For three flange box beams (D-cell) Perry and Bruhn always show the stringers (flanges) as geometric points with no area or moment of inertia. No neutral axis is ever mentioned and moments and all forces are usually assumed to act exactly at stringers. The stringers at the rear, vertical web are assumed to take moment equally as a force couple meaning that the neutral axis is exactly in the middle between the stringers. I can't help but assume that as soon as a stringer is assigned an actual area with an actual moment of inertia and as soon as stringers (flanges) are optimized to take higher positive than negative G's, the neutral axis (which is never indicated in an example) will shift and the flanges will receive bending forces unequally as opposed to the examples. Unfortunately, no real life design examples are worked out or flanges sized. What is the correct way to go about this? Should the moments be assumed to be acting about a neutral axis rather than at a stringer? The former makes sense to me.

BDD, I do not know the text that you are citing, but I got my Mechanical Engineering Structures from Timoshenko and Gere and my Composites from Tsai and Hahn. And they cover the contributions of the lamina, but then they go ahead and neglect the same things as your authors. Technically, the flange and shear web elements do have moments of inertia, but they are puny compared to what you get by their being spread to the top and bottom surfaces of the wing. The result of the omission is that the real structure will be ever so slightly stiffer and stronger than the calculations show. The same is also true for the shift in nuetral axis when you build an asymetric beam because most wing and tail spars are only slightly asymetric. Remember also that you will be adding in some additional material in both the shear web and the flanges for a couple more reasons: First, in virtually all materials, you will have to have some excess material just to build it; Second, fasteners and stress concentrations around them will require that some more material will have to be put in to make up for it. Think about an aluminum web. You have several thicknesses available, and you will not take the one that is too thin, and the next one, the next one is too thick but that is the one that goes in. Then the spar caps, or flanges have the same problem. Similar things happen in wood, and if you think that composites beat this, you are wrong. You end up with increments of whole plies. In my spars, that is 0.007" in shear webs and 0.025" in caps, and then the last couple inches of each ply can not be counted on to carry load anyway. But, let's do the whole calculation: Xn = Sum (EAx)/Sum(EA). Now if E is the same for all of your material, (building in aluminum?)this becomes SUM(Ax)/SUM(A). In wood and composites, you have different stiffness (E) depending upon grain/fiber orientations; For a thin, wide rectangle, like a spar cap lamina, A=bh, I=(bh^3)/12, It = I + A(x-Xn)^2 where b is width (and large), h is thickness (and small), and (x-Xn) is the distance from neutral axis (and large). I almost disappears, and A(x-Xn)^2 dominates; For a tall thin rectangle, like a shear web lamina, same equations, except that b is small, h is large, I dominates and (x-Xn) almost vanishes; Anyway, calculate the A's, x's, I's, then Xn's, then Itotal's, then sum up the Itotals. If you are working with plywood shear webs or composites, multiply each I by its E in the direction of the spar before adding them. Do it both ways on one section, and see if it really makes a significant difference. Clear as mud? Basically, the errors are small, and almost always in the conservative direction. Billski

One of the first types of beam I was going to check out would be a wooden box beam. This would have spar caps, flanges or stringers of some dimension and area. Lamina are thin and non-consequential. I take it you are referring to composite design terminology?

I read what you asked, and gave you the correction to include finite thickness lamina. Call them elements if you feel a need to distiguish from infinitesimally thin lamina. The same nomenclature can apply for any spar, from a composite beam with differing materials in the caps and webs (many lamina of small but finite thickness) to a simple rectangular wooden beam (one lamina of very substantial thickness). When you work in wood, yeah, the lamina are thicker, in fiberglass, thinner, but they are still lamina. And the notation I gave you work for all of them. So, if you want to take into account the bending stiffness of the caps and the offset nuetral axis, go ahead and do so. Now remember, if you are using the same plywood for caps and webs, they will have the same material stiffnesses (E), but if you work with solid clear spruce for the caps and plywood at +/-45 degrees for the webs, they have different stiffnesses, and thus different contributions to overall stiffness Sum(EI). Billski

