Modeling for Structural Analysis

Transcription

1 Modeling for Structural Analysis Behavior and Basics Graham H. Powell Professor Emeritus of Structural Engineering University of California at Berkeley Computers and Structures, Inc., Berkeley, California, USA

2 Chapter 1 Introduction Overview The Phases of Structural Analysis Modeling Interpretation Computation Relative Importance ofthe Three Phases Demand and Capacity Performance Assessment Direct Design Elastic vs. Inelastic Analysis Behavior of a Structural Component Elastic vs. Inelastic Behavior StrengthBased Design Using Elastic Analysis StrengthBased Design Using Inelastic Analysis DeformationBased Design for Earthquake Loads StrengthBased Design for Earthquake Loads Capacity Design Using Elastic Analysis Static vs. Dynamic Analysis Small vs. Large Displacements Analysis Overview Equilibrium Compatibility (Continuity) Analysis Types ; Catenary Effect Demand Analysis vs. Capacity Analysis Overview Lateral Load at First Yield Earthquake Intensity at Collapse Bending Strength of a Beam Plastic Hinge Rotation Capacity Conclusion for this Section Conclusion for this Chapter Topics for the Following Chapters 26 ix

3 x Chapter 2 What is an Analysis Model? Actual Structure vs. Analysis Model Two Types ofanalysis Model Features of NodeElement Model Some Element Types Connection between Nodes and Elements Overview Connection for Bar Elements Rigid End Zones Gaps and Overlaps between Elements Surface Elements ElementtoElement Contact Equilibrium between Elements Equilibrium at Element Boundaries Equilibrium at Connections Discrete Model with Finite Size Nodes and Zero Length Elements Continuum Model Frame Structure Wall Structure Elements and Components 43 Chapter 3 The Direct Stiffness Method Element Stiffness and Flexibility Overview Bar Element Rigid Body Displacements and Deformation Modes Beam Element Surface and Solid Elements Slab and Shell Elements Stiffness and Flexibility Analysis Methods The Direct Stiffness Method Overview Degrees of Freedom Addition of Stiffness Assembly of Structure Stiffness 60

4 Uniaxial Rigid xi Addition of Flexibility Optimal Node Numbering Equilibrium Equations Load Matrix Equation Solving Element Deformations and Forces Equilibrium Check Cause of Equilibrium Unbalance Reactions at Rigid Supports Forces Corresponding to Slaving Constraints Slaving Constraints Floor Diaphragm Equilibrium Errors in Linear Analysis Overview Unstable Structure Stiffness Coefficient Mismatch Programming Error Detection of Unstable Mode Element Loads Overview Element ForceDisplacement Relationship Structure Equilibrium Equations Element Initial Deformations Imposed Displacements at Rigid Supports Dynamic and Nonlinear Analysis Dynamic Loads Material Nonlinearity Geometric Nonlinearity Substructures and Superelements 86 Chapter 4 Component Behavior FD Relationships Overview Components and Elements Modeling Goals for Components Component ForceDeformation Relationships A Common FD Relationship FD Relationship for Monotonic Deformation Complications for Cyclic Deformation Elastic and Plastic Deformations Ductility Ratio RigidPlastic Hinges Other Nonlinear FD Relationships 101

5 The xii Summary for this Section What Type of FD Relationship is Needed? Stiffness for Elastic Analysis Bending Stiffness for Beams, Columns and Walls Shear Deformation in Reinforced Concrete Connections Summary for this Section FD Relationships for Inelastic Analysis Overview Amount of Inelastic Behavior Practical FD RelationshipGeneric Form FD Relationships in ASCE Backbone Relationship Hysteresis Loops for Inelastic Analysis Loop Anchored to Backbone Relationship Hysteresis Loop in CSI PERFORMED Practical Modeling of Cyclic Degradation "InCycle" and "BetweenCycle" Strength Loss Conclusion for this Chapter 121 Chapter 5 Component Behavior MultiAxial FD Relationships with Interaction Overview Stiffness Interaction Strength Interaction Inelastic Interaction : Behavior after Yield Plasticity Theory for Yield of Metals Overview Yield of ElasticPerfectlyPlastic Metals Strain Hardening Interaction Surface for Friction Bearing Component with Friction Cohesion vs. Friction Extension to PMM Interaction Overview Steel Section Analogy Works Sharp Peak in Yield Surface 141

