Velocity Discontinuity Surfaces Research Articles

An improved discretization‐based kinematic approach for stability analyses of nonuniform c‐φ soil slopes

AbstractThis paper proposes an improved discretization‐based kinematic approach (DKA) with an efficient and robust algorithm to investigate slope stability in nonuniform soils. In an effort to ensure rigorous upper‐bound solutions which may be not satisfied by the initial DKA based on a forward difference method (DKA‐FD), a central and backward difference “point‐to‐point” method (DKA‐CD and DKA‐BD) is proposed to generate discretized points to form a velocity discontinuity surface. Varying (including constant) soil frictional angles along depth are discussed, which can be readily considered in the improved DKA‐CD. Work rate calculations are performed to derive upper‐bound formulations of slope stability number, and critical failure surface is correspondingly obtained at limit state. The comparison with forward and backward difference methods clearly reveals that the improved DKA‐CD could significantly reduce the mesh‐dependency issue and enhance efficacy of slope stability analyses in nonuniform soils.

A New Kinematical Admissible Translational–Rotational Failure Mechanism Coupling with the Complex Variable Method for Stability Analyses of Saturated Shallow Square Tunnels

Tunnels are commonly constructed in water-bearing zones, which necessitates stability analyses of saturated tunnels based on the upper bound of the plastic theory. Previous kinematical approaches have the following drawbacks: (1) using an empirical approach to estimate pore-water pressure distributions; (2) using failure mechanisms that are not rigorously kinematically admissible. To overcome these shortcomings, we proposed a rigorously kinematically admissible translational–rotational failure mechanism for an underwater shallow square tunnel where velocity discontinuity surfaces were derived. Then, the pore-water pressure field surrounding the tunnel under the boundary of constant water pressure is analytically generated based on the complex variable method and imported into the kinematically admissible velocity field. Work rates performed by external forces and the internal dissipation rate are numerically computed to formulate the power balance equation, followed by a mixed optimization algorithm to capture the critical states of the surrounding soils of tunnels. The outcomes of pore-water pressure distributions, safety factors, and failure mechanisms are in tandem with those given by the numerical simulation but show higher computational efficiency than the numerical simulation. In the end, we highlight the advantages of the proposed model over the empirical approach, where soil properties and water table elevation effects are analyzed.

Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable

Expanded anchor cables have been widely used in the rapid development of underground engineering. However, there are still some deficiencies in the computation of the ultimate bearing capacity of expanded anchor cables. Based on the upper-bound theorem of limit analysis, the upper-bound solution of the ultimate bearing capacity of an expanded anchor cable was derived. For this calculation, the instability mechanism of the soil at the front surface of the anchorage segment of the expanded anchor cable was assumed to satisfy the logarithmic spiral failure model, its 3D velocity discontinuity surface was generated using the spatial discretization technique, and the optimal solution was determined through the particle swarm optimization (PSO) algorithm. The breakage mechanism of the anchorage side surface was assumed to appear at the interface between the anchorage body and the soil, and its velocity field satisfied the requirements of the associated flow rule. The accuracy of the proposed analytical solution was well-verified through a comparison with three-dimensional numerical simulations, and its superiority was also well-demonstrated in comparison to the existing theoretical calculation method. Subsequently, the influence of the key parameters of the anchor cable on the ultimate lateral resistance, end resistance, and total pullout capacity was discussed. The results showed that: the anchorage segment diameter, anchorage segment length, and buried depth of the expanded anchor cable had a great influence on the ultimate lateral resistance, ultimate end resistance, and total ultimate pullout capacity; however, the anchorage segment length had little influence on the ultimate end resistance, and the inclination angle of the anchor cable had relatively little influence on the resistance.

