February 2012 

NASA/TP–2012-217342 

The Exact Solution for Linear Thermoelastic 

Axisymmetric Deformations of Generally 

Laminated Circular Cylindrical Shells 

Michael P. Nemeth and Marc R. Schultz 

Langley Research Center, Hampton, Virginia 
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 peerreviewed formal professional papers, but having 

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. Englishlanguage translations of foreign scientific and 

technical material pertinent to NASA’s mission. 

 Specialized services also include creating custom 

thesauri, building customized databases, and 

organizing and publishing research results. 

 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
National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23681-2199 

 

February 2012 

NASA/TP–2012-217342 

The Exact Solution for Linear Thermoelastic 

Axisymmetric Deformations of Generally 

Laminated Circular Cylindrical Shells 

Michael P. Nemeth and Marc R. Schultz 

Langley Research Center, Hampton, Virginia
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1

Summary

A detailed exact solution based on the Sanders-Koiter linear shell theory is presented for 

laminated-composite circular cylinders that undergo axisymmetric deformations. The overall 

solution is formulated in a general, systematic way and is obtained from the solution of a single 

fourth-order, nonhomogeneous ordinary differential equation with constant coefficients in which 

the radial displacement is the dependent variable. Moreover, the full response-coupling effects 

associated with generally laminated wall construction are included in the solution. A key element 

of the analysis is the use of the positive-definiteness of the strain energy to define uniquely the 

form of the basis functions spanning the solution space of the ordinary differential equation. 

Loading conditions are considered that include axisymmetric edge loads, surface tractions, and 

temperature fields. Likewise, all possible axisymmetric displacement and traction boundary 

conditions are considered.

Results are presented for five examples that demonstrate a wide range of behavior for 

specially orthotropic and fully anisotropic cylinders. Several of the results illustrate how 

laminated-wall construction, loading and support conditions, and cylinder length affect the size 

of localized zones of bending deformations that typically occur near the cylinder ends. For some 

of the examples, the results indicate highly localized bending deformations that attenuate rapidly. 

For other examples, the bending deformations exhibit very little attenuation. Approximate 

solutions are also given for several of the examples that add credibility to the corresponding 

results obtained from the exact solution and that provide insight into the possible pitfalls of 

making certain types of simplifications in a preliminary-design process. For example, several 

results are presented that illustrate conditions for which sizing a laminated-composite cylinder 

based on an approximate linear membrane analysis is inappropriate.

Results obtained from the exact solution presented herein also have the potential to serve as 

fundamental benchmarks for the validation of numerical solution methods such as the finiteelement method. Likewise, results obtained from the exact solution have the potential to provide 

insight into the extent of discretization that might be needed for analyzing accurately shells with 

zones of localized bending caused by cutouts, stiffeners, or other types of discontinuities and 

gradients.

Symbols

 modified compliances defined by equations (15)

A11, A12, A16, laminate membrane stiffnesses

A22, A26, A66

 modifed laminate stiffnesses defined by equations (12)

b1

(x), b2(x), solution basis functions defined by equations (45)

b3(x), b4(x)

a 11, a 16, a 66

A16, A26, A66
2

 modified compliances defined by equations (15)

B11, B12, B16, laminate membrane-bending coupling stiffnesses

B22, B26, B66

 modified laminate stiffnesses defined by equations (12) and (87)

c1

, c2 shell-theory constants defined in Appendix A

 constants defined by equations (47)

 constants defined by equations (67)

 constants defined by equations (85)

 constants defined by equations (91)

 constants defined by equations (81)

 constants defined by equations (60)-(61)

 thermal constitutive parameters defined by equation (B17)

 modified thermal constitutive parameter defined by equation (101b)

C1, C2, C3 constants defined by equations (19)-(21)

C4(x) function defined by equation (22)

modified compliances defined by equations (15)

D11, D12, D16, laminate bending stiffnesses

D22, D26, D66

e eccentricity parameter defined by equation (25)

E1

, E2 major and minor principal lamina moduli, respectively

G0, G1, Gs

, Gc temperature field constants defined by equation (97b)

G12 lamina shear modulus

b 12, b 62

B16, B26, B66

cN, cT

c1

M

, c2

M

c3

M

, c4

M

c5

M

, c6

M

c3

N

, c4

N

c1

u

, c2

u

, c1

v

, c2

v

cx

Th

, c

Th

, cx

Th

cx

Th

c 12, c 62
3

h cylinder wall thickness

hply lamina ply thickness

H0

, H1, Hs, Hc temperature field constants defined by equation (97a)

K1, K2, K3, K4 unknown solution constants defined by equation (44)

L cylinder length

 thermal constitutive parameters defined by equation (B17)

Mx, M, Mxaxial and circumferential bending and twisting stress resultants,

 respectively

 thermal stress resultants defined by equation (B15)

 stress resultant functions defined by equations (67)

 stress resultant constants defined by equations (67)

 stress resultant functions defined by equations (85)

 stress resultant constants defined by equations (85)

 stress resultant functions defined by equations (91)

 stress resultant constants defined by equations (91)

 applied uniform end moment stress resultant defined by equation (B21)

 thermal constitutive parameters defined by equation (B17)

 modified thermal constitutive parameter defined by equation (101a)

Nx

, N, Nxaxial, circumferential, and shear membrane stress resultants,

respectively

thermal stress resultants defined by equation (B15)

mx

Th

, m

Th

, mx

Th

Mx

Th, MTh, MxTh

Mxk x

Mx5, Mx6

Mk x

M5, M6

Mxk x

Mx5, Mx6

M

nx

Th

, n

Th

, nx

Th

nx

Th

Nx

Th, NTh, NxTh
4

 stress resultant functions defined by equations (80)

 stress resultant constants defined by equations (80)

 constant of integration defined by equation (1a)

 function defined by equation (1b)

applied uniform axial stress resultant defined by equation (B18)

pu

 uniform ullage pressure used in Example 4

P(x) loading function appearing in differential equation (26) and defined by

equations (29)

P0, P1(x) P(x) terms defined by equations (29)

qx(x), q(x), qn

(x) applied axial, circumferential, and normal surface tractions, respectively

 surface traction amplitudes defined by equations (96)

Qx, Qaxial and circumferential transverse-shear stress resultants, respectively

Qxk(x) stress resultant functions defined by equations (69)

Qk

(x) stress resultant functions defined by equations (95)

R mid-surface cylinder radius

Q, S, T constant coefficients of differential equations (26) and (40), defined by 

equations (27), (28), and (39)

 modified shear stress resultant defined by equation (4)

 thermal stress resultant defined by equation (12f)

 value of applied at a cylinder end (see equations (6) and (B19))

Nk x

N5, N6

N0

N1 x

N ̃

qx

0

, q

0

, qn

0

,

qx

1

, q

1

, qn

1

,

qx

s

, q

s

, qn

s

,

qx

c

, q

c

, qn

c

,

Tˆ

T

Th

T

 ̃ Tˆ
5

 constant of integration defined by equation (7a)

 function defined by equation (7b)

U strain-energy density defined by equation (30)

u, v axial and circumferential displacements, respectively

 applied end displacements defined by equations (B18) and (B19)

 constants of integration defined by equations (56) and (57)

 displacement functions defined by equations (56), (58a), and (60)

 displacement constants defined by equations (50)

 displacement functions defined by equations (57), (58b), and (61)

 displacement constants defined by equations (52)

 applied uniform transverse-shear stress resultant defined by equation (B20)

w radial displacement

wH(x) homogeneous solution to equation (40), defined by equation (44)

wp(x) particular solution to equation (40), defined by equations (46) and (47)

 constant and variable parts of wp

(x) defined by equations (46) and (47)

 coefficients of particular solution defined by equations (104)

 coefficients of approximate solution defined by equation (E4)

applied radial end displacement defined by equations (B20)

x axial coordinate of cylinder

T0

T1 x

u, v

u, v

Uk x

U5, U6

Vk x

V5, V6

V

wp

0

, wp x

wp

0

, wp

1

, wp

2

, wp

s1

wp

s2

, wp

c1

, wp

c2

W0, W1

w ̃
6

Y0, Y1, Y2, Ys1, coefficients of P1(x) defined by equations (102)

Ys2, Yc1, Yc2

 rotations defined by equations (B9)-(B11)

 applied end rotation defined by equation (B21) 

 specific weight of a fluid

T(x, z) temperature change function defined by equation (B16)

axial, circumferential, and shear membrane strains, respectively

circumferential, angular coordinate

 temperature change functions defined by equation (B16)

 axial and circumferential bending and twisting strains, respectively

 parameter appearing in equations (41)-(43)

μ constant defining different shell theories, given by equation (3)

12 lamina major Poisson’s ratio

 lamina fiber angle

Introduction

Weight and structural performance are key elements in the design of aircraft and spacecraft. 

For example, weight directly affects fuel costs incurred by airlines, which affects consumers and 

commerce. In addition, structural performance affects the safety of the flying public. Similarly, 

the present cost for placing a spacecraft in Earth orbit ranges from approximately $5,000 to 

$10,000 per pound, depending on the altitude. Thus, structural weight is a significant design 

driver in launch-vehicle structures that operate near the Earth, such as the Space Shuttle external 

tank and solid rocket boosters shown in figure 1, and in vehicles that operate far from Earth such 

as the conceptual lunar lander shown in figure 2. In recent studies of newer vehicles like a lunar 

lander, fiber-reinforced, laminated-composite materials have received significant attention for use 

in primary structural members. These advanced composite materials have higher stiffness-todensity ratios, higher strength-to-density ratios, and very low coefficients of thermal expansion, 

compared to common isotropic materials. Additionally, because laminated composites are 

composed of orthotropic layers, oriented at various angles, the potential exists to tailor a laminate 

	x

o

, 	

o

, 	n

o

	

 ̃


x

o

, 


o

,  x

o

0 x , 1 x

x

o

, 

o

, x

o
7

to meet a specific set of high-performance, or special purpose, structural requirements. Thus, 

laminated composites offer a design space that is much wider than that for isotropic materials. 

However, along with the potential structural performance benefits of laminate tailoring comes a 

much more complicated analysis and design process.

 Efforts to reduce structural weight while maintaining, or even enhancing, structural 

preformance through the use of fiber-reinforced composite materials have been underway for 

approximately 50 years. Although many advancements have been made, and several large-scale 

structures have been built and tested, advances in understanding the mechanics of composite 

structures, especially those with extreme orthotropies and anisotropies, need to continue. One 

common, fundamental structural element is the circular cylinder. For example, understanding the 

buckling response of stiffened cylinders is fundamental to the design of aircraft fuselages and 

launch-vehicle fuel tanks and interstages. Likewise, understanding how edge effects associated 

with end fittings on the tubular members of a lunar lander affect the structural performance is 

significant. When dealing with generally laminated cylinders, deformation modes associated with 

bending, twisting, and stretching can be highly coupled, and failure to fully understand these 

effects on a design can have dangerous and costly unintended consequences. Thus, continued 

advancement of understanding the basic behavioral characteristics of generally laminated circular 

cylinders is an important part of the quest to develop high performance, or special purpose, 

structures by exploiting orthotropies and anisotropies. Moreover, advancing this understanding 

can be beneficial to postmortem analysis of failures that always occur when the limits of design 

technology are pushed, and should help to mitigate risks. Furthermore, advancing this 

understanding may be beneficial to the development of material-characterization test 

methodology for future exotic, advanced materials. Improved understanding of fundamental 

structural elements also provides insight into finite element modeling of complex flight-vehicle 

structures. 

Several analyses for predicting the stresses, strains, and displacements of laminatedcomposite cylinders that are relevant to the present study can be found in the literature. Perhaps, 

one of the first significant papers to address anisotropic cylinders was given by Dong et al.1

 In 

this paper, equations for the linear static response of general shells are given that are based on the 

simplifications attributed to Donnell, Mushtari, and Vlasov (see Brush and Almroth2). In addition, 

the general-shell equations are specialized to circular cylinders and a stress-function formulation 

is given. Moreover, a solution is presented for the axisymmetric bending of a semi-infinite 

cylinder subjected to edge moments. Similarly, an analysis for anisotropic cylinders undergoing 

axisymmetric deformations was given by Gulati and Essenburg3

 in 1967 that is based on a shell 

theory more refined than Donnell-type theories and that includes the effects of transverse shear 

deformation.

In 1970, Stavsky and Smolash4

 presented a detailed thermoelastic analysis of orthotropic 

composite cylinders. In particular, numerous results are presented for fixed-end cylinders 

undergoing axisymmetric deformations. Also in that year, Vicario and Rizzo5

 presented an exact 

solution for the axisymmetric response of symmetrically laminated cylinders with coupling 

between extension and shear (unbalanced). Later, in 1972, Reuter6

 presented an analytical 

solution for axisymmetric deformation of unbalanced symmetrically laminated circular 

cylindrical shells and balanced unsymmetrically laminated circular cylindrical shells that are 
8

subjected to internal pressure and thermal loads. Reuter’s solution is based on Donnell-type linear 

equations and numerical results are presented for filament-wound cylinders made of heat-treated 

carbon-carbon material.

In 1975, Hennemann7 presented an analytical solution for the axisymmetric response of 

homogeneous orthotropic cylinders and unsymmetrically laminated cross-ply cylinders subjected 

to uniform axial-compression and lateral pressure loads. Similar results were presented by Jones 

and Henemann8

 later in 1980. In 1984, Cheng and He9 derived a pair of complex conjugate, fourthorder equations, that are based on Flugge’s10 corresponding equations, for specially orthotropic 

cylinders that can be solved in closed form. Moreover, eigenfunction solutions were presented 

that include the solution for the axisymmetric bending boundary layer and several simplifed 

equations were presented and their relative accuracy was analyzed. 

Chaudhuri et al.11 presented an analytical solution for unbalanced unsymmetrically laminated 

circular cylindrical shells that are subjected to internal pressure in 1986. Their analysis is based 

on a variant of the Love-Kirchhoff shell theory,12 which uses an expression for the change in 

surface twist that was given by Timoshenko and Gere.13 Numerical results are also presented for 

a two-ply shell that demonstrate the coupling effects of the shell anisotropies. Later, in 1989, AbuArja and Chaudhuri14 presented similar closed-form solutions for unbalanced, symmetrically 

laminated cylinders and balanced unsymmetrically laminated angle-ply cylinders subjected to 

pressure loads that include the effects of transverse-shear deformation. In 1989, Butler and Hyer15

presented an axisymmetric-shell analysis for a general unbalanced unsymmetrically laminated 

circular cylindrical shell that is subjected to axial compression, torsion, or thermal loading. 

Several results are presented that demonstrate the coupling effects of the shell anisotropies on the 

structural response. 

In the early 1990s, Hyer and Paraska16, 17 presented studies of the axisymmeric response of 

balanced symmetrically laminated and balanced unsymmetrically laminated cylinders subjected 

to axial compression and thermal loads. These studies are in the context of nonlinear prebuckling 

deformations that occur as a result of the compression and thermal loads considered. In particular, 

the effects of laminate stacking sequence on the extent and character of the bending boundary 

layers are presented for two groups of three similar laminates. Two of the laminates are 

unsymmetric. Birman18 presented equations for axisymmetric thermal-stress analysis of ringstiffened orthotropic cylinders in 1991 that include the first-order effects of transverse-shear 

deformation. In 1992, Birman19 extended this work to symmetrically laminated cross-ply and 

specially orthotropic cylinders, and developed axisymmetric solutions based on a refined sheardeformation shell theory. Additionally, Birman20 extended his work to include nonlinear 

axisymmetric deformations of generally laminated cylinders in 1993. 

In 1997, Vinson21 presented a linear analysis that focuses mainly on balanced, symmetrically 

laminated cylinders, and an expression is given for the length of the bending boundary layers near 

the cylinder ends that is based on the Love-Kirchhoff shell equations. In contrast, McDevitt and 

Simmonds22 used Goldenveizer’s static-geometric duality principle23 to reduce the Sanders-Koiter 

equations24, 25 for fully anisotropic, right-circular cylindrical shells to two coupled fourth-order 

equations that use a stress and a curvature function as the unknown, primary field variables. The 
9

reduction is accomplished by adding certain negligibly small terms to the stress-strain relations, 

which are intrinsically in error because they must be established experimentally. The approach 

demonstrates how the static-geometric duality principle can be used to reduce greatly the amount 

of algebra needed to obtain results. Eigenfunction solutions are also presented for specially 

orthotropic cylinders that are in agreement with corresponding results presented by Cheng and 

He.9

 Moreover, asymptotic formulas that can be used to determine axisymmetric bending 

boundary layer attenuation lengths and the decay of other non-axisymmetric, self-equilibrated 

edges loads are given.

An in-depth analytical, parametric study of the attenuation of bending boundary layers in 

symmetrically and unsymmetrically laminated thin cylindrical shells, including balanced and 

unbalanced constructions, was presented for nine contemporary material systems by Nemeth and 

Smeltzer26 in 2000. Herein, and in reference 26, the term "balanced" refers to laminates that do not 

exhibit coupling between extension or contraction and shearing. The analysis is based on the 

linear Sanders-Koiter shell equations and specializations to the Love-Kirchhoff shell equations 

and Donnell’s equations are included. Two nondimensional parameters are indentified that 

characterize and quantify the effects of laminate orthotropy and laminate anisotropy on the 

bending boundary-layer decay length in a very general and encompassing manner. In this 

particular study, herein, and in much of the recent literature; the term "orthotropy" refers to 

orthogonally anisotropic laminates that exhibit deformation and stress states in which dilatation 

is completely uncoupled from distortion, with respect to the given coordinate system. In contrast, 

the term "anisotropy" implies the presence of material-induced coupling between dilatation and 

distortion. By using these nondimensional parameters, a substantial number of structural design 

technology results are presented in reference 26 that apply to a wide range of laminated-composite 

cylinders. Also in 2000, Paris and Costello27 presented equations for analyzing cord-reinforced 

composite cylindrical shells and closed-form solutions are given for the axisymmetric response. 

