In our previous topics, we have seen some important
concepts such as Concepts of direct and bending stresses, Middle third rule for rectangular section, Middle quarter rule for circular section, shear
stress distribution diagram and basic
concept of shear force and bending moment in our previous posts.

Now we will start here, in this post, another
important topic i.e. Derivation for deflection and slope of beam under action
of load. If a beam will be loaded with point load or uniformly distributed
load, beam will be bent or deflected from its initial position.

We have already discussed terminologies and various terms used in deflection of beam with the help of our previous post. Now we
will determine here the amount by which beam will be deflected from its initial
state and also slope of the deflected beam under the action of load in this
post.

So let us come to the main topic without wasting the
time and let us see here the method to secure the amount by which beam will be
deflected from its initial state and also slope of the deflected beam under the
action of load.

Let us consider a beam AB of length L is simply
supported at A and B as displayed in following figure. Let us think that one
load P is acting at the midpoint of the beam. As beam is loaded with load P at
the center of the beam and therefore beam will be subjected with bending moment
and hence beam will be bent or deflected from its initial position under this
load P.

We have shown the initial position of the beam i.e.
before loading as ACB and deflected position of the beam i.e. after loading as
ADB.

We have following information from above figure as
mentioned here

AB = Beam

P = Load applied at the center of the beam AB

L = Length of the beam AB

E = Young’s modulus of elasticity of the material of
the beam

I = Moment of inertia of the beam section

ACB = Initial position of the beam i.e. before
loading

ADB = Deflected position of the beam i.e. after
loading

y = CD, deflection of the beam under the action of
the load P

R = Radius of the curvature or elastic curve of the
deflected beam

θ= Slope of the beam which is basically angle, measured
in radian, between tangent drawn at the elastic curve and original axis of the
beam as displayed in above figure.

###
**
Deflection of beam **

Let us recall the concept of geometry of circle and
we will have following equation as mentioned here.

AC x CB = DC x CE

L/2 x L/2 = y x (2R-y)

L

^{2}/4= 2yR- y^{2}
We can neglect the value of y

^{2 }as deflection will be quite small practically and its square value will be too small and could be neglected.
L

^{2}/4= 2yR
y = L

^{2}/8R
Let us recall the bending equation and we can write
as mentioned here

M/I = E/R

R = (E x I)/M

Now we will use the above value of radius R in
equation of deflection and we will have the formula or expression for the
deflection of the beam y as mentioned here.

###
*y
= ML*^{2}/8EI

*y = ML*

^{2}/8EI###
**Slope
of the deflected beam **

Slope in deflection of beam will be
basically defined as the angle made between the tangent drawn at the elastic
curve and original axis of the beam.

Slope will be measured in radian and
will be indicated by dy/dx or θ.

Let us consider the triangle AOB and
write here the value for Sin θ

Sin θ = AC/AO

Sin θ = L/2R

As we know that practically slope angle “θ”
will be quite small and hence we can write here

θ = L/2R

###
*θ = ML/2EI*

*θ = ML/2EI*

We will see another topic in the category of
strength of material in our coming post. Please
comment your feedback and suggestions in comment box provided at the end of
this post.

###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

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