NASA/TP—2012–215976

First- and Second-Order Displacement 

Transfer Functions for Structural Shape 

Calculations Using Analytically Predicted 

Surface Strains

William L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

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March 2012
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NASA/TP—2012–215976

First- and Second-Order Displacement 

Transfer Functions for Structural Shape 

Calculations Using Analytically Predicted 

Surface Strains

William. L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

 Insert conference information, if applicable; otherwise delete 

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National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, CA 93523-0273

March 2012
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.

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

the National Aeronautics and Space Administration
v

TABLE OF CONTENTS

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

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

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

BASICS OF DISPLACEMENT TRANSFER FUNCTIONS ....................................................................3

Beam Differential Equations..................................................................................................................3

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

Basic Assumptions.................................................................................................................................4

Linearity of Bending Moment................................................................................................................5

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

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

ALTERNATIVE DISPLACEMENT TRANSFER FUNCTIONS.............................................................7

FIRST-ORDER DISPLACEMENT TRANSFER FUNCTION.................................................................7

First-Order Slope Equations...................................................................................................................8

First-Order Deflection Equations...........................................................................................................9

SECOND-ORDER DISPLACEMENT TRANSFER FUNCTION..........................................................11

Second-Order Slope Equations............................................................................................................11

Second-Order Deflection Equations ....................................................................................................12

SUMMARY OF ALL DISPLACEMENT TRANSFER FUNCTIONS...................................................13

Previous Displacement Transfer Functions .........................................................................................14

New Displacement Transfer Functions................................................................................................14

Characteristics of Displacement Transfer Functions ...........................................................................15

ANALYTICAL SHAPE PREDICTIONS ................................................................................................15

COMPARISON OF SHAPE PREDICTION ACCURACY.....................................................................16

Tapered Cantilever Tubular Beams .....................................................................................................16

Surface Bending Strains.......................................................................................................................16

Deflection Data ....................................................................................................................................17

Error Curves.........................................................................................................................................20

DOMAIN DENSITY ................................................................................................................................20

Improved Strain Curves .......................................................................................................................20

Improved Deflection Data....................................................................................................................20

Error Comparisons...............................................................................................................................22

DISCUSSION ...........................................................................................................................................24

SUMMARY..............................................................................................................................................24

FIGURES ..................................................................................................................................................26
vi

APPENDIX A: DERIVATIONS OF THE FIRST-ORDER SLOPE AND 

DEFLECTION EQUATIONS .............................................................................................................34

APPENDIX B: DERIVATIONS OF THE SECOND-ORDER SLOPE AND 

DEFLECTION EQUATIONS .............................................................................................................42

REFERENCES..........................................................................................................................................51
1

ABSTRACT

New first- and second-order displacement transfer functions have been developed for deformed shape 

calculations of nonuniform cross-sectional beam structures such as aircraft wings. The displacement 

transfer functions are expressed explicitly in terms of beam geometrical parameters and surface strains 

(uniaxial bending strains) obtained at equally spaced strain stations along the surface of the beam 

structure. By inputting the measured or analytically calculated surface strains into the displacement 

transfer functions, one could calculate local slopes, deflections, and cross-sectional twist angles of the 

nonuniform beam structure for mapping the overall structural deformed shapes for visual display. The 

accuracy of deformed shape calculations by the first- and second-order displacement transfer functions 

are determined by comparing these values to the analytically predicted values obtained from finiteelement analyses. This comparison shows that the new displacement transfer functions could quite 

accurately calculate the deformed shapes of tapered cantilever tubular beams with different tapered 

angles. The accuracy of the present displacement transfer functions also are compared to those of the 

previously developed displacement transfer functions.

NOMENCLATURE

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

ci depth factor (vertical distance from neutral axis to i-th strain station on bottom surface of 

nonuniform beam), in.

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

cn / c0 beam depth ratio, dimensionless 

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

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

deg degree

E Young’s modulus, lb/in2

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

I moment of inertia, in4

j index

l length of beam, in.

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

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

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

P force, lb

SPAR Stuctural Performance And Resizing

t wall thickness of tubular beam, in.

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

xi axial coordinate associated with 

 

i-th surface strain station, in. 

yi ≡ y(xi) , beam deflection in y-direction at axial location, x = xi , in. 

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

α ≡ tan−1 [(c0 − cn )/ l] taper angle of beam, deg

Δl ≡ (xi − xi−1) = l / n , distance between two adjacent strain stations { xi−1, xi }, in.

εi ≡ ε(xi) , surface bending strain at strain station, xi , in/in

ε(x) surface bending strain (axial strain) at axial location, x , in/in
2

θi ≡ θ(xi), slope of deformed beam at axial location, x = xi , rad or deg

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

σi ≡ Eεi , axial stress associated with surface bending strain, εi , lb/in2

σ(x) ≡ Eε(x) , axial stress associated with surface bending strain, ε(x), lb/in2

INTRODUCTION

Recently, several displacement transfer functions were formulated for structural deformed shape 

calculations using surface strains (refs. 1–4). In the formulations of the displacement transfer functions, 

the beam-like structure (such as an aircraft wing) was discretized evenly into multiple small domains of 

the same length. Through such discretization, linear-variation assumptions could be made for both beam 

depth and surface strain within each small domain. This approach enabled the integrations of the classical 

beam differential equation in closed forms to yield beam slopes and deflections for each small domain. 

By combining the slope and deflection equations, one can write the final forms of the deflection 

equations, called displacement transfer functions, in terms of beam geometrical parameters and surface 

strains evaluated at evenly distributed inboard strain stations (at domain junctures). Thus, the 

displacement transfer functions can be used to convert the surface strains into displacements (deflections 

and angular rotations) at multiple domain junctures so that one can map out overall deformed shapes of 

the beam structures for visual display. Thus, the displacement transfer functions, with an accompanying 

surface strain–sensing system, has created a revolutionary new structural shape–sensing technology 

(ref. 5, Method for Real-Time Structure Shape-Sensing, Ko-Richards, U.S. Patent No. 7,520,176) for inflight deformed shape monitoring of flexible wings and tails. In addition, the wing shape monitored in 

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

the structural deformed shape calculations, the distributed surface strains also can be used to predict the 

structural operational loads (ref. 6, Process for Using Surface Strain Measurements to Obtain 

Operational Loads for Complex Structures, Richards-Ko, U.S. Patent No. 7,715,994) for flight vehicle 

operational load monitoring.

When the structural shape–sensing technology is applied, using the conventional strain gage system 

for sensing the surface strains of flight vehicles is impractical because of excessive lead wire weight. The 

most attractive candidate for a flight vehicle surface strain–sensing system, an alternative to the 

conventional strain gage system, is the fiber-optic strain-sensing system, because the fiber optics are

lightweight, fine, and flexible filaments (approximately the size of human hairs), and they can be highly 

multiplexed (Bragg gratings) to define the strain-sensing points at desired sensing intervals (refs. 7–9). 

Another powerful characteristic of the fiber-optic strain-sensing system is that, for sensing a given 

structure, the needed number of actual strain data extraction points can be easily specified by means of a 

single command.

The displacement transfer functions were originally developed for deformed shape predictions of 

straight beams and were successfully validated for accuracy by finite-element analyses of different 

sample structures, such as cantilever tubular beams (uniform, tapered, slightly tapered, stepwise tapered), 

two-point supported tapered tubular beams, flat panels, and tapered wing boxes (unswept and swept). The 

straight-beam displacement transfer functions were further extended to the shape predictions of slender 

curved structures by introducing curvature-effect collections (ref. 10).

In this report, the first- and second-order displacement transfer functions are developed for deformed 

shape predictions of nonuniform cantilever beam structures (such as aircraft wings). Analytically 

calculated surface strains are then input to the transfer functions, and structural shape calculations are

made. The shape calculation accuracy of the newly formulated displacement transfer functions are then 
3

compared with those of the earlier displacement transfer functions (ref. 2) developed for nonuniform and 

slightly nonuniform cantilever structures. 

BASICS OF DISPLACEMENT TRANSFER FUNCTIONS

The basics of the displacement transfer functions previously formulated (refs. 1–4) for shape 

predictions of cantilever straight beams using distributed surface strains are briefly described in this 

section. The piecewise-linear assumptions made in the formulation of the displacement transfer functions 

also are discussed.

Beam Differential Equations

The formulation of the displacement transfer functions stems from the classical beam differential 

equation (elastic curvature of the deformed straight uniform beam) given by (refs. 11–12)

d2y

dx2 = M(x)

EI (1)

in which y is the vertical deflection, x is the beam axial coordinate, M (x) is the bending moment, E is 

the Young’s modulus, and I is the cross-sectional moment of inertia. Equation (1) can be applied to a 

nonuniform beam segment with sufficient accuracy if the beam cross sections change gradually (weak 

nonuniform beam) (ref. 11, p. 143). For the weak nonuniform beam case, I is a weak function of x , and 

over a short beam segment, the average value can be used. 

At the cross-section, x, the bending moment, M (x), and associated bending strain (axial strain), 

ε(x)[= σ(x)/ E], at the bottom surface point of the nonuniform straight beam can be written as

M (x) = I

σ(x)

c(x)

= EI

ε(x)

c(x) (2)

in which c(x) is the beam depth factor (vertical distance from the neutral axis to the bottom surface of 

the nonuniform beam, or half depth if the neutral axis is located at the half depth of the beam). For a 

tubular beam case, c(x) will be the outer radius. The sign convention of equation (2) is for the positive 

bending moment, M (x) (bending the cantilever beam upward; fig. 1), to induce a positive bending strain, 

ε(x), on the bottom surface of the beam. Combining equations (1) and (2), one obtains the modified 

differential equation for the nonuniform beam as

d2y

dx2 = ε(x)

c(x) (3)

Note that the modified beam differential equation (3) contains only the beam depth factor, c(x), and 

the surface bending strain, ε(x). The flexural rigidity,

 

EI, is eliminated. The formulation of the 

displacement transfer functions for the nonuniform straight beam is based upon the modified beam 

differential equation (3), which could be sufficiently accurate if the nonuniform beam cross sections 
4

change gradually (ref. 11, p. 143). If the mathematical functional forms of {c(x), ε(x)} are known, 

equation (3) can be integrated once to yield the beam slope, dy / dx[≡ tanθ(x)] , and then integrated a

second time to yield the beam deflection, y(x), for each small domain needed for the structural deformed 

shape predictions.

Discretization

To carry out the single and double integrations of equation (3) in closed forms, the nonuniform 

cantilever beam of length l was discretized into n number of small domains of equal length, Δl(= l / n)

(fig. 1). The i-th (i = 0, 1, 2, 3, ..., n) strain stations are to be located on the bottom surface of the 

domain junctures, x = xi (i = 0, 1, 2, 3, ..., n) (called strain stations, xi ). Note from figure 1 that the 

first and last strain stations { x0 , xn } are located at the fixed end (x = x0 = 0) and at the free end 

(x = xn = l) , respectively. Within each small domain, xi−1 ≤ x ≤ xi , between the two adjacent strain 

stations, { xi−1, xi }, the following piecewise linear assumptions can be made with sufficient accuracy for 

slowly changing functions. 

