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Review
Journal of Composite Materials
2023, Vol. 0(0) 1–29
© The Author(s) 2023
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DOI: 10.1177/00219983231215863
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Review of elasto-static models for
three-dimensional analysis of thick-walled
anisotropic tubes
Andreas Kastenmeier
1
, Marco Siegl
1
, Ingo Ehrlich
1
and Norbert Gebbeken
2
Abstract
Most shell or beam models of anisotropic tubes under bending have no validity for thick-walled structures. As a result, the
need to develop three-dimensional formulations which allow a change in the stress, strain and displacement distributions
across the radial component arises. Basic formulations on three-dimensional anisotropic elasticity were made either stress-
or displacement-based by Lekhnitskii or Stroh on plates. Lekhnitskii also was the first to expand these analytical for-
mulations to tubes under various loading conditions. This paper presents a review of the stress and strain analysis of tube
models using three-dimensional anisotropic elasticity. The focus lies on layered structures, like fiber-reinforced plastics,
under various bending loads, although the basic formulations and models regarding axisymmetric loads are briefly dis-
cussed. One section is also dedicated to the determination of an equivalent bending stiffness of tubes.
Keywords
Analytical modelling, anisotropy, elasticity, layered structrues, composite tube bending
Introduction
This paper provides a review of analytical and semi-analytical
models and enhancements for the three-dimensional analysis
of stresses, strains and displacements of cylindrical tubes with
linear-elastic, anisotropic material behavior. Since the main
focus is on composite materials and their multi-layer design,
special cases of material symmetry like monotropy, ortho-
tropy and transverse-isotropy are considered. In general, all
formulations are basicly developed for a single-layered tube
and the layering is achieved by a multiple use of the model
description and fulfillment of continuity conditions at the
interphases. Even though the material description is three-
dimensional, the stress and strain distributions are usually
functions of only one or two coordinate directions to obtain a
solvable system of governing equations. However, there are
models with approximate solutions for the remaining di-
mensions using a series expansion or numeric methods.
The tube is globally described by a cylindrical co-
ordinate system, where the principal directions are
called radial r, circumferential θand axial z,aswellasa
cartesian coordinate system x,yand z. As illustrated in
Figure 1, both have their origin in the center of the
circular cross section. Therefore, x-andy-positions
could be expressed as x=rsin φand y=rcos φ.In
some cases a third global coordinate system is
established, with the only difference being that the
radial component, later refered to as ~
r, originates in the
center of the laminate and not the tube cross-section.
The associated displacements are indicated as u
x
,u
y
,u
z
or rather u,v,wfor cartesian coordinates and as u
r
,u
θ
,u
z
for cylindrical coordinates, respectively. Stresses and
strains are described by the related 3×3 tensor, which
could be transferred to a 6×1 vector in notation by
Voigt,
1
for the cylindrical coordinate system.
It should be noted here that the global cartesian tube
coordinate system in Figure 1 is taken from the basic
definition of Lekhnitskii,
2
who defines the basis for further
bending models. For the material description, however, the
local and global coordinate systems must be used to ensure a
consistent transformation of the ply to laminate properties.
For the transformation from the global cartesian laminate
1
Ostbayerische Technische Hochschule Regensburg, Neustadt an der
Donau, Germany
2
University of the Armed Forces Munich, Neubiberg, Germany
Corresponding author:
Andreas Kastenmeier, Ostbayerische Technische Hochschule
Regensburg, Technology Campus Neustadt a. d. Donau, Raffineriestraße
8, Neustadt an der Donau 93333, Germany.
Email: andreas.kastenmeier@oth-regensburg.de
coordinate system to the cylindrical tube coordinate system
according to Lekhnitskii,
2
reference should be made to Siegl
and Ehrlich,
3
who ensure the use of the correct transformed
material parameters for the use of tube bending models via
an introduced permutation matrix. Lekhnitskii
2
relates his
cartesian coordinate system to generally anisotropic ma-
terials and does not take into account the fiber orientation,
which according to his definition would lie in the radial
direction in the tube cross-section and thus has no technical
application.
3
If consistent coordinate systems for the ma-
terial description as well as of the global tube based on the
fiber orientation is desired, reference should be made to
Almeida et al.,
4
nevertheless the relations of Lekhnitskii
(1963) are necessary to use the bending models, which is
why they are used in this paper.
