Slip Line Method Research Articles

Considerations about Load Inclination Effect on Seismic Bearing Capacity Factors of Strip Foundations

Assessment of seismic bearing capacity of strip foundations has been the concern of only few researchers in the field of geotechnical earthquake engineering during the last decades. The previous works have mostly assumed that in the presence of a horizontal earthquake acceleration (kh.g) besides the body force due to gravity, the ratio of the shear stresses to the normal stresses along the ground surface is equal to kh and the load inclination beneath the footing with respect to vertical equals to tan-1(kh). But this assumption does not seem to be always correct. For example in the case of multi span frames where only some spans may be braced and the support conditions at the end of the columns may not be uniform, the load inclination beneath some footings may not be equal to tan-1(kh). This paper presents the seismic bearing capacity factors Nγ, Nq and Nc of a rigid strip foundation resting on a horizontal ground surface, when the load inclination beneath the footing is not equal to tan-1(kh). All calculations are executed by the well known stress characteristic method, commonly referred to as the slip-line method. It is assumed that the soil behaves rigid perfectly plastic and obeys the well known Mohr-Coulomb failure criteria under plane strain conditions. It is assumed that the state of the stress everywhere in the neighborhood of the foundation is at failure. It is also assumed that everywhere along the surface of the foundation, the ratio of the shear to the normal stress is equal to the horizontal earthquake acceleration coefficient (kh). The Nγ parameter was calculated based on the both-sides failure mechanism, whereas the Nq and Nc parameters were calculates based on the single-side failure mechanism. The Nc parameter for cohesive media was calculated using the rule of equivalent states. Some graphs and tables are presented that could be used in order to assess the seismic bearing capacity factors of rigid strip foundations obtained by the stress characteristic method for various load inclinations. It is shown that the seismic bearing capacity factors, just like the static bearing capacity factors, increase considerably with reduction in the load inclination and vice versa. Further, this important outcome of some previous research works has been confirmed once again that the earthquake's effect on each of the body force as well as the overburden is as important as the earthquake's effect on the load inclination which should not be neglected. It is also shown that because of the non-symmetric nature of the seismic loading, the disposal of the characteristic lines as well as the pressure distribution below the foundation is non-symmetric. In other words, the foundation would be subjected to a moment in addition to the vertical and horizontal reactions, as a result of the non-symmetric soil pressure distribution during earthquakes.

Design of geocell reinforcement for supporting embankments on soft ground

The methods of design available for geocell-supported embankments are very few. Two of the earlier methods are considered in this paper and a third method is proposed and compared with them. In the first method called slip line method, plastic bearing failure of the soil was assumed and the additional resistance due to geocell layer is calculated using a non-symmetric slip line field in the soft foundation soil. In the second method based on slope stability analysis, general-purpose slope stability program was used to design the geocell mattress of required strength for embankment. In the third method proposed in this paper, geocell reinforcement is designed based on the plane strain finite element analysis of embankments. The geocell layer is modelled as an equivalent composite layer with modified strength and stiffness values. The strength and dimensions of geocell layer is estimated for the required bearing capacity or permissible deformations. These three design methods are compared through a design example. It is observed that the design method based on finite element simulations is most comprehensive because it addresses the issue of permissible deformations and also gives complete stress, deformation and strain behaviour of the embankment under given loading conditions.

The Bearing Capacity Analyses of Soil beneath the Blade of Circular Cassion

Based on the assumption of pointed blade of circular cassion, the formulas of the bearing capacity of soil beneath the blade are presented by the slip line method. The values of Nγ are not too sensitive to change for the soil’s internal friction angle, and are significantly greater than those obtained by Terzaghi’s and Berezantsev’s formulas. They decrease with an increase in the value of blade embedment depth and the taper angle.

Analytical study of maximal tapping torque during forming screw process

Forming screws lead to a bolted assembly forming the threads of the work-piece by displacing material. The automotive industry is one of the largest markets for forming screws, which have been developed in the last ten years. One of the problems facing the designer of bolted assemblies using forming screws is the knowledge of the forming torque during the process. This study proposes an analytical model based on a previous study concerning radial penetration of a rigid acute wedge into a perfectly plastic material. The method uses the slip-line method with a model taking into account the interaction between two consecutive formed threads and enables the mean pressure on the thread flank to be obtained as a function of the formed thread height. From these results and a knowledge of the forming screw end geometry, an analytical method is proposed for the maximum forming torque, the physical phenomena of the displacement of material in the tapping process is illustrated and the significantly influential parameters highlighted. An experimental study seeks to determine the optimal lead hole on the work piece for an M8 screw. This sets limits on the forming torque and conserves a sufficiently well formed thread shape for good stripping resistance. The experimental results are compared favorably with the results of the analytical study in order to validate the forming screw model.

