NASA/TP-2009-214643

Further Development of Ko Displacement Theory 

for Deformed Shape Predictions of Nonuniform 

Aerospace Structures

William L. Ko and Van Tran Fleischer

NASA Dryden Flight Research Center

Edwards, California

September 2009

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NASA/TP-2009-214643

Further Development of Ko Displacement Theory 

for Deformed Shape Predictions of Nonuniform 

Aerospace Structures

William L. Ko and Van Tran Fleischer

NASA Dryden Flight Research Center

Edwards, California

September 2009

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273
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.

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

(443) 757-5802

The structural shape prediction method described in this report is protected under U.S. Patent No. 7,520,176, 

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

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

Cover art: NASA Dryden Flight Research Center, art number 090014.
TABLE OF CONTENTS

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

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

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

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

NONUNIFORM CANTILEVER BEAMS....................................................................................................... 4

Discretization ................................................................................................................................................ 4

Slope Equations............................................................................................................................................. 5

Deflection Equations..................................................................................................................................... 6

FURTHER MATHEMATICAL DEVELOPMENT ........................................................................................ 6

Slope Equation .............................................................................................................................................. 7

Deflection Equations..................................................................................................................................... 7

Limit-Case Mathematical Problems.............................................................................................................. 8

SLIGHTLY NONUNIFORM CANTILEVER BEAMS .................................................................................. 9

Slope Equations............................................................................................................................................. 9

Deflection Equations................................................................................................................................... 10

UNIFORM CANTILEVER BEAMS ............................................................................................................. 12

Slope Equations........................................................................................................................................... 12

Deflection Equations................................................................................................................................... 12

Relative Magnitude of Strain Terms........................................................................................................... 13

TWO-POINT SUPPORTED BEAMS............................................................................................................ 15

Left End Simply Supported......................................................................................................................... 15

Left End Fixed ............................................................................................................................................ 17

TWO-LINE STRAIN-SENSING SYSTEM................................................................................................... 17

Sensor Installation Examples...................................................................................................................... 17

Wing Box ................................................................................................................................................ 18

Cantilever Tubular Beam........................................................................................................................ 18

Deflections .................................................................................................................................................. 18

Cross-Sectional Twists................................................................................................................................ 19

APPLICATION TO FREE-FREE BEAMS ................................................................................................... 19

APPLICATION TO STRUCTURAL PANELS............................................................................................. 20

VALIDATION OF KO DISPLACEMENT THEORY .................................................................................. 20

Input Strains................................................................................................................................................ 20

Tapered Tubular Beams.............................................................................................................................. 21

Cantilever Beams.................................................................................................................................... 21
iv

Bending Only ...................................................................................................................................... 21

Bending and Torsion........................................................................................................................... 22

Two-Point Supported Beams .................................................................................................................. 23

Simply Supported Ends....................................................................................................................... 23

Fixed Ends........................................................................................................................................... 24

Tapered Wing Boxes................................................................................................................................... 24

Depth-Tapered Wing Boxes.................................................................................................................... 25

Unswept Wing Boxes.......................................................................................................................... 25

Swept Wing Boxes.............................................................................................................................. 26

Width-Tapered Wing Boxes ................................................................................................................... 27

Double-Tapered Wing Boxes.................................................................................................................. 27

EXACT AND EXPANDED DEFLECTION EQUATIONS.......................................................................... 28

CONCLUDING REMARKS.......................................................................................................................... 29

FIGURES ........................................................................................................................................................ 31

APPENDIX A: DERIVATION OF DEFLECTION EQUATIONS FOR NONUNIFORM CANTILEVER BEAMS...58

APPENDIX B: EXPANSION OF LOGARITHMIC TERMS.........................................................................................64

APPENDIX C: DERIVATIONS OF DEFLECTION EQUATION FOR SLIGHTLY NONUNIFORM

CANTILEVER BEAMS...................................................................................................................................................67

APPENDIX D: DERIVATIONS OF DEFLECTION EQUATIONS FOR UNIFORM CANTILEVER BEAMS .........74

APPENDIX E: RELATIVE MAGNITUDES OF STRAIN TERMS FOR UNIFORM CANTILEVER BEAMS .........77

REFERENCES................................................................................................................................................ 79
ABSTRACT

The Ko displacement theory previously formulated for deformed shape predictions of nonuniform beam 

structures is further developed mathematically. The further-developed displacement equations are expressed 

explicitly in terms of geometrical parameters of the beam and bending strains at equally spaced strainsensing stations along the multiplexed fiber-optic sensor line installed on the bottom surface of the beam. 

The bending strain data can then be input into the displacement equations for calculations of local slopes, 

deflections, and cross-sectional twist angles for generating the overall deformed shapes of the nonuniform 

beam. The further-developed displacement theory can also be applied to the deformed shape predictions of 

nonuniform two-point supported beams, nonuniform panels, nonuniform aircraft wings and fuselages, and 

so forth. The high degree of accuracy of the further-developed displacement theory for nonuniform beams is 

validated by finite-element analysis of various nonuniform beam structures. Such structures include tapered 

tubular beams, depth-tapered unswept and swept wing boxes, width-tapered wing boxes, and double-tapered 

wing boxes, all under combined bending and torsional loads. The Ko displacement theory, combined with 

the fiber-optic strain-sensing system, provide a powerful tool for in-flight deformed shape monitoring of 

unmanned aerospace vehicles by ground-based pilots to maintain safe flights.

NOMENCLATURE

 

c half depth of uniform beam (distance from neutral surface to bottom or top surface), in.

CG center of gravity

ci half depth of nonuniform beam at strain-sensing station, xi (distance from neutral surface 

to strain-sensing station, xi ), in.

cn value of ci at free end (beam tip) strain-sensing station, xn , in.

cn c0 beam tip–root depth ratio 

c(x) half depth of nonuniform beam at point x, in.

c0 value of ci at fixed end (beam root) strain-sensing station, x0 , in.

d chord-wise distance between two span-wise parallel strain-sensing lines, in.

deg degree

di chord-wise distance between tapered front and rear strain-sensing stations at { xi , x′

i }, in.

E Young modulus, lb/in2

e calculated nonzero deflection error at right fixed end of two-point supported beam, in. 

G shear modulus, lb/in

2

I moment of inertia, in4

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

j index

l length of beam, in.

M bending moment, in-lb

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

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

P applied load, lb

SPAR Structural Performance And Resizing

T applied torque, in-lb

t thickness of wall, in.

w width of uniform wing box, in.

wn width of wing box tip, in.

w0 width of wing box root, in.
2

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

xi axial coordinate (or symbol) associated with 

 

i-th strain-sensing station, in. 

yi deflection in y-direction at xi , in. 

yi

B deflection in y-direction at xi for two-point supported beam, in.

y(x) deflection in y-direction at axial location, x, in.

α half of angle, sustained by d, separating two strain-sensing lines on tubular beam, deg

β wing box swept angle, deg

γ i surface shear strain, rad

Δl = l n , distance between two adjacent strain-sensing stations, in.

εi surface bending strain at strain-sensing station, xi , in/in.

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

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

θ(x) slope of deformed beam at axial location, x, rad or deg

ν Poisson’s ratio

σi axial stress at strain-sensing station, xi , lb/in2

σ(x) surface bending stress at axial location, x, lb/in2

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

( )′ quantity associated with rear strain-sensing line

INTRODUCTION

Conventional aerospace instrumentation systems, such as those required for in-flight deformed shape 

monitoring of aircraft, are subject to strict weight and size restrictions. The existing electro-optical flight 

deflection measurement system (ref. 1) and conventional strain gage measurement system are too heavy 

(because of strain gage lead wires) for lightweight flying vehicles. The most attractive candidate for an inflight deformed shape measurement system is the fiber Bragg grating (FBG) sensing system, because the

sensing lines are extremely small and lightweight (approximately the size of human hairs), possess excellent 

flexibility, and can be highly multiplexed with hundreds of sensors located on single fibers at desired 

sensing intervals.

When these highly multiplexed sensors are installed on beam-like structures (such as aircraft wings), the 

measured strain data can be input into special algorithms to calculate the deflections and cross-sectional 

twist angles of the beam-like structure for deformed shape predictions. Special algorithms based on the Ko 

displacement theory are described in detail in reference 2. In the mathematical formulations, the beam is 

first discretized into multiple domains. Based on the classical beam theory, the displacement theory (ref. 2) 

was formulated to predict the deformed shapes (slopes, deflections, and cross-sectional twist angles) of 

cantilever beams (uniform, tapered, slightly tapered, stepwise tapered) using the fiber-optic measured strain 

data as inputs. 

The Ko displacement equations, combined with the onboard fiber-optic strain-sensing system (called 

Method For Real-Time Structure Shape-Sensing, U.S. Patent No. 7,520,176, issued April 21, 2009; ref. 3),

provide a powerful tool for in-flight deformed shape monitoring of unmanned aerospace vehicles by 

ground-based pilots to maintain safe flights. In addition, the wing shape that is monitored in real time could 

then be input into the aircraft control system for aeroelastic wing shape control. 
3

In the original formulation of the Ko displacement theory (refs. 2, 3) for nonuniform cantilever beams 

(for example, tapered, slightly tapered), the displacement equations are written in indicial recursion 

relationships because of multiple beam domains, and they need to be fully developed into more compact 

mathematical forms. In this report, the displacement equations for nonuniform beams (cantilever and twopoint supported beams) formulated previously (refs. 2, 3) are further developed into more compact 

mathematical forms so that the beam deflections at each strain-sensing station can be expressed exclusively 

in terms of inboard geometrical parameters of the beam and inboard surface bending strains. The accuracy 

of the further-developed displacement theory for nonuniform beams is validated by finite-element analysis 

of various nonuniform beam structures (tapered tubular beams, such as cantilever and two-point supported 

beams, depth-tapered unswept and swept wing boxes, width-tapered wing boxes, and double-tapered wing 

boxes) under different loading conditions. 

BASICS OF THE KO DISPLACEMENT THEORY

The formulation of the displacement theories is built upon the classical beam differential equation 

(elastic curvature of a deformed uniform beam) given by (refs. 4, 5)

d2y

dx2 = M (x)EI (1)

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

Young modulus, and I is the moment of inertia. 

At the cross section of the beam, x, the bending moment, M (x) (eq. (1)), induces the bending stress, 

σ(x) , at the bottom (or top) fiber of the beam (outermost point from the neutral axis) (refs. 2–5) with the 

magnitude proportional to M (x) according to

σ(x) = M (x)c

I (2)

in which c is the half depth (distance from the neutral axis to the bottom or top surface) of the uniform 

beam. The bending stress, σ(x) , is related to the bending strain, ε(x), at the same point on the bottom (or 

top) surface of the beam through Hooke’s law as 

ε(x) = σ(x)

E (3)

When equations (2) and (3) are combined, the bending moment, M (x), can be expressed in terms of the 

bending strain, ε(x), as

M (x) = EI

ε(x)

c

(4)
4

In light of equation (4), beam differential equation (1) becomes

d2y

dx2 = ε(x)

c

(5)

Note that under the present strain formulation, beam differential equation (5) contains only the half depth of 

the beam, c , and the bending strain, ε(x). The flexural rigidity, EI, is eliminated.

Beam differential equation (5) for the uniform beam could be used with sufficient accuracy for 

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

half depth of the beam, c, is no longer constant but is a function of x (that is, c = c(x) ). Hence, for the 

nonuniform beam, equations (4) and (5) can be modified, respectively, to

M (x) = EI

ε(x)

c(x) (6)

and

d2y

dx2 = ε(x)

c(x) (7)

The formulation of the Ko displacement theory (ref. 2) for nonuniform beams is based upon modified 

beam differential equation (7). By discretizing the beam into multiple domains, and assuming { c(x), ε(x) } to 

be piecewise linear, one could integrate equation (7) over each domain to yield local slope and deflection 

equations for generating the overall deformed shape of the beam (ref. 2). The Ko displacement theory 

requires further mathematical development as described in the subsequent sections.

NONUNIFORM CANTILEVER BEAMS

The nonuniform cantilever beam is defined as a cantilever beam in which the cross section varies slowly 

along the axis of the beam. This type of beam can also be called a weak nonuniform beam.

Discretization

Figure 1 shows a nonuniform cantilever beam with a length, l, and varying depth, 2 c(x) (for a tubular 

beam, c(x) is the local radius). Let the fiber-optic strain-sensing line be multiplexed with n +1

(0, 1, 2, 3, ..., n) number of equally spaced strain-sensing stations, xi (i = 0, 1, 2, 3, ..., n), at axial 

locations, x = xi , on the bottom surface of the beam, and let the measured bending strains be denoted by εi

(i = 0, 1, 2, 3, ..., n). Note from figure 1 that the first and last strain-sensing stations, { x0, xn }, are located 

at the fixed end ( x = 0 ) and at the free end ( x = l ), respectively. The index, i (= 0, 1, 2, 3, ..., n), is also 

used as the identification number of the strain-sensing stations. The nonuniform cantilever beam is now 

divided into n sections with identical domain length, Δl = l / n . The nonuniform beam can then be idealized 

as a piecewise linearly tapered beam (either tapering down or tapering up) between any two adjacent 
5

strain-sensing stations, { xi−1, xi }. Under this assumption, the half depth of the beam, c(x), in the region, 

xi−1 ≤ x ≤ xi , between the two adjacent strain-sensing stations, { xi−1, xi }, can be expressed as a linear 

function of x as

c(x) = ci−1 − (ci−1 − ci)

x − xi−1

Δl ; xi−1 ≤ x ≤ xi (8)

in which { ci−1, ci } are the values of c(x) at the strain-sensing stations, { xi−1, xi }, respectively. Likewise, 

the bending strain, ε(x), in the region, xi−1 ≤ x ≤ xi , between the two adjacent strain-sensing stations, 

{ xi−1, xi }, can be assumed to change linearly with x as

ε(x) = εi−1 − (εi−1 − εi)

x − xi−1

Δl ; xi−1 ≤ x ≤ xi (9)

in which { εi−1, εi } are the values of ε(x) at the strain-sensing stations, { xi−1, xi }, respectively.

If c(x) is a weak linear function of x (small slope term in eq. (8)), then the bending moment for the 

nonuniform beam, M (x) = EIε(x)/ c(x) (eq. (6)), will vary almost linearly between the two adjacent strainsensing stations, { xi−1, xi }, because the higher order terms in the binomial expansion of the denominator, 

[c(x)]

−1 , can be neglected. This argument may not hold, however, if c(x) is not a weak linear function of x, 

because the higher order terms in the binomial expansion of the denominator [c(x)]

−1 must be retained. 

For the uniform beam case, [ c(x) = c = constant], the bending moment, M(x), is directly proportional to 

the bending strain, ε(x), according to equation (4). Therefore, if the strain, ε(x), is a linear function of x, 

then M(x) is also a linear function of x.

Figure 2 shows the slopes, θi (i = 0, 1, 2, 3, ..., n), and deflections, yi (i = 0, 1, 2, 3, ..., n), at the 

strain-sensing stations, xi (i = 0, 1, 2, 3, ..., n), on the lower surface of the deformed tubular cantilever 

beam (using the uniform tubular beam as an example).

Slope Equations 

The slope, tanθ(x) , of the nonuniform beam at the axial location, x, between the two adjacent strainsensing stations, xi−1 ≤ x ≤ xi , can be obtained by integrating equation (7) once and enforcing the 

continuity of slope at the next inboard strain-sensing station, xi−1. Namely,

tanθ(x) = d2

y

dx2 dx

xi−1

x

∫

Slope increment 



+ tanθi−1

Slope at xi−1

 = ε(x)

c(x)

dx

xi−1

x

∫ + tanθi−1 ; xi−1 ≤ x ≤ xi (10)

in which tanθi−1 is the slope at xi−1.
6

Substituting beam half depth equation (8) and strain equation (9) into slope equation (10), and carrying 

out the integration to the upper limit, x = xi , causes equation (10) to yield the slope, tanθi [ ≡ tanθ(xi) ], at 

the strain-sensing station, xi , as (see Appendix D of ref. 2) 

tanθi = Δl εi−1 − εi

ci−1 − ci

+

εi−1ci − εici−1

(ci−1 − ci)

2 loge

ci

ci−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ tanθi−1 (11)

When i = 1, the slope, tanθ1−1 = tanθ0 , will be the slope at the left fixed end, which is zero ( tanθ0 = 0 ) for 

the cantilever beam. If the reference left end is simply supported (as in the two-point supported beam case), 

then the slope, tanθ0 , is nonzero ( tanθ0 ≠ 0 ).

