ABSTRACT

In general, the direction of the maximum horizontal stress in the 

eastern United States is fairly well defined. However, the variation 

of the magnitudes of the horizontal stresses is not very well

understood. Because the horizontal stresses cause severe ground 

control problems in underground coal and limestone mines

throughout the eastern United States a more complete

understanding of how the magnitudes vary would be useful for 

developing mine design strategies to combat horizontal stress 

related ground control problems. Therefore, in this National

Institute for Occupational Safety and Health (NIOSH) study, the 

variation of the magnitude of the horizon stresses in sedimentary 

deposits in the eastern and Midwestern United States are examined 

with respect to two factors, the elastic modulus of the rock and the 

site depth. Stress measurements from thirty-seven sites are used in 

the evaluation. 

Examining the applied excess strains indicates that the eastern 

United States can be, separated into high and low strain zones. For 

most of the eastern United States the maximum applied excess or 

tectonic strain ranges from only 300 to 550 micro strains. However, 

there is one area, a portion of the Beckley seam in the central 

Appalachian region where the strains are significantly higher than 

the other regions. In this higher strain zone, the maximum applied 

tectonic strains range from 700 to 1,000 micro strains. Regression 

models for each zone based on the elastic modulus can explain 

between 83 to 85% of the variation of the maximum horizontal 

stress. Because one region, the northern Appalachian district, has 

strains that are about 20% higher than the other regions in the low 

strain zone, multiple strain models based on geographic regions 

were developed for the low strain zone that can explain 87 to 91% 

of the maximum horizontal stress variation with the elastic modulus. 

Depth was found not to be a significant causal factor in any 

increase in the horizontal stress even though the site depths ranged 

from 275 to 2,500 ft. Beyond a theoretical increase, based on 

Poisson’s effect and gravity, no other increase in the horizontal 

stress with depth can be justified with this data. The most 

significant factor controlling the variation of the maximum

horizontal stress is the elastic modulus of the rock, not the

overburden depth. 

INTRODUCTION

In coal mines in the eastern United States, the horizontal stress 

direction and pattern has been recognized and is fairly well

understood with the direction of the maximum horizontal stresses 

apparently related to plate tectonics (1). From a standpoint of mine 

design and layout to control ground control problems due to 

horizontal stress, the direction of the maximum horizontal stress is 

very important (2). However, other important horizontal stress 

parameters, such as the magnitude and the magnitude variation, are 

not as well understood. The magnitudes, in general, from the stress 

measurements used in this analysis though fit the stress model for 

the North American mid-plate region developed from the World 

Stress Map Project (3). Essentially, the largest stress component is 

the maximum horizontal stress. Therefore, the resultant maximum 

horizontal stress magnitude is caused, to a large degree, by applied 

loads at the tectonic plate boundaries.

In the past, some linkage has been shown between the depth 

and an increase in the magnitude of the horizontal stress in North 

America (4, 5). In a recent study, equations have been developed 

for the increase in the maximum horizontal stress with depth in coal 

mines in the eastern United States (6). However, although there is a 

statistical relationship between the maximum horizontal stress and 

depth, the correlation is weak. 

Recent studies in the United Kingdom have shown that a very 

strong correlation exists between the maximum horizontal stress 

and the elastic properties of the rock (7). Coefficients of

determination as high as 0.95 indicate that there is a strong linear 

relationship between the elastic modulus and the maximum

horizontal stress. Obviously, the theory of elasticity provides a 

direct relationship between the elastic modulus and the stress at a 

point. The implication of such a strong linear correlation as seen in 

the United Kingdom, is that the region is being subjected to a fairly 

uniform strain field at least from the maximum horizontal stress 

component. However, the linear correlation between the minimum 

horizontal stress and the elastic modulus is much weaker (8). Site 

depths in these studies ranged from 300 to 2,500 ft and therefore 

the results cover the general zone of mining. A relationship 

between the elastic modulus and the horizontal stress has also been 

observed in China (9). 

In this study, stress measurements in the eastern United States 

are examined to determine if there is a relationship between the 

VARIATION OF HORIZONTAL STRESSES AND STRAINS IN MINES IN BEDDED 

DEPOSITS IN THE EASTERN AND MIDWESTERN UNITED STATES

Dennis R. Dolinar, Mining Engineer

National Institute for Occupational Safety and Health

Pittsburgh Research Laboratory

Pittsburgh, Pennsylvania USA
elastic modulus and the horizontal stress. Further, the effect of 

depth on the magnitude of the horizontal stress is reexamined. 

