1. Structural Analysis UNIT 4 MOMENT DISTRIBUTION METHOD
Dr. Mohammad Ahmed Hussain, 2nd Year 2nd Sem

2. MOMENT DISTRIBUTION METHOD - INTRODUCTION AND BASIC PRINCIPLES
(Method developed by Prof. Hardy Cross in 1932)
The method solves for the joint moments in continuous beams and
rigid frames by successive approximation.
Moment distribution method is basically a displacement method of analysis.
This method side steps the calculation of the displacement and instead makes it possible to apply a series of converging corrections that allow direct calculation of the end moments.

3. Basic Principles and definitions
Member stiffness factor
A clockwise moment MA is applied at A to produce a +ve bending in beam AB.
Case 1: Far End Fixed
Case 2: Far End Hinged B
A MA MB A RA RB L A
Stiffness factor
Stiffness factor

4. 2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Fixed end moments

5. 2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Fixed end moments

6. 2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Fixed end moments

7. Distribution Factor ( k / ∑k )
2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Distribution Factors
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
4EI/6 BC
4E2I/5 C CB CD
4EI/5

8. Distribution Factor ( k / ∑k )
2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Distribution Factors
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
4EI/6
5/17 BC
4E2I/5
12/17 C CB
2/3 CD
4EI/5
1/3

9. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
+5.0

10. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-1.06
-2.54
+0.93
+0.47

11. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-1.06
-2.54
+0.93
+0.47

12. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23

13. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42

14. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42

15. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21

16. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21
-0.12
-0.30
0.11
0.05

17. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21
-0.12
-0.30
0.11
0.05

18. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21
-0.12
-0.30
0.11
0.05
-0.06
+0.05
-0.15
+0.025

19. A B C D
F E M
Balance
C. O
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21
-0.12
-0.30
0.11
0.05
-0.06
+0.05
-0.15
+0.025
-0.02
-0.03
0.1
0.05

20. A B C D
F E M
Balance
C. O
C.O to fixed end
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21
-0.12
-0.30
0.11
0.05
-0.06
+0.05
-0.15
+0.025
-0.02
-0.03
0.1
0.05

21. A B C D
F E M
Balance
C. O
C.O to fixed end
Final Moments
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21
-0.12
-0.30
0.11
0.05
-0.06
+0.05
-0.15
+0.025
-0.02
-0.03
0.1
0.05
-0.01
+0.025

22. A B C D
F E M
Balance
C. O
C.O to fixed end
Final Moments
5/17
12/17
2/3
1/3
-6.0
6.0
-2.4
3.6
-5.0
5.0
-1.06
-2.54
+0.93
+0.47
-0.53
+0.46
-1.27
+0.23
-0.14
-0.32
0.85
0.42
-0.07
+0.42
-0.16
+0.21
-0.12
-0.30
0.11
0.05
-0.06
+0.05
-0.15
+0.025
-0.02
-0.03
0.1
0.05
-0.01
+0.025
-6.67
+4.66
-4.66
+4.01
-4.01
5.49

23. 2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Fixed end moments

24. 2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Fixed end moments

25. Distribution Factor ( k / ∑k )
2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Distribution Factors
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
3EI/6 BC
4E2I/5 C CB CD
3EI/5

26. Distribution Factor ( k / ∑k )
2 kN/m
8 kN
5 kN
6 m 5m
2.5 m A B C D I 2I
3 m
Distribution Factors
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
3EI/6
5/21 BC
4E2I/5
16/21 C CB
8/11 CD
3EI/5
3/11

27. A B C D
F E M
Release Jt . A and D
C. O
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0

28. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5

29. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5

30. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06

31. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06

32. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52

33. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69

34. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69

35. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54

36. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14

37. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35

38. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1

39. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1

40. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1
+0.12
-0.07

41. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1
+0.12
-0.07
-0.03
-0.09
0.05
0.02

42. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1
+0.12
-0.07
-0.03
-0.09
0.05
0.02
+0.02
-0.05

43. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1
+0.12
-0.07
-0.03
-0.09
0.05
0.02
+0.02
-0.05
-0.01
-0.01
0.03
0.02

44. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
Final Moments
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1
+0.12
-0.07
-0.03
-0.09
0.05
0.02
+0.02
-0.05
-0.01
-0.01
0.03
0.02

45. A B C D
F E M
Release Jt . A and D
C. O
Initial Moment
Balance
Final Moments
5/21
16/21
8/11
3/11
-6.0
6.0
-2.4
3.6
-5.0
+5.0
-5.0
6.0
3.0
-2.5
9.0
-2.4
3.6
-7.5
-1.56
-5.04
2.84
1.06
+1.42
-2.52
-0.34
-1.08
1.83
0.69
+0.92
-0.54
-0.22
-0.70
0.40
0.14
+0.20
-0.35
-0.05
-0.15
0.25
0.1
+0.12
-0.07
-0.03
-0.09
0.05
0.02
+0.02
-0.05
-0.01
-0.01
0.03
0.02
6.79
-6.79
5.46
-5.46

