Question

Derive the expression of depression at the loaded end of a cantilever.​

Answer 1

Norm, thanks for the A2A.

Cantilever beams are simply beams supported at one end with the other end free. In this configuration the one end takes all of the load thus a good anchor is needed to resist the bending moment. (Unless the beam continues on to something else.). This configuration is typically called Encastre. At times you do not have the ability to avoid using a cantilever beam because of access requirement. Sometime it is used for aesthetic purposes in structures. By carful design you can make a beam with little deflection but it will have varying values of second moment of ar along its length.

Examples of a cantilever is a spring board. Clearly they are using the fact the board deflects a lot to provide lift to the diver. Other example are many, you just need to look around. Quiet often a bus shelter has a catekever roof, or car parks, or railway stations. They are practical answers to practical problems.

I hope my answer will help you...☺

Answer 2

Answer:

The expression of depression at the loaded end of the cantilever is

BENDING MOMENT=$YIg/R$

Explanation:

Cantilever is thin uniform bar, so the weight can be neglected.

Let,

The bean has x and y point in one line drawn straight from one point to another  before bending and another line also considered as x' and y' respectively before bending.

After bending the x and y it became x'' and y'' with an angleФ from an imaginary point o.

If 1/R =Rate of change of slope and xo=R

Bending moment of bean

xy=RФ

x'y' = (R+x)Ф

Elongation of x'y' = x''y'' -xy

Elongation of strain =(R+x)Ф -RФ

                                 =RФ+xФ -RФ

Strain =xФ/RФ

Longitudinal stress =Y*strain ; Y*x/R     Y=young's modulus

dA is considered as a small area in beam.

FORCE on dA= stress* area

                          Y*x/R*dA

Moment of force =$Yx^{2} /R*dA$

Total moment of force or depression =∑$Yx^{2} /R*dA$

                                                               =Y/R∑$x^{2} dA$

                                                               =$Y/R*ax^{2}$

                                                               =$YIg/R$

Ig is the geometrical moment of gyration intertia of beam.

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