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Linked Interpolation in Higher-Order Triangular Mindlin Plate Finite Elements
Dragan RIBARIĆ, Gordan JELENIĆ[email protected], [email protected]
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
14
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.
21
Thank you
for your kind attention
22