**Composite Beam Theory**

We have had people thinking about composite beams and talking about advantageous or non-advantageous schemes for building beams or spars. This is intended to help folks get a little closer to what really happens in a composite beam under load and how to analyze it.

Prerequisites

Prerequisites

If you do not have a handle on Beam Theory, then you should do that first. Then revisit Matrix Algebra, also called Linear Algebra. Then get into this.

Ground Rules and Assumptions

Ground Rules and Assumptions

Let’s talk about what is the same. For a wing or tailplane or control surface or a fuselage or… you still have to figure out the loading, find shear and bending moment diagrams, and apply them to the structure. It is still wise, once you have loading and strains in the parts, to check for buckling and crippling and local strength around hard points, and so on, and to make sure that you have conceived a structure that you can build and maintain.

- So, like I did in the Beam Theory article, we need some ground rules and assumptions:
- We will be working in what is called composite plate theory;
- Plate theory is where we imagine a plate that has thickness in the z axis and goes off in the x and y directions, and we analyze an element of that plate that is small in the x and y directions;
- Plate theory can be easily made into beam theory, and we will convert our plate theory into a composite beam or composite wing at the end;
- The plate has a long axis in each of the x and y axis directions that is straight when the plate is unloaded;
- These long axes run through the neutral axis in each direction of the plate;
- The external loads on any one plate element are comprised of three linear loads and three moments. The linear loads are tensile or compressive along the x axis, tensile or compressive along the y axis, and shear from loads in the z axis. Moments are bending in the x axis (the plate curves visibly in x-z plane), bending in the y axis (the plate curves visibly in the y-z plane), and torsion ( two corners opposite each other go down, while the other two corners go up, twisting the plate element);
- The plate element is a laminate made up of a number of separate lamina;
- Each lamina may have its own stiffness and strengths, which may be different from all of the other lamina;
- The entire plate curves and deflects together when loaded;
- The plate thickness is small compared to the other dimensions;
- The materials all obey Hooke’s law and have a modulus in each direction that relates deflection and load.

**Basic of Composites**

Composites as we use them in airplanes consist of strong, stiff fibers that are embedded in a matrix of some sort of resin or plastic. The fibers in each lamina are generally in layers called lamina and are either all in one direction or are in some fixed proportions in two or more directions. The first is called a unidirectional lamina, while the second is usually made from a woven or knitted cloth of some sort. One of the neat things about composites is that they can be tailored to give more strength or stiffness in one direction than in other directions.

Homogenous materials such as metals are modeled as being the same stiffness and strength in all directions, and that allows us to go pretty directly from loads through cross section characteristics then to stresses. One little bit of area is the same as all others. Since the material has different strength and stiffness with different fibers, with different directions of the fibers, and in different places within your parts, we must do our analysis differently. We must figure out the stiffness contributions of each lamina separately and combine them, then find the strains to check if the material will fail. The stiffnesses and strengths of each lamina must either be known or generated from other known lamina.

The basic way to analyze composites is to use plate theory, which works really well and can be applied with to wings and fusealges. A more complicated part, such as a beam or wing is usually a laminate made by building up a succession of lamina until the shape we want is present and sufficient strength and stiffness has been obtained for the duty of the part. Likewise to the process of making the part, we shall have to build up a stiffness matrix by obtaining the stiffness matrix for each lamina and then build up the stiffness matrix for the laminate. Plate theory also makes a rather complete check for failure possible.

Plates are made up by stacking up any number of different lamina. In each lamina, several things can be unique: Lamina thickness, fiber type, and fiber direction. There can also be low density cores or other materials. All materials will move together, and strengths must stand the deformation applied to the laminate. In many cored plates, cores are omitted at the edges and sometimes in the interior of the plate as well. This glass-to-glass (graphite-to-graphite?) bond is used to carry shear loads that might be too high for low density cores. Plates are generally thin, on the order of an inch and less. In little airplanes, plates form fuselage and wing skins, and are usually on the order of 1” and less.

Spars and Beams are simple extensions of plates, in that you will still have any number of lamina with varying thickness, fiber type, and fiber direction but now instead of simply stacking lamina that extend off a ways, the lamina have finite width and can take on any shape and position. You can build up a spar with caps, a web bridging the outside of the caps, another bridging the inside of the caps, foam cores, etc, than analyze it as one structure

To do an entire wing, you add in more general lamina for the drag spar and the wing skins. Do not put in ribs, they are sparsely used in composite wings and do not actually participate in carrying shear, bending and torsion.

