v

TABLE OF CONTENTS

ABSTRACT...................................................................................................................................................... 1

NOMENCLATURE.......................................................................................................................................... 1

INTRODUCTION ............................................................................................................................................ 2

BASICS OF THE KO DISPLACEMENT THEORY ...................................................................................... 3

Discretization ....................................................................................................................................... 3

Slope equation...................................................................................................................................... 4

Deflection equation.............................................................................................................................. 4

IKHANA UNMANNED AIRCRAFT.............................................................................................................. 5

IKHANA WING GEOMETRY........................................................................................................................ 5

TWO–LINE STRAIN–SENSING SYSTEM .................................................................................................. 6

Strain–Sensing Line Locations ............................................................................................................ 6

Deflections........................................................................................................................................... 7

Cross–Sectional Twist Angle............................................................................................................... 7

FINITE–ELEMENT ANALYSIS..................................................................................................................... 8

Ikhana Wing Model ............................................................................................................................. 8

Finite–Element Strains......................................................................................................................... 8

CROSS–SECTIONAL NEUTRAL AXIS........................................................................................................ 9

ACCURACY OF THE KO DISPLACEMENT THEORY ............................................................................ 10

CONCLUDING REMARKS.......................................................................................................................... 11

FIGURES ........................................................................................................................................................ 12

APPENDIX A: DERIVATION OF DEFLECTION EQUATIONS FOR WEAK NONUNIFORM 

CANTILEVER BEAMS................................................................................................................................. 17

REFERENCES................................................................................................................................................ 23

1

ABSTRACT

The Ko displacement theory, formulated for weak nonuniform (slowly changing cross sections) 

cantilever beams, was applied to the deformed shape analysis of the doubly-tapered wings of the Ikhana 

unmanned aircraft. The two-line strain-sensing system (along the wingspan) was used for sensing the

bending strains needed for the wing-deformed shapes (deflections and cross-sectional twist) analysis. The 

deflection equation for each strain-sensing line was expressed in terms of the bending strains evaluated at 

multiple numbers of strain-sensing stations equally spaced along the strain-sensing line. For the preflight 

shape analysis of the Ikhana wing, the strain data needed for input to the displacement equations for the 

shape analysis were obtained from the nodal-stress output of the finite-element analysis. The wing 

deflections and cross-sectional twist angles calculated from the displacement equations were then compared 

with those computed from the finite-element computer program. The Ko displacement theory formulated for 

weak nonlinear cantilever beams was found to be highly accurate in the deformed shape predictions of the 

doubly-tapered Ikhana wing.

NOMENCLATURE

c distance from neutral axis to outermost point of a cross section of uniform beam, in

 

c(x) distance from neutral axis to outermost point of 

 

x-cross section of nonuniform beam, in

 

ci = c(xi), distance from neutral axis to i-th surfaced strain sensor, in

 

cn value of 

 

ci at wing tip, 

 

x = xn = l, in

 

c0 value of 

 

ci at wing root, 

 

x = x0 = 0, in

 

di strain-sensing lines separation distance at strain-sensing station, 

 

xi

, in

 

dn value of 

 

di

 at wing tip, 

 

x = xn = l, in

 

d0 value of 

 

di

 at wing root, 

 

x = x0 = 0, in

E Young’s modulus, lb/in2

 

h0 wing root depth at front strain-sensing line, in

 

hn wing tip depth at front strain-sensing line, in

 

(h0)max wing root maximum depth, in

 

(hn )max wing tip maximum depth, in

 

h ′

0 wing root depth at rear strain-sensing line, in

 

h ′

n wing tip depth at rear strain-sensing line, in

I moment of inertia, in4

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

j dummy index

l length of wing, in

M bending moment, in-lb

n index associated with the last span-wise strain-sensing station 

P applied load, lb

SPAR Structural Performance And Resizing

T applied torque, in-lb

 

wn wing tip chord length (width), in

 

w0 wing root chord length (width), in

x, y Cartesian coordinates (x in span-wise direction, y in vertical direction), in

 

xi axial coordinate (

 

x = xi

) associated with 

 

i-th surface strain sensor (called 

strain sensing station), in

xn strain sensing station at wing tip (

 

x = xn = l), in

x0 strain sensing station wing root (

 

x = x0 = 0), in

 

yi vertical deflection at 

 

x = xi

, in 
2

y0 vertical deflection at wing root (

 

x = x0 = 0), in 

Δl 

 

= l n , distance between any two adjacent strain-sensing stations (domain length), in

 

εi bending strain associated with i-th surface strain sensor, in/in

ε0 bending strain associated with wing root (

 

i = 0) surface strain sensor, in/in

θi slope of deformed wing at 

 

x = xi

, rad or deg 

θ0 slope of deformed wing at wing root (

 

x = x0 = 0), rad or deg 

σi stress associated with 

 

