September 2014 

NASA/TM–2014-218517 

Aeroelastic Tailoring via Tow Steered 

Composites

Bret K. Stanford 

Langley Research Center, Hampton, Virginia 

Christine V. Jutte 

Craig Technologies, Inc., Cape Canaveral, Florida 
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Septemeber 2014 

NASA/TM–2014-218517 

Aeroelastic Tailoring via Tow Steered 

Composites

Bret K. Stanford 

Langley Research Center, Hampton, Virginia 

Christine V. Jutte 

Craig Technologies, Inc., Cape Canaveral, Florida 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
Aeroelastic Tailoring via Tow Steered Composites

Bret K. Stanford 

NASA Langley Research Center, Hampton, VA, 23681 

Christine V. Jutte 

Craig Technologies, Inc., Cape Canaveral, FL, 32920 

The use of tow steered composites, where fibers follow prescribed curvilinear paths 

within a laminate, can improve upon existing capabilities related to aeroelastic tailoring of 

wing structures, though this tailoring method has received relatively little attention in the 

literature. This paper demonstrates the technique for both a simple cantilevered plate in 

low-speed flow, as well as the wing box of a full-scale high aspect ratio transport 

configuration. Static aeroelastic stresses and dynamic flutter boundaries are obtained for 

both cases. The impact of various tailoring choices upon the aeroelastic performance is 

quantified: curvilinear fiber steering versus straight fiber steering, certifiable versus noncertifiable stacking sequences, a single uniform laminate per wing skin versus multiple 

laminates, and identical upper and lower wing skins structures versus individual tailoring. 

I. Introduction 

Historically, the most prominent type of aeroelastic tailoring is arguably the use of composite laminates in wing 

structures [1]. The basic tailoring concepts can be seen in the laminate stiffness matrix of Eq. 1, where  is the 

vector of in-plane (membrane) loads on a laminated plate- or shell-like structure,  is a vector of bending/twisting 

moments,  is the resulting mid-plane strains, and  the mid-plane curvatures. By altering the stacking sequence of 

a laminate, the members of the stiffness matrix (, , and  are the extensional stiffness matrix, flexural stiffness 

matrix, and the bending-extension coupling matrix) will change, and various couplings become available. From an 

aeroelasticity/wing structures perspective, some of these couplings are desirable, other undesirable. An example of 

the latter is embedded in the  matrix, where transverse loads along a wing lead to both the typical bending 

curvatures () as well as an atypical in-plane extension (), which can be large and nonlinear. This matrix is easily 

set to zero by utilizing a symmetric stacking sequence. An example of the former is the bend-twist coupling found 

within the  matrix: this is the most well-known and heavily-utilized strategy for composites-based aeroelastic 

tailoring. 



      

 (1) 

Bend-twist coupling in a straight (non-swept) plate-like wing structure is seen in Figure 1, taken from Ref. [1]. 

The  matrix is a 3×3 matrix, where the first column is associated with chord-wise bending of the wing, the second 

column with span-wise bending, and the final column with wing torsion. The figure provides 3 scenarios for the 

coupling terms (traditionally called and , though located in the (1, 3) and (2, 3) cells of the  matrix). For 

an isotropic material or strictly orthotropic material, these terms are exactly zero. In response to a transverse 

aerodynamic loading, a straight wing will bend but not twist (notwithstanding any pitching moments inherent within 

the flight loads during straight and level flight). Alternatively, a negative bend-twist coupling (wash-out) may exist, 

which is characterized by an alleviation of the aerodynamic loads. This is known to be beneficial for leading-edge 

control effectors, static divergence, maneuver load/drag alleviation, etc. Finally, if these coupling terms are positive 

(along with the rest of the matrix), positive bend-twist coupling exists (wash-in), along with an augmentation of the 

flight loads: beneficial for trailing edge control effectors, lift effectiveness, and flutter. The aeroelastic benefits 

obtained in either coupling scenario comes at the expense of the benefits in the opposite scenario, necessitating a 

series of compromises during the tailoring process. The largest bend-twist coupling exists for unbalanced laminates. 

For balanced laminates, and are small, but non-zero: mathematically they will tend to zero, however, as the 

number of layers in the laminate increases. 
 2

Figure 1. Various bend-twist coupling strategies, adapted from Ref. [1]. 

In conventional laminate structures, all fibers in a ply are straight and parallel, with a given orientation. 

Alternatively, one may consider tow steered laminates, where the fiber orientation within a ply varies continuously 

throughout the structure. This provides tailored load paths within the laminate, which may now be termed “variablestiffness”. Tow steering is enabled by computer-numerically-controlled (CNC) fiber tow placement machines, as 

drawn in Figure 2. These machines place precise courses (or tows) of pre-impregnated carbon fiber along 

commanded curvilinear paths, along a tooling surface. If the paths are strongly curved, adjacent courses will 

overlap at certain points in the structure. These overlaps can either be allowed to occur (providing a build-up in the 

local laminate thickness), or tow cutters can be used to adaptively “drop” portions of the current course being laid 

down on the surface, to maintain a uniform thickness. For all of the results given in this report, the latter situation is 

assumed. 

Figure 2. CNC fiber placement machine, courtesy of Cincinnati Machine. 

A significant amount of work can be found in the existing literature regarding the mechanics and design of tow 

steered structures (see for example Ref. [2], and the additional papers cited therein), including recent work 

conducted at NASA LaRC [3], though only limited work has been done for aeroelastic tailoring (see recent Refs. [4] 

and [5]), which is the goal of the current work. Two aeroelastic problems are developed: a cantilevered flat plate, 

and a built-up wing box structure of the Common Research Model (CRM) concept [6], a high aspect ratio subsonic 

transport configuration. Various methods to parameterize the composite laminate structures will be discussed, as 

well as methods to sample the design space. The resulting framework can then be used to answer a series of 

questions pertaining to the efficacy, feasibility, and limits of tow steered aeroelastic tailoring. First, what are the 

benefits of using non-steered, straight-fiber laminates for wing structures? The answer to this question will not 

necessarily be a novel one: as noted above, these methods have been in use for decades. But for each new wing 

configuration that is discussed (and there are two used here), the exact quantitative benefits of non-steered tailoring 

need to be obtained before the answer to the second question can be put into perspective: what additional benefit can 

be obtained by introducing tow steering into the problem? 

In assessing the benefits of tow steering, the relevant comparison is not between tow steered wing structures and 

metallic structures, but between tow steered structures and non-steered (conventional laminate) structures. Tow 

steering is probably best seen as an augmentation to conventional laminate-based tailoring, and so the questions 

need to be answered in the order given above. A third question is: if the available ply angle choices were limited to 
 3

a few discrete choices (i.e., required for certification), what penalty would be paid in terms of aeroelastic 

performance? Final issues that are discussed: what benefits can be obtained if the upper and lower skins of a wing 

box have different laminates? What if more than one laminate is used per skin? What is the impact of using loworder steering concepts from wing root to tip, as opposed to higher-order interpolation schemes? A thorough 

examination of each of these questions, provided here, should provide a clear understanding of the various tradeoffs, benefits, and limitation associated with tow steered aeroelastic tailoring. 

II. Flat Plate Aeroelastic Tailoring 

The first case to be discussed is for a simple aeroelastic flat plate problem. This problem’s low computational 

cost, simple structural configuration, and small number of designable plies, lends it to a rapid and nearcomprehensive sampling of the design space. The resulting insight will be thorough, but limited in the sense that this 

problem is of an academic nature. 

