Minimal Supports reaction loads 1

[smyth13]

When using balanced loads and the 3-2-1 minimal support scheme, what type of reaction loads would be considered acceptable? The goal, at least in my opinion, is to have zero reaction forces and moments at these nodes, but that would seem impossible due to the necessity of having a reaction force provide the DOF constraint. Am I missing something? Thanks.

 

[prost]

First, some background and notation. In the FE method, you set up the following matrix system, Ku=f. You are looking for "u" given the K and f. The reason you need to put in at least a 3-2-1 support scheme (AKA "rigid body constraint") is that without the 3-2-1 scheme, the K matrix is singular---in 3D, there are at least 6 equations that are dependent on there other N-6 equations in the Ku=f equations (after all, the Ku=f is really just a handy notation and solution scheme for solving N equations, is it not?). That's the 25 cent answer why you need to do this 3-2-1 rigid body contraint.

Perhaps each FE software does it differently? Here's what I would do to constrain the K matrix in Ku=f--in particular, how is this 3-2-1 constraint typically handled? You are constraining 6 degrees of freedom and forcing them to be zero. Your solution is 'u', the displacement vector. You are saying 6 displacements are forced to be zero (remember the displacements "u" is 3 displacements, on each in direction x, y and z, for each node--if you have "M" nodes, you have 3*M=N degrees of freedom). You have N total degrees of freedom (DOFs) in the K matrix; say as part of your 3-2-1 scheme, you are saying displacement no. 383 is zero. You cause your K matrix to be constrained for no. 383 equal to zero by putting a bunch of zeroes on row 383, with a zero in the "f" vector for the f(383). Now put a "1" (one) in the k(383,383)--that is, the diagonal element of matrix K in the 383rd row, 383rd column. If you wrote out the 383rd equation, it would look like u(383)=0. And that's what you should 'compute' for u(383) when you do the inversion, u=K(-1)*f. So there is no 'reaction force' other than zero for this 383rd element of the "f" vector. In fact, if you compute a reaction force much bigger than zero for f(383), then your body is probably not properly constrained. (remember that the "f" vector is a force vector, with three components, one each in the x,y and z directions, at each node).

In my explanation, the "3-2-1" scheme is not the same as a symmetry constraint. With a symmetry constraint, of course you DO expect forces in the direction perpendicular to the edge you are constraining with symmetry constraint.

 

[johnhors]

With balanced loads and 3-2-1 minimal support, you would expect to see negligible reaction forces, at the level of round off errors in the data input (for coordinates and forces).

As prost said the 6 total supports removes all rigid body rotations and translations, this is the minimum necessary to make the stiffness matrix non-singular and thus solveable (for a 3D solid with no symmetry restraints).

The fact that the supports react zero or near zero forces doesn't mean that they are not doing their job. If you do see significant forces reacted then this is an indication that something is wrong ! That is one of the strengths of this approach.

 

[smyth13]

prost and johnhors,

Thanks for the quick replies. I initially believed that there should be a negligible reaction at these 3 nodes, assuming my loading was balanced. It was only after running some analyses that I began to question myself.

As it were, I was using some symmetry boundary conditions along with the 3-2-1 constraints and the balanced loading. I believe this is where my problems are arising. How do you guys handle symmetry? Do you apply the appropriate loading on the symmetry planes? Or do you find it easier to just model the entire geometry?

As a note, I am really interested in this method and I was wondering if you have any references, etc. I always thought it was artificial to create these "rigid" supports and I tried to steer clear of the results found near these areas.

 

[johnhors]

smyth13 - please see this thread where you will find the answers to your questions:-

Minimal support - 2D Axisymmetric Model
thread727-215433: Minimal support - 2D Axisymmetric Model

http://www.eng-tips.com/viewthread.cfm?qid=215433&page=3

When used correctly the results at and near the nodes used for the 3-2-1 are in no way affected by the restraints and are perfectly valid.

 

[prost]

A symmetry constraint is a constraint on displacements--say the symmetry edge is parallel to the x-axis. A symmetry constraint on this edge makes all y-displacements on this edge set to zero. You use the same technique---for every y-displacement degree of freedom (or node for all you h-version types) on this edge, you put a bunch of zeroes in the rows assigned to those DOFs in the "K" matrix, and the "f" vector, and put a one (1) in the diagonal for that DOF.

How does one compute the force at a node, then? Using the displacement field on that symmetry edge, you can compute the spatial derivatives to compute the strains, then use the Elasticity matrix to compute the stresses. You can then integrate the stresses to get the forces.

 

[feajob]

As Johnhors said negligible reaction forces, at the level of round off errors in the data input are acceptable. I calculate the resultant at all interface loads and I accept 0.5% of the maximum resultant load as acceptable level.

A.A.Y.

 

[smyth13]

Thanks. Your replies have clarified many of my questions. When choosing the plane that you are going to apply the 3-2-1 constraints on, what types of variations will you see depending on your plane location and the location of the nodes that you are constraining? Basically how sensitive are the results to these choices?

 

[GBor]

I believe selecting the locations of the minimal constraints is not unlike selecting locations for boundary conditions: avoid areas of high stress gradient, avoid areas which your engineering judgement says should be "of interest", extend your model outward of the first two statements can't be achieved. As far as "sensitivity" goes, every model will be different. Geometric complexity will play a role, load types and how they are applied...there are many variables that would affect the answer to this question...

 

[johnhors]

The only real rule to apply when selecting the three points used for 3-2-1 is that they should all be well seperated (and not in line!). Any convenient plane will suffice. With a wide seperation any small moment imbalance has a relatively large moment arm between the supports, further minimising any reactions.