Shear Force and Bending Moment. Fundamentals and Problem Solving

SHEAR FORCE AND BENDING MOMENT

Example 2.6 Cantilever Beam 2.0

A cantilever beam loaded as shown in Figure 2.6. If reaction at support A
are RA = 75kN and MA = 165kNm.
i. Calculate the value of shear force and bending moment.
ii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iii. Determine the maximum bending moment.

Figure 2.6

Solution:

Step 1: Calculate Shear Force, kN.

Point/ Figure Shear Force
Interval = R = +75

A

A–B = − 1
C
= 75 − 20 3 = 15


= − 2
= 15 − 15 = 0

44

SHEAR FORCE AND BENDING MOMENT

Example 2.6 Cantilever Beam

Step 2: Illustrate the Shear Force Diagram (SFD) .

2.0

Step 3: Based on SFD in Figure 2.6, calculate Bending Moment, kNm.

Point/ Figure Bending Moment
Interval

A = −165 m

= + 1 + Area of
2 trapezium

A–B = −165kNm + 1 (75 + 15 ) 3
B–C 2

= −30kNm

= + 2m
= −30 + 15
= 0

45

SHEAR FORCE AND BENDING MOMENT

Example 2.6 Cantilever Beam

Step 4: Illustrate the Bending Moment Diagram (BMD) .

MA = 165kNm

Things to keep in 2.0
mind

At fixed support of a
cantilever beam, the
bending moment is
always maximum

The maximum bending
moment, Mmax = 165kNm

At free end of the
cantilever beam, the
bending moment is
always zero.

The bending moment can also be calculated using Figure 2.6.

Point/ Figure Bending Moment
Interval

A = −165 m

= + − 1
2

A–B = −165kNm + 75 3 − 20 3 3
A–C 2

= −30kNm

= + − 1 +
2

= −165 + 75 3m + 2m

−20 3 3 + 2
2

= 0

46

SHEAR FORCE AND BENDING MOMENT

Example 2.7 Cantilever Beam 2.0

A cantilever beam loaded as shown in Figure 2.7. If reaction at support A
are RA = 75kN and MA = 165kNm.
i. Calculate the value of shear force and bending moment.
ii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iii. Determine the maximum bending moment.

Figure 2.7

Solution:

Step 1: Calculate Shear Force, kN.

Point/ Figure Shear Force
Interval

A = −20
B
B–C =
= −20

= − 2

= −20 − 15 2 = −50kN


D = +
= −50 + 50kN = 0

47

SHEAR FORCE AND BENDING MOMENT

Example 2.7 Cantilever Beam

Step 2: Illustrate the Shear Force Diagram (SFD) .

2.0

Step 3: Based on SFD in Figure 2.7, calculate Bending Moment, kNm.

Point/ Figure Bending Moment
Interval

A = 0

A–B = +
= 0kNm + (−20 ) 2
= −40kNm

B ′ = + M
B–C ′ = −40 + 10kNm = 0
′ = −30kNm

= ′ + 1 +
2

= −30 + 1 (−20 + (−50 ) 2 =0
2

= −100kNm

48

SHEAR FORCE AND BENDING MOMENT

Example 2.7 Cantilever Beam

Point/ Figure Bending Moment
Interval

C–D = + 2.0
= −100kNm + (−50 ) 1
= −150kNm

′ = +
D ′ = −150 + 150kNm

′ = 0

Step 4: Illustrate the Bending Moment Diagram (BMD) .

The maximum
bending moment,
Mmax = 150kNm

49

SHEAR FORCE AND BENDING MOMENT

Example 2.7 Cantilever Beam

The bending moment can also be calculated using Figure 2.7.

