The holiday season brings unique opportunities to see engineering principles in action—even in unexpected places, like decorating a Christmas tree. In fact, Altair expert Matthew Sauer faced such an opportunity while decorating his family’s tree. What began as a simple seasonal tradition quickly turned into a fascinating case study in nonlinear analysis. (Ever think you’d hear that sentence?)

Read this edition of Digital Debunking to see if nonlinear analysis can help you keep your star on top of your Christmas tree.

The Challenge: A Slipping Star

As his family was decorating their Christmas tree, Sauer’s oldest child tried to place the star on the tree’s slender top branch. However, the star wouldn’t stay put. Instead, the branch bent and buckled under the star’s weight. 

“Why is this happening?” she cried. The star problem quickly turned the family’s holiday cheer into frustration. Luckily, their father—an Altair engineer—knew he was qualified to find a solution. He also knew this was the perfect time to show his daughter that her father does much more exciting things than “meetings” in his job. He began thinking of how a simple structural analysis model could illustrate this problem and help him solve it. Thus, our Debunking question was born: Can nonlinear analysis stop your Christmas tree from buckling?

The Analysis Model: Simulating a Christmas Tree

To start, Sauer built a simplified model of a Christmas tree. The zone of interest was the tree’s trunk, which he modeled as a 3-meter tall tapered beam with a 50 millimeter dia meter base and a 1.5 millimeter tip. But at this stage, this model didn’t resemble a tree as his daughter knew it. “Why are there no branches and no star?” she asked. Sauer needed to satisfy his daughter’s demands – he needed to decorate. 

To do this, he created a simple 2D mesh comprised of decorative quadrilateral/triangular elements that were inactive within the model; they wouldn’t contribute structurally and thus wouldn’t affect any calculations. He used this same strategy for the star and rigidly connected the load application joint to the tip of the tree trunk.

Loading of the Christmas Tree

Sauer was on a roll now, excited by the heart-pounding action. He had to tell his daughter what came next. “Now we need to think about the loading!” he said. She didn’t seem to share his enthusiasm. No matter – he had a tree to model. 

For the loading, Sauer considered the self-weight of the trunk as well as a point load representing the star’s mass. The illustration used an applied “star” load with a value of 100 Newtons; this value was large enough to ensure buckling. Though this isn’t representative of the actual star’s mass, it illustrated the concepts he was trying to explain to his 5-year-old.

After he combined the loads, Sauer had to accept that the tree’s construction lacked the precision commonly found in other types of structures. It was anything but plumb, and this influenced the lateral deflection he observed. To account for the disparity between his model and the real-life tree, he applied a 1% notional load factor consistent with ASCE 7 recommendations (although this may have been a generous estimate). Put simply, a notional load factor takes a portion of your vertical reaction and applies it laterally.

Setting Up the Nonlinear Analysis

Since the tree became less stable as its shape contorted, Sauer next ran a nonlinear analysis to capture this behavior. The nonlinear analysis would update both the stress-stiffness matrix and the geometry matrix as the loads were applied incrementally. This captured large deformation effects and buckling instabilities more effectively than analytical methods. A total of 100 evenly distributed load increments were used, and he used Altair® S-FRAME® to include the incremental results in the output, creating a detailed view of the tree's behavior up to its failure.

Analysis Results

With all this talk of Newtons and matrices and analyses, Sauer knew he was losing his daughter’s interest. To keep her excitement alive, he plotted the incremental results of the joint at the tip of the tree for the load combination he solved.

To his delight, the incremental results showed a sudden change in deformation – the bending observed in real life – as the incremental load ramped up to around 11% of the total applied load.

Though this was an exciting result, Sauer’s daughter was still unimpressed. He knew he needed to make the process more interactive. How could they stabilize the star? He considered a few ideas:

• Insert two branches to support the star
• Improve tree construction to reduce its lack of plumbness
• Support the star with strings hooked to a neighboring wall
• Buy a newer, lighter star

Time to capture his daughter’s attention was running out. He chose to focus on the first item on the list. He varied the diameter of the tree’s tip and compared the observed point of buckling. He did this by observing the sudden change in deformation due to a small increase in load. He plotted the data into the graph below.

Conclusion

This data gave him the answer he needed. With it, he could determine that a tree tip diameter of less than 2 millimeters greatly increases the tree’s chances of buckling.   In short, the thicker the tree, the more resilient it is. To answer our Debunking question: Nonlinear analysis can’t stop your tree from buckling, but it can help you determine how stable your tree will be – and help you decide how heavy your star should be.

Having completed his grand analysis, Sauer was left to consider two practical takeaways. The first takeaway was that it’s probably not a good idea to choose a 100-Newton star – which is equivalent to about 22 pounds, or 10 kilograms – unless you have a gargantuan tree. For your average tree, it’s best to stick with the stars that weigh a couple of ounces.

The second takeaway – and perhaps the most important conclusion – is that most 5-year-olds aren’t all that interested in nonlinear analysis. But they are interested in putting the star on the Christmas tree and seeing their father solve problems – something you don’t need nonlinear analysis to validate.

So before you polish off your Christmas tree’s final detail, take a moment to consider how its weight might impact the tree—it might just save your holiday spirit. From all of us at Altair, happy holidays!