The considered load types for the static analysis are
differentiated into axisymmetric and antisymmetric loads.
The former includes axial tension and compression, internal
and external pressure, torsion as well as in-plane shearing,
the latter mainly consists of bending, transverse shearing
and local transverse load. This review is focusing on tubes
subjected to different bending loads and boundary condi-
tions, but will also give an overview of general formulations
regarding three-dimensional anisotropic elasticity and tubes
under axisymmetric loads. Even though, some models in
this review are capable of enabling dynamic analyses due to
the physical description in form of an eigenvalue problem, it
is focused on models under static and mechanical loads.
Three-dimensional anisotropic elasticity also implies that
the tubes are thick-walled and shell or beam formulations
are not considered. A distinction between thin-walled and
thick-walled tubes is made by the ratio of radius to thick-
ness, which is strongly dependend on the used material and
load case.
Constitutive law
Regardless of the respective model or approach, the
continuum is described by three fundamental equations:
constitutive law, equilibrium equations and kinematic re-
lationships. In case of a cylindrical tube, the general an-
isotropic material law in global form becomes
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
εr
εθ
εz
γθz
γrz
γrθ
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
¼
2
6
6
6
6
6
6
4
S11 S12 S13 S14 S15 S16
S21 S22 S23 S24 S25 S26
S31 S32 S33 S34 S35 S36
S41 S42 S43 S44 S45 S46
S51 S52 S53 S54 S55 S56
S61 S62 S63 S64 S65 S66
3
7
7
7
7
7
7
5
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
σr
σθ
σz
τθz
τrz
τrθ
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;(1)
Figure 1. General cartesian and cylindrical coordinate systems of the tube used according to the definition of Lekhnitskii
2
and the local
and global cartesian coordinate system of the material in the wound fiber-reinforced tube with specification of the angle definition for
the transformation of the material properties from the ply to laminate.
2Journal of Composite Materials 0(0)
or
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
σr
σθ
σz
τθz
τrz
τrθ
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
¼
2
6
6
6
6
6
6
4
C11 C12 C13 C14 C15 C16
C21 C22 C23 C24 C25 C26
C31 C32 C33 C34 C35 C36
C41 C42 C43 C44 C45 C46
C51 C52 C53 C54 C55 C56
C61 C62 C63 C64 C65 C66
3
7
7
7
7
7
7
5
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
εr
εθ
εz
γθz
γrz
γrθ
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
(2)
in notation of the compliances [S
ij
] or stiffnesses ½Cij¼
½Sij1with i,j= 1-6. Due to the presence of a symmetry
plane and the reflection of all characteristic values on it, the
stiffness and compliance matrices can be simplified to a
monotropic or monoclinic material behavior. The compli-
ance matrix, as an example, is then expressed by
Sij, mono¼
2
6
6
6
6
6
6
4
S11 S12 S13 S14 00
S21 S22 S23 S24 00
S31 S32 S33 S34 00
S41 S42 S43 S44 00
0000S55 S56
0000S65 S66
3
7
7
7
7
7
7
5
:(3)
If a second or even third symmetry plane exists or-
thogonal to the first one, a further simplification to ortho-
tropic material behavior is possible and
Sij, ortho¼
2
6
6
6
6
6
6
4
S11 S12 S13 000
S21 S22 S23 000
S31 S32 S33 000
000S44 00
0000S55 0
00000S66
3
7
7
7
7
7
7
5
:(4)
A material is called transversely isotropic or in some
literature cylindrically anisotropic, if there are infinite
symmetry planes around one coordinate direction. The
matrix occupancy does not change in comparison to or-
thotropy, but the individual entries can be expressed by less
characteristic values. The elastic state of the material can be
expressed by 13 independent mechanical properties in case
of monotropy, 9 in case of orthotropy and 5 in case of
transversely isotropy. Note that there are many different
notations for constitutive law and, in particular, the com-
pliances in the literature.
