Upper Bound Theorem Of Limit Analysis Research Articles

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.

Upper bounds for the three-dimensional seismic active earth pressure coefficients

A numerical implementation of the upper-bound theorem of limit analysis is applied to determine two-dimensional (2D) and three-dimensional (3D) active horizontal earth pressure coefficients considering seismic actions through a horizontal seismic coefficient. Results are obtained for vertical wall, horizontal soil, different friction angles of the soil, soil-to-wall friction ratios, horizontal seismic coefficients and wall width-to-height ratios. The few cases for which 3D active earth pressure coefficients are available in the literature using upper-bound methods were used for comparison with the corresponding earth pressure coefficients obtained in this study. This showed a general improvement of these results, which allows expecting a good accuracy for the set of cases studied. The ratios between the 3D and 2D horizontal active earth pressure coefficients are found to be practically independent of the soil-to-wall friction ratio. An equation is proposed for calculating these ratios. This equation can be easily used in the design of geotechnical structures requiring the determination of 3D active earth pressure coefficients.

Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable

Expanded anchor cables have been widely used in the rapid development of underground engineering. However, there are still some deficiencies in the computation of the ultimate bearing capacity of expanded anchor cables. Based on the upper-bound theorem of limit analysis, the upper-bound solution of the ultimate bearing capacity of an expanded anchor cable was derived. For this calculation, the instability mechanism of the soil at the front surface of the anchorage segment of the expanded anchor cable was assumed to satisfy the logarithmic spiral failure model, its 3D velocity discontinuity surface was generated using the spatial discretization technique, and the optimal solution was determined through the particle swarm optimization (PSO) algorithm. The breakage mechanism of the anchorage side surface was assumed to appear at the interface between the anchorage body and the soil, and its velocity field satisfied the requirements of the associated flow rule. The accuracy of the proposed analytical solution was well-verified through a comparison with three-dimensional numerical simulations, and its superiority was also well-demonstrated in comparison to the existing theoretical calculation method. Subsequently, the influence of the key parameters of the anchor cable on the ultimate lateral resistance, end resistance, and total pullout capacity was discussed. The results showed that: the anchorage segment diameter, anchorage segment length, and buried depth of the expanded anchor cable had a great influence on the ultimate lateral resistance, ultimate end resistance, and total ultimate pullout capacity; however, the anchorage segment length had little influence on the ultimate end resistance, and the inclination angle of the anchor cable had relatively little influence on the resistance.

Analytical Stability Analysis of Rainfall-Infiltrated Slopes Based on the Green–Ampt Model

Rainfall infiltration can alter the pore-water pressure and decrease matric suction, which can lead to slope instability and landslides. Most of the previous numerical studies conducted on rain infiltration have assumed a two-dimensional (2D) failure mechanism, which neglects the three-dimensional (3D) characteristics of slopes. An analytical assessment framework was developed using the upper-bound theorem of limit analysis to determine the factor of safety and failure pattern in the case of a 3D vertical cut slope under diverse rainfall situations combined with the unsaturated soil mechanics. The Green–Ampt model was adopted to depict the infiltration process and the progression of the wetting front into the soil. The slope factor of safety was determined using the energy balance equation by adopting the strength reduction approach. The validity of the framework was verified using the numerical simulation method. The effects of duration, intensity, and patterns of rainfall and the 3D traits of slope on the slope failure pattern and stability were determined. A set of design charts was included for the preliminary design. The present study quantitatively determines the stability of slopes susceptible to rainfall infiltration.

Unified solution of safety factors for three-dimensional compound slopes considering local and global instability

Local instability of a compound slope is commonplace in practice. Extant analytical solutions only considered the single global failure pattern, which results in an inaccurate assessment of slope stability. This paper develops a unified analytical framework to account for the local and global instability of a compound slope. A failure pattern coefficient is proposed for the determination of instability behaviors. On this basis, the kinematically admissible three-dimensional (3D) failure mechanism is modified to describe the collapse of slopes. Under the framework of the upper-bound theorem of limit analysis, the external work rates and the internal dissipation rates can be calculated from the unified failure mechanism. The strength reduction technique is introduced into the work rate balance equation, and then the explicit solution of the safety factor is derived. The critical safety factor and slip surface are obtained through a hybrid optimization of the objective function. By way of example, the present solutions are compared with previous solutions and the results of the finite element method. Good agreements of the safety factor and slip surface show the good capability of this study to implement the stability analysis of compound slopes. A detailed parametric study is carried out to reveal the influence of slope geometry, soil strength, and external excitation on slope stability and failure pattern. The method presented in this paper provides a more rigorous solution for the stability assessment of compound slopes.

