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Failure constellations in and around underground openings
Engineering Fracture Mechanics Vol. 35, No. l/2/3, pp. 451-471, 1990
0013-7944/90 $3.00 + 0.00 Pergamon Press plc.
Printed in Great Britain.
FAILURE
CONSTELLATIONS IN AND AROUND UNDERGROUND OPENINGS HELMUT
HABENICHT
Montanuniversitiit
Leoben, Austria
Abstract-The paper enters into the matter of fracture and failure development from the starting view point that the rock mass is already present as a fracture damaged substance. The engineering viewpoints are being emphasized and examples are stated demonstrating the variety of potential failure zones in a number of different underground openings. The interference of the strength parameters of the rock mass, of the stress field, and of the opening is illustrated. On the basis of the influential variations of input parameters, of complicated mechanical formulations, and of requirements of precision in engineering work for successful decisions it is shown why the macroscopic and phenomenological approach in engineering practice through the performance of field measurements is relatively successful.
INTRODUCTION dealing with the rock mass as a structural substance have always tended to detect the minute and basic components of strength inherent in the substance. By employing the methods and the knowledge of physics great progress has been achieved in the understanding of that strength which is provided by lattice bonds and surface bonds. Cracks and their influence on the strength of mineral matter or ceramic materials have been the subject of wide and intensive research. Engineers in mining and in underground construction are, however, faced with goals-and questions-that lie at a different level of considerations. Their problems as well as their means and methods are of a rather macroscopic nature. Although knowledge and understanding of the physical nature of mineral matter contributes favourably to engineering thought the logical chain from the basic physical parameters and properties to the presentable results of engineering efforts appears rather long and minutely subdivided. This situation and the capabilities developed at the engineering level permit solutions in some of the engineering work which needs not to be based on the microstructural approach. This fact will be illustrated by this paper. ENGINEERS
PHYSICAL
STRENGTH
AND ROCK MASS STRENGTH
The basic effort of this conference concentrates on the fracture phenomena with interpreting the same as the initial process in which the inherent strength of the matter is overcome. This inherent strength is typically understood to be built up by microscopic bonding such as would occur for example in lattice structures or at crystal contacts. In this view a fracture constitutes the break down of the ideal strength of the quasi-continuous matter. For this one the physical strength can only be proven in relatively complicated laboratory experiments, or can be discussed and evaluated only on theoretical principles. The many questions yet unsolved in this respect and the fundamental importance of this problem no doubt justify continuous efforts in this direction. Looking beyond this it is the intention of this paper to present insight into the possibilities which are offered to engineers in the technical branch of rock mechanics for their practical approach to fracture and failure problems. The difference is that engineers have to deal with matter of that status which constitutes an essentially damaged system in contrast to the continuity and/or uniformity of ideal matter[l, 21. The tensile strength of an ideal substance if represented by pure rock salt for example would amount to approximately lo5 MPa, and in compression its strength would even be for multiples higher. The real mineral rock salt, however, as it is present in the earth crust exhibits strength parameters (of technical definition) in the order of magnitude of for example 10 MPa in tension and 100 MPa in uniaxial compression[3]. Thus the strength available in real substances for engineering applications in rock mechanics is by multiple orders of magnitude smaller than the one of ideal matter. 457
458
HELMUT
HABENICHT
With the strength properties being so far apart in terms of magnitude the techniques of their evaluation are naturally different and much coarser than the ones in physics research. In fact, the formulation of molecular or atomic forces does not play a significant role in strength evaluation experiments. The practical approach of engineers is a rather macroscopic one concentrating on mechanical testing of specimens. In addition to that, even strength parameters have been defined from phenomenology such as the ones of the Mohr-Coulomb failure condition, which have been introduced through combining a loading analysis with interpretational results of testing data. This condition is defined through the tangent line to the Mohr’s circles of the limit stress constellation shown in Fig. 1. A stress constellation above the tangent line (Mohr’s envelope) means failure and anyone below means stability. The deviation of the engineering approach from the one of physics and of continuum mechanics goes even so far that under certain conditions and for certain rock mass types a tensile strength is not more accounted for, and loading of the rock mass in tension must be prohibited by the designer. In continuation of this transition of engineering from the quasi-continuum to the damaged matter the practice in rock mechanics has even recognized and defined a multiple body system (Vielk~~rsystem) given by the joint-body system (Kluftk~~ersystem) in a rigorously jointed rock mass[l]. Again from here it is conceivable that by breaking down the joint body size (and shape) one can arrive at particles and phenomenological properties which constitute the subject matter of soil mechanics, i.e. of a science that has provided much input to the one of rock mechanics. The sciences of rock mechanics and soil mechanics particularly applied to civil engineering, to ~derground construction and to mining involve a greater number of additional factors influencing the strength and the fracture of rock and rock like materials. Of these only such typical properties will be considered here which pertain to the solid state and to quasi-brittle fracture, for simplicity reasons. By restricting the discussion only to these prominent features a short view will be presented on the strength characteristics, the loading patterns, and on such design criteria which safeguard against failure. A few examples out of the engineering practice will illustrate the conditions and means of solution prevailing in the macroscopic scale. No attempts will be made at this time to integrate the contents of fracture physics into such engineering treatment since cases appear rather rare and of particular setting where the material defects would be of such minor influence as to permit a significant role of the microscopic strength terms. THE ROCK MASS AS AN ENG~EERING MATERIAL The earth’s crust-in recent years again recognized as being the stage for the continuous process of plate tectonics-is characterized by rupture surfaces of vast extension and of varying orientation as it is symbohzed by Fig. 2. These tectonic planes or surfaces occur in different scale including for example such as would be crossing the floor of the Atlantic to the ones mapped in central Europe shown in Fig. 3, and to such small ones[2] as may be evident in the foundation rock of a highway bridge illustrated in Fig. 4. Manifold examples could be stated for this feature. It plainly persists throughout the majority of the earth’s rocks. It ranges over all sizes of sampling
Fig. 1. Mohr’s circles and Mohr’s envelope showing the shear stress failure conditions under combined normal and shear stresses.
Strain-softening
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Fig. 2. Topography of the ocean floor along the Atlantic Rift Zone illustrating fracture systems in the earth’s crust in a large scale.
Fig. 4. Jointed rock surface of approx. 2.5 x 1.5 m size illustrating the fragmented structure of a rock mass in a small scale.
