1

ECCM 2010IV European Conference on Computational Mechanics

Palais des Congrès, Paris, France, May 16-21, 2010

A New Quadrilateral Thick Plate Element Based on a Pure Displacement Linked Interpolation

G. Jelenić, D. Ribarić University of Rijeka, Civil Engineering Faculty, Rijeka, Croatia, {gordan.jelenic,dragan.ribaric}@gradri.hr

Abstract

In this work the use of linked interpolation in the design of plate finite elements is analysed. Benefits of the linked interpolation are well known in the Timoshenko (thick) beam finite elements and the basis for development of a family of the Mindlin plate elements presented in this paper is found in the analogy between the Timoshenko beam theory and the Mindlin plate theory. In contrast to the beam elements, the linked interpolation in plate elements cannot completely reproduce the analytical solution, not even for the relatively simple load cases. The results obtained in this are compared and numerically assessed against the reference results from literature using various mesh densities and various order of interpolation.

Key words

Finite element method, Mindlin plate theory, linked interpolation, higher-order interpolation

1. Introduction

Many finite elements have been developed for the Mindlin–Reissner moderately thick plates and in a number of them the idea of linking the displacement field to the rotations of the cross-sections has been thoroughly investigated and exploited [1-9]. As a general conclusion it has been realised that this idea on its own cannot eliminate the problem of shear locking, especially for coarse meshes, which is in stark contrast to the results obtained by applying the idea to the Timoshenko beam elements [7,10,11]. Consequently, a number of remedies have been proposed, which are mostly based on the application of the assumed or enhanced strain concepts [2,12], or the use of mixed or hybrid approaches [1,3,9].

In this presentation we re-visit this classic topic and study the possibility of eliminating the shear-locking problem while remaining firmly in the framework of the standard displacement-based finite-element design technique. Within this approach, the kinematic and constitutive equations of the problem are satisfied in the strong sense, while the equilibrium equations are satisfied in the weak sense with the unknown displacement and rotation fields as the only interpolated quantities.

In Section 2 we illustrate the relevant results for the Timoshenko beam elements from [11] giving a family of interpolation functions which follow a very structured pattern and provide the exact solution for arbitrary polynomial loadings. In many senses the Mindlin theory of thick plates may be regarded as a 2D generalisation of the Timoshenko theory of thick beams and, as a consequence, a number of attempts have been made to apply the linked-interpolation beam methodology [7,10] to thick plates [1-3,5-8]. In stark contrast to thick beams, however, the differential equations of equilibrium for thick plates cannot be solved in terms of a finite number of parameters and

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consequently no exact finite-element interpolation can be found in this way. Nonetheless, in [1,3,13] such interpolation has been used to formulate three-node triangular and four-node quadrilateral thick plate elements, while in [2,6] a six-node triangular and a nine-node quadrilateral elements have been proposed.

In Section 3 we firstly consider a quadrilateral four node plate element, for which the constant shear strain condition imposed on the element edges is known to lead to an interpolation for the displacement field which is dependent not only on the nodal displacements, but also on the nodal rotations around the in-plane normals to the element edges. In this way the displacement interpolation becomes linked to the nodal rotations via shape functions which are linear in one direction and quadratic in the other. Such linked interpolation for plates may be also obtained as a 2D generalization of the linked interpolation for beams [11]. This approach enables an easy generalization of the linked interpolation concept to higher-order rectangular plate elements. A related idea has been pursued in [8] where a family of displacement-based linked-interpolation triangular elements has been derived on a premise of the shear strain variation being of a prescribed order. Secondly, we note that generalizing this idea to arbitrary quadrilateral shapes is non-trivial and special care needs to be taken for such elements to satisfy the standard patch tests. We present a manner in which arbitrarily shaped elements may be properly developed, which involves an additional internal displacement degree of freedom, and conduct some numerical tests which demonstrate the potential of this approach, in particular for the higher-order elements.

2. Family of linked-interpolation elements for Timoshenko beams

In the Timoshenko beam theory the initially planar cross section of a beam remains planar after the deformation, but the angle which it closes with the centroidal axis may change during the deformation resulting in the shear angle

θγ += 'w (1)

x

w

w

γ-θ

dwdx

dwdx

q

S M

Figure 1: Co-ordinate system and initial configuration of a thick beam

where w is the lateral displacement of the beam shown in Fig. 1, the dash (') indicates a differentiation with respect to the co-ordinate x, and θ is the rotation of a cross section.

