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Pillar design for mines in saltrock—2
125
Mining Science and Technology, 9 (1989) 125-135 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Pillar design for mines in saltrock-2 DOUGLAS F. HAMBLEY Department of Earth Sciences, University of Waterloo, Waterloo, Ont. N2L 3G1 (Canada) (Received January 11, 1989; accepted March 15, 1989)
Abstract Hambley, D.F., 1989. Pillar design for mines in saltrock--2. Min. Sci. Technol., 9: 125-135. A method for designing pillars based on the stress distribution at steady-state is developed. The concept of an equivalent elliptical opening to account for the zone of brittle failure immediately surrounding an opening is examined and a more precise formulation is presented. Formulae are developed for the stress distributions at the midpoints of the roof and walls of openings in viscoelastic and viscoplastic zones using a Von Mises yield criterion and a circular approximation to the effective ellipse. Examples are provided that demonstrate the use of the method.
Introduction
The accompanying paper demonstrates that existing pillar design methods for saltrock are, for a variety of reasons, unsatisfactory. Rational pillar design can be performed with the aid of numerical models. However, although computer-based numerical analyses are the most exact means of determining the stresses and deformations, viscoelastic-viscoplastic computer codes can require large computer memory and considerable computer time for realistic analyses, and thus may not be cost effective for routine mine planning unless performed on a dedicated computer. Hence, it would be valuable to have available design methods that can be carried out on hand-held calculators or personal computers. This paper presents a method that meets this requirement. It should be noted that strain rate and cumulative closure are also important con0167-9031/89/$03.50
siderations in designing pillars in saltrock. However, prediction of these parameters is beyond the scope of this contribution, as is consideration of the behavior in the transient stage of creep. This paper is restricted to steady-state stress conditions which can be derived using elastic-plastic calculation methods as the stresses in steady-state are assumed to be constant. As noted in the accompanying paper, saltrock in pillars deforms in a ductile manner. However, as a result of the high stresses generated by the excavation process, there will be a brittle failure zone immediately surrounding the opening. Before we can evaluate the stress distribution outside this zone, we require a means of delineating the extent of the brittle failure. Mraz [1,2] has shown that this zone can be approximately represented by an effective equivalent ellipse. This assumption will be followed in this paper; however, some refinements of his method are made.
© 1989 Elsevier Science Publishers B.V.
126
Representation of the failed zone by an equivalent elliptical opening Preliminary considerations Before we can discuss the stress distributions around openings of the shapes found in salt and potash mines we must first consider the expected stress state prior to excavation. In saltrock, the primitive stress state tends to be lithostatic in all directions (hydrostatic stress state) due to the tendency of saltrock to flow under deviatoric stress--i.e., the salt will tend to flow until the deviatoric stress is eliminated. At lateral contacts between dome salt and surrounding rock, however, the horizontal stress must be continuous, and, therefore, a hydrostatic stress state would not occur at these locations. However, the hydrostatic state would obtain in the interior of the dome away from its boundaries. Thus, in a dome, there is a stress gradient in both the horizontal and vertical directions. (For similar reasons, a thermal gradient would also exist in both directions in a dome.) The assumption of a hydrostatic stress state in undisturbed saltrock has been confirmed at the WIPP site in New Mexico using hydraulic fracturing stress measurements [3]. (Actually, all that could be estimated was the least principal stress, which was found to be vertical. Because, however, the WIPP site is not in a zone of high horizontal stress, one can then conclude that the three stresses must be equal.) The shape of openings in salt and potash mines depends on the means of excavation used. Openings excavated by blasting or by roadheaders are square or rectangular in cross section (with rounded corners in the case of roadheaders). Openings excavated by full-face continuous miners are rectangular, in the sense that the width is greater than the height, and the roof and floor are flat; the walls, however, are semicircular or rounded. Assuming that instantaneous redistribution of stress occurs in the manner described by elastic theory, the
stress distributions for the two cases would be somewhat different. However, in both cases some brittle failure will occur in a skin surrounding the opening as a result of the large dynamic stresses and strain rates induced by excavation. As noted previously, Mraz [1,2] accounts for this failure by assuming that the limit of the failed zone can be represented by an ellipse circumscribed around the actual opening.
Shape of the effective equivalent ellipse The method used by Mraz [1,2] to specify the shape and dimensions of the equivalent ellipse is based on geometric considerations. The present author, however, decided to investigate shape by theoretical means by calculating the stress distribution around rectangular openings with rounded corners using the methods of Muskhelishvili [4]. The extent of the failed zone was then determined using a suitable three-dimensional failure criterion and typical saltrock failure properties augmented to account for the dynamic nature of the stress state at excavation. It was found that the length of the horizontal semiaxis of the effective ellipse could be expressed as a function of the W / H ratio using the following relationship:
d = W / 2 - 0.466 + 0 . 5 I S ( W / H )
(1)
where d = semiwidth of the effective ellipse, W / H = width-to-height ratio and W = width of the rectangular opening. The correlation coefficient was found to be 0.982. The semiheight of the effective ellipse can then be found from:
b2 = d 2 y 2 / ( d2 - x 2)
(2)
where b = semiheight of the effective ellipse, and d, x and y are the rectangular coordinates of the ellipse. The coordinates are those of the corner of the opening, allowing for a failed zone at that point (generally about 0.5 m thick but depen-
127 dent on the shape of the opening). Having defined the ellipse, the radii of curvature of the ellipse at the ends of the semiaxes can be found from [5]: b2/d
(3)
a v = d2/b
(4)
a h =
where a h = horizontal radius of curvature at the wall, a v = vertical radius of curvature at the midspan of the roof, and b and d as defined previously. It should be noted that eqn. (1) is based on an opening in an infinite body of saltrock. If a competent layer exists in the roof at a relatively short distance from the opening, it will probably be necessary to modify the dimensions of the effective ellipse. (This analysis assumed that the longer axis of the opening cross section was in the horizontal direction. For openings whose longer axis is in the vertical direction, the same procedure can be followed with the width and height transposed.)
