This document provides examples and explanations for determining internal forces like shear force and bending moment in structural members like beams and frames. It begins by introducing sign conventions and the procedure for analysis, which involves determining support reactions, drawing free body diagrams, and using equilibrium equations. Numerous step-by-step examples are then provided to demonstrate how to calculate and graph shear force and bending moment diagrams for beams and frames with different loading and support conditions.

1. Chapter 4Chapter 4
Internal Loading Developed in
Structural Members

2. Chapter 4
Internal Loading
Developed in Structural
Members

3. Internal loading at a specified Point
In General
• The loading for coplanar structure will
consist of a normal force N, shear force V,
and bending moment M.
• These loading actually represent the resultants
of the stress distribution acting over the member’s
cross-sectional are

4. Sign Convention
+ve Sign

5. Procedure for analysis
• Support Reaction
• Free-Body Diagram
• Equation of Equilibrium

6. Example 1
Determine the internal shear and moment acting in the
cantilever beam shown in figure at sections passing through
points C & D

7. mkNM
MM
kNV
VF
c
cC
C
Cy
.50
020)3(5)2(5)1(50
15
05550







8. mkNM
MM
kNV
VF
D
DC
C
Dy
.50
020)3(5)2(5)1(50
20
055550







9. Example 2
kNRR BA 9
mkNM
MM
kNV
VF
D
y
.12
0)2(9)1(60
3
0690
sectionat






6kN
Determine the internal shear and moment acting in section 1 in the
beam as shown in figure
18kN

10. Example 3
Determine the internal shear and moment acting in the
cantilever beam shown in figure at sections passing through
points C

11. ftkM
MM
kV
VF
D
c
Cy
.48
0)6(9)2(30
6
0390
c







12. Shear and Moment function
Procedure for Analysis:
1- Support reaction
2- Shear & Moment Function
 Specify separate coordinate x and associated origins, extending
into regions of the beam between concentrated forces and/or
couple moments or where there is a discontinuity of distributed
loading.
 Section the beam at x distance and from the free body diagram
determine V from , M at section x

13. Example 4
Determine the internal shear and moment Function

14. Example 5
Determine the internal shear and moment Function

15. 15
1
30
2
x
w
2
1
2
2
2
1
2
3
0 30 0
15
30 0.033
0 30( ) 600 0
15 3
600 30 0.011
y
S
x
F V
V x
x x
M M x
M x x
     
 
 
      
 
   


w 2
30
x

16. Example 6
Determine the internal shear and moment Function

17.  
2
11
211S
1
1
1
21081588
0410815880
4108
041080
120
1
xxM
xxMM
xV
xVF
x
x
y








18.  
130060
064810815880
60
0481080
2012
2
22S
2







xM
xxMM
V
VF
x
y

19. Example 7
Determine the internal shear and moment Function

20. 9
20
x
w
 
1
2
2
1
2 2
2 3
0 75 10 (20) 0
9
75 10 1.11
0 75 10 (20) 0
9 3
75 5 0.370
y
x
S
x
F V x x
V x x
x x
M M x x x
M x x x
 
       
 
  
  
       
  
  


w 20
9
x

21. Shear and Moment diagram for a Beam

22.  
 2
O
)(
0)()(0
)(
0)()(0
xxwxVM
MMxxxwMxVM
xxwV
VVxxwVFy













dxxVMV
dx
dM
dxxwVxw
dx
dV
xfor
)(
)()(
0

23. When F acts downward on the beam, ∆ is negative so that the
shear diagram shows a “jump” downward.
Likewise, if F acts upward, the jump is upward.
Internal Shear due to concentrated Load

24. If an external couple moment M’ is applied clockwise, ∆ is positive, so
that the moment diagram jumps downward,
and
when M’ acts counterclockwise, the jump must be upward.
Internal Moment due to concentrated moment

