Sunday, 28 May 2017

DERIVE THE DIFFERENTIAL EQUATION OF ELASTIC CURVE OF BEAM

DERIVE THE DIFFERENTIAL EQUATION OF ELASTIC CURVE OF BEAM


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

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.

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Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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