Structural Engineering: Models and Methods for Statics, Instability and Inelasticity

Preface
Contents
1 Introduction
	1.1 Motivation and Objectives
	1.2 Main Topics Outline
	1.3 Further Studies Recommendations
	1.4 Summary of Main Notations
2 Truss Model: General Theorems and Methods of Force, Displacement and Finite Elements
	2.1 Truss Model—Strong Form and Weak Form
		2.1.1 Strong or Differential Form: Analytic Solution
		2.1.2 Weak or Integral Form
	2.2 General Theorems of Structural Mechanics on Truss Model
		2.2.1 Principle of Virtual Work
		2.2.2 Principle of Complementary Virtual Work
		2.2.3 Principle of Minimum of Total Potential Energy
		2.2.4 Applied General Theorems
	2.3 Castigliano's Theorems, Force and Displacement Methods
		2.3.1 Castigliano's Theorems—Stiffness and Flexibility
		2.3.2 Force and Displacement Methods
	2.4 Finite Element Method Implementation for Truss Model
		2.4.1 Local or Elementary Description
		2.4.2 Consistence of Finite Element Approximation
		2.4.3 Equivalent Nodal External Load Vector
		2.4.4 Higher Order Finite Elements
		2.4.5 Role of Numerical Integration
		2.4.6 Finite Element Assembly Procedure
3 Beam Models: Refinement and Reduction
	3.1 Reduced Models of Solid Mechanics: Planar Beams  of Euler, Timoshenko and Reissner
		3.1.1 Euler-Bernoulli Planar Beam Model
		3.1.2 Solid Mechanics Versus Beam Model Accuracy for Planar Cantilever Beam
		3.1.3 Timoshenko Planar Beam Model
		3.1.4 Brief on Reissner Planar Beam Model
	3.2 Beam Model Refinement and Reduction
		3.2.1 Method of Direct Stiffness Assembly for 3D Beam Elements
		3.2.2 Beam Model Refinement: Flexibility Approach for Reduced Model in Deformation Space
		3.2.3 Beam Model Reduction: Joint Releases and Length Invariance
	3.3 Curved Shallow Beam and Non-locking FE Interpolations
		3.3.1 Two-Dimensional Curved Shallow Beam: Linear Kinematics
		3.3.2 Non-locking Finite Element Interpolation for Shallow Beam
		3.3.3 Illustrative Numerical Examples and Closing Remarks
4 Plate Models: Validation and Verification
	4.1 Finite Elements for Analysis of Thick and Thin Plates
		4.1.1 Motivation: Timoshenko Beam Element Linked Interpolations
		4.1.2 Reissner-Mindlin Plate Model and FE Discretization
		4.1.3 Illustrative Numerical Examples and Closing Remarks
	4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes
		4.2.1 Reissner-Mindlin Plate Model and Enhanced FE Interpolations
		4.2.2 Illustrative Numerical Examples and Closing Remarks
	4.3 Validation or Model Adaptivity for Thick or Thin Plates Based on Equilibrated Boundary Stress Resultants
		4.3.1 Thick and Thin Plate Finite Element Models
		4.3.2 Model Adaptivity for Plates
		4.3.3 Illustrative Numerical Examples and Closing Remarks
	4.4 Verification or Discrete Approximation Adaptivity for Discrete Kirchhoff Plate Finite Element
		4.4.1 Kirchhoff Plate Bending Model
		4.4.2 Kirchhoff Plate Finite Elements
		4.4.3 Error Estimates for Kirchhoff Plate Elements Based Upon Equilibrated Boundary Stress Resultants
		4.4.4 Implementation of Equilibrated Element Boundary Tractions Method For DKT Plate Element
		4.4.5 Examples on Error Indicators Comparison and Closing Remarks
5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures
	5.1 Solids with Drilling Rotations: Variational Formulation
		5.1.1 Strong Form of the Boundary Value Problem
		5.1.2 Variational Formulation, Stability Analysis and Regularization
		5.1.3 Alternative Variational Formulations, Extension to Nonlinear Kinematics and Closing Remarks
	5.2 Membranes with Drilling Rotations: Discrete Approximation
		5.2.1 Discrete Approximations with Quadratic and Cubic Displacement Fields
		5.2.2 Illustrative Numerical Examples and Closing Remarks
	5.3 Shells with Drilling Rotations: Linearized Kinematics
		5.3.1 Geometrically Linear Shallow Shell Theory
