Engineering Geology 91 (2007) 117 – 134

www.elsevier.com/locate/enggeo

Plan curvature and landslide probability in regions

dominated by earth flows and earth slides
Gregory C. Ohlmacher ⁎
Kansas Geological Survey, The University of Kansas, 1930 Constant Ave, Lawrence, Kansas 66047, USA
Received 21 July 2006; received in revised form 20 December 2006; accepted 16 January 2007
Available online 27 January 2007

Abstract

Damaging landslides in the Appalachian Plateau and scattered regions within the Midcontinent of North America highlight the
need for landslide-hazard mapping and a better understanding of the geomorphic development of landslide terrains. The Plateau
and Midcontinent have the necessary ingredients for landslides including sufficient relief, steep slope gradients, Pennsylvanian and
Permian cyclothems that weather into fine-grained soils containing considerable clay, and adequate precipitation. One commonly
used parameter in landslide-hazard analysis that is in need of further investigation is plan curvature. Plan curvature is the curvature
of the hillside in a horizontal plane or the curvature of the contours on a topographic map. Hillsides can be subdivided into regions
of concave outward plan curvature called hollows, convex outward plan curvature called noses, and straight contours called planar
regions. Statistical analysis of plan-curvature and landslide datasets indicate that hillsides with planar plan curvature have the
highest probability for landslides in regions dominated by earth flows and earth slides in clayey soils (CH and CL). The probability
of landslides decreases as the hillsides become more concave or convex. Hollows have a slightly higher probability for landslides
than noses. In hollows landslide material converges into the narrow region at the base of the slope. The convergence combined with
the cohesive nature of fine-grained soils creates a buttressing effect that slows soil movement and increases the stability of the
hillside within the hollow. Statistical approaches that attempt to determine landslide hazard need to account for the complex
relationship between plan curvature, type of landslide, and landslide susceptibility.
© 2007 Elsevier B.V. All rights reserved.

Keywords: Engineering geology; Geologic hazards; Geomorphology; Landslides; Statistical analysis; Mechanical analysis

1. Introduction of McMechen, West Virginia damaged 56 houses on a

colluvial hillside (Gray and Gardner, 1977). Geologi-
The Appalachian Plateau in Pennsylvania, West Vir- cally, this portion of the Appalachian Plateau features
ginia, and Ohio, USA (Fig. 1) is an area of high Pennsylvanian and Permian cyclothems. Cyclothems
landslide incidence in the eastern United States are rhythmic sedimentary sequences consisting of alter-
(Radbruch-Hall et al., 1982). Landslides in this region nating transgressive and regressive deposits (Heckel,
have destroyed houses, closed highways, and damaged 1977). In southeastern Ohio, the dip of the units is about
infrastructure. In March of 1975, a landslide in the town 6 m/km (∼ 0.3°) to the southeast (Fisher et al., 1968).
These cyclic sedimentary deposits contain paleosols and
⁎ Tel.: +1 785 864 2194; fax: +1 785 864 5317. shales that weather to form clayey residual soils that are
E-mail address: ohlmac@kgs.ku.edu. prone to mass movement. This region is incised by the
0013-7952/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.enggeo.2007.01.005
118 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

and Topal, 2005), factor weighting (Lee and Min, 2001;