6 xiii RC Section The Analogy Does Not Work So Wei! Ductile Limit and Strength Loss Hysteresis Loops and Stiffness Degradation Cyclic Degradation Other Cross Section Shapes Is Plasticity Theory Useful for PM Interaction? Overview Assumptions and Approximations Importance of Axial Deformation Steel Column Is the Axial Deformation in a Plastic Hinge Correct? Accumulated Axial Deformation Concrete Column Can the Axial Extension be Assumed to be Zero? Axial Deformation in Static PushOver Analysis Conclusion for this Section Axial Extension in Concrete Beams Fiber Sections for PM interaction Overview Fiber Sections for Beams Fiber Sections for Columns Fiber Sections for Walls Fiber Segments RigidPlastic Fiber Hinge Limitations of Fiber Models Inelastic Shear in Beams and Columns Overview MV Interaction in Steel Beams PMV Interaction in Steel Columns MV Interaction in Reinforced Concrete Beams PMV Interaction in Reinforced Concrete Columns PMV Interaction in Connections Analysis vs. Design Shear in Concrete Walls MultiAxial Material Models for Plain Concrete Motivation 173

7 xiv Plasticity Theory Compression Field Theory Simple Models Based on Uniaxial StressStrain Possible Model with PV Interaction Plain Concrete Models for 3D Stress OverReliance on Analysis A Note on Demand and Capacity Analyses Capacity Interaction Overview Effect of Axial Force on Bending Ductility Effect of Shear Force on Bending Ductility Effect of Hinge Rotation on Shear Strength Plastic Deformation and Deformation D/C Ratios Summary for this Chapter Conclusion for this Chapter 189 Chapter 6 PA Effects, Stability and Buckling Overview Causes of Geometric Nonlinearity Types of Analysis for Geometric Nonlinearity PA Effect Load Types and Corresponding Analyses Elastic and Inelastic Analysis Topics for this Chapter PA and P5 Contributions in a Single Column Overview Cantilever Column Approximations in PA Analysis PA vs. Small Displacements Effects Relative Importance of PA and P8 Contributions Cantilever Column Column in an Unbraced Frame Modeling of PA and P8 Contributions Overview Model for a Cantilever Column Behavior of P5 Cable Behavior of Elastic Cantilever Column Moments and Shears for Column Design Elastic Column in an Unbraced Frame 210

8 Inelastic xv Inelastic Column in an Unbraced Frame Element Models 213 Lateral Load Behavior of Frames Overview PA Struts and PA Columns Review of Terminology Main Structure and PA Column Effect of PA Column on Stiffness and Strength Load Increase or Strength Decrease? Load on PA Column vs. Load on Main Structure PA Effect in Beams PA Effect in Braced Frames Summary for this Section 222 Buckling Behavior of Frames Overview Stability of Equilibrium State Bifurcation of Equilibrium Inelastic Behavior after Buckling Inelastic Behavior before Buckling Effect of Initial Imperfection Effect of Initial Imperfection Elastic Case 228 Case Effect of Gravity Sway Are Buckling Loads Useful? Summary for this Section 231 PA Columns in MultiStory and 3D Buildings PA Column in Multistory Buildings PA Column in 3D Buildings Buildings with Multiple PA Columns 235 Buckling of MultiStory and 3D Frames 236 Buckling of an Axially Loaded Column Overview Ideal Elastic Column Imperfect and Inelastic Column Steel Column Reinforced Concrete Column Theories for Buckling Strength Combined Material and Geometric Nonlinearity Analysis Model for Buckling Strength Summary for this Section 247