εi

, lb/in2

σ 0 bending stress at wing root upper front strain sensing station, lb/in2

σ 0 bending stress at wing root lower front strain sensing station, lb/in2

 

φi cross-sectional twist angle at axial location, 

 

x = xi

, deg

φn cross-sectional twist angle at wing tip (

 

x = xn = l), deg

 

( ) ′ quantity associated with rear strain-sensing line

INTRODUCTION

In-flight wing-deformed shape monitoring of manned or unmanned aircraft is particularly difficult 

because of instrumentation weight restrictions. Sensor wiring for conventional strain gage instrumentation is 

too heavy for many applications where wing shape sensing is desired. Fiber optic sensors offer many 

advantages over conventional sensor technology because they are lightweight, possess a fine flexible 

filament form factor (approximately the size of a human hair), and can be highly multiplexed at desired 

sensing intervals. If multiple strain sensors, such as light-weight fiber optic strain-sensing systems, are 

installed at discrete sensing stations on a beam-like structure (e.g., aircraft wings), it is possible to use those 

strain sensor data to input to special displacement equations in order to calculate the deflections and

cross-sectional twist angles of a beam-like structure, and thereby predict its deformed shape. The above 

approach was originally used to develop strain-data-dependent displacement theories for the in-flight 

deformed shape predictions of the highly flexible, long-span, lightweight, unmanned Helios flying wing 

(247 ft wingspan; up to 40 ft wing tip deflection).

Based on the classical beam theory, Ko (refs. 1–3) formulated several strain-data-dependent 

displacement equations for the predictions of deformed shapes (deflections and cross-sectional twist) of 

cantilever beams (uniform, tapered, slightly-tapered, and step-wisely tapered). The Ko displacement 

equations were formulated in terms of bending strains evaluated at multiple strain-sensing stations 

embedded on the surface of the beam. The measured (or finite-element generated) bending strain data are 

then input to the displacement equations for the calculations of deflections and the cross-sectional twist 

angle of the beam. When applied to the unmanned aircraft, it is possible for the ground-based pilot to 

visually monitor the deformed shapes of the aircraft during the flights. The Ko displacement theory together 

with the onboard fiber optics strain-sensing system could form a powerful tool for in-flight monitoring of 

the wing shapes by the ground-based pilot for maintaining safe flights of the unmanned aircraft. In addition, 

the real-time shape monitoring of flexible structures also represents an important step toward the eventual 

goal of actively controlling the shape of these structures so that they can quickly adapt to their environment 

while in operation.

This report deals with the preflight deformed shape analysis of the doubly-tapered wings of the Ikhana 

unmanned research aircraft. Because measured strain data are not yet available, finite-element analysis was 

used to generate the desired bending strains for inputs to the displacement equations (developed for 

nonuniform cantilever beams) for the Ikhana wing deformed shape predictions. The Ikhana wing deflections 

and cross-sectional twist angles calculated from the Ko displacement theory will be compared with those 

calculated from finite-element analysis, and the accuracy of the displacement theory will be examined.
3

BASICS OF THE KO DISPLACEMENT THEORY

The formulation of the Ko displacement theory for nonuniform cantilever beams stems from the 

classical differential equation for the uniform beams given by (refs. 4–5) 

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 

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

Using the bending-stress/moment relationship and Hooke’s law, the bending moment M(x) may be 

written as 

M (x) = I

σ(x)

c = EI

ε(x)

c

(2)

in which 

 

c is the uniform beam half depth and 

 

ε(x) is the bending strain at the beam bottom (or top) 

surface. Substitution of equation (2) into equation (1) yields the following modified beam differential 

equation 

d2y

dx2 = ε(x)

c

(3)

Note that, under the present strain formulation, the beam differential equation (3) contains only the 

beam half depth 

 

c and the bending strain 

 

ε(x). The flexural rigidity term EI is eliminated.

The beam differential equation (3) for the uniform beam could be used with sufficient accuracy for 

nonuniform beams for which the cross-section is assumed to change gradually (ref. 4, p. 143). Replacing the 

constant half depth c in equation (3) with the varying 

 

c(x), one obtains the following modified beam 

differential equation for nonuniform beams. 

d2y

dx2 = ε(x)

c(x) (4)

Equation (4) is the basic equation from which the Ko displacement theory for nonuniform beams was 

formulated (refs. 1–3). The beam slope and deflection may be obtained by carrying out single and double 

integrations of equation (4). 

