Plates and FEM

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; 5.4 Twist-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 Detailing; 11.6.1 Steel; 11.6.2 Reinforced Concrete; 11.7 Message of the Chapter; 12 Understanding FEMPlate Bending; 12.1 Intended Goal and Chosen Structure; 12.2 Bending Moments and Equilibrium; 12.2.1 Discussion ofMoment Diagrams; 12.2.2 Equilibrium Check for Moments; 12.3 Shear Forces, Support Reactions and Equilibrium; 12.3.1 Discussion of Shear Force Diagrams; 12.3.2 Equilibrium Check for Shear Forces; 12.4 Message of the Chapter; 13 FE Analysis for Different Supports; 13.1 Simply-Supported Plat; 13.1.1 Distributed Load; 13.1.2 Point Load; 13.2 Corner Supports; 13.2.1 Distributed Load; 13.2.2 Point Load; 13.3 Edge Beams; 13.3.1 Rigid Beams;13.3.2 Flexible Beams; 13.4 Pressure-Only Support; 13.5 Message of the Chapter; 14 Handling Peak Moments; 14.1 Peaks at Columns; 14.2 Column Reaction Distribution; 14.3 Application; 14.4 Cast-Connected Column; 14.5 Dependence on Program; 14.5.1 Review of FEMResults; 14.5.2 Program Comparison; 14.6 Dependence on User; 14.7 Impact of Support Flexibility and Concrete Cracking; 14.7.1 Application of Finite Element Program; 14.8 Message of the Chapter; 15 Sense and Nonsense of Mindlin; 15.1 Result Dependence on Analyst and Program; 15.1.1 Invitation; 15.1.2 Submitted Results; 15.2 Explanation of the Differences; 15.3 Supporting Side Study; 15.3.1 Thin Plate Results; 15.3.2 Thick Plate Results; 15.4 Comparison in Hindsight; 15.5 Message of the Chapter; 15.5.1 Lessons for Thin Plates; 15.5.2 Lessons for Thick Plates; 16 Reinforcement Design Using Linear Analysis; 16.1 Design ofMembrane States; 16.2 Design of Slabs -- Normal Moment Yield Criterion; 16.3 Slab and Shell Elements -- BasicModel; 16.3.1 Basic Model -- No Cracking Due to Transverse Shear; 16.3.2 BasicModel -- Cracking Due to Transverse Shear; 16.3.3 Evaluation; 16.4 Formulation of the Advanced Three-layer Model; 16.5 Applications on Element Level; 16.5.1 Element with Membrane Force and Bending Moment; 16.5.2 Slab Element with TwistingMoment; 16.6 Applications on Structural Level; 16.6.1 Deep Beam; 16.6.2 Slab; 16.7 Message of the Chapter; 17 Special Slab Systems; 17.1 Wide-Slab Floor; 17.2 Reinforced Floor Unit; 17.2.1 Serviceability Limit State; 17.2.2 Ultimate Limit State; 17.3 Pre-stressed Floor Unit without Lateral Reinforcement; 17.3.1 Serviceability Limit State; 17.3.2 Ultimate Limit State; 17.4 Pre-stressed Floor Unit with Lateral Reinforcement; 17.4.1 Serviceability Limit State; 17.4.2 Ultimate Limit State; 17.5 Strengthened Strip Floor; 17.6 Message of the Chapter; 17.6.1 Wide-slab Floor; 17.6.2 Strengthened Strip Floor; 18 Special Topics and Trends; 18.1 Stringer-Panel Method; 18.1.1 Beam with Dapped Ends; 18.1.2 ShearWall with Opening; 18.2 Membrane Plates with Concrete Pressure Only; 18.2.1 ShearWall with Opening; 18.3 Advanced Orthotropy; 18.3.1 Bridge with Point Load; 18.4 Plates on Soil Foundation; 18.4.1 Two Close Slabs; 18.5 Message of this Chapter; 19 Case History of Cable-Stayed Wide-Box Bridge; 19.1 Introduction; 19.2 Calculation of a Construction Phase; 19.2.1 Problem Definition and Finite Element Mesh; 19.2.2 Results of the FE Analysis and the Model Test; 19.3 Interpretation of the Results; 19.3.1 Stress sxx in Span Direction; 19.3.2 Stress syy in Transverse Direction;19.3.3 Shear Stress sxy; 19.3.4 Deflection Diagram; 19.3.5 Evaluation; 19.4 Message of the Chapter; Part 4 Shape Orthotropy; 20 Shape-Orthotropic Membrane Rigidities; 20.1 Problem Statement; 20.2 Occasion of the Chapter; 20.3 Membrane Plate with Stiffeners; 20.4 Plate Strips of Different Thickness; 20.4.1 Extensional Rigidity; 20.4.2 Shear Rigidity the y-Direction; 20.5 Plate with Hat Stiffeners; 20.5.1 Extensional Rigidity; 20.5.2 Shear Rigidity; 20.6 Message of the Chapter; 21 Orthotropic Plates in Bending and Shear; 21.1 Problem Statement; 21.2 Plate with I-Sections; 21.2.1 Flexural Rigidity; 21.2.2 Torsional Rigidity; 21.2.3 Shear Rigidity; 21.3 Multi-Cell Bridge; 21.3.1 Flexural Rigidity; 21.3.2 Torsion Rigidity; 21.3.3 Shear Rigidity; 21.3.4 Combination of Shear Force and Twisting Moment; 21.4 Plate with Separate Boxes; 21.4.1 Flexural Rigidity; 21.4.2 Torsional Rigidity; 21.4.3 Shear Rigidity; 21.5 Message of the Chapter; References; Index