CC BY
38
Математика. Физика
УДК 539.3
HYBRID SOLID-PLATE QUADRILATERALS. AN ASSESSMENT AND NEW DEVELOPMENTS G.M. Kulikov, S.V. Plotnikova
Department of Applied Mathematics and Mechanics, TSTU
Key words and phrases: assumed natural strain method; four-node solid-plate element; hybrid finite element models.
Abstract: On the basis of the first-order solid-plate theory, proposed in earlier authors’ papers, the hybrid stress, strain and stress-strain quadrilateral ANS four-node plate elements are developed. All hybrid solid-plate elements are based on the unified technique that gives an opportunity to assess their advantages and disadvantages. For instance, a hybrid stress-strain quadrilateral ANS element permits the analytical integration, which leads to a very simple structure of its stiffness matrix. At the same time the hybrid stress ANS quadrilateral exhibits a superior performance in all plate problems considered but requires a numerical large size matrix inversion.
1. Introduction
A displacement-based four-node solid-plate element formulation based on the assumed natural strain (ANS) method has been proposed in [1]. To improve the computational efficiency of the displacement-based ANS low-order plate/shell elements, a hybrid method can be applied. This method is based on the robust finite element formulation pioneered by Pian [2]. In such a formulation the displacements on the element boundary are assumed to provide displacement compatibility between elements, whereas internal stresses are assumed so as to satisfy the differential equilibrium equations. The Pian’s work was originally based upon the principle of the stationary complementary energy. Later, an alternative assumed stress method was proposed by applying the Hellinger-Reissner variational principle that simplifies the evaluation of the element stiffness matrix [3].
However, herein we do not use this terminology referring to Gallagher’s proposal (see paper [4]), where it is said that “the hybrid method in structural mechanics is defined at the one which is formulated by multivariable variational functional, yet the resulting matrix equations consist of only the nodal values of displacements as unknown”. Independently, the hybrid strain [5] and hybrid stress-strain [6] shell elements were developed. The former is based on the modified Hellinger-Reissner functional in which displacements and strains are utilized as fundamental shell unknowns, whereas the latter departs from the Hu-Washizu functional depending on displacements, stresses and strains.
In the present paper, all three hybrid ANS four-node plate elements are studied by using a unified technique. This allows one to assess their advantages and disadvantages and to compare them with a displacement-based ANS plate element [1]. In this context, we notice that a hybrid stress-strain ANS plate element permits the analytical integration of some matrices that yields the simple analytical matrix inversion [1]. At the same time the hybrid stress and hybrid strain aNs plate elements exhibit an excellent performance in all benchmark problems considered but require expensive numerical matrix inversions.
2. Hybrid stress-strain ANS solid-plate element
To develop the hybrid stress-strain finite element formulation, we have to invoke the Hu-Washizu variational principal in which displacements, strains and stresses are utilized as independent variables. Inserting distributions of displacements, displacement-dependent and displacement-independent strains through the thickness of the plate [1] into the 3D Hu-Washizu functional, one finds
Jj[sET (DE - H) + 5HT (E - E) + 5ETH - 5uTP]dx1 dx2 - 5Wf1 = 0:
(1)
u = [Ml U2 M+ M3 u+
P = [-Pl p+ -p2 p2 -p3 p3+] > (2)
]T, P=[
E = [sl1 S+1 s22 s+2 2s12 2s+2 2s13 2s+3 2s 23 2s+3 ^33]
E = |EnET1 E-2 E+2 2E
42 2E12 2E13 2E13
2E1+3 2E
23 2E+3 E33
H = [h- H+1 H22 H+2 H-2 H+2 H-3 H+3 H~-3 H +3 H33]
where u~ and ut are the components of displacement vectors u- and u of the bottom and top planes; p- and p+ are the components of surface traction vectors p-and p + applied to the bottom and top planes; is the work done by external loads acting on the boundary surface E e[; sap, ea3 and e33 are the in-plane, transverse shear and transverse normal components of the displacement-dependent strain tensors; Eo±p, Ea±3 and E33 are the in-plane, transverse shear and transverse normal
components of the displacement-independent strain tensors; Hap, H03 and H33 are
the components of the stress resultant tensors; D is the constitutive stiffness matrix of order 11 x 11 [7] whose components are found in accordance with the simplest remedy of S.W. Lee et al. [8] to prevent thickness locking; indices i, j take the values 1, 2 and
3, whereas indices a, p take the values 1 and 2. We refer to (1) as a Hu-Washizu variational equation. A short discussion on that is presented in Appendix A.
