Plates and FEM

This textbook aims to be the bridge between plate theory and FE-software.

Structural engineers must translate elastic analysis results into economic structure dimensions and reinforcement, which raises difficulties. Not all engineers are well enough equipped for the increasingly easy-to-use powerful programs. The problem is not lack of FE-knowledge, but rather ignorance of plate behaviour. Therefore, this book starts with classical plate theory for membrane and bending states, and proceeds to FE-practice.

This volume can be used for university courses, serving as practical preparation for the engineering profession, and as a guide to structural designers.

Klappentext

The Finite Element Method, shortly FEM, is a widely used computational tool in structural engineering. For basic design purposes it usually suf ces to apply a linear-elastic analysis. Only for special structures and for forensic investigations the analyst need to apply more advanced features like plasticity and cracking to account for material nonlinearities, or nonlinear relations between strains and displacements for geometrical nonlinearity to account for buckling. Advanced analysis techniques may also be necessary if we have to judge the remaining structural capacity of aging structures. In this book we will abstain from such special cases and focus on everyday jobs. Our goal is the worldwide everyday use of linear-elastic analysis, and dimensioning on basis of these elastic computations. We cover steel and concrete structures, though attention to structural concrete prevails. Structural engineers have access to powerful FEM packages and apply them intensively. Experience makes clear that often they do not understand the software that they are using. This book aims to be a bridge between the software world and structural engineering. Many problems are related to the correct input data and the proper interpretation and handling of output. The book is neither a text on the Finite Element Method, nor a user manual for the software packages. Rather it aims to be a guide to understanding and handling the results gained by such software. We purposely restrict ourselves to structure types which frequently occur in practise.

Inhalt

Preface ; Acknowledgements ; Free Software ; Conversion of SI Units to Imperial Units ; Conversion of SI Units to US Customary System ; Part 1 Theory of Plates; 1 Plate Membrane Theory ; 1.1 Introduction: Special Case of a Plate, the Truss; 1.2 Membrane Plate Problem Statement ; 1.2.1 Kinematic Equations ; 1.2.2 Constitutive Equations ; 1.2.3 Equilibrium Equations ; 1.2.4 The Displacement Method ; 1.3 Boundary Conditions ; 1.4 Message of the Chapter ; 2 Applications of the Plate Membrane Theory ; 2.1 Trial Solutions in the Form of Polynomials ; 2.1.1 Homogeneous Stress States ; 2.1.2 Constant BendingMoment in Beam; 2.1.3 Constant Shear Force in Beam; 2.2 Solution for aWall ; 2.2.1 Beam Intermezzo ; 2.2.2 Solution for theWall ; 2.2.3 Practical Application2.3 Stresses, Transformations and Principal Stresses ; 2.4 Other Applications ; 2.5 Message of the Chapter ; 3 Thick Plates in Bending and Shear ; 3.1 Introduction - Beam as Special Case ; 3.1.1 Illustration ; 3.1.2 Simplification for Slender Beam; 3.1.3 Suppositions of Timoshenko Beam in Hindsight ; 3.2 Outline of Thick Plates ; 3.2.1 Suppositions ; 3.3 Basic Equations ; 3.3.1 Kinematic Equations; 3.3.2 Constitutive Equations ; 3.3.3 Equilibrium Equations ; 3.4 Differential Equations for Thick Plates ; 3.5 Orthotropic Plate ; 3.6 Twisted Plate Strip ; 3.7 Message of the Chapter ; 4 Thin Plates in Bending ; 4.1 Theory for Thin Plates ; 4.2 Transformation Rules and PrincipalMoments; 4.3 Principal Shear Force ; 4.4 Boundary Conditions for Thin Plates ; 4.4.1 Clamped Edge ; 4.4.2 Simply-Supported Edge ; 4.4.3 Free Edge ; 4.4.4 Discontinuity in Thickness; 4.5 Message of the Chapter ; 5 Rectangular Plate Examples ; 5.1 Basic Bending Cases ; 5.1.1 Cylindrical Deflection; 5.1.2 Cylindrical Deflection of Arbitrary Shape ; 5.1.3 Omni-Directional Bending; 5.2 Torsion Panel ; 5.3 Two-Way Sine Load on Square Plate ; 5.3.1 Displacement ; 5.3.2 Moments and Shear Forces ; 5.3.3 Support Reactions ; 5.3.4 Stiff Edge Beams ; 5.4Twist-Less Plate ; 5.5 Edge Load on Viaduct ; 5.6 Message of the Chapter ; 6 Circular Membrane Plates ; 6.1 Axisymmetric Circular Membrane Problems ; 6.1.1 Thick-Walled Tube ; 6.1.2 Circular Hole in a Homogeneous Stress State ; 6.1.3 Curved Beam Subjected to ConstantMoment ; 6.2 Non-Axisymmetric Circular Membrane Problems; 6.2.1 Point Load on a Half Plane ; 6.2.2 Brazilian Splitting Test ; 6.2.3 Hole in Plates with Shear and Uniaxial Stress ; 6.3 Message of the Chapter ; 7 Circular Thin Plates in Bending; 7.1 Derivation of the Differential Equation ; 7.2 Simply-Supported Circular Plate with Edge Moment ; 7.3 Clamped Circular Plate with Distributed Load ; 7.4 Simply-Supported Circular Plate with Distributed Load ; 7.5 Clamped Circular Plate with Point Load ; 7.6 Simply-Supported Circular Plate with Point Load ; 7.7 Circular Plate Part on Top of Column; 7.8 Message of the Chapter; Part 2 Didactical Discrete Models; 8 Discrete Model for Membrane Analysis ; 8.1 TrussModel ; 8.2 Membrane PlateModel ; 8.2.1 Example. Deep Beam Subjected to OwnWeight ; 8.3 Message of the Chapter; 9 Discrete Model for Plate Bending ; 9.1 BeamModel ; 9.1.1 Example. Cantilever Beam ; 9.2 Plate BendingModel ; 9.2.1 Example 1. Rectangular Simply-Supported Plate ; 9.2.2 Example 2. Lift-Slab in Office Building ; 9.3 Didactical Model for Simply-Supported Plate ; 9.4 DiscreteModel for Plate on Flexible Edge Beams ; 9.5 Message of the Chapter ; Part 3 FE-Based Design in Daily Practice10 FEM Essentials ; 10.1 Elements and Degrees of Freedom ; 10.2 Stiffness Matrix and Constraints ; 10.3 Model Input ; 10.4 Output Selection ; 10.5 Message of the Chapter ; 11 Handling Membrane FEMResults; 11.1 Surprising Stresses ; 11.1.1 Effect of Poisson's Ratio ; 11.1.2 Effect of Kink in Beam Flange ; 11.2 Stress Singularities in FEM ; 11.3 FEM-Supported Strut-and-Tie Modeling ; 11.4 Re-entrant Corner ; 11.5 TallWall with Openings; 11.5.1 Modeling withMembrane Elements; 11.5.2 Modeling as Frame ; 11.6 Checking and