*6.1.1. Stresses developed along the bolt*

When a beam with a straight longitudinal axis is loaded laterally, its longitudinal axis is deformed into a curve, and the resulting stresses and strains are directly related to the deflection curve, which is affected by the surrounding materials. Figure 11 shows a quarter of the model with induced loads along the shear joint.

When the beam was bent there was deflection and rotation at each point. The angle of rotation is the angle between the bolt axis and the tangent to the deflection curve, shown as point o. was measured for the bolts tested. The deflection trend in 20 MPa concrete is shown in Figure 12.

**Figure 11.** Numerical model (s = symmetric planes, c = compression zone, T = tension zone

Also to find the relationship between deflection and each point along the axis of the bolt, raw output data from the numerical simulation were classified and entered as input data to Maple software. Equation 3 and Figure 13 were established.

$$
\tan \alpha = \frac{dv}{d\mathbf{x}}\tag{1}
$$

Numerical Simulation of Fully Grouted Rock Bolts 619

where;

Shear displacement (mm)



*dx*

*dv*

Distance from centre to end (mm)

**A B** 

**o** 

Shear joint location

**Figure 12.** Bolt displacement in 20 MPa, without Pre-tension

Effective height

**Figure 13.** Shear displacement as a function of bolt length sections in 20 MPa concrete

$$
\alpha = \arctan \frac{dv}{d\mathbf{x}} \tag{2}
$$

$$\mathcal{U}\_y = -40.76 + 26 \, A \text{rc\,\text{tan}\,(e^{(0.05 \times -7.2)})} \tag{3}$$

where;

618 Numerical Simulation – From Theory to Industry

*6.1.1. Stresses developed along the bolt* 

Shear load

of the model with induced loads along the shear joint.

When a beam with a straight longitudinal axis is loaded laterally, its longitudinal axis is deformed into a curve, and the resulting stresses and strains are directly related to the deflection curve, which is affected by the surrounding materials. Figure 11 shows a quarter

When the beam was bent there was deflection and rotation at each point. The angle of rotation is the angle between the bolt axis and the tangent to the deflection curve, shown as point o. was measured for the bolts tested. The deflection trend in 20 MPa concrete is

**Figure 11.** Numerical model (s = symmetric planes, c = compression zone, T = tension zone

Maple software. Equation 3 and Figure 13 were established.

S

Concrete

Bolt

S

Also to find the relationship between deflection and each point along the axis of the bolt, raw output data from the numerical simulation were classified and entered as input data to

Confining pressure

C C

T

Shear joint

T

S

S

Tensioning load

tan , *dv dx*

arctan

*dv dx*

(1)

(2)

(0.05 7.2) 40.76 26 tan( ) *<sup>x</sup> U A <sup>y</sup> rc e* (3)

**6.1. Bolt behaviour** 

shown in Figure 12.

Grout

S


**Figure 12.** Bolt displacement in 20 MPa, without Pre-tension

**Figure 13.** Shear displacement as a function of bolt length sections in 20 MPa concrete

The relationship between vertical displacement at the bolt-joint intersection and hinge point is:

Numerical Simulation of Fully Grouted Rock Bolts 621

**Figure 16.** Trend of stress built up along the bolt axis 20 MPa concrete with 80 kN pre-tension

Bolt axis

strain occurs between them.

It can be seen that induced stresses at these tensile and compression zones are high and the bolt appears to be in a state of yield. At the two hinges the yield limit of the bolt is reached quickly. However, a further increase in the shear load has no apparent influence on the stress built up at the hinge point. From this stage afterwards, only tensile stresses are developed and expanded between the hinge points, and may lead the bolt to fail at some distance between the hinge points located near the shear joint, as the maximum stress and

Distance from centre to end (mm)