Süzen and Doyuran, 2004a,b), decision trees (Keefer,
1993), fuzzy set theory (Chung and Fabbri, 2001),
Bayesian statistics (Chung and Fabbri, 1998; Lee and
Choi, 2004), and other approaches. The discussion of
the advantages and disadvantages of the various
methods is not germane to this paper. Here, the primary
focus is the datasets used in the analysis. Some re-
searchers use every dataset available regardless of how
it relates to landslide susceptibility. These individuals
collect data from a variety of sources, analyze the data
with a statistical method, and get a set of values that
measure the relative susceptibility. This approach may
give accurate results for conditions that fall within the
bounds of the data; however, it does not provide further
insight into landslide processes.
Some datasets contain variables that change over
time. For example, land use may be highly correlated
with landslide susceptibility, but land use patterns change
in space and time. If land use is a parameter in landslide
Fig. 1. Location map. Study area 1 is a region with a high incidence of susceptibility analysis, a commitment is being made
landslides within the Appalachian Plateau Province (AP) and includes to reevaluate the landslide susceptibility as land use
parts of Ohio (OH), West Virginia (WV), and Pennsylvania (PA).
changes. Additionally, the initial landslide susceptibility
Study area 2 is located in northeastern Kansas (KS). Other states
shown include Alabama (AL), Missouri (MO), Maryland (MD), New map may encourage some people to alter design and
York (NY), Vermont (VT), Virginia (VA), and Kentucky (KY). construction practices in areas where the hazard is higher.
Newly developed areas may have a different suscepti-
bility than older developed areas with otherwise similar
Ohio River and its tributaries creating the necessary characteristics. In such a case it would not be sufficient to
slope gradients and relief, and receives sufficient simply substitute a new land-use map; instead, the entire
precipitation for almost annual landslide activity. statistical analysis will need to be repeated.
Other areas in the central United States have con- In Kansas, the choice of parameters used in the sta-
ditions analogous to the Appalachian Plateau and have tistical analysis has been guided by engineering mech-
varying degrees of landslide susceptibility. One analo- anics. Slope failure is controlled by landslide geometry,
gous region is the Kansas City Metropolitan Area in material properties, and pore-water pressure. Pore-water
Missouri and Kansas (Fig. 1) where nearly horizontal pressure, in general, varies considerably over short
Pennsylvanian cyclothems are incised by the Missouri distances and short increments of time, and the change in
and Kansas rivers. The dip of the units is about 3 m/km pore pressure in response to rainfall is related to the
(∼0.2°) to the west-northwest (Ohlmacher, 2000b). The intensity and duration of the rainfall, the time since the
mean annual precipitation for Kansas City (∼ 97 cm/ last rainfall, and the pre-storm moisture in the soil
year) is about the same as the Appalachian Plateau (97– (Haneberg and Gökce, 1994). Because pore-water
112 cm/year) (National Climatic Data Center, On-line pressure is highly variable, it is ignored in the landslide
Data). The local relief around Kansas City is less than susceptibility analysis used in Kansas. Slope geometry
that found in the Appalachian Plateau, but this has not and material properties are less variable. A recent land-
prevented the occurrence of major landslides. In 1995, slide hazard analysis in Kansas used the slope gradient
two new houses were destroyed by a landslide in Over- and the surface geology as the parameters for failure-
land Park, Kansas, a suburb of Kansas City. surface geometry and material properties, respectively
The development of quantitative landslide-suscepti- (Ohlmacher and Davis, 2003). This approach has pro-
bility and landslide-hazard maps is a topic of active duced usable landslide-hazard maps for the Kansas
research (Soeters and Van Westen, 1996). Many dif- portion of the Kansas City metropolitan area.
ferent approaches are being used, including multiple Another factor that can be considered in landslide-
logistic regression (Ohlmacher and Davis, 2003), arti- hazard analysis is topographic curvature. Mathematically,
ficial neural networks (Lee et al., 2003, 2004; Yesilnacar curvature is defined as the change in slope angle along a
 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134 119

where: q = f x2 + f y2 + 1, fx = ∂f/∂x, fy = ∂f /∂y, fxx = ∂2f/∂x2,

fyy = ∂2f/∂y2, fxy = ∂2f /∂x∂y, β is the angle between the
tangent of the curve and the x-axis in the horizontal
plane and, φ is the angle between the normal to the
surface and the section plane (the imaginary plane of
intersection) (Kepr, 1969; Mitášová and Hofierka, 1993;
Moore et al., 1993a). For profile curvature, κprofile, the
normal to the surface is always in the section plane; the
angular relationships are

