anisotropic kirchoff plates.pdf
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Comput Mech (2012) 49:629641
DOI 10.1007/s00466-011-0666-6
O R I G I N A L PA P E R
Analysis of anisotropic Kirchhoff plates using a novelhypersingular BEM
M. Wnsche F. Garca-Snchez A. Sez
Received: 7 June 2011 / Accepted: 16 November 2011 / Published online: 7 December 2011
Springer-Verlag 2011
Abstract In this article a hypersingular boundary element
method (BEM) for bending of thin anisotropic plates is pre-sented. A new complex variable fundamental solution is
implemented in the algorithm. For spatial discretization a
collocation method with discontinuous quadratic elements is
adopted. The domain integrals arising from the transversely
applied load are transformed analytically into boundary inte-
grals by means of the radial integration technique. The
considered numerical examples prove that the novel BEM
formulation presented in this study is much more efficient
than previous formulations developed for the analysis of this
kind of problems.
Keywords Complex fundamental solutionKirchhoff plate bending Anisotropic materials
Boundary element method
1 Introduction
Plate bending problems are a classical and crucial task in
the engineering design of thin structures. Different numeri-
calmethods such as thewell developed finite element method
(FEM) (e.g., [31]) and various meshless methods (e.g.,
[15,24,26]) have been investigated for Kirchhoff plates in
M. Wnsche A. Sez
Departamento de Mecnica de Medios Continuos, Universidad de
Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain
e-mail: [email protected]
A. Sez
e-mail: [email protected]
F. Garca-Snchez (B)
Departamento de Ingeniera Civil, de Materiales y Fabricacion,
Universidad de Mlaga, C/ Dr. Ortiz Ramos s/n, 29071 Mlaga, Spain
e-mail: [email protected]
the last years. The boundary element method (BEM) has
been successfully applied to isotropic Kirchhoff plate bend-ing problems (e.g.,[3,7,10,20,21,25,27,28]).
A more detailed review of the different BEMformulations
for plate bending problems can be found in [2].
In contrast to isotropic materials, the number of papers
devoted to Kirchhoff plate bending problems in anisotropic
materials is rather limited. The extension and applications of
the BEM to generally anisotropic thin plates has been suc-
cessfully done by Shi and Bzine[23]. In this study, a real
variable fundamental solution given by Wu and Altiero [30]
is implemented. This fundamental solution involves cumber-
some expressions and the resulting BEM approach loses part
of its advantages in comparison to other numerical meth-
ods. Additionally, it is worth to mention that this formula-
tion involves domain integrals arising from the transversely
applied loads. Such domain integrals are computed directly
by cell integration in the work of Shi and Bzine [23]. In this
case, an additional mesh must be defined in the domain.
Rajamohan and Raamachandran [22] developed an alter-
native algorithm by the use of particular solutions to avoid
the domain discretization. However, depending on the prob-
lem, the use of particular solutions can be a complex task.
Albuquerque et al.[1] used the radial integration technique,
introduced by Gao [8], to transform the domain integrals
into boundary ones. Here again, the implementation of the
fundamental solution given by Wu and Altiero [30] leads to
somehow cumbersome expressions. To avoid the use of the
complicated real variable fundamental solution, Dong et al.
[5] developed a Trefftz boundary collocation method. In this
method, two arbitrary complex variable analytical functions
are expressed in the form of power series to solve the result-
ing boundary value problem.
Lu and co-workers[18,16,17] and Hwu and co-workers
[11,12] applied theStrohformalismto bending of anisotropic
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plates and obtained complex variable fundamental solutions
for this kind of problems. Recently, Hwu [13] has presented
a BEM approach based on his fundamental solution [11].
In that study complex forms are converted in real forms
through the use of some identities getting rather laborious
expressions.
Maksimenko and Podruzhin [19] presented a complex
variable fundamental solution, based on the classical theoryof bending of thin anisotropic plates following the Lekhnit-
skii elastic complex potentials [14]. This complex variable
fundamental solution has an expression more compact and
easierto handlethan theones above mentioned. Furthermore,
its structure allows us to use the regularization technique,
over singular and hypersingular integrals, implemented by
Garca-Snchez et al. [9] for crack problems in anisotropic
2D solids.
In this article, we present a more general and powerful
hypersingular BEM approach for bending problems of plates
with generally anisotropic and linear elastic material behav-
ior based on Maksimenko and Podruzhin [19] fundamentalsolution.
To solve the strongly singularand the hypersingularbound-
ary integral equations(BIEs), a collocation method is adopted
with quadratic discontinuous elements in order to fulfil the
C1-continuity requirement of the transverse displacements
in the hypersingular BIE. By using quadratic discontinuous
elements the required boundary integrations are carried out
numerically by applying a regularization technique based on
a suitable changeof variable[9]. Thedomain integralsarising
from the transversely applied load are transformed analyti-
cally into boundary ones by using the radial integration tech-
nique [8] and therefore no domain discretization is needed.
