Limit Analysis Research Articles

Intrados FRCM-strengthening of a masonry bridge: Experimental and analytical investigations

This paper will present the results of an experimental campaign aimed to study the structural behavior of solid clay brick masonry arch models strengthened at the intrados with Fiber Reinforced Cementitious Matrix (FRCM). The arch models’ peculiarities are: i) the presence of the haunching made of hydraulic lime mortar-based conglomerate, and ii) the geometry that reproduces a 1:2 scaled longitudinal section of a typical Italian railway single span bridge. The arch models are tested, imposing loading-unloading cycles to analyze the effectiveness of the strengthening system. The results are expressed in terms of failure mode, load carrying capacity and ductility, which shows the increase in the strength and displacement capacity of the strengthened model. Finally, limit analysis was performed to validate the experimental results and to assess the maximum fiber strain and the corresponding stress at the peak load.

Probabilistic Analysis of a Three-Dimensional Slope Based on Limit Analysis

Probabilistic Analysis of a Three-Dimensional Slope Based on Limit Analysis

Pseudo-dynamic stability analysis of 3D rock slopes considering tensile strength-modified Hoek–Brown failure criterion: Seismic UBLA implementations

The Hoek-Brown (HB) criterion has found extensive application in the stability analysis of intact or jointed rock slopes, in both two-dimensional (2D) and three-dimensional (3D) contexts. In the early days, the assessment of Hoek-Brown slopes rarely incorporated tensile strength until, recently, limited studies examined the tension-controlled toppling failure at the sliding head. Nevertheless, the adapted tensile strength, which is extrapolated from the HB envelope the tensile regime, has been found overestimate experimental results. Thus, for more realistically representing the influence of rock tensile strength, the present study introduces a novel procedure for assessing the seismic stability of 3D HB slopes. In this study, the tensile strength-modified HB envelopes considering cut-off effects are derived in the principal stress space and Mohr space. To assess the slope stability, the obtained HB envelope is employed in a multi-cone mechanism based on upper bound limit analysis (UBLA). Specifically, the counter of this mechanism is determined by 10 optimized dependent rupture angles, highly considering the nonlinearity of the Hoek-Brown envelope compared to the traditional linearization strategies. As an improved seismic stability analysis, the seismic load is characterized by a modified pseudo-dynamic method. Analytical expressions for two stability indicators, i.e., the stability number and safety factor, are derived. Comparisons with numerical simulations demonstrate the rationality of the proposed procedure. Parametric analysis indicates the significant effect of the tension cut-off on the HB rock slope stability and the contour of the sliding surface.

Fast Limit Analysis of Domes Belonging to the Architectural Heritage Under Different Hypotheses on Masonry Strength

ABSTRACT The load-bearing capacity of selected case studies of domes belonging to the architectural heritage is evaluated by means of two different semi-analytical approaches framed in the context of limit analysis. The first method is a static approach known as the “Durand-Claye method” or “stability area method”, and the second is a kinematic approach that combines the virtual power principle with the upper bound theorem of limit analysis. The fundamentals of these two methods were presented in previous works by the authors, under different hypotheses about the strength of masonry. Building on this theoretical background, the current study introduces novel analytical solutions for domes of varying shapes, specifically ogival domes and segmental domes with non-uniform thickness. These solutions are applied to real case studies of significant architectural importance: the ogival dome of the Church of Anime Sante in L’Aquila (Italy); the ogival dome of Escuelas Pías in Valencia (Spain), and the main segmental dome of the Global Vipassana Pagoda in Mumbai (India). The varying sizes of the examined domes allow for an analysis of the impact of limited tensile and compressive strength on collapse behavior, by comparing the results obtained through the static and kinematic methods with those derived from Heyman’s simplified assumptions.

Three-dimensional stability of a fill slope reinforced by a frame beam anchor plate

The frame beam anchor plate is a new supporting structure reinforcing fill slopes. At present, only a few studies have evaluated its static stability based on the two-dimensional plane assumption. In this paper, the limit analysis method is combined with the three-dimensional spiral failure mechanism to evaluate the factor of safety (Fs) of a slope supported by a frame beam anchor plate. The pseudo-static approach is used to express the seismic effects. When calculating the internal energy dissipations, both the pull-out failure of anchor plates and tensile failure of tie rods are considered. The accuracy of the approach is verified by comparison with calculation results from the finite element method. Parameter analysis summarizes the influence law of each supporting structure design parameter on the Fs. Therefore, this study develops a theoretical basis for evaluating slope stability reinforced by frame beam anchor plates, which can better guide the design of slopes reinforced by this supporting structure.

