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The reaction of earth to pull-out process of frictional rock bolts was here modeled by the distinct element method (DEM). Ten frictional bolts were prepared; the expanding shells of five bolts included convex edges and the others had the shells with concave bits. The strength of bolts was measured by applying a standard pull-out test; the results confirmed that the strength of shells with convex edges was remarkably more than the strength of other shells. Furthermore, a two-dimensional DEM model of the test was developed by a particle flow code; the obtained results showed that the reaction of rock particles to the contacts occurring between the convex edges and earth was considerably more than those of the concave bits. In the other words, the convex edges transferred the pull-out force into a large area of the surrounded rock, causing these bolts to have the highest resistance against earth movements.

In order to increase the stability of rock structures, rock bolt systems have been developed dramatically for 50 years; they were designed for variety ranges of underground spaces and ground slopes. A rock bolt consists of a steel rod that is located in a hole and stuck mechanically or chemically to the inner wall of hole, so it is tensioned to transfer the applied loads to rock particles using the established contacts during earth movements (

contact strength must be high enough to tolerate dead weights of rocky roofs [

The DEM was developed for modeling discrete systems that include all numerical methods treating the problem domain as an assemblage of independent units. It is mainly applied to the problems of fractured rocks, granular media, concrete and multi-body systems in mechanical engineering. The Particle Flow Code in two Dimensions (PFC2D) based on DEM was adopted to model the corresponding pull-out tests; the rocky wall of holes was modeled as an assembly of rigid particles bonded together. Also, to verify the obtained results, the pull-out strengths acquired from DEM modeling were compared to those estimated from the experimental tests [

To simulate the rocky environment surrounding rock bolts, a cubic block of concrete with 0.5 × 0.7 × 1 m in dimensions was prepared and 10 holes with 45 mm in diameter and 0.85 m in depth were drilled as shown in

Property | Rock bolt with shells include convex edges (RB-M24) | Rock bolt with shells include concave bits (R-DCA-A4) |
---|---|---|

Inner diameter (mm) | 20 | 20 |

Outer diameter (mm) | 25 | 25 |

Length (mm) | 1000 | 1000 |

Tensile strength of steel road (MPa) | 500 | 500 |

Based on the D 4435-84 ASTM standard [

Rock bolts with shells include convex edges | Rock bolts with shells include concave bits | ||
---|---|---|---|

Bolt number | Maximum pull-out force (KN) | Bolt number | Maximum pull-out force (KN) |

1 | 59.1 | 1 | 0.23 |

2 | 89.3 | 2 | 0.15 |

3 | 19.6 | 3 | 0.13 |

4 | 10.2 | 4 | 0.23 |

5 | 10.1 | 5 | 0.26 |

penetrate into the inner wall holes and transfer pull out forces to the surrounded environment very well, so a high frictional resistance occurs remarkably. However, this process did not happen for the shells with concave bits.

In order to examine the reaction between the earth and shells of bolts, a granular model has been developed by the particular flow code in two dimensions (PFC2D), which is based on the distinct-element-method (DEM). The software models the mechanical behaviors of rigid particle assemblies; particle interactions are treated as dynamic processes with states of equilibrium developing whenever the internal forces reach a balance condition. The contact forces and displacements of particles are traced by a time-stepping algorithm; the chosen time step is very small to prevent the spread of disturbance among the particles. In each calculation step, the forces acting on each particle are determined in the point contacts using the force-displacement law, and then Newton’s law is used

to determine the motion of each particle arising from the contact and body update the new position and contacts of particles. The properties of a distinct system which are called micro-mechanical properties are divided into two categories related to contacts and bonds between particles. As shown in

If the relative displacement increment at the contact during a time step is given by Δ δ n (the normal component) and Δ δ s (the shear component), the liner normal and shear contact force ( F n and F n , respectively) will be equal to:

F n = { F 0 n + K n Δ δ n if Δ δ n ≤ 0 F 0 n + 0 if Δ δ n ≥ 0 F s = F 0 s − K s Δ δ s (3.1)

where F 0 n and F 0 s are the linear normal and shear forces at the beginning of the time step, respectively. The micro-mechanical parameters input to the code

are Young modulus (E), friction coefficient ( μ , for Coulomb limit), and normal-to-shear stiffness ratio ( α ) of contacts. The stiffnesses are calculated by [

K n = A E L K s = α K n (3.2)

where:

A = { 2 r t ( 2 D , p l a n e s t r a i n s a t e t , t = 1 ) π r 2 ( 3 D ) L = { R ( 1 ) + R ( 2 ) b a l l - b a l l c o n t a c t R ( 1 ) w a l l - b a l l c o n t a c t r = { m i n ( R ( 1 ) , R ( 2 ) ) b a l l - b a l l c o n t a c t R ( 1 ) w a l l - b a l l c o n t a c t (3.3)

