a locking-free three-node shell finite element ... - Semantic Scholar

COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O˜ nate and E. Dvorkin (Eds.) c

CIMNE, Barcelona, Spain 1998

A LOCKING-FREE THREE-NODE SHELL FINITE ELEMENT FORMULATION R. J. Marczak1 and A. M. Awruch2 1 Department

of Mechanical Engineering - Federal University of Rio Grande do Sul Rua Sarmento Leite, 425 - Porto Alegre - RS, Brazil - 90050-170 2 Graduate Program in Civil Engineering - Federal University of Rio Grande do Sul Av. Osvaldo Aranha, 99, 3o andar, Porto Alegre - RS, Brazil - 90035-190

Key Words: Finite element method, plates and shells, shear locking. Abstract. A simple triangular shell finite element with fifteen degrees of freedom for the analysis of general shell structures is presented in this work. The element uses a substitute transverse shear strain field to avoid locking, and is formulated on the basis of RSDS-element (Resultant Stress Degenerated Shell Element) approach, which leads to very simple strain-displacement expressions. The substitute shear strain field is derived assuming constant shear strains along the sides of the element, similar to the MITC (Mixed Interpolation in Tensorial Componentes) procedure. All integrations can be performed using only one point Gaussian rule, leaving the element with one non-comunicable zero energy mode, which does not affect the results. This procedure allows the use of symbolic engines to derive analytic expressions for stiffness matrices, reducing computational cost of the analysis by a significative amount.

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R. J. Marczak and A. M. Awruch

1

INTRODUCTION

The study and applications of shell finite elements has been subject of intense development during the last three decades. Still today this is a very fruitful research area, with several related subjects yet to be explored. The reason why there are so many formulations for shell finite element analysis is intrinsically related to the complex nature of shell equations, arguably the most challenging among the usual structural theories. The difficulty (or the impossibility) of finding analytical solutions for practical cases of geometry, material and loading, even with the aid of modern symbolic algebra programs, leaves the use of computer based discrete solutions as the natural trend. In addition, practical needs as negative gaussian curvature, the influence of shear strains and the extensibility of thickness fibers, among other aspects, bring the governing equations to an even higher degree of complexity. Moreover, the accurate solution of practical cases requires the use of suitable meshes that can be obtained only with intensive use of mesh generators. In spite of the the quality which can be achieved by modern mesh generators, the interference of the user is still necessary for practical cases. Stress concentration points and rapidly varying geometry areas always need semi-automatic mesh adaptation, and this can be very cumbersome when higher order elements are used. These are among the main reasons why low order elements have been preferred in many situations. In addition, the code for a low order element (like three-node or four-node element) can be easily vectorized, which enable the use of modern supercomputers and huge meshes. In spite of the large number of shell elements formulations already published, several of them lack one or more features which would be necessary to allow their general use in complex engineering problems commonly found in industry. A short list of requirements that would be present in any general shell formulation is (we restrict ourself to some arguments which are valid for low order shell elements in linear elastic analysis): (a) The element has to pass in the patch test for constant membrane, constant bending and constant shear. (b) If possible, the element should have only its middle surface nodes coordinates as input data. (c) The stiffness matrix of the element should achieve the correct rank, specially for triangular elements, which usually presents severe shear locking and can not be solved only with reduced integration. (d) If reduced integration is to be used, all the resulting spurious zero energy deformation modes should be controlled. (e) The numerical efficiency of the element can be much improved by the use of analytical or semi-analytical integration of the stiffness matrix. (f) The shear and bending energy norms of the element should be bounded, specially for very thin shells. An attempt to incorporated partially the requirements listed above in a general threenode shell element is the main goal of this work. The element is formulated on a basis similar to the MITC (Mixed Interpolation in Tensorial Components) family. Because of the low number of nodes, the present approach leads to very simple strain-displacement expressions. All integrations are carried out using only one Gauss point rule, leaving the

2

R. J. Marczak and A. M. Awruch

element with one non-communicable zero energy mode, which does not affect the results. This procedure allows the use of symbolic engines to generate analytic expressions for stiffness matrices, reducing by a significant amount the computational cost of the analysis. 2

