Knee

Soe vies miles

Figure 10.—Contour map of 99% total stress in study area (contour Interval, 40 pel).



8

RADIUS=PYRAMID RADIUS(VALUES(COUNT,1))
STRESS=TOTAL STRESS CALCULATED FOR GIVEN RADIUS(VALUES(COUNT,2))
X1=SUM OF X**(-3)
X2=SUM OF X**(-6)
Y1=SUM OF Y
YX=SUM OF YX**(-3)
SE=RESIDUAL MEAN SQUARE
SY=CORRECTED MEAN SUM OF SQUARES
CALCULATE NECESSARY SUMS
DO 100 COUNT=5,12
Y1=Y1+VALUES(COUNT,2)
X1=X1+VALUES(COUNT,1)**(-3)
X2=X2+ VALUES(COUNT,1)**(-6)
YX = YX + VALUES(COUNT,2)*VALUES(COUNT,1)**(-3)
100 CONTINUE
C CALCULATE BO, THE INTERCEPT, AND B1, THE SLOPE OF THE EQUATION
CD, D0, AND D1 ARE 2x2 DETERMINANTS
D=X1*X1-NUM*X2
D0=X1*YX-Y1*°X2
D1=Y1*X1-NUM*YX
C BO, THE INTERCEPT, IS THE ESTIMATE OF THE FULL TOPOGRAPHIC LOAD
BO=D0/D
Bi=D1/D
C CALCULATE A CORRELATION COEFFICIENT SQUARED (RSQU)
DO 300, COUNT=5,12
SE=SE+ ((VALUES(COUNT,2)-(B0+B1*VALUES(COUNT,1)**(-3)))**2)/
C (NUM-2)
SY=SY+((VALUES(COUNT,2)-Y1/NUM)**2)/(NUM-1)
300 CONTINUE
C THE CORRELATION COEFFICIENT SQUARED
RSQU=1-SE/SY
PRINT (301), ESTIMATED STRESS =’,B0, PS?’
PRINT (302)SLOPE=’,B1
PRINT (303)’CORRELATION COEFFICIENT SQUARED =’,RSQU
301 FORMAT (A17,F8.2,A4)
302 FORMAT (A6,F13.2)
303 DO 300, COUNT=5,12
SE=SE+ ((VALUES(COUNT,2)-(B0+B1*VALUES(COUNT,1)**(-3)))**2)/
C (NUM-2)
SY=SY + ((VALUES(COUNT,2)-Y1/NUM)**2)/(NUM-1)
300 CONTINUE
C THE CORRELATION COEFFICIENT SQUARED
RSQU=1-SE/SY
PRINT (301), ESTIMATED STRESS =’,B0,’ PS?’
PRINT (302),SLOPE=’,B1
PRINT (303)’CORRELATION COEFFICIENT SQUARED =’,RSQU
301 FORMAT (A17,F8.2,A4)
302 FORMAT (A45,F8.6)
C CALCULATE PYRAMID SIZE THAT WILL PROVIDE 95% AND 99% OF TOTAL STRESS
B095=(0.95)*(B0)
B099=(0.99)*(B0)
L=10/30
X95 = ((B095-B0)/B1)**(-L)
X99 = ((B099-B0)/B1)**(-L)

aaaaaaaaa


average distance for each square along the perimeter. The
total stress beyond the pyramid of influence can be cal-
culated by integration of equation 19 from the pyramid
load application half width to infinity:S

So - [ow"aw P, (20)

where Sp = stress at point of calculation due to to-

pography beyond zone of influence.

Integration of equation 20 yields

Sg =CH?P. (21)

The total stress due to topography at the point of cal-
culation can be calculated by adding together the stress
calculated from the pyramid of influence and the stress

beyond the pyramid (equation 22),
S. = Sy + Sp, (2)
where S, = total stress from topography above calcu-

lation surface,

and Sjy_= stress calculated from pyramid of influ-
ence,

Substituting equation 21 into equation 22 and rearranging
the terms yields

Sy = S,- CH? @). (3)

Although the equation was originally applied to a flat
surface, when the pyramid of influence is large, the varia-
tion of the loads beyond the pyramid width will not have
a large effect on the stresses at the point of calculation.
The number of cells beyond the pyramid base is very
large, but because the stress value that each would con-
tribute is very small, an average load for each cell can be
used. This average load (Q,) is the total sum of the ap-
plied loads on the dA’s divided by the total number of
cells beyond the pyramid of influence. Equation 23 can
then be written in terms of a new constant (K).

Sy = S,-KH*, (24)

SThere is a difference of one grid cell between the width of the
pyramid base and the width of the area where the point loads are ap-
plied, the load application width. ‘This occurs because the point loads
are applied at the center of the grid cells with no loads actually being
applied at the perimeter.

where K =CQ,,
and Q, = average load that can be applied to each
dA beyond pyramid of influence.

By calculating the various values of the stress from
topographic features within different sizes of pyramid
bases, a regression analysis can be performed to calculate
the total stress from equation 24. Through the regression
analysis, the variation of topography within the zone of the
pyramid is used to determine the stress due to topography
beyond the pyramid of influence. Essentially, an assump-
tion is made that the same type of topographic variation
exists both inside and outside the pyramid of influence.
Even if this assumption is not entirely correct, the error is
greatly mitigated by the large distances between these to-
pographic variations outside the pyramid zone and the
point of calculation at depth. Because of the assumptions
made in developing the equation, it is not accurate for H
values for pyramid base widths less than twice the distance
from the calculation surface to the coal seam surface.
Further, to provide a good estimate of the total stress
from the regression analysis, the H values should range
from at least 3 to 10 times the distance from the calcu-
lation surface to the point of calculation on the coal seam
surface.

LOCATION OF STUDY AREA

The study area includes the Jim Walters Resources,
Tnc., No. 5 Mine in the Black Warrior Basin of Alabama.
The No. 5 is an underground mine in Tuscaloosa County
extracting the Mary Lee coal seam. The mine property
boundaries and the study area encompass portions of six
7.5- by 7.5-min quadrangles as shown in figure 6.

The Black Warrior Basin is located within a subdivi-
sion of the Appalachian Plateau Province, called the Cum-
berland Plateau. The sedimentary strata within this prov-
ince are predominately flat lying, showing only minor
structural deformation (6-7). The Cumberland Plateau,
however, has been highly dissected by erosion into a con-
figuration of isolated, flat-topped plateaus separated by
deep valleys displaying large vertical relief, making the lo-
cation a good study area for analyzing the stress effects of
steep topography.

ELEVATION DATA
Surface Topography Elevations
The surface topography elevations for the study area
are provided by DEM’s. A DEM consists of profiles or

arrays of topographic elevation values corresponding to
regularly spaced ground positions every 30 by 30 m. Each


Ke JM
RES,
F . aa

N

0 1
el
Scale, miles

Figure 14—Contour map of topographic elevations In study area (contour Interval, 100 ft).



UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT

ft foot min minute
m meter psi pound (force) per square inch
m? square meter psi/ft pound (force) per square inch per foot

DISCLAIMER OF LIABILITY

The U.S. Bureau of Mines expressly declares that there are no warranties expressed or implied that apply to the
software described herein. By acceptance and use of said software, which is conveyed to the user without consideration
by the Bureau of Mines, the user hereof expressly waives any and all claims for damage and/or suits for or by reason
of personal injury, or property damage, including special, consequential, or other similar damages arising out of or in any
way connected with the use of the software described herein.


310

320

330

350

OBDN=TOPO(COALRO + OFFSET,COALCL+ OFFSET)-COAL(COALRO,COALCL)
IF (OBDNMN .GT. OBDN) GOTO 320
GOTO 330
OBDNMN=OBDN
OBMNRO=COALRO+OFFSET
OMNCOL=COALCL+OFFSET
IF (OBDNMX .LT. OBDN) GOTO 340
GOTO 350
OBDNMX=OBDN
OBMXRO=COALRO+OFFSET
OBMXCL=COALCL+ OFFSET
IF (COALMN .GT. COAL(COALRO,COALCL)) GO TO 360
GOTO 370
COALMN=COAL(COALRO,COALCL)
CMNROW=COALRO+ OFFSET
CMNCOL=COALCL+ OFFSET
IF (COALMX LT. COAL(COALRO,COALCL)) GOTO 380
GOTO 390
COALMX = COAL(COALRO,COALCL)
CMXROW=COALRO+OFFSET
CMXCOL=COALCL+ OFFSET
CONTINUE

CONTINUE

FORMAT (F9.2)

CLOSE (102)

PRINT (202),'COALMN=",COALMN, AT ROW=’,CMNROW, COL=’,CMNCOL
PRINT (202),'COALMX=’,COALMX, AT ROW=’,CMXROW,’ COL=’,CMXCOL
PRINT (202),,0BDNMN=’,OBDNMN; AT ROW=’,OBMNRO,’ COL=’,OMNCOL
PRINT (202),,0BDNMX=’,OBDNMX, AT ROW=’,OBMXRO,’ COL=’,OBMXCL

498 CONTINUE

PRINT °(A30),7ENTER TEST POINT NAME?

READ (1,'(A6)’) PTINAME

PRINT °(A30),,ENTER CENTER CELL COORDINATES’
PRINT *(A4),,ROW=’

READ (1,'(16)’) MIDROW

PRINT ’(A4)’COL=’

COUNT=0

READ (1,°(I6)’) MIDCOL

PRINT °(A7,F9.2)’,, TOPOEL=’,TOPO(MIDROW,MIDCOL)

PRINT ’(A7,F9.2)’, *COALEL=’,COAL(MIDROW-OFFSET,MIDCOL-OFFSET)

Z1=MINEL-COAL(MIDROW-OFFSET,MIDCOL-OFFSET)
INTBDN = Zi * 1.100000

499 CONTINUE

DO 800, COUNT=1,12

CUMVEC=0

RADIUS=INT(ARRAY(COUNT,1))
PRINT °(A32,13)’; CALCULATING STRESSES FOR RADIUS=",RADIUS
DO 600, PYRROW =MIDROW-RADIUS,MIDROW+RADIUS

DO 500, PYRCOL=MIDCOL-RADIUS,MIDCOL+RADIUS

C CALCULATE STRESS VECTOR FROM EACH TOPOGRAPHY CELL DIRECTED TO THE
C CENTRALLY LOCATED POINT ON THE COAL SEAM SURFACE




22

influence on the calculated P, than the surface topographic
features, the influence of which is modified by the distance
to the calculation surface.