Their analysis is based on Flugge’s equations. 

 In 2001, Ye28, 29 presented a three-dimensional elasticity formulation for the decay rates of 

stresses and displacements in cross-ply and angle-ply laminated composite cylinders subjected to 

self-equilibrated end loads. Analyses for the axisymmetric response of semi-infinite laminated 

cylinders that are based on three-dimensional elasticity theory, on a refined shear-deformation 

shell theory, and on classical Love-Kirchhoff shell theory were presented by Bhaskar and 

Balasubramanyam30 in 2002. Somewhat unique to their study is a sizeable collection of tabular 

results in addition to figures. In 2003, Paris and Costello31 extended their previous work on cordreinforced composite cylindrical shells to include generally laminated wall constructions. This 

later analysis is also based on Flugge’s equations and closed-form solutions are given for the 

axisymmetric response. Also in 2003, Kollar and Springer32 gave a relatively terse presentation 

of the general closed-form solution for the axisymmetric response of fully anisotropic cylinders 

subjected to mechanical and thermal loads that are also based on Flugge’s equations.

The literature described previously herein contains many elements of a general formulation 

for the axisymmetric behavior of cylindrical shells. However, none of the studies is complete and 

many details of the solutions are omitted. The objective of the present study is to give a detailed, 

complete formulation for axisymmetric deformations of a generally laminated circular cylindrical 

shell subjected to combined mechanical and thermal loads. As a first element of a building block 
10

process to obtain improved theories, the analysis presented herein is based on the linear SandersKoiter shell equations, which neglects transverse-shear deformations. Moreover, the analysis is 

an extension of that presented by Nemeth and Smeltzer26 in that it uses positive-definiteness of the 

strain energy density to determine the specific form of the analytical solution. This approach 

results in a significant simplification of the analysis procedure. To accomplish this objective, the 

Sanders-Koiter equations for axisymmetric deformations are given and then manipulated into a 

convenient form that facilitates obtaining the analytical solution. Then, a systematic presentation 

of the exact solution is given, followed by examples that demonstrate the typical solution 

processes. Approximate solutions are also presented to add credibility to the present study and to 

illustrate potential pitfalls of using approximate solutions in the analysis and design of laminatedcomposite cylinders.

Equations Governing Axisymmetric Behavior

The ordinary differential equation that governs the axisymmetric bending behavior of a rightcircular cylinder that is subjected to edge loads or displacements and surface tractions is obtained 

by first specializing the linear Sanders-Koiter shell equations, that are given in Appendix A, for 

axial symmetry. For the equations presented herein, x and denote the axial and 

circumferential coordinates of a right-circular cylinder, respectively, as shown in figure 3. The zcoordinate shown in the figure is positive in the radially outward direction, and z = 0 corresponds 

to the middle surface of the shell. The cylinder has length L, radius R, and uniform thickness h. 

The specialization to axial symmetry is obtained by eliminating all terms in the equations that are 

differentiated with respect to the circumferential coordinate, . The resulting set of equations for 

axisymmetric behavior is given in Appendix B.

The particular equations used in the present study to obtain the exact solution for 

axisymmetric behavior are obtained by first noting that integration of equation (B1) yields

 (1a)

where is a constant of integration and 

 (1b)

Next, equations (B2) and (B5) are combined to get

 (2)

where the prime mark denotes differentiation with respect to the independent variable, x. For 

convenience, the parameter 

Nx x = N0 + N1 x

N0

N1 x = 
 qx x dx

Nx x + 1

R c1 + 1

2c2 Mx x + qx = 0
11

 (3)

is introduced such that the Sanders-Koiter equations are given by μ = 3/2 and the Love-Kirchhoff 

equations are given by μ = 1. Donnell’s equations are given by μ = 0. Similarly, the function

 (4)

is introduced so that equation (2) becomes

 (5)

and the corresponding boundary condition given by equation (B19) becomes

 or (6)

at x = 0 and x = L. Integration of equation (5) yields

 (7a)

where is a constant of integration and 

 (7b)

Next, equations (B3) and (B4) are combined to get

 (8)

The next step in the analysis is the simplification of the constitutive equations. First, by using 

equations (B8) and (3), equation (B14) is expressed as

 (9)

By using equations (4) and (9), the constitutive equations given by matrix equation (B15) are 

expressed as

 (10)

μ = c1 + 1

2c2

T x = Nxx + μ

RMxx

T x + q(x) = 0

T = T v=v

T x = T0 + T1 x

T0

T1 x = 
 qx dx

Mx x – Nx

R + qn x = 0

x

o

= μ

R x

o

Nx

N

T

Mx

=

A11 A12 A16 B11

A12 A22 A26 B12

A16 A26 A66 B16

B11 B12 B16 D11


x

o




o

 x

o

x

o




Nx

Th

N

Th

T

Th

Mx

Th
12

and

 (11)

where

 (12a)

 (12b)

 (12c)

 (12d)

 (12e)

 (12f)

where the subscripted A, B, and D terms appearing in these equations are the usual laminate 

stiffnesses of classical plate and shell theory (see reference 34), in which the fiber orientation of 

a typical ply is denoted by the angle shown in figure 3. The motivation for writing the 

constitutive equations in this form is that the matrix equation given by equation (10) is the only 

part of the full constitutive equations that appears in the strain energy density function, which is 

used in the present paper to determine the corresponding positive-definiteness conditions. Noting 

that and are the only strains in equation (10) that depend on the axial 

displacement u(x) and the circumferential displacement v(x), it is useful to extract 

 (13a)

from equation (10). Using equations (1a), (7a), (B6) - (B8), and (B12) with equation (13a) gives

 (13b)

Solving for the derivatives of the axial displacement u(x) and the circumferential displacement 

v(x) gives

M = B12x

o

+ B22

o

+ B26 x

o

+ D12x

o

 M

Th

A16 = A16 + μ h

R

B16

h

A26 = A26 + μ h

R

B26

h

A66 = A66 + 2μ h

R

B66

h + μ2 h

R

2 D66

h2

B16 = B16 + μ h

R

D16

h

B26 = B26 + μ h

R

D26

h

T

Th

= Nx

Th

+ μ

RMx

Th

x

o

= u x  x

o

= v x

Nx + Nx

Th

T + TTh = A11 A16

A16 A66

x

o

 x

o +

A12

A26



o

+

B11

B16

x

o

N0 + N1 x + Nx

Th

x

T0 + T1 x + TTh

x

= A11 A16

A16 A66

u x

v x +

A12

A26

w x

R  B11

B16

w x
13

 (14)

where

 (15a)

 (15b)

 (15c)

and where w(x) is the radial deflection that is positive-valued when outward. Equations (14) and 

(15) indicate that the circumferential displacement v(x) becomes uncoupled from the axial 

displacement u(x) and the radial displacement w(x) when , which implies 

that A16 = A26 = B16 = D16 = 0. In addition, the constitutive equation, equation (10), indicates that 

Nx, N, and Mx become uncoupled from the torsional, shear strain when , 

and that , that is defined by equation (4), becomes uncoupled from Furthermore, 

equation (11) indicates that Mbecomes uncoupled from when , which implies B26 = 

D26 = 0.

 Next, equations (14) are then substituted into and , and the resulting 

expressions for and , along with equations (B7) and (B12) are substituted into the 

constitutive equations given by equation (10). This action converts the strains and stress 

resultants in equation (10) into functions of the radial displacement w(x). In particular,

 (16a)

becomes

 (16b)

and

 (17a)

u x

v x = a 11 a 16

a 16 a 66

N0 + N1 x + Nx

Th

x

T0 + T1 x + TTh

x

+

b 12

b 62

w x

R 
 c 12

c 62

w x

a 11 a 16

a 16 a 66

= A11 A16

A16 A66


 1

= 1

A11A66 
 A16

2

A66 
 A16


 A16 A11

b 12

b 62

= 
 a 11 a 16

a 16 a 66

A12

A26

= 1

A11A66 
 A16

2

A16A26 
 A12A66

A12A16 
 A11A26

c 12

c 62

= 
 a 11 a 16

a 16 a 66

B11

B16

= 1

A11A66 
 A16

2

A16B16 
 B11A66

B11A16 
 A11B16

A16 = A26 = B16 = 0

 x

o A16 = A26 = B16 = 0

T 
x

o

, 


o

, and x

o

.

 x

o B26 = 0


x

o

= u x  x

o

= v x


x

o  x

o

Nx = A12u x + A22

w x

R + A26 v x 
 B12 w x 
 N

Th

x

Nx = A12a 11 + A26a 16 N0 + N1 x + Nx

Th

x + A12a 16 + A26a 66 T0 + T1 x + TThx

+ A22 + A12b 12 + A26b 62

w x

R 
 B12 + A12c 12 + A26c 62 w x 
 N

Th

x

Mx x = B11u x + B12

w x

R + B16 v x 
 D11 w x 
 Mx

Th

x
14

becomes

 (17b)

Substituting equations (16b) and (17b) into equation (8) yields 

 (18)

The constant coefficients are given by

 (19)

 (20)

 (21)

The function C4

(x) is given by

 

 (22)

Equations (19) - (21) are expressed in terms of elements appearing in matrix equation (10) further 

by using equations (15); that is,

 (23)

 (24)

Mx x = B11a 11 + B16a 16 N0 + N1 x + Nx

Th

x + B11a 16 + B16a 66 T0 + T1 x + TThx

+ B12 + B11b 12 + B16b 62

w x

R 
 D11 + B11c 12 + B16c 62 w x 
 Mx

Th

x

C1 w x + C2 w x + C3 w x = C4 x

C1 = D11e and e =1+ B11c 12 + B16c 62

D11

C2 = 
 1

R 2B12 + A12c 12 + A26c 62 + B11b 12 + B16b 62

C3 = 1

R2 A22 + A12b 12 + A26b 62

C4 x = qn x 
 Mx

Th x + 1

R N

Th

x


 1

R A12a 11 + A26a 16 N0 + N1 x + Nx

Th

x 
 1

R A12a 16 + A26a 66 T0 + T1 x + TTh

x

+ B11a 11 + B16a 16 N1 x + Nx

Th x + B11a 16 + B16a 66 T1 x + TTh x

C2 = – 2

R B12 +

A16A26 – A12A66 B11 + A12A16 – A11A26 B16

A11A66 – A16

2

C3 = 1

R2

A11A22 – A12

2

A66 – A11A26

2

– A22A16

2

+ 2A12A16A26

A11A66 – A16

2
15

 (25)

Next, equation (18) is divided by the coefficient C1

 given by equation (19) to get

 (26)

where the constants S and Q are given by

 (27)

 (28)

The function P(x) is given by

 (29a)

where

 (29b)

 (29c)

e =1– A66B11

2

+ A11B16

2

– 2A16B11B16

A11A66 – A16

2

D11

w x + 4S w x + 4Q w x = P x

S =

C2

4C1

= – 1

2RD11e

B12 +

A16A26 – A12A66 B11 + A12A16 – A11A26 B16

A11A66 – A16

2

Q =

C3

4C1

=

A11A22 – A12

2

A66 – A11A26

2

– A22A16

2

+ 2A12A16A26

4R2D11 e A11A66 – A16

2

P(x) =

C4(x)

C1

 P0 + P1(x)

P0 = 
 1

RD11 e

A12a 11 + A26a 16 N0 + A12a 16 + A26a 66 T0

P1(x) = 1

D11 e

qn(x) 
 Mx

Th x + 1

R N

Th

x


 1

RD11 e

A12a 11 + A26a 16 N1 x + Nx

Th

x + A12a 16 + A26a 66 T1 x + TThx

+ 1

D11 e

B11a 11 + B16a 16 N1 x + Nx

Th x + 1

D11 e

B11a 16 + B16a 66 T1 x + TTh

 x
16

Exact Solution

Equation (26) is a linear, nonhomogeneous, fourth-order ordinary differential equation with 

constant coefficients, and the form of its solution depends on the numerical values of the constants 

S and Q. To determine the specific form of the solution, it is useful to examine the positivedefiniteness conditions on the strain-energy density function. The strain energy density function 

U for this problem is given by (e.g., see Paraska,17 p. 16) 

 (30)

By using equations (4) and (9), the strain-energy density function is expressed as

 (31)

The strain energy density is expressed in terms of the total strains, mechanical plus thermal, and 

constitutive terms by using the constitutive equation given by equation (10); that is,

 (32)

The stiffness terms in equation (32) that have overbars are defined by equations (12) and are 

functions of the shell wall thickness-to-radius parameter, h/R. Because the strain energy density 

function is positive valued, the matrix appearing in equation (32) is positive definite. As a result 

(e.g., see Zwillinger33, pp. 133-134), the diagonal terms A11, A22, , and D11 are positive-valued, 

and the following determinants are positive valued

 (33)

 (34)

in addition to the determinant of the constitutive matrix in equation (32). Moreover, by 

rearranging the strain energy density function into the form

2U = Nx 
 Nx

Th


x

o

+ N
 N

Th




o

+ Nx
 Nx

Th

 x

o

+ Mx 
 Mx

Th

x

o

+ Mx
 Mx

Th

x

o

2U = Nx 
 Nx

Th


x

o

+ N
 N

Th




o

+ T 
 T

Th

 x

o

+ Mx 
 Mx

Th

x

o

U = 1

2


x

o




o

 x

o

x

o

T

A11 A12 A16 B11

A12 A22 A26 B12

A16 A26 A66 B16

B11 B12 B16 D11


x

o




o

 x

o

x

o

A66

A11 A12

A12 A22

= A11A22 
 A12

2

> 0

A11 A12 A16

A12 A22 A26

A16 A26 A66

= A11A22 – A12

2

A66 
 A11A26

2


 A22A16

2

+ 2A12A16A26 > 0
17

 (35)

the following additional positive-definiteness condition is obtained

 (36)

 The homogeneous solution for equation (26) involves the square root of the quantity Q - S2

.

By using equations (27) and (28), this quantity is given by

 (37)

Substituting equations (19) and (23) - (25) into equation (37) and simplifying, the quantity Q - S2

is found to be given by

 (38)

It follows logically, that Q - S2

 > 0 because the positive definiteness of the strain energy density 

function requires that the determinants in equations (36) and (38) be positive valued. Moreover, 

Q - S2

 > 0 implies that Q > 0, and Q > 0 implies that C3 /C1 > 0. Equations (34), (36), and (24) 

indicate that C3 > 0. Thus, C3 /C1 > 0 yields the condition that C1 = D11 e > 0 (see equations (19) 

and (25)). Because D11 > 0, e > 0. To reinforce the postive valuedness of Q, it is convenient to 

introduce the expression

 (39)

such that T2

 - S2 > 0, and to express equation (26) as

 (40)

U = 1

2


x

o

 x

o




o

x

o

T A11 A16 A12 B11

A16 A66 A26 B16

A12 A26 A22 B12

B11 B16 B12 D11


x

o

 x

o




o

x

o

A11 A16

A16 A66

= A11A66 
 A16

2

> 0

Q 
 S

2

= 4C1C3 
 C2

2

16C1

2

Q 
 S

2

= 1

4C1

2

R2A11A66 
 A16

2

A11 A12 A16 B11

A12 A22 A26 B12

A16 A26 A66 B16

B11 B12 B16 D11

T2=Q=

A11A22 – A12

2

A66 – A11A26

2

– A22A16

2

+ 2A12A16A26

4R2D11 e A11A66 – A16

2

w x + 4S w x + 4T2w x = P0 + P1(x)
18

Equation (40) is a linear, fourth-order, nonhomogeneous ordinary differential equation with 

constant coefficients. The characteristic equation of equation (40) is given by

 (41)

Using the knowledge that T2

 - S2 > 0, the roots of the characteristic equation are obtained from the 

quadratic formula; that is,

 (42)

where . Solution of this equation for  yields four roots of equation (41) that are pairs 

of complex conjugates that are given by

 (43)

The homogeneous solution of equation (40) can be written as follows

 (44)

where

 (45a)

 (45b)

 (45c)

 (45d)

are basis function spanning the solution space and where 0  x  L. The symbols K1

, K2, K3, and 

K4

 are real-valued constants that are determined from the boundary conditions. The solution 

given by equation (44) represents a damped, oscillatory response that decays from each end of the 

cylinder. The regions near the edges of the cylinder, where the amplitude of wH(x) typically varies 

the most, are called the bending boundary layers. This particular form for the basis functions was 

selected to prevent numerical issues associated with evaluating exponential functions with a large 

positive-valued argument.

