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Accepted Manuscript
Nonlinear large deformation analysis of shells using the variational
differential quadrature method based on the six-parameter shell theory
R. Ansari, E. Hasrati, A.H. Shakouri, M. Bazdid-Vahdati, H. Rouhi
PII:
DOI:
Reference:
S0020-7462(18)30078-7
https://doi.org/10.1016/j.ijnonlinmec.2018.08.007
NLM 3065
To appear in:
International Journal of Non-Linear Mechanics
Received date : 6 February 2018
Revised date : 11 August 2018
Accepted date : 16 August 2018
Please cite this article as: R. Ansari, E. Hasrati, A.H. Shakouri, M. Bazdid-Vahdati, H. Rouhi,
Nonlinear large deformation analysis of shells using the variational differential quadrature method
based on the six-parameter shell theory, International Journal of Non-Linear Mechanics (2018),
https://doi.org/10.1016/j.ijnonlinmec.2018.08.007
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Nonlinear large deformation analysis of shells using the variational differential
quadrature method based on the six-parameter shell theory
a
a
b
R. Ansari*, a E. Hasrati, a A. H. Shakouri, a M. Bazdid-Vahdati, b H. Rouhi
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan,
University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran
Abstract
The present work is concerned with the application of the variational differential quadrature
(VDQ) method [Appl. Math. Model., vol. 49, pp. 705–738], in the area of computational
mechanics, to the nonlinear large deformation analysis of shell-type structures. To this end,
based on the six-parameter shell model, the functional of energy in quadratic form is derived
based on Hamilton’s principle which is then directly discretized by the VDQ technique. The
formulation of article is presented in a general form so that it can be readily used for different
structures such as beams, annular plates, cylindrical shells and hemispherical shells under
various loading conditions. In order to reveal the accuracy of developed solution strategy, it is
tested in several popular benchmark problems for the geometric nonlinear analysis of shells.
The results show that the present numerical method is capable of yielding highly accurate
solution in the nonlinear large deformation analysis of shells. It is also easy to implement due
to its compact and explicit matrix formulation.
Keywords: Large deformation analysis; Six-parameter shell model; Variational differential
quadrature method
1. Introduction
Shells are fundamental engineering structures with various applications in different fields [13]. Hence, studying their mechanical behaviors has been the subject of many research works
up to now. In particular, the large deformation analysis of shells is a challenging topic in the
related literature. One can follow this field of research from the work of Sabir and Lock in
*Corresponding
author. Tel. /fax: +98 13 33690276.
E-mail address: r_ansari@guilan.ac.ir (R. Ansari).
1
1972 [4] to the recently published papers [5-15]. Using different shell theories and proposing
different solution methods are the main focus of these works. In the following, some of the
important papers are reviewed.
Simo and co-workers [16-19] developed the six-parameter shell model which is known as
the geometrically exact shell model. They proposed the continuum foundation of the
geometrically exact shell theory obtained by reduction of the three-dimensional theory using
the one inextensible director kinematic assumption [16]. Atluri [20] focused on alternate
stress and conjugate strain measures and mixed variational formulations involving rigid
rotations for the analysis of finitely deformed shells. Pietraszkiewicz [21] derived the
equilibrium equations together with four geometric and static boundary conditions for an
entirely Lagrangian nonlinear theory of shells. Schmidt [22] studied the geometrically
nonlinear first approximation Kirchhoff-Love type theories for thin elastic shells undergoing
small strains accompanied by moderate, large or unrestricted rotations.
Braun et al. [23] and Büchter et al. [24] developed the seven-parameter shell theory. In [23],
arbitrary complete three-dimensional constitutive equations without manipulation in
nonlinear shell analysis were used, and the seven-parameter formulation was extended to
study the response of laminated structures with arbitrarily large displacements and rotations.
The research of Bischoff and Ramm [25] revealed that using the six-parameter shell
formulation leads to the unexpected Poisson locking problem.
A large number of studies are also associated with different finite element formulations and
shell elements [4, 8, 9, 10-12, 26-31]. For example, Sansour and Kollmann [9] compared
various kinds of four-node and nine-node elements of the finite element method (FEM)
including hybrid stress, hybrid strain and enhanced strain elements for investigating the large
deformation of shells within the framework of the seven-parameter shell theory. Arciniega
and Reddy [28, 29] studied the large deformation responses of shells made of isotropic,
laminated composites and functionally graded materials (FGMs). They utilized high-order
elements (e.g. Q25, Q49 and Q81) to avoid the locking issue as well as using mixed
interpolation methods like enhanced assumed strain elements.
Amabili [32] developed a geometrically nonlinear theory for shells of generic shape
considering third-order thickness and shear deformation and rotary inertia including eight
parameters. Alijani and Amabili [33] investigated the geometrically nonlinear static and
vibrations of rectangular plates taking full nonlinear terms corresponding to Green–Lagrange
strain–displacement relations into account based on the second-order thickness deformation
theory. Gutiérrez Rivera et al. [34] proposed a twelve-parameter shell finite element for large
2
deformation analysis of composite shell structures via third-order thickness stretch
kinematics. Also, Amabili [35] developed an eight-parameter theory with higher-order shear
deformation considering geometric non-linearities in the in-plane and transverse
displacements.
