Numerical Limit Analysis Research Articles

Three-dimensional lower-bound solutions for the stability of plate anchors in sand

Soil anchors are commonly used as foundation systems for structures that require uplift or lateral resistance. These types of structure include transmission towers, sheet pile walls and buried pipelines. Although anchors are typically complex in shape (e.g. drag or helical anchors), many previous analyses idealise the anchor as a continuous strip under plane strain conditions. This assumption provides numerical advantages, and the problem can be solved in two dimensions. In contrast to recent numerical studies, this paper applies three-dimensional numerical limit analysis and axisymmetrical displacement finite element analysis to evaluate the effect of anchor shape on the pullout capacity of horizontal anchors in sand. The anchor is idealised as either square or circular in shape. Results are presented in the familiar form of break-out factors based on various anchor shapes and embedment depths, and are also compared with existing numerical and empirical solutions.

Applications of numerical limit analysis (NLA) to stability problems of rock and soil masses

This work presents formulations and results obtained with computer implementations of an alternative to the more standard techniques for the determination of the state of collapse of geotechnical structures in rock or soil masses. Examples of normally available and used techniques for those purposes are limit equilibrium based procedures and elasto-plastic finite elements. As an alternative to these techniques, the present paper describes Numerical Limit Analysis (NLA). The fundamentals for limit analysis, summarized in the so-called bound theorems, have been known for decades. Analytical solutions obtained with limit analysis are however limited in scope and are seldom used in the engineering practice. NLA on the other hand, by solving the limit analysis equations through numerical methods are general and applicable to a wide range of problems. The paper presents a discussion on available alternatives for the formulation of NLA specialized for the determination of collapse load factors of geotechnical structures in/on rock (fractured or not) and soil masses. Rock masses in particular are modelled as standard continua, Cosserat equivalent continua and true discontinua formed by discrete blocks. Finite elements are used for the solution of NLA equations of standard continua and Cosserat continua. The paper presents derivation of the pertinent equations, the numerical formulations used and details of their numerical implementation in computer programs. Attempt was made to validate all the implementations through existing analytical solutions. The obtained results permit to state that NLA is a promising and very often advantageous numerical technique to establish collapse states of geotechnical structures in rock and soil masses.

Stability of Inclined Strip Anchors in Purely Cohesive Soil

Soil anchors are commonly used as foundation systems for structures requiring uplift resistance such as transmission towers, or for structures requiring lateral resistance, such as sheet pile walls. To date, most anchor studies have been concerned with either the vertical or horizontal uplift problem. In many instances, anchors are placed at inclined orientations depending on the type of application and loading (e.g., transmission tower foundations). However, the important effect of anchor inclination has received very little attention by researchers. This paper applies numerical limit analysis and displacement finite-element analysis to evaluate the stability of inclined strip anchors in undrained clay. Results are presented in the familiar form of breakout factors based on various anchor geometries.

Numerical limit analysis solutions for the bearing capacity factor Nγ

Numerical limit analyses are applied to evaluate the self-weight bearing capacity factor for a rigid surface footing with a smooth or rough interface. To isolate the effect of the self-weight on the bearing capacity, the soil is modeled as a cohesionless frictional Mohr–Coulomb material. Assuming an associated flow rule, the true collapse load is bracketed to within 3.42% by computing rigorous lower and upper bound solutions using recently developed limit analysis methods [Lyamin and Sloan, 2002a. Upper bound limit analysis using linear finite elements and non-linear programming. International Journal for Numerical and Analytical Methods in Geomechanics 26 (2), 181; Lyamin and Sloan, 2002b. Lower bound limit analysis using linear finite elements and non-linear programming. International Journal for Numerical Methods in Engineering 55 (5), 573]. With these sharp bounds, the average bearing capacity factor provides an excellent estimate of the exact bearing capacity factor that can be used with confidence for design purposes. The numerical formulations of both the upper and the lower bound theorems are obtained by means of finite elements and solved using efficient non-linear mathematical programming schemes. The solutions, presented in the form of graphs, are compared against existing semi-empirical expressions and numerical solutions. An approximate analytical expression for N γ is suggested at the end of the paper.

