NASA/TP-2010 -214656

Methods for In-Flight Wing Shape Predictions of 

Highly Flexible Unmanned Aerial Vehicles: 

Formulation of Ko Displacement Theory 

William L. Ko and Van Tran Fleischer

NASA Dryden Flight Research Center

Edwards, California 

August 2010
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NASA/TP-2010-214656

Methods for In-Flight Wing Shape Predictions of 

Highly Flexible Unmanned Aerial Vehicles: 

Formulation of Ko Displacement Theory 

William L. Ko and Van Tran Fleischer

NASA Dryden Flight Research Center

Edwards, California 

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273

August 2010
The structural shape prediction method described in this report is protected under U.S. Patent 

No. 7,520,176, issued April 21, 2009. Therefore, those interested in using the method should contact the 

NASA Innovative Partnership Program Office at the Dryden Flight Research Center for more 

information.

 Cover art: NASA Dryden Flight Research Center, NASA Photograph: ED01-0209-5

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Available from:

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443-757-5802
ABSTRACT

Displacement equations are developed for a cantilever tubular wing spar under bending, torsion, and 

combined bending and torsion loading. The displacement equations are expressed in terms of strains 

measured at multiple sensing stations equally spaced on the surface of the wing spar. The bending and 

distortion strain data can then be input to the displacement equations to calculate slopes, deflections, and 

cross-sectional twist angles of the wing spar at the strain-sensing stations for generating the deformed 

shapes of flexible aircraft wing spars. The displacement equations have been successfully validated for 

accuracy by finite-element analysis. The strain-displacement theory that has been developed could also be 

applied to calculate the deformed shape of simple and tapered beams, plates, and tapered cantilever wing 

boxes. The displacement equations and associated strain-sensing system (such as fiber optic sensors) form 

a powerful tool for in-flight deformation monitoring of flexible wings and tails, such as those often 

employed on unmanned aerial vehicles. Ultimately, the calculated displacement data can be visually 

displayed in real time to the ground-based pilot for monitoring the deformed shape of unmanned aerial 

vehicles during flight. 

NOMENCLATURE

A cross-sectional area of uniform cantilever beam, in2

a mean radius of four-ply composite tube wall and aluminum tube wall, or mean radius of Helios 

31-ply reinforced spar cap wall, in.

c depth factor of beam (distance between neutral axis to bottom surface of beam), in.

deg degree

E Young’s modulus, lb/in2

EL Young’s modulus of lamina in fiber direction, lb/in2

ET Young’s modulus of lamina in direction transverse to fiber direction, lb/in2

e finite-element span-wise length, in.

G shear modulus, lb/in2

GLT shear modulus of lamina, lb/in2

I moment of inertia of tube cross section, in4

i identification number for strain-sensing station (= 0, 1, 2, 3,..., n)

J polar moment of inertia, in4

j index

l length of cantilever tube, in.

M moment, in-lb

Mi moment at strain-sensing station, xi , in-lb

n identification number for the last strain-sensing station, xn(= l)

P applied load, lb

SPAR Structural Performance And Resizing

Ti twisting moment at distortion-sensing station, xi , in-lb

t thickness of four-ply composite (or aluminum) wall, in.

thickness of 31-ply composite reinforced wall region, in.
2

UAV unmanned aerial vehicle

x, y Cartesian coordinates, in.

xi i -th strain-sensing station, or its axial coordinate, x = xi , in. 

y deflection in y-direction at any point, x, between two adjacent strain-sensing 

stations, xi-1 ≤ x ≤ xi , in. 

yi deflection in y-direction at strain-sensing station, xi , in. 

γ i surface distortion angle (shear strain) at distortion-sensing station, xi , rad

Δl = l n , distance between two adjacent strain-sensing stations (equally spaced), in.

δ1 ,δ 2 displacement at two ends of finite element, in.

ε bending strain at any point, in/in.

εi bending strain at strain-sensing station, xi , due to bending only, in/in.

εi true bending strain at strain-sensing station, xi , due to bending and torsion, in/in.

ε i

p principal tensile strain at distortion-sensing station, xi , in the 45-deg helical direction, in/in.

beam slope at any point, x, between two adjacent strain-sensing stations, xi-1 ≤ x ≤ xi , rad or deg

θi beam slope at strain-sensing station, xi , rad or deg 

ν Poisson’s ratio

ν LT Poisson’s ratio of lamina

σ bending stress of outermost fiber, lb/in2

σ i

p principal tensile stress at twist-sensing station, xi , in the 45-deg helical direction, in/in.

