This equation now shows that the peak abutment
stress for the laminated model is proportional to the
square root of the depth (H), similar to the peak
abutment stress for the empirical ALPS formula as
shown in equation 10.

Equation 14 still indicates that the peak abutment
stress for the laminated model is directly
proportional to the square root of the seam modulus
(E,) and inversely proportional to the square root of
the overburden modulus (E) and the extraction
thickness (M). For most coal measure rocks, the
ratio of the seam modulus to the overall overburden
modulus will probably remain fairly constant
between different mines and not greatly affect the
relative value of the peak abutment stress.
Conversely, the extraction thickness (M) distinctly
varies between different mines and it is shown to be
inversely proportional to the peak abutment stress
for the laminated model. This means that thicker
seams will have lower peak abutment stresses and
tend to spread the abutment load over a larger
distance from the panel. This result (derived from
the laminated model) suggests that the seam
thickness is an important factor to consider in
determining the distribution of the abutment stress at
the edge of an extraction panel.

ABUTMENT LOADING - MAGNITUDE

In addition to the load distribution, the other factor
to determine regarding the abutment load is the
magnitude. In ALPS, a simple geometric
conceptualization utilizing an abutment angle (B) is
used (see Figure 1). The magnitude of the side
abutment load (L,) is calculated as the weight of the
overburden within a wedge defined by this abutment
angle and a vertical line from the edge of the panel.
For a supercritical panel the formula is:

L, = H? (tan) (2) as)

and for a subcritical panel the formula is:
Hp) _{_??
2 Stan) |”

- where: P = the panel width

L = (16)

A panel is subcritical when:

a7

461

With the recommended value of B = 21° (Mark
1992), the transition from supercritical to subcritical
occurs at a depth-to-width ratio (H/P) of about 1.3.

In order to investigate the change in the magnitude
of abutment loading with depth, the equations for the
side abutment load can be normalized by dividing
through by half the total load (L,) over the longwall
panel in order to calculate a percentage load

= YHP
a)

¢

(18)

Then, the side abutment load equations can be
further normalized by writing them as a function of
the ratio of panel depth to panel width (H/P). For
the supercritical case (equation 15), the result is:

H
P

zr (tan B) (19)

For the subcritical case (equation 16), the
normalization result is:
L 1

ell -
L
‘ 3) 4 (tan)

These last two equations show that the percentage of
the total panel cover load that results in the side
abutment load is linear with relation to the depth-to-
width ratio up to the transition point from
supercritical to subcritical. After the transition, the
percentage of side abutment load for subcritical

(20)

Abutment Load / Total Load (%)

13
Pane! Depth / Panel Width

Figure 5. A comparison of the percentage
magnitude of the abutment load between the ALPS
method and the laminated overburden model.


program. Proceedings of the 3rd North American
Rock Mechanics Symposium. 2:381-390. Mexico
Sociedad Mexicana del Mecdnica de Rocas.

Holland, C.T. 1964. The strength of coal in mine
pillars. 6 U.S. Symposium on Rock Mechanics:
450-466. Missouri: University of Missouri.

Hustrulid, W. 1976. A review of coal pillar strength
formula. Rock Mechanics. 8:115-145.

Jaeger, J.C. & N.G.W. Cook 1979. Fundamentals
of Rock Mechanics. London: Chapman and
Hall.

Mark, C. 1990. Pillar design methods for longwall
mining. BuMines: IC 9247.

Mark, C. 1992. Analysis of longwall pillar stability
(alps): an update. Proceedings of the Workshop
on Coal Pillar Mechanics and Design. BuMines:
IC 9315:238-249.

Mark, C. & F.E. Chase 1997. Analysis of retreat
mining pillar stability (armps). Proceedings: New
Technology for Ground Control in Retreat
Mining. NIOSH: IC 9446:17-34. ©

Mark C. & A.T. Iannacchione 1992. Coal pillar
mechanics: theoretical models and field
measurements compared. Proceedings of the
Workshop on Coal Pillar Mechanics and Design,
BuMines: IC 9315:78-93.

