PART 2: HOW TO DRAW THE BENDING MOMENT DIAGRAM USING MOMENT DISTRIBUTION METHOD|DISTRIBUTION TABLE.

Basic Concepts

The moment distribution method of analysis of beams and frames was developed by Hardy Cross and formally presented in 1930. Although this method is a deformation method like the slope-deflection method, it is an approximate method and, thus, does not require solving simultaneous equations, as was the case with the latter method. The degree of accuracy of the results obtained by the method of moment distribution depends on the number of successive approximations or the iteration process.

Sign Convention

The sign convention for the moment distribution method is similar to the one established for the slope-deflection method; that is, the moment at the end of a member is considered positive if it tends to turn the end of the member clockwise and negative if it tends to turn it counterclockwise.

Definitions

Unbalanced moments:
This method of analysis assumes that the joints in a structure are initially clamped or locked and then released successively. Once a joint is released, a rotation takes place, since the sum of the fixed end moments of the members meeting at that joint is not zero. The value of the sum of the end moments obtained is the unbalanced moment at that joint.

Carry-over moments:
The distributed moments in the ends of members meeting at a joint cause moments in the other ends, which are assumed to be fixed. These induced moments at the other ends are called carry-over moments.

Carry-over factor: The ratio of the induced moment to the applied moment is referred to as the carry-over factor

Distribution factor (DF):
The distributed factor is a factor used to determine the proportion of the unbalanced moment carried by each of the members meeting at a joint.

Distributed moments:
Upon the release of the imaginary clamp at a joint, the unbalanced moment at that joint causes it to rotate. The rotation twists the end of the members meeting at the joint, resulting in the development of resisting moments.

Analysis of Indeterminate Beams

The procedure for the analysis of indeterminate beams by the method of moment distribution is briefly summarized as follows:

Procedure for Analysis of Indeterminate Beams by the Moment Distribution Method

Calculate the fixed-end moments for members, assuming that the joints are clamped against rotation.

Calculate the distribution factor for each of the members connected at the joint

Calculate the unbalanced moment at each joint and distribute the same to the ends of members connected at that joint.

Carry over one-half of the distributed moment to the other ends of members.

Add or subtract these latter moments (moments obtained in steps three and four) to or from the original fixed-end moments.

Apply the determined end moments at the joints of the given structure.

Draw the free-body diagram of each span of the given beam, showing the loads and moments at the joints obtained by the moment distribution method.

Determine the support reactions for each span.

Compute and construct the shearing force and bending moment diagrams for each span.

Draw one bending moment and one shearing force diagram for the given beam by combining the diagrams in step 9

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