To illustrate the accuracy and the efficiency of the present
BEM approach, several numerical examples are presented
and discussed.
It should be stressed here that the novelty of the present
paper lies in the combination and implementation of algo-
rithms and ideas that had been previously presented in the
literature together with a new fundamental solution, so that
the resulting BEM approach is clearly more efficient in terms
of implementation and computational costthan previous BEM
formulations.
2 Problem statement and boundary integral equations
for anisotropic plate bending
Let us consider a homogeneous, anisotropic and linear elas-
tic thin plate denoted by with boundary , see Fig.1. A
cartesian system of reference with x3 perpendicular to the
plate is used so that a point of the plate can be designated as
x =(x1,x2).
Fig. 1 Sketch of thin anisotropic plate indicating the reference system
A general distributed load, p (x), is applied over a zone of
the domain denoted byp with boundaryp .
In the most general case, the boundary can be consid-
ered decomposed into three parts: a clamped part denoted
by c, a simply supported part denoted by s and a free
part denoted by f . Considering so, we can write: =
c s f, c s = , c f = ands f = .
Under the Kirchhoff assumptions, the above mentioned
boundary conditions can be defined as follows:
Clamped boundary conditions:
w(x)= 0,w (x)
n=0, x c. (1)
Simply-supported boundary conditions:
w(x)= 0,Mn(x)= 0, x s . (2)
Free edge boundary conditions:
Mn(x)= 0, Vn (x)= 0, x f. (3)
In Eqs. (13) w(x),Mn (x) and Vn(x) are, respectively,
the transverse displacement, the bending moment and the
Kirchhoff equivalent shear force at a point x = (x1,x2) of
the plate and n denotes the outward unit normal to the bound-
ary.
The transverse displacement field satisfies the following
governing equation, [14]:
D11 4w(x)
x41+ 4D16
4w(x)
x31 x2+ 2(D12+ 2D66)
4w(x)
x21 x22
+ 4D26
4w(x)
x1x32 + D22
4w(x)
x42 = p(x), (4)
where Di j are the flexural rigidities of the anisotropic plate
defined as:
Di j = t3Ci j
12, (5)
withCi j (i, j =1, 2, 6)being the elasticity matrix andt the
thickness of the plate.
In order to find the transversal displacements field satisfy-
ingEq. (4), a boundary valueproblem canbe setout governed
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by the boundary integral equation (BIE) previously derived
byShiandBezine[23]. In their study Shiand Bezineestablish
the BIE for Kirchhoffs anisotropic plate bending problems
by using the generalized Rayleigh-Green identity. Such rec-
iprocity relation is applied between the solution to the bend-
ing problem under study and the fundamental solution, i.e.,
the bending response of an infinite plate under a transversal
unit point load. To this end, the only requirement is that thefundamental solution holds a sufficient degree of continuity
that satisfy both the real-valued fundamental solution in Shi
and Bezines work[23] and the complex variable solution
resulting from Maksimenko and Podruzhins approach [19],
as derived in next section. Therefore, the resulting BIE coin-
cides with the following, as obtained by Shi and Bezine[23]
12
w(x) +
VGn (x, y)w(y) M
Gn (x, y)
w
n(y)
dy
+
Kc
j =1
RGcj (x, y)wcj (y)
=
wG (x, y)Vn(y)
wG
n(x, y)Mn(y)
dy
+
Kcj =1
wGcj (x, y)Rcj (y)+
p
wG (x, y)p(y)d. (6)
The notation and terms in Eq. (6)should be understood as
follows:
()/n is the directional derivative along the outward
normal to the boundary, defined by
()
n=
()
y1n1+
()
y2n2. (7)
The termswcj and Rcj are, respectively, the transversal
displacement and the thin-plate reaction at node j.
Kc is the number of corners of the plate.
()G stands for the fundamental solution [19]. wG (x, y),
MGn (x, y), VG
n (x, y),RGcj
(x, y) are, respectively, the
fundamental solution transversal displacement, bending
moment, shear force and thin-plate j th corner reaction.