Besides the stresses, the horizontal strains are also evaluated. This 

analysis uses only those sites from sedimentary deposits including 

both coal and limestone. 

STRESS MEASUREMENTS

Stress measurements from 37 sites are used in this study with 

figure 1 showing the site locations. The stress measurements are 

from the northern (7 sites) and central (19 sites) Appalachian 

regions as well as from the eastern Mid-Continent region (11 sites). 

Most of the stress measurement results have been previously 

reported (1, 10). The site depths range from 275 to 2,500 feet and 

encompass the range of coal and limestone mining depths in the 

eastern United States. 

A requirement for the inclusion of a stress measurement in the 

analysis was that not only the magnitudes of the horizontal stress 

had to be known but also the site depth and the elastic modulus of 

the rock. Requiring the elastic properties of the rock limited the 

published stress measurement data that could be used in this 

analysis. 

APPLIED EXCESS HORIZONTAL STRAIN VARIATION 

FOR THE EASTERN UNITED STATES

For a strong relationship or significant correlation to exist 

between the horizontal stress and the elastic modulus, a necessary 

condition is that the strain field must be sufficiently uniform. 

Therefore, the applied excess strains for the sites are examined. 

Excess strains are the strains after the effects of gravity have been 

removed and are calculated from the excess horizontal stresses. 

Using the excess strains minimizes the expected influence of depth 

in the comparisons between sites. These strains are used to 

evaluate the strain variation regionally and across the eastern 

United States.

Excess horizontal stresses are calculated by subtracting the 

expected effects of gravity from the measured horizontal stresses. 

The following equations can b e used to calculate the excess stresses 

(10, 11):

 (1a)

and

 (1b)

where Pe = maximum excess horizontal stress, psi,

P = maximum horizontal stress, psi, 

í = Poisson’s ratio,

D = depth, ft,

Qe = minimum excess horizontal stress, psi, and

Q = minimum horizontal stress, psi.

These stresses have also been refered to as tectonic stresses. To 

calculate the excess stress, a Poisson’s ratio of 0.25 was used for all 

sites.

The applied excess horizontal strains can then be calculated 

from the excess stresses by the following equations (12). 

 (2a)

and 

 (2b)

where åpA = maximum excess applied strain, micro strain,

åqA = minimum excess applied strain, micro strain, and

E = elastic modulus of rock, million psi.

Again, a Poisson’s ratio of 0.25 is used to calculate the strains. The 

strain components in equation 2 are the actual strains that are being 

applied in excess of the gravity load. 

Figure 2 shows a histogram of the maximum excess applied 

horizontal strain by site across the eastern United States. Fifteen of 

the sites are between 300 to 400 micro strains and 23 of the 37 sites 

have a maximum excess strain between 300 and 550 micro strains. 

These sites are spread across all the main geographic regions. 

Therefore, the typical maximum excess strain field across the 

eastern United States appears be between 300 to 550 micro strains. 

The maximum excess strains above 550 micro strains were all 

measured in the central Appalachian region. There are 12 sites in 

this higher strain group. Those strains above 700 micro strains are 

all found in the Beckley seam where an extensive stress

measurement program was conducted (13). 

From the histogram, the strain field is skewed with only two 

sites below 300 micro strains. Both these sites are in the Illinois 

basin of the eastern Mid -Continent region. Because the mode is 

between 300 and 400 micro strains with the sites in this group from 

all the regions and from a range of depths, those sites below 300 

micro strains may be in partial strain relief. One site is near a fault 

and the other is at a depth of only 275 ft. This is the shallowest site

in the Illinois basin located in a region with valleys filled with 

unconsolidated material. This shallow site was excluded from 

x D

1

Pe P 1.1 



























−

= −

n

n

IN

IL OH

PA

NY

VA

WV

NC

KY

TN

MS AL

GA

SC

FL

x

x

x

x

x

x(3) x o(15)

(2)o

o

o

x

x

+(3)

+

+

+

+

o-Central Appalachian

+-Northern Appalachian

x-Eastern Mid-Continent

MI

Figure 1. Location of stress measurement sites used in this 

study, numbers in parentheses indicate multiple sites.

