Proceedings th FIG Symposim on Deformation Measrements Santorini Greece 00. ABSOUTE DEFORMATION PROFIE MEASUREMENT IN TUNNES USING REATIVE CONVERGENCE MEASUREMENTS Mahdi Moosai and Saeid Khazaei Mining Engineering Department The Uniersity of Tehran Tehran Iran Abstract Using conergence tape for measring rock moement in a tnnel dring constrction to control the stability of the excaation is a common practice. In this techniqe a nmber of reference points (conergence pins) are installed arond the tnnel periphery. After each face adance the distance between pairs of pins is measred by 0.0 mm accracy. Stability of the tnnel can then be ealated according to the recorded deformations. Usally in this method the exact locations of pins after deformations are not determined and jst changes in length are measred. In this paper methods of calclating exact coordinates of pins are presented by sing conergence tape as the main instrment and leeling tools and single point extensometers as complementary. Recommendations are also made to optimize the installation location of the pins to maximize obtained information from the readings and ease of interpretations. In the second part of the paper error sorces in Sakrai s formla for back analysis of tnnel deformation is detected and eliminated by sing the offered generalized formla.. Introdction Instrmentation of excaations dring and after constrction is the best method for monitoring the stability of ndergrond strctres. Many different types of instrment are aailable for this prpose sch as tape and rod extensometers load cell and pressre cell etc. Tape extensometer is the most common type of instrment for displacement measrement of rock mass arond tnnel periphery. This is a relatiely inexpensie portable tool and ery easy to se. Many researchers sch as Sakrai (98) and Hisatake (999) hae proposed methods of stability assessment for tnnels based on conergence measring reslts. The general practice with tape extensometer is to obtain relatie displacement of tnnel periphery and asses the stability based on the slope of displacement time graph. If rock moement accelerates with time it can be a case of instability otherwise the tnnel is stable. In this type of analysis the absolte displacement of each pin is not calclated bt instead relatie change in length between a pair of pins is determined. If one can determine the absolte displacement of each pin arond tnnel it can proide alable information for frther analysis. This paper is focsed on methods which make possible the determination of absolte displacement of pins dring conergence measrement with assistance of some peripheral accessories which are sally aailable at site. Special arrangements for pins are also proposed which can facilitate determination of the aforementioned parameters with minimizing any interference with other actiities.
. Problem definition Determination of exact location of pins in space is reqired for most back analysis methods. This can theoretically be done by sreying tools. Problems associated with this approach are limited accracy of the typical tools existence of dst and limited access to the points in the roof. The main aim in here is to make se of the accrate measrements that sally take place for stability prposes by tapes and to eliminate the need for sing sreying tools for locating the pins. The main problem in obtaining absolte displacement of pins in conergence operations is de to the fact that the nmber of nknowns is more than the nmber of eqations in analytical approach. In figre for example 5 pins are installed arond the tnnel. To determine the exact location of each pin parameters (x and y coordinates) mst be defined therefore 0 parameters altogether are nknown. In this example 0 diameters can be measred sing conergence tape bt these sets of eqations can still not be soled since the determinant of coefficient matrix is zero. Practically this means that conergence measrements can only gie relatie displacements not absolte ones. To oercome this problem graphical and analytical methods are proposed as follows. 4 6 7 5 9 0 8 Figre : Possible diagonal readings with 5 pins.. Graphical method To determine the exact coordinates of pins after deformation of tnnel one of the pins is assmed as reference point (i.e. pin 5 then x 5 y 5 00). It is also assmed that the diagonal between pins 5 and 4 is horizontal (meaning y 5 y 4 ). By these assmptions three nknowns are omitted and 7 remains to be determined. It will be seen that sing 7 diagonals are enogh to determine the exact coordinates of the points. The arrangement of these 7 diagonals among 0 possibilities is a delicate choice. In figre a non proper arrangement (case a) and two sitable arrangements (cases b and c) are shown for comparison. The first arrangement for diagonal readings preents application of this graphical method bt the other two will facilitate this approach. Other facilities in the tnnel also exist sch as air and water pipelines or electricity cables that determine which arrangement case (b or c) is the most sitable one with minimm interference with other operations. The first step after deciding on the most sitable pin arrangement is to find their initial locations (before tnnel conergence) in space by diagonal measrements initially performed by conergence tape. According to figre the steps for finding the location of pins are smmarized as follows. - Draw a horizontal line with length 7 (between pins 5 and 4) - Then from pin 5 draw a circle with radis 5 (the location of pin is somewhere on this circle). The same shold be done from pin 5 bt with radis for pin nmber. - From pin 4 draw a circle with radis 4 to cross the loci of pin. At this point the location of pin is determined.
(a) (b) (c) Figre : Non-sitable arrangement (a) sitable arrangements based on the location of facilities (b and c). 4- From point draw a circle with radis to cross the loci of pin and find the location of this pin. 5-From pin 4 draw another circle with radis 6 and find the location of pin sing the diagonal from pin the same way it was performed for pin. At this stage the exact location of all pins at installation time (zero time) is determined. It is notable that in this method the only measring tool sed was tape extensometer and no sreying instrments were reqired (apart from the fact that for installation of diagonal 7 horizontally one might se a leeling tool). The whole steps can be done sing a simple graphical tool sch as ACAD program. This can also be performed sing analytical soltions by crossing two circles with known radii (i.e. and 4) and known center points (pins 5 and pin4 respectiely) to find the coordinates the other pins. 4 7 4 5 6 Figre : s for finding the coordinates of pins in graphical method.
