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Friday, 26 May 2017

DERIVATION FOR SLOPE AND DEFLECTION OF BEAM


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)
L2/4= 2yR- y2

We can neglect the value of y2 as deflection will be quite small practically and its square value will be too small and could be neglected.
L2/4= 2yR
y = L2/8R

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 = ML2/8EI

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

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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