Billski: Thanks for the replies. I know that the equations you gave apply to all structures. Since aircraft structures are designed to be lightweight I wouldn't really want much error built in. I was wondering if there is some reason that makes these solutions for 3 flange box beams by considering the flanges to be geometric points with no recognition that actual flanges have area, moment of inertia about their own axes which can move the neutral axis and thereby change the actual flange loads due to bending. Perry says outright that the areas of the flanges are irrelevant. Both sources say that the forces in 3 flange box beams can be found by simple statics and are statically determinate. They never imply that such solutions are in any way preliminary or inaccurate. Yet Bruhn later says that such beams should be proportioned such that the c.g. of the box beam is near the airfoils aerodynamic center. They also say that the location of the geometric points that describe the flanges should be located at the c.g (or n.a?) of the flanges. That's true, of course and makes the solution more accurate but it also brings into discussion the areas and moment of inertia of the flanges and should also bring in the fact that the n.a. of the flanges is important to locate correctly. I was wondering if the n.a. is irrelevant for some reason for these beams, and if they can be solved exactly by statics alone (with no reference to a neutral axis) and why? BDD

BDD, I do not have their derivations to refer to so I do not know what your authors had in mind. It does sound like they gave sizing methods without worrying over fine tuning. Perhaps they had some fudge factors built in too. The amount of dependance upon structural test to reveal weak spots has surprised me. Yet there are authors who lean on "correction based upon experience". I can tell you that the simple method I cited reflects standard theory, and is broadly used in structures. The method I gave allows you to obtain neutral axis from the areas and locations of each lamina or element, calculate the second moment of inertia of each lamina about its centroid, and then shift each lamina from the neutral axis to obtain its full contribution. It does correct for the true location of the neutral axis and includes all of the contributions to bending stiffness. All that any of these methods are doing is allowing a simple method of adding up dA*x^2 for all of the dA to cover bending. This allows you to most accurately represent the actual part in pure bending. Rigorously, you should also calculate second moment in the for-aft plane and in torsion, and figure out what the drag moments and pitching moments are, then check the total stress state, although the contributions from these are small. Now for the mechanics lesson. The rigorous solution is completely accurate only in pure bending, where plane sections remain plane when bending has occurred and the sections are not distorted. In reality, the flanges change shape, and the shear web has to carry shear at the same time that one spar cap is longer and the other is shorter, taking the shear web with it... Also the D tube portions that form skins bulge due to aero loads too. Then, as I pointed out earlier, the structure has to be attached together, and so you have to get loads from one piece to another. Many metal spars are built up with strips and rivets. Wooden box spars are glued and screwed... By the time the structure will stay together, it must be beefed up atleast a little. Then the last surprise. The lightest way to get to a structure that holds together is actually to beef the spar caps more than you beef the shear web, keep checking your von Mises failure criteria for the caps and the shear web (with bending, torsion, and shear loads imposed) and pretty soon, it will get to required load carrying capacity. That is fine for aero loads. Then you have to do landing loads, and if your landing gear attaches to the wings, they have to carry that too, which designed my center section as much the aero loads... When you get done, you will probably find that the spar really can be quite light, even if it does not look efficient. But you can not save big weight by designing within a gnat's ass of the perfect strength. Weight is saved by sharply tapering the caps and webs towards the tips (yes, multiple analysis). Weight is saved by using thicker sections, so that the 2nd moment requires less cross sectional area. Weight is saved by keeping control surfaces light which keeps balance weights light too. Remember that engineering is about making good estimates. Good luck. Billski

