ARTICLE IN PRESS
Engineering Analysis with Boundary Elements 32 (2008) 494–511
www.elsevier.com/locate/enganabound
An EFG method for the nonlinear analysis of plates undergoing
arbitrarily large deformations
Carlos Tiagoa,, Paulo M. Pimentab
a
Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
b
Escola Politécnica, Universidade de São Paulo Av. Prof. Almeida Prado, trav. 2, 83 São Paulo, Brazil
Received 2 February 2007; accepted 10 October 2007
Available online 28 January 2008
Abstract
The applicability of a meshfree approximation method, namely the EFG method, on fully geometrically exact analysis of plates is
investigated. Based on a unified nonlinear theory of plates, which allows for arbitrarily large rotations and displacements, a Galerkin
approximation via MLS functions is settled. A hybrid method of analysis is proposed, where the solution is obtained by the independent
approximation of the generalized internal displacement fields and the generalized boundary tractions. A consistent linearization
procedure is performed, resulting in a semi-definite generalized tangent stiffness matrix which, for hyperelastic materials and conservative
loadings, is always symmetric (even for configurations far from the generalized equilibrium trajectory). Besides the total Lagrangian
formulation, an updated version is also presented, which enables the treatment of rotations beyond the parameterization limit. An
extension of the arc-length method that includes the generalized domain displacement fields, the generalized boundary tractions and the
load parameter in the constraint equation of the hyper-ellipsis is proposed to solve the resulting nonlinear problem. Extending
the hybrid-displacement formulation, a multi-region decomposition is proposed to handle complex geometries. A criterium for the
classification of the equilibrium’s stability, based on the Bordered–Hessian matrix analysis, is suggested. Several numerical examples are
presented, illustrating the effectiveness of the method. Differently from the standard finite element methods (FEM), the resulting
solutions are (arbitrary) smooth generalized displacement and stress fields.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Plates; Geometrically exact; Meshfree; Moving least squares; Hybrid weak form; Multi-region
1. Introduction
1.1. Historical background
The research on geometrically exact shell models was
initiated by Simo and co-workers. The formulation and
parameterization of the model was presented in [1], where
the hypothesis of one inextensible director—used in the
present work—was already considered. In the subsequent
papers the linear and nonlinear computational aspects
of the theory were dealt. Other perspectives were later
considered, like the through-the-thickness stretch, a
plasticity constitutive model, time-stepping conserving
Corresponding author.
E-mail addresses: [email protected] (C. Tiago),
[email protected] (P.M. Pimenta).
0955-7997/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.enganabound.2007.10.014
algorithms for dynamical analysis and shell intersections
problems.
Nevertheless, some drawbacks were still present, like the
need for complex configuration updates and the use of
assumed strain methods to avoid the shear and membrane
locking effects.
On the twin papers [2,3] a unified theory for beams and
shells, respectively, was presented. Here, the fundamental
variable for parameterizing the rotation tensor is the
rotation vector, delivering an expression for the tangent
stiffness which is always symmetric for hyperelastic
materials and conservative loads, even far from the
equilibrium path.
Implementation of this theory for beams was presented
in [4], which was latter generalized to curved rods [5] and to
accommodate warping and a genuine moderate finite strain
constitutive relation [6].
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
In the shell model implementation [7] a constitutive
relation was derived based on a true plane stress condition.
The generalization presented in [8] accommodates the
thickness variation of the shell, thus allowing the use of a
full three-dimensional (3D) finite strain constitutive model.
The first geometrically exact analysis using meshfree
approximations was presented in [9]. The solution of beam
problems was performed by using moving least squares
(MLS) approximation to discretize the generalized displacements fields. Hence, the procedure can be considered
an extension of the element-free Galerkin (EFG) method
[10] for the geometrically exact analysis of structures.
1.2. Scope of the present work
The numerical solution of the governing equations of the
geometrically exact models has been done, invariably, by
the traditional version of the finite element method1
(FEM). Accordingly, some problems commonly associated
with the FEM are also present in the encountered
solutions, like (i) the need to explicitly set up incidence
relations between nodes (in order to shape elements),
(ii) the lack of equilibrium on the static boundary and in
the interelement interface and (iii) the shear and membrane
locking effects.
In the recent years a new class of methods for the
solution of sets of differential equations has emerged:
the so-called meshless methods. These may be classified in
classical or weak form based methods. Among the first
group we can cite the generalized finite-differences, the
radial basis functions collocation or the method of
fundamental solutions. The latter set includes, e.g., the
diffuse element method, the element-free Galerkin method,
the h-p clouds method, the partition of unity method, the
reproducing kernel particle method, the finite spheres
method and the meshless local Petrov–Galerkin method.
For reviews see, e.g., [11,12].
In the present work an alternative method for the
numerical analysis of plates is presented. Instead of the
traditional FEM approach, the EFG is now extended to
plate analysis. Hence, a fresh approximation method is
applied to the numerical solution of a, also recent, shell
model. In this way we try to overcome some of the
inconveniences of the FEM by the use of meshfree
discretization, like (i) preclude the use of structured grids,
(ii) the obtainment of smooth static (generalized stresses)
and kinematic (generalized displacements) fields and (iii)
avoid locking effects. Furthermore, it is well known that
the excellent rates of convergence of the MLS functions
both for approximation of data and for the solution of
linear differential equations, see [13,14].
1
By traditional version of the finite element method we refer to the wellknown displacement model using nodal shape functions for approximation of both the geometry and the generalized displacements fields and
imposition of the essential boundary conditions through collocation. Nonconventional formulations (like hybrid, mixed or equilibrium) are thus
excluded from this set.
495
The present work is a part of a global framework to
perform geometrically exact analysis using meshfree
approximations. This research activity was initiated by
[9], with the analysis of straight beams. In the present work,
the plates problem is addressed. Currently the initially
curved beams and shells are under development. The final
step will be the consideration of thin shells. In this latter
case the meshfree approximations have a clear advantage
over the FEM, because, at least, C 1 continuity is required
for the generalized displacement fields. This constraint is
easily fulfilled by the former but is practically impossible in
the latter.
The only kinematical assumption is the Reissner–
Mindlin plane section hypothesis. The internal virtual
work is expressed by the first Piola–Kirchhoff stress tensor
and the deformation gradient. The exact parametrization
of the rotation tensor is made through Euler–Rodrigues
formula. As all vectorial parameterizations of the rotation
tensor, this closed-form solution has a limited range
of application, beyond which a singularity occurs. To
circumvent this problem, an update Lagrangian formulation can be used. Here the technique presented in [9] is
generalized for the present case.
In order to circumvent the non-interpolation character
of the approximations—which impairs the construction of
trial and test spaces which satisfy a priori the kinematic
boundary conditions and the homogeneous kinematic
boundary conditions, respectively—a hybrid-displacement
weak form suitable for meshless approximations is
presented, which includes the internal virtual work, the
external virtual work, the external complementary virtual
work arising from the kinematic boundary and a weak
statement of the kinematic boundary conditions.
Besides assuring a quadratic convergence rate in the
solution of the nonlinear incremental/iterative algorithm,
the knowledge of the exact form of the generalized tangent
equilibrium2 contains all the relevant information to
classify the stability of the equilibrium. A criteria for the
classification of the equilibrium’s stability, based on the
Bordered–Hessian matrix analysis and the static criteria of
the principle of potential energy, is applied to the present
analysis, enabling to distinguish the several solution
branches.
In order to pave the way to the analysis of plates with
complex geometries, the extension of the hybrid-displacement weak form to interface boundaries is also presented,
rendering a multi-region method.
The inextensibility of the director is complemented by a
plane stress condition. This is imposed over constitutive
model, which is the neo-Hookean material.
2
Unlike the traditional FEM, in the present work the equations of the
system matrix do not merely express (a weak statement of) the
equilibrium, but also the kinematic and interfaces compatibility. Hence
the denomination exact form of the tangent equilibrium is incomplete and is
replaced by exact form of the generalized tangent equilibrium.
ARTICLE IN PRESS
496
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
An extended nonlinear arc-length constraint, which
includes the generalized internal displacement fields, the
generalized boundary and interface tractions and the load
parameter, is proposed to solve the resulting nonlinear
problem.
1.3. Notation and text organization
Throughout the text italic Latin or Greek lowercase
letters ða; b; . . . a; b; . . .Þ denote scalar quantities, bold italic
Latin or Greek lowercase letters (a; b; . . . a; b; . . .) denote
vectors, bold italic Latin or Greek capital letters (A; B; . . .)
denote second-order tensors, bold Calligraphic Latin
capital letters ðA; B; . . .Þ denote third-order tensors and
bold blackboard italic Latin capital letters
. . .Þ denote
forth-order tensors in a 3D Euclidian space. The same
letter is used to identify the skew-symmetric second order
tensors (A; B; . . . X; H; . . .) and their associated axial vector
ða; b; . . . x; h; . . .Þ. Vectors and matrices built of tensor
components on orthogonal frames (e.g., for computational
purposes) are expressed by boldface upright Latin letters
ðA; B; . . . a; b; . . .Þ. Greek indices range from 1 to 2, while
Latin indices range from 1 to 3.
