**3. A review of numerical modelling in rock bolts**

A number of computer programmes have been developed for modelling civil and geotechnical problems. Some of them can be partially used to design and analyse roof bolting systems. It is noted that 3D software is necessary to simulate the whole characters of a model, such as modelling the joints, bedding planes, contact interface and failure criterion. Several numerical methods are used in rock mechanics to model the response of rock masses to loading and unloading. These methods include the method (FEM), the boundary element method (BEM), finite difference method (FDM) and the discrete element method (DEM).

A number of studies were carried out on bolt behaviour in the FE field, including those by *Coats and Yu* (1970), *Hollingshead* (1971), *Aydan* (1989), *Saeb and Amadei* (1990), *Aydan and Kawamoto* (1992), *Swoboda and Marence* (1992), *Moussa and Swoboda* (1995), *Marence and Swoboda* (1995), *Chen et al*. (1994, 1999, 2004), and *Surajit* (1999).

One of the earliest attempts to use standard FEs to model the bolt and grout was done by *Coats and Yu* (1970). The study was carried out on the stress distribution around a cylindrical hole with the FEmodel either in tension or compression. It was found that the stress distribution was a function of the bolt and rock moduli of elasticity. The presence of grout between the bolt and the rock was not considered and there was no allowance for yielding. The analysis was only carried out in linear elastic behaviour with two-phase materials, which limited the model. *Hollingshead* (1971) solved the same problem using a three phase material (bolt-grout and rock) and allowed a yield zone to penetrate into the grout using an elastic, perfectly plastic criterion, according to the Tresca yield criterion, for the three materials (Figure 1). How the interface behaved was not considered in the model.

*John and Dillen* (1983) developed a new one-dimensional element passing through a cylindrical surface to which elements representing the surrounding material are attached (Figure 2). They considered three important modes of failure for fully grouted bolts, a bilinear elasto-plastic model for axial behaviour, elastic- perfectly plastic, and residual plastic model for bonding material, was assumed. Although this model eliminated many previous limitations and agreed with the experimental results, it neglected rock stiffness and in-situ stress around the borehole. They claimed that critical shear stress occurred at the grout rock interface, which is not always the case in the field or laboratory. Aydan (1989) presented a FE model of the bolt. He assumed that a cylindrical bolt and grout annulus is connected to the rock with a three-dimensional 8-nodal points.

Two nodes are connected to the bolt and six to the rock mass. The use of boundary element and FE techniques to analyse the stress and deformation along the bolt was conducted by Peng and Guo (1992) (Figure 3). The effect of the face plate was replaced by a boundary element. The effect of reinforcement because of the assumption of perfect bonding was overestimated.

**Figure 1.** FE Simulation of bolted rock mass (after Hollingshead, 1971)

608 Numerical Simulation – From Theory to Industry

elasticity, contact surfaces, and creep behaviour.

**3. A review of numerical modelling in rock bolts** 

*Swoboda* (1995), *Chen et al*. (1994, 1999, 2004), and *Surajit* (1999).

connected to the rock with a three-dimensional 8-nodal points.

comprehensive list of analytical capabilities including linear static analysis, multiple non-

In this chapter only structural analysis is considered. Structural analyses are available in the ANSYS Multiphysics, ANSYS Mechanical, ANSYS Structural, and ANSYS Professional programmes only. Statistical analysis is used to determine displacement and stress and strain under static loading conditions (both linear and non-linear statistical analyses). Nonlinearity can include plasticity, stress stiffening, large deflection, large strain, hyper-

A number of computer programmes have been developed for modelling civil and geotechnical problems. Some of them can be partially used to design and analyse roof bolting systems. It is noted that 3D software is necessary to simulate the whole characters of a model, such as modelling the joints, bedding planes, contact interface and failure criterion. Several numerical methods are used in rock mechanics to model the response of rock masses to loading and unloading. These methods include the method (FEM), the boundary element method (BEM), finite difference method (FDM) and the discrete element method (DEM).

A number of studies were carried out on bolt behaviour in the FE field, including those by *Coats and Yu* (1970), *Hollingshead* (1971), *Aydan* (1989), *Saeb and Amadei* (1990), *Aydan and Kawamoto* (1992), *Swoboda and Marence* (1992), *Moussa and Swoboda* (1995), *Marence and* 

One of the earliest attempts to use standard FEs to model the bolt and grout was done by *Coats and Yu* (1970). The study was carried out on the stress distribution around a cylindrical hole with the FEmodel either in tension or compression. It was found that the stress distribution was a function of the bolt and rock moduli of elasticity. The presence of grout between the bolt and the rock was not considered and there was no allowance for yielding. The analysis was only carried out in linear elastic behaviour with two-phase materials, which limited the model. *Hollingshead* (1971) solved the same problem using a three phase material (bolt-grout and rock) and allowed a yield zone to penetrate into the grout using an elastic, perfectly plastic criterion, according to the Tresca yield criterion, for the three