(τ max )i maximum shear stress at distortion-sensing station, i, lb/in2

φi cross-sectional twist angle at strain-sensing station, xi , rad or deg

( )i+ outboard side of strain-sensing station, xi

( )i- inboard side of strain-sensing station, xi

INTRODUCTION

Many unmanned aerial vehicles (UAVs) currently in service today must be ultralightweight and 

designed with highly flexible structural control surfaces. One such vehicle, the Helios flying wing, has a 

wingspan of 247 ft. The wingspan of the Helios is longer than the wingspans of the U.S. Air Force C-5 

military transport (Lockheed Martin Aeronautics Company, Marietta, Georgia), which is 222 ft, and the 

Boeing 747 commercial jetliner (Boeing Company, Seattle, Washington), which ranges from 195 to 215 

ft, depending on the model. Because of an extremely long wingspan, the wings of vehicles such as the 

Helios will undergo extra-large deformations into high dihedrals during flight, with wingtip deflections 

reaching as high as 40 ft. This type of deformed shape is similar to a 72-deg circular arc with a 197-ft 

radius.

Recently the Helios wing broke up in midair while flying at an altitude of approximately 3000 ft. The 

primary cause of the mishap is believed to have resulted from undamped pitch oscillations of the highly 

deformed wing, resulting in the breakup of the wing. To avoid future mishaps, a method must be 

developed in which the wing deformation can be visually monitored during flight. This way, if the 
3

wingtip deflections approach the design limit, the ground-based pilot could execute timely emergency 

maneuvers to avoid further deformations. 

The major load-carrying main spar of the Helios wing is a carbon fiber composite tube (4-in. radius), 

which is thicker on the top and bottom (spar caps) to improve bending stiffness. The spar also is wrapped 

with Nomex® and Kevlar® (both registered trademarks of E.I. duPont de Nemours and Company, 

Wilmington, Delaware) for additional strength. If multiple strain sensors, such as lightweight fiber optic 

strain-sensing systems, are installed at discrete sensing stations on a beam-like structure (such as the longspan highly flexible Helios flying wing), the strain sensor data can be used to calculate the deflections and 

cross-sectional twist angles of the beam-like structure and thereby predict its deformed shape during 

flight.

This report discusses the development of theoretical displacement equations for uniform cantilever 

beams subjected to bending, torsion, and combined bending and torsion. The displacement equations are 

expressed in terms of strains, the output of multiple strain sensors embedded on the wing spar surface, for 

predicting wing deflections and cross-sectional twist angles during flight. The displacement equations and 

onboard strain-sensing system could form a powerful tool for in-flight wing deformations monitoring by 

the ground-based pilot for maintaining safe flights of highly flexible flying wings.

HELIOS UNMANNED AERIAL VEHICLE (UAV) 

FLEXIBLE STRUCTURE

The Helios prototype (fig. 1) is an ultralightweight, unmanned, solar-powered flying wing aircraft 

(1,600-lb weight) designed to fly at altitudes up to 100,000 ft. Fourteen electric motors (2 hp each) are 

required to drive 79-in. diameter composite propellers and other systems. The power required for these 

motors is generated by high-efficiency solar cells spread over the upper surface of the wing. The cruising 

speed of the Helios is in the range of 19 to 27 mi/h. 

The Helios wing has an 8-ft chord length uniform over the 247-ft wingspan. As discussed previously, 

the wingspan of the Helios is longer than the wingspans of the U.S. Air Force C-5 military transport (222 

ft) and the Boeing 747 commercial jetliner (195–215 ft, depending on the model). The maximum wing 

loading on the Helios 1,976-ft2 wing is 0.81 lb/ft2. 

IN–FLIGHT WING SHAPE MONITORING

In-flight wing deformation monitoring of ultralightweight flying wing aircraft such as the Helios is 

particularly difficult because of weight restrictions and highly flexible nature of the extremely long-span 

structure. Several methods for in-flight monitoring of wing deflections exist for use on conventional 

aircraft.

One method is the electro-optical flight deflection measurement system, which is composed of 

onboard optical receivers and several wing-mounted light-emitting targets (ref. 1). This system provides 

wing displacement information during flight but is too heavy for lightweight flying wing applications. 

Another in-flight deflection measurement method is the use of conventional strain gages to collect 

local strains for wing deflection information. Numerous strain-sensing stations are required to capture the 

higher displacement modes of the flexible wings. When multiple strain-sensing stations are used, the 

weight of the strain gage lead wires alone is too heavy and impractical for most weight-conscious 

lightweight flying wing aircraft. 
4

Fiber optic sensors offer the most attractive alternative to conventional strain gages. They are 

lightweight, fine, and flexible filaments (approximately the size of human hairs), and they can be highly 

multiplexed at desired sensing intervals.

