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.

Three-dimensional upper bound limit analysis of underground cavities using nonlinear Baker failure criterion

A generalized nonlinear Baker failure criterion is employed with the upper bound limit analysis to study the surrounding rock stability of underground cavities. A three-dimensional(3D) failure mode is established by extending the two-dimensional(2D) failure mode, which offers an upper bound expression of the surrounding rock pressure. This method is validated with a series of examples before the influence of four parameters of scale parameter, curvature parameter, shift parameter and lateral pressure coefficient, on the surrounding rock pressure is analyzed. According to these results, failure ranges of the underground cavities are determined. The following conclusions are reached:(1) the proposed approach is more accurate to predict surrounding rock pressure than the Mohr-Coulomb failure criterion;(2) the surrounding rock with large scale parameter, curvature parameter, shift parameter, and lateral pressure coefficient can lead to a more stable underground cavity;(3) the failure range in 3D mode can be predicted according to the upper bound solutions.

Upper-bound limit analysis based on the natural element method

The natural element method （NEM） is a newly- developed numerical method based on Voronoi diagram and Delaunay triangulation of scattered points, which adopts natural neighbour interpolation to construct trial functions in the framework of Galerkin method. Owing to its distinctive advantages, the NEM is used widely in many problems of computational mechanics. Utilizing the NEM, this paper deals with numerical limit analysis of structures made up of perfectly rigid-plastic material. According to kinematic the- orem of plastic limit analysis, a mathematical programming natural element formulation is established for determining the upper bound multiplier of plane problems, and a direct iteration algorithm is proposed accordingly to solve it. In this algorithm, the plastic incompressibility condition is handled by two different treatments, and the nonlinearity and nons- moothness of the goal function are overcome by distinguishing the rigid zones from the plastic zones at each iteration. The procedure implementation of iterative process is quite simple and effective because each iteration is equivalent to solving an associated elastic problem. The obtained limit load multiplier is proved to monotonically converge to the upper bound of true solution. Several benchmark examples are investigated to validate the significant performance of the NEM in the application field of limit analysis.

LOWER BOUND LIMIT ANALYSIS OF THREE-DIMENSIONAL ELASTOPLASTIC STRUCTURES BY BOUNDARY ELEMENT METHOD

Based on the lower bound theorem of limit analysis, a solution procedure for limit analysis of three_dimensional elastoplastic structures was established using conventional boundary element method (BEM). The elastic stress field for lower bound limit analysis was computed directly by three_dimensional boundary element method (3_D BEM). The self_equilibrium stress field was constructed by the linear combination of several self_equilibrium “basis vectors” which can be computed by elastic_plastic incremental iteration of 3_D BEM analysis. The lower bound limit analysis problem was finally reduced to a series of nonlinear programming sub_problems with relatively few optimal variables. The complex method was used to solve the nonlinear programming sub_problems. The numerical results show that the present solution procedure has good accuracy and high efficiency.

Combined influence of nonlinearity and dilation on slope stability evaluated by upper-bound limit analysis

The combined influence of nonlinearity and dilation on slope stability was evaluated using the upper-bound limit analysis theorem.The mechanism of slope collapse was analyzed by dividing it into arbitrary discrete soil blocks with the nonlinear Mohr–Coulomb failure criterion and nonassociated flow rule.The multipoint tangent(multi-tangent) technique was used to analyze the slope stability by linearizing the nonlinear failure criterion.A general expression for the slope safety factor was derived based on the virtual work principle and the strength reduction technique,and the global slope safety factor can be obtained by the optimization method of nonlinear sequential quadratic programming.The results show better agreement with previous research result when the nonlinear failure criterion reduces to a linear failure criterion or the non-associated flow rule reduces to an associated flow rule,which demonstrates the rationality of the presented method.Slope safety factors calculated by the multi-tangent inclined-slices technique were smaller than those obtained by the traditional single-tangent inclined-slices technique.The results show that the multi-tangent inclined-slices technique is a safe and effective method of slope stability limit analysis.The combined effect of nonlinearity and dilation on slope stability was analyzed,and the parameter analysis indicates that nonlinearity and dilation have significant influence on the result of slope stability analysis.