A. Problem Description 

This problem is taken directly from Ref. [7]: an un-swept rectangular flat plate of chord 76.2 mm and a semispan of 305 mm (aspect ratio of 8) which is fully-clamped at its root, and subjected to low-speed flow over both 

surfaces. The plate is comprised of 6 layers of uni-directional graphite/epoxy, whose material properties are given 

in Table 1. Aeroelastic analysis is conducted in NASTRAN, via two modules. First, static aeroelastic behavior is 

explored. The plate-wing is placed at a fixed angle of attack of 5° and a flow velocity of 20 m/s. SOL144 is used to 

find the deformed configuration at which internal elastic forces balance external aerodynamic loads. Tsai-Wu 

failure indices are then computed for each ply within each finite element, using the strength metrics in the right 

column of Table 1. Next, the aeroelastic stability boundary of the plate is determined using SOL145, by marching 

the flow velocity from 1 m/s to 100 m/s, and identifying the critical speed at which the plate becomes statically 

unstable (divergence) or dynamically unstable (flutter). For SOL144 and SOL145 analysis, the finite element mesh 

is discretized into 12×48 squares, each of which is divided into 2 linear triangular elements (CTRIAR) elements. A 

12×48 aerodynamic doublet lattice mesh is used, and an 18×72 finite plate spline (FPS) mesh is used to 

communicate force and displacement data between the two modules. Every third finite element node communicates 

with the FPS mesh (i.e., is used for splining). 

Table 1. Material properties for the aeroelastic flat plate problem. 

Property Value Property Value 

 98.0 GPa  1314 MPa 

 7.9 GPa  1220 MPa 

0.28  43 MPa 

5.6 GPa  168 MPa 

Ply thickness 0.134 mm 48 MPa 

Density 1520 kg/m3 

The laminates for this flat plate case (like all of the laminates in this work) are symmetric, leaving 3 designable 

layers. Steering concepts are illustrated in Figure 3, where the fiber paths for a single layer of the 6-ply laminate are 

shown. Simply, the fiber angle at the wing root  and at the wing tip  are specified, with a linear variation in 

between these two control points. This results in a curvilinear fiber path, explicitly given in Ref. [2]. A minor 

addition to this parameterization is given in the tilting parameter Φ, which defines a local coordinate system   . 

Fiber angle is a linear function of , and constant along the 

 direction. This tilting rotation varies continuously 

between 0° (span-wise steering from root to tip), and 90° (chord-wise steering from leading edge to trailing edge). 

Each of these parameters    can be defined independently for each layer in the laminate, resulting in 9 total 

design variables. If steering is not considered ( = 0°,  = ), then there are only 3 total design variables for this 

conventional, straight-fiber laminate design problem. The fiber angle convention noted in Figure 3 is used for the 

entirety of this work: a 0° fiber is parallel to the flow velocity vector (from leading to trailing edge), and a 90° fiber 

travels from wing root to tip. 

B. Solution Methodology 

The method used here to explore the design space is a multi-objective genetic algorithm: NSGA-II [8]. Given 

appropriate resources, the algorithm will find the global Pareto front within the design space, which is the set of non-
 4

dominated designs. Two objectives are considered: large flutter speeds (SOL145), and low values for the peak TsaiWu failure metric found throughout the finite element mesh during static aeroelastic deformations (SOL144). These 

two metrics are expected to conflict, creating a Pareto front. The GA ranks designs, not based explicitly on 

aeroelastic performance metrics, but by non-domination. A design is non-dominated (Pareto optimal) if no other 

design exists which out-performs it in every metric. For a given set of designs, the Pareto front is rank 1, the next 

layer is rank 2, etc. Low-rank designs have the greatest probability of propagating their characteristics onto the next 

generation. 

Figure 3. Steering concepts for a single layer of the flat plate laminate. 

Though this method is a true optimizer, care must be taken in interpreting many of the results given here as 

“optimal”. This is partly due to a lack of computational resources (a true limitation for the CRM work given below, 

but less so for the flat plate problem, which is relatively inexpensive and operates in a smaller design space), and 

partly due to the fact that the current work is more concerned, at this stage, with identifying trends and limitations of 

the tailoring scheme, rather than locating the global optimum. The NSGA-II algorithm merely provides a diverse 

sampling throughout the design space, with clustering towards the non-dominated portion of that space (and less 

wasted samples in the heavily-dominated portion). As the number of design variables decreases and/or the 

computational cost drops (or the computational resources are increased) our confidence in actually obtaining the true 

global Pareto front increases as well. The interested reader is referred to Ref. [9] for additional details of using these 

algorithms with limited resources and interpreting the results. 

It is further noted that the current work does not represent true “design” work in any sense either. This is 

obviously true for the flat plate problem, owing to the academic nature of the problem. Even for the CRM problem, 

however, a large number of load cases (longitudinal and lateral maneuvers, gust loads, taxi bumps, etc.) would need 

to be added to the 2 cases already considered (flutter and a single static aeroelastic configuration), as well as 

additional geometric detail, drag-based metrics, etc., for this work to qualify as a design effort. As noted, the current 

focus is on identifying underlying trends, the limits of a tailoring tool, and comparing disparate tailoring methods. 

Results for this flat plate problem are organized as follows. First, aeroelastic tailoring for un-steered fibers is 

considered, where fiber angles can take any continuous value. The study is then repeated with steered fibers. This 

is done for aeroelastic flutter metrics (section II-C) and aeroelastic strength (II-D). Next, fiber angles are restricted 

to a few discrete choices in section II-E, with and without tow steering, for direct comparison with the previous 

continuous ply angle problem. 

C. Flutter Results 

First, the GA seeks to find the design with the highest flutter speed, for un-steered laminates (i.e., laminates 

with straight fibers). This design space is defined by only 3 design variables, so there is a high likelihood that the 

final design is the true global optimum. This design is seen in Figure 4, where layer 1 is the top layer, and layer 6 is 

the bottom, and is defined as [-50.7° / 43.2° / 39.3°]s. The laminate is symmetric, by design, and unbalanced. The 

resulting eigenvalue migration behavior for this structure is given in Figure 5. The migration, with increasing flow 

velocity, of the first four aeroelastic modes is shown, which roughly correspond to a first bending mode (1B), 

second bending (2B), first torsion (1T), and second torsion (2T). For a flow velocity close to zero, the imaginary 

portions of each eigenvalue (frequency) are equal to the natural vibration frequencies. Even at this point, each mode 

is only roughly identified with the labels given above, due to the stiffness coupling of the unbalanced laminate 

which alters the typical bending and torsion mode shapes. As the flow velocity increases and the aerodynamic 

interactions strengthen, the mixing of mode shapes increases. 
 5

The first bending mode becomes non-oscillatory at a speed of 45.02 m/s, but does not destabilize (diverge). 

The system flutters shortly after, at 45.98 m/s. Both the second bending mode and the first torsion mode flutter 

when the real parts of their eigenvalues (damping) go from negative damping (stable) to positive damping (unstable, 

un-bounded growth). The first flutter mechanism is brought about by a mild approach of the 1B and 2B frequencies, 

the second mechanism by a mild approach of the 2B and 1T frequencies. These two flutter mechanisms occur at 

nearly the same speed, which is not a coincidence. It is well-known that the most efficient structures are those that 

engage as many failure modes as possible, at the same time (e.g., a fully-stressed design). The optimal design in 

Figure 4 is the best performer found by the GA, in terms of flutter, presumably for this reason. The second bending 

mode flutters more aggressively than the torsion mode, the latter of which becomes a hump mode, and eventually restabilizes. 

Figure 4. Un-steered flat plate laminate with the highest flutter speed. 

Figure 5. Flutter behavior for the flat plate laminate of Figure 4. 

The above exercise is repeated, this time including tow steering in the structural parameterization, as seen 

graphically in Figure 3. The best flutter result returned by the GA of the 9-design variable problem is shown in 

Figure 6. Even though the design space has been given a wide-range of steering paths to take advantage of, it does 

not: the best flutter result utilizes only mild steering from root to tip. The greatest spatial variation in steering angle 

is seen for the inner layers 3 and 4, but the stiffness of these layers is only of minor importance for a plate 

undergoing bending deformations (i.e., dominated by the , rather than the  matrix). The flutter behavior of this 
 6

structure is shown in Figure 7, and bears many similarities to that of the previous case. One distinct difference is 

that the bending mode becomes non-oscillatory much sooner (38.2 m/s), and the resulting real roots are less stable 

(but still non-critical) than before. Otherwise, dual flutter modes are still seen, occurring at a speed of 46.83 m/s. 

