NASA/TM—2015–218358

 

Wing Shape Sensing from Measured Strain 

Chan-gi Pak

Armstrong Flight Research Center, Edwards, California

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January 2015

PATENT PROTECTION NOTICE

Patent is pending for System and Method for Monitoring the Deflection and Slope of a ThreeDimensional Structure such as a Wing Using Strain Measurements at Discrete Locations, Patent 

Application Number 14/482784. Therefore, those interested in using the system and method should 

contact the NASA Technology Transfer Office at NASA Armstrong Flight Research Center, 

Edwards, California, for more information. 
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NASA/TM—2015–218358

 

Wing Shape Sensing from Measured Strain 

Chan-gi Pak

Armstrong Flight Research Center, Edwards, California 

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National Aeronautics and

Space Administration 

Armstrong Flight Research Center

Edwards, CA 93523-0273 

January 2015
Acknowledgments 

The work presented in this report was funded by the Aero Science and Fixed Wing Projects under the 

ARMD Fundamental Aeronautics Program. The author also thanks an undergraduate student research 

program student, Mr. David McWilliams, from Rensselaer Polytechnic Institute (Troy, New York) for his 

support during the summer intern session in 2013. 

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Available from:

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NASA Center for AeroSpace Information

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1

Abstract

A new two-step theory is investigated for predicting the deflection and slope of an entire structure 

using strain measurements at discrete locations. In the first step, a measured strain is fitted using a 

piecewise least-squares curve fitting method together with the cubic spline technique. These fitted strains 

are integrated twice to obtain deflection data along the fibers. In the second step, computed deflection 

along the fibers are combined with a finite element model of the structure in order to interpolate and 

extrapolate the deflection and slope of the entire structure through the use of the System Equivalent 

Reduction and Expansion Process. The theory is first validated on a computational model, a cantilevered 

rectangular plate wing. The theory is then applied to test data from a cantilevered swept-plate wing 

model. Computed results are compared with finite element results, results using another strain-based 

method, and photogrammetry data. For the computational model under an aeroelastic load, maximum 

deflection errors in the fore and aft, lateral, and vertical directions are -3.2 percent, 0.28 percent, and 

0.09 percent, respectively; and maximum slope errors in roll and pitch directions are 0.28 percent

and -3.2 percent, respectively. For the experimental model, deflection results at the tip are shown to be 

accurate to within 3.8 percent of the photogrammetry data and are accurate to within 2.2 percent in most 

cases. In general, excellent matching between target and computed values are accomplished in this study. 

Future refinement of this theory will allow it to monitor the deflection and health of an entire aircraft in 

real time, allowing for aerodynamic load computation, active flexible motion control, and active induced 

drag reduction.

Nomenclature

AFRC Armstrong Flight Research Center

cp pressure coefficient

ܿ(s) vertical distance from neutral axis to surface at s, inch

DOF degrees of freedom

ܧ Young’s modulus, Msi

FBG fiber Bragg grating

FE finite element

FOSS fiber optic strain sensing

NASA National Aeronautics and Space Administration

{ࢗ෥ࡹ {measured master degrees of freedom

{ࢗ)࢚ {(generalized displacement vector at time (t)

{ࡹࢗ)࢚ {(master degrees of freedom at time (t)

{ࡿࢗ)࢚ {(slave degrees of freedom at time (t)

SEREP System Equivalent Reduction Expansion Process

s distance along the fiber, inch

[ࢀ [transformation matrix

ߥ Poisson’s ratio

ݓ)ݏ (deflection at s, inch

߳௔(s) axial strain at s

߳(s) strain at s

ߢ)s) curvature at s, in-1

)ࡹ eigen-matrix corresponds to master degrees of freedom

)ࡿ eigen-matrix corresponds to slave degrees of freedom

(*)T transpose of a matrix

(*)-1 inverse of a matrix
2

Introduction

Real-time measurement of the deflection and slope of aerospace structures in flight is a valuable tool 

for improving aircraft performance, safety, and efficiency. Potential applications include structural health 

monitoring, flexible motion control, induced drag reduction through active aerodynamic tuning, real-time 

aerodynamic load calculation, and virtual display of structure motion. Flight-test methods of deflection 

measurement can use cameras (ref. 1) and while these methods can be effective for test flight, they are 

impractical for permanent installation and long term reliability. In addition, they are only able to measure 

deflection at discrete locations.

