NASA/TP—2013–216550

Large-Deformation Displacement Transfer 

Functions for Shape Predictions of Highly 

Flexible Slender Aerospace Structures

William L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

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December 2013
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NASA/TP—2013–216550

Large-Deformation Displacement Transfer 

Functions for Shape Predictions of Highly 

Flexible Slender Aerospace Structures

William L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

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National Aeronautics and

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Dryden Flight Research Center

Edwards, CA 93523-0273

December 2013
PATENT PROTECTION NOTICE

The method for structural deformed shape predictions using rectilinearly distributed surface strains and 

embodied Ko Displacement Transfer Functions for conversions into out-of-plane deflections described in 

this NASA technical publication is protected under Method for Real-Time Structure Shape-Sensing

(Invented by William L. Ko and William Lance Richards), U.S. Patent No. 7,520,176 issued 

April 21, 2009. Therefore, those interested in using the method should contact The NASA Innovative 

Partnership Program Office, NASA Dryden Flight Research Center, Edwards, California for more 

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v 

TABLE OF CONTENTS

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

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

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

CURVATURE-STRAIN RELATIONSHIP.............................................................................................. 4

DIFFERENT CURVATURE EQUATIONS............................................................................................. 4

Classical (Eulerian) Curvature Equation............................................................................................... 4

Physical (Lagrangian) Curvature Equation........................................................................................... 5

Shifted Curvature Equation................................................................................................................... 6

CURVATURE-STRAIN DIFFERENTIAL EQUATIONS ...................................................................... 7

NONUNIFORM AND UNIFORM BEAMS............................................................................................. 8

SHIFTED DISPLACEMENT TRANSFER FUNCTIONS....................................................................... 8

Based on Piecewise Linear Strain Representations............................................................................... 8

Nonuniform Beams............................................................................................................................ 9

Slightly Nonuniform Beams .............................................................................................................. 9

First Order Expansion.......................................................................................................................10

Second Order Expansion ..................................................................................................................11

Based on Piecewise Nonlinear Strain Representations........................................................................11

Improved Case..................................................................................................................................12

Log-expanded Case ..........................................................................................................................13

Depth-expanded Case .......................................................................................................................13

Characteristics of Displacement Transfer Functions ...........................................................................14

EULERIAN FORMULATION OF DISPLACEMENT TRANSFER FUNCTIONS ..............................15

Projected Domain Lengths...................................................................................................................15

Slope Equation.....................................................................................................................................16

Deflection Equation .............................................................................................................................17

Log-Expanded Slope Equation ............................................................................................................18

Log-Expanded Deflection Equation.....................................................................................................19

Simplified Deflection Equation ...........................................................................................................19

LAGRANGIAN FORMULATION OF DISPLACEMENT TRANSFER FUNCTIONS........................20

Slope Equation.....................................................................................................................................21

Deflection Equation .............................................................................................................................22

Simplified Deflection Equation ...........................................................................................................23

STRUCTURE USED FOR SHAPE PREDICTION ACCURACY STUDIES ........................................23

Finite-Element Analysis.......................................................................................................................23

Surface Strains .....................................................................................................................................24

COMPARISONS OF SHAPE PREDICTION ACCURACIES ...............................................................24

Slope Predictions..................................................................................................................................24
vi 

Slope Curves ........................................................................................................................................27

Deflection Predictions..........................................................................................................................27

Deflection Curves ................................................................................................................................30

SHAPE PREDICTION ERROR CURVES ..............................................................................................30

Slope-Error Curves ..............................................................................................................................30

Deflection-Error Curves.......................................................................................................................31

EXPERIMENTAL VALIDATIONS OF SHAPE-PREDICTION ACCURACIES.................................32

DISCUSSIONS.........................................................................................................................................33

CONCLUDING REMARKS....................................................................................................................34

FIGURES..................................................................................................................................................36

APPENDIX A: ALTERNATIVE DERIVATION OF THE PHYSICAL (LAGRANGIAN)

CURVATURE EQUATION................................................................................................................47

APPENDIX B: GRAPHICAL DERIVATION OF THE SHIFTED CURVATURE EQUATION .........50

APPENDIX C: DERIVATION OF PROJECTED DOMAIN LENGTH FOR 

EULERIAN FORMULATION............................................................................................................52

APPENDIX D: DERIVATIONS OF THE EULERIAN SLOPE EQUATIONS .....................................53

APPENDIX E: DERIVATIONS OF LAGRANGIAN SLOPE EQUATION ..........................................57

REFERENCES .........................................................................................................................................58
1

ABSTRACT

Displacement Transfer Functions were formulated for deformed shape predictions of structures under 

large deformations. By discretization of the embedded beam (depth-wise cross section of structure along the 

surface strain-sensing line), the distributions of surface bending strains could be represented with piecewise 

linear functions. Such a piecewise approach enabled piecewise integrations of the curvature equations 

(Eulerian, Lagrangian, and Shifted curvature equations) of the deformed embedded-beam to yield 

closed-form Displacement Transfer Functions, which are expressed in terms of embedded beam geometrical 

parameters and surface strains. By inputting the surface strain data, the Displacement Transfer Functions 

can then transform the surface strains into deflections for mapping out the overall structural deformed 

shapes for visual display. A long tapered cantilever tubular beam was chosen for studying the 

large-deformation shape predictions. The input surface strains were analytically generated from 

finite-element analysis. Also, the finite-element generated slopes and deflections were used as reference 

yardsticks to study the theoretical shape prediction accuracies. The Displacement Transfer Functions based 

on the Shifted curvature equation were found to be amazingly accurate beyond expectation. However, the 

Displacement Transfer Functions based on the Eulerian and Lagrangian curvature equations, resulted in 

very complex mathematical expressions, and gave very poor shape predictions at large deformations.

NOMENCLATURE

c depth factor (vertical distance from neutral axis to bottom surface of uniform beam), in

c(s) depth factor (vertical distance from deformed neutral axis to bottom surface of nonuniform 

embedded beam), in

c(x) depth factor (vertical distance from undeformed neutral axis to bottom surface of nonuniform 

embedded beam), in

i c depth factor (vertical distance from neutral axis to i-th strain-sensing station on bottom 

surface of nonuniform embedded beam), in

0 c value of i c at fixed end (beam root), x  x0  0 , in

n c value of i c at free end (beam Math equationsp), x x l  n  , in

DLL design limit load

E Young’s modulus, lb/in2

i  0,1,2,3,....,n , strain-sensor identification number, or an imaginary unit, i  1

l length of embedded beam, in

n index for the last span-wise strain-sensing station (or number of domains) 

R radius of curvature, in

R

averaged radius of curvature for domain, i i x  x  x 1 , in 

RRF Rotated reference frame

SPAR Structural Performance And Resizing finite element computer program 

STS Space Transport System for the Space Shuttle flights

s curved coordinate along deformed embedded beam elastic curve, in

u axial displacement, in

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

axial coordinate of -th strain sensor (or strain-sensing station at ), in 

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

y

 curved deflection (curved distance traced by the same material point from its initial undeformed position to its final deformed position, in

i y ( )i  y x , beam deflection at axial location, i x  x , in 
2

V  dy / dx  tan (x) , beam slope, in/in

Vi ( )i  V x , value of V at strain-sensing station, in/in

 (xi  xi1 )  l / n , domain length (distance between two adjacent strain-sensing stations 1 { i x , }i x ,

in

c l chord length of arc formed by bent , in

Li projected domain length on the x-axis of bent and tilted , in 

(s) surface bending strain at curved location , in/in

(x) surface bending strain at axial location , in/in

i 	 , surface bending strain at strain-sensing station, , in/in

s 	 axial strain in s-direction, in/in

 (s) beam slope in reference to s-system, rad or deg

 (x) beam slope in reference to x-system, rad or deg

i ( )i   x , slope at strain-sensing station, i x  x , rad or deg

 n ( ) n   x , slope at beam tip strain-sensing station, x x l  n  , rad or deg


   i1 x x , local axial coordinate for domain, i i x  x  x 1 , in

INTRODUCTION

For structure (for example, aircraft wings) deformed shape predictions using discretely distributed 

surface strains (bending strains), the Displacement Transfer Functions (refs. 1–5) are needed to convert the 

surface strains into out-of-plane deflections so that one can plot the overall structural deformed shape for 

visual display. The surface strains are to be obtained from the strain-sensing stations evenly distributed 

along each strain-sensing line on the surfaces of the structure. The depth-wise cross section of the structure 

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

formulations of earlier Displacement Transfer Functions (refs. 1–5), the embedded beam was first evenly 

divided into multiple small domains, with domain junctures matching the strain-sensing stations. Thus, 

within each small domain, the beam depth factor could be described with a linear function, and the surface

strain variation could be described with either linear or nonlinear function. Such piecewise approach enabled 

the piecewise integrations of the curvature equation [classical (Eulerian), physical (Lagrangian), and Shifted 

curvature equations] for the elastic curve of each deformed embedded beam to yield closed-form slope and 

deflection equations in recursive forms. Those recursive slope and deflection equations were then combined 

into a single deflection equation in dual summation form (called the Displacement Transfer Function), 

which is expressed in terms of embedded beam geometrical parameters and discretely distributed surface 

strains. By inputting the surface strain data into the Displacement Transfer Function, one can then calculate 

deflections along each embedded beam. By using multiple strain-sensing lines, the overall structural 

deformed shape (induced by bending and torsion) can be geometrically mapped out for visual display. 

The Displacement Transfer Functions (in different mathematical forms for different types of structural 

geometry) developed by Ko (refs.1–5), can be used to calculate deflections from strain data obtained from 

the fiber optic strain-sensing method introduced by Richards (ref. 6) to create a revolutionary structureshape-sensing technology called; “Method for Real-Time Structure Shape-Sensing,” U.S. Patent Number 

7,520,176 (Ko-Richards), (ref. 6). This patented technology is quite attractive to the in-flight deformed 

shape monitoring of highly flexible wings of unmanned flight vehicles by the ground-based pilot for 

maintaining safe flights. In addition, the real time wing shape monitored could then be input to the aircraft 

control system for aero-elastic wing shape control. 

The surface strain data for inputs to the Displacement Transfer Functions (ref. 1–5) for shape 

predictions can be obtained from the conventional strain gages, from fiber optic strain sensors, or 
3

analytically calculated from finite-element analysis. It is important to mention that without the use of the 

Displacement Transfer Functions, any type of surface strain sensors can only sense the strains, but not the 

out-of-plane deflections nor the cross-sectional rotations of the structure. 

The distributed fiber optic surface strains can also be input to the Displacement Transfer Functions, the 

Stiffness and Load Transfer Functions to calculate structural stiffness (bending and torsion), and operational

loads (bending moments, shear loads, and torques) for monitoring the flight-vehicle’s operational loads. 

This patented method is called “Process for Using Surface Strain Measurements to Obtain Operational 

Loads for Complex Structures,” U.S. Patent No. 7,715,994 (Richards-Ko), (ref. 7). The accuracy of this

patented method for estimating operational loads on structures was analytically confirmed by using 

finite-element analysis of different aerospace structures (tapered cantilever tubular beam, depth-tapered 

un-swept wing box, depth-tapered swept wing box, and doubly-tapered generic long-span wing) (ref. 8).

The earlier Displacement Transfer Function (refs. 1–5) were analytically validated for prediction 

accuracies by finite-element analysis of different sample structures such as cantilever tubular beams 

(uniform, tapered, slightly tapered, and step-wisely tapered), two-point supported tapered tubular beams, flat

panels, and tapered wing boxes (un-swept and swept). 

To eliminate mathematical indeterminacy at the limit of uniform beam depth, additional First-Order and 

Second-Order displacement transfer functions (ref. 9) were formulated for structural deformed shape 

predictions. The prediction accuracies of the two new Displacement Transfer Functions were found to be 

comparable with those of the earlier Displacement Transfer Functions 

(refs. 1–5). 

Also, by using piecewise nonlinear (instead of piecewise linear) strain representations in the

formulations of the improved Displacement Transfer Functions (ref. 10), the shape prediction accuracies 

could be improved greatly especially for highly tapered slender structures.

The earlier Displacement Transfer Functions (refs. 1–5) were formulated for the straight beams. 

However, by introducing empirical curvature-effect correction terms, the resulting modified Displacement 

Transfer Functions could be applied to the shape predictions of slender curved structures (including 

full-circle structures), with great success (ref. 11).

All the past Displacement Transfer Functions were formulated by integrating the beam curvature 

equation in reference to the undeformed fixed coordinate system, and were found to be amazingly accurate 

for the shape predictions of various slender structures under small bending deformations.

For certain new type of unmanned aircraft such as Global Observer, with highly flexible wing structures 

having long wingspan reaching 175 ft, the wing tip deflection could reach up to 32 ft during flights. For the 

shape predictions of such type of highly flexible structures, there is a need to formulate large-deformation 

Displacement Transfer Functions.

The present technical publication discusses different curvature equations [classical (Eulerian), physical 

(Lagrangian), and Shifted curvature equations] for the embedded beam elastic curves, referred to deformed 

or undeformed coordinate systems. This technical publication describes the formulations of different 

large-deformations Displacement Transfer Functions through beam discretization and piecewise integrations 

of the said different curvature equations. 

A long tapered cantilever tubular beam was used in the shape prediction accuracy analysis of the newly 

formulated Displacement Transfer Functions. The input surface strains in the present technical publication

were analytically generated from the finite-element analysis of the tapered cantilever tubular beam. For the 
4

shape prediction accuracy studies of the newly formulated Displacement Transfer Functions, the slopes and 

deflections calculated from the finite-element analysis were used as reference yardsticks. 

CURVATURE-STRAIN RELATIONSHIP

Figure 1 shows the deformed state of a nonuniform beam with changing depth factor, c(s). The 

curvature-strain relationship can be established graphically from figure 1. The beam elastic curve has local 

radius of curvature, R(s), within a small beam segment subtended by d . The un-deformed curve length, 

AB, lies on the beam neutral surface, and the deformed curve length, 	
(s)]} [where (s) is the 

surface strain], lies on the beam bottom surface. From the two similar slender sectors, OAB and OAB ,

the following relationship can be established:

(1)

From equation (1) the following curvature-strain equation is obtained:

(2)

Equation (2) geometrically relates the local curvature, 1/R(s), of the deformed beam elastic curve to the 

associated surface strain, (s), and the beam depth factor, c(s). Equation (2) was the basis for the 

formulations of the earlier Displacement Transfer Functions (ref. 1–5), and is also used in the present 

technical publication to formulate the Displacement Transfer Functions for large deformations. 