fx fy
cosu ¼ 1; sinb ¼ pffiffiffiffiffi; and sinb ¼ pffiffiffiffiffi; and
pq pq

Fig. 2. Mathematical definition of curvature. Curvature is the change in Eq: ð1Þ becomes
slope of a curve over a small increment ds along the curve at point a.
The curvature is the inverse of the radius of a circle ρ that is tangent to
the curve over the same increment ds. fxx fx2 þ 2fxy fx fy þ fyy fy2
jprofile ¼ pffiffiffiffiffi
p q3
very small arc of the curve, ds, (Fig. 2) (Thomas, 1968).
Curvature is the inverse of the radius of a circle that is where p = f x2 + f y2 (Mitášová and Hofierka, 1993; Moore
tangent over a small arc, at least three points, of the curve et al., 1993a). For plan curvature, κplan, the normal to the
(Kepr, 1969). A straight line has a curvature of zero curve is only in the section plane when the ground
(infinite radius of the tangent circle) and as the curve surface is vertical. The angular relationships are
becomes more acute the curvature value increases. At a
point on a three-dimensional surface — for example, a rffiffiffi
p fy fx
hillslope, an infinite number of curvature values can cosu ¼ ; cosb ¼ pffiffiffi; and sinb ¼ −pffiffiffi; and
q p p
exist, because the curvature is a function of the ori-
entation of the increment ds and the orientation of the the equation is
plane of intersection with the surface. A special case
exists where the hillslope at a point is straight in all
fxx fy2 −2fxy fx fy þ fyy fx2
directions and the curvature is equal to zero in all jplan ¼ pffiffiffiffiffi
directions. The three curvature values used for hillslope p3
and landslide analysis are profile, plan, and tangential
curvature (Dikau, 1989; Moore et al., 1993a,b; Ayalew (Mitášová and Hofierka, 1993; Moore et al., 1993a). For
and Yamagishi, 2004). Profile curvature is the curvature tangential curvature, κtangential, the normal to the curve is
in the downslope direction (aspect) along a line formed only in the section plane when the ground surface is
by the intersection of an imaginary vertical plane with horizontal. Thus, the angular relationships are
the ground surface. For profile curvature the increment
1 fy fx
ds has the same bearing and dip angle as the maximum cosu ¼ pffiffiffi; cosb ¼ pffiffiffi; and sinb ¼ −pffiffiffi; and
slope gradient at the point of interest. Plan curvature q p p
is the curvature of the topographic contours or the
the equation is
curvature of a line formed by the intersection of an
imaginary horizontal plane with the ground surface.
Tangential curvature is the curvature in a vertical plane fxx fy2 −2fxy fx fy þ fyy fx2
jtangential ¼ :
that is tangent to the contour at the point of interest. The p
orientation of the increment ds is horizontal and normal
to the aspect for both the plan and tangential curvatures. This corrects the error in the angle φ found in
The curvature of a line formed by the intersection of a published tangential curvature equations by Mitášová
surface and an imaginary plane is given by: and Hofierka (1993) and Gallant and Wilson (1996).
These curvature equations were tested on mathemat-
ical surfaces created from a series of functions Z(x,y). A
fxx cos2 b þ 2fxy cosbsinb þ fyy sin2 b
j¼ pffiffiffi ð1Þ 3 × 3 grid was created with the point of interest at the
ð q Þcosu center of the grid, and the required derivatives of Z(x,y)
120 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

