# Moderately thick plates: theory and finite element formulations

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## Model error. Discretization error. The optimal model.

In order to find the optimal model for the analysis of a structure behavior – in our case elastic plate bending – the criteria have to be set first. If the accuracy is the only criteria, the best possible model is the exact fully three-dimensional model of elastic plate bending. Such model is of course prohibitively complicated for usual plate bending problems. The plate model is therefore proclaimed as optimal in the engineering sense: besides the necessary (desired) accuracy, the model must be as simple as possible.
The plates are basic structural elements and several models of their bending behavior exist. Since the third dimension of the plate is significantly smaller than the other two, the plate models are subject to dimensional reduction. The computational advantage of using two-dimensional model instead of fully three-dimensional model is obvious. Although the three dimensional model should asymptotically converge to two-dimensional plate model (for decreasing plate thickness), not many plate models are strictly build on the principles of asymptotic analysis. Various important plate models are rather based on a-priori assumptions coming from the engineering intuition. Nevertheless, a hierarchy of such plate models can still be established with respect to the convergence to the full three-dimensional model, [Babuˇska Li, 1990]. It is thus possible to form a family of hierarchically ordered plate models – a series of models, whose solutions converge to the exact three-dimensional solution of plate bending problem. The convergence to the exact three-dimensional solution manifests itself, for example, in the ability to describe the boundary layer effects, which are typical for the three-dimensional solution.

### Further details of the Kirchhoff model

The biharmonic equation (2.2.26) is a partial differential equation of the fourth order. It can only have two boundary conditions at each boundary point. In the moderately thick plate model three independent quantities appear naturally at a boundary point: the transverse displacement w, the normal rotation n and the tangential rotation s. The conjugate quantities are the transverse shear force q, twisting moment mn and tangential (bending) moment ms respectively. Integration per partes in (2.2.43), however, shows that only two pairs of conjugate quantities exist for the Kirchhoff model: (1) displacement w and effective shear qef , and (2) tangential rotation s and tangential moment ms. This is the direct consequence of the Kirchhoff assumption (2.2.6) which relates displacement to the rotations.

#### An overview of thin plate elements

Among the first successful conforming plate elements was the rectangular element with 16 DOFs ( ˆ w, ˆ w,x, ˆ w,y and ˆ w,xy at each node) developed by [Bogner et al., 1966]. The element uses Hermitian interpolation functions with bi-cubic shape functions. The triangular elements offer more flexibility in the design of the conforming elements that the rectangular ones. It was shown in [Felippa, 2009] that a complete fifth order polynomial is the first which satisfy the C1 continuity requirements. From the Pascal’s triangle it follows that the complete quintic polynomial contains 21 terms. Conforming triangular plate element with 21 DOFs was independently developed by several authors ([Argyris et al., 1968] et al., [Bell, 1969], [Bosshard, 1968], [Visser, 1968]). Conforming quadrilateral elements can be produced from the conforming triangular elements. If the combined element has an additional internal degrees of freedom they are eliminated by static condensation. A direct derivation of conforming quadrilateral element is also possible. The derivation was first proposed by [Sander, 1964] and [De Veubeke, 1968].
Although the conforming plate elements exhibit excellent accuracy and performance they have some important drawbacks. The C1 continuity presents a difficulty when discontinuous variation of material properties occurs. It is also difficult to impose the right boundary conditions since ’nodal forces’ as energy conjugates to chosen degrees of freedom can no longer be interpreted intuitively.

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Natural coordinate systems

We have thought of shape function Ni in section 2.3.1 as of a function defined over Pi h . In practice, the shape functions are defined over a finite element e h.
The description of the element shape functions in the Cartesian coordinates (x, y) is not very convenient. We therefore seek for coordinates i, i = 1, 2 such that the location of a point P(x, y) inside an element defined by nodes (xi, yi), i = (1, 2, 3)(triangle) or i = (1, 2, 3, 4)(quadrilateral) would be described in a ”natural” way; i.e. as x = P i Ni(k)xi where Ni is now an element shape function, defined by i over e h. The coordinate system which emerges naturally is therefore called a natural coordinate system.