DIFFERENT CURVATURE EQUATIONS

This section discusses three types of curvature equations for deformed beam elastic curve: 1) classical 

(Eulerian) curvature equation, 2) physical (Lagrangian) curvature equation, and 3) Shifted curvature 

equation. These three curvature equations will be geometrically related to surface strains and beam depth 

factor for establishing three curvature-strain differential equations for the formulations of the 

large-deformation displacement transfer functions.

Classical (Eulerian) Curvature Equation

The derivation of the classical curvature equation for a plane curve can be found in any standard 

calculus textbook, and are presented in the following paragraphs for the purpose of refreshing memory.

From figure 1, the infinitesimal curved segment, , and the local slope, 

tan ( / ) 1 dy dx    , can be used to express the curvature, 1/ R  d / ds, in the following forms (see slender 

sector, OAB in figure 1) (ref. 12):

 (3)
5

Equation (3) is the classical mathematical curvature equation for a plane curve, and can be found in any 

standard calculus textbook. 

When equation (3) is applied to describe the curvature of the deformed-beam elastic curve, the 

x-variable in equation (3) will now denote the abscissa of any material point on the elastic curve in its final 

deformed position (Eulerian description), (fig. 1). Thus, based on equation (3), a certain x-coordinate of a 

material point in the deformed configuration no longer represents the same material point in the undeformed 

state. Namely, the deformed material point has no memory of its initial undeformed x-location. As will be 

seen shortly, the slope equation formulated based on equation (3) gave erroneous slope data, and caused

deflection predictions to be poor at large deformations. For more detailed discussions of the classical 

curvature equation (3), please see references 1316.

It must be mentioned that the classical Euler-Bernoulli beam theory was formulated based on the 

linearized classical (Eulerian) curvature equation (3) [that is, the term 2 (dy / dx) neglected], which was 

referred to the deformed x-coordinate. For small deflections, the difference between the deformed and 

undeformed x-coordinates is negligible; however, for large bending deformations the two x-systems can be 

quite different.

Physical (Lagrangian) Curvature Equation

As shown in figure 2, the infinitesimal curved segment, ds, of the deformed-beam elastic curve in the 

limit could be considered as an infinitesimal straight-line segment. Then, the local slope, (s) , along the 

s-curve can be expressed as ( ) sin ( / ) 1

s dy ds    . The local curvature expressed in reference to the s-system, 

can then be written as equation (4):

(4)

In general, under bending deformation, the original undeformed length, dx, along the beam x-axis will 

be deformed into arc length, ds, along the s-curve (fig. 2). Then, one can write:

(5)

in which is the axial strain in s-direction. For an in-extensional beam, (s)=0, and equation (5) becomes:

(6)

Equation (6) shows that for the in-extensional beam, the s-coordinate of any material point in the 

deformed configuration is always equal to its initial undeformed x-coordinate. In view of equation (6), the 

curvature equation (4) can be rewritten in reference to the undeformed x-coordinate as:

(7)
6

Equation (7) is the physical (Lagrangian) curvature equation for the elastic curve of an in-extensional 

beam expressed in reference to the undeformed x-coordinate (Lagrangian description). Equation (7) was 

also derived by Kopmaz (ref. 13) and Hodge (ref. 14) based on kinematics of a deformed beam (fig. 2). The 

alternative derivation of equation (7) is described in Appendix A. 

It is important to mention that for an in-extensional beam ( ds  dx ), (fig. 2), when the s-system is 

converted into the undeformed x-system, sin (s)( dy / ds) in the s-system will become tan (x)( dy / dx)

in the x-system [that is, sin (s)  tan (x) ]. Thus, the slope,  (x) , in the x-system will be slightly smaller 

than the slope,  (s) in the s-system. Therefore, the line AE drawn from point A (fig 2) to define the 

slope,  (x) , in the x-system will be slightly tilted away from the true tangent to the s-curve at point A

(fig. 2). However, this slope-tilt is miniscule because the axial displacements induced by the lateral bending 

are quite small.

Shifted Curvature Equation

The formulations of the Shifted Displacement Transfer Functions are based on the undeformed 

x-coordinate. Namely, the deformed material points are shifted back in the x-direction and plotted at their 

respective initial undeformed x-coordinates (fig. 3). Thus, the Shifted curvature equation to be formulated 

will be slightly different from the physical curvature equation (7) for which the deformed configuration of 

the beam is described with the un-deformed fixed coordinate system (that is, no horizontal shifting of 

deformed material points).

From a small right triangle ABC (fig. 2), the following relationship holds: 

2 2 2 ds  (dx  du)  dy (8)

in which u (= AA ), (fig 2) is the x-displacement (axial direction) of a material point A. In view of equation 

(8), the physical curvature, equation (4), in the s-system can be rewritten as:

(9)

Because in the beam bending deformations, the axial displacement, u, of a material point is quite small, 

it is reasonable to set u 
 0 , which is equivalent to horizontal shifting of the deformed material points like 

{A , B} (fig. 3) toward the right to the positions {A , B} with x-coordinates respectively matching the 

initial x-coordinates of the un-deformed positions {A, B}.

Under the condition, u 
 0 , equation (9) becomes:

(10)
7

For an in-extensional beam, the curved coordinate, s, of any deformed material point is always equal to 

its initial undeformed coordinate x (fig. 3); namely, s  x and ds  dx [ eq. (6)]. Then equation (10) can be 

written in the un-deformed x-system as:

(11)

Equation (11) is called the Shifted curvature equation for the in-extensional beam ( ds  dx ), referred to the 

undeformed x-coordinate. Equation (11) can also be established graphically in view of figure 3 (see 

Appendix B).

The mathematical form of equation (11) is equivalent to the linearized form of the Lagrangian curvature 

equation (7) [the term 2 (dy / dx) dropped] because both equations (7) and (11) are referred to the 

undeformed x-coordinate (figs. 2, 3). It is important to mention that, for an in-extensional beam, setting 

u 
 0 (that is, shifting deformed material points and plot at their respective undeformed x-locations) causes 

2 (dy / dx) 0 according to equation (A8) in Appendix A and, therefore, implying linearization of the 

Lagrangian curvature equation (7).

However, equation (11) cannot be equivalent to the linearized Eulerian curvature equation (3) [the term

2 (dy / dx) removed] even with similar mathematical forms because, equation (11) is referred to the 

undeformed x-coordinate, while equation (3) is referred to the deformed x-coordinate (fig. 1). For small 

bending deformations, the difference between the deformed and un-deformed x-coordinates could be 

negligible; however, for large bending deformations, the two x-systems could be quite different.

As will be seen shortly, the Displacement Transfer Functions formulated based on the Shifted curvature 

equation (11) turned out to be amazingly accurate beyond expectation for large-deformation shape 

predictions (for example, cantilever beam-tip slope reaching even as large as 66° under bending).

The unexpected discovery in the present technical publication is that the linearized curvature equation 

used in the formulations of earlier Displacement Transfer Functions (refs. 1–5) was not referred to the 

deformed x-coordinate, but to the undeformed x-coordinate. Therefore, the earlier linearized curvature 

equation was actually the Shifted curvature equation (11), and the Displacement Transfer Functions 

formulated based on equation (11) were found to be amazingly accurate beyond expectation in the structural 

deformed shape predictions.

CURVATURE-STRAIN DIFFERENTIAL EQUATIONS

In view of equation (2), the Eulerian, Lagrangian, and the Shifted curvature equations (3), (7), and (11) 

can be written respectively in the forms of curvature-strain differential equations for the deformed beam as: 

1. Eulerain curvature-strain differential equation:

(12)
8

2. Lagrangian curvature-strain differential equation: 

(13)

3. Shifted curvature-strain differential equation:

(14)

The formulations of the large-deformation displacement transfer functions stemmed from the 

integrations of the curvature-strain differential equations (12)–(14). Note that, equations (12)–(14) are 

purely geometrical relationships, containing no material properties. Likewise, the Displacement Transfer 

Functions formulated by integrating equations (12)–(14) will also contain no material properties. In fact, the 

material properties will affect the outputs of the surface strains, 	 (x) (refs. 1–5).

NONUNIFORM AND UNIFORM BEAMS

In the present technical publication, the term “beam” implies the imaginary embedded beam (the 

depth-wise cross section of the structure along the surface strain-sensing line), and not the classical isolated 

Euler-Bernoulli beams. The nonuniform beam in the present technical publication is defined as the 

embedded beam with a varying depth factor, c(x) , a vertical distance from the neutral axis to the bottom 

surface of the nonuniform beam at axial location, . For a nonuniform tubular beam, c(x) will be the 

outward radius. The uniform beam in the present technical publication is defined as the embedded beam 

with constant depth factor [that is, c(x)  c ].

SHIFTED DISPLACEMENT TRANSFER FUNCTIONS

In the formulations of earlier Displacement Transfer Functions (refs. 1–5) for structural deformed shape 

predictions, the curvature-strain differential equation used turned out to be equivalent to the 

Shifted-curvature-strain differential equation (14). Therefore, there is no need to reformulate the 

large-deformation Displacement Transfer Functions based on the Shifted curvature-strain differential 

equation (14). The Displacement Transfer Functions formulated earlier can be called Shifted Displacement 

Transfer Functions, which, as will be shown shortly, were found to be amazingly accurate beyond 

expectations for the shape predictions not only for small deformations, but also for very large deformations. 

The Shifted Displacement Transfer Functions formulated earlier for equal domain length ( =constant) 

(refs. 1–5) are reproduced in the following sections. Keep in mind that the slopes, [ ( ) ( / ) ] i i dy dx i    x 

(i 1,2,3,....,n), at the strain-sensing stations, , appearing in the following Shifted Displacement 

Transfer Functions are referred to the initial undeformed x-coordinates. 

Based on Piecewise Linear Strain Representations

The following four sets of Shifted slope and deflection equations [eqs. (15)–(18)] were formulated by 

using piecewise-linear functions to describe the distributions of both surface strains and depth factors, 

{	 (x), c(x)}.
9

1. Nonuniform Beams (refs. 1, 3)

Slope equation:

1

1

2

1

1 1

1

1 log tan

( )

tan 

 

 



  



 







  

   i

i

i

i i

i i i i

i i

i i

i

c

c

c c

c c

c c

l  	 	 	 	  ; (i 1,2,3,....,n) (15a)

Deflection equation:

a. In recursive form: 

1 1 1

1

3

1

1 1

1

2 1 log ( ) tan

2( ) ( ) ( )   

 

 



    



 





















  

  

   i i i i

i

i

i

i i

i i i i

i i

i i

i c c y l

c

c

c

c c

c c

c c

y l  	 	 	 	

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

(15b)

b. In dual summation form:

   

     

     

=0 for cantilever beams

0 0

Contribution from slope terms

1

1 1

2

1

1 1

1

2 1

Contributionsfrom deflection terms

1

1

1

3

1

1 1

1

2 1

log ( ) tan

( ) ( ) ( )

log ( ) 2( ) ( ) ( )



	 	 	 	

	 	 	 	

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

  















 



















  

   















 

















  

  

  

!

!



  

 









 

 





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

(15c)

Equation (15c) is called the Nonuniform Displacement Transfer Function.

2. Slightly Nonuniform Beams [ e log terms in equations (15a)–(15c) expanded] (refs. 1, 3)

Slope equation:

1 1

1 1

2 tan

2

tan  

 

 



 





 













    i i i

i

i

i

i

c

c

c

l  	 	  ; (i 1,2,3,....,n) (16a)

Deflection equation:

a. In recursive form:

1 1 1

1 1

2

3 tan

6

( )   

 

   



 





 













    i i i i

i

i

i

i y l

c

c

c

l

y 	 	  ; (i 1,2,3,....,n) (16b)
10

b. In dual summation form:

   

           

=0 for cantilever beams

0 0

Contributionsfrom slope terms

1

1

1

1 1

2

Contribution from deflection terms

1

1

1 1

2

tan

2 ( )

2

( ) 3 1

6

( )



	 	 	 	

y l

c

c

c

l i j

c

c

c

l

y

i

j

j j

j

j

j

i

j

j j

j

j

j

i

  















 

















 













    















 

















 













    ! !







  



 

(16c)

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

Equation (16c) is called the Slightly-Nonuniform Displacement Transfer Function.

3. First Order Expansion of 1/ c(x)(ref. 8)

Slope equation:

1

1

1

1 1

4 5 2 tan

6

tan 





 

 



 





















  













    i i

i

i

i

i

i

i

i c

c

c

c

c

l  	 	  ; (17a)

Deflection equation:

a. In recursive form:

1 1

1

1

1 1

2

5 3 tan

12

( )  





 

   



 





















  













    i i i

i

i i

i

i

i

i y l

c

c

c

c

c

l

y 	 	  ; (i 1,2,3,....,n) (17b)

b. In dual summation form:

   

     

     

=0 for cantitever beams

0 0

Contributionsfrom slope terms

1

1 1

1

1 1

2

Contributionsfrom deflection terms

1 1

1

1 1

2 2

4 5 2 tan ( )

6

( )

5 3 ( )

12

( )

	 	 

	 	

y l

c

c

c

c

c

l i j

c

c

c

c

c

l l

y

i

j

j

j

j

j

j

j

i

i

j

j

j

j

j

j

j

j

i

  

































  













    

































  













    

!

!



 



 

 



 

(17c)

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

Equation (17c) is called the First-Order Displacement Transfer Function.
11

4. Second Order Expansion of 1/ c(x) (ref. 8)

Slope equation:

1

2

1 1

1

2

1 1 1

9 4 13 10 3 tan

12

tan 

 



  

















 

































  































  

  i i

i

i

i

i

i

i

i

i

i

i

i

c

c

c

c

c

c

c

c

c

l  	 	  (18a)

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

Deflection equation:

a. In recursive form:

1 1

2

1 1

1

2

1 1 1

2

27 9 2 18 11 3 tan

60

( )  

 



  

  















 

































  































  

  i i i

i

i

i

i

i

i

i

i

i

i

i y l

c

c

c

c

c

c

c

c

c

l

y 	 	  (18b)

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

b. In dual summation form:

   

     

     

=0 for cantilever beams

0 0

Contributionsfrom slope terms

1

1

2

1 1

1

2

1 1 1

2

Contributionsfrom deflection terms

1

2

1 1

1

2

1 1 1

2

( ) tan

9 4 13 10 3 ( )

12

( )

27 9 2 18 11 3 1

60

( )



	 	

	 	

y i l

c

c

c

c

c

c

c

c

c

l i j

c

c

c

c

c

c

c

c

c

l

y

i

j

j

j

j

j

j

j

j

j

j

j

j

i

j

j

j

j

j

j

j

j

j

j

j

j

i

  



















 

































   





























  

  



















 

































   





























  

 

!

