Stability and Bifurcation of Structures: Statical and Dynamical Systems

Preface
Contents
1 Introduction
	1.1 Basic Concepts
	1.2 Overview of the Book
	1.3 Book Style
2 Phenomenological Aspects of Bifurcation of Structures
	2.1 Introduction
	2.2 Stability and Bifurcation
		2.2.1 Equilibrium Points
		2.2.2 Stability of Equilibrium
			Lagrange-Dirichlet Theorem
		2.2.3 Bifurcation
			Bifurcation of Equilibrium
			Static and Dynamic Bifurcations
	2.3 An Example of Static Bifurcation: The Euler Beam
	2.4 Static Bifurcations of Elastic Structures
		2.4.1 Fork and Transcritical Bifurcations
		2.4.2 Snap-Through Phenomenon
		2.4.3 Interaction Between Simultaneous Modes
			An Example of a Two-Parameter Family: The Compressed Truss
			Structural Optimization in the Linear Optics
			Nonlinear Interaction Between Simultaneous Modes
	2.5 Dynamic Bifurcations of Elastic Structures Subject to Nonconservative Forces
		2.5.1 Flutter Induced by Follower Forces
		2.5.2 Galloping Induced by Aerodynamic Flow
		2.5.3 Parametric Excitation Induced by Pulsating Loads
	References
3 Stability and Bifurcation Linear Analysis
	3.1 Introduction
	3.2 Dynamical Systems
	3.3 Mechanical Systems
	3.4 Linear Stability Analysis
		3.4.1 Conservative Systems
		3.4.2 Circulatory Systems
		3.4.3 Influence of Damping
			Damped Conservative Systems
			Damped Circulatory Systems
	3.5 An Illustrative Example: The Planar Mathematical Pendulum
		3.5.1 Equation of Motion and the Phase Portrait
			Equilibrium Points
			Phase Portrait
		3.5.2 Local Stability Analysis
			Center Point (Lower Equilibrium Position)
			Saddle Point (Upper Equilibrium Position)
		3.5.3 Energy Criterion of Stability
		3.5.4 Effect of Damping
			Equation of Motion
			Local Stability Analysis
	3.6 Bifurcations of Autonomous Systems
		3.6.1 Equilibrium Paths
		3.6.2 Bifurcations from a Trivial Path
		3.6.3 Bifurcations from a Non-trivial Path
			Linearized Equation of Motion
			Bifurcation Analysis
		3.6.4 Bifurcation Mechanisms for Conservative and Circulatory Systems, without or with Damping
			Conservative Systems
			Circulatory Systems
			Damped Conservative Systems
			Damped Circulatory Systems
	References
4 Buckling and Postbuckling of Conservative Systems
	4.1 Introduction
	4.2 Static Analysis of Conservative Systems
	4.3 Classification of the Equilibrium Points
	4.4 Numerical Continuation Methods
		4.4.1 Newton-Raphson Method
		4.4.2 Sequential Continuation
			Sequential Continuation Failure
		4.4.3 Arclength Method
	4.5 Asymptotic Analysis of Bifurcation from Trivial Path
		4.5.1 Linear Stability Analysis
			Adjacent Equilibrium Criterion
		4.5.2 Nonlinear Bifurcation Analysis
			Asymptotic Expression of the Bifurcated Path
			Normalization
			Perturbation Equations
			Solution to the Perturbation Equations
			Case of Symmetric Systems
	4.6 Effect of Imperfections
		4.6.1 Equilibrium Equations
			Geometric Imperfections
			Load Imperfections
			Equilibrium Equation for Imperfect System
		4.6.2 Asymptotic Construction of the Imperfect Equilibrium Paths
			Non-symmetric Systems
			Symmetric Systems
	4.7 Stability of the Equilibrium Paths
	4.8 Systems with Precritical Deformations
		4.8.1 Asymptotic Construction of the Non-trivial Fundamental Path
		4.8.2 Bifurcation from Non-trivial Path
			Incremental Variable
			Critical Load
	References
5 Paradigmatic Systems of Buckling and Postbuckling
	5.1 Introduction
	5.2 Single Degree of Freedom Systems with Trivial Fundamental Path
