Limit State of the Plate Elements of Steel Structures

The necessity to save steel leads to a marked tendency towards thin-walled structures. Such structures are made of thin plating, the behaviour - and, of course, design - of which is very significantly affected by stability phenomena. In fact, with up-to-date thin-walled steel plated structures, it is very frequently the point of view of stability that governs the design. So it is not astonishing that the attention of a great number of research teams in various parts of the world has been for a good many years directed to investigations into numerous aspects of the buckling behaviour of steel plated structures. However, the current problems of buckling research, which require to account for the effect of initial imperfections, post-buckled behaviour and plastic reserve of strength (this leading in theoretical research to the necessity to solve boundary value problems of geometrically and physically non-linear partial differential equations, and in experimental studies to conduct experiments on full-size test girders) are very complex and time-consuming. Then it is beyond the means of one investigator, or even of one research team, to deal successfully with such problems and, conse­ quently, effective cooperation is indispensable. This was also the reason for the initiation of a fruitful collaboration between the first author of this book (Assoc. Prof. J. Djubek, D. Sc. ) and the third author (Assoc. Prof. M. Skaloud, D. Sc.

1 Basic Assumptions of Theory of Slender Webs.- 1.1 Basic Notions and Formulation of Basic Assumptions.- 1.2 Relationships between Web Deformations and Displacements in Its Middle Plane.- 1.3 Relationships between Deformations and Stresses.- 1.31 Hook’s Law.- 1.32 Incremental Theory of Plasticity.- 1.33 Ilyushin’s Deformation Plasticity Theory.- References.- 2 Mathematical Problems of the Fundamental Equations.- 2.1 Introductory Remarks.- 2.2 Some Types of Equations from Considerations about the Flexure of Plates.- 2.3 Basic Boundary Value Problems.- References.- 3 Approximate Methods of Solution.- 3.1 Compactness Method.- 3.2 Topological Method.- 3.3 Variational Methods.- 3.4 Uniqueness of Solution.- 3.5 Two-Sided Estimations of Approximate Solutions.- References.- 4 Bifurcation Problems of Basic Equations.- 4.1 Properties of the Homogeneous Problem.- 4.2 Problem of Bifurcation Points.- 4.3 One Property of Problem (4.10, 4.11).- References.- 5 Problems of Solution of a System of Non-Linear Algebraic Equations.- 5.1 Simple Iteration Methods.- 5.2 The Newton—Raphson Method.- 5.3 The Booth Method.- 5.4 An Improvement of the Initial Approximation by Means of Extrapolation.- 5.5 Method of Prolongation of Solution with Respect to Parameter.- 5.6 The Perturbation Method.- 5.7 Perturbation Method for an Ideally Plane Web.- 5.71 Boundary Conditions, System of Algebraic Equations.- 5.72 Numerical Results.- References.- 6 Large Deflections of Elastic Isotropic Webs.- 6.1 Introduction.- 6.2 A Slender Rectangular Web with Flanges Flexible in the Web Plane and Subject to Compression.- 6.21 Formulation of the Problem.- 6.22 Boundary Conditions.- 6.23 Solution to the Equilibrium Equation.- 6.24 Numerical Results.- 6.3 A Slender Web Subjected to Compression and to Combined Compression and Bending, with Boundary Members Flexible in the Web Plane. Unsymmetrical Cross-Section.- 6.31 Formulation of the Problem, Equations and Boundary Conditions of the Web.- 6.32 Numerical Results.- 6.4 Slender Webs Loaded in Shear with Boundary Members Flexible or Inflexible in the Web Plane.- 6.41 Theoretical Solution.- 6.42 Boundary Value Problems.- 6.43 Numerical Solution to the Boundary Value Problem A.- 6.44 Numerical Solution to the Boundary Value Problem B.- 6.45 Comparison of the Theoretical Solution with Experimental Results.- References.- 7 Large Deflections of Orthotropic Webs.- A. Orthotropic Webs in Compression with Imperfections.- 7.1 Equations of Equilibrium.- 7.11 Rigidities.- 7.12 Cross-Sectional Forces.