In those expressions,x stands for the source or colloca-tion point, i.e., the point where the load is applied and
y is a general point of the plate where the response of
the plate is calculated. The relations among them are as
follows
MGn (x, y) =
f1
2wG (x, y)
y21+ f2
2wG (x, y)
y1y2
+f32wG (x, y)
y22
, (8)
VGn (x, y) =
h1
3wG (x, y)
y31
+ h23wG (x, y)
y21 y2+h3
3wG (x, y)
y1y22
+ h43wG (x, y)
y32
1
Rh5
2wG (x, y)
y21
+ h62wG (x, y)
y1y2+h7
2wG (x, y)
y22
, (9)
RGcj (x, y) =
g1
2wG (x, y)
y21
+ g2 2wG (x, y)
y1y2+g3
2wG (x, y)
y22
,
(10)
where the terms fi , hi andgi are defined in AppendixAandR is the radius of curvature in a smooth point of the
boundary.
The expression ofw G (x, y)will be detailed in epigraph
3.
By considering the boundary conditions (13) it is easy
to see that Eq. (6) involves two known and two unknown
boundary values, so an additional BIE is needed to get a
closed system of equations.
This new BIE can be set up by differentiating Eq. (6) in
the direction of the outward unit normal vector at source
point x. Calling n0this normal vector and differentiating we
obtain
1
2
w (x)
n0
+
VGn (x, y)
n0w(y)
MGn (x, y)
n0
w(y)
n
dy
+
Kcj =1
RGcj (x, y)
n0wcj (y)
=
wG (x, y)
n0Vn(y)
2wG (x, y)
n0nMn(y)
dy
+
Kcj =1
wGcj (x, y)
n0Rcj (y) +
p
wG (x, y)
n0p(y)d.
(11)
From a numerical point of view, the kernels of integrals
in Eqs. (6) and (11) can be classified as shown in Table1.
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Table 1 Numerical behavior of the fundamental solution and its deriv-
atives when the integration point tends to the source point
FS term Numerical behavior
wG , w G /n, w G /n0 Regular
MGn , 2wG /n0n Weakly singular: O (ln (r))
VGn , MGn /n0 Strongly singular:O (r
1)
VGn /n0 Hypersingular: O (r2)
3 Complex fundamental solution for anisotropic
materials
The fundamental solutions play an essential role in solving
problems with the BEM since they have a strong influence
on the efficiency of the solution algorithm.
In this study a new complex variable fundamental solution
has been derived following the guidelines given in the study
by Maksimenko and Podruzhin [19]. In doing so, the expres-
sion for the transversal displacements at a general point y of
an infinite anisotropic plate, under a point force applied on a
pointx can be written as
wG (x, y)=
2m=1
Am
dm (y x)2
log
dm (y x)
3
2
. (12)
In Eq.(12) stands for real part, dm = (1, m )and Amis defined by
Am =(1)m1
(m 1)(m 2), (13)
being = i1122
2D11(1 2), i the roots of the charac-
teristic equation of the material and denoting the complex
conjugate.
For the case we are analyzing the characteristic equation
of the material takes the form:
D224 +4D26
3 + 2(D12+ 2D66)2
+4D16+ D11 = 0. (14)
This equation has four complex roots that appear as two
pairs of complex conjugates. Throughout the article, the con-
sidered roots are only the ones with positive imaginary part.
The concise mathematical expression (in cartesian coor-
dinates) of this fundamental solution makes the required
derivations, according to Eqs. (6)and (11), very simple. The
final expressions are much more compact than the ones (in
polar coordinates) used, for instance, by Sih and Bzine [23].
Moreover, this fundamental solution has a quite
similar mathematical structure as the one developed by
Eshelby et al. [6] and Cruse [4] for plane anisotropic prob-
lems and, therefore, the algorithms developed in [9] to deal
with singular and hypersingular boundary integrals can be
applied.
The rest of the terms of the fundamental solution appear-
ing in BIEs(6)and (11)are given in AppendixA.
4 Transformation of the domain integrals
into boundary integrals
The boundary integral Eqs. (6) and (11) involve domain inte-
grals arising from the transversely applied loads.
These domain integrals can be computed directly by cell
integration over p [23]. By doing so, an additional mesh
has to be defined in the domain and the BEM would lose its
basic idea and main advantage.
In the present study, the domain integrals are transformed
analytically into boundary integrals by using the radial inte-
gration technique [1,8]. In order to do this, polar coordinates
must be introduced, see Fig.2:
y1 =rcos + x1 := cos ,
y2 =rsin + x2 := sin . (15)
Using these definitions, the domain integrals in Eqs. (6)
and (11)can be expressed as
P =
p
wG (x, y)p(y)d
=
r0
wG ( , )p(, )dd, (16)
Pn =
p
wG (x, y)
n0p(y)d
d
rd
Fig. 2 Sketch supporting the radial integration technique for the trans-
formation of the domain integrals into boundary ones
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=
r0
wG ( , )
n0p(, )dd, (17)
where p(y)is a general distributed transversal load.
