to various members meeting at the joint proportional to their distribution factors. Do a similar exercise for all joints.

- Carryover half the distributed moment to the far ends of the members. This upsets the balances of the joints.
- Repeat steps 3 and 4 till distributed moment are negligible.
- Sum up all the moments at a particular end of the member to get a final moment.

**Analyse the continuous beam shown in the figure by moment distribution method and draw bending moment and shear force diagrams.**

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