Higher-Order Differential Equations and Elasticity

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Diagrams, Lists, Notes, and Tables
Preface
Acknowledgments
About the Author
Physical Quantities
5.
Special, Second, and Higher-Order Equations
	5.1.
C-, p-discriminants and Special Curves and Integrals
		5.1.1.
General and Special Integrals of a Differential Equation
		5.1.2.
C-discriminant, Envelope, Node, and Cusp Loci
		5.1.3.
p-discriminant, Envelope, Tac, and Cusp Loci
		5.1.4.
Four Special Curves and One Special Integral
		5.1.5.
Special Integrals of Higher-Order Differential Equations
	5.2.
Quadratic in the Slope or Arbitrary Constant
		5.2.1.
First-Order Differential Equation—Quadratic in the Slope
		5.2.2.
A Family of Parabolas with the Real Axis as Envelope
		5.2.3.
A Family of Circles with Two Envelopes and a Tac-Locus
		5.2.4.
Integral Curves with Envelope and Node and Tac Loci
	5.3.
Clairaut (1734) and D’Alembert (1748) Special Equations
		5.3.1.
Special Integral of Clairaut (1734) Differential Equation
		5.3.2.
Parabola as the Envelope of a Family of Straight Lines
		5.3.3.
Generalization of the Clairaut to the D’Alembert (1748) Differential Equation
		5.3.4.
Family of Parabolas with Envelope Parallel to Diagonals of Quadrants
	5.4.
Equations Solvable for the Variables or Slope
		5.4.1.
Equation Solvable for the Dependent or Independent Variable
		5.4.2.
Special Integral of Equation Solvable for the Dependent Variable
		5.4.3.
Equation Solvable for the Dependent Variable 
and Its Special Integral
		5.4.4.
Equation of Degree N Solvable for the Slope
	5.5.
Dual Differentials and Thermodynamics
		5.5.1.
Legendre Transformation of an Exact Differential
		5.5.2.
Original and Dual Differential Equations
		5.5.3.
Solution of a Differential Equation from Its Dual
		5.5.4.
Work of the Pressure in a Volume Change
		5.5.5.
Simple and General Thermodynamic Systems
		5.5.6.
Extensive and Intensive Thermodynamic Variables
		5.5.7.
Mechanical and Thermal Equilibrium of a System
		5.5.8.
Internal Energy, Enthalpy, Free Energy, and Free Enthalpy
		5.5.9.
Specific Volume, Pressure, Temperature, and Entropy
		5.5.10.
Equivoluminar, Isobaric, Isothemal, and Adiabatic Processes
		5.5.11.
Heat and Work in a Thermodynamic Process
		5.5.12.
Specific Heats at Constant Volume and Pressure
		5.5.13.
Adiabatic Exponent and Thermodynamic Relations
		5.5.14.
Thermodynamic Derivatives and Equation of State
		5.5.15.
Adiabatic Sound Speed and Non-Adiabatic Coefficient
		5.5.16.
Perfect Gas Consisting of Non-Interacting Molecules
		5.5.17.
Internal Energy and Equation of State
		5.5.18.
Properties of Perfect and Ideal Gases
		5.5.19.
Adiabatic Perfect and Ideal Gases
		5.5.20.
Entropy of an Ideal Gas
		5.5.21.
Microscopic Statistics of Perfect/Ideal Gases
		5.5.22. Monatomic, Diatomic, and Polyatomic Molecules
		5.5.23. Evaluation of Thermodynamic Derivatives Using Jacobians
		5.5.24. Arbitrary Substances versus Perfect/Ideal Gases
		5.5.25. Adiabatic and Isothermal Sound Speeds
		5.5.26. Atmosphere of the Earth at Sea Level
	5.6. Equation of the Second-Order Missing Slope or One Variable
		5.6.1.
Reduction of Second-Order to First-Order Differential Equation
		5.6.2.
Second-Order Differential Equation with Independent Variable Missing
		5.6.3.
Both Independent Variable and Slope Missing
		5.6.4.
Second-Order Differential Equation with Dependent Variable Missing
	5.7.
Equation of Order N Depressed to Lower Order
		5.7.1.
Differential Equation of Order N Involving Only the Independent Variable
		5.7.2.
Differential Equation Not Involving Derivatives 
up to Order N – 2
		5.7.3.
Differential Equation not Involving the Independent Variable
		5.7.4.
Differential Equation Involving Only Derivatives 
of Orders N and N – 2
		5.7.5.
Lowest-Order Derivative Appearing is N – P
	5.8.
Factorizable and Exact Differential Equations
		5.8.1.
Differential Equation with Factorizable Operator
		5.8.2.
Exact Differential Equation of Any Order
	5.9.
Two Kinds of Non-Linear Homogeneous Differential Equations
		5.9.1.
Homogeneous Differential Equation of the First Kind
		5.9.2.