sumed that away from the ends of the opening, the stress state is one of plane strain. It should be noted that because saltrock is viscoelastic, the Poisson's ratio near the radius of yield on the viscoelastic side of the boundary will approach 0.5. Moreover, the value in the viscoelastic zone will in general be greater than the value measured in compressive strength tests--the value changes during transient creep. These preliminaries aside, we can proceed with determining the stress state. It has been shown that the m i n i m u m principal stress beyond the failed zone can be approximated by the radial stress for a circular opening having as its radius the radius of curvature of the effective ellipse [6]. Thus, we can perform the stress analysis in polar coordinates. In addition, for a circular opening in a uniform stress field, the shear stresses are zero. Assuming that the Von Mises criterion holds, we have the following relationship for the principal stresses at yield [7]:
(0-1 Stress distribution around a single opening
Having defined the equivalent effective opening that can be used to account for the failed region, we can now proceed to determine the stress distribution around the opening after yield (more correctly, timedependent, low strain-rate failure) has occurred. Before proceeding, however, we should clarify some of the assumptions implicit in the stress analysis. First, the stress distribution at steady-state is assumed to be constant by definition. Thus, even though saltrock behaves as a viscoelastic-viscoplastic material, it can be taken to be an elastic-plastic one in steady-state. Second, the rocks surrounding the opening are assumed to be homogeneous. Thus, the effects of clay partings, anhydrite layers, and other inhomogeneities are ignored. Finally, it is as-
_
02
)2
+ (0"2--03) 2 q- (O3--0-1
)2
=0-)
(5)
where % = material constant. For laboratory creep testing using a conventional triaxial apparatus, the intermediate and m i n i m u m principal stresses are equal, and eqn. (5) reduces to: 0-1 - % = %
(6)
Equation 6 is readily recognized as a form of the Tresca criterion. Hence, for conventional laboratory triaxial testing, the Von Mises and Tresca criteria are equivalent. However, such is not the case for the general stress state underground. It should also be noted that eqn. (6) corresponds to the conventional M o h r - C o u l o m b criterion for an angle of internal friction of zero. Hence, the angle of internal &iction for long-term time-dependent strength is zero, and % in eqns. (5) and (6) is the time-dependent, low-strain-rate uniaxial compressive strength, and twice the maxim u m shear stress (k). The maximum shear
128 stress has been called the Prandtl limit by Mraz and Dusseault [8]. Proceeding with the analysis, upon substitution of the plane strain relationship for o2 (in terms of a I and 03) into eqn. (5), and after simplification, we obtain: (012 + o32)(1 - v - u 2)
--
+ (4k/vrS )( rf/r ) 2 ln( rr/a ) at = o0(1 + r2/r 2)
(15)
- (4k/vC3 )( rf/r ) 2 ln( rr/a ) (7)
In the yielded zone, v -- 0.5, and by substituting this value into eqn. (7) we obtain: -
or = 0 0 ( 1 - r ? / r 2 )
O103(1 q- 2v - 202)
=4k 2
0.7502
plate, we obtain for the major and minor principal stresses in the viscoelastic zone:
1.500103 q-0.7502 = 4k 2
(8)
where r r is the radius of the interface between the viscoelastic and viscoplastic zones. For plane strain, the intermediate principal stress (the longitudinal stress) in the viscoelastic zone is given by:
which upon simplification gives:
aI - a 3 = 4 k / f 3
o 1 = 2v% (9)
Letting a t = ol, and a r = a 3, and substituting into the equilibrium equation in polar coordinates, we obtain:
Oar/Or = 4 k / ( rv/3 )
(10)
A f t e r integration and insertion of the b o u n d a r y condition that or = 0 at r = a, we obtain: o r = ( 4 k / f 3 - ) ln(r/a)
(11)
where r = radius to the point at which the stress is calculated, a = radius of the curvature of the opening and k -- the Prandtl limit (shear stress at zero mean stress). The tangential stress in the yielded zone is found to b e : a t = (4k/v/3-)[1 + ln(r/a)]
(12)
(16)
Having determined the formulae for the stresses in both the viscoelastic and viscoplastic zones, we can now find the formula for the radius of the viscoelastic-viscoplastic interface. At the interface, the radial and tangential stresses on the viscoelastic side are: o r = ( 4 k / f 3 ) ln(rf/a) a t = 2o 0 - (4k/vFJ -) ln(rf/a)
(17)
As the radial and tangential stresses at the interface are continuous, upon substituting eqn. (17) into eqn. (7), simplifying, and rearranging, we find the radius of the viscoplastic-viscoelastic interface to be:
rf=a exp[(v/-3oo/ZK - 1)/2]
(18)
For plane strain, the intermediate principal stress in the yielded zone is given by:
Using eqn. (18), the depths of the interface at the midpoint of the roof and in the walls can be shown to be given by:
a , = o2 = (2k/v~-)[1 + 2 ln(r/a)]
mpv=av{eXp[(VCSOo/2K - 1 ) / 2 ] - 1}
(13)
where o t = longitudinal stress. Turning now to the viscoelastic zone, we know that for a circular opening in a uniform stress field: a t -1- O r =
200
(14)
Using this relationship and Airy's stress function for a circular hole in a semi-infinite
+ b - H/2
(19a)
mph=ah{exp[(v~Oo/2K-- 1)/21 -- 1} +d - W / 2
(19b)
mpv and mph a r e the depths of the viscoelastic-viscoplastic interface at the midpoint of the roof and the midheight of the
where
129 walls respectively and a v and a h are as given previously. These results complete the stress analysis for a single opening. However, as indicated previously, the radii of curvature and semiaxis values calculated thus far assume a semiinfinite saltrock body surrounding the opening. If such is not the case, mpv c a n be no greater than the distance to the nearest competent strata in the b a c k . The horizontal semiaxis must be reduced and the vertical semiaxis increased if the calculated mpv exceeds that distance. However, the point at the corner remains fixed. If we know the slope of the ellipse at that point, we can calculate the required ratio of the semiaxes from:
and (20) we obtain the required values for the semiaxes. The radii of curvature can then be calculated from eqns. (3) and (4). Finally, the depths of the viscoplastic-viscoelastic interfaces and the stresses can be calculated as before. Stress distribution in pillars between openings
q = - xZyA2.
F r o m a mining standpoint, the stress distribution in pillars is critical to stability of the workings, and hence worker safety. Thus, pillar design is of great importance--oversized pillars prevent valuable reserves from being mined, whereas undersized pillars result in local or regional instability. W h e n there are several openings in close proximity to each other, the apparent initial vertical stress in the pillars between the openings can be considered to increase to some value Crv'.If the pillars were completely elastic, and there were no failed zone around the opening, the value would be defined by tributary area, based on the pillar dimensions and the spacing between the openings if the rooms were in the middle of a semi-infinite mining panel. This scenario, however, applies only to relatively narrow pillars, and in saltrock could only be considered at shallow depths. A more realistic scenario would be one where there are failed and yielded zones around the rooms and where the increased stress occurs in the portion of the pillar between the limits of the failed zone surrounding each opening. The increased stress also results in enlarged yielded zones compared to those around single openings. In addition, it can be shown that at the center of the pillar:
mpv a = actual distance to nearest competent
% = ( v/(1-
strata.
where v = Poisson's ratio, ox = horizontal stress in the direction perpendicular to the axes of the openings and %' = vertical stress, as the horizontal strain in the middle of the pillar must be zero in that direction. If we assume that the rooms are driven simultaneously, as well as closure, we can
b2/d 2 = -y
tan O / x
(20)
where ¢? = slope of the curve and d, b, x and y are as defined previously. Using the definition of a v (eqn. (4)), we can rearrange eqn. (20) to: a v / b =- - x / ( y
tan q,)
(21)
Substituting eqn. (21) into eqn. (19a), solving for b, and equating the result to the equation for an ellipse in terms of b 2, we obtain the following relationship for the slope of the curve: m tan3q, + n tan2q5 + p tan q5 + q = 0
(22)
where: m=xy n
=
2.
(rFlpv a + H / 2 ) 2 y -- 2 x 2 y A _ y 3 .
p = x3A 2 + 2 x y 2 A .
A = {exp[(vr3oo/Zk - 1)/2]-
1}.
