1. Dragan RIBARIĆ, Gordan JELENIĆ
Linked Interpolation in Higher-Order Triangular Mindlin Plate Finite Elements
Dragan RIBARIĆ, Gordan JELENIĆ
University of Rijeka, Civil Engineering Faculty, Rijeka, Croatia m

2. Outline
Motivation: Linked interpolation for straight thick beams (Timoshenko beam)
Generalisation to the 2D problem of thick plates (Mindlin theory of moderately thick plates).
Triangular elements with 3, 6 and 10 nodes
Comparable elements from literature
The patch test
Test examples
Application on facet shell elements
Conclusions

3. 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
q is the rotation of a cross section
- constitutive equations: and
- combined with equilibrium equations give: and
- differential equations to solve are: and

4. 1. Linked interpolation for thick beams
General solution for Timoshenko’s equations:
For polynomial loading of n-4 order the following interpolation completely reproduces the above exact results
L - beam length, wi , θi - node displacements and rotations (equidistant)
Inj – Lagrangian polynomials of n-1 order

5. 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

6. 2. Linked interpolation for thick plates
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 G
Equilibrium conditions (will not be used in the strong form):

7. 2. Linked interpolation for thick plates
From the stationarity condition on the functional of the total potential energy, a system of algebraic equations is derived:
fw, fq 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 wbk will be condensed

8. 2. Linked interpolation for thick plates
2.1 Triangular plate element with three nodes
Interpolation 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

9. 2. Linked interpolation for thick plates
2.2 Triangular plate element with six nodes
Interpolation functions:
for displacement
for rotations
The transverse displacement interpolation is
a complete cubic polynomial
The rotations are quadratic
The interpolations are conforming

10. 2. Linked interpolation for thick plates.
2.3 Triangular plate element with ten nodes
Interpolation 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.

11. 2. Linked interpolation for thick plates.
2.3 Triangular plate element with ten nodes
Interpolation functions:
for rotations
The rotations are complete cubic polynomials.
All interpolations are conforming.

12. 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.

13. 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:

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

15. 4. Test examples: clamped square plate
Table 3: Clamped square plate: displacement and moment at the centre using mesh pattern b), L/h = 1000.
Element
T3-U2 = MIN3
T6-U3
T10-U4
mesh w* M*
1x1
2x2
4x4
8x8
16x16
32x32
64x64
Ref. sol. [11]
T3-U2 3-node plate element with linked interpolation
T6-U3 6-node plate element with linked interpolationm
T10-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-LIM (using FEAP)
MIN6
T3BL [12 ]
mesh w* M*
1x1
2x2
1.735
4x4
2.209
8x8
2.275
16x16
2.287
32x32
2.290
64x64
Ref. sol. [11]
The dimensionless results w*= w / (qL4/100D) and M*=M / (qL²/100)

16. 5. Test examples: simply supported skew plate
Element
T3-U2 =MIN3
T6-U3
T10-U4
mesh w*
M22*
M11*
2x2
4x4
8x8
12x12
16x16
24x24
32x32
48x48
Ref. [31]
0.423
E=10.92, L=100.
n=0.30, h=1.0, q=1.0
T3-U2 3-node plate element with linked interpolation
T6-U3 6-node plate element with linked interpolation
T10-U4 10-node plate element with linked interpolation
T3-LIM Auricchio-Taylor mixed plate
element (FEAP)
MIN6 Liu-Riggs element with linear
shear
Elementm
T3-LIM [27 ]
MIN6
mesh w*
M22*
M11*
2x2
0.9207
1.7827
4x4
1.0376
1.8532
8x8
1.1008
1.9247
12x12
16x16
1.1233
1.9376
24x24
32x32
1.1284
1.9344
48x48
Ref. [31]
0.423
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

17. 4. Test examples: simply supported skew plate
Element
T3-U2 = MIN3
T6-U3
T10-U4
mesh w*
M22*
M11*
2x2
4x4
8x8
12x12
16x16
24x24
32x32
48x48
Ref. [36]
0.4080
1.08
1.91
E=10.92, L=100.
n=0.30, h=0.1, q=1.0
T3-U2 3-node plate element with linked interpolation
T6-U3 6-node plate element with linked interpolationm
T10-U4 10-node plate element with linked interpolationm
MIN6 Liu-Riggs element with linear
shear
Element
D.o.f.
MIN6
mesh w*
M22*
M11*
2x2 59
4x4
211
8x8
803
12x12
1779
16x16
3139
24x24
7011
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
[36] L.S.D. Morley, Bending of simply supported rhombic plate under uniform normal loading,
Quart. Journ. Mech. and Applied Math. Vol. 15, , 1962.

18. 4. Test examples: simply supported skew plate
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

19. 5. Application on shells
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

20. 5. Example of a folded plate structure
Table 8: Vertical and horizontal displacements at the control points of the folded plate structure (one quarter of the model)
T3-U2
T6-U3
T10-U4
Mesh wc we M8 M1
Q4-U2
SHELL from FEAP
Mesh wc we M8 M1

21. 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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