Slip-line Field Solutions Research Articles

A Slip-Line Field Solution for the Ultimate Bearing Capacity of a Pipeline Under Oblique Loading on a Clayey Seabed

A Slip-Line Field Solution for the Ultimate Bearing Capacity of a Pipeline Under Oblique Loading on a Clayey Seabed

Plane strain wedge indentation revisited

For a plane strain wedge indentation problem, detailed comparisons between slip-line field theory (SLFT) solutions, experimental observations and finite-element analysis results are carried out. Several past remaining questions on this problem are explored. The finite-element computations appear to account for the nature of a modified slip-line field solution (Petryk H. 1980 Journal de Mécanique Appliquée 4 , 255–282) that assumes the existence of an additional fan-shaped slip-line field just above the boundary between yielding and unyielding regions. For obtuse-angled wedges, finite-element computations show that the indentation load could be smaller for a rough wedge than for a smooth wedge. The question that was raised in the literature with respect to the reversed order of required loads for smooth and rough wedges is not necessarily a paradox. It is revealed that with increasing friction coefficient, a distinct deformation mode resembling the dead metal cap solution in the SLFT is established immediately after passing the critical friction coefficient value at which the contact mode theoretically changes from frictional sliding to perfect sticking. The profiles of the bulged-out lip material remain straight in all the finite-element computations for non-hardening materials. Nonlinearity of the lip profiles frequently observed in experiments appears in the presence of strain hardening or strain-rate sensitivity.

Flow-pipe-soil coupling mechanisms and predictions for submarine pipeline instability

The stability of a submarine pipeline on the seabed concerns the flow-pipe-soil coupling, with influential factors related to the ocean waves and/or currents, the pipeline and the surrounding soils. A flow-pipe-soil coupling system generally has various instability modes, including the vertical and lateral on-bottom instabilities, the tunnel-erosion of the underlying soil and the subsequent vortex-induced vibrations (VIVs) of free-spanning pipelines. This paper reviews the recent advances of the slip-line field solutions to the bearing capacity, the flow-pipe-soil coupling mechanism and the prediction for the lateral instability, the multi-physical coupling analysis of the tunnel-erosion, and the coupling mechanics between the VIVs and the local scour. It is revealed that the mechanism competition always exists among various instability modes, e.g., the competition between the lateral-instability and the tunnel-erosion. Finally, the prospects and scientific challenges for predicting the instability of a long-distance submarine pipeline are discussed in the context of the deep-water oil and gas exploitations.

Temperature estimation and slip-line force analytical models for the estimation of the radial forming force in the RARR process of flat rings

In this study, a mathematical model for the prediction of the temperature evolution in the ring during the radial-axial ring rolling process is developed and used, together with the authors’ previous results, to determine analytically the flow stress of the material during process. These results, combined with Hill's slip-line field solution adapted to the RARR process, allow a fast and reasonably precise calculation of the radial forming force, a key parameter at the preliminary stage of the process design. The approach is validated by applying the proposed model to a case available in the literature and comparing the analytical results with those of the laboratory experiment and FEM simulation. Following the successful comparison, the models were applied to a large variety of flat rings, comparing analytical predictions with the results of FEM simulations. The accuracy of the analytical calculation and the reliability of the proposed models, for different ring configuration and process parameters, are presented and discussed.

A general slip-line field solution for the ultimate bearing capacity of a pipeline on drained soils

An accurate evaluation of the ultimate bearing capacity of a cylindrical foundation is crucial for predicting pipe–soil interaction behaviors. A general slip-line field solution is derived for the ultimate bearing capacity of a pipeline on the drained soil obeying Mohr–Coulomb criterion. The slip-line field around the pipeline matches well with the corresponding plastic incremental-displacement field simulated by utilizing finite element analysis. Parametric studies indicate that as the internal friction angle of the soil approaches zero, the derived bearing capacity factors for the pipeline on the drained soil limit to those for the pipeline on the undrained soil obeying Tresca criterion. The bearing capacity factors for a fully-smooth pipeline then limit to those for a conventional rectangular strip-footing while the pipeline embedment approaches zero. Moreover, the dimensionless collapse load increases with increasing the pipeline embedment and the pipe–soil interfacial friction.