Next, it is important to observe that the constant-valued term P0 in equation (29a) contains 

4 + 4S 2 + 4T2 = 0



2

1,2 = 2 
 S ± i T2


 S

2

i = 
 1

1,2,3,4 = ± T 
 S ± i T+S

wH x = K1b 1 x + K2b 2 x + K3b 3 x + K4b 4 x

b 1 x = e


 T 
 S x

sin T+S x

b 2 x = e


 T 
 S x

cos T+S x

b 3 x = e


 T 
 S L 
 x

sin T+S x

b 4 x = e


 T 
 S L 
 x

cos T+S x
19

two of the unknown constants of integration, Thus, it is convenient to partition the 

particular solution of equation (40) into

 (46)

where is a constant obtained as the particular solution associated with P0 and is the part 

of the particular solution associated with P1(x). Moreover, it is convenient to express

 (47a)

where

 (47b)

 (47c)

The form of depends on the specific functional representations of the applied surface 

tractions and thermal fields. By using equations (44), (46), and (47a), the general solution to 

equation (40) is expressed as

 (48)

With the form of w(x) known, the first of equations (14) is expressed as

 (49)

where

 (50a)

 with k = 1, 2, 3, and 4 (50b)

 (50c)

N0 and T0.

wp x = wp

0

+ wp x

wp

0

wp x

wp

0

=R cNN0 + cTT0

cN = A16A26 
 A12A66

A11A22 
 A12

2

A66 
 A11A26

2


 A22A16

2

+ 2A12A16A26

cT = A16A12 
 A11A26

A11A22 
 A12

2

A66 
 A11A26

2


 A22A16

2

+ 2A12A16A26

wp x

w x = K1b 1 x + K2b 2 x + K3b 3 x + K4b 4 x +R cNN0 + cTT0 + wp x

u x = U0 x + U1 x K1 + U2 x K2 + U3 x K3 + U4 x K4 + U5N0 + U6T0

U0 x = a 11 N1(x) + Nx

Th

x + a 16 T1(x) + TThx + b 12

wp x

R 
 c 12wp x

Uk x = b 12

b k x

R 
 c 12 b k x

U5 = a 11 + b 12cN
20

 (50d)

Likewise, the second of equations (14) is expressed as

 (51)

where

 (52a)

 with k = 1, 2, 3, and 4 (52b)

 (52c)

 (52d)

Next, the derivatives of the basis functions, given by equations (45), are expressed as

 (53a)

 (53b)

 (53c)

 (53d)

 (53e)

 (53f)

 (53g)

 (53h)

U6 = a 16 + b 12cT

v x = V0 x + V1 x K1 + V2 x K2 + V3 x K3 + V4 x K4 + V5N0 + V6T0

V0 x = a 16 N1(x) + Nx

Th

x + a 66 T1(x) + TThx + b 62

wp x

R 
 c 62wp x

Vk x = b 62

b k x

R 
 c 62 b k x

V5 = a 16 + b 62cN

V6 = a 66 + b 62cT

b 1 x = T+S b 2 x 
 T 
 S b 1 x

b 2 x = 
 T+S b 1 x 
 T 
 S b 2 x

b 3 x = T+S b 4 x + T 
 S b 3 x

b 4 x = 
 T+S b 3 x + T 
 S b 4 x

b 1 x = 
 2S b 1 x 
 2 T

2


 S

2

b 2 x

b 2 x = 2 T

2


 S

2

b 1 x 
 2S b 2 x

b 3 x = 
 2S b 3 x + 2 T

2


 S

2

b 4 x

b 4 x = 
 2 T

2


 S

2

b 3 x 
 2S b 4 x
21

By using these expressions for the derivatives, equations (50b) are expressed as

 (54a)

 (54b)

 (54c)

 (54d)

Likewise, equations (52b) are expressed as

 (55a)

 (55b)

 (55c)

 (55d)

Next, equations (49) and (51) are integrated directly to get

 (56)

U1 x = b 12

R + 2Sc 12 b 1 x + 2c 12 T

2


 S

2

b 2 x

U2 x = 
 2c 12 T

2


 S

2

b 1 x +

b 12

R + 2Sc 12 b 2 x

U3 x = b 12

R + 2Sc 12 b 3 x 
 2c 12 T

2


 S

2

b 4 x

U4 x = 2c 12 T

2


 S

2

b 3 x +

b 12

R + 2Sc 12 b 4 x

V1 x = b 62

R + 2Sc 62 b 1 x + 2c 62 T

2


 S

2

b 2 x

V2 x = 
 2c 62 T

2


 S

2

b 1 x +

b 62

R + 2Sc 62 b 2 x

V3 x = b 62

R + 2Sc 62 b 3 x 
 2c 62 T

2


 S

2

b 4 x

V4 x = 2c 62 T

2


 S

2

b 3 x +

b 62

R + 2Sc 62 b 4 x

u x = U0 x + U1 x K1 + U2 x K2 + U3 x K3 + U4 x K4 + U5N0 + U6T0 x+u
22

and

 (57)

where

 (58a)

 (58b)

 (58c)

 (58d)

for values of k = 1, 2, 3, and 4; and and are the corresponding constants of integration. 

Next, the integrals of the basis functions, given by equations (45), are expressed as

 (59a)

 (59b)

 (59c)

 (59d)

By using these expressions for the integrals, equations (58c) and (58d) are expressed as

 (60a)

 (60b)

v x = V0 x + V1 x K1 + V2 x K2 + V3 x K3 + V4 x K4 + V5N0 + V6T0 x+v

U0 x = a 11 N1(x) + Nx

Th

x dx + a 16 T1(x) + TThx dx +

b 12

R wp x dx 
 c 12wp x

V0 x = a 16 N1(x) + Nx

Th

x dx + a 66 T1(x) + TThx dx +

b 62

R wp x dx 
 c 62wp x

Uk x = Uk x dx

Vk x = Vk x dx

u v

b 1 x dx = 
 1

2T T 
 S b 1 x + T+S b 2 x

b 2 x dx = 1

2T 
 T 
 S b 2 x + T+S b 1 x

b 3 x dx = 1

2T T 
 S b 3 x 
 T+S b 4 x

b 4 x dx = 1

2T T 
 S b 4 x + T+S b 3 x

U1 x = c1

u

b 1 x 
 c2

u

b 2 x

U2 x = c2

u

b 1 x + c1

u

b 2 x
23

 (60c)

 (60d)

with

 (60e)

 (60f)

and

 (61a)

 (61b)

 (61c)

 (61d)

with

 (61e)

 (61f)

Equations (48), (56), and (57) define completely the displacement fields in terms of eight 

constants that are determined from the boundary conditions. These constants are

 (62)

To determine the unknown constants given by equation (62), the boundary conditions stated 

U3 x = 
 c1

u

b 3 x 
 c2

u

b 4 x

U4 x = c2

u

b 3 x 
 c1

u

b 4 x

c1

u

= T 
 S

2T 2c 12T 
 b 12

R

c2

u

= T+S

2T 2c 12T +

b 12

R

V1 x = c1

v

b 1 x 
 c2

v

b 2 x

V2 x = c2

v

b 1 x + c1

v

b 2 x

V3 x = 
 c1

v

b 3 x 
 c2

v

b 4 x

V4 x = c2

v

b 3 x 
 c1

v

b 4 x

c1

v

= T 
 S

2T 2c 62T 
 b 62

R

c2

v

= T+S

2T 2c 62T +

b 62

R

K1, K2, K3, K4, u, v, N0, T0
24

by equations (B18) - (B21) must be enforced. To enforce these equations, expressions for the 

rotation and the stress resultants are needed. First, the expression for the slope is obtained by 

differentiating equation (48) and using the expressions for the derivatives of the basis functions 

given by equations (53). This task yields

 (63)

where

 (64a)

 (64b)

 (64c)

 (64d)

 (64e)

Next, equations (48), (49), and (51) are substituted into equation (17a) to get

 (65)

where

 (66a)

 with k = 1, 2, 3, and 4 (66b)

 (66c)

 (66d)

Using equation (50a) for U0(x), equation (52a) for V0(x), equations (54) for U1(x) through U4(x), 

equations (55) for V1(x) through V4(x), equation (50c) for U5, equation (50d) for U6, equation 

(52c) for V5, and equation (52d) for V6 , and equations (53) for the second derivatives of the basis 

functions gives

	x

o

	x

o

x = 	0 x + K1	1 x + K2	2 x + K3	3 x + K4	4 x

	0 x = 
 wp x

	1 x = T 
 S b 1 x 
 T+S b 2 x

	2 x = T+S b 1 x + T 
 S b 2 x

	3 x = 
 T+S b 4 x 
 T 
 S b 3 x

	4 x = T+S b 3 x 
 T 
 S b 4 x

Mx = Mx0 x + Mx1 x K1 + Mx2 x K2 + Mx3 x K3 + Mx4 x K4 + Mx5N0 + Mx6T0

Mx0 x = B11U0 x + B12

wp x

R + B16V0 x 
 D11wp x 
 Mx

Th

x

Mxk x = B11Uk x +

B12

R b k x + B16Vk x 
 D11b k x

Mx5 = B11U5 + B12cN + B16 V5

Mx6 = B11U6 + B12cT + B16 V6
25

 (67a)

 (67b)

 (67c)

 (67d)

 (67e)

 (67f)

 (67g)

where

 (67h)

 (67i)

Next, the transverse shear stress resultant Qx(x) is obtained by using equation (65) with equation 

(B4) to get

 (68)

where for values of k = 1, 2, 3, and 4. Differentiating equation (67a) gives

 (69a)

Similarly, differentiating equations (67b) - (67e) and using equations (53) gives

Mx0 x = B11a 11 + B16a 16 N1(x) + Nx

Th

x + B11a 16 + B16a 66 T1(x) + TThx

+ B12 + B11b 12 + B16b 62

wp x

R 
 D11 + B11c 12 + B16c 62 wp x 
 Mx

Th

x

Mx1 x = c1

M

b 1 x + c2

M

b 2 x

Mx2 x = 
 c2

M

b 1 x + c1

M

b 2 x

Mx3 x = c1

M

b 3 x 
 c2

M

b 4 x

Mx4 x = c2

M

b 3 x + c1

M

b 4 x

Mx5 = cN B12 + B11b 12 + B16b 62 + B11a 11 + B16a 16

Mx6 = cT B12 + B11b 12 + B16b 62 + B11a 16 + B16a 66

c1

M

= B11b 12 + B12 + B16b 62

R + 2S B11c 12 + B16c 62 + D11

c2

M

= 2 T

2


 S

2

B11c 12 + B16c 62 + D11

Qx x = Mx x  Qx0 x + Qx1 x K1 + Qx2 x K2 + Qx3 x K3 + Qx4 x K4

Qxk x = Mxk x

Qx0 x = Mx0 x = B11a 11 + B16a 16 N1(x) + Nx

Th x + B11a 16 + B16a 66 T1(x) + TTh

 x

+ B12 + B11b 12 + B16b 62

wp x

R 
 D11 + B11c 12 + B16c 62 wp x 
 Mx

Th x
26

 (69b)

 (69c)

 (69d)

 (69e)

All expressions needed to enforce 8 of the 16 boundary conditions given by equations (B18)-

(B21) have been obtained. In particular, equations (1a) and (56) are used to enforce the boundary 

conditions stated by equation (B18). Equations (7a) and (57) are used to enforce the boundary 

conditions stated by equation (6), and equivalently equation (B19). Equations (68) and (48) are 

used to enforce the boundary conditions stated by equation (B20), and equations (63) and (65) are 

used to enforce the boundary conditions stated by equation (B21). The resulting 16 possible 

boundary conditions, in which one must be selected from each pair, are listed as follows:

 or (70a)

 (70b)

 or (71a)

 (71b)

 or (72a)

 (72b)

 or (73a)

Qx1 x = 
 c1

M

T 
 S + c2

M

T+S b 1 x + c1

M

T+S 
 c2

M

T 
 S b 2 x

Qx2 x = c2

M

T 
 S 
 c1

M

T+S b 1 x 
 c2

M

T+S + c1

M

T 
 S b 2 x

Qx3 x = c1

M

T 
 S + c2

M

T+S b 3 x + c1

M

T+S 
 c2

M

T 
 S b 4 x

Qx4 x = c2

M

T 
 S 
 c1

M

T+S b 3 x + c2

M

T+S + c1

M

T 
 S b 4 x

u 0 = u0: U1 0 K1 + U2 0 K2 + U3 0 K3 + U4 0 K4 + u = u0 
 U0 0

Nx 0 = N0: N0 = N0 
 N1 0

v 0 = v0: V1 0 K1 + V2 0 K2 + V3 0 K3 + V4 0 K4 + v = v0 
 V0 0

T 0 = T0: T0 = T0 
 T1 0

w 0 = w0: K1b 1 0 + K2b 2 0 + K3b 3 0 + K4b 4 0 +R cNN0 + cTT0 = w0 
 wp 0

Qx 0 = V0: Qx1 0 K1 + Qx2 0 K2 + Qx3 0 K3 + Qx4 0 K4 = V0 
 Qx0 0

	x

o

0 = 	0: K1	1 0 + K2	2 0 + K3	3 0 + K4	4 0 = 	0 
 	0 0
27

 (73b)

 or (74a)

 (74b)

 or (75a)

 (75b)

 or (76a)

 (76b)

 or (77a)

 (77b)

For certain combinations of boundary conditions, rigid-body displacements may be present that 

must be eliminated in order to have a unique solution. 

After determining the unknown constants given by equation (62), the displacements, strains, 

and stress resultants are known. The stress resultant Nis obtained from equation (16a). 

Substituting equations (48), (49), and (51) into equation (16a) gives

 (78)

Mx 0 = 0:

Mx1 0 K1 + Mx2 0 K2 + Mx3 0 K3 + Mx4 0 K4 + Mx5N0 + Mx6T0 = 0 
 Mx0 0

u L = uL:

U1 L K1 + U2 L K2 + U3 L K3 + U4 L K4 + U5N0 + U6T0 L+u = uL 
 U0 L

Nx L = NL: N0 = NL 
 N1 L

v L = vL:

V1 L K1 + V2 L K2 + V3 L K3 + V4 L K4 + V5N0 + V6T0 L+v = vL 
 V0 L

T x = TL: T0 = TL 
 T1 L

w L = wL:

K1b 1 L + K2b 2 L + K3b 3 L + K4b 4 L +R cNN0 + cTT0 = wL 
 wp L

Qx x = VL: Qx1 L K1 + Qx2 L K2 + Qx3 L K3 + Qx4 L K4 = VL 
 Qx0 L

	x

o

L = 	 L: K1	1 L + K2	2 L + K3	3 L + K4	4 L = 	 L 
 	0 L

Mx L = L:

Mx1 L K1 + Mx2 L K2 + Mx3 L K3 + Mx4 L K4 + Mx5N0 + Mx6T0 = L 
 Mx0 L

N= N0 x + N1 x K1 + N2 x K2 + N3 x K3 + N4 x K4 + N5N0 + N6T0
28

where

 (79a)

 with k = 1, 2, 3, and 4 (79b)

 (79c)

 (79d)

Using equation (50a) for U0

(x), equation (52a) for V0(x), equations (54) for U1(x) through U4(x), 

equations (55) for V1(x) through V4(x), equation (50c) for U5, equation (50d) for U6, equation 

(52c) for V5, and equation (52d) for V6 , and equations (53) for the second derivatives of the basis 

functions gives

 (80a)

 (80b)

 (80c)

 (80d)

 (80e)

 (80f)

 (80g)

where

 (81a)

 (81b)

N0 x = A12U0 x + A22

wp x

R + A26V0 x 
 B12wp x 
 N

Th

x

Nk x = A12Uk x +

A22

R b k x + A26Vk x 
 B12b k x

N5 = A12U5 + A22cN + A26V5

N6 = A12U6 + A22cT + A26V6

N0 x = A12a 11 + A26a 16 N1(x) + Nx

Th

x + A12a 16 + A26a 66 T1(x) + TThx

+ A22 + A12b 12 + A26b 62

wp x

R 
 B12 + A12c 12 + A26c 62 wp x 
 N

Th

x

N1 x = c3

N

b 1 x + c4

N

b 2 x

N2 x = 
 c4

N

b 1 x + c3

N

b 2 x

N3 x = c3

N

b 3 x 
 c4

N

b 4 x

N4 x = c4

N

b 3 x + c3

N

b 4 x

N5 = cN A22 + A12b 12 + A26b 62 + A12a 11 + A26a 16

N6 = cT A22 + A12b 12 + A26b 62 + A12a 16 + A26a 66

c3

N

= A12b 12 + A22 + A26b 62

R + 2S A12c 12 + A26c 62 + B12

c4

N

= 2 T

2


 S

2

A12c 12 + A26c 62 + B12
29

Similarly, The stress resultant Mis obtained from equation (11) by using equations (B6) - (B8) 

and (B12); that is

 (82)

Substituting equations (48), (49), and (51) into equation (82) gives

 (83)

where

 (84a)

 with k = 1, 2, 3, and 4 (84b)

 (84c)

 (84d)

Again, using equation (50a) for U0(x), equation (52a) for V0(x), equations (54) for U1(x) through 

U4

(x), equations (55) for V1(x) through V4(x), equation (50c) for U5, equation (50d) for U6, 

equation (52c) for V5, and equation (52d) for V6 , and equations (53) for the second derivatives 

of the basis functions gives

 (85a)

 (85b)

 (85c)

 (85d)

 (85e)

