A limit equilibrium method for analyzing progressive failure in slopes

Abstract

A limit equilibrium method is proposed for analyzing progressive failure in the stability of brittle soil slopes. In the proposed method, the failure surface is firstly assumed and the whole failure mass is divided into many vertical slices. By satisfying force and moment equilibrium for each slice as well as for the total mass, the overall factor of safety for the assumed slip surface can be calculated. This factor of safety is then used to check the state of stress along the bottom of each slice. If the state of stress exceeds the soil strength, then local failure occurs and the residual strength is assigned. The whole procedure will be repeated unless the propagation of local failure zone stops. Finally, a more reasonable definition for factor of safety is determined. This analysis is illustrated for well documented case records, and the results are quite consistent with the field observations.     

Improvement of semi‐analytical design sensitivities of non‐linear structures using equilibrium relations

The accuracy problem of the semi‐analytical method for shape design sensitivity analysis has been reported for linear and non‐linear structures. The source of error is the numerical differentiation of the element internal force vector, which is inherent to the semi‐analytical approach. Such errors occur for structures whose displacement field is characterized by large rigid body rotations of individual elements. This paper presents a method for the improvement of semi‐analytical sensitivities. The method is based on the element free body equilibrium conditions, and on the exact differentiation of the rigid body modes. The method is efficient, simple to code, and can be applied to linear and non‐linear structures. The numerical examples show that this approach eliminates the abnormal errors that occur in the conventional semi‐analytical method. Copyright © 2001 John Wiley & Sons, Ltd.     

Size Sensitivity for the Reliability Index in Stochastic Finite Element Analysis of Damage

The direct differentiation method is applied to the estimation of statistical size effect behaviour in quasi-brittle solids. The scale factor is included in the finite element model and the autocorrelation function. Particular attention is paid to the proper differentiation of the Nataf transformation, which has been chosen to convert the basic random variables into a set of uncorrelated, standard normal variables. The predictive possibilities of the presented algorithm provide a valuable insight in the actual mechanisms responsible for failure. It can be evaluated to what extent the scale factor sensitivity of the failure probability is influenced by the phenomena related to the material disorder or the deterministic size effect.     

Element differential method and its application in thermal‐mechanical problems

In this paper, a new numerical method, element differential method (EDM), is proposed for solving general thermal‐mechanical problems. The key point of the method is the direct differentiation of the shape functions of Lagrange isoparametric elements used to characterize the geometry and physical variables. A set of analytical expressions for computing the first‐ and second‐order partial derivatives of the shape functions with respect to global coordinates are derived. Based on these expressions, a new collocation method is proposed for establishing the system of equations, in which the equilibrium equations are collocated at nodes inside elements, and the traction equilibrium equations are collocated at interface nodes between elements and outer surface nodes of the problem. Attributed to the use of the Lagrange elements that can guarantee the variation of physical variables consistent through all elemental nodes, EDM has higher stability than the traditional collocation method. The other main features of EDM are that no mathematical or mechanical principles are required to set up the system of equations and no integrals are involved to form the coefficients of the system. A number of numerical examples of 2‐ and 3‐dimensional problems are given to demonstrate the correctness and efficiency of the proposed method.     

基于多航线产品的航空公司市场价格竞争研究

为深入分析航空运输市场中同时存有直飞航线与经停航线的航空公司市场价格竞争问题,在研究中国航空运输市场结构特征的基础上,构建了基于多航线产品的航空公司价格竞争博弈模型,开展了市场价格竞争及其均衡分析,探讨了航空公司差异化与航线产品替代性对多航线产品市场竞争与演化均衡的影响。研究结果表明：在多航线产品市场中,航空公司采取联合定价模式将比单独定价模式获取更大的市场收益;航空公司间差异化对航空公司价格竞争均衡、市场收益等有显著影响;航空公司拥有资源的情况不同,差异化系数对航空公司价格竞争的影响演化也不同;多航线产品间的替代性对航空公司价格竞争均衡也有显著影响。     

Tracing postbuckling paths of structures containing multi-loops

Most incremental-iterative methods for tracing the postbuckling paths of structures have failed to achieve convergent solutions when the paths make sharp turns in regions where nearby equilibrium paths exist. This is largely due to the fact that iterations are not performed along a correct direction in these regions. To overcome this issue, two parameters for detecting the change in direction of loading and for guiding the direction of iteration are proposed. Also, a unit conversion factor is introduced into the constraint (arc length) equation to make it physically consistent. With these modifications, the present method is capable of tracing the postbuckling paths of structures with multi-loops. A two-member truss well known for its looping characteristics in the post-buckling response has been solved as an illustration.     