Bearing capacity of a cohesive-frictional soil under non-eccentric inclined loading

This paper applies numerical limit analysis to evaluate the bearing capacity of a strip footing, subjected to a non-eccentric inclined load, resting on a ponderable cohesive-frictional soil. Accurate lower and upper bounds are calculated rigorously using finite elements and nonlinear programming. By adopting typical values for the friction angle, the inclination angle, and a dimensionless parameter related to the self-weight, most cases of practical interest are treated. The results are presented in the form of tables. As the gap between the bounds does not exceed 3%, the average limit load provides a good estimate of the exact ultimate load and can be used with confidence for design purposes. The numerical results are compared with ultimate loads predicted by the theories of Meyerhof, Hansen and Vesić. The comparison shows that the Meyerhof and Vesić theories results are unconservative for inclined loading. In particular, the inclination factors from the Meyerhof theory appear to be inaccurate, whilst the Vesić theory does not take proper account of the self-weight. For a ponderable soil under vertical or inclined loading, the collapse mechanism from the rigorous upper bound analysis is different to that assumed by previous authors.

Lower bound solutions for bearing capacity of jointed rock

This paper applies numerical limit analysis to evaluate the bearing capacity of strip footings on an anisotropic, homogenous material. While solutions exist for footings on jointed rock [Proceedings of the Third International Conference on Computational Plasticity 2 (1992) 935; Proceedings of the International Conference on Structural Foundations on Rock 3 (1980) 83] little work has been done on the effect of variation of joint strength and relative joint orientation (for cases with two or more joint sets). From a macroscopic point of view, many jointed materials such as rock may be assumed to have anisotropic homogenous properties. The overall behaviour of a jointed rock mass is controlled by the mechanical properties of the intact rock as well as the strength and orientation of the discontinuities. The formulation presented here assumes plane strain conditions, and makes use of the Mohr–Coulomb failure criterion. In order to utilise the lower bound theorem of classical plasticity two basic assumptions must be made. Firstly the material is assumed to exhibit perfect plasticity and obey an associated flow rule without strain hardening or softening. Secondly, it is assumed that the body undergoes only small deformations at the limit load so that the effect of geometry changes is small. By using a Mohr–Coulomb approximation of the yield surfaces, the proposed numerical procedure computes a statically admissible stress field via linear programming and finite elements. The stress field is modelled using linear three-noded triangular elements and allows statically admissible stress discontinuities at the edges of each triangle. By imposing equilibrium, yield and stress boundary conditions on the unknown stresses, an expression of the collapse load is formed which can be maximized subject to a number of linear constraints on the nodal stresses. As all the requirements are met for a statically admissible stress field, the solution obtained is a rigorous lower bound on the actual collapse load. An extensive parametric analysis is presented to investigate the effects of joint orientation and strength properties on the overall bearing capacity of jointed rock. The analysis overcomes the limitations of previous solutions in that non-orthogonal joint sets are considered. Consequently this work represents an invaluable tool for designers.

Stability of seismically loaded slopes using limit analysis

Numerical limit analysis is used to assess the stability of slopes subjected to seismic loading. The soil is assumed to follow the Mohr–Coulomb failure criterion. The lower and upper bound theorems are formulated as linear problems to be solved using linear programming techniques. Based on finite element discretisation of the slope, the velocity field is optimised to find the lowest upper bound, and the stress field is optimised to obtain the highest lower bound. Limit equilibrium computations and log-spiral upper bound solutions were also performed for comparison purposes. Additionally, finite element analyses were done for selected cases. Results from the limit equilibrium and finite element methods are in excellent agreement with the rigorous lower and upper bounds for all cases studied. The slip surfaces obtained from both the limit equilibrium and log-spiral upper bound methods lie within the plastic zones obtained for the slopes from both finite element and numerical limit analysis. Plots are presented of the horizontal pseudo-static acceleration ratio kc = ac/g required to cause collapse of simple homogeneous slopes as a function of the slope inclination and shear strength parameters.

