Stability Charts Research Articles

Comparison of stability analysis methods for safe design of volcanic rock slope

The performance of the stability chart (SC), limit equilibrium (LE), and finite element (FE) methods for stability analysis of weathered volcanic rock slopes under static and earthquake loads is not fully understood. This research aimed to evaluate the performance of the SC, LE, and FE methods for slope stability analyses by studying a case of a weathered volcanic rock slope at the Leuwikeris Dam diversion tunnel in Indonesia. Site characterisations were carried out to obtain the input parameters for the static and pseudostatic slope stability analyses. The results showed that using various search methods of failure surface and the suggested optimum number of finite elements, multiple non-circular failure surfaces were predicted from the LE and FE methods that were not anticipated in the initial slope design based on the SC method. The location and shape of the failure surfaces from the LE method agreed with those from the FE method. The difference between the Fs values from the static LE analysis and those from the static FE analyses was less than 14%, but the discrepancy increased up to 78% in the pseudostatic analyses. This study highlights the importance of validating the results from one method with the other methods to prevent slope failures.

Dynamic stability and response of morphing wing structure with time-varying stiffness

Abstract Morphing, adaptable or smart structures are being used in mechanical and aerospace applications in recent years. These structures often have the property of time-varying stiffness or inertial properties, which can cause parametric instability issues that are not well understood. This paper examines the dynamic stability and response of a morphing aircraft wing with periodically time-varying structural stiffness. The wing is modeled as a beam with coupled bending-torsion motion, and parametrically excited stiffness. Aerodynamic loads introduce aerodynamic damping and aerodynamic stiffness to the wing structure. The dynamic and aeroelastic equation of motion resembles a coupled, damped Mathieu-type equation but differs with asymmetric damping and stiffness matrices, and symmetric inertial matrix. Further, these equations are functions of airspeed, magnitude and frequency of parametric excitation. Initially, dynamic stability of the wing is analyzed using Floquet theory, and the instability regions are numerically quantified by stability charts. Subsequently, dynamic responses in the stable and unstable regions are investigated with a Floquet-based Harmonic balance method. The findings reveal that, at zero airspeed, the combination instabilities are eliminated by varying the bending and torsional stiffness with equal magnitudes and frequency. However, as airspeed increases, instability regions shift unevenly, leading to the emergence of new instabilities. Furthermore, the response analysis within stable regions uncovers several unfavorable zones for operating the variable stiffness, where response decay is slower. The results clearly show that parametric excitation can cause unusual phenomena that significantly impact the operation of morphing wings with variable stiffness, which needs to be thoroughly investigated for successful implementation.

Seismic stability charts for three-dimensional rock slopes using Hoek–Brown criterion

Seismic excitation serves as a crucial trigger for slope collapse in seismically active regions. This study employs the non-linear shear strength reduction (SSR) technique in conjunction with the pseudo-static method to analyse the seismic stability of three-dimensional (3D) rock slopes following the generalised Hoek–Brown (GHB) criterion. Considering the non-linear characteristics of the GHB envelope, the instantaneous Mohr–Coulomb (MC) parameters for rock masses are identified by locating the tangent at the intersection of the GHB envelope with the shear and normal stress planes. These parameters are subsequently incorporated into the non-linear SSR analysis for seismic stability based on the MC criterion. The non-linear SSR technique is then validated by comparing the calculated factor of safety (FOS) with those from other methods and used to assess the impact of geometric and GHB parameters on the FOS of 3D rock slopes. Finally, a set of seismic stability charts for 3D rock slopes in GHB media has been developed and applied to evaluate the seismic stability of two practical rock slope cases following the GHB failure envelope.

Roof stability analysis for a shallow-buried cylindrical tunnel in Hoek-Brown rock strata

Roof stability of shallow-buried cylindrical tunnels in rock strata following the generalized Hoek-Brown criterion (GHB) is analyzed in this work. Based on the limit analysis method, a three-dimensional (3D) shallow-buried collapse mechanism for tunnel roofs is developed to derive various stability measures. Optimization codes are programmed to capture the optimal stability measure solutions. The effects of the 3D geometric characteristics, the rock mass strength parameters, and the buried depth on tunnel roof stability are investigated by a parametric analysis. The critical buried depth ratios corresponding to the transition between the shallow-buried and deep-buried tunnel roof collapse mechanisms are also presented. Thereafter a series of stability charts of the required supporting pressure and the factor of safety (FoS) is proposed for design purposes. This study provides a theoretical basis for design and engineering of tunnels with limited buried depth.