Obert, L. & W.I. Duval 1967. Rock mechanics and
the design of structures in rock: 531-545. New
York: Wiley.

Pappas, D.M. & C. Mark 1993. Behavior of
Simulated Longwall Gob Material. BuMines: RI
9458.

Park, D.W. 1992. Numerical modeling as a tool for
mine design. Proceedings of the Workshop on
Coal Pillar Mechanics and Design. BuMines: IC
9315:250-268.

Peng, S.S., & H.S. Chiang 1984. Longwall mining.
New York: Wiley.

Salamon, M.D.G. 1992. Strength and ‘stability of
coal pillars. Proceedings of the Workshop on
Coal Pillar Mechanics and Design. BuMines: IC
9315:94-121.

Salamon, M.D.G. 1989. Subsidence prediction using
a laminated linear model. Proceedings of the 30th
U.S. Symposium of Rock Mechanics: 503-510.
Rotterdam: Balkema.

Salamon, M.D.G. & A.H. Munro 1967. A study of
the strength of coal pillars. J. S. Afr. Inst. Min.
Metall. 68:55-67.

Salamon, M.D.G. 1964. Elastic analysis of
displacements and stresses induced by the mining
of seam or reef deposits, part ii. J. S. Afr. Inst.
Min. Metall. 64 (6):197-218.

Wagner, H. 1974. Determination of the complete
load deformation characteristics of coal pillars.
3” ISRM Congress on Rock Mechanics, Denver:
1076-1081.

Wilson, A.H. 1973. An hypothesis concerning pillar

stability. Min. Eng. London. 131(141) June:409-
417.

Yang, G. 1992. Numerical approach to the prediction
of subsidence due to longwall coal mining using
a laminated model. Ph.D. Thesis. Colorado:
Colorado School of Mines.

Zipf, R.K. 1992a. Mulsim/nl application and
practitioner's manual. BuMines: IC 9322.

Zipf, R.K. 1992b. Mulsim/nl theoretical and
programmer's manual. BuMines: IC 9321.


20.29 y 1 EH 8)
t = 20: -v =u (8)

For a typical seam modulus (E,) of 2 GPa in the
laminated model, equation 8 provides a fitted
lamination thickness of 157 m. The plot of the
abutment stress curve for the laminated overburden
model with a fitted lamination thickness of 157 m is
shown together with the empirically determined
abutment stress in Figure 4. The degree of
agreement between the two curves is very good, but
the lamination thickness of 157 m needed to provide
this degree of fit is unrealistic.

Additional insight about abutment loading can be
gained by taking a even closer look at the abutment
load. distribution (the shape of the curves) as
represented by the empirical formula (equation 1)
and the laminated model (equation 5). First, assume
that the total overburden load is carried on the
abutments (there is no gob load), then:

L, = ql = yHL (9)

and if equation 1 is expanded using equation 2
andonly the stress at the edge of the panel is
considered, the peak abutment load is:

o(L) = (0.5843)y VAL (10)

This equation shows that the empirically-determined
peak abutment stress at. the edge of the panel is
directly proportional to the overburden density (y)
and the panel half-width (L). These relationships
seem intuitively logical. Equation 10 also shows
that the magnitude of the empirical abutment stress
is proportional to the square root of the depth. This is
an interesting result. However, if one considers
equation 2 which shows that the extent of the
empirical abutment stress is also proportional to the
square root of the depth, then the combination of
these two relationships would give the logical result

that the total abutment load is directly proportional

to the depth.