Ultimate bearing capacity of a pipeline on clayey soils: Slip-line field solution and FEM simulation

The ultimate bearing capacity of a cylindrical foundation (e.g. a submarine pipeline, circular mooring lines) on clayey soils is analyzed analytically and numerically by employing the slip-line field theory and the plane-strain finite element method respectively. A slip-line field solution is presented, taking into account of circular configuration of the pipe, the pipe embedment, and the pipe–soil interfacial cohesion. The derived bearing capacity factors for a smooth rigid pipe may limit to those for the conventional rectangle-shaped strip footing while the pipe embedment is approaching zero. A plane-strain finite element model is further proposed to simulate the quasi-static process of the pipeline penetrating into the clay soil, in which the contact-pair algorithm and adaptive meshing technique are employed, and the Drucker–Prager constitutive model is used for modeling the soil plasticity. Comparison indicates that the present numerical results match well with the derived slip-line solutions. According to the obtained vertical load–displacement curves, concurrently referring to the plastic strain field and the soil incremental-displacement vector field, the shear failure types and the corresponding collapse loads can be thereby determined for the pipeline foundations on clayey soils.

Slip-line field solution for ultimate bearing capacity of a pipeline on clay soils

A slip-line field solution is presented for the ultimate bearing capacity of the pipeline on a purely-cohesive clay soil, taking into account of circular configuration of the pipe, the pipe embedment, and the pipe-soil interfacial cohesion. The derived bearing capacity factors for a smooth rigid pipe may degenerate into those for the conventional rectangular-shaped strip footing while the pipe embedment approaches zero. Parametric studies indicate that, the pipe-soil interfacial properties have much influence on the bearing capacity for the pipe foundation on clayedy soils.

Slip Line Field Solution for Second Pass in Lubricated 4-High Reversing Cold Rolling Sheet Mill

The development of a possible slip line field (slf) for theoretical calculations of the deforming pressure (load) in a second pass of a lubricated cold rolling sheet mill and validation using values from an aluminium sheet rolling mill was done in this work. This will be relevant in the manufacturing industries providing an easy method for determining necessary applied rolling load. Experimental rolling was carried out to observe the shear lines in the deformation field. Construction of possible slip line field model was developed adhering strictly to assumptions of rigid plastic model. Calculation of the deforming force/load was achieved using Hencky’s equation. Results showed that the load calculations for constructed slip line field using aluminium sheet rolling as an example tallied with values obtained from Tower Aluminium rolling mill. Slip line fields constructed for the second pass described adequately the rolling pressure in the cold rolling process, giving a valid solution of the exact load estimates on comparison with the industrial load values. Roll pressure along the arc of contact rose fairly linearly from the entrance to a maximum at the exit point. This work showed that slf for the first pass in a cold rolling mill cannot be used for subsequent passes; it requires construction of slfs for each pass in the cold rolling process.

表計算によるすべり線場解析

A method of spreadsheet calculation of plane strain slip line field solutions by the matrix operator method is presented. Excel and its add-in software ‘solver′ are used for determining the values of variables satisfying the boundary value problems of slip line fields. Two examples of known slip line fields of smooth die compression and indentation problems are calculated to determine if the Excel calculation can give correct results, by comparing with the previous data obtained by Collins and Dewhurst. Our results are in agreement with Collins and Dewhurst's data, confirming that the Excel calculated values are correct. CAD software is used to draw slip line fields and hodographs on the basis of spreadsheet numerical data and found to be suitable for drawing slip line fields and hodographs easily and rapidly.

