NASA/TP—2013–216518

Method for Estimating Operational Loads on 

Aerospace Structures Using Span-wisely 

Distributed Surface Strains

William L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

 

April 2013
NASA STI Program ... in Profile

Since its founding, NASA has been dedicated 

to the advancement of aeronautics and space science. 

The NASA scientific and technical information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI program operates under the auspices 

of the Agency Chief Information Officer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and its 

public interface, the NASA Technical Reports Server, 

thus providing one of the largest collections of 

aeronautical and space science STI in the world. 

Results are published in both non-NASA channels and 

by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase of 

research that present the results of NASA 

Programs and include extensive data or theoretical 

analysis. Includes compilations of significant scientific and technical data 

and information deemed to be of continuing 

reference value. NASA counter-part of peerreviewed formal professional papers but has less 

stringent limitations on manuscript length and 

extent of graphic presentations.

• TECHNICAL MEMORANDUM. 

Scientific and technical findings that are 

preliminary or of specialized interest, 

e.g., quick release reports, working 

papers, and bibliographies that contain minimal 

annotation. Does not contain extensive analysis.

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. 

Collected papers from scientific and technical 

conferences, symposia, seminars, or other 

meetings sponsored or 

co-sponsored by NASA.

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from NASA 

programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission.

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and feeds, 

providing information desk and personal search 

support, and enabling data exchange services.

For more information about the NASA STI program, 

see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question to help@sti.nasa.gov

• Fax your question to the NASA STI Information

Desk at 443-757-5803

• Phone the NASA STI Information Desk at 

443-757-5802

• Write to:

STI Information Desk

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

This page is required and contains approved text that cannot be changed.
NASA/TP—2013–216518

Method for Estimating Operational Loads on 

Aerospace Structures Using Span-wisely 

Distributed Surface Strains

William L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, CA 93523-0273

April 2013
Patent Notice 

The method for structure operational load estimations using distributed surface strains described 

in this technical paper is protected under “Process for Using Surface Strain Measurements to 

Obtain Operational Loads for Complex Structures”, U.S. Patent No. 7,715,994, issued May 11, 

2010. Therefore, those interested in using the invention should contact the NASA Innovative 

Partnership Program Office at the Dryden Flight Research Center for more information. 

Click here: Press F1 key (Windows) or Help key (Mac) for help 

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802
v

TABLE OF CONTENTS

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

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

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

STRUCTURE DISCRETIZATION ...........................................................................................................3

TWO-LINE STRAIN-SENSING SYSTEM...............................................................................................4

METHOD FOR OPERATIONAL LOAD ESTIMATIONS......................................................................4

Step One-Determinations of Structural Stiffness, EI i, GK i ..........................................................4

Step Two-Estimations of Operational Loads, Mi, Pi, Ti .....................................................................5

BASIC EQUATIONS FOR BENDING .....................................................................................................5

Determination of Bending Stiffness, EI i.............................................................................................5

Estimations of Operational Moments and Shear Loads, {Mi, Pi} .........................................................6

BASIC EQUATIONS FOR TORSION......................................................................................................6

Determination of Torsion Stiffness, GK i ............................................................................................7

Estimations of Operational Torques, Ti .................................................................................................8

FLOW CHART FOR OPERATIONAL LOADS ESTIMATIONS...........................................................8

ANALYTICAL OPERATIONAL LOAD ESTIMATIONS ......................................................................8

EXAMPLES OF OPERATIONAL LOAD ESTIMATIONS.....................................................................9

Tapered Cantilever Tubular Beam.........................................................................................................9

Determination of Bending Stiffness, EI i .......................................................................................10

Estimations of Operational Moments and Shear Loads, Mi, Pi .....................................................11

Current Bending Stiffness, EI i.......................................................................................................12

Depth-Tapered Un-swept Wing Box ...................................................................................................12

Determination of Bending Stiffness..................................................................................................13

Estimations of Operational Moments and Shear Loads, Mi, Pi .....................................................14

Current Bending Stiffness, EI i.......................................................................................................15

Determination of Torsion Stiffness, GK i .......................................................................................16

Estimations of Operational Torques, Ti ............................................................................................17

Current Torsion Stiffness, GK i ......................................................................................................17

Depth-Tapered Swept Wing Box .........................................................................................................18

Determination of Bending Stiffness, EI i .......................................................................................19

Estimations of Operational Moments and Shear Loads, Mi, Pi .....................................................20

Bending Only (P = 200 lb) ............................................................................................................20

Combined bending and Torsion (P = 200 lb, T = − 1,700 in-lb) .................................................22

Bending-induced Components, Mi

P, PiP ......................................................................................23

Torsion-induced Components, Mi

T, PiT .......................................................................................24

Determination of Torsion Stiffness, GK i .......................................................................................26

Estimations of Operational Torques, Ti

P+T........................................................................................27
vi

T-Induced Torques, Ti

T ..................................................................................................................27

P-Induced Torques, Ti

P...................................................................................................................28

Combined-Load-Induced Torques, Ti

P+T .......................................................................................29

Generic Long-Span Wing ....................................................................................................................30

Determination of Bending Stiffness, EI i .......................................................................................31

Method 1-Averaged Geometry and Strains...................................................................................31

Method 2-Averaged Bending Stiffness .........................................................................................32

Estimations of Operational Moments and Shear Loads, Mi, Pi .....................................................32

Concentrated Loads.......................................................................................................................33

Distributed Loads ..........................................................................................................................34

Determinations of Torsion Stiffness GK i ......................................................................................35

Estimations of Operational Torque, Ti..............................................................................................36

CONCLUDING REMARKS....................................................................................................................37

FIGURES ..................................................................................................................................................39

APPENDIX A: SUMMARY OF DIFFERENT DISPACEMENT TRANSFER FUNCTIONS..............60

APPENDIX B: FLOW CHART FOR STRUCTURAL STIFFNESS AND OPERATIONAL 

LOADS ESTIMATIONS USING DISTRIBUTED SURFACE STRAINS........................................62

REFERENCES..........................................................................................................................................64
1

ABSTRACT

This report presents a new method for estimating operational loads (bending moments, shear loads, 

and torques) acting on slender aerospace structures using distributed surface strains (unidirectional 

strains). The surface strain-sensing stations are to be evenly distributed along each span-wise 

strain-sensing line. A depth-wise cross section of the structure along each strain-sensing line can then be 

considered as an imaginary embedded beam. The embedded beam was first evenly divided into multiple 

small domains with domain junctures matching the strain-sensing stations. The new method is comprised 

of two steps. The first step is to determine the structure stiffness (bending or torsion) using surface strains 

obtained from a simple bending (or torsion) loading case, for which the applied bending moment (or 

torque) is known. The second step is to use the strain-determined structural stiffness (bending or torsion), 

and a new set of surface strains induced by any other loading case to calculate the associated operational 

loads (bending moments, shear loads, or torques). Performance of the new method for estimating 

operational loads was studied in light of finite-element analyses of several example structures subjected to 

different loading conditions. The new method for estimating operational loads was found to be fairly 

accurate, and is very promising for applications to the flight load monitoring of flying vehicles with 

slender wings.

NOMENCLATURE

c depth factor (vertical distance from neutral surface to outermost fiber of bottom surface of 

uniform beam), in

!(!) depth factor (vertical distance from neutral surface to bottom surface outermost fiber of 

nonuniform beam) at axial location, x, in

ci ≡ c xi , value of c (x) at strain-sensing station, xi, in

c0 value of ci at fixed end (wing root), x = x0 = 0, in

cn value of ci at free end (wing tip), x = xn = l, in

di separation distance of front and rear strain sensors for εi, εi

' at x = xi

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

EI i bending stiffness at strain-sensing station, xi, lb-in2

Fi resultant force of distributed load lying outboard of wing cross section at x = xi, lb

G shear modulus, lb/in2

GK i torsional stiffness of the i-th domain lying between two adjacent strain-sensing stations 

xi-1, xi , lb-in2

GK i

N torsional stiffness referred to xN-axis system, lb-in2

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

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

I moment of inertia, in4

i = 0,1,2,3....,n, strain sensor identification number 

j dummy index

J polar moment of inertia, in4

K twisting moment of inertia (K = J, polar moment of inertia for circular cross section), in4

l length of slender structure, in

M(x) bending moment at axial location, x, in-lb

Mi ≡ M xi , bending moment at strain-sensing station, xi , in-lb 

Mi

P+T bending moment at strain-sensing station, xi, induced by bending and torsion, in-lb

Mi

P bending moment at strain-sensing station, xi, induced by bending only, in-lb
2

Mi

T bending moment at strain-sensing station, xi, induced by torsion only, in-lb

n index for the last span-wise strain-sensing station (or discretized domain number) 

P applied point load, lb

Pi operational shear load at strain-sensing station, xi, lb

Pi

P+T total shear load at strain-sensing station, xi, induced by bending and torsion, lb

Pi

P shear load at strain-sensing station, xi, induced by bending only, lb

Pi

T shear load at strain-sensing station, xi, induced by torsion only, lb

T applied torque, in-lb

Ti torque at strain-sensing station, xi, in-lb

Ti

P+T torque at strain-sensing station, xi, induced by combined bending and torsion, in-lb

Ti

P bending-induced torque at strain-sensing-station, xi, in-lb

Ti

T torsion-induced operational torque at strain-sensing station, xi, in-lb

t wall thickness, in

wi width of wing box at strain-sensing station, xi, in

wn wing tip chord length (width), in

w0 wing root chord length (width), in

x, y Cartesian coordinates (x in beam axial direction, y in lateral direction), in

xi axial coordinate of i-th strain sensor (called strain-sensing station), in 

y(x) beam deflection in y direction at axial location, x, in 

yi ≡ y (xi), value of y(x) at strain-sensing station, xi, in 

Δl = xi − xi!1 = l/n, domain length [distance between two adjacent strain-sensing stations,

xi!1, xi ], in

β swept angle of wing-box front edge, deg

εi ≡ ε xi , value of ε(x) at strain-sensing station xi, in/in

εi ≡ εi + εi

' /2 , averaged surface strain at strain-sensing station xi, in/in

εi

P+T value of εi associated with combined bending and torsion loading, in/in

εi

T value of εi associated torsion loading only, in/in

θ slope of embedded beam, rad or deg

v Poisson's ratio

φi cross sectional twist angle at axial location, x = xi rad or deg

φi

P+T

cross-sectional twist angle at axial location, x = xi, due to combined bending and torsion, rad or 

deg.

φi

P

cross-sectional twist angle at axial location, x = xi, due to bending only, rad or deg

φi

T

cross-sectional twist angle at axial location, x = xi, due to torsion only, rad or deg

σi surface stress associated with surface strain, εi, lb/in2

' quantity associated with rear strain-sensing line

INTRODUCTION

The strain sensors (for example, conventional strain gages or fiber optic sensors) installed on the 

surface of a structure (for example, aircraft wings), can only sense the surface strains, and cannot sense 

the out-of-plane deflections and cross-sectional rotation of a structure. However, after the formulations of 

the Displacement Transfer Functions (refs. 1–9), one can now convert the surface strains (unidirectional 

bending strains) into out-of-plane deflections for mapping out the overall structure deformed shapes for 

visual display. The displacement theory, combined with any strain-sensing system, thus created a 

revolutionary powerful new structure deformed shape-sensing technology, “Method of Real-Time 

Structure Shape-Sensing” [U.S. Patent No. 7,520,176, (ref. 10)], which is very attractive for applications 
3

to the in-flight structure deformed shape monitoring of unmanned and manned flight vehicles, and other 

aerospace structures.

For flight vehicles, the structure operational load estimations are as important as the structure 

deformed shape predictions. The flight loads can come from air loads and inertia loads, and the 

deformation is the response to external loads based on cross-sectional stiffness, which in turn is a function 

of mass, moment of inertia, and material properties. 

Similar to the structure deformed shape calculations, the span-wisely distributed surface strains can 

also be used to calculate operational loads (bending moments, shear loads, and torques) acting on 

complex slender structures (for example, high-aspect-ratio aircraft wings). This new method is called, 

“Process for Using Surface Strain Measurements to Obtain Operational Loads for Complex Structures” 

[U.S. Patent No 7,715,994, (ref. 11)]. To apply this new method to estimate operational loads, structural 

stiffness (bending and torsion) must be determine first. 

For nonuniform cantilever slender structures such as high-aspect-ratio aircraft wings with complex 

internal structural configurations, the structural stiffness varies in the span-wise direction. One method to 

determine the structural stiffness is to perform finite-element analysis of the structure. Because of internal 

structural complexity (for example, aircraft wings), it is extremely cumbersome and time-consuming to 

generate an accurate finite-element model for the calculations of stiffness at multiple span-wise locations. 

Keep in mind that finite-element analysis is conventionally used for classical load calibrations and finding 

the load paths. 

An alternative simple way to determine the structural stiffness (bending or torsion) is to conduct a 

simple bending (or torsion) test, and use the induced surface strains (obtained at span-wisely distributed 

strain-sensing stations) to calculate the structural stiffness at each strain-sensing station. Once the bending 

and torsion stiffness are determined, one can then use those strain-determined stiffness and a new set of 

surface strains, induced by another type of loading condition, to calculate the associated operational loads 

(bending moments, shear loads, and torques) at all the strain-sensing stations for mapping out the overall 

operational load profile.

This report describes the new method for estimating the operational loads (bending moments, shear 

loads, and torques) acting on slender structures (beam-like structures) using span-wisely distributed 

surface strains. Several slender structures were chosen to demonstrate the new method. The structures 

considered were: 1) tapered cantilever tubular beam; 2) depth-tapered un-swept wing box; 

3) depth-tapered swept wing box; and 4) doubly-tapered generic long-span wing. The new method of 

estimating the operational load was found to be reasonably accurate, and is very promising for 

applications to flight load monitoring of flying vehicles with slender wings.

STRUCTURE DISCRETIZATION

To use the distributed surface strains (unidirectional strains) to estimate structural operational loads 

(moments, shear loads, and torques), the strain-sensing stations are to be evenly distributed in the 

span-wise direction of a slender structure (fig. 1). A line along which the strain-sensing stations are to be 

distributed is called a strain-sensing line (fig. 1). A narrow strip of structure beneath each surface 

strain-sensing line can then be considered as an imaginary embedded beam. In developing the stiffness 

transfer functions and load transfer functions, the embedded beam was first evenly divided into multiple 

small domains, with domain junctures matching the strain-sensing stations. 

The surface strains can be obtained from the conventional strain sensors, from fiber optic 

strain-sensors (light weight), or generated from finite-element analysis. In the earlier finite-element shape 
4

predictions of Ikhana wings [360-inch long per wing, (ref. 4)], a two-line strain-sensing system was used 

for each wing. Using only nine strain-sensing stations for each strain-sensing line, quite accurate 

deformed-shape predictions could be achieved (ref. 4). If the conventional strain gages are used, the 

lead-wires for just 18 strain gages per wing are unlikely to add significant weight to the Ikhana vehicle.

TWO-LINE STRAIN-SENSING SYSTEM

For the operational load estimations of a slender structure under combined bending and torsion 

loading, a two-line strain-sensing system can be used if the location of the neutral surface is known. 

Figure 2 shows a typical two-line strain-sensing system located on the lower surface of a double-tapered 

wing box (neutral surface at half depth) subjected to combined bending and torsion. 

If the location of the neutral surface is unknown, then a four-line strain-sensing system (two lines on 

lower skin, two lines on upper skin) must be used to locate the neutral surface based on pairs of lower and 

upper surfaces strains at the same strain-sensing cross section (ref. 4). 

The surface strains, εi, εi

' , of the front and rear strain-sensing lines can be used to sense both 

bending and torsion (fig. 2). However, for sensing the torsion, the Displacement Transfer Functions 

(duplicated in Appendix A) developed earlier (ref. 1–9), are needed to convert the surface strains, εi, εi

' , 

into out-of-plane deflections, yi, yi

' , of the front and rear strain-sensing line for determining the crosssectional twist angles, φi (see Appendix B). Thus, the two-line strain-sensing system with embodied 

Displacement Transfer Functions totally eliminates the need for the conventional distortion (shear) strain 

sensors to sense torsions.

For wider structures (low aspect ratios), multiple strain-sensing lines are needed on both upper and 

lower surfaces because the bending and torsion stiffness can vary, not only in the span-wise direction, but 

also in the chord-wise direction. The feasibility of this approach is currently under investigation.

METHOD FOR OPERATIONAL LOAD ESTIMATIONS

The method for estimating operational loads using distributed surface strains requires the 

foreknowledge of structural stiffness (bending and torsion). The method consists of two steps as described 

below. 

Step One–Determinations of Structural Stiffness, EI i, GK i

Step one is to determine the bending and torsion stiffness, EI i, GK i . It must be understood that 

in the discretized cantilever beam system (figs. 1, 2), the bending stiffness, EI i, is evaluated at the 

strain-sensing station, xi (i=0,1,2,3,....,n), but the torsion stiffness, GK i, is constant over the i-th domain 

lying between the two adjacent strain-sensing stations xi!1, xi (fig 1). Keep in mind that there is no i = 0

domain (figs. 1, 2). 

When using a two-line strain-sensing system (fig. 2), the values of EI i can be determined by using 

the surface strains induced by a specified bending loading test (or finite-element generated) for which

εi = εi

' . The values of GK i can be determined from the cross-sectional twist angle, φi

, induced by a 

specified torsion test (or finite-element generated) for which εi ≠ εi

' . For torsion, the Displacement 

Transfer Functions must be introduced to convert the surface strains, εi = εi

' , into deflections, yi

, yi

' , 

from which the cross-sectional twist angles, φi, can be determined.
5

The simplest loading case to determine the values of EI i is to apply a point load, P, at the beam tip 

(no torsion). Also, the simplest loading case to find the values of GK i is to apply a torque, T, at the 

beam tip (no bending). As will be seen shortly, the values of { EI i, GK i} calculated from different 

loading conditions remain practically the same (that is, invariant to loading conditions). 

Step Two–Estimations of Operational Loads, Mi, Pi, Ti

Step two for bending is to use the values of EI i, pre-determined in step one, and a new set of 

averaged surface strains, εi = εi + εi

' /2 (fig. 2), induced by other type of loading condition to estimate 

the associated operational loads (moments, shear loads), {Mi, Pi}, at the strain-sensing stations, xi

(i = 0,1,2,3,....,n).

