杆系结构非线性后屈曲分析的增量割线刚度法

NONLINEAR POST-BUCKLING ANALYSIS OF TRUSS STRUCTURES USING AN INCREMENTAL SECANT STIFFNESS APPROACH

基于结构构件刚体运动与其变形抗力无关原理,假定构件经历与经典Updated-Lagrangian列式隐含的"微小自然变形-刚体运动"顺序相反的运动过程,建立空间杆系结构几何非线性分析的势能增量列式,推导了适用于典型增量步有限位移、有限应变分析的割线刚度矩阵。克服了Updated-Lagrangian列式下高阶非线性刚度矩阵推导过程繁琐及表达式不唯一等问题。该文提出的增量割线刚度既能预测位移(与协同转动法使用的割线刚度相比),又能较精确校正变形恢复力,列式简便而易于实际应用(与拉格朗日列式使用的割线刚度相比)。为了提升数值追踪算法追踪各类型平衡路径的通过能力及计算效率,提出非线性方程求解的增量割线刚度法:应用增量割线刚度矩阵作为非线性分析"预测"和"校正"算子,建立基于柱面弧长约束的直接迭代策略,提出适应多回路路径的荷载因子自动调整算法实现自动加载。经典算例验证了增量割线刚度法能有效防止路径追踪"回溯",快速收敛到正确解,可靠地反映杆系结构受力全过程行为。

Based on the basic principle that the deformational resistance of a structural member is independent of the rigid body motion, an incremental potential energy formulation has been redeveloped for geometrically nonlinear analysis of spatial truss structures by assuming that a truss member undergoes a reversed process of ＂natural deformation-rigid body motion＂ implied by the Updated-Lagrangian formulation. A secant stiffness matrix has been derived to solve finite displacement and finite strain problem in a typical incremental step of the nonlinear post-buckling analysis using the proposed energy method. Such a procedure effectively overcomes the issues in the Updated Lagrangian approaches related to the tediously obtained, inconsistent high-order stiffness matrices. Alternatively, the recommended secant stiffness matrix is capable of predicting nonlinear responses that are impossible by using the secant stiffness operator defined in the co-rotated configuration and accurately, yet in a simple manner, recovering member forces as compared with the high-order stiffness matrices. In this sense, to improve the robustness of numerical algorithms for a reliable path tracking as well as iteration efficiency, an incremental secant stiffness approach is presented for solving nonlinear problems in which the proposed secant stiffness matrix plays a common role in both the ‘predictor＇ and ‘corrector＇ in a typical iterative step. As a result, a direct iteration scheme with the cylindrical arc-length constraint is established to ensure a converged solution in arbitrary regions of the equilibrium path. An automatic loading technique that can suitably adjust the step size is therefore put forward for tracing the path with multiple loops. The results from benchmark examples demonstrate the capabilities of the secant stiffness approach for completely removing the ‘turning back＇ of the solution direction, making a fast convergence to correct solutions and reliably indicating the full-range behavior of truss structures.