In our previous topics, we have seen some important
articles such as derivation for deflection and slope of beam under action of
load, basic
concept of shear force and bending moment, concepts
of direct and bending stresses, middle
third rule for rectangular section, middle
quarter rule for circular section, and shear
stress distribution diagram in our previous posts.

Now we will start here, in this post, another
important topic i.e. Derivation for differential equation of elastic curve of a
beam. 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 derive here the differential equation of elastic curve of a beam in this
post.

Let us consider a beam and its bending deformation as displayed in following figure. Let us think that load is applied over the beam in such a way that stresses developed in the beam will be within the elastic limit i.e. beam will retain its shape and dimensions after removal of the load and therefore deflection as well as slope will be very small practically.

Before going ahead we must recall the definition of
elastic curve as we will assume here one portion of elastic curve in order to
derive the differential equation and therefore first we will have to recall the
basic definition of elastic curve in deflection of beam.

If a beam will be loaded with point load or
uniformly distributed load, beam will be bent or deflected from its initial
position in the form of a curvature or in to a
circular arc. Curvature or circular arc of the beam, formed under the
action of load, will be termed as elastic curve.

Let us consider the curve PQ as elastic curve of the
beam i.e. curve PQ is representing here the deflection of the beam. Now we will
consider here one infinitesimal portion AB of this beam as displayed here in
following figure.

We have following information from above figure.

θ = Angle made by tangent at A with X axis

θ + dθ = Angle
made by tangent at B with X axis

C = Centre of curvature of the curve PQ.

y = Deflection of point A

y + dy = Deflection of point B

dx = Length of the infinitesimal portion AB

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__
Additional information __

__Additional information__

M = Bending moment acting over the
infinitesimal portion AB

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

I = Moment of inertia of the beam
section

EI = Flexural rigidity of the beam and
it will be remain constant through the beam

We can write from above figure

As θ is quite small and therefore we have mentioned
above Tan θ = θ

We can also write following equation by considering
the angle ACB,

Now we will use the value of θ in above equation and
we will have following equation

Let us recall the flexural formula for beam and we can write here the bending equation as mentioned here

Let us consider the above two equations and we can
write the differential equation of elastic curve for a beam as displayed here
in following figure.

Above equation is termed as differential equation of elastic curve for a beam. We will see deflection and slope of simply supported beam with point load at center 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.

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**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google