If equation 5 for the laminated overburden model
is similarly expanded using equation 6, and only the
stress at the edge of the panel is considered, then the
following relationship results:

o,(L) = (2.632)yHL at)

EMt

In equation 11, the peak abutment stress at the edge

460

of the panel for the laminated model is also directly
proportional to the overburden density (y) and the
panel half-width (L) as was the peak abutment stress
for the empirical formula. However, equation 11
shows that the peak stress for the laminated model is
directly proportional to the depth (H) as opposed to
being proportional to the square root of the depth, as
was the case for the empirical formula. Equation 11
also shows that the peak abutment stress for the
laminated model is directly proportional to the
square root of the seam modulus (E,) and inversely
proportional to the square root of the overburden
modulus (E), extraction thickness (M) and
lamination thickness (t). These last relationships all
seem reasonable considering that elastic plate theory
was used to derive the laminated model. However,
if these relationships are reasonable, why are the
relationships not sufficiently evident in the empirical
data to have been incorporated into equation 1?

The effect of depth needs to be more closely
examined. In previous work, it was determined that
for optimum subsidence prediction with the
laminated model, the lamination thickness should
increase with depth (Yang 1992, Salamon 1989).

To formalize this relationship, Yang proposed an
inherent overburden constant, w.

o- H/s0-¥)

a2)

In later work with subsidence prediction using the
laminated model, the same trend toward increasing
lamination thickness with increasing was also
found (Heasley & Barton 1998). Similarly, when
modeling stresses with the laminated model, it was
found that the optimum lamination thickness is
proportional to the area-of-interest, such that thinner
laminations were best for modeling the small scale
inter-seam pillar stresses, but thicker laminations
were optimum for modeling wide area longwall
abutment stresses (Heasley 1998).

If the value of w is indeed a constant for the given
strata, then equation 12 implies that the lamination
thickness should be directly proportional to the

depth (rearranging equation 12):
1 = 2G) (13)
a

If equation 13 is then substituted back into equation
11:

6,(L) = 2y/HLo (4)



panels asymptotically approaches 100% as the
depth-to-width ratio increases. The relationship
between the percentage of total panel load that goes
into the side abutment load and the panel depth-to-
width ratio is shown in Figure 5. The ALPS method
essentially implies that, for a given panel width, the
gob load is constant for a seam deeper than a depth-
to-width ratio (H/P) of about 1.3. Therefore, the gob
load in a 200 m wide panel at a depth of 240 m
would be identical to the gob load in a 200 m wide
panel at a depth of 600 m. This relationship does not
seem intuitively reasonable and conflicts with the
behavior implied by the homogeneous-elastic and
the laminated models as will be shown below.

With the homogeneous elastic or the laminated
overburden model, the magnitude of the abutment
load is determined by a complex interaction between
the displacement of the overburden, and the
compaction and support provided by the gob
material. However, some intuition into the depth-
related response of the abutment load can be gained
by closely examining the analytical displacement
functions for we overburden and the gob.

If there is no support provided by the gob, the

‘convergence of the seam in the panel for the
homogeneous elastic model (s,) can be determined
as (Jaeger & Cook 1979):

30) = 40 -v) YO

(21)

This equation states that the convergence of the seam
in the panel is proportional to the depth (H) and to
the distance from the edge of the panel (L? - x”), and
inversely proportional to the overburden modulus
().

For the laminated overburden model, the formula
for the convergence of the seam in the open panel
(s) is very similar (Heasley 1998):

s(x) = WRG =) 1H 2x2)

(22)

This equation states that the convergence of the seam
in the panel is proportional to the depth (H) and to
the square of the distance from the edge of the panel
(lL? - x’), and inversely proportional to the
overburden modulus (E) and lamination thickness
(t). If @ is a constant as in equation 12; and
therefore, the realistic lamination thickness is a
function of the depth as in equation 13, then
equation 22 can be written as:

2
s(x) = 22% @?- 2%) @3)

462

In this equation, the convergence in the panel is now
only proportional to the square of the distance from
the edge of the panel (L? - x?) and to the overburden
constants @ and y, and inversely proportional to the
overburden modulus (E). Thus, once w is
considered to be a constant, the convergence across a
panel in a given geology is also constant, regardless
of the depth.

