NASA/TP—2012–216060

 

Improved Displacement Transfer Functions for 

Structure Deformed Shape Predictions Using 

Discretely Distributed Surface Strains 

William L. Ko and Van Tran Fleischer 

 

Dryden Flight Research Center, Edwards, California

November 2012
NASA STI Program ... in Profile

Since its founding, NASA has been dedicated 

to the advancement of aeronautics and space 

science. The NASA scientific and technical 

information (STI) program plays a key part in 

helping NASA maintain this important role. 

The NASA STI program operates under the 

auspices of the Agency Chief Information 

Officer. It collects, organizes, provides for 

archiving, and disseminates NASA’s STI. The 

NASA STI program provides access to the 

NASA Aeronautics and Space Database and its 

public interface, the NASA Technical Report 

Server, thus providing one of the largest 

collections of aeronautical and space science 

STI in the world. Results are published in both 

non-NASA channels and by NASA in the 

NASA STI Report Series, which includes the 

following report types: 

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant 

phase of research that present the results of 

NASA Programs and include extensive data 

or theoretical analysis. Includes compilations of significant scientific and technical 

data and information deemed to be of 

continuing reference value. NASA counterpart of peer-reviewed formal professional 

papers but has less stringent limitations on 

manuscript length and extent of graphic 

presentations. 

• TECHNICAL MEMORANDUM. 

Scientific and technical findings that are 

preliminary or of specialized interest, 

e.g., quick release reports, working 

papers, and bibliographies that contain 

minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. 

Collected papers from scientific and 

technical conferences, symposia, 

seminars, or other meetings sponsored 

or co-sponsored by NASA. 

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, 

often concerned with subjects having 

substantial public interest.

• TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission.

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and feeds, 

providing help desk and personal search 

support, and enabling data exchange services. 

For more information about the NASA STI 

program, see the following: 

• Access the NASA STI program home page 

at http://www.sti.nasa.gov

• E-mail your question via the Internet to 

help@sti.nasa.gov 

• Fax your question to the NASA STI Help 

Desk at 443-757-5803 

• Phone the NASA STI Help Desk at 

443-757-5802 

• Write to:

NASA STI Help Desk

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

This page is required and contains approved text that cannot be changed.
NASA/TP—2012–216060

 

Improved Displacement Transfer Functions for 

Structure Deformed Shape Predictions Using 

Discretely Distributed Surface Strains 

William. L. Ko and Van Tran Fleischer 

 

Dryden Flight Research Center, Edwards, California 

National Aeronautics and

Space Administration 

Dryden Flight Research Center

Edwards, CA 93523-0273 

November 2012
PATENT NOTICE

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

embodied Displacement Transfer Functions for conversion into deflections described in this technical 

paper is protected under “Method for Real-Time Structure Shape-Sensing’, U.S. Patent No. 7,520,176 

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

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

more information. 

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

NOTICE

Use of trade names or names of manufacturers in this document does not constitute an 

official endorsement of such products or manufacturers, either expressed or implied, by 

the National Aeronautics and Space Administration 
v

TABLE OF CONTENTS

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

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

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

NONUNIFORM AND UNIFORM BEAMS............................................................................................. 3

EMBEDDED BEAM CURVATURE EQUATIONS................................................................................ 3

DISCRETIZATION OF NONUNIFORM EMBEDDED BEAM............................................................. 4

SUMMARY OF PREVIOUS DISPLACEMENT TRANSFER FUNCTIONS ........................................ 5

CHARACTERISTICS OF DISPLACEMENT TRANSFER FUNCTIONS............................................. 6

PREVIOUS PIECEWISE LINEAR REPRESENTATIONS..................................................................... 6

BEHAVIOR OF STRAIN CURVES FOR DIFFERENT TAPERED BEAMS........................................ 7

Tapered Cantilever Tubular Beams ...................................................................................................... 7

Analytical Surface Strains..................................................................................................................... 7

Shapes of Strain Curves........................................................................................................................ 8

FORMULATIONS OF IMPROVED DISPLACEMENT TRANSFER FUNCTIONS ............................ 9

Piecewise Nonlinear Strain Representations......................................................................................... 9

Strain Extrapolation .............................................................................................................................. 9

Piecewise Nonlinear Strain Curves......................................................................................................10

UNIFORM BEAMS .................................................................................................................................10

Improved Slope Equation.....................................................................................................................10

Improved Deflection Equation.............................................................................................................11

Improved Displacement Transfer Function .........................................................................................12

NONUNIFORM BEAMS.........................................................................................................................12

Improved Slope Equation.....................................................................................................................12

Improved Deflection Equation.............................................................................................................13

Improved Displacement Transfer Function .........................................................................................14

LOG-EXPANDED CASE ........................................................................................................................15

Log-Expanded Slope Equation ............................................................................................................15

Log-Expanded Deflection Equation.....................................................................................................16

Log-Expanded Displacement Transfer Function .................................................................................17

Log-Expanded Slope Equation for Uniform Beams............................................................................17

Log-Expanded Deflection Equation for Uniform Beams ....................................................................17
vi

DEPTH-EXPANDED CASE....................................................................................................................18

Depth-Expanded Slope Equation .........................................................................................................18

Depth-Expanded Deflection Equation .................................................................................................20

Depth-Expanded Displacement Transfer Function..............................................................................21

NEW STRUCTURE SHAPE-SENSING TECHNOLOGY .....................................................................21

SHAPE-PREDICTION ACCURACIES...................................................................................................22

Tapered Cantilever Tubular Beams .....................................................................................................22

Comparisons of Predicted Deflections.................................................................................................22

Shape-Prediction Errors.......................................................................................................................26

Normalized Shape-Prediction Errors...................................................................................................27

DISCUSSION...........................................................................................................................................28

CONCLUDING REMARKS....................................................................................................................28

FIGURES..................................................................................................................................................30

APPENDIX A: DERIVATIONS OF IMPROVED SLOPE AND DEFLECTION EQUATIONS FOR 

UNIFORM BEAMS ............................................................................................................................40

APPENDIX B: DERIVATIONS OF IMPROVED DISPLACEMENT TRANSFER FUNCTION FOR 

UNIFORM BEAMS ............................................................................................................................42

APPENDIX C: DERIVATIONS OF IMPROVED SLOPE AND DEFLECTION EQUATIONS FOR 

NONUNIFORM BEAMS....................................................................................................................44

APPENDIX D: DERIVATIONS OF IMPROVED DISPLACEMENT TRANSFER FUNCTION FOR 

NONUNIFORM BEAMS....................................................................................................................50

APPENDIX E: DERIVATIONS OF LOG-EXPANDED SLOPE AND DEFLECTION EQUATIONS 

FOR NONUNIFORM BEAMS...........................................................................................................54

APPENDIX F: DERIVATIONS OF LOG-EXPANDED DISPLACEMENT TRANSFER FUNCTION 

FOR NONUNIFORM BEAMS...........................................................................................................61

APPENDIX G: DERIVATIONS OF DEPTH-EXPANDED SLOPE AND DEFLECTION EQUATIONS 

FOR NONUNIFORM BEAMS...........................................................................................................64

APPENDIX H: DERIVATIONS OF DEPTH-EXPANDED DISPLACEMENT 

TRANSFER FUNCTION FOR NONUNIFORM BEAMS ................................................................70

APPENDIX I: FLOW CHART FOR STRUCTURE DEFORMED SHAPE VISUALIZATIONS .........74

REFERENCES .........................................................................................................................................78
1

ABSTRACT

In the formulations of earlier Displacement Transfer Functions for structure shape predictions, the 

surface strain distributions, along a strain-sensing line, were represented with piecewise linear functions. 

To improve the shape-prediction accuracies, Improved Displacement Transfer Functions were formulated 

using piecewise nonlinear strain representations. Through discretization of an embedded beam 

(depth-wise cross section of a structure along a strain-sensing line) into multiple small domains, 

piecewise nonlinear functions were used to describe the surface strain distributions along the discretized 

embedded beam. Such piecewise approach enabled the piecewise integrations of the embedded beam 

curvature equations to yield slope and deflection equations in recursive forms. The resulting Improved 

Displacement Transfer Functions, written in summation forms, were expressed in terms of beam 

geometrical parameters and surface strains along the strain-sensing line. By feeding the surface strains 

into the Improved Displacement Transfer Functions, structural deflections could be calculated at multiple 

points for mapping out the overall structural deformed shapes for visual display. The shape-prediction 

accuracies of the Improved Displacement Transfer Functions were then examined in view of 

finite-element-calculated deflections using different tapered cantilever tubular beams. It was found that by 

using the piecewise nonlinear strain representations, the shape-prediction accuracies could be greatly 

improved, especially for highly-tapered cantilever tubular beams.

NOMENCLATURE

A , 1/in.

B , 

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

 uniform beam), in.

, depth factor of nonuniform beam at strain-sensing station, , in.

value of at free end (beam tip), , in.

/ depth ratio, no dimension

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

nonuniform beam) at axial location, , in.

value of at fixed end (beam root), , in.

C , in/in

CG center of gravity

di chord-wise distance between front and rear strain-sensing stations at { , }, in.

D , 1/in.

DLL design limit load

i , strain-sensing station identification number 

j dummy index

l length of beam, in.

n index for the last span-wise strain-sensing station (or strain-sensing station density) 

P force, lb

RRF rotated reference frame, aligns with the wing’s out-of-plane direction

R(x) radius of curvature of beam elastic curve, in.

SPAR Structural Performance And Resizing (finite-element computer program)

t tubular beam wall thickness, in.

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

axial coordinate or symbol of -th strain-sensing station, in.
2

beam deflection in y-direction at axial location, , in.

, beam deflection in y-direction at axial location, , in.

beam-tip deflection calculated from SPAR, in.

, distance between two adjacent strain-sensing stations { , }, in.

 | |, beam-tip shape-prediction error (difference between predicted and 

SPAR-generated beam-tip deflections), in.

, beam taper angle, deg

surface bending strain at axial location, , in/in

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

slope of deformed beam at axial location, , rad or deg

, slope of deformed beam at axial location, , rad or deg

, local axial coordinate with origin at , in.

cross-sectional twist angle, rad or deg

 quantities associated with rear strain-sensing line

INTRODUCTION

In the formulations of earlier Displacement Transfer Functions (refs. 1-5) for structure shape 

predictions using distributed surface strains, the embedded beam (depth-wise cross-section of the

structure along a strain-sensing line) was first discretized into multiple small domains of equal length. 

