Limit Analysis Approach Research Articles

3D limit analysis of reinforced concrete with sliding along smeared cracks

Limit analysis (LA) is successfully used for investigating the bearing capacity of reinforced concrete (RC) structures. Some cautions must be taken when using this method for RC since the concrete component exhibits a softening behavior with decreasing strength and limited ductility. A commonly adopted provision consists of considering isotropic reduced values of concrete strength to be input in the analysis (empirical effectiveness factors). In this paper, an alternative and completely new approach is proposed and investigated in which concrete strength is weakened in a more targeted manner. To that purpose, the commonly used 3D truncated Mohr-Coulomb (TMC) criterion is adopted to classically describe the compressive, tensile, and shear failure of concrete. However, TMC is here in an original way enriched by additional constraints that allow to account for weakness and anisotropy induced by preferential failure patterns, assumed a priori. The limit analysis approach is then formulated for two dual analyses in the convex optimization framework, making it possible to quantify the numerical error and obtain a lower and an upper bound of the limit load. Numerical examples illustrate the agreement of the formulation with academic results and laboratory tests.

Discrete macro - models of nonlinear interlocking mechanisms in the out-of-plane failure of masonry walls

AbstractHistorical unreinforced masonry (URM) constructions are generally vulnerable to out-of-plane (OOP) failures due to the absence of rigid floors and poor connections between orthogonal walls. That leads to the activation of rocking mechanisms of external walls, whose ultimate force and displacement are affected by complex nonlinear interactions with sidewalls. These interactions are often neglected in the engineering practice, potentially leading to significant approximations, as demonstrated by experimental and numerical studies available in the literature. As a novel contribution to the field, this paper presents an upgraded discrete macro-element model (DMEM) to predict the rocking capacity of OOP loaded URM walls interacting with sidewalls. Considering both the onset and the evolution of the rocking mechanism of the front wall, interlocking effects with the sidewalls are first simulated through frictional resistances using the macro-block model (MBM) and the nonlinear kinematic approach of limit analysis. Then, the upgraded DMEM is implemented on the basis of the equivalence between the continuous distribution of these forces, introduced as a further novelty of the paper, and the discrete distribution of lateral elastic-plastic links, accounting for mechanical and geometrical nonlinearities. The results of the two models are discussed in terms of both frictional resistance-displacement and pushover curves, referring to a case study of a front wall belonging to a two-storey URM building. The wall response is also compared with the results derived from the original source of the case study and analysed by changing the number of nonlinear links to define different levels of accuracy.

Face stability assessment of a longitudinally inclined tunnel considering pore water pressure

AbstractThe face stability analysis of a longitudinally inclined shield tunnel using an analytical approach in water‐rich areas is still a research gap. To solve this face stability problem, a numerical simulation based on the FLAC3D is first conducted to calculate the seepage field behind the inclined tunnel face. An improved rotational failure mechanism is developed to make it possible to investigate the face stability of inclined tunnels using analytical approaches. In the framework of the kinematic approach of limit analysis, the limit support pressures and corresponding failure surfaces of the inclined tunnel face are determined to analyze the face stability issue. The interpolation tool (griddata) in MATLAB is adopted to involve the obtained numerical values of pore water pressures into the analysis of the stability issue. The analytical solutions obtained from the proposed method are validated by comparisons with existing results from published literatures and numerical results. For a quick estimation of the inclined tunnel face stability in water‐rich areas, a series of design charts are then presented for various soil strength parameters, water tables, and inclined angles. Finally, an application of the proposed method to a practical tunneling case is provided, which further illustrates the effectiveness of the proposed method.

A computed tomography-based limit analysis approach to investigate the mechanical behavior of the human femur prone to fracture

The paper proposes a refined CT-based FE modelling strategy that implements a limit analysis numerical procedure, namely the Elastic Compensation Method (ECM), to estimate a lower bound to the collapse load of a human femur. In particular, the model geometry was obtained from CT images by segmentation of a fresh-frozen human cadaveric femur that was discretized with second-order tetrahedral 3D finite elements. A yield criterion of Tsai–Wu-type, expressed in principal stress space, was adopted to model the bone tissues for which the strength limit values in tension, compression and shear are computed locally from the femoral density distribution also derived from CT images. The developed CT-based numerical technique showed the ability to predict, at least for the examined femur for which the experimental collapse load is available, a lower bound to the collapse load. The proposed approach seems a promising and effective tool that could be adopted into clinical practice to predict the fracture risk of human femur starting from patient-specific data given by medical imaging.