459
Strain-softening
of concrete
461
Fig. 3. Major tectonic faults mapped in the central European area demonstrating the discontinuous nature of the earth’s crust in an intermediate scale.
down to the handpiece and to the thin section prepared for microscopic analysis which exhibits cracks and flaws in the structure. Thus, discontinuities as planes or surfaces of weakness or of zero strength are prevailing in rock mechanical problems. Therefore in rock mass stability considerations a tensile strength cannot be attributed to the rock mass except for specially isolated bodies or for rare zones of high quality and undisturbed rock volumes. It is due to this fact that the design of structures in rock is built on criteria and configurations which involve loading in shear and compression only. An additional condition for such constellations to permit failure is the proximity of a free surface to which the overstressed portions can yield. This condition is satisfied in most underground openings or structural members. In these applications the predominant law is given by the Mohr-Coulomb criterion as it is shown in the stress envelope of Fig. 1, or by extended formulations such as for example by the Leon-parabola[ 11. In some respect one could apply these criteria to the rock mass in a generalized way by ascribing to the latter the strength parameters as if it would represent a continuum, i.e. as if the parameters would prevail evenly and uniformly throughout the mass. Advanced considerations have resolved the rock mass into such proportions that are regarded continuous matter of some technical strength value and into such ones that are given by preexisting failure or weakness planes. This subdivision of the rock mass would more closely represent the effect of joints which do usually transgress the rock mass intermittently. This would then represent an analogy to the ideal physics approach in which the continuous matter or the lattice is interrupted by missing bonds or cracks or flaws. The scale is however much larger. It is, furthermore, doubtful whether failure of the rock mass would be regularly promoted by stress concentrations at the joint boundary (analogous to the crack boundary in Griffith’s Theory) or otherwise by stress build up in zones of high stiffness (resistance) within the solid portions as the rock mass is usually composed rather heterogenuously, or even whether failure would originate from relatively weak zones inside the substance which have not been critically loaded before.
HELMUT
462
HABENICHT
There is still another dimension to the considerations of strength: i.e. the resistance parameters introduced through surface roughness. Along crack surfaces the negative and positive protrusion points may match together such that they generate a formfit which by itself would resist any displacement in tangential direction. A magnified surface structure[4] is symbolized in Fig. 5 which would explain that sliding resistance is on one hand influenced by the slopes and magnitudes of asperities and on the other hand by confinement against dilatation. Cracks, joints and fault planes in the rock mass offer thus some property of strength as long as the contact planes are pressed against each other. The resistance has been experienced to be of fluctuating nature as is indicated by the slip curves of Fig. 6. Particularly, curves Al and A2 as well as Bl and B2 display the stick-slip character of resistance built up and subsequent displacement. As this form of resistance cannot simply be explained by friction it represents rather an expression of the surface properties. The task to study stability conditions against failure development is, however, for engineers restricted to known zones of limited extent, i.e. these zones in which peak loading conditions occur or where weakest components are situated. For this reason the selection of any particular method of analysis may be made depending on the nature of the rock mass, dimensions of the critical zone and the loading conditions.
UNIAXIAL
COMPRESSION
AND PILLAR
LOADING
The uniaxial compression test which appears to represent one of the most simple forms of loading a test specimen would promote the expectation that the stress trajectories run parallel to the line of action of the applied forces, and would show constant density independent of the considered cross section of the specimen. This latter feature has not yet been studied in detail. A deviation from this simple geometry may well cause irregularities in the results of different tests. The major fact that can be recognized from uniaxial testing results and specimen failure shapes is that although the test is designated to be one for compressive strength, the failure occurs in a combined stress constellation whereby even the shear component could play the major role. Nevertheless the uniaxial testing procedure is considered to provide sufficient consistency in geometry and loading conditions that it is considered a sufficiently reliable configuration for testing all kinds of rocks, and for comparing the results with even using them for design work. In contrast to this the test shows some inconsistencies as indicated in Fig. 7 since the failure zone appears of irregular pattern, and the correlation of load to area of the carrying cross-section is not precisely expressed[5].
Fig. 5. Surface roughness condition causing sliding-up or stick-slip behaviour along a pre-existing contact plane.
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463
-Fractured
Mocrocrock
0
1
?
a ¶lB&&,-,’
. .
1 I/
8
lo
e
Fig. 6. Examples of shear resistance vs displacement from testing various surface types with recording at high resolution levels.
0
rock
rock --
- - - _
--
J , . J~y~$
solid
,.-
STRAIN.
Ultimote
strength
Proportional
limit
Engineering
stress-strain
curve
q
Fig. 7. Uniaxial test sample showing nonuniformity of the failure zone, and so called true stress-strain curve against the engineering stress-strain curve.
From the conical development of the failure plane[6] as is illustrated in Fig. 8, and the varying cross-section of the remaining rock body along the pillar or specimen axis it appears rather that every cross-section exhibits a different normal stress magnitude distribution. Beyond this, the loading conditions in a regular mine pillar[7] are far more complicated than in a uniaxial compression specimen, as is indicated schematically in Fig. 9. There, a coal pillar of 80 ft (24 m) width is displayed in a IO-ft (3-m) thick seam whereby only stress and strength components are shown which act perpendicular to the seam. The indicated initial elastic stress profile cannot be resisted by the coal material such that (after three steps of computational iteration in a finite element model) the final equilibrium stress profile is reached after certain portions of the pillar have yielded according to local strength properties and criteria. Inside this stress profile the “safe” volume is shown, i.e. that volume in which uniaxial strength at the pillar boundary and triaxial compression and confinement in the interior zone supply sufficient strength. Portions of this volume could well be already fragmented to the residual strength level, while other parts such as the centre region could still be below the failure condition. Thus the example illustrates that rather complicated constellations of loading and strength exist simultaneously within the structural member (pillar). This makes the definition of its structural failure extremely difficult for the engineer even though in large portions fracture processes of various nature have occurred. TYPICAL FAILURE IN A ROCK TUNNEL A failure constellation typical for tunnels and drifts is the shear failure condition shown in Fig. 10. The case shown[8] corresponds to a primary stress field with the major compressive stress
HELMUT
464
HABENICHT LEGEND Stmngth
envelop*
Initior
elastic
stress
Strsrr
profile
oftrr
first
profile
Str888
profile
after
second
iteration iteration
Final
~~l~br~rn steady-state stress profile Unfrocturrd Fmcturrd
t
SMt-wide
chain
Fig. 9. The load distribution (axial compressive stress) in a mine pillar is primarily arranged in accordance with the strength distribution inherent in the pillar.
component oriented vertically, and some magnitude of the lateral compressive stress. From theory of elasticity as well as also from field measurements and from model studies it is known that in the rock neighbou~ng the sidewalls of the tunnel the stress concentration factor for the vertical component reaches a maximum. This generates a relatively high shear stress condition which then causes eventually the failure pattern as illustrated in the figure. The location of this failure zone may actually change depending on a possible rotation of the primary stress field in the ground. It has been the recognition by Rabcewicz[9] that this form of stress combination initiates the collapse of longitudinal openings, and that therefore the critical design consideration has to be the one on the shear failure which itself occurs under some combination of normal compressive stress. The mechanical analysis for the stability condition is illustrated in Fig. 11. The relationships shown there form the result of advanced knowledge of the process of fracturing and failure which is understood to commence at one of those maximum shear trajectories which run through the zone of most critical loading. In the drawing of Fig. 11 point A symbolizes an initiation point and the force polygon in the detail below[lO] exhibits the stress and the resistance components. The balance of forces is being expressed by summation of all forces along the potential failure line S. If the activating forces would be in excess the designer still has the possibility to apply more resistance forces through setting up structural support elements at the contour of the opening. These are not shown in the figure.