Three-Dimensional Lower Bound Solutions for Stability of Plate Anchors in Clay

Soil anchors are commonly used as foundation systems for structures that require uplift or lateral resistance. These types of structures include transmission towers, sheet pile walls, and buried pipelines. Although anchors are typically complex in shape (e.g., drag or helical anchors), many previous analyses idealize the anchor as a continuous strip under plane strain conditions. This assumption provides numerical advantages and the problem can be solved in two dimensions. In contrast to recent numerical studies, this paper applies three-dimensional numerical limit analysis to evaluate the effect of anchor shape on the pullout capacity of horizontal anchors in undrained clay. The anchor is idealized as either square, circular, or rectangular in shape. Estimates of the ultimate pullout load are obtained by using a newly developed three-dimensional numerical procedure based on a finite-element formulation of the lower bound theorem of limit analysis. This formulation assumes a perfectly plastic soil model with a Tresca yield criterion. Results are presented in the familiar form of break-out factors based on various anchor shapes and embedment depths, and are also compared with existing numerical and empirical solutions.

Stability of plate anchors in undrained clay

Soil anchors are commonly used as foundation systems for structures requiring uplift resistance, such as transmission towers, or for structures requiring lateral resistance, such as sheet pile walls. To date, the design of these anchors has been largely based on empiricism. This paper applies numerical limit analysis to rigorously evaluate the stability of vertical and horizontal strip anchors in undrained clay. Rigorous bounds on the ultimate pull–out capacity are obtained by using two numerical procedures that are based on finite element formulations of the upper and lower bound theorems of limit analysis. These formulations follow standard procedure by assuming a rigid perfectly plastic clay model with a Tresca yield criterion, and generate large linear programming problems. By obtaining both upper and lower bound estimates of the pull–out capacity, the true pull–out resistance can be bracketed from above and below. Results are presented in the familiar form of break–out factors based on various soil strength profiles and geometries, and are compared with existing numerical and empirical solutions.

Stability of plate anchors in undrained clay

Soil anchors are commonly used as foundation systems for structures requiring uplift resistance, such as transmission towers, or for structures requiring lateral resistance, such as sheet pile walls. To date, the design of these anchors has been largely based on empiricism. This paper applies numerical limit analysis to rigorously evaluate the stability of vertical and horizontal strip anchors in undrained clay. Rigorous bounds on the ultimate pull–out capacity are obtained by using two numerical procedures that are based on finite element formulations of the upper and lower bound theorems of limit analysis. These formulations follow standard procedure by assuming a rigid perfectly plastic clay model with a Tresca yield criterion, and generate large linear programming problems. By obtaining both upper and lower bound estimates of the pull–out capacity, the true pull–out resistance can be bracketed from above and below. Results are presented in the familiar form of break–out factors based on various soil strength profiles and geometries, and are compared with existing numerical and empirical solutions.

Rigorous plasticity solutions for the bearing capacity of two-layered clays

This paper applies numerical limit analysis to evaluate the undrained bearing capacity of a rigid surface footing resting on a two-layer clay deposit. Rigorous bounds on the ultimate bearing capacity are obtained by employing finite elements in conjunction with the upper and lower bound limit theorems of classical plasticity. Both methods assume a perfectly plastic soil model with a Tresca yield criterion and generate large linear programming problems. The solution to the lower bound linear programming problem is obtained by modelling a statically admissible stress field, whereas the upper bound problem is solved through modelling a kinematically admissible velocity field. Results from the limit theorems typically bracket the true collapse load to within approximately 12%, and have been presented in the form of bearing capacity factors based on various layer properties and geometries. A comparison is made between existing limit analysis, empirical and semi-empirical solutions. This indicates that the latter can overestimate or underestimate the bearing capacity by as much as 20% for certain problem geometries. Dans cet exposé, nous servons d'analyses limites numériques pour évaluer la capacité porteuse non drainée d'une assise à surface rigide reposant dur un dépöt argileux à deux couches. On obtient des limites rigoureuses pour la capacité porteuse ultime en employant des é1éments finis en conjonction avec les théorèmes de limite à borne supérieure et inférieure de la plasdcité classique. Les deux méthodes supposent un modéle de sol parfaitement plastique avec un critère d'élasdcité Tresca et entraïnent de gros problèmes de programmation linéaire. On obtient la solution au problème de programmation linéaire de limite inférieure en faisant la maquette d'un champ de contrainte admissible en statistique tandis que le problème de limite supérieure est résolu par la mise en maquette d'un champ de vélocité admissible en cinétique. De manière typique, les résultats de théorèmes limites cernent dans une marge d'environ 12% la vraie charge d'affaissement et sont présentés sous forme de facteurs de capacité porteuse basés sur les propriétés et géométries des diverses couches. Nous comparons les analyses limites existantes, les solutions empiriques et les solutions semiempiriques. Cette comparaison indique que ces dernières peuvent surévaluer ou sousestimer de 20% la capacité porteuse dans le cas de certaines géométries à problème.