In all cases, we are assuming that the whole plate/beam/wing bends, shears, and twists as a unit. If for some reason your structure has the various parts disconnected from each other such that they do not move together under load, the standard methods will under-represent (perhaps greatly) deflections, stresses and the likelihood of failure. The pieces of the structure that move separately must be analyzed separately as to stiffness and then as to load distributions, separately for deflections and then for failure checks. These two options are the only ones that exist – either the parts are connected and move together or some parts move separately and are analyzed as separate parts.

In the rest of this paper, I will talk about matrices and vectors and about concepts in use of composite materials. Try not to get hung up on exactly how the matrix algebra is done unless you intend to actually do the analysis yourself. You do need to know that we have to analyze the structure using all of its stiffness terms, all of it load terms, and all of its deflection or strain terms. Skipping terms can create significant errors and make your calculations much less reliable. Try to get through so you can get the concepts I am trying to help you with.

Analysis Approach for Plates, Spars, and Wings

Analysis Approach for Plates, Spars, and Wings

How different are various lamina? The strongest unidirectional graphite lamina are more than ten times stronger than the weakest square weave glass cloth at =/-45 degrees. Unidirectional lamina in the direction of the fibers can be 20 to 40 times as strong as the strength across the width of the fibers. These methods are needed to have a chance at getting things right.

Now for a summary of the analytical process: Do not worry; no actual linear algebra is needed to get through this. The fiber type determines how stiff and how strong the individual lamina is. The fiber orientation, weave, etc further modifies the stiffness and strength. Lamina are layered into a bigger laminate. Start by determining the axial and bending stiffnesses for each lamina, then combine them for the laminate, define the load cases for the laminate, find the deflections of the laminate for each load case, decompose strains back to lamina strains, and then check failure criteria for each lamina. Sounds a little more involved than with metal, does it not? Inevitably, the initial design will need iterating to weight where you can or to preclude failures. That is a whole bunch of things that all have to be done right in a sequence. Let me tell you, Excel or MatLab is your friend when doing this.

What I am summarizing for you is the high points of a senior year/graduate level ME/AE course that is taught over a full semester. The course started with the prerequisite that you have Senior standing in engineering school or greater. That implies knowledge of calculus, linear algebra, statics, mechanics of materials (beam theory and failure criteria of homogenous solids), but not plate theory. I learned it using Tsai and Hahn’s book

__Introduction to Composite Materials__. Jones’ book is good too, and has a slightly more complete failure criteria discussion.

Material Elastic Characteristics

Material Elastic Characteristics

Fibers of high strength and stiffness are what make most laminated composites have advantages over homogenous materials. These fibers are in a matrix of some resin, typically epoxy, and the two materials move under load together. Loads are applied to the bulk of the product, which resolve as stresses and deflections all way down to the fiber and matrix level of the part. In any unidirectional ply or lamina, there will be one modulus for tension and compression in the direction of the fibers, another for tension and compression across the direction of the fibers, and one for shear of the lamina. These moduli are similar to Young’s Modulus and Shear Modulus in homogeneous materials, relate stress and strain in the materials, and are Ex, Ey, and G.

When we apply tension along the fibers (the x direction), we are making the lamina element a little longer in that direction, the length in the x direction increases and the length across the fibers (the y direction) decreases. Poisson’s Ratio is involved in being able to calculate this, similar to homogenous materials. Wouldn’t the element also get thinner in the z direction? Yeah, except that we made that little assumption that the thickness is small compared to the other part dimensions, so thickness change is assumed to be pretty small. The math gets much more complicated and not enough more accurate to bother when we include that third direction in dimension changes, so it is routinely skipped in composite work.

When we apply shear, other things happen. We need to describe shear so that it makes some sense. If we have a cube, and we pull down on the left face and push up on the right face, and then apply a force to the left on the bottom face and a force to the right on the top face, and all four of these forces are the same magnitude, the little element is in pure shear. Yes, all four are there when you apply vertical forces to a beam or plate or pin or…

For our work here, we can have force extending or compressing the element in two directions perpendicular to each other and we can have shear of the element. What if the world provides loads that are not perpendicular to each other? When you learned statics you learned how to resolve in-plane loads into other in-plane loads that are more convenient for whatever it is you want to do. Like turning any combination of loads and directions into two loads perpendicular to each other, plus shear…

Further on pure shear. If you rotate that little element 45 degrees, the edge shears transform into pure compression one way and pure tension on the other. The tool for doing all this is Mohr’s Circle – very useful tool taught in Mechanics of Materials courses. Since we are glueing our fibers to each other and maybe to some core, the fiber sets in both compression and in tension will work to resist this shear load, and the deformation that our little element undergoes will be a product of both the stiffness of the fiber in compression and the fiber in tension.

