Note that linear elastic buckling theory is ruled by the sudden lost of elastic stiffness due to internal stresses effect, in general formulated as (K-P*Kg)*fi=0.

Mathematically, it is "similar" to the resonance problem where the inertia effects cancel the classic stiffness K at certain frequencies wi  [ (K-wi^2*M)*fi=0]. Physically it is different because buckling (here a bifurcation) and appears as a jump to a new equilibrium position.

This bifurcation behavior in the case of elastic slender beams, subject to compression axial load, to be an eigenvalue problem (means non trivial solutions of an homogeneous system AX = 0) requires among other aspects a perfectly axis aligned beam, i.e. no initial transversal deformation, and load applied centrally (aligned with axis, i.e. no eccentric load), otherwise it is not an eigenvalue problem but a non-linear elastic problem (KK(u)*u=P).

In the case of finite elements, the eigenvalue problem (K-Pi*Kg)*fi=0, with Pi or P_i, is a discretization of the buckling problem, and its response is associated with the stress stiffning effect as well as the buckled configuration.

The answer is YES, in general one must indicate that this problem requires: 1. convergence in the prebuckling analysis (required for stresses);  as well as 2. convergence in the buckling analysis where should be verified the eigenvalue and eigenvector convergence with respect to the mesh (and to the highest eigenvector of the subspace if using an iterative solver).

Note that, a negative eigenvalue Pi from (K-Pi*Kg)*fi=0 means that the load factor Pi require opposite direction of the initially assumed proportional load direction.E.g. if you analize a beam in axial traction F you will obtain that its critical Euler Load is Pcr=Pi*F with Pi negative.

Also, as K is positive definite in the problem (K-P*Kg)*fi=0 then the eigenvalues must be real.

You can see more details in http://www.colorado.edu/engineering/cas/courses.d/NFEM.d/NFEM.Ch29.d/NFEM.Ch29.index.html as well as at any good book in the field.

Hope it helps.

Note that linear elastic buckling theory is ruled by the sudden lost of elastic stiffness due to internal stresses effect, in general formulated as (K-P*Kg)*fi=0.

Mathematically, it is "similar" to the resonance problem where the inertia effects cancel the classic stiffness K at certain frequencies wi  [ (K-wi^2*M)*fi=0]. Physically it is different because buckling (here a bifurcation) and appears as a jump to a new equilibrium position.

This bifurcation behavior in the case of elastic slender beams, subject to compression axial load, to be an eigenvalue problem (means non trivial solutions of an homogeneous system AX = 0) requires among other aspects a perfectly axis aligned beam, i.e. no initial transversal deformation, and load applied centrally (aligned with axis, i.e. no eccentric load), otherwise it is not an eigenvalue problem but a non-linear elastic problem (KK(u)*u=P).

In the case of finite elements, the eigenvalue problem (K-Pi*Kg)*fi=0, with Pi or P_i, is a discretization of the buckling problem, and its response is associated with the stress stiffning effect as well as the buckled configuration.

The answer is YES, in general one must indicate that this problem requires: 1. convergence in the prebuckling analysis (required for stresses);  as well as 2. convergence in the buckling analysis where should be verified the eigenvalue and eigenvector convergence with respect to the mesh (and to the highest eigenvector of the subspace if using an iterative solver).

Note that, a negative eigenvalue Pi from (K-Pi*Kg)*fi=0 means that the load factor Pi require opposite direction of the initially assumed proportional load direction.E.g. if you analize a beam in axial traction F you will obtain that its critical Euler Load is Pcr=Pi*F with Pi negative.

Also, as K is positive definite in the problem (K-P*Kg)*fi=0 then the eigenvalues must be real.

You can see more details in http://www.colorado.edu/engineering/cas/courses.d/NFEM.d/NFEM.Ch29.d/NFEM.Ch29.index.html as well as at any good book in the field.

Hope it helps.

Dear Dr. Neves,

Thanks a lot for your very useful answer, that gave me a comprehensive view to the problem. Recently I could discover the issue which has been causing the complex and negative eigenvalues. In my modeling and FEM discretization I get five stiffness matrices, which four of them are symmetric and one of them is asymmetric. I discovered that this asymmetric matrix causes those troubles. Would you please help me to solve this problem?

Thanks in advance.

Can you explain why your matrix is asymmetric?

Yes. Im considering both nonlocal effects and Winkler foundation for my model. In my weak form I have this term: (d2 w/dx2)*w , actually I have 5 terms, but one of them is like this, which cause non-symmetrical matrix. I use Galerkin method for solving this.

Is it too bad too have an asymmetric matrix?!!

Okay, I see that you have an unsymmetric term d2w/dx2*v.

Is there any reason to not be able to perform one more integration by parts as e.g. the following?

Int(d2w/dx2*v)=[dw/dx*v] - Int(dw/dx*dv/dx)

Can you recheck it again and comment?

Concerning your question, and from my experience, it is to avoid as much as possible asymmetric matrices. For example, note that symmetry in the weak formulation in general results (advantages) in the use of less smooth approximations, simplify the introduction of the boundary conditions, allows to use efficient symmetric linear algebra tools, etc. For that reason I also am not a specialist in use of solvers for unsymmetric systems.

Dear Dr. Miguel, 

You are totally right! One more integration by parts solves the problem and I get a symmetric matrix.

Thanks a million for your great help.

Okay. It is good to know that it helped.

Dear Dr. Neves,

I need your help if its possible.

I have this term Int(d2w/dx2*dv/dx), which results in unsymmetric matrix. How can I solve this problem?

As far as I see, it has no direct way to avoid the unsymmetry via integration by parts (only switches the derivative from the solution function to the test function). It remember me the cases with weak formulation of non-self-adjoint systems e.g. advection-diffusion problems (e.g. -u'' + ku' = 0).

But, I am not an expert on that, however if you need some direction try to ask to experts in Discontinuous Galerkin as well as who uses UMFPACK. After, if possible give us here some feedback.