Previous research describes at least two methods of calculating internal loads on an aircraft during 

flight. The Ko Displacement theory (ref. 2), set forth in (U.S. Patent Number 7,520,176) (ref. 3) and 

(U.S. Patent Number 7,715,994) (ref. 4), uses closed-form equations to calculate real-time operational 

loads on complex structures using surface strain data. Another calculation method uses a matrix equation 

based on the displacement and strain mode shape matrices (ref. 5). Both of the above-described prior art 

methods are suitable only for calculating deflection and load values in cross-sectional lumps based on the 

location of the sensors used to measure strain on the surface.

The method described by Moore (ref. 6), (U.S. Patent Number 7,813,599), includes the steps of 

deducing curvature and bend radius data for the fiber from the strain data; curve-fitting to obtain 

curvature and bending direction functions pertaining to the fiber; calculating a torsion function using the

bending direction function; and calculating the three-dimensional shape of the optical fiber from the 

combination of the curvature, and bending and torsion functions. Park et al. (ref. 7) uses beam theory and 

fits a curvature function to a bending needle using a simple second order polynomial fit appropriate for 

just two sensor locations.

The National Aeronautics and Space Administration (NASA) Helios Prototype was an ultra-light 

flying wing with a 247-foot wingspan that was developed and tested by NASA in the late 1990s and early 

2000s. Due to the light weight and large wingspan of the Helios, it experienced a large dihedral while in 

flight. During a test flight in 2003, the Helios broke up into the Pacific Ocean after the deformation 

became too large. Stability analysis near the terminal condition indicated a strong tendency to exhibit 

over-damped, integrator-type static deflection in the structure due to external loading. This accident 

emphasized the need for monitoring real-time deflection information on lightweight aircraft (ref. 8).

The light weight and accuracy of fiber optic sensors, such as fiber Bragg gratings (FBGs), make them 

an ideal choice for aerospace applications. This statement is especially true on lightweight aerospace 

structures, where conventional strain-gauges and associated cabling would present an issue in both weight 

and bulk. Recently developed fiber optic strain sensing (FOSS) technology uses a series of FBGs to 

obtain measurements at intervals as small as every half inch along a fiber and at frequencies of several 

kHz (ref. 9). The ability of FBGs to operate at such high frequencies makes them useful for both static

and dynamic aerospace applications. 

The availability of wing deflection and slope data, and the corresponding internal and external load 

values, at all element grid points across the structure will allow engineers to undertake more accurate, 

real-time analyses of both internal and external loads at any point on the structure, and will enable 

real-time virtual display of wing motion. Real-time internal and external load values over the entire 

surface of a structure may also find application in structural health monitoring, active flexible motion 

control, and active induced drag reduction. 

At present the above-described prior art methods are suitable only for calculating deflection and load 

values along singular fibers or in localized areas based on the location of the sensors used to measure 
3

strain on the surface. What is needed is a way to compute the deflected shape of an entire three 

dimensional structure using strain measurements from discrete locations. This would vastly improve the 

ability to monitor the health and behavior of a complex three dimensional structure such as an aircraft in 

flight.

It is, therefore, an object of the present study to provide a system and method for calculating wing 

deflection and slope over the entire surface of a three dimensional structure, as opposed to finite points 

thereon as in previous research. This paper focuses on a strain-based method that integrates strain data to 

compute out of plane deflection along fibers, and then uses a finite element (FE) model to interpolate and 

extrapolate all the deflections and slopes of the entire structure. Two experiments are completed in this 

study. The first experiment investigates the accuracy of the theory by applying it to a cantilevered 

rectangular wing model analyzed using the MSC/NASTRAN code (MSC Software Corporation, Newport 

Beach, California) (ref. 10). The second experiment applies the theory to experimental data collected 

from a test plate fabricated and tested at the NASA Armstrong Flight Research Center (AFRC) in 

Edwards, California.

Mathematical Background of the Two-step Theory

The structural dynamics group at NASA AFRC have developed a two-step theory and accompanying 

software to predict the deflection and slope of an entire structure using strain measurements from discrete 

locations. Figure 1 shows a block diagram of the overall system architecture. This paper is mainly focused

on the deflection analyzer and expansion module. As indicated in the introduction the wing is equipped 

with FOSS segments for measuring strain at multiple locations on the wing structure. The strain is 

computed at each (x, y, z) location of the FBG sensors in real time. The system architecture adapted for 

an aircraft is essentially a computerized feedback loop capable of executing the various software modules,

including a strain assembler software module for accumulating, indexing, and storing measured strain 

values in a strain matrix organized by the FOSS segment from which they were derived. Assembler 

software then passes the strain matrix to a deflection analyzer which calculates deflection along the 

optical fibers by the curve fitting techniques in section A.

