结构稳定极限承载力分析的力素增量方法
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摘要
本文研究结构稳定极限承载力分析的简便方法。根据物理概念,提出了系统(结构)静、动力平衡状态稳定性分析的力素增量方法。建立了结构单元弹塑性增量平衡方程及结构单元弹塑性增量割线刚度矩阵,推导了五种常见结构单元的弹塑性增量割线刚度矩阵的显式表达式,避免了数值积分运算。提出了一致显式增量割线刚度直接迭代算法,提高迭代效率。根据虚位移原理,采用Updated-Lagrange列式及本文提出的力素增量方法建立了结构单元大变位分析的虚功增量平衡方程,揭示了该式左边各项的物理意义。以平面梁单元为例,得到了反映单元大变形抗力增量与荷载增量相互平衡的代数式,将复杂的积分方程改造成为代数方程,简化了计算。根据以下物理事实：单元大变形过程可分解为自然变形阶段和整体刚性运动阶段,单元只在自然变形过程中才会改变节点抗力大小,而刚体运动不会改变单元抗力的大小,只会改变其方向,导出了结构单元大变位分析的增量割线刚度方法并推导了四种结构单元大变位分析的增量割线刚度矩阵。主要的研究内容和取得的成果如下：
(1)提出了结构静力平衡状态稳定性分析的力素增量评判准则：当抗力增量大于荷载增量时,平衡状态稳定；小于时,平衡状态不稳定；等于时,结构处于临界失稳状态。建立了系统运动稳定性分析的位移变分法,解决了非线性系统稳定性简便分析以及系统稳定性分布区域的确定问题。
(2)推演了杆单元、抛物线索单元、平面梁单元、空间梁单元、板壳单元等的弹塑性刚度矩阵计算表达式,无需数值积分运算,简化了计算过程,提高运算效率。
(3)提出了一致显式增量割线刚度直接迭代算法,上一迭代步“校正”阶段形成的结构单元增量割线刚度矩阵可以用于本次迭代步“预测”阶段计算结构单元增量位移,迭代过程简单,提高求解效率。
(4)以平面梁单元为例,揭示了结构单元大变形三维虚功增量平衡方程中虚应变能增量的物理意义：分别表示单元抵抗自然变形的抗力虚功,单元刚体变位引起的单元自然变形抗力增量虚功以及单元初始节点力增量虚功,从而建立了单元大变形力素增量平衡方程。
(5)建立了结构单元几何非线性分析的增量割线刚度列式、四种常见结构单元增量割线刚度计算表达式,以及提出了这些单元大变形分析的增量割线刚度方法。
(6)提出了基于增量割线刚度的柱面弧长算法,求解结构弹塑性大变形非线性平衡方程,追踪结构从开始加载直到顺利跨越极值点全过程的平衡路径。该算法可以有效地避免传统柱面弧长法中由于荷载增量因子选取不当引起求解路径回溯并最终导致迭代过程发散的问题,求解过程简单,可以收敛于正确解。
(7)基于本文理论研究成果,编制了结构稳定极限承载力分析的计算程序。分析了文献中选取的11例结构非线性响应计算的经典考题,得到的结果均与解析解或者与其它学者提出的数值解非常吻合,计算了三汊矶湘江大桥在全桥满布荷载工况下的稳定极限承载力,从而验证了本文分析方法的正确性和有效性。
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.
(1)提出了结构静力平衡状态稳定性分析的力素增量评判准则：当抗力增量大于荷载增量时,平衡状态稳定；小于时,平衡状态不稳定；等于时,结构处于临界失稳状态。建立了系统运动稳定性分析的位移变分法,解决了非线性系统稳定性简便分析以及系统稳定性分布区域的确定问题。
(2)推演了杆单元、抛物线索单元、平面梁单元、空间梁单元、板壳单元等的弹塑性刚度矩阵计算表达式,无需数值积分运算,简化了计算过程,提高运算效率。
(3)提出了一致显式增量割线刚度直接迭代算法,上一迭代步“校正”阶段形成的结构单元增量割线刚度矩阵可以用于本次迭代步“预测”阶段计算结构单元增量位移,迭代过程简单,提高求解效率。
(4)以平面梁单元为例,揭示了结构单元大变形三维虚功增量平衡方程中虚应变能增量的物理意义：分别表示单元抵抗自然变形的抗力虚功,单元刚体变位引起的单元自然变形抗力增量虚功以及单元初始节点力增量虚功,从而建立了单元大变形力素增量平衡方程。
(5)建立了结构单元几何非线性分析的增量割线刚度列式、四种常见结构单元增量割线刚度计算表达式,以及提出了这些单元大变形分析的增量割线刚度方法。
(6)提出了基于增量割线刚度的柱面弧长算法,求解结构弹塑性大变形非线性平衡方程,追踪结构从开始加载直到顺利跨越极值点全过程的平衡路径。该算法可以有效地避免传统柱面弧长法中由于荷载增量因子选取不当引起求解路径回溯并最终导致迭代过程发散的问题,求解过程简单,可以收敛于正确解。
(7)基于本文理论研究成果,编制了结构稳定极限承载力分析的计算程序。分析了文献中选取的11例结构非线性响应计算的经典考题,得到的结果均与解析解或者与其它学者提出的数值解非常吻合,计算了三汊矶湘江大桥在全桥满布荷载工况下的稳定极限承载力,从而验证了本文分析方法的正确性和有效性。
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.
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