Step two for torsion is to use the values of GK i pre-determined in step one, and the values of the 

cross-sectional twist angles, φi, calculated from the Displacement Transfer Functions using a new set of 

surface strains, εi, εi

' (fig. 2), induced by another type of loading condition to estimate the associated 

operational torques, Ti , at the strain-sensing stations, xi (i = 0,1,2,3,....,n) .

BASIC EQUATIONS FOR BENDING

For a cantilever beam structure discretized into n small domains (fig. 1), the bottom surface bending 

strain, !!, induced by the bending moment, Mi, at strain-sensing station, xi, can be related through the 

following moment equation (refs. 11, 12):

Mi = (EI)i

! i

ci

 ; 

!

(i = 0,1,2,3,....,n) (1)

In equation (1), ci is the depth factor (distance from the neutral surface to beam bottom surface 

outermost fiber) at strain-sensing station, xi (fig 1). Equation (1) is written in such a way that the positive 

bending moment, Mi, (bending the cantilever beam upward) will induce positive (tensile) strain, εi, on the 

bottom surface (fig.1). 

Equation (1) can be rewritten in the following form (called bending-stiffness equation):

(EI)i = Mi

ci

! i

 ; 

!

(i = 0,1,2,3,....,n) (2)

 

If the right-hand side of equation (2) is known, one can then use equation (2) to determine the 

unknown bending stiffness, EI i, at strain-sensing station, xi. For a given slender structure, the depth 

factor, ci, can be determined by using pairs of strains obtained from lower and upper surfaces (ref. 4). If 

the bending moment, Mi, is specified in a simple bending loading case, then the Mi-induced surface 

strains, εi, will be known from actual measurements, or εi can be calculated analytically from finiteelement analysis (used in the present report). Thus, based on the known values of {Mi, εi}, the values of 

bending stiffness, EI i, can be calculated from equation (2) as described in the following section. 

Determination of Bending Stiffness, EI i

To determine the bending stiffness, EI i, the bending moment, Mi, must be specified. The simplest 

way to specify Mi is to consider a simple bending case of a cantilever beam of length, l, subjected to an 
6

upward point load, P, at the beam tip. The bending moment, Mi, for such a bending case is a linearly 

decreasing function of xi (x0 = 0 at fixed end), and can be can be expressed as:

Mi = M0 1! xi

l

"

#

$ %

&

' = M0 1! i

n

"

#

$ %

&

' ; 

!

(i = 0,1,2,3,....,n) (3)

in which M0 = Pl is the maximum bending moment at the fixed end, x = x0 = 0.

With the bending moment, Mi, thus specified from equation (3), the induced surface strains, εi, will be 

known. Then the known values of Mi,εi,ci can be input to the bending stiffness equation (2) to 

determine EI i, which is needed for the calculations of operational moments and shear loads, {Mi, Pi}, as 

described in the following section. 

Estimations of Operational Moments and Shear Loads, {Mi, Pi}

Once the cross-sectional bending stiffness, EI i, is determined from equation (2), the straindetermined EI i and a new set of surface strains, εi, induced by another type of loading condition, can be 

used as inputs to calculate the current operational moment, Mi, at strain-sensing station, xi, from equation 

(1), and the current operational shear loads, Pi, at strain-sensing station, xi, from the following shear-load 

equation (or load-transfer function) (ref. 11): 

Pi = ! dM

dx

"

#

$ %

&

'

i

( ! Mi ! Mi!1

)l = 1)l (EI)i!1

* i!1

ci!1

! (EI)i

* i

ci

+

,

- .

/

0 ; 

!

(i =1,2,3,....,n) (4)

in which the moment equation (1) was used. Because the x-axis originated from the fixed end, 

equation (4) is written in such a way that decreasing bending moment in the span-wise direction (x

direction) will produce positive shear load, Pi. 

If the two-line strain-sensing system is used for sensing bending moments and shear loads (fig. 2), 

then ci, εi in equations (1), (2), and (4) must be replaced respectively with averaged depth factors, 

ci ≡ ci + ci

' /2 , and averaged strains, ε! ≡ εi + εi' /2 , where ci, εi , ci

'

, εi

' are the depth factor

and strain associated respectively with the front and rear strain-sensing lines (see Appendix B).

BASIC EQUATIONS FOR TORSION

In the estimations of operational torques, Ti, at the strain-sensing station, xi, the two-line (or four-line) 

strain-sensing system (fig. 2) can be used to sense the cross-sectional rotations. The front and rear strainsensing line strains, εi, εi

' , at the strain-sensing station, xi, can be first converted into front and rear 

strain-sensing-line deflections, yi, yi

' , using the Displacement Transfer Functions (Appendix A). Then 

the differentials of yi, yi

' , can be used to calculate cross-sectional twist angles, φi

, for the estimations of 

operational torques, Ti. 

Before the estimations of operational torques, Ti, using the cross-sectional twist angles, φi, one must 

first determine the torsion stiffness, GK i, for the small domain, xi!1 ≤ x ≤ xi, bounded by the two 

adjacent strain-sensing stations, {xi!1 , xi}. Here, G is the shear modulus and K is the twisting moment of 

inertia for a non-circular cross sections [for a circular cross section, K = J, the polar moment of inertia 

(ref. 11)].
7

Determination of Torsion Stiffness, GK i

The torsion stiffness, GK i , for a small domain, xi!1 ≤!!x ≤ xi, can be expressed in terms of the 

torque, !!, and cross-sectional twist angle differential, φi − φi!1 , at the strain-sensing station, !!, using 

the following torsion-stiffness equation (ref. 12):

!

(GK)i = Ti("l)

#i $ #i$1

 ; 

!

(i =1,2,3,....,n) (5)

In equation (5), !" ≡ !/! is the small domain length, and φ!

, φ!!1 are the cross-sectional twist 

angles respectively at the strain-sensing stations, !!, !!!! , and which can be calculated from the 

following cross-sectional twist-angle equation (refs. 1 and 2) (see Appendix B):

!

"i = sin#1 yi # y $

i

di

%

&

'

(

)

* ;

!

(i = 0,1,2,3,....,n) (6)

In equation (6), di is the chord-wise separation distance between the front and rear strain-sensing lines 

at x = xi, and yi

, yi

' are respectively the deflections of the front and rear strain-sensing lines at x = xi (fig. 

2). Using the surface strains, εi, εi

' , as inputs, the deflections yi

, yi

' , in equation (6) can be calculated 

from any of the Displacement Transfer Functions (listed in Appendix A) depending on the type of 

structure under consideration. For example, for the case of a doubly-tapered slender structure like a long 

span aircraft wing, the following Nonuniform Displacement Transfer Functions [eqs. (7) and (8)] can be 

used:

For front strain-sensing line:

 

!

yi = ("l)

2 # j$1 $# j

2(c j$1 $ c j)

$

# j$1c j $# jc j$1

(c j$1 $ c j)

3 c j loge

c j

c j$1

+ (c j$1 $ c j)

%

&

'

(

)

*

+

,

-

.

-

/

0

-

1

- j=1

i

2

Contributions from deflection terms

1 4 4 4 4 4 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 4 4 4 3 

 + ("l)

2 (i $ j)

# j$1 $# j

c j$1 $ c j

+

# j$1c j $# jc j$1

(c j$1 $ c j)

2 loge

c j

c j$1

%

&

'

'

(

)

*

*

+

,

-

.

-

/

0

-

1

- j=1

i$1

2

Contribution from slope terms

1 4 4 4 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 4 3 

+ y0 + (i)"ltan3 0

=0 for cantilever beams

1 4 4 2 4 4 3 

(7)

!

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

For rear strain-sensing line:







=0 for cantilever beams

0 0

Contribution fromslope terms

1

1 1 2

1

1 1

1

2 1

Contributionsfromdeflection terms

1

1

1 3

1

1 1

1

2 1

log ( ) tan

( ) ( ) ( )

log ( )

( ) 2( ) ( )

θ

ε ε ε ε

ε ε ε ε

+ ′ + Δ ′

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

′

′

′ − ′

′ ′ − ′ ′ + ′ − ′′ − ′ + Δ −

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ ′ − ′ ′

′ ′ ′ − ′′ ′ − ′ ′ − ′ − ′′ − ′ ′ = Δ

∑

∑

−

= − −

− −

−

−

=

−

− −

− −

−

−

y i l

c

c

c c

c c

c c

l i j

c c

c

c

c

c c

c c

c c

y l

i

j j

j e

j j

j j j j

j j

j j

i

j

j j

j

j

j e

j j

j j j j

j j

j j

i

(8)

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

By considering a simple torsion loading case of applying a torque, T, at the beam tip, the crosssectional torque, Ti, at the strain-sensing station, xi, will be known (that is, Ti = T), and the cross-sectional 

twist angle, φi, can be calculated from equation (6) using the T-induced surface strains εi, εi

' . Then, the 

torsion stiffness, GK i, for the small domain, xi!1 ≤ x ≤ xi, can be calculated from the torsion stiffness 

equation (5).

Estimations of Operational Torques, Ti

Once the torsion stiffness, GK i, is determined from a simple torsion loading case mentioned above, 

a new set of cross-sectional twist angles, φi, induced by other types of torsion loading cases, can be used 

to calculate the operational torque, Ti, at strain-sensing station, !!, using the following torque equation 

[rewritten form of equation (5)]:

!

Ti = (GK)i

"i # "i#1

$l ; 

!

(i =1,2,3,....,n) (9)

In equation (9), the values of φi are to be calculated from the twist angle equation (6) using the 

deflections, !!, !!

! , converted from the new sets of surface strains, εi, εi

' , induced by other types of 

torsion loading cases using equations (7) and (8). Keep in mind that the operational torque, Ti, remains 

constant over domain, xi!1 ≤ x ≤ xi, of the discretized system.

FLOW CHART FOR OPERATIONAL LOADS ESTIMATIONS

For easy understanding of the new method of using distributed surface strain data to estimate both 

structure stiffness (bending, torsion) and operational loads (bending moments, shear loads, torques), a 

flow chart was constructed and is presented in Appendix-B for graphical illustration of the new method.

ANALYTICAL OPERATIONAL LOAD ESTIMATIONS

The new method of structural operational load estimations presented in this report is called the 

analytical method of operational load estimations. Namely, without using experimentally measured 

surface strains, the Structural Performance And Resizing (SPAR) finite-element computer program 

(ref. 13) was used to analytically calculate the surface strains and deflections. The surface strains, !!

(i = 1,2,3....,n), at the strain-sensing station, xi, needed for inputs to calculate deflections [eqs. (7) and

(8)] were obtained by converting the SPAR nodal stresses at the strain-sensing station, xi, into bending 

strains, εi, through stress-strain law. Alternatively, the bending strains, εi, could also be calculated from 
9

the axial length changes of the finite elements where the bending strains, εi, are evaluated. However, this 

method was found to lose accuracy near the highly bent regions of the deformed beam and, therefore, it 

was used only in the low slope region near the fixed end (refs. 1 and 2).

The SPAR-generated surface strains, εi, εi

' , can then be input to the Displacement Transfer 

Functions [eqs. (7) and (8)] for the calculations of the deflections, yi, yi

' , from which the cross-sectional 

twist angles, φi, could be calculated from equation (6). However, in the present report, the values of φi

were calculated directly from equation (6) using the already existing SPAR-generated deflections, yi, yi

' , 

instead of using the deflections calculated from equations (7) and (8). It is important to mention that the 

deflections calculated from equations (7) and (8), and from the SPAR-program were found to be 

amazingly close beyond expectations (refs. 1– 9). This high degree of accuracies of the Displacement 

Transfer Functions [for example, equations (7) and (8)], were also experimentally verified by large-scale 

ground loads tests of a full-scale long span wing (175-ft wingspan) (ref. 14). 

EXAMPLES OF OPERATIONAL LOAD ESTIMATIONS

To show how to actually use the new method of operational load estimations, several example 

structures were chosen. The structures considered were: 

1. Tapered cantilever tubular beam, 

2. Depth-tapered un-swept wing box, 

3. Depth-tapered swept wing box (bending-torsion coupling), 

4. Doubly-tapered generic long-span wing.

Tapered Cantilever Tubular Beam

The first example structure used in the operational load estimation analysis is the aluminum- tapered 

cantilever tubular beam (fig. 3) with dimensions listed in table 1. 

Table 1. Dimensions of aluminum tapered cantilever tubular beam.

l, in c0, in cn, in t, in

(length) (root radius) (tip radius) (wall thickness)

100.5 4.0 1.0 0.02296

As shown in figure 1, a single strain-sensing line is located on the bottom generatrix of the tapered 

cantilever tubular beam. The single-line strain-sensing system can only sense the bending-induced 

moments and shear loads, Mi, Pi . The strain-sensing line has n = 8 evenly spaced strain-sensing

domains (that is, n + 1 = 9 strain-sensing stations). Figure 3 shows the un-deformed and deformed shapes 

of the SPAR model generated for the tapered cantilever tubular beam subjected to upward point load of 

P = 100 lb at the beam-tip disk center. This SPAR model has:

3,673 nodes----------------------- 101 nodes in axial direction 

 36 nodes in circumferential direction 

 37 nodes on the beam-tip disk 

3,636 four-nodes elements----3,600 elements for tube wall 

 36 elements for beam-tip disk 

 36 three-nodes elements----- beam-tip disk central region
10

1. Determination of Bending Stiffness, EI i

In the first step of determining the bending stiffness, EI i, the tapered cantilever tubular beam 

(SPAR model) was subjected to an upward point load of P=100 lb at the beam tip (applied at disk center,

fig. 3). This point load will produce the applied bending moments of Mi = Pl 1 − i/n = 10,050× 1 − i/n

in-lb [eq. (3)].

Figure 4 shows the plot of the SPAR-generated surface strains, εi, induced by the present loading 

case. Unlike the straight-line strain distribution for a uniform cantilever beam (ref. 2), the strain curve for 

the tapered beam is highly nonlinear and bow shaped.

Using the known depth factor, ci, the known applied bending moments, 

Mi = 10,050× 1 − i/n in-lb [eq. (3)], and the induced surface strains, εi, shown in figure 4, the values of 

the bending stiffness, EI i, at strain-sensing station, i x , were calculated from bending stiffness equation 

[eq. (2)], and are listed in table 2. 

Table 2. Bending stiffness, EI i, for tapered cantilever tubular beam determined from surface strains, εi, 

induced by point load, P = 100 lb, at beam tip; n = 8, l = 100.5 in, ∆l = 12.5625 in, M0 = Pl = 10,050 in-lb.

i ci,

in

εi , ×10!3

in/in

Mi/M0 (EI)i

, ×107

lb-in2

Mi = EI i εi/ci ,

in-lb

Pi,

lb

(SPAR) (known) [from

eq. (2)]

(elasticity) (error, 

%)

[calculated, eq. (1)

= applied]

[calculated, eq. (4)

= applied]

0 4.000 0.837143 1.0 4.802047 (4.847201) (0.9316) 10,050.00 100.00

1 3.625 0.889524 7/8 3.583641 (3.607740) (0.6680) 8,793.75 100.00

2 3.250 0.948571 6/8 2.582503 (2.599928) (0.6702) 7,537.50 100.00

3 2.875 1.010476 5/8 1.787137 (1.799802) (0.7037) 6,281.25 100.00

4 2.500 1.068571 4/8 1.175635 (1.183399) (0.6561) 5,025.00 100.00

5 2.125 1.109524 3/8 0.721804 (0.726755) (0.6812) 3,768.75 100.00

6 1.750 1.090476 2/8 0.403207 (0.405906) (0.6649) 2,512.50 100.00

7 1.375 0.882857 1/8 0.195654 (0.196888) (0.7307) 1,256.25 100.00

8 1.000 0.000000 0/8 0.099145* (0.072131) (37.4513) 0.00 100.00

 *Three-points extrapolation [eq. (10)]. 

In table 2 the values of EI i calculated from the theory of elasticity are also listed (in parenthesis) for 

comparison. In the first column of EI i, the strain-determined EI 8 at beam-tip i = 8 was extrapolated 

using the following three-points extrapolation formula:

(EI)8 = (EI)5 ! 3(EI)6 + 3(EI)7 (10)

Note from table 2 that the percent errors of strain-determined EI i are in the negligible range of 

(0.6561 ~ 0.9316) percent except for the extrapolated value of EI 8. The large 37.4513 percent error of 

EI 8 at beam-tip i = 8 can be attributed to a small number divided by another small number. 

For checking the accuracy of the strain-determined bending stiffness, EI i, the moment equation (1) 

and shear-load equation (4) were used to calculate the operational moments and shear loads, Mi, Pi , 

associated with the applied point load of, P=100 lb. The results are listed in the last two columns of 

table 2. Note that the values of Mi, Pi are all identical to the applied moments, 

Mi = 10,050× 1 − i/n in-lb and applied shear loads, Pi = P = 100 lb, confirming the accuracies of the 

strain-determined EI i. 
11

Figure 5 shows the plots of both strain-determined EI i and elasticity-calculated EI i based on the 

data of table 2. The strain-determined and elasticity-calculated values of EI i are practically coincidental, 

giving confidence in the new method for determining EI i from equation (2) using surface strains as 

inputs. 

2. Estimations of Operational Moments and Shear Loads, Mi, Pi

Step 2 is to calculate the operational moments and shear loads, Mi, Pi , using the strain-determined

EI i, and the surface strains, εi, associated with a new loading case, for which the tapered cantilever 

tubular beam was subjected to point load, P!!= 100 lb, at the beam midpoint, and a second point load, 

P!!= 100 lb, at the beam tip. For the present two-points loading case, the applied bending moment can be 

expressed as:

For inboard region:

Mi = Pl

2

+ Pl 1! 2 i

n

"

#

$ %

&

' = 5,025 +10,050 ( 1! 2 i

n

"

#

$ %

&

' (in-lb) ; (0 ! i ! n / 2) (11)

For outboard region:

Mi = Pl 1! i

n

"

#

$ %

&

' = 10,050 ( 1! i

n

"

#

$ %

&

' (in-lb) ; (n / 2 ! i ! n) (12)

Figure 6 shows the SPAR-generated surfaced strains, εi, induced by the present two-points loading 

case. The strain curve is consisted of two bent curves because of two-points loading.

