AROUND UNDERGROUND OPENINGS
ANATOLY LIBERMAN
I TR - GSI / 5 / 95 I I February 1995, Jerusalem I
GEOLOGICAL SURVEY 30 Malkhe Yisrael St.,
Jerusalem 95501, Israel
AROUND UNDERGROUND OPENINGS
ANATOLY LIBERMAN
I TR - GSI / 5 / 95 I I February 1995, Jerusalem I
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CONCLUSIONS ................................................................ 18
~~Itl£~<:~~ .................................................................. 1~
FIGURE 1. Scheme of pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
FIGURE 2. Isolines of the difference in principal stresses within a photo-elastic model. 9
FIGURE 3. Photo-elastic model. Grid for calculations. . . . . . . • • . • . • . • . . • .• 10
FIGURE 4. Photo-elastic model. Grid with support lines. .•.•.........•... 11
FIGURE 5. Scheme of calculations: a) difference in principal stresses; b) sum of principal
stresses. ........................................ 14
FIGURE 6. Limiting state area (LSA) near tunnels. (black color - LSA, points - stable
area, spaces - tunnels). .............................. 17
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ABSTRACT
Detennination of the limiting state area (LSA) around underground openings is carried out
by means of optical simulation and computer processing. Optical simulation represents isolines
of difference in principal stresses around openings; computer processing gives their sum and
thus complete infonnation on stresses is obtained. This is necessary to fmd the limiting state
area.
Using the stresses in a model (in relative units) and considering the depth of openings and
average volume weight of rocks one can compute in-situ stresses. Comparing in-situ stresses and
limiting stresses of rocks (which are determined in laboratory tests) a picture of limiting state
area can be obtained.
This picture may be used in tunnel planning to detennine: a) the fonn of openings, b)
~
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INTRODUCTION
The optical simulation method is based on the property of some translucent materials to
decompose under pressure a monochromatic polarized light to the trajectory of principal stresses.
These materials consist of epoxy resins, plasticizer and hardener. The components of optical
experiment includes: a source of monochromatic light, polarizer, model of the object to be
tested, analyzer and a photocamera. Planes of polarizer and analyzer must be perpendicular.
Without a mod~l one can see only a black field in the analyzer. The polarizer polarizes a light:
the vector of electric intensity of a ray oscillates only on one plane - the plane of polarization.
Passing through the model the ray is decomposed to the trajectory of principal stresses. In the
analyzer the ray is collected in the polarized plane. Following, an interference picture can be
seen - the isolines of difference in the principal stresses.
This method was first used in building bridges (Gordon, 1968). Then it was applied to the
solving many problems in machine constructions (Parton, 1990). In recent years this method
began to be used in underground mining, tunnel building and geology (Muller, 1963). However,
it was not widespread because he gave only a picture of isolines of difference in principal
stresses, but could not give a sum of the stresses. Thus it does not represent an entire field of
stresses. There were other methods of obtaining the principal stresses in some cross-sections
(Frocht, 1948), but they were not useful because they gave stresses only in selected lines (for
example, four instead of eighty); and accuracy of calculaion was so low that it was impossible
to use these methods.
Libman (Frocht, 1948) attempted to solve this problem and suggested the use of an electric
conductive paper for determination of the sum of the principal stresses. However, this takes so
much time (current problems require knowing stresses at 5000 - 10000 points) that its use is not
practical. Even if one had the time to determine stresses at all points, it would be necessary to
compare them with limiting stresses in order to obtain the limiting state area (LSA). Therefore
it seems impossible to carry this out without automatization.
Liberman (1985) suggested a method that completely automatizes the process of obtaining
the limiting state area.
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This method, described below, can be used in tunnel building, as shown here for the case
of Mount Scopus tunnel.
GEOLOGICAL APPLICATION
The Mount Scopus tunnel (Arkin et aI., 1993) represents a test case for the photo-elastic and
associated numerical techniques. The tunnel site is located between the university campus and
the Augusta Victoria hospital. The general direction of the tunnel is EW and part of the
university road leading to Ma' ale Adumim will run through it.
The tunnel will be excavated in chalk, marl and some clay of the Senonian Menuha
Formation. The strata dip 5°_10° SE, and are part of the eastern monoclinal flank of the Judean
Anticline leading towards the Dead Sea.