For the isoparametric quadrilateral four-node solid-plate element (Fig. 1) the position vector in the initial configuration and the displacement vector are approximated according to the standard C° interpolation
Q
el
]
T
T
x = Ix1 x2 0
xr = [xr xr? 0]
ur = u1r U'2r u1r u3r u3r
(3)
(4)
where ur are the displacement vectors of the element nodes; Nr fe1,1?) are the
bilinear shape functions of the element; |a are the natural coordinates; the index r runs from 1 to 4 and denotes a number of nodes.
In order to avoid shear locking, we employ the ANS method using its non-conventional treatment [1] that leads in conjunction with displacement interpolation (4) to
(5)
where U is the displacement vector of the solid-plate element; B is the strain-displacement transformation matrix of order 11 x 24 .
In order to fulfill a patch test, the assumed stress resultants and displacement-independent strains are interpolated throughout the element following ideas [3, 9]
H = PhV V = [1 V2 ••• V22^ (6)
E = Pe9 9 = [1 92 ••• 922]T, (7)
where PH and PE are the matrices of order 11 x 22 depending on natural coordinates
^ [1].
Using interpolations (3) - (7) in the variational equation (1), the following equilibrium equations are obtained:
Q V = GE9, Q9 = RhU,
Here, F is the force vector and
R h V = F.
1 1
1 1
-1-1
GE = J JPjTDPeAd^d2, RH = J JPjTBAd^d2
11
Q = JJpH Pe AdE,1d^2
(8)
(9)
-1-1
-1-1
where A is the determinant of the Jacobian matrix defined as
A = co + q^1 + c2^2,
c0 = ^[1 - x3 W - x[ )- (x2 - x4K2 - x3
1 [( 1 1 V 2 2 ) ( 1 1 )( 2 2
C1 = ^ LvX1 - X2 Ax3 - x4)- \x3 - x4 Ax1 - x2
1 [( 1 1 V 2 2 ) ( 1 1)( 2 2
c2 = ^ L\x1 - x4 Ax2 - x3j- \x2 - x3 Ax1 - x4
(10)
r
Eliminating assumed stress and strain parameter vectors v and 9 from elemental equations (8), we arrive at the governing equations
K EH U = F, (11)
where KEH denotes the element stiffness matrix given by
K EH = RHQ-1G eQ-1R h. (12)
Because of a simple structure of the matrix Q, its inversion can be readily fulfilled in a closed form [1]. Thus, no expensive numerical matrix inversion is needed if one uses the hybrid stress-strain method.
3. Hybrid stress ANS solid-plate element
The hybrid stress finite element formulation is based on the Hellinger-Reissner variational principle, which can be expressed in our notations (2) as
[[[sHT (( - D-1H)+8ETH -8uTp]dx1dx2 -SW^ = 0. (13)
Qel
We refer to (13) as a Hellinger-Reissner variational equation (see Appendix A).
The use of interpolations (3) - (6) in the mixed variational equation (13) leads to equilibrium equations
R h U = G hV, R hV = F, (14)
where
11
Gh = J [pHD-1PhAdE/dE2. (15)
-1-1
Due to the fact that stress resultants (6) are interpolated discontinuously across element boundaries, the assumed stress parameter vector v can be eliminated at the element level and we arrive at the pure displacement-based problem
K H U = F, (16)
where KH denotes the element stiffness matrix given by
K H = R Hg H1R h. (17)
It is seen that the hybrid stress plate element formulation requires a numerical inversion of the matrix of order 22 x 22 .
