Lower Bound Limit Analysis Research Articles

Finite elements limit analysis formulation using the power type yield criterion for plane strain and axisymmetric stability problems

The current research employs a power type yield criterion which accounts for the dependency of the shear strength parameters, namely, cohesion and internal friction, on stress level which becomes quite effective in obtaining very accurately the solutions of different stability problems in geomechanics involving soil and rock media. The present manuscript introduces plane strain and axisymmetric formulations which deal with the power type yield criterion and are based on the lower and upper bound finite elements limit analysis in conjunction with the power cone programming (PCP). To demonstrate the efficacy of the proposed formulations, the problems of the determination of the bearing capacity of strip and circular foundations in a cohesive-frictional soil medium have been addressed. In the case of a strip footing, the earthquake inertial forces in a horizontal direction have also been incorporated. The adaptive mesh pattern has been employed to improve the accuracy of the solution at the expense of minimal computational effort. The results have been thoroughly validated by making the comparison of the obtained solutions with that reported in literature. The proposed formulations are expected to be quite useful in solving different stability problems in geomechanics.

Developing soft-computing regression model for predicting bearing capacity of eccentrically loaded footings on anisotropic clay

In this investigation, the bearing capacity solution of a strip footing in anisotropic clay under inclined and eccentric load is analyzed using the numerical simulation model. The lower and upper bound finite element limit analysis (FELA) approaches are utilized to establish precise modeling and derive the numerical outcomes of a strip footing's bearing capacity. All analyses use effective automated adaptive meshes with three iteration stages to enhance the accuracy of the outcomes. The parametric analysis is performed to examine the influence of four dimensionless parameters which are taken into account in this study, namely the anisotropic strength ratio, the dimensionless eccentricity, the load inclination angle, and the adhesion factor to the bearing capacity factor. Furthermore, a new model has been proposed to predict the bearing capacity factor for the calculation of the undrained bearing capacity for footings resting on an anisotropic clay using an advanced data-driven method (MOGA-EPR). The new model takes into account the anisotropy, eccentricity, and inclination of the applied load and could be used with confidence in routine designs of shallow foundations in undrained conditions with the consideration of the anisotropic strengths of clays.

Evaluation of vertical bearing capacity factors for conical footing with varying base roughness using FELA and MARS model

ABSTRACT The conical base of spudcan foundation supporting offshore structures is also called as conical footing contributes to the overall capacity of the foundation. In the study, the vertical bearing capacity factors for the conical footing considering different base roughness (α = 0 and 1), varying friction angles of soil (f = 0–40°), and different apex angles (b = 30–180°) are evaluated by performing two–dimensional lower and upper bound finite element limit analysis (FELA). The base roughness α = 0 represents a smooth base footing and α = 1 represents a perfectly rough base footing and the apex angle b = 180° represents a flat base circular footing. Multivariate adaptive regression splines models are developed, influence of investigated parameters (α, f, and b) on the bearing capacity factors are examined and the empirical equations are proposed for the evaluation of the bearing capacity factors of conical footing.

Finite element limit analysis: fundamentals and extensions

The basic finite element formulations of upper and lower bound limit analysis are first reviewed with an emphasis on arriving at a unified formulation applicable to arbitrary yield criteria. A number of extensions are then reviewed: elastoplasticity (perfect and hardening), dynamics, and consolidation. In all cases, it is emphasized how these follow as direct extensions from limit analysis and may be treated using the same solution algorithms.

Lower Bound Limit Analysis of Non-Persistent Jointed Rock Masses Using Mixed Numerical Discretization

The bearing capacity of a non-persistent jointed rock mass containing a rock bridge is investigated by combining the lower bound limit analysis theory, a mixed numerical discrete method, and linear mathematical programming. A mixed numerical discrete method is proposed to divide non-persistent jointed rock masses in which rigid block elements are used to simulate the rock blocks, whereas the finite element method is used to simulate the intact rock bridges. A linear mathematical programming model for the ultimate bearing capacity is constructed and solved using the interior point algorithm. The proposed formulation is validated by application to three rock slopes.