The advent of tow steering for this problem has brought about only a mild improvement in the performance: a 

1.84% increase in the flutter speed. This is presumably a configuration-dependent result, in the sense that with two 

modes fluttering simultaneously, any design change which improves one flutter point (e.g., 2B) will make the other 

worse (e.g., 1T), and vice-versa. This is a hard corner of the design space, and the introduction of new variables 

(tow steering) cannot improve the situation much. It is also noted that none of the steering patterns in Figure 6 make 

use of chord-wise steering, with the bulk of the weak fiber gradient in the span-wise direction (i.e., Φ is nearly 0°). 

Figure 6. Steered flat plate laminate with the highest flutter speed. 

Figure 7. Flutter behavior for the steered flat plate laminate of Figure 6. 

D. Strength Results 

Attention is now turned to static aeroelastic behavior (SOL144), specifically the Tsai-Wu failure index for each 

layer (6 in total) of each finite element. This is a simpler problem in the sense that statically-deformed wing 

behavior is simpler than a dynamic flutter mechanism, but potentially more complex due to the chosen objective 

function. The location of the peak failure index may change from one element to another (or even change from one 

layer to another within a given element, although the outer layers are usually expected to be critical), and is thus a 

discontinuous function of the design space, slowing the convergence of the genetic algorithm. The un-steered 
 7

design with the lowest peak failure index (flutter is ignored in this section), as located by the GA, is seen in Figure 8, 

and is defined by: [94.3° / 102.2° / 82.4°]s. The failure index for each layer of each element is also shown. 

As before, this laminate is symmetric and unbalanced, but only mildly. Layers 2 and 3 form a near ±θ pair, and 

layer 1 is almost parallel to the wing itself (94.3°). Indeed, all of the layers are relatively close to 90°, which 

suggests that this design metric places less of an emphasis on aeroelastic tailoring via coupling for load alleviation, 

and more upon just providing the stiffest possible plate. That there are some deviations from 90° is either indicative 

of a secondary need for tailored coupling, or due to the non-uniform aerodynamic pressure distribution over the 

plate. The failure metrics are small: the largest value is 0.0346. Any value over unity is considered to have 

elastically failed, not including any safety factor. As would be expected, stresses grow as one travels from the inner 

layers to the outer layers, and peak stress occurs at the wing root. Asymmetries between the top and the bottom 

halves of the plate (layer 1 vs. layer 6, for example) are indicative of the mixed bending/torsion loading. 

Figure 8. Un-steered flat plate laminate with the highest strength, and the corresponding failure indices. 

Repeating this exercise with tow steering does finally obtain a design that utilizes heavily-steered laminates, 

Figure 9. Fiber angles at the root are nearly parallel to the leading edge, but curve upwards by about 45° as one 

travels to the wing tip. This results in a significant reduction in the peak failure index to 0.0260, a drop of 24.85%. 

The steering is able to draw the peak stresses out from the root over a greater portion of the wing, similar to a fullystressed design methodology. Though both the steered result and the un-steered result have failure indices far below 

unity, the dynamic pressure over the wing was chosen somewhat arbitrarily. If that value were increased by a factor 

of 10, the stresses would increase similarly; the difference between the two strategies is of importance, not the actual 

value. Furthermore, even for the dynamic pressure used in the current results, there are several laminates in the 

design space which do fail. This is either because they are too weak in the span-wise direction (0° plies, for 

example), or because they are statically unstable: the flow velocity chosen is above the divergence speed. Finally, it 

is noted that most of the steering seen in Figure 9 is span-wise ( = 0°). Some mild chord-wise steering is noted at 

the root of layers 3 and 4, but these inner layers have a minor effect on the aeroelastic strength. 

E. Structural Certification Considerations 

The previous results of this section have allowed the fiber angle for each layer to be a continuous design 

variable, capable of taking on any value between specific bounds. Historically however, aerospace design with 

composite materials has made use of only four discrete choices: 0°, 90°, 45°, and -45°. There are several reasons for 

this, related to convenience and manufacturability, but the main reason is due to certification: most available 

laminate data (via experimental testing) is limited to these ply angles. Certifiable laminates not only are limited to 

these discrete ply angle choices, but must also be symmetric and balanced. While this is a very real limitation in 

present-day aircraft structures, it does not necessarily need to be adhered to for this work, whose aeroelastic wing 

tailoring strategies may not be fully realized for many years (by which time, certification-by-analysis may be 
 8

possible, rather than relying on costly and therefore limited experimentation). It is worthwhile, however, to 

ascertain the penalty that is paid, in terms of aeroelastic performance, when one restricts the design space to only a 

few choices. 

Figure 9. Steered flat plate laminate with the highest strength, and the corresponding failure indices. 

Tow steering is still possible when limited to only 0°, 90°, and ±45° ply angles, but each layer in the stacking 

sequence can no longer be steered individually, as done in Figure 3. Now, a core laminate is defined (some 

combination of 0°, 90°, and ±45° plies), as well as a single steering path. The core laminate is merely rotated with 

the steering path, as seen in the top of Figure 10. The “90° fibers” remain parallel to the steering path, the “0° 

fibers” are always perpendicular to this path, etc. Globally, the fiber angles at every point in the plate wing can take 

any value, but locally (i.e., with respect to the other plies), they only take values of 0°, 90°, and ±45°. For a core 

laminate without any 90° plies (a stacking sequence of [45° / 0° / 0°]s for example), none of the fibers in the 

laminate will actually follow the prescribed steering path. The total number of design variables for this scenario is 

6: the core laminate is defined by 3 discrete variables (one for each designable layer), and the single steering path is 

defined by 3 continuous variables: , , and . 

Figure 10. Tow steering concepts when limited to 0°, 90°, and ±45° fiber angles. 
 9

 The flutter results of Figure 4 through Figure 7 are repeated here using these discrete ply concepts, and 

summarized in Table 2. The previous best flutter result without steering was [-50.7° / 43.2° / 39.3°]s. The new best 

result within the discrete design space, is [-45° / 45° / 45°]s which is simply the previous laminate rounded to the 

nearest available choice (though the design is actually found via an exhaustive search through the GA). In moving 

from the continuous design space to the discrete one, a 6.41% penalty is paid in flutter performance. This may not 

be an acceptable decrease in flutter, depending on how close the original design’s flutter boundary was to some 

conceptual flight envelope. The previous best result with tow steering included is seen in Figure 6. When this 

exercise is repeated using the steering concepts of Figure 10, the best design found by the GA uses the same core 

laminate as the non-steered case [-45° / 45° / 45°]s, and steers this laminate along the mildly-curved path given in 

Figure 11. The performance penalty (due to the restricted design space) between the two cases is very small 

(0.17%), which would indicate that the discrete ply angle problem benefits from tow steering far more than the 

continuous ply angle problem. It is also noted that because the core laminate in Figure 11 contains no 90° plies, 

none of the fibers in the plate follow the shown steering path, but instead intersect these curves at angles of ±45°. 

Table 2. Penalties in the flutter speed when ply angles are limited to discrete choices. 

Case Laminate Steering Objective Penalty 

Continuous ply 

angles w/o steering [-50.7° / 43.2° / 39.3°]s - 45.98 m/s 

6.41% Discrete ply angles 

w/o steering [-45° / 45° / 45°]s - 43.03 m/s 

Continuous ply 

angles with steering Figure 6 Figure 6 46.83 m/s 

0.17 % Discrete ply angles 

with steering [-45° / 45° / 45°]s Figure 11 46.75 m/s 

Figure 11. Steering path for the laminate with the highest flutter speed: core laminate = [-45° / 45° / 45°]s. 