Deflection Equations 

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

strain-sensing stations, { xi−1, xi }, can be obtained by integrating slope equation (10) and enforcing the 

continuity of deflection at the adjacent inboard strain-sensing station, xi−1. Namely,

y(x) = tanθ(x)dx

xi−1

x

∫

Integration of slope



+ yi−1

Deflection

at xi−1

 = ε(x)

c(x)

dx dx

xi−1

x

∫ xi−1

x

∫

Deflection increment



+ tanθi−1 xi−1

x

∫ dx

Deflection at x

due to tanθi−1



+ yi−1

Deflection

at xi−1



 ( xi−1 ≤ x ≤ xi )

(12)

in which yi−1 is the deflection at xi−1.

Carrying out the double and single integrations of deflection equation (12) in light of beam depth 

equation (8) and strain equation (9), we obtain the deflection, yi [ ≡ y(xi) ], at the strain-sensing station, xi , 

as (see Appendix D of ref. 2) 

yi = (Δl)

2 εi−1 − εi

2(ci−1 − ci)

− εi−1ci − εici−1

(ci−1 − ci)

3 ci loge

ci

ci−1

+ (ci−1 − ci) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ yi−1 + Δl tanθi−1 (13)

When i = 1, the deflection and slope, { y1−1 = y0 , tanθ1−1 = tanθ0 }, become the deflection and slope at the 

reference left fixed end where { y0 = 0, tanθ0 = 0 }. If the left end is simply supported (as in the two-point 

supported beam case), the deflection, y0 , is zero ( y0 = 0 ), but the slope, tanθ0 , is nonzero ( tanθ0 ≠ 0 ).

FURTHER MATHEMATICAL DEVELOPMENT

Equations (11) and (13) are the recursion formulae. In this section these equations are further developed 

to more compact mathematical forms expressed explicitly in terms of beam geometry, { Δl , ci }, and 

bending strains, εi (fig. 1).
7

Slope Equation

Equation (11) is a recursion formula showing that the slope, tanθi , at the strain-sensing station,

 

xi , is

expressed in terms of the slope, tanθi−1 , at the adjacent inboard strain-sensing station, xi−1. Applying the 

descending indices relationship causes slope equation (11) to become the summation form as 

(see Appendix A for details)

tanθi = Δl

ε j−1 − ε j

c j−1 − c j

+

ε j−1c j − ε jc j−1

(c j−1 − c j)

2 loge

c j

c j−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i

∑ + tanθ0 (14)

in which tanθ0 is the slope at the left fixed end (reference strain-sensing station, x0 ) and is zero 

( tanθ0 ≠ 0 ). For a simply supported left end (as in the two-point supported beam case), however, the slope, 

tanθ0 , is nonzero ( tanθ0 ≠ 0 ). Note that the slope, tanθi , is determined from the geometrical and strain 

data evaluated at all the inboard strain-sensing stations ( x0,, x1, x2,..., xi ), including the current strainsensing station, xi .

Deflection Equations 

Under the present discretization formulation, deflection equation (13) is also a recursion formula 

showing that the deflection, yi , at the strain-sensing station, xi , is expressed in terms of the deflection, 

yi−1 , and the slope, tanθi−1 , at the adjacent inboard strain-sensing station, xi−1. By applying the 

successive descending recursion relationships for yi−1 in light of equation (13), and writing equation (14) 

for tanθi−1 , we could express deflection equation (13) for 

 

yi

 with two series summations (with different 

summation limits) containing only the geometrical parameters and bending strains (see Appendix A for 

derivations). Namely,

yi = (Δl)

2 ε j−1 − ε j

2(c j−1 − c j)

−

ε j−1c j − ε jc j−1

(c j−1 − c j)

3 c j loge

c j

c j−1

+ (c j−1 − c j)

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭ j=1 ⎪

i

∑

Contributions from deflection terms



 + (Δl)

2 (i − j)

ε j−1 − ε j

c j−1 − c j

+

ε j−1c j − ε jc j−1

(c j−1 − c j)

2 loge

c j

c j−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i−1

∑

Contributions from slope terms



+ y0 + (i)Δl tanθ0

(15)

in which y0 = tanθ0 = 0 at the fixed end (reference strain-sensing station, x0 ). For a simply supported left 

end (as in the two-point supported beam case), however, the slope, tanθ0 , is nonzero ( tanθ0 ≠ 0 ). Note 

that the deflection, yi , is expressed in terms of the geometrical and strain data evaluated at all the inboard 

strain-sensing stations ( x0,, x1, x2,..., xi ), including the current strain-sensing station, xi . 
8

After grouping the terms, we can rewrite equation (15) in an alternative form (see Appendix A for 

details) as

yi = (Δl)

2 [2(i − j) +1] ε j−1 − ε j

2(c j−1 − c j)

−

ε j−1c j − ε jc j−1

(c j−1 − c j)

3 c j loge

c j

c j−1

+ (c j−1 − c j)

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭ j=1 ⎪

i

∑

 + (Δl)

2 (i − j)

ε j−1c j − ε jc j−1

(c j−1 − c j)

2 loge

c j

c j−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i−1

∑ + y0 + (i)Δl tanθ0

(16)

in which y0 = tanθ0 = 0 at the fixed end (reference strain-sensing station, x0 ).

By inputting the geometrical parameters of the beam and the bending strain data into deflection equation 

(16), we can calculate the deflection of the beam, yi , at any strain-sensing station, xi . Therefore, the overall 

deformed shape of the nonuniform cantilever beam can be generated without the need to know the flexural 

rigidity of the beam, EI. The ability to generate deformed shapes without this information is a powerful 

characteristic of the Ko displacement theory, because it does not require the tedious and time-consuming 

calculations of the moment of inertia I(x), which changes along the span of a complex beam-like structure 

such as an aircraft wing.

Note that the fixed-end condition, y0 = tanθ0 = 0 , will cause the calculated deflection, yi (or slope,

tanθi ), to be relative to the tangent line passing through the reference strain-sensing station, x0 , at the fixed 

end. Thus, deflection equation (16) is applicable to both stationary and moving cantilever beams such as 

aircraft wings during flight.

Furthermore, deflection equation (16) is formulated for the nonuniform cantilever beam under lateral 

loading only (bending). If the beam is also subjected to axial loading, the measured strains, εi

(i = 0, 1, 2, 3, ..., n), will contain both bending and axial-load strain components. To remove the axial-load 

strain components, an additional strain-sensing line is needed on the upper surface along the axis of the 

beam. If the neutral axis is located at the half depth of the beam, the axial-load strain components can be 

removed by averaging the magnitudes of the lower and upper surface strains measured at the same strainsensing cross section to yield the true magnitude of the bending strains.

Limit–Case Mathematical Problems

The mathematical forms of slope equation (14) and deflection equation (16) cannot be applied directly 

to the uniform beam, because at the limit of ( ci−1 = ci ), the logarithmic terms and the denominators with 

( ci−1 − ci ) factors in equations (14) and (16) will go to zero (that is, loge(ci ci−1) = 0 , ci−1 − ci = 0), 

causing equations (14) and (16) to be indeterminate. The way to circumvent this limit-case mathematical 

problem is to first consider the case of a slightly nonuniform cantilever beam and use the perturbation 

method to expand the logarithmic terms in series form. The resulting series expansion will contain 

( ci−1 − ci ) factors in the numerators, which will cancel out the ( ci−1 − ci ) factors in the denominators. 

Thus, the modified mathematical forms for equations (14) and (16) will be able to degenerate into the limit-
9

case expressions for uniform beams, ( ci−1 = ci ). The modified slope and deflection equations for slightly 

nonuniform cantilever beams are derived in the subsequent sections. 

SLIGHTLY NONUNIFORM CANTILEVER BEAMS

When the cantilever beam is slightly tapered (that is, (ci ci−1)→1 ), the logarithmic terms in the slope 

and deflection equations (eqs. (14) and (16)) will approach zero (that is, loge(ci ci−1)→ 0 ). Therefore, the 

perturbation method must be used to expand the logarithmic term, loge(ci ci−1) , in the neighborhood of 

(ci ci−1) ≈ 1 to obtain nonzero mathematical expressions, so that the slope and deflection equations can 

degenerate into those for the uniform cantilever beam case, (ci ci−1) = 1 .

Slope Equations 

For a slightly tapered beam, the logarithmic term, loge(ci ci−1) , in slope equation (11) can be 

expanded in the neighborhood of (ci ci−1) ≈ 1 (that is, loge(ci ci−1) ≈ 0 ). Carrying out the expansion up to 

the second-order terms in (ci−1 − ci)

2 causes the logarithmic term, loge(ci ci−1) , to take on the following 

form as 

 

ci approaches ci−1 [(ci ci−1)→1 ] (see Appendix B for details of mathematical expansions;

refs. 2, 6):

loge

ci

ci−1

≈ ci−1 − ci

2ci−1

2 (ci − 3ci−1) ; 

ci

ci−1

→1 (17)

Substitution of equation (17) into slope equation (11) yields the following slope equation for the slightly 

nonuniform cantilever beam (ref. 2), with the ( ci−1 − ci ) factor in the denominators eliminated: 

tanθi = Δl

2ci−1

2 − ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + εi

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθi−1 ; ci−1 → ci (18)

Applying the descending recursion relationship causes slope equation (18) to become

tanθi = Δl

2

1

c j−1

2 − c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i

∑ + tanθ0 ; ci−1 → ci (19)

Note that slope equations (18) and (19) do not contain the ( ci−1 − ci ) factor and can now be used for the 

limit case ( ci−1 = ci ) of the uniform beam.
10

Deflection Equations 

When the beam becomes slightly tapered (that is, ci → ci−1), the term, ci loge(ci ci−1) + ci−1 − c ⎡ ( i) ⎣ ⎤

⎦ , 

in equation (13) can be expanded in the neighborhood of (ci ci−1) ≈ 1 (that is, loge(ci ci−1) ≈ 0 ). Carrying 

out the expansion up to the third-order terms in (ci−1 − ci)

3 causes the term, 

ci loge(ci ci−1) + ci−1 − c ⎡ ( i) ⎣ ⎤

⎦ , to take on the following form as ci approaches 

 

ci−1[(ci ci−1)→1 ] (see 

Appendix B for details of mathematical expansions; refs. 2, 6):

ci loge

ci

ci−1

+ (ci−1 − ci) ⎡

⎣

⎢ ⎤

⎦

⎥ ≈ (ci−1 − ci)

2

6ci−1

3+ (ci−1 − ci)

ci−1

⎡

⎣

⎢ ⎤

⎦

⎥ ; 

ci

ci−1

→1 (20)

In light of equation (20), deflection equation (13) becomes

yi = (Δl)

2

6ci−1

3− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + εi

⎡

⎣

⎢ ⎤

⎦

⎥ + yi−1 + Δl tanθi−1 ; 

ci

ci−1

→1 (21)

which contains no ( ci−1 − ci ) factor in the denominators.

Applying the descending recursion relationship of equation (21) for yi−1 , and writing slope

equation (19) for tanθi−1 causes deflection equation (21) to take on a form that has two series summations 

(see Appendix C), as

(22)

in which y0 = tanθ0 = 0 at the fixed end (reference strain-sensing station, x0 ) of the cantilever beam. Note 

from equation (22) that the two summation upper limits are different. 

Through grouping similar terms in the two series summations, we can write equation (22) in a final 

compact form consisting of only one series summation (see the derivation in Appendix C). Namely,

yi = (Δl)

2

6

1

c j−1 j=1

i

∑ 3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + (i)Δl tanθ0 (23)

In equation (23), y0 = tanθ0 = 0 for the cantilever beam; therefore, the terms, y0 + (i)Δl tanθ0 , can be 

dropped. The y0 + (i)Δl tanθ0 terms are purposely kept, however, to make equation (23) applicable to other 

yi = (Δl)

2

6

1

c j−1

3− c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i

∑

Contributions from deflection terms



+ (Δl)

2

2

(i − j)

c j−1

2 − c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i−1

∑

Contributions from slope terms



+ y0 + (i)Δl tanθ0
11

beam cases, such as simply supported beams, for which {y0 = 0, tanθ0 ≠ 0} at the reference strain-sensing 

station, x0 (see the section entitled, “Two-Point Supported Beams”). Note that equation (23) is applicable 

to both stationary and moving cantilever beams (for example, airborne aircraft wings), because the

deflection, yi , calculated from equation (23) is relative to the tangent line ( tanθ0 = 0 ) stemming from the 

reference strain-sensing station, x0 , which can be either stationary or moving (fig. 2). 

For easy visualization of the functional progression with ascending indices, equation (23) can be written 

out explicitly for different indices, i ( = 1,2,3, ..., n), as 

y1 = (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + Δl tanθ0 (23a)

y2 = (Δl)

2

6

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

9 − 4 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + 2Δl tanθ0 (23b)

y3 = (Δl)

2

6

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

9 − 4 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 4ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

15 − 7 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + y0 + 3Δl tanθ0

(23c)

y4 = (Δl)

2

6

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 4ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε2

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

+

 +

1

c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +10ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+ y0 + 4Δl tanθ0

(23d)

y5 = (Δl)

2

6

1

c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c3

9 − 4 c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 4ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

15 − 7 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 7ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c1

21−10 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 +10ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

27 −13 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +13ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+ y0 + 5Δl tanθ0

(23e)

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

(23f)

in which the fixed-end condition, y0 = tanθ0 = 0 , holds.

Equation (23) explicitly shows that the deflection, yi , at the strain-sensing station, xi , is the function of 

the inboard half depth of the beam ( c0,, c1, c2,..., ci ) and the associated inboard strains ( ε0,, ε1, ε2,..., εi ), 

yn = (Δl)

2

6

1

cn− j

3(2 j −1) − (3j − 2)

cn− j+1

cn− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εn− j + (3j − 2)εn− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

n

∑ + y0 + nΔl tanθ0
12

including the values of { ci, εi } at the current strain-sensing station, xi , in which the deflection, yi , is 

evaluated. By inputting the geometry of the beam and the measured strain data into deflection equation (23), 

we can calculate the deflection, yi , at any strain-sensing station and thereby generate the deformed shape of 

the slightly nonuniform cantilever beam such as an aircraft wing. Note that deflection equation (23), 

developed for the slightly nonuniform beam, can now be used for the uniform beam case by setting 

ci = ci−1 = c as a limit case without creating the mathematical breakdown problem encountered in

deflection equations (15) and (16).

UNIFORM CANTILEVER BEAMS 

When the nonuniformity of the beam diminishes (that is, ci = ci−1 = c ; fig. 2), slope equations (18) and 

(19) and deflection equations (21) and (23), developed for the slightly nonuniform cantilever beam, can 

degenerate into the limit forms for the uniform cantilever beam.