Equilibrium Equations
The equilibrium equations are derived using the cartesian
coordinate system for cylindrical stresses under the as-
sumption of small angles on a segment of the hollow
cylinder. On the basis of these assumptions, the curved
surfaces of the segment can be approximated by plane
surfaces and the cross section becomes trapezoidal. For all
stresses, a Taylor series expansion is applied according to
σ
r
(r)=σ
r
and σ
r
(r+dr)=σ
r
+(∂σ
r
/∂r) dr. All terms
multiplied by the finite dimensions dr,dθor dz can be
neglected because of the low values. By also neglecting
inner forces of the continuum, the equilibrium equations
result to
∂σr
∂rþ1
r
∂τrθ
∂θþ∂τrz
∂zþσrσθ
r¼0,
∂τθr
∂rþ1
r
∂σθ
∂θþ∂τθz
∂zþ2τθr
r¼0,
∂τzr
∂rþ1
r
∂τzθ
∂θþ∂σz
∂zþτzr
r¼0:
(5)
The indices ij (i,j=r,θ,z) correspond to the notation of
the section plane iand the direction of action j. Equation (5)
are describing the general form of equilibrium equations. In
dependence of the specific model the stresses are inde-
pendent of one or two coordinates, thus eliminating the
derivatives in the respective directions.
Kinematic relationships
The strains can be distinguished in normal strains {ε
i
}and
shear strains {γ
ij
} for i,j=r,θ,z. Mathematically, they are
described by changes in the length and angular ratios of the
continuum, which can be expressed in terms of displace-
ments. Due to the contiguous ring cross-section of the tube,
a normal strain in tangential direction also produces an
additional displacement in the radial direction. The kine-
matic relationships, in form of the strain-displacement re-
lations, follow from vectorial considerations on a segment
of the hollow cylinder with small-angle approximations, a
Taylor series expansion according to the predescribed use
for the stresses and the assumption that only a minimal
change over the radius occurs. In general terms, they are
εr¼∂ur
∂r,γθz¼∂uθ
∂zþ1
r
∂uz
∂θ,
εθ¼1
r
∂uθ
∂θþur
r,γrz ¼∂uz
∂rþ∂ur
∂z,
εz¼∂uz
∂z,γrθ¼1
r
∂ur
∂θþ∂uθ
∂ruθ
r:
(6)
In equation (6) the derivatives for the respective directions
disappear, if the strains and displacements are considered
invariabel along these directions.
Further damage and
failure-considering investigations
In addition to the elasto-static models for three-dimensional
analysis of thick-walled anisotropic tubes summarised here
Kastenmeier et al. 3
in detail, there are also numerous analytical, numerical and
experimental studies that deal with damage, failure and
buckling of filament wound composite tubes for further load
cases besides bending, as well as considering the aero-
elasticity and buckling of plates that could be used for an
application on fiber-reinforced tubes.
Further experimental, numerical and analytical studies on
composite tubes under different loading scenarios. If, at higher
loads, the material behaviour can no longer be adequately
described with the linear-elastic approaches, reference
should be made to the investigations on the damage and
failure of filament wound tubes under different loading
scenarios. Experimental and numerical approaches under
external pressure loads are described in Almeida et al.
5
For
radial compression of the composite tubes, damage
modelling can be found in Almeida et al.
6
and the in-
fluence of the winding pattern of the composite tubes is
described in Lisboa et al.
7
Based on genetic algorithm
accounting progressive damage, Almeida et al.
8
developed
an optimization of the stacking sequence of composite
tubes under internal pressure. Regarding the internal
pressure loading of the composite tubes, in Azizian and
Almeida
9
surrogate models are used for stochastic,
probabilistic and relaibility analyses have using artificial
neural network metamodels. Further, optimisations and
effects on the manufacturing of variable-angle composite
cylinders can be obtained from experiments using digital
image correlation (DIC) to capture the strains on com-
posite tubes under axial compression in Almeida et al.
10
Here, the measurement of imperfections is also used to
perform non-linear numerical model along with a pro-
gressive damage analysis to describe the occurring
buckling mechanisms more precisely. Further findings and
fundamentals on buckling of composite tubes under axial
compression are presented in Almeida et al.
4
regarding
buckling and post-buckling as linear, nonlinear, damage
and experimental analysis and in Almeida et al.
11
the basic
design methodology for optimising the tube with variable-
axial fiber layout can be found. Furthermore, in Wang
et al.
