- Email: [email protected]
Roof tensile failures in underground excavations
International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Contents lists available at SciVerse ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Roof tensile failures in underground excavations J. Alcalde-Gonzalo a, M.B. Prendes-Gero b, M.I. A´lvarez-Ferna´ndez c, A.E. A´lvarez-Vigil d, C. Gonza´lez-Nicieza c,n a
Civil Engineer. Dept. Exploitation and Prospecting Mines. University of Oviedo - Spain Department of Construction and Manufacture Engineering, Polytechnic School of Mieres, Mieres, 33600 Asturias, Spain c Department of Mining Engineering, Mining Engineering School, University of Oviedo, Independencia 13, 33004 Asturias, Spain d Department of Mathematics, Mining Engineering School, University of Oviedo, Independencia 13, 33004 Asturias, Spain b
a r t i c l e i n f o Article history: Received 26 March 2012 Received in revised form 25 July 2012 Accepted 3 October 2012 Available online 28 November 2012
1. Introduction The continuous depletion of the surface deposits of minerals and concern for the environment have led to increasing depths of extraction, resulting to an increase in production costs. This has led many mining companies studying the feasibility of transforming their open pit mines into underground mines. In this transformation it is crucial to determine the mechanical properties of the rock mass to ensure its adequate exploitation and a selfsupporting underground operation. During the last few decades, the Geotechnical Engineering Group of the University of Oviedo has determined that, for competent rock masses, the most adequate method of exploitation is room-and-pillar. In this process, the material is removed in some areas to create subterranean rooms and left in other ones to construct pillars or columns that support the roof weight. In this case, the dimensions of the rooms and pillars ensure, or not, the economic viability of the project. In general, the roof of these underground excavations is usually composed of layers of rocks with parallel laminations, called bedded rock, in which the bed thickness is small compared to the roof span and the bond between the layers is weak [1]. In this situation the thinner strata near openings tends to detach from the main rock mass and form separate beams, allowing the theory of elastic beams to be applied. Failure occurs when the thinner beam, in the immediate roof, flexes downward and cracks on its upper surface, at the ends, and centrally on the lower surface [2]. The theory of elastic beams assumes that the layers are loaded by their own weight, and thus designs the roof span for a specified
n
Corresponding author. Tel.: þ34 985 10 42 66; fax: þ 34 985 10 42 67. E-mail address: [email protected] (C. Gonza´lez-Nicieza).
1365-1609/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2012.10.003
allowable tensile stress of the rock. Besides which, it is assumed that: the rock in an individual layer is perfectly elastic, homogeneous and isotropic; the neutral axis coincides with the centre line of the layer; the stress of the longitudinal fibres is proportional to the distance from the neutral axis for each layer; and there are no joints in the strata. In 1972, Barker and Hutt examined the mechanical behaviour of various elastic, layered unfractured and fractured roof beams. They concluded that for unfractured roofs, simple elastic beam theory is adequate for determining the maximum tensile stress of the rock, but not the roof’s deflection which is highly dependent on shear deformation and it is not taken into account in simple beam theory [3]. However, a year earlier, Stephansson [4] applied elastic beam theory to elastic supports. He divided thin single-layer roofs into two types depending upon the ratio between thickness T and roof span L, and concluded that in layers with a ratio T/Lo0.2 the deformations due to built-up shear stresses can be disregarded. The existence of joints crossing the rock mass encourages the consideration of the roof as a Voussoir beam [5–8]. This model supposes that when the beam is incapable of sustaining tensile stresses, vertical tensile fractures are formed at the abutments and the beam becomes simply supported, and a compression arch develops within the beam, rising from the abutments to a high point at midspan [9]. In the two models, elastic theory and Voussoir beam analogy, the strain distribution in each section is assumed to be triangular (Fig. 1a), with a symmetrical distribution of compression and tensile stresses about the horizontal midline. But there are studies that refute the hypothesis of a symmetrical distribution of compression and tensile stresses [10–12]. In this case, the distribution of stresses adopts a form similar to that depicted in Fig. 1b, where the tensile strength of the rock is approximately 10% of its compressive strength.
142
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Fig. 1. Variation of the stresses along the height of a beam.
In this work, a non-linear failure criterion for rock mass is proposed, based on the double theory of elasticity, and from it the determination of the maximum span that can support the tensile stresses through the roof. Besides, the behaviour of the roof when the tensile stresses increase the tensile strength is analysed. The method is applied on two real examples: Kampanzar Quarry situated in the municipality of Arrasate in Gipuzkoa (Spain) and Calzada Quarry situated in Villamartı´n de Vadeorras in Orense (Spain). The numerical results have been compared with those ones obtained from the Stephansson formulation, and the behaviour after failure is compared with the compression arch assumed by the Voussoir beam analogy.
Table 1 Mechanical properties of rocks [13–15]. Rock
st (MPa)
sc (MPa)
g (kg/m3 103)
E (GPa)
Granite Slate Quartzite Marble Limestone
2.94–7.84 10.00–30.00 4.00–23.00 2.94–8.83 4.20–5.80
97–310 138–207 207–627 69–241 14–255
2.60–2.80 2.60–2.70 2.60–2.65 2.60–2.70 2.20–2.60
22–59 10–34 10–44 3–78 3–78
each section (x) of the roof layer are given by: P PL PL2 MðxÞ ¼ x2 þ x 2 2 12 P 4 PL 3 PL2 2 x x þ x 24EI 12EI 24EI
ð2Þ
2. Tensile failure criteria
yðxÞ ¼
There are numerous examples of open pit mines, exploiting very different rock masses that have been transformed into underground mines. The diversity of the materials of these mines is reflected in the enormous list of values that identify their physical properties (Table 1) and that establish the technical and economic viability of their exploitation. The full-size structure considered in this paper is a horizontally bedded roof on a long underground opening in rock. In this approximate structure, each layer is assumed to act as a bi-fixed ends beam, to be of rectangular cross-section, and to support a uniformly distributed load that is equal to its own weight. Along with these assumptions the following restrictions are imposed: the length of the excavation is more than twice the roof span; for each layer the deflection is small in comparison to the layer’s thickness; within each layer the thickness is uniform; the rock in an individual layer is perfectly elastic, homogeneous and isotropic; the neutral axis coincides with the centre line of the layer; both abutments of each opening have the same properties and can be considered homogeneous, isotropic and perfectly elastic; the layer is considered a bi-fixed ends beam. Considering a straight bi-fixed ends beam subjected to a uniformly distributed load, P, acting in the principal plane of the symmetric cross-section, the differential equation of a beam in bending that describes the deflection of the roof is
These Eqs. (1–3) assume a symmetrical distribution of the compression and tensile stresses about the horizontal midline. However in Table 1 it is observed that in all cases the tensile strength presents very low values against the compressive strength. Therefore, under these conditions it is possible to say that the rock is weaker against tensile stresses and that failure will take place when these stresses are greater than the tensile strengths st. When this situation is reached, the thickness in the tensile zone (T/2) can be divided in two (Fig. 1b): the former into the failure status yf(x) and the second supporting the tensile stress yt(x). Under these conditions, the thickness that is really working or ‘‘useful’’ T1(x) is the sum of the thicknesses operating under compression (T/2) and under tensile stress (yt(x)):
4
EIð
d y Þ¼P dx4
ð1Þ
where I is the moment of inertia. Differentiating Eq. (1) twice and four times with the boundary conditions at the ends of the free span (deflection and rotation null), the bending moment, M(x), and the deflection line, y(x), in
T 1 ðxÞ ¼
T þ yt ðxÞ 2
ð3Þ
ð4Þ
Applying the law of Navier sðxÞ ¼ ðMðxÞ=IÞyðxÞ, on the assumption that the stress, s(x), is equal to the maximum tensile strength, st, and y(x) is the thickness under tensile stress, yt(x), the useful thickness of the beam T1(x) is: T 1 ðxÞ ¼
T st I T st ¼ þ 2 MðxÞ 2 ðP=2IÞx2 ðPL=2IÞx þðPL2 =12IÞ
ð5Þ
Replacing the uniformly distributed load, P, in Eq. (5) for its expression in terms of specific weight, g, width, H, and thickness, T, (P ¼ gHT) and considering that the most critical zones are over the abutments (x ¼0 and x ¼L) where the tensile stresses are greater, Eq. (6) is obtained. This equation demonstrates that for a certain rock mass, with a tensile strength st and a specific weight g, the thickness under tensile stress and consequently the useful thickness T1 (in the more critical sections) is a function of the real
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
143
Fig. 2. Ratio thickness/span for different specific weight of the rock masses.
