The distribution of stress around a single underground opening in a layered medium under gravity loading

Int. J. Rock Mech. Min. Sci. Vol. 9, pp. 127-154. Pergamon Press 1972. Printed in Great Britain

THE DISTRIBUTION OF STRESS A R O U N D A SINGLE U N D E R G R O U N D OPENING IN A LAYERED MEDIUM U N D E R GRAVITY L O A D I N G G. BARLA* Henry Krumb School of Mines, Columbia University, New York

(Received 31 August 1970) Abstract--The finite-element method is used in order to compute the stresses in horizontally

layered structures. The distribution of stress around openings of several shapes, located in a center layer of a three-layer medium, is studied in detail. Consideration is given to the influence of the geometry of the situation (i.e. thickness of the center layer and shape of the opening) and of the contrast in the mechanical properties between the center layer and the confining homogeneous medium (i.e. ratio of the elastic moduli). The method of solution and the numerical results presented prove its worth in the analysis of repeatable situations encountered in rock mechanics practice (e.g. mining and development workings in layered deposits). INTRODUCTION THE effect due to the presence o f non-homogeneities in a rock mass results in a change of the general stress distribution expected from the corresponding solution for a homogeneous medium. Different types o f non-homogeneities occur in a rock mass. Some o f these nonhomogeneities can often be reduced to a simple geometrical form and the results o f the stress analysis can be used in repeatable situations. The type o f n o n - h o m o g e n e o u s structure considered in this paper is a rock medium with horizontal layers. Previous work on this subject has been carried out by various investigators. The stresses a r o u n d a circular tunnel in either a two-layer [l] or a three-layer medium [2], [3] were c o m p u t e d by the finite-element method. In the first case, the opening has been located in either the 'soft' or the ' h a r d ' layer. In the second case, consideration has been given only to the opening in the soft layer. Further, the results were derived by subjecting the structure to a uniform stress field, obtained by applying a system o f concentrated loads on the outer boundaries o f the corresponding finite-element model. Also, mention should be made o f an alternative a p p r o a c h used, which substitutes the layered structure with a transversely isotropic medium and applies concepts of the theory of elasticity for anisotropic media to effect a solution (see, for example, [4]). METHOD OF SOLUTION In nature, a layered rock mass m a y occur in several forms. A typical example is illustrated in Fig. 1, where a typical opening is also shown. Such a mass m a y be viewed as a composite structure where the individual layers are the homogeneous components with assigned material properties. At the layers interface, different conditions for the stresses and displacements can be written out in order to represent a realistic situation. It m a y be assumed that the layers * Present address: Istituto di Arte Mineraria del Politecnico, Torino, Italy. 127

128

G. BARLA

remain in contact during the deformation and that the vertical components of stress and displacement be continuous at the interface. Additionally, the adjacent layers may either be bonded so that no slip occurs (i.e. shear stresses and displacements are continuous) or slip over each other (i.e. the shear stresses on both sides of the interface are zero). Only the first situation will be studied here. Therefore, the method reported is applicable if the conditions of continuity of the normal and shear components of stress and displacement at the interface are satisfied. It may be noted that in the case of an actual layered rock mass, this condition implies that the individual layers be perfectly welded.

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Fio. 1. Exampleof a layeredrock mass with arched opening.

The usefulness of a method which allows one to determine the distribution of stress around openings located in a layered rock mass and for any geometry of the situation is recognized in both mining and civil engineering practice. Such a method is easily developed by the finite-element technique which is well suited to the analysis of non-homogeneous structures. A finite-element program [5] is easily adapted to perform the required computations for any horizontally layered rock structure. A subprogram was set up to generate automatically the required finite-element models by a change of the mechanical properties of the layers, for any chosen combination of horizontal layers which is to be represented. Within the elastostatic theory, the distribution of stress around the openings depends upon the geometry of the situation and the ratios of the elastic constants for the individual layers, EdEl+l, *l/~l + 1. Thus, the subprogram was developed to generate data for any chosen ratio of the elastic constants. A limitation to this method of solution is found in the expected decrease of accuracy for the finite-element technique in the near vicinity of the interfaces between adjacent layers. If the number of layers in the rock structure is increased considerably, the above limitation may also resuR in only an approximate estimate of the value for the stresses in the individual layers.

DISTRIBUTION OF STRESS AROUND A SINGLE UNDERGROUND OPENING

129

NUMERICAL RESULTS FOR A THREE-LAYER ROCK M A S S

While the foregoing finite-element program allows one to treat any layered rock structure, only the stress distribution around openings located in the center layer of a three-layer medium is considered in this paper. The geometrical symmetry with respect to the vertical axis of the structure makes finiteelement models similar to those described in a previous paper [5] also applicable in the present case. As the previous results show, gravity loading reproduces the initial stress field in the rock structure more appropriately than a uniform loading applied on the outer boundary. Therefore, gravity loading is used and the boundary conditions are set equal to those described by the author [5]. The distribution of stress around an opening located in a three-layer rock mass is a function of the geometry of the problem (opening shape and thickness of the middle layer), the contrast in the material properties between the middle layer and that of the confining homogeneous medium, and the initial stress field. The influence of these different parameters is studied in the following. The numerical results are reported in two parts, considering first openings with aspect ratio equal to 1. In each part, the stress distribution relates to the dimensionless quantities: t/a, where t is the thickness of the middle layer and a is the half of the opening height; E2/E1 where El and Ez are respectively the elastic moduli of the confining homogeneous medium and of the middle layer. The values of t/a are taken as varying from 1 to 6. The lower value corresponds to a horizontal layer of a thickness equal to the half of the opening height. The upper value is chosen, because the distribution of stress in this case is approximately that of the corresponding homogeneous solution. The values of E2/EI are set to vary from 0.1 to 10. Values of E2/EI ~< 1 correspond to a 'soft' middle layer, values of E2/EI >/ I to a 'hard' one. The lower and higher values of E2/E~ were chosen as the contrast in the mechanical properties of two geologic strata is found to be seldom out of this range. Values of t/a and E 2 / E t , respectively equal to 0 and 1, correspond to a homogeneous structure. Both uniaxial and biaxial stress fields are considered. These are derived (as in BARLA [5]) from displacements properly assigned on the outer boundary of the finite-element model and the assumed plane-strain conditions. The same Poisson's ratio is taken for the middle layer and the confining medium, i.e. cr2 = ~ . The depth h of the opening center below the horizontal traction-free surface is set in each case equal to 10a. As observed by the author [5], the distribution of stress around the opening is, under this condition and for the shapes considered, practically independent of the ratio h/a. The numerical results for all the stresses have been normalized to the value of the vertical stress at the center of the opening. The boundary stresses refer to three positions on the opening contour (R, B and F) and the principal stresses around the opening are plotted along the following axes: horizontal, vertical positive and vertical negative. (a) The circle, the arched opening and the square The numerical values of the boundary stresses for circle, arched opening and square are reported in Tables 1-3. For each opening, Table (a) refers to the uniaxial and Table (b) to the biaxial stress field. The results are given for different values of t/a and E2/E1. In order to illustrate the typical features of the numerical solution, the boundary stresses at the rib, back and floor of the openings--all normalized to the vertical stress expected at the center, r o pgh--are plotted in Figs 2-4 for different values of t/a, E z / E l --~ 10, aO(?K 9/I