THEORETICAL APPROACH

If multiple strain sensors are installed at discrete sensing stations on a cantilever wing spar, sensor 

data can be used to calculate the deflections and twists of the wing spar during flight. Classical beam 

theory can be used to develop theoretical slope, deflection, and cross-sectional twist angle equations for 

the uniform cantilever beam. These displacement equations are written in terms of the strains for 

calculating the deformed shape of the beam. In the present report, the input strains are calculated from the 

displacement output of finite-element stress analysis. The accuracy of the newly developed displacement 

equations (for bending and torsion) for the uniform cantilever beam is then verified by the finite-element 

displacement output. With slight modifications, the displacement equations developed for the uniform 

cantilever beam could be applied to predict the deformed shape of simple beams (two-point supported), 

tapered beams, plates, and tapered cantilever wing boxes.

UNIFORM CANTILEVER BEAMS

In this section, basic displacement equations are developed for a uniform cantilever beam (for 

example, the tubular spar of the Helios wing) subjected to bending, torsion, and combined bending and 

torsion. Although the highly flexible wing of the Helios is capable of undergoing extraordinarily large 

deformations (wingtip deflections up to 40 ft), the local strains on the wing spar surface remain small. 

Therefore, the classical small strain theory can be used to derive the slope and displacement equations for 

the uniform cantilever beam.

Bending

In this section, displacement equations for bending only are developed. These equations are used to 

describe the local slopes and deflections at points along the strain-sensing line on the bottom surface of a 

uniform cantilever beam of arbitrary cross-sectional shape (for example, circular tubes of the Helios spar).

Moment-Strain Relationship

The classical bending equation for the uniform beam is given by (refs. 2, 3) 

d2y

dx2 = M(x)

EI (1)

in which y is the vertical displacement, x is the span-wise coordinate, M(x) is the bending moment, E is 

the Young’s modulus, and I is the moment of inertia.
5

The bending moment, M(x) , in equation (1) may be related to the associated bending strain, ε(x), at 

the bottom or top fiber (for example, generatrix of the Helios spar tube) through the outermost fiber 

bending stress equation, σ(x) = [M(x)c] I (refs. 2, 3):

ε(x) = σ(x)

E = M (x)cEI ; σ(x) = M (x)cI (2)

In equation (2), c is the depth factor of the beam (distance between the neutral axis and the bottom 

surface of the beam). In light of equation (2), the beam bending equation (1) can be written in term of 

ε(x) as

(3)

Note that under the strain formulation, beam bending equation (3) contains only the beam geometrical 

parameter, depth factor, c, and that the flexural rigidity term, EI, is eliminated. 

Consider the case of a highly flexible, extremely long tubular spar of the Helios wing that carries 

concentrated weight (solar-powered motors, pods, hydrogen storage tanks, and so forth) at different wing 

stations. Because the wing is subjected to aerodynamic lift forces during flight, the wing spar tube 

bending moment, M (x), could be a complicated nonlinear function of the span-wise coordinate, x. 

Strain-Sensing Stations

Figure 2 shows the uniform cantilever tubular spar with a length of l and an outer radius of c. The 

beam is installed with n + 1 equally spaced bending strain sensors at the sensing station, xi (i = 0, 1, 2, 3, 

..., n), along the bottom or top generatrix, which is a straight line on the tubular surface parallel to the 

axis, x, of the tubular spar. The first and final strain-sensing stations { x0(x=0), xn(x=l) } are located at 

the built-in end and the beam tip, respectively. The beam is now discretized into n sections with equal 

length, Δl = l/n . Although the bending strain is zero at the spar tip, the nth bending strain sensor is 

installed there because of mathematical convenience in the derivation of the displacement equations. 

The bending moment, M (x), can be assumed to be a piecewise-linear function along the axial 

coordinate, x. Namely, in the region, xi−1 < x < xi , between any two adjacent strain-sensing stations, 

{ xi−1, xi }, M (x)is considered as a linear function of (x − xi−1) as

M (x) = Mi−1 − (Mi−1 − Mi)

x − xi−1

Δl ; xi−1 ≤ x ≤ xi (4)

in which {Mi−1, Mi} are the bending moments at the two adjacent strain-sensing stations, {xi−1, xi} , 

respectively, and Δl ( = xi − xi−1 = l n ) is the axial distance between the two adjacent strain-sensing 

stations, {xi−1, xi} .
6

In light of equation (2), local moment equation (4) can be written in terms of the local bending strain, 

ε(x), for the region, xi−1 < x < xi , as

ε(x) = εi−1 − (εi−1 − εi)

x − xi−1

Δl ; xi−1 < x < xi ; Δl = ln

(5)

in which { εi−1 , εi } are the bending strains measured at the two adjacent sensing stations, { xi−1, xi }, 

respectively.

