Theorem Of Limit Analysis Research Articles

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.

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.

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.

Ultimate Support Pressure Determination for Shield Tunnel Faces in Saturated Strata Based on Seepage Flow Control

Existing methods for calculating the ultimate support pressure of tunnel faces do not consider the control of seepage flow. Therefore, a model for calculating the ultimate support pressure under seepage conditions was established based on a two-dimensional water head distribution model and the upper bound theorem of limit analysis. The reliability of this method was verified through comparisons with other studies. Subsequently, the influence of water level and tunnel face water pressure coefficient on stability was analyzed. The results indicate that the ultimate support pressure is linearly positively correlated with the water level and tunnel face water pressure coefficient; as the water level increases and the water pressure coefficient decreases, the failure area extends and enlarges. Finally, an existing seepage flow calculation formula was introduced, and a method for calculating the ultimate support pressure based on seepage control was proposed. The appropriate tunnel face water pressure coefficient is determined through the seepage flow calculation formula, and the corresponding ultimate support pressure is then calculated. The results demonstrate that this method can provide better theoretical guidance for seepage control in tunnel faces in practical engineering.

Upper Bound Analysis of Two-Layered Slopes Subjected to Seismic Excitations Using the Layer-Wise Summation Method

Due to natural sedimentation and artificial filling, slopes exhibit heterogeneity in the form of multi-layer soils, namely, layered slopes. Compared with homogenous slopes, the failure mechanism of layered slopes is more complex owing to the different shear strengths of each soil layer. Therefore, it is of great importance to gain insight into the stability of layered slopes. In this study, the upper bound theorem of limit analysis incorporated with a pseudo-static approach is utilized to investigate the seismic stability of two kinds of two-layered slopes: one with a stiff lower soil layer and the other with a weak lower soil layer. Three failure patterns, namely face failure, toe failure and base failure, are taken into account. A depth coefficient (Δ) is introduced to describe the distribution of two soil layers. The layer-wise summation method is adopted to calculate the safety factor and yield acceleration coefficient more conveniently. Based on Newmark’s method, the earthquake-induced horizontal displacement is estimated. The calculated results are validated by comparisons with published literature and the numerical method in terms of safety factor, critical failure surface and yield acceleration coefficient. The results show that the depth coefficient has a significant influence on the failure mechanism of two-layered slopes by determining whether the stability of upper-layered soil is dominant in the overall slope stability or not. Inaccurately identifying the failure patterns will overestimate the seismic performance of two-layered slopes in the aspects of safety factor and yield acceleration coefficient, leading to an underestimation of earthquake-induced horizontal displacement.

The three-dimensional stability analysis of piled slopes in unsaturated and inhomogeneous soils with nonlinear failure criterion

Under the nonlinear Mohr-Coulomb (MC) failure criterion, based on the upper-bound limit analysis theorem, the three-dimensional(3D) stability of piled unsaturated inhomogeneous slopes is analyzed. By incorporating the nonlinear strength parameters c t and φ t, the energy balance equation is established, and the explicit expression of slope safety factor (F S) is derived by the gravity increase method (GIM). In addition, a genetic algorithm (GA) is used to calculate the minimum F S. The results obtained were compared with those of existing literature to verify the effectiveness of the method used in this paper. The influence of nonlinear parameters on slope stability was studied through parameter analysis. The results show that the nonlinear parameters of soil have a significant influence on the stability of piled slopes. The F S increases with the increase of nonlinear coefficient and uniaxial tensile strength and decreases with the increase of initial cohesion of soil. In addition, the critical slip surface of the slope becomes shallower with the weakening of soil nonlinearity. Furthermore, ignoring the suction effect will overestimate the slope stability, and the slope stability intensified with decreased soil inhomogeneity.