!



  



  

  



  

(18c)

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

Equation (18c) is called the Second-Order Displacement Transfer Function.

Based on Piecewise Nonlinear Strain Representations

The following three sets of Shifted slope and deflection equations [eqs. (19)–(21)] were formulated by 

using piecewise-nonlinear function to describe the actual distribution of the surface strain, 	 (x), but using 

piecewise-linear function to describe the actual distribution of the depth factor, c(x) .
12

1. Improved Case (ref. 9)

Slope equation:

(19a)

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

Deflection Equations:

a. In recursive form:

" #

" # 2 1 1 1 1 1 1 1

1

2

1

1

4 1 1 1 1 1

1

2

(8 5 ) 2(5 2 ) (2 ) tan

12( )

( )

(2 )( 2 ) log ( ) 2( )

( )

      







    



        

 





 



      

 

i i i i i i i i i i i

i i

i i

i

i

i i i i i i i i i i e

i i

i

c c c c c c y l

c c

l

c c

c

c

c c c c c c c

c c

l

y

	 	 	 

	 	 	

(19b)

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

b. In dual summation form:

" #

" #       Contribution from deflection terms

1

2 1 1 1 1 1

1

1

1

4 1 1 1 1 1

2 1

(8 5 ) 2(5 2 ) (2 ) 12( )

1

(2 )( 2 ) log ( ) 2( )

1

( ) !

    







    





























 

      















        

i

j

j j j j j j j j j

j j

j j

j

j

j j j j j j j j j j e

j j

i

c c c c c c

c c

c c

c

c

c c c c c c c

c c

y l

	 	 	

	 	 	

" #

" #

   

     

=0 for cantilever beams

0 0

Contributionsfrom slope terms

1

1

2 1 1 1 1 1

1

1

3 1 1 1 1 1

2 1

( ) tan

(5 3 ) 2(3 ) ( ) 4( )

1

(2 )( 2 ) log 2( )

1

( ) ( )



	 	 	

	 	 	

y i l

c c c c c c

c c

c

c

c c c c c c

c c

l i j

i

j

j j j j j j j j j

j j

j

j

j j j j j j j j j e

j j

  



























 

      

      ! 





    





    



(19c)

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

Equation (19c) is called the Improved Displacement Transfer Function.
13

2. Log-expanded Case [ terms in equation (19a, b, c) expanded] (ref. 9)

Slope equation:

(20a)

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

Deflection equation:

a. In recursive form:

" #

" #" #

1 1

1

2

1

2

1 1 1

2

1

1

3

1

2 3

1 1

3

1

4

1

2

tan

2 (3 ( )(3 8 ) 2( )

7 ( )(8 3 ) 3( )

24

( )

 

     

    



  















 

      

      

i i

i i i i i i i i i i i i

i i i i i i i i i

i

i

y l

c c c c c c c c c c

c c c c c c c c

c

l

y



	 	

	

(20b)

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

b. In dual summation form:

(20c)

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

Equation (20c) is called the Log-Expanded Displacement Transfer Function.

3. Depth-expanded Case [expansion of 1/ c(x)] (ref. 9)

Slope equation:

1 ( 12 3 ) tan

2(5 8 ) 1 (3 10 ) 24tan

1 1 1

2

1

1 1

1

1 1

1

  



 



 













  













  







  















     

i i i i

i

i

i i i

i

i

i i i

i

i

c

c

c

c

c

l

	 	 	 

 	 	 	 	 	 	

(21a)

(i 1,2,3,....,n)
14

Deflection equation:

a. In recursive form:

$ % $ %

$ % 1 1 1 1

2

1

1 1

1

1 1

1

2

1 12 3 tan

5

1

1 4 7

5

2 7 6

24

( )

   



 



 



  









  















 







  















     

i i i i i

i

i

i i i

i

i

i i i

i

i

y l

c

c

c

c

c

l

y

	 	 	 

	 	 	 	 	 	

(21b)

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

b. In dual summation form: 

$ % $ %

$ %

      Contributionsfrom deflection terms

1

1 1

2

1

1 1

1

1 1

1

2

1 12 3

5

1

1 4 7

5

2 7 6

1

24

( ) !

 



 



 



































  















 

  















   

 

i

j

j j j

j

j

j j j

j

j

j j j

j

i

c

c

c

c

c

l

y

	 	 	

	 	 	 	 	 	

   

     

=0 for cantilever beams

0 0

Contributionsfrom slope terms

1 1

2

1

1 1

1

1 1

1

1 1

2

( ) tan

1 ( 12 3 )

2(5 8 ) 1 (3 10 )

( )

24

( )



	 	 	

	 	 	 	 	 	

y i l

c

c

c

c

c

l i j

j j j

j

j

j j j

j

j

j j j i

j j

  

































  















 

  















   

  

 



 



  

 

!

(21c)

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

Equation (21c) is called the Depth-Expanded Displacement Transfer Function.

Characteristics of Displacement Transfer Functions

In each of the Displacement Transfer Functions [eqs. (15c)(21c)], the deflection, , at the 

strain-sensing station, , is expressed in terms of the inboard beam depth factors, , and

the associated inboard surface strains, , including the values of at the 

strain-sensing station where deflection, , is calculated. It is important to mention that equations

(15c)(21c) are purely geometrical relationships, containing no material properties. In fact, the values of the 

surface strains, , can be affected by the material properties and internal structural configurations.

Thus, in using equations (15c)(21c) for shape predictions of complex structures such as long-span 

aircraft wings, there is no need to know the material properties, nor the complex geometries of the internal 

structures. As will be seen shortly, the Shifted Displacement Transfer Functions [eqs. (15c)(21c)] turned 

out to be amazingly accurate beyond expectations for the shape predictions of a long cantilever tapered 

tubular beam under large bending deformations with beam tip slope even reaching as large as n  deg.
15

EULERIAN FORMULATION OF DISPLACEMENT TRANSFER 

FUNCTIONS 

The next attempt is to formulate the large-deformations displacement transfer functions based on the 

Eulerian curvature-strain differential equation (12). Because equation (12) is referred to the deformed 

coordinate system (Eulerian description), the x-coordinate used in this section must be considered as the 

deformed x-coordinate (no change in y-coordinate). It is natural to expect that by retaining the term

2 (dy / dx) , the shape prediction accuracy could be improved; however, as will be seen shortly, the results 

turned out to be entirely negative at large deformations.

Projected Domain Length

Under large deformations, strain-sensing stations, x (i 1,2,3,...., n) i  , on the un-deformed configuration 

will move along the x-axis to their deformed positions. Therefore, in the present Eulerian formulation, one 

must keep in mind that now denotes the deformed locations of the strain-sensing stations on the x-axis.

Similar to the earlier formulations of the Displacement Transfer Functions (refs. 1–5), the undeformed 

beam was first divided evenly into n number of small domains (domain length, l  l / n ), so that under 

deformation, the surface strains and depth factors, {(x), c(x)}, within each deformed small domain, 

i i x  x  x 1 , could be represented with linear functions (piecewise linear assumptions) of the following 

forms: 

; (22)

; (23)

In equations (22) and (23), 1 { i 	 , ci1} and , are respectively the values of {	 (x), c(x)} at the 

deformed strain-sensing stations, 1 { i x , }i x ; and ( )  i  i  i1 L x x is the projected domain length along the xaxis of the bent and tilted domain length, (unchanged for in-extensional beam), (fig. 2), and can be 

calculated from equation (24) (see Appendix C):



















 

 

















 



















  















i i

i i

i

i i

i i

i i

i i

i

c c

l

c c

c c l L

1

1

1

1

1

1

1

2

cos

2

2 sin 	 	  	 	

	 	

(24)

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

in which [ tan ( / ) ]1

1

1 

 i  dy dx i is the slope at the deformed strain-sensing station, (fig. 4) and can be 

calculated from the yet-to-be formulated slope equations (29) or (34) in the following section.

The piecewise linear assumptions [eqs. (22) and (23)] enabled the piecewise integrations of the Eulerian 

curvature-strain differential equation (12) to yield the slope and deflection equations for each small domain. 
16

Slope Equation

Let V  dy / dx  tan (x) , then the Eulerian curvature-strain differential equation (12) can be written as:

(25)

In view of piecewise linear assumptions [eqs. (22) and (23)], the piecewise-integration of equation (25) 

within each deformed small deformed domain, , can be written as:

; (26)

in which the lower limit of integral, Vi1 , is the value of V at the deformed strain-sensing station, ,

namely, 1 1 1 1 i  ( i )  tan ( i )  tan i V V x  x  .

Integration of equation (26) to obtain the slope equation in closed form is relatively easy; however, 

subsequent integration of the slope equation to obtain the deflection equation in neat closed-form turned out 

to be a nightmare, and an alternative simplified approach had to be introduced as discussed in the 

subsequent sections.

After carrying out integration of equation (26), one obtains equation (27) (ref. 17, and Appendix D):





 

























  

    

 









 

  







 log 1 1

( ) ( )

1 1

1

1

2

1

1 1

1

1

1

2

1

1

2

i

i

i

i e

i i

i i i i

i i

i i

i i

i

i

L

x x

c

c

c c

c c

x x L

V c c

V

V

V 	 	 	 	

( ) i 1 i x  x  x 

(27)

By squaring both sides, rearranging, then taking the square root, and writing V (x)  tan (x) , one obtains 

the slope equation in the following form (see Appendix D):

2

1

1

2

1

1 1

1

1

1

1

2

1

1

1

2

1

1 1

1

1

1

1

2

1

log 1 1

( ) ( )

1 tan

tan 1

log 1 1

( ) ( )

1 tan

tan

tan ( )















 





 

























  

    

 









 

























  

    

 







 

  











 

  









i

i

i

i e

i i

i i i i

i i

i i

i i

i

i

i

i

i

i e

i i

i i i i

i i

i i

i i

i

i

L

x x

c

c

c c

c c

x x L

c c

L

x x

c

c

c c

c c

x x L

c c

x

	 	 	 	





	 	 	 	







( ) i 1 i x  x  x 

(28)

in which, when i 1 , for a cantilever beam we have tan11  tan 0  0 at the fixed end. 
17

At, the deformed strain-sensing station, , we have i i Li x  x 1   , i i tan (x )  tan , and equation (28)

becomes:

2

1

2

1

1 1

1

1

1

2

1

1

2

1

1 1

1

1

1

2

1

log ( ) 1 tan

tan 1

log ( ) 1 tan

tan

tan















 





 







  

  









 







  

  

 

 

 









 

 









i

i e

i i

i i i i

i i

i i

i

i

i

i

i e

i i

i i i i

i i

i i

i

i

i

i

c

c

c c

c c

c c

L

c

c

c c

c c

c c

L

	 	 	 	





	 	 	 	







(29)

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

Equation (29) is the Eulerian slope equation for large deformations of nonuniform beams, but is not 

applicable to uniform beams ( ) 1 c c c i  i  , because the logarithmic terms, log ( / ) e i i1 c c , and the 

denominators containing factor, ( ) i  i1 c c , will cause mathematical indeterminacy (that is, 0/0). The way to 

avoid such mathematical indeterminacy is to expand the logarithmic function in series form as described in 

the subsequent sections.

At, the deformed strain-sensing station, , we have xi1  xi1  0 , 1 1 tan ( i )  tan i  x  , then 

equation (28) becomes an identity equation [eq. (30)] as:

1 2

1

2

1

1

2

1

1 tan

1 tan

tan 1

1 tan

tan

tan 









 















 





  i

i

i

i

i

i 











tan 0  0 ; (i 1,2,3,....,n)

(30)

which confirms the mathematical accuracy of equation (28).

Deflection Equation

The deflection equation can be obtained by integrating the slope equation (28) as:

(31)
18

in which ( ) i1  i1 y y x is the deflection at the deformed strain-sensing station, . When i 1 , we have 

y11  y0  0 at the fixed end of a cantilever beam.

The powerful Wolfram Mathematica Online Integrator (http://integrals.wolfram.com/index.jsp) 

indicated that no integrated formula existed for the mathematical form of equation (31), which contains 

logarithmic functions.

Log-Expanded Slope Equation

As indicated earlier that in the slope equation (28), the logarithmic terms and the denominators 

containing factor, ( ) i  i1 c c , will cause mathematical indeterminacy (that is, 0/0) for the uniform beam case

( ) 1 c c c i  i  . The way to avoid such a mathematical breakdown problem is to expand the logarithmic 

function in series form as described below.

Note that in equation (28), the two conditions, 0 1 1 



  

i

i

L

x x and 1 1 0

1

&













  & 

i

i

c

c , cause the 

required condition, 1 1 1 1

1

 &



 























  &  

 i

i

i

i

L

x x

c

c , to hold for series expansion of the logarithmic term in 

equation (28) in terms of ( ) i  i1 c c (ref. 16 and Appendix D) as: 





 

























  



log 1 1 1

1 i

i

i

i e L

x x

c

c .... 2

1

2

1

2

1

1 1

1

1  

































   

   



 





i

i

i

i i

i

i

i

i i

L

x x

c

c c

L

x x

c

c c (32)

Substitution of equations (32) into the slope equation (31), causing the factor, ( ) i  i1 c c , to be canceled out 

to yield (see Appendix D):

2

2

1

2

1

1 1

1

1

1

1

2

1

2

1

2

1

1 1

1

1

1

1

2

1

2

( ) ( )

1 tan

tan 1

2

( ) ( )

1 tan

tan

tan ( )















 



    







    

 





  













  









i

i

i

i i i i

i

i

i

i

i

i

i

i

i i i i

i

i

i

i

i

L

x x

c

c c

x x

c

L

x x

c

c c

x x

c

x

	 	 	





	 	 	







(33)

At the deformed strain-sensing station, i x  x , we have i i Li x  x 1   , and equation (33) takes on the form 

(see Appendix D):

2

1

1 1 1

2

1

1

1 1 1

2

1

2

2 1 tan

tan 1

2

2 1 tan

tan

tan















 





 





 













  











 





 













  



 



  





  



i i

i

i

i

i

i

i

i i

i

i

i

i

i

i

i

c

c

c

L

c

c

c

L

	 	





	 	







(34)

(i 1,2,3,....,n)
19

Equation (34) is the log-expanded Eulerian slope equation for large deformations of nonuniform beams, 

including uniform beams as special cases because the logarithmic functions and the factor, ( ) i 1 i c  c  in the 

denominator of equation (28) have been eliminated.