Basic Assumptions

The following two basic assumptions are needed for deriving the displacement transfer functions:

1. Within the small domain, xi−1 ≤ x ≤ xi , between the two adjacent strain stations, { xi−1, xi }, the 

depth factor, c(x), can be represented with a linear function of (x − xi−1), as

c(x) = ci−1 − (ci−1 − ci)

x − xi−1

Δl ; xi−1 ≤ x ≤ xi (4)

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

2. Within the small domain, xi−1 ≤ x ≤ xi , between the two adjacent strain stations, { xi−1, xi }, the

bottom surface bending strain, ε(x), also can be assumed to vary linearly with (x − xi−1), as

ε(x) = ε i−1 − (ε i−1 − ε i)

x − xi−1

Δl ; xi−1 ≤ x ≤ xi (5)

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

If ε(x) is a strong nonlinear function of x, the domain length, 

 

Δl, should be reduced so that the piecewise 

linear assumption of equation (5) can remain as a good approximation.
5

Linearity of Bending Moment

Substituting equations (4) and (5) into equation (2), and expanding the factor, 1/ c(x) , in binomial 

series form because of a small coefficient (ci − ci−1) of the slope term (weak nonuniform beam), one 

obtains 

M (x) = EI

ε(x)

c(x)

= EI

ε i−1

ci−1

1− 1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2 (x − xi−1)

2

(Δl)

2

+ ....

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

(6)

For a weak nonuniform beam, (1− ci / ci−1) is small (that is, (1− ci / ci−1) << 1 ), and within the small 

domain, xi−1 ≤ x ≤ xi , if the average value of I is used, the bending moment, M (x), within the domain, 

xi−1 ≤ x ≤ xi , will vary almost linearly because higher order terms in (1− ci / ci−1) can be neglected. For 

a moderate nonuniform beam, however, the value of (1− ci / ci−1) is no longer very small, and the higher 

order terms must be retained. This retention of the higher order terms will cause M (x) to be slightly 

nonlinear over the domain, xi−1 ≤ x ≤ xi .

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

the bending strain, εi(x) , according to equation (2). Therefore, if the bending moment, M (x), is a linear 

function of x , then the bending strain, ε(x), also will be a linear function of x .

Basic Slope Equations

The slope, tanθ(x)(≡ dy / dx), of the nonuniform beam at the axial location, x, within the domain, 

xi−1 ≤ x ≤ xi (fig. 1), can be obtained by integrating equation (3) once, and enforcing the continuity of 

the slope at the inboard strain station, xi−1, as

tanθ(x) = d2

y

dx2 dx

xi−1

x

∫

Integration of eq. (3) 



+ tanθi−1

Slope at xi−1

 = ε(x)

c(x)

dx

xi−1

x

∫

Slope increment

above tanθi−1



+ tanθi−1

Slope at xi−1

 ; ( xi−1 ≤ x ≤ xi )

(7)

in which tanθi−1 is the slope at the inboard strain station, xi−1. In light of the piecewise linear 

assumptions of {c(x), ε(x)} given by equations (4) and (5), respectively, equation (7) can be written in 

the form

tanθ(x) =

εi−1 − (εi−1 − εi)

x − xi−1

Δl

ci−1 − (ci−1 − ci)

x − xi−1

Δl

dx

xi−1

x

∫ + tanθi−1 ; ( xi−1 ≤ x ≤ xi ) (8)
6

Note that the direct integration of equation (8) will result in a slope equation of the form (ref. 1, eq. 

(23))

tanθi = Δl εi−1 − εi

ci−1 − ci

+

εi−1ci − εici−1

(ci−1 − ci)

2 loge

ci

ci−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ tanθi−1 ; (i = 1, 2, 3, ..., n) (9)

in which tanθi[≡ tanθ(xi)] is the slope at the strain station, xi . Note from equation (9) that in the limit, 

ci / ci−1 = 1, for the uniform beam, the first term, (εi−1 − εi) (ci−1 − ci), will go to infinity, and the 

second term with the factor, loge(ci / ci−1 [ )] (ci−1 − ci)

2 , will become mathematically indeterminate

(that is, 

 

0/0). To circumvent this limit-case mathematical breakdown problem, the logarithmic term must 

be expanded in terms of a (ci−1 − ci) series in the vicinity of ci / ci−1 ≈ 1 , so that the denominators 

containing the factor, (ci−1 − ci), can be cancelled out. After mathematical manipulations are carried out, 

equation (9) can be reduced to the form (ref. 1, eq. (27))

tanθi = Δl

2ci−1

2 − ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ ε i−1 + ε i

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ tanθi−1 ; (i = 1, 2, 3, ..., n) (10)

which is now applicable to the limit case, ci / ci−1 = 1, of the uniform beam.

Basic Deflection Equations

The deflection, y(x), of the nonuniform beam within the domain, xi−1 ≤ x ≤ xi (fig. 1), can be 

obtained by integrating the slope equation (7) and enforcing the continuity of the deflection at the inboard 

adjacent strain station, xi−1, as

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 above yi−1



+ tanθi−1 xi−1

x

∫ dx

Deflection at x

due to tanθi−1



+ yi−1

Deflection

at xi−1



(xi−1 ≤ x ≤ xi) (11)

in which yi−1 is the deflection at the inboard strain station, xi−1. 

 

In light of the piecewise linear assumptions of { c(x), ε(x) } given by equations (4) and (5), 

respectively, the deflection equation (11) takes on the form

y(x) =

1− 1− εi

εi−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

1− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

dx

xi−1

x

∫ xi−1

x

∫ dx + tanθi−1 dx

xi−1

x

∫ + yi−1 ; (xi−1 ≤ x ≤ xi) (12)
7

Note that direct integration of equation (12) will yield the deflection equation in the form (ref. 1, 

eq. (25)) 

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

(i = 1, 2, 3,..., n) (13)

in which yi[≡ y(xi)] is the deflection at the strain station, xi . Note from equation (13) that in the limit, 

ci / ci−1 = 1, for a uniform beam case, the first and third terms with the factor, (ci−1 − ci), in the 

denominators will go to infinity, and the second term with the factor, loge(ci / ci−1 [ )] (ci−1 − ci)

3 , will 

become mathematically indeterminate (that is, 

 

0/0). To circumvent this limit-case mathematical 

breakdown problem, the logarithmic term must be expanded in terms of a (ci−1 − ci) series in the 

neighborhood of ci / ci−1 ≈ 1 so that the denominators containing the factor, (ci−1 − ci), can be cancelled 

out. After the mathematical manipulations are carried out, equation (13) can be reduced to the form 

(ref. 1, eq. (30))

yi = (Δl)

2

6ci−1

3− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + εi

⎡

⎣

⎢ ⎤

⎦

⎥ + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3, ..., n) (14)

which is now applicable to the limit case, ci / ci−1 = 1, of the uniform beam.

ALTERNATIVE DISPLACEMENT TRANSFER FUNCTIONS

To circumvent the aforementioned limit-case mathematical problems, the development of alternative 

displacement transfer functions for the nonuniform cantilever beam is needed. The new alternative 

displacement transfer functions also must be applicable to the limit case of a uniform beam without 

creating limit-case mathematical indeterminacy problems. Because the cantilever beam is assumed to be a 

weak nonuniform beam (that is, 1− (ci / ci−1) << 1 ), the perturbation method can be applied to formulate 

the new displacement transfer functions by expanding the factor, 1/ c(x) , in the integrand of equations (8) 

or (12) in binomial series form up to the first- or second-order terms. The formulations of the new firstand second-order displacement transfer functions are presented in the following sections. Appendices A

and B present the detailed mathematical derivations for readers to easily follow.

FIRST-ORDER DISPLACEMENT TRANSFER FUNCTION

In this section, the first-order slope equations and first-order deflection equations are formulated for 

the nonuniform cantilever beam. Appendix A presents the detailed mathematical derivations.
8

First-Order Slope Equations

Under the assumption of the present weak nonuniform cantilever beam for which (1− ci / ci−1) << 1 , 

the factor, 1− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢ ⎤

⎦

⎥

−1

, in the integrand of the slope equation (8) can be expanded in 

binomial series form up to the first-order term to yield

tanθ(x) = ε i−1

ci−1

1− 1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

+ ... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

dx

xi−1

xi ∫ + tanθi−1

( xi−1 ≤ x ≤ xi ) (15)

in which tanθi−1 [≡ tanθ(xi−1)] is the slope at x = xi−1 .

 

After carrying out the integration of equation (15) (ref. 13), one obtains the first-order slope equation 

for the domain, xi−1 ≤ x ≤ xi (see Appendix A for detailed derivations),

tanθ(x) = εi−1

ci−1

(x − xi−1) +

εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

2Δl − 1− εiεi−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

3(Δl)

2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ tanθi−1

( xi−1 ≤ x ≤ xi ) (16)

At the strain station, xi , we have (xi − xi−1) = Δl , and equation (16) takes on the form (see 

Appendix A)

tanθi = Δl

6ci−1

4 − ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + 5− 2 ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθi−1 ; (i = 1, 2, 3, ..., n) (17)

in which tanθi [≡ tanθ(xi)] is the slope at x = xi . Equation (17) is the first-order slope equation in a 

descending recursion relationship. Applying the descending recursion relationships, one can rewrite 

equation (17) in a series summation form for the strain station, xi , as

tanθi = Δl

6

1

c j−1

4 − c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j−1 + 5− 2

c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟ ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i

∑ + tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (18)
9

Equation (18) is the final form of the first-order slope equation in series summation form for the 

nonuniform cantilever beam.

For the uniform beam case ( ci = ci−1 = c ), equation (18) degenerates into the form

tanθi = Δl

2c

(ε j−1 + ε j)

j=1

i

∑ + tanθ0

=0 for canti- lever beams

 ; (i = 1, 2, 3, ..., n) (19)

which agrees with previously established equation (25) of reference 2.

First-Order Deflection Equations

The deflection, y(x), of the nonuniform cantilever beam within the domain, xi−1 ≤ x ≤ xi , can be 

obtained by integrating the slope equation (16) (ref. 13), and enforcing the continuity of the deflection at 

the inboard adjacent strain station, xi−1, as (see Appendix A)

y(x) = tanθ(x

eq. (16)

)

xi−1

x

∫ dx + yi−1 

= (x − xi−1)

2

12

εi−1

ci−1

6 + 2 εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

Δl − 1− εiεi−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

(Δl)

2

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

 + (x − xi−1)tanθi−1 + yi−1

( xi−1 ≤ x ≤ xi ) (20)

in which yi−1 is the deflection at the inboard strain station, xi−1. 

At the strain station, xi , we have (xi − xi−1) = Δl , and equation (20) yields the deflection in the form 

(see Appendix A) 

yi = (Δl)

2

12ci−1

5− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + 3− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi

⎡

⎣

⎢ ⎤

⎦

⎥ + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3, ..., n) (21)

in which yi[≡ y(xi)] is the deflection at the strain station, xi . Equation (21) is the first-order deflection 

equation in a descending recursion relationship. 
10

For a uniform beam ( ci = ci−1 = c ), equation (21) degenerates into the form

yi = (Δl)

2

6c

2εi−1 + ε ( i) + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3, ..., n) (22)

which agrees with the previously established equation (26) of reference 2.

Substituting the slope equation (18) into the deflection equation (21), with descending recursion 

relationships applied, yields (see Appendix A for derivations)

yi = (Δl)

2

12

1

ci− j

5+ 8(j −1) − 1+ 2(j −1)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j

⎧

⎨

⎪

j=1 ⎩⎪

i

∑

 + 3+10(j −1) − 1+ 4(j −1)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j+1

⎫

⎬

⎪

⎭⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (23)

Equation (23) is the final form of the first-order deflection equation in series summation form (called the 

first-order displacement transfer function) for the nonuniform cantilever beam.

For the uniform beam case ( ci = ci−1 = c ), equation (23) degenerates into

yi = (Δl)

2

6c

(3j −1)εi− j + (3j − 2)εi− j+1 ⎡

⎣ ⎤⎦

j=1

i

∑ + y0 + (i)Δl tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (24)

which is identical to equation (27) of reference 2.