Using the known beam depth factors, ci, the surface strains, εi, of figure 6, and the pre-determined 

values of EI i listed in table 2, operational moments and shear loads, Mi, Pi , for the two-points loading 

case were calculated from equations (1) and (4), and are listed in table 3(a). 

Table 3(a). Estimated operational moments and shear loads, Mi, Pi for tapered cantilever tubular beam 

subjected to point load, P = 100 lb, at beam midpoint and second point load, P = 100 lb, at beam tip;

n = 8; l = 100.5 in, ∆l = 12.5625 in.

i ci, 

in

εi, ×10!3

in/in

(EI)

i

,×107

lb-in2

Mi = EI i εi/ci , 

in-lb

Pi, 

lb

(SPAR) (from table 2) (estimated) (applied) (error, %) (estimated) (applied) (error, %)

0 4.000 1.253333 4.802047 15,046.41 (15,075.00) (0.1897) 197.94 (200.00) (1.0300)

1 3.625 1.270476 3.583641 12,559.81 (12,562.50) (0.2141) 197.94 (200.00) (1.0300)

2 3.250 1.264762 2.582503 10,050.01 (10,050.00) (0.0001) 199.79 (200.00) (0.1050)

3 2.875 1.212381 1.787137 7,536.32 (7,537.50) (0.0157) 200.09 (200.00) (0.0450)

4 2.500 1.073810 1.175635 5,049.63 (5,025.00) (0.4902) 197.95 (200.00) (1.0250)

5 2.125 1.109524 0.721804 3,768.75 (3,768.75) (0.0000) 101.96 (100.00) (1.9600)

6 1.750 1.090476 0.403207 2,512.50 (2,512.50) (0.0000) 100.00 (100.00) (0.0000)

7 1.375 0.882857 0.195654 1,256.25 (1,256.25) (0.0000) 100.00 (100.00) (0.0000)

8 1.000 0.000000 0.099145* 0.00 (0.00) (0.0000) 100.00 (100.00) (0.0000)

 * Three-points extrapolation [eq. (10)].

In table 3(a), the applied moments, !! (induced by point load, P = 100 lb at beam midpoint, and 

second point load, P = 100 lb, at the beam-tip, fig. 6), were calculated from equations (11) and (12). In 
12

table 3(a), the estimation errors of Mi, Pi are also listed within the parentheses. The estimation error is 

defined as the percent difference between the estimated and applied values. The estimation errors of 

Mi, Pi are extremely small, in the ranges of (0 ~ 0.4902) percent for Mi, and (0 ~ 1.9600) percent for Pi. 

The estimated shear loads, Pi, are nearly equal to the applied load of P = 200 lb in the inboard half region 

(0 ≤ i ≤ 4), and suddenly dropped to almost P = 100 lb in the outboard half region (4 < i ≤ 8) because of 

decreasing loading. 

Figure 7 shows the plots of the estimated and applied Mi, Pi data listed in table 3(a) for graphical 

comparisons. The estimated Mi, Pi curves (dashed curves with open circular and diamond symbols) 

agreed nicely with the applied Mi, Pi curves (solid lines with solid circular and diamond symbols). The 

high proximity of the estimated and applied moments and shear-loads gives confidence in the accuracies 

of the new method for structural operational load estimations.

3. Current Bending Stiffness, !" !

For the current two-point loading (first point load, P = 100 lb, at beam midpoint, second point load, 

P = 100 lb, at beam tip), the bending moments, Mi, are known, and are expressed by equations (11) and 

(12). Therefore, the values of EI i can be re-calculated and compared with the pre-determined EI i

listed in table 2. The results are compared in table 3(b). 

Table 3(b). Bending stiffness, EI i, for tapered cantilever tubular beam determined from surface strains, 

ε!, induced by point load, P = 100 lb, at beam midpoint and second point load, P = 100 lb, at beam tip; 

n = 8, l = 100.5 in, ∆l = 12.5625 in, M0 = Pl = 15,075 in-lb.

i ci, 

in

εi, ×10!3

in/in

Mi, 

in-lb

 (EI)

i

, ×107

lb-in2

Mi = EI i εi/ci , 

in-lb

Pi, 

lb

(SPAR) (known) [from

eq. (2)]

(from

table 2)

(diff., %) [from eq. (1)

= applied]

[from eq. (4)

= applied]

0 4.000 1.253333 (15,075.00) 4.811171 (4.802047) (0.1900) 15,075.00 200.00

1 3.625 1.270476 (12,562.50) 3.584410 (3.583641) (0.0215) 12,562.50 200.00

2 3.250 1.264762 (10,050.00) 2.582502 (2.582503) (0.0000) 10,050.00 200.00

3 2.875 1.212381 (7,537.50) 1.787418 (1.787137) (0.0157) 7,337.50 200.00

4 2.500 1.073810 (5,025.00) 1.169900 (1.175635) (0.4878) 5,025.00 200.00

5 2.125 1.109524 (3,768.75) 0.721804 (0.721804) (0.0000) 3,768.75 100.00

6 1.750 1.090476 (2,512.50) 0.403207 (0.403207) (0.0000) 2,512.50 100.00

7 1.375 0.882857 (1,256.25) 0.195654 (0.195654) (0.0000) 1,256.25 100.00

8 1.000 0.000000 (0.00) 0.099145* (0.099145*) (0.0000) 0.00 100.00

 * Three-points extrapolation [eq. (10)].

Note from table 3(b) that the values of EI i re-calculated from equation (2) based on the current 

two-points loading condition are extremely close to the pre-determined EI i values of table 2, with 

percent differences in the negligible range of (0 ~ 0.4878) percent. This proximity implies that EI i is 

practically invariant to the loading conditions, and giving confidence in the values of the 

pre-determined EI i. If the current EI i is used, the estimation errors will become zero, and the 

estimated and the applied moments and shear loads, Mi, Pi , will be identical [last two columns of table 

3(b)].

Depth-Tapered Un-swept Wing Box

The second example structure used in the operational load estimation analysis is the aluminum 

depth-tapered wing box with dimensions listed in table 4. 
13

Table 4. Dimensions of aluminum depth-tapered un-swept wing box.

l, in c0, in cn, in w0, in wn, in t, in

(length) (root half depth) (tip half depth) (root width) (tip width) (wall thickness)

100.0 2.0 0.5 17.0 17.0 0.02296

Figure 8 shows the SPAR model generated for the depth-tapered wing box, with size indicated. This 

wing-box model was generated from the tubular beam model (fig. 3) by distorting the circular cross 

sections into rectangular cross sections. Each skin (upper or lower) was modeled with 1,700 four-node 

elements (100 elements in span-wise direction and 17 elements in chord-wise direction). 

Because the neutral surface for the present depth-tapered wing box is located at half depth, a two-line 

strain-sensing system (inset of figure 8) was used for sensing the surface strains induced by bending and 

torsion. The front and rear strain-sensing lines are respectively located along the wing box lower front and 

rear edges. Each strain-sensing line has n = 8 evenly spaced domains (that is, n + 1 = 9 strain-sensing 

stations (inset of figure 8).

1. Determination of Bending Stiffness, EI i 

In step 1 of determining the bending stiffness, EI i, the depth-tapered wing box was subjected to 

point load, P = 100 lb, at the wing-box tip (50 percent chord point). For the present loading case, the 

applied moment, Mi, at the strain-sensing station, xi, in view of equation (3), is given by 

Mi = M0 1 − i/n = 10,000× 1 − i/n in-lb.

Figure 9 shows the plot of surface strains, εi = εi

' , and the associated deflections, yi = yi

' , for the 

current loading case. Note that the bow-shaped strain curves and the deflection curves associated with the 

front and rear strain lines are identical. 

Using the given depth factor, ci, the known cross-sectional moments, Mi = 10,000× 1 − i/n in-lb,

and the strains, εi = εi

' , shown in figure 9, the values of the bending stiffness, EI i, were calculated 

from equation (2), and are listed in table 5.

Table 5. Bending stiffness, EI i, for depth-tapered un-swept wing box determined from surface strains,

εi = εi

' , induced by point load, P = 100 lb, at wing-box tip; n = 8, l = 100 in, ∆l = l/n = 12.5 in, 

M0 = Pl = 10,000 in-lb.

i ci, 

in

εi, ×10!3

in/in

Mi/M0 (EI)

i

,×107

lb-in2

Mi = EI i εi/ci , 

lb-in

Pi, 

lb

(SPAR) (known) [from 

eq.(2)]

(elasticity) (error, %) [calculated, eq. (1) 

= applied]

[calculated, eq. (4) 

= applied]

0 2.000 0.358095 1.0 5.585110 (5.476832) (1.9038) 10,000.00 100.00

1 1.8125 0.363810 7/8 4.359247 (4.316765) (0.9893) 8,750.00 100.00

2 1.625 0.362857 6/8 3.358761 (3.323538) (1.0598) 7,500.00 100.00

3 1.4375 0.356190 5/8 2.522355 (2.485710) (1.4742) 6,250.00 100.00

4 1.25 0.341905 4/8 1.827993 (1.791836) (2.0179) 5,000.00 100.00

5 1.0625 0.316190 3/8 1.260120 (1.230478) (2.4090) 3,750.00 100.00

6 0.875 0.269048 2/8 0.813052 (0.790190) (2.8932) 2,500.00 100.00

7 0.6875 0.187619 1/8 0.458043 (0.459531) (0.3238) 1,250.00 100.00

8 0.500 0.000000 0/8 0.195093* (0.227059) (14.0783) 0.00 100.00

 * Three-points extrapolation [eq. (10)].
14

In table 5 the values of EI i calculated from the theory of elasticity are also listed (in parenthesis) for 

comparison. Note that the percent errors of strain-determined EI i are in the small range of 

(0.9893 ~ 1.9038) percent, except for the extrapolated value of EI 8. The large 14.0783 percent error of 

EI 8 at beam-tip i = 8 can be attributed to a small number divided by another small number. Note also 

from table 5 that the values of Mi, Pi , calculated respectively from equations (1) and (4) using the 

strain-determined values of EI i are identical to the applied moments, Mi = Pl 1 − i/n = 10,000× 1 −

i/n in-lb [eq. (3)] and applied shear loads, Pi = P = 100 lb. This agreement implies that the values of the 

strain-determined EI i are correct based on the present strain data.

Figure 10 shows the plots of both strain-determined EI i (dashed curve with open circular symbols) 

and elasticity-calculated EI i (solid curve with solid circular symbols) using data listed in table 5. Note 

from figure 10 that the strain-determined EI i curve and the elasticity- calculated EI i curve are almost 

coincidental except for the wing-box root where the strain-determined EI i is only 1.9038 percent larger 

than the elasticity-calculated EI i. 

2. Estimations of Operational Moments and Shear Loads, Mi, Pi

Step 2 is to use the strain-determined values of EI i listed in table 5 to calculate the operational 

moments and shear loads, Mi, Pi , associated with other loading case, for which the depth-tapered wing 

box was subjected to point load, P = 200 lb and clockwise torque, T = 100×w = 1,700 in-lb at wing-box 

tip. The clockwise direction (looking toward the wing-box root from the wing-box tip) is defined as a 

positive torque in the present report. For the present loading condition, the applied moments, Mi, can be 

calculated from equation (3) by setting M0 = Pl = 200×100 = 20,000 in-lb in the form of equation (13):

⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ = × − ⎠⎞ ⎜

⎝

⎛ = − n

i

n

i Mi M 1 20,000 1 0 (in-lb) ; (i = 1,2,3,....,n) (13)

Figure 11 shows the SPAR-generated surface strains, εi, εi

' , and the associated deflections, yi

, yi

' , 

induced by the present combined bending and torsion loading case. The shapes of the two strain-curves 

are quite similar, and diverged in opposite directions toward the wing-box root due to intensified torsion

effect at the fixed end.

Using the known beam depth factor, c!, the averaged strains, εi ≡ εi + εi

' /2 (to eliminate torsion 

effect) obtained from strains εi, εi

' given in figure 11, and the strain-determined values of EI i, listed in 

table 5, the operational moments and shear loads, Mi, Pi , were calculated respectively from 

equations (1) and (4). The results are listed in table 6(a).
15

Table 6(a). Estimated operational moments, shear loads Mi, Pi , using pre-determined EI i for 

depth-tapered un-swept wing box subjected to combined point load, P = 200 lb, and torque, 

T = 100×w = 1,700 in-lb, at wing-box tip; n = 8, l = 100 in, ∆l = 12.5 in, M0 = Pl = 20,000 in-lb.

i ci, 

in

εi,×10!3

in/in

εi

'

,×10!3

in/in

εi,×10!3

in/in

EI i,×107

lb-in2

Mi = EI i εi/ci , 

in-lb

Pi, 

lb

(SPAR) (SPAR) (averaged) (from table 5) (estimated) (applied) (error, %) (estimated) (error, %)

0 2.000 0.822857 0.570476 0.696667 5.737712 19,986.37 (20,000.00) (0.0682) 197.99 (1.0050)

1 1.8125 0.749524 0.706667 0.728096 4.359247 17,511.44 (17,500.00) (0.0654) 197.99 (1.0050)

2 1.625 0.740952 0.710000 0.725476 3.358761 14,995.08 (15,000.00) (0.0328) 201.31 (0.6550)

3 1.4375 0.730476 0.693333 0.711905 2.522355 12,491.67 (12,500.00) (0.0666) 200.27 (0.1350)

4 1.25 0.708095 0.659048 0.683572 1.827993 9,996.51 (10,000.00) (0.0666) 199.61 (0.1950)

5 1.0625 0.664762 0.598095 0.631429 1.260120 7,488.71 (7,500.00) (0.1505) 200.62 (0.3100)

6 0.875 0.587619 0.490000 0.538810 0.813052 5,006.63 (5,000.00) (0.1326) 198.57 (0.7150)

7 0.6875 0.447619 0.301905 0.374762 0.458043 2,496.83 (2,500.00) (0.1268) 200.78 (0.3900)

8 0.500 0.000000 0.000000 0.000000 0.195093* 0.00 (0.00) (0.0000) 199.75 (0.1250)

 * Three-points extrapolation [eq. (10)].

Note from Table 6(a) that the estimation errors are in the negligible ranges of (0 ~ 0.1505) percent for Mi, 

and (0.1250 ~ 1.0050) percent for Pi. 

In figure 12, the estimated operational moments and shear loads, Mi, Pi , listed in Table 6-a are 

plotted for graphical comparisons with the applied Mi, Pi , values. The estimated Mi, Pi , curves 

(dashed curve with open circular and diamond symbols) practically fell on top of the respective applied 

Mi, Pi , curves (solid lines with solid circular and diamond symbols). Again, the high degree of 

proximity of the predicted and applied values of Mi, Pi , implies the accuracies of the new method for 

operational load estimations.

3. Current Bending Stiffness, EI i

For the current combined bending and torsion loading case, the applied bending moments, Mi, are 

known and given by equation (13). Therefore, the values of EI i can be re-calculated and compared with 

the pre-determined EI i listed in table 5. The results are compared in table 6(b). 

Table 6(b). Comparison of pre-determined EI i and current EI i for depth-tapered un-swept wing box

subjected to combined point load, P = 200 lb, and torque, T = 100×w = 1,700 in-lb, at wing-box tip; n = 8,

l = 100 in, ∆l = 12.5 in, M0 = Pl = 20,000 in-lb.

i ci, 

in

εi,×10!3

in/in

εi

'

,×10!3

in/in

εi,×10!3

in/in

Mi /M0 EI i,×107

lb-in2

Mi = EI i εi/ci ,

in-lb

Pi,

lb

(SPAR) (SPAR) (averaged) (current) (from 

table 5)

(diff., %) (estimated 

= applied)

(estimated

= applied)

0 2.000 0.822857 0.570476 0.696667 8/8 5.741624 (5.737712) (0.0682) 20,000.00 200.00

1 1.8125 0.749524 0.706667 0.728096 7/8 4.356397 (4.359247) (0.0654) 17,500.00 200.00

2 1.625 0.740952 0.710000 0.725476 6/8 3.359863 (3.358761) (0.0328) 14,000.00 200.00

3 1.4375 0.730476 0.693333 0.711905 5/8 2.524038 (2.522355) (0.0667) 12,500.00 200.00

4 1.25 0.708095 0.659048 0.683572 4/8 1.828630 (1.827993) (0.0348) 10,000.00 200.00

5 1.0625 0.664762 0.598095 0.631429 3/8 1.262018 (1.260120) (0.1506) 7,500.00 200.00

6 0.875 0.587619 0.490000 0.538810 2/8 0.811975 (0.813052) (0.1325) 5,000.00 200.00

7 0.6875 0.447619 0.301905 0.374762 1/8 0.458624 (0.458043) (0.1268) 2,500.00 200.00

8 0.500 0.000000 0.000000 0.000000 0/8 0.201965* (0.195093)* (3.5224) 0.00 200.00

* Three-points extrapolation [eq. (10)]. 
16

Note from table 6(b) that the values of EI i re-calculated from equation (2) based on the current loading 

condition are extremely close to the pre-determined EI i values, with percent differences in the 

negligible range of (0.0328 ~ 3.5224) percent. This proximity implies that EI i is practically invariant to 

the loading conditions, and one can feel confident in using the pre-determined EI i. If the current EI i is 

used, table 6(b) shows that the estimation errors will become zero, causing the estimated operational 

moments and shear loads, Mi, Pi , to have exactly the same values as the applied Mi, Pi . 

For complex loading conditions (for example, during flights), the applied moments, Mi, are unknown 

and constantly changing. Therefore, one has to rely upon the pre-determined EI i for estimating the 

operational moments and shear loads, Mi, Pi . 

4. Determination of Torsion Stiffness, GK i

To determine the domain torsion stiffness, GK i, a clock-wise torque (positive, leading edge up) of

T = 50×w = 850 in-lb was applied at the un-swept wing-box tip. Figure 13 showed the plots of the front 

and rear deflections, yi, yi

' , and the cross-sectional twist angles, φi [calculated from equation (6)]. Using 

the given beam depth factor, ci, the known cross-sectional torques, Ti[ = T = 50×w = 850 in-lb], and the 

cross-sectional twist angles, φi, induced by the current torsion loading (fig. 13), the domain torsion 

stiffness, GK i, are calculated from equation (5), and are listed in table 7.