If you have vertical and horizontal fibers and put the lamina in shear, you are resisting shear by bending individual fibers and distorting the resin in shear. Not the best, but it can work by putting more plies.

It turns out that the ideal orientation (lightest way to make strength) of composites for pure shear is fibers at plus and minus 45 degrees. Convenient… One more point – whether built of woven cloth or lamina of unidirectional cloth, the fiber layers in each direction are glued to the fiber layers running the other way, and thus support each other up to first fiber failure stresses of the laminate.

Interestingly, the typical failures include buckling of the fibers in compression and parting of fibers in tension, with buckling being much more common. Note - these are NOT strings in space, with only the tensioned strings carrying load. Fibers in compression and in tension are both resisting shear deformation right up to the load where fibers start to buckle or crack.

Individual E’s are not so useful by themselves, because loads frequently get applied from more than one direction at a time, which changes things. Well, once you do some math involving Poisson’s Ratio of your material, you transform the E’s into Q’s. This comes right out of conventional plate and shell theory, and has been established for about a century. Now we can have three equations that express how stresses in x, y, and shear directions resolve into strains in the x y, and z directions, if you solve them simultaneously. Yep, matrix algebra starts here. {stress} = [Q]*{e} is the matrix equation sort of like stress = Q*e, except that {stress} and {e} are 1x3 vectors and [Q] is a 3x3 stiffness matrix. Relax, I won’t get into the detail of how to make the matrices here, but you have to know they are out there in the books, and your results are not reliable if you do not work with them.

Effects of Fiber Orientation

Effects of Fiber Orientation

Now we rarely build anything with all of the fibers of the same type and aligned in the same direction. We rotate the fiber orientation for any particular lamina to a new angle and transform the [Q] from the x and y axes to the 1 and 2 axes. To do this, we apply trigonometry as another 3x3 matrix, multiply it by the basic [Q] and we get a transformed [Q] matrix that now represents the stiffness of a lamina with the fibers at an angle to the axes of the part. It is still a 3x3 matrix, but now it is applicable to a lamina with angled fibers. {stress} = [Q]*{e} still applies, but now the axes are aligned to some reference line we put on the part, our [Q] has had all of the terms changed, and we must account for this or our results will not be reliable.

Material Strength Characteristics

Material Strength Characteristics

If any single lamina fails, the remaining lamina carry the load, deformation increases, more lamina are in jeopardy of failure, and complete failure of the laminate is likely. At some point in our analysis we will want to know if lamina failure will occur under each load case. To do this, we start with the tensile and compressive strength of the lamina in the x direction, in the y direction, and in shear across the x and y directions. Like the stiffness terms, we do a transform to get strength in strain terms to make them easily used in checking failure criteria.

One thing to know, many composites have different strengths in compression and in tension, and they can be particularly large differences across the fibers. The failure envelope in composites is similar in concept to the von Mises criteria in metals, but can be highly asymmetric, allowing very large stresses when both x and y directions are in compression and somewhat modest stresses when loads in both x and y directions are positive. This is why we cannot simply check stress, but must check failure criteria.

Loading, Shear and Moments

Loading, Shear and Moments

If we have a wing structure with a beam or two in it and skins around it, we can carry lift and pitching moment distributed along the wing, and these result in a distribution of shear, bending moment, and torsional moment applied to the wing. The calculation of these loads is arrived at the same way as for conventional beam theory. At any span wise position along the wing, we have shear and moments, F6, M1, and M6 in a 1x6 vector we call {FM}. The other three positions (F1, F2, and M2) could also be present, but in cantilever wings, they are usually quite small and simply called zero. This load vector will be used further on.

Stiffness Matrix

Stiffness Matrix

So far, we know that we can apply loads along the plane of the lamina and compute strains in the lamina - {stress} = [Q]*{e}, but this is not useful if we layer up lamina into a laminate and if we apply those moments mentioned above. So let’s get into layers and moments.

First thing to know is that we can formulate {F} = t*[Q]*{e} for each lamina where t is cross sectional thickness of the lamina. Let’s get our terms right. {F} is a 1x3, with axial force per unit length in the 1,2, and 6 directions, t is just the lamina thickness, [Q] is still the 3x3 transformed axial and shear stiffness matrix for this lamina, and {e} is still strain in the 1,2, and 6 directions.

