Shear Force and Bending Moment Diagram

CE6306 - Strength of Materials
Shear Force and Bending Moment
Diagrams
[SFD & BMD]
Presented by
Mr. Amos Gamaleal David
Assistant Professor
Department of Mechanical Engineering
Panimalar Institute of Technology

Shear force is a force that tends to cause sliding
of one cross-section against another. When you
slice bread , you are applying a shear force with
your knife.
Shear Force

Bending Moment is a force that tends to cause
a member to rotate or “bend” so that the
straight member tends to assume a curved
profile.
Bending Moment

Shear Force and Bending Moments
Consider a section x-x at a distance 6m from left hand support A
5kN 10kN 8kN
4m 5m 5m 1m
A
C D
B
RA = 8.2 kN RB=14.8kN
E
x
x
6 m
Imagine the beam is cut into two pieces at section x-x and is separated, as
shown in figure

To find the forces experienced by the section, consider any one portion of the
beam. Taking left hand portion
Transverse force experienced = 8.2 – 5 = 3.2 kN (upward)
Moment experienced = 8.2 × 6 – 5 × 2 = 39.2 kN-m (clockwise)
If we consider the right hand portion, we get
Transverse force experienced = 14.8 – 10 – 8 =-3.2 kN = 3.2 kN (downward)
Moment experienced = - 14.8 × 9 +8 × 8 + 10 × 3 = -39.2 kN-m = 39.2 kN-m
(anticlockwise)
5kN
A
8.2 kN
10kN 8kN B
14.8 kN
4 m
6 m
9 m
1 m
5 m

5kN
A
8.2 kN
10kN 8kN B
14.8 kN
3.2 kN
3.2 kN
39.2 kN-m
39.2 kN-m
Thus the section x-x considered is subjected to forces 3.2 kN and
moment 39.2 kN-m as shown in figure. The force is trying to shear off
the section and hence is called shear force. The moment bends the
section and hence, called bending moment.

Shear force at a section: The algebraic sum of the vertical forces
acting on the beam either to the left or right of the section is
known as the shear force at a section.
Bending moment (BM) at section: The algebraic sum of the
moments of all forces acting on the beam either to the left or
right of the section is known as the bending moment at a section
3.2 kN
3.2 kN
F
F
Shear force at x-x
M
Bending moment at x-x
39.2 kN

Moment and Bending moment
Bending Moment (BM): The moment which causes the
bending effect on the beam is called Bending Moment. It is
generally denoted by ‘M’ or ‘BM’.
Moment: It is the product of force and perpendicular
distance between line of action of the force and the point
about which moment is required to be calculated.

Sign Convention for shear force
F
F
F
F
+ ve shear force - ve shear force

The bending moment is considered as Sagging Bending
Moment if it tends to bend the beam to a curvature
having convexity at the bottom as shown in the Fig.
given below. Sagging Bending Moment is considered
as positive bending moment.
Sign convention for bending moments:
Fig.Sagging bending moment[Positive bending moment ]
Convexity

Similarly the bending moment is considered as
hogging bending moment if it tends to bend the
beam to a curvature having convexity at the top
as shown in the Fig. given below. Hogging
Bending Moment is considered as Negative
Bending Moment.
Sign convention for bending
moments:
Fig. Hogging bending moment [Negative bending moment ]
Convexity

Shear Force Diagram (SFD):
The diagram which shows the variation of
shear force along the length of the beam is
called Shear Force Diagram (SFD).
Bending Moment Diagram (BMD):
The diagram which shows the variation of
bending moment along the length of the
beam is called Bending Moment Diagram
(BMD).
Shear Force and Bending Moment Diagrams
(SFD & BMD)

Point of Contra flexure [Inflection point]:
It is the point on the bending moment diagram
where bending moment changes the sign from
positive to negative or vice versa.
It is also called ‘Inflection point’. At the point of
inflection point or contra flexure the bending
moment is zero.

Variation of Shear force and bending moments
Variation of Shear force and bending moments for various standard
loads are as shown in the following Table
Type of load
SFD/BMD
Between point
loads OR for no
load region
Uniformly
distributed load
Uniformly
varying load
Shear Force
Diagram
Horizontal line Inclined line Two-degree curve
(Parabola)
Bending
Moment
Diagram
Inclined line Two-degree curve
(Parabola)
Three-degree
curve (Cubic-
parabola)
Table: Variation of Shear force and bending moments

Steps Involved
 Draw the Free body diagram
 Calculate the Reaction forces
 Calculate the Shear Force at various points
 Draw the Shear force Diagram
 Calculate the Moment at various points
 Draw the Bending moment Diagram

Example Problem 1
E
5N 10N 8N
2m 2m 3m 1m
A
C D
B
1. Draw shear force and bending moment diagrams [SFD
and BMD] for a simply supported beam subjected to
three point loads as shown in the Fig. given below.

