摘要
在土木工程中,结构物的极限承载力和破坏模式的确定是一项重要的研究题和工程问题。分析此类问题的方法大致分为两类:一类是弹塑性的增量分;另一类则是塑性极限分析方法。极限分析的上限和下限方法以塑性极限定为理论基础,是工程结构的设计和分析中直接而又严格的极限状态分析方法。工程的实际应用中通常采用极限分析的数值方法。其中,在工程结构的极限析中较为常用。另外,极限分析方法最终需要求解一个数学规划问题,根据体的情况大致可分为线性和非线性规划问题。随着问题维数的增加,数学规问题将可能成为大规模优化问题,其求解成为了一个难题。因此本文从经典性极限分析理论出发,进一步改进运动许可速度场的构造方法,并将数值优领域中提出的新算法应用于数值极限分析上限的数学规划问题的求解中,取的主要成果如下。
在刚体有限元上限分析中,如果将安全系数定义为目标函数,则数学规划题就成为了带有约束的非线性规划问题。本文首次采用一种新型的优化算法P-free方法求解此非线性规划问题。求解非线性规划的常用算法为序列二次规(SQP)方法。然而,在初始点任意的情况下,传统的SQP在求解刚体有限上限分析法的非线性规划模型时出现了子问题不相容的问题而导致得不到最解,且在每个迭代步中都要花费大量计算来求解一个二次规划(QP)问题。对这一非线性规划问题,文中采用了一种新型非线性优化算法——QP-free法来求解刚体有限元上限分析法的非线性规划问题。该方法的转轴操作可以免子问题不相容的问题,并且在每个迭代步中将求解QP问题转化为求解三具有相同系数矩阵的线性方程组。而根据虚功率方程将安全系数表示为运动可速度场的函数,目的就是使得非线性规划问题的目标函数避免了其导数成常数向量,便于采用QP-free算法进行求解。通过两类算法对经典边坡稳定题的对比分析,QP-free算法比传统的SQP算法则更为有效。
在上述的刚体有限元上限分析法中,数学规划模型的非线性是由于采用安系数作为评价指标而引起的。为了避免求解非线性规划问题,文中采用了加系数的定义来衡量结构的极限载荷,并且根据临界加速度的概念可由加载系计算安全系数,实现了利用线性规划模型来求解结构的安全系数。在此基础,通过在刚体单元之间的界面上的两个端点施加相关联流动法则,使得在构造运动许可速度场时可以考虑刚体单元形心速度的转动分量,因而得到了考虑转动破坏模式的刚体有限元上限法的线性规划模型。进而将该方法应用于边坡稳定分析和地基承载力计算中。根据边坡稳定和地基承载力的经典问题的分析验证了本章中所提出的刚体有限元上限法及其线性规划模型的正确性。另外,通过单纯形方法和原对偶内点算法对线性规划问题求解的对比发现,后者比前者的计算效率要高,尤其是针对大规模问题的情况。
然而,对于大多数的岩土材料而言,相关联流动法则并不能反映其真实的力学特性。需要在塑性极限分析中考虑非相关联流动法则。因此,在考虑相关联流动法则的刚体有限元上限法中,应当将该方法推广到考虑非相关联流动法则的情况。在构造考虑转动模式运动许可速度场时,在刚体单元之间界面的两端点处施加非相关联流动法则,使得刚体有限元上限法可以在非相关联流动法则的情况下建立求解结构极限承载力的线性规划模型。并将考虑非相关联流动法则的刚体有限元上限法应用于膨胀系数对边坡稳定安全系数的影响研究中。另外,还可以将这一方法推广到研究土体剪胀性对土钉极限抗拔力的影响。
为了克服调整网格划分带了的困难,在径向基点插值法构造运动许可速度场的基础上,并结合Cartesian积分变换数值积分法对能量耗散功率的近似计算,推导了无需任何网格的数值极限分析上限法的非线性数学规划模型。为了求解此类无网格上限法,采用了一种逐步区分刚性和塑性区域的直接迭代算法。针对摩擦型材料而言,无论强度参数取何值时,直接迭代算法的迭代控制因子都等于零,因此该方法不能应用于求解服从Mohr-Coulomb或Drucker-Prager破坏准则的摩擦型材料的极限承载力问题,而只能求解服从椭圆型屈服函数的非模型材料的极限承载力的问题。通过对垂直边坡的临界高度、带孔板的极限承载力以及厚壁圆筒的极限扩张压力的计算和分析,验证了本章所提出的无网格上限法及其无搜索迭代算法。
The collapse load and failure mechanism of solid or structure play an important role in engineering structure design and safety assessment in civil engineering. In general, the methods used to determine the collapse load and failure mechanism of a structure can be classified into two types: a) elastoplastic incremental approaches, and b) upper and lower bound approaches of limit analysis. The upper and lower bound approaches which based on limit theorem are more direct and rigorous method for designing engineering structures. In general, the numerical upper bound limit analysis is more popular in pracitcal engineering due to the facility of constructing the kinematically admissble velocity field. In the upper bound limit analysis, the linear or nonlinear mathematical programming problems will be solved ultimately. With the increasing of dimension of kinematically admissble velocity, these mathematical programming problems will become too difficult to be solved. Therefore, in this paper, the research focuses on the numerical theories of plastic upper bound approach and the application of new algorithm of optimization. The summary of this paper is illustrated as follows:
If the safety factor is treated as the objective function in RFEM (rigid finite element method)-based upper bound approach, the mathematical programming will be a nonlinear programming problem with constraints. In genearl, the so-called sequential quadratic programming (SQP) method is used to solve the constrained nonlinear programming. However, with an arbitrary starting point, it is quite time consuming and difficult to search the optimum based on the SQP-type algorithms because the sub-QP problem is infeasible at every iteration step. Fortunately, a QP-free algorithm based on the so call pivoting operation and active-set strategy can be convergent toward the optimal points with arbitrary start point. The infeasibility of sub-QP problem can be avoided using poviting operation, and the QP can be transformed into three linear systems of equations with the same coefficient matrixs. For overcoming the constant derivatives of objective function, the objective function (safety factor) was reformulated as a function of the kinematically admissible velocity field based the virtual work rate equation. Two classical problems of slope stability are solved by this QP-free algorithm, the results show that QP-free method is more efficiency than the SQP method.