Homogeneous Differential Equation of the Second Kind
		5.9.3.
Depression of the Order of Homogeneous Differential Equations
	Conclusion 5
6.
Buckling of Beams and Plates
	6.1. Non-Linear Buckling of a Beam under Compression
		6.1.1.
Shape, Slope, and Curvature of the Elastica
		6.1.2.
Stiff Beam under Traction/Compression
		6.1.3.
Linear Bending for Small Slope
		6.1.4.
Non-Linear Buckling with Large Slope
		6.1.5.
Critical Buckling Load for a Clamped Beam
		6.1.6.
Buckling of a Beam with Both Ends Pinned
		6.1.7.
Beam with One Clamped and One Pinned Support
		6.1.8.
Linear and Non-Linear Boundary Conditions at a Free End
		6.1.9.
Buckling of a Clamped-Free or Cantilever Beam
		6.1.10.
Non-Linear Elastica of a Cantilever Beam
		6.1.11.
Linear Approximation and Non-Linear Corrections of All Orders
		6.1.12.
Comparison of Linear and Non-Linear Effects
		6.1.13.
Coincidence of Linear and Non-Linear Critical Buckling Loads
		6.1.14.
Non-Linear Effect on the Shape of the Buckled Elastica
		6.1.15.
Non-Linear Generation of Harmonics
	6.2.
Opposition or Facilitation of Buckling of Beams (Campos & Marta 2014)
		6.2.1.
Resilience Parameter for a Translational Spring
		6.2.2.
Cantilever Beam with a Translational Spring at the Tip
		6.2.3.
Clamped Beam with a Translational Spring at the Middle
		6.2.4.
Pinned Beam with a Translational Spring at the Middle
		6.2.5.
Clamped-Pinned Beam with Spring at the Middle
		6.2.6.
Force (Moment) Due to a Translational (Rotary) Spring
		6.2.7.
Rotary Spring at the Free End of a Cantilever Beam
		6.2.8. Rotary Spring at the Middle of a Clamped Beam
		6.2.9.
Pinned Beam with Rotary Spring at the Middle
		6.2.10.
Clamped-Pinned Beam with a Rotary Spring
		6.2.11.
Combinations of Four Supports and Two Types of Springs
		6.2.12.
Buckling Modes for a Cantilever Beam
		6.2.13.
First Three Modes of Buckled Cantilever Beam
		6.2.14.
Infinite Roots for the Critical Buckling Load
		6.2.15.
Rotary Spring Favoring or Opposing Buckling
		6.2.16.
Transition Resilience for Buckling without an Axial Load
		6.2.17.
Strengthening a Structure or Facilitating Its Demolition
	6.3.
Beam under Traction/Compression with Attractive/Repulsive Springs
		6.3.1.
Beam Supported on Attractive or Repulsive Springs
		6.3.2.
Biquadratic Differential Equation with Constant Coefficients
		6.3.3.
Beam under Traction without Spring Support
		6.3.4. Bar Continuously Supported on Attractive Springs
		6.3.5.
Bar Supported on Translational Repulsive Springs
		6.3.6.
Beam with Axial Compression Balanced by Attractive Spring
		6.3.7.
Beam with Axial Traction Balanced by Attractive Springs
		6.3.8.
Beam with Axial Traction Dominating Attractive Spring
		6.3.9.
Beam with Axial Compression Dominating Attractive Springs
		6.3.10.
Beam with Axial Tension Dominating Repulsive Springs
		6.3.11 Beam with Attractive Springs Dominating the Axial Tension
		6.3.12.
Generalized Critical Buckling Condition
		6.3.13.
Combined Tension-Spring Buckling Load
		6.3.14.
Deflection by a Uniform Shear Stress
	6.4.
Axial Traction/Compression of a Straight Bar
		6.4.1.
Analogy with the Deflection of an Elastic String
		6.4.2.
Analogy with Steady Heat Conduction
		6.4.3.
Linear/Non-Linear Deformation of Homogeneous/Inhomogeneous Media
		6.4.4.
Axial Contraction/Extension of an Inhomogenous Bar
	6.5.
In-Plane Loads in a Flat Plate
		6.5.1.
Linear Relation between Stresses and Strains in Elasticity
		6.5.2.
Thin Elastic Plate without Transverse Load
		6.5.3.
Comparison with Stresses and Strains in Plane Elasticity
		6.5.4.
Second-Order Differential Equation for the Displacement 
Vector
		6.5.5.
Biharmonic Equation for the Stress Function
		6.5.6.
Stresses Due to the Drilling of a Screw
		6.5.7.
Plate with a Circular Hole Subject to In-Plane Stresses
		6.5.8.
Stress Concentration Near a Hole in a Plate
		6.5.9.
Disk under Axial Compression
	6.6.
Deflection of a Membrane under Anisotropic Tension
		6.6.1.