Enhancing slope stability prediction through integrated PCA-SSA-SVM modeling: a case study of LongLian expressway

China is one of the regions most frequently affected by landslides, which have significant socio-economic impacts. Traditional slope stability analysis methods, such as the limit equilibrium method, limit analysis method, and finite element method, often face limitations due to computational complexity and the need for extensive soil property data. This study proposes a novel approach that combines Principal Component Analysis (PCA), Sparrow Search Algorithm (SSA), and Support Vector Machine (SVM) to improve the accuracy of slope stability prediction. PCA effectively reduces data dimensionality while retaining critical information. SSA optimizes SVM parameters, addressing the limitations of traditional optimization methods. The integrated PCA-SSA-SVM model was applied to a dataset of 257 slope stability samples and validated using five-fold cross-validation to ensure the model’s generalization capability. The results show that the model exhibits superior performance in prediction accuracy, precision, recall, and F1-score, with the test set achieving an accuracy of 84.6%, a recall of 84.7%, a precision of 83.1%, and an F1-score of 84.6%. The model’s robustness was further validated using slope data from the LongLian Expressway, demonstrating high consistency with the actual stability status. These findings indicate that the PCA-SSA-SVM-based slope stability prediction model has significant potential for practical engineering applications, providing a reliable and efficient tool for slope stability forecasting. Classify the training samples through cross-validation, using the accuracy of cross-validation as the fitness of the sparrow individual. Retain the optimal fitness value and position information.

A New Method for Determining the Safety Distance Between Irregular Karst Cave and Circumferential Shield Tunnel

ABSTRACTWhen a shield tunnel passes through a karst area, the water‐filled cave can easily make the surrounding rock metamorphic, resulting in water inrush, ground collapse, and shield machine failure and other engineering hazards. Natural cavities have a significant degree of geometric irregularity due to groundwater alteration and soluble rock erosion. Considering the difficulties in describing the shape of a natural irregular cavity, circular, rectangular, and elliptical geometries have been simplified in most related studies. Based on the upper bound theorem of limit analysis, we established a three‐dimensional failure model including the karst caves located directly above and below the circumferential side of the tunnel. Then, we deduced the corresponding analytical solution of the critical safety distance (CSD). Furthermore, the effects of rock mass parameters, cave parameters, and geometric parameters on the CSD were analyzed. Then we designed the numerical simulation considering the irregular geometry shape at the circumferential side of tunnel using the Fourier descriptors. In addition, we estimated the CSDs for two failure models using the revised dichotomy and failure criterion. The findings demonstrated a quantifiable association between CSD and Fourier descriptors of irregular cave shape, resulting in the development of a CSD prediction model. These test results can provide a theoretical foundation and direction for predicting water inrush caused by the constrained irregular cave.

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.

Physics-guided neural network-based framework for 3D modeling of slope stability

Physics-informed neural networks (PINN) have gradually attracted attention in the field of geotechnical engineering. This paper proposes a novel PINN-based framework for the three-dimensional (3D) stability analysis of soil slopes. Based on the fundamental theorem of plasticity and limit analysis, the partial differential equations (PDE) with regard to slope collapse are derived and integrated into the physics-guided loss function. A kinematically admissible failure mechanism that rigorously satisfies the Mohr-Coulomb associated flow rule is obtained by minimizing the loss function, thereby circumventing complex mathematical calculations. The discrete points generated by PINN are selected and refined through a series of procedures to represent the failure block of slopes using a two-dimensional matrix. The entire training process of the PINN-based framework is conducted without the need for any labeled data. The resulting discretized failure mechanism is meshfree and capable of accommodating spatially discrete data. A validation exercise is performed to verify the proposed framework by comparing it with the classical 3D rotational failure mechanism. To further consider the impact of external excitation on slope stability, a hybrid PINN framework is developed to assess the stability of slopes subjected to complex external environments. In addition to the PINN to generate a failure mechanism, a parallel PINN is employed to acquire the corresponding spatially discrete data of specified external excitations. The hybrid PINN framework for seismic stability assessment of slopes is demonstrated by way of example, indicating favorable feasibility and applicability of the developed approach. The proposed PINN-based framework provides innovative and promising avenues for 3D slope stability analysis.