After the terms are grouped, equation (24) can be rewritten in the alternative form (see Appendix A 

for derivations)

 

yi = (Δl)

2

6c

(3i −1)ε0 + 6 (i − j)ε j

j=1

i−1

∑ + εi

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + (i)Δl tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (25)

which agrees with equation (28) of reference 2.
11

SECOND-ORDER DISPLACEMENT TRANSFER FUNCTION

In this section, the second-order slope equations and second-order deflection equations are formulated 

for the nonuniform cantilever beam. Appendix B presents the detailed mathematical derivations.

Second-Order Slope Equations

For the weak nonuniform beam with a slowly changing depth (that is, (1− ci / ci−1) << 1 ), accuracy of 

the displacement transfer functions can be improved by expanding the factor, 1− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢ ⎤

⎦

⎥

−1

, 

in the integrand of the slope equation (8) in binomial series form up to the second-order term as 

tanθ(x) = εi−1

ci−1

1− 1− εi

εi−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢ ⎤

⎦

⎥ 1+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2 (x − xi−1)

2

(Δl)

2 + ....

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

dx

xi−1

x

∫

+ tanθi−1

( xi−1 ≤ x ≤ xi ) (26)

in which tanθi−1 is the slope at the inboard strain station, xi−1. 

 

After carrying out the integration of equation (26), one obtains the second-order slope equation for 

the domain, xi−1 ≤ x ≤ xi , as (see Appendix B for details)

tanθ(x) = εi−1

ci−1

(x − xi−1) +

εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

2Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

3(Δl)

2

⎧

⎨

⎪

⎩⎪

− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− εi

εi−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

4

4(Δl)

3

⎫

⎬

⎪

⎭

⎪

+ tanθi−1

( xi−1 ≤ x ≤ xi ) (27)

 

At the strain station, x = xi , we have (xi − xi−1) = Δl , and equation (27) becomes (see Appendix B)

tanθi = Δl

12ci−1

9 − 4 ci

ci−1

+

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi−1 + 13−10 ci

ci−1

+ 3 ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ tanθi−1

(i = 1, 2, 3, ..., n) (28)

 
12

in which tanθi[≡ tanθ(xi)] is the slope at the strain station, xi . Equation (28) is the second-order slope 

equation in a descending recursion relationship. Applying the descending recursion relationships, one can 

write equation (28) in series summation form as 

tanθi = Δl

12

1

c j−1

9 − 4

c j

c j−1

+

c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε j−1 + 13−10

c j

c j−1

+ 3

c j

c j−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε j

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭ j=1 ⎪

i

∑ + tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (29)

Equation (29) is the final form of the second-order slope equation in series summation form for the 

nonuniform cantilever beam. 

For the uniform beam case ( ci = ci−1 = c ), the second-order slope equation (29) degenerates into 

tanθi = Δl

2c

(ε j−1 + ε j)

j=1

i

∑ + tanθ0

=0 for canti- lever beams

 ; (i = 1, 2, 3, ..., n) (30)

which agrees with equation (25) of reference 2.

Second-Order Deflection Equations

The deflection, y(x), of the nonuniform beam within the domain, xi−1 ≤ x ≤ xi , can be obtained by 

integrating the slope equation (27), and enforcing the continuity of the deflection at the inboard adjacent 

strain station, xi−1, as

y(x) = tanθ(x)

eq. (27)

xi−1 

x

∫ dx + yi−1

= εi−1

2ci−1

(x − xi−1)

2 +

1

3

εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

Δl

+

1

6

1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

4

(Δl)

2

⎧

⎨

⎪

⎩⎪

− 1

10

1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− εi

εi−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

5

(Δl)

3

⎫

⎬

⎪

⎭

⎪

+ yi−1 + (x − xi−1)tanθi−1

(xi−1 ≤ x ≤ xi) (31)

in which yi−1 is the deflection at the inboard strain station, xi−1. 
13

At the strain station, xi , we have (xi − xi−1) = Δl , and equation (31) becomes (see Appendix B)

yi = (Δl)

2

60ci−1

27 − 9

ci

ci−1

+ 2

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i−1 + 18 −11

ci

ci−1

+ 3

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ yi−1 + Δl tanθi−1

(i = 1, 2, 3, ..., n) (32)

which is the second-order deflection equation in a descending recursion relationship. 

For the uniform beam case ( ci = ci−1 = c ), equation (32) degenerates into

yi = (Δl)

2

6c

(2εi−1 + εi) + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3, ..., n) (33)

which is identical to equation (26) of reference 2. 

Substituting the slope equation (29) into the deflection equation (32), with descending recursion 

relationships applied, yields (see Appendix B)

yi = (Δl)

2

60

1

ci− j

27 + 45(j −1) − 9 + 20(j −1)

ci− j+1

ci− j

+ 2 + 5(j −1)

ci− j+1

ci− j

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

εi− j

⎧

⎨

⎪

⎩ j=1 ⎪

i

∑

 + 18 + 65(j −1) − 11+ 50(j −1)

ci− j+1

ci− j

+ 3+15(j −1)

ci− j+1

ci− j

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

εi− j+1

⎫

⎬

⎪

⎭

⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (34)

Equation (34) is the final form of the second-order deflection equation in series summation form (called 

the second-order displacement transfer function) for the nonuniform cantilever beam. 

For the uniform beam case ( ci = ci−1 = c ), equation (34) degenerates into

yi = (Δl)

2

6c

(3j −1)εi− j + (3j − 2)εi− j+1 ⎡

⎣ ⎤⎦

j=1

i

∑ + y0 + (i)Δl tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (35)

which is identical to equation (27) of reference 2.

SUMMARY OF ALL DISPLACEMENT TRANSFER FUNCTIONS

This section summarizes all the displacement transfer functions previously developed (refs. 1, 2, 4). 

These transfer functions are listed for comparison of mathematical functional forms.
14

Previous Displacement Transfer Functions

The displacement transfer functions previously developed for nonuniform and slightly nonuniform 

straight cantilever beams have the following mathematical forms (refs. 1, 2, 4):

1. The displacement transfer function for the nonuniform cantilever straight beam ( ci ≠ ci−1) 

(derivations in ref. 2, Appendix A) is expressed 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 log

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 log

c j

c j−1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i−1

∑ + y0 + (i)Δl tanθ0

=0 for cantilever beams



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

Equation (36) is not applicable to the uniform beam because of mathematical indeterminacy 

problems at ci = ci−1 = c .

2. The displacement transfer function for the slightly nonuniform cantilever straight beam

( ci / ci−1 →1) is expressed as

yi = (Δl)

2

6

1

ci− j j=1

i

∑ 3(2 j −1) − (3j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j + (3j − 2)εi− j+1

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (37)

which was obtained from equation (36) by expanding the logarithmic terms in series form in the 

neighborhood of ci / ci−1 ≈ 1 (derivations in ref. 2, Appendix C).

New Displacement Transfer Functions

The two new displacement transfer functions (23) and (34) developed in this report are duplicated as 

equations (38) and (39), respectively, as follows:

1. The first-order displacement transfer function (eq. (23)) is expressed as

yi = (Δl)

2

12

1

ci− j

5+ 8(j −1) − 1+ 2(j −1)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j

⎧

⎨

⎪

j=1 ⎩⎪

i

∑

 + 3+10(j −1) − 1+ 4(j −1)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

εi− j+1

⎫

⎬

⎪

⎭⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (38)
15

2. The second-order displacement transfer function (eq. (34)) is expressed as

yi = (Δl)

2

60

1

ci− j

27 + 45( j −1) − 9 + 20( j −1)

ci− j+1

ci− j

+ 2 + 5( j −1)

ci− j+1

ci− j

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i− j

⎧

⎨

⎪

⎩ j=1 ⎪

i

∑

+ 18 + 65( j −1) − 11+ 50( j −1)

ci− j+1

ci− j

+ 3+15( j −1)

ci− j+1

ci− j

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i− j+1

⎫

⎬

⎪

⎭

⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (39)

3. The displacement transfer function for the uniform straight cantilever beam ( ci = ci−1 = c ) 

(eq. (25)) is expressed as

yi = (Δl)

2

6c

(3i −1)ε 0 + 6 (i − j)ε j

j=1

i−1

∑ + ε i

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + (i)Δl tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (40)

Equation (40) is the degenerated form of equations (37), (38), and (39) for the uniform beam case 

( ci = ci−1 = c ) (see Appendices A and B).

Characteristics of Displacement Transfer Functions

Except for equation (36), equations (37), (38), and (39) could degenerate into equation (40) for the 

uniform beam case ( ci = ci−1 = c ) without a mathematical indeterminacy problem (see Appendix A and 

Appendix B). Note that in each of the displacement transfer functions (36)–(40), the deflection, yi , at the 

current strain station, xi , is expressed in terms of domain length, Δl , inboard beam half depths 

( c0 , c

1 , c2 , c3 ,..., ci), and the associated inboard strains ( ε 0 , ε1, ε 2 , ε 3,..., ε i ), including the values of ci

and ε i at the current strain station, xi , where the deflection, yi , is calculated. Because deflection 

equations (36)–(40) contain no structural properties, in the deformed shape predictions of complex 

structures (such as aircraft wings), one can avoid tedious computations of bending stiffness, EI , at 

different strain stations (cross sections). In fact, the effect of EI is absorbed implicitly by the surface 

bending strains, εi . Thus, not needing to know EI is the powerful characteristic of the displacement 

transfer functions for structural deformed shape predictions.

ANALYTICAL SHAPE PREDICTIONS

The shape calculations in this report are referred to as an analytical shape prediction study. Namely, 

instead of using actual measured surface bending strains (which require actual flights or time-consuming 

ground tests), the Structural Performance And Resizing (SPAR) finite-element computer program 

(ref. 14) was used to generate the surface bending strains and beam deflections. The SPAR-generated 

deflection curves (assumed to be the correct deflection curves) were then used as reference points to 

check the accuracy of the corresponding deflection curves calculated from different displacement transfer 

functions.
16

The surface bending strains, ε i (i = 0, 1, 2, 3, ..., n) , at the i-th strain stations needed for inputs to 

calculate the beam deflections were generated by converting the SPAR-generated axial nodal stresses, 

σi (i = 0, 1, 2, 3, ..., n) , at the i-th strain station into the associated bending strains, ε i , through Hooke’s 

law (that is, εi = σi /E), because the lateral stresses are practically zero for beam structures. Alternatively, 

the surface bending strains, εi , also can be generated from the axial length changes of the SPAR 

elements where the bending strains, εi , are to be obtained. This method was found to lose accuracy, 

however, near the highly bent beam tip region and therefore was not used. For accuracy verification, the 

SPAR-generated bending strains were then input to different displacement transfer functions for 

calculations of theoretical deflection curves for comparison with the corresponding SPAR-generated 

deflection curves.

COMPARISON OF SHAPE PREDICTION ACCURACY

In the following section, the shape prediction accuracy of the first- and second-order displacement 

transfer functions are compared with those of previous displacement functions developed for nonuniform 

and slightly nonuniform cantilever beams. For the comparative shape prediction accuracy studies, tapered 

cantilever tubular beams with different taper angles were chosen. 

Tapered Cantilever Tubular Beams 

For the tapered cantilever tubular beam (fig.1), the equally spaced strain stations will be along the 

bottom generatrix of the beam. Each of the tapered cantilever tubular beams analyzed has a radius that 

linearly decreases from the fixed end toward the free end. Table 1 lists the dimensions and taper angles, 

α , of the aluminum tapered tubular beams considered.

Table 1. Dimensions of aluminum tapered cantilever tubular beams.

l, in. 

(length)

t, in.

(wall thickness)

c0 , in.

(root depth)

cn , in.

