重积分的“线、点、面”定限法——高等数学教学札记

本文将二重积分与三重积分的定限方法概括为“线、点”与“线、点、面”

二维连续型随机变量线性函数的概率密度的新算法

针对二维连续型随机变量的线性函数的积分定限和计算问题,提出了一种更加简单的不等式组定限方法和系统的计算步骤.

Upper bound solution of surrounding rock pressure of shallow tunnel under nonlinear failure criterion

The nonlinear Baker failure criterion is introduced into the upper-bound limit analysis to examine the face stability of a shallow tunnel. The tunnel face under the ultimate limit state is analyzed from the perspective of energy balance. The work rates of external forces and internal energy dissipation are calculated. An analytical solution of necessary face pressures is derived. The optimal upper-bound solution of the face pressures is obtained by optimization. The results show that the three dimensionless parameters A, T, n of nonlinear Baker failure criterion have different effects on the necessary face pressures and the pattern failure mechanisms ahead of tunnel face. A is the most important one;n takes the second place, and T is the least one. The computed necessary face pressures are nonlinearly increasing when A is reduced. Combined with the actual monitoring data of Taxia tunnel, the calculation results in this paper is verified. It is suggested that the tunnel face supports should be strengthened timely in soft rocks to prevent the occurrence of face collapse.

Limit equilibrium stability analysis of slopes under external loads

Two calculation modes for the effect of external load on slope stability, i.e., mode I in which the external load is thought to act on slope surface, and mode II in which the external load is thought to act on slip surface along the force action line, were considered. Meanwhile, four basic distribution patterns of external load were used, of which complex external loads could be composed. In analysis process, several limit equilibrium methods, such as Swedish method, simplified Bishop method, simplified Janbu method, Spencer method, Morgenstern-Price(M-P) method, Sarma method, and unbalanced thrust method, were also adopted to contrast their differences in slope stability under the external load. According to parametric analysis, some conclusions can be obtained as follows:(1) The external load, with the large magnitude, small inclination angle, and acting position close to the slope toe,has more positive effect on slope stability;(2) The results calculated using modes I and II of external load are similar, indicating that the calculation mode of external load has little influence on slope stability;(3) If different patterns of external loads are equivalent to each other, their slope stability under these external loads are the same, and if not, the external load leads to the better slope stability,as action position of the resultant force for external load is closer to the lower sliding point of slip surface.

A strength reduction method based on double reduction parameters and its application

In the traditional strength reduction method,the cohesion and the friction angle adopt the same reduction parameter,resulting in equivalent proportional reduction.This method does not consider the different effects of the cohesion and friction angle on the stability of the same slope and is defective to some extent.Regarding this defect,a strength reduction method based on double reduction parameters,which adopts different reduction parameters,is proposed.The core of the double-parameter reduction method is the matching reduction principle of the slope with different angles.This principle is represented by the ratio of the reduction parameter of the cohesion to that of the friction angle,described as η.With the increase in the slopeangle,ηincreases; in particular,when the slope angle is 45°,tηis 1.0.Through the matching reduction principle,different safety margin factors can be calculated for the cohesion and friction angle.In combination with these two safety margin factors,a formula for calculating the overall safety factor of the slope is proposed,reflecting the different contributions of the cohesion and friction angle to the slope stability.Finally,it is shown that the strength reduction method based on double reduction parameters acquires a larger safety factor than the classic limit equilibrium method,but the calculation results are very close to those obtained by the limit equilibrium method.

Study on the FEM design of reinforced earth retaining wall with geogrid

At present,limit equilibrium method is often adopted in the design of reinforced earth retaining wall. Geotechnical engineers home and abroad have done a lot of work to improve the traditional calculation methods in recent years,while there are lots of defects. This paper first identifies the location of failure surface and safety factor through the finite element program of PLAXIS and then analyses the influencing factors of the stability of reinforced earth retaining wall with geogrid. The authors adopt strength reduction FEM (finite element method)in the design and stability analysis of reinforced earth retaining wall and have achieved some satisfying results. Without any assumptions,the new design method can automatically judge the failure mode of reinforced earth retaining wall,consider the influence of axial tensile stiffness of the reinforcement stripe on the stability of retaining wall,identify reasonable distance and length of the reinforcement stripe,and choose suitable parameters of reinforcement stripe,including strength,stiffness and pseudo-friction coefficient which makes the design optimal. It is proved through the calculation examples that this method is more reasonable,reliable and economical in the design of reinforced earth retaining wall.

Three-dimensional stability analysis of anisotropic and non-homogeneous slopes using limit analysis

A method of three-dimensional loaded slope stability for anisotropic and nonhomogeneous slopes was presented based on the upper-bound theorem of the limit analysis approach. The approach can be considered as a modification and extension of the solutions. The influences of friction angle, anisotropy factor, nonhomogeneous factor, slope angle, ratio of width to depth, and load on the slope crest were investigated. The results show that solutions are suitable to deal with the purely cohesive soils and frictional/cohesive soils, isotropic and anisotropic, homogeneous and nonhomogeneous, loaded and unloaded cases.