Point/ Figure Bending Moment
Interval
2.0
A
= 0

A–B =
= −20 2
= −40kNm

B ′ = +
A–C = −20 2 + 10
= −30kNm
A–D
D = 1 + − 2
2

= (−20 )(4m) + 10kNm

− 15 2 2
2

= −100

= 1 + − 2 +
2

= −20 5m + 10kNm

− 15 2 2 + 1
2

= −150

′ = +
′ = −150kNm + 150
′ = 0

50

SHEAR FORCE AND BENDING MOMENT

Example 2.8 Cantilever Beam 2.0

A cantilever beam loaded as shown in Figure 2.8.
i. Calculate reactions at the support of beam
ii. Calculate the value of shear force and bending moment.
iii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iv. Determine the maximum bending moment.

Figure 2.8

Solution:
Step 1: Sketch Free Body Diagram.

Calculate the resolution force of
32kN at an angle of 450 above the
horizontal using trigonometric.

= 32kN
= . 450

=
= .

Step 2: Determine reactions at support A by using principle of equilibrium forces.

→ = ←; Therefore, the direction of

+ 22.627 = 0 is to the leftward (←).
= −22.627

↑= ↓;

= 10 1.5 + 22.627


= 37.627

= 0; 1.5
2
(22.627 )(1.5 ) + 10 1.5 + 1.5 − = 0

= 67.691

51

SHEAR FORCE AND BENDING MOMENT

Example 2.8 Cantilever Beam

Step 3: Calculate the Shear Force.

= = 37.627 2.0

= −
= 37.627 − 22.627kN
= 15kN

= −

= 15 − 10 1.5

=0

Step 4 : Calculate the Bending Moment.

= −67.691kNm

= +
= −67.691kNm + 37.627 1.5m
= −11.25kNm

= + 1
= 2
1
−11.25 m − 2 −15 1.5

=0

The maximum

bending moment,
Mmax = 67.691kNm

52

SHEAR FORCE AND BENDING MOMENT

SELF-TEST 2B Cantilever Beam

A cantilever beam loaded as shown in figure below. 2.0
i. Calculate the reaction at the support of beam.
ii. Calculate the value of shear force and bending moment.
iii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iv. Determine the maximum bending moment.

1.

2.
3.

4.

53

SHEAR FORCE AND BENDING MOMENT

Example 2.9 Overhanging Beam 2.0

An overhanging loaded as shown in Figure 2.9. If reaction at support A
are RA = 32.5kN and RD = 62.5kN.
i. Calculate the value of shear force and bending moment.
ii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iii. Determine the maximum bending moment.

Figure 2.9

Solution:

Step 1: Calculate Shear Force, kN.

Point/ Figure Shear Force
Interval

A = = +32.5

= − BC where, = = 32.5kN

C = 32.5 − 10 1.5 = 17.5


′ = − 1
C’ ′ = 17.5 − 35 = −17.5kN

= ′ −

D = −17.5 − 10 3 = −47.5kN


54

SHEAR FORCE AND BENDING MOMENT

Example 2.9 Overhanging Beam

Point/ Figure Shear Force
Interval
′ = + 2.0
D’ ′ = −47.5 + 62.5 = 15kN

= ′ − w BC

E = 15 − 10 1.5 = 0


Step 2: Illustrate the Shear Force Diagram (SFD) .

55

SHEAR FORCE AND BENDING MOMENT

Example 2.9 Overhanging Beam

Step 3: Based on SFD in Figure 2.9, calculate Bending Moment, kNm.

Point/ Figure Bending Moment
Interval
2.0
A
= 0

A–B =
B–C = (32.5kN) 1.5
C–D = 48.75kNm
D–E
= + 1 ( + )
2

= 48.75 m + 1 32.5 + 17.5 1.5m
2

= 86.25kNm

= + 1 ( ′ + )
2

= 68.25 m + 1 −17.5 + (−47.5 3m
2

= −11.25kNm

= + 1 ( ′)
2

= −11.25 m + 1 15 1.5m
2

= 0kNm

56

SHEAR FORCE AND BENDING MOMENT

Example 2.9 Overhanging Beam

Step 4: Illustrate the Bending Moment Diagram (BMD) .