(tip depth)

cn / c0

(depth ratio)

α {= tan−1

[(c0 − cn) / l]}, 

deg (taper angle)

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

100.5 0.02296 4 3 3/4 (0.75) 0.57

100.5 0.02296 4 2 2/4 (0.5) 1.14

100.5 0.02296 4 1 1/4 (0.25) 1.71

100.5 0.02296 4 0.5 0.5/4 (0.125) 1.99

100.5 0.02296 4 0.25 0.25/4 (0.0625) 2.14

Figure 2 shows the finite-element model of a typical tapered cantilever tubular beam (depth ratio of

cn / c0 = 2/4) generated from the SPAR finite-element computer program (ref. 14). The size of the SPAR 

model (identical for all tapered beam cases) also is indicated in figure 2. The tapered cantilever tubular 

beam is subjected to an upward load of 

 

P =100 lb at the beam tip. Figure 3 shows the SPAR-generated 

deformed shapes of the tapered cantilever tubular beams with different taper angles. For the higher taper

angles (figs. 3d–f), the tapering down region near each beam tip is highly bent, and therefore, high 

bending strains are induced in this region (see next section).

Surface Bending Strains

Figure 4 shows the SPAR-generated surface bending strain curves for different tapered cantilever 

tubular beams under bending, using the domain density, n = 8 (the total number of strain stations is 
17

1+ n = 9 ). The bending strain curve for the uniform beam case ( cn / c0 = 4/4) is a classical straight line. 

For the tapered beams, however, the strain curves are bow shaped, and the bow bend becomes deeper as 

the depth ratio, cn / c0 , decreases. At the depth ratio, cn / c0 = 0.5/4, the strain curve is deeply bent by 

nearly 130° toward the beam tip. When the depth ratio reaches the minimum of cn / c0 = 0.25/4, the strain 

curve changes to a “cat ear” shape with a nearly 160° sharp bend near the beam tip region where the 

strain reaches a maximum, and then sharply drops toward the beam tip. Note from figure 4 that the 

SPAR-generated strains are not exactly zero (theoretically zero) at the free beam tip (especially for the 

cn / c0 = 0.5/4 and cn / c0 = 0.25/4 cases) because of the discretization of the beam with finite elements 

(fig. 2).

Keep in mind that in the formulation of the present first- and second-order displacement transfer 

functions and all the previous displacement transfer functions, the distribution of the bending strain along 

the discretized cantilever beam was assumed to be piecewise linear (eq. (5)). Therefore, the strain curves 

used in the formulation should be piecewise-linear curves (connecting the two adjacent strain data points 

with straight lines) instead of the smooth strain curves shown in figure 4. For moderately bent strain 

curves (fig. 4), the piecewise-linear approximation could be quite accurate, but in a strongly bent region 

of the strain curve (that is, cn / c0 = 0.25/4, fig. 4), the piecewise linear approximation may lose some 

accuracy locally using the domain density of n = 8. Thus, for sharply bent strain curves, more strain data 

points (that is, increasing domain density, n) are needed to make the piecewise-linear strain curve 

approach the true smooth strain curves for improving the shape prediction accuracy (see the section 

entitled, “Domain Density”).

Deflection Data

The strain data shown in figure 4 were input to deflection equations (36)–(40) for calculations of 

deflections of the tapered cantilever tubular beams with six depth ratios: cn / c0 = 4/4, 3/4, 2/4, 1/4, 0.5/4, 

0.25/4. Tables 2a–f compare the resulting predicted deflections with the corresponding SPAR-generated 

deflections. The prediction errors produced by deflection equations (36)–(39) are indicated in parentheses 

for comparison. The percent prediction error at a strain station, xi , produced by each deflection equation 

is defined as the difference between the predicted and SPAR-generated deflections divided by the 

associated SPAR-generated beam tip deflection.

Table 2a. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; uniform cantilever tubular beam; cn / c0 = 4/4; n = 8.

Theories

y0

(error, 

percent)

y1

(error, 

percent)

y2

(error, 

percent)

y3

(error, 

percent)

y4

(error, 

percent)

y5

(error, 

percent)

y6

(error, 

percent)

y7

(error, 

percent)

y8

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01787

(0.0000)

0.06340

(0.0000)

0.13368

(0.0000)

0.22450

(0.0000)

0.33181

(0.0000)

0.45174

(0.0000)

0.57933

(0.0000)

0.71162

(0.0000)

Nonuniform,

eq. (36) -------- -------- -------- -------- -------- -------- -------- -------- --------

Slightly

nonuniform,

eq. (37)

0.00000

(0.0000)

0.01573

(0.3007)

0.06027

(0.4398)

0.12952

(0.5846)

0.21936

(0.7223)

0.32568

(0.8614)

0.44435

(1.0385)

0.57127

(1.1326)

0.70230

(1.3097)

First-order,

eq. (38)

0.00000

(0.0000)

0.01573

(0.3007)

0.06027

(0.4398)

0.12952

(0.5846)

0.21936

(0.7223)

0.32568

(0.8614)

0.44435

(1.0385)

0.57127

(1.1326)

0.70230

(1.3097)

Second-order,

eq. (39)

0.00000

(0.0000)

0.01573

(0.3007)

0.06027

(0.4398)

0.12952

(0.5846)

0.21936

(0.7223)

0.32568

(0.8614)

0.44435

(1.0385)

0.57127

(1.1326)

0.70230

(1.3097)
18

Table 2b. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 3/4; n = 8.

Theories

y0

(error, 

percent)

y1

(error, 

percent)

y2

(error, 

percent)

y3

(error, 

percent)

y4

(error, 

percent)

y5

(error, 

percent)

y6

(error, 

percent)

y7

(error, 

percent)

y8

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01823

(0.0000)

0.06697

(0.0000)

0.14572

(0.0000)

0.25204

(0.0000)

0.38294

(0.0000)

0.53428

(0.0000)

0.70092

(0.0000)

0.87634

(0.0000)

Nonuniform, 

eq. (36)

0.00000

(0.0000)

0.01623

(0.2282)

0.06407

(0.3309)

0.14180

(0.4473)

0.24707

(0.5671)

0.37676

(0.7052)

0.52686

(0.8467)

0.69210

(1.0065)

0.86568

(1.2164)

Slightly

nonuniform,

eq. (37)

0.00000

(0.0000)

0.01624

(0.2271)

0.06409

(0.3286)

0.14185

(0.4416)

0.24718

(0.5546)

0.37699

(0.6790)

0.52724

(0.8033)

0.69269

(0.9391)

0.86659

(1.1126)

First-order,

eq. (38)

0.00000

(0.0000)

0.01623

(0.2282)

0.06405

(0.3332)

0.14175

(0.4530)

0.24699

(0.5763)

0.37665

(0.7178)

0.52669

(0.8661)

0.69187

(1.0327)

0.86538

(1.2507)

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01623

(0.2282)

0.06407

(0.3309)

0.14179

(0.4485)

0.24706

(0.5683)

0.37676

(0.7052)

0.52686

(0.8467)

0.69209

(1.0076)

0.86567

(1.2176)

Table 2c. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 2/4; n = 8. 

Theories

y0

(error, 

percent)

y1

(error, 

percent)

y2

(error, 

percent)

y3

(error, 

percent)

y4

(error, 

percent)

y5

(error, 

percent)

y6

(error, 

percent)

y7

(error, 

percent)

y8

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01864

(0.0000)

0.07113

(0.0000)

0.16057

(0.0000)

0.28831

(0.0000)

0.45510

(0.0000)

0.65965

(0.0000)

0.89759

(0.0000)

1.15801

(0.0000)

Nonuniform, 

eq. (36)

0.00000

(0.0000)

0.01681

(0.1580)

0.06855

(0.2228)

0.15709

(0.3005)

0.28385

(0.3851)

0.44936

(0.4957)

0.65236

(0.6295)

0.88815

(0.8152)

1.14592

(1.0440)

Slightly 

nonuniform,

eq. (37)

0.00000

(0.0000)

0.01680

(0.1589)

0.06849

(0.2280)

0.15693

(0.3143)

0.28357

(0.4093)

0.44895

(0.5311)

0.65188

(0.6710)

0.88778

(0.8471)

1.14549

(1.0812)

First-order,

eq. (38)

0.00000

(0.0000)

0.01679

(0.1598)

0.06848

(0.2288)

0.15690

(0.3169)

0.28348

(0.4171)

0.44873

(0.5510)

0.65137

(0.7150)

0.88672

(0.9387)

1.14334

(1.2668)

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01681

(0.1580)

0.06855

(0.2228)

0.15708

(0.3014)

0.28383

(0.3869)

0.44932

(0.4991)

0.65230

(0.6347)

0.88807

(0.8221)

1.14518

(1.1079)

Table 2d. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 1/4; n = 8.

Theories

y0

(error, 

percent)

y1

(error, 

percent)

y2

(error, 

percent)

y3

(error, 

percent)

y4

(error,

percent)

y5

(error, 

percent)

y6

(error, 

percent)

y7

(error, 

percent)

y8

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01912

(0.0000)

0.07598

(0.0000)

0.17923

(0.0000)

0.33816

(0.0000)

0.56506

(0.0000)

0.87454

(0.0000)

1.28196

(0.0000)

1.78417

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01742

(0.0953)

0.07372

(0.1267)

0.17627

(0.1659)

0.33448

(0.2063)

0.56030

(0.2668)

0.86823

(0.3537)

1.27201

(0.5577)

1.76228

(1.2269)

Slightly 

nonuniform,

Eq. (37)

0.00000

(0.0000)

0.01738

(0.0975)

0.07344

(0.1424)

0.17549

(0.2096)

0.33280

(0.3004)

0.55718

(0.4417)

0.86290

(0.6524)

1.26376

(1.0201)

1.75295

(1.7498)

First-order,

eq. (38)

0.00000

(0.0000)

0.01739

(0.0970)

0.07355

(0.1362)

0.17577

(0.1939)

0.33339

(0.2674)

0.55820

(0.3845)

0.86444

(0.5661)

1.26542

(0.9270)

1.75143

(1.8350)

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01742

(0.0953)

0.07371

(0.1272)

0.17623

(0.1681)

0.33440

(0.2107)

0.56013

(0.2763)

0.86788

(0.3733)

1.27135

(0.5947)

1.76105

(1.2958)
19

Table 2e. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 0.5/4; n = 8.

Theories

y0

(error, 

percent)

y1

(error, 

percent)

y2

(error, 

percent)

y3

(error, 

percent)

y4

(error, 

percent)

y5

(error, 

percent)

y6

(error, 

percent)

y7

(error, 

percent)

y8

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01938

(0.0000)

0.07872

(0.0000)

0.19047

(0.0000)

0.37082

(0.0000)

0.64535

(0.0000)

1.05517

(0.0000)

1.67290

(0.0000)

2.59712

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01775

(0.0627)

0.07665

(0.0797)

0.18790

(0.0990)

0.36790

(0.1124)

0.64197

(0.1371)

1.05147

(0.1425)

1.66633

(0.2530)

2.54871

(1.8640)

Slightly 

nonuniform,

eq. (37)

0.00000

(0.0000)

0.01768

(0.0656)

0.07620

(0.0970)

0.18656

(0.1506)

0.36482

(0.2310)

0.63559

(0.3758)

1.03867

(0.6353)

1.64049

(1.2479)

2.50844

(3.4146)

First-order,

eq. (38)

0.00000

(0.0000)

0.01771

(0.0643)

0.07640

(0.0893)

0.18714

(0.1282)

0.36615

(0.1798)

0.63829

(0.2718)

1.04386

(0.4355)

1.64986

(0.8871)

2.51231

(3.2655)

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01775

(0.0628)

0.07663

(0.0805)

0.18783

(0.1017)

0.36774

(0.1186)

0.64159

(0.1448)

1.05057

(0.1771)

1.66394

(0.3435)

2.54197

(2.1235)

Table 2f. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 0.25/4; n = 8.