Log-Expanded Deflection Equation

The log-expanded deflection equation can be obtained by the integration of the log-expanded slope 

equation (33). 

Let A, B, C, and be defined respectively in equation (35) as: 

1

2

1

1 tan

tan









i

i A





; ; 













 

  



 

2

1

1 1

2

1

i

i i i i

i c

c c

L

C 	 	

; (35)

then equation (33) can be written in a compact form as:

2 2

2

1 ( )

tan ( )


 



 
 

A B C

A B C

x

  

   (36)

The deflection, , can be obtained by integrating equation (36) as:

(37)

The powerful Wolfram Mathematica Online Integrator (http://integrals.wolfram.com/index.jsp) can 

now carry out the integration of equation (37) to yield the log-expanded Eulerian deflection equation. 

However, the resulting integrated formula is nearly ten pages long, containing various mathematical 

functions including inverse sine functions, elliptic integrals of the first, the second, and the third kinds 

(ref. 17). Because a very accurate alternative simplified deflection equation could be introduced (see the

following section), no attempt was made to program and use the complicated ten-page-long integrated 

formula of the log-expanded Eulerian deflection equation (37).

Simplified Deflection Equation

To bypass direct integration of equation (37), one can introduce a simplified deflection equation by an 

alternative approach with high accuracy. Because the slope, tan (x) , changes very slowly within each 

deformed small domain, , it is reasonable to represent tan (x) with a linear function as:

; (38)

in which  i tan can be obtained from the exact slope equation (29) (with logarithmic terms) or from the 

log-expanded slope equation (34).
20

Equation (38) can now be easily integrated to yield the following closed-form alternative deflection 

equation:

; (39)

in which ( ) i1  i1 y y x is the deflection at the deformed strain-sensing station, i1 x .

At the deformed strain-sensing station, i x , we have i i Li x  x 1   , and equation (39) gives the 

deflection, i i y(x )  y , in the following simple form:

1 1 (tan tan ) 2    

  i i i

i

i y

L

y   ; (i 1,2,3,....,n) (40)

Equation (40) is the simplified Eulerian deflection equation for the large-deformation shape predictions, and 

turned out to be very accurate. The high accuracy of equation (40) was verified in the following way.

Using the slope data calculated from the Shifted slope equation (15a) (very accurate) as inputs to 

equation (40), with Li replaced with , the calculated beam-tip deflection of a tapered cantilever beam 

(table 1) was found to be only 0.0277 percent off from the deflections calculated from the Shifted 

Displacement Transfer Function [eq. (15c)] (very accurate). The high accuracy of the simplified deflection 

equation (40) thus eliminated the need to program and use the ten-page long integrated formula of equation 

(37). 

Table 1. Geometries of the long tapered cantilever tubular beam.

l, in. t, in. c0, in. cn, in. cn / c0 {= tan[(  )/	]}, deg

(Length) (Wall 

thickness)

(Root depth 

factor)

(Tip depth 

factor)

(Depth 

ratio)

(Taper angle)

300 0.02296 4 1 1/4 0.5729

LAGRANGIAN FORMULATION OF DISPLACEMENT 

TRANSFER FUNCTIONS

Lastly, the Lagrangian formulation of the displacement transfer function for large deformations is based 

on the Lagrangian curvature-strain differential equation (13), which is referred to the undeformed 

x-coordinate. Like Eulerian formulation, by retaining the term 2 (dy / dx) in equation (13), one will 

naturally expect more accurate shape predictions; however, as will be seen in this report, the results turned 

out to be very discouraging.
21

Slope Equation

Writing V  dy / dx  tan (x) (fig. 2), the Lagrangian curvature-strain differential equation (13) can be 

written as:

(41)

In light of the piecewise linear assumption of the surface strain and the depth factor, {	 (x), c(x)} [ Li

in eqs. {(22), (23)} replaced with ], integration of equation (41) within the small domain, i i x  x  x 1 ,

can be written as:

; (42)

On the left hand side of equation (42), the lower limit of integration, Vi1 , is the value of V at the 

strain-sensing station, i1 x [that is, 1 1 1 ( / ) tan ( ) tan i  x 1  i  i V dy dx x i   ].

Carrying out integrations of both sides of equation (42), one obtains the slope equation [eq. (43)] in the 

following form (see ref. 1 and Appendix E):





 



  



  

    

   







  



 

  ( ) 1 ( ) log ( ) ( ) ( )

sin sin 1

1

1

2

1

1 1

1

1

1

1

1 1

i

i

i i

i i

i i i i i

i i

i i i x x

lc

c c

c c

c c

x x l

c c

V V 	 	 	 	 (43)

from which the slope, tan (x)[V ] can be expressed explicitly as:



























 





 



  



  

  

 

















 







( ) 1 sin (tan ) ( ) log ( )

( ) ( )

tan ( ) sin

1

1

1

1

1

2

1

1 1

1

1

1

i i

i

i i

i i

i i i i

i

i i

i i

x x

lc

c c

c c

c c l

x x

c c

x

 	 	

	 	

 (44)

At the strain-sensing station, i x , we have x x l i  i1   , i i tan (x )  tan , and equation (44) becomes:















 

 



 







     



 

 



 log sin (tan ) ( ) ( )

tan sin 1

1

1

2

1

1 1

1

1

i

i

i

i i

i i i i

i i

i i

i

c

c

c c

c c

c c

l  	 	 	 	  (45)

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

As will be seen in this technical publication, the values of the slopes, i , calculated from equation (45) did 

not turn out to be as accurate as the corresponding slopes calculated from the Shifted slope equation (15a) 

for large deformations. 
22

Deflection Equation

The deflection equation can be obtained by integrating the slope equation (44) as:

(46)

in which is the deflection at the strain-sensing station, .

Let A , B , C , D , and be defined respectively in equation (47) as: 

sin (tan )1

1



 A  i ; ( ) 1

1

i i

i i

c c

B 

 



 	 	

; 2

1

1 1

( ) i i

i i i i

c c

c c C l 

  



  	 	

;

1

1 ( )







  

i

i i

lc

c c D ;   i1 
 x x (47)

then equation (46) can be written in a compact form as:

; (0  
  l) (48)

Using the Wolfram Mathematica Online Integrator (http://integrals.wolfram.com/index.jsp), equation (48) 

can be integrated to yield the following complex form of the Lagrangian deflection equation:

1

Incomplete gamma function

2

Incomplete gamma function

2

2

( ) ( 1) cos sin 1 ,

( ) 1 ,

cos sin ( 1) ( ) ( 1) 2

1 ( )











 



   ( ' 



 











 







  





 





   











 





 



 

(  ' 















 







 





   



 











 





  



 



 





 



    

i

iC

iC iC

iC

y D

i BD B iC

D

B i A

D

B D A

D

i BD B iC

D

B i A

D

B A

D

BD B

D

iB D D

B y

     

     


 






 
 
 


(0  
  l)

(49)

In equation (49), except for the term (i is an integer), the symbol, i, appearing in the rest of the 

terms is not an integer, but an imaginary unit, i  1 . Equation (49) contains imaginary unit,i  1 ,

and incomplete gamma functions of the general form: , [where 

a 1  iC,)  i(BD
  B)/ D ], and is quite cumbersome. Because a very accurate alternative simplified 
23

deflection equation could be introduced (see the following section) no attempt was made to program and use 

the cumbersome Lagrangianm deflection equation (49). 

Simplified Deflection Equation

If is replaced with , the simplified Eulerian deflection equation (40) can be converted into the 

following simplified Lagrangian deflection equation: 

1 1 (tan tan ) 2    



i  i i i y

l

y   ; (i 1,2,3,....,n) (50)

in which 1 {tani , tan } i are to be calculated from the slope equation (45). 

Like the simplified Eulerian deflection equation (40), the simplified Lagrangian deflection equation (50) 

can also calculate extremely accurate deflections if accurate input slope data were used. Therefore, the need 

to program and use the extremely cumbersome equation (49) is avoided.

STRUCTURE USED FOR SHAPE PREDICTION ACCURACY STUDIES

For the shape-prediction accuracy studies of the newly formulated slope and deflection equations, a long 

aluminum tapered cantilever tubular beam (fig. 5) was used. This structure has geometries listed in table 1.

As shown in figure 5, the strain-sensing stations [indicated with i 	 (i 1,2,3,....,n)] are equally spaced 

along the bottom strain-sensing line, with strain-sensing stations, , , respectively located at the fixed 

end (x  0) and free end (x  l) . An upward point load of P = 300 lb (or P = 600 lb) was applied at the 

beam free end (fig. 5). 

Finite-Element Analysis

For the shape prediction accuracy studies, reference slopes and deflections are needed as yardsticks. 

Therefore, finite-element analyses were first performed using SPAR (Structural Performance And Resizing) 

finite-element computer program (ref. 18) to generate the reference slopes and deflections. The SPAR 

program was developed by National Aeronautics and Space Administration Langley Research Center 

(Hampton, Virginia) in 1978 for preflight and subsequent flights (STS-1~STS-5) reentry heat transfer 

analyses of the Space Shuttle Orbiter Columbia until its loss on February 1, 2003 during STS-107 reentry 

flight (12,500 miles per hour) at 203,000 feet altitude above North Central Texas. 

The structural part of SPAR can handle only linear analysis. The SPAR model was used to analytically 

generate surface strains for input to the Displacement Transfer Functions to calculate slopes and deflections 

for comparison with SPAR-calculated reference slopes and deflections for shape prediction accuracy 

analysis. Since aircraft wings are practically operating within linear range, the current technical publication 

is for linear analysis only. Structural buckling, collapsing failures, and geometrical nonlinearity were not 

considered. Keep in mind that small strains can produce large deflections for long span wings such as 

Global Observer (see: “Experimental Validations of Shape-Prediction Accuracies” section).

Figure 6 shows the undeformed and deformed shapes of the SPAR model generated for the tapered 

cantilever tubular beam subjected to beam-tip load of P  300 lb (or P = 600 lb). In the figure, the size of 

the SPAR model is also indicated. Note that, for the loading case of P  300 lb, the beam-tip deflection 

reached 	


-tip slope reached as large as n  deg. For the doubled loading case 
24

of P = 600 lb, the beam-tip deflection was doubled to yn 
 283.2 in (because of linear elasticity), however, 

the beam-tip slope was not doubled, but only increased by 1.375 times to n  deg (= 1.375 48 deg).

Surface Strains

For the present technical publication, the surface strains were analytically calculated by converting the 

nodal stresses or element stresses of the SPAR outputs into strains through stress-strain law. Because of 

finite-element modeling, the strain-sensing stations shown in figure 5 can be coincidental with the SPAR 

model nodal points or lying between the two adjacent nodal points along the bottom generatrix. 

Figure 7 shows the plots of SPAR-generated surface strains for the two loading cases P = {300, 600} lb. 

Note that the strains for the two loading cases increased practically linearly from the fixed ends and reached 

their respective peaks in the outboard region, then decreased rapidly toward the beam tip (because of 

decreasing depth factor), and finally became nearly zero at the beam tip. Theoretically, the strain at the 

beam tip (free end) should be zero. However, because of the finite-element discretization, the 

SPAR-generated beam-tip strain for each loading case showed a slightly non-zero value, which is 

proportional to the applied load, P, due to linear elasticity. Keep in mind that for the uniform cantilever 

beam, the strain curve is a straight line with maximum strain at the fixed end and zero strain at the free end 

(refs. 1–5).

The SPAR-generated strain data of figure 7 were then input to the newly developed displacement 

transfer functions for the shape predictions. The results were then compared with the corresponding SPAR 

reference values for the shape prediction accuracy analysis. 

COMPARISONS OF SHAPE PREDICTION ACCURACIES

In the comparisons of the shape prediction accuracies, the Shifted slope and deflection equations

[eqs. (15)–(21)]; the Eulerian slope and deflection equations [eqs. (29)–(40)]; and the Lagrangian slope and 

deflection equations [eqs. (45)–(50)] were used. The slope and deflection prediction errors are respectively 

defined as the percent differences (deviations) from the SPAR-generated slope and deflection data, which 

were used as reference yardsticks.

Slope Predictions

Tables 2 and 3 respectively compare the slopes predicted from Shifted slope equations (15a)–(21a), the 

Eulerian slope equation (29), Lagrangian slope equation (45) for the loading cases of P = 300 lb and 

P = 600 lb. 
25

Table 2. Comparisons of slopes calculated from SPAR and from different slope equations; long tapered tube 

( = 300 in, 0 c = 4 in, n c = 1 in) subjected to tip load of P = 300 lb; n = 16; l = 18.75 in. 

Slope, deg 0 2 4 6 8 10 12 14 16

(% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.)

SPAR 

(reference)

0.0000 4.3526 9.4255 15.299 21.9903 29.3683 37.0656 44.2216 48.2679

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)

Based on Shifted curvature equation (11)

Nonuniform

eq. (15a)

0.0000 4.3461 9.4213 15.2919 21.9831 29.3545 37.0367 44.1651 48.0961

(0.0000) (0.1493) (0.0466) (0.0523) (0.0327) (0.0470) (0.0780) (0.1278) (0.3559)

Slightly 

tapered

eq. (16a)

0.0000 4.3405 9.4084 15.2690 21.9481 29.3059 36.9773 44.1141 48.1723

(0.0000) (0.2780) (0.1814) (0.2020) (0.1919) (0.2125) (0.2382) (0.2431) (0.1981)

1st order eq. 