- 7.2 The Basic Differential Equations of Large Deflections of a Web with Structural Orthotropy.- 7.21 Compatibility Equations (Membrane Equations).- 7.22 Equilibrium Equations (Slab Equations).- 7.23 Solution to the Compatibility Equation.- 7.24 Boundary Conditions.- 7.25 Solution to the Equilibrium Equation.- 7.3 An Isotropic Web.- B. Design of Longitudinally Stiffened Compression Flanges of Steel Box-Girder Bridges.- 7.4 The Limit Load of a Compression Flange Plate with Structural Orthotropy.- 7.41 Reduction Factor m1N.- 7.42 Reduction Factor m2N.- 7.5 Numerical Results.- 7.51 A Parametric Study of the Problem.- 7.6 Approximate Relationship for the Determination of the Reduction Factor m1N.- 7.7 Comparison of Theoretical and Experimental Values.- References.- 8 Large Deflections of Elasto-Plastic Webs.- 8.1 Introduction.- 8.2 Cyclic Plasticity.- 8.3 Equations Equivalent to the Prandtl—Reuss Equations.- 8.4 Relationships between Forces on Unit Length and Deformations.- 8.41 Incremental Theory of Plasticity.- 8.42 Parameters fbi.- 8.43 Deformation Theory of Plasticity.- 8.44 Parameters cii.- 8.5 Residual Stresses.- 8.6 Basic System of Differential Equations.- 8.7 Solution to the Equilibrium Equations.- 8.8 Numerical Results.- References.- 9 Ultimate Load Theories of Webs.- 9.1 Introduction.- 9.2 Ultimate Load Behaviour of Webs in Shear.- 9.3 The Rockey and Skaloud Theory for the Ultimate Load of Webs in Shear.- 9.31 Three Stages of the Behaviour of a Shear Girder.- 9.32 Stage 1: Web Operates in Pure Shear.- 9.33 Stage 2: Web Operates as a Tension Band.- 9.34 Stage 3: Failure Mechanism.- 9.35 Ultimate Load.- 9.36 Extension of the Theory to Unsymmetrical Girders, to Webs Fitted with Both Transverse and Longitudinal Stiffeners and to Webs Subject to Combined Shear and Bending.- 9.4 Other Ultimate Load Theories for Webs in Shear.- References.- 10 Large Deflections of Slender Webs Fitted with Ribs.- 10.1 Introduction.- 10.2 Solution to the Problem.- 10.21 Differential Equations and Boundary Conditions.- 10.22 Assumption for the Deflection Surface of the Web.- 10.23 Stress Function and Membrane Stresses.- 10.24 Relative Displacements of the Opposite Web Edges, Displacements u and v, and Shear Deformation ?.- 10.25 Evaluation of the Parameters wij by the Energy Method.- 10.26 Critical Stress of the Stiffened Web.- 10.27 Deformation of the Stiffened Web in the Post-Buckled Range.- 10.28 Efficiency of the Stiffener in the Post-Buckled Range.- References.- 11 Buckling of the Compression Flanges of Steel Box-Girder Bridges.- 11.1 Linear Buckling Theory of Compression Flanges.- 11.11 Definition of the Optimum Rigidity of Stiffeners.- 11.12 Solution to the Stability Problems of Thin-Walled Plated Structures.- 11.13 Application of Folded Plate Theory to the Solution to the Stability Problems of Thin-Walled Plated Structures.- 11.14 Stability Problem of a Compression Flange Panel Stiffened by Numerous Longitudinal Ribs.- 11.15 Stability Problem of a Compression Flange Panel Stiffened by a Small Number of Longitudinal Ribs.- 11.2 Post-Buckled Behaviour of Compression Flanges.- 11.21 General Solution.- 11.22 Numerical Solution for a Compression Flange Stiffened by Two Longitudinal Closed-Section Ribs.- References.- 12 Interaction of the Buckling of Thin-Walled Bars with the Buckling of Their Plate Elements.- 12.1 Critique of the Classical Concept of the Design of the Plate Elements of Compressed Bars.- 12.11 Classical Concept of the Design of the Plate Elements of Compressed Bars.- 12.12 Critique of the Classical Concept.- 12.2 Theoretical Investigation into the Interaction between Overall and Local Buckling of “Actual” Thin-Walled Bars.- 12.21 Stability Solution to the Problem.- 12.22 Mathematical Solution to the Problem.- 12.23 Critical Loads of “Ideal” Thin-Walled Columns Calculated for an Effective Cross-Section.- 12.24 Results of the Investigation.- 12.3 Eccentrically Loaded Thin-Walled Columns.- 12.31 Stability Solution to the Problem.- 12.32 Results of the Investigation.- References.