Recently, Faghih Shojaei and Ansari [36] presented a numerical method in the area of
computational mechanics called as variational differential quadrature (VDQ). The main idea
of this technique is based upon the accurate and direct discretization of the energy functional
in the structural mechanics. In VDQ, by introducing an efficient matrix formulation and via
an accurate integral operator, the discretized governing equations are directly obtained from
the weak form of equations with no need for the analytical derivation of the strong form. The
VDQ method provides an alternative way to discretize the energy functional, which avoids
the local interpolation and the assembly process existing in similar methods like FEM. The
VDQ method is general so that it can be used for any engineering structure that is modeled
through energy methods in the context of weak form formulation. Also, it is easy to
implement due to its compact and explicit matrix formulation. Accordingly, the VDQ
method, due to its simplicity and accuracy, has been utilized in several research works for
solving various problems of beams, plates and shells [37-44].
In the present article, the VDQ method is adopted to analyze the large deformation
behaviors of shell-type structures in the context of the six-parameter shell theory [16-19]. It is
worth mentioning that the seven-parameter shell model is used in the finite element analyses
to overcome the locking problem. But, the six-parameter shell model is used herein as the
VDQ method is locking-free. Also, it should be mentioned that this work focuses on thin
shells. The article is organized as follows.
In Section 2, the nonlinear strain-displacement relations and constitutive relations for the
large deformation analysis of six-parameter shells are derived. In Section 3, the obtained
relations in Section 2 are matricized. In Section 4, the quadratic representation of energy
functional is derived based on Hamilton’s principle. In Section 5, the VDQ method is
introduced for the discretization of quadratic representation of energy functional in a direct
way. Finally in Section 6, several well-known benchmark problems given by Sze et al. [10]
are considered to show the accuracy and efficiency of the VDQ method for geometric
nonlinear analysis of shells.
3
2. Six-parameter shell model
In this section, the basic relations of the six-parameter shell model are presented. The reader
is referred to [45, 46] for more details on the six-parameter shell theory.
Since the coordinates of a point in the convective coordinate system remain constant, the
deformation gradient tensor can be written as
⊗
⊗
where
⊗
(1)
is the normal vector of mid-surface,
1,2,3 and
1,2. In addition,
and
denote spatial coordinate system base vectors and material coordinate system base vectors,
respectively. According to Eq. (1), the right Cauchy-Green deformation tensor ( ) and Greenst. Venant strain tensor ( ) are obtained as
⊗
1
2
(2)
1
2
in which
(3)
⊗
and
are the metric tensors in spatial coordinate system and material
coordinate system, respectively. Considering the mid-surface displacement vectors in the
spatial coordinate system ( ) and material coordinate system ( ) as follows
,
, 0,
(4)
,0
,0
, 0,0
(5)
and the base vectors of the material coordinate system and spatial coordinate system on the
mid-surface as
,
,
│
│
,
,
,
│
,
│
,
│
(6)
(7)
one can obtain the displacement vectors of material coordinate system as
,
(8)
where
denotes curvilinear surficial coordinates,
is time, and the coordinate in the
thickness direction is denoted as .
Based on the six-parameter shell theory, the displacement vector in the spatial coordinate
system can be defined as
,
(9)
in which
is an unknown vector which is called “director”. Now, the material coordinate
system base vectors can be rewritten as
,
,
,
(10)
,
4
,
where
is the shifter tensor which makes a transformation between the base vectors
and
;
⊗
(11)
Using the following relations
,
(12)
2
(13)
and according to Eq. (12) and defining
│
,
⊗
(14)
⊗
⊗
(15)
the deformation gradient tensor ( ) can be recast as
(16)
Introducing the displacement vector ( ) and difference vector ( ) as follows
,
(17)
one can obtain
,
,
⊗
⊗
(18)
⊗
(19)
in order to achieve strains, by substituting Eq. (16) into (2), the right Cauchy-Green
deformation tensor ( ) can be attained as
(20)
Accordingly, the strain tensor (E) is obtained as follows
1
2
1
2
1
2
By ignoring the terms including
(21)
, Eq. (21) can be rearranged as
(22)
Based on Eq. (13) one has
1
2
1
2
(23)
(24)
After decomposing Eqs. (23) and (24) one can write
⊗
⊗
⊗
⊗
5
(25)
⊗
⊗
⊗
⊗
(26)
where
1
2
,
⋅
,
1
2
1
2
⋅
,
⋅
,
(27)
⋅
,
⋅
⋅
,
1
2
⋅
,
⋅
1
2
⋅
⋅
,
,
⋅
⋅
,
,
,
⋅
,
,
⋅
⋅
,
,
⋅
,
,
⋅
,
(28)
0
Considering a Cartesian coordinate system with base vectors
, the components of the
displacement vector and difference vector can be presented as
,
(29)
and the base vectors in curvilinear coordinate can be given as
,
,
⋅
,
(30)
Consequently, one can represent the components of nonlinear strains of six-parameter
shells in the Cartesian coordinate as follows
(31)
where
1
2
,
1
2
,
,
,
,
,
(32)
1
2
6
1
2
,
,
,
1
2
,
,
,
,
,
,
,
,
,
(33)
,
0
where ,
1,2 and
1,2,3. In addition, the displacement vector is given by
(34)
in which
,
directions, and
,
are the displacement projections on the mid-surface along
,
,
denote the rotations about
, ,
, ,
axis, respectively.
which based on Eq. (31) can be recast as
Also, the constitutive relation is
(35)
where
(36)
2
2
2
(37)
(38)
,
,
0,
,
0
2
(39)
It is worth mentioning that considering some extensions and modifications, the presented
shell theory can be applied in different fields [47]. Surface stress phenomena [48], thinwalled structures made of micropolar materials [49], thin-walled structures made of
viscoelastic materials [50-53], shells with initial stresses [54] and shells/plates with phase
transition [55, 56] are some examples.