Slope Equations

For the limit case of a uniform beam, ci = ci−1 = c , slope equations (18) and (19) developed for the 

slightly nonuniform cantilever beam degenerate, respectively, into the following limit forms for the uniform 

cantilever beam:

tanθi = Δl

2c

(εi−1 + εi) + tanθi−1 (24)

tanθi = Δl

2c

(ε j−1 + ε j)

j=1

i

∑ + tanθ0 (25)

Deflection Equations

Also, as the nonuniformity of the beam diminishes (that is, ci = ci−1 = c ), deflection equations (21) and 

(23) for the slightly nonuniform cantilever beam degenerate, respectively, into limit forms for the uniform 

cantilever beam as

yi = (Δl)

2

6c

(2εi−1 + εi) + yi−1 + Δl tanθi−1 (26)

and

yi = (Δl)

2

6c

(3j −1)εi− j + (3j − 2)εi− j+1 ⎡

⎣ ⎤⎦

j=1

i

∑ + y0 + (i)Δl tanθ0 (27)

in which y0 = tanθ0 = 0 holds at the left fixed end.
13

After grouping the strain terms of identical indices, we can modify equation (27) into the following 

more compact form (see details of derivation in Appendix D):

yi = (Δl)

2

6c

(3i −1)ε0 + 6 (i − j)ε j

j=1

i−1

∑ + εi

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + (i)Δl tanθ0 (28)

Equation (28) can be written out explicitly for different indices, i (= 1, 2, 3, ..., n), so that the functional 

progression of equation (28) with the increasing value of i can easily be seen:

y1 = (Δl)

2

6c

(2ε0 + ε1) + y0 + Δl tanθ0 (28a)

y2 = (Δl)

2

6c

5ε0 + 6ε1 + +ε ( 2 ) + y0 + 2Δl tanθ0 (28b)

y3 = (Δl)

2

6c

8ε0 + 6(2ε1 + ε2 ) + ε [ 3] + y0 + 3Δl tanθ0 (28c)

y4 = (Δl)

2

6c

11ε0 + 6 3ε1 + 2ε2 + ε ( 3) + ε 4 ⎡

⎣ ⎤⎦ + y0 + 4Δl tanθ0 (28d)

y5 = (Δl)

2

6c

14ε0 + 6 4ε1 + 3ε2 + 2ε 3 + ε ( 4 ) + ε5 ⎡

⎣ ⎤⎦ + y0 + 5Δl tanθ0 (28e)

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

yn = (Δl)

2

6c

(3n −1)ε0 + 6 (n − j)ε j

j=1

n−1

∑ + εn

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + nΔl tanθ0 (28f)

If the terms, y0 and Δl tanθ0 , are removed, then deflection equation (28f) will be identical to deflection 

equation (12e) of reference 2 for the uniform cantilever beam. Equation (28) explicitly shows that the 

deflection, yi , at the strain-sensing station, xi , is obtained by summing up only the inboard strains 

( ε0,, ε1, ε2,..., εi ), including the strain, εi , at the current strain-sensing station, xi . 

Relative Magnitude of Strain Terms

The tip deflection, yn , expressed in equation (28f) is the summation of the strain data obtained at all the 

inboard strain-sensing stations. This section examines the relative magnitude of those strain terms, because 

each strain term has a different coefficient. The example used is the classical uniform cantilever beam under 
14

tip vertical load, P. For this case, the bending strain, ε(x), is a linearly decreasing function of the axial 

coordinate, x, and becomes zero at the beam tip (refs. 4, 5). Namely,

ε(x) = 1− x

l

⎛

⎝

⎜ ⎞

⎠

⎟ ε0 (29)

in which l is the length of the beam, and ε0 is the maximum bending strain at the fixed end ( x = x0 ). Based 

on the strain function given in equation (29), the beam tip deflections, yn (eq. (28f)), for different values of 

n can be calculated as follows (see Appendix E for details):

For n = 1:

y1 = (Δl)

2

6c

(2ε0 + ε1) = l

2

3c

ε0 (30a)

For n = 2:

y2 = (Δl)

2

6c

5ε0 + 6ε1 + ε ( 2 )

 = l

2

3c

ε0 0.625

maximum

 + 0.375 + 0 ⎛

⎝

⎜

⎞

⎠

⎟ = l

2

3c

ε0

(30b)

For n = 4: 

y4 = (Δl)

2

6c

11ε0 + 6 3ε1 + 2ε2 + ε ( 3) + ε 4 ⎡

⎣ ⎤⎦

 = l

2

3c

ε0 0.34375 + 0.421875

maximum

 + 0.1875 + 0.046875 + 0

⎛

⎝

⎜

⎞

⎠

⎟ = l

2

3c

ε0

(30c)

For n = 8:

(30d)

In equations (30b)–(30d), the dominant terms are underlined and labeled, “maximum.”

y8 = (Δl)

2

6c

23ε0 + 6 7ε1 + 6ε2 + 5ε 3 + 4ε 4 + 3ε5 + 2ε6 + ε ( 7 ) + ε8 ⎡

⎣ ⎤⎦ 

 = l

2

3c

ε0 0.17969 + 0.28711

maximum

 + 0.21094 + 0.14648 + 0.09375 + +0.05273+ 0.02344 + 0.00586 + 0

⎛

⎝

⎜

⎞

⎠

⎟

 = l

2

3c

ε0 
15

Figure 3 compares the relative magnitude of the strain terms, inside the brackets of equations

(30a)–(30d), for different values of n. For n = 1–2, the maximum strain term is located at the fixed end ( x0 ). 

When n is greater than 2 ( n > 2 ), the maximum strain term shifts toward the next outboard strain-sensing 

station, x1 , which becomes the most dominant strain-sensing station. Thus, the accuracy of measured strain 

at the dominant strain-sensing station greatly affects the accuracy of the calculated beam deflections using 

equation (28).

TWO-POINT SUPPORTED BEAMS

For the case of a nonuniform cantilever beam, the fixed end ( y0 = tanθ0 = 0 ) is located at the left-hand 

side at the reference strain-sensing station, x0 . For a two-point supported beam (simple beam), the two 

ends can be under any support condition (fixed, simply supported, or elastically supported). 

Left End Simply Supported

For a simple beam with a simply supported left end, the deflection is zero ( y0 = 0 ), but the slope is 

nonzero ( tanθ0 ≠ 0 ). If deflection equation (23) ( y0 = tanθ0 = 0 ) for the cantilever beam is used to 

calculate the deflection curve of the simple beam case, the deflection curve will have a zero slope 

( tanθ0 = 0 ) at the left support, and a maximum deflection ( yn ≠ 0 ) at the right end (deflection curve at 

position A, fig. 4). Thus, to apply cantilever beam deflection equation (23) for the simple beam case, the 

unknown slope, tanθ0 , at the left support point must be determined first. By setting the right-end deflection, 

yn , to zero ( yn = 0 ) in equation (23f), we can determine the unknown slope, tanθ0 , at the simply 

supported left end as

tanθ0 = − Δl

6n

1

c j−1 j=1

n

∑ 3(2 j −1) − (3j − 2)

cn− j+1

cn− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εn− j + (3j − 2)εn− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

≡ − 1

nΔl

yn (31)

in which the yn is the deflection of the cantilever beam tip given by 

yn = (Δl)

2

6

1

c j−1

3(2 j −1) − (3j − 2)

cn− j+1

cn− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εn− j + (3j − 2)εn− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

n

∑

Equation (23-f); y0 =tanθ0 =0

 (32)

which is equation (23f) with the terms, y0 + nΔl tanθ0 , removed, because y0 = tanθ0 = 0 for the cantilever 

beam. 
16

With tanθ0 determined in equation (31), deflection equation (23) for the cantilever beam can be 

modified to calculate the deflection, yi

B , of the simple beam with a simply supported left end (the right end 

can be under any support condition) as 

yi

B = (Δl)

2

6

1

c j−1 j=1

i

∑ 3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

yi [equation (23); y0 =tanθ0 =0]



− i

n

yn

Shift factor

 (33)

In equation (33), the last term, (i / n)yn (i = 1, 2, 3, ..., n), is a linearly increasing shifting factor (with 

increasing i) for vertical downward shifting of the cantilever beam deflection curve from position A to 

position B, to convert it into the simple beam deflection curve (fig. 4; zero shift at the left end). Namely, the 

downward shifting is equivalent to the clockwise rotation of the inclined reference line 0A, with respect to 

the left support point 0, by θ0 deg to the horizontal position 0B with the length slightly shrunk (fig. 4). For 

easy visualization of the functional progression with increasing i, equation (33) is written out for different 

strain-sensing stations as

y1

B = (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ − 1

n

yn (33a)

y2

B = (Δl)

2

6

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

9 − 4 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

− 2

n

yn (33b)

y3

B = (Δl)

2

6

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

9 − 4 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 4ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

15 − 7 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

− 3

n

yn (33c)

y4

B = (Δl)

2

6

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 4ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε2

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +10ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

− 4

n

yn

(33d)

y5

B = (Δl)

2

6

1

c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c3

9 − 4 c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 4ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

15 − 7 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 7ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c1

21−10 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 +10ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

27 −13 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +13ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

− 5

n

yn

(33e)

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

yn

B = (Δl)

2

6

1

c j−1

3(2 j −1) − (3j − 2)

cn− j+1

cn− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εn− j + (3j − 2)εn− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

n

∑ − n

n

yn =0 (33f)
17

If the right end is also simply supported, the right-end slope, tanθn , at the strain-sensing station, xn , is 

nonzero ( tanθn ≠ 0 ), and slope equation (19) can be used to determine the unknown right-end slope, 

tanθn , in terms of the left-end slope, tanθ0 , calculated from equation (32) as

tanθn = Δl

2

1

c j−1

2 − c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

n

∑ + tanθ0

 = Δl

2

1

c j−1

2 − c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

n

∑ − 1

nΔl

yn

(34)

Note that the right-end slope, tanθn , does not appear in deflection equation (33); therefore, the knowledge 

of tanθn is not needed in the deflection calculations.

Left End Fixed

Deflection equation (23) for the nonuniform cantilever beam is also applicable to the nonuniform simple 

beam with the left end fixed (the right end can be either fixed or simply supported). The deflection, yn , at 

the right end, calculated from cantilever beam deflection equation (23f) with exact strain inputs, 

theoretically should be zero. If the calculated right-end deflection, 

 

yn , should show a small nonzero error, e

( yn = e << 1 ), however, then deflection equation (23) for the cantilever beam ( y0 = tanθ0 = 0 ) must be 

corrected for the present case as

yi

B = (Δl)

2

6

1

c j−1 j=1

i

∑ 3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

yi [equation (23); y0 =tanθ0 =0]



− i

n

e

 Correction

 term

 (35)

Similar to figure 4, equation (35) is used to proportionally shift the calculated deflection curve slightly 

downward or upward depending on the sign of e to make the right-end deflection zero ( yn

B = 0). Keep in 

mind that the degree of shifting for the present case is minuscule compared with the previous case (left end 

simply supported, eq. (33)). 

TWO–LINE STRAIN–SENSING SYSTEM

The two-line strain-sensing system proposed by Ko (ref. 2) is a powerful method for simultaneously 

monitoring the bending and torsional deformation of nonuniform cantilever beams under combined bending 

and torsion. The two-line strain-sensing system eliminates the need for installing the shear strain sensors to 

measure the surface distortions through which the cross-sectional rotations of the beam can be determined. 

Sensor Installation Examples

Installation of the two-line strain-sensing system for typical beam-like structures is discussed in the 

succeeding sections. These typical beam-like structures include a wing box and cantilever tubular beam.
18

Wing Box

Figure 5 shows the first example of applying the two-line strain-sensing system to a wing box structure 

under combined bending and torsion. The two strain-sensing lines (front and rear) can be installed along the 

lower edges (front and rear) of the wing box with chord-wise separation distance, di , at the strain-sensing 

cross section, x = xi . The two strain-sensing lines can be parallel ( di = constant) or tapered with the 

separation distance, di , decreasing toward the wingtip according to the horizontal taper rate of the wing.

Cantilever Tubular Beam

Figure 6 shows another example of the application of the two-line strain-sensing system to a tapered 

cantilever tubular beam under bending and torsion. The front and rear strain-sensing lines can be installed at 

a symmetrical angular location of ±α measured from the bottom of the tube. To obtain larger bending strain 

data, a smaller value of α is desired. But to obtain more accurate cross-sectional twist data, a larger α (that 

is, a larger separation distance) is desired. Therefore, α = ±30 ° may be a good compromise, because the 

separation distance, di , between the two strain-sensing lines will be equal to the tube radius. The magnitude 

of the strains measured at α = ±30 ° location (numerators of equation (23)) will be cosα times the strains 

(numerators of equation (23)) measured at the bottom of the beam. The vertical distance, ci , from the 

neutral axis to the strain-sensing stations (denominators of equation (23)), however, will be cosα times the 

tube radius, thereby canceling out the effect of cosα . Therefore, the averaged deflection curve generated by 

using the two-line strain-sensing system at α = ±30 ° will be equivalent to the deflection curve generated by 

using a single strain-sensing line along the bottom surface of the tube.

Deflections

The deflections, { yi , y′

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

x = xi (i = 0, 1, 2, 3, ..., n) (figs. 5, 6), can be calculated from deflection equation (23), which is rewritten 

(terms y0 + (i)Δl tanθ0 removed) in the following forms for the front and rear strain-sensing stations, 

respectively:

For the front strain-sensing line,

yi = (Δl)

2

6

1

ci− j j=1

i

∑ 3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 ; (i = 1, 2, 3, ..., n) (36)

For the rear strain-sensing line,

y′

i = (Δl)

2

6

1

ci− j

3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε′

i− j + (3j − 2)ε′i− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

i

∑ ; (i = 1, 2, 3, ..., n) (37)

By averaging the front and rear deflections, { yi , y′

i }, calculated from equations (36) and (37), respectively, 

we can eliminate the torsional effect to yield the true bending deflections of the wing box.
19

Cross-Sectional Twists

If di denotes the chord-wise separation distance between the front and rear strain-sensing cross section 

at x = xi (figs. 5, 6), then the cross-sectional twist angle, φi , at the strain-sensing cross section, x = xi , can 

be calculated from the cross-sectional twist angle equation

φi = sin−1 yi − y′

i

di

⎛

⎝

⎜

⎞

⎠

⎟ ; (i = 0, 1, 2, 3, ..., n) (38)

in which φ0 = 0 at the wing root. The deflections, { yi , y′

i }, can be calculated from deflection equations 

(36) and (37), respectively, using the measured bending strain data as inputs. 

For an aircraft wing (fig. 5), when the value of di is much larger than the wing depth, equation (38) 

could give sufficiently accurate φi without the need to install distortion sensors (see the section entitled, 

“Tapered Wing Boxes”), because the effect of surface shear strains on the bending strains is negligible. For 

the tubular structure (fig. 6) for which di is in the order of the radius, however, equation (38) may lose 

accuracy, because the surface shear strains will influence the bending strains (see the section entitled, 

“Tapered Tubular Beams”). An alternative method to obtain the cross-sectional twist angles,φi , for a 

tubular structure is to install a surface shear strain-sensing line for measurement of surface distortion angles, 

from which φi can be calculated. This alternative method is discussed in reference 2. 

APPLICATION TO FREE–FREE BEAMS

The fuselage of an aircraft during flight is a good example of a free-free unsupported beam (fig. 7). The 

fuselage is a tubular structure with varying cross sections and can be considered as a nonuniform beam so 

that deflection equations (16) and (23), developed for nonuniform and slightly nonuniform cantilever beams, 

respectively, can be applied. For monitoring the deflections of the airborne fuselage under either vertical or 

horizontal bending, the strain-sensing line can be installed axially on the bottom (belly) surface or on the 

side of the fuselage (fig. 7). 

Before deflection equation (16) or (23) is applied for the deformed shape predictions of the fuselage, 

some basics of equations (16) and (23) must be understood. Remember that in the formulation of deflection 

equations (16) and (23), the reference strain-sensing station, x0 , is located at the built-in end. With the 

built-in condition, y0 = tanθ0 = 0 , imposed, equations (16) and (23) are also applicable to the moving 

cantilever beams, because the deflection, yi , calculated from equation (23) is relative to the tangent line 

( tanθ0 = 0 ) stemming from the reference strain-sensing station, x0 , at the built-in end (fig. 7). Equations 

(16) and (23) can be applied to the free-free unsupported moving fuselage case using the following 

approach.

