1) You cannot have a negative eigenvalue
2) You cannot have a negative buckling factor
3) Even if you did have a negative eigenvalue, it wouldn't indicate a negative buckling factor
Could you mean you have a buckling factor of less than unity? This would indicate a buckling load factor that is lower than the design load (thus no safety factor). Euler buckling is not terribly useful in TRW where non-linearities, eccentricities, and other realities need to be studied.
Best regards,
Matthew Ian Loew "I don't grow up. In me is the small child of my early days" -- M.C. Escher
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.
A negative buckling factor simply means that the structure will buckle when the directions of the applied loads are all reversed.
A classic case is a pressure vessel. If you set up a typical pressure vessel model (eg cylindrical shell, semi-spherical ends), and apply an internal pressure case, then run a buckling check, you should find numerous negative Eigen factors (buckling factors). This is quite sensible - it simply tells you that the shell would buckle if subject to excessive external pressure / internal vacuum, and it may not have ANY positive buckling factors.
In fact, it is quite possible to have structures for which ALL Eigen Factors are negative. (e.g a structure which is in tension throughout under normal loading, so is not prone to buckling failure, unless all loads reversed.)
In the more general case, your structure may have both positive and negative buckling factors. Consider a water tank supported on columns. The shell will have numerous negative buckling factors (the shell wall would buckle if subject to a vacuum), but the columns would have positive buckling factors (the columns will buckle under increasing total load).
Your job as the analyst / designer is to determine which buckling load cases are sensible in your case, and then determine whether the load factors are high enough. If your loads cannot be reversed, you can probably ignore the negative load factors. If the loads are reversible, you need to check the negative buckling factors.
if jsboy could state the problem in more detail, perhaps folks could settle out on solutions. it reads like lots of good answers but to different problems.
A very simple example:
A cantilever with first end fixed and second end free. Euler 2 I think it is. Load the cantilever with a tensile load and you vill probably get a negative eigenvalue. Change sign om the load (from tension to compression) and the eigenvalue vill be positive.
Since you say "modal terms", you might mean negativ frequency and that would be strange. But I don't find negative load for buckling very odd.
Julian Hardy is quite right. Whilst both modal and buckling analyses involve solving for eigen vectors and eigen values and will use the same solution routines, they do differ in the range of possible solutions. Yes modal analysis will only yield eigen values greater than or equal to zero. In the buckling problem the mass matrix of the modal problem is replaced with an element geometric stiffness matrix. Most mechanics books should give the derivation and examples for this matrix.
I analysis aircraft landing gears which are far more complex than most people realise. A buckling analysis of these structures throws up both positive and negative eigen values. The negative eigen values are generally mathematical solutions of the eigen problem for which there are no real world scenarios (i.e. the wheels being pulled down instead of being pushed back up into the wings)
ThomasH and johnhors have already responded very well, saving me the trouble of giving you an example. I suggest that if you have access to an FEA package with buckling capability, you try exactly the exercise ThomasH suggests. If you put a tensile load on a column, you will get a negative Eigen Factor; if you reverse the sign of the load, you should get a positive Eigen Factor of the same magnitude.
As johnhors says, natural frequency solvers and buckling solvers typically use the same solution methodology. In both cases, you are solving for Eigen Values which make the stiffness matrix singular (i.e. the structure has zero stiffness). In the case of a buckling solver, you are looking for values of the load multiplier which cause the structure to have no stiffness (i.e. it buckles). In the case of a natural frequency solver, you are looking for values of frequency for which the structure has no stiffness (i.e. its natural frequencies and mode shapes). Buckling and natural frequencies can be thought of as two cases of the same problem - one is dependent upon the magnitude of the load at zero frequency, while the other is dependent only upon the frequency of application of the load.
You are correct that you cannot get a negative Eigen Value in a natural frequency solver - this would be meaningless, and if it came about, it would indicate some problem with either the solver or the stiffness matrix. However, you most definitely CAN get negative Eigen Values in buckling solvers.
What this tells us is that the negative solution is equal in magnitude to the positive solution for frequency. The physical meaning of this is that the system s still in resonance whether the clock is running in forward time or reverse time.