Seven exploratory boreholes were made to test the rock mass along the proposed tunnel
alignment. Chalk is the main lithology and generally occurs in the upper and lower parts of the
Menuha Formation. Clayey chalk and marl occur in the central part of the formation. The clayey
component within the marl and clay are smectites and some kaolinite and are sensitive to water.
In general the rock mass represents poor to good quality rock; in order to determine the type
of preliminary and permanent support one must know the distribution of the potential limiting
state area (LSA). LSA is the area of intensive redistribution of in-situ stresses (Liebowitz, 1976).
If the LSA is not strengthened it could be transformed into a failure area and could become a
source of a failure event.
The present research is aimed at determining the LSA around underground openings. For
this purpose it is necessary to know the limiting and real stresses in the rock mass. The
comparison of these stresses defmes the LSA.
To determine the limiting stresses samples of the rock mass from the researched area must
be tested in a laboratory. Laboratory tests include:
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OPTICAL SIMULATION
The distribution of stresses around underground openings can be determined in a model
composed from optical sensitive material (Frocht, 1948). A translucent model representing a
rock mass with openings is placed under pressure and illuminated by circular polarized
monochromatic light. The material for this model was epoxy-elastic (Malkis, 1984), it size was
240 x 160 x 18 IDm. The model was in plane strain conditions and was pressed in the vertical
plane with the pressure of 2 KPa (Fig. 1).
In order to perform the stress analysis, a photograph of the photo-elastic model (Fig. 2) must
be overlain the with a square grid, so that the boundary of openings and the model will conform
to points on the grid. This operation gives the initial data required for computer calculations
(Fig. 3) (Liberman, 1985).
For obtaining initial data one needs to selectc support lines - lines where difference in
principal stresses V = 0'1 - 0'2 is well known. In Figure 4 one can see their numbers (numbered
lines> lenJ. In support lines one needs to mark the support points - points of crossing of the
isolines and support lines and all points along the right boundary of the model. After that it is
possible to begin computer calculations.
COMPUTER CALCULATIONS
By using the linear interpolation method it is possible to spread the values of V throughout
:. the whole model (stripes occur only in about 5 % of the model). In order to determine the LSA
thier sum U = 0'1 + 0'2 must be calculated.
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Computer calculations are made by five programs in GWBASIC for mM PC written by
A.Liberman. These are:
DIF - calculation of the values V = al - a2
SUM - calculation of the values U = al + a2
LSAP - calculation of principal stresses and drawing LSA.
Initial data are entered by the program INDATA. It is stored on a disk and transmitted to
the printer. After checking it is possible to correct the data using program CORDATA.
Initial data include:
- numbers of support lines;
- numbers of support points and the value of the difference in principal stresses
at these points;
(Appendix I)
If the support point lies on the left boundary of the openings after which there is a space -
it is marked with "-", if it lies on the other points of the opening boundary it is marked with
" + ". If a point of an opening boundary is in the tensile condition, the value of stress is marked
with "-", if it is in the compressive condition it is marked with "+".
-.
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F8 must be pressed, if it lies on the support line, then F7.
The program DIF calculates V = 0'1 - 0'2 using a linear interpolation method. The
formula for interpolation is:
where V. --------- calculated value of V I .
Vb and V f - known values of V in points with coordinates of B and F
correspondingly (points B and F lie on isolines);
Xi ---------- point where V is calculated (B < Xi < F) (Fig. 5).
The results of calculations between 19 and 31 lines and between 21 and 65 columns are
shown in Appendix ll.
Calculations of the sum of principal stresses is made by the program SUM. Calculations are
based on the numerical solution of the Laplace equation:
where U = 0'1 + 0'2 at the points with coordinates X and Y
Initial data for the solution are:
- U on the boundary of the model (is known)
- U on the boundary of openings (is known because U = V on the free boundary). All
these data were entered in the program INDATA.
The numerical solution is based on the approximation method. An initial approximation is
made for all points necessary to calculate; it is made automatically as the average value of U at
boundary points.
'.