If the field point is located on the boundary of the loaded
zone p, therelationship between thearc lengthand theinfin-
itesimal length of the boundary d, see Fig. 2, can be written
as [8,1]
d = cos
rd =
er
rd , (18)
where is the angle between the outward unit normal to
p, , and the unit vector er.
Substituting the fundamental solution (12)inEqs. (16)and
(17), taking into account Eq. (18) and the following relation
dm (y x)= r(cos + msin ), (19)
we obtain
P =p
er
r
r0
p(, )
2m=1
Amr3(cos + msin )
2
log
r(cos + msin )
3
2
dd, (20)
and
Pn =
p
er
r
r0
p(, )
2
m=1
2Am (n01+ m n02)r2(cos + msin )
log
r(cos + msin )
1
dd . (21)
To derive Eq. (21) the expression ofwG (x, y)/n0given
by Eq.(35), is considered.
The inner integrals in Eqs.(20) and(21) can be computed
analytically, without difficulty, for the most common loading
configurations. By doing so, the domain integrals involved
in Eqs. (16) and (17) are transformed into boundary ones.
AppendixB shows the analytical integration for uniformly,
linearly and quadratic distributed loads.
It is worth to mention here again that this analytical trans-
formation becomes very simply due to the use of the consid-
ered complex fundamental solution.
5 Numerical solution algorithm
To solve the BIEs (6) and(11), a collocation method with
discontinuous quadratic elements is developed.
The use of discontinuous elements is adopted in order
to fulfill theC1-continuity requirement of the transverse dis-
placements necessary to obtain the hypersingular BIE (11). A
detailed description on discontinuous elements can be found,
for instance, in Garca-Snchez et al.[9] and Aliabadi [2].
To compute numerically the strongly singular and hyper-
singular integrals involved in BIEs (6) and(11) numerically
a regularization technique, based on a suitable change ofvariable [9], is applied in this article. This technique is inde-
pendent of the shape of the elements, straight or curved. Nev-
ertheless, this formulation allows analytical integration for
straight elements.
Unlike BEM formulations published so far, e.g. [1,23],
that can only be used over straight elements, this technique
can be used indistinctly over straight or curved elements.
Regarding the radial integration technique, implemented
to transform volume integrals in boundary ones for the load
terms, the applied load is approximated by piecewise qua-
dratic functions according to the geometry of the plate and
the boundary values.After spatial discretization, the BIEs(6) and (11)lead to
a system of linear algebraic equations that can be written as
V M RVn Mn Rn
V M R
ww
nwc
=
W W
n W
Wn Wnn Wn
W Wn W
VnMnRc
+
P
Pn
Pc
. (22)
Note that for the corners expressed in the third line of
Eq. (22)only the first BIE (6) is required. By invoking theboundary conditions (13), Eq. (22) can be rearranged to
yield a system of linear algebraic equations
A =b, (23)
whereis the vector of the unknown boundary values, A is
the system matrix and b is the vector containing the known
boundary values.
6 Numerical examples
In the following, some examples are presented to show the
efficiency and accuracy of the developed formulation. As
benchmark analytical results have been used when they exist,
otherwisenumericalresults afterconvergence have beencon-
sidered as reference.
For comparison purposes, numerical results obtained
using the commercial FEM program ANSYS are included.
The element used for all FEM computations is the
SHELL63 (ANSYS nomenclature). This element has four
nodes and six degrees of freedom at each one.
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Fig. 3 Plate with simply supported edges under uniformly distributed
load
6.1 Square plate under uniformly distributed load, simply
supported edges
In the first example, see Fig.3,we consider a homogeneous
square plate of thickness t = 0.01 m with simply sup-
ported edges under a uniformly distributed load p(x1,x2)=
0.01 MPa.The geometry is defined byh =l =0.5 m. The material
is considered orthotropic with properties
E1 = 206.8 G Pa , E2 = E1/15,
G12 =0.6055 G Pa , 12 = 0.3. (24)
The principal directions of the material are considered
parallel to thex1 x2 axis.
Analytical results obtained by Timoshenko and Woinows-
ki-Krieger [29] are used as the benchmark solution. Accord-
ing to this reference, the transversal displacements at points
A(0, 0)and B (l/2, h/2)are w(A)= 8.1258 mm andw(B)
=4.5211 mm.
In Table2 these reference results are compared with the
ones obtained, for several meshes, using the present formu-
lation and ANSYS.
It can be seen that errors of order 0.1% are obtained using
only one element by side (four elements in total) for the pres-
ent BEM formulation. In order to obtain an equivalent level
of accuracy for FEM results more than 16 elements by side
(256 elements in total) are needed.
It is evident that the accuracy of the present formulation
is, for the analyzed case, much higher than the one of the
FEM and less mesh-depending.