Let us also define the linear-elastic constitutive equations

'θ⋅= EIM , γ⋅= GAS (2)

3

where M and S are the cross-sectional stress-couple and shear stress resultants, while EI and GA are the bending and shear stiffness respectively, as well as the equilibrium equations

SM =' , qS −=' (3)

where q is the distributed loading per unit of length of the beam. This results in the following differential equations of the Timoshenko beam:

qEI −=⋅ '''θ , ,)'''( qwGA −=+⋅ θ (4)

which have the following general solutions:

322

1211 CxCxCqdxdxdx

EI+++−= ∫ ∫ ∫θ (5)

.21

6111

542

23

1 CxCxCxCqdxdxdxdxEI

qdxdxGA

w ++−−−−= ∫ ∫ ∫ ∫∫ ∫ (6)

For a polynomial loading of order n-4 it has been shown in [11] that the following interpolation completely reproduces the above exact results:

∑=

=n

iiiI

1θθ , ,

11

)1(1 1 1

1i

n

i

n

j

n

i

ijii i

nN

nLwIw θ∑ ∏ ∑

= = =

−

−−

−−= (7)

where L is the beam length, iθ and iw are the values of the displacements and the rotations at the n nodes equidistantly spaced between the element ends, iI are the standard Lagrangian polynomials of

order n-1, and LxN j = for j=1 and

Lx

jnN j 1

11−−

−= otherwise.

2.1 Two-node beam element

Introducing a dimensionless co-ordinate )(21 1xxL

−+−=ξ , for n=2 equations (6) and (7) give

.2

12

1

),(22

12

12

12

1

21

2121

θξθξθ

θθξξξξ

++

−=

−+−

−+

+−

=Lwww

(8)

This interpolation is illustrated in Fig. 2.

4

θξ

θθ

−(θ1−θ2)/2+(θ1−θ2)/2

(w-w )/2

∆w=−(θ1−θ2)L/8

Lagrangean

Lagrangean

Linked

x -x =L-1 +10

x x

-1 +10

x x

2

2 1

1

2 1w1

w2

2

1

x

w

21x

Figure 2: Interpolation functions for w and θ for a two-node beam element

Using an isoparametric mapping for the coordinate variable x

21 21

21 xxx ξξ −

+−

= (9)

and from equations (1), (2) and (8) the bending moment and the shear force in the beam cross section can be computed as

LEIM 12 θθ −

= (10)

and

.2

2112

+

+−

=θθ

Lww

GAS (11)

Obviously, interpolation (8) is capable of describing the state of constant shear. For a constant moment, the shear force must be zero over the element, so the term in the parentheses must be equal to zero i.e.

Lww2

2112

θθ +−=− (12)

meaning that the displacement change over the element is the integral of the rotational function over the element, which is a standard engineering result. Since the displacement interpolation now also involves the rotational contributions, the displacement at the midpoint of the element is not just the

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mean value of the end-node displacements, but includes the following additional contribution shown in Fig. 2:

21

2221 Lw

θθ −−=∆ (13)

2.2 Three-node beam element

For n=3 equations (6) and (7) give

321

321321

21

21

214

21

),2(32

12

12

12

12

142

1

ξθξθξξθξξθ

θθθξξξξξξξξξ

++

−++

−−=

−+−−+

−+

+−+

+−

−=Lwwww

(14)

This interpolation is illustrated in Fig. 3.

θ

ξ

θ1θ3

Lagrangean

Lagrangean

L-1 +10

x1 x3

x1 x3

x2

θ2

x2 x

x

w

w1 w3w2

x1 x3x2

(−θ1+2 θ2−θ3)/3 (−θ1+2 θ2−θ3)/3−(−θ1+2 θ2−θ3)/6

Linked

Figure 3: Interpolation functions for displacement w and rotation θ for a three-node beam element

Using an isoparametric mapping for the coordinate variable x

332211321 21

21

214

21 xIxIxIxxxx ξξξξξξξξξ ++=⋅

+

+

−

+

+

−

−= (15)

and from equations (2) and (14) the bending moment in the beam cross section can be computed as

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−+−−

−= ξ

θθθθθLL

EIM 32113 22 (16)

Shear deformation is derived from (1), (2) and (14). The form of it generally allows for a quadratic polynomial, but a scalar multiplying ξ² becomes zero by the appropriate form of the linked part in w eventually giving

−+

+−++

−= ξ

θθθ

22

2 133212

13

Lwww

Lww

GAS (17)

By equilibrium, the shear force is a derivative of the moment (16) with respect to x, here therefore giving a constant value. It can be found that the term multiplying ξ in the above equation equals to zero by noting from Figure 3 that

231

2ww

ww+

−=∆ (18)

and realising that the dashed curve in this figure is a quadratic polynomial, for which w∆ has been already given in (13), with 2θ in this equation now becoming 3θ .