4. Hybrid strain ANS solid-plate element
The hybrid strain finite element formulation is based on the modified Hellinger-Reissner functional in which displacements and strains are utilized as fundamental shell unknowns. Therefore, taking into account the distribution of strains through the
thickness of the shell [1] and using constitutive equations into the aforementioned 3D mixed functional, we represent the 2D modified Hellinger-Reissner variational equation in the following form:
[[[sEtD(E-E) + SETDE-SuTp]dx1dx2 -SW^ = 0. (18)
Qel
Substituting interpolations (3) - (5) and (7) in the mixed variational equation (18) and recalling notations (9), we derive the elemental equilibrium equations
R EU = G E9 R E^ = F, (19)
where
11
RE = J [PjTDBAdE1dE2. (20)
-1-1
Since the assumed strain interpolations (7) are discontinuous at the element
boundaries, a static condensation on the element level yields the element stiffness
matrix
K E = R EG E1r E. (21)
Thus, again a pure displacement-based problem
K E U = F (22)
has to be solved.
One can see that the hybrid strain plate element formulation also leads to the numerical inversion of the matrix of order 22 x 22 .
Remark. Using a link between transverse components of the displacement-dependent strain tensor and following a technique [10], one can derive four coupling conditions for the transverse components of the displacement-independent strain tensor. These conditions imply that only 18 assumed strain parameters are independent of 22 ones introduced by approximation (7). As a result, the elemental stiffness matrices of all hybrid methods developed, namely, KEH, K H and K E have six, and only six, zero eigenvalues as required for satisfaction of the general rigid-body motion representation, since 24 displacement degrees of freedom are introduced. Note that integrals are calculated by applying a Gauss numerical integration scheme with 2 x 2 integration points.
5. Numerical examples
The performance of the proposed quadrilateral four-node solid-plate elements is evaluated with several problems extracted from the literature. A listing of these elements and the abbreviations used to identify them are contained in Table 1.
5.1. Cook’s membrane problem
A trapezoidal plate is clamped on one side while the opposite side is subjected to a distributed in-plane load as shown in Fig. 2. This test is an excellent ability to verify a proper representation of the membrane dominated stress state with skewed meshes.
I MNorm
Table 2 lists the normalized in-plane tip displacement of the midplane U2 ) .
For this purpose a converged finite element solution of 23.91 [9] has been used. As turned out, the SPQ4e element finished with the lowest rank among all mixed solid-plate elements developed.
5.2. Simply supported square plate under central load
This problem is invoked to test the effect of mesh distortion on the performance of proposed solid-plate elements. A quarter of the simply supported square plate subjected to a concentrated load is modeled by a coarse 2 x 2 mesh. The geometrical and
Table 1
Listing of quadrilateral four-node solid-plate elements
Name
Description
SPQ4
SPQ4ct6
SPQ4ct
SPQ4e
Solid-plate quadrilateral based on the displacement-based ANS formulation [1]
Solid-plate quadrilateral based on the consistent hybrid stress-strain ANS formulation [1]
Solid-plate quadrilateral based on the hybrid stress ANS formulation (section 3)
Solid-plate quadrilateral based on the hybrid strain ANS formulation (section 4)
Fig. 2. Cook’s membrane problem:
h = 1; E = 1; v = 0.33; F = 1
Normalized in-plane tip displacement of trapezoidal plate
Table 2
Mesh SPQ4 SPQ4CT6 SPQ4ct SPQ4e Simo [9]
2 x 2 0.4954 0.7332 0.8835 0.5555 0.883
4 x 4 0.7654 0.9126 0.9627 0.8120 0.963
8 x 8 0.9234 0.9760 0.9906 0.9421 0.991
16x16 0.9798 0.9948 0.9987 0.9853 0.999
mechanical data of the problem are displayed in Fig. 3. The mesh is distorted by moving the inner node along the diagonal of the upper right square. In such situation a Jacobian may be written according to (10) as
Л -1 f1-A + -A^1 + -A^2 41 2 2
(23)
Taking into account (23), one can represent Jacobian values at the crucial Gauss point as a function of the parameter A:
ЛІ -
1
1
positive for A - 0.6,
zero for A - 73/ (3+1)- 0.6340,
negative for A - 0.7.