Table 7. Torsion stiffness, GK i, for depth tapered un-swept wing box determined from loading case of 

clock-wise torque (positive), T = 50×w = 850 in-lb., at the wing box tip; n = 8, l = 100 in, 

∆l = l/n = 12.5 in.

i ci, 

in

φi,×10!3

rad

φi − φi!1 ,×10!3

rad

GK i,×107

lb-in2

Ti, 

in-lb

[eq. (6)] [calculated, eq. (5)] [calculated, eq. (9) = applied]

0 2.00 0.000000 ------------ 5.059234 850

1 1.8125 0.210012 0.210012 5.059234 850

2 1.625 0.504714 0.317650 3.344876 850

3 1.4375 0.914116 0.409402 2.595249 850

4 1.25 1.442358 0.528242 2.011389 850

5 1.0625 2.151171 0.708813 1.498985 850

6 0.875 3.155293 1.004122 1.058138 850

7 0.6875 4.686488 1.531195 0.693902 850

8 0.5 7.203585 2.517097 0.422113 850

In table 7, the value of GK i at i = 1 was used for i = 0 (not calculated) because GK i stays constant 

within the domain x0 ≤ x ≤ x1. In table 7, the torques, Ti, (last column) were calculated from equation (9) 

for the purpose of checking the accuracy of the strain-determined GK i. Note that the calculated values 

of Ti are all exactly the same as the applied beam tip torque (that is, Ti = T = 850 in-lb), implying that the 

values of the strain-determined GK i are all correct.

In figure 14, the strain-determined values of GK i listed in table 7 are plotted for graphical display of 

the shape of the GK i curve. Because GK i is constant within each small domain, a dashed stepwise 

plot was used for the present finite number of domains. In the limit when the domain density becomes 

infinity (that is, n → ∞), the stepwise curve will degenerate into a smooth solid curve shown in figure 14. 
17

5. Estimations of Operational Torques, Ti

The strain-determined values of GK i listed in table 7, are to be used to estimate the operational 

torques, Ti, for a combined loading case of the depth-tapered wing box subjected to point load, 

P = 200 lb, and clock-wise torque, T = 100×w = 1,700 in-lb, at the wing-box tip. 

Figure 15 shows the plots of the front and rear deflections, yi, yi

' , and cross-sectional twist angles, 

φi, for the current combined loading case. Using the strain-determined values of GK i listed in table 7, 

and the values of φi, shown in figure 15, the operational torques, Ti, were calculated from the torque 

equation (9), and are listed in table 8(a). 

Table 8-a. Estimated operational torques, Ti, using strain-determined GK i for depth-tapered un-swept 

wing box subjected to combined point load, P = 200 lb., and clockwise torque, T = 100×w = 1,700 in-lb, 

at wing tip; n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci, 

in

φi,×10!3

rad

φi − φi!1 , ×10!3

rad

GK i,×107

lb-in2

Ti, 

in-lb

[eq. (6)] (from table 7) (estimated) (applied) (error, %)

0 2.000 0.000000 0.000000 5.059234 1,699.89 (1,700.00) (0.0065)

1 1.8125 0.419996 0.419996 5.059234 1,699.89 (1,700.00) (0.0065)

2 1.625 1.055296 0.635300 3.344876 1,700.00 (1,700.00) (0.0000)

3 1.4375 1.873524 0.818228 2.595249 1,698.80 (1,700.00) (0.0706)

4 1.25 2.930006 1.056482 2.011389 1,700.00 (1,700.00) (0.0000)

5 1.0625 4.348243 1.418237 1.498985 1,700.73 (1,700.00) (0.0429)

6 0.875 6.355337 2.007094 1.058138 1,699.03 (1,700.00) (0.0571)

7 0.6875 9.418076 3.062739 0.693902 1,700.20 (1,700.00) (0.0118)

8 0.5 14.454031 5.035955 0.422113 1,700.59 (1,700.00) (0.0347)

In table 8(a), the values of GK i, Ti at i = 1 were used for i = 0 (not calculated) because GK i, Ti

at i = 1 is for the domain x0 ≤ x ≤ x1. The estimation errors of operational torques, Ti, are in the

negligible range of (0.000 ~ 0.0706) percent.

In figure 16, the estimated operational torques, Ti, listed in table 8(a) are plotted for comparison with 

the applied torque, T = 1,700 in-lb. Note that, the estimated operational torques, Ti (dash curve with open 

circular symbols), practically fell on top of the applied torque line of T = 1,700 in-lb (solid line with solid 

circular symbols), showing the new method is promising for the structure operational load estimations. 

6. Current Torsion Stiffness, GK i

For the current loading condition, values of applied torques and cross-sectional twist angles, Ti, φi ,

are known; therefore, the values of GK i can be recalculated from equation (5) for comparison with the 

pre-determined GK i listed in table 7. The results are compared in table 8(b). 
18

Table 8-b. Comparison of pre-determined GK i and current GK i for depth-tapered un-swept wing box 

subjected to combined point load, P = 200 lb, and clockwise torque, T = 100×w = 1,700 in-lb, at wing tip; 

n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci,

in

φi,×10!3

rad

φi − φi!1 ,×

10!3, rad

GK i,×107

lb-in2

Ti,

in-lb

[eq. (6)] [current, eq. (5)] (from table 7) (diff., %) (estimated = applied)

0 2.000 0.000000 0.000000 5.059572 (5.059234) (0.0067) 1,700.00

1 1.8125 0.419996 0.419996 5.059572 (5.059234) (0.0067) 1,700.00

2 1.625 1.055296 0.635300 3.344876 (3.344876) (0.0000) 1,700.00

3 1.4375 1.873524 0.818228 2.597076 (2.595249) (0.0704) 1,700.00

4 1.25 2.930006 1.056482 2.011393 (2.011389) (0.0002) 1,700.00

5 1.0625 4.348243 1.418237 1.498339 (1.498985) (0.0431) 1,700.00

6 0.875 6.355337 2.007094 1.058745 (1.058138) (0.0574) 1,700.00

7 0.6875 9.418076 3.0062739 0.693823 (0.693902) (0.0114) 1,700.00

8 0.5 14.454031 5.035955 0.421966 (0.422113) (0.0348) 1,700.00

Note from table 8(b) that the values of GK i re-calculated from equation (5) based on φi of the current 

loading condition are amazingly close to the pre-determined GK i of table 7, with percent differences 

only in the negligible range of (0.0000 ~ 0.0704) percent. This proximity implies that, similar to EI i, the 

pre-determined GK i are also practically invariant to the loading conditions, and thus confirming the 

accuracy of the pre-determined GK i. 

Depth-Tapered Swept Wing Box

The third example structure used in the operational load estimation analysis was the aluminum 

depth-tapered swept wing box (fig. 17) with dimensions listed in table 9. 

Table 9. Dimensions of depth-tapered swept wing box.

l, in c0, in cn, in w0, in wn, in t, in β, deg

(length) (root half depth) (tip half depth) (root width) (tip width) (wall thickness) (swept angle)

100.00 2.0 0.5 17.0 17.0 0.02296 10.0

 

Except for the swept angle, the dimensions of the swept wing box are identical to those of the 

un-swept wing box (table 4). The SPAR model (figs. 17, 18) of the depth-tapered swept wing box was 

generated from the SPAR model of the depth-tapered un-swept wing box (fig. 8) by simply distorting the 

rectangular skin shape of the un-swept wing box into a parallelogram skin shape by the swept angle, β.

Similar to the un-swept wing box case (fig. 8), the front and rear strain-sensing lines for the swept 

wing box are located respectively along the front and rear lower edges of the wing box (inset of fig.17). 

Each strain-sensing line has n = 8 evenly spaced domains (that is, n + 1 = 0 strain-sensing stations) (inset 

of fig. 17). Thus, the strain-sensing cross sections (formed by the pairs of front and rear strain-sensing 

stations) are parallel to the wing-box root chord and are, therefore, slanted with respect to the x-axis of the 

wing box by swept angle, β (in deg, fig. 17). Keep in mind that for the swept wing box, bending will 

induce torsion and vice versa (that is, bending and torsion are coupled).
19

1. Determination of Bending Stiffness, EI i

For the swept wing box, the strain-sensing cross sections are slanted by β deg with respect to the wing 

box axis (x-axis, fig. 17). Therefore, the strain-determined EI i will not be the true EI i, but must be 

considered as apparent EI i associated with the current particular strain-sensing station configurations. 

Because the values of the apparent EI i are to be used only as stepping-stones to calculate the operational 

moments and shear loads, Mi, Pi , it is not required to obtain the true EI i by re-arranging the strainsensing cross sections to be normal to the x-axis (fig. 17).

In step 1 of determining the apparent bending stiffness, EI i, the depth-tapered swept wing box was 

subjected to upward point load of P = 100 lb at the wing-box tip (50 percent chord, fig. 18). For the 

current loading condition, the applied moments will be Mi = Pl 1 − i/n = 10,000× 1 − i/n in-lb 

[eq. (3)]. 

As shown in figure 18, the upward bending of the swept wing box induced slight counterclockwise 

rotation (defined as negative = decreasing angle of attack) when looking toward the wing box root from 

the wing-box tip. Keep in mind that for the swept wing box, bending and torsion are coupled. Namely, 

bending will induce torsion and vice versa.

Figure 19 shows the plots of surface strains εi, εi

' , deflections yi

, yi

' , and cross-sectional twist 

angle, φi [calculated from eq. (6)] induced by a point load, P = 100 lb at the swept wing-box tip (50

percent chord). Note that, under bending, the rear deflections, yi

'

, are slightly larger than the associated 

front deflections, yi (that is, yi

' > yi

), indicating slight counterclockwise cross-sectional rotations (that is,

φi < 0), with maximum cross-sectional twist angle of only φi = φn = − 0.2642 deg at the wing-box tip. 

Note also from figure 19 that the differences between the pairs of front and rear strains, εi, εi

' , are 

graphically inconspicuous except for the wing box root region where the rear train, εi

'

, is markedly larger 

than the front strain, εi, because of increasing stress differentials caused by the swept effect.

In order to eliminate the bending-induced torsion components containing in the surface strains, 

εi , εi

' (fig.19), averaged strains, εi ≡ εi + εi' /2 , were used in the calculations of the apparent EI i. 

Using the given depth factor, ci, the averaged strains, !!, and the known bending moments, 

Mi = Pl(1 − l/n) in-lb, the apparent bending stiffness, EI i, were calculated from equation (2) and are 

listed in table 10.
20

Table 10. Apparent bending stiffness, EI i, for depth-tapered swept wing box determined from surface 

strains, εi, εi

' , induced by point load, P = 100 lb, at wing-box tip; n = 8, l = 100 in, ∆l = l/n = 12.5 in, 

M0 = Pl = 10.000 in-lb.

i ci,

 in

εi,×10!3

in/in

εi

'

,×10!3

in/in

εi,×10!3

in/in

!!/!! EI i,×107

lb-in2

Mi = EI i εi/ci , 

in-lb

Pi, 

lb

(SPAR) (SPAR) (averaged) (known) [eq. (2), 

swept]

(table 5, 

un-swept)

(diff., 

%)

[from eq. (1)]

= applied

[from eq. (4)

= applied]

0 2.000 0.260000 0.448571 0.354286 1.0 5.645157 (5.737712) (1.6131) 10,000.00 100.00

1 1.8125 0.345714 0.377143 0.361429 7/8 4.387964 (4.359247) (0.6588) 8,750.00 100.00

2 1.625 0.356190 0.368095 0.362143 6/8 3.365383 (3.358761) (0.1972) 7,500.00 100.00

3 1.4375 0.351429 0.360952 0.356191 5/8 2.522348 (2.522355) (0.1972) 6,250.00 100.00

4 1.25 0.338095 0.345714 0.341905 4/8 1.827993 (1.827993) (0.0000) 5,000.00 100.00

5 1.0625 0.313333 0.318095 0.315714 3/8 1.262020 (1.260120) (0.1508) 3,750.00 100.00

6 0.875 0.270952 0.267619 0.269286 2/8 0.812333 (0.803152) (1.1431) 2,500.00 100.00

7 0.6875 0.194286 0.180952 0.187619 1/8 0.461558 (0.458043) (0.7674) 1,250.00 100.00

8 0.500 0.000000 0.000000 0.000000 0/8 0.209695* (0.195093) (7.4877) 0.00 100.00

 * Three-points extrapolation [eq. (10)].

In table 10 the values of EI i for the un-swept wing box (taken from table 5) are also listed (in 

parenthesis) for comparison. The two sets of EI i are quite close with percent differences in the 

negligible range of (0.0000 ~ 1.6131) percent, except wing-box tip. The large difference (7.4877 percent) 

at the wing-box tip is due to a small number divided by another small number. In the last two columns of 

table 10, the values of operational moments and shear loads, Mi, Pi , were calculated respectively from 

equations (1) and (4) using the strain-determined apparent EI i. The calculated values of Mi, Pi are 

exactly the same as the applied values of Mi, Pi . This perfect agreement indicates that the calculated

apparent EI i associated with the present strain-sensing station configuration can be used with 

confidence as stepping-stones in the estimations of the operational loads on the swept wing-box. 

Figure 20 shows the plots of the strain-determined apparent EI i curve (solid curve with solid 

circular symbols) for the swept wing box and the elasticity-calculated EI i curve (dashed curve with 

open circular symbols) for the un-swept wing box (table 5). The two EI i curves are quite close except 

for the slight difference near the wing-box root (see table 10).

2. Estimations of Operational Moments and Shear Loads, !!, Pi

Based on the pre-determined apparent bending stiffness, EI i, listed in table 10, the operational 

moments and shear loads, Mi, Pi , were calculated for two different loading cases: 

1) Bending load, P = 200 lb, applied at wing-box tip (to study bending-induced torsion)

2) Combined bending load, P = 200 lb, and counterclockwise torque, 

T = − 100×w = − 1,700 in-lb, applied at the wing-box tip (to study bending-torsion coupling).

2.1. Bending Only (P!=!200 lb)

For step 2, the pre-determined apparent EI i listed in table 10 were used to calculate the operational 

moments and shear loads, Mi

P, PiP , for the depth-tapered swept wing box subjected to point load of 

P = 200 lb at the wing-box tip (50 percent chord). For the current loading condition, the applied moments 

will be Mi

P = Pl 1 − i

n = 20,000(1 − i/n) in-lb.

Figure 21 shows the plots of SPAR-generated strains, εi, εi

' , deflections, yi

, yi

' , and cross-sectional 

twist angle, φi

P [eq. (6)], induced by the point load of P = 200 lb at the wing-box tip. Note that the values 
21

of the rear strains, εi

'

, are slightly larger than the corresponding front strains, εi, indicating that the upward 

bending induced counterclockwise rotation (φi

P < 0). The data of φiP

are to be used to calculate the 

bending-induced torque components, TP, in the subsequent combined bending and torsion analysis. The 

difference between the two strains is greatly magnified toward the wing-box root because of swept effect. 

Using the known depth factor, ci, the averaged strains, εi

P ≡ εi + εi

' /2 (to eliminate torsion effect) 

based on the strains, εi, εi

' , of figure 21, and the strain-determined apparent EI i listed in table 10, the 

operational moments and shear loads, Mi

P, PiP were calculated respectively from equations (1) and (4), 

and are listed in table 11. 

Table 11. Estimated operational moments and shear loads, Mi

P, PiP , based on strain-determined apparent 

EI i for depth-tapered swept wing box subjected to point load, P = 200 lb, at wing-box tip; n = 8, 

l = 100 in, ∆l = l/n = 12.5 in, M0 = Pl = 20,000 in-lb.

i ci, 

in

εi,

×10!!

in/in

εi

'

,

×10!!

in/in

εi

P,

×10!!

in/in

φi

P

,

×10!3

rad

EI i,

×107

lb-in2

Mi

P = EI i εi

P/ci , 

in-lb

Pi

P, 

lb

(SPAR) (SPAR) (averaged) [eq. (6)] (from 

table 10)

(estimated) (applied) (error, 

%)

(estimated) (applied) (error, 

%)

0 2.000 0.520000 0.898095 0.709048 0.000000 5.645157 20,013.42 (20,000)(0.0671) 206.61 (200) (3.3050)

1 1.8125 0.688571 0.751429 0.720000 -0.602941 4.387964 17,430.81 (17,500)(0.3954) 206.61 (200) (3.3050)

2 1.625 0.712381 0.736667 0.724524 -1.589412 3.365383 15,004.93 (15,000)(0.0329) 194.07 (200) (2.9650)

3 1.4375 0.702587 0.720952 0.711770 -2.717941 2.522348 12,489.26 (12,500)(0.0859) 201.25 (200) (0.6250)

4 1.25 0.675714 0.691429 0.683572 -3.961765 1.827993 9,996.52 (10,000)(0.0048) 199.42 (200) (0.2900)

5 1.0625 0.627619 0.636190 0.631905 -5.310882 1.262020 7,505.66 (7,500)(0.0755) 199.27 (200) (0.3650)

6 0.875 0.541429 0.535714 0.538572 -6.725294 0.812333 5,000.00 (5,000)(0.0000) 200.45 (200) (0.2250)

7 0.6875 0.387619 0.362857 0.375238 -8.071176 0.461558 2,519.19 (2,500)(0.7676) 198.46 (200) (0.7700)

8 0.500 0.000000 0.000000 0.000000 -9.214706 0.209695* 0.00 (0.00)(0.0000) 201.54 (200) (0.7700)

The errors of the estimated Mi

P, PiP listed in table 11 are in reference respectively to the applied 

moments, Mi

P = M0 1 − i/n = 20,000× 1 − i/n in-lb, and applied shear loads, PiP = 200 lb. Note from 

table 11 that the errors of estimated Mi

P are in the negligible ranges of (0 ~ 0.7676) percent, and the errors 

of estimated Pi

P are in the small range of (0.2250 ~ 3.3050). Table 11 also lists the cross-sectional twist 

angles, φi

P (< 0), induced by the bending load of P = 200 lb because of swept effect. The values of φi

P

listed in table 11 will be used to calculate the bending-induced torque components, TP, in the subsequent 

combined bending and torsion analysis after the values of apparent torsion stiffness, GK i have been 

determined.

