The beam has unsymmetrical section and is simply supported at its both ends and subjected to forces and moments in all 3 directions at a point.
Dear Kirti Singh
As suggested by Claudio Pedrazzi
you should first evaluate the cross sectional properties of such asymmetrical section. To this end, I suggest you to use VABS or SwiftComp of Prof. Wenbin Yu to get the cross sectional properties. Once you know them, you can calculate the tube displacements. Be aware of the fact that, due to asymmetry, the tube can be coupled and hence you cannot solve the linear equations one at a time but the system as whole.Regards,
Francesco
Claudio Pedrazzi
Francesco DanziThe aim is to calculate the Moment of Inertia required to limit the deflection to a certain value. Therefore, the idea was to calculate deflection analytically for the unsymmetrical section and the beam idealization is acceptable.
Do you have at least one vertical plane of symmetry? Otherwise vertical load would give not only bending but also torsion (bending-torsion coupling)
As far as I understand, the beam geometry and loads are known (input). Therefore, if your system is linear, you can apply the superposition principle: decompose the overall loading in simple loading cases (axial force, bending, shear, torsion), for each one compute the corresponding stress (sigma or tau) and finally compute the resultant stress by summing up each computed stress in each point of your cross section.
Sorry, please ignore my previous post. I forgot that you asked about displacement not stress...
You need to be more specific, which "source" of asymmetry you have? Is this due to geometry of the cross section? due to a different lamination (in case you are dealing with composites)? or both? In the most general case, considering the Timoshenko's beam model, you may end-up with a fully coupled configuration, i.e. 21 unknowns (cross sectional stiffnesses). As far I understood, you need to change those to satisfy a (or some) prescribed constraint(s) given in terms of maximum deflection(s).
As for the equation for the displacement, you can use the principal of virtual work to evaluate the displacements but for a fully coupled configuration it is a long way to go. You can certainly use superposition as far as the tube works in the linear regime. Seemingly you have a pre-defined set of applied loads, so you need to systematically "link" the geometrical parameters to the the cross sectional stiffnesses (exactly as you do for a simple rectangular section where you know that I=1/12ab^3), and change those parameters to get the response you're looking for. If I understood correctly your problem, you may want to use parametrical analysis or optimization to get meaningful results. In fact, if you just play with the values of the cross sectional stiffnesses, you might have a numerical solution which satisfies your constraint but which in turn is unfeasible.
The cross section has no symmetry.
Inputs are: The forces and the moments applied on the beam.
Desired output: Required MOI to limit the deflection of beam to a certain value.
Dear Kirti Singh ,
I'm now assuming that your beam is made with isotropic material. If this is the case, I suggest to approximate the second moment of area considering a piecewise approximation of the cross section you've posted. You need to calculate also the shear center and all the relevant cross sectional properties. Then calculate the resultant moment and forces for your simply supported beam and apply the PVW to calculate the dispalcements. At first, in writing the PVW I would neglect the effect of the shear, limiting the derivation to moments and normal forces (tension/compression). Good luck with your calculation.
Cheers
If I replace the above the general cross section with a rectangular cross section for approximation, how will i ensure that the results are correct?
Is there any concept of equivalent beam?
Kirti Singh,
I'm afraid it is not possible to identify an equivalent beam with all the cross sectional properties (Ix, Iy, Ixy, Jt, position of the shear center and so forth) equal to those of your beam simultaneously, otherwise you'll end up with your initial cross section. Hence, as for the results, I would say the closer the approximation of your geometry the better the results.
Note also that to get the equivalent structure, you need to know already the cross sectional properties, which, I believe, are unknown so far.
Finally, if your are looking for a closed-form equation, you need at least to approximate your cross section with piecewise straight elements or, even better, use integral definition of cross sectional stiffnesses; otherwise, if you're looking for a quick answer, use FE and either parametric analysis or optimization to get your solution, as suggest by Claudio Pedrazzi
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