NASA/TP—2011–214657

Extension of Ko Straight-Beam Displacement 

Theory to Deformed Shape Predictions of 

Slender Curved Structures

William L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

April 2011
NASA STI Program ... in Profile

Since its founding, NASA has been dedicated 

to the advancement of aeronautics and space 

science. The NASA scientific and technical 

information (STI) program plays a key part in 

helping NASA maintain this important role.

The NASA STI program operates under the 

auspices of the Agency Chief Information 

Officer. It collects, organizes, provides for 

archiving, and disseminates NASA’s STI. The 

NASA STI program provides access to the 

NASA Aeronautics and Space Database and its 

public interface, the NASA Technical Report 

Server, thus providing one of the largest 

collections of aeronautical and space science 

STI in the world. Results are published in both 

non-NASA channels and by NASA in the 

NASA STI Report Series, which includes the 

following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant 

phase of research that present the results of 

NASA Programs and include extensive data 

or theoretical analysis. Includes compilations of significant scientific and technical 

data and information deemed to be of 

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

papers but has less stringent limitations on 

manuscript length and extent of graphic 

presentations.

• TECHNICAL MEMORANDUM. 

Scientific and technical findings that are 

preliminary or of specialized interest, 

e.g., quick release reports, working 

papers, and bibliographies that contain 

minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. 

Collected papers from scientific and 

technical conferences, symposia, 

seminars, or other meetings sponsored 

or co-sponsored by NASA.

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, 

often concerned with subjects having 

substantial public interest.

• TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission.

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and feeds, 

providing help desk and personal search 

support, and enabling data exchange services.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page 

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

• E-mail your question via the Internet to 

help@sti.nasa.gov

• Fax your question to the NASA STI Help 

Desk at 443-757-5803

• Phone the NASA STI Help Desk at 

443-757-5802

• Write to:

NASA STI Help Desk

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

This page is required and contains approved text that cannot be changed.
NASA/TP—2011–214657

Extension of Ko Straight-Beam Displacement 

Theory to Deformed Shape Predictions of 

Slender Curved Structures

William L. Ko and Van Tran Fleischer

Dryden Flight Research Center, Edwards, California

 Insert conference information, if applicable; otherwise delete 

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

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, CA 93523-0273

April 2011
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.

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

 Available from:

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

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802

NOTICE

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

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

the National Aeronautics and Space Administration
v

TABLE OF CONTENTS

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

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

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

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

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

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

Linearity of Bending Moments..............................................................................................................5

Beam Slopes and Deflections.................................................................................................................5

Straight-Beam Deflection Equations......................................................................................................6

UNIFORM CANTILEVER STRAIGHT BEAMS.....................................................................................7

TWO-POINT SUPPORTED STRAIGHT BEAMS...................................................................................8

Left End Simply Supported....................................................................................................................8

Left End Clamped ..................................................................................................................................9

BASICS OF CURVED BEAMS ..............................................................................................................10

BASICS OF ANALYTICAL SHAPE PREDICTIONS ...........................................................................11

SHAPE PREDICTIONS OF CANTILEVER CURVED BEAMS...........................................................12

Geometry..............................................................................................................................................12

Uncorrected Deflections.......................................................................................................................13

Shape Prediction Errors........................................................................................................................13

Curvature-Effect Corrections...............................................................................................................14

Corrected Deflections...........................................................................................................................14

Reduced Shape Prediction Errors.........................................................................................................15

SHAPE PREDICTIONS OF TWO-POINT SUPPORTED CURVED BEAMS......................................15

Geometry..............................................................................................................................................15

Curvature-Effect Corrections...............................................................................................................16

Both Ends Clamped..............................................................................................................................17

Strain Curves.....................................................................................................................................17

Deflection Equations.........................................................................................................................17

Deflection Curves .............................................................................................................................18

Both Ends Simply Supported ...............................................................................................................19

Strain Curves.....................................................................................................................................19

Deflection Equations.........................................................................................................................19

Deflection Curves .............................................................................................................................21

GENERIC CREW EXPLORATION VEHICLE CURVED STRUCTURES..........................................22
vi

SHAPE PREDICTIONS OF CREW EXPLORATION VEHICLE CANTILEVER 

CURVED BEAMS ...................................................................................................................................22

Strain Curves........................................................................................................................................23

Deflection Curves.................................................................................................................................23

SHAPE PREDICTIONS OF CREW EXPLORATION VEHICLE TWO-POINT SUPPORTED 

CURVED BEAMS ...................................................................................................................................24

Both Ends Clamped..............................................................................................................................24

Case of Both Ends Clamped and Free to Move Along Beam Curve (CC-Free) ..............................25

Case of Both Ends Clamped and Stationary (CC-Fixed)..................................................................26

Both Ends Simply Supported ...............................................................................................................27

Case of Both Ends Simply Supported and Free to Move Along Beam Curve (SS-Free).................27

Case of Both Ends Simply Supported and Stationary (SS-Fixed)....................................................29

SINGLE-POINT COLLOCATIONS........................................................................................................30

Clamped Curved Beams.......................................................................................................................30

Simply Supported Curved Beams........................................................................................................32

STRAIN-SENSING STATION DENSITY..............................................................................................33

Clamped Curved Beams.......................................................................................................................33

Simply Supported Curved Beams........................................................................................................34

DISCUSSION ...........................................................................................................................................35

SUMMARY..............................................................................................................................................36

FIGURES ..................................................................................................................................................38

REFERENCES..........................................................................................................................................76
1

ABSTRACT

The Ko displacement theory originally developed for shape predictions of straight beams is extended 

to shape predictions of curved beams. The surface strains needed for shape predictions were analytically 

generated from finite-element nodal stress outputs. With the aid of finite-element displacement outputs, 

mathematical functional forms for curvature-effect correction terms are established and incorporated into 

straight-beam deflection equations for shape predictions of both cantilever and two-point supported 

curved beams. The newly established deflection equations for cantilever curved beams could provide 

quite accurate shape predictions for different cantilever curved beams, including the quarter-circle 

cantilever beam. Furthermore, the newly formulated deflection equations for two-point supported curved 

beams could provide accurate shape predictions for a range of two-point supported curved beams,

including the full-circular ring. Accuracy of the newly developed curved-beam deflection equations is

validated through shape prediction analysis of curved beams embedded in the windward shallow spherical 

shell of a generic crew exploration vehicle. A single-point collocation method for optimization of shape 

predictions is discussed in detail.

NOMENCLATURE

 

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

CC both ends clamped

CEV crew exploration vehicle

ci ≡ c(xi) distance from neutral axis to i-th strain sensor, in.

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

c(x) distance from neutral axis to outermost surface point of nonuniform beam (or radius

of tapered tubular beam) at axial location,

 

x, in.

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

deg degree

E Young’s modulus, lb/in2

ei ≡ yi − yi

C , shape prediction error at strain-sensing station, xi

en ≡ yn − yn

C , shape prediction error at cantilever curved-beam tip, in.



en ≡ 

yn − yn

C , reduced shape prediction error at cantilever curved-beam tip, in.

h depth of beam, in.

i index associated with i-th strain sensor at xi (i = 0, 1, 2, 3, ..., n)

I moment of inertia, in4

j index, 1, 2, 3, ... 

kin ≡ σ in σ , bending stress ratio on concave side

kout ≡ σ out σ , bending stress ratio on convex side

l length of beam, in.

M bending moment, in-lb

Mi bending moment at xi , in-lb

n index of last strain sensor (also used to indicate strain-sensing station density)

P force, lb

R radius of curvature of curved beam, in.
2

SPAR Structural Performance And Resizing

SS both ends simply supported

TPS thermal protection system 

UAV unmanned aerial vehicle

 

x axial coordinate of beam measured from left end, in.

xi x-coordinate (or symbol) of strain-sensing station (cross section) where i-th strain sensor is 

located, in. 

 

y beam deflection normal to x-axis, in.

yi deflection of cantilever straight beam at xi , in.

yi

B deflection of two-point supported straight beam at xi , in.

yi

C radial deflection of curved beam at i-th strain-sensing station calculated from SPAR, in.



yi radial deflection of curved beam at i-th strain-sensing station, in.



yi

B radial deflection of two-point supported curved beam at i-th strain-sensing station, in.

 

δ neutral axis offset (distance between neutral axis and centroidal axis), in.

∆l

 

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

 

Δφ ≡ φn n , angular coordinate differential between two adjacent strain-sensing stations, deg 

ε i surface bending strain at i-th strain-sensing station, in/in.

 

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

 

η ≡ yn/2

C − yn/2B , amplitude of sine correction function, in.

θi deformed beam slope at axial location, xi , rad or deg

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

 

σ magnitude of surface bending stress of equivalent straight beam, lb/in2

σ in magnitude of bending stress on concave side of curved beam, lb/in2

σ out magnitude of bending stress on convex side of curved beam, lb/in2

φi angular coordinate (measured from left end of beam) of i-th strain-sensing station, deg 

φn angular coordinate (measured from left end of beam) of n-th (last) strain-sensing station (called 

curved-beam angle), deg

INTRODUCTION

By installing a series of strain sensors at discrete points along the surface of a beam-like structure 

(such as an aircraft wing), one can use the measured surface strains as input to the Ko displacement 

theory to calculate structural deflections and cross-sectional rotations at discrete points for mapping the 

structural deformed shapes. The most attractive strain-sensing system, an alternative to the conventional 

strain gage system (which contains too much lead wire weight), 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 sensing stations at desired sensing 

intervals (refs. 1–3). In the formulation of the Ko displacement theory (refs. 4–7), the beam was first 

divided into multiple domains of equal length, so that variations of both bending strain and beam depth 

could be assumed to be linear within in each small domain. This approach enabled piecewise integrations 

of the classical beam differential equation to yield beam slope and deflection equations for each domain.

The resulting displacement equations were then expressed in terms of strains measured at multiple equally 
3

spaced strain-sensing stations (at domain junctures) on the surface of the beam structures. The surface 

strain data could then be input to the displacement equations for the calculation of slopes, deflections, and 

cross-sectional twist angles of the beam structures at the strain-sensing cross sections for generating the 

overall deformed shapes of the beam structures.

The Ko displacement theory (refs. 4–7), combined with the onboard fiber-optic strain-sensing system,

form a powerful tool (called Method for Real-Time Structure Shape-Sensing, U.S. Patent No. 7,520,176;

ref. 8) for in-flight deformed shape monitoring of flexible wings and tails (such as those often employed 

on unmanned aerial vehicles, UAVs) by the ground-based pilot for maintaining safe flights. In addition, 

the wing shape, monitored in real time, could be input to the aircraft control system for aeroelastic wing 

shape control. The fiber-optic strain-sensing technique (refs. 4–8) has been further applied to a new 

reverse process (called Process for Using Surface Strain Measurements to Obtain Operational Loads for 

Complex Structures, U.S. Patent No. 7,715,994; ref. 9) for structural operational load monitoring.

The Ko displacement theory (refs. 4–8) was originally developed for predicting the deformed shapes

of straight beams (cantilever and two-point supported beams). In addition, the theory can also be applied 

to shape predictions of unsupported free-free beam-like structures, such as aircraft wings and fuselages,

during flight. The displacement theory (refs. 4–5) has been successfully validated for its accuracy by 

finite-element analysis of different sample structures such as cantilever tubular beams (uniform, tapered, 

slightly tapered, step-wise tapered), two-point supported tapered tubular beams, flat panels, and tapered 

wing boxes (unswept and swept).

By introducing curvature-effect corrections, one can modify the straight-beam displacement theory 

(refs. 4–7) for shape predictions of curved beams. Mathematical functional forms of the curvature-effect 

correction terms could be established, with the aid of finite-element displacement solutions, and 

incorporated into the straight-beam deflection equations for the formulation of curved-beam deflection 

equations.

This technical report examines the prediction errors arising from the use of straight-beam deflection 

equations for shape predictions of curved beams, and demonstrates the methods used in the formulation 

of curved-beam deflection equations for shape predictions of both cantilever and two-point supported 

curved beams. The accuracy of the newly formulated curved-beam deflection equations has been 

validated through shape prediction analysis of curved beams embedded in the windward shallow spherical 

wall of a generic crew exploration vehicle (CEV). This report also discusses the single-point collocation 

method for optimizing the accuracy of the shape predictions of curved beams.

BASICS OF THE KO DISPLACEMENT THEORY

The basics of the Ko displacement theory for the straight beams (refs. 4–5) are briefly described in

this section. The principal assumptions made in the formulation of the theory also are discussed.

Beam Differential Equations

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

(elastic curvature of a deformed straight beam) given by (refs. 10–11)

d2y

dx2 = M(x)

EI (1)
4

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

Young’s modulus, and I is the moment of inertia. 

At the cross section, x, the bending moment,

 

M(x), and associated bending strain,

 

ε(x), induced at 

the bottom (or top) fiber (outermost point from the neutral axis) of the nonuniform straight beam can be 

written as (refs. 4–5) 

M(x) = EI

ε(x)

c(x) (2)

in which 

 

c(x) is the local half depth (distance from the neutral axis to the bottom surface) of the 

nonuniform beam. When equations (1) and (2) are combined, the modified differential equation for the

nonuniform beam is obtained as

 

d2y

dx2 = ε(x)

c(x) (3)

Note that under the present strain formulation, the modified beam differential equation (3) contains 

only the beam half depth,

 

c(x), and the bending strain,

 

ε(x). The flexural rigidity, EI, is eliminated. 