Theories

y0

(error, 

percent)

y1

(error, 

percent)

y2

(error, 

percent)

y3

(error, 

percent)

y4

(error, 

percent)

y5

(error, 

percent)

y6

(error, 

percent)

y7

(error, 

percent)

y8

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01952

(0.0000)

0.08017

(0.0000)

0.19668

(0.0000)

0.38984

(0.0000)

0.69566

(0.0000)

1.18157

(0.0000)

2.00286

(0.0000)

3.58289

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01793

(0.0444)

0.07826

(0.0533)

0.19445

(0.0622)

0.38766

(0.0608)

0.69340

(0.0631)

1.17872

(0.1103)

2.00100

(0.0519)

3.43492

(4.1299)

Slightly 

nonuniform,

eq. (37)

0.00000

(0.0000)

0.01785

(0.0466)

0.07771

(0.0687)

0.19276

(0.1094)

0.38359

(0.1744)

0.68447

(0.3123)

1.15879

(0.6358)

1.95019

(1.4700)

3.32381

(7.2310)

First-order,

eq. (38)

0.00000

(0.0000)

0.01789

(0.0455)

0.07796

(0.0617)

0.19354

(0.1216)

0.38546

(0.1222)

0.68850

(0.1998)

1.16756

(0.3910)

1.97136

(0.8792)

3.34045

(6.7666)

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01793

(0.0444)

0.07824

(0.0539)

0.19436

(0.0648)

0.38743

(0.0673)

0.69284

(0.0787)

1.17719

(0.1222)

1.99555

(0.2040)

3.40914

(4.8494)

Note from tables 2a–f that the deflections predicted from the four deflection equations (36)–(39) are 

fairly close. Note also that the SPAR program provided slightly larger deflections than the corresponding 

deflections predicted from the four deflection equations. The reason for this difference is that the SPAR 

deflections contain additional deflection components of transverse shear, which are neglected in the 

formulation of deflection equations (36)–(39) (see graphical demonstrations in reference 2, figures 17 

and 19). Also note that the prediction accuracy of deflection equations (36) and (39) are slightly better 

than the comparable accuracy of deflection equations (37) and (38).

The deflection data of tables 2a–f calculated from deflection equations (36)–(40) are plotted in 

figures 5a–d, respectively, for different tapered cantilever tubular beams, and are compared with the 

corresponding deflection curves calculated from the SPAR program. The small prediction errors shown in 

tables 2a–f are almost inconspicuous for depth ratio range, 4 / 4 ≥ cn / c0 ≥ 1/ 4 . For the cases of low 

depth ratios, cn / c0 = (0.5/4, 0.25/4), however, the predicted deflection curves near the beam tip regions 

show slight deviations from the corresponding SPAR deflection curves because of the insufficient strain 

data points used. As discussed in the section entitled, “Domain Density,” the shape prediction accuracy 

near the beam tip regions could be greatly improved by increasing the domain density (increasing the 
20

number of strain stations). Overall, the good graphical correlations between the predicted and SPARgenerated deflection curves provide confidence in the high accuracy of deflection equations (36)–(39) 

formulated for the nonuniform cantilever beam. 

Error Curves

Figure 6 shows the beam tip prediction errors (tables 2a–f) based on different deflection equations 

plotted as functions of depth ratio, cn / c0 . For the uniform beam case ( cn / c0 = 4/4), the deflections 

calculated from deflection equations (37)–(39) are identical, indicating the mathematical accuracy of 

deflection equations (37)–(39) at the limit of the uniform beam case. For the tapered beam cases, all the 

deflection equations (36)–(39) predicted reasonably close deflections (a prediction error of approximately 

1 percent) in the region of the depth ratios, 3/4

 

≥ cn / c0

 

≥2/4. As the beam depth ratio decreases from 

cn / c0 = 2/4, the prediction errors continue to increase and reach a respective maximum at the lowest 

depth ratio, cn / c0 = 0.25/4. The beam tip shape prediction errors at cn / c0 = 0.25/4 of both the 

nonuniform deflection equation (36) and second-order deflection equation (39) are comparable, and are 

nearly 60 percent of the prediction errors of both the slightly nonuniform deflection equation (37) and the 

first-order deflection equation (38). 

DOMAIN DENSITY

As shown in figure 4, as the beam depth ratio, cn / c0 , decreases, the strain curve continues to bend 

more and reaches a sharp bend near the beam tip region for the lowest depth ratio, cn / c0 = 0.25/4. To 

obtain more accurate strain curves with a piecewise linear approximation (eq. (5)), the strain data points 

must be increased. Increasing the domain density, n, implies increasing the number (1 + n) of strain 

stations. Therefore, the domain density, n, was increased from n = 8 to n = 16 to obtain additional strain 

data points for generating more accurate strain curves, thereby achieving better shape predictions. 

Improved Strain Curves

Figure 7 compares the SPAR-generated strain curves based on the domain densities of n = 8 and 

n = 16. For the depth ratios, cn / c0 = 4/4–1/4, the effect of increasing the domain density, n, is practically 

inconspicuous. Increasing the value of n, however, has a marked effect on improving the strain curves 

near the beam tip regions especially for the lower depth cases, cn / c0 = 0.5/4 and cn / c0 = 0.25/4. Again, 

the nonzero strains at the free beam tip are caused by discretization of the beam (theoretically, the free 

beam tip strain should be zero). 

Improved Deflection Data

As shown in figure 7, the beam deflections were recalculated from only deflection equations (36) 

and (39) (fig. 6) for different tapered tubular beams using the strains based on n = 16. Tables 3a–f list the 

results, with prediction errors indicated in parentheses.
21

Table 3a. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; uniform cantilever tubular beam; cn / c0 = 4/4; n = 16.

Theories

y0

(error, 

percent)

y2

(error, 

percent)

y4

(error, 

percent)

y6

(error, 

percent)

y8

(error, 

percent)

y10

(error, 

percent)

y12

(error, 

percent)

y14

(error, 

percent)

y16

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01787

(0.0000)

0.06340

(0.0000)

0.13368

(0.0000)

0.22450

(0.0000)

0.33181

(0.0000)

0.45174

(0.0000)

0.57933

(0.0000)

0.71162

(0.0000)

Nonuniform,

eq. (36) ------ ------ ------ ------ ------ ------ ------ ------ ------

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01576

(0.2965)

0.06034

(0.4300)

0.12962

(0.5705)

0.21948

(0.7055)

0.32580

(0.8446)

0.44448

(1.0202)

0.57139

(1.1158)

0.70243

(1.2914)

Table 3b. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 3/4; n = 16.

Theories

y0

(error, 

percent)

y2

(error, 

percent)

y4

(error, 

percent)

y6

(error, 

percent)

y8

(error, 

percent)

y10

(error, 

percent)

y12

(error, 

percent)

y14

(error, 

percent)

y16

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01823

(0.0000)

0.06697

(0.0000)

0.14572

(0.0000)

0.25204

(0.0000)

0.38294

(0.0000)

0.53428

(0.0000)

0.70092

(0.0000)

0.87634

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01627

(0.2237)

0.06417

(0.3195)

0.14199

(0.4256)

0.24738

(0.5318)

0.37726

(0.6482)

0.52758

(0.7645)

0.69312

(0.8901)

0.86711

(1.0532) 

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01627

(0.2237)

0.06417

(0.3195)

0.14199

(0.4256)

0.24738

(0.5318)

0.37726

(0.6482)

0.52759

(0.7634)

0.69313

(0.8889)

0.86712

(1.0521)

Table 3c. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 2/4; n = 16.

Theories

y0

(error, 

percent)

y2

(error, 

percent)

y4

(error, 

percent)

y6

(error, 

percent)

y8

(error, 

percent)

y10

(error, 

percent)

y12

(error, 

percent)

y14

(error, 

percent)

y16

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01864

(0.0000)

0.07113

(0.0000)

0.16057

(0.0000)

0.28831

(0.0000)

0.45510

(0.0000)

0.65965

(0.0000)

0.89759

(0.0000)

1.15801

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01682

(0.1572)

0.06859

(0.2193)

0.15719

(0.2919)

0.28407

(0.3661)

0.44983

(0.4551)

0.65325

(0.5527)

0.88981

(0.6718)

1.14828

(0.8402) 

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01682

(0.1572)

0.06859

(0.2193)

0.15719

(0.2919)

0.28407

(0.3661)

0.44983

(0.4551)

0.65324

(0.5535)

0.88980

(0.6727)

1.14826

(0.8420)

Table 3d. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 1/4; n = 16.

Theories

y0

(error, 

percent)

y2

(error, 

percent)

y4

(error, 

percent)

y6

(error, 

percent)

y8

(error, 

percent)

y10

(error, 

percent)

y12

(error, 

percent)

y14

(error, 

percent)

y16

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01912

(0.0000)

0.07598

(0.0000)

0.17923

(0.0000)

0.33816

(0.0000)

0.56506

(0.0000)

0.87454

(0.0000)

1.28196

(0.0000)

1.78417

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01739

(0.0970)

0.07364

(0.1312)

0.17612

(0.1743)

0.33427

(0.2180)

0.56011

(0.2774)

0.86835

(0.3469)

1.27375

(0.4602)

1.77114

(0.7303) 

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01739

(0.0970)

0.07364

(0.1312)

0.17611

(0.1749)

0.33426

(0.2186)

0.56009

(0.2786)

0.86829

(0.3503)

1.27365

(0.4658)

1.77124

(0.7247)
22

Table 3e. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 0.5/4; n = 16.

Theories

y0

(error, 

percent)

y2

(error, 

percent)

y4

(error, 

percent)

y6

(error, 

percent)

y8

(error, 

percent)

y10

(error, 

percent)

y12

(error, 

percent)

y14

(error, 

percent)

y16

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01938

(0.0000)

0.07872

(0.0000)

0.19047

(0.0000)

0.37082

(0.0000)

0.64535

(0.0000)

1.05517

(0.0000)

1.67292

(0.0000)

2.59712

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01770

(0.0647)

0.07649

(0.0859)

0.18752

(0.1136)

0.36721

(0.1390)

0.64077

(0.1763)

1.04963

(0.2133)

1.66517

(0.2984)

2.57675

(0.7843) 

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01770

(0.0647)

0.07648

(0.0862)

0.18751

(0.1140)

0.36719

(0.1398)

0.64071

(0.1787)

1.04949

(0.2187)

1.66478

(0.3134)

2.57544

(0.8348)

Table 3f. Comparison of deflections calculated from SPAR with those calculated from different 

deflection equations; tapered cantilever tubular beam; cn / c0 = 0.25/4; n = 16.

Theories

y0

(error, 

percent)

y2

(error, 

percent)

y4

(error, 

percent)

y6

(error, 

percent)

y8

(error, 

percent)

y10

(error, 

percent)

y12

(error, 

percent)

y14

(error, 

percent)

y16

(error, 

percent)

SPAR

(reference)

0.00000

(0.0000)

0.01952

(0.0000)

0.08017

(0.0000)

0.19668

(0.0000)

0.38984

(0.0000)

0.69566

(0.0000)

1.18157

(0.0000)

2.00286

(0.0000)

3.58289

(0.0000)

Nonuniform,

eq. (36)

0.00000

(0.0000)

0.01786

(0.0463)

0.07802

(0.0600)

0.19387

(0.0784)

0.38651

(0.0929)

0.69143

(0.1181)

1.17563

(0.1658)

1.99425

(0.2403)

3.53790

(1.2557) 

Secondorder,

eq. (39)

0.00000

(0.0000)

0.01786

(0.0463)

0.07801

(0.0603)

0.19385

(0.0790)

0.38648

(0.0938)

0.69135

(0.1203)

1.17539

(0.1725)

1.99331

(0.2665)

3.53116

(1.4438)

The deflection data shown in tables 3a–f are plotted in figure 8. Notice from figures 8a and 8b that the 

beam tip prediction errors using n = 16 did indeed decrease from those using n = 8 (tables 2a–f)), 

especially for the lower depth ratio cases, cn / c0 = 0.25/4, 0.5/4. 