Shear joint

From analysing the results in different pre-tension loads it was found there are no significant changes in induced stresses along the bolt with an increase in pre-tension load in the tension zone. However there is a slight reduction in compressive stress with an increasing pre-tension load. Induced stresses are higher than the yield point and less than the maximum tensile strength of the steel bolt in both situations (with and without pretension in all strength concrete). Moreover, in different strength concrete it was observed that the strength of the concrete affects shear displacement and bolt contribution. However there were no meaningful changes in induced stress beyond the yield point along the bolt axis with increasing rock strength but stress was reduced slightly with high pre-tension loading and strength of concrete. The Von Mises stress trend along the bolt axis perpendicular to the shear joint in 20 MPa concrete is plotted in Figure 17. Comparing the results in 20 MPa concrete with and without pre-tension, Von Mises stress decreased

slightly, with an increase in bolt pre-tension. However, this difference is insignificant.

the bolt in one side of the joint surrounded with soft concrete.

Figure 18 shows the concentration of shear stress along the bolt and the rate of change along the axis is shown in Figure 19. Figure 20 shows the trend of shear stress along the length of

$$\text{Uy (hinge)} = (0.15 \text{--} 0.2) \text{ Uy (joint)}$$

Which is consistent with the laboratory results. Figure 14 shows the bolt deflection in 40 MPa concrete.

**Figure 14.** Bolt deflection at the moving side and hinge point versus loading process, in 40 MPa concrete without pre-tension load

Figure 15 shows the contours of stress developed along the bolt in 20 MPa concrete, where the stress in the top part of the bolt and towards the perimeter are tensile and compressive at the centre. However, the stress conditions at the lower half section of the bolt are reversed. In addition, the shape of the bolt between the hinges can be considered as linear. The rate of stress changes in the post failure region is plotted in Figure 16.

**Figure 15.** Stress built up along the bolt axis in 20 MPa concrete without pre-tension

concrete without pre-tension load

MPa concrete.

Shear displacement (mm)

**O** 

The relationship between vertical displacement at the bolt-joint intersection and hinge point is:

Uy (hinge) = (0.15-0.2) Uy (joint) Which is consistent with the laboratory results. Figure 14 shows the bolt deflection in 40

**Figure 14.** Bolt deflection at the moving side and hinge point versus loading process, in 40 MPa

The rate of stress changes in the post failure region is plotted in Figure 16.

Compression zone

**Figure 15.** Stress built up along the bolt axis in 20 MPa concrete without pre-tension

Figure 15 shows the contours of stress developed along the bolt in 20 MPa concrete, where the stress in the top part of the bolt and towards the perimeter are tensile and compressive at the centre. However, the stress conditions at the lower half section of the bolt are reversed. In addition, the shape of the bolt between the hinges can be considered as linear.

Tensile zone

Bolt deflection at moving side

Loading steps

Tensile zone Shear joint

Compression zone

**A** 

Bolt deflection around hinge point

**Figure 16.** Trend of stress built up along the bolt axis 20 MPa concrete with 80 kN pre-tension

It can be seen that induced stresses at these tensile and compression zones are high and the bolt appears to be in a state of yield. At the two hinges the yield limit of the bolt is reached quickly. However, a further increase in the shear load has no apparent influence on the stress built up at the hinge point. From this stage afterwards, only tensile stresses are developed and expanded between the hinge points, and may lead the bolt to fail at some distance between the hinge points located near the shear joint, as the maximum stress and strain occurs between them.

From analysing the results in different pre-tension loads it was found there are no significant changes in induced stresses along the bolt with an increase in pre-tension load in the tension zone. However there is a slight reduction in compressive stress with an increasing pre-tension load. Induced stresses are higher than the yield point and less than the maximum tensile strength of the steel bolt in both situations (with and without pretension in all strength concrete). Moreover, in different strength concrete it was observed that the strength of the concrete affects shear displacement and bolt contribution. However there were no meaningful changes in induced stress beyond the yield point along the bolt axis with increasing rock strength but stress was reduced slightly with high pre-tension loading and strength of concrete. The Von Mises stress trend along the bolt axis perpendicular to the shear joint in 20 MPa concrete is plotted in Figure 17. Comparing the results in 20 MPa concrete with and without pre-tension, Von Mises stress decreased slightly, with an increase in bolt pre-tension. However, this difference is insignificant.