M= B12u x + B22

w x

R + B26v x 
 D12w x 
 M

Th

x

M= M0 x + M1 x K1 + M2 x K2 + M3 x K3 + M4 x K4 + M5N0 + M6T0

M0 x = B12U0 x + B22

wp x

R + B26V0 x 
 D12wp x 
 M

Th

x

Mk x = B12Uk x +

B22

R b k x + B26Vk x 
 D12b k x

M5 = B12U5 + B22cN + B26V5

M6 = B12U6 + B22cT + B26V6

M0 x = B12a 11 + B26a 16 N1(x) + Nx

Th

x + B12a 16 + B26a 66 T1(x) + TThx

+ B22 + B12b 12 + B26b 62

wp x

R 
 D12 + B12c 12 + B26c 62 wp x 
 M

Th

x

M1 x = c3

M

b 1 x + c4

M

b 2 x

M2 x = 
 c4

M

b 1 x + c3

M

b 2 x

M3 x = c3

M

b 3 x 
 c4

M

b 4 x

M4 x = c4

M

b 3 x + c3

M

b 4 x
30

 (85f)

 (85g)

where

 (85h)

 (85i)

The stress resultant Mx(x) is obtained by using equation (B15) and (9) to get 

 (86)

where

 (87)

Then, using equations (B6) - (B8) and (B12), equation (86) becomes

 (88)

Substituting equations (48), (49), and (51) into equation (88) gives

 (89)

where

 (90a)

 with k = 1, 2, 3, and 4 (90b)

 (90c)

 (90d)

M5 = cN B22 + B12b 12 + B26b 62 + B12a 11 + B26a 16

M6 = cT B22 + B12b 12 + B26b 62 + B12a 16 + B26a 66

c3

M

= B12b 12 + B22 + B26b 62

R + 2S B12c 12 + B26c 62 + D12

c4

M

= 2 T

2


 S

2

B12c 12 + B26c 62 + D12

Mxx = B16
x

o

+ B26


o

+ B66 x

o

+ D16x

o


 Mx

Th

B66 = B66 + μ h

R

D66

h

Mxx = B16u x + B26

w x

R + B66v x 
 D16w x 
 Mx

Th

x

Mx= Mx0 x + Mx1 x K1 + Mx2 x K2 + Mx3 x K3 + Mx4 x K4 + Mx5N0 + Mx6T0

Mx0 x = B16U0 x + B26

wp x

R + B66V0 x 
 D16wp x 
 Mx

Th

x

Mxk x = B16Uk x +

B26

R b k x + B66Vk x 
 D16b k x

Mx5 = B16U5 + B26cN + B66V5

Mx6 = B16U6 + B26cT + B66V6
31

Again, using equation (50a) for U0

(x), equation (52a) for V0(x), equations (54) for U1(x) through 

U4(x), equations (55) for V1(x) through V4(x), equation (50c) for U5, equation (50d) for U6, 

equation (52c) for V5, and equation (52d) for V6 , and equations (53) for the second derivatives 

of the basis functions gives

 (91a)

 (91b)

 (91c)

 (91d)

 (91e)

 (91f)

 (91g)

where

 (91h)

 (91i)

Then, equations (4) and (7a) give

 (92)

The transverse shear stress resultant Qis obtained from equation (B5); that is,

 (93)

Differentiating equations (89) and (91), and using equations (53) gives

Mx0 x = B16a 11 + B66a 16 N1(x) + Nx

Th

x + B16a 16 + B66a 66 T1(x) + TThx

+ B26 + B16b 12 + B66b 62

wp x

R 
 D16 + B16c 12 + B66c 62 wp x 
 Mx

Th

x

Mx1 x = c5

M

b 1 x + c6

M

b 2 x

Mx2 x = 
 c5

M

b 1 x + c6

M

b 2 x

Mx3 x = c5

M

b 3 x 
 c6

M

b 4 x

Mx4 x = c5

M

b 3 x + c6

M

b 4 x

Mx5 = cN B26 + B16b 12 + B66b 62 + B16a 11 + B66a 16

Mx6 = cT B26 + B16b 12 + B66b 62 + B16a 16 + B66a 66

c5

M

= B16b 12 + B26 + B66b 62

R + 2S B16c 12 + B66c 62 + D16

c6

M

= 2 T

2


 S

2

B16c 12 + B66c 62 + D16

Nxx = T0 + T1(x) 
 μ

RMxx

Qx = Mx x
32

 (94)

where

 (95a)

 (95b)

 (95c)

 (95d)

 (95e)

Particular Solutions

Whenever P1(x) given by equation (29c) is nonzero, the particular solution must be 

determined. To illustrate the solution process, consider a cylinder subjected to the surface 

tractions

 (96a)

 (96b)

 (96c)

and the temperature-change distributions

 (97a)

 (97b)

Qx = Q0 x + Q1 x K1 + Q2 x K2 + Q3 x K3 + Q4 x K4

Q0 x = B16a 11 + B66a 16 N1(x) + Nx

Th x + B16a 16 + B66a 66 T1(x) + TTh

 x

+ B26 + B16b 12 + B66b 62

wp x

R 
 D16 + B16c 12 + B66c 62 wp x 
 Mx

Th x

Q1 x = 
 c5

M

T 
 S + c6

M

T+S b 1 x + c5

M

T+S 
 c6

M

T 
 S b 2 x

Q2 x = c5

M

T 
 S 
 c6

M

T+S b 1 x 
 c5

M

T+S + c6

M

T 
 S b 2 x

Q3 x = c5

M

T 
 S + c6

M

T+S b 3 x + c5

M

T+S 
 c6

M

T 
 S b 4 x

Q4 x = c5

M

T 
 S 
 c6

M

T+S b 3 x + c5

M

T+S + c6

M

T 
 S b 4 x

wp x

qx x = qx

0

+ qx

1

x+qx

s

sin x

L + qx

c

cos 2x

L

qx = q

0

+ q

1

x+q

s

sin x

L + q

c

cos 2x

L

qn x = qn

0

+ qn

1

x+qn

s

sin x

L + qn

c

cos 2x

L

0 x = H0 + H1x+Hs sin x

L + Hc cos 2x

L

1 x = G0 + G1 x+Gs sin x

L + Gc cos 2x

L
33

where are specified 

constants. The first step in constructing the particular solution is to obtain all the input quantities 

that appear in equation (29c) for P1(x). Equations (1b) and (96a) give

 (98)

and equations (7b) and (96b) give

 (99)

Equations (B17) and (97) give

 (100a)

Likewise, 

 (100b)

 (100c)

In addition, equations (B17), (12f), and (97) give

 (100d)

where

 (101a)

 (101b)

Using equations (96c), (98), (99), and (100) with equation (29c) yields

qx

0

, qx

1

, qx

s

, qx

c

, q

0

, q

1

, q

s

, q

c

, qn

0

, qn

1

, qn

s

, qn

c

, H0, H1, Hs, Hc, G0, G1, Gs, and Gc

N1 x = 
 qx

0

x 
 qx

1

2 x

2

+ qx

s

L

 cos x

L 
 qx

c

L

2

sin 2x

L

T1 x = 
 q

0

x 
 q

1

2 x

2

+ q

s

L

 cos x

L 
 q

c

L

2

sin 2x

L

Nx

Th

x = H0nx

Th + G0cxTh + H1nxTh + G1cxTh

x

+ Hsnx

Th + GscxTh sin x

L + Hcnx

Th + GccxTh

cos 2x

L

N

Th

x = H0n

Th + G0cTh + H1nTh + G1cTh

x

+ Hsn

Th + GscTh sin x

L + Hcn

Th + GccTh

cos 2x

L

Mx

Th

x = H0cx

Th + G0mxTh + H1cxTh + G1mxTh

x

+ Hscx

Th + GsmxTh sin x

L + Hccx

Th + GcmxTh

cos 2x

L

T

Th

x = H0nx

Th + G0cxTh + H1nxTh + G1cxTh

x

+ Hsnx

Th + GscxTh sin x

L + Hcnx

Th + GccxTh

cos 2x

L

nx

Th

= nx

Th

+ μ

Rcx

Th

cx

Th

= cx

Th

+ μ

Rmx

Th
34

 (102a)

where

 (102b)

 (102c)

 (102d)

 (102e)

 (102f)

P1(x) = Y0 + Y1 x+Y2 x2

+ Ys1 sin x

L + Ys2 sin 2x

L + Yc1 cos x

L + Yc2 cos 2x

L

Y0 = 1

D11 e

qn

0


 qx

1

B11a 11 + B16a 16 
 q

1

B11a 16 + B16a 66

+

H0

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+

G0

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

Y1 = 1

D11 e

qn

1

+ qx

0

R A12a 11 + A26a 16 + q

0

R A12a 16 + A26a 66

+

H1

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+

G1

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

Y2 = 1

2RD11 e

qx

1

A12a 11 + A26a 16 + q

1

A12a 16 + A26a 66

Ys1 = 1

D11 e

qn

s

+

Hs

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+ Hs

2

L2 cx

Th


 nx

Th B11a 11 + B16a 16 
 nx

Th B11a 16 + B16a 66

+

Gs

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

+ Gs

2

L2 mx

Th


 cx

Th B11a 11 + B16a 16 
 cx

Th B11a 16 + B16a 66

Ys2 = 1

D11 e

2

L qx

c

B11a 11 + B16a 16 + q

c

B11a 16 + B16a 66

+ L

2R qx

c

A12a 11 + A26a 16 + q

c

A12a 16 + A26a 66
35

 (102g)

 (102h)

By applying the method of undetermined coefficients to equation (40) with P0 = 0, the particular 

solution is found to be given by

 (103)

where

 (104a)

 (104b)

 (104c)

 (104d)

 (104e)

Yc1 = 
 1

D11 e



L qx

s

B11a 11 + B16a 16 + q

s

B11a 16 + B16a 66

+ L

R qx

s

A12a 11 + A26a 16 + q

s

A12a 16 + A26a 66

Yc2 = 1

D11 e

qn

c

+

Hc

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+ Hc

42

L2 cx

Th


 nx

Th B11a 11 + B16a 16 
 nx

Th B11a 16 + B16a 66

+

Gc

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

+ Gc

42

L2 mx

Th


 cx

Th B11a 11 + B16a 16 
 cx

Th B11a 16 + B16a 66

wp x

wp x = wp

0

+ wp

1

x+wp

2

x

2

+ wp

s1 sin x

L + wp

s2 sin 2x

L + wp

c1

cos x

L + wp

c2

cos 2x

L

wp

0

= 1

4T2 Y0 
 2SY2

T2

wp

1

= Y1

4T2

wp

2

= Y2

4T2

wp

s1 = Ys1

4

L4 
 42L2 S + 4T2


 1

wp

s2 = Ys2

164

L4 
 162L2 S + 4T2


 1
36

 (104f)

 (104g)

Examples

Five examples are presented subsequently to illustrate the utility of the exact solution 

presented herein and some of the behavioral characteristics of laminated-composite cylinders that 

exhibit general forms of anisotropy. For each example, results are presented for length-to-radius 

ratios L/R = 0.5 and 2, and for fixed values of the radius-to-thickness ratio R/h. Cylinders with 

these two L/R values exhibit the full range of variations in behavior; that is, behavior in which 

zones of localized bending are isolated near the ends of a cylinder and behavior in which the 

bending behavior is not localized at all. Zones of localized bending are referred to herein as 

bending boundary layers. For every laminate considered, the lamina material properties are given 

by the major principal modulus E1 = 18.5 x 106

 psi, the minor principal modulus E2 = 1.64 x 106

psi, the shear modulus G12 = 0.87 x 106

 psi, the major Poisson’s ratio 12 = 0.30, the major principal 

coefficient of thermal expansion 1

 = 0.25 x 10- 6 /oF, and the minor principal coefficient of thermal 

expansion 2

 = 16.2 x 10- 6 /oF. For all laminates considered, the ply thicknesses are identical and 

the ply thickness is taken as the total thickness h divided by the number of plies. From a practical 

standpoint, atypically thick plies are considered to be composed of groups of thinner plies with 

the same orientation. For all results presented, based on the exact solution, the Sanders-Koiter 

equations were used, which corresponds to the value μ = 3/2. For several of the examples, 

simplified approximate solutions are also presented. Each simplified solution provides a 

necessary condition for establishing the correctness, and hence credibility, of the corresponding 

more complicated exact solution. Several examples also include the effects of temperature-change 

fields. For convenience, the temperature-change fields on the outside and inside of a cylinder are 

denoted by and respectively (see equations (B16)).

Example 1: Self-Equilibrated End Loads

Perhaps one of the most fundamental problems of shell theory is the attenuation behavior of 

a circular cylinder subjected to self-equilibrated end loads. For the example considered herein, the 

boundary conditions specified in the present study are

 (105a)

wp

c1 = Yc1

4

L4 
 42L2 S + 4T2


 1

wp

c2 = Yc2

164

L4 
 162L2 S + 4T2


 1

To = T x, h

2 Ti = T x, 
 h2 ,

Nx 0 = 0
37

 (105b)

 (105c)

 (105d)

at x = 0 and

 (106a)

 (106b)

 (106c)

 (106d)

at x = L. These boundary conditions correspond to a cylinder that is completely clamped around 

the circumference at x = L and subjected to a uniform transverse shear stress resultant and a 

uniform bending moment resultant, as depicted in figure 4. For these loading conditions, the 

particular solution given by equation (103) is identically zero-valued.

Results were obtained for geometrically identical cylinders with two very different 

laminated-wall constructions, and for the applied loads given by = 100 lbs and = 100 lbs/

in. The geometries considered are characterized by the radius-to-thickness ratio R/h = 100 and the 

length-to-radius ratios L/R = 0.5 and 2. In particular, results are presented in figures 4-7 for 

cylinders with laminated walls composed of all 90-degree plies and a total wall thickness h = 0.1 

in. This wall construction is referred to herein as specially orthotropic because the principal axes 

of each orthotropic ply are aligned with the x and coordinate directions shown in figure 3. 

Therefore, these laminates exhibit no form of anisotropy associated with coupling between 

extension and shear, coupling between bending and twisting, or coupling between any membrane 

deformation and any bending deformation. As a result, the cylinders with this wall construction 

do not exhibit any circumferential displacements. The values of the nonzero constitutive matrices 

computed for this laminate are given by 

 (107a)

T 0 = 0

Qx 0 = V0

Mx 0 = 0

u L = 0

v L = 0

w L = 0

	x

o

L = 0

M ̃ 0 V ̃ 0

A11 A12 A16

A12 A22 A26

A16 A26 A66

= 0.165319 0.0495957 0

0.0495957 1.86488 0

0 0 0.087000

 106 lbs

in.
38

 (107b)

Similarly, the constitutive expressions defined by equations (15) are given by

 (108a)

 (108b)

 (108c)

The values of S and T, defined by equations (27) and (39), respectively, that are used to 

determine the basis functions of the solution space are given by S = 0 and T = 5.79409 in.-2. 

In contrast to the specially orthotropic wall, results are presented in figures 8-13 for cylinders 

with laminated walls composed of all 10-degree plies and a total wall thickness h = 0.1 in. This 

wall construction exhibits anisotropy in the form of coupling between membrane expansion or 

contraction and shearing deformations, and coupling between pure bending and twisting 

deformations. The values of the nonzero constitutive matrices computed are given by 

 (109a)

 (109b)

Similarly, the constitutive expressions defined by equations (15) are given by

 (110a)

 (110b)

D11 D12 D16

D12 D22 D26

D16 D26 D66

= 0.137766 0.0413297 0

0.0413297 1.55407 0

0 0 0.0725000

 103in.
lbs

a 11 a 16

a 16 a 66

= 6.04891 0

0 11.494  10
 6 in.

lbs

b 12

b 62

= 
 0.3

0

c 12

c 62

= 0

0 in.

A11 A12 A16

A12 A22 A26

A16 A26 A66

= 1.76734 0.0958898 0.272513

0.0958898 0.170273 0.0181288

0.272513 0.0181288 0.133294

 106 lbs

in.

D11 D12 D16

D12 D22 D26

D16 D26 D66

= 1.47278 0.0799082 0.227094

0.0799082 0.141894 0.0151073

0.227094 0.0151073 0.111078

 103in.
lbs

a 11 a 16

a 16 a 66

= 0.826304 
 1.6893


 1.6893 10.9577  10
 6 in.

lbs

b 12

b 62

= 
 0.0486091

0.0366266
39

 (110c)

For this anisotropic-wall construction, S = 4.23574 x 10-5 in.-2 and T = 0.529147 in.-2. 

Graphs of the normalized radial displacement w/h, bending stress resultant transverseshear stress resultant and circumferential stress resultant for the cylinders with all 90-

degree plies are shown in figures 4-7, respectively, as a function of the axial distance x normalized 

by the cylinder length L. Two curves appear in each figure that correspond to values of L/R = 0.5 

and 2. These results exhibit the common trend in which the effects of the edge loads attenuate as 

x/L increases, with the longer cylinder exhibiting the larger attenuation rate. Likewise, graphs of 

the normalized axial displacement u/h, circumferential displacement v/h, radial displacement 

w/h, bending stress resultant transverse-shear stress resultant and circumferential 

stress resultant for the cylinders with all 10-degree plies are shown in figures 8-13, 

respectively, as a function of the normalized axial distance x/L. The results for these cylinders 

with L/R = 2 also exhibit the expected response-attenuation behavior. It is noteworthy to point out 

that the response quantities for the shorter cylinder with all 10-degree plies do not attenuate as x/

L increases, unlike the corresponding cylinders with all 90-degree plies. This difference in 

attenuation behavior is due to the higher axial stiffness of the cylinders with all 10-degree plies. 

The results in figure 9 show relatively small circumferential displacements, as expected, that are 

generated by the wall anisotropy. For both wall constructions, circumferential stresses (figure 13) 

are generated that are significantly larger in magnitude than the corresponding value of the 

applied transverse-shear load. Moreover, the largest magnitude of the circumferential stresses in 

the cylinders with all 90-degree plies (figure 7) are approximately six times larger than those in 

the cylinders with all 10-degree plies.