Stability Charts for Limiting Horizontal Normal Pressure on Cohesive-frictional Backfill for Deep Contiguous Piled Walls

Background: This study presents a numerical solution for determining the limiting uniform normal pressure acting horizontally behind cohesive-frictional backfill material in a deep contiguous piled wall. At this limiting pressure, the soil tends to flow out in gaps between a series of vertical piles placed at a certain uniform horizontal spacing. Methods: The lower and upper bound plane strain finite element limit analysis (FELA) has been carried out for this purpose. The Mohr-Coulomb failure criterion, using an associated flow rule, was employed to impose the yield condition in the soil mass. Results: A parametric study was carried out to obtain the magnitude of the non-dimensional limiting lateral resistance (F/cD) as a function of normalized pile spacing (S/D), the friction angle of soils (ϕ), and the adhesion factor (δ) at the soil-pile interface; here, F refers to the lateral normal resistance (force) per unit length offered by the pile, S forms the clear spacing between piles, D is the diameter of the pile and c is soil cohesion. The impact of the different parameters on the failure mechanisms has been examined comprehensively. Conclusion: The lateral resistance (F/cD) offered by the piles increases generally with a decrease in the spacing between the piles. The magnitude of F/cD increases further with an increase in the values of ϕ and δ.

Estimation of seismic active earth pressure on reinforced retaining wall using lower bound limit analysis and modified pseudo-dynamic method

Present study estimates seismic active earth pressure on the reinforced retaining wall by combining the lower bound finite element limit analysis and the modified Pseudo-dynamic method. A series of parametric analyses are performed by varying seismic acceleration coefficient, time period of seismic loading, soil friction and dilation angles, reinforcement spacing, length of reinforcement, soil-reinforcement interface, damping ratio of soil, soil-wall interface, wall inclination, and ground inclination. Maximum active earth pressure is exerted when natural time period of reinforced soil matches with the time period of an earthquake. Reinforcement is found to be effective in terms of reducing active earth pressure significantly on the wall subjected to seismic loading. Effectiveness of reinforcement depends upon two factors, namely vertical spacing and soil-reinforcement interface friction angle. For relatively smaller reinforcement spacing, soil-reinforcement behaves like a composite block, which helps to constraint stresses within a small area behind the wall. Maximum tensile resistance is developed when fully rough interface condition is assumed between soil and reinforcement layer. Failure patterns are provided to understand the behaviour of reinforced retaining wall under different time of seismic loading.

A machine learning regression approach for predicting the bearing capacity of a strip footing on rock mass under inclined and eccentric load

In this study, the Multivariate Adaptive Regression Splines (MARS) model is employed to create a data-driven prediction for the bearing capacity of a strip footing on rock mass subjected to an inclined and eccentric load. The strengths of rock masses are based on the Hoek-Brown failure criterion. To develop the set of training data in MARS, the lower and upper bound finite element limit analysis (FELA) is carried out to obtain the numerical results of the bearing capacity of a strip footing with the width of B. There are six considered dimensionless variables, including the geological strength index (GSI), the rock constant/yield parameter (mi), the dimensionless strength (γB/σci), the adhesion factor (α), load inclined angle from the vertical axis (β), and the eccentricity of load (e/B). A total of 5,120 FELA solutions of the bearing capacity factor (P/σciB) are obtained and used as a training data set. The influences of all dimensionless variables on the bearing capacity factors and the failure mechanisms are investigated and discussed in detail. The sensitivity analysis of these dimensionless variables is also examined.

Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D

The stability of unsupported rectangular excavations in undrained clay is examined under the influence of anisotropy and heterogeneity using the three-dimensional finite element upper and lower bound limit analysis with the Anisotropic Undrained Shear (AUS) failure criterion. Three anisotropic undrained shear strengths are considered in the study, namely triaxial compression, triaxial extension, and direct simple shear. Special considerations are given to the study of the linearly-increased anisotropic shear strengths with depth. The numerical solutions are presented by an undrained stability number that is a function of four dimensionless parameters, i.e., the excavated depth ratio, the aspect ratio of the excavated site, the shear strength gradient ratio, and the anisotropic strength ratio. To the authors’ best knowledge, this is the first of its kind to present the stability solutions of 3D excavation considering soil anisotropy and heterogeneity. As such, this paper introduces a novel approach for predicting the stability of unsupported rectangular excavation in undrained clays in 3D space, accounting for soil anisotropy and non-homogeneity. Notably, it develops a basis to formulate a mathematical equation and design charts for estimating the stability factor of such type of excavation, which should be of great interest to engineering practitioners.

A novel Lower Bound Limit Analysis model with hexahedron elements for the failure analysis of laboratory and thin infill masonry walls in two-way bending

A novel Lower Bound (LB) Limit Analysis (LA) Finite Element (FE) model for the study at failure of laboratory and thin infill masonry walls in two-way bending is presented. In the model, a masonry plate is discretized into infinitely resistant hexahedrons and quadrilateral interfaces where all plastic dissipation occurs. Three internal static variables act on interfaces, namely bending moment, out-of-plane Kirchhoff shear and torque. Equilibrium is imposed on hexahedrons, whereas admissibility is enforced exclusively on interfaces between adjoining elements. The admissibility of the internal actions on masonry interfaces is imposed with a strength domain obtained by means of an already existing LB homogenization technique where joints are reduced to interfaces. The resultant Linear Programming (LP) problem - which allows to estimate collapse loads and distribution of internal actions at collapse - is characterized by a number of variables and constraints relatively limited, which requires little computational burden to be solved through standard interior point LP software. Failure mechanisms are obtained solving the dual LP problem. The procedure is validated against two existing experimental datasets of walls in two-way bending, namely three series of panels with and without perforations tested at collapse at the University of Adelaide Australia and four series of solid and perforated panels tested at the University of Plymouth UK. Comparing the results in terms of collapse loads, active failure mechanisms and distribution of internal actions with both experimental evidence and previously presented numerical models, excellent results are found, an outcome showing the reliability of the procedure proposed.

Simplified Approach for In-Plane Strength Capacity of URM Walls by Using Lower-Bound Limit Analysis and Predefined Damage Patterns

In this study, a two-phase simplified approach is proposed to predict the in-plane strength capacity of unreinforced masonry (URM) walls. In the first phase, in-plane damage and failure patterns of URM walls are determined from available observational (field) data, experimental data and also from numerical analysis data. Then, a set of rules are proposed to estimate damage and failure patterns of URM wall panels. In the second phase, this valuable information is employed to develop a simplified numerical model with a coarse mesh for the masonry wall, which is consistent with the crack formation at the ultimate state. Then, lower-bound limit analysis approach is used to predict the failure load of the wall without any detailed micro-element analysis. At the final stage, the proposed approach is verified by comparing the numerical results with experimental data from URM wall tests. By the assistance of this approach, it becomes possible to estimate the lateral capacities of ordinary, non-engineered URM walls and buildings from damage patterns at failure state. As an ultimate goal, this structural information can be used for seismic risk assessment of regions where the building typology considered in this study governs the building stock.

Numerical study on the bearing capacity of strip footing resting on partially saturated soil subjected to combined vertical-horizontal-moment loading

This study investigates the bearing capacity of strip footings resting on unsaturated soils subjected to combined loading using the lower-bound limit analysis coupled with the finite element discretization approach. In this regard, second-order cone programming (SOCP) is exploited to simulate the nonlinear form of the universal Mohr-Coulomb yield criterion during the process of stress field optimization. The significant influence of matric suction induced below the surface footing was accounted for by adopting the suction stress concept under the no-flow and steady-state infiltration/evaporation flow conditions. The eccentricity and inclination of the foundation loading are introduced into the equilibrium equations along the strip footing so as to render various combinations of moment (M), vertical (V) and shear (H) forces. The results stemming from the lower-bound finite element limit analysis are compared with several previous studies throughout the literature for verification of the model. The substantial contribution of suction stress to the evolution of failure loci and the distribution of subsurface stresses for the shallow foundation subjected to inclined and eccentric loadings is thoroughly discussed. A general three-dimensional failure envelope is presented for shallow foundations resting on partially saturated soils under combined vertical, horizontal and moment loadings.