The problem is presented in Section 2, where the relevant
kinematics and statics are briefly presented, followed by the
proposed variational formulation of the problem in Section
3. We emphasize the need to develop this custom made
hybrid statement to be able to discretized the domain using
approximations (like MLS) instead of interpolations (like
Lagrange polynomials). The results of the linearization of
the weak form are reported in Section 4 without proof. In
order to accommodate non-smooth plates, the generalization to multi-regions is presented in Section 5 followed
by the suggested meshfree discretization in Section 6, with
the corresponding residual and exact tangent generalized
stiffness. Some essential aspects for the success of the
implementation are discussed in Section 7, as the constitutive model, a rotation update procedure, the classification of the stability of the equilibrium and a nonlinear
constraint suitable for a continuation method. Some
numerical applications are presented in Section 8 and
conclusions are extracted in Section 9.
2. A summary of a geometrically exact plate model
2.1. The model problem
Consider the plate exhibited in Fig. 1, where two
orthonormal right-handed coordinate systems are represented, namely, eri for the reference configuration and ei for
the current configuration.
The reference plane is denoted by Or R2 . The contour
of Or is denoted by Gr , i.e., Gr ¼ qOr and can be
decomposed as Grt [ Gru ¼ Gr and Grt \ Gru ¼ ;, where Grt
and Gru identify the static and kinematic boundaries. The
volume of the body is identified by V r and plate thickness is denoted by H r ¼ ½hrb ; hrt , both on the reference
Fig. 1. The reference and current configurations of the plate.
configuration. The endpoints of H r are collected in the set
C r ¼ fhrb ; hrt g, thus C r ¼ qH r .
We assume the applied loads vary linearly with a loading
parameter, l. Nevertheless, for simplicity, this dependance
will be omitted in the following. The plate is under
r
the effect of body forces, b , per unit volume of the
r
reference configuration and traction forces, t , per unit
area of the reference configuration on the top, bottom
and lateral surfaces.3 Eventually, configuration dependant
loads may be included. In the lateral surfaces the plate is
r
subjected either to prescribed tractions, t , per unit area of
the reference configuration, or to imposed displacements.
The precise definition of the quantities to be imposed in
order to explicitly prescribe the displacements is introduced
latter.
2.2. Kinematics
The reference configuration can be described by n, which
can be written as
n ¼ f þ ar ,
(1)
xa era
where f ¼
defines the position of a material point over
the middle plane of the reference configuration, Or , and
ar ¼ zer3 represents the component along the normal,
z 2 H r.
From Fig. 1 it can be concluded that, on the deformed
configuration,
x ¼ z þ a,
(2)
where
a ¼ Qar .
(3)
Q is a rotation tensor which, in the present work, is
parameterized by the Euler–Rodrigues formula [15].
Accordingly, the rotational variables are collected in the
rotation vector, h. Its skew-symmetric tensor, H, is defined
as H ¼ SkewðhÞ and the rotation is given by y ¼ khk.
3
The distinction in the notation used for traction forces on the top,
bottom and lateral surfaces is made through the superscripts t, b and l,
respectively.
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
Hence,
and
Q ¼ I þ h1 ðyÞH þ h2 ðyÞH2 ,
(4)
where h1 ðyÞ and h2 ðyÞ are the trigonometric functions
sin y
1 sinðy=2Þ 2
and h2 ðyÞ ¼
h1 ðyÞ ¼
.
(5)
y
2
y=2
In order to meet certain objectivity requirements all the
necessary quantities are expressed in the reference configuration. This transformation is made by means of the
rotation tensor, Q. For example, we refer to the first order
tensor vr as being the back-rotated counterpart of v if
v ¼ Qvr ; 8v; vr 2 R3 . Of course, this transformation is fully
generalizable to tensors of an arbitrary order.
For evaluating the internal power it is chosen to use, in
the present work, the energy conjugated pair formed by the
(back-rotated counterparts of) the first Piola–Kirchhoff
stress tensor and the rate of the deformation gradient.
The back-rotated counterpart of the deformation gradient, F r , can straightforwardly be derived from the
displacement field and may be written as
r
F ¼Iþ
cra
era .
(6)
The generalized strain vectors, cra , are given by
cra ¼ gra þ jra ar ,
(7)
where
gra
497
T
¼ Q z;a era ,
jra ¼ CT h;a .
(8a)
(8b)
Tensor C relates spin like variables with the appropriate
variation of the rotation vector. In particular, in expression
(8b) the tensor C maps the spatial variation of the rotation
tensor into a generalized curvature vector and is given by
C ¼ I þ h2 ðyÞH þ h3 ðyÞH2 ,
(9)
being
1 h1 ðyÞ
.
(10)
y2
Expressions (8) may be regarded as the compatibility
equations, as they (nonlinearly) relate generalized strains
with generalized displacements.
The strain variations will eventually be relevant within a
weak statement of the problem. These are given by
h3 ðyÞ ¼
dcra ¼ dgra þ djra ar ,
(11)
being
dgra ¼ QT ðdu;a þ Z ;a CdhÞ,
(12a)
djra ¼ QT ðC;a dh þ Cdh;a Þ.
(12b)
Here
C;a ¼ h2 ðyÞH;a þ h3 ðyÞðHH;a þ H;a HÞ
þ h4 ðyÞðh h;a ÞH þ h5 ðyÞðh h;a ÞH2
ð13Þ
h1 ðyÞ 2h2 ðyÞ
and
y2
h2 ðyÞ 3h3 ðyÞ
.
h5 ðyÞ ¼
y2
h4 ðyÞ ¼
ð14Þ
The generalized strains of the plate model can be collected
in the vector
" r#
" r#
e1
ga
r
r
e ¼ r
where ea ¼
.
(15)
e2
jra
Introducing the generalized displacements vector, d, given
by
" #
u
d¼
,
(16)
h
the variation of the generalized strains can be recast in the
compact form
der ¼ WDdd,
where
2
QT
6
6 O
W¼6
6 O
4
O
2
(17)
O
QT C
O
O
O
O
O
QT
O
O
O
QT C
q
6 qx1
6
6
6 O
6
6
6
D¼6 q
6I
6 qx2
6
6
6 O
4
I
O
3
QT Z ;1 C
7
QT C;1 7
7,
QT Z ;2 C 7
5
QT C;2
(18a)
3
O
7
7
q 7
7
I
qx1 7
7
7
7.
O 7
7
7
q 7
7
I
qx 5
(18b)
2
I
2.3. Statics
F being chosen as a measure of the deformation, it
becomes necessary to use the first Piola–Kirchhoff stress
tensor, P, to characterize the state of stress at each point.
Again, the back-rotated counterpart fulfils the necessary
rigid-body invariance requirements. Hence,
Pr ¼ sri eri ,
(19)
where sri are back-rotated stress vectors.
As the present plate model does not take into
consideration changes in the plate director length it is
convenient to assume, at the constitutive level, a plane
stress state. The associated stress vector is then designated
by esri . The generalized stresses—after the plane stress
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
498
imposition—can be collected in the vector
" r#
" r #
r1
na
r
r
r ¼
where ra ¼
r
r2
mra
where
(20)
assembles the generalized stresses acting on cross-sectional
planes whose normal on the reference configuration is era
being
Z
Z
r
r
r
r
esa dH and ma ¼
na ¼
ar esra dH r .
ð21Þ
Hr
T T
PF ¼ ðPF Þ ,
(22)
is locally satisfied by the constitutive law.
"
¼
lO
r
;
q
Grt
"
¼
r
n Gt
r
lGt
#
and
q
Gru
"
¼
r
n Gu
r
lGu
#
(26)
n
Grt
n
Gru
Z
lr
¼
Hr
t dH r
Z
¼
Hr
and
(27b)
rr dH r
(27c)
collect the forces and
m
3. Variational formulation of the problem
3.1. A hybrid weak form
Grt
m
t
Using the FEM it is trivial to construct a trial ftestg
space for d fddg which satisfy a priori the kinematic
boundary conditions fthe homogeneous kinematic boundary conditionsg. On the contrary, meshfree approximations
do not satisfy the well-known Kronecker–delta property
Z
(23)
fi ðxj Þ being the value at node j of the nodal function
centered at node i. Accordingly, the task of setting the
appropriate trial and test spaces of the weak statement in
order to locally enforce the kinematic boundary terms is
considerably more difficult than in the FEM interpolants
case.5
To alleviate the conditions over the trial and test spaces
we impose the kinematic boundary conditions in weak
sense, setting in this way a hybrid-displacement model. For
a review on the subject, see [16]. It can be proved that the
internal virtual work may be expressed as
Z
Z
dW int ¼
P : dF dV r ¼
rr der dOr
(24)
Or
and the external virtual work may be written as
Z
Z
r
r
Or
q dd dO þ
qGt dd dGrt
dW ext ¼
m
Z
br
b
lr
¼
Hr
Gru
tr
¼a t þa t þ
4
Vr
#
Hr
Or
fi ðxj Þ ¼ dij ,
r
nO
are cross-sectional generalized resultants on the reference
configuration.
Z
r
tr
br
Or
n ¼t þt þ
b dH r ,
(27a)
Hr
It is assumed that the second equilibrium equation in the
reference configuration, i.e.