The stress calculation program also offers advantages
over the methods of Hooker (7) and Savage (2-3) by al-
lowing calculation of topographic features in two dimen-
sions along the surface of the earth rather than along

a profile representing a cross section of the topography
and by permitting analysis of real, asymmetrical topo-
graphy. Another advantage is that P, can be calculated
over large areas encompassing several square miles. Thus,
P, maps can easily be developed for an entire coal mine
and adjacent regions.

REFERENCES

1. Hooker, V. E., D. L, Bickel, and J. R. Aggson. In Situ Determina-
tion of Stresses in Mountainous Topography. BuMines RI 7654, 1972,
19 pp.

2. Savage, W. Z., and H. 8. Swolfs, Tectonic and Gravitational Stress
in Long Symmetric Ridges and Valleys. J. Geophys. Res., v. 91, 1986,
P. 3677-3685.

3. Savage, W. Z, H. S. Swolfs, and P, S. Powers. Gravitational
Stresses in Long Symmetric Ridges and Valleys, Int. J. Rock Mech. Min.
Sci. & Geomech. Abstr., v. 22, 1985, pp. 291-302.

4, Adler, L., and M. Sun. Ground Control in Bedded Formations.
VA Polytech. Inst., Res. Div., Blacksburg, VA, Bull. 28, 1968, 266 pp.

5. Terzaghi, K, and R. B. Peck. Soil Mechanics in Engineering Prac-
tice. Wiley, 1956, 566 pp.

6. Murrie, G. W., W. P. Diamond, and S. W. Lambert. Geology of
the Mary Lee Group of Coalbeds, Black Warrior Coal Basin, Alabama.
BuMines RI 8189, 1976, 49 pp.

7. Adams, G, 1, C. Butts, L. W. Stephenson, and W. Cooke. Geol-
ogy of Alabama. AL Geol. Survey Spec. Rep. 14, 1926, 312 pp.

8 U.S. Geological Survey (Dep. Interior). Digital Elevation Models.
Data Users Guide 5, 1987, 38 pp.




c P1 = (column height) * (1.1 psi/vertical ft) * area
P1=(TOPO(PYRROW,PYRCOL)-MINEL)*1.1000*((98.400)**2.0)
R1=SQRT(FLOAT(PYRROW-MIDROW)**2) +

C (FLOAT(PYRCOL-MIDCOL)**2)) * 98.4
VECTOR =(P1/(Z1**2))*((3/(2*PD))/((1+ ((R1/Z1)**2))**(5.0/2.0)))

C KEEP A RUNNING TOTAL OF VECTOR VALUES
CUMVEC=CUMVEC+VECTOR

500 CONTINUE

600 CONTINUE

601 FORMAT (A7,Fi2.2)

SUM = CUMVEC + INTBDN
C STORE SUM OF ALL VECTORS AND RADIUS VALUE TO ARRAY
C — ARRAY(COUNT,1)=RADIUS

ARRAY(COUNT,2)=CUMVEC

PRINT ’(A19,F9.2)’’STRESS FOR PYRAMID =’,ARRAY(COUNT,2)

602 FORMAT (A6,F9.2,48,3)

603 FORMAT (A15,F15.4,A14,F154)

800 CONTINUE

805 CONTINUE

CALL REGRES(ARRAY,INTBDN,PTNAME)

PRINT ’(A45)’;IF YOU WISH TO CHANGE TEST POINT, ENTER 1’
PRINT (A45),IF YOU WISH TO EXIT PROGRAM, ENTER 2
READ (1,'(I1)’) REPLY

IF (REPLY .EO. 1) GOTO 498

CLOSE (103)

PRINT ’(A3)’,,BYE’

END

SUBROUTINE REGRES(VALUES,IBSTRS, TESTPT)

C THIS Is A LEAST SQUARES PROGRAM TO FIT THE CURVE Y= BO + Bi * X**(-3)

CAS A STEP IN DETERMINING THE OPTIMUM RADIUS FOR THE PYRAMID OF INFLUENCE

Cc FOR THE TOPOGRAPHY OF THE STUDY AREA

c

C THE DUMMY ARGUMENT ’VALUES’ CONTAINS TWELVE RADIUS VALUES AND
C _ CORRESPONDING TOTAL STRESS VALUES FOR TEST POINT

C THE LAST 8 VALUES ARE USED IN THE REGRESSION ANALYSIS

c

C THE OUTPUT IS TABLE OF INTERCEPT VALUES, SLOPE VALUES, AND A RADIUS
C VALUE AT WHICH 99% TOTAL STRESS IS CALCULATED
Cc
INTEGER COUNT,RADIUS,NUM
REAL IBSTRS,STRESS,X1,X2, Y1, YX,SY,SE,RSQU,B095,B099,L,X95,X99
CHARACTER*15 OUTPUT
CHARACTER"6 TESTPT
DIMENSION VALUES(12,2)
COMMON/ARGS/ARRAY,INTBDN,PTNAME
X1=0
X2=0
Y1=0
YX=0
SY=0
SE=0
NUM=8

31


RESULTS

A number of test points were selected within the study
area to determine the size of the pyramid of influence
from the regression analysis, The test points included the
minimum and maximum topographic elevations, the mini-
mum and maximum coal seam elevations, the minimum
and maximum overburden thickness, the center point of
the study area, and the center points of each quadrant of
the study area. The maximum topographic elevation in the
study was 800 ft above sea level. The minimum elevation
and, therefore, the elevation of the calculation surface was
187 ft. The minimum and maximum coal seam elevations
were 1,641 and 1,305 ft below sea level, respectively.

The stress at each test point was calculated at increas-
ingly larger half width values to establish a relationship
between the half width of the pyramid base and the cal-
culated stress value. Through regression analysis, the half
width and stress values were used to extrapolate the total
stress at each test point. The half width values calculating
99% total stress at each test point were examined, and the
maximum half width value was selected as the half width
for the entire study area. A half width value calculating
95% total stress was also selected.

The half width value for the pyramid base calculating
99% total stress for this study area was determined to be
70 grid cells. The base of the pyramid encompassed
141 by 141 grid cells, A half width yielding 95% total
stress calculation was 42 grid cells, forming a pyramid base
of 85 by 85 grid cells.

The stress calculation was run twice, once with the pyr-
amid base width set to calculate 95% total stress and once
with the pyramid base width set to calculate 99% total
stress, For comparison, direct stress was calculated for the
study area, which used only overburden thickness directly
above the coal seam grid cell, multiplied by the approxi-
mate value of 1.1 psi/ft of overburden to calculate vertical
stress, An example of the format of the output files re-
sulting from these three calculations is included in figure 9.
The output files were converted by contouring programs
into contour maps representing stress values, allowing the
data files to be more easily analyzed and compared.

The comparison of the 99% and 95% total stress cal-
culations, represented in both the stress contour maps in
figures 10 and 11, respectively, and the test points in ta-
ble 3, shows only minimal difference in P, values cal-
culated for cach point. The savings of computer time war-
rants the decrease in accuracy involved in calculating only
95% total stress. In comparing the direct stress map
(fig. 12) to the other two stress maps, it can be seen that
the direct stress method creates artificial zones of abrupt
stress change across relatively small horizontal distances.

The abrupt stress changes, if they were real, would indi-
cate great stress differentials in these zones that might
cause ground control problems in an underground mine
workings intersecting these zones. The abrupt stress
changes are not observed in the maps produced from the
pyramid of influence calculation method for two reasons.
The first reason is that this method takes into account the
effect of stress from topographic features lying adjacent to
a point directly overlying a point on the coal seam. Be-
cause the calculation involves several topographic elevation
cells and assigns the final value to one stress cell, the
calculation acts as a filter to the effects of abrupt changes
in topographic elevation. The second reason is that Z is
directly dependent on the coal seam elevation values. The
calculated stress contour maps, therefore, more closely
resemble the patterns of the coal seam elevation map
(fig. 13) than those of the topographic elevation map
(Gg. 14).

Table 3.—Stress values calculated at study area test
pointe, In pounds (force) per equare Inch

Test point Direct Total 99% total 95% total
stress' stress” _stress__stress
Topographic elevation:
Minimum + 19176 2,084.3 2,082.0 2,072.7
+ 2520.0 2,429.2 2,423.3 2,400.0
+ 24143 24164 24123 2,996.1
+ 22332 21915 21860 2,164.0
+ 1,913.1 2,080.1 2,086.7 2,077.1
Maximum + 26238 2,580.3 25835 2,560.4
Center point:
Study area....... 2,228.2 2,284.5 2,280.2 2,262.9
Northwest quadrant 1,045.0 2,008.8 2,095.6 2,083.0
Northeast quadrant 2,318.0 2,237.2 2,231.9 2,210.4
Southwest quadrant 2,269.1 22746 2,271.1 2,257.0
Southeast quadrant 2,366.9 2,255.4 2,251.6 2,236.4
11.1 psi/ft x overburden.
Based on equation 22.

As seen by comparing stress values for the test points
in table 3, the direct stress method overestimates vertical
stress values at some points and underestimates stress val-
ues at others. Figure 15 is a contour map showing the dif-
ference between the direct stress and the pyramid of in-
fluence methods for the Alabama site. On the map, the
stress calculated from the pyramid of influence method is
subtracted from the stress calculated from the direct stress
method. The direct method overestimates the vertical
stress under hills and ridges, resulting in a positive stress
differential, and underestimates the stress under valleys
and drainages, resulting in a negative stress differential.


Because the configuration of the calculation surface and
the point at depth resembles that of an inverted pyramid
and each point on the calculation surface influences the
stress on the point at depth, the arrangement is called "the
pyramid of influence."

TEST TOPOGRAPHY AND COAL
SEAM SURFACES

The stress calculation program was tested first with a
flat semi-infinite topographic surface overlying a flat coal
seam top. Because there was no variation in topography,
the total stress at any point on the coal seam could be
calculated as direct stress (also known as "tributary area”
method) using the standard average value of 1.1 psi/ft of
overburden depth for coal-bearing strata. For this study,
direct stress was defined as the stress at a point on the
coal seam exerted by the topographic elevation lying di-
rectly above it. Direct stress did not take into account any
stresses exerted by topographic features lying at a distance
away from the point directly above.