Through discretization, the axial distributions of both beam depth and surface strain within each small 

domain could be represented with piecewise linear functions. The discretization approach enabled the 

piecewise integrations of the beam curvature equation (second-order differential equation) over the small 

domains in closed forms to yield beam slope and deflection equations in recursive forms. Each set of 

recursive slope and deflection equations was then combined into a single dual summation-form deflection

equation called a Displacement Transfer Function (refs. 1-5). The Displacement Transfer Function was 

then written in terms of geometrical parameters of discretized embedded beam and surface strains

obtained at strain-sensing stations (at domain junctures) evenly spaced along the strain-sensing line. By 

inputting the surface strains from multiple strain-sensing lines, the associated Displacement Transfer 

Functions can convert the surface strains into deflections at multiple points, enabling the mapping of 

overall structural deformed shapes for visual display. Thus, the Displacement Transfer Functions with an 

accompanying surface strain-sensing system [for example, fiber optic strain sensors (refs. 6-8)], has 

created a new structure shape-sensing technology, called Method for Real-Time Structure Shape-Sensing

(U.S. Patent Number 7,520,176) (ref. 9), which could be very useful for in-flight deformed shape 

monitoring of flexible wings and tails, such as those often employed on unmanned flight vehicles by the 

ground-based pilot for maintaining safe flights. In addition, the wing shape being monitored in real time

could then be input to the aircraft control system for aeroelastic wing shape control.

Similar to the structure deformed shape predictions, the distributed surface strains can also be fed into 

the special stiffness and Load Transfer Functions described in Process for Using Surface Strain 

Measurements to Obtain Operational Loads for Complex Structures (U.S. Patent No. 7,715,994) (ref. 10), 

for predicting the operational loads on a flight vehicle. 
3

The previously-developed Displacement Transfer Functions were successfully validated for their 

accuracies in view of 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) (refs. 1, 2, 4, and 5); and the 

doubly-tapered wing of the NASA Ikhana Unmanned Science and Research Aircraft System (a modified 

Predator-B unmanned aerial system, General Atomics Aeronautical Systems Inc., San Diego, California).

(ref. 3).

The earlier Displacement Transfer Functions were formulated for straight embedded beams. With the 

introduction of empirical curvature-effect correction terms, however, those Displacement Transfer 

Functions could be used for the shape predictions of slender curved structures with different arc-angles up 

to 360 deg (a complete circle) (ref. 11).

All the earlier Displacement Transfer Functions were formulated using piecewise linear 

representations of the actual strain distributions. For the shape predictions of tapered beam structures, for 

which the strain gradients change rapidly (highly nonlinear strain distributions) (ref. 3), the use of 

piecewise linear strain representations could result in a loss of accuracy. One way to avoid this loss of

shape-prediction accuracy is to increase the domain density (that is, use increasing strain-sensing stations) 

so that the piecewise linear strain representation can maintain a good approximation. An alternative 

approach is to introduce piecewise nonlinear representations of the actual strain distributions.

This technical paper discusses the mathematical formulations of the Improved Displacement Transfer 

Functions using the piecewise nonlinear strain representations. A family of tapered cantilever tubular 

beams with different taper angles was used to study the shape-prediction accuracies of the Improved

Displacement Transfer Functions. The shape-prediction accuracies were then analyzed by comparing the 

predicted values with the analytically-predicted values obtained from the finite-element analyses. The 

results showed that by using the piecewise nonlinear strain representations in the formulation of the 

Displacement Transfer Functions, the shape-prediction accuracies were greatly improved, especially for 

highly-tapered beam structures.

NONUNIFORM AND UNIFORM BEAMS

In the present technical paper, the term “beam” implies the imaginary embedded beam, which is 

defined as the depth-wise cross-section of the structure along the strain-sensing line. The “nonuniform 

beam” in the present technical paper is defined as the embedded beam with a varying depth factor, , 

a vertical distance from the neutral surface to the bottom surface of the nonuniform beam at axial 

location, x. For a nonuniform tubular beam, will be the outward radius. The “slightly nonuniform 

beam” in the present technical paper is defined as the embedded beam with slowly changing depth factor, 

. The “uniform beam” in the present technical paper is defined as the embedded beam with constant 

depth factor [that is, ].

EMBEDDED BEAM CURVATURE EQUATIONS

Figure 1 shows a deformed uniform embedded beam with constant depth factor, c. In a small beam 

segment bounded by , the small un-deformed curve length, , is on the beam neutral surface, 

and the deformed curve length, , lies on the beam bottom surface 

under bending strain, . From the two similar slender sectors, and , the local radius of 
4

curvature, , of the beam elastic curve, can be related to the bottom surface bending strain, , 

through the beam depth factor, c, as:

(1)

Applying the Lagrangian curvature equation, (refs. 12 and

13), neglecting the term, (negligible axial displacements), equation (1) can be written in the 

form of a second-order differential equation as:

(2)

In equation (2), x is the undeformed axial coordinate, and y is the out-of-plane deflection. Because 

equation (2) is referred to the un-deformed x-coordinate (Lagrangian formulation), the term, , in 

equation (2) is not the simplified form of the classical curvature equation,

= , which is referred to the deformed x-coordinate (Eulerian 

formulation) (refs. 12, 13, and 14). 

Notice that curvature equation (2) is purely a geometrical relationship, containing no material or 

structural properties. In fact, the effect of the material or structural properties is felt by the surface 

bending strains, ; the value of is influenced by the material or structural properties. 

The curvature equation (2) could also be extended to the nonuniform embedded beam without losing 

accuracy if the embedded beam depth factor, , changes slowly along the beam axis (ref. 15). 

Namely, 

(3)

For given beam geometries, the depth factors {c, c(x)} are known. If the functional form of the 

surface strain, , is specified, equation (2) or (3) can be integrated once to yield the beam slope 

equation, and the second integration will yield the beam deflection equation. To integrate the curvature 

equations (2) or (3), the piecewise integration method can be used. (refs. 1-5).

DISCRETIZATION OF NONUNIFORM EMBEDDED BEAM 

The integrations of curvature equation (2) or (3) can be achieved by means of piecewise integrations 

through beam discretizations (refs. 1, 2, and 5). As shown in figure 2, the embedded beam of length, l, is 

first discretized evenly into n multiple small domains, , with equal domain 

length, . The strain-sensing stations where the bending strains will be measured (or 

analytically calculated) are located on the beam bottom surface at the domain junctures (including at both 

ends of the beam), (fig. 2). Through discretization, both beam depth factor and 

surface bending strain, , , within each small domain, , could be expressed with 
5

simple linear functions. Such an approach enables the piecewise integrations of curvature equation (2) or 

(3) in closed forms for each small domain to yield beam slope and deflection equations in recursive

forms. These recursive slope and deflection equations can be combined into a dual summation-form 

deflection equation called the Displacement Transfer Function (refs. 1-5) and expressed in terms of 

domain length, , beam depth factors, and surface bending strains, , (fig. 2).

SUMMARY OF PREVIOUS DISPLACEMENT TRANSFER FUNCTIONS

All the earlier Displacement Transfer Functions were formulated through piecewise integrations of 

equation (2) or (3). Four earlier Displacement Transfer Functions formulated for nonuniform, slightly 

nonuniform, and uniform straight embedded beams, are shown below for comparison of their 

mathematical functional forms (refs. 1, 2, and 5).

1. Displacement Transfer Function for nonuniform beams ( ) (derivations in Appendix A of 

ref. 2):

(4)

 

2. Displacement Transfer Function for slightly nonuniform beams ( ) (derivations in

Appendix C of ref. 2):

(5)

 

which was obtained from equation (4) by expanding the logarithmic terms in the neighborhood of 

.

3. Second-order Displacement Transfer Function for nonuniform beams ( ) [ in 

equation (3) expanded in series up to the second-order term] (ref. 5):
6

(6)

 

4. Displacement Transfer Function for uniform beams ( ) (derivations in ref. 1; Appendix 

D of ref. 2; Appendix A of ref. 5):

; (7)

Equation (7) is the degenerated form of equations (5) and (6) for the uniform beam case,

( ).

CHARACTERISTICS OF DISPLACEMENT TRANSFER FUNCTIONS 

Except for equation (4), equations (5) and (6) could degenerate into equation (7) for the uniform beam 

case ( ) without incurring mathematical indeterminacy problems (refs. 1 and 2). Note that in 

each of the deflection equations (4)–(7), the deflection, , at the current strain-sensing station, , is 

expressed in terms of domain length, , inboard beam depth factors, ( , , , ,...., ), and the 

associated inboard surface strains, ( , , , ,...., ), including the values of ( , ) at the current 

strain-sensing station , where deflection, , is calculated. Because equations (4)−(7) contain no 

material or structural properties, when equations (4)−(7) are used in the deformed shape calculations of 

complex structures (for example, aircraft wings), there is no need to know the material or structural 

properties. In fact, the effect of the material or structural properties is felt by the surface strains, 

. This is the key characteristic of the Displacement Transfer Functions (refs. 1−5).

PREVIOUS PIECEWISE LINEAR REPRESENTATIONS

The above four displacement transfer functions, equations (4)–(7), were formulated through 

discretization of the embedded beam into multiple small domains (fig. 2). The variation of the beam depth 

factor, , and the surface bending strain, , within each small domain, , between the 

two adjacent strain-sensing stations, , , was then expressed with linear functions as:

 

; (8)
7

 ; (9)

Equation (8) can be a good piecewise linear representation of the slowly-varying depth factor, , 

of the nonuniform beams. The accuracy of equation (9) for piecewise linear representation of varying

surface strain, , however, needs to be examined when the beam taper angle increases and causes

strain distributions to be highly nonlinear.

BEHAVIOR OF STRAIN CURVES FOR DIFFERENT TAPERED BEAMS

This section is to examine the accuracy of piecewise linear strain representation equation (9) using 

tapered beams with different taper angles, and to show that equation (9) can lose accuracy in representing 

nonlinear strain distributions for the cases of highly-tapered beams.

Tapered Cantilever Tubular Beams

To study the accuracy degradation of the piecewise linear strain representations equation (9), a family 

of tapered cantilever tubular beams with different taper angles was chosen. The dimensions and taper 

angles, , of the aluminum tapered cantilever tubular beams considered are listed in Table 1.

Table 1. Dimensions of aluminum tapered cantilever tubular beams.

l, in. 

(length)

t, in.

(wall thickness)

, in.

(root depth)

, in.

(tip depth)

/

(depth ratio)

, 

deg (taper angle)

100.5 0.02296 4 4 (uniform) 4/4 (1.0) 0.00

100.5 0.02296 4 3 3/4 (0.75) 0.57

100.5 0.02296 4 2 2/4 (0.5) 1.14

100.5 0.02296 4 1 1/4 (0.25) 1.71

100.5 0.02296 4 0.5 0.5/4 (0.125) 1.99

100.5 0.02296 4 0.25 0.25/4 (0.0625) 2.14

The imaginary embedded beam shown in figure 2 can also represent the embedded beam of a typical 

tubular beam, which is defined as the span-wise cross section through vertical diameters along the 

strain-sensing line. For each tapered tubular beam, the depth factor, , of the embedded beam is the 

known local outer radius. Therefore, only one strain-sensing line on the bottom surface oriented in the 

span-wise direction is needed for shape predictions (fig. 2).