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FIGURE 5. Scheme of calculations: a) difference in principal stresses; b) sum of principal
stresses
~ .a;:..
by the following fonnula:
where U1 , U2 , U3 , U4 - values of U at neighboring points (Fig. 5):
Uo - the calculated value of U.
When calculations at all points are completed they are compared to the preliminary values.
Calculations are complete if:
a) the deviation from preliminary values is not greater than the accuracy of the
results;
These conditions are entered by the researcher.
The obtained values are the solution of the Laplace equation. If these conditions are not met
the obtained values are used as the initial values of the next approximations until one of them
is satisfied.
Values of the sum of principal stresses are presented in Appendix III.
After obtaining the stresses it is possible to obtain in-situ stresses (Sedov, 1973) (equation
4).
(4)
0 - average volume weight of rocks
H - depth of openings
The in-situ stresses Uis are real stresses and must be compared with limiting stresses for
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determining the LSA.
An estimation of the LSA can be made using the values of V and U by the program '
LSAP.
Firstly this program calculates in-situ stresses by equation 1; then it compares real and
limiting stresses using the Mohr theory and it gives a graphic representation of the LSA (Fig .
6).
- Uniaxial compressive strength - ac
- Average volume weight of rocks - J
In the present case of the Mount Scopus tunnel these parameters are taken from Arkin et al.
(1993) .
'0 = 18 KN/m3
Once a distribution of stresses is obtained, one can obtain a range of LSA for various values
of ac' apf' H, O. Figure 6 shows the black color area - the LSA, the area with points - stable area and spaces ,
which represent tunnels. This picture shows that the LSA propagates from the horizontal line
. . . .
. .. . ..
. . . .
. . . . . .... . . . .
..
. , . .
. . .
,',
.. . . , . . .
.. .
" I ••• , .. , ....
FIGURE 6. Limiting state area (LSA) near tunnels. (black color - LSA, points - stable
area, spaces - tunnels)
-....l
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the characteristic linear size of a tunnel), has the width .... O.5L and the LSA also occurs in all
squares of the pillar between the tunnels.
Consequently, it is neccesary:
a) to increase the width of the pillar and to plan a support
considering the LSA in the pillar;
or
b) to choose another design, for example one tunnel instead of two in order to eliminate
the interaction of them.
1. The suggested method of optical simulation together with computer calculations gives a
possibility of obtaining the LSA quickly and with a high accuracy.
2. In tunnel planning knowing the LSA is prerequisite to choosing the forms of tunnels, the
-.
Arkin, Y., Michaeli, L. and Wolfson, N. (1993). Har Hazofnn
tunnel geological and geotechnical site investigation.
Usrael Geological Survey. Rep. GSI 18/93, Jerusalem, 62 p.
Frocht, M. (1948). Photoelasticity, New York, Wiley, 2 vol., 535 p.
Gordon, J. (1968). The New Science of Strong Materials or Why We
don't Fall Through the Floor. Penguin Books
Harmondsworth, 272 p.
(in Russian).
Liebowitz, H. (1976). Fracture. An Advanced Treatise. School of Engineering and Aplplied
Science. The George Washington University, Washington, D. C.
Academic Press. New York and London, 7 vol., 4235 p.
Malkis, N. (1984). Epoxy-elastic - the New Matherial for the Photo-elastic Experiment.
Skochinsky Mining Institute, Moscow, 25 p., (in Russian).
Muller, I. (1963). Der Felsbau. Salzburg, Enster Band. Theoretischer Teel.
Ferdinand Enke Verlag Stuttgart, 256 p., (in German).
Parton, V. (1990). Fracture Mechanics. From Theory to Ptactice. "NAUKA", Moscow, 240 p,
(in Russian).
Sedov, L. (1973). Mechanics of Continuous Medium. "NAUKA", 2 vol.,
Moscow, 1125 p., (in Russian) .
-.
!