The effect of the intermediate principal stress on the ultimate bearing capacity of a foundation on rock masses

In this paper, the strength envelope of rock masses is considered to follow a non-linear unified strength criterion that considers the effect of the intermediate principal stress. A unified slip line method for resolving the differential equation system that governs the stress field is established to research the ultimate bearing capacity, which can be adapted for a wide variety of rock masses. Through this new theory, a suitable characteristic method for rock masses of interest can be obtained, and the relationships between different types of characteristic methods are revealed. The characteristic methods are based on different strength criteria, such as the Hoek–Brown criterion, modified Hoek–Brown criterion, or non-linear twin-shear strength criterion; each of these criteria is a special case of the proposed theory. Moreover, a series of new characteristic methods can be easily derived from it. In terms of the proposed theory, the ultimate bearing capacity with the Hoek–Brown criterion forms the lower bound; the ultimate bearing capacity with a non-linear twin-shear strength criterion forms the upper bound; and a series of the ultimate bearing capacities ranging between these two bounds may be derived from a non-linear unified strength criterion. A comparison is made between existing numerical analyses and non-linear finite element analysis solutions with the Drucker–Prager strength criterion. It is shown that the present results are in good agreement with existing numerical analyses and non-linear finite element analysis solutions with the Drucker–Prager strength criterion.

Axi-symmetric active earth pressure for layered backfills obtained by the slip line method

The axi-symmetric active earth pressure for layered backfills was investigated using the slip line method. Due to different soil properties, the interfaces of the soil layers were considered as discontinuity surface. Accordingly, the change of the major principal direction was obtained and a new computation scheme was proposed to deal with the discontinuity in the calculation, finally the slip line method was extended to layered backfills. Results indicate that, the major principal direction, as well as the earth pressure, has a finite jump on passing the soil interfaces. The magnitude of the jump depends on the soil properties mainly. Generally, the active earth pressure when a strong layer is overlying a weak layer is much larger than that when the weak layer is lying on the strong layer. The present solution can be reduced to plane strain case and it has been compared with Rankine’s and Coulomb’s results, and a good agreement is observed.

What is the final shape of a viscoplastic slump?

The final shape of a two-dimensional viscoplastic slump is constructed assuming that the fluid yielded everywhere in its passage to the final state and that the yield stress dominates except in a viscous boundary layer at the base. These assumptions reduce the problem to a related one in plasticity theory. Two methods are presented. First, an asymptotic expansion based on small aspect ratio, ϵ , is used to build an analytical solution valid to third order in ϵ . Second, a slipline method is used to construct slump shapes for arbitrary aspect ratio. The slipline theory exposes flaws in the assumption that the fluid yields everywhere, and rigid plug zones must be inserted in the solution to match all the boundary conditions. The results are compared to a set of experiments with Carbopol in which fluid slumps down a channel. Care is taken to ensure that the width of the channel, any slip over the walls, and the mechanism of emplacement are not important. Despite this, the experiments and theory are not in particularly good agreement, suggesting that some of the theoretical assumptions are invalid.

Analytical solution of general axisymmetric active earth pressure

AbstractThe active earth pressure of circular excavation is expected to be smaller than the plane strain active earth pressures. Using the plane strain active earth pressure coefficients in design is obviously a conservative approach. In this paper, an analytical solution is developed to determine the active earth pressure on inclined rough circular walls with sloping ground surface based on the results of slip line method. Two types of failure profiles are assumed for the slip lines behind the circular wall. The active earth pressures are then obtained by solving the differential equation in the direction of slip lines. The analytical solutions are compared with the results using the numerical formulation by Cheng et al. (Int. J. Geomech. 2007; 7(1):1–15). It is found that the difference between the analytical solution and the numerical solution is small when the inclination of the ground surface and the wall friction angle are small. Finally, design tables of the active earth pressure coefficients for vertical circular retaining walls with various angles of ground surface and wall friction angle are presented. Copyright © 2008 John Wiley & Sons, Ltd.