Long-term measurements of radon equilibrium factor in Greek dwellings

The time variation of the radon equilibrium factor F was measured every four hours in the Nuclear Technology Laboratory of the Aristotle University of Thessaloniki (in northern Greece) during October 1998-April 1999. The time dependence of the mean weekly value of radon equilibrium factor F is relatively small. During October 1999-May 2000 the radon equilibrium factor was measured in 25 apartments randomly distributed in Thessaloniki. The mean value of the equilibrium factor F is 0.47 +/- 0.09, close to the typical value of 0.4 adopted by UNSCEAR.     

Calculating osmotic pressure according to nonelectrolyte Wilson nonrandom factor model

The osmotic pressure of NaCl solutions was determined by the air humidity in equilibrium (AHE) method. The relationship between the osmotic pressure and the concentration was explored theoretically, and the osmotic pressure was calculated according to the nonelectrolyte Wilson nonrandom factor (N-Wilson-NRF) model from the concentration. The results indicate that the calculated osmotic pressure is comparable to the measured one.     

基于Woodbury非线性方法的迭代算法对比分析

在结构局部非线性求解过程中,刚度矩阵仅部分元素发生改变,此时切线刚度矩阵可写成初始刚度矩阵与其低秩修正矩阵和的形式,每个增量步的位移响应可用数学中快速求矩阵逆的Woodbury公式高效求解,但通常情况下迭代计算在结构非线性分析中是不可避免的,因此迭代算法的计算性能也对分析效率有重要影响。该文以基于Woodbury非线性方法为基础,分别采用Newton-Raphson （N-R）法、修正牛顿法、3阶两点法、4阶两点法及三点法求解其非线性平衡方程,并对比分析5种迭代算法的计算性能。利用算法时间复杂度理论,得到了5种迭代算法求解基于Woodbury非线性方法平衡方程的时间复杂度分析模型,定量对比了5种迭代算法的计算效率。通过2个数值算例,从收敛速度、时间复杂度和误差等方面对比了各迭代算法的计算性能,分析了各算法适用的非线性问题。最后,计算了5种算法求解基于Woodbury非线性方法平衡方程的综合性能指标。     

Taguchi optimization approach for Pb(II) and Hg(II) removal from aqueous solutions using modified mesoporous carbon

Using the Taguchi method, this study presents a systematic optimization approach for removal of lead (Pb) and mercury (Hg) by a nanostructure, zinc oxide-modified mesoporous carbon CMK-3 denoted as Zn-OCMK-3. CMK-3 was synthesized by using SBA-15 and then oxidized by nitric acid. The zinc oxide was loaded to the modified CMK-3 by the equilibrium adsorption of Zn(II) ions from aqueous solution followed by calcination to convert zinc nitrate to zinc oxide. The CMK-3 had porous structure and high specific surface area which can accommodate zinc oxide in a spreading manner, the zinc oxide connects to the carbon surface via oxygen atoms. The controllable factors such as agitation time, initial concentration, temperature, dose and pH of solution have been optimized. Under optimum conditions, the pollutant removal efficiency (PRE) was 97.25% for Pb(II) and 99% for Hg(II). The percentage contribution of each controllable factor was also determined. The initial concentration of pollutant is the most influential factor, and its value of percentage contribution is up to 31% and 43% for Pb and Hg, respectively. Our results show that the Zn-OCMK-3 is an effective nanoadsorbent for lead and mercury pollution remediation. Langmuir and Freundlich adsorption isotherms were used to model the equilibrium adsorption data for Pb(II) and Hg(II).     