Next thing to know is that t*[Q] is called the [A] matrix. You take the [A] matrices for each of the lamina and simply add the terms at each position to get the terms for the laminate [A] matrix.

What about bending and torsional moments? Well, in plain beams, EI represents bending stiffness and GJ is torsional stiffness. E and G are axial and torsional moduli of the materials. I is a cross section characteristic that relates to bending resistance computed by taking the integral of da*x^2. J is similar but it relates to torsional stiffness, and is computed by integrating da*r^2. Do the same thing mathematically and multiply it times [Q] and obtain [D].

We are beginning to get there. One other thing goes on with composites. A single lamina with fibers angled one way, when pulled on not only extends but shifts to the side because it is softer one way than the other. Take two pieces of this lamina, one with the angle going one way, and the other with the angle going the other way, and bond the two together. When you pull on it, it twists… There are several forms of this axial-bending coupling. There is some more dedicated math to multiply by [Q] and this makes up what is called the

**matrix. If we make the build of off-axis laminate symmetric about the neutral axis, the B terms become zeros. Unless you want axial-bending coupling, symmetric laminates are wise and simplify the math.**

Now we have A, B, and D terms that we assemble into the [ABBD] matrix, first for the lamina, then sum them up to make the [ABBD] matrix for the laminate. This is a 6x6 matrix and we call it the stiffness matrix.

Back to the loads. We have had an axial and shear stress vector {stress} and then an axial and shear force vector {F}. But we also have bending moments in x and y directions and a torsional moment. Well, that gets assembled into a 1x6 vector called {FM}.

If we have a 1x6 load vector, shouldn’t we also end up with a strain vector? You bet, it is a 1x6 called {ek} where e’s are axial strains, k’s are curvatures. Now, unless some deflection of the element is prevented or imposed, we usually have {FM} and [ABBD], and we solve for {ek}. You could do matrix inversion to find [ABBDinv], or you could just solve for {ek} by using Gaussian elimination, which is easily performed in Excel or MatLab.

We are getting there, we know how much the element extends or contracts, shears, bends, and twists, which are useful for computing the deformed shape of the laminate. But if we are interested in lamina failures, what we need is how much strain is in each lamina… If we just take each of the axial strains and add in the curvature in that axis times the distance from neutral axis to the lamina, we will have strains for each lamina in x, y and shear directions.

While we are dealing in only plates (beams come later), we have bending and shear developed from the applied loads. Most composite plate design is started by educated guesses, then iterating the design. There are other ways:

Now, remember those strength terms worked out before? Well, they were transformed into strain space, and a failure criteria equation is written that uses the strains and the failure terms (in strain space). If the sum of all the terms is equal to or greater than unity, the lamina failed. If the sum is less than unity, the lamina is fine. Check all of the lamina. Again, Excel is your friend.

Once you have run the analysis and have the failure criteria for all of the lamina, you can see which lamina are close to trouble and which have a lot of margin. You can make adjustments to the lamina, add or remove lamina, change materials, change lamina thickness, change fiber orientation. Serious optimization may require some application of Designed Experiments and Response Surface techniques.

First thing that happens when you move to beams is the plate lamina become narrower, some much narrower. Do not worry; you get it back with wing skins later. Next thing is that instead of t, you’re [A] matrix uses cross section area of the various lamina,

Now we have A, B, and D terms that we assemble into the [ABBD] matrix, first for the lamina, then sum them up to make the [ABBD] matrix for the laminate. This is a 6x6 matrix and we call it the stiffness matrix.

Load VectorLoad Vector

Back to the loads. We have had an axial and shear stress vector {stress} and then an axial and shear force vector {F}. But we also have bending moments in x and y directions and a torsional moment. Well, that gets assembled into a 1x6 vector called {FM}.

Strain VectorStrain Vector

If we have a 1x6 load vector, shouldn’t we also end up with a strain vector? You bet, it is a 1x6 called {ek} where e’s are axial strains, k’s are curvatures. Now, unless some deflection of the element is prevented or imposed, we usually have {FM} and [ABBD], and we solve for {ek}. You could do matrix inversion to find [ABBDinv], or you could just solve for {ek} by using Gaussian elimination, which is easily performed in Excel or MatLab.

Lamina StrainsLamina Strains

We are getting there, we know how much the element extends or contracts, shears, bends, and twists, which are useful for computing the deformed shape of the laminate. But if we are interested in lamina failures, what we need is how much strain is in each lamina… If we just take each of the axial strains and add in the curvature in that axis times the distance from neutral axis to the lamina, we will have strains for each lamina in x, y and shear directions.

