Engineering Analysis with Boundary Elements 23 (1999) 405–418
Plate bending boundary element formulation considering
variable thickness
E.W.V. Chaves, G.R. Fernandes, W.S. Venturini*
São Carlos School of Engineering, University of São Paulo, Av. Carlos Botelho 1465, São Carlos, Brazil
Abstract
In this article the plate bending boundary element method formulation, based on the Kirchhoff’s hypothesis, is extended to take into
account possible variation in the plate thickness or stiffness. For this particular case, an appropriate form of the Betti’s reciprocal work
theorem was taken to derive integral representations of displacements and internal forces, starting by assuming that the plate domain exhibits
variable thickness or stiffness. Some alternative integral representations to treat this problem are discussed pointing out the basic differences
among them. The domain integrals, remaining in all integral equations and required to take into account the stiffness variation, are properly
handled leading to small and reduced number of internal unknowns and their corresponding integral representations. As usual for this
problem, only deflection integral representations related to boundary collocations are taken to write the necessary algebraic relations, written
for singular and nonsingular points. 䉷 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Plate bending; Boundary elements; Varying thickness
1. Introduction
The boundary element method (BEM) is already a wellestablished numerical technique to deal with an enormous
number of engineering complex problems. Analysis of plate
bending problems using BEM techniques has attracted the
attention of many researchers during the past years, proving
to be a particularly adequate field of applications for that
technique. For instance, BEM is particularly suitable to
evaluate internal force concentrations owing to point
loads, that very often occur in these kinds of structures.
Thin and moderately thick plates in bending can exhibit
concentration of moments and shear forces either in the
vicinity of concentrated loads or owing to loads distributed
over small regions. Line loads owing to the presence of
walls or reactions of linear supports cause high internal
force gradients as well. Moreover, BEM can deal with
deflections, slopes, moments and shear forces approaching
them by using the same order polynomials. For instance,
shear forces are better evaluated using BEM when
compared with other numerical method techniques; it
depends only on the assumed boundary value approximation.
Since 1978, when the first general direct formulation
based on the Kirchhoff’s hypothesis appeared, the technique
* Corresponding author. Tel.: 00 55 162739455; fax: 00 55 162739482.
E-mail address: [email protected] (W.S. Venturini)
has had a rather large growth, being nowadays applied to
several practical engineering problems. The first works
discussing the use of boundary element direct formulation,
in conjunction with the Kirchhoff’s theory, were by Bezine
[1,2], Stern [3] and Tottenhan [4]. It is also important
mentioning some previous works, dealing with plate bending problems as well, but in the context of indirect methods
[5–7]. Those works, as well as several other more recent
publications, have pointed out the capability of the method
to model plates in bending, pointing out its accuracy and
confidence. Gou-Shu [8] and Hartmann and Zotemantel [9]
have presented interesting approaches, where displacement
restrictions at internal points and the use of hermitian interpolations were discussed in detail. Venturini and Paiva [10]
have shown several alternatives to define the algebraic
system of equations by placing the collocations either on
the boundary or at particular outside points.
The works reported before basically describe formulations of the BEM linear plate bending problem, for which
the Kirchhoff hypothesis was assumed, and their applications in engineering. After commenting those works, other
developments must be pointed out for a wider overview
concerning the general use of BEM to deal with plate bending problems. For instance, it is important to mention the
plate bending formulation proposed, in 1982, to deal with
moderate thick plates [11] by incorporating the Reissner’s
hypothesis [12]. BEM formulations related with plate
bending analysis for problems where large displacement
0955-7997/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved.
PII: S0955-799 7(98)00097-6
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E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
assumptions must be taken into account have also appeared
in several works. Kamiya and Sawary [13] have proposed a
so-called DBEM direct formulation for finite deflections of
thin plates; Tanaka [14] has developed coupled boundary
and inner domain integral equations written in terms of
stresses and displacements. Boundary element formulation
dedicated to nonlinear plate bending problems has already
appeared as well. For instance, in the work proposed by
Moshaiov and Vorus [15], one of the first nonlinear
formulations of BEM to deal with plate bending is
presented. Chueri and Venturini [16] have also studied
this problem proposing a nonlinear procedure based on the
initial moment technique to analyse reinforced concrete
slabs.
Although the field of applications of boundary methods is
nowadays rather large as demonstrated above, formulations
to treat problems characterised by plates with varying thickness rarely appeared in the literature. The importance of this
kind of formulation is not restricted only to plates that exhibit varying thickness, but can also be easily adapted to other
situations. Plate stiffness variation over the domain can be
assumed to model some complex behaviour, orthotropic
plates and stiffness deterioration owing to damage among
others. The formulation can also be adopted to analyse
plates that exhibit zoned domain very appropriate to deal
with building floor structures [17].
The first works reporting formulations proposed to deal
with varying thickness plate bending problems were by
Katsikadelis and Sapountzakis [18,19]. It is also important
to point out some other works developed following the same
strategy of the ones reported before ([20–22]). Although
proposed in the context of zoned domains, it is also convenient to mention the work developed by Venturini and Paiva
[17]. In this work the authors have derived a particular BEM
formulation to consider plate zones with different thickness,
without treating them by the classical sub-region technique.
They have avoided dividing the domain into several
constant thickness sub-regions to write the corresponding
algebraic equations.
