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  • 8/14/2019 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

    1 3

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    630 Comput Mech (2012) 49:629641

    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

    1 3

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    Comput Mech (2012) 49:629641 631

    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.

    1 3

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    632 Comput Mech (2012) 49:629641

    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

    1 3

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    Comput Mech (2012) 49:629641 633

    =

    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.

    1 3

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    634 Comput Mech (2012) 49:629641

    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.

    1 3

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    Comput Mech (2012) 49:629641 635

    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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    2

    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.

    1 3

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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].

    1 3

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    Comput Mech (2012) 49:629641 639

    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.

    1 3

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    Comput Mech (2012) 49:629641 641

    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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