连续介质力学行为分析的静态拉格朗日方法

STATIC LAGRANGIAN METHOD FOR ANALYSIS OF CONTINUUM MECHANICAL BEHAVIORS

通过对FLAC的研究,得出一个结论:拉格朗日方法无论从计算收敛的角度还是从物理意义的角度,基于运动方程求解都是不必要的。提出了一种新的求解思想,建立了所谓静态拉格朗日方法:(1)由节点不平衡力和节点刚度系数确定节点位移增量;(2)由节点位移增量确定单元应变和应力增量;(3)由单元应力增量确定单元应力状态,进而确定新的节点不平衡力。步骤(1)～(3)的过程重复进行,体系将达到平衡状态。给出了静态拉格朗日方法的基本公式和基本处理方法,包括节点不平衡力、节点刚度系数、单元应变和应力公式以及介质离散、单元应变和应力求解、介质开挖与支护、大变形问题等的处理方法。基于静态拉格朗日方法编制了计算软件,能模拟多荷载步、介质开挖与支护等复杂问题以及介质线弹性、弹塑性、流变等多种力学行为。软件可视化的后处理功能,可给出介质位移矢量图、塑性区分布图及节点位移、单元应力求解过程线。给出了静态拉格朗日方法算例,并与FLAC计算结果作了比较,通过算例说明了静态拉格朗日法的特点。将静态拉格朗日方法应用于实际工程,对黄河小浪底水利枢纽地下厂房大型洞室群围岩进行了力学分析。基于现场压缩试验,将岩体作为黏弹性介质,采用广义Kelvin模型,并确定了岩体力学参数;阐述了黏弹性模型静态拉格朗日方法的计算过程;根据地下洞室群开挖施工过程确定了数值计算的荷载步;建立了三维网格;基于实测计算了原始地应力场;数值计算给出了考虑流变效应的岩体位移和应力状态,给出了岩体流变稳定时间,为评价岩体的稳定性和设计支护方案提供了依据。概述连续介质静态拉格朗日分析方法的特点,该方法属于一种松弛或迭代方法,不需要通过节点运动方程求解,也不需要节点质量、阻尼力、惯性力、节点运动速度、加速度、时间步长等这些物理力学量,使求解方法简化。较FLAC方法的优点是,求解过程基本上是静态的,介质没有振动,或只有很小的振动,计算精度提高,克服了FLAC求解过程中介质振动、计算结果物理意义不明确这一缺点;算例的结果表明,对于较小的几何模型(单元和节点数较少),静态拉格朗日方法具有更快的计算速度,而对于较大的几何模型(单元和节点数较多),FLAC方法计算速度更快,在计算速度方面静态拉格朗日方法是否存在固有缺陷,尚需要进一步研究。

Basic characteristics and solving processes of FLAC are summarizedt;he advantages and disadvantages of FLAC are pointed out.Based on FLAC,a conclusion is obtained that it is not necessary for Lagrange method to solve through motion equation from both the angle of calculation convergence and physical meaning.A new solving idea is put forward and a so-called“static Lagrangian method”（SLM） is founded as follows：（1） to determine node displacement increment by node unbalanced force and nodal stiffness coefficient;（2） to determine element strain and stress increment by nodal displacement increment;and（3） to determine element stress state by stress increment,and thus to determine new node unbalanced force.If the procedures from（1）–（3） are carried out continuously,the system could reach to its equilibrium state.Basic formulae and treatment methods of SLM are given and explained,including formulae of nodal unbalanced force,nodal stiffness coefficient,element strain and stress,and treatment method of medium discretization,element strain and stress solving,medium excavation and support,large deformation problem,etc..Software of SLM is written to simulate complex problems such as multi-load-step,medium excavation and supporting,and medium linear-elastic,elasto-plastic,rheological behaviors.Post-treatment function of the software could give medium displacement vector figure,plastic zone distribution figure,curve of nodal displacement and element stress during solving processes.An example is calculated by SLM,and the result is compared with that of FLAC^3D.Characteristics of SLM are illustrated through the example.SLM is applied to practical engineering：a mechanical analysis of underground caverns surrounding rock mass of the Xiaolangdi key water control project at the Yellow River is carried out.Based on in-situ compression test,the rock is regarded as viscoelastic medium,and the generalized Kelvin model is adopted,and the rock mechanical parameters are determined.SLM calculation processes for viscoelastic model are expounded.Load steps in numerical calculation are determined according to the underground caverns practical excavation process.The three-dimensional meshes are founded.The geostress field is calculated based on in-situ test.The numerical calculation gives rock displacement and stress state and time for rock to reach stability during excavation,which could offer basis for rock stability evaluation and supporting scheme design.Finally,characteristics of SLM are summarized;and the advantages of SLM compared with those of FLAC are as follows：（1） SLM does not solve through motion equation,and the mechanical quantities of nodal mass,damping force,inertia force,nodal velocity and acceleration,time step,etc.are not needed;so the Lagrange method is simplified;and（2） the solving processes of SLM are basically static,and the medium has no vibration or only a slight vibration.The calculation precision is improved;and the defects of FLAC of medium vibration in solving process and physical meaning of calculation results that are not clear are overcome.Some examples show that SLM has higher calculation speed in comparatively small geometric model（having less elements and nodes）,whereas in large geometric model,FLAC has higher calculation speed.It needs to be studied that if SLM has inherent defect in calculation speed.