Reliability analysis of shallow-buried tunnel construction adjacent to karst cave

In this paper, based on the upper-bound theorem of limit analysis and the reliability theory, the reliability performance function of the rock and soil mass around the tunnel and the karst cave should be constructed. Through Matlab software programming, the instability probability and reliability index of the rock and soil mass around the tunnel and the karst cave are calculated by applying the Monte Carlo method. Applying the method to practical engineering, the reliability of the rock and soil in each damage zone in the karst area is studied. The study shows that when the ratio of tunnel buried depth to tunnel diameter is greater than 2, the tunnel passes through the moderately weathered dolomite layer or the slightly weathered dolomite layer, the self-supporting capacity of the rock and soil mass in front of the tunnel face has been significantly enhanced. The stability of the rock and soil mass around the left side of the karst cave has been significantly enhanced when the ratio of the clear distance between the cave and the bottom of the tunnel to cave diameter is greater than 2.

Theoretical Analysis on Safety Thickness of the Water-Resistant Rock Mass of Karst Tunnel Face Taking Into Account Seepage Effect

We took into account the adverse influence of the karst water seepage effect on the water-resistant rock mass of karst tunnel face. Based on the upper-bound theorem of limit analysis and the Hoek–Brown failure criterion, we obtained the calculation method of the critical safety thickness of the water-resistant rock mass. We carried out a feasibility analysis, an analysis of influencing factors and a comparative analysis with previous related research achievements of this method. The results showed that: (1) With the decrease of surrounding rock grade, the safety thickness of water-resistant rock mass gradually increased, and the safety thickness of surrounding rock at all grades remained within a reasonable range. (2) The safety thickness decreased as the compressive strength, the tensile strength and parameter A increased, and it increased as the karst water pressure, the tunnel excavation height, and parameter B increased. (3) The change trend of the safety thickness with the influencing factors was completely consistent under the two conditions of considering and without considering the seepage effect, and the safety thickness with considering the seepage force was greater than that without considering the seepage force. Taking the Yunwushan Tunnel of Yiwan railway as an example, the critical safety thickness of the water-resistant rock mass was calculated and the calculated value was in good coincidence with the safety thickness adopted in the actual project. The research results are of great significance to prevent the occurrence of high pressure filling karst geological disasters such as water inrush in tunnels.

Investigation on the Stability of Fissured Slopes Reinforced with Anchor Cables under Seismic Action

Based on the upper bound theorem of limit analysis (UBLA) combined with the pseudostatic methods, this paper elaborates on a calculated procedure for evaluating fissured slope stability under seismic conditions reinforced with prestressed anchor cables. An existing simple slope case is presented as a case study in this work. The comparison is given to verify that the solution derived from this study is correct and feasible. By means of a numerical optimization procedure, the critical location of the crack is determined from the best upper bound solutions. The results demonstrate a significant influence of the depth of crack and seismic acceleration coefficient on the critical location distribution of the cracks. Meanwhile, the axial force of anchor cables is investigated via parametric studies. It is shown that the variation of the crack depth has little effect on the axial force of anchor cables. Moreover, this paper also illustrates the variation in the axial force of anchor cables under the impact of five marked factors (crack depth, anchor arrangement, anchor inclination angle, slope angle, and seismic conditions). Finally, the required critical length of the free section of anchor cables is determined to ensure the stability of fissured slopes subjected to seismic action.

