Statics and Influence Functions: From a Modern Perspective

Preface to the Second Edition
Preface to the First Edition
Contents
1 Foundation
	1.1 Introduction
		1.1.1 Integration by Parts
		1.1.2 Principle of Virtual Displacements
		1.1.3 Betti\'s Theorem
		1.1.4 Influence Functions
		1.1.5 Identities
	1.2 Green\'s Identities
		1.2.1 Longitudinal Displacement u(x) of a Bar
		1.2.2 Shear Deformation wS(x) of a Beam
		1.2.3 Deflection w of a Rope
		1.2.4 Deflection w of a Beam
		1.2.5 Deflection w of a Beam, 2nd Order Theory
		1.2.6 Beam on an Elastic foundation
		1.2.7 Tensile Chord Bridge
		1.2.8 Torsion
	1.3 Variational Principles of Structural Analysis
	1.4 Zero Sums
	1.5 Examples
		1.5.1 The Principle of Virtual Displacements
		1.5.2 Conservation of Energy
		1.5.3 The Principle of Virtual Forces
	1.6 Frames
	1.7 Spring Support
	1.8 Single Forces and Moments
	1.9 Support Settlements
	1.10 Springs
	1.11 Temperature
	1.12 The Complete Equation
	1.13 Shortcuts
	1.14 Duality
	1.15 Mohr Versus Betti
	1.16 Weak and Strong Influence Functions
	1.17 The Canonical Boundary Values
	1.18 The Reduction of the Dimension
	1.19 Boundary Element Method
	1.20 Finite Elements and Boundary Elements
	1.21 Test Functions
	1.22 Do Virtual Displacements Have to Be Small?
	1.23 Only When in Equilibrium?
	1.24 What Counts as Displacement and What as Force?
	1.25 The Number of Force and Displacement Terms
	1.26 Why the Minus in -Hw\'\' = p?
	1.27 The Virtual Internal Energy
	1.28 Castigliano\'s Theorem(s)
	1.29 Equilibrium
	1.30 The Mathematics Behind the Equilibrium Conditions
	1.31 Balance and Second-Order Theory
	1.32 Sources and Sinks
	1.33 The Principle of Minimum Potential Energy
		1.33.1 Minimum or Maximum?
		1.33.2 Cracks
		1.33.3 The Size of the Trial Space mathcalV
	1.34 Infinite Energy
	1.35 Sobolev\'s Embedding Theorem
	1.36 Reduction Principle
	1.37 The Force Method
	1.38 Where Does It Run To?
	1.39 Finite Elements and Green\'s Identity
	References
2 Betti\'s Theorem
	2.1 Basics
	2.2 Influence Functions for Displacements
		2.2.1 Formulation
	2.3 Influence Functions for Forces
		2.3.1 Influence Function for N(x)
		2.3.2 Influence Function for M(x)
		2.3.3 Influence Function for V(x)
		2.3.4 Settlement of a Support
		2.3.5 Temperature Effects
		2.3.6 Single Moments Differentiate the Influence Functions
	2.4 Statically Determinate Structures
		2.4.1 Pole-Plans
		2.4.2 Construction of Pole-Plans
		2.4.3 How to Determine the Magnitude of Rotations
		2.4.4 Influence Function for a Shear Force, Fig. 2.17
		2.4.5 Influence Function for a Normal Force, Fig. 2.18
		2.4.6 Influence Function for a Moment, Fig. 2.19
		2.4.7 Influence Function for a Moment, Fig. 2.20
		2.4.8 Influence Function for a Shear Force, Fig. 2.21
		2.4.9 Influence Function for Two Support Reactions, Fig. 2.22
		2.4.10 Abutment Reaction, Fig. 2.23
	2.5 Free Ends
	2.6 Statically Indeterminate Structures
	2.7 Influence Functions for Support Reactions
	2.8 Jumps in Internal Forces
	2.9 The Zeros of the Shear Force
	2.10 Dirac Deltas
	2.11 Dirac Energy
	2.12 Point Values in 2-D and 3-D
	2.13 Duality
	2.14 The Adjoint Operator
	2.15 Monopoles and Dipoles
	2.16 The Leaning Tower of Pisa
	2.17 Influence Functions for Integral Values
	2.18 Influence Functions Integrate
	2.19 St. Venant\'s Principle
	2.20 Second-Order Theory
	References
3 Finite Elements
	3.1 The Minimum
	3.2 Why the Nodal Values of the Rope Are Exact
	3.3 Adding the Local Solution
	3.4 Projection
	3.5 Equivalent Nodal Forces
	3.6 Fixed End Forces
	3.7 Shape Forces and the FE-load
	3.8 How the Ball Got Rolling
	3.9 Assembling the Element Matrices
	3.10 Equivalent Stress Transformation
	3.11 Calculation of Influence Functions with Finite Elements
	3.12 Functionals
	3.13 Generalized Influence Functions
	3.14 Weak and Strong Influence Functions
	3.15 The Local Influence Function
	3.16 The Central Equation
	3.17 Representation of an FE-solution
	3.18 Frame Structures and J = gTf
	3.19 State Vectors and Measurements
	3.20 Maxwell\'s Theorem