3. Matrix representation
The relations defined in the previous section will be represented in a novel matrix form in the
current section in order to be used in Hamilton's principle. The elasticity matrix can be
rewritten as the following form based on Eq. (38)
7
(40)
.
in which by calculating
using Eqs. (38) and (39), the elasticity matrix can be recast for
different geometries (see the Appendix). Moreover, based on Eq. (37), the strain vector is
given by
2
2
2
(41)
Based on Eqs. (31) and (32), the strain-displacement relations can be defined as
,
(42)
where
1
2
,
in which
and
1
2
,
(43)
stand for linear and nonlinear strains, respectively. The vector of
kinematic parameters is expressed as
,
,
(44)
and the displacement vector can be recast as
,
in which
(45)
stands for an 3
⋯
,
1,2, … ,18
⋯
,
1,2, … ,45
0
0
0
(46)
0
0
(47)
0
where ∂ denotes
(
3 identity matrix. By defining the following relations
and
stands for the Kronecker delta, one can obtain the linear strains
) as follows
(48)
,
,
,
,
,
,
,
8
(49)
⊗
⊗
(50)
in which ⊗ denotes the Kronecker tensor product,
stands for an 6
is defined in Eq. (30). Also, the nonlinear strains (
6 identity matrix and
) can be obtained as follows
∘
(51)
in which ∘ denotes the Hadamard product and
〈
∘
in which 〈∎〉
1
2
〉
〈
〉
∎ . Accordingly,
〉
〈
〈
1
〈
2
∘
can be written as
〉
〈
〉
(52)
can be expressed as follows
〉
(53)
where
〈
in which
〉
〈
,
and
〉
(54)
are defined as follows
(55)
1
2
1
2
,
(56)
1
2
;
;
;
;
;
;
;
,
(57)
;
;
;
;
;
In addition, using Eq. (47)
;
,
;
,
1,2
1,2, … ,6 can be introduced as
;
(58)
;
,
;
9
,
;
;
;
,
;
;
,
;
,
;
;
4. Hamilton's Principle
According to Hamilton’s principle for a deformable body, one can write
0
where ,
and
(59)
are the kinetic energy, strain energy and the work of external forces,
respectively. The kinetic energy vanishes for the bending problem. The strain energy is
formulated as
(60)
in which the strain energy density
1
⋅
2
is proposed in a quadratic matrix representation as
1
2
(61)
Based upon Eqs. (35) and (42), Eq. (61) can be recast as follows
1
2
1
2
1
2
(62)
Inserting Eq. (43) into (62) leads to
1
2
1
2
1
2
Therefore, the variation of strain energy becomes
1
2
1
2
10
(63)
(64)
It should be noted that
(65)
√
where
and
stand for the determinant of the shifter tensor and base vectors, respectively,
which are defined for each geometry, specifically. The integration on the volume
reduced to surface
is
through introducing following matrices
,
,
(66)
1,2
So, Eq. (63) can be rewritten as
1
2
1
2
(67)
The variation of work of external loads is given by
⋅
in which
(68)
shows the force vector. Substituting Eq. (45) into (68) gives
(69)
where
.
5. VDQ discretization
In the VDQ method [36], the quadratic form of energy functional is directly discretized using
matrix differential and integral operators. The matrix operators are presented in the
Appendix. Calculating the components of the vector
at the grid points which are selected
from the Chebyshev-Gauss-Lobatto grid [57] yields the discretized displacements vector .
Following the technique of VDQ discretization, the discretized form of Eqs. (67) and (65) on
the space domain can be derived as
1
2
1
2
⊗
where
coordinates and
(70)
(71)
⊗ ,
,
1,2 . Let
〈 〉, where
be the integral operator along the in-plane
is defined in the Appendix. One can write the discretized
form of Eq. (54) as follows
11
〈
〉
where
〈
and
〉
(72)
are the discretized forms of
and
, respectively, and are presented by
Eqs. (85-87). By defining following block matrices
1
2
1
3
(73)
(74)
2
(75)
3
2
(76)
⊗
in which
(77)
and
denote the discretized linear stiffness matrix and the discretized external
forces vector, respectively. In addition,
and
are the discretized nonlinear parts of the
stiffness matrix. Accordingly, Eqs. (70) and (71) are rewritten as
(78)
(79)
By inserting Eqs. (78) and (79) into Eq. (59) (Hamilton’s principle), one arrives at
0
(80)
Solving Eq. (80) using the Newton-Raphson scheme results in finding
which is the
vector of unknown degrees of freedom and the displacements values at grid points. It should
0, is used as the
be noted that, the solution of the linear part of Eq. (80), which is
initial guess for the Newton-Raphson method. Only un-symmetric matrix-part of above
that is defined using Eq. (75). The symmetric counterpart of
relation is
〉
〈
Substituting
with
leads to
is derived as
〉
〈
(81)
which is a symmetric matrix. The Jacobian of
stiffness parts of Eq. (80) is obtained as
(82)
Furthermore, the linear and nonlinear discretized strain matrices are given by
;
;
,
;
⊗
,
⊗
;
,
;
⊗
(83)
;
in which
⨂
,
⨂
,
⨂
12
(84)
where
and
are the differentiation matrix operators in
and
Appendix). Besides,
where
and
and
indicate identity matrices with the size
are the number of grid points in
and
directions (see the
and
,
directions, respectively.
