Analysis of thin isotropic rectangular and circular plates with multiquadrics
A computational method based on radial basis functions has been applied to the linear solution of thin plates. This meshless numerical methodgives high flexibility in the analysis of irregular geometries, due to its insensivity to spatial dimension. The multiquadrics approach is used in this pa...
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| Cite this: | Analysis of thin isotropic rectangular and circular plates with multiquadrics / A.J.M. M. Ferreira, C.M.C. Roque, P.A.L.S. Martins // Проблемы прочности. — 2005. — № 2. — С. 72-84. — Бібліогр.: 13 назв. — англ. |
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Ferreira, A.J.M. Roque, C.M.C. Martins, P.A.L.S. 2013-07-25T08:47:36Z 2013-07-25T08:47:36Z 2005 Analysis of thin isotropic rectangular and circular plates with multiquadrics / A.J.M. M. Ferreira, C.M.C. Roque, P.A.L.S. Martins // Проблемы прочности. — 2005. — № 2. — С. 72-84. — Бібліогр.: 13 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/47674 539.4 A computational method based on radial basis functions has been applied to the linear solution of thin plates. This meshless numerical methodgives high flexibility in the analysis of irregular geometries, due to its insensivity to spatial dimension. The multiquadrics approach is used in this paper. The numerical solution is compared with Kirchhoff theory for plates. Виконано лінійний розрахунок напружено-деформованого стану тонких пластин числовим методом, що базується на використанні мультіквадратичних радіальних базисних функцій. Показано, що даний метод є досить гнучким при розрахунках об’єктів зі складною геометрією, оскільки не потребує сіткового розбиття і нечутливий до їх просторових координат. Отримані числові результати порівнюються з даними розв’язків на основі теорії пластин Кірхгофа. Выполнен линейный расчет напряженно-деформированного состояния тонких пластин численным методом, базирующемся на использовании мультиквадратических радиальных базисных функций. Показано, что данный метод оказывается весьма гибким при расчетах объектов со сложной геометрией, поскольку не требует сеточного разбиения и нечувствителен к их пространственным координатам. Полученные численные результаты сравниваются с данными решений на основе теории пластин Кирхгофа. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Analysis of thin isotropic rectangular and circular plates with multiquadrics Анализ тонких изотропных прямоугольных и круглых пластин методом мультиквадратических функций Article published earlier |
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Analysis of thin isotropic rectangular and circular plates with multiquadrics |
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Analysis of thin isotropic rectangular and circular plates with multiquadrics Ferreira, A.J.M. Roque, C.M.C. Martins, P.A.L.S. Научно-технический раздел |
| title_short |
Analysis of thin isotropic rectangular and circular plates with multiquadrics |
| title_full |
Analysis of thin isotropic rectangular and circular plates with multiquadrics |
| title_fullStr |
Analysis of thin isotropic rectangular and circular plates with multiquadrics |
| title_full_unstemmed |
Analysis of thin isotropic rectangular and circular plates with multiquadrics |
| title_sort |
analysis of thin isotropic rectangular and circular plates with multiquadrics |
| author |
Ferreira, A.J.M. Roque, C.M.C. Martins, P.A.L.S. |
| author_facet |
Ferreira, A.J.M. Roque, C.M.C. Martins, P.A.L.S. |
| topic |
Научно-технический раздел |
| topic_facet |
Научно-технический раздел |
| publishDate |
2005 |
| language |
English |
| container_title |
Проблемы прочности |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| format |
Article |
| title_alt |
Анализ тонких изотропных прямоугольных и круглых пластин методом мультиквадратических функций |
| description |
A computational method based on radial basis
functions has been applied to the linear solution
of thin plates. This meshless numerical methodgives high flexibility in the analysis of irregular
geometries, due to its insensivity to spatial dimension.
The multiquadrics approach is used in
this paper. The numerical solution is compared
with Kirchhoff theory for plates.
Виконано лінійний розрахунок напружено-деформованого стану тонких пластин
числовим методом, що базується на використанні мультіквадратичних
радіальних базисних функцій. Показано, що даний метод є досить гнучким
при розрахунках об’єктів зі складною геометрією, оскільки не потребує сіткового розбиття і нечутливий до їх просторових координат. Отримані
числові результати порівнюються з даними розв’язків на основі теорії
пластин Кірхгофа.