Example 2: Uniform Axial End Load

Another fundamental problem of shell theory is the response-attenuation behavior of a 

circular cylinder subjected to a uniform axial end load. This problem is particularly important in 

the study of cylinder buckling. For the corresponding example considered in the present study, 

the boundary conditions are given by

 (111a)

 (111b)

c 12

c 62

= 0.575447

3.731960  10
 4

in.

Mx

M0

,

Qx

V0

, N

V0

Mx

M0

, Qx

V0

,

N

V0

u 0 = 0

v 0 = 0
40

 (111c)

 (111d)

at x = 0 and

 (112a)

 (112b)

 (112c)

 (112d)

at x = L. For these loading conditions, the particular solution given by equation (103) is 

identically zero-valued.

Results for this example were also obtained for cylinders with radius-to-thickness ratio R/h 

= 100 and the length-to-radius ratios L/R = 0.5 and 2. Specifically, results are presented in figures 

14-19 for cylinders with laminated walls composed of two inner 90-degree plies, two outer 10-

degree plies, and a total wall thickness h = 0.1 in. This wall construction exhibits the full extent 

of anisotropy possible in the exact solution; that is, coupling between membrane expansion or 

contraction and shearing deformations, coupling between pure bending and twisting 

deformations, and coupling between membrane and bending-twisting deformations. The values 

of the constitutive matrices that were computed are given by 

 (113a)

 (113b)

 (113c)

Similarly, the constitutive expressions defined by equations (15) are given by

w 0 = 0

Mx 0 = 0

Nx L = L

v L = 0

w L = 0

Mx L = 0

A11 A12 A16

A12 A22 A26

A16 A26 A66

= 0.966328 0.0727428 0.136257

0.0727428 1.01758 0.00906441

0.136257 0.00906441 0.110147

 106 lbs

in.

B11 B12 B16

B12 B22 B26

B16 B26 B66

= 2.00252 0.0578677 0.340641

0.0578677 
 2.11826 0.022661

0.340641 0.022661 0.0578677

 104lbs

D11 D12 D16

D12 D22 D26

D16 D26 D66

= 8.05273 0.60619 1.13547

0.60619 8.4798 0.0755367

1.13547 0.0755367 0.917892

 102in.
lbs
41

 (114a)

 (114b)

 (114c)

For this laminate, S = 0.119157 in.-2 and T = 2.55869 in.-2. 

Graphs of the normalized axial displacement u/h, radial displacement w/h, bending stress 

resultants twisting stress resultant and circumferential stress resultant 

are shown in figures 14-19, respectively, as a function of the axial distance x normalized by the 

cylinder length L. Two pairs of blue and red curves appear in each figure that correspond to values 

of L/R = 0.5 and 2. The red curves correspond to an approximate solution, given in Appendix C, 

that is based on the equations of linear membrane theory, in which the bending boundary layer is 

ignored such that the radial displacement is presumed to be constant valued. In the figures with 

only one red curve, the curve corresponds to both values of L/R. For all results, a value of 

 lbs/in. was used for the end load. The blue curves correspond to exact solution 

presented herein.

The results in figure 14 show a nearly linear variation in the axial displacement, with the 

largest displacements being exhibited by the longer cylinder. Moreover, the results indicate that 

the approximate solution yields nearly the same results as the exact solution for both values of 

L/R. The normalized axial displacements at x = L were computed as u/h = 0.216 and 0.060 for 

L/R = 2 and 0.5, respectively. In contrast, the results in figures 15-19 show the presence of 

bending boundary layers in the longer cylinders with L/R = 2 that attenuate to the approximate 

solution, as expected, at distances from the cylinder ends that are about 10% of the cylinder 

length. The results also indicate that most of the the response quantities for the shorter cylinders 

with L/R = 0.5 do not exhibit zones of bending attenuation and, as a result, the approximate 

solution is inadequate for estimating the interior response of the cylinders. The normalized radial 

displacements at x = L/2 were computed as w/h = -0.007 and -0.004 for L/R = 2 and 0.5, 

respectively.

Example 3: Uniform Temperature and Internal Pressure Fields

A very practical problem of shell theory deals with the behavior of a circular cylinder 

subjected to uniform internal pressure and different, uniform internal and external temperature 

fields. For the corresponding example considered herein, the boundary conditions are given by

a 11 a 16

a 16 a 66

= 1.25506 
 1.5559


 1.5559 10.9932  10
 6 in.

lbs

b 12

b 62

= 
 0.0771401

0.0131601

c 12

c 62

= 
 1.98063

0.64773  10
 2

in.

Mx

Lh

and M

Lh

, Mx

Lh

, N

L

L = 1000
42

 (115a)

 (115b)

 (115c)

 (115d)

at x = 0 and

 (116a)

 (116b)

 (116c)

 (116d)

at x = L. Thus, the cylinder is completely clamped around the circumference at each end. The 

applied internal pressure given by and the applied thermal load is given by an 

external temperature increase T0 = 300 o

F and an internal temperature increase Ti = 100 oF. 

From equations (B16) and (97) it follows that 0(x) = H0 = 200 o

F and 1(x) = G0 = 2000 oF/in. 

For these loading conditions, the particular solution given by equation (103) reduces to

 (117a)

where Y0 given by equation (102b) reduces to

 (117b)

For this example, results were obtained for cylinders with radius-to-thickness ratio R/h = 100 

and the length-to-radius ratios L/R = 0.5 and 2. Selected results are presented in figures 20-27 for 

the previous generally anisotropic cylinders with laminated walls composed of two inner 90-

degree plies, two outer 10-degree plies, and a total wall thickness h = 0.1 in. The values of the 

u 0 = 0

v 0 = 0

w 0 = 0

	x

o

0 = 0

u L = 0

v L = 0

w L = 0

	x

o

L = 0

qn x = qn

0

= 100 psi

wp x = wp

0

= Y0

4T2

Y0 = 1

D11 e

qn

0

+

H0

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+

G0

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66
43

constitutive matrices computed are given by equations (113) and (114) and, as before, S = 

0.119157 in.-2 and T = 2.55869 in.-2. The values of the thermal constitutive parameters in equation 

(B17) that were computed are given by

 (118a)

 (118b)

 (118c)

Graphs of the normalized axial displacement u/h, circumferential displacement v/h, radial 

displacement w/h, circumferential stress resultant bending stress resultants 

 twisting stress resultant and transverse-shear 

stress resultant are shown in figures 20-27, respectively, as a function of the axial 

distance x normalized by the cylinder length L. Two pairs of blue curves and coincident red lines 

appear in each figure that correspond to values of L/R = 0.5 and 2. The red lines correspond to an 

approximate solution given in Appendix D that is based on the equations of linear membrane 

theory, in which the bending boundary layer is ignored such that the radial displacement is 

presumed to be constant valued.

The results in figures 20 and 21 show relatively small-magnitude antisymmetric distributions 

of the axial and circumferential displacements that attenuate to a value of zero in the central region 

of the cylinders, with the larger magnitudes occuring near the ends. The corresponding 

displacements predicted by the approximate solution are zero valued, as indicated by the red line 

in each figure. The exact solution gives the constant values Nx = -288.68 lbs/in. and -270.10 lbs/

in. for the cylinders with L/R = 0.5 and 2, respectively. In addition, the exact solution gives the 

constant values = 40.15 lbs/in. and 42.47 lbs/in. for the cylinders with L/R = 0.5 and 2, 

respectively. The approximate solution gives Nx = -263.90 lbs/in. and = 43.25 lbs/in., 

regardless of the value for L/R. 

nx

Th

n

Th

nx

Th

= 2.00154

1.9587


 0.121494

lbs

F

0


in.

cx

Th

c

Th

cx

Th

= 
 0.0172256

0.0172256


0.00303735

lbs

F

0

mx

Th

m

Th

mx

Th

= 0.00166795

0.00163225


 0. 000101245

in.
lbs

F

0

N

qn

0

R+N

Th ,

Mx

qn

0

R+N

Th h

and M

qn

0

R+N

Th h

, Mx

qn

0

R+N

Th h

,

Qx

qn

0

R+N

Th

T

T
44

The results in figures 22-27 show the presence of bending boundary layers in the longer 

cylinders with L/R = 2 that attenuate to the approximate solution, as expected, at distances from 

the cylinder ends that are about 10% of the cylinder length. The results also indicate that most of 

the response quantities for the shorter cylinder with L/R = 0.5 exhibit small centrally located 

zones of attenuation and, as a result, the approximate solution is inadequate for estimating the 

interior response of these cylinders. The approximate solution yields the values w/h = 0.14, 

and 

with lbs/in. The values agree well with the exact solution 

in the central, attenuated region of the longer cylinders. However, even in the central region of the 

shorter cylinders, there are significant differences between the approximate and exact solutions 

for several of these quantities. 

Example 4: Linearly Varying Temperature and Internal Pressure Fields

This example corresponds to cylinder proportions and loading conditions that are typical of 

a large-scale launch vehicle. Specifically, a circular cylinder subjected to an axial end load, an end 

circumferential shear load, a linearly varying internal pressure field, associated with a liquid fuel, 

and different, linearly varying internal and external temperature fields. For this example, the 

boundary conditions specified in the present study are

 (119a)

 (119b)

 (119c)

 (119d)

at x = 0 and

 (120a)

 (120b)

 (120c)

N

qn

0

R+N

Th = 0.701, Mx

qn

0

R+N

Th h

= 0.006, M

qn

0

R+N

Th h

= 
 0.255,

Mx

qn

0

R+N

Th h

= 0.008 N

Th = 426.19

Nx 0 = N0

T 0 = T0

w 0 = 0

Mx 0 = 0

u L = 0

v L = 0

w L = 0
45

 (120d)

at x = L. The values of the applied end loads used in the present example are given by 

lbs/in. and lbs/in. The applied pressure load is given by where pu

= 30 psi represents an ullage pressure and  = 0.04123 lb/in.3

 is the specific weight of the fluid 

inside the cylinder. Thus, equation (96c) reduces to where and 

 The applied thermal load is given by an external temperature rise T0

(x) = - 500 + 100 

x/L o

F and an internal temperature rise Ti(x) = - 500 + 50 x/L oF. From equations (B16) and (97) 

it follows that, and where H0 = - 500 o

F, G0 = 0 oF/in., 

H1 = 75/L o

F/in., and G1 = 50/(hL) oF/in.2. For these loading conditions, the particular solution 

given by equation (103) reduces to where and In 

addition, Y0 and Y1, given by equations (102), reduce to

 (121a)

and

 (121b)

For this example, results were obtained for cylinders with radius-to-thickness ratio R/h = 330 

and the length-to-radius ratios L/R = 0.5 and 2. Selected results are presented in figures 28-35 for 

the previously considered generally anisotropic cylinders with laminated walls composed of two 

inner 90-degree plies, two outer 10-degree plies, but with a total wall thickness h = 0.5 in. The 

values of the constitutive matrices computed are given by 

 (122a)

Mx L = 0

N0 = 1000

T0 = 200 qn x = pu +  L 
 x ,

qn x = qn

0

+ qn

1

x, qn

0

= pu + L

qn

1

= 
 .

0 x = H0 + H1 x 1 x = G0 + G1 x,

wp x = wp

0

+ wp

1

x, wp

0

= Y0

4T2 wp

1

= Y1

4T2 .

Y0 = 1

D11 e

qn

0

+

H0

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+

G0

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

Y1 = 1

D11 e

qn

1

+

H1

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+

G1

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

A11 A12 A16

A12 A22 A26

A16 A26 A66

= 4.83164 0.363714 0.681283

0.363714 5.08788 0.045322

0.681283 0.045322 0.550735

 106 lbs

in.
46

 (122b)

 (122c)

and the constitutive expressions defined by equations (15) are given by

 (123a)

 (123b)

 (123c)

The values of S and T, defined by equations (27) and (39), respectively, that are used to 

determine the basis functions of the solution space were found to be given by S = 0.001444 in.-2

and T = 0.031012 in.-2, and the values of the thermal constitutive parameters in equation (B17) 

are given by

 (124a)

 (124b)

 (124c)

Graphs of the normalized axial displacement u/h, circumferential displacement v/h, radial 

B11 B12 B16

B12 B22 B26

B16 B26 B66

= 5.00630 0.144669 0.851603

0.144669 
 5.29564 0.0566525

0.851603 0.0566525 0.144669

 105lbs

D11 D12 D16

D12 D22 D26

D16 D26 D66

= 100659 7577.37 14193.4

7577.37 105997 944.209

14193.4 944.209 11473.7

in.
lbs

a 11 a 16

a 16 a 66

= 2.50793 
 3.10445


 3.10445 2.19917  10
 6 in.

lbs

b 12

b 62

= 
 0.0771309

0.0131291

c 12

c 62

= 
 0.0990770

0.0321475 in.

nx

Th

n

Th

nx

Th

= 10.0077

9.79348


 0.607469

lbs

F

0


in.

cx

Th

c

Th

cx

Th

= 
 0.430641

0.430641


0.0759336

lbs

F

0

mx

Th

m

Th

mx

Th

= 0.208494

0.204031


 0.0126556

in.
lbs

F

0
47

displacement w/h, circumferential stress resultant bending stress resultants 

 twisting stress resultant and transverse-shear 

stress resultant are shown in figures 28-35, respectively, as a function of the axial 

distance x normalized by the cylinder length L. The denominator term pu + L represents the 

maximum pressure in the cylinder. Two pairs of blue and red curves appear in each figure that 

correspond to values of L/R = 0.5 and 2. The red curves correspond to an approximate solution 

given in Appendix E that is found by applying the equations of linear membrane theory, ignoring 

the bending boundary layer and presuming that the radial displacements vary like the pressure and 

temperature fields. In the figures with only one red curve, the curve corresponds to both values of 

L/R.

The results in figures 28 and 29 show a nearly linear variation in the axial and circumferential 

displacements, respectively, with the largest displacement magnitudes being exhibited by the 

longer cylinder. Moreover, the results indicate that the approximate solution yields nearly the 

same results as the exact solution for both values of L/R. In contrast, the results in figures 30-35 

show the presence of bending boundary layers that attenuate to the linearly varying approximate 

solution at distances from the cylinder ends that are about 10% and 20% of the cylinder length for 

L/R = 2 and 0.5, respectively. The linear variation in the response quantities is generally more 

pronounced in the longer cylinders because the pressure increases with increasing cylinder length.

Example 5: Sinusoidal Pressure and Temperature Fields

This example illustrates the behavior of a circular cylinder subjected to sinusoidal internal 

pressure and different, sinusoidal internal and external temperature fields. The cylinder is fixed at 

one end and free to expand axially and twist circumferentially at the other end. The corresponding 

boundary conditions specified in the present study are

 (125a)

 (125b)

 (125c)

 (125d)

N

pu + L R

,

Mx

pu + L Rh

and M

pu + L Rh, Mxpu + L Rh,

Qx

pu + L R

u 0 = 0

v 0 = 0

w 0 = 0

	x

o

0 = 0
48

at x = 0 and

 (126a)

 (126b)

 (126c)

 (126d)

at x = L. The applied mechanical load given by where psi. The 

applied thermal load is given by an external temperature rise T0(x) = 300 sin(x/L) o

F and an 

internal temperature rise Ti

(x) = 100 sin(x/L) oF. Equations (B16) and (97) give 0(x) = Hs

sin(x/L) and 1(x) = Gs

 sin(x/L) where Hs = 200 oF and Gs = 200/h oF/in. For these loading 

conditions, the particular solution given by equation (103) reduces to 

 (127)

where is given by equation (104d).

For this example, results were obtained for cylinders with radius-to-thickness ratio R/h = 100 

and the length-to-radius ratios L/R = 0.5 and 2. Selected results are presented in figures 36-45 for 

generally anisotropic cylinders with laminated walls composed of two inner 90-degree plies, two 

outer 10-degree plies, and a total wall thickness h = 0.1 in. The values of the constitutive matrices 

are given by equations (113) and (114), and the values of the thermal constitutive parameters in 

equation (B17) are given by equations (118). Like before, S = 0.119157 in.-2 and T = 2.55869 in.-2. 

Graphs of the normalized axial displacement u/h, circumferential displacement v/h, and 

radial displacement w/h are shown in figures 36-38, respectively, as a function of the axial 

distance x normalized by the cylinder length L, and the axial rotation is shown in figure 39 as 

a function of x/L. Graphs of the normalized circumferential stress resultant bending stress 

resultants twisting stress resultant and transverse-shear stress resultants 

 are shown in figures 40-45, respectively, as a function of x/L. In these 

nondimensional quantities, is given by equation (F10b), is given by equation (F7b), and 

 is given by equation (F11b). One blue curve and one red curve is shown in each part of the 

Nx 0 = 0

T 0 = 0

w L = 0

	x

o

L = 0

qn x = qn

s

sin x

L , qn

s

= 1000

wp x = wp

s1 sin x

L

wp

s1

	x

0

N

N

,

Mx

Mx

and M

M

, Mx

Mx

,

Qxh

Mx

and Qh

Mx

NMx

M49

two-part figures. The red curves correspond to a simplified exact trigonometric solution, given in 

Appendix F, for the corresponding cylinder with simply supported ends instead of clamped ends. 

The blue curves represent the exact solution obtained herein.