Thanks, Billski: Do you have any opinions about the loads that should get factored in my earlier question at this site? I was wondering if drag or torsion (pitching moment) forces should ever get factored due to....say....horizontal wind gusts? Normally pitching moments don't change with angle of attack, change in lift coefficient, etc, but they technically would change with accellerated changes of velocity per the formulas for drag and pitching moment. I only seem to see references to g loads (maneuvering forces) being applied to lift loads. I gather that, at the very least, drag and pitching moment loads are not usually factored. BDD

I designed in composites, so I was designing a wing with a large conventional spar, two small shear spars, and a full torsion box. The horizontal and vertical tails were similar, but much smaller. I designed to 6 g and FOS of 2.0 for all of my structures, per FAR Part 23. When I wrote my spreadsheets, I included lift, drag, and pitching moments, which in combination with angle of attack, reacts in the wing to give bending and shear in vertical plane, bending and shear in horizontal plane, and torsion. Because I was working in composites, I had to define my full structure loads, then my full structure strains, then lamina strains, then check for failure. And I had to do it for all of the lamina. So, I had two 3x3 matrices for in-plane and out-of-plane stiffnesses, and a 1x6 strain matrix for the whole structure and one set for each lamina. Once you get all of this in there, including all of the loads is not a big stretch. Drag components to bending and shear ended up being pretty small, but torsion was significant, particularly at 6g and Vd. I never did check to see if I really had to consider torsion By the time my wing skins were sturdy enough to prevent some fly-in attendee from putting a pencil point through them, and the spar caps were beefed enough to beat shear web failure, stresses were pretty reasonable. There are several defining load states: 1g Stall, 6g Stall/ Vne/ Vd/ Va (50 ft/s veritcal gust), and so on. Horizontal gusts would not seem to be a big deal if those others are covered. In general, I would get a hold of a copy of the FAR for light aircraft and live with its limitations. It covers a bunch of conditions, is conservative, but is not excessive. Billski

Billski: The books I was referring to are older and based on hand calculation techniques of course. Peery (which I was mispelling) 1950 and Bruhn, 1949 revised 1952, I believe. Older classic texts for aeronautical engineers of that time. The older books tend to have some info. on wood structures before that was completely abandoned by the aircraft industry (for commercial and militery planes). BDD

Let's see if I can throw in a few cents worth. The texts by Bruhn and Peery are the classics of airplane design and are considered by many to still be the best examples of hand calculations that one can find. Both books have many worked out problems and thus are an excellent refernce to practical approaches to aircraft structural design. I use them from time to time to author the examples into a MathCAD format so as to be able to do quick trade-offs, without the extensive work that is generally needed to do a good FEA model. The format of the methods of analysis that are presented is really optimized for doing the design through hand work or slide rule solutions. It is the classical method of analysis that concetrates on dividing a structural makeup into the major subcomponents, then solving for the loads in each of the components so that each section can be designed independantly of the others. As such, the wing's primary structure is divided up into a series of points that can be solved through realtively straight forward algebraic methods. This is of course similar to entry level algebra, where one is working with a's, b's, x's and y's, instead of numerical values. In the case of the structures, this allows for the basic solution of simultaneous equations, using the various forces and shears, rather than working with stresses, gauges or any other physical entities. Although not really explained anywhere, the examples do displace the points analyzed so as to approximate the displaced localized neutral axees. The problem is then run two or three times, each time getting a better idea of where the points need to be to get a better representaton of where the final hardware configuration ends up. Although it goes only through one iteration, the example on page 168 in Peery shows some of this in its rather basic format. By dividing up the structure into the basic components, the designer can then easily solve for the different requiremetns. Axial loads simply drive the cap thicknesses and geometry - this is eventually combined with the requiremetns of manufacture, to arrive at the manufacturable configuration. However to get there, several other layers of analysis have to be done in order to account for the additional stresses created by the fastener holes, as well as any localized instability forces such as are caused by things like semi-tension shear beams, or any other semi-stable structural elements. Shear flow drives the skin gauges and rivet spacing, and can be used to derive the panel shear stresses and the subsequent issues of panel stability and reinforcement that may be needed, if any. Finally, I should reinforce what Billski mentined above - there is nothing more risky than trying to design your wing structure to within the gnat's ass of the theoretical material strength or any design guidelines. The weakest point of any structure is the weakest link of the structural chain. There can be several issues within a structure, or within the materials you use, that will make it beneficial to design with a "broader pen". The last thing you need in a flight structure is a surprise element. The weight penalty for this is nearly negligible and the subsequent peace of mind is priceless. Concentrating on things such as the deflection of the neutral axis when the structure is under load is silly at best. Furthermore, if your neutral axis shifts sufficiently when your airplane is under load, to afffect the structural nature of the wing, you've got other, more basic and serious problems.