12
a reliability-based buckling optimisation with an
accelerated kriging metamodel can be found for in the
winding process using variable angles. Based on this,
further developments of the Kriging-based metamodel in
combination with particle swarm optimisation can be
found in Wang et al.
13
Further insights into the com-
pression of composite tubes can be found in Stedile Filho
et al.,
14
who also investigate the torsion load of the
structure, which is used as a drive shaft. Furthermore, the
influences of mosaic pattern on hygrothermally-aged
composite tubes under axial compression have been in-
vestigated in Azevedo et al.
15
Further investigations on aeroelasticity and buckling of composite
plates. Aeroelasticity considers the phenomena resulting
from the interaction of aerodynamic (especially transient),
inertial and elastic forces that occur during the relative
motion of a fluid (air) and a flexible body (aircraft).
16
Based
on the approaches to aeroelasticity and buckling of com-
posite plates, certain material properties can be derived that
can be used for further investigations on fiber-reinforced
tubes that could be used in the aerospace industry. For this
purpose, the recent studies by Sharma et al.
17
on stochastic
frequency analysis of composite plates with curvilinear fiber
and Sharma et al.
18
on stochastic aeroelastic analysis of
laminated composite plates with variable fiber spacing can
be considered. The aeroelastic analysis of plates made of
lightweight materials with material uncertainty is described
in Swain et al.
19
Methods for quantifying the uncertainty in
the free vibration and aeroelastic response of an angularly
adjustable laminates are given in Sharma et al.
20
The
aeroelastic control of delaminated angle tow laminated
composite plates using piezoelectric patches is given in
Sharma et al.
21
The use of piezoelectric patches is also
described in the study by Sharma et al.
22
to investigate active
flutter suppression of damaged variable stiffness laminated
rectangular plate. Further investigations in the field of free
vibration can be found in Sharma et al.,
23
who studied the
static and free vibration analysis of smart variable stiffness
composite plates with delaminations. On the other hand, in
Sharma et al.
20
a study is made for uncertainty quantification
in free vibration and aeroelastic response of variable angle
tow laminated composite plates. Uncertainty quantification
under thermal loading in buckling strength of variable
stiffness laminated composite plates can be found in Sharma
et al.
24
For functionally graded sandwich plates using
layerwise theory, a vibration and certainty analysis has been
carried out in Sharma et al.
25
Three-Dimensional anisotropic elasticity
Three-dimensionality implies that the examined continua
are thick-walled and thus their properties change in
thickness direction. Theories which reduce the material
properties of a laminate to its median plane are therefore
excluded. Models regarding three-dimensional anisotropic
elasticity can be distinguished into formulations using
complex variables as well as formulations using the state-
space approach. Although some of these theories provide
exact solutions in all three coordinate directions for simple
continua, most models need to limit the stresses, strains and
displacements to functions of only one or two coordinates to
obtain a solveable system of governing equations. There-
fore, some approaches use approximate solutions along one,
two or even all three directions.
4Journal of Composite Materials 0(0)
General formulation using complex variables
There are two fundamental formulations of general aniso-
tropic elasticity, the stress- or compliance-based according
to Lekhnitskii
2
and the displacement- or stiffness-based
according to Stroh,
26
Stroh
27
and Eshelby et al.
28
Both
methods share similar approaches to describe the elastic
state of the homogenous continuum, although different state
variables are used. Constitutive law, equilibrium equations
and strain-displacement relations are employed as funda-
mental equations to generate a system of governing
equations. Stresses, strains and displacements are existent in
three dimensions, but do not vary along one coordinate. For
most of the tube models, there is no variation along the tube
axis z.
General formulation by Lekhnitskii. Lekhnitskii
2
is ex-
pressing the strains and displacements as functions of the
stresses, which leads to the necessity of performing
compatibility conditions for the three displacements of
the motion field. By applying the stress functions of Airy
and Prandtl, the stresses are reduced to two unknowns. A
system of partial differential equations is formed for the
unknown stress functions using the aforementioned
fundamental equations in terms of reduced compliances
as well as the geometrical parameters and their deriva-
tions. The solution is then obtained in form of a sixth
order polynomial, the so-called sextic equation, by in-
serting predefined initial functions for the stress functions
and a subsequent integration. Henceforth, it is possible to
describe stresses, strains and displacements as functions
of the unknown integration constants C
i
(i=1-6)inthe
polynominal equation, the compliances, reduced com-
pliances and coordinate positions. For the respective load
condition and continuum, simplifications of the funda-
mental equations can be made beforehand and the inte-
gration constants are determined through boundary
conditions.
