Solving indeterminate frame by moment distribution method
Problem 8-2
Use Moment distribution method to find the resultant end moments for the non-sway frame shown in figure 8-2(a). Also draw bending moment diagram. Take EI as constant for all the members of the frame.
Figure 8-2(a)
Solution:
Step 1: The given rigid frame is non-sway because of lateral support at D. Consider each span (AB, BC, CD and CE) with both ends fixed and calculate the fixed-end moments as follows;
M_{AB} = 0
M_{BA}_{ }= 0
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
M_{CE} = 0
M_{EC}_{ }= 0
Step 2: Stiffness Coefficients
k_{AB} = 4EI/L = 4EI/5 =0.8EI
k_{BA} = 4EI/L = 4EI/5 =0.8EI
k_{BC} = 4EI/L = 4EI/4 = EI
k_{C}_{B} = 4EI/L = 4EI/4 = EI
k_{CD} = 4EI/L = 4EI/4 = EI
k_{DC} = 4EI/L = 4EI/4 = EI
k_{CE} = 4EI/L = 4EI/4 = EI
k_{EC} = 4EI/L = 4EI/4 = EI
Step 3: Distribution factors
D_{AB} = 0
D_{BA} = 0.8EI/(0.8EI+EI) = 0.44
D_{BC} = EI/(0.8EI+EI) = 0.56
D_{CB} = EI/(EI+EI+EI) =0.33
D_{CD} = EI/(EI+EI+EI) = 0.33
D_{CE} = EI/(EI+EI+EI) = 0.33
D_{DC} = 0
D_{EC} = 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 carry-over moments (half of the balancing moment at near end) are applied on the far end, e.g. carry-over 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 | E | |||
Members |
AB |
BA |
BC |
CB |
CE |
CD |
DC |
EC |
Distribution Factors | 0 | 0.44 | 0.56 | 0.33 | 0.33 | 0.33 | 0 | 0 |
Initial Fixed-end Moments | 0 | 0 | -6.67 | +6.67 | 0 | -10 | +10 | 0 |
Balancing |
6.67x0.44 2.94 |
6.67x0.56 3.74 |
+3.33x0.33 1.11 |
+3.33x0.33 1.11 |
+3.33x0.33 1.11 |
|||
Carry-over | 1.47 | 0.555 | 1.87 | 0.555 | 0.555 | |||
Balancing |
-0.555x0.44 -0.242 |
-0.555x0.56 -0.312 |
-1.87x0.33 -0.62 |
-1.87x0.33 -0.62 |
-1.87x.33 -0.62 |
|||
Carry-over | -0.121 | -0.31 | -0.156 | -0.31 | -0.31 | |||
Balancing |
0.31x0.44 0.14 |
0.31x0.56 0.17 |
0.156x0.33 0.052 |
0.l56x0.33 0.052 |
0.156x0.33 0.052 |
|||
Carry-over | 0.07 | 0.026 | 0.09 | 0.026 | 0.026 | |||
Balancing |
-0.026x0.44 -0.011 |
-0.026x0.56 -0.015 |
-0.09x0.33 -0.03 |
-0.09x0.33 -0.03 |
-0.09x0.33 -0.03 |
|||
Carry-over | -0.006 | -0.015 | -0.007 | -0.015 | -0.015 | |||
Final End-Moments | 1.41 | 2.83 | -2.83 | 8.98 | 0.51 | -9.49 | 10.26 | 0.26 |
Resultant bending moment diagram in figure 8-2(b) is plotted by superposing the span moment diagram and support moment diagram.
Note: Please use our 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 8-2(b)
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