Figure 3 shows the finite-element model of a typical tapered cantilever tubular beam (depth ratio, 

/ = 2/4) generated from the SPAR (Structure Performance And Resizing) finite-element computer 

program (ref. 16). The size of the SPAR model (identical for all tapered beams) is also indicated in figure 

3. The tapered cantilever tubular beam is subjected to upward load of =100 lb at the beam tip.

Analytical Surface Strains

In the present technical paper, the surface strains at the strain-sensing stations (fig. 2) were calculated 

analytically (not actually measured) using the SPAR program. Namely, finite elements were used to 

simulate actual strain gages. The surface bending strains, , at the i-th strain-sensing 
8

stations (fig. 2) were obtained by converting SPAR-generated nodal or element axial stresses into axial 

strains, (bending strains), through stress-strain law. 

Also, the surface strain, , can be obtained from the span-wise nodal displacement differentials of 

the SPAR elements. It was found, however, that in the beam-tip regions of highly-bent tapered beams, the

SPAR element strains became negative because the output nodal displacements were the projected 

displacements along the original beam axis, and not along the deformed beam axis to reflect true spanwise displacements. Therefore, this displacement method was not used.

Shapes of Strain Curves

The SPAR-generated strain curves were approximated with piecewise linear functions using domain 

densities of and .

For n = 8

Figure 4 shows the shapes of piecewise linear strain curves (solid lines with solid circular symbols) 

using domain density, , to approximate the SPAR-generated strain curves (dashed curves with open 

circular symbols) for different tapered cantilever tubular beams. Note that the SPAR-generated strain data 

are not exactly zero (theoretically zero) at the free beam tip (especially for the / = 0.5/4, 0.25/4 

cases); the reason being that the beam-tip finite element (simulated strain gage) has one end at the beam 

tip, but the other end is slightly off from the beam tip because of finite length. By reducing the element 

sizes, the beam-tip strain should approach zero. If an actual strain gage were installed at the beam tip, the 

strain output could also be nonzero because of finite length of the strain gage.

For moderately-bent strain curves (for example, / = 3/4, 2/4, 1/4) (fig. 4), the piecewise linear

strain representations could be reasonably good, but for highly-bent regions of the strain curves (for 

example, / = 0.5/4, 0.25/4), the piecewise linear strain representations could lose accuracy. Note 

from figure 4 in the highly-bent regions of the strain curves (for example, / = 0.5/4, 0.25/4), the 

typical piecewise linear points, calculated from equation (9), lie slightly off from the corresponding SPAR 

data points. 

Keep in mind that for the low depth ratios, (for example, / = 0.5/4, 0.25/4), each beam will bend 

like a fishing pole in the outboard regions. These unrealistic very low depth ratios were included in the 

current analysis solely for theoretical interest to examine the shape-prediction accuracies of the 

Displacement Transfer Functions. For real aircraft wings, the depth ratios are much higher. For example, 

the Ikhana wing has a depth ratio of / = 0.4444, for which the strain curve is bow-shaped (ref. 3).

For n = 16

To improve piecewise linear strain representations using equation (9), the domain density was 

increased from to to cause the piecewise linear strain curves to approach the corresponding 

SPAR-generated smooth strain curves. 

Figure 5 shows that by using domain density , the piecewise linear strain representations 

greatly improved except for the sharply-bent regions. The peak strain point at strain-sensing station 

, which was missed at for the / = 0.25/4 case, is now included. In the sharply-bent 

region of the strain curve of the / = 0.25/4 case, the typical piecewise linear point calculated from 
9

equation (9) lies below the corresponding SPAR data point; however, further increasing of the domain 

density, n, could cause the piecewise linear strain curves to approach the corresponding SPAR-generated 

smooth strain curves. 

An alternative approach is to use piecewise nonlinear strain representations (described below) to 

approximate the nonlinear strain curves and thereby improve the shape-prediction accuracies.

FORMULATIONS OF IMPROVED DISPLACEMENT TRANSFER 

FUNCTIONS

In the formulation of the Improved Displacement Transfer Functions, the piecewise linear assumption 

according to equation (9) for strain distribution was replaced with the piecewise nonlinear strain 

representations described below. Equation (8) was used without modification, however, because

equation (8) is a good piecewise linear representation of the slowly-varying depth factor, , of the 

nonuniform beams.

Piecewise Nonlinear Strain Representations

In the formulation of the Improved Displacement Transfer Functions, the distribution of surface 

bending strain, , within the dual domain, , was described with piecewise nonlinear 

(second-order) functions in terms of three strain values, { , , }, respectively at three adjacent 

strain-sensing stations, { , , }, (fig. 6). The resulting piecewise nonlinear strain equation has the 

following form:

(10)

 

Equation (10) is the piecewise nonlinear strain equation to replace the piecewise linear strain equation 

(9) for formulating the Improved Displacement Transfer Functions. 

Strain Extrapolation

Note that equation (10) is for the strain-sensing stations , , , ,...., . At the beam tip 

strain-sensing station , however, an extra strain point, , is needed for three-points strain curve 

approximation. A simple way to estimate is through extrapolation using the three known strains, 

{ , , }, respectively at the three adjacent strain-sensing stations, { , , }, (fig. 7). The 

three-point extrapolation equation established for calculating has the following form:

(11)

Equation (11) was obtained from equation (10) by setting and .
10

Piecewise Nonlinear Strain Curves

Figure 8 compares the piecewise nonlinear strain curves (solid curves with solid circular symbols) 

based on equation (10) using the domain density , and the corresponding SPAR-generated strain 

curves (dashed curves with open circular symbols) for different tapered cantilever tubular beams. In the 

sharply-bent segment of the strain curve of the / = 0.25/4 case (fig. 8), the typical piecewise 

nonlinear point, calculated from equation (10), has now moved above the corresponding SPAR data point. 

Except for the / = 0.25/4 case, the piecewise nonlinear strain representations based on equation (10) 

could provide very good approximations of the bow-shaped SPAR strain curves. As will be seen, by 

changing from piecewise linear to piecewise nonlinear strain representations (see figs. 5 and 8), shapeprediction accuracies could be greatly improved.

UNIFORM BEAMS

The formulation of the Improved Displacement Transfer Functions will start from the mathematically 

simpler case of uniform beams, and progress to the mathematically more complex case of nonuniform

beams. As mentioned above, the uniform beam considered in the present technical paper is the embedded 

beam with constant depth. In the formulations of the Displacement Transfer Functions, dimensions of 

beam width are not needed. Using the piecewise linear depth equation (8), and the piecewise nonlinear 

strain equation (10), the Improved Displacement Transfer Functions were formulated first for the uniform 

beams. The slope and deflection equations are to be obtained by piecewise integrations of the curvature 

equation (2) for the uniform beams. 

Improved Slope Equation

The slope,  of the uniform beam at axial location within the small domain, 

, bounded by the two adjacent strain-sensing stations, , , can be obtained by 

integrating the uniform-beam curvature equation (2) once, and enforcing the continuity of slope at the 

adjacent inboard strain-sensing station, . Namely,

 ; (12)

in which is the slope at . Substituting equation (10) into equation (12) and carrying out the 

integration (ref. 17), one obtains the slope equation for the small domain, (see Appendix A):

(13)
11

At the strain-sensing station, , we have , and equation (13) gives the slope, 

,at the strain-sensing station, , as:

 ; (14)

Equation (14) is the Improved slope equation for the uniform beams in recursive form. Applying the 

recursive relationship, equation (14) can be written as:

 ; (15)

Equation (15) is the Improved slope equation for the uniform beams in summation form.

Improved Deflection Equation

The deflection, , of the uniform cantilever beam within the small domain, , can be 

obtained by integrating the slope equation (13) and enforcing the continuity of deflection at the inboard 

adjacent strain-sensing station, , as: 

 ; (16)

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

In view of equation (13), equation (16) can be integrated (ref. 17) to yield the following deflection 

equation for the small domain, , as (see Appendix A):

(17)

 

At the strain-sensing station, , we have , and equation (17) gives the deflection,

, at the strain-sensing station, , as: 

 ; (18)

Equation (18) is the Improved deflection equation for the uniform beams in recursive form. 
12

Improved Displacement Transfer Function

In view of equation (15), and applying recursive relationships, equation (18) can be written in dual 

summation form as (see Appendix B):

(19)

Equation (19) is called the Improved Displacement Transfer Function for the uniform beams because 

equation (19) transforms the surface strains into deflections. Note that the first summation is contributed 

from the deflection terms, and the second summation is contributed from the slope terms. Note also that a 

factor appeared in the second summation term due to cumulative summations of identical terms 

(see Appendix B). Note that the second summation ends at (that is, one term shorter than the 

first summation) (see Appendix B).

NONUNIFORM BEAMS

As defined earlier, the nonuniform beam has a variable depth factor, . Using the piecewise linear 

depth equation (8), and the piecewise nonlinear strain equation (10), the Improved Displacement Transfer 

Functions were formulated for the nonuniform beams. The slope and deflection equations are obtained by 

piecewise integrations of the curvature equation (3) for the nonuniform beams.

Improved Slope Equation

The slope,  of the nonuniform beam at axial location x within the small domain, 

, bounded by the two adjacent strain-sensing stations, , , can be obtained by 

integrating the nonuniform-beam curvature equation (3) once, and enforcing the continuity of slope at the 

adjacent inboard strain-sensing station, . Namely,

 ; (20)

in which , is the slope at . Substituting equations (8) and (10) into equation (20), and carrying 

out integration, one obtains the slope equation for the small domain, , as (see Appendix C):
13

(21)

At the strain-sensing station, , we have , and equation (21) gives the slope, 

, at the strain-sensing station, , as:

(22)

Equation (22) is the Improved slope equation for the nonuniform beams in recursive form. Applying the 

recursive relationship, equation (22) can be rewritten as:

(23)

Equation (23) is the Improved slope equation for the nonuniform beams in summation form.

Improved Deflection Equation

The deflection, , of the nonuniform cantilever beam within the small domain, , can 

be obtained by integrating the slope equation (21), and enforcing the continuity of deflection at the 

inboard adjacent strain-sensing station, , as:

 ; (24)
14

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

In view of equation (21), equation (24) can be integrated (ref. 17) to yield the following deflection 

equation for the nonuniform beam for the small domain, , as (see Appendix C):

(25)

At the strain-sensing station, , we have , and equation (25) gives the deflection, 

at the strain-sensing station, , as:

(26)

Equation (26) is the Improved deflection equation for the nonuniform beam in recursive form.

Improved Displacement Transfer Function

In view of equation (22), and applying recursive relationships, equation (26) can be written in dual 

summation form as (see Appendix D):
15

(27)

Equation (27) is called the Improved Displacement Transfer Function for the nonuniform beam.

Notice that the first summation is contributed from the deflection terms, and the second summation is 

contributed from the slope terms.

LOG-EXPANDED CASE

Note that the slope equations (22) and (23) and the deflection equations (26) and (27) contain 

logarithmic terms, , in the numerators, and terms in the denominators, causing 

mathematically indeterminacy (0/0) when the beam depth becomes uniform (that is, ). This 

mathematical indeterminacy problem can be eliminated if the logarithmic terms are expanded in terms of 

series to cancel the terms in the denominators (refs. 1, 2). 