Orion: Thanks for the input. This has nothing whatever to do with looking for changes to the neutral axis while a beam is under load. The neutral axis would be determined when the flange areas and moments of inertia are determined for the flanges acting together. This would be established relatively soon as a design progresses. Just like a normal box beam or two flange beam with shear web connecting them, etc. the location of the neutral axis wouldn't change or would never be assumed to change while under load. That's just part of the basic geometry and inherent section properties of the beam. They don't change. But the section properties WOULD be required to know how the loads are distributed to the various flanges. What I was wondering was, is the solution shown for three flange beams, without regard to neutral axis, an "exact" solution for that type of beam structure? Again, it seems to me that a more accurate (and safe) solution would look for the neutral axis of the flanges and to distribute bending loads to the flanges via the flexure formula fb=Mc/I....or M/S. Peery is the only one who specifically states (when he introduces the concept of this type of beam) that the flexure formula and knowledge of the flange areas is NOT REQUIRED. He doesn't clearly say why this is true though. Why this type of beam can be analyzed differently from OTHER statically determinate beams. He seems to say that this is BECAUSE the three flange beam has no redundant members and is statically determinate. But normal I beams are also statically determinate (depending on how they are supported) and stresses are found for them with the flexure formula. He says that because three flange beams are statically determinate, all forces can be found by knowing that the sum of all moments and forces about the three axes is equal to zero. I don't know why this means the neutral axis isn't relevant as seems to be implied. I also don't know why this means we don't need to know the flange areas (because that would determine the location of the neutral axis and that would determine how much load is carried by each flange). The flexure formula is used when a beam has more than three flanges and is therefore statically indeterminate. In a wing design I would want to know what the true forces are. Not to design to the gnat's ass but with safety in mind.

Jeez, You have me thinking now. All beams are pretty much analyzed the same way, but I do have to admit that I do not recognize the term three-flange box beam. Originally, I had interpreted it to mean a D-tube with three primary elements: A shear web at about the 25% chord point with upper and lower caps that extend to form most of the forward part of the airfoil. Another way to do this is to place two smaller webs at say 10% and 50% with caps between the webs, which makes for four elements... I could see how that would be called a three-flange box: One as a shear web, one on the top as a cap, and one on the bottom as a cap. If this is the case, yeah, you compute the centroid and fix the neutral axis, then Ixx and shear web area, apply moments and shears, calculate stresses and von Mises stresses, and check for failures... And yes, the nuetral axis should be computed and used to compute the second moments and torsional moment, and then stresses. Please clear up my knowledge on what you are calling a three flange box. And in any case, I am with you - I do not see how the neutral axes can go anywhere but through the centroid... All that I can figure (without knowing the derivations used) is that they neglected the neutral axis location and the thickness effects of components figuring that the effects are small. Billski

I'll be brief now. May add more later. The three flange box beam described IS a d-cell structure. It has three flanges to take all bending loads (the thin skin is usually neglected, I guess but it is an issue of whether it buckles and a thick skin would tend to have permanent wrinkles). The third flange makes the beam stable against fore and aft loads. The other two flanges occur near the top and bottom of the rear, vertical shear web. These flanges resist vertical loads in bending. The flanges take bending loads and the three shear webs (two of them curved) take shear and torsional loads. BDD