This exercise can be repeated for the strength-based metrics in Figure 8 and Figure 9, and is summarized in 

Table 3. The previous best strength result as found in [94.3° / 102.2° / 82.4°]s, which has a peak failure index of 

0.0346. If one simply rounds each of these ply angles to the nearest discrete choice [90° / 90° / 90°]s, this laminate 

has a failure index of 0.0631. A cursory search of the design space via the GA returns a better design with a lower 

failure index of 0.0527: [90° / -45° / 0°]s. The outer layer ply angle (and the most influential for bending 

deformations) for both wings is 90°, but some minor differences are seen in the inner layers. Regardless, the penalty 

paid when restricting the design space to discrete choices (0°, 90°, ±45°) is significant: a 52.3% increase in the peak 

stress. While the failure index for both wing structures is much lower than unity (and therefore very safe), it is the 

delta between the two which is of greater interest, as these absolute performance metrics will scale with the 

arbitrarily-chosen dynamics pressure over the wings. 

Table 3. Penalties in the peak Tsai-Wu failure index when ply angles are limited to discrete choices. 

Case Laminate Steering Objective Penalty 

Continuous ply 

angles w/o steering [94.3° / 102.2° / 82.4°]s - 0.0346 

52.3% Discrete ply angles 

w/o steering [90° / -45° / 0°]s - 0.0527 

Continuous ply 

angles with steering Figure 9 Figure 9 0.02603 

~ 0% Discrete ply angles 

with steering [90° / 90° / 90°]s Figure 12 0.02604 
 10 

Finally, the best steering result for Tsai-Wu failure through the continuous design space is given in Figure 6, 

while the best result for the discrete design space uses a core laminate of [90° / 90° / 90°]s (which, unlike the flutter 

results in Table 2., is not the same core laminate as used by the un-steered case) and a steering path of Figure 12. 

This aeroelastic structure has practically the same peak failure index as the previous best result: there is no 

appreciable penalty. As noted above, the discrete problem benefits more from tow steering than the continuous 

problem. Put another way, while there is a penalty associated with moving from continuous ply angles to discrete 

ply angles, the introduction of tow steering can make up for most, if not all, of this gap. Even for the discrete ply 

angle problem, the use of tow steering re-introduces an element of spatial continuousness into the problem, with 

significant performance benefits. It is finally noted that, because each layer of the laminate in Figure 12 is 90°, 

every fiber in the structure follows the path shown. There are many practical reasons not to use a [90° / 90° / 90°]s 

laminate (it is very weak in the matrix direction, and it is also inadvisable to use so many identical plies stacked 

adjacently, owing to matrix cracking issues [10]), but the GA is wholly unaware of these issues, and should not 

distract from the demonstrated benefits of tow steered laminates. 

Figure 12. Steering path for the laminate with the highest strength: core laminate = [90° / 90° / 90°]s. 

III. CRM Tailoring 

The second case discussed here is a more complex built-up wing box problem, specifically for the CRM 

configuration. This problem has a much larger computational cost and more designable plies than the above flat 

plate problem, and so the GA results will present a less-comprehensive picture of the design space, owing to 

computational resource limitations. 

A. Problem Description 

The CRM model used here is a full-scale 777-derivative wing with span of 192.8 ft, an aspect ratio of nine, a 

taper ratio of 0.275, a leading edge sweep angle of 35°, and a break along the trailing edge at 37% of the semi-span 

[6]. The wing box lies between 10% and 70% of the local chord: an exploded view is given in Figure 13. It is 

important to note which portions of this structure are being tailored in this work, and which are not. The allaluminum internal support structure (ribs, spars, stringers, stiffeners) is left un-touched, aeroelastic tailoring is only 

considered in the composite laminate upper and lower wing skins. While there is certainly some benefit to the use 

of composite materials for the internal substructure (ribs, spars, stringers, etc.) with or without tow steering, these 

concepts are not explored here. 

Figure 13. Exploded view of the CRM wing box structure. 

Full-depth aluminum spars (2024-T3 alloy) are placed at the box leading edge, trailing edge, and one-third of 

the distance between the two. The thickness of the spars is linearly tapered from 0.75 inches at the root to 0.1 inches 

at the tip. Thirty-seven straight aluminum ribs are evenly distributed from root to tip, each aligned with the airflow, 
 11 

and linearly tapered from a thickness of 0.12 inches to 0.063 inches. Seven pairs of aluminum stringers (one on 

each skin, not shown in the figure) travel from root to tip: two pairs are evenly distributed between the leading edge 

spar and the inner spar, and five pairs between the inner and trailing edge spar. These stringers have a rectangular 

cross section, with a depth of 2.95 inches and a thickness of 0.18 inches. A full-depth aluminum rib stiffener exists 

at each stringer-rib intersection, each with a depth of 2.64 inches and a thickness of 0.18 inches. 

The skins, spars, and ribs are modeled with triangular shell finite elements, stringers and rib stiffeners are 

modeled with beam elements, and the displacement degrees of freedom of all nodes at the wing-root of Figure 13 

are fixed to zero. The inertial impact of leading and trailing edge control effectors are modeled as lumped masses, 

connected to the leading and trailing edge spars via un-weighted interpolation elements. Six 320 lb masses are used 

along the inboard leading edge, and three additional of 240 lbs outboard. Similarly, six 840 lb masses and three 140 

lb masses are used along the trailing edge. These mass values were calculated by scaling data from a similar 

commercial transport. 

All of the composite wing skin tailoring results presented below are compared against the performance of an 

all-metallic baseline wing. This wing box consists of the metallic substructure described above (ribs, spars, 

stringers, stiffeners), as well as aluminum wing skins. This thickness of these skins linearly tapers from 0.25 inches 

to 0.05 inches at the tip. The flutter dynamic pressure and the weight of this wing are both used to normalize values 

obtained during composite tailoring. Material properties for these tailorable composite skins are given in Table 4. 

The static aeroelastic analysis and flutter analysis are conducted in NASTRAN. MATLAB scripts generate 

input files for the analyses and extract the data from the output files to compare performance metrics and assess the 

aeroelastic tailoring concepts. Flat-plate aerodynamic paneling is utilized for both steady and unsteady air loads, 

with a 10×10 mesh of boxes for the inboard section of the wing, and a 10×40 mesh outboard. Finite element nodes 

located at intersections of the upper skins and ribs, or the upper skins and spars, are used to interpolate between the 

structural and aerodynamic meshes. 

Static aeroelastic wing deformation is computed at an angle of attack of 6°, a Mach number of 0.85, and an 

altitude of 35 kft. A flutter analysis (p-k method) is then performed with 20 structural dynamic modes at a Mach 

number of 0.85, and using the speed of sound at sea level, the velocity is computed and fixed. The dynamic pressure 

varies from 0 to 14.8 psi, divided into 250 increments by varying the flow density: zero-damping cross-over points 

indicate flutter. 

Table 4. Material properties for the Common Research Model wing skins. 

Property Value Property Value 

 20.44 Msi  19.06 Msi 

 1.41 Msi  17.76 Msi 

0.35  0.62 Msi 

0.68 Msi  2.43 Msi 

Ply thickness 0.005 in 0.69 Msi 

Density 0.058 lb/in3 

B. Solution Description 

The previous flat plate laminate problem had only 3 designable layers, which meant that one design variable (or 

more, if tow steering was considered) could be assigned to each layer. The resulting design space is relatively lowdimensional, and is well-explored with the GA. A transport wing skin, on the other hand, may have 50-100 layers 

of carbon fiber, particularly at the root (where the bending moments are high) of the upper skin (where buckling is a 

concern): attaching design variables to each layer is no longer feasible. Such a large-variable design space can only 

be explored with gradient-based optimization; see Ref. [11], for example. Otherwise, two alternatives are 

commonly seen in the literature. The first directly uses members of the  and  matrices as design variables 

(assuming a symmetric laminate:   ). Care must be taken that these stiffness matrices actually correspond to 

some stacking sequence (an inverse process), and so the feasible design space must be derived a-priori. An example 

of this approach for aeroelastic wing design can be seen in Ref. [12]. 

