design of rock bolt and shotcrete support of tunnel

Australian Geomechanics – June 2002 81

DESIGN OF ROCK BOLT AND SHOTCRETE SUPPORT OF TUNNELROOFS IN SYDNEY SANDSTONE

Robert Bertuzzi and Philip J N PellsPells Sullivan Meynink

ABSTRACTThe analytical design of rock bolt and shotcrete support of roofs of wide span tunnels under moderate cover in Sydney’sHawkesbury Sandstone is described. The simple process allows rapid parametric studies to be carried out prior todetailed numerical modelling. Design graphs for a typical three-lane road tunnel are presented.

1 INTRODUCTIONThis paper details the design of rock bolt and shotcrete support of roofs of wide span tunnels in Sydney’s HawkesburySandstone at depths to approximately 60m. It provides a simplified analytical process for the method developed for theSydney Opera House Carpark (18 m spans) and which was used for the Eastern Distributor Tunnels (spans up to 24 m).The methodology is essentially an extension to the linear arch theory.

Hawkesbury Sandstone is typically characterised by: horizontal bedding with typical spacing of 1 to 3 m; dominantnorth-south and east-west trending steeply dipping joints; and a horizontal stress that is 1.5 to 2 times the vertical stress.In the unweathered to slightly weathered state, the rock mass is typically medium strength (average rock mass UCSabout 20MPa), with an elastic modulus in the range 1000 to 3000 MPa. Bedding planes are often infilled with sandyclays up to 20mm thick with typical friction angles of 20 to 30°.

2 DESIGN METHODOLOGYThe design methodology described in this paper is based on the work developed in Pells & Best (1991) and Pells et al(1994). The underlying philosophy is to use the displacements derived from a jointed rock mass analysis to design rockbolt reinforcement that provides greater capacity than the stresses derived from an elastic continuum analysis. Shotcretethickness is calculated based on techniques adapted from the work of Barrett & McCreath (1995) once the rock boltspacing has been determined.

The steps involved comprise calculating the following:

(i) Roof Beam Thickness. The minimum roof beam thickness for the tunnel span in question is assessed to limitroof sag to acceptable levels. This is used to set the rock bolt length. As a first pass, the roof sag may be set between10 to 50mm. Linear arch theory is used to calculate roof beam thickness.

(ii) Stresses. The induced stresses acting along and across the bedding planes are calculated as if the roof beamwas acting as an elastic continuum. Stresses can be calculated using beam theory and/or numerical solutions.

(iii) Excess Shear Stress. The excess shear stress is the difference between the induced shear stress from step (ii)and the shear capacity of the bedding planes. This excess needs to be resisted by the rock bolts.

(iv) Shear. The induced relative horizontal movement or shear between beds is calculated as if the roof beam wasallowed to de-laminate and sag into the tunnel opening. This can be calculated using beam theory and/or numericalsolutions.

(v) Rock Bolt Contribution. The contribution of the rock bolt as a result of shear displacement calculated in step(iv) can be calculated using published graphs such as Pells et al (1994), analytical solutions such as Pellet & Egger(1996) and/or numerical solutions.

(vi) Factor of Safety. A FoS is calculated as the force resisting shear divided by the force causing shear calculatedin step (ii). If the shear induces unacceptable rock bolt loads, then either the bolt spacing can be reduced or the roofbeam thickness increased.

(vii) Shotcrete Thickness. The thickness of shotcrete required to support loosened blocks of sandstone between therock bolts is calculated with and without relying on adhesion of the shotcrete to the sandstone.

ROCK BOLT AND SHOTCRETE SUPPORT BERTUZZI & PELLS

Australian Geomechanics – June 200282

Figure 1: Failure modes considered by Evans (1941)

3 STEP (i) - ROOF BEAM THICKNESSEvans (1941) introduced the concept of the voussoir or linear arch to explain the source of strength in underminedhorizontally bedded strata (Figure 1). The lateral thrust generated by deflection of the beam against the abutmentsdefines the span of the beam. Several researches, notably Sterling (1977), Beer and Meek (1982), Sofianos (1996) andHutchison & Diederichs (1996), developed the theory further.