Development of Finite Element Limiting Analysis Method and Its Applications in Geotechnical Engineering

The Finite Element Limiting Analysis Method(LELAM) has the advantage of combining a numerical analysis method with traditional limiting equilibrium methods.It is particularly applicable to the analysis and design of geotechnical engineering.In the early 20th century,FELAM has been developed vigorously in domestic geotechnical engineering over international common finite element procedures.It has made great achievements in basic theory research and computational precision,thus broadening the application fields in practical projects.In order to gradually make innovations in geotechnical design methods,some of our research results are presented,mainly including geotechnical safety factor definitions,the principles for use of the method concerned,the overall failure criterion,the deduction and selection of the yield criterion,and the measurement to improve the computational precision,etc..The application field has been broadened from two-dimensional to three-dimensional,from soil slope to jointed rock slope and foundation,from stable seepage to non-stable seepage,from slope and foundation to tunnel.This method has also been used in search of many hidden sliding surfaces of complex landslides,conducting the structural support design considering the interaction between the soil and the structure,and computing simulation foundation bearing plates load tests,etc..

Upper bound limit analysis of roof collapse in shallow tunnels with arbitrary cross sections under condition of seepage force

The analytical solutions for predicting the exact shape of collapse mechanisms in shallow tunnels with arbitrary excavation profiles were obtained by virtue of the upper bound theorem of limit analysis and variation principle according to Hoek-Brown failure criterion. The seepage force was included in the upper bound limit analysis, and it was computed from the gradient of excess pore pressure distribution. The seepage was regarded as a work rate of external force. The numerical results of roof collapse in square and circular tunnels with different rock parameters were derived and discussed, which proves to be valid in comparison with the previous work. The influences of different parameters on the shape of collapsing blocks were also discussed.

Upper Bound Limit Analysis of Anisotropic Soils

In this paper,a novel discretization method inσ-τspace is developed to calculate the upper bound limit loads and failure modes of anisotropic Mohr-Coulomb materials.To achieve this objective,the Mohr-Coulomb yield criterion is linearized inσ-τspace,which allows for upper bound solution of soils whose cohesion and friction coefficient varying with direction.The finite element upper limit analysis formulation using the modified anisotropic yield criterion is then developed.Several examples are given to illustrate the capability and effectiveness of the proposed numerical procedure for computing rigorous upper bounds for anisotropic soils.

THE APPLICATION OF WEINSTEIN-CHIEN′S METHOD——THE UPPER AND LOWER LIMITS OF FUNDAMENTAL FREQUENCY OF RECTANGULAR PLATES WITH EDGES ARE THE MIXTURE OF SIMPLY SUPPORTED PORTIONS AND CLAMPED PORTIONS

In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz' method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.

A Method Combining Numerical Analysis and Limit Equilibrium Theory to Determine Potential Slip Surfaces in Soil Slopes

This paper describes a precise method combining numerical analysis and limit equilibrium theory to determine potential slip surfaces in soil slopes. In this method, the direction of the critical slip surface at any point in a slope is determined using the Coulomb’s strength principle and the extremum principle based on the ratio of the shear strength to the shear stress at that point. The ratio, which is considered as an analysis index, can be computed once the stress field of the soil slope is obtained. The critical slip direction at any point in the slope must be the tangential direction of a potential slip surface passing through the point. Therefore, starting from a point on the top of the slope surface or on the horizontal segment outside the slope toe, the increment with a small distance into the slope is used to choose another point and the corresponding slip direction at the point is computed. Connecting all the points used in the computation forms a potential slip surface exiting at the starting point. Then the factor of safety for any potential slip surface can be computed using limit equilibrium method like Spencer method. After factors of safety for all the potential slip surfaces are obtained, the minimum one is the factor of safety for the slope and the corresponding potential slip surface is the critical slip surface of the slope. The proposed method does not need to pre-assume the shape of potential slip surfaces. Thus it is suitable for any shape of slip surfaces. Moreover the method is very simple to be applied. Examples are presented in this paper to illustrate the feasibility of the proposed method programmed in ANSYS software by macro commands.