Alternatively, one may exploit the geometry of the wing box. If the depth of the wing box (airfoil thickness) is 

much larger than the thickness of the upper and lower wings skins, then it is known that skin stresses due to midplane strains  are much larger than those due to the mid-plane curvatures  (referencing Eq. 1). This would mean 

the extensional stiffness matrix  is far more important than the flexural stiffness matrix . The preceding is true 

for global deformations, such as wing bending and twisting: for local deformations (panel buckling, e.g.), both terms 
 12 

contribute. As  is only computed from the number of plies at each orientation (rather than the exact stacking 

sequence, which is used to compute ), this can enable significant reductions in the number of design variables. 

Under these conditions, permutations of the same stacking sequence for the skins will give very similar 

aeroelastic results for the CRM: e.g., [-30° / 70° / 20° / -30°]s, [70° / -30° / -30° / 20°]s, etc. On a top level, only the 

discrete variables associated with each available ply angle need to be stored: = 2, = 1, = 1. At the 

NASTRAN level (within the PCOMP cards) one can use any stacking sequence, as long as the expected wing 

deformations are of a global nature. One possible parameterization would be to discretize the ply angles by steps of 

5°:        . This results in 28 discrete design variables per laminate, which is still a 

high-dimensional design space. Alternatively, one could discretize into steps of 45°:     , which 

is only 4 design variables. Finally, one could use just 3 variables:    , where each variable 

creates twice the number of plies in the laminate for balancing purposes. 

This final 3-design variable space will be used for the CRM tailoring work. While its limited nature is not 

expected to provide as highly-performing designs as the 28-variable option, several considerations apply. First, the 

28-variable problem is too large of a design space to be considered by any method except gradient-based 

optimization. This issue is compounded by the likely situation that more than one laminate is used per wing. 

Secondly, the 3-variable problem creates designs that are automatically balanced (due to the  term), and thus 

certifiable. Finally, the previous work on the flat plate problem indicated that the aeroelastic performance of a 

laminate with limited ply angle choices could be greatly improved (compared to a laminate with unlimited choices) 

by the introduction of tow steering, a trend which may continue in the CRM problem. Once the number of each ply 

angles is decided upon in each laminate, the permutation problem could be solved at the local panel level to 

determine a stacking sequence that will satisfy buckling constraints (whose local deformations will depend on the 

matrix, and hence the stacking sequence). This is not considered here, but the technique is demonstrated in Ref. 

[13]. 

As only tow steering was explored for the flat-plate problem, and not the possible addition or deletion of entire 

plies, the mass of the plate-wing was assumed constant. This is no longer true for the current parameterization, and 

so within the context of the multi-objective NSGA-II algorithm, a Pareto front spanning 3 objectives is sought: 

minimum mass, maximum flutter speed, and minimum stress-based failure index. Previously, the largest Tsai-Wu 

failure index across the plate was identified and exposed to the GA as an objective. A more refined approach is used 

here: the Kreisselmeier-Steinhauser (KS) function [14]. The KS function is a stress aggregate function which 

compresses each failure index throughout the finite element model (every Tsai-Wu failure index from each ply of 

each composite element, as well as each Von Mises failure index from each metallic element) into a single scalar via 

a nonlinear weighting function. Highly-stressed elements have a greater weight than others, which is controlled via 

a tuning parameter. For the purposes of this work, a KS value greater than unity would indicate that some element 

in the mesh has failed elastically, notwithstanding a safety factor (though a degree of conservatism is built into the 

KS function, as dictated by the same tuning parameter). 

The remainder of this section utilizes eight different design scenarios, with increasing levels of tailorability. The 

first four scenarios are for straight, non-steered laminates, splitting the wing into multiple segments to vary the 

locations of the different tailored laminates on the wings. First, the same un-steered laminate is applied to both the 

upper and lower wing skins (section III-C). Then different un-steered laminates are used for the upper and lower 

skins (section III-D). Next, each of the 4 skin segments have different laminates from root to tip but the upper and 

lower skins are the same; and finally each of the 4 skin segments have different laminates from root to tip with 

different upper and lower skins. The next four scenarios (section III-E) follow the same approach of keeping 

laminates the same on top and bottom and splitting the wing into multiple segments for more refined span-wise 

tailoring, but also introduce the ability to steer the tow angles within the plane of the laminate instead of holding the 

fibers straight and parallel. Thus, the tow steered scenarios are: same laminates for upper and lower skins, steered 

from root to tip; different laminates for upper and lower skins, steered from root to tip; 5 control points steered from 

root to tip but same laminates on upper and lower skins; and 5 control points steered from root to tip and different 

laminates on upper and lower skins. 

C. Single Laminate Design 

The first design scenario considered here is where the entire upper skin of the CRM is covered with a single 

laminate, and this same laminate covers the entire lower skin as well. This leaves 3 total design variables, as 

described above. The terms  and  are both allowed to lie between 0 and 4, whereas  must lie 

between 1 and 4. In order to retain the presence of a wing skin, one of these 3 variables must have a lower bound 

greater than 0: the ±45° plies are a natural choice, as these plies are typically utilized on the exposed surface of wing 
 13 

structures [10]. Given that each of these variables creates 2 plies, and further multiplying by 2 to create a symmetric 

laminate, the thinnest structure  has 4 layers total, while the thickest  has 48 total (where the first 

number is , the second is , and the third is ). The Pareto front returned by the GA for this design 

space is shown in Figure 14. The three-dimensional surface has been plotted in two separate figures, where wing 

weight and flutter are both normalized by the baseline parameters. 

Several data points in Figure 14, when viewed in a two-dimensional space, appear to be dominated (i.e., nonPareto optimal). On the left side of the plot, for example, several points appear to have higher KS values and higher 

weight than others. When viewed in a true three-dimensional context however, each of these data points will have 

higher flutter boundaries than the others, and so are indeed Pareto optimal. For this case, there is a strong trade-off 

between wing weight and wing flutter. The heaviest design  is the safest, in terms of the normalized flutter 

dynamic pressure, while the lightest design  has the worst flutter behavior, with a very aggressive instability 

boundary. While there are bands of constant weight (which are due to the fact that each variable is discrete, and so 

there are only 12 possible weight values) within which the flutter speed can be moderately increased by making 

changes to each variable, gross changes to the flutter boundary necessitate the addition of more weight to the wing. 

Figure 14. CRM Pareto front: single laminate for entire skin, identical upper and lower skin structures. 

The flutter results for design  can be seen in Figure 15, where dynamic pressure is normalized by the 

flutter dynamic pressure of the baseline structure. Several interesting discrepancies exist between this vibration 

behavior and those of the flat plate in Figure 5. First, a predominately edge-wise bending mode (1E, or fore-aft 

bending vibration) can be seen at a relatively low frequency, though its aeroelastic interactions with the unsteady 

flow are minimal: its frequency does not change with increasing dynamic pressure, and (more telling) its 

aerodynamic damping is nearly zero. Work on high-aspect ratio wings (see Ref. [15], for example) has identified 

interactions between edgewise and flapwise (1B, 2B) bending, as well as torsional modes (1T) as the main driver for 

post-flutter limit cycle oscillations, which may be of interest for future work. Secondly, the flutter mechanism in 

Figure 15 is the typical bending-torsion coalescence, and only a single mode flutters. This mechanism is remarkably 

consistent for a wide variety of structural configurations, though the actual flutter dynamic pressure can vary 

significantly, as shown in Figure 14. Though two hump modes appear, both driven by the second bending mode, 

neither is ever seen to actually destabilize, regardless of the tailoring strategy. This is in stark contrast to the flat 

plate problem: different laminates can lead to 1B divergence, 2B flutter, or 1T flutter, as also seen in Ref. [7]. 