Table 2 Geometry and properties of the rock mass in Kampanzar and Calzada quarries.
Real thickness (T) Width (H) Specific weight (g) Tensile strength (st) Elastic modulus (E) Friction angle (j) Cohesion (c) Poisson’s ratio (n)
Fig. 3. Graphic scheme of the criterion.
thickness of the beam T and its span L: yt ¼
st T 2 ð Þ g L
ð6Þ
By plotting Eq. (6) it is possible to determine, in a graphical way (Fig. 2), the span permitted to avoid failure for a real thickness of beam. This span decreases with an increase in the specific weight of the rock mass, a decrease in the tensile strength of the rock mass, an increase in the thickness layer. From Fig. 2, failure will occur when the tensile thickness is less than half the real thickness (yt oT/2), so by introducing in Fig. 2 the location of half of the real thickness and drawing a horizontal line to cut the curve with the tensile strength of the rock mass the ratio T/L is obtained, and therefore the maximum span that can support the stresses without failure. The scheme of this process is represented in Fig. 3. Of special interest is a comparison between the length of the span obtained graphically and the value obtained numerically and analytically. In this work, two real examples have been assessed, Kampanzar Quarry, where limestone is extracted, and Calzada Quarry, when the rock mass is slate (Table 2). Both of them present similar specific weights, although the tensile strength is very different. 3. First application: Kampanzar quarry 3.1. Calculating maximum span without failure 3.1.1. Numerical analysis In 2003, for environmental reasons, studies of the economic and technical feasibility for transforming the limestone quarry
Kampanzar
Calzada
1m 1m 27 kN/m3 5.57 MPa 100 GPa 301 2 MPa 0.25
2m 1m 27 kN/m3 15 MPa 64 GPa 44.31 10.9 MPa 0.15
into an underground mine began. One of the first steps in the transformation process into an underground mine was the use of informatics simulations. Specifically ALMEC 3D software (Fig. 4) was used to calculate the length of the span inside the rooms and the height of the columns. From this analysis it was concluded that, from a point of view of safety, the maximum span of the rooms and the maximum height of the columns had to be less than 20 m and 25 m, respectively. 3.1.2. Graphical analysis Nevertheless, and in spite of the use of more and more powerful computers, the numerical simulations require great amounts of data and long periods of calculation. In contrast, the use of failure curves (Fig. 2) allows the very fast determination, and with only two mechanical properties, the maximum length of the span that the roof can have to avoid cracking. In the Kampanzar quarry, by introducing into Fig. 2 (right) the position of half of the real thickness (0.5 m) and drawing a horizontal line to cut the curve with the tensile strength of the rock mass (5.57 MPa), a ratio of T/L¼0.05 is obtained. Therefore, with a thickness of 1 m, the maximum span that can support the roof without breaking is about 20 m. This value is similar to the span calculated by the numerical simulations. 3.1.3. Analytical analysis When the theory of elasticity is applied, the value of the maximum span that can support the tensile stress is equal to the value obtained by the informatics simulations. A longer span produces a value of tensile stress (Table 3) greater than the tensile strength. In Table 3, the values from the Stephansson formulation are also presented. In this case, a single-layer roof configuration has been employed, case I (T/Lo1/5) [4]. The values presented in Table 3 establish failure with spans greater than 23 m.
144
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Fig. 4. Distribution of vertical stresses. ALMEC 3D.
Table 3 Kampanzar quarry: analytical results of moment, deflection and tensile stress. Parameter
Span (m)
Elasticity theory
Stephansson formulation
x¼ 0
x¼ L/2
x ¼0
x¼ L/2
Moment (Nm)
15 20 23.5 25 30
5.06eþ 05 9.00eþ 05 1.24eþ 06 1.41eþ 06 2.02eþ 06
2.53eþ 05 4.50eþ 05 6.21eþ 05 7.03eþ 05 1.01eþ 06
2.84e þ05 6.11e þ05 9.07eþ 05 1.05eþ 06 1.60eþ 06
4.75e þ05 7.39e þ05 9.57e þ05 1.06eþ 06 1.44e þ06
Deflection (mm)
15 20 23.5 25 30
0.00 0.00 0.00 0.00 0.00
0.43 1.35 2.57 3.30 6.83
0.05 0.08 0.11 0.12 0.16
0.19 0.43 0.71 0.86 1.57
Tensile stress (N/m2)
15 20 23.5 25 30
3.04eþ 06 5.40eþ 06 7.46e þ06 8.44e þ06 12.12eþ 06
1.15e þ 06 2.70e þ06 3.73e þ 06 4.22e þ 06 6.07e þ 06
1.70e þ06 3.67e þ 06 5.44e þ 06 6.30e þ06 9.60e þ06
2.85eþ 06 4.43eþ 06 5.74eþ 06 6.35eþ 06 8.64eþ 06
In all cases (Table 3) the greatest values of tensile stresses appear over the abutments and with the theory of elasticity, while with the Stephansson formulation the stresses present very similar values over the abutments and at midspan. In the former case, failure begins over the abutments and afterwards occurs at midspan, in accordance with [2]. In the second case, failure is generalised across the span. The difference presented (Table 3) is only due to the fact that two different situations are being compared. The first case (theory of elasticity) is considering a bi-fixed ends beam, while Stephansson considers flexible supports. These distribute the moment along the beam, increasing it at midspan and decreasing it over the abutments.
