Aircraft wings, inherently flexible structures, are susceptible to catastrophic failure under certain aerodynamic conditions. This study seeks to breakdown the complex aeroelastic behavior of wings, focusing on the critical concept of shear center and its implications on structural integrity. The investigation unveils a phenomenon known as divergence, where the interaction between bending and twisting modes leads to structural failure. To enhance understanding and engage students in aerospace engineering, the project introduces an innovative approach using 3D printing technology. The core components include an easily replicable test rig, 3D printed beams and plates, controls and data measurement, and a model wind tunnel. The experiments conducted with these components aim to complement undergraduate and graduate courses in Mechanics of Solids, Control Systems, Aerospace Structures, Flight Dynamics, and Aeroelasticity. The educational objective of the project is to provide students with tangible, 3D printed models and experiments that serve as practical tools for exploring and quantifying theoretical concepts related to aerospace structures, with a specific focus on the shear center. Drawing inspiration from previous work by Professors Lawrence N. Virgin and Earl H. Dowell at Duke University, the project extends their research by integrating modern 3D printing materials into the study of aeroelastic phenomena. The foundation of solid mechanics is thus bridged with cutting-edge technology to create replicable experiments, fostering a deeper understanding of the intricate relationship between structures and aerodynamics in aircraft.


Aircraft wings are inherently flexible structures meaning that under load they can bend and twist and if certain conditions are met these modes can couple as shown in the video below. 

In some cases this can lead to catastrophic failure. A prime example is the Fokker D8 built by Anthony Fokker for the Germans in WWI. Unfortunately, in intense combat the wings of the D8 began snapping off. 

To localize the cause, German engineers suspended the plane upside down and added weights six times the weight of the aircraft to replicate aerodynamic loading. When further increasing the weight, they noticed a failure of the aft spar. In response, all aft spars were made thicker and stronger. However, the planes began failing more frequently! But wait, would this imply that adding more material made the wing weaker? When repeating the tests, the Germans observed that the wings were not only bending, but twisting simultaneously even though no torsional loads were applied. This created a detrimental loop where the twisting would increase the angle of attack which would increase the aerodynamic loading thereby further increasing the twisting until the wings eventually twist off. This phenomenon is known as divergence.

Spars in a wing can be represented simplistically as two parallel beams. If these beams are loaded exactly between them pure bending occurs. The point where loading results in pure bending is the shear center. 

Wing loaded equidistant between two spars. Deformation is pure bending. Source: Gordon J. E. Structures or Why Things Don’t Fall Down

However, in reality, wings are loaded at the center of pressure which is located closer to the leading spar (about a quarter of the chord from the leading edge). As such, there will naturally be twisting because the loading is off the shear center. The distance between the shear center and the center of pressure is known as the elastic margin and represents the lever arm for the lifting force moment. If one of the beams is thicker and stronger, the shear center will move aft and the elastic margin will increase creating a greater lifting moment leading to divergence. 

Wing loaded at center of pressure with a thicker rear spar. Deformation is a coupled bending and twisting. Source: Gordon J. E. Structures or Why Things Don’t Fall Down

This phenomenon is so critical that today’s aircraft designers spend hundreds of millions to ensure that this doesn’t occur. A fantastic example is Boeing’s bend test:

Boeing 787 wing flex test. Source: Boeing

We see that to understand the complex structural and aeroelastic behavior of planes we must build on the foundations of solid mechanics. We aim to leverage 3D printing to develop an intuition for these concepts while providing students with replicable experiments.

Project Decomposition

Having introduced the academic and industrial context of our project, we breakdown its key aspects:

Module Scaffold

We divide the theoretical and experimental components of our project into a module scaffold. The beginner section is intended for undergraduates in a solid mechanics course and is designed to build intuition for theoretical concepts. The intermediate section is intended for undergraduates who have taken courses in solid mechanics and serves as an additional primer for electronics and controls. It is a primarily experimental section that allows students to quantify the theory from the beginner section and couple it with additional important engineering skills. The advanced section is intended for advanced undergraduate and graduate students in Dynamics and Aeroelasticity. It contains both a theoretical and experimental component.

Each module below has a link that will take you to a separate page containing all the requisite information and files.

Meet the team!


1. Virgin LN. A shear center demonstration model using 3D-printing. International Journal of Mechanical Engineering Education. 2022;50(3):739-748. doi:10.1177/03064190211057429

2. Dowell, E. H., Scanlan, R. H., Sisto, F., Curtiss, H. C., Jr., and Saunders, H. (April 1, 1981). “A Modern Course in Aeroelasticity.” ASME. J. Mech. Des. April 1981; 103(2): 261–262. https://doi.org/10.1115/1.3254897 

3. Gere James M and Barry J Goodno. Mechanics of Materials. 7th ed. SI version ed. Cengage Learning 2009. 

4. D’Amore, Pierluigi, et al. Calculating Shear Center for C-Shaped and L-Shaped Cantilevered Columns 


6. Gordon J. E. Structures or Why Things Don’t Fall Down 

7. T.H.G. Megson, In Aerospace Engineering, Aircraft Structures for Engineering Students (Seventh Edition), Butterworth-Heinemann, 2022, ISBN 9780128228685, https://doi.org/10.1016/B978-0-12-822868-5.09993-9 

8. K. C. Hall. “Eigenanalysis of unsteady flows about airfoils, Cascades, and Wings”. AIAA Journal. Vol. 32, No.12, pp.2426-2432. Dec 1994.

9. D. M. Tang, et al. “Flutter and limit cycle oscillations of two-dimensional panels in three-dimensional axial flow”. Journal of Fluid and Structures. Vol. 17, No. 2, pp.225-242. Feb. 2003.

10. I. Wang. “Aeroelastic and Flight Dynamics Analysis of Folding Wing Systems”. Dissertation, Duke University. 2013.

11. McMahon & Graham, “Introduction to Engineering Materials: The Bicycle & the Walkman,” Merion (1992)

12. Mississippi State. (2007). Shear Center. https://www.ae.msstate.edu/tupas/SA2/chA14.2_text.html

13. Ungureanu, Viorel. “Torsion of Members.”

14. Hibbeler R. C. Mechanics of Materials. 5th ed. Pearson Education 2003

15. Theodorsen, Theodore, et.al. “Mechanism of Flutter: A Theoretical and Experimental Investigation of the Flutter”. NASA-TR-685. Jan 1940.