Error Comparisons

Tables 4a–f compare the beam tip prediction errors based on n = 8 and n = 16.

Table 4a. Comparison of beam tip prediction errors based on different deflection equations using strain 

station densities (n = 8, 16); tapered cantilever tubular beam; cn / c0 = 4/4.

n = 8 n = 16

Theories yn , in. Error, percent yn , in. Error, percent Error, percent of 

n = 8

SPAR 0.71162 0.00 0.71162 0.00 0

Nonuniform,

eq. (36)

--- --- --- --- ---

Second-order,

eq. (39)

0.70230 1.31 0.70243 1.29 98
23

Table 4b. Comparison of beam tip prediction errors based on different deflection equations using strain 

station densities (n = 8, 16); tapered cantilever tubular beam; cn / c0 = 3/4.

n = 8 n = 16

Theories yn , in. Error, percent yn , in. Error, percent Error, percent of 

n = 8

SPAR 0.87634 0.00 0.87634 0.00 0

Nonuniform, 

eq. (36)

0.86568 1.22 0.86711 1.05 86

Second-order,

eq. (39)

0.86567 1.22 0.86712 1.05 86

Table 4c. Comparison of beam tip prediction errors based on different deflection equations using strain 

station densities (n = 8, 16); tapered cantilever tubular beam; cn / c0 = 2/4.

n = 8 n = 16

Theories yn , in. Error, 

percent

yn , in. Error, percent Error, percent of 

n = 8

SPAR 1.15801 0.00 1.15801 0.00 0

Nonuniform,

eq. (36)

1.14592 1.04 1.14828 0.84 81

Second-order,

eq. (39)

1.14518 1.1 1.14827 0.84 76

Table 4d. Comparison of beam tip prediction errors based on different deflection equations using strain 

station densities (n = 8, 16); tapered cantilever tubular beam; cn / c0 = 1/4.

n = 8 n = 16

Theories yn , in. Error, 

percent

yn , in. Error, percent Error, percent of 

n = 8

SPAR 1.78417 0.00 1.78417 0.00 0

Nonuniform,

eq. (36)

1.76228 1.23 1.77144 0.71 58

Second-order,

eq. (39)

1.76105 1.30 1.77124 0.72 55

Table 4e. Comparison of beam tip prediction errors based on different deflection equations using strain 

station densities (n = 8, 16); tapered cantilever tubular beam; cn / c0 = 0.5/4.

n = 8 n = 16

Theories yn , in. Error, 

percent

yn , in. Error, percent Error, percent of 

n = 8

SPAR 2.59712 0.00 2.59712 0.00 0

Nonuniform,

eq. (36)

2.54871 1.86 2.57675 0.78 42

Second-order,

eq. (39)

2.54197 2.12 2.57544 0.83 39
24

Table 4f. Comparison of beam tip prediction errors based on different deflection equations using strain 

station densities (n = 8, 16); tapered cantilever tubular beam; cn / c0 = 0.25/4.

n = 8 n = 16

Theories yn , in. Error, 

percent

yn , in. Error, percent Error, percent of 

n = 8

SPAR 3.58289 0.00 3.58289 0.00 0

Nonuniform,

eq. (36)

3.43492 4.13 3.53790 1.26 31

Second-order,

eq. (39)

3.40914 4.85 3.52116 1.44 30

Figure 9 plots the beam tip deflection prediction errors based on n = 8 and n = 16 (tables 4a–f) for 

comparison. Note that the effect of increasing the domain density, n, on the reduction of prediction errors 

becomes more conspicuous as the depth ratio, cn / c0 , decreases. At the lowest depth ratio, 

cn / c0 = 0.25/4, the beam tip prediction errors based on n = 16 were reduced to almost 30 percent of those 

based on n = 8. 

DISCUSSION

For the present shape predictions of the tapered cantilever tubular beam, all four displacement transfer 

functions (deflection equations (36)–(39)) were found to provide quite accurate shape predictions. The 

slightly nonuniform deflection equation (37) and the first-order deflection equation (38) have comparable 

prediction accuracy and are recommended for shape predictions of slightly nonuniform beam cases. For 

shape predictions of nonuniform beam structures, both the exact deflection equation (36) and secondorder deflection equation (39) can be used, because both deflection equations can provide comparable 

shape prediction accuracy.

Note that when the fiber-optic strain-sensing system is used, off-the-shelf sensor density is high, 

thereby improving shape prediction accuracy. This high sensor density can be decimated by an integer 

factor to reduce computational requirements while sacrificing prediction accuracy. Thus, for improving 

shape prediction accuracy, the previously specified number (n + 1) of strain data extraction points can be 

increased as desired, up to a level corresponding to the off-the-shelf sensor density, by means of a simple 

command. By increasing the domain density, n, the assumed piecewise-linear strain curves will approach 

the associated true smooth strain curves (especially for the highly tapered beam tip region where the 

bending strains reach a maximum and then drop rapidly toward the beam tip), and thus greatly improve 

the shape prediction accuracy. 

SUMMARY

The first- and the second-order displacement transfer functions were formulated for the nonuniform 

cantilever beam through binomial series expansions of the beam depth function. The principal results are 

as follows:

1. Through binomial series expansions of the axially varying beam depth function, the mathematical 

indeterminacy problem at the limit case of a uniform beam was successfully eliminated.

2. The deflections calculated from all four displacement transfer functions were reasonably close and 

were slightly lower than the corresponding deflections generated by the Structural Performance 
25

And Resizing (SPAR) program. The SPAR-generated deflections contained additional 

components induced by the transverse shear effect, which was ignored in the analysis.

3. For the uniform beam case ( cn / c0 = 4/4), the deflections predicted from different displacement 

transfer functions were identical, confirming the mathematical accuracy of all the displacement 

transfer functions formulated.

4. The shape prediction accuracy of the nonuniform and second-order displacement transfer functions 

were slightly better than the comparable accuracy of both the slightly nonuniform and first-order 

displacement transfer functions.

5. For the tapered beam shape predictions, using the second-order displacement transfer function 

instead of the first-order deflection equation did not change the beam tip prediction error for the 

uniform beam (zero taper angle). For the tapered beam, however, the prediction error continued to 

decrease with increasing taper angle, reaching a 28-percent error reduction at the peak taper angle.

6. For the tapered beam shape predictions, using the nonuniform displacement transfer function 

instead of the slightly nonuniform deflection equation decreased the beam tip prediction errors

with increasing taper angle, reaching a 43-percent error reduction at the peak taper angle.

7. For the tapered beam, the shape prediction accuracy could be improved by increasing the domain 

density. By doubling the domain density, one could reduce the average beam tip prediction errors

by 2 percent at a zero taper angle (uniform beam). The error reduction continued to improve with 

increasing taper angle, reaching a 69-percent error reduction at the peak taper angle.
26

FIGURES

Figure 1. Nonuniform cantilever beam instrumented with equally spaced surface bending strain sensors.

Figure 2. Finite-element model for tapered cantilever tubular beam subjected to a tip vertical load of 

P = 100 lb. 
27

Figure 3. Deformed shapes, generated from the Structural Performance And Resizing (SPAR) program, 

of different tapered cantilever tubular beams; P = 100 lb.
28

Figure 3. Concluded.

Figure 4. Surface bending strains calculated from the Structural Performance And Resizing (SPAR) 

element stress outputs for different tapered cantilever tubular beams; n = 8; P = 100 lb.
29

(a) Deflections calculated from deflection equation (36), for nonuniform beams, compared with those 

calculated from SPAR (deflection equation (40) was used for the uniform beam).

(b) Deflections calculated from deflection equation (37), for slightly nonuniform beams, compared with 

those calculated from SPAR (deflection equation (40) was used for the uniform beam).

Figure 5. Comparison of deflections calculated from deflection equations with those calculated from the 

Structural Performance And Resizing (SPAR) program for different tapered cantilever tubular beams;

n = 8; P = 100 lb.
30

(c) Deflections calculated from first-order deflection equation (38) compared with those calculated from 

SPAR (deflection equation (40) was used for the uniform beam).

(d) Deflections calculated from second-order deflection equation (39) compared with those calculated 

from SPAR (deflection equation (40) was used for the uniform beam).

Figure 5. Concluded.
31

Figure 6. Beam tip deflection prediction errors associated with different displacement transfer functions 

plotted as functions of depth ratio, cn / c0 .

Figure 7. Comparison of bending strain curves for different tapered cantilever tubular beams based on 

n = 8 and n = 16; P = 100 lb.
32

(a) Deflections calculated from deflection equation (36), for nonuniform beams, compared with those 

calculated from SPAR (deflection equation (40) was used for the uniform beam).

(b) Deflections calculated from second-order deflection equation (39) compared with those calculated 

from SPAR.

Figure 8. Comparison of deflections calculated from deflection equations with those calculated from the 

Structural Performance And Resizing (SPAR) program for different tapered cantilever tubular beams; 

n = 16; P = 100 lb.
33

Figure 9. Comparison of beam tip deflection prediction error curves associated with different 

displacement functions based on n = 8 and n = 16.
34

APPENDIX A

DERIVATIONS OF THE FIRST-ORDER SLOPE

AND DEFLECTION EQUATIONS

Appendix A presents the detailed mathematical derivations of the first-order slope equations and 

first-order deflection equations for the nonuniform cantilever beam.

Basic Slope Equations

The slope, tanθ(x)(≡ dy / dx), of the nonuniform beam at the axial location, x, within the domain, 

xi−1 ≤ x ≤ xi (fig. 1), can be obtained by integrating the beam differential equation (3) once, and 

enforcing the continuity of the slope at the inboard strain station, xi−1, as

tanθ(x) = d2

y

dx2 dx

xi−1

x

∫

Integration of eq. (3) 



+ tanθi−1

Slope at xi−1

 = ε(x)

c(x)

dx

xi−1

x

∫

Slope increment

above tanθi−1



+ tanθi−1

Slope at xi−1

 ; (xi−1 ≤ x ≤ xi)

(A1)

in which tanθi−1 is the slope at the inboard strain station, xi−1. 

In light of the following linear assumptions of both 

 

c(x) and 

 

ε(x) within the domain, xi−1 ≤ x ≤ xi ,

c(x) = ci−1 − (ci−1 − ci)

x − xi−1

Δl ; (xi−1 ≤ x ≤ xi) (A2)

ε(x) = εi−1 − (εi−1 − εi)

x − xi−1

Δl ; (xi−1 ≤ x ≤ xi

) (A3)

equation (A1) becomes

tanθ(x) = εi−1

ci−1

1− 1− εi

εi−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

1− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

dx

xi−1

x

∫ + tanθi−1 ; (xi−1 ≤ x ≤ xi

) (A4)
35

First-Order Slope Equations

Because the coefficient, 1− (ci / ci−1) , of the slope term in the denominator of equation (A4) is 

considered small, one can expand the factor, 1− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

−1

, in binomial series up to the 

first-order term as

tanθ(x) = ε i−1

ci−1

1− 1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

+ .... ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

dx

xi−1

x

∫ + tanθi−1

= ε i−1

ci−1

1+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ − 1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

x − xi−1

Δl − 1− ε iε i−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

(Δl)

2

+ ....