3.1.4. Comparison between numerical, graphical and analytical approaches The comparison between the different approaches (Table 4) presents very similar values of the maximum span. While the value is equal with numerical and graphical analysis (20 m), this value is slightly larger with the theory of elasticity, but only by 0.5 m. In contrast, the Stephansson formulation displays a maximum span of 23 m. This can be explained because Stephansson’s
method is the only one that considers elastic supports, leading to a more uniform distribution of the tensile stresses along the span.
3.2. Behaviour of the roof with larger spans than the maximum calculated When lengths of span longer than those calculated are considered, the beam breaks over the abutments in the top surface. But with even longer lengths the beam also breaks at midspan in the bottom surface (Fig. 5). In these conditions, when failure is produced, the beam is not a prism but also a section with variable thickness. Following [16] it is possible to calculate the new distribution of moments employing the theory of elasticity, with the hypothesis that in the breakage zones the maximum tensile stress is equal to the tensile strength st. In this case (Fig. 5), and denoting xi (i¼1–6), the points where the limiting surfaces cut the initial section, the new distribution of moments M(x) (Eq. (7)) allows the new deflection (Eq. (8)) to be
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Table 4 Kampanzar quarry: comparison between results. Numerical analysis
Maximum span (m)
Graphical analysis
20
20
145
Table 5 Kampanzar quarry: analytical results of moment, deflection and tensile stress.
Analytical analysis Elasticity theory
Stephansson formulation
20.5
23
Parameter
Moment (Nm) Deflection (mm) Tensile stress (N/m2)
Span (m)
30 30 30
Conjugate beam method x¼ 0
x ¼ L/2
4.94eþ 05 0.00 5.57e þ06
8.53e þ 05 6.84 5.57e þ06
pffiffiffi 4st x Q ðxÞ ¼ 16 3A½ln RðxÞln SðxÞ þ ET pffiffiffi 4st x Q ðxÞ ¼ yðxÞ16 3 A½ln RðxÞln SðxÞ ET pffiffiffi 2st 2 x Q ðxÞx ZðxÞ ¼ 16 3A½ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ þ ET pffiffiffi 2st 2 Z ðxÞ ¼ yðxÞ16 3 A½ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ x Q ðxÞx ET Fig. 5. Kampanzar quarry: useful section after failure.
obtained by means of the conjugate beam method. MðxÞ ¼ s6t T 1 ðxÞ2
if
ðx1 rx rx2 Þ or ðx3 o x r x4 Þ
PL P 2 MðxÞ ¼ PL 12 þ 2 x 2 x
if
ðx2 rx rx5 Þ or ðx6 o x r x3 Þ
MðxÞ ¼ s6t T 1 ðxÞ2
if
ðx5 ox rx6 Þ
2
ð7Þ
pffiffiffi st 2 yðxÞ ¼ 16 3A ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ 2ET x þ Q ðx1 Þ þZ ðx1 Þ 2
PL PL 3 P yðxÞ ¼ 24EI x2 þ 12EI x 24EI x4 i pffiffiffi h st 2 yðxÞ ¼ 16 3A lnRðxÞ1 RðxÞ lnSðxÞ1 SðxÞ þ 2ET x þ Q ðx5 Þ þ Z ðx5 Þ 2
PL P yðxÞ ¼ 24EI x2 þ PL x3 24EI x4 pffiffiffi 12EI st 2 yðxÞ ¼ 16 3A ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ 2ET x þ Q ðx4 Þ þZ ðx4 Þ
By applying Eqs. (7) and (8), the new moment and deflection of a beam with a span of 30 m (greater than the safe length of 20 m) is shown numerically in Table 5 and graphically in Fig. 6. Fig. 6 shows a new distribution of moments and deflections. The shape of the moment distribution is non-continuous, and this can be explained because the second derivate of the deflection (continuous) has to be multiplied, in each section, for the inertia moment. This is directly proportional to the useful thickness, which is variable near to the abutments and at midspan (Fig. 5). if
x1 rx rx2
if
x2 ox rx5
if
x5 ox rx6
if
x6 ox rx3
if
x3 ox rx4
ð8Þ
3.3. Compression arch R(x), RðxÞ, S(x), SðxÞ, Q(x), Q ðxÞ, Z(x), ZðxÞ where the parameters A, A, are defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I2 s4t A¼ 2 3 2 E PT ðL PT48Ist Þ
A¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I2 s4t E2 PT 3 ðL2 PT þ48Ist Þ
RðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ þ 3ETAðL2 PT48Ist Þ 6Is2t
RðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ þ 3ETAðL2 PT þ48Ist Þ 6Is2t
SðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ 3ETAðL2 PT48Ist Þ 6Is2t
SðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ 3ETAðL2 PT þ 48Ist Þ 6Is2t
The Voussoir model and the theory of elasticity are two totally different concepts. Voussoir assumes that a beam with fixed ends and distributed loading yields when the maximum tensile stress exceeds the tensile strength of the rock [9]. In this case, vertical tensile fractures form at the abutments, and the beam becomes simply supported (assuming no slip at the abutments). Then the stress is higher, and a progressive process of cracking at the abutments and at midspan begins. At this point, the development of a compression arch and the generation of a moment between the reaction force at misdpan and at the abutments which acts to resist the moment imposed by self-weight are assumed. To resolve the loss of horizontal stress symmetry, the model considers an average thickness NT for this arch, where N is a reduction coefficient of T. The beam will be at equilibrium when N is closer to 0.75. Although the theory of elasticity does not consider a compression arch, the variable section (Fig. 5) has been adjusted with an arch where all the sections, along the height, work. For that, and with the simplification that the neutral axis coincides with the centre line of the layer, the new neutral axis has adjusted to a second-degree polynomial (a þbxþcx2 ¼0). Parameters a, b and c, which depends on the length and bending of the beam, are
146
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Fig. 6. Kampanzar quarry: distribution of moments and deflections (top); tensile thickness (bottom).
Fig. 7. Distribution of vertical stresses. FLAC 2D.
4. Second analysis: Calzada quarry
calculated by least squares and are shown in Eq. (9). 2
neutral axisðxÞ ¼ 0:0897 þ0:0009x0:00002x
ð9Þ
This curve will be the centre line of the arch, and the distance between the curve and the point of failure x1 will be half of the thickness. In the example (Fig. 6 bottom) the new thickness is 64% of the initial thickness. This percentage is close to 70%, a value considered the equilibrium value according to the Voussoir model [5].