T
q
Or
Z
¼
Hr
al t dH r
and
a rr dH r
r
Hr
a b dH r ,
(28a)
(28b)
(28c)
collect the moment resultants. Here rr are the tractions
on the kinematic boundary and at , ab and al are the
directors—on the current configuration—of the top,
bottom and lateral applied tractions.6 Notice the inclusion
of the virtual work term arising from the kinematic
boundary in (25), given by the projection of the generalized
reactions on the virtual displacements.
The weak form of the equilibrium of the plate can be
recast by the following virtual work principle:
dW int dW ext ¼ 0
8dd,
(29)
where dd stands for an infinitesimal perturbation of the
generalized displacements field.
Due to the kinematical assumption, the displacements of
a given point, f, on the lateral surface are not independent
along ar . Let us assume that the prescribed displacements
are given as
" #
u
d¼
,
(30)
h
Grt
Or
Z
r
þ
Gru
qGu dd dGru ,
ð25Þ
4
In fact, sometimes this task is not possible due the kinematic boundary
form of the models, like in circular shaped ones. In those cases, the
boundary configuration is merely approximated and an error in the
geometry configuration is introduced.
5
With an appropriate change of coordinates this could also be
accomplished, see [12, p. 116]. Nevertheless, an error still exists in between
the particles arranged along the kinematic boundary.
i.e., we assume that the prescribed orientation of the
kinematic part of the contour of the plate is already
expressed in terms of the Euler–Rodrigues parameters. In
general, a rotation tensor can be used to prescribe the
displacements. In this case an extraction procedure should
be applied.
6
No specific notation was introduced for the directors of the body forces
and the tractions on the kinematic boundary, as the obvious designations,
ab and ar , would introduce ambiguities in the notation.
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
The weak imposition statement of the kinematic
boundary conditions reads7
Z
r
r
dqGu ðd dÞ dGru ¼ 0 8dqGu .
(31)
Gru
The combination of the principle of virtual work (29) and
the weak constraint imposition (31) gives the final weak
form, which is the following hybrid functional:
in Or ,
dW ¼ 0
where
(32)
r
Or
r
r
Z
r de dO Z
Gru
q
Gru
dd
dGru
Or
q
Or
Z
r
dd dO Grt
q
Grt
dd
r
Gru
dqGu ðd dÞ dGru .
dGrt
ð33Þ
3.2. Recovering the governing equations
In this section the governing equations that are imposed
in weak sense will be obtained from the derived weak form.
Substituting (17) in (33) and integrating by parts in du;a and
ðCdhÞ;a yields
Z
r
r
ððna;a þ nO Þ du þ CT ðma;a þ z;a na þ mO Þ dhÞ dOr
r
O
Z
r
r
þ
ððna na nGt Þ du þ ðna la lGt Þ dhÞ dGrt
Z
Grt
r
Gru
Gru
r
ððna na nGu Þ du þ ðna la lGu Þ dhÞ dGru
ððu uÞ dn
Gru
Gru
þ ðh hÞ dl
Þ dGru
¼ 0,
r
n a n a n Gu ¼ o
8du,
(37a)
na la lGu ¼ o 8dh
(37b)
r
plus
r
u u ¼ o 8dnGu ,
(38a)
r
na;a þ nO ¼ o 8du,
r
ma;a þ z;a na þ mO ¼ o
(35b)
r
in the domain, O ,
r
n a n a n Gt ¼ o
8du,
4. Linearization of the weak form
For the solution of the weak form of the problem, stated
by (33), within a Newton/Raphson’s type of incremental/
iterative process it is crucial to explicitly know the exact
tangent operator. This can be achieved by the consistent
linearization of the weak form. Here this process must be
performed not only on the generalized displacements, d, as
usually is done, but also in the generalized reaction forces,
r
qGu . As usual, it is assumed in the following that Ddu;a ¼ o,
Ddh ¼ o and Ddh;a ¼ o.
The incremental/iterative perturbation, D, of the hybrid
weak form, given by (33), is
Z
DdW ¼
ððWDddÞ ðDWDDdÞ þ ðDddÞ ðGDDdÞ
Or
Z
r
r
dd ðLO DdÞÞ dOr dd ðLGt DdÞ dGrt
Z
r
Gru
dd DqGu dGru Z
Grt
r
Gru
dqGu Dd dGru .
ð39Þ
Here
(35a)
8dh
(38b)
on the kinematic boundary, Gru .
The sets of Eqs. (35) and (36) simply express the
equilibrium in the domain and on the static boundary
between the applied generalized forces and the internal
generalized forces. These are the usual set of equations
imposed in a weak sense in the traditional FEM (besides the
pointwise imposition of essential boundary values). Set (37) is
the equilibrium equations on the kinematic boundary. This
apparent contradiction is, in fact, what is being imposed: the
equilibrium between the internal generalized forces and the
independently approximated generalized reaction forces. Set
(38) is the compatibility at the kinematic boundary.
ð34Þ
where na denotes the outward normal components.
The Euler–Lagrange equations of (34) are
7
on the static boundary, Grt , and
h h ¼ o 8dlGu
Z
Combinations of variational statements were extensively used
for generating generalized principles for linear analysis. Here
the extension for nonlinear analysis of the hybrid-displacement model is accomplished. If the problem under analysis is
conservative, the variational form could be derived from a
constrained stationary potential energy principle.
Besides the usual requirements in order the integrals in
(33) make sense, no additional restrictions are demanded.
In particular, the usual conditions d ¼ d and dd ¼ o on the
kinematic boundary, Gru , are absent in order to be able to
use approximations not fulfilling the Kronecker-delta
property (23).
þ
(36b)
8dh
r
Z
dW ¼
Z
r
na la lGt ¼ o
499
(36a)
The convenience of the introduction of the minus sign is associated
with (i) the attainment of a symmetric linearized weak form and (ii) the
r
possibility of identifying qGu with the generalized reaction force.
2
qnr1
6 qgr1
6
6 qmr
6 1
6 r
r
6 qg1
qr
D¼ r ¼6
6 qnr2
qe
6
6 qgr
6 1
6 r
4 qm2
qgr1
qnr1
qjr1
qmr1
qjr1
qnr2
qjr1
qmr2
qjr1
qnr1
qgr2
qmr1
qgr2
qnr2
qgr2
qmr2
qgr2
3
qnr1
qjr2 7
7
qmr1 7
7
7
qjr2 7
7
qnr2 7
7
qjr2 7
7
7
qmr2 5
qjr2
(40)
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
500
is the generalized constitutive tangent stiffness,
3
2
0
O
O
O
O
G 1u y
7
6
0
6 O
O
O
O
G 1y y 7
7
6
7
6
u0 y
7
O
O
O
O
G
G ¼6
2
7
6
7
6
0
yy
7
6 O
O
O
O
G2
5
4
G yu
1
0
0
G yy
1
G yu
2
0
G yy
2
0
5. Multi-region decomposition
(41)
yy
G yy
1 þ G2
is the generalized geometric tangent stiffness,
2
3
O
O
r
tr
br
r
t
b 7
T
T
O
6
LO ¼ 4 O Vðh; m Þ þ C T A C þ C T A C 5
R
r
þCT H r B A dH r C
(42)
is the generalized tangent stiffness due the loading in the
domain and
"
#
O
O
r
R
r
r
LGt ¼
(43)
O Vðh; mGt Þ þ CT H r T A dH r C
is the generalized tangent stiffness due the loading in the
static boundary. The terms appearing in definition of (41)
are
0
0
T
¼ N a C,
G ua y ¼ G yu
a
0
(44a)
0
y yT
G yy
¼ Vðh; ma Þ and
a ¼ Ga
(44b)
yyT
¼ CT Z ;a N a C
G yy
a ¼ Ga
Vðh; z;a na ÞV ;a ðh; h;a ; ma Þ CT;a M a C,
ð44cÞ
where N a ¼ Skewðna Þ, M a ¼ Skewðma Þ. In the former
expressions the operators Vðh; tÞ and V ;a ðh; h;a ; tÞ are
defined by
Vðh; tÞ ¼ h2 ðyÞT þ h3 ðyÞðTH 2HTÞ
h4 ðyÞðHt hÞ þ h5 ðyÞðH2 t hÞ
and
ð45aÞ
V ;a ðh; h;a ; tÞ ¼ h3 ðyÞðTH;a 2H;a TÞ h4 ðyÞðH;a t h
þ Ht h;a Þ þ h5 ðyÞ½ðH;a H þ HH;a Þt h
þ H2 t h;a þ ðh h;a Þ½h4 ðyÞT þ h5 ðyÞðTH 2HTÞ
þ ðh h;a Þ½h6 ðyÞðHt hÞ þ h7 ðyÞðH2 t hÞ, ð45bÞ
being
h3 ðyÞ h2 ðyÞ 4h4 ðyÞ
and
y2
h4 ðyÞ 5h5 ðyÞ
h7 ðyÞ ¼
.
ð46Þ
y2
The linearization carried out only took into consideration
the possibility of configuration independent body forces,
r
r
b , and applied tractions, t . For other load types the
linearization terms (42) and (43) will, eventually, be
different.
Notice the two last terms on (39) do not depend on the
generalized displacements themselves, but only on their
virtual and incremental/iterative counterparts.
h6 ðyÞ ¼
Until now it was assumed that the plate was modeled
using a single region or, equivalently, the plate was
discretized in a single finite element of arbitrary shape.