General formulation by Stroh. Stroh
26
on the other hand
operates with already compatible displacements of an ar-
bitrary solid, that are independent of one dimension (here z)
of an cartesian coordinate system (x,y,z). According to
Eshelby et al.
28
the compatible displacement vector {u
c
}
can be expressed by the summation of three complex
functions composed of an unknown vector {A
c,i
} and a
function of an unknown complex coefficent p
i
as well as the
two dependent coordinates xand ywith
fucg¼X
3
i¼1fAc,igfðxþpiyÞ, (7)
By utilization of the fundamental equations, an equation
of sixth order with the unknown roots p
i
(i=1,2,3)is
obtained. These roots are necessarily complex, which was
proven by Eshelby et al.,
28
and correspond to the eigen-
values of the continuum.
26
Based on the fact, that the co-
efficients are real and elastic stability must be met, these
solutions appear in three complex conjugate pairs.
29
For the
real displacements all imaginary parts vanish and only the
real parts have to be considered.
28
For the respective
continuum and load case simplifications can be made and
the particular solutions are found through boundary con-
ditions. The general solution can be superposed from the
particular solutions.
Comparison of Lekhnitskii and Stroh. For a long time it was
only assumed that both formulations are equivalent re-
garding their sextic equations. It was finally proven by
Barnett and Kirchner
29
by reducing the six-dimensional
formulation of Stroh into two homogeneous, linear alge-
braic equations in terms of the reduced compliances. A
more direct comparison of the coefficents, depending on
the formulation as functions of the stiffnesses or reduced
compliances, isn’t possible. According to Tarn and
Wan g ,
30
the Lekhnitskii-formulation facilitates the rep-
resentation of the stresses and the Stroh-formulation those
of the displacements. Furthermore, the approach of
Lekhnitskii
2
is not feasible for static motions like the
determination of eigenvalues and eigenforms.
31
Stroh
26
enables this by reducing the deformation problem to the
determination of the eigenvalues and eigenvectors of the
system and connecting them to the constitutive law
through special eigenrelations.
32
Barnett and Kirchner
29
favorate the formulation of Stroh,
26
because of a more
direct computation and the already met compatibility
conditions. But they also point out that the choice must be
made in regards to the specific case.
General formulation of state-space approach
In addition to these two formulations, a mixed approach
known as state-space approach exists, where the gov-
erning equations are derived from the fundamental
equations in terms of stresses and displacements. This so-
called state equation in general form is expressed by the
derivation of a state vector {R}, usually consisting of
three displacements and three selected stresses. For most
models the stresses in radial direction are chosen and the
state equation is
∂
∂rfRg¼½AðrÞfRg:(8)
A second equation {S}=[B(r)] {R} is used after solving
the state equation for the computation of the vector {S},
containing the remaining three stress components. Matrices
[A(r)] and [B(r)] are linear differential operators, which
Kastenmeier et al. 5
depend on one coordinate, here r, but only consist of
derivations of the other two coordinates.
33
Although state-
space models use elastic equations in three-dimensional
form, they must discretize in at least one coordinate di-
rection for a unique solution because of the fact that, for the
usual boundary conditions, the state equation becomes a
partial differential equation with infinite order.
33
As a rule,
an approximation method by node subdivision is applied,
for example, by means of the finite-difference method, the
finite-element method or a series development in the
laminate plane (here z-θ). In this way, the state equation is
converted into a system of linear differential equations that
can be solved using standard methods by integrating the
boundary conditions into the matrices [A(r)] and [B(r)].
However, the boundary conditions at the end faces of the
continuum must again be formulated by simplifications in
certain coordinate directions. Exact solutions using three-
dimensional elastic equations are only possible in special
caseslikeRogersetal.