Balancing riderless electric scooters at zero speed in the presence of a feedback delay

AbstractThe nonlinear dynamics of electric scooters are investigated using a spatial mechanical model. The equations of motion are derived with the help of Kane’s method. Two control algorithms are designed in order to balance the e-scooter in a vertical position at zero forward speed. Hierarchical, linear state feedback controllers with feedback delay are considered. In the case of a delay-free controller, the linear stability properties are analyzed analytically, with the help of the Routh–Hurwitz criteria. The linear stability charts of the delayed controllers are constructed with the help of the D-subdivision method and semi-discretization. The control gains of the controllers are optimized with respect to the robustness against perturbations. The effects of the feedback delay of the controllers, the rake angle, the trail, and the center of gravity of the handlebar on the linear stability are shown. The performance of the control algorithms is verified by means of numerical simulations.

Stability and characteristics of kerosene-in-water emulsions with xanthate surfactants: Influence of hydrophilic-lipophilic balance and molecular weight

The understanding of emulsion stability is important for several industrial applications including foods, pharmaceuticals, petroleum, personal care products, and in our case mineral processing. Xanthate functional surfactants may be useful to improve mineral recovery through froth flotation by delivering oil to valuable mineral surfaces as they may be used to stabilize kerosene-in-water emulsions. The eco-friendly nature of naturally derived reagents such as vitamin E-based xanthates would be beneficial to the mineral processing industry. In this study, conventional (potassium amyl xanthate (PAX)) and novel xanthate functional surfactants such as α-tocopherol polyethylene glycol 400 succinate (TPGS) xanthates which we synthesized from Vitamin E were investigated as emulsifiers. The creaming and oil phase separation of the kerosene-in-water emulsions produced with the xanthate surfactants were studied. The influence of hydrophilic-lipophilic balances (HLBs) and molecular weights (MWs) of the water-soluble xanthates and other commercial surfactants including polyoxyethylene fatty acid esters (Tweens), sorbitan fatty acid esters (Spans), and polyoxyethylene fatty ethers (Brijs) on the characteristics of the kerosene-in-water emulsions was examined. The experimental results showed that the TPGS xanthate formed the most stable emulsion among the xanthate surfactants considered. The results indicated that HLB is not the only factor influencing the emulsifying properties of surfactants. The chemical structure, particularly the molecular weight of the surfactant has a significant impact on its emulsifying properties. Thus, an emulsion stability chart incorporating the HLB and molecular weight of surfactants is proposed. This stability chart can be used for a better classification of the emulsifying properties of commercially available surfactants and novel reagents.

Efficient approximation of stochastic turning process based on power spectral density

Turning is one of the most important material removal processes in manufacturing, where the proper understanding of the process is crucial for the quality of the final product. In this study, the stochastic cutting force is utilized to enhance the existing 1-degree-of-freedom turning model. A stochastic model is adopted to address the stochastic resonance phenomenon occurring near stability boundaries. Additionally, a novel simplified stochastic model is introduced with additive noise only. The comparison of the two models reveals that, with the recommended noise intensity of 0.1 to 1%, there is no significant difference in the stability charts and mean square characteristics between the two models. As a result, the time-consuming numerical methods can be bypassed, as the simplified model offers a computationally more efficient analytical approach to compute variance based on power spectral density (PSD). By combining techniques such as D-separation to determine stability boundaries and the PSD-based variance calculation, it only takes a minute instead of hours to construct a heatmap using the introduced simplified stochastic turning model that clearly outlines dangerous stochastic resonance regions inside the stable domain.

A practical stability/instability chart analysis for slope large deformations using the material point method

Slope instability and failure may lead to severe geological disasters, which are a primary concern of geotechnical engineering. To estimate the safety of a slope, Taylor (1937) proposed a stability chart that evaluates the factor of safety (FOS) of a homogeneous slope using several simple geometric and material parameters. This study extends the stability chart further into the field of failure, summarising and discussing the failure mode and affected area. A few hundred slope failure simulations have been conducted using the material point method (MPM), which reproduces similar FOSs to traditional stability charts. Three major failure modes, i.e., face failure, toe failure, and base failure, are categorized according to the different positions of the failure bands, and a failure mode distribution spectrum chart is established according to the material parameters of the slope. The affected area of slope failure is characterized by four indicators: the influence distance, run-out distance, sliding depth, and sliding volume. An instability chart is proposed to describe the affected area of slope failure based on simple geometry and material parameters. Further analysis indicates that the internal friction angle plays the most prominent role in the failure modes and slope failure volumes.