H, x, y, o0 and k are as defined previously. Solving eqn. (22) for tan e~, substituting the result in eqn. (20), and combining eqns. (2)
v))cr"
(23)
130 ignore the initial yielded zone around an isolated opening. While this assumption is not completely realistic, it simplifies the analysis, and the closure will probably not be sufficiently great to significantly affect the results, especially if the time lag between driving the openings is not long. Because of symmetry, we need consider only half the pillar. The stress distribution can be thought of as that due to a hole in a plate of finite size, for which the general formulae for the stress c o m p o n e n t s are given by T i m o s h e n k o and Goodier [9] and Southwell [10]. Solving these equations, we obtain:
X (b 4 + 4b2a 2 - 3 b n a 4 / r 4 +
The next task is to determine the radius of the viscoelastic-viscoplastic interface. At the m i d p o i n t of the roof of the opening, the formula remains the same as for a single opening (previous section). Substitution of eqn. (24) into eqn. (7) results in an eighth-order polynomial equation in r that m u s t be solved to obtain the radius to the interface. Such an equation cannot be solved directly unless the polynomial fortuitously happens to factor into a quadratic or quartic. However, the equation can be solved iteratively (by trial-and-error, or the m e t h o d of successive approximations) by taking, for an initial value, the radius to the pillar center. If this radius remains within the yielded zone, then the pillar should be enlarged. (Similarly, if secondary mining is planned, and the width of the viscoelastic zone is less than the anticipated secondary r o o m width, a larger pillar should be considered.) Once the radius of the viscoelastic-viscoplastic interface has been determined, the stresses in the viscoelastic and viscoplastic zones can be determined. The stresses in the viscoplastic zone are given by eqns (11)-(13). We can determine the stresses in the viscoelastic zone as before except that instead of a b o u n d a r y condition at r = a we have:
2bZa2(b2 + aZ)/r 2) sin 2t9
at
Ot= { 0:/[2(1 - v)(b 4 - a4)]} X [b2(1 + a2/r2)(b 2 Jr- a 2) -k-
(1 - 2 v ) ( b 4 + 4bia 2 +
3b4a4/r 4)
x cos 2 0 ] Or= { Ova/[2( ] -- 0 ) ( b 4 - a 4 ) ] }
a2/r2)(b 2 + a 2) - (1 - 2v) x (b 4 + 4b2a2-ab2a2(b2 + ae)/r 2 + X [b2(1 -
3b4a4/r 4) cos 2 0 ]
(24)'
'rrt = { o:(1 -- 2 U ) / [ 2 ( 1 - - v ) ( b 4 - a 4 ) ] }
where av' = effective vertical stress, b = radius from center of curvature to pillar center, a = radius of curvature of effective opening and v = Poisson's Ratio. Inspection of the above formulae indicates that it would be necessary to determine the principal stresses before we can evaluate the yield criterion at any point and hence find the radius of the viscoelastic-viscoplastic interface. However, we are interested in the stresses on the horizontal plane where the shear stress disappears. Thus, the formulae for ot and o, give us the principal stresses o I and 03 respectively. For a rib pillar, 02 is found using the assumption of plane strain.
r=rf, Or=4kln(rt/a)/v/3; "/'rt=0
Solving the equations, we obtain: or= -4k
ln(rf/a)(b 2 + r2)rf2/
[¢~(bZ-rZ)r2] + (o:/[2(1-v)(bZ-r2)]) × {b2(l+r2/r2)+[(1-2v)/ (b2 - Ff2)] × (b 4 - 262V + 2,'?rVb
bar~/r 4) cos 2 0 )
2 -
(25)
131 or = 4 k l n ( r r / a )(b 2 - r 2 ) r ~ / [ v ~ - ( b 2 - r ? ) r 2]
+ (o:/[2(1 - v)(b2- r )l ) × {b2(1 - rfa/r 2) - [(1 - 2 v ) /
we now present, in point form, a step-by-step method for selecting mine room-and-pillar dimensions. Rib pillars
(b 2 - rf2)](b 4 - 2b2r 2 + 2 r 4 6 2 / r 2 -
b4r;/r 4) cos 20) "/'rt
:
{O~ (1
--
2 v ) / [ 2 ( 1 - v)(b 2 - r 2)2]}
× (b 4 _ 2b2rf 2 + 4 r 2 / b 2 + rf4b2/r 2 b 4 4 / r 4) sin 2 0
(26)
Equation 26 completes the analysis for a rib pillar with viscoelastic and viscoplastic zones. However, there is still the question of rectangular and square pillars. For a square pillar, the intermediate stress at any point on a centerline is the radial stress at the center of the pillar. Hence, a single set of calculations using the formulae derived previously will give the stress distributions on both center lines if the cross-cuts have the same width as the rooms. Otherwise, it will be necessary to determine the effective opening and extraction ratio separately for the crosscuts. For a rectangular pillar, the extraction ratio will be different in the longer direction and hence the radial stress at the center along the second centerline must be calculated separately. However, the same formulae are used. If the cross-cuts have the same width as the rooms, it is simply a matter of determining Ov' for the different pillar thickness. If the crosscuts are a different width, it will be necessary to determine the effective opening before proceeding with the remaining calculations. The next, and final, step is to define a step-by-step calculation procedure. This method will be presented for each of the three cases--rib, rectangular and square pillars--in the next section.
Pillar design method Having presented the analytical formulae for determining pillar dimensions and stresses,
(1) Determine the shape of the equivalent effective opening assuming a semi-infinite cover of saltrock. (The effective opening is a circle if the opening has a square cross section.) (2) Determine the thickness of the yielded zone in the back and compare with the distance to the nearest overlying competent layer. If the actual distance is smaller, determine the modified effective opening geometry. (3) Calculate the stresses in the roof. (4) Select a pillar thickness and determine the thickness of the yielded zone, and hence the thickness of the viscoelastic core of the pillar. (5) Repeat step 4 until the pillar width is adequate. (6) Repeat the procedure for the barrier pillars (assuming no load is taken by the intervening yield pillars) in the case of panels whose pillars are designed to ultimately yield. Square pillars The method is the same as for rib pillars, except that the intermediate principal stress in steps 3 and 4 is the radial stress at the center of the pillar. If the cross-cuts do not have the same width as the rooms, it will be necessary to calculate the effective opening and radial stress in the second direction first. Rectangular pillars (1) Determine the shape of the effective opening for sections through both centerlines of the pillar (longitudinal and transverse). (2) Determine the thickness of the yielded zone in the back for both cases and compare with the distance to the nearest overlying competent stratum. If the latter distance is smaller, determine the modified geometries.
132 (3) Calculate the stresses in the roof. (4) Select a pillar thickness and determine the thickness of the yielded zone, and hence the thickness of the viscoelastic core of the pillar. This calculation is carried out for both directions. (5) Repeat step 4 until the pillar size is adequate. (6) Repeat the procedure for barrier pillars if any are provided.
Commentary on the method Computer programs were written in BASIC for a Macintosh Plus for the rib pillars with rectangular and square openings. (A 0.5 m thickness of broken material at the corners of the opening was assumed based on the dynamic failure analysis.) For square and rectangular pillars, initial trial configurations can be obtained by first performing the calculations for a rib pillar, and using the dimensions for that case as a first approximation. Use of the computer programs allows a large number of trial layouts t o be evaluated in a matter of minutes. For detailed evaluation of stresses on a mine-wide basis (or panel-wide basis), more sophisticated finite element or other numerical methods should be used.
Examples of pillar design using this method To illustrate the use of the method, we now provide three hypothetical examples. However, resemblance to existing or former operations, insofar as the depth of the workings and the size and shape of the rooms are concerned, is intentional in order to make the examples realistic and representative of actual operating conditions. Nevertheless, the values of the yield stress (Prandtl limit) and Poisson's ratio used are estimates and not based on actual mine property data. Hence, similarity
to exact pillar dimensions at mines resembling these examples is strictly fortuitous. Except where noted, Poisson's ratio has been taken as 0.4 in all example analyses.