Выполнен линейный расчет напряженно-деформированного состояния тонких пластин
численным методом, базирующемся на использовании мультиквадратических радиальных
базисных функций. Показано, что данный метод оказывается весьма гибким при расчетах
объектов со сложной геометрией, поскольку не требует сеточного разбиения и нечувствителен
к их пространственным координатам. Полученные численные результаты сравниваются
с данными решений на основе теории пластин Кирхгофа.
|
| issn |
0556-171X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/47674 |
| citation_txt |
Analysis of thin isotropic rectangular and circular plates with multiquadrics / A.J.M. M. Ferreira, C.M.C. Roque, P.A.L.S. Martins // Проблемы прочности. — 2005. — № 2. — С. 72-84. — Бібліогр.: 13 назв. — англ. |
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2025-11-25T20:39:13Z |
| last_indexed |
2025-11-25T20:39:13Z |
| _version_ |
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| fulltext |
UDC 539.4
Analysis of Thin Isotropic Rectangular and Circular Plates with
Multiquadrics
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins
Departamento de Engenharia Mecânica e Gestäo Industrial Faculdade de Engenharia,
Universidade do Porto, Porto, Portugal
УДК 539.4
Анализ тонких изотропных прямоугольных и круглых пластин
методом мультиквадратических функций
А. Ж. М. Феррейра, С. М. С. Рок, П. А. Л. С. Мартинс
Университет г. Порто, Португалия
Выполнен линейный расчет напряженно-деформированного состояния тонких пластин
численным методом, базирующемся на использовании мультиквадратических радиальных
базисных функций. Показано, что данный метод оказывается весьма гибким при расчетах
объектов со сложной геометрией, поскольку не требует сеточного разбиения и нечувст
вителен к их пространственным координатам. Полученные численные результаты сравни
ваются с данными решений на основе теории пластин Кирхгофа.
Ключевые слова : численный метод, напряженно-деформированное состоя
ние, тонкие пластины, мультиквадратические функции, радиальные базис
ные функции, несимметричная коллокация.
Introduction. The purpose of this paper is the application of radial basis
functions, to the analysis of thin plates. Hardy [1] was the first to develop the
multiquadrics method (MQ) as a general scattered data approximation algorithm
to approximate two-dimensional geographical surfaces. The MQ method was
found to be one of the best interpolation schemes for scattered data by Franke [2].
The method used in this paper is based on the work of Kansa [3] where a
modified MQ scheme was proposed for the solution o f partial differential
equations. The numerical solution of partial differential equations (PDEs) is
traditionally dominated by finite element methods, finite volume methods or finite
difference methods. All of these methods are based on local interpolation
strategies and depend on a mesh for local approximation. In these methods
although the function is continuous across meshes, its partial derivatives are not
[4-6]. T
form g (
ie MQ method depends on the distance to a center point x j and is of the
). The radial basis functions (RBF) may also depend on a shapex — x
parameter c, in which case g( ) is replaced by g ( c) [3, 7].
The multiquadrics solution depends o f the choice o f the shape parameter, a
user-defined parameter. This choice is still an open discussion. This method was
© A. J. M. FERREIRA, C. M. C. ROQUE, P. A. L. S. MARTINS, 2005
72 ISSN 0556-171X. Проблемы прочности, 2005, N 2
Analysis o f Thin Isotropic Rectangular and
previously applied with success to some engineering problems [8] and to
composite materials by Ferreira et al. [9-11].
In this paper it is investigated the application of the MQ method in the
solution of thin plates, which is a relevant problem for mechanical, civil and
structural engineers. The numerical solutions are compared with Kirchhoff theory.
1. The Multiquadric Method. Consider a set of nodes x 1, x 2 ,.. ., x N E
Q C f i " . The radial basis functions centered at x ,• are defined as
g j ( x ) = g ( x — x ) e R n, j = 1,..., N, (1)
where x — x ,• is the Euclidian norm. Some of the most common RBFs are
multiquadrics: g j (x ) = (
inverse multiquadrics: g j ( x ) = (
Gaussians: g j ( x ) = e
x — x j + c 2) 1/2,
x — x
—c II x —x .
+ c 2)
2N — 1/2
(2)
(3)
(4)
thin-plate splines: g j (x ) = x — x
2m
log x — x m = 1,2,..., n , (5)
where c is a shape (user-defined) parameter.