The flat test surface lay at an elevation of 600 ft above
sea level, with the calculation surface located at 200 ft

Topographic surface

Coal seam surface

KEY

JL Point toad (a;)

HH Incremental area (dA)

HM Center cell of pyramid base
H

Pyronid base half width measured in nusber of grid cells
away from center cell of pyranid base

R Horizontal distance from load to point N
Z Vertical distance between point N and calculation surface

Figure 3.—Pyramid of Influence showing topographic feature
Tepresented as collection of point loads resting on calculation
surface exerting stress at point N at depth.

above sea level. The coal seam lay at an elevation of
400 ft below sea level, creating an overburden thickness of
1,000 ft. The stress calculation was performed on the
same point on the coal seam surface with increasingly
larger pyramid bases, thus encompassing an increasingly
larger topographic area surrounding the point at depth.
Table 1 lists the calculated stresses with the number of
grid cells defining the half width of the pyramid base used
in each calculation. The direct stress value of 1,100 psi is
the total stress to which the calculated values were com-
pared. The stress calculation values approached total
stress quickly at first with increasing pyramid base sizes.
After an initial rapid approach to total stress, the calcu-
lated stress increased more slowly toward the total stress
value. Figure 4 is a graph illustrating the percentage of
total stress calculated using increasingly larger pyramid
half widths. As the pyramid base grew larger, more cells
were added to the perimeter and were included in the
summation of point stresses. Even though more point
loads were being included, they were progressively farther
away from the center point of the pyramid, and each cell
contributed a lesser amount of stress to the total stress.
As the curve illustrates, the stress values calculated with
larger and larger pyramid bases approached total stress
asymptotically, Total stress would be achieved only with
an infinite pyramid base.

100
98 - 4
98H 4
94H 4
921 4
90 Fy 4
88 4
86 4
84 4
82 4

Fe
0 20 40 60 80 100 120 140 160 180 200

HALF WIDTH OF PYRAMID BASE,
30-m by 30-m grid celis

TOTAL STRESS, %

Figure 4.—Percentage of calculated stress versus number of
grid cells comprising half width of pyramid base.



2B

C COMPILED ON PRIME 6350, PRIMOS OPERATING SYSTEM, Rev. 22.1.2
CEFF EERE EEO EEE EERE EE EEE HEHE EEE EEE EEE EEE EE
c
Cc Disclaimer of Liability Clause
c
C The Bureau of Mines expressly declares that there are no warranties
C expressed or implied that apply to the software contained herein.
C By acceptance and use of said software, which is conveyed to the
C user without consideration by the Bureau of Mines, the user hereof
C expressly waives any and all claims for damage and/or suits for
C or by reason of personal injury, or property damage, including
C special, consequential, or other similar damages arising out of or
C in any way connected with the use of the software contained herein.
CEE EERE REE H REET H EEE EEE EE ETE EEE EERE TEE EEE
CHARACTER*11 TOPOIN,COALIN
CHARACTER*15 OUTPUT
CHARACTER*6 PTINAME
INTEGER TOPROW, TOPCOL, COALRO, COALCL, RADIUS, MIDROW, MIDCOL,
C PYRROW, PYRCOL, REPLY, ROWS, COLS, OFFSET,
C COUNT, TMXROW, TMXCOL, TMNROW, TMNCOL, CMXROW, CMXCOL, CMNROW,
C CMNCOL, OBMXRO, OBMXCL, OBMNRO, OMNCOL
REAL COAL, TOPO, COALMN, COALMX, TOPOMN, TOPOMX, PYRBAS, ARRAY,
C MINEL, OBDNMN, OBDNMX, INTBDN, Z1, R1, P1, VECTOR, CUMVEC, OBDN
C edit array dimensions
DIMENSION COAL (163,133), TOPO(463,433), ARRAY(12,2)
PARAMETER (PI=3.14159)
COMMON/BCOAL/COAL
COMMON/BTOPO/TOPO
COMMON/PYRMID/PYRBAS
C fill array with radius values
DATA (ARRAY(COUNT,1),COUNT = 1,12)/10.0,20.0,30.0,40.0,50.0,60.0,
C 70.0,80.0,90.0,100.0,110.0,120.0/
C these values will be calculated in program
COALMN = 10000.0
MINEL=10000.0000
TOPOMN = 10000.0
COALMX=-10000.0
TOPOMX=0.0
OBDNMN= 10000.0
OBDNMX=0.0
C edit these values
OFFSET =120
ROWS =163
COLS=133
C these variables will be input by user
C TOPOIN
C COALIN
C OUTPUT
C prompt for output file name
PRINT ’(A45),,ENTER OUTPUT FILE NAME (max 15 characters)?
READ (1,(A15)’) OUTPUT
OPEN (UNIT=103, FILE=OUTPUT, STATUS=’UNKNOWN’)


C user edits values

RADIUS=45

OFFSET =45
C topographic file name

TOPOIN=’THREE.TOPO’
C coal seam file name

COALIN=’THREE.COAL’
C output file name

TOTOUT=’THREE.STRESS’
10 FORMAT (A45)
C READ COAL SEAM ELEVATION FILE INTO ARRAY

PRINT (10), (READING COAL SEAM TOP ELEVATION FILE’

OPEN (UNIT=101, FILE=COALIN, STATUS=’OLD’, ERR=20,

C IOSTAT=ERSTAT, ACTION=’READ’)
IF (ERSTAT.EQ.0) GOTO 30
20 PRINT ’(A50)’, PROBLEM WITH OPEN FOR COAL TOP FILE’
30 CONTINUE

DO 200, COALRW=1, ROWS
DO 100, COALCL=1, COLS
READ (101, 201) COAL(COALRW, COALCL)
100 CONTINUE
200 CONTINUE
201 FORMAT(F9.2)
CLOSE (101)

c
C READ IN TOPOGRAPHY ARRAY & DETERMINE MINEL
PRINT (10)READING TOPO ELEVATION FILE’
c
OPEN (102, FILE=TOPOIN, STATUS=’OLD’, ERR=202,
C IOSTAT=ERSTAT, ACTION=’READ”)
IF (ERSTAT.EQ.0) GOTO 210
202 PRINT ’(AS0)’, PROBLEM WITH OPEN FOR TOPO FILE’
210 CONTINUE
DO 400, TOPROW=1, ROWS + (OFFSET * 2)
DO 300, TOPCOL=1, COLS + (OFFSET * 2)
READ(102,201) TOPO(TOPROW, TOPCOL)
IF (TOPO(TOPROW,TOPCOL) LT. MINEL) THEN
MINEL=TOPO(TOPROW,TOPCOL)
ENDIF
300 CONTINUE
400 CONTINUE
CLOSE (102)
PRINT °(A6,F7.2)’MINEL=’,MINEL
c
Cc
PRINT '(A40)’,BEGIN STRESS CALCULATIONS’
C OPEN OUTPUT FILE
OPEN(103,FILE=TOTOUT, STATUS="UNKNOWN’)
c

LPP eee eee ee eee ee eee eee ee eee eee ee eS eee eT ee ee

Cc


INTRODUCTION

Ground control problems are inherent to mine develop-
ment. Mine planning attempts to alleviate or avoid obvi-
ous hazardous areas that would cause unstable ground
conditions during mining. Problems are encountered when
mine entries intersect hazardous areas that were unfore-
seen prior to mining. Vertical (gravitational) stress is an
important factor contributing to unstable ground surround-
ing mine entries, The distribution of vertical stress at the
mining level is influenced by the overlying topographic
features. Because of the rugged topography present in
many coal-bearing regions, there was a need to develop a
method to calculate the stress on a coal seam surface in-
duced by the effect of topographic features overlying the
coal seam.

When a coal seam is located in an area in which the
topography is rugged with steep mountains, plateaus, or
ridges, the abrupt change in overburden thickness above
the coal affects the stress induced on a coal seam surface.
When the coal seam is mined, stress and changes in stress
will have to be considered in the design of the mine en-
tries and the strata control plan. A method of assessing
the stress at the level of the coal seam surface result-
ing from changes in topography would aid in identifying
high stress areas and stress gradients before they are
encountered in mining and could provide a helpful tool in

effective mine design. As part of its program to improve
mining safety, the U.S. Bureau of Mines has developed a
method to calculate the vertical stress on a surface at
depth that takes into account the variation of the overlying
topography.

Hooker (1)? and Savage (2-3) investigated methods for
calculating gravitational stresses induced by topographic
features. Savage (2-3) developed models describing the
effects of gravitational stresses induced by long symmet-
tic ridge and valley topography. Hooker (1) developed a
model for mountainous topography. Both of these meth-
ods use a cross-sectional plane through the topographic
feature in question to calculate the stress effects of the
feature at depth. These methods, therefore, cannot be
used effectively in areas of more complex topography.
The calculation method discussed in this report differs
from the previous approaches by including in the stress
calculation the effects of topographic features extending in
two dimensions along the surface of the earth rather than
a cross-sectional plane through the feature. Another ad-
vantage of this proposed method is that the topographic
features do not have to match any specifications for sym-
metry or form. This technique allows for the evaluation of
the vertical stress and topography for areas encompassing
several square miles,

STRESS CALCULATION METHODOLOGY

The objective of the stress program is to calculate the
distribution of stresses induced by a topographic feature
upon a surface at depth at points that are located at vary-
ing horizontal distances away from the topographic fea-
ture. The accomplishment of this objective requires a
method to. quantify the magnitude of a stress yector from
any point of origin on the topographic surface'to any point
of termination on the coal seam surface. Such a method
is supplied by Boussinesg, who designed a model for con-
tinuous stress distribution from a vertical load to hori-
zontal surfaces located beneath the load (4).

BOUSSINESQ’S EQUATION
Theory

Boussinesq’s theory applies to the effect that a con-
centrated vertical load (Q) has on point N at depth. As
shown in figure 1, Q is resting on the surface of a semi-
infinite mass and induces a vertical stress (dP,) at point N
within the mass. dP, is quantified by Boussinesq to be

(s/2)
a,-22}_ 1 | mn)
2Z? | 1+ (R/Z?
where Z = vertical distance between point N and
surface mass,

and R_ = horizontal distance from applied load to
a point at depth within mass.