Recent advances in stability analysis and design of 3D slopes

Slope failures in nature and engineering are typically three-dimensional (3D). The rotational failure mechanism derived from the variational limit equilibrium (LE) method shows superior performance in the stability analysis of the 3D slope. In contrast to the traditional LE methods, it avoids arbitrary kinematical and statical assumptions. Stability charts obtained by the variational LE method are used to derive explicit expression equations of the safety factor, also known as the stability equations, for both 3D reinforced and unreinforced slopes. These equations are highly accurate and can provide a convenient means to assess slope stability in practical engineering. An example of a convex reinforced slope with a turning arc is illustrated in this study to investigate the effect of the 3D effects on the required reinforcement length for design. The results indicate that the 2D method underestimates the required reinforcement length when dealing with a 3D reinforced slope problem. Furthermore, a forensic analysis of the Yeager Airport reinforced slope is conducted within the framework of the variational LE method. The required strength for stability is found to be significantly less than the allowable strength of reinforcements without considering the decrease in soil shear strength. However, the required strength greatly exceeds the allowable strength when the decrease in soil shear strength is considered. The results verify that the decrease in shear strength of the weak layer is responsible for the collapse.

Critical Face Pressure of a Tunnel Driven by a Shield Machine Considering Seismic Forces and Tunnel Shape Influence

Evaluating critical face pressure with high accuracy is an important topic for shield tunneling. The existing research focuses more on the influence of complex geological conditions or environmental factors, and there are few reports on the influence of tunnel shape on critical face pressure. In reality, the tunnel shape has a significant impact on the deformation zone in front of the tunnel face, which may lead to non-negligible differences in shield pressure. To fill this gap, an efficient analytical approach is proposed to estimate the critical face pressure considering different tunnel shapes in the framework of limit analysis. As earthquakes are potential threats to the tunnel face stability in seismically active regions, the seismic effect is also taken into account with the help of the pseudo-static method. Several cases with the same area but different tunnel shapes are investigated using the limit analysis method. A comparison with static analysis is given to highlight the influence of seismic forces on the tunnel face stability. The results show that the critical face pressures increase by 15.3% when the tunnel face shape changes from rectangular to circular, and by 23.5% when the horizontal seismic coefficient varies from 0 to 0.1. A further validation with a 3D finite difference method is performed with respect to four typical tunnel shapes considered in this study. Lastly, several stability charts are provided for a quick estimation of the critical tunnel face pressure subjected to seismic forces. It is concluded that the proposed method can be applied to a tunnel stability assessment of various cross-sections and is highly efficient compared with numerical simulations.

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This article addresses the bifurcation characteristics and vibration reduction of a 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2$$\\end{document}-DOF dynamical system simulating the nonlinear oscillation of an asymmetric rotor model subjected to simultaneous multiparametric and external excitations. To suppress the system's vibrations, two 1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/2$$\\end{document}-DOF active dampers are attached to the system in linear and cubic nonlinear forms via a magnetic coupling actuator. The closed-loop system model is derived as two differential equations with multi-control terms, including cubic, quantic, and septic, coupled nonlinearly to two first-order systems. Applying perturbation theory, the system model is solved, and the autonomous system describing the closed-loop slow-flow dynamics is obtained. Through numerical algorithms, the motion bifurcation is analyzed using various tools such as 2D and 3D bifurcation diagrams, two-parameter stability charts, basins of attraction, orbit plots, and time response profiles. The analytical investigations confirm that the uncontrolled model behaves like a hardening Duffing oscillator with multistability characteristics, displaying simultaneous mono-stable, bi-stable, tri-stable, or quadri-stable periodic oscillations depending on both the asymmetric nonlinearities and angular velocity. Subsequently, the influence of different control parameters is analyzed to determine the threshold between mono and multi-stability conditions. Finally, optimal control parameters are designed to eliminate multistability characteristics and achieve minimum and safe vibration levels.