		5.2.1 Inverted Elastic Pendulum
			Exact Analysis of the Perfect System
			Exact Analysis of the Imperfect System
		5.2.2 Inverted Pendulum with Sliding Spring
			Exact Analysis of the Perfect System
			Exact Analysis of the Imperfect System
		5.2.3 Cable-Stayed Inverted Pendulum
			Exact Analysis of the Perfect System
			Exact Analysis of the Imperfect System
	5.3 Two Degrees of Freedom Systems with Trivial Fundamental Path
		5.3.1 Reverse Elastic Double Pendulum
			The Equilibrium Equations
			Bifurcation Analysis
			Critical Loads and Modes
			Bifurcated Path
			Nonlinear Deformation
		5.3.2 Spherical Inverted Elastic Pendulum
			Equilibrium Equations
			Linearized Bifurcation Analysis
			Bifurcation Analysis in the Degenerate Case
			Solution to the Perturbation Equations
			Bifurcated Paths
			Stability of the Bifurcated Paths
	5.4 Euler Beam as a Paradigm of Continuous Systems
		5.4.1  Inextensible and Shear-Undeformable Planar Beam Model
			Kinematics
			Total Potential Energy
			Equilibrium Equation Expanded in Series
		5.4.2 Linear Boundary Conditions: Simply Supported Beam
			Perturbation Equations
			Solution to the ε1 Order Problem
			Solution to the ε3 Order Problem
			Bifurcation Diagram
		5.4.3 Nonlinear Boundary Conditions: Cantilever Beam
			Perturbation Equations
			Solution to the ε1 Order Problem
			Solution to the ε3 Order Problem
			Bifurcation Diagram
	5.5 Systems with Non-trivial Path: The Snap-Through of the Three-Hinged Arch
		5.5.1 Exact Analysis
		5.5.2 Perturbation Analysis
	5.6 Bifurcation from Non-trivial Path: The Extensible Pendulum
	References
6 Linearized Theory of Buckling
	6.1 Introduction
	6.2 Variational Formulation of the Equilibrium of Prestressed Bodies
		6.2.1 Discrete Systems
			Elastic Law and Total Potential Energy
			Nonlinear Kinematics
			TPE Truncated at the Second Degree
			Equilibrium
		6.2.2 Continuous Systems
	6.3 Adjacent Equilibrium Through the Virtual Work Principle
	6.4 Direct Equilibrium of Prestressed Bodies
	6.5 Linearized Effects of Imperfections
	6.6 An Illustrative Example: The Extensible Inverted Pendulum
	References
7 Elastic Buckling of Planar Beam Systems
	7.1 Introduction
	7.2 Extensible Beam Model
	7.3 Critical Loads of Single-Span Beams
	7.4 Beams Transversely Loaded: Second Order Effects
		7.4.1 Simply Supported Beam Under Sinusoidal Transverse Load
		7.4.2 Simply Supported Beam Under Generic Transverse Load
	7.5 Stepped Beams
		7.5.1 Exact Analysis
		7.5.2 Ritz Analysis
	7.6 Beams Under Piecewise Variable Compression
		7.6.1 Partially Compressed Beam
		7.6.2 Beam Under Independent Compressive Forces: The Domain of Interaction
	7.7 Beams Under Distributed Longitudinal Loads
		7.7.1 Power Series Solution
		7.7.2 Ritz Solution
	7.8 Elastically Constrained Beams
		7.8.1 Beam Elastically Supported at One End
		7.8.2 Beam Elastically Supported in the Span
	7.9 Beam on Winkler Soil
		7.9.1 Model
		7.9.2 Beam on Elastic Soil Simply Supported at the Ends
		7.9.3 Beam on Elastic Soil Arbitrarily Constrained at the Ends
	7.10 Prestressed Reinforced Concrete Beams
		7.10.1 Externally Cable-Prestressed Beams
		7.10.2 Internally Cable-Prestressed Beams
	7.11 Local and Global Instability of Compressed Truss Beams
	7.12 Finite Element Analysis of Buckling
		7.12.1 Polynomial Finite Element
		7.12.2 Exact Finite Element
	References
8 Elasto-Plastic Buckling of Planar Beam Systems
	8.1 Introduction
	8.2 Elasto-Plastic Buckling of a Single Beam