Efficient meta-heuristic mesh adaptation strategies for NURBS upper–bound limit analysis of curved three-dimensional masonry structures

We propose a new adaptive NURBS upper-bound limit analysis approach for the assessment of general three-dimensional curved masonry structures, based on different meta-heuristic mesh adaptation schemes. The method, which can easily be integrated within CAD modeling environments, allows to establish the actual failure mechanism and load bearing capacity of a masonry structure by iteratively adjusting a tentative pattern of yield lines, defined upon a suitable initial mesh of NURBS rigid elements, by means of a suitable meta-heuristic algorithm which searches for the minimum collapse load multiplier, thus enforcing the upper-bound theorem of limit analysis. In particular, we investigate and discuss the efficiency of several meta-heuristic algorithms in delivering the optimal solution: a specifically devised Prey Predator Algorithm (PPA) is compared with the Particle Swarm Optimization (PSO) Algorithm, the Firefly Algorithm (FA) and a suitable Genetic Algorithm (GA). Four masonry vaults have been chosen as case studies. In particular, the modified PPA proves to be the most efficient mesh-adjustment scheme for the proposed adaptive NURBS-based limit analysis procedure.

Seismic stability analysis of tunnel face in purely cohesive soil by a pseudo-dynamic approach

To give a solution for seismic stability of tunnel faces subjected to earthquake ground shakings, the pseudo-dynamic approach is originally introduced to analyze tunnel face stability in this study. In the light of the upper-bound theorem of limit analysis, an advanced three-dimensional mechanism combined with pseudo-dynamic approach is proposed. Based on this mechanism, the required support pressure on tunnel face can be obtained by equaling external work rates to the internal energy dissipation and implementing an optimization searching procedure related to time. Both time and space feature of seismic waves are properly accounted for in the proposed mechanism. For this reason, the proposed mechanism can better represent the actual influence of seismic motion and has a remarkable advantage in evaluating the effects of vertical seismic acceleration, soil amplification factor, seismic wave period and initial phase difference on tunnel face stability. Furthermore, the pseudo-dynamic approach is compared with the pseudo-static approach. The difference between them is illustrated from a new but understandable perspective. The comparison demonstrates that the pseudo-static approach is a conservative method but still could provide precise enough results as the pseudo-dynamic approach if the value of seismic wavelengths is large or the height of soil structures is small.

Evolutive and Kinematic Limit Analysis Algorithms for Large-Scale 3D Truss-Frame Structures: Comparison Application to Historic Iron Bridge Arch

Two new computational algorithms for the Limit Analysis (LA) of large-scale 3D truss-frame structures recently proposed by the authors are reconsidered and adapted for a comparison prediction of the elastoplastic response of a strategic beautiful historic infrastructure, namely the Paderno d’Adda bridge (or San Michele bridge), a riveted wrought iron railway viaduct that was built in northern Italy in 1889. The first LA algorithm traces a fully exact evolutive piece-wise linear elastoplastic response of the structure, up to plastic collapse, by reconstructing the true sequence of activation of made-available plastic joints (as a generalization of plastic hinges), in the true spirit of LA. The second LA algorithm develops an independent kinematic iterative approach apt to directly determine the plastic collapse state, in terms of collapse load multiplier and plastic mechanism, based on the upper-bound theorem of LA. Specifically, the marvelous doubly built-in parabolic arch of the bridge is analyzed, under a static loading configuration at try-out stage, and its elastoplastic response is investigated, in terms of evolutive load-displacement curve, collapse load multiplier and plastic collapse mechanism. The two LA algorithms are found to much effectively run and perform, despite the rather large size of the computational model, with a number of dofs in the order of four thousands, by achieving good corresponding matches in terms of the estimate of the load-bearing capacity and of the collapse characteristics of the arch substructure, showing this to constitute a well-set structural element. Moreover, the direct kinematic method displays a rather dramatic performance, in truly precipitating from above onto the collapse load multiplier and rapidly adjusting to the collapse mode, in very few iterations, by a considerable saving of computational time, with respect to the complete evolutive elastoplastic analysis. This shall open up the way for further adoption of such advanced LA tools, with LA regaining a new momentum within the modern optimization analysis of structural design and form-finding problems.