The fuselage can be considered two nonuniform cantilever beams joined together at the center-ofgravity (CG) cross section (fig. 7). During flight, the CG cross section has the least movement compared 

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

station, x0 , at the CG cross section and divide the strain-sensing line into two segments (fig. 7). One 

segment is for the fore section and the other for the aft section of the fuselage. Thus, by imposing 
20

y0 = tanθ0 = 0 at the reference strain-sensing station, x0 (considered as a built-in end), deflection 

equations (16) and (23) for the cantilever beam can be used to calculate the deflections of the fore and aft 

sections of the fuselage. The calculated deflection, yi , will then be the relative deflection with respect to the 

tangent line passing though the reference strain-sensing station, x0 , which moves (translates and rotates) 

with the fuselage (fig. 7).

APPLICATION TO STRUCTURAL PANELS

Deflection equations (33) and (35) developed for slightly nonuniform simple beams can also be used for 

deformed shape predictions of structural panels (for example, flat, cylindrical, shallow spherical, and wing 

panels). For the shape sensing of structural panels, multiple biaxial strain-sensing lines are required (fig. 8). 

The panel strip region lying beneath each strain-sensing line will behave as an embedded beam (cantilever 

or simple beam, depending on the panel edge support conditions); therefore, deflection equations (33) and 

(35) are applicable. Each surface strain sensor will then sense the bending strain of the fictitious beam 

embedded beneath the strain-sensing line and not sense the bending strain of an isolated beam. As such, 

each strain output is the true panel bending strain with a biaxial effect. Keep in mind that the bending strain 

of the beam that is embedded in the panel will increase if the embedded beam is isolated. Therefore, the 

one-dimensional beam strain does not need to be corrected into a two-dimensional panel strain. This feature 

is a powerful characteristic of the Ko displacement theory, which is applicable whether the structure is onedimensional and beam-like or two-dimensional and panel-like. The deflection equations previously 

developed for uniform beams (cantilever and two-point supported; ref. 2) were used for the deformed shape 

predictions of uniform-thickness square panels (four edges simply supported or four edges clamped), and 

were quite accurate in the panel shape predictions.

VALIDATION OF KO DISPLACEMENT THEORY

To examine the accuracy of the further-developed Ko displacement theory for nonuniform beams, 

several beam structure cases were chosen: (1) tapered tubular beams (cantilever and two-point supported), 

and (2) wing boxes (depth-tapered unswept, depth-tapered swept, width-tapered unswept, and doubletapered unswept). The Structural Performance And Resizing (SPAR) finite-element computer program

(ref. 7) was then used to generate both the nodal strains and nodal displacements for these example 

structures. All the structures considered were made of aluminum material.

Input Strains

To test the accuracy of the displacement theory for nonuniform beams, the fiber-optic measured strain 

data were needed for input to the deflection equations for the calculations of beam deflections. Because the 

measured bending strain data were not yet available from the experiments, the SPAR program was used to 

generate the bending strains. The SPAR bending strains were generated as follows.

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

then the corresponding bending strain, εi , for input to the deflection equations can be calculated from 

Hooke’s law, 

εi = σi

E (39)
21

Using the finite-element generated strains (eq. (39)), we can calculate the structural deflections and then 

compare them with the corresponding nodal displacements obtained from the SPAR displacement outputs 

for validating the accuracy of the Ko displacement theory. 

Alternatively, using the SPAR displacement outputs, we can also obtain the bending strain, εi , by 

dividing the element axial length change by the undeformed length of the element at the strain-sensing 

station, xi (ref. 2). This element displacement method gives bending strains that are practically identical to 

those obtained from the nodal stress method (eq. (39)) in the region of small deflections, but it starts to lose 

accuracy in the highly bent region near the free end. The SPAR displacement outputs are referred to the 

undeformed beam axis, and at increasing deflections near the highly bent beam tip, the true tensile strains 

(positive value) along the deformed beam axis, when projected on the undeformed beam axis, may appear as 

compressive strains (negative value). Thus, in the calculations of strains using the element displacement 

method, y-displacement components must be considered. Therefore, the simpler nodal stress method

(eq. (39)) was used in generating the input strains.

Tapered Tubular Beams

Each of the tapered tubular beams (cantilever, two-point supported) used in the SPAR analysis has a 

radius that linearly decreases from one end toward the other end. The tapered tubular beam tapers down not 

only in the vertical and horizontal planes, but also in any radial plane because of circular cross sections.

Table 1 lists the dimensions of each aluminum tapered tubular beam considered. 

Table 1. Dimensions of aluminum tapered tubular beams.

l, in. c0 , in. cn , in. cn / c0

100.5 4 4 1.00 (uniform)

100.5 4 3 0.75

100.5 4 2 0.50

100.5 4 1 0.25

Cantilever Beams

Figure 9 shows the finite-element model for a tapered cantilever tubular beam. The beam is subjected to 

a tip vertical load of P = 100 lb (fig. 9a), and a combined tip vertical load and tip torque of P = 100 lb and

T = 400 ×cn in-lb, respectively (fig. 9b).

Bending Only

Figure 10 shows the strain curves for different tapered cantilever tubular beams under bending only,

calculated from equation (39), using the SPAR element stress outputs. The strain curve is a straight line for 

the classical uniform beam case ( cn / c0 = 4 / 4 ) but becomes bow shaped as the depth ratio, cn / c0 , 

decreases. When the depth ratio reaches a minimum of cn / c0 = 1/ 4 , the strain curve is deeply bent by 

nearly 100°.

Figure 11 shows the beam deflection curves for different tapered cantilever tubular beams, calculated 

from deflection equation (16) (eq. (23) for the uniform beam), using the strain data at the strain-sensing 

stations shown in figure 10 (solid curves with solid circular symbols). For comparison, the deflections 

calculated from the SPAR program are also plotted (dashed curves with open circular symbols). The 

excellent correlation between the two sets of deflection curves shows the high degree of accuracy of the 

displacement theory (eqs. (16) and (23)) formulated for the nonuniform cantilever beams.
22

Keep in mind that in the formulation of the Ko displacement theory (ref. 2), the distribution of bending 

strains along the discretized cantilever beam was assumed to be piecewise linear as described by

equation (9). Therefore, the strain curves associated with this theory are the piecewise linear approximations 

(connecting the strain data points at the strain-sensing stations with straight lines) of the smooth strain 

curves shown in figure 10. Note from figure 10 that for slightly bent strain curves, the piecewise linear 

approximation could be quite accurate, but in a highly bent region of the strain curve for the lowest depth 

ratio, cn / c0 = 1/ 4 , the piecewise linear approximation may lose some accuracy locally. In spite of this 

shortcoming, the Ko displacement theory (eqs. (16) and (23)) still predicts highly accurate deflections for 

the whole range of depth ratios, 4 / 4 ≥ cn / c0 ≥ 1/ 4 (fig. 11). The same arguments also hold for the 

remaining examples presented in the succeeding sections.

Bending and Torsion

Figure 12 shows the SPAR-generated strain curves for the front (fig. 12a) and rear (fig. 12b) strainsensing lines for different tapered cantilever tubular beams under combined bending and torsion. The front 

(left) and rear (right) strain-sensing lines are 60° apart as shown in figure 6. The front and rear strain curves 

are very similar because of the proximity of the two strain-sensing lines. Note from figure 12 that the strain 

curves for the tapered tubes ( cn / c0 < 4 / 4 ) are bow shaped, and the degree of bend increases as the depth 

ratio, cn / c0 , decreases. When the depth ratio reaches a minimum of cn / c0 = 1/ 4 , the front and rear strain 

curves are deeply bent by nearly 100°, similar to the strain curves in the bending only cases (fig. 10). For the 

classical uniform cantilever beam case ( cn / c0 = 4 / 4 , fig. 12), the front and rear bending strain curves are 

linear except in the beam tip region. The abnormal deviation of bending strains (especially for the front 

strain curve) near the beam tip region could be attributed to the increasing local element distortions caused 

by the concentrated tangential loads applied at selected points at the beam tip to create torsion. Keep in mind 

that the bending strains have been calculated from the finite-element stresses.

If torsion is removed, the front and rear strain curves of the classical uniform cantilever beam case 

( cn / c0 = 4 / 4 ) will be perfectly straight lines as shown with dashed lines in figure 12 (see also fig. 10 for 

the cn / c0 = 4 / 4 case). With torsion added, each bending strain curve for the cn / c0 = 4 / 4 case (fig. 12) 

gradually deviates from the corresponding strain curve for bending only (dashed lines, fig. 12), and the 

deviation increases slightly toward the beam tip. 

Figure 13 shows the deflection curves along the front and rear strain-sensing lines for different tapered 

cantilever tubular beams, calculated from deflection equation (16) (eq. (23) for the uniform beam), using the 

strain data shown in figure 12 (solid curves with solid circular symbols). For comparison, the deflections 

calculated from the SPAR program are also plotted (dashed curves with open circular symbols. The 

excellent correlation between the two sets of deflection curves demonstrates the high accuracy of the 

displacement theory (eqs. (16) and (23)) formulated for nonuniform cantilever beams.

Figure 14 shows the cross-sectional twist angles, φi , for different tapered cantilever tubular beams,

calculated from equation (38) using the deflection data, { yi,, y′

i }, along the front and rear strain-sensing 

lines shown in figure 13. Graphically, the agreement between the twist angles calculated from the 

displacement theory and those calculated from the SPAR program does not appear as good as the agreement 

between the deflection curves (fig. (13)). It must be understood that the values of φi are quite small (less 

than 1°) because of the proximity (1− 4 in. apart) of the front and rear strain-sensing lines, and because of a 

low twisting moment, T = 400 ×cn in-lb. Therefore, slight differences (almost graphically inconspicuous) 

between the Ko displacement theory and SPAR program deflection data of figure (13) have a strong effect 

on the values of φi calculated from equation (38). Table 2 lists the beam tip deflections, { yn , y′

n }, of the 
23

front and rear strain-sensing lines, and the cross-sectional rotation angles, φi , calculated from the SPAR 

program and Ko displacement theory.

Table 2. Comparison of displacement data calculated from SPAR program with those calculated

from Ko displacement theory for different tapered cantilever tubular beams; l = 100.5 in., P = 100 lb,

T = 400 × cn in-lb.

SPAR Ko SPAR Ko SPAR Ko

4.0* 4.0* 0.72027 0.72677 0.70249 0.71001 0.25468 0.24007

4.0 3.0 0.88400 0.88706 0.86854 0.87291 0.29526 0.27025

4.0 2.0 1.16457 1.16199 1.15133 1.15148 0.37930 0.30109

4.0 1.0 1.78962 1.77362 1.77859 1.76628 0.63197 0.42056

* Uniform beam.

Note from table 2 that the differences between the φn values calculated from the SPAR program and 

those calculated from the Ko theory are minuscule, in the range of {0.0146, 0.0250, 0.0782, 0.2114} deg, 

for the depth ratios, cn c0 = {4 / 4, 3 / 4, 2 / 4, 1 / 4} , respectively. As seen in the subsequent wing box cases, 

in which the values of φi are larger because of greater distances between the front and rear strain-sensing 

lines, the prediction of φi turns out to be quite good.

Two–Point Supported Beams

Two cases of simple beams (two-point supported beams) were considered. Figure 15 shows the SPAR 

models of the two cases of tapered tubular simple beams, one with both ends simply supported (fig. 15a), 

and one with both ends fixed (fig. 15b). Each simple beam is subjected to a downward load of P = –200 lb 

at the center of the beam.

Simply Supported Ends 

Figure 16 shows the strain curves, calculated from equation (39) using the SPAR element stress outputs,

for different tapered tubular simply supported beams subjected to a downward force of P = –200 lb at the 

center of the beam. The strain curve is roof shaped for the classical uniform beam ( cn / c0 = 4 / 4 ). For the 

depth ratio, cn / c0 < 4 / 4 , the strain curves are slightly concave in the left-hand region but convex (bow 

shaped) in the right-hand region with the degree of bow bend increasing with the decreasing depth ratio, 

cn / c0 . At the lowest depth ratio of cn / c0 = 1/ 4 , the strain curve in the right-hand region (decreasing 

radius) is deeply bent by nearly 90°.

Figure 17 shows the beam deflection curves for different tapered tubular simply supported beams,

calculated from simply supported beam deflection equation (33) using the strain data shown in figure 16

(solid curves with solid circular symbols). These deflection curves are compared with the deflection curves 

calculated from the SPAR program (dotted curves with open circular symbols). The SPAR program gives 

slightly larger deflections, because the transverse shear effect is accounted for in the SPAR program. Note

from figure 17 that the transverse shear effect (difference between the two sets of deflection curves) is quite 

small for the uniform beam, cn / c0 = 4 / 4 , and gradually grows larger as the depth ratio, cn / c0 , decreases. 

c0,in. cn,in.

yn, in. y′

n, in. φn, deg
24

At the lowest depth ratio, cn / c0 = 1/ 4 , the transverse shear effect at the center of the beam amounts to 

3.63 percent of the beam center deflection. Remember, for the tapered cantilever tubular beam cases

(fig. 11), the transverse shear is practically inconspicuous for the whole range of depth ratios,

4 / 4 ≥ cn / c0 ≥ 1/ 4 .

To show that the difference between the deflections calculated from the SPAR program and those 

calculated from the displacement theory is caused by the transverse shear effect, the shear modulus, G, was

set very large ( G → ∞ ) in the SPAR program to eliminate the transverse shear effect. As expected, the 

resulting SPAR deflection curves (dashed curves with open square symbols, fig. 17) practically fall on top 

of the corresponding Ko deflection curves (solid curves with solid circular symbols, fig. 17), thus proving 

the argument. The excellent deflection predictions based on deflection equation (33) for the simply 

supported beam validate the accuracy of the Ko displacement theory developed for slightly nonuniform 

beams. 

Fixed Ends

Figure 18 shows the strain curves generated from the SPAR program (equation (39)), for different 

tapered tubular simple beams with two ends fixed, subjected to a central downward load of

P = –200 lb. The strain curve is again roof shaped for the classical uniform beam ( cn / c0 = 4 / 4 ). For the 

depth ratio, cn / c0 < 4 / 4 , the strain curves are slightly concave in the left-hand region but convex in the 

right-hand region, with the degree of convexity increasing as the depth ratio, cn / c0 , decreases. At the 

lowest depth ratio of c0 / cn = 1/ 4 , the strain curve in the right-hand region is deeply bent.

Figure 19 shows the beam deflection curves for the fixed-end simple beams, calculated from cantilever 

beam deflection equation (16) (eq. (23) for the uniform beam), using the strain data shown in figure 18

(solid curves with solid circular symbols). These deflection curves are compared with the deflection curves 

calculated from the SPAR program (dotted curves with open circular symbols). Note that equations (16) and 

(23) give yn = 0 at the right end, and therefore it is not necessary to use the error correction equation (35) 

for the yn ≠ 0 case. Note from figure 19 that the correlation between the two sets of deflection curves is 

reasonably good. Similar to the simply supported case, the SPAR program gives slightly larger deflections, 

because the transverse shear effect is considered in the SPAR program. Note that the transverse shear effect 

(difference between each set of SPAR and Ko deflection curves) increases slightly as the depth ratio, 

cn / c0 , decreases. At the lowest depth ratio, cn / c0 = 1/4, the transverse shear effect at the center of the 

beam amounts to 10.29 percent of the beam center deflection. Again, by setting the shear modulus, G, very 

large ( G → ∞ ) in the SPAR program, the transverse shear effect was removed, and the resulting SPAR 

deflection curves (dashed curves with open square symbols) fall nearly on top of the corresponding Ko 

deflection curves (solid curves with solid circular symbols) for the whole range of depth ratios,

4 / 4 ≥ cn / c0 ≥ 1/ 4 . Thus, if the transverse shear effect is ignored, then cantilever beam deflection 

equations (16) and (23) are quite accurate when applied to the fixed-end tapered simple beam, even down to 

the lowest depth ratio of c0 / cn = 1 / 4 .