Limit equilibrium method for slope stability based on assumed stress on slip surface

In the limit equilibrium framework, two- and three-dimensional slope stabilities can be solved according to the overall force and moment equilibrium conditions of a sliding body. In this work, based on Mohr-Coulomb(M-C) strength criterion and the initial normal stress without considering the inter-slice(or inter-column) forces, the normal and shear stresses on the slip surface are assumed using some dimensionless variables, and these variables have the same numbers with the force and moment equilibrium equations of a sliding body to establish easily the linear equation groups for solving them. After these variables are determined, the normal stresses, shear stresses, and slope safety factor are also obtained using the stresses assumptions and M-C strength criterion. In the case of a three-dimensional slope stability analysis, three calculation methods, namely, a non-strict method, quasi-strict method, and strict method, can be obtained by satisfying different force and moment equilibrium conditions. Results of the comparison in the classic two- and three-dimensional slope examples show that the slope safety factors calculated using the current method and the other limit equilibrium methods are approximately equal to each other, indicating the feasibility of the current method; further, the following conclusions are obtained: 1) The current method better amends the initial normal and shear stresses acting on the slip surface, and has the identical results with using simplified Bishop method, Spencer method, and Morgenstern-Price(M-P) method; however, the stress curve of the current method is smoother than that obtained using the three abovementioned methods. 2) The current method is suitable for analyzing the two- and three-dimensional slope stability. 3) In the three-dimensional asymmetric sliding body, the non-strict method yields safer solutions, and the results of the quasi-strict method are relatively reasonable and close to those of the strict method, indicating that the quasi-strict method can be used to obtain a reliable slope safety factor.

Stability analysis for natural slope by kinematical approach

The stability of natural slope was analyzed on the basis of limit analysis. The sliding model of a kind of natural slope was presented. A new kinematically admissible velocity field for the new sliding model was constructed. The stability factor formulation by the upper bound theorem leads to a classical nonlinear programming problem, when the external work rate and internal energy dissipation were solved, and the constraint condition of the programming problem was given. The upper bound optimization problem can be solved efficiently by applying a nonlinear SQP algorithm, and stability factor was obtained, which agrees well with previous achievements.

Formability improvement and blank shape definition for deep drawing of cylindrical cup with complex curve profile from SPCC sheets using FEM

In the current work, to predict and improve the formability of deep drawing process for steel plate cold rolled commercial grade （SPCC） sheets, three parameters including the blanking force, the die and punch comer radius were considered. The experimental plan according to Taguchi＇s orthogonal array was coupled with the finite element method （FEM） simulations. Firstly, the data from the test of stress-strain and forming limit curves were used as input into ABAQUS/Explicit finite element code to predict the failure occurrence of deep drawing process. The three parameters were then validated to establish their effects on the press formability. The optimum case found via simulation was finally confirmed through an experiment. In order to obtain the complex curve profile of cup shape after deep drawing, the anisotropic behavior of earring phenomenon was modeled and implemented into FEM. After such phenomenon was correctly predicted, an error metric compared with design curve was then measured.

Finite-Time Stability of Nonautonomous Delayed Systems

The finite-time stability to linear discontinuous time-varying delayed system was investigated. By applying the method of upper and lower solutions, some sufficient conditions of this kind of stability were obtained. Furthermore, it also developed a monotone iterative technique for obtaining solutions which are obtained as limits of monotone sequences

Settling Time Design and Parameter Tuning Methods for Finite-Time P-PI Control

High precision position control and high speed control of the robot manipulators are fundamental and important control problems. The effectiveness of finite-time P-PI control was confirmed by end-effector position control of robot manipulators. However, parameter tuning method has not been proposed to finite-time P-PI control. In this paper, we propose a settling time design method and a parameter tuning method for the finite-time P-PI control. The effectiveness of the proposed parameter tuning method is confirmed by experiments of end-effcctor position control of a robot manipulator.

Measurement of Infinite Diluted Activity Coefficient of Solvents in Polymer by Inverse Gas Chromatography Method

The infinite diluted activity coefficients of solvents in polyisopropyl methylacrylate was measured using inverse gas chromatography. The solvents used were benzene, toluene, ethyl benzene, methyl acetate, ethyl acetate, propyl acetate, butyl acetate, methanol, ethanol isopropyl alcohol, butyl alcohol, 1,2-dichloroethane, and chloroform. It was observed that the infinite diluted activity coefficient of alcohols are well above those of the other solvents investigated.

A Locking-free Nonconforming Finite Element for Planar Linear Elasticity

We propose a locking-free nonconforming finite element method to solve for the displacement variation in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is robust and optimal, in the sense that the convergence estimate in the energy is independent of the Lame parameter λ.

