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About the Book

The book is a theoretical treatise on the subject of higher order moments. The theory of moments developed into a specific branch of mathematics which became a useful tool for solving com­plicated problems in structural mechanics. A number of Soviet scientists developed the so-called moment-operational method, which has proved to be extremely efficient for solving various pro­blems of modern engineering. These problems arise when non-linear elasticity must be taken into account, when precise designing of non-uniform structural elements is required, when the loading of a structure is essentially non-uniform, etc.

The moment-operational method has been widely employed for compiling manuals containing numerous tables and formulas for designing beams, arches and frames. A number of scientific research works and textbooks present the theory of moments of higher order methods examples of problems solved by the moment-operational method.

The knowledge of the theory of higher moments as well as of the moment-operational method will serve to extend the field of their application to new problems.

The book was translated from the Russian by N. Lebedinsky and was published by Mir in 1974.

Table of Contents

Part 1 Moments of Higher Order: Theory and Application 9

Chapter 1. Theory of Moments 9

  1. General Concept of Moments of Area 9
  2. Moments as Geometric Characteristics of Beam Cross Sections 18
  3. Uniaxial Moments of Point Forces and Couples 36
  4. Uniaxial Moments of Balanced and Unbalanced Systems 40
  5. Uniaxial Moments of Areas of Simple Figures 45
  6. Uniaxial Moments of Areas of More Complicated Figures 52
  7. Uniaxial Moments of Compound Loads 56
  8. Moments of Higher Order and Generalized Forces 64

Chapter II. Application of Theory of Moments 73

  1. Rigidity or Uniform Beams 73
  2. Geometrical Interpretation of Moments 74
  3. Calculation of Displacement Integrals in Rod Systems 83
  4. Application of Higher Moments to Loading of Parabolic Influence Lines 92
  5. Formulas for Statically Indeterminate Structures 94
  6. Rigidity of Beams Composed of Prismatic Parts 140
  7. Moments of Area of Flexibility Diagram for Non-Uniform Beams 145

Part II Moment-Operational Method: Theory and Application 160

Chapter III. Moment-Operational Method 160

  1. Principles of Moment-Operational Method 160
  2. Bimoments 162
  3. Differential and Integral Bimoments 163
  4. Application of Moment-Operational Method for Solving
    Linear Differential Equations 170

Chapter IV. Rigidity of Non-Uniform Beams 173

  1. Determination of Displacements by Coefficients of Flexibility Polynomial Expression 173
  2. Determination of Displacements by Derivatives of Flexibility Analytical Expression 178
  3. Determination of Displacements by Flexibility Integrals 182
  4. Dermination of Displacements when Rigidity Follows Power 194
  5. Determination of Displacement Integrals by Coefficients of Flexibility Polynomial 196
  6. Determination of Displacement Integrals by Derivatives of Flexibility Analytical Expression 199

Chapter V. Multispan Non-Uniform Beams 202

  1. Equation of Three Moments in Flexibility Polynomial Coeffi­cients 202
  2. Equation of Three Moments in Derivatives of Analytically Expressed Flexibility 207
  3. Equation of Three Moments in Flexibility Integrals 210
  4. Mohrs Integrals for Non-Uniform Beams 219

Chapter VI. Beams on Elastic Foundation 223

  1. General 223
  2. Prismatic Beams on Foundation of Constant Rigidity 224
  3. Prismatic Beams on Foundation of Linear Rigidity 227
  4. Beams on Foundation of Hyperbolic Rigidity 229
  5. Beams on Elastic Foundation with Moment Reaction 236

Chapter VII. Beams Under Combined Flexure and Compression 238

  1. General 238
  2. Prismatic Beams Under Arbitrary Transverse Loads and Constant Axial Forces 239
  3. Rotating Rod of Constant Rigidity Under Compression and Flexure 255
  4. Prismatic Beam Under Arbitrary Transverse Load and Uni­formly Distributed Axial Forces 260
  5. Prismatic Beam Under Arbitrary Transverse Load and Linear Axial Forces 270
  6. Prismatic Beam Under Arbitrary Transverse Load and Axial Law Distributed Along the Beam According to Polynomial 283

Chapter VIII. Application of Moment-Operational Method to Certain Complex Problems 285

  1. Stability of Bars Under Axial Compression 285
  2. Beams on Elastic Foundation Under Combined Compression and Bending 290
  3. Non-Uniform Beam Under Axial Force 294
  4. Higher Moments of Vector Quantities in Space 306
  5. Rigidity of Beams: General Case of Non-Linear Stress-Strain Relationship 320

Chapter IX. Application of Moment-Operational Method to Structural Mechanics of Ships 331

  1. Flexure of Irregular Decks 331
  2. Non-Uniform Beams on Elastic Foundation 341
  3. Stability and Vibration of Irregular Decks 346