A topographic feature can be represented as a distributed
load (q) resting on a semi-infinite surface of a mass. This
is analogous to having a footing with q lying on the sur-
face. The distribution of stress throughout the material
underlying such a footing takes the configuration of a
bulb-shaped stress dome as shown in figure 2, The area
affected by the footing increases in horizontal extent with

3Italic numbers in parentheses refer to items in the list of references
preceding the appendix at the end of this report.


CALCULATION OF VERTICAL STRESS EXERTED
BY TOPOGRAPHIC FEATURES

By Valois R. Shea-Albin,' Dennis R. Dolinar,? and Douglas C. Peters”

ABSTRACT

An accurate assessment of the vertical stress on a coal seam at depth is important for mine design.
Vertical stress calculation techniques presently available either are not sufficiently accurate or cannot
handle complex surface topography. Therefore, the U.S. Bureau of Mines developed a computerized
method to calculate vertical stress exerted on surfaces at depth that includes the effect of topography.
Two input data sets are required: a digital elevation model containing topographic elevations and a coal
seam file defining coal seam elevations at depth. Boussinesq’s equation then quantifies the vertical
stress resulting from topography.

The computerized method was tested on a coal seam overlain by a complex topography where a verti-
cal stress map encompassing several square miles was produced. Results show that depth mitigates the
affects of surface topography, while the coal seam topography has a direct effect on the vertical stress.
In comparison to the computerized method, the direct stress method overestimates stress under hills
and underestimates stress under valleys. The largest difference between the two methods occurs under
the steepest topographic relief. A limitation of the computerized method is that stresses cannot be
accurately determined near an outcrop.

‘Materials engineer.
2Mining engineer.
Denver Research Center, U.S. Bureau of Mines, Denver, CO.


14

om 30 m 60 m 90 m 120m 150m 180m 210m 240m 270m 300m
1 2 3 4 5 6 7 8 9 10
20.94 | 20.94 | 20.93} 20.93 | 20.94] 20.94] 20.94] 20.95 | 20.95} 20.96
30m
306 307 308 309 310 311 312 313 314 315
20.98 | 20.97 | 20.97] 20.96 | 20.96] 20.96} 20.96] 20.96] 20.99] 20.97
60 m
611 612 613 614 615 616 617 618 619 620
21.02} 21.01] 21.00) 21.00] 29.99} 20.99] 20.99} 20.99] 20.99] 20.99
90 m
916 917 918 919 920 921 922 923 924 925
21.07] 21.06} 21.05] 21.04 | 21.03] 21.03] 21.02] 21.02] 21.01] 21.01
120 m|
1,221 | 1,222 | 1,223 | 1,224 |1,225 | 1,226 | 1,227 | 1,228 |1,229 | 1,230
21.12] 21.10] 21.09] 21.08) 21.07] 21.06] 21.06] 21.05] 21.04] 21.03
150 mi
1,526 | 1,527 | 1,528 | 1,529 |1,530 | 1,531 | 1,532 | 1,533 | 1,534 | 1,535
21.12) 21.15] 21.14] 21.13 | 21.12] 21.11] 21.10] 21.08] 21.07] 21.06
180 m|
1,831 | 1,832 | 1,833 | 1,834 |]1,835 | 1,836 | 1,837 | 1,838 | 1,839 | 1,840
21.22) 21.21} 21.19] 21.18 | 21.17] 21.15) 21.14] 21.12] 21.11] 21.09
210 m|
2,136 | 2,137 | 2,138 | 2,139 | 2,140 | 2,141 | 2,142 | 2,143 | 2,144 | 2,145
21.27] 21.26) 21.24) 21.23] 21.21] 21.20) 21.18} 21.16] 21.15] 21.13
240 m
2,441 | 2,442 | 2,443 | 2,444 12,445 | 2,446 | 2,447 | 2,448 | 2,449 |-2,450
21.32) 21.31) 21.29] 21.28] 21.26] 21.24) 21.22] 21.20] 21.19] 21.17
270 m
2,746 | 2,747 | 2,748 | 2,749 | 2,750 | 2,751 | 2,752 | 2,753 | 2,754 | 2,755
21.37 | 21.36) 21.34) 21.33] 21.31] 21.29] 21.27] 21.25] 21.23] 21.20
300 m!
KEY |
1 point number N
20.94 stress value, psi |

Figure 9.—Format of output file containing calculated stress values related to locations within study area.

The differential stress contours tend to follow the valley,
hill, and ridge pattern (compare with figures 12, 14, and
15). The greatest stress differential exists in the areas
such as the stream channels, where the most rapid change
in topography within the study area occurs. Essentially,
the greater the change in elevation for a given horizontal
distance, the greater the stress differential. The average
absolute difference in stress for all the cells is 28 psi per
cell. This number indicates that over much of the map
there is not a large difference in the stresses calculated by
the two methods. The algebraic maximum and minimum
stress differences are 146 and -187 psi and are located in
those areas on the map that are enclosed by the +100-
and -100-psi contours. The minimum stress difference of
-187 psi is located in the valley, while the maximum stress
difference of 146 psi is located on a hill. At the Alabama

site, the pyramid of influence method could, therefore, be
used with the greatest benefit in those areas where the
largest and most rapid change in topography occurs, in the
drainage systems and around the highest hills. For the
rest of the area, where the change in topography is less
steep, the direct stress method could be used without a
large error. The pyramid of influence method is still use-
ful to check the stress from overburden at any point to
determine if the direct method is sufficiently accurate.
However, the overburden stress pattern established by the
direct method is very different from the stress pattern
developed from the pyramid of influence method. If any
mine design decisions are based on the pattern of over-
burden stress and not just the magnitude of the stress, the
pyramid of influence method must be used to establish the
most accurate pattern.


The limitation still exists where the stress within the mass
cannot be evaluated within a depth from a point load
equal to three times the width of the dA.

Boussinesq’s equation applies to load-bearing masses
that are homogeneous, elastic, and isotropic, and extend
infinitely in all directions below the level at which the mass
rests. This application of Boussinesq’s equation involves
the calculation of only vertical stresses. It has been estab-
lished that the distribution of vertical stresses throughout
the substrata is independent of the physical properties of
the substrata (5). These assumptions will, therefore, hold
true for the distribution of vertical stresses through coal-
bearing strata.

Calculation Surface

Three surfaces are required for the calculation of P,
due to a topographic feature. These are the coal seam,
topographic, and calculation surfaces. The coal seam
surface is the surface on which P, will be calculated at
various points, while the topographic surface is the terrain
of the earth overlying the coal seam. The calculation sur-
face is a semi-infinite plane upon which the vertical load
due to the topography rests. The calculation surface ex-
tends parallel to a flat earth surface and lies between the
coal seam and topographic surfaces. The vertical load dis-
tribution on this plane can be approximated as columns
that have square bases equal to dA and have a height
equal to the difference in elevation between the topo-
graphic and calculation surfaces,

In this application of Boussinesq’s theory, the dA’s cor-
respond to the 30- by 30-m‘ grid cells containing the topo-
graphic elevations and the coal seam elevations. The
surface topographic features, represented as various eleva-
tion values distributed within the grid pattern, are con-
sidered to rest upon the calculation surface. The 30- by
30-m grid is also projected onto the calculation surface.
The elevation values assigned to the grid cells on the cal-
culation surface are all the same value because, by defi-
nition, the calculation surface is a surface of constant
elevation. The thickness of overburden above the calcula-
tion surface at each 30- by 30-m grid cell can be calculated
by subtracting the elevation value within each grid cell
from the surface topography elevations in the overlying
grid cells. From this overburden thickness and the den-
sity of the material, the stress exerted by each column of

‘Thirty-meter-square grid cells are used because that is the standard
spatial distribution for digital elevation model (DEM) data,

overburden over each grid cell of the calculation surface
can be determined.

Stress induced on points lying on the top of the coal by
overlying topographic features can be calculated using this
concept of a calculation surface. The body of distributed
mass is the topographic feature, such as a mountain or
ridge, resting on the horizontal calculation surface. Ideal-
Iy, the calculation surface should be at the highest possible
elevation. The plane of highest elevation over which the
topographic features of the study area are considered to
distribute their load is a plane lying at the same elevation
as the topographic low for the study area. The point N
upon which the stress is induced exists on the surface of
a coal seam lying at depth. The dA’s in this application
are grid cells corresponding to the grid cells of the sur-
face topography file, sectioned into 30- by 30-m squares
as measured along the calculation surface. Because the
dA’s are square cells, the total area bearing the load is a
square, rather than a circular area. The stress on each
grid cell is assumed to be a point load located at the cen-
ter of each grid cell. The stress directed toward point N
on the coal seam surface, corresponding to the center cell
of the grid, is calculated by summing all the stress vectors
originating from each dA. The number of dA’s or grid
cells involved in the summation is determined through re-
gression analysis of the asymptotic relationship between
calculated stress and an increasingly larger area of the
topographic surface included in the calculation. The topo-
graphic surface area is measured in grid cells away from
the point on the surface topography that directly overlies
the point on the coal seam surface corresponding to point
N, the point bearing the stress. Equation 3 can be used to
calculate the stress for cells up to a given distance from
the point of calculation:

, 2
p22 30, 1 (s/2)
. 227 | 1+ (R\/Z?

H = half width of surface load application
area, the shortest horizontal distance in
grid cells from dA where P, is to be
calculated to perimeter of square area
over which dP, is calculated for each
cell,

i=1

where

For a distance of H grid cells away from the center
point, the total load-bearing area is (H-2 + 1)?- 900 m?.
Figure 3 illustrates the spatial relationships described.


LEGEND N

Residual Stress, psi 0 1
Less than — ce
-100 5

Scale, miles
0 to 100

Figure 15.—Contour map of direct stress values minus calculated stress values over study area.