Slope Equations 

The slope, tanθ(x) , of the uniform beam in the region, xi−1 ≤ x ≤ xi , between the two adjacent 

strain-sensing stations, { xi−1, xi }, can be obtained by integrating equation (3), with the constant of 

integration determined by enforcing the continuity of the slope at the adjacent inboard strain-sensing 

station, xi−1, as

tanθ(x) = d2

y

dx2 dx

xi−1

x

∫

Slope increment 



+ tanθi−1

Slope at xi−1

 = ε(x)

c

dx

xi−1

x

∫ + tanθi−1 ; xi−1 ≤ x ≤ xi (6)

in which tanθi−1 (constant of integration) is the slope at the adjacent inboard strain-sensing station, 

xi−1. Substituting strain equation (5) into slope equation (6), and carrying out the integration, one obtains 

the slope, tanθi ≡ [tanθ(xi)], at the strain-sensing station, xi , as (ref. 4)

tanθi = Δl

2c

(εi−1 + εi) + tanθi−1 ; (i = 1, 2, 3, ..., n) (7)

Equation (7) is a recursion formula showing that the slope, tanθi , at the strain-sensing station, xi , is 

related to the slope, tanθi−1, at the adjacent inboard strain-sensing station, xi−1. Applying the 

descending indices relationship causes slope equation (7) to become 

tanθi = Δl

2c

ε0 + 2 ε j

j=1

i−1

∑ + εi

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 ; (i = 1, 2, 3, ..., n) (8)

Note that the slope, tanθi , is determined from the geometrical and strain data evaluated at all the 

inboard and current strain-sensing stations, ( x0,, x1, x2, ..., xi ). The first objective is to calculate tanθi (i

= 1, 2, 3, ..., n) for all strain-sensing stations. If the last strain-sensing station, xn , is located at the 

wingtip, then the slope, tanθn , becomes the wingtip slope. 

Deflection Equations

The deflection, y(x), of the uniform beam in the region, xi−1 ≤ x ≤ xi , between the two adjacent 

strain-sensing stations, { xi−1, xi }, can be obtained by integrating slope equation (6) with the constant of 
7

integration determined by enforcing the continuity of deflection at the adjacent inboard strain-sensing 

station, xi−1 , as

(9)

in which yi−1 (integration constant) is the deflection at the adjacent inboard strain-sensing station, xi−1. 

Carrying out the integration of equation (9) in light of strain equation (5), one obtains the deflection, 

yi ≡ [y(xi)], at the strain-sensing station, xi , as (ref. 4)

yi = (Δl)

2

6c

2εi−1 + ε ( i) + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3,..., n) (10)

Deflection equation (10) is also a recursion formula showing that the deflection, yi , at the strainsensing station, xi , is expressed in terms of the deflection, yi−1 , and the slope, tanθi−1 , at the adjacent 

inboard strain-sensing station, xi−1. Applying the descending recursion relationship for yi−1 in light of 

equation (10), and applying expression (8) for tanθi−1 , one could express deflection equation (10) for yi

as a summation of all the inboard strains including the strain at the current sensing station, xi :

yi = (Δl)

2

6c

(3i −1)ε0 + 6 (i − j)ε j + εi

j=1

i−1

∑

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 ; (i = 1, 2, 3,..., n) (11)

By plotting the deflections, yi (i = 1, 2, 3,..., n), at the strain-sensing stations, xi , one can construct the 

deformed shape of the tubular wing spar, which could be displayed on the screen so that the ground-based 

pilot can view the real-time wing deformation status.

The contribution of deflection caused by shear effect usually is very small for a long cantilever beam. 

If shear effect is desired, however, an additional shear effect term must be added to equation (11) (see 

Appendix B of reference 5).

Torsion

In this section, displacement equations are developed for a tubular cantilever spar under torsion. 