3. Family of linked-interpolation elements for Mindlin plates

3.1 Mindlin plate theory

The Mindlin plate theory may be regarded as a generalisation of the Timoshenko beam theory to two-dimensional structures. The plate analysed is assumed to be of a uniform thickness h with a mid-surface lying in the horizontal co-ordinate plane of Fig. 1. and a distributed loading q assumed to act on the plate mid-surface in the direction perpendicular to it. The angles which the vertical material line elements close with the mid-surface are not necessarily retained during the deformation process, which results in the following shear angles [3]:

∂∂

+−

∂∂

+=

yw

xw

x

y

yz

xz

θ

θ

γγ

(19)

where xθ and yθ are the rotations of these line elements around the respective horizontal co-ordinate

axes. Here, we shall be using the constitutive equations for a linear elastic material

∂∂

−∂

∂∂

∂−

∂

∂

−−=

xy

y

xEh

MMM

xy

x

y

xy

y

x

θθ

θ

θ

νν

ν

ν2

1000101

)1(12 2

3

,

=

yz

xz

y

x kGhSS

γγ

(20)

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where xM , yM , and xyM are the bending and twisting moments around the respective co-ordinate

axes, xS and yS are the shear-stress resultants, E and G are the Young and shear moduli, while ν

and k are Poisson’s coefficient and the shear correction factor usually set to 5/6. The differential equations of equilibrium are given as

xxyx S

yM

xM

=∂

∂+

∂∂

, yyxy S

yM

xM

=∂

∂+

∂

∂, q

yS

xS yx −=

∂

∂+

∂∂

, (21)

where q is now the distributed loading per unit of area of the plate. The above results in the following differential equations of the linear elastic Mindlin plate:

qyxyx

Eh xy −=

∂

∂−

∂

∂

∂∂

+∂∂

−θθ

ν 2

2

2

2

2

3

)1(12, (22)

qyw

yxw

xkGh xy −=

∂∂

−∂∂

−

∂∂

+∂∂ θθ (23)

0)1(12 2

3

2

3

3

3

3

3

2

3

=

∂∂

−∂∂

+

∂∂

+∂∂

−

∂∂

∂+

∂∂∂

−∂∂

+∂

∂

− yw

yxw

xkGh

yxyxyxEh

xyyxxy θθ

θθυθθν

(24)

which, in contrast to the earlier Timoshenko beam case, cannot be solved in terms of a finite number of parameters, hence the finite-element solution cannot give the exact result. However, an idea to extend the results from Section 2 to the plate case is still attractive as a design tool to derive more accurate Mindlin plate elements.

3.2 Linked interpolation for a four-node quadrilateral plate element

The linked interpolation as defined in equation (7) may be applied to 2D situation in its original form written for arbitrary number of nodal points, but such a form would not be very illustrative. Instead, here we shall first apply it to a quadrilateral element with four nodal points at the element vertices depicted in Figure 4. At each nodal point i (i=1,2,3,4) there exist three degrees of freedom (displacement iw , and rotations xiθ , and yiθ ). In analogy to (8), the linked interpolation for the displacement field may be now given as

4321*

21

21

21

21

21

21

21

21 wwwww

+

−

+

+

+

+

−

+

+

−

−

=ηξηξηξηξ

)(2

14

12

)(2

14

12 43

343

42

4312

212

12

12nnnn

ll θθηξθθηξ−

+

−−−

−

−−

)(4

12

12

)(4

12

12 23

323

22

2314

414

12

14nnnn

ll θθηξθθηξ−

−

+

+−

−

−

+ (25)

where ξ and η are the standard natural co-ordinates, ijl are the lengths of the element sides between

nodes i and j and θin,ij denotes a rotation component perpendicular to the element side between the

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nodes i and j at node i (see Figure 5).

x

y

x2-x1

x4-x3

1

2

3

4θx

θy

θx2

θy2

θx3

θy3

θx4

θy4

θx1θy1

x3-x2

x1-x4

y2-y1

y3-y2y4-y3

y1-y4

ξ

η

Figure 4: Four-node quadrilateral plate element and its nodal rotations. The nodal displacements are perpendicular to the element plane

The interpolation functions for the rotations are given in a usual manner using the standard Lagrangian polynomials:

4321 21

21

21

21

21

21

21

21

xxxxx θηξθηξθηξθηξθ

+

−

+

+

+

+

−

+

+

−

−

=

4321 21

21

21

21

21

21

21

21

yyyyy θηξθηξθηξθηξθ

+

−

+

+

+

+

−

+

+

−

−

= (26)

w

)(2

14

12 12

212

12

12nn

l θθηξ−

−

−−

Figure 5. Linking shape function for normal rotations on the side “1-2” of the element

All the interpolation functions are expressed in natural coordinates ξ and η, which are related to the

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Cartesian co-ordinates x and y via