Also, one has
;
;
;
;
;
;
;
,
(85)
;
;
;
;
;
;
,
;
;
,
;
;
,
;
;
,
;
;
,
;
,
;
(86)
;
where
⊗
,
⊗
,
⊗
(87)
It is worth mentioning that the present numerical approach can be extended to study the
behavior of shells based on the micropolar theory as the studies presented in [58, 59].
6. Benchmark tests
In this section, the accuracy of developed solution methodology is revealed through eight
popular benchmark problems on large deformation behaviors of cantilevers, plates,
cylindrical shells and spherical shell. It should be noted that the cantilever is modeled as a
long rectangular plate. Comparisons are made with the results reported in [10]. The results
13
are represented in the form of load–defection curves. Base vectors, strains and elasticity
matrix for various geometries are given in the Appendix.
6.1. Cantilever subjected to end shear force and uniformly distributed load
As the first benchmark problem, a cantilever under the end shear force
distributed load
and uniform
is considered as shown in Figs. 1a and 1b. It should be noted that the cases
of uniformly distributed loads on beams and plates are very difficult to be treated numerically
as shown in [60, 61]. The material and geometrical properties are assumed as
1.2
where
10 Pa ,
, , ,
and
respectively. Also,
0,
10 m ,
1 m ,
0.1 m
are Young’s modulus, Poisson’s ratio, length, width and thickness,
4 N⁄m and
4 N⁄m . In Fig. 1c, the end shear force
and uniformly distributed load are plotted versus the vertical and horizontal tip defections. As
can be seen, there is an excellent agreement between the present results and those given by
Sze et al. [10]. Moreover, the deformed configurations of cantilever subjected to the
maximum end shear force and uniformly distributed load are indicated in Figs. 1d and 1e,
respectively.
Moreover, Table 1 provides a comparison between the convergence rate of present
approach and that of [10]. In this table, the number of load increments (NINC) and the total
number of iteration (NITER) to reach the converged solution are presented. It is observed that
the NITER of the present approach is smaller than that of [10].
6.2. Cantilever subjected to end moment
In this case study, a cantilever subjected to end moment
is considered as shown in Fig. 2a.
It is assumed that
1.2
10 Pa ,
12 m ,
0,
1 m ,
0.1 m ,
50 ⁄3 N⁄
The end moment-vertical/horizontal tip defections curves and the deformed configuration
of cantilever under the maximum moment are shown in Figs. 2b and 2c, respectively. Fig. 2b
clearly indicates the reliability of the VDQ method.
14
Similar to Table 1, Table 2 is given to compare the convergence characteristics of present
approach with those of [10] in the case of cantilever subjected to end moment. Again, it is
seen that the VDQ method needs lesser computational effort.
6.3. Slit annular plate subjected to lifting line force
According to Fig. 3a, a slit annular plate is selected in this benchmark test. The line force
is
applied at one end of the slit, whereas its other end is fully clamped. The material and
geometrical parameters are chosen as
21
10 Pa ,
6 m ,
0,
10 m ,
0.3 m
0.8 N force⁄length . Fig. 3b shows the variation
Besides, it is considered that
of load parameter with the vertical defections at the tips of the slit, A and B. The deformed
configuration of the slit annular plate subjected to the maximum force is also illustrated in
Fig. 3c.
6.4. Hemispherical shell subjected to alternating radial forces
As the fourth benchmark problem, the large deformation of a hemispherical shell subjected to
alternating radial forces is analyzed. In Fig. 4a, a hemispherical shell with an 18◦ circular
cutout at its pole is shown. As indicated, alternating radial point forces s at 90◦ intervals are
applied to the hemispherical shell. The parameters are assumed as
6.825
10 m ,
10 Pa ,
0.04 m ,
0,
400 N
It should be noted that owning to symmetry, one-quarter of the shell is modeled.
According to Fig. 4b which shows the load-deflection curves of the shell, the present results
agree well with those of [10].
15
6.5. Pullout of an open-ended cylindrical shell
Fig. 5a shows an open-ended cylindrical shell pulled by a pair of radial forces
s. It is
assumed that
10.5
10 Pa ,
0.3125,
4.953 m ,
10.35 m ,
0.094 m ,
40000 N
In Figs. 5b and 5c, the load-deflection curves and deformed configuration are shown. Due
to symmetry, one-eighth of the shell is modeled.
6.6. Pinched semi-cylindrical shell
In this case study, the large deformation problem of a pinched semi-cylindrical shell is
solved. Fig. 6a shows the semi-cylindrical shell subjected to an end pinching force at the
middle of the free-hanging circumferential periphery. Note that half of the shell is modeled
owning to symmetry. It is considered that
2068.5 Pa ,
101.6 m ,
0.3,
304.8 m ,
3 m ,
2000 N
The results of analysis are shown in Figs. 6b and 6c.