In this work the BEM formulation based on the Kirchhoff’s hypothesis, derived to deal with the varying plate
thicknesses, was generalised. The formulation is prepared
to consider any case of plate with varying thickness, including zoned problems, where thickness or the equivalent stiffness is constant over particular zones, and the orthotropic
case. An appropriate form of the Betti’s reciprocal work
theorem was taken to derive the required integral’s representations of displacements and bending and twisting
moments, starting by assuming that the plate domain is
characterised by exhibiting varying thickness or stiffness.
According to the way chosen to derive the integral equation’s unknown values of displacements, rotations or
moments at internal points can be left inside domain integrals. Those domain integrals represent the necessary terms
to account for the stiffness variation. They can be properly
handled to reduce the number of internal unknowns and
consequently require less corresponding integral representations.
Regarding the algebraic system definition, as usual for the
plate-bending problem, only deflection integral representations are taken to write the necessary algebraic relations.
Thus, as two representations are needed for each node
resulting from the discretisation, one must write complementary integral equations as well for outside collocations.
In addition, new integral representations have also to be
added to the system owing to the extra values left inside
the domain integrals. If only displacements are left in the
domain integral term, only one extra equation is necessary
to be written for each internal node. Other appropriate relations must be written for internal nodes if rotations or bending and twisting moments are values remaining in the
domain integral terms. For any case, these domain integrals
are evaluated by approaching the domain values and the
plate stiffness over internal cells. Classical numerical examples are used to verify the accuracy of the proposed formulation.
2. Basic equations
A flat plate of thickness h, referred to a Cartesian system
of co-ordinates with axes x1 and x2 lying on its middle
surface and axis x3 perpendicular to that plane, is considered. It is assumed that the plate supports distributed load g
acting in the x3 direction on the plate middle plane, in
absence of any distributed external moments. For this
plate, the equilibrium equations are given by the following
expressions:
mij;j ⫺ qi ˆ 0
…1a†
and
qi;i ⫹ g ˆ 0
…1b†
where mij are bending and twisting moments, while qi
represents shear forces, with subscripts taken in the range
i; j ˆ {1; 2}:
Eqs. (1a) and (1b) can be joined together to give the wellknown plate bending differential equation, as follows:
mij;ij ⫹ g ˆ 0:
…2†
The plate domain is denoted by V, while its boundary is
represented by G. Over G, the following boundary conditions may be assumed: ui ˆ ui on G1 (generalised displacements, deflections and rotations) and pi ˆ p i on G2
(generalised tractions, normal bending moment and effective shear forces), where G1 傼 G2 ˆ G:
As usual, the generalised internal force × displacement
relations can be expressed in the following form:
mij ˆ D…ndij w;kk ⫹ …1 ⫺ n†w;ij †;
…3a†
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
407
referred to the boundary normal and tangential directions,
respectively; no summation is implied.
3. Integral representations
Fig. 1. Plate domain inserted into the infinite space.
qi ˆ ⫺Dw;jji
…3b†
where D ˆ Eh =…1 ⫺ n † is the plate flexural rigidity.
By replacing the internal force definition, Eqs. (3a) and
(3b), into the differential Eq. (2) and assuming constant
stiffness D, ones has,
g
…i; j ˆ 1; 2†
…4†
w;iijj ˆ
D
3
2
where w;iijj ˆ 72 w, the Laplacian operator applied on the
deflections.
For the case to be seen in this work the stiffness is
assumed varying over the plate domain, therefore Eq. (4)
is modified owing to the presence of derivatives of D(s). As
shown in Timoshenko and Krieger [23], for this case, the
differential equation written in its explicit form is given by,
D72 72 w ⫹ 2
2D 2 2
2D 2 2
7 w⫹2
7 w ⫹ 72 D72 w
2x 1 2x 1
2x 2 2x 2
22 D 22 w
22 D
22 w
22 D 22 w
⫺
2
⫹
⫺…1 ⫺ n†
2x 1 2x 2 2x 1 2x 2
2x21 2x22
2x22 2x21
ˆ g:
!
…5†
The variable thickness plate bending problem can also be
solved by assuming an appropriate initial moment field, m0ij ,
applied over the plate domain. Thus, by incorporating into
Eqs. (3a), (3b) and (4) the corresponding initial moment
field term, for the constant thickness problem, one obtains
the following modified relations, respectively:
mij ˆ ⫺D‰nw;kk dij ⫹ …1 ⫺ n†w;ij Š ⫺ m0ij ;
…6†
qj ˆ ⫺Dw;kkj ⫺ m0ij;i ;
…7†
w;iijj ˆ …g ⫺ m0ij;ij †=D:
…8†
It is worth remembering that owing to the assumed
Kirchhoff hypothesis only four boundary values are independent. Twisting moments and shear forces along the
boundary can be combined to give the well-known effective
shear force, given by:
Vn ˆ qn ⫹ 2mns =2s
…9†
where (n, s) are the local co-ordinate system, with n and s
As is well known, the direct formulation of the boundary
element method for Kirchhoff’s plates can be derived from a
reciprocity relation written in terms of bending moments
and curvatures of two independent mechanical states. The
first state is represented by the actual plate bending problem
valid over the domain V, for which curvatures w;ij …q† and
moments mij …q† at a field point q are defined, as well as the
associated boundary values: two generalised displacements,
ui(Q), and two generalised tractions, pi(Q), referred to a
boundary field point Q. The second elastic state is given
by the well-known fundamental solution w*(s, q), the
deflection at the field point q owing to an unit load applied
at a source point s. This elastic state is defined over the
infinite space of domain V∞ that contains V (see Fig. 1).