4321 21

21

21

21

21

21

21

21 xxxxx

+

−

+

+

+

+

−

+

+

−

−

=ηξηξηξηξ

4321 21

21

21

21

21

21

21

21 yyyyy

+

−

+

+

+

+

−

+

+

−

−

=ηξηξηξηξ (27)

Interpolation functions have been already reported in the literature and applied to the development of thick plate finite elements [1,3,7]. The normal rotations about an element edge are shown in Figure 6 and expressed by the global node rotations as

12

121

12

12112112112

1 )()(cossina

xxa

yyyxyxn

−−

−=−= θθαθαθθ (28)

and

( ) ( )12

1221

12

122112

212

1 )()(a

xxa

yyyyxxnn

−−−

−−=− θθθθθθ (29)

1

2

3

4

θ

θ

θθ

ξ

α12

α12α12

α12

x2

y2

y1

x1

x

y

Figure 6: Components of the nodal rotations to the normal on the side between nodes 1 and 2

As shown for the Timoshenko beam with two nodes, the linked interpolation can exactly

reproduce the quadratic displacement function, and the same should be expected for the 2D interpolation considered here. Furthermore, this must be true for any direction which interpolation (24) cannot satisfy unless enriched by a bi-quadratic bubble term

bw4

14

1 22 ηξ −− (30)

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eventually giving

bwww4

14

1 22* ηξ −−

+= (31)

where *w is given in equation (25). The displacement and rotational interpolations (25) and (31) can reproduce a constant bending moment and zero shear distribution throughout the arbitrary quadrilateral element.

3.3 Linked interpolation for a nine-node quadrilateral plate element

For a nine-node quadrilateral element the linked interpolation for the displacement field may be defined correspondingly (written in positional order as shown in Figure 7), i.e.

533632731* wIIwIIwIIw ⋅+⋅+⋅= ηξηξηξ

423922821 wIIwIIwII ⋅+⋅+⋅ ηξηξηξ

313212111 wIIwIIwII ⋅+⋅+⋅ ηξηξηξ

)2(2

12

1443

)2(2

143

4983

485673

57ξξξξξξ θθθηηξξθθθηηξξ

nnnnnnll

+−

−

+

−++−

+

−+

)2(2

143

3213

13ξξξ θθθηηξξ

nnnl

+−

−

−−

)2(42

12

143

)2(42

13

6923

267813

17ηηηηηη θθθηηξξθθθηηξξ nnnnnn

ll+−

−

−

+

−+−

−

−

+

)2(42

13

5433

35ηηη θθθηηξξ

nnnl

+−

−

+

− (32)

where ξ1I , ξ2I , ξ3I are the interpolation functions defined in equation (15), η1I , η2I , η3I are the

corresponding interpolation functions with natural co-ordinate η in place of ξ , and ijl is the distance between nodes i and j. Again, θi

nξ and θin η are the components of the nodal rotations to the normal of

the respective nodal lines.

x

y

1

ξ

θx

θy

2

3

97

6

5

l35

l 17

l48

l13

l 26

l57

8

4

Figure 7: Nine-node quadrilateral plate element and its geometry.

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The interpolations for rotations field, in positional order reads:

533632731 xxxx IIIIII θθθθ ηξηξηξ ⋅+⋅+⋅=

423922821 xxx IIIIII θθθ ηξηξηξ ⋅+⋅+⋅ (33)

313212111 xxx IIIIII θθθ ηξηξηξ ⋅+⋅+⋅

533632731 yyyy IIIIII θθθθ ηξηξηξ ⋅+⋅+⋅=

423922821 yyy IIIIII θθθ ηξηξηξ ⋅+⋅+⋅ (34)

313212111 yyy IIIIII θθθ ηξηξηξ ⋅+⋅+⋅

where θxi and θyi denote rotations in the global directions at the node i. The isoparametric mapping from natural coordinates ξ and η to the global variables x and y follows the same rule:

533632731 xIIxIIxIIx ⋅+⋅+⋅= ηξηξηξ

423922821 xIIxIIxII ⋅+⋅+⋅ ηξηξηξ (35)

313212111 xIIxIIxII ⋅+⋅+⋅ ηξηξηξ

533632731 yIIyIIyIIy ⋅+⋅+⋅= ηξηξηξ

423922821 yIIyIIyII ⋅+⋅+⋅ ηξηξηξ (36)

313212111 yIIyIIyII ⋅+⋅+⋅ ηξηξηξ

In these expressions, the coordinates for nodes 6, 8, 4 and 2 are exactly in the middle between the adjacent nodes and node 9 is at the element centroid.

As before, in order to reproduce exactly the polynomial of the third order in arbitrary direction, an additonal bi-cubic term involving an internal bubble displacement wb has to be added to displacement interpolation (32), i.e. eventually the displacement interpolation reads

bwww ⋅

−

−+=

44

33* ηηξξ (37)

The displacement and rotational interpolations (33), (34) and (37) can reproduce the states of linear bending and constant shear throughout the arbitrary quadrilateral element.