6.7. Pinched cylindrical shell mounted over rigid diaphragms
In this benchmark test, a pinched cylindrical shell mounted on rigid end diaphragms, over
which the in-plane displacements
and
are restrained, is considered (see Fig. 7a). It is
assumed that
30
10 Pa ,
100 m ,
0.3,
200 m ,
1 m ,
12000 N
Because of symmetry, one-eighth of the shell is modeled. The results of analysis are
shown in Figs. 7b and 7c.
16
6.8. Hinged cylindrical isotropic roof
Fig. 8a indicates a hinged semi-cylindrical roof subjected to a central pinching force. Onequarter of the roof is modeled due to symmetry. The considered parameters are as follows
3102.75 Pa ,
2540 m ,
0.3,
254 m ,
12.7 m ,
0.1 rad ,
3000 N
The variation of load parameter versus deflection and deformed configuration are
portrayed in Figs. 8b and 8c.
6.9. Comparison to theories with thickness deformation
In order to compare the present six-parameter shell theory to theories with thickness
deformation, the present results are compared to those of Alijani and Amabili [33] for the
case of a simply-supported aluminum square plate under uniformly distributed load in Fig. 9.
They studied the geometrically nonlinear static and vibrational behaviors of rectangular
plates by considering all of the nonlinear terms corresponding to the Green–Lagrange strain–
displacement relations based on the second-order thickness deformation theory. They also
carried out another analysis using the ANSYS three-dimensional modeling.
The geometrical parameters, material properties and magnitude of uniform load for the
comparison study are considered as follows:
70 GPa ,
0.1 m ,
1.05
0.3,
0.001 m ,
10 Pa
The essential boundary conditions at the edges for an immovable simply-supported plate
can be presented as:
at
0,
0
at
0,
0
17
The schematic of the plate and deformed configuration for the maximum uniformly
distributed load are shown in Figs. 9a and 9b, respectively. Also, comparisons are made
between the present results and those given in [33] for the non-dimensional maximum
transvers deflection at the center of the plate,
⁄2 , ⁄2 ⁄ , the maximum rotation,
0, ⁄2 , and the dimensionless thickness deformation, Δ ⁄ , in Figs. 9c-9e, respectively.
Fig. 9c shows the non-dimensional maximum transvers deflection at the center of the
plate,
⁄2 , ⁄2 ⁄ , versus the non-dimensional uniformly distributed load,
, for
different theories. It can be seen that the results of six-parameter shell theory agree well with
those reported in [33]. Fig. 9d indicates that the maximum rotation,
0 and
at
⁄2, versus the non-dimensional uniformly distributed load based on different theories. One
can see that, the present maximum rotation curve is in reasonable agreement with those of
2TS-NL and ANSYS 3D. It is worth noting that based on the definitions given in [33], 2TS
denotes the second-order thickness stretching theory and NL stands for the retaining
nonlinearities and thickness deformation which can be defined as Δ
. According to
Fig. 9e, the non-dimensional thickness deformation at the center of the plate (
/2) for
1428.6 is converged for different theories, while it is divergent close to the edges. Also,
it should be noted that the present six-parameter shell theory can only take the first-order
thickness deformation parameter into account.
7. Conclusion
In this paper, in the context of the six-parameter shell theory, an efficient solution strategy
was proposed to analyze the nonlinear large deformation of shells. The idea of paper was
based on the VDQ method, a recently developed technique to simplify numerical analysis of
structures. First, the nonlinear strain-displacement relations and constitutive relations for the
large deformation analysis of the six-parameter shell model were derived and represented in
suitable matrix forms. Then, according to Hamilton’s principle and by means of the VDQ
discretization technique, the quadratic form of energy functional was directly discretized. In
order to show the accuracy of developed formulation, eight well-known benchmark problems
were considered, and comparisons were made between the present results and those of FEM
reported in the literature. Also, comparisons were made between the results of six-parameter
shell theory and those of theories with thickness deformation. It was revealed that the VDQ
method is capable of solving various problems on the large deformation analysis of shell-type
18
structures with high accuracy. Easy implementation, general formulation, absence of locking
problem, accuracy and low computational cost are the main advantages of approach proposed
in this paper.
19
Appendices:
A. Differential and integral operators
The differential and integral operators used in the VDQ method are presented here. First, the
operators are given in general form using the differential quadarature technique and then they
are defined for each geometry, specifically.
A.1. Differential operator
Consider the following column vector
⋯
where
is the value of
(A-1)
at each grid point
, and
denotes the number of grid
points.
The values of -th derivative of
at each point is defined as
(A-2)
⋯
By means of the generalized differential quadarature (GDQ) method, it is rewritten as
(A-3)
in which the differentiation matrix operator
is evaluated at grid point values through the
following relation [62]
0
,
,
,
1
2, … ,
1,2, … ,
1
(A-4)
1
,
where ,
1,2, … ,
and is
identity matrix, and
is
(A-5)
,
The grid points distribution is selected according to the Chebysev-Gauss-Lobatto grid
because of its high convergent in the GDQ method [57] which can be expressed as
20
1
1
2
1
1
cos
,
1,1 which can be mapped onto
This distribution gives points on
2
2
(A-6)
1,2, ⋯ ,
,
as
(A-7)
,
A.2. Integral operators
Using the Taylor series and the GDQ technique, an accurate row vector integral operator can
be constructed as [63]
ξ
ξ ξ
,
ξ
ξ
ξ
ξ
(A-8)
ξ
with
ξ
2
ξ
1 !