Sizing of PlateSizing of Plate

While we are dealing in only plates (beams come later), we have bending and shear developed from the applied loads. Most composite plate design is started by educated guesses, then iterating the design. There are other ways:

- You can make a first estimate of how thick your laminate will be, then selecting the biggest M from the {M} vector, back calculating how much of the applicable D term is needed for the top and bottom lamina, then back calculate the lamina area and z dimension. Then pick up the next biggest M, and repeat for the 2
^{nd}lamina in from top and bottom. We include the effects of the 1^{st}lamina in the 2 direction in this look at the 2^{nd}largest M. Usually this does the job for a starting point, and you then can iterate the ply count in each direction, the core thickness, etc; - If you are clever, you can configure the fiber orientations or the ratios of plies in each orientation such that D11 and D22 of the laminate have a ratio similar to the ratios of M1 and M2, and then adjust lamina and core thicknesses to make strength.

Failure Criteria CheckFailure Criteria Check

Now, remember those strength terms worked out before? Well, they were transformed into strain space, and a failure criteria equation is written that uses the strains and the failure terms (in strain space). If the sum of all the terms is equal to or greater than unity, the lamina failed. If the sum is less than unity, the lamina is fine. Check all of the lamina. Again, Excel is your friend.

Design IterationDesign Iteration

Once you have run the analysis and have the failure criteria for all of the lamina, you can see which lamina are close to trouble and which have a lot of margin. You can make adjustments to the lamina, add or remove lamina, change materials, change lamina thickness, change fiber orientation. Serious optimization may require some application of Designed Experiments and Response Surface techniques.

Converting from Plates to BeamsConverting from Plates to Beams

First thing that happens when you move to beams is the plate lamina become narrower, some much narrower. Do not worry; you get it back with wing skins later. Next thing is that instead of t, you’re [A] matrix uses cross section area of the various lamina,

**and [D] use moment of inertia terms. And the {FM} vector goes from forces and moments per linear length to the edge and just becomes force and moments. Plates are generally not really thick, a half inch would be a lot, but our spars in airplanes will be several inches thick. The last thing is that you generally get shear webs of width much less than the caps, and the webs carry the bulk of the shear loads. Design iteration changes too.**

Now a few things to know about building beams in composites:

Foam – how do we apply it in the [ABBD]? Usually we do not. Its unit stiffnesses are pretty small, and the applied area is usually pretty modest, so it is routinely ignored in the analysis. If we have a lot of it, (massive foam cores per Rutan et al) you can find its E’s and Q’s and area and moments of inertia and include it. What about failure criteria? Typical foam deformation at failure is several times that of the composites that encase it and move with it. A couple of cautions:

So how do we do initial sizing of composite beams? Well, pretty much like we do in Beam Theory. And that initial sizing has similar shortcomings when you actually make a part and load it up.

The caps are usually made of fibers running in the spar x direction, and as far up and down in the space for the spar as you can. Get an initial estimate of the distance between the caps, divide the bending moment by the distance, and you have the first guess at the force in each cap. Divide the force by the strength of the material, and you have a first estimate of the area required in that cap.

Shear webs are usually made of square woven cloth and applied at +/-45 degrees, wrapping onto the caps to help transfer load from cap to cap. But the amount of shear web that carries the beam’s shear load is only the part between the two caps. All the rest contributes to bending and torsional stiffness and serves to help attach the parts into a unitized structure. Take the shear load and divide it by the shear strength of the web material, and you have a first estimate of shear web area needed between the caps.

Design guidance here: You can usually make a lighter spar if the caps are as wide as practical and thus as thin as you can make them and not have them buckle or cripple. Then the web can be a little thinner for the same area which makes for a little less total weight.

If your beam is carrying much torsion, all by itself, it will need significant width. Most wings do not… If your channel or wide flange beam does not become adequate easily, consider a hollow box of foam as the core, with big width for the core and flanges, and fibers at + and – 45 degrees all the way around. It works great for LongEz’s and their derivatives.

Wings analyze just like beams, because they are beams, except that your beam now includes the drag spar and the skin.

First thing to know is that you size the main spar to carry shear and bending, the drag spar reacts only the moments, and the skin can be initially sized to carry the pitching moment. That will do for your starting point. In reality, everything is at least a little involved in carrying everything, but that comes out when you do the analysis.

