Solving indeterminate beam by moment distribution method
Problem 81
Use Moment distribution method to find the resultant end moments for the continuous beam shown in figure 81(a). Take EI as constant for the beam. Also draw bending moment diagram.
Figure 81(a)
Solution:
Step 1: The given continuous beam has three spans. Consider each span (AB, BC, and CD) with both ends fixed and calculate the fixedend moments as follows;
M_{AB} = {wa^{2} (6L^{2}  8aL + 3a^{2} )}/(12L^{2} ) = 17.1 kNm
M_{BA}_{ }= {wa^{3} (4L  3a}/(12L^{2} ) = +9.9 kNm
M_{BC} = wL^{2} /12 = 6.67 kNm
M_{CB} = wL^{2} /12 = +6.67 kNm
M_{CD} = PL/8 = 10 kNm
M_{DC} = PL/8 = +10 kNm
Step 2: Stiffness Coefficients
k_{AB} = 4EI/L = 4EI/5 =0.8EI
k_{BA} = 3EI/L = 3EI/5 =0.6EI
k_{BC} = 4EI/L = 4EI/4 = EI
k_{CB} = 4EI/L = 4EI/4 = EI
k_{CD} = 4EI/L = 4EI/4 = EI
k_{DC} = 4EI/L = 4EI/4 = EI
Step 3: Distribution factors
D_{AB} = 1
D_{BA} = 0.6EI/(0.6EI+EI) = 0.375
D_{BC} = EI/(0.6EI+EI) = 0.625
D_{CB} = EI/(EI+EI) =0.5
D_{CD} = EI/(EI+EI) = 0.5
D_{DC} = 0
Calculation for moment distribution is shown in a systematic way in a tabular form. Each joint is balanced by distributing the unbalanced moment into the connected members in proportion of distribution factors. These calculation steps for balancing of joints are shown with grey colour. The carryover moments (half of the balancing moment at near end) are applied on the far end, e.g. carryover half of the balancing moment from BC to CB.
(In case of mobile device please scroll horizontally to view the full width of table)
Table for Moment Distribution  
Joints 
A 
B 
C 
D 

Members  AB 
BA 
BC 
CB 
CD 
DC 
Distribution Factors  1  0.375  0.625  0.5  0.5  0 
Initial Fixedend Moments  17.1  +9.9  6.67  +6.67  10  +10 
Balancing  +17.1 
3.23x0.375 1.21 
3.23x0.625 2.02 
+3.33x0.5 1.665 
+3.33x0.5 1.665 

Carryover  8.505  0.833  1.01  0.833  
Balancing 
9.34x0.375 3.5 
9.34x0.625 5.84 
1.01x0.5 0.505 
1.01x0.5 0.505 

Carryover  0.253  2.92  0.253  
Balancing 
0.253x0.375 0.095 
0.253x0.625 0.158 
2.92x0.5 1.46 
2.92x0.5 1.46 

Carryover  0.73  0.079  0.73  
Balancing 
0.73x0.375 0.274 
0.73x0.625 0.456 
0.079x0.5 0.04 
0.079x0.5 0.04 

Carryover  0.02  0.23  0.02  
Balancing 
0.02x0.375 0.007 
0.02x0.625 0.012 
0.23x0.5 0.12 
0.23x0.5 0.12 

Carryover  0.06  0.006  0.06  
Final EndMoments  0  13.28  13.27  6.2  6.2  11.95 
Resultant bending moment diagram in figure 81(b) is plotted by superposing the span moment diagram and support moment diagram.
Note: Please use bending moment calculator to get the values of span moments
(all values in kNm)
(span moment in blue colour, support moment in red colour)
Figure 81(b)
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