The 12th International Conference of

International Association for Computer Methods and Advances in Geomechanics (IACMAG)

1-6 October, 2008

Goa, India

A Numerical Analysis of Non-destructive Tests for the Maintenance

and Assessment of Corrosion of Rockbolts and Rock Anchors

Ö. Aydan

Tokai University, Dept. of Marine Civil Engineering, Shizuoka, Japan

S. Tsuchiyama, T. Kinbara, F. Uehara

Chubu Electric Power Company, Nagoya, Japan

N. Tokashiki

Ryukyu University, Faculty of Engineering, Okinawa, Japan

T. Kawamoto

Nagoya University, Faculty of Engineering, Emeritus Professor, Nagoya, Japan

Keywords: underground, powerhouse, corrosion, assessment, rockbolt, rockanchor

ABSTRACT: The assessment of support systems such as rockbolts and rock anchors are very important for the

maintenance of existing rock-engineering structures. The utilization of non-destructive tests is preferable to

destructive tests such as pullout tests. The dynamic responses of rock anchors and rockbolts could be used for

non-destructive tests. These tests should generally provide some information on the constructional conditions,

corrosion and axial force conditions. In the first part of this study, some theoretical models are developed for the

axial and trasverse dynamic tests and then results of numerical simulations are presented by considering

possible conditions in-situ. These conditions are the bonding quality, the existence of corroded parts or couplers

and pre-stress. The results of the numerical simulations could be very valuable for interpretation of the wave

responses to be measured in non-destructive dynamic tests in-situ.

1 Introduction

Rockbolts and rock anchors are commonly used as support members in underground and surface excavations.

They are generally made of steel, which is resistant against corrosion. Nevertheless, the geo-environmental

conditions imposed on rockbolts and rock anchors may be very adverse and they may cause some corrosion in a

short period of time. While rockbolts are generally made of steel bars, rock anchors may be either high-strength

steel bar or cables consisting of several wires. Since the surface area of cable rock anchors are much larger they

are much more prone to corrosion as compared with steel bars.

The maintenance of existing rock engineering structures such as underground powerhouses, dams and slopes

requires the re-assessment of support members. Although in-situ pullout tests would be the best option to assess

the state of the rockbolts and rock anchors, the cost is quite high and the labourship is tremendous to perform

such tests. Furthermore, they may require re-installation of rockbolts and rock anchors if they are pulled-out or

broken. Therefore, the use of non-destructive tests is desirable in view of the cost and labourship. Although there

are several attempts to assess the state of support members, it is very difficult to say that such techniques are

satisfactory for practical purposes. Furthermore, the structural part to be used during the non-destructive tests is

quite limited as compared with the overall size of the support system.

The authors have been now attempting to develop a non-destructive assessment system for rock anchors and

rockbolts. Since the structural part to be used during the non-destructive tests is quite limited, it will be necessary

to utilize the numerical techniques to study the fundamental features of the expected responses, which may be

used for the interpretation of measured responses in-situ non-destructive tests. With this in mind, some

theoretical models are developed for the axial and traverse dynamic tests and then numerical simulations are

carried out based on these fundamental equations by considering possible conditions in-situ. Specifically, the

bonding quality, the existence of corroded parts or couplers and pre-stress are simulated through numerical

experiments. The results of the numerical simulations are then used if the interpretation of the dynamic responses

to be measured in non-destructive dynamic tests in-situ is possible or not.

40

2 Mechanical and Finite Element Modelling of Dynamic Responses of Rockbolts and

Rockanchors

Aydan (Aydan, 1989, Aydan et al. 1987,1988; Kawamoto et al. 1994) studied the static response of rockbolts and

rock anchors and developed some theoretical and numerical models. However, the dynamic responses to be

used in non-destructive tests require dynamic equilibrium equations for axial and traverse responses and their

numerical representations and solutions. These dynamic equilibrium equations and their numerical

representations are given in next sub-sections.