INITIAL DATA FILENAME - - - - - - - -indatal.dat DIFFERENCE IN STRESSES DATA FILENAME dif1.dat SUM OF STRESSES DATA FILENAME - - -suml.dat ADDITIONAL DATA FILENAME - - - - - - ad1.dat
INITIAL DATA FILENAME - indata1.dat
QUANTITY OF LINES = 55 QUANTITY OF COLUMNS = 85 QUANTITY OF SUPPORT LINES = 33 QUANTITY OF SUPPORT POINTS = 932 BOUNDARY DIFFERENCE IN STRESSES = 1
SUPPORT LINES
ORDINARY NUMBER
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
SUPPORT LINE
2 6 9 12 15 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 42 45 49 52
SUPPORT POINTS AND STRESSES
ORDINARY NUMBER SUPPORT POINT STRESS
1 26 1 2 43 2 3 60 1 4 F7 5 F8 6 Fa 7 Fa 8 8 4 9 . 14 3 10 20 2
.u ";0 1 12 33 2 13 40 3 14 43 3 15 46 3 16 53 2 17 60 1 18 66 2 19 72 3 20 78 4 21 F7 22 FB 23 F8 24 12 4 25 ]8 3 26 22 2 27 27 1 28 32 2 29 35 3 30 41 4 31 43 4 32 45 4 33 51 3 34 54 2 35 59 1 36 64 2 37 68 3 38 74 4 39 F7 40 Fa 41 F8 42 3 4 43 8 4.5 44 17 4 45 21 3 46 23 2 47 27 1 48 31 2 49 33 3 50 36 4 51 41 5 52 43 5 53 45 5 54 SO 4 55 53 3 56 55 2 57 59 1 58 63 2 59 65 3 60 69 4 61 78 4.5 62 83 4 63 F7 64 FB 65 F8 .. 66 4 4 67 12 5 68 16 5 69 21 4 70 23 3 71 25 Z 72 26 1 73 28 0 74 30 1 75 31 2 76 -. ::12 3
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672 22 5 673 25 2.7 674 26 2 675 28 .2 676 29 0 677 30 .4 678 31 1.1 679 32 2 680 34 3.9 681 35 4.8 682 37 6 683 40 6.5 684 43 6.7 685 46 6.5 686 49 6 687 51 4.8 688 52 3.9 689 54 2 690 55 1.1 691 56 .4 692 57 0 693 58 .2 694 60 2 695 61 2.7 696 64 5 697 66 5.9 698 69 6 699 74 5 700 79 4 701 83 ,3 702 F7 703 3 3 704 7 4 705 12 5 706 15 5.5 707 18 5.8 708 21 5 709 23 4 710 26 1.9 711 27 1 712 29 .3 713 31 1.2 714 32 2 715 33 2.9 716 34 3.7 717 35 4.2 718 36 5 719 39 6 720 43 6.45 721 47 6 722 50 5 723 51 4.2 724 52 3.7 725 53 2.9 726 54 2 727 55 1.2 728 57 .3 729 59 1 730 60 1.9 731 63 4 732 65 5 733 68 5.8 734 71 5.5 735 74 5 736 79 4
7'J7 03 3 738 F7 739 3 3 740 7 4 741 12 5 742 15 5.3 743 20 5 744 21 4.9 745 27 1.1 746 29 .5 747 30 .9 748 32 2 749 35 4 750 37 5 751 41 6 752 43 6.2 753 45 6 754 49 5 755 51 4 756 54 2 757 56 .9 758 57 .5 759 59 1.1 760 65 4.9 761 66 5 762 71 5.3 763 74 5 764 79 4 765 83 3 766 F7 767 3 3 768 7 4 769 12 5 770 19 5 771 22 4 772 23 3.25 773 24 2.8 774 25· 2.1 775 26 1.8 776 27 1.1 777 29 .9 778 30 1.1 779 32 2 780 35 3.9 781 38 5 782 40 5.4 783 43 5.9 784 46 5.4 785 48 5 786 51 3.9 787 54 2 788 56 1.1 789 57 .9 790 59 1.1 791 60 1.8 792 61 2.1 793 62 2.8 794 63 3.25 795 64 4 796 67 5 797 74 5 798 79 4 799 83 3 800 F7 801 3 3 802 -, 6 4
a03 13 5 804 17 5 805 21 4 806 23 3.1 807 24 2.8 808 25 2.3 809 26 1.9 810 27 1.4 811 29 1 812 30 1.1 813 32 2 814 34 3 815 36 4 816 39 5 817 42 5.5 818 43 5.7 819 44 5.5 820 47 5 821 50 4 822 52 3 823 54 2 824 56 1.1 825 57 1 B26 S9 1.4 827 60 1.9 828 61 2.3 R29 62 2.8 830 63 3.1 831 65 4 832 69 S 833 73 5 834 80 4 835 83 3 836 F7 837 Fa 838 3 3 839 6 " 840 12 5 841 19 4 842 22 3.1 843 23 2.9 844 25 2 845 29 1.3 846 32 2 847 34 2.9 848 35 3.1 849 38 4.1 850 40 4.1 851 43 5 852 46 4.5 853 48 4.1 854 51 3.1 855 52 2.9 856 54 2 857 57 1.3 858 61 2 859 63 2.9 860 64 3.1 861 67 4 862 74 5 863 80 4 864 83 J 865 F7 866 F8 867 Fa 868 . 2 3
-.