were calculated using published approximations (Mitá- slumps), earth flows, and combination earth slide-flow
šová and Hofierka, 1993; Moore et al., 1993a; Gallant (Sharpe and Dosch, 1942; Fisher et al., 1968; Mills
and Wilson, 1996). Additionally, the curvature was cal- et al., 1987; Ohlmacher, 2000b). Other types of land-
culated from the derivatives of function Z(x,y) evaluated slides, especially rock falls and rock topples, occur in
at the center point of the grid, and graphically by deter- these regions but are not considered in this analysis.
mining the radius of the tangent curve. The results from The Appalachian Plateau (Fig. 1) is a dissected
the three approaches compare favorable indicating that plateau that stretches from central New York State to
the curvature equations presented above produce rea- northern Alabama in the eastern United States. The
sonable results. As the spacing between grid points region of particular interest to this study is the portion
decreased, the accuracy of the approximations of the with a high-incidence of landslides labeled 1 in Fig. 1
derivatives improves along with the values for the cur- (Radbruch-Hall et al., 1982). Jacobson and Pomeroy
vature. One function Z(x,y) was for a series of concentric provide an overview of landslides in the Appalachian
circular contours, and the plan curvature equations using Plateau (Mills et al., 1987). Along the Ohio River near
the approximations of the derivatives correctly calcu- Weirton, West Virginia, the local relief is about 128 m
lated the inverse of the radius of the circular contours. (D'Appolonia et al., 1967). The U.S. Geological Survey
Several authors (Dietrich et al., 1995; Heimsath et al., Wheeling 7½-minute topographic quadrangle has a total
1999, 2005) use the del2 operator (∇2z = ∂2z/∂x2 + ∂2z/ relief of 231 m and the local relief along the Ohio River
∂y2) as a curvature value. By examining Eq. (1), the del2 is up to 212 m. Further south, the Marietta, Ohio 7½-
operator can be a curvature value if the following minute topographic quadrangle has a total relief of
conditions are met: ∂ 2 z/∂ x∂ y = 0, β = 45°, and 135 m and bluffs along the Ohio and Muskingum rivers
pffiffiffi
cosu ¼ 2= q. If the term ∂2z/∂x∂y is not equal to can have local relief values of up to 85 m. The Marietta
zero and the del2 operator is used, then the magnitude and Wheeling areas were chosen as examples of the
and possibly the sign of the curvature value will be Appalachian Plateau, because of landslide-inventory
incorrect. The sign of the curvature value is important mapping by Pomeroy (1984) and Lessing et al. (1976),
for determining concavity or convexity of the curve. respectively.
Points exist within a digital elevation model where ∂2z/ The Midcontinent region in the United States is
∂x∂y is not equal to zero and where use of the del2 an area of generally low relief that extends from the
operator will give erroneous results. Appalachian Plateau to the Rocky Mountains and which
Curvature equations also have been developed by is locally dissected by rivers and streams. Ohlmacher
Zevenbergen and Thorne (1987). The plan curvature (2000b) gives a description of landslides in the vicinity
equation published in Zevenbergen and Thorne is cal- of the City of Atchison in northeastern Kansas (Fig. 1).
culating tangential curvature.
pffiffiffiffiffi The profile-curvature As an example of the Midcontinent region, the Leaven-
equation is missing the q3 term in the denominator, worth, Kansas 7½-minute topographic quadrangle has a
and thus the values are incorrect. total relief of 114 m and the bluffs along the Missouri
Both profile and plan curvature will affect the sus- River and its tributaries can have local relief values up to
ceptibility to landslides. Profile curvature affects the 85 m. It should be noted that the Oread Escarpment, a
driving and resisting stresses within a landslide in the cuesta where the southeast-facing slopes are steeper
direction of motion. Plan curvature controls the con- and have higher relief than the northwest-facing
vergence or divergence of landslide material and water slopes, intersects the Missouri River valley at the City
in the direction of landslide motion (Carson and Kirkby, of Leavenworth creating locally higher relief. Away
1972). This paper examines the role of plan curvature in from the escarpment, the bluffs of the Missouri River
landslide susceptibility and the effect of convergence of have modest relief values of about 45 m.
material in the direction of landslide motion. The bedrock of both areas consists of cyclically de-
posited Pennsylvanian and Permian sedimentary rocks
2. Landslide background called cyclothems (Heckel, 1977). An ideal cyclothem
contains a transgressive and regressive sequence with
This paper uses the landslide classification scheme of sandstones, siltstones, shales, and limestones (Fig. 3).
Varnes (1978) and Cruden and Varnes (1996) that sub- The ideal sequence begins with such non-marine sedi-
divides landslides into types based on material and mentary rocks as sandstones, siltstones, gray and red
movement. The study areas are covered predominantly shales, and coal. The sequence proceeds into marine
by fine-grained soils referred to as earth, and the pre- limestones and shales with black shales representing the
dominant landslide types are earth slides (including end of the transgression. The regression is represented
 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134 121