爆破设计中推广的体积平衡法及优化

文章把定向爆破抛掷堆积过程分为抛掷和滑移阶段，分别用体积平衡法和滑移方程进行计算，并在一定的条件下进行优化，使目前的体积平衡法在理论和应用上都提高了一步。     

Lower‐bound limit analysis by using the EFG method and non‐linear programming

Intended to avoid the complicated computations of elasto‐plastic incremental analysis, limit analysis is an appealing direct method for determining the load‐carrying capacity of structures. On the basis of the static limit analysis theorem, a solution procedure for lower‐bound limit analysis is presented firstly, making use of the element‐free Galerkin (EFG) method rather than traditional numerical methods such as the finite element method and boundary element method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced‐basis technique is adopted to solve the mathematical programming iteratively in a sequence of reduced self‐equilibrium stress subspaces with very low dimensions. The self‐equilibrium stress field is expressed by a linear combination of several self‐equilibrium stress basis vectors with parameters to be determined. These self‐equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto‐plastic incremental analysis. The Complex method is used to solve these non‐linear programming sub‐problems and determine the maximal load amplifier. Numerical examples show that it is feasible and effective to solve the problems of limit analysis by using the EFG method and non‐linear programming. Copyright © 2007 John Wiley & Sons, Ltd.     

Interaction of a wedge disclination dipole with interfacial cracks

The elastic interaction between a wedge disclination dipole and collinear interfacial cracks in bimaterials is investigated. The general solutions of complex potentials to this problem are presented by using complex potential theory. As illustrative examples, the closed-form solutions for a wedge disclination dipole interacting with a finite interfacial crack and a semi-infinite interfacial crack are obtained. The stress intensity factors at the tips of the crack and the force acting on the disclination dipole center are also given. The shield and anti-shield effect of the wedge disclination dipole upon the stress intensity factors is evaluated, and the equilibrium position of the disclination dipole is discussed for various crack geometries and material mismatch. The results indicate that the shielding or anti-shielding effect to the stress intensity factors increases acutely when the disclination dipole approaches the tip of the crack. If the center of the dipole is fixed, there always exists a critical value of angle of the dipole arm which the shielding or anti-shielding effect to the stress intensity factor is maximal. In addition, the length of the dipole arm and the material mismatch have significant influence on the stress intensity factors. The results also show that the interfacial crack always attracts the wedge disclination dipole and an equilibrium position of the disclination dipole may be available near the interface, which differs from the case of a perfect bonded interface, when the dipole approaches the surface of the crack from infinity. The present solutions contain a series of new and previously known results which can be shown to be special cases.     

压电喷墨液滴撞击光滑壁面铺展行为的数值研究

目的 研究典型流体相关无量纲参数对墨滴在光滑承印物表面铺展行为的影响,确定各无量纲参数对铺展直径、铺展因子和稳定铺展时间的影响规律。方法 利用Ansys软件,建立墨滴撞击光滑壁面的数值计算模型,采用VOF模型追踪液滴形状,采用PISO算法计算压力速度耦合。引入韦伯数、雷诺数、奥内佐格数来分析墨滴撞击光滑承印物表面的铺展行为。结果 计算获得不同韦伯数、雷诺数、奥内佐格数下墨滴的最大铺展直径、最终平衡铺展直径、最大铺展因子和最终铺展时间。结论 韦伯数和雷诺数对墨滴最大铺展直径的影响较大,对最终平衡直径的影响较小。韦伯数或雷诺数越小,回缩阶段越短,越快达到平衡。韦伯数、雷诺数与最大铺展因子呈明显正相关。奥内佐格数对墨滴的最大铺展直径、最终平衡直径的影响都较小。奥内佐格数越小,回缩阶段越短,越快达到平衡,奥内佐格数与液滴最大铺展因子呈不明显的正相关性。     

Load increment procedure for post‐buckling analysis of laminated plates under in‐plane loads by finite strip method

A load increment procedure has been presented to integrate with the finite strip method for the post‐buckling analysis of laminated plates when subjected to uniform end shortening. In‐plane loads are introduced to reflect the end shortening effect. The Newton–Raphson procedure is implemented to attend a solution that satisfies the equilibrium condition and at the same time meets the loading requirements. Error associated with loading condition is minimised by adjusting the load factor to preserve the rate of convergence. The enhanced capability can be easily incorporated into the context of both classical and shear deformation plate theories. A range of application has been described. Convergence test and numerical results are presented for isotropic plate and laminates with general lay‐up arrangement. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of rebar/concrete interface debonding of FRP strengthened RC beams under fatigue load