A potash mine at intermediate depth This hypothetical mine employs rooms 3.05 m high and 14.63 m wide, excavated by drilling and blasting. The workings are found at a depth of 305 m and the nearest competent strata occur 152 m above the workings. The radius of curvature of the corners of the openings is 0.15 m and the thickness of the failed zone at the corners is assumed to be 0.5 m. The primitive stress and the yield stress (Prandtl limit) are estimated to be 6.9 and 5.2 MPa, respectively. The parameters of the effective opening were found to be as follows: b = 3.18 m. d = 9.33 m. a h = 1.08 m. a v = 27.39 m. mpv = 3.78 m. The mine uses square pillars. Analysis showed that the yielded zones in the pillars just merge if the pillar width were 21.93 m, giving an extraction ratio of 64%. Inspection of the pressure arch width data of Abel and Djahanguiri [11] showed that for a mine at this depth the distance would be no greater than 106 m. Based on the room-and-pillar dimensions, ~ this transfer distance would allow only three rooms per panel. Therefore, pillars with elastic cores were chosen rather than a yield-and-barrier pillar arrangement. Further analysis showed that by increasing the pillar thickness to 25 m the depth of the yielded zone in the pillars would be reduced to 4.36 m, giving an elastic core 17.3 m thick. This pillar size would seem sufficient to ensure stability and results in an extraction of 60%. N o t e that an increase of only 3 m in pillar thickness makes a considerable difference in the stability of the pillar.
133
A salt mine at intermediate depth This hypothetical mine employs rooms 7.62 m high and 15.24 m wide, excavated by drilling and blasting. The workings are at a depth of 345 m and the primitive stress was estimated to be 8.57 MPa. The yield stress (Prandtl limit) was assumed to be 10.4 MPa. (This relatively high value of the Prandtl limit results from the fact that the ore is almost pure halite.) The radius of curvature of the corners of the openings and the thickness of the failed zone in the corners were assumed to be 0.15 and 0.5 m, respectively. The nearest competent layer occurs 0.765 m above the workings so that a modified shape of the effective ellipse must be used. The parameters of the effective openings were found to be as follows: b -- 6.63 m. d - - 10.12 m. a h = 4.35 m. a v = 15.43 m. mpv = 0.77 m. This mine also uses square pillars. Analysis showed that the yielded zones in the pillars would just coalesce if the pillar width were 24.38 m, giving an extraction ratio of 62.1%. In this case also the workings are too shallow for a practical pressure arch to develop. Hence, pillars with elastic cores were chosen rather than yield pillars. Further analysis showed that increasing the pillar width to 26 m would result in an elastic core in the pillar 8.8 m thick and an extraction ratio of 60.3%. This shows that the equilibrium in correctly sized pillars is quite delicate and indiscriminant pillar robbing could easily disrupt it.
A deep potash mine This mine uses continuous miners to drive rooms 3.05 m high and 9.15 m wide. The walls of the openings are semicircular and
because of the geometry the thickness of the failed zones in the corners is assumed to be 0.25 m. The workings are at a depth of 1040 m and the assumed primitive stress is 23.45 MPa. The Prandtl limit varies according to the carnallite content of the ore, being as high as 9.5 M P a when the carnallite content is negligible, and as low as 5.2 MPa when the carnallite content is relatively high. The mine employs rib pillars and uses barrier pillars after every seven rooms. The distance to the nearest competent strata in the back is 18 m. To show the effect of the different yield stresses (Prandtl limit values) on the mine layout, analyses were performed for 5.2, 7.4 and 9.5 MPa. The effective opening parameters, the m i n i m u m pillar size, and the m a x i m u m extraction ratio for the three cases are given in Table 1. Another parameter that has a considerable effect on the required pillar size is Poisson's r a t i o - - t h e horizontal stress at the pillar center is v / ( 1 - v) times the effective vertical stress. In viscoelastic materials, Poisson's ratio is not a constant with time and furthermore will approach 0.5 as the limit of viscoelastic behavior is reached (i.e., as the material yields). Thus, in the weaker material in this mine, it is likely that the Poisson's ratio is greater than the 0.4 value hitherto assumed. Similarly, in the strongest material, Poisson's ratio may be less
TABLE 1 Effect of variation in yield stress on mine layout in a deep potash mine (k = MPa) k~5.2
k=7.4
k=9.5
b (m) d (m) a h (m) a v (m) mpv (m) Min. pillar width (m)
4.36 4.49 4.23 4.63 18.00
2.20 5.24 0.93 12.44 18.00
1.93 5.66 0.66 16.60 13.13
49.0
15.20
11.38
Max. extraction ratio (%)
15.7
37.6
44.5
134 TABLE 2 Effect of variation in Poisson's ratio on pillar geometry in a deep potash mine v=0.35 v=0.40 v=0.45 v=0.50 Min. pillar width (m) 18.10 Max. extraction ratio (%) 33.6
11.38 44.5
7.40 55.3
5.24 63.6
than 0.4. (In high strain-rate laboratory tests on saltrock, it is often about 0.35.) Thus, the effect on the pillar size of varying Poisson's ratio was examined using a constant Prandtl limit of 9.5 M P a . The results are given in Table 2. The conclusion to be drawn is that the higher Poisson's ratio to be expected in weaker material will partially compensate for the smaller yield stress so that the minimum pillar size is reduced and the m a x i m u m extraction ratio is increased. Thus, referring back to Table 1, mining the weaker material will b e more economical than the initial analysis led us to expect. Recalculating the case for k = 5.2 M P a using a Poisson's ratio of 0.45, we obtain a minimum pillar width of 23.0 m, corresponding to an extraction ratio of 28.5%. Even so, the extraction ratio is still sufficiently small that mining this lowstrength, low-grade (due to the carnallite) material could be questionable economically. At the beginning of this example, it was indicated that barrier pillars were to be left after every seven rooms. Hence, an approximation analysis was also carried out to estimate the required width of the barrier pillars. This analysis was approximate from two standpoints. First, it was assumed that all the area between the barrier pillars was mined out whereas in reality there would be yield pillars taking a small portion of the load; second, because of the very long axis of the assumed opening between the barrier pillars, the yielded zone in the back did not correspond to the distance to the first competent
layer b u t was considerably closer to the surface. It was found that the yielded zones in the pillars just merged for a pillar width of 94.25 m. However, extending the pillar width to 95 m gave a viscoelastic core of 41.8 m. (Both these cases were for k = 9.5 M P a and a Poisson's ratio of 0.4.) These barrier pillar widths should not, however, be accepted without verification b y numerical analysis (e.g., finite element analysis).
Summary This paper presents a method for the design of salt and potash mines. A calculation procedure was developed for determining the shape of the effective opening. Procedures for calculating the stresses around single openings and in pillars and the locations of the viscoelastic-viscoplastic interfaces, using the effective opening and the Von Mises yield criterion, were presented. Additionally, examples were presented of the use of the method for three hypothetical cases.
References 1 Mraz, D.Z., Plastic behaviour of salt rock utilized in designing a mining method. Can. Inst. Min. Bull., 73 (1980): 115-123. 2 Mraz, D.Z., Solutions to pillar design in plastically behaving rocks. Can. Inst. Min. Bull., 77 (1984): 55-62. 3 Wawersik, W.R. and Stone, C.M., Experience with hydraulic fracturing tests for stress measurements in the WIPP. In: H.L. Hartman (Editor), Rock Mechanics: Key to Energy Production. Soc. Min. Eng., Littleton, Colo. (1986), pp. 940-947. 4 Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Leiden (1977). 5 Tuma, J.J., Engineering Mathematics Handbook. McGraw-Hill, New York (1987), 3rd Ed. 6 Hambley, D.F., A methodology for designing pillars in salt and potash mines. Univ. Waterloo, Ont., Grad. Course ES 692 (1988), 90 pp.