Consider a cantilever tubular spar of length l, with n number of equally spaced distortion-sensing stations 

(fig. 2), subjected to span-wise varying twisting moments.

y(x) = tanθ(x)dx

xi−1

x

∫

Integration of slope



+ yi−1

Deflection

at xi−1

 = ε(x)

c

dx dx

xi−1

x

∫ xi−1

x

∫

Deflection increment



+ tanθi−1 xi−1

x

∫ dx

Deflection at x

due to tanθi−1



+ yi−1

Deflection

at xi−1



 xi−1 ≤ x ≤ xi
8

Torque-Twist Relationship

Let Ti be the twisting moment at the distortion-sensing station, xi , and let γ i be the associated 

surface distortion angle (shear strain) at the distortion-sensing station, xi (fig. 3). The classical torquedistortion 

relationship can then be written as (refs.1, 2) 

γ i = Tic

GJ (12)

in which G is the shear modulus, J is the polar moment of inertia, and c is the outer radius of the wing 

spar. The surface distortion angle, γ i (shear strain), in equation (12) is obtained from the distortion

sensor output using

γ i = (τ max )i

G = σ i

p

G = EG

ε i

p = 2(1+ν)ε ip (13)

in which (τ max )i is the maximum shear stress at the distortion-sensing station, i, and σ i

p [= (τ max )i ] 

and εi

P are the principal tensile stress and strain, respectively, in the 45-deg helical direction at the 

distortion-sensing station, i (fig. 3).

Cross-Sectional Twists

The total cross-sectional twist angle, φi , at the strain-sensing station, xi , can be obtained by 

summing up all surface distortion angles, γ j , at all the inboard strain-sensing stations, j (= 1, 2, 3,..., i –

1), as (fig. 3) 

(14)

Notice that when equation (14) is written for the beam tip (i = n), the resulting expression of the beam 

tip cross-sectional twist angle, φn , does not contain the tip distortion term, γ n . Therefore, installation of 

the tip distortion sensor for measuring γ n is not required (fig. 2).

Combined Bending and Torsion

In this section, displacement equations are developed for a cantilever tubular spar under combined 

bending and torsion. The calculations of true bending strains are presented, from which modified slope 

and deflection equations are established. 

True Bending Strain

As shown in figure 3, if the tubular spar is subjected to torsion in addition to bending, the bending 

strain-sensing station, xi (for measuring bending strain, εi ), will move tangentially by a cross-sectional 

twist angle, φi , and the bending strain-sensing axis will be tilted by a local surface distortion angle, γ i . 

Thus, the strain output, εi , obtained at the sensing station, xi , no longer gives the true bending strain. To 
9

obtain the true bending strain, εi , the value of the measured strain, εi , must be corrected by using the 

bending strain correction equation,

εi = εi

cosφi cosγ i

(15)

in which the surface distortion angle, γ i , and the cross-sectional twist angle, φi , are calculated from 

equations (13) and (14), respectively.

Modified Slope and Deflection Equations

With the effect of torsion, slope equation (8) and deflection equation (11) for the bending case must 

be modified by replacing εi with εi :

tanθi = Δl

2c

ε0 + 2 ε j

j=1

i−1

∑ + εi

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ (16)

yi = (Δl)

2

6c

(3i −1)ε0 + 6 (i − j)ε j + εi

j=1

i−1

∑

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ (17)

When the true bending strain data, {ε0,ε1,ε2,ε 3,...,εi} , are substituted, the slopes and deflections of the 

tubular spar at the strain-sensing stations can be calculated from equations (16) and equations (17), 

respectively, for the combined bending and torsion loading case.

FINITE–ELEMENT ANALYSIS

To test the prediction accuracy of deflection equation (11), strain sensor data and measured deflection 

data are needed. Before the experimental data are available, an alternative approach is to use finiteelement analysis to generate the desired bending strain and deflection data. The Structural Performance 

And Resizing (SPAR) finite-element computer program (ref. 6) was used for this purpose. The following

section describes the method for generating the bending strains, εi , from the SPAR output.

Generation of Bending Strains

From the SPAR nodal displacement output, the axial displacements, {δ1, δ 2} , at the two ends of the

element lying in the lower outermost surface can be used to calculate the bending strain for that particular 
10

element. If the strain-sensing station, xi , is located in the middle of the element, the displacement 

differential, (δ 2 −δ1) , can be divided by the element axial length, e, to generate the desired bending 

strain, εi , at the sensing cross section, xi . Namely,

εi = δ 2 −δ1

e

⎛

⎝

⎜ ⎞

⎠

⎟

i

(18)

If the sensing station, xi , is located at the juncture of two adjacent elements, then the following 

average strain value is used for the strain, εi , at the sensing station, xi : 

εi = 1

2

δ 2 −δ1

e

⎛

⎝

⎜ ⎞

⎠

⎟

i+

+

δ 2 −δ1

e

⎛

⎝

⎜ ⎞

⎠

⎟

i−

⎡

⎣

⎢ ⎤

⎦

⎥ (19)

In equation (19), { ( )i+ , ( )i− } denote the elements on the outboard and inboard sides, respectively, of 

the strain-sensing cross section, xi . 