RI| 9409 REPORT OF INVESTIGATIONS/1992

Calculation of Vertical Stress Exerted
by Topographic Features

By Valois R. Shea-Albin, Dennis R. Dolinar,
and Douglas C. Peters

UNITED STATES DEPARTMENT OF THE INTERIOR

BUREAU OF MINES


KEY
-—-— Centerlines of cells

| Distance from middle of center cell of a perimeter
side to middle of selected cell along same

perimeter side

R Horizontal distance from applied load to a point within the mass

r Width of individual grid cells

W Half width of extended perimeter

Figure 5.—Square perimeter surrounding pyramid base.

r = width of individual cell,
j = lr,

= distances in number of grid cells from
middle of center cell of perimeter side
to middle of another cell along same
perimeter side,

where

and h = W/r,

= half width of perimeter in number of
grid cells,

Canceling terms in equation 9,

R; = W(1 + j2/h?)°5, (10)

Substituting equation 10 into equation 6,

30,23 2 5/2
o 82 5 |[1- 3] .

Because the load applied to each incremental cell (Q,)
does not vary for a flat surface, this incremental load can
be replaced by one constant load. Equation 11 becomes

(at)

3 a )s2
a2 [2] a)

P = constant vertical load applied to each
dA.

where


Table 1.—Stross values calculated for point on flat test surface
with Increasing pyramid base sizes, In pounds
(force) per square inch

Pyramid base Calculated || Pyramid base Calculated
half wiatht stress half width stress

1. 803.6 1,099.6

10 1,058.8 4,098.6

20 1,091.5 4,099.7

30 4,097.1 4,099.7

40 1,098.7 1,099.7

50 1,009.3, 4,099.7

60 1,099.5

1Half width measured in number of grid celis away from
center cell of pyramid base.

NOTE.—Topographic elevation = 600 ft above sea level; cal-
culation surface = 200 ft above sea level; coal seam elevation
= 400 ft below sea level; and total stress = 1,000 ft overburden
X 1.1 psl/ft overburden = 1,100 pel.

CALCULATION OF VERTICAL STRESS FOR A COAL MINE

‘When topographic variations occur as in the case of an
actual coal mine, the total stress which the calculated
stress approaches asymptotically is unknown. Once there
are topographic variations at the surface, the direct stress
method does not calculate the correct stress values at an
underlying point. Using the relationship established with
the flat test surface, the total stress can be estimated even
though it is unknown. The calculated stresses follow the
same relationship as before, approaching asymptotically a
maximum value for stress. This maximum can be esti-
mated through regression analysis techniques.

Given the relationship between the pyramid base size
and the calculated stress observed at the test points using
a flat topographic test surface, it is possible to estimate the
total topographic stress at a point for a flat surface by de-
veloping an equation that will predict the amount of stress
contributed by the topographic load beyond a selected
base size of the inverted pyramid of influence. Rewriting
equation 1 yields

@ - ae [rez]? w

If Z is held constant, and R gets much greater than Z,
then equation 5 becomes

@ ~ 22 [es], ©

Equation 6 provides an approximation for the stress in-
duced by a point load located beyond the bounds of the

pyramid base and, therefore, is not included in the stress
calculations, Figure 5 shows a square perimeter consisting
of a single row of cells that is centered around but lies
beyond the pyramid base. The stress induced beneath the
center of the pyramid base by a cell in this perimeter can
be written in terms of the perimeter dimensions. The dis-
tance from the center of the pyramid to the cell is

R= (W? +17), y)

W = half width of extended perimeter load
application, the horizontal distance
from center of pyramid to load applica-
tion point of nearest extended perim-
eter cell,

where

and 1 = distance in meters from middle of center

cell of perimeter side to middle of
another cell along same perimeter side.

Rearranging terms in equation 7,
R, = W(1 + Pw?y5, (8)

Writing the distances inside the parentheses in terms of an
individual cell width dimensions,

Ry = WL + (ja)? /(be)?)8, ©


2

DO 1500,CONROW =1,RADIUS*2+1
DO 1400,CONCOL=1,RADIUS*2+1
Cc
PSeESETESETILESESSSSSSTICES SSS STT TTT TSS T Teer Tee eres
C _ BOUSSINESQ’S EQUATION
Prererevereverrerrrrserr Terre rere Terre eer reer Tyee eye
c
c P1 = (column height) * (1.1 psi/vertical ft) * area
P1=(PYRMID(CONROW,CONCOL)-MINEL)*1.1*((98.4)**2)
Ri=SQRT((CONROW - (RADIUS + 1.0))**2 + (CONCOL -
C (RADIUS + 1.0))**2) * (98.4)
VECTOR = (P1/(Z1**2))*((3/(2*PD)/((1+ ((R1/Z1)**2))*"(5.0/20)))

ACCUMULATE STRESS COMPONENTS FROM PYRAMID
CUMVEC = CUMVEC + VECTOR

CONTINUE
CONTINUE

ADD STRESS DUE TO INTERBURDEN TO TOTAL STRESS FROM PYRAMID
SUM = CUMVEC + INTBDN

WRITE TOTAL STRESS FROM PYRAMID OF INFLUENCE TO OUTPUT FILE AT POINT
CORRESPONDING TO CENTER OF PYRAMID CELL TO WHICH THE STRESS IS
DIRECTED ON COAL SEAM TOP

WRITE (103,1501) SUM
FORMAT (F122)

CONTINUE
CONTINUE
CLOSE (103)
END

gs g aaaqaaa ooogge aaa

PROGRAM WIDTHCALC
CHEER R EEE EEE EE EEE EEE
C PROGRAM TO CALCULATE A PROGRESSIVELY LARGER PYRAMID BASE AROUND A
C CENTRAL POINT LOCATED ON THE TOP OF THE UNDERLYING COAL SEAM, THEN
C PASSES RADIUS VALUES AND RESPECTIVE TOTAL STRESS VALUES TO THE
Cc REGRESSION ANALYSIS SUBROUTINE
C PROGRAMMER: VALOIS R. SHEA-ALBIN
BUREAU OF MINES
DENVER RESEARCH CENTER
BUILDING 20, DENVER FEDERAL CENTER
DENVER, COLORADO 80225
(303)-236-0775
C PROGRAMMING LANGUAGE: FORTRAN 77
Rev. T2.1-22.1 Copyright (c) 1990
Prime Computer, Inc.


&R

C BEGIN CALCULATING PYRAMID
c
CHEER
c
C — CCNTRY=CENTER CELL IN COAL SEAM TOP GRID (Y CONTROL)
C  CCNTRX=CENTER CELL IN COAL SEAM TOP GRID (X CONTROL)
c TCNTRY=CENTER CELL IN TOPOGRAPHY GRID (Y CONTROL)
C TCNTRX=CENTER CELL IN TOPOGRAPHY GRID (X CONTROL)
Cc CONROW=PYRAMID OF INFLUENCE Y GRID CELL CONTROL
c CONCOL=PYRAMID OF INFLUENCE X GRID CELL CONTROL
Cc
DO 3000,CCNTRY=1, ROWS

DO 2000,CCNTRX=1, COLS
Cc

TCNTIRY=CCNTRY +OFFSET

TCNTRX=CCNTRX + OFFSET
Cc
C BEGIN PYRAMID
c PRINT ’(A30,16)’, "CENTER COAL CELL Y CRD =’,CCNTRY
c PRINT ’(A30,I6)’, "CENTER COAL CELL X CRD =’,CCNTRX
Cc PRINT ’(A30,16)’, "CENTER TOPO CELL Y CRD =’, TCNTRY
c PRINT ’(A30,16)’, "CENTER TOPO CELL X CRD =’,TCNTRX

C FILL PYRAMID WITH ZEROES
DO 1000, CONROW=1,RADIUS*2+1
DO 900, CONCOL=1,RADIUS*2+1
PYRMID(CONROW,CONCOL)=0
900 CONTINUE
1000 CONTINUE
C FILL PYRAMID WITH TOPOGRAPHIC VALUES
CONROW=1
DO 1200, TOPROW=(TCNTRY-RADIUS),(TCNTRY +RADIUS)
CONCOL=1
DO 1100, TOPCOL= (TCNIRX-RADIUS),(TCNTRX +RADIUS)
PYRMID(CONROW,CONCOL)=TOPO(TOPROW,TOPCOL)
CONCOL=CONCOL+1
1100 CONTINUE
CONROW=CONROW+1
1200 CONTINUE

CALCULATE INTERBURDEN THICKNESS FOR CENTER CELL
Z1 = FLOAT(MINEL - COAL(CCNTRY,CCNTRX))
CALCULATE STRESS DUE TO INTERBURDEN THICKNESS
INTBDN = Zi * 11

INITIALIZE TOTAL STRESS COUNTER

CUMVEC=0

aq aagq aaa ann

BEGIN ISOLATING AND CALCULATING STRESS VECTORS


KEY
dh, Vertical stress induced at point N

Q Load applied to a point
R Horizontal distance from load to point N at depth
Z Vertical distance between point N and surface mass

Figure 1.Stress exerted at point N at depth by Q resting on
‘semb-infintte horizontal surface.

depth from the elevation of the applied load. As a result,
the amount of load at any point on a horizontal surface
within the stress dome decreases with depth so that the
total stress remains constant and equal to that of the ap-
plied load (4). To calculate the vertical stress induced by
the footing, the point load theory is used at distances into
the medium greater than three times the footing width (b),
while inside this distance, the loads must be integrated
over the footing surface. Because of the horizontal extent
of topographic features, the area of interest, in general, is
less than twice the width of the feature, and therefore, the
surface loads must be integrated.

To integrate, the surface area A bearing q produced by
the topographic feature can be divided into smaller in-
cremental areas (dA’s) each with a q. These q’s can be
replaced by Q at the center of each incremental area
where the concentrated load is Q = q-dA. Equation 1
can then be used to calculate the vertical stress (dP,)
caused by each point load at a point N within the mass,
Equation 1 becomes

(5/2)
, 0

a, = 2% ft
“22? 11+ @/BP

1.0b

1.5b

01g

KEY
b Footing width
q Distributed load

Figure 2.—Contours of vertical stress distribution underlying
distributed load resting on semi-infinite surtace. [From Adler (4)]

where Q; = q;dA,
= vertical point load applied to dA,

i = identity of given dA,

R, = horizontal distance from dA to point of
calculation in mass,
and = dP,;_ = vertical stress induced at a point within
mass by Q on dA.