A similar interpolation for the displacements has been applied to six node triangular plate elements in [6], while a serendipity-type formulation of a similar sort written in a framework of the hierarchical interpolation has been presented in [2]. A family of triangular elements based on the linked interpolation has been derived in [8] stemming from the requirement of the shear strain in the element being of a prescribed order. The above methodology, in contrast, generates the linked interpolation from the underlying interpolation functions developed for the beam elements and may be consistently applied to quadrilateral plate elements of arbitrary order. The formation of the element stiffness matrix and the external load vector for the interpolation functions defined in this way follows the standard finite-element procedure described in text-books e.g. [14-16].

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4. Test examples

4.1. Patch test

Consistency of the interpolation functions for the developed elements is tested for the constant strain and stress condition on the patch example with five irregular elements, covering a rectangular domain of a plate as shown in Figure 8 and defined in [9]. The displacements and rotations for the four internal nodes within the patch are tested given the specific displacements and rotations for the four external nodes. The plate properties are chosen as E=106 , ν=0.3, k=5/6 and two values of the plate thickness are considered: h= 1.0 and h= 0.01, coresponding to a thick and a thin plate, respectively.

Two strain-stress states are tested:

• constant shear strain/stress.

Displacements and rotations are expressed respectively by

)(10 3 yxw += − , 310−−=xθ , 310−=yθ ,

with uniformly distributed couples as body forces in the rotational directions

hmx ⋅−= 026.641 , hmy ⋅= 026.641 ,

• constant bending strain/stress

)(10 223 yxyxw ++= − , )2(10 3 yxx +−= −θ , )2(10 3 yxy += −θ ,

with no body forces.

The exact displacements and rotations at the internal nodes are expected and the exact strains/stresses at every integration point. For the state of constant shear, the bending moments and the torsional moment vanish (Mx= My= Mxy= 0) and the shear forces are Sx= Sy= 641.026·h . For the state of constant bending, the moments are constant Mx= My= 238.095·h³, Mxy=64.103·h³ and the shear forces vanish (Sx= Sy= 0) [9].

x

y

(0,0)

(4,2)

(24,0)

(18,3)

(24,12)

(16,8)(6,8)

(0,12)

Figure 8: Patch for consistency assessment

The four-node element is tested on the patch given in Figure 7. For the values of the displacements and rotations at the external nodes calculated from the above data, the displacements and rotations at

13

the internal nodes as well as the bending and torsional moments and the shear forces at the integration points are calculated and correspond exactly to the analytical results given above. The results are not affected if the internal nodes are defined at a different set of co-ordinates.

The nine-node element is tested on the same patch examples (the mesh is given in Figure 8). Only the displacements and rotations at the boundary nodes are given (8 displacements and 16 rotations). All the other nodal displacements and rotations are to be calculated and they are calculated exactly for both the constant shear and the constant bending tests. The moments and shear forces at the integration points are again exact.

x

y

(0,0)

(4,2)

(24,0)

(18,3)

(24,12)

(16,8)(6,8)

(0,12)

Figure 9: Patch for consistency assessment of nine node element

4.2 Clamped square plate

In this example a quadratic plate with clamped edges is considered. Only one quarter of the plate is modelled with symmetric boundary conditions imposed on the connection lines with the other quarters. Two ratios of span versus thickness are analysed, L / h = 10 representing a relatively thick plate and L / h = 1000 representing its thin counterpart. The loading on the plate is uniformly distributed of magnitude q = 1. The plate material properties are E = 1092.0 and ν = 0.3 .

L/2

(4x4 mesh)

E= 10.92

ν= 0.3

k= 5/6

x

y

L/2

w=0

, θ =

0, θ

=0

x

w=0, θ =0, θ =0x

L/2

(4x4 mesh)

x

y

L/2

w=0

, θ =

0x

w=0, θ =0y

(SS2

)

(SS2)y

y

L= 10.0

h= 1.0 and 0.01

q= 1.0

clamped simply supported

Figure 10: A quarter of the square plate under uniform load (16-element mesh)

Numerical results are given in Tables 1 and 2 and compared to the elements presented in [3,9] based on the mixed approach and with the reference solutions taken from [3]. The dimensionless results w*= w / (qL4/100D) and M*=M / (qL²/100), where D=Eh³/(12(1-ν²)), and L is the plate span (Lx=Ly=L), given in these tables are related to the central displacement of the plate and the bending moment at the integration point nearest to the centre of the plate. The number of elements per mesh in

14

these tables is intended for one quarter as shown in Figure 10 for the mesh of 16 elements with the exception of the nine-noded element .