Figure 18 shows the concentration of shear stress along the bolt and the rate of change along the axis is shown in Figure 19. Figure 20 shows the trend of shear stress along the length of the bolt in one side of the joint surrounded with soft concrete.

Numerical Simulation of Fully Grouted Rock Bolts 623

**Figure 19.** The rate of shear stress change along the bolt axis in concrete 20 MPa without pre-tension

y = 430.07e-0.1052x

= 0.9399

R2

**Figure 20.** The rate of shear stress along the bolt axis in concrete 20 MPa without pre-tension in one

Figure 21 shows the trend of changes in shear stress profile with the shear stress tapering off to a stable state past the yield point. It shows the shear stress trend is not exceeded during

0 10 20 30 40 50 60 Distance from joint (mm)

side of the joint plane

Shear stress (MPa)

further loading after the yield point.

Joint plane

**Figure 17.** Von Mises stress trend in 20 MPa concrete without pre-tension

**Figure 18.** Shear stress contour in the concrete 20 MPa without pre-tension

As it shows the maximum shear stress is concentrated in the vicinity of the joint plane, and according to structural analysis, the bending moment at this point is zero. These stress slowly increase, beginning with plastic deformation, and end with a stable situation. The shear stress reduces dramatically from the shear joint towards the bolt end. This trend reaches zero at the hinge point. In the two hinges, the yield limit of the steel is reached quickly, at about 0.3 P and 0.4 P in concrete 20 and 40 MPa respectively, (P is the maximum given applied load). Further increase in the shear force has no apparent influence on stress in the hinges. The distance between the hinge points is reduced as the strength of the concrete is increased.

**O**

Von Mises stress along the bolt (MPa)

**O** 

**Figure 17.** Von Mises stress trend in 20 MPa concrete without pre-tension

**Figure 18.** Shear stress contour in the concrete 20 MPa without pre-tension

between the hinge points is reduced as the strength of the concrete is increased.

As it shows the maximum shear stress is concentrated in the vicinity of the joint plane, and according to structural analysis, the bending moment at this point is zero. These stress slowly increase, beginning with plastic deformation, and end with a stable situation. The shear stress reduces dramatically from the shear joint towards the bolt end. This trend reaches zero at the hinge point. In the two hinges, the yield limit of the steel is reached quickly, at about 0.3 P and 0.4 P in concrete 20 and 40 MPa respectively, (P is the maximum given applied load). Further increase in the shear force has no apparent influence on stress in the hinges. The distance

**Shear joint** 

Distance from centre to end (mm)

**Shear joint A** 

**Max Stress concentration**  **A** 

**Figure 19.** The rate of shear stress change along the bolt axis in concrete 20 MPa without pre-tension

**Figure 20.** The rate of shear stress along the bolt axis in concrete 20 MPa without pre-tension in one side of the joint plane

Figure 21 shows the trend of changes in shear stress profile with the shear stress tapering off to a stable state past the yield point. It shows the shear stress trend is not exceeded during further loading after the yield point.

Eventually, a combination of this stress with induced tensile stress at the bolt - joint intersection leads the bolt to fail. By increasing the initial tensile load on the bolt, the shear stress was decreased, which was seen in different strength concrete but there was no significant changes with increasing shear load after the yield point. Any reduction in shear stress causes an increase in the resistance of the bolt to shear. It can be noted that the shear stress increased slightly with an increasing strength of concrete.

Numerical Simulation of Fully Grouted Rock Bolts 625

**A** 

**Figure 22.** Deformed bolt shape in post failure region in 20 MPa concrete

Compression strain

+13


Shear load

**Figure 23.** Plastic strain contour along the bolt axis in concrete 20 MPa without pre-tension

Tensile strain

deflection.