3. Plan curvature and landslides

Plan curvature can be used to subdivide hillslopes

into three regions: hollows, noses, and relatively planar
regions (Fig. 4). Hollows are regions in which the plan
curvature of the contours is concave in the downslope
direction and where surface water would converge as it
moves downslope (Hack and Goodlett, 1960; Reneau
and Dietrich, 1987). The term “coves” was used by
Sharpe and Dosch (1942) and Patton (1956) and is
analogous to “hollows”. Noses are regions where the
plan curvature of the contours is convex in the down-
slope direction, and surface water will diverge (Hack
and Goodlett, 1960). Relatively planar regions (Jacob-
son et al., 1993) have plan curvature values around zero.
Hack and Goodlett (1960) used the term “side slopes”
for these regions. For the purpose of this paper, no cutoff
values of curvature were established between the
hollows, planar regions, and noses.

Fig. 3. Idealized cyclothem simplified from (Heckel, 1977).

by marine limestones and shale grading into non-marine

sedimentary rocks. Although the basic sequence of
layers is relatively constant, cyclothems can vary both
spatially and temporally; for example, the non-marine
sequence can be thicker in some regions or during some
time intervals or cyclothems can be incomplete missing
one or more layers. Of particular interest to landslide
studies are the shales that can be zones of weakness
within the bedrock and can weather to provide clayey
colluvium (Fisher et al., 1968; Hamel and Flint, 1972).
Slope stability problems of the Appalachian Plateau are
related to the varied physical characteristics of the cy-
clothems with the non-marine shales being the weakest
units (D'Appolonia et al., 1967).
Shales, disaggregated in the laboratory, and soil
samples from Kansas are classified as CH, CL, and ML
soils and contain expansive clays including mixed-layer
illite/smectite, mixed-layer chlorite/smectite, and Ca-
montmorillonite along with non-expansive illite, kao-
linite, and chlorite (Ohlmacher, 2000b). The red mud-
stones and shales in eastern Ohio have a tendency to
slake rapidly; however, other shales will slake to varying
degrees (Fisher et al., 1968). Soil developed from the Fig. 4. Hollows, noses and landslide features. Hollows are regions with
shale and sampled in the Appalachian Plateau region concave topographic contours; while noses are regions of convex
have peak cohesion values of 0–38 kPa, peak angle of contours. The black polygons are recent landslides having fresh
internal friction of 19°–30°, residual cohesion values landslide features (scarps and fissures). The gray polygons are older
landslides having muted features. The landslides in this area are earth
of 0–3 kPa, and residual angle of internal friction of flows and earth slides. The area labeled A is a region of planar plan
8°–25° (D'Appolonia et al., 1967; Gray and Gardner, curvature with a recent and an older landslide. This landslide inventory
1977; Hamel and Adams, 1981). map is located within the City of Leavenworth, Kansas (Fig. 1).
122 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