2.0

The maximum

bending moment,
Mmax = 86.25kNm

57

SHEAR FORCE AND BENDING MOMENT

Example 2.9 Overhanging Beam

The bending moment can also be calculated using Figure 2.9.

Point/ Figure Bending Moment
Interval
= 0 2.0
A

A–B =
= + 32.5 1.5
= 48.75kNm

= − 2
2

A–C = + 32.5 3 − 10 1.5 1.5
A–D 2
A–E
= 86.25kNm

= − 1 −
2

= + 32.5 6 − (35 )(3 )

− 10 4.5 4.5
2

= −11.25kNm

= − 1 − +
2

= + 32.5 7.5 − (35 )(4.5 )

− 10 6 6 + (62.5 )(1.5 )
2

= 0

58

SHEAR FORCE AND BENDING MOMENT

Example 2.10 Overhanging Beam 2.0

An overhanging loaded as shown in Figure 2.10. If reaction at support A
are RA = 54.6kN and RE = 25.4kN.
i. Calculate the value of shear force and bending moment.
ii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iii. Determine the maximum bending moment.

Figure 2.10

Solution:

Step 1: Calculate Shear Force, kN.

Point/ Figure Shear Force
Interval

A = − 1= −15

B = + B
= −15 + 54.6kN = 39.6

C = B − 2
= 39.6 − 25 = 14.6kN

= C − w CD

C–D = 14.6 − 10 3m = −15.4kN


59

SHEAR FORCE AND BENDING MOMENT

Example 2.10 Overhanging Beam

Point/ Figure Bending Moment 2.0
Interval
E = D +
E + = −15.4 + 25.4kN = 10kN
RE = 25.4kN
F = E − 3
F3 = 10 − 10kN = 0
10k
N−

E

Step 2: Illustrate the Shear Force Diagram (SFD) .

hh1 1==291k4N.6kN l2 = 3 – x
1 2

ll11 == xx h2 = 15.4kN

1 = 2
ℎ1 ℎ2

= 3 −
14.6 15.4

= 3(14.6)
(14.6 + 15.4)

= 1.46

Therefore;
3 − = 3 − 1.46 = 1.54

60

SHEAR FORCE AND BENDING MOMENT

Example 2.10 Overhanging Beam

Step 3: Based on SFD in Figure 2.10, calculate Bending Moment, kNm.

Point/ Figure Bending Moment
Interval

A = 0 2.0

A–B =
= (−15kN) 2
= −30kNm

B–C = +
C–x = −30kN + (39.6 ) 1
x–D = 9.6kNm

D = + 1
D–E 2
E–F
= 9.6kN + 1 (14.6 ) 1.46
2

= 20.258kNm

= + 1
2

= 20.258kN + 1 (−15.4 ) 1.54
2

= 8.4kNm

′ = − M
′ = 8.4kN − 13kNm
′ = −4.6kNm

= ′ +
= −4.6kNm + (−15.4kN) 1
= −20kNm

= +
= −20kNm + (10kN) 1
= 0

61

SHEAR FORCE AND BENDING MOMENT

Example 2.10 Overhanging Beam

Step 4: Illustrate the Bending Moment Diagram (SFD) .

2.0

The maximum
bending moment,
Mmax = 20.258kNm

62

SHEAR FORCE AND BENDING MOMENT

Example 2.10 Overhanging Beam

The bending moment can also be calculated using Figure 2.10.

