Solving indeterminate beam by Moment Distribution Method

(To know more Please Watch our Video Lecture on Moment distribution method)

Problem 8-1

Use Moment distribution method to find the resultant end moments for the continuous beam shown in figure 8-1(a). Take EI as constant for the beam. Also draw bending moment diagram.

Figure 8-1(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 fixed-end moments as follows;

M_AB = {wa² (6L² - 8aL + 3a² )}/(12L² ) = -17.1 kNm

M_BA = -{wa³ (4L - 3a}/(12L² ) = +9.9 kNm

M_BC = wL² /12 = -6.67 kNm

M_CB = -wL² /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 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
Members	AB	BA	BC	CB	CD	DC
Distribution Factors	1	0.375	0.625	0.5	0.5	0
Initial Fixed-end 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
Carry-over		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
Carry-over			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
Carry-over			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
Carry-over			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
Carry-over			0.06	-0.006		0.06
Final End-Moments	0	13.28	-13.27	6.2	-6.2	11.95

Resultant bending moment diagram in figure 8-1(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 8-1(b)