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

dx xi−1

x

∫ + tanθi−1 (A5)

After the integration is carried out, equation (A5) becomes

tanθ(x) = εi−1

ci−1

(x − xi−1) +

εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

2Δl − 1− εiεi−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

3(Δl)

2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ tanθi−1

(xi−1 ≤ x ≤ xi) (A6)

which is the first-order slope equation for the domain, xi−1 ≤ x ≤ xi .

At the strain station, x = xi , we have (xi − xi−1) = Δl , and equation (A6) becomes

tanθi ≡ tanθ(xi) = Δlεi−1

ci−1

1+

1

2

εi

εi−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ − 1

3

1− εi

εi−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢ ⎤

⎦

⎥ + tanθi−1

= Δl

6ci−1

6 − 3 ci

ci−1

− 2 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢ ⎤

⎦

⎥εi−1 + 3+ 2 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢ ⎤

⎦

⎥εi

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ tanθi−1

(i = 1, 2, 3, ..., n) (A7)

After the terms are grouped, equation (A7) becomes

tanθi = Δl

6ci−1

4 − ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ ε i−1 + 5− 2

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ ε i

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ tanθi−1 ; (i = 1, 2, 3, ..., n) (A8)
36

Equation (A8) is equation (17) in the main section of this report, and the first-order slope equation is 

written in descending recursion form. Applying the descending recursion relationships, one can write 

equation (A8) in summation form as

tanθi = Δl

6

1

c j−1

4 − c j

c j−1

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

ε j−1 + 5− 2

c j

c j−1

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

ε j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ j=1

i

∑ + tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (A9)

Equation (A9) is equation (18) in the main section of this report, and the first-order slope equation is 

written in series summation form. 

For the uniform beam ( ci = ci−1 = c ), equation (A9) degenerates into the form

tanθi = Δl

2c

(ε j−1 + ε j)

j=1

i

∑ + tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (A10)

which is identical to equation (25) of reference 2.

First-Order Deflection Equations

The deflection, y(x) , at the axial location, x, within the domain, xi−1 ≤ x ≤ xi (fig. 1), can be 

obtained by integrating equation (A6) and enforcing the continuity of the deflection at the inboard strain 

station, xi−1. Namely,

y(x) = tanθ(x)

Eq. (A6)

xi−1 

x

∫ dx + yi−1

 = ε i−1

ci−1

(x − xi−1) +

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

2Δl − 1− ε iε i−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

3(Δl)

2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

dx

xi−1

x

∫

 + tanθi−1 dx

xi−1

x

∫ + yi−1

(xi−1 ≤ x ≤ xi) (A11)
37

in which yi−1 is the deflection at the inboard strain station, xi−1. After the integration is carried out, 

equation (A11) becomes

y(x) = ε i−1

12ci−1

6(x − xi−1)

2 + 2

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

Δl − 1− ε iε i−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

4

(Δl)

2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 + (x − xi−1)tanθi−1 + yi−1

(xi−1 ≤ x ≤ xi) (A12)

Equation (A12) is the first-order deflection equation for the domain, xi−1 ≤ x ≤ xi .

At the strain station, x = xi , we have (xi − xi−1) = Δl , and equation (A12) takes on the form

yi ≡ y(xi) = (Δl)

2ε i−1

12ci−1

6 + 2

ε i

ε i−1

− 2

ci

ci−1

− 1− ε i

ε i−1

− ci

ci−1

+

ci

ci−1

ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ yi−1 + Δl tanθi−1

(i = 1, 2, 3, ..., n) (A13)

After the terms are grouped, equation (A13) becomes

yi = (Δl)

2

12ci−1

5− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi−1 + 3− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ εi

⎡

⎣

⎢ ⎤

⎦

⎥ + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3, ..., n) (A14)

Equation (A14) is equation (21) in the main section of this report, and the first-order deflection equation 

is written in descending recursion form.

For the uniform beam ( ci = ci−1 = c ), equation (A14) degenerates into

yi = (Δl)

2

6c

2εi−1 + ε ( i) + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3, ..., n) (A15)

which is identical to equation (26) of reference 2.

Final Form of Equation (A14)

Substituting the slope equation (A8) into deflection equation (A14), and applying the descending 

indicial recursion relationships, one can write the resulting deflection equation for each index, 

i(= 1, 2, 3, ..., n), as follows:
38

For i = 1,

y

1 = (Δl)

2

12

1

c0

5− c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 3− c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + Δl tanθ0 (A16)

For i = 2,

y2 = (Δl)

2

12

1

c

1

5− c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 3− c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

12

1

c0

5− c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 3− c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + Δl tanθ0

y

1



+ (Δl)

2

6

1

c0

4 − c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 5− 2

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ Δl tanθ0

Δl tanθ1

 (A17)

After the terms are grouped, equation (A17) becomes

y2 = (Δl)

2

12

1

c

1

5− c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 3− c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+

1

c0

13− 3

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 13− 5

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+ y0 + 2Δl tanθ0 (A18)

For i = 3,

y3 = (Δl)

2

12

1

c2

5− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + 3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 3

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 + (Δl)

2

12

1

c

1

5− c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 3− c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

12

1

c0

13− 3

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 13− 5

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 2Δl tanθ0

y2



+ (Δl)

2

6

1

c

1

4 − c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 5− 2

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

6

1

c0

4 − c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 5− 2

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ Δl tanθ0

Δl tanθ2

 (A19)

After the terms are grouped, equation (A19) becomes

y3 = (Δl)

2

12

1

c2

5− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + 3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 3

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

12

1

c

1

13− 3

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 13− 5

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 + (Δl)

2

12

1

c0

21− 5

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 23− 9

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 3Δl tanθ0

(A20)
39

For i = 4,

y4 = (Δl)

2

12

1

c3

5− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 4

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+

(Δl)

2

12

1

c2

5− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + 3− c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 3

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

12

1

c

1

13− 3

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 13− 5

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 + (Δl)

2

12

1

c0

21− 5

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 23− 9

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 3Δl tanθ0

⎧

⎨

⎪

⎪⎪

⎩

⎪

⎪

⎪

⎫

⎬

⎪

⎪⎪

⎭

⎪

⎪

⎪

y3



+

(Δl)

2

6

1

c2

4 − c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + 5− 2

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 3

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

6

1

c

1

4 − c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 5− 2

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

6

1

c0

4 − c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 5− 2

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ Δl tanθ0

⎧

⎨

⎪

⎪⎪

⎩

⎪

⎪

⎪

⎫

⎬

⎪

⎪⎪

⎭

⎪

⎪

⎪

Δl tanθ3



(A21)

After the terms are grouped, equation (A21) becomes

y4 = (Δl)

2

12

1

c3

5− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 3 + 3− c4

c3

⎛

⎝

⎜

⎞

⎠

⎟ ε 4

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

12

1

c2

13− 3

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 2 + 13− 5

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟ ε 3

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

12

1

c

1

21− 5

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε1 + 23− 9

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟ ε 2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ (Δl)

2

12

1

c0

29 − 7

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε 0 + 33−13

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟ ε1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + 4Δl tanθ0 (A22)

Observing the indicial behavior in equations (A16), (A18), (A20), and (A22), one can write the 

generalized first-order deflection equation for any index, i(= 1, 2, 3,..., n) , in final summation form as

yi = (Δl)

2

12

1

ci− j

5+ 8( j −1) − 1+ 2( j −1)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε i− j

⎧

⎨

⎪

⎩ j=1 ⎪

i

∑

 + 3+10( j −1) − 1+ 4( j −1)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε i− j+1

⎫

⎬

⎪

⎭

⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (A23)

which is equation (23) in the main section of this report (called the first-order displacement transfer 

function). 
40

For the uniform beam ( ci = ci−1 = c ), equation (A23) degenerates into

yi = (Δl)

2

12c

⎡5+ 8( j −1) − 1+ 2( j −1) ⎣ ⎤

⎦ε { i− j

j=1

i

∑

 + ⎡3+10( j −1) − 1+ 4( j −1) ⎣ ⎤

⎦ε i− j+1} + y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (A24)

After the terms are grouped, the deflection equation (A24) for the uniform beam takes on the form

yi = (Δl)

2

6c

(3 j −1)ε i− j + (3 j − 2)ε i− j+1 ⎡

⎣ ⎤⎦

j=1

i

∑ + y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (A25)

which agrees with equation (27) of reference 2.

One can write out equation (A25) for typical indices, i (= 1, 2, 3, 4, ..., n), as follows:

For i = 1,

y

1 = (Δl)

2

6c

(2ε 0 + ε1) + y0 + Δl tanθ0 (A26)

For i = 2,

y2 = (Δl)

2

6c

(2ε1 + ε 2 + 5ε 0 + 4ε1) + y0 + 2Δl tanθ0

 = (Δl)

2

6c

(5ε 0 + 6ε1 + ε 2) + y0 + 2Δl tanθ0

(A27)

For i = 3,

y3 = (Δl)

2

6c

(2ε 2 + ε 3 + 5ε1 + 4ε 2 + 8ε 0 + 7ε1) + y0 + 3Δl tanθ0

 = (Δl)

2

6c

8ε 0 + 6(2ε1 + ε 2) + ε 3 ⎡

⎣ ⎤⎦ + y0 + 3Δl tanθ0

(A28)
41

For i = 4,

y4 = (Δl)

2

6c

(2ε 3 + ε 4 + 5ε 2 + 4ε 3 + 8ε1 + 7ε 2 +11ε 0 +10ε1) + y0 + 4Δl tanθ0

 = (Δl)

2

6c

11ε 0 + 6(3ε1 + 2ε 2 + ε 3) + ε 4 ⎡

⎣ ⎤⎦ + y0 + 4Δl tanθ0

(A29)

Observing the functional behavior of equations (A26)–(A29) with an increasing index, i, one can write 

the generalized deflection equation for any index, i(= 1, 2, 3, ..., n) , for the uniform cantilever beam as

yi = (Δl)

2

6c

(3i −1)ε 0 + 6 (i − j)ε j

j=1

i−1

∑ + ε i

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + (i)Δl tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (A30)

which agrees with equation (28) of reference 2.
42

APPENDIX B

DERIVATIONS OF THE SECOND-ORDER SLOPE

AND DEFLECTION EQUATIONS

Appendix B presents the detailed mathematical derivations of the second-order slope equations and 

second-order deflection equations for the nonuniform cantilever beam.

Second-Order Slope Equations

Expanding the factor, 1− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

−1

, in the integrand of equation (A4) in binomial 

series form up to the second-order term results in

tanθ(x) = ε i−1

ci−1

1− 1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2 (x − xi−1)

2

(Δl)

2 ....

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

dx xi−1

x

∫ + tanθi−1

= ε i−1

ci−1

1+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2 (x − xi−1)

2

(Δl)

2

⎡

⎣

⎢

⎢

⎢ xi−1

x

∫

− 1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl − 1− cici−1

⎛

⎝

⎜

⎞

⎠

⎟ 1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

(Δl)

2 − 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

(Δl)

3 ....

⎤

⎦

⎥

⎥

⎥

dx + tanθi−1

= ε i−1

ci−1

1+

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

x − xi−1

Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

(Δl)

2

⎡

⎣

⎢

x ⎢ i−1

x

∫

− 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

(Δl)

3 ....