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130

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0.10 and the uniaxial stress field. The values of the boundary stresses for t/a = 0 are those obtained by the author [5] for h/a = 10 (homogeneous structure). Figure 2 shows that the effect of the presence of a middle layer is that of reducing the values of the rib stresses for E2/E~ = 0-10 and increasing them for E2/Et ----10.0, whenthe results are compared with those corresponding to the homogeneous structure, The rib stress exhibits for each opening shape a similar trend of behavior with increasing values of

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DISTRIBUTION OF STRESS AROUND A SINGLE UNDERGROUND OPENING

131

t/a. For t/a -- 4, this stress is approximately equal to that obtained for the homogeneous structure. For E2/EI = 0.10, the minimum values of the rib stresses correspond to t/a ~-- 1, while, for Ee/E~ = 10-0 the maximum values are obtained for t/a ~ 2. The highest values of the rib stress occur always for the circular opening, while the lowest values are obtained for the square. It may also be noticed that larger deviations from the rib stresses prevailing for the homogeneous structure occur for E2/E~ -- 10.0 and for each opening shape than those for E2/Et = 0" 10. In Fig. 3, the curves for the back stresses show, as t/a increases, a characteristic behavior. For E2/E~ = 0.10, the stresses increase from the values which correspond to the homogeneous structure and reach a maximum. Then, they decrease to reach a minimum and gradually increase toward the same values obtained for the homogeneous structure. For E2/E l 10"0, the behavior is reversed. =:

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It should be observed that the maximum values in the tensile stresses occur for E 2 / E t = 10.0. Also, a clear tendency is that the tensile stresses are always located in the hard layer, the circular and the arched openings showing the highest values. Figure 4 shows that, as t/a increases, the floor stresses exhibit a trend of behavior similar to that of the back stresses. The highest values of the tensile stresses occur for the arched and the square openings and still for E2/EI = 10.0. Similar observations with appropriate changes in the sign of stresses can be made for the biaxial stress field. However, in this case, the deviations from the values for the homogeneous structure are in general smaller than those corresponding to the uniaxial stress field. The effect of changing the contrast in the material properties of the middle layer and that of the confining homogeneous medium is illustrated in Fig. 5, where the results refer to the arched opening and to the uniaxial stress field. Only the results for this opening shape are plotted as the observations which can be made in this case are typical of openings with aspect