In general, gob material is considered to be strain-
hardening such that the stiffness of the material
increases with increasing strain. Basically, the
gtanular/blocky gob material becomes stiffer as it is
compacted and the void ratio decreases (Zipf 1992a,
b). This exact response was documented by Pappas
and Mark (1993) and incorporated into the
MULSIM (Zipf 1992a, b) and the LAMODEL
(Heasley 1998) programs using the following
formula, where the gob stress (0,) is related to the
gob strain (e) by the equation:

Es,

(2)

where: E; is the initial tangent modulus at zero stress,
Ey is the final tangent modulus at the ultimate
stress (0,), and
nis the gob height factor (after Zipf 1992a b).

5-8,

(24)

A schematic of the stress-strain curve for the
material defined by equation 24 is given in Figure 6.

If the overburden displacement is considered to be
linearly proportional to the depth (as with the
homogeneous elastic overburden, equation 21, and
the laminated overburden,equation 22) and the gob
material is strain-hardening, then the gob should
support an increasing percentage of load as the panel
gets deeper. Therefore the percentage of the
abutment load should decrease with depth (see curve
2 in Figure 5). If, on the other hand, the overburden
displacement is constant with depth (as in equation
23 where the lamination thickness is proportional to
the depth), then the change in the gob load will be

o
KEY

E; Initial modulus
E E¢ Final modulus
J Oy Ultimate stress

Strain-Hardening

Figure 6. A schematic of the stress-strain
relationship for the strain-hardening gob material.


Pacific Rocks 2000, Girard, Liebman, Breeds & Doe (eds) © 2000 Balkema, Rotterdam, ISBN 90 5809 155 4

The forgotten denominator, pillar loading

K.A. Heasley
National Institute of Occupational Safety and Health, Pittsburg, Pa., USA

ABSTRACT: In the last few decades, a considerable amount of effort was directed at accurately determining
the coal/pillar strength to use for safely designing coal mines. The outcome of this early work was the well
known Obert-Duvall, Holland-Gaddy, Bieniawski, and Salamon-Munro equations for coal pillar strength. All
of these equations were developed for, and were calibrated with, pillars in a large room-and-pillar area such
that the loading could be determined using the "tributary-area" theory. In order to account for the abutment
loads associated with full-extraction mining, the empirical methods have typically adopted a simple
conceptualization of the abutment load through use of an "abutment angle.” Short of a complete numerical or
analytical analysis of the coal seam and surrounding strata, very little work has been ¢‘rected toward refining
the empirical analysis of pillar loading. It appears that the vast majority of the research has been directed at
determining the pillar strength, the numerator of the safety factor equation, when the denominator of the
safety factor equation, the pillar loading, plays an equally important role in pillar design. This paper will
address the deficit of pillar loading research by exploring the accuracy of the empirical abutment load
calculations using insight provided by an elastic overburden model, a laminated overburden model and field
observations. Ultimately, it is determined that a constant abutment angle probably over-predicts the abutment
load as the mining depth increases.

INTRODUCTION actual use and encapsulate considerable knowledge
of coal pillar behavior into a simple practical form
(Mark & Iannacchione 1992). All of these equations
were developed for, and were calibrated with, pillars
in a large room-and-pillar area such that the loading
could be determined using the “tributary-area”

In the last few decades, a considerable amount of
effort has been directed at accurately determining the
coal pillar strength to use for safely designing coal
mines. Some of the first scientific research in this

area consisted of determining the relationship
between coal strength and the shape and size of
laboratory and field specimens (Gaddy 1956,
Holland 1964, Obert & Duvall 1967, Bieniawski
1968, Wagner 1974, Hustrulid 1976). These
experimentally-derived relationships between coal
strength, and specimen shape and size were then
used to develop formulas for estimating pillar
strength in the field. A further refinement of this
empirical technique for determining pillar strength
was to use a statistical analysis of pillar performance
in the field in order to enhance the parameters of the
experimentally-determined size and shape equation
or to provide a factor of safety for practical
application of the equation (Holland 1964, Salamon
& Munro 1967, Bieniawski 1983). The outcome of
this early work was the well-known Obert-Duvall,
Holland-Gaddy, Bieniawski, and Salamon-Munro
equations for coal pillar strength. These empirical
equations have been validated through many years of

theory.

