**7.1. Bolt behaviour**

From the simulations it was found that there will be an increase in grout - bolt surface debonding, and this decrease in diameter due to Poisson's effect in the steel, contributes to an axial elongation of about 0.084 mm at the top collar where the load is applied. This value in push test is around 0.05 mm as shown in Figure 37.

Numerical Simulation of Fully Grouted Rock Bolts 635

**Figure 39.** Shear strain in bolt ribs in push test

and

Where, *<sup>t</sup>* 

satisfied:

**7.2. Grout behaviour** 

Maximum tensile stress along the bolt is 330 MPa. This is one half of the strength of the elastic yield point of 600 MPa. This means the bolt behaves elastically and is unlikely to

*t*

2 4

(4)

(5)

(6)

(7)

is the

*b T D*

2 \* 4 *Db t <sup>T</sup>* 

is the tensile stress, *T* is the axial load, Db is the bolt diameter and *<sup>y</sup>*

yield strength of the bolt. The bolt behaves elastically as long as the following expression is

*t* < *y* 

The behaviour of interface grout annulus is assumed to be elastic, softening, residual, plastic

So in this situation with failure along the bolt-grout interface will not yield.

flow type. This behaviour was developed by Aydan (1989), and is given as:

 *G* max 

 

reach a yield situation. Axial stress developed along the bolt is given by:

**Figure 37.** The bolt movement in pulling test

Figure 38 show the maximum induced strain near the applied load position in both the pull and push results. The strain is around the elastic strain and therefore the bolt is unlikely to yield.

**Figure 38.** Bolt displacement contour in Bolt Type T1 in case of push test

**Figure 39.** Shear strain in bolt ribs in push test

Maximum tensile stress along the bolt is 330 MPa. This is one half of the strength of the elastic yield point of 600 MPa. This means the bolt behaves elastically and is unlikely to reach a yield situation. Axial stress developed along the bolt is given by:

$$
\sigma\_t = \frac{4T}{\pi \text{D}\_b} \tag{4}
$$

and

634 Numerical Simulation – From Theory to Industry

**Figure 37.** The bolt movement in pulling test

Grout Outer plate

**Figure 38.** Bolt displacement contour in Bolt Type T1 in case of push test

yield.

grout

Push load

Pull

Figure 38 show the maximum induced strain near the applied load position in both the pull and push results. The strain is around the elastic strain and therefore the bolt is unlikely to

Outer plate

b lt

Bolt

Shear and tensile

*Bolt*

*Rock*

*Grout*

$$T = \frac{\pi D\_b^{-2} \ast \sigma\_t}{4} \tag{5}$$

Where, *<sup>t</sup>* is the tensile stress, *T* is the axial load, Db is the bolt diameter and *<sup>y</sup>* is the yield strength of the bolt. The bolt behaves elastically as long as the following expression is satisfied:

$$
\sigma\_t \prec \sigma\_y \tag{6}
$$

So in this situation with failure along the bolt-grout interface will not yield.

#### **7.2. Grout behaviour**

The behaviour of interface grout annulus is assumed to be elastic, softening, residual, plastic flow type. This behaviour was developed by Aydan (1989), and is given as:

$$
\tau = G\gamma \quad \tau < \tau\_{\text{max}} \tag{7}
$$

$$
\tau = \tau\_{\text{max}} - \frac{\gamma - \gamma\_{\text{max}}}{\gamma\_r - \gamma\_{\text{max}}} (\tau\_{\text{max}} - \tau\_r) \tag{8}
$$

$$
\boldsymbol{\pi} = \boldsymbol{\pi}\_r \tag{9}
$$

Numerical Simulation of Fully Grouted Rock Bolts 637


**Figure 40.** Shear stress contours along the grout interface

= Shear stress along the bolt grout interface

Using the *Farmer* (1975) equation the shear strength was equal to 27 MPa.

0.2 ( )

*x <sup>a</sup> e*

(12)

0.1

During shearing the outer plate of the bolt was influenced by the stresses and strains of the resin. From the analyses it was found that induced stress along the surface of the outer plate was insignificant at about 30 % of the yield stress, which is not sufficient to cause the outer plate to yield. In addition, grout de-bonding occurred around 50 to 60 kN at different levels

Numerical analysis of the grout – concrete - bolt interaction has demonstrated that:

 There were no significant changes in induced stresses along the bolt with increasing pre-tension load, particularly in the tension zone. However, there was a small reduction

 The yield limit of the bolt at the hinge point depends on the strength of the concrete. In 20 MPa concrete the yield limit was 0.3P and in 40 MPa concrete it increased to 0.4P. A


where;


of applied load.

**8. Summary** 

= Axial stress

in compression stress.



where;


The grout material is in elastic conditions if the following expression is satisfied;

$$T\_t < T\_y \tag{10}$$

where;


From the strain generated along the grout interface it was found that the surface of the grout was disturbed by shear stress induced at the interface and this strain is higher than the elastic strain that damaged the grout at the contact surface. Figure 39 shows the shear stress contour at the grout interface. The whole contact area of the grout was affected by the shear stress and consequently the induced shear strain dominated. The maximum bonding stress was approximately 38% of the uniaxial compressive strength of the resin grout. The stress produced along the grout contact interface was greater than the yield strength of the grout of 16 MPa, and beyond the yield point only a slight increase in load is enough to damage the whole contact surface. Shear displacement increased as a result bonding failure. The shear stress at the bolt - grout interface can be calculated by Equation (11), which agrees with the results from the numerical simulation.

Thus,

$$
\tau = \frac{f}{A} = \frac{\sigma \pi D^2}{8 \pi r l} = 23.2 MPa \tag{11}
$$

where;




results from the numerical simulation.


= Shear strain at any point in the interface

= Shear strain at residual shear strength

= Shear strain at peak shear strength

= Residual shear strength of the interface

= Peak shear strength of interface

= Shear stress at any point in interface

where;





where;

Thus,

where;




max max max max

> *r*

 

*r* 

 

The grout material is in elastic conditions if the following expression is satisfied;

From the strain generated along the grout interface it was found that the surface of the grout was disturbed by shear stress induced at the interface and this strain is higher than the elastic strain that damaged the grout at the contact surface. Figure 39 shows the shear stress contour at the grout interface. The whole contact area of the grout was affected by the shear stress and consequently the induced shear strain dominated. The maximum bonding stress was approximately 38% of the uniaxial compressive strength of the resin grout. The stress produced along the grout contact interface was greater than the yield strength of the grout of 16 MPa, and beyond the yield point only a slight increase in load is enough to damage the whole contact surface. Shear displacement increased as a result bonding failure. The shear stress at the bolt - grout interface can be calculated by Equation (11), which agrees with the

2

*<sup>f</sup> <sup>D</sup> MPa*

8

*A rl* 

= Shear stress in the grout - bolt interface (MPa)

23.2

(11)

  ( )*<sup>r</sup>*

(8)

(9)

*t y T T* (10)

**Figure 40.** Shear stress contours along the grout interface

Using the *Farmer* (1975) equation the shear strength was equal to 27 MPa.

$$\frac{\pi}{\sigma} = 0.1e^{-(\frac{0.2\times}{a})} \tag{12}$$

where;


During shearing the outer plate of the bolt was influenced by the stresses and strains of the resin. From the analyses it was found that induced stress along the surface of the outer plate was insignificant at about 30 % of the yield stress, which is not sufficient to cause the outer plate to yield. In addition, grout de-bonding occurred around 50 to 60 kN at different levels of applied load.
