Higher-order Linked Interpolation in Thick Plate Finite Elements Gordan JELENIĆ, Dragan RIBARIĆ...

Higher-order Linked Interpolation in Thick Plate Finite Elements

Gordan JELENIĆ, Dragan RIBARIĆUniversity of Rijeka, Faculty of Civil Engineering, V.C. Emina 5,

51000 Rijeka

International Congress of Croatian Society of MechanicsSeptember 30 –October 2, 2009, Dubrovnik, Croatia

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1. Introduction

In this presentation:• A family of linked interpolation functions for straight Timoshenko beam is generalised

to 2D plate problem of solving Reissner-Mindlin equations for moderately thick plates• Resulting solutions are just approximation to the true solution problem• Displacement field and rotational field for plate behaviour are interdependent • Linked interpolation has been used in formulation of 3-node and 4-node thick plate

elements, often combined with same kind of internal degrees of freedom.• Avoiding shear locking effects and reduced integration• Here we propose a structured family of linked interpolation functions for thick plate

elements

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2. Linked interpolation for thick beams

Timoshenko theory of beams:

- hypothesis of planar cross sections after the deformation (Bernoulli),

- but not necessarily perpendicular to the centroidal axis of the deformed beam:

= w’+ w is lateral displacement with respect to arc-length co-ordinate x. w’ is it’s derivative respect x is the rotation of a cross section

- constitutive equations: M= -EI and T= GA - combined with equilibrium equations give:

M’= T and T’= q - differential equations to solve are:

EI’’’=q and GA(w’’+’)= -q

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2. Linked interpolation for thick beams

General solution for Timoshenko’s equations:

322

12

11CxCxCqdxdxdx

EI (5)

.2

1

6

11154

22

31 CxCxCxCqdxdxdxdx

EIqdxdx

GAw (6)

For polynomial loading of n-4 order the following interpolation completely reproduces the above exact results

n

ii

inI

1

, ,1

1)1(

1 1 1

1i

n

i

n

j

n

i

ijni

in i

nN

n

LwIw

(7)

L - beam length, wi , θi - node displacements and rotations

(equidistant)In

j – Lagrangian polynomials of n-1 order

L

xN j

n for j=1 and L

x

j

nN j

n 1

11

otherwise

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3. Linked interpolation for thick plates

3.1 Reissner-Mindlin theory of moderately thick plates

,,

,

,,

,

,

y

xy

yz

xz

xxyy

yx

xy

xy

yy

xx

w

w K

Kinematics of the plate gives relations for curvature vector and shear strain vector

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3.1 Reissner-Mindlin theory of moderately thick plates

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dz

dz

S

S

dzz

dzz

dzz

M

M

M

yz

xz

y

x

xy

y

x

xy

y

x

S,

M

xy

y

x

xy

y

xEt

M

M

M

1

2

100

01

01

)1(12 2

3

y

x

y

x Etk

S

S

10

01

)1(2

Stress resultants can be derived by integration over thickness of the plate

and constitutive relations are

or in matrix form:

M = Db K S = Ds

Equilibrium follows from minimisation of the functional of the total potential energy

3. Linked interpolation for thick plates

3.2 Linked interpolation for plates is based on the generalisation of the linked interpolation for beams

Displacement and rotational fields are described as:

wi are nodal transverse displacements, x

i and yi (or just i) are nodal rotations and

wbis optional internal bubble parameter (just one)

bi

wi

w wKKwNw and

iN ,

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From stationary condition for the strain energy functional, a system of algebraic equations is derived:

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3. Linked interpolation for thick plates

f

f

f

w

w

KsKsKs

KsKsKbKs

KsKsKs w

byixi

i

w

Twwww

Tw

Twwww

,,

dALNDLNKb bT

A)()(

dAKeNDKKs

dANDKKs

dAKDKKs

dAKeNDKeNKs

dANDNKs

wsT

A w

wsT

A ww

wsT

A w

wsT

A www

wsT

A www

)()(

)()(

)()(

)()(

)()(

• fw, f and f are terms due to load and boundary conditions.