From this state, fundamental values for curvatures,
w;ij *…s; q†, and moments, mij *…s; q† can be easily derived
(see appendix). Taking into account both states and assuming that the stiffness D is constant, the following reciprocity
relation is easily found:
Z
Z
mij *…s; q†w;ij …q† dV…q† ˆ
mij …q†wij *…s; q† dV…q†: …10†
V
V
From Eq. (10), it is very easy to derive the integral representation of displacements. For a plate subjected to a transversal load g(s) applied over a particular region Vg, one can
achieve,
Z 2w
…Q† dG…Q†
Vn *…S; Q†w…Q† ⫺ Mnn *…S; Q†
C…S†w…S† ⫹
2n
G
⫹
NC
X
RC *…S; C†wC …C†
Cˆ1
ˆ
Z 2w*
…S; Q† dG…Q†
Vn …Q†w*…S; Q† ⫺ Mnn …Q†
2n
G
⫹
NC
X
Cˆ1
RC *…C†wC *…S; C† ⫹
Z
Vg
…g…q†w*…S; q†† dV…q†
…11†
where Q and q are field points taken along the boundary G
and inside the domain respectively, while s is source point
that can be placed anywhere.
In Eq. (11), w, 2w/2n, Mnn, Vn and RC are actual plate
boundary values, deflections, rotations, normal bending
moment, effective shear forces and corner reactions, respectively. The values indicated by the symbol * are the corresponding ones derived from the infinite domain problem
with the source point defined anywhere inside the domain,
outside or along the boundary. C(S) represents the wellknown free term that is equal to the unit when S is an
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E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
internal point, zero when outside defined and b C/2p for
boundary points, b C being the internal of G angle at S.
From Eq. (11) one can derive slope, moment and shear
force integral representations by differentiating it and applying appropriately the Hooke’s law. Those possible integral
representations can be properly derived for internal, boundary or external collocations when required. Any of them can
be chosen and transformed to algebraic ones to define a
convenient set of equations to solve the problem in terms
of boundary values. However, it was already verified that
Eq. (11) alone is enough to give a very stable set of algebraic
equations for plate bending analysis [10]. By adopting that
strategy one has to define two collocation points to each
discretisation node and one extra node corresponding to
each geometrical corner. It is important to remember that
one can find the second algebraic representation for each
node using any other representation obtained from Eq. (11).
Adopting deflection and slope representations is the usual
way followed to solve plate bending problems by BEM [1–
4].
Let us consider a more general case of plate bending
problems. The plate is now characterised by exhibiting variable thickness, i.e., t ˆ t(q). All relations presented in the
previous section are still valid, excepting Eq. (4) that now
must be replaced by Eq. (5). Deriving the necessary integral
representations from Eq. (5) is the way followed by
Sapountzakis and Katsikadelis [18] to formulate BEM for
varying thickness plate bending problem. Herein, we have
preferred to derive a modified reciprocity relation by consid_
ering the stiffness given in terms of rates, D…q†.
For simplicity, the stiffness can be written incrementally using a
reference value D0, i.e. D(q) ˆ D0 ⫹ DD(q). The same
steps used to derive the reciprocity relation (10) can be
followed to achieve:
Z
V
mij …q†wij *…s; q† dV…q† ˆ
⫹
Z
V
mij *…s; q†w;ij …q† dV…q†
Z DD…q†
mij *…s; q†w;ij …q† dV…q†:
D0
V
…12†
From Eq. (12) one can write a reciprocity relation applied
to plate exhibiting any kind of stiffness or thickness variation. For that, the actual stiffness D(q) can be introduced to
replace DD(q). Then, Eq. (12) becomes:
Z
V
mij …q†w;ij *…s; q† dV…q†
Z D…q†
mij *…s; q†w;ij …q† dV…q†
ˆ
V D0
…13†
where D(q) is the field point stiffness and D0 is the reference
value used to compute the fundamental solution.
As for the constant stiffness case, assuming that the stiffness at the source point D(s) is adopted as the reference
value D0, one can integrate Eq. (13) by parts to obtain the
deflection integral representation, as follows:
1 Z
2D…Q†
Mns *…S; Q†
D…Q†Vn *…S; Q† ⫹ 2
D…S† G
2s
!
2D…Q†
1
Mn *…S; Q† w…Q† dG…Q† ⫺
⫹
2n
D…S†
C…S†w…S† ⫹
Z
G
⫹
D…Q†Mn *…S; Q†
nc
X
DC
1
RC *…S; C†wC …C† ⫺
D…S†
D…S†
Cˆ1
Z
ˆ
⫹
2w
…Q† dG…Q†
2n
V
Z
G
Nc
X
Cˆ1
!
2D…q†
22 D…q†
q *…S; q† ⫹
m *…S; q† w…q† dV…q†
2
2x i i
2xi 2sj ij
!
2w*
…S; Q† dG…Q†
Vn …Q†w*…S; Q† ⫺ Mn …Q†
2n
RC …C†wC *…S; C† ⫹
Z
Vs
…g…q†w*…S; q†† dV…q†:
…14†
As it has already been said for the constant thickness case,
the integral representation of displacements derived before
for the varying thickness problem is enough to give the
necessary algebraic relations, since a proper selection of
collocation points is made.
It is worth noting that Eq. (14) exhibits only deflections as
internal values, which necessarily are unknowns of the
problem. Rotations and curvatures were eliminated by
appropriate integration by parts. This integral representation
was derived for a single region plate, where the stiffness
D(q), deflections, w(q), and rotations, 2w…q†=2x;k , were
assumed to be continuous functions over the whole domain.
