Linked Interpolation in Higher-Order Triangular Mindlin Plate Finite Elements

Dragan RIBARIĆ, Gordan JELENIĆdragan.ribaric@gradri.hr, gordan@gradri.hr

University of Rijeka, Civil Engineering Faculty, Rijeka, Croatia

1

Outline

1. Motivation: Linked interpolation for straight thick beams (Timoshenko beam)

2. Generalisation to the 2D problem of thick plates (Mindlin theory of moderately thick plates). o Triangular elements with 3, 6 and 10 nodeso Comparable elements from literature

3. The patch test

4. Test examples

5. Application on facet shell elements

6. Conclusions

2

1. Linked interpolation for thick beams

(Bernoulli’s limiting case for thin 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 is lateral displacement with respect to arc-length co-ordinate x. w’ is its derivative respect x is the rotation of a cross section

- constitutive equations: and

- combined with equilibrium equations give: and

- differential equations to solve are: and

3

'wdx

dw

dx

dEIM

sGAS

0 dx

dw

qEI ''' qwGAs )'''( dx

dMS

dx

dSq

1. Linked interpolation for thick beams

General solution for Timoshenko’s equations:

322

12

11CxCxCqdxdxdx

EI

.2

1

6

11154

22

31 CxCxCxCqdxdxdxdx

EIqdxdx

GAw

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

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

4

2. Linked interpolation for thick plates

Mindlin theory of moderately thick plates

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

5

ww

y

x

y

wx

w

y

x

x

y

yz

xz

eθΓ

01

10Lθκ

y

x

xy

x

y

xy

y

x

yx

y

x

xy

y

x

0

0

2. Linked interpolation for thick plates

6

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 conditions (will not be used in the strong form):

xxyx S

y

M

x

M

yyxy S

y

M

x

M

q

y

S

x

S yx

7

From the stationarity condition on the functional of the total potential energy,

a system of algebraic equations is derived:

2. Linked interpolation for thick plates

b

w

b

yixi

i

bbbbw

Tbwwww

Tbw

Twwww

f

f

f

w

w

KsKsKs

KsKsKbKs

KsKsKs

,,

dALIDLIKb bT

A)()(

dAKeIDKKs

dAIDKKs

dAKDKKs

dAKeIDKeIKs

dAIDIKs

wsT

A wbb

wsT

A wbbw

wbsT

A wbbb

wsT

A www

wsT

A www

)()(

)()(

)()(

)()(

)()(

• fw, f and fb are the terms due to load and boundary conditions.• Of all the blocks in the stiffness matrix only one depends on the bending strain energy and all others are derived from the shear strain energy:

Internal bubble parameter wb will be condensed

extTT

extTT

yx dAdAdAdAw )(2

1)(

2

1)(

2

1)(

2

1),,( ΓDΓκDκΓSκM sb

2.1 Triangular plate element with three nodesInterpolation functions:• for displacement

• and rotations

Area coordinates of an interior point

• The transverse displacement interpolation is

a complete quadratic polynomial and • The rotations are linear• The interpolations are conforming

8

2. Linked interpolation for thick plates

332211 xxxx

332211 yyyy

3211

32132121332211 2

1abwwww yyxx

213213132

1ab yyxx

332211 xxxx

332211 yyyy

132132322

1ab yyxx

2. Linked interpolation for thick plates

2.2 Triangular plate element with six nodesInterpolation functions:• for displacement

• for rotations

• The transverse displacement interpolation is

a complete cubic polynomial • The rotations are quadratic• The interpolations are conforming

9

613532421333222111 444121212 wwwwwww

324132411221 223

1ab yyyxxx

135213522332 223

1ab yyyxxx

216321633113 223

1ab yyyxxx

613532421333222111 444121212 xxxxxxx

613532421333222111 444121212 yyyyyyy

bw321

2. Linked interpolation for thick plates.

2.3 Triangular plate element with ten nodesInterpolation functions:• for displacement

• The transverse displacement interpolation is a complete cuartic polynomial (15 terms from Pascal’s triangle).

• The third term that appears to be missing in expression to complete the cyclic triangle symmetry, namely is actually linearly dependent on the two other added terms and the 10th term in w.

10

522141211111 2

913

2

913

2

11323 wwww

733262322222 2

913

2

913

2

11323 www

10321911383133333 272

913

2

913

2

11323 wwww

32541325412121 33338

11313 ab yyyyxxxx

13762137623232 33338

11313 ab yyyyxxxx

21983219831313 33338

11313 ab yyyyxxxx

232321121321 )()( bb ww

313321 )( bw

2. Linked interpolation for thick plates.

2.3 Triangular plate element with ten nodesInterpolation functions:• for rotations

• The rotations are complete cubic polynomials.• All interpolations are conforming.

11

522141211111 2

913

2

913

2

11323 xxxx

733262322222 2

913

2

913

2

11323 xxx

10321911383133333 272

913

2

913

2

11323 xxxx

522141211111 2

913

2

913

2

11323 yyyy

733262322222 2

913

2

913

2

11323 yyy

10321911383133333 272

913

2

913

2

11323 yyyy

2. Linked interpolation for thick plates – elements from literature

2.4 MIN3 - Triangular plate element with three nodes (Tessler, Hughes, 1985.)Is derived to have linear shear expression in every direction crossing the element.

