Buckling

Possible a deeper insightn into the problem is possible, but i can offer following explanation.

Buckling is caused by bifurcation which is the existence of two different (but energetically equivalent) configurations under the same load. It means that when the compressive force ahieves critical value, solution of euqillibrium equations becomes nonunique (you have to consider finite strains case to see this). At this point the process can continue in one of the two possible ways: original deformation mode (uniaxial compression) or buckled mode. In reality buckled mode has a priority over the original mode due to imperfections of different nature (it is impossible to apply a force precisely in the center of gravity, also a real column has some small defects).

For a naturally strainght and slender column (e.g., hinged-hinged column), if you apply a compressive load, this load tries to reduce the stiffness of the system. When the applied load increases to critical load or buckling load, the stiffness of the system becomes zero and the stability can be exchanged at this point. This point is what we call the buckling of the column.

Buckling is essentially an instability phenomenon.  Given a configuration (like the column under a compressive load) think of small perturbations (changes to this configuration), and analyze its effect.  If the system would return to its original configuration, then the original configuration was stable, and buckling would not occur.  This is what happens, when the applied load is below the critical load, since elastic restoring force is capable of bringing the system back to the original configuration.

However, above the critical buckling load, on perturbing, the tendency of the system will be to attain an energetically favorable new configuration, and we say that buckling has occurred.

Essentially at the buckling load there is a change from a stable equilibrium configuration to an unstable equilibrium.  This happens by passing through a Neutrally stable equilibrium, and the analytical methods like Euler's formula try to find this neutrally stable configuration so as to get the buckling load.  For all practical purposes this is sufficient, as there will always be some disturbances to upset the equilibrium, and it is almost impossible to retain a structure in the unstable configuration (above the critical load).

However, please remember that buckling is more general than column buckling, and need not always be bifurcation type (eg. snap-through buckling).  Also, to say that compressive loading reduces stiffness of column is misleading at the best, since axial stiffness (which is the stiffness usually referred to) remains essentially constant.  It could be possible that bending stiffness becomes zero (though I am not sure of it!).

For an interesting case, see this discussion in iMechanica itself. 

I hope my comment was helpful,

Regards,

Jayadeep

Thanks Jayadeep. For the case of the slender column, it is bending stiffness where it can be reduced by compressive load. A simple example is a rigid-link column of two rigid segments connected with a flexible rotational spring subjected to compression at its end. The total potential energy of the rigid-link column is 2Θ^2β-PL(1-cosθ) where θ is the end angle of the column, β is stiffness of the column, L is span length of the column and P is the compressive load. After taking the first derivative w.r.t. θ , we have 4θβ-PLsinθ where it relates to equilibrium if 4θβ-PLsinθ=0. And if we further take the second derivative on it w.r.t. θ, we can get 4β-PLcosΘ where the system is stable when 4β-PLcosΘ>0, the system becomes critical when 4β-PLcosΘ=0 and the system becomes unstable when 4β-PLcosΘ<0.  And if we consider at a straight column (θ=0), we can have 4β-PL where it may be positive, zero or negative depending on load P. This can be seen that if the compressive load increases the stiffness of the system decreases until 4β-PL=0, the system becomes critical and the applied load at this stituation is buckling load.

Hi,

I wish to add one more point here. Start applying gradually an axial comporessive load on a column. Axial strain energy stored in the column will keep increasing. At some load, the column will no longer take any extra compressive load and becomes unstable. At this load, the column goes to another state (benidng/flexural), where the axial strain energy is converted to bending strain energy. The load at which this happens is the buckling (critical) load.

With regards,

- Ramadas

Hi 

Rajnish.....When a column is subjected to harmonic load....it under goes a instability called parametric instability or dynamic instability. when the frequency of external load becomes integral multiple of natural frequency  of the column, this instability takes place. The degree of freedom which is normal to the excitation load keeps on increasing exponentially w.r.t time. Solution is unbounded and amplitude keeps on increasing exponentially. In this case we get a equation called Hills equation. It is a second order differential equation with periodic coefficients. In this instability the external loading term get into the system and changes any of the property periodically. The quation goes like this:

                                    d2X/dt2+CdX/dt+B(t)X = 0 

 It could be put in perturbed form as follows:

d2X/dt2+εCdx/dt+(B(o)+εB(t))X = 0

Where ε>0 is a small parameter (perturbation). If the periodic load is harmonic the Hills equation gives Mathieu's equation.

     This equation is a nonlinear equation. This equation have two solutions which are periodic with period T and 2T. There are ways to solve this equation. Floquet's theory is used. But it goes comlicated when the sustem is multiple degree of freedom. So technique's like Perturbation methods, theory of multiple scales, harmonic methods etc,.

     There is book on this kind of instability studies. It is bible infact for Dynamic Instability studies. The book follows:

The Dynamic Stability of Elastic Syatems- V. V. Bolotin.

Very old one, 1964.

Hope the discussion was usefull....

Sreenivas 

Hi

    The EOM you got should be Mathieu's Equation. For this kind of instabilitity study, one is not interested in finding the exact solution. Most of the literature gives boundary frequencies. with this one can plot stability and instability charts. These charts gives us the range of excitation frequency and excitation amplitudes for which response is bounded and unbounded. If you are specific to find the response, the response is known: I mean, the response inside the instability region is unbounded and increases exponentially w.r.t time, the response on the region (line which seperates stability and instability zone) is periodic; have a beat like behavior and the response in the stability region is periodic. 

    If you want to solve for the response, you should decide how you want to solve? I mean numerically or analytically. Numerically it is very easy. Use FEM, use some time integration methods like Newmark time integraion , Harmonic acceleration method or successive symmetric quadratue method etc,. If you want to solve analytically then you need to work, there is a book on Mathieu's Equation go through the book, it may help you.

Theory and Application of Mathieu Functions - N. W. McLachlan..

Old one 1947, but the only book dedicated fully to Mathieu's Equation.

   There are many papers on this study. The below paper will be usefull for you. 

Dynamic stability of elastic structures: a finite element approach - L. Briseghella

Link : http://www.sciencedirect.com/science/article/pii/S0045794998000844

      I have seen your profile, I think you are in IISc. People there have done lot of work in this field. As you are into Helicopter Lab, there is a straight application of this instability to Helicopter blades in turbulent atmosphere. There are papers on this study as well. I will quote one for reference:

Numerical methods for the treatment of periodic systems with applications to structural dynamics and helicopter rotor dynamics - 

P.P. Friedmann 

Link : http://www.sciencedirect.com/science/article/pii/004579499090059B

Hope it will be usefull. Any more discussion please feel free to contact. 

Sreenu