In figure 22, the estimated operational moments and shear loads, Mi

P, PiP , listed in the last two 

columns of table 11 are plotted for graphical comparison with the applied moments and applied shear 

loads, Mi

P, PiP (table 11). Note that the estimated MiP curve (dashed line with open circular symbols) is 

practically coincidental to the applied Mi

P curves (solid straight line with solid circular symbols). On the 

other hand, the estimated Pi

P curve (dashed curve with open diamond symbols) agreed nicely with the 

applied Pi

P curve (solid straight line with solid diamond symbols) in the outboard region, but slightly off 

near the wing box root. The good correlations between the estimated and applied Mi

P, PiP thus give the 

confidence in the new method for operational load predictions using surface strains. 
22

2.2. Combined bending and Torsion (P!=!200 lb, T!=! − 1,700 in-lb)

For step 2, the values of strain-determined apparent EI i listed in table 10 were used to estimate the 

operational moments and shear loads, Mi

P+T, PiP+T , for the combined loading case of the depth-tapered 

swept wing box subjected to point load, ! = 200 lb, and counterclockwise torque, T = − 1,700 in-lb, at 

the wing-box tip. For the current combined bending and torsion loading, the applied moment, Mi

P+T, 

induced by the applied point load , P, and torque, T, can be expressed as:

Mi

P+T = Pl 1! i

n

"

#

$ %

&

'

P-induced

bending moment

!#"#$

+ T sin(

T -induced 

bending 

moment

!"$ = 20,000 ) 1! i

n

"

#

$ %

&

'

P-induced

!##"##$

+ 295.20

T -induced

!"$

 

(14)

(i = 0,1,2,3,....,n)

Note from equation (14) that the torque-induced bending moments are constant along the x-axis of the 

swept wing box axis (fig. 17), implying that the applied torque, T, induces only pure bending for which 

shear loads are zero. The estimations of operational torque, Ti

P+T, induced by the current combined 

bending and torsion loading case will be delayed until the values of torsion stiffness, GK i, are

determined. 

Figure 23 shows the plots of strains, εi, εi

' , the total averaged strains, εi

P+T ≡ εi + εi

' /2 , 

deflections, yi, yi

' , and cross-sectional twist angle, φi

P+T < 0 , induced by the current combined bending 

and torsion loading. Also in figure 23, the averaged surface strains, εi

T, induced by the counterclockwise 

torque, T = − 1,700 in-lb only are also plotted for comparison. Note that the values of εi

T are almost 

negligible in comparison with the values of εi

P+T. Because of counterclockwise twist, the rear bending 

strains and deflections, {εi

'

, yi

'

}, are somewhat larger than the front bending strains and deflections,{εi, yi}. 

Note that the difference between εi and εi

' is greatly magnified (greater than bending case in fig. 21) 

toward the swept wing-box root because of added torsion and swept effect. 

In view of the known depth factor, ci, and the values of strain-determined apparent EI i listed in 

table 10, the operational moments and shear loads, Mi

P!T, PiP!T , were calculated respectively from 

equations (1) and (4) using the averaged strains, εi

P!T. The resulting values of the estimated total 

operational bending moments and shear loads Mi

P!T, PiP!T for the current combined bending and torsion 

loading case are listed in table 12.

 

Note in table 12 that the errors of estimated Mi

P!T are in the negligible range of (0.0437 ~ 0.8765)

percent except for the wing-box tip (22.5847 percent error). On the other hand, the errors of estimated 

Pi

P!T are also in the negligible range (0.025 ~ 2.7450) percent except for the fixed end region 

(3.7550 percent error).
23

Table 12. Estimated operational moments and shear loads, Mi

P!T, PiP!T , based on strain-determined 

apparent EI i for depth tapered swept wing box subjected to point load, P = 200 lb, and 

counterclockwise torque, T = − 100×w = − 1,700 in-lb, at wing box tip; n = 8, l = 100 in, 

∆l = l/n = 12.5 in.

i ci, 

in

εi,×10!3

in/in

εi

'

,×10!3

in/in

εi

P+T

,×10!3

in/in

EI i,×107

lb-in2 Mi

P+T = EI i εi

P+T

/ci , 

in-lb

Pi

P+T, 

lb

(SPAR) (SPAR) (averaged) (from 

table 10)

(estimated) (applied) (error, %) (estimated) (applied) (error, %)

0 2.000 0.393333 1.047619 0.720476 5.645157 20,336.00 (20,295.20) (0.2010) 207.51 (200) (3.7550)

1 1.8125 0.680000 0.785714 0.732857 4.387964 17,742.07 (17,795.20) (0.2986) 207.51 (200) (3.7550)

2 1.625 0.711429 0.767143 0.739286 3.365383 15,310.65 (15,295.20) (0.1007) 194.51 (200) (2.7450)

3 1.4375 0.700952 0.758095 0.729524 2.522348 12,800.79 (12,795.20) (0.0437) 200.79 (200) (0.3950)

4 1.25 0.674286 0.737619 0.705953 1.827993 10,323.82 (10,295.20) (0.2780) 198.16 (200) (0.9240)

5 1.0625 0.619048 0.697143 0.658096 1.262020 7,816.76 (7,795.20) (0.2766) 200.56 (200) (0.2800)

6 0.875 0.525238 0.620000 0.572619 0.812333 5,316.08 (5,295.20) (0.3943) 200.05 (200) (0.0250)

7 0.6875 0.358095 0.481905 0.420000 0.461558 2,819.70 (2,795.20) (0.8765) 199.71 (200) (0.1450)

8 0.500 0.021952 0.150619 0.086286 0.209695* 361.87 (295.20)(22.5847) 196.63 (200) (1.6850)

In figure 24, the estimated operational moments and shear loads, listed in table 12 are plotted for 

graphical comparison with the applied moments [eq. (14)] and applied shear loads, Pi

P+T = PiP = 200 lb. 

Note that the estimated Mi

P+T curve (dashed curve with open circular symbols) are graphically 

coincidental with applied Mi

P+T curve (solid straight lines with solid circular symbols). On the other hand, 

the estimated Pi

P+T curve (dashed curve with open diamond symbols) agreed with the applied PiP+T curves 

(solid straight lines with solid diamond symbols), but is slightly off in the fixed and free end regions. 

Such satisfactory performance gives confidence in the new method for estimating the operational loads 

using surface strains. The accuracies of the components contained in the total estimated loads 

Mi

P+T, PiP+T will also be investigated in the following sections.

2.3. Bending-induced Components, Mi

P, PiP

To calculate the P-induced operational load components, Mi

P, PiP , the averaged strains, εi

T, induced 

by the counterclockwise torque, T = − 1,700 in-lb (fig. 23) must be abstracted from the total averaged 

strains, εi

P+T. Namely, to eliminate T-induced components, Mi

T, PiT , the strain differentials, εi

P+T − εiT, 

must be used to calculate Mi

P, PiP .

In view of the known depth factor, ci, and the values of strain-determined apparent EI i listed in 

table 10, the P-induced operational load components, Mi

P, PiP were calculated respectively from 

equations (1) and (4) using the averaged strain differentials, εi

P+T − εiT. The resulting estimated values of 

Mi

P, PiP are listed in table 13. 
24

Table 13. Bending induced components Mi

P, PiP , of estimated operational moments and shear loads, 

Mi

P+T, PiP+T , based on strain-determined apparent EI i for depth-tapered swept wing-box subjected to 

point load, P = 200 lb, and counter-clockwise torque, T = − 100×w = − 1,700 in-lb, at wing-box tip; 

n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci, 

in

εi

P+T

,

×10!!

in/in

εi

T

,

×10!!

in/in

εi

P+T − εiT ,

×10!!

in/in

EI i,

×107

lb-in2

Mi

P = EI i εi

P+T − εiT /ci , 

in-lb

Pi

P, 

lb

(P+T) (T only) (differential) (from

table 10)

(estimated) (applied) (error, 

%)

(estimated) (applied) (error, 

%)

0 2.00 0.720476 0.011905 0.708571 5.645157 20,001.44 (20,000.00)(0.0072) 200.12 (200) (0.0600)

1 1.8125 0.732857 0.010000 0.722857 4.387964 17,499.98 (17,500.00)(0.0001) 200.12 (200) (0.0600)

2 1.625 0.739286 0.014762 0.724542 3.365383 15,005.30 (15,000.00)(0.0353) 195.57 (200) (2.2150)

3 1.4375 0.729524 0.017619 0.711905 2.522348 12,491.63 (12,500.00)(0.0670) 201.09 (200) (0.5450)

4 1.25 0.705953 0.021190 0.684763 1.827993 10,013.94 (10,000.00)(0.1394) 198.22 (200) (0.8900)

5 1.0625 0.658096 0.026667 0.630667 1.262020 7,490.96 (7,500.00)(0.1206) 201.84 (200) (0.9200)

6 0.875 0.572619 0.034286 0.538333 0.812333 4,997.78 (5,000.00)(0.0444) 199.45 (200) (0.2750)

7 0.6875 0.420000 0.038573 0.381429 0.461558 2,560.75 (2,500.00)(2.4000) 194.96 (200) (2.5200)

8 0.500 0.086286 0.042857 0.043429 0.209695* 182.14 (0.00) ---------- 190.29 (200) (4.8550)

* Three-points extrapolation [eq. (10)].

In table 13, the applied moments, Mi

P, listed in parentheses, were calculated from MiP = M0 1 −

i/n = 20,000× 1 − i/n in-lb [eq. (3)]. Note from table 13 that the estimation errors of P-induced 

operational moment, M!

!, are in the small range of (0.0001 ~ 2.4000) percent. The estimation errors of 

P -induced operational shear loads, Pi

P, are also in the small range of (0.0600 ~ 2.5200) percent except for 

the free end (4.8550 percent error). Note also that the values of {Mi

P, PiP} extracted from combined 

bending and torsion loading (table 13) are reasonably close to the values of {Mi

P, PiP} determined from 

bending loading only (table 11).

In view of equation (14), the torsion-induced applied moment, Mi

T = 295.20 in-lb, is constant along 

the span, and such pure bending does not induce any shear loads. Therefore, the applied shear loads for 

both Pi

P+T (table 12) and PiP (table 13) are identical to the applied point load, P = 200 lb. 

2.4. Torsion-induced Components, Mi

T, PiT

The torsion-induced components of operational moments and shear loads, Mi

T, PiT , can be calculated 

using the combined-load induced averaged surface strains, εi

P+T (table 12). However, the bending-induced 

averaged strains, εi

P (table 11), must be subtracted from εiP!T. 

In view of the known depth factor, ci, and the values of strain-determined apparent EI i listed in 

table 10, the torsion-induced operational moments and shear loads, Mi

T, PiT were calculated respectively 

from equations (1) and (4) using the averaged strain differentials, εi

P+T − εiP , to eliminate P-induced 

components, Mi

P, PiP . The resulting estimated values of MiT, PiT are listed in table 14. 
25

Table 14. Torsion induced components Mi

T, PiT of estimated operational moments and shear loads, 

Mi

P+T, PiP+T , based on strain-determined apparent EI i for depth-tapered swept wing box subjected to 

point load, P = 200 lb, and counterclockwise torque, T = − 100×w = − 1,700 in-lb, at wing-box tip; 

n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci, 

in

εi

P+T

,×10!3

in/in

εi

P,×10!3

in/in

εi

P!T − εiP ,

×10!3, in/in

EI i,×10!

lb-in2 Mi

T = EI i εi

P+T − εiP /ci , 

in-lb

Pi

T

, 

lb

(combined 

load)

(P-induced, 

from 

table 11)

(differential) (from table 10) (estimated) (applied) (error, %) (estimated)(applied)

0 2.000 0.720476 0.709048 0.011428 5.645157 322.56 (295.20) (9.2683) 0.90 (0.00)

1 1.8125 0.732857 0.720000 0.012857 4.387964 311.26 (295.20) (5.4404) 0.90 (0.00)

2 1.625 0.739286 0.724524 0.014762 3.365383 305.72 (295.20) (3.5637) 0.44 (0.00)

3 1.4375 0.729524 0.711770 0.017754 2.522348 311.53 (295.20) (5.5318) -0.46 (0.00)

4 1.25 0.705953 0.683572 0.022381 1.827993 327.30 (295.20) (10.8740) -1.26 (0.00)

5 1.0625 0.658096 0.631905 0.026191 1.262020 311.09 (295.20) (5.3828) 1.29 (0.00)

6 0.875 0.572619 0.538572 0.034047 0.812333 316.09 (295.20) (7.0766) -0.40 (0.00)

7 0.6875 0.420000 0.375238 0.044762 0.461558 300.51 (295.20) (1.7988) 1.25 (0.00)

8 0.500 0.086286 0.000000 0.086286 0.209695* 361.87 (295.20) (22.5847) -4.91 (0.00)

 * Three-points extrapolation [eq. (10)].

In table 14, the torsion-induced applied moments are constant and were calculated from 

Mi

T = T sin β = 295.20 in-lb [last term in equation (14)]. The errors of estimated torque-induced moment, 

Mi

T, are in the range of (1.7988 ~ 9.2683) percent excluding the wing-box tip (22.5947 percent error). 

Because the torsion-induced bending moments are constant (pure bending) along the wing box span, no 

applied shear loads were induced (that is, Pi

T = 0). Note that the torque-induced components of the 

estimated operational shear loads, Pi

T, in the last column are very close to the theoretical zero value. 

Alternatively, the torsion-induced components Mi

T, PiT can be obtained from the total estimated 

operational moments and shear loads, Mi

P+T, PiP+T (table 12) by subtracting the P-induced components, 

Mi

P, PiP (table 11). Namely, using the formulae, MiT= MiP+T − MiP and PiT= PiP+T − PiP. The results are 

listed in table 15.

Table 15. Estimated operational moments and shear loads, Mi

T, PiT , based on strain-determined apparent 

EI i for depth tapered swept wing box subjected to point load, P = 200 lb, and counterclockwise torque, 

T = − 100×w = − 1,700 in-lb, at wing box tip; n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci, 

in

Mi

P+T

,

in-lb

Mi

P, 

in-lb

Mi

T = Mi

P+T − MiP , 

in-lb

Pi

P+T

, 

lb

Pi

P,

lb

Pi

T = Pi

P+T − PiP ,

lb

(table 12) (table 11) (estimated) (applied) (error, %) (table 12) (table 11) (estimated) (applied)

0 2.000 20,336.00 20,013.42 322.58 (295.20) (9.2752) 207.51 206.61 0.90 (0.00)

1 1.8125 17,742.07 17,430.81 311.26 (295.20) (5.4404) 207.51 206.61 0.90 (0.00)

2 1.625 15,310.65 15,004.93 305.72 (295.20) (3.5637) 194.51 194.07 0.44 (0.00)

3 1.4375 12,800.79 12,489.26 311.53 (295.20) (5.5318) 200.79 201.25 -0.46 (0.00)

4 1.25 10,323.82 9,996.52 327.30 (295.20) (10.8740) 198.16 199.42 -1.26 (0.00)

5 1.0625 7,816.76 7,505.66 311.10 (295.20) (5.3862) 200.56 199.27 1.29 (0.00)

6 0.875 5,316.08 5,000.00 316.08 (295.20) (7.0732) 200.05 200.45 -0.40 (0.00)

7 0.6875 2,819.70 2,519.19 300.51 (295.20) (1.7988) 199.71 198.46 1.25 (0.00)

8 0.50 361.87 0.00 361.87 (295.20) (22.5847) 196.63 201.54 -4.91 (0.00)
26

Note that the estimated values of the torsion-induced components, Mi

T, PiT , listed in table 15 are 

practically identical to the values of Mi

T, PiT listed in table 14. This result shows that either method can 

be used to obtain the torsion-induced components, Mi

T, PiT .

3. Determination of Torsion Stiffness, GK i

For determination of the cross-sectional torsion stiffness, GK i, a clockwise (positive) torque of

T = 50×w = 850 in-lb was applied at swept wing-box tip. It is important to mention that the 

counterclockwise (negative) torque of T = − 50×w = − 850 in-lb did also produce identical values 

of GK i. 

Figure 25 shows the plots of the front and rear deflections, yi, yi

' , averaged deflection, 

yi = yi + yi

' /2, and cross-sectional twist angles, φi

T > 0 , induced by the clockwise torque of 

T = 50×w = 850 in-lb. The non-zero averaged deflection, yi, indicates the magnitude of bending 

deflections induced by torsion because of the swept effect. 

Using the given beam depth factor, c!, the applied torques, Ti[ = T = 50×w = 850 in-lb], and the crosssectional twist angles, φi

T (in reference to surfaces slanted by β deg with respect to x-axis (fig. 18) given 

in figure 25, the apparent torsion stiffness, GK i were calculated from equation (5), and are listed in table 

16.

Table 16. Apparent torsion stiffness GK i for depth-tapered swept wing box determined from the 

loading condition of clockwise torque, T = 50×w = 850 in-lb, at wing box tip; n = 8, l = 100 in, 

∆l = l/n = 12.5 in.

i ci,

 in

φ×10−3,

rad

φi − φi-1 ×10!3, 

rad

GK i×107,

lb-in2

Ti, 

in-lb

[eq. (6)] [eq. (5),

 swept]

(table 7, 

un-swept)

(diff., %) [from eq. (9)]

= applied

0 2.000 0.000000 ----------- 5.073052 (5.059234) (0.2731) 850

1 1.8125 0.209440 0.209440 5.073052 (5.059234) (0.2731) 850

2 1.625 0.537649 0.328209 3.237266 (3.344876) (3.2172) 850

3 1.4375 0.950594 0.412945 2.572982 (2.595249) (0.8580) 850

4 1.25 1.485293 0.534699 1.987099 (2.011389) (1.2076) 850

5 1.0625 2.206183 0.720890 1.473873 (1.498985) (1.6753) 850

6 0.875 3.230011 1.023828 1.037772 (1.058138) (1.9247) 850

7 0.6875 4.798556 1.568545 0.677379 (0.693902) (2.3812) 850

8 0.5 7.379479 2.580923 0.411674 (0.422113) (2.4730) 850

In table 16 the values of GK i for un-swept wing box taken from table 7 are also listed (in 

parentheses) for comparison. Note that the values of the apparent GK i of the swept wing box are quite 

close to the corresponding values of the strain-determined GK i of the un-swept wing box. Note also 

from table 16 that the operational torques, T!, listed in the last column were calculated from equation (9) 

using the apparent GK i. The estimated values of T! are exactly the same as the applied beam tip torque 

(that is, Ti = T = 850 in-lb), implying that one can have confidence in using the apparent GK i for 

estimating the operational torques. 
27

Figure 26 shows the stepwise plot of the apparent GK i listed in table 16, because of a finite number 

of small domains. The continuous curve shown in figure 26 is for the limit case when the domain number 

reaches infinity (that is, n → ∞).

