Int.,l. Rock Mech. Mm. Sci.& Geomech. Abstr. Vol. 12, pp. 347 351. Pergamon Press 1975. Printed in Great Britain

Stress Distribution along a Resin Grouted Rock AnchorI. W. FARMER* The paper compares theoretical shear-stress distribution along a loaded resin,qrouted rock anchor, with computed shear-stress distributions obtained j?om tests on instrumented anchors in concrete, limestone and chalk. The results show that whilst at lower anchor loads in the concrete, the observed shearstress distribution is similar to the theoretical shear-stress distribution; in the weaker limestone and chalk there is e~,idence (?[ si.qnificant dehondin,q. It is concluded that anchor resistance in these rocks comprised mainly fully mobilised shear resistance.

1. INTRODUCTIONThe recent introduction and widespread utilisation[1,2] of resin grouted rock bolts and anchors in civil and mining engineering practice has often been based on quite uncritical acceptance of design data largely derived from simple short term"pull-out" tests[3,4]. This approach, of necessity, ignores the stress distribution along the fixed resin anchor which can have important implications for the resultant stress distribution in the anchored rock, and for the overall stability of the anchor. Although some good theoretical analyses of stress distributions along anchorages are available[5 8] there is a shortage of corroborative experimental data. The preseni paper presents a simple analysis of anchor stress distribution, and compares this with experimental data from tests on instrumented resin anchors in concrete, limestone and chalk.

ordG x -- 2: -c~.

dx

a

-

(2)

But since, provided the deformation is elastic, a x -Ea6~x/3x, where~ is the extension of the bar, equation (2) becomes: d2C_, 2 rx= . dxa E~(3)

If the grout annulus is thin (R - a< a) then the shear stress (r~.) at the steel/resin interface will be representative of the shear stress in the annulus:)2rx--

(R

~-~"

a) G,,.

(4)

If the annulus is thicker (R - a> a) then r, will be affected by radial changes in shear stress and it can

2. THEORETICAL STRESS DISTRIBUTIONFor the purposes of simple analysis, a steel rod grouted into a rock borehole by a filled polyester or epoxy resin grout may be regarded as an elastic anchor (moduhls of elasticity E,) surrounded by a shearable grout (modulus of rigidity G~) symmetrically positioned in a rigid socket (Fig. 1). The modulus of elasticity of the rock will, in fact, be about an order of magnitude greater than the resin. If a tensile force is applied to the rod, this will be transferred to the grout, through bond or shear stresses at the rod/grout interface, causing differential rod extension and grout shear along the anchor. At a thin diametrical slice between x and x+ 6x (Fig. 1) this transfer may be represented by~a 2

fo I

rocRT~grout

/

~cr~ -

--

27rar~ 6x,

(1)

i_

¢"

,~

* Engineering Geology Laboratories, University of Durham, Durham, England.

Fig

. 1. Stress situation m a groutcct anchor.

347

348 be shown that:z.~x-

I.W. Farmer The transfer length is equivalent to the optimum design length for the fixed anchorage. An approximate indication of shear-stress distribution along a typical resin anchor is given in Fig. 2. For typical resin/steel anchor combinations K~ 0.01, and (R - a)= 0.25 a, reducing~ to 0.2/a, in equation (14), and r x to:Oo

a In R/ a

Gg.

(5)

In either case, equation (3) will by substitution take the form of a standard differential:d2~x_ _

dx 2

_

0(2~x= 0 .

(6)

zx= O-1 exp - (0.2 x/a).

(16)

with the standard solution:~= A exp 0(x+ B exp~x where0( 2

(7)3. E X P E R I M E N T A L STRESS DISI~AilUTION

= E~t(g - a )

2Gg

2Gg or~ In R/ a,

(8)

depending on the annulus thickness. Equation (7) can be solved for any given boundary conditions; in this case: (rx= ao when x= 0; (r~= 0 when x= L, whence: A= ao exp -~L Ea~ exp 0(L- exp - 0(LB=ao exp~ E=0( exp 0(L - exp - 0(L

(9) (10)

which gives when substituted back into equation (7):~=

go cosh 0((L - x)

E~0(

sinh 0(L

(ll)

If L is much larger than i/0( (a likely proposition in most anchorages)then equation (11) becomes a simple exponential decay:~= Ea0( andzx=½ a0(ao exp - 0(x.ao e x p - c~ x

(12)

