Research On Equilibrium Matrix Analysis Theory Of Kinematically Indeterminate Structures

Kinematically indeterminate structures, which are in contrast to kinematically determinate structures, are special structures that consist of internal mechanism. Internal mechanism has the same physical meaning as the self-vibration mode of zero-frequency. It can be divided into infinitesimal mechanism (Type I) and finite mechanism (Type II), respectively. Type I, which is just movement trend, can be stiffened by self-stress. This kind of structures can usually be imposed by prestress, such as tensegrity, cable dome, etc. Type II can undergo non-strain motion. There are several different mechiansm phenomenons exist in structures, for instance, retractable structures, folding structures, rapid assembly structures and erection technology in engineering constructions. Unlike kinematically determinate structures, static and dynamic responses are no longer the only concerned issues for kinematically indeterminate structures. As the geometry and topology of them are sensitive to structural performance, research on their structural system and configurations appears very important, which had been widely concerned by scholars at home and abroad. Based on equilibrium matrix, thesis founds structural system and configurations analysis theory of kinematically indeterminate structures systematically.Thesis describes the research status quo of kinematically indeterminate structures and their applications in engineering. Several common issues and key scientific problems are proposed.The physical meaning of equilibrium matrix is discussed and matrix singular value decomposition(SVD) and reduction theory are introduced. Equilibrium matrices of various types of elements are deduced, including bar, beam, cable, and macroelements are also generated, such as sliding cable element, multi-angulated beam element, pantograph element, radial-type angulated beam element. In general, the basic numerical analysis tools of this thesis are established.Classification of structural assemblies are summarized based on the literatures of Pellegrino, Calladine and other scholars. The kinematic properties, load-carrying path and working principle are analyzed and more generalized criteria of structural system are proposed. Based on first and second variations of the potential energy function of structures under conservative force field, new criteria for the mobility and equilibrium stability of mechanisms are concluded by analyzing the equilibrium matrix. Together with the criteria for geometrical stability, they constitute the integrated theoretical analysis system of kinematically indeterminate structures.Base on the nonlinear modification of equilibrium matrix and iterative algorithm, thesis proposes geometrically non-linear force method (NFM). The accuracy of NFM is validated by experiment. Relationships between equilibrium matrix and tangent stiffness matrix, linear stiffness matrix, initial stress stiffness matrix are discussed. NFM separates stiffness into two parts, which are derived from topology and material. The advantages of NFM dealing with kine- matically indeterminate structures are mentioned. Several different incremental loading strategies are introduced in the analysis of pre- and post-buckling behavior. The dynamic response of Type II can be calculated by introducing inextensional mechanism mode into dynamic equation.Form-finding methods of kinematically indeterminate structures with equilibrium matrix are proposed, including form-finding of Type I under self-stress and Type II under loads. Equilibrium state finding of multi-degree-of-freedom mechanism will be analyzed by the product of forces and the modes of inextensional mechanisms, and modification of second order is considered for accuracy results. Further more, method of reasonable form finding of network kinematically indeterminate assemblies under different types of loads is analyzed. One new form-finding method of tensegrity, which is carried out with iteration between equilibrium matrix and force density matrix, is put forward. Numerical algorithm only needs topoloy and prototype of cable and strut. Tensegrities with single or multi self-stress modes are found. Expanded octahedron, truncated tetrahedron, and also topology-diagram-mapping-based tensegrities are illustrated to validate the robustness of algorithm.Finally, compatible path and kinematic property of Type II are analyzed. Thesis proposes two kinds of trajectory tracking simulation methods based on NFM, i.e. active and positive control. Corresponding algorithm procedures are listed. It is able to produce much more accurate solutions when both rigid-body and elastic deformations exist in motions of mechanisms. Numerical design method of over-constraint mechanism based on minimum non-zero singular value is studied, and two types of circular radially retractable structures are used as examples. In addition, numerical design of adaptive deployable mechanisms formed by pantograph elements is studied, and corresponding trajectory are simulated by NFM.According to the theory and algorithm aforementioned, a computer aided design program named configuration analysis of structures, CASCAD for short, is developed. CASCAD is programmed by VC++ language that based on Object Oriented Programming (OOP) idea. It has friendly pre- and post-processing functions, graphical interfaces, animation, and functional modals such as structural system analysis, NFM analysis, form-finding, trajectory simulation, etc.Numerical examples and experiment prove that the theory presented is correct, and the algorithm is effective. It can be theory basis of kinematically indeterminate structures. The conclusions and problems that should be studied further are summarized at the end of thesis.