Lower Bound Finite Element Limit Analysis of Geo-Structures with Non-Associated Flow Rule

The use of finite elements and limit analysis in the stability analysis of geo-structures is not uncommon nowadays, with the assumption of associated flow rule and rigid perfectly plastic constitutive soil model. Nevertheless, the simplified assumption may lead to misevaluation of the objective function for the nonlinear optimisation solution, thus giving rise to uncertainty in the design of geo-structures. In this study, the influence of non-associated flow rule on the stability evaluations for obliquely- and eccentrically-loaded shallow foundations, as well as for retaining structures with cohesionless granular backfill, is examined using lower bound finite elements and second-order cone programming (SOCP). A wide range of dilation angles (ψ) is considered for the study (e.g., from zero to a maximum of soil internal friction angle φ), and it is generally observed that failing to consider the non-associated flow rule in the soil constitutive model would yield non-conservative designs of geo-structures. These inaccurate predictions and subsequent unreliable stability analysis are more pronounced in cases of large soil internal friction angles. The findings in this study would be of great importance for practical design applications.

Estimation of bearing capacity of shallow footings on layered sand using finite elements analysis

PurposeThis paper aims to estimate the bearing capacity of a surface strip and circular footings lying on layered sand using numerical limit analysis.Design/methodology/approachLower and upper bound limit analysis, as well as finite elements and second-order conic programming (SOCP), are used in this analysis. The yield criterion of Mohr-Coulomb is used to model soil behavior. Using this technique, stringent lower and upper bounds on ultimate bearing capacity can be achieved by assuming an associated flow law.FindingsThe obtained results indicate that the exact collapse load is typically being bracketed to within 6% about a mean of both the bounds. The obtained results are compared with the existing literature wherever applicable.Originality/valueTo the best of the authors’ knowledge, no study has used lower and upper bound limit analysis, as well as finite elements and SOCP, to estimate the bearing capacity of a surface strip and circular footings lying on layered sand.

Numerical and analytical evaluation of stirrup-induced confinement on resistance of RC beams using lower bound limit analysis

In this paper, a modified three-dimensional element is introduced for limit analysis of RC structures, which considers more aspects of structural behaviour. Using the introduced element, the influence of stirrup confinement on the shear and moment resistance of RC beams is evaluated by lower bound limit analysis. To analyse and solve the relevant optimization problem, the semidefinite programming method, SDP, is employed. The modified Mohr–Coulomb yield criterion is utilized to constrain the concrete stress state. Beams with a constant amount of shear reinforcement per unit length, with different values of stirrup spacing, are studied. It means that they only differed in the effect of confinement induced by stirrup reinforcements. The performance of the proposed element is verified by comparing the results of numerical limit analyses with experimental results. This comparison revealed the efficiency and accuracy of the proposed model and the capability of the introduced element to take the effect of stirrup-induced confinements into account. After modelling and investigating the effect of stirrup confinement in load bearing of RC beams, a simplified equation for the design and practical purposes is proposed. Using this equation, the shear resistance of the beams could be modified for different stirrup space choices. The results obtained from this simplified equation are compared with numerical limit analysis and experimental results. The comparisons indicate the reasonable accuracy of the proposed equation.

