Learn about snap-through buckling analysis – why is it hard and how can we analyze this behavior in Simcenter 3D? Take advantage of the Arc-Length method in Simcenter Nastran to solve a seemingly unsolvable problem.

### Challenges:

- Predicting snap-through behavior of structures
- Finding a way to solve complex nonlinear problems that isn’t readily apparent
- How to accurately capture critical deformation data using analysis

### Values:

- Arc-Length methods enable solution of snap-through problems
- Avoid buckling failure of snap through structures by using simulation
- Easily visualize loads and displacements using Simcenter 3D with Simcenter Nastran

# Snap-Through Buckling

In order to understand what snap-through buckling is, consider the simple problem illustrated below. As a force P is applied, compressive load is applied to the two bars. The system will remain stable so long as the bars are above a horizontal line formed by the two pinned ends. As soon as the load application point moves below this horizontal line, the structure is no longer stable and the compressive load in the bars actually contributes to buckling which takes place rapidly (snap). Eventually the system becomes stable again as additional applied load is reacted by tension in the bars. However, in many structures, this snap-through behavior will most likely lead to failure.

*Simple structure illustrating snap-through behavior*

# Snap-Through Instability is Hard

As you can imagine, this type of snap-through instability is difficult to model and analyze. Take for example the plot below which shows the load-deflection curve for a hinged cylindrical shell undergoing snap-through buckling. Nonlinear methods using Newton’s method will try to trace a path that looks something like the dashed red line. Little problem: the actual deformation takes a completely different path as the structure undergoes a very sudden and quick “snap” behavior. Sure, it’s possible to wait long enough and luck into a solution. But even if a solution is found, we certainly can never visualize the snap through loads and displacements accurately using these analysis methods (notice that there are three possible solutions at the displacement defined by the red vertical line).

*Load-deflection curve for hinged cylindrical shell, typical nonlinear solver approach shown with dashed red line, vertical red line illustrates three possible solutions at the defined displacement [1]*

How can we solve this?

# Arc-Length Method to the Rescue

The Arc-Length method offers an efficient and versatile scheme for solving these exceptionally difficult, nonlinear problems. What differentiates the arc-length method? Let’s start by looking at Newton’s method first with the figure below:

*Schematic representation of the Newton solver [2]*

Notice that Newton’s method traces the nonlinear problem by computing the tangent stiffness. As the curve flattens (i.e. the tangent stiffness approaches zero), the step size used by the solver becomes extremely slow. Additionally, the method fails to solve the problem once a limit point is reached. This is exactly what happens in snap-through buckling. Now let’s compare this method to the arc-length method shown below. For the Arc-Length method, the increment is defined by a circle of radius Δl as shown.

*Schematic representation of the arc length solver [2]*

The advantage of this is that multiple solutions can be found and the correct ones identified as the solution path is traversed. See the figure below for example. Additionally, the step size does not need to become infinitely small like it had to for Newton’s method when a horizontal tangent stiffness was encountered. This method enables us to solve complex problems that have both horizontal and vertical tangents.

*The Arc-Length method in action [2]*

In this blog, I will walk you through snap-through analysis in Simcenter 3D with NX Nastran using the Arc-Length method.

## Step 1: Define the Problem

We will be walking through the hinged cylindrical shell problem defined by the figure shown below and previously discussed.

*Problem definition for hinged cylindrical shell snap-through problem*

The geometry and material behavior are shown on this figure.

## Step 2: Build the Finite Element Model

The finite element model was built in Simcenter 3D using CQUAD4 elements. Only linear material properties were used. The figure below illustrates the model and boundary conditions.

*Finite element model of hinged cylindrical shell snap-through problem*

The blue lines represent the hinged boundary condition. The red arrow represents the applied load (0.6 kN).

## Step 3: Set up the Solution

The analysis will be performed using NX Nastran solution sequence 106. The Arc-Length method will be used. The following nonlinear controls and arc-length controls were defined:

*Nonlinear parameters for snap-through problem*

*Arc-Length method parameters for snap-through problem*

It should be noted that these parameters need to be defined for both the solution sequence as well as the subcase. Finally, large displacements need to be enabled:

*Enabling large displacements in Simcenter 3D*

## Step 4: Solve and Post-Process

Now comes the fun part: reviewing the results.

First, let’s look at an animation of the snap-through behavior.

*Animation showing snap-through buckling of hinged cylindrical shell*

Note that the animation is at 10% model scale. We would also like to look at the deflection and applied load vs iteration.

*Displacement vs. iteration for hinged cylindrical shell snap-through problem*

*Applied load vs. iteration for hinged cylindrical shell snap-through problem*

Finally, we would like to reproduce the plot from the reference. To do this, we will plot displacement vs. load in Simcenter 3D. We will first save the two graphs as Afu’s. Afu’s allow us to store XY data on our computer so that we can reference it when we create additional plots.

*Exporting a graph to Afu in Simcenter 3D*

Next, we create a Two Function plot as shown.

*Creating a Two Function plot in Simcenter 3D*

The result is shown below. The general behavior appears to match closely with what we expected.

*Two Function plot of force vs displacement for hinged cylindrical shell snap-through problem*

This plot is shown overlaid above the original plot below:

*Comparison of NX Nastran results vs. literature for hinged cylindrical shell snap-through problem*

Notice that there is very good agreement with the expected result. We were able to accurately capture snap-through buckling behavior for a hinged cylindrical shell.

# Summary

Problems that exhibit snap-through buckling can be very difficult to solve. As such, they require a different solution approach. The Arc-Length method is perfectly suited for these types of problems. In this blog we walked through a snap-through buckling analysis in Simcenter 3D with NX Nastran. We were able to solve a difficult nonlinear problem and match literature very well.

## Written by Vitaliy Rezekulov

Vitaliy Rezekulov is a Senior Engineer with a background in structural analysis. He has worked on some interesting linear and nonlinear problems. Outside of work he enjoys spending time with his family and friends.