A final note of interest from Figure 15 is the appearance of an aerodynamic lag root at a normalized dynamic 

pressure of 1.05. This root is a manifestation of the unsteady doublet-lattice aerodynamics through the structural 

vibration modes; the first-order aerodynamic terms translate into a non-oscillatory root. The root itself is not of 

significant interest, as it never destabilizes to become the critical divergence mode, but NASTRAN’s SOL145 is 

unable to track all the existing modes as well as this new mode. The algorithm abandons 1B to track the lag root 

instead, then loses 1E, and then loses 1T. This final torsion mode is permanently lost, which is probably related to 

the fact that an oscillatory root also appears with the non-oscillatory lag root shown in Figure 15 (but NASTRAN 

does not capture), which eventually collides with the torsional root, annihilating each other [16]. Given that the root 

which ultimately flutters (1B) is temporarily dropped by the flutter algorithm, there may be some doubt as to the 
 14 

robustness of the method for computing flutter points across a wide variety of structural configurations. For the 

current work however, quality flutter points were always obtained, each via the typical 1T-1B coalescence 

mechanism. 

While the trade-off between flutter and weight is very strong (shown in Figure 14), this is not the case for KS 

and weight. The heaviest design does not have the lowest stress, and vice-versa (though  is very flexible with 

a large KS value): indeed, relatively low-weight designs can be found with low KS values. The design with the 

lowest KS value (and therefore the lowest failure indices throughout the mesh) is  , a 32-layer laminate 

without any 0° plies. The ply-angle configurations are the same as in Figure 3: 0° plies are parallel to the flow 

vector, and a 90° ply would have its fibers travel from root to tip of an un-swept wing. For the swept CRM 

configuration, a 60° ply (not an option in the current parameterization, of course, although possible with tow 

steering) would align with the wing leading edge, which is swept back from the horizontal by 35°. 

Figure 15. Eigenvalue migration behavior for the design with the best flutter boundary in Figure 14. 

The distribution of the Tsai-Wu failure indices throughout the wing skins for the best design  is shown in 

Figure 16. Stresses develop in the metallic substructure as well, but failure indices in the skins tend to be higher 

(and thus have a greater weight in the KS function), and will be the focus here. Indices are given throughout the 90° 

plies, 45° plies, and -45° plies, for both the upper and lower wing skins. The stress contour plot for the 90° plies on 

the upper skin, for example, is representative of the stresses in every 90° ply in this laminate (there are 16), 

regardless of where it may lay in a stacking sequence. As explained above, this is due to the large difference 

between wing box depth and skin thickness. Owing to the wing taper, this disparity may be smaller towards the 

wing tip, but stresses are very low here, and less consequential. Peak stresses occur at the yehudi break in the wing, 

within the -45° plies. The peak failure index is 0.721, though the KS function for this design, as seen in Figure 14, 

is 0.767, owing to the conservatism of the metric. 

D. Multiple Laminate Design 

Attention is now turned to the more realistic case where separate laminates are used for the upper and lower 

skins. This may be particularly important for the current parameterization, which only uses balanced laminates. 

Given the known importance of unbalanced laminates for aeroelastic tailoring [1], these global bend-twist coupling 

effects may be somewhat recaptured by having disparate upper and lower surface laminates. For the previous 3-

design variable case, the design space is sufficiently limited that the GA is able to evaluate every single design, and 

the Pareto front is exact. This is no longer true for the current 6-design variable case, whose Pareto front is 

compared with the old front in Figure 17: the design space is too large (though not so large that the GA cannot make 

considerable progress in converging to this front). Regardless, several interesting comparisons can be made in the 

figure, where the first three numbers of each highlighted design are the laminate of the upper skin, and the last three 

are for the lower skin. 

The bands of constant weight have doubled, from 12 to 24, due to the increase in the number of discrete design 

variables. There is a significant reduction in the KS values for a given wing weight, when comparing to the case 

where upper and lower skin laminates were identical. The best design  is similar to the previous best, 
 15 

except  on the upper skins is decreased by two. Reviewing Figure 16, the stresses in these plies are relatively 

low, and so removing them is in accordance with a fully stressed design strategy. The failure indices for this design 

(Figure 18) do show a greater distribution in peak stress over more of the layers (especially in the upper skin), but 

peak values are still concentrated at the break. Unlike in the flat plate case (Figure 9), where peak stresses are 

spread out over more of the wing, it appears that the details of the CRM’s planform geometry are a greater driver of 

stress distribution than detailed tailoring can overcome. 

Figure 16. Skin stress distribution for the design with the lowest KS function in Figure 14. 

Figure 17. CRM Pareto front: single laminate for entire skin, separate upper and lower skin structures. 

More spatially-detailed laminate concepts may be pursued, as summarized in Figure 19. The upper left of this 

plot shows the design with the lowest KS function in Figure 14, and the upper right the lowest KS function in Figure 

17. As already noted, the latter design is able to decrease both the KS function and the weight, though the flutter 
 16 

boundary becomes worse. Alternatively, the wing skins may be divided into four laminates, but the upper and lower 

skins still constrained to be identical. In this situation, the best KS result as returned by the GA is given in the lower 

left of Figure 19: peak thickness is near the yehudi break, where stresses are highest, and the typical thickness taper 

is evident towards the root. The final result in this figure (lower right) has different laminates between upper and 

lower skins. Between these four results, the number of design variables increases from 3 to 6 to 12 to 24, and the 

performance of the best KS result found by the GA progressively drops from 0.767 to 0.714 to 0.709 to 0.694, as 

would be expected (though the movement of flutter dynamic pressure and weight of these designs is inconsistent). 

As the dimensionality of the design space increases, the computational resources available to the GA become 

less adequate. Indeed, the decrease in KS function value between the last 3 designs is not as large as might be 

expected with a progressive doubling of the number of design variables: it seems likely that far superior designs 

exist, especially for the lower two structures in Figure 19. An extreme case of the approach taken in this figure may 

be to identify a separate laminate for each panel (where a panel is defined as wing skin bordered by rib and spar 

intersections), as done in Ref. [13]. This may be expected to give the best Pareto fronts, but would require formal 

gradient-based optimization, and is beyond the scope of the current work. Finally, it is noted that the best flutter 

design is always found to be the heaviest 48-ply laminate, regardless of the ply orientations in the laminate. The 

lightest design, of course, is always the 4-ply laminate ( = 1). 

Figure 18. Skin stress distribution for the design with the lowest KS function in Figure 17. 

E. Steering Results 

The tow steering concepts used for the CRM are similar to those used above for the flat plate in Figure 3 and 

Figure 10. A steering angle is specified at the wing root () and the tip (), with a linear variation in between. As 

above, this results in a curvilinear tow path. As the tilting angle  was found above to have a minor impact on the 

problem, it is omitted from this section. Then, the core laminate, as defined by , , and , is rotated 

with the steering path, where 90° fibers (if any) are those that will remain parallel to the path, as in Figure 10. A 

steering angle ( or ) of 90° means than a laminate is locally un-steered, and an angle of 60° would align the 90° 

fibers (if any) approximately with the leading edge of the swept CRM planform. The wing structure in the upper left 

of Figure 19 is used for all of the initial results in this section. Though better results have been obtained (as seen by 

the progression in that figure),  will be used here for its simplicity, and to aid the interpretation of results. 

Using this laminate for both the upper and lower skins, Figure 20 displays the KS function (contour plot) when 

and  are independently swept from 30° to 150°. Steering patterns are shown for various data points along the 

perimeter of the plot moving in a counter-clockwise direction: (30°,-30°), (90°,-30°), (150°,-30°), (150°,-90°), 

(150°,-150°), (90°,-150°), (30°,-150°), and (30°,-90°), starting at the lower left. It is further noted that each of these 
 17 

designs has the same weight (equal to the normalized value of 0.888 from the un-steered structure in Figure 19): tow 

steering merely re-orients existing material, and is not additive. 

Figure 19. Designs with the lowest KS function under various spatial laminate parameterizations. 

The data point at (90°,-90°) which corresponds to the center of the figure is entirely un-steered, and gives the 

exact same performance as the structure in the upper left of Figure 19 (KS = 0.767). Any design along the line 

   (the diagonal) is un-steered in the sense that there are no curvilinear fibers, but the entire stacking sequence 

is rotated uniformly from root to tip, as drawn in the lower left (30°,-30°) and upper right (150°,-150°) subplots. 