9 Initial xvi 6.10 Simple Structure with PinEnded Members Overview Example Structure and Analysis Model StrengthBased Design Using Elastic Analysis DeformationBased Design Using Inelastic Analysis Modeling of Diagonal Brace Behavior PinEnded Elastic Column with Bending Overview Elastic Column with Sinusoidal Lateral Load Elastic Column with Other Lateral Loads Elastic Column with End Moments Direct Calculation of Amplified Moments Elastic Column in a Frame BeamColumn Strength Overview ElasticPerfectlyPlastic Behavior Behavior with Progressive Yield Amplification Using Tangent Modulus Theory Tangent, Reduced or Secant Modulus? StrengthBased Design of BeamColumns Overview Steel BeamColumns Reinforced Concrete BeamColumns DeformationBased Design of BeamColumns Compression Members in Braced Frames Overview Frame Braced by a Wall StrengthBased Design Using Elastic Analysis DeformationBased Design Using Inelastic Analysis Diagonally Braced Frame Columns in Unbraced Frames Overview Frame and Column Buckling StrengthBased Design Using Elastic Analysis DeformationBased Design Using Inelastic Analysis A Complication Drifts Overview Modeling Methods 275

10 Geometric Stiffness xvii 6.18 A Second Complication Reduction Overview Effect on Lateral Load Analysis Modeling of Stiffness Reduction Effect on the P8 Contribution Effect on Buckling Analysis Is a Reduced Stiffness Analysis Necessary? Some Theory Stiffness Overview Geometric Stiffness Matrix for PA Strut Structure Stiffness Matrix P8 Contribution Methods for Elastic Lateral Load Analysis Overview Sway and NonSway Amplification Methods for Sway Amplification Methods for NonSway Amplification Sway and NonSway Analysis Similarities 293 Methods "DoubleB" Analysis "SingleB" Analysis "ZeroB" Analysis End Moments in ZeroB Analysis Initial Imperfections Stiffness Reduction and Sway Amplification Stiffness Reduction and NonSway Amplification Demand/Capacity Calculation Conclusion for this Section Direct Analysis Method for Steel Frames Overview Acceptable SecondOrder Analysis Benchmark for Sway Amplification Benchmark for NonSway Amplification Acceptable Methods Initial Drifts Basic Stiffness Reductiqn Advanced Stiffness Reduction Alternative to Advanced Stiffness Reduction Conclusion for this Section Inelastic Lateral Load Analysis of Frames Buckling Analysis 315

11 xviii Overview Analysis Method with Equilibrium Bifurcation Buckling Analysis With Imperfections Some Other Structures Overview Walls With OutofPlane Bending Long Span Roofs Bridge Columns Buckling of Pipe With Zero Axial Foree Buckling of Buried Pipe Large Displacements of Buried Pipe Structures With "Follower" Forces LateralTorsional Buckling of Beams Overview Causes of LateralTorsional Buckling Analysis Models Torsional Behavior of lsection Beams Compression Flange as a Column Effect of Shear Force on Beam Buckling Practical Modeling : Bracing to Prevent Buckling PA Effects in Seismic Isolators Overview FrictionPendulum Isolator with Flat Sliding Surface.' FrictionPendulum Isolator with Curved Surface RubberType Isolator Alternative Model for RubberType Isolator Which Model is Correct? Some Other Types of Buckling True Large Displacements Conclusion for this Chapter 343 Chapter 7 Some Other Aspects of Behavior Plastic Mechanisms Collapse Mechanism vs. Plastic Mechanism PushOver Analysis for Earthquake Loads Desirable and Undesirable Mechanisms 348

12 xix Lateral Strength Calculation Given a Mechanism Other Causes of a Mechanism Mechanism Control Using Capacity Design Concept Examples of Mechanism Control Higher Mode Effects in Tall Buildings Static Indeterminacy and Redundancy Statically Determinate Structure Statically Indeterminate Structure Redundancy Nonstructural Components Work and Energy Energy Balance in a Real Structure Energy Balance in an Elastic Analysis Model Energy Balance in an Inelastic Analysis Model Energy Balance Check During Analysis Amount and Distribution of Dissipated Energy Dissipated Energy as a DemandCapacity Measure Living With Uncertainty 363