In all the examples to follow the elements presented in this paper are denoted as Q4-U02 for the four-node element and Q9-U03 for the nine- node element. The results are compared with the mixed-type element of Auricchio and Taylor [3] denoted as Q4-LIM and with the hybrid-type element of de Miranda and Ubertini [9] denoted as 9βQ4.

Table 1: Clamped square plate: displacement and moment at the centre with regular meshes, L/h = 10

Mesh Q4-LIM 9βQ4 Q4-U02 Q9-U03 w* M* w* M* w* M* w* M*

1x1 0.12107 0.02679 0.15059 1.971 2x2 0.14211 1.660 0.1625190 2.83817 0.11920 1.63801 0.15046 2.229 4x4 0.14858 2.157 0.1534432 2.44825 0.14361 2.16385 0.15044 2.297 8x8 0.14997 2.279 0.1511805 2.35119 0.14876 2.28107 0.15046 2.314 16x16 0.15034 2.310 0.1506379 2.32770 0.15004 2.31025 0.15046 2.31857 32x32 0.15043 2.317 0.1505061 2.32191 0.15036 2.31755 0.15046 2.31963 64x64 0.15045 2.319 0.15044 2.31938

Ref. sol. 0.1499 2.31 0.1499 2.31 0.1499 2.31 0.1499 2.31

Table 2: Clamped square plate: displacement and moment at the centre with regular meshes,

L/h = 1000 Mesh Q4-LIM 9βQ4 Q4-U02 Q9-U03 w* M* w* M* w* M* w* M*

1x1 0.090741 0.0000027 0.0002699 0.0041 2x2 0.11469 0.13766768 2.745544 0.0001284 0.099183 1.998 4x4 0.12362 2.120 0.12938531 2.423885 0.0046933 0.10106 0.12112 2.224 8x8 0.12584 2.248 0.12725036 2.323949 0.059881 1.17024 0.12621 2.282 16x16 0.12637 2.280 0.12671406 2.298876 0.11899 2.16797 0.12653 2.28902 32x32 0.12649 2.288 0.12657946 2.292605 0.12600 2.28115 0.12653 2.29016 64x64 0.12652 2.290 0.12648 2.28948

Ref. sol. 0.1265319 2.29051 0.1265319 2.29051 0.1265319 2.29051 0.1265319 2.29051

Even though the displacement-based elements Q4-U02 and Q9-U03 still suffer from shear locking for the thin plate case when the meshes are coarse (especially element Q4-U02), they converge to the correct result and, in case of the nine-node element Q9-U03 even faster than the other elements.

4.3 Simply supported square plate

In this example the same quadratic plate as before is considered, but this time with the simply supported edges of the type SS2 (displacements and rotations around the normal to the edge set to zero). The same elements as before are tested and the results are given in Tables 3 and 4 for the thick and the thin plate, respectively.

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Table 3: Simply supported square plate (SS2): displacement and moment at the centre with regular meshes, L/h = 10

Mesh Q4-LIM 9βQ4 Q4-U02 Q9-U03 w* M* w* M* w* M* w* M*

1x1 0.40562 2.078 0.26102 1.62404 0.42983 4.443 2x2 0.42626 4.125 0.4286943 5.264135 0.41163 4.140 0.42749 4.700 4x4 0.42720 4.623 0.4276333 4.905958 0.42448 4.636 0.42730 4.766 8x8 0.42727 4.747 0.4273690 4.817645 0.42664 4.751 0.42729 4.783 16x16 0.42728 4.778 0.4273052 4.795864 0.42713 4.779 0.42728 4.78722 32x32 0.42728 4.786 0.4272895 4.790443 0.42725 4.78629 0.42728 4.78828 64x64 0.42728 4.788 0.42727 4.78805

Navier series

0.427284 4.78863 0.427284 4.78863 0.427284 4.78863 0.427284 4.78863

Table 4: Simply supported square plate (SS2): displacement and moment at the centre with regular meshes, L/h = 1000

Mesh Q4-LIM 9βQ4 Q4-U02 Q9-U03 w* M* w* M* w* M* w* M*

1x1 0.37646 2.018 0.0000008 0.00053 0.35527 3.531 2x2 0.40365 4.119 0.4063653 5.241563 0.0031093 0.03396 0.39807 4.609 4x4 0.40586 4.623 0.4063062 4.904153 0.055621 0.66535 0.40475 4.751 8x8 0.40616 4.747 0.4062559 4.817513 0.29658 3.52362 0.40615 4.782 16x16 0.40622 4.778 0.4062421 4.795855 0.39706 4.67709 0.40624 4.78720 32x32 0.40623 4.786 0.4062386 4.790442 0.40562 4.77974 0.40624 4.78828 64x64 0.40624 4.788 0.40619 4.78764

Navier series

0.406237 4.78863 0.406237 4.78863 0.406237 4.78863 0.406237 4.78863

For these boundary conditions, the similar conclusions may be drawn for the displacement-based elements Q4-U02 and Q9-U03 as before.