Three‐dimensional stability analysis for a deep‐buried tunnel roof considering soil stratum strength nonlinearity

AbstractA three‐dimensional (3D) collapse mechanism was employed in this work to investigate the roof stability of deep‐buried cylindrical tunnels in soil considering strength nonlinearity. Based on the kinematic approach of limit analysis, three tunnel roof stability measures, namely, the stability number, the required support pressure, and the factor of safety solutions were derived to provide quantitative indicators for evaluating tunnel roof stability. Optimal upper bound solutions for these measures were obtained through optimization. Subsequently, the effects of soil strength nonlinearity and 3D geometry characteristics on roof stability and the profiles of collapsing blocks were investigated: The increase in the nonlinear coefficient m and the tensile strength σt, as well as the decrease in the initial cohesion c0, will lead to a deterioration in the stability of the tunnel roof, and also result in a larger range of collapse blocks of the tunnel roof. A 3D stability analysis of the tunnel roof can provide more reasonable stability estimates than a 2D stability analysis. When the length‐to‐radius ratio L/R → ∞, the 2D solution results.

Experimental and numerical evaluation of geometrical imperfection effects on the load bearing capacity of small-scaled voussoir arches

The study of the behaviour of masonry arches represents a complex problem, which involves different aspects that can significantly affect the load bearing capacity of such structures. In this regard, geometrical imperfections, such as the irregular shape of the arch blocks, the initial position of the first and last voussoirs (boundary conditions), the alignment between the adjoining blocks and the rounded corners of each block, play an important role. An experimental campaign was conducted on a toy arch made by eleven irregular blocks, with the aim of evaluating the effect of such irregularities. A numerical model was developed basing on lower bound (LB) and upper bound (UB) limit analysis approaches (referred to as the static and the kinematic theorem, respectively). A linear programming (LP) solving algorithm was implemented in the software MATLAB and used to solve the minimization/maximization problems. The probability distribution of each considered nonlinearity was estimated, the results of the numerical analysis were correlated with the experimental outcomes and a Montecarlo simulation was carried out to validate the hypothesised probability distributions.

A limit analysis approach to uplift bearing capacity of shallow plate anchors in marine environments

AbstractOffshore oil and gas exploration activities involve the installation of various subsea infrastructures on the ocean floor, often supported by shallow foundations such as anchor plates or mudmats. These foundations must be designed to withstand applied service loads along the structure lifecycle, including decommissioning loads required for subsequent removal. In this context, this work is dedicated to the evaluation of uplift bearing capacity of shallow anchors within the theoretical framework of limit analysis and its related kinematic approach. Based on the implementation of 3D failure mechanisms, rigorous upper bound estimates for the uplift force viewed as ultimate load for the material system are derived from either total or effective stress analyses. In that respect, emphasis is given to the formulation of limit analysis problem in the context of effective stresses, showing how the effect of seepage flow may be accounted for by means of driving body forces derived from the gradient of excess pore pressure distribution and how suction forces arise from pore pressure difference between upper and lower anchor surfaces. Evaluation of suction forces developed during anchor lifting is addressed by resorting to a simplified modeling applied to circular and rectangular anchors. The accuracy of the upper bound predictions derived for uplift force is assessed by comparison with available experimental data. Several numerical results are also provided to investigate the effects of relevant problem parameters on the uplift bearing capacity.

A limit analysis-based CASS approach for the in-plane seismic capacity of masonry façades

The present paper proposes a new methodology for the load-bearing capacity analysis of 2D masonry constructions by extending and reformulating the Continuous Airy-based for Stress Singularities (CASS) method, with a particular focus on the incorporation of volume forces, crucial to solve mechanical problems involving masonry constructions accurately. The masonry material is modelled using a Normal, Rigid, No-Tension (NRNT) model, largely adopted by the scientific community as material parameters, often unknowable, are not needed.The load-bearing capacity problem is derived from an energy-based formulation and framed as a classic limit analysis approach, allowing for the direct determination of the maximum incremental load that a masonry structure can withstand, along with the corresponding internal stress pattern.The structural domain is discretised with a simple finite element mesh, and the Airy stress potential is adopted to enforce the internal equilibrium directly. The boundary value problem is then solved as second-order cone programming (SOCP).The CASS method is shown to offer modelling and computational advantages. Indeed, it accurately describes the mechanical response, including the ability to capture singular stress fields typically exhibited by masonry structures, particularly where cracks appear. The adoption of the Airy potential allows for a reduction in the number of explicit constraints from the problem. Moreover, the formulation of the boundary value problem as a SOCP ensures the existence of a unique load multiplier while offering computationally fast solutions even for large problems.Several numerical examples are presented to demonstrate the CASS potential. Specifically, the ability to capture singular stress patterns diagonally crossing the finite elements showcases the CASS mesh independence, providing a straightforward approach to modelling complex geometries and loading conditions.