( )

x D

1

Qe Q 1.1 











−

= −

n

n

( )

E

P Q

å

e e

pA

− n

=

( )

E

Q

å

e e

q A

− n P

=
further analysis in this study. Essentially, the distribution is 

skewed because the strains cannot realistically fall below zero and 

the location of the mode indicates that those sites below about 

300 micro strains are possibly in partial strain relief. 

For the central Appalachian region the strain field appears to be 

bimodal. Because of the strain distribution and the geographic 

location of the sites, the sites can be grouped into high and low 

strain zones. Those sites with the maximum strain above

550 micro-strains are considered in the high strain group. In the 

Beckley seam, eight of the high strain sites are in adjoin ing areas of 

three mines (Bonny mine, Beckley Mining Co. mine, and northeast 

section of Maple Meadows mine) covering a distance of about 18 

miles (figure 3). Five of the low strain sites are in a separate but 

adjacent geographic area to the high strain zone stretching across 

two mines (southwest section of Maple Meadows Mine and

Beckley No. 1 Mine) for a distance of about 6 miles. Therefore, 

geographically distinct strain zones exist in the Beckley seam

encompassing areas that are tens to hundreds of square miles. Two 

other high strain sites, including one in the Beckley No. 2 mine, are 

adjacent to the low strain zone but are geographically separated 

from the high strain zone. 

In the rest of the central Appalachian region, the two remaining 

high stain sites are in a mine 25 miles to the southwest of the 

Beckley seam study area. The remaining two low strain sites are 

10 miles to the west and 100 miles to the southwest of the Beckley 

study area. The relationship of these high and low strain sites to 

those in the Beckley area and the extent of the high and low strain 

zones is not known. However, because of their geographic

locations these sites are grouped in the central Appalachian region 

with the Beckley sites.

For both the eastern Mid -Continent and low strain central

Appalachian regions, the sites are concentrated between 300 to 400 

micro strains with only one or two sites for each region between 

400 to 550 micro strains. For the northern Appalachian region, the 

sites are distributed fairly evenly across this 300 to 550 micro strain 

range. 

Figure 4 shows a histogram of the minimum excess applied 

horizontal strains across the eastern United States. Most of the 

sites (22) have strains between 100 to 400 micro strains. Nine of 

the sites have stra ins that are below 100 micro strains and almost 

all are found in the eastern Mid -Continent region. All the sites with 

strains above 400 micro strains are found in the Beckley seam and 

central Appalachian region. However, although the distribution is 

again skewed, it is not as well defined as that for the maximum 

horizontal strain. Further, because several of the sites have

relatively low strains, some of these sites could be in a partial strain 

relief situation regarding the minimum strain. 

Table 1 gives the regional average excess applied strain. The 

data from the central Appalachian region is separated into a low 

and high strain zones. The average strain for the high strain zone is 

statistically greater than the average for any of the low stain g roups 

at a 0.05 significance level. Clearly, the central Appalachian region 

has areas with substantially higher maximum excess applied strains 

than the other regions. In the Beckley seam the low and high strain 

sites are in distinct geographic groups. However, there is not 

sufficient data from the rest of the central Appalachian region to 

establish a clear pattern to the regional strain field.

0

2

4

6

8

10

12

14

16

0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 800-900 900-

1000

Strain, micro strain

Number of Sites

Eastern Mid-Continient 

Northern Appalachian

Central Appalachian (LS)

Central Appalachian (HS)

Figure 2. Average maximum excess applied horizontal 

strain distribution across eastern United States, by site.

Figure 3. Beckley seam mines indicating the high and low 

strain zones, the low strain zone is shaded.

0

1

2

3

4

5

6

7

8

9

10

0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 800-900 900-

1000

Strain, micro strain

Number of Sites

Eastern Mid-Continent

Northern Appalachian

Central Appalachian (LS)

Central Appalachian (HS)

Figure 4. Average minimum excess applied horizontal strain 

distribution across eastern United States, by site.
Table 1. Average excess applied horizontal strain by region for the eastern United States.