4.4 Simply supported skew plate

In this example the rhombic plate is considered with the simply supported edges, but this time of the so-called soft type SS1 (only displacements of the edge set to zero). The same elements as before are tested and the results are given in Tables 5 and 6 for the thick and the thin plate, respectively. The problem geometry and material properties are given in Figure 11, where an example of a 16-element mesh is also shown. In contrast to the earlier examples, it must be noted that here the new displacement-based element perform worse than the elements given in [3,9].

16

30

L

0

L

(4x4 mesh)

E= 10.92

ν= 0.3

L= 100.

q= 1.

Figure 11: A simply supported (SS1) skew plate under uniform load

Table 5: Simply supported skew plate (SS1): displacement and moment at the centre with regular meshes, L/h = 100

Mesh Q4-LIM 9βQ4 Q4-U02

Q9-U03

w* M1* M2* w* M1* M2* w* M1* M2* w* M1* M2*

2x2 0.52729 0.745 1.42 0.06153 0.21493 0.480 0.949 4x4 0.42571 1.068 1.775 0.502900 1.354215 2.054527 0.16287 0.382 0.850 0.32974 0.8478 1.5611 8x8 0.42096 1.140 1.773 0.443176 1.082957 1.956494 0.29165 0.7263 1.3934 0.38719 1.0507 1.7822 16x16 0.42307 1.172 1.858 0.432211 1.149921 1.966940 0.37449 0.9996 1.7273 0.40892 1.1396 1.8524 32x32 0.42619 1.190 1.889 0.426964 1.148682 1.959293 0.40532 1.1260 1.8380

Ref. [9]

0.423 0.423 0.423 0.423

Table 6: Simply supported skew plate (SS1): displacement and moment at the centre with regular

meshes, L/h = 1000

Mesh Q4-LIM

9βQ4 Q4-U02

Q9-U03

w* M1* M2* w* M1* M2* w* M1* M2* w* M1* M2*

2x2 0.52154 0.962 1.927 0.00087 0.15198 0.366 0.782 4x4 0.41851 1.083 1.830 0.501994 1.352235 2.056188 0.00940 0.0240 0.0742 0.24245 0.6775 1.4209 8x8 0.40941 1.128 1.772 0.442198 1.079937 1.955570 0.08101 0.2116 0.4828 0.31048 0.8237 1.5397 16x16 0.40725 1.134 1.819 0.430832 1.149342 1.964564 0.20304 0.5148 1.0856 0.35196 0.9553 1.7073 32x32 0.40803 1.138 1.843 0.424505 1.143222 1.953153 0.31271 0.8110 1.5325

Ref. [9]

0.4080 1.08 1.91 0.4080 1.08 1.91 0.4080 1.08 1.91 0.4080 1.08 1.91

4.5 Simply supported circular plate

As our last example, the circular plate with the simply supported edges (type SS1) is analysed. The results are given in Tables 7 and 8 for the thick and the thin plate, respectively. The problem geometry and material properties are given in Figure 12 (only one quarter of the plate is analysed), where an example of a 12-element mesh is also shown. Again, here the displacement-based elemnent coverge

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very quickly towards the exact solution, in particular the nine-node element Q9-U03, which provides better results than the other elements for the comparable number of the degrees of freedom.

R

R

x

y

w=0, (12 quad mesh)simplysupported

Figure 12: A simply supported (SS1) skew plate under uniform load

Table 7: Simply supported circular plate (SS1): displacement and moment at the centre, R/t = 5 (t=1.0):

Mesh Q4-LIM 9βQ4 Q4-U02 Q9-U03 wc* Mc* wc* Mc* wc* Mc* wc* Mc*

1 3 0.42030 4.585 0.2812290 5.11686 0.40106 4.57904 0.41547 5.05346 12 0.41888 5.070 0.4133307 5.16833 0.41446 5.09176 0.41596 5.13258 48 0.41774 5.133 0.4153994 5.16188 0.41561 5.13937 192 0.4158536 5.15782 768 0.41602 5.1548 0.4159633 5.15669

Ref. 0.415994 5.1563 0.415994 5.1563 0.415994 5.1563 0.415994 5.1563

Table 8: Simply supportd circular plate (SS1): displacement and moment at the centre, R/t = 50

(t=0.1):

Mesh Q4-LIM 9βQ4 Q4-U02 Q9-U03 wc* Mc* wc* Mc* wc* Mc* wc* Mc*

1 3 0.39595 4.614 0.2692074 5.10642 0.35930 4.12421 0.39025 4.94879 12 0.40020 5.086 0.3959145 5.14593 0.38282 4.81201 0.39814 5.12874 48 0.39880 5.134 0.3977919 5.16070 0.39681 5.11504 192 0.3981935 5.15773 768 0.3982894 5.15668

Ref. 0.398315 5.1563 0.398315 5.1563 0.398315 5.1563 0.398315 5.1563

Acknowledgement. The results shown here have been obtained within the scientific project ‘Improved accuracy in non-linear beam elements with finite 3D rotations’ financially supported by the Ministry of Science, Education and Sports of the Republic of Croatia.

