Determination of the ground reaction curve using the hodograph method

Int J Rock Mech Mm S c l & Geomech Abstr Vol 22 No 3, pp 173-176. 1985

0148-9062/85 $300+000 Copyright ~' 1985 Pergamon Press Ltd

Pnnted m Great Britain All rights reserved

Determination of the Ground Reaction Curve Using the Hodograph Method E DETOURNAY*t I VARDOULAKIS* Thts paper describes a consistent method for computation o f the Ground Reactton Curve m deep tunnels The techmque o f calculatton ts based on Btot' s hodograph method which makes use o f fimte stress-strata relatwns for the rock

INTRODUCTION The Ground ReacUon Curve (GRC), the relatmn between the support pressure and the ground displacement at the tunnel, is recogmzed as a useful engineering tool for the design of deep tunnels [1, 2]. Calculatmn of the G R C (see Brown et al [3]), for an exhaustwe hst of references) is usually based on the following slmphfylng assumpUons namely 0) the tunnel is cyhndncal, (n) the grawty force is neglected, (m) the m suu stress field Is hydrostauc 0sotroplc), and 0v) there ~s a plane strain condmon m a plane perpendicular to the tunnel ax~s The object of this paper is to describe a quick method for calculauon of the Ground Reacuon Curve This techmque is based on Brat's [4,5] hodograph method which makes use of fintte stress-stram laws for the rock. PROBLEM STATEMENT Consider a c~rcular tunnel of radms R driven m a homogeneous and ~sotroptc rock mass, subject to a uniform hydrostauc far field stress ao (see Fig. l a) Assuming a monotomc unloading h~story for the radial stress aR apphed at r = R, one seeks to calculate the relation between the radml stress a R and the reduced radml d~splacement uR at the tunnel boundary. STRESS-STRAIN RELATION FOR THE ROCK

The fimte stress-stram relations of the rock are defined for plane strata condltmns Let (x~,x2) be a cartesmn co-ordinate system defined m the plane of deformation Denoting, respecUvely, by a~a and e~a(ct, fl = 1, 2) the plane cartesmn components of the Cauchy stress tensor and of the mfimteslmal strata tensor, the following stress and stram mvartants can

be defined (compression and contracuon are taken posmve): mean stress P = try/2 shear stress intensity.

T = (s,as~#/2) ~/2

dilatatmn:

A = e~

shear strata intensity

F = (2e~#e~#)~/"

(l)

where,

e~ = s~ - A6~B/2 Consider then the result of a plane-strata experiment, where the shear strata intensity F ts imposed to increase monotomcally from zero Coheswe-fnctlonal, dilatant geological matenal can be described by stress-strata relauons of the form

T = T(P, F)

(2)

A =

(3)

+

The constant ~ ~s related to the elasuc bulk modulus o f the rock. The function ~ ( r ) describes the "plastic" volume increase caused by shearing of the rock (the funcUon ~ ( r ) is ldenucally zero for r < re, re being the shear strata intensity at first yteldmg, for rock with an mmal hnear elasuc response). Both funcuons iV(p, r ) and ~ ( r ) are only defined for loading processes characterized by monotonically mcreasing shear strata intensity. %

% %

action

UR

* Department of Cwd and Mineral Engmeenng, Umverslty of Minnesota, 500 Pillsbury Drive S E , Mmneapohs, MN 55455, U.S A t Currently with Agbabmn Assocmtes, 250 North Nash Street, El Segundo, CA 90245, U S A

%

(b)

Fig 1 (a) Circular tunnel under a hydrostatic far field stress, a0 (b) Ground Reaction Curve

173

174

D E T O U R N A Y and V A R D O U L A K I S

THE H O D O G R A P H M E T H O D

The hodograph method [4,5] is a techmque of stress analysis apphcable to plane strata axmlly symmetric problems, for materml behavlour described by fimte stress-stram laws The central idea of the hodograph method is to ehmmate the radml co-ordinate r between the e q m h b r m m equaUon and the compaUbfllty equaUon, itself expressed m terms of the stress by means of the constltuuve equauons The result of th~s operauon is a first order ordinary &fferentml equation for the radml and tangenual stress components, where one of them ~s considered as a function of the other [5]. In terms of the stress and strain m v a n a n t s (1), this ordinary dlfferentml equauon becomes T(?F

dr

r+

d--P = r-

?A)

\aP

rfev CA) \tiT +

(4)

The soluuon of equauon (4) is a one-parameter family with respect to a constant of mtegratlon C T = F(e,

(5)

C)

The constant C ~s ehmmated by taking into account the c o n d m o n at lnfimty for

P=a 0

T=0,

(6)

GROUND REACTION CURVE

~s mmally under a state of hydrostauc stress a0 care must be taken to replace P m equauon (3) by P - a,, Thus. m case of an lnltml ~sotroplc stress ao. the reverted stress-strain relations are symbohcally written F = F'(P, T), A =/~(P, T. a0)

