Numerical analysis of undercut anchor effect on rock

The paper presents the results of a numerical analysis using the Finite Element Method (FEM) of the friction issue in the contact between the undercut anchor head and rock during anchor pull-out. Formation of failure zone of rock medium was analysed assuming different Coulomb friction coefficients in the contact zone of conical anchor head with a rock. The problem is interesting as regards practical aspects of rock mass loosening during anchor pull-out. The analysis revealed a significant effect of the friction coefficient on the propagation and extent of the failure zone. Increasing the friction factor significantly decreases the extent of the failure zone measured on a free rock surface.


Introduction
The use of numerical modelling in combination with the results obtained from experimental studies enabled a detailed understanding of the actual behaviour of engineering structures and their optimisation [1-6]. The issue related to the estimation of load capacity of anchors typically used in the installation of equipment, in concrete engineering structures is topical and constantly developed due to its importance in strategic facilities such as nuclear power plants and other constructions located in the earthquake zone [7][8][9]. To date, several empirical models have been developed to describe the mechanism of cone failure formation in concrete under the action of anchors of common designs and installation technologies [10][11][12][13][14][15][16]. The main focus has been on determining the minimum pull-out force of an anchor given its intended load-carrying capacity (load capacity) [17][18][19][20]. Using neural networks, the load capacity of anchors is predicted (e.g. [10]) and using FEM (Finite Element Method) systems, models are built to predict the propagation of failure of the medium under the action of the anchor, to estimate its load capacity depending on the structural parameters of the anchor or anchor installation technology [21][22][23]. The authors of this paper try to apply the technology of installing undercut anchors and then pulling them out (along with the resulting cone failure), to potential rock loosening in unusual conditions of mining works. An example of such can be tunnelling during rescue operations in collapsed galleries when there is an increased concentration of methane and a risk of explosion. Another potential application can be the modernisation of engineering structures, where mining with explosives cannot be performed, or mining with mechanised systems. The subject of the study concerns the technology of loosening potentially large rock fragments with the use of undercut anchors (Figure 1) [7].
From the point of view of the proposed loosening technology, for a given depth of embedment, it is important to obtain the maximum range of loosening. This translates into the volumes of the detached solids, which in turn affects the overall evaluation of the efficiency of the process. In the light of existing knowledge, undercut anchors are generally used for equipment installation technology in engineered concrete structures [24][25][26]. The failure zone of concrete is roughly described, among other things, as a cone of the height of hef and base diameter of 3hef (measured on the free surface of the concrete), e.g. [25,27,28], as in Figure 2.
For sandstones, it has been found [29,30] that the extent of the failure zone is much larger than in the case of concretes. Numerical analyses of the formation of the failure zone under the action of a single anchor [31], two anchors [32] and an assembly of three anchors [33] have been carried out. In the case of an anchor assembly [32,33], the interaction effect of "cone failure" that then appears, depending on the anchor spacing, has been analysed more closely. For the same embedment depth and the same anchor pull-out force, this effect can lead to an increase in the volume of the loosened elements (with a suitably chosen anchor spacing).  The extent and course of the failure zone depend on a plethora of factors. In addition to the effective embedment depth hef, they depend on the physical and mechanical parameters of the rock (e.g. Young's modulus, Poisson's ratio, tensile strength) [34], the value of the angle of the undercut head  [35,36] and the value of the friction coefficient in the contact of the head with a rock.

FEM simulation using ABAQUS
The action of the i-th undercut element of the anchor head on the rock (Figure 1) can be considered as a typical contact issue with friction. As a result of the force F, there is a visible deformation in the rock structure and relative movement of the rock along the surface of the conical part of the head with the angle of the cone . Fi component of F per undercut element can therefore be decomposed into a normal force (Fn) to the conical surface and the longitudinal (friction) component Ff. As a result, the i-th undercut element acts on the rock with a resultant force Fr, at an angle  concerning the axis X of the adopted coordinate system. This angle is the sum of the angle of friction  of rock against the head and the angle of the head  (i.e., =). It should also be noted that the action of the horizontal component  The problem was considered axisymmetric. The analysis used a finite element mesh discretised model as in Figure 4a. The mesh was compacted along the potential failure surface, inclined at an angle  to axis X (approx. 25°, e.g. [31]). The method of restraining the boundary nodes of the model is illustrated in Figure 4b. Forcing in the form of controlled displacement of the anchor along its axis (along the Y-axis of the adopted coordinate system) was applied. Only for this analysis, the friction coefficient value = 0.0, 0.015, 0.25, 0.75, 1.0, 1.5 was used in the simulation. The actual value of the contact friction coefficient depends on several factors, i.e., rock grain size, type of grain bonding medium, rock moisture content, type of anchor material, etc. According to various literature sources, it may vary from 0.2 to 0.4. Embedment depth was assumed to be hef=80mm.

Analysis of results
It is clear from Figure 5 that the coefficient of contact friction  has a significant impact on the course of rock medium failure. For minute values of this coefficient (Figure 5a, b) in the initial stage of penetration, the crack develops deep into the material (angle  takes negative values in this phase). In the next phase, the crack begins to move towards the free surface of the model. As the value of the friction coefficient increases (Figure 5 c, d)

Conclusions
The analysis showed an incredibly significant influence of the friction coefficient in the contact zone of the undercut anchor head with the rock. Under the conditions of field investigations, one may expect, even within the same rock formation, varied loosening, depending on local rock mass conditions, e.g., degree of rock moisture. Moisture may significantly affect the value of the friction coefficient and thus the extent of the failure zone in the rock medium.