Region Number of 

Sites

Maximum 

Strain Micro 

strain

Standard 

Deviation

Range of Site 

Maximum Strain 

Micro strain

Minimum 

Strain Micro 

strain

Standard 

Deviation

Strain Ratio 

Max/Min

Northern Appalachian 7 440 90 300-540 260 70 1.69

Central Appalachian

 Low Strain Zone 7 370 60 300-480 170 70 2.17

 High Strain Zone 12 760 130 550-970 410 130 1.85

Eastern Mid -Continent 10 370 90 240-530 80 90 4.6

Table 2. Coefficients and regression statistics for maximum and minimum stress models (linear regressions fit through zero).

Maximum Stress Minimum Stress

Region Number of 

Sites Coefficient, KP

Coefficient of 

Determination

Coefficient, 

KQ

Coefficient of 

Determination

Low Strain Models

Northern Appalachian/Central Appalachian/ 

Eastern Mid -Continent 24 460 0.83 260 0.48

Eastern Mid -Continent/Central Appalachian 17 410 0.88 200 0.33

Northern Appalachian 7 530 0.87 350 0.73

Eastern Mid -Continent 10 390 0.91 120 0.02

High Strain Model

Central Appalachian 12 920 0.85 610 0.43

Except for the central Appalachian high strain zone, the other 

regions appear to have similar strain fields. The northern

Appalachian region though has strains that are about 20 percent 

higher than those in the other low strain regions. This has 

broadened the range of the lower strain zones as seen in the 

histograms. Further, the strain fields in the eastern Mid -

Continent region are more biaxial than the other regions as 

signified by the maximum to minimum strain ratio. The standard 

deviations are about 20% or less of the average regional strains 

indicating the uniformity of the strain fields across these

geographic areas. 

HORIZONTAL STRESS VARIATION WITH THE 

ELASTIC MODULUS FOR THE EASTERN UNITED 

STATES

For the variation of the horizontal stress with the elastic 

modulus in the eastern Untied States, a separate analysis needs to 

be conducted for both the low and high strain sites. This can be 

done because the high and low strain sites can be grouped to a 

large extent geographically. Since the high strain sites are all 

found in the central Appalachian region, the high strain model is 

restricted to this region. However, the low strain model could be 

in general applicable to much of the eastern United States. Again, 

to minimize the expected depth effects, the excess horizontal 

stresses are used. 

To examine the variation of the excess stress with the elastic 

modulus, a linear regression can be fit through the data with a 

zero intercept. The resulting equation is

 (3a)

and

 (3b)

where KP = maximum excess strain coefficient, micro strain, 

and

KQ = minimum excess strain coefficient, micro strain.

Figures 5 and 6 show the site average maximum and

minimum excess horizontal stress versus the elastic modulus for 

both the high and low strain sites. Table 2 gives the resulting 

coefficients for the elastic modulus in equation 3 for the low and 

high strain groups. These coefficients can be considered the 

strain from the designated excess or tectonic stress component. 

The low strain group is further separated geographically 

because the northern Appalachian region has strains that are 

about 20 percent higher than the other regions. The results of 

these separate models based on geographic regions or groupings

are also shown in table 2. The improved correlation when 

separate low strain models are used clearly reflects the higher 

strains in the northern Appalachian region. Although the low 

strain area of the central Appalachian region is geographically 

adjacent to the other two regions, the limitation is that the 

distribution of the low and high strain zones is not completely 

known across this region. 

Qe = KQ x E

Pe = Kp x E
For the maximum horizontal stress, there are clearly two 

groups of data, with no overlap between the lo w and high strain 

groups while the correlations for the models are extremely good. 

Therefore, the low strain model or models explain between 83 to 

Low Strain

y = 460x 

R

2

 = 0.84

High Strain

y = 920x

R

2

 = 0.83

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 2 4 6 8 10 12 14

Elastic modulus, 106 psi

Stress, psi

Northern Appalachian

Central Appalachian (LS)

Eastern Mid-Continent

Central Appalachian (HS)

Figure 5. Maximum excess horizontal stress versus elastic modulus for the eastern United States.