THE G R O U N D REACTION CURVE

The integral curve (7) can also be interpreted as the stress h~story of a single material point during the unloading process Using this mterpretaUon of (7) for a point at the boundary r = R, the ground reacuon curve can readdy be calculated as follows The &splacement at the boundary, m terms of the strata lnvanants F and A, Is gwen by uR = R(FR + AR)/2,

In mfimte domain problems (Fig. la), the constant C Is thus ehmmated by the condition at mfimty only (m finite problems, two boundary condmons are needed because the soluuon (7) is not explicit m the radml co-ordinate r) The boundary c o n d m o n at r = R is only used to define the "starting" point of the stress profile (7) Furthermore, since aR/> 0 and trR = P -

(8)

7",

the stress soluUon (7) needs to be determined up to T = P only, see Fig 2 In order to formulate exphcltly the &fferentml equauon (4), the dflataUon A and the shear strata intensity F must be expressed as a funcUon of T and P, by reverting the stress-strata relaUon (2). Since the me&urn

(1 i)

where the subscript R indicates that the quantity is evaluated at r = R The strata mvarlants FR and A R are computed as a funcuon of tr R, using the boundary condmon (8), the stress soluUon (7), and the stress-stram relaUons (2) and (3) or (9) and (10). FR = I=R(aR),

(7)

(10)

If the strata-stress relauons (9) and (10) cannot be formulated exphotly, it would be necessary to solve the &fferenual equauon (4) numerically using (2) and (3)

thus resulting m the stress soluuon T = 7"(P)

(9)

Ak = ~R(trR)

(12)

The Ground Reacuon Curve u R = fiR(as) is then constructed from equaUons (11) and (12). ApphcaUon of Blot's hodograph method to the determmaUon of the G R C ~s thus strmghtforward It should be noted, however, that computaUon of the radial &splacement at any point r > R ~s not as simple A c o n d m o n of the form g ( t r R, P, T, r) = 0,

(13)

~s needed, together with the stress soluUon (7) and the stress-stram relauons (2) and (3) or (9) and (10), to determine the funcUons F = r ' ( a R, r),

A = 7((tr R, r )

(14)

The condmon (13), which reduces to equation (8) at r = R, is obtamed by integrating the e q m h b r m m equauon m the radml dlrecUon using the stress soluUon (7) and the boundary c o n d m o n (8) APPLICATION

/ ~'/

,,

6*//

/

Apphcatton of the hodograph method to the calculauon of the G R C is illustrated for the case of an mcompresslble elastoplastlc material, characterized by hardenmg cohesive-frictional properties Other examples are outhned m Vardoulakls and Detournay [6] Stress-strata

%

%

Fig 2 Stress profile

relatton6

In a plane stram experiment, w~th a loading process characterized by a constant P and a m o n o t o m c increase of F, the shear response of the material is mmally hnear,

DETOURNAY and VARDOULAKIS GROUND REACTION CURVE

In the plastm region, the governing dlfferenual equatmn (22) becomes, using equatmn (15)

ttien non=hnear of the power law type

TIT~(P)

f F/F,(P); 0 <, T <~ T~(P) =

-.~[i..li~e(p)]n;

T

>

175

(15)

dT

T~(P) '

F).~

~l

1~--7~, + ~----7--~,| T = 2, le[l')] Lt, t r ;

dP

(25)

where the hardening coefficient n is In the range where, 0~
(16)

The symbols Te(P) and F , ( P ) represent the values of the shear stress and shear strain lntensmes at first yielding in a loading process where P is kept constant In this model, the proportlonahty limit T~(P) and the elastic shear modulus G ( P ) = T~(P)/F~(P) are assumed to be hnear functions of P T~(P) = To+ #0P,

(17)

G(P)

(18)

=

Go + [3P,

where To 1S the cohesion, ~ the frictional coetficmnt at first yielding, Go the shear modulus at zero pressure, and fl a material constant red,caring pressure sensmv~ty of the elasUc shear modulus. The material constants, Go, fl, To and go are all posmve The coefficient fl must fulfill the lnequahty

3 < #oGo/To,

dT

F + T OF OP

T(P)

=

(26)

(27)

T,(P)F(P. tro),

x [G-~-----))]'

D = ~ o - ~To. and In G(X)

2 = 1

L(x)'

in

I(X) =

r,(x-----) -

~.=l ~

C-')[ k

l (-qF k ]L

(24)

which ~s one of the properties of the Lam6 solution.

l'

D

_]'

2 non integer The hardening function F(P, ao) can actually be expressed as a function of the ratio P ' = P/ao, provided that the normalized parameters

TO = To/ao

(30)

are introduced (the prime ~s exclusively used here to denote normalization of a stress quantity by ao) The stress solution (24, 28) can thus be cast in dimensionless form, respectwely, as P ' = l,

P = a0,

,L(x)

Z-gT -Sj ;

2 =2,3,

- l n Te(X)-k=,-k

(23)

the considered problem IS characterized by the existence o f a plasuc region (R <<.r <~Re) surrounded by an elastic region (R, ~< r < oo) In the infinite elasUc region, the solution (7) of the dlfferentml equation (22) is

