Statics of Rods

The volume is devoted to mechanics of rods, which is a branch of mechanics of deformable bodies. The main goal of the book is to present systematically theoretical fundamentals of mechanics of rods as well as numerical methods used for practical purposes. The monograph is concerned with the most general statements of the problems in mechanics of rods. Various types of external loads that a rod may be subject to are discussed. Advanced technique that includes vector is used in the derivation of linear analysis, linear algebra, and distributions and nonlinear equilibrium equations. The use of this technique helps us to make transformations and rearrangement of equations more transparent and compact. Theoretical basics of rods interacting with external and internal flows of fluid and the derivation of the formulas for the hydrodynamic and aerody­ namic forces are presented. The book consists of six chapters and appendices and may be convention­ ally divided into two parts. That is, Chapters 1 to 3 contain, in the main, theoretical material, whereas Chapters 4 to6 illustrate the application of the theoretical results to problems of practical interest. Problems for self-study are found in Chapters 1, 3, 4, and 5. The solutions to most of the problems are given in Appendix B. The monograph is addressed to scientists, institutional and industrial re­ searchers, lecturers, and graduate students.

1. Equilibrium Equations.- 1.1 Vector Equilibrium Equations.- 1.1.1 Basic Definitions and Hypothesis.- 1.1.2 Vector Equilibrium Equations.- 1.1.3 Relationship Between the Vectors M and æ.- 1.1.4 Relationship Between the Vectors æ and ?.- 1.1.5 Displacement of an Axial Line.- 1.1.6 Nondimensional Form of Equations.- 1.1.7 Boundary Conditions.- 1.2 External Loads.- 1.2.1 Types of External Loads.- 1.2.2 Increments of External Loads.- 1.3 Equilibrium Equations in the Attached and Cartesian Coordinate Systems.- 1.3.1 Vector Equilibrium Equations in the Attached Coordinate System.- 1.3.2 Equilibrium Equations in the Attached Coordinate Frame.- 1.3.3 Special Cases of Equilibrium Equations in the Attached Coordinate Frame.- 1.3.4 Vector Equilibrium Equations in the Cartesian Coordinate System.- 1.3.5 Equilibrium Equations in the Cartesian Coordinate System.- 1.4 Equilibrium Equations for Small Displacements and Angles of Rotation.- 1.4.1 Equilibrium Equations in the Attached Coordinate System.- 1.4.2 Equilibrium Equations of the Zeroth Approximation in the Attached Basis.- 1.4.3 Equilibrium Equations of the Zeroth Approximation in the Cartesian Coordinate System.- 1.4.4 Increments of External Loads.- 1.4.5 Equilibrium Equations of the First Approximation in the Attached Coordinate System.- 1.5 Problems.- 2. Integration of Equilibrium Equations.- 2.1 Integration of Linear Equilibrium Equations.- 2.1.1 Equilibrium Equations of the Zeroth Approximation.- 2.1.2 Picard Iteration Method for Determination of the Fundamental Matrix K (?).- 2.2 Equilibrium Equations for Rods with Lateral Supports.- 2.2.1 Rods with Lateral Hinge Supports.- 2.2.2 Rods with Lateral Elastic Supports.- 2.2.3 Rods with Predetermined Displacement of Some Cross Sections.- 2.3 Method of Step-by-Step Loading.- 2.3.1 Equilibrium Equations for One Step of Loading.- 2.3.2 Integration of the Equilibrium Equations.- 2.3.3 Method of Successive Approximations.- 3. Static Stability of Rods.- 3.1 Basic Concepts.- 3.1.1 State of Equilibrium.- 3.1.2 Examples.- 3.2 Equilibrium Equations for a Rod After Loss of Stability.- 3.2.1 Vector Equilibrium Equations in the Attached Coordinate System.- 3.2.2 Increments of Forces and Moments.- 3.2.3 Equations in the Form Suitable for Integration.- 3.3 Plane Curvilinear Rods.- 3.3.1 Rods of Plane Axial Line Before Loss of Stability.- 3.3.2 Stability of Plane Configuration of a Ring.- 3.3.3 Stability of a Plane Configuration of a Rod with Lateral Supports.- 3.4 Increments of Loads at Loss of Stability.- 3.4.1 Forces Directed at a Fixed Point.- 3.4.2 Forces Which Follow a Straight Line.- 3.4.3 Increments of Concentrated Forces Which Follow a Straight Line: Small Deflections of a Rod.- 3.4.4 Increments of Concentrated Forces Directed at a Fixed Point: Large Deflections of a Rod.- 3.4.5 Increments of Concentrated Forces Directed at a Fixed Point: Small Deflections of a Rod.- 3.5 Computer-Oriented Methods.- 3.5.1 Natural and Critical Configurations Coincide.- 3.5.2 Natural and Critical Configurations Differ.