Supporting Action Of Anchors Bolts

15 Supporting action of anchors/bolts

Anchors or rockbolts are reinforcements (usually made of steel) which are inserted into the ground to increase its stiffness and strength. There are various sorts of reinforcement actions and the corresponding terminology is not uniform.1 The following terminology is used in soil mechanics: 1. If the reinforcement bar is fixed only at its both ends, then it is called an ’anchor’. Anchors can be pre-stressed or not, in the latter case they assume force only after some extension (e.g. due to convergence of the tunnel). 2. If the reinforcement bar is connected to the surrounding ground over its entire length, then it is called a ’nail’ or ’bolt’. The connection can be achieved with cement mortar (Fig. 15.1).

Fig. 15.1. Nail

In jointed rock, reinforcement bars are placed ad hoc to prevent collapse of individual blocks (Fig. 15.2). Anchoring or bolting in a regular array is called ’pattern bolting’. 1

see also C.R. Windsor, A.G. Thompson: Rock Reinforcement - Technology, Testing, Design and Evaluation. In: Comprehensive Rock Engineering, Vol. 4, Pergamon Press 1993, 451-484

308

15 Supporting action of anchors/bolts

Fig. 15.2. Individual application of anchors to prevent downfall of blocks

15.1 Impact of pattern bolting It is generally believed that reinforcing improves the mechanical behaviour of ground. Despite several attempts however, the reinforcing action of stiff inlets is not yet satisfactorily understood and their application is still empirical. In some approaches, reinforced ground is considered as a two-phase continuum in the sense that both constituents are assumed to be smeared and present everywhere in the considered body. Thus, their mechanical properties prevail everywhere, provided that they are appropriately weighed (Appendix F). The stiffening action of inlets can be demonstrated if we consider a conventional triaxial test on a soil sample containing a thin pin of, say, steel (Fig. 15.3).

Fig. 15.3. Steel inlet in triaxial sample, distribution of vertical displacements and stress trajectories for two different orientations of the inlet.

The stiff inlet is here assumed as non-extendable (i.e. rigid). Therefore, its vertical displacement is constant as shown in Fig. 15.3. This implies a relative slip of the adjacent soil, which is oriented downwards in the upper half and upwards in the lower half. Being stiffer, the pin ’attracts’ force and, thus, the adjacent soil is partly relieved from compressive stresses. As a result, the triaxial sample, viewed as a whole, is now stiffer. This effect is closely related to ’tension stiffening’ known in concrete engineering.

15.1 Impact of pattern bolting

309

Another way to increase the stiffness of reinforced soil is given by increasing the pressure level. As known, the stiffness of granular materials increases almost linearly with stress level. The latter can be increased by pre-stressing an array of anchors, i.e. of reinforcing inlets that transmit the force to the surrounding ground only at their ends ad not over their entire length. An analysis of this mechanism is presented in the next section. 15.1.1 Ground stiffening by pre-stressed anchors The strengthening effect of pre-stressed pattern bolting will be considered for the case of a tunnel with circular cross section within a hydrostatically stressed elastoplastic ground. The primary hydrostatic stress is σ∞ . If the spacing of the anchors is sufficiently small, their action upon the ground can be approximated with a uniform radial stress σA (Fig. 15.4).

Fig. 15.4. Idealised pattern bolting

The radial stress σA is obtained by dividing the anchor force with the pertaining surface. Let n be the number of anchors per one meter of tunnel length. We then obtain σA0 =

nA 2πr0

,

σAe =

nA 2πre

or σAe = σA0 ·

r0 re

.

It is, thus, reasonable to assume the following distribution of σA within the range r0 < r < re σA = σA0 ·

r0 r

.

(15.1)

We consider the entire stress in the range r0 < r < re . Pre-stressing of the anchors increases the radial stress from σr to σr + σA (Fig. 15.5).

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15 Supporting action of anchors/bolts

We now assume that in the range r0 < r < re the shear strength of the ground is fully mobilised. For this case we will determine the support pressure p. For simplicity, we consider a cohesionless ground (c = 0) and obtain σθ = Kp (σr + σA ) with Kp =

1 + sin ϕ 1 − sin ϕ

(15.2)

.

Fig. 15.5. Limit stress in pre-stressed region

Equilibrium in radial direction reads d(σr + σA ) σr + σA − Kp (σr + σA ) + =0 dr r

.

(15.3)

Introducing (15.1) into (15.3) yields dσr 1 r0  + σr (1 − Kp ) − Kp σA0 =0 . dr r r

(15.4)

The solution of the differential equation (15.4) is obtained as σr = const · rKp −1 − σA0

r0 r

. !

The integration constant is obtained from the boundary condition σr (r0 ) = p where p is the pressure exerted by the ground upon the lining. We finally obtain  σr = (p + σA0 ) ·

r r0

Kp −1

− σA0

r0 r

.