Only when  and  are unequal do curvilinear paths emerge, the best of which (in terms of the lowest KS value, 

and hence the lowest stresses) is (50°,-150°), which has a performance improvement of 9.6% over the un-steered 

case (90°,-90°). As may be expected, this high-performing steering path is nearly aligned with the CRM leading 

edge at the wing root, and perpendicular to it at the wing tip. 

Similar data is given in Figure 21 for the normalized flutter dynamic pressure of each case. The best flutter 

point is located at (150°,-150°) (a 52.3% increase in the flutter speed), which uniformly rotates all the fibers 

counterclockwise by 60°, nearly perpendicular to the leading edge. Like the flat plate problem discussed in Figure 

7, aeroelastic flutter metrics for this CRM case are largely impervious to the advent of curvilinear tow paths. For the 

flat plate, the likely reason was due to a bi-modal flutter point that presented a hard constraint boundary in the 

design space. The flutter mechanism for this CRM case is similar in nature to Figure 15: its uni-modal nature should 

not present a similarly hard boundary. Though some beneficial flutter behavior due to tow steering will be shown 

below, the insensitivity noted to this point may be simply due to the fact that curvilinear steering is a spatially-local 

tailoring scheme with a minor impact upon a global flutter mode. The wing stresses, on the other hand, are a local 

metric (albeit due to global wing deformations) which can bear improvement from steering. 
 18 

Figure 20. KS function for different root () and tip () steering:  = 4,  = 4, and  = 0 for 

both the entire upper and the lower skin laminate. 

Figure 21. Normalized flutter dynamic pressure for different root () and tip () steering:  = 4, 

 = 4, and  = 0 for both the entire upper and the lower skin laminate. 
 19 

Next, different steering patterns are considered for the upper and lower skins, though the core laminate for each 

is the same as before. The root steering angle  is fixed at an arbitrary value of 90° (to reduce the design space), 

while the tip steering angles  for the upper and lower skins are independently varied through 10° increments. 

Results for the KS function and the normalized flutter dynamic pressure are given in Figure 22 and Figure 23, 

respectively. As before, select steering patterns are drawn for various designs along the perimeter, where the red 

lines are the paths along the upper skin, and the blue lines are for the lower skin. These paths are identical at the 

wing root, of course, because  is fixed at 90° for both. The design at (90°,-90°) is entirely un-steered, as before, 

and identical to the case in the upper left of Figure 19 (KS = 0.767, normalized flutter = 0.614). 

Any design in these figures along the unit slope (US   LS ) has no difference in the steering patterns 

between the upper and lower skins. Any design along the vertical line US   90° has no steering in the upper 

skins, and any design along the horizontal line LS   90° has no steering in the lower skins. The design with the 

lowest KS function (Figure 22) is found at (140°, 30°), which presents a 13.5% drop as compared to the baseline. 

This is superior to what was found in Figure 20; presumably even better designs exist if one drops the restriction that 

  90°, and conducts an exhaustive search through the 4-dimensional space. The best flutter result (Figure 23) is 

found at (30°,-30°), a design which improves the flutter point by 40.4%. This design has no differences in the 

steering between upper and lower skins, and a better, entirely un-steered result, was found in Figure 21. 

Figure 22. KS function with a root  of 90°, and different upper skin tip (US ) and lower skin tip (LS ) 

steering:  = 4,  = 4, and  = 0 for both the entire upper and the lower skin laminate. 

As may be expected, the trends shown in these four figures are strongly dependent upon the underlying core 

laminate. The previous 4 figures are repeated for the laminate where  = 4,  = 4, and  = 4, which is 

the structure with the best flutter point in Figure 14 and Figure 17. KS and flutter results are given in Figure 24 and 

Figure 25 for identical upper and lower skin steering, and in Figure 26 and Figure 27 for separate steering. For 

identical upper-lower steering, the KS steering results show some similarities between the two laminates, though 

tow steering becomes less advantageous, as the best design is only mildly steered. The flutter results between the 

two laminates are completely out of phase, though steered laminates still have no benefit over un-steered. Similar 

observations may be made for the case where the upper and lower skins have different steering patterns. Wing 

stresses show less benefit from curvilinear paths, and flutter trends are largely opposite, as compared to the previous 

results. This reinforces the overall tailoring issues discussed above: the impact of tow steering is heavily dependent 

on the un-steered laminate design, and can only be put in context if conducted sequentially. 
 20 

Figure 23. Normalized flutter dynamic pressure for a root  of 90°, and different upper skin tip (US ) 

and lower skin tip (LS ) steering:  = 4,  = 4, and  = 0 for the upper and the lower skins. 

Figure 24. KS function for different root () and tip () steering:  = 4,  = 4, and  = 4 for 

both the entire upper and the lower skin laminate. 
 21 

Figure 25. Normalized flutter dynamic pressure for different root () and tip () steering:  = 4, 

 = 4, and  = 4 for both the entire upper and the lower skin laminate. 

Figure 26. KS function with a root  of 90°, and different upper skin tip (US ) and lower skin tip (LS ) 

steering:  = 4,  = 4, and  = 4 for both the entire upper and the lower skin laminate. 
 22 

Figure 27. Normalized flutter dynamic pressure for a root  of 90°, and different upper skin tip (US ) 

and lower skin tip (LS ) steering:  = 4,  = 4, and  = 4 for the upper and the lower skins. 

All of the previous work in this section have used steering concepts with only two control points (wing root and 

tip), and a linear variation in-between. The final result in this section expands on this idea, with 5 control points 

total, and a piece-wise linear variation along the span. The core laminate chosen for this exercise is shown in the 

lower right corner of Figure 28, a Pareto-optimal design (in terms of weight, KS, and flutter) found above, when 

each wing skin was broken into 4 laminates, but the upper and lower skin laminates were chosen to be identical. 

This design shows the expected taper towards the wing tip, and the inner two panels happen to have the same 

thickness (though different laminates). For this exercise, the core laminate is fixed, and a steering control point 

(design variable) is placed at the edge of each panel. The NSGA-II algorithm is then used to find the 2-D Pareto 

front between KS and flutter (as above, steering leaves the wing weight unchanged) when the upper and lower skin 

steering is identical (5 continuous design variables), or distinct (10 continuous design variables). 

An asterisk in Figure 28 marks the performance of the un-steered structure, which is dominated (i.e., higher 

stress and lower flutter point) by both Pareto fronts. For both sets of GA results, steering patterns are drawn into the 

plot along various regions of the fronts; for the case where upper and lower skin steering is distinct, red lines 

correspond to the upper skin and blue to the lower skin. The results of Figure 28 provide the only results in this 

work that show a flutter-based benefit to curvilinear tow steering. The steering path for these high-flutter designs do 

not take strong advantage of the higher-ordered curves available to it (i.e., the steering angle variation from wing 

root to tip is weakly nonlinear), and so the flutter-benefit seen here, but not above, may be largely due to the tapered 

underlying laminate (i.e., thinner shell at the tip than at the root). 

Furthermore, there is little separation between the two Pareto fronts in the high-flutter region, which does echo 

previous results: flutter has only minor benefit from separate steering strategies between the upper and lower skins. 

Of course, this conclusion may be shifted by a situation where different core laminates are used in the two skins. 

The 10-design variable space is also large enough to cast doubt that the Pareto front given in Figure 28 is the true 

front (due to computational limitations), and further GA iterations may provide some separation between the two 

fronts. In the low-stress (and thus low flutter, due to the strong trade-off) region, however, there is noticeable 

benefit to using different steering strategies between the upper and lower skins. By allowing separate steering, the 

flutter point may be improved with no penalty in wing stresses, or vice-versa. It is also noted that these designs with 

separate steering and low stresses are the only ones that show a noticeable nonlinear steering angle variation from 

root to tip. Significant undulations are particularly seen in the lower skins. It may be that increasing the rib 
 23 

separation within the wing box, and thus increasing the level of local panel vibration/deformation seen during 

aeroelastic behavior, may increase the need for significant variations in the local steering angle along the wing. Of 

course, this set of conclusions, as already noted, may be heavily biased by the core laminate chosen for this exercise. 