Axi-symmetric active earth pressure obtained by the slip line method with a general tangential stress coefficient

The axi-symmetric active earth pressure problem has been investigated by several researchers. Most have suggested that it is reasonable to adopt the Haar & von Karman hypothesis in the slip line method. In this study, a general tangential stress coefficient is used instead of the Haar & von Karman hypothesis in the slip line method, and its effect on the earth pressure is discussed. We find that the hypothesis is reasonable when the soil behind the wall fails completely. Using earth pressures when the soil behind the wall begins to fail is better in practice. The earth pressure is expressed with earth pressure coefficients and the coefficients are presented in figures for practical use.

A generalized slip line solution to the active earth pressure on circular retaining walls

The slip line method was extended to calculate the active earth pressure on circular retaining walls in general cases. The slope of the backfill, wall friction and wall inclination were taken into account in the calculation. Two solutions were proposed and a computer program was coded for each one. By parametric study, it is observed that, the slope of the backfill, wall friction and wall inclination influences the results greatly, especially for the wall inclination. The comparison with existing solutions was made and a good agreement was observed. The generalized slip line solution is found useful in determining the active earth pressure in general cases, and can be referred to for design purpose.

Solution of the bearing capacity problem by the slip line method

The authors have derived the bearing capacity factor Nqon sloping ground based on the slip line method and have verified it using finite difference analysis. The bearing capacity factor Nγon sloping ground is obtained by an iterative finite difference solution of the slip line equations, and the results are close to those from laboratory testing. Design figures for Nγon sloping ground are prepared for rapid engineering use. For earthquake loading, the authors have applied the concept of equivalent horizontal acceleration and have derived Nqand Nγbased on slip line analysis.Key words: bearing capacity, slip line, finite difference, sloping ground, earthquake.

Analysis of the Ductile Strength of Welds Weakened by Notches in Longitudinally Welded Pipes of Large Diameter by the Method of Slip Lines

We study the stressed state of an inhomogeneous welded joint containing an interlayer of a more high-strength material under the action of tensile loads. It is assumed that the contact boundary has a cracklike defect. We deduce analytic dependences of the fracture load on the mechanical inhomogeneity of the joint, thickness of the interlayer, and depth of the defect. The maximum depth of the defect to which the joint is insensitive is established. The proposed theoretical results are innovative and agree with the experimental data obtained for specimens cut out from a pipe of 14KhGS steel.

Seismic lateral earth pressure coefficients for c– φ soils by slip line method

In this paper, the author proposes a rotation of axes in the solution of slip line equations for determination of lateral earth pressure with the presence of seismic loading under general conditions. A general c– φ soil system under a general condition can be considered so that K pγ, K pc, K pq, K aγ, K ac and K aq can all be determined. For passive pressure, K pc is found to be much greater than K pγ and the effect of seismic coefficient on passive pressure is relatively small as compared with the corresponding situation for active pressure. The use of iterative analysis is found to be useful for passive pressure determination but unimportant for active pressure determination.

Influence of anisotropic damage on ultimate load of plane strain problems for foundation

Abstract The solutions of ultimate load problems are derived by the slip line method coupling the anisotropic damage under plane strain condition. Application is made to flat obtuse-angle wedge. Bearing capacity for the flat obtuse-angle wedge is obtained. Influences of anisotropic damage on bearing capacity factors of shallow and deep foundations, earth pressure of retaining wall are analyzed in this paper. For deep and shallow foundations, bearing capacity factors, N q decrease with the increase in damage variables, D 1 and D 2 . When D 1 is constant, the factor of active earth pressure, N aq increases with the increase in D 2 while the factor of passive earth pressure, N pq decreases with the increase in D 2 . When D 2 is constant, N aq decreases with the increase in D 1 while N pq increases with the increase in D 1 .

Slip line solutions, deformation mode and loading history during equal channel angular extrusion

Equal channel angular extrusion (ECAE) is analyzed by the slip line method. Full solutions are considered in function of contact friction, backpressure and tool design. It is shown that variations in boundary conditions have a moderate effect on equivalent strains but the strain distribution depends noticeably on friction uniformity and the channel geometry. The characteristic deformation mode corresponds to single shear or to sequence of shears along slip lines. The processing mechanics plays an especially important role in the loading history of material elements. During one pass, monotonic loading is realized only in the cases of zero and maximum friction. For intermediate friction, the deformation history is composed by cross loading into three divergent directions. Multi pass ECAE via different routes is linked with a grain refinement mechanism under severe plastic deformation.