4. Estimations of Operational Torques, Ti

P+T

In this section, the pre-determined apparent shear stiffness, GK i listed in table 16, were used to 

estimate the operational torques, Ti

P+T, for combined loading case of bending load, P = 200 lb, and 

counterclockwise torque, T = − 100×w = − 1,700 in-lb, applied at the wing-box tip.

The combined-load-induced operational torques, Ti

P+T, contain the P-induced torques, TiP, and 

T-induced torques, Ti

T [that is, TiP+T = TiP + TiT]. In reality, TiP+T are actually felt by the structure, however, 

in checking the accuracies the estimated operational torques, Ti

P+T, the accuracies of component torques, 

Ti

P, TiT were also checked.

4.1. T-Induced Torques, Ti

T

In this section, the pre-determined apparent GK i (table 16) were used to calculate the operational 

torque components, Ti

T, induced by the torque, T = − 100×w = − 1,700 in-lb.

Figure 27 shows the plots of the total cross-sectional twist angles, φi

P+T < 0 induced by the current 

combined bending and torsion loading (taken from figure 23), and the cross-sectional twist angles, 

φi

P < 0 , induced by the upward bending load, P = 200, only (taken from figure 21). 

Because of the bending-induced cross-sectional twist angle, φi

P (fig. 27), in the calculations of the 

torsion-induced torque components, Ti

T, for the current combined loading case, the bending-induced 

cross-sectional twist angle, φi

P

, must be removed from the total cross-sectional twist angle, φi

P+T. Namely 

the twist angles differential, φi

T = φiP+T − φiP < 0 were used in equation (9) to calculate the values of

Ti

T. The results are listed in table 17. 

Table 17. Estimated T-induced operational torques, Ti

T, based on pre-determined apparent GK i for 

depth-tapered swept wing box subjected to counterclockwise torque, T = − 100×w = − 1,700 in-lb, at 

wing tip; n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci,

in

φi

P!T

, 

×10!!

rad

(P+T)

φi

P

,

×10!!

rad

(P only, 

table 11)

φi

T = φi

P+T − φiP ,

×10!!

rad

φi

T − φi!1T ,

×10!!

rad

GK i,×107

lb-in2

[from 

table 16]

Ti

T

, 

in-lb

(estimated) (applied) (error, 

%)

0 2.0 0.000000 0.000000 0.000000 0.000000 5.073052 -1,766.62 (-1,700) (3.9188)

1 1.8125 -1.038235 -0.602941 -0.435294 -0.435294 5.073052 -1,766.62 (-1,700) (3.9188)

2 1.625 -2.665297 -1.589412 -1.075885 -0.643291 3.237266 -1,666.00 (-1,700) (2.0000)

3 1.4375 -4.619428 -2.717941 -1.901487 -0.825602 2.572982 -1,699.41 (-1,700) (0.0347)

4 1.25 -6.932997 -3.961765 -2.971232 -1.069745 1.978099 -1,700.55 (-1,700) (0.0324)

5 1.0625 -9.723388 -5.310882 -4.412506 -1.441274 1.473873 -1,699.40 (-1,700) (0.0353)

6 0.875 -13.185676 -6.725294 -6.460382 -2.047876 1.037772 -1,700.18 (-1,700) (0.0106)

7 0.6875 -17.669449 -8.071176 -9.598273 -3.137891 0.677379 -1,700.43 (-1,700) (0.0253)

8 0.5 -23.976415 -9.214706 -14.761709 -5.163436 0.411674 -1,700.52 (-1,700) (0.0306)
28

Note from the last column of table 17 that the errors (in view of applied torque, T = − 1,700 in-lb) for the 

estimated T-induced operational torques, Ti

T, are in the negligible range of (0.0106 ~ 3.9188) percent. The 

values of Ti

T listed in table 17 are plotted in figure 28 for graphical comparison with the applied torque, 

T = − 1,700 in-lb. Note that, except for the fixed end region where the estimated data points are slightly 

off, the rest of the data points of the estimated T-induced operational torques Ti

T (dashed curve with open 

circular symbols) pictorially fell on top of the T = − 1,700 in-lb applied torque line (solid line with solid 

circular symbols), giving confidence in using the new method for structure operational loads (torques) 

estimations.

4.2. P-Induced Torques, Ti

P

As shown in figure 21, the swept effect caused the point load, P = 200 lb, at wing-box tip to induce 

counterclockwise cross-sectional rotations (that is, φi

P < 0). Therefore, in this section, the pre-determined 

apparent GK i (table 16) will be used to calculate the P-induced operational torques, Ti

P. The P-induced 

applied torques, Ti

P, can be expressed as equations (15) and (16): 

For Ti

P at domain junctures:

Ti

P = !P(lsin") 1! i

n

#

$

% &

'

( = !3,472.96 ) 1! i

n

#

$

% &

'

( (in-lb)

(i = 0,1,2,3,....,n)

(15)

For Ti

P at domain centers (TiP constant over the domain):

Ti

P = !P(lsin")

1

2

1! i

n

#

$

% &

'

( + 1! i !1

n

#

$

% &

'

( )

*

+ ,

-

. = !3,472.96 /

1

2

1! i

n

#

$

% &

'

( + 1! i !1

n

#

$

% &

'

( )

*

+ ,

-

. (in-lb)

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

(16)

Using the strain-determined apparent GK i of table 16 and the P-induced cross-sectional twist angle, 

φi

P

, of figure 21, one can calculate the P-induced operational torques, Ti

P, using equation (9). The results 

are listed in table 18.

Table 18. Estimated P-induced operational torques, Ti

P, induced by point load, P = 200 lb, at tip of 

depth-tapered swept wing box; n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci,

in

φi

P

,×10!3

rad

φi

P − φi!1P ,×10!3

rad

GK i,×107

lb-in2 Ti

P, 

in-lb

(P only, table 11) (differential) (from table 16) (estimated) [applied, 

eq. (16)]

(error, %)

0 2.000 0.000000 0.000000 5.073052 -------- (-3,472.96) (--------)

1 1.8125 -0.602941 -0.602941 5.073052 -2,447.00 (-3,255.90) (24.8441)

2 1.625 -1.589412 -0.986471 3.237266 -2,554.78 (-2,821.78) (9.4621)

3 1.4375 -2.717941 -1.128529 2.572982 -2,322.95 (-2,387.66) (2.7102)

4 1.25 -3.961765 -1.243824 1.987099 -1,977.28 (-1,953.54) (1.2152)

5 1.0625 -5.310882 -1.349117 1.473873 -1,590.74 (-1,519.42) (4.6939)

6 0.875 -6.725294 -1.414412 1.037772 -1,174.28 (-1,085.30) (8.1987)

7 0.6875 -8.071176 -1.345882 0.677379 -729.34 (-651.18) (12.0028)

8 0.5 -9.214706 -1.143530 0.411674 -376.61 (-217.06) (73.5050)
29

Note from table 18 that the estimation errors for Ti

P (last column) are somewhat discouraging near the 

wing-box root and tip regions.

4.3. Combined-Load-Induced Torques, Ti

P+T

In this section, the pre-determined apparent GK i (table 16) will be used to estimate the operational 

torques, Ti

P+T, for a loading case of the depth-tapered wing box subjected to combined point load, 

! = 200 lb, and counterclockwise torque, T = − 100×w = − 1,700 in-lb, at the wing-box tip. The 

operational torques, Ti

P+T, contain P-induced torque components, TiP, and T-induced torque components, 

Ti

T (that is, TiP+T = TiP + TiT). For a current combined loading case, the combined applied torques can be 

expressed as equations (17) and (18) :

For Ti

P+T at domain junctures:



 torque induced

-induced torque

 (sin ) 1

−

+

⎟ +

⎠

⎞ ⎜

⎝

⎛ = − −

= +

T

T

i

P

T

i

P

i

P T

i

T

n

i Pl

T T T



β = !3,472.96 " 1! i

n

#

$

% &

'

( !1,700 (in-lb) (17)

(i = 0,1,2,3,....,n) 

For Ti

P+T at domain centers (!!

!!! constant over the domain):

Ti

P+T = TiP + TiT = !Pl(sin")

1

2

1! i

n

#

$

% &

'

( + 1! i !1

n

#

$

% &

'

( )

*

+ ,

-

.

P-induced torque

!######"######$

+ Ti

T

T !induced

torque

%

 = !3,472.96 "

1

2

#

$

% &

'

( 1! i

n

#

$

% &

'

( + 1! i !1

n

#

$

% &

'

( )

*

+ ,

-

. !1,700 (in-lb) 

(18)

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

For the current combined loading case of bending load, P = 200 lb, and counter-clock-wise torque, 

T = − 100×w = 1,700 in-lb, at the swept wing-box tip, the values of the strain-determined apparent GK i

of table 16 and the cross-sectional twist angle, φi

P+T < 0 , of figure 23 can be used to calculate the total 

operational torque, Ti

P+T, using equation (9). The results are listed in table 19. 
30

Table 19. Estimated operational torque, Ti

P+T, based on strain-determined GK i for depth-tapered swept 

wing box subjected to combined bending load, ! = 200 lb, and counter-clock-wise torque, 

T = − 100×w = − 1,700 in-lb, at wing tip; n = 8, l = 100 in, ∆l = l/n = 12.5 in.

i ci,

in

φi

P!T

,

×10!! rad

φi

P!T − φi!1P!T ,

×10!! rad

!" !,

×10! lb-in2 Ti

P!T

, 

in-lb

(P+T) (differential) (from table 16) (estimated) [applied, eq. (18)] (error, %)

0 2.0 0.000000 0.000000 5.073052 ---------- (-5,172.96) (--------)

1 1.8125 -1.038235 -1.038235 5.073052 -4,213.62 (-4,955.90) (14.9777)

2 1.625 -2.665297 -1.627062 3.237266 -4,213.79 (-4,521.78) (6.8113)

3 1.4375 -4.619428 -1.954131 2.572982 -4,022.36 (-4,087.66) (1.5975)

4 1.25 -6.932997 -2.313569 1.987099 -3,677.83 (-3,653.54) (0.6648)

5 1.0625 -9.723388 -2.790391 1.473873 -3,290.15 (-3,219.42) (2.1970)

6 0.875 -13.185676 -3.462288 1.037772 -2,874.45 (-2,785.30) (3.2007)

7 0.6875 -17.669449 -4.483773 0.677379 -2,429.77 (-2,351.18) (3.3426)

8 0.5 -23.976415 -6.306966 0.411674 -2,077.13 (-1,917.06) (8.3498)

Note from table 19 that the errors of estimated operational torques, Ti

P+T, are fairly good except for the 

regions near the fixed and free ends.

Figure 29 shows the plots of estimated Ti

P+T (table 18) and estimated TiP (table 17). The applied 

Ti

P+T, TiP curves (straight lines) (tables 17, and 18) are also shown for comparison. The estimated values 

of Ti

P+T, TiP compare fairly well with the applied TiP+T, TiP curves (straight lines) in the outboard 

region. However, the two estimated Ti

P+T, TiP curves started to divert from the applied TiP+T, TiP curves 

as the wing root is approached. This data deviation could be attributed to the increasing errors of φi near

the wing root. Remember that φi is calculated from equation (6) using the deflection-differential, 

yi − yi

' , between the front and rear strain-sensing lines. As the wing root is approached, both yi

, yi

'

become very small, causing the error of yi − yi

' (small number minus another small number) to increase 

and, therefore, affect the accuracy of φi. Increasing the strain-sensing stations could improve the 

accuracies of φi. More study of this accuracy problem is currently under way.

Note from figure 29 that the vertical distances between Ti

P and TiP+T curves will graphically give the 

T-induced component torque, Ti

T1,700 in-lb.

From tables 17, 18, and 19, one can verify that the summations of the component torques, Ti

P, TiT , 

indeed, equal to the total operational torques, Ti

P+T (that is, TiP+T = TiP + TiT).

Generic Long-Span Wing

The fifth structure considered was a generic long-span wing, which is a doubly- (horizontally and 

vertically) tapered long cantilever beam-like structure (fig. 30). The dimensions of the generic long-span 

wing with length, l, chord-wise separation distances of strain-sensing stations at wing root and wing tip, 

w0, wn (fig. 31), and wing root and wing tip maximum depths, h0 max, hn max , are listed in table 20.

Table 20. Dimensions of generic long-span wing.

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

360.0 64.8 28.8 12.96 5.76
31

Figure 30 shows the SPAR finite-element model generated for the double-tapered generic long-span 

wing. This SPAR model has 3,636 nodes and 6,917 four-node bending elements. For each skin (lower or 

upper), there are 101 nodes in the span-wise direction, and 18 nodes in the chord-wise direction. The 

model is subjected to a typical combined loading condition of bending load, P = 200 lb, and clockwise 

torque, T = 1,000×wn = 1,000×28.8 = 28,800 in-lb, at the wing tip (fig. 30). 

As shown in figure 31, the two-line strain-sensing system is on the upper convex surface. The 

dimensions of the depth factors, ci, ci

' , respectively along the front and rear strain-sensing lines are 

assumed to be linearly decreasing from the wing root toward the wing tip. Each strain-sensing line has 

n = 8 evenly spaced strain-sensing intervals. The dimensions of the wing root and wing tip depth factors 

c0,c0

' , cn,cn

' are listed in table 21. 

Table 21. Depth factors associated with front and rear strain-sensing stations at wing root and wing tip 

for generic long-span wing.

Wing root Wing tip

c0, in c0

' , in cn, in cn' , in

(front) (rear) (front) (rear)

6.4051 7.3899 2.8467 3.2844

(= 0.5380 h0) (= 0.5912 h0

'

) (= 0.5380 hn) (= 0.5912 hn

'

)

In table 21, h0, h0

' and hn, hn' are the wing root depth and wing tip depths associated respectively 

with the front and rear strain-sensing stations (fig. 31). 

1. Determination of Bending Stiffness, EI i

In step 1 of determining the bending stiffness, EI i, a point load, P = 200 lb, was applied at the 

generic wing tip (between front and rear strain-sensing lines, fig. 30). For the present loading case, the 

applied moments, Mi, will be Mi = M0 1 − i/n = 72,000× 1 − i/n in-lb [eq. (3)]. Because the strainsensing lines are located on the upper surface, the positive moment, Mi, will induce negative surface 

strains. In order to use equations (1) – (4) in the subsequent operational load estimations, negative signs 

of the surface strains were neglected. Figure 32 shows the plots of the induced surface strains, εi, εi

' , and 

deflections, yi, yi

' , along the two strain-sensing lines. The magnitudes of the front and rear strains, 

εi, εi

' , are slightly different even under bending loading because of different depth factors for the front 

and rear strain-sensing lines (that is, ci ≠ ci

'

, table 21). However, the deflections, yi, yi

' , respectively 

along the front and rear strain-sensing lines are practically coincidental. Two methods were used to 

determine the values of EI i for the generic long-span wing as described in the following sections. 

1.1. Method 1–Averaged Geometry and Strains

Method 1 is comprised of using the averaged depth factor, ci ≡ ci + ci

' /2, and the averaged strains, 

εi ≡ εi + εi

' /2 (absolute values), obtained from figure 32, and the known cross-sectional moment, 

Mi = 72,000× 1 − i/n in-lb. The values of the bending stiffness, EI i, were then calculated from 

equation (2) by replacing εi, ci with εi, ci . The results are listed in table 22.
32

Table 22. Bending stiffness, EI i, calculated from eq. (2) using averaged depth factors, ci, and averaged 

strains, εi, for generic long-span wing subjected to bending load, P = 200 lb, at wing tip; n = 8, l = 360 in, 

∆l = l/n = 45 in, M0 = Pl = 72,000 in-lb.

i ci,

in

ci

'

,

in

ci,

in

εi, ×10!3

in/in

εi

'

, ×10!3

in/in

εi,×10!3

in/in

Mi/M0 EI i = Mi ci/εi ,

×107, lb-in2

(averaged) (SPAR) (SPAR) (averaged) (known) [eq. (2)]

0 6.4051 7.3899 6.8975 0.551429 0.643810 0.597620 8/8 83.0997

1 5.9693 6.8767 6.4230 0.564762 0.639048 0.601905 7/8 67.2281

2 5.5155 6.3635 5.9395 0.566190 0.641429 0.603810 6/8 53.1182

3 5.0707 5.8503 5.4605 0.558095 0.632381 0.595238 5/8 41.2814

4 4.6259 5.3371 4.9815 0.536190 0.607143 0.571667 4/8 31.3704

5 4.1811 4.8240 4.5026 0.492381 0.557143 0.524762 3/8 23.1667

6 3.7363 4.3108 4.0236 0.410000 0.464762 0.437381 2/8 16.5587

7 3.2915 3.7976 3.5446 0.263810 0.299048 0.281429 1/8 11.3355

8 2.8467 3.2844 3.0656 0.000000 0.000000 0.000000 0/8 7.4971*

 *Three-points extrapolation [eq. (10)].

1.2. Method 2–Averaged Bending Stiffness

Method 2 is to determine the individual bending stiffness, { EI i, EI i

'

} from equation (2) using 

[ εi, ci , εi

'

, ci

' ] respectively for the front and rear strain-sensing lines, and the known cross-sectional 

moments, Mi = 72,000× 1 − i/n . The calculated individual bending stiffness, { EI i, EI i

'

} and the 

averaged bending stiffness EI i ≡ [ EI i + EI i

'

]/2 are listed in table 23.