Tapered Wing Boxes

For numerical validation of Ko displacement theory, three different types of wing boxes were 

considered: (1) depth-tapered wing boxes (unswept and swept), (2) width-tapered unswept wing boxes, and 

(3) double-tapered unswept wing boxes. The two strain-sensing lines for each wing box case are installed at 

the front and rear lower edges of the wing box.
25

Depth–Tapered Wing Boxes

The depth-tapered wing boxes considered are unswept (fig. 20a) and swept (swept angle, β, deg; 

fig. 20b). All depth-tapered wing boxes have a constant width, with the depth tapering off toward the free 

end (vertically tapered). Table 3 lists the dimensions of the depth-tapered unswept and swept wing boxes. 

Each wing box is subjected to vertical load of P = 100 lb and a torque of T = 50 ×w in-lb (clockwise) at the 

tip of wing box (fig. 20). For the swept wing boxes, a counterclockwise torsion of T = –50 ×w in-lb was

also considered.

Table 3. Dimensions of depth-tapered unswept and swept wing boxes.

l, in. c0 , in. cn , in. w , in. β, deg

(unswept) (swept)

100.0 2.0 2.0 (uniform) 17.0 0.0 10.0

100.0 2.0 1.5 17.0 0.0 10.0

100.0 2.0 1.0 17.0 0.0 10.0

100.0 2.0 0.5 17.0 0.0 10.0

Unswept Wing Boxes

Figure 21 shows the plots of the strains along the front and rear strain-sensing lines, calculated from the 

SPAR element stress outputs, for different depth-tapered unswept wing boxes subjected to a vertical load of 

P = 100 lb and a torque of T = 50 ×w in-lb at the wing box tip (fig. 20). The front-edge and rear-edge strain 

curves for the nontapered uniform wing box ( cn / c0 = 2 / 2 ; fig. 21) are practically straight lines like those 

in the classical cantilever beam case. For the tapered wing boxes, the front-edge strain curves

(fig. 21a) become bow shaped with the bend increasing as the taper depth ratio, cn / c0 , decreases (very 

similar to the tapered cantilever beam cases, fig. 10). When the taper depth ratio reaches a minimum of 

cn / c0 = 0.5/2, the strain curve becomes deeply bent by more than 90°. The rear-edge strain curves for the 

tapered wing boxes are almost straight except in the wing root and wingtip regions. As the taper depth ratio, 

cn / c0 , decreases, the strain curves become more tilted. 

Figure 22 shows the front-edge and rear-edge deflections, { yi,, y′

i }, for different depth-tapered unswept 

wing boxes, calculated from deflection equations (36) and (37), respectively, using the strain data shown in 

figure 21 as inputs (solid lines with solid circular symbols). For comparison, the deflections obtained from 

the SPAR nodal displacement outputs are also plotted (dashed lines with open circular symbols). The 

excellent correlation between the two sets of deflection curves reinforces the high degree of accuracy of the 

Ko displacement theory (eqs. (16) and (23)) formulated for nonuniform cantilever beams.

Note that under the combined bending and torsional loading, the front-edge and rear-edge deflection 

curves shown in figure 22 contain torsional components. To obtain an overall bending deflection curve for 

the wing box, the torsional effect must be removed by averaging the front-edge and rear-edge deflection 

curves of figure 22.

Figure 23 shows the cross-sectional twist angles, φi , for different depth-tapered unswept wing boxes,

calculated from equation (38) using the deflection data, { yi,, y′

i }, along the front and rear strain-sensing 

lines. Again the agreement between the displacement theory and SPAR program is quite good. Interestingly, 

at the taper depth ratios of cn / c0 = 0.5/2 and cn / c0 = 1/2, the cross-sectional twist angle, φi (under 

combined loading), in the inboard region has the opposite sign of the wingtip torque, T. This sign change 
26

of φi could result from the lack of cross-sectional shear stiffness, because the SPAR wing box models are 

hollow with no interior shear reinforcements. It was found that if rib and spar reinforcement elements were 

added to the hollow SPAR models, the cross-sectional twist angle, φi , at any point along the wingspan, 

would follow the sign of the wingtip torque, T.

Swept Wing Boxes

Figure 24a shows the plots of the strains along the front strain-sensing line, calculated from the SPAR 

element stress outputs, for different depth-tapered swept wing boxes subjected to a vertical load of

P = 100 lb and a torque of T = 50 × wn in-lb (clockwise, looking toward the wing root) at the wing box tip

(fig. 20b). Overall the shapes of the front-edge strain curves are similar to those for the unswept wing boxes 

cases (fig. 21) except in the wing root regions where the strain curves have saddle-shaped areas. The

front-edge strain curve for the uniform wing box ( cn / c0 = 2 / 2 ; fig. 24a) is a straight line except in the 

wing root region. For the depth-tapered swept wing boxes, the front-edge strain curves become bow shaped 

(except in the wing root region), with the bend increasing as the taper depth ratio, cn / c0 , decreases (very 

similar to the depth-tapered unswept wing box cases, fig. 21). When the taper depth ratio reaches a 

minimum of cn / c0 = 0.5/2, the strain curve becomes deeply bent by more than 90°.

Figure 24b shows the plots of the strains along the rear-edge strain-sensing line, calculated from the 

SPAR element stress outputs, for different depth-tapered swept wing boxes. Overall the rear-edge strain 

curves are almost linear like those for the unswept wing boxes cases (fig. 21) except in the wing root regions 

where the strains increase sharply. The rear-edge strain curve for the uniform wing box ( cn / c0 = 2 / 2 ) is 

almost a straight line except in the wing root and wingtip regions. For the tapered swept wing boxes, the 

strain curves are nearly straight except in the wing root and wingtip regions. As the taper depth ratio, 

cn / c0 , decreases, the strain curves become more tilted.

Figure 25 shows the front-edge and rear-edge deflections, { yi,, y′

i }, for different depth-tapered swept 

wing boxes under bending and clockwise torsion, calculated from deflection equations (36) and (37), using 

the strain data shown in figure 24 as inputs (solid lines with solid circular symbols). For comparison, the 

deflections obtained from the SPAR nodal displacement outputs are also plotted (dashed lines with open 

circular symbols). Note that the correlation between the Ko and SPAR deflection curves is excellent, 

providing confidence in the Ko displacement theory (eqs. (16) and (23)) formulated for nonuniform 

cantilever beams.

Figure 26 shows the cross-sectional twist angles, φi (with respect to airstream), for different 

depth-tapered swept wing boxes (clockwise torsion cases), calculated from equation (38) using the 

deflection data, { yi,, y′

i }, along the front-edge and rear-edge strain-sensing lines. Similar to the unswept 

wing cases (fig. 23), at the taper depth ratios, cn / c0 = 0.5/2 and cn / c0 = 1/2, the cross-sectional twist 

angle, φi , in the inboard region has the opposite sign of the wingtip torque, T, because the SPAR wing box 

models are hollow, having no interior reinforcements (no cross-sectional shear stiffness). The peculiar 

behavior of φi (under combined loading) disappeared if the interior shear reinforcement elements were 

added in the hollow SPAR models. Overall the agreement between the Ko displacement theory and the 

SPAR program is excellent for the swept wing boxes.

Figures 27, 28, and 29 show sets of strain curves, deflection curves, and cross-sectional twist angle 

curves, respectively, for the swept wing boxes under combined bending and counterclockwise torsion. 

Notice that by reversing the direction of torsion, the shapes of the front and rear strain and deflection curves 

do not become the interchanged versions of the corresponding curves for the clockwise torsion cases
27

(figs. 24, 25, 26) but have slight variations resulting from the swept effect near the wing root and wingtip. 

The most conspicuous difference between the two torsion cases is that the front-edge strain curve of

cn = 0.5 in. for the counterclockwise torsion case (fig. 27b) does not have a saddle-shaped area near the 

wing root like the rear-edge strain curve of cn = 0.5 in. for the clockwise torsion case (fig. 24a). The 

excellent agreement shown in figures 28 and 29 reinforces the high degree of accuracy of the Ko 

displacement theory (eqs. (16) and (23)) formulated for nonuniform cantilever beams.

Width–Tapered Wing Boxes 

Figure 30 shows the unswept width-tapered wing box with a constant half depth, c0 = cn = 0.5 in., and a 

decreasing width toward the free end (horizontally tapered). The strain-sensing lines are located at the lower 

front and rear edges. Each beam is subjected to a vertical load of P = 100 lb and a torque of T = 50 × wn

in-lb (clockwise) at the wing box tip (fig. 30). Table 4 lists the dimensions of each width-tapered unswept 

wing box.

Table 4. Dimensions of width-tapered unswept wing boxes.

l, in. c0 , in. cn , in. w0 , in. wn , in.

 

β , deg

60.0 0.5 0.5 30.0 30.0 (uniform) 0.0

60.0 0.5 0.5 30.0 20.0 0.0

60.0 0.5 0.5 30.0 10.0 0.0

61.0 0.5 0.5 30.0 5.0 0.0

Figure 31 shows the strain curves generated from the SPAR element stress outputs for different widthtapered wing boxes subjected to a vertical load of P = 100 lb and a torque of T = 50 × wn in-lb (clockwise) 

at the wing box tip (fig 30).

Figure 32 shows the deflection curves for different width-tapered unswept wing boxes, calculated from 

deflection equation (23), using the associated input strain data of figure 31 (solid lines with solid circular 

symbols). For comparison, the deflection curves obtained from the SPAR nodal displacement outputs are 

also plotted (dashed lines with open circular symbols). The excellent correlation between the two sets of 

deflection curves validates the high degree of accuracy of the Ko displacement theory (eq. (23)) formulated 

for nonuniform cantilever beams. Figure 33 compares the cross-sectional twist angles, φi , calculated from 

equation (38) with those generated from the SPAR program, for different width-tapered wing boxes. Again 

the agreement is excellent.

Double–Tapered Wing Boxes

The double-tapered wing box (fig. 34) is unswept and has both a width and depth (horizontally and 

vertically) that taper down toward the free end. The front and rear strain-sensing lines are located at the front 

and rear lower edges, respectively, of the wing box. Table 5 lists the dimensions of the double-tapered wing 

box. The wing box is subjected to a vertical load of P = 100 lb and a torque of T = 50 ×wn in-lb (clockwise) 

at the wing box tip.
28

Table 5. Dimensions of double-tapered unswept wing box.

l, in. c0 , in. cn , in. w0 , in. wn , in.

 

β , deg

100.0 2.0 1.0 34.0 17.0 0.0

Figure 35 shows the strain curves generated from the SPAR element stress outputs for the front and rear 

oblique strain-sensing lines at the lower edges of the double-tapered wing box. The strain curves are deeply 

bent because of the double tapering effect. Remember that for the classical uniform cantilever beam, the 

strain curve is a straight line.

Figure 36 shows the deflection curves along the two oblique strain-sensing lines, calculated from 

deflection equation (16), using the strain data generated from the SPAR program shown in figure 35 (solid 

lines with solid circular symbols). For comparison, the deflections calculated from the SPAR program are 

also plotted (dashed lines with open circular symbols). The excellent correlation between the two sets of 

deflection curves validates the high degree of accuracy of the Ko displacement theory (eq. (16)) formulated 

for nonuniform cantilever beams.

Figure 37 compares the cross-sectional twist angles, φi , calculated from equation (38) with those 

calculated from the SPAR program, for the double-tapered wing box. The Ko and SPAR curves for φi

practically fall on top of each other, showing the surprisingly high degree of accuracy of the Ko 

displacement theory.

EXACT AND EXPANDED DEFLECTION EQUATIONS

All the deflection curves for vertically tapered beams predicted from the Ko displacement theory, as 

shown in the preceding sections, were calculated using the exact deflection equation (16) formulated for 

nonlinear beams (for example, tapered beams). Because deflection equation (16) breaks down for the 

uniform beams, the deflection curves for the uniform beam cases had to be calculated from the expanded 

deflection equation (23), which was obtained from the exact deflection equation (16) by expanding the 

logarithmic term, loge(ci / ci−1) , in the neighborhood of (ci / ci−1) ≈ 1. The preceding examples demonstrate 

that equation (16) provides quite accurate shape predictions for tapered beams with all the taper depth ratios 

down to cn / c0 = 1/ 4 . 

To examine the deflection prediction accuracy, expanded deflection equation (23) was used to calculate 

the deflections of tapered cantilever tubular beams with different taper depth ratios, as shown in figure 11. 

Table 6 compares the beam tip deflections calculated from the SPAR program with those calculated from 

the exact deflection equation (16) and those calculated from the expanded deflection equation (23). 

Note from table 6 that the two sets of deflection data calculated from deflection equations (16) and (23)

are very close, with a peak difference of only 0.5293 percent for the lowest depth ratio, cn / c0 = 1 / 4 . Thus, 

if the deflection data of equation (23) are plotted out in figure 11, the deflection curve will fall practically on 

top of the deflection curve calculated from equation (16) (fig. 11). This observation indicates that the 

expanded deflection equation (23) is quite accurate for the whole range of depth ratios, 

4 / 4 ≥ cn / c0 ≥ 1/ 4 .
29

Table 6. Comparison of beam tip deflections, yn , of tapered cantilever tubular beams calculated from SPAR 

program with those calculated from displacement equations (16) and (23); l =100.5 in., P =100 lb, n = 8. 

yn , in.

c0 , in. cn , in. SPAR Eq. (16)

(exact)

Eq. (23)

(expanded)

Percent difference 

between

eq. (16) and eq. (23)

4.0 4.0 (uniform) 0.711620 Not applicable 0.702304 --------

4.0 3.0 0.876340 0.865681 0.866592 0.1052

4.0 2.0 1.158010 1.145292 1.145487 0.0170

4.0 1.0 1.784170 1.762277 1.752950 –0.5293

CONCLUDING REMARKS 

The Ko displacement theory, originally formulated (in recursion formats) for shape predictions of 

nonuniform cantilever beams, was further developed into more compact mathematical forms. The accuracy 

of the further-developed Ko displacement theory was validated by finite-element analyses of various beam 

structures. The principal findings are as follows:

1. In the further-developed Ko displacement theory for the nonuniform cantilever beam, the 

deflections at the strain-sensing stations were expressed explicitly in terms of geometrical 

parameters of the beam and bending strains at all the inboard strain-sensing stations. 

2. The exact deflection equations for nonuniform cantilever beams were expanded for slightly 

nonuniform cantilever beams so that the expanded deflection equations could degenerate into the 

limit case for uniform cantilever beams without encountering mathematical breakdown problems. 

3. For tapered cantilever tubular beams with different taper rates, the deflections calculated from the 

expanded deflection equations were only 0.017–0.529 percent off from those calculated from the 

exact deflection equations formulated for nonuniform cantilever beams.

4. The further-developed displacement equations for nonuniform cantilever beams were used to 

formulate the displacement equations for nonuniform two-point supported beams and could be used 

for shape predictions of nonuniform structural panels. 

5. The proposed two-line strain-sensing system provided a powerful method for simultaneously 

monitoring the bending and torsional deformations of nonuniform cantilever beams such as wing 

boxes. The two-line strain-sensing system eliminated the need to install shear strain sensors to 

measure the surface distortions for monitoring cross-sectional rotations of the beam.

6. A finite-element computer program was used to generate simulated strain data for input to the 

further-developed displacement equations. The displacement equations were then used to calculate 

deflections and cross-sectional twist angles, at discreet strain-sensing stations of the beam-like 

structure, for constructing the overall deformed shape of the whole structure. 

7. The high degree of accuracy of the further-developed displacement theory for nonuniform beams 

was validated by finite-element analyses of various types of nonuniform beam structures (tapered 

tubular beams, such as cantilever and two-point supported beams; depth-tapered unswept and swept 
30

wing boxes; width-tapered wing boxes; and double-tapered wing boxes) under combined bending 

and torsion. 