- 3.5.3 Concentrated Loads Applied to Arbitrary Cross Sections: Determination of Critical Loads..- 3.6 Problems.- 4. Straight Rods.- 4.1 Rods of Straight Natural Configuration.- 4.1.1 Traditional Routines of Derivation of Equilibrium Equations.- 4.1.2 General Equilibrium Equations in the Case of Straight Rods.- 4.2 Equilibrium Equations for Small Displacements and Angles of Rotation.- 4.2.1 Vector Equations.- 4.2.2 Equilibrium Equations in the Attached Coordinate System.- 4.2.3 Equilibrium Equations in the Cartesian Coordinate System.- 4.3 Naturally Twisted Straight Rods.- 4.3.1 Nonlinear Vector Equations of Equilibrium.- 4.3.2 Linear Vector Equations of Equilibrium.- 4.3.3 Equilibrium Equations in the Attached Coordinate System.- 4.4 Straight Rods on Elastic Foundation.- 4.4.1 Forces Acting on a Rod.- 4.4.2 Equilibrium Equations.- 4.4.3 Krylov’s Functions.- 4.4.4 Equilibrium Equations for Rods of Constant Cross Section.- 4.4.5 Equilibrium Equations for Rods with Lateral Supports.- 4.4.6 Equilibrium Equations for Rods of Varying Cross Section.- 4.5 Application of Approximate Methods.- 4.5.1 Principle of Virtual Displacements.- 4.5.2 Principle of Minimum of Potential Energy.- 4.5.3 Ritz Method.- 4.5.4 Approximating Methods Based on Lagrangian Multipliers.- 4.6 Stability of Compressed-Twisted Rods.- 4.7 Stability of Straight Rods with Local Constraints.- 4.8 Problems.- 5. Curvilinear Rods.- 5.1 Plane Rods.- 5.1.1 Equilibrium Equations for a Rod Whose Axial Line Remains a Plane Curve During Deformation.- 5.1.2 Nonlinear Equilibrium Equations in the Cartesian Coordinate System.- 5.1.3 Equilibrium Equations for a Rod Whose Axial Line is a Spatial Curve in a Deformed Configuration.- 5.1.4 Equilibrium Equations in the Case of Small Displacement of Axial Points.- 5.2 Elementary Theory of Cylindrical Springs.- 5.2.1 Helical Rods.- 5.2.2 Linear Theory of Cylindrical Springs.- 5.2.3 Basics of Nonlinear Theory of Cylindrical Springs.- 5.3 General Theory of Cylindrical Springs.- 5.3.1 Linear Equilibrium Equations.- 5.3.2 Cylindrical Springs of Variable Angle of Helix.- 5.4 Flexible Rods in a Rigid Conduit.- 5.4.1 Statement of the Problem.- 5.4.2 Equilibrium Equations.- 5.4.3 Equilibrium Equations for Friction-Free Case.- 5.4.4 Specialization of Equilibrium Equations (5.151) for Rods of Different Bending Stiffnesses (A22 ? A33).- 5.4.5 Specialization of Equilibrium Equations for Rods with Equal Bending Stiffnesses (A22 = A33).- 5.4.6 Determination of Twisting Moments for Rods with Equal Bending Stiffnesses (A22 = A33).- 5.5 Stability of Plane Curvilinear Rods.- 5.6 Problems.- 6. Rods Interacting with Liquid or Air Flows.- 6.1 Introduction.- 6.2 Basic Concepts of Aerohydrodynamics.- 6.2.1 Eulerian and Lagrangian Representations.- 6.2.2 Basic Principles of Aerodynamics.- 6.3 Experimental Results.- 6.4 Aerodynamic Forces Acting on Rods of Circular Cross Section.- 6.5 Stress-Strain State of a Rod Interacting with an Air Flow.- 6.6 Aerodynamic Forces Acting on Rods of Noncircular Cross Section.- 6.6.1 Components of qn1 and q1 in the Cartesian Coordinate System.- 6.6.2 Components of qn1, and q1 in the Attached Coordinate System.- 6.7 Increments of Aerodynamic Forces at Small Displacements of Axial Points.- 6.7.1 Rods of Noncircular Cross Section.- 6.8 Rods Containing Internal Liquid Flows.- A. Appendices.- A.1 Elements of Vector Algebra.- A.1.1 Vector Bases; Coordinates of Vectors.- A.1.2 Scalar Product.- A.1.3 Vector Product.- A.1.4 Scalar Triple Product.- A.1.5 Vector Triple Product.- A.1.6 Transformation of Base Vectors.- A.2 Basics of Differential Geometry.- A.2.1 The Derivative of a Radius Vector.- A.2.2 Spatial Curves.- A.2.3 Derivatives of the Base Vectors.- A.2.4 Geometrical Meaning of the Components of the Vector æ.- A.2.6 Derivatives of a Vector in the Attached Coordinate System.- A.3 Increments of the Components of a Vector under Transformation of the Attached Coordinate System.- A.4 Distributions.- A.4.1 The ?-function.- A.4.2 The Nondimensional ?-function.- A.4.3 The Heaviside Function.- A.4.4 Applications of the ?-function.- A.4.5 Integrals Containing Derivatives of the ?-function.- A.5 Direction Cosines of the Unit Vector Tangent to a Rod Axis.- A.5.1 Plane Curve.- A.5.2 Spatial Curve.- A.6 Equations of the First and Higher Approximation.- B. Solution of the Problems.- B.1 To Chapter 1.- B.2 To Chapter 3.- B.3 To Chapter 4.- B.4 To Chapter 5.- References.