Pells and Best (1991) published a description of the “cracked beam theory” which was developed to remove Evans’assumption of a line of thrust in the voussoir arch and to explicitly include modelling of abutment stiffness, initialhorizontal stresses and surcharge loading. This is achieved through the use of an iterative one-dimensional finiteelement model (Figure 2).

Figure 2: Example of cracked beam (Pells & Best, 1991)



For the purpose of this paper however, the closed-form solution of Sofianos (1996) has been entered into an Excelspreadsheet to provide a quick estimate of the roof deflection for various beam thicknesses of an 18 m span tunnel intypical Hawkesbury Sandstone at a depth of about 35 m. The appeal of Sofianos’ solution is that checks againstcrushing, shear and buckling are also made. Numerical analyses could be used for this and subsequent steps.

Figure 3 suggests a 5 to 6 m thick roof beam is required to limit maximum sag to between 10 and 15mm. A rock boltlength of about 7m ensures a 1 m embedment length.

0

10

20

30

40

50

10 11 12 13 14 15 16 17 18 19 20

Span (m)

Def

lect

ion

(mm

)

2m bed 3 00 4 00 5 00 6 00

Figure 3: Roof sag as a function of tunnel span and bedding thickness for 35m depth

4 STEP (ii) – STRESSES IN PSEUDO ELASTIC BEAMBeam theory can be used to provide estimates of the induced stresses acting within the roof beam assuming it acts as apseudo-elastic continuum and insitu stresses are not considered. However, to take advantage of the high insituhorizontal stresses to support the roof beam, the elastic solution of Obert & Duvall as reproduced by Poulos & Davis(1974) has been entered into an Excel spreadsheet to calculate the induced stresses around a rectangular tunnel. Figure4 presents the induced shear stress acting along bedding planes spaced every metre above the tunnel crown for the 6 mthick roof beam.

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.00 2 4 6 8 10

Distance across span (m)

Indu

ced

Shea

r Str

ess

(MPa

)

1m above crown 2m 3m 4m 5m 6m

Figure 4: Induced shear stress acting on bedding



5 STEP (iii) - EXCESS SHEAR STRESSBedding planes in Hawkesbury Sandstone vary in characteristics from a simple change in lithology to an erosionalfeature. Quite frequently the bedding horizons comprise silty clay seams between 2 and 20 mm thick. For the presentexample it is assumed that the shear strength can be represented by zero cohesion and a friction angle of 20°.

The elastic solutions provided by Poulos & Davis (1974) can also be used to calculate the shear capacity of the beddingplanes. For our example, at 35m depth the shear capacity of a plane 1m above the crown varies from 6kPa at midspanto 0.8MPa over the abutment (Figure 5). The shear capacity (Figure 5) is subtracted from the induced shear (Figure 4)to obtain the excess shear stress that needs to be resisted by rock bolts (Figure 6).

0.0

0.1

0.2

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0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10


Bed

ding

Pla

ne S

hear

Cap

acity

(MPa

)


Figure 5: Shear capacity of bedding planes

Figure 6: Excess shear stress acting on bedding

6 STEP (iv) - SHEAR DISPLACEMENTSInduced shear displacements can be calculated using the analogy of a laminated beam. The relative horizontalmovement, or shear displacement, between the top of one bed and the bottom of the overlying bed, is the differencebetween the displacements of the top and bottom fibres of a laminated beam.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 2 4 6 8 10


Exce

ss S

hear

Str

ess

(MPa

)




The shear movement along a bedding plane can therefore be given by:

( ) ( ) ( )1246IE

LdD Lx2

Lx3

Lx3

+= -ω

whered = bed thickness (m)E = Young’s Modulus (kPa)ω= vertical force acting on beam (weight plus surcharge) (kN)L = span (m)I= second moment of area (m4)x = position along beam (distance from wall)

For our example, we have assumed that the tunnel roof is fully fixed but is free to displace axially a point 1.8m into theabutment. Figure 7 presents the shear movement that occurs along a bedding plane 1m above the tunnel roof. As canbe seen the shear is greatest about a fifth of the span from the abutment.