Shear resistance of retrofitted castellated link beams: Numerical and limit analysis approaches

This paper deals with the cyclic behavior and shear capacity of non-compact steel castellated link beams (CLBs) using finite element analysis. To enhance the shear resistance and inelastic rotation capacity of such vulnerable link beams, some retrofitting strategies are explored and examined. To this end, a three dimensional finite-element model using ABAQUS is developed for the inelastic nonlinear analysis of the unstiffened and stiffened CLBs. It was found that unstiffened short CLBs cannot reach the required inelastic rotation capacity of 0.08 rad as per the Seismic Provisions for shear links. However, two strengthening strategies were found that can be used to make their cyclic behavior more stable and increase their inelastic rotation capacity to an acceptable value as per the Provisions. Furthermore, in order to determine the ultimate shear resistance of unstiffened and stiffened CLBs, a theoretical formula based on the upper-bound theorem of limit analysis and a refined collapse mechanism was proposed. It was shown that the limit analysis results are in good agreement with the FEA predictions.

Failure modes and face instability of shallow tunnels under soft grounds

For shallow tunnels under soft grounds, it is very difficult to gain a complete understanding on all the load-bearing arch effects during excavation, since such tunnels can easily deform or even collapse. Among these effects, the tunnel face stability plays the most important role and has not been understood and handled very well; therefore, the design and construction of such tunnels under soft grounds remain as a challenge. In this work, the different failure models of face instability of the shallow tunnels were established based on the upper-bound theorem of limit analysis for the formation of clayey and sandy soils, respectively, and the analytical equations for determining the limit support force for the face stability were also derived. Furthermore, the influences of various factors on the stability were analyzed and compared. For the sandy soil formation, an equivalent pressure coefficient was proposed for the tunnel faces of bolt reinforcement. Finally, taking the face stability of the shallow tunnels in sandy soils as an example, a complete numerical simulation was conducted to verify the accuracy and effectiveness of the analytical results based on the theoretical models as well as their application conditions.

Upper-Bound Multi-Rigid-Block Solutions for Seismic Performance of Slopes with a Weak Thin Layer

The presence of a weak layer has an adverse influence on the seismic performance of slopes. The upper-bound solution serves as a rigorous method in the stability analysis of geotechnical problems. In this study, a multi-rigid-block solution based on the category of the upper-bound theorem of limit analysis is presented to examine the seismic performance of nonhomogeneous slopes with a weak thin layer. Comparison of the static factors of safety is conducted with various solutions (i.e., limit analysis with a different failure mechanism, limit equilibrium solution, and numerical method), and the results exhibit reasonable consistency. An analytical solution in estimating the critical yield acceleration coefficient is derived, and the influence of slope angle, slope height, and soil strength on the critical yield acceleration coefficient and failure mechanism is analyzed. Subsequently, Newmark’s analytical procedure is employed to evaluate cumulative displacement with various real earthquake acceleration records as input motion. Results show that the strength and geometric parameters have a remarkable influence on the critical yield acceleration coefficient, and the cumulative displacement increases with the increasing slope angle.

Joined Influences of Nonlinearity and Dilation on the Ultimate Pullout Capacity of Horizontal Shallow Plate Anchors by Energy Dissipation Method

The ultimate pullout capacity (UPC) and the shape modification factors of horizontal plate anchors were calculated by using upper-bound limit analysis, in which the assumptions of both a nonlinear failure criterion and the nonassociated flow rule were made upon the soil mass above the anchor plate. Three types of anchor plates, including strip anchors, circle anchors, and rectangle anchors, and the cor- responding failure mechanisms are taken into consideration. The anchor breakout factors were obtained according to the principle of virtual power, which was realized numerically by the nonlinear sequential quadratic programming algorithm. The shape modification factors for different kinds of anchors were given through a multiple nonlinear regression method. Numerical experiments demonstrate the validity of the solutions by reducing the solutions (nonlinear criterion and nonassociated flow rule) into their special cases (linear criterion and associated flow rule), which matches well with existing work. The dilation and nonlinearity of soil mass should be considered because it plays a remarkable role in the UPC of anchor plates. DOI: 10.1061/(ASCE)GM.1943-5622.0000075. © 2011 American Society of Civil Engineers. CE Database subject headings: Failures; Plates; Anchors; Pullout; Energy dissipation. Author keywords: Nonlinear failure criterion; Nonassociated flow rule; Plate anchors; Ultimate pullout capacity; Upper-bound theorem of limit analysis.

