Nonlinear Mechanics of Structures

The aim of this book is to provide a unified presentation of modern mechanics of structures in a form which is suitable for graduate students as well as for engineers and scientists working in the field of applied mechanics. Traditionally, students at technical universities have been taught subjects such as continuum mechanics, elasticity, plates and shells, frames or finite element techniques in an entirely separate manner. The authors' teaching experience clearly suggests that this situation frequently tends to create in students' minds an incomplete and inconsistent picture of the contemporary structural mechanics. Thus, it is very common that the fundamental laws of physics appear to students hardly related to simplified equations of different "technical" theories of structures, numerical solution techniques are studied independently of the essence of mechanical models they describe, and so on. The book is intended to combine in a reasonably connected and unified manner all these problems starting with the very fundamental postulates of nonlinear continuum mechanics via different structural models of "engineer­ ing" accuracy to numerical solution methods which can effectively be used for solving boundary-value problems of technological importance. The authors have tried to restrict the mathematical background required to that which is normally familiar to a mathematically minded engineering graduate.

I: Theoritical Foundations.- 1 General notions and principles.- Purpose of the chapter.- 1.1 Introductory concepts.- 1.1.1 Space and time.- 1.1.2 Motion.- 1.1.3 Mass.- 1.1.4 Forces.- 1.1.5 Temperature, energy and entropy.- 1.1.6 Heat supply.- 1.2 Fundamental principles.- 1.2.1 Balance of momentum.- 1.2.2 Balance of moment of momentum.- 1.2.3 Balance of energy.- 1.2.4 Dissipation principle.- 1.3 Basic fields.- 1.3.1 Displacements.- 1.3.2 Strains and strain rates.- 1.3.3 Stresses and stress rates.- 1.3.4 Heat fluxes.- 1.3.5 Matrix notation for the basic fields.- 1.4 Resulting relations.- 1.4.1 Equation of motion.- 1.4.2 Energy equation.- 1.4.3 Dissipation inequality.- 1.4.4 Matrix form of the resulting equations.- Summary.- Problems.- 2 Materials, loadings and constraints.- Purpose of the chapter.- 2.1 Materials.- 2.1.1 Real materials and ideal materials.- 2.1.2 Thermo-visco-elastic materials.- 2.1.3 Linear thermo-elastic materials.- 2.1.4 Thermo-elastic-plastic materials.- 2.1.5 Matrix form of constitutive relations.- 2.2 Loadings and heat supply.- 2.2.1 External agents.- 2.2.2 Body loadings and internal heat supply.- 2.2.3 Boundary loadings.- 2.2.4 Boundary heat supply.- 2.3 Constraints.- 2.3.1 Boundary and internal constraints.- 2.3.2 Reactions.- 2.3.3 Bilateral constraints.- 2.3.4 Constraint functions.- 2.3.5 Constraints for stresses and heat fluxes.- Summary.- Problems.- 3 Formulation of engineering theories.- Purpose of the chapter.- 3.1 General form of governing relations.- 3.1.1 Finite variational formulation.- 3.1.2 Incremental variational formulation.- 3.1.3 Variational equations for stresses and heat fluxes.- 3.1.4 Calculation of reactions.- 3.1.5 Matrix form of variational equations.- 3.2 Method of constraints: finite formulation.- 3.2.1 Foundations.- 3.2.2 Shell and plate theories.- 3.2.3 Rod and beam theories.- 3.2.4 Plane problems.- 3.3 Method of constraints: incremental formulation.- 3.3.1 General equations.- 3.3.2 Shells and rods.- 3.3.3 Plane problems.- 3.4 Method of constraints: discrete description of elements and structures.- 3.4.1 Discretization concept.- 3.4.2 Finite formulation.- 3.4.3 Incremental formulation.- 3.4.4 Shell and plate elements.- 3.4.5 Rod and beam elements.