	3.21 The Inverse Stiffness Matrix
	3.22 The Trial Space mathcalVh Has Two Bases
	3.23 General Form of an FE-Influence Function
	3.24 The Dominance of the Columns gi of the Inverse
	3.25 Nature Makes No Jumps, but Finite Elements Do
	3.26 The Path from the Source Point to the Load
	3.27 The Columns of K and K-1
	3.28 The Inverse as an Analysis Tool
	3.29 Local Changes and the Inverse
	3.30 Mohr and the Flexibility Matrix K-1
	3.31 Non-uniform Plates
	3.32 Sensitivity Plots
	3.33 Support Reactions
	3.34 If a Support Settles
	3.35 Influence Function for a Rigid Support
	3.36 Shear Forces
	3.37 Influence Function for an Elastic Support
	3.38 Elasticity Theory and Point Supports
	3.39 Point Supports Are Hot Spots
	3.40 The Amputated Dipole
	3.41 A Dipole at the Edge
	3.42 Single Force at a Node
	3.43 Predeformations
	3.44 The Limits of FE-Influence Functions
	3.45 Checking Results
	3.46 Transient Problems
	3.47 The Intelligence of Functions
	References
4 Betti Extended
	4.1 Proof
	4.2 At Which Points Is the FE-Solution Exact?
	4.3 Exact Values
	4.4 One-Dimensional Problems
	4.5 Isogeometric Analysis
	4.6 Planar Problems
	4.7 Point Supports
	4.8 If the Solution Lies in mathcalVh
	4.9 Patch Test
	4.10 Adaptive Refinement
	4.11 Pollution
		4.11.1 Causes
		4.11.2 Details
	4.12 Super Convergence
	References
5 Stiffness Changes and Reanalysis
	5.1 Parameter Identification
	5.2 Introductory Remarks
	5.3 Adding or Subtracting Stiffness
	5.4 Dipoles and Monopoles
	5.5 Displacements and Forces
	5.6 Symmetry and Antisymmetry
	5.7 Orthogonality
	5.8 The Effects Fade Away
	5.9 The Relevance of These Results
	5.10 Frames
	5.11 Forces j+
	5.12 Replacement as Alternative
	5.13 The Derivative of the Inverse K-1
	5.14 The Derivatives u/fk and u/kij
	5.15 Integrating over the Defective Element
	5.16 Mohr or the Weak Form of Influence Functions
	5.17 Near and Far
	5.18 Supports
	5.19 Integral Bridges
	5.20 Retrofitting
	5.21 Classical Formulation
	5.22 Calculation of uc
		5.22.1 Iteration
		5.22.2 Direct Solution
		5.22.3 Support Stiffness
	5.23 Sherman-Morrison-Woodbury
	5.24 One-Click Reanalysis
		5.24.1 When the Load ``Is Hit\'\'
		5.24.2 Singular Stiffness Matrices
	5.25 Subsequent Installation of Joints
	5.26 Buckling Loads
	5.27 Dynamic Problems
		5.27.1 Antisymmetry in the Compensating Motions
	5.28 The Continuum
		5.28.1 Potential Theory
		5.28.2 Kernels j+
		5.28.3 The Two Approaches
	References
6 Singularities
	6.1 Singular Stresses
	6.2 Singular Support Reactions
	6.3 Single Forces
	6.4 Decay of Stresses
	6.5 Infinite Stresses
	6.6 Symmetry of Adjoint Effects
	6.7 Cantilever Wall Plate
	6.8 Standard Situations
	6.9 Singularities in Influence Functions
7 Mixed Formulations
	7.1 Bernoulli Beam
	7.2 Timoshenko Beam
	7.3 Poisson Equation
	7.4 The Plate Equation
	7.5 Kirchhoff Plate
	7.6 Reissner-Mindlin Plate
	7.7 Influence Functions
	7.8 Betti Extended
	Reference
8 Nonlinear Problems
	8.1 Introduction
	8.2 Gateaux Derivative
	8.3 Nonlinear Bar
		8.3.1 Newton\'s Method
	8.4 Geometrically Nonlinear Beam
		8.4.1 Conservation of Energy
	8.5 Geometrically Nonlinear Kirchhoff Plate
	8.6 Nonlinear Elasticity Theory
		8.6.1 Linearization
		8.6.2 A Truss Element in 3-D
		8.6.3 Planar Problem
	8.7 Nonlinear Functionals
	References
9 Addenda
	9.1 Basics
	9.2 Notation
	9.3 FE-Notation
	9.4 Vectors and Functions
	9.5 The Algebra of the Identities
	9.6 The Algebra of Finite Elements
	9.7 Eigenvalues and Eigenvectors
	9.8 mathcalVh and mathcalVh+
	9.9 Galerkin
	9.10 Weak Solution
	9.11 Variation and Green\'s First Identity
	9.12 The Basic Functional (Hu-Washizu)
	9.13 Force Method and Slope Deflection Method
	9.14 The Adjoint Operator and Green\'s Function
	9.15 Rope
	9.16 Filter
	9.17 Poisson Equation
	9.18 Potentials and Potentials
	9.19 Single Force Acting on a Plate
	9.20 Multipoles
	9.21 The Dimension of the fi
	9.22 Weak and Strong Influence Functions
	9.23 How the Embedding Theorem Got Its Name
	9.24 Point Loads and Their Energy
	9.25 Early Birds
	References
10 Software
References
Index