The SPAR bending strains calculated from equations (18) and (19) can be input to deflection equation 

(11) to calculate the beam deflections. These calculated deflections can then be compared with the 

deflections obtained from the SPAR displacement output to check the prediction accuracy of deflection 

equation (11). 

Cantilever Tube Models

Figure 4 shows a typical finite-element model for a cantilever tube with a length of l =100.5 in. The 

model consists of 100 four-node shell elements in the axial direction and 36 in the circumferential 

direction. The end disk was added so that the tip load can be applied at the disk center. The cantilever 

tube is fixed in space at the left end and is subjected to an upward vertical load of P = 100 lb at the tip. 

The following three types of tubes, fabricated with different materials, were considered:

1. Aluminum tube 

2. Four-ply composite tube: carbon fiber composite wall

3. Helios composite tube: carbon fiber composite wall with additional carbon fiber composite 

reinforcements on the top and bottom of the spar caps. 

Table 1 lists the dimensions and table 2 lists the material properties of these three types of tubes.

Table 1. Dimensions of three types of tubular spars.

Tube type l, in. a, in. c, in. t, in. 

Aluminum 100.5 4.0 4.01148 0.02296

Four-ply composite 100.5 4.0 4.01148 0.02296

Helios composite 100.5 4.0* (4.07749)** 4.16646** 0.02296* (0.17794)**

* Four-ply regions ** Reinforced regions
11

Table 2. Material properties of carbon fiber composite and aluminum tubes.

Carbon fiber composite Aluminum

EL = 16.3×106 lb/in2 E = 10.5 ×106 lb/in2

ET = 0.98 ×106 lb/in2 NA

GLT = 0.71 lb/in2 G = 4.0 ×106 lb/in2

vLT = 0.34 v = 0.33

NUMERICAL EXAMPLES – CANTILEVER TUBES

The SPAR displacement output and strain equations (18) and (19) were used to calculate the bending 

strain, εi , for the sensing stations in all cases. These strain values were then input to the appropriate 

deflection equations to calculate the deflections at all the strain-sensing stations of all the beam cases.

This section presents examples of three types of cantilever tubes (aluminum, four-ply composite, and 

Helios composite). The deflections calculated from the Ko displacement theories are compared with those 

calculated from the SPAR finite-element analysis.

Aluminum Cantilever Tube 

Figure 5 shows the bending strains, εi , at the sensing stations on the aluminum tube, calculated from 

the SPAR nodal displacement output. The strain data from figure 5 were used to calculate the deflections, 

yi , from deflection equation (11). Table 3 compares the deflections, yi , calculated from deflection 

equation (11) with those obtained from the SPAR displacement output for the case of n = 8 (n+1= 9 

strain-sensing stations). Note that the difference between the deflection values calculated from the SPAR 

program and those calculated from equation (11) is minimal for the present uniform cantilever aluminum 

tube.

Table 3. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (11); aluminum cantilever tube; n = 8.

Deflection ( yi ), in.

y0 y1 y2 y3 y4 y5 y6 y7 y8

SPAR 0.0000 0.0179 0.0636 0.1341 0.2253 0.3330 0.4530 0.5814 0.7138

Deflection 

equation (11) 0.0000 0.0157 0.0605 0.1302 0.2205 0.3273 0.4465 0.5738 0.7051

Difference, 

percent

0.0000 12.2142 4.9356 2.9223 2.1262 1.7208 1.4524 1.3072 1.2230

The data in table 3 are plotted in figure 6. The two sets of deflection curves are quite close, indicating 

the accuracy of deflection equation (11). The deflection curves calculated from equation (11) lie slightly 

below those obtained from the SPAR output. 
12

Four-Ply Composite Cantilever Tube 

Figure 7 shows the bending strain curve, εi , calculated from the SPAR output, for the strain-sensing 

stations on the four-ply composite tube. In this case the SPAR strain curve is practically linear. The strain 

data from figure 7 were used to calculate the deflections, yi , from deflection equation (11) for the case of 

n = 8. Table 4 compares the deflections, yi , calculated from deflection equation (11) with those obtained 

from the SPAR displacement output. Again, the difference between the values calculated from the SPAR 

program and those calculated from equation (11) is minimal. 