Figure 6. Minimum excess horizontal stress versus elastic modulus for the eastern Un ited States.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 2 4 6 8 10 12

Elastic modulus, 106 psi

Stress, psi

Eastern Mid-continent

Northen Appalachian 

Central Appalachian (LS)

Central Appalachian (HS)

Low Strain

Y= 260X 

R

2

=0.48

High Strain

Y=610X 

R

2

=0.43
91% of the variation of the maximum excess horizontal stress 

across much of the eastern United States. The high strain model 

explains 85% of the variation in the high strain zones of the 

central Appalachian region. However, there are no definitive 

boundaries over where the model should be applied. In general, 

the low strain models reflect the strain field applied across most 

of the eastern United States, including a portion of the central 

Appalachian region, while the high strain model reflects a high 

strain region that has only been observed in the central

Appalachian region. 

For the minimum horizontal stress, the correlation with the 

elastic modulus is much lower. The elastic modulus explains 

only between 2 and 73% of the variation (table 2). This lower 

correlation results from the difference in the biaxial nature of the 

strain fields between regions (table 1). Essentially, to have a 

minimum stress that is as highly correlated to the elastic modulus 

as the maximum horizontal stress, the ratio of the maximum to 

minimum stress or strain must be similar between the sites and 

across the regions. Apparently there is a difference between the 

natures of the two stress components where there may be other 

factors that are controlling the minimum horizontal stress

magnitude. 

Based upon the calculated coefficients for the elastic modulus, 

equations or models fo r estimating the maximum horizontal

stress that includes depth and the elastic modulus can be

developed. Combining equation 3 with the expected increase of 

the horizontal stress with depth results in the following type of 

equation:

 

 (4)

This equation calculates the maximum horizontal stress in a 

given region or strain zone and explains between 83 to 91% of 

the variation of the maximum horizontal stress between sites. 

A similar equation can be developed for the minimum

horizontal stress. However, much lower correlations make

estimations less reliable and subject to the regional and site 

differences in the biaxial nature of the stress field. 

EFFECTS OF DEPTH ON HORIZONTAL STRESS FOR 

THE EASTERN UNITED STATES

Because there is such a strong correlation between the

maximum horizontal stress and the elastic modulus, what, if any, 

of the remaining variation of the horizontal stress can be

explained by the depth? Figure 7 shows a graph of the maximum 

horizontal stress versus depth for the sites in this study. The 

resulting linear regression equation is:

 (5)

The coefficient of determination is only 0.18 while the tstatistic indicates that the depth factor is significantly greater than 

zero at a 0.05 significance level. Therefore, in general there 

appears to be a significant increase in the maximum horizontal 

stress of about 1.5 psi per foot of depth in the eastern United 

States, though the depth can only explain about 18 percent of the 

variation. Further, depth cannot explain the high maximum 

stresses at shallow depths (figure 7). The highest maximum 

horizontal stress in the study is at a depth of only 400 ft. For the 

maximum stresses above 5,000 psi, the elastic modulus ranges 

from 6.32 to 11.2 million psi. These high stresses are explained 

by the elastic modulus in combination with the strain field and 

reflects the strong relationship with the elastic modulus. 

x D K x E

1

P 1.1 + 1

































−

=

n

n

P = 1.48 x D + 1700

y = 1.48x + 1700

R

2

 = 0.18

0

1000

2000

3000

4000

5000

6000

7000

8000

0 500 1000 1500 2000 2500

Depth, ft

Stress, psi

Figure 7. Maximum horizontal stress versus depth in eastern United States.
To more clearly evaluate the depth effects, the influence of 

the elastic properties can be removed by normalizing the

maximum horizontal stress by the elastic modulus. This

normalization produces the strain from the maximum horizontal 

stress. The resulting regression equation is 

 (6)

where åp = strain from maximum horizontal stress, micro

strain.

The coefficient of determination is only 0.05 while the depth 

coefficient is not statistically greater than zero. Therefore, there 

is no correlation between strain and depth and no significant 

increase in strain with depth. This change in significance implies 

that there could be a relationship between the elastic modulus and 

the depth. The correlation between the elastic modulus and depth 

is only 0.2. Essentially there is very little correlation between the 

two parameters. However, there is sufficient relationship

between the elastic modulus and the depth that the stresses

normalized by the modulus show no significant increase with 

depth.

Both the increase in the maximum horizontal stress and the 

elastic modulus could be related to regional differences in the 

depth, elastic modulus and strain from the maximum horizontal 

stress. Table 3 shows various average parameters for each region. 

