As I wrote previously in many cases nonlinear analysis of buckling yield better results than linear buckling. Setting up an analysis like that however requires certain knowledge. Today I will discuss steering of numerical analysis that is crucial in proper task definition.

To start you need to have a defined and correctly restrained (supported) model. Also you need to know what the loads are. This article do not cover those topics, I assume everything in that regard is already made.

#### Today’s topic: choice of steering

When a model is already defined a choice needs to be done: which type of “steering” you wish to have in your analysis. “Steering” as I call it is your choice on how the solver should approach incrementation in nonlinear analysis (since loads must be applied in increments in those cases).

Types of “steering”

• Force – I would say that steering with force is the weakest possible strategy, still in many cases it is perfectly fine. Most of the time, you will introduce loads in your model as forces, moments, pressures etc. (in general I will refer to them as “forces”). Steering with forces means that you will divide the “forces” you applied into small increments, and apply them one after the other. Such solution have a certain negative impact on the analysis: If the defined loads are higher than capacity of the model, at maximal load increment analysis will not converge (since higher load cannot be obtained), so a post-collapse path cannot be traced.
• Displacement – This is a second possibility that can trace threw collapse mechanisms like snap-threw. The difference is that you introduce loads not as “forces” but rather as enforced deformations. In “force” steering model capacity is important, since there is a point at which you cannot increase loads more, because model is collapsing already. In displacement steering it doesn’t really matter how big the capacity of the model is since even if it is “destroyed” you can always enforce higher deformations to the model. Downside of this approach is, that not every load can be easily “transformed” into enforced deformation. Think about a simple beam pinned at both ends: if we put a force in the middle we can easily change this into enforced deformation in the direction of this force. But if the same beam would be loaded with uniform load on its length, then we would have to know the solution in order to implement enforced deformations that would fit the solution (not too mention that would be a horrible amount of work).
• Arc-Length – the last “most powerful” strategy. Solver in increments do not increase loads to search “fitting” displacements nor the other way around. In each increment displacements and forces are increased both, and then convergence is searched in a rather complex manner (I will write about it someday for sure). This gives the advantage that all stability failure modes including snap-threw and shell buckling can be traced with this method. Downside is however that it is harder to define correctly, as more parameters are involved in the process.

Below a stability path of shell buckling. It should also be motioned that displacement controlled analysis took more time to converge, since the “drop” after collapse required many iterations in order to find a different point on the stability path (the after-collapse part). This is however relative, depending on solver parameters and solver possibilities, so the difference may not be very big in some cases.

On the chart above difference in steering methods are easily seen.

• Analysis with force steering ended (no convergence) at load around 30kN in point [A]. This is because a correct stability path would have a drop in load at this stage, but force steering requires constant force increase in each increment which in this case results in lack of convergence. Changing solver parameters (more iterations, line search, bisections etc.) wont change this, even threw such things usually help with convergence problems. Here however lack of convergence is caused by the choice of the method: it does not matter how well you search for solution fulfilling convergence criteria – since there is no solution (as the loads drop and you increase it instead) you can’t find a solution.
• Analysis with displacement steering resulted with stability path that is incorrectly traced in the drop of capacity region. There are only two outcome points there: on the top (where capacity is reached) in point [A], and at the end of the drop (once again on stability path) in point [B]. “During” the drop (between points [A] and [B]) not a single point can be found since that would require convergence, and that is not possible outside of stability path. This is why it takes so much time to converge in many cases: solver must “guess” the correct answer many times before the new point on stability path is found with the increased deformations (since it is nowhere “near” the previous point on stability path). Note that shell decreases load and displacement after failure, but displacement steering constantly increase deformation, meaning that certain part of the curve (where displacements decreases) was cut off.
• Analysis with arc-length steering allowed me to obtain a correct solution for this problem.

Usually arc length steering gives a proper solution to a problem. There are however certain limitations: if for example a laboratory test of our shell is made, and deformations are enforced constantly without a possibility of “relaxation” in shell, solution with displacement steering is actually a correct one, since arc length steering follows the “natural stability path” of the structure, but with our way of performing laboratory tests we made it impossible for the model to follow that “natural stability path”.

Summary:

If you wish to perform a nonlinear analysis you need to decide which “steering” strategy will you choose. If in doubt pick arc length method, but if you know for sure that displacements will be enforced in reality without a chance of decreasing imposed displacement during the loading displacement steering might be a better option. If that is the case be prepare for convergence problems just after collapse.

This is getting long for one post. I will continue this topic next week. If you wish, you can subscribe – I will send you an email about an update so you won’t miss it.

Have a good one!

Łukasz