One of the most important and long-standing applications of tabular layout analysis programs based on the displacement discontinuity boundary element method (for example, MINSIM, MSCALC, MAP3D, TEXAN) is to determine average pillar stress values in narrow stope width excavations in coal, gold and platinum mines. One aspect of the use of these programs that is not always appreciated is that field point stress estimates will depend on the chosen element grid size when using constant strength displacement discontinuity elements.
Read more to find out about a simple extrapolation technique for the numerical estimation of average pillar stress values, as explained by John Napier and Francois Malan of the Department of Mining Engineering of the University of Pretoria.
Ryder and Napier^{1} have noted previously that the error bias in elastic convergence estimates can be eliminated by introducing special stiffness adjustments to mined elements adjacent to the edges of un-mined pillars. This adjustment, unfortunately, does not address the problem of how to correct the grid-size bias that arises in the calculation of average pillar stress values. This issue has been investigated recently in a paper to be published by Napier and Malan^{2}. In particular, it appears that a simple extrapolation technique can be employed to obtain robust estimates of average pillar stress values.
In order to illustrate this procedure, it is assumed that a base solution is established in which the element grid size is relatively uniform (or, ideally, constant) and has a characteristic dimension, g. The average stress in a given pillar, denoted by A(g), is then computed by taking the arithmetic mean of the stress values in the grid elements covering the pillar region. An additional analysis is then carried out in which the grid size is set equal to g/2 and the average pillar stress value arising in the pillar of interest is re-computed and is designated to be A(g/2). The grid-halving procedure implies that for planar mine layouts, four times as many elements are used in the finer scale analysis. The re-meshing can be accomplished by a simple splitting procedure that is applied to each coarse-sized element in the base layout. The average pillar stress value, A(g), that is associated with a grid size, g, may now be estimated by assuming an explicit functional relationship of the form.
(1)
where k, c_{0} ,c_{1},c_{2} etc. are unknown parameters that have to be estimated. In equation (1), A (0) is also not known and represents the ultimate estimate of the “exact” average pillar stress when the grid size is extrapolated to zero. The unknown value of A (0) and the parameters k, c_{0} ,c_{1},c_{2} can now be determined from equation (1) if successive estimates of A(g), A(g/2), A(g/4), A(g/8) etc. are available. For example, if A(g) and A(g/2) are known and if it is assumed that k=1 and c_{1}=0, c_{2}=0 etc. then the “exact” average pillar stress value is given by the linear extrapolation formula
(2)
Examples
In order to illustrate the effectiveness of the extrapolation procedure, consider initially the case of a single strip pillar of width 20 m centrally located between two parallel-sided panels each having a span of 120 m. The layout is assumed to be horizontal and may be solved in plane strain. If the primitive stress at the panel horizon is 100 MPa, then the average pillar stress can be determined analytically^{2} and in this case is equal to 516.24 MPa. Figure 1 shows a plot of successive estimates the average pillar stress values as a function of the element grid size. Applying equation (2) to the two finest grid size estimates with A(g) = 509.94 MPa and A(g/2) = 513.10 MPa respectively yields an extrapolated estimate of A(0) = 516.26 MPa which has an error of less than 0.004 %. It is clear from Figure 1 that the values of the estimated average pillar stress in this example follow a nearly perfect linear trend. If equation (2) is applied to the two coarsest grid estimates with A(g) = 464.56 MPa and A(g/2) = 490.57 MPa respectively, the extrapolated estimate is A(0) = 516.53 MPa which is still extremely accurate with an error of less than
0.07 %. The linear trend of the average pillar stress as a function of the grid size is unfortunately not found to be true for general planar layouts as illustrated in the example depicted in Figure 1.
As a second example, consider the case of a circular pillar of radius 30 m located at the centre of an annular excavation with an outer radius of 150 m. The primitive stress at the reef horizon is assumed to be 100 MPa. Figure 2 shows a plot of the average pillar stress values corresponding to each assumed grid size. It is apparent that in this case, the average stress does not yield a linear trend as a function of the grid size. Employing equation (1) with but with two non-zero parametersand, admits a quadratic extrapolation formula of the form
(3)
Using the best three estimates shown in Figure 2 with g = 5 m and A(g) = 424.6, A(g/2) = 430.9 and A(g/4) = 432.9, yields the extrapolated average pillar estimate A(0) = 434.1 MPa. The consistency of the quadratic extrapolation formula (3) can be tested by considering the coarser average stress sequence starting at g = 10 m and setting A(g) = 388.9, A(g/2) = 424.6 and A(g/4) = 430.9 respectively. In this case the extrapolated estimate using equation (3) is A(0) = 429.6 MPa. This value appears to be incorrect in relation to the sequence of estimates shown in Figure 2 as it now falls below the value A(2.5) = 430.9 MPa.
It is evident that the extrapolated value of A(0) that is obtained from equation (1) could also be determined by employing k andas free parameters. Using this scheme, implies a power-law relationship between A(g) and g. In this case, it may be shown that
(4)
and the power exponent k is given by the formula
(5)
Employing the finest grid size sequence with A(g) = 424.6, A(g/2) = 430.9 and A(g/4) = 432.9, in equation (4), yields the extrapolated average pillar estimate A(0) = 433.8 MPa. The consistency of this estimate may in turn be checked by employing the average stress value sequence, A(g) = 388.9, A(g/2) = 424.6 and A(g/4) = 430.9 which yields the extrapolated estimate A(0) = 432.3 MPa which can be seen to be in satisfactory agreement with the estimate of 433.8 MPa. (The exact value of the average pillar stress in this example is not known; an independent numerical estimate of the value of A(0) for the circular pillar was found to be A(0) ~ 435 MPa). Care should obviously be exercised when using equation (4) to ensure that the denominator is not close to zero as would occur when considering a nearly linear sequence of values as shown in Figure 1.
Conclusions
The examples illustrated in this short note illustrate that average pillar stress values estimated using standard, constant strength displacement discontinuity boundary element codes are dependent on the element grid size. A simple correction strategy is presented to allow the extrapolation of the estimated pillar stress values to achieve an accurate asymptotic estimate of average pillar stress that is grid size independent. The procedure discussed here is essentially equivalent to the numerical method known as Richardson’s extrapolation technique^{3}. The application of the extrapolation formula requires at least two estimates of the average pillar stress evaluated with a chosen element size, g, and with the grid size set to g/2 in order to apply the linear extrapolation formula represented by equation (2). For more accurate estimates of average pillar stress (and to confirm whether linear extrapolation is justified), at least three values of average stress have to be estimated using successive grid sizes g, g/2 and g/4. It appears that the power law formula expressed by equation (4) provides a useful representation of the limiting error behaviour unless explicit knowledge is available concerning the choice of the exponent parameter k.
References
1. Ryder, J.A. and Napier, J.A.L. Error analysis and design of a large-scale tabular mining stress analyser. 5th Int. Conf on Num Methods in Geomech., Nagoya, Japan, 1985, pp. 1549–1555.
2. Napier, J.A.L. and Malan, D.F. Numerical computation of average pillar stress and implications for pillar design. Paper accepted for publication, J. S. Afr. Inst. Min. Metall., 2011.
3. Kopal, Z. Numerical Analysis. John Wiley & Sons, Second Edition, 1961.
Figure 1. Single strip pillar average pillar stress magnitude estimated as a function of the element grid size. The pillar has a width of 20 m and is located between two parallel-sided panels each having a span of 120 m. | |
Figure 2. Effect of element grid size on the estimates of average stress in a circular pillar of radius 30 m located at the centre of an annular excavation with an outer radius of 150 m. (See also Napier and Malan^{2}). |