E
5N 10N 8N
2m 2m 3m 1m
A
C D
B
Solution:
Using the condition: ΣMA = 0
- RB × 8 + 8 × 7 + 10 × 4 + 5 × 2 = 0  RB = 13.25 N
Using the condition: ΣFy = 0
RA + 13.25 = 5 + 10 + 8  RA = 9.75 N
RA RB
[Clockwise moment is Positive]

Shear Force at the section 1-1 is denoted as V1-1
Shear Force at the section 2-2 is denoted as V2-2 and so on...
V0-0 = 0; V1-1 = + 9.75 N V6-6 = - 5.25 N
V2-2 = + 9.75 N V7-7 = 5.25 – 8 = -13.25 N
V3-3 = + 9.75 – 5 = 4.75 N V8-8 = -13.25
V4-4 = + 4.75 N V9-9 = -13.25 +13.25 = 0
V5-5 = +4.75 – 10 = - 5.25 N (Check)
5N 10N 8N
2m 2m 3m 1m
RA = 9.75 N RB=13.25N
1
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8 9
8 9
0
0
Shear Force Calculation:

5N 10N 8N
2m 2m 3m 1m
A
C D E
B
9.75N 9.75N
4.75N 4.75N
5.25N 5.25N
13.25N 13.25N
SFD

Bending moment at A is denoted as MA
Bending moment at B is denoted as MB
and so on…
MA = 0 [ since it is simply supported]
MC = 9.75 × 2= 19.5 Nm
MD = 9.75 × 4 – 5 × 2 = 29 Nm
ME = 9.75 × 7 – 5 × 5 – 10 × 3 = 13.25 Nm
MB = 9.75 × 8 – 5 × 6 – 10 × 4 – 8 × 1 = 0
or MB = 0 [ since it is simply supported]
Bending Moment Calculation

5N 10N 8N
2m 2m 3m 1m
19.5Nm
29Nm
13.25Nm
BMD
A B
C D E

E
5N 10N 8N
2m 2m 3m 1m
A
C D
B
BMD
19.5Nm
29Nm
13.25Nm
9.75N 9.75N
4.75N 4.75N
5.25N 5.25N
13.25N 13.25N
SFD
Example Problem 1
VM-34

BMD
19.5Nm
29Nm
13.25Nm
E
5N 10N 8N
2m 2m 3m 1m
A
C D
B
9.75N 9.75N
4.75N 4.75N
5.25N 5.25N
13.25N 13.25N
SFD

2. Draw SFD and BMD for the double side overhanging
beam subjected to loading as shown below. Locate points
of contraflexure if any.
5kN
2m 3m 3m 2m
5kN 10kN
2kN/m
A B
C D E
Example Problem 2

2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
A B
C D E
Solution:
Calculation of Reactions:
Due to symmetry of the beam, loading and boundary
conditions, reactions at both supports are equal.
.`. RA = RB = ½(5+10+5+2 × 6) = 16 kN
RA RB

2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
1
1 3
4
2
3
2
4 6
6
5
5
9
9
8
7
7
8
Shear Force Calculation: V0-0 = 0
V1-1 = - 5kN V6-6 = - 5 – 6 = - 11kN
V2-2 = - 5kN V7-7 = - 11 + 16 = 5kN
V3-3 = - 5 + 16 = 11 kN V8-8 = 5 kN
V4-4 = 11 – 2 × 3 = +5 kN V9-9 = 5 – 5 = 0 (Check)
V5-5 = 5 – 10 = - 5kN
RA=16kN RB = 16kN
0
0

2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
A B
C D E
5kN
+
+
5kN 5kN
5kN 5kN 5kN
11kN
11kN
SFD

2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
A B
C D E
Bending Moment Calculation:
MC = ME = 0 [Because Bending moment at free end is zero]
MA = MB = - 5 × 2 = - 10 kNm
MD = - 5 × 5 + 16 × 3 – 2 × 3 × 1.5 = +14 kNm
RA=16kN RB = 16kN

2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
A B
C D E
10kNm
10kNm
14kNm
BMD

2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
A B
C D E
10kNm 10kNm
14kNm
BMD
+
+
5kN 5kN
5kN 5kN 5kN
11kN
11kN
SFD

10kNm 10kNm
Let x be the distance of point of contra flexure from support A
Taking moments at the section x-x (Considering left portion)
0
2
2
16
)
2
(
5
2







x
x
x
M x
x
x = 1 or 10
.`. x = 1 m
x
x
x
x
Points of contra flexure
2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
A B
C D E