4.1. Calculating maximum span without failure 4.1.1. Numerical analysis In this case, the height of the banks and the volume of the slag heaps were the main reasons for the consideration of the transformation of the quarry into one underground mine. The stability of the exploitation was studied with FLAC 2D software
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
147
Table 6 Calzada Quarry: analytical results of moment, deflection and tensile stress. Parameter
Span (m)
Elasticity theory
Stephansson formulation
x¼ 0
x ¼L/2
x¼ 0
x ¼L/2
Moment (Nm)
47 47.5 51 57.5 67
9.94eþ 06 10.15eþ 06 11.70eþ 06 14.88eþ 06 20.20eþ 06
4.97e þ 06 5.08e þ06 5.85e þ 06 7.44e þ 06 10.10e þ06
6.04eþ 06 6.12eþ 06 7.50eþ 06 10.19eþ 06 14.80eþ 06
8.87e þ06 9.01e þ06 10.05e þ06 12.12e þ 06 15.51e þ 06
Deflection (mm)
47 47.5 51 57.5 67
0 0 0 0 0
16.083 16.778 22.30 36.03 66.416
0.21 0.21 0.23 0.28 0.36
0.81 0.84 1.03 1.48 2.38
Tensile stress (N/m2)
47 47.5 51 57.5 67
14.911e þ06 15.23eþ 06 17.557e þ06 22.32eþ 06 30.301e þ06
7.455e þ 06 7.615e þ 06 8.778e þ 06 11.16e þ 06 15.15eþ 06
9.062e þ06 9.324e þ06 11.25e þ06 15.28eþ 06 22.192e þ06
13.304e þ06 13.52e þ06 15.08eþ 06 18.20eþ 06 23.259e þ06
Table 7 Calzada quarry: comparison between results. Numerical analysis
Maximum span (m)
40
Graphical analysis
47
Analytical analysis Elasticity theory
Stephansson formulation
47
50
Fig. 8. Calzada quarry: distribution of moments and deflections (top); tensile thickness (bottom).
148
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Table 8 Calzada quarry: analytical results of moment, deflection and tensile stress. Parameter
Moment (Nm) Deflection (mm) Tensile stress (N/m2)
Span (m)
50 50 50
Conjugate beam method x ¼0
x¼ L/2
8.92e þ06 0.00 15.00e þ06
5.63eþ 06 20.60 8.44e þ 06
(Fig. 7), and although instability was found for spans over 40 m, the rooms and columns were designed with a 20 m width and 25 m height respectively. These dimensions were selected after consideration of the technical and economic viability of the project. 4.1.2. Graphical analysis Similarly to the Kampanzar quarry, it is possible to calculate in a graphical and very fast manner the maximum span of the roof without failure. In this case, introducing a tensile thickness of 1 m (half the real thickness) and drawing a horizontal line to cut the curve with the tensile strength of the rock mass (15 MPa), a ratio T/L¼0.0425 is obtained (Fig. 2 right). From which a maximum span without failure of 47 m is determined. 4.1.3. Analytical analysis The analytical results (Table 6) show how in the previous example the worst values were obtained over the abutments and with the theory of elasticity. An increase in the span produces failure as much over the abutments as at midspan. The Stephansson formulation is less conservative, with values of stresses very similar over the abutments and at midspan. In this case the maximum span to avoid failure with the theory of elasticity is 47 m. While the maximum span with the Stephansson formulation is 50 m. 4.1.4. Comparison between numerical, graphical and analytical approaches In the Calzada quarry there is a greater difference between the results of the three approaches (Table 7). The numerical analysis results in a maximum span of 40 m, while the graphical and theory of elasticity result in 47 m. Again the Stephansson formulation is less conservative, with a maximum span of 50 m due to the distribution of the tensile stresses along the roof. The difference between the results of the numerical evaluation and the theory of elasticity or the graphical analysis is explained for the process of the numerical analysis. In a first step, an approximation is calculated with a mesh not very small. In this case the result (40 m) was considered so good that the evaluation was stopped, and the model was not recalculated. 4.2. Behaviour of the roof with larger spans than the maximum calculated compression arch The new moments and deflections for a beam with a span of 50 m have two inflection points very near the abutments. In this case the length employed is close to that considered safe (47 m), so failure is only over the abutments (Fig. 8). The numerical results (Eqs. (7) and (8)) are collected in Table 8, and the new
distribution of the moments and deflections are depicted in Fig. 8 (top). In these conditions the neutral axis descends very gradually over the abutments, and the neutral axis is adjusted by the curve represented in Eq. (10). neutral axisðxÞ ¼ 0:0092 þ0:0009x0:00002x2
ð10Þ
In this case the compression arch has a thickness about 90% of the initial thickness (Fig. 8 bottom).
5. Conclusions
It is possible to determine the maximum design span in a very
fast manner and with only two variables, allowing a safe underground mine to be designed. The inability of a roof to withstand tensile stress produces failure zones, first over the abutments and later at the midspan of the roof. The useful thickness (that is the thickness effectively supporting the stress) is variable throughout the length of the beam, reaching a minimum at the extremes and the midpoint. The new compression arch can be fitted to a parabola with a high correlation coefficient. It is possible to employ the double theory of elasticity to analyse the stability of the roof beam.
References [1] Adler L., Sun M.C. Ground control in bedded formations. Virginia Polytechnic Institute, Blacksburg; 1968. [2] Goodman RE. Introduction to Rock Mechanics. New York: Wiley; 1989. [3] Barker R.M., Hutt F. Joint effects in bedded formation roof control. In: Proceedings of the 14th US symposium on rock mechanics. Pennsylvania; 11–14 June 1972. p. 247–61. [4] Stephansson O. Stability of single openings in horizontally bedded rock. Eng Geol 1971;5:5–71. [5] Evans WH. The strength of undermined strata. Trans Inst Miner Metall 1941;50:475–532. [6] Seedsman RW. Geotechnical sedimentology – its use in underground coal mining. Int J Coal Geol 2001;45:147–53. [7] Brady BHG, Brown ET. Rock Mechanics for Underground Mining. London: George Allen & Unwin; 1985. [8] Swart AH, Handley MF. The design of stable stope panels for near-surface and shallow mining operations. In: Proceedings of the first international platinum conference ‘Platinum adding value’. Sun City, South Africa; 3–7 October 2004. p. 275–86. [9] Diederichs MS, Kaiser PK. Stability of large excavations in laminated hard rock masses: the voussoir analogue revisited. Int J Rock Mech Miner Sci 1999;36:97–117. [10] Exadaktylos GE, Vardoulakis I, Kourkoulis SK. Influence of nonlinearity and double elasticity on flexure of rock beams – I. Technical theory. Int J Solids Struct 2001;38:4091–117. [11] Exadaktylos GE, Vardoulakis I, Kourkoulis SK. Influence of nonlinearity and double elasticity on flexure of rock beams – II. Characterization of Dionysos marble. Int J Solids Struct 2001;38:4119–45. [12] Vardoulakis I, Exadaktylos GE, Kourkoulis SK. Bending of marble with intrinsic length scales: a gradient theory with surface energy and size effects. J Phys IV 1998;8:399–406. [13] Touloukian YS, Ho CY. Physical Properties of Rocks and Minerals. New York: McGraw-Hill; 1981. [14] Winkler EM. Stone-Properties, Durability in Man’s Environment. New York: Springer; 1973. [15] Farmer IW. Engineering Properties of Rock. London: E&FN Spon; 1968. [16] Timoshenko S, Goodier JN. The´orie de l’e´lasticite´. Paris: McGraw-Hill; 1961.