This is the traditional way in which meshless approximations are used. Although the use of more than one region
departs from the original meshless spirit and creates
difficulties in reestablishing the continuity of the generalized displacements and tractions, the generalization to
multi-region can present advantages in certain particular
situations.
This concept was explored by Cordes and Moran [17]
and by Tiago and Leitão [18] to introduce discontinuities.
In the former these were caused by material heterogeneity
and the displacements continuity was imposed in weak
sense using a generalization of the modified principle [10]
for eliminating the interface tractions from the weak
statement. In the latter the discontinuity was due the
application of a punctual load and a full hybrid-displacement formulation was used to impose the continuity of the
generalized displacement fields.
In the present case the motivation to introduce
discontinuities is posed from the geometrical point of view:
(i) handle with complex plates or (ii) the existence of nonsmooth shells, like folded plates.
We will start by dealing with the case (i). Let the
subscripts ‘‘þ’’ and ‘‘’’ be associated with two distinct
regions with a common interface, Gri , in the undeformed
configuration. In this case the hybrid weak form includes
the same terms as (33) written each of the two regions and
additional interface terms. For simplicity, only these latter
are explicitly represented in the following. The weak form,
in this situation, renders
Z
r
dW ¼ qGi ðdd þ dd Þ dGri
Gr
i
Z
r
Gri
dqGi ðd þ d Þ dGri
Gr
r
r
8dd þ ; dqþu ; dd ; dqGu ; qGi ,
ð47Þ
r
where qGi collects the independently approximated interface generalized tractions.
The linearization of (47) yields
Z
Z
r
r
dd þ DqGi dGri þ
dd DqGi dGri
DdW ¼ Gr
i
Z
Gr
i
r
dqGi
Dd þ dGri þ
Gr
i
Z
r
Gr
i
dqGi Dd dGri .
ð48Þ
The Euler–Lagrange equations of (47) are (35)–(38) for the
regions þ and plus additional interface boundary terms,
namely, the equilibrium between the generalized interface
traction and the stresses on region Orþ ,
r
naþ naþ nGi ¼ o
8duþ ,
(49a)
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
501
Fig. 2. C and I rods as assembly of flat panels.
r
naþ laþ lGi ¼ o
8dhþ ,
(49b)
the equilibrium between the generalized interface traction
and the stresses on region Or ,
r
na na þ nGi ¼ o
r
na la þ lGi ¼ o
8du ,
hþ h ¼ o
(50b)
8dh
and the compatibility of displacements between
uþ u ¼ o
(50a)
r
8dnGi ,
Gri
8dl .
Orþ
and Or
(51a)
(51b)
Eliminating the independent field of generalized boundary
tractions from the sets of equilibrium equations (49) and
(50), we conclude that
naþ naþ þ na na ¼ o,
(52a)
naþ laþ þ na la ¼ o,
(52b)
hence confirming that the Euler–Lagrange equations
indeed include the enforcement of the continuity of the
generalized stresses between both regions.
The obvious similarities between the set of Eqs. (52) and
their elasticity analogs, see [19, Eq. (8), p. 847], disguises
the specific needs that a general structural theory, like the
one developed here, demands. Contrary to the 3D fplaneg
elasticity cases, where only two bodies can flow into a
common interface—formed by a surface fcurveg—in the
present case it is natural that more than two panels share a
mutual straight line, e.g., rods with I, C and T crosssections. This fact motivates the discussion of case (ii).
The previously presented method in this section for
dealing with a single interface between two regions can be
generalized to the imposition of the continuity conditions
of the displacements and rotational parameters to those
cases where n regions converge into a single line, being
nX3. For this purpose it is necessary to enforce a set of
ðn 1Þ weak form continuity constraints between the n
regions. The remaining constraint is linearly dependent
from the mentioned set and, consequently, is excluded. It
follows that ðn 1Þ independently generalized interface
tractions fields will be generated at that interface. For
generating the set of constraints there will be C nn1 ¼ n
possible combinations. The weak form and the respective
governing equations are not explicitly given as they are an
ðn 1Þ times repetition of the two regions case. This
situation is illustrated in Fig. 2 for rods with C and I crosssections idealized as a assemble of flat panels. The former
requires the use of three regions and two interface
boundaries, while the latter resources five regions with
four interface boundaries.
6. The meshless discretization
6.1. The MLS approximation
The approximation of the six generalized displacements
fields, d, over the (plane) reference configuration, Or , is
made in the present work through MLS nodal functions.
This approximation was developed by Lancaster and
Šalkauskas [20] and is briefly outlined in the following.
Consider a continuous function, uðxÞ : O ! R and let
the known values of u at a set of points fxi gN
i¼1 ; xi 2 O, be
denoted by ui where N is the total number of points.
At each point within the domain a local approximation
of uðxÞ, here denoted by uh ðxÞ, is defined as
uh ðxÞ ¼ pT ðxÞaðxÞ,
(53)
where
pðxÞ ¼ ½ p1 ðxÞ p2 ðxÞ . . .
pm ðxÞ T
(54)
gathers a basis of m functions and
aðxÞ ¼ ½ a1 ðxÞ a2 ðxÞ . . .
am ðxÞ T
(55)
collects the weights of the approximation.
Let wi ðx xi Þ 2 C l0 ðoi Þ be a compactly supported weight
function with continuity given by l on a neighborhood oi
of xi . Due to its properties this is usually called a bellshaped function. The compact supports, also known as
clouds, form a opening covering of the domain, O, i.e.,
O
N
[
oi .
(56)
i¼1
Let the number of non-zero weight functions at a given
point, x, be denoted by nðxÞ, being mpnðxÞpN.
At each point x the determination of the vector aðxÞ is
done by the minimization of a weighted discrete least
squares norm of the error, L2 , [20]. The obtained
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
502
approximation can be written in the usual FEM shape
function format, i.e.,
uh ðxÞ ¼ Uu,
(57)
where
U ¼ ½ f1
f2
...
fN and
(58a)
uT ¼ ½ u1
u2
...
uN (58b)
collect the approximation functions and the prescribed
values, respectively. The former are evaluated through
fi ¼ pT ðxÞA1 ðxÞBi ðxÞ,
(59)
being
AðxÞ ¼
nðxÞ
X
wi ðx xi Þpðxi Þpðxi ÞT
and
(60a)
i¼1
Bi ðxÞ ¼ wi ðx xi Þpðxi Þ.
(60b)
The evaluation of the MLS approximation can be
implemented in a very efficient way by avoiding the
inversion of the moment matrix, A, at each sample point,
as described in [21]. The same reasoning can be applied to
the derivatives.
6.2. The discretization
We start this section by justifying the options made
regarding the specific choice of the approximation functions either for the domain and the kinematic/interface
boundary. The geometrically exact theories are especially
interesting in the analysis of slender structures, where the
change of the structural response due the variation of the
configuration is important. As the shear deformation was
taken into account, the shear-locking presence can be
anticipated. In meshless methods, particularly for the ones
relying on the use of MLS approximation, there is no, in
general, such concept as reduced integration, as the closed
form solutions for the integrals appearing in the generalized residual vector and the generalized tangent stiffness
matrix are unknown (even for linear problems).
There are several ways to overcome the locking effects in
meshless methods. The most representative [22] are the
(i) increase of the degree of basis functions, (ii) resource to
the consistency paradigm, (iii) use of nodal integration,
(iv) adoption of a change of variables and (v) employ of a
mixed formulation.
The increase in the degree of the polynomial basis (54)—
as suggested by Garcia et al. [23] in the h-p cloud context—
is, in practice, the counterpart of a p-refinement in the
FEM context for alleviating the shear-locking. In general,
the use of basis with degree higher than three is sufficient to
eliminate the shear-locking effects, even for extremely high
values of the slenderness ratio.
Resorting the facilities of the meshless approximations to
generate arbitrarily continuous functions, it is very easy to
chose approximations in such a way that the Kirchh-
off limit is exactly reproduced, the so-called consistent
approximation [24]. Here the approximations of the
rotation fields are established by the derivatives of the
displacement field approximation. However, it was recently
proved [22] that this procedure necessarily leads to a
singular equation system, due to linear dependencies
presence in the approximation functions for the rotations.
Moreover, except for the one-dimensional (1D) case,8 the
number of dependencies grows with the order of the
basis (in the common case of polynomial basis are used).
Nevertheless, if appropriate solvers are employed this
problem can be easily overcome.
The nodal integration procedure are usually associated
with spurious singular modes [25] and requires the addition
of ad hoc stabilization schemes.
The change of variables [26] has to be eliminated a priori
in this work, as the rotational parameters would have to be
replaced by the shear strains of the middle plane as primary
variables. However, on the one hand, the relations involved
in the change of variables is nonlinear and, on the other
hand, the rotational parameters play the principal part in
the geometrically exact act.
The mixed formulation were already used for eliminating
volumetric locking [27] in meshless methods, but the
extension to shear and membrane locking, even for the
linear case, is still an open question.
Hence, the former option was the chosen one in the
present work due to (i) its generality—the extension for
geometrically nonlinear analysis is straightforward—and
(ii) facility of implementation.