Development of stability charts for double salience reluctance machine modeled using hill’s equation

The paper presents a novel algorithm for the development of stability charts. The second-order differential homogeneous equation describing a double salient reluctance machine with a capacitance connected to its stator winding is transformed into hill’s equation. The circuit components are the stator coil time-varying inductance of a double salient reluctance machine, capacitance and resistance. All these are modeled by hill’s equation. The double salient reluctance machine acts as an energy conversion system. The maximum and minimum inductance of the energy conversion system is measured in laboratory by inductance, capacitance, and resistance (LCR) meter. These values help to determine the inductance modulation index. The inductance modulation indetx, the characteristic constant and the characteristic parameter obtained from modeling equations are used in the MATLAB/Simulink model. The MATLAB/Simulink simulations generate stable and unstable oscillations to form stability charts. The proposed stability charts are in good agreement with the Ince-Stritt stability chart, which is widely applied in physics, mechanics and in electrical engineering, especially where the state of stability of a system or an electric oscillatory circuit is to be determined.

Integrating Mathews Stability Chart into the Stope Layout Determination Algorithm

Integrating Mathews Stability Chart into the Stope Layout Determination Algorithm

Design stability charts for construction procedure of basement walls using staged bermed excavation: a parametric study

The construction of basement walls using discontinuous staged berms is based on excavating the central zone of a lot and leaving a lateral berm – which is then removed in phases with unexcavated sections (buttresses) remaining until a concrete wall is completed in the excavated areas. It is a commonly used technique in many nations, but its use is unsupported by regulations or scientific studies. This paper addresses the need for a analysis of this technique and makes a study of the geotechnical parameters of the subsoils where it is applied, as well as the commonly used practices. The research involved over 4000 finite element method calculations integrating geotechnical parameters with construction geometry. The results have enabled the preparation of four stability design charts based on linear polynomial surface adjustment for two project scenarios: with and without surcharge load. This paper proposes the use of these stability design charts for staged bermed excavations in a broad spectrum of soil types and the incorporation of a designer-defined safety level to ensure temporary stability. Additional charts are provided to assess the safety factor of projects once the geometries and geotechnical parameters of the subsoil are known.

A novel stability equation for the estimation of the factor of safety for homogeneous dry finite slopes

AbstractThis paper introduces a novel closed‐form equation (surrogate model) for approximating the Morgenstern–Price estimate of the factor of safety of homogeneous dry finite slopes with circular failure surfaces. Unlike typically used methods, the proposed equation does not require the definition of a critical failure surface, splitting the soil mass into slices, or the iterative reduction of soil resistance to the limit state. It can be easily programmed into calculators or computers and accurately determines the minimum factor of safety based on the shear strength parameters and slope geometry—meaning that it is ideally suited for integration into reliability calculations. The equation is determined parametrically for various soil parameters and slope geometry using the Morgenstern–Price method and compares favorably with conventional techniques, such as slope stability charts, limit equilibrium, and strength reduction methods (SRM). From a practical perspective, the proposed equation greatly simplifies the analysis of the slope stability as the creation of a numerical model is not required and is suited to use in the feasibility stage of a project, for example, for large linear infrastructure projects with differing slope geometries or in situations where a quick assessment is desired.

Supercritical Hopf bifurcations in the stage-structured model of housefly populations

Insect populations, which are diverse and widespread, provide a principal area of utilization of the stage-structured modeling approach. In this paper, housefly populations incorporating a stage-structured model are investigated theoretically and graphically. First, stability charts and rightmost characteristic roots of the positive equilibrium are elucidated analytically and numerically. Furthermore, the Hopf bifurcation at the positive equilibrium is derived employing geometric stability switch criterion. Second, the properties of Hopf bifurcation are determined using the center manifold theorem and by reducing the equation to the Poincaré normal form. Finally, the correctness of the theoretical derivation is confirmed using a numerical simulation based on specific parameter values. Our results show that with an increase in delay [Formula: see text], the unique positive equilibrium may undergo two stability switches: from stable to unstable, and from unstable to stable. Interestingly, the characteristic equation has pure imaginary roots at the second pair and subsequent critical values. However, Hopf bifurcation theorem is not satisfied because all other characteristic roots of the characteristic equation at these critical values do not have strictly negative real parts, except the pure imaginary roots. We also simulate the unstable periodic solutions at the second pair of critical values through a bifurcation diagram. Therefore, a pair of supercritical Hopf bifurcations appear around the positive equilibrium of the housefly population stage-structured model.