0"10 0"20 0"33 0"66 1"50 3"00 5"00 10"00

E2/E1

0" 10 0"20 0"33 0"66 1"50 3,00 5"00 10"00

EdE1

0"18 0"21 0"24 0"26 0'27 0"26 0"24 0"21

B

1"41 1"58 1"74 2"03 2"44 2"83 3"10 3"42

R

1"0

B

R

F

--1"00 --0-94 --0"88 --0"78 --0"65 --0"54 --0'45 --0"32 R

1"76 1.90 2"03 2"28 2"66 3"05 3"36 3.77 B

--0"71 --0"70 --0"68 --0"63 --0"51 --0"36 --0"24 --0"13

2"0

F

--0"45 --0"41 --0"36 --0'28 --0"18 --0"10 --0"04 --0"03 R

1"96 2"05 2.16 2"33 2"60 2"86 3"06 3"25 B

--0-23 --0"31 --0"39 --0-55 --0"76 --0"96 --1"12 --1"25

3"0

t/a

F

--0"25 --0"35 --0"45 --0"61 --0"82 --0"98 --1"08 --1 "08 R

2.12 2"19 2"26 2"38 2-55 2"70 2"81 2"86 B

--0"40 --0"45 --0"50 --0"59 --0"71 --0"81 --0"88 --0"86

4"0

F

--0"42 --0"48 --0"55 --0"65 --0"78 --0"89 --0"95 --0"96

F

0"09 0"12 0"16 0"20 0"24 0"26 0"27 0"28 R

1"58 1"70 1"83 2"06 2"42 2"81 3"13 3"56 B

0"13 0"16 0"20 0"27 O" 39 O' 50 0"58 0"66

2"0

F

0"28 0"32 0"35 0'41 O"49 O" 54 0'56 0"55 R

1"76 1"84 1'94 2"10 2"37 2"63 2"82 3"00

B

0"31 0"30 O" 30 0"28 0"24 O" 17 0"13 0"13

3"0

~a

F

0"31 0"29 0"28 0"25 0"20 O" 17 0"17 0"17

R

1"91 1"97 2"03 2"15 2"32 2"47 2"57 2"63

B

0"32 0"31 0"30 0"28 0"25 0"19 0"16 0"16

4"0

F

0"33 0"31 0'30 0"25 0"20 0"15 0"13 0,60

R

2,08 2'11 2'13 2"19 2"30 2'34 2"39 2'41

R

2"30 2" 33 2.36 2"42 2"50 2"58 2"63 2"60

TABLE l(b). STRESS CONCENTRATIONS FOR CIRCLE; BIAXIAL STRESS FIELD UNDER GRAVITY LOADING-THREE-LAYER ROCK MASS

--0"82 --0.78 --0"74 --0'68 --0"61 --0'55 --0.52 --0"47

1"57 1-75 1"94 2-25 2"67 3"02 3.24 3"45

1"0

TABLE 1 (a). STRESS CONCENTRATIONS FOR CIRCLE; UNIAXIAL STRESS FIELD UNDER GRAVITY LOADING-THREE-LAYER ROCK MASS

B

0"30 0"30 0'29 0"28 0"25 0"23 0"21 0"41

6"0

B

--0"53 --0"56 --0"58 --0"62 --0"67 --0"72 --0"72 --0"69

6"0

F

0"29 0"28 0"27 0'24 0"21 0"17 0"16 0"16

F

--0"59 --0"62 --0"64 --0"69 --0"74 --0"79 --0"79 --0"82

to

--0"90 --0"84 --0"78 --0"69 --0"58 --0"51 --0"46 --0"41

B

1"42 1'58 1'72 1"92 2"14 2"29 2"37 2"46

R

0"10 0'20 0"33 0"66 1"50 3"00 5"00 10'00

F

--0'87 --0'87 --0"87 --0"87 --0"86 --0"84 --0"83 --0"81 R

1"47 1"60 1"70 1"90 2"18 2"45 2"64 2"88 B

--0"76 --0"74 --0"71 --0'64 --0"48 --0"30 --0"18 --0-07

2'0

F

-0"98 --0"96 --0"94 --0"90 --0"83 --0'77 --0"73 --0"69 R

1"58 1"70 1'76 1"92 2"16 2-40 2"~0 2"83

t/a

B

--0"19 --0"28 --0"37 --0"53 --0'74 --0"92 --1"06 --1"23

3"0

F

--0"28 --0"39 --0"50 --0"72 --1"02 --1"34 --1"60 --2"00 R

1"70 1'76 1"82 1"95 2"13 2"30 2"42 2"56 B

--0"32 --0'39 --0"46 --0"56 --0"71 --0"82 --0-91 --l'01

4"0

F

--0"50 --0"57 --0"65 --0"72 --0"95 --1"10 --1"19 --1"31 R

1"83 1-90 1"92 1"99 2'08 2"17 2"23 2"29

1 "0

0'00 0"11 0'16 0'21 0-25 0"24 0"22 0"18

B

1"26 1"40 1"52 1'73 2"0l 2"27 2"46 2"70

R

E2/EI

0"10 0"20 0"33 0"66 1"50 3'00 5"00 10'00

F

--0"22 --0"23 --0"23 --0"25 --0"27 --0"28 --0'30 --0"31 R

1"30 1"4l 1"52 1"72 2"03 2"34 2"57 2"88 B

0"01 0"13 0"16 0"24 0"39 0-54 0'64 0"74

2"0

F

--0'28 --0"28 --0"27 --0"26 --0'25 --0"26 --0'28 --0'30 R

1"42 1"50 1"59 1"75 2"00 2"26 2"45 2"68 B

0"32 0"28 0"29 0'26 0"20 0"13 0"12 0"14

3"0

t/a

F

0"00 --0"02 --0'06 --0"17 --0"37 --0'65 --0"94 --1"42

R

1"53 1"60 1"65 1"78 1"97 2"15 2'27 2"40

B

0"36 0"39 0"32 0"27 0"19 0"11 0"08 0'07

4"0

F

0"04 --0'06 --0"10 --0"19 --0"34 --0"51 --0"64 --0"77

R

1"68 1"71 1"75 1"82 1-92 2'01 2"07 2"12

TABLE 2 ( b ) . STRESS CONCENTRATIONS FOR ARCHED O P E N I N G ; BIAXIAL STRESS FIELD UNDER GRAVITY L O A D I N G - THREE-LAYER ROCK MASS

1"0

E2/E~

TABLE 2 ( a ) . STRESS CONCENTRATION~S FOR ARCHED OPENING ; UNIAXIAL STRESS FIELD UNDER GRAVITY L O A D I N G - THREE-LAYER ROCK MASS

B

0"33 0"51 0"29 0"26 0"20 0'16 0" 13 0"10

6"0

B

--0"48 --0"51 --0"55 --0"60 --0"67 --0"72 --0-76 --0"79

6"0

F

--0"10 --0"16 --0"18 --0"22 --0"29 --0'35 --0"40 --0"44

F

--0"73 --0'76 --0"79 --0"84 --0"89 --0"92 --0-94 --0"96

k.a

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--0"78 --0"69 --0"66 --0"61 --0"55 --0-51 --0"48 --0"44

B

1"36 1.54 1"66 1"83 1-98 2,03 2"03 1"99

R

0"10 0"20 0-33 0"66 1.50 3"00 5"00 10"00

1 "0

0"19 0"22 0"24 0-27 0"28 0;28 0.:27 0.27

B

1"13 1.25 1"37 1,55 1"80 2.01 2.16 2.32

R

E2/EI

0"10 0"20 0"33 0"66 1'50 3-00 5,00 10.00

F

--0"24 -0"24 --0'25 --0"25 --0"26 --0.27 --0-28 --0.29

3(b).

F

--0"94 --0"85 --0"85 --0"84 --0"83 --0"81 --0"79 --0"76

TABLE

1 "0

E2/E1

B

--0"17 --0"75 --0"71 --0"64 --0"51 --0'36 --0"26 --0"16 F

--0"24 --0"92 --0-91 --0-87 --0"80 --0"75 --0"71 --0"68 R

1"58 1-55 1-64 1.80 2-04 2"28 2"47 2"71 B

--0-30 --0.25 --0"33 --0"48 --0"68 --0"85 --0"98 --1"15

3-0

t/a

F

--0-45 --0"34 --0"46 --0"67 --1"02 --1"40 --1"72 --2"20 R

1"78 1"64 1-71 1"83 2"01 2-18 2"30 2"44 B

--0"46 --0"36 --0"41 --0.51 --0"65 --0-76 --0"84 --0"94

4"0

F

--0"68 --0"53 --0"60 --0-74 --0-94 --1-14 --1"29 --1"47

R

1"12 1-23 1"34 t'53 1"83 2"13 2.36 2-70 B

0"12 0-16 0"18 0-23 0,33 0,45 0.54 0'65

2"0

F

--0"28 --0"28 -0-27 --0-26 --0-25 --0.26 --0"28 -O.31 R

1"23 1"31 1-40 t'56 1"80 2.06 2:26 2.50

B

0"23 0"22 0"22 0"24 0.32 0"45 0,56 0"71

3"0

t/a

F

0"04 0-00 -0"05 --0"16 -0-38 --0.71 1-03 ---1.50

R

1-34 1"40 1.46 1"58 1-77 1.95 2.07 2-21

B

0"29 0"28 0"27 0-27 0"29 0.33 0-37 0,42

4"0

F

0-01 0-04 --0-09 ....0"18 --0,35 -0.53 --0.68 -0"89

R

1"52 1"55 1"58 1'63 1.72 1-80 1,85 1;90

R

1-84 1'80 1"82 1"88 1.96 2-04 2"09 2"14

FOR SQUARE; UNIAXIAL STRESS FIELD UNDER GRAVITY L O A D I N G - THREE-LAYER ROCK MASS

STRESS CONCENTRATIONS FOR SQUARE; BIAXIAL STRESS FIELD UNDER GRAVITY L O A D I N G - THREE-LAYER ROCK MASS

R

1-46 1-48 1"59 1"78 2"05 2"30 2"48 2"73

2-0

TABLE 3(a). STRESsCONCENTRATIONS

B

0"30 0,29 0"28 0"28 0"27 0.27 0.27 0-27

6"0

B

--0.52 --0"48 --0"50 --0-55 --0"61 --0"66 --0"70 --0"73

6"0

F

--0"13 -0"15 -0.17 --0"22 0-29 ....0-36 0-40 0'45

F

--0-77 --0"71 --0-74 --0"80 --0"88 --0"95 --1"00 --1"05

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136

G. BARLA

ratio equal to 1. The stresses at the rib (~'0g, the back (rt)8, and the floor (zt)~ - -all normahzc to the vertical stress ro -- pgh are plotted in (a), (b) and (c). Reported on the horizontal axis and in logarithmic scale are the corresponding values of E2/EI. Each curve refers to a different value of t/a, i.e. to different thicknesses of the middle layer. Numerical results for other opening shapes and the two stress fields considered can be found in the tables, and from which detailed observations should also be derived in each case. Figure 5 shows in (a), where for clarity only the curves for t/a ~ 1, 3, and 6 have been plotted, that a reduction in the modulus of elasticity of the middle layer produces a decrease in the rib stress, when this stress is compared with the corresponding value for the homogeneous structure. This value may be obtained from the same figure by setting E2/Et -- 1. In addition, it should be observed that this is also an indirect check of the numerical values reported as the rib stress for Ez/E~ -- 1 gives approximately the same value obtained from the homogeneous solution. Conversely, the effect of increasing the modulus of elasticity of the middle layer is that of increasing the rib stresses. Figure 5 illustrates in (b) the curves for the back stress. These curves show for t/a =. 1, 2 a behavior pattern which differs from that for t/a -- 3, 4 and 6. For Ez/EI < 1, the back stress increases for t/a -- 1, 2 and decreases for t/a -- 3, 4 and 6. For E2/E1 > 1, the back stress decreases for t/a -- 1, 2 and increases for t/a -- 3, 4 and 6. The change in behavior which occurs for t/a -- 3 is explained as it is for this value that the back of the opening is located either in the 'hard' or in the 'soft' medium depending on the middle layer being, for t/a > 2, either 'hard' or 'soft'. For t/a increasing, the value of the back stress tends toward that corresponding to the homogeneous solution. Larger deviations always occur, for a constant value of E2/EI and a given t/a, for E2/E~ > 1 than for E2/E~ < I. Figure 5 illustrates in (c) the behavior of the floor stress. Observations similar to those for the back stress can also be made in this case. In addition, it should be observed that. due to the difference in the geometry of the back and floor contours for both the arched opening and the square, the values of this stress are higher than that of the back stress. Also, the deviation with respect to the corresponding homogeneous solution is. for t/a ~-~ 1, 6 and any value of E2/E~, smaller than that for the back stress. Next, consideration is given to the distribution of the two principal stresses around the openings. Except for some obvious differences in the near vicinity of the opening contour, due to the known difference in the boundary stresses, all the openings with aspect ratio equal to 1 show a similar trend of behavior. This has to be so, as the same observation was already made in a previous paper [5], where detailed plottings were given for each opening shape. In fact, it is clear that the problem there considered can be viewed, at least above the opening, as a limiting case of the present one, for which E2/EL -- O. The traction-free surface would take the place of the interface between the middle layer and the confining homogeneous medium. Therefore, only the results for a typical opening with aspect ratio equal to 1 are presented in diagrams. As the arched opening possesses the geometrical characteristics of both the circle and the square, the numerical results for this case are given in detail. Also, the following discussion refers only to the highest contrast between the mechanical properties of the middle layer and of the confining homogeneous medium, i.e. E2/E~ --- 10.0, 0.10. In Figs 6-9 the distribution of the principal stresses for the arched opening is plotted for both stress fields. In each figure, (a) refers to the biaxial and (b) to the uniaxial stress field. The following observations are limited to the stresses away from the opening contour. In Fig. 6 the m a x i m u m principal stress has been plotted for E2/E1 -- 10.0. It is observed that the stress above the opening increases almost linearly for t/a -- 3. 4. It reaches a