For the approximation of the generalizedr kinematic and
r
interface boundary tractions, qGu and qGi , respectively,
several options are available, e.g., (i) 1D Lagrange
polynomials, (ii) 1D MLS or even (iii) the restriction to
the boundary of the same MLS approximation functions
used in the domain. Numerical tests show that (i) using
the first option the chosen degree of the polynomials has
little influence in the results for the same number of nodes,
(ii) the 1D MLS nodal functions are computationally more
expensive (relatively to the Lagrange polynomials) and do
not significantly improve the results [28] and (iii) the use of
two-dimensional (2D) MLS locally enforces the kinematic
boundary condition (apart from a limitable integration
error), which in turn, may introduce static dependencies in
the global system [29]. In the present work the first
option—the linear Lagrange polynomials—is chosen.
Hence, consider the following approximations:
d ¼ Ud;
dd ¼ Udd;
Dd ¼ U Dd;
8
r
r
qGu ¼ WqGu ;
r
r
r
r
(61a)
r
dqGi ¼ YdqGi ,
dqGu ¼ WdqGu ;
r
r
qGi ¼ YqGi ,
r
DqGu ¼ WDqGu ;
r
(61b)
r
DqGi ¼ YDqGi ,
(61c)
In 1D approximation only one dependency, per field, is introduced.
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
for the real, virtual and incremental/iterative generalized
displacements, generalized boundary tractions9 and generalized interface tractions, respectively. In the previous
expansions we deliberately omitted the identification of the
considered region though the subscripts þ or . The
explicit form of the operators collecting the approximations is
" u
#
O
funOr I
f1 I O
U¼
...
,
(62a)
O fy1 I
O
fynOr I
2
W¼4
cu1 I
O
O
cy1 I
2
ju1 I
Y¼4
O
r
...
O
jy1 I
...
cunGru I
O
O
cynGru I
juGr I
O
O
jyGr I
n i
n
i
3
5 and
(62b)
3
5
(62c)
r
r
nO , nGu and nGi being the number of approximation
functions of the generalized displacements, boundary and
interface tractions. Notice the possibility of using different
functions for displacements and parameters of the Euler–Rodrigues formula.
Substituting the approximations (61), in the hybrid weak
form (47), yields, after some algebraic manipulations,
(63)
rðxÞ ¼ 0 8dx,
where rðxÞ is a generalized vector of residuals and x collects
the unknowns, i.e.,
3
2
Gr Gr
3
2
Gr Gr
sþ þ Bþu qþu þ Bþi qþi
Ddþ
7
6
r
r 7
r
r
7
6
6
6 s þ BGu qGu þ BGi qGi 7
6 Dd 7
7
7
6
6
r
7
6
6 Gu 7
Gr T
r¼6
Bþu dþ vþ
7 and Dx ¼ 6 Dqþ 7.
7
6
6 Gr 7
7
6
6 Dqu 7
Gru T
B
d v 7
5
6
4
5
4
Gri
r
r
Gi T
Dq
Bþ dþ BGi T d
(64)
Here
Z
Z
r
s¼
ðDUÞT WT rr dOr UT qO dOr
Or
Or
Z
r
UT qGt dGrt ,
ð65aÞ
Grt
r
BGu
Gri
B
Z
¼
Gru
UT W dGru ,
Z
¼
Gr
U
T
Y dGri
(65b)
and
Z
v¼
Gru
503
WT d dGru .
(65d)
The use of the same approximations in the linearized form
(48) renders
K Dx
(66)
8dx,
where
2
6
6
6 O
6 r
6 GT
K ¼ 6 Bþu
6
6 O
6
4 Gr T
Bþi
Gr
O
Bþu
O
S
O
BGu
O
O
O
Gru T
B
O
O
Gri T
B
O
O
Sþ
r
Gr
Bþi
3
7
r 7
BGi 7
7
7
O 7,
7
O 7
7
5
O
(67)
S being the generalized stiffness matrix
Z
S¼
ððDUÞT WT DWðDUÞ
Or
r
þ ðDUÞT GðDUÞ UT LO UÞ dOr
Z
r
UT LGt U dGrt .
ð68Þ
Grt
By the inspection of the generalized vector of residuals,
rðxÞ, Eq. (64)1 and the generalized tangent form, K,
Eq. (67), readily it is perceived that only part of these
operators are function of the
generalized displacements. In
r
r
fact, the operators BGu , BGi and v are constant along the
analysis and all the nonlinearity is concentrated on s and S.
7. Implementation remarks
7.1. Constitutive model
The constitutive model is derived from a pure 3D large
strain formulation law, more specifically, the neo-Hookean
material [30]. The respective strain energy function is
cðJ; I 1 Þ ¼ 12 lð12 ðJ 2 1Þ ln JÞ þ 12 mðI 1 3 2 ln JÞ,
(69)
where l and m are the Lamé constants, J is the Jacobian
of the deformation gradient and I 1 is the trace of
the Cauchy–Green tensor. A detailed derivation of the
constitutive operator can be found elsewhere [7]. Here only
the final results are summarized. Let us start by defining the
following quantities
f ra ¼ era þ cra ,
(70a)
J ¼ er3 f r1 f r2 ,
(70b)
gra ¼ ab f rb er3 ,
(70c)
(65c)
i
ab ¼ er3 era erb
9
Notice the subtle difference between the matrix differential operator W
defined in Eq. (18a) and the matrix W that collects the independent
approximation functions of the generalized kinematic boundary tractions.
jðJÞ ¼ m
and
l þ 2m
3
lJ þ 2mJ
.
(70d)
(70e)
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
504
f ra are columns of the deformation gradient, J is deformation gradient Jacobian at middle surface level and ab is a
permutation symbol. With the previous definitions the rfirst
e ¼
Piola–Kirchhoff stresses and the tangent tensors C
ab
r
r
qesa =qcb are given by
esra
¼
jðJÞgra
þ
mf ra
and
(71a)
e r ¼ qjðJÞ gr gr jðJÞab Skewðer Þ þ mdab I.
(71b)
C
b
3
ab
qJ a
Thus, the constitutive tensors gathered by the D components in (40) are given by
Z
qnra
e r dH r ,
¼
(72a)
C
ab
r
qgb
Hr
qnra
¼
qjrb
qmra
¼
qgrb
rotation tensor, Qo , and the back-rotated curvature vector,
jor
a , are evaluated and collected. If an update in the
configuration is made, Qe , the current configuration
rotation tensor can be expressed by
Q ¼ Qe Qo .
The current curvature tensors, K a ¼ Skewðja Þ, are given by
K a ¼ Q;a QT ¼ K ea þ Qe K oa QeT ,
(74)
where
K ea ¼ Qe;a QeT
and
K oa ¼ Qo;a QoT .
(75)
Consequently, the current curvature vectors are
ja ¼ Ce he;a þ Qe joa .
(76)
The back-rotated counterpart of (76) is
Z
Hr
e r Ar dH r ,
C
ab
(72b)
jra ¼ QoT CeT he;a þ jor ,
(77)
being
Z
e r dH r
Ar C
ab
Hr
qmra
¼
qjrb
Z
Hr
and
e r Ar dH r .
Ar C
ab
(72c)
(72d)
The through the thickness integrations in (72) are evaluated
numerically using three Gauss–Legendre sampling points.
Contrary to other constitutive models [31], no drill
values are assigned to the generalized tangent stiffness or
residual. To prevent the possibility of obtaining a global
singular matrix, a value of Ehr3 , hr ¼ hrb þ hrt being the
plate thickness, is added to the drill stiffnesses. Likewise, a
value of mra er3 ¼ Ehr3 jra er3 is added to the residual
vector.
7.2. Rotation update
The formulation presented in Section 2 is intrinsically
general, because it can, in principle, handle arbitrary large
displacements and rotations. However, due to the adopted
type of parameterization for the rotation tensor—a
vectorial parameterization—the rotation angle, y, is
restrained to belong to a limited interval. For the
Euler–Rodrigues formula, this interval is ½0; 2p½.10
In the following an update scheme is presented which
allows to overcome this limitation. The rotation angle
increment, within a single load step, does not approach the
singularity limit due to the bound imposed by the basins of
attraction of Newton’s method. Hence, an update rotation
scheme can be devised if the total rotation tensor of a given
interest point—typically a integration point—is conveniently stored. The same applies to the curvature vectors.
Consider a given configuration, which is the solution of
the discretized problem of the last point found during the
loading path of a structure. At each point the current
jor ¼ QoT jo .
In fact, numerical tests indicate that the presence of the singularity
deteriorates the solution in a relatively large interval.
(78)
The physical meaning of the update formula (77) for the
back-rotated curvature vectors is clear. These vectors are
given by the sum of the back-rotated curvature vectors of
the computed configuration and the back-rotated curvature vectors of the update.
The basic idea, as latter was found, is already present
in the original work of [32], under the designation of
‘‘formulation in T Rref SO ð3Þ’’ or updated Lagrangian
formulation. The differences between the result in (77)
and [32] are due to (i) the update formula for the rotation
tensor11 and (ii) the definition of the tensor C.
This update procedure can be performed after each
converged configuration or only when, at given point, the
rotation angle ye ¼ khe k approaches 2p.