Upper-Bound Analysis for Equal Channel Angular Pressing (ECAP) with an Intersecting Channel Angle of 120°

Abstract The exact calculations of the stress and strain distributions based on the controlling equations for a forming process with large deformation are often difficult. To circumvent such difficulties, some analytical methods such upper-bound analysis and slip-line field theory have been established by making a number of simplifying assumptions regarding the material properties and deformation modes. In this work an analytical model based on the upper-bound theory was successfully developed to predict material flow pattern and maximum process loads for an Equal Channel Angular Pressing (ECAP) die with circular cross-section and an intersecting channel angle of 120°. Based on the model, the power dissipated on all frictional and velocity discontinuity surfaces were determined and optimized in order predict the maximum process force as function of the channel geometry and the material plastic behavior. To validate the developed model, the ECAP die were produced and used to determine experimental load-displacement curves of AA6061-T6 specimens with different lengths. A good correlation between theoretical and experimental results was observed. In addition, the constant friction factor demonstrated to have a strong effect on the relative extrusion pressure.

Stability analysis of 3D cracked slope reinforced with piles

The presence of cracks in soil slopes is of major concern to researchers and engineers, as this phenomenon is typically considered to be an early indication of instability and potential landslides. In this study, the kinematical approach of limit analysis was applied to evaluate the three-dimensional (3D) stability of soil slopes with pre-existing cracks and the reinforcement effects of piles. A vertical crack was accommodated in the classical 3D rotational failure mechanism by introducing a velocity discontinuity surface. Formulas for the external work rates and internal energy dissipation rates of the failure mechanism were derived within the framework of limit analysis. An optimization program was developed in combination with the strength reduction method to search for the most critical failure mechanism and the corresponding upper-bound factor of safety. A new parameter, the maximum crack depth coefficient, was introduced for evaluating the sensitivity of slopes to the presence of cracks. The numerical results indicated that slopes with large inclination angles are extremely sensitive to natural cracks, whereas gentle slopes are less likely to be affected. It was also found that stabilizing piles could effectively restrict the depth of the crack opening and thus, improve the stability of fissured slopes; the reinforcement effect was extremely dependent on the installation location and the density of the pile group.

Limit variation analysis of shallow rectangular tunnels collapsing with double-layer rock mass based on a three-dimensional failure mechanism

In the framework of upper bound theorem of limit analysis, the progressive collapse of shallow rectangular tunnels with double-layer rock mass has been theoretically analyzed based on the three-dimensional (3D) velocity discontinuity surfaces. According to the virtual work principle, the difference theorem and the variation method, the collapse surface of double-layer rock mass is determined based on the Hoek-Brown failure criterion. The formula can be degenerated to a single-layer rock collapsing problem when the rock mass is homogeneous. To estimate the validity of the result, the numerical simulation software PLAXIS 3D is used to simulate the collapse of shallow tunnels with double-layer rock mass, and the comparative analysis shows that numerical results are in good agreement with upper-bound solutions. According to the results of parametric analysis, the potential range of collapse of a double-layer rock mass above a shallow cavity decreases with a decrease in A1/A2, σci1/σci2 and σtm1/σtm2 and an increase in B1/B2, γ1/γ2. The range will decrease with a decrease in support pressure q and increase with a decrease in surface overload σ;s. Therefore, reinforced supporting is beneficial to improve the stability of the cavity during actual construction.

LOADING REGIME OPTIMIZATION FOR HIGH-SPEED IMPACT EXTRUSION OF BIMETALLIC FLAT-STEP ROD PRODUCTS

The work presents a physicomechanical model developed for calculating the force action on the punch with high-speed impact extrusion of bimetallic stepped rod products under conditions of plane deformation. In order to obtain the result, the process of impact loading of the workpiece is divided into two stages – acceleration and braking. At the acceleration stage, a linear dependence on the graph P n (h n ) “force on the punch – the deformation line” is adopted. For the braking stage, a procedure is given for calculating the force acting on the punch with the plastic flow of the bimetallic workpiece in a stepped narrowing cavity with three deformation centers. Based on the method of upper evaluation for the case of plastic flow at the final stage of the process, an equation is derived for calculating the force acting on the punch through deformation centers. By solving the problem in a quasistatic formulation (the action of dynamic tensions on the surfaces of velocity discontinuity and inertia forces does not affect the type and shape of the velocity and acceleration hodographs constructed), starting from the condition of the minimum power of plastic shaping, the dependences for calculating the optimum angles of the matrix cavity α опт , β опт , γ опт depending on the stretching λ and the coefficient of friction μ were determined. The use of a matrix with the optimum taper angles will allow us to realize the process of high-speed impact extrusion with minimum load acting on the punch. On the basis of the developed model, an equation for calculating the minimum upper force P n,min acting on the punch under high-speed impact plastic flow of metals through the deformation centers was obtained within the framework of the adopted assumptions. The equation presents the rheological characteristics of the deformed main part of the workpiece (k, ρ), technological parameters (λ 1 , λ 2 , λ 3 , V), contact friction coefficients μ for different parts of the surface of the matrix cavity, the impacting masses of the punch and the workpiece. The developed model for calculating the optimal power regime and equation can be used in engineering practice to develop a technology for high-speed impact extrusion of flat-step bimetallic products for various purposes.