,9 gh

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1,0

2.0~

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T_L, pgh

(E2/Ez = 10"0).

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FIG. 6. M a x i m u m principal stress for arched o p e m n g in a three-layer rock mass

STRESS

Comlxessio~ 1,0

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2

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FIG. 9. M i n i m u m principal stress for arched o p e n i n g in a three-layer rock m a s s

STRESS

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DISTRIBUTION OF STRESS AROUND A SINGLE UNDERGROUND OPENING

141

maximum at the interface between the 'hard' middle layer and the 'soft' medium. Then, on crossing this interface and in both cases, a jump in the value of the stress takes place, which explains the change in the mechanical properties between one medium and the other one. The highest change in the stress at the interface occurs for t/a = 3 and in the biaxial stress field. For t/a > 4, this same change in the stress decreases to practically disappear when t/a = 6. The trend of behavior is similar in both (a) and (b) with a slightly lower value in the stress at the interface, under uniaxial stress field. It is interesting at this point to observe also the similarity, even in the values of the stresses, between this behavior and that encountered previously [5] in the vicinity of the traction-free surface and for h]a = 1 • 5, 2. This observation let one infer that an increase in Ez/E~ over 10.0 would not result in a proportional increase of the stress at the interface. Rather, this stress is expected to reach an asymptotic value as Ez/E~ increases. The maximum principal stress below the opening shows a similar behavior pattern. However, in this case, the stress in the 'hard' layer and at the interface attains values higher than those above the opening. The maximum principal stress along the horizontal axis, i.e. in the 'hard' layer, is higher than the corresponding stress for the homogeneous structure. Figure 7 shows the minimum principal stress for Ez/E1 = 10.0. As could be expected from the results for the boundary stresses, it is seen that the effect of the 'hard' layer is that of increasing the values of the tensile stress in the near vicinity of the opening. This phenomenon is clearly noticed at the floor, where the geometry of the arched opening is seen to play in this respect a major role. Additionally, it is observed that for t/a = 3, 4 and under uniaxial stress field, the tensile zone extends above and below the opening and involves a major portion of the layer. The minimum principal stress along the horizontal axis is affected by the presence of the 'hard' layer. The value of this stress increases with respect to that obtained for the corresponding homogeneous structure. As already observed for the maximum principal stress, also for the minimum principal stress, a jump in the values takes place at the interfaces. Figures 8 and 9 give the maximum and minimum principal stresses for the arched opening where Ez/EI = 0.10. The influence of the 'soft' layer is that of transferring the load from the near vicinity of the opening to the confining 'hard' homogeneous medium. The minimum principal stress is seen to decrease above and below the opening with respect to the corresponding stress obtained for the homogeneous structure. The zone in which tensile stresses occur in the back and floor regions is reduced in its extension. A jump in the values of the principal stresses still occurs at the interfaces between the 'soft' middle layer and the homogeneous medium. The two principal stresses along the horizontal axis are reduced in value due to the presence of the 'soft' layer. It should be observed that for either E2/E t = 10"0 o r E2/E 1 = 0" 10 the distribution of stress around the opening and in both stress fields approaches, as the ratio t/a increases, the distribution of stress obtained for the homogeneous structure. Additionally, similar features shown by the plotted diagrams for the distribution of stress would be revealed if the results for all values ofE2/E1 > 1 and E2/E~ < 1 were considered. (b) The ellipses and the rectangles The numerical results for the boundary stresses of the ellipses and the rectangles (for both openings the aspect ratio is equal to 0- 66 and 0.50) are given in Tables 4-7. For each opening shape, Table (a) again refers to the uniaxial and Table (b) to the biaxial stress field.