ROCK LOADING OPENING
CONDITIONS AROUND ON UNDERGROUND OF RE~ANG~AR CROSS SECTION
Rectangular cross sections exhibit a rather complicated stress field. If for simplicity reason the floor and the roof are assumed horizontal the stress trajectories from gravity loading may develop a rather symmetrical pattern as is illustrated in Fig. 12. Assuming a primarily vertically oriented stress field[l2] the high intensity stress concentrations occur in the sidewalls (exactly around the corners of the opening) such that a shear failure in the mode indicated in Fig. 10 or another one in the corner points could develop if the strength would be overcome there first. Another rather sensible failure can occur in the roof strata in bending if horizontal compression would be lacking to an extent that would cause bending or tensile failure. The arch shaped, dashed curve in the roof strata would represent eventually a stable line along which an eventual failure process due to bending, tension, or shear can stop such that a so-called natural arch would form if the immediate roof strata could not be stabilized, and if failure would not occur in the sidewalls.
Strain-softening
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Fig. 8. Partially failed pillar exhibiting the primary failure planes of conical form (in situ compression test on a cubical coal specimen of 1.5m edge length).
Fig. 10. Tunnel of arched cross-section with shear failure in the spring line.
465
Strain-softening
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461
DISTRIBUTION IN THE
VICINITY
OF FORCES
OF A NARROW
OPENING
rO.dadius of fractured zone radius of excavation R, ..radius of rock carrying ring a..starting
angle of shear failure plane
s ..length of shear olane Y. .friction angle
Y..englc
between shear plane at A
and x-axis b. .portion of opening width which is under rock load to be resisted along
length “s’*
Fig. 11. Stability analysis involving shear failure for a tunnel sidewall.
Fig. 12. Loading conditions in compression, shear, bending and tension around a rectangular opening.
Beyond this, in such openings even floor failure can result if the floor strata would form the weakest member. For understanding this possible process (not shown in the figure) one must realize that the stress trajectory pattern essentially continues in the mode of a mirror image below the horizontal axis of the opening with the only small difference that gravity loading there tends to keep the floor layers in place. The shear and compressive stresses below the sidewalls in the floor strata can, however, occasionally cause the sidewalls to punch into the floor and to initiate the migration of floor material into the free opening, which can be observed there as a floor upheaval.
FAILURE OF UNDERMINED STRATA If a coal seam or another mineral stratum is mined out entirely the overburden strata will lose their carrying base and will fail in the fashion that they develop a pattern of relatively fine rupture and fragmentation in the immediate roof with the fragment size increasing towards higher elevations, and with finally changing into a quasi-plastic deformation at greater distance from the mined out seam[l2], with no or little fractures appearing. The technical process leading into such a development resembles much the one as if an opening of rectangular cross section (Fig. 12) would be enlarged by excavating continuously one of its sidewalls such that the unsupported span of the roof layers would grow until failure occurs. A graphic display of such phenomena is given in Fig. 13. Whereas fracture and failure initiation in this kind of mining is not a major point of concern, and as a matter of fact can relatively easily be determined with technical precision by practical observation, the problem setting for the engineers here is to determine the gradation curve of fragments, the depth of harmless effects to the ground surface, the amount of settlement, and the development of the settlement with the advance of the mining front or caving front. A set of parameters[l2] used to establish engineering relationships for the evaluation of the subsidence is
HELMUT
HABENICHT
KEY S
S
~a~~~,~klecr
ZI
Caved strata
f? D
Eatractkn Thkhners
subshd *tmta thkhners of ovrrburden
Fig. 13. Schematic drawing illustrating the transition of fragment size with distance from the mined seam, and the quasi-continuous subsidence deformation of the surface in greater distance.
displacement
Sa
Maximum
E
Slope
at
2%
Vertical
D
Depth
H
Thickness
at surface
point A
subsidence
of seam of seam
2W
Panel
width
B
Angle
of drow
Fig. 14. Geometry and parameters explaining the overburden settlement above mined out seams.
shown in Fig. 14. From this, for example, it is being derived that the maximum amount of subsidence occurring in the subsidence trough at the ground surface can be expressed as s.=Hn[l-exp(--.:)I, wherein u is a proportionality zone, due to the voids.
(1)
factor representing the increase of volume within the fragmented
CONDITIONS
AT THE EXCAVATION FRONT
Various means of support are available to support the roof strata at the excavation front in a coal seam such as has been outlined in the previous chapter. For their design or selection the actual load expected for them can only be evaluated from the rock mechanics considerations on the roof strata stability and on the stress conditions above the coal seams and above the mine opening. Basically, the loading of the coal seam and of the roof strata at the extraction front is characterized by a high stress concentration which loads the roof strata and the coal seam prominently in shear[l3] as is indicated in Fig. 15. The quantitative determination of loading conditions applicable to the mine support in the working area (extraction face) is not only complicated because of the irregularity of the roof strata composition and of existing planes of weakness but also by irregularities in the failure- and caving-process. Strata may vary with respect to their inclination to collapse[l4] to such an extent that a severe loading condition can exceed a standard one by several multiples in the order of magnitude. Figure 16 illustrates variations of this kind. Under these conditions great difficulty exists for providing the suitable support. Practically in Europe a compromise is approached between support types which are easy to handle and relatively low in costs whereas in other continents such as in America engineers tend towards selecting the support of such quality that it would suffice under most extreme load developments.
Spin-so~en~ng
of concrete
D
Fig. 15. Schematic pattern of shear trajectories at the excavation face with: A. . . zone of radial shear displacements, B. . . zone of transversal compression, C. , , zone of transversal extension, D. sI zone of undisturbed abutment, (b. . I angle af internal friction.