25. Example 8
Draw shear force
and Bending
moment Diagram
B.M.D
S.F.D

26. Example 9
Draw shear force
and Bending
moment Diagram
S.F.D
B.M.D

27. Example 10
Draw shear force
and Bending moment
Diagram
18 kN
Max. moment at x = L/2
then
8
2222
2
max
2
wL
M
LwLwL
M















28. Example 11
Draw shear force and Bending moment Diagram

29. S.F.D
B.M.D

30. Draw shear force
and Bending
moment Diagram
Example 12
49
)7(14)5.3(14
7
142





M
MM
x
x
S

31. Example 13a
Draw shear force
and Bending
moment Diagram
S.F.D
B.M.D

32. Example 13b
Draw shear force
and Bending
moment Diagram
S.F.D
B.M.D

33. Example 13c
Draw shear force
and Bending
moment Diagram
S.F.D
B.M.D

34. Example 13d
Draw shear force
and Bending
moment Diagram

37. Draw shear force and Bending moment Diagram
Example 14

38. 32

39. Draw shear force and Bending moment Diagram
Example 15

40. V(kN)

41. Example 16
Draw shear force and Bending moment Diagram

42. +

43. Example 17
Draw shear force
and Bending
moment Diagram
99

44. Example 18
Draw shear force and Bending moment Diagram

46. Sketch BMD

47. Sketch BMD

48. Sketch BMD

49. Example 19
Draw shear force and Bending moment Diagram

50. +
+
+
+

51. Problem 1
Draw shear force and Bending moment Diagram

52. Example 20
Draw shear force and Bending moment Diagram

53. 486
30.5 23.5
+
+
-
-

54. Example 21
Draw shear force and Bending moment Diagram

55.  
kE
E
EF
kC
CM
kA
AM
y
y
xx
y
yE
y
yleftB
6
0454205180F
00
45
060)32(4)27(20)16(5)6(18)12(0
4
060)5(20100
y











Reaction Calculation

57. Frame Structures
(Example 1)
Draw Bending moment Diagram

58. Support reaction & Free Body diagram

60. _
_
S.F.D B.M.D

61. +
-
S.F.D
B.M.D

62. -
B.M.D
-
15k.ft
15
+
-
S.F.D
3
1
-
N.F.D
3
-

63. Frames (Example 2)
Draw shear force and Bending moment Diagram

65. N.F.D
S.F.D
B.M.DN.F.D S.F.D
B.M.D
N.F.D
+
+
+
_
+
+
-
-

66. Frames (Example 3)
Draw shear force and Bending moment Diagram

68. B.M.DN.F.D S.F.D
-
--

69. _
+
+
N.F.D
S.F.D
B.M.D
251.6
64
26

70. B.M.D
N.F.D
S.F.D
168

71. S.F.D
B.M.D
168
432 139.3
251.6
432
36
64
26
13.22
31.78
+
_
_
+
_
_
+

72. Frames (Example 4)
Draw shear force and Bending moment Diagram

75. +
+
S.F.D
B.M.D

76. +
_ S.F.D
B.M.D

77. Frames (Example 5)
Draw shear force and Bending moment Diagram

79. Frames (Example 6)
Draw shear force and Bending moment Diagram

81. _
_
_
N.F.D S.F.D B.M.D

82. +
_
+
_
__
+
N.F.D
S.F.D
B.M.D

83. _
+
_
N.F.DS.F.DB.M.D

84. Problem 1
Draw Normal force, shear force and Bending moment Diagram

85. Problem 2
Draw Normal force, shear force and Bending moment Diagram

86. Problem 3
Draw Normal force, shear force and Bending moment Diagram

87. Frames (Example 5)
Draw Normal force, shear force and Bending moment Diagram

88. 10kN/m
60kN
o26.56
53.7
26.8
47.7
43.2
10.5
20.8
110

89. N.F.D S.F.D B.M.D
S.F.D
B.M.D

90. N.F.D
S.F.D
B.M.D

91. B.M.D

92. Moment diagram constructed by the
method of superposition
Example 1

95. Example 2.a

97. Example 2.b

99. Problem 4
Draw Normal force, shear force and Bending moment Diagram

100. Problem 5
Draw Normal force, shear force and Bending moment Diagram

101. Problem 6
Draw Normal force, shear force and Bending moment Diagram