Collapse analysis of masonry arches and domes considering finite friction and uncertainties in compressive strength

Within the context of the lower bound theorem of classical limit analysis, the collapse of stone masonry arches and domes characterized by finite friction and uncertain finite compressive strength is addressed. Self-weight is considered, along with a live point load applied at the crown. A multi-constrained funicular approach based on the concept of force densities is implemented to provide the maximum static load multiplier for any assigned value of the friction coefficient and of the compressive strength. The numerical method, which takes into full account the stereotomy of blocky stone structures, is validated by using a novel version of the semi-analytical Durand-Claye’s stability area method. Two different types of brickwork are considered in the investigation, addressing the compressive strength as a random variable with prescribed log-normal distributions. Due to the limited availability of data to elaborate probabilistic models for the angle of friction, the effect of this mechanical parameter on the probability of failure of the blocky structures is assessed by developing a set of fragility curves. For each discrete value in a representative range of friction coefficients, a Monte Carlo investigation is performed by iteratively applying the funicular method considering samples of the random variable that describes the compressive strength. The retrieved load multipliers are processed to estimate the probability of failure for any given magnitude of live load, which depends on the value of the friction coefficient. An insight into the collapse modes occurring in the masonry arches and domes under examination is provided, as well.

Revisiting the face stability of rock tunnels in the Hoek–Brown strength criterion with tension cutoff

AbstractIn this work, the three‐dimensional stability of deep tunnel faces is evaluated in rock masses characterized by the generalized Hoek–Brown (H–B) criterion from the perspective of the limit analysis theorem. Considering that underground engineering is gradually developing towards larger burial depths and larger sizes, and the tensile strength of rocks is usually overestimated, the concept of tension cutoff (T‐C) is introduced to substitute the parabolic form of the H–B envelope in the tensile region with a circular arc. The multitangent technique is used to piecewise approximate the H–B envelope, instead of the conventional linear substitution, accounting for the nonlinear dependence of the shear strength on the pressure. Meanwhile, the earthquake loading is considered by the classic pseudostatic method. The tunnel face stability is measured by a stability number, which is calculated based on a newly developed multicone failure mechanism. Parameter studies indicate that the T‐C has almost no impact on the critical support pressure, while has a significant effect on the critical stability number, especially in the presence of earthquakes. Nearly all stress points back‐calculated from the rupture angles fall within the T‐C region, except for the first stress point associated with the first segment. Interestingly, the influence of the geological strength index (GSI) on the face stability is inverse to that of mi due to the brittleness of rocks. Moreover, the critical stability number is not sensitive to mi, but sensitive with the introduction of earthquake loadings.

A Method for Calculating the Instability Probability in a Deeply Buried Tunnel Construction with a Hidden Karst Cave at the Bottom

In order to quantitatively assess the risk of deeply buried tunnel construction in the case of a hidden karst cave at the bottom of the tunnel, the first deeply buried tunnel construction failure model (PH−1) is firstly proposed based on the upper bound theorem of limit analysis and Protodyakonov’s theory. Thereafter, based on the upper bound theorem of limit analysis and the energy dissipation theory, the second deeply buried tunnel construction failure model (PH−2) is proposed. Furthermore, on the basis of the reliability theory, the dimensionless performance functions of each damage region under these two deeply buried tunnel construction damage modes are constructed. Finally, the instability probability of each damage region under the two damage modes is calculated using the Monte Carlo sampling method. This study suggests that the PH−1 failure model is not suitable for analysing the probability of instability in tunnel construction in hard plastic red clay strata, while the PH−2 failure model is appropriate for analysing the probability of instability in tunnel construction in hard plastic red clay strata. The worse the geological conditions, the larger the diameter of the cavern, the smaller the clearance between the cavern and the tunnel, and the greater the threat to safe tunnel construction.

Combined bearing capacity of footings on geogrid-reinforced granular fill over soft clay

The current study investigates the ultimate bearing capacity of obliquely/eccentrically loaded shallow strip foundations resting on a geogrid-reinforced granular fill with limited thickness over a very soft to soft clay layer. To this end, the lower bound theorems of the finite element limit analysis (FELA) and second-order cone programming (SOCP) are effectively exploited to simulate the underlying clay deposit, geogrid layer, and granular fill along with the inclined/eccentric loading exerted on the overlying footing. The accuracy of the adopted formulations is rigorously examined through several comparisons with the results of a well-established analytical approach in the literature. A comprehensive parametric study is then conducted to properly examine the influences of the geosynthetic layer characteristics and soft clay properties on the overall bearing capacity and failure envelope of the strip footing subjected to wide ranges of inclined and eccentric loadings. The results show that placement of a geogrid-reinforced granular fill layer over the soft clayey soil significantly increases the bearing capacity of the shallow foundation. The ultimate tensile strength of the reinforcement layer and the cohesion of the underlying very soft to soft clay deposit are also observed to have considerable effects on the size of failure envelopes.