Effective yield strength of a saturated porous medium with a spheroidal meso-pore and spherical micro-pores

A macroscopic yield criterion has been derived in the present work for a double saturated porous medium with a spheroidal pore at the mesocale and spherical pores at the microscale. These two types of pores are well separated at two different scales. The meso spheroidal pore saturated by a pore pressure which is different from the one in the micro spherical pores. A Drucker-Prager type criterion is adopted for the solid phase at the microscopic scale to describe its asymmetric behavior between tension and compression. The methodology to formulate this criterion is based on the limit analysis approach of a spheroidal volume containing a confocal spheroidal pore subjected to a uniform strain rate boundary conditions. The matrix at the mesoscopic scale obeys to a general elliptic yield criterion. Based on a two-step homogenization step, the influence of meso-pore shape (spherical, prolate or oblate), micro-porosity, meso-porosity and the effect of pore pressures at different scales are taken into account explicitly by this macroscopic yield criterion.

Roof stability analysis of cylindrical tunnels in hard soil/soft rock with reduced tension strength

Three-dimensional roof stability analyses of cylindrical tunnel segments limited along the tunnel direction were performed based on the modified Mohr–Coulomb failure criterion with reduced tensile strength. A wedge-shaped region of the classical Mohr–Coulomb envelope, where unrealistic tensile and shear strengths are predicted, is truncated by a Mohr circle associated with the true tensile strength. This revised failure criterion with partial nonlinearity allows geomaterials deep into the roof to deform with large velocity gradients under the normality plastic flow rule, observed in field investigations. Using the kinematic approach of limit analysis, rigorous bounds for three quantitative stability measures–stability number, factor of safety, and supporting pressure–were calculated. The roof failure mechanism adopted is relatively elaborate, involving three-dimensional geometry intersecting with a cylindrical tunnel segment; thus, an efficient approach was presented for tunnels with long length constraints. With respect to extremely short tunnel lengths, a modified roof failure mechanism with distinct ridges was proposed. The factor of safety with accommodation of the nonlinear pressure dependency was estimated by a consistent procedure associated with a reduced elliptic failure criterion. The results show that roof stability distinctly depends on the magnitude of the true tensile strength, internal friction angle, and length constraint.

Stability assessment of a non-circular tunnel face with tensile strength cut-off subject to seepage flows: A comparison analysis

This work aims to develop a numerical-analytical framework within the kinematic approach of limit analysis to assess the stability of non-circular tunnel faces under seepage conditions with tensile strength cut-off from a discretization-based perspective. Considering the overestimation of the tensile strength of the soil by the Mohr-Coulomb criterion, a modification based on the concept of tensile strength cut-off is recommended to reduce or eliminate the tensile strength. A modified 3D rotational discretization-based failure mechanism incorporating the tension cut-off is generated point-to-point starting from a real non-circular tunnel face to model the face failure, where the failure surface on the top intersects more rapidly due to the increasing rupture angle in the tensile regime. The seepage towards the non-circular tunnel face is numerically simulated, and then the numerical seepage field is extracted into the 3D discretization-based failure mechanism. Within the kinematic approach, the seepage effect is introduced from two aspects: pore pressure and seepage force. A hybrid optimization routine is employed to search for the maximum required face pressure. The developed framework is validated by comparisons with numerical fluid-mechanical simulations and previous studies. Afterward, parametric analyses are carried out on different seepage conditions, different face pressure distributions, different soil strengths, and different tension cut-off conditions. The developed framework can give insight into the combined effect of the tension cut-off and seepage on the stability of non-circular tunnel faces under various conditions.

The basis for ductility evaluation in SFRC structures in MC2020: An investigation on slabs and shallow beams

AbstractThe paper presents a synthesis of an extensive experimental campaign on linear and two‐dimensional steel fiber reinforced concrete (SFRC) structural elements carried out to check the ductility requirements aimed at guaranteeing limit analysis approaches for the computation of ultimate load‐bearing capacity of SFRC structures; special attention is devoted to the role of the degree of redundancy of the structure. In particular, full‐scale shallow beams and slabs reinforced with steel fibers (with or without conventional longitudinal reinforcement) were tested in two different laboratories: the Politecnico di Milano (PoliMI) and the University of Brescia (UniBS). In this experimental campaign, two different fiber contents and fiber types were considered. The experimental investigation, carried out within the activities to support Annex L of Eurocode 2, was fundamental also for developing the design rules included in the fib Model Code 2020 and allowed to formulate conclusions regarding optimization of the mix design, ductility, and design prediction at the ultimate capacity.