Interpolation functions:

for i=1,2,3

The interpolation for MIN3 is transformed T6-U3 interpolation.

12

yiixiiii MLwNw

xiix N

yiiy N

iiN

ikjjiki bbL 2

1

jikikji aaM 2

1

sincos yxs

i

2. Linked interpolation for thick plates – elements from literature

2.5 MIN6 –triangular plate element with six nodes (Liu, Riggs, 2005)

Is derived to have linear shear expression in every direction crossing the element.

Interpolation functions:

for i=1,2,…6 for i=1,2,3 for i=4,5,6

The rigid body mode conditions should be satisfied for functions N, L and M:

Liu–Riggs interpolation for MIN6 should coincide with the T6-U3 interpolation, if for wb is taken:

13

yiixiiii MLwNw

xiix N

yiiy N

12 iiiN jiiN 43

kjjkiii bbL 3

112

kjjkiii aaM 3

112

2

1

2

1

3

143 jiijjii bbL

2

1

2

1

3

143 jiijjii aaM

16

1

i

iN 06

1

i

iL 06

1

i

iM

3123122312311231233

1yxyxyxb aabbaabbaabbw

6316315235234124123

2yxyxyx aabbaabbaabb

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4. Test examples: clamped square plate

Clamped square plate uniformly loaded – Mx distribution along the centreline (y=0) obtained with 4x4 mesh for one quarter of the plate

For T3-U2:• Mx along x and y axes is constant for any value of ν.

For higher order elements:• Mx along x and y axes is a function proportional to higher order

CL

15

4. Test examples: clamped square plate

T3-U2 3-node plate element with linked interpolationT6-U3 6-node plate element with linked interpolationmT10-U4 10-node plate element with

linked interpolationm

T3BL and T3-LIM Auricchio-Taylor mixed plate element (FEAP)MIN6 Liu-Riggs 6 node plate element

Element T3-U2 = MIN3 T6-U3 T10-U4

mesh w* M* w* M* w* M*

1x1 0.000069 0.00206 0.130198 4.05972

2x2 0.000062 0.00127 0.097869 2.03760 0.126738 2.74752

4x4 0.001793 0.03807 0.121256 2.44562 0.126527 2.34533

8x8 0.028267 0.55913 0.125905 2.38599 0.1265340 2.29246

16x16 0.104428 2.12079 0.1265121 2.31332 0.1265344 2.29055

32x32 0.124820 2.32200 0.1265341 2.29420

64x64 0.126403 2.29435

Ref. sol. [11] 0.126532 2.29051 0.126532 2.29051 0.126532 2.29051

Element T3-LIM (using FEAP) MIN6 T3BL [12 ]

mesh w* M* w* M* w* M*

1x1 0.000061 0.00182

2x2 0.093098 1.735 0.097850 2.03613 0.093098 1.40767

4x4 0.118006 2.209 0.121205 2.43983 0.118006 2.10245

8x8 0.124616 2.275 0.125878 2.38437 0.124616 2.24825

16x16 0.126092 2.287 0.1265107 2.31409 0.126092 2.28031

32x32 0.126429 2.290 0.1265341 2.29440 0.126429 2.28798

64x64 0.126509 2.290 0.126509 2.28987

Ref. sol. [11] 0.126532 2.29051 0.126532 2.29051 0.126532 2.29051

Table 3: Clamped square plate: displacement and moment at the centre using mesh pattern b), L/h = 1000.

CL

C L

CL

C L

The dimensionless results w*= w / (qL4/100D) and M*=M / (qL²/100)

16

5. Test examples: simply supported skew plate

E=10.92, L=100.=0.30, h=1.0, q=1.0

T3-U2 3-node plate element with linked interpolationT6-U3 6-node plate element with linked interpolationT10-U4 10-node plate element with

linked interpolation

T3-LIM Auricchio-Taylor mixed plate element (FEAP)MIN6 Liu-Riggs element with linear shear Table 6: Simply supported skew plate (SS1): displacement and