‘As longwall mining became more prevalent, the
classic pillar design equations (Mark &
Tannacchione 1992) were extended to designing
pillars for gateroads. In order to account for the
abutment loads associated with full-extraction
mining, the empirical methods have typically

Hen B al PI2
< — KE
tbe / Vo tbee| |

al / \ I _ a
4s \ |
5 \t \/ B\\
Mined out panel net pe
Supercritical Subcritical
Figure 1. The coi ization of a side abutment

load angle (after Mark, 1992).

457


Proportional to the ratio between the gob modulus at
the equilibrium stress and the modulus of the coal as
in the third curve in Figure 5. For this third curve,
the equilibrium gob modulus and the modulus of the
coal are fairly equal; therefore the percentage of
abutment load is fairly constant with changing
depth. (For the LAMODEL plots in Figure 5, the
input parameters were: a panel width of 200 m (L =
100), an extraction thickness (M) of 2 m, an elastic
modulus of the rock mass (E) of 20 GPa, a Poisson’s
Ratio (v) of the rock mass of 0.25, a constant
lamination thickness (t) of 5.5 m or a constant w of
7.0, an initial tangent modulus (E;) of 0.69 MPa, a
final tangent modulus (E,) of 1.65 GPa, an ultimate
stress (o,) of 27.6 MPa, and a gob height factor (n)
of 1. For the various depth-to-width ratios, the depth
was varied between 80 and 640 m.)

Within realistic limits for the equilibrium gob
modulus, the third curve in Figure 5 could easily
bend a little either down or up. However, it would
have to be a very “soft” gob for the abutment load
calculated using either the homogeneous elastic
model or the laminated model to behave similarly to
the behavior of the abutment load as calculated by
the ALPS method in the first curve. This result
derived from the analytical models, that the
percentage of the overburden load carried on the
abutments should be constant or even decrease with
depth, suggests that the constant abutment angle
used by the ALPS method over predicts the amount
of the abutment load as the depth increases. A
possible correction would be to decrease the ALPS’s
abutment as the overburden increases.

This actual trend was recently observed in
Australian longwall panels (Colwell et al. 1999),
where the apparent abutment angle was back
calculated from field measurements and found to be
much smaller than the ALPS default (21°) for the
deeper panels. In fact, it was noted that: “the
abutment angles calculated for the two deepest
mines, ..., are the smallest of any in the database,”
5.9° and 8.5°. A smaller abutment angle at depth
would also help explain a conundrum with the
“Analysis of Retreat Mining Pillar Stability”
(ARMPS) method which also uses a constant

* abutment angle for calculating the pillar load (Mark
& Chase 1997). In the ARMPS database, 70% of
the case histories deeper than 300 m are successful
with a Stability Factor less than 1.0.

CONCLUSIONS

Based on an analysis of the abutment stress implied
by the empirically-derived ALPS method, the
. abutment stress derived from the homogeneous
elastic model and the abutment stress derived from
the laminated overburden model, a number of
interesting results can be emphasized. First, using

463

realistic parameters, the abutment load distribution
used in the ALPS method is not very close to the
load distributions determined from either the
homogeneous elastic model or the laminated model.
The abutment load distribution from the laminated
model can be adjusted to fit the empirical abutment
load distribution, but the parameter values needed to
provide this degree of fit are unrealistic. Second, the
laminated model analysis did suggest that the seam
thickness, which is not included in the ALPS
abutment load determination, may be an important
factor to consider in determining the distribution of
the abutment stress at the edge of an extraction
panel. Finally, in comparing the response of the
total magnitude of the abutment load to changes in
depth as computed by the ALPS method, the
homogeneous elastic model and the laminated
model, it appears that the constant abutment angle
used by the ALPS method probably over predicts the
amount of the abutment load as the depth increases.
This result suggests that some type of systematic
decrease in the abutment angle with increasing depth
may be more realistic.