In order to derive the deflection representation for a general
case, where some discontinuity may appear (for instance,
abrupt changes in the plate stiffness can be introduced
owing to the abrupt thickness variation), the required integration by parts must be properly performed. Assuming any
discontinuity case, the necessary integration by parts will
give corresponding new internal line integrals. The simple
case of a plate exhibiting several sub-regions with different
constant thickness, therefore assuming abrupt changes of
D(q), has already been shown by Venturini and Paiva
[17], where accurate results were computed using that
formulation. Instead integrals over G, the general expression
written to replace Eq. (14) will contain integrals over G ⫹
Gi, with Gi representing the abrupt stiffness change lines or
interfaces. For that case, rotations along the internal lines
cannot be eliminated, therefore two values are associated to
each node, w and 2w=2n: Then, two integral representations
must be written for each internal line node; one can choose
the corresponding slope integral representations or write
again the deflection integral equation for another collocation
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
taken along the discontinuity line. Moreover, internal corner
reactions must be taken into account at nonsmooth interface
nodes.
As only deflections are kept as internal values, in Eq.
(14), that integral representation is strongly recommended
to build the final set of algebraic equations for analysing
plates with varying thickness. In this case, only one degree
of freedom per each internal node has to be considered,
leading therefore to a rather small number of internal
unknowns. Although keeping only domain integrals characterised by having deflections as the density is a convenient
way to treat plate bending problems with varying thickness,
one can modify Eq. (13) in a different way to have other
more appropriate domain values as problem unknowns. We
could also keep rotations and curvatures as internal values,
what may be recommended for other particular situations.
For instance, if curvatures are kept in the final integral
representation, the degree of freedom owing to internal
nodes increases three times.
Three kinds of domain integrals characterised by their
densities and corresponding kernels are clearly possible to
be left in the final integral equation. Two integrations by
parts were performed leaving only deflections. By integrating by parts only once one may achieve a representation that
exhibits rotations as internal unknowns, which characterises
a second alternative formulation. The formulation that exhibits curvatures, or moments if proper modifications are
introduced in Eq. (13) before integrating it by parts, is easily
achieved. The reciprocity relation has to be properly rearranged replacing curvatures by moments by using the
Hooke’s law, before transforming it into an integral equation. It is worth noting that the second domain integral in the
right hand side of Eq. (13) is already written in terms of
curvatures, which makes it easy to keep this density as an
internal value.
In order to complete the necessary integral relations for
the analysis of plate bending problem with variable thickness, one can derive, from Eq. (14) after making C(S) ˆ 1,
moment and shear force integral representations for internal points by carrying out its second and third derivatives
and applying appropriately the definitions given by Eqs.
(3a) and (3b), respectively. One must pay attention how
to carry out those derivatives; for any case, free terms will
be formed when performing derivatives of strong and
hypersingular integral terms. By differentiating Eq. (14)
twice and introducing the definitions given by Eq. (3a),
one obtains,
mij …s† ˆ ⫺
Z
G
D…Q†V nij *…s; Q† ⫹ 2
2D…Q† M nsij *…s; Q†
2s
!
2D…Q† M nij *…s; Q† w…Q† dG…Q†
⫹
2n
⫺
Z
nij *…s; Q† 2w …Q† dG…Q†
D…Q†M
2n
G
⫺
Nc
X
409
D…C†R Cij *…s; C†wC …C†
Cˆ1
⫹
Z G
Vn …Q†wij *…s; Q† ⫺ Mn …Q†
× dG…Q† ⫹
Nc
X
2wij *
…s; Q†
2n
RC …C†wCij *…s; C†
Cˆ1
⫹
Z
Vs
g…q†wij *…s; q† dV…q†:
…15†
Analogously, the shear force integral representation is
obtained from Eq. (3b) together with the third derivative
of the deflection integral Eq. (14), as follows:
qi …s† ˆ ⫺
Z
D…Q†Vni *…s; Q† ⫹ 2
G
2D…Q† M nsi *…s; Q†
2s
!
2D…Q†
Mni *…s; Q† w…Q† dG…Q†
⫹
2n
⫹
⫹
Z
G
D…Q†Mnij *…s; Q†
Nc
X
Cˆ1
2w
…Q† dG…Q†
2n
D…C†RCij *…s; C†wC …C† ⫹
Z
G
Vn …Q†wi *…s; Q†
!
Nc
X
2wi *
…s; Q† dG…Q† ⫹
RC …C†wCk *…s; C†
⫺ Mn …Q†
2n
Cˆ1
!
Z
g…q†wi *…s; q† dV…q†:
…16†
⫹
Vs
Owing to the order of singularities exhibited by the
domain integrals, no free term arises when performing
the required derivatives. Moreover, as Eqs. (15) and (16)
were obtained for the particular case of continuous variations of stiffness and rotations, no domain integral is left.
Note that those integral representations are only valid for
collocations defined at cell centres (after assuming the
domain is divided into cells), to avoid dealing with discontinuities of the second and third deflection derivatives. All
kernels appearing in Eqs. (15) and (16) are obtained by
differentiating the corresponding values in Eq. (14) and
applying the internal force definitions given in Eqs. (3a)
and (3b). For convenience, those kernels are given in the
appendix.
The alternative of obtaining integral representation by
performing one integration by parts, leading therefore a
domain term with internal rotations, is not analysed here.
We prefer discussing the third possibility, where the final
representation exhibits a domain integral with curvature
density. One can deal directly on the last integral, in Eq.