,…,
ξ
2,3, … ,
ξ
ξ
1 !
2
ξ
,…,
ξ
2
ξ
1 !
,
1
(A-9)
where
denotes the GDQ differential operator or the weighting coefficients matrix of the
th-order derivative, and
signals a column vector with
values of the arbitrary function
at
,
components containing the nodal
1,2, … , . The integral operators for different
geometries are given in the following.
Integral operator in the
direction ( ) and
direction ( ) for cylindrical shell can be
presented as follows
(A-10)
(A-11)
in which
denotes the radius of shell. Also, integral operator in the
direction ( ) and
direction ( ) for annular plate are defined as follows
∘
(A-12)
(A-13)
where is the grid points in
direction
⋯
In addition, for the hemispherical shell, integral operator in the
direction ( ) can be given as
21
direction ( ) and
∘
,
cos
(A-14)
(A-15)
in which
stands for the radius of shell and
is the vector of grid points in
direction
⋯
and integral operator in the
and
directions can be introduced as
(A-16)
(A-17)
A.3. Generalization to N-Dimensional space
,
The nodal value of function
⋯
, , ,…,
,
, ,…,
where
, , ,…,
,…,
,…,
in the N-Dimensional space can be given by
⋯
, , ,…,
,
, ,…,
, , ,…,
⋯
,
1, … ,
1, … ,
⋮
1, … ,
,
,
,…,
(A-20)
is the number of grid points in each dimension and
is the number of dimensions.
By means of matrix differential operator of Eq. (A-3), the partial derivative of function in
-
dimensional space is approximated by the following relation
⋯
,
,…,
⊗
…
Additionally, the integral operator of
⋯
,
,…,
⊗ ⋯⊗
⊗
(A-21)
-dimensional space is formulated as
…
⊗
⊗ ⋯⊗
⊗
…
(A-22)
Thus, for a two-dimensional problem, the integral operator can be presented as
⊗
(A-23)
22
B. Base vectors, strains and elasticity matrices for various geometries
The relations of base vectors, strains and elasticity matrices are given for each geometry here.
B.1. Cantilever
The base vectors in the Cartesian coordinate system ( ) and curvilinear coordinates system
(
) for a cantilever can be defined as follows
,
,
,
,
,
,
,
(B-1)
,
It should be noted that the Cartesian ( , , ) and curvilinear coordinates ( ) coincide in
this geometry. Also, the determinant of the shifter tensor is given as
1
(B-2)
Based on Eq. (30) and using the given base vectors in this geometry, the coefficients
can be calculated as
1,
0,
(B-3)
,
,
,
,
,
0
,
(B-4)
and accordingly, the components of nonlinear strains of six-parameter cantilever in the
Cartesian coordinate can be represented as follows
1
2
1
2
,
,
,
,
,
,
,
,
,
,
2
,
2
,
,
,
,
2
,
,
,
,
,
,
,
,
,
,
,
(B-5)
1
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2
,
2
,
,
,
,
,
,
,
,
(B-6)
,
,
23
,
,
,
,
,
,
,
2
,
,
,
,
0
Furthermore, the components of metric tensor (
1,
) for a cantilever can be presented as
1,
1
(B-7)
using Eq. (38), on can attain the elasticity matrix as
Consequently, by calculating
follows
2
0
0
2
0
0
0
0
0
0
0
0
0
0
.
(B-8)
2
B.2. Annular plate
The base vectors in the Cartesian coordinate system ( ) and curvilinear coordinates system
(
) for an annular plate can be presented as
,
,
,
,
,
cos
where
,
sin
,
sin
cos
,
(B-9)
denotes polar coordinate system and the determinant of the shifter tensor ( ) is
defined as follows
1
(B-10)
Similarly, the coefficients
for annular plate are given as
cos
sin
sin
cos
0
,
0
0
,
Using
0
,
,
1
,
(B-11)
0
,
and based on Eqs. (32) and (33), the strain components for an annular plate can
be presented as
cos
sin
,
sin
,
,
cos
1
2
,
,
1
2
,
,
,
(B-12)
,
,
24
,
,
2
sin
cos
,
,
2
,
2
,
cos
,
sin
,
,
,
,
,
,
,
cos
sin
sin
,
,
cos
,
,
,
,
1
2
cos
sin
,
sin
2
,
,
cos
,
sin
,
cos
,
,
,
,
,
,
,
,
,
,
2
,
,
,
,
2
,
,
,
,
,
,
cos
,
,
,
,
,
,
,
sin
,
,
,
,
,
,
,
(B-13)
,
0
Also, the components of metric tensor (
) and the elasticity matrix ( ) in this geometry
can be presented as follows
1
1,
2
2
,
1
0
0
0
0
0
0
0
0
0
0
0
(B-14)
(B-15)
0
.
