The Mathematical Foundation of Structural Mechanics

This book attempts to acquaint engineers who have mastered the essentials of structural mechanics with the mathematical foundation of their science, of structural mechanics of continua. The prerequisites are modest. A good working knowledge of calculus is sufficient. The intent is to develop a consistent and logical framework of theory which will provide a general understanding of how mathematics forms the basis of structural mechanics. Emphasis is placed on a systematic, unifying and rigorous treatment. Acknowledgements The author feels indebted to the engineers Prof. D. Gross, Prof. G. Mehlhorn and Prof. H. G. Schafer (TH Darmstadt) whose financial support allowed him to follow his inclinations and to study mathematics, to Prof. E. Klingbeil and Prof. W. Wendland (TH Darmstadt) for their unceasing effort to achieve the impossible, to teach an engineer mathematics, to the staff of the Department of Civil Engineering at the University of California, Irvine, for their generous hospitality in the academic year 1980-1981, to Prof. R. Szilard (Univ. of Dortmund) for the liberty he granted the author in his daily chores, to Mrs. Thompson (Univ. of Dortmund) and Prof. L. Kollar (Budapest/Univ. of Dortmund) for their help in the preparation of the final draft, to my young colleagues, Dipl.-Ing. S. Pickhardt, Dipl.-Ing. D. Ziesing and Dipl.-Ing. R. Zotemantel for many fruitful discussions, and to cando ing. P. Schopp and Frau Middeldorf for their help in the production of the manuscript. Dortmund, January 1985 Friedel Hartmann Contents Notations ........................................................... XII Introduction ........................................................ .

1 Fundamentals.- 1.1 Vectors and Matrices.- 1.2 Differentiation.- 1.3 Domains.- 1.4 Integrals.- 1.5 Integration by Parts.- 1.6 Gateaux Differentials.- 1.7 Functionals.- 1.8 Sobolev Spaces.- 1.9 The Differential Equations.- 1.9.1 The Straight Slender Frame Element.- 1.9.1.1 The Axial Displacement u (x).- 1.9.1.2 The Rotation ? (x) of a (circular) bar.- 1.9.1.3 The Lateral and Vertical Deflections v and w.- 1.9.1.4 Shear Deformations.- 1.9.2 The Kirchhoff Plate.- 1.9.3 The Elastic Body.- 1.9.4 Elastic Plates.- 1.9.5 The Membrane.- 1.9.6 Reissner’s Plate.- 2 Work and Energy.- 2.1 Integral Identities.- 2.1.1 The Beam.- 2.1.2 The Kirchhoff Plate.- 2.1.3 The Elastic Plate and Body.- 2.2 Summary.- 2.3 Three Corollaries.- 2.4 A Beam.- 2.4.1 Principle of Virtual Displacements.- 2.4.2 Principle of Virtual Forces.- 2.4.3 Betti’s Principle.- 2.5 Eigenwork = Internal Energy.- 2.6 Equilibrium.- 2.7 Summation of Work and Energy.- 2.8 Rigid Supports and free Boundaries.- 2.9 Elastic Supports.- 2.10 The Mathematical Basis of Structural Mechanics.- 2.11 The Space $${C^{2m}}(\bar \Omega)$$ and its Limitations.- 3 Continuous Beams, Trusses and Frames.- 3.1 Continuous Beams.- 3.2 Trusses.- 3.3 Frames.- 3.3.1 The Method of Reduction.- 3.4 Stiffness Matrices.- 3.4.1 The Axial Displacement.- 3.4.2 Shear Deformations.- 3.4.3 Rotation.- 3.4.4 Deflections v and w.- 4 Energy Principles.- 4.1 The Basic Principle.