135 7 Jaeger, J.C., Elasticity, Fracture and Flow. Chapman and Hall, London (1969), 3rd Ed. 8 Mraz, D.Z. and Dusseault, M.B., Long term behaviour of a fully-instrumented test room in potash. In: H.L. Hartman (Editor), Rock Mechanics: Key to Energy Production. Soc. Min. Eng., Littleton, Colo. (1986), pp. 405-414.
9 Timoshenko, S.P. and Goodier, J.N., Theory of Elasticity. McGraw-Hill, New York (1970), 3rd Ed. 10 Southwell, R.V., An Introduction to the Theory of Elasticity. Dover Publ., New York (1969). 11 Abel, J.F., Jr. and Djahanguiri, F., Mining inputs to salt nuclear waste repository design. In: U.S. Rock Mech. Symp., 25th (Evanston, Ill.) (1984), 22 pp.
Mining Science and Technology, 9 (1989) 125-135 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Pillar design for mines in saltrock-2 DOUGLAS F. HAMBLEY Department of Earth Sciences, University of Waterloo, Waterloo, Ont. N2L 3G1 (Canada) (Received January 11, 1989; accepted March 15, 1989)
Abstract Hambley, D.F., 1989. Pillar design for mines in saltrock--2. Min. Sci. Technol., 9: 125-135. A method for designing pillars based on the stress distribution at steady-state is developed. The concept of an equivalent elliptical opening to account for the zone of brittle failure immediately surrounding an opening is examined and a more precise formulation is presented. Formulae are developed for the stress distributions at the midpoints of the roof and walls of openings in viscoelastic and viscoplastic zones using a Von Mises yield criterion and a circular approximation to the effective ellipse. Examples are provided that demonstrate the use of the method.
Introduction
The accompanying paper demonstrates that existing pillar design methods for saltrock are, for a variety of reasons, unsatisfactory. Rational pillar design can be performed with the aid of numerical models. However, although computer-based numerical analyses are the most exact means of determining the stresses and deformations, viscoelastic-viscoplastic computer codes can require large computer memory and considerable computer time for realistic analyses, and thus may not be cost effective for routine mine planning unless performed on a dedicated computer. Hence, it would be valuable to have available design methods that can be carried out on hand-held calculators or personal computers. This paper presents a method that meets this requirement. It should be noted that strain rate and cumulative closure are also important con0167-9031/89/$03.50
siderations in designing pillars in saltrock. However, prediction of these parameters is beyond the scope of this contribution, as is consideration of the behavior in the transient stage of creep. This paper is restricted to steady-state stress conditions which can be derived using elastic-plastic calculation methods as the stresses in steady-state are assumed to be constant. As noted in the accompanying paper, saltrock in pillars deforms in a ductile manner. However, as a result of the high stresses generated by the excavation process, there will be a brittle failure zone immediately surrounding the opening. Before we can evaluate the stress distribution outside this zone, we require a means of delineating the extent of the brittle failure. Mraz [1,2] has shown that this zone can be approximately represented by an effective equivalent ellipse. This assumption will be followed in this paper; however, some refinements of his method are made.
© 1989 Elsevier Science Publishers B.V.
126
Representation of the failed zone by an equivalent elliptical opening Preliminary considerations Before we can discuss the stress distributions around openings of the shapes found in salt and potash mines we must first consider the expected stress state prior to excavation. In saltrock, the primitive stress state tends to be lithostatic in all directions (hydrostatic stress state) due to the tendency of saltrock to flow under deviatoric stress--i.e., the salt will tend to flow until the deviatoric stress is eliminated. At lateral contacts between dome salt and surrounding rock, however, the horizontal stress must be continuous, and, therefore, a hydrostatic stress state would not occur at these locations. However, the hydrostatic state would obtain in the interior of the dome away from its boundaries. Thus, in a dome, there is a stress gradient in both the horizontal and vertical directions. (For similar reasons, a thermal gradient would also exist in both directions in a dome.) The assumption of a hydrostatic stress state in undisturbed saltrock has been confirmed at the WIPP site in New Mexico using hydraulic fracturing stress measurements [3]. (Actually, all that could be estimated was the least principal stress, which was found to be vertical. Because, however, the WIPP site is not in a zone of high horizontal stress, one can then conclude that the three stresses must be equal.) The shape of openings in salt and potash mines depends on the means of excavation used. Openings excavated by blasting or by roadheaders are square or rectangular in cross section (with rounded corners in the case of roadheaders). Openings excavated by full-face continuous miners are rectangular, in the sense that the width is greater than the height, and the roof and floor are flat; the walls, however, are semicircular or rounded. Assuming that instantaneous redistribution of stress occurs in the manner described by elastic theory, the
stress distributions for the two cases would be somewhat different. However, in both cases some brittle failure will occur in a skin surrounding the opening as a result of the large dynamic stresses and strain rates induced by excavation. As noted previously, Mraz [1,2] accounts for this failure by assuming that the limit of the failed zone can be represented by an ellipse circumscribed around the actual opening.
Shape of the effective equivalent ellipse The method used by Mraz [1,2] to specify the shape and dimensions of the equivalent ellipse is based on geometric considerations. The present author, however, decided to investigate shape by theoretical means by calculating the stress distribution around rectangular openings with rounded corners using the methods of Muskhelishvili [4]. The extent of the failed zone was then determined using a suitable three-dimensional failure criterion and typical saltrock failure properties augmented to account for the dynamic nature of the stress state at excavation. It was found that the length of the horizontal semiaxis of the effective ellipse could be expressed as a function of the W / H ratio using the following relationship:
d = W / 2 - 0.466 + 0 . 5 I S ( W / H )
(1)
where d = semiwidth of the effective ellipse, W / H = width-to-height ratio and W = width of the rectangular opening. The correlation coefficient was found to be 0.982. The semiheight of the effective ellipse can then be found from:
b2 = d 2 y 2 / ( d2 - x 2)
(2)
where b = semiheight of the effective ellipse, and d, x and y are the rectangular coordinates of the ellipse. The coordinates are those of the corner of the opening, allowing for a failed zone at that point (generally about 0.5 m thick but depen-
127 dent on the shape of the opening). Having defined the ellipse, the radii of curvature of the ellipse at the ends of the semiaxes can be found from [5]: b2/d
(3)
a v = d2/b
(4)
a h =
where a h = horizontal radius of curvature at the wall, a v = vertical radius of curvature at the midspan of the roof, and b and d as defined previously. It should be noted that eqn. (1) is based on an opening in an infinite body of saltrock. If a competent layer exists in the roof at a relatively short distance from the opening, it will probably be necessary to modify the dimensions of the effective ellipse. (This analysis assumed that the longer axis of the opening cross section was in the horizontal direction. For openings whose longer axis is in the vertical direction, the same procedure can be followed with the width and height transposed.)