Formulation of the Ko displacement theory (refs. 4–5) for nonuniform straight beams is based upon the 

modified beam differential equation (3), which could be sufficiently accurate if the nonuniform beam 

cross sections change gradually (ref. 10, p. 143).

If the mathematical functional forms of { c(x), ε(x)} are given, equation (3) can be integrated once to 

yield the beam slope,

 

dy dx . The equation can then be integrated a second time to yield the beam 

deflection,

 

y(x), for the shape predictions.

Basic Assumptions

When the single and double integrations of equation (3) were carried out, the following assumptions

were made:

1. The nonuniform beam of length l (fig. 1) is discretized into n domains of equal length, Δl(= l n) , 

and the beam section within each domain is assumed to linearly taper down (or linearly taper up). 

The strain sensors, i (= 0, 1, 2, 3,..., n), are to be installed at the bottom (or top) of the domain 

junctures, x = xi (i = 0, 1, 2, 3,..., n) (also called strain-sensing stations, xi ). Thus, in the domain,

xi−1 ≤ x ≤ xi , between any two adjacent strain-sensing stations, { xi−1, xi }, the beam half depth,

c(x), can be expressed as 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-sensing stations, { xi−1, xi }, respectively.
5

2. The distribution of the bending strain,

 

ε(x), in the domain, xi−1 ≤ x ≤ xi , between the two 

adjacent strain-sensing stations, { xi−1, xi }, is also assumed to change 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 { ε i−1, ε i } are the values of 

 

ε(x) at the strain-sensing stations, { xi−1, xi }, respectively.

Linearity of Bending Moments

Note that for a slightly nonuniform beam, c(x) is a weak linear function of (x − xi−1) (the small 

slope term in eq. (4)), and the associated bending moment, M(x) (eq. (2)), will vary almost linearly 

between the two adjacent strain-sensing stations, { xi−1, xi }, because the higher-order terms in the 

binomial expansion of the denominator [c(x)]

−1 may be neglected. This argument may not hold, 

however, if c(x) is not a weak linear function of x, because the higher-order terms in the binomial 

expansion of the denominator [c(x)]

−1 must be retained.

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

bending strain, ε(x), according to equation (2). Therefore, if the strain, ε(x), is a linear function of x

given by equation (5), then M(x) will also be a linear function of x.

Beam Slopes and Deflections 

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-sensing 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 ) (6)
6

in which tanθi−1 is the slope at the inboard strain-sensing station, xi−1. The deflection,

 

y(x), of the 

nonuniform beam within the domain, xi−1 ≤ x ≤ xi (fig. 1), can be obtained by integrating slope equation 

(6), and enforcing the continuity of deflection at the inboard adjacent strain-sensing station, xi−1, as

y(x) = tanθ(x) dx

xi−1

x

∫

Integration of

eq. (6)



+ 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 ) (7)

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

In light of the linearity assumptions of {

 

c(x), ε(x)} given by equations (4) and (5), respectively, 

equations (6) and (7) can now be integrated once and twice, respectively, to yield the indicial forms of the 

slope and deflection equations for each domain. These indicial equations are then combined to yield the 

final mathematical forms of the deflection equations expressed in terms of beam geometrical parameters 

and surface strains presented in the following section (refs. 4–5).

Straight-Beam Deflection Equations 

The deflection equations (integrations of equation (7)) previously developed for nonuniform, slightly 

nonuniform, and uniform straight cantilever beams, have the following mathematical forms (refs. 4–5):

1. For nonuniform straight cantilever beams ( ci ≠ ci−1) (derivations in ref. 5, Appendix A),

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

2. For slightly nonuniform straight cantilever beams ( ci ci−1 →1),

yi = (Δl)

2

6

1

ci− j j=1

i

∑ 3(2 j −1) − (3 j − 2)

ci− j+1

ci− j

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ε i− j + (3 j − 2)ε i− j+1

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

+ y0 + (i)Δl tanθ0

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

which was obtained from equation (8) by expanding the logarithmic terms in the neighborhood of 

ci ci−1 ≈ 1 (derivations in ref. 5, Appendix C).
7

3. For uniform straight cantilever beams ( ci = ci−1 = c ),

 yi = (Δl)

2

6c

(3i −1)ε 0 + 6 (i − j)ε j

j=1

i−1

∑ + ε i

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + (i)Δl tanθ0 ; (i = 1,2,3,...,n) (10)

which was obtained by grouping the terms after setting ci = ci−1 = c in equation (9) (derivations 

in ref. 4 and Appendix D of ref. 5).

In each of the deflection equations (8)–(10), the deflection,

 

yi

, at the sensing station,

 

xi

, is expressed 

in terms of the inboard beam half depths ( c0, c

1, c2,..., ci ) and the associated inboard strains 

( ε 0, ε1, ε 2,..., ε i ), including the values of { ci

, ε i } at the strain-sensing station,

 

xi

, where the deflection,

 

yi

, is calculated. Because equations (8)–(10) contain no structural properties (they actually are contained 

implicitly in the strains), in the shape predictions of complex structures (for example, aircraft wings), one 

can avoid tedious computations of bending stiffness, EI, at different strain-sensing stations (cross 

sections). This feature is a characteristic of the Ko displacement theory.

UNIFORM CANTILEVER STRAIGHT BEAMS

In this report, only the uniform beams ( ci = ci−1 = c = constant) are considered; therefore, only 

equation (10) is needed. To show the progression of the functional form of equation (10) with an 

increasing value of the index, i, deflection equation (10) is written out explicitly for different indices, i

(strain-sensing station, xi ), as

y

1 = (Δl)

2

6c

(2ε 0 + ε1) + y0 + Δl tanθ0 (10a)

y2 = (Δl)

2

6c

5ε 0 + 6ε1 + ε 2 ( ) + y0 + 2Δl tanθ0 (10b)

y3 = (Δl)

2

6c

8ε 0 + 6(2ε1 + ε 2) + ε 3 ⎡

⎣ ⎤⎦ + y0 + 3Δl tanθ0 (10c)

y4 = (Δl)

2

6c

11ε 0 + 6 3ε1 + 2ε 2 + ε 3 ( ) + ε 4 ⎡

⎣ ⎤⎦ + y0 + 4(Δl)tanθ0 (10d)

y5 = (Δl)

2

6c

14ε 0 + 6 4ε1 + 3ε 2 + 2ε 3 + ε 4 ( ) + ε5 ⎡

⎣ ⎤⎦ + y0 + 5(Δl)tanθ0 (10e)

y6 = (Δl)

2

6c

17ε 0 + 6 5ε1 + 4ε 2 + 3ε 3 + 2ε 4 + ε5 ( ) + ε 6 ⎡

⎣ ⎤⎦ + y0 + 6(Δl)tanθ0 (10f)
8

y7 = (Δl)

2

6c

20ε 0 + 6 6ε1 + 5ε 2 + 4ε 3 + 3ε 4 + 2ε5 + ε 6 ( ) + ε 7 ⎡

⎣ ⎤⎦ + y0 + 7(Δl)tanθ0 (10g)

y8 = (Δl)

2

6c

23ε 0 + 6 7ε1 + 6ε 2 + 5ε 3 + 4ε 4 + 3ε5 + 2ε 6 + ε 7 ( ) + ε8 ⎡

⎣ ⎤⎦ + y0 + 8(Δl)tanθ0 (10h)

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

yn = (Δl)

2

6c

(3n −1)ε 0 + 6 (n − j)ε j

j=1

n−1

∑ + ε n

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+ y0 + n(Δl)tanθ0 (10i)

Equation (10) explicitly shows that the deflection, yi , at the strain-sensing station, xi , is obtained by 

summing up all the inboard strains ( ε 0, ε1, ε 2,..., ε i ), including the strain, ε i , at the current strain-sensing 

station, xi . For the cantilever beam, both the deflection, y0 , and slope, tanθ0 , at the fixed end 

( x = x0 = 0 ) in equation (10) are zero (that is, y0 = tanθ0 = 0 ). The terms, y0 + (i)Δl tanθ0 , however, 

are purposely retained for completeness so that equation (10) can be extended to the deflection equations 

for the simply supported beam, for which the left-end deflection, y0 , is zero ( y0 = 0 ), but the left-end 

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

TWO–POINT SUPPORTED STRAIGHT BEAMS

Deflection equation (10) for uniform cantilever beams also can be used for shape predictions of 

uniform two-point supported beams (simple beams). Depending on the support conditions, equation (10) 

can be used directly or with modifications.

Left End Simply Supported

If the uniform two-point supported straight beam is simply supported at the left end (the right end 

may be either simply supported or clamped), the slope, tanθ0 , at the left support ( x = x0 = 0 ) is no 

longer zero ( tanθ0(≠ 0) ). Also, the unknown nonzero slope, tanθ0 , must be determined so that 

equation (10) can be modified for shape predictions of left-end simply supported beams.

Let yi

B be the deflection at the strain-sensing station, xi , of the left-end simply supported straight 

beam. Then yi

B can be calculated from equation (10) ( yi replaced by yiB ) with the nonzero slope term, 

tanθ0(≠ 0) , retained:

 yi

B = (Δl)

2

6c

(3i −1)ε 0 + 6 (i − j)ε j

j=1

i−1

∑ + ε i

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

Cantilever beam deflection equation



+ i(Δl) tanθ0

Nonzero 

 ; (i = 1,2,3,...,n) (11)
9

By setting the deflection, yi

B , at the right support ( x = xn = l) to zero (that is, yiB = ynB = 0 ), one can 

determine the unknown slope, tanθ0 , at the left support ( xi = x0 = 0 ) from equation (11) as

tanθ0 = − 1

n(Δl)

yn (12)

in which yn is the deflection at the right end of the beam ( xi = xn = l) calculated from equation (10i), 

neglecting { y0 + (i)Δl tanθ0 } terms. Namely,

yn ≡ (Δl)

2

6c

(3n −1)ε 0 + 6 (n − j)ε j

j=1

n−1

∑ + ε n

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

Cantilever beam tip deflection (eq. (10i) neglecting 

{y0 +n(Δl)tanθ0} terms)



 (13)

In light of equation (12), deflection equation (11) for the left-end simply supported straight beam 

becomes

 yi

B = yi − i

n

yn

To enforce

yn

B = 0 

at right

support

(i = n)



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

in which yi is the cantilever beam deflection at the strain-sensing station, xi , given by equation (10),

setting y0 = 0 and tanθ0 = 0 (or from equation (13), setting the index n = i). 

Left End Clamped

If the uniform two-point supported straight beam is clamped at the left end (the right end can be either 

clamped or simply supported), the deflections, yi

B , can be calculated from the cantilever-beam deflection 

equation (10) (replacing yi with yi

B and setting y0 = 0 and tanθ0 = 0 ) for the shape predictions 

(refs. 4–5). The calculated deflection, yn

B , at the right support ideally should be zero ( yn = 0 ). If the 

calculated value of yn

B is not exactly zero ( yn ≠ 0 ), however, and it has a small value, a correction term
10

must be added to equation (10) to enforce yn

B = 0 . To enforce ynB = 0 at the right end, and also to 

maintain a zero slope ( tanθ0 = 0 ) at the clamped left end, equation (14) can be modified by squaring the

second term as

 yi

B = yi − i

n

⎛

⎝

⎜

⎞

⎠

⎟

2

yn

To enforce

yn

B = 0 

at right support

(i = n)



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

for the calculations of deflections, yi

B , of the left-end clamped straight beam. 

BASICS OF CURVED BEAMS

The purpose of this section is to examine the deflection prediction errors involved in using the 

straight-beam deflection equation (10) for shape predictions of curved beams with different curvatures, 

and then to show how the prediction errors can be practically eliminated by introducing the curvatureeffect correction terms to equation (10). Table 1 lists the geometrical parameters and associated bending 

stresses for different curved beams with rectangular cross sections (taken from ref. 10, p. 148). 

Table 1. Curvature-related data for curved beams (ref. 10).

R/c kin kout δ / R

1.2 2.89 0.57 0.305

1.4 2.13 0.63 0.204

1.6 1.79 0.67 0.149

1.8 1.63 0.70 0.112

2.0 1.52 0.73 0.090

3.0 1.30 0.81 0.041

4.0 1.20 0.85 0.021

6.0 1.12 0.090 0.0093

8.0 1.09 0.92 0.0052

10.0 1.07 0.94 0.0033

In table 1, {R, c} are the radius of curvature and the half depth of the uniform curved beam,

respectively,δ is the neutral axis offset (distance between the neutral axis and centroidal axis) given by 

(ref. 10, p. 148; ref. 11, p. 183)

 δ = R − 2c

ln

R + c

R − c

⎛

⎝

⎜

⎞

⎠

⎟

 (16)
11

and { kin , kout } are the magnitudes of the normalized bending stresses on the concave and convex sides, 

respectively, of the curved beam defined as 

 kin = σ in

σ (17)

 kout = σ out

σ (18)

in which {σ in,σ out } are the magnitudes of the bending stresses on the concave and convex sides, 

respectively, and σ is the magnitude of the surface bending stress of the equivalent straight beam.