As it has been indicated, the domain integral arising from
the load terms are transformed into boundary integrals by
the radial integration technique. All BEM computations are
done using two elements per side for the boundary of the
loaded domain, p , which in this case coincides with the
whole plate.
In order to show how the discretization of the boundary of
the loaded zone influences the solution, different numbers of
divisions per edge ofp are investigated. Numerical results
at points Aand B for the used meshes are shown in Table3.
Table 2 BEM and FEM results versus analytical solution
El./edge w(A)(m) dif. (%) w(B)(mm) dif. (%)
BEM
1 8.1360 0.126 4.5360 0.330
2 8.1297 0.048 4.5264 0.117
4 8.1272 0.017 4.5225 0.031
8 8.1261 0.004 4.5214 0.007
16 8.1259 0.001 4.5211 0.0
32 8.1258 0.0 4.5211 0.0
FEM
4 7.8321 3.614 4.1366 8.505
8 8.0615 0.791 4.3998 2.683
16 8.1117 0.174 4.4890 0.710
32 8.1224 0.042 4.5130 0.179
64 8.1250 0.010 4.5190 0.046
128 8.1256 0.002 4.5205 0.013
188 8.1257 0.001 4.5208 0.007
Results for transversal displacements at points A(0, 0) andB(l/2, h/2)for several meshes
Table 3 Transversal displacements at points A(0, 0)and B (l/2, h/2)
using different meshes for the integration of load terms
Divisions/edge w(A)(mm ) w(B)(mm )
1 8.1259 4.5210
2 8.1259 4.5211
4 8.1259 4.5211
8 8.1259 4.5211
16 8.1259 4.5211
32 8.1259 4.5211
The transversal displacements obtained for several divi-
sions of the loaded zone show a very stable behavior regard-
ing this parameter. The differences between 1 and 32 ele-
ments per side are about 0.002% or even smaller. This result
confirms the expected [8] robustness and accuracy of the
present approach regarding the transformation of the domain
integrals to the boundary ones.
Figure4shows the agreement for the transverse displace-
ment fields obtained by the present BEM (left) and by the
FEM (right) for the whole plate.
6.2 Square plate under uniformly distributed load,
clamped edges
In the second example, a homogeneous square plate with
clamped edgesunder uniformly distributedload (p(x1,x2)=
0.01 MPa) is investigated, see Fig.5.
The dimensions and the material of the plate are the same
as in the previous case.
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Fig. 4 Transverse displacement fields obtained by the present BEM
(lef t) and the FEM using ANSYS (right)
Fig. 5 Plate with clamped edges under uniformly distributed load
Table 4 Comparison BEM versus FEM
El./edge w(A)(mm) dif. (%) w(B)(mm) dif. (%)
BEM
1 1.6332 1.359 0.7156 2.9788
2 1.6100 0.081 0.6938 0.158
4 1.6113 0.0 0.6950 0.014
8 1.6113 0.0 0.6949 0.0
16 1.6113 0.0 0.6949 0.0
32 1.6113 0.0 0.6949 0.0
FEM
4 1.8953 17.626 0.7137 2.705
8 1.6961 5.263 0.6984 0.504
16 1.6333 1.365 0.6957 0.115
32 1.6169 0.348 0.6951 0.029
64 1.6127 0.087 0.6949 0.0
128 1.6117 0.025 0.6949 0.0188 1.6115 0.012 0.6949 0.0
Transversal displacements results at points A(0, 0) and B(l/2, h/2),
several meshes
BEM results versus FEM ones using several meshes are
shown in Tables 4 and 5. The formerone is for transversal dis-
placements at points A(0, 0)and B(l/2, h/2)and the latter
one for bending moments at points C(0, h)and D (l, 0).
Table 5 Comparison BEM versus FEM
Mx2 (C) 100 Mx1 (D) 100
El./edge (N m/m) dif. (%) (N m/m) dif. (%)
BEM
1 1.4408 0.153 9.0756 2.192
3 1.4391 0.03 8.8866 0.064
7 1.4386 0.0 8.8811 0.002
11 1.4386 0.0 8.8810 0.001
21 1.4386 0.0 8.8809 0.0
33 1.4386 0.0 8.8809 0.0
FEM
4 1.6484 14.584 8.6789 2.275
8 1.5877 10.364 8.7798 1.138
16 1.4789 2.801 8.8612 0.222
32 1.4485 0.688 8.8762 0.053
64 1.4410 0.167 8.8798 0.012
128 1.4392 0.042 8.8807 0.002
188 1.4388 0.014 8.8808 0.001
Bending moments results at pointsC(0, h) andD(l, 0), several meshes
In this example, the constant value reached for the present
approach are considered as benchmark:
w(A)= 1.6113 mm
w(B)= 0.6949 mm
Mx2 (C) 100= 1.4386N m/m (25)
Mx1 (D) 100= 8.8809N m/m
As in the previous case, transversal displacements and
bending moments of the present BEM are quite stable for all
investigated boundary divisions. The maximum difference in
the transversal displacements is about 3% and in the bending
moments 2%. Less than 10 elements per side are enough to
get results with differences smaller than 0.001%. This fact
confirms again the accuracy and efficiency of the present
formulation.
