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

Indeterminate structures of Moment Distribution Method

• These are the structures that cannot be fully analysed by the condition of equilibrium.
• The equations of statical equilibrium which are based on Sir Isaac Newton’s law governing the motion of bodies 1687 are

1. The summation of all forces in any axis is zero.
2. The summation of all moments in any axis is zero.

If summation is taken about the x, y and z axes, symbolically this becomes Moment Distribution Method.

• ∑Fx = 0;    ∑Fy = 0;  and  ∑Fz = 0;
• ∑Mx = 0;   ∑My = 0; and  ∑Mz = 0;

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

• ∑Fx = 0, ∑Fy =  0, and  ∑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) – (3j +c)
• Or statically determinate if 3j = 3m +r

• Degree of Indeterminacy of Plane Truss: i=(2j -3) – m
• Or statically determinate if m=2j -3

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 the carryover moment

Carryover factor

• The ratio of the carryover moment to the applied moment is called the carryover factor.
• Carryover factor = M’/M

Stiffness

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

OR

• When a structural member of a uniform section is subjected to a moment at one end only, then the moment required to rotate that end to produce a unit slope is called the stiffness of the member.

Distribution factor of Moment Distribution Method

• If a moment is connected to a rigid joint where several 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

Application of Moment Distribution Method to Continuous Beams with Fixed Ends

• Assuming all ends are fixed, find the fixed end moments 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 balance 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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