Limit analysis of masonry domes with oculus and lantern: A comparison between different approaches

This paper aims to illustrate and compare different approaches for the limit analysis of masonry domes of revolution with an oculus at the top. It specifically focuses on the influence of the limited compressive and tensile strength of masonry on the bearing capacity of the dome. The first assessment is conducted through a semi-analytical formulation that revisits Durand-Claye’s method within the framework of the lower bound theorem of limit analysis. The second investigation is performed using a kinematic approach, which allows for closed-form solutions by exploiting the upper bound theorem of limit analysis and the virtual work theorem. These approaches are applied to a case study that was experimentally tested, involving a hemispherical dome with an oculus. The dome is subjected to its self-weight and a vertical load at the crown. Parametric investigations are presented to compare the results obtained through the proposed methods with those derived from the classical limit analysis of masonry structures, based on Heyman’s hypotheses.

Modified pseudo-dynamic bearing capacity of reinforced soil foundations

In engineering practice, horizontal layers of reinforcement geosynthetics are widely used to improve the bearing capacity of foundations. However, there are few studies on the seismic bearing capacity of foundations on reinforced soil, and there is a lack of derivation of theoretical formulas in this direction. In this paper, based on the upper bound theorem of limit analysis, the seismic bearing capacity of foundations on reinforced soil is studied for the first time using the modified pseudo-dynamic approach. Seismic forces are calculated using a slicing method that adapts to spatial and temporal variations. Based on three different failure mechanisms, three different formulas and constraints are used. Each formula contains 61 variables, and the bearing capacity value is optimized by MATLAB program. By comparing the calculation results of three failure mechanisms, the exact upper bound solution of seismic bearing capacity for sliding failure is obtained. Considering the temporal as well as spatial characteristics of seismic forces, the effects of the coefficient of seismic acceleration, the internal friction angle, the damping ratio, the normalized frequency, and the reinforcement burial depth on the seismic bearing capacity are analyzed, providing a theoretical basis for the application of reinforced soil foundations in engineering practice.

Revisiting the face stability of circular tunnels driven in strength nonlinearity soils

The face stability of shield-driven tunnels in soil materials with distinct strength nonlinearity is re-evaluated, based on the kinematic limit analysis theorem. The nonlinear Power-Law failure criterion is piecewise approximated by a multi-tangent technique to improve the conventional single straight-line substitution. A new multi-cone rotational mechanism formed by n curvilinear cones is proposed to accommodate the nonlinearity of the strength envelope considering the associative flow rule. The analytical solution of the critical support pressure on tunnel faces is derived from the energy equilibrium equation. Calculation results show that the multi-tangent method can improve the calculated values by up to 20% compared to the generalized tangent method. According to the back-calculated stress vectors, it is interestingly found that the stress level is not the critical factor influencing the tunnel face stability, while the shear strength parameters are the direct factor. Although the upper-bound analysis is a kinematic method without involving stress analysis, the parametric form of the strength criterion gives a chance to infer the possible stress distribution on the rupture surface and explain the variational trend of the result based on the distribution of back-calculated stress vectors.