Once again, in the FEM model it is necessary to use a
number of elements three orders higher than the number of
elements used in the BEM model to get a comparable level
of accuracy.
6.3 Quasi-isotropic clamped square plate subjected
to linearly distributed load
In the next example, Fig.6,a homogeneous quasi-isotropic
square plate with clamped edges is investigated. A linearly
distributed load is considered. This load is defined as
p(x1,x2)= p0x1+ l
2l; p0 =0.01M Pa . (26)
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Fig. 6 Plate with clamped edges under linearly distributed load
Table 6 Comparison BEM versus FEM
El./edge Mx2 (A)a dif. (%) Mx1 (B)
a dif. [%]
BEM
1 0.0254 1.550 0.0336 0.599
3 0.0258 0.0 0.0336 0.599
7 0.0258 0.0 0.0336 0.599
11 0.0258 0.0 0.0336 0.59921 0.0258 0.0 0.0336 0.599
33 0.0258 0.0 0.0336 0.599
FEM
4 0.0221 14.341 0.0272 18.563
8 0.0250 3.101 0.0322 3.593
16 0.0256 0.775 0.0332 0.599
32 0.0257 0.388 0.0335 0.299
64 0.0257 0.388 0.0335 0.299
128 0.0257 0.388 0.0335 0.299
188 0.0258 0.0 0.0336 0.599
a [M/(p0
l2) 102]
Dimensionless bending moments results at pointsA(0, h)and B (l, 0),
several meshes
The geometry is determined by t =0.01 m andh =l =
0.5 m.
The following quasi-isotropic material properties are con-
sidered
E1 = 210.0G Pa , E2 = 209.9G Pa ,
G12 = 76.92G Pa , 12 = 0.3. (27)
The bending moments at points A(0, h) and B(l, 0) arepresented in Table6while transversal displacements at point
C(0, 0)are given in Table7.
The following dimensionless analytical results, obtained
by Timoshenko and Woinowski-Krieger [29], are used as
benchmarks
w(C)E2h3/(p0l
4) 103 =6.8776
Mx2 (A)/(p0l2) 102 =0.0258 (28)
Mx1 (B)/(p0l2) 102 =0.0334.
Table 7 Comparison BEM versus FEM
w(C)
El./edge [wE2h3/(p0l
4) 103] dif. (%)
BEM
1 6.9371 0.865
3 6.9624 1.233
7 6.9616 1.22111 6.9616 1.221
21 6.9616 1.221
33 6.9616 1.221
FEM
4 7.5167 9.292
8 7.1112 3.397
16 6.9988 1.762
32 6.9709 1.357
64 6.9639 1.255
128 6.9622 1.230
188 6.9619 1.226
Dimensionless transversal displacements results at pointC(0, 0), sev-
eral meshes
Again, bending moments and transversal displacements
obtained by the present BEMagree very well with the analyt-
ical solution [29] for all investigated meshes. The maximum
difference is about 0.6% for the bending moments using only
two elements per side and 1.2% for the transversal displace-
ments using seven element per boundary.
In contrast, the numerical results from the FEM show a
stronger sensitivity to the mesh. For this reason a significant
smaller element is needed to obtain similar accuracies.
6.4 Rectangular plate subjected to linearly distributed load
In the next example, a rectangular plate with different bound-
ary conditions is considered, Fig.7. The linearly distributed
load is defined as in [26]. The numerical calculations are
carried out for the geometrical parameters l = 2.0 m and
h = 1.0 m and thickness t=0.02 m.
To investigate the effects of the anisotropy behavior, the
orthotropic properties given in Eq.(24) are adopted consid-
ering different angles between the orthotropic directions
Fig. 7 Plate with mixed boundary conditions subjected to a linearly
distributed load
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1
Fig. 8 Transverse displacements along the line x2 = 0 (lef t) and
x1 = 0 (right) obtained by the present BEM and by the FEM
and the reference system, see Fig.7. The analyzed angles are
=0, 45 and 90.
In the BEM mesh 10 elements are used for the shorter
side and 20 for the longer one. A mapped mesh using 100
and 200 elements, for the respective sides, is utilized for the
FEM computations made for comparison purposes.