3D Semi‐Analytical and Numerical Upper‐Bound Homogenisation Approaches to the Out‐of‐Plane Strength Domain of a Running‐Bond Masonry Wall

ABSTRACTThe present contribution proposes a new semi‐analytical homogenisation approach to determine a running‐bond masonry wall's in‐ and out‐of‐plane strength domain based on the yield design framework. The main novelty of such an approach is that it does not rely on simplifying assumptions such as infinitely thin joints or plane stress state within the blocks, by making use of 3D virtual failure mechanisms in the kinematic approach. The new semi‐analytical approach is then compared to a state‐of‐the‐art numerical implementation of the kinematic approach of yield design, relying on semi‐definite programming. Several comparisons show a good agreement between the semi‐analytical and the numerical approaches and outline the computational efficiency of the semi‐analytical approach as well as the fact that it is very well suited for engineering design purposes. Both proposed approaches are then compared to existing approaches based on the limit analysis or yield design framework.

3D seismic bearing capacity of rectangular foundations near rock slopes using upper bound method

The analysis of foundation's bearing capacity is generally conducted under the assumption of plane strain. However, the damage of rectangular foundation usually presents obvious three-dimensional (3D) effect. This article considers this 3D effect, which, for the first time, provides a theoretical framework for assessing the bearing capacity of rectangular foundations placed on rock slopes under seismic condition. In order to apply the kinematic method of limit analysis, a new 3D kinematically admissible collapse mechanism is first constructed. Owing to the point-to-point discretized technique, this 3D mechanism can avoid complex surface integration and effectively reduce spatial coordinate iterative calculations while maintaining high accuracy. The pseudo-static method is adopted to simulate the effects of earthquakes. Generalized multi‑tangential technique is employed to derive the Mohr-Coulomb constants from Hoek-Brown criterion. The comparison between the present results and results of previous literatures proves the validity of the theoretical framework in this paper. Shape factor and reduction factor are introduced in the parametric study, showing that the 3D effect is more obvious when the rectangular foundation aspect ratio is reduced. As the rectangular foundation location moves away from the slope, its bearing capacity gradually converges towards that of the horizontal foundation. The critical collapse mechanism is also explored, demonstrating that the collapse extent expands with increasing internal friction angle of the rock.

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.

Effects of inclined loads on strip footings with an underlying tunnel

The self-developed Finite Element Limit Analysis (FELA) code was utilized to examine the ultimate bearing capacity of strip footings positioned above tunnels affected by inclined loads. Bearing capacity factors were predicted using both upper bound (UB) and lower bound (LB) solutions, with a variance of less than 3 %. The primary focus of this study lies in assessing the influences of underlying tunnels and inclined loads on potential failure modes. In particular, the concept of failure envelopes was introduced, by which the load properties (inclination angle), the tunnel location (relative lateral distance and longitudinal distance from the footing) and the material properties of rock masses (GSI, mi, sci, g) were involved in to facilitate preliminary designs. In addition to envelopes, the transition among failure patterns was summarized, resting with any possible factors.

Constraints on the Geometry of Peripheral Faults above Mafic Sills in the Tarim Basin, China: Kinematic and Mechanical Approaches

Host rock deformation associated with sill emplacement is used to constrain magma transfer and storage within the upper crust. In contrast to classic models suggesting that the host rock above mafic sills is dominated by elastic bending, recent studies show that bounding faults that limit the uplift area can occur at the peripheries of a mafic sill. However, the accurate dip of this type of fault, named peripheral faults here, is still not well constrained. Their origin is also controversial in some cases. In this study, kinematic modeling and limit analysis are performed to better constrain the structure and mechanical properties of the peripheral faults based on seismic interpretation of a mafic sill from the Tarim Basin, China. The trishear kinematic model successfully reproduces peripheral faulting and associated folding of the host rock by performing a displacement of 58 m on a vertical fault plane with a fault propagation (P) to fault slip (S) ratio of 2.5. The limit analysis also predicts vertical damage at the sill tip by sill inflation. These results suggest that the dip angle of the fault in the case study is 90°, which is more accurate than that from the seismic interpretation with an 88° inward dip. This value may vary in other cases as it depends on the sill geometry (such as diameter and inclination), thickness, depth, and mechanical properties of the host rock. The study supports that peripheral faulting and associated folding can occur at the tips of the mafic sill due to the vertical uplift of the host rock caused by sill inflation. It is also suggested that trishear kinematic modeling and limit analysis are effective methods for studying the geometry of peripheral faults.