Temperature effect on the stability of 3D reinforced slopes under unsaturated flow conditions

Slopes are increasingly exposed to elevated temperatures owing to extreme climates or human activities, such as droughts, heat waves, and shallow geothermal technologies. This phenomenon has worsened with global warming. However, the influence of temperature changes on slope stability has not been fully investigated from the perspective of safety. This study introduces an analytical framework for calculating the safety of three-dimensional reinforced slopes which considers the effects of temperature changes and different unsaturated flow conditions. With temperatures ranging from 10 °C to 40 °C, the matric suction of unsaturated soil is thermally sensitive, which causes change in the shear strength of unsaturated soil. To incorporate the temperature effect into the reinforced slope stability assessment, the temperature dependence of the apparent cohesion of unsaturated soil was quantified using a temperature-related effective stress model and soil–water characteristic curves. Thus, the safety factors and reinforcement strength of the slopes were deduced based on the upper-bound theorem of limit analysis. For validation, the proposed method was compared to two previously reported analytical methods, and its effectiveness was further verified through numerical simulations. Finally, the temperature effect was fully investigated and discussed by performing a series of parametric analyses, and suggestions were provided for practical engineering reference.

Theory of Plastic Mechanism Control for Linked Column Frames

AbstractThe main purpose of this work is to define a design procedure for dissipative replaceable link frames (LCFs). These structural schemes are characterized by the columns connected with link elements representing the plastic zones of the structure and, consequently, they assure the maximum dissipation of seismic energy in input. The design rules, proposed in this paper, are based on the Theory of Plastic Mechanism Control (TPMC). This procedure is based on the extension of the kinematic theorem of limit analysis to the concept of the equilibrium curves of the failure mechanisms and it guarantees the achievement of the global collapse mechanism under severe seismic actions. In order to evaluate the accuracy of this procedure, a case study is presented. In particular, a LCF with four storeys and three bays is designed by means of the proposed design rules. Finally, the same steel frame is implemented in SAP2000 program, and a pushover analysis is performed to evaluate the ultimate behaviour of the modelled structure.

Analytical Solution for Bearing Capacity of Reinforced Strip Footings on Unsaturated Soils under Steady Flow

The study of analytical solutions for the bearing capacity of reinforced soil foundations is a very important topic in engineering mathematics. Existing evaluations of the foundation-bearing capacity on reinforced soils are based on dry conditions, while many foundations are located on unsaturated soils in real engineering. In this paper, a new formula for the bearing capacity of reinforced strip footings on unsaturated soils is presented. Two sliding failure mechanisms are constructed based on the position of the reinforcement layer relative to the sliding surface. The distribution of apparent cohesion in the depth direction is calculated by considering the effect of matrix suction. By additionally considering the work conducted by the reinforcement and the contribution of the apparent cohesion, the bearing capacity formula is obtained using the upper bound theorem of limit analysis. The bearing capacity solution is obtained by adopting the sequential quadratic programming (SQP) algorithm. Comparing the results under two failure mechanisms, the optimal bearing capacity and the optimal embedment depth of reinforcement are obtained. The results of this paper are consistent with those of the existing literature. Finally, the effects of reinforcement embedment depth, effective internal friction angle, uniform load, and unsaturated soil parameters on the optimal bearing capacity are investigated through parametric analysis. This paper provides useful recommendations for the engineering application of reinforced strip footings on unsaturated soils.

Seismic Stability Limit Analysis of Unsaturated Soil Slopes Reinforced by Frame Beam Anchor Plates

The frame beam anchor plate (FBAP) reinforced slope has been popularized and applied in some fill slope projects. Nevertheless, the existing design method ignores the influence of earthquake action and the suction effect of unsaturated soils. The stability of unsaturated soil slopes reinforced with FBAPs under an earthquake is analyzed based on the upper bound theorem of limit analysis and pseudo-dynamic method to solve the above problems. The influence of the suction effect on the effective weight and strength of unsaturated soils under one-dimensional steady seepage conditions was considered in the derivation of the energy balance equation. A simplified method based on the volume integral of the rotating body is used to calculate the external work rate. Two possible failure modes of anchorage structures are considered when solving internal energy dissipation. Results show that the influence of suction and seepage conditions on slope stability decreases with the increasing soil pore size distribution parameter n and saturated hydraulic conductivity ks. Also, the stability of the slopes is significantly improved with the increase of total rows of anchor plates. Likewise, the stability of slopes decreases with the increase in the horizontal seismic coefficient kh and amplification coefficient fa.