2
B.3. Hemispherical Shell
One can define the base vectors in the Cartesian coordinate system ( ) and curvilinear
coordinates system (
) for a hemispherical shell as
25
,
,
,
,
,
sin
,
cos
sin
sin
cos
cos
cos
cos
, ,
where
cos
,
(B-16)
,
sin
sin
denote spherical coordinates and the determinant of the shifter tensor ( ) can
be presented as
(B-17)
1
Using the relations of the base vectors (
and
) and Eq. (30), the coefficients
in this
geometry are given as
sin
cos
cos
sin
cos
sin
cos
cos
cos
,
sin
cos
,
,
cos
sin
,
sin
cos
cos
0
sin
sin
sin
sin
cos
cos
(B-18)
,
cos
,
0
and accordingly, the nonlinear strains of six-parameter spherical shell in the Cartesian
coordinate can be attained as
sin
cos
1
2
cos
2
cos
,
,
sin
cos
cos
,
cos
cos
,
,
sin
2
cos
cos
cos
cos
,
cos
cos
,
1
2
,
,
cos
,
,
sin
,
sin
,
,
cos
,
,
,
,
sin
sin
,
sin
sin
,
cos
,
sin
,
,
sin
,
(B-19)
,
cos
cos
cos
sin
,
cos
,
,
sin
sin
,
cos
cos
cos
,
sin
,
,
sin
2
sin
,
,
26
cos
,
,
sin
cos
,
1
2
,
sin
sin
cos
sin
,
sin
cos
sin
sin
cos
sin
sin
,
2
sin
2
cos
,
cos
sin
sin
sin
,
,
,
,
,
(B-20)
,
cos
,
,
cos
,
sin
,
,
,
cos
,
,
,
cos
,
cos
,
,
,
sin
,
cos
cos
sin
,
,
cos
,
,
sin
cos
cos
sin
,
,
,
cos
,
cos
,
,
sin
sin
,
,
sin
,
,
,
sin
,
sin
,
,
cos
cos
cos
,
,
cos
,
cos
sin
,
,
,
,
cos
,
cos
cos
cos
cos
cos
,
,
cos
,
cos
2
sin
,
cos
,
sin
,
,
0
Also, the components of metric tensor (
) and the elasticity matrix (
) are written as
follows
1
1
cos
,
2
cos
2
cos
,
1
0
0
0
0
0
0
0
0
0
0
0
cos
cos
.
(B-21)
cos
(B-22)
0
2
B.4. Cylindrical shell
The base vectors in the Cartesian coordinate system ( ) and curvilinear coordinates system
(
) for a cylindrical shell can be given as
27
,
,
,
,
,
,
(B-23)
,
sin
cos
cos
,
sin
in which ( , , ) denote the cylindrical coordinate system and the determinant of the shifter
tensor ( ) can be defined as
1
(B-24)
Using Eq. (30) one can write
1
0
0
0
sin ,
0
cos
cos
,
0
,
,
0
,
sin
0
sin
(B-25)
,
0
,
cos
and the six-parameter nonlinear strains for a cylindrical shell can be written as follows
,
1
2
,
,
,
2
,
,
1
2
,
,
,
,
2
,
,
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(B-26)
,
,
,
,
1
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(B-27)
,
,
,
2
,
,
,
2
,
,
,
,
,
,
,
,
,
,
,
,
28
,
,
,
,
,
,
,
2
,
,
,
,
,
0
In addition, one can write the components of metric tensor (
1
1,
,
1
and based on Eqs. (38) and (40 ) the elasticity matrix (
2
2
0
0
0
0
0
0
0
0
0
0
0
(B-28)
) can be presented as follows
(B-29)
0
.
) as
2
29
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[58] Altenbach, J., Altenbach, H., and Eremeyev, V. A., 2010, “On generalized Cosserat-type
theories of plates and shells: a short review and bibliography,” Arch. Appl. Mech., 80,
pp. 73-92.
[59] Altenbach, H., and Eremeyev, V. A., 2009, “On the linear theory of micropolar
plates,” ZAMM, 89, pp. 242-256.
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clamped Euler beam (Elastica) with distributed load: large deformations,” Math. Model.
Meth. Appl. Sci., 27, pp. 1391-1421.
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of planar extensible beams and pantographic lattices: heuristic homogenization,
35
experimental and numerical examples of equilibrium,” Proc. Royal Soc. London A, 472,
(2185).
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[63] Ansari, R., Shojaei, M. F., Rouhi, H., and Hosseinzadeh, M., 2015, “A novel variational
numerical method for analyzing the free vibration of composite conical shells,” Appl.
Math. Model., 39, pp. 2849-2860.