Contents lists available at SciVerse ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Roof tensile failures in underground excavations J. Alcalde-Gonzalo a, M.B. Prendes-Gero b, M.I. A´lvarez-Ferna´ndez c, A.E. A´lvarez-Vigil d, C. Gonza´lez-Nicieza c,n a
Civil Engineer. Dept. Exploitation and Prospecting Mines. University of Oviedo - Spain Department of Construction and Manufacture Engineering, Polytechnic School of Mieres, Mieres, 33600 Asturias, Spain c Department of Mining Engineering, Mining Engineering School, University of Oviedo, Independencia 13, 33004 Asturias, Spain d Department of Mathematics, Mining Engineering School, University of Oviedo, Independencia 13, 33004 Asturias, Spain b
a r t i c l e i n f o Article history: Received 26 March 2012 Received in revised form 25 July 2012 Accepted 3 October 2012 Available online 28 November 2012
1. Introduction The continuous depletion of the surface deposits of minerals and concern for the environment have led to increasing depths of extraction, resulting to an increase in production costs. This has led many mining companies studying the feasibility of transforming their open pit mines into underground mines. In this transformation it is crucial to determine the mechanical properties of the rock mass to ensure its adequate exploitation and a selfsupporting underground operation. During the last few decades, the Geotechnical Engineering Group of the University of Oviedo has determined that, for competent rock masses, the most adequate method of exploitation is room-and-pillar. In this process, the material is removed in some areas to create subterranean rooms and left in other ones to construct pillars or columns that support the roof weight. In this case, the dimensions of the rooms and pillars ensure, or not, the economic viability of the project. In general, the roof of these underground excavations is usually composed of layers of rocks with parallel laminations, called bedded rock, in which the bed thickness is small compared to the roof span and the bond between the layers is weak [1]. In this situation the thinner strata near openings tends to detach from the main rock mass and form separate beams, allowing the theory of elastic beams to be applied. Failure occurs when the thinner beam, in the immediate roof, flexes downward and cracks on its upper surface, at the ends, and centrally on the lower surface [2]. The theory of elastic beams assumes that the layers are loaded by their own weight, and thus designs the roof span for a specified
n
Corresponding author. Tel.: þ34 985 10 42 66; fax: þ 34 985 10 42 67. E-mail address: [email protected] (C. Gonza´lez-Nicieza).
1365-1609/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2012.10.003
allowable tensile stress of the rock. Besides which, it is assumed that: the rock in an individual layer is perfectly elastic, homogeneous and isotropic; the neutral axis coincides with the centre line of the layer; the stress of the longitudinal fibres is proportional to the distance from the neutral axis for each layer; and there are no joints in the strata. In 1972, Barker and Hutt examined the mechanical behaviour of various elastic, layered unfractured and fractured roof beams. They concluded that for unfractured roofs, simple elastic beam theory is adequate for determining the maximum tensile stress of the rock, but not the roof’s deflection which is highly dependent on shear deformation and it is not taken into account in simple beam theory [3]. However, a year earlier, Stephansson [4] applied elastic beam theory to elastic supports. He divided thin single-layer roofs into two types depending upon the ratio between thickness T and roof span L, and concluded that in layers with a ratio T/Lo0.2 the deformations due to built-up shear stresses can be disregarded. The existence of joints crossing the rock mass encourages the consideration of the roof as a Voussoir beam [5–8]. This model supposes that when the beam is incapable of sustaining tensile stresses, vertical tensile fractures are formed at the abutments and the beam becomes simply supported, and a compression arch develops within the beam, rising from the abutments to a high point at midspan [9]. In the two models, elastic theory and Voussoir beam analogy, the strain distribution in each section is assumed to be triangular (Fig. 1a), with a symmetrical distribution of compression and tensile stresses about the horizontal midline. But there are studies that refute the hypothesis of a symmetrical distribution of compression and tensile stresses [10–12]. In this case, the distribution of stresses adopts a form similar to that depicted in Fig. 1b, where the tensile strength of the rock is approximately 10% of its compressive strength.
142
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Fig. 1. Variation of the stresses along the height of a beam.
In this work, a non-linear failure criterion for rock mass is proposed, based on the double theory of elasticity, and from it the determination of the maximum span that can support the tensile stresses through the roof. Besides, the behaviour of the roof when the tensile stresses increase the tensile strength is analysed. The method is applied on two real examples: Kampanzar Quarry situated in the municipality of Arrasate in Gipuzkoa (Spain) and Calzada Quarry situated in Villamartı´n de Vadeorras in Orense (Spain). The numerical results have been compared with those ones obtained from the Stephansson formulation, and the behaviour after failure is compared with the compression arch assumed by the Voussoir beam analogy.
Table 1 Mechanical properties of rocks [13–15]. Rock
st (MPa)
sc (MPa)
g (kg/m3 103)
E (GPa)
Granite Slate Quartzite Marble Limestone
2.94–7.84 10.00–30.00 4.00–23.00 2.94–8.83 4.20–5.80
97–310 138–207 207–627 69–241 14–255
2.60–2.80 2.60–2.70 2.60–2.65 2.60–2.70 2.20–2.60
22–59 10–34 10–44 3–78 3–78
each section (x) of the roof layer are given by: P PL PL2 MðxÞ ¼ x2 þ x 2 2 12 P 4 PL 3 PL2 2 x x þ x 24EI 12EI 24EI
ð2Þ
2. Tensile failure criteria
yðxÞ ¼
There are numerous examples of open pit mines, exploiting very different rock masses that have been transformed into underground mines. The diversity of the materials of these mines is reflected in the enormous list of values that identify their physical properties (Table 1) and that establish the technical and economic viability of their exploitation. The full-size structure considered in this paper is a horizontally bedded roof on a long underground opening in rock. In this approximate structure, each layer is assumed to act as a bi-fixed ends beam, to be of rectangular cross-section, and to support a uniformly distributed load that is equal to its own weight. Along with these assumptions the following restrictions are imposed: the length of the excavation is more than twice the roof span; for each layer the deflection is small in comparison to the layer’s thickness; within each layer the thickness is uniform; the rock in an individual layer is perfectly elastic, homogeneous and isotropic; the neutral axis coincides with the centre line of the layer; both abutments of each opening have the same properties and can be considered homogeneous, isotropic and perfectly elastic; the layer is considered a bi-fixed ends beam. Considering a straight bi-fixed ends beam subjected to a uniformly distributed load, P, acting in the principal plane of the symmetric cross-section, the differential equation of a beam in bending that describes the deflection of the roof is
These Eqs. (1–3) assume a symmetrical distribution of the compression and tensile stresses about the horizontal midline. However in Table 1 it is observed that in all cases the tensile strength presents very low values against the compressive strength. Therefore, under these conditions it is possible to say that the rock is weaker against tensile stresses and that failure will take place when these stresses are greater than the tensile strengths st. When this situation is reached, the thickness in the tensile zone (T/2) can be divided in two (Fig. 1b): the former into the failure status yf(x) and the second supporting the tensile stress yt(x). Under these conditions, the thickness that is really working or ‘‘useful’’ T1(x) is the sum of the thicknesses operating under compression (T/2) and under tensile stress (yt(x)):
4
EIð
d y Þ¼P dx4
ð1Þ
where I is the moment of inertia. Differentiating Eq. (1) twice and four times with the boundary conditions at the ends of the free span (deflection and rotation null), the bending moment, M(x), and the deflection line, y(x), in
T 1 ðxÞ ¼
T þ yt ðxÞ 2
ð3Þ
ð4Þ
Applying the law of Navier sðxÞ ¼ ðMðxÞ=IÞyðxÞ, on the assumption that the stress, s(x), is equal to the maximum tensile strength, st, and y(x) is the thickness under tensile stress, yt(x), the useful thickness of the beam T1(x) is: T 1 ðxÞ ¼
T st I T st ¼ þ 2 MðxÞ 2 ðP=2IÞx2 ðPL=2IÞx þðPL2 =12IÞ
ð5Þ
Replacing the uniformly distributed load, P, in Eq. (5) for its expression in terms of specific weight, g, width, H, and thickness, T, (P ¼ gHT) and considering that the most critical zones are over the abutments (x ¼0 and x ¼L) where the tensile stresses are greater, Eq. (6) is obtained. This equation demonstrates that for a certain rock mass, with a tensile strength st and a specific weight g, the thickness under tensile stress and consequently the useful thickness T1 (in the more critical sections) is a function of the real
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
143
Fig. 2. Ratio thickness/span for different specific weight of the rock masses.