Discretization

Using the piece-wise linear assumption and discretizing the beam domain into n sections, and integrating 

equation (4) (within each beam section) once to obtain the beam slopes, and twice to obtain the beam 

deflections; Ko (refs. 1–3) formulated the beam slope equations and beam deflection equations for each 

piece-wise linear section in terms of geometry and bending strains described below.
4

Let the bottom (or top) surface of the nonuniform beam [length l and varying beam depth 2

 

c(x)] be 

installed with n +1 bending strain sensors at equally spaced strain-sensing stations 

 

xi

 (axial coordinate, 

x = xi ), including the fixed end x0(x = 0), and the free end xn(x = l). The nonuniform cantilever beam 

domain is thus equally divided into n sections with sectional length, Δl = l / n . The beam is then idealized as 

a piecewise linearly tapered (either tapering down or tapering up) beam between any two adjacent 

strain-sensing stations { xi−1, xi }. In the region xi−1 ≤ x ≤ xi between the two adjacent strain-sensing 

stations { xi−1, xi }, the vertical distance c(x) from the neutral surface to the beam bottom (or top) surface, 

and the bending strain { ε(x) } may be assumed to be linear functions of x expressed as

c(x) = ci−1 − (ci−1 − ci)

x − xi−1

Δl ; xi−1 < x < xi (5)

ε(x) = εi−1 − (εi−1 − εi)

x − xi−1

Δl ; xi−1 < x < xi (6)

where { ci−1, ci } and { εi−1, εi } are respectively the values of c(x) and ε(x) at the strain-sensing stations 

{ xi−1, xi }. 

Slope Equation

In light of equations (5) and (6), equation (4) may be integrated once (refs. 1, 2, 6) to yield the beam 

slope 

 

tanθi

 expressed in equation (7) (see Appendix A for derivation):

tanθi = Δl

2

1

c j−1

2 − c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i

∑ + tanθ0

=0

 ; (i = 1, 2, 3, ..., n) (7)

Deflection Equation

In light of equations (5) and (6), equation (4) may be integrated twice (refs. 1, 2, 6) to yield the beam 

deflection 

 

yi

 expressed in equation (8) (see Appendix A for derivation): 

yi = (Δl)

2

6

1

c j−1 j=1

i

∑ 3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beam



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

(8)

For the cantilever beam, both the deflection 

 

y0 and the slope tanθ0 at the built-in end are zero 

(i.e., y0 = tanθ0 = 0 ). The deflection equation (8) will now be used for the Ikhana wing deformed shape 

analysis with the aid of finite-element analysis.
5

For the uniform beams { ci(x) = c }, equation (8) degenerates into equation (9) (refs. 1–2)

yi = (Δl)

2

6c

(3i −1)ε0 + 6 (i − j)ε j

j=1

i−1

∑ + εi

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + (i)Δl tanθ0

=0 for cantilever beam



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

(9)

IKHANA UNMANNED AIRCRAFT

Ikhana (fig. 1) is an unmanned science and technology development research aircraft for civilian 

missions, built by General Atomics Aeronautical Systems Inc., San Diego, California (ref. 7). The aircraft 

has been named “Ikhana” (ee-kah-nah), a Native American word from the Choctaw Nation meaning 

intelligent, conscious or aware. The aircraft structure is fabricated with lightweight composite materials, and 

has a wingspan of 66 ft and a fuselage length of 36 ft. More than 400 lb of sensors can be carried internally 

and over 2,000 lb in external wing pods. Ikhana is powered by a Honeywell TPE 331-10T turbine engine 

(Honeywell Aerospace, Phoenix, Arizona) and is capable of reaching altitudes well above 40,000 ft. This 

aircraft is the first production Predator B equipped with an upgraded digital electronic engine controller 

(developed by Honeywell and General Atomics Aeronautical Systems Inc.) that will make Ikhana 5-10 

percent more fuel-efficient.

The Ikhana aircraft, which is designed for long-endurance and high-altitude flight, will be used for Earth 

science studies. A variety of atmospheric and remote sensing instruments (e.g., a multi-spectral wildfire 

sensor), including duplicates of those sensors on orbiting satellites, can be installed to collect data for up to 

30 hours. 

Ikhana will also be used for advanced aircraft systems research and technology development. Initial 

experiments will look into the use of fiber optics for wing shape sensing and control, and structural loads 

measurements.

IKHANA WING GEOMETRY

Figure 1 shows the Ikhana unmanned research aircraft during flight. The Ikhana wing (fig. 2) is a 

doubly (horizontally and vertically) tapered long cantilever beam. The dimensions of wing length l, wing 

root and wing tip chord lengths {w0,wn }, and wing root and wing tip maximum depths 

{ (h0 )max,(hn )max } are listed in table 1.

Table 1. Dimensions of the Ikhana wing.

l , in w0 , in wn , in (h0 )max , in (hn )max , in

360.00 64.80 28.80 12.96 5.76
6

TWO–LINE STRAIN–SENSING SYSTEM

The two-line strain-sensing system for bending strain sensing (fig. 2), may be used for the deformed 

shape analysis of the Ikhana wing. Each sensing line contains n+1 number of equally spaced strain-sensing 

stations 

 

xi

 (i = 0, 1, 2, 3, ..., n), including the wing root x0(x = 0) and the wing tip xn(x = l). 