The implementation of this updated Lagrangian formulation removes the singularity problem, but, unfortunately, originates a path-dependent formulation. A
formulation is called path-independent if the final solution
is solely a function of the final configuration and hence
independent of the path followed by the incremental/
iterative procedure. In the present work as long as the total
rotation is directly approximated the solution is, indeed,
path-independent. However, if some update of the configuration is necessary, then the path-dependence is introduced. We emphasize that this imperfection is introduced
by the specific parameterization chosen for the rotation
tensor and is by no means related with the geometrically
exact theory.
There exists more efficient ways to store of the current
rotation tensor, Qo , than just collect its nine components.
Simple alternatives are the extraction of the rotation vector
from the rotation tensor or the resource to quaternions.
The former was used in the present work.
Instead of Q ¼ Qe Qo used in the present work, [32] employed
Q ¼ Qo Qe . In their terminology this updates are denominated via right
translation (material rotation) and via left translation (spatial rotation).
11
10
(73)
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
7.3. On the stability of the equilibrium
A reliable criteria to establish the stability of a given
equilibrium configuration is essential to identify bifurcation and limit points and thus, understand the structural
r
behavior. Assume that (i) body forces, b , and applied
r
tractions, t , are configuration independent and (ii) that a
hyperelastic material law was adopted. If the traditional
version of the FEM was used a static stability criteria based
on the second variation of the potential energy would be
sufficient. In the present context there also must exists a
potential function from where the weak form (32), either
specified by (33) or (47), can be derived. Contrary to the
traditional version of the FEM, this potential is not
suitable for classifying the stability of the equilibrium.
Indeed, the extremum associated with the undeformed
configuration (which is assumed to be stable) is already a
saddle point.
The classification of the local constrained extremum is a
relatively new subject. To the authors knowledge, the first
approach to the problem was presented by Mann [33] but a
general test, based on the determinant of the principal
minors of the bordered Hessian matrix, was only to be
published by Spring [34]. This matrix is, in the present
context, the linearized form of the hybrid functional, i.e.,
the full generalized tangent stiffness. This test is of less
practical importance for the present purposes due the
limiting conditions under which the classification criteria
holds [34, Theorem 1]. Also the conclusions drawn with
this test may depend on permutations of rows and columns
of the bordered Hessian (which may be interpreted as
relabeling the particles and their degrees of freedom’s
(dof’s)). A much more useful test, based on the eigenvalues
of the bordered Hessian, was latter derived by Hassell and
Rees [35]. An alternative treatment of the subject is
provided by Shutler [36] and a generalization for certain
infinite dimensional calculus of variational problems is
given by Greenberg et al. [37].
Let Iðf ðaÞÞ be the number of independent directions in
which function f decreases at a certain point, a. For
practical purposes, if this number is zero, then a is a
minimum of f. Also let P and PH be the potential energy
function and the constrained (hybrid) potential energy
function, respectively, and nc the number of independent
constraints associated with the imposition of the essential
and continuity boundary conditions. Restated for the
present context, the criterium derived by Hassell and Rees
[35] establishes that
IðPÞ ¼ IðPH Þ nc .
(79)
Based on this simple expression, it is possible to generalize
the standard stability criterium based on the second
variation of the potential energy to the present hybriddisplacement approach. If the structure is stable in the
undeformed configuration, the number of negative eigenvalues associated with IðPH Þ in the beginning of the
analysis is always equal to nc and, consequently, IðPÞ
505
equals zero. After each bifurcation point is crossed IðPH Þ
increases one unit and so does IðPÞ.
In the tested applications it was found that the number
of negative eigenvalues may be erroneously identified due
to the very different absolute values of matrix K. While
matrices S depend
on the material parameters, matrices
r
r
BGu and BGi are only an integral of approximation
functions products. An efficient way to handle this
numerical problem—which was used in the present
work—was to resort to a change of variables, in which
r
the generalized kinematic boundary tractions,r qGu , and the
generalized interface boundary tractions, qGi , are divided
r
r
by a factor, $, and, consequently, matrices BGu and BGi are
multiplied by this same factor. In this work we use
$ ¼ Ehr3 .
The computation of critical loads can be efficiently
implemented using a simple bisection method. The convergence is extremely fast as the number of iterations, within
each incremental step, is very low. In general, to locate a
generalized equilibrium point belonging to the fundamental
trajectory one or two iterations are enough.
7.4. A nonlinear constraint
If the solution of the resulting nonlinear system of
equations (63) cannot be achieved by the use of the
standard incremental/iterative approach of the Newton/
Raphson method, then a combination with some (nonphysical) constraint is used in order to trace the full
trajectory of the structural model. To be consistent with the
approximations made, this constraint should include not
only generalized displacements and loads, but also should
render the generalized boundary tractions on the kinematic
and interface boundaries.
From now on we assume the applied loads and imposed
displacements vary linearly with the load parameter, l.
Therefore, the linearization of the generalized weak form
(33) in l is trivial and will be omitted. Hence, (63) is
replaced by
rðx; lÞ ¼ 0.
(80)
The chosen arc-length constraint, that nonlinearly relates
the incremental generalized displacements, boundary tractions and load parameter, is
2
Ds2 Ds ¼ 0,
(81)
where
Ds2 ¼ DxT Wx Dx þ a2 Dl2 .
(82)
Ds is a preassigned value of the arc-length within a
incremental step, a is a scaling parameter and Wx is a
weighting matrix which is, at least, positive semi-definite
diagonal
Wx ¼ Wx ¼ diag½Wu ; Wy ; Wn ; Wm .
(83)
ARTICLE IN PRESS
506
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
Moreover, Dx and Dl are the incremental variations
Dx ¼ x xE ,
b
Geometrical data:
h
(84a)
b = 0.1
h = 0.1
l = 1.0
A
(84b)
where E denotes a certain point in the load-generalized
configuration space. The usual option is to identify point E as
the last previously known point belonging to the generalized
equilibrium path. In this way Eq. (81) renders a constraint on
the distance that the next sought point should be from E. The
metric of this distance on the load-configuration space is given
by Wx and a. It can be very useful to consider non-centered
constraints, i.e., identify E with a point outside the generalized
equilibrium path, see [38]. In particular, if we consider that
point E is halfway between the last previously known point
belonging to the generalized equilibrium path and the point
generated by the predictor step and use, in the corrector steps,
Ds=2 instead of Ds, the solutions of the quadratic equations
are either the correct solution or coincides with the last
previously known point belonging to the generalized equilibrium path. In the latest, the arc-length distance, Ds, should
be reduced (e.g., by half) and the step should be repeated,
hence avoiding track-back of the solution.
The presented constraint can be used either with the
Crisfield [39] version of the arc-length, where the iterative
solutions are constrained to remain in the interior of the
hyper-ellipsis, or with the consistent linearized version of
Schweizerhof and Wriggers [40]. Hence, the arc-length
method was generalized in order to include the essential
and interface boundary reactions, resulting in a robust and
fast procedure.
8. Numerical examples
8.1. Introductory remarks
The generalized kinematic and boundary tractions approximations, (61)2;3 , are always interpolated using linear
Lagrange polynomials. In general, we used a number of
nodes similar to the number of particles used for the domain
discretization in the vicinity of the side. This is, in principle,
the most reasonable choice. In this way the generalized
boundary traction parameters are not only weights of an
expansion but also can be identified with the nodal values
of the generalized boundary tractions, hence justifying the
denomination interpolation.
The Gauss–Legendre rule is always adopted. The
integration in the thickness direction was performed using
three integration points. Within each integration cell 3 3
sample points were used. Each boundary cell resorts to a
five point integration rule.
In the present work we use the weight function described
in [41] with the parameter value s ¼ 3 for the MLS
approximation setup. As only polynomial basis are used
(which are C 1 ) the final approximation always possesses
x2
l
Material properties:
x3
x1
E = 1.0.107
v = 0.3
B
Fig. 3. Data for plane cantilever beam subjected to an end load.
ξ2
Dl ¼ l lE ,
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
ξ1
Fig. 4. The particle distribution used for the approximations on the
reference configuration, Or , of plane cantilever beam subjected to end
force.
C 2 continuity. The supports are always circular. Let the
support dimension parameter of the node i, Ri , denotes
the ratio between the actual support radius of node i and
the minimum support radius that node i should have in
order that the moment matrix, M, can be inverted, see [11].
Here we always used Ri ¼ 1:5 for the all N particles set.
8.2. Cantilever beam
Consider the cantilever beam subjected to an end load
with resultant equal l represented in Fig. 3. This classical
problem has been studied by several researchers.
Here we test the hybrid meshless formulation for both
plane elasticity and plate bending behavior. The former
model is obtained by considering a plane structure
contained in the ðx1 ; x2 Þ plane acted by forces in that same
plane while the latter is achieved by considering a plate
contained in the ðx1 ; x3 Þ plane acted by forces perpendicular to that same plane. The same parameters are used for
both analysis.
A pseudo-random distribution including 63 nodes was
used in order to generate an admissible particle distribution. The distribution on the reference configuration, Or , is
displayed in Fig. 4. Notice the existence of particles located
outside the domain. Although this is not an usual
practice,12 the particles can be located anywhere as long
as the supports are sufficiently large to generate an
admissible particle distribution.