REFERENCES

Barron, K. & Y. Pen 1992: Revised model for coal
pillars. Proceedings of the Workshop on Coal
Pillar Mechanics and Design. BuMines: IC
9315:144-157.

Bieniawski, Z.T 1968. In-situ strength and
deformation characteristics of coal. Eng. Geol.
2(5):325-340.

Bieniawski, Z.T. 1983. New design approach for
room-and-pillar coal mines in the USA. 5“ ISRM
Congress on Rock Mechanics: E27-E36.
Rotterdam: Balkema.

Colwell, M., R. Frith & C. Mark 1999. Analysis of
longwall tailgate serviceability (ALTS): A chain
pillar design methodology for australian
conditions. Proceedings of the Second
International Workshop on Coal Pillar
Mechanics and Design. NIOSH: IC 9448:33-48.

Gaddy, F.L. 1956. A study of the ultimate
compressive strength of coal. Bulletin of the
Virginia Polytechnic Institute. Engineering
experiment Station. 112(August):1-27.

Hasenfus, G.J. & D.W.H. Su 1992. Comprehensive
integrated approach for longwall development
design. Proceedings of the Workshop on Coal
Pillar Mechanics and Design. BuMines: IC
9315:225-237.

Heasley, K.A. 1998. Numerical modeling of coal
mines with a laminated displacement
discontinuity code. Ph.D. Thesis. Colorado:
Colorado School of Mines.

Heasley, K.A. & T.M. Barton 1998. Subsidence
prediction using a laminated, boundary-element


adopted a simple conceptualization of the abutment
load through the use of an “abutment angle”. In this
concept, the abutment load is the weight of the
wedge of overburden material defined by the
abutment angle (B) and a vertical line at the edge of
the panel (see Figure 1, after Mark 1992). Also, in
response to the gateroad pillar design problem, a
new and more complex concept of pillar behavior
based on a confined core surrounded by a crushed or
yielded coal zone was developed and advocated
(Wilson 1973, Barron & Pen 1992, Salamon 1992).
In the original confined core design method, the
abutment loads were calculated using the abutment
angle technique. More recently, the complete
deformation of the rock mass-coal system has been
analyzed for determining the pillar behavior and
loading conditions (Salamon 1992, Hasenfus & Su
1992, Park 1992),

Short of a complete numerical or analytical
analysis of the coal seam and surrounding strata,
very little work has been directed toward refining the
empirical abutment angle analysis of pillar loading.
It appears that the vast majority of the research has
been directed at determining the pillar strength, the
numerator of the safety factor equation, when the
denominator of the safety factor equation, the pillar
loading, plays an equally important role in pillar
design. Recently, some field measurements in
Australia (Colwell et al. 1999) and observations in
the United States (Mark & Chase 1997) have
indicated that the constant abutment, angle concept
for pillar loading may need some refinement. This
paper will address the deficit of pillar loading
research by exploring the accuracy of the empirical
abutment load calculations using insight provided by
an elastic overburden model, a laminated overburden
model and field observations.

ABUTMENT LOADING - DISTRIBUTION

It is informative to investigate abutment loading and
the differences in abutment loading calculations used
by various empirical pillar design and analytical
modeling techniques. First, let’s examine the
empirical Analysis of Longwall Pillar Stability
(ALPS) method (Mark 1990) which is widely used
to help design pillars in longwall gateroads. For the
ALPS method, Mark (1990) tabulated numerous
field measurements. of abutment loads where he
found’ that the measured distribution of induced
abutment stress (o,) follows the equation:

ee” OD

5(x) @,-x)? 0)

where: x = the distance from the center of the panel,
L = the half width of the panel,
L, ¢ total side abutment load,
D, = the maximum horizontal extent of the
abutment stress from the panel edge.
(x>Land x <D,)
The maximum horizontal extent of the abutment
stress was also determined from field measurements
(Peng & Chiang 1984, Mark 1990) to be equal to (in
metric units):

D, = L +93 030

(2)

where: H = the depth of overburden (see Figure 2,
after Mark 1992).