- 4.2 Examples.- 4.2.1 An Elastic Plate.- 4.2.2 A Bar.- 4.2.3 A Cantilever Beam.- 4.2.4 A Beam on a Spring.- 4.2.5 A Kirchhoff Plate.- 4.3 The Principle of Minimum Potential Energy.- 4.4 The Complementary Principle.- 4.5 The Formulation of ?1 (u) and ?2 (u).- 4.6 The Sign of the Total Energy.- 4.7 The Point ? (w) and the Classes R1 and R2.- 4.8 Displacement Method and Force Method.- 4.9 Energy Principles for Cont. Beams, Trusses and Frames.- 4.10 The Formulation of the Functional “by hand”.- 4.11 Lagrange Multipliers.- 4.12 The Algebra of Structural Mechanics.- 5 Concentrated Forces.- 5.1 Fundamental Solutions.- 5.1.1 The Bar, — EAd2/dy2.- 5.1.2 The Beam, EId4/dy4.- 5.1.3 The Kirchhoff Plate, K??w.- 5.1.4 The Elastic Plate and the Elastic Body.- 5.1.5 Summary.- 5.2 Fundamental Solutions and Integral Identities.- 5.3 Results.- 5.3.1 Bars.- 5.3.2 Beams.- 5.3.3 The Kirchhoff Plate.- 5.3.4 Elastic Plates and Bodies.- 5.4 Summary.- 5.5 An Extension.- 5.6 Theorems “eigenwork = int. energy”.- 5.7 The Characteristic Functions.- 5.7.1 Their Origin.- 5.7.2 A Mechanical Interpretation.- 5.7.3 Integral Representation of c(x).- 5.8 An Alternative.- 5.9 Castigliano’s Theorem.- 5.10 Castigliano’s Generalized Theorem.- 5.11 Concentrated Forces or Disturbances on the Boundary.- 6 Influence Functions.- 6.1 Integral Representations.- 6.1.1 The Bar, — EAd2/dy2.- 6.1.2 The Beam, Eld4/dy4.- 6.1.3 The Kirchhoff Plate.- 6.1.4 Elastic Plates and Bodies.- 6.1.5 The Integral Representation of ?w(x)/?n.- 6.2 Green’s function.- 6.3 Compatibility on the Boundary.- 6.3.1 The Order of the Integral Operators.- 6.3.2 The Essential Compatibility Conditions.- 6.4 Summary.- 6.4.1 Bars.- 6.4.2 Beams.- 6.4.3 Kirchhoff Plates.- 6.4.4 Elastic Plates and Bodies.- 6.5 An Example.- 6.6 Stiffness Matrices and Compatibility Conditions.- 6.7 The Boundary Element Method.- 6.8 The Trace Theorem.- 6.9 Elastic Potentials.- 7 The Operators A.- 7.1 The Systems.- 7.1.1 Bars.- 7.1.2 Elastic Plates and Bodies.- 7.1.3 Beams.- 7.1.4 Kirchhoff Plates.- 7.2 Identities.- 7.2.1 Elastic Plates and Bodies.- 7.2.2 Kirchhoff Plates.- 7.2.3 Bars.- 7.2.4 Beams.- 7.3 Energy Principles.- 7.3.1 Elastic Plates and Bodies.- 7.4 Sufficient Conditions.- 7.5 Other Mixed Formulations.- 7.6 The Babuška-Brezzi Condition.- 8 Shells.- 8.1 Shells as Surfaces.- 8.2 Statics.- 8.3 Koiter’s Model.- 8.4 The first Identity.- 9 Second-Order Analysis.- 9.1 Beams.- 9.2 Stability.- 9.3 Lateral Buckling of Beams.- 9.4 The Kirchhoff Plate.- 9.5 Nonconservative Problems.- 9.6 Initial Value Problems.- 9.7 Vibrations.- 9.8 Hamilton’s Principle.- 10 Nonlinear Theory of Elasticity.- 10.1 The Differential Equations.- 10.2 The first Identity.- 10.3 Energy Principles.- 10.4 Incremental Procedures.- 10.5 Large Displacement Analysis of Beams.- 10.6 Large Displacement Analysis of Plates.- 10.7 The Principle “eigenwork = int. energy”.- 10.8 Influence Functions.- 11 Finite Elements.- 11.1 Shape Functions.- 11.2 The Error in Finite Elements.- 11.3 Nonconforming Shape Functions.- 11.4 The Patch Test.- 11.5 Hybrid Energy Principles.- 11.6 Hybrid Energy Principles for Operators A.- 12 References.- 13 Subject Index.