0.10 0.20 0.33 0.66 1.50 3.00 5.00 lO+IO

&I&

0.10 0.20 0.33 0.66 1.50 3.00 5.00 1om

&I.%

B

R

-0.74 -0.71 -0.69 -0.65 -0.54 -0.40 -0.32 -0.27

B

2.16 2.37 2.63 3.07 3.70 4.28 4.97 5.82 R

-0.90 -0.88 -0.85 -0.80 -0.73 -0.68 -0.64 --o-59

F

2.0

F

-0.58 -0.53 -0.49 -0.43 -0.35 -0.30 -0.13 -0.07

R

2.40 2.51 2.74 3.12 3.64 4.10 4.65 5.03

t/a

B

-0.20 -0.29 -0.43 -0.67 -1.05 -1.43 -1.69 -2.05

3.0

F

-0.16 -0.94 -0.89 -0.81 -0.72 -0.66 -1.54 -1.83

R

2.60 2.62 2.82 3.16 3.60 3.98 4.34 4.48 B

-0.35 -0.39 -0.51 -0.71 -1.01 -1.32 -1.43 -1.57

4.0

F

-0.29 -0.96 -0.91 -0.82 -0.72 -0.65 -1.16 -1.26

R

2.90 2.77 2.94 3.21 3.55 3.81 3.93 4.00

-0.26 -0.22 -0.19 -0.15 -0.12 -0.11 -0~11 -0.11

B

1.73 2.03 2.32 2.83 3.59 4-34 4.93 5.67

R

1.0 1~98 2.15 2.41 2.87 3.55 4.25 4.80 5.89

R

-0.16 -0.15 -0.14 -0.12 -0.11 -0.10 -0.10 -0.10

F

B

-0.16 -0.15 -0.13 -0.10 0.15 0.26 0.33 0.37

2.0

F

0.09 0.15 0.18 0.20 0.00 0.24 0.25 0.46

R

2.20 2.27 2.50 2.91 3.51 4.10 4-52 4.99

B

0.14 0.10 0.07 -0.06 -0.24 -0.48 -0.70 -.1*23

3.0

tla

2.40 2.38 2.58 2.94 3.47 3.97 4.30 4.37

R

0.13 ---0.14 -0.13 --09 12 -0.11 -O*ll -0.11 m-o.79 F

B

0.20 0.15 0.10 --0.06 -0.24 -0.46 --0.63 -0.87

4.0 2.70 2.54 2.70 2.99 3.42 3.83 4”lO 4.15

R

0.29 -0.14 -0.13 --0.12 -0.11 -0.11 --0.11 -0.69 F

TABLE 4(b). STRESSCONCENTRATIONS FORELLIPSE(b/a = 0.66); BIAXIALSTRESS FIELDUNDER GRAVITYLOADINGTHREE-LAYERROCK MASS

-1.05 -1.01 -0.97 -0.90 -0.81 -0.73 -0.68 -0.62

1.87 2.19 2.50 3.02 3.74 4.37 4.80 5.24

1.0 6.0

B

0.10 0.09 0.06 ---o-o9 ---0.20 -0.35 --0.47 --0.48

6.0

B

-0.60 -0.55 -0.63 -0.76 -0.95 -1.14 -1.12 -1.15

TABLE 4(a). STRESSCONCENTRATIONS FORELLIPSE(b/a = O-66); UNIAMAL STRESS FIELDUNDER GRAVITYLOALXNGTHREE-LAYERROCK MASS

0.23 -0.16 -0.14 ---0.12 -0.11 ---0.10 ---0. 10 -0.10

F

-0.51 -0.99 -0.93 -0.83 -0.71 -0.63 -0.87 -0.85

P

0

--0-94 --0"91 --0"88 --0"83 --0-75 --0-69 --0"64 --0"58

B

2"10 2"40 2"74 3'31 4'15 4"88 5"39 5"97

R

0"10 0"20 0'33 0"66 1"50 3"00 5"00 10"00

1"0

--0-32 --0"29 --0"27 --0"24 --0-20 --0-17 --0"16 --0"15

B

1"93 2'27 2"60 3"18 4"05 4"90 5'54 6'35

R

E2/E1

0"10

0"20 0"33 0"66 1"50 3"00 5"00 10"00

F

I'00 --0"98 --0"96 --0"92 --0"88 --0"78 0"73 --0"67 R

2.33 2"58 2.85 3"35 4-13 4"95 5"64 6"70 B

--0"85 --0"84 0"84 --0.81 --0"75 --0"63 --0"50 --0"26

2"0

F

F

--0-25 --0"25 --0"24 --0"22 --0"19 --0"17 --0"16 --0"15 --0"24 --0-17 --0"06

3"21 4"04 4"94 5"70 6"86 R

--0-28

2-44

2"71

B

0"20

0"06

--0-32 --0"30

2"18

2"0

0"00 0"14 0"34

--0"13

--0"20

--0"23

--0"25 --0"24

F

R

2"58 2"78 3"00 3"41 4"07 4"78 5"36 5"88 B

--0"14 --0"25 --0"38 --0"62 --0"99 --1"38 --1-73 --2"14

3'0

t/a

F

--0"16 --0"28 --0"42 --0"68 --1"10 --1"54 --1-92 --2"43 R

2"79 2"95 3"13 3"46 4"01 4"59 5"04 5"31 B

--0"23 --0"33 --0"44 --0"65 --0"96 --1"29 --1"57 --1"77

4-0

F

--0"27 --0"38 --0'49 --0"72 --1"07 --1"44 --1"75 --2"08 R

3"14 3"24 3"35 3"56 3"90 4"26 4"51 4"66

R

2-42 2"61 2"83 3"26 3"99 4"78 5"41 5"94 B

--0"62 --0"95 --1"31

--0"33

0-04 0"01 --0"04 --0-14

3"0

t/a

F

--0"64 --0"98 --1"35

--0"32

--0'12

--0"02

0'07 0'03

R

4-58 5"06 5"34

3"93

2"62 2"78 2"96 3"32

B

0'12 0"08 0"03 --0"11 --0"36 --0"67 --0"94 --1"08

4-0

F

--0'09 --0"35 --0"70 --1"01 --1"20

0"06

0'15 0"11

R

2"97 3"07 3"18 3-41 3"81 4"20 4"46 4"60

(b/a = O. 50); BIAXIAL STRESS FIELD UNDER GRAVITY L O A D I N G - THREE-LAYER ROCK MASS

-0.88 --0"88 --0"89 --0.88 --0"82 --0"70 --0-54 --0"27

TABLE 5(b). STRESS CONCENTRATIONS FOR ELLIPSE

1 "0

E2/E1 6"0

B

0"06 0'03 0'02 --0"14 --0"31 --0"47 --0"59 --0"61

6"0

B

0"45 --0"52 --0"59 --0"71 --0"89 --1"06 --1"18 --1"24

TABLE 5 ( a ) . STRESS CONCENTRATIONS FOR ELLIPSE (b/a ~ O" 5 0 ) ; UNIAXIAL STRESS FIELD UNDER GRAVITY L O A D I N G - THREE-LAYER ROCK MASS

F

0"08 0"05 0"03 --0"12 --0"30 --0"50 --0"64 --0'69

F

--0"53 --0"59 --0"66 --0"79 --0"98 --1"18 --1-32 --1-45

z

z Z

©

Z

©

7~ C~

:Z

Z

t2

0

,.q

Z ©

C ~d

--0-76 --0"74 --0,72 --0"70 --0"68 --0"65 --0"63 --0"60

B

1"55 1"73 1"88 2"06 2"19 2"22 2"19 2"12

R

0"10 0"20 0"33 0"66 1"50 3"00 5.00 10.00

F

--0"86 --0"87 --0"86 --0"86 --0"85 --0"83 --0"81 -0"78 R

1"67 1.72 1-81 2"00 2.29 2"54 2"72 2"95 B

--0"77 --0"77 --0"76 --0-72 --0-65 --0-56 --0"49 --0"41

2"0

F

--0"96 --0"94 --0"91 --0"88 --0-83 --0"77 --0"74 --0"70 R

1"74 1"77 1.84 2"01 2"30 2"62 2"89 3-29 B

--0-13 --0"23 --0.31 --0"54 --0"86 --1.23 --1 "58 --2"21

3"0

t/a

F

--0"16 --0"28 --0"41 --0"66 --1"08 --1.58 --2-08 --2"97 R

1"83 1"85 1"90 2-03 2-27 2"53 2"75 3-02 B

--0"25 --0-32 --0"41 --0"57 --0"82 --1"10 --1-34 --1"68

4"0

F

--0"31 --0"40 --0"50 --0-70 --1"03 --1"40 --1"73 --2-19 R

1"99 1-99 2"01 2"08 2"21 2"36 2"46 2"58

B

--0"48 --0"51 --0"55 --0"63 --0-75 --0"87 --0-95 --1"04

6"0

1 "0

--0"26 --0"22 0"22 --0"21 --0,21 --0'20 --0'21 "0"22

B

1"25 1"42 1"54 1"76 2"04 2"29 2"45 2"59

R

E2/EI

0-10 0-20 0"33 0"66 1"50 3"00 5"00 10;00

F

--0-30 --0"32 --0"32 --0"33 --0-34 --0'35 --0-35 --0"36 R

1.39 1"43 1.52 1"73 2"08 2"44 2"70 3"05 B

--0"23 --0-24 --0"25 --0"23 --0"19 --0"14 -0"12 -0"11

2"0

F

--0"37 --0"35 --0"34 --0-34 --0"33 --0"34 --0"35 --0-36 R

1 "49 1"52 1"57 1-75 2"08 2.44 2"74 3"15

B

0"05 0"01 --0'04 -0"13 --0"32 --0.60 --0"96 --1.63

t/a 3-0

F

0"01 -0"04 --0"09 -0"22 --0"49 -0-93 --1"44 --2-42

R

1"58 1 '60 1"63 1"77 2"05 2-35 2"58 2"85

B

0"10 0"05 --0"01 --0'12 --0"32 --0"58 .....0"82 --1"15

4"0

F

0"05 0.03 --0"07 --0"21 --0"49 --0,87 -1"22 --1"70

R

1"74 1"74 1"74 1"82 1"99 2"15 2"26 2"39

B

0"02 --0"04 --0"08 --0"16 --0"27 --0"37 .-0"45 -0"54

6"0

TABLE 6(b). STRESS CONCENTRATIONSFOR RECTANGLE(b/a = 0" 66); BIAXIALSTRESSFIELD UNDER GRAVITY LOADING-THREE-LAYER ROCK MASS