A LOWER BOUND LIMIT ANALYSIS OF DUCTILE COMPOSITE MATERIALS

The plastic load-bearing capacity of ductile composites such as metal matrix composites is studied with an insight into the microstructures. The macroscopic strength of a composite is obtained by combining the homogenization theory with static limit analysis, where the temperature parameter method is used to construct the self-equilibrium stress field. An interface failure model is proposed to account for the effects of the interface on the failure of composites. The static limit analysis with the finite-element method is then formulated as a constrained nonlinear programming problem, which is solved by the Sequential Quadratic Programming （SQP） method. Finally, the macroscopic transverse strength of perforated materials, the macroscopic transverse and off-axis strength of fiber-reinforced composites are obtained through numerical calculation. The computational results are in good agreement with the experimental data.

The limit analysis in soil and rock:a mature discipline of geomechanics

The solution of a slope stability problem can be approached by its least upper-bound and maximum lower-bound with high accuracy. The limit equilibrium methods that employ vertical slices imply a lower bound of the factor of safety. It has been successfully extended to the area of active earth pressure analysis that accounts for different input of locations of earth pressure applications. Those methods that employ slices with inclined interfaces give an upper-bound approach to the stability analysis. It enjoys a sound mechanical background and is able to provide accurate solutions of soil plasticity. It has been successfully extended to the area of bearing capacity analysis in which various empirical coefficients are no longer necessary. The 3D upper- and lower-bound methods under this framework have been made possible and show great potential for solving various engineering problems.

Limit variation analysis of shallow rectangular tunnels collapsing with double-layer rock mass based on a three-dimensional failure mechanism

In the framework of upper bound theorem of limit analysis, the progressive collapse of shallow rectangular tunnels with double-layer rock mass has been theoretically analyzed based on the three-dimensional (3D) velocity discontinuity surfaces. According to the virtual work principle, the difference theorem and the variation method, the collapse surface of double-layer rock mass is determined based on the Hoek-Brown failure criterion. The formula can be degenerated to a single-layer rock collapsing problem when the rock mass is homogeneous. To estimate the validity of the result, the numerical simulation software PLAXIS 3D is used to simulate the collapse of shallow tunnels with double-layer rock mass, and the comparative analysis shows that numerical results are in good agreement with upper-bound solutions. According to the results of parametric analysis, the potential range of collapse of a double-layer rock mass above a shallow cavity decreases with a decrease in A1/A2,σci1/σci2 and σtm1/σtm2 and an increase in B1/B2,γ1/γ2. The range will decrease with a decrease in support pressure q and increase with a decrease in surface overload σs. Therefore, reinforced supporting is beneficial to improve the stability of the cavity during actual construction.

Limit analysis method for active earth pressure on laggings between stabilizing piles

Stabilizing pile is a kind of earth shoring structure frequently used in slope engineering. When the piles have cantilever segments above the ground,laggings are usually installed to avoid collapse of soil between piles. Evaluating the earth pressure acting on laggings is of great importance in design process.Since laggings are usually less stiff than piles,the lateral pressure on lagging is much closer to active earth pressure. In order to estimate the lateral earth pressure on lagging more accurately,first,a model test of cantilever stabilizing pile and lagging systems was carried out. Then,basing the experimental results a three-dimensional sliding wedge model was established. Last,the calculation process of the total active force on lagging is presented based on the kinematic approach of limit analysis. A comparison is made between the total active force on lagging calculated by the formula presented in this study and the force on a same-size rigid retaining wall obtained from Rankine's theory. It is found that the proposed method fits well with the experimental results.Parametric studies show that the total active force on lagging increases with the growth of the lagging height and the lagging clear span; while decreases asthe soil internal friction angle and soil cohesion increase.

PLASTIC LIMIT ANALYSIS OF DISCONTINUOUS LINING UNDERGROUND PRESSURE

Discontinuous lining is a special form of support in underground excavation. Based on the method of plastic limit analysis, it is found the upper and the lower bound solution of the pressure of circular discontinuous lining and discussed support parameter of discontinuous lining and its applicable conditions , which provides theoretical basis for the design and calculation of discontinuous lining.

Upper bound analysis of slope stability with nonlinear failure criterion based on strength reduction technique

Based on the upper bound limit analysis theorem and the shear strength reduction technique, the equation for expressing critical limit-equilibrium state was employed to define the safety factor of a given slope and its corresponding critical failure mechanism by means of the kinematical approach of limit analysis theory. The nonlinear shear strength parameters were treated as variable parameters and a kinematically admissible failure mechanism was considered for calculation schemes. The iterative optimization method was adopted to obtain the safety factors. Case study and comparative analysis show that solutions presented here agree with available predictions when nonlinear criterion reduces to linear criterion, and the validity of present method could be illuminated. From the numerical results, it can also be seen that nonlinear parameter rn, slope foot gradient ,β, height of slope H, slope top gradient a and soil bulk density γ have significant effects on the safety factor of the slope.