The data listed in table 1 are plotted in figure 2. Note that the magnitude of the bending stress on the 

concave side ( kin ) is larger than the magnitude of the bending stress on the convex side ( kout ), and that 

the difference between kin and kout decreases as the radius of curvature factor, R c , increases. Also 

notice that the neutral axis of the curved beam does not coincide with the centroidal axis, but it is shifted 

away from the centroidal axis toward the concave side by an offset distance, δ . As shown in figure 2, the 

curvature effect gradually decreases with an increasing radius of curvature factor, R c , and practically 

diminishes at approximately R/c = 17 (δ / R ≈ 0.001). In the region R/c >17, one observes that kin ≈ kout

and δ ≈ 0 , and the curved beam approaches the straight beam. Therefore, for the slightly curved beam

with a radius of curvature factor, R/c, greater than 17 (R/c >17, fig. 2), the curved beam can be treated like 

a straight beam. Thus the straight-beam deflection equation (10) can be applied for shape predictions 

without introducing too many errors. The next section discusses the prediction errors of using the straightbeam deflection equation (10) for shape predictions of curved beams with different curvatures.

BASICS OF ANALYTICAL SHAPE PREDICTIONS

The present shape prediction study of curved beams is called an analytical shape prediction study. 

Instead of using the strain data measured by fiber optics, the Structural Performance And Resizing 

(SPAR) finite-element computer program (ref. 12) was used to generate the surface bending strain data 

and beam deflection curves (to be used as reference deflection curves).

The bending strains, ε i (i = 0, 1, 2, 3,..., n), at the i-th strain sensors needed for deflection 

calculations (eq. (10)) were generated by converting the SPAR axial nodal stresses, σ i

(i = 0, 1, 2, 3,..., n), at the i-th strain sensor point, into the bending strains, ε i , through Hooke’s law (that 

is, ε i = σ i E ), in which E is the Young’s modulus. Alternatively, the bending strains, ε i , can also be 

generated from the axial length changes of the SPAR elements where the bending strains, ε i , are to be 

measured. This method was found to lose accuracy near the high slope regions of the deformed beam,

however, and therefore was not used (ref. 4).

The bending strains generated by the SPAR program were then input to the straight-beam deflection 

equations to calculate the theoretical deflection curves for comparison with the corresponding SPAR 

deflection curves (used as reference). The prediction errors caused by curvature effect also were 

examined. The final mathematical functional forms for the curvature-effect correction terms were 
12

determined when the trial deflection curves best fit the corresponding SPAR deflection curves. The 

curvature-effect correction terms thus determined were then introduced to the straight-beam deflection 

equations to formulate the curved-beam deflection equations.

SHAPE PREDICTIONS OF CANTILEVER CURVED BEAMS

Before the straight-beam deflection equation (10) is applied to the shape predictions of cantilever

curved beams, the prediction errors resulting from equation (10) first must be examined. For the 

prediction error studies, cantilever curved beams with different curvatures, listed in the next section, were

analyzed.

Geometry

Table 2 lists the dimensions of five types of cantilever curved beams (the straight cantilever beam is 

listed as a limit case) for the shape prediction error studies. All the cantilever curved beams considered 

have a unit width, an identical curved length of l = 100 in., and an identical half depth of c = h/2 = 0.25 in. 

To maintain the same curved-beam length, the radius of curvature, R, was changed with the change of the 

curved-beam angle, φn .

Table 2. Dimensions of cantilever curved beams analyzed; width = 1 in.

l, in. φn , deg R, in. h, in. c, in. R/c

100 0.0 (straight) ∞ 0.50 0.25 ∞

100 22.5 254.65 0.50 0.25 1,019

100 45.0 127.32 0.50 0.25 509

100 67.5 84.88 0.50 0.25 340

100 90.0 (1/4 circle) 63.66 0.50 0.25 255

Based on the range of the radius of curvature factor, R/c, listed in table 2, the neutral axis offset, δ ,

calculated from equation (16) was negligible (fig. 2). Therefore, the neutral axis offset effect was ignored 

in the shape prediction analysis.

Figure 3 shows the undeformed and deformed shapes, generated from the SPAR program, of a typical 

cantilever curved beam with an inward radial load of P = 1 lb at the beam tip. There are n + 1 (n = 8) 

equally spaced strain sensors (i = 0, 1, 2, 3,..., n) installed on the lower surface of the beam, with the 

strain sensors (0, n) located at the fixed end ( x = x0 = 0 ) and at the free end ( x = xn = l ), respectively.

Figure 4 shows the strain curves, generated from the SPAR program, for the cantilever curved beams 

with different curvatures. Note from figure 4 that as the curvature increases (that is, an increasing curvedbeam angle, φn ), the strain curve near the left fixed end continues to bend further downward from the 

straight strain line of the straight beam (φn = 0 ). Also note from figure 4 that all the strain curves for the 

cantilever curved beams have very small curvatures, and any segment of the strain curve between the two 

adjacent strain-sensing stations is very close to a straight line, reinforcing the assumption of piecewise 

linear distributions of bending strains (eq. (5)) used in the formulation of the Ko displacement theory 

(refs. 4–6). The piecewise linear strain curves will converge to the corresponding actual curvilinear strain 

curves (fig. 4) as the number of strain-sensing stations is increased. For the fiber-optic strain sensors, the 
13

number of strain-sensing stations can be increased very easily at will by simply reducing the sensing 

intervals of the Bragg gratings along the optical fiber (refs. 1–3).

Uncorrected Deflections

To use the straight-beam deflection equation (10) for shape predictions of a cantilever curved beam, 

the axial coordinate, x, will now represent the tangential coordinate, x = Rφ , of the curved beam, and 

 

y

will represent the radial displacement, which is considered positive when pointed inward. Because the 

radius of curvature factors, R/c = 255–1,019 (table 2), are much greater than 17 (beyond which the 

curvature effect diminishes), the neutral axis and centroidal axis of the cantilever curved beams are almost 

coincidental (fig. 2). Therefore, when equation (10) was used, the half depth, c, was not modified,

because the neutral axis offset, δ , was practically zero (δ ≈ 0 ).

Figure 5 shows the deflection curves calculated for all the cantilever curved beams. The deflection 

curves calculated from equation (10) are represented by solid lines with solid circular symbols, and the 

deflection curves calculated from the SPAR program are represented by dotted lines with open circular 

symbols. Note that equation (10) overpredicts the curved-beam deflections calculated from the SPAR 

program, and the prediction error increases progressively with the increasing curved-beam angle, φn . 

Namely, as the curvature increases (that is, increasing φn ), the deflection curve calculated from 

equation (10) continues to bend upward further away from the corresponding SPAR deflection curve 

(considered as the reference deflection curve).

Shape Prediction Errors

Let yi

C (i = 1, 2, 3, ..., n) be the radial deflections calculated from the SPAR program (assumed to 

be the true radial deflections), and let yi (i = 1, 2, 3, ..., n) be the deflections calculated from the straightbeam deflection equation (10). Then the difference, ei

(≡ yi − yi

C ) , will be considered as the shape

prediction error at the strain-sensing station, xi . Table 3 lists the beam tip deflections, { yn

C, yn } 

(maximum), of the cantilever curved beams together with the normalized beam tip shape prediction 

errors, en yn

C , for different cantilever curved beams. Note that the prediction error, en ynC , at the 

curved-beam tip based on the straight-beam deflection equation (10) increases progressively with the 

increase in the curved-beam angle φn , and reaches a maximum of 26.6437 percent at φn = 90° for a 

quarter-circle beam.
14

Table 3. Comparison of beam tip deflections calculated from SPAR with those calculated from straightbeam deflection equation (10) for different cantilever curved beams; l = 100 in.; c = 0.25 in.; P = 1 lb; 

n = 8.

φn , deg yn

C , in.

(SPAR)

yn , in.

(eq. (10))

en / yn

C ×100%

(eq. (10) prediction error)

0.0 (straight) 9.4714 9.4793 0.0834

22.5 9.1834 9.3209 1.4951

45.0 8.3693 8.8805 6.1000

67.5 7.1645 8.1891 14.3011

90.0 (1/4 circle) 5.7582 7.2924 26.6437*

*Peak prediction error.

Curvature-Effect Corrections

To reduce (or eliminate) the prediction errors from using the straight-beam deflection equation (10) 

for shape predictions of cantilever curved beams, a curvature-effect correction term must be introduced to 

modify equation (10). A trial mathematical function for the curvature-effect correction term was added to 

equation (10) to calculate the corrected deflection curve for each cantilever curved beam, which was then

compared with the corresponding SPAR deflection curve (used as reference). After extensive searching, 

the most accurate mathematical functional form for the curvature-effect correction term was introduced to 

equation (10) to formulate the following modified deflection equation for calculations of deflections, 

yi (i = 1, 2, 3, ..., n), of the cantilever curved beams (established September 24, 2008). For 0 ≤ φn ≤ 90 °

arcs,

 

yi = yi − 1

4.5

φn

90

⎛

⎝

⎜

⎞

⎠

⎟

2

i

n

⎛

⎝

⎜

⎞

⎠

⎟

4

yn

Curvature-effect

 correction term



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

in which yi (i =1, 2, 3, ..., n) is to be calculated from the cantilever straight-beam deflection 

equation (10) (setting y0 + (i)Δl tanθ0 = 0). Equation (19) is called the cantilever curved-beam deflection 

equation and is applicable for the whole range ( 0 ≤ φn ≤ 90 °) of cantilever curved beams analyzed, 

including the special case of a straight beam (φn = 0°), for which the curvature-effect correction term 

vanishes (that is, 

yi = yi ). 

Corrected Deflections

The corrected deflection curves calculated from the cantilever curved-beam deflection equation (19) 

are plotted in figure 6 for different curved beams. Note that the predicted deflection curves practically 

match the corresponding SPAR deflection curves, with pictorially inconspicuous prediction errors. This 

good agreement confirms the accuracy of the newly developed cantilever curved-beam deflection 

equation (19) for shape predictions of a whole range ( 0 ≤ φn ≤ 90 °) of cantilever curved beams analyzed.
15

Reduced Shape Prediction Errors

Table 4 lists the corrected beam tip deflections, 

yn , together with the reduced prediction errors,



en(≡ yn − yn

C ) . The uncorrected beam tip deflections, yn , calculated from equation (10) and the 

associated prediction errors, en(≡ yn − yn

C ) , taken from table 3 also are listed for comparison. Note that

the corrected beam tip deflection errors, 

en yn

C , are merely 0.0849–1.4987 percent, which are almost 

negligible compared with the uncorrected beam tip deflection errors of en yn

C of 1.4951–26.6437 

percent.

Table 4. Comparison of curved-beam tip deflections calculated from SPAR with those calculated from 

deflection equations (10) and (19) for different cantilever curved beams; l = 100 in.; c = 0.25 in.; P = 1 lb;

n = 8.

φn , deg yn

C , in.

(SPAR)

yn , in.

(eq. (10))

en / yn

C ×100%

(eq. (10) prediction error)



yn

(eq. (19))



en / yn

C ×100%

(eq. (19) prediction error)

0.0 9.4714 9.4793 0.0834 9.4793 0.0834

22.5 9.1836 9.3209 1.4951 9.1914 0.0849

45.0 8.3693 8.8805 6.100 8.3871 0.2127

67.5 7.1645 8.1891 14.3011 7.1654 0.0126

90.0 5.7582 7.2924 26.6437 5.6719 1.4987

For easy visualization, figure 7 shows the uncorrected and corrected curved-beam tip deflection 

prediction errors { en yn

C ,



en / yn

C } plotted as functions of the curved-beam angle, φn , based on the data 

of table 4. Note that the prediction errors of using equation (10) increases progressively with increasing 

φn . With curvature-effect corrections (eq. (19)), the beam tip prediction error curve is nearly a horizontal 

line because of a negligible error range (table 4).

SHAPE PREDICTIONS OF TWO–POINT SUPPORTED CURVED BEAMS

The straight-beam deflection equations (14) and (15) may be reasonably accurate for shape 

predictions of simply supported and clamped shallow curved beams, respectively. For shape predictions 

of deeper curved beams, however, equations (14) and (15) could lose prediction accuracy. Therefore, 

curvature-effect correction terms are needed to modify both equations (14) and (15). The mathematical 

functional forms for the curvature-effect correction terms were determined with the aid of SPAR finiteelement analysis of different two-point supported curved beams listed in the next section.

Geometry

Table 5 lists the dimensions of the two-point supported curved beams that were analyzed for 

establishing the mathematical functional forms of the curvature-effect correction terms. All the two-point 

supported curved beams that were analyzed have a unit width, an identical curved length of l = 200 in.,

and an identical half depth of c = h/2 = 0.25 in. To maintain the same curved-beam length, the radius of 

curvature, R, was changed with the change of the curved-beam angle, φn . For the range of radius of
16

curvature factor, R/c, listed in table 5, the neutral axis offset factor, δ / R , is practically zero according to 

the δ / R curve of figure 2.