The I and J of the skin laminations need to be found relative to the centroid of the wing relative to M1. Yes you use the whole wing. The spars are straightforward, but the wing skins can be a bit of a handful. If you are working a constant chord wing, you can set it up once, get the centroid of the skin, its area and moments of inertia. But if we are working in composites, why accept a Hershey Bar wing? We can taper it nicely… To do the equivalents of EA and EI and GJ at every station could be a pain in the butt. So, what I do is run my skin math at both ends of the wing, and at 1/3 and 2/3 between the ends, then curve fit the outputs. So now you can have an equation for each of chord, spar depth, [ABBD] elements, and {FM} based upon the wing station, which makes it easy to fill out for each station. All you have to do is drag down the station specific numbers including basic sizing, look at the failure criteria, and iterate the number of plies in each of the lamina, and find the lightest section for each station.

Just like in homogenous material beams, the webs and caps from the main spar are both likely to need iterating, but not in an intuitive manner. What is likely is that the shear web will be seriously under strength, just like in homogenous beams. The reason is that near the caps, the shear web extends or contracts with the caps while also carrying shear, but we only sized it for shear.

Since the web is what failed, won’t we just thicken the web until all lamina pass? Well, you can, but the cross section weight goes way up by the time you get to strength. So I propose that you bump both cap laminas by several plies, and adjust the web to just pass, then note the ply counts and weight. Bump the cap ply count again, reset the web again, noting ply counts and weights. The cross section weight will go down for a while and eventually start back up. Refine your search to find the number of plies that pass at minimum weight. This optimization usually results in a significant thickening of both caps and web, but is the min weight spar section for that load case and station.

Something interesting happens part way out the wing: The bending moment drops below a couple plies of spar cap material. I do not know about you, but I cannot conscience going thinner than 2 plies of UNI tape in my caps, so that is where I stop tailoring the cap thickness. Same thing happens further out on the shear webs, and I stop tailoring at 4 UNI in the webs. What I do is I start narrowing the flanges and the core in the web at the next station out after I stop thinning my caps.

Once you have optimized the spar at all sections, you can build your schedule of cap and web plies for building your spars.

One important thing to remember about long slender spars: They do not carry wing pitching moments very well all by themselves unless you make them pretty wide. We already know this: Rag and tube wings have struts supporting each spar while wings with structural skins are either cantilevered or supported by a single strut. Those wings with structural skins are really large torque tubes, and the spars carry little torsion with a structural skin present.

Do yourself a favor if you are designing fabric covered wings, design structures other than the spars to react off pitching moments, make M66 zero for the spars in that case.

Usually, the drag spars are pretty small in terms of [ABBD] contributions, but the skin and maybe a massive foam core can get significant. If you really want to search out min weight, you might want to look into extra plies of unidirectional material in the skins near the point of max section thickness, and play the number of these plies against the number of plies in the spar caps. Unidirectional lamina in the skin can have a fair amount of area at larger distance from the neutral axis than even the spar caps, and may result in a lighter overall structure. Most of us won’t bother, as we go glass-to-glass in the skins at the main spar, so the system is already pretty darned good.

The Rutan Long Ez and its derivatives use three plies of 7 oz UNI cloth, one each at +45. -45, and 0 degrees. Yep, some for torsion, some for bending. I vacuum bagged my skins using 22 oz TRIAX cloth on the outside. The stuff I get has 50% of its weight at 0 degrees, and the other 50% at + and – 45 degrees. Something to think about to help get bending and torsional stiffness.

With structural skin wings, the whole wing is what should be analyzed over most of the span, and pitching moment (M66) will be the accumulated pitching moment from the tip to the wing station you are analyzing. When you get close to mounting points, try to remember that the main spar reacts almost all of the lift and bending moment, the drag spar reacts pitching moment, and the skins carry nothing to the next structure. Adjust your [ABBD] so that the skin is contribution goes near zero as you approach the mounting station.

If you are really chasing the lightest structure, go for all graphite, using Graphlite rod caps, graphite cloth for the shear web, and start the skins with two plies of 5-6 oz graphite cloth on each side of ¼” foam cores for the wing skins. Vacuum bag all parts, keep glue lines thin, and you will have close to the lightest wings possible. Do a good job of optimizing the spars, and it will be hell for strong, monster stiff, and light!

So, now you know the basic path through figuring out where to start with composite parts, how to bring them up to strength, why the designs out there look the way that they do. If you already have the training equivalent to undergraduate engineering coursework I described near the beginning, you could buy Jones or Tsai and Hahn, get into the details, write a program or three in Excel or MatLab, and go to town designing wings. Or you could buy someone else’s design and be able to appreciate the work they did to arrive at their design, whether it be light and efficient, or heavy and clumsy.