In this paper we will concentrate on the multiquadrics approach. Radial basis
functions are insensitive to spatial dimension, making the implementation of this
method much easier than, e.g., finite elements. Illustration of a network of nodes
is given in Fig. 1.
Fig. 1. Illustration of a network of nodes.
2
2
2
An important feature of the RBF method is that is does not require a grid.
The only geometric properties needed in an RBF approximation are the pairwise
distances between points. Working with higher dimensional problems is not
difficult as distances are easy to compute in any number of space dimensions.
ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N 2 73
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins
In this paper, it is proposed to use Kansa’s unsymmetric collocation method
[3]. Consider a boundary-valued problem with a domain Q C fi" and a linear
elliptic partial differential equation of the form
Lu(x) = s(x) C R n,
Bu( x ) dQ = f ( x ) e Rn ,
(6)
(7)
where dQ represents the boundary of the problem. We use points along the
boundary (x j , j = 1,..., N B ) and in the interior (x j , j = N B + 1,..., N ).
Let the RBF interpolant to the solution u(x ) be
N
s( x , c ) = 2 aj g (
j=i
x — x , c ). (8)
Collocation with the boundary data at the boundary points and with PDE at
the interior points leads to equations
N
sb ( x , c ) = 2 aj Bg (
j=i
x —x j c) = A(x i ), i = 1,..., N b , (9)
N
SL(x ,c) = 2 ajLg(\ x — x j ||, c)= 0 ( x f ), i = N b + 1,..., N , (10)
j=i
where A(x i ) and 0 (x i ) are the prescribed values at the boundary nodes and the
function values at the interior nodes, respectively.
This corresponds to a system of equations with an unsymmetric coefficient
matrix, structured in matrix form as
'Bg" T
[a ] = O (11)
2. Thin Plate Analysis.
2.1. Thin Plate Theory. It is considered a thin plate, subjected to lateral
transverse loads q = q(x , y ). The symbol D is used to represent the flexural
rigidity of the plate:
D =
Eh :
12(1— v 2 )
(12)
In the plate problem, the unknown transverse displacement, w, is calculated
by solving the boundary-valued problem, defined by the differential equation:
74 ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N9 2
Analysis o f Thin Isotropic Rectangular and
d 4 w d 4 w d 4 w q(x , y )
dx4 dx 2dy2 dy4 D
(13)
The boundary conditions are defined, for example, for a simply-supported
edge parallel to the x-axis at y = b as
[w] y=b = 0
d 2 w
dx 2
= 0.
y=b
(14)
The flexural moments are obtained by
M x = - D
2d2w
+ v
2d2w
dx
(15)
M y = - D
I -.2 -.2 ̂d w d w
dx 2 + dy 2
(16)
Mxy = D(1 — v )
d 2 w
dxdy (17)
2.2. M ultiquadrics Interpolation for the Governing Equation. Applying
the multiquadrics method previously explained, the differential equation (13) is
now interpolated for each node i, as
N XN
2 <p J
j=1
-, 4 -.4 -.4
d g j d g j d g
+ 2 --- :rj +
dx 4 - . 2 ^ 2 dx dy dy
q( x, y )
D (18)
where g i was defined in Eq. (1).
2.3. M ultiquadrics Interpolation for B oundary Conditions. For each
boundary node, the multiquadric interpolation for each boundary node follows the
approach in Eq. (8). A simply-supported condition in x = a edge imposes two
boundary conditions (see Fig. 2)
2 2 d w d w
wx=a = 0 and —r + v T T = 0 (19)dx 2 dy 2
In this collocation method, we chose to impose the first boundary condition
on node with x = a and the second boundary condition on node with x = a ± d
when d is a very small number. This approach was previously applied by Jang
et al. [13] to differential quadrature method on plates. It is really important to
apply this methodology due to the improvement of the quality of solution when
ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N 2 75
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins
ô is used. The collocation points, after the inclusion of Ô, can be seen on Fig. 3,
for various point cloud density.
1
0.a
06 * ■
0.4
0
0.2 0 4 D.B O.e I * 0 D.E £1,4 0.6 o ,a 1
D.B
Q.fi
0Я
: о.»