Figures8and 9show, respectively, transverse displace-
ments and bending moments along the lines x1 = 0 and
x2 = 0. In all investigated cases the agreement between the
BEM and FEM results is excellent.
Figure10 reveals the influence of the used angles and
the considered mixed boundary conditions over the trans-verse displacement fields. In the case of =0 and =90
the behavior is orthotropic whilst the transverse displace-
ments obtained for = 45 show a clear influence of the
anisotropic behavior.
6.5 Rectangular plate with a central hole
In the last example, a rectangular plate with a central hole
of radiusr is considered, Fig.11.The external boundary is
x2
x1
2
1
Fig. 9 Bending moments along the boundariesx2 = h (lef t) and
x1 = l (right) obtained by the present BEM and by the FEM
assumed clamped and a uniformly distributed load of
0.005M Pa is defined.
The geometry of the plate is defined byl = 2.0 m,h =
1.0 m,t = 0.02 m andr =0.5 m. The same material prop-
erties as in the previous case for =45 is chosen.
For the external boundaries, a mesh of 10 elements for
the shorter side and 20 for the longer one is used. For the
hole, three different discretizations are compared with the
aim of showing the benefit of curved elements in the pres-
ent approach. The first mesh (M1) is composed of 10 curved
elements, the second one (M2) of 4 curved elements and the
third one (M3) of 4 straight elements.
One more time for comparison purposes FEM results are
obtained. The used mapped mesh has 100 elements in the
shorter side and 200 for the longer one. The comparison are
done by mean of the displacements along the lines x1 = 0
andx2 = 0, Fig.12,as well as the bending moments along
straight boundaries, Fig.13.
As in all previous examples, the agreement between BEM
(using some orders of magnitude less in the number of ele-
ments) and FEM results is evident.
Figures 12 and 13 show that 4 curved elements are enough
to obtain as accurate results as using 10 curved elements.
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Fig. 10 Transverse displacements fields for = 0, 45 and 90
obtained by the present BEM
Fig. 11 Plate with a central hole under uniformly distributed load
This means that the convergence of the presented approach
using curved elements is as quick as the one observed, in the
preceding analyzed cases, using only straight elements.
The important differences obtained using the meshes M2
and M3 reveal the advantages of using curved elements with
no additional meshing effort.
2
1
Fig. 12 Transverse displacements along x2 = 0 (lef t) and x1 = 0
(right) obtained by the present BEM and by the FEM
7 Conclusions
This article presents a novel hypersingular boundary element
method for anisotropic thin plates bending problems based on
the complex variable fundamental solution of Maksimenko
and Podruzhin [19]. The use of that fundamental solution
leads to a BEM approach where the kernels of the integrals
are much more simple than in previous BEM formulations
[1,23]. The result is a quick, efficient and robust solution
algorithm including an exact transformation of the domain
integrals into boundary ones.
To solve the strongly singular and hypersingular integrals
involved in the BIEs a collocation method, with discontin-
uous quadratic elements is implemented. In this way, it is
possible to adopt a special regularization procedure which is
independent of the shape of the element (straight or curved).
According to this technique, only regular integrals must be
computed numerically because the strongly singular and
hypersingular behaviors are shifted to integrals with well
known analytical solutions [9].
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x2
x1
2
1
Fig. 13 Bending moments along the boundariesx2 = h (lef t) and
x1 = l (right) obtained by the present BEM and by the FEM
Several numerical examples with different boundary and
loading conditions are shown to evaluate the suitability and
the efficiency of the present BEM.
For the analyzed cases, the comparisons of the results
obtained by the present formulation show a very good agree-
ment with the results used as benchmarks: analytical ones
when they exist and FEM results, using ANSYS, when
they do not.
According to our experiences, using equal boundary divi-
sions the present BEM requires less computational time for
solving the problem than the FEM using ANSYS. From
the point of view of the convergence, the presented approach
has shown a much better behavior than the FEM. In order
to get similar accuracies, for FEM models it has been neces-sary to use meshes with a number of elements several orders
of magnitude higher than the number of elements of the
BEMmeshes. This is an important feature in computing large
models.