2.1 Mechanical Models

By modification of the static equilibrium equations developed for rockbolts and rock anchors by Aydan (1989) The

equation of motion for the axial responses of rockbolts and rock anchors together with the consideration of inertia

component can be written in the following form (Figure 1a)

b

b

b

t x r

u τ

σ

ρ 2

2

2

+

∂

∂

=

∂

∂

(1)

where b b ρ ,u ,σ ,r and b τ are density, axial displacement, axial stress and radius of tendon and shear stress

along the tendon-grout interface. As for traverse response with the assumption of the existence of the axial

tensile force, the following equation may be written (Figure 1b)

2

2

2

2

x

v

A

T

t

v b b

∂

∂

=

∂

∂

ρ

(2)

where v A b , and T are traverse displacement, area and acting axial tensile stress on tendon .

(a) Axial Response

(b) Traverse Response

Figure 1. Mathematical modelling of axial and traverse response of tendons

2.2 Finite Element Models

(a) Weak Form Formulation

The integral form of Eq.(1) may be written as follows

41

dx

r

dx u

x

dx u

t

u u b

b

b b

b

∫ b ∫ + ∫

∂

∂

=

∂

∂

δ τ

σ

δ ρ δ 2

2

2

(3)

Introducing the following identity into Eq.(3)

( ) σ

δ

δ σ

σ

δ

x

u u

x x

u b

b b ∂

∂

−

∂

∂

=

∂

∂

one can easily obtaines the weak form of Eq.(3) as follows

∫ ∫ ∫ ∫ =

= − =

∂

∂

+

∂

∂ x b

b b n x a

b

b

b b

b dx u

r

dx u

x

dx u

t

u u σ δ τ δ σ

δ

δ ρ 2

2

2

(4)

(b) Finite Element Formulation

Displacement field of rockbolts and rock anchors is discretised in space in a classical finite element form as

[ ]{ } b b u = N U (5)

With the use of Eq. (5), one can write the followings

[ ]{ } b

b

b N u

t

a u = &&

∂

∂

= 2

2

(6a); ([ ]{ }) [ ]{ } b b

b

b N x u B u

x x

u =

∂

∂

=

∂

∂

ε = ( ) (6b)

If Eqs. (5) and (6) are inserted into Eq. (4), one can easily obtain the following expression

[ ]{ } [ ]{ } [ ]{ } { } e b e b e b e M u&& + C u& + K u = f (7)

where

M [N]T [N]dx

[ e ] = ∫ρ ; C [B]T [B]dx

[ e ] = ∫η ; [ ] [ ] [N] [N]dx

r

K E B B dx Kg T

b

T

[ e ] = ∫ − ∫ 2

Similarly, the finite element form of Eq.(2) may be written as follows:

[ ]{ } [ ]{ } { } e b e b e M v&& + T v = t (8)

where

M [N]T [N]dx

[ e ] = ∫ρ ; [B] [B]dx

A

T T T

[ e ] = ∫

3 Properties of Rockbolts and Rockanchors

The authors have been performing some destructive and non-destructive tests on non-corroded and corroded

iron and steel bars. Various relavent parameters are given in Table 1. This table also includes the parameters of

steel bars sampled from 70 years old reinforced concrete structures.

Table 1. Physical and mechanical parameters of various steel bars and cables

Material Unit

Weight

(kN/m3)

P-wave

velocity

(km/s)

S-wave

velocity

(km/s)

Elastic

Modulus

(GPa)

Tensile

Strength

(MPa)

PC Steel Bar 7.56 6.05 200

PC Wire Cable 6.96 167

Ikejima-Corroded smooth bar 4.83

Shimizu Bridge, Smooth Bar (70years) 7.54-7.85 196-213 425-458

Deformed Bar 5.55

Steel 5.95 3.23

Iron 5.91

Theoretical 7.56 5.97 3.19 200

4 Evaluation of Corrosion of Rockbolts and Rock anchors

The evaluation of corrosion from the measured responses is one of the most important items in the interpretation

of non-destructive test investigations. This will definitely require some mechanical models for interpretation. The

mechanical properties of corroded part of the steel are much less than those of steel. Furthermore, the corrosion

may be limited to a certain zone where the anti-corrosive protection may be damaged due to either relative

motions at rock discontinuities or chemical attacks of corrosive elements in the ground water. If the corrosion is

assumed to be taking place uniformly around the steel bar for a certain length as illustrated in Figure 3, the

equivalent elastic, shear moduli and density of the tendon may be obtained using the micro-structure theory

(Aydan et al 1996) as follows:

42

(1 )((1 ) / )

* (1 ) /

c b

c b

b

b

E E

E E

E

E

λ λ α α

α α

+ − − +

− +

= ,

(1 )((1 ) / )

* (1 ) /

c b

c b

b

b

G G

G G

G

G

λ λ α α

α α

+ − − +

− +

= , (9a)

b

c

b

b

ρ

ρ

λ λ α α

ρ

ρ

= (1− ) + ((1− ) +

*

,

L

λ = lc ,

b

c

A

A α = (9b)

where Ec ,Gc and ρ c are properties of corroded part. Ac is corrosion area. If the properties of the corroded part

are negligible, then the above equations take the following form

(1 )(1 )

* (1 )

λ λ α

α

+ − −

−

=

b

b

E

E

,

(1 )(1 )

* (1 )

λ λ α

α

+ − −

−

=

b

b

G

G

(10)

If corrosion is uniformly distributed over the total length of the tendon, then one may write the following equation

* 2

1 ⎟

⎟

⎠

⎞

⎜ ⎜

⎝

⎛

= − o

p

p

v

v

α (11)

The dynamic parameters to be obtained from in-situ non-destructive tests may be used to obtain the dimensions

of corrosion using the models presented above.

Figure 2. Geometrical illustration for the evaluation of effect of corroded part

5 Numerical Analyses and Discussions

Numerical experiments presented in this section are performed to clarify the dynamic responses of rock anchors

and rockbolts for various conditions to be encountered in actual situations. Specifically the following conditions

are considered.

Case 1: Unbonded and non-corroded bar

Case 2: Bonded and non-corroded bar

Case 3: Unbonded bar with corrosion

Case 4: Bonded bar with corrosion

Case 5: Unbonded and non-corroded bar under prestress

Case 6: Unbonded bar with corrosion under prestress

Rock anchors were assumed to be 10m long with a 26mm diameter and they are elastic. When rock anchors are

bonded it is bonded along its entire length. The shear modulus of grouting material and rock are assumed to be 2

and 0.5 GPa respectively. The effect of the bonding is taken into account according to Aydan’s model for

rockbolts. Figure 3 shows the axial dynamic response of rock anchors for Case 1 and Case 2. As noted from the

figure, the wave travels according to p-wave velocity and the bonding has little influence on the arrival time

response of the reflected wave. Nevertheless, some noise-like responses are noted following the fundamental

wave and the amplitude of the reflected waves differs

43

4 8 12 16

-400

-200

200

400

0

TIME (ms)

DISPLACEMENT (μm)

PC; Db=25mm; D h=70mm; L b=10m

Eb=210 GPa; G r=0.0 GPa; G g=0 GPa

Unbonded PC

4 8 12 16

-500

-250

250

500

0

TIME (ms)

DISPLACEMENT (μm)

PC; Db=25mm; Dh=70mm; Lb=10m

Eb=210 GPa; Gr=0.5 GPa; Gg=2 GPa

Bonded PC

4 8 12 16

-500

-250

250

500

0

TIME (ms)

DISPLACEMENT (μm)

PC; Db=25mm; Dh=70mm; Lb=10m

Eb=210 GPa; Gr=0.0 GPa; Gg=0 GPa

Unbonded PC

corrosion at middle section (0.8)

4 8 12 16

-500

-250

250

500

0

TIME (ms) DISPLACEMENT (

μm)

PC; Db=25mm; Dh=70mm; Lb=10m

Eb=210 GPa; Gr=0.5 GPa; Gg=2 GPa

Bonded PC

corrosion at middle section (0.8)

(a) Case 1 (b) Case 2

Figure 3. Axial dynamic displacement reponses of unbonded and bonded rock anchors without corrosion

Next two numerical experiments were concerned with the effect of corrosion on the dynamic response of

unbonded and bonded rock anchors (Case 3 and Case 4). Figure 4 shows the computed dynamic displacement

responses for corroded tendon at the middle with a cross sectional reduction of 20%. Compared with the

responses in previous cases, there are some reflected waves before the arrival of the reflected main shock.