Shear performance of horizontally curved steel box bridge girders under hydrocarbon fire exposure conditions: Numerical investigation and design implications

Recent fire incidents indicated that fire-induced bridge collapse is commonly associated with local shear failures occurred in web plates. However, current studies provided failure modes only related to deflection arose from steel bridge girders, irrespective of the underlying shear mechanism. To bridge the knowledge gap, this study presents an investigation on structural behavior of horizontally curved steel box bridge girders (HCSBBGs) under the combined effects of hydrocarbon fire exposure and shear-dominant loadings. The numerical models are developed utilizing the ANSYS program, incorporating a validated thermal-structural analysis model and a shear limit analysis model that accounts for web initial imperfections. These models are further employed to examine effects of fire-shear load scenarios, curvature radius, web slenderness ratio, and aspect ratio on shear performance of HCSBBGs. Herein, deflections, degradation of shear capacity, and web out-of-plane displacement (WOPD) of the steel box bridge girders are discussed in detail. Failure modes of steel box bridge girders can be determined via moment-shear interaction diagrams as specified by Eurocode 3 (EC3). The findings indicate that decreasing web slenderness ratio and web aspect ratio can enhance shear capacity of steel box girders. Further, shear failure modes are predominant when fire occurs near side support of HCSBBGs. Current fire-resistant design practices relying on deflection or flexural limit based failure criteria, cannot be applicable in predicting fire resistance of steel box bridge girders under shear failure mode. Therefore, a failure criterion based on the ultimate web out-of-plane displacement ratio (UWOPDR), related to the web shear limit state, can be more suitable for predicting fire resistance of HCSBBGs.

Simplified discontinuous field model for RC beams with a medium shear span ratio: Experimental study and shear strength formula

It is generally believed that the shear span ratio is one of critical factors governing the shear failure of reinforced concrete beams. For a certain shear span scenario, the shear transfer mechanism is often attributed to one of the two typical categories: ‘beam-action’ for slender beams with a large shear span ratio, and ‘arch-action’ in deep beams with a small shear span ratio. However, in the medium shear span ratio range, neither the beam action nor the arch action dominates. In this study, an alternative model based on the notion of discontinuous fields is established under the framework of limit analysis. A shear strength formula for medium shear spans is derived subsequently by solving a combined action of the smeared-arch subfield and the vertical stirrups linking the fan subfield. After that, the least shear strength is obtained via the kinematic analysis of these discontinuous fields. To examine the proposed model, an experimental study is conducted through two T-beam specimens with artificially preset cracks and two benchmark specimens. In comparison with existing approaches and test results collected from this study and other twenty-two specimens in the literature, the proposed discontinues field model has a fairly good agreement in the shear strength predictions. A further discussion on the advantages and limitations of the simplified discontinuous field model also demonstrates that it provides a better transition of the shear strength verifications between slender beams and deep beams.

Computational modeling of three-dimensional slope reinforcement by anchor cable micropile group considering coordinated deformation

The application of anchor cable micropile group (ACMPG) in small and medium-sized slope support is gradually widespread. However, the coordination deformation of the structure and the resistance of the soil in front of the piles are less considered in the existing calculation models. Therefore, based on the load transfer approach and according to the structural characteristics of the ACMPG, the calculation method was provided for the resistance of the soil in front of the pile and the tension of the anchor after deformation. And the equations for calculating the resistance of the ACMPG were established. Based on the method of upper bound limit analysis, the display equation had been established for the safety factor of reinforcing the 3D slope of the structure. And finite element models were developed to validate the theoretical model. Finally, a parametric study was conducted to investigate the influence of slope angle, soil friction angles, location, 3D effects, on the slope safety factor and the support effect of piles and anchor cables supporting the structure. The results showed that the theory proposed in this paper can accurately calculate the safety factor of slopes, which is helpful for the design and construction of slope reinforcement.