 

Table 5. Dimensions of two-point supported curved beams; width = 1 in.

l, in. φn , deg R, in. h, in. c, in. R/c

200 0 (straight) ∞ 0.50 0.25 ∞

200 45 254.65 0.50 0.25 1,019

200 90 (1/4 circle) 127.32 0.50 0.25 509

200 135 84.88 0.50 0.25 340

200 180 (1/2 circle) 63.66 0.50 0.25 255

200 225 50.93 0.50 0.25 204

200 270 (3/4 circle) 42.44 0.50 0.25 170

200 315 36.38 0.50 0.25 146

200 360 (full circle) 31.83 0.50 0.25 127

Figure 8 shows a typical two-point supported curved beam under two types of support conditions: 

(1) both ends clamped, and (2) both ends simply supported. Each two-point supported curved beam was

subjected to an upward (inward) radial force of P = 2 lb at the curved beam midpoint. As shown in 

figure 8, n + 1 (n = 16) equally spaced strain sensors were installed along the outer surface of the curved 

beam. The corresponding strain-sensing stations (not shown) for the inner surfaces of the curved beam 

also are needed so that the outer and inner strains at the same beam cross sections can be averaged to 

eliminate possible axial strain components caused by beam curvature and support constraints.

 Curvature-Effect Corrections

For shape predictions of curved beams, the straight-beam equation (14) (for both ends simply 

supported) or equation (15) (for both ends clamped) will generate too many prediction errors when the 

beam curvature increases. Therefore, the curvature-effect correction terms must be introduced to 

equations (14) and (15) to formulate the modified deflection equations for shape predictions of two-point 

supported curved beams.

The mathematical functional forms for the curvature-effect correction terms were extensively 

searched with the aid of the SPAR computer program. Namely, the SPAR program was used to generate 

both the bending strains and deflection curves of various two-point supported curved beams. The SPAR 

bending strains were then input to the straight-beam deflection equation (14) (for both ends simply 

supported) or equation (15) (for both ends clamped) to calculate the respective predicted deflection curves 

for comparison with the corresponding SPAR deflection curves (considered as reference deflection 

curves). The curvature correction terms expressed in trial mathematical functional forms were added to 

equation (14) or (15) to improve the curved-beam shape predictions. The curvature correction 

mathematical functional forms were continually modified until the predicted deflection curves converged 

toward the corresponding SPAR deflection curves. The reformulated equation (14) or (15) containing the 

final forms of the curvature-effect correction terms were then considered as the deflection equations for 

the two-point supported curved beams.
17

Both Ends Clamped

Figure 9 shows the deformed shapes, generated from the SPAR program, of various clamped curved 

beams including the straight beam ( 0 ≤ φn ≤ 360 °), each of which was subjected to an upward central 

force of P = 2 lb. Note that in the range of 45 ≤ φn ≤ 270 °, the central regions of the curved beams caved

in, but both the left and right regions buckled out because of stationary supports. The central inward 

bending diminished for the curved-beam angles, φn = 315° and φn = 360°.

Strain Curves

Figure 10 shows the strain curves, generated from the SPAR program, for various clamped curved 

beams including the straight beam ( 0 ≤ φn ≤ 360 °). The strain curve for the clamped straight beam 

(φn = 0) has the classical V-shape consisting of two inclined straight lines. The strain curves for the 

clamped curved beams, however, have a rounded M-shape and stay quite close. Slight strain differences 

are visible in the two submit regions, and the strain curves fan out toward the left and right supports.

Deflection Equations

After the mathematical functions were searched extensively for the best curvature-effect correction

terms in light of the SPAR deflection curves, the modified deflection equations were formulated for

calculations of the deflections, 

yi

B (i =1, 2, 3, ..., n), of the clamped curved beams with curved-beam 

angles up to φn = 360° (complete circle). The modified deflection equations for the whole range 

( 0 ≤ φn ≤ 360 °) of clamped curved beams are presented in the next paragraph. For completeness, the 

degenerated deflection equation for the clamped straight beam is written out separately.

For φn = 0° (straight beam),



yi

B ≡ yiB = yi − i

n

⎛

⎝

⎜

⎞

⎠

⎟

2

yn

To enforce 

yn

B = 0 at

right support

(i = n).



 ; (i = 1,2,3,...,n) (20a)

For 0 ≤ φn ≤ 225 ° arcs (established July 14, 2008) (Chino Hills earthquake, July 29, 2008,

11:40 a.m., scale 5.4),



yi

B = yi − i

n

⎛

⎝

⎜

⎞

⎠

⎟

2

+

4

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

2

sin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0 at

right support (i = n). 



 ; (i = 1,2,3,...,n) (20b)
18

For 225 < φn ≤ 360 ° arcs (established July 17, 2008), 



yi

B = yi − i

n

⎛

⎝

⎜

⎞

⎠

⎟

2

+ 2.25 φn

360

⎛

⎝

⎜

⎞

⎠

⎟

3

sin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0 at

right support (i = n). 



 ; (i = 1,2,3,...,n) (20c)

In equation (20), yi is the deflection of the cantilever straight beam given by equation (10) 

(setting y0 = 0 and tanθ0 = 0 ). Note that the functional forms of the curvature-effect correction terms are 

slightly different for different ranges of the curved-beam angle, φn . For shape predictions of the clamped 

curved-beam case, only two deflection equations are needed to cover the whole range of curved-beam 

angles ( 0 ≤ φn ≤ 360 °).

In equations (20b–c), the first correction term, (i / n)

2 , within the brackets is to cause the 

yi

B = yi

curve to rotate with respect to the left support point to nullify the deflection at the right support (that is, 



yn

B = 0 ), thereby causing the left and right regions of the rotated theoretical deflection curve to fit the 

corresponding regions of the SPAR deflection curve. The second correction term with a sine function 

(eqs. (20b–c)) is to bend the central region of the rotated theoretical deflection curves upward to match 

the corresponding central regions of the SPAR deflection curves as shown in the following section. 

Deflection Curves

The strain data generated from the SPAR program (fig. 10) were input to the clamped curved-beam 

deflection equation (20) to generate the theoretical deflection curves. Figure 11 shows the deflection 

curves generated from the SPAR program compared with the theoretical deflections curves calculated 

from the straight-beam deflection equations (10) and (15), and also from the modified deflection 

equation (20), for the whole range of curved-beam angles ( 0 ≤ φn ≤ 360 °).

For the straight beam (φn = 0, fig. 11a), the deflection curves, yi

B = yi (cantilever straight-beam 

deflection equation (10)) and yi

B = yi − (i / n)

2

yn (clamped straight-beam deflection equation (19a)),

agree quite nicely with the SPAR deflection curve. Equation (10) ( yi

B = yi ), however, gives a pictorially 

inconspicuous nonzero deflection of yn = 0.014285 in. at the right end. Keep in mind that the term,

(i / n)

2

yn , in the deflection equation, yi

B = yi − (i / n)

2

yn , is to rotate (with slight distortions) the 

yi

B = yi curve (eq. (10)), with respect to the left support, to nullify the right-end deflection, 

yn

B , of the 

clamped straight beam (that is, to set 

yn

B = 0 ). For the curved beams (figs. 11b–i), the deflection curves,

yi

B = yi (eq. (10)), produce marked prediction errors (in the range of yn = –0.10952 to –0.15475) at the 

right ends of the beam because of the curvature effect. With the term, (i / n)

2

yn , included, the 
19

yi

B = yi − (i / n)

2

yn curve (eq. (20a)) compares nicely with the corresponding SPAR deflection curve for 

the curved-beam angle, φn = 45° (fig. 11b). For curved-beam angles,

 

φn ≥ 90° (figs. 11c–i), however, 

the yi

B = yi − (i / n)

2

yn curve can fit only the regions near both ends of the corresponding SPAR 

deflection curve, and the central region of the yi

B = yi − (i / n)

2

yn curve starts to diverge from the 

corresponding SPAR curve as φn increases, and reaches a maximum level at φn = 360°. When the sine 

function correction terms (eqs. (20b–c)) are introduced, the central region of each final theoretical 

deflection curve can be bent upward to converge toward the central region of the corresponding SPAR 

deflection curve (figs. 11b–i).

Both Ends Simply Supported

Figure 12 shows the deformed shapes, generated from the SPAR program, of various simply 

supported curved beams including the straight beam ( 0 ≤ φn ≤ 360 °), each of which was subjected to an 

upward central force of P = 2 lb. Similar to the clamped curved beams (fig. (9)), for the simply supported 

curved beams with the curved-beam angle, 45 ≤ φn ≤ 270 °, the central region of the curved beam bent 

inward, but both the left and right regions bulged out because of stationary supports. For the curved-beam 

angles, φn = 315° and φn = 360°, both the left and right regions bulged out because of stationary 

supports, but the central region did not cave in.

Strain Curves

Figure 13 shows the strain curves, generated from the SPAR program, for different simply supported 

curved beams with a whole range of curved-beam angles (φn = 0–360°). The strain curve for the straight 

beam (φn = 0) again has the classical V-shape, consisting of two inclined straight lines. The strain curves 

for the curved beams, however, have a rounded M-shape and are very similar. Slight differences in the 

strain curves are magnified in the left and right summit regions. 

Deflection Equations

The strain data generated from the SPAR program (fig. 13) were used to calculate the theoretical 

deflection curves for comparison with the corresponding reference SPAR deflection curves. After 

mathematical functions were searched extensively for the curvature-effect correction terms in light of the 

reference SPAR deflection curves, modified deflection equations were established for calculations of 

deflections, 

yi

B (i =1, 2, 3, ..., n), of simply supported curved beams with a whole range of curved-beam 

angles ( 0 ≤ φn ≤ 360 °). The modified deflection equations for simply supported curved beams are 

presented in this section. For completeness, the deflection equation for the simply supported straightbeam case (φn = 0°) is written out separately. 
20

For φn = 0° (straight-beam case),



yi

B ≡ yiB = yi − i

n

yn

To enforce

yn

B = 0 at

right support

(i = n)



 ; (i = 1,2,3,...,n) (21a)

For φn = 45° arc,



yi

B = yi − i

n

+

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

2

sin

i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



 ; (i = 1,2,3,...,n) (21b)

For φn = 90° arc,



yi

B = yi − i

n

+

1

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

3

sin

i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



 ; (i = 1,2,3,...,n) (21c)

For 135 ≤ φn ≤ 225° arcs,



yi

B = yi − i

n

− 1

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

3

sin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



 ; (i = 1,2,3,...,n) (21d)

For φn = 270° arc,



yi

B = yi − i

n

− 0.9

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

3

sin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



 ; (i = 1,2,3,...,n) (21e)
21

For φn = 315° arc,



yi

B = yi − i

n

− 0.8

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

3

sin3 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



 ; (i = 1,2,3,...,n) (21f)

For φn = 360° ring,



yi

B = yi − i

n

− 0.7

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

3

sin3 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



 ; (i = 1,2,3,...,n) (21g)

Note that six different deflection equations are needed for the simply supported curved beams to 

cover the whole range of curved-beam angles ( 0 ≤ φn ≤ 360 °). Remember that for the clamped curved 

beams, only two deflection equations are needed to cover the whole range of curved-beam angles 

( 0 ≤ φn ≤ 360 °).

Deflection Curves

Figure 14a shows the deflection curve, 

yi

B = yiB = yi − (i n) yn , for the simply supported straight 

beam (eq. (21a)) compared with the deflection curve generated from the SPAR program. The two 

deflection curves are extremely close, showing the prediction accuracy of equation (21a). Figures 14b–i

compare the theoretical and SPAR deflection curves for the curved-beam angles, φn = 45–360°. For the 

simply supported curved beams, the deflection curves, 

yi

B = yiB = yi − (i n) yn , calculated from the 

straight-beam equation (21a) are fairly close to the corresponding SPAR deflection curves up to 

φn = 135°, and the prediction error (underprediction) gradually increases with increasing φn , reaching a 

maximum at φn = 360°.

Remember that the term, (i / n) yn , in the deflection equation, 

yi

B = yiB = yi − (i n) yn (eq. (21a)), is 

to shift the 

yi

B = yi curve upward proportionally with respect to the left support point to bring the right 

support deflection, 

yn

B , to zero ( 

yn

B = 0 ). For curved beams, this shift causes the 



yi

B = yiB = yi − (i n) yn curves to fit only the left and right regions of the corresponding SPAR 

deflection curves. The central regions remain to be corrected. When the sine correction terms 

(eqs. (21b–g)) are added, the central regions of the final deflection curves could be bent upward to fit the 

central regions of the corresponding SPAR deflection curves quite well.
22

GENERIC CREW EXPLORATION VEHICLE CURVED STRUCTURES

The new deflection equations developed for cantilever curved beams (eq. (19)) and for two-point 

supported curved beams (eqs. (20) and (21)) will now be applied to shape predictions of the generic crew 

exploration vehicle (CEV) curved structures. As shown in figure 15, the windward shallow spherical wall 

of the generic CEV is fabricated with dual honeycomb sandwich walls (exterior composite wall and 

interior aluminum wall). Both walls have an identical depth of 2c = 0.5 in. and are separated by a 2-in. 

gap. The exterior wall region chosen for the shape prediction analysis has a radius of curvature of 

R = 245.75 in., and subtends an azimuth angle of 22°, which excludes the rounded toroidal shoulder

regions. The shape predictions of the CEV outer shell (with the thermal protection system removed) can

start from the shape predictions of the CEV curved beam embedded in the exterior wall of the CEV. Two 

types of CEV curved beams were analyzed: (1) cantilever curved beam, and (2) two-point supported 

curved beam.