Log-Expanded Slope Equation

For , the logarithm term, , in equations (23) and (24) can be expanded in 

series form up to the third-order term as (ref. 17):

(28)

Substitution of equation (28) into equations (22) and (23), and after grouping terms, yields the 

Log-expanded slope equations given below (see Appendix E):
16

1. In recursive form:

(29)

2. In summation form:

(30)

Log-Expanded Deflection Equation

For developing the Log-expanded deflection equations for , the logarithm term, 

, in equations (26) and (27) must be expanded in series form up to the fourth-order term as 

(ref. 17):

(31)

Substitutions of equation (31) into equations (26) and (27), and after grouping terms, yields the 

resulting Log-expanded deflection equations given as (see Appendix E):

(32)

Equation (32) is the Log-expanded deflection equation for the nonuniform beam in recursive form.
17

Log-Expanded Displacement Transfer Function

In view of equation (29), and applying recursive relationships, equation (32) can be written in dual 

summation form as (see Appendix F):

 

(33)

Equation (33) is called the Log-expanded Displacement Transfer Function for nonuniform beams.

Log-Expanded Slope Equation for Uniform Beams

For uniform beams, we have , and the above Log-expanded slope equations (29) and (30)

degenerate respectively into the following simplified forms:

1. In recursive form:

 ; (34)

2. In summation form:

 ; (35)

Equations (34) and (35) agree respectively with equations (14) and (15) formulated for the uniform 

beams, and thus confirm the derivation accuracy of the Log-expanded slope equations (29) and (30).

Log-Expanded Deflection Equation for Uniform Beams

For uniform beams, we have , and the above Log-expanded deflection equations (32) and

(33) degenerate respectively into the following simplified forms:

1. In recursive form:

 ; (36)
18

2. In summation form:

(37)

Note that equations (36) and (37) agree respectively with equations (18) and (19) formulated for the 

uniform beams, and thus confirm the derivation accuracy of the Log-expanded deflection equations (32)

and (33).

DEPTH-EXPANDED CASE

This section describes the formulation of the Depth-expanded slope and deflection equations for 

nonuniform beams. The purpose of this formulation is to avoid the logarithmic terms after carrying out 

direct integration of curvature equation (3). The depth factor of the nonuniform beam is assumed to 

change slowly along the beam axis.

Depth-Expanded Slope Equation

The slope,  of the nonuniform beam at axial location, x, within the small domain,

, between the two adjacent strain-sensing stations, , , can be obtained by carrying 

out the integration of the nonuniform beam curvature equation (3). Namely,

 ; (38)

in which is the slope at . 

An alternative way to avoid the logarithmic terms after direct integration of equation (38) is to expand 

the term, in the integrand of equation (38) in binomial series form because is a 

slowly-varying function of x (ref. 17). Namely, 

 

(39)

Note that in equation (39), the variable term in the denominator was moved to the 

numerator. This approach avoids the appearance of logarithmic term, , after direct 

integration of equation (38). Substituting equations (10) and (39) into equation (38) and carrying out 
19

integration, one obtains the Depth-expanded slope equation for the small domain, , as (see 

Appendix G):

(40)

 

At the strain-sensing station, , we have , and equation (40) gives the slope, 

at the strain-sensing station, , as:

 

(41)

Equation (41) is the Depth-expanded slope equation for the nonuniform beams in recursive form.

For the uniform beams, we have , and equation (41) degenerates into the following

simplified forms:

 ; (42)

which agrees with equation (14) formulated for the uniform beams.

Applying the recursive relationship, equation (41) can be rewritten as:

 

(43)
20

Equation (43) is the Depth-expanded slope equation for the nonuniform beams in summation form, 

and which can be degenerated into equation (15) for uniform beams ( ).

Depth-Expanded Deflection Equation

The deflection, , of the nonuniform cantilever beam within the small domain, , can 

be obtained by integrating the slope equation (40), and enforcing the continuity of deflection at the 

inboard adjacent strain-sensing station, , as:

 ; (44)

in which is the deflection at the inboard strain-sensing station, . In light of equation (40), 

equation (44) can be integrated (ref. 17) to yield the following deflection equation for the small domain, 

, (see Appendix G):

(45)

At the strain-sensing station, , we have , and equation (45) gives the deflection, 

at the strain-sensing station, , as:

(46)

Equation (46) is the Depth-expanded deflection equation for the nonuniform beams in recursive form.
21

For the uniform beams, we have , and equation (46) degenerates into the following 

simplified form:

 ; (47)

Equation (47) agrees with the deflection equation (18) for the uniform beams.

Depth-Expanded Displacement Transfer Function

In view of equation (41), and applying recursive relationships, equation (46) can be written in dual 

summation form as (see Appendix H):

(48)

Equation (48) is the Depth-expanded Displacement Transfer Function for the nonuniform beams, and 

which can degenerate into equation (19) for the uniform beams ( ). Again, the first summation 

in equation (48) is the contributions from the deflection terms, and the second summation is the 

contributions from the slope terms.

NEW STRUCTURE SHAPE-SENSING TECHNOLOGY

The Displacement Transfer Functions with the accompanying strain-sensing system thus formed a 

new technology for structure deformed shape predictions. This new technology is called Method for 

Real-Time Structure Shape-Sensing (U.S. Patent Number 7,520,176, ref. 9), which could be very useful

for in-flight deformed shape monitoring of flexible wings and tails, such as those often employed on 

unmanned flight vehicles by the ground-based pilot for maintaining safe flights.

A typical flow chart of the new structure shape-sensing technology using the two-line strain-sensing 

system is shown in Appendix I. In the flow chart, only the Depth-expanded Displacement Transfer 

Function (48) was used as the example for each strain-sensing line. Other Displacement Transfer 

Functions may also be selected, depending on the types of structures (uniform, nonuniform, slightly 

nonuniform, et cetera) under consideration. 
22

The two-line strain-sensing system can be used for the shape-sensing of the tapered wing box for 

which the neutral surface is located at the wing box half-depth (fig. I-1), If, however, the neutral surface is 

not located at exactly half the depth of a structure, such as aircraft wings, a second set of the two-line 

sensing system must be installed on the upper surface, right on top of the lower sensing system. Using 

pairs of lower and upper strains, one can determine the actual location of the neutral surface for finding 

the values of the depth factors, , needed for the calculations of deflections (see the 

Ikhana wing shape predictions, ref. 3).

It is important to mention that when using the two-line strain-sensing system for the wing structures, 

the Displacement Transfer Functions can not only calculate the deflections but also the cross-sectional 

rotations. Thus, the need to use surface distortion strains can be eliminated.

Appendix I also describes the application of the two-line strain-sensing system to a free-free beam 

structure such as an airborne aircraft fuselage (free-free nonuniform tubular beam structure) (fig. I-2) for 

sensing the horizontal and vertical bending strains for inputs to the Displacement Transfer Functions for 

the calculations of in-flight deformed shapes in horizontal and vertical planes (ref. 2). 

SHAPE-PREDICTION ACCURACIES

As mentioned above, the surface strains were analytically calculated from the SPAR program, and 

were not directly measured. In the shape-prediction accuracy studies, the deflections calculated from the 

SPAR program were used as reference yardsticks (0% error) to evaluate the accuracies of the 

corresponding deflections calculated from the new Displacement Transfer Functions. The 

shape-prediction accuracies of the newly-formulated Displacement Transfer Functions were also 

compared with those of the earlier Displacement Transfer Functions.

Tapered Cantilever Tubular Beams

The sample structures chosen in the shape-prediction accuracy studies were again a family of 

aluminum tapered cantilever tubular beams having the dimensions listed in table 1. The same loading 

condition of upward load, lb, was applied at each beam tip.

Comparisons of Predicted Deflections

Tables 2(a)-2(f) compare the accuracies of deflections calculated from different Displacement

Transfer Functions based on piecewise linear and piecewise nonlinear strain representations for the six 

tapered cantilever tubular beams listed in table 1. Note from tables 2(a)-2(f) that only two typical

Displacement Transfer Functions [the nonlinear Displacement Transfer Function (4); and the secondorder Displacement Transfer Function (6)], based on piecewise linear strain representations were added to 

the accuracy comparisons. In tables 2(a)-2(f), the shaped prediction errors (in percent) are indicated in 

parentheses.
23

Table 2(a). Comparisons of deflections (in inches) calculated from SPAR and from different 

Displacement Transfer Functions; uniform cantilever tubular beam; / = 4/4; .

Theories y0 y2 y4 y6 y8 y10 y12 y14 y16

(% error)

SPAR

(reference)

0.00000

(0.0000)

0.01787

(0.0000)

0.06340

(0.0000)

0.13368

(0.0000)

0.22450

(0.0000)

0.33181

(0.0000)

0.45174

(0.0000)

0.57933

(0.0000%)

0.71162

(0.0000)

Piecewise linear strain representations

Nonuniform,

eq. (4)* ------ ------ ------ ------ ------ ------ ------ ------ ------

Secondorder,

eq. (6)

0.00000

(0.0000)

0.01576

(0.2965)

0.06034

(0.4300)

0.12962

(0.5705)

0.21948

(0.7054)

0.32580

(0.8446)

0.44448

(0.9752)

0.57139

(1.1158)

0.70243

(1.2914)

Piecewise nonlinear strain representations

Improved,

eq. (27)* ------ ------ ------ ------ ------ ------ ------ ------ ------

Logexpanded,

eq. (33)

0.00000

(0.0000)

0.01577

(0.2951)

0.06036

(0.4272)

0.12965

(0.5663)

0.21951

(0.7012)

0.32585

(0.8375)

0.44454

(1.0118)

0.57146

(1.1059)

0.70249

(1.2830)

Depthexpanded,

eq. (48)

0.00000

(0.0000)

0.01577

(0.2951)

0.06036

(0.4272)

0.12965

(0.5663)

0.21951

(0.7012)

0.32585

(0.8375)

0.44454

(1.0118)

0.57146

(1.1059)

0.70249

(1.2830)

* No solutions for uniform beams because of mathematical indeterminacy. 

Note from Table 2(a) that both equations (4) and (27) produced no solutions for uniform beams 

because of mathematical indeterminacy problems. 

Table 2(b). Comparisons of deflections (in inches) calculated from SPAR and from different 

Displacement Transfer Functions; tapered cantilever tubular beam; / = 3/4; .

