Underconstrained Structural Systems

Mechanical engineering, an engineering discipline born of the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solu­ tions, among others. The Mechanical Engineering Series is a new series, featuring graduate texts and research monographs, intended to address the need for information in contemporary areas of mechanical engineering. The series is conceived as a comprehensive one that will cover a broad range of concentrations important to mechanical engineering graduate edu­ cation and research. We are fortunate to have a distinguished roster of consulting editors, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the first page of the volume. The areas of concentration are applied mechanics, biomechanics, computa­ tional mechanics, dynamic systems and control, energetics, mechanics of materials, processing, thermal science, and tribology. Professor Marshek, the consulting editor for dynamic systems and con­ trol, and I are pleased to present this volume of the series: Underconstrained Structural Systems by Professor Kuznetsov. The selection of this volume underscores again the interest of the Mechanical Engineering Series to pro­ vide our readers with topical monographs as well as graduate texts.

I Fundamentals.- 1 Statical-Kinematic Analysis.- 1.1 Basic Concepts and Definitions.- Rigid and Underconstrained Structural Systems.- Intrinsic Properties of Structural Systems.- 1.2 External Loads and Constraint Variations. Statical-Kinematic Duality.- Equilibrium and Perturbation Loads.- Constraint Variations.- Statical-Kinematic Duality.- 1.3 Statical-Kinematic Types of Structural Systems.- Relations Among Statical and Kinematic Properties.- A Statical-Kinematic Classification.- 1.4 Structural Indeterminacies.- Statical Determinacy and Indeterminacy.- Kinematic and Virtual Indeterminacies.- 2 Systems with Infinitesimal Mobility.- 2.1 Simple First-Order Mechanisms.- Analytical Criterion for a Simple First-Order Mechanism.- Matrix Test for First-Order Infinitesimal Mobility.- Second-Order Statical-Kinematic Duality.- 2.2 Multiple Singularity. Compound Systems.- Complex and Compound Systems.- Synthesis of First-Order Mechanisms.- 2.3 Higher-Order Infinitesimal Mobility.- Higher-Order Virtual Displacements.- Prestressability and Kinematic Immobility.- 3 Systems Involving Unilateral Constraints.- 3.1 Invariant Systems with Unilateral Constraints.- Constraint Conditions, Their Linearization and Reduction.- Analytical Criterion for Invariance.- Statical Criterion of Invariance.- 3.2 Systems Other Than Invariant.- Analytical Criteria for Noninvariant Systems.- Structural Properties of Systems with Unilateral Constraints.- 3.3 Constraint Imposition and Removal.- Imposition and Removal of Bilateral Constraints.- Imposition and Removal of Unilateral Constraints.- Limit Equilibrium.- 4 Kinematic and Elastic Mobility.- 4.1 The Statical-Kinematic Stiffness Matrix.- Load Resolution and Incremental Analysis.- Statical-Kinematic Stiffness Matrix.- 4.2 Applications of the Matrix.- Variant Systems.- Quasi-Variant Systems.- 4.3 Interplay of Kinematic and Elastic Mobility.- Elastic Displacements in Underconstrained Systems.- Systems Involving Undeformable and Flexible Components.- Globally Statically Indeterminate Finite Mechanisms.- II Applications to Particular Systems.- 5 Cables and Cable Systems.- 5.1 Some Problems of Statics of Elastic Cables.- A Cable or a Membrane Trough Under Hydrostatic Pressure.- More Comprehensive Solutions.- 5.2 Shallow Cables.- Modal Discretization and Load Resolution.- Stiffness Matrix for a Shallow Cable.- 5.3 Radial Cable Systems.- One-Layer Radial Cable Systems.- Two-Layer Radial Systems.- 5.4 Contour Ring.- Statics and Geometry of a Momentless Ring.- Interaction of the Radial Cable System with the Support Ring.- 6 Nets.- 6.1 Statical-Geometric Theory of Singular Nets.- Information from Tensorial Differential Geometry.- Statical Vector of Net.- Existence and the Number of Singular Nets.- 6.2 Particular Classes of Singular Nets.- Orthogonal Singular Nets.- Conjugate Singular Nets.- Singular Nets of Principal Curvatures.- 6.3 Equilibrium Loads and Configurations.- Singular Versus Nonsingular Nets.- Geodesic and Semigeodesic Nets.- 6.4 Statics of Elastic Nets.- Small Deformations of a Net.- Statics of Shallow Nets.- 7 Axisymmetric Nets.- 7.1 Singular Axisymmetric Nets.- General Statical-Kinematic Relations.- Torque-Free Singular Axisymmetric Nets.- Geodesic Nets.- Chebyshev Nets.- Other Singular Axisymmetric Nets.- 7.2 Equilibrium Loads and Configurations.- General Statical-Geometric Relations.- Particular Axisymmetric Nets.- 7.3 Kinematic and Elastic Deformations.- 8 Membranes.- 8.1 The Membrane Shell as an Underconstrained Structural System.- Implications for the Linear Membrane Theory.- Infinitesimal Mobility of Toroidal Membrane.- 8.2 Net of Principal Force Trajectories.- Homogeneous Problem.- Inhomogeneous Problem.- 8.3 Wrinkling Membranes: Fundamentals.- Preservation of Geodesies.- Pressurized Axisymmetric Membranes with Axial Forces.- Pressurized Membranes Without Axial Forces.- 8.4 Wrinkling Membranes: Advanced Problems.- Membrane Compressed or Indented by Rigid Disks.- Pressurized Toroidal Membrane.- Axisymmetric Membrane with Torque.- 9 Other Underconstrained Systems and Contact Problems.- 9.1 Band and Cable-Band Systems.- The Band as a Structural Component.- Kinematics of Systems with Unrestrained or Partially Restrained Intersections.- Equilibrium of Two-Array Systems.- 9.2 Axisymmetric 3-Nets.- General Statical-Geometric Relations.- Singular Axisymmetric 3-Nets.- 9.3 Elastic Systems Stiffened by Nets.- Axisymmetric Elastic Membrane Stiffened by an Inextensible Net.- A Composite Elastic Cord.- 10 Related Applications.- 10.1 Kinematic Adaptivity and Statically Controlled Geometry.- Concept Description and Applications.- Case Study: Kinematic Reconfiguration of a Mesh Reflector.- 10.2 Bending Instability of Improperly Supported Plates and Shallow Shells.- Problem Formulation and Analysis.- Bending Bifurcation of Plates.- Imperfect Plates and Shallow Shells.- 10.3 Shape Optimization of a Membrane Shell of Revolution.- References.