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10


Shea

r Dis

plac

emen

t (m

m)

Figure 7: Shear movement along bedding plane 1 m, above crown

7 STEP (v) - ROCK BOLT CONTRIBUTIONIn Hawkesbury Sandstone and other horizontally bedded, vertical jointed rock masses a rock bolt contributes to thestrength in two ways.

1. By providing additional shear resistance across a bedding plane via:

- dowel action (R1),- the component of the bolt axial force acting against shear movement (R5).

2. By providing additional normal force across a bedding plane via increased defect friction by virtue of increasednormal stress induced by the bolt:

- from tensioning (R2),- dilatancy of the defect during shearing (R3),- axial rock movement (R4).

If the rock bolt spacing is y then over a bedding plane area of s2 the bolt provides an increase in cohesion ∆c’ and anincrease in normal stress ∆σ’n of:

∆c’ = (R1 + R5) ÷ y2

∆σ’n = (R2 + R3 + R4) ÷ y2

and the resultant increase in shear strength over that of the bedding plane whose strength can be given in terms of c’j

and φ’j is:Sj = (c’j + ∆c’) + (σ’n +∆σ’n)tanφ’j



Figure 8 reproduces a figure taken from Pells et al (1994) which summarises the resistance provided by two types ofbolts as a result of shear movement along a bedding plane which has a dilatancy angle of 15°, typical of HawkesburySandstone. In our example, we can expect shear movements of up to 7mm, (Figure 7) which would mobilise a shearresistance of 110kN in a 24mm diameter steel bar installed normal to the bedding plane; and 460kN in a 23mmdiameter high tensile capacity steel bar installed at 60° to the bedding plane. Note that the bolts in Figure 8 are fullycement grout encapsulated. The bolt contribution can also be calculated using the technique of Pellet & Egger (1996).

Figure 8: Bolt contribution as a result of shear movement (Pells et al. 1994)

8 STEP (vi) - FACTOR OF SAFETYThe final step in the rock bolt design process is to calculate the Factor of Safety against shear. The optimum boltspacing and orientation is an iterative process. However, in the example presented here, we have assumed systematicbolt spacing of 1.75m as per Figure 9. Each bolt must contribute to providing a shear resistance greater than the excessshear stress over its area of influence. The total excess shear stress is the area under the curves shown in Figure 6between bolts. The bolt’s shear resistance is mobilised by the shear movement that would occur along the beddingplane at that point where the bolt crosses the bedding plane.

Let us assume that high tensile capacity bars will be used for the bolting. Then from Figures 7 and 8, the additionalshear resistance along a bedding plane 1m above the crown provided by each bolt in the layout shown in Figure 9(starting from the abutment) is 370, 420, 400, 290 and 230kN, which is a total of 1710kN. This additional shearresistance is approximately 500kN more than the excess shear stress for this “patch” of rock (area under the graph inFigure 6). Hence, we can say that the proposed bolt layout provides a FoS of 1.4 against shear along a bedding plane1m above the tunnel crown. The same process is followed for each of the bedding planes. In our case this would showthat FoS against shear along the other bedding planes is greater than 1.5. Hence, the proposed bolt layout in Figure 9would maintain the voussoir or linear arch.



Figure 9: Bolting layout

Practical recommendations in regard to types of rock bolts for up to 100 year design life are given in Pells & Bertuzzi(1999). Now we focus our attention to supporting any “loosened” sandstone, or sandstone that may deteriorate overtime, with shotcrete.

9 STEP (vii) - SHOTCRETEThe majority of shotcrete is used to prevent small, loosened blocks falling from between the bolts in tunnels inHawkesbury Sandstone. In good ground the adhesion of the shotcrete to the rock can be relied on to achieve this goal.In poorer ground, it cannot. Hence there is a case for using two shotcrete mixes. The first is used when adhesion can berelied upon and is lightly reinforced to control shrinkage cracks. The second requires substantial reinforcement to carrythe rock load between the rock bolts in flexure. A connection between the shotcrete and the rock bolts needs to be madefor this second mix.