Limit analysis of cantilever retaining walls with spaced piles

A modified method is proposed for designing cantilever retaining walls with spaced piles. The approach used is based on a new translational three-dimensional failure mechanism within the framework of the upper-bound theorem of limit analysis. Numerical optimisation of the mechanism is performed by a minimisation procedure for the critical three-dimensional passive earth pressure coefficient. Internationally accepted analytical methods for designing these types of retaining structure assume a simplified rectilinear earth pressure distribution, not only for non-spaced piles but also for spaced piles, where the earth pressure distribution is multiplied by the pile spacing. The results of the proposed method indicate that the pile-to-pile spacing, depending on the input soil parameters, strongly influences the shape of the three-dimensional failure mechanism and the passive pressure along the embedment depth to the point of moment equilibrium. Distribution of the passive earth pressure is no longer rectilinear, but the earth mass contributing to the soil–pile interaction on the dredged side of the embedment depth is reduced, which leads to longer piles and higher inner forces in the retaining structure.

Force analysis of pile foundation in rock slope based on upper-bound theorem of limit

Based on the characteristic that the potential sliding surfaces of rock slope are commonly in the shape of either line or fold line, analysis thought of conventional pile foundation in the flat ground under complex load condition was applied and the upper-bound theorem of limit analysis was used to compute thrust of rock layers with all possible distribution shapes. The interaction of slope and pile was considered design load in terms of slope thrust, and the finite difference method was derived to calculate inner-force and displacement of bridge pile foundation in rock slope under complex load condition. The result of example shows that the distribution model of slope thrust has certain impact on displacement and inner-force of bridge pile foundation. The maximum displacement growth rate reaches 54% and the maximum moment and shear growth rates reach only 15% and 20%, respectively, but the trends of inner-force and displacement of bridge pile foundation are basically the same as those of the conventional pile foundation in the flat ground. When the piles bear the same level lateral thrust, the distribution shapes of slope thrust have different influence on inner-force of pile foundation, especially the rectangle distribution, and the triangle thrust has the smallest displacement and inner-force of pile foundation.

Active and passive earth pressure coefficients by a kinematical approach

A simple method is proposed for calculating the active and passive earth pressure coefficients in the general case of an inclined wall and a sloping backfill. The approach used is based on rotational log-spiral failure mechanisms in the framework of the upper-bound theorem of limit analysis. It is shown that the energy balance equation of a rotational log-spiral mechanism is equivalent to the moment equilibrium equation about the centre of the log-spiral. Numerical optimisation of the active and passive earth pressure coefficients is performed automatically by a spreadsheet optimisation tool. The implementation of the proposed method is illustrated using an example. The predictions by the present method are compared with those given by other authors.

Three-Dimensional Analysis of Ice Sheet Indentation: Limit Analysis Solutions

A method is presented for determining the collapse pressures of an ice sheet subjected to a uniformly distributed edge load by applying the upper-bound theorem of limit analysis. The ice sheet is idealized as a semi-infinite layer of elastic-perfectly plastic material. A quadratic anisotropic yield criterion is used to calculate the indentation pressures. The ice sheet consists of columnar ice and is assumed isotropic in the plane of the ice sheet. Upper-bound solutions are found by optimizing a three-dimensional discontinuous velocity field representing an assumed collapse pattern of the ice sheet. Solutions are based on various ratios of indentor width to ice thickness, thereby providing an envelope of indentation pressures over a range of aspect ratios, from conditions of plane strain to plane stress. Solutions are then compared with corresponding two and three-dimensional lower-bound analyses.

Kinematical compatibility problems in yield-line theory

Summary An attempt is made to classify assumptions made in yield-line theory in connexion with the upper-bound theorem of limit analysis. Both classical theory and theory taking into consideration membrane forces are examined for reinforced concrete slabs. The complete set of generalized stresses and strain rates is considered, and compatibility conditions imposed by the associated flow law are discussed, including those for the configuration of yield lines and for internal compatibility of the cross-section. In the case of Johansen's yield-line theory, the general type of fan pattern generated from the envelope line is considered, and some statical restrictions applied in the upper-bound technique are discussed. Considerations are limited to the case of isotropy, unless otherwise specified, but conclusions can be more or less directly extended to the case of orthotropic slabs.