GENERAL FORMULATION OF A FOSD FLAT SHELL ELEMENT

This section discusses some general characteristics of a three node shell element based on Ahmad’s kinematics.2 In the context of a displacement based formulation, the degenerated element concept leads to a very simple and robust way to formulate elements with any number of nodes. In this section, we follow very closely the approach proposed by Liu et alli 10 for the kinematic description of the element, particularized for the case of a three node element. The element account for shear deformability through the first order shear deformable (FOSD) plate/shell theories of Mindlin and Reissner.12, 15 Throughout this text, latin letters denote variation from 1 to 3 (e.g. i = 1 . . . 3) while greek letters vary from 1 to 2 (e.g. α = 1 . . . 2) The element has three nodes, each one associated with thickness hi . The coordinates T  . The of the nodes in the global coordinate system are denoted xi = x1i x2i x3i T  thickness are measured over an normal vector pi = p1i p2i p3i . The purpose of a fiber vector is merely to define an axis along which the zero normal stress hypotesis will be invoked. Thus, in the case of an element with variable fiber vector one may expect a more sensitive behaviour to curvature changes. The element is mapped  to a normalized N N N domain (ξ = , ξ , ξ ) to allow the use of standard linear shape functions: 1 2 3 1 2 3  1 − ξ1 ξ1 ξ2 , in such a way that each coordinate of the position vector at a point on the middle surface of the element in a given time t is interpolated in the usual way: x (ξ1 , ξ2 , t) = Ni (ξ1 , ξ2 ) xi (, t)

;

Inside an element, the fiber vector is interpolated in the same way:10 p (ξ1 , ξ2 ) =

3 X

Ni (ξ1 , ξ2 )

i=1

hi ξ3 pi (t) 2

,

as well as the thickness: h (ξ1 , ξ2, t) = Ni (ξ1 , ξ2 ) hi (t) . Then, the fiber vector accounts for the position of a general point not belonging to the middle surface of the element, which allows one to write the position vector of a general point as: X (ξ1 , ξ2 , ξ3 , t) = x (ξ1 , ξ2 , t) + p (ξ1 , ξ2 , ξ3 , t)

.

In addition to the usual global coordinate system, most shell formulations employ several coordinate systems attached to the element:9 3

R. J. Marczak and A. M. Awruch

• Global coordinate system (GCS): The GCS is simply a non-inertial coordinate system defined by a canonical base. • Normalized coordinate system (NCS): The NCS stands for the base vectors of the normalized domain used for interpolation and integration of the variables. In this system ξ1 ,ξ2 = [0, 1], while ξ3 = [−1, 1]. • Lamina coordinate system (LCS): The LCS is the system used to employ the kinematic description of the Mindlin-Reissner plate/shell models. The element stiffness matrix will be defined first in this system, and then rotated to appropriate systems. The transformation of a tensor Vl in the LCS to the GCS is accomplished by the following operator: R : Vl → V where Rij = ei · elj

.

The LCS is obviously constant for three-node flat elements, but in the case of higher order elements the LCS vectors are interpolated by the standard shape functions: elj (ξ1 , ξ2 , t) = Ni (ξ1 , ξ2 ) elji . • Fiber coordinate system (FCS): The FCS is used to define the rotational degrees of freedom of the element. It is choosen in such a way that ef3i ≡ pi at each node i. Several diferent conventions can be used to generate the FCS. In general, it is defined as a system following the shell middle surface. In this work, we have used the algorithm proposed by Liu et alli 10 Inside the element, the FCS vectors are interpolated in the standard way: efj (ξ1 , ξ2 , t) = Ni (ξ1 , ξ2) efji . The transformation from FCS to the GCS is given by the linear transformation S, given by: S : Vf → V where Sij = ei · efj

.

It is worth to note that el3 6= ef3 , in general. This algorithm generates an orthogonal FCS, uniquely defined at each node, and ensures always one of the axis (ef1 or ef2 ) to be tangent to a continuous boundary of the shell, making easier to prescribe boundary conditions. On the other hand, it can turn difficult to the user to visualize which of the FCS axes is tangent to a given boundary, specially for complex geometries. Because the stiffness matrix will be derived in the LCS, but the rotational degrees of freedom will stay attached to the FCS, the following relation is useful in order to perform the transformations from one system to the other: Q : Vl → Vf

where Q = ST R .