1 Introduction
1.1 Motivation
1.1.1 Verification and Validation
1.1.2 Discretization
1.1.3 Model error. Discretization error. The optimal model.
1.1.4 Adaptive modeling. Error estimates and indicators
1.2 Goals of the thesis
1.3 Outline of the thesis
2 Thin plates: theory and finite element formulations
2.1 Introduction
2.2 Theory
2.2.1 Governing equations
2.2.2 Further details of the Kirchhoff model
2.3 Finite elements
2.3.1 Preliminary considerations
2.3.2 An overview of thin plate elements
2.3.3 Natural coordinate systems
2.3.4 Conforming triangular element
2.3.5 Discrete Kirchhoff elements
2.4 Examples
2.4.1 Uniformly loaded simply supported square plate
2.4.2 Uniformly loaded clamped square plate
2.4.3 Uniformly loaded clamped circular plate
2.4.4 Uniformly loaded hard simply supported skew plate
2.5 Chapter summary and conclusions
3 Moderately thick plates: theory and finite element formulations
3.1 Introduction
3.2 Theory
3.2.1 Governing equations
3.2.2 Strong form
3.2.3 Weak form
3.2.4 Boundary layer and singularities
3.3 Finite elements
3.3.1 Elements with cubic interpolation of displacement
3.3.2 Elements with incompatible modes
3.4 Examples
3.4.1 Uniformly loaded simply supported square plate
3.4.2 Uniformly loaded simply supported free square plate
3.4.3 Uniformly loaded simply supported skew plate
3.4.4 Uniformly loaded soft simply supported L-shaped plate
3.5 Chapter summary and conclusions
4 Thick plates: theory and finite element formulations
4.1 Introduction
4.2 Theory
4.2.1 Governing equations
4.2.2 Principle of virtual work
4.3 Finite elements
4.3.1 Higher-order plate elements
4.4 Hierarchy of derived plate elements
4.5 Examples
4.5.1 Uniformly loaded simply supported square plate
4.5.2 Uniformly loaded soft simply supported L-shaped plate
4.5.3 Simply supported square plate with load variation
4.6 Chapter summary and conclusions
5 Discretization error estimates
5.1 Introduction. Classification of error estimates
5.2 Definitions. Linear elasticity as a model problem
5.3 Recovery based error estimates
5.3.1 Lumped projection
5.3.2 Superconvergent patch recovery (SPR)
5.4 Residual based error estimates
5.4.1 Explicit
5.4.2 Implicit
5.5 Illustration of SPR and EqR methods on 1D problem
5.6 Chapter summary and conclusions
6 Discretization error for DK and RM elements
6.1 Introduction
6.2 Discretization error for DK elements
6.2.1 Discrete approximation for error estimates
6.2.2 Formulation of local boundary value problem
6.2.3 Enhanced approximation for the local computations
6.2.4 Numerical examples
6.2.5 An example of adaptive meshing
6.3 Discretization error for RM elements
6.4 Chapter summary and conclusions
7 Model error concept
7.1 Introduction
7.2 Model error indicator based on local EqR computations
7.2.1 Definition of model error indicator
7.2.2 Construction of equilibrated boundary tractions for local problems .
7.3 Chapter summary and conclusions
8 Model error indicator for DK elements
8.1 Introduction
8.2 Model error indicator
8.2.1 Regularization
8.2.2 The local problems with RM plate element
8.2.3 Computation of the model error indicator
8.2.4 Numerical examples
8.3 Chapter summary and conclusions
9 Conclusions
10 Razˇsirjeni povzetek
10.1 Motivacija
10.1.1 Verifikacija in validacija
10.1.2 Prilagodljivo modeliranje – optimalni model
10.1.3 Napaka diskretizacije, modelska napaka
10.2 Cilji