Appendix A: Fundamental solutions
Introducing for convenience the relation
zm =dm (y x)= (y1 x1) +m (y2 x2), (29)
the transversal displacement fundamental solution [19] can
be expressed, as
wG (zm )=
2m=1
Amz2m
log(zm )
3
2
. (30)
The rest of the fundamental solution terms taking part in
the BIE(6) are the following:
wG (zm )
n
=
2m=1
2Am (n1+ m n2)zm
log(zm ) 1
, (31)
MG (zm )
=
2m=1
2Am
f1+ mf2+ 2mf3
log(zm )
, (32)
VG (zm )
=
2m=1
2Am
h1+ m h2+
2m h3 +
3m h4
zm
1
R
h5+ m h6+
2m h7
log(zm )
, (33)
RGcj (zm )
=
2m=1
2Am
g1+ m g2+ 2m g3
log(zm )
. (34)
And the ones taking part in the BIE(11) are, consideringEq. (7):
wG (zm )
n0
=
2m=1
2Am (n01+m n02)zm
log(zm ) 1
, (35)
2wG (zm )
n0n
=
2m=1
2Am (n01+m n02)(n1+ m n2) log(zm )
,
(36)
MG (zm )
n0
=
2m=1
2Am(n01+ m n02)
f1+ mf2+
2mf3
zm
,
(37)
VG (zm )
n0=
2m=1
2Am (n01+ mn02)
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h1+ m h2+
2m h3+
3m h4
z2m
1
R
h5 + m h6+
2m h7
zm
, (38)
RGcj (zm )
n0=
2m=1
2Am(n01+ mn02)
g1+ m g2+
2m g3
zm
. (39)
The constants fi in Eqs. (32) and (37) are defined by
f1 = D11n21 +2D16n1n2+ D12n
22, (40)
f2 =2(D16n21 + 2D66n1n2+ D26n
22), (41)
f3 = D12n21 +2D26n1n2+ D22n
22, (42)
wheren i are the components of the outward unit vector.
The constantsh i , Eqs.(33) and(38), are given by
h1 = D11n1(1+ n22) +2D16n32 D12n1n22, (43)h2 =4D16n1+ D12n2(1+ n
21) +4D66n
32
D11n21n22D26n1n
22, (44)
h3 =4D26n2+ D12n1(1+ n22) +4D66n
31
D22n1n222D16n
21n2, (45)
h4 = D22n2(1 + n21) + 2D26n
31 D12n
21n2, (46)
h5 =(D12 D11) cos 2 4D16sin 2, (47)
h6 =2(D26 D16) cos 2 4D66sin 2, (48)
h7 =(D22 D12) cos 2 4D26sin 2, (49)
where is the angle between the global coordinate systemand the local coordinate system in the field point y .
Finally, the constantsgi , Eqs.(34) and(39), are
g1 =(D12 D11) cos sin + D16(cos2 sin2 ), (50)
g2 =2(D26 D16) cos sin + 2D66(cos2 sin2 ),
(51)
g3 =(D22 D12) cos sin + D26(cos2 sin2 ). (52)
Appendix B: Analytical integration with respect
to of the domain integrals
The results of the analytical integrations are given in the fol-
lowing for a quadratic distributed load of the form
p(y1,y2)= C0 + C1y1+ C2y2 + C3y21 + C4y1y2+ C5y
22 .
(53)
For convenience of the presentation we introduce the rela-
tion
zm =cos + msin . (54)
By substituting Eqs. (53)and (54) in Eq.(20) and inte-
grating them analytically with respect to leads to
P(r, )=
2m=1
AmIunif +Ili n+ Iquad
, (55)
where the uniformly, linearly and quadratic parts are defined
by
Iunif = 1
4r3
p
Cz2m
log(r zm )
7
4
(n1r1 + n2r2)d ,
(56)
Ili n = 1
5r4
p
(C1cos + C2sin )z2m
log(r zm )
17
10
(n1r1+ n2r2)d , (57)
Iquad = 1
6r5
p
(C3cos2 + C4cos sin + C5sin
2 )z2m
log(r zm )
5
3
(n1r1+ n2r2)d . (58)
The new constantC follows from the transformation of
Eq. (53) in polar coordinates according Eq. (15).
In the same way, the substitutions of Eqs.(53) and (54) in
Eq.(21), the analytical integration with respect to results
in
Pn(r, )
=
2m=1
Am (n01+m n02)Iunif + Ili n + Iquad
,
(59)
where
Iunif = 1
3r2
p
Czm
log(r zm )
4
3
(n1r1+ n2r2)d ,
(60)
Ili n = 1
4r3
p
(C1cos + C2sin )zm
log(r zm )
5
4
(n1r1 + n2r2)d , (61)
Iquad = 15
r4
p
(C3cos2 + C4cos sin + C5sin2 )
zm
log(r zm )
6
5
(n1r1 +n2r2)d. (62)
It should be noted that the results of the analytical integra-
tions canbe further simplifiedby summarizing theuniformly,
linearly and quadratic parts according the applying loading
configuration to reduce the numerically computational cost
of the boundary integration.
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Acknowledgements This study is supported by the Spanish Minis-
try of Science and Innovation under project DPI2010-21590-C02-02
and by the Junta de Andaluca under project P09-TEP-5054. The finan-
cial support is gratefully acknowledged.
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