8. For tapered beams, the bending strain distribution along the axis of the beam was nonlinear, and the 

nonlinearity increased with increasing taper rate. The piecewise linear approximation of strain 

distributions was adequate to obtain highly accurate deflection curves for the whole range of beam 

taper rates.

9. The Ko displacement theory, combined with the onboard strain-sensing system, form a powerful 

tool for in-flight deformed shape monitoring of unmanned aerospace vehicles (by ground-based 

pilots) to maintain safe flights. In addition, the real-time wing shape could be input to the aircraft 

control system for aeroelastic wing shape control. 
31

FIGURES

Figure 1. Nonuniform cantilever beam instrumented with bending and distortion strain sensors.

Figure 2. Deformed cantilever tubular beam showing slopes, θi , and deflections, yi , at strain-sensing 

stations, xi (i = 0, 1, 2, 3, ..., n).
32

Figure 3. Relative magnitude of strain terms in deflection equation for a uniform cantilever beam.

Figure 4. Linearly increasing downward shifting of deflection curve from cantilever beam position A to 

simple beam position B; zero shift at left end.
33

Figure 5. Tapered wing box instrumented with a two-line strain-sensing system.

Figure 6. Tapered cantilever tubular beam instrumented with a two-line strain-sensing system.
34

Figure 7. Aircraft fuselage instrumented with a two-line strain-sensing system.

Figure 8. Structural panel instrumented with multiple biaxial strain-sensing lines.
35

(a) Tip vertical load of P = 100 lb; tip twisting moment of T = 0.

(b) Tip vertical load of P = 100 lb; tip twisting moment of T = 400 × cn in-lb.

Figure 9. Finite-element model for a tapered cantilever tubular beam subjected to different tip loading 

conditions.
36

Figure 10. Bending strains calculated from SPAR element stress outputs for different tapered cantilever 

tubular beams; n = 8; P = 100 lb.

Figure 11. Comparison of deflections calculated from Ko displacement theory with those calculated from 

SPAR program for different tapered cantilever tubular beams; n = 8; P = 100 lb.
37

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 12. SPAR-generated strains, { εi,εi

′ }, along front and rear strain-sensing lines on different tapered 

cantilever tubular beams; n = 8; P = 100 lb, T = 400 × cn in-lb.
38

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 13. Comparison of deflections along strain-sensing lines calculated from Ko displacement theory 

with those calculated from SPAR program for different tapered cantilever tubular beams; n = 8; P = 100 lb, 

T = 400 × cn in-lb.
39

Figure 14. Comparison of cross-sectional twist angles calculated from Ko displacement theory with those 

calculated from SPAR program for different tapered cantilever tubular beams; n = 8 ; P = 100 lb ,

T = 400 × cn in-lb .
40

(a) Both ends simply supported.

(b) Both ends fixed.

Figure 15. Finite-element model for tapered tubular simple beam subjected to central downward vertical 

load of P = –200 lb.
41

Figure 16. Bending strains calculated from SPAR element stress outputs for different tapered tubular simple 

beams; n = 8; two ends simply supported; P = 100 lb.

Figure 17. Comparison of deflections calculated from Ko displacement theory with those calculated from 

SPAR program for different tapered tubular simple beams; two ends simply supported; n = 8; P = 100 lb.
42

Figure 18. Bending strains calculated from SPAR element stress outputs for different tapered tubular simple 

beams; two ends fixed; n = 8; P = 100 lb.

Figure 19. Comparison of deflections calculated from Ko displacement theory with those calculated from 

SPAR program for different tapered tubular simple beams; two ends fixed; n = 8; P = 100 lb.
43

(a) Unswept wing box.

(b) Swept wing box.

Figure 20. Finite-element models for depth-tapered wing boxes subjected to bending and torsion; 

P = 100 lb; T = 50 × w in-lb.
44

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 21. SPAR-generated strains, { εi ,εi

′ }, along front and rear strain-sensing lines on different depthtapered unswept wing boxes; n = 8; P = 100 lb; T = 50 × w in-lb (clockwise).
45

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 22. Comparison of deflections, { yi, yi

′ }, along front and rear strain-sensing lines, calculated from 

Ko displacement theory with those calculated from SPAR program for different depth-tapered unswept wing 

boxes; n = 8; P = 100 lb; T = 50 × w in-lb (clockwise).
46

Figure 23. Comparison of cross-sectional twist angles calculated from Ko displacement theory with those 

calculated from SPAR program for different depth-tapered unswept wing boxes; n = 8; P = 100 lb; 

T = 50 × w in-lb (clockwise).
47

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 24. SPAR-generated strains, { εi,εi

′ }, along front and rear strain-sensing lines on different depthtapered swept wing boxes; n = 8; P = 100 lb; T = 50 × w in-lb (clockwise).
48

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 25. Comparison of deflections, { yi, yi

′ }, along front and rear strain-sensing lines calculated from Ko 

displacement theory with those calculated from SPAR program for different depth-tapered swept wing 

boxes; n = 8; P = 100 lb; T = 50 × w in-lb (clockwise).
49

Figure 26. Comparison of cross-sectional twist angles calculated from Ko displacement theory with those 

calculated from SPAR program for different depth-tapered swept wing boxes; n = 8; P = 100 lb;

T = 50 × w in-lb (clockwise).
50

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 27. SPAR-generated strains, { εi,εi

′ }, along front and rear strain-sensing lines on different depthtapered swept wing boxes; n = 8; P = 100 lb; T = −(50 × w) in-lb (counterclockwise).
51

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 28. Comparison of deflections, { yi, yi

′ }, along front and rear strain-sensing lines calculated from Ko 

displacement theory with those calculated from SPAR program for different depth-tapered swept wing 

boxes; n = 8; P = 100 lb; T = –(50 × w) in-lb (counterclockwise).
52

Figure 29. Comparison of cross-sectional twist angles calculated from Ko displacement theory with those 

calculated from SPAR program for different depth-tapered swept wing boxes; n = 8; P = 100 lb;

T = –(50 × w) in-lb (counterclockwise).

Figure 30. Finite-element model for a width-tapered unswept wing box subjected to bending and torsion;

P = 100 lb; T = 50 × wn in-lb.
53

(a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 31. SPAR-generated strains, { εi,εi

′ }, along front and rear strain-sensing lines on different widthtapered unswept wing boxes; n = 8; P = 100 lb; T = 50 × wn in-lb (clockwise).
54

 (a) Front strain-sensing line.

(b) Rear strain-sensing line.

Figure 32. Comparison of deflections, { yi, yi

′ }, along front and rear strain-sensing lines calculated from Ko 

displacement theory with those calculated from SPAR program for different width-tapered unswept wing 

boxes; n = 8; P = 100 lb; T = 50 ×wn in-lb (clockwise).
55

Figure 33. Comparison of cross-sectional twist angles calculated from Ko displacement theory with those 

calculated from SPAR program for different width-tapered unswept wing boxes; n = 8 ; P = 100 lb ;

T = 50 × wn in-lb (clockwise).
56

Figure 34. Finite-element model for a double-tapered wing box subjected to bending and torsion;

P = 100 lb, T = 50 × wn in-lb.

Figure 35. SPAR-generated strains, { εi,εi

′ }, along front and rear strain-sensing lines on a double-tapered 

unswept wing box; n = 8; P = 100 lb; T = 50 × wn in-lb (clockwise).
57

Figure 36. Comparison of deflections, { yi, yi

′ }, along front and rear strain-sensing lines calculated from Ko 

displacement theory with those calculated from SPAR program, for a double-tapered unswept wing box; 

n = 8; P = 100 lb; T = 50 ×wn in-lb (clockwise).

Figure 37. Comparison of cross-sectional twist angles calculated from Ko displacement theory with those 

calculated from SPAR program for a double-tapered unswept wing box; n = 8; P = 100 lb; T = 50 × wn in-lb 

(clockwise).
58

APPENDIX A

DERIVATION OF DEFLECTION EQUATIONS FOR

NONUNIFORM CANTILEVER BEAMS

Slope equation (11) and deflection equation (13) for nonuniform cantilever beams are duplicated here as 

follows:

Slope equation,

tanθi = Δl εi−1 − εi

ci−1 − ci

+

εi−1ci − εici−1

(ci−1 − ci)

2 loge

ci

ci−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ tanθi−1 (A1)

Deflection equation,

yi = (Δl)

2 εi−1 − εi

2(ci−1 − ci)

− εi−1ci − εici−1

(ci−1 − ci)

3 ci loge

ci

ci−1

+ (ci−1 − ci) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ yi−1 + Δl tanθi−1 (A2)

Writing out equation (A2) for different indices, i, and making use of the indicial relationships expressed 

in equations (A1) and (A2), we obtain the following:

For i = 1,

y1 = (Δl)

2 ε0 − ε1

2(c0 − c1)

− ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + Δl tanθ0 (A2-a)
59

For i = 2,

y2 = (Δl)

2 ε1 − ε2

2(c1 − c2 )

− ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y1 + Δl tanθ1 

 = (Δl)

2 ε1 − ε2

2(c1 − c2 )

− ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 

 + (Δl)

2 ε0 − ε1

2(c0 − c1)

− ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + Δl tanθ0

y 1

 

 + (Δl)

2 ε0 − ε1

c0 − c1

+

ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ Δl tanθ0



Δl ta



nθ

1 

 

(A2-b)

After the terms are grouped, equation (A2-b) becomes

y2 = (Δl)

2 ε1 − ε2

2(c1 − c2 )

+ 3 ε0 − ε1

2(c0 − c1)

⎡

⎣

⎢ ⎤

⎦

⎥

 − (Δl)

2 ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 2Δl tanθ0 

(A2-

 

b ′)

= (Δl)

2 ε1 − ε2

2(c1 − c2 )

− ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

+

ε0 − ε1

2(c0 − c1)

− ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 ε0 − ε1

c0 − c1

+

ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 2Δl tanθ0 
60

For i = 3,

y3 = (Δl)

2 ε2 − ε 3

2(c2 − c3)

− ε2c3 − ε 3c2

(c2 − c3)

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y2 + Δl tanθ2 

 = (Δl)

2 ε2 − ε 3

2(c2 − c3)

− ε2c3 − ε 3c2

(c2 − c3)

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+

(Δl)

2 ε1 − ε2

2(c1 − c2 )

− ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε0 − ε1

2(c0 − c1)

− ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 ε0 − ε1

c0 − c1

+

ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 2Δl tanθ0

⎧

⎨

⎪

⎪⎪

⎩

⎪

⎪

⎪

⎫

⎬

⎪

⎪⎪

⎭

⎪

⎪

⎪

y 2

 + (Δl)

2 ε1 − ε2

c1 − c2

+

ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

+

ε0 − ε1

c0 − c1

+

ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ Δl tanθ0

Δ

l t



an



θ2

= (Δl)

2 ε2 − ε 3

2(c2 − c3)

− ε2c3 − ε 3c2

(c2 − c3)

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε1 − ε2

2(c1 − c2 )

− ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

ε0 − ε1

2(c0 − c1)

− ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 

 + (Δl)

2 ε1 − ε2

c1 − c2

+

ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

+ 2 ε0 − ε1

c0 − c1

+ 2 ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 3Δl tanθ0 

(A2-c)

After the terms are grouped, equation (A2-c) becomes

y3 = (Δl)

2 ε2 − ε 3

2(c2 − c3)

+ 3 ε1 − ε2

2(c1 − c2 )

+ 5 ε0 − ε1

2(c0 − c1)

⎡

⎣

⎢ ⎤

⎦

⎥

 − (Δl)

2 ε2c3 − ε 3c2

(c2 − c3)

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 

 + (Δl)

2 ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

+ 2 ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 3Δl tanθ0 

(A2-

 

c ′)
61

For i = 4, 

(A2-d)

y4 = (Δl)

2 ε 3 − ε 4

2(c3 − c4 )

− ε 3c4 − ε 4c3

(c3 − c4 )

3 c4 loge

c4

c3

+ (c3 − c4 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y3 + Δl tanθ3 

 = (Δl)

2 ε 3 − ε 4

2(c3 − c4 )

− ε 3c4 − ε 4c3

(c3 − c4 )

3 c4 loge

c4

c3

+ (c3 − c4 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+

(Δl)

2 ε2 − ε 3

2(c2 − c3)

− ε2c3 − ε 3c2

(c2 − c3)

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε1 − ε2

2(c1 − c2 )

− ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

ε0 − ε1

2(c0 − c1)

− ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+

+(Δl)

2 ε1 − ε2

c1 − c2

+

ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

+ 2 ε0 − ε1

c0 − c1

+ 2 ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 3Δl tanθ0

⎧

⎨

⎪

⎪

⎪

⎪⎪

⎩

⎪

⎪

⎪

⎪

⎪

⎫

⎬

⎪

⎪

⎪

⎪⎪

⎭

⎪

⎪

⎪

⎪

⎪

y 3

 

 +

(Δl)

2 ε2 − ε 3

c2 − c3

+

ε2c3 − ε 3c2

(c2 − c3)

2 loge

c3

c2

+

ε1 − ε2

c1 − c2

+

ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

⎡

⎣

⎢

⎢

 +

ε0 − ε1

c0 − c1

+

ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎤

⎦

⎥

⎥

+ Δl tanθ0

⎧

⎨

⎪

⎪⎪

⎩

⎪

⎪

⎪

⎫

⎬

⎪

⎪⎪

⎭

⎪

⎪

⎪



Δ



l t



an



θ3

= (Δl)

2 ε 3 − ε 4

2(c3 − c4 )

− ε 3c4 − ε 4c3

(c3 − c4 )

3 c4 loge

c4

c3

+ (c3 − c4 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

+

ε2 − ε 3

2(c2 − c3)

− ε2c3 − ε 3c2

(c2 − c3)

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥

 +

ε1 − ε2

2(c1 − c2 )

− ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε0 − ε1

2(c0 − c1)

− ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 ε2 − ε 3

c2 − c3

+

ε2c3 − ε 3c2

(c2 − c3)

2 loge

c3

c2

+ 2 ε1 − ε2

c1 − c2

+ 2 ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

⎡

⎣

⎢

⎢

+3

ε0 − ε1

c0 − c1

+ 3

ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎤

⎦

⎥

⎥

 + y0 + 4Δl tanθ0 
62

After the terms are grouped, equation (A2-d) becomes

(A2- d′ )

Based on the indicial progression patterns in equations (A2-a)–(A2-d), the deflection, yi , can be written 

in a generalized form with two summations (with different summation limits) as 

yi = (Δl)

2 ε j−1 − ε j

2(c j−1 − c j)

−

ε j−1c j − ε jc j−1

(c j−1 − c j)

3 c j loge

c j

c j−1

+ (c j−1 − c j)

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭ j=1 ⎪

i

∑

 + (Δl)

2 (i − j)

ε j−1 − ε j

c j−1 − c j

+

ε j−1c j − ε jc j−1

(c j−1 − c j)

2 loge

c j

c j−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i−1

∑ + y0 + (i)Δl tanθ0

(A2-e)

which is equation (15).