		5.3.2 Incompatible Modes Based Finite Element Approximation
		5.3.3 Illustrative Numerical Examples and Closing Remarks
6 Large Displacements and Instability: Buckling Versus Nonlinear Instability
	6.1 Large Displacements and Deformations in 1D Truss with Instabilities
		6.1.1 Large Strain Measures for 1D Truss
		6.1.2 Strong and Weak Forms for 1D Truss in Large Displacements
		6.1.3 Linear Elastic Behavior for 1D Truss in Large Displacements
		6.1.4 Finite Element Method for 1D Truss in Large Displacements
		6.1.5 Buckling, Nonlinear Instability and Detection Criteria
	6.2 Geometrically Nonlinear Curved Beam and Nonlinear Instability
		6.2.1 Curved Reissner's Beam: Nonlinear Kinematics
		6.2.2 Finite Element Implementation for Curved Reissner's Beam and Comment on Objectivity
		6.2.3 Control of Nonlinear Instability
	6.3 Buckling of (Heterogeneous) Euler's Beam
		6.3.1 Euler Instability Problem: Two Alternative Formulations
		6.3.2 Analytic Solution Based on Strong Form
		6.3.3 Numerical Solution Based on Reduced Models with Finite Element Method
	6.4 Buckling Analysis of Complex Structures with Refined Models of Plates and Shells
		6.4.1 Buckling Problems for Plates and Shells
		6.4.2 Finite Element Shell Approximation Including Drilling Rotations
		6.4.3 Illustrative Numerical Examples and Closing Remarks
	6.5 Buckling Problems for Coupled Thermomechanical Extreme Conditions
		6.5.1 Linear Thermoelasticity 1D
		6.5.2 Linearized Instability for Thermoelastic Coupling
		6.5.3 General Linear Eigenvalue Problem Solution Procedure
		6.5.4 Thermomechanical Coupling Model: Illustrative  and Validation Examples
		6.5.5 Brief on Instability for Thermomechanical Coupling  in 1D Finite Elasticity
7 Inelasticity: Ultimate Load and Localized Failure
	7.1 Stress Resultants Finite Element Model for Reinforced-Concrete Plates
		7.1.1 Plate Element for Reinforced-Concrete Slabs
		7.1.2 Stress-Resultants Constitutive Model for Reinforced-Concrete Plates
		7.1.3 Illustrative Numerical Examples and Closing Remarks
	7.2 Stress Resultants Plasticity for Metallic Plates
		7.2.1 Variational Formulation and Discrete Approximation for Metallic Plates
		7.2.2 Stress Resultants Plasticity Formulation for Metallic Plates
		7.2.3 Illustrative Numerical Examples and Closing Remarks
	7.3 Plasticity Criterion with Thermomechanical Coupling in Folded Plates and Non-smooth Shells
		7.3.1 Theoretical Formulation of Shell Model for Folded Plates and Non-smooth Shells
		7.3.2 Finite Element Implementation with Shell Element
		7.3.3 Stress Resultants Constitutive Model of Saint-Venant Plasticity
		7.3.4 Thermomechanical Coupling
		7.3.5 Operator Split Solution Procedure with Variable Time Steps
		7.3.6 Illustrative Numerical Examples and Closing Remarks
	7.4 Stress Resultants Plasticity and Localized Failure  of Reissner's Beam
		7.4.1 Reissner's Beam with Localized Elastoplastic Behavior
		7.4.2 Stress Resultant Plasticity Discrete Approximations and Computations
		7.4.3 Illustrative Numerical Examples and Closing Remarks
8 Brief on Mulitscale, Dynamics and Probability
	8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics
		8.1.1 Geometrically Exact Shear Deformable Beam as Cohesive Link
		8.1.2 Micro and Macro Constitutive Models for Dynamic Fracture
		8.1.3 Dynamics of Lattice Network and Time-Stepping Schemes
		8.1.4 Illustrative Numerical Examples and Closing Remarks
	8.2 Stochastic Upscaling, Size Effect and Damping Replacement
		8.2.1 Stochastic Upscaling in Localized Failure
		8.2.2 Probability-Based Size Effect in Ductile Failure
		8.2.3 Damping Model Replacement of Rayleigh Damping
	8.3 Reduced Stochastic Models for Euler Beam Dynamic Instability
		8.3.1 Duffing Oscillator: Reduced Model for Euler Instability
		8.3.2 Instability Studies in Dynamics Framework
		8.3.3 Stochastic Solution to Euler Instability Problem
References