Hi, How can i derive maximum deflection of tapered cone mounted with a cylinder, both having circular area of cross section when a point force is applied at the tip? (kindly look at the diagram attached)
You have used fixed support for the base . Can the structure be supported on elastic foundation ( Springs or cables ). I think that support type has to be changed in order to achieve maximum deflection at tip
Could you elaborate the problem
Support for the base is fixed (No spring and cables). My aim is not to achieve maximum deflection but just to derive and check how much deflection will be there in this composite structure (truncated conical bottom +cylindrical beam at top). Height of the truncated cone can be taken h1 and for full structure h. and radius can rb (bottom radius) and rt (top radius). I already have a derivation but I think its not exact. Thank you.
It is quite simple... Consider the entire structure to be a Bernoulli-Euler cantilevered beam. Draw the shear and moment diagram. Then compute the rotations and finally deflection field (everywhere in the structure).
You have 4 boundary conditions: 2 essentials at the fixed end (deformation and slope are both zero), and you have 2 natural boundary conditions at the free end.
There is no hard math in this solution... We are assuming slender structures energy due to bending only.
At the end, you should get your solution like PL^3/(3EI), which is the solution for a cantilever beam.
Thank you for your reply. I don't think that will work and u might not find a simpler version of solution similar to a cantilever beam because for the truncated conical part, the moment of inertia changes according to the radius in x direction and also when a point load acts on the cylindrical part, there is a moment caused by that and also another point load on the conical tip (superimposition method). Plus there is also an additional deflection caused by the angle of rotation factor. I think I found the right derivation now as it converges for both cylindrical and conical part, and it doesn't look too simple :D
Dear Ashish Rathore,
your problem seems similar to the one proposed in exercise # 11 at page 126 of the book "Theory of elasticity" by Timoshenko. In particular the author highlights that stress distribution within the tapered and the prismatic parts of the structure are substantially different.
Furthermore, the attached paper demonstrates that the non trivial stress distribution in a tapered beam deeply influences also the beam constitutive relations and the beam stiffness.
As a consequence, for planar problems the application of the model proposed in the attached paper could be an effective solution whereas, for more complex geometries, my suggestion is to avoid beam models but use 3D FE.
Article Non-prismatic Beams: a Simple and Effective Timoshenko-like Model
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