ISBN: 3540674527
TITLE: Statics of Rods
AUTHOR: Svetlitsky, V.A.
TOC:

Introduction 1
1. Equilibrium Equations 11
1.1 Vector Equilibrium Equations 11
1.1.1 Basic Definitions and Hypothesis 11
1.1.2 Vector Equilibrium Equations 13
1.1.3 Relationship Between the Vectors M and  15
1.1.4 Relationship Between the Vectors  and vartheta 18
1.1.5 Displacement of an Axial Line 18
1.1.6 Nondimensional Form of Equations 20
1.1.7 Boundary Conditions 22
1.2 External Loads 23
1.2.1 Types of External Loads 23
1.2.2 Increments of External Loads 29
1.3 Equilibrium Equations in the Attached and Cartesian Coordinate Systems 33
1.3.1 Vector Equilibrium Equations in the Attached Coordinate System 33
1.3.2 Equilibrium Equations in the Attached Coordinate Frame 35
1.3.3 Special Cases of Equilibrium Equations in the Attached Coordinate Frame 36
1.3.4 Vector Equilibrium Equations in the Cartesian Coordinate System 40
1.3.5 Equilibrium Equations in the Cartesian Coordinate System 41
1.4 Equilibrium Equations for Small Displacements and Angles of Rotation 42
1.4.1 Equilibrium Equations in the Attached Coordinate system 42
1.4.2 Equilibrium Equations of the Zeroth Approximation in the Attached Basis 45
1.4.3 Equilibrium Equations of the Zeroth Approximation in the Cartesian Coordinate System 47
1.4.4 Increments of External Loads 49
1.4.5 Equilibrium Equations of the First Approximation in the Attached Coordinate System 51
1.5 Problems 54
2. Integration of Equilibrium Equations 57
2.1 Integration of Linear Equilibrium Equations 57
2.1.1 Equilibrium Equations of the Zeroth Approximation 57
2.1.2 Picard Iteration Method for Determination of the Fundamental Matrix K (eta) 65
2.2 Equilibrium Equations for Rods with Lateral Supports 72
2.2.1 Rods with Lateral Hinge Supports 72
2.2.2 Rods with Lateral Elastic Supports 74
2.2.3 Rods with Predetermined Displacement of Some Cross Sections 76
2.3 Method of Step-by-Step Loading 77
2.3.1 Equilibrium Equations for One Step of Loading 77
2.3.2 Integration of the Equilibrium Equations 81
2.3.3 Method of Successive Approximations 83
3. Static Stability of Rods 89
3.1 Basic Concepts 89
3.1.1 State of Equilibrium 89
3.1.2 Examples 89
3.2 Equilibrium Equations for a Rod After Loss of Stability 93
3.2.1 Vector Equilibrium Equations in the Attached Coordinate System 93
3.2.2 Increments of Forces and Moments 95
3.2.3 Equations in the Form Suitable for Integration 96
3.3 Plane Curvilinear Rods 98
3.3.1 Rods of Plane Axial Line Before Loss of Stability 98
3.3.2 Stability of Plane Configuration of a Ring 101
3.3.3 Stability of a Plane Configuration of a Rod with Lateral Supports 108
3.4 Increments of Loads at Loss of Stability 111
3.4.1 Forces Directed at a Fixed Point 111
3.4.2 Forces Which Follow a Straight Line 112
3.4.3 Increments of Concentrated Forces Which Follow a Straight Line: Small Deflections of a Rod 114
3.4.4 Increments of Concentrated Forces Directed at a Fixed Point: Large Deflections of a Rod 115
3.4.5 Increments of Concentrated Forces Directed at a Fixed Point: Small Deflections of a Rod 117
3.5 Computer-Oriented Methods 117
3.5.1 Natural and Critical Configurations Coincide 117
3.5.2 Natural and Critical Configurations Differ 122
3.5.3 Concentrated Loads Applied to Arbitrary Cross Sections: Determination of Critical Loads 123
3.6 Problems 125
4. Straight Rods 129
4.1 Rods of Straight Natural Configuration 129
4.1.1 Traditional Routines of Derivation of Equilibrium Equations 129
4.1.2 General Equilibrium Equations in the Case of Straight Rods 135
4.2 Equilibrium Equations for Small Displacements and Angles of Rotation 138
4.2.1 Vector Equations 138
4.2.2 Equilibrium Equations in the Attached Coordinate System 139
4.2.3 Equilibrium Equations in the Cartesian Coordinate System 152
4.3 Naturally Twisted Straight Rods 155
4.3.1 Nonlinear Vector Equations of Equilibrium 155
4.3.2 Linear Vector Equations of Equilibrium 155
4.3.3 Equilibrium Equations in the Attached Coordinate System 156