A part of the question you ask above sort of caught my attention - this is regarding whether this is an "exact solution". Assuming you mean whether the presented process will give you a precise answer, the answer is not really, at least, not in the first go around. However, based on your questions and comments, I maybe should assume you probably know that. As I said above, many of the techniques presented within the two classic reference texts were structured in such a way as to be able to be utilized using only the simplest of analysis and calculation means. Due to this approach, a bit of accuracy is lost. It can be regained by going through the problem in several iterations, each time positioning the local neutral axis of the spar cap (flange) closer and closer to where it will eventually end up. But keep in mind that these excercises are designed for the student of the time and for the purpose of presenting the overall process of the design discipline. The books try to simplify the approach to the analysis of these type of structures. But one of the nice aspects of this approach is that it demonstrates that the wing structure is in a sense additive - in other words, it shows that the different aspects of the structure are each designed to take care of a particualr aspect of the anticipated loading and each can be analyzed and/or designed almost seperately ie. the spar web and caps can be designed for the span loads to resist bending and the skin and ribs can be designed to take care of the torsional loads of the airfoil section and of the control deflections. No, this apporach is not precise since in its pure form it really does not take into account that the skin provides some resistance to bending nor does it take into account the contribution to torsional resistance offered by a fixed aft spar, as well as a couple of other aspects of the design process we now can analyze for very simply using tools like FEA modeling. Working with the specific cross sectional properties is of course more accurate and if you know how you want your spar to look (what the material makeup is), then that particular approach makes much more sense. Personally, I of course prefer this methodology since it better matches my visulasation of any structure. In a sense, this is a good example of the difference of approaches to structural design as presented by aero engineers and disciplines like mechanical engineering. When I was in school and took my first aero structural engineering course, one the fist things that the text introduced was a tensor. Not having had the advanced linear algebra course behind my belt at that point in time, I could only answer with the ever intelligent comment "Huh???" And so, in the first week of the quarter, I was already half a year behind - so much fun! When I took my mechanical engineeing structural courses, those texts took a bit more practical approach - the first thing they introduced was an "I" beam. Now that I can relate to! And although I did OK in the aero structural disciplines, it wasn't until I dug deep into the structures as presented in the mechanical discipline did I start to get an understanding of what the aero work was trying to accomplish. In a sense therefore, I can understand your comments when questioning the approaches taken by Bruhn and Peery, and why you are looking to go directly to analyzing physical structures with fixed properties, rather than the more ethereal (and maybe a hair less precise) approach taken by the aero structures texts. But both have their benefits and both approaches are acceptable, delivering in the end, very similar results.

By "exact" I was meaning is the approach shown a preliminary one or are the results accurate enough for final design? No where is the approach shown (ignoring neutral axis, etc.) called preliminary.

The techniques used in the texts were not considered preliminary although it was assumed that the designer would do the calculations in a repeated sequence and optimize each one as more information became available. However do keep in mind that both texts were designed for students. It is unrealistic that one can read all this and be able to go out and design an entire airplane without having any real and practical expereince doing this behind his belt. This takes time and experience and requires that each designer/engineer becomes a part of a team where those with longer practical experience and backgrounds will teach the new designer(s) the so called "tricks of the trade" and pass on the practicalities of the design process. In my opinion, one of the more potentially dangerous trends today is the series of "cookbook" texts being offered to our industry that claim that by following a few simple procedures and formulas, someone can go out and design a flight structure. The really questionable ones propose that by following their procedures even a non-engineer can do this with little ristk. In my experience, nothing is further from the truth. Now, that is not to say that any engineer can do all this safely. But having that background I think is very necessary since that bit of education does enable one to better evaluate 1) what he or she does not know and 2) what he or she needs to know to accomplish a certain task. Aviation is one arena where ignorance is not bliss.