sumed that away from the ends of the opening, the stress state is one of plane strain. It should be noted that because saltrock is viscoelastic, the Poisson's ratio near the radius of yield on the viscoelastic side of the boundary will approach 0.5. Moreover, the value in the viscoelastic zone will in general be greater than the value measured in compressive strength tests--the value changes during transient creep. These preliminaries aside, we can proceed with determining the stress state. It has been shown that the m i n i m u m principal stress beyond the failed zone can be approximated by the radial stress for a circular opening having as its radius the radius of curvature of the effective ellipse [6]. Thus, we can perform the stress analysis in polar coordinates. In addition, for a circular opening in a uniform stress field, the shear stresses are zero. Assuming that the Von Mises criterion holds, we have the following relationship for the principal stresses at yield [7]:
(0-1 Stress distribution around a single opening
Having defined the equivalent effective opening that can be used to account for the failed region, we can now proceed to determine the stress distribution around the opening after yield (more correctly, timedependent, low strain-rate failure) has occurred. Before proceeding, however, we should clarify some of the assumptions implicit in the stress analysis. First, the stress distribution at steady-state is assumed to be constant by definition. Thus, even though saltrock behaves as a viscoelastic-viscoplastic material, it can be taken to be an elastic-plastic one in steady-state. Second, the rocks surrounding the opening are assumed to be homogeneous. Thus, the effects of clay partings, anhydrite layers, and other inhomogeneities are ignored. Finally, it is as-
_
02
)2
+ (0"2--03) 2 q- (O3--0-1
)2
=0-)
(5)
where % = material constant. For laboratory creep testing using a conventional triaxial apparatus, the intermediate and m i n i m u m principal stresses are equal, and eqn. (5) reduces to: 0-1 - % = %
(6)
Equation 6 is readily recognized as a form of the Tresca criterion. Hence, for conventional laboratory triaxial testing, the Von Mises and Tresca criteria are equivalent. However, such is not the case for the general stress state underground. It should also be noted that eqn. (6) corresponds to the conventional M o h r - C o u l o m b criterion for an angle of internal friction of zero. Hence, the angle of internal &iction for long-term time-dependent strength is zero, and % in eqns. (5) and (6) is the time-dependent, low-strain-rate uniaxial compressive strength, and twice the maxim u m shear stress (k). The maximum shear
128 stress has been called the Prandtl limit by Mraz and Dusseault [8]. Proceeding with the analysis, upon substitution of the plane strain relationship for o2 (in terms of a I and 03) into eqn. (5), and after simplification, we obtain: (012 + o32)(1 - v - u 2)
--
+ (4k/vrS )( rf/r ) 2 ln( rr/a ) at = o0(1 + r2/r 2)
(15)
- (4k/vC3 )( rf/r ) 2 ln( rr/a ) (7)
In the yielded zone, v -- 0.5, and by substituting this value into eqn. (7) we obtain: -
or = 0 0 ( 1 - r ? / r 2 )
O103(1 q- 2v - 202)
=4k 2
0.7502
plate, we obtain for the major and minor principal stresses in the viscoelastic zone:
1.500103 q-0.7502 = 4k 2
(8)
where r r is the radius of the interface between the viscoelastic and viscoplastic zones. For plane strain, the intermediate principal stress (the longitudinal stress) in the viscoelastic zone is given by:
which upon simplification gives:
aI - a 3 = 4 k / f 3
o 1 = 2v% (9)
Letting a t = ol, and a r = a 3, and substituting into the equilibrium equation in polar coordinates, we obtain:
Oar/Or = 4 k / ( rv/3 )
(10)
A f t e r integration and insertion of the b o u n d a r y condition that or = 0 at r = a, we obtain: o r = ( 4 k / f 3 - ) ln(r/a)
(11)
where r = radius to the point at which the stress is calculated, a = radius of the curvature of the opening and k -- the Prandtl limit (shear stress at zero mean stress). The tangential stress in the yielded zone is found to b e : a t = (4k/v/3-)[1 + ln(r/a)]
(12)
(16)
Having determined the formulae for the stresses in both the viscoelastic and viscoplastic zones, we can now find the formula for the radius of the viscoelastic-viscoplastic interface. At the interface, the radial and tangential stresses on the viscoelastic side are: o r = ( 4 k / f 3 ) ln(rf/a) a t = 2o 0 - (4k/vFJ -) ln(rf/a)
(17)
As the radial and tangential stresses at the interface are continuous, upon substituting eqn. (17) into eqn. (7), simplifying, and rearranging, we find the radius of the viscoplastic-viscoelastic interface to be:
rf=a exp[(v/-3oo/ZK - 1)/2]
(18)
For plane strain, the intermediate principal stress in the yielded zone is given by:
Using eqn. (18), the depths of the interface at the midpoint of the roof and in the walls can be shown to be given by:
a , = o2 = (2k/v~-)[1 + 2 ln(r/a)]
mpv=av{eXp[(VCSOo/2K - 1 ) / 2 ] - 1}
(13)
where o t = longitudinal stress. Turning now to the viscoelastic zone, we know that for a circular opening in a uniform stress field: a t -1- O r =
200
(14)
Using this relationship and Airy's stress function for a circular hole in a semi-infinite
+ b - H/2
(19a)
mph=ah{exp[(v~Oo/2K-- 1)/21 -- 1} +d - W / 2
(19b)
mpv and mph a r e the depths of the viscoelastic-viscoplastic interface at the midpoint of the roof and the midheight of the
where
129 walls respectively and a v and a h are as given previously. These results complete the stress analysis for a single opening. However, as indicated previously, the radii of curvature and semiaxis values calculated thus far assume a semiinfinite saltrock body surrounding the opening. If such is not the case, mpv c a n be no greater than the distance to the nearest competent strata in the b a c k . The horizontal semiaxis must be reduced and the vertical semiaxis increased if the calculated mpv exceeds that distance. However, the point at the corner remains fixed. If we know the slope of the ellipse at that point, we can calculate the required ratio of the semiaxes from:
and (20) we obtain the required values for the semiaxes. The radii of curvature can then be calculated from eqns. (3) and (4). Finally, the depths of the viscoplastic-viscoelastic interfaces and the stresses can be calculated as before. Stress distribution in pillars between openings
q = - xZyA2.
F r o m a mining standpoint, the stress distribution in pillars is critical to stability of the workings, and hence worker safety. Thus, pillar design is of great importance--oversized pillars prevent valuable reserves from being mined, whereas undersized pillars result in local or regional instability. W h e n there are several openings in close proximity to each other, the apparent initial vertical stress in the pillars between the openings can be considered to increase to some value Crv'.If the pillars were completely elastic, and there were no failed zone around the opening, the value would be defined by tributary area, based on the pillar dimensions and the spacing between the openings if the rooms were in the middle of a semi-infinite mining panel. This scenario, however, applies only to relatively narrow pillars, and in saltrock could only be considered at shallow depths. A more realistic scenario would be one where there are failed and yielded zones around the rooms and where the increased stress occurs in the portion of the pillar between the limits of the failed zone surrounding each opening. The increased stress also results in enlarged yielded zones compared to those around single openings. In addition, it can be shown that at the center of the pillar:
mpv a = actual distance to nearest competent
% = ( v/(1-
strata.
where v = Poisson's ratio, ox = horizontal stress in the direction perpendicular to the axes of the openings and %' = vertical stress, as the horizontal strain in the middle of the pillar must be zero in that direction. If we assume that the rooms are driven simultaneously, as well as closure, we can
b2/d 2 = -y
tan O / x
(20)
where ¢? = slope of the curve and d, b, x and y are as defined previously. Using the definition of a v (eqn. (4)), we can rearrange eqn. (20) to: a v / b =- - x / ( y
tan q,)
(21)
Substituting eqn. (21) into eqn. (19a), solving for b, and equating the result to the equation for an ellipse in terms of b 2, we obtain the following relationship for the slope of the curve: m tan3q, + n tan2q5 + p tan q5 + q = 0
(22)
where: m=xy n
=
2.
(rFlpv a + H / 2 ) 2 y -- 2 x 2 y A _ y 3 .
p = x3A 2 + 2 x y 2 A .