Table 2 Geometry and properties of the rock mass in Kampanzar and Calzada quarries.
Real thickness (T) Width (H) Specific weight (g) Tensile strength (st) Elastic modulus (E) Friction angle (j) Cohesion (c) Poisson’s ratio (n)
Fig. 3. Graphic scheme of the criterion.
thickness of the beam T and its span L: yt ¼
st T 2 ð Þ g L
ð6Þ
By plotting Eq. (6) it is possible to determine, in a graphical way (Fig. 2), the span permitted to avoid failure for a real thickness of beam. This span decreases with an increase in the specific weight of the rock mass, a decrease in the tensile strength of the rock mass, an increase in the thickness layer. From Fig. 2, failure will occur when the tensile thickness is less than half the real thickness (yt oT/2), so by introducing in Fig. 2 the location of half of the real thickness and drawing a horizontal line to cut the curve with the tensile strength of the rock mass the ratio T/L is obtained, and therefore the maximum span that can support the stresses without failure. The scheme of this process is represented in Fig. 3. Of special interest is a comparison between the length of the span obtained graphically and the value obtained numerically and analytically. In this work, two real examples have been assessed, Kampanzar Quarry, where limestone is extracted, and Calzada Quarry, when the rock mass is slate (Table 2). Both of them present similar specific weights, although the tensile strength is very different. 3. First application: Kampanzar quarry 3.1. Calculating maximum span without failure 3.1.1. Numerical analysis In 2003, for environmental reasons, studies of the economic and technical feasibility for transforming the limestone quarry
Kampanzar
Calzada
1m 1m 27 kN/m3 5.57 MPa 100 GPa 301 2 MPa 0.25
2m 1m 27 kN/m3 15 MPa 64 GPa 44.31 10.9 MPa 0.15
into an underground mine began. One of the first steps in the transformation process into an underground mine was the use of informatics simulations. Specifically ALMEC 3D software (Fig. 4) was used to calculate the length of the span inside the rooms and the height of the columns. From this analysis it was concluded that, from a point of view of safety, the maximum span of the rooms and the maximum height of the columns had to be less than 20 m and 25 m, respectively. 3.1.2. Graphical analysis Nevertheless, and in spite of the use of more and more powerful computers, the numerical simulations require great amounts of data and long periods of calculation. In contrast, the use of failure curves (Fig. 2) allows the very fast determination, and with only two mechanical properties, the maximum length of the span that the roof can have to avoid cracking. In the Kampanzar quarry, by introducing into Fig. 2 (right) the position of half of the real thickness (0.5 m) and drawing a horizontal line to cut the curve with the tensile strength of the rock mass (5.57 MPa), a ratio of T/L¼0.05 is obtained. Therefore, with a thickness of 1 m, the maximum span that can support the roof without breaking is about 20 m. This value is similar to the span calculated by the numerical simulations. 3.1.3. Analytical analysis When the theory of elasticity is applied, the value of the maximum span that can support the tensile stress is equal to the value obtained by the informatics simulations. A longer span produces a value of tensile stress (Table 3) greater than the tensile strength. In Table 3, the values from the Stephansson formulation are also presented. In this case, a single-layer roof configuration has been employed, case I (T/Lo1/5) [4]. The values presented in Table 3 establish failure with spans greater than 23 m.
144
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Fig. 4. Distribution of vertical stresses. ALMEC 3D.
Table 3 Kampanzar quarry: analytical results of moment, deflection and tensile stress. Parameter
Span (m)
Elasticity theory
Stephansson formulation
x¼ 0
x¼ L/2
x ¼0
x¼ L/2
Moment (Nm)
15 20 23.5 25 30
5.06eþ 05 9.00eþ 05 1.24eþ 06 1.41eþ 06 2.02eþ 06
2.53eþ 05 4.50eþ 05 6.21eþ 05 7.03eþ 05 1.01eþ 06
2.84e þ05 6.11e þ05 9.07eþ 05 1.05eþ 06 1.60eþ 06
4.75e þ05 7.39e þ05 9.57e þ05 1.06eþ 06 1.44e þ06
Deflection (mm)
15 20 23.5 25 30
0.00 0.00 0.00 0.00 0.00
0.43 1.35 2.57 3.30 6.83
0.05 0.08 0.11 0.12 0.16
0.19 0.43 0.71 0.86 1.57
Tensile stress (N/m2)
15 20 23.5 25 30
3.04eþ 06 5.40eþ 06 7.46e þ06 8.44e þ06 12.12eþ 06
1.15e þ 06 2.70e þ06 3.73e þ 06 4.22e þ 06 6.07e þ 06
1.70e þ06 3.67e þ 06 5.44e þ 06 6.30e þ06 9.60e þ06
2.85eþ 06 4.43eþ 06 5.74eþ 06 6.35eþ 06 8.64eþ 06
In all cases (Table 3) the greatest values of tensile stresses appear over the abutments and with the theory of elasticity, while with the Stephansson formulation the stresses present very similar values over the abutments and at midspan. In the former case, failure begins over the abutments and afterwards occurs at midspan, in accordance with [2]. In the second case, failure is generalised across the span. The difference presented (Table 3) is only due to the fact that two different situations are being compared. The first case (theory of elasticity) is considering a bi-fixed ends beam, while Stephansson considers flexible supports. These distribute the moment along the beam, increasing it at midspan and decreasing it over the abutments.