Furthermore, the amplitude of the reflected main shocks are no longer the same and the noise-like waves are

noted following the main shock. The amplitude and duration of these noise-like responses become larger as time

passes. However, the results from these numerical analyses indicate that it is possible to locate the corrosion

location. The amplitude of the reflection from the corroded part depends upon the geometrical and mechanical

characteristics of the corroded part.

(a) Case 3 (b) Case 4

Figure 4. Axial dynamic displacement reponses of unbonded and bonded rock anchors with corrosion

The effect of the pre-stress in rock anchors on the traverse response of rock anchors is investigated (Case

5). In the numerical tests, the prestress value are varied. Figure 5 shows the dynamic displacement

responses for two different values of prestress. As noted from the figure, the travel time of the reflected

traverse wave becomes shorter as the value of prestress increases. These results indicate that the

traverse wave responses should provide valuable information on the prestress state of actual rock anchors.

The final example is concerned with the effect of the prestressed tendon with a corroded part in the middle

section (Case 6). Figure 6 shows the dynamic displacement responses for non-corroded and corroded

tendons subjected to the same prestress. The corroded part is assumed to have the 80% of the original

cross section. When there is no corrosion, the travel time of the reflected displacement wave is 80ms.

Since the behaviour of the tendon is assumed to be elastic the reflected wave arrives at each 80ms

interval. However, when it is corroded at the middle section, it is noted that a wave is reflected before the

arrival of the main wave. Furthermore, the amplitude of the reflected main shocks are no longer the same

and the noise-like waves are noted following the main shock. The amplitude and duration of these noiselike

responses become larger as time passes. However, the results from these numerical analyses

indicate that it is possible to locate the corrosion location. The amplitude of the reflection from the

corroded part is associated with the geometrical and mechanical characteristics of the corroded part.

44

Figure 5. The effect of pre-stress on axial wave responses of pre-stressed tendons (Case 5)

(a) No corrosion (b) With corrosion

Figure 6. The effect of corrosion on axial wave responses of pre-stressed tendons (Case 6)

6 Conclusions

The assessment of support systems such rockbolts and rockanchors are very important for the maintenance of

existing rock engineering structures. The utilization of non-destructive tests utilizing dynamic responses of

support members is very preferable compared to destructive tests. These tests should generally provide some

information on the constructional conditions, corrosion and axial force conditions. In the first part of this study,

some theoretical and finite element models for the axial and trasverse dynamic tests of rockbolt and rock anchors

are presented. Then results of numerical tests are explained by considering possible conditions in-situ such as

the bonding quality, the existence of corroded parts and pre-stress. The numerical experiments indicate that it is

possible to evaluate the length of the rockbolts and rock anchors, the value of the applied pre-stress and the

location of corrosion. Therefore, the results of the numerical simulations presented in this paper could be very

valuable for interpretation of the dynamic responses to be measured in non-destructive dynamic tests in

laboratory and in-situ.

References

Aydan, Ö. (1989): The stabilisation of rock engineering structures by rockbolts. Doctorate Thesis,Nagoya University, 205pages.

Aydan, Ö., Kyoya T., Ichikawa Y., and Kawamoto T. 1987. Anchorage performance and reinforcement effect of fully grouted

rockbolts on rock excavations. The 6th Int. Congress on Rock Mechanics, ISRM, Montreal, 2, 757-760.

Aydan, Ö., Kyoya T., Ichikawa Y., and Kawamoto T. Ito T., and Shimizu Y. 1988. Three-dimensional simulation of an advancing

tunnel supported with forepoles, shotcrete, steel ribs and rockbolts. The 6th Int. Conf. on Num. Meths. in Geomechanics,

Innsbruck, 2, 1481-1486.

Aydan, Ö., Tokashiki N., Seiki T. 1996. Micro-structure models for porous rocks to jointed rock mass. APCOM'96, 3, 2235-

2242.

T. Kawamoto, Kyoya T., Aydan Ö. 1994. Numerical models for rock reinforcement by rockbolts. Int. Conf. on Computer

Methods and Advances in Geomechanics, IACMAG, Morgantown, 1, 33-45.

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A Numerical Analysis of Non-destructive Tests for the Maintenance and Assessment of Corrosion of Rockbolts and Rock Anchors

# A Numerical Analysis of Non-destructive Tests for the Maintenance and Assessment of Corrosion of Rockbolts and Rock Anchors

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