Alternatively, based on the progression of indices in equations (A2-

 

a ′)–(A2-

 

d ′), the generalized 

deflection equation can be rewritten in an alternative form as

yi = (Δl)

2 [2(i − j) +1] ε j−1 − ε j

2(c j−1 − c j)

−

ε j−1c j − ε jc j−1

(c j−1 − c j)

3 c j loge

c j

c j−1

+ (c j−1 − c j)

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭ j=1 ⎪

i

∑

 + (Δl)

2 (i − j)

ε j−1c j − ε jc j−1

(c j−1 − c j)

2 loge

c j

c j−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i−1

∑ + y0 + (i)Δl tanθ0

(A2-f)

which is equation (16).

y4 = (Δl)

2 ε 3 − ε 4

2(c3 − c4 )

+ 3 ε2 − ε 3

2(c2 − c3)

+ 5 ε1 − ε2

2(c1 − c2 )

+ 7 ε0 − ε1

2(c0 − c1)

⎡

⎣

⎢ ⎤

⎦

⎥ 

 − (Δl)

2 ε 3c4 − ε 4c3

(c3 − c4 )

3 c4 loge

c4

c3

+ (c3 − c4 ) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε2c3 − ε 3c2

(c2 − c3)

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 

 +

ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 

 + (Δl)

2 ε2c3 − ε 3c2

(c2 − c3)

2 loge

c3

c2

+ 2 ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

+ 3

ε0c1 − ε1c0

(c0 − c1)

2 loge

c1

c0

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 4Δl tanθ0
63

The general deflection equation (A2-f) is checked for accuracy using i = 4 as follows:

y4 = (Δl)

2 [2(4 −1) +1] ε1−1 − ε1

2(c1−1 − c1)

− ε1−1c1 − ε1c1−1

(c1−1 − c1)

3 c1 loge

c1

c1−1

+ (c1−1 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 [2(4 − 2) +1] ε2−1 − ε2

2(c2−1 − c2 )

− ε2−1c2 − ε2c2−1

(c2−1 − c2 )

3 c2 loge

c2

c2−1

+ (c2−1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 [2(4 − 3) +1] ε 3−1 − ε 3

2(c3−1 − c3)

− ε 3−1c2 − ε 3c3−1

(c3−1 − c2 )

3 c3 loge

c3

c3−1

+ (c3−1 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 [2(4 − 4) +1] ε 4−1 − ε 4

2(c4−1 − c4 )

− ε 4−1c4 − ε 4c4−1

(c4−1 − c4 )

3 c4 loge

c4

c4−1

+ (c4−1 − c4 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + 3(Δl)

2 ε1−1c1 − ε1c1−1

(c1−1 − c1)

2 loge

c1

c1−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ 2(Δl)

2 ε2−1c2 − ε2c2−1

(c2−1 − c2 )

2 loge

c2

c2−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 + (1)(Δl)

2 ε 3−1c3 − ε 3c3−1

(c3−1 − c3)

2 loge

c3

c3−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 4Δl tanθ0

Finally,

y4 = (Δl)

2 (1) ε 3 − ε 4

2(c3 − c4 )

+ (3) ε2 − ε 3

2(c2 − c3)

+ (5) ε1 − ε2

2(c1 − c2 )

+ (7) ε0 − ε1

2(c0 − c1)

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2 ε0c1 − ε1c0

(c0 − c1)

3 c1 loge

c1

c0

+ (c0 − c1) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε1c2 − ε2c1

(c1 − c2 )

3 c2 loge

c2

c1

+ (c1 − c2 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

ε2c2 − ε 3c2

(c2 − c2 )

3 c3 loge

c3

c2

+ (c2 − c3) ⎡

⎣

⎢ ⎤

⎦

⎥ +

ε 3c4 − ε 4c3

(c3 − c4 )

3 c4 loge

c4

c3

+ (c3 − c4 ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 + (Δl)

2 3

ε0c1 − ε1c1−1

(c0 − c1)

2 loge

c1

c0

+ 2 ε1c2 − ε2c1

(c1 − c2 )

2 loge

c2

c1

+

ε2c3 − ε 3c2

(c2 − c3)

2 loge

c3

c2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 + y0 + 4Δl tanθ0

(A3)

which agrees with equation (A2-

 

d ′).
64

APPENDIX B

EXPANSION OF LOGARITHMIC TERMS

This appendix discusses the mathematical processes of expanding the logarithmic terms (taken from 

Appendix E of reference 2). 

When the cantilever beam is slightly tapered (that is, ci → ci−1), the logarithmic terms in slope equation 

(14) and deflection equation (16) will approach zero (that is, loge(ci ci−1)→ 0 ). Therefore, the 

perturbation method must be used to expand the logarithmic term, loge(ci ci−1) , in the neighborhood of 

(ci ci−1) ≈ 1 to yield nonzero mathematical expressions, which are derived in the following sections.

Expansion of loge (ci ci-1 ) for Slope Equation 

When the beam is slightly tapered (that is, ci → ci−1), the logarithmic term, loge(ci ci−1) , can be 

expanded in the neighborhood of (ci ci−1) ≈ 1 (that is, loge(ci ci−1) ≈ 0 ) (ref. 6) as follows:

loge

ci

ci−1

≈ ci

ci−1

−1 ⎛

⎝

⎜

⎞

⎠

⎟ − 1

2

ci

ci−1

−1 ⎛

⎝

⎜

⎞

⎠

⎟

2

+

1

3

ci

ci−1

−1 ⎛

⎝

⎜

⎞

⎠

⎟

3

−....

 ≈ ci − ci−1

ci−1

1− ci − ci−1

2ci−1

+ (ci − ci−1)

2

3ci−1

2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

−....

 ; ci

ci−1

→1 (B1)

Carrying out the expansion and retaining up to the second-order terms in (ci−1 − ci)

2 causes the 

logarithmic term, loge(ci ci−1) , to take on the following form when ci approaches ci−1 ( ci → ci−1):

loge

ci

ci−1

≈ ci−1 − ci

2ci−1

2 (ci − 3ci−1) ; ci

ci−1

→1 (B2)

Equation B2 is equation (17).
65

Expansion of ci

loge (ci ci-1 ) + ci-1 - c ⎡ ( i) ⎣ ⎤

⎦ for Deflection Equation

When the beam is slightly tapered (that is, ci → ci−1), the term, ci loge(ci ci−1) + ci−1 − c ⎡ ( i) ⎣ ⎤

⎦ , in 

deflection equation (16) can be expanded in the neighborhood of (ci ci−1) ≈ 1 (that is, loge(ci ci−1) ≈ 0 ). 

In light of the loge(ci ci−1) expansion equation (B1), the expansion, ci loge(ci ci−1) + ci−1 − c ⎡ ( i) ⎣ ⎤

⎦ , can 

be carried out up to the third-order terms in (ci−1 − ci)

3 when ci approaches 

 

ci−1( ci → ci−1) (ref. 6) as

ci loge

ci

ci−1

+ ci−1 − c ( i) ⎡

⎣

⎢ ⎤

⎦

⎥ ≈ ci

ci

ci−1

−1 ⎛

⎝

⎜

⎞

⎠

⎟ − 1

2

ci

ci−1

−1 ⎛

⎝

⎜

⎞

⎠

⎟

2

+

1

3

ci

ci−1

−1 ⎛

⎝

⎜

⎞

⎠

⎟

3

−....

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ ci−1 − c ( i)

 ≈ − ci

ci−1

(ci−1 − ci) + (ci−1 − ci)

2

2ci−1

+ (ci−1 − ci)

3

3ci−1

2 +....− ci−1

ci

ci−1 − c ( i) ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

≈ − ci

ci−1

(ci−1 − ci) 1− ci−1

ci

⎛

⎝

⎜

⎞

⎠

⎟ + (ci−1 − ci)

2

2ci−1

+ (ci−1 − ci)

3

3ci−1

2 +.... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

≈ − ci

ci−1

− (ci−1 − ci)

2

ci

+ (ci−1 − ci)

2

2ci−1

+ (ci−1 − ci)

3

3ci−1

2 +.... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

≈ − (ci−1 − ci)

2

6ci−1

−6 + 3 ci

ci−1

+ 2 ci

ci−1

2 (ci−1 − ci)+.... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

≈ (ci−1 − ci)

2

6ci−1

3+ 3− 3 ci

ci−1

− 2 ci

ci−1

2 (ci−1 − ci) −.... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

≈ (ci−1 − ci)

2

6ci−1

3+ 3

(ci−1 − ci)

ci−1

− 2 ci

ci−1

2 (ci−1 − ci) −.... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

≈ (ci−1 − ci)

2

6ci−1

3+ (ci−1 − ci)

ci−1

+ 2 (ci−1 − ci)

ci−1

− 2 ci

ci−1

2 (ci−1 − ci) −.... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

≈ (ci−1 − ci)

2

6ci−1

3+ (ci−1 − ci)

ci−1

+ 2 (ci−1 − ci)

2

ci−1

2 −.... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 ; ci

ci−1

→1 (B3)
66

Neglecting the fourth-order term in (ci−1 − ci)

4

, we obtain the final expression for 

ci loge(ci ci−1) + ci−1 − c ⎡ ( i) ⎣ ⎤

⎦ when ci approaches ci−1 ( ci → ci−1) as

loge

ci

ci−1

+ (ci−1 − ci) ⎡

⎣

⎢ ⎤

⎦

⎥ ≈ (ci−1 − ci)

2

6ci−1

3+ (ci−1 − ci)

ci−1

⎡

⎣

⎢ ⎤

⎦

⎥ ; ci

ci−1

→1 (B4)

which is equation (20).
67

APPENDIX C

DERIVATIONS OF DEFLECTION EQUATION FOR

SLIGHTLY NONUNIFORM CANTILEVER BEAMS

Deflection equation (21) for the slightly nonuniform cantilever beam contains the slope term, 

Δl tanθi−1 , as follows:

yi = (Δl)

2

6ci−1

3− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + εi

⎡

⎣

⎢ ⎤

⎦

⎥ + yi−1 + Δl tanθi−1 (C1)

In light of equation (18), the slope, 

 

tanθi−1, can be written as 

tanθi−1 = Δl

2ci−2

2 − ci−1

ci−2

⎛

⎝

⎜

⎞

⎠

⎟ εi−2 + εi−1

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθi−2 (C2)

Writing out both equations (C1) and (C2) for different indices (i = 1, 2 , 3, 4, 5), we obtain

 

y1 = (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥+ y0 + Δltanθ0 ;

 

y0 = 0 ; Δltanθ0 = 0 (C3-a)

 

y2 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+ y1 + Δltanθ1 ;

 

tanθ1 = Δl

2c0

2 − c1

c0

⎛

⎝

⎜ ⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥+ tanθ0 (C3-b)

 

y3 = (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+ y2 + Δltanθ2 ;

 

tanθ2 = Δl

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+ tanθ1 (C3-c)

 

y4 = (Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ε3 + ε4

⎡

⎣

⎢

⎤

⎦

⎥+ y3 + Δltanθ3 ;

 

tanθ3 = Δl

2c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+ tanθ2 (C3-d)

 

y5 = (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ε4 + ε5

⎡

⎣

⎢

⎤

⎦

⎥+ y4 + Δltanθ4 ;

 

tanθ4 = Δl

2c3

2 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ε3 + ε4

⎡

⎣

⎢

⎤

⎦

⎥+ tanθ3 (C3-e)

Based on equations (C3-a)–(C3-e), the deflection, 

 

yi

, can be rewritten for different values of i as 

follows:
68

For i = 2, equation (C3-b) becomes

(C4-a)

(C4- a′ )

For i = 3, equation (C3-c) becomes

y3 = (Δl)

2

6c2

3 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + y2 + Δl tanθ2

 = (Δl)

2

6c2

3 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

 + (Δl)

2

6c1

3 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

3 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε `1

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 2Δl tanθ0

y 2

 + (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0

Δ

l t

an

θ2

 

 

= (Δl)

2

6

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+

1

c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

2

1

c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+ 2 1

c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪ + y0 + 3Δltanθ0 

(C4-b)

 

= (Δl)

2

6

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+

1

c1

9 − 4 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + 4ε2

⎡

⎣

⎢

⎤

⎦

⎥+

1

c0

15 − 7 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + 7ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + y0 + 3Δltanθ0

(C4- b′)

y2 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + y1 + Δl tanθ1

 = (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε `1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + Δl tanθ0

y 1

+ (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0

Δ

l t



an



θ1

 

 

 = (Δl)

2

6

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε `1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ 

 + y0 + 2Δl tanθ0

 = (Δl)

2

6c

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

9 − 4 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + 4ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + 2Δl tanθ0 
69

For i = 4, equation (C3-d) becomes

(C4-c)

y4 = (Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + y3 + Δl tanθ3

 = (Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥

 +

(Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ 

 + (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + 2 (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 3Δl tanθ0

⎧

⎨

⎪

⎪

⎩

⎪

⎪

⎫

⎬

⎪

⎪

⎭

⎪

⎪

y 3

 + (Δl)

2

2c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0



Δ



l t



an



θ3

 = (Δl)

2

6

(Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 + (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 

 + (Δl)

2

2

1

c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

2

c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

3

c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + y0 + 4Δl tanθ0 

 

 = (Δl)

2

6c3

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 4ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε2

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +10ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+ y0 + 4Δl tanθ0 

 

(C4- c′)
70

For i = 5, equation (C3-e) becomes

(C4-d)

y5 = (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ + y4 + Δl tanθ4 

 = (Δl)

2

6c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥

 +

(Δl)

2

6c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

6c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥

+ (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + 2 (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥

+3

(Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 4Δl tanθ0

⎧

⎨

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎫

⎬

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

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

 +

(Δl)

2

2c3

2 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ + (Δl)

2

2c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥

+ (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + Δl tanθ0

⎧

⎨

⎪

⎪

⎩

⎪

⎪

⎫

⎬

⎪

⎪

⎭

⎪

⎪



Δ



l t



an



θ4

 

 = (Δl)

2

6

1

c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

 

 + (Δl)

2

2

1

c3

2 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

2

c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

3

c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε2

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

4

c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+ y0 + 5Δl tanθ0 (C− 4d) 

 = (Δl)

2

6

1

c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ ε 4 + ε5

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c3

9 − 4 c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 4ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

15 − 7 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε2 + 7ε 3

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c1

21−10 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 +10ε2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

27 −13 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε0 +13ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+ y0 + 5Δl tanθ0 (C-4d′)

 

(C4- d′)
71

In summary,

 

y1 = (Δl)

2

6c0

3− c1

c0

⎛

⎝

⎜ ⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥+ y0 + Δltanθ0 (C5-a)

(C5-b)

or,

 

y2 = (Δl)

2

6

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+

1

c0

9 − 4 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + 4ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + 2Δltanθ0 (C5-

 

b ′)

 

y3 = (Δl)

2

6

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+

1

c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

2

1

c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+

2

c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪ + y0 + 3Δltanθ0 

(C5-c)

or,

 

y3 = (Δl)

2

6

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+

1

c1

9 − 4 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + 4ε2

⎡

⎣

⎢

⎤

⎦

⎥+

1

c0

15 − 7 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + 7ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + y0 + 3Δltanθ0

(C5-

 

c ′)

 

y4 = (Δl)

2

6

1

c3

3− c4

c3

⎛

⎝

⎜ ⎞

⎠

⎟ε3 + ε4

⎡

⎣

⎢

⎤

⎦

⎥+

1

c2

3− c3

c2

⎛

⎝

⎜ ⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+

1

c1

3− c2

c1

⎛

⎝

⎜ ⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

 +

1

c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

2

1

c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+

2

c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+

3

c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + y0 + 4Δltanθ0

(C5-d)

y2 = (Δl)

2

6

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε 2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + ε `1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + (Δl)

2

2c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥ + y0 + 2Δl tanθ0 
72

or,

 

y4 = (Δl)

2

6

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ε3 + ε4

⎡

⎣

⎢

⎤

⎦

⎥+

1

c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + 4ε3

⎡

⎣

⎢

⎤

⎦

⎥+

1

c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + 7ε2

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

 +

1

c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 +10ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎫

⎬

⎪

⎭

⎪

+ y0 + 4Δltanθ0

(C5-

 

d ′)

 

y5 = (Δl)

2

6

1

c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ε4 + ε5

⎡

⎣

⎢

⎤

⎦

⎥+

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ε3 + ε4

⎡

⎣

⎢

⎤

⎦

⎥+

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

 +

1

c1

3− c2

c1

⎛

⎝

⎜ ⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥+

1

c0

3− c1

c0

⎛

⎝

⎜ ⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

2

1

c3

2 − c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ε3 + ε4

⎡

⎣

⎢

⎤

⎦

⎥+

2

c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + ε3

⎡

⎣

⎢

⎤

⎦

⎥+

3

c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 + ε2

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

 +

4

c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 + ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎫

⎬

⎪

⎭

⎪

+ y0 + 5Δltanθ

(C5-e)

or,

 

y5 = (Δl)

2

6c4

1

c4

3− c5

c4

⎛

⎝

⎜

⎞

⎠

⎟ε4 + ε5

⎡

⎣

⎢

⎤

⎦

⎥+

1

c3

9 − 4 c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ε3 + 4ε4

⎡

⎣

⎢

⎤

⎦

⎥+

1

c2

15 − 7 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ε2 + 7ε3

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

 +

1

c1

21−10 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ε1 +10ε2

⎡

⎣

⎢

⎤

⎦

⎥+

1

c0

27 −13 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ε0 +13ε1

⎡

⎣

⎢

⎤

⎦

⎥

⎫

⎬

⎪

⎭

⎪

+ y0 + 5Δltanθ0

(C5-

 

e ′)

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

Observing the indicial behavior in equations (C5-a)–(C5-e), we can establish a generalized displacement 

equation for any index, i (= 1, 2, 3, ..., n) as

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

(C5-f)

which is deflection equation (22).

yi = (Δl)

2

6

1

c j−1

3− c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i

∑ + (Δl)

2

2

(i − j)

c j−1

2 − c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i−1

∑ + y0 + iΔl tanθ0
73

Also, based on the indicial behavior in equations (C5-b′)–(C5-

 

e ′), the generalized deflection equation 

can be written in a more compact form with single summation as

 

yi = (Δl)

2

6

1

ci− j

3(2 j −1) − (3j − 2)

ci− j +1

ci− j

⎡

⎣

⎢

⎤

⎦

⎥εi− j + (3 j − 2)εi− j +1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪ j=1

i

∑ + y0 + iΔltanθ0

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

(C5-

 

f ′)

which is deflection equation (23).