Strain–Sensing Line Locations

The installations of front and rear strain-sensing lines are planned on the upper convex surface of the 

Ikhana wing. Unlike a rectangular cross section with flat lower and upper surfaces, the installations of the 

two-line strain-sensing system on the upper convex surface of the Ikhana wing require the following 

technical considerations.

It is desirable to install the strain-sensing lines near the summit of the upper convex surface of the 

Ikhana wing (small sensing-line-separation distance 

 

di

) in order to obtain larger outputs of bending strains 

 

εi

 because the value of 

 

ci

, the vertical distance from the neutral surface to the sensing stations, reaches a 

maximum at the upper surface summit. However, to obtain more accurate data for the cross-sectional twist 

angle φi , a larger 

 

di

 is desired. But, the larger 

 

di

 will decrease the value of 

 

ci

 because of decreasing wing 

depth from the summit. The decreased value of 

 

ci

 will reduce the output of strain 

 

εi

 at the same rate under 

bending moment M according to equation (2). As shown in equation (8), the depth term 

 

c j−1 is in a 

denominator, but the strain terms {εi− j, εi− j+1} are in a numerator. Therefore, simultaneous reductions of 

[ c j−1, {εi− j, εi− j+1} ] at the same rate of increasing 

 

di

 may have little effect on the resulting value of 

deflection 

 

yi

. After exploratory finite-element trade-off studies using different 

 

di

, suitable locations were 

chosen for the two strain-sensing lines as discussed in the next section.

Figure 2 shows the two strain-sensing lines to be installed at the suitable locations on the upper curved 

surface of the wing. Each sensing line is to sense the deflections along the sensing line direction. Note that 

the two strain-sensing lines are not parallel to each other, but separated according to the wing taper rate. The 

dimensions of sensing-line-separation distances { d0 , dn } at the wing root and wing tip finally determined 

are given by equation (10), 

d0

dn

⎧

⎨

⎩

⎫

⎬

⎭

= 18

8

⎧

⎨

⎩

⎫

⎬

⎭

 in (10)

which has the taper rate dn = 0.4444

 

d0 identical to the wing taper rate (table 1). The sensing-line-separation 

distance

 

di

 at any sensing cross section, 

 

xi

 (i = 0, 1, 2, 3, ..., n) is then given by

di = d0 − (d0 − dn )

xi

l (11)

The wing depths under the two sensing lines at the wing root and wing tip are given in table 2. Note 

that, from table 2, the wing depths under the two strain-sensing lines are slightly less than the associated 

maximum wing depths, and that the wing depths under the rear strain-sensing line are much closer to the 

associated maximum wing depths as compared with the wing depths under the front strain-sensing line.
7

Table 2. Wing depths under two sensing lines.

Wing root Wing tip

h0 , in h′

0 , in hn , in h′n , in

11.9059

= 0.9187 (h0 )max

12.5008

= 0.9646 (h0 )max

5.2915

= 0.9187 (hn )max

5.5559

= 0.9646 (hn )max

Deflections

The deflections { yi , y′

i } of the front and rear sensing stations lying at the sensing cross section

 

xi

 (i = 0,1, 2, 3, ..., n) (fig. 2) may be calculated from the deflection equation (8), which is rewritten 

separately for the front and rear strain-sensing lines as,

Front strain-sensing line:

yi = (Δl)

2

6

1

ci− j j=1

i

∑ 3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beam



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

(12)

Rear strain-sensing line:

y′

i = (Δl)

2

6

1

c′

i− j

3(2 j −1) − (3j − 2)

c′

i− j+1

c′

i− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε′

i− j + (3j − 2)ε′i− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

i

∑ + y′

0 + (i)Δl tanθ′0

=0 for cantilever beam



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

(13)

Equations (12) and (13) respectively give the deflections along the front and rear strain-sensing lines, 

which for the Ikhana case, are not parallel to each other according to equation (11), and therefore are not in 

the exact span-wise direction. Because actual measured strain data are not yet available, for the present 

pre-flight shape analysis, finite-element generated strains will be used as inputs to equations (12) and (13) 

for the calculations of deflections { yi, y′

i } (see FINITE-ELEMENT ANALYSIS section). It must be 

mentioned that for upward bending of the Ikhana wing, the induced upper surface bending strains { εi , ε′

i } 

must be used in equations (12) and (13) to yield positive upward deflections { yi , y′

i }.

Cross–Sectional Twist Angle 

Using the values of { yi , y′

i , di }, the cross-sectional twist angle φi at the strain-sensing cross section 

x =

 

xi

 may be calculated from the following cross-sectional twist angle equation,

φi = sin−1 yi − y′

i

di

⎛

⎝

⎜

⎞

⎠

⎟ ; (i = 0, 1, 2, 3, ..., n) (14)

which gives zero twist angle φ0 = 0 (i = 0) at the wing root where y0 = 0 and y′

0 = 0 . The deflections 

{ yi , y′

i } appearing in equation (14) are to be calculated from deflection equations (12) and (13), and 

 

di

 is 
8

calculated from equation (11). With the use of equation (14), installations of surface distortion sensors for 

torsion displacement measurements are not required. 