An approximate load estimation method for three dimensional metal forming problems

Slip line field solutions have been presented only for plane strain and axisymmetric problems in the literature. In fact due to the very nature of the differential equations for the slip line field theory, it has never been possible to apply the theory to three dimensional problems. In this paper a three dimensional solution is presented for metal forming processes using an approximate load estimation method based on the slip line field theory. This has been done by extending the axisymmetric solution and modifying it in order to apply it to the three dimensional case. Using the idea of stream lines and stream surfaces, the deforming region has been defined. A generic stream surface was developed which was a ruled surface with a three dimensional shape. The slip line field theory could only be applied to flat surfaces in the case of axisymmetric problem. Here using some assumptions an approximate load estimation method was developed so that it was applied to the three dimensional stream surfaces and hence to the deforming region under considerations. This method was applied to the forward extrusion of elliptical sections from round billets. Experiments were also carried out by authors to verify the theory. The results obtained from the new formulations were compared to the results obtained from other analytical, numerical and experimental methods. It was shown that there exist good agreements between these results.

Mathematical slip-line field solutions for ploughing a hard particle over a softer material

Wear mechanisms and friction in metals can be investigated by the analysis of the unit event represented by the interaction of a hard particle or asperity with a softer surface. Effective friction is the result of the interaction of many such asperities which constitute the roughness of the harder of the solid surfaces. Three types of plastic deformation at the metal surface can be identified: ploughing, edge formation and chip formation. Each mode of plastic deformation can be analysed using the slip-line field plasticity theory which requires as inputs the geometry of the hard particle and some information on the interface between the harder and the softer surfaces. The classical and the recent chord solution by Oxley assumes a sharp edge sliding against a metal surface but does not consider a curved roughness profile. However, the profiles of real asperities are more like waves with rounded summits. In the present work a new model for the asperities interaction is shown, using the slip-line field theory to calculate the friction forces, depth of sheared layer, average contact pressure and friction coefficient for a cylindrical hard particle sliding over a softer surface. The theoretical results are presented as friction graphs and maps in which the regions of elastic deformations are shown using the Hertz theory while the region of plastic strains is obtained from the present analysis. Present model results are in good agreement with experimental data obtained by Busquet et al. and are quite different from the Oxley chord model for sliding a circular particle.

Design analysis of ductile failure in dovetail connections

The static plastic collapse of ductile dovetail structures is investigated by three analysis methods: slip-line field (SLF) theory based on a sheet drawing model, finite element limit analysis, and linear elastic finite element analysis with adapted pressure vessel design stress linearization and categorization methods. A range of angles and heights are considered in the investigation. Three experimental test cases are also presented. The limit analysis results are found to give the best comparison with the limited experimental results, indicating similar collapse loads and modes of ductile collapse. The SLF solution is found to give conservative but useful failure loads for small dovetail angles but, at angles greater than 30°, the solution is not generally conservative. The pressure vessel design by the analysis stress categorization procedure was adapted for dovetail analysis and was found to give reasonably conservative collapse loads in most cases. However, the procedure requires the designer to consider a number of different stress classification lines to ensure that a conservative collapse load is identified. It is concluded that the finite element limit analysis approach provides the best and most direct route to calculating the allowable load for the joint and is the preferred method when appropriate finite element analysis facilities are available.

Slipline field modeling of orthogonal machining for a worn tool with elastic effects and adhesion friction at the contact regions