(Equations 1 and 2 were modified from Mark’s
original formula by shifting the x origin from the
edge of the panel to the center of the panel to be
consistent with the following derivations for the
homogeneous and laminated overburden models.
Also, the units were converted to metric units.)

LL

|} Pillar A ——|=- Pillar B —-|-— Barrier pillar —~

Figure 2. Distribution of the side abutment
load (after Mark, 1992).

Next, consider the analytical abutment stress
predicted at the edge of a two-dimensional slot in an
infinite, homogeneous, isotropic, elastic model (0,)

as given by (Salamon 1964):
9, (2) = 4 (for x > L)
i ers ®

where the virgin in situ stress (q) can be written as:
q*1H @)

with: =the overburden density.

458


This equation provides the stress distribution at the
edge of a theoretical longwall panel where: (1) the
coal seam and overburden are all one elastic
material, (2) the width of the panel is negligible in
comparison to the depth, and (3) the thickness of the
seam is negligible in comparison to the width of the
panel. It is interesting to note that equation 3 is
independent of material properties, and because of
the assumptions in the derivation, the seam thickness
does not appear in the equation.

Finally, consider the analytical abutment stress
predicted at the edge of a two-dimensional slot in the
laminated overburden model (o,) with
“homogeneous stratifications” as represented by
LAMODEL (Heasley 1998).

EL -Yab-k
sg VExue (5)

o(*) = qlee

where: E, = the elastic modulus of the seam,
E = the elastic modulus of the overburden,
M = the extraction thickness,
(for x > L)

and the lamination constant, i, is defined as:

Pe ee
{ 12(1- v4)

where: t = the lamination thickness,
v= the Poisson’s Ratio of the overburden.

©)

In order to plot and compare the abutment stresses
computed from equations 1, 3 and 5, some “typical”
values were assumed for the geometric and rock
mass parameters: a panel width of 200 m (L = 100
m), an overburden depth (H) of 160 m (q = 4 MPa),
an extraction thickness (M) of 2 m, an elastic
modulus of the rock mass (E) of 20 GPa, a Poisson’s
Ratio (v) of the rock mass of 0.25, a lamination
thickness (t) of 15 m, and no gob load (L, = qL).
Using these values for the parameters, the abutment
stresses at the edge of a simulated panel for the
empirical ALPS formula (equation 1), the
homogeneous elastic overburden formula (equation
3), and the laminated overburden formula (equation
5) are plotted in Figure 3. (It should be noted that
equations 1 and 5 calculate induced stresses, and that

_ equation 3 calculates the total stress. Therefore, in
the plots, the virgin overburden stress (q) has been
added to the results from equations 1 and 5 to
provide a direct comparison with the total abutment

« stress values from equation 3.)

In Figure 3, it can be seen that the homogeneous
elastic abutment stress has a relatively sharp, infinite

10 20 30 40 50
Distance from Edge of Panel (m)

Figure 3. Comparison of the longwall abutment
stress computed from the homogeneous elastic model,
the laminated model and the empirical formula.

peak at the edge of the panel and approaches virgin
vertical stress asymptotically with increasing
distance from the panel. In contrast, the abutment
stress in the laminated overburden is finite at the
panel edge and asymptotically approaches the virgin
overburden stress (q) less rapidly. Neither of these

Stress (MPa)
8 8 8 8 8
\ }

Empirical Abutment Stress

10 i
° Laminated (t=187) Overburden
ry 1

10 20 30 40 80
Distance from Edge of Panel (m)

Figure 4. Plot of the laminated abutment stress
fitted to the empirical formula.

mathematical models
parameters) comes very
close to matching the empirical abutment stress
curve which is much flatter.

However, if the abutment stress level in the
laminated model and the stress level obtained from
the empirical formula are equated at the edge of the
seam (x=L):

2E, 32,
9(L) = aL) ay = =o(L) (7)

Then, by expanding 4, L,, and D,, and solving for the
lamination thickness (t), the value of t which ensures
this equality can be determined as:

(using the assumed

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