Another matter of cuncem in this luading ~onstellatiun at a coal face is the one of rock or coal bursts. Material hke coal can store elastic energy under these loading conditions where abutment pressures amount to 3-8 times the primary vertical stress. However, upon reaching the limits of strength failure processes can develop in violent nature, during which the coal (ur sometimes the roof or the floor strata) can burst into the opening with devastating effect. This form af faiiure is subject to manifoId investigations. A great variety of measures have been found through research to recognise the danger and to decrease or avoid the damage. No relationship has
b
Fig. 16. Three different configurations of roof strata action in a caving process, Strata may: (a) collapse shortly behind the working area; [b) bridge over in partially Fragmented condition; or (c) develop a relatively long overhang which GM cause intensive loading on the coal seam and on the support.
410
HELMUT HABENICHT
however been established as yet between basic crack or fracture theories and the engineering phenomena. Technically, the macroscopic approach has yielded simpler, faster and more practical methods.
CONCLUSIVE
REMARKS
The examples of failure constellations presented above have been selected from the great variety of problems in rock mechanics and underground engineering to confer an impression of the scale and level at which failure situations are being treated in engineering practice. No effort was employed to present the quantitative formulations of the mechanical interrelationships-which even sometimes are still disputed-as these would require by far more space to be treated. However, a major point to be stated is the question of precision which by itself frequently limits the depth of logics used in analysing or in proving stability conditions. Even when technical parameters are being used for calculations the input quantities do usually exhibit error margins or standard deviations[l4, 151 which, together with an eventually greater number of such parameters involved in a calculation, generate an inaccuracy of the result, which may be as great in order of magnitude as the resulting quantity itself, or even greater. The success of the engineering approach is thus limited by the nature of input quantities and by the degree of complication of the quantitive formulations. It is exactly this basic problem which has caused many engineers to work with the instrument of field measurements[l6, 17, 18, 191 conducted before, during and after the technical process such that the overall behaviour of the created structures or structural members can be observed. Judgement and progress decisions are then made on an empirical basis whereby educated and experienced engineering is employed as a guideline, and for explanation of phenomena as well as for decisions on subsequent steps. With this trend the engineering practice is presently pointing exactly into the opposite direction from the one at which the precise physics solutions of microstructure studies and of analytic formulations of elementary mineral matter can be found. This fact is stated here not to point out a divergence or an insufficiency but to stress the difficult nature of the task of dealing with geological materials. The microstructural approach of fracture mechanics and of strength considerations-although it has been more successful in other areas such as in processing of materials and in ceramics-still forms a scientific section capable of providing much input and much explanation even to the macrostructural considerations and to engineering thought. This is the more valid as rock mechanics and rock engineering has a distinct interdisciplinary character, and has as a science, drawn many benefits from being procured under interdisciplinary concepts.
REFERENCES [I] L. Miiller, Der Felsbau, Bd. I. Ferdinand Enlce, Stuttgart 624 pp. (1963). [2] H. Habenicht, Auf den Spuren der Spanmmgen in der Erdkruste, Unser Befrieb. Werk.zeitschri# der C. Deilmonn AG, Bud Benrheim, B.R.D. H.3, S. 1-7 und H.4, S. 4-10 (1977). [3] W. Dreyer und H. Borchert, Zur Druckfestigkeit von Salzgesteinen. Kali und Steins& 3 234-241 (1961). [4] E. Fecker and N. Rengers, Measurements of large scale roughness of rock planes by means of profilograph and geological compass. Proc. Symp. ZSRM Rock Fracture, Nancy paper 1-18 (1971). 151 _ _ B. T. Brady and W. I. Duvall, Strenatheninn of fractured rock nillars bv use of small radial reinforcement oressures. United S&es Department of-the Inlerior, USBM, RI 7755 (1973). _ 161 MEG _ _ Z. T. Bieniawski. Mechanism of brittle fracture of rock. Council of Scientific and Industrial Research. Reoort . 580, Pretoria, R.S.A. (1967). [7] N. P. Kripakov, Analysis of pillar stability on steeply pitching seam using finite element method. United States Department of the Interior, USBM, RI 8579 (1981). [8] F. Poisel, Ein Beitrag zur Wirkungsweise von Systemankerungen bei tietliegenden Gebirgshohlraumbauten. Rock Mechanics, Suppl. 11, pp. 173-186. Springer, Wien (1981). (91 L. V. Rabcewicz, Theorie und Praxis bei Untertagebauten eines gro5en Dammbauvorhabens. Rock Mechanics, Suppl. 2, pp. 193-224 Springer, Wien (1973). [lo] H. Habenicht, A Seminar on Anchoring. Unpublished lecture notes (1983). [ll] N. N., Roof and Rib Control. USBM Instruction Guide 17. U.S. Department of the Interior, USBM, Pittsburgh, Pennsylvania (1973). [12] S. Tandanand and L. R. Powell, Assessment of subsidence data from the Northern Appalachian Basin for subsidence prediction. U.S. Department of the Interior, USBM, RI 8630 (1982).
Strain-softening
of concrete
471
31 R. A. L. Black and A. M. Starfield, Versuche einer dynamischen Erkliirung der Gebirgsbeherrschung in Theorie und Praxis. Vierte Intemat. Konferenz fiber Schichtenkontrolle und Gebirgsmechanik, Columbia University, New York (1964). [14] 0. Jacobi, Praxis der Gebirgsbeherrschung, 2. Aufl. Gliickauf GmbH, Essen (1981). ]15] H. Habenicht, Stand und Miiglichkeiten der Dimensionierung von Rergfesten.‘Teil~I: Entwickelte Verfahren. BHM 121.1 W-126 (1976). Teil II: Systematik und Einteilung der Verfahren. BHM 121,229-235 (1976). Teil III: Reurteilung und Ausblick. BHM 121, 287-294 (1976). [16] H. Habenicht, Measuring for feedback. Consult Engr 44, 46-50 (January 1980). [17] L. V. Rabcewicz, J. Golser and E. Hack& Die Redeutung der Messung im Hohlraumbau. Der Bauingenieur 47, Teil I-225-234 and Teil 11-278-287 (1972). [18] L. Miiller, Grundtitze der MeDtechnik im Felsbau. Interfels MeDtechnik Information, pp. 3-l 1. Fa. Interfels Ges. m.b.H., Salzburg (1973). [19] E. Hackl, Messungen im Tunnelbau und ihr Einflug auf das Baugeschehen. Interfels MeDtechnik Information, pp. 12-15 Interfels Ges. m.b.H., Salzburg (1973). (Received for publication 16 November 1988)
0013-7944/90 $3.00 + 0.00 Pergamon Press plc.