Dring gradal adance of the tnnel face the conergence measrements are performed progressiely. When the face is far enogh from the conergence station so that no frther adance can affect rock moement the displacement time graphs will leel off and displacements reach a constant magnitde. At this stage we can proceed with determination of new pin locations. The seqence for this part is exactly the same as one described before bt this time the new diagonal lengths are employed. Once this is done the deformed shape of the tnnel is determined (figre 4a). It is clear that this new shape might be displaced from its initial location and rotated as well. To determine the exact location and direction of the new deformed shape of the tnnel the followings needed to be done; - Before any adance in tnnel excaation install long enogh rebars which are anchored to the rock at their end point (debonded elsewhere) at points 5 and 4 according to figre 4. Rebars are long enogh to garantee no rock moement at the end points. In this configration the ertical displacement of pin and horizontal displacement of points 4 and 5 are determined. a b Figre 4: Finding deformed profile (a) and measrement of total tnnel displacement at three points (b). - on the original configration of pins three lines (two ertical lines close to the pins 4 and 5 and one horizontal line close to pin with the distance measred from the rebar moements) are drawn (figre 5a). The new location of pins 4 5 shold be somewhere on these three lines. - Drag and rotate the deformed configration of pins so that pins 4 and 5 lie on these three lines. At this stage there are sally two soltions aailable based on the rotation direction of the deformed grid (figres 5b and 5c). The final soltion between these two possibilities is decided based on the direction of diagonal 7 (whether this is inclined to the right or left). Now the exact location of the pins in the space after deformation of tnnel is determined and exact coordinates of each pin is now measred easily from the graphical program. a b c Figre 5: Finding the final shape of the deformed profile graphically.
4. Analytical approach As Sakrai (98) has discssed if the initial location of two pins (with coordinates of ( ) and ( )) with length () is change to a new position ( ) and ( ) the length change (l ) can be written as: () ( ) ( ) l Cosθ + Sinθ è Figre 6: ength change after moing two points of a diagonal. To simplify the discssion we limit the analysis to a three pin sitation with three diagonals of reading. There are some simplifying assmptions inherent in this eqation which can be sorces of error if one is not aware of those. These assmptions are: The displacements of the pins are small. Rotation of the diagonals dring deformation is negligible The ertical displacements of points and are neglected. The horizontal displacement of points and are known. Diagonal - is assmed horizontal. To eliminate some of these limitations especially the rotation component we proceed as follows: If the diagonal (between pins and ) is moed and rotated in space (figre 6) the original and new coordinates of the pin can be written as: New coordinates after relocation are The displacements of each point can then be written as Replacing (4) and (5) in eqation (6) yields θ Sinθ Cosθ θ Cos Sin + Sin θ + or Sin θ + θ Cosθ Cos + Cosθ Cos ( ) ( ) θ Sin θ Sinθ Sin θ Sinθ Sinθ If the two sides of eqation (9) is mltiplied by Cos t and eqation (0) by Sin t we hae ( ) ( ) θ Cos Cos θ Cos θ θ θ Sin Cos è (4) (5) (6) (7) (8) (9) (0) () ()
Adding two sides of eqations () and () retrns ( ) ( ) ( ) ' θ Cosθ + Sinθ Cos θ () Comparing eqations () and () shows that in Sakrai s assmption the rotation effect is neglected and if we assme then eqation () trns into eqation (). θ θ ' Haing extended the Sakrai s eqation we can now proceed with or analytical soltion to find the exact locations of the deformed pins in a D space with three diagonals. In this approach we reqire to know The angle of one diagonal with horizontal line before and after moement (or assming this diagonal to be horizontal) as discssed in graphical method preiosly. Assme coordinates of a pin as reference point and determine the new coordinates after moement (for example sing long rebars). For point we can write - ( ) ( ) + In eqation (4) points ( ) and ( ) are coordinates of points and after deformation. We also hae (4) (5) in which and are known parameters. Combining eqations (4) and (5) and sing eqation () a system of two eqations with two nknowns is formed from which the coordinates and can be determined. This wold yield two sets of answers which mean that the new diagonal can either be rotated positie or negatie with respect to the original diagonal direction. The same as graphical method the real answer can be jdged based on the inclination of the new diagonal with respect to its original direction. 4. Conclsions De to the wide spread se of conergence tape in monitoring tnnel stability it is important to try and get as mch information as possible from this simple operation. Methods are proposed in here enabling calclation of exact location of pins de to tnnel adance based on conergence tape measrements and simple leeling tools. Knowing exact location of pins in tnnel can proide alable information which is reqired for a proper back analysis operation. In this research the sorce of limitation inherent in Sakrai s eqation is pin pointed and eliminated. The simplified assmption in that eqation is not too important in rigid rock masses in which the deformations are small bt for softer or less intact rock masses these errors will rise and can affect the correctness and accracy of or analysis. The proposed method has oercome this shortcoming. References Hisatake M. (999). Direct estimation of initial stresses of the grond arond a tnnel. Proceedings of the Nmerical Methods in Geomechanics Balkema Rotterdam 7-77. Sakrai S. (98). Interpretation of displacement measrements. Proceeding of the International Symposim on Weak Rock Tokyo 75-756.