Reappraisal of Size Effect of Bearing Capacity from Plastic Solution

A simplified method for estimating the ultimate bearing capacity of surface footings on sand is described with special attention to the dependency of the angle of internal friction of sand on confining stress. An extended slip line method is developed, in which the dependency of the angle of internal friction on the confining stress is formulated from results of conventional triaxial compression tests for various sands. Based on results from a comprehensive series of calculations employing the extended slip line method, the writers reappraise size effects on bearing capacity and investigate the relationship between strength parameters of sand and size effects on bearing capacity. A modified formula and several diagrams that provide a simple estimation method are proposed to consider size effects on bearing capacity. A comparison between estimations using the formula and ultimate bearing capacities measured from several series of centrifuge tests demonstrates the practicability of the proposed method for both strip and circular footings.

Kinematics of the surface layer in the process of drawing with deformation

By the method of hardness testing, it is experimentally demonstrated that, in the process multi-cycle treatment, the gradient of hardening of the surface layer of a workpiece is practically perpendicular to the surface. The engineering solution of the problem of stressed state in the contact zone is obtained by the method of slip lines. The accuracy of this solution is estimated. The strained state is studied by the method of dividing grids. On the basis of an analysis of the history of deformation, we deduce a simple engineering formula for the evaluation of the elapsed plasticity limit according to the Kolmogorov criterion.

A method for calculating the stress–strain state in the general boundary-value problem of metal forming—part 1

To simulate metal-forming processes, one has to calculate the stress–strain state of the metal, i.e. to solve the relevant boundary-value problems. Progress in the theory of plasticity in that respect is well known, for example, via the slip-line method, the finite element method, etc.) , yet many unsolved problems remain. It is well known that the slip-line method is scanty. In our opinion the finite element method has an essential drawback. (No one is against the idea of the discretization of the body being deformed and the approximation of the fields of mechanical variables.) The results of calculation of the stress state by the FEM do not satisfy Newtonian mechanics equations (these equations are said to be softened, i.e, satisfied approximately) and stress fields can be considered poor for solution of the subsequent fracture problem. We believe that it is preferable to construct an approximate solution by the FEM and soften the constitutive relations (not Newtonian mechanics equations) , especially as, in any event, they describe the rheology of actual deformable materials only approximately. We seem to have succeeded in finding the solution technique. Here we present some new results for solving rather general boundary-value problems which can be characterized by the following: the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; non-isothermal flow; rapid flow, with inertial forces; a non-stationary state; movable boundaries; alternating and non-classical boundary conditions, etc. Solution by the method proposed can be made in two stages: (1) integration in space with fixed time, with an accuracy in respect of some parameters; (2) integration in time of certain ordinary differential equations for these parameters. In the first stage the method is based on the principle of virtual velocities and stresses. It is proved that a solution does exist and that it is the only possible one. The approximate solution softens (approximately satisfies) the constitutive relations, all the rest of the equations of mechanics being satisfied precisely. The method is illustrated by some test examples.

A general algorithm for plastic flow simulation by finite element limit analysis

Limit analysis has been rendered versatile in many structural and metal forming problems. In metal forming analysis, the slip-line method and the upper bound method have filled the role of limit analysis. As a breakthrough of the previous work, a computational approach to limit solutions is considered as the most challenging area. In the present work, a general algorithm for limit solutions of plastic flow is developed with the use of finite element limit analysis. The algorithm deals with a generalized Hölder inequality, a duality theorem, and combined smoothing and successive approximation in addition to a general procedure for finite element analysis. The algorithm is robust such that from any initial trial solution, the first iteration falls into a convex set which contains the exact solution (s) of the problem. The idea of the algorithm for limit solutions is extended from rigid⧹perfectly plastic materials to work-hardening materials by the nature of the limit formulation, which is also robust with numerically stable convergence and highly efficient computing time.

Optimum Blank Shapes for Prismatic Cup Drawing—Consideration of Friction and Material Anisotropy

A theory employing the concept of potential flow and an easily implemented numerical method are developed. It can offer a preliminary prediction of the optimum blank shape with a little computational effort. The effects of the material anisotropy and the interfacial friction between the workpiece and blank holder on the material flow can be modeled by superimposing a “sink” or “source” term in the potential field. Variations in the cup wall thickness are considered. Both convex and non-convex punch profiles are studied. Generally speaking, the contours of the present theory are smoother than the solutions from the slip-line method. The discrepancy of the optimum blank contours between the plane strain and anisotropic cases is more significant for polygon and irregular cup drawing. The influence of friction on the optimum blank contour is not negligible.