vicinity of the face
for a tunnel in the Mingtam Power Cavern Project．The measured data are plotted as dots in
Fig．4．Based on this data，Hock(1999)suggested the following empirical best-fit relationship
between Ur and distance X from the face： Ur
rITlax - l+expl l·Iul j it 喜’： l — 。
一一——广一一——— 一—百 E— xc — avatin— f ； 耋 f g dir
Fig．3 Profile of radial displacements U for an unsuppo~ed tunnel 90 Van·hung DAO．Water Sconce and Engineering，Sep．2009，Vo1．2，No．3，87—95
● ， ● ， 1．00 O．75 0．50 0．25 0 广— ＼口 E
xcavating direc~ ?
Pan et 995 “ “ ( 1 ) ‘ ＼ 、：
8 6 4 2 0 —2 _4
Fig．4 profiles derived from elastic models(Panet 1 995)，measurements in tunnel(Chem et a1．1 998)，and
best fit tO measurements(Hoek 1 999)
2．3 Characteristic curve of supporting structure
The characteristic curve shows the working capacity of the supporting structures
(concrete，gunite，rock anchor or form stee1)．It is based on the linear relation between
supporting pressure Pi and radial displacement“，，and it is applied to a supporting section for
a unit length along the tunnel axis．
Assuming the stifness of supporting stru ctures to be ，the elastic section of the
support characteristic curve can be calculated using the following formula：
= U (8)
The stiffness of concrete or gunite stru ctures iS K s = 1+Vc
ri 一(ri— ) (1—2vc)ri +( 一 )
where is the elastic modulus of gunite(concrete)，Vc is the Poisson coeficient of gunite
(concrete)，and tc is the lining thickness．
The stiffness of a steel support stru cture is calculated with the following form ula：
： ( +sinOcos0) Is
2sin 0 + E (10。 )
where S iS the distance between the supports along the tunnel axis(m)， is the half of：：the
angle between the tamping bars(。)，W is the width of the tamping blocks(m)，A is the
cross-sectional area of the section(m )，I iS the moment of inertia of the section(m4)，巨iS
the Young’S modulus for the steel(MPa)，tB is the thickness of the block(m)，and EB is the
Young’S modulus for the block matedal(MPa)．
The stiffness of a supporting structure using a mechanical anchor or chemical bonding
anchor with a length of lh and a diameter of d can be calculated as follows：
1 一 S f ’
where S is the distance between the anchors 石 ：Eb along +Q] (11)
the tunnel circumference，Sl is the
Van。hung DAO．Water Science and Engineering,Sep．2009，Vo1．2，No．3
， 87—95 9 1
distance between the anchors along the tunnel axis，a is the anchor pulling force，Eb is the
elastic modulus of anchor materials，and，is the free length of the bolt or cable．
W hen composite supporting structures are used，the components of the composite
supporting structures are all assumed to be installed at the same time，and the stifness of the
composite supporting structures is assumed to be the sum of the stifness of each of the
structure’S components： Ks= l+ K s2 (12)
where 1 is the stifness of the first supporting structure，and Ks2 is the stifness of the
second supporting structure．
Therefore，the characteristic curve of the supporting structure is specified by the
following equation： 罢 (13)
where“p is the displacement component of supporting structures and compressed rock，and
U0 is the initial displacement component of the tunnel before the lining is installed(defined by
means of the stress release efect)． 3 Example study
3．1 Description of example and design parameters
A survey of the intake tunnel of the Ban Ve Hydroelectric Power Plant(Nghe An
Province，Vietnam)was carried out．The material parameters ofthe tunnel are shown inTable 1．
Table I Physical and mechanical parameters of tunnel
Th e applied supporting structure was a combination of Gunite M 300 with a thickness of
10 cm an d steel anchors with diameters of 20 mm and lengths of 2 m．Anchor spacing along
the tunnel circumference an d along the tunnel axis was 1．5 m．The Matlab programming
language was used for the computation． 3．2 Calculation results and analysis
The stress value of the ground base P0：7．300 8 MPa．Figs．5 and 6 show the stresses
within the plastic an d elastic regions，respectively．It can be seen from Fig．5 that the
maximum plastic region radius =1．145 1 m．Therefore，the stress at the elasto—plastic
boundary ：4．019 5 MPa．This iS the maximum pressure value that the supporting structure
is able to bear．Th e maximum displacement“ ：0．111 8 m，which corresponds to Pi=0
(without support)．Fig．7 shows the stress release coeficient of the tunnel boundary without
support along the tunnel axis．The interactive curves between the ground base and supporting
structures at diferent initial displacements are shown in Fig．8．