Furthermore, parallel bonds can be created between the balls; it provides the mechanical behavior of a finite-sized piece of cement-like material deposited between the pieces. The parallel bonds can be imagined as a set of elastic springs with constant normal and shear stiffness ( K ¯ n and K ¯ s ) that are uniformly distributed over the contact surfaces; these springs act in parallel with the springs of the linear component (

The parallel-bond force is resolved into a normal and shear force ( F ¯ n and F ¯ s ), and the parallel-bond moment is resolved into a twisting and bending moment ( M ¯ b and M ¯ t ) that can be calculated by:

{ F ¯ n = F ¯ 0 n + K ¯ n A ¯ Δ δ n F ¯ s = F ¯ 0 s − K ¯ s A ¯ Δ δ s { M ¯ b = M ¯ 0 b − K ¯ n I ¯ Δ θ b M ¯ t = M ¯ 0 t − K ¯ s J ¯ Δ θ t (3.4)

where Δ δ n , Δ δ s , Δ θ b and Δ θ t are the relative normal, shear displacement, bend-rotation, and twist-rotation increment, respectively. Also, A ¯ , I ¯ , and J ¯ are the area, moment of inertia, and the polar moment of inertia of the bond cross-section, respectively [

I ¯ = { 2 3 t R ¯ 3 ( 2 D , p l a n e s t r a i n s a t e t , t = 1 ) 1 4 π R ¯ 4 ( 3 D ) J ¯ = { 0 ( 2 D ) 1 2 π R ¯ 4 ( 3 D ) A ¯ = { 2 R ¯ t ( 2 D , p l a n e s t r a i n s a t e t , t = 1 ) π R ¯ 2 ( 3 D ) R ¯ = { m i n ( R ( 1 ) , R ( 2 ) ) b a l l - b a l l c o n t a c t R ( 1 ) w a l l - b a l l c o n t a c t (3.5)

The maximum normal stress ( σ max ) and shear stress ( τ max ) are calculated as following [

σ max = ‖ F ¯ n ‖ A ¯ + ‖ M ¯ b ‖ R ¯ I ¯ τ max = ‖ F ¯ s ‖ A ¯ + { 0 ( 2 D ) ‖ M ¯ t ‖ R ¯ J ¯ ( 3 D ) (3.6)

PFC2D software simulates macro-scale material behavior from the interactions of micro-scale components whose parameters are micro-mechanical properties of constituents, which are listed in

Micro-mechanical properties | |
---|---|

Young modulus of contacts | 0.3 GPa |

Friction coefficient of contacts | 0.2 |

Normal-to-shear stiffness ratio of contacts | 1.0 |

Young modulus of bonds | 0.3 |

Tensile strength of bonds | 1.6 MPa |

Cohesion of bonds | 2.3 MPa |

Normal-to-shear stiffness ratio of bonds | 1.0 |

The studied rock bolts were modeled by wall elements in the PFC2D; the wall elements are rigid and can only move and interact with the particles depending on the stiffness and friction at the contact points. The concrete was simulated by 29,047 disc-shaped rigid particles with range of 0.5 to 0.75 mm in radius. The particles of concrete were bonded using the parallel method; the micro parameters listed in

Furthermore, the reactions of earth to pull out the bolts were shown in

than the concrete region reacting to pull out the shell with concave bits. Consequently, the convex edges create higher frictional resistances than those developed by concave bits in rock structures against earth movements. On the other hand, frictional rock bolts with edges penetrating rocks improve rock strength well in active-support systems.

In this study, the pull-out strength of ten frictional rock bolts with expanding shells was investigated in the laboratory and then modeled using distinct element method. Five bolts included shells with convex edges while the shells of other bolts had concave bits. The experimental results showed that the strength of the rock bolts with convex edges against the pulling out process was more than that of the shells with concave bits. The DEM model confirmed that the strength depended on the shape of the shell structure; the convex edges could transfer the pull-out load to the area that is larger than the concrete region reacting to pull out the shell with concave bits. Because the shells with concave bits could not penetrate to the inner wall of holes and they couldn’t create a high frictional resistance against earth movements. However, the expansion-shell-rock bolts with convex edges are so suitable to improve rock strength in an active-supporting system, because they penetrated to the inner wall and could resist against the pulling out process remarkably. Furthermore, The PFC models confirmed that the DEM modeled the reaction of earth to pulling out process of bolts and transferring applied forces on the bolt roads into the inner wall of holes very well.

The authors would like to acknowledge The University of Zanjan for the financial support.

The authors declare no conflicts of interest regarding the publication of this paper.

Ayyoobi, M.S. and Refahi, A. (2020) Investigating Earth Reaction to Pull-Out Process of Frictional Rock Bolts Using Distinct Element Method. Open Journal of Geology, 10, 851-862. https://doi.org/10.4236/ojg.2020.108038