M
Table 3 displays the transverse central displacement of the midplane u3
normalized with respect to the analytical solution [11] of 0.01160Fa2 /d based on the Kirchhoff plate theory. It is seen that all hybrid solid-plate elements perform excellently even in the case of a negative Jacobian. As mentioned in [9], it is an unacceptable situation for practical implementations of course.
Fig. 3. Simply supported square plate under central load:
a = 4; h = 0.004; E = 1.092x106; v = 0.3; F = 10-5
Table 3
Normalized transverse center point displacement of square plate
A SPQ4 SPQ4CT8 SPQ4ct SPQ4e Simo [9]
0.0 0.9922 1.0042 1.0074 1.0042 1.002
0.1 0.9871 0.9990 1.0026 0.9986 0.998
0.2 0.9768 0.9871 0.9917 0.9886 0.988
0.3 0.9609 0.9676 0.9739 0.9739 0.972
0.4 0.9378 0.9395 0.9483 0.9536 0.949
0.5 0.9048 0.9007 0.9139 0.9257 0.918
0.6 0.8511 0.8480 0.8697 0.8865 0.879
0.7 fail 0.7802 0.8126 0.8256 -
5.3. Simply supported rhombic plate under uniform pressure
A simply supported rhombic plate with a skew angle 30° is subjected to uniformly distributed transverse loading. This problem is particularly challenging due to the fact that moments at the obtuse corners are singular. The plate is discretized with regular meshes as shown in Fig. 4.
i M^Norm
The normalized transverse central displacement of the midplane u3 is listed in Table 4. The analytical Morley’s solution [13] of 0.04455 has been used to normalize the derived results. All proposed ANS solid-plate elements exhibit a superior performance except for the hybrid stress-strain SPQ4ae element, which lags behind in the case of coarse mesh configurations.
5.4. Circular plate under central load
Consider a thin circular plate subjected to a concentrated load at the center point. The mechanical and geometrical characteristics of the plate are given in Fig. 5. The standard meshes are used to model due to symmetry only one quarter of the plate.
Table 5 lists the transverse central midplane displacements u3M of the simply
supported and clamped plates normalized with respect to the analytical solutions [11]
SS (3 + v)FR
u3 -■
16n(1 + v) D
CL FR and m3 -
16nD
based on the Kirchhoff plate theory. As can be seen, the developed SPQ4ct and SPQ4e elements are the best performers again although the other ANS solid-plate elements [1] work well.
Fig. 4. Simply supported rhombic plate under uniform pressure:
a = 100; h = 1; E = 107; v = 0.3; p = 1
Table 4
Normalized transverse center point displacement of rhombic plate
Mesh SPQ4 SPQ4CT6 SPQ4ct SPQ4e Simo [9] MITC4 [12]
4 x 4 0.8791 0.6269 0.9463 1.0176 0.961 0.879
8x8 0.8714 0.7724 0.9540 0.9840 0.957 0.871
16 x16 0.9382 0.8895 0.9911 0.9678 0.985 0.933
32 x 32 0.9952 0.9681 1.0188 1.0054 1.009 0.985
2
Fig. 5. Circular plate under central load:
R = 1000; h = 1; E = 2x106; v = 0.3 ; F = 1
Table 5
Normalized transverse center point displacement of circular plate
Nel Simply supported plate Clamped plate
SPQ4 SPQ4CT6 SPQ4ct SPQ4e SPQ4 SPQ4CT6 SPQ4ct SPQ4e
3 0.9831 0.9892 0.9905 0.9894 0.7865 0.8022 0.8049 0.8022
12 0.9956 0.9964 0.9982 0.9986 0.9484 0.9507 0.9551 0.9559
48 0.9984 0.9988 0.9992 0.9992 0.9860 0.9869 0.9881 0.9881
Fig. 6. Interconnection of used variational equations
6. Conclusions
On the basis of the first-order solid-plate theory [1, 14] accounting for thickness stretching the hybrid stress and hybrid strain ANS quadrilateral solid-plate elements have been developed. It is worth noting that stiffness matrices of the proposed hybrid solid-plate elements have six zero eigenvalues as required for satisfaction of the general rigid-body motion representation. The extension of the hybrid ANS quadrilateral solid-plate elements to the geometrically exact solid-shell quadrilaterals is currently under development and will be reported in the next papers.