Elasto-plastic ground settlement response and stability of single and twin circular unsupported and supported tunnels

An attempt has been made to examine numerically, based on an elasto-plastic finite elements (FE) analysis, soil unit weight-ground displacement response due to laying of single and twin circular tunnels which are supported by an internal isotropic pressure (σi); the case σi= 0 implies an unsupported tunnel. The soil mass has been assumed to be linearly elastic and perfectly plastic obeying the Mohr-Coulomb (MC) yield criterion and an associated flow rule. The stability numbers, corresponding to ultimate collapse, based on lower and upper bound finite elements limit analysis (FELA) have also been separately evaluated. The effects of cover (H) to diameter (D) ratio, spacing (S) to diameter ratio of the tunnels and soil internal friction (ϕ) angle on the results have been examined in detail. The study clearly reveals that the value of the non-dimensional loading parameter, γDc +σi, corresponding to given surface vertical settlement (sv) and that at ultimate collapse, with an assumption that the elastic and plastic material parameters do not vary with depth, increases continuously with an increase in the values of S/D as well as ϕ, but decreases with an increase in the magnitude of H/D. The obtained computational results were found to compare well with the data reported in literature. It is expected that the study will be useful for stability evaluation and estimating vertical ground settlements for single and twin tunnels.

Bearing capacity of conical footings on Hoek–Brown rock masses using finite element limit analysis

New plastic solutions for the bearing capacity factors of conical footings on rock masses obeying the Hoek–Brown yield criterion are presented in the paper by using the lower and upper bound finite element limit analysis. A conical footing has a cone apex angle and is penetrated into a rock mass. The considered dimensionless parameters include the cone apex angle, the yield parameter, and the geological strength index of a rock mass, where the effects of these dimensionless parameters on the bearing capacity factor are compressively investigated. The collapse mechanisms of conical footings on rock masses are examined and discussed in the paper.

Automatic Adaptive Recovery Stress ES-FEM for Lower-Bound Limit Load Analysis of Structures

The paper proposes a novel automatic adaptive recovery stress edge-smoothed finite element method (ES-FEM) that determines the maximum load capacity of inelastic structures at plastic collapse. This approach performs solely a series of elastic ES-FEM analyses with systematic modulus variations (considering the influences of stress singularity) to converge the collapse load solutions. The smoothed [Formula: see text]-continuous recovery stress field ensures the satisfaction of static admissible stress and yield conformity conditions underpinning lower-bound limit analysis theorems. A modified modulus error function within the newest node bisection algorithm enables automatic mesh refining and coarsening constructions, and fast convergence to the lower-bound collapse limit.

Limit Analysis Solutions for Bearing Capacity of Ring Foundations on Rocks Using Hoek–Brown Failure Criterion

In this paper, new limit analysis solutions for the bearing capacity of ring foundations on rock masses are presented, where the Hoek–Brown yield criterion is used as a failure criterion for rock masses. The lower and upper bound finite element limit analysis is employed to derive the bearing capacity solutions of ring foundations on rock masses. The ring foundation has internal and external radii. The considered dimensionless parameters include the ratio between the internal radius and the external radius, the yield parameter, and the geological strength index of rock, where the effects of these dimensionless parameters on the bearing capacity factor are investigated. It is found that a high yield parameter or a high geological strength index yields a high value of the bearing capacity factor. When the ratio between the internal radius and the external radius is around 0.25, the bearing capacity factor becomes the largest. The collapse mechanisms of this problem are also examined and discussed in the present study.

End Bearing Capacity Factor for Annular Foundations Embedded in Clay Considering the Effect of the Adhesion Factor

New limit analysis solutions for the end bearing capacity of annular foundations in clay with linearly increasing shear strength are presented in the paper. The strength profile of clay corresponds to a typical case of a normally consolidated clay in deep water, where the strength at the ground surface is defined as zero and increases linearly with the depth. The annular foundation has internal and external radii and is embedded in clay, where the adhesion factor representing the strength at the soil-structure interface is also taken into account. Using the lower and upper bound finite element limit analysis, the end bearing capacity factor of this problem can be obtained based on three considered dimensionless parameters, which are the ratio between the internal radius and the external radius, the embedded depth ratio, and the adhesion factor. The collapse mechanisms of annular foundations in clay are also examined and discussed in the paper to portray the effects of the radius ratio, the depth ratio, and the adhesion factor.