The strain-displacement matrix considered in RSDS elements are obtained directly from the Ahmad’s kinematic relations for shells, for linear static analysis. In view of 4

R. J. Marczak and A. M. Awruch

the absence of transverse normal stresses, the 6×6 constitutive matrix can be contracted to a 5×5 form. In addition, we are not considering drilling degrees of freedom. From these assumptions, and making use of the Green strain tensor definition for infinitesimal displacements, the strain vector can be approximated by the following relation: εl = Bli dli

,

with 

∂Ni ∂xl1

 0   ∂Ni l Bi =   ∂xl2  0  0

0 ∂Ni ∂xl2 ∂Ni ∂xl1

0 0

or

0 0 0 ∂Ni ∂xl2 ∂Ni ∂xl1

 Bli

=

i ξ3 h2i ∂N 0 ∂xl1 hi ∂Ni 0 ξ3 2 ∂xl 2 hi ∂Ni i ξ3 2 ∂xl ξ3 h2i ∂N ∂xl1 2 Ni h i 0 hl Ni h i 0 hl

Blmi Blbi 0 Blsi

0 0 0 0 0

       

,

 ,

(1)

Note that the last row of Bli results in zero entries because it refers to the drilling rotation stiffness, not considered here. The associated jacobian is written as: l ∂x ∂xl 1 l ∂ξ11 ∂ξ12 1 l l J = h ∂xl ∂xl = h A . (2) 2 2 1 2 ∂ξ2

∂ξ2

The stiffness matrix block relating the nodes i and j now can be evaluated by integration over the thickness and over the area of the element:  Z Z T l l l l ¯ ij = K Bi D Bj dh dA4 . 4

h

¯ lij to the GCS, the following transformation is performed: In order to transform K ¯ lij QT Kij = QK 3 3.1

.

ANOTHER SOLUTION FOR SHEAR LOCKING Common alternatives to solve shear locking

The main problem of standard isoparametric three or four node plate and shell elements with full integration is their inability to predict correct values for shear deflection. The use of reduced or selective integration alliviates the problem for quadrilateral elements, although leading to the possible manifestation of hourglass modes, which necessarily have 5

R. J. Marczak and A. M. Awruch

to be controlled for a reliable analysis. In the case of triangular elements, even the use of reduced integration may not solve this problem. Several remedies other than reduced integration has been proposed in the literature to solve the shear locking problem. Particularly in the case of three or four node plate and shell elements, some solutions are equivalent from the variational point of view, others are purely tentatives. But in any case, the main purpose of the corrections is to improve the ability of the element to reproduce constant shear situations and still pass in the patch test. Among the usual solutions for the shear locking in low order plate/shell elements we ¯ methods: these methods are based in the substitution of the shear strainhave: (a) B ¯ s able to adequately represent shear displacement matrix Bs with a substitute matrix B strains. (b) Bubble functions: this method consists in the addition of some high order terms in the interpolation function to allow an enriched displacement representation. (c) EAS methods: The enhanced assumed strain methods are based on the Hu-Washizu principle. The key point of the method lies in the use of a strain field composed of a compatible strain field and an enhanced strain field. (d) Mixed methods: Are based on hybrid models derived from the Hellinger-Reissner principle with displacement and stress assumptions. (e) Energy correction: An energy correction is accomplished by adding a complementary energy term which neutralizes the locking effect. 3.2

A vectorial derivation for the substitute shear strain field

The most successful approaches to achieve correct rank in a locking-free element have the same key point: to evaluate the shear strains at each node using the mean value of the shear strains evaluated along the sides adjacent to the node (whose values are considered constants). The nodal values of the shear strains are interpolated using the same shape functions used for the displacements, thus ensuring a consistent formulation. In this section we will develop the same ideas in a slighly different way, in order to demonstrate that in fact these methods are only based in the use of substitute shape functions for the shear strains, and investigate if these shape functions hold the necessary conditions for a consistent interpolation. We used an approach similar of the work of Aalto.1 It is worth to point out that we have deriving a new shear stiffness matrix to replace Blsi in eq.(1), using the LCS as reference. Considering the LCS arbitrarily oriented in the plane of the element, the shear strain components simply read (the index l will be supressed for clarity purposes): 2εα3 =

∂u3 + θα ∂xα

.

(3)

At each nodal point i, define a shear vector given by: γi = 2εα3i elα

.