Appendix A
Finally, we describe an interconnection between displacement-based [1] and mixed variational equations employed in the present paper for development of the hybrid four-node solid-plate quadrilaterals. They are displayed in Fig. 6.
The present research was supported by Russian Foundation for Basic Research (GrantNo. 08-01-00373).
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Гибридные четырехугольные трехмерные элементы пластины. Оценка и новые подходы
Г.М. Куликов, С.В. Плотникова
Кафедра «Прикладная математика и механика», ГОУ ВПО «ТГТУ»
Ключевые слова и фразы: гибридные конечно-элементные модели; метод введенных локальных деформаций; четырехузловой трехмерный элемент пластины.
Аннотация: На основе трехмерной теории пластин первого порядка, предложенной ранее авторами, построены гибридные четырехузловые АКБ элементы пластины с введенными распределениями напряжений и деформаций. Все гибридные трехмерные элементы пластины основаны на общей методологии, что дает возможность оценить их достоинства и недостатки. Например, гибридный четырехугольный АКБ элемент с введенными распределениями напряжений и деформаций позволяет использовать аналитическое интегрирование, которое приводит к очень простой структуре его матрицы жесткости. В то же время гибридный четырехугольный АКБ элемент с введенным распределением напряжений демонстрирует превосходную производительность во всех рассмотренных тестовых задачах, но требует численного обращения матрицы большой размерности.
Hybride viereckige dreidimensionale Plattenelemente.
Einschatzung und neue Behandlungen
Zusammenfassung: Auf Grund der dreidimensionalen von den Autoren fruher vorgeschlagenen Plattentheorie der ersten Ordnung sind die hybriden vierknotigen ANS Plattenelemente mit den eingefuhrten Spannungsverteilungen und mit der Deformierung gebaut. Alle hybriden dreidimensionalen Plattenelemente sind auf die gesamten Metodologie gegrundet, was die Moglichkeit ihre Vorteile und Nachteile einzuschatzen erlaubt. Zum Beispiel, das hybride viereckige ANS Element mit den eingefuhrten Verteilungen der Spannungen und der Deformationen erlaubt, die analytische Integration zu verwenden, die zur sehr einfachen Struktur seiner Matrix der Harte bringt. Gleichzeitig demonstriert das hybride viereckige ANS Element mit der eingefuhrten Verteilung der Anstrengungen die ausgezeichnete Produktivitat in allen betrachteten Prufungsaufgaben, aber fordert die numerische Behandlung der Matrix der grofien Dimension.
Elements quadrilateraux hybrides tridimensionels des plaques.
Evaluations et nouvelles approches
Resume: A la base de la theorie des plaques du premier ordre proposee par les auteurs auparavant sont construits des elements hybrides a quarte noeuds ANS avec l’introduction des repartitions des tensions et des deformations. Tous les elements hybrides tridimensionels de la plaque sont fondes sur une methode commune ce qui permet d’evaluer leurs qualites et leurs defauts. Par exemple, l’element quadrilateral ANS avec l’introduction des repartitions des tensions et des deformations permet d’utiliser l’integration analytique qui aboutit a une tres simple structure de sa matrice de rigidite. En meme temps, l’element hybride quadrilateral ANS avec l’introduction des repartitions des tensions montre une exelente productivire dans tous les problemes de test examines, mais demande une conversion numerique de la matrice de grande dimension.