Roof collapse of shallow tunnel in layered Hoek-Brown rock media

Collapse shape of tunnel roof in layered Hoek-Brown rock media is investigated within the framework of upper bound theorem. The traditional collapse mechanism for homogeneous stratum is no longer suitable for the present analysis of roof stability, and it would be necessary to propose a curve failure mode to describe the velocity discontinuity surface in layered media. What is discussed in the paper is that the failure mechanism of tunnel roofs, consisting of two different functions, is proposed for layered rock media. Then it is employed to investigate the impending roof failure. Based on the nonlinear Hoek-Brown failure criterion, the collapse volume of roof blocks are derived with the upper bound theorem and variational principle. Numerical calculations and parametric analysis are carried out to illustrate the effects of different parameters on the shape of failure mechanism, which is of overriding significance to the stability analysis of tunnel roof in layered rock media.

Upper bound analysis for bearing capacity of nonhomogeneous and anisotropic clay foundation

The bearing capacity calculations for nonhomogeneous and anisotropic clay foundation are often employed the methods of limit equilibrium, limit analysis and characteristic. In this paper, an approach of upper bound theorem in conjunction with new failure mechanism is adopted to estimate the bearing capacity of foundation in nonhomogeneous and anisotropic clays. The proposed failure mechanism is composed of a series of rigid blocks, which is constructed in a manner to respect the normality condition of limit analysis theory at every point of velocity discontinuity surfaces, considering the property of spatially varying materials. Using a spatial discretization technique, the slip surface of failure mechanism is determined point by point. Based on upper bound theory of limit analysis, the internal energy dissipation and the work rate of external forces are calculated for each block. Coding the corresponding computer program, the solutions of bearing capacity of nonhomogeneous and anisotropic clay foundation are obtained. The results are compared with those previously published solutions using the characteristic and limit equilibrium methods. From the comparisons, it is found that the approach of this paper is effective to analyze the bearing capacity of nonhomogeneous and anisotropic clays. Then, the solutions are presented, and the influence of nonhomogeneity and anisotropy on the bearing capacity is discussed.

An analytical approach for simple shear extrusion process with a linear die profile

Abstract In the present work, for the first time, an analytical approach based on upper-bound theorem is proposed to analyze the simple shear extrusion process. In this regard new die parameter named maximum inclination angle is introduced. By this model, the power dissipated on all frictional and velocity discontinuity surfaces is determined and the total power is optimized for two types of die, fixed and movable inlet channel die. To check the validation of the upper-bound model, the process is simulated by the commercial finite element code, Deform-3D. To compare the theoretical results with experiments, a fixed inlet channel die and two dies with movable inlet channel are used to determine the processing force for different cross sections. The developed model predicts that the relative extrusion pressure increases with increasing the constant friction factor; also, for a given value of the constant friction factor and the maximum distortion angle, there is an optimized maximum inclination angle which minimizes the power. Comparing the fixed inlet channel die with the movable inlet channel one, it is seen that the optimized maximum inclination angle is higher in the FIC die.