(17a)

0.0000 4.3426 9.4131 15.2771 21.9594 29.3191 36.9864 44.0967 48.0164

(0.0000) (0.2297) (0.1316) (0.1433) (0.1405) (0.1675) (0.2137) (0.2824) (0.5211)

2nd order 

eq. (18a)

0.0000 4.3459 9.4210 15.2912 21.9820 29.3526 37.0337 44.1603 48.0899

(0.0000) (0.1539) (0.0477) (0.0569) (0.0377) (0.0535) (0.0861) (0.1386) (0.3688)

Improved

eq. (19a)

0.0000 4.3447 9.4201 15.2903 21.9837 29.3607 37.0588 44.2497 48.2910

(0.0000) (0.1815) (0.0573) (0.0627) (0.0300) (0.0259) (0.0183) (0.0635) (0.0479)

Logexpanded 

eq. (20a)

0.0000 4.3444 9.4196 15.2892 21.9822 29.3590 37.0582 44.2597 48.3375

(0.0000) (0.1884) (0.0626) (0.0699) (0.0368) (0.0317) (0.0200) (0.0862) (0.1442)

Depthexpanded 

eq. (21a)

0.0000 4.3446 9.4198 15.2897 21.9826 29.3589 37.0560 44.2461 48.2888

(0.0000) (0.1838) (0.0605) (0.0667) (0.0350) (0.0320) (0.0259) (0.0544) (0.0433)*

Based on Eulerian curvature equation (3)

Eulerian 

nonuniform

eq. (29)

0.0000 4.3544 9.5072 15.6656 23.1294 32.2247 43.2335 55.6503 63.8446

(0.0000) (0.0414) (0.8668) (2.3902) (5.1800) (9.7261) (16.6405) (25.8442) (32.2713)

Lagrangian

nonuniform

eq. (45)

0.0000 4.3419 9.3789 15.1108 21.4453 28.0683 34.4098 39.5432 41.9122

(0.0000) (0.2458) (0.4944) (1.2360) (2.4784) (4.4265) (7.1651) (10.5794) (13.1679)^
26

Table 3. Comparisons of slopes calculated from SPAR and from different slope equations; long tapered tube

( = 300 in, 0 c = 4 in, n c = 1 in) subjected to tip load of P = 600 lb; n = 16; l = 18.75 in.

Slope, deg (% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.) (% diff.)

SPAR

(reference)

0.0000

(0.0000)

8.6553

(0.0000)

18.3669

(0.0000)

28.6842

(0.0000)

38.9263

(0.0000)

48.3787

(0.0000)

56.4976

(0.0000)

62.8071

(0.0000)

65.9637

(0.0000)

Based on Shifted curvature equation (11)

Nonuniform

eq. (15a)

0.0000

(0.0000)

8.6426

(0.1467)

18.3691

(0.0120)

28.6713

(0.0450)

38.9162

(0.0259)

48.3626

(0.0333)

56.4700

(0.0489)

62.7611

(0.0732)

65.8349

(0.1953)

Slightly

tapered

eq. (16a)

0.0000

(0.0000)

8.6319

(0.2704)

18.3351

(0.1731)

28.6335

(0.1768)

38.8669

(0.1526)

48.3061

(0.1501)

56.4131

(0.1496)

62.7196

(0.1393)

65.8921

(0.1085)

1st order

eq. (17a)

0.0000

(0.0000)

8.6360

(0.2230

18.3439

(0.1252)

28.6468

(0.1304)

38.8829

(0.1115)

48.3215

(0.1182)

56.4218

(0.1342)

62.7054

(0.1619)

65.7749

(0.2862)

2nd order

eq. (18a)

0.0000

(0.0000)

8.6424

(0.1490)

18.3585

(0.0457)

28.6703

(0.0485)

38.9147

(0.0298)

48.3605

(0.0376)

56.4672

(0.0538)

62.7572

(0.0794)

65.8303

(0.2022)

Improved

eq. (19a)

0.0000

(0.0000)

8.6400

(0.1768)

18.3569

(0.0544)

28.6686

(0.0544)

38.9171

(0.0236)

48.3698

(0.0184)

56.4911

(0.0115)

62.8300

(0.0365)

65.9811

(0.0264)

Logexpanded

eq. (20a)

0.0000

(0.0000)

8.6394

(0.1837)

18.3559

(0.0599)

28.6669

(0.0603)

38.9150

(0.0290)

48.3678

(0.0225)

56.4907

(0.0122)

62.8381

(0.0494)

66.0159

(0.0791)

Depthexpanded

eq. (21a)

0.0000

(0.0000)

8.6397

(0.1802)

18.3563

(0.0577)

28.6676

(0.0579)

38.9156

(0.0275)

48.3678

(0.0225)

56.4885

(0.0161)

62.8271

(0.0318)

65.9795

(0.0240)*

Based on Eulerian curvature equation (3)

Eulerian

nonuniform

eq. (29)

0.0000

(0.0000)

8.7088

(0.6181)

19.0144

(3.5254)

31.3314

(9.2288)

46.2593

(18.8382)

64.4514

(33.2227)

86.4839

(53.0754)

86.2696

(37.3564)

88.1214

(33.5907)

Based on Lagrangian curvature equation (7)

Lagrangian

nonuniform

eq. (45)

0.0000

(0.0000)

8.6099

(0.2461)

18.0458

(1.7483)

27.4737

(4.2201)

35.8469

(7.9108)

42.0569

(13.0673)

44.9455

(20.4471)

44.9390

(28.4492)

44.9879

(31.7990)^

 |<---------Acceptable range----------->|--------------------------------Poor range-------------------------------------------->|

* Most accurate at beam tip ^ Worst case

Note from tables 2 and 3 that the slope data predicted from the Shifted slope equations (15a) – (20a) are 

amazingly accurate beyond expectation. For the loading case of P = 300 lb (table 2), the beam-tip slope 

prediction errors are in the negligible range of 0.0433 to 0.5211 percent, and for the loading case of 

P = 600 lb (table 3), the beam-tip slope prediction errors decreased to the range of 0.0240 to 0.2862 percent

(a nearly 45 percent reduction). The Depth-expanded slope equation (21a) produced the most accurate 

beam-tip slope data with infinitesimal prediction errors of only {0.0433, 0.0240} percent respectively for 

the loading cases of P = 300 lb (table 2) and P = 600 lb (table 3). 

In tables 2 and 3, the slopes calculated from the Eulerian slope equation (29) are referred to the 

deformed x-coordinate and are therefore somewhat larger than the corresponding SPAR slopes, which are 

referred to the un-deformed x-coordinate [that is,  (s) *  (x) , (fig. 2)]. The numbers listed in the 

parentheses under the slope data based on Eulerian formulation do not represent prediction errors, but are 

the percent differences because two different x-systems were used. When the load was doubled from 

P = 300 lb to P = 600 lb, the percent slope differences at the beam-tip based on Eulerian formulation 

increased from 32.2713 to 33.5907 percent (a nearly 4 percent increase). 
27

The Lagrangian slope equation (45) predicted acceptable slope data in the inboard region, but the 

predicted slopes became lower than the corresponding SPAR reference slopes in the outboard region. When 

the load was doubled from P = 300 lb to P = 600 lb, the percent slope differences (errors) based on 

Lagrangian formulation increased from 13.1676 to 31.7990 percent (a nearly 141 percent increase) at the 

beam tip.

Slope Curves

The slope data listed in tables 2 and 3 are plotted in figures 8 and 9 respectively for the loading cases of 

P = {300, 600} lb for visual comparisons of the slope curves based on different formulations. As shown in 

figures 8 and 9, the slope curves based on the Shifted formulation [eqs. (15a) –(21a)] almost formed a single 

curve, which practically fell on top of the corresponding SPAR reference slope curve for the two loading 

cases P = {300, 600} lb, showing the high degrees of accuracies of the Shifted slope equations (15a)–(21a).

In figure 8 (loading case, P = 300 lb), the slope curve of Eulerian formulation [eq. (29)] practically lies 

on top of the SPAR reference slope curve only in the inboard region, but branched out upward away from 

the SPAR reference slope curve in the outboard region because SPAR slopes are based on the undeformed 

x-system. Keep in mind that the Eulerian slopes [ ] are always larger than the corresponding Lagrangian 

slopes [ (x) ] [that is,  (s) *  (x) , (fig. 2)].

The slope curve of Lagrangian formulation [eq. (45)] is quite close to the SPAR reference slope curve 

only in the inboard region, but branched out downward slightly away from the SPAR reference slope curve 

in the outboard region (fig. 8).

In figure 9 (loading case, P = 600 lb), the slope curve based on the Eulerian formulation [eq. (29)], 

except for a small inboard region, continued to deviate upward further away from the SPAR reference slope 

curve in the outboard region. Keep in mind that the Eulerian slopes [ ] are always larger than the 

corresponding Lagrangian slopes [ (x) ] [that is,  (s) *  (x) , (fig. 2)].The peculiar wavy behavior of the 

Eulerian slope curve near the beam tip region could be attributed to the mathematical functional behaviors 

of equations (29) and (24) at large slopes approaching 90 deg (table 3) as a result of the magnified effect of 

the term 2 (dy / dx) .

At increasing load, P = 600 lb (fig. 9), the slope curve based on the Lagrangian formulation [eq. (45)], 

except for the small inboard region, deviated downward further away from the SPAR reference slope curve 

in the outboard region, causing the slope prediction errors to be greatly magnified. The rather poor slope 

predictions at increasing load (increasing deflections) could be attributed to the intensified effect of the term

2 (dy / dx) in the Lagrangian curvature equation (7).

Deflection Predictions

Tables 4 and 5 respectively compare the deflections calculated from the Shifted deflection equations 

(15b, c)–(21b, c), from the Eulerian deflection equation (40), and from Lagrangian deflection equation (50), 

for the loading cases of P = 300 lb and P = 600 lb. 
28

Table 4. Comparisons of deflections calculated from SPAR and from different deflection equations; long 

tapered tube ( = 300 in, 0 c = 4 in, n c = 1 in) subjected to tip load of P = 300 lb; n = 16; l = 18.75 in.

Deflection, 

in.

y0

(% diff.)

y2

(% diff.)

y4

(% diff.)

y6

(% diff.)

y8

(% diff.)

y10

(% diff.)

y12

(% diff.)

y14

(% diff.)

y16

(% diff.)

SPAR

(reference)

0.0000

(0.0000)

1.4029

(0.0000)

5.8910

(0.0000)

14.0737

(0.0000)

26.6907

(0.0000)

44.7190

(0.0000)

69.3198

(0.0000)

101.7093

(0.0000)

141.6016

(0.0000)

Based on Shifted curvature equation (11)

Non-uniform

eq. (15b, c)

0.0000

(0.0000)

1.3871

(1.1262)

5.8746

(0.2784)

14.0501

(0.1667)

26.6660

(0.0925)

44.6814

(0.0841)

69.2690

(0.0733)

101.6065

(0.1011)

141.3056

(0.2090)

Slightly

tapered

eq. (16b, c)

0.0000

(0.0000)

1.3859

(1.2118)

5.8679

(0.3921)

14.0321

(0.2956)

26.6282

(0.2342)

44.6117

(0.2399)

69.1504

(0.2444)

101.4226

(0.2819)

141.1004 

(0.3540)

1st order

eq. (17b, c)

0.0000

(0.0000)

1.3863

(1.1833)

5.8704

(0.3497)

14.0387

(0.2487)

26.6416

(0.1840)

44.6348

(0.1883)

69.1847

(0.1949)

101.4585

(0.2466)

141.0562 

(0.3852)

2nd order

eq. (18b, c)

0.0000

(0.0000)

1.3871

(1.1262)

5.8744

(0.2818)

14.0496

(0.1712)

26.6650

(0.0963)

44.6794

(0.0886)

69.2649

(0.0792)

101.5985

(0.1089)

141.2901

(0.2200)

Improved

eq. (19b, c)

0.0000

(0.0000)

1.3865

(1.1690)

5.8732

(0.3022)

14.0476

(0.1855)

26.6632

(0.1030)

44.6810

(0.0850)

69.2812

(0.0557)

101.6760

(0.0327)

141.5683

(0.0235)*

Logexpanded

eq. (20b, c)

0.0000

(0.0000)

1.3865

(1.1690)

5.8731

(0.3039)

14.0471

(0.1890)

26.6620

(0.1075)

44.6787

(0.0901)

69.2772

(0.0615)

101.6684

(0.0402)

141.5643

(0.0263)

Depthexpanded

eq. (21b, c)

0.0000

(0.0000)

1.3865

(1.1690)

5.8731

(0.3039)

14.0472

(0.1883)

26.6622

(0.1068)

44.6789

(0.0897)

69.2773

(0.0613)

101.6686

(0.0400)

141.5561

(0.0321)

Based on Eulerian curvature equation (3)

Eulerian

non-uniform

eq. (40)

0.0000

(0.0000)

1.3965

(0.4562)

5.8862

(0.0815)

14.0146

(0.4199)

26.4191

(1.0176)

43.7890

(2.0797)

66.7253

(3.7428)

95.3525

(6.2500)

128.0932

(9.5397)

Based on Lagrangian curvature equation (7)

Lagrangian

non-uniform

eq. (50)

0.0000

(0.0000)

1.3958

(0.5061)

5.8823

(0.1477)

14.0025

(0.5059)

26.3874

(1.1364)

43.7123

(2.2512)

66.5437

(4.0048)

94.9396

(6.6559)

127.5099

(9.9517)^

 |<----------Acceptable range --------->|<--------------------------------Poor range-------------------------------------------->|

* Most accurate at beam tip ^ Worst case
29

Table 5. Comparisons of deflections calculated from SPAR and from different deflection equations; long 

tapered tube ( = 300 in, 0 c = 4 in, n c = 1 in) subjected to tip load of P = 600 lb; n l = 18.75 in.

Deflection, 

in.

y0

(% diff.)

y2

(% diff.)

y4

(% diff.)

y6

(% diff.)

y8

(% diff.)

y10

(% diff.)

y12

(% diff.)

y14

(% diff.)

y16

(% diff.)