Printed in Great Britain.
FAILURE
CONSTELLATIONS IN AND AROUND UNDERGROUND OPENINGS HELMUT
HABENICHT
Montanuniversitiit
Leoben, Austria
Abstract-The paper enters into the matter of fracture and failure development from the starting view point that the rock mass is already present as a fracture damaged substance. The engineering viewpoints are being emphasized and examples are stated demonstrating the variety of potential failure zones in a number of different underground openings. The interference of the strength parameters of the rock mass, of the stress field, and of the opening is illustrated. On the basis of the influential variations of input parameters, of complicated mechanical formulations, and of requirements of precision in engineering work for successful decisions it is shown why the macroscopic and phenomenological approach in engineering practice through the performance of field measurements is relatively successful.
INTRODUCTION dealing with the rock mass as a structural substance have always tended to detect the minute and basic components of strength inherent in the substance. By employing the methods and the knowledge of physics great progress has been achieved in the understanding of that strength which is provided by lattice bonds and surface bonds. Cracks and their influence on the strength of mineral matter or ceramic materials have been the subject of wide and intensive research. Engineers in mining and in underground construction are, however, faced with goals-and questions-that lie at a different level of considerations. Their problems as well as their means and methods are of a rather macroscopic nature. Although knowledge and understanding of the physical nature of mineral matter contributes favourably to engineering thought the logical chain from the basic physical parameters and properties to the presentable results of engineering efforts appears rather long and minutely subdivided. This situation and the capabilities developed at the engineering level permit solutions in some of the engineering work which needs not to be based on the microstructural approach. This fact will be illustrated by this paper. ENGINEERS
PHYSICAL
STRENGTH
AND ROCK MASS STRENGTH
The basic effort of this conference concentrates on the fracture phenomena with interpreting the same as the initial process in which the inherent strength of the matter is overcome. This inherent strength is typically understood to be built up by microscopic bonding such as would occur for example in lattice structures or at crystal contacts. In this view a fracture constitutes the break down of the ideal strength of the quasi-continuous matter. For this one the physical strength can only be proven in relatively complicated laboratory experiments, or can be discussed and evaluated only on theoretical principles. The many questions yet unsolved in this respect and the fundamental importance of this problem no doubt justify continuous efforts in this direction. Looking beyond this it is the intention of this paper to present insight into the possibilities which are offered to engineers in the technical branch of rock mechanics for their practical approach to fracture and failure problems. The difference is that engineers have to deal with matter of that status which constitutes an essentially damaged system in contrast to the continuity and/or uniformity of ideal matter[l, 21. The tensile strength of an ideal substance if represented by pure rock salt for example would amount to approximately lo5 MPa, and in compression its strength would even be for multiples higher. The real mineral rock salt, however, as it is present in the earth crust exhibits strength parameters (of technical definition) in the order of magnitude of for example 10 MPa in tension and 100 MPa in uniaxial compression[3]. Thus the strength available in real substances for engineering applications in rock mechanics is by multiple orders of magnitude smaller than the one of ideal matter. 457
458
HELMUT
HABENICHT
With the strength properties being so far apart in terms of magnitude the techniques of their evaluation are naturally different and much coarser than the ones in physics research. In fact, the formulation of molecular or atomic forces does not play a significant role in strength evaluation experiments. The practical approach of engineers is a rather macroscopic one concentrating on mechanical testing of specimens. In addition to that, even strength parameters have been defined from phenomenology such as the ones of the Mohr-Coulomb failure condition, which have been introduced through combining a loading analysis with interpretational results of testing data. This condition is defined through the tangent line to the Mohr’s circles of the limit stress constellation shown in Fig. 1. A stress constellation above the tangent line (Mohr’s envelope) means failure and anyone below means stability. The deviation of the engineering approach from the one of physics and of continuum mechanics goes even so far that under certain conditions and for certain rock mass types a tensile strength is not more accounted for, and loading of the rock mass in tension must be prohibited by the designer. In continuation of this transition of engineering from the quasi-continuum to the damaged matter the practice in rock mechanics has even recognized and defined a multiple body system (Vielk~~rsystem) given by the joint-body system (Kluftk~~ersystem) in a rigorously jointed rock mass[l]. Again from here it is conceivable that by breaking down the joint body size (and shape) one can arrive at particles and phenomenological properties which constitute the subject matter of soil mechanics, i.e. of a science that has provided much input to the one of rock mechanics. The sciences of rock mechanics and soil mechanics particularly applied to civil engineering, to ~derground construction and to mining involve a greater number of additional factors influencing the strength and the fracture of rock and rock like materials. Of these only such typical properties will be considered here which pertain to the solid state and to quasi-brittle fracture, for simplicity reasons. By restricting the discussion only to these prominent features a short view will be presented on the strength characteristics, the loading patterns, and on such design criteria which safeguard against failure. A few examples out of the engineering practice will illustrate the conditions and means of solution prevailing in the macroscopic scale. No attempts will be made at this time to integrate the contents of fracture physics into such engineering treatment since cases appear rather rare and of particular setting where the material defects would be of such minor influence as to permit a significant role of the microscopic strength terms. THE ROCK MASS AS AN ENG~EERING MATERIAL The earth’s crust-in recent years again recognized as being the stage for the continuous process of plate tectonics-is characterized by rupture surfaces of vast extension and of varying orientation as it is symbohzed by Fig. 2. These tectonic planes or surfaces occur in different scale including for example such as would be crossing the floor of the Atlantic to the ones mapped in central Europe shown in Fig. 3, and to such small ones[2] as may be evident in the foundation rock of a highway bridge illustrated in Fig. 4. Manifold examples could be stated for this feature. It plainly persists throughout the majority of the earth’s rocks. It ranges over all sizes of sampling
Fig. 1. Mohr’s circles and Mohr’s envelope showing the shear stress failure conditions under combined normal and shear stresses.
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Fig. 2. Topography of the ocean floor along the Atlantic Rift Zone illustrating fracture systems in the earth’s crust in a large scale.
Fig. 4. Jointed rock surface of approx. 2.5 x 1.5 m size illustrating the fragmented structure of a rock mass in a small scale.
