# Plates & Shells - Special Issues in Finite Element Analysis

## Course Description:

The first part of the course aims at understanding the behavior of the plate, which is a basic structural element in the civil engineering. The plate equation is derived and several plate problems are solved. In the second part, finite elements (FE) used for plate analysis are presented. A simple introduction to the theory of shells is then given. Static behavior of cylindrical and axisymmetric shells as well as their numerical simulation through FE is attempted next. The discretization error in the FE method and ways to handle it are then discussed. Finally mixed types of FEs are presented.

### Prerequisite Knowledge

All courses on Mechanics and Structural Analysis

### Course Units

# Title Description Hours
1 Introduction to mathematical theory of elasticity Elements of theory of elasticity. Stress, displacements and strains. Constitutive equations, equations of equilibrium, compatibility equations of strains. 4
2 Thin plates Thin plates with small deflections. Basic Kirchhoff theory assumptions. Bending surface and its geometrical properties. 4
3 Differential equation of the plate Stress resultants. Differential equation of the plate and boundary support conditions for straight and curvilinear boundaries. 6
4 Rectangular plates Rectangular plates. Analytical and approximate methods for solving thin plates with small deflections. 3
5 Circular plates Plate equation and stress resultants in polar coordinates. Circular and annular plates under axisymmetric and arbitrary loading. 3
6 Thick plates Shear effect on plate behavior. 4
7 Recalling general features of the finite element method. Principles of discretization. Different finite element (FE) types and fields of their application. Discretization and numerical errors. 4
8 FE elements for plate analysis (Kirchhoff) Boundary conditions. 4-noded rectangular plate elements. Patch test. 6-noded triangular FE with 6 degrees of freedom 4
9 Πεπερασμένα στοιχεία Πλακών (Mindlin) Rectangular isoparametric Reissner-Mindlin plate elements. Shear locking. Selective and reduced integration. Triangular plate element with selective application of the Reissner-Mindlin theory. Isoparametric Timoshenko beam element. 2x3
10 Shells. Cylindrical and axisymmetric shells. Introduction to the theory of shells. Menbrane and bending behavior. Analytical and numerical treatment by FEs of cylindrical and axisymmetric shells under axisymmetric loading. 2x4
11 Modelling of surface structures Modelling of a real surface structure using a FE program. (e. g. modelling of a bridge, cylindrical shell roof, etc.) 2
12 Adaptive Finite Elements A priori error estimate. H and p refinement. Hierarchical FEs. 2
13 Mixed type elements Mixed type elements 2

### Learning Objectives

The first part of the course aims at understanding the behavior of the plate, which is a basic structural element in the civil engineering. The plate equation is derived and several plate problems are solved. In the second part, finite elements (FE) used for plate analysis are presented. A simple introduction to the theory of shells is then given. Static behavior of cylindrical and axisymmetric shells as well as their numerical simulation through FE is attempted next. The discretization error in the FE method and ways to handle it are then discussed. Finally mixed types of FEs are presented.

### Teaching Methods

 Teaching methods Presentation of the theory and applications. Examples -problems to solve. Use of software Teaching from the black board and auxiliary use of slides. Solution of plate problems using software codes Simple problems to solve Simulation of real structures with finite elements. Evaluation of results. Presentation project by the students

### Student Assessment

• Final written exam: 70%
• Assignments (projects, reports): 30%

### Textbooks - Bibliography

1. Θεωρία Πλακών, Ε. Σαπουντζάκης, Εκδόσεις Ε.Μ.Π., 2005.
2. The Boundary Element Method For Plate Analysis, J. Katsikadelis, Academic press, Elsevier, 2014.
3. Ανάλυση Φορέων Με Τη Μέθοδο Των Πεπερασμένων Στοιχείων, Μ. Παπαδρακάκης, 1996.
4. Μέθοδος Πεπερασμένων Στοιχείων Ι, Γ. Τσαμασφύρος, Ε. Θεοτόκογλου, Εκδόσεις Συμμετρία, 2005.
5. Πεπερασμένα Στοιχεία Στην Ανάλυση Κατασκευών, Χ. Προβατίδης, Εκδόσεις Τζιόλα, 2016.
6. Στατική των Δομικών Φορέων, Τόμος Β', Λ. Σταυρίδης, Εκδόσεις Κλειδάριθμος, 2008.

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