Non-Linear Strain Tensor and Elastic Energy of a Membrane
		6.6.2.
Non-Linear Deflection of a Membrane under Anisotropic Stresses
		6.6.3.
Diagonalization of the Anisotropic Laplace Operator
		6.6.4.
Green’s Function for the Anisotropic Laplace Operator
		6.6.5.
Characteristics as the Lines of Constant Deflection
		6.6.6.
Deflection of an Elliptic Membrane by Its Own Weight
		6.6.7.
Deflection by Arbitrary Loads for an Unbounded Membrane
	6.7.
Linear Bending of a Thin Plate
		6.7.1.
Linear/Non-Linear and Weak/Strong Bending of Plates
		6.7.2.
Bending Moment and Stiffness and Principal Curvatures
		6.7.3.
Twist and Curvatures in Rotated Axis
		6.7.4.
Twist and Stress Couples on a Plate
		6.7.5.
Elastic Energy of Bending
		6.7.6.
Balance of Turning Moments and Transverse Forces
		6.7.7.
Bending Displacement, Strain, and Stress
		6.7.8.
Balance of Elastic Energy and Work of the Normal Forces
		6.7.9.
Principle of Virtual Work and Bending Vector
		6.7.10.
Deformation Vector and Balance Equation
		6.7.11.
Comparison of the Bending of Bars and Plates
		6.7.12.
Clamped, Pinned, and Free Boundaries
		6.7.13.
Normal Stress Couple and Turning Moment
		6.7.14.
Rectangular, Circular, and Arbitrarily Shaped Plates
		6.7.15.
Bending Due to a Concentrated Force at the Center
		6.7.16.
Bending of a Clamped/Supported Circular Plate
		6.7.17.
Constant Forcing of a Biharmonic Equation with Radial Symmetry
		6.7.18.
Bending of a Circular Plate by Its Own Weight
		6.7.19.
Bending of a Heavy Suspended Circular Plate
	6.8.
Elastic Stability of a Stressed Orthotropic Plate
		6.8.1.
Linear Buckling of a Plate under Tension
		6.8.2.
Membranes and Weak/Strong Bending of Plates
		6.8.3.
Inertia, Volume, and Surface Forces
		6.8.4.
Balance of Moments and Symmetry of the Stresses
		6.8.5.
Kinetic Energy and Work of Deformation
		6.8.6.
Residual, Elastic, and Inelastic Stresses
		6.8.7.
Components and Symmetries of the Stiffness Tensor
		6.8.8.
Isotropy as the Maximum Material Symmetry
		6.8.9.
Homoclinic and Orthotropic Elastic Materials
		6.8.10.
Stiffness and Compliance Matrices as Inverses
		6.8.11.
Elastic Moduli of Orthotropic Materials
		6.8.12.
Elastic Energy of Bending of an Orthotropic Plate
		6.8.13.
Plate Made of a Pseudo-Isotropic Orthotropic Material
		6.8.14.
Comparison of Isotropic and Pseudo-Isotropic Orthotropic Plates
		6.8.15.
Generalized Bending Stiffness for Pseudo-Isotropic Orthotropic Plates
		6.8.16.
Boundary Conditions for Bending of a Plate
		6.8.17.
Rectangular Plate Supported on All Four Sides
		6.8.18.
Mixed Supported and Free Boundary Conditions
		6.8.19.
Critical Buckling Stress for a Plate
		6.8.20.
Shape of the Directrix of a Buckled Plate
	6.9.
Non-Linear Coupling in a Thick Plate (Foppl 1907, Von
Karman 1910)
		6.9.1.
Transverse Displacement with Non-Uniform In-Plane Stresses
		6.9.2.
Linear and Non-Linear Terms in the Strain Tensor
		6.9.3.
Relation between the Stresses and the In-Plane and Transverse Displacements
		6.9.4.
Elastic Energies of Deflection, Bending, and In-Plane Deformation
		6.9.5.
Balance Equations and Boundary Conditions
		6.9.6.
Bending and Stretching of an Elliptic Plate
		6.9.7.
Slope, Stress Couples, and Turning Moments
		6.9.8.
Perturbation Expansions for the Transverse Displacement and Stress Function
		6.9.9.
Non-Linear Coupling of Bending and Stretching with Axial Symmetry
		6.9.10.
Strong Bending of a Circular Plate by Transverse Loads
		6.9.11.
Radially Symmetric Biharmonic Equation Forced by a Power
		6.9.12.
Heavy Circular Plate under Axial Compression
		6.9.13.
Non-Linear Coupling of Bending and Compression
		6.9.14.
Transverse Displacement and Stress Function with Clamping
		6.9.15.
Stresses, Strains, and Radial Displacement
		6.9.16.
Slope, Stress Couple, and Augmented Turning Moment
		6.9.17.
Method for Axisymmetric Strong Bending
	Conclusion 6
Bibliography
Index