3.1.4. Comparison between numerical, graphical and analytical approaches The comparison between the different approaches (Table 4) presents very similar values of the maximum span. While the value is equal with numerical and graphical analysis (20 m), this value is slightly larger with the theory of elasticity, but only by 0.5 m. In contrast, the Stephansson formulation displays a maximum span of 23 m. This can be explained because Stephansson’s
method is the only one that considers elastic supports, leading to a more uniform distribution of the tensile stresses along the span.
3.2. Behaviour of the roof with larger spans than the maximum calculated When lengths of span longer than those calculated are considered, the beam breaks over the abutments in the top surface. But with even longer lengths the beam also breaks at midspan in the bottom surface (Fig. 5). In these conditions, when failure is produced, the beam is not a prism but also a section with variable thickness. Following [16] it is possible to calculate the new distribution of moments employing the theory of elasticity, with the hypothesis that in the breakage zones the maximum tensile stress is equal to the tensile strength st. In this case (Fig. 5), and denoting xi (i¼1–6), the points where the limiting surfaces cut the initial section, the new distribution of moments M(x) (Eq. (7)) allows the new deflection (Eq. (8)) to be
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Table 4 Kampanzar quarry: comparison between results. Numerical analysis
Maximum span (m)
Graphical analysis
20
20
145
Table 5 Kampanzar quarry: analytical results of moment, deflection and tensile stress.
Analytical analysis Elasticity theory
Stephansson formulation
20.5
23
Parameter
Moment (Nm) Deflection (mm) Tensile stress (N/m2)
Span (m)
30 30 30
Conjugate beam method x¼ 0
x ¼ L/2
4.94eþ 05 0.00 5.57e þ06
8.53e þ 05 6.84 5.57e þ06
pffiffiffi 4st x Q ðxÞ ¼ 16 3A½ln RðxÞln SðxÞ þ ET pffiffiffi 4st x Q ðxÞ ¼ yðxÞ16 3 A½ln RðxÞln SðxÞ ET pffiffiffi 2st 2 x Q ðxÞx ZðxÞ ¼ 16 3A½ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ þ ET pffiffiffi 2st 2 Z ðxÞ ¼ yðxÞ16 3 A½ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ x Q ðxÞx ET Fig. 5. Kampanzar quarry: useful section after failure.
obtained by means of the conjugate beam method. MðxÞ ¼ s6t T 1 ðxÞ2
if
ðx1 rx rx2 Þ or ðx3 o x r x4 Þ
PL P 2 MðxÞ ¼ PL 12 þ 2 x 2 x
if
ðx2 rx rx5 Þ or ðx6 o x r x3 Þ
MðxÞ ¼ s6t T 1 ðxÞ2
if
ðx5 ox rx6 Þ
2
ð7Þ
pffiffiffi st 2 yðxÞ ¼ 16 3A ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ 2ET x þ Q ðx1 Þ þZ ðx1 Þ 2
PL PL 3 P yðxÞ ¼ 24EI x2 þ 12EI x 24EI x4 i pffiffiffi h st 2 yðxÞ ¼ 16 3A lnRðxÞ1 RðxÞ lnSðxÞ1 SðxÞ þ 2ET x þ Q ðx5 Þ þ Z ðx5 Þ 2
PL P yðxÞ ¼ 24EI x2 þ PL x3 24EI x4 pffiffiffi 12EI st 2 yðxÞ ¼ 16 3A ðln RðxÞ1ÞRðxÞðln SðxÞ1ÞSðxÞ 2ET x þ Q ðx4 Þ þZ ðx4 Þ
By applying Eqs. (7) and (8), the new moment and deflection of a beam with a span of 30 m (greater than the safe length of 20 m) is shown numerically in Table 5 and graphically in Fig. 6. Fig. 6 shows a new distribution of moments and deflections. The shape of the moment distribution is non-continuous, and this can be explained because the second derivate of the deflection (continuous) has to be multiplied, in each section, for the inertia moment. This is directly proportional to the useful thickness, which is variable near to the abutments and at midspan (Fig. 5). if
x1 rx rx2
if
x2 ox rx5
if
x5 ox rx6
if
x6 ox rx3
if
x3 ox rx4
ð8Þ
3.3. Compression arch R(x), RðxÞ, S(x), SðxÞ, Q(x), Q ðxÞ, Z(x), ZðxÞ where the parameters A, A, are defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I2 s4t A¼ 2 3 2 E PT ðL PT48Ist Þ
A¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I2 s4t E2 PT 3 ðL2 PT þ48Ist Þ
RðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ þ 3ETAðL2 PT48Ist Þ 6Is2t
RðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ þ 3ETAðL2 PT þ48Ist Þ 6Is2t
SðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ 3ETAðL2 PT48Ist Þ 6Is2t
SðxÞ ¼
pffiffiffi 3Is2t ðL2xÞ 3ETAðL2 PT þ 48Ist Þ 6Is2t
The Voussoir model and the theory of elasticity are two totally different concepts. Voussoir assumes that a beam with fixed ends and distributed loading yields when the maximum tensile stress exceeds the tensile strength of the rock [9]. In this case, vertical tensile fractures form at the abutments, and the beam becomes simply supported (assuming no slip at the abutments). Then the stress is higher, and a progressive process of cracking at the abutments and at midspan begins. At this point, the development of a compression arch and the generation of a moment between the reaction force at misdpan and at the abutments which acts to resist the moment imposed by self-weight are assumed. To resolve the loss of horizontal stress symmetry, the model considers an average thickness NT for this arch, where N is a reduction coefficient of T. The beam will be at equilibrium when N is closer to 0.75. Although the theory of elasticity does not consider a compression arch, the variable section (Fig. 5) has been adjusted with an arch where all the sections, along the height, work. For that, and with the simplification that the neutral axis coincides with the centre line of the layer, the new neutral axis has adjusted to a second-degree polynomial (a þbxþcx2 ¼0). Parameters a, b and c, which depends on the length and bending of the beam, are
146
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Fig. 6. Kampanzar quarry: distribution of moments and deflections (top); tensile thickness (bottom).
Fig. 7. Distribution of vertical stresses. FLAC 2D.
4. Second analysis: Calzada quarry
calculated by least squares and are shown in Eq. (9). 2
neutral axisðxÞ ¼ 0:0897 þ0:0009x0:00002x
ð9Þ
This curve will be the centre line of the arch, and the distance between the curve and the point of failure x1 will be half of the thickness. In the example (Fig. 6 bottom) the new thickness is 64% of the initial thickness. This percentage is close to 70%, a value considered the equilibrium value according to the Voussoir model [5].