A = {exp[(vr3oo/Zk - 1)/2]-
1}.
H, x, y, o0 and k are as defined previously. Solving eqn. (22) for tan e~, substituting the result in eqn. (20), and combining eqns. (2)
v))cr"
(23)
130 ignore the initial yielded zone around an isolated opening. While this assumption is not completely realistic, it simplifies the analysis, and the closure will probably not be sufficiently great to significantly affect the results, especially if the time lag between driving the openings is not long. Because of symmetry, we need consider only half the pillar. The stress distribution can be thought of as that due to a hole in a plate of finite size, for which the general formulae for the stress c o m p o n e n t s are given by T i m o s h e n k o and Goodier [9] and Southwell [10]. Solving these equations, we obtain:
X (b 4 + 4b2a 2 - 3 b n a 4 / r 4 +
The next task is to determine the radius of the viscoelastic-viscoplastic interface. At the m i d p o i n t of the roof of the opening, the formula remains the same as for a single opening (previous section). Substitution of eqn. (24) into eqn. (7) results in an eighth-order polynomial equation in r that m u s t be solved to obtain the radius to the interface. Such an equation cannot be solved directly unless the polynomial fortuitously happens to factor into a quadratic or quartic. However, the equation can be solved iteratively (by trial-and-error, or the m e t h o d of successive approximations) by taking, for an initial value, the radius to the pillar center. If this radius remains within the yielded zone, then the pillar should be enlarged. (Similarly, if secondary mining is planned, and the width of the viscoelastic zone is less than the anticipated secondary r o o m width, a larger pillar should be considered.) Once the radius of the viscoelastic-viscoplastic interface has been determined, the stresses in the viscoelastic and viscoplastic zones can be determined. The stresses in the viscoplastic zone are given by eqns (11)-(13). We can determine the stresses in the viscoelastic zone as before except that instead of a b o u n d a r y condition at r = a we have:
2bZa2(b2 + aZ)/r 2) sin 2t9
at
Ot= { 0:/[2(1 - v)(b 4 - a4)]} X [b2(1 + a2/r2)(b 2 Jr- a 2) -k-
(1 - 2 v ) ( b 4 + 4bia 2 +
3b4a4/r 4)
x cos 2 0 ] Or= { Ova/[2( ] -- 0 ) ( b 4 - a 4 ) ] }
a2/r2)(b 2 + a 2) - (1 - 2v) x (b 4 + 4b2a2-ab2a2(b2 + ae)/r 2 + X [b2(1 -
3b4a4/r 4) cos 2 0 ]
(24)'
'rrt = { o:(1 -- 2 U ) / [ 2 ( 1 - - v ) ( b 4 - a 4 ) ] }
where av' = effective vertical stress, b = radius from center of curvature to pillar center, a = radius of curvature of effective opening and v = Poisson's Ratio. Inspection of the above formulae indicates that it would be necessary to determine the principal stresses before we can evaluate the yield criterion at any point and hence find the radius of the viscoelastic-viscoplastic interface. However, we are interested in the stresses on the horizontal plane where the shear stress disappears. Thus, the formulae for ot and o, give us the principal stresses o I and 03 respectively. For a rib pillar, 02 is found using the assumption of plane strain.
r=rf, Or=4kln(rt/a)/v/3; "/'rt=0
Solving the equations, we obtain: or= -4k
ln(rf/a)(b 2 + r2)rf2/
[¢~(bZ-rZ)r2] + (o:/[2(1-v)(bZ-r2)]) × {b2(l+r2/r2)+[(1-2v)/ (b2 - Ff2)] × (b 4 - 262V + 2,'?rVb
bar~/r 4) cos 2 0 )
2 -
(25)
131 or = 4 k l n ( r r / a )(b 2 - r 2 ) r ~ / [ v ~ - ( b 2 - r ? ) r 2]
+ (o:/[2(1 - v)(b2- r )l ) × {b2(1 - rfa/r 2) - [(1 - 2 v ) /
we now present, in point form, a step-by-step method for selecting mine room-and-pillar dimensions. Rib pillars
(b 2 - rf2)](b 4 - 2b2r 2 + 2 r 4 6 2 / r 2 -
b4r;/r 4) cos 20) "/'rt
:
{O~ (1
--
2 v ) / [ 2 ( 1 - v)(b 2 - r 2)2]}
× (b 4 _ 2b2rf 2 + 4 r 2 / b 2 + rf4b2/r 2 b 4 4 / r 4) sin 2 0
(26)
Equation 26 completes the analysis for a rib pillar with viscoelastic and viscoplastic zones. However, there is still the question of rectangular and square pillars. For a square pillar, the intermediate stress at any point on a centerline is the radial stress at the center of the pillar. Hence, a single set of calculations using the formulae derived previously will give the stress distributions on both center lines if the cross-cuts have the same width as the rooms. Otherwise, it will be necessary to determine the effective opening and extraction ratio separately for the crosscuts. For a rectangular pillar, the extraction ratio will be different in the longer direction and hence the radial stress at the center along the second centerline must be calculated separately. However, the same formulae are used. If the cross-cuts have the same width as the rooms, it is simply a matter of determining Ov' for the different pillar thickness. If the crosscuts are a different width, it will be necessary to determine the effective opening before proceeding with the remaining calculations. The next, and final, step is to define a step-by-step calculation procedure. This method will be presented for each of the three cases--rib, rectangular and square pillars--in the next section.
Pillar design method Having presented the analytical formulae for determining pillar dimensions and stresses,
(1) Determine the shape of the equivalent effective opening assuming a semi-infinite cover of saltrock. (The effective opening is a circle if the opening has a square cross section.) (2) Determine the thickness of the yielded zone in the back and compare with the distance to the nearest overlying competent layer. If the actual distance is smaller, determine the modified effective opening geometry. (3) Calculate the stresses in the roof. (4) Select a pillar thickness and determine the thickness of the yielded zone, and hence the thickness of the viscoelastic core of the pillar. (5) Repeat step 4 until the pillar width is adequate. (6) Repeat the procedure for the barrier pillars (assuming no load is taken by the intervening yield pillars) in the case of panels whose pillars are designed to ultimately yield. Square pillars The method is the same as for rib pillars, except that the intermediate principal stress in steps 3 and 4 is the radial stress at the center of the pillar. If the cross-cuts do not have the same width as the rooms, it will be necessary to calculate the effective opening and radial stress in the second direction first. Rectangular pillars (1) Determine the shape of the effective opening for sections through both centerlines of the pillar (longitudinal and transverse). (2) Determine the thickness of the yielded zone in the back for both cases and compare with the distance to the nearest overlying competent stratum. If the latter distance is smaller, determine the modified geometries.
132 (3) Calculate the stresses in the roof. (4) Select a pillar thickness and determine the thickness of the yielded zone, and hence the thickness of the viscoelastic core of the pillar. This calculation is carried out for both directions. (5) Repeat step 4 until the pillar size is adequate. (6) Repeat the procedure for barrier pillars if any are provided.
Commentary on the method Computer programs were written in BASIC for a Macintosh Plus for the rib pillars with rectangular and square openings. (A 0.5 m thickness of broken material at the corners of the opening was assumed based on the dynamic failure analysis.) For square and rectangular pillars, initial trial configurations can be obtained by first performing the calculations for a rib pillar, and using the dimensions for that case as a first approximation. Use of the computer programs allows a large number of trial layouts t o be evaluated in a matter of minutes. For detailed evaluation of stresses on a mine-wide basis (or panel-wide basis), more sophisticated finite element or other numerical methods should be used.