Theories y0 y2 y4 y6 y8 y10 y12 y14 y16

(% error)

SPAR

(reference)

0.00000

(0.0000)

0.01823

(0.0000)

0.06697

(0.0000)

0.14572

(0.0000)

0.25204

(0.0000)

0.38294

(0.0000)

0.53428

(0.0000)

0.70092

(0.0000)

0.87634

(0.0000)

Piecewise linear strain representations

Nonuniform,

eq. (4)

0.00000

(0.0000)

0.01627

(0.2237)

0.06417

(0.3195)

0.14199

(0.4256)

0.24738

(0.5310)

0.37726

(0.6482)

0.52758

(0.7645)

0.69312

(0.8901)

0.86711

(1.0532) 

Secondorder,

eq. (6)

0.00000

(0.0000)

0.01627

(0.2237)

0.06417

(0.3195)

0.14199

(0.4256)

0.24738

(0.5310)

0.37726

(0.6482)

0.52759

(0.7634)

0.69313

(0.8888)

0.86712

(1.0521)

Piecewise nonlinear strain representations

Improved,

eq. (27)

0.00000

(0.0000)

0.01627

(0.2237)

0.06419

(0.3172)

0.14204

(0.4199)

0.24748

(0.5203)

0.37741

(0.6310)

0.52781

(0.7383)

0.69345

(0.8524)

0.86754

(1.0042)

Logexpanded,

eq. (33)*

0.00000

(0.0000)

0.01627

(0.2237)

0.06419

(0.3172)

0.14205

(0.4188)

0.24748

(0.5203)

0.37742

(0.6299)

0.52782

(0.7372)

0.69346

(0.8513)

0.86755

(1.0030)*

Depthexpanded,

eq. (48)

0.00000

(0.0000)

0.01627

(0.2237)

0.06419

(0.3172)

0.14204

(0.4199)

0.24748

(0.5203)

0.37741

(0.6310)

0.52781

(0.7383)

0.69345

(0.8524)

0.86754

(1.0042)

* Most accurate prediction (closest to SPAR). 
24

Table 2(c). Comparisons of deflections (in inches) calculated from SPAR and from different displacement 

transfer functions; tapered cantilever tubular beam; / = 2/4; .

Theories y0 y2 y4 y6 y8 y10 y12 y14 y16

(% error)

SPAR

(reference)

0.00000

(0.0000)

0.01864

(0.0000)

0.07113

(0.0000)

0.16057

(0.0000)

0.28831

(0.0000)

0.45510

(0.0000)

0.65965

(0.0000)

0.89759

(0.0000)

1.15801

(0.0000)

Piecewise linear strain representations

Nonuniform,

eq. (4)

0.00000

(0.0000)

0.01682

(0.1572)

0.06859

(0.2193)

0.15719

(0.2919)

0.28407

(0.3661)

0.44983

(0.4551)

0.65325

(0.5527)

0.88981

(0.6718)

1.14828

(0.8402) 

Secondorder,

eq. (6)

0.00000

(0.0000)

0.01682

(0.1572)

0.06859

(0.2193)

0.15719

(0.2919)

0.28407

(0.3661)

0.44983

(0.4551)

0.65324

(0.5535)

0.88980

(0.6727)

1.14826

(0.8420)

Piecewise nonlinear strain representations

Improved,

eq. (27)

0.00000

(0.0000)

0.01682

(0.1572)

0.06861

(0.2176)

0.15723

(0.2884)

0.28417

(0.3575)

0.45002

(0.4387)

0.65359

(0.5233)

0.89042

(0.6192)

1.14930

(0.7522)

Logexpanded,

eq. (33)*

0.00000

(0.0000)

0.01682

(0.1572)

0.06861

(0.2176)

0.15723

(0.2884)

0.28417

(0.3575)

0.45002

(0.4387)

0.65359

(0.5233)

0.89044

(0.6174)

1.14935

(0.7478)*

Depthexpanded,

eq. (48)

0.00000

(0.0000)

0.01682

(0.1572)

0.06861

(0.2176)

0.15723

(0.2884)

0.28417

(0.3575)

0.45001

(0.4395)

0.65358

(0.5242)

0.89041

(0.6200)

1.14929

(0.7530)

* Most accurate prediction (closest to SPAR). 

Table 2(d). Comparisons of deflections (in inches) calculated from SPAR and from different displacement 

transfer functions; tapered cantilever tubular beam; / = 1/4; .

Theories y0 y2 y4 y6 y8 y10 y12 y14 y16

(% error)

SPAR

(reference)

0.00000

(0.0000)

0.01912

(0.0000)

0.07598

(0.0000)

0.17923

(0.0000)

0.33816

(0.0000)

0.56506

(0.0000)

0.87454

(0.0000)

1.28196

(0.0000)

1.78417

(0.0000)

Piecewise linear strain representations

Nonuniform,

eq. (4)

0.00000

(0.0000)

0.01739

(0.0970)

0.07364

(0.1312)

0.17612

(0.1743)

0.33427

(0.2180)

0.56011

(0.2774)

0.86835

(0.3469)

1.27375

(0.4602)

1.77144

(0.7135) 

Secondorder,

eq. (6)

0.00000

(0.0000)

0.01739

(0.0970)

0.07364

(0.1312)

0.17611

(0.1749)

0.33426

(0.2186)

0.56009

(0.2786)

0.86829

(0.3503)

1.27365

(0.4658)

1.77124

(0.7247)

Piecewise nonlinear strain representations

Improved,

eq. (27)*

0.00000

(0.0000)

0.01738

(0.0976)

0.07362

(0.1323)

0.17609

(0.1760)

0.33424

(0.2197)

0.56010

(0.2780)

0.86850

(0.3385)

1.27462

(0.4114)

1.77473

(0.5291)*

Logexpanded,

eq. (33)

0.00000

(0.0000)

0.01738

(0.0975)

0.07362

(0.1323)

0.17608

(0.1766)

0.33422

(0.2208)

0.56007

(0.2797)

0.86845

(0.3413)

1.27452

(0.4170)

1.77468

(0.5319)

Depthexpanded,

eq. (48)

0.00000

(0.0000)

0.01738

(0.0975)

0.07362

(0.1323)

0.17608

(0.1766)

0.33422

(0.2208)

0.56008

(0.2791)

0.86845

(0.3413)

1.27453

(0.4164)

1.77458

(0.5375)

* Most accurate prediction (closest to SPAR). 
25

Table 2(e). Comparisons of deflections (in inches) calculated from SPAR and from different

Displacement Transfer Functions; tapered cantilever tubular beam; / = 0.5/4; .

Theories y0 y2 y4 y6 y8 y10 y12 y14 y16

(% error)

SPAR

(reference)

0.00000

(0.0000)

0.01938

(0.0000)

0.07872

(0.0000)

0.19047

(0.0000)

0.37082

(0.0000)

0.64535

(0.0000)

1.05517

(0.0000)

1.67292

(0.0000)

2.59712

(0.0000)

Piecewise linear strain representations

Nonuniform,

eq. (4)

0.00000

(0.0000)

0.01770

(0.0620)

0.07649

(0.0859)

0.18752

(0.1136)

0.36721

(0.1390)

0.64077

(0.1763)

1.04963

(0.2133)

1.66517

(0.2984)

2.57675

(0.7843) 

Secondorder,

eq. (6)

0.00000

(0.0000)

0.01770

(0.0620)

0.07648

(0.0862)

0.18751

(0.1140)

0.36719

(0.1398)

0.64071

(0.1787)

1.04949

(0.2187)

1.66478

(0.3134)

2.57544

(0.8348)

Piecewise nonlinear strain representations

Improved,

eq. (27)*

0.00000

(0.0000)

0.01768

(0.0655)

0.07644

(0.0878)

0.18741

(0.1178)

0.36700

(0.1417)

0.64038

(0.1914)

1.04905

(0.2356)

1.66527

(0.2946)

2.58806

(0.3488)* 

Logexpanded,

eq. (33)

0.00000

(0.0000)

0.01769

(0.0651)

0.07644

(0.0878)

0.18740

(0.1182)

0.36697

(0.1482)

0.64030

(0.1944)

1.04887

(0.2426)

1.66475

(0.3146)

2.58595

(0.4301)

Depthexpanded,

eq. (48)

0.00000

(0.0000)

0.01768

(0.0655)

0.07644

(0.0878)

0.18740

(0.1182)

0.36698

(0.1479)

0.64032

(0.1937)

1.04891

(0.2410)

1.66489

(0.3092)

2.58719

(0.3823)

* Most accurate prediction (closest to SPAR). 

Table 2(f). Comparisons of deflections (in inches) calculated from SPAR and from different 

Displacement Transfer Functions; tapered cantilever tubular beam; / = 0.25/4; . 

Theories y0 y2 y4 y6 y8 y10 y12 y14 y16

(% error)

SPAR

(Reference)

0.00000

(0.0000)

0.01952

(0.0000)

0.08017

(0.0000)

0.19668

(0.0000)

0.38984

(0.0000)

0.69566

(0.0000)

1.18157

(0.0000)

2.00286

(0.0000)

3.58289

(0.0000)

Piecewise linear strain representations

Nonuniform,

Eq. (4)

0.00000

(0.0000)

0.01786

(0.0463)

0.07802

(0.0600)

0.19387

(0.0784)

0.38651

(0.0929)

0.69143

(0.1181)

1.17563

(0.1658)

1.99425

(0.2403)

3.53790

(1.2557) 

Secondorder,

eq. (6)

0.00000

(0.0000)

0.01786

(0.00463)

0.07801

(0.0603)

0.19385

(0.0790)

0.38648

(0.0938)

0.69135

(0.1203)

1.17539

(0.1725)

1.99331

(0.2665)

3.53116

(1.4398)

Piecewise nonlinear strain representations

Improved,

eq. (27)*

0.00000

(0.0000)

0.01784

(0.0469)

0.07795

(0.0620)

0.19369

(0.0835)

0.38615

(0.1030)

0.69068

(0.1390)

1.17400

(0.2113)

1.99125

(0.3240)

3.57152

(0.3173)*

Logexpanded,

eq. (33)

0.00000

(0.0000)

0.01784

(0.0469)

0.07794

(0.0622)

0.19368

(0.0837)

0.38609

(0.1047)

0.69057

(0.1421)

1.17372

(0.2191)

1.99044

(0.3466)

3.56071

(0.6191)

Depthexpanded,

eq. (48)

0.00000

(0.0000)

0.01784

(0.0469)

0.07794

(0.0622)

0.19368

(0.0837)

0.38611

(0.1041)

0.69058

(0.1418)

1.17373

(0.2188)

1.99026

(0.3517)

3.56740

(0.4323)

* Most accurate prediction (closest to SPAR).

The deflections calculated from equations (4), (6), (27), (33), and (48), and listed in tables 2(a)-2(f)

are plotted respectively in figures 9-13 (solid curves with solid circular symbols) for visual comparisons 

with the corresponding SPAR deflection curves (dashed curves with open circular symbols). Note that by 

using the piecewise nonlinear strain representations (figs. 11-13) instead of the piecewise linear strain 

representations (figs. 9 and 10), the shape-prediction accuracies, especially for the smaller depth ratios, 

/ ={0.25/4, 0.5/4}, were greatly improved. Pictorially, the deflection curves calculated from 
26

equations (27), (33), and (48), and shown respectively in figures 11-13, fell extremely close to the 

corresponding SPAR deflection curves because the piecewise nonlinear strain representations were used

[see the actual values of the prediction errors indicated within the parentheses in tables 2(a)-2(f)]. 