SHAPE PREDICTIONS OF CREW EXPLORATION VEHICLE 

CANTILEVER CURVED BEAMS

Figure 16 shows the SPAR model of the CEV cantilever curved beam, which is a cutout of the 

generic CEV windward exterior wall along its radius (fig. 15). An inward radial load of P = 1 lb is 

applied at the beam tip. There are n + 1 (n = 8) equally spaced strain sensors (i = 0, 1, 2, 3,..., n) (fig. 16) 

designated on the lower surface of the CEV cantilever curved beam, with strain sensors (0, n) located at 

the fixed end ( x = x0 = 0 ) and at the free end ( x = xn = l ), respectively.

The CEV cantilever curved beam has a curved length of l = 94.36 in., a curved-beam angle of

φn = 22°, and a unit width. Table 6 provides the complete dimensions of the CEV cantilever curved beam, 

including the dimensions of an equivalent cantilever straight beam (same length, depth, and width as the 

CEV cantilever curved beam). 

Table 6. Dimensions of CEV cantilever curved beam and equivalent straight beam.

Beam l, in. φn , deg R, in. h, in. c, in.

CEV cantilever 94.3612 22.0 254.75 0.50 0.25

Equivalent straight 94.3612 0.0 ∞ 0.50 0.25

Based on the geometrical data provided in table 6, the radius of curvature factor, R c , for the CEV 

curved beam is R c =

 

983, which is far greater than the curvature-effect limit of R c = 17 (fig. 2),

beyond which the curvature effect diminishes. Thus in light of figure 2, the CEV curved beam can be 

classified as a very shallow curved beam.

Also based on the data in table 6, the neutral axis offset, δ , of the CEV curved beam was calculated 

from equation (16) to be δ = 0.049754 in., which gives the normalized neutral axis offset of 

 

δ /R = 

0.000202. This value is far too small to appear in table 1. Thus the neutral axis of the CEV curved beam 

is practically coincidental with the centroidal axis (see fig. 2). Therefore, in the calculations of theoretical 

deflection curves, the neutral axis offset effect was neglected.
23

Strain Curves

Figure 17 shows the curves for the bending strains, ε i , generated from the SPAR program, for the 

lower surface of the CEV cantilever curved beam and the equivalent straight beam. The strain curve for 

the CEV cantilever curved beam is very close to that of the equivalent straight beam, for which the strain 

curve is a straight line with maximum strain at the built-in end and tapering down to zero at the beam tip.

Deflection Curves

The strain data, generated from the SPAR program, of figure 17 were used as inputs to the cantilever

curved-beam deflection equation (19) for deflection calculations for the CEV cantilever curved beam. 

Table 7 compares the calculated deflections with the deflections generated from the SPAR program for 

the CEV cantilever curved beam. 

Table 7. Comparison of deflections calculated from SPAR with those calculated from curved-beam 

deflection equation (19) for the CEV cantilever curved beam.



yi

B



y0

B

(fixed 

end)



y

1

B 

y2

B 

y3

B 

y4

B 

y5

B 

y6

B 

y7

B 

y8

B

(free end)

SPAR 0.0000 0.1747 0.6687 1.4375 2.4335 3.6096 4.9166 6.3055 7.7265

Eq. (19), 

curved 0.0000 0.1743 0.6675 1.4343 2.4289 3.5986 4.9171 6.3183 7.7624

Difference, 

percent 0.0000 0.2290 0.1795 0.2225 0.1890 0.3047 0.0102 0.2030 0.4646

For the prediction error comparison, table 8 compares the deflections calculated from the straightbeam deflection equation (10) with those calculated from the SPAR program for the equivalent cantilever 

straight beam.

Table 8. Comparison of deflections calculated from SPAR with those calculated from straight-beam 

deflection equation (10) for the equivalent straight cantilever beam.

yi

y0

(fixed 

end)

y

1 y2 y3 y4 y5 y6 y7

y8

(free end)

SPAR 0.0000 0.1787 0.6834 1.4684 2.4863 3.6910 5.0355 6.4734 7.9578

Eq. (10), 

straight 0.0000 0.1788 0.6841 1.4694 2.4881 3.6934 5.0388 6.4776 7.9631

Difference, 

percent 0.0000 0.0560 0.1024 0.0681 0.0724 0.0650 0.0655 0.0649 0.0666

Note from table 7 that based on the cantilever curved-beam deflection equation (19), the prediction 

error at the tip of the CEV cantilever curved beam is only 0.4646 percent. For the equivalent straight 

cantilever beam (table 8), the prediction error at the beam tip based on equation (10) is even smaller, in 

the amount of 0.0666 percent.
24

The deflection data of tables 7 and 8 are plotted in figure 18 for visual comparison. The deflection 

curves, calculated from the cantilever curved-beam deflection equation (19) and from the straight-beam 

deflection equation (10), fall practically on top of the corresponding SPAR deflection curves. The small 

number of prediction errors listed in tables 7 and 8 are pictorially inconspicuous in figure 18. The 

excellent agreement between the theoretical deflection curves and the corresponding SPAR deflection 

curves validates the high accuracy of both the cantilever curved-beam deflection equation (19) and the 

cantilever straight-beam deflection equation (10).

SHAPE PREDICTIONS OF CREW EXPLORATION VEHICLE 

TWO–POINT SUPPORTED CURVED BEAMS

Figure 19 shows the CEV two-point supported curved beam, which is a cutout of the generic CEV 

windward exterior wall along its diameter (rounded shoulder regions excluded). The CEV two-point 

supported curved beam is a union of the CEV cantilever curved beam and its mirror image. An inward 

radial load of P = 2 lb is applied at the beam center (symmetrical loading condition). The CEV two-point 

supported curved beam has a curved length of l = 188.72 in., which subtends a curved-beam angle of 

φn = 44°, and has a unit width. Table 9 provides the complete dimensions of the CEV two-point 

supported curved beam and an equivalent straight beam (same length and depth as the CEV curved 

beam).

Table 9. Dimensions of CEV two-point supported curved beam and equivalent straight beam.

Beam l, in. φn , deg R, in. h, in. c, in.

CEV full 188.72 44.0 254.75 0.50 0.25

Straight 188.72 0.0 ∞ 0.50 0.25

There are n + 1 (n = 16) equally spaced strain sensors (i = 0, 1, 2, 3, ..., n) installed on the curvedbeam convex surface. The corresponding strain-sensing stations (not shown) for the curved-beam concave 

surface also are needed so that a pair of the convex and concave surface strains can be averaged to 

eliminate possible axial strain components caused by beam curvature and stationary supports.

In the shape prediction analysis of the CEV two-point supported curved beam, the following four 

types of support conditions were considered (fig. 19):

1. Both ends clamped and free to move along beam curve (CC-free)

2. Both ends clamped and stationary (CC-fixed)

3. Both ends simply supported and free to move along beam curve (SS-free) 

4. Both ends simply supported and stationary (SS-fixed) 

As discussed previously, the neutral axis of the CEV curved beam is practically coincidental with the 

centroidal axis (that is, δ ≈ 0 , fig. 2). Therefore, like the CEV cantilever curved-beam case, the neutral

axis offset effect was ignored in the calculations of theoretical deflection curves of the CEV two-point 

supported curved beam.

Both Ends Clamped 

Figure 20 shows the deformed shapes of the CEV clamped curved beam subjected to an inward radial 

central force of P = 2 lb. For the CC-free case (fig. 20a), the deformed shape is slightly wavy, because the 

supports are allowed to move tangentially. For the CC-fixed case (fig. 20b), the deformed shape is wavier

because of stationary support constraints.
25

Because the CEV curved-beam angle, φn = 44°, falls within the range, 0 ≤ φn ≤ 225°, the clamped 

curved-beam deflection equation (20b) was used for the shape predictions of the CEV curved beam with 

both CC-free and CC-fixed support conditions. Equation (20b) is rewritten as



yi

B = yi − i

n

⎛

⎝

⎜

⎞

⎠

⎟

2

+

4

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

2

sin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0 at right

support (i = n). 



 ; (i = 1,2,3,...,n) (22)

Case of Both Ends Clamped and Free to Move Along Beam Curve (CC-Free)

Figure 21 shows the strains, ε i , generated from the SPAR program, for the CEV two-point supported 

curved beam and the equivalent straight beam under the CC-free support condition. The two strain curves 

are V-shaped and very close, indicating that the CEV curved beam behaves almost like a straight beam 

under the CC-free support condition. The strain data, generated from the SPAR program, of figure 21 

were used to calculate the deflections for the CEV CC-free curved beam and CC-free equivalent straight 

beam using deflection equations (22) and (10), respectively. Table 10 compares the deflections 

calculated from the SPAR program with those calculated from the curved-beam deflection equation (22)

for the CEV CC-free curved beam.

Table 10. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (22) for the CEV CC-free curved beam.

 



yi

B 

y0

B

(left end)



y

1

B 

y2

B 

y3

B 

y4

B 

y5

B 

y6

B 

y7

B 

y8

B

(center)

SPAR 0.0000 0.0882 0.3150 0.6351 1.0021 1.3693 1.6898 1.9164 2.0025

Eq. (22), 

curved 0.0000 0.0838 0.3056 0.6200 0.9812 1.3427 1.6581 1.8811 1.9655

Difference, 

percent 0.0000 4.9887 2.9841 2.3776 2.0856 1.9426 1.8760 1.8420 1.8477

|<----------------------------------------------Left half of CEV CC-free curved beam--------------------------------------->

Table 11 compares the deflections calculated from the SPAR program with those calculated from the 

straight-beam deflection equation (15) for the CC-free equivalent straight beam. Note from tables 10 and 

11 that the prediction error at the center ( y8 ) of the CEV CC-free curved beam is only 1.8477 percent 

compared with the miniscule prediction error of 0.2471 percent at the center ( y8 ) of the CC-free 

equivalent straight beam.
26

Table 11. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (15) for the CC-free equivalent straight beam.



yi

B 

y0

B

(left end)



y

1

B 

y2

B 

y3

B 

y4

B 

y5

B 

y6

B 

y7

B 

y8

B

(center)

SPAR 0.0000 0.0868 0.3121 0.6298 0.9936 1.3571 1.6741 1.8982 1.9833

Eq. (15), 

straight 0.0000 0.0852 0.3103 0.6287 0.9939 1.3593 1.6779 1.9031 1.9882

Difference, 

percent 0.0000 1.8433 0.5767 0.1747 0.0302 0.1621 0.2270 0.2581 0.2471

 |<---------------------------------------------Left half of CEV CC-free straight beam--------------------------------------->

Figure 22 shows the deflection curves calculated from the data of table 10 for the CEV CC-free

curved beam. Because of graphical proximity, and to avoid plot congestion, the deflection curve for the 

equivalent straight beam was not plotted from the data in table 11. The predicted deflection curve based 

on equation (22) is represented by a solid line with solid circular symbols, whereas the deflection curve 

calculated from the SPAR program is represented by a dotted line with open circular symbols. Figure 22 

also shows the deflection curve, 

yi

B = yi , calculated from the straight-beam deflection equation (10), 

which is represented by a dashed-dotted line with solid triangular symbols. Note that the deflection curve,



yi

B = yi , gives relatively good shape prediction on the left-half region of the beam. The prediction error, 

however, gradually increases beyond the midpoint of the beam and reaches a maximum value of 

yn = 0.241682 in. at the right support. With the curvature effect introduced (eq. (22)), the predicted 

deflection curve converges to the SPAR deflection curve. The accuracy of the shape predictions may be 

optimized by slightly altering the coefficient (4/p) of the sine correction function in equation (22).

Case of Both Ends Clamped and Stationary (CC-Fixed)

Figure 23 shows the strains, ε i , generated from the SPAR program, for the CEV two-point supported 

curved beam under the CC-fixed support condition. Because of stationary supports, the strain curve has a 

rounded M-shape. The strain data of figure 23 were used to calculate the deflections using the curvedbeam deflection equation (22). Table 12 compares the deflections calculated from the SPAR program 

with those calculated from the curved-beam deflection equation (22) for the CEV CC-fixed curved beam.

Note that the prediction error at the center ( 

y8

B ) (maximum deflection point) of the CEV CC-fixed curved 

beam is only 0.2349 percent, showing the degree of accuracy of deflection equation (22) formulated for 

clamped curved beams.
27

Table 12. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (22) for the CEV CC-fixed curved beam.



yi

B 

y0

B

(left end)



y

1

B 

y2

B 

y3

B 

y4

B 

y5

B 

y6

B 

y7

B 

y8

B

(center)

SPAR 0.0000 –0.0168 –0.0457 –0.0621 –0.0529 –0.0157 –0.0421 0.0998 0.1277

Eq. (22) 0.0000 –0.0167 –0.0454 –0.0619 –0.0525 –0.0149 0.0425 0.1000 0.1274

Difference, 

percent 0.0000 0.5952 0.6565 0.3221 0.7561 2.6144 0.9501 0.2004 0.2349

 |<-------------------------------------------Left half of CEV CC-fixed curved beam -------------------------------------->

Note: Inward deflection is considered positive; outward deflection is negative.

Figure 24 shows the deflection curves for the CEV CC-fixed curved beam plotted from the data listed 

in table 12. The deflection curve calculated from equation (22) is represented by a solid line with solid 

circular symbols, whereas the deflection curve calculated from the SPAR program is represented by a 

dotted line with open circular symbols. Also, the deflection curve, 

yi

B = yi , calculated from the straightbeam deflection equation (10) is represented by a dashed-dotted line with solid triangular symbols. Note 

that the straight-beam deflection equation (10) induces a maximum prediction error of yn = –0.097526 in. 

at the right support. With the curvature effect considered, the deflection curve calculated from 

equation (22) pictorially agrees with the SPAR deflection curve, confirming the high accuracy of the 

curved-beam deflection equation (22) formulated for clamped curved beams. 