**O** 

Figure 25 shows the beginning of plastic strain during shearing and a trend of strain developing as a function of load stepping. It notes that both the tensile and compression strain around the bolt started approximately 27-30 % after loading began and increased with an increasing shearing load. However, the rate of increase in the tensile zone is higher than the compression zone. It also showed these strains appeared in the early stage of loading with a small displacement (around 3 mm), which increased with increase in shear

Shear joint


+14.6

Loading steps

**Figure 21.** Shear stress trend in bolt –joint intersection in concrete 20 MPa at post failure region without pre-tension load

#### *6.1.2. Strain developed along the bolt*

Strain was generated along all the surrounding materials as the shear load increased, particularly along the axis of the bolt. As deflection increased, plastic strain is induced in the critical locations in all three materials (bolt - resin and concrete). Figure 22 shows the location of maximum plastic deformation along the bolt while bending. It shows there are two hinge points around the shear plane approximately 50 mm from the shear joint in 20 MPa concrete.

However an increasing pre-tension load has not affected hinge point distances, which are around 2.3 Db (Db is bolt diameter). This value in the laboratory test is around 44 mm that is 2 Db. The strain and the rate of strain changes along the bolt in 20 MPa concrete are shown in Figures 23 and 24.

As Figure 23 shows that the outer layer of the bolt yielded, whereas the middle section remained in an elastic state.

**Figure 22.** Deformed bolt shape in post failure region in 20 MPa concrete

pre-tension load

Shear stress (MPa)

MPa concrete.

in Figures 23 and 24.

remained in an elastic state.

*6.1.2. Strain developed along the bolt* 

Eventually, a combination of this stress with induced tensile stress at the bolt - joint intersection leads the bolt to fail. By increasing the initial tensile load on the bolt, the shear stress was decreased, which was seen in different strength concrete but there was no significant changes with increasing shear load after the yield point. Any reduction in shear stress causes an increase in the resistance of the bolt to shear. It can be noted that the shear

**Figure 21.** Shear stress trend in bolt –joint intersection in concrete 20 MPa at post failure region without

Loading steps

Nearly constant

Strain was generated along all the surrounding materials as the shear load increased, particularly along the axis of the bolt. As deflection increased, plastic strain is induced in the critical locations in all three materials (bolt - resin and concrete). Figure 22 shows the location of maximum plastic deformation along the bolt while bending. It shows there are two hinge points around the shear plane approximately 50 mm from the shear joint in 20

However an increasing pre-tension load has not affected hinge point distances, which are around 2.3 Db (Db is bolt diameter). This value in the laboratory test is around 44 mm that is 2 Db. The strain and the rate of strain changes along the bolt in 20 MPa concrete are shown

As Figure 23 shows that the outer layer of the bolt yielded, whereas the middle section

stress increased slightly with an increasing strength of concrete.

**Figure 23.** Plastic strain contour along the bolt axis in concrete 20 MPa without pre-tension

Figure 25 shows the beginning of plastic strain during shearing and a trend of strain developing as a function of load stepping. It notes that both the tensile and compression strain around the bolt started approximately 27-30 % after loading began and increased with an increasing shearing load. However, the rate of increase in the tensile zone is higher than the compression zone. It also showed these strains appeared in the early stage of loading with a small displacement (around 3 mm), which increased with increase in shear deflection.

Numerical Simulation of Fully Grouted Rock Bolts 627

**A** 

**6.2. Concrete behaviour** 

strength concrete.