A series of tests were carried out on instrumented 20 mm dia steel bars grouted into 28mm dia holes in concrete, limestone and chalk. Laboratory moduli and strengths are summarised in Table 1. The grouted lengths were 350 and 500 mm in-concrete, 500 m.m in limestone and 700 mm in chalk. Each bar was instrumented with six or seven axially directional e.r.s, gauges equally spaced on a machined surface (Fig. 3). The grout comprised a slate-dust filled epoxy resin, pre. mixed and poured into the hole before insertion of the instrumented bar. The anchor holes were drilled into freshly exposed surfaces i n the limestone and chalk, and were formed around a liner in the concrete, which was cast into a 100 mm dia steel tube in the laboratory. The holes were grout filled to the surface and the resin allowed to cure for 24 hr before testing. During testing the anchors were loaded against a surface bearing plate 300 mm sq. at a force rate of 5 kN/min. Surface extension was monitored continuously and strain readings monitored at 5 or 10 kN intervals. Typical results are presented in Figs. 4 and 5 (concrete) Fig. 6 (limestone) and Fig. 7 (chalk L as: (a) load-displacement curves at the grouted anchor end--not corrected for bolt extension,TW

(13)

If for an elastic material E is assumed equal to 2G, then 0(, equation (8), can be expressed in terms of a modular ratioK= 2G o _ Eg Ea Ea

o.q¢

o-o4

0.o6

o-oe

oi=

between the shearing grout and extending rod, thus:8

K 0(2 _ or a(R -- a) a2(ln R/a)

K

(14)

F

~=o.i., p ( o e÷~-) J=o.i e,,p (-oe

X

One implication of the exponential decay in equations (12) and (13) is that when 0(x is equal t o 4"6, exp -0(x= 0-01 and~x,%

are reduced to 1% of their magnitude at the top of the anchor. In other words, the load on the anchor is effectively dissipated and the anchor length is equivalent to a transfer lenoth, L T given by:LT=

20

L=23oFig. 2. Theoretical stress distribution along a resin anchor in a rigid socket and having a thin resin annulus.

4-6

(15)

Stress Distribution Along a Resin Grouted Rock Anchor Table 1. Anchor material propertiesE Strength

349

~/m 2Steel rod Filled expoxy resin ( 2 4 hours) 1.$O x 108 2.25 x 1C6

~/m 25 x 105 6,OOO$5,000 160,OOO 33,000 13,000 3,500 (tensile) (tensile) (shear) (compressive) (compressive) (compressive) (compressive)

Concrete Limestone Chalk

2Cx

106

3.(5 x 10~ 3 x 10 5

*

Regularly bedded,

silt size calcite dolomite from the lower Magnesian Co. Durham.

Limestone, Houghton-le-Spring,

**

Fissured fine-grained si]iceous chalk from the Lower Chalk, Chinnor, Oxfordshire.

(b) strain distribution curves over the anchor length, and (c) computed shear-stress distribution curves; the mean shear stress between two gauges being computed as h, e= E,,/a(~ -~2)21 where 1 is the gauge separation. It is important to remember before any discussion of the results that although the shear-stress distributions represent the shear stress at the anchor-grout interface, the grout annulus in the present case is relatively thin[R - a= 0-4a, see equation (4)]. It is therefore reasonable to assume that the computed shear stress is representative of the shear stress throughout the grout, and by implication at the grout-rock interface. It is therefore reasonable to assume on the basis of equation (13) that some failure will occur in the weaker

rock substrate at the grout-rock interface at higher shear stress levels. Choice of the rock types and anchor lengths was based on the desire to investigate the effect of this failure on anchor reactions, and to determine the extent and type of failure. The results throughout were remarkably similar for each set of 4 chalk, 4 limestone and 4 (of each length)reo GaugeC

/ 12G od

2500

:/80

2000

8::L

G~500m

4o

EGougeI000 2 4 6 8 I0 mm GQuge<92 '\00 500?(3~N

Dlsplocement,~)Geu Gouge~)

Goug

500

400

300 Distance along

200 rodx,

I O0 rn m

0

I

io,o0o

.ooo6OO0

E

4000m

20¢)0

500

400

300Distance along

200rod

I00 x, mm

0

Fig. 3. Instrumented anchors.

Fig. 4. Load displacement, strain distribution, and computed shear stress distribution curves- 500ram resin anchors in concrete. Top: each curve represents the strain distribution at the specified anchor load. Bottom: the broken lines are theoretical shear-stress distribution curves. The solid lines are computed from the strain distributioncurves.

350Pull out failure

I.W. Farmer~oo P u l l oul" fo)lure

z

80 6o Gouge(])

~

4o-

l--

3C z20 -Gau ge~)

600

_= a"::L

2000

400

:i

=o

~0%~/~ A -- 1500

o

I

I

I

2 4 6 DlSplocerne~%,

I 8 I0 mm

o~

Gouge@/~o2",~/I~~~.~000

G

700

600

500Distance

400along

300rod

200x,mm

IO0

0

~ o .~@/ _ . . - ' U . --/~Ir,

_.>C.~ Ii i I I I00

~oo

w

3EK)

300

250Distance

1

1 200 along

11 150 rod

|

I

I

0

50

8OO~E

x,

mrn

)

I0,000

25kN 2OkN

--- 6 0 0

8000

15kNI O k N~

8200~,---""-

6OOO

5kN J

/74000

l 700

, 600

I 500Distance

4 O0along

500

200

I00

="2000

a

rod

x,

mm

Fig. 7. Load displacement, strain distribution, and computed shear stress distribution curves--70Omm resin anchors in chalk,

350

300

250Distance

200along

150rod

IOOX, mm

50

C

Fig. 5. Load displacement, strain distribution, and computed shear stress distribution curves--350mm resin anchors m concrete. Top: each curve represents the strain distribution at the specified anchor load. Bottom: the broken lines are theoretical shear-stress distribution curves. The solid lines are computed from the strain distribution curves.