The total vertical stress at a point at depth caused by the
topography (P,) can be determined by summing each dP,
induced from all the point loads applied to the surface.

n D 39, (5/2)
P= Py; = — |———_ | .@
. x, “ x 22? i + a |
where n= total number of dA’s,


vy

| tua

am)
é Si

Scale, miles

Figure 12.-Contour map of direct stress in study area (contour Interval, 100 psi).


APPENDIX.—USER’S GUIDE FOR RUNNING COMPUTER PROGRAMS
TO CALCULATE VERTICAL STRESS ON A COAL SEAM

RUNNING THE PROGRAMS

As discussed in the text, the purpose of this computer
program is to calculate P, on a coal seam surface induced
by the weight of the overlying rock mass. The program
STRESS accomplishes this by calculating a number of dP,
vectors originating from various points within a square
area of the topographic surface. These stress vectors are
summed to create a P, value at a point on the coal seam
surface below. This procedure is performed for a number
of points on the coal seam surface. The end result is an
array of P, values that represent the vertical stress induced
by overburden at regularly spaced points along the coal
seam surface. The configuration described has been
named the pyramid of influence, which is discussed in the
main text in the section entitled "Calculation Surface" and
illustrated in figure 3.

The size of the pyramid of influence necessary to cal-
culate the P, varies for different topographic and coal
seam configurations. To determine the size of the pyra-
mid needed for a given minesite, program WIDTHCALC
is run. In program WIDTHCALC, pyramids of progres-
sively larger sizes are used to calculate P, at a number of
test points. This information is given to a regression
analysis subroutine to determine the size of pyramid
required to calculate 99% and 95% of the total P,. The
two percentages are included to give the user the option
for calculating only 95% of the stress to reduce computer
tun time. The size of the pyramid is determined by the
"radius" value, which represents the half width of the
pyramid base in number of grid cells.

Before running the programs WIDTHCALC and
STRESS, two American Standard Code for Information
Interchange (ASCII) input files must be available and in
the correct format. The first file contains topographic
elevation values of the ground surface overlying the mine.
The second file contains coal seam top elevations. The
input files are discussed in detail in the main text in the
section entitled "Elevation Data.”

WIDTHCALC prompts the user for the names of the
two input files. After reading the two files, sorting
through elevation values, and calculating overburden thick-
ness at each point, WIDTHCALC determines six test
points, The test points include the minimum and maxi-
mum topographic elevations, the minimum and maximum
coal seam elevations, and the locations of minimum and
maximum overburden thickness in the area of interest.
The coordinates for each test point are listed on the
screen, It is helpful for the user to make note of the

coordinates for each test point for input into the program.
The program then prompts the user for the coordinates of
each test point. In addition to the six selected test points,
the user has the option to enter the coordinates of any
additional points of particular interest in the study area,
As the coordinates for each test point are entered, the
program calculates P, for radius values from 10 to 120 in
increments of 10 to create a series of progressively larger
pyramids. After calculating the stress for 12 progressively
larger pyramids, the stress values are passed to the regres-
sion analysis subroutine, REGRESS. In the subroutine,
the stresses calculated for each radius value are entered
into a regression analysis algorithm, which calculates
radius values for which 95% and 99% stress would be cal-
culated for these topographic and coal seam top files. The
radius values are listed on the screen and are written to a
user-named output file. Once all test points of interest
have been entered, the user exits the program. After the
user looks at the output file, the largest radius for either
95% or 99% total stress is selected to include in program
STRESS. Calculating only 95% stress will save computer
time as compared with calculating 99% stress.

Program WIDTHCALC will need to be edited for each
study area, The values that might require editing include
names and array dimensions for the input and output files
and variables OFFSET, ROWS, and COLS. These vari-
ables are noted by comments in the FORTRAN code.
ROWS and COLS are the number of rows and columns,
respectively, in the coal seam top file. The topography file
is larger than the coal seam file because a buffer zone is
included in the topographic surface around the outside of
the mine boundaries to accommodate the base of the pyr-
amid of influence. The size difference of the topographic
and coal seam files is accounted for by the OFFSET value.
OFFSET = 1/2-(number of ROWS in topographic file -
number of ROWS in coal seam file).

After running WIDTHCALC and selecting the
maximum radius value to calculate either 95% or 99% P,,
edit program STRESS to input the radius value. Other
values that may require editing for a particular study
include input file and output file names and array dimen-
sions, input and output file names, ROWS, COLS, and
OFFSET. These variables are noted by comments in the
FORTRAN code.

Because the STRESS program will run a long time, it
might be best to run it as a batch file.

The output file is an ASCII file containing a list of
stress values in pounds (force) per square inch. The for-
mat resembles that of the coal seam file in dimensions


CONTENTS

Abstract .
Introduction
Stress calculation methodology
Boussinesq’s equation
Calculation surface .
Test topography and coal seam surfaces.
Calculation of vertical stress for a coal mine ..
Location of study area ....
Elevation data
Surface topography elevations . . .
Coal seam top elevations
Results .....

‘Appendix.—User’s guide for running computer programs to calculate vertical stress on a coal seam .

ILLUSTRATIONS

1, Stress exerted at point N at depth ....

2. Contours of vertical stress distribution .

3, Pyramid of influence .. seeeee
4, Percentage of calculated stress versus number of grid cells .

5. Square perimeter surrounding pyramid base eee e eee eeeee
6. Study area map .......... 0005 weeewe .
7. Format of input file containing topographic elevations»...

8, Format of input file containing coal seam top elevations .

9. Format of output file containing calculated stress values ....
10. Contour map of 99% total stress ............00005

11. Contour map of 95% total stress ....
12. Contour map of direct stress .....
13. Contour map of coal seam top elevations .
14. Contour map of topographic elevations .
15. Contour map of direct stress values minus

TABLES

Stress values calculated for point on flat test surface with increasing pyramid base sizes .
Summation of cell distances according to increasing perimeter widths .
Stress values calculated at study area test points ........ oe

PRr

BES auuew

Boa


10

87.500° 87.125°
33.375° 33.375 °
Lake Nicol Brookwood Abernant
N
Cottondale Coating Woodstock
33.125 ° 33.125"
87.500° 87.125°
LEGEND 0 5
Study area Ml Mine site Scale, miles

Figure 6.—Study area map of Jim Walter Resources, Inc., No. 5 Mine in Black Warrior Basin, Tuscaloosa County, AL.

DEM provides coverage for a 7.5- by 7.5-min block corre-
sponding to the standard U.S. Geological Survey (USGS)
1:24,000 scale map series quadrangles.

In the DEM file, the elevation values are contained in
profiles, which are one-dimensional arrays containing a
series of elevation values, called nodes. The first node in
the profile is the elevation of a point at the south edge of
the quadrangle. The subsequent values represent eleva-
tions at successive points to the north, spaced 30 m apart.
The profiles are listed in the file in order from cast to
west. Each profile begins with header information de-
scribing the profile. The profiles within each DEM do not
have the same number of elevations. Because there is an
angle of varying size between true north and the north
direction of the Universal Transverse Mercator (UTM)
grid, the 7.5-min quadrangle boundaries clip profiles,
which parallel the UTM grid system at some angle to true
north,

The elevation data contained in a DEM are produced
in 7.5- by 7.5-min blocks either by digitizing contour maps
or scanning high-altitude photographs. The data are then

processed and formatted in a 30- by 30-m grid. Additional
information on DEM data collection and file format is
included in the "Data Users Guide 5” entitled "Digital Ele-
vation Models" provided by the USGS National Mapping
Program (8).

To obtain the coverage of elevation values for the study
area, computer programs were developed to extract from
each DEM file only the elevation values falling within the
boundaries of the study area. The header information for
each profile provides the UTM northing and easting of the
first point in the profile and the number of elevations con-
tained in each profile. The easting of the profile is com-
pared with the minimum (westernmost) and maximum
(casternmost) easting of the study area. Once a profile is
determined to be within the easting range of the study
area, the program calculates the northing of each elevation
point and extracts only the points falling within the north-
ing range of the study area.

Because of the size of each DEM file and the fact that
it was necessary to read in six files to obtain the coverage


n

data base structure expedites the storage and extraction of
geographically referenced drill log information. A file con-
taining only geographical coordinates of the drill logs and
the elevation of the top of the Mary Lee coal seam was
created within the data base and entered into a gridding
program to interpolate regularly spaced coal seam top ele-
vation points between the irregularly spaced drill log lo-
cations. The gridding program used was Gridzo, a product
of RockWare, Inc.§

The coal seam elevation coverage for the study was too
large for the Gridzo gridding program to handle as one
file. The area was divided into four contiguous areas, each
gridded separately by the Gridzo gridding program and
stored as separate grid files, The files were transferred
from the PC to a Prime 6350 super-minicomputer and
merged into one large file containing the coal seam ele-
vation coverage for the study area.

“Reference to specific products does not imply endorsement by the
USS. Bureau of Mines.

The Gridzo gridding program provides a number of al-
gorithms to interpolate between irregularly spaced data
points. The method chosen for this study was the tri-
angulation method, because it uses the points outside the
designated grid boundaries to interpolate up to the bound-
aries and, therefore, acted to "link together" the four areas
that were gridded separately.

The resulting coverage containing the Mary Lee coal
seam top elevations for the study area extends from
465,615 m easting on the west side to 474,765 m easting on
the east side. The southern boundary is 3,678,135 m
northing, and the northern boundary is 3,687,285 m north-
ing. The maximum coal seam elevation within the study
area is -1,305 ft below mean sea level, and the minimum
elevation is -1,641 ft. The grid cell coordinates for both
the topographic elevation coverage and the Mary Lee coal
seam top coverage were designed to correspond geograph-
ically. An example of the file is shown in figure 8.

