An Incremental Force Component Approach To Ultimate Load Carrying Capacity Of Structures

This doctoral thesis is mainly focused on the development of a simplified method for the evaluation of ultimate load carrying capacity of structures. According to the basic concepts of structural stability also unveiled herein, an incremental force component approach is put forward to investigating the stability behavior of static and dynamic equilibrium state of mechanical systems(structures). A general procedure for deriving incremental equilibrium equations of elasto-plastic structural element and its incremental secant stiffness matrix becomes available. By following this procedure, one can explicitly formulate the incremental inelastic secant stiffness matrix for five types of structural elements that often used in practice, which renders the numerical integration of high cost unnecessary. A consistent direct iterative solution scheme consistent with present incremental secant stiffness formulation is developed to solve the path-dependent nonlinear equilibrium equations with a comparable high level of efficiency. By using the principle of virtual displacements, the Updated Lagrange formulation for large deformation analysis, and the proposed incremental force component approach, the incremental virtual work equations describing the large deformation behavior of structural elements has been established. From these equations, the physical sense of each integral at the left hand side has extensively been investigated. With such physical significance bearing in mind and by considering the large displacement process experienced by a planar beam element, a set of algebraic equations representing the equilibrium relationships between the resistance of the element due to large deformations and applied load increments can be derived as an alternative to the conventional more complicated integral equations. By doing so, the analysis process is greatly simplified. Recall the following reality that the large deformation can usually be divided into two stages, namely, the natural deformation and the purely rigid body motion, only the natural deformation will generate nodal resisting forces, while according to the rigid body motion rule the nodal forces shall remain their magnitude, but rotate through a definite angle when rigid body motion takes place, an incremental secant stiffness formulation for the large deformation analysis of structural elements can consequently be derived. At this point, the incremental secant stiffness matrix for use in the geometric nonlinear analysis of four types of typical structural elements is explicitly given. According to the above brief discussion, the main characteristics of this dissertation can be summarized as follows:(1) A general criterion in terms of incremental force components has been presented for checking the stability of the static equilibrium state of structures. The global structural form will remain stable if a structure has sufficient capacity to accommodate the externally applied forces when being disturbed. The structure becomes unstable when the applied loads dominate the structural resistance. A critical state can be obtained as the applied loads fulfill the residual load carrying capacity of the structures. Following the concepts of instability analysis of systems at rest, a procedure called the displacement-based variational method is proposed to solve the stability problems of nonlinear systems in motion in a more efficient way and to mark out the stable zone and unstable zone.(2) The governing incremental equilibrium equations of inelastic structures are formulated through a novel incremental secant stiffness approach, by which the element and therefore the structural global secant stiffness matrices are explicitly calculated. Such procedure facilitates the task of formulation and implementation of finite element model in that the cumbersome numerical integration is completely unnecessary. For the purpose of illustration, the incremental secant stiffness matrices are derived for five types of commonly used structural elements, i.e., truss, parabolic cable, planar beam, spatial beam and flat plate or shell, respectively.(3) A consistent direct iterative solution scheme consistent with the present formulation is presented. In a typical iterative step, the structural stiffness matrix does not need to be reformed at the very beginning of the 'predictor'stage. Rather, it will be replaced by the one obtained from the 'corrector' stage of last iteration. Consequently, the iteration process itself is efficient in its implementation and minimize the amount of computation time.(4) From large deformations of a planar beam element, the physical meaning of three integrals for each representing the corresponding incremental virtual strain energy in the incremental virtual work equation of equation (4-17) can be individually interpreted as the virtual work of element end forces induced by structural natural deformations, the virtual work of element end force increments caused by rigid body motion of the element, and the virtual work of initial nodal force increments also due to element moves as a rigid body. In this work, the foregoing physical interpretations serve as a valid basis for establishing the incremental equilibrium equations when element undergoing large deformations.(5) An element formulation using the incremental secant stiffness properties is presented for analyzing the geometric non-linear behavior of structures. As an illustration, explicit expressions for the incremental secant stiffness matrices of four typical types of structural elements are derived for future reference.(6) An incremental secant stiffness based cylindrical arc-length method is proposed as an attempt to solving material and geometric non-linear incremental equilibrium equations and to tracking the equilibrium path throughout whole loading history. The proposed solution algorithm can effectively tackle the problem of 'tracing back'when predicting a correct equilibrium path by conventional cylindrical arc-length method, as a misleading applied load factor may occasionally be selected. Meanwhile, the solution scheme is quite simple for implementation, and can always be converged to the correct equilibrium path.(7) Based on the proposed method in current study, a computer program for calculating the ultimate load carrying capacity of structures and components has been written. Subsequently, this program is applied to solving 11 benchmark problems of analyzing the response of nonlinear structures under static loads of gradually increasing magnitude. The obtained numerical results are in closely agreement with the available analytical solutions or the numerical results reported by other researchers. In addition, as an attempt to solving practical problems, the ultimate load carrying capacity of a self-anchored suspension bridge subject to full-span uniformly distributed live loads has been evaluated. All of the above works successfully demonstrate the capabilities and effectiveness of the present formulation.