		8.2.1 Tangent Elastic Modulus Theory
		8.2.2 Reduced Elastic Modulus Theory
	8.3 Elasto-Plastic Analysis of Beam Systems
		8.3.1 Geometric Effects on the Elasto-Plastic Response of Planar Frames
			First Order Elasto-Plastic Analysis
			Second Order Elasto-Plastic Analysis
		8.3.2 Column Subjected to a Constant Compression and Monotonically Increasing Transverse Forces
			First Order Push-Over Response
			Second Order Push-Over Response
		8.3.3 Elastic Beam with Elasto-Plastic Bracing
			Elasto-Plastic Evolution of the Structure
			Response to Transverse Loads
			Second Order Push-Over Curve
	References
9 Buckling of Open Thin-Walled Beams
	9.1 Introduction
	9.2 Elastic Stiffness Operator
		9.2.1 Kinematics
			In-plane Displacements
			Out-of-Plane Displacements
			Strains
		9.2.2 Equilibrium Equations
			Elastic Potential Energy
			Load Potential Energy
			Equilibrium Equations
	9.3 Geometric Stiffness Operator
	9.4 Uniformly Compressed Thin-Walled Beams
		9.4.1 Formulation
			Geometric Stiffness Operator
			Equilibrium Equations
		9.4.2 Uniformly Compressed Beam, Simply Resting on Warping-Unrestrained TorsionalSupports
			Non-symmetric Cross-Section
			Mono-symmetric Cross-Section
			Bi-symmetric Cross-Section
	9.5 Uniformly Bent Thin-Walled Beams
		9.5.1 Formulation
			Geometric Stiffness Operator
			Equilibrium Equations
		9.5.2 Uniformly Bent Beam, Simply Resting on Warping-Unrestrained Torsional Supports
			Mono-axial Bending of a Generic Cross-Section
			Mono-axial Bending of a Symmetric Cross-Section with Respect to the Moment Axis
	9.6 Eccentrically Compressed Thin-Walled Beams
		9.6.1 Formulation
		9.6.2 Eccentrically Compressed Beam, Simply Resting on Warping-Unrestrained Torsional Supports
			Instability Due to an Eccentric Tensile Force
			Solicitation Center Coincident with the Torsion Center
			Solicitation Center Belonging to the Symmetry Axis of a Mono-symmetric Cross-Section
			Solicitation Center Belonging to One of the Two Symmetry Axes of a Bi-symmetric Cross-Section
	9.7 Non-uniformly Bent Thin-Walled Beams
		9.7.1 Formulation
			Prestress and Load Energies
			Normal Prestress Energy
			Tangential Stress Energy
			Total Prestress Energy
			Quadratic Load Energy
			Geometric Stiffness Operator
			Equilibrium Equations
			Bending in a Plane of Symmetry
		9.7.2 Fixed-Free Beam with Thin Rectangular Cross-Section Subject to a Transverse Load Applied at the Free End
			Reduction of the System to a Single Equation
			Solution by Power Series
		9.7.3 Ritz Method
	9.8 Finite Element Buckling Analysis of Thin-Walled Beams
		9.8.1 Polynomial Finite Element
			Total Potential Energy
			Interpolation Functions
			Stiffness Matrices
			Matrix Assembly
		9.8.2 Numerical Examples
	References
10 Buckling of Plates and Prismatic Shells
	10.1 Introduction
	10.2 Kirchhoff Plate Model
		10.2.1 Kinematics
		10.2.2 Internal Forces and Elastic Law
		10.2.3 Elastic Potential Energy and Equilibrium Equations
	10.3 In-Plane Prestressed Plate
	10.4 Plate Simply Supported on Four Sides and Compressed in One Direction
	10.5 Plate Simply Supported on Four Sides and Subject to Bi-Axial Stress
	10.6 Separation of Variables and Exact Finite Element
		10.6.1 Transverse Elastic Line Equation
		10.6.2 Exact One-Dimensional Finite Element
		10.6.3 Critical Load of Single Plates, Simply Supported on Two Opposite Sides
	10.7 Plate Otherwise Solicited or Constrained
	10.8 Compressed Plate Stiffened by a Longitudinal Rib