First Step (Deflection Analyzer)

In the first step, a structure is fit with fiber optic cables that take strain measurements at several points 

along the length using AFRC FOSS technology. For experimental data, strain measurements along each 

fiber are divided into segments, and each segment is curve-fit using a piecewise least-squares method in 

order to minimize measurement noise. After noise reduction, an Akima spline is run through the strain 

data and is numerically integrated twice to compute the slope and deflection along each fiber. The first 

step can be summarized as follows:

1. Fit the strain/offset (or curvature) data using the piecewise least-squares curve fitting method 

to minimize noise in measured data. A coordinate system used in this study is shown in 

figure 2.

2. Obtain a cubic spline function using re-generated curvature data points as shown in figure 3,

(ݏ)ݓଶ݀

(1) (ݏ)/ܿ(ݏ)߳െ) = ݏ)ߢ = ଶݏ݀

where, ߳(ݏ (and c(ݏ (are strain and vertical offset values at s, respectively, and s is along the 

fiber direction. A small motion assumption is used in equation (1). In figure 2, the radius of 

the fiber is assumed to be small enough compared with offset distance ܿ(ݏ(.
4

3. Integrate the fitted spline function to get slope data: ௗ௪(௦)

ௗ௦ (slope in fiber direction).

4. Obtain a cubic spline function using computed slope data.

5. Integrate the fitted spline function to get the deflection data: ݓ)ݏ(

The calculated deflection data ݓ)ݏ (along each FOSS segment is then passed to an expansion module 

which expands the calculated deflection data to all points along the three dimensional wing surface. This 

expansion procedure is explained in section B.

Second Step (Expansion Module)

In the second step, computed deflection along the fibers is combined with a FE model of the structure 

in order to interpolate, and extrapolate the deflection and slope of the entire structure through the use of 

System Equivalent Reduction and Expansion Process (SEREP) (ref. 11). Re-arrange all the degrees of 

freedom (DOF) in the FE model as equation (2):

൜ = {(࢚)ࢗ}

(࢚)ࡹࢗ

(࢚)ࡿࢗ

ൠ (2)

where, {ࡹࢗ)࢚ {(is the master DOF. In this approach, deflections and slopes along the fiber obtained from 

the first step are defined as the master DOF. The rest of the deflections and slopes all over the structure 

are defined as slave DOF, {ࡿࢗ)࢚{(. Consider the following coordinate transformation for the model 

reduction and expansion techniques as equation (3):

൜

(࢚)ࡹࢗ

(࢚)ࡿࢗ

(3) {(࢚)ࡹ෥ࢗ}[ࢀ] = ൠ

where, {ࢗ෥ࡹ)࢚ {(is the measured master DOF which was computed from the first step. From SEREP, the 

transformation matrix [ࢀ [can be expressed as equation (4):

൥ = [ࢀ]

ࡹ(൫ࡹ(

൯ࡹ( ୘

ିଵ

ࡹ(

୘

ࡹ(൫ࡿ(

൯ࡹ( ୘

ିଵ

ࡹ(

୘ ൩ (4)

where, )ࡹ and )ࡿ are eigen-matrices corresponding to master and slave DOFs, respectively. These 

eigen-matrices are computed from the FE model of the test article using the MSC/NASTRAN code. From 

equations (2) through (4), all the DOF in the structure can be computed using equation (5):

൜ = {(࢚)ࢗ}

(࢚)ࡹࢗ

(࢚)ࡿࢗ

ൠ = ൥

ࡹ(൫ࡹ(

൯ࡹ( ୘

ିଵ

ࡹ(

୘

ࡹ(൫ࡿ(

൯ࡹ( ୘

ିଵ

ࡹ(

(5) {(࢚)ࡹ෥ࢗ}൩୘ 

The first and second steps are summarized in figure 4. While several methods have been developed to 

compute deflection along single fibers, the innovative ability of the method proposed in this study to 

predict the deflection and slope of an entire structure makes it an excellent candidate for monitoring the 

health and behavior of a complex aircraft in flight.
5

Results and Discussions

The theory is first validated on a computational model, a cantilevered rectangular plate. The theory is 

then applied to test data from a cantilevered swept-plate model. Computed results are compared with 

finite element results, results using another strain-based method, and photogrammetry data.