Pull OUl foilure 6O

IO00

z

800 40 600:::t

}" 2o _1o

Gauqe(~

400200

.{ 2to

500

4010

300

200

I00

D stance

along

rod

x,

mm

~1/,-,~1 i~

SO00

%,~

concrete anchors, and the specimen set of results in each case is entirely typical. The anchors in concrete (Figs. 4 and 5) most closely simulate the assumed theoretical conditions of a rigid rock boundary, and the longer anchors{Fig. 4) were arranged so that the bolt would fail rather than the anchor. It is nevertheless interesting to note that although the experimental shear stress distribution at a pull out force of 20 kN (stress --- 7× 104 kN/m 2) is very close to the theoretical stress distribution, there is a substantial difference at 40 kN pull out force and this difference increases with increasing force. It is evident therefore, that at higher stresses, the whole of the fixed anchor length has debondedand that pull out resistance is largely attributable to a h~ level of skin friction. The non-linearity of the load-Zdisplacement curve is illustrative of this. In the case of the shorter anchors .in concrete (Fig. 5), this pattern is repeatedat lower stress levels, but at higher stress levels, the greater movement allowed as the anchor restraint is overcome, reduced skin friction to residual levels over the top h a l f o f the anchor. dolomi theoret: betweel 1.6. It lesser d and th early st _ but a rapid reductio~ levels in the chalk anchors (Fig. 7). It is evident that shear resistance and constraint are so low at the grout-rock interface

Z/2OO0I000

.

;300 Disl-once along

500

400

200rod

IJ~--~~'"x,mm

II00

o

Fig. 6, Load displacement, strain distribution, and computed shear stress distribution curves--500mm resin anchors in limestone. Top: each curve represents the strain distribution at the specified anchor load. Bottom: the broken lines are theoretical shear-stress distribution curves. The solid lines are computed from the strain distributionCUrVeS.

Stress Distribution Along a Resin Grouted Rock Anchor that debonding occurs at very low stress l

evels, and also that residual skin friction or shear resistance is mobilised at low pull out lorces. The implications of the results as regards the design and use of resin anchor systems are quite important. The almost universally accepted anchor"pull-out" formula on which design charts are based: P= 0-1S~rcRL (17)

351

by A. Dick as part of his M.Sc. Advanced Course m Engineering Geology.

Received 3 March 1975.

REFERENCES1. Askey A. Rock bolting with polyester resins. J. Irish1 ttiohw. Engrs, 18, 28 32 (197(I). 2. Carr F. Full hole resin anchored strata bolting in France. National Coal Board, Prod. Dept. Rept. (1969). 3. Franklin J. A.& Woodfield P. F. Comparison of a polyester resin and mechanical rockbolt anchor. 7}'a~2s. hTst. Mill. Memll. 80, A91 A101 (197l). 4. Dunham R. K. Some aspects of resin anchored rockbolting. 7imnels and Tumwlling 5, 376 385 11973}. 5. Hawkes J. M.& Evans R. H. Bond stresses in rcinforced columns and beans. Struct. Engr, 29, 323 327 (1951). 6. Coates D. F.& Yu H. S. Three dimensional stress distribution around a cylindrical hole and anchor. Prin. 2rot Corot. Ira. Soc. Rock Mech., Belqrade 3, 175 182 (1970). 7. Coates D. E.& Yu H. S. Rock anchor design mechanics. Mining Research Centre, Report R233, Dept. of Energy Mines and Resources, Ottawa ( 1971 ). 8. Hollingshead G. W. Stress distribution in rock anchors. Ca~l. Geozech. J. 8, 588 592 (1971).

where P is the"pull-out" torce S,. is the rock compressive strength gives a reasonable indication of anchor resistance in all three cases considered. The results show, however, that whilst in the stronger rocks this may comprise some residual shear resistance and some partially mobilised shear resistance; in weaker rocks it comprises largely fully mobilised shear resistance. Short or medium term tests on this basis are an unreliable guide to long term resistance, particularly where the rock may be subject to environmental change.Acknowledgemen~ The theoretical analysis in Section 2 was developed by Dr. R. A. Scott. The experimental work was carried out