The results in figures 36-45 show that the difference between the exact solution (blue 

response curves) for the cylinders with rotation restrained at the ends and the exact trigonometric 

solution for simply supported ends is much smaller for the cylinders with L/R = 2 than for those 

with L/R = 0.5, as expected. For the longer cylinders, significant differences between most of the 

corresponding response curves appear to be confined to edge zones that are about 10% of the 

cylinder length. For the shorter cylinders, the zones of significant difference grow to about 20% 

of the cylinder length. For both cylinder lengths, the greatest overall discrepancies between the 

exact and corresponding trigonometric solutions are exhibted by the axial displacements and the 

bending stress resultant Mx.

Concluding Remarks

A detailed exact solution has been presented for generally laminated circular cylinders that 

undergo axisymmetric deformations. The cylinders exhibit the full range of anisotropy that is 

possible in first-approximation shell theories that neglect high-order effects such as transverseshear deformation. The overall solution has been formulated in a general, systematic way and is 

based on the solution of a single fourth-order, nonhomogeneous ordinary differential equation 

with constant coefficients in which the radial displacement is the dependent variable. Moreover, 

positive-definiteness of the strain energy has been used to define uniquely the form of the basis 

functions spanning the solution space of the ordinary differential equation. This approach appears 

to be a unique feature of the analysis presented herein.

Axisymmetric loading conditions have been considered that include edge loads, surface 

tractions, and temperature fields. In addition, all possible corresponding boundary conditions 

have been considered, and several particular solutions to the fourth-order, nonhomogeneous 

ordinary differential equation governing the response have been presented. Results have been 

presented for five examples that demonstrate a wide range of behavior. Several of the results 

illustrate how laminated-wall construction and cylinder length affect the size of localized zones 

of bending deformations that occur near the cylinder ends. For some of the examples, the results 

indicate highly localized bending deformations that attenuate rapidly. For other examples, the 

bending deformations exhibit very little attenuation. In addition, approximate solutions for 

several of the examples have been presented that add credibility to the corresponding results 

obtained from the exact solution and that provide insight into the possible pitfalls of making 

certain types of simplifications in a preliminary-design process. For example, several results are 

presented that illustrate conditions for which sizing a laminated-composite cylinder based on an 

approximate linear membrane analysis is inappropriate.

Results obtained from the exact solution presented herein also have the potential to serve as 

fundamental benchmarks for the validation of numerical solution methods such as the finiteelement method. Likewise, results obtained from the exact solution have the potential to provide 
50

insight into the extent of discretization that might be needed for analyzing accurately shells with 

zones of localized bending caused by cutouts, stiffeners, or other types of discontinuities and 

gradients.

References

1. Dong, S. B.; Pister, K.. S.; and Taylor, R. L.: On the Theory of Laminated Anisotropic Shells 

and Plates. Journal of the Aerospace Sciences, vol. 29, no. 8, 1962, pp. 969-975.

2. Brush, D. O.; and Almroth, B.: Buckling of Bars, Plates, and Shells. McGraw-Hill, 1975.

3. Gulati, S. T.; and Essenburg, F.: Effects of Anisotropy in Axisymmetric Cylindrical Shells. 

Journal of Applied Mechanics, vol. 34, September, 1967, pp. 659-666.

4. Stavsky, Y.; and Smolash, I.: Thermoelasticity of Heterogeneous Orthotropic Cylinders. 

International Journal of Solids and Structures, vol. 6, 1970, pp. 1211-1231.

5. Vicario, A. A.; and Rizzo, R. R.: Effect of Length on Laminated Thin Tubes Under Combined 

Loading. Journal of Composite Materials, vol. 4, 1970, pp. 273-277.

6. Reuter, R. C., Jr.: Analysis of Shells Under Internal Pressure. Journal of Composite Materials, 

Vol. 6, 1972, pp. 94-113.

7. Hennemann, J. C. F.: Effect of Prebuckling Deformations on Buckling of Laminated 

Composite Circular Cylindrical Shells. Ph. D. Dissertation, Civil and Mechanical Engineering 

Dept., Southern Methodist University, Dallas, Texas, 1975.

8. Jones, R. M.; and Hennemann, J. C. F.: Effect of Prebuckling Deformations on Buckling of 

Laminated Composite Circular Cylindrical Shells. Proceedings of the AIAA/ASME 19th 

Structures, Structural Dynamics, and Materials Conference, Bethesda, Maryland, April 3-5, 

1978, pp. 370-379.

9. Cheng, S.; and He, F. B.: Theory of Orthotropic and Composite Cylindrical Shells, Accurate 

and Simple Fourth-Order Governing Equations. Journal of Applied Mechanics, vol. 51, 1984, 

pp. 736-744.

10. Flügge, W.: Stresses in Shells, Springer-Verlag, Berlin, 1967.

11. Chaudhuri, R. A.; Balaraman, K.; and Kunukkasseril, V. X.: Arbitrarily Laminated, 

Anisotropic Cylindrical Shell Under Internal Pressure. AIAA Journal, vol. 24, no. 11, 1986, 

pp. 1851-1858.

12. Kraus, H.: Thin Elastic Shells - An Introduction to the Theoretical Foundations and the 

Analysis of Their Static and Dynamic Behavior, John Wiley and Sons, Inc., 1967.
51

13. Timoshenko, S. P.; and Gere, J. M.: Theory of Elastic Stability. Second ed., McGraw-Hill, 

1961.

14. Abu-Arja, K. R.; and Chaudhuri, R. A.: Moderately Thick Angle-ply Cylindrical Shells 

Under Internal Pressure. Journal of Applied Mechanics, vol. 56, 1989, pp. 652-657.

15. Butler, T. A.; and Hyer, M. W.: The Structural Response of Unsymmetrically Laminated 

Composite Cylinders. Dept. of Mechanical Engineering, Technical Report 89-2, The 

University of Maryland, August 1989.

16. Hyer, M. W.; and Paraska, P. J., "Innovative Design of Composite Structures: Axisymmetric 

Deformations of Unsymmetrically Laminated Cylinders Loaded in Axial Compression," 

Dept. of Engineering Science and Mechanics, VPI-E-90-10 Report, VA Tech, June 1990.

17. Paraska, P. J.: Axisymmetric Deformations and Stresses of Unsymmetrically Laminated 

Composite Cylinders in Axial Compression with Thermally-Induced Preloading Effects. 

CARDEROCKDIV-U-SSM-65-93/03, Carderock Division, Naval Surface Warfare Center, 

March 1993.

18. Birman, V.: Thermal Bending of Shear-Deformable Orthotropic Cylindrical Shells 

Reinforced by Cylindrically-Orthotropic Rings. International Journal of Solids and 

Structures, vol. 28, no. 7, 1991, pp. 819-830.

19. Birman, V.: Exact Solution of Axisymmetric Problems of Laminated Cylindrical Shells with 

Arbitrary Boundary Conditions - Higher-Order Theory. Mechanics Research 

Communications, vol. 19, no. 3, 1992, pp. 219-225.

20. Birman, V.: Axisymmetric Bending of Generally Laminated Cylindrical Shells. Journal of 

Applied Mechanics, vol. 60, 1993, pp. 157-162.

21. Vinson, J. R.: Composite Shell Structures and Their Behavior. Proceedings of the AIAA/

ASME/ASCE/AHS/ASC 38th Structures, Structural Dynamics, and Materials Conference, 

Kissimmee, Florida, April 7-10, 1997, pp. 2400-2406. 

22. McDevitt, T. J.; and Simmonds, J. G.: Reduction of the Sanders-Koiter Equations for Fully 

Anisotropic Circular Cylindrical Shells to Two Coupled Equations for a Stress and a 

Curvature Function. Journal of Applied Mechanics, vol. 66, September, 1999, pp. 593-597.

23. Goldenveizer, A. L.: Theory of Thin Elastic Shells, Pergamon, New York, 1961.

24. Sanders, J. L., Jr.: An Improved First-Approximation Theory for Thin Shells. NASA TR R-24, 

1959.

25. Koiter, W. T.: A Consistent First Approximation in the General Theory of Thin Elastic Shells. 

The Theory of Thin Elastic Shells, Proceedings of the IUTAM Symposium, Delft, NorthHolland, Amsterdam, The Netherlands, 1960, pp. 12-33.
52

26. Nemeth, M. P.; and Smeltzer, S. S., III: Bending Boundary Layers in Laminated-Composite 

Circular Cylindrical Shells. NASA/TP-2000-210549, 2000.

27. Paris, A. J.; and Costello, G. A.: Bending of Cord Composite Cylindrical Shells. Journal of 

Applied Mechanics, vol. 67, March, 2000, pp. 117-127.

28. Ye, J.: Decay Rate of Edge Effects in Cross-Ply-Laminated Hollow Cylinders. International 

Journal of Mechanical Sciences, vol. 43, 2001, pp. 455-470.

29. Ye, J.: Edge Effects in Angle-Ply Laminated Hollow Cylinders. Composite Structures, vol. 

52, 2001, pp. 247-253.

30. Bhaskar, K.; and Balasubramanyam, G.: Accurate Analysis of End-Loaded Laminated 

Orthotropic Cylindrical Shells. Composite Structures, vol. 58, 2002, pp. 209-216.

 

31. Paris, A. J.; and Costello, G. A.: Bending of Cord Composite Laminate Cylindrical Shells. 

Journal of Applied Mechanics, vol. 70, May 2003, pp. 364-373.

32. Kollar, L. P.; and Springer, G. S.: Mechanics of Composite Structures. Cambridge University 

Press, 2003, pp. 368-380.

33. Standard Mathematical Tables and Formulae, D. Zwillinger, ed., CRC Press, 1996.

34. Jones, R. M.: Mechanics of Composite Materials, Second ed., Taylor and Francis, 1999.
53

USA

United States

External tank

Solid rocket booster

Orbiter

Aft orbiter

attachment

fitting

Aft SRB

attachment

 point

Liquid-oxygen tank

Intertank

Liquid-hydrogen tank

attachment fitting

Forward orbiter

Figure 1. Space Shuttle external tank components.

Foreward SRB

attachment

 point
54

Figure 2. Conceptual lunar lander.

R

h



x

Typical lamina fiber

L

Figure 3. Cylinder geometry, surface coordinate system, and lamina fiber orientation.



z
55

Radial

displacement,

w/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.2

0

- 0.2

- 0.4

- 0.6

- 0.8

- 1.0

Figure 4. Radial displacement of laminated cylinder with all 90-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

L/R = 0.5

L/R = 2

M0 = 100 lbs

V0 = 100 lbs

in.

Bending stress

resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

1.2

1

0.8

0.6

0.4

0

- 0.2

Figure 5. Bending stress resultant for laminated cylinder with all 90-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

M0

~

Mx

0.2

L/R = 0.5

L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.
56

Transverse shear

stress resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

1.5

1

0.5

0

- 0.5

- 1.5

- 2

Figure 6. Transverse shear stress resultant for laminated cylinder with all 90-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

V0

~

Qx

-1

L/R = 0.5

L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.

Circumferential

stress resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

50

0

-50

-100

- 150

- 200

Figure 7. Circumferential stress resultant for laminated cylinder with all 90-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

V0

~

N

L/R = 0.5

L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.
57

Axial

displacement,

u/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.01

0.005

0

- 0.005

- 0.01

Figure 8. Axial displacement of laminated cylinder with all 10-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

L/R = 0.5

L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.

Circumferential

displacement,

v/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.01

0.005

0

- 0.005

- 0.01

Figure 9. Circumferential displacement of laminated cylinder with all 10-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

L/R = 0.5

L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.
58

Radial

displacement,

w/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.5

0

- 0.5

- 1.0

- 2.0

Figure 10. Radial displacement of laminated cylinder with all 10-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

L/R = 0.5 L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

- 1.5

M0 = 100 lbs

V0 = 100 lbs

in.

Bending stress

resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

1.5

1.0

0.5

0

- 0.5

Figure 11. Bending stress resultant for laminated cylinder with all 10-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

M0

~

Mx

L/R = 2 L/R = 0.5

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.
59

Transverse shear

stress resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

1.5

1.0

0.5

0

- 1.0

Figure 12. Transverse shear stress resultant for laminated cylinder with all 10-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

V0

~

Qx

- 0.5 L/R = 0.5

L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.

Circumferential

stress resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

5

0

-5

-10

-15

- 25

- 30

Figure 13. Circumferential stress resultant for laminated cylinder with all 10-deg plies, subjected to edge loads at x = 0,

and clamped at x = L (R/h = 100).

V0

~

N

- 20

L/R = 0.5

L/R = 2

2R

h

x

x = L

x = 0

V0

~

M0

~

M0

~

V0

~

M0 = 100 lbs

V0 = 100 lbs

in.
60

Axial

displacement,

u/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.25

0.20

0.15

0.10

0

Figure 14. Axial displacement of [902/102]T laminated cylinder subjected to an axial edge load at x = L,

with u, v, and w fixed at x = 0, and with v and w fixed at x = L (R/h = 100).

L/R = 0.5

L/R = 2

0.05

Approximate solution 

2R

h

x

x = 0 x = L

NL

~

NL = 1000 lb/in. ~

Radial

displacement,

w/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.01

0

- 0.01

L/R = 0.5

L/R = 2

2R

h

x

x = 0 x = L

NL

~

Figure 15. Radial displacement of [902/102]T laminated cylinder subjected to an axial edge load at x = L,

with u, v, and w fixed at x = 0, and with v and w fixed at x = L (R/h = 100).

- 0.02

- 0.03

- 0.04

- 0.05

NL = 1000 lb/in. ~

Approximate solution 
61

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.4

0.3

0.2

0

L/R = 0.5

L/R = 2 0.1

2R

h

x

x = 0 x = L

NL

~

Bending stress

resultant,

NLh ~

Mx

Figure 16. Bending stress resultants of [902/102]T laminated cylinder subjected to an axial edge load at x = L,

with u, v, and w fixed at x = 0, and with v and w fixed at x = L (R/h = 100).

NL = 1000 lb/in. ~

Approximate solution 

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.15

0.10

0.05

- 0.05

L/R = 0.5

L/R = 2

0

Bending stress

resultant,

NLh ~

M

2R

h

x

x = 0 x = L

NL

~

Figure 17. Bending stress resultants of [902/102]T laminated cylinder subjected to an axial edge load at x = L,

with u, v, and w fixed at x = 0, and with v and w fixed at x = L (R/h = 100).

NL = 1000 lb/in. ~

Approximate solution 
62

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.20

0.15

0.10

0

L/R = 2 L/R = 0.5

0.05

Approximate solution 

Twisting stress

resultant,

NLh ~

Mx

2R

h

x

x = 0 x = L

NL

~

Figure 18. Twisting stress resultants of [902/102]T laminated cylinder subjected to an axial edge load at x = L,

with u, v, and w fixed at x = 0, and with v and w fixed at x = L (R/h = 100).

NL = 1000 lb/in. ~

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.2

0

- 0.2

- 0.6

L/R = 0.5

L/R = 2

- 0.4

Approximate solution 

Figure 19. Circumferential stress resultants of [902/102]T laminated cylinder subjected to an axial edge load at x = L,

with u, v, and w fixed at x = 0, and with v and w fixed at x = L (R/h = 100).

NL

~

N

Circumferential

stress resultant,

NL = 1000 lb/in. ~
63

Axial

displacement,

u/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.015

0.010

0.005

- 0.005

Figure 20. Axial displacement of [902/102]T laminated cylinder clamped at both ends and subjected internal

pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

0

Approximate solution 

2R

h

x

x = 0 x = L

qn

0

T0

Ti

T0 = 300 oF

Ti = 100 oF

qn

0

= 100 psi

Circumferential

displacement,

v/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.015

0.010

0.005

- 0.005

Figure 21. Circumferential displacement of [902/102]T laminated cylinder clamped at both ends and subjected internal

pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

0

Approximate solution 

2R

h

x

x = 0 x = L

qn

0

T0

Ti

T0 = 300 oF

Ti

 = 100 oF

qn

0

= 100 psi
64

Radial

displacement,

w/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.4

0.3

0.2

0

Figure 22. Radial displacement of [902/102]T laminated cylinder clamped at both ends and subjected internal

pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

0.1

Approximate solution 2R

h

x

x = 0 x = L

qn

0

T0

Ti

T0 = 300 oF

Ti = 100 oF

qn

0

= 100 psi

Circumferential

stress resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

1.5

1.0

0.5

Figure 23. Circumferential stress resultant of [902/102]T laminated cylinder clamped at both ends and

subjected internal pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

Approximate solution 

0

- 0.5

N

qn

0

R+N

Th

2R

h

x

x = 0 x = L

qn

0

T0

Ti

T0 = 300 oF

Ti = 100 oF

qn

0

= 100 psi
65

Bending stress

resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.2

0.1

0

Figure 24. Bending stress resultant of [902/102]T laminated cylinder clamped at both ends and subjected to

internal pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

Approximate solution 

2R

h

x

x = 0 x = L

qn

0

T0

Ti

Mx

qn

0

R+N

Th h

- 0.1

- 0.2

T0 = 300 oF

Ti

 = 100 oF

qn

0

= 100 psi

Bending stress

resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.1

0

Figure 25. Bending stress resultant of [902/102]T laminated cylinder clamped at both ends and subjected to

internal pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

- 0.1

Approximate solution 

2R

h

x

x = 0 x = L

qn

0

T0

Ti

- 0.2

- 0.3

M

qn

0

R+N

Th h

T0 = 300 oF

Ti = 100 oF

qn

0

= 100 psi
66

Twisting stress

resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.03

0.02

0.01

Figure 26. Twisting stress resultant of [902/102]T laminated cylinder clamped at both ends and subjected to

internal pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

0

Approximate solution 

- 0.01

- 0.02

Mx

qn

0

R+N

Th h

2R

h

x

x = 0 x = L

qn

0

T0

Ti

T0 = 300 oF

Ti

 = 100 oF

qn

0

= 100 psi

Transverse-shear

stress resultant,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.10

0.05

Figure 27. Transverse-shear stress resultant of [902/102]T laminated cylinder clamped at both ends and

subjected to internal pressure and uniform interior and exterior temperature fields (R/h = 100).