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5 Conclusion In this work the use of purely displacement-based linked interpolation in the design of Mindlin plate finite elements is presented and numerically assessed.

We have specifically considered quadrilateral four-node and nine-node plate elements, for which the shear strain condition of a certain order imposed on the element edges leads to the so-called linked interpolation for the displacement field whereby the nodal rotations around the normal to the element edges also contribute to the element out-of-plane displacements. In contrast to the mixed-type approaches, here it has been found that additional internal degrees of freedom are required in order to satisfy the standard patch tests. The elements developed in this way are capable of reproducing the exact analytical result for the case of cylindrical bending of certain order (quadratic for the four-node elements, cubic for the nine-node elements), but still suffer from the shear-locking effect for very coarse meshes. However, these elements perform competitively for a number of standard benchmark tests as the finite-element mesh is refined, one notable exception being the skewed plate, for which the presented elements still do not perform so well. The nine-node element, in general, is surprisingly successful when compared to the lower-order elements for the problems with the same total number of the degrees of freedom.

The work is in progress on the development of a family of linked interpolation elements designed along the lines presented here, which also include the 16-node element capable of reproducing the exact results for the case of cylindrical bending caused by the application of a uniformly distributed loading.

References

[1] A new family of quadrilateral thick plate finite elements based on linked interpolation, F. Auricchio, R.L. Taylor, Report No. UCB/SEMM-93/10, University of California at Berkeley, Department of Civil Engineering, Berkeley, 1993.

[2] Quadrilateral finite-elements for analysis of thick and thin plates, A. Ibrahimbegović, Computer Methods in Applied Mechanics and Engineering, 110: 195-209, 1993.

[3] A shear deformable plate element with an exact thin limit, F. Auricchio, R.L. Taylor, Computer Methods in Applied Mechanics and Engineering, 118: 393-412, 1994.

[4] Numerical analysis of some mixed finite element methods for Reissner-Mindlin plate, C. Chinosi, F. Lovadina, Computational Mechanics, 16: 36-44, 1995.

[5] Analysis of kinematic linked interpolation methods for Reissner-Mindlin plate problem, F. Auricchio, C. Lovadina, Computer Methods in Applied Mechanics and Engineering, 190: 2465-2482, 2001.

[6] A quadratic linked plate element with an exact thin plate limit, R.L. Taylor, S. Govindjee, Report No. UCB/SEMM-2002/10, University of California at Berkeley, Department of Civil and Environmental Engineering, Berkeley, 2002.

[7] The Finite Element Method for Solid and Structural Mechanics, O.C. Zienkiewicz, R.L. Taylor, Elsevier Butterworth-Heinemann, Oxford, 2005.

[8] The MIN-N family of pure-displacemnet, triangular, Mindlin plate elements, Liu,Y.J., Riggs H.R., Structural Engineering and Mechanics 19 (3), pp. 297-320, 2005.

[9] A simple hybrid stress element for shear deformable plates, de Miranda S., Ubertini F., International Journal for Numerical Methods in Engineering 65 (6), pp. 808-833, 2006.

[10] On a hierarchy of conforming Timoshenko beam elements, A. Tessler, S.B. Dong, Computers and Structures, 14:335-344, 1981.

[11] Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order, G. Jelenić, E. Papa, Archive of Applied Mechanics, DOI: 10.1007/s00419-009-0403-1, 2009.

[12] A class of mixed assumed-strain methods and the method of incompatible modes, J.C. Simo, M.S. Rifai, International Journal for Numerical Methods in Engineering, 29:1595–1638, 1990.

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[13] A triangular thick plate finite element with an exact thin limit, F. Auricchio, C. Lovadina, Finite Elements in Analysis and Design, 19:57-68, 1995.

[14] Finite Element Procedures, K.-J. Bathe, Prentice Hall, New Jersey, 1989. [15] The Finite Element Method. Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes, Dover

Publications inc., Mineola, New York, 2000. [16] The Finite Element Method. Its Basis and Fundamentals, O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, Elsevier

Butterworth-Heinemann, Oxford, 2005.