(15.5)

At the boundary of the elastic region (at r = re ) it must be σr = σe , where σe is obtained from equation 14.21:

15.1 Impact of pattern bolting

 (p + σA0 )

re r0

Kp −1

− σA0 ·

r0 2 · σ∞ = re Kp + 1

311

(15.6)

We meet the simplifying assumption that the plastified zone coincides with the anchored ring, i.e. we introduce re = r0 + l, where l is the theoretical anchor length, into Equ. 15.6 and eliminate p. We thus obtain the support pressure in dependence of the pre-stressing force A of the anchors, their number n per tunnel meter, the theoretical anchor length l, the tunnel radius r0 , the primary stress σ∞ and the friction angle ϕ:  p=

2σ∞ nA r0 + · Kp + 1 2πr0 r0 + l



r0 r0 + l

Kp −1

−

nA 2πr0

(15.7)

The real anchor length L should be greater than the theoretical one, in such a way that the anchor force can be distributed along the boundary r = re (Fig. 15.6). In practice, the anchor lengths are taken as 1.5 to 2 times the thickness of the plastified zone.

Fig. 15.6. Theoretical (l) and real (L) anchor lengths

15.1.2 Pre-stressed anchors in cohesive soils To consider cohesion, equation 15.2 is replaced by

σθ = Kp (σr + σA ) + 2c Thus, equilibrium in radial direction reads

cos ϕ 1 − sin ϕ

.

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15 Supporting action of anchors/bolts

  1 dσr r0 cos ϕ + · σr (1 − Kp ) − 2c − Kp σA0 =0 dr r 1 − sin ϕ r

.

The solution of this differential equation reads: σr = const · rKp −1 − σA0

r  0

r

−c

2 cos ϕ (Kp − 1)(1 − sin ϕ)

!

With the boundary condition σr (r0 ) = p and with finally obtains  σr = (p + σA0 + c · cot ϕ)

r r0

2 cos ϕ (Kp −1)(1−sin ϕ)

Kp −1 + σA0

. = cot ϕ one

r0 − c · cot ϕ . r

From the requirement σr (re ) = σe with σe according to equation 14.25 it is obtained: 

Kp −1 re r0 − σA0 − c · cot ϕ (p + σA0 + c · cot ϕ) r0 re = σ∞ (1 − sin ϕ) − c · cos ϕ , With re = r0 + l it finally follows:   Kp −1 Kp r0 r0 nA p = σ∞ (1 − sin ϕ) − 1− r0 + l 2πr0 r0 + l    Kp −1 Kp −1 r0 r0 −c · cot ϕ 1 − . − c · cos ϕ r0 + l r0 + l 

(15.8)

If the ground pressure is to be taken solely by the anchors (i.e. p = 0), then:

nA ≥



2πr0

! ·

 σ∞ (1 − sin ϕ)

r0 r0 + l

Kp −1

Kp r0 r0 + l    Kp −1 Kp −1 " r0 r0 −c · cot ϕ 1 − − c · cos ϕ r0 + l r0 + l 1−

.

In case of large convergences, support by anchors is preferable to shotcrete which is not sufficiently ductile and may fracture. However, adjustable anchors should be used.

15.1 Impact of pattern bolting

313

15.1.3 Stiffening effect of pattern bolting In this section we consider the stiffening effect of arrays of bolts, i.e. reinforcing elements that are not pre-stressed and transmit shear forces to the surrounding ground over their entire length. Considering equilibrium of the normal force N and the shear stress τ applying upon the periphery of a bolt element of the length dx (Fig. 15.7) we obtain dN = τ πddx. With N = σπd2 /4, σ = Eε and ε = dus /dx we obtain d2 us 4τ = 2 dx Ed

,

with us being the displacement of the bolt. Obviously, the shear stress τ acting between bolt and surrounding ground is mobilised with the relative displacement, τ = τ (s), s = us − u, where u is the displacement of the ground.2

Fig. 15.7. Forces upon a bolt element

Of course, u depends on τ : In a first step of simplified (uncoupled) analysis we assume that u does not depend on τ and is given by the elastic solution (cf. Equ. 14.16): u=

σ∞ − p r0 2 2G r

.

Herein, r is the radius with respect to the tunnel axis. Furthermore, we assume a rigid-idealplastic relation τ (s), i.e. τ achieves immediately its maximum value τ0 . Thus, the total force transmitted by shear upon a bolt of the length l is lτ0 πd. This force is applied via the top platen upon the tunnel wall. Assuming n bolts per m2 tunnel wall we obtain thus the equivalent support pressure pbolt = nlτ0 πd. If the arrangement of bolts is given by the spacings a and b (Fig. 15.8), then n = 1/(ab). Thus, 2

Consider e.g. the relations used in concrete engineering: K. Zilch and A. Rogge, Grundlagen der Bemessung von Beton-, Stahlbeton- und Spannbetonbauteilen nach DIN 1045-1. In: Betonkalender 2000, BK1, 171-312, Ernst & Sohn Berlin, 2000

314

15 Supporting action of anchors/bolts

pbolt =

1 τo πdl ab

(15.9)

modifies the support line as shown in Fig. 15.9.

Fig. 15.8. Array of bolts

Fig. 15.9. Ground reaction line and support line affected by idealised bolts (Assumptions: rigid bolts, rigid-idealplastic shear stress transmission to the ground, ground displacement not influenced by the bolts, installation of bolts is instantaneous).

An alternative approach based on the multiphase model of reinforced ground is given in Appendix F.