lecture mechanical engg

1. Shear Force and Bending Moment Diagrams
[SFD & BMD]
Dr. Mukesh Kumar
Department of Mechanical
Engineering
IIT GUWAHATI

2. A beam is a structural element that
primarily resists loads applied laterally to the
beam's axis.
The following are the important types of beams:
1. Cantilever beam,
2. Simply supported beam,
3. Overhanging beam,
4. Fixed beams, and
5. Continuous beam.
Beams

3. 1. Cantilever beam
A beam which is fixed at one of its end and
the other end is free is called a cantilever beam.
Beams

4. 2. Simply Supported Beam
A beam which is freely supported at both
ends is called a simply supported beam.
This support allow to horizontal movement
of beam.
Beams

5. 3. Overhanging Beam
The beam freely supported at any two
points and having one or both ends projected
beyond these supports is called an overhanging
beam.
Beams

6. 4. Fixed Beams
A beam, whose both ends are fixed or built-
in walls, is known as fixed beam.
It is only under shear stress and no moment
produces in this beams.
Beams

7. 5. Continuous Beam
A beam which is provided more than two
supports.
This beam is similar to simply supported
beam except more than two support are used on
it.
Beams

8. 1. Concentrated Load or Point Load
This type of load acts relatively on a smaller
area. For example, the force exerted by a chair or a
table leg on the supporting floor or load exerted by a
beam on a supporting column are both considered
to be concentrated.
Types Of Loads

9. 2. Uniformly Distributed Load
A distributed load is a load which is spread on some
length of a beam, i.e. The reason it is measured in intensity
with units newton/meter. If the intensity is constant along
the length then it is named as uniformly distributed load

10. 3.Uniformly Varying Load
Whenever the load distributed along the length of the
beam varies in intensity uniformly, according to some law.
Then it is named as uniformly varying load and we can see
some conventional figures below which are representing
uniformly varying loads

13. Beams are supported on roller, hinged or
fixed supports as shown in fig
1. Simple Support:
If one end of the beam rests in a fixed
support, the support is known as simple support.
The beam is free to move in any direction
and also to rotate about the support.
Types of Supports:

14. 2. Roller Support:
Here one end of the beam is supported on a
roller.
Types of Supports:

15. 3. Fixed Support
This support keeps the end of the beam fixed, i.e. The
beam end resists to take any kind of translation or bending
moment. In the below figures we are going to see how this
support will be.

16. 4. Hinged Support:
At the hinged support the beam does not move
either along or normal to its axis.
A Hinged support restricts the movement of the
beam in any directions but it will allow the beam to
rotate about the support just like a door which is an
best example of hinged support.
Types of Supports:

18. 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

19. 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.

20. Sign Convention For Shear Force
F
F
F
F
+ ve shear force - ve shear force

21. Sign Convention For Bending Moments:
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.
Fig. Sagging bending moment [Positive bending moment ]
Convexity

22. Sign Convention For Bending Moments
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.
Fig. Hogging bending moment [Negative bending moment ]
Convexity

23. Shear Force and Bending Moment Diagrams
(SFD & BMD)
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).

24. 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.

25. Relationship Between Load, Shear Force And
Bending Moment
Fig. A simply supported beam subjected to general type loading
L
w kN/m
x
x
x1
x1
dx
The above Fig. shows a simply supported beam subjected
to a general type of loading. Consider a differential
element of length ‘dx’ between any two sections x-x and
x1-x1 as shown.

26. dx
v
V+dV
M M+dM
Fig. FBD of Differential element of the beam
x
x x1
x1
w kN/m
O
Taking moments about the point ‘O’ [Bottom-Right corner of the
differential element ]
- M + (M+dM) – V.dx – w.dx.dx/2 = 0
V.dx = dM 
dx
dM
v 
It is the relation between shear force and BM
Neglecting the small quantity of higher order

27. dx
v
V+dV
M M+dM
Fig. FBD of Differential element of the beam
x
x x1
x1
w kN/m
O
Considering the Equilibrium Equation ΣFy = 0
- V + (V+dV) – w dx = 0  dv = w.dx 
dx
dv
w 
It is the relation Between intensity of Load and
shear force

28. 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