15kN 25kN 13kNm 10kN

10kN/m 2.0

RB = 54.6kN RE = 25.4kN

2m 1m 3m D 1m E 2m F
A BC

Figure 2.10

Point/ Bending Moment
Interval

A = 0
A–B = 1 = − 15 2 = −30kNm
A–C
A–D = 1 − = − 15 3 + 54.6 1 = 9.6

D = 1 + − 2 −
A–E 2

A–F = − 15 6 + 54.6 4 −(25 ) 3 − 10 3 3
2

= 8.4kNm

′ = + = 8.4 − 13 = −4.6kNm

= 1( ) + ( ) − 2 − + −
2

= − 15 7 + 54.6 5 − 25 4 − 10 3 3 + 1
2

−13kNm

= −20kNm

= 1 + − 2 − + − +
2

= − 15 9 + 54.6 7 − 25 6 − 10 3 3 + 3
2

−13kNm + (25.4kN)(2m)

= 0kNm

63

SHEAR FORCE AND BENDING MOMENT

Example 2.11 Overhanging Beam 2.0

An overhanging loaded as shown in Figure 2.11.
i. Calculate reactions at the support of beam
ii. Calculate the value of shear force and bending moment.
iii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iv. Determine the maximum bending moment.

Figure 2.11

Solution:
Step 1: Sketch Free Body Diagram.

Step 2: Determine reaction at support B and D by using principle of equilibrium forces.

→ = ←;
= 0

= 0;

− 12 2 2 + 12 7 7 + (35 )(5 ) − (7 ) = 0
2 2

= 63.571

↑= ↓;

+ 63.571 = 12 9 + 35


= 79.429

64

Example 2.11 Overhanging Beam SHEAR FORCE AND BENDING MOMENT

Step 3: Calculate the Shear Force. 2.0

= 0

=

=− 12 2

= −24

′ = +
= −24 + 79.429
= 55.429kN

= ′ −

= 55.429 − 12 5

= −4.571kN

′ = −
= −4.571N − 35kN
= −39.571kN

= ′ −

= −39.571 N − 12 2

= −63.571kN

′ = +
= −63.571 N − 63.571 N
= 0kN

Step 4 : Calculate the Bending Moment.

= 0

= 1 −24 2m
2
= −24kNm

= + 1 ( ′)
2
1
= −24kNm + 2 (55.429 ) 4.619m

= 104.013kNm

= + 1 ( )
2
1
= 104.013k m + 2 −4.571 0.381m

= 103.142kNm The maximum

= + 1 ( ′ + ) bending moment,
2 Mmax = 104.013kNm
1
= 103.142 Nm + 2 −39.571 − 63.571 (2m)

=0

65

SHEAR FORCE AND BENDING MOMENT

SELF-TEST 2C Overhanging Beam

An overhanging beam loaded as shown in figure below. 2.0
i. Calculate the reactions at the support of beam.
ii. Calculate the value of shear force and bending moment.
iii. Illustrate the Shear Force Diagram (SFD) and Bending Moment

Diagram (BMD) that indicate all the values at important points.
iv. Determine the maximum bending moment.

1.

2.

3.

4.

66

SHEAR FORCE AND BENDING MOMENT

SUMMARY 2.0

It is important to know how the shear
forces and bending moments vary along
the length of a beam that is being
designed.

The Employing these diagrams, the
maximum and minimum shear force and
bending moment are easily identified and

located.

Test your knowledge and Digital QUIZ
understanding here

67

PROBLEM SOLVING

A SIMPLY SUPPORTED BEAM

Consider a simply supported beam as shown in figure below.
i. Calculate the reactions at support of the beam.
ii. Draw the shear force diagram (SFD) and bending moment

diagram (BMD).
iii. Hence, determine the value of maximum bending moment

Question 1 35kN 40kN

3m 3m 3m
A BC D

Question 2 30kN

15kN/m

6m 3m
A BC

68

PROBLEM SOLVING

Question 3 Simply Supported Beam

30kN 20kN

25kN/m

1m 1m 1m 3m
A B CD E

Question 4 51kN

20kN/m 3kNm

3m 2m 1m
A B CD

Question 5

5kNm 30kN
15kN/m

2m 2m 2m
A B CD

69

PROBLEM SOLVING

Question 6 Simply Supported Beam

22kN/m 8kNm 16kN/m

3m B 3m C
A

Question 7 10kN/m 30kN
6kNm 300

1m 4m 1m
AB CD

Question 8

12kN/m 60kN
450

A 4m 1m 2m
BC D

70

PROBLEM SOLVING

B CANTILEVER BEAM

Consider a cantilever beam subjected to a load as shown in figure
below.
i. Calculate the reactions at support of the beam.
ii. Draw the shear force diagram (SFD) and bending moment