Shape-Prediction Errors

In the shape-prediction error studies, the SPAR-generated deflections were used as the reference 

yardsticks with which to measure the shape-prediction errors. The shape-prediction error studies focused 

only on the beam tips because of peak deflections. The beam-tip shape-prediction error, 

( | |) (in.) for each type of beam is defined as the magnitude of difference between the 

SPAR-generated beam-tip deflection, , and the corresponding beam-tip deflection, , calculated 

from equations (4), (6), (27), (33), and (48). From data listed in Table 2(a)-2(f), the calculated beam-tip 

shape-prediction errors, , are compared in table 3. 

Table 3. Comparisons of beam-tip shape-prediction errors, (in.) of different Displacement Transfer 

Functions; tapered cantilever tubular beams with depth ratios 4/4 / 0.25/4.

Beam-tip shape-prediction error, , in.

Piecewise linear Piecewise nonlinear

/ , 

in.

Nonuniform, Second -

order, Improved, Logexpanded, Depth-expanded,

eq. (4) eq. (6) eq. (27) eq. (33) eq. (48)

4/4 0.71162 ---- 0.00919 ---- 0.00913 0.00913

3/4 0.87634 0.00923 0.00922 0.00880 0.00879* 0.00880

2/4 1.15801 0.00973 0.00975 0.00871 0.00866* 0.00872

1/4 1.78417 0.01273 0.01293 0.00944* 0.00949 0.00959

0.5/4 2.59712 0.02037 0.02168 0.00906* 0.01117 0.00993

0.25/4 3.68289 0.04499 0.05173 0.01137* 0.02218 0.01549

* Most accurate.

The beam-tip shape-prediction errors, , listed in table 3 are plotted in figure 14 for visual 

comparisons of the shape-prediction accuracies of the Displacement Transfer Functions (4), (6), (27), 

(33), and (48). One observes from figure 14 that, for the uniform beam ( / = 4/4), changing from 

piecewise linear to piecewise nonlinear strain representations has no effect in reducing the shapeprediction errors because it is immaterial to use either piecewise linear or piecewise nonlinear function to 

represent the linear (straight-line) strain curve for the uniform beam (figs. 5 and 8). Note also that the 

values of the beam-tip shape-prediction errors, , are practically insensitive to the change of the depth 

ratio, / , and stayed almost unchanged in the region, 4/4 / 2/4, but started to increase 

rapidly as the depth ratio, / , decreases, and reached the maximum values of 0.04499, 0.05173, 

0.01137, 0.02218, and 0.01549 in. at the minimum depth ratio, / = 0.25/4, respectively for equations 

(4), (6), (27), (33), and (48) (table 3). Because at lower depth ratios, / 1/4, the strain curves are 

highly nonlinear (fig. 8), and using the piecewise nonlinear strain representations [equations (27), (33), 

and (48)], instead of using the piecewise linear strain representations [equations (4) and (6)], greatly 

reduced the beam-tip shape-prediction errors (fig. 14). Note also in figure 14 that all the error curves bent 
27

upward (increasing errors) when the depth ratio decreased toward the minimum depth ratio, 

/ 0.25/4. This upward bend is most severe for the piecewise linear strain representations 

[equations (4) and (6)], and to a lesser degree for the piecewise nonlinear strain representations [equations 

(27), (33), and (48)]. The increasing errors at low depth ratios, / {0.25/4, 0.5/4}, can be attributed 

to the relatively poor strain representations of the sharply-bent segments of the strain curves (figs. 5 and

8). The beam-tip shape-prediction errors plotted in figure 14, however, are very small and under 0.06 

inches. The Improved Displacement Transfer Function (27) has the smallest error (0.01137 in.).

Normalized Shape-Prediction Errors

The beam-tip shape-prediction errors, , of different Displacement Transfer Functions can be 

better compared if each is normalized by the corresponding SPAR-generated beam-tip deflection, 

, (prediction-error measuring yardsticks). The normalized beam-tip shape-prediction errors, 

, of different Displacement Transfer Functions are compared in table 4.

Table 4. Comparisons of normalized beam-tip shape-prediction errors ( 100%) of different 

Displacement Transfer Functions; tapered cantilever tubular beams with depth ratios, 4/4 / 1/4.

Normalized beam-tip shape-prediction error, , 100%

Piecewise linear Piecewise nonlinear

/ Nonuniform, Secondorder, Improved, Logexpanded, Depth-expanded,

eq. (4) eq. (6) eq. (27) eq. (33) eq. (48)

4/4 ---- 1.2914 ---- 1.2830 1.2830

3/4 1.0532 1.0521 1.0042 1.0030* 1.0042

2/4 0.8402 0.8420 0.7522 0.7478* 0.7530

1/4 0.7135 0.7147 0.5291* 0.5319 0.5375

0.5/4 0.7843 0.8248 0.3488* 0.4301 0.3823

0.25/4 1.2557 1.4398 0.3173* 0.6191 0.4323

 *Most accurate.

The normalized beam-tip shape-prediction errors, (in percent), listed in table 4 are plotted 

in figure 15 as functions of depth ratio, , for visual comparisons of the shape prediction accuracies 

of different Displacement Transfer Functions. Note from figure 15 that the values of 

associated with all types of Displacement Transfer Functions decreased almost linearly with decreasing 

depth ratio, / , and then [except for the Improved Displacement Transfer Function (27)] increased 

rapidly with further decreasing depth ratio, / (see fig. 14). The reason for the values of 

decreasing with decreasing / in the region, 4/4 / 1/4, is dividing the almost-unchanging 

values of (see fig. 14) by the associated values of , which increase with decreasing depth ratio, 

/ (see column 2 of table 3).
28

At the low depth ratios, / ={0.5/4, 0.25/4}, the Improved Displacement Transfer Function (27) 

gave the lowest value of among all of the five Displacement Transfer Functions that were

considered (fig. 15). At the lowest depth ratio, / = 0.25/4, the beam-tip shape-prediction errors of the 

Improved, Depth-expanded, and Log-expanded Displacement Transfer Functions (27), (48), and (33) are 

respectively 25%, 34%, and 49% of the error produced by the Nonuniform Displacement Transfer 

Function (4) (the most accurate equation, based on piecewise linear strain representations, see table 4). 

This is equivalent to error reductions of 75%, 66%, and 51%, respectively, by using the Displacement 

Transfer Functions (27), (48), and (33). Note from figure 15 that in the depth ratio region, 

4/4 / 1/4, all three of the Displacement Transfer Functions (27), (48), and (33) exhibited nearly 

equal shape-prediction accuracies. Similarly, in the same depth ratio region, 4/4 / 1/4, the error 

curves of the Displacement Transfer Functions (4) and (6) (based on piecewise linear strain 

representations), also collapsed into a single curve. Overall, the prediction errors of the five Displacement 

Transfer Functions are very small: only in the ranges of (0.7135~1.4398)% based on piecewise linear 

strain representations and (0.3173~1.2830)% based on piecewise nonlinear strain representations.

DISCUSSION

For shape predictions of complex aircraft wings (which are constructed with stiffened panels) using 

the Displacement Transfer Functions, one must first locate the local cross-sectional neutral axis using 

pairs of lower and upper surface strains, and then determine the local depth factors, 

(see Appendix I), which are needed for the deflection calculations. The surface strain values will change 

with the material or structural properties. The above approach was applied in the shape predictions of an 

Ikhana wing (ref. 3) and a Global Observer wing (ref. 18), for which the local neutral axes are not at the 

half-depth of the wing cross sections.

Concurrently with the writing of this technical paper, large-scale ground load tests of the Global 

Observer (175-ft wingspan) were carried out at the NASA Dryden Flight Loads Laboratory (ref. 18). The 

measured strain test data were later used in the previously listed Displacement Transfer Function (5),

(suitable for the slightly-tapered Global Observer wing) to calculate the wing deflections. The calculated 

values were then compared with the real-time photogrammetry data as shown in figures 16 and 17 (taken 

from ref. 18). Note that equation (5) proved to be very accurate, having prediction errors in the range of 

(0.09-1.72)% at the wingtip, which was at nearly 14 ft of deflection.

CONCLUDING REMARKS

New Improved Displacement Transfer Functions were formulated using the piecewise nonlinear 

representations of surface strain distributions. Four different Displacement Transfer Functions were 

formulated for both uniform and nonuniform beams:

• Improved Displacement Transfer Function for uniform beams

• Improved Displacement Transfer Function for nonuniform beams

• Log-expanded Displacement Transfer Function for nonuniform beams

• Depth-expanded Displacement Transfer Function for nonuniform beams.

The key findings resulting from using the new Displacement Transfer Functions to improve the 

accuracies of structure shape predictions of tapered cantilever tubular beams are:

1. By using the piecewise nonlinear strain representations, beam-tip prediction errors could be 

greatly reduced by up to 75% for the peak taper angle case.
29

2. For smaller taper angles, changing the strain representations from piecewise linear to piecewise 

nonlinear improved the shape-prediction accuracies along the entire beam.

3. For larger taper angles, changing the strain representations from piecewise linear to piecewise 

nonlinear improved the prediction accuracies only in the beam tip regions where the strain 

gradients change signs, but did not improve the prediction accuracies in the inboard regions 

where the strain gradients change very slowly.

4. For larger taper angles, among the three new Displacement Transfer Functions formulated for 

nonuniform cantilever beams, the Improved Displacement Transfer Function provided the most 

accurate shape predictions.

5. For smaller taper angles, the shape-prediction accuracies of the Improved, Depth-expanded, and 

Log-expanded Displacement Transfer Functions are practically the same.

6. For the uniform beams, there is no benefit in the prediction error reductions by changing from 

linear strain representations to nonlinear strain representations. As the depth ratio decreases (that 

is, increasing taper angles), however, the benefits of prediction error reductions gradually 

increased and finally reached the peaks at the minimum depth ratio (the highest taper angle). 
30

FIGURES

Figure 1. Small segment of deformed uniform embedded beam for relocating local curvature R(x) to 

associated surface bending strain ε(x).

Figure 2. Embedded beam (cross section of any structure, including tubular beam) along the 

strain-sensing line with evenly distributed strain-sensing stations.
31

Figure 3. Finite-element model for tapered cantilever tubular beam subjected to tip vertical load 

P = 100 lb.
32

Figure 4. SPAR-generated surface strain data points fitted with piecewise linear strain curves calculated 

from equation (9) for different tapered cantilever tubular beams discretized with n = 8 domains; 

P = 100 lb.

Figure 5. SPAR-generated surface strain data points fitted with piecewise linear strain curves calculated 

from equation (9) for different tapered cantilever tubular beams discretized with n = 16 domains; 

P = 100 lb.
33

Figure 6. Local second-order curve fitting of three adjacent strain points.

Figure 7. Extrapolation of three adjacent strains {εn−2, εn−1, εn} to obtain fictitious strain, εn+1.
34

Figure 8. SPAR-generated surface strain data points fitted with piecewise linear strain curves calculated 

from equation (10) for different tapered cantilever tubular beams discretized with n = 16 domains; 

P = 100 lb.
35

Figure 9. Comparisons of deflections calculated from SPAR program and from Nonuniform 

Displacement Transfer Function [equation (4)] (based on piecewise linear strain representations) for 

tapered cantilever tubular beams; P = 100 lb.