SPAR

(reference)

0.0000

(0.0000)

2.8058

(0.0000)

11.7821

(0.0000)

28.1474

(0.0000)

53.3814

(0.0000)

89.4380

(0.0000)

138.6400

(0.0000)

203.4154

(0.0000)

283.2032

(0.0000)

Based on Lagrangian curvature equation (11)

Log-Nonuniform

eq. (15b, c)

0.0000

(0.0000)

2.7742

(1.1262)

11.7492

(0.2792)

28.1001

(0.1680)

53.3319

(0.0927)

89.3628

(0.0841)

138.5379

(0.0736)

203.2131(

0.0995)

282.6112

(0.2090)

Slightly

tapered

eq. (16b, c)

0.0000

(0.0000)

2.7717

(1.2153)

11.7359

(0.3921)

28.0643

(0.2952)

53.2565

(0.2340)

89.2234

(0.2399)

138.3008

(0.2447)

202.8453

(0.2803)

282.2007

(0.3540)

1st order

Eq. (17b, c)

0.0000

(0.0000)

2.7726

(1.1833)

11.7409

(0.3497)

28.0774

(0.2487)

53.2832

(0.1840)

89.2696

(0.1883)

138.3694

(0.1951)

202.9170

(0.2450)

282.1124

(0.3852)

2nd order

eq. (18b, c)

0.0000

(0.0000)

2.7742

(1.1262)

11.7489

(0.2818)

28.0993

(0.1709)

53.3300

(0.0963)

89.3587

(0.0887)

138.5299

(0.0794)

203.1971

(0.1073)

282.5803

(0.2199)

Improved

eq. (19b, c)

0.0000

(0.0000)

2.7731

(1.1654)

11.7465

(0.3022)

28.0952

(0.1855)

53.3264

(0.1030)

89.3619

(0.0851)

138.5624

(0.0560)

203.3521

(0.0311)

283.1367

(0.0235)*

Logexpanded

eq. (20b, c)

0.0000

(0.0000)

2.7731

(1.1654)

11.7462

(0.3047)

28.0943

(0.1886)

53.3240(

0.1075)

89.3573

(0.0902)

138.5543

(0.0617)

203.3368(

0.0386)

283.1285

(0.0264)

Depthexpanded

eq. (21b, c)

0.0000

(0.0000)

2.7730

(1.1690)

11.7462

(0.3047)

28.0944

(0.1883)

53.3244

(0.1068)

89.3579

(0.0896)

138.5545

(0.0617)

203.3372

(0.0384)

283.1123

(0.0321)

Based on Eulerian curvature equation (3)

Eulerian 

non-uniform

eq. (40)

0.0000

(0.0000)

2.79213

(0.4887)

11.7129

(0.5866)

27.6060

(1.9232)

51.1125

(4.2505)

82.4108

(7.8571)

131.4861

5.1598)

130.7083

(36.7442)

151.6729

(46.4438)^

Based on Lagrangian curvature equation (7)

Lagrangian

non-uniform

eq. (50)

0.0000

(0.0000)

2.7865

(0.6879)

11.6815

(0.8538)

27.5015

(2.2947)

50.8036

(4.8290)

81.3751

(9.0151)

117.2992

(15.3930)

154.5417

(24.0265)

191.8715

(32.2495)

 |<----------Acceptable range---------->|<----------------------------------Poor range------------------------------------------>|

* Most accurate at beam tip ^Worst case

Note from tables 4 and 5 that the deflections calculated from all the Shifted deflection equations 

(15b, c)–(21b, c) are amazingly accurate beyond expectations with negligible beam-tip prediction errors in 

the range of 0.0235 to 0.3852 percent for both loading cases, P = {300, 600} lb. Note also from tables 4 and 

5 that under linear analysis, the predicted deflections and associated predicted deflection errors (in inch) are

linear functions of the applied load. Because of linearity, the deflection prediction errors (in percentage) are 

invariant to the loading magnitude. The improved deflection equations (19b, c) turned out to be the most 

accurate deflection equation, which produced only infinitesimal amount of prediction error of 

0.0235 percent at the beam-tip for both loading cases, P = {300, 600} lb. 

The Eulerian deflection equation (40) predicted acceptable deflection data in the inboard region, but the 

prediction errors continued to magnify toward the beam tip. Because the slope and deflections are not 

directly proportional to the surface strains in the Eulerian formulation [eqs. (34) and (40)], when the applied 

load, P = 300 lb, was doubled to P = 600 lb (surface strains doubled), the deflection prediction errors were 

greatly magnified from 9.5397 to 46.4438 percent (a nearly 387 percent increase) at the beam tip.

The Lagrangian deflection equation (50) also predicted acceptable deflection data in the inboard region, 

beyond which the prediction errors continue to magnify toward the beam tip. Because the slope and 
30

deflections are not directly proportional to the surface strains in the Lagrangian formulation [eqs. (45) and

(50)], when the applied load, P = 300 lb, was doubled to P = 600 lb, the deflection prediction errors 

increased nonlinearly from 9.9517 to 32.2495 percent (a nearly 224 percent increase) at the beam tip.

Deflection Curves

The deflection data listed in tables 4 and 5 are plotted in figures 10 and 11 respectively for the loading 

cases of P = {300, 600} lb for visual comparisons of the deflection curves based on different formulations. 

As shown in figure 10 (loading case of P =300 lb), all the deflection curves based on the Shifted 

formulation [eqs. (15b, c)–(21b, c)] graphically fell on top of the associated SPAR reference deflection 

curves, showing exceptional accuracy of the Shifted deflection equations (15b, c)–(21b, c).

Note from figure 10 that for loading case of P = 300 lb, the deflection curves based on the Eulerian 

formulation [eq. (40)] and Lagrangian formulation [eq. (50)] are practically coincidental, and agreed with 

the SPAR reference deflection curve only in the inboard region, but deviated slightly downward from the 

SPAR reference deflection curve in the outboard region (fig. 10).

Figure 11 shows that at double load, P = 600 lb, the deflection curves based on the Shifted formulation 

[eqs. (15-b, c)–(21-b, c)] are again extremely close to the SPAR-reference deflection curve, showing 

amazing high degrees of prediction accuracies (tables 4, 5). The deflection curve of the Eulerian formulation 

agreed with the SPAR reference deflection curve only in the inboard region and deviated downward from 

the SPAR reference deflection curve in the outboard region, and became wavy near the beam tip, showing 

poor prediction performances. Again, the peculiar wavy behavior of the Eulerian deflection curve near the 

beam tip region is associated with the wavy behavior of the slope curve shown in figure 9, and could be 

attributed to the mathematical functional behaviors of equations (40) and (24) at large slopes approaching 

90 deg (table 3).

As shown in figure 11, at doubled load P = 600 lb, the deflection curves based on the Lagrangian

formulation [eq. (50)] fell on top of the SPAR reference deflection curve only in a small inboard region, and 

then branched out downward further away from the SPAR reference deflection curve toward the beam 

outboard region, giving discouraging shape predictions. 

SHAPE PREDICTION ERROR CURVES

The slope and deflection predictions errors (percent differences from associated SPAR data) listed in the 

parentheses in tables 2–5 are plotted in the following sections for visual comparisons.

Slope-Error Curves

Figure 12 shows the slope-error curves for P = 300 lb loading case associated with different 

Displacement Transfer Functions. The slope differential curve based on Eulerian formulation shown in 

figure 12 is not the slope error curve, but the curve showing the difference in slopes referred to the deformed 

and un-deformed x-coordinate. The slope differential for the Eulerian formulation steadily increased 

span-wisely and reached as large as 32.2713 percent at the beam tip (table 2).

The slope-prediction errors based on the Lagrangian formulation continued to increase almost linearly 

span-wisely and reached a peak value of 13.1676 percent at the beam tip (table 2). 
31

The slope-error curves (fig. 12) based on the Shifted formulation practically formed a single curve along 

the 0 percent-error line and was indistinguishable; therefore, those slope error curves had to be re-plotted 

independently with a different scale.

Figure 13 shows the slope-error curves for the P = 300 lb loading case, calculated from seven Shifted 

slope equations (15a)–(21a). For the present example structure, the Depth-expanded case [eq. (21a)] 

produced the lowest slope-prediction errors near the beam tip region, with a slope prediction error of merely 

0.0433 percent at the beam tip (table 2).

Figure 14 shows the slope-error curves for the P = 600 lb loading case associated with nine different 

displacement transfer functions. The lofty curve of the Eulerian formulation shown in figure 14 is not the 

slope-error curve, but a slope-differential curve for showing the difference in slopes based on the deformed 

x-coordinate (Eulerian) and un-deformed x-coordinate (SPAR). The peak slope differential of 

53.0754 percent occurred at the strain-sensing station i =12 (table 3). 

The slope-error curve of the Lagrangian formulation (fig. 12) is slightly s-shaped and tilted away from 

the 0 percent-error axis with peak error of 31.7990 percent at the beam tip (table 3). The seven slope-error 

curves based on the Shifted formulation (fig. 14) are crowded together near the 0 percent-error axis and 

practically indistinguishable. Therefore, for better visual comparisons, those seven slope-error curves based 

on the seven Shifted formulations had to be re-plotted with a different scale. 

Figure 15 shows the slope-error curves for the P = 600 lb loading case calculated from the seven Shifted 

slope equations (15a)–(21a). For the present example structure, the Depth-expanded case [eq. (21a)] 

produced the lowest slope-error of 0.0240 percent at the beam tip ( table 3). Because the beam slopes are not 

directly proportional to the applied load, increasing the load from P = 300 lb to P = 600 lb caused the 

overall beam tip slope errors to come down by nearly 50 percent (compare figures 13 and 15).

Deflection-Error Curves

Figure 16 shows the deflection-error curves for the P = 300 lb loading case associated with nine

different displacement transfer functions. Note that both the Eulerian and Lagrangian formulations produced 

similar concave upward deflection-error curves lying far above the 0 percent-error axis, with respective 

deflection prediction errors as large as {9.5397, 9.9517}percent at the beam tip (table 4). Because the 

deflection-error curves based on the Shifted formulation are so close to the 0 percent-error axis, and are 

almost indistinguishable in figure 16, those deflection-error curves were re-plotted with a different scale for 

better visual comparisons.

Figure 17 shows the deflection-error curves for the P = 300 lb loading case, calculated from the seven 

sets of the Shifted deflection equations. (15b, c)–(21b, c)]. For the present example structure, the Improved 

case [eq. (19b, c)] produced the lowest deflection-prediction errors in the outboard region, with negligible 

deflection-prediction error of 0.0235 percent at the beam tip (table 4). The increasing deflection errors at 

strain-sensing station, i = 2, is due to a small predicted deflection error divided by a small SPAR reference 

deflection.

Figure 18 shows the deflection-error curves for the P = 600 lb loading case associated with nine

different displacement transfer functions. The deflection errors based on the Shifted formulation are very 

small as compared to those of both the Eulerian and Lagrangian formulations. The dent in the Eulerian 

deflection-error curve (fig. 18) is associated with the hump in the Eulerian slope-error curve (fig. 12). 

Because the deflection-error curves based on the Shifted formulation are so close and are almost 

indistinguishable in figure 18, those deflection-error curves were re-plotted using an enlarged scale for 

better visual comparisons. 
32

Figure 19 shows the deflection-error curves for the P = 600 lb loading case calculated from the seven 

sets of Shifted deflection equations (15b, c)–(21b, c). Because the beam deflections are directly proportional 

to the strains (or applied load), by increasing the load from P = 300 lb to P = 600 lb, the overall deflection

errors remain practically the same (tables 4 and 5). Again, the increasing deflection errors at strain-sensing 

station, i  2 , is due to a small deflection error divided by a small SPAR reference deflection. For the 

present example structure, the Improved case [eq. (19b, c)] produced the lowest deflection errors in the 

outboard region, with a negligible deflection prediction error of 0.0235 percent at the beam tip (same as the 

P = 300 lb case) (that is, figs. 17 and 19 are practically identical).

EXPERIMENTAL VALIDATIONS OF SHAPE-PREDICTION 

ACCURACIES

In the process of writing the present technical publication, large-scale ground loads tests were conducted 

on a long span (high aspect ratio) full-scale tapered wing (175-ft wingspan) at the NASA Dryden Flight

Loads Laboratory. These ground tests were used to validate the shape prediction accuracies of the 

Displacement Transfer Functions. The results were reported in a separate NASA technical publication 

authored by Jutte, Ko, et al (ref. 19). 

In the ground tests, the whole wing was loaded from 0 percent up to 100 percent design limit load 

(DLL) at which the wing-tip slope exceeded 20 deg. Surfaces strains were measured by using four 

fiber-optic strain-sensing lines installed on the wing surfaces (two on the wing upper surface, two on the 

wing lower surface). Wing deflections were measured by means of a photogrammetry system, and were 

used as yardsticks to evaluate the accuracies of the predicted wing deflections. Using the measured surface 

strains as inputs, theoretical wing deflections were calculated from the recursive Displacement Transfer 

Functions [eqs. (16a, b)], which were formulated for slightly tapered beams (based on piecewise-linear 

strain representations) such as the current slightly tapered wing structure. 

For completeness of the present technical report, wing deflection plots in figures 11, and 12 of reference 

19 were duplicated respectively as figures 20, and 21 in the present technical report. Figures 20, and 21

respectively show the plots of the predicted and experimentally measured wing deflections along the 

number 3 (FWD) and number 4 (AFT) strain-sensing lines. Amazingly, the measured data points practically 

fell on top of the associated predicted deflection curves for the whole range of loading levels 

[(0~100) percent DLL]. The wing-tip prediction errors associated with the number 3 and number 4

strain-sensing lines are respectively in the negligible ranges of (0.09~1.72) percent and (0.25~1.45) percent

at 100 percent DLL depending on the sensor-to-sensor distances used. For details of the experimental 

shape-prediction studies, please see reference 19.

Note from figures 20 and 21 that the maximum wing-tip deflections of the Global Observer reached 

approximately 160 in. Hence, the ratio of (maximum wing-tip deflection) to (half wingspan) can be 

calculated as:

0.1524 = 15.24% (51)

The 15.24 percent normalized wing-tip deflection from equation (51) indicates that the ground test of 

the Global Observer was operating in a borderline region between the linear and nonlinear regimes.

Keep in mind that each Shifted Displacement Transfer Function was formulated by geometrically 

relating the surface strains to curvatures, and then to deflections. Therefore, one could expect that the 

Shifted Displacement Transfer Function can faithfully compute the right deflections whether the input 
33

surface strains come from linear or nonlinear analysis. To find out whether a good correlation can be 

obtained with realistic truly large deformation cases, nonlinear analysis will be carried out separately, and 

the results will be published in a subsequent NASA publication.

The Global Observer ground test results (ref. 19) thus validated the mathematical accuracies of the 

Shifted Displacement Transfer Functions, and established confidence in using the Shifted 

Displacement Transfer Functions for large-deformation shape predictions.

DISCUSSIONS

The vitally important lesson learned in the present technical publication is that for large deformations, 

one cannot just naively integrate the classical Eulerian and Lagrangian curvature equations {(12), (13)} 

(written in the x-y system) without considering the true nature of large deformations. After spending long 

hours in the lengthy mathematical derivations because of keeping the nonlinear term (dy/dx)

2

, both Eulerian 

and Lagrangian formulations resulted in poor shape predictions at large deformations, and contrary to the

hopeful expectation, no better shape prediction accuracies could be achieved. 