Om 30 m 60m 90m 120m 150m 180m 210m 240m 270m 300m
1 2 3 4 5 6 7 8 9 10
1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434 | -1,434 | -1,434 |-1,434 |-1,434

30m
306 307 308 309 310 311 312 313 314 315
1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434
60 m
611 612 613 614 615 616 617 618 619 620
1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434 | -1,434 | -1,434 [-1,434 | -1,434
90 m
916 917 918 919 920 921 922 923 924 925
1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434
120
1221 1222 1223 1224 1225 1226 1227 1228 1229 1230
1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434
150
1526 1527 1528 1529 1530 1531 1532 1533 1534 1535
=1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434] -1,434 | -1,434 |-1,434 | -1,434
180 m|
1831 1832 1833 1834 1835 1836 1837 1838 1839 1840
1,434 | -1,434 | -1,434 | -1,434 |-1,434 | -1,434] -1,434 | -1,434 | -1,434 | -1,434
210 m|

2,136 | 2,137 | 2,138 | 2,139 |2,140 | 2,141 | 2,142 | 2,143 [2,144 | 2,145

71,434 | -1,434 | -1,434 | -1,434 |-1,434 |-1,434] -1,434 | -1,434 | -1,434] -1,434
240 m|

2,441 | 2,442 [2,443 | 2,444 12,445 [2,446 | 2,447 | 2,448 | 2,449 | 2,450

1,434 | -1,434 | -1,434 | -1,434 }-1,434 |-1,434] -1,434] -1,434 | -1,434| -1,434
270 m|

2,746 | 2,747 | 2,748 | 2,749 |2,750 |2,751 | 2,752 | 2,753 | 2,754 | 2,755

1,434] -1,434 | -1,434 | -1,434 |-1,434 |-1,434] -1,434| -1,434 | -1,434] -1,434
300 m

KEY
1 point number |
1,434 elevation value, ft '

Figure 8.—Format of input file containing coal seam top elevations.


Library of Congress Cataloging in Publication Data:

Shea-Albin, V. R.
Calculation of vertical stress exerted by topographic features / by Valois R. Shea-
Albin, Dennis R. Dolinar, and Douglas C. Peters.

P. cm. — (Report of investigations; 9409)
Includes bibliographical references (p. 22).
1. Ground control (Mining). 2. Rock pressure—Measurement—Data Processing.
3. Geomorphology. 4. Coal mines and mining. I. Dolinar, Dennis R. Il. Peters,

Douglas C. Ill. Title. IV. Series: Report of investigations (United States. Bureau
of Mines); 9409.

TN23.U43 [TN288] 622 s—dc20 [622°.28] 91-30851 CIP



PRINT (304), RADIUS SIZE FOR 95% TOTAL STRESS=’,NINT(X95)
PRINT (304), RADIUS SIZE FOR 99% STRESS=’,NINT(X99)

304 FORMAT (A35,F6.1)
WRITE (103,(2A12)’) "TEST POINT=’,TESTPT
PRINT ’(A40)’, WRITE INTERCEPT SLOPE CORRELATION DATA’
WRITE (103,FMT=405) INTERCEPT SLOPE CORRELATION’
WRITE (103,FMT=405) ’ ,
WRITE (103,FMT=406)’ BO BI’,

C R22 95%B0 RADIUS 99%B0 RADIUS’
WRITE (103,FMT=407) BO,B1,RSQU,B095,X95,B099,X99
CONTINUE

FORMAT (A50)

FORMAT (A15,A60)

FORMAT (F7.,F12.2,F9.2,2F12.2,2F11.2)

WRITE (103,FMT=405)?

WRITE (103,FMT=405)’

PRINT (A40)’, WRITE STRESS TABLE DATA’

WRITE (103,FMT=501) STRESS TABLE’

WRITE (103,FMT=405)’ ’

WRITE (103,FMT=501)
ce CALCULATED STRESS ESTIMATED STRESS’

WRITE (103,FMT=502) RADIUS © FROM PYRAMID’,
C  FROMPYRAMID TOTAL STRESS’
C ESTIMATED STRESS TOTAL STRESS’

DO 500, COUNT=1,12
WRITE (103,FMT=503) VALUES(COUNT, 1), VALUES(COUNT,2),B0,
CBO+IBSTRS
500 CONTINUE
501 FORMAT (AS50)
502 FORMAT (A27,A37)
503 FORMAT (15,3F19.2)
END

8888

33

INT.BU.OF MINESPGH.PA 29534


21

RESTRICTIONS PLACED ON STRESS CALCULATION METHODOLOGY

Boussinesq’s equation calculates the stress distribution
for a point load acting on the flat surface of a semi-infinite
solid. Where an outcrop of the calculation surface or the
surface at depth exists, the solid cannot be considered
infinite in the outcrop direction. Theoretically, the stress
distribution given by equation 1 no longer holds true be-
cause of the different boundary conditions imposed by the
outcrop. From a practical standpoint, however, the cal-
culation method can be used to within a certain distance
of the outcrop without serious error. This distance is di-
rectly related to the size of the pyramid of influence used
to calculate a given percentage of P,. At a point, to de-
termine a certain percentage of P, requires a pyramid of
influence of a specific size. The pyramid of influence is
centered around the point of calculation. If the point is
close to an outcrop where the pyramid of influence ex-
tends beyond the outcrop, the stress calculated at the point
will be less than the predicted percentage of stress because
no P, is being calculated from that portion of the pyramid
that is beyond the outcrop. The greater the portion of the
pyramid of influence beyond the outcrop, the smaller the
calculated P,. An example for the test case given pre-
viously in the report, where the topography was flat and
the overburden thickness was 1,000 ft with a resulting
stress of 1,100 psi, a pyramid base with a half width of
100 30- by 30-m cells will result in a stress of 1,099.7 psi
(table 2) being determined, of which 660 psi is direct over-
burden and 339.7 psi is calculated based on the topog-
raphy. If an outcrop is parallel to a side of the pyramid
base and is located 100 cells (3,000 m) from the cell where
the stress is to be calculated, the resulting stress at the
point will still be 1,099.7 psi. For an outcrop that passes
next to the point with half the pyramid of influence beyond
the outcrop, the calculated stress will be approximately
880 psi. The stress at the point is still 1,100 psi, where
only 220 psi is calculated based on the pyramid of influ-
ence or 50% of the actual value (440 psi), since no stress
is being calculated from half the pyramid of influence.
The pyramid of influence can therefore touch, but should
not go beyond, the outcrop to calculate a stress at a known
percentage of the actual value. (The closest distance a

point of calculation can be to an outcrop is then either half
the pyramid base width or the square root of two times
half the pyramid base width.)

The shape of the outcrop and how the pyramid base
edge lines up with the outcrop will determine the appro-
priate distance to be used. For computational purposes,
it is possible that an average distance could be used with-
out incorporating a large error into the calculation results.
Tf the distance between the calculation surface and the
surface at depth is decreased, the size of the pyramid base
required to calculate the stress at a point is also decreased.
By artificially decreasing this distance, the stress at points
closer to the outcrop can be calculated. As this distance
is reduced, however, the calculated stress approaches the
stress determined by the direct or tributary method. There
will be some loss in the accuracy of the calculated stress,
If the decrease in distance between the surface at depth
and the calculation surface occurs as a natural phenome-
non (for example, a steeply dipping coal seam approaching
the calculation surface), the change in distance between
the two surfaces does not pose a problem. The closest
distance between the calculation surface and the surface at
depth should not be less than three times the width of dA
or square grid cell at which the load is applied (approxi-
mately 100 ft when a DEM file area 30 by 30 m is used).
At less than this distance, the difference between q and Q
is no longer minimal. If the grid cell size is reduced, the
minimum required distance between the two surfaces can
be reduced accordingly, and the calculations surface and
the surface at depth could be even closer to the outcrop.
However, because more grid cells would be required to
cover the study area, more computational time would be
required for the stress calculation. As the outcrop is ap-
proached, it may be more practical at some point to use
the direct stress calculation method. For the best results,
the calculation surface should be as far from the surface
at depth as possible without violating the outcrop condi-
tion. For this reason, the location of the calculation sur-
face in the Alabama study area was the minimum topo-
graphic elevation.

CONCLUSIONS

The stress calculation program appears to provide an
effective method to calculate vertical stress values induced
on a surface at depth by overlying topography. This cal-
culation method is more reliable than a direct overburden
stress calculation method because it takes into account the

influence of adjacent topographic features overlying the
coal seam. The effect that changes in topography have on
P, at the surface below is mitigated with depth, which the
direct stress method does not take into account. The var-
iation of the coal seam topography has a more direct


Report of Investigations 9409

Calculation of Vertical Stress Exerted
by Topographic Features

By Valois R. Shea-Albin, Dennis R. Dolinar,
and Douglas C. Peters

UNITED STATES DEPARTMENT OF THE INTERIOR
Manuel Lujan, Jr., Secretary

BUREAU OF MINES
TS Ary, Director


C prompt for topography file name
10 PRINT ’(A30)’,ENTER TOPOGRAPHY FILE NAME’
READ (1,'(A11)’) TOPOIN
C open topography file
PRINT ’(A20)’,;OPENING TOPO FILE’
OPEN (UNIT=101, FILE=TOPOIN, ACTION=’READ’, ERR =20)
GOTO 30
20 PRINT ’(A45)’,PROBLEM WITH OPENING FILE; CHECK FILE NAME.’
GOTO 10
C prompt for coal seam file name
30 PRINT ’(A30)’,, ENTER COAL SEAM FILE NAME;’
READ (1,(A11)) COALIN
C open coal seam file
PRINT ’(A20)’,,OPENING COAL FILE’
OPEN (UNIT=102, FILE=COALIN, ACTION=’READ’, ERR=40)
GOTO 50
40 PRINT ’(A45)’,, PROBLEM WITH OPENING FILE; CHECK FILE NAME.’
GOTO 30
C read topo file into array, and determine test points
50 PRINT ’(A30)’,,/ READING TOPO FILE’
DO 200, TOPROW=1, ROWS + (OFFSET * 2)
DO 100, TOPCOL=1, COLS + (OFFSET * 2)
READ (101,201) TOPO(TOPROW,TOPCOL)
IF (MINEL LT. TOPO(TOPROW,TOPCOL)) GOTO 60
MINEL=TOPO(TOPROW,TOPCOL)
60 IF ((TOPROW .GT. OFFSET) AND. (TOPROW .LT. ROWS+OFFSET))
C GOTO 70
GOTO 100
70 IF ((TOPCOL .GT. OFFSET) AND. (TOPCOL LT. COLS+OFFSET))
C GOTO 80

GOTO 100
80 IF (TOPOMN .GT. TOPO(TOPROW,TOPCOL)) GO TO 90
GOTO 91
90 TOPOMN=TOPO(TOPROW,TOPCOL)

TMNROW=TOPROW
TMNCOL=TOPCOL

1 IF (TOPOMX LT. TOPO(TOPROW,TOPCOL)) GOTO 93

2 GOTO 100

8 TOPOMX=TOPO(TOPROW,TOPCOL)
TMXROW=TOPROW

TMXCOL=TOPCOL

100 CONTINUE
200 CONTINUE
201 FORMAT (F9.2)

CLOSE (101)

PRINT *(A6,F12.4),MINEL=’,MINEL

PRINT (202), TOPOMN=’,TOPOMN,’ AT ROW=’, TMNROW,’ COL=’,TMNCOL

PRINT (202)TOPOMX=’,TOPOMX, AT ROW=’,TMXROW,’ COL=’,TMXCOL
202 FORMAT (A7,F10.2,A8,16,A5,16)
C read coal file into array, determine test points

PRINT °(A30), READING COAL FILE”

DO 400, COALRO=1,ROWS

DO 39,COALCL=1,COLS

READ (102,401) COAL(COALRO,COALCL)


mA

and data point spacing. Processing the stress values with
a contouring package will present the data in a more
visible format. The list of stress values can be observed as
stress value contours, which is more easily comprehended
by the user.