Now define an orthogonal coordinate system (ti , ni ) aligned to the side i of the element, whose end nodes are i and j. Using the nodal convention j(i = 2) = 1, j(i = 3) = 2, j(i = 6

R. J. Marczak and A. M. Awruch

1) = 3 one can write, for each side gi = γ i · ti gi = γ j · t i

(4) (5)

,

where gi is a mean value of the shear strains on side i. Noting that, in the LCS tj =

xij l yij l e + e lj 1 lj 2

where xki = xl1k − xl1i , yki = xl2k − xl2i and lj is the length of the side j, eqs.(4-5) can be written for each node and inverted in the following form:   g1 γ 1 = T1 for node 1 g3  γ 2 = T2  γ 3 = T3

g2 g1 g3 g2

 for node 2  for node 3

Grouping the variables in an element-wise vector results: γ e = Pγ

,

where  γ e = 2 ε131 ε231 ε132 ε232 ε133 ε233  T g1 g2 g3 γ = and

   1  P= l A   

y31 l1 0 −y21 l3 −x31 l1 0 x21 l3 −y32 l1 y21 l2 0 x32 l1 −x21 l2 0 0 −y13 l2 y32 l3 0 x13 l2 −x32 l3

where Al is given by eq.(2).

7

       

,

R. J. Marczak and A. M. Awruch

Now it is possible to interpolate γi using the linear shape functions:    l  2ε13i 2ε13 = Ni , 2εl23 2ε23i

(6)

or, in matrix form: 

with

 ¯ = N

2εl13 2εl23

 ¯γ = NPγ = N

N1 0 N2 0 N3 0 0 N1 0 N2 0 N3

,

(7)

 P = NP ,

¯ is the substitute shape functions used for the shear strain components. In a where N mixed finite element formulation, eq.(6) plays the role of a constraint equation for the shear strains with respect to the parameters gi . Finally, it is necessary to relate the mid-edge shear strain gi with the element nodal displacements. Recalling eq.(3), it is clear that, for each side i, the following relations hold: gi =

∂u3 ¯ +θ , ∂l

where: u3 − u3j ∂u3j = i ∂l lj

θn + θnj θ¯ = i 2

and

Then the shear constraint can be written as gj =

ul3 i − ul3j lj

+

  nj · θjl + θil 2

,

(8)

where ni is the vector normal to the side i. Recovering eq.(8) for all nodes in matrix form results: γ = A db

,

(9)

where db stands only for the bending/shear nodal displacements at the element level, in such a way that: d = { dm db }T = { u11 u21 u31 θ11 θ21 · · · θ23 }T dm = { u11 u21 · · · u23 }T db = { u31 θ11 θ21 · · · θ23 }T 8

.

R. J. Marczak and A. M. Awruch

and



 −1/l1 n11 /2 n21 /2 1/l2 n11 /2 n21 /2 0 0 0 0 0 −1/l2 n12 /2 n22 /2 1/l2 n12 /2 n22 /2  A= 0 1/l3 n13 /2 n23 /2 0 0 0 −1/l3 n13 /2 n23 /2

Rewriting eq.(7) using A and remembering that we are using the LCS leads to:  l    l 2ε13 ¯ ls B ¯ ls dlb . ¯ ls dlb = B ¯s B = N P A dlb = B (10) l 1 2 3 2ε23 ¯ ls replaces Bls in eq.(10). If we write The matrix B i i   i i i N M M l w θ θ x 1 2 ¯s = B i Nwi y Liθ1 Liθ2

(11)

it becomes clear that 2εl13 = Nwi x u3i + Mθi1 θ1i + Mθi2 θ2i

2εl23 = Nwi y u3i + Liθ1 θ1i + Liθ2 θ2i

.

At this point it is interesting to note that the following relations can be easily verified: 3 X i=1 3 X

Mθi1 = 1 Mθi2 = 0

i=1 3 X i=1 3 X

Liθ1 = 0 Liθ2 = 1 ,

i=1

and so the fundamental constant strain criterion is guaranteed.6 3.3

A tensorial derivation for the substitute shear strain field

The development of the equations presented above is merely a formalization of the idea of considering a shear deformable beam attached to each side of the element. A more rigorous way to derive the substitute shear strain field was presented by Boisse et alli,5 using the Green-Lagrange strain definitions written in a covariant base along the sides of the element. In this section, we summarize the tensorial derivation of the substitute 9