459
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Fig. 3. Major tectonic faults mapped in the central European area demonstrating the discontinuous nature of the earth’s crust in an intermediate scale.
down to the handpiece and to the thin section prepared for microscopic analysis which exhibits cracks and flaws in the structure. Thus, discontinuities as planes or surfaces of weakness or of zero strength are prevailing in rock mechanical problems. Therefore in rock mass stability considerations a tensile strength cannot be attributed to the rock mass except for specially isolated bodies or for rare zones of high quality and undisturbed rock volumes. It is due to this fact that the design of structures in rock is built on criteria and configurations which involve loading in shear and compression only. An additional condition for such constellations to permit failure is the proximity of a free surface to which the overstressed portions can yield. This condition is satisfied in most underground openings or structural members. In these applications the predominant law is given by the Mohr-Coulomb criterion as it is shown in the stress envelope of Fig. 1, or by extended formulations such as for example by the Leon-parabola[ 11. In some respect one could apply these criteria to the rock mass in a generalized way by ascribing to the latter the strength parameters as if it would represent a continuum, i.e. as if the parameters would prevail evenly and uniformly throughout the mass. Advanced considerations have resolved the rock mass into such proportions that are regarded continuous matter of some technical strength value and into such ones that are given by preexisting failure or weakness planes. This subdivision of the rock mass would more closely represent the effect of joints which do usually transgress the rock mass intermittently. This would then represent an analogy to the ideal physics approach in which the continuous matter or the lattice is interrupted by missing bonds or cracks or flaws. The scale is however much larger. It is, furthermore, doubtful whether failure of the rock mass would be regularly promoted by stress concentrations at the joint boundary (analogous to the crack boundary in Griffith’s Theory) or otherwise by stress build up in zones of high stiffness (resistance) within the solid portions as the rock mass is usually composed rather heterogenuously, or even whether failure would originate from relatively weak zones inside the substance which have not been critically loaded before.
HELMUT
462
HABENICHT
There is still another dimension to the considerations of strength: i.e. the resistance parameters introduced through surface roughness. Along crack surfaces the negative and positive protrusion points may match together such that they generate a formfit which by itself would resist any displacement in tangential direction. A magnified surface structure[4] is symbolized in Fig. 5 which would explain that sliding resistance is on one hand influenced by the slopes and magnitudes of asperities and on the other hand by confinement against dilatation. Cracks, joints and fault planes in the rock mass offer thus some property of strength as long as the contact planes are pressed against each other. The resistance has been experienced to be of fluctuating nature as is indicated by the slip curves of Fig. 6. Particularly, curves Al and A2 as well as Bl and B2 display the stick-slip character of resistance built up and subsequent displacement. As this form of resistance cannot simply be explained by friction it represents rather an expression of the surface properties. The task to study stability conditions against failure development is, however, for engineers restricted to known zones of limited extent, i.e. these zones in which peak loading conditions occur or where weakest components are situated. For this reason the selection of any particular method of analysis may be made depending on the nature of the rock mass, dimensions of the critical zone and the loading conditions.
UNIAXIAL
COMPRESSION
AND PILLAR
LOADING
The uniaxial compression test which appears to represent one of the most simple forms of loading a test specimen would promote the expectation that the stress trajectories run parallel to the line of action of the applied forces, and would show constant density independent of the considered cross section of the specimen. This latter feature has not yet been studied in detail. A deviation from this simple geometry may well cause irregularities in the results of different tests. The major fact that can be recognized from uniaxial testing results and specimen failure shapes is that although the test is designated to be one for compressive strength, the failure occurs in a combined stress constellation whereby even the shear component could play the major role. Nevertheless the uniaxial testing procedure is considered to provide sufficient consistency in geometry and loading conditions that it is considered a sufficiently reliable configuration for testing all kinds of rocks, and for comparing the results with even using them for design work. In contrast to this the test shows some inconsistencies as indicated in Fig. 7 since the failure zone appears of irregular pattern, and the correlation of load to area of the carrying cross-section is not precisely expressed[5].
Fig. 5. Surface roughness condition causing sliding-up or stick-slip behaviour along a pre-existing contact plane.
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-Fractured
Mocrocrock
0
1
?
a ¶lB&&,-,’
. .
1 I/
8
lo
e
Fig. 6. Examples of shear resistance vs displacement from testing various surface types with recording at high resolution levels.
0
rock
rock --
- - - _
--
J , . J~y~$
solid
,.-
STRAIN.
Ultimote
strength
Proportional
limit
Engineering
stress-strain
curve
q
Fig. 7. Uniaxial test sample showing nonuniformity of the failure zone, and so called true stress-strain curve against the engineering stress-strain curve.
From the conical development of the failure plane[6] as is illustrated in Fig. 8, and the varying cross-section of the remaining rock body along the pillar or specimen axis it appears rather that every cross-section exhibits a different normal stress magnitude distribution. Beyond this, the loading conditions in a regular mine pillar[7] are far more complicated than in a uniaxial compression specimen, as is indicated schematically in Fig. 9. There, a coal pillar of 80 ft (24 m) width is displayed in a IO-ft (3-m) thick seam whereby only stress and strength components are shown which act perpendicular to the seam. The indicated initial elastic stress profile cannot be resisted by the coal material such that (after three steps of computational iteration in a finite element model) the final equilibrium stress profile is reached after certain portions of the pillar have yielded according to local strength properties and criteria. Inside this stress profile the “safe” volume is shown, i.e. that volume in which uniaxial strength at the pillar boundary and triaxial compression and confinement in the interior zone supply sufficient strength. Portions of this volume could well be already fragmented to the residual strength level, while other parts such as the centre region could still be below the failure condition. Thus the example illustrates that rather complicated constellations of loading and strength exist simultaneously within the structural member (pillar). This makes the definition of its structural failure extremely difficult for the engineer even though in large portions fracture processes of various nature have occurred. TYPICAL FAILURE IN A ROCK TUNNEL A failure constellation typical for tunnels and drifts is the shear failure condition shown in Fig. 10. The case shown[8] corresponds to a primary stress field with the major compressive stress
HELMUT
464
HABENICHT LEGEND Stmngth
envelop*
Initior
elastic
stress
Strsrr
profile
oftrr
first
profile
Str888
profile
after
second
iteration iteration
Final
~~l~br~rn steady-state stress profile Unfrocturrd Fmcturrd
t
SMt-wide
chain
Fig. 9. The load distribution (axial compressive stress) in a mine pillar is primarily arranged in accordance with the strength distribution inherent in the pillar.
component oriented vertically, and some magnitude of the lateral compressive stress. From theory of elasticity as well as also from field measurements and from model studies it is known that in the rock neighbou~ng the sidewalls of the tunnel the stress concentration factor for the vertical component reaches a maximum. This generates a relatively high shear stress condition which then causes eventually the failure pattern as illustrated in the figure. The location of this failure zone may actually change depending on a possible rotation of the primary stress field in the ground. It has been the recognition by Rabcewicz[9] that this form of stress combination initiates the collapse of longitudinal openings, and that therefore the critical design consideration has to be the one on the shear failure which itself occurs under some combination of normal compressive stress. The mechanical analysis for the stability condition is illustrated in Fig. 11. The relationships shown there form the result of advanced knowledge of the process of fracturing and failure which is understood to commence at one of those maximum shear trajectories which run through the zone of most critical loading. In the drawing of Fig. 11 point A symbolizes an initiation point and the force polygon in the detail below[lO] exhibits the stress and the resistance components. The balance of forces is being expressed by summation of all forces along the potential failure line S. If the activating forces would be in excess the designer still has the possibility to apply more resistance forces through setting up structural support elements at the contour of the opening. These are not shown in the figure.