(29)

[I(P)-I(~o)],

w~th

Provided that 0 ~< aR ~< (1 -- #0)a0 -- To,

(28)

l' . LG(ao)j + ,.u~-'

Go = Go/ao,

F - T OF OT

n)

where F(P, ao) is the hardening funcuon given by

(21)

(22)

-

Hence, in the plasuc regmn, the solution of (22) with the condltmn (27) is

(20)

Integral curve T = T(P) Since the material IS incompressible, the dlfferentml equation (4) reduces to

n/(l

T = Te(ao)

AS for the second stress-strain law (3) or (10), the assumpUon of ~sochonc deformaUon Is made m the present case a=0

=

Due to equauon (24), the condition at the interface is gwen by

(19)

which constitutes a necessary and sufficmnt con&tion for F~(P) to be an Increasing function of P This restriction imposes small possible values for/~, whmh is m accordance w~th the experimental evidence at h~gh stress levels The coefficient of fnctmn # must also obey the following lnequahty

# < #oF"/FT(P) < #max < 1

2

T'(P') = (TO + # o P ' ) f (P'),

(elasnc)

(31)

(plastic)

(32)

with

f (P /ao) = F(P, ¢ro)

(33)

Equauon of the Ground Reacuon Curve Since the radial displacement at the tunnel is given by uR = RFR/2, (34)

176

DETOURNAY and VARDOULAKIS

GROUND REACTION CURVE

then n o n - h n e a r , m which case the f u n c t i o n rR(a;~i is given m p a r a m e t r i c form by

\

,o/¢

To+ rap, r . - G~ +/~P' [f(P')]'"

!

0 <~ a~ ~< 1 - T~ - lt~

(36)

i

//"

,f

a~ = P ' - (T(~ + # 0 P ' )

oo.o/ i/

.,.o / / / / / /

oo

o

F~gure 3 shows a graphical c o n s t r u c t i o n of the normahzed G r o u n d R e a c u o n Curve FR = I~R(a ~), the curve is constructed m the fourth q u a d r a n t using the stress solution (31) a n d (32) m the first q u a d r a n t a n d stress-strata law m the third q u a d r a n t The influence of the h a r d e n i n g coefficient o n the shape of the G r o u n d Reaction Curve Js illustrated m Fig 4 for a particular set of the parameters G~, fl, To, /a,~

'

/ CONCLUDING REMARKS It has been shown m th~s paper that the h o d o g r a p h m e t h o d , a technique developed by Blot for the stress analysis of plane a x l s y m m e t n c problems, provides a s~mple a n d consistent m e a n s of calculating the G r o u n d Reaction Curve m deep t u n n e l s Because ~t allows use of experimental stress-strata laws, th~s m e t h o d of calculation of the G r o u n d Reaction Curve ~s likely to be of practmal slgmficance

Fig 3 Graphtcal construction of the curve F R= l'R(a~) lo o8

06

¢ t~ 0 4

Recewed 6 Januar) 1984, rellsed 27 September 1984

n.O

02

REFERENCES

I 0

002

I 'g-N 004

006

I"~ 008

J 010

2U R / R

Fig 4 Normahzed Ground Reaction Curve F R= rR(aj~) (TO= 0 1, /~0=0 15, G0 = 20, ,B =5)

for an incompressible material, the curve FR = r'R(a~) represents a n o r m a l i z e d measure of the G r o u n d Reaction Curve The f u n c t m n l'R(a~) IS simply d e t e r m i n e d from the stress-strata relation (15) a n d the stress solution (31) a n d (32) Imtmlly, the response is hnear, 1 --a~

FR----

Go+

1-

fl '

T o-#o~a~
(35)

1 Daemen J J K Tunnel support loadmg caused by rock failure Ph D dtssertatton, Umv of Mmnesota (1975) Available as Technical Rept MRD-3-75, Corps of Engineers, Omaha, Nebraska 2 Kaiser P K A new concept to evaluate tunnel performance-influenceof excavation procedure Rock Mechanics from Research to Apphcatton (22rid U S Syrup on Rock Mechantcs ). pp 264-271 MIT Press. Cambridge (1981) 3 Brown E T, Bray J W, Ladanyl B and Hoek E Groundresponse curves for rock tunnels J Geote~h Engng 109(1), 12-32 (1983) 4 Blot M A Exact slmphfied non-linear stress and fracture analysts around cavttms in rock Int J Rock Mech Mm Sct & Geomech Abstr I1, 261-266 (1974) 5 Blot M A Hodograph method of nonhnear stress analyms of thtck-walled cyhnders and spheres, including porous materials lnt J Sohds Structures 12, 613-618 (1976) 6 Vardoulakls I and Detournay E Determmatlon of the ground reactton curve m deep tunnels using Blot's hodograph method Proc 4th lnt Conf on Numertcal Methods in Geomechamcs (Edited by Z Etsensteln), pp 619-623 Balkema. Edmonton (1982)