Both Ends Simply Supported

Figure 25 shows the deformed shapes, generated from the SPAR program, of the CEV simply 

supported curved beam under an inward radial central load of P = 2 lb. For the SS-free case (fig. 25a), the 

deformation is a shallow bow shape, because the supports are allowed to move tangentially. For the 

SS-fixed case (fig. 25b), the deformed shape is wavier because of the stationary support points.

Because the CEV two-point supported curved beam has a curved-beam angle (φn = 44°) very close 

to φn = 45°, in the calculations of the deflections for both the SS-free and SS-fixed cases, the curvedbeam deflection equation (20b) (φn = 45°), rewritten as equation (23), was used:



yi

B = yi − i

n

+

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

2

sin

i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



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

Case of Both Ends Simply Supported and Free to Move Along Beam Curve (SS-Free)

Figure 26 shows the strains, ε i , generated from the SPAR program, for the CEV two-point supported 

curved beam and for the equivalent straight beam under the SS-free support condition. The two strain 

curves are V-shaped and are fairly close, but not as close as those in the CC-free support case (fig. 21). 

The strain data of figure 26 were used to calculate the deflection curves for the CEV SS-free curved beam 
28

using deflection equation (23). Table 13 compares the deflections calculated from the SPAR program 

with those calculated from the curved-beam deflection equation (23) for the CEV SS-free curved beam.

Table 13. Comparison of deflections calculated from SPAR with those calculated from deflection 

equation (23) for the CEV SS-free curved beam.



yi

B 

y0

B

(left end)

 

y

1

B 

y2

B 

y3

B

 



y4

B 

y5

B 

y6

B 

y7

B 

y8

B

 (center)

SPAR 0.0000 1.6723 3.2909 4.8022 6.1534 7.2916 8.1664 8.7264 8.9242

Eq. (23), 

straight 0.0000 1.6566 3.2600 4.7573 6.0960 7.2242 8.0911 8.6465 8.8419

Difference, 

percent 0.0000 0.9388 0.9390 0.9350 0.9328 0.9244 0.9221 0.9156 0.9222

 |<-------------------------------------------Left half of CEV SS-free curved beam---------------------------------------->

The strain data of figure 26 were used as inputs to calculate the deflections for the SS-free equivalent 

straight beam, using both the SPAR program and deflection equation (14) (for simply supported straight 

beams). Table 14 compares these deflections. Note from tables 13 and 14 that the prediction error of 

0.9222 percent at the center ( 

y8

B ) of the SS-free CEV curved beam is slightly larger than the 

corresponding prediction error of 0.8214 percent for the SS-free equivalent straight beam.

Table 14. Comparison of deflections calculated from SPAR with those calculated from deflection

equation (14) for SS-free equivalent straight beams.



yi

B 

y0

B

(left end)

 

y

1

B 

y2

B 

y3

B 

y4

B 

y5

B 

y6

B 

y7

B 

y8

B

 (center)

SPAR 0.0000 1.4737 2.9011 4.2356 5.4319 6.4425 7.2220 7.7232 7.9010

Eq. (14) 0.0000 1.4865 2.9262 4.2721 5.4779 6.4969 7.2823 7.7874 7.9659

Difference, 

percent 0.0000 0.8686 0.7652 0.8617 0.8468 0.8444 0.8349 0.8313 0.8214

 |<-----------------------------------------------Left half of SS-free equivalent straight beam------------------------------>

Figure 27 shows the deflection curves based on the data listed in table 13 for the CEV SS-free curved 

beam. Because of data proximity and to avoid plot congestion, deflection curves for the SS-free 

equivalent straight beam were not plotted from the data in table 14. The predicted deflection curve based 

on the curved-beam deflection equation (23) is represented by a solid line with solid circular symbols, 

whereas the deflection curve calculated from the SPAR program is represented by a dotted line with open 

circular symbols. The proximity of the predicted and SPAR deflection curves validates the accuracy of 

the deflection equation (23) formulated for simply supported curved beams.

For comparison, figure 27 also shows the deflection curve, 

yi

B = yiB = yi − (i n) yn (eq. (14) for 

simply supported straight beams), represented by a dashed-dotted line with solid triangular symbols, to 

demonstrate the prediction accuracy. Note that for the SS-free support case, the deflection curve 

calculated from deflection equation (14) lies slightly under the SPAR deflection curve in the central 

region of the beam. 
29

Case of Both Ends Simply Supported and Stationary (SS-Fixed)

Figure 28 shows the strains, ε i , generated from the SPAR program, for the CEV two-point supported 

curved beam under the SS-fixed support condition. The strain curve has a distorted M-shape instead of a 

V-shape (as in the straight-beam case) because of the curvature effect and support constraints. The strain 

data of figure 28 were used to calculate the deflections for the CEV SS-fixed curved beam using the 

SPAR program and the curved-beam deflection equation (23). Table 15 presents the resulting deflection 

data. The radial inward deflections are defined as positive, and the outward deflections are negative.

Table 15. Comparison of deflections calculated from SPAR with those calculated from curved-beam 

deflection equation (23) for the CEV SS-fixed curved beam.



yi

B 

y0

B

(left end)



y

1

B 

y2

B 

y3

B 

y4

B 

y5

B 

y6

B 

y7

B 

y8

B

(center)

SPAR 0.0000 0.0882 0.3150 0.6351 1.0021 1.3693 1.6898 1.9164 2.0025

Eq. (22),

curved 0.0000 0.0838 0.3056 0.6200 0.9812 1.3427 1.6581 1.8811 1.9655

Difference, 

percent 0.0000 439887 2.9841 2.3776 2.0856 1.9426 1.8760 1.8420 1.8477

 |<----------------------------------------------Left half of CEV SS-fixed curved beam-------------------------------------->

Note: Inward deflection is considered positive; outward deflection is negative.

Figure 29 shows the deflection curves based on the data listed in table 15 for the CEV SS-fixed

curved beam. The theoretical deflection curve based on the curved-beam deflection equation (23) (solid 

line with solid circular symbols) is fairly close to the SPAR deflection curve (dotted line with open 

circular symbols). In figure 29, the deflection curve, 

yi

B = yiB = yi − (i n) yn , calculated from the 

straight-beam deflection equation (14) (dashed-dotted line with solid triangular symbols) somewhat 

overpredicts the deflections in the central region of the beam.

To improve the shape predictions for the SS-fixed case, the curved-beam deflection equation (23) can

be modified to the following form: 



yi

B = yi − i

n

+

6

π

φn

360

⎛

⎝

⎜

⎞

⎠

⎟

2

sin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

yn

Correction term to enforce yn

B = 0

at right support (i = n). 



 ; (i = 1,2,3,...,n) (24)

Based on the modified curved-beam deflection equation (24), the deflections of the CEV SS-fixed curved 

beam were recalculated, and the resulting deflection data are listed in table 16. 
30

Table 16. Comparison of deflections calculated from SPAR with those calculated from modified 

curved-beam deflection equation (24) for the CEV SS-fixed curved beam.



yi

B 

y0

B

(left end)



y

1

B 

y2

B 

y3

B 

y4

B 

y5

B 

y6

B 

y7

B 

y8

B

(center)

SPAR 0.0000 –0.0568 –0.0913 –0.0925 –0.0581 0.0059 0.0849 0.1560 0.1880

Eq. (24),

curved 0.0000 –0.0570 –0.0918 0.0930 –0.0588 0.0046 0.0832 0.1545 0.1868

Difference, 

percent 0.0000 0.3521 0.5476 0.5405 1.2048 22.0339* 2.0024 0.9615 0.6383

|<--------------------------------------------Left half of CEV SS-fixed curved beam---------------------------------------->

Note: Inward deflection is positive; outward deflection is negative.

*Caused by a small number divided by a small number.

Note from table 16 that when the modified deflection equation (24) is used, the prediction error at the 

beam center ( y8 ) for the SS-fixed case could be reduced to nearly one-tenth (1/10) that when equation 

(23) is used (table 15). In table 16, the large error of 22.0339 percent for the deflection, 

y5

B , is caused by 

a small number divided by another small number, and it is insignificant.

Figure 30 compares the deflection curves calculated from the modified curved-beam deflection 

equation (24) with those calculated from the SPAR program using the data listed in table 16. The 

predicted deflection curve practically falls on top of the SPAR deflection curve, demonstrating the 

accuracy of the modified curved-beam deflection equation (24) established for the CEV SS-fixed curved 

beam. 

SINGLE–POINT COLLOCATIONS

The curved-beam deflection equations (20) and (21) formulated for two-point supported curved 

beams could provide sufficiently accurate shape predictions. For further optimization of shape prediction 

accuracy, however, the coefficients of the sine correction functions appearing in the curved-beam 

deflection equations (20) and (21) can be modified slightly by applying the single-point collocation 

method discussed in this section.

Clamped Curved Beams

As shown in figure 11, the deflection curves generated from the curved-beam deflection equation (20) 

practically agree with the respective SPAR deflection curves (references). For optimum prediction 

accuracy, however, a single-point collocation method can be applied to further improve the correction 

functions in equation (20). Equations (20b–c) can be unified into a general form for 0 ≤ φn ≤ 360 ° arcs as

 (25) 

yi

B = yi − i

n

⎛

⎝

⎜

⎞

⎠

⎟

2

yn

= yi

B(eq. (20a))



+ ηsin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

Curvature-effect

correction term



= yi

B +ηsin4 i

n

π ⎛

⎝

⎜

⎞

⎠

⎟ ; (i = 1,2,3,...,n)
31

in which equation (20a) is used, and 

 

η (in.) is the amplitude of the sine correction function. When the 

single-point collocation method was applied, the collocation point was chosen to be at the midpoint of the 

beam (load application point) where the deflection is at a maximum.

Let { yn/2

C , yn/2B } be the deflections at the center of the two-point supported curved beam calculated 

from the SPAR program and from the two-point supported straight-beam deflection equation (20a) or 

(21a), respectively, then the midpoint deflection differential ( yn/2

C − yn/2B ) will give the value of 

 

η. 

Namely, 

η ≡ yn/2

C − yn/2B (26)

Table 17 lists the values of 

 

η calculated from equation (26) for all the clamped curved beams. 

Table 17. Central deflections and amplitude of sine correction functions, η , for different clamped curved 

beams; n = 16.

φn , deg yn/2

C , in.

(SPAR)

yn/2

B , in.

( = yi − (i / n)

2

yn )

η , in.

( yn/2

C − yn/2B ) η / yn/2

C , percent

0 (straight) 2.35078 2.365080 –0.014300* –0.6083**

45 0.15144 0.147520 0.003920 2.5885

90 (1/4 circle) 0.15317 0.143866 0.009304 6.0743

135 0.15974 0.137798 0.021942 13.7361

180 (1/2 circle) 0.17012 0.126289 0.043831 25.7648

225 0.18499 0.105753 0.079237 42.8331

270 (3/4 circle) 0.20548 0.087697 0.117783 52.4543

315 0.23336 0.053076 0.180284 77.2557

360 (full circle) 0.27090 –0.001092 0.271992 100.4031

*Negative for straight beam.

** Negligible.

Note in table 17 that the value of η for the straight beam (φn = 0 ) is negative, because the right-end 

deflection, yn , is positive (fig. 11a). Also note that the magnitude of η for the straight beam (φn = 0 ) is 

greater than those for the φn = 45 –90° curved beams. When η is normalized by the corresponding SPAR 

central deflection, yn/2

C , however, the magnitude of η / yn/2C for the straight beam (φn = 0 ) becomes a 

negligible minimum among all cases.

Figure 31 shows the plot of the η data listed in table 17 as a function of the curved-beam angle, φn . 

When the η -curve of figure 31 is used, the value of η for any curved-beam angle, φn , can be established 

graphically. When the values of η listed in table 17 are used, the deflection curves calculated from the 

general curved-beam deflection equation (25) match perfectly with the corresponding SPAR deflection 
32

curves of figure 11 for all the curvature angles ( 0 ≤ φn ≤ 360 °). For graphical visualization of the 

deflection curves, only a typical case of a φn = 180 ° clamped curved beam is shown in figure 32. Note 

that the deflection curve, 

yi

B = yiB = yi − (i n)

2

yn (eq. (20a) for clamped straight beams), lies somewhat 

below the SPAR deflection curve in the center region of the beam. With the single-point collocation, the 

corrected deflection curve calculated from equation (25) matches precisely with the SPAR deflection 

curve. Note that the values of η can also be determined with the aid of experimentally generated 

deflection curves instead of using the deflection curves generated from the SPAR program.

Simply Supported Curved Beams

The displacement equations (21b–g) for the simply supported curved beam can also be unified into a

general form, with η representing the amplitude of sine correction functions for all the curved-beam 

angles ( 0 ≤ φn ≤ 360 ), as



yi

B = yi − i

n

yn

= yi

B (eq. (21a))



+ ηsinm i

n

π ⎛

⎝

⎜

⎞

⎠

⎟

Curvature-effect

correction term 



= yi

B +ηsinm i

n

π ⎛

⎝

⎜

⎞

⎠

⎟ ; (i = 1,2,3,...,n) (27)

in which equation (21a) was used, and the values of the exponent m of the sine correction function is 

m = 1, 3, or 4, depending on the value of φn (eqs. (21b–g)). The values of η for the simply supported 

curved beams were calculated from equation (26) using the single-point collocation method described in 

the clamped curved-beam cases. Table 18 lists these values.