1'0

E2/Ex

TABLE 6(a). STRESS CONCENTRATIONSFOR RECTANGLE(b/a = 0" 66); UNIAXIALSTRESSFIELD UNDER GRAVITY LOADING-THREE-LAYER ROCK MASS

f

--0"12 --0"15 --0"18 --0"27 .....0"42 -0-57 0"69 -0"81

F

--0"60 --0-64 --0"69 --0"78 --0"93 --1"09 --1"20 .....1"32

> Im r" >

4>-

X

--0"76 --0"75 --0"74 --0'72 --0"69 --0"66 --0"63 --0"60

B

1"65 1.86 2'03 2'26 2"43 2"46 2'41 2'30

R

0"I0 0"20 0'33 0"66 1-50 3"00 5"00 10"00

F

--0"81 --0"82 --0"82 --0.82 --0"81 --0"79 --0"77 --0"74 R

1.59 1.75 1.91 2.18 2.56 2.88 3-11 3"40 B

--0.83 --0"80 --0'78 --0"74 --0-66 --0"59 --0"53 --0"47

2"0

F

--0"88 --0"87 --0"86 --0"83 --0"79 --0'75 --0"72 --0"68 R

1.69 1"81 1"94 2"19 2"57 2"99 3"35 3"90 B

--0"12 0"21 --0"32 --0"54 --0"89 --1"28 --1"68 --2"44 F

--0"13 --0.23 --0"36 --0"62 --1.05 --1"56 --2"08 --3"15 R

1.79 1.89 2.00 2.21 2.54 2.91 3.22 3"64 B

--0"19 --0-28 --0"38 --0"56 --0-86 --1.22 --1"56 --2"11

4"0

F

--0"20 --0-30 --0.42 --0"64 --1"02 --1"49 --1"94 --2"68 R

1-98 2"05 2"12 2.26 2"49 2"73 2"92 3"14

1"0

--0"02 --0"29 --0"28 --0'27 --0'26 --0"26 --0"26 --0"26

B

1.42 1'54 1'74 1.97 2"31 2"58 2'75 2"93

R

E2/E1

0"10 0"20 0-33 0"66 1"50 3"00 5"00 10'00

F

--0"02 --0"23 --0-33 -0"34 --0"35 --0"36 --0"36 --0.36 R

1-54 1-47 1.63 1.93 2.38 2.81 3.15 3.59 B

0"00 --0"33 --0"32 --0"29 --0"25 --0"22 --0"20 --0"19

2"0

F

0"01 --0"34 --0"34 --0-35 --0"35 --0"35 --0"36 --0"37 R

1"72 1"55 1"68 1"94 2.37 2"84 3-23 3"82 B

0"00 --0"05 --0"10 --0"19 --0"37 --0"66 --1-03 --1"85

/'/a 3-0

F

--0"05 --0.07 --0"12 --0'24 --0"49 --0"89 --1"40 --2"56

R

1"50 1.64 1.75 1"97 2,34 2.76 3"09 3.54

B

0'05 0'00 --0"05 --0'16 --0-40 --0.75 --1"10 --1.67

4"0

F

0.05 0"00 --0"07 --0-21 --0"53 --1.00 --l.48 --2"28

R

1"68 1"79 1" 86 2.02 2-28 2"55 2"75 2"96

B

0"00 --0-05 --0'10 --0"19 --0.35 --0.52 --0"64 --0'79

6"0

B

--0"38 0.44 --0"51 --0'62 --0.79 --0"95 --1"08 --1"23

6"0

TABLE 7(b). STRESS CONCENTRATIONSFOR RECTANGLE(b/a = O. 50); BIAXIALSTRESS FIELD UNDER GRAVITYLOADING-THREE-LAYER ROCK MASS

1 "0

E2/E1

t/a 3"0

TABLE 7(a). STRESS CONCENTRATIONSFOR RECTANGLE(b/a = O" 50); UNIAXIALSTRESS FIELD UNDER GRAVITY LOADING-THREE-LAYER ROCK MASS

F

--0-02 --0.09 --0"14 --0.25 --0'46 --0'71 --0"90 --1"15

F

--0-45 --0'51 --0"58 --0.72 --0"93 --1.16 --1.35 --1"56

Z

z

©

:Z

7~ ©

7~

r"

Z

),.

Z

©

>

© Z ©

-t

146

G. BARLA

I00 90 80 70

Openicg

Ez IIEI: I 0 0

--.---',---o ...... --A--~--

EtJiptic~l (b/a =0 50) Elliptical (b/a=O 6G) Rectangular { b / a : 0 5 0 } Rectangular (b/el=0.66)

--a----a--

E z IE(:O-IO

0

2

I

3

5

4

--*~--*---o~--~-. --i----~---~----~-

Elliptical (b/a:O.50} EHiptical (b/a=O,66) Rectaflg u tar ( b/¢1: O- 50) RectangulrJr( b / a = 0 . 6 6 )

6

f/a

FIG. 10. Rib stress for opening shapes with aspect ratio smaller than 1. Three-layer rock O"10; uniaxial stress field).

mass

(Ez/E,

=

10" O,

The results are reported for different values of t/a and E2/Ex. As for the openings with aspect ratio equal to 1, only the boundary stresses for E2/E~ -- O. 10, 10.0 and in uniaxial stress field are plotted for increasing values of t/a. These cases are deemed to be representative of the typical features revealed by a detailed analysis of the data in the tables. Figure 10 gives the rib stress, for all opening shapes with aspect ratio smaller than 1, as a function of t/a. Expectedly, the general behavior patterns observed in the solution for openings with aspect ratio equal to 1 are now shown. The influence of a 'hard' middle layer is that of increasing the rib stress. The value of this stress is, for a given E2/E~, function of the opening shape and of t/a. The highest compressive stress occurs for the ellipse with

-30

-25 ¸

-20 '~\~

Opening

~ ~\`

-I.5

[ ...........

"--:%

: ~..

Ez/EI =100[ . . . . . . .

EilipticaE (b/a : 0.50) E l l i p t i c a l ( b / a = O 66} Rectangular (b/a =066)

......

Elliptical (b/a = 0 50) Ell i ptical (b/a = 0 66)

Ez/Ei=O IO ~

0

I

2

~

4

=

g

5

_

[ _= ......