Non-Symmetrical Limit Loads on Strip Footings

Limit loads on footings (bearing capacity) can be conveniently analyzed using the kinematic approach of limit analysis. Nonsymmetrical loads on strip footings include the horizontal component of the load and the moment. They can be also represented as load inclination and eccentricity. For the analysis to be sensitive to nonsymmetrical loads, the collapse mechanisms also need to be nonsymmetrical, i. e., they need to allow for footing rotation and horizontal component of displacement. Based on such mechanisms, inclination coefficients for surface strip footings are derived. These coefficients fit the numerical limit analysis results better than the coefficients suggested earlier in the literature. It is also shown that for cohesive-frictional soils the method often used to account for the load eccentricity (where the footing width is reduced by twice-the-eccentricity) yields a bearing capacity lower than that obtained from the kinematic limit analysis. However, this method overestimates the bearing capacity for purely frictional soils, particularly when the surcharge load is small.

Numerical limit analysis of cylindrical shells with part-through slots

A numerical limit analysis is performed for cylindrical shells with part-through slots using a general computational method for limit analysis of 3-D structures. The upper bounds for the limit pressures are given for a comprehensive range of geometric parameters. Some of the calculated results are compared with the results of detailed 3-D elastoplastic finite element analysis and where possible, with existing theoretical, experimental or numerical solutions. The effect of various shapes and sizes of part-through slots on the load-carrying capacity of cylindrical shells is investigated and evaluated. Two kinds of typical failure modes corresponding to different dimensions of slots are studied. Based on the numerical results, a geometric parameter G which combines the slot dimensions and the cylinder geometry is presented. It reflect reasonably the overall effect of slot on the limit load of a cylinder. An empirical formula for estimating the limit pressure of a cylindrical shell with part-through slot is obtained.

Limit analysis and design of containment vessels

In the introduction, the theory of plastic analysis of shells is briefly recalled. Minimum-volume design for assigned load factor at plastic collapse is then considered and optimality criteria are derived for plates and shells of continuously varying or piecewise-constant thickness. In the first part, containers made of metal are examined. Analytical and numerical limit analysis solutions and corresponding experimental results are considered for various types of vessels, including intersecting shells. Attention is given to experimental post-yield behavior. Some tests up to fracture are discussed. New theoretical and experimental results of limit analysis of stiffened cylindrical vessels are presented, in which reinforcing rings are treated as discrete structural element (no smearing out) and due account is taken of their strong curvature. Cases of collapse by instability under internal pressure are pointed out. Minimum-volume design of circular plates and cylindrical shells is then formulated and various examples are presented of sandwich and solid metal structures. Containers of piecewise-constant thickness are given particular attention. Available experimental evidence on minimum-volume design of plates and shells is reviewed and commented upon. The second part deals with reinforced concrete vessels. Cylindrical containers are studied, from both points of view of limit analysis and of limit design with minimum volume of reinforcement. The practical use of the latter solutions is discussed. A third part reviews other loading cases (including cyclic and impact loads) and gives indications on corresponding theories, formulations and solution methods. The last part is devoted to a discussion of the limitations of the methods presented, within the frame of the “limit states” design philosophy, which is first briefly recalled. Considerations on further research in the field conclude the paper.

Numerical limit analysis of perforated plates

AbstractFull limit analysis is preseted of a thin plate under plane stress, perforated with circular holes arranged in a regular penetration pattern. The material is assumed to be elastic–perfectly plastic and to obey the Huber–Mises yield condition together with its associated flow law. The finite element tangential stiffness method is employed. Triangular constant‐strain elements are used, the load being generated by means of constant edge displacements at the periphery of a suitably selected subregion. Load‐displacement diagrams are obtained for various edge displacement programs and then an interaction curve is constructed for the perforated plate considered. Each computational step is associated with a certain plastic zone which develops until the limit state configuration is reached. The plate is provided with equilateral triangular configuration of holes with arbitrary cutout coefficients. The interaction curves are plotted against the Huber–Mises yield condition for the plate without openings.