There are differences in the average depth, elastic modulus, and 

maximum average strain for each region. In this case the Illinois 

basin data has been separated from the eastern Mid -Continent 

region. Although the average strain in the Illinois basin is similar 

to the other regions except for the high strain zone, the elastic 

modulus is much lower and results in a much lower maximum 

horizontal stress than the other regions. Combine these lower 

stresses with the lowest regional average depth and some increase 

in the horizontal stress and elastic modulus will probably be 

observed across the eastern United States. Therefore the

statistical increase may be due in part to regional differences 

where the cause may not be the depth. 

Table 3. Summary of average parameters for each region.

Region Number 

of Sites

Depth, 

ft

Elastic 

modulus, 

106 psi

Maximum1 

train Micro 

strain

Northern 

Appalachian 839 6.31 600

Central Appalachian

 Low strain 7 831 6.09 490

 High Strain 12 996 3.29 1040

Eastern Mid -

Continent Region 10 636 4.09 480

Illinois Basin 7 600 2.60 510

1Strain from maximum horizontal stress.

Further, there are at least two strain fields in the eastern 

United States that are apparently related to the geographic

location rather than depth. To eliminate the affects of this 

regional strain difference only those sites from the lower strain 

regions can be examined. 

The resulting regression equation for the twenty-four low 

strain sites is 

 (7).

The coefficient of determination is only 0.0004 while the strain 

actually decreases with depth though not significantly. 

Essentially, without the high strain zone in the central

Appalachian region, there is no increase in the strain with depth. 

This is an indication of the effect of the high strain zone having 

the greatest average depth. However, development of the high 

strain zone does not appear to be related to an increase in depth 

but to the geographic location. 

Regional differences result in a statistically significant

relationship between the depth and the maximum horizontal 

stress. Because the strains do not increase, depth cannot be 

considered a major causal factor in this increase. However, a 

theoretical increase based on gravity and Poisson’s ratio is used 

in this study. This effect, if it exists, is small and could easily be 

masked in the data. 

CONCLUSIONS

In general, the strain field across the eastern United States can 

be separated into a high and low strain zones based on

geographic location. The high strain zone is geographically 

limited to the Beckley coal seam and central Appalachian region 

though the exact extent of this zone or zones is not known. The 

high and low strain zone distribution in the central Appalachian 

region is also not fully recognized. The lower strain zone 

encompasses much of the remaining eastern United States. The 

lower strain zone itself can be further subdivided because of the 

higher strains in the northern Appalachian region. Within each 

zone or region there is still some variation in the maximum 

horizontal strains between sites, though the standard deviations 

are 20% or less of the average strain. 

Even though there is some variation between sites, the

maximum horizontal strains are sufficiently uniform in each zone 

where the elastic modulus is the main factor controlling the 

magnitude of the horizontal stress. The elastic modulus can 

explain between 83 to 91% of the variation in the maximum 

horizontal stress between sites for each region. 

Developed regression models that include the elastic modulus 

and the effects of gravity and Poisson’s ratio can estimate the 

maximum horizontal stress magnitude. The low strain models 

are applicable to much of the eastern United States. The high 

strain model is confined to the central Appalachian region,

though the exact area of application is not known. 

For the minimum horizontal stress, the correlation to the 

elastic modulus is much weaker. In general, the minimum

horizontal stress magnitude is less dependent on the elastic 

properties of the rock. This indicates a difference in the nature of 

the two stress components and is reflected by the regional

variations in the ratio between the maximum and minimum

horizontal stress or strain. 

For this data set, depth was found not to be a significant 

causal factor in any increase in the horizontal stress. The 

statistically significant increase apparently results from the

gp = 0.16 x D + 570

gp = -0.005 x D + 520.
regional differences between other parameters such as the strain, 

elastic modulus, and average depth and a small correlation 

between the elastic modulus and depth.

Results of this study for the eastern United States are similar 

to those found in the United Kingdom studies and can be 

summarized by the following statement “Unlike the vertical

stress, the horizontal stress component is not related to depth but 

to the rock stiffness” (14). It must be emphasized that this 

conclusion applies only to the range of depths and the geographic 

areas investigated. 

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