Rectangmar (b/a =0 50) Rectengular ( b/a =0 66)

6

t/a

Fro. 11. Back stress for opening shapes with aspect ratio smaller than 1. Three-layer rock mass 10.0, O. 10; uniaxial stress field).

(E2/Et =

DISTRIBUTION OF STRESS AROUND A SINGLE UNDERGROUND OPENING

2.5

147

''\

"\,\ Opening

-,5

:.~

i. -I0 I

o

E2/EI= 100 / - - ' - - ' - -

Ez/E I = 0 tO [ . . . . . .

Elliptical (bZa-O E Iliptica[ ( b / a - O Rectangular ( b/a Recta ngular [ b/a

~ - - *-- -~-"-'~-- a l /

- o ~1

Oq

°--~

Elliptical [b/a: 0'50] Elliptical (b/a =0 66) Rectangula r ( b/a =0.50) Rectangular (b/a=O 66]

"':,

[

"X'/ ~',

2

/~s~-~

3

4

5

-

50) 66] = 0,50) = 0 66)

6

t/a

FIG. 12. Floor stress for opening shapes with aspect ratio smaller than 1. Three-layer rock mass (E2/E~ 10"0, 0" 10; uniaxial stress field). aspect ratio equal to 0.50. It may be observed that the effect of the finite radii of curvature at the rib, which are characteristic of the ellipses, greatly influences the value of the maximum stress which occurs for t/a = 2. In comparison, the increase in the stress is, for the rectangular openings, quite small. The presence of a 'soft' middle layer is that of decreasing the rib stress. However, the range of variation is much smaller than that obtained for a 'hard' middle layer. For either a 'hard' or a 'soft' middle layer, the values of the appropriate stresses for the homogeneous structure are gradually reached as the ratio t/a increases. Figures 11 and 12 give the back and floor stresses for openings with aspect ratio smaller than I. Again, the same characteristic behavior obtained for openings with aspect ratio equal to 1 is revealed. However, one may now notice that the absolute values for the appropriate stresses are considerably higher than that obtained in the previous case. For E z / E 1 = 10.0, the back and floor stresses decrease from the values corresponding to the homogeneous structure, reach a minimum and suddenly increase to attain a maximum. Then, these stresses gradually decrease toward the same values. The highest back and floor stresses are obtained for the rectangular opening with aspect ratio equal to 0.50. Additionally, it should be noticed that for each value of t/a, the difference in the floor stress, between ellipses and rectangles with equal aspect ratio, is much greater than that in the back stress. This can be explained as due to the highly different geometrical features that rectangles and ellipses possess at the floor. For Ez/E~ = 0-10, the behavior for the back and floor stresses is reversed. However, the range of variation in the stresses is much smaller than that for E2/E~ = 10.0. A comparison of the results obtained in the two cases let one ascertain that the difference in the geometrical features of the opening contour becomes an important factor in determining the values of the back and floor stresses, particularly when the back and floor regions are located in the hard medium (t/a > 2). Figure 13 gives the rib, back and floor stresses for a typical opening with aspect ratio smaller than 1, the rectangle with b/a = O. 50. The numerical results refer to the uniaxial stress field and are plotted in the same way as Fig. 5. Again, it is intended to illustrate, for a typical

148

G. BARLA

/ ( rt )R

', fl i r

~gh

~/,,/o

./y.J

3-0

!

f

z

2-0

f'0

Io,

I

I0 E~/E I

E2/E (

( a )

R*b

(¢

$l,reol

Flo4f

Stress

/

! reJ e Pll

.

Q

.......:

-I.0

~:

..~,N-N~NN~,-~,,N-~NN\,N~,,-~__~

~'~'~" "~

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~ o ~ I

I0

~ e ~ * ~

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Floor 80Ok

E2/E I LEGENO (b)

OoCk

Stress

FIe. 13. Stresses at (a) rib, (b) back and (¢) floor for a typical opening shape with aspect ratio smaller than 1 (rectangular opening with b/a = 0.50; ~ a l stress field).

opening with aspect ratio smaller than 1, the influence, on the boundary stresses, of the contrast in mechanical properties between the middle layer and the confining homogeneous medium. A detailed analysis similar to that previously carried on for openings with aspect ratio equal to 1 is now possible, appropriate modifications being necessarily introduced because of the differences in the absolute value of the appropriate stresses. The distribution of the principal stresses around the openings with aspect ratio smaller than 1 and away from the near vicinity of the contour is essentially a function of the aspect ratio. Some observations on the general behavior patterns of these stresses are made by plotting them at least in one case, the rectangle with aspect ratio equal to 0.50. Figures 14

P~

(a)

-I,0

STRESS

0 ]

4

'~i

r/.

~.0

-

FIELD

l~i~n

(

~-0

~

~/~rrlm

X~\, [ /

0

~-o

2,0

3:0

4

...........

te~io~ )

2

..........

L EOENO

7

1

6

..........

5

rnmmmmru~nmm~,3

F

~~ ¢ - ~

(b)

......

~_z~ _ _ i

UNIAXIAL

-bo

I

i.o

5

S

7

6

3

{~ ",

STRESS

+o

o

/iill!,

~,~

;

7 rlo

I i.o

I~

i

.....

FIELD

-

2.0

4-0

2.o

F~G. 14. Maximum principal stress for a rectangular opening (b/a = O" 50) in a three-layer rock mass (Ez/Ez = I0.0).

81AXIAL

Com~r Illh:I~

.

j

6

7

8

.~

II

Z

©

0

t~

z

0

>.

0

z

0

C

&e

(o)

'BIAXIAL

Tr - Dgh -

"'

STRESS

2,0

FIELD

-f'O

i-o

~r

i

?

5

4

'

o

I

0,0

2

I

I'0

- i.o

~"

I

[ 2

- 2.0

3

"4-

"C.\

Comp~essio~ 2.0 ---+--- . . . . .

Tension

i',/

I 4

l

7

...........

- 3"0

(b/a =

IS

4

3

2

. . . . . . . .

..........

LEGEND

rlo

I

6

..........

pgh

5

-----4:-

F

FIG: 15. M i n i m u m principal stress for a r e c t a n g u l a r o p e n i n g

Tension

4-. 2-0

"',

o.fi

I.O

1"5

4

r/a

JOgh

'__L.,

B

rz

- 2,0

Tensio~

Gompres~on

-I,0

~ 1,0

6

5

4

3

-~O,

0 l

o

L

10.0).

UNIAX A L

2~0

-2,0

T. pgh

STRESS

[ C*mp,.~*. ~o

-I.o

I-0

l'5

(E2/E1 :=

( b)

0" 50) in a three-layer rock m a s s

pgh

\

r/o

I

6

El

F!ELO

r'w

(o)

STRESS

3

FIELD

t'O

-vO j Tm~m

bo

4

2.0

( Curves

~.~..~

,eg,~

4 6

.......

LEGEND

3

6

5

---+---7 r/e

.........

J

I-

~ ] i ~ m l l t l m l m f f ,

.oI

5

6

7

8 ?la

F

I,: 2~/) . / . . .

}

tla

r.2_~ Ogh

"zo~ ., ~

foil il

/

(b)

fhi%

T41niim

]

1*10

UNtAXIAL

COlllpre$1aon

[ ,

7

6

5

4

3

?

I

STRESS

0

+

I

Z

3

4

r/o

6

2

~

p9h

\

Tonston

FIELD

i-O

-I'@

0-0 ~ _ q

r

i.o T

l

r)

4 "0 *-

2.0

3

J

~

4

Flo, 16, M a x i m u m principal stress for a rectangular o p e n i n g (b/a-= 0"50) in a three-layer rock mass (E2/E~ = 0' 10).