An application of stability charts to prediction of buckling instability in tapered columns via Galerkin’s method

This work tackles the mathematical modeling of buckling problem to obtain their critical loads in tapered columns subjected to concentrated and axial distributed loads. The governing model is a general eigenvalue problem that has no exact solution due to some new terms included. A semi-analytical technique satisfying the boundary conditions is proposed for the solution procedure. The minimum residual Galerkin’s method is suggested due to its effectiveness as a semi-analytical tool for the buckling problem to obtain the buckling shape modes by using admissible periodic functions. The study investigates the buckling instability and the responses of tapered columns with different periodic trial shape functions as approximations to the exact solutions. Based on the eigenvalue problem, Galerkin’s method is employed to obtain the transition curves to represent the critical loads. The stability charts (Ince–Strutt diagrams) among the parameters of the problem are proposed to explain the elastic stability of different tapered columns subjected to different shapes of cross sections and distributed weights. Consequently, the influences of the included parameters on the critical buckling loads are discussed. Among the different tapered columns presented, some parameters in the proposed distributions have a big influence on the critical buckling load and the creation of the instability regions in the chart for the clamped-clamped boundary conditions. The results are verified using the analytical solutions for some specific known problems.

Nonreciprocal energy transmission in short discrete systems with strong spatiotemporal modulations

Introducing spatiotemporally varying properties in a phononic lattice can enable nonreciprocal transmission of energy. The hallmark of this phenomenon is the unidirectional energy transmission in infinitely long systems. In short lattices, however, identifying nonreciprocity through differences in transmitted energies (norm bias) becomes challenging because the primary contributor to nonreciprocity is the transmitted phases. Although stronger modulation can achieve a higher norm bias, it can also result in parametric instabilities and trigger large-amplitude oscillations that lead to device failure. To better understand the tradeoff between stability and norm bias in strongly modulated systems, we investigate the parametric stability of a finite one-dimensional lattice with spatiotemporally modulated elasticity. We use Floquet theory to compute the stability charts for strongly modulated systems. Thus, we can identify operating conditions that allow for a relatively large norm bias while maintaining a stable, bounded response.

Bifurcation analysis of thin-walled structures trimming process with state-dependent time delay

Our previous study [1] revealed that trimming of thin-walled structures has a strongly nonlinear nature, hence this investigation delves into nonlinear dynamic analysis. Specifically, a perturbation method was applied to analyze the linear stability, whilst a direct numerical integration approach was employed to construct bifurcation diagrams, where unsafe cutting caused by multiple stabilities had been found. Basins of attraction were computed to evaluate the cutting safety. Finally, the stability charts for both linearized and fully nonlinear models were constructed showing a good degree of similarity but the linear one has a larger area of stable trimming. The data from our focused experimental studies was then superimposed on both diagrams and it is clear that more refinements in models’ calibration are needed. This work has laid firm foundations for further advanced modelling investigations and experimental studies to gain more insights into nonlinear dynamics of the machining process.

Assessing the stability of underground caves through iSUMM (innovative, straightforward, user-friendly, mechanically-based method)

A huge number of sinkhole events has been recorded in different Italian urban areas, with an occurrence frequency largely increasing in the last decades, sometimes even causing loss of human lives. The main reason for such catastrophic events is the presence of man-made underground cavities, excavated within soft rocks, several decades ago and then abandoned, at shallow depths. Here, the possibility of interaction with overlying buildings and infrastructures and the corresponding sinkhole hazard is relatively high. In such contexts, the low mechanical properties of the soft rock formations where the cavities have been excavated, like those formed of calcarenites, which outcrop in large areas of Southern Italy, and their high susceptibility to weathering processes, represent one of the most important predisposing factors for instability. Therefore, assessing the stability of underground cavities is crucial for land management and planning purposes. The mechanically-based stability charts developed by Perrotti et al. (Int J Geomech 18(7):04018071, 2018) have proved to be a valid tool for preliminary stability assessment and, although allow to identify an eventual proneness of the cave to instability, they do not provide quantitative assessment about the safety margin itself. In that regard, this study intends to present the most recent outcomes obtained in the development of the methodology and is aimed at promoting an enhanced way for their application, so that the charts can become an operative tool for preliminary sinkhole hazard assessment in similar regions in the world.