Dislocation-based finite element method for homogenized limit domain characterization of structured metamaterials

PurposeHeterogeneous and micro-structured materials have been the object of multiscale and homogenization techniques aimed at recognizing the elastic properties of the equivalent continuum. The proposed investigation deals with the mechanical characterization of the heterogeneous material structured metamaterials through analyzing the ultimate strength using the limit analysis of the Representative Volume Element (RVE). To get the desired material strength, a novel finite element formulation based on the derivation of self-equilibrated solutions through the finite elements devoted to calculating the lower bound theorem has been implemented together with the limit analysis in Melàn’s formulation.Design/methodology/approachThe finite element formulation is based on discrete mapping of Volterra dislocations in the structure using isoparametric representation. Using standard finite element techniques, the linear operator V, which relates the self-equilibrated internal solicitation to displacement-like nodal parameters, has been built through finite element discretization of displacement and strain.FindingsThe proposed work presented an elastic homogenization of the mechanical properties of an elementary cell with a geometry known in the literature, the isotropic truss. The matrix of elastic constants was calculated by subjecting the RVE to numerical load tests, simulated with a commercial FEM calculation code. This step showed the dependence of the isotropy properties, verified with Zener theory, on the density of the RVE. The isotropy condition of the material is only achieved for certain section ratios between body-centered cubic (BCC) and face-centered cubic (FCC), neglecting flexural effects at the nodes. The density that satisfies Zener’s conditions represents the isotropic geomatics of the isotropic truss.Originality/valueFor the isotropic case, the VFEM procedure was used to evaluate the isotropy of the limit domain and was compared with the Mises–Schleicher limit domain. The evaluation of residual ductility and dissipation energy allowed a measurement parameter for the limit anisotropy to be defined. The novelty of the proposal consisted in the formulation of both the linearized and the nonlinear limit locus of the material; hence, it furnished the starting point for further limit analysis of the structures whose elementary volume has been described through the proposed approach.

Kinematic bearing capacity analysis of strip footings on reinforced soils using discretisation technique

The kinematic discretisation technique of upper bound limit analysis is adopted in this study to evaluate the limit load of a strip footing located on the soils reinforced with different geosynthetic layers. The reinforcement is assumed to withstand the tension but not bending moment. The failure model of the reinforced foundation soil is generated using the discretisation method, a ‘point by point’ technique. Two failure modes of the reinforcement, the tensile rupture and sliding, are considered in the analysis. This article proposed the formulas for the first time to evaluate the effect of soil friction angle φ on the ultimate bearing capacity and on the failure mechanism of foundation. The calculation results are given considering different reinforcement layers, which describe the impact of different reinforcement depth and length on the ultimate bearing capacity. The optimum positions of different reinforcement layers and the optimum reinforcement length are obtained. The bearing capacity ratios are also presented in figures combing impact of reinforcement depths, soil cohesion and friction angle. The present method was verified by the results reported in literature. The present method can provide a reference for the footing design on the reinforced soils.

Impact of porosity on free vibration and limit analysis of power-law-based functionally graded disks

PurposeThis study aimed to analyze the free vibration of a radially graded Ni-Al2O3-based functionally graded (FG) disk with uniform thickness.Design/methodology/approachUsing the energy method, natural frequencies of rotating and non-rotating disks were determined at the limit elastic angular speed. Material properties were estimated using a modified rule of mixture. Both even and uneven porosity variation effects were considered in the material modeling. Finite element analysis validated the analytical approach.FindingsThe study explored limit angular speeds and natural frequencies across various grading indices, investigating the impact of porosity types and grading indices on these parameters.Practical implicationsInsights from this research are valuable for researchers and design engineers involved in modeling and fabricating porous FG disks, aiding in more effective design and manufacturing processes.Originality/valueThis study contributes to the field by providing a comprehensive analysis of free vibration behavior in radially graded Ni-Al2O3-based FG disks. The incorporation of material modeling considering both even and uneven porosity variation adds originality to the research. Additionally, the validation through finite element analysis enhances the credibility of the findings.