RIGID FINITE ELEMENT AND LIMIT ANALYSIS

According to the lower-bound theorem of limit analysis the Rigid Finite Element Meth-od(RFEM)is applied to structural limit analysis and the linear programmings for limit analysis are deducedin this paper.Moreover,the Thermo-Parameter Method(TPM)and Parametric Variational principles(PVP)are used to reduce the computational effort while maintaining the accuracy of solutions.A better solution isalso obtained in this paper.

A New Curve Fitting Method for Forming Limit Experimental Data

The forming limit curve （FLC） can be obtained by means of curve fitting the limit strain points of different strain paths. The theory of percent regression analysis is applied to the curve fitting of forming limit experimental data.Forecast intervals of FLC percentiles can be calculated. Thus reliability and confidence level can be considered. The theoretical method to get the limits of limit strain points distributing region is presented, and the FLC position can be adjusted according to practical requirement. Method for establishing FLC with high reliability using small samples is presented at the same time. This method can make full use of the current experimental data and the previous data.Compared with the traditional method that can only use current experimental data, fewer specimens are required in the present method to obtain the same precision and the result is more accurate with the same number of specimens.

Upper bound analysis of ultimate pullout capacity of shallow 3-D circular plate anchors based on nonlinear Mohr-Coulomb failure criterion

Based on the nonlinear Mohr-Coulomb failure criterion and the associated flow rules,the three-dimensional(3-D)axisymmetric failure mechanism of shallow horizontal circular plate anchors that are subjected to the ultimate pullout capacity(UPC)is determined.A derivative function of the projection function for projecting the 3-D axisymmetric failure surface on plane is deduced using the variation theory.By using difference principle,the primitive function of failure surface satisfying boundary condition and numerical solution to its corresponding ultimate pullout capacity function are obtained.The influences of nonlinear Mohr-Coulomb parameters on UPC and failure mechanism are studied.The result shows that UPC decreases with dimensionless parameter m and uniaxial tensile strength increases but increases when depth and radius of plate anchor,surface overload,initial cohesion,geomaterial density and friction angle increase.The failure surface is similar to a symmetrical spatial funnel,and its shape is mainly determined by dimensionless parameter m;the surface damage range expands with the increase of radius and depth of the plate anchor as well as initial cohesion but decreases with the increase of dimensionless parameter m and uniaxial tensile strength as well as geomaterial density.As the dimensionless parameter m=2.0,the numerical solution of UPC based on the difference principle is proved to be feasible and effective through the comparison with the exact solution.In addition,the comparison between solutions of UPC computed by variation method and those computed by upper bound method indicate that variation method outperforms upper bound method.

The height-width ratio limited value for rubber bearing isolated structure computed by uniform design method

Rubber isolation is the most mature control technology in practical application, and is widely used by short rigid buildings. However, many high isolation buildings have been built around the world in recent years, which do not follow the existing criterions and codes. Many researchers began to research the special problems caused by larger height-width ratio isolation structures. The overturning effect of high height-width ratio structures with rubber bearing is firstly studied. Considering the main factors, such as the height-width ratio of structures, type of site, the designed basic acceleration of ground motion and the decouple factor in horizon, computing experiment is defined with the Uniform Design Method, which is also known as designing isolation structure. The forces of the bearing under edge of structures based on the position of the rubber bearing are calculated. The result indicates that the rubber bearings will lose its functionality under very high tension and compressing force of earthquake motion in horizon and vertical, when the height-width ratio is over a certain value. Thus, based on the calculation result of isolation structures defined in the uniform design method, regression analysis is conducted, and also the rubber edge force regression formula are gotten, which has higher correlation and smaller standard deviation. This formula can be used to roughly calculate whether the pull force occurs at the edge of the building. By the edge bearings of isolation structure minimum force formula, the height-width ratio limited value of the isolation structure is deducted when rubber bearing has minimum force of zero.