Table 23. Averaged bending stiffness, EI i, for generic long-span wing subjected to bending load, 

P = 200 lb, at the wing tip; l = 360 in, ∆l = l/n = 45 in, M0 = Pl = 72,000 in-lb. 

i ci, 

in

ci

'

, 

in

 εi, ×10!3

 in/in

εi

'

, ×10!3

in/in

Mi/M0 EI i, ×107

lb-in2

EI i

'

, ×107

lb-in2

EI i, ×107

lb-in2

 (SPAR) (SPAR) (known) |------calculated from eq. (2)------| (averaged)

0 6.4051 7.3899 0.551429 0.643810 8/8 83.6313 82.6444 83.1379

1 5.9693 6.8767 0.564762 0.639048 7/8 66.5884 67.7934 67.1909

2 5.5155 6.3635 0.566190 0.641429 6/8 52.6037 53.5724 53.0881

3 5.0707 5.8503 0.558095 0.632381 5/8 40.8858 41.6305 41.2582

4 4.6259 5.3371 0.536190 0.607143 4/8 31.0585 31.6459 31.3522

5 4.1811 4.8240 0.492381 0.557143 3/8 22.9273 23.3778 23.1526

6 3.7363 4.3108 0.410000 0.464762 2/8 16.4033 16.7234 16.5634

7 3.2915 3.7976 0.263810 0.299048 1/8 11.2291 11.4291 11.3291

8 2.8467 3.2844 0.000000 0.000000 0/8 7.4047* 7.4949* 7.4498*

*Three-points extrapolation [eq. (10)].

The values of strain-determined bending stiffness, EI i, listed in table 22 and the averaged bending 

stiffness, EI i, listed in table 23 are plotted in figure 33. Note that the bending stiffness { EI i, EI i}

decreased nonlinearly toward the wingtip because of the doubly-tapered structure. The two bending 

stiffness curves are graphically coincidental. Therefore, in the following operational load predictions, 

only the values of EI i determined by method 1 (table 22) were used.

2. Estimations of Operational Moments and Shear Loads, Mi, Pi

In the following sections, the strain-determined bending stiffness, (EI)i, will be used to estimate 

operational loads, Mi, Pi , associated with other types of loading conditions. The loading conditions

considered were concentrated loads and distributed loads.
33

2.1. Concentrated Loads

The strain-determined EI i of table 22 were used in the estimations of the operational moments and 

shear loads, Mi, Pi , for the loading case of combined bending load, P = 200 lb, and clockwise torque, 

T = 1000×28.8 = 28,800 in-lb, at the wingtip. For the present loading case, the applied moments, Mi, will 

be Mi = 72,000× 1 − i/n in-lb. 

Figure 34 shows the plots of the magnitudes of the induced surface strains, εi, εi

' , and deflections, 

yi, yi

' , for the current combined loading case. 

Using the known depth factors, ci, ci

' , the surface strains, εi, εi' , of figure 34, and the straindetermined EI i in table 22, the estimated operational bending and shear loads, Mi, Pi , were calculated 

respectively from equations (1) and (4). The results are listed in table 24.

Table 24. Estimated operational moments and shear loads Mi, Pi calculated from eq. (4) based on the 

strain-determined EI i for generic long-span wing subjected to combined bending load, P = 200 lb, and 

torque, T = 1,000×wn = 28,800 in-lb, at wing tip; n = 8, l = 360 in, ∆l = l/n = 45 in, applied Mi = 72,000×

1 − i/n in-lb.

i ci, 

in

ci

'

, 

in

ci, 

in

εi,× 10!3

in/in

εi

'

,×x10!3

in/in

εi, ×10!3

in/in

EI i,×107

lb-in2 Mi = EI i

εi

ci

,

×104 in-lb

Pi, 

lb

(SPAR) (SPAR) (averaged) (from 

table 22)

(predicted)

(error, %)

 (applied) (predicted)

(% error)

(applied)

0 6.40517.3899 6.8795 0.592000 0.623810 0.607905 83.0997 7.3239

(1.7200)

(7.2000) 214.67

(7.3350)

(200.00)

1 5.96936.8767 6.4230 0.582071 0.632724 0.607398 67.2281 6.3575

(0.9127)

(6.3000) 214.67

(7.3350)

(200.00)

2 5.51556.3635 5.9395 0.573333 0.646667 0.610000 53.1182 5.4554

(1.0259)

(5.4000) 200.47

(0.2350)

(200.00)

3 5.07075.8503 5.4605 0.568571 0.637143 0.602857 41.2814 4.5576

(1.2800)

(4.5000) 199.52

(0.2400)

(200.00)

4 4.62595.3371 4.9815 0.551429 0.612857 0.582143 31.3704 3.6660

(1.8333)

(3.6000) 198.13

(0.9350)

(200.00)

5 4.18114.8240 4.5026 0.507619 0.563810 0.535715 23.1667 2.7564

(2.0889)

(2.7000) 202.13

(1.0650)

(200.00)

6 3.73634.3108 4.0236 0.429524 0.473333 0.451429 16.5587 1.8578

(3.2111)

(1.8000) 199.69

(0.1550)

(200.00)

7 3.29153.7976 3.5446 0.300000 0.300952 0.300476 11.3355 0.9609

(6.7667)

(0.9000) 199.31

(0.3460)

(200.00)

8 2.84673.2844 3.0656 0.000000 0.000000 0.000000 7.4971* 0.0000

(0.0000)

(0.0000) 213.53

(6.7650)

(200.00)

 * Three-points extrapolation [eq. (10)].

Note from table 24 that the estimation errors of Mi are in the negligible ranges of (0 ~ 1.7200) percent

except for near tip region (6.7667 percent error). The estimation errors of Pi are also in a small range of

(0.2400 ~ 2.1550) percent excluding the wing root (7.3350 percent error) and wing tip (6.7650 percent

error).

In figure 35, the values of the estimated operational moments and shear loads, Mi, Pi , listed in table 

24 are plotted for graphical visualizations. The estimated operational moments, Mi (dashed curve with 

open circular symbols), compared quite well with the applied moments, Mi (solid line with solid circular 

symbols). The estimated operational shear loads, Pi (dashed curve with open diamond symbols),
34

compared nicely with the applied load, P = 200 lb (horizontal solid line with solid diamond symbols),

except for the free end and the fixed end regions where the !! data points are slightly off.

2.2. Distributed Loads

The values of strain-determined EI i from table 22 were also used to estimate the operational 

moments and shear loads, Mi, Pi , for the case of a distributed load of 1 lb at each node of the wing 

lower skin. The distributed load-induced applied moment and shear load, Mi, Pi , at wing cross-section, 

i, can be written as equations (19) and (20):

    Moment arm Moment arm

Force

1 1,818 1 180 1

2 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ × × − ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ = × − ⎠⎞ ⎜

⎝

⎛ = × − n

i

n

i

n

l i M F

Fi

i i (in-lb) (19)

(i = 0,1,2,3,....,n)

⎟

⎠

⎞ ⎜

⎝

⎛ × = × − ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ = × − n

i

n

i Pi 1,818 1 1.0lb 1,818 1

force

Nodal

Number of nodes





(lb) (20)

(i = 0,1,2,3,....,n)

In figure 36, the induced surface strains, εi, εi

' , and deflections, yi

, yi

' , for the distributed loading 

case are plotted. Similar to the bending under concentrated load (fig. 32), the rear strains, εi

'

, are slightly 

larger than the associated front strains, εi, because of unequal depth ratios (that is, ci

' > ci). The values of 

the operational moments and shear loads, Mi, Pi , calculated based on the strains, εi, εi

' , from figure 36, 

are listed in table 25. For the distributed load case, the operational moments and shear loads, Mi, Pi , are 

located at the midpoint of each domain.
35

Table 25. Estimated moments and shear loads, Mi, Pi , calculated from eq. (4) based on 

strain determined EI i for generic long-span wing subjected to distributed load of P = 1 lb at each lower 

surface node; n = 8, l = 360 in, ∆l = l/n = 45 in.

i ci, 

in

εi,×10!3

in/in

εi

'

,×10!3

in/in

εi,×10!3

in/in

EI i,×107

lb-in2

Mi = EI i εi/ci ,×104

in-lb

Pi, 

lb

(SPAR) (SPAR) (averaged) (from 

table 22)

(estimated) (applied) (error, %) (estimated) (applied) (error, %)

0 6.8975 2.447619 2.910476 2.679084 83.0997 32.2771 (32.7240) (1.3657) ---------

1,614.78

(1,818.00)

(1,704.38)

(--------)

(5.2570)

1 6.4230 2.240952 2.538095 2.389524 67.2281 25.0106 (25.0643) (0.2142)

1,467.22 (1,477.13) (0.6709)

2 5.9395 1.928571 2.188095 2.058333 53.1182 18.4081 (18.4073) (0.0043)

1,243.09 (1,249.88) (0.5433)

3 5.4605 1.588095 1.801905 1.695000 41.2814 12.8142 (12.7828) (0.2456)

1,018.02 (1,022.63) (0.4508)

4 4.9815 1.224762 1.390000 1.307381 31.3704 8.2331 (8.1810) (0.6368)

795.38 (796.38) (0.1256)

5 4.5026 0.847143 0.961905 0.904524 23.1667 4.6539 (4.6018) (1.1322)

569.09 (568.13) (0.1690)

6 4.0236 0.476191 0.540952 0.508572 16.5587 2.0930 (2.0453) (2.3322)

345.31 (340.88) (1.2996)

7 3.5446 0.158095 0.179048 0.168572 11.3355 0.5391 (0.5113) (5.4371)

119.80 (113.63) (5.4299)

8 3.0656 0.000952 0.001905 0.001429 7.4971* 0.0003 (0.0000) (--------) -------- (0.00) (--------)

 * Three-points extrapolation [eq. (10)].

In table 25, the values of the applied moments and applied shear loads, Mi, Pi , were calculated 

respectively from equations (19) and (20). Note from table 25 that the estimation errors of operational Mi

P

are in the ranges of (0.0043 ~ 5.4371) percent, and the estimation errors of operational Pi

P are in the small 

range of (0.6709 ~ 1.2996) percent excluding the wing root region (5.2570 percent error) and the wing tip 

region (5.4299 percent error).

Figure 37 shows the plots of the estimated and applied operational moments and shear loads, Mi, Pi , 

for graphical comparisons. Note from figure 37 that the estimated Mi curve (dashed curve with open 

circular symbols) compares nicely with the applied Mi curve (solid curve with solid circular symbol). The 

estimated Pi curve (dashed curve with open diamond symbols) also compares nicely with the applied Pi

curve (straight line with solid diamond symbols) except for the wing root region where the estimated Pi

data point are slightly lower than the corresponding applied Pi value. 

3. Determinations of Torsion Stiffness GK i

To determine the torsional stiffness GK i of the generic long span wing, a clockwise torque, 

T = 1,000×wn = 1,000×28.8 = 28,800 in-lb, was applied at the wing tip. 

Figure 38 shows the plots of the deflections, yi, yi

' , and cross-sectional twist angle, φi

, induced by 

the current torsion loading. For the current torsion loading case, the operational torque, Ti, is equal to the 

applied torque (that is, Ti = T). 

Using the known torque, Ti = T , and the cross-sectional twist angles, φi (fig. 38), the values of 

GK i were calculated from equation (5) and are listed in table 26.
36

Table 26. Domain torsional stiffness, GK i, of generic long-span wing determined from loading 

condition of clockwise torque, T = 1,000×wn = 28,800 in-lb, at wing tip; n = 8, l = 360 in, ∆l = l/n = 45 in.

i φi

,×10!3

rad

φi − φi!1 ,×10!3

rad

GK i,×107

lb-in2

Ti, 

in-lb

[eq. (6)] [eq. (5)] [calculated, eq. (9) 

= applied]

0 0.000000 ----------- 47.915314 28,800

1 2.704772 2.704772 47.915314 28,800

2 6.572264 3.867492 33.510089 28,800

3 11.209831 4.637567 27.945688 28,800

4 17.206154 5.996323 21.613245 28,800

5 25.224999 8.018845 16.161929 28,800

6 36.266667 11.041668 11.737357 28,800

7 51.924330 15.657663 8.277097 28,800

8 74.440002 22.515672 5.755991 28,800

In the last column of table 26, the values of Ti = 28,800 in-lb were calculated from equation (9) using 

the strain-determined GK i, and are exactly the same as the applied torques, T = 28,800 in-lb, confirming 

the accuracies of the strain-determined GK i.

Figure 39 shows the plot of the strain-determined GK i data listed in table 26. The GK i curve for 

the generic long-span wing is a concave downward because of the doubly-tapered structure.

4. Estimations of Operational Torque, Ti

The pre-determined GK i listed in table 26 were used to estimate the operational torques, Ti, for the 

combined loading case, for which the generic long-span wing was subjected to bending load, P = 200 lb, 

and clockwise torque, T = 1,000×wn = 28,800 in-lb, at the wing tip. 

Figure 40 shows the plots of the displacements, yi, yi

' , and the cross-sectional twist angles, φi

, 

induced by the current combined loading.

Using the strain-determined GK i listed in table 26, and the values of φi shown in figure 40, the 

operational torque, Ti, were calculated from equation (9). The results are listed in table 27.
37

Table 27. Estimated operational torques Ti based on strain-determined GK i for generic long-span wing 

subjected to combined bending load, P = 200 lb, and torque, T = 1,000×wn = 28,800 in-lb, at wing tip; 

n = 8, l = 360 in, ∆l = l/n = 45 in.

i ci, 

in

φi, ×10!3

rad

φi − φi!1 , ×10!3

rad

GK i, ×107

lb-in2

Ti, 

in-lb

[eq. (6)] (from table 26) (estimated) (applied) (% error)

0 4.000 0.000000 ------------ 47.915314 28,161.27 (28,800) (2.2178)

1 3.625 2.644785 2.644785 47.915314 28,161.27 (28,800) (2.2178)

2 3.250 6.550378 3.905593 33.510089 29,083.72 (28,800) (0.9851)

3 2.875 11.239536 4.689158 27.945688 29,120.39 (28,800) (1.1125)

4 2.500 17.291623 6.052087 21.613245 29,067.83 (28,800) (0.9300)

5 2.125 25.374644 8.083021 16.161929 29,030.49 (28,800) (0.8003)

6 1.750 36.493822 11.119178 11.737357 29,002.17 (28,800) (0.7020)

7 1.375 52.249712 15.755890 8.277097 28,980.67 (28,800) (0.6273)

8 1.000 74.821043 22.571331 5.755991 28,871.19 (28,800) (0.2472)

Note from table 27 that the estimated operational torques, Ti, have negligible errors in the range of 

(0.2472 ~ 2.2178) percent. 

In figure 41, the estimated and applied operational torque, Ti, listed in table 27 are plotted for 

graphical comparisons. Note the estimated data points of Ti (dashed curve with open circular symbols) 

practically fell on top of the applied torque line, T = 28,800 in-lb (solid horizontal line with solid circular 

symbols). The proximity between the estimated and applied torques validated the accuracies of the new 

method for operational load estimations using distributed surface strains. 

CONCLUDING REMARKS

A new method for estimating the operational loads (moments, shear loads, and torques) using 

distributed surface strains (unidirectional) was presented. The embodied stiffness and load transfer 

functions were developed for converting surface strains into structural stiffness (bending, torsion) and 

operational loads. The operational loads estimations require pre-determinations of structural stiffness (true 

or apparent). Several slender structures (under different loading conditions) were chosen for finiteelement accuracy studies of the new method. The key results are listed below:

1. For the slender structures, the bending stiffness determined using distributed surface strains (step 

one of the new method) agreed quite nicely with the values calculated from the theory of elasticity. 

2. The strain-determined stiffness is merely a stepping stone needed for the calculations of the 

operational loads (moments, shear loads, and torques), and it can be a true stiffness, or an apparent 

stiffness associated with a particular strain-sensing station configuration. 

3. The same strain-sensing configuration must be used to determine both the structural stiffness (step 

one) and the operational loads (step 2). 

4. By using the two-line strain-sensing system with embodied Displacement Transfer Functions, it is 

possible to determine torsion stiffness and operational torques, and thus eliminate the need for 

distortion strain gages to sense torsions.

5. For a slender structure with known neutral surface location, the two-line strain-sensing system can 

be used to estimate all components of the operational loads (bending moments, shear loads, and 

torques) with sufficient accuracies. 

6. For the tapered tube (using a single strain-sensing line) the strain-determined bending stiffness and 

the estimated operational loads (moments, shear loads) agreed quite nicely with the associated 

theoretically calculated values. 
38

7. For the un-swept wing box, and the generic long-span wing, the estimated operational loads

(bending moments, shear loads, and torques) agreed fairly well with actual applied loads 

(theoretically calculated loads).

8. For the swept wing box, bending and torsion are coupled (that is, bending induces torsion, and 

vice versa), and the coupling effect increases with the swept angle. Bending induced a 

considerable amount of torsion, however, torsion induced only a low level of pure bending, which 

generates no shear loads.

9. Overall, the new method for estimating operational loads (moments, shear loads, and torques) was 

found to be fairly accurate, and is very promising for applications to flight load monitoring of 

aircraft with slender wings. 
39

FIGURES

Figure 1. Embedded beam (cross section of any structure) along the strain-sensing line with evenly 

distributed strain-sensing stations.

Figure 2. Two-line strain-sensing system for sensing multiple surface strains for deformed shape 

calculations of tapered cantilever wing box subjected to bending and torsion.
40

Figure 3. SPAR finite-element model generated for tapered cantilever tubular beam subjected to upward 

vertical load, P = 100 lb, at beam tip.

Figure 4. SPAR-generated surface bending strains, εi, for tapered cantilever tubular beam subjected to 

point load, P = 100 lb, at the beam tip.
41

Figure 5. Span-wise plots of strain-determined bending stiffness, (EI)i, and elasticity-calculated bending 

stiffness, (EI)i, for tapered cantilever tubular beam.

Figure 6. SPAR-generated surface bending strains, εi, for tapered cantilever tubular beam subjected to 

point load, P = 100 lb, at the beam tip and second point load, P = 100 lb, at mid-span.
42

Figure 7. Plots of estimated operational moments and shear load, {Mi , Pi}, for tapered cantilever tubular 

beam subjected to point load, P = 100 lb, at the beam tip and second point load, P = 100 lb, at mid-span.

Figure 8. SPAR finite-element model generated for depth-tapered wing box subjected to typical bending 

load, P = 100 lb, and clockwise torque, T = 50×w = 850 in-lb.
43

Figure 9. SPAR-generated surface bending strains, εi( = εi

'

), and deflections, yi( = yi

'

), for depth-tapered 

un-swept wing box subjected to point load, P − 100 lb, at wing box tip for determination of bending 

stiffness, (EI)i.

Figure 10. Span-wise plots of strain-determined bending stiffness, (EI)i, for depth-tapered un-swept wing 

box.
44

Figure 11. SPAR-generated surface bending strain, {εi, εi

'

}and deflections, {yi, yi

'

}, for depth-tapered 

un-swept wing box subjected to point load, P = 200 lb, and clockwise torque, T = 100 × w = 1,700 in-lb,

at un-swept wing box tip.