Table 4. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (11); four-ply composite cantilever tube; n = 8.

Deflection ( yi ), in.

y0 y1 y2 y3 y4 y5 y6 y7 y8

SPAR 0.0000 0.0749 0.2577 0.5421 0.9110 1.3475 1.8347 2.3558 2.8946

Deflection 

equation (11) 0.0000 0.0658 0.2519 0.5413 0.9167 1.3610 1.8570 2.3877 2.9356

Difference, 

percent

0.0000 12.2264 2.2545 0.1531 0.6301 1.0000 1.2176 1.3526 1.4154

The data from table 4 are plotted in figure 8 for visual comparison. The two deflection curves are 

quite close, indicating the accuracy of deflection equation (11). The deflection curves calculated from 

deflection equation (11) for the four-ply composite tube lie slightly above those obtained from the SPAR 

output. 

Helios Composite Cantilever Tube

Figure 9 shows the bending strain curve, εi , calculated from the SPAR output, for the strain-sensing 

stations on the Helios composite tube. In this case, the SPAR strain curve is nearly linear. The strain data 

from figure 9 were used to calculate the deflections, yi , from deflection equation (11) for the case of n = 

8. Table 5 compares the deflections, yi , calculated from deflection equation (11) with those obtained 

from the SPAR displacement output. Again, the difference between the beam-tip displacement values 

calculated from the SPAR program and those calculated from deflection equation (11) is minimal. 
13

Table 5. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (11); Helios composite cantilever tube; n = 8. 

Deflection ( yi ), in.

y0 y1 y2 y3 y4 y5 y6 y7 y8

SPAR 0.0000 0.0240 0.0673 0.1324 0.2167 0.3166 0.4284 0.5479 0.6721

Deflection 

equation (11) 0.0000 0.0152 0.0577 0.1235 0.2089 0.3099 0.4227 0.5432 0.6676

Difference, 

percent

0.0000 36.7126 14.2263 6.6853 3.5818 2.1381 1.3307 0.8669 0.6725

The data from table 5 are plotted in figure 10 for visual comparison. The two deflection curves are 

quite close, indicating the accuracy of deflection equation (11). The deflection curves calculated from 

deflection equation (11) for the Helios composite tube lie slightly below those obtained from the SPAR

output.

DISCUSSION

In-flight predictions of the deformed shape of the ultralightweight flying wing require the installation 

of multiple bending and distortion strain sensors, equally spaced on the surface of the tubular wing spar, 

for measuring the bending and distortion strains of the wings. The strain sensor data can then be input to 

theoretical displacement equations for calculating slopes, deflections, and twist angles of the wing spar at 

all strain-sensing stations, including the sensing station on the spar tip. 

Because the accuracy of the displacement (bending and torsion) equations has been validated by 

finite-element analysis, the equations are ready for application. To solidify confidence, however, the 

accuracy of the equations must be checked extensively by testing several types of continuous and 

composite cantilever tubular beams under bending, torsion, and combined bending and torsion. The 

validation experiments are currently under way. 

When strain sensors are installed on the wing spar (or wing box), the displacement equations together 

with strain sensor data can be used to construct the in-flight deformed shape of an aircraft such as the 

Helios flying wing. Because they are lightweight, fiber optic sensors may be the optimal choice for the 

embedded strain sensors on the wing spar surface. The use of a conventional strain gage sensing system is 

impractical, because it has numerous lead wires that add weight, thus it is too heavy for the lightweight 

flying wing to carry.

The displacement theory that has been developed can also be applied to calculate the deformed shape 

of simple beams and plates (ref. 5). The displacement theories can be extended to cases of nonuniform 

beams, such as tapered cantilever wing boxes and aircraft fuselages (varying cross sections) under 

combined bending and torsion (ref. 5).

CONCLUDING REMARKS

Displacement equations were developed for a cantilever tubular wing spar under bending, torsion, and 

combined bending and torsion loading. These displacement equations were expressed in terms of strains 
14

measured at multiple sensing stations equally spaced on the surface of the wing spar. The principal results 

are as follows:

1. The displacement equations developed for the uniform cantilever beam were successfully validated 

for accuracy by classical beam theory and finite-element analysis. 

2. The deflections predicted from the deflection equations for the three types of cantilever tubular 

spars

agreed quite well with the deflections calculated from the finite-element analysis.

