boundary conditions in 1D FEA of beam deflection
boundary conditions in 1D FEA of beam deflection
(OP)
Hello everyone. I am pogramming some software for a 1D FEA of a thin beam with multiple pinned supports and loads. It's ends are always free.
I have managed to assemble the global stiffness matrix, but I am not sure how to set the boundary conditions which is needed to make it non-singular. The matrix form is:
F1 v1
M1 u1
F2 = K * v2
M2 u2
. .
. .
. .
where K is the stiffness matrix, F is force, M momentum, v displacement and u rotation angle. Obviously I want to solve it for v and u.
A pinned support should have the deflection v = 0, so I guess that I could at least remove these columns? But what rows should I remove so that the system is not over-constrained? Does it matter?
I am a total newbie when it comes to finite elements, so forgive me if the question is badly formulated. Thanks in advance.
I have managed to assemble the global stiffness matrix, but I am not sure how to set the boundary conditions which is needed to make it non-singular. The matrix form is:
F1 v1
M1 u1
F2 = K * v2
M2 u2
. .
. .
. .
where K is the stiffness matrix, F is force, M momentum, v displacement and u rotation angle. Obviously I want to solve it for v and u.
A pinned support should have the deflection v = 0, so I guess that I could at least remove these columns? But what rows should I remove so that the system is not over-constrained? Does it matter?
I am a total newbie when it comes to finite elements, so forgive me if the question is badly formulated. Thanks in advance.





RE: boundary conditions in 1D FEA of beam deflection
Your beam has two degrees of freedom, it can deflect (v) and rotate (u) using your nomenclature. Therefore to make the stiffness matrix non-singular you require at least two restraints, one of which must be v = 0. More than two restraints and the beam is over constrained. And yes it does matter which nodes are restrained. Restraints effectively apply a set of forces/moments to balance out your applied loading. Changing position of applied loads and restraints will of course generate different deflections.
RE: boundary conditions in 1D FEA of beam deflection
RE: boundary conditions in 1D FEA of beam deflection
The matrix is now non-singular and can be solved. The actual reaction at your supported nodes is equal to the (very small) deflection at the node multiplied by the (very large) thumping stiffness value.
If you wish one of your load cases to include a defined displacement at a supported node, then you convert that to a force whose magnitude would cause the required displacement in the thumping spring.
This approach does run the risk of introducing numerical errors, so: (1) store your numbers with the highest available numerical precision; (2) do not use your program on problems with a large number of degrees of freedom; and (3) always check your results with an even larger setting on your scepticism-o-meter than you would usually use. In particular, check that your calculated reactions are in good equilibrium with your applied loads.
The more rigorous approach is to rearrange the entire equation system so that it is partioned between those degrees of freedom that are free and those that are constrained. You will find this described in numerous (advanced) books on strucural analysis.
Good luck.
RE: boundary conditions in 1D FEA of beam deflection
RE: boundary conditions in 1D FEA of beam deflection
RE: boundary conditions in 1D FEA of beam deflection
RE: boundary conditions in 1D FEA of beam deflection
when i first read the post i thought 1D ??? ... but then i visualised drawing a line to represent the beam (a la hand calc) and thought 1D was appropiate.