Deflection of a beam with non-uniform section

[vichu_vv]

Hello all,
How to find deflections at force acting points in a non-uniform beam?. Please look at the figure. Thank you.

Best regards

 

[Mech_Engineer]

Is this homework? We cannot ethically solve the problem for you, but knowing more about what you have tried already will help us see what you need help with.

 

[vichu_vv]

No, this is not a homework. I have tried with uniform diameter to the whole length of the beam. I am applying loads simultaneously at positions of 1 and 2. I have gone thro the following formulas to find the deflections at 1 and 2 with only one force at a time.

fFi = fFi1 + fFi2

i - load position
1 - deflection at position 1
2 - deflection at position 2

Deflection formula when loads at midspan: (used uniform dia.)
y(x) = (Fx^2/6EI)*(3a-x) for 0<x<=a
y(x) = (Fa^276EI)*(3x-a) for a<x<=l

 

[FredGarvin]

Have you looked into singularity functions? Here is a link with a quick example that shows a single stepped shaft.

http://books.google.com/books?id=j8...X&oi=book_result&resnum=2&ct=result#PPA197,M1

 

[vichu_vv]

Thank you. Anymore links or supports from you to make me more clear. Thanks

 

[Dr.D]

The easy way to solve this problem is with beam elements. Failing that, the way to solve it is to consider two beam problems, where each beam is of constant cross section. The loads from the beam on the right can be transferred to produce and equivalent shear and moment at the interface to load the beam on the left. Then the boundary conditions between the two problems have to be matched to assure displacement and slope continuity across the interface. It is a messy problem, but it is entirely doable.