Analysis and FEM simulation of extrusion process of bimetal tubes through rotating conical dies

Bimetal tube extrusion process through rotating conical dies was studied analytically and numerically. A kinematically admissible velocity field was developed to evaluate the internal power and the power dissipated on frictional and velocity discontinuity surfaces. By balancing the moment applied by the rotary die with the moments caused by the circumferential frictions in the container and on the mandrel, the twisting length of the material in the container was determined. By equating the total power with the required external power, the extrusion pressure was determined by optimizing with respect to the slippage parameter between the die and the outer material. It is shown that the extrusion pressure is decreased by about 20% by the die rotation. The bimetal tube extrusion process through rotating die was also simulated using the finite element code, ABAQUS. Analytical results were compared with the results given by the finite element method. These comparisons show a good agreement.

Formulation of a new generalized kinematically admissible velocity field with a variable axial component for the forward extrusion of shaped sections

For most three-dimensional analytical solutions proposed for the extrusion of shaped sections, the axial component of the velocity vector has been assumed to be constant at each cross section throughout the deforming zone. This shortcoming means that these velocity fields are not in accordance with the reality of the extrusion problem, and hence, the upper bounds based on such fields give high values for the extrusion pressure. To overcome this, a new formulation has been presented in this paper for which a kinematically admissible velocity field has been developed using a variable axial velocity component. For this purpose, curved surfaces of velocity discontinuities at the entry and exit have been proposed and incorporated into the formulation for the extrusion of shaped sections from circular billets. As an example, a square profiled section has been chosen for the extrusion problem. The upper bound on extrusion pressure was computed using the new formulation. It was shown that the initial velocity discontinuity surface at the entry to the deforming region was flat, and as one travels into the deforming region towards the exit section, the velocity discontinuity surface gradually became convex, having the highest convexity at the exit section. This was contrary to what has been suggested in the literature so far. The measure of convexity depends on the extrusion parameters which have been investigated in this work. Experiments were also carried out to verify the theoretical results, and good agreements were observed between the two. Comparison of the present results with similar previous works showed good improvements as well.

Upper bound analysis of thick wall tubes extrusion process through rotating curved dies

In this study, extrusion process of thick wall tubes through rotating curved dies is investigated by the method of upper bound. Total deformation region is divided into four deformation zones and a velocity field is developed for each deformation zone. The twist moments generated on container and mandrel surfaces are calculated and by equating them with the twist moment exerted by rotating die, the twisting length of tube inside the container is determined. Then, the internal powers, the powers dissipated on frictional and velocity discontinuity surfaces for a rigid-perfectly plastic material are evaluated and they are used in upper bound model. By optimizing the total power with respect to the slippage parameter between die and the tube material, the required relative extrusion pressure for a given process conditions and die angular velocity is determined. The results of finite element simulations are also presented and satisfactory agreement between the calculated and FEM results are demonstrated.

An upper bound analysis of tube extrusion process through rotating conical dies

In this paper, an upper bound approach is used to analyse the tube extrusion process through steadily rotating conical dies. The material under deformation in the die and inside the container is divided into four deformation zones. In each deformation zone, a kinematically admissible velocity field is developed to evaluate the internal power, the power dissipated on frictional and velocity discontinuity surfaces and the twist moments generate on frictional surfaces. By equating the total power with the required external power, the extrusion pressure is determined. The tube extrusion process through rotating die is also simulated by using the finite element code, ABAQUS. Analytical results are compared with the results given by the finite element method. These comparisons show a good agreement.

Limit Analysis of Supporting Pressure for Failure Surface of Deep-Buried Tunnel with Nonlinear Failure Criterion

The collapse surface of surrounding rock induced by the excavation has significant influence on the stability and supporting structures of deep-buried tunnel since located in high in-situ stress field. Based on the assumption of curved velocity discontinuity surface, the internal dissipated power along the surface was calculated in the framework of Power-Law nonlinear failure criterion. In considering the role of supporting forces, an objective function composed of external rate of work and the internal energy rate of dissipation was constructed by using the upper bound theorem of limit analysis, and the analytical expression of collapse surface was obtained with the help of variational calculation. The effect of parameters on the collapse surface was analyzed. It is found that the height and width of the collapse block increased with the rise of nonlinear coefficient m, which illustrate that the nonlinear coefficient m has significant influence on the shape of collapse block. In the limit state, the height and width of the collapse block increase with the growth of supporting pressure.