diagram (BMD).
iii. Hence, determine the value of maximum bending moment

Question 1 10kN

5kN

1m 4m 1m
AB CD

Question 2 20kN

15kNm 35kN/m

A 2m 2m C 2m D 2m E
B

71

Question 3 PROBLEM SOLVING Cantilever Beam

25kN 20kN/m
300

1m 1m 4m
ABC D

Question 4
45kN

20kN/m 10kNm

A 1m B 5m C

Question 5 10kN

25kN/m 15kNm

3m 1m 3m D
A BC

72

PROBLEM SOLVING

Question 6 20kN Cantilever Beam
14kN/m
16kN/m

2.5m 2.5m C
AB

Question 7 20kN

4kNm
15kN/m

1m 1m 1.5m D
ABC

Question 8

10kN 14kN

5kN/m

1m 1m 1m
A B CD

73

PROBLEM SOLVING

C OVERHANGING BEAM

An overhanging beam is subjected to a load as shown in figure
below.
i. Calculate the reactions at support of the beam.
ii. Draw the shear force diagram (SFD) and bending moment

diagram (BMD).
iii. Hence, determine the value of maximum bending moment

and the point at which it occurs.

Question 1 10kN

5kN/m

A 1m B 3m C 1m D 2m E

Question 2 45kN

60kN/m

A 2m B 3m C 2m D

74

Question 3 PROBLEM SOLVING

20kN/m 50kN Overhanging Beam
10kN/m
6m
A 2m
BC

Question 4 10kN 10kNm
5kN/m 600

A 3m B 3m C 2m D

Question 5 1kN

2kN/m

2m 1m
A BC

75

PROBLEM SOLVING

Question 6 5kN Overhanging Beam
3kN

2kN/m

2m 6m 2m
AB CD

Question 7 20kN
30kN

45kN/m

1m 4m 2m
AB CD

Question 8

60kN 35kNm 15kN/m
300

1m 5m 2m 4m
AB CD E

76

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International Edition.

James M. Gere 2013. Mechanics of Materials. 8th Edition: Stamford, Ct
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Keith M. Walker. 2014. Applied Mechanics for Engineering Technology. 8th
Edition: Pearson New International Edition, United State, Pearson.

Pant M K. 2016. Fundamental of Structural Analysis. 7th Edition. New Delhi S K
Kataria Sons.

Rajput, R. K. (2010). Strength of Materials: Mechanics of Solids by R.K. Rajput
(2007–12-01) (Revised Edition). S Chand & Co Ltd.

Namita (2017). Loads on Building for Modeling | Structure Load Calculation |
Analyze Building | Excel Spreadsheet. https://civildigital.com/calculation-
loads-building-modeling-structure-load-calculation-design-building-excel-
spreadsheet/

N. (2017, May 4). Loads on Building for Modeling | Structure Load Calculation
| Analyze Building | Excel Spreadsheet. Civildigital.Com.
https://civildigital.com/calculation-loads-building-modeling-structure-load-
calculation-design-building-excel-spreadsheet/

Terri Meyer, B., & Vincent, H. (2020). Fun is in the Details: Innovation in Steel
Connections. Canadian Institute of Steel Construction.
http://www.tboake.com/SSEF1/pin.shtml

Lesics. (2013, July 16). Beams| Bending Moment and Shear Force Diagram
[Video]. YouTube. https://youtu.be/5K27dJqGpf8

The Efficient Engineer. (2019, November 20). Understanding shear force and
bending moment diagrams [Video]. YouTube. https://youtu.be/C-
FEVzI8oe8