46. Since no external loads coming on to the B C I I I
3 m 2m
2 m
Support B sinks by 30 mm
I = 3 X 106 mm4
E =2 X 105 N/mm2
Since no external loads coming on to the
Structure the FEM comprise only due to the
Moment induced due to sinking of supports
F E M

47. Support B sinks by 30 mm I = 3 X 106 mm4 E =2 X 105 N/mm2 3 m 2m 2 m AC D I
Support B sinks by 30 mm
I = 3 X 106 mm4
E =2 X 105 N/mm2

48. Distribution Factor ( k / ∑k )
3 m 2m
2 m A B C D I
Distribution Factors
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
4EI/3 BC
4EI/2 C CB CD

49. Distribution Factor ( k / ∑k )
3 m 2m
2 m A B C D I
Distribution Factors
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
4EI/3
0.4 BC
4EI/2
0.6 C CB
0.5 CD

50. A B C D
F E M
Balance
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27

51. A B C D
F E M
Balance
C. O
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

52. A B C D
F E M
Balance
C. O
-3.0
-6.75
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

53. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

54. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

55. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
1.13
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

56. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

57. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
-0.51
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

58. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
0.2
-0.51
0.17
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

59. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
0.2
-0.51
0.17
0.1
0.08
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

60. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
0.2
-0.51
0.17
0.1
-0.03
0.08
-0.07
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

61. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
0.2
-0.51
0.17
0.1
-0.03
0.08
-0.07
-0.02
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

62. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
0.2
-0.51
0.17
0.1
-0.03
0.08
-0.07
-0.02
0.02
C. O to A and D
0.01
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

63. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
0.2
-0.51
0.17
0.1
-0.03
0.08
-0.07
-0.02
0.02
C. O to A and D
0.01
Final Moments
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

64. A B C D
F E M
Balance
C. O
-3.0
2.7
-6.75
+2.25
+1.35
-0.45
1.13
-1.02
-0.22
0.2
-0.51
0.17
0.1
-0.03
0.08
-0.07
-0.02
0.02
C. O to A and D
0.01
Final Moments
0.4
0.6
0.5
0.5
-12.0
-12.0 27 27
-6.0
-9.0
-13.5
-13.5

65. Support B sinks by 10 mm
I = 1.2 X 10-4 m4
E =2 X 105 N/mm2

66. Support B sinks by 10 mm
I = 1.2 X 10-4 m4
E =2 X 105 N/mm2

67. Distribution Factor ( k / ∑k )
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
4EI/6 BC
4EI/5 C CB CD
3EI/4

68. Distribution Factor ( k / ∑k )
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
4EI/6
0.46 BC
4EI/5
0.54 C CB
0.51 CD
3EI/4
0.49

78. Fixed End Moments

79. Fixed End Moments

80. Distribution Factor ( k / ∑k )
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
3E3I/4 BC
4E10I/8 C CB CD
3E2I/4
Note: Since support ‘A’ is simply supported end the absolute stiffness was calculated by (3EI/L) and also since ‘D’ can be considered as simply supported with a definite
moment , therefore the absolute stiffness of CD has also been calculated using the formula (3EI/L) .

81. Distribution Factor ( k / ∑k )
Joint
Member
Absolute Stiffness
Distribution Factor ( k / ∑k ) B BA
3E3I/4
0.31 BC
4E10I/8
0.69 C CB
0.77 CD
3E2I/4
0.23

92. Find The Final Moments from slope deflection method and
Assignment -1
Find The Final Moments from slope deflection method and
Moment Distribution method and Draw Bending Moment diagram
2 kN/m
10 kN
1.5 m 3m
1 m A B C D I 2I
2 m
12kN
2 kN/m
9 kN
1.5 m 3m
1 m A B C D I 2I
2 m
8 kN
8 kN
1.5 m

93. Support C sinks by 20 mm I = 2.4 X 106 mm4 E =2 X 105 N/mm2
4 kN/m
5 kN
1.5 m 3m
1 m A B C D I
2 m
6 kN
Support C sinks by 20 mm
I = 2.4 X 106 mm4
E =2 X 105 N/mm2
2 kN/m
9 kN
1.5 m 3m
1 m A B C D I
2 m
8 kN
Support A sinks by 10 mm
Support B sinks by 30 mm
Support C sinks by 20 mm
I = 2.4 X 106 mm4
E =2 X 105 N/mm2