ROCK LOADING OPENING
CONDITIONS AROUND ON UNDERGROUND OF RE~ANG~AR CROSS SECTION
Rectangular cross sections exhibit a rather complicated stress field. If for simplicity reason the floor and the roof are assumed horizontal the stress trajectories from gravity loading may develop a rather symmetrical pattern as is illustrated in Fig. 12. Assuming a primarily vertically oriented stress field[l2] the high intensity stress concentrations occur in the sidewalls (exactly around the corners of the opening) such that a shear failure in the mode indicated in Fig. 10 or another one in the corner points could develop if the strength would be overcome there first. Another rather sensible failure can occur in the roof strata in bending if horizontal compression would be lacking to an extent that would cause bending or tensile failure. The arch shaped, dashed curve in the roof strata would represent eventually a stable line along which an eventual failure process due to bending, tension, or shear can stop such that a so-called natural arch would form if the immediate roof strata could not be stabilized, and if failure would not occur in the sidewalls.
Strain-softening
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Fig. 8. Partially failed pillar exhibiting the primary failure planes of conical form (in situ compression test on a cubical coal specimen of 1.5m edge length).
Fig. 10. Tunnel of arched cross-section with shear failure in the spring line.
465
Strain-softening
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461
DISTRIBUTION IN THE
VICINITY
OF FORCES
OF A NARROW
OPENING
rO.dadius of fractured zone radius of excavation R, ..radius of rock carrying ring a..starting
angle of shear failure plane
s ..length of shear olane Y. .friction angle
Y..englc
between shear plane at A
and x-axis b. .portion of opening width which is under rock load to be resisted along
length “s’*
Fig. 11. Stability analysis involving shear failure for a tunnel sidewall.
Fig. 12. Loading conditions in compression, shear, bending and tension around a rectangular opening.
Beyond this, in such openings even floor failure can result if the floor strata would form the weakest member. For understanding this possible process (not shown in the figure) one must realize that the stress trajectory pattern essentially continues in the mode of a mirror image below the horizontal axis of the opening with the only small difference that gravity loading there tends to keep the floor layers in place. The shear and compressive stresses below the sidewalls in the floor strata can, however, occasionally cause the sidewalls to punch into the floor and to initiate the migration of floor material into the free opening, which can be observed there as a floor upheaval.
FAILURE OF UNDERMINED STRATA If a coal seam or another mineral stratum is mined out entirely the overburden strata will lose their carrying base and will fail in the fashion that they develop a pattern of relatively fine rupture and fragmentation in the immediate roof with the fragment size increasing towards higher elevations, and with finally changing into a quasi-plastic deformation at greater distance from the mined out seam[l2], with no or little fractures appearing. The technical process leading into such a development resembles much the one as if an opening of rectangular cross section (Fig. 12) would be enlarged by excavating continuously one of its sidewalls such that the unsupported span of the roof layers would grow until failure occurs. A graphic display of such phenomena is given in Fig. 13. Whereas fracture and failure initiation in this kind of mining is not a major point of concern, and as a matter of fact can relatively easily be determined with technical precision by practical observation, the problem setting for the engineers here is to determine the gradation curve of fragments, the depth of harmless effects to the ground surface, the amount of settlement, and the development of the settlement with the advance of the mining front or caving front. A set of parameters[l2] used to establish engineering relationships for the evaluation of the subsidence is
HELMUT
HABENICHT
KEY S
S
~a~~~,~klecr
ZI
Caved strata
f? D
Eatractkn Thkhners
subshd *tmta thkhners of ovrrburden
Fig. 13. Schematic drawing illustrating the transition of fragment size with distance from the mined seam, and the quasi-continuous subsidence deformation of the surface in greater distance.
displacement
Sa
Maximum
E
Slope
at
2%
Vertical
D
Depth
H
Thickness
at surface
point A
subsidence
of seam of seam
2W
Panel
width
B
Angle
of drow
Fig. 14. Geometry and parameters explaining the overburden settlement above mined out seams.
shown in Fig. 14. From this, for example, it is being derived that the maximum amount of subsidence occurring in the subsidence trough at the ground surface can be expressed as s.=Hn[l-exp(--.:)I, wherein u is a proportionality zone, due to the voids.
(1)
factor representing the increase of volume within the fragmented
CONDITIONS
AT THE EXCAVATION FRONT
Various means of support are available to support the roof strata at the excavation front in a coal seam such as has been outlined in the previous chapter. For their design or selection the actual load expected for them can only be evaluated from the rock mechanics considerations on the roof strata stability and on the stress conditions above the coal seams and above the mine opening. Basically, the loading of the coal seam and of the roof strata at the extraction front is characterized by a high stress concentration which loads the roof strata and the coal seam prominently in shear[l3] as is indicated in Fig. 15. The quantitative determination of loading conditions applicable to the mine support in the working area (extraction face) is not only complicated because of the irregularity of the roof strata composition and of existing planes of weakness but also by irregularities in the failure- and caving-process. Strata may vary with respect to their inclination to collapse[l4] to such an extent that a severe loading condition can exceed a standard one by several multiples in the order of magnitude. Figure 16 illustrates variations of this kind. Under these conditions great difficulty exists for providing the suitable support. Practically in Europe a compromise is approached between support types which are easy to handle and relatively low in costs whereas in other continents such as in America engineers tend towards selecting the support of such quality that it would suffice under most extreme load developments.
Spin-so~en~ng
of concrete
D
Fig. 15. Schematic pattern of shear trajectories at the excavation face with: A. . . zone of radial shear displacements, B. . . zone of transversal compression, C. , , zone of transversal extension, D. sI zone of undisturbed abutment, (b. . I angle af internal friction.