Seismic vulnerability prediction of masonry aggregates: Iterative Finite element Upper Bound limit analysis approximating no tensile resistance

A novel 3D Finite Element Upper Bound limit analysis approach for the evaluation of the seismic vulnerability of masonry aggregates when the collapse may occur both locally and globally is presented.The model relies on a 3D discretization of the aggregate by means of infinitely resistant hexahedron elements and rigid plastic quadrilateral interfaces, where all plastic dissipation occurs.The interface behavior is assumed isotropic, with friction, cohesion, and tension cutoff. The collapse horizontal acceleration is obtained in the discretized model solving large-scale linear programming problems with standard numerical routines. According to a consolidated literature, the vulnerability prediction should be done with pre-assigned failure mechanisms formed by large macro-blocks mutually roto-translating, and assuming a material unable to withstand tensile stresses without mutual sliding between macro-blocks. However, such assumptions may be responsible for an overestimation of the load-carrying capacity and the determination of a realistic mechanism is neither obvious nor possible in any case. Discretizing the aggregate into 3D elements and interfaces, the failure mechanism is automatically found in the solution of the limit analysis problem, but enforcing the no-tension material hypothesis may be responsible for the activation of parasite sliding between contiguous elements or stalling issues of the numerical algorithm. In the first case, the failure mechanism changes considerably and the computational evaluation of the horizontal acceleration at collapse can be affected by unsafe inaccuracy.The present approach iteratively approximates the no-tension material model at low computational cost, progressively shrinking the failure surface and evaluating (i) collapse accelerations, (ii) failure mechanisms active, (iii) power dissipated on interfaces and (iv) power expended by loads not dependent on the collapse multiplier. By analyzing the failure mechanisms obtained, it is possible to select where spurious sliding is not dominant. Enforcing to zero the power dissipated on interfaces it is possible to estimate the collapse multiplier corresponding to the twofold assumption that (i) parasite sliding is negligible and (ii) the material approximates in a technically acceptable way the no-tension one.The model is first benchmarked on a masonry portal, where a reference solution is available, and then applied to two historical aggregates located in Italy.

Parametric analysis of masonry arches following a limit analysis approach: Influence of joint friction, pier texture, and arch shallowness

Among the most characteristic structures in historical constructions for crossing large spans are the masonry vaulted structures by utilizing their geometric stability to safely transfer the loads to supports with regard to their negligible tensile strength. The ability of masonry piers to bear such transferred stresses and safely convey them to the support is directly related to their structural integrity, as well as to a number of other factors. Using an in-house limit analysis code, a study on the crucial parameters impacting the safety level of piers under the thrust of arches is performed. Parameters such as pier texture, joint friction angle, and arch shallowness, namely, shallow, semi-circular, and pointed arches, were investigated under three load scenarios: horizontal and concentrated vertical live load applied at mid-span and quarter-span. The main findings of this work show that all studied parameters have a significant influence on the structure response. Higher friction values, sharper arches, and piers that follow the rule of art result in higher collapse multipliers. Furthermore, this work emphasizes the importance of accounting for the sliding mechanism and masonry texture, parameters that are often neglected.

Seismic Bearing Capacity Solution for Strip Footings in Unsaturated Soils with Modified Pseudo-Dynamic Approach

In engineering mathematics, the unsaturated nature of soil has a significant impact on the seismic bearing capacity solution. However, it has generally been neglected in the published literature to date. Based on the kinematic approach of limit analysis, the present study proposes a method for calculating the bearing capacity of shallow strip footings located in unsaturated soils, taking four common types of soils as examples. The modified pseudo-dynamic (MPD) approach is used to calculate the seismic forces varying with time and space, and the layerwise summation method is used to derive the power generated by the seismic forces. In the calculation of internal energy dissipation, this paper introduces the effective stress based on the suction stress to derive the cohesion expression at different depths. The analytical formula of bearing capacity is obtained by the principle of virtual work, and its value is optimized by the Sequential Quadratic Programming (SQP) algorithm. In order to verify the validity of the proposed method, the present results are compared with the solutions published so far and a good agreement is obtained. Finally, a parametric study is performed to investigate the influence of different types of parameters on the bearing capacity.