In this paper, two slipline field models are presented for orthogonal machining with a worn tool with a finite flank wear land. Friction at tool–chip and tool–work piece interfaces are assumed to be governed by the adhesion friction law as suggested by [Maekawa, K., Kitagawa, T., Childs, T.H.C., 1997. Friction characteristics at chip–tool interface in steel machining. 23rd Leeds Lyon Symposium in Tribology] It is further assumed that the natural contact length consists of a zone of elastic contact and a zone of plastic contact such that the forces in the elastic contact region and those at the rigid-plastic chip boundary keep the chip in a state of static equilibrium. The relation between the angular range of α- and β-lines within the secondary shear zone is assumed to be given by the linear equation β = m 0 α. The fields are analyzed by the matrix operational procedure developed by [Dewhurst, P., Collins, I.F., 1973. A matrix methods for constructing slipline field solutions to a class of plain strain plasticity problems. Int. J. Numer. Methods Engineering 7, 357–378; Dewhurst, P., 1984. The Coulomb friction boundary value problem in plain-strain slipline field theory. In: First Int. Conf. on Technology of Plasticity (ICTP) Proc. (Advanced Technology of Plasticity), Tokyo, Japan, pp. 1085–1090; Dewhurst, P., 1985. A general matrix operator for linear boundary value problems in slip-line field theory. Int. J. Numer. Methods engineering 21, 169–182]. The limits of the validity of the proposed fields are established using Hill's overstressing criteria. Effect of flank wear on ploughing and cutting forces is studied and their variation with friction parameters examined. It is shown that the peak tool tip pressure for a worn tool is much lower than that for a sharp tool. The results of computation further demonstrate that the force ratio and average tool–chip Coulomb coefficient of friction for a sharp tool decrease with rake angle. But presence of even a small flank wear land may reverse this trend. The results from theoretical analysis are also compared with experiment.

Foundations of cable car towers upon alpine glaciers

This paper presents a design approach for strip footings upon glacier ice. Safety against ultimate limit state is proved by the geotechnical slip-line field solution by Prandtl. Glacier ice at 0°C can be modelled as purely cohesive material. Statistical evaluation of uniaxial compression tests with high strain rate revealed a mean value of the cohesion of 600 kPa and a characteristic value c k = 355 kPa (5% fractile). With a coefficient of variation V c = 0.3, the partial safety factor turns out to be γ c = 1.9. An approximate solution for estimating the creep settlement rate $${\dot{u}}$$ is presented to check the serviceability limit state: $${\dot{u}}=\beta_1 b^{\beta_2}p^{\beta_3},$$ with the width b of the strip foundation, p the foundation pressure and $$\beta_1=8.6 \times 10^{{-5}}\,\hbox{m\, kPa}^{-\beta_3} \hbox{a}^{-1}\,\hbox{m}^{1-\beta_2}, \beta_2=0.92, \beta_3=1.74$$ for ice at 0°C. Experiences on Stubai glacier with grate shaped footings showed that creep settlements occurring per year due to maximum foundation pressures 250 kPa did not influence the operation and the maintenance of the cable cars.

A class of slipline field solutions for metal machining with sticking–slipping zone including elastic contact

In the present investigation, slipline field solutions for orthogonal machining with elastic contact are presented when the plastically stressed region in the chip/tool contact length consists of both slipping and sticking zones. The interface friction in the elastic as well as plastic contact zone is assumed to obey Coulomb’s law. Solutions are obtained both for parabolic and exponential normal stress distribution in the elastic contact region. A constant ratio of elastic to plastic contact length is assumed. Results are presented for variation of cutting forces, cutting ratio, chip-curl radius, contact length, the sticking length and sticking ratio with variation in rake angle and interface friction. The variation of normal and shear stresses at the chip/tool interface with variation in coefficient of friction as well as rake angles is predicted. Results from the theoretical analysis are compared with experimental values reported in the literature.

A modified upper bound approach to limit analysis for plane strain deeply cracked specimens

The classical upper bound approach of limit analysis is based on assumption of rigid blocks of deformation that move between lines of tangential displacement discontinuity. This assumption leads to considerable simplification but often at cost of higher estimate of the actual load. Moreover, in many cases, it does not give a correct shape of the plastic field. In order to overcome these limitations a modified upper bound approach is proposed in this article. The proposed approach is basically an energetic approach but unlike the classical upper bound approach it is capable of including presence of statically governed stress field. As an application, of proposed approach, theoretical plane strain solutions are presented for deeply cracked fracture mechanics specimens (single edge cracked specimen in pure bending – SE (PB), single edge cracked specimen in three-point bending – SE (B), and compact tension – C (T) specimens). Plane strain plasticity problem in rigid elastic–plastic mono-material (homogeneous) was solved to evaluate useful parameters like limit load, plastic eta function ( η p) and plastic rotation factor ( r p) and in bi-material (mismatch welds) to evaluate mismatch limit load, for deeply cracked specimens. New kinematically admissible velocity fields are proposed for SE (B) and C (T) specimens. Proposed theoretical solutions were confirmed by classical slip-line field solutions, wherever available, and by detailed elastic–plastic finite element analysis with Von-Mises yield criterion. Good agreement was found between proposed solutions and results obtained from the classical slip-line field theory and finite element analysis.