0% 0,3 ci 4 as o.a i L\ 0.2 0,4 Qt o.s 1
Fig. 3. Regular discretization for N = 9, 13, 17, and 21.
The multiquadrics interpolation of equations (19) leads to a change in the
global equations system. As an example, for each node i where w = 0, the
following multiquadric equation is imposed:
NX N ___________________________
2 ^ W ( x i ~ x j )2 + (y i ~ У j )2 + c 2 = 0 (20)
j=1
76 ISSN 0556-171X. Проблемы прочности, 2005, N 2
Analysis o f Thin Isotropic Rectangular and
3. Discussion of Results. The analysis of square and circular Kirchhoff
plates is performed, for simply-supported and clamped conditions. All plates are
subjected to a uniform unit load. Square plates have unit side and circular plates
have unit radius.
In figures below, N is considered as the number of nodes per side in square
plates and the number of nodes per radius in circular plates. In all problems a unit
bending stiffness is considered. The objective of the numeric studies is to evaluate
the accuracy of the model with exact Kirchhoff results, as presented by
Timoshenko [12].
The influence of various parameters in the quality of the solutions is
investigated. In particular the effect of the shape-parameter, c, and the 6
parameter in the solution error are considered.
The root-mean-square error (RMS) was calculated using:
N x N -NB
RMS = ------------------ У
N x N - N b \
I \ 2wexact wMQ
we
(21)
Solutions for the transverse displacement curve are presented.
In Fig. 4 it is illustrated the deformed shape of a simply-supported square
plate, for N = 23, and d = 1-10_7. It can be seen a very smooth deformation
pattern, that presents very accurate forms when compared with exact solution as
can be seen in Fig. 5 where an error study is conducted. In this figure it is
presented the evaluation of the root-mean-square error with N . The convergence
to exact results is exponential. After about 20 nodes per side the solution is almost
exact, with very small RMS error.
Fig. 4. Transverse displacement for a simply supported square plate with N = 21, 6 = 10 7, and
c = 2/4N .
ISSN 0556-171X. Проблемы прочности, 2005, N2 2 77
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins
Further calculation for square plates considered N = 19 due to the good
behavior as seen in Fig. 5.
The influence of shape-parameter c in the RMS error is plotted in Fig. 6. All_3
values of c G [0,1] produce errors around 10 . However, values in range
[0.4-1.0] should be preferred. Our usual choice is c = 2 /V N , which matches this
interval and in fact produces one of the smallest error. This choice was previously
proposed by Fasshauer [7].
Fig. 5. RMS error dependence on N for a simply supported square plate with ^ = 1-10 5 and
c = 2/VN.
Fig. 6. RMS error dependence on c for a simply supported square plate with N = 19 and
(5 = 1-10_5.
78 ISSN 0556-171X. npodxeMbi npounocmu, 2005, N9 2
Analysis o f Thin Isotropic Rectangular and
Figure 7 plot the variation of RMS error with d parameter. This parameter is
necessary to adequately introduce the second set of boundary conditions and
needs to be a very small number. The variation of RMS error with d is in general
linear, with a few exceptions.
The evaluation of RMS error for the transverse displacement with N is
plotted in Fig. 8. A very fast convergence is obtained. A good solution quality is
obtained after 18 points per side.
Fig. 7. RMS error dependence on d for a simply supported square plate with N = 19 and
c = 2/VN.
Fig. 8. RMS error dependence on N for a clamped square plate with d = 1 • 10 5 and c = 2/4 N .
ISSN 0556-171X. Проблемы прочности, 2005, N2 2 79
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins
In Figs. 9 and 10 the evaluation of RMS with c and RMS with d are
plotted. The results follow the same good pattern as for simply-supported plates.
In Fig. 11 are illustrated some meshes of circular plates for various ‘mesh’
densities. Near the external boundary nodes a new set of nodes is placed in order
to apply a second set of boundary conditions. This ‘d-approach’ already used by
Jang et al. [13] will affect significantly the quality of the solution as will be seen
in figures below.
Fig. 9. RMS error dependence on c for a simply supported square plate with N = 19.
RM S (w)
Fig. 10. RMS error dependence on d for a clamped square plate with N = 19 and c = 2/vN.