Figure 10. Comparisons of deflections calculated from SPAR program and from Second-order 

Displacement Transfer Function [equation (6)] (based on piecewise linear strain representations) for 

tapered cantilever tubular beams; P = 100 lb.
36

Figure 11. Comparisons of deflections calculated from SPAR program and from Improved Displacement 

Transfer Function [equation (27)] (based on piecewise nonlinear strain representations) for tapered 

cantilever tubular beams; P = 100 lb.

Figure 12. Comparisons of deflections calculated from SPAR program and from Log-expanded 

Displacement Transfer Function [equation (33)] (based on piecewise nonlinear strain representations) for 

tapered cantilever tubular beams; P = 100 lb.
37

Figure 13. Comparisons of deflections calculated from SPAR program and from Depth-expanded 

Displacement Transfer Function [equation (48)] (based on piecewise nonlinear strain representations) for 

tapered cantilever tubular beams; P = 100 lb.

Figure 14. Comparisons of beam-tip deflection prediction errors of different Displacement Transfer 

Functions based on piecewise linear and piecewise nonlinear strain representations.
38

Figure 15. Comparisons of percent beam-tip deflection prediction errors of different Displacement 

Transfer Functions based on piecewise linear and piecewise nonlinear strain representations.
39

Figure 16. Comparison of predicted and measured wing deflections along No. 3 strain-sensing line for 

different loading levels (ref. 18).

Figure 17. Comparison of predicted and measured wing deflections along No. 4 strain-sensing line for 

different loading levels (ref. 18).
40

APPENDIX A

DERIVATIONS OF IMPROVED SLOPE AND DEFLECTION 

EQUATIONS FOR UNIFORM BEAMS

Appendix A presents the details of integrations of the slope equation (12) and the deflection equation 

(16) for the uniform beams to obtain the final mathematical forms given respectively by equations (14)

and (18).

Slope Equations

The slope,  of the uniform beam in the small domain, , between the two 

adjacent strain-sensing stations, , , can be obtained by carrying out the integration of equation 

(12), which is reproduced below:

 ; (A-1)

The nonlinear distribution of strain, , in the domain, , between the two adjacent 

strain-sensing stations, { }, described by equation (10), is duplicated below:

(A-2)

In view of equation (A-2), equation (A-1) can be integrated to yield (ref. 17):

 (A-3)

which is equation (13) in the text. At the strain-sensing station , one can write , and 

equation (A-3) yields the slope, , at the strain-sensing station, as:

 ; (A-4)

After grouping terms, equation (A-4) takes on the following form:
41

 ; (A-5)

which is equation (14) in the text, the Improved slope equation for the uniform beams.

Deflection Equations

The deflection, , of the uniform-depth beam in the small domain, , between the two 

adjacent strain-sensing stations, { }, can be obtained by integrating the slope equation (A-3) 

[or equation (13)] with constant of integration determined by enforcing the continuity of deflection at the 

inboard adjacent strain-sensing station, , as:

 ; (A-6)

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

Substituting equation (A-3) into equation (A-6), and carrying out integrations, one obtains (ref. 17):

(A-7)

which is equation (17) in the text. At the strain-sensing station, , one can write , and 

equation (A-7) gives the deflection,  at the strain-sensing station, , as:

(A-8)

After grouping terms, equation (A-8) takes on the form:

 ; (A-9)

which is equation (18) in the text, the Improved deflection equation for the uniform beams.
42

APPENDIX B

DERIVATIONS OF IMPROVED DISPLACEMENT TRANSFER 

FUNCTION FOR UNIFORM BEAMS 

Appendix B presents the derivation of the final dual summation form of the deflection equation for

the uniform cantilever beams. Equations (14) [or equation (A-5)] and (18) [or equation (A-9)] are 

duplicated below as equations (B-1) and (B-2), respectively, and can be combined into one equation with 

two summation terms as derived below.

 ; (B-1)

 ; (B-2)

In view of equation (B-1), equation (B-2) may be written for different indices as:

For :

(B-3)

For :

 

(B-4)

After grouping terms, equation (B-4) becomes:

(B-5)
43

For :

 

 

(B-6)

After grouping terms, equation (B-6) becomes:

(B-7)

Observing the indicial behavior, equation (B-7) can be generalized for index i, and the deflection, , 

can be expressed in dual summation form as:

(B-8)

which is equation (19) in the text, the Improved Displacement Transfer Function for the uniform beams.
44

APPENDIX C

DERIVATIONS OF IMPROVED SLOPE AND DEFLECTION 

EQUATIONS FOR NONUNIFORM BEAMS

Appendix C presents the details of integrations of the slope equation (20) and the deflection equation 

(24) for the nonuniform beams to obtain the final mathematical forms given respectively by equations 

(22) and (26).

Improved Slope Equations

The slope,  of the nonuniform beam in the small domain, , between the two 

adjacent strain-sensing stations, { }, can be obtained by carrying out the integration of equation 

(20) which is reproduced below as equation (C-1):

 ; (C-1)

The distributions of beam depth factor, [equation (8)], and bending strain, [equation (10)], in 

the small domain, , between the two adjacent strain-sensing stations, { }, are 

respectively duplicated below as equations (C-2) and (C-3):

 ; (C-2)

 ; (C-3)

Substituting equations (C-2) and (C-3) into equation (C-1) yields:

 

(C-4)

Let
45

(C-5)

 

(C-6)

 

(C-7)

 (C-8)

then equation (C-4) takes on the following simplified form:

(C-9)

After carrying out integration of equation (C-9), one obtains (ref. 17):

 

 

 

 

 

(C-10)
46

In view of the definitions from equations (C-5), (C-6), (C-7), and (C-8), equation (C-10) takes on the 

following form:

 (C-11)

In equation (C-11), the two terms, { , }, can be simplified through grouping terms as follows:

 

 

 

(C-12)

 
47

 

(C-13)

Substitutions of equations (C-12) and (C-13) into equation (C-11) yields:

(C-14)

At the strain-sensing station, , one can write , and equation (C-14) gives the slope, 

, at the strain-sensing station, , as

 

(C-15)

The term in equation (C-15) can be simplified as follows:
48

 

 

(C-16)

 

In view of equation (C-16), equation (C-15) takes on the final form as:

 

(C-17)

which is equation (22) in the text, the Improved slope equation for the nonuniform beams.

Improved Deflection Equations

The deflection, , of the uniform beam in the small domain, , between the two 

adjacent strain-sensing stations, { }, can be obtained by integrating the slope equation (C-14) with 

the constant of integration determined by enforcing the continuity of deflection at the inboard adjacent 

strain-sensing station, , as

 ; (C-18)

Substituting equation (C-14) into equation (C-18), and carrying out the integrations (ref. 17), one obtains:
49

 

(C-19)

At the strain-sensing station, , one can write , and equation (C-19) gives the 

deflection, , at the strain-sensing station, , as follows:

 

 

 

(C-20)

After grouping terms, equation (C-20) becomes:

 (C-21)

which is equation (26) in the text, the Improved deflection equation for the nonuniform beams.
50

APPENDIX D

DERIVATIONS OF IMPROVED DISPLACEMENT TRANSFER 

FUNCTION FOR NONUNIFORM BEAMS 

Appendix D presents the derivations of the Improved slope and deflection equations (22) [or equation 

(C-17)] and (26) [or equation (C-21)] [duplicated below as equations (D-1) and (D-2), respectively] that 

will be combined into one equation in dual summation form.

(D-1)

(D-2)

In view of equation (D-1), equation (D-2) may be written for each index, i, as:

For i = 1:

(D-3)

For i = 2:

 
51

 

 

(D-4)

After grouping terms, equation (D-4) becomes:

 

(D-5)

For :

 
52

(D-6)

After grouping terms, equation (D-6) becomes:

 
53

(D-7)

.........................

Observing the indicial behavior, equation (D-7) can be generalized for index i, and the deflection, , 

can be expressed in dual summation form as: 

 (D-8)

which is equation (27) in the text, the Improved Displacement Transfer Function for the nonuniform 

beams. 
54

APPENDIX E

DERIVATIONS OF LOG-EXPANDED SLOPE AND DEFLECTION 

EQUATIONS FOR NONUNIFORM BEAMS

Appendix E presents the mathematical derivations of the Log-expanded slope equation (29) and the 

Log-expanded deflection equation (32). The depth factor, , was considered to be a slowly-changing 

function of x. 

Log-Expanded Slope Equations

For slowly-changing [that is, ], the logarithm term, , in the slope 

equation (22) may be expanded in series form up to the third-order term as (ref. 17):

 

(E-1)

In view of equation (E-1), the Improved slope equation (22) [or equation (C-17)] can be expressed as:

 

 
55

(E-2)

After the final grouping of terms, equation (E-2) becomes:
56

(E-3)

which is equation (29) in the text, the Log-expanded slope equation for the nonuniform beams.

For the uniform beam case ( ), equation (E-3) degenerates into:

 ; (E-4)

which is identical to equation (14) in the text.

Log-Expanded Deflection Equations

For the case of the Log-expanded deflection equation, the logarithm term, , had to be 

expanded up to the fourth-order term as (ref. 17):

(E-5)

The term, , can then be written as:

 

(E-6)

In view of equation (E-6), equation (26) [or equation (C-21)] in the text can be rewritten as:
57

 

 

 (E-7)

The calculations of the terms, , , in equation (E-7) are shown below:

For the term:
58

 

 

 

 

 

 

 (E-8)

After grouping terms, one obtains the final form of as:

(E-9)

For the term:

 

 
59

 

 

 

 

 

 

 (E-10)

After grouping terms, one obtains the following final form of as:

(E-11)

For the term:

 

 

 

 

 

(E-12)

After grouping terms, equation (E-12) becomes:

(E-13)
60

In view of equations (E-9), (E-11), and (E13), the displacement equation (E-7) becomes:

(E-14)

After eliminating the factor in the denominator, there results:

(E-15)

 

which is equation (32) in the text, the Log-expanded deflection equation for the nonuniform beams.
61

APPENDIX F

DERIVATIONS OF LOG-EXPANDED DISPLACEMENT TRANSFER 

FUNCTION FOR NONUNIFORM BEAMS 

Appendix F presents the derivation of the final Log-expanded Displacement Transfer Function 

written in dual summation form. The Log-expanded slope equation (29) [or equation (E-3)] and the

Log-expanded deflection equation (32) [or equation (E-15)] are duplicated below as equations (F-1) and 

(F-2), respectively:

(F-1)

 

(F-2)

In view of equation (F-1), equation (F-2) can be written out explicitly for different values of the

index, i, as follows.

For :

(F-3)

For :

(F-4)

Substitutions of equations (F-1) (for ) and (F-3) into equation (F-4) yield:
62

 

 

(F-5)

After grouping terms, equation (F-5) becomes:

(F-6)

For :

(F-7)

Substitutions of equations (F-1) (for ) and (F-6) into equation (F-7) yield:

 
63

(F-8)

After grouping terms, equation (F-8) becomes:

 

(F-9)

.........................