• Of all partitions of the stiffness matrix only one depends on bending strain energy and all others are derived from shear strain energy:

Internal bubble parameter wb will be condensed

3.3 Plate element with four nodes

bni

nj

ii

wii

i

wi wKLKwNw

)(4

1

4

1

xii

ix N

4

1, yi

iiy N

4

1,

The transverse displacement interpolation is bi-linear in the nodal parameters wi

enriched with linked quadratic-linear and linear-quadratic functions in terms of

n and n , and a bi-quadratic function for internal bubble parameter w

b (just one)

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3.3 Plate element with four nodes

Where ijN , w

iK ,w

jK and wK are:

)1)(1(4

1

)1)(1(4

1

)1)(1(4

1

)1)(1(4

1

4

3

2

1

N

N

N

N

)1)(1(

16

1

)1)(1(16

1

)1)(1(16

1

)1)(1(16

1

24

23

22

21

w

w

w

w

K

K

K

K

and the bubble shape function is

4

)1(

4

)1( 22

wK

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3.3 Plate element with four nodes

Bi-linear shape functions for displacement and rotational field

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3.3 Plate element with four nodes

Quadratic-linear and linear-quadratic shape functions for displacement fieldlinked functions of one order higher then shape function for rotational field:

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3.3 Plate element with four nodes

• Bubble shape function:

f 1

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3.4 Plate element with nine nodes

byj

yj

yj

j

yj

wj

xi

xi

xi

i

xwiij

x

ji

wij wKaKbKwNw

i

)2()2( 321

3

1321

3

1

33

1,

xij

x

jiij

x N

33

1,

yij

x

jiij

y N

33

1,

The transverse displacement interpolation is biquadratic in the nodal parameters w ij

enriched with linked quadratic-cubic and cubic-quadratic functions in terms of

ij andijand bicubic function for internal bubble parameter w

b (just one)

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3.4 Plate element with nine nodes

Where Nwij =

ijN =

ijN ,

wiK ,

wjK and wK are:

)(2

)1()(

2

)1(2

)1(

2

)1()(

2

)1(4

2

)1())((

2

)1(

32

21

11

w

w

w

N

N

N

)(2

)1(

4

)1()(

3

2

)1(

2

)1(

4

)1()(

3

2

)1()(

4

)1()(

3

2

31

2

21

2

11

aK

aK

aK

w

w

w

(just the first column) (just the first column in )

and 4

)1())((

4

)1( 22

wK

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3.4 Plate element with nine nodes

Bi-quadratic shape functions Nwij

ijfor displacements field and

rotational field:

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3.4 Plate element with nine nodes

a b

K1

ab

K2

ba

K3

ba

K1

a b

K2

a

K3

Quadratic-cubic and cubic-quadratic shape functions Kwij for

displacements field – linked functions of one order higher then shape function for rotational field:

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3.4 Plate element with nine nodes

Bubble shape function Kw for the 9-node element

f

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Example 1: Cylindrical bending

Q4-LIM is Auricchio-Taylor mixed plate element9βQ4 is De Miranda-Ubertini hybrid stress plate elementQ4-U02 4-node plate element with linked interpolationQ9-U03 9-node plate element with linked interpolation

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E=1000.0, ν=0, k=5/6, L=1.0, L/h=1, F=400.0 (at the midspan)

Table I: Cylindrical bending and shear of simply supported strip Mash Q4-LIM Q4-U02 wb wtot S M wb wtot S M 1x10 0.099829 0.33983 200.0 90.0 0.099000 0.33900 200.0 90.0 1x20 0.099958 0.33996 200.0 95.0 0.099750 0.33975 200.0 95.0 1x50 0.099994 0.33999 200.0 98.0 0.099960 0.33996 200.0 98.0 1x100 0.099999 0.34000 200.0 99.0 0.099990 0.33999 200.0 99.0

