Theory And Application Of Rigid Block Limit Analysis

Limit analysis of plasticity theory has merits in stability of slopes and tunnels.Based on the upper bound and lower bound theorem,the limit load or safety factor of structures is derived by establishing a kinematically admissible velocity field and statically admissible stress field in the form of analytical or semi-analytical solutions,thereby providing guidance for engineers in the design and construction phase.The core work in limit analysis is to construct a feasible kinematically admissible velocity field and statically admissible stress field,which are usually based on two main approaches:1)explicit failure pattern;and 2)numerical discretization constitution such as finite or rigid elements.According to the classical limit analysis of plasticity theory,this paper aims to investigate stability problems like tunnel face and slopes from the perspective of formation of field variables.The main content is summarized as:(1)In this paper,a new rigorous procedure based on the block element method for computing the lower bound of three dimensional(3D)slope stability has been proposed.Based on the lower bound limit theorem and block element method,the statically admissible stress fields satisfying the equilibrium,stress boundary and yield conditions are constructed.Six equilibrium conditions including force and moment equilibrium conditions are strictly satisfied for every block element.Additionally,Two possibilities in the direction of resultant shear resistance on the slip surface,including unique direction and non-unique direction,are taken into account.In contrast with the previous studies adopting the assumptions about the inter-block forces,inter-block forces are explicitly calculated in this work.Moreover,excellent numerical characteristics are obtained by using the proposed linear programming optimization technique for both unique and non-unique direction models.The simplicity and general applicability of the rigorous method proposed in this study for analyzing different 3D slope stability problems have been verified by comparing the calculated results with its counterparts obtained by the various existing theories on the stability analysis for four types of slopes.Finally,its application is further demonstrated in the stability analysis of the left bank rock slope of Jinping-I Hydropower Station in detail.(2)A lower bound limit analysis approach based on the block element method is proposed.The search for the maximum value of the factor of safety is set up as a nonlinear programming problem.Sequential quadratic programming(SQP)algorithm from a reasonable initial value is applied to obtain the optimal solution.This approach provides a strict lower bound solution considering the sliding mode and rotation effect simultaneously to analyze wedge stability problem.Then,the "Normal resolution assumption" in the traditional limit equilibrium method(TLE)is discussed.Finally,the stability of six typical wedges in the Three Gorges project is studied.(3)The lower bound limit analysis approach based on the block element method subjected to pore water pressure is proposed.The pore water pressure is considered as an external force and then is exerted to the centroid of block element.Several numerical examples are presented to illustrate the validation and potential applications of the present method,and the influence of the crack depth and the depth of water in the tension crack and the change of reservoir water level on the slope stability is analyzed.In addition,the lower bound limit analysis approach based on the block element method also can investigate the seismic stability analysis of slopes.The earthquake inertia force can be simulated with the pseudo-static method and then is exerted to the centroid of block element.In addition,the influence of the seismic coefficient on the slope stability is analyzed.(4)A rigid block based lower bound limit analysis method for analyzing stability of fractured rock mass in 2D and 3D conditions is proposed.Based on the the main idea of the "element-block-assembling approach",the interest domain is firstly subdivided into a finite number of convex rigid blocks,the statically admissible stress field is then constructed.The rock bridge effects are considered in the general formation.The proposed method is theoretically rigorous and simple.The task of finding a statically admissible stress field to maximize Fs is mathematically stated as as a nonlinear programming problem.The proposed method was verified by considering three typical examples in 2D and 3D conditions,which demonstrate that proposed method is efficient to find the unstable blocks.Additionally,the results suggest that,in addition to the fractures themselves,rock bridge plays a key role in stabilizing the rock blocks,which should be greatly concerned in the stability analysis of rock mass.(5)Based on the intact double log-spirals failure mechanism,the kinematic solution of supporting pressure of tunnel face is obtained in closed form.The optimal support pressure necessitated for maintaining the face stability is then determined at critical state using the optimization technique,and some specific examples validates the correctness and efficiency of the proposed approach and the formula derived in this paper.Within this intact double log-spirals collapse mechanism,the effect of each parameter on the optimal supporting pressure and failure areas is investigated,with the numerical results presented in figures,tables and the empirical formulas under multi-factors.(6)Employing the novel truncated double log-spirals failure mechanism,the upper bound solution of the supporting pressure for maintaining the tunnel face stability is derived considering a longitudinal cavity at a short distance existed above tunnel roof in karst areas,and the optimal solutions are sought through the exhaustive method combined with SQP.The validity of the proposed approach is verified through specific examples,and the results show that the use of truncated log-spiral collapse mechanism aids to improve the accuracy of upper bound solutions.Based on this,a parametric study is presented to estimate the influence of the corresponding parameters on support pressure exerted on tunnel face.It is noted that it is feasible to employ the double log-spirals failure pattern for stability analysis of shallow tunnels.