Beams are structural members which are most commonly used in buildings
in a beam transverse load is acted, which in fact comes from the slabs to the
column or walls
for analysis beam can be separated out from beam column system
length of a beam
is much higher than its lateral dimensions
axial strain developed in a beam is very small compared to shear strain
or strain induced due to bending
So for design purposes beams, analysis of
shear force and bending moment induced are of atmost importance
The interesting thing is that you can draw a sheer force and bending moment
distribution along any beam
by understanding what exactly is sheer force and bending moment
first shear force. Shear force is the resistance created in beam crosssection
in order to balance transverse external load acting on beam
Consider this beam. It does not matter from where you take a section
when you add forces acting on it it, it should be in equilibrium
shear force is induced exactly for this purpose
to bring the section to equilibrium in vertical direction
It acts parallel to cross section
Usual sign convention is as follows:
So just by applying force balance
in vertical direction on the free body diagram, we can determine value of
shear force at a particular cross-section
now we can apply same concept in different cross-section and find how
shear force varies along the length of the beam
But balance of transverse forces alone does not guarantee equilibrium of a section
there is
another possibility a beam rotation, if moment acting on it is not balanced
If this is the case a bending moment will be induced in cross-sectional beam
to arrest this rotation
It will be induced as normal forces acting on fiber cross-section
as shown
Resultant
forces were
be zero, but it will produce a moment to counterbalance the external moment
sign convention a bending moment is as follows:
so we can calculate moment induced
at any cross-section by balancing the external moment acting on the free body
diagram
with these concepts developed we can easily calculate distribution and shear
force
and bending moment along the length of the beam
we will see some examples
For this cantilever carryng three loads we can start
analysis from the free end
so
between the A&B if you take a section the only external force
acting on it is F1
so a sheer force should induce in section to balance this force
so value of here force between A and B is F1
but force balance alone
does not guarantee equilibrium of the section
there is an external moment on the section
so a bending moment will
be induced in Section in order to balance the external moment
since value of external moment is F1 into X
bending moment will vary in the nearly
Between B&C effect
of F2 also comes
So sheer force becomes F1 plus F2
in bending moment
effect f2 also gets added
similar analysis is done between section C and D
also
so SFD in BMD have this problem would look like this
Now consider this problem
a simply supported beam
with uniformly distributed load
first step here would be determination of reaction forces
since the problem is symmetrical reaction forces will be equal
and will be have have total load acting on beam
Lets start analysis from point A
if you take
in between points A&B it should be in equilibrium
so shear force will have equal magnitude of reaction force
so bending moment also gives a linear variation
but after point B effect a point load and
distributed load comes
effect have distributed load is something interesting
take a section
in BC
in this section along with two point loads there is a distributed load also
this distributed load can be assumed as a point load passing through centroid of
distributed load
value of this is U into X minus L by three
and it
is at a distance (X-L/3)/2
from section line
so shear force will have one more term which comes from distributed load
from the equation it's clear that shear force varies linearly
you can easily predict how bad
a moment varies along length from the same force diagrams
since this equation is quadratic it will have a parabolic shape
same procedure is repeated in remaining section
since this problem is symmetrical bending moment and shear force are
having
a symmetrical distribution
hope you got a good intern
analysis beams
thank you