L/R = 0.5

L/R = 2

Approximate solution 

0

- 0.10

2R h

x

x = 0 x = L

qn

0

T0

Ti

Qx

qn

0

R+N

Th

- 0.05

T0 = 300 oF

Ti = 100 oF

qn

0

= 100 psi
67

Axial

displacement,

u/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

1.0

0.8

0.6

0.4

0

Figure 28. Axial displacement of [902/102]T laminated cylinder subjected to edge loads at x = 0, linearly varying

pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 0.5

L/R = 2

0.2

Approximate solution 2R

x

x = L x = 0

h

qn

N0 

~

T0 = - 500 + 100 x/L oF

Ti = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3

T0

Ti

Circumferential

displacement,

v/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.5

0

-0.5

-1.5

Figure 29. Circumferential displacement of [902/102]T laminated cylinder subjected to edge loads at x = 0, linearly 

varying pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 2

L/R = 0.5

-1.0

Approximate solution 2R

x

x = L x = 0

h

qn

N0 

~

T0

Ti

T0 = - 500 + 100 x/L oF

Ti

 = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3
68

Radial

displacement,

w/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.4

0.3

0.2

0

Figure 30. Radial displacement of [902/102]T laminated cylinder subjected to edge loads at x = 0, linearly 

varying pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 0.5

L/R = 2

0.1

Approximate solution 

2R

x

x = L x = 0

h

qn

N0 

~

T0

Ti

T0 = - 500 + 100 x/L oF

Ti

 = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2.0

1.5

1.0

0

Figure 31. Circumferential stress resultant of [902/102]T laminated cylinder subjected to edge loads at x = 0, 

linearly varying pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 0.5

L/R = 2

0.5

Approximate solution 

Circumferential

stress resultant,

N

pu + L R

2R

x

x = L x = 0

h

qn

N0 

~

T0

Ti

T0 = - 500 + 100 x/L oF

Ti

 = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3
69

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.4

0.2

0

-0.4

Figure 32. Bending stress resultant of [902/102]T laminated cylinder subjected to edge loads at x = 0, linearly 

varying pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 0.5

L/R = 2

-0.2

Approximate solution 

Bending stress

resultant,

Mx

pu + L Rh

2R

x

x = L x = 0

h

qn

N0 

~

T0

Ti

T0 = - 500 + 100 x/L oF

Ti

 = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.3

0.2

0.1

-0.1

Figure 33. Bending stress resultant of [902/102]T laminated cylinder subjected to edge loads at x = 0, linearly 

varying pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 0.5

L/R = 2

0

Approximate solution 

Bending stress

resultant,

M

pu + L Rh

2R

x

x = L x = 0

h

qn

N0 

~

T0

Ti

T0 = - 500 + 100 x/L oF

Ti = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3
70

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.02

0

-0.02

-0.06

Figure 34. Twisting stress resultant of [902/102]T laminated cylinder subjected to edge loads at x = 0, linearly 

varying pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 0.5

L/R = 2

-0.04

Approximate solution 

Twisting stress

resultant,

Mx

pu + L Rh

2R

x

x = L x = 0

h

qn

N0 

~

T0

Ti

T0 = - 500 + 100 x/L oF

Ti

 = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.03

0

0.01

-0.02

Figure 35. Transverse-shear stress resultant of [902/102]T laminated cylinder subjected to edge loads at x = 0, 

linearly varying pressure and temperatures, with u, v, and w fixed at x = L, and with w fixed at x = 0 (R/h = 330).

L/R = 0.5

L/R = 2

-0.01 Approximate solution 

Transverse-shear 

stress resultant,

0.02

Qx

pu + L R

2R

x

x = L x = 0

h

qn

N0 

~

T0

Ti

T0 = - 500 + 100 x/L oF

Ti = - 500 + 50 x/L oF

qn = pu + (L - x) psi 

N0 = 1000 lb/in. ~

T0 = 200 lb/in. ~

pu = 30 psi

 = 0.04123 lb/in.3
71

Axial

displacement,

u/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.02

0

-0.06

-0.04

-0.02

Axial

displacement,

u/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.02

0

-0.06

Figure 36. Axial displacement of [902/102]T laminated cylinder subjected to sinusoidal pressure and temperatures,

fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential motion (R/h = 100).

-0.04

Trigonometric solution,

-0.02

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 

h = 0.10 in.

h = 0.10 in.

simply supported ends 
72

Circumferential

displacement,

v/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.05

0

-0.15

-0.10

-0.05

Circumferential

displacement,

v/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.05

0

-0.15

Figure 37. Circumferential displacement of [902/102]T laminated cylinder subjected to sinusoidal pressure

and temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

-0.10

-0.05

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

h = 0.10 in.

h = 0.10 in.

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 
73

Radial

displacement,

w/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2.0

1.5

0

0.5

1.0

Radial

displacement,

w/h

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2.0

1.5

0

Figure 38. Radial displacement of [902/102]T laminated cylinder subjected to sinusoidal pressure

and temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

0.5

1.0

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

h = 0.10 in.

h = 0.10 in.

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti

 = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 
74

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

10

5

- 10

- 5

0

Axial

rotation,

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

10

5

- 10

Figure 39. Axial rotation of [902/102]T laminated cylinder subjected to sinusoidal pressure

and temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

- 5

0

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

h = 0.10 in.

h = 0.10 in.

	 x

o

, degrees

Axial

rotation,

	 x

o

, degrees

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 
75

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2.0

1.5

0

0.5

1.0

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2.0

1.5

0

Figure 40. Circumferential stress resultant of [902/102]T laminated cylinder subjected to sinusoidal pressure

and temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

0.5

1.0

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

Circumferential

stress resultant,

N

N

Circumferential

stress resultant,

N

N

h = 0.10 in.

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 

N= 10000.3 lb

in.

h = 0.10 in.

N= 10000.3 lb

in.
76

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2

0

-6

-4

-2

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

15

10

-5

Figure 41. Bending stress resultant of [902/102]T laminated cylinder subjected to sinusoidal pressure and

temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

0

5

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

Bending stress

resultant,

Mx

Mx

Bending stress

resultant,

Mx

Mx

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 

h = 0.10 in.

h = 0.10 in.

Mx = 
 1.33 lbs

Mx = 13.36 lbs
77

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2.0

1.5

0

0.5

1.0

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2.0

1.5

0

Figure 42. Bending stress resultant of [902/102]T laminated cylinder subjected to sinusoidal pressure and

temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

0.5

1.0

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

Bending stress

resultant,

Bending stress

resultant,

M

M

M

M

h = 0.10 in.

h = 0.10 in.

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti

 = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 

M= 
 224.60 lbs

M= 
 222.41 lbs
78

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2

1

-2

-1

0

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

2

1

-2

Figure 43. Twisting stress resultant of [902/102]T laminated cylinder subjected to sinusoidal pressure and

temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

-1

0

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

Twisting stress

resultant,

Twisting stress

resultant,

Mx

Mx

Mx

Mx

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 

h = 0.10 in.

h = 0.10 in.

Mx = 
 1.33 lbs

Mx = 13.36 lbs
79

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

3

1

-3

-2

0

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

3

1

-3

Figure 44. NTransverse-shear stress resultant of [902/102]T laminated cylinder subjected to sinusoidal pressure

and temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

-1

0

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

Transverse-shear

stress resultant,

Qxh

Mx

Transverse-shear

stress resultant,

Qxh

Mx

-2

2

-1

2 Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 

h = 0.10 in.

h = 0.10 in.

Mx = 
 1.33 lbs

Mx = 13.36 lbs
80

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.4

0.2

-0.4

-0.2

0

0 0.2 0.4 0.6 0.8 1.0

Axial distance, x/L

0.4

-0.4

Figure 45. Transverse-shear stress resultant of [902/102]T laminated cylinder subjected to sinusoidal pressure

and temperatures, fixed at x = 0, and partially fixed at x = L to permit unrestrained axial and circumferential 

0

(a) Length-to-radius ratio, L/R = 0.5

(b) Length-to-radius ratio, L/R = 2

2R h

x

x = L x = 0

T0

Ti qn

motion (R/h = 100).

Transverse-shear

stress resultant,

Transverse-shear

stress resultant,

-0.2

0.2

Qh

Mx

Qh

Mx

h = 0.10 in.

h = 0.10 in.

Trigonometric solution,

simply supported ends 

T0 = 300 sin(x/L) oF

Ti = 100 sin(x/L) oF

qn = 1000 sin(x/L) psi 

Mx = 
 1.33 lbs

Mx = 13.36 lbs
81

Appendix A

Sanders-Koiter Equations

The linear Sanders-Koiter shell equations24,25 are presented in this appendix for a rightcircular cylinder with a radius that is given by R. For these equations, x is the axial coordinate, 

is the circumferential coordinate, and z is the radial coordinate, as shown in figure 3. First, the 

equilibrium equations are presented, then the kinematic equations and the constitutive equations 

are presented. Last, the boundary conditions are given for a complete right-circular cylinder at an 

edge that is given by a constant value of x.

Equilibrium Equations

The equilibrium equations are given in a form similar to those found in Ref. 24; that is,

 (A1)

 (A2)

 (A3)

 (A4)

 (A5)

where Nx, N, and Nxare the membrane stress resultants; Qx and Qare the transverse shearstress resultants; Mx, M, and Mxare the bending stress resultants; qx, q, and qn are the applied 

surface tractions; and c1 and c2 are constants that identify the equations of other shell theories that 

are considered herein. In particular, the Sanders-Koiter equations are given by c1 = c2 = 1 and the 

Love-Kirchhoff equations are give by c1 = 1 and c2 = 0. Donnell’s equations are given by c1

 = c2

= 0. This convention is used throughout the present study.

Kinematic Equations

The kinematic equations are given by

 (A6)

Nx

x

+ 1

R

Nx


 c2

2R2

Mx

+ qx = 0

Nx

x

+ 1

R

N

+

c1

R Q+

c2

2R

Mx

x + q= 0

Qx

x

+ 1

R

Q


 N

R + qn = 0

Mx

x

+ 1

R

Mx


 Qx = 0

Mx

x

+ 1

R

M


 Q= 0


x

o = u

x
82

 (A7)

 (A8)

 (A9)

 (A10)

 (A11)

 (A12)

 (A13)

 (A14)

where u, v, and w are the axial, circumferential, and radial displacements of a point of the shell 

middle surface; , , and are the membrane strains; , , and are the rotations; and , 

, and are the bending strains. The displacement w is positive when it is outward from the 

cylinder reference surface.

Constitutive Equations

 The isothermal constitutive equations are given in matrix form by

 (A15)



o = 1

R

v

+w

R

 x

o

= v

x

+ 1

R

u



x

o

=  w

x



o

= c1

Rv  1

R

w



n

o

= c2

2

v

x  1

R

u



x

o

= x

o

x =  

2

w

x

2



o

= 1

R



o

= c1

R2

v

 1

R2



2

w

2

x

o

= 1

R

x

o

+ n

o

+ 

o

x =  2

R



2

w

x+ 1

R c1 + 1

2c2

v

x  c2

2R2

u



x

o o  xo x

o o no x

o



o xo

Nx

N

Nx

Mx

M

Mx

=

A11 A12 A16 B11 B12 B16

A12 A22 A26 B12 B22 B26

A16 A26 A66 B16 B26 B66

B11 B12 B16 D11 D12 D16

B12 B22 B26 D12 D22 D26

B16 B26 B66 D16 D26 D66

x

o



o

 x

o

x

o



o

x

o



Nx

Th

N

Th

Nx

Th

Mx

Th

M

Th

Mx

Th
83

where the subscripted A, B, and D terms of the matrix are the stiffnesses of laminated composite 

shells that are obtained from the Love-Kirchhoff shell theory. Moreover, the constitutive terms 

in Eq. (A15) are identical to those for laminated-composite plates that are given in Ref. 34 on pp. 

198 and 243-244. The terms with the superscript "Th" are the ficticious thermal stress resultants.

Boundary Conditions

The boundary conditions for an edge that is defined by a constant value of the axial 

coordinate x are given by

 or (A16)

 or (A17)

 or (A18)

 or (A19)

where are applied edge displacements; is an applied edge rotation; and 

are applied edge loads.

Nx = N() u=u()

Nx+ 1

R c1 + 1

2c2 Mx= T() v=v()

Qx + 1

R

Mx

= V() w=w()

Mx = M() 	x

o = 	()

u(), v(), and w() 	()

N(), T(), V(), and M()
84

Appendix B

Equations for Axisymmetry

The linear Sanders-Koiter shell equations that are presented in Appendix A for a rightcircular cylinder with a radius R are specialized in this appendix for the case of axisymmetric 

behavior. For these equations, x and denote the axial and circumferential coordinates, 

respectively. The specialization to axial symmetry is conducted by eliminating all terms in the 

equations of Appendix A that are differentiatied with respect to the circumferential coordinate, . 

First, the equilibrium equations, the kinematic equations, and the constitutive equations are 

presented. Then, the boundary conditions are given for a complete right-circular cylinder at an 

edge that is given by a constant value of x.

Equilibrium Equations

The equilibrium equations for axisymmetric behavior are given by

 (B1)

 (B2)

 (B3)

 (B4)

 (B5)

where the membrane stress resultants Nx

, N, and Nx; the transverse shear-stress resultants Qx and 

Q; the bending stress resultants Mx

, M, and Mx; and the applied surface tractions qx, q, and qn

are functions of only the axial coordinate, x.

Kinematic Equations

The kinematic equations are given by

 (B6)

 (B7)

dNx

dx + qx(x) = 0

dNx

dx +

c1

R Q+

c2

2R

dMx

dx + q(x) = 0

dQx

dx 
 N

R + qn(x) = 0

dMx

dx 
 Qx = 0

dMx

dx 
 Q= 0


x

o = du

dx




o = w

R
85

 (B8)

 (B9)

 (B10)

 (B11)

 (B12)

 (B13)

 (B14)

where the middle-surface displacements u, v, and w; the membrane strains the 

rotations and the bending strains are functions of only the axial 

coordinate, x.

Constitutive Equations

The constitutive equations reduce to

 (B15)

For thin shells, it is convenient to introduce a temperature-change function

 (B16a)

such that 

 x

o = dv

dx

	x

o

= 
 dw

dx

	

o

= c1

Rv

	n

o = c2

2

dv

dx

x

o

= d	x

o

dx = 
 d

2

w

dx2



o = 0

x

o = 1

R	n

o +

d	

o

dx = 1

R c1 + 1

2c2

dv

dx


x

o

, 


o

, and  x

o

;

	x

o

, 	

o

, and 	n

o

; x

o

, 

o

, and x

o

Nx

N

Nx

Mx

M

Mx

=

A11 A12 A16 B11 B12 B16

A12 A22 A26 B12 B22 B26

A16 A26 A66 B16 B26 B66

B11 B12 B16 D11 D12 D16

B12 B22 B26 D12 D22 D26

B16 B26 B66 D16 D26 D66


x

o




o

 x

o

x

o

0

x

o




Nx

Th

N

Th

Nx

Th

Mx

Th

M

Th

Mx

Th

T x, z = 0 x + z 1 x
86

 (B16b)

 (B16c)

and

 (B17)

where the two vectors on the right-hand-side of the equal sign are the constants that result from 

substituting equation (B16) into expressions for the ficticious thermal stress resultants, like those 

given on pp. 243-244 of reference 34. 

Boundary Conditions

The boundary conditions for an edge that is defined by a constant value of the axial 

coordinate, x, are given by

 or (B18)

 or (B19)

 or (B20)

 or (B21)

where the applied edge displacements the applied edge rotation and the applied 

edge loads are all constants.