The aim of this paper is to simulate the rebar/concrete interface debonding of FRP strengthened RC beams under fatigue load and also, to ascertain the influence of design parameters such as the elastic modulus, thickness and length of the FRP plate on the debonding performance. In order to simplify the simulation, some basic equilibrium equations are formulated and then the stresses of the rebar and FRP plate are numerically solved, and stress intensity factor is avoided in the simulation by fundamentals of fracture mechanics because of its complexity around the crack tip of bi-material interface. With the combination of finite element method and difference approximation, authors program the degradation model of coefficient of friction, debond criterion, propagation law and loop of load process into a commercial finite element code to investigate the fatigue debonding. The relationships between the debond length as well as other fatigue parameters and number of cyclic load are obtained and discussed.     

Solving nonlinear equilibrium equations of deformable systems by method of embedded polygons

Solving of nonlinear algebraic equations is an obligatory stage of studying the equilibrium paths of nonlinear deformable systems. The iterative method for solving a system of nonlinear algebraic equations stated in an explicit or implicit form is developed in the present work. The method consists of constructing a sequence of polygons in Euclidean space that converge into a single point that displays the solution of the system. Polygon vertices are determined on the assumption that individual equations of the system are independent from each other and each of them is a function of only one variable. Initial positions of vertices for each subsequent polygon are specified at the midpoints of certain straight segments determined at the previous iteration. The present algorithm is applied for analytical investigation of the behavior of biaxially compressed nonlinear-elastic beam-column with an open thin-walled cross-section. Numerical examples are made for the I-beam-column on the assumption that its material follows a bilinear stress-strain diagram. A computer program based on the shooting method is developed for solving the problem. The method is reduced to numerical integration of a system of differential equations and to the solution of a system of nonlinear algebraic equations between the boundary values of displacements at the ends of the beam-column. A stress distribution at the beam-column cross-sections is determined by subdividing the cross-section area into many small cells. The equilibrium path for the twisting angle and the lateral displacements tend to the stationary point when the load is increased. Configuration of the path curves reveals that the ultimate load is reached shortly once the maximal normal stresses at the beam-column fall outside the limit of the elastic region. The beam-column has a unique equilibrium state for each value of the load, that is, there are no equilibrium states once the maximum load is reached.     

Computation of the equilibrium position of a liquid with surface tension inside a tank of complex geometry and extension to sloshing dynamic cases

To study the vibrations of a tank partially filled with a liquid in low-gravity environment, we first have to find the static position of the liquid. In this paper, we present a three-dimensional finite element approach to find this equilibrium configuration for any tank geometry. Both gravity and capillary effects are taken into account. The nonlinear equations of this problem are derived from the differentiation of the total potential energy of the system, then the problem is transformed into a liquid free surface form-finding. The well-known singularity of this kind of problems is regularized using the updated reference strategy. The equations of the regularized problem are discretized using the finite element method and solved by the Newton–Raphson algorithm. Several examples illustrate the effectiveness of this method, even for complex cases, and two validation tests are presented. The linear sloshing vibrations of the liquid are finally studied near this equilibrium position and two validation cases are proposed for the eigenvalue dynamic problem.     

Incremental finite element sensitivity analysis for non-linear mechanics applications

Finite element formulations for structural sensitivity analysis of non-linear systems with fixed overall shape are discussed. Both the direct differentiation and adjoint variable methods are employed. The resulting sensitivity algorithms are consistent with time integration scheme adopted for solving equilibrium problem. Effectiveness and computational aspects of the procedures are discussed and compared. Numerical algorithms are shown to be readily implemented in existing finite element codes. Large-scale examples illustrate the paper.     

局部面积搜索法在高压相平衡中的应用

用最小Gibbs 能曲线积分的正面积最大方法(AM)进行相平衡的计算和预测,可以比传统的闪蒸计算法更好地计算高压下和近临界区的相平衡组成。但是这一方法由于使用了在全组成范围的"穷举"方法搜索最大正面积对应的相平衡组成,使得收敛速度慢,计算和搜索时间过长;将其修改成多次局部面积搜索法(LSAM),极大提高了收敛速度和减少了计算时间,并利用所测定的苯与丙烯高压平衡数据对其进行了有效性检验。