A SLIPLINE FIELD ANALYSIS OF FREE-CHIP ORTHOGONAL MACHINING WITH ADHESION FRICTION AT RAKE FACE

The paper presents slipline field solutions for metal machining assuming adhesion friction at the chip-tool interface. The field is of “indirect” type and is analyzed by the matrix method suggested by Dewhurst, Dewhurst and Collins. The range of validity of the proposed solutions is examined from the consideration of overstressing of rigid vertices in the assumed rigid regions. Rake angle and rake friction are found to be the most important variables that influence the deformation process in machining. Variation of cutting forces, chip thickness ratio, chip curvature and contact length with rake angle and friction parameters is investigated. It is observed that cutting and thrust forces and cutting ratio decrease as rake angle increases but increase as coefficient of friction increases. However, tool-chip contact length decreases as rake angle increases. As a result the average normal and shear stresses on the tool face increases as rake angle increases though, the cutting and thrust forces decrease. Results indicate that friction coefficient cannot be uniquely determined by the rake angle alone, but may have a range of allowable values for a particular value of rake angle. The theoretical results are compared with experimental data available in literature and also with those obtained by the authors from orthogonal cutting tests.

굽힘 하중이 작용하는 비대칭노치시편의 완전소성해석

This paper proposes slip line fields for bending of unequally notched specimens in plane strain that have a sharp crack in one side and a sharp V-notch in the other side. Depending on the back angle, two slip line fields are proposed, from which the limit moment and crack tip stress fields are obtained as a function of the back angle. Excellent agreement between slip line field solutions with those from detailed finite element limit analysis based on non-hardening plasticity provides confidence in the proposed slip line fields. One interesting point is that, for the unequally notched specimen, the difference between the crack tip triaxial stress for tension and that for bending increases significantly with increasing the back angle. This suggests that such a specimen could be potentially useful to investigate the crack tip constraint effect on fracture toughness of materials. In this respect, the possibility of designing a new toughness testing specimen with varying crack tip constraint is discussed.

Slip Line Field Analysis of Unequally Grooved Specimens in Bending and Its Implication to Constraint Toughness Testing

This paper proposes slip line fields for bending of unequally grooved specimens that has a sharp crack in one side and a sharp V-notch in the other side. Depending on the back angle, two slip line fields are proposed, from which the limit moment and crack tip stress fields are obtained as a function of the back angle. Excellent agreements between slip line field solutions with those from detailed finite element limit analysis based on non-hardening plasticity provide confidence in the proposed slip line fields. Based on these results, possibility of designing a new toughness testing specimen with varying crack tip constraint is discussed.

Fully plastic analyses of unequally notched specimens in bending

This paper proposes slip line fields for bending of unequally notched specimens in plane strain, that have a sharp crack in one side and a sharp V-notch in the other side. Depending on the back angle, two slip line fields are proposed, from which the limit moment and crack tip stress fields are obtained as a function of the back angle. Excellent agreement between slip line field solutions with those from detailed finite element limit analysis based on non-hardening plasticity provides confidence in the proposed slip line fields. One interesting point is that, for the unequally notched specimen, the difference between the crack tip triaxial stress for tension and that for bending increases significantly with the increase in the back angle. This suggests that such a specimen could be potentially useful to investigate the crack tip constraint effect on fracture toughness of materials. In this respect, the possibility of designing a new toughness testing specimen with varying crack tip constraint is discussed.