R. J. Marczak and A. M. Awruch

shear strain field given by eq.(11) as proposed by Boisse et alli .5 As a matter of fact, the work of Boisse et alli 5 is intrinsically based in the MITC (mixed interpolation in tensorial components) element family.3, 4 In the present work, the MITC approach will be used only to derive a new Bls matrix in eq.(1), although the method can be used to derive membrane and bending strain-displacement matrices too. Because of the close resemblance with the approach used to derive the MITC plate/shell elements, we called the present element MITC3. ¯ methods are rooted in the fact that they use a The equivalence between several B constraint equation for the shear strains somewhere in the element. The simplest form of the shear constraint can be written as eq.(3) along the sides of the element. The use of this constraint can be justified by a simple analysis of locking in Timoshenko beam elements, and then it is assumed that the plate element edges behave like beams. Considering a LCS (x1 , x2 , x3 ) arbitrarily oriented in the plane of the element, definition of the Green-Lagrange strain tensor in a general coordinate system reads:7 εl = εeij gi ⊗ gj

,

(12)

where gi are the contravariant base vectors related to (x1 , x2 , x3 ) and εeij are the strain components written in the covariant base (only the linear part will be considered here):  
1 ∂u 0 1 1 ∂u 0 1 0 0 εeij = gi · gj − gi · gj = gj + gi , (13) 2 2 ∂ξi ∂ξj The aim is to interpolate the transverse shear strains in the local coordinate system LCS, from the nodal shear strain values, which are evaluated by: 2εα3 = Ni εeiα3

.

The nodal components εeiα3 are obtained under the assumption that the transverse shear strains along the sides are constant, and evaluated at the mid-side of each edge. With this purpose in mind, it is better to define a nodal covariant base vector (f1i , f2i , f3i ) with its axis parallel to the edges of the element which intercept the node i. Using the covariant frame (f1i , f2i , f3i ), the Green-Lagrange strains can be expressed as: εlij = e εeij f i ⊗ f j

,

where e εeij are the components of the strain components related to the (f1i , f2i , f3i ) frame. Thus,
1 1 e , εeα3 = fα · 1 g3 − 0 fα · 0 g3 2 and the linear part results, for each side i:   1 ∂u i ∂u 0 e εeα3 = · fα − · g3 . (14) 2 ∂ξ3 ∂rαi at midside 10

R. J. Marczak and A. M. Awruch

Eq.(14) is evaluated at the middle of the element edges because of the assumption that e εeα3 is constant. Calling each mid-side position by j ∈ [4, 5, 6], we have: h ∂u = (θi + θi+1 ) . (15) ∂ξ3 4 j

The second term on the right hand side of eq.(14) results: ∂u = wi+1 − w =i ∂r1i j ∂u = wi−1 − wi . ∂r2i j

(16) (17)

Finally, eq.(14) can be recovered for each mid-side position j ∈ [4, 5, 6] as a function of nodal displacements only using eq.(15), eq.(16) and eq.(17): j e εe = Gi di

(18)

Now we can use the same rule given by eq.(14) to write: i ∂u 0 ∂u 0 i ∂u 0 i ∂u 0 i 2e εeα3 = · gα + · g3 = · fα + · f3 ∂ξ3 ∂rαi ∂ξ3 ∂rαi

.

Using the relationship between 0 fαi and 0 gα in last equation, it is possible to relate the nodal shear strain components in the (g1i , g2i , g3i ) system with the shear strains evaluated along the edges of the element in the (f1i , f2i , f3i ) system, which may be written as follows: i εei = Fie εe

(no sum on i) .

Using eq.(18), we have εei = Fi Gi di

(no sum on i)

(19)

Eq.(19) is now used to interpolate εe in any point inside the element: i

εe =

3 X

Ni Fi Gi di

(no sum on i)

(20)

i=1

where Ni is the shape function matrix associated to node i. The reader should note that eq.(20) expresses the shear strain components for each node i written in the (g1i , g2i , g3i ) system. In order to transform eq.(20) to the local coordinate system attached to the element, we recall eq.(12) in the following form: ε = Hi εei

.

Finally, one obtains ε=

3 X

¯ i di Hi Ni Fi Gi di = B

i=1

11

.