3. The deflection equations and associated strain-sensing system (such as fiber optic sensors) form a

powerful tool for in-flight calculations of slopes, deflections, and cross-sectional twist angles of the 

wing spar at any strain-sensing station. These calculated displacements can be used to construct the 

deformed shape of the long-span flying wing. Ultimately, the calculated deformation data can be 

visually displayed in real time for the ground-based pilot to monitor the in-flight deformed shape of 

unmanned aerial vehicles. 
15

FIGURES

ED01-0209-2

Figure 1. Solar powered Helios prototype flying wing during a test flight over the Pacific Ocean. 

Figure 2. Cantilever tubular spar with equally spaced bending and torsion strain sensors. 
16

Figure 3. Instrumented cantilever tubular spar under combined bending and torsion. 

Figure 4. Helios tubular spar finite-element model subjected to a vertical tip load of P = 100 lb. 
17

Figure 5. Bending strains, εi, at different sensing stations, xi, calculated from the SPAR program; 

aluminum cantilever tube; n = 8; P = 100 lb. 

Figure 6. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (11); aluminum cantilever tube; n = 8; P = 100 lb. 
18

Figure 7. Bending strains, εi, at different sensing stations, xi, calculated from the SPAR program; four-ply 

composite cantilever tube; n = 8; P = 100 lb. 

Figure 8. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (11); four-ply composite cantilever tube; n = 8; P = 100 lb. 
19

Figure 9. Bending strains, εi, at different sensing stations, xi, calculated from the SPAR program; Helios 

composite cantilever tube; n = 8; P = 100 lb. 

Figure 10. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (11); Helios composite cantilever tube; n = 8; P = 100 lb. 
20

REFERENCES

1. DeAngelis, V. Michael, and Robert Fodale, “Electro-Optical Flight Deflection Measurement 

System,” Society of Flight Test Engineers 18th Annual Symposium, Paper No. 22, Amsterdam, 

Netherlands, Sept. 1987.

2. Roark, Raymond J., Formulas for Stress and Strain, Third Edition, McGraw-Hill Book 

Company, Inc., New York, 1954.

3. Faupel, Joseph H., Engineering Design: A Synthesis of Stress Analysis and Materials 

Engineering, John Wiley and Sons, Inc., New York, 1964.

4. Hodgman, Charles D., Standard Mathematical Tables, Seventh Edition, Chemical Rubber 

Publishing Company, Cleveland, 1957.

5. Ko, William L., W.L. Richards, and Van T. Tran, Displacement Theories for In-Flight Deformed 

Shape Predictions of Aerospace Structures, NASA/TP-2007-214612, 2007.

6. Whetstone, W.D., SPAR Structural Analysis System Reference Manual, System Level 13A, Vol. 

1, Program Execution, NASA CR-158970-1, 1978.
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

1. REPORT DATE (DD-MM-YYYY)

01-08-2010

2. REPORT TYPE 

Technical Publication

 4. TITLE AND SUBTITLE

Methods for In-Flight Wing Shape Predictions of Highly Flexible Unmanned 

Aerial Vehicles: Formulation of Ko Displacement Theory

5a. CONTRACT NUMBER

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Ko, William L. and Fleischer, Van Tran

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NASA Dryden

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Ko and Fleischer, NASA Dryden Flight Research Center

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14. ABSTRACT

The Ko displacement theory is formulated for a cantilever tubular wing spar under bending, torsion, and combined bending and 

torsion loading. The Ko displacement equations are expressed in terms of strains measured at multiple sensing stations equally 

spaced on the surface of the wing spar. The bending and distortion strain data can then be input to the displacement equations to 

calculate slopes, deflections, and cross-sectional twist angles of the wing spar at the strain-sensing stations for generating the 

deformed shapes of flexible aircraft wing spars. The displacement equations have been successfully validated for accuracy by 

finite-element analysis. The Ko displacement theory that has been formulated could also be applied to calculate the deformed shape 

of simple and tapered beams, plates, and tapered cantilever wing boxes. The Ko displacement theory and associated strain-sensing 

system (such as fiber optic sensors) form a powerful tool for in-flight deformation monitoring of flexible wings and tails, such as 

those often employed on unmanned aerial vehicles. Ultimately, the calculated displacement data can be visually displayed in real 

time to the ground-based pilot for monitoring the deformed shape of unmanned aerial vehicles during flight. 

15. SUBJECT TERMS

Beam curvature equation, Displacement equations, Distributed surface strains, Ko Displacement Theory, Structural shape 

predictions

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