FINITE–ELEMENT ANALYSIS

Before the installations of the two strain-sensing lines to gather the actual strain data for inputs to the 

displacement equations (12), (13), and (14); either from the ground tests or from the actual Ikhana flight 

tests, finite-element-generated input strains will be used in the present pre-flight wing shape analysis. The 

SPAR (Structural Performance And Resizing) finite-element computer program (ref. 8) was used in the 

finite-element analysis of the Ikhana wing. The SPAR finite-element outputs will provide both nodal 

stresses and nodal displacements (deflections). Through Hooks’s law, the nodal stresses can be converted 

into nodal strains for input to equations (12), (13), and (14) for the calculations of deflections, which may 

then be compared with the corresponding deflections (nodal displacements) obtained from the SPAR 

program.

Ikhana Wing Model

Figure 3 shows the SPAR finite-element model generated for the Ikhana wing, which has 3,639 nodes 

and 3,632 four-node bending elements. The Ikhana wing model was subjected to combined bending and 

torsion loading at the wing tip with upward vertical load of P = 200 lb and clockwise (increasing 

angle-of-attack) torque of T = 1,000 × wn (figs. 2-3).

Finite–Element Strains

If σi is the axial stress at the strain-sensing station xi obtained from the SPAR nodal stress outputs, 

then the corresponding bending strain 

 

εi

 for the input to the deflection equations (12) and (13) may be 

calculated from Hooke’s law

εi = σi

E (15)

where E is the Young’s modulus.

The deflections calculated from equations (12) and (13) using the SPAR generated finite-element strains 

(eq. (15)) can then be compared along with the nodal displacements obtained from the SPAR outputs to 

validate the accuracy of the Ko displacement theory.

Alternatively, using the SPAR displacement outputs, the bending strain 

 

εi may also be obtained through 

dividing the element axial nodal displacement differentials by the length of the element at the strain-sensing 

station xi (ref. 2). This element-displacement method gave practically identical bending strains as the 

element-stress method described above (eq. (15)). However, the element-displacement method was found to 

lose accuracy as the slope of the deflection curve increases toward the wing tip. The SPAR displacement 

outputs are in reference to the un-deformed wing axis. Therefore, at an increasing slope near the highly bent 

wing tip, the true tensile strains along the deformed wing axis, when projected on the un-deformed wing 

axis, may appear as apparent compressive strains near the wing tip. Therefore, the element-displacement 

method was not used.
9

CROSS–SECTIONAL NEUTRAL AXIS

To apply the deflection equations (12) and (13), the input strains εi and the values of the vertical 

distances { ci , c′

i } from the front and rear strain-sensing stations { xi, x′i } to the cross-sectional neutral axis 

must be known for wing shape predictions. The airfoil geometry (cross-sectional shape) of the Ikhana wing 

is nonsymmetrical with respect to the neutral axis (fig. 4). If the lower strain sensors were also installed in 

addition to upper strain sensors, the values of { ci , c′

i } may also be determined experimentally by converting 

the measured bending strains into surface stresses (positive and negative) through which the zero-stress 

point may be located (fig. 4). Alternatively, the vertical distances { ci , c′

i } could be determined from the 

SPAR nodal stress outputs. In the present report, the latter method was used.

Let {σ 0 , σ 0 } respectively be the bending stresses (finite-element calculated, or measured) at the upper 

and lower front strain-sensing stations at the wing root (built-in end), and let h0 be the vertical distance 

between the upper and lower wing-root strain-sensing stations, then the values of ci = c(xi) at the wing 

root, c0 = c(0) may be determined from

c0 = σ 0

σ 0 + σ 0

h0 (16)

Equation (16) may be written for the rear sensing line using 

 

( ) ′ symbol as 

c′

0 = σ′

0

σ′

0 + σ′0

h′

0 (17)

From the SPAR nodal stress outputs (or actually measured), the values of {σ 0 , σ 0 } and {σ′

0 , σ′0 } 

can be found, and the values of { c0 , c′

0 } may be calculated respectively from equations (16) and (17). The 

wing tip values { cn, c′

n } may then be obtained by using the wing taper rate as shown in equation (18).

cn

c′

n

⎧

⎨

⎩

⎫

⎬

⎭

= 0.4444

c0

c′

0

⎧

⎨

⎩

⎫

⎬

⎭ (18)

The values of {

 

c0,c ′

0} and {

 

cn,c ′

n } determined from the SPAR outputs are listed in Table 3.

Table 3. Distance of strain-sensing stations measured from the neutral surface of the wing.

