1. Higher-order Linked Interpolation in Thick Plate Finite Elements
International Congress of Croatian Society of Mechanics
September 30 –October 2, 2009, Dubrovnik, Croatia
Higher-order Linked Interpolation in Thick Plate Finite Elements
Gordan JELENIĆ, Dragan RIBARIĆ
University of Rijeka, Faculty of Civil Engineering, V.C. Emina 5,
51000 Rijeka

2. 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

3. 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:
g= w’+q
w is lateral displacement with respect to arc-length co-ordinate x.
w’ is it’s derivative respect x
q is the rotation of a cross section
- constitutive equations: M= -EIq and T= GAg
- combined with equilibrium equations give:
M’= T and T’= q
- differential equations to solve are:
EIq’’’=q and GA(w’’+q’)= -q

4. 2. 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. 3. Linked interpolation for thick plates
3.1 Reissner-Mindlin theory of moderately thick plates
Kinematics of the plate gives relations for curvature vector and shear strain vector

6. 3.1 Reissner-Mindlin theory of moderately 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 follows from minimisation of the functional of the total potential energy

7. 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,
q xi and q yi (or just q i) are nodal rotations and
wkb is optional internal bubble parameter (just one)

8. 3. Linked interpolation for thick plates
From stationary condition for the strain energy functional, a system of algebraic equations is derived:
fw, fq and fk 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 wbk will be condensed

9. 3.3 Plate element with four nodes
The transverse displacement interpolation is bi-linear in the nodal parameters wi enriched with linked quadratic-linear and linear-quadratic functions in terms
of qxn and qhn , and a bi-quadratic function for internal bubble parameter wkb (just one)

10. 3.3 Plate element with four nodes

11. 3.3 Plate element with four nodes
Bi-linear shape functions for displacement and rotational field

12. 3.3 Plate element with four nodes
Quadratic-linear and linear-quadratic shape functions for displacement field - linked functions of one order higher then shape function for rotational field:

13. 3.3 Plate element with four nodes
Bubble shape function:

14. 3.4 Plate element with nine nodes
The transverse displacement interpolation is biquadratic in the nodal parameters wij enriched with linked quadratic-cubic and cubic-quadratic functions in terms
of qxij and qhij , and bicubic function for internal bubble parameter wkb (just one)

15. 3.4 Plate element with nine nodes

16. 3.4 Plate element with nine nodes
Bi-quadratic shape functions Nwij = N qij for displacements field and rotational field:

17. 3.4 Plate element with nine nodes
Quadratic-cubic and cubic-quadratic shape functions Kwqij for displacements field – linked functions of one order higher then shape function for rotational field:

18. 3.4 Plate element with nine nodes
Bubble shape function Kwk for the 9-node element

19. Example 1: Cylindrical bending
Q4-LIM is Auricchio-Taylor mixed plate element
9βQ4 is De Miranda-Ubertini hybrid stress plate element
Q4-U02 4-node plate element with linked interpolation
Q9-U03 9-node plate element with linked interpolation

20. Example 2: Clamped square plate
Square plate under uniform load: a) clamped; b) simply supported SS1; c) simply supported SS2

21. Clamped square plate Q4-LIM is Auricchio-Taylor mixed plate element
9βQ4 is De Miranda-Ubertini hybrid stress plate element
Q4-U02 4-node plate element with linked interpolation
Q9-U03 9-node plate element with linked interpolation

22. Clamped square plate Q4-LIM is Auricchio-Taylor mixed plate element
9βQ4 is De Miranda-Ubertini hybrid stress plate element
Q4-U02 4-node plate element with linked interpolation
Q9-U03 9-node plate element with linked interpolation

23. Simply supported square plate
Q4-LIM is Auricchio-Taylor mixed plate element
9βQ4 is De Miranda-Ubertini hybrid stress plate element
Q4-U02 4-node plate element with linked interpolation
Q9-U03 9-node plate element with linked interpolation

24. Simply supported square plate
Q4-LIM is Auricchio-Taylor mixed plate element
9βQ4 is De Miranda-Ubertini hybrid stress plate element
Q4-U02 4-node plate element with linked interpolation
Q9-U03 9-node plate element with linked interpolation

25. 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.

26. for your kind attention
Thank you
for your kind attention