(12), which gives the stiffness variation influences, keeping
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E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
therefore curvature as problem unknowns. In this case, one
has to write the curvature integral representation to have
equal numbers of equations and unknowns.
Although working with curvatures are possible as related
before, a similar representation is obtained by modifying
expression (12) to have in the last integral an initial moment
field, m0ij …s†, written in terms of the stiffness variation D(s)/
D0. This alternative leads to the well-known BEM plate
bending formulation used for nonlinear analysis [16]. In
this case, the final moment mij …s† is represented by two
parts as usual, the elastic values, meij …s†, and the initial
moment, m0ij …s†, as follows:
mij …s† ˆ meij …s† ⫺ m0ij …s†
V
wij *…s; q†g…q† dV…q† ⫺
Z
V
e;ijk` *…s; q†m0k` …q† dV…q†
p
…q† ⫺ m0ij …q†;
⫺ gijk` …q†Mk`
qb …s† ˆ ⫺
"
Z
⫺
G
Nc
X
#
2w
…Q† dG…Q†
Vnb *…s; Q†w…Q† ⫺ Mnb *…s; Q†
2n
Rcb *…s; Q†wc …Q† ⫹
cˆ1
⫹
⫹
⫺
G
Z
Vs
Z
V
Nc
X
wcb *…s; Q†Rc …Q†
cˆ1
"
Z
…20†
#
2wb *
…q; P†Mn …P† dG…P†
wb *…s; Q†Vn …Q† ⫺
2n
wb *…q; p†g…p† dV…p†
e;bk` *…q; p†m0k` …p† dV…p† ⫺ q pb …q†:
…21†
V
⫺
Z
V
w;ij *…s; q†m0ij …q† dV…q†:
…18†
By integrating by parts Eq. (18) one transforms it into the
standard integral equation form for deflections, as follows:
Z 2w
…Q† dG…Q†
Vn *…S; Q†w…Q† ⫺ Mn *…S; Q†
C…S†w…S† ⫹
2n
G
⫹
Z
…17†
which allows us to recover the basic Eqs. (6)–(8).
After modifying appropriately the last integral in Eq.
(12), one can write the standard reciprocity relation usually
adopted to deal with temperature or nonlinear problems
based on the initial moment technique, as follows [16]:
Z
Z
mij …q†w;ij *…s; q† dV…q† ˆ
mij *…s; q†w;ij …q† dV…q†
V
⫹
Nc
X
RC *…S; C†wC …C†
Note that for this case a domain integral is left in the
moment integral representation, Eq. (20); this term exhibits
singularity of order 1/r 2, owing to the kernel eijk` *…q; p†,
requiring therefore proper care to be evaluated. All the
necessary kernels and the free term gijk` …s†, appearing in
Eqs. (20) and (21), are given in the appendix as well.
Usually Eqs. (19)–(21) are the integral representations
necessary to formulate nonlinear numerical algorithms
[16]. In this work, however, they are properly arranged to
build appropriate schemes to deal with varying thickness
plate bending problems, as shown in the next section.
Cˆ1
ˆ
Z G
⫹
Nc
X
RC …C†wC *…S; C† ⫹
Cˆ1
⫺
2w*
…S; Q† dG…Q†
2n
Vn …Q†w*…S; Q† ⫺ Mn …Q†
Z
V
Z
Vs
4. Plate bending BEM algebraic equations
…g…q†w*…S; q†† dV…q†
w;kl *…S; q†Mkl0 …q† dV…q†:
…19†
Similar to the previous case, Eq. (19) can be differentiated
to give moment and shear force integral representations, as
follows:
#
"
Z
2w
…Q†
Vnij *…s; Q†w…Q† ⫺ Mnij *…s; Q†
mij …s† ˆ ⫺
2n
G
⫺
Nc
X
cˆ1
wcij *…s; Q†Rc …Q† ⫹
"
Z
G
#
2wij *
…s; Q†Mn …Q† dG…Q†
⫺
2n
wnij *…s; Q†Vn …Q†
As usual for any BEM formulation, the integral representations (14)–(16), or Eqs. (19)–(21) when the initial stress
formulation is adopted, can be transformed into algebraic
expressions after discretising the boundary into elements,
the domain into cells and approaching appropriately boundary and internal values. For the present case, the plate
boundary was discretised into geometrically linear elements
over which the boundary values were approximated
by quadratic shape functions. The internal values are
approximated as well (only deflections or initial moments
are approximated, depending on the group of integral equations taken), now using linear shape functions defined over
triangular cells.
The approximations of the boundary and internal values
enable us to write algebraic representations of deflections,
rotations, moments and shear forces for any collocation
point taken inside the domain, along the boundary or even
out of the plate, using the integral expressions derived in the
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
411
Fig. 2. Boundary discretisation and collocation points.
last section. After selecting an appropriate number of
algebraic equations, one can assemble a convenient set of
equations to solve the problem in terms of boundary values.
Note that there are four boundary values at each node
defined by the discretisation, plus two extra values at each
corner to take into account corner reactions. As it has
already been observed in previous studies [10], it is better
to use only the deflection representations to define the final
set of equations taking the collocation points either along
the boundary or outside the body. It is worth noting that
algebraic equations for collocations defined inside the
domain are also required. For the first group of Eqs. (14)–
(16), one needs relations corresponding to the number of
internal deflections unknown. Equations to compute
moments and shear forces can also be written if those values
were required. For the second group of equations, only
moment algebraic expressions at internal points are necessary to complete the total amount of relations to solve varying thickness plate bending problems. Deflection and shear
force algebraic equations at internal points are written only
for the result output.