⎤

⎦

⎥

⎥

⎥

dx + tanθi−1

(xi−1 ≤ x ≤ xi) (B1)

Integration of equation (B1) results in

tanθ(x) = ε i−1

ci−1

(x − xi−1) +

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

2Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

3(Δl)

2

⎡

⎣

⎢

⎢

 − 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

4

4(Δl)

3

⎤

⎦

⎥

⎥

⎥

+ tanθi−1

(xi−1 ≤ x ≤ xi) (B2)

 

which is the general second-order slope equation for the domain, xi−1 ≤ x ≤ xi .
43

At the strain station, x = xi , we have (xi − xi−1) = Δl , and equation (B2) becomes

tanθi = tanθ(xi) = Δl

ε i−1

ci−1

1+

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ 1

2

+

1

3

1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎧

⎨

⎪

⎩⎪

− 1

4

1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

⎫

⎬

⎪

⎭

⎪

+ tanθi−1

= Δl

ε i−1

ci−1

1+

1

6

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ 5− 2

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎧

⎨

⎪

⎩⎪

 − 1

4

1− ε i

ε i−1

− 2

ci

ci−1

+ 2

ci

ci−1

ε i

ε i−1

+

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

− ε i

ε i−1

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎫

⎬

⎪

⎭

⎪

+ tanθi−1

= Δl

12

ε i−1

ci−1

12 +10

ε i

ε i−1

−10

ci

ci−1

− 4

ci

ci−1

ε i

ε i−1

+ 4

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎧

⎨

⎪

⎩⎪

2

 −3+ 3

ε i

ε i−1

+ 6

ci

ci−1

− 6

ci

ci−1

ε i

ε i−1

− 3

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

+ 3

ε i

ε i−1

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2 ⎫

⎬

⎪

⎭

⎪

+ tanθi−1

(i = 1, 2, 3, ..., n) (B3)

After the terms are grouped, equation (B3) becomes

tanθi = Δl

12ci−1

9 − 4

ci

ci−1

+

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i−1 + 13−10

ci

ci−1

+ 3

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ tanθi−1

(i = 1, 2, 3, ..., n) (B4)

which is equation (28) in the main section of this report, the second-order slope equation in descending 

recursion form. Applying the descending recursion relationships, one can write equation (B4) in 

summation form as

tanθi = Δl

12

1

c j−1

9 − 4

c j

c j−1

+

c j

c j−1

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε j−1 + 13−10

c j

c j−1

+ 3

c j

c j−1

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε j

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭ j=1 ⎪

i

∑ + tanθ0

=0 for canti- lever beams



(i = 1, 2, 3, ..., n) (B5)

which is the second-order slope equation in series summation form.
44

For the uniform beam, ci = ci−1 = c , equation (B5) degenerates into

tanθi = Δl

2c

(ε j−1 + ε j)

j=1

i

∑ + tanθ0

=0 for cantilever beams

 ; (i = 1, 2, 3, ..., n) (B6)

which is identical to equation (25) of reference 2.

Second-Order Deflection Equations

The second-order deflection equation is obtained by carrying out the integration of the second-order 

slope equation (B2) as

y(x) = tanθ(x)

Eq. (B2)

xi−1 

x

∫ dx + yi−1

 = ε i−1

ci−1

(x − xi−1) +

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

2

2Δl

+ 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

3(Δl)

2

⎡

⎣

⎢

x ⎢ i−1

x

∫

 − 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

4

4(Δl)

3

⎤

⎦

⎥

⎥

⎥

dx + yi−1 + tanθi xi−1

x

∫ dx

(xi−1 ≤ x ≤ xi) (B7)

After integration of equation (B7), one obtains

y(x) = ε i−1

2ci−1

(x − xi−1)

2 +

1

3

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

3

Δl

+

1

6

1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

4

(Δl)

2

⎡

⎣

⎢

⎢

 − 1

10

1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

(x − xi−1)

5

(Δl)

3

⎤

⎦

⎥

⎥

⎥

+ yi−1 + (x − xi−1)tanθi−1

(xi−1 ≤ x ≤ xi) (B8)

which is the general second-order deflection equation for the domain, xi−1 ≤ x ≤ xi .
45

At the strain station, x = xi , we have (xi − xi−1) = Δl , and (B8) becomes

yi ≡ y(xi) = (Δl)

2

60

ε i−1

ci−1

30 +10

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟ + 5 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

ε i

ε i−1

− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

 − 3 1− ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

1− ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

⎤

⎦

⎥

⎥

⎥

+ yi−1 + Δl tanθi−1

= (Δl)

2

60

ε i−1

ci−1

30 +10

ε i

ε i−1

−10

ci

ci−1

+ 5

ε i

ε i−1

− 5

ci

ci−1

− 5

ci

ci−1

ε i

ε i−1

+ 5

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎧ 2

⎨

⎪

⎩

⎪

 − 3+ 6

ci

ci−1

− 3

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

2

+ 3− 6

ci

ci−1

+ 3

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i

ε i−1

⎛

⎝

⎜

⎞

⎠

⎟

⎫

⎬

⎪

⎭

⎪

+ yi−1 + Δl tanθi−1

(i = 1, 2, 3, ..., n) (B9)

After the terms are grouped, equation (B9) becomes

yi = (Δl)

2

60ci−1

27 − 9

ci

ci−1

+ 2

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i−1 + 18 −11

ci

ci−1

+ 3

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ yi−1 + Δl tanθi−1

(i = 1, 2, 3, ..., n) (B10)

which is equation (32) in the main section of this report, the second-order deflection equation in 

descending recursion form.

For the uniform beam, ci = ci−1 = c , equation (B10) degenerates into

yi = (Δl)

2

6c

2ε i−1 + ε i ( ) + yi−1 + Δl tanθi−1 ; (i = 1, 2, 3, ..., n) (B11)

which agrees with equation (26) of reference 2.

Final Form of Equation (B10)

Substituting the slope equation (B5) into the deflection equation (B10), and applying the descending 

indicial recursion relationships, one can write out the deflection equation (B10) explicitly for each index,

i(= 1, 2, 3, ..., n), as follows:
46

For i = 1,

y

1 = (Δl)

2

60c0

27 − 9

ci

c0

+ 2

ci

ci−1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 18 −11

c

1

c0

+ 3

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + Δl tanθ0 (B12)

For i = 2,

y2 = (Δl)

2

60c

1

27 − 9

c2

c

1

+ 2

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 18 −11

c2

c

1

+ 3

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

(Δl)

2

60c0

27 − 9

c

1

c0

+ 2

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 18 −11

c

1

c0

+ 3

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + Δl tanθ0

y

1



(Δl)

2

12c0

9 − 4

c

1

c0

+

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 13−10

c

1

c0

+ 3

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ Δl tanθ0

Δl tanθ1



(B13)

After the terms are grouped, equation (B13) can be written as follows:

y2 = (Δl)

2

60c

1

27 − 9

c2

c

1

+ 2

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 18 −11

c2

c

1

+ 3

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c0

72 − 29

c

1

c0

+ 7

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 83− 61

c

1

c0

+18

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + 2Δl tanθ0

(B14)
47

For i = 3,

y3 = (Δl)

2

60c2

27 − 9

c3

c2

+ 2

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2 + 18 −11

c3

c2

+ 3

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 3

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 +

(Δl)

2

60c

1

27 − 9

c2

c

1

+ 2

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 18 −11

c2

c

1

+ 3

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c0

72 − 29

c

1

c0

+ 7

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 83− 61

c

1

c0

+18

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + 2Δl tanθ0

⎧

⎨

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎫

⎬

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

y2



+

(Δl)

2

12c

1

9 − 4

c2

c

1

+

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 13−10

c2

c

1

+ 3

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

12c0

9 − 4

c

1

c0

+

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 13−10

c

1

c0

+ 3

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ Δl tanθ0

⎧

⎨

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎫

⎬

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

Δl tanθ2



(B15)

After the terms are grouped, equation (B15) can be written as follows:

y3 = (Δl)

2

60c2

27 − 9

c3

c2

+ 2

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2 + 18 −11

c3

c2

+ 3

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 3

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c

1

72 − 29

c2

c

1

+ 7

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 83− 61

c2

c

1

+18

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c0

117 − 49

c

1

c0

+12

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 148 −111

c

1

c0

+ 33

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + 3Δl tanθ0

(B16)
48

For i = 4,

y4 = (Δl)

2

60c3

27 − 9

c4

c3

+ 2

c4

c3

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 3 + 18 −11

c4

c3

+ 3

c4

c3

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 4

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 +

(Δl)

2

60c2

27 − 9

c3

c2

+ 2

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2 + 18 −11

c3

c2

+ 3

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 3

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c

1

72 − 29

c2

c

1

+ 7

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 83− 61

c2

c

1

+18

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ (Δl)

2

60c0

117 − 49

c

1

c0

+12

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 148 −111

c

1

c0

+ 33

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + 3Δl tanθ0

⎧

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎪

⎪

⎫

⎬

⎪

⎪

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

⎪

⎪

y3



+

(Δl)

2

12c2

9 − 4

c3

c2

+

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2 + 13−10

c3

c2

+ 3

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 3

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

12c

1

9 − 4

c2

c

1

+

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 13−10

c2

c

1

+ 3

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

12c0

9 − 4

c

1

c0

+

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 13−10

c

1

c0

+ 3

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ Δl tanθ0

⎧

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎪

⎪

⎫

⎬

⎪

⎪

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

⎪

⎪

Δl tanθ3



(B17)
49

After the terms are grouped, equation (B17) becomes

y4 = (Δl)

2

60c3

27 − 9

c4

c3

+ 2

c4

c3

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 3 + 18 −11

c4

c3

+ 3

c4

c3

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 4

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c2

72 − 29

c3

c2

+ 7

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2 + 83− 61

c3

c2

+18

c3

c2

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 3

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c

1

117 − 49

c2

c

1

+12

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1 + 148 −111

c2

c

1

+ 33

c2

c

1

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 + (Δl)

2

60c0

162 − 69

c

1

c0

+17

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε 0 + 213−161

c

1

c0

+ 48

c

1

c0

⎛

⎝

⎜

⎞

⎠

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + 4Δl tanθ0

(B18)

Observing the indicial progressions in equations (B12), (B14), (B16), and (B18), one can write the 

generalized second-order deflection equation for any index, i(= 1, 2, 3, ..., n) , in series summation form 

as

yi = (Δl)

2

60

1

ci− j

27 + 45( j −1) − 9 + 20( j −1)

ci− j+1

ci− j

+ 2 + 5( j −1)

ci− j+1

ci− j

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i− j

⎧

⎨

⎪

⎩ j=1 ⎪

i

∑

 + 18 + 65( j −1) − 11+ 50( j −1)

ci− j+1

ci− j

+ 3+15( j −1)

ci− j+1

ci− j

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎡ 2

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ε i− j+1

⎫

⎬

⎪

⎭

⎪

+ y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (B19)

which is equation (34) in the main section of this report, called the second-order displacement transfer 

function.
50

For the uniform beam, ci = ci−1 = c , equation (B19) degenerates into

yi = (Δl)

2

60c

⎡27 + 45( j −1) − 9 + 20( j −1) + 2 + 5( j −1) ⎣ ⎤

⎦ε { i− j

j=1

i

∑

 + ⎡18 + 65( j −1) − 11+ 50( j −1) + 3+15( j −1) ⎣ ⎤

⎦ε i− j+1} + y0 + (i)Δl tanθ0

= (Δl)

2

60c

⎡20 + 30( j −1) ⎣ ⎤

⎦ε { i− j

j=1

i

∑ + ⎡10 + 30( j −1) ⎣ ⎤

⎦ε i− j+1} + y0 + (i)Δl tanθ0

(i = 1, 2, 3, ..., n) (B20)

which can be written in a compact form as

yi = (Δl)

2

6c

(3 j −1)ε i− j + (3 j − 2)ε i− j+1 ⎡

⎣ ⎤⎦

j=1

i

∑ + y0 + (i)Δl tanθ0

=0 for cantilever beams



(i = 1, 2, 3, ..., n) (B21)

which is identical to equation (A25) of Appendix A.
51

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