Upper strut joint on aircraft – pinned joint; strut experiences axial and transverse combined load [1]

## Beam Column Effect – Simcenter 3D vs. Hand Calculations

The beam column effect can have a highly detrimental impact on the strength of members subjected to combined axial and transverse loads. In this post you will learn what the beam column effect is, see a hand calculation to estimate its impact, and finally compare the hand calculation results against Simcenter 3D.

Challenges:

• Beam column effect can greatly reduce the load carrying capacity of beams
• Accurate results are necessary to prevent failure of loaded components

Values:

• Simcenter 3D takes advantage of the NASTRAN nonlinear solver, enabling accurate analysis of the beam column effect
• Hand calculations allow for simple and quick model validation

### Beam Column Effect

A beam column is a member subjected to both transverse loads (or end moments) and axial loads. The beam column effect is most pronounced when the cross-sectional dimensions of the member are small compared to its length. As the beam deflects under transverse loading, the axial load at the ends induces a moment in the beam. This additional moment can have a significant effect on the strength of the structure.

As an example, consider a strut that has pinned ends. The strut carries compressive loading as well as a transverse pressure distributed loading. The scenario is summarized below.

In this technical blog, we will use the above scenario to look at the beam column effect. We will start by looking at the loads individually and superimpose them to predict a linear solution combination stress. We will then perform a hand calculation using the Bell Structural Design Manual [2] to understand the impact of the beam column effect. Finally, we will compare the hand calculation against NASTRAN SOL 106 nonlinear stress results.

### Linear Solution Combination

It’s fairly straightforward to compute the linear stress for this problem. We have two load components. The first is an axial compression. The stress due to axial compression is given by:

S_a=P/A

The second component of stress is a bending stress due to transverse load. This stress is given as follows:

S_t=(wl^2\ D_0)/16I

The combined stress would be the sum of these two components:

S_c=S_t+S_a

The result of this calculation is shown below:

We would expect the linear combination stress to match the linear finite element analysis in Simcenter 3D. A simple shell finite element model was built in Simcenter 3D to replicate the model setup described above. Pressure and axial compression were applied to the model. The results of the linear finite element analysis are shown below.

We see that the difference between the linear finite element analysis and the hand calculation is less than 1%.

### Hand Calculation using Bell Structural Design Manual

In order to take into account the beam column effect, we will use the Bell Structural Design Manual [2]. Table 11.3 of this manual has the following:

The load case under consideration matches the third scenario in the table above. Using the equations in the table, the following beam stress is computed.

By comparing this stress to the previously calculated stress, we can see that the beam column effect has a huge contribution to the peak moment and maximum beam stress – on the order of 60% higher.

### Simcenter 3D Nonlinear Stress Analysis

Now we will see if we can reproduce this result in Simcenter 3D. This time we will run nonlinear (SOL 106) NASTRAN using the exact same boundary conditions and loads. The resulting stress plot is shown below.

Comparing the obtained result to the beam column hand calculation, we again find that our stress is within 1% of the expected value. We can thus conclude that the nonlinear solution captured the behavior we were looking for.

The table below summarizes our findings. By looking at the results for the force only and pressure only cases, it is apparent that it is precisely the beam column effect that is driving the stresses. While the compressive force only contributes 1ksi in compressive stress individually, the beam column effect produces a 10ksi increase in stress, a ten-fold increase.

A few takeaways:

Q: The hand calculation matches the FEA stress so well. Why waste time on FEA?

A: Because not all problems are going to have such simple geometry and loads. What if you wanted to consider nonlinear material behavior? Etc. It’s always great to have hand calculations in your back pocket for validation and sanity checks!

Q: How do I know if my problem needs a nonlinear analysis?

A: Sometimes it’s not obvious and you need to run it just in case. In this case, we had a small cross-section and a long beam, which meant we could expect a large deflection if a transverse load is applied. Since the deflection is large, it’s likely the analysis would behave differently in the nonlinear domain.

## Summary

As can be seen, the beam column effect can have a profound and detrimental effect on structural behavior. In the problem we considered a 60% increase in stress was observed if this effect was ignored. In this blog we presented a few ways to quantify the beam column effect. First, we presented hand calculations that can be used for simple geometries and loading scenarios. Second, we showed that this behavior can be captured using the Simcenter 3D nonlinear solver capabilities. If you would like to learn more about nonlinear analysis in Simcenter 3D or structural analysis in general, feel free to contact us today!

## 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, playing and watching sports, and being outdoors.

References:
[1] https://www.kitplanes.com/stressing-structure-18/
[2] Bell Structural Design Manual