BillskiNow a few things to know about building beams in composites:

- The very lightest way to carry bending loads is with the strongest material placed as far from the neutral axis as you can. This makes the I needed achievable at relatively small cross section area and weight. And your strongest and stiffest form of any fiber is a unidirectional lamina with the fibers running the long way in the beam. So your lightest beams will have bending carried by unidirectional lamina in wide spar caps at the top and bottom of a spar that fully fills the vertical space we have available;
- Shear is best carried by a web of material with fibers at plus and minus 45 degrees. The web has to not only carry shear but hold the caps very firmly in position and be able to transfer loads between caps and web, so the web material usually wraps onto the caps or between elements of the caps. In channels and wide flange beams, much of the shear web is between the caps, with a few plies around the outside. In foam box section spars, the shear web wraps around the outside;
- All lamina in the spar contributes to axial and bending terms except the shear term. Why? In a beam with a wide set of flanges and a relatively thin web, only the web area between the caps is carrying much shear, and so we only tally up the part of the shear web between caps for the stiffness term applicable to direct shear, A66.
- Web buckling - We rarely need webs that are thick, except that thin webs are subject to buckling/crippling unless we do something about it. Two paths are taken: we can apply stiffeners to the webs or we can build the web on both sides of foam cores. Both work, but cores are easy and light, particularly if you laminate the web using vacuum bag techniques. Some of us transition cores to laminated plywood at hardpoints to take fastener tightening loads.

Foam – how do we apply it in the [ABBD]? Usually we do not. Its unit stiffnesses are pretty small, and the applied area is usually pretty modest, so it is routinely ignored in the analysis. If we have a lot of it, (massive foam cores per Rutan et al) you can find its E’s and Q’s and area and moments of inertia and include it. What about failure criteria? Typical foam deformation at failure is several times that of the composites that encase it and move with it. A couple of cautions:

- Avoid foams that have low deformation at failure or known tendencies to go to powder under long term use. Some bad choices do exist out there;
- If the design has only foam between the spar caps, the top and bottom skins should connect with each other and make a more-than-adequate shear web by calculation. In most cases the foam all by itself is not an adequate shear web.

Sizing of Spar Caps and Shear WebsSizing of Spar Caps and Shear Webs

So how do we do initial sizing of composite beams? Well, pretty much like we do in Beam Theory. And that initial sizing has similar shortcomings when you actually make a part and load it up.

The caps are usually made of fibers running in the spar x direction, and as far up and down in the space for the spar as you can. Get an initial estimate of the distance between the caps, divide the bending moment by the distance, and you have the first guess at the force in each cap. Divide the force by the strength of the material, and you have a first estimate of the area required in that cap.

Shear webs are usually made of square woven cloth and applied at +/-45 degrees, wrapping onto the caps to help transfer load from cap to cap. But the amount of shear web that carries the beam’s shear load is only the part between the two caps. All the rest contributes to bending and torsional stiffness and serves to help attach the parts into a unitized structure. Take the shear load and divide it by the shear strength of the web material, and you have a first estimate of shear web area needed between the caps.

Design guidance here: You can usually make a lighter spar if the caps are as wide as practical and thus as thin as you can make them and not have them buckle or cripple. Then the web can be a little thinner for the same area which makes for a little less total weight.

If your beam is carrying much torsion, all by itself, it will need significant width. Most wings do not… If your channel or wide flange beam does not become adequate easily, consider a hollow box of foam as the core, with big width for the core and flanges, and fibers at + and – 45 degrees all the way around. It works great for LongEz’s and their derivatives.

Analyzing WingsAnalyzing Wings

Wings analyze just like beams, because they are beams, except that your beam now includes the drag spar and the skin.

First thing to know is that you size the main spar to carry shear and bending, the drag spar reacts only the moments, and the skin can be initially sized to carry the pitching moment. That will do for your starting point. In reality, everything is at least a little involved in carrying everything, but that comes out when you do the analysis.