	10.9 Plate Subject to Uniform Shear Force
		10.9.1 Infinitely Long Plate: Exact Solution
		10.9.2 Infinitely Long Plate: Ritz Approximate Solutions
		10.9.3 Plate of Finite Dimensions
	10.10 Local Instability of Uniformly Compressed Thin-Walled Members
		10.10.1 Finite Strip Method
		10.10.2 Finite Element Sectional Model
		10.10.3 Illustrative Examples of Local and Distortional Buckling
	References
11 Dynamic Bifurcations Induced by Follower Forces
	11.1 Introduction
	11.2 Nonconservative Nature of the Follower Forces
	11.3 Ziegler Column
		11.3.1 Linearized Equations of Motion
		11.3.2 Undamped System
		11.3.3 Damped System
	11.4 Limit Cycles of the Ziegler Column
		11.4.1 Nonlinear Model
		11.4.2 Lindstedt-Poincaré Method
		11.4.3 Numerical Results
	11.5 Viscoelastic Beck Beam
		11.5.1 Linearized Model
		11.5.2 Undamped Beam
		11.5.3 Damped Beam
	References
12 Aeroelastic Stability
	12.1 Introduction
	12.2 Aerodynamic Forces
	12.3 Galloping of Single Degree of Freedom Systems
		12.3.1 Model
			Aeroelastic Force
		12.3.2 Linear Stability Analysis
			Numerical Values of the Galloping Aerodynamic Coefficient
			Influence of the Orientation of the Cross-Section with Respect to the Flow
		12.3.3 Nonlinear Analysis: The Limit Cycle
			Nonlinear Aeroelastic Forces
			Nonlinear Equation of Motion
			Lindstedt-Poincaré Method
			Solution to the Perturbation Equations
	12.4 Galloping of Strings and Beams
		12.4.1 Strings
		12.4.2 Euler-Bernoulli Beams
	12.5 Planar Aeroelastic Systems
		12.5.1 Three Degrees of Freedom Model
		12.5.2 Aeroelastic Forces
			First-Level Quasi-steady Theory
			Second-Level Quasi-steady Theory: The Mean Radius Conjecture
			Linearized Aeroelastic Forces
		12.5.3 Linear Stability Analysis
			Cross-Sections Symmetric with Respect to the Flow Direction
			Dynamic Bifurcations
	12.6 Unidirectional Motions: Galloping and Rotational Divergence
	12.7 Two Degrees of Freedom Translational Galloping
	12.8 Roto-translational Flutter and Galloping
		12.8.1 Steady Aeroelasticity
			Case Cm0=0
		12.8.2 Quasi-steady Aeroelasticity
	12.9 Unsteady Aeroelasticity
	References
13 Parametric Excitation
	13.1 Introduction
	13.2 Introductory Examples
	13.3 Theory of Linear Ordinary Differential Equations with Periodic Coefficients
		13.3.1 Floquet Theorem
		13.3.2 Periodic Systems as Discrete-Time Systems: The Poincaré Map
	13.4 Characteristic Multipliers of Single Degree of Freedom Systems
		13.4.1 General Systems
		13.4.2 Undamped and Damped Hill Equation
	13.5 Mathieu Equation
		13.5.1 Strutt Diagram
		13.5.2 Asymptotic Construction of the Transition Curves
		13.5.3 Influence of Damping
	13.6 Instability Regions of a Physical System: The Bolotin Beam
		13.6.1 Transformation into Canonical Form and Use of the Strutt Diagram
		13.6.2 Direct Construction of the Transition Curves
	13.7 Nonlinear Single Degree of Freedom Systems: The Mathieu-Duffing Oscillator
		13.7.1 Principal Resonance
		13.7.2 Undamped System
		13.7.3 Damped System
	13.8 Linear Systems with Multiple Degrees of Freedom
		13.8.1 Flip and Divergence Bifurcations
		13.8.2 Neimark-Sacker Bifurcation
		13.8.3 Evaluation of the Combination Resonances by Straightforward Expansions
		13.8.4 Combination Resonance and Transition Curves in a Two Degree of Freedom System
	References
14 Solved Problems
	14.1 Introduction
	14.2 Elastic Buckling of Planar Beam Systems
		14.2.1 Stepped Beam
		14.2.2 Clamped-Free Beam Under Distributed and Concentrated Axial Loads