Exact 0.10000 0.34000 200.0 100.0 0.100000 0.34000 200.0 100.0 Mash 9βQ4 Q9-U03 wb wtot S M wb wtot S M 1x1 or 1x2 0.10000 0.34000 200.0 100.0 0.10000 0.34000 200.0 100.0 1x10 0.10000 0.34000 200.0 100.0 0.10000 0.34000 200.0 100.0 1x100 0.10000 0.34000 200.0 100.0 0.10000 0.34000 200.0 100.0

Exact 0.10000 0.34000 200.0 100.0 0.100000 0.34000 200.0 100.0

Example 2: Clamped square plate

Square plate under uniform load: a) clamped; b) simply supported SS1; c) simply supported SS2

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Clamped square plate

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

Mash Q4-LIM 9βQ4 Q4-U02 Q9-U03 w* M* w* M* w* M* w* M* 2x2 0.12107 0.16252 2.838 0.02679 4x4 0.14211 1.660 0.15344 2.448 0.11920 1.638 8x8 0.14858 2.157 0.15118 2.351 0.14361 2.164 16x16 0.14997 2.279 0.15064 2.328 0.14876 2.281 0.14957 2.290 32x32 0.15034 2.310 0.15051 2.322 0.15004 2.310 0.15039 2.313

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

w*= w / (qL4/100D)

M*=M / (qL²/100)

with D=Eh³/(12(1-ν²)) and L is a span (Lx=Ly=L)

Q4-LIM is Auricchio-Taylor mixed plate element9βQ4 is De Miranda-Ubertini hybrid stress plate elementQ4-U02 4-node plate element with linked interpolationQ9-U03 9-node plate element with linked interpolation

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Clamped square plate

Q4-LIM is Auricchio-Taylor mixed plate element9βQ4 is De Miranda-Ubertini hybrid stress plate elementQ4-U02 4-node plate element with linked interpolationQ9-U03 9-node plate element with linked interpolation

Table III: Clamped square plate: displacement and moment at the centre with regular mashes, L/h = 1000:

Mash Q4-LIM 9βQ4 Q4-U02 Q9-U03 w* M* w* M* w* M* w* M* 2x2 0.09074 0.13767 2.7455 0.0000027 4x4 0.11469 0.12939 2.4239 0.0001284 0.0000131 8x8 0.12362 2.120 0.12725 2.3239 0.0046933 0.1011 0.098505 1.762 16x16 0.12584 2.248 0.12671 2.2989 0.059881 1.1702 0.12055 2.222 32x32 0.12637 2.280 0.12658 2.2926 0.11899 2.1680 0.12553 2.277

Ref. sol. 0.12653 2.2905 0.12653 2.2905 0.12653 2.2905 0.12653 2.2905

w*= w / (qL4/100D)

M*=M / (qL²/100)

with D=Eh³/(12(1-ν²)) and L is a span (Lx=Ly=L)

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Simply supported square plate

Q4-LIM is Auricchio-Taylor mixed plate element9βQ4 is De Miranda-Ubertini hybrid stress plate elementQ4-U02 4-node plate element with linked interpolationQ9-U03 9-node plate element with linked interpolation

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Simply supported square plate

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Q4-LIM is Auricchio-Taylor mixed plate element9βQ4 is De Miranda-Ubertini hybrid stress plate elementQ4-U02 4-node plate element with linked interpolationQ9-U03 9-node plate element with linked interpolation

Conclusions

• Elements developed on the linked interpolation model are defined by just displacements and rotations, with no other requirements.

• They are reasonably competitive to the elements based on mixed approaches in designing thick plates.

• In the limiting case of thin plates they do not behave so well as mixed elements and they require denser mashes. In coarse mashes they are subject to shear locking.

• The bubble interpolation function for the displacement field (not present in beam) is important for satisfying standard patch tests, especially for irregular meshes.

• Elements with any higher number of nodes, for example could be constructed following the same approach.

• Overcoming the above difficulties is the aim of our further research.

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Thank you

for your kind attention

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