Lecture -5

1. Lecture 5
Bending Moment & Shear force

2. Types of beam
• A structural member which carries lateral or
transverse forces (forces at right angle to the axis
of the member) is termed as a beam or joist.
• Depending upon the end conditions the various
types of beam are:
Cantilever beam
Simply supported beam
Overhanging beam
Fixed beam
Continuous beam

3. Cantilever beam
• A beam which is fixed at
one end and free at the
other end, is known as
cantilever beam.

4. Simply supported beam
• A beam supported or
resting freely on
supports at its ends, is
known as simply
supported beam or
knife edge supported
beam.

5. Overhanging beam
• A beam in which its end
portion (or portions) is
extended beyond the
support, is known as
overhanging beam.
• A beam may be
overhanging on one
side or on both sides

6. Fixed beam
• A beam whose both
ends fixed or built in
walls is known as fixed
beam.

7. Continuous beam
• A beam supported on
more than two supports
is known as continuous
beam.
• The extreme left and
right supports are
called end supports and
all the remaining
supports are known as
intermediate supports.

8. • Cantilevers, simply supported beams and
overhanging beams are statically determinate
beams. In such beams, support reactions can be
determined by using the equations of static
equilibrium. The equations of static equilibrium
are:
ΣH = 0, ΣV = 0, ΣM = 0
• Fixed end beams and continuous beam are
statically indeterminate beams. In such beams,
the support reactions can not be determined by
using the equation of static equilibrium alone.

9. Types of end supports of beams
• The following are the important types of
supports for the beams:
Simple or free support
Roller support
Hinged or pinned support
Fixed support

10. Roller support
• In this case, one of the
end of beam is
supported on rollers in
order to permit free
movement in horizontal
direction.
• Roller reaction is always
at right angles to the
roller base as shown in
figure.

11. Hinged or Pinned support
• In such case, the end of
a beam is hinged to the
support as shown in
figure.
• The reaction on such an
end may be either
vertical or horizontal
depending upon type of
loading on the beam.

12. Simple or free support
• It is that support at
which beams rests
freely.
• In such a case, the
reaction is always
vertical as shown in
figure.

13. Fixed support
• Fixed support is that in
which the beam is fixed
in position as well as in
direction which means
the beam neither move
horizontally and
vertically nor rotated.

14. • When a beam is loaded, the applied load
induce stresses in the fibers of beam cross-
section.
• To avoid failure of beam it is to be ensured
that the induced stresses do not exceed the
safe allowable stress for the material of which
beam is made.

15. Contd.
• Load on beam tend to
cause failure in two
ways:
 By shearing the beam
across its cross section
as shown in figure.
 By excessive bending of
beam as shown in
figure.

16. Definitions
• Bending Moment (BM)
Bending moment at any section of a beam is the resultant
moment about that section of all the forces acting on one side
of the section.
• Shear Force
Shear Force at any section of beam will be equal to the algebraic
sum of vertical forces (including reactions) to the left or right of
section.
Note: Firstly it should be decided which side of the given beam Section is
going to be taken. Whichever side is taken the answers will be the same. In
cantilevers it is better to take the free side. SF values are algebraic sum of the
vertical forces therefore their position does not matter provided these are on
one side of the section

17. Concave upward Convex upward
Positive bending
caused by a positive
BM (Sagging BM)
Positive SF Negative SF
Negative bending
caused by a negative
BM (Hogging BM)
Shown above are the two types of curvature that a given bending moment
may impose on a beam. These are distinguished by employing ‘plus’ or
‘minus’ signs.
Shear force may cause the section to slide either upward or downward.
If the section to the right slides downward it will be positive SF and if the
section to the right upward it will be negative SF.
Convention of Signs for BM and SF

18. BM and SF Diagrams
It is sometimes necessary to draw a graph showing the variation of the
bending moment along the span of the beam. Such a graph is known as
bending moment diagram. A BM diagram has two different scales:
i) a linear scale for the span(e.g. 1cm = 1m)
ii) A BM scale for the vertical coordinate (e.g. 1cm = 1Knm)
In the same way the SF diagram is constructed to linear and force scales.

19. Important points for drawing SF & BM
diagrams
• The positive values of shear force and bending moment are
plotted above the base line, and negative values below the
base line.
• If there is a vertical load (including reactions) at a sections,
SF diagrams will decrease or increase suddenly i.e. by a
vertical straight line at the section.
• The shear force between any two vertical load is constant
and hence shear force diagram between vertical loads will
be horizontal. The BM diagram will be inclined.
• If there is a UDL between two sections, the shear force
diagram will be an inclined straight line, and the BM
diagram will be a parabolic curve.

20. Contd.
• If there is a triangular or trapezium load
distribution acting between two sections, the
SF diagram will be parabolic.
• The BM at the supports of a simply supported
beam and at the free end of a cantilever will
be zero.