The general deflection equation (C5-f) is checked for correctness by using i = 4. For i = 4,

equation (C5-f) becomes

(C6)

which agrees perfectly with equation (C5-d) and thus confirms the accuracy of general deflection equation 

(C5-f).

Likewise, the alternative general deflection equation (C5-

 

f ′) is also checked for accuracy by using i = 4. 

For i = 4, equation (C5-

 

f ′) becomes

y4 = (Δl)

2

6

1

c4− j

3(2 j −1) − (3j − 2)

c4− j+1

c4− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε 4− j + (3j − 2)ε 4− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

j=1 ⎭⎪

4

∑ + y0 + 4Δl tanθ0

= (Δl)

2

6

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c2

9 − 4 c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + 4ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

15 − 7 c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 7ε 2

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

 +

1

c0

21−10 c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 +10ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎪

⎭⎪

+ y0 + 4Δl tanθ0

(C7)

which agrees with equation (C5-

 

d ′) and thus confirms the accuracy of the alternative general deflection 

equation (C5-

 

d ′).

y4 = (Δl)

2

6

1

cj−1

3− cj

cj−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

j=1 ⎥

4

∑ + (Δl)

2

2

(4 − j)

cj−1

2 − cj

cj−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

j=1 ⎥

4−1

∑ + y0 + 4Δl tanθ0

= (Δl)

2

6

1

c3

3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + ε 4

⎡

⎣

⎢ ⎤

⎦

⎥

1

c2

3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c1

3− c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε 2

⎡

⎣

⎢ ⎤

⎦

⎥ +

1

c0

3− c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ (Δl)

2

2

1

c2

2 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + ε 3

⎡

⎣

⎢ ⎤

⎦

⎥ +

2

c1

2 − c2

c1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + ε 2

⎡

⎣

⎢ ⎤

⎦

⎥ +

3

c0

2 − c1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + ε1

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + 4Δl tanθ0
74

APPENDIX D

DERIVATIONS OF DEFLECTION EQUATIONS FOR

UNIFORM CANTILEVER BEAMS

When the nonuniformity of the beam diminishes (that is, 

 

ci = ci−1 = c, as in the uniform cantilever 

beam case), slope equations (18) and (19) degenerate into the following forms:

 

tanθi = Δl

2c

(εi−1 + εi) + tanθi−1 (D1)

 

tanθi = Δl

2c

(ε j−1 + ε j)

j=1

i

∑ + tanθ0 (D2)

Furthermore, deflection equations (21) and (23) take on the following forms:

 

yi = (Δl)

2

6c

(2εi−1 + εi) + yi−1 + Δltanθi−1 (D3)

 

yi = (Δl)

2

6c

(3j −1)εi− j + (3 j − 2)ε [ i− j +1]

j=1

i

∑ + y0 + iΔltanθ0

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

 

tanθ0 = 0

(D4)

Writing out equation (D4) for indices, i = 1–5, we obtain

 

y1 = (Δl)

2

6c

(2ε0 + ε1) + y0 + Δltanθ0 (D4-a)

 

y2 = (Δl)

2

6c

(3−1)ε2−1 + (3− 2)ε2−1+1 + (3× 2 −1)ε2−2 + (3× 2 − 2)ε [ 2−2+1]

 + y0 + 2Δltanθ0

 = (Δl)

2

6c

2ε1 + ε2 + 5ε0 + 4ε ( 1) + y0 + 2Δltanθ0 

 = (Δl)

2

6c

5ε0 + 6ε1 + +ε ( 2 ) + y0 + 2Δltanθ0

(D4-b)
75

y3 = (Δl)

2

6c

(3 − 1)ε 3−1 + (3 − 2)ε [ 3−1+1 ] + (3 × 2 − 1)ε 3−2 + (3 × 2 − 2)ε [ 3−2+1 ]

+ (3 × 3 − 1)ε 3−3 + (3 × 3 − 2)ε [ 3−3+1 ]

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + y0 + 3Δl tanθ0

 = (Δl)

2

6c

2ε 2 + ε ( 3 ) + 5ε1 + 4ε ( 2 ) + 8ε 0 + 7ε ⎡ ( 1 ) ⎣ ⎤

⎦ + y0 + 3Δl tanθ0

 = (Δl)

2

6c

8ε 0 + 12ε1 + +6ε 2 + ε ( 3 ) + y0 + 3Δl tanθ0

 = (Δl)

2

6c

8ε 0 + 6(2ε1 + ε 2 ) + ε [ 3 ] + y0 + 3Δl tanθ0 (D4-c)

y4 = (Δl)

2

6c

(3 − 1)ε 4−1 + (3 − 2)ε [ 4−1+1 ] + (3 × 2 − 1)ε 4−2 + (3 × 2 − 2)ε { [ 4−2+1 ]

 + (3 × 3 − 1)ε 4−3 + (3 × 3 − 2)ε [ 4−3+1 ] + (3 × 4 − 1)ε 4−4 + (3 × 4 − 2)ε [ 4−4+1 ]} + y0 + 4Δl tanθ0

 = (Δl)

2

6c

2ε 3 + ε ( 4 ) + 5ε 2 + 4ε ( 3 ) + 8ε1 + 7ε ( 2 ) + 11ε 0 + 10ε ⎡ ( 1 ) ⎣ ⎤

⎦ + y0 + 4Δl tanθ0

 = (Δl)

2

6c

11ε 0 + 18ε1 + 12ε 2 + 6ε 3 + ε ( 4 ) + y0 + 4Δl tanθ0

= (Δl)

2

6c

11ε 0 + 6 3ε1 + 2ε 2 + ε ( 3 ) + ε 4 ⎡

⎣ ⎤⎦ + y0 + 4Δl tanθ0 (D4-d)

y5 = (Δl)

2

6c

(3 − 1)ε 5−1 + (3 − 2)ε [ 5−1+1 ] + (3 × 2 − 1)ε 5−2 + (3 × 2 − 2)ε { [ 5−2+1 ]

 + (3 × 3 − 1)ε 5−3 + (3 × 3 − 2)ε [ 5−3+1 ] + (3 × 4 − 1)ε 5−4 + (3 × 4 − 2)ε [ 5−4+1 ]

 + (3 × 5 − 1)ε 5−5 + (3 × 5 − 2)ε [ 5−5+1 ]} + y0 + 5Δl tanθ0 

 = (Δl)

2

6c

2ε 4 + ε ( 5 ) + 5ε 3 + 4ε ( 4 ) + 8ε 2 + 7ε ( 3 ) + 11ε1 + 10ε ( 2 ) + 14ε 0 + 13ε ⎡ ( 1 ) ⎣ ⎤

⎦

 + y0 + 5Δl tanθ0

 = (Δl)

2

6c

14ε 0 + 24ε1 + 18ε 2 + 12ε 3 + 6ε ( 4 ) + y0 + 5Δl tanθ0

= (Δl)

2

6c

14ε 0 + 6 4ε1 + 3ε 2 + 2ε 3 + ε ( 4 ) + ε 5 ⎡

⎣ ⎤⎦ + y0 + 5Δl tanθ0 (D4-e)
76

Based on these indicial progressions, deflection equations (D4-a)–(D4-e) can be written in generalized 

forms for the indices, i, as

yi = (Δl)

2

6c

(3i −1)ε0 + 6 (i − j)ε j

j=1

i−1

∑ + εi

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + iΔl tanθ0 (D4-f)

which is identical to equation (28).
77

APPENDIX E

RELATIVE MAGNITUDES OF STRAIN TERMS FOR

UNIFORM CANTILEVER BEAMS

For the study of relative magnitudes of the strain terms in uniform beam deflection equation (28), beam 

tip deflection equation (28f), duplicated as follows, is used:

 

yn = (Δl)

2

6c

(3n −1)ε0 + 6 (n − j)ε j

j=1

n−1

∑ + εn

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + nΔltanθ0 (E1)

The classical case of a uniform cantilever beam of a length, l, subjected to a tip vertical load, P, is 

considered in the analysis. For this case, the bending strain, 

 

ε(x), is a linearly decreasing function of x, with 

a maximum at the fixed end and zero at the free end according to the expression

 

ε(x) = 1− x

l

⎛

⎝

⎜ ⎞

⎠

⎟ε0 (E2)

in which 

 

ε0 is the maximum bending strain at the built-in end. 

The beam tip deflection, 

 

yn , for different values of n (= 1, 2, 4, 8) are provided in the remainder of this 

appendix.

For n = 1, the strain at the beam tip is ε1 = 0 × ε 0 , and equation (E1) yields

 

y1 = (Δl)

2

6c

(2ε0 + ε1) = l

2

3c

(1+ 0)ε0

 = l

2

3c

ε0 = l

2

3c

M(0)c

EI = l

2

3c

(Pl)c

EI = Pl33EI

(E3)

in which the strain, 

 

ε0 , is expressed in terms of the moment, 

 

M(0), using the strain-moment relationship 

(eq. (4)).

For n = 2, the strains at strain-sensing stations, i = 1–2, are 

 

ε1 = 1

2

ε0, ε2 = 0

2

ε0 , and equation (E1) 

yields

y2 = (Δl)

2

6c

5ε0 + 6ε1 + +ε ( 2 ) = 1

6c

l

2

22

⎛

⎝

⎜

⎞

⎠

⎟ 5ε0 + 6 1

2

ε0 +

0

2

ε0 ⎛

⎝

⎜ ⎞

⎠

⎟

 = l

2

3c

ε0 0.625

maximum

 + 0.375 + 0 ⎛

⎝

⎜

⎞

⎠

⎟

 = l

2

3c

ε0(1) = Pl

3

3EI

(E4)
78

For n = 4, the strains at strain-sensing stations, i = 1–4, are 

 

ε1 = 3

4

ε0, 

 

ε2 = 2

4

ε0 , 

 

ε3 = 1

4

ε0 ,

 

ε4 = 0

4

ε0, 

and equation (E1) yields

y4 = (Δl)

2

6c

11ε0 + 6 3ε1 + 2ε2 + ε ( 3) + ε 4 ⎡

⎣ ⎤⎦

 = 1

6c

l

2

42

⎛

⎝

⎜

⎞

⎠

⎟ 11ε0 + 6 3 3

4

ε0 + 2 2

4

ε0 +

1

4

ε0 ⎛

⎝

⎜ ⎞

⎠

⎟ +

0

4

ε0

⎡

⎣

⎢ ⎤

⎦

⎥

 = l

2

3c

ε0 0.34375 + 0.421875

maximum

 + 0.1875 + 0.046875 + 0

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

 = l

2

3c

ε0 (1) = Pl

3

3EI (E5)

For n = 8, the strains at strain-sensing stations, i = 1–8, are 

 

ε1 = 7

8

ε0 , 

 

ε2 = 6

8

ε0 , 

 

ε3 = 5

8

ε0 , 

 

ε4 = 4

8

ε0, 

 

ε5 = 3

8

ε0 , 

 

ε6 = 2

8

ε0 , 

 

ε7 = 1

8

ε0, 

 

ε8 = 0

8

ε0 , and equation (E1) yields 

(E6)

y8 = (Δl)

2

6c

23ε 0 + 6 7ε1 + 6ε 2 + 5ε 3 + 4ε 4 + 3ε 5 + 2ε 6 + ε ( 7 ) + ε 8 ⎡

⎣ ⎤⎦

= 1

6c

l

2

82

⎛

⎝

⎜

⎞

⎠

⎟ 23ε 0 + 6 7 7

8

ε 0 + 6 6

8

ε 0 + 5 5

8

ε 0 + 4 4

8

ε 0 + 3 3

8

ε 0 + 2 2

8

ε 0 +1

1

8

ε 0

⎛

⎝

⎜ ⎞

⎠

⎟ +

0

8

ε 0

⎡

⎣

⎢ ⎤

⎦

⎥ 

= l

2

3c

ε 0 0.17969 + 0.28711

maximum

 + 0.21094 + 0.14648 + 0.09375 + 0.05273+ 0.02344 + 0.00586 + 0

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

= l

2

3c

ε 0 (1) = Pl

3

3EI

 
79

REFERENCES

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

Society of Flight Test Engineers 18th Annual Symposium, Paper No. 22, Amsterdam, Netherlands, Sept.

1987. 

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

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

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

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

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

New York, 1954.

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

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

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

Company, Cleveland, Ohio, 1957.

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

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

OMB No. 0704-0188

1. REPORT DATE (DD-MM-YYYY)

01-09-2009

2. REPORT TYPE 

Technical Publication

 4. TITLE AND SUBTITLE

Further Development of Ko Displacement Theory for Deformed Shape 

Predictions of Nonuniform Aerospace Structures

5a. CONTRACT NUMBER

 6. AUTHOR(S)

William L. Ko and Van Tran Fleischer

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

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

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

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

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H-2889

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NASA

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Ko and Fleischer, NASA Dryden Flight Research Center. 

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

19b. NAME OF RESPONSIBLE PERSON

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14. ABSTRACT

The Ko displacement theory previously formulated for deformed shape predictions of nonuniform beam structures is further developed mathematically. The furtherdeveloped displacement equations are expressed explicitly in terms of geometrical parameters of the beam and bending strains at equally spaced strain-sensing stations 

along the multiplexed fiber-optic sensor line installed on the bottom surface of the beam. The bending strain data can then be input into the displacement equations for 

calculations of local slopes, deflections, and cross-sectional twist angles for generating the overall deformed shapes of the nonuniform beam. The further-developed 

displacement theory can also be applied to the deformed shape predictions of nonuniform two-point supported beams, nonuniform panels, nonuniform aircraft wings 

and fuselages, and so forth. The high degree of accuracy of the further-developed displacement theory for nonuniform beams is validated by finite-element analysis of 

various nonuniform beam structures. Such structures include tapered tubular beams, depth-tapered unswept and swept wing boxes, width-tapered wing boxes, and 

double-tapered wing boxes, all under combined bending and torsional loads. The Ko displacement theory, combined with the fiber-optic strain-sensing system, 

provide a powerful tool for in-flight deformed shape monitoring of unmanned aerospace vehicles by ground-based pilots to maintain safe flights.

15. SUBJECT TERMS

Deflection equations, Displacement theories, Embedded strain sensing, In-flight shape predictions, Slope equations

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