Wing root Wing tip

c0 , in c′

0 , in cn , in c′n , in

6.4051

= 0.5380 h0

7.3899

= 0.5912 h′

0

2.8467

= 0.5380 hn

3.2844

= 0.5912 h′

n

Note that, from table 3, because of the nonsymmetrical wing cross section with respect to the 

corresponding neutral axis, the magnitudes of { c0 , c′

0 , cn , c′n } are slightly larger than one half of the 

respective local wing depth (neutral surface is not the mid-surface). 
10

Finally, the values of { ci , c′

i } at any strain-sensing cross section may then be calculated from the 

following equations:

ci = c0 − (c0 − cn )

xi

l (19)

c′

i = c′0 − (c′0 − c′n )

xi

l (20)

ACCURACY OF THE KO DISPLACEMENT THEORY

Table 3 lists the bending strains calculated from equation (15) using the SPAR nodal stress outputs (eq. 

(15)), and the wing deflections calculated from deflection equations (12) and (13), and also from the SPAR 

program. The bending strains on the upper surface of the wing are compressive strains with negative signs 

in the SPAR outputs; however, the absolute values of the strain data were used for inputs to the deflection 

equations (12) and (13) to calculate the upward deflections.

Table 4. Ikhana wing deflections calculated from deflection equations (12), (13), and from the SPAR 

program; P= 200 lb, T= 1,000 × wn in-lb.

Front sensing line Rear sensing line

i εi(SPAR),in / in yi(Ko),in yi(SPAR),in ε′

i(SPAR),in / in y′i(Ko),in y′i(SPAR),in

0 0.643810-3 0.000000 0.000000 0.623810-3 0.000000 0.000000

1 0.607619-3 0.102220 0.130155 0.634289-3 0.087930 0.085855

2 0.599048-3 0.412410 0.445820 0.646667-3 0.362560 0.344290

3 0.593333-3 0.942570 0.971055 0.637143-3 0.841590 0.810895

4 0.574762-3 1.708610 1.723180 0.612857-3 1.540140 1.498400

5 0.527619-3 2.723990 2.716775 0.563810-3 2.469490 2.418655

6 0.444762-3 3.991970 3.956860 0.473333-3 3.632580 3.573760

7 0.317143-3 5.497200 5.430535 0.300952-3 4.947445 4.947445

8 0.000000-3 7.178920 7.095200 0.000000-3 6.497190 6.497190

Figure 5 shows the plots of bending strains (absolute values) along the front and rear strain-sensing lines 

using the SPAR calculated bending strain data listed in table 4. The strain curves are strongly nonlinear and 

bow shaped because of the effect of double tapering and the cross-sectional shape of the Ikhana wing. Keep 

in mind that for the case of the classical uniform cantilever beam under tip vertical load, the strain curve is a 

straight line tapering down toward zero at the beam tip (refs. 4 and 5).

Figure 6 shows the plots of wing deflections using the deflection data presented in table 4. The 

excellent agreements between the deflection curves calculated from deflection equations (12) and (13), and 

from the SPAR displacement outputs, implies the high accuracy of the Ko displacement theory. 

Table 5 lists the wing cross-sectional twist angles 

 

φi

 calculated from the cross-sectional angle twist 

equation (14) and also from the SPAR program.
11

Table 5. Ikhana wing cross-sectional twist angles φi calculated from the cross-sectional twist angle equation 

(14) and from the SPAR program; P= 200 lb, T= 1,000 × wn in-lb.

i φi(Ko), deg φi(SPAR), deg

0 0.00000 0.00000

1 0.04890 0.15154

2 0.18429 0.37531

3 0.40600 0.64399

4 0.74256 0.99077

5 1.24112 1.45392

6 1.96162 2.09106

7 3.00807 2.99392

8 4.57254 4.28736

Figure 7 shows the plots of the cross-sectional twist angle φi (fig. 3) using the data given in table 5. The 

correlation between the Ko displacement theory and the SPAR program is again quite good, but is not as 

close as the deflection curves (fig. 6). The SPAR program gives slightly higher values of φi than the Ko 

displacement theory up to the sixth (i = 6) strain-sensing station and then gives slightly lower values beyond 

the sensing station i = 7. Keep in mind that φi is extremely sensitive even to the graphically inconspicuous 

changes in deflections { yi , y′

i }. 

CONCLUDING REMARKS

The Ko displacement theory formulated for nonlinear cantilever beams was used to predict the 

deformed shape of the doubly-tapered unmanned Ikhana aircraft wing. Some important conclusions are 

given as follows:

1. The two-line strain-sensing system (bending sensor system) is a powerful method for 

simultaneously sensing the wing bending and torsion displacements without the need of installing 

surface distortion sensors.

2. The two strain-sensing lines do not have to be parallel to each other, but follow the wing taper rate

because each sensing line senses the deflections along the sensing line.

3. To use the Ko displacement theory for shape predictions of a wing with an airfoil cross section, it is 

necessary to locate the wing cross-sectional neutral axis by means of finite-element analysis (or by

direct measurements of top and bottom surface strains), and thereby determine the vertical distance 

from the sensing stations to the neutral axis (for input to the displacement equations).