4.1. Calculating maximum span without failure 4.1.1. Numerical analysis In this case, the height of the banks and the volume of the slag heaps were the main reasons for the consideration of the transformation of the quarry into one underground mine. The stability of the exploitation was studied with FLAC 2D software
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
147
Table 6 Calzada Quarry: analytical results of moment, deflection and tensile stress. Parameter
Span (m)
Elasticity theory
Stephansson formulation
x¼ 0
x ¼L/2
x¼ 0
x ¼L/2
Moment (Nm)
47 47.5 51 57.5 67
9.94eþ 06 10.15eþ 06 11.70eþ 06 14.88eþ 06 20.20eþ 06
4.97e þ 06 5.08e þ06 5.85e þ 06 7.44e þ 06 10.10e þ06
6.04eþ 06 6.12eþ 06 7.50eþ 06 10.19eþ 06 14.80eþ 06
8.87e þ06 9.01e þ06 10.05e þ06 12.12e þ 06 15.51e þ 06
Deflection (mm)
47 47.5 51 57.5 67
0 0 0 0 0
16.083 16.778 22.30 36.03 66.416
0.21 0.21 0.23 0.28 0.36
0.81 0.84 1.03 1.48 2.38
Tensile stress (N/m2)
47 47.5 51 57.5 67
14.911e þ06 15.23eþ 06 17.557e þ06 22.32eþ 06 30.301e þ06
7.455e þ 06 7.615e þ 06 8.778e þ 06 11.16e þ 06 15.15eþ 06
9.062e þ06 9.324e þ06 11.25e þ06 15.28eþ 06 22.192e þ06
13.304e þ06 13.52e þ06 15.08eþ 06 18.20eþ 06 23.259e þ06
Table 7 Calzada quarry: comparison between results. Numerical analysis
Maximum span (m)
40
Graphical analysis
47
Analytical analysis Elasticity theory
Stephansson formulation
47
50
Fig. 8. Calzada quarry: distribution of moments and deflections (top); tensile thickness (bottom).
148
J. Alcalde-Gonzalo et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 141–148
Table 8 Calzada quarry: analytical results of moment, deflection and tensile stress. Parameter
Moment (Nm) Deflection (mm) Tensile stress (N/m2)
Span (m)
50 50 50
Conjugate beam method x ¼0
x¼ L/2
8.92e þ06 0.00 15.00e þ06
5.63eþ 06 20.60 8.44e þ 06
(Fig. 7), and although instability was found for spans over 40 m, the rooms and columns were designed with a 20 m width and 25 m height respectively. These dimensions were selected after consideration of the technical and economic viability of the project. 4.1.2. Graphical analysis Similarly to the Kampanzar quarry, it is possible to calculate in a graphical and very fast manner the maximum span of the roof without failure. In this case, introducing a tensile thickness of 1 m (half the real thickness) and drawing a horizontal line to cut the curve with the tensile strength of the rock mass (15 MPa), a ratio T/L¼0.0425 is obtained (Fig. 2 right). From which a maximum span without failure of 47 m is determined. 4.1.3. Analytical analysis The analytical results (Table 6) show how in the previous example the worst values were obtained over the abutments and with the theory of elasticity. An increase in the span produces failure as much over the abutments as at midspan. The Stephansson formulation is less conservative, with values of stresses very similar over the abutments and at midspan. In this case the maximum span to avoid failure with the theory of elasticity is 47 m. While the maximum span with the Stephansson formulation is 50 m. 4.1.4. Comparison between numerical, graphical and analytical approaches In the Calzada quarry there is a greater difference between the results of the three approaches (Table 7). The numerical analysis results in a maximum span of 40 m, while the graphical and theory of elasticity result in 47 m. Again the Stephansson formulation is less conservative, with a maximum span of 50 m due to the distribution of the tensile stresses along the roof. The difference between the results of the numerical evaluation and the theory of elasticity or the graphical analysis is explained for the process of the numerical analysis. In a first step, an approximation is calculated with a mesh not very small. In this case the result (40 m) was considered so good that the evaluation was stopped, and the model was not recalculated. 4.2. Behaviour of the roof with larger spans than the maximum calculated compression arch The new moments and deflections for a beam with a span of 50 m have two inflection points very near the abutments. In this case the length employed is close to that considered safe (47 m), so failure is only over the abutments (Fig. 8). The numerical results (Eqs. (7) and (8)) are collected in Table 8, and the new
distribution of the moments and deflections are depicted in Fig. 8 (top). In these conditions the neutral axis descends very gradually over the abutments, and the neutral axis is adjusted by the curve represented in Eq. (10). neutral axisðxÞ ¼ 0:0092 þ0:0009x0:00002x2
ð10Þ
In this case the compression arch has a thickness about 90% of the initial thickness (Fig. 8 bottom).
5. Conclusions
It is possible to determine the maximum design span in a very
fast manner and with only two variables, allowing a safe underground mine to be designed. The inability of a roof to withstand tensile stress produces failure zones, first over the abutments and later at the midspan of the roof. The useful thickness (that is the thickness effectively supporting the stress) is variable throughout the length of the beam, reaching a minimum at the extremes and the midpoint. The new compression arch can be fitted to a parabola with a high correlation coefficient. It is possible to employ the double theory of elasticity to analyse the stability of the roof beam.
References [1] Adler L., Sun M.C. Ground control in bedded formations. Virginia Polytechnic Institute, Blacksburg; 1968. [2] Goodman RE. Introduction to Rock Mechanics. New York: Wiley; 1989. [3] Barker R.M., Hutt F. Joint effects in bedded formation roof control. In: Proceedings of the 14th US symposium on rock mechanics. Pennsylvania; 11–14 June 1972. p. 247–61. [4] Stephansson O. Stability of single openings in horizontally bedded rock. Eng Geol 1971;5:5–71. [5] Evans WH. The strength of undermined strata. Trans Inst Miner Metall 1941;50:475–532. [6] Seedsman RW. Geotechnical sedimentology – its use in underground coal mining. Int J Coal Geol 2001;45:147–53. [7] Brady BHG, Brown ET. Rock Mechanics for Underground Mining. London: George Allen & Unwin; 1985. [8] Swart AH, Handley MF. The design of stable stope panels for near-surface and shallow mining operations. In: Proceedings of the first international platinum conference ‘Platinum adding value’. Sun City, South Africa; 3–7 October 2004. p. 275–86. [9] Diederichs MS, Kaiser PK. Stability of large excavations in laminated hard rock masses: the voussoir analogue revisited. Int J Rock Mech Miner Sci 1999;36:97–117. [10] Exadaktylos GE, Vardoulakis I, Kourkoulis SK. Influence of nonlinearity and double elasticity on flexure of rock beams – I. Technical theory. Int J Solids Struct 2001;38:4091–117. [11] Exadaktylos GE, Vardoulakis I, Kourkoulis SK. Influence of nonlinearity and double elasticity on flexure of rock beams – II. Characterization of Dionysos marble. Int J Solids Struct 2001;38:4119–45. [12] Vardoulakis I, Exadaktylos GE, Kourkoulis SK. Bending of marble with intrinsic length scales: a gradient theory with surface energy and size effects. J Phys IV 1998;8:399–406. [13] Touloukian YS, Ho CY. Physical Properties of Rocks and Minerals. New York: McGraw-Hill; 1981. [14] Winkler EM. Stone-Properties, Durability in Man’s Environment. New York: Springer; 1973. [15] Farmer IW. Engineering Properties of Rock. London: E&FN Spon; 1968. [16] Timoshenko S, Goodier JN. The´orie de l’e´lasticite´. Paris: McGraw-Hill; 1961.