For the trace of the response we use, for the MLS
approximation on the domain, a complete quadratic basis.
The domain integrations were carried out using a
uniform cell structure of 20 2 integration cells.
12
To the authors’ knowledge, this is the first time when, in a MLS
approximation, the points are located outside the domain.
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
0
x2
x3
1000
x1
1
3
6
10
x2
x3
x1
0
1
3
10
6
Fig. 6. Deformed shapes for plane cantilever beam subjected to end force
at selected steps. (a) Plane elasticity model. (b) Plate model.
ξ2
The kinematical boundary integrations extend over the
side characterized by x1 ¼ 0 of the reference configuration.
These integrations were carried out using two integration
cells.
The solution was generated using the Newton/Raphson
method with constant load increments equal to 100.
The obtained response for the displacements u1 and u2
and the rotation y3 in the center of the cross-section B, see
Fig. 3, are presented in Fig. 5. The results are compared
with those obtained by numerical evaluation of elliptic
integral solutions of a large deflection beam model, see [42].
Notice that no rotation y3 is obtained when the beam is
idealized as a plane elasticity model. This fact is a direct
consequence of the constitutive model used for the drill
rotational dof.
The beam model is based on the Euler–Bernoulli
formulation, hence disregarding the shear deformation.
The plane elasticity model does not, in practice, use any
rotational parameters. The plane elasticity and the plate
model resources a nonlinear large strain constitutive
model, while the beam model uses a linear small strain
model. Although there exists these huge differences in the
models, the results presented in Fig. 3 are extremely close
to each other.
The deformed shapes at selected steps are shown in
Fig. 6.
Consider now the replacement of the end load by a
moment (parallel to the x3 direction). In this case, the
analytical solution for the equivalent beam model, assuming a linear elastic (small strain) model, is known, see, e.g.,
[43]. To be consistent, Poisson’s ratio is equalized to zero
and the neo–Hookean material its replaced by its first order
approximation in cra [7]. The applied moment possess a
507
0.1
0.05
0
900
0
0.2
0.4
0.6
0.8
1
ξ1
800
700
Fig. 7. The particle distribution used for the approximations on the
reference configuration, Or , of plane cantilever beam subjected to end
moment.
600
500
400
300
200
100
0
0
0.25
u1,
u1,
u1,
u2,
0.5
0.75
u1, u2,
EFG (elastostatics)
EFG (plate)
[42] (beam)
EFG (elastostatics)
1
1.25
1.5
3
u 2 , EFG (plate)
u 2 , [42] (beam )
1 , EFG (plate)
1 , [42] (beam )
Fig. 5. EFG response for plane cantilever beam subjected to end force at
the center of cross-section B, see Fig. 3.
constant direction, which is always orthogonal to the plane
where motion occurs. Hence, there is no distinction
r
between the applied moment, mGt , and an applied
r
pseudo-moment, lGt , the contribution for the generalized
stiffness matrix, in both cases, being null. The latter applied
moment was used in this numerical test.
The applied moment per unit length was chosen
in such way that the beam rolls up into two complete
circles, i.e.,
r
lGt ¼ 4p
EJ
,
lb
where J ¼ bh3 =12.
(85)
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
508
In this case the final deformed shape demands a more
refined discretization in order to perfectly capture the
motion. Hence the mesh with 124 particles displayed in
Fig. 7 is used. As only plane motion takes place, the total
rotation formulation could, in principle, supply a singularity-free parameterization. However, this is not entirely
true, as it was numerically verified. The motion is only
approximately plane and, after the 2p rotation is computed, the convergence properties progressively deteriorate: the number of steps to attain a converged solution
increases and the load increment has to be decreased. Even
so, sometimes the all load-path could be traced out. On the
contrary, the updated rotation formulation preserves the
asymptotic quadratic convergence rate and always converges to the desired solution. The convergence of the
(normalized) Euclidian norm of the residual is illustrated in
Table 1 for steps 1, 10, 20. The solution presented in Fig. 8
was obtained with a complete quartic polynomial basis.
The deformed shapes at selected steps are shown in Fig. 9.
The determination of the first lateral buckling load of
model represented in Fig. 3 is now performed. For
buckling to take place, the beam characteristics have to
be changed. Hence, consider b ¼ 0:01164, h ¼ 0:1, l ¼ 2:4,
E ¼ 210:0 106 and n ¼ 0:3125. The applied load remains
unchanged.
Using the criteria presented in Section 7.3 the values for
the lateral buckling critical load, presented in Table 2, were
obtained. The analytic solution for the equivalent beam
model, given by [44], is also reproduced. Four random
particle distributions were used together with three different
MLS nodal functions, obtained with the complete quadratic,
cubic and quartic polynomial basis. Notice the present
solutions are not strictly compatible and, accordingly, the
convergence of the critical load does not have to be
monotonic. Even so, the convergence is, for this example,
monotonic. The reference beam solution should not be taken
as the exact solution, and merely as an indicative value.
1000
900
800
700
600
500
u1 ,EFG
400
u2 , EFG
3
300
200
100
3
0
0.5
1
0
1
2
3
4
5
1.5
2
u1 , u2,
Fig. 8. EFG response for plane cantilever beam subjected to end moment
at the center of cross-section B, see Fig. 3.
4
6
3
8
2
10
20
1
x3
x2
x1
0
3
2
4
This is one of the most popular benchmark test for
geometrically nonlinear analysis of frame and shell
Iteration number
, Exact
0
8.3. Lateral post-buckling of a cantilever right-angle frame
Table 1
Convergence of the normalized Euclidian norm of the residual for plane
cantilever beam subjected to end moment.
, EFG
u 1 , Exact
u 2 , Exact
1
0
6
Step number
1
10
20
3:266 10þ02
6:304 10þ00
1:478 10þ00
2:063 1002
5:639 1004
6:275 1009
3:353 10þ02
6:431 10þ00
1:499 10þ00
2:119 1002
5:809 1004
6:491 1009
3:613 10þ02
6:729 10þ00
1:551 10þ00
2:231 1002
5:849 1004
6:011 1009
x3
x2
8
10
x1
20
Fig. 9. Deformed shapes for plane cantilever beam subjected to end
moment at selected steps. (a) First view. (b) Second view.
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
7
Table 2
Critical loads for cantilever plate
m
m= 3
m= 4
6
Number of nodes
2
3
4
Beam model [44]
509
5
50
147
292
485
3.952675
2.651493
2.625371
2.517747
2.443864
2.463233
2.396334
2.443391
2.394138
2.373608
2.407034
2.387951
2.383334
4
3
2
1
0
A
0
10
20
Cross section:
b
B
40
50
60
Fig. 12. EFG solution for point C obtained with a total of 2 165
particles and m ¼ 3 and 4 for L cantilever beam subjected to end load.
h
l
30
uC
2
Geometrical data:
b = 0.6
h = 30.0
l = 240.0
x3
l
x1
x2
Material properties:
E = 71240.0
v = 0.31
C
Pdist
Fig. 10. Data for cantilever right-angle frame.
50
40
30
Fig. 13. Deformed shapes for L cantilever beam subjected to end load at
four selected steps.
20
0
10
500
region 1 particles
region 2 particles
0
0
10
20
30
40
50
Fig. 11. Particle distribution for cantilever right-angle frame.
structures. The data of the problem is depicted in Fig. 10.
The cross-section A is clamped and Pdist is a small
disturbance load that enables the branch switching from
the fundamental trajectory to the secondary branch.
Two regions with 2 165 particles were used, connected
by the appropriate continuity conditions. The particle
distribution is represented in Fig. 11.
The disturbance load, Pdist , is set to 1 106 and kept
constant throughout the analysis. The results for the displacements at the center of the cross-section at point C are displayed
in Fig. 12, where, for the MLS approximation, complete cubic
1000
1500
2000
0
500
1000
1500
2000
Fig. 14. Structure of the system of equations for the two region
discretization of L cantilever beam subjected to end load.
and quartic basis were used. Deformed shapes at steps 0, 32, 52
and 82 are shown in Fig. 13 for the quartic basis case. The
whole structure and the displacements are presented in true
ARTICLE IN PRESS
510
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
scale. The large slenderness of the rectangular cross-section is
responsible for the dominant out-of-plane displacements.
Although the connection between the two legs is imposed in
a weak sense, the obtained displacements along the interface
boundary are nearly the same.
A plot of the structure’s system of equations can be
found in Fig. 14. In this case, the MLS approximation is
based on a complete quadratic base. From this
figure is evident the reduced increase in the number of
unknowns due the independent boundary/continuity tractions. The density of the matrix13 in this case is equal to
0:1103.
9. Conclusions
A meshless method for the nonlinear analysis of plates
was presented. A geometrically exact approach was
incorporated in a hybrid-displacement functional, so the
essential boundary and interface conditions are imposed
via Lagrange multipliers. The consistent linearization of
the weak form rendered a symmetric tangent bilinear form
which, for hyperelastic materials and conservative loadings, is always symmetric, even for configurations far from
the generalized equilibrium trajectory. The MLS nodal
functions were used for the discretization of the generalized
displacements on the domain, rendering a truly meshfree
method. Several novel aspects, which are necessary for the
present meshfree analysis, were then discussed: (i) the
generalization of the hybrid-displacement formulation to a
multi-region decomposition in order to handle complex
geometries, together with the respective recovering of the
governing equations, (ii) a criteria for the classification of
the equilibrium’s stability, based on the Bordered–Hessian
matrix eigenvalues, is established and (iii) a nonlinear
constraint, necessary for the extension of the arc-length
continuation method, which relates generalized internal
displacement fields, the generalized (kinematic and interface) boundary tractions and the load parameter, is
proposed. The limitation of the rotations parameterization
was circumvented by the use of an intermediate configuration between the reference and the current ones, hence
rendering an updated Lagrangian approach. Several
numerical examples are presented, illustrating the effectiveness of the method.