*John and Dillen* (1983) developed a new one-dimensional element passing through a cylindrical surface to which elements representing the surrounding material are attached (Figure 2). They considered three important modes of failure for fully grouted bolts, a bilinear elasto-plastic model for axial behaviour, elastic- perfectly plastic, and residual plastic model for bonding material, was assumed. Although this model eliminated many previous limitations and agreed with the experimental results, it neglected rock stiffness and in-situ stress around the borehole. They claimed that critical shear stress occurred at the grout rock interface, which is not always the case in the field or laboratory. Aydan (1989) presented a FE model of the bolt. He assumed that a cylindrical bolt and grout annulus is

materials (Figure 1). How the interface behaved was not considered in the model.

linear analyses, modal analysis, contact interface analyses and many other types.

**Figure 2.** Three-Dimensional rock bolt element (after John and Dillen, 1983)

Numerical Simulation of Fully Grouted Rock Bolts 611

In this chapter, three-dimensional formulations and non-linear deformation of rock, grout, bolt, and two interfaces are taken into account in the reinforced system. A description of the

The FE method is the most suitable computational method to evaluate the real behaviour of the bolt, grout, and surrounding rock when there are composite materials with different interfaces. A three dimensional FE model of a reinforced structure subjected to shear loading was used to examine the behaviour of bolted rock joints. Three governing materials (steel, grout, and concrete) with two interfaces (bolt-grout and grout-concrete) were considered. To create the best possible mesh, symmetry rules should be applied. To reduce computing demand and time (when a fine mesh is used) the density of the mesh has been optimised during meshing. The division of zones into elements was such that the smallest elements were used where details of stress and displacement were required. The process of

Care was taken to develop the best model for concrete and grout that could offer appropriate behaviour. 3D solid elements, Solid 65 that has 8 nodes was used with each node having three translation degrees of freedom that tolerates irregular shapes without a significant loss in accuracy. Solid 65 is used for the 3-D modeling of solids with or without reinforcing bars (rebar). The geometry and node locations for this type of element are shown in Figure 5 a. The solid element is capable of plastic deformation, cracking in tension, crushing in compression, creep non-linearity, and large deflection geometrical non-linearity, and also includes the failure criteria of concrete Fanning (2001), Feng et al. (2002) and Ansys (2012). Concrete can fail by cracking when the tensile stress exceeds the tensile strength, or by crushing when the compressive stress exceeds the compressive strength. A FE mesh for concrete is shown in Figure 5 b. Figure 6 shows the FE mesh for grout. Due to symmetry

The steel bar, which resists axial and shear loads during loading, due to rock movement, is the main element within the rock bolt system,. The steel bar was modelled appropriately, particularly with regard to the type of element designed and bolt behaviour, in the linear and non-linear region. 3D solid elements, solid 95 with 20 nodes, was used to model the steel bar, with each node having three translation degree of freedom. The approach adopted is to reveal that the experimentally verified shear resistance of fully grouted bolt can be investigated by numerical design. Elastic behaviour of the elements was defined by Young's Modulus and Poisson's ratio of various materials. The stress, strain relationship of steel is assumed as the bi-linear kinematic hardening model and the modulus of elasticity of strain hardening after yielding, is accounted as a hundredth of the original one, *Cha et al.* (2003),

numerical model developed is presented below.

**4. Materials design model** 

FE analysis is shown in Figure 4.

**4.1. Modelling concrete and grout** 

only a quarter of the model needed to be treated.

**4.2. Modelling the bolt** 

**Figure 3.** Bolt-Rock interaction model (after Peng and Guo, 1992)

*Stankus and Guo* (1996) investigated that in bedded and laminated strata, point anchor and fully grouted bolts are very effective, especially if quickly installed at high tension after excavation. They used three lengths 3300, 2400, and 1500 mm and three tensions, 66, 89, and 110 kN and found that:


They developed a method for achieving the optimum beaming effect (OBE). However there were some assumptions in their methodology such as, the problem with the gap element, which is not flexible for any kind of mesh, especially with thin grout. Many relevant parameters about the contact interface cannot be defined in gap element. All materials were modelled in the elastic region.

*Marence and Swoboda* (1995) developed the Bolt Crossing Joint (BCJ) element that connects the elements on both sides of the shear joint. It has two nodes, one each side of the discontinuity. The model cannot predict the de-bonding length along the bolt, grout interface and hinge point position.

It was realised that to further facilitate data analysis and the stress and strain build up along a bolt surrounded by composite material and their interaction, a powerful computer simulation was needed. FE modelling is considered to be the only tool able to accomplish this goal. There is still a lack of an adequate global models of grouted bolts to analyse bolt behaviour properly, particularly at the contact interfaces.

In this chapter, three-dimensional formulations and non-linear deformation of rock, grout, bolt, and two interfaces are taken into account in the reinforced system. A description of the numerical model developed is presented below.