Although other alternatives concerning the collocation
point selection may be proposed to achieve the appropriate
algebraic representations, two strategies were tested and
proved to give accurate results. The first scheme consists
of defining one collocation point coincident with the boundary node (Q) and another one (A) externally placed at a
convenient small distance d from the boundary. For the
second strategy two outside collocation points (A1 and
A2) are adopted, as shown in Fig. 2. Defining two independent boundary collocation points for each node is possible,
but this strategy leads to singular systems of equations; it
can be adopted, however if at least one external collocation
is defined to guarantee that the algebraic equations are linearly independent.
As a rule, continuous elements are adopted to keep
boundary value continuity between boundary elements,
but discontinuities are assumed at corners and at other
points where boundary conditions may change abruptly.
For those cases, it is convenient to define five collocation
points associated with five unknowns (ten boundary values:
w, 2w/2n, Vn and Mn related to the adjacent element ends
plus wc and Rc, the corner values). Four collocations points
are related with the element end unknowns, while the extra
collocation makes corner reactions and displacements independent values.
Fig. 2 illustrates the strategies described before to select
the collocation points. The distance di is given by:
di ˆ ai `m , `m being the average adjacent element length
and ai a parameter defined within the range {0.1, 0.5} for
both collocation point selection schemes described before.
A sub-element technique is always required to perform
accurately the integrals along the adjacent elements.
In order to perform the integrals over internal cells, they
were transformed to its boundary and the sub-element
scheme was also adopted to assure their accurate evaluation.
Let us first deal with the varying thickness problem using
the integral representation that contains the deflection
domain term, Eq. (14). After discretising the plate boundary
and dividing its domain into cells over which deflections
will be approximated, one can write the algebraic form of
Eq. (14). Writing it for the necessary number of collocation
points, taken along the boundary and out of the domain (or
only out of domain according to the second scheme
described before), one finds the BEM standard matrix equation. This algebraic representation, now modified by terms
resulting from the domain integrals which give the stiffness
variation influences, is written in the following matrix form:
‰Hb ⫹ Sbb ŠUb ⫹ Sbi Ui ˆ Gb Pb ⫹ Tb
…22†
where the matrices S are obtained by integrating the domain
term related with boundary (Ub) and internal (Ui) deflections, respectively; the subscripts b and i indicate boundary
and internal values, H and G are the usual matrices for BEM
formulations and T represents the domain load influences.
Eq. (22) contains domain (deflections only) and boundary
412
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
relating the boundary values is found [16],
HU ˆ GP ⫹ T ⫹ EM0
…27†
where U and P are generalised boundary displacement and
traction nodal values, T represents the loading and M 0 gives
the initial moment values at internal and boundary nodes; H
and G are the standard square matrices achieved by integrating all boundary elements, while the matrix E is
obtained by performing the domain integrals over all cells.
The moment and shear representations, Eqs. (20) and
(21), can also be written into their algebraic form, using a
similar procedure,
Fig. 3. Moment × curvature relations for different stiffnesses.
(deflections, rotations, normal bending moments and
effective shear forces) unknowns and algebraic relations
referred to boundary nodes only. In order to complete the
required number of relations, similar algebraic equations
must be written for all internal nodes defined by the cell
discretisation. Representing those equations in its matrix
form one has:
M ˆ ⫺H 0 U ⫹ G 0 P ⫹ T 0 ⫹ …E 0 ⫺ I†M0 ;
…28†
Q ˆ H 00 U ⫹ G 00 P ⫹ T 00 ⫹ E 00 M0
…29†
with I being the identity matrix.
As usual in boundary element formulations [16,24,25],
the matrix Eqs. (27)–(29) can be solved and conveniently
arranged to express the solution in terms of boundary
values, moments and shear forces, as follows:
…23†
X ˆ L ⫹ RM0 ;
…30a†
where the same symbols used in Eq. (22) were adopted,
modifying the subscript when necessary.
Joining together Eqs. (22) and (23) gives,
"
#( ) "
#
( )
Tb
Gb
Sbi
Ub
Hb ⫹ Sbb
ˆ
: …24†
Pb ⫹
Hi ⫹ Sib I ⫹ Sii
Ui
Gi
Ti
M ˆ N ⫹ SM0 ;
…30b†
Q ˆ N 0 ⫹ S 0 M0 :
…30c†
Ui ˆ ⫺Hi Ub ⫹ Gi Pb ⫹ {Sib Sii }{Ub Ui } ⫹ Ti
In Eqs. (22)–(24), corner values, reactions and deflections, are being considered together with the corresponding
boundary vector indicated by the subscript b. Sub-blocks are
used only to identify boundary and internal values.
Eq. (24) is the algebraic representation of plates in bending for which the variable thickness or stiffness was
assumed. After solving this set of equations, one has
computed all boundary values and internal deflections
required to continue the analysis. When required those
computed values are used in the discretised forms of Eqs.
(15) and (16) to compute moments and shear forces, i.e.
M ˆ H 0U ⫹ G 0P ⫹ S 0U ⫹ T 0;
…25†
Q ˆ H 00 U ⫹ G 00 P ⫹ S 00 U ⫹ T 00
…26†
where, U and P are vectors now used to represent the
complete set of values, either computed at boundary nodes
or at internal points.