Acknowledgments
We wish to thank professor Eduardo M.B. Campello for
many helpful discussions. This work was carried out in the
framework of the research activities of ICIST, Instituto de
Engenharia de Estruturas, Território e Construc- ão, and was
partially funded by Fundac- ão para a Ciência e Tecnologia
through Research Grant POCTI/ECM/33066/99.
13
The density of a matrix is here defined as the number of non-zero
elements divided by the total number of the matrix elements.
The first author would like to express his gratitude to
Fundac- ão Calouste Gulbenkian for supporting his staying
in São Paulo, Brasil (Grant 63179).
The second author acknowledges the support by CNPq
(Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico) and of DFG (Deutsche Forschungsgemeinschaft)
during this research work, partly done at the IBNM of the
Leibniz University of Hannover.
References
[1] Simo JC, Fox DD. On a stress resultant geometrically exact shell
model. Part I: formulation and optimal parametrization. Comput
Methods Appl Mech Eng 1989;72(3):267–304.
[2] Pimenta PM. On the geometrically-exact finite-strain rod model. In:
Proceeding of the third Pan-American congress on applied mechanics, PACAM III. Brazil: Escola Politécnica da Universidade de
São Paulo; January 4–8, 1993. p. 279–82.
[3] Pimenta PM. On the geometrically-exact finite-strain shell model. In:
Proceeding of the third Pan-American congress on applied mechanics, PACAM III. Brazil: Escola Politécnica da Universidade de
São Paulo, January 4–8, 1993. p. 616–19.
[4] Pimenta PM, Yojo T. Geometrically exact analysis of spatial frames.
Appl Mech Rev 1993;46(11):118–28.
[5] Pimenta PM. Geometrically exact analysis of initially curved rods, In:
Topping BHV, editor. Advances in computational techniques for
structural engineering; 1996. p. 99–108.
[6] Pimenta, PM, Campello, EMB. Geometrically nonlinear analysis of
thin-walled space frames, In: Waszczyszyn Z, Stein E, editors. Second
European conference on computational mechanics. Cracow, Poland;
2001.
[7] Campello EMB, Pimenta PM, Wriggers P. A triangular finite shell
element based on a fully nonlinear shell formulation. Comput Mech
2003;31(6):505–18.
[8] Pimenta PM, Campello EMB, Wriggers P. A fully nonlinear multiparameter shell model with thickness variation and a triangular shell
finite element. Comput Mech 2004;34(3):181–93.
[9] Tiago C, Pimenta P. Geometrically exact analysis of space frames by
a meshless method. In: Leitão V, Alves C, Duarte C, editors.
ECCOMAS thematic conference on meshless methods. Lisbon,
Portugal; 2005. p. C441–8.
[10] Lu YY, Belytschko T, Gu L. A new implementation of the element
free Galerkin method. Comput Methods Appl Mech Eng
1994;113(3–4):397–414.
[11] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless
methods: an overview and recent developments. Comput Methods
Appl Mech Eng 1996;139(1–4):3–47.
[12] Li S, Liu WK. Meshfree and particle methods and their applications.
Appl Mech Rev 2002;55(1):1–34.
[13] Belytschko T, Lu YY, Gu L, Element-free Galerkin methods. Int J
Numer Methods Eng 1994;37(2):229–56.
[14] Duarte CA, Tinsley Oden J. H-p clouds—an h-p meshless method.
Numer Methods Partial Differ Equations 1996;12(6):673–705.
[15] Argyris J. An excursion into large rotations. Comput Methods Appl
Mech Eng 1982;32(1–3):85–155.
[16] Teixeira de Freitas JA, Moitinho de Almeida JP, Ribeiro Pereira
EMB. Non-conventional formulations for the finite element method.
Comput Mech 1999;23(5–6):488–501.
[17] Cordes LW, Moran B. Treatment of material discontinuity in the
element-free Galerkin method. Comput Methods Appl Mech Eng
1996;139(1–4):75–89.
[18] Tiago C, Leitão V. Development of a EFG formulation for damage
analysis of reinforced concrete beams. Comput Struct 2004;82
(17–19):1503–11.
ARTICLE IN PRESS
C. Tiago, P.M. Pimenta / Engineering Analysis with Boundary Elements 32 (2008) 494–511
[19] Moitinho de Almeida JP, Teixeira de Freitas JA. Continuity
conditions for finite element analysis of solids. Int J Numer Methods
Eng 1992;33(4):845–53.
[20] Lancaster P, Šalkauskas K. Surfaces generated by moving least
squares methods. Math Comput 1981;37(155):141–58.
[21] Belytschko T, Krongauz Y, Fleming M, Organ D, Liu WKS.
Smoothing and accelerated computations in the element free
Galerkin method. J Comput Appl Math 1996;74(1–2):111–26.
[22] Tiago C, Leitão VMA. Eliminating shear-locking in meshless
methods: a critical overview and a new framework for structural
theories. In: Leitão VMA, Alves C, Armando Duarte C, editors.
Advances in meshfree techniques. Computational methods in applied
sciences, vol. 5. Berlin: Springer; 2007. p. 123–45.
[23] Garcia O, Fancello EA, de Barcellos CS, Duarte CA. Hp-clouds in
Mindlin’s thick plate model. Int J Numer Methods Eng 2000;47(8):
1381–400.
[24] Donning BM, Liu WK. Meshless methods for shear-deformable
beams and plates. Comput Methods Appl Mech Eng 1998;152(1–2):
47–71.
[25] Beissel S, Belytschko T. Nodal integration of the element-free
Galerkin method. Comput Methods Appl Mech Eng 1996;139(1–4):
49–74.
[26] Cho JY, Atluri SN. Analysis of shear flexible beams, using the
meshless local Petrov–Galerkin method, based on a locking-free
formulation. Eng Comput 2001;18(1–2):215–40.
[27] Dolbow J, Belytschko T. Volumetric locking in the element free
Galerkin method. Int J Numer Methods Eng 1999;46(6):925–42.
[28] Tiago C, Leitão V. Further developments on GMLS: plates on elastic
foundation. In: Atluri SN, Tadeu AJB, editors. Advances in
computational and experimental engineering and sciences. CDROM Proceedings. Madeira, Portugal: Tech Science Press; July
2004. p. 1378–83.
[29] Tiago C, Leitão VMA. Local imposition of essential boundary
conditions in meshfree methods. In: VII World congress on
computational mechanics. Los Angeles, USA; July 2006.
511
[30] Simo JC, Hughes TJR. Computational inelasticity. Interdisciplinary
applied mathematics. Berlin: Springer; 2000.
[31] Ibrahimbegović A. Stress resultant geometrically nonlinear shell
theory with drilling rotations—Part I. A consistent formulation.
Comput Methods Appl Mech Eng 1994;118(3–4):265–84.
[32] Cardona A, Géradin M. A beam finite-element non-linear theory
with finite rotations. Int J Numer Methods Eng 1988;26(11):2403–38.
[33] Mann HB. Quadratic forms with linear constraints. Am Math Mont
1943;50(7):430–3.
[34] Spring D. On the second derivative test for constrained local extrema.
Am Math Mont 1985;92(9):631–43.
[35] Hassell C, Rees E. The index of a constrained critical point. Am
Math Mont 1993;100(8):772–8.
[36] Shutler P. Constrained critical points. Am Math Mont 1995;102(1):
49–52.
[37] Greenberg L, Maddocks JH, Hoffman KA. The bordered operator
and the index of a constrained critical point. Math Nachr 2000;
219(1):109–24.
[38] Qi Z, Tianqi Y. The adaptive parameter incremental method for the
analysis of snapping problems. Appl Math Mech 1995;16(9):851–8.
[39] Crisfield MA. A fast incremental/iterative solution procedure that
handles ‘‘snap–through’’. Comput Struct 1981;13(1–3):55–62.
[40] Schweizerhof KH, Wriggers P. Consistent linearization for path
following methods in nonlinear FE analysis. Comput Meth Appl
Mech Eng 1986;59(3):261–79.
[41] Atluri SN, Cho JY, Kim H-G. Analysis of thin beams, using the
meshless local Petrov–Galerkin method, with generalized moving
least squares interpolations. Comput Mech 1999;24(5):334–47.
[42] Mattiasson K. Numerical results from large deflection beam and
frame problems analysed by means of elliptic integrals. Int J Numer
Meth Eng 1981;17(1):145–53.
[43] Liu XH, Sze KY, Lo SH. Popular benchmark problems for geometric
nonlinear analysis of shells. Finite Elem Anal Des 2004;40(11):1551–69.
[44] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed.
New York: McGraw-Hill; 1961.