34
Therefore, most models in the
state-space only deal with axisymmetric loads and, like the
previous approaches, only allow changes of the dis-
placements, strains and stresses in two coordinate
directions.
Stress-Based approaches
In addition to the basic formulation of three-dimensional
anisotropic elasticity, Lekhnitskii deals in his books
‘Theory of Elasticity of an Anisotropic Elastic Body’,
2
and ‘Anisotropic Plates’
35
as well as numerous
publications
36–38
with tasks regarding infinite plates,
bars, beams, cylinders, pipes, and plates with elliptical
defects or inclusions under differing axisymmetric or
bending loads. Each further described literature is based
on these formulations.
General formulation of the tube according
to Lekhnitskii
For the single-layered cylindrical tube, the fully populated
constitutive law, corresponding to equation (1), is used
initially. The tube is in the state of generalized plane strain,
which means that a strain ε
z
is present. However, like all
other strains, displacements and stresses, it does not alter
along the axial component. This allows a simplification of
the contitutive law by a linear approach for the axial strain
with
εz¼S33 ðAxþByþCÞ
¼S33 ðArsin θþBrcos θþCÞ:(9)
It is assumed that the axial strain only results from the
axial stress σ
z
and consists of one component for each of the
bending moments about the x- and y-axis as well as one
component for the axisymmetric loads. The unknowns A,B,
and Crepresent the magnitudes of these stresses that in-
crease with distance xor yto the neutral fiber for a moment
load (Aand B) and are constant for an axial load (C). By
substituting the equation (9) into the third equation of (1),
the stress in zdirection can be computed and inserted into
the remaining equations of (1). The result is the reduced
material law
εr¼β11 σrþβ12 σθþβ14 τθzþβ15 τrz þβ16 τrθ
þS33 ðArsin θþBrcos θþCÞ,
εθ¼β21 σrþβ22 σθþβ24 τθzþβ25 τrz þβ26 τrθ
þS33 ðArsin θþBrcos θþCÞ,
γθz¼β41 σrþβ42 σθþβ44 τθzþβ45 τrz þβ46 τrθ
þS33 ðArsin θþBrcos θþCÞ,
γrz ¼β51 σrþβ52 σθþβ54 τθzþβ55 τrz þβ56 τrθ
þS33 ðArsin θþBrcos θþCÞ,
γrθ¼β61 σrþβ62 σθþβ64 τθzþβ65 τrz þβ66 τrθ
þS33 ðArsin θþBrcos θþCÞ,
(10)
with the reduced compliances
βij ¼Sij Si3S3j
S33
for i,j¼1;2;4;5;6:(11)
Furthermore an approach according to the stress func-
tions of Airy F(r,θ) and Prandtl Ψ(r,θ) for the remaining
five stresses is used. These stress functions meet the given
equilibrium equation (5) and allow the boundary value
problem to be converted to only two stress variables.
Stresses can then be written as
σr¼1
r
∂Fðr,θÞ
∂rþ1
r2
∂2Fðr,θÞ
∂θ2,
σθ¼∂2Fðr,θÞ
∂r2,
τrθ¼ ∂2
∂r∂θFðr,θÞ
r,
τrz ¼1
r
∂Ψðr,θÞ
∂θ,
τθz¼
∂Ψðr,θÞ
∂r:
(12)
The six strains are described by only three displacements
in the strain-displacement relations (6), which implies that
they can’t be independent of one another and must meet
compatibility requirements. The equation (6) are therefore
inserted into the material law (2). By integration of the 3rd,
