The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal. The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
Introduction In the moment distribution method, every joint of the structure to be analysed is fixed so as to develop the fixed-end moments. Then each fixed joint is sequentially released and the fixed-end moments (which by the time of release are not in equilibrium) are distributed to adjacent members until equilibrium is achieved. The moment distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration. The moment distribution method falls into the category of displacement method of structural analysis.
Implementation In order to apply the moment distribution method to analyse a structure, the following things must be considered.
Fixed end moments Fixed end moments are the moments produced at member ends by external loads. Spanwise calculation is carried out assuming each support to be fixed and implementing formulas as per the nature of load ,i.e. point load (mid span or unequal), udl, uvl or couple.
Bending stiffness The bending stiffness (EI/L) of a member is represented as the flexural rigidity of the member (product of the modulus of elasticity (E) and the second moment of area (I)) divided by the length (L) of the member. What is needed in the moment distribution method is not the specific values but the ratios of bending stiffnesses between all members.
Distribution factors When a joint is being released and begins to rotate under the unbalanced moment, resisting forces develop at each member framed together at the joint. Although the total resistance is equal to the unbalanced moment, the magnitudes of resisting forces developed at each member differ by the members' bending stiffness. Distribution factors can be defined as the proportions of the unbalanced moments carried by each of the members. In mathematical terms, the distribution factor of member k {\displaystyle k} framed at joint j {\displaystyle j} is given as:
D j k = E k I k L k ∑ i = 1 i = n E i I i L i {\displaystyle D_{jk}={\frac {\frac {E_{k}I_{k}}{L_{k}}}{\sum _{i=1}^{i=n}{\frac {E_{i}I_{i}}{L_{i}}}}}}
where n is the number of members framed at the joint.
Carryover factors When a joint is released, balancing moment occurs to counterbalance the unbalanced moment. The balancing moment is initially the same as the fixed-end moment. This balancing moment is then carried over to the member's other end. The ratio of the carried-over moment at the other end to the fixed-end moment of the initial end is the carryover factor.
Determination of carryover factors Let one end (end A) of a fixed beam be released and applied a moment M A {\displaystyle M_{A}} while the other end (end B) remains fixed. This will cause end A to rotate through an angle θ A {\displaystyle \theta _{A}} . Once the magnitude of M B {\displaystyle M_{B}} developed at end B is found, the carryover factor of this member is given as the ratio of M B {\displaystyle M_{B}} over M A {\displaystyle M_{A}} :
C A B = M B M A {\displaystyle C_{AB}={\frac {M_{B}}{M_{A}}}}
In case of a beam of length L with constant cross-section whose flexural rigidity is E I {\displaystyle EI} ,
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