As the complexity increases with each of the methods discussed above, so does the accuracy of the results. However, not every stage of the design requires the same precision in results. Since this discussion focusses on how the loads approach impacts the design of the aircraft, let us first compare the calculated loads for all three methods.
Below is a table outlining the loads for each method and the percentage difference when compared to the aeroelastic model.
When comparing the lifting force of upper and lower wings of the aircraft, the aerodynamic loading from method 1 underestimates the lift on the upper wing by 10.2% and over estimates the lift on the lower wing by 22.5%. Applying Simple Biplane Theory in method 2 captures the interference effects and estimates the wing loading much more accurately with the upper wing lift being only 2.6% less and the lower wing lift being 6.3% greater, when compared to the aeroelastic model.
A more detailed way to compare the resulting design impact of the three different load generation methods is to look at the internal shear and bending moment within the spars. Below is the shear force diagram for the upper wing leading edge spar.
Starting with the simplest method, the 2D non-interfering lift generates the highest shear force.
As one would expect the variation in the maximum internal shear force is small (at most 7% difference) over all the models since the total lift generated by the plane was set to be constant. The differences are partially due to how the lift is distributed between the upper and lower wings. The spanwise distribution also clearly has an impact on the internal shear force.
Interestingly, the model that matches the Aeroelasticity model shear force the closest is the Biplane Theory model using Schrenk’s approximation to capture span wise loading effects. The differences produced by these aerodynamic models becomes even more apparent when inspecting the internal bending moment as one might expect.
The internal bending moment clearly shows how the differences in aerodynamic models can propagate. The most basic model (2D Non-interfering Lift) produces the highest bending moment, 33% higher than the Aeroelastic solution (the most accurate loads generation approach). While it is better to be over conservative than under-conservative, this kind of inaccuracy could lead to substantially higher weight in the structure, limiting the performance.
Again, of all the purely analytical models, the Biplane Theory using Schrenk’s approximation does the best job at matching the Aeroelastic model with only a 2.3% higher prediction in the maximum bending moment. The Biplane Theory using an elliptical distribution produces a substantially lower maximum bending moment, 32% lower than that predicted by the Aeroelastic model.
Finally, it should be noted that since the Rigid Aeroelasticity and Aeroelasticity curves are so similar for both the leading-edge spar shear force and bending moment, it can be inferred that at the max dynamic pressure of the Fokker (125 mph at sea level), the flexibility of the structure is not great enough to impact the loading.
A similar narrative is replicated in the upper wing trailing edge spar. The maximum shear force produced by the 2D non-interfering lift was 30% larger, the maximum shear force produced by the Biplane theory using an elliptical lift distribution predicted was 33% lower, and the maximum shear force produced by the Biplane Theory using Schrenk’s approximation was 1.3% greater, producing the closest correlation when compared with the most accurate Aeroelasticity generated aerodynamic model.
Much as the leading-edge spar exhibited disparate internal bending moments depending on the aerodynamic model used, the trailing edge spar internal bending moment varied greatly. In this case, the max internal bending moment experienced by the trailing edge spar was 49% larger using the 2D non-interfering lift, 34% lower using the biplane theory with an elliptical distribution, and 13% higher when using the Biplane theory using Schrenk’s approximation.
The shear and bending moment diagrams are often excellent indicators for simple structures of the internal stress state within the structure. As a stress engineer, comparing stress plots is the most meaningful way to compare how the different aerodynamics models impact the stress throughout the vehicle.
Below is a picture of the upper wing leading and trailing edge spars Von Mises stress under 4G pull up when using the Aeroelasticity model. The max spar cap Von Mises stress is 11.4 ksi.
In comparison, the same stress contour is presented below, only for the case using the Biplane theory using Schrenk’s approximation, which exhibited a max Von Mises stress of 11.5 ksi, a 0.8% difference from that produced using the aeroelastic aerodynamic model.
In contrast, the Von Mises stress state for the upper wing under the 2D non-interfering lift can be seen below as a gross overprediction in the stress within the wing, predicting a max Von Mises stress of 15.2 ksi, 33% higher than the aeroelastic aerodynamic model.
Having exhaustively explored the impact of different aerodynamic models on the final stress results, several conclusions have become clear.
First and foremost, as laid out at the beginning of the paper, none of these approaches are inherently bad. However their mileage does vary significantly.
Requiring the least technical background, the 2D non-interfering lift model provides a good approximation of the stress state in the leading and trailing edges, but is decently over-conservative producing internal stresses 33% greater than the most accurate model.
As expected, including the interference effects between the upper and lower wing using the simple Biplane Theory and applying finite span effects has the potential to predict stresses within 0.8% of the most accurate model. Unfortunately, this relies on the user correctly calculating the span wise loading and interference effects which can be done but often requires complex analytical methods or the use of a potential flow method. Furthermore, there are ample opportunities for an engineer to make a mistake when taking this approach, and it could be difficult to detect without the results of a more accurate model to compare to.
Now, given that fairly accurate stress distribution using semi-analytical methods can be achieved, you might be asking yourself, why might anyone want to spend the money to use the NX Nastran Aeroelasticity module?
Why use the Nastran Aeroelasticity module?
First, it removes substantial uncertainty in the accuracy of the aerodynamic model. The Nastran Aeroelasticity module can account for interfering lifting surfaces, slender fuselage effects, ground effect, compressibility, wing sweep and taper, as well as a slew of other factors accurately.
Additionally, once implemented, the Nastran Aeroelasticity module is more flexible than generating the loads from an outside source and then applying those loads within FEMAP (the pre/post environment). Nastran can generate the loads for any flight condition such as steady level flight, a 4g pull up, or a 3g coordinated turn, requiring only high level information from the user.
Finally, in doing so, the user is also provided with additional information such as the trim angle of attack, control surface deflection angles, and vehicle stability derivatives.
As with most problems, there is rarely a single correct approach, but when high accuracy and case generation flexibility are desired, then using NX Nastran’s Static Aeroelastic Solution 144 is the way to go.
If you are working on a budget, can take on additional mass, or do not have the technical background to employ Solution 144, then using an analytical method or generating the loads some other way externally is probably the way to go.