Computational Validation

For initial validation and refinement of this theory, three test load cases are run on a rectangular plate

FE model created and analyzed using the MSC/NASTRAN and ZAERO (Zona Technology Inc., 

Scottsdale, Arizona) (ref. 12) codes. The model has a uniform chord length of 4.56 inches, a span length 

of 11.5 inches, a thickness of 0.065 inches as shown in figure 5(a). The material property is assigned a 

Young’s modulus, ܧ, of 9.848 Msi; a Poisson’s ratio, ߥ, of 0.3529; and a mass density of 0.1117 lb/in3

(ref. 13). To represent fiber optic strain sensors, the model is fit with 22 beam elements that compute axial 

strain along the length of the wing. Because strain measurements come directly from the FE model 

solution, no piecewise least squares curve-fitting is necessary to eliminate noise.

Three load cases are used: a uniform 1g load, shown in figure 5(b); a wing tip torsion load, shown in 

figure 5(c); and an aerodynamic load corresponding to 1° angle of attack at Mach 0.715 and dynamic 

pressure of 1.4 psi, shown in figure 5(d). For each configuration, deflection and slope are computed over 

the entire structure and compared with the deflection and slope calculated by the MSC/NASTRAN code 

directly. In case of the last load case, ZAERO code is used to create trimmed aerodynamic load vectors at 

Mach 0.715.

Under 1g Load

Figure 6(a) shows the curvature distribution for fibers 1 and 11 computed by the MSC/NASTRAN

code and the two-step approach after the first step under a 1g load. Corresponding wing deflections along 

the fibers 1 and 11, and over the FE model are shown in figures 6(b) and 6(c), respectively. Wing tip 

deflections along the fibers 1, 3, 5, 7, 9, and 11 are given in table 1. Target data in this table are computed 

using the MSC/NASTRAN code. As shown in table 1, the wing tip deflection errors in this sample 

problem under a 1g load are equal to and less than 0.06 percent and 0.8 percent after the first and the 

second steps, respectively. In this study, strain sensors are located quadratically toward the root of the 

wing in order to yield a more accurate result for a given number of sensors. For each fiber, a total of 50 

strain sensors are used in this rectangular wing problem.

Figure 6(d) shows wing slope in roll direction along the fibers 1 and 11, respectively. Wing tip slope 

along the fibers 1, 3, 5, 7, 9, and 11 are given in table 2. Similar trends occurred in the slope data as had in 

the deflection data. Errors of the wing tip slope under a 1g load are 0.1 percent and 1.3 percent after the 

first and the second steps, respectively. In general, normalized wing tip slope error for a given setup is

roughly twice larger than the corresponding deflection error. Relatively small computation errors are 

observed in the case of the 1g load condition.

Under Wing Tip Torsion Load (1 lb upward at the leading-edge and 1 lb downward at the 

trailing-edge)

Figure 7(a) shows the curvature distribution for fibers 1, 3, 5, 7, 9, and 11 computed by the 

MSC/NASTRAN code and the two-step approach after the first step under the wing tip torsion condition. 

Corresponding wing deflections after the first step and over the FE model after the second step are shown 

in figures 7(b) and 7(c), respectively. Figure 7(d) shows wing slope in roll direction along the fibers 1, 3, 

5, 7, 9, and 11 after the first step. Wing slopes are in excellent agreement with the target

MSC/NASTRAN results after the first step.
6

Wing tip deflections and roll slopes along the fibers 1, 3, 5, 7, 9, and 11 after the first and the second 

steps are given in tables 3 and 4, respectively. The second step in these tables are based on the first 50 

modes. The number of modes in the second step is selected based on the convergence study in this 

section. The wing tip deflection errors are 1.9 percent and 1.6 percent after the first and the second steps, 

respectively.

The effect of the number of modes on the wing deflection and slopes are studied and results are shown 

in figures 8 through 10. Wing deflections are not affected too much on the number of modes as shown in 

figure 8. However in figure 9, wing slopes in the roll direction near the leading-edge of the wing tip 

section are noticeably affected. Torsional deformation due to a concentrated load at the wing tip is 

difficult to fit with a small number of modes. When the total number of modes for the second step 

increases, then this wiggling near the leading-edge of the wing tip becomes smoother. Due to the same 

reason, wing slopes in the pitch direction near the leading-edges of the wing tip section in figure 10 are 

affected by the number of modes for the second step. Wing slopes in the pitch direction are not computed 

during the first step. Compare the figure 10(a) spread of data near the leading-edge of the wing tip section 

to that of figure 10(d).