Figure 28. CRM Pareto front: multiple laminates per skin with higher-order steering concepts. 

IV. Conclusions 

Several interesting conclusions may be drawn from the above studies. For the flat-plate problem (section II), 

the non-steered laminate (allowing each fiber angle to take any value) with the best flutter behavior experiences two 

modes which flutter at nearly the same speed. This is consistent with known attributes of efficient structures, which 

engage as many failure modes as possible at the same time. Any substantial design changes to the structure to make 

one flutter point better will degrade the other, leading to a hard corner in the design space. The advent of tow 

steering is unable to make significant improvements to this situation: in other words, increasing the design space to 

include curvilinear tow paths has a muted effect. Tsai-Wu failure stresses found in plate deformations due to steady 

aeroelastic loads, contrastingly, do show a significant improvement when tow steering is included in the problem. 

The overall stress level can be decreased, and peak stresses are spread out over a greater portion of the plate (similar 

to a fully stressed design methodology), whereas previously they had been located largely at the wing root. 

Restricting the available ply angles to only four discrete choices (0°, 90°, 45°, and -45°) may be considered for 

flight certification. For the un-steered case, it is found that there is a penalty (degradation in aeroelastic 

performance) within this restricted design space. Tow steering may still be considered, but when restricted to 

discrete fiber angle choices, it is no longer possible to steer each layer independently. Instead, a core laminate 

(composed of only the four available angles) is simply rotated along a single steering path throughout the wing. For 

both flutter and stress metrics, it is found that, while there is a penalty when the ply angles are restricted in the unsteered case, the penalty can be made up for (in some cases, almost entirely), when tow steering is introduced into 
 24 

the problem. In other words, when comparing the un-steered discrete ply angle problem with the un-steered 

continuous ply angle problem, the former benefits far more from tow steering. 

For the CRM wingbox (section III), it is found that, due to the large disparity between the depth of the wingbox 

and the thickness of the skins, the stacking sequence of the skins is largely unimportant. Given that the laminate 

parameterization used for the flat plate problem (where a few design variables are attached to each layer) is only 

possible due to the 3 designable layers (as opposed to the 50-100 which are typical of transport wing problems), a 

different parameterization was used which exploited the insensitivity to the stacking sequence. The number of 

plies at each angle is sufficient to describe the structure, rather than the exact stacking sequence. This work downselects to just 3 variables per laminate:    . This provides an automatically-balanced laminate 

and a reasonably-sized design space for sampling purposes. 

Progressively better designs, in terms of the stresses that develop throughout the wing (via the KS function) are 

obtained as more geometric detail is given to the wing structure. Specifically, different laminates may be considered 

between the upper and lower wing skins, and more than one laminate may be considered per skin. The peak stress is 

typically seen near the yehudi break of the planform, and this value does not have a strong trade-off with the weight 

of the wing. Flutter, on the other hand, is strongly conflicting with wing weight; the flutter mechanism itself is fairly 

consistent however (though the actual flutter point can shift substantially), typically consisting of the usual bendtwist coalescence. The best flutter design found during this process was always the heaviest, independent of how 

many laminates were allowed per wing. 

In many cases, both KS and (to a lesser extent) flutter were shown to benefit from the addition of tow steering. 

As above, different steering strategies for the upper and lower skins, or higher-ordered steering curves from root to 

tip, showed enhanced improvements. The trends and conclusions drawn from these CRM studies, however, are 

strongly dependent upon the core laminate used to construct the wing skins. In several cases, changing this core 

laminate in only moderate ways led to substantial changes in the trends, if not a complete reversal. The obvious 

solution is to replicate the strategy used for the flat plate problem: provide a GA with the ability to change both the 

core laminate and the steering path at the same time. The high cost of each CRM evaluation, as well as the large 

design space encountered in that scenario, make this undertaking largely impossible under current computational 

resource limitations. However, future work may consider it within the context of gradient-based optimization, 

which scales well under problems with large numbers of design variables. It may also be possible to relax the 

requirement that each ply angle take only discrete values, and quantify the associated penalty, as done in Table 3 for 

the flat plate. 

Acknowledgements 

This work is funded by the Fixed Wing project under NASA’s Fundamental Aeronautics Program. 

References 

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[2] Alhajahmad, A., Abdalla, M., Gürdal, Z., “Optimal Design of Tow-Placed Pressurized Fuselage Panels for 

Maximum Strength with Buckling Considerations,” Journal of Aircraft, Vol. 47, pp. 775-782, 2010. 

[3] Wu, K., Stanford, B., Hrinda, G., Wang, Z., Martin, R., Kim, H., “Structural Assessment of an Advanced 

Composite Tow-Steered Shell,” AIAA Structures, Structural Dynamics, and Materials Conference, Boston, 

MA, April 2013. 

[4] Haddadpour, H., Zamani, Z., “Curvilinear Fiber Optimization Tools for Aeroelastic Design of Composite 

Wings,” Journal of Fluids and Structures, Vol. 33, pp. 180-190, 2012. 

[5] Stodiek, O., Cooper, J., Weaver, P., Kealy, P., “Improved Aeroelastic Tailoring Using Tow-Steered 

Composites,” Composite Structures, Vol. 106, pp. 703-715, 2013. 

[6] Vassberg, J., DeHaan, M., Rivers, S., Wahls, R., “Development of a Common Research Model for Applied 

CFD Studies.” AIAA Applied Aerodynamics Conference, Honolulu, HI, August, 2008. 

[7] Hollowell, S., Dugundji, J., “Aeroelastic Flutter and Divergence of Stiffness Coupled, Graphite/Epoxy 

Cantilevered Plates, Journal of Aircraft, Vol. 21, No. 1, pp. 69-76, 1984. 

[8] Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., “A Fast and Elitist Multiobjective Genetic Algorithm,” IEEE 

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432, 2010. 
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[10]Niu, M., Airframe Structural Design, Conmilit Press Ltd., Hong Kong, 1988. 

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Design Optimization,” AIAA Paper 2012-5475. 

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Aeroelastic Constraints,” Journal of Aircraft, Vol. 50, No. 4, pp. 1159-1168, 2013. 

[13]Liu, B., Haftka, R., Akgün, M., “Two-Level Composite Wing Stuctural Optimization Using Response 

Surfaces,” Structural and Multidisciplinary Optimization, Vol. 20, No. 3, pp. 87-96, 2000. 

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International Federation of Active Controls Symposium on Computer-Aided Design of Control Systems, Zurich, 

Switzerland, 1979. 

[15]Tang, D., Dowell, E., “Experimental and Theoretical Study on Aeroelastic Response of High-Aspect Ratio 

Wings,” AIAA Journal, Vol. 39, pp. 1430-1441, 2001. 

[16] van Zyl, L., Maserumule, M., “Divergence and the p-k Flutter Equation,” Journal of Aircraft, Vol. 38, No. 3, 

pp. 584-586, 2001. 
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Aeroelastic Tailoring via Tow Steered Composites 

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14. ABSTRACT

The use of tow steered composites, where fibers follow prescribed curvilinear paths within a laminate, can improve upon 

existing capabilities related to aeroelastic tailoring of wing structures, though this tailoring method has received relatively little 

attention in the literature. This paper demonstrates the technique for both a simple cantilevered plate in low-speed flow, as well

as the wing box of a full-scale high aspect ratio transport configuration. Static aeroelastic stresses and dynamic flutter 

boundaries are obtained for both cases. The impact of various tailoring choices upon the aeroelastic performance is quantified:

curvilinear fiber steering versus straight fiber steering, certifiable versus noncertifiable stacking sequences, a single uniform 

laminate per wing skin versus multiple laminates, and identical upper and lower wing skins structures versus individual 

tailoring. 

15. SUBJECT TERMS

Aeroelasticity; High aspect ratio; Laminate; Steering; Wings

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