Adhesion (Mix 1)

Barrett & McCreath (1995) show that the critical failure mode is the loss of adhesion and propose the capacity ofshotcrete to resist debonding is equal to:

Ca = 4σasawhereσa = is the adhesive strength of the shotcrete to the substrate (MPa),s = is the spacing between rock bolts (m),a = is the distance around the perimeter of the shotcrete panel over which adhesive forces act (m).

(Barrett & McCreath suggest a is between 30 and 50 mm)

The authors are aware of recent testing for a tunnelling project, that suggest σa=1 MPa is readily achieved in goodquality Hawkesbury Sandstone with a 50 mm thick shotcrete layer. Therefore, for a bolt spacing of 1.75 m, Ca = 210kN, which is equivalent to a 1.5 to 2.8 m thick block of sandstone 1.75 x 1.75 m square (see Figure 10, assuming σa=0.5to 1 MPa).

Flexure (Mix 2)

This can be designed as a two-way thin slab. The maximum moment is at the edge of the slab and is limited by theflexural strength of the shotcrete by the relationship:

22max 6

1)1(81 stfsM y=+−= νω

whereω = is the uniformly distributed load (MN/m)ν = is the Poisson’s Ratio of the shotcrete (assumed 0.15)ƒy= is the residual flexural strength of the shotcrete (MPa)t= is the shotcrete thickness (m)



0

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40

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0 0.5 1 1.5 2 2.5 3

Thickness of Rock Load (m)

Con

tact

Wid

th (m

m)

0.25MPa 0.5MPa 1MPa 2MPa

Figure 10: Contact widths required as a function of rock load for various adhesive strengths and 1.75 m bolt spacing– adhesive failure.

Figure 11 shows the relationship between rock load and shotcrete thickness. For a bolt spacing of 1.75m, a 130mmthick layer of shotcrete can support approximately a 1m thick layer of loosened sandstone.

Figure 11: Thickness of SRFS as a function of rock load for 1, 1.5 and 2 m bolt spacings – flexural failure

9.1 MATERIALThe design of a shotcrete is largely determined by its material composition. The following is given as a starting point.

9.2 STRENGTHA compressive strength target of 30MPa should be sought. A maximum of 50 to 60MPa should be specified to avoidbrittle failure of the steel fibres (Naaman & Sujivorakul 2001).

Mix 1 does not require a residual flexural strength because it functions in adhesion and then shear. Mix 2 requires aresidual flexural strength equivalent to 2MPa at 2mm deflection of the ASTM beam, or approximately 400 J at 40mmdeflection as measured by the Round Determinate Panel test.

0

50

100

150

0 0.5 1 1.5

Thickness of Rock Load (m)

SFR

S th

ickn

ess

(mm

)

1m bolt spacing 1.5m 2m



9.3 GRADINGGrading of the aggregate is one of the biggest issues to achieve a pumpable, shootable and durable shotcrete. TheEFNARC grading envelopes are proposed as guidelines.

9.4 FIBRESFibre type is particularly important in Mix 2. Not all fibres can achieve the residual flexural strength. As there is somuch development being carried out in terms of fibres at this stage, specifications should only state that fibres are to beused to achieve the desired performance. It appears that the direction the industry is taking is to use two types of fibres(or perhaps more) to provide toughness at small and large deflections.

9.5 CEMENTITIOUS Typically, tunnels in Hawkesbury Sandstone will not require very high strength shotcrete therefore flyash can be usedas a cement replacement. Flyash reduces shrinkage, improves workability and durability. EFNARC recommendsflyash can be a maximum of 30% of OPC although, Morgan et al (2001) suggests this typically should be 10%. Anexample mix is: 400 kg/m3 of OPC cement, 60 kg/m3 of fly ash (15% of cement), and 200 litres/m3 of water. This mixgives w/c ratio of 0.43.