Target wing tip slopes computed using the MSC/NASTRAN code are compared with the 

corresponding computed values in tables 4 and 5. As shown in table 4, the maximum of 0.12 percent error 

in the wing tip slope in the roll direction becomes a second step maximum of -7.9 percent error. A

maximum error of 6.1 percent is observed for the wing slope in the pitch direction as shown in table 5. In 

this study, 1 lb upward and 1 lb downward loads are applied to simulate wing tip torsion. Relatively large 

computation errors compared to the 1g load condition are observed near the applied load point. It should 

be noted that only the second step is a FE model dependent approach.

Under 1° Angle of Attack at Mach 0.715

In this study, the ZAERO code is used to have aeroelastic wing deformations. Figure 11(a) shows 

pressure distribution over the rectangular wing under 1° angle of attack at Mach 0.715 and a dynamic 

pressure of 1.4 psi. Pressure in figure 11(a) is splined to have an applied aerodynamic load vector at the 

structural grid points as shown in figure 11(b). The MSC/NASTRAN code is used to compute target 

strain distributions, as shown in figure 5(d); and the target and computed curvature distributions for fibers

1, 5, 9, 13, 17, and 21 after the first step are shown in figure 12(a). In the case of a two nodes beam 

element, strains are calculated at the center of each beam, and therefore, extrapolation of curvature is 

needed at two end points of each fiber. The length of a beam element near the wing tip area is relatively 

large, and therefore, small extrapolation errors are observed at the wing tip section.

The Z deflections along each fiber from the first step are used in the second step to compute wing 

deflections and slopes at all the grid points in the FE model. Three dimensional wing deformation shape 

is shown in figure 12(b). Figure 13 shows a comparison between target and computed wing deformation. 

In figures 13(a) through 13(f), it can be concluded that not only Z deflection, but also X and Y 

deflections; and roll, pitch, and yaw slopes all have excellent matching with target values. Correlations at 

the wing tip section are summarized in table 6. The maximum relative error for Z deflection after the 

second step is 0.09 percent. On the other hand, maximum relative errors for Y and X deflections are 

0.28 percent and -3.2 percent, respectively. A similar relative error trend is also observed for slope results 

in table 6. The maximum relative error for slope in roll is 0.28 percent as with the Y deflection; and the 

maximum relative error for slope in pitch is -3.2 percent, the same as the X deflection. In yaw direction, 

figure 13(f), target and computed results are essentially zero. It should be noted in table 6 that the small Z 

deflection errors caused by extrapolation at the wing tip section after the first step are smoothed after the 

second step. Therefore, the average relative error 0.07 percent after the first step becomes 0.04 percent

after the second step.
7

Experimental Testing

Once the theory has been validated on the computational model in the previous section, the theory is

applied to experimental data from a test plate fabricated and tested at NASA AFRC. The test plate has a 

uniform chord length of 12 inches, leading- and trailing-edge lengths of 50 inches, a thickness of 

0.19 inches, and is backward-swept at a 45° angle. Test plate is made of 6061-T6 aluminum with a 

Young’s modulus of 10 Msi, a Poisson’s ratio ߥ of 0.3, and a mass density of 0.098 lb/in3

. A FE model

with two load cases and the test setup are shown in figures 14 and 15, respectively.

The top and bottom surfaces of the test plate are each fitted with three fiber optic cables. Each surface 

has one fiber located 0.5 inches from the leading-edge, one located 0.5 inches from the trailing edge, and 

one located at the mid-chord. Along each fiber are white dots used for photogrammetry imaging as shown 

in figure 15. Load cases are applied by hanging calibrated weights to different points along the leading 

and trailing edges of the plate. Test data under a leading-edge and uniform load cases, as shown in figure

14, are used in this study.

It is recommended that the fibers on an experimental model should be extended all the way to the root 

to minimize the computation error since the boundary conditions are applied at the root section of a wing. 

One correction that needs to be made to the model is due to large deflection. For a beam in pure bending, 

the curvature is given by equation (6) where ߳(ݏ (is the surface strain, and ܿ(ݏ (is the distance from the 

neutral axis to the surface.

(6) (ݏ)/ܿ(ݏ)߳െ) = ݏ)ߢ

For the experimental model, the large deflection created an axial load in addition to a bending load. 

The resulting strain is given by equation (7).

(7) (ݏ)௔߳ + (ݏ)ܿ(ݏ)ߢെ) = ݏ)߳

To account for the non-linear effect, the curvature yielded by equation (6) cannot be integrated directly 

to compute deflection and slope. Averaging the curvatures calculated by using each fiber individually 

eliminates the effect of the axial load. This computation is performed after curve-fitting each set of data 

individually to minimize noise. 