moment at the centre with regular meshes, L/h = 100

w*= w / (qL4/10000D) M*=M / (qL²/100)with D=Eh³/(12(1-ν²)) and L is a span

Elementm T3-LIM [27 ] MIN6

mesh w* M22* M11* w* M22* M11*

2x2 0.63591 0.9207 1.7827 0.442458 1.61239 2.53454

4x4 0.45819 1.0376 1.8532 0.386472 1.36333 2.10434

8x8 0.43037 1.1008 1.9247 0.405862 1.16617 1.95019

12x12 0.414750 1.13335 1.94337

16x16 0.42382 1.1233 1.9376 0.418385 1.13270 1.94618

24x24 0.421307 1.13604 1.94954

32x32 0.42183 1.1284 1.9344

48x48

Ref. [31] 0.423 0.423

Element T3-U2 =MIN3 T6-U3 T10-U4

mesh w* M22* M11* w* M22* M11* w* M22* M11*

2x2 0.425288 0.65647 1.35584 0.442337 1.59547 2.48908 0.259711 0.67991 1.29292

4x4 0.393156 1.00823 1.72050 0.391393 1.38415 2.10533 0.410136 1.17851 1.92258

8x8 0.376569 1.11747 1.84630 0.409028 1.18349 1.96100 0.419818 1.12774 1.94013

12x12 0.416692 1.14172 1.94918

16x16 0.403524 1.09072 1.87753 0.419769 1.13814 1.95024 0.423207 1.13774 1.95080

24x24 0.412799 1.10360 1.92291 0.422181 1.13934 1.95222

32x32 0.416390 1.11361 1.93165

48x48 0.419306 1.12368 1.93948

Ref. [31] 0.423 0.423 0.423

17

4. Test examples: simply supported skew plate

[36] L.S.D. Morley, Bending of simply supported rhombic plate under uniform normal loading, Quart. Journ. Mech. and Applied Math. Vol. 15, 413-426, 1962.

E=10.92, L=100.=0.30, h=0.1, q=1.0

T3-U2 3-node plate element with linked interpolationT6-U3 6-node plate element with linked interpolationmT10-U4 10-node plate element with

linked interpolationm

MIN6 Liu-Riggs element with linear shear

Element T3-U2 = MIN3 T6-U3 T10-U4

mesh w* M22* M11* w* M22* M11* w* M22* M11*

2x2 0.421115 0.64790 1.33819 0.443104 1.62431 2.52722 0.246108 0.60215 1.16324

4x4 0.393999 1.01757 1.67013 0.348698 1.30423 1.98429 0.356469 1.05704 1.76663

8x8 0.305318 1.11745 1.58532 0.326564 0.86335 1.71994 0.365434 0.95047 1.78325

12x12 0.343720 0.91364 1.76582 0.390331 1.02111 1.85543

16x16 0.285607 1.05519 1.61422 0.358165 0.97179 1.80192

24x24 0.309866 1.05211 1.68162 0.376441 0.99480 1.82813

32x32 0.330657 1.00984 1.72183

48x48 0.360440 0.96062 1.77377

Ref. [36] 0.4080 1.08 1.91 0.4080 1.08 1.91 0.4080 1.08 1.91

Element D.o.f. MIN6

mesh w* M22* M11*

2x2 59 0.443109 1.62421 2.52671

4x4 211 0.348478 1.30308 1.98405

8x8 803 0.324182 0.83373 1.70398

12x12 1779 0.339881 0.86754 1.74022

16x16 3139 0.354328 0.93747 1.78247

24x24 7011 0.373069 0.97850 1.81691

32x32

48x48

Ref. [36] 0.4080 1.08 1.91

Table 7: Simply supported skew plate (SS1): displacement and moment at the centre with regular meshes, L/h = 1000

w*= w / (qL4/10000D) M*=M / (qL²/100)with D=Eh³/(12(1-ν²)) and L is a span

Figure 19: Simply supported skew plate under uniform load – b) principal moment in D-C-E direction (M22) distribution along diagonal A-C - a) perpendicular principal moment (M11) distribution along diagonal A-C

18

4. Test examples: simply supported skew plate

CLCL

5. Application on shells

19

• Basic triangular elements T3U2, T6U3 and T10U4 can be applied on facet shell elements to approximate folded plate structures and shells.

• Inplane stiffness is added to the transverse stiffness of the element

• Straight element sides insure constant shear along the element

5. Example of a folded plate structure

20

T3-U2 T6-U3 T10-U4Me

sh

wc we wc we wc we

M8 -0.05137 0.06109 -0.055587 0.06240 -0.056273 0.06280

M1 -0.05265 0.06302 -0.057022 0.06394 -0.056426 0.06022

Q4-U2 SHELL from FEAPMe

sh

wc we wc we

M8 -0.05558 0.06355 -0.056056 0.06378

M1 -0.05690 0.06479 -0.057189 0.06479

Table 8: Vertical and horizontal displacements at the control points of the folded plate structure (one quarter of the model)

7. Conclusions

• A family of linked interpolation functions for straight Timoshenko beam is generalized to 2D plate problem of solving Mindlin equations for moderately thick plates

• Resulting solutions are just approximations to the true solution problem unlike straight Timoshenko beam where exact solution is achieved

• Displacement field and rotational field for plate behavior are interdependent . Only first derivatives are needed

• The bubble term for the displacement field (not present in beam element) is important for satisfying standard patch tests, especially for higher order elements and higher order patch tests.

• Linked interpolation formulations for 3-node thick plate elements, often combined with additional internal degrees of freedom, were proposed earlier in the literature. Here we propose a structured family of thick plate elements based on the interpolation of just displacements and rotations (displacement based approach).

• They are reasonably competitive to the elements based on mixed approaches in designing thick and thin plates and folded plate structures.

• In the limiting case of thin plates, depending on type of loading, low order elements exhibit locking due to inadequate shear interpolation and they require denser meshes to completely overcome this effect.

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Thank you

for your kind attention

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