4. The wing deflections and twist angles calculated from the Ko displacement theory compared very

nicely with the wing deflections and twist angles calculated from the finite-element program.

5. The Ko displacement theory was found to be highly accurate in the deformed shape predictions of 

the doubly-tapered cantilever beam such as the Ikhana wing.
12

FIGURES

ED07-0186-01

Figure 1. Ikhana unmanned science and technology development aircraft.
13

Figure 2. Tapered Ikhana wing installed with a two-line strain-sensing system on the upper surface.
14

Figure 3. The SPAR finite-element model generated for the Ikhana wing subjected to combined bending and

torsion: P = 200 lb; T = 1,000 × wn in-lb.

Figure 4. Determination of neutral axis for Ikhana wing root cross section using SPAR bending

stress outputs. 
15

Figure 5. Bending strain curves generated form SPAR element-stress outputs.

Figure 6. Comparison of Ikhana wing deflections predicted by the Ko displacement theory and calculated by 

the SPAR analysis. 
16

Figure 7. Comparison of the Ikhana wing cross-sectional twist angles predicted by the Ko displacement 

theory and calculated by the SPAR analysis. 
17

APPENDIX A

DERIVATION OF DEFLECTION EQUATIONS FOR WEAK 

NONUNIFORM CANTILEVER BEAMS

The deflection equation Ko (ref.1) formulated for weak nonuniform cantilever beams has the following 

form:

yi = (Δl)

2

6ci−1

3− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + εi

⎡

⎣

⎢ ⎤

⎦

⎥ + yi−1 + Δl tanθi−1 (A1)

in which the slope term tanθi−1 is expressed as (ref. 1)

tanθi−1 = Δl

2ci−2

2 − ci−1

ci−2

⎛

⎝

⎜

⎞

⎠

⎟ εi−2 + εi−1

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθi−2 (A2)

Write out equation (A1) and (A2) for different indices (i = 1-5) as follows:

y1 = (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + Δl tanθ0 ; y0 = 0 ; Δl tanθ0 = 0 (A3a)

y2 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + y1 + Δl tanθ1 ; tanθ1 = Δl

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθ0 (A3b)

y3 = (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + y2 + Δl tanθ2 ; tanθ2 = Δl

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθ1 (A3c)

y4 = (Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + y3 + Δl tanθ3 ; tanθ3 = Δl

2c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθ2 (A3d)

y5 = (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ + y4 + Δl tanθ4 ; tanθ4 = Δl

2c3

2 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθ3 (A3e)

Using the recursion relationships above, one can write out the deflection equation (A1) for different 

indices as 
18

y2 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + y1 + Δl tanθ1

 

 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + Δl tanθ0

y 1

+ (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0



ta



nθ

1 

 

 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

9 − 4 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 2Δl tanθ0 

(A4a)

y3 = (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + y2 + Δl tanθ2

 

 = (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

9 − 4 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 2Δl tanθ0

y 2

 + (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0



Δ



l t



an



θ2

 

 = (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

9 − 4 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 4ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

15 − 7 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 7ε1

⎡

⎣

⎢ ⎤

⎦

⎥ 

 + y0 + 3Δl tanθ0

(A4b)
19

y4 = (Δl)

2

6c3

3 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε3 + ε4

⎡

⎣

⎢ ⎤

⎦

⎥ + y3 + Δl tanθ3

 

 = (Δl)

2

6c3

3 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε3 + ε4

⎡

⎣

⎢ ⎤

⎦

⎥ 

 + (Δl)

2

6c2

3 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

9 − 4 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 4ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

15 − 7 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 7ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 3Δl tanθ0



y3

 

 + (Δl)

2

2c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0



Δ



l t



an



θ3



 

 

 = (Δl)

2

6c3

3 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε3 + ε4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 4ε3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε2

⎡

⎣

⎢ ⎤

⎦

⎥ 

 + (Δl)

2

6c0

21− 10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 10ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 4Δl tanθ0

(A4c)
20

y5 = (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ + y4 + Δl tanθ4

= (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥

+

(Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 4ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε2

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2

6c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +10ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 4Δl tanθ0

⎧

⎨

⎪

⎪

⎩

⎪

⎪

⎫

⎬

⎪

⎪

⎭

⎪

⎪

y 4

+

(Δl)

2

2c3

2 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥

+ (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0

⎧

⎨

⎪

⎪

⎩

⎪

⎪

⎫

⎬

⎪

⎪

⎭

⎪

⎪



Δ



l t



an



θ4

= (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥

(Δl)

2

6c3

9 − 4 c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 4ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

15 − 7 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 7ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ 

 + (Δl)

2

6c1

21−10 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 +10ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

27 −13 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +13ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 5Δl tanθ0

(A4d)

In summary,

y1 = (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + Δl tanθ0 (A5a)

y2 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

9 − 4 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 2Δl tanθ0 (A5b)

y3 = (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

9 − 4 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 4ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