Observing the indicial behavior, equation (F-9) can be generalized for index i, and the deflection, , 

can be expressed in dual summation form as: 

(F-10)

which is equation (33) in the text, the Log-expanded Displacement Transfer Function for the nonuniform 

beams.
64

APPENDIX G

DERIVATIONS OF DEPTH-EXPANDED SLOPE AND DEFLECTION 

EQUATIONS FOR NONUNIFORM BEAMS

Appendix G presents the details of the derivations of the Depth-expanded slope equation (41) and the 

Depth-expanded deflection equation (46) for nonuniform beams. The reciprocal of the depth factor, 

1/ , was first expanded in series form before carrying out the integrations of the curvature 

equation (3) in the text.

Depth-Expanded Slope Equation

The slope,  of the nonuniform beam in the region, , between the two adjacent 

strain-sensing stations, { }, can be obtained by integrating the nonuniform-beam curvature 

equation (3) with the constant of integration determined by enforcing the continuity of slope at the 

inboard adjacent strain-sensing station, , as

 ; (G-1)

The distributions of beam depth factor, , and bending strain, , in the domain, , 

between the two adjacent strain-sensing stations, { }, described by equation (10) are rewritten 

respectively below:

 ; (G-2)

(G-3)

Substituting equations (G-2) and (G-3) into equation (G-1) yields:

(G-4)

Because the slope term, , of the depth factor, , in equation (G-2) is small, its 

reciprocal, , can be expanded in binomial series form as (ref. 17):
65

 

(G-5)

Let 

(G-6)

 

(G-7)

 

(G-8)

 (G-9)

then, equation (G-4) takes on the following simplified form:

 

 

(G-10)

After carrying out the integration of equation (G-10), one obtains (ref. 17):

(G-11)

In view of the definitions from equations (G-6), (G-7), (G-8), and (G-9), equation (G-11) takes on the 

following form:
66

(G-12)

At the strain-sensing station , one can write , and equation (G-12) gives the 

slope, , at the strain-sensing station, as:

 

 
67

 

(G-13)

After rearrangements, equation (G-13) becomes:

(G-14)

which is equation (41) in the text, the Depth-expanded slope equation for the nonuniform beams.

Depth-Expanded Deflection Equations

The deflection, , of the uniform beam in the small domain, , between the two 

adjacent strain-sensing stations, { }, can be obtained by integrating the slope equation (G-12) with 

constant of integration determined by enforcing the continuity of deflection at the inboard adjacent strainsensing station, , as

 ; (G-15)

Substituting equation (G-12) into equation (G-15), and carrying out integrations, one obtains (ref. 17):

(G-16)
68

At the strain-sensing station , one can write , and equation (G-16) yields the 

deflection, , at the strain-sensing station, , as follows: 

 

 

 

 

(G-17)
69

After rearranging equation (G-17), there results:

(G-18)

which is equation (46) in the text, the Depth-expanded deflection equation for the nonuniform beams.
70

APPENDIX H

DERIVATIONS OF DEPTH-EXPANDED DISPLACEMENT TRANSFER 

FUNCTION FOR NONUNIFORM BEAMS

Appendix H presents the derivation of the final Depth-expanded Displacement Transfer Function 

written in dual summation form. The Depth-expanded slope equation (41) [or equation (G-14)] and 

Depth-expanded deflection equation (46) [or equation (G-18)] are duplicated below as equations (H-1) 

and (H-2) respectively:

(H-1)

(H-2)

In view of equation (H-1), equation (H-2) may be written out explicitly for different values of index, 

i, as follows:

For :

(H-3)

For :

 
71

 

 (H-4)

After grouping terms equation (H-4) becomes:

(H-5)

For :

 
72

 (H-6)

After grouping terms, equation (H-6) becomes:

 

 (H-7)

.........................

Observing the indicial behavior, equation (H-7) can be generalized for index i, and the deflection, , 

can be expressed in dual summation form as: 
73

 

(H-8)

which is equation (48) in the text, the Depth-expanded Displacement Transfer Function for the 

nonuniform beams.
74

APPENDIX I

FLOW CHART FOR STRUCTURE DEFORMED SHAPE 

VISUALIZATIONS

(cf., U.S. Patent No. 7,520,176, issued April, 2009)

(Two-line strain-sensing system, refs. 1-3).

Displacement Transfer Functions (Depth-Expanded):

For the front strain-sensing line:

 

(I-1)
75

For the rear strain-sensing line:

 

(I-2)

Cross-Sectional Twist Angle Equation:

 ; (I-3)

( = strain-sensing lines separation distance, fig. I-1)

Tapered Wing Box

Figure I-1 shows the application of the two-line strain-sensing system to a tapered wing box structure 

(the neutral surface is located at half depth) to sense the surface bending strains induced by combined 

bending and torsion loadings.
76

Figure I-1. Tapered cantilever wing box instrumented with two-line strain-sensing system for deformed 

shape predictions of structure under combined bending and torsion (ref. 2). 

The two strain-sensing lines (front and rear) are located along the lower edges (front and rear) of the 

wing box with chord-wise separation distance, , which decreases toward the wing box tip according to 

the wing box horizontal taper rate. The two strain-sensing lines may also be aligned parallel to each other 

(that is, = constant) along the span of the wing box.

As shown in the flow chart, the surface strains, are to be input to the left box of 

the chart, and the Displacement Transfer Functions {equation (I-1), equation (I-2)} (the second box from

the left) are to convert the surface strains into out-of-plane deflections, , (the third box from the

left), from which the cross-sectional twist angles, , can be calculated from the cross-sectional twist 

angle equation, equation (I-3). The resulting data could then be used to map the overall structural 

deformed shapes for visual display using the deformed-shape-visualization computer program (the fourth 

box from the left), and for control feedback.

If the neutral surface is not at the half depth, a second two-line strain-sensing system must be installed 

on the upper surface, right on top of the lower sensing lines. Using pairs of lower and upper strains, the 

depth factor, , needed for the deflection calculations, can be determined (see the 

Ikhana wing shape predictions, ref. 3).

Free-Free Beam Structure 

A good example of the free-free (unsupported) beam structure is an aircraft fuselage during flight 

(fig. I-2).
77

Figure I-2. Two-line strain-sensing system installed on an aircraft fuselage for horizontal and vertical 

bending strain sensing (ref. 2). 

The fuselage may be considered as an union of two nonuniform cantilever tubular beams (that is, a 

fore- and an aft-fuselage section) that are joined together at the center-of-gravity (CG) cross section 

(fig. I-2). For monitoring the deflections of the airborne fuselage under vertical bending and horizontal 

bending, two strain-sensing lines can be installed axially on the bottom (belly) surface and on the side of 

the fuselage (fig. I-2). During the flight, the CG cross section has the least movement compared with the 

other fuselage cross sections. For practical purposes, it is best to choose the reference strain-sensing 

station, , at the CG cross section, and divide each strain-sensing line into two segments (fig. I-2). One 

segment is for the fore-fuselage section and the other for the aft-fuselage section. Thus, by imposing 

at the reference sensing station (considered as a moving built-in end), deflection 

equation (I-1) can be used for the bottom strain-sensing line to calculate the vertical deflections of the 

fore- and aft-fuselage sections. The vertical deflection, , calculated from equation (I-1) will then be the 

relative deflection with respect to the tangent line passing though the reference sensing station, , which 

moves with the fuselage during flight (fig. I-2). The same arguments hold for the strain-sensing line for 

the horizontal bending.
78

REFERENCES

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

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

2. Ko, William L., and Van Tran Fleischer, Further Development of Ko Displacement Theory for 

Deformed Shape Predictions of Nonuniform Aerospace Structures, NASA/TP-2009-214643, 

September 2009.

3. Ko, William L., W. L. Richards, and Van Tran Fleischer, Applications of Ko Displacement Theory to 

the Deformed Shape Predictions of the Doubly-Tapered Ikhana Wing, NASA/TP-2009-214652, 

November 2009.

4. Ko, William L., and Van Tran Fleischer, Methods for In-Flight Wing Shape Predictions of Highly 

Flexible Aerospace Structures: Formulation of Ko Displacement Theory, NASA/TP-2010-214656, 

August 2010.

5. Ko, William L., and Van Tran Fleischer, First- and Second-Order Displacement Transfer Functions 

for Structural Shape Predictions Using Analytically Predicted Surface Strains, NASA/TP2012-215976, March 2012.

6. Bang, Hyung-Joon, Hyun-Kyu Kang, Chang-Sun Hong, and Chung-Gon Kim, “Optical Fiber Sensor 

Systems for Simultaneous Monitoring of Strain and Fractures in Composites,” Smart Materials and 

Structures, Vol. 14, pp. N52-N58, 2005.

7. Wehrle, Gunther, Percy Nohama, Hypolito Jose Kalinowski, Pedro Ingacio Torres, and Luiz Carlos 

Guedes Valente, “A Fiber Optic Bragg Grating Strain Sensor for Monitoring Ventilatory 

Movements,” Measurement Science and Technology, Vol. 12, pp. 805-809, 2001.

8. Baldwin, Chris S., Toni J. Salter, and Jason S. Kiddy, “Static Shape Measurements Using 

Multiplexed Fiber Bragg Grating Sensor System,” Smart Structures and Materials 2004: Smart 

Sensor Technology and Measurement Systems, Vol. 5384, pp. 206-217, 2004.

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

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

10. Richards, W. Lance, and William L. Ko, Process for Using Surface Strain Measurements to Obtain 

Operational Loads for Complex Structures, U.S. Patent No. 7,715,994, issued, May 11, 2010.

11. Ko, William L., and Van Tran Fleischer, Extension of Ko Straight-Beam Displacement Theory to 

Deformed Shape Predictions of Slender Curved Structures, NASA/TP-2011-214657, April 2011. 

12. Kopmaz, Osman, and Omer Gundogdu, “On the Curvature of an Euler-Bernoulli Beam,”

International Journal of Mechanical Engineering Education, Vol. 31, Issue 2, p. 132, 2004.

13. Hodges, Dewey H., “Proper Definition of Curvature in Nonlinear Beam Kinematics,” AIAA Journal, 

Vol. 22, No. 12, pp. 1825-1827, December 1984.

14. Faupel, Joseph, H., Engineering Design, John Wiley and Sons, New York, 1964.
79

15. Roark, Raymond J., Formulas for Stress and Strain, McGraw-Hill Book Company, Inc., New York, 

1954.

16. Whetstone, W. D., SPAR Structural Analysis System Reference Manual, System Level 13A, Vol. 1, 

Program Execution, NASA CR-158970-1, December 1978.

17. Hodgeman, Charles D., Standard Mathematical Tables, 7-th edition, Chemical Rubber Publishing 

Co. Cleveland, Ohio, 1957.

18. Jutte, Christine, William L. Ko, Craig Stephens, John A. Bakalyar, W. Lance Richards, and Allen R. 

Parker, Deformed Shape Calculation of a Full-Scale Wing Using Fiber Optic Strain Data from a 

Ground Loads Test, NASA/TP-2011-215975, December 2011.