To reduce noise, each set of raw strain measurements is divided into five span-wise segments which 

are each fit with cubic polynomials calculated using piecewise least squares curve fitting. An actual fit 

along the upper and lower leading-edge fibers are shown in figure 16. Cubic polynomials are used 

because they are capable of following the data without creating too much wiggling. Five segments are 

chosen because it is the lowest number of segments that allows the cubic polynomials to reasonably 

follow the data.

Complete results can be found in figures 17 and 18 which show the deflection computed after the first 

step and the second step with 10 modes. Results are compared to photogrammetry data and also to the 

solution computed by Bakalyar and Jutte (ref. 14) along each fiber. Results along the fibers are similar in 

accuracy to the referenced results (ref. 14), and the general results smoothly interpolate the entire surface. 

This smoothness is especially apparent in figure 19, which shows a close-up of the result from the FE 

model after the second step with 10 modes. Along the fibers, computed results after the first step are as or 

more accurate than the referenced results. Table 7 shows the computed tip deflection along each fiber 

compared to the deflection measured using photogrammetry. In general, the computed results are least 

accurate along the trailing-edge fiber. The maximum absolute error of about 0.20 inches occurred on the 
8

trailing-edge fiber during the leading-edge and uniform load cases. The maximum relative error 

of -3.8 percent occurred on the trailing-edge fiber during the leading-edge load case.

Most of the error is experienced on the trailing-edge fiber, and it is clear from the data that the error 

begins near the root and is propagated down the length of the wing. A possible explanation for this error 

at the root is the steep gradient in strain that occurs in this region as shown in figure 20. It should be noted 

that the largest strain gradient exists near the trailing-edge of the wing root section. Due to the limited 

number of strain sensors that are used, it is likely that the data obtained near the root are too coarse to 

accurately integrate the strain along the trailing edge fiber.

Conclusion

The two-step approach for computing all the degrees of freedom in a structural FE model from 

measured strain along the FOSS is successfully applied to a cantilevered rectangular wing model and a 

test plate. The first experiment investigates the accuracy of the theory by applying it to a cantilevered 

rectangular wing model analyzed using the MSC/NASTRAN and ZAERO codes. The wing tip deflection 

errors under a 1g load are equal to or less than 0.06 percent and 0.8 percent after the first and second 

steps, respectively. Errors of the wing tip slope in roll direction under a 1g load are 0.1 percent and 

1.3 percent after the first and second steps, respectively.

On the other hand, the wing tip deflection errors under a wing tip torsion load are 1.9 percent and 

1.6 percent after the first and second steps, respectively. If we exclude the leading-edge fiber case, then 

the maximum errors are -0.61 percent and 0.30 percent after the first and second steps, respectively. Wing 

slope errors in the pitch and roll directions are usually twice larger than the wing deflection errors.

Maximum errors of 0.12 percent and 7.9 percent are observed for errors in roll direction after the first and 

second steps, respectively. The larger error for the results from the second step is mainly because the 

torsional deformation, due to a concentrated load at the leading-edge of wing tip, is difficult to fit with a

small number of modes. Therefore, a large number of modes are needed to fit the plate deflection during 

the second step. This statement is also true for wing slope in the pitch direction which has a maximum 

error of 6.1 percent.

The two-step approach is tested with aeroelastic wing deformation in the last load case. Target 

aeroelastic deformations are obtained from MSC/NASTRAN and ZAERO codes. All six DOF computed 

and compared with target values. Maximum deflection errors in X, Y, and Z directions are -3.2 percent,

0.28 percent, and 0.09 percent, respectively, and maximum slope errors in the roll and pitch directions are 

0.28 percent and -3.2 percent, respectively. It should be noted that all DOF in the FE model of the 

rectangular wing are computed based only on the wing deflection in the Z direction computed from the 

first step. These computed DOFs have excellent matching with target values.

The second experiment applies the theory to experimental data collected from a test plate fabricated 

and tested at the NASA AFRC. The deflections calculated from the experimental model are extremely 

accurate. Average absolute errors at the tip are less than 1.7 percent and 1.9 percent after the first and 

second steps, respectively.
9

Tables

Table 1. Target data versus computed wing deflection in Z direction at the wing tip section under 1g 

load.