15 − 7 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 7ε1

⎡

⎣

⎢ ⎤

⎦

⎥

 + y0 + 3Δl tanθ0 (A5c)
21

y4 = (Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 4ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε2

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2

6c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +10ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 4Δl tanθ0

(A5d)

y5 = (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c3

9 − 4 c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 4ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

15 − 7 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 7ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2

6c1

21−10 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 +10ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

27 −13 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +13ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 5Δl tanθ0

(A5e)

Based on the indicial progressions in equations (A5a) to (A5e), the following generalized deflection 

equation for any index i (= 1, 2, 3, ..., n) may be established:

yi = (Δl)

2

6

1

ci− j

3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

i

∑ + y0 + iΔl tanθ0

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

 

y0 = tanθ0 = 0

(A5f)

Equation (A5f) is the deflection equation (8) in the text.

Accuracy Checks

The general deflection equation (A5f) will be checked for its accuracy by using i = 4, 5. 

For i = 4, equation (A5f) takes on the following form,

y4 = (Δl)

2

6

1

c4− j

3(2 j −1) − (3j − 2)

c4− j+1

c4− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε 4− j + (3j − 2)ε 4− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

4

∑ + y0 + 4Δl tanθ0 (A6)

which after writing out the summation yields,

y4 = (Δl)

2

6

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 4ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε2

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +10ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+ y0 + 4Δl tanθ0

(A7)

which agrees with equation (A5d) and thus, confirms the accuracy of the general deflection equation (A5f) 

formulated for slightly nonlinear cantilever beams.
22

For i = 5, equation (A5f) takes on the following form:

y5 = (Δl)

2

6

1

c5− j

3(2 j −1) − (3j − 2)

c5− j+1

c5− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε5− j + (3j − 2)ε5− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

5

∑ + y0 + 5Δl tanθ0 (A8)

Writing out the summations of equation (A8) results in,

y5 = (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c3

9 − 4 c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 4ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

15 − 7 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 7ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2

6c1

21−10 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 +10ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

27 −13 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +13ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 5Δl tanθ0

(A9)

which is identical to equation (A5e), and thus validates the accuracy of the general deflection equation (A5f) 

developed for slightly nonlinear cantilever beams.
23

REFERENCES

1. 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. 

2. Ko, William L., Further Development of Ko Displacement Theory for Deformed Shape Predictions 

 of Nonuniform Aerospace Structures, NASA/TP-2009-214643, 2009.

3. Ko, William L., and W. Lance Richards, Method for Real-time Structure Shape-Sensing, U.S. Patent 

 No. 7,520,176, issued April 21, 2009. 

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

 Inc., New York, 1954.

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

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

6. Hodgeman, Charles D., Standard Mathematical Tables, Eleventh Edition, Chemical Rubber 

 Publishing Company, Cleveland, Ohio, 1957.

7. Cobleigh, Brent R., Ikhana: A NASA UAS Supporting Long Duration Earth Science Missions, 

 NASA/TM-2007-214614, 2007.

8. 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)

01-10-2009

2. REPORT TYPE 

Technical Publication

 4. TITLE AND SUBTITLE

Applications of Ko Displacement Theory to the Deformed Shape Predictions

of the Doubly-Tapered Ikhana Wing

5a. CONTRACT NUMBER

 6. AUTHOR(S)

William L. Ko, W. Lance Richards, and Van Tran Fleischer

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

H-3006

10. SPONSORING/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

Ko, Richards, and Fleischer, NASA Dryden Flight Research Center.

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -- Unlimited

Subject Category 39 Availability: NASA CASI (443) 757-5802 Distribution: Standard

19b. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov)

14. ABSTRACT

The Ko displacement theory, formulated for weak nonuniform (slowly changing cross sections) cantilever beams, was applied to the deformed

shape analysis of the doubly-tapered wings of the Ikhana unmanned aircraft. The two-line strain-sensing system (along the wingspan) was used for

sensing the bending strains needed for the wing-deformed shapes (deflections and cross-sectional twist) analysis. The deflection equation for each

strain-sensing line was expressed in terms of the bending strains evaluated at multiple numbers of strain-sensing stations equally spaced along the

strain-sensing line. For the preflight shape analysis of the Ikhana wing, the strain data needed for input to the displacement equations for the shape

analysis were obtained from the nodal-stress output of the finite-element analysis. The wing deflections and cross-sectional twist angles calculated

from the displacement equations were then compared with those computed from the finite-element computer program. The Ko displacement theory

formulated for weak nonlinear cantilever beams was found to be highly accurate in the deformed shape predictions of the doubly tapered Ikhana

wing.

15. SUBJECT TERMS

Ko displacement theory, Deflection equations, Cross-sectional rotation equations, Distributed surface strains, Ikhana wing deformed

shape predictions

18. NUMBER

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 PAGES

30

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802

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