(21)

R. J. Marczak and A. M. Awruch

4

NUMERICAL RESULTS

The present element was tested for some problems and results are presented in the sequel. Figures 1 and 2 show the convergence curves for clamped and simply supported square plates, respectively, under uniform loading. Due to symmetry of the problem, only the third quadrant of the plate was analyzed. The results are compared with some recent quadrilateral and triangular elements. The displacements are normalized (w ∗) with the classical Kirchhoff solution for a/h = 1000 and with series (thick plate) solution for a/h = 10, where a is the side length and h is the plate thickness. 1.20

T3BL (Taylor& Auricchio [1993]) Q 4BL (Zienkiewicz etalli[1993])

1.15

w* 1.10 1.05

1.00

0.95

0.90

0.85

0.80

0.75 2

4

6

8

10

# ofelem ents perside

Figure 1: Convergence results for simply supported (soft) plates under uniform loading. a/h = 1000.

The results show the convergence to the correct solution from below for thin as well as moderately thick cases. The convergence rate is acceptable in all cases. In order to show the absence of locking, a cantilever beam was analyzed in accordance to Figure 4. The length of the beam is L, with width b and thickness h. The results presented in Figure 3 refers to dimensionless tip displacement (with respect to the EulerBernoulli beam solution) for different h/L ratios. Clearly, good agreement was achieved for a very large range of h/L ratio.

12

R. J. Marczak and A. M. Awruch

1.15 T3BL (Taylor& Auricchio [1993]) Q 4BL (Zienkiewicz etalli[1993])

1.10

w* 1.05 1.00

0.95

0.90

0.85

0.80

0.75

0.70 2

4

6

8

10

# ofelem ents perside

Figure 2: Convergence results for simply supported (soft) plates under uniform loading. a/h = 10. 1.20 BEM -Linearelem ent FEM -9 node elem ent(selective integration)

1.15

FEM -8 node elem ent(selective integration) M ITC3

1.10

w/w

Analyticalsolution (Tim oshenko and W oinowski-Krieger[1970]

1.05

max 1.00 0.95 0.90 0.85 0.80 1E-1

1E-2

1E-3

1E-4

1E-5

1E-6

h/a

Figure 3: Behavior of the MITC3 element in the thin plate limit.

Some shell problems were also analyzed using the present formulation. Figure 4 shows the results for a curved beam under tip load. The maximum analytical displacement 13

R. J. Marczak and A. M. Awruch

solution for this problem is wmax = 0.0752 , indicating a normalized solution equal to 0.0996.

wmax = 0.0749

Figure 4: The curved beam problem.

The spherical cap under concentrated loads is a classical benchmark for shell problems. Figure 5 shows the mesh pattern used and results for four different meshes (the analytical solution for the maximum displacement is 0.094).

Mesh

wmax

6x6 10x10 16x16 20x20

0.0650 0.0814 0.0875 0.0900

Figure 5: The spherical cap problem.

14

R. J. Marczak and A. M. Awruch

Another important benchmark analyzed in this work was is the cylindrical shell under concentrated loads. This is a very severe test because of the localized bending strains around the point of application of the forces.9 Figure 6 illustrates the results obatined using the MITC3 element for two regular meshes and a irregular mesh. The analytical solution of this problem is: wmax = 0.183 × 10−4 .

wmax = 0.124 x10-4

wmax = 0.150 x10-4

wmax = 0.175 x10-4

Figure 6: Results for the pinched cylinder.

5

CONCLUSIONS

The development of a triangular finite element with substitute shear strain field to avoid locking is presented in this work. This version of the MITC3 element seems to be the three-node counterpart of the MITC4 element presented by Bathe & Dvorkin.3, 4 The shear constrained elements proposed by O˜ nate et alli13 also uses similar ideas, but in their approach the shear strains are additional degrees of freedom which should be condensed before the stiffness matrix assembly phase, usually requiring matrix inversions. Furthermore, no bubble function was introduced in the present approach. The approach used in this work generates a stiffness matrix with the correct rank when integrated by three Gauss stations. A very attractive explicit computer code can be generated using only one centroidal Gauss station, but in this case the element has one in-plane torsional non-comunicable spurious mode, which does not appear in an assembly of two or more elements. As a general rule, the MITC3 element can be competitive when used with cross diagonal meshes. The results for some shell problems showed a slow convergence, indicating that some improvements should be done in the formulation, particularly in the membrane stiffness (we are using the constant strain triangle). On the other hand, some further improvements on the behaviour of the element can be achieved by means of the residual bending flexibility but, as proved by Prathap,14 this is an extravariational trick, and 15