Table 18. Central deflections, amplitudes, and exponents {η,m } of the sine correction functions for 

different simply supported curved beams; n = 16.

φn , deg yn/2

C , in.

(SPAR)

yn/2

B , in.

( = yi − (i / n)

2

yn )

η , in.

( yn/2

C − yn/2B )

m η / yn/2

C , 

percent

0 (straight) 9.40818 9.479265 –0.071085* ----- –0.7556**

45 0.21423 0.229521 –0.015291* 1 –7.1377

90 (1/4 circle) 0.22723 0.233778 –0.006548* 1 –2.8817

135 0.24530 0.225642 0.019658 4 8.0139

180 (1/2 circle) 0.27246 0.213297 0.059163 4 21.7144

225 0.31359 0.186855 0.126735 4 40.4142

270 (3/4 circle) 0.37707 0.135262 0.241808 4 64.1281

315 0.47909 0.039780 0.439310 3 91.6968

360 (full circle) 0.65333 –0.157195 0.810525 3 124.0606

*Negative for φn = 0–90°.

**Negligible.
33

The values of η listed in table 18 are plotted in figure 33 for visualization of the shape of the η

curve from which the values of η for other unlisted curved-beam angles, φn , can be easily obtained 

graphically. In table 18, the magnitude of η for the straight beam (φn = 0 ) appears to be greater than 

those for the φn = 45 –180° curved beams. When η is normalized by the corresponding SPAR central 

deflection, yn/2

C , however, the magnitude of η / yn/2C for the straight beam ( φn = 0 ) becomes a 

negligible minimum among all cases.

When the values of η listed in table 18 for the general curved-beam deflection equation (27) are 

used, the calculated deflection curves practically coincide with the corresponding SPAR deflection curves 

of figure 14. For graphical visualization of the deflection curves, only a typical case of φn = 180 simply 

supported curved beam is shown in figure 34. Note that the deflection curve, 

yi

B = yiB = yi − (i n) yn

(eq. (21a) for simply supported straight beams), lies somewhat below the SPAR deflection curve in the 

center region of the beam. With the single-point collocation, the corrected deflection curve based on 

equation (27) matches precisely with the SPAR deflection curve. In actual shape predictions, the values of 

η can be determined in the light of actual measured deflection curves instead of using the SPAR 

deflection curves.

STRAIN–SENSING STATION DENSITY

It is important to mention that when fiber-optic strain-sensing systems are used, the strain-sensing

station density (value of n) can be easily increased at will by simply increasing the multiplexed Bragg 

grating density (that is, reducing the sensing intervals; ref. 3) without the need for time-consuming 

installations of additional strain sensors, as is the case with using conventional strain gages. The effect of 

strain-sensing station density on the shape prediction accuracy was investigated by increasing the value of 

n from 16 to 32. Increasing the number of strain-sensing stations implies the shrinking of the divided 

beam subdomains (that is, shrinking the sensing intervals), causing the theoretical piece-wise linear beam 

to approach the actual curved beam. Therefore, one might expect the shape prediction accuracy to 

improve, but the overall benefit was found to be miniscule.

For the cantilever curved beams, the shape prediction accuracy remained practically the same when 

the value of n was increased from 16 to 32. Likewise, for the two-point supported curved beams, the 

prediction accuracy was insensitive to the change in the n value from 16 to 32 as described in detail in the 

following section.

Clamped Curved Beams

Figure 35 shows the bending strain curves, generated from the SPAR program, for the clamped 

curved-beam cases ( 0 ≤ φn ≤ 360 ) based on n = 32. These strain curves are very similar to the strain 

curves based on n = 16 (fig. 10). The strain data of figure 35 were used to calculate the amplitude, η , of 

the sine correction functions for n = 32 from the single-point collocation equation (26) for different 

clamped curved-beam cases. Table 19 lists these values and also lists the corresponding values of η for 

n = 16 (taken from table 17) for comparison.
34

Table 19. Central deflections and amplitudes of the sine correction function, η , calculated for different 

clamped curved beams; n = 32.

φn , deg yn/2

C , in.

(SPAR)

yn/2

B , in.

( ≡ yi − (i / n)

2

yn )

η , in.

( ≡ yn/2

C − yn/2B )

(n = 32)

η , in.

(table 17)

(n = 16)

0 (straight) 2.35078 2.370436 –0.019656* –0.014300*

45 0.15144 0.144950 0.006490 0.003920

90 (1/4 circle) 0.15317 0.144379 0.008791 0.009304

135 0.15974 0.137835 0.021905 0.021942

180 (1/2 circle) 0.17012 0.127108 0.043012 0.043822

225 0.18499 0.110216 0.074774 0.079237

270 (3/4 circle) 0.20548 0.089372 0.116108 0.117783

315 0.23336 0.053559 0.179801 0.180284

360 (full circle) 0.27090 –0.001129 0.272029 0.271992

* Negative for the straight beam.

Table 19 shows that η is nearly insensitive to the change in strain-sensing station density (value of n). 

Figure 36 shows the η curves based on the data listed in table 19 for the clamped curved beams. The 

η curves for n = 16 and n = 32 are extremely close, indicating that prediction accuracy is insensitive to 

the change in strain-sensing station density (value of n).

Figure 37 shows the deflection curves of different clamped curved beams based on n = 32. The 

corresponding deflection curves based on n = 16 (taken from fig. 33) also are plotted for comparison. 

Note that increasing the value of n from 16 to 32 did move the deflection curves, 

yi

B = yi (eq. (10) for 

cantilever straight beams), upward and thereby reduced the magnitudes of the right-end deflections, yn . 

The deflection curves,



yi

B = yiB = yi − (i n)

2

yn (eq. (20a) for clamped straight beams), based on n = 16 

and n = 32 for each curved-beam angle, however, turned out to be extremely close, and their difference is 

graphically indistinguishable. Note in figure 37 that the single-point collocation deflection curves 

calculated from equation (25) based on n = 32 are practically coincidental with the corresponding SPAR 

deflection curves for the whole range of curved-beam angles ( 0 ≤ φn ≤ 360 ) (see fig. 32 for the n = 16

case). Based on these observations, using n = 16 could be considered adequate in the present shape 

prediction analysis of the clamped curved beams (see ref. 4).

Simply Supported Curved Beams

Figure 38 shows the bending strain curves, generated from the SPAR program, for the simply 

supported curved-beam cases ( 0 ≤ φn ≤ 360 ) for n = 32. Again, these strain curves are practically 

identical to the strain curves for n = 16 (fig. 13). The strain data of figure 38 were used to calculate the 

amplitude of the sine correction function, η , for n = 32 from the single-point collocation equation (26) 

for different simply supported curved beams. Table 20 lists these values and also lists the values of η

based on n = 16 (taken from table 18) for comparison.
35

Table 20. Central deflections, amplitudes, and exponents {η , m} of the sine correction function for 

different simply supported curved beams; n = 32.

φn , deg yn/2

C

, in.

(SPAR)

yn/2

B

, in.

( ≡ yi − (i / n)yn )

η , in.

( ≡ yn/2

C − yn/2B )

(n = 32) (n = 16)

η , in.

(table 18)

0 (straight) 9.40818* 9.41767 –0.066570** –0.071085**

45 0.21423 0.216002 –0.001772** –0.015291**

90 (1/4 circle) 0.22723 0.217717 –0.009513** –0.006548**

135 0.24530 0.211281 0.034019 0.019658

180 (1/2 circle) 0.27246 0.197559 0.074901 0.059163

225 0.31359 0.170597 0.142992 0.126735

270 (3/4 circle) 0.37707 0.120870 0.256200 0.241808

315 0.47909 0.026165 0.452925 0.439310

360 (full circle) 0.65333 –0.172672 0.826002 0.810525

*Very much larger compared with curved-beam cases.

**Negative for φn = 0–90°. 

Figure 39 compares the η curves for n = 16 and n = 32, plotted from the data listed in table 20, for 

the simply supported curved beams. The two η curves are quite close, indicating that the prediction 

accuracy is insensitive to the change in strain-sensing station density (value of n).

In figure 40, the deflection curves calculated from 

yi

B = yiB = yi − (i n) yn (eq. (21a) for simply 

supported straight beams), based on n = 32 for different simply supported curved beams, are compared 

with the corresponding deflection curves, 

yi

B = yiB = yi − (i n) yn , based on n = 16 (taken from fig. 14). 

The two deflection curves, 

yi

B = yiB = yi − (i n) yn , based on n = 16 and n = 32 for each curved-beam 

angle are fairly close, but not as close as those in the clamped support cases (fig. 37). Figure 40 also 

shows that the single-point collocation deflection curves calculated from equation (27) based on n = 32 

are pictorially coincidental with the corresponding SPAR deflection curves for curved-beam angles in the 

range 0 ≤ φn ≤ 180 °. For larger curved-beam angles, 225 ≤ φn ≤ 360 , however, the left and right regions 

of the deflection curves based on equation (27) become slightly off from the corresponding SPAR 

deflection curves (see fig. 34 for the n = 16 case). Again, based on these observations, using n = 16 could 

be considered adequate for the present shape prediction analysis of the simply supported curved beams 

(see also ref. 4).

DISCUSSION

In the shape prediction analysis of the CEV curved wall and other curved aerospace structures, using 

the clamped support condition is more realistic than using the simply supported condition. Under the 

clamped support condition, only two mathematical forms of deflection equations (20b–c) are needed for 

the shape predictions of clamped two-point supported curved beams, including a full circular ring.

The values of η listed in tables 19 and 20 are for the central load of P = 2 lb only. For any other 

central load, P, the associated η values (for φn = 0–360°) can be obtained by simply scaling up (or 
36

scaling down) the values of η in tables 19 and 20 in proportion to the magnitude of the new central 

load, P. The present curved-beam shape analysis is for the symmetrical loading condition, which occurs 

in most of the actual curved structures, such as an aircraft fuselage subjected to landing impacts, or a 

bomber fuselage subjected to a side blast wave. For measurements of the radial displacements of the 

fuselage cross section, either a semicircumferential or full-circumferential fiber-optic strain-sensing line 

system can be used (see figure 41 for the landing impact case).

For rare nonsymmetrical loading cases, the mathematical functional forms of the curvature-effect 

correction terms must be properly modified accordingly. Furthermore, the present curved-beam shape 

prediction method can also be applied to shape predictions of irregular-shaped structures such as a generic 

CEV shell, which is the union of a windward shallow spherical shell, a leeward deep spherical shell, a 

shoulder toroidal shell, and a side conical shell. The shape prediction analysis of the CEV shell (circular 

cross section normal to the axis of symmetry, and rounded triangular cross section along the axis of 

symmetry) is currently under way, and the results are expected to be presented in a subsequent report. 

SUMMARY

The Ko displacement theory originally developed for shape predictions of straight beams was further 

extended to shape predictions of curved beams with different curvatures. The surface strains for input to 

the deflection equations for shape predictions were analytically generated from finite-element nodal stress 

outputs. The shape prediction errors resulting from the use of straight-beam deflection equations for 

curved beams were examined in light of the finite-element deflection curves. Mathematical functional 

forms for curvature-effect correction terms were established and incorporated into the straight-beam 

deflection equations for shape predictions of both cantilever curved beams and two-point supported 

curved beams. The newly developed curved-beam deflection equations were applied to shape predictions 

of crew exploration vehicle (CEV) curved beams for accuracy validation. The principal findings of the 

present analytical shape prediction analysis of curved beams are as follows:

1. For both cantilever straight beams (limit case of a curved beam) and two-point supported straight 

beams under tip load, the straight-beam deflection equation is very accurate in shape predictions.

2. When the straight-beam deflection equation was used for shape predictions of cantilever curved 

beams, the beam tip prediction errors were {0.0834, 1.4951, 6.1000, 14.3011, 26.6437} percent 

for cases of curved-beam angles of {0.0, 22.5, 45.0, 67.5, 90.0} degrees, respectively.

3. When the newly established cantilever curved-beam deflection equation was used, the beam tip 

prediction errors could be brought down to {0.0834, 0.0849, 0.2127, 0.0126, 1.4978} percent for

cases of curved-beam angles of {0.0, 22.5, 45.0, 67.5, 90.0} degrees, respectively.

4. For two-point clamped curved beams, two deflection equations were formulated for accurate 

shape predictions of a whole range of curved beams, including the circular ring.

5. For two-point simply supported curved beams, six deflection equations were formulated for 

accurate shape predictions of a whole range of curved beams, including the circular ring.

6. The newly formulated cantilever curved-beam deflection equation was applied to shape 

predictions of the CEV cantilever curved beam with a miniscule prediction error of 0.4646 

percent at the beam tip.
37

7. The modified deflection equations for two-point supported curved beams were applied to shape 

predictions of the CEV two-point supported curved beam with negligible beam center prediction 

errors of {1.8477, 0.2349, 0.9222, 0.6383} percent, respectively, for the support cases of both 

ends clamped and free to move along the beam curve, both ends clamped and stationary, both 

ends simply supported and free to move along the beam curve, and both ends simply supported 

and stationary. 