Figure 12. Plots of estimated operational moments and shear loads, {Mi , Pi}, for depth- tapered un-swept 

wing box subjected to point load, P = 200 lb, and clockwise torque, T = 100 × w = 1,700 in-lb, at wing 

box tip.
45

Figure 14. Span-wise plot of strain-determined torsional stiffness, (GK)i, for depth-tapered un-swept wing 

box.

Figure 14. Span-wise plot of strain-determined torsional stiffness, (GK)i , for depth-tapered un-swept 

wing box.
46

Figure 15. SPAR-generated deflections, {yi, yi

'

}, and cross-sectional twist angle, φi, for depth-tapered 

un-swept wing box subjected to point load, P = 200 lb, and clockwise torque, T = 100 × w = 1,700 in-lb, 

at wing box tip.

Figure 16. Plots of estimated operational torques, Ti, for depth-tapered un-swept wing box subjected to 

point load, P = 200 lb, and clockwise torque, T = 100 × w = 1,700 in-lb, at wing box tip.
47

Figure 17. SPAR finite-element model generated for depth-tapered swept wing box subjected to a typical 

combined bending load, P = 100 lb, and clockwise torque, T = 50 × w = 850 in-lb at wing box tip.

Figure 18. Counterclockwise rotation (decreasing angle of attack) of depth-tapered swept wing box 

induced by bending load, P = 100 lb at wing box tip.
48

Figure 19. SPAR-generated surface bending strains, {εi, εi

'

}, deflections, {yi, yi

'

}, and bending induced 

cross-sectional twist angle, φn(< 0), for depth tapered swept wing box subjected to bending load, 

P = 100 lb, at wing box tip for determination of bending stiffness, (EI)i.

Figure 20. Span-wise plots of strain-determined apparent bending stiffness, (EI)i , for depth-tapered unswept wing box.
49

Figure 21. SPAR-generated surface bending strains, {εi, εi

'

}, deflections, {yi, yi

'

}, and bending induced 

cross-sectional twist angles, φi

P

(< 0), for depth-tapered swept wing box subjected to point load, 

P = 200 lb, at wing box tip.

Figure 22. Plots of estimated operational moments and shear load, {Mi , Pi}, for depth-tapered swept wing 

box subjected to point load, P = 200 lb, at wing box tip.
50

Figure 23. SPAR-generated surface bending strains {εi, εi

'

, εi

P+T} deflections, {yi

, yi

'

}, and cross-sectional 

twist angles, φi

P+T, for depth-tapered swept wing box subjected to point load, P = 200 lb, and 

counterclockwise torque, ! = −100×! = −1,700 in-lb, at wing box tip εi

T; is induced by T = −

100 × w = − 1,700 in-lb only.

Figure 24. Plots of estimated operational moments and shear loads, {Mi, Pi}, for depth-tapered swept 

wing box subjected to point load, P = 200 lb, and counterclockwise torque, 

! = −100×! = −1,700 in-lb, at wing box tip.
51

Figure 25. SPAR-generated deflections, {yi, yi

'

}, and cross-sectional twist angle, φi

T

, for depth-tapered 

swept wing box subjected to clockwise torque, T = 50×w = 850 in-lb, at wing box tip for determination of 

torsion stiffness, (GK)i.

Figure 26. Span-wise plots of strain-determined torsional stiffness, (GK)i, for depth-tapered swept wing 

box.
52

Figure 27. SPAR-generated cross-sectional twist angles, φi

P!T

, for depth-tapered swept wing box 

subjected to point load, P = 200 lb, and counterclockwise torque, T = − 100×w = − 1,700 in-lb, at wing 

box tip; φi

P is induced by P = 200 in-lb only.

Figure 28. Plots of estimated operational torques, Ti , for depth-tapered swept wing box subjected to point 

load, P = 200 lb, and clockwise torque, T = − 100×w = − 1,700 in-lb, at wing box tip.
53

Figure 29. Plots of estimated operational torques, Ti

P!T, for depth-tapered swept wing box subjected to 

point load, P = 200 lb, and clockwise torque, T = − 100×w = − 1,700 in-lb, at wing box tip; operational 

torque components, Ti

P, induced by P = 200 lb only.

Figure 30. SPAR finite-element model generated for doubly-tapered generic long-span wing subjected to 

combined bending and torsion; P = 200 lb, T=2,000×wn in-lb.
54

Figure 31. Two-line strain-sensing system on upper surface of doubly-tapered generic long-span wing for 

sensing surface bending strains.
55

Figure 32. SPAR-generated surface bending strains, {εi, εi

'

}, deflections, {yi, yi

'

}, for doubly-tapered 

generic long-span wing subjected to point load, P = 200 lb, at wing tip for determination of bending 

stiffness, (EI)i.

Figure 33. Span-wise plots of strain-determined bending stiffness, {(EI)i, (EI)i} for doubly-tapered 

generic long-span wing.
56

Figure 34. SPAR-generated surface bending strains, {εi, εi

'

}, and deflections, {yi, yi

'

}, for doubly-tapered 

generic long-span wing subjected to point load, P = 200 lb, and clockwise torque, 

T = 1,000×wn = 28,800 in-lb, at wing tip.

Figure 35. Plots of estimated operational moments and shear loads, {Mi, Pi}, for doubly-tapered generic 

long-span wing subjected to point load, P = 200 lb, and clockwise torque, T = 1,000×wn = 28,800 in-lb, at 

wing tip.
57

Figure 36. SPAR-generated surface bending strains, {εi, εi

'

}, and deflections, {yi, yi

'

}, for doubly-tapered 

generic long-span wing subjected to distributed load of P = 1 lb at each lower surface node of SPAR 

model.

Figure 37. Comparison of applied and estimated operational moments and shear loads, {Mi, Pi}, for 

doubly-tapered generic long-span wing subjected to distributed load of P = 1 lb at each lower surface 

node of SPAR model.
58

Figure 38. SPAR-generated deflections, {yi, yi

'

}, and cross-sectional twist angles, φi, for doubly-tapered 

generic long-span wing subjected to clockwise torque, T = 1,000×wn = 28,800 in-lb, at wing tip for 

determination of torsion stiffness, (GK)i.

Figure 39. Span-wise plot of strain-determined torsion stiffness, (GK)i, for doubly-tapered generic 

long-span wing.
59

Figure 40. SPAR-generated deflections, {yi, yi

'

}, and cross-sectional twist angles, φi, for doubly-tapered 

generic long-span wing subjected to point load, P = 200 lb, and clockwise torque, 

T = 1,000×wn = 28,800 in-lb, at wing tip.

Figure 41. Comparison of applied and estimated operational torques, Ti, for doubly-tapered generic longspan wing subjected to point load, P = 200 lb, and clockwise torque, T = 1,000×wn = 28,800 in-lb, at 

wing tip.
60

APPENDIX A

SUMMARY OF DIFFERENT DISPLACEMENT TRANSFER FUNCTIONS

Different Displacement Transfer Functions developed earlier (refs. 1 and 2) for nonuniform and 

slightly nonuniform and uniform straight cantilever beams, are summarized in the following for 

comparison of mathematical functional forms: 

1. Nonuniform Displacement Transfer Functions (ci ≠ ci!1) (ref. 2):







=0 for cantilever beams

0 0

Contributionsfromslope terms

1

1 1 2

1

1 1

1

2 1

Contributionsfromdeflection terms

1

1

1 3

1

1 1

1

2 1

log ( ) tan

( ) ( ) ( )

log ( )

( ) 2( ) ( )

θ

ε ε ε ε

ε ε ε ε

y i l

c

c

c c

c c

c c

l i j

c c

c

c

c

c c

c c

c c

y l

i

j j

j

e

j j

j j j j

j j

j j

i

j

j j

j

j

j e

j j

j j j j

j j

j j

i

+ + Δ

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

−

− + −− + Δ −

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ − −

− − −− = Δ

∑

∑

−

= − −

− −

−

−

=

−

− −

− −

−

−

(A1)

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

Equation (A1) is not applicable to uniform beams because of a mathematical breakdown problem 

(that is, 0/0 at ci = ci!1 = c).

2. Slightly Tapered Displacement Transfer Functions (ci/ci!1 → 1) (ref. 2):

yi = (!l)

2

6

1

cj"1

3" cj

cj"1

#

$

%

%

&

'

(

(! j"1 +! j

)

*

+

+

,

-

.

.

/

0

1

21

3

4

1

j=1 51

i

6

Contributions from deflection terms

!######"######$

+ (!l)

2

2

(i " j)

cj"1

2 " cj

cj"1

#

$

%

%

&

'

(

(! j"1 +! j

)

*

+

+

,

-

.

.

/

0

1

21

3

4

1

j=1 51

i"1

6

Contributions from slope terms

!######"######$

+ y0 + !l tan!0

=0 for cantilever beams

!##"##$

 (A2)

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

which was obtained from equation (A1) by expanding the logarithmic terms in the neighborhood of

ci/ci!1 ≈ 1.

3. First Order Displacement Transfer Function (ref. 7):

yi = (!l)

2

12

(!l)

2

cj"1

5" cj

cj"1

#

$

%

%

&

'

(

(! j"1 + 3" cj

cj"1

#

$

%

%

&

'

(

(! j

)

*

+

+

,

-

.

. j=1

i

/

Contributions from deflection terms

!########"########$

+ (!l)

2

6

(i " j)

ci"1

4 " cj

cj"1

#

$

%

%

&

'

(

(! j"1 + 5" 2

cj

cj"1

#

$

%

%

&

'

(

(! j

)

*

+

+

,

-

.

. j=1

i"1

/

Contributions from slope terms

!########"########$

+ y0 + !l tan!0

=0 for cantitever beams

!##"##$

(A3)

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

4. Second Order Displacement Transfer Function (ref. 7):
61

yi = (!l)

2

60

1

cj"1

27" 9

cj

cj"1

+ 2

cj

cj"1

#

$

%

%

&

'

(

(

) 2

*

+

+

,

-

.

.

! j"1 + 18"11

cj

cj"1

+3

cj

cj"1

#

$

%

%

&

'

(

(

) 2

*

+

+

,

-

.

.

! j

/

0

1

2

1

3

4

1

5 j=1 1

i

6

Contributions from deflection terms

!#############"#############$

+ (!l)

2

12

(i " j)

cj"1

9 " 4

cj

cj"1

+

cj

cj"1

#

$

%

%

&

'

(

(

) 2

*

+

+

,

-

.

.

! j"1 + 13"10

cj

cj"1

+3

cj

cj"1

#

$

%

%

&

'

(

(

) 2

*

+

+

,

-

.

.

! j

/

0

1

2

1

3

4

1

5 j=1 1

i"1

6

Contributions from slope terms

!#############"#############$

+ y0 +(i)!l tan!0

=0 for cantilever beams

!##"##$

(A4)

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

5. Uniform Displacement Transfer Function (ci = ci!1 = c) (refs. 1, 2):

yi = (!l)

2

6c

(2! j"1 +! j)

j=1

i

#

Contributions from deflection terms

!###"###$

+ (!l)

2

2c

(i " j)(! j"1 +! j)

j=1

i"1

#

Contributions from slope terms

!####"####$

+ y0 + !l tan!0

=0 for cantilever beams

!##"##$ (A5)

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

Equation (A5) is the degenerated form of equations (A2), (A3), and (A4) for the uniform beam case 

(ci = ci!1 = c). 
62

APPENDIX B

FLOW CHART FOR STRUCTURAL STIFFNESS AND OPERATIONAL 

LOADS ESTIMATIONS USING DISTRIBUTED SURFACE STRAINS

(cf., U.S. Patent No. 7,715,994, issued May 11, 2010)

Flow chart for the determinations of bending and torsion stiffness, EI i, GK i , for estimations of 

operational moments, shear loads, and torques, Mi, Pi, Ti , using two-line strain-sensing system (fig. B1) 

is presented below. 

Step 1–Determinations of EI i, GK i Using Strains Induced by Simple (Bending, Torsion) Tests: 

Simple bending test Averaged surface strains and depth factors Determination of (EI)i

Mi (specified) ! !i = !i +!!

i

2

"

#

$ ; ci = ci + c!

i

2

"

#

$

 ! i

i

i i

c EI M

ε

( ) =

Simple torsion test Surface strains Deflections Cross-sectional twist angles Determination of 

GK i ( )

Ti (specified) ! !{ i ,!!

i} ! y{ i , y!i} * ! !i = sin!1 yi ! y"

i

di

#

$

% &

'

( ! (GK)i = Ti(!l)

!i "!i"1

 

Step 2--Estimations of {Mi , Pi ,Ti

} for Other Types of Loading Based on Strain-Determined 

{(EI)i , (GK)i} : 

 Estimations of Mi Estimations of Pi

For 

Mi

Pi

!

"

#

$

%

&

:

ci,

!i,

!

"

#

c!

i

!!

i

"

#

$

! ci = ci + c!

i

2 ; !i = !i +!!

i

2

"

#

$

%

&

' ! Mi = (EI)i

!i

ci

!

Pi = 1

!l (EI)i"1

!i"1

ci"1

"(EI)i

!i

ci

#

$

% &

'

(

Calculations of !i Estimations of Ti

For Ti : !{ i ,!!

i} ! y{ i , y!i} * ! !i = sin!1 yi ! y"

i

di

#

$

% &

'

( ! Ti = (GK)i

!i !!i!1

"l

* Deflections, yi, yi

' can be calculated from any of the Displacement Transfer Functions (Appendix A) 

using surface strains, εi, εi

' , as inputs. For example, when the Nonuniform Displacement Transfer 

Functions are used, yi, yi

' have the following forms:
63

For front strain-sensing line:







=0 for cantilever beams

0 0

Contributionsfromslope terms

1

1 1 2

1

1 1

1

2 1

Contributionsfromdeflection terms

1

1

1 3

1

1 1

1

2 1

log ( ) tan

( ) ( ) ( )

log ( )

( ) 2( ) ( )

θ

ε ε ε ε

ε ε ε ε

y i l

c

c

c c

c c

c c

l i j

c c

c

c

c

c c

c c

c c

y l

i

j j

j

e

j j

j j j j

j j

j j

i

j

j j

j

j

j e

j j

j j j j

j j

j j

i

+ + Δ

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

−

− + −− + Δ −

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ − −

− − −− = Δ

∑

∑

−

= − −

− −

−

−

=

−

− −

− −

−

−

(B1)

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

For rear strain-sensing line:







=0 for cantilever beams

0 0

Contribution fromslope terms

1

1 1 2

1

1 1

1

2 1

Contributionsfromdeflection terms

1

1

1 3

1

1 1

1

2 1

log ( ) tan

( ) ( ) ( )

log ( )

( ) 2( ) ( )

θ

ε ε ε ε

ε ε ε ε

+ ′ + Δ ′

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

′

′

′ − ′

′ ′ − ′ ′ + ′ − ′′ − ′ + Δ −

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ ′ − ′ ′

′ ′ ′ − ′′ ′ − ′ ′ − ′ − ′′ − ′ = Δ

∑

∑

−

= − −

− −

−

−

=

−

− −

− −

−

−

y i l

c

c

c c

c c

c c

l i j

c c

c

c

c

c c

c c

c c

y l

i

j j

j

e

j j

j j j j

j j

j j

i

j

j j

j

j

j e

j j

j j j j

j j

j j

i

(B2)

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

Figure B1. Two-line strain-sensing system for sensing multiple surface strains for deformed shape 

calculations of a tapered cantilever wing box subjected to combined bending and torsion (ref. 2).
64

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, October 2007.

2. Ko, William L., and Van Tran Fleischer, Further Development of Ko Displacement Theory for 

Deformed Shape Predictions of Nonuniform Aerospace Structures, NASA/TP-2009-214643, 

September 2009.

3. Ko, William L., Collection of Memoranda on Applications of Ko Displacement Theory to Deformed 

Shape Predictions of Aerospace Structures, NASA BR-119, Aerostructures Branch, Dryden Flight 

Research Center, Edwards, CA, April 2008.

4. Ko, William L., W. L. Richards, and Van Tran Fleischer, Applications of Ko Displacement Theory to 

the Deformed Shape Predictions of Doubly-Tapered Ikhana Wing, NASA/TP-2009-214652, 

October 2009.

5. Ko, William L., and Van T. Fleischer, Methods for In-Flight Wing Shape Predictions of Highly 

Flexible Unmanned Aerial Vehicles: Formulation of Ko Displacement Theory,

NASA/TP-2010-214656, April 2010.

6. Ko, William L., and Van Tran Fleischer, Extension of Ko Straight-Beam Displacement Theory to 

Deformed Shape Predictions of Slender Curved Structures, NASA/TP-2011-214567, April 2011.

7. Ko, William L., and Van Tran Fleischer, First- and Second- Order Displacement Transfer Functions 

for Structural Shape Calculations Using Analytically Predicted Surface Strains, 

NASA/TP-2011-215976, March 2012.

8. Ko, William L., and Van Tran Fleischer, Improved Displacement Transfer Functions for Structure 

Deformed Shape Predictions Using Discretely Distributed Surface Strains, NASA/TP-2012-216060, 

November 2012.

9. Ko, William L., and Van Tran Fleischer, Large-Deformation Displacement Transfer Functions for 

Shape Predictions of Highly Flexible Slender Aerospace Structures, NASA/TP-2013-unnumbered (to 

be published).

10. 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.

11. Richards, W. Lance, and William L. Ko, Process for Using Surface Strain Measurements to Obtain 

Operational Loads for Complex Structures, U.S. Patent No. 7,715,994, issued May 11, 2010.

12. Roark, Raymond J., Formulas for Stress and Strain, McGraw-Hill Book Company, Inc., New York, 

1954.

13. Whetstone, W. D., SPAR Structural Analysis System Reference Manual, System Level 13A,Vol. 1, 

Program Execution, NASA CR-158970-1, December 1978.

14. Jutte, Christine, William L. Ko, Craig A. Stephens, John A. Bakalyar, W. Lance Richards, and Allen R. 

Parker, Deformed Shape Calculation of a Full-Scale Wing Using Fiber Optic Strain Data from a 

Ground Loads Test, NASA/TP-2011-215975, December 2011.