R. J. Marczak and A. M. Awruch

should not be included in the element formulation without a strong mathematical basis. The inclusion of drilling degrees of freedom is another interesting feature to be added to the present element. Finally, it is interesting to say that whichever the approach used to constrain the shear strains, the same procedure should be followed to derive the consistent mass matrix and the geometrical stiffness matrix. This is sometimes overlooked when extending the application range of an element to dynamic as well as to non-linear analysis. As a final remark, it is worth to note that the present element, as well as most shear constrained plate elements, the bending L2 energy norm is bounded, but not the shear L2 energy norm. Then, bad results can be expected for shear forces in exceptionally thin plates.4 REFERENCES [1] Aalto, J.: From Kirchhoff to Mindlin Plate Elements, Comm.Appl.Num.Meth., 4, pp. 231-241 (1988). [2] Ahmad, S., Irons, B. and Zienkiewicz, O.C.: Analysis of Thick and Thin Shell Structures by Curved Finite Elements, Int.J.Num.Meth.Engng., 2, pp. 419-451 (1970). [3] Bathe,K.-J. and Dvorkin, E.N.: A Formulation of General Shell Elements - The Use of Mixed Interpolation of Tensorial Components, Int.J.Num.Meth.Engng., 22, pp. 697-722 (1986). [4] Bathe,K.-J. : Finite Element Procedures, Prentice-Hall (1996). [5] Boisse, P., Daniel, J.L. and Gelin, J.C.: A Simple Isoparametric Three-Node Shell Finite Element, Computers & Structures, 44 (6), pp. 1263-1273 (1992). [6] Donea, J. and Lamain, L.G.: A Modified Representation of Transverse Shear in C o Quadrilateral Plate Elements, Comp.Meth.Appl.Mech.Engng., 67, pp. 183-207 (1987). [7] Green, A.E. and Zerna, W.: Theoretical Elasticity, Dover (1992). [8] Hughes, T.J.R. and Taylor, R.L.: The Linear Triangular Bending Element, in: The Mathematics of Finite Elements and Applications IV - MAFELAP 1981, pp. 127142, Academic Press (1982). [9] Hughes, T.J.R.: The Finite Element Method - Linear Static and Dynamic Analysis, Prentice-Hall (1987). [10] W.K. Liu, E.S. Law, D. Lam and T. Belytschko: Resultant-Stress Degenerated-Shell Element, Comp. Meth. Appl. Mech. Engng., 55, pp. 259-300 (1986). [11] MacNeal, R.H.: The Evolution of Lower Order Plate and Shell Elements in MSC/Nastran, Finite Elements in Analysis and Design, 5, pp. 197-222 (1989). [12] Mindlin, R. D., Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates, Journal of Applied Mechanics 18, 31-38 (1951). [13] O˜ nate, E., Zienkiewicz, O.C., Suarez, B. and Taylor, R.L.: A General Methodology fro Deriving Shear Constrained Reissner-Mindlin Plate Elements, Int. J. Num. Meth. Engng., 33, pp. 345-367 (1992). 16

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[14] G. Prathap: The Variationally Correct Rate of Convergence for a Two-Noded Beam Element, or Why Residual Bending Flexibility Correction is an Extra Variational Trick, Comm.Num.Meth.Engng., 11, pp. 403-407 (1995). [15] Reissner, E., On the Theory of Bending of Elastic Plates, Journal of Mathematical Physics 23, 184-191 (1944). [16] Taylor, R.L. and Auricchio, F.: Linked Interpolation for Reissner-Mindlin Plate Elements: Part II - A Simple Triangle, Int. J. Num. Meth. Engng., 36, pp. 3057-3066 (1993). [17] A. Tessler and T.J.R. Hughes: A Three-Node Mindlin Plate Element With Improved Transverse Shear, Comp. Meth. Appl. Mech. & Engng., 50, pp. 71-101 (1985). [18] Timoshenko, S. P. and Woinowski-Krieger S., Theory of Plates and Shells, 2nd ed., McGraw-Hill (1970). [19] Zienkiewicz, O.C., Xu, Z., Zeng, L.F., Samuelsson, A. and Wiberg, N.-E.: Linked Interpolation for Reissner-Mindlin Plate Elements: Part I - A Simple Quadrilateral, Int. J. Num. Meth. Engng., 36, pp. 3043-3056 (1993).

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