01-09-1994
10:19:09
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.)- 0 33 0 34 5 35 6 36 6.125 37 6.25 38 6.375 39 6.5 40 6.625 41 6.75 42 6.875 43 7 44 6.875 45 6.75 46 6.625 47 6.5 48 6.375 49 6.25 50 6.125 51 6 52 5 53 0 54 0 55 0 56 0 57 0 58 0 59 0 60 0 61 0 62 3.5 63 5 64 6 65 6
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-.
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- 41 -
SUM OF PRINCIPAL STRESSES DATA FILENAME -sum1.dat
ACCURACY = .000001 MAXIMUM NUMBER OF ITERATIONS = 100 MAXIMUM TIME FOR ITERATIONS = 1800 sec
NUMBER OF ITERATION ACCURACY
1 .2579671 7 .1253407 3 8.385832E-02 4 6.438735E-02 5 .0528715 6 4.521566E-02 7 .0397608 8 3.568638E-02 9 .0325324 10 3.001802E-02 11 2.796171E-02 12 2.624168E-02 13 2.477359E-02 14 2.349809E-02 15 2.237258E-02 16 2.136609E-02 17 2.045558E-02 18 1.962392E-02 19 1.885802E-02 20 1.814786E-02 21 1. 748548E-02 22 1. 686468E -02 23 1.6<.'8046E-02 ?I! ~ ~7~!379E-O~
25 1.520625E-02 26 1.4710]3E-02 27 1.423805E-02 28 1.378806E-02 29 .0133584 30 1. 2911767E-02 31 .0125545 32 1. 217779E -02 .,., 1. 181653E-02 oJJ
.
37470.72 37492.96 31515.26 37537.61 31559.97 31582.32 37604.68 31627.03 37649.44 37671.8 37694.15 37716.56 37738.92 37761.27 37783.63 37806.04 37828.39 37850.75 37873.1 37895.45 37917.81 37940.16 37962.52 37984.87 38007.23 38029.58 38051.88 38074.24 38096.59 38118.95 38141. 3 38163.66 38185.96 38208.31 38230.61 38252.96 38275.32 38297.62 38319.97 38342.27 38364 .63 38386.93 38409.28 38431.58 38453.88 38476.24 38498.54 38520.84 38543.19 38565.49 38587.79 38610.09 38632.39 38654.74
55 0.531623£-03 38677.04 ~6 6.J75626E-03 38699.34 57 6.224759E-03 38721.64 S8 6.078807E-03 38743.94 59 5.937601E-03 38766.24 GO 5.800948E-03 38788.54 61 5. 668677E -03 38810.84 62 5.540639E-03 38833.14 63 5.416642E-03 38855.44 64 5.296559E-03 38877.74 65 5.180246E-03 38899.99 66 5.067561E-03 38922.29 67 4.958372E-03 38944.59 68 4.852533E-03 38966.89 69 4.749945[-03 38989.19 70 4.65048E-03 39011.43 71 4.554024E-03 39033.73 72 4.460457E-03 39056.03 73 4.369697E-03 39078.33 74 4.281623E-03 39100.57 75 4.196143E-03 39122.87 76 4.113183E-03 39145.12 77 4.01264E-03 39167.42 78 3.9S441E-03 39189.66 79 3.878434E-03 39211.96 80 3.80464E-03 39234.21 81 3.732925E-03 39256.51
, LINE= 19
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-.
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