9.6 ADDITIVESA super performing shotcrete mix is usually not needed in Hawkesbury Sandstone and hence additives can easily bekept to a minimum. The additives suggested are:

• silica fume – mainly to increase bond strength and reduce permeability and rebound, say 20 kg/m3 silica fume(5% of cement content);• air entraining admixture – to improve pumping, say admixture sufficient to achieve initial air content of up to15%, although this may be reduced depending on the cost of the supplied mix; and• alkali-free accelerator.

9.7 CURINGIn the past, others proposed the high humidity of tunnels in Sydney as a means of ‘self-curing’. However, partial curingproduces much the same results as not curing (Bernard & Clements 2001). The authors recommend curing.

10 DESIGNThe above process leads to a roof support design comprising high capacity bolts at 1.75m centres and shotcrete of either50mm thickness in good quality rock (Mix 1) or 130mm thickness (Mix 2) in poorer quality rock. The simplicity of theprocess allows parametric studies to be rapidly completed. Detailed numerical models can now be carried out toconfirm the design.

11 CONCLUSIONA analytical method of designing rock bolt and shotcrete support of wide span tunnels in horizontally bedded sandstoneis described. The beauty of the methodology is that it follows engineering principles and arrives at a Factor of Safetythat can be used objectively to rank various bolting layouts. A solution using only closed-form equations is provided toshow how simple and fast a very good first pass roof support can be designed. The same methodology can be applied tonumerical models, such as FLAC and UDEC, to refine the design.

12 REFERENCESBarrett, S.V.L. & McCreath, D.R. 1995. Shotcrete support design in blocky ground: towards a deterministic approach.

Tunnelling and Underground Space Technology, Vol 10 No 1, pp 79-89.Beer, G. & Meek, J.L. 1982. Design curves for roofs and hanging-walls in bedded rock based on ‘voussior’ beam and

plate solutions. Trans IMM, 91, ppA18-A22.Bernard, E.S. & Clements, M.J.K. 2001. The influence of curing on the performance of fibre reinforced shotcretepanels. Proc. Int. Conf. on engineering developments in shotcrete.Brady, B.H.G. & Brown, E.T. 1985. Rock mechanics for underground mining (page 217) George Allen & Unwin Evans, W.H. 1941.The strength of undermined strata Trans IMM, 50, pp475-532.



Hutchinson, D.J. & Diederichs, M.S. 1996. Cablebolting in underground mines §2.18.12, BiTech Publishers.Morgan, D.R., Heere, R., Chan, C., Buffenbarger, J.K. & Tomita, R. 2001. Evaluation of shrinkage-reducing

admixtures in wet and dry-mix shotcretes. Proc. Int. Conf. on engineering developments in shotcrete. .Naaman, Ae & Sujivorakul, C. 2001.Pull-out mechanisms of twisted steel fibres embedded in concrete. Proc. Int. Conf.

on engineering developments in shotcrete.Pellet, F. & Egger, P. 1996. Analytical model for the mechanical behaviour of bolted rock joints subjected to shearing

Rock Mech Rock Engineering No.29 (2), pp73-97.Pells, P.J.N. & Bertuzzi, R. 1999. Permanent rockbolts – the problems are in the detail. 10th Australian Tunnelling

Conf.Pells, P.J.N. & Best, R.J. 1991. Aspects of primary support design for tunnels in the Sydney Basin Trans IEAus.Pells, P.J.N., Best, R.J. & Poulos, H.G. 1994. Design of roof support of the Sydney Opera House underground parking

station .Tunnelling and Underground Space Technology, Vol 9, No 2, pp201-207.Poulos, H.G. & Davis, E.H. 1974. Elastic solutions for soil and rock mechanics Wiley & Sons.Sofianos, A.I. 1996. Analysis and design of an underground hard rock voussoir beam roof. Int J Rock Mech Sci &

Geomech Abstr. 33 No2, 153-166.Sterling, R.S. 1977. Roof design for underground openings in near surface bedded rock formations. PhD thesis.

design of rock bolt and shotcrete support of tunnel

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