		14.2.3 Clamped-Sliding Beam on Partial Elastic Soil
		14.2.4 Free-Free Beam on Elastic Soil
		14.2.5 Beam Elastically Restrained Against Rotation
		14.2.6 Braced Frame
	14.3 Buckling of Open Thin-Walled Beams
		14.3.1 Uniformly Compressed Clamped-Free Beam
		14.3.2 Uniformly Bent Clamped-Free Beam
		14.3.3 Compressed and Bent Clamped-Free Beam
		14.3.4 Simply Supported Beam, Bent by a Uniformly Distributed Load
	14.4 Buckling of Plates and Prismatic Shells
		14.4.1 Plate Simply Supported on Four Sides and Subject to Bi-axial Stress
		14.4.2 Clamped-Free Plate Elastically Supported at a Vertex, Equally Compressed in Two Directions
		14.4.3 Square Plate on Elastic Soil, Simply Supported on Four Sides and Subject to Bi-axial Stress
		14.4.4 Uniformly Compressed Rectangular Tube with Wings
	14.5 Dynamic Bifurcations Induced by Follower Forces
		14.5.1 Triple Pendulum Subjected to Follower Forces
		14.5.2 Planar Beam Braced at the Tip, Subjected to a Follower Force
		14.5.3 Foil Beam in 3D, Eccentrically Braced at the Tip, Subjected to a Follower Force
	14.6 Aeroelastic Stability
		14.6.1 Nonlinear Galloping of a Base-Isolated Euler-Bernoulli Beam
		14.6.2 Linear Galloping of a Base-Isolated Shear Beam
		14.6.3 Galloping of a Pipeline Suspension Bridge
		14.6.4 Two Degrees of Freedom Translational Galloping
		14.6.5 Flutter in the Steady Theory
		14.6.6 Roto-Translational Galloping in the Quasi-Steady Theory
	14.7 Parametric Excitation
		14.7.1 Exact Stability Analysis of the Mathieu Equation
		14.7.2 Computation of the Characteristic Exponents of the Mathieu Equation via the Hill Infinite Determinant
		14.7.3 Pendulum with Motion Impressed at the Base
		14.7.4 Pendulum with Moving Mass
	References
A Calculus of Variations
	A.1 The Concept of Functional via a Structural Example
	A.2 First Variation of a Functional
	A.3 Euler-Lagrange Equations and Natural Conditions
	References
B Ritz Method
	B.1 Discretization Method
	B.2 Algorithm
	B.3 Ritz Method for Rectangular Plates
		B.3.1 Stiffness Matrices
		B.3.2 Choice of the Trial Functions
		B.3.3 Exploiting the Orthogonality Properties of the Buckling Modes
	References
C Non-uniform Torsion of Open Thin-Walled Beams
	C.1 Mechanics of Torsion
		C.1.1 Effects of the Torsional Warping on the State of Stress
			Uniform Torsion
			Non-uniform Torsion
		C.1.2 Introductory Examples: The I- and C-Cross-Sections
			I-Cross-Section
			C-Cross-Section
	C.2 Vlasov Theory of Non-uniform Torsion
		C.2.1 Kinematics
			In-plane Displacements
			Warping
		C.2.2 Center of Torsion
			Normal Stresses
			Coordinates of the Center of Torsion
			Coincidence Between Torsion and Shear Centers
			Principal Origin of the Sectorial Area
	C.3 One-Dimensional Shaft Model
		C.3.1 Formulation
			Generalized Strains
			Generalized Stresses
			Equilibrium Equations
			Constitutive Law
			Elastic Problem in Terms of Displacements
		C.3.2 Solution to the Problem
		C.3.3 Normal and Tangential Stresses
			Normal Stresses
			Tangential Stresses
	C.4 Illustrative Example: The Open Circular Tube
	C.5 Finite Element Analysis
		C.5.1 Exact Finite Element
			Displacement Field
			Nodal Forces
		C.5.2 Polynomial Finite Element
		C.5.3 Numerical Examples
	References
D Extended Hamilton Principle and Lagrange Equations of Motion
	D.1 Variational Principles for Nonconservative Systems
	D.2 Extended Hamilton Principle
	D.3 Lagrange Equations of Motion
	References
Index