Another matter of cuncem in this luading ~onstellatiun at a coal face is the one of rock or coal bursts. Material hke coal can store elastic energy under these loading conditions where abutment pressures amount to 3-8 times the primary vertical stress. However, upon reaching the limits of strength failure processes can develop in violent nature, during which the coal (ur sometimes the roof or the floor strata) can burst into the opening with devastating effect. This form af faiiure is subject to manifoId investigations. A great variety of measures have been found through research to recognise the danger and to decrease or avoid the damage. No relationship has
b
Fig. 16. Three different configurations of roof strata action in a caving process, Strata may: (a) collapse shortly behind the working area; [b) bridge over in partially Fragmented condition; or (c) develop a relatively long overhang which GM cause intensive loading on the coal seam and on the support.
410
HELMUT HABENICHT
however been established as yet between basic crack or fracture theories and the engineering phenomena. Technically, the macroscopic approach has yielded simpler, faster and more practical methods.
CONCLUSIVE
REMARKS
The examples of failure constellations presented above have been selected from the great variety of problems in rock mechanics and underground engineering to confer an impression of the scale and level at which failure situations are being treated in engineering practice. No effort was employed to present the quantitative formulations of the mechanical interrelationships-which even sometimes are still disputed-as these would require by far more space to be treated. However, a major point to be stated is the question of precision which by itself frequently limits the depth of logics used in analysing or in proving stability conditions. Even when technical parameters are being used for calculations the input quantities do usually exhibit error margins or standard deviations[l4, 151 which, together with an eventually greater number of such parameters involved in a calculation, generate an inaccuracy of the result, which may be as great in order of magnitude as the resulting quantity itself, or even greater. The success of the engineering approach is thus limited by the nature of input quantities and by the degree of complication of the quantitive formulations. It is exactly this basic problem which has caused many engineers to work with the instrument of field measurements[l6, 17, 18, 191 conducted before, during and after the technical process such that the overall behaviour of the created structures or structural members can be observed. Judgement and progress decisions are then made on an empirical basis whereby educated and experienced engineering is employed as a guideline, and for explanation of phenomena as well as for decisions on subsequent steps. With this trend the engineering practice is presently pointing exactly into the opposite direction from the one at which the precise physics solutions of microstructure studies and of analytic formulations of elementary mineral matter can be found. This fact is stated here not to point out a divergence or an insufficiency but to stress the difficult nature of the task of dealing with geological materials. The microstructural approach of fracture mechanics and of strength considerations-although it has been more successful in other areas such as in processing of materials and in ceramics-still forms a scientific section capable of providing much input and much explanation even to the macrostructural considerations and to engineering thought. This is the more valid as rock mechanics and rock engineering has a distinct interdisciplinary character, and has as a science, drawn many benefits from being procured under interdisciplinary concepts.
REFERENCES [I] L. Miiller, Der Felsbau, Bd. I. Ferdinand Enlce, Stuttgart 624 pp. (1963). [2] H. Habenicht, Auf den Spuren der Spanmmgen in der Erdkruste, Unser Befrieb. Werk.zeitschri# der C. Deilmonn AG, Bud Benrheim, B.R.D. H.3, S. 1-7 und H.4, S. 4-10 (1977). [3] W. Dreyer und H. Borchert, Zur Druckfestigkeit von Salzgesteinen. Kali und Steins& 3 234-241 (1961). [4] E. Fecker and N. Rengers, Measurements of large scale roughness of rock planes by means of profilograph and geological compass. Proc. Symp. ZSRM Rock Fracture, Nancy paper 1-18 (1971). 151 _ _ B. T. Brady and W. I. Duvall, Strenatheninn of fractured rock nillars bv use of small radial reinforcement oressures. United S&es Department of-the Inlerior, USBM, RI 7755 (1973). _ 161 MEG _ _ Z. T. Bieniawski. Mechanism of brittle fracture of rock. Council of Scientific and Industrial Research. Reoort . 580, Pretoria, R.S.A. (1967). [7] N. P. Kripakov, Analysis of pillar stability on steeply pitching seam using finite element method. United States Department of the Interior, USBM, RI 8579 (1981). [8] F. Poisel, Ein Beitrag zur Wirkungsweise von Systemankerungen bei tietliegenden Gebirgshohlraumbauten. Rock Mechanics, Suppl. 11, pp. 173-186. Springer, Wien (1981). (91 L. V. Rabcewicz, Theorie und Praxis bei Untertagebauten eines gro5en Dammbauvorhabens. Rock Mechanics, Suppl. 2, pp. 193-224 Springer, Wien (1973). [lo] H. Habenicht, A Seminar on Anchoring. Unpublished lecture notes (1983). [ll] N. N., Roof and Rib Control. USBM Instruction Guide 17. U.S. Department of the Interior, USBM, Pittsburgh, Pennsylvania (1973). [12] S. Tandanand and L. R. Powell, Assessment of subsidence data from the Northern Appalachian Basin for subsidence prediction. U.S. Department of the Interior, USBM, RI 8630 (1982).
Strain-softening
of concrete
471
31 R. A. L. Black and A. M. Starfield, Versuche einer dynamischen Erkliirung der Gebirgsbeherrschung in Theorie und Praxis. Vierte Intemat. Konferenz fiber Schichtenkontrolle und Gebirgsmechanik, Columbia University, New York (1964). [14] 0. Jacobi, Praxis der Gebirgsbeherrschung, 2. Aufl. Gliickauf GmbH, Essen (1981). ]15] H. Habenicht, Stand und Miiglichkeiten der Dimensionierung von Rergfesten.‘Teil~I: Entwickelte Verfahren. BHM 121.1 W-126 (1976). Teil II: Systematik und Einteilung der Verfahren. BHM 121,229-235 (1976). Teil III: Reurteilung und Ausblick. BHM 121, 287-294 (1976). [16] H. Habenicht, Measuring for feedback. Consult Engr 44, 46-50 (January 1980). [17] L. V. Rabcewicz, J. Golser and E. Hack& Die Redeutung der Messung im Hohlraumbau. Der Bauingenieur 47, Teil I-225-234 and Teil 11-278-287 (1972). [18] L. Miiller, Grundtitze der MeDtechnik im Felsbau. Interfels MeDtechnik Information, pp. 3-l 1. Fa. Interfels Ges. m.b.H., Salzburg (1973). [19] E. Hackl, Messungen im Tunnelbau und ihr Einflug auf das Baugeschehen. Interfels MeDtechnik Information, pp. 12-15 Interfels Ges. m.b.H., Salzburg (1973). (Received for publication 16 November 1988)