Concrete displacement (mm)

**O** 

*6.2.1. Stress developed in concrete* 

The behaviour of the centre concrete under shear load in double shearing assembly was analysed in different strength concrete and different pre-tension loads. During shearing the middle part of the assembled system was displaced downwards with increasing shear load. Figure 26 shows the deflection rate after failure. Reaction forces are developed during the middle concrete block displacement, which increased in critical locations (at the vicinity of the shear joint), affected by the bolt. The reaction forces induce and propagate stress and strain in sheared zones. Figure 27 shows the high-induced stress near the shear joint as the maximum reaction forces are expected there. When induced stress is larger than the ultimate stress the concrete will be crushed. Figure 28 displays the rate of induced stress at the interface near the shear joint. It shows that induced stresses are much higher than the compressive strength, and the concrete at this location would be severely crushed. From the figure it can be seen that the high stress is approximately 60 mm from the shear plane. At an early stage of loading, the concrete was crushed and stresses propagated throughout, with bolt yield to start at around 2 mm from the edge of the intersection. Beyond this point stresses increased quickly near the joint intersection and reaction zones. Induced stresses near the shear joints were reduced slightly with increase in the pre-tension load on the bolt. In addition the trend of induced stresses and strains built up along the concrete interface in 40 MPa concrete was the same as with 20 MPa concrete. However, the value of stresses and strains were slightly reduced in higher

**Figure 26.** Concrete displacement in non-pretension condition in 20 MPa concrete

Distance from centre to end (mm)

**Figure 24.** Strain trend along the bolt axis in concrete 20 MPa without pre-tension in upper fibre of the bolt

**Figure 25.** Yield strain trend as a function of time stepping concrete 20 MPa in 20 kN pre-tension load

With an increase in loading, shear displacement was increased. There was a significant increase in shear displacement after 35% of loading time. Bending of the bolt is predominant at a low loading time. plastic strain begins at the hinge point around 35 % of loading. A comparison of the data (with and without pre-tension) shows that the intensity of the strain along the axis of the bolt is slightly reduced with an increase in pretension load. However the affected area in the tensile zone expands towards the shear joint. The strains in the compression and tension zones were reduced in higher strength concrete.

#### **6.2. Concrete behaviour**

626 Numerical Simulation – From Theory to Industry

Bolt axis

bolt

Compression & tension strain

concrete.

**Figure 24.** Strain trend along the bolt axis in concrete 20 MPa without pre-tension in upper fibre of the

Tensile strain trend

Shear joint

**Figure 25.** Yield strain trend as a function of time stepping concrete 20 MPa in 20 kN pre-tension load

Loading streps

Compression strain

With an increase in loading, shear displacement was increased. There was a significant increase in shear displacement after 35% of loading time. Bending of the bolt is predominant at a low loading time. plastic strain begins at the hinge point around 35 % of loading. A comparison of the data (with and without pre-tension) shows that the intensity of the strain along the axis of the bolt is slightly reduced with an increase in pretension load. However the affected area in the tensile zone expands towards the shear joint. The strains in the compression and tension zones were reduced in higher strength

#### *6.2.1. Stress developed in concrete*

The behaviour of the centre concrete under shear load in double shearing assembly was analysed in different strength concrete and different pre-tension loads. During shearing the middle part of the assembled system was displaced downwards with increasing shear load. Figure 26 shows the deflection rate after failure. Reaction forces are developed during the middle concrete block displacement, which increased in critical locations (at the vicinity of the shear joint), affected by the bolt. The reaction forces induce and propagate stress and strain in sheared zones. Figure 27 shows the high-induced stress near the shear joint as the maximum reaction forces are expected there. When induced stress is larger than the ultimate stress the concrete will be crushed. Figure 28 displays the rate of induced stress at the interface near the shear joint. It shows that induced stresses are much higher than the compressive strength, and the concrete at this location would be severely crushed. From the figure it can be seen that the high stress is approximately 60 mm from the shear plane. At an early stage of loading, the concrete was crushed and stresses propagated throughout, with bolt yield to start at around 2 mm from the edge of the intersection. Beyond this point stresses increased quickly near the joint intersection and reaction zones. Induced stresses near the shear joints were reduced slightly with increase in the pre-tension load on the bolt. In addition the trend of induced stresses and strains built up along the concrete interface in 40 MPa concrete was the same as with 20 MPa concrete. However, the value of stresses and strains were slightly reduced in higher strength concrete.

