Moment distribution method offers a convenient way to analyse statically indeterminate beams and rigid frames.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.
Moreover, the moment distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration. Added to that, it falls into the category of displacement method of structural analysis.
Finally, moments distribution method of structural analysis is thoroughly presented in the following sections in addition to provide solved example.
Basic definitions of terms in moment distribution method of structural analysis
Fixed end momentsFixed end moments are the moments produced at member ends by when the joints are fixed. Table 1 provides equations for fixed end moment computations. Table 1: Fixed end moment equations
Flexural stiffnessThe flexural stiffness (EI/L) of a member is represented as the product of the modulus of elasticity (E) and the second moment of area (I) divided by the length (L) of the member. Additionally, what is needed in the moment distribution method is not the exact value but the ratio of flexural stiffness of all members.
Distribution factorsDistribution factors can be defined as the proportions of the unbalanced moments carried by each of the members.
Carryover factorsUnbalanced moments are carried over to the other end of the member when the joint is released. Added to that, the ratio of the carried-over moment at the other end to the fixed-end moment of the initial end is the carryover factor. Lastly, For prismatic members, the carryover moment in each span has the same sign as the distribution end moment, but is one-half as large.
Sign conventionAny moment acting clockwise is considered to be positive. This differs from the usual engineer's sign convention, which employs a Cartesian coordinate system with positive x-axis to the right and positive y-axis up, resulting in positive moment about the z-axis being counterclockwise.
Framed structuresFramed structures with or without sides way can be analysed using the moment distribution method.
Moment distribution analysis procedure for beams1. Restrain all possible displacements. 2. Then, calculate Distribution Factors: The distribution factor DFi of a member connected to any joint J is
Me: external moment applied to the joint (if any)
Mo: total out of balance moment at the joint
FEMi: fixed-end moment
Mi: moment distributed to any member
DFi: distribution factor of member i7. Calculation of the carry over moment at the far end of each member. The procedure is stopped when, at all joints, the out of balance moment is a negligible value. In this case, the joints should be balanced and no carry-over moments are calculated. 8. Finally, calculate the final moment at either end of each member. This is the sum of all moments (including FEM) computed during the distribution cycles.
Analyzing statically indeterminate beam using moment distribution method
- Members AB, BC, CD have the same length
- Flexural rigidities are EI, 2EI, EI respectively.
- Concentrated load of magnitude
acts at a distance from the support A.
- Uniform load of intensity
acts on BC.
- Member CD is loaded at its mid-span with a concentrated load of magnitude
Fixed-end momentsIn the following calculations, counterclockwise moments are positive:
Carryover factorsThe carryover factors are
Details of calculationsTable 2: details of moment distribution calculations
|Sum of moments||0||-11.569||11.569||-10.186||10.186||-13.657|
ResultsMoments at joints determined by the moment distribution method
Shear force and bending moment diagramTable 3 provides shear and moment diagram for the analyzed beam. Note that the moment distribution method only determines the moments at the joints. Moreover, developing complete bending moment diagrams require additional calculations using the determined joint moments and internal section equilibrium. Table 3: shear force and moment diagram for the analyzed indeterminate beam
|Shear force diagram
||Bending moment diagram