Initial Displacement Methodology Discussion

[MDSmith526]

We have been looking at how the -disp tag works with nodes for assigning initial nodal displacements. For the elements that are based in a basic (natural) frame, not all of the initial displacements are taken into consideration. PDelta and Corotational correctly take nodal translations into account when they form the initial rotation matrices, and in the case of corotational, the initial length. However, nodal rotations and any other DOFs (such as warping or local plate buckling) are not handled properly because the coordinate transformations only deal with rigid body effects. There is the possibility that the coordinate triads at node i and node j may have an effect, but I am having a hard time seeing it.

First, let's consider one of the force-based elements formulated in the natural frame (lacking rigid body modes). I believe I am correct in that it does not have any components that consider the Wagner effect. And, if the Wagner effect is not considered, a force-based element will not ever experience torsional buckling. There are some alternative elements called "mixed" elements in the literature, but I don't see any of those in OpenSees. Therefore, since the force-based elements do not have forces that interact with their displacement fields, I do not think initial rotations will ever affect them.

Next, let's consider a disp-based element. Suppose this element has both a linear portion of the stiffness and a geometric stiffness. K = Ke + Kg. There are formulations that include wagner effects in the Ke portion. However, since the system being solved is (K^-1)*Del_P = Del_u , in general, only the forces are affecting the first solution iteration (and the translational geometric effects from the rigid body considerations of the crdtransf). So, unless Kg includes initial displacement effects, I don't see how initial rotations or twist could affect the solution. Therefore, I am thinking of a solution, but I need some input from people smarter than me.

Kg is a function of the total internal forces. Initial displacements are stress-free, so they are not included when calculating the internal forces for Kg. This is why twist won't cause buckling. Here is my strategy. What if a set of effective initial forces are determined by the following. At initialization, take the total nodal displacements, including initial, and send those to the section and ask for the corresponding forces, given the initial section stiffness. These could be used to formulate a Kg2 which is an initial geometric stiffness matrix that takes into account the effects from the initial displacements. This matrix would stay constant throughout all analyses and the total tangent stiffness would become K = Ke + Kg + Kg2, where only Ke and Kg are updated using incremental displacements and the total real forces. That means for the initial iteration, Ke is still the initial linear stiffness and Kg is (assuming no initial stresses are present). This seems like it would work, except for one problem.

For elements with material nonlinearity, does my solution hold? It feels like superposition a little bit. And, superposition doesn't hold for nonlinear systems. Does anyone have any thoughts on this? We have a real need to be able to model initial twist to show that our element can experience torsional buckling that agrees with the theoretical solution. As of now, we cannot do this because we need to be able to impose an initial twist without applying an initial twisting force.

[MDSmith526]

Also, if you know a professor or researcher who would be a good resource on this matter, their name would be appreciated.

[MDSmith526]

This isn't going to work. Kg alone does not cause interaction between the elongation DOF and the twisting DOFs. Right now, the only thing I can think of is to set up one of these:

Fo = K*delta_o

and use it as an effective force to get the structure to deform. I have problems with that though, one, what if I wanted, say, a horizontally curved beam with a fairly large curvature. These would be huge forces that would have to be equilibrated during analysis. It would be horrible if the material model was inelastic...my initial curvature would yield the beams and then the structure would never be in equilibrium. This would be a problem for post-buckling response as well.

I'd love some ideas....

[fmk]

sorry i thought i had replied to this .. only a few of the elements will have a Kg component (force and disp beam coluns with a pdelta or corot transformation, NLShell and corot truss) .. all other ele as far as i know assume small disp in stiffness matrix computation and hence there would be no Kg component added .. if your model was using these elements then you could do as you propose for the elastic case.

the NL case is complicated. most analysts (and other codes) use a modal approach to get some form of initial imperfections to apply and then apply some sort of incremental load analysis until some failure criteria is met. except for the simple buckling column type of models, this can and does fail quite often to correctly predict the buckling mode when material nonlinearies have a great impact on the mode of failure.

as for a person to contact, you could try Prof. Filippou at Berkeley.

[MDSmith526]

Thank you, sir!