Node Reaction

[edwardal]

I was using the 2D cantilever example (Ex1a.Canti2D.EQ.modif.tcl with A10000.tcl ground-motion file) to investigate the node reaction recorder and I can't figure out what the recorder is outputting for the top node.

Looking through the forum, my understanding is that the node reaction is the sum of the end forces of all elements framing into the node. However, when I use the example 2D cantilever and do nothing but add the following recorder to get the top node reaction

recorder Node -file Data/RTop.out -time -node 2 -dof 1 2 3 reaction

I seem to get an unexpected result. I expect the reaction in the X direction would be equal to the beam end force from the recorder "recorder Element -file Data/FCol.out -time -ele 1 force" (already included in the example), but it is not.

What is causing the discrepancy between the element end force and node reaction at the top of the cantilever? (there is no discrepancy at the bottom node reaction)

Any insight would be appreciated.

[edwardal]

Upon further investigation, it appears that the discrepancy is equal to the base acceleration times the mass at the node. I will give a brief explanation of what I think is going on, and if someone could confirm this or tell me where I am mistaken that would be great.

Considering the equation of motion from Dynamics of Structures 3rd Edition (Chopra) eqn 1.7.4:

m*a + c*v + k*x = -m*(a_g)

Where:
m = nodal mass
a = relative acceleration (between the node and the ground)
c*v = Fd = damping force
k*x = Fs = static force
(a_g) = ground acceleration

Substituting and rearranging I get

Fs + m*(a_g) = - m*a - Fd

And based on the model output I see that this is equal to the reaction force, R, meaning

[eqn 1]: Fs + m*(a_g) = - m*a - Fd = R

Ultimately I am looking to obtain the forces due to modal damping (unfortunately there seems to be no "modalForces" argument similar to "rayleighForces" arguement for the Node recorder). It seems as though I could just rearrange [eqn 1] and get the modal damping forces at each analysis step as

(Fd_i) = -(R_i) - m*(a_i)

where
Fd_i = the damping force at analysis step i
R_i = the node reaction at analysis step i (output from "recorder Node ... reaction")
a_i = the node relative acceleration at step i (output from "recorder Node ... accel")

Again, any insight would be appreciated.

[fmk]

no,
1) the 'reaction' command only records the "stataic element forces", to include damping contribution from element connected to the base you would use different recorder options, reactionIncInertia. if using the other option, and modal damping is employed, the modal forces are not included in nodal reaction forces, the reason is that they are not an element or node contribution and are added to the rhs of the matrix equation only. to calculate the modal forces requires recording nodal accelerations and element forces and using equlibrium .. a royal pain (i know from experience especially for larger models) .. note when doing so you would need to ensure that the convergence test is an Unbalance one (which it should be for dynamic analysis) otherwise some of the forces you are attributing to modal damping will include some of the nodal unbalance (which can sometimes be quite large if you loook at them when using NormDispIncr)

[edwardal]

Thanks for the reply.

Is there a way to check the unbalance at a node (while employing modal damping)? Or is it just assumed that using an unbalance test (i.e. NormUnbalance) will result in no unwanted force unbalance? I did come across the 'unbalance' option for the node recorder, but I can't find any documentation on it. From the models I have run with base excitation, it seems to output:

('unbalance' recorder output) = (Applied Load) - (nodal mass) * (ground acceleration)

which I don't think helps me very much in determining the force "unbalance" that I am looking for here. Do you have any methods to find the force unbalance you mentioned could be quite large when using NormDispIncr?

[EDIT: removed comments about the 'unbalance' recorder that I realized were wrong]

[fmk]

the unbalance will include everything but the modal damping forces .. if you have a simple model with no equalDOF constraints probably the best way to look at the unbalance would be to put a 4 at the end of the convergence test, that will show the rhs of the residual vector at each analysis step.. looking at the node id's (obtained when you print the nodes) you can see which dof corresponds to which matrix equation.

[edwardal]

Unfortunately, my model is fairly complex and does have EqualDOF constraints...

Based on the wiki page for NormUnbalance, the convergance test is comparing the norm of the "right hand side of the matrix equation" which "Usually, though not always, it is equal to the unbalanced forces in the system" to the specified tolerance, $tol. Based on this information, it seems to me that the norm would be the maximum force unbalance across all of the DOF's, so $tol would be the upper limit on the force unbalance at all nodes?

Please let me know if this is the correct logic.

PS - for the "though not always" case, it seems that this is affected by the the integrator and constraint handler. For reference, I am using:
integrator Newmark 0.5 0.25
constraints Plain

EDIT: Basically, I am not concerned about the actual unbalance at each node as long as the unbalance is limited to some quantifiable amount (related to $tol).

[fmk]

the norm is the 2 norm of the rhs .. if you want to look at the numbers in the actual vector use a 4 at the end of the test

[edwardal]

Is there a reason that the 2-norm is default over another type of norm?

[fmk]

it is the one used most often in other codes .. you can always edit to do something else and recompile