CHAPTER #3
SHEAR FORCE & BENDING
MOMENT
 Introduction
 Types
of beam and load
 Shear force and bending Moment
 Relation between Shear force and
Bending Moment
INTRODUCTION




Devoted to the analysis and the design of beams
Beams – usually long, straight prismatic members
In most cases – load are perpendicular to the axis of the
beam
Transverse loading causes only bending (M) and shear
(V) in beam
Types of Load and Beam

The transverse loading of beam may consist of


Concentrated loads, P1, P2, unit (N)
Distributed loads, w, unit (N/m)
Types of Load and Beam



Beams are classified to the way they are supported
Several types of beams are shown below
L shown in various parts in figure is called ‘span’
Determination of Max stress in
beam
m 
Mc
m 
M
I
S
1 2
S  bh
6
1 3
I  bh
12
SHEAR & BENDING MOMENT DIAGRAMS


Shear Force (SF) diagram – The
Shear Force (V) plotted against
distance x Measured from end of
the beam
Bending moment (BM) diagram –
Bending moment (BM) plotted
against distance x Measured
from end of the beam
DETERMINATIONS OF SF & & BM

The Shear & bending moment
diagram will be obtained by
determining the values of V
and M at selected points of
the beam
DETERMINATIONS OF SF & & BM


The Shear V & bending moment M at a given point of a beam are said
to be positive when the internal forces and couples acting on each
portion of the beam are directed as shown in figure below
The shear at any given point of a beam is positive when the external
forces (loads and reactions) acting on the beam tend to shear off the
beam at that point as indicated in figure below
DETERMINATIONS OF SF & & BM

The bending moment at any given point of a beam is positive when the
external forces (loads and reactions) acting on the beam tend to bend
the beam at that point as indicated in figure below
Relation between Shear force and
Bending Moment

When a beam carries more than 2 or 3 concentrated
load or when its carries distributed loads, the earlier
methods is quite cumbersome

The constructions of SFD and BMD is much easier if
certain relations existing among LOAD, SHEAR &
BENDING MOMENT

There are 2 relations here:
Relations between load and Shear

Relations between Shear and Bending Moment
Relations between load and Shear

Let us consider a simply supported beam AB carrying distributed
load w per unit length in figure below

Let C and C’ be two points of the beam at a distance Δx from each
other

The shear and bending moment at C will be denoted as V and M
respectively; and will be assumed positive, and

The shear and bending moment at C’ will be denoted as V+ ΔV and
M + ΔM respectively
Relations between load and Shear (cont.)

Writing the sum of the vertical components
of the forces acting on the F.B. CC’ is zero
V  V  V   wx  0
V  wx

Dividing both members of the equation by
Δx then letting the Δx approach zero, we
obtain
dV
 w
dx
Relations between load and Shear (cont.)

The previous equation indicates that, for a beam loaded as figure,
the slope dV/dx of the shear curve is negative; the numerical value of
the slope at any point is equal to the load per unit length at that point

Integrating the equation between point C and D, we write
xD
VD  VC    w dx
xC
VD  VC  (area underload curve between C and D)
Relations between Shear and Bending
moment

Writing the sum of the moment about C’ is
zero, we have
x
 M  M   M  V  x  w x ( )  0
2
1
M  V x  w(x) 2
2

Dividing both members of the eq. by Δx and
then letting Δx approach zero we obtain
dM
V
dx
Relations between Shear and Bending
moment (cont.)

The equation indicates that, the slope dM/dx of the bending moment
curve is equal to the value of the shear

This is true at any point where a shear has a well-defined value i.e.
at any point where no concentrated load is applied.

It also show that V = 0 at points where M is Maximum

This property facilitates the determination of the points where the
beam is likely to fail under bending

Integrate eq. between point C and D, we write
xD
M D  M C   V dx
xC
M D  M C  area under shear curve between C and D)
Relations between Shear and Bending
moment (cont.)

The area under the shear curve should be considered positive where
the shear is positive and vice versa

The equation is valid even when concentrated loads are applied
between C and D, as long as the shear curve has been correctly
drawn.

The eq. cease to be valid, however if a couple is applied at a point
between C and D.
xD
M D  M C   V dx
xC
M D  M C  area under shear curve between C and D)