81AXIAL

C©mp?~s~on

t

t

r~ pgh

B

7,

5

I

6

t/a

t

?

rj )

0 m Z

0

0

Z 0

>

Z

0

>

r./) rJ )

tll

,.4

,11

0

ze

-]

-1

r=

210

{O)

5

~

i

I "0

- 1,0

"~

--

Gomprels~o~

2

STRESS F.IELD

I'~:'

O.O

0.5

I.o 1

3

__ 4

,Ogh

5

-6 r/o

LEGEND

7

6

4

3

2

1

T=

l/a

~gh

l

Tens~

(b)

Co~rl$1io~

.......

6

5

+ 0

UN~AXtAL STRESS

l

- ~-0

hO

rle

1.5

T,

Te~sp~

FIELD

hO

-l,O

k_ O"10).

2-0

i"°[

7

a

FIG. 17. Minimum principal stress for a rectangular opening (b/a = O"50) in a threeAaycr rock mass ( E z / E i

B|AXIAL

r/o

L_

0 , ~

÷

i

- I •0

O

I.O

rfa

~;~__

4

5

6

7

#gh

B

r~

>

tJ

DISTRIBUTION OF STRESS AROUND A SINGLE UNDERGROUND OPENING

153

and 15 give the principal stresses for E2/E1 = 10.0 and Figs 16 and 17 for Ez/EI = 0.10. Again, in each figure (a) refers to the biaxial and (b) to the uniaxial stress field. In the presence of a 'hard' middle layer, the maximum principal stress along the horizontal axis is seen to increase when compared with the values obtained for the homogeneous structure. This observation relates to the fact already revealed through the analysis of the boundary stresses, i.e. a transfer of load takes place from the 'soft' to the 'hard' medium. The trend of behavior for the same stress is reversed when the middle layer is 'soft'. However, it should be observed that the range of variation of the stress is in the first case much higher than in the latter. The minimum principal stress along the same horizontal axis increases, in the vicinity of the opening and for t/a > 2, when the middle layer is 'hard'. This stress decreases when the same layer is 'soft'. Some of the most interesting features of the solution are revealed, when consideration is given to the principal stresses above and below the opening. For a 'hard' middle layer and in both stress fields, the maximum principal stress above and below the opening is seen to undergo a sudden increase, the highest value being reached at the layer interface, where a jump in the stress takes place. The highest compressive stress is always located in the middle layer and occurs when t/a = 4, and for the biaxial stress field. The minimum principal stress, above and below the opening and for t/a > 2, is seen to attain tensile values in the vicinity of the opening. These values are much higher than those expected in the corresponding solution for the homogeneous structure. It should be observed that the zone of tensile stresses above and below the opening extends to a major portion of the layer when t/a = 3. In the presence of a 'soft' middle layer and in either stress fields the maximum principal stress above and below the opening is not greatly affected by the layer interface. In biaxial stress field and t/a > 2, the minimum principal stress in the near vicinity of the opening is seen to decrease considerably. In uniaxial stress field the same observation holds true. However, in this case, the presence of the interface has the effect of inducing, in the 'hard' confining homogeneous medium, a tensile stress. Consequently, the tensile zone extends considerably above and below the opening. The distribution of stress in the near vicinity of the opening shows a clear tendency, as t/a increases, to become equal to that obtained in the corresponding homogeneous structure. This fact is ascertained in the presence of either a 'hard' or a 'soft' middle layer.

CONCLUDING REMARKS

In this paper, consideration has been given to the determination of the distribution of stress around singular openings of several geometrical shapes and located in a horizontally layered rock mass. The numerical results here reported refer to a three-layer medium with a middle layer, which is either 'harder' or 'softer' than the confining homogeneous medium. On the basis of the results previously reported in [5], gravitational loading has been introduced in order to approach more closely the conditions expected in nature. The presence of a 'hard' middle layer greatly influences the stress distribution around the opening. In this case, the rib stress is increased considerably when compared with the values expected in the corresponding homogeneous structure. The highest values for this stress occur in uniaxial stress field, for the elliptical opening with aspect ratio equal to 0.50, and for the highest contrast in the mechanical properties. A comparison of the numerical values for this stress, between ellipses and rectangles with equal aspect ratio, reveals the major

154

G. BARLA

role of the geometrical features of the contour where the opening is located in the 'hard' middle layer. The presence of corners causes a localized concentration of compressive stress which is much higher than that derived from the corresponding solution for a homogeneous structure. For openings with aspect ratio equal to 1, the highest rib stresses occur for the circle, in uniaxial stress field, and again for the highest contrast in the mechanical properties. The interfaces between the 'hard' middle layer and the confining 'soft' homogeneous medium are seen to greatly affect the stresses in the back and floor regions, when they are in the near vicinity of the opening. In uniaxial stress field, the back and floor stresses (tensile) are increased with higher values pertaining to: (a) for openings with aspect ratio equal to 1~ the floor of the arched and the square openings; (b) for openings with aspect ratio smaller than 1, the rectangular openings with smaller aspect ratio. In biaxial stress field, the same phenomenon takes place for all openings with aspect ratio smaller than 1 and for the square opening. However, the back and floor stresses for the circle and the back stress for the arched opening becomes compressive. The floor stress for the arched opening is tensile and increases. Therefore, it is concluded that the presence of a flat roof causes, when the interface is located in the near vicinity of it, and the middle layer is a 'hard' one, a concentration of high tensile stress which is expected to endanger the general stability of the opening. A 'soft' middle layer allows a relief of the stress concentration around the opening. A transfer of the stresses from the near vicinity of the contour to the 'hard' confining homogeneous medium takes place. This phenomenon is of particular interest in the uniaxiat stress field and for all the openings. The tensile zone is greatly extended in the regions above and below the opening. The importance of this need not be emphasized in practice. It certainly relates to the tensile-type failures which are observed in the roof of some underground openings located in layered deposits. The methods of solution investigated may provide some insights into the problem of determining the distribution of stress around openings located in a rock mass. Some of the most important factors have been included in the analysis: geometrical features of the opening, gravity loading and presence of horizontal layers with different mechanical properties. However, common to the foregoing approaches is the fundamental assumption of linearly elastic behavior for the rock material. REFERENCES

1. GOODMANR. E. On the distribution of stress around circular tunnels in non-homogeneous rocks. Proceedings of the First Congress of the International Society of Rock Mechanics, Lisbon, Vol. 1, pp. 249255 (1966). 2. A6ARWALR. K. and BosIaKOVS. H. Stresses and displacements around a circular tunnel in a three-layer medium. Int. J. Rock Mech. Min. Sci. 6, 519-540 (1969). 3. BARLAG. The Distribution of Stress around Underground Openings--Effects of Some Geologic and Mechanical Features of the Rock Mass, Proe. il Primo Convegno lnternazionale sui Problemi Tecnici nella Costruzione di Gallerie, Torino (1969). 4. KAWAMOTOT. On the Calculation of the Orthotropic Elastic Properties from the State of Deformation around a Circular Hole Subjected to Internal Pressure in Orthotropic Elastic Medium, Proceedings of the First Congress of the International Society of Rock Mechanics, Lisbon, Vol, 1, pp. 269-272 (t966). 5. BARLAG. This issue, p. 103.