Distance from centre to end (mm)

**Figure 26.** Concrete displacement in non-pretension condition in 20 MPa concrete

Numerical Simulation of Fully Grouted Rock Bolts 629

*6.2.2. Strain developed in concrete* 

**Figure 29.** Strain contours in 20 MPa concrete without pre-tension

tension and 27 mm diameter hole

Strain in grout-

Strain in grout and concrete

**Figure 30.** Induced strain in concrete 20 MPa in grout and concrete versus loading without a pre-

Loading steps

Concrete Grout

The highest level of induced stress was near the shear joint, so it is expected that strain would be highest around this zone. Figure 29 shows the induced strain contours at the high pressure zone. Figure 30 shows induced strain in terms of loading time in grout and concrete. It shows that the strain generation begins in the concrete before it is seen in the resin grout because lower strength concrete is one third the strength of grout. There is an approximate exponential relationship in the strain trend as loading increases. After 20% of loading steps, plastic strain is induced along the contact interface near the shear joint. This value in soft concrete (20 MPa) is at an earlier stage, which is around 15% of loading step. This shows the strain built up along the axis of the bolt is lower than in the shear direction.

**Figure 27.** Yield stress induced in 20 MPa concrete without pre-tension condition

Distance from centre to end (mm)

**Figure 28.** Induced stress and displacement trend in 20 MPa concrete without pre-tension

#### *6.2.2. Strain developed in concrete*

628 Numerical Simulation – From Theory to Industry

**O**

**O**

Dis

pl. (mm)- induced stress (MPa)

Maximum reaction stresses

**Figure 27.** Yield stress induced in 20 MPa concrete without pre-tension condition

**Shear joint location** 

Stress trend

Deflection trend

**A** 

**A**

**Figure 28.** Induced stress and displacement trend in 20 MPa concrete without pre-tension

Distance from centre to end (mm)

The highest level of induced stress was near the shear joint, so it is expected that strain would be highest around this zone. Figure 29 shows the induced strain contours at the high pressure zone. Figure 30 shows induced strain in terms of loading time in grout and concrete. It shows that the strain generation begins in the concrete before it is seen in the resin grout because lower strength concrete is one third the strength of grout. There is an approximate exponential relationship in the strain trend as loading increases. After 20% of loading steps, plastic strain is induced along the contact interface near the shear joint. This value in soft concrete (20 MPa) is at an earlier stage, which is around 15% of loading step. This shows the strain built up along the axis of the bolt is lower than in the shear direction.

**Figure 29.** Strain contours in 20 MPa concrete without pre-tension

Loading steps

**Figure 30.** Induced strain in concrete 20 MPa in grout and concrete versus loading without a pretension and 27 mm diameter hole

A comparison of induced strain along the joint interface with and without pre-tension found that the strain in the shear direction is reduced (around 15%) with increasing pre-tension. In the axial and shear direction strain was concentrated near the shear joint.

Numerical Simulation of Fully Grouted Rock Bolts 631

**Figure 32.** Maximum induced stress contours in grout layer without pre-tension and 20 MPa

High stress zone

**Figure 33.** Gap formation in post failure region in 20 MPa concrete in the Numerical simulation

Created gap

**Figure 34.** Gap formation in post failure region in 20 MPa concrete in the laboratory test

Figure 31 shows the deformation behaviour of both concrete medium and bolt. Plastic deformation of concrete occurs nearly 15 % of the maximum shear load while the deformation of the bolt occurs at 33% of the loading steps. From the graphs it can be inferred that in very low values of bolt deflection and time steps, fractures happen in the concrete, which is in the elastic range of the bolt. Any further increase in shearing does not influence the stress at the hinge points, however induced stress in the concrete blocks causes extensively fractures and eventually leads to failure.

**Figure 31.** Induced strain in concrete and bolt as a function of loading steps in 20 MPa concrete with 80 kN pre-tension