Upper Bound Solution Analysis of the Ultimate Bearing Capacity of Two–Layered Strip Foundation Based on Improved Radial Movement Optimization

Considering the change of different soil layer parameters of a two–layered strip foundation, a planar kinematically permissible multi–block failure mechanism of a two–layered strip foundation under vertical uniform load is formulated. Based on upper–bound limit analysis theorem, the solution process of ultimate bearing capacity of a two–layered strip foundation is proposed. The improved radial movement optimization (IRMO) could search the critical slip surface of the foundation and calculate associated ultimate bearing capacity. On this basis, analyze the influence parameters. The results show that the IRMO algorithm is feasible, stable, and efficient in solving the ultimate bearing capacity. With the increase in the internal friction angle and cohesion of the two–layered strip foundation, the critical slip surface will expand more deeply, and the ultimate bearing capacity will increase. The influence of upper soil parameters on the calculation results is greater than that of lower soil. For the case with upper–hard soil, the ultimate bearing capacity of it increases gradually with the increase in the H/B ratio. For the upper–soft case, the ultimate bearing capacity of it decreases gradually with the increase in the H/B ratio. Moreover, the increase of ground overload will also cause a linear–increasing in the ultimate bearing capacity.

Bearing capacity of shallow strip foundations on reinforced soil subjected to combined loading using upper bound theorem of finite element limit analysis and second-order cone programming

Bearing capacity of shallow strip foundations placed over both unreinforced and reinforced granular soils and subjected to combined loading is calculated by implementation of upper bound finite element limit analysis in conjunction with second-order cone programming technique. The soil mass is considered to be a granular medium reinforced with the inclusion of horizontal single and double layers of geogrid elements. To represent the impact of tensile resistance of reinforcement elements, an axillary variable is utilized to simulate the plastic energy dissipation of reinforcing elements without the application of any stress variables. General loading conditions are considered to profile the expansion of the failure envelopes in the V-H and V-M spaces for shallow strip foundations resting on reinforced soils. Results of this study are compared and validated against three different cases including the bearing capacity of shallow strip foundations subjected to vertical loading in the case of perfectly smooth and fully rough interfaces, limit load of shallow strip foundations subjected to combined loading, and vertical bearing capacity of shallow strip footings over a soil deposit reinforced with a single-layer of geosynthetic. Consistency of the results with those reported in the literature demonstrates the efficiency of proposed extended finite element upper bound formulations. The results are reported in the form of an efficiency factor which can be multiplied by the vertical bearing capacity of the shallow strip footing over unreinforced soil to find the ultimate bearing capacity of the reinforced foundation. The results show that the load inclination angle and eccentricity have significant impacts on the limit load and the optimum embedment depths of the reinforcement layer. It is further noted that the effect of reinforcement layer(s) in the improvement of bearing capacity in the V-M space is more pronounced than in the V-H space. The efficiency of reinforcement elements inclusion in improving the expanse of safe loading domain in the V-H and V-M spaces enhances with an increase in the soil internal friction angle.

Research on Upper Bound Limit Stability Analysis of Multi-Step High Steep Slope

In order to explore the stability of multi-step high steep slope, this paper considers layer distribution of soil and builds a slope failure mechanism which meets requirements of velocity separation based on plastic mechanics principles, then derives the calculation method of external force power and internal energy dissipation power in failure area using numerical analysis theory, and proposes a loop algorithm for slope stability analysis combining upper bound limit analysis theorem and strength reduction principle. At the same time, in order to enhance the applicability of the upper bound limit analysis method, this paper develops a slope stability calculation system of upper bound limit analysis combined with computer programming technology and evaluates the accuracy of the calculation system through slope engineering practice in open-pit mine. The results show that stability factor by calculation system of upper bound limit analysis is an upper limit solution compared with limit equilibrium method, but the error rate of two methods is less than 5%, which indicates that the slope stability calculation system established in this paper has high accuracy in stability analysis of multi-step high steep slope in the practice of open-pit mine. On the other hand, the most dangerous sliding surface searched by upper bound limit analysis can fully solve velocity problems of multi-step high steep slope failure, so it has stronger reference values in engineering practice.