Stability and reinforcement analysis of rock slope based on elasto-plastic finite element method

The rigid body limit equilibrium method(RBLEM) and finite element method(FEM) are two widely used approaches for rock slope's stability analysis currently. RBLEM introduced plethoric assumptions; while traditional FEM relied on artificial factors when determining factor of safety(FOS) and sliding surfaces. Based on the definition of structure instability that an elasto-plastic structure is not stable if it is unable to satisfy simultaneously equilibrium condition, kinematical admissibility and constitutive equations under given external loads, deformation reinforcement theory(DRT) is developed. With this theory, plastic complementary energy(PCE) can be used to evaluate the overall stability of rock slope, and the unbalanced force beyond the yield surface could be the identification of local failure. Compared with traditional slope stability analysis approaches, the PCE norm curve to strength reduced factor is introduced and the unbalanced force is applied to the determination of key sliding surfaces and required reinforcement. Typical and important issues in rock slope stability are tested in TFINE(a three-dimensional nonlinear finite element program), which is further applied to several representatives of high rock slope's stability evaluation and reinforcement engineering practice in southwest of China.

New methods of safety evaluation for rock/soil mass surrounding tunnel under earthquake

The objective of this work is to obtain the seismic safety coefficient and fracture surface and proceed with the seismic safety evaluation for the rock mass or soil mass surrounding a tunnel,and the limitation of evaluating seismic stability is considered using the pseudo-static strength reduction.By using the finite element software ANSYS and the strength reduction method,new methods of seismic safety evaluation for the rock mass or soil mass surrounding a tunnel are put forward,such as the dynamic finite element static shear strength reduction method and dynamic finite element shear strength reduction method.In order to prove the feasibility of the proposed methods,the results of numerical examples are compared with that of the pseudo-static strength reduction method.The results show that 1) the two methods are both feasible,and the plastic zone first appears near the bottom corners; 2) the safety factor of new method Ⅱ is smaller than that of new method I but generally,and the difference is very small.Therefore,in order to ensure the safety of the structure,two new methods are proposed to evaluate the seismic stability of the rock mass or soil mass surrounding a tunnel.A theoretical basis is provided for the seismic stability of the rock mass or soil mass and the lining surrounding a tunnel and also provided for the engineering application.

Verified Solution for a Statically Determinate Truss Structure with Uncertain Node Locations

We consider a statically determinate structural truss problem where all of the physical model parameters are uncertain： not just the material values and applied loads, but also the positions of the nodes are assumed to be inexact but bounded and are represented by intervals. Such uncertainty may typically arise from imprecision during the process of manufacturing or construction, or round-off errors. In this case the application of the finite element method results in a system of linear equations with numerous interval parameters which cannot be solved conventionally. Applying a suitable variable substitution, an iteration method for the solution of a parametric system of linear equations is firstly employed to obtain initial bounds on the node displacements. Thereafter, an interval tightening （pruning） technique is applied, firstly on the element forces and secondly on the node displacements, in order to obtain tight guaranteed enclosures for the interval solutions for the forces and displacements.

Analysis of Unbalances Forces Using Methods of Identification and Finite Elements

The study of damage in rotating machineries is of fundamental interest in the fields of machine and structure design. A rotating system, supported by bearings and under some dynamic conditions, can generate a variety of problems that are encountered in many different types of rotating machines. One of these problems is the unbalance due to non-homogeneous mass distribution along the shaft. One of the techniques which are widespread today is the identification of parameters and excitation forces that may well followed by monitoring the evolution and change of possible variations of these parameters. Although several methods for the identification of unbalance excitation force are available in the literature, none of them can be considered unrestricted to be applied for all rotating systems. In this study, two methodologies to identify unknown excitations, such as unbalance, have been proposed. This project refers to the analysis of unbalanced forces from displacement parameters and speed by using methods of identification by Fourier series and Legendre polynomials together with the finite element method, state observers in reasons of the problem of absence of signs of rotational displacement, bandpass filter were used to noise suppression of the data collected from the experimental part, Quasi-Newton method to minimize a function in which the bearing stiffness and its damping are unknowns, and also the experimental verification of the methodology, using for this system owned by a rotary mechanical vibrations of the Department of Mechanical Engineering of Faculty of Engineering, campus of llha Solteira.

A Class of Two-level High-order Accuracy Explicit Difference Scheme for Solving 3-D Parabolic Partial Differential Equation

A class of two-level high-order accuracy explicit difference scheme for solving 3-D parabolic P.D.E is constructed. Its truncation error is (Δt2+Δx4) and the stability condition is r=Δt/Δx2=Δt/Δy2=Δt/Δz2≤1/6.

Numerical Study of Unsteady Behavior of Partial Cavitation on Two Dimensional Hydrofoils

This paper deals with time dependent performance characteristics of cavitating hydrofoils, the flow around which has been simulated using pressure-based finite volume method. A bubble dynamics cavitation model was used to investigate the unsteady behavior of cavitating flow and describe the generation and evaporation of vapor phase. For choosing the turbulence model and mesh size a non cavitating study was conducted. Three turbulence models such as Spalart-Allmaras, Shear Stress Turbulence （SST） κ-ω model, Re-Normalization Group （RNG） κ-ε model with enhanced wall treatment are used to capture the turbulent boundary layer along the hydrofoil surface. The cavitating study presents an unsteady behavior of the partial cavity attached to the foil at different time steps for σ = 0.8 and σ = 0.4. Moreover, this study is focused on cavitation inception, the shape and general behavior of sheet cavitation, lift and drag forces for different cavitation numbers.