In the above RFEM-based upper bound approach, the nonlinearity of mathematical programming is caused by the definition of the safety factor. For avoiding this nonlinearity, the limit load multiplier is used to evaluate the limit load of the structures. And according to the concept of critical acceleration, the safety factor of stablility of strucutres can be computed iteratively from the limit load multiplier using linear programming. In addtion, the rotational failure mechanism can be considered in constructing kinematically admissible velocity field by using enforce the associated flow rule at two points along the interface between two adjacent rigid elements. Therefore, the presented RFEM-based upper bound approach can be earlier applied into slope stability and bearing capacity analysis. In addition, the linear programming problem of RFEM-based upper bound approach can be solved by using simplex algorithm or primal-dual interior point method. The numerical results of several test problems show that the primal-dual interior point method is very suit for the large-scale linear programming problem with sparse coefficient matrix.
For rock or soil materials, the non-associated flow rule should be satisfied when the kinematically admissible velocity field was constructed. Therefore, in RFEM-based upper bound approach, the non-associated flow rule was enforced at two end points along interface between two adjacent rigid elements. And then, the linear programming problem of upper bound limit analysis was formulated considering the non-associated flow rule. Therefore, the influence of non-associated flow rule on safety factor of slope stability can be solved based on the present method. Furthermore, the pull out resistance of soil nailing within the dilative soils can also be computed by using the present method.
For overcoming the difficulty of adaptive mesh, the kinematically admissible velocity field can be constructed by using radial point interpolation method (RPIM). Then, the external work rate and internal energy dissipation rate are computed by using Cartesian transformation method (CTM). As a result, the nonlinear programming problem of upper bound approach is presented based on meshless method (MM). For solving the presented MM-based upper bound approach, a direct iterative algorithm based on the distinguishing rigid/plastic zone is adopted. Regarding the direct iterative algorithm for frictional materials, the iteration control parameter is identically vanishing for all the shear strength parameters. Therefore, the direct iterative algorithm can‘t be used to calculate the limit load of frictional materials which follow the Mohr-Coulomb or Drucker-Prager yield criterion, and the direct iterative algorithm can be only used to compute the limit load of non-frictional materials which follow the ellipsoid yield function. To verify and extend the presented MM-based upper bound approach, the limit loads of some classical problems including stability of slopes, plate and thick-walled cylinder are calculated by using the presented method.
引文
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