80 ISSN 0556-171X. Проблемы прочности, 2005, N2 2
Analysis o f Thin Isotropic Rectangular and
In Fig. 12 the vertical (transverse) displacement for each node is plotted,
corresponding to a cloud of nodes representing the deformation of the plate. The
deformation of the plate is presented for N = 23 for a simply-supported plate.
N represents the number of nodes in a single radius of the plate.
- 0 . 5 0 0.5 - -0.5 C 0.5
" ■- ̂ Î / >
0.5
\ *
' \ \
■ X \ \
/
/ ,
O $ ¥ :■
.........*
\ \
■ * i \ ‘‘ / /
\
-ax> O' 95 1 -1 -0.6 C 0-5
Fig. 11. Regular discretization for a circular plate with N = 8, 12, 16, and 20.
Fig. 12. Regular discretization for a simply supported circular plate with N = 13, d = kj 100, and
c = 2/VN.
In Fig. 13 the evolution of root-mean-square curve (transverse displacement)
with the shape parameter c is presented for simply-supported circular plates.
Similar curves are obtained for N = 11, 19, 25. There is a fast convergence to
final results for every N .
The RMS error was calculated only for node where maximum displacement
is taking place using:
ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N 2 81
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins
NX N- N b
RMS = 2N X N - N B ]j
wmax, exact wmax, MQ
wmax, exact
(22)
Keeping d = k /100, the evolution of RMS error with the shape-parameter
for clamped circular plate is illustrated in Fig. 14, for various grids. Significant
dependence of RMS error is noticeable. However, for N > 9, c > 0.2 very small
errors are produced. In particular, c = 2 /V N is included in such range which
confirms the results of Fasshauer [7].
RM S <w)
Fig. 13. RMS error dependence on c for a simply supported circular plate d = k/100.
RM S (w)
82
Fig. 14. RMS error dependence on c for a clamped circular plate b = к/100.
ISSN 0556-171X. Проблемы прочности, 2005, № 2
1
Analysis o f Thin Isotropic Rectangular and
In Tables 1 and 2, the results from the present model and the exact
(Kirchhoff) values [12] are presented. In Table 1, the results are presented for
simply-supported and clamped square plates. In Table 2, the results are presented
for simply-supported and clamped circular plates. The present results are in very
good agreement with exact values.
T a b l e 1
Model and Exact Values of w and Mx for Thin Square Plates
Simply supported
д = 10“ 7
Clamped
д = 10_7
N w(0,0) Mx (0,0) w(0,0) Mx (0,0)
10 0.0039329 0.046740 0.00100 0.0196
15 0.0040110 0.047431 0.00110 0.0890
21 0.0040104 0.047436 0.00120 0.0225
Exact 0.0041600 0.047900 0.00126 0.0231
T a b l e 2
Model and Exact Values of w and Mx for Thin Circular Plates
Simply supported
д=к / 100 д=
Cla
kk
!
00
ed
N li 0) Mx (r = 0) r li 0) Mx (r = 0)
6 0.0510 0.1727 0.0132 0.0740
10 0.0599 0.1962 0.0150 0.0796
14 0.0622 0.2023 0.0154 0.0807
16 0.0627 0.2036 0.0155 0.0809
Exact 0.0637 0.2062 0.0156 0.0813
Conclusions. In this paper, the multiquadrics method was applied to the
structural analysis of square and circular plates. The method is most adequate to
analyze any geometry with the same governing equations provided adequate
boundary conditions are imposed. The quality of the solution converges
exponentially for both square and circular plates. Given the collocation procedure,
a ‘д-approach’ was used in order to impose a second set of boundary conditions.
This procedure provides a significant improvement in the quality of the solution.
In general, the model agrees very well with the exact Kirchhoff theory for plates.
This method can be a good alternative to the finite element method and to the
finite difference method in the analysis of such structures.
Р е з ю м е
Виконано лінійний розрахунок напружено-деформованого стану тонких плас
тин числовим методом, що базується на використанні мультіквадратичних
радіальних базисних функцій. Показано, що даний метод є досить гнучким
при розрахунках об’єктів зі складною геометрією, оскільки не потребує
ISSN 0556-171X. Проблеми прочности, 2005, № 2 83
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins
сіткового розбиття і нечутливий до їх просторових координат. Отримані
числові результати порівнюються з даними розв’язків на основі теорії
пластин Кірхгофа.
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(1989).
Received 27. 10. 2003
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