Limit Equilibrium Equations Research Articles

Closed system of limit equilibrium equations for axisymmetric shells

Closed system of limit equilibrium equations for axisymmetric shells

Symmetric wedge in the roof of a tunnel excavated in an inclined stress field

An analytical method is presented for the calculation of the stability of a two-dimensional symmetric rock wedge which is formed in the roof of an underground circular opening within an inclined biaxial stress field. The stability of the wedge is evaluated using a two-stage relaxation procedure. In the first stage, the forces acting on the joint faces of the wedge are calculated analytically from the elastic stress distribution around the opening. Thus, formulae are provided that calculate the horizontal and vertical force components on the wedge faces exerted by the surrounding rock mass. In the second stage, a steadily increasing pull out force is applied to the wedge which strains its joints. Failure is assumed to occur due to simultaneous yield along the face of one of the joints. The stability of the wedge, which is expressed by its pull out resistance, is calculated from limit equilibrium equations at failure. Finally, analytically calculated values of the pull out resistance of the wedge are compared with numerically obtained ones using a discrete element code.

A strain-based design method for the collapse limit state of reinforced soil walls or slopes

A new design methodology was developed to predict the collapse of reinforced soil walls or slopes caused by failure of the reinforcing elements. The rupture surface was modelled by a generalized log spiral selected by numerical optimization, and the conventional assumption of rigid–plastic deformation at wall collapse was not made. The limit equilibrium equations were established by examining the equilibrium of slices of soil parallel to the reinforcing elements. A strain-based criterion was used to model the initiation of wall collapse by breakage of reinforcement. The tensile forces in the reinforcing elements were determined based on, approximately, strain compatibility along the rupture surface. The proposed method can model reinforcement with nonlinear load extension response that depends on embedment, non-uniform distribution of reinforcement, and dilatancy of soil. An expedient numerical scheme for performing the analysis is presented to enable the routine use of the analysis in a design office. The proposed method was validated by comparing the predictions with published observations and finite element simulation of a reinforced soil wall. Key words : collapse, kinematics, limit equilibrium, numerical optimization, reinforced soil, strain.

La compression triaxiale des sols armés

For a better understanding of the mechanical behaviour of reinforced soils, the paper presents a study of triaxial compression tests on soils reinforced by identical, equally spaced, plane ties. Soil–reinforcement interactions are considered by using a method similar to the one that resulted in the introduction of the cohesion effect concept in reinforced earth retaining walls. The material's limit-equilibrium equation is established for any tie orientation. These ties give to a reinforced soil a strength higher than the unreinforced soil strength. The difference may be considered as a cohesion term, which usually depends on the strength, concentration, and inclination of the ties, as well as on the radial stress level. The cohesion term given by triaxial tests is different from the cohesion effect term already introduced in reinforced earth walls, as there is no cohesion effect in a triaxial test. The paper also presents a study of adhesion conditions. By examining stress distribution in the soil between the ties, it is established that beyond a given inclination, no traction can be developed in the ties.

Limit equilibrium of granular medium with variable weight

The present paper is devoted to the investigation of plane limit equilibrium equations of a granular medium, when its volume weight is a function of mean pressure. The differential equations of characteristics for a certain form of the above mentioned function have integrated combinations which they do not have when the volume weight is constant. The problems of the forms of stable slopes and the problems of pressure on retaining walls are also considered here. The solutions of these problems have a closed form, while the corresponding solutions for constant volume weight can be found in a numerical form only.