- 3.4.6 Special cases and matrix notation.- 3.4.7 Interaction between elements.- 3.5 Evaluation and correction of solutions.- 3.5.1 Residual reactions.- 3.5.2 Evaluation of solutions.- 3.5.3 Correction of solutions.- 3.5.4 On adaptive solution refinements.- Summary.- Problems.- 4 Extensions and specifications of the general theory.- Purpose of the chapter.- 4.1 Domain discretization.- 4.2 Solving differential equations by weighted residual methods.- 4.3 Assembly of element matrices and vectors.- 4.4 Elastic pin-jointed element in space.- 4.5 Elastic frame element in space.- 4.6 Elastic lattice-type structures.- 4.6.1 Discrete models.- 4.6.2 Continuous models.- 4.7 Engineering shell theories.- 4.7.1 Foundations.- 4.7.2 General form of equations of motion.- 4.7.3 Special theories.- 4.7.4 Example: axisymmetric deformations of shells.- 4.8 Thin shell finite element for nonlinear axisymmetric analysis.- 4.9 Linear strain triangular element for nonlinear plane stress analysis.- 4.10 Constitutive matrices for thermo-elastic-plastic materials.- 4.11 On inelastic analysis under non-proportionally varying loads..- 4.11.1 Elastic-plastic structures subjected to mechanical loads..- 4.11.2 Thermo-elastic-plastic structures subjected to mechanical and thermal loads.- Summary.- Problems.- 5 Numerical algorithms and software concepts.- Purpose of the chapter.- 5.1 Introductory comments.- 5.2 Nonlinear quasi-statics.- 5.3 Integration of elastic-plastic constitutive law.- 5.4 Initial and linearized buckling.- 5.5 Nonlinear dynamics.- 5.6 Dynamic stability under non-periodic loads.- 5.7 Nonlinear heat transfer.- 5.8 Development of software.- Summary.- Problems.- II: SELECTED APPLICATIONS.- 6 Trusses, frames, lattice-type shells.- Purpose of the chapter.- 6.1 Trusses.- 6.1.1 Nonlinear effects in truss analysis — a model problem.- 6.1.2 Accuracy of computations and further examples.- 6.1.3 Elastic-plastic truss under non-proportionally varying loads.- 6.2 Frames.- 6.2.1 Elastic-plastic constitutive matrices for beam elements.- 6.2.2 Limit state conditions.- 6.2.3 Nonlinear analysis of curved beams.- 6.2.4 Examples of numerical frame analysis up to collapse.- 6.2.5 Elastic-plastic beam under non-proportionally varying loads..- 6.2.6 Elastic buckling of plane grid.- 6.2.7 Approximate large displacement analysis of frames using buckling mode superposition.- 6.3 Lattice-type structures.- 6.3.1 Lattice shells — linear and “second-order” theories.- 6.3.2 Buckling of elastic grid.- 6.3.3 Perfectly-plastic lattice-type plates — discrete model.- 6.3.4 Limit load of polar grids — continuous model.- 6.3.5 Minimum weight design of ideally plastic rectangular dense grid.- Summary.- Problems.- 7 Thin plates loaded in-plane.- Purpose of the chapter.- 7.1. Elastic-plastic bending of a cantilever beam.- 7.2 Limit analysis of perforated plates.- 7.3 Elastic-plastic analysis of a cantilever under non-proportionally varying loads.- 7.4 In-plane buckling of an elastic-plastic strip.- 7.5 Necking of an elastic-plastic strip.- 7.6 Extension of a thin rectangular plate with a central hole.- Summary.- 8 Plate and shell problems..- Purpose of the chapter.- 8.1 Thin elastic-plastic axisymmetric shells.- 8.2 Elastic-viscoplastic analysis of axisymmetric shells.- 8.3 Buckling of elastic-plastic spatial plate assemblies.- Summary.- 9 Heat conduction and thermal stress problems.- Purpose of the chapter.- 9.1 Heat flow in flash and friction welding.- 9.2 Thermally induced stresses in elastic-plastic plate.- 9.3 Elastic-plastic cylindrical shell under thermo-mechanical load..- 9.4 Elastic-plastic truss under nonproportionally varying temperature and mechanical load.- Summary.