36
(a)
(b)
1
-u 1 (tip) - Present
0
0.9
u3 (tip) - Present
-u 1 (tip) - [10]
u3 (tip) - [10]
0
0.8
-u 1 (tip) - uni. load
u3 (tip) - uni. load
0
0.7
0
0.6
0
0.5
0
0.4
0
0.3
0
0.2
0
0.1
0
0
1
2
3
4
5
deflectio
on
(c)
37
6
7
8
9
(d)
(e)
Figure 1: (a) Cantilev
ver subjecteed to end shear force, (b
b) Cantileveer subjected
d to
unifoormly distribbuted load (c)
( Load–deefection curv
ves, (d) Defformed connfiguration for
f the
maaximum endd shear forcce, (e) Deforrmed config
guration forr the maxim
mum uniform
mly
diistributed lo
oad
38
(a)
1
0
0.9
0
0.8
0
0.7
0
0.6
0
0.5
0
0.4
-u 1 (tip)) - Present
0
0.3
u3 (tip) - Present
0
0.2
-u 1 (tip)) - [10]
0
0.1
0
u3 (tip) - [10]
0
3
6
9
deflectioon
(b)
39
12
15
(c)
Figurre 2: (a) Canntilever sub
bjected to ennd bending moment, (b
b) Load–deffection curv
ves, (c)
D
Deformed
co
onfigurationn for the maaximum ben
nding momeent
40
(a)
1
0
0.9
u 3 (A) - Preseent
0
0.8
u 3 (A) - [10]
u 3 (B) - Preseent
u 3 (B) - [10]
0
0.7
0
0.6
0
0.5
0
0.4
0
0.3
0
0.2
0
0.1
0
0
2
4
6
8
10
12
14
16
18
deflectioon
(b)
(c)
Figgure 3: (a) Slit
S annularr plate underr the line fo
orce , (b) Load–defect
L
tion curves,, (c)
Deformed configuraation for thee maximum line force
41
(a)
1
0
0.9
u 1 (A) - Pressent
0
0.8
u 1 (A) - [10]]
-u 2 (B) - Present
-u 2 (B) - [10]]
0
0.7
0
0.6
0
0.5
0
0.4
0
0.3
0
0.2
0
0.1
0
0
1
2
3
4
5
deflectio
on
(b)
42
6
7
8
9
(c)
mispherical shell subjeected to alterrnating radiial forces, (bb) Load–defection
Figure 4: (a) Hem
cuurves, (c) Deformed coonfiguration
n for the max
ximum forcces
43
(a)
1
0
0.9
u 2 (A) , Pressent
0
0.8
-u 3 (C) , Pressent
-u 3 (B) , Pressent
0
0.7
u 2 (A) , [10]]
0
0.6
-u 3 (C) , [10]]
-u 3 (B) , [10]]
0
0.5
0
0.4
0
0.3
0
0.2
0
0.1
0
0
0.5
5
1
1.5
2
2.5
deflectioon
(b)
44
3
3
3.5
4
4.5
5
(c)
Figu
ure 5: (a) Open-end cyllindrical sheell under raadial pulling
g forces, (b)) Load–defeection
c
curves,
(c) Deformed
D
coonfiguration
n for the maaximum loaad
45
(a)
(b)
46
(c)
Figuree 6: (a) Sem
mi-cylindrical shell undder an end pinching
p
force, (b) Loadd–defection
n curve,
(c) Defo
ormed confiiguration forr the maxim
mum load
47
load parameter
(a)
(b)
48
(c)
Figuree 7: (a) Cyliindrical shelll under raddial pinching
g forces resttrained withh rigid diaph
hragms,
(b) Load–ddefection cu
urves, (c) Deeformed con
nfiguration for the maxximum load
d
49
(a)
1
-u 2 (A) - Pressent
0
0.9
-u 2 (A) - [10]]
0
0.8
0
0.7
0
0.6
0
0.5
0
0.4
0
0.3
0
0.2
0
0.1
0
0
5
10
15
deflectioon
(b)
50
20
255
30
(c)
Figgure 8: (a) Hinged
H
cylindrical rooff under a ceentral pinchiing force, (bb) Load–defection
c
curve,
(c) Deformed
D
coonfiguration
n for the maaximum loadd
51
(a)
(b)
52
16
14
12
10
8
6
4
HSDT, 2TS, 2TS-NL [33]
ANSYS-3D [33]
6-parameter
2
0
0
3
6
9
P
10
12
15
4
(c)
0
HSDT, 2TS [33]
ANSYS-3D [33]
2TS-NL [33]
6-parameter
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0
3
6
9
P
(d)
53
10
4
12
15
10-3
-1
-2
-3
-4
-5
-6
-7
ANSYS-3D [33]
2TS-NL [33]
HSDT & 2TS [33]
6-parameter
-8
-9
-10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/L
(e)
Figure 9: (a) Simply-supported square plate subjected to uniformly distributed load, (b)
Deformed configuration for the maximum load, (c) Non-dimensional maximum transverse
deflection curve at the center of the plate versus the non-dimensional uniformly distributed
load, (d) Maximum rotation
versus the non-dimensional uniformly distributed load, (e)
Non-dimensional thickness deformation (Δ ⁄ ) at
1428.6
54
⁄2 for the non-dimensional pressure
Table 1: Comparison of the number of load increments (NINC) and the number of iterations (NITER) required
to obtain the ultimate solution for the cantilever subjected to end shear force
Ref. [10]
8
1 S4R elements
16
present
1 S4R elements
15
5 nodes
NINC
15
15
15
NITER
78
80
57
Table 2: Comparison of the number of load increments (NINC) and the number of iterations (NITER) required
to obtain the ultimate solution for the cantilever subjected to end bending moment
Ref. [10]
8
1 S4R elements
16
present
1 S4R elements
15
5 nodes
NINC
125
125
103
NITER
715
714
394
55
Highlights
A new approach is proposed for the nonlinear large deformation analysis of shells
based on the six-parameter shell theory.
The proposed method is implemented in the context of weak form formulation and
can be easily used for various geometries.
Easy implementation, general formulation, absence of locking problem, accuracy and
low computational cost are the main advantages of proposed approach.
56
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