0 x = 1

2 T x, h

2 + T x, 
 h2

1 x = 1

h T x, h

2 
 T x, 
 h2

Nx

Th

N

Th

Nx

Th

Mx

Th

M

Th

Mx

Th

= 0 x

nx

Th

n

Th

nx

Th

cx

Th

c

Th

cx

Th

+ 1 x

cx

Th

c

Th

cx

Th

mx

Th

m

Th

mx

Th

Nx = N u=u

Nx+ 1

R c1 + 1

2c2 Mx= T v=v

Qx = V w=w

Mx = M 	x

o

= 	

u, v, and w; 	;

N, T, V, and M
87

Appendix C

Approximate Solution for Uniform Axial End Load

For this example, the boundary conditions are specified as

 (C1a)

 (C1b)

 (C1c)

 (C1d)

at x = 0 and

 (C2a)

 (C2b)

 (C2c)

 (C2d)

at x = L. The only applied load is that defined by equation (C2a). An approximate solution is 

found by applying the equations of linear membrane theory, in which the bending boundary layer 

is ignored such that the radial displacement is presumed to be constant valued. First, equation (B1) 

gives a constant value for the axial stress resultant Nx

. Thus, the boundary condition given by 

equation (C2a) yields the result Similarly, equation (5) yields where is 

an unknown constant. However, because this problem does not have a boundary condition 

specified for the problem is statically indeterminate. Next, the transverse-shear stress 

resultant Qx, that arises from variations in the bending stress resultant Mx (see equation (B4)), is 

neglected in equation (B3) to get Similarly, the derivatives of the radial displacement 

w vanish and, as a result, the rotation the bending strain The matrix constitutive 

equation given by equation (14) reduces to

 (C3)

u 0 = 0

v 0 = 0

w 0 = 0

Mx 0 = 0

Nx L = NL

v L = 0

w L = 0

Mx L = 0

Nx x = NL. T x = T0, T0

T x ,

Nx = 0.

	x

o

= 0 x

o

= 0.

u x

v x = a 11 a 16

a 16 a 66

NL

T0

+

b 12

b 62

w

R
88

In addition, equation (16a) reduces to

 (C4)

Substituting equation (C4) into (C3) yields

 (C5)

Integration of these equations reveals that u(x) and v(x) are linear functions. The only possible 

way that the boundary conditions given by (C1b) and (C2b) can be satisfied is for v(x) = 0. Thus, 

equation (C5) can be expressed as (see equations (13)) 

 (C6)

In addition, equation (C4) gives

 (C7)

Substituting equation (C7) into the first of equations (C6) gives the result

 (C8)

Next, substituting equation (C8) into equation (C7), integrating, and applying boundary-condition 

equation (C1a) gives the result

 (C9)

Substituting equations (C8) and (C9) into the second of equations (C6) gives

 (C10)

Approximations to the bending stress resultants are obtained by substituting v(x) = 0 and 

equations (C8) and (C9) into equations (17a), (82), and (88) to get

A22

w

R = 
 A12u 
 A26 v

A22 + A12b 12 A26b 12

A12b 62 A22 + A26b 62

u

v = A22

a 11 a 16

a 16 a 66

NL

T0

NL

T0

= A11 A16

A16 A66

u x

0 +

A12

A26

w

R

u = 
 A22

A12

w

R

w

R = 
 A12 NL

A11A22 
 A12

2

u x = A22 NL x

A11A22 
 A12

2

T0 = A22A16 
 A12A26

A11A22 
 A12

2 NL
89

 (C11)

 (C12)

 (C13)

Because these bending stress resultants are constants, it follows that the transverse-shear stress 

resultants Qx and Qare zero valued (see equations (B4) and (B5)). An approximation to Nxis 

obtained by substituting and equation (C13) into equation (4).

Mx x = A22B11 
 A12B12

A11A22 
 A12

2 NL

Mx = A22B12 
 A12B22

A11A22 
 A12

2 NL

Mxx = A22B16 
 A12B26

A11A22 
 A12

2 NL

T x = T0
90

Appendix D

Approximate Solution for Uniform Pressure and Temperatures

For this example, the boundary conditions are specified as

 (D1a)

 (D1b)

 (D1c)

 (D1d)

at x = 0 and

 (D2a)

 (D2b)

 (D2c)

 (D2d)

at x = L. The applied mechanical load is given by and the applied thermal load is given 

by For these loading conditions, equations (1b) and (7b) yield 

 and Likewise, equations (100) reduce to 

 (D3a)

 (D3b)

 (D3c)

 (D3d)

In addition, equation (B17) yields

u 0 = 0

v 0 = 0

w 0 = 0

	x

o

0 = 0

u L = 0

v L = 0

w L = 0

	x

o

L = 0

qn x = qn

0

0 x = H0 and 1 x = G0.

N1 x = 0 T1 x = 0.

Nx

Th

x = H0nx

Th + G0cxTh

N

Th

x = H0n

Th + G0cTh

Mx

Th

x = H0cx

Th + G0mxTh

T

Th

x = H0nx

Th + G0cxTh
91

 (D3e)

 (D3f)

 (D3g)

An approximate solution is found by applying the equations of linear membrane theory, in which 

the bending boundary layer is ignored such that the radial displacement is presumed to be constant 

valued. First, equations (1) and (7) give constant values for the axial stress resultant , 

and the torsional stress resultant, where and are unknown constants. Because this 

problem does not have any stress-resultant boundary conditions, the problem is statically 

indeterminate. Next, the transverse-shear stress resultant Qx

, that arises from variations in the 

bending stress resultant Mx (see equation (B4)), is neglected in equation (B3) to get 

Similarly, the derivatives of the radial displacement w vanish and, as a result, the 

rotation and the bending strain The matrix constitutive equation given by equation 

(14) reduces to

 (D4)

In addition, equation (16a) reduces to

 (D5)

Substituting equation (D5) into (D4) yields

 (D6)

Integration of these equations reveals that u(x) and v(x) are linear functions. The only possible 

way that the boundary conditions given by (D1a), (D1b), (D2a), and (D2b) can be satisfied is for 

u(x) = 0 and v(x) = 0. Thus, the left-hand-side of equation (D4) vanishes and the resulting 

equations are expressed as (see equations (13)) 

Nx

Th

x = H0nx

Th + G0cxTh

M

Th

x = H0c

Th + G0mTh

Mx

Th

x = H0cx

Th + G0mxTh

Nx x = N0,

T x = T0, N0 T0

Nx = qn

0

R.

	x

o

= 0 x

o

= 0.

u x

v x = a 11 a 16

a 16 a 66

N0 + Nx

Th

x

T0 + TTh

x

+

b 12

b 62

w

R

A22

w

R = qn

0

R+N

Th


 A12u 
 A26 v

A22 + A12b 12 A26b 12

A12b 62 A22 + A26b 62

u

v = A22

a 11 a 16

a 16 a 66

N0 + Nx

Th

T0 + TTh +

b 12

b 62

qn

0

R+N

Th
92

 (D7)

In addition, equation (D5) reduces to

 (D8)

Substituting equation (D8) into (D7) gives the results

 (D9)

Approximations to the bending stress resultants are obtained by substituting u(x) = 0, v(x) = 0, 

and equations (D8) into equations (17a), (82), and (88) to get

 (D10)

 (D11)

 (D12)

Because these bending stress resultants are constants, if follows that the transverse-shear stress 

resultants Qx

 and Qare zero valued (see equations (B4) and (B5). An approximation to Nxis 

obtained by substituting and equation (D12) into equation (4).

N0

T0

= A12

A26

w

R 
 Nx

Th

T

Th

A22

w

R = qn

0

R+N

Th

N0

T0

= qn

0

R+N

Th

A22

A12

A26


 Nx

Th

T

Th

Mx = B12

A22

qn

0

R+N

Th


 Mx

Th

M= B22

A22

qn

0

R+N

Th


 M

Th

Mx= B26

A22

qn

0

R+N

Th


 Mx

Th

T x = T0
93

Appendix E

Approximate Solution for Linearly Varying Pressure and Temperatures

For this example, the boundary conditions are specified as

 (E1a)

 (E1b)

 (E1c)

 (E1d)

at x = 0 and

 (E2a)

 (E2b)

 (E2c)

 (E2d)

at x = L. The applied pressure load given by and the applied thermal load is given 

by and For these loading conditions, equations (1b) and 

(7b) yield and Likewise, equations (100) reduce to 

 (E3a)

 (E3b)

 (E3c)

 (E3d)

An approximate solution is found by applying the equations of linear membrane theory, in which 

Nx 0 = N0

T 0 = T0

w 0 = 0

Mx 0 = 0

u L = 0

v L = 0

w L = 0

Mx L = 0

qn x = qn

0

+ qn

1

x

0 x = H0 + H1 x 1 x = G0 + G1 x.

N1 x = 0 T1 x = 0.

Nx

Th

x = H0nx

Th + G0cxTh + H1nxTh + G1cxTh

x

N

Th

x = H0n

Th + G0cTh + H1nTh + G1cTh

x

Mx

Th

x = H0cx

Th + G0mxTh + H1cxTh + G1mxTh

x

T

Th

x = H0nx

Th + G0cxTh + H1nxTh + G1cxTh

x
94

the bending boundary layer is ignored such that the radial displacement is presumed to vary like 

the pressure and temperature fields; that is,

 (E4)

First, equations (1) and (7) give constant values for the axial stress resultant , and the 

torsional stress resultant, where and constants determined from applying 

boundary-condition equations (E1a) and (E1b). Applying these boundary conditions yields

 (E5a)

 (E5b)

By using equations (E3), (E4), and (E5), the matrix constitutive equation given by equation (14) 

is reduced to

 (E6)

Integrating these equations and applying boundary-condition equations (E2a) and (E2b) gives

 (E7)

The remaining part of the solution is the determination of the constants W0

 and W1. These 

constants are found by first noting that P0

 appearing equation (40), and given by equation (29b) 

reduces to the constant

 (E8)

Likewise, equation (102a) reduces to

 (E9)

w x = W0 + W1 x

Nx x = N0,

T x = T0, N0 T0

Nx x = N0 = N0

T x = T0 = T0

u x

v x = a 11 a 16

a 16 a 66

N0 + H0nx

Th + G0cxTh + H1nxTh + G1cxTh

x

T0 + H0nx

Th + G0cxTh + H1nxTh + G1cxTh

x

+

b 12

b 62

W0 + W1 x

R

u x

v x = a 11 a 16a 16 a 66

N0 + H0nx

Th + G0cxTh

x 
 L + 1

2 H1nx

Th + G1cxTh

x

2


 L2

T0 + H0nx

Th + G0cxTh

x 
 L + 1

2 H1nx

Th + G1cxTh

x2
 L2

+

b 12

b 62

2W0 x 
 L + W1 x2
 L2

2R

P0 = 
 1

RD11 e

A12a 11 + A26a 16 N0 + A12a 16 + A26a 66 T0

P1(x) = Y0 + Y1 x
95

for the given loading conditions, where

 (E10a)

 (E10b)

Substituting equations (E4), (E7), and (E9) into equation (40) yields

 (E11a)

 (E11a)

Approximations to the stress resultants N(x), Mx

(x), M(x), and Mx(x) are determined by 

substituting equations (E4) and (E7) into equation (16a), (17a), (82), and (88), respectively. 

Approximations to the transverse-shear stress resultants are then obtained by substituting the 

resulting expressions for Mx

(x) and Mx(x) into equations (B4) and (B5), respectively. An 

approximate expression for Nxis obtained by substituting equation (E5b) and the expression for 

Mx(x) into equation (4).

Y0 = 1

D11 e

qn

0

+

H0

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+

G0

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

Y1 = 1

D11 e

qn

1

+

H1

R n

Th


 nx

Th A12a 11 + A26a 16 
 nx

Th A12a 16 + A26a 66

+

G1

R c

Th


 cx

Th A12a 11 + A26a 16 
 cx

Th A12a 16 + A26a 66

W0 = P0 + Y0

4T2

W1 = Y1

4T2
96

Appendix F

Exact Solution for Sinusoidal Pressure and Temperatures

For this example, the boundary conditions are specified as

 (F1a)

 (F1b)

 (F1c)

 (F1d)

at x = 0 and

 (F2a)

 (F2b)

 (F2c)

 (F2d)

at x = L. It should be noted that these boundary conditions are for simply supported ends rather 

than the clamped ends considered in the corresponding example presented herein. The applied 

mechanical load given by and the applied thermal load is given by 

 and For these loading conditions, equations (1b) and (7b) 

yield and Likewise, equations (100) reduce to 

 (F3a)

 (F3b)

 (F3c)

 (F3d)

u 0 = 0

v 0 = 0

w 0 = 0

Mx 0 = 0

Nx L = 0

T L = 0

w L = 0

Mx L = 0

qn x = qn

s

sin x

L

0 x = Hs sin x

L 1 x = Gs sin xL .

N1 x = 0 T1 x = 0.

Nx

Th

x = Hsnx

Th + GscxTh sin x

L

N

Th

x = Hsn

Th + GscTh sin x

L

Mx

Th

x = Hscx

Th + GsmxTh sin x

L

T

Th

x = Hsnx

Th + GscxTh sin x

L
97

In addition, equation (B17) yields

 (F3e)

 (F3f)

 (F3g)

Applying boundary-condition equations (F2a) and (F2b) to equations (1a) and (7a) gives 

and Thus, and Next, the radial displacement is presumed to be given by

 (F4)

where W is a constant to be determined. By using equations (F3) and (F4), equations (14) 

become

 (F5)

Integrating these two equations and applying boundary-condition equations (F1a) and (F1b) gives

 (F6)

Substituting equations (F3)-(F5) into equation (17a) gives

 (F7a)

where

 (F7b)

Inspection of equation (F7) reveals that boundary-condition equations (F1d) and (F2d) are also 

satisfied.

Nx

Th

x = Hsnx

Th + GscxTh sin x

L

M

Th

x = Hsc

Th + GsmTh sin x

L

Mx

Th

x = Hscx

Th + GsmxTh sin x

L

N0 = 0

T0 = 0. Nx x = 0 T x = 0.

w x = Wsin x

L

u x

v x = a 11 a 16

a 16 a 66

Hsnx

Th + GscxTh

Hsnx

Th + GscxTh + W

R

b 12

b 62

+ W2

L

2

c 12

c 62

sin x

L

u x

v x = a 11 a 16a 16 a 66

Hsnx

Th + GscxTh

Hsnx

Th + GscxTh + W

R

b 12

b 62

+ W

2

L

2

c 12

c 62

L

 1 
 cos x

L

Mx x = Mx sin x

L

Mx = Hs B11 a 11nx

Th

+ a 16nx

Th + B16 a 16nx

Th

+ a 66nx

Th


 cx

Th

+ Gs B11 a 11cx

Th

+ a 16cx

Th + B16 a 16cx

Th

+ a 66cx

Th


 mx

Th

+ W

R

R

2

L

2 D11 + B12 + B11 b 12 + R

2

L

2 c 12 + B16 b 62 + R

2

L

2 c 62
98

The remaining part of the solution is the determination of the constant W. This constant is 

found by first noting that P0 = 0 in equation (40) and equation (102a) reduces to

 (F8)

for the given loading conditions, where Ys1 is given by equation (102e). Substituting equations 

(F4) and (F8), and P0 = 0, into equation (40) yields

 (F9)

The remaining stress resultants are determined by substituting equations (F3)-(F5), (97), and 

(B17) into equation (16a), (82), and (88). The results are:

 (F10a)

with

 (F10b)

 (F11a)

with

 (F11b)

 (F12a)

with

 (F12b)

P1(x) = Ys1 sin x

L

W = Ys1L

4



4


 4S 

2

L

2

+ 4T2

L

4

Nx = Nsin x

L

N= Hs A12 a 11nx

Th

+ a 16nx

Th + A26 a 16nx

Th

+ a 66nx

Th


 n

Th

+ Gs A12 a 11cx

Th

+ a 16cx

Th + A26 a 16cx

Th

+ a 66cx

Th


 c

Th

+ W

R

R2

L

2 B12 + A22 + A12 b 12 + R2

L

2 c 12 + A26 b 62 + R2

L

2 c 62

Mx = Msin x

L

M= Hs B12 a 11nx

Th

+ a 16nx

Th + B26 a 16nx

Th

+ a 66nx

Th


 c

Th

+ Gs B12 a 11cx

Th

+ a 16cx

Th + B26 a 16cx

Th

+ a 66cx

Th


 m

Th

+ W

R

R

2

L

2 D12 + B22 + B12 b 12 + R

2

L

2 c 12 + B26 b 62 + R

2

L

2 c 62

Mxx = Mxsin x

L

Mx= Hs B16 a 11nx

Th

+ a 16nx

Th + B66 a 16nx

Th

+ a 66nx

Th


 cx

Th

+ Gs B16 a 11cx

Th

+ a 16cx

Th + B66 a 16cx

Th

+ a 66cx

Th


 mx

Th

+ W

R

R2

L

2 D16 + B26 + B16 b 12 + R

2

L

2 c 12 + B66 b 62 + R

2

L

2 c 62
99

Likewise, equations (B4) and (B5) give

 (F13)

and

 (F14)

An expression for Nxis obtained by substituting and equations (F12b) into equation (4).

Qx x = 

L Mx cos xL

Qx = 

L Mxcos xL

T x = 0
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Publication

 4. TITLE AND SUBTITLE

The Exact Solution for Linear Thermoelastic Axisymmetric Deformations 

of Generally Laminated Circular Cylindrical Shells 

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Nemeth, Michael P.; Schultz, Marc R.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20112

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified Unlimited

Subject Category 39

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

A detailed exact solution is presented for laminated-composite circular cylinders with general wall construction and that 

undergo axisymmetric deformations. The overall solution is formulated in a general, systematic way and is based on the 

solution of a single fourth-order, nonhomogeneous ordinary differential equation with constant coefficients in which the radial

displacement is the dependent variable. Moreover, the effects of general anisotropy are included and positive-definiteness of 

the strain energy is used to define uniquely the form of the basis functions spanning the solution space of the ordinary 

differential equation. Loading conditions are considered that include axisymmetric edge loads, surface tractions, and 

temperature fields. Likewise, all possible axisymmetric boundary conditions are considered. Results are presented for five 

examples that demonstrate a wide range of behavior for specially orthotropic and fully anisotropic cylinders. 

15. SUBJECT TERMS

Anisotropic shells; Bending boundary layer; Cylindrical shells; Edge effects

18. NUMBER

 OF 

 PAGES

104

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

869021.04.07.01.13

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

NASA/TP-2012-217342

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.

1. REPORT DATE (DD-MM-YYYY)

01- 02 - 2011