The I and J of the skin laminations need to be found relative to the centroid of the wing relative to M1. Yes you use the whole wing. The spars are straightforward, but the wing skins can be a bit of a handful. If you are working a constant chord wing, you can set it up once, get the centroid of the skin, its area and moments of inertia. But if we are working in composites, why accept a Hershey Bar wing? We can taper it nicely… To do the equivalents of EA and EI and GJ at every station could be a pain in the butt. So, what I do is run my skin math at both ends of the wing, and at 1/3 and 2/3 between the ends, then curve fit the outputs. So now you can have an equation for each of chord, spar depth, [ABBD] elements, and {FM} based upon the wing station, which makes it easy to fill out for each station. All you have to do is drag down the station specific numbers including basic sizing, look at the failure criteria, and iterate the number of plies in each of the lamina, and find the lightest section for each station.

Iterating BeamsIterating Beams

Just like in homogenous material beams, the webs and caps from the main spar are both likely to need iterating, but not in an intuitive manner. What is likely is that the shear web will be seriously under strength, just like in homogenous beams. The reason is that near the caps, the shear web extends or contracts with the caps while also carrying shear, but we only sized it for shear.

Since the web is what failed, won’t we just thicken the web until all lamina pass? Well, you can, but the cross section weight goes way up by the time you get to strength. So I propose that you bump both cap laminas by several plies, and adjust the web to just pass, then note the ply counts and weight. Bump the cap ply count again, reset the web again, noting ply counts and weights. The cross section weight will go down for a while and eventually start back up. Refine your search to find the number of plies that pass at minimum weight. This optimization usually results in a significant thickening of both caps and web, but is the min weight spar section for that load case and station.

Something interesting happens part way out the wing: The bending moment drops below a couple plies of spar cap material. I do not know about you, but I cannot conscience going thinner than 2 plies of UNI tape in my caps, so that is where I stop tailoring the cap thickness. Same thing happens further out on the shear webs, and I stop tailoring at 4 UNI in the webs. What I do is I start narrowing the flanges and the core in the web at the next station out after I stop thinning my caps.

Once you have optimized the spar at all sections, you can build your schedule of cap and web plies for building your spars.

One important thing to remember about long slender spars: They do not carry wing pitching moments very well all by themselves unless you make them pretty wide. We already know this: Rag and tube wings have struts supporting each spar while wings with structural skins are either cantilevered or supported by a single strut. Those wings with structural skins are really large torque tubes, and the spars carry little torsion with a structural skin present.

Do yourself a favor if you are designing fabric covered wings, design structures other than the spars to react off pitching moments, make M66 zero for the spars in that case.

Iterating WingsIterating Wings

Usually, the drag spars are pretty small in terms of [ABBD] contributions, but the skin and maybe a massive foam core can get significant. If you really want to search out min weight, you might want to look into extra plies of unidirectional material in the skins near the point of max section thickness, and play the number of these plies against the number of plies in the spar caps. Unidirectional lamina in the skin can have a fair amount of area at larger distance from the neutral axis than even the spar caps, and may result in a lighter overall structure. Most of us won’t bother, as we go glass-to-glass in the skins at the main spar, so the system is already pretty darned good.

The Rutan Long Ez and its derivatives use three plies of 7 oz UNI cloth, one each at +45. -45, and 0 degrees. Yep, some for torsion, some for bending. I vacuum bagged my skins using 22 oz TRIAX cloth on the outside. The stuff I get has 50% of its weight at 0 degrees, and the other 50% at + and – 45 degrees. Something to think about to help get bending and torsional stiffness.

With structural skin wings, the whole wing is what should be analyzed over most of the span, and pitching moment (M66) will be the accumulated pitching moment from the tip to the wing station you are analyzing. When you get close to mounting points, try to remember that the main spar reacts almost all of the lift and bending moment, the drag spar reacts pitching moment, and the skins carry nothing to the next structure. Adjust your [ABBD] so that the skin is contribution goes near zero as you approach the mounting station.

If you are really chasing the lightest structure, go for all graphite, using Graphlite rod caps, graphite cloth for the shear web, and start the skins with two plies of 5-6 oz graphite cloth on each side of ¼” foam cores for the wing skins. Vacuum bag all parts, keep glue lines thin, and you will have close to the lightest wings possible. Do a good job of optimizing the spars, and it will be hell for strong, monster stiff, and light!

ConclusionConclusion

So, now you know the basic path through figuring out where to start with composite parts, how to bring them up to strength, why the designs out there look the way that they do. If you already have the training equivalent to undergraduate engineering coursework I described near the beginning, you could buy Jones or Tsai and Hahn, get into the details, write a program or three in Excel or MatLab, and go to town designing wings. Or you could buy someone else’s design and be able to appreciate the work they did to arrive at their design, whether it be light and efficient, or heavy and clumsy.

Billski