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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 International Congress of Croatian Society of Mechanics September 30 –October 2, 2009, Dubrovnik, Croatia 1

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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 2

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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 3

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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, w i, θ i - node displacements and rotations (equidistant) I n j – Lagrangian polynomials of n-1 order 4

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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 5

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3.1 Reissner-Mindlin theory of moderately thick plates 6 Stress resultants can be derived by integration over thickness of the plate and constitutive relations are or in matrix form: M = D b K S = D s Equilibrium follows from minimisation of the functional of the total potential energy

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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: w i are nodal transverse displacements, x i and y i (or just i ) are nodal rotations and w b is optional internal bubble parameter (just one) 7

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From stationary condition for the strain energy functional, a system of algebraic equations is derived: 8 3. Linked interpolation for thick plates f w, 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 w b will be condensed

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3.3 Plate element with four nodes The transverse displacement interpolation is bi-linear in the nodal parameters w i 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) 9

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3.3 Plate element with four nodes 10

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

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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: 12

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3.3 Plate element with four nodes Bubble shape function: 13

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3.4 Plate element with nine nodes 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 and ij and bicubic function for internal bubble parameter w b (just one) 14

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3.4 Plate element with nine nodes 15

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3.4 Plate element with nine nodes Bi-quadratic shape functions N w ij ij for displacements field and rotational field: 16

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

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3.4 Plate element with nine nodes Bubble shape function K w for the 9-node element 18

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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 19

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

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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 21

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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

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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 23

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Simply supported square plate 24 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

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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. 25

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Thank you for your kind attention 26

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