10kNm 10kNm
0
2
2
16
)
2
(
5
2







x
x
x
M x
x
x
x
x
x
Points of contra flexure
2m 3m 3m 2m
5kN 10kN 5kN
2kN/m
A B
C D E
1
10
0
10
11
0
16
5
10
2
2














x
or
x
x
x
M
x
x
x
M
x
x
x
x

3. Draw SFD and BMD for the single side overhanging beam
subjected to loading as shown below. Determine the
absolute maximum bending moment and shear forces and
mark them on SFD and BMD. Also locate points of contra
flexure if any.
4m 1m 2m
2 kN 5kN
10kN/m
A
B
C D
Example Problem
Example Problem 3

4m 1m 2m
2 kN 5kN
10kN/m
A B
RA RB
Solution : Calculation of Reactions:
ΣMA = 0
- RB × 5 + 10 × 4 × 2 + 2 × 4 + 5 × 7 = 0  RB = 24.6 kN
ΣFy = 0
RA + 24.6 – 10 x 4 – 2 + 5 = 0  RA = 22.4 kN

4m 1m 2m
2 kN 5kN
10kN/m
RA=22.4kN
RB=24.6kN
Shear Force Calculations:
V0-0 =0; V1-1 = 22.4 kN V5-5 = - 19.6 + 24.6 = 5 kN
V2-2 = 22.4 – 10 × 4 = -17.6kN V6-6 = 5 kN
V3-3 = - 17.6 – 2 = - 19.6 kN V7-7 = 5 – 5 = 0 (Check)
V4-4 = - 19.6 kN
1
1
2
2
3
3
4
4
5
5
6
6
7
7
0
0

4m 1m 2m
2 kN 5kN
10kN/m
RA=22.4kN
RB=24.6kN
22.4kN
19.6kN 19.6kN
17.6kN
5 kN 5 kN
SFD
x = 2.24m
A
C B D

Max. bending moment will occur at the section where the shear force is
zero. The SFD shows that the section having zero shear force is available
in the portion AC. Let that section be X-X, considered at a distance x
from support A as shown above.
The shear force at that section can be calculated as
Vx-x = 22.4 - 10. x = 0  x = 2.24 m
4m 1m 2m
2 kN 5kN
10kN/m
A
B
C D
RA=22.4kN
RB=24.6kN
X
X
x

Calculations of Bending Moments:
MA = MD = 0
MC = 22.4 × 4 – 10 × 4 × (4/2) = 9.6 kNm
MB = 22.4 × 5 – 10 × 4 × (4/2 +1) – 2 × 1 = - 10kNm
(Considering Left portion
of the section)
Alternatively
MB = -5 × 2 = -10 kNm (Considering Right portion of the section)
Absolute Maximum Bending Moment is at X- X ,
Mmax = 22.4 × 2.24 – 10 × (2.24)2 / 2 = 25.1 kNm
4m 1m 2m
2 kN 5kN
10kN/m
A
B
C D
RA=22.4kN
RB=24.6kN

4m 1m 2m
2 kN 5kN
10kN/m
A
B
C D
RA=22.4kN
RB=24.6kN
X
X
x = 2.24m
9.6kNm
10kNm
BMD
Point of
contra flexure
Mmax = 25.1 kNm

9.6kNm
10kNm
BMD
Point of
contra flexure
4m 1m 2m
2 kN 5kN
10kN/m
A
B
C D
RA=22.4kN
RB=24.6kN
X
X
x = 2.24m
22.4kN
19.6kN 19.6kN
17.6kN
5 kN 5 kN
SFD
x = 2.24m

Calculations of Absolute Maximum Bending Moment:
Max. bending moment will occur at the section where the shear force is
zero. The SFD shows that the section having zero shear force is available
in the portion AC. Let that section be X-X, considered at a distance x
from support A as shown above.
The shear force at that section can be calculated as
Vx-x = 22.4 - 10. x = 0  x = 2.24 m
Max. BM at X- X ,
Mmax = 22.4 × 2.24 – 10 × (2.24)2 / 2 = 25.1 kNm
4m 1m 2m
2 kN 5kN
10kN/m
A
B
C D
RA=22.4kN
RB=24.6kN
X
X
x

4m 1m 2m
2 kN 5kN
10kN/m
A
B
C D
RA=22.4kN
RB=24.6kN
X
X
x = 2.24m
Mmax = 25.1 kNm
9.6kNm
10kNm
BMD
Point of
contra flexure

Mmax = 25.1 kNm
9.6kNm
10kNm
BMD
Point of
contra flexure
a
Let a be the distance of point of contra flexure from support B
Taking moments at the section A-A (Considering left portion)
A
A
0
6
.
24
)
2
(
5 





a
a
M A
A
a = 0.51 m

Shear Force and Bending Moment Diagram

Shear Force and Bending Moment Diagram

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Shear Force and Bending Moment Diagram