## 1032

Substituting for 0B, 0C, and C from Equations 2.32 into Equations 2.23 to 2.28, we get

Mab = 11.03 kN m Mba = 125.3 kN m Mbc = -125.3 kN m Mcb = 121 kN m Mcd = -121 kN m Mdc = -83 kN m

### 2.4.3 Moment Distribution Method

The moment distribution method involves successive cycles of computation, each cycle drawing closer to the exact answers. The calculations may be stopped after two or three cycles, giving a very good approximate analysis, or they may be carried on to whatever degree of accuracy is desired. Moment distribution remains the most important hand-calculation method for the analysis of continuous beams and frames and it may be solely used for the analysis of small structures. Unlike the slope deflection method, this method does require the solution to simultaneous equations.

The terms constantly used in moment distribution are fixed-end moment, unbalanced moment, distributed moment, and carry-over moment. When all of the joints of a structure are clamped to prevent any joint rotation, the external loads produce certain moments at the ends of the members to which they are applied. These moments are referred to as fixed-end moments. Initially, the joints in a structure are considered to be clamped. When the joint is released, it rotates if the sum of the fixed-end moments at the joint is not equal to zero. The difference between zero and the actual sum of the end moments is the unbalanced moment. The unbalanced moment causes the joint to rotate. The rotation twists the ends of the members at the joint and changes their moments. In other words, rotation of the joint is resisted by the members and resisting moments are built up in the members as they are twisted. Rotation continues until equilibrium is reached — when the resisting moments equal the unbalanced moment — at which time the sum of the moments at the joint is equal to zero. The moments developed in the members resisting rotation are the distributed moments. The distributed moments in the ends of the member cause moments in the other ends, which are assumed to be fixed, and these are the carry-over moments.

### 2.4.3.1 Sign Convention

The moments at the end of a member are assumed to be positive when they tend to rotate the member clockwise about the joint. This implies that the resisting moment of the joint would be counterclockwise. Accordingly, under gravity loading conditions the fixed-end moment at the left end is assumed to be counterclockwise (negative) and at the right end, clockwise (positive).

### 2.4.3.2 Fixed-End Moments

Fixed-end moments for several cases of loading may be found in Figure 2.8. Application of moment distribution may be explained with reference to a continuous beam example as shown in Figure 2.23. Fixed-end moments are computed for each of the three spans. At joint B the unbalanced moment is obtained and the clamp is removed. The joint rotates, thus distributing the unbalanced moment to the

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