The specifics of generating the input files are not dis-
cussed in this report. Since the files are large, to save
space, the X and Y coordinates are omitted and only the
Z values, or elevation values, are included in the files. As
described in the main text, the first point in the file

FORTRAN CODE LISTING
PROGRAM STRESS

corresponds to the northwestern-most point in the study
area, and subsequent values are arranged point-wise at
30-m increments from west to east in rows arranged from
north to south in 30-m increments. The topography, coal
seam, and output stress files each follow this format. To
reformat the files to include X and Y values, it is
necessary only to know the coordinates of the northwest
corner of the study area and to assign the rest of the
coordinates in order based on the 30-m increments.

Re ee eee eee eee eee SECC Eee CC eee SCCSSSOSOCSSSOSSS SOE

C PROGRAM TO CALCULATE PYRAMID OF INFLUENCE OVER A COAL SEAM

C PROGRAMMER: VALOIS R. SHEA-ALBIN
BUREAU OF MINES
DENVER RESEARCH CENTER

BUILDING 20, DENVER FEDERAL CENTER

DENVER, COLORADO 80225
(303)-236-0775
C PROGRAMMING LANGUAGE: FORTRAN 77
Rev. T2.1-22.1 Copyright (c) 1990
Prime Computer, Inc,

C COMPILED ON PRIME 6350, PRIMOS OPERATING SYSTEM, Rev. 22.1.2

CEPR E ROOK EEE THEE AE EEE E REE EEE

c
c Disclaimer of Liability Clause
c

C The Bureau of Mines expressly declares that there are no warranties
C expressed or implied that apply to the software contained herein.

C By acceptance and use of said software, which is conveyed to the

C user without consideration by the Bureau of Mines, the user hereof

C expressly waives any and all claims for damage and/or suits for

C or by reason of personal injury, or property damage, including

C special, consequential, or other similar damages arising out of or

C in any way connected with the use of the software contained herein,

(CLEP Pee eee Pee eS Sree CesT ee eee eee eee ee eee eee eee Ty

CHARACTER*11 TOPOIN,COALIN,TOTOUT
REAL TOPO, COAL
INTEGER RADIUS, ROWS, COLS, ERSTAT,

C OUTFIL, ROW, COL, YCOORD, XCOORD, IOSTAT, CCNTRY, CCNTRX,
C TCNTRY, TCNTRX, CONROW, CONCOL, TOPROW, TOPCOL, COALRW,

C COALCL, OFFSET

REAL MINEL, MAXEL, INTBDN, P1, Z1, Ri, CUMVEC, PI, VECTOR, SUM
DIMENSION PYRMID(100,100), TOPO(253,230), COAL(163,140)

COMMON/COALBL/COAL
COMMON/TOPOBL/TOPO
PARAMETER (PI=3,1415926536)
MINEL = 100000.00

ROWS=163

COLS=140


The stress at the point at depth due to the perimeter cells
can be determined by summing up the stress due to the in-
dividual cells along the perimeter. From a center cell to
a corner cell along a perimeter side, the summation is

5/2
Py = 2 wt z [3 , (3)
i=0

where P., = vertical stress at point at depth due to
cells from center cell to corner cell of
perimeter side.

For the adjacent side, the stress due to elements from the
cell next to the corner cell to the perimeter side center cell
is

h-1 2-5/2
Pa = 22 ws) by [+4] 4)
f

where Pig = vertical stress at point at depth due to
cells beginning with cell adjacent to
corner cell through center cell of pe-

rimeter side.

Equations 13 and 14 must be multiplied by 4, then added
together to calculate the total stress at the point at depth
due to the perimeter.

2)
Py = sz ws) © a + z
s/t
+ 5 (15)
mal #

where P, = total vertical stress at point at depth due
to all cells around perimeter.

Multiplying both numerator and denominator by 8 m,
the number of cells along the perimeter, equation 15
becomes

h py bie 2 ys
4 1+— +4 1+—
Ultra XE ittg

Each summation in equation 16 divided by h in the de-
nominator can be calculated for a given perimeter width.
Table 2 shows these values for various perimeter widths
and illustrates that the summations converge. An average
distance can be used for all perimeter cells. Replacing the
summations in equation 16 by the number on which the
summations converge yields

P ey)

v

= ZW oy, [4(0.59) + 4(0.59)]
2 h .

Table 2.Summation of cell distances

according to increasing
perimeter widths

Width of h het

perimeter (in = =

grid cells) i=0 1=1
100 . 0.595 0.583
150 593 585,
200 592 586
1,000 590 589
10,000 . 589 589
589 589

20,000 .

h Half width of perimeter in num-
ber of grid cells,

Replacing h by W/r and canceling terms in equation 17
gives

3pz3w4
P, = == [@ 59]. (18)
Combining the constants in equation 18 yields
Py = cwTMP, (19)
3
where C = 32° (4) (059) |
t
= a constant.

The summation terms represent the distance to each small
square along the perimeter. Dividing through by 8 h, the
number of grid cells along the perimeter will result in an

3 ayy 5 :
p, = Fz w ) g, Li=0

oh (6)


for the study area, it was more expedient to separate the
process into two steps. One program extracts the ele-
vation from one DEM file at a time and saves the extract-
ed elevations to a smaller file. Then, a second program
reads in the six smaller files and combines them to provide
the one large study area file. Because the UTM coordi-
nate system is not parallel to the boundaries of the 7.5-min
quadrangle, DEM profiles vary in length. It is, therefore,
necessary to keep track of the northing and easting values
of the first point in each of the extracted profiles to match
them properly with adjacent profiles in the contiguous cut
files, The resulting elevation coverage for this study area
is a file of 445 by 445 elevation points arranged in the 30-
by 30-m grid pattern of the original DEM. The elevation
points are ordered from cast to west in rows, with the
tows ordered from north to south, as illustrated in fig-
ure 7. This format was chosen to be compatible with the
display program used for remote sensing data.

i

The study area extends from 463,515 m easting in the
west to 476,865 m easting in the east. The southern
boundary is at 3,676,035 m northing, and the northern
boundary is 3,689,385 m northing. The maximum surface
elevation is 800 ft above mean sea level, and the minimum
elevation is 187 ft. (Feet, instead of meters, were chosen
for the unit of measure of elevation in the data files to be
compatible with the coal seam elevation values, which
were in feet, and to facilitate comparison to topographic
maps.)

Coal Seam Top Elevations

The mining company provided logs of drill holes
throughout the mine property. Coal seam and overburden.
strata elevations were entered into a personal computer
(PC) data base along with the geographical (northing and
easting) coordinates for cach drill hole location. The

Om 30m 60m 90m 120m 150m 180m 210m 240m 270m 300m

1 2 3 4 5 6 7 8 9 10

429.7 | 449.4) 465.8) 485.4 | 501.8]! 515.0] 515.0} 508.4 | 492.0] 485.4
30m

306 307 308 309 310 311 312 313 314 315

426.4 | 436.2] 449.4] 472.3 | 492.0] 505.1} 498.6| 492.0) 475.6| 475.6
60 m

611 612 613 614 615 616 617 618 619 620

449.4) 459.2] 475.6] 485.4 | 488.7] 501.8) 488.7] 485.4] 469.0] 469.0
90 m

916 917 918 919 920 921 922 923 924 925

452.6) 452.6] 469.0) 472.3} 469.0] 478.9; 462.5] 465.8] 455.9] 455.9
120

1,221 | 1,222 | 1,223 | 1,224 |1,225 | 1,226 | 1,227 | 1,228 | 1,229 | 1,230

462.5] 452.6] 459.2] 462.5 | 452.6] 455.9] 442.8] 452.6] 462.5] 452.6
150

1,526 | 1,527 | 1,528 | 1,529 |1,530 | 1,531 | 1,532 | 1,533 | 1,534 | 1,535

449.4) 439.5) 449.4) 449.4 | 436.2} 436.2} 423.1) 433.0] 446.2] 436.2
180

1,831 | 1,832 | 1,833 | 1,834 {1,835 | 1,836 | 1,837 | 1,838 | 1,839 | 1,840

429.7] 426.4) 439.5) 446.1] 429.7) 423.1] 413.3) 413.3] 419.8] 429.7
210 me

2,136 | 2,137 | 2,138 | 2,139 |2,140 | 2,141 | 2,142 | 2,143 | 2,144 | 2,145

410.0} 413.3) 423.1) 429.7] 410.0] 410.0] 403.4) 396.9] 400.2] 419.8
240 ny

2,441 | 2,442 | 2,443 | 2,444 |2,445 | 2,446 | 2,447 | 2,448 | 2,449 | 2,450

400.2] 410.0) 410.0) 406.7 | 406.7} 419.8) 413.3] 396.9] 393.6] 413.3
270 mi

2,746 | 2,747 | 2,748 | 2,749 | 2,750 | 2,751 | 2,752 | 2,753 | 2,754 | 2,755

380.5{ 390.3} 390.3) 387.0} 400.2] 419.8} 429.7} 419.8] 416.6] 433.0
300 ml

KEY
1 Point number |
429.7 Elevation value, ft

Figure 7.—Format of Input file containing topographic elevations.