The upper right and lower right points on a Vn diagram correspond to maximum positive and negative n factors and maximum dive speed; which is assumed to be 1.25 to 1.5 of maximum level flight speed; which could change depending on your design. It also corresponds to the positive low angle of attack and negative low angle of attack load cases. What velocities are assumed for the upper left and lower left "corners" of the diagram? BDD

The upper and lower points at left of the V-n diagram are where the limit load line crosses the line that represents the lift the wing can actually achieve for the lower airspeeds before stalling. The curved line, which starts at 1G and at the stall speed, is represented by the lift equation. In other words, flying at the speed of stall, the airplane can only fly at 1G - if you try to turn and actually increase the G loading, the wing will stall. Flying a bit above the stall speed, the wing can now generate a higher G loading, but still below the 4.4 G point (or whatever you're using). At some point, the airplane reaches a speed where the wing can just pull 4.4 Gs, and that's where the lines cross. The equation for the curved line extending from the stall speed to the upper left corner of the plot is: n = 1.25 x CLmax x [(rho x S x Vsquared)/(2 x W)] If I recall right the factor of 1.25 accounts for dynamic lift. "CLmax" is the maximum unflapped lift coefficient; "rho" is the air density; "S" is the gross wing area; "Vsquared" is the velocity, squared; and "W" is the anticipated gross weight. Classicaly, the airspeed of the upper left corner is called the maneuvering airspeed and theoretically it is where you would not be able to overstress the airplane no matter what maneuver you did or how fast you did it.

So to find that velocity I would just solve for V in the formula. The Clmax I believe is sometimes taken as being 25% or so above the normal stall Cl to account for dynamic effects. I would probably use this value in the formula for positive angles of attack. Aain, this is in older engineering textbooks. It wasn't usually assumed for negative angles of attack because it was assumed such maneuvers wouldn't be performed as suddenly as positive G ones. I'm thinking that this point on the Vn curve corresponds to the positive and negative (lower left corner) high angle of attack conditions. At such a high angle of attack, the chordwise component of the factored lift force would actually point forward. The relatively slow speed and therefore relatively low drag in this condition helps maximize the net forward lift component. In WWI, I remember reading, some early airplane designers (like Fokker?) were puzzled that the wings tore off in a forward direction. This is probably what was happening. This was before they understood aerodynamic loads and the wings were built much lighter then than they are now (closer to the beginning of the war). In fact, wings were always tearing off of WWI fighter aircraft in dives and other maneuvers. Yikes! Add to that, the allied generals never allowed the use of parachutes as they would "reduce the will to fight".

Orion: When I see a version of this formula, it might tend to have a Cz (or vertical force coefficient?) for the entire airplane listed instead of Cl. Does this matter much? I was kind of avoiding using that form of the formula because the examples I see show coefficients like this being found and plotted. I wanted to avoid that and use a more straignt forward approach for these calc's. if possible. I was going to set the + and - maneuvering force (n) factors at a proper value for an aerobatic airplane. I was thinking of +6 and -4. I would use a larger negative load factor if that's standard for aerobatic planes. This is partly to account for the higher effect of gust loads on a lightly loaded wing. I was also going to check what n fcator resulted from standard gust velocities (30 fps, vertical, hard edged gust). I was then going to set the velocities to use for the +haa, -haa, +laa and -laa load conditions. With that, I would calculate unfactored drag forces and pitching moments. The lift load was going to be the max. gross airplane weight x =/- n factor for all conditions. Would the simpler n value calc. (to find the velocity for the +/- haa conditions) that you quoted work for this? Thanks, BDD

In my experience, in application to the general aviation product the use of Cl versus Cz is OK. This results in a slightly simpler approach and as long as you understand what the diffences are in the Cl versus CZ calculations, you should be fine. Since Cz is net vertical force on the wing (in relation to the flight path) and is derived from a combination of the lift coefficient and drag coefficient (CLcos(theta) + CDsin(theta)), where CL and CD are in reference to the chord plane and "theta" is the angle of attack (or more accurately the flight path angle in relation to the chord plane), then use of the Cl should be more conservative since CDsin(theta) will act downward and actualy result in a lower Cz than if you use Cl alone. (somebody double check this - I'm in a bit of a hurry this morning and I might have mixed up my signs)

I'm also thinking that Cz is an overall vertical force coefficient used in calculations to balance tail loads etc. For a simplified approach, I'm just wanting to use Cl, etc. if it's acceptable practice. Similarly, I'm assuming that a 6g load factor is a good one to use for a lightplane. It should be somewhat conservative in most cases and is simple to use; just multiply gross weight by 6 and distribute that lift along the wing in a conservative way (to produce conservative moments and shear forces).

FAR Part 23 specifies a positive "limit" load factor of 6G for the design of aerobatic aircraft. This then has to be multiplied by 1.5 for any metal structural component and 2.0 for composite, to achieve the "ultimate" load factor. By definition, at the "limit" condition no part of the structure is allowed to achieve any form of permanent set other than what is started out with. In other words, no yielding of any part of the structure is allowed. Between the "limit" load and "ultimate" load, yielding is allowed but no component can totally fail until past the "ultimate" load rating. When testing a physical structure to ultimate, the structure has to withstand this magnitude of loading for at least 10 seconds before it is allowed to fail. Care must be taken in areas of the structure that are subject to stress risers (such as holes or changes in geometry) since classical structural predictions will fail to provide safe solutions for the prescribed factors listed above.

I'm also using the safety factor of 1.5. I've mentioned it in previous posts at this site. Classical analysis methods are used every day in building structural steel design. That engineering field also uses a safety factor of 1.5 and bases it's methods on allowable stresses. Yielding is considered failure in building design and the allowable loads (bending, shear, compression, combined loads, etc.) take into account empirical results after years of experience with steel as a structural material. There is another method called load factor design but that wouldn't apply at all here. It involves applying load factors that take into account the probablility of certain load combinations and durations. I suppose that steel connections might be designed using steel design methods and allowable stresses like these with consideration of cyclic loading. The fastener holes in the wing spars, etc. would still cause areas of stress concentration in those members though. Hand calculations are a perfectly acceptable method of analysis. That's how all of the advanced, high performance aircraft in WWII were designed. If certain rules are followed in connection design, stress concentrations and certain types of shear and bearing failure should be avoidable when using classical analysis methods. One thing though is that some interactions and behaviors of thin webs weren't pefectly understood at the time that these methods were developed. This is were static load testing would be useful. Also, If you are designing a non-aerobatic plane for a conservative loading of six G's (plus the standard safety factor) that goes a long way towards accounting for certain unknown or highly indeterminate (by hand calculation) material behaviors. I could also get a F.E.M. program for use in connection design. In general though, I prefer hand calculations because you are constantly checking your calc's. and have to methodically think through any analysis as you work. What can really be dangerous is computer software that will "spit out" "solutions" that really don't make sense or which are somehow misleading. Computer derived results have to be checked. What is an acceptable negative G loading for an aerobatic aircraft?

For the latter question, the answer is that it really depends on how you picture the aircraft is to be used. Most folks I've worked with, or that are designing their own, tend to design to 6 Gs for the positive and the negative case. In the case of wings this is a default since most designers tend to make symmetrical spars, despite the stated difference between positive and negative requirements. However, you can also argue that since you will most likely not pull as many negative Gs as positive ones, then the use of negative 4 Gs would be more than sufficient. I guess it's all in how you approach your design, how confident you are in your work, materials and building techniques, and how you realistically plan on using the airplane.