Fiber number Target (inch) Computed deflection (inch) Relative error (percent)

Step 1 Step 2 Step 1 Step 2

1 -0.06638 -0.06634 -0.06624 -0.06 -0.21

3 -0.06641 -0.06641 -0.06616 0.00 -0.38

5 -0.06643 -0.06644 -0.06608 0.02 -0.53

7 -0.06645 -0.06646 -0.06602 0.02 -0.65

9 -0.06646 -0.06647 -0.06597 0.02 -0.74

11 -0.06647 -0.06648 -0.06596 0.02 -0.77

Table 2. Target data versus computed wing slope in roll direction at the wing tip section under 1g load.

Fiber number Target Computed slope Relative error (percent)

Step 1 Step 2 Step 1 Step 2

1 -0.007812 -0.007817 -0.007759 0.06 -0.68

3 -0.007811 -0.007818 -0.007752 0.10 -0.75

5 -0.007809 -0.007817 -0.007737 0.10 -0.91

7 -0.007807 -0.007815 -0.007721 0.10 -1.10

9 -0.007806 -0.007813 -0.007709 0.09 -1.20

11 -0.007806 -0.007813 -0.007704 0.09 -1.30

Table 3. Target data versus computed wing deflection in Z direction at the wing tip section under wing 

tip torsion.

Fiber 

number

Target 

(inch)

Computed deflection (inch) Relative error (percent)

Step 1 Step 2 with 50 modes Step 1 Step 2 with 50 modes

1 0.07163 0.07024 0.07050 -1.90 -1.60

3 0.05727 0.05709 0.05692 -0.30 -0.61

5 0.04293 0.04292 0.04295 -0.02 0.06

7 0.02861 0.02862 0.02869 0.05 0.28

9 0.01430 0.01431 0.01433 0.07 0.22

11 0.00000 0.00000 0.00000 0.00 0.00

Table 4. Target data versus computed wing slope in roll direction at the wing tip section undergoing 

wing tip torsion.

Fiber 

number Target Computed wing slope Relative error (percent)

Step 1 Step 2 with 50 modes Step 1 Step 2 with 50 modes

1 0.006769 0.006775 0.006457 0.093 -4.60

3 0.005483 0.005490 0.005712 0.120 4.20

5 0.004126 0.004131 0.004337 0.110 5.10

7 0.002754 0.002757 0.002734 0.120 -0.71

9 0.001377 0.001379 0.001268 0.120 -7.90

11 0.000000 0.000000 0.000000 0.000 0.00
10

Table 5. Target data versus computed wing slope in pitch direction after the second step at the wing tip 

section undergoing wing tip torsion.

Fiber number Target Computed wing slope with 50 modes Relative error 

(percent)

1 0.03148 0.02956 -6.10

3 0.03144 0.03014 -4.10

5 0.03138 0.03099 -1.20

7 0.03135 0.03141 0.19

9 0.03134 0.03143 0.32

11 0.03133 0.03138 0.17
11

Table 6. Target data versus computed wing deformation at the wing tip section under 1° angle of attack 

at Mach 0.715.

Fiber number Target (inch) Computed X deflection (inch) Relative error (percent)

Step 1 Step 2 with 10 modes Step 1 Step 2 with 10 modes

1 0.0006927 N/A 0.0006703 N/A -3.2

5 0.0006975 N/A 0.0006776 N/A -2.8

9 0.0007029 N/A 0.0006936 N/A -1.3

13 0.0007047 N/A 0.0007081 N/A 0.48

17 0.0007035 N/A 0.0007144 N/A 1.6

21 0.0007023 N/A 0.0007143 N/A 1.7

Fiber number Target (inch) Computed Y deflection (inch) Relative error (percent)

Step 1 Step 2 with 10 modes Step 1 Step 2 with 10 modes

1 -0.003280 N/A -0.003282 N/A 0.08

5 -0.003272 N/A -0.003278 N/A 0.18

9 -0.003260 N/A -0.003268 N/A 0.26

13 -0.003248 N/A -0.003257 N/A 0.28

17 -0.003239 N/A -0.003246 N/A 0.23

21 -0.003235 N/A -0.003242 N/A 0.20

Fiber number Target (inch) Computed Z deflection (inch) Relative error (percent)

Step 1 Step 2 with 10 modes Step 1 Step 2 with 10 modes

1 0.9280 0.9269 0.9275 -0.12 -0.06

5 0.9085 0.9091 0.9087 0.07 0.02

9 0.8889 0.8894 0.8895 0.06 0.08

13 0.8691 0.8696 0.8699 0.05 0.09

17 0.8493 0.8497 0.8498 0.05 0.06

21 0.8296 0.8300 0.8297 0.04 0.01

Fiber number Target Computed roll slope Relative error (percent)