The alternative procedure to analyse the varying thickness plate bending problem is given by the integral representations obtained by assuming an initial moment field
applied on plate surface, i.e., Eqs. (19)–(21). Similarly to
the previous case, after discretising the boundary into
elements and the domain into cells, and approaching the
problem values, the following set of algebraic equations
In Eqs. (30a)–(30c) L, N and N 0 give the elastic solution
owing to prescribed loads acting along the boundary or over
the domain, while the M 0 effects are represented by the
matrices R, S and S 0 .
In order to adopt those algebraic equations to solve varying
thickness plate bending problem, let us consider a sub-region
of a plate with stiffness D inside a domain characterised by
the stiffness D0. The moment value M can be obtained by
correcting the value M e (computed by assuming the plate
reference stiffness D0 for the whole domain) by subtracting
an initial moment M 0, i.e. M ˆ M e ⫺ M 0, as schematically
illustrated in Fig. 3. According to the local elastic relations
written for a particular point, the value M 0 can be written in
Fig. 4. Plate definition.
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
413
Fig. 5. Deflection convergence at the plate centre (Example 1).
terms of M or M e, i.e., M 0 ˆ (D0/D ⫺ 1)M. Replacing this
relation into Eq. (31) gives:
M ˆ …I ⫺ Sa†⫺1 N
some numerical examples, demonstrating to be an alternative procedure, but more dependent on the discretisation
adopted.
…31†
where a ˆ (D0/D ⫺ 1).
Although this procedure is general and can be adopted to
solve plates with any kind of thickness or stiffness variation,
it requires the use of three internal values for each node,
which increases too much the size of the algebraic set of
equations required to be solved. In fact, this procedure is
equivalent to the scheme that works with curvatures. Moreover, approaching moments means that more values are
approximated to solve the problem that might lead to
worse numerical solution in comparison with the previous
case. For this reason, we will concentrate our numerical
examples to illustrate the first scheme, for which only one
value is taken at each defined internal node. The second
procedure was implemented as well and used to solve
5. Numerical example
In order to illustrate the formulation presented in the
previous sections several plate-bending problems with
different thickness variation and several boundary conditions were analysed. Before presenting the examples and
consequently obtaining the corresponding numerical results,
it is important to mention that an extensive study was
carried out concerning the position of the collocation points.
It was found that the outside collocation position recommended should be defined at a position less than 0.5 to
assure accurate results. For instance, all numerical analyses
presented here were carried out making a ˆ 0.25.
The first example to be analysed consists of a square plate
Fig. 6. Moment mx convergence at the plate centre (Example 1).
414
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
Fig. 7. Moment my convergence at the plate centre (Example 1).
exhibiting varying stiffness. Two opposite sides are
assumed simply supported, while the other two have no
deflection or slope value prescribed. A uniform distributed
load g acting over the whole plate, is the only one considered for this analysis. For this plate, the elastic modulus E is
assumed, while the Poisson ration, n , was taken to be equal
to 0.3. The side length is taken equal to a and the stiffness is
assumed to vary from D at the left side to D/2 at the right
side. For simplicity, no variation is assumed along the opposite direction (Fig. 4).
This problem was analysed using the two alternative
schemes presented in this paper. Many boundary and internal discretisations were adopted to verify the convergence
of each proposed scheme. The first analysis that we have
carried out is concerning the convergence of the results
against the degrees of freedom (DOF). We have started
with poor meshes, for instance using only four boundary
elements and we finished the analysis using 400 DOF.
Even for the poorest mesh using only one internal point
the results were satisfactory when the approach that
employs internal displacement unknowns was adopted. In
contrast, the initial moment formulation technique
accomplished with poor meshes does not allow achieving
acceptable results. For both procedures the numerical
answers go quickly to very accurate results when the
mesh is refined. In order to verify the accuracy of the
BEM results, the problem was also solved using a finite
element code based on the DKQ elements [26], whose
results were used for comparison.
Figs. 5–7 show the result improvement when the mesh
(given by the adopted boundary and internal discretisations)
is uniformly refined. The BEM solutions, computed by
using the two proposed schemes, are compared with the
DKQ finite element results [26]. In those figures, captions
’’This work-D‘‘ and ’’This work-M‘‘ are used to identify
results obtained by using the two proposed schemes, the first
one dealing with internal deflections and the other that uses
initial moment field, respectively. Fig. 5 illustrates the
Fig. 8. Deflection convergence analysis at the plate centre (Example 2).
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
415
Fig. 9. mx convergence analysis at the plate centre (Example 2).
deflection improvement at the plate centre, while Figs. 6 and
7 exhibit how bending moments, mx and my, converge at the
same point as well. As already mentioned all these results
show that the BEM formulations are very accurate, even
when using a rather coarse mesh. As expected, it worth
noting that numerical moments and defections exhibit
more or less the same accuracy as usual for boundary
elements. Therefore, the domain discretisation and approximation, required for those schemes, do not change this
characteristic of the method.
The second example given to illustrate the proposed
BEM formulations was only analysed by using the first
technique, i.e., the one based on internal deflections only.
This technique has been shown to be more accurate,
mainly for coarse meshes and requires much less internal
degrees of freedom.
The same square plate given in Fig. 4 was analysed
assuming the stiffness varying along both directions. The
stiffness was linearly defined based on three thickness
values: 0.2 and 1.0 at the two corners in the left side and
0.5 at the first corner of the right side. For simplicity, we
assume the stiffness varying following that defined plane.