A coupled stream function-finite element analysis for wire drawing processes

A numerical approach has been developed based on the stream function technique and the finite element analysis to predict required power and temperature rise in wire drawing processes. An admissible velocity field is first proposed using a stream function and then power consumption in the wire drawing is optimized to achieve sensible and unique deformation geometry. In addition, the finite element method together with axi-symmetric Petrov–Galerkin algorithm is coupled with the deformation model to assess the temperature distribution in both the deforming wire and the die during the process. The work hardening effects are also considered in the model both in the deformation zone and on the velocity discontinuity surfaces. The model can estimate the effects of various process parameters such as drawing velocity and die geometry. In order to evaluate the results of the model, the predictions are compared with the established models including force equilibrium as a lower-bound approach and an upper-bound solution based on the spherical velocity field.

A generalized spherical velocity field for bi-metallic tube extrusion through dies of any shape

In this paper, an upper-bound approach is used to analyze the extrusion process of bi-metallic tubes through dies of any shape with moving cylindrical shaped mandrel. A generalized kinematically admissible velocity field using spherical coordinate system is developed to evaluate the internal power and the power dissipated on frictional, velocity discontinuity surfaces and the total power. The extrusion process is also simulated using the finite element code, ABAQUS. Analytical results are compared with the results given by experiments of other researchers and also by the finite element method. These comparisons show a good agreement.

Validation of a New 2D Failure Mechanism for the Stability Analysis of a Pressurized Tunnel Face in a Spatially Varying Sand

A new two-dimensional (2D) limit analysis failure mechanism is presented for the determination of the critical collapse pressure of a pressurized tunnel face in the case of a soil exhibiting spatial variability in its shear strength parameters. The proposed failure mechanism is a rotational rigid block mechanism. It is constructed in such a manner to respect the normality condition of the limit analysis theory at every point of the velocity discontinuity surfaces taking into account the spatial variation of the soil angle of internal friction. Thus, the slip surfaces of the failure mechanism are not described by standard curves such as log-spirals. Indeed, they are determined point by point using a spatial discretization technique. Though the proposed mechanism is able to deal with frictional and cohesive soils, the present paper only focuses on sands. The mathematical formulation used for the generation of the failure mechanism is first detailed. The proposed kinematical approach is then presented and validated by comparison with numerical simulations. The present failure mechanism was shown to give results (in terms of critical collapse pressure and shape of the collapse mechanism) that compare reasonably well with the numerical simulations at a significantly cheaper computational cost.

Theoretical and Experimental Research on Rolling Force for Rail Hot Rolling by Universal Mill

For rail rolling by universal mill, a simplified three-dimensional theoretical model was built firstly. The kinematically admissible velocity field of the web, head, and base of rail was determined respectively; moreover, the corresponding strain rate field and the strength of shear strain rate were obtained. Then, the plastic deformation power of corresponding deformation zone, the power consumed on the velocity discontinuity surface, and the power generated by backward slip and forward slip were proposed. According to the upper-bound method, the rolling force of horizontal roll and two vertical rolls could be obtained. For verifying the theoretical model, the universal rolling experiments of 18 kg/m light rail was accomplished in Yanshan University Rolling Laboratory, and the experimental data of 60 kg/m heavy rail universal rolling were obtained from the Anshan Iron and Steel Group Corporation. Compared with the experimental data, the theoretical results of rolling force for 18 kg/m light rail and 60 kg/m heavy rail universal rolling were somewhat greater than experimental data, but in general did not exceed them by 15%. Thus, the simplified model was reliable and feasible for presetting and optimizing the parameters of rolling technology according to the upper-bound method.

Ray expansions of solutions around fronts as a shock-fitting tool for shock loading simulation

In shock loading computations based on an implicit finite-difference scheme, the surfaces of velocity discontinuity and the discontinuity sizes are determined by computing an asymptotic (ray) expansion of the solution behind the front surfaces at every time step. The method for constructing ray expansions is based on a recurrence formulation of the geometric and kinematic consistency conditions for discontinuities of the derivatives of functions that are discontinuous on a moving surface. The algorithm is illustrated by computing a simple example of the out-of-plane motion of an incompressible elastic medium.