4th and 5th equation of the resulting system over the z-axis
and conversion to the displacements, the following equa-
tions can be deduced
6Journal of Composite Materials 0(0)
By using the displacements due to strains U
r
,U
θ
,U
z
in
the cylindrical coordinate system as well as the translatory
u
0
,v
0
,w
0
and rotatory ω
1
,ω
2
,ω
3
rigid-body motions in the
cartesian coordinate system, equations for the unknown
displacement functions, resulting from the integration, can
be established. Due to the small angles, the rigid-body
motions can be transformed into the cylindrical system
by means of the trigonometric functions. The displacement
field is thus given by Lekhnitskii
2
in the form
Ur,0 ¼Urþu0cos θþv0sin θ,
Uθ,0 ¼Uθu0sin θþv0cos θþω3r,
Uz,0 ¼Uzþω1rsin θω2rcos θþw0:
(14)
For compatibility, the equation (13)mustbe
substituted into the other three equations of the con-
stitutive law (2). The governing partial differential
equation system can be established as a function of
stress functions using the reduced constitutive law (10),
the stress function approach (12), the displacements
(13) and the displacements resulting from integration
and expressed by the displacement field (14). For
brevity, the system results in
2
L0
4FþL0
3Ψ¼2½ðS13 S23ÞAS36 Bsin θ
r
þ2½S36 AþðS13 S23 ÞBcos θ
r,
(15a)
L00
3FþL0
2Ψ¼ðS35 Aþ2S34 BÞcosθ
þð2S34 AþS35 BÞsinθþCS
34
1
r
:(15b)
The differential operators of second order L0
2, third order
L0
3,L00
3and fourth order L0
4are
L0
2¼β44
∂2
∂r22β45
1
r
∂2
∂r∂θþβ55
1
r2
∂2
∂θ2þβ44
1
r
∂
∂r,
L0
3¼β24
∂3
∂r3þðβ25 þβ46 Þ1
r
∂3
∂r2∂θ
ðβ14 þβ56 Þ1
r2
∂3
∂r∂θ2þβ15
1
r3
∂3
∂θ3
þðβ14 2β24Þ1
r
∂2
∂r2þðβ46 β15 Þ1
r2
∂2
∂r∂θ
þβ15
1
r3
∂
∂θ,
L00
3¼β24
∂3
∂r3þðβ25 þβ46 Þ1
r
∂3
∂r2∂θ
ðβ14 þβ56 Þ1
r2
∂3
∂r∂θ2þβ15
1
r3
∂3
∂θ3
ðβ14 þβ24 Þ1
r
∂2
∂r2þðβ15 β46 Þ1
r2
∂2
∂r∂θ
þðβ14 þβ56 Þ1
r3
∂2
∂θ2þβ46
1
r3
∂
∂θ,
L0
4¼β22
∂4
∂r42β26
1
r
∂4
∂r3∂θþð2β12 þβ66 Þ1
r2
∂4
∂r2∂θ2
2β16
1
r3
∂4
∂r∂θ3þβ11
1
r4
∂4
∂θ4þ2β22
1
r
∂3
∂r3
ð2β12 þβ66 Þ1
r3
∂3
∂r∂θ2þ2β16
1
r4
∂3
∂θ3
β11
1
r2
∂2
∂r22ðβ16 þβ26Þ1
r3
∂2
∂r∂θ
þð2β11 þ2β12 þβ66Þ1
r4
∂2
∂θ2
þβ11
1
r3
∂
∂rþ2ðβ16 þβ26Þ1
r4
∂
∂θ:
(16)
uz¼zðS31 σrþS32 σθþS33 σzþS34 τθzþS35 τrz þS36 τrθÞþUz,0ðr,θÞ,
uθ¼zðS41 σrþS42 σθþS43 σzþS44 τθzþS45 τrz þS46 τrθÞ
z2
2
1
r
∂ðS31 σrþS32 σθþS33 σzþS34 τθzþS35 τrz þS36 τrθÞ
∂θz
r
∂Uz,0
∂θþUθ,0ðr,θÞ,
ur¼zðS51 σrþS52 σθþS53 σzþS54 τθzþS55 τrz þS56 τrθÞ
z2
2
∂ðS31 σrþS32 σθþS33 σzþS34 τθzþS35 τrz þS36 τrθÞ
∂rz∂Uz,0
∂rþUr,0ðr,θÞ:
(13)
Kastenmeier et al. 7
Pure bending of the tube according to Lekhnitskii
For the load case of pure bending around the y-axis, some
simplifications can be made. The orthotropic constitutive law, see
(4), is used and the following variables are set equal to zero
∂
∂z¼0; B¼C¼0; Ψ¼0:(17)
Band Cas described in section General formulation of
the tube according to Lekhnitskii are coupled to bending
around the x-axis and axisymmetric loads. The Prandtl
stress function Ψis only considered for the load case of
torsion. This simplifies the equation system (15a), simul-
taneously eliminating