4.4 Straight Rods on Elastic Foundation 158
4.4.1 Forces Acting on a Rod 158
4.4.2 Equilibrium Equations 160
4.4.3 Krylov's Functions 161
4.4.4 Equilibrium Equations for Rods of Constant Cross Section 162
4.4.5 Equilibrium Equations for Rods with Lateral Supports 167
4.4.6 Equilibrium Equations for Rods of Varying Cross Section 168
4.5 Application of Approximate Methods 169
4.5.1 Principle of Virtual Displacements 169
4.5.2 Principle of Minimum of Potential Energy 181
4.5.3 Ritz Method 184
4.5.4 Approximating Methods Based on Lagrangian Multipliers 185
4.5 Stability of Compressed-Twisted Rods 186
4.6 Stability of Straight Rods with Local Constraints 202
4.8 Problems 208
5. Curvilinear Rod's 211
5.1 Plane Rods 211
5.1.1 Equilibrium Equations For a Rod Whose Axial Line Remains a Plane Curve During Deformation 211
5.1.2 Nonlinear Equilibrium Equations in the Cartesian Coordinate System 212
5.1.3 Equilibrium Equations for a Rod Whose Axial Line is a Spatial Curve in a Deformed Configuration 219
5.1.4 Equilibrium Equations in the Case of Small Displacement of Axial Points 220
5.2. Elementary Theory of Cylindrical Springs 228
5.2.1 Helical Rods 228
5.2.2 Linear Theory of Cylindrical Springs 229
5.2.3 Basics of Nonlinear Theory of Cylindrical Springs 235
5.3 General Theory of Cylindrical Springs 236
5.3.1 Linear Equilibrium Equations 236
5.3.2 Cylindrical Springs of Variable Angle of Helix 249
5.4 Flexible Rods in a Rigid Conduit 252
5.4.1 Statement of the Problem 252
5.4.2 Equilibrium Equations 253
5.4.3 Equilibrium Equations for Friction-Free Case 255
5.4.4 Specialization of Equilibrium Equations (5.151) for Rods of Different Bending Stiffnesses (A_22 non= A_33) 255
5.4.5 Specialization of Equilibrium Equations for Rods with Equal Bending Stiffnesses (A_22 = A_33) 257
5.4.6 Determination of Twisting Moments for Rods with Equal Bending Stiffnesses (A_22 = A_33) 257
5.5 Stability of Plane Curvilinear Rods 262
5.6 Problems 269
6. Rods Interacting with Liquid or Air Flows 271
6.1 Introduction 271
6.2 Basic Concepts of Aerohydrodynamics 273
6.2.1 Eulerian and Lagrangian Representations 273
6.2.2 Basic Principles of Aerodynamics 276
6.3 Experimental Results 281
6.4 Aerodynamic Forces Acting on Rods of Circular Cross Section 283
6.5 Stress-Strain State of a Rod Interacting with an Air Flow 292
6.6 Aerodynamic Forces Acting on Rods of Noncircular Cross Section 298
6.6.1 Components of q_n1 and q_1 in the Cartesian Coordinate System 300
6.6.2 Components of q_n1 and q_1 in the Attached Coordinate System 303
6.6.7 Increments of Aerodynamic Forces at Small Displacements of Axial Points 305
6.7.1 Rods of Noncircular Cross Section 307
6.8 Rods Containing Internal Liquid Flows 310
A. Appendices 321
A.1 Elements of Vector Algebra 321
A.1.1 Vector Bases; Coordinates of Vectors 321
A.1.2 Scalar Product 324
A.1.3 Vector Product 325
A.1.4 Scalar Triple Product 327
A.1.5 Vector Triple Product 327
A.1.6 Transformation of Base Vectors 325
A.2 Basics of Differential Geometry 334
A.2.1 The Derivative of a Radius Vector 334
A.2.2 Spatial Curves 335
A.2.3 Derivatives of the Base Vectors 337
A.2.4 Geometrical Meaning of the Components of the Vector  338
A.2.5 Relationship Between _i and vartheta_j 342
A.2.6 Derivations of a Vector in the Attached Coordinate System 346
A.3 Increments of the Components of a Vector under Transformation of the Attached Coordinate System 347
A.4 Distributions 349
A.4.1 The J-function 349
A.4.2 The Nondimensional Function 351
A.4.3 The Heaviside Function 352
A.4.4 Applications of the J-function 353
A.4.5 Integrals Containing Derivatives of the J-function 353
A.5 Direction Cosines of the Unit Vector Tangent to a Rod Axis 354
A.5.1 Plane Curve 354
A.5.2 Spatial Curve 356
A.6 Equations of the First and Higher Approximation 356
B. Solution of the Problems 361
B.1 To Chapter 1 361
B.2 To Chapter 3 367
B.3 To Chapter 4 374
B.4 To Chapter 5 381
References 385
Index 387
END