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.

Tunnel Face Stability Considering the Influence of Excess Slurry Pressure

With excess slurry pressures exerted on the tunnel face, slurry particles tend to infiltrate into the soil in front of the tunnel. There will be excess pore pressure ahead of the tunnel in the case of infiltration, leading to an impairment in the supporting effect contributed by the excess slurry pressure. Corresponding to three slurry infiltration scenarios distinguished by the forms of the filter cake, different pressure transfer models are employed to describe the pore pressure distribution. Using the kinematic approach of limit analysis and the numerically simulated seepage field, the study of tunnel face stability under different slurry infiltration cases is extended by employing a 3D discretization-based failure mechanism. In addition, two simple empirical formulas describing the pore pressure distributions above the tunnel and in advance of the tunnel are established and verified. Combined with the dichotomy method and strength reduction method, the safety factors yielding rigorous upper-bound solutions are obtained by optimization. The proposed method is validated by a comparative analysis. The developed framework allows considering the influence of excess pore pressure on the whole failure mechanism and the three-dimensional characteristics of seepage. A parameter analysis is performed to study the effect of the excess slurry pressure, hydraulic conditions, soil strength properties, and pressure drop coefficient. The results show that the steady-state flow model leads to much more conservative results than the full-membrane model. The safety factor increases with the increasing excess slurry pressure and the decreasing pressure drop coefficient. The present work provides an effective framework to quickly assess the face stability of tunnels under excess slurry pressure considering different filter cake scenarios.

Limit analysis and design-oriented approach for out-of-plane loaded masonry walls strengthened by grouted anchors

Masonry structures without box-like behaviour are often prone to the complete or partial out-of-plane failure of their walls under horizontal actions, i.e. mainly the seismic ones. Steel grouted anchors or tie-rods have been widely used in the past to improve connections between orthogonal walls and avoid out-of-plane mechanisms. However, reliable analytical approaches to design such strengthening systems are lacking, whereas the evaluation of their effectiveness is still a matter of ongoing research. The paper presents a force-based design approach to assess the performance of three-wall masonry systems strengthened by grouted anchors, using a frictional macro-block model and a non-standard limit analysis, previously developed for unreinforced masonry structures. The extension of this model to strengthened masonry systems, with specific reference to the use of grouted anchors, allows assessing their horizontal capacity in terms of minimum load multiplier and expected failure. The novel aspects introduced in the updated model are: (1) a simplified closed-form formulation for the calculation of the stabilising moments provided by the grouted anchors at the onset of the failure mechanism, (2) the possibility of placing the grouted anchors with variable inclinations, which may improve their stabilising contribution and optimise the intervention. The effectiveness of the simplified formulation is firstly proved by the comparison with literature results; then, several parametric analyses are carried out to highlight the influence of some variables on the capacity of both the unreinforced and the strengthened three-wall system. Finally, a literature example related to a traditional intervention with steel tie-rods is used to apply the proposed procedure and investigate the performance of different design solutions with innovative grouted anchors.

Seismic Bearing Capacity of Strip Footing Placed on Sand Layer Over Hoek–Brown Media using Finite Element Limit Analysis and Machine Learning Approach

Seismic Bearing Capacity of Strip Footing Placed on Sand Layer Over Hoek–Brown Media using Finite Element Limit Analysis and Machine Learning Approach

Seismic stability analysis of heterogeneous slopes reinforced by inclined soil nails

This study aims to propose an effective approach to evaluate the seismic stability of heterogeneous slopes reinforced by inclined soil nails. The modified pseudo dynamic approach is applied to properly describe the characteristics of seismic loads with time and space. The discretization-based failure mechanism generated from the slope crest is divided horizontally into many blocks to incorporate the heterogeneous soil properties and depth-dependent seismic accelerations into the analysis of the stability issue. The critical seismic acceleration coefficient is then determined to assess the slope stability using the kinematic approach of limit analysis. The proposed method is validated by comparisons with existing solutions and numerical results for some available cases. A systematic parametric analysis is conducted to discuss the effects of model parameters on the slope stability. Results show that there exist optimal relative nail lengths and optimal incline angles of soil nails to obtain the maximum critical seismic acceleration coefficient for reinforced slopes. Finally, an application of the proposed model to spatially variable soil strength properties is performed to illustrate that the proposed method has the potential to serve as a benchmark for the slope stability under complex geological conditions.