Step 1* Step 2 with 10 modes Step 1 Step 2 with 10 modes

1 0.10090 0.1010 0.10100 0.12 0.08

5 0.10070 0.1016 0.10090 0.96 0.18

9 0.10030 0.1012 0.10060 0.94 0.26

13 0.09993 0.1009 0.10020 0.93 0.28

17 0.09966 0.1006 0.09989 0.92 0.23

21 0.09954 0.1004 0.09974 0.88 0.20

Fiber number Target Computed pitch slope Relative error (percent)

Step 1 Step 2 with 10 modes Step 1 Step 2 with 10 modes

1 0.02131 N/A 0.02063 N/A -3.2

5 0.02146 N/A 0.02085 N/A -2.9

9 0.02163 N/A 0.02134 N/A -1.3

13 0.02168 N/A 0.02179 N/A 0.5

17 0.02165 N/A 0.02198 N/A 1.5

21 0.02161 N/A 0.02198 N/A 1.7

Fiber number Target Computed yaw slope Absolute error

Step 1 Step 2 with 10 modes Step 1 Step 2 with 10 modes

1 2.2e-31 N/A 7.9e-18 N/A 0.0000

5 1.9e-31 N/A 6.6e-18 N/A 0.0000

9 1.7e-31 N/A 6.1e-18 N/A 0.0000

13 1.7e-31 N/A 5.7e-18 N/A 0.0000

17 1.4e-31 N/A 5.9e-18 N/A 0.0000

21 1.2e-31 N/A 3.3e-18 N/A 0.0000

*: Roll slope without effect of X and Y deflections (These slopes are not used during step 2 computation.)
12

Table 7. Computed wingtip deflection versus photogrammetry data.

Measured 

(inch)

Computed (inch) Relative error (percent)

Bakalyar 

and Jutte* Step 1

Step 2 

with 10 

modes

Bakalyar 

and Jutte Step 1

Step 2 

with 10 

modes

Leading-edge load

Leading edge fiber -4.525 -4.500 -4.542 -4.569 -0.55 0.38 0.97

Middle fiber -4.912 -4.952 -4.880 -4.843 0.81 -0.65 -1.40

Trailing edge fiber -5.300 -5.067 -5.091 -5.097 -4.40 -3.90 -3.80

Uniform load

Leading edge fiber -6.541 -6.546 -6.630 -6.684 0.08 1.40 2.20

Middle fiber -7.256 -7.408 -7.313 -7.238 2.10 0.79 -0.25

Trailing edge fiber -7.971 -7.667 -7.750 -7.763 -3.80 -2.80 -2.60

*: extrapolated result
13

Figures

Figure 1. Block diagram of the overall system architecture.

Figure 2. Coordinate system used in this study.
14

Figure 3. Effect of piecewise least-squares curve fitting before cubic spline curve fitting.

Figure 4. Steps used to compute deflection and slope.
15

Figure 5. Cantilevered rectangular wing and strain (y component) distributions over the upper surface of 

the wing model.

Figure 6. Curvature, wing deflection, and slopes of the rectangular wing model along Fibers 1 and 11 

under a 1g load.
16

Figure 7. Curvature distribution, wing deflection, and slopes of the rectangular wing model under a wing 

tip torsion load.
17

Figure 8. Effect of the number of modes on the wing deflection during the second step.
18

Figure 9. Effect of the number of modes on the wing slope in roll direction during the second step.
19

Figure 10. Effect of the number of modes on the wing slope in pitch direction during the second step.

Figure 11. Pressure distribution and corresponding splined load vector under 1° angle of attack at 

Mach 0.715.
20

Figure 12. Curvature distribution and deformed wing shape of the rectangular wing under 1° angle of 

attack at Mach 0.715.
21

Figure 13. Six DOF wing deformation comparison between MSC/NASTRAN and the two-step approach 

under 1° angle of attack at Mach 0.715.
22

Figure 14. FE model with two load cases for the test plate.

Figure 15. Test setup for the test plate.
23

Figure 16. Piecewise least squares curve fit of curvature data along the leading-edge fiber for the test 

plate.
24

Figure 17. Deflection results under the leading-edge load case for the test plate.
25

Figure 18. Deflection results under a uniform load case for the test plate.
26

Figure 19. Deformed wing shapes (step 2 results with 10 modes) for the test plate.

Figure 20. Stress distributions near the root under the leading-edge load case for the test plate.
27

References

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