In the piecewise formulations, the embedded beam is discretized into multiple small domains. Within 

each small domain, the deformations are relative to the inboard domain juncture, and can be considered 

small. Therefore, in the piecewise integrations of Eulerian or Lagrangian curvature equations {(12), (13)},

the nonlinear term (dy/dx)

2

, could be neglected to achieve high accuracy shape predictions like the Shifted

case.

For large bending of the beam (for example, fishing pole), the true deflection, y

 , is actually the curved 

distance traced by the same material point from its initial undeformed position to its final deformed position 

(fig. 3). Thus, the conventional vertical deflection, y, is merely the vertical component of the curved 

deflection, y

 . For small deformations, the curved deflection, y , can be approximated with the vertical 

deflection, . However, for large deformations, the difference between y

 and y becomes larger ( y > y).

For large deformations, one could write the curvature equation in the s- y

 system (instead of x-y system) as

equation (52):

; (52)

which contains no classical nonlinear term (dy/dx)

2 (see also Appendix B).

In view of the incremental length relationship, ds2 = dx2

+dy2 (fig. 1), the original Eulerian curvature 

equation (3) can be rewritten in the following revised forms for the in-extensional beam (ds = dx) as:

a. Revised Eulerian Curvature equation (changing deformed x-coordinate into undeformed x-coordinate, 

and then setting ds = dx):

(53)
34

Also, in view of the incremental length relationship, ds2 = (dx–du)

2 + dy2 (fig. 2), the original 

Lagrangian curvature equation (4) can be rewritten in the following revised forms for the in-extensional 

beam (ds = dx) as: 

b. Revised Lagrangian curvature equation (setting du = 0, ds = dx)

 (54)

Note that, for the in-extensional beams (ds = dx), both equations (53) and (54), written in undeformed

x-system (x = s), degenerated into the Shifted curvature equation (11), containing no nonlinear term

(dy / dx)

2

. Thus, the formulations based on the revised Eulerian and Lagrangian curvature equations 

{(53) or (54)}, become the Shifted formulation, and can provide accurate shape predictions. 

CONCLUDING REMARKS

Large-Deformation Displacement Transfer Functions were formulated through piecewise integrations of 

the Eulerian curvature-strain equation, the Lagrangian curvature-strain equation, and the Shifted 

curvature-strain equation. A long tapered cantilever tubular beam was chosen in the shape-prediction 

accuracy studies of the newly developed Displacement Transfer Functions. Surface strains for input to the 

Displacement Transfer Functions were analytically calculated from the finite-element analysis of the tapered

cantilever tubular beam. Also, the finite-element-generated slopes and deflections were used as reference 

yardsticks in the prediction accuracy studies. The results are itemized below: 

1. Changing from piecewise-linear strain representations to piecewise-nonlinear strain representations, 

the shape prediction accuracies could be improved considerably for the tapered cantilever tubular beam.

2. Based on the Shifted curvature equation, the resulting seven sets of the Shifted Displacement Transfer 

Functions were found to be amazingly accurate beyond expectation for large-deformation shape predictions, 

even for the case with a beam-tip slope reaching as large as 66 deg. The beam-tip shape prediction errors of 

the seven sets of the Shifted slope and deflection equations, are in the negligible ranges of 

{0.0240 – 0.5211} percent for the slopes, and {0.0235 – 0.3852} percent for the deflections.

3. For the Shifted formulation (based on linear analysis), the deflections are directly proportional to the 

surface strains, and therefore, for the same type of loading condition, the negligible deflection prediction 

errors are theoretically independent of the loading magnitude (that is, degree of deformation).

4. The amazing accuracy of the Shifted formulation could be attributed to: 1) shifting of the deformed 

material points back to their respective initial undeformed x-positions; and 2) piecewise integrations of the 

Shifted curvature equation, enabled by dicretization of the embedded beam. 

5. Based on the classical Eulerarian curvature equation, the resulting slope and deflection equations 

gave poor shape predictions at large deformations because of retaining the term (dy / dx)

2 in the discretized 

approach. The beam tip deflection prediction errors are in the poor ranges of {9.5397 – 46.4438} percent.
35

6. Based on Eulerian formulation, when the applied load was doubled (that is, doubling deflections), 

because the deflections are not directly proportional to the surface strains, the deflection prediction errors 

were greatly magnified from 9.5397 percent to 46.4438 percent at the beam-tip (a 386.8476 percent

increase). 

7. Based on the physical (Lagrangian) curvature equation, the resulting slope and deflection equations 

gave discouraging shape predictions at large deformations because of retaining the term (dy / dx)

2 in the 

discretized approach. The beam-tip prediction errors are in the poor range of {13.1676 – 31.7990} percent

for the slopes, and {9.9517 – 32.2495} percent for the deflections. 

8. Based on the Lagrangian formulation, when the applied load was doubled (that is, doubling 

deflections), because the slopes and deflections are not directly proportional to the surface strains, the 

beam-tip slope prediction errors were magnified from 13.1676 percent to 31.7990 percent (a 141.4943

percent increase), and the beam-tip deflection prediction errors increased from 9.9517 percent to 32.2485

percent (a 224.0602 percent increase). 

9. If the deformed material points of an in-extensional beam are shifted back to their respective initial 

undeformed x-locations, then the term (dy / dx)

2 in the Lagrangian curvature equation will become zero, 

implying linearization. The resulting linearlized Lagrangian formulation will become the Shifted 

formulation, and provides equally accurate shape predictions.

10. The linearized Eulerian curvature equation [the term (dy / dx)

2 neglected], even with similar 

mathematical form, cannot be equivalent to the Shifted curvature equation because, the former is referred to 

the deformed x-coordinate, while the latter to the undeformed x-coordinate. The two x-systems will become 

quite different at large deformations.

11. The simplified alternative deflection equations developed for the Eulerian and Lagrangian

formulations could not provide good shape predictions at large deformations because of the poor input slope 

data at large deformations.

12. Retaining the term (dy / dx)

2 in the piecewise integrations of curvature equations resulted in rather 

poor shape prediction performances of both the Eulerian and Lagrangian formulations. 

13. For in-extensional beams (ds = dx), by writing the original Eulerian and Lagrangian curvature

equations in s-system, the formulations based on the revised Eulerian and Lagrangian curvature equations

can degenerate into the Shifted formulation and, therefore, could provide accurate shape predictions. 

14. The Global Observer ground test results validated the mathematical accuracies of the Shifted 

Displacement Transfer Functions formulated, and giving confidence in using the Displacement Transfer 

Functions for large-deformation shape predictions.
36

FIGURES

Figure 1. Small segment of the deformed nonuniform embedded beam for geometrically relating local radius 

of curvature, R(s), to associated surface bending strain.

Figure 2. Small segment of the deformed-beam elastic curve used for deriving the physical (Lagrangian) 

curvature equation.
37

Figure 3. Small segment of the deformed-beam elastic curve used for deriving the Shifted curvature 

equation.


 	   ! "
#"#

 !  $&" 

 # Li, along with -axis in the 

Eulerian description.
38

Figure 5. Tapered cantilever tubular beam with strain-sensing stations evenly distributed along the bottom 

strain-sensing line.

Figure 6. Large deformation of the tapered cantilever tubular beam (SPAR finite-element model) subjected 

to a vertical tip load of P = 300 lb (or P = 600 lb); l = 300 in; c0 = 4 in; cn = 1 in. 
39

Figure 7. SPAR-generated surface-strain curves for the tapered cantilever tubular beam subjected to tip 

loads of P = {300,600} lb; l = 300 in; c0 = 1 in.

Figure 8. Slopes of the tapered cantilever tubular beam calculated from slope equations based on the 

Eulerian, Lagrangian, and Shifted formulations; l = 300 in; c0 = 4 in; cn = 1 in; P = 300 lb.
40

Figure 9. Slopes of the tapered cantilever tubular beam calculated from slope equations based on the

Eularian, Lagrangian, and Shifted formulations; l = 300 in; c0 = 4 in; cn = 1 in; P = 600 lb.

Figure 10. Deflections of the tapered cantilever tubular beam calculated from deflection equations based on 

the Eularian, Lagrangian, and Shifted formulations; l = 300 in; c0 = 4 in; cn = 1 in; P = 300 lb.
41

Figure 11. Deflections of the tapered cantilever tubular beam calculated from deflection equations based on 

the Eulerian, Lagrangian, and Shifted formulations; l = 300 in; c0 = 4 in; cn = 1 in; P = 600 lb.

Figure 12. Comparisons of i prediction errors of different displacement transfer functions; long tapered 

cantilever tubular beam; l = 300 in; c0 = 4 in; cn = 1 in; P = 300 lb at tip.
42

Figure 13. Comparisons of i prediction errors of Shifted displacement transfer functions; long tapered 

cantilever tubular beam; l = 300 in; c0 = 4 in; cn = 1 in; P = 300 lb at tip.

Figure 14. Comparisons of i prediction errors of different displacement transfer functions; long tapered 

cantilever tubular beam; l = 300 in; c0 = 4 in; cn = 1 in; P = 600 lb at tip.
43

Figure 15. Comparisons of i prediction errors of Shifted displacement transfer functions; long tapered 

cantilever tubular beam, l = 300 in; c0 = 4 in; cn = 1 in; P = 600 lb at tip.

Figure 16. Comparisons of yi prediction errors of different displacement transfer functions; long tapered 

cantilever tubular beam; l = 300 in; c0 = 4 in; cn = 1 in; P = 300 lb at tip.
44

Figure 17. Comparisons of yi predicition errors of Shifted displacement transfer functions; long tapered 

cantilever tubular beam; l = 300 in; c0 = 4 in; cn = 1 in; P = 300 lb at tip.

Figure 18. Comparisons of yi prediction errors of different displacement transfer functions; long tapered 

cantilever tubular beam; l = 300 in; c0 = 4 in; cn = 1 in; P = 600 lb at tip.
45

Figure 19. Comparisons of yi prediction errors of Shifted displacement transfer functions; long tapered 

cantilever tubular beam; l = 300 in; c0 = 4 in; cn = 1 in; P = 600 lb at tip.

Figure 20. Comparison of predicted and measured wing deflections along the number 3 strain-sensing line 

for different loading levels (duplication of figure 11, ref. 18). 
46

Figure 21. Comparison of predicted and measured wing deflections along the number 4 strain-sensing line 

for different loading levels (duplication of figure 12, ref.18). 
47

APPENDIX A

ALTERNATIVE DERIVATION OF THE PHYSICAL (LAGRANGIAN)

CURVATURE EQUATION

The alternative method used by Kopmaz, et al (ref. 12) and Hodges (ref. 13) in deriving the physical 

(Lagrangian) curvature equation (7) is described in the following paragraphs.

In view of figure 2, when the beam is bent, the undeformed material points, {A(x,0) , B(x  dx,0)} , will 

be moved to respective deformed positions, {A[(x  u), y] , B[(x  dx)  (u  du),( y  dy)]}, in which {u,

y} respectively denote the horizontal and vertical displacement of material point A (moving to point ). 

Then, in view of the curved triangle ABC , the infinitesimal curved length, ds( AB), along the elastic 

curve of the deformed beam can be expressed in terms of {dx, du, dy} as:

2 2 ds  (dx  du)  (dy) (A1)

Dividing through equation (A1) by dx , one obtains equation (A2):

2 2

1 



 



    



   dx

dy

dx

du

dx

ds (A2)

In view of curved triangle ABC (fig. 2), the slope, tan (s) , at point A can be expressed as equation 

(A3):

du dx

dy dx

dx du

dy s

1 /

/

tan ( )      (A3)

from which, one obtains equation (A4):





 





  

du dx

dy dx

s

1 /

/ ( ) tan 1  (A4)

Differentiating equation (A-4) with respect to x, one obtains equation (A5):

2

2

2 2 2 2

1 /

/ 1

(1 / )

( / )(1 / ) ( / )( / )

( )





 





 



 



du dx

dy dx

du dx

d y dx du dx dy dx d u dx

dx

d s

2 2

2

2

2

2

1

1





 



  













 

 



 



 



dx

dy

dy

du

dx

d u

dx

dy

dx

du

dx

d y

(A5)

The curvature, 1/ R , of the elastic curve of the deformed beam can be written as:

ds

dx

dx

d s

ds

d s

R

1  ( )  ( )   (A6)
48

Substitutions of equations (A2) and (A5) into equation (A6) yields equation (A7):

3 2 2 2

2

2

2

2

/

2 2

/

2 2

2

2

2

2

1

1

1

1

1

1

1















 



 



    



 

 



 



 







 



   

 



   

 



    



 

 



 



 



dx

dy

dx

du

dx

d u

dx

dy

dx

du

dx

d y

dx

dy

dx

du

dx

dy

dx

du

dx

d u

dx

dy

dx

du

dx

d y

R

dx ds

  

d dx

        

(A7)

For the in-extensional beam, the curved coordinate, s, of any deformed material point is always equal to 

its initial un-deformed x-coordinate (fig. 2). Namely, s  x and ds  dx , and equation (A2) can be written 

as:

2 2

1 1 



 



    



   dx

dy

dx

du (A8)

Rearranging equation (A8) as equation (A9):

2 0

2 2

  

 



    

 





dx

dy

dx

du

dx

du (A9)

Solving for du / dx one obtains:

2

1 1 



 



  +  dx

dy

dx

du (A10)

which can be rewritten as: 

2

1 1 



 



    dx

dy

dx

du  (A11)

Differentiating equation (A10), 2 2 du / dx can be expressed as:

2

2

2

2

2

1 



 



 



dx

dy

dx

d y

dx

dy

dx

d u  (A12)
49

Substitutions of equations (A11) and (A12) into the curvature equation (A7) yields equation (A13):

2 3 2 2 2

2

2 2 2

2

2

3 2

2

2

2

2

2

2

2

2

2

1 1

1

1

1

1

1





 



 















 



 



    



 





 



















 



 



 





















 





 



  



















 



 





 



  



















 



 



dx

dy

dx

dy

dx

dy

dx

d y

dx

dy

dx

dy

dx

d y

dx

dy

dx

dy

dx

dy

dx

d y

dx

dy

dx

dy

dx

dy

dx

d y

R

 



 

(A13)

After cancelling terms, and choosing the positive sign, one obtains equation (A14):

$ %2

2 2

1 /

1 /

dy dx

d y dx

R 

 (A14)

which is the physical (Lagrangian) curvature equation (7) for the in-extensional beam. 

