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