For this example, the convergence of deflections and
bending moments was analysed as well, now plotting
BEM results up to 1000 DOF (see Figs. 8–10). As it can
be seen no result change can be seen in the BEM solution
after 400 DOF. These results have just stressed what was
observed in the first example: The proposed BEM scheme to
analyse varying thickness plate bending is accurate and
quickly converges to the stable solution.
In order to complete the numerical results of this
example, deflections and bending moments along the central
line in the x direction were computed as well. Those results,
illustrated in Figs. 11 and 12, respectively, were obtained
using a rather fine mesh when very little answer variation
was observed. These results were computed assuming
several selected boundary conditions indicated by the
symbols: S – simply supported side; C – Clamped side; F
– Free side. For instance, the symbol SSSC means three
simply supported sides plus one clamped side.
6. Conclusions
A general BEM formulation to deal with Kirchhoff’s
plate bending problems exhibiting thickness or stiffness
Fig. 10. mx convergence analysis at the plate centre (Example 2)
416
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
Fig. 11. Central axis deflections (Example 2).
variable was developed. Two alternative schemes to deal
with this problem were discussed, basically showing what
are the possible transformations that can be performed in the
domain integrals to have convenient domain degrees of freedom. After successfully implementing these two schemes,
examples were analysed to illustrate the formulations.
Those examples and several others performed to check the
formulation show that the proposed approaches are very
efficient and accurate. In particular, the scheme dealing
only with unknown deflections at internal points has
shown to be particularly very accurate, therefore recommended for study of varying thickness plate bending
problems. This procedure is accurate enough even for
very coarse meshes, emphasising that the domain
approximations have little influence in the final numerical solutions. The scheme based on the initial moment
technique, although requiring finer meshes in comparison with the first scheme, is also accurate for coarse
meshes. For both models, it was observed that the
numerical solutions are not too sensitive to internal
discretisations, which allow the use of domain meshes
given by few internal points, therefore leading to quite
small matrices, without losing significantly the accuracy. It is also important to observe that the way
adopted to formulate the problem can be easily
followed to derive other orthotropic plate bending
problems.
Appendix
The displacement fundamental value computed at the
field point q owing to a unit load applied at the source
point, is expressed by,
w*…s; q† ˆ r2 …`nr ⫺ 1=2†=…8pD0 †
…A:1†
where r ˆ r(s, q) is the distance between the points s and q,
and D0 is a reference plate stiffness that, for convenience, is
taken to be equal to the plate source point stiffness D(s).
By differentiating Eq. (A.1) and applying the appropriate
their definitions one can find the other required fundamental
values:
2w*
r
ˆ
ln…r†…r;i ·ni †;
2n
4pD0
Mn * ˆ ⫺
…A:2†
i
1 h
…1 ⫹ n† ln…r† ⫹ …1 ⫺ n†…r;i ·ni †2 ⫹ n ; …A:3†
4p
Fig. 12. Central axis moment mx. Example 2.
E.W.V. Chaves et al. / Engineering Analysis with Boundary Elements 23 (1999) 405–418
Mns * ˆ ⫺
1
…1 ⫺ n†…r;i ·ni †…r;j ·sj †;
4p
i
r; ·n h
Vn * ˆ i i 2…1 ⫺ n†…r;j ·sj †2 ⫺ 3 ⫹ n
4pr
i
1⫺nh
1 ⫺ 2…r;i ·si †2 ;
⫹
4pR
RC * ˆ Mns * ⫹ ⫺Mns *⫺
…A:4†
…A:5†
…A:6†
where n and s represent the outward and tangent unit
vectors, respectively, and the superscripts ⫹ and ⫺ are
used to indicate values taken before and after the corner.
All kernels found when deriving moment and shear force
representations, Eqs. (15) and (16), are easily computed
following the internal force definitions, Eqs. (3a) and (3b).
For a symbolic fundamental value F* the kernels
nsij *; V nij * and R Cij * are given by the following
nij *; M
M
expression:
!
22 F*
22 F*
…s; Q† ⫹ …1 ⫺ n†
…s; Q† :
F ij *…s; Q† ˆ ⫺ ndij
2xv 2xv
2x i 2x j
…A:7†
Similarly, the kernels appearing in Eq. (16) are given by,
!
2
22 F*
…s; Q† :
…A:8†
F i *…s; Q† ˆ ⫺
2xi 2xk 2xk
The remaining kernels in both equations wij *; wi *,
wCij * and wCi * are directly obtained from the fundamental
value w*, assuming the stiffness D equal the unit and applying formulas (A.7) and (A.8). Their final expressions can be
conveniently derived:
!
22 w*
22 w*
…s; Q† ⫹ …1 ⫺ n†
…s; Q† ;
wij *…s; Q† ˆ ⫺ ndij
2xv 2xv
2x i 2x j
…A:9†
2
wi *…s; Q† ˆ
2x i
!
22 w*
⫺r;i
:
…s; Q† ˆ
2xk 2xk
2pr
…A:10†
The kernels presented in the initial moment integral
representations, Eqs. (20) and (21), can be easily derived
using the second and third derivatives of the fundamental
values Vn *…s; Q†; Mn *…s; Q†; Rc *…s; C†; wc *…s; C†; w*…s; C†;
2w*…s; Q†=2n and w;k` *…s; q†; applying the operator (A.9)
and (A.10), now multiplied by the source point stiffness
D(s).
The free term is Eq. (20), appeared owing to the differentiation of a singular integral, is given by,
i
1h
gijk` …q† ˆ ⫺ …1 ⫺ n†…dik d`j ⫹ dkj di` † ⫹ …1 ⫹ 3n†dij dk` :
8
…A:11†
417
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