Keep in mind that dy / dx in equation (A14) is referred to the undeformed x-system. From right triangle, 

(ADE) (fig. 2), the slope,  (x) , in the x-system can be expressed as equation (A15):





 



  

dx

dy x

1  ( ) tan (A15)

Comparing equations (A15) and (A4), one notes that  (x) in the undeformed x-system is slightly 

smaller than in the s-system [that is, ] (fig. 2).

In the bending deformations, the axial displacement, u, of a material point is very small. If the axial 

displacement, u, is neglected (that is, u 
 0 ) in equation (A4), then  (s) will become  (x) of equation 

(A15). The condition ( u 
 0 ) implies that the deformed material points {A , B} (fig. 3) are horizontally 

shifted back to points {A , B} at their respective initial undeformed x-locations. It is important to mention 

that u 
 0 will make dy / dx 
 0 according to equation (A8).
50

APPENDIX B

GRAPHICAL DERIVATION OF THE SHIFTED CURVATURE EQUATION

For the in-extensional beam, the curved coordinate, s, of any deformed material point is always equal to 

its initial un-deformed coordinate x (fig. 3). Namely, and , then the basic curvature equation 

can be written as equation (B1):

(B1)

In the Eulerian or Lagrangian curvature equation (3) or (7), the term 2 (dy / dx) appeared because, the 

slope,  (s) , of the beam elastic curve was expressed in term of either or sin ( / ) 1 dy ds  . It 

was found that term 2 (dy / dx) caused the formulated Displacement Transfer Functions to give rather poor 

shape predictions at large deformations. To avoid the appearance of the term 2 (dy / dx) , one can 

alternatively express the slope,  (s) , in term of the curved deflection, y

 , (fig. 3), which is the curved 

length AA actually traced by the same material point from its initial un-deformed position to its final 

deformed position . Thus, in view of the infinitesimal curved triangle, ABC [C will converge to 

BB -curve in the limit, d 
 0 , (fig. 3)], the slope,  (s) , can then be expressed as:

(B2)

By combining equations (B1) and (B2), one obtains the following curvature equation (B3) associated 

with the curved deflection, y

 , as:

(B3)

In the formulations of the Shifted displacement transfer functions, the deformed material points 

{ } A B (fig. 3) are shifted horizontally to points {A , B} with x-coordinates respectively matching the

original un-deformed x-coordinates of points {A, B}. The curve connecting the shifted deformed material 

points will be slightly detached from the true elastic curve of the actual deformed beam. Keep in mind that, 

in the bending deformations, the axial displacements are miniscule, and so also the amounts of horizontal 

shifting.

Graphically, the horizontal shifting (fig. 3) will cause the curved deflection, , to distort into a straight 

vertical deflection, y, and also causing the curved triangle, ABC , to distort into a right triangle, ABC .

Thus, the shifted-slope, ( )[ tan ( / )] 1

x dy dx    , in the x-system will be slightly smaller than the original 

slope, ( )[ sin ( / )] 1

s dy ds    in the s-system [that is,  (x) &  (s) , (fig. 3)]. After the horizontal shifting, and 

ignoring the resulting small perturbation of R(x), equation (B-3) can be rewritten as equation (B-4):

(B4)
51

which is similar to equation (11) in the main text, and is called the Shifted curvature equation for the 

in-extensional beam .
52

APPENDIX C

DERIVATION OF PROJECTED DOMAIN LENGTH FOR THE

EULERIAL FORMULATION

In view of curvature-strain equation (2), the averaged curvature, 1/ R (fig. 4), for the elastic curve 

bounded by  can be approximated by using the averaged strains and the averaged depth 

factor as equation (C1):

( ) / 2

1 ( ) / 2

1

1

1

1

i i

i i

i i

i i

R c c c  c

  

 







 	 	 	 	

(C1)

In view of figure 4, for the in-extensional beam, one can relate the chord length, c l , between points 

{ i 1 ,i}, to the curved domain length, l , as:

R

l lc R R

2

2 sin

2

2 sin       (C2)

The deformed domain length, Li , [projection of c l along the x-axis] can be obtained by taking cosine 

of the chord length, c l [eq. (C-2)] as: 





 



 



  



 



      



 R

l

R

l L l R i

l

i c i

c

2

cos

2

2 sin

2

cos 1  1

 



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

(C3)

Substitution of equations (C1) and (C2) into equation (C3) yields the deformed domain length, ,

along the deformed x-axis as equation (C4):



















 

 

















 



















  















i i

i i

i

i i

i i

i i

i i

i

c c

l

c c

c c l L

1

1

1

1

1

1

1

2

cos

2

2 sin 	 	  	 	

	 	

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

(C4)

which is equation (24) in the main text. 
53

APPENDIX D

DERIVATIONS OF THE EULERIAN SLOPE EQUATIONS 

Equation (26) is reproduces as equation (D1) in the following:

; (D1)

On the left hand side of equation (D1), the lower limit of the integration, Vi1 , is defined as the value of 

V at i1 x , namely, 1 1 1 i  ( i )  tan ( i)  tan i V V x  x  .

Integrations of both sides of equation (D1) yield (refs.1, 17,): 























  



 



    



  



     

1

1 1

1

2

1

1 1 1 1 1 1

( ) ( )

log ( )

( ) ( )

i

i i

i

i i

e

i i

i i i i i i i i

i

c

x x c

L

c c

c c

c c c c L 	 	 	 	

( ) i 1 i x  x  x 

(D2)

After simplification, equation (D2) becomes equation (D3):















 





 

























  

    

 



  

 

 









 log 1 1

( ) ( )

1

1 1

1

2

1

1 1

1

1

1

2

1

2 1

i

i

i

i

e

i i

i i i i

i i

i i

i i

i

i

L

x x

c

c

c c

c c

x x L

V c c

V V V 	 	 	 	

( ) i 1 i x  x  x 

(D3)
54

Squaring both sides and rearranging, and then taking the square root, the slope, V (x)  tan (x) can be 

expressed as:

2

1

1

2

1

1 1

1

1

1

1

2

1

1

1

2

1

1 1

1

1

1

1

2

1

log 1 1

( ) ( )

1 tan

tan 1

log 1 1

( ) ( )

1 tan

tan

tan ( )



























 





 

























  

    

 









 

























  

    

 

 



 

 













 

 











     

 
   	 

M

i

i

i

i

e

i i

i i i i

i i

i i

i i

i

i

M

i

i

i

i

e

i i

i i i i

i i

i i

i i

i

i

L

x x

c

c

c c

c c

x x L

c c

L

x x

c

c

c c

c c

x x L

c c

x

	 	 	 	





	 	 	 	







( ) i 1 i x  x  x 

(D4)

At the deformed strain-sensing station, i x  x , we have i i Li (x  x 1 )   , and equation (D4) becomes

equation (D5):

2

1

2

1

1 1

1

1

1

2

1

1

2

1

1 1

1

1

1

2

1

log ( ) 1 tan

tan 1

log ( ) 1 tan

tan

tan















 





 







  

  









 







  

  

 

 

 









 

 









i

i

e

i i

i i i i

i i

i i

i

i

i

i

i

e

i i

i i i i

i i

i i

i

i

i

i

c

c

c c

c c

c c

L

c

c

c c

c c

c c

L

	 	 	 	





	 	 	 	







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

(D5)

which is identical to the Eulerian slope equation (29) for large deformations of nonuniform beams (uniform 

beams excluded).

Log-Expanded Slope Equation 

Note from equation (C4) that, the logarithmic terms and the denominators containing factor, ( ) i  i1 c c ,

will cause mathematical indeterminacy (that is, 0/0) for the uniform beam case ( ) 1 c c c i  i  . The way to 

bypass such a mathematical indeterminacy problem is to expand logarithmic function in series form in terms 

of ( ) i  i1 c c . This approach enables the canceling out of the factor, ( ) i  i1 c c , in the denominator of 

equation (D4). Keep in mind that in equation (D4), the two conditions, 0 1 1 



  

i

i

L

x x and 

1 1 0

1

&













  & 

i

i

c

c , caused the required condition, 1 1 1 1

1

 &



 























  &  

 i

i

i

i

L

x x

c

c , to hold for series 

expansion of the logarithmic term, 



 

























  



log 1 1 1

1 i

i

i

i e L

x x

c

c , expressed in terms of ( ) i  i1 c c (ref. 17)

as:
55





 

























  



log 1 1 1

1 i

i

i

i e L

x x

c

c 1 .... 2

1 1

2

1

2

1

1

1

 

































   

















   





 i

i

i

i

i

i

i

i

L

x x

c

c

L

x x

c

c

.... 2

1

2

1

2

1

1 1

1

1  

































   

   



 





i

i

i

i i

i

i

i

i i

L

x x

c

c c

L

x x

c

c c (D6)

In light of equation (D6), the term M in equation (D4) can be written as equation (D7):





 

























  

    

  

 

 





 log 1 1

( ) ( ) 1

1

2

1

1 1

1

1

1

i

i

i

i

e

i i

i i i i

i i

i i

i i

L

x x

c

c

c c

c c

x x L

c c

M 	 	 	 	

      Log-expanded terms

2

1

2

1

1 1

1

1

2

1

1 1

1

1

1 .... 2

1

( ) ( ) 

















































   

 



    

  



 







 







i

i

i

i i

i

i

i

i i

i i

i i i i

i i

i i

i i

L

x x

c

c c

L

x x

c

c c

c c

c c

x x L

c c

	 	 	 	

.... 2

( ) ( ) ( ) ( )

2

1

2

1

1 1

1

1 1

1 1

1

1

1 



    

   

  



 



 

 







i

i

i

i i i i

i

i i i

i i i i

i

i i

i i

L

x x

c

c c

x x

c c c

c c

x x

c c

	 	 	 	 	 	





 









      

  



 

    

 

 .... 2

( ) ( ) ( )

( ) 2

1

2

1

1 1

1 1 1 1 1

1 1

1

i

i

i

i i i i

i i i i i i i i

i i i

i

L

x x

c

c c

c c c c

c c c

x x 	 	

	 	 	 	





 









    

  



 

 

 

 .... 2

( ) [ ( )] ( )

( ) 2

1

2

1

1 1

1 1

1 1

1

i

i

i

i i i i

i i i

i i i

i

L

x x

c

c c

c c

c c c

x x 	 	

	 (D7)

After canceling out the factor, ( ) i 1 i c  c  , the term M becomes:





 









     



 





 .... 2

( ) ( )

2

1

2

1

1 1

1

1

1

i

i

i

i i i i

i

i

i

L

x x

c

c c

x x

c

M 	 	 	 (D8)

In view of equation (D8), the slope equation (D4) takes on the following form:

2

2

1

2

1

1 1

1

1

1

1

2

1

2

1

2

1

1 1

1

1

1

1

2

1

2

( ) ( )

1 tan

tan 1

2

( ) ( )

1 tan

tan

tan ( )















 



    







    

 





 















 











i

i

i

i i i i

i

i

i

i

i

i

i

i

i i i i

i

i

i

i

i

L

x x

c

c c

x x

c

L

x x

c

c c

x x

c

x

	 	 	





	 	 	







( ) i 1 i x  x  x 

(D9)
56

At the deformed strain-sensing station, i x  x , we have i i Li (x  x 1 )   , and equation (D9) becomes:

2

2

1

1 1

1

1

1

2

1

2

1

1 1

1

1

1

2

1

2 1 tan

tan 1

2 1 tan

tan

tan















  

  





 

  

 



 











 









i

i

i i i i

i

i

i

i

i

M

i

i

i i i i

i

i

i

i

i

i

L

c

c c L

c

L

c

c c L

c

	 	 	





	 	 	







 
   	 

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

(D10)

The term M in equation (D10) can be further simplified as:

2 2

1

1 1

1

1 i

i

i i i i

i

i

i L

c

c c L

c

M     



 



 	 	 	















    

    



   



 





2

1

1 1 1 1

2

1

1 1

1

1

2

2

2

i

i i i i i i

i

i

i i i i i

i

i

i

c

c c c L

c

L c c L

c

	 	 	

	 	 	





 





 













    

 

i i

i

i

i

i

c

c

c

L

	 	 1

1 1

2

2 (D11)

In view of equation (D11), equation (D10) takes on the final form as:

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

(D12)

Equation (D12) is the log-expanded Eulerian slope equation (34) for large deformations of nonuniform 

beams including uniform beams as special cases.
57

APPENDIX E

DERIVATION OF THE LAGRANGIAN SLOPE EQUATION

Equation (42) is duplicated as equation (E1) in the following:

 ; (E1)

In equation (E1), V  dy / dx  tan (x) , and the lower limit of integration, Vi1 , on the left hand side is the 

value of V at the strain-sensing station,  i1 x x [that is, 1 1 ( / ) Vi  dy dx i tan ( )  i1  x 1  tani ]. 

Integrations of both sides of equation (E1) yield (refs. 1, 17): 

1

1 1 sin sin   V  Vi ( ) ( ) 1

1

1 



  

  i

i i

i i x x

c c

	 	

 











 



 



  



 







  



    



   

1 1 1

1

2

2

1

1 1 1 1

( ) log ( ) log

( )

( )

( ) ( )

i i e i

i i e

i i

i i i i i i

x x c c

l

c c

l

c c

l

c c

l

c 	 	 	

( ) ( ) 1

1

1 



  

  i

i i

i i x x

c c

	 	



















  



 



    



 





   

1

1 1

1

2

1

1 1 1 1

( ) ( )

log ( )

( ) ( )

i

i i

i i

e

i i

i i i i i i

c

x x c

l

c c

c c

c c c l 	 	 	 (E2)

After simplifications, equation (E2) becomes:





 



  



      

   







 









  ( ) 1 ( ) log ( ) ( ) ( )

sin sin 1

1

1

2

1

1 1

1

1

1

1

1 1

i

i

i i

e

i i

i i i i

i

i i

i i

i x x

lc

c c

c c

c c

x x l

c c

V V 	 	 	 	

( ) i 1 i x  x  x 

(E3)

Equation (E3) is equation (43) in the text.
58

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

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Shape Predictions of Aerospace Structures, NASA/BR-119, April 2008.

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Deformed Shape Predictions of Non-Uniform Aerospace Structures, NASA/TP-2009-214643,

September 2009.

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the Deformed Shape Predictions of Doubly Tapered Ikhana Wing, NASA/TP-2009-214652, November 

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Flexible Aerospace Structures: Formulation of Ko Displacement Theory,

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

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Operational Loads for Complex Structures, U.S. Patent No. 7,715,994, issued May 11, 2010.

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Aerospace Structures Using Span-wisely Distributed Surface Strains, NASA TP/2013-216518, 

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59

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