8. For both cantilever and two-point supported curved beams, the prediction accuracy was found to 

be insensitive to the change in strain-sensing station density from 16 to 32.

9. The single-point collocation method could be used to optimize the shape prediction accuracy of 

two-point supported curved beams. 
38

FIGURES

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

Figure 2. Plots of stress ratios, { kin , kout }, and neutral axis offset factor, δ / R , for curved beams as 

functions of curvature factor, R / c .
39

Figure 3. Cantilever curved beam subjected to a tip load of P = 1 lb.

Figure 4. Bending strains, generated from SPAR, for different cantilever curved beams; P = 1 lb; n = 8.
40

Figure 5. Comparison of deflection curves calculated from SPAR with those calculated from 

straight-beam deflection equation (10) for different cantilever curved beams; P = 1 lb; n = 8.

Figure 6. Comparison of deflection curves calculated from SPAR with those calculated from cantilever 

curved-beam deflection equation (19) for different cantilever curved beams; P = 1 lb; n = 8.
41

Figure 7. Uncorrected and corrected deflection prediction errors, { en / yn

C ,



en / yn

C }, at curved-beam tip 

plotted as functions of curved-beam azimuth angle, φn ; P = 1 lb; n = 8.

Figure 8. Two-point supported curved beam under different support conditions subjected to a central 

upward load of P = 2 lb.
42

Figure 9. Deformed shapes, generated from SPAR, of two-point supported curved beams with different 

curvatures; both ends clamped; P = 2 lb.
43

Figure 10. Bending strains, generated from SPAR, for two-point supported curved beams with different 

curvatures; both ends clamped; P = 2 lb; n = 16.

(a) φn = 0 ̊

Figure 11. Deflection curves calculated for two-point supported curved beams with different curvatures; 

both ends clamped; P = 2 lb; n = 16.
44

(b) φn = 45 ̊

(c) φn = 90 ̊

Figure 11. Continued.
45

(d) φn = 135 ̊

(e) φn = 180 ̊

Figure 11. Continued.
46

(f) φn = 225 ̊

(g) φn = 270 ̊

Figure 11. Continued.
47

(h) φn = 315 ̊

(i) φn = 360 ̊

Figure 11. Concluded.
48

Figure 12. Deformed shapes, generated from SPAR, of two-point supported curved beams with different 

curvatures; both ends simply supported; P = 2 lb.
49

Figure 13. Bending strains, generated from SPAR, for two-point supported curved beams with different 

curvatures; both ends simply supported; P = 2 lb; n = 16.

(a) φn = 0 ̊

Figure 14. Deflection curves calculated for two-point supported curved beams with different curvatures; 

both ends simply supported; P = 2 lb; n = 16.
50

(b) φn = 45 ̊

(c) φn = 90 ̊

Figure 14. Continued.
51

(d) φn = 135 ̊

(e) φn = 180 ̊

Figure 14. Continued.
52

(f) φn = 225 ̊

(g) φn = 270 ̊

Figure 14. Continued.
53

(h) φn = 315 ̊

(i) φn = 360 ̊

Figure 14. Concluded.
54

Figure 15. Generic crew exploration vehicle fabricated with dual sandwich wall structures.

Figure 16. Crew exploration vehicle cantilever curved beam subjected to a tip load of P = 1 lb.
55

Figure 17. Bending strains, generated from SPAR, for crew exploration vehicle cantilever curved beam 

and equivalent straight beam; P = 1 lb; n = 8.

Figure 18. Comparison of SPAR deflection curves with corresponding predicted deflection curves 

calculated from deflection equations (10) and (19) for crew exploration vehicle cantilever curved beam 

and equivalent straight beam, respectively; P = 1; n = 8 lb.
56

Figure 19. Crew exploration vehicle two-point supported curved beam under different support conditions 

subjected to a central load of P = 2 lb.

(a) Both ends clamped and free to move along beam curve (CC-free).

(b) Both ends clamped and stationary (CC-fixed).

Figure 20. Deformed shapes, generated from SPAR, of crew exploration vehicle two-point supported 

curved beams; both ends clamped; P = 2 lb.
57

Figure 21. Comparison of SPAR bending strains for crew exploration vehicle two-point supported curved 

beam and equivalent straight beam; both ends clamped and free to move along beam curve (CC-free);

P = 2 lb; n = 16.

Figure 22. Comparison of SPAR deflection curve with predicted deflection curve based on curved-beam 

deflection equation (22) for crew exploration vehicle two-point supported curved beam; both ends 

clamped and free to move along beam curve (CC-free); P = 2 lb; n = 16.
58

Figure 23. Bending strains, generated from SPAR, for crew exploration vehicle two-point supported 

curved beam; both ends clamped and stationary (CC-fixed); P = 2 lb; n = 16.

Figure 24. Comparison of SPAR deflection curve with predicted deflection curve based on curved-beam 

deflection equation (22) for crew exploration vehicle two-point supported curved beam; both ends 

clamped and stationary (CC-fixed); P = 2 lb; n = 16.
59

(a) Both ends simply supported and free to move along beam curve (SS-free).

(b) Both ends simply supported and stationary (SS-fixed).

Figure 25. Deformed shapes, generated from SPAR, of crew exploration vehicle two-point supported 

curved beams; both ends simply supported; P = 2 lb.

Figure 26. Comparison of SPAR bending strains for crew exploration vehicle two-point supported curved 

beam and equivalent straight beam; both ends simply supported and free to move along beam curve (SSfree); P = 2 lb; n = 16.
60

Figure 27. Comparison of SPAR deflection curve with predicted deflection curve based on curved-beam 

deflection equation (23) for crew exploration vehicle two-point supported curved beam; both ends simply 

supported and free to move along beam curve (SS-free); P = 2 lb; n = 16.

Figure 28. Bending strains, generated from SPAR, for crew exploration vehicle two-point supported 

curved beam; both ends simply supported and stationary (SS-fixed); P = 2 lb; n = 16.
61

Figure 29. Comparison of SPAR deflection curve with predicted deflection curve based on curved-beam 

deflection equation (23) for crew exploration vehicle two-point supported curved beam; both ends simply 

supported and stationary (SS-fixed); P = 2 lb; n = 16.

Figure 30. Comparison of SPAR deflection curve with predicted deflection curve based on curved-beam 

deflection equation (24) for crew exploration vehicle two-point supported curved beam; both ends simply 

supported and stationary (SS-fixed); P = 2 lb; n = 16.
62

Figure 31. Plot of amplitude of sine correction function, η, as a function of curved-beam angle, φn , for 

the clamped curved beam; n = 16.

Figure 32. Comparison of SPAR deflection curve with single-point collocation deflection curves for the 

clamped curved beam; φn = 180°; P = 2 lb; n = 16.
63

Figure 33. Plot of amplitude of sine correction function, η, as a function of curved-beam angle, φn , for 

the simply supported curved beam; n = 16.

Figure 34. Comparison of SPAR deflection curve with single-point collocation deflection curves for the 

simply supported curved beam; φn = 180°; P = 2 lb; n = 16.
64

Figure 35. Bending strains, generated from SPAR, for the clamped curved beam; P = 2 lb; n = 32.

Figure 36. Plot of amplitude of sine correction function,η, as a function of curved-beam angle, φn , for 

the clamped curved beam; n = 32.
65

(a) φn = 0 ̊

(b) φn = 45 ̊

Figure 37. Deflection curves calculated for two-point supported curved beams with different curvatures; 

both ends clamped; P = 2 lb; n = 32.
66

(c) φn = 90 ̊

(d) φn = 135 ̊

Figure 37. Continued.
67

(e) φn = 180 ̊

(f) φn = 225 ̊

Figure 37. Continued.
68

(g) φn = 270 ̊

(h) φn = 315 ̊

Figure 37. Continued.
69

(i) φn = 360 ̊

Figure 37. Concluded.

Figure 38. Bending strains, generated from SPAR, for the simply supported curved beam; P = 2 lb; 

n = 32.
70

Figure 39. Plot of amplitude of sine correction function, η, as a function of curved-beam angle for the 

simply supported curved beam; n = 32.

(a) φn = 0 ̊

Figure 40. Comparison of SPAR deflection curve with single-point collocation deflection curves for the 

simply supported curved beam; φn = 180°; P = 2 lb; n = 32.
71

(b) φn = 45 ̊

(c) φn = 90 ̊

Figure 40. Continued.
72

(d) φn = 135 ̊

(e) φn = 180 ̊

Figure 40. Continued.
73

(f) φn = 225 ̊

(g) φn = 270 ̊

Figure 40. Continued.
74

(h) φn = 315 ̊

(i) φn = 360 ̊

Figure 40. Concluded.
75

(a) Semicircumferential strain-sensing line system.

(b) Full-circumferential strain-sensing line system.

Figure 41. Strain-sensing systems for measurements of radial displacements of aircraft fuselage cross 

sections caused by landing impacts.
76

REFERENCES

1. Bang, Hyung-Joon, Hyun-Kyu Kang, Chang-Sun Hong, and Chun-Gon Kim, “Optical 

Fiber Sensor Systems for Simultaneous Monitoring of Strain and Fractures in Composites,” 

Smart Materials and Structures, Vol. 14, Sept. 2005, pp. N52–N58.

2. Wehrle, Günther, Percy Nohama, Hypolito José Kalinowski, Pedro Ingácio Torres, and Luiz 

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

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

3. Baldwin, Christopher S., Toni J. Salter, and Jason S. Kiddy, “Static Shape Measurements Using 

A Multiplexed Fiber Bragg Grating Sensor System,” Proceedings of Smart Structures and 

Materials 2004: Smart Sensor Technology and Measurement Systems, edited by Eric Udd and 

Daniele Inaudi, Vol. 5384, San Diego, March 15, 2004, pp. 206–217.

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

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

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

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

2009.

6. Ko, William L., Collection of Memoranda on Applications of Ko Displacement Theory to 

Deformed Shape Predictions of Aerospace Structures, NASA BR-119, Aerostructures Branch, 

Dryden Flight Research Center, Edwards, CA, April 2008.

7. Ko, William L., W. Lance Richards, and Van Tran Fleischer, Applications of Ko Displacement 

Theory to the Deformed Shape Predictions of the Doubly-Tapered Ikhana Wing,

NASA/TP-2009-214652, 2009.

8. Ko, William L., and William Lance Richards, Method for Real-Time Structure Shape-Sensing, 

U.S. Patent No. 7,520,176, issued April 21, 2009. 

9. Richards, William Lance, and William L. Ko, Process for Using Surface Strain Measurements to 

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

May 11, 2010.

10. Roark, Raymond J., Formulas for Stress and Strain, Third Edition, McGraw-Hill Book 

Company, Inc., New York, 1954.

11. Faupel, Joseph, H., Engineering Design: A Synthesis of Stress Analysis and Materials 

Engineering, John Wiley and Sons, Inc., New York, 1964.

12. Whetstone, W.D., SPAR Structural Analysis System Reference Manual, System Level 13A,

Vol. 1, Program Execution, NASA CR-158970-1, 1978.
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

1. REPORT DATE (DD-MM-YYYY)

01-04-2011

2. REPORT TYPE 

Technical Publication

 4. TITLE AND SUBTITLE

Extension of Ko Straight-Beam Discplacement Theory to Deformed Shape 

Predictions of Slender Curved Structures

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Ko, William L. and Fleischer, Van Tran

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, CA 93523-0273

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

H-3073

10. SPONSORING/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

Ko and Fleischer, NASA Dryden Flight Research Center

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -- Unlimited

Subject Category 39 Availability: NASA CASI (443) 757-5802 Distribution: Standard

19b. NAME OF RESPONSIBLE PERSON

STI Help Desk at email: help@sti.nasa.gov

14. ABSTRACT

The Ko displacement theory originally developed for shape predictions of straight beams is extended to shape predictions of curved 

beams. The surface strains needed for shape predictions were analytically generated from finite-element nodal stress outputs. With 

the aid of finite-element displacement outputs, mathematical functional forms for curvature-effect correction terms are established 

and incorporated into straight-beam deflection equations for shape predictions of both cantilever and two-point supported curved 

beams. The newly established deflection equations for cantilever curved beams could provide quite accurate shape predictions for 

different cantilever curved beams, including the quarter-circle cantilever beam. Furthermore, the newly formulated deflection 

equations for two-point supported curved beams could provide accurate shape predictions for a range of two-point supported curved 

beams, including the full-circular ring. Accuracy of the newly developed curved-beam deflection equations is validated through 

shape prediction analysis of curved beams embedded in the windward shallow spherical shell of a generic crew exploration vehicle. 

A single-point collocation method for optimization of shape predictions is discussed in detail.

15. SUBJECT TERMS

Curvature-effect corrections, Curved-beam shape predictions, Deflection equations, Distributed surface strains, Ko Displacement 

Theory

18. NUMBER

 OF 

 PAGES

83

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802

a. REPORT

U

c. THIS PAGE

U

b. ABSTRACT

U

17. LIMITATION OF 

 ABSTRACT

UU

Prescribed by ANSI Std. Z39-18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TP-2011-214657

16. SECURITY CLASSIFICATION OF:

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing 

data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or 

any other aspect of this collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, 

Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware 

that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a 

currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.