Examples of pillar design using this method To illustrate the use of the method, we now provide three hypothetical examples. However, resemblance to existing or former operations, insofar as the depth of the workings and the size and shape of the rooms are concerned, is intentional in order to make the examples realistic and representative of actual operating conditions. Nevertheless, the values of the yield stress (Prandtl limit) and Poisson's ratio used are estimates and not based on actual mine property data. Hence, similarity
to exact pillar dimensions at mines resembling these examples is strictly fortuitous. Except where noted, Poisson's ratio has been taken as 0.4 in all example analyses.
A potash mine at intermediate depth This hypothetical mine employs rooms 3.05 m high and 14.63 m wide, excavated by drilling and blasting. The workings are found at a depth of 305 m and the nearest competent strata occur 152 m above the workings. The radius of curvature of the corners of the openings is 0.15 m and the thickness of the failed zone at the corners is assumed to be 0.5 m. The primitive stress and the yield stress (Prandtl limit) are estimated to be 6.9 and 5.2 MPa, respectively. The parameters of the effective opening were found to be as follows: b = 3.18 m. d = 9.33 m. a h = 1.08 m. a v = 27.39 m. mpv = 3.78 m. The mine uses square pillars. Analysis showed that the yielded zones in the pillars just merge if the pillar width were 21.93 m, giving an extraction ratio of 64%. Inspection of the pressure arch width data of Abel and Djahanguiri [11] showed that for a mine at this depth the distance would be no greater than 106 m. Based on the room-and-pillar dimensions, ~ this transfer distance would allow only three rooms per panel. Therefore, pillars with elastic cores were chosen rather than a yield-and-barrier pillar arrangement. Further analysis showed that by increasing the pillar thickness to 25 m the depth of the yielded zone in the pillars would be reduced to 4.36 m, giving an elastic core 17.3 m thick. This pillar size would seem sufficient to ensure stability and results in an extraction of 60%. N o t e that an increase of only 3 m in pillar thickness makes a considerable difference in the stability of the pillar.
133
A salt mine at intermediate depth This hypothetical mine employs rooms 7.62 m high and 15.24 m wide, excavated by drilling and blasting. The workings are at a depth of 345 m and the primitive stress was estimated to be 8.57 MPa. The yield stress (Prandtl limit) was assumed to be 10.4 MPa. (This relatively high value of the Prandtl limit results from the fact that the ore is almost pure halite.) The radius of curvature of the corners of the openings and the thickness of the failed zone in the corners were assumed to be 0.15 and 0.5 m, respectively. The nearest competent layer occurs 0.765 m above the workings so that a modified shape of the effective ellipse must be used. The parameters of the effective openings were found to be as follows: b -- 6.63 m. d - - 10.12 m. a h = 4.35 m. a v = 15.43 m. mpv = 0.77 m. This mine also uses square pillars. Analysis showed that the yielded zones in the pillars would just coalesce if the pillar width were 24.38 m, giving an extraction ratio of 62.1%. In this case also the workings are too shallow for a practical pressure arch to develop. Hence, pillars with elastic cores were chosen rather than yield pillars. Further analysis showed that increasing the pillar width to 26 m would result in an elastic core in the pillar 8.8 m thick and an extraction ratio of 60.3%. This shows that the equilibrium in correctly sized pillars is quite delicate and indiscriminant pillar robbing could easily disrupt it.
A deep potash mine This mine uses continuous miners to drive rooms 3.05 m high and 9.15 m wide. The walls of the openings are semicircular and
because of the geometry the thickness of the failed zones in the corners is assumed to be 0.25 m. The workings are at a depth of 1040 m and the assumed primitive stress is 23.45 MPa. The Prandtl limit varies according to the carnallite content of the ore, being as high as 9.5 M P a when the carnallite content is negligible, and as low as 5.2 MPa when the carnallite content is relatively high. The mine employs rib pillars and uses barrier pillars after every seven rooms. The distance to the nearest competent strata in the back is 18 m. To show the effect of the different yield stresses (Prandtl limit values) on the mine layout, analyses were performed for 5.2, 7.4 and 9.5 MPa. The effective opening parameters, the m i n i m u m pillar size, and the m a x i m u m extraction ratio for the three cases are given in Table 1. Another parameter that has a considerable effect on the required pillar size is Poisson's r a t i o - - t h e horizontal stress at the pillar center is v / ( 1 - v) times the effective vertical stress. In viscoelastic materials, Poisson's ratio is not a constant with time and furthermore will approach 0.5 as the limit of viscoelastic behavior is reached (i.e., as the material yields). Thus, in the weaker material in this mine, it is likely that the Poisson's ratio is greater than the 0.4 value hitherto assumed. Similarly, in the strongest material, Poisson's ratio may be less
TABLE 1 Effect of variation in yield stress on mine layout in a deep potash mine (k = MPa) k~5.2
k=7.4
k=9.5
b (m) d (m) a h (m) a v (m) mpv (m) Min. pillar width (m)
4.36 4.49 4.23 4.63 18.00
2.20 5.24 0.93 12.44 18.00
1.93 5.66 0.66 16.60 13.13
49.0
15.20
11.38
Max. extraction ratio (%)
15.7
37.6
44.5
134 TABLE 2 Effect of variation in Poisson's ratio on pillar geometry in a deep potash mine v=0.35 v=0.40 v=0.45 v=0.50 Min. pillar width (m) 18.10 Max. extraction ratio (%) 33.6
11.38 44.5
7.40 55.3
5.24 63.6
than 0.4. (In high strain-rate laboratory tests on saltrock, it is often about 0.35.) Thus, the effect on the pillar size of varying Poisson's ratio was examined using a constant Prandtl limit of 9.5 M P a . The results are given in Table 2. The conclusion to be drawn is that the higher Poisson's ratio to be expected in weaker material will partially compensate for the smaller yield stress so that the minimum pillar size is reduced and the m a x i m u m extraction ratio is increased. Thus, referring back to Table 1, mining the weaker material will b e more economical than the initial analysis led us to expect. Recalculating the case for k = 5.2 M P a using a Poisson's ratio of 0.45, we obtain a minimum pillar width of 23.0 m, corresponding to an extraction ratio of 28.5%. Even so, the extraction ratio is still sufficiently small that mining this lowstrength, low-grade (due to the carnallite) material could be questionable economically. At the beginning of this example, it was indicated that barrier pillars were to be left after every seven rooms. Hence, an approximation analysis was also carried out to estimate the required width of the barrier pillars. This analysis was approximate from two standpoints. First, it was assumed that all the area between the barrier pillars was mined out whereas in reality there would be yield pillars taking a small portion of the load; second, because of the very long axis of the assumed opening between the barrier pillars, the yielded zone in the back did not correspond to the distance to the first competent
layer b u t was considerably closer to the surface. It was found that the yielded zones in the pillars just merged for a pillar width of 94.25 m. However, extending the pillar width to 95 m gave a viscoelastic core of 41.8 m. (Both these cases were for k = 9.5 M P a and a Poisson's ratio of 0.4.) These barrier pillar widths should not, however, be accepted without verification b y numerical analysis (e.g., finite element analysis).
Summary This paper presents a method for the design of salt and potash mines. A calculation procedure was developed for determining the shape of the effective opening. Procedures for calculating the stresses around single openings and in pillars and the locations of the viscoelastic-viscoplastic interfaces, using the effective opening and the Von Mises yield criterion, were presented. Additionally, examples were presented of the use of the method for three hypothetical cases.
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