3.1. Field observations with 25% occurring on planar slopes. None of the papers
that are based on work in the Appalachian Plateau
Sharpe and Dosch (1942) observed in Ohio that earth describes how curvature was measured. East of the
flows tend to occur in valley-head coves (hollows). Appalachian Plateau (Fig. 1) in the Valley and Ridge
Hollows concentrate ground and surface water, and the Province of West Virginia and Virginia, Jacobson et al.
concentration of water probably leads to increased earth- (1993) observed that hillsides with planar plan curvature
flow activity (Patton, 1956; Carson and Kirkby, 1972). have the highest susceptibility to landslides. Their study
Carrara et al. (1977) did a statistical analysis of land- area included debris slides, earth slides, debris flows,
slides and morphologic features for two study areas in and earth flows.
Calabria, Italy, and reported that in the first study area Field observations reveal that earth flows can occur
hillslopes with planar plan curvature had a higher in any area of the slope regardless of plan curvature
percentage of landslides than hollows, 55% to 34% (Fig. 4). As observed in earlier studies, earth flows do
respectively. In the second study area hollows had a occur in hollows-for example, in Kansas (Fig. 4) and
higher percentage of landslides than hillsides with Ohio (Fig. 5a). Examples are presented of recent earth
planar plan curvature, 37% to 53% respectively. In flows in regions of plan curvature that are broadly
both study areas noses had the lowest percentage of concave (Fig. 5b) and broadly convex (Fig. 5c) in
landslides. In Appalachian Plateau of West Virginia, the eastern Ohio. Label A in Fig. 4 indicates a region of
frequency of landslides is greatest in concave landforms planar plan curvature with both recent and older earth
with almost twice as many landslides in concave areas flows. Numerous hollows exist without any recent or
as in planar regions (Lessing et al., 1976; Lessing and older landslide features (Fig. 5d). What is the distribu-
Erwin, 1977; Lessing et al., 1994). For the Greater tion of earth flows with respect to plan curvature? Are
Pittsburgh, Pennsylvania region, Pomeroy (1982) stated hollows an end-member landform in a terrain dominated
that 60% of the landslides occur in concave landforms by earth flows and earth slides?

Fig. 5. Photographs of landslides and plan curvature. (a) A recent earth flow that occurred in a hollow (concave plan curvature) located eastern Ohio.
(b) A recent earth flow that occurred in a broadly concave (almost planar plan curvature) located across the Ohio River from Wheeling, West Virginia.
(c) A recent earth flow that occurred on nose (convex plan curvature) located in eastern Ohio. (d) A hollow located northeast of Wheeling, West
Virginia with no evidence of recent or older landslide activity.
 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134 123

3.2. Statistics background datasets are only available for a portion of the study
area. A comparison of the 30-m and 10-m DEM datasets
Plan curvature was calculated from U. S. Geological was conducted for a 64-km2 region consisting of the
Survey Digital Elevation Model (DEM) data using a Kansas portion of the U. S. Geological Survey Parkville
computer program based on the curvature equations and North Kansas City 7½-minute topographic quad-
presented above. The program input is a DEM that was rangles. As would be expected, the values for the grid
converted to an ESRI raster in ASCII format. The ESRI cells vary between the 10-m and 30-m data because
raster is subdivided into 3 × 3 clusters of grid cells, and derivatives calculated from the 10-m data are based on
the required derivatives are calculated using the more closely spaced data than those calculated from 30-
equations found on page 445 of Moore et al. (1993a). meter data. The range of curvature values is larger for
The derivatives are then entered into the equation for the 10-m DEM dataset with curvature values from − 8.8
plan curvature. If one or more elevation values are to 9.2 in units of 1/m whereas the values for the 30-m
missing from a 3 × 3 cluster, no curvature value is cal- dataset range from −1.5 to 2.7. The relative distribution
culated for that cluster, and no curvature values are of the plan curvature values is roughly the same for
calculated for the grid cells along the perimeter of the both data sets (Fig. 7a). Only a few data points exist
database. The output is an ESRI raster in ASCII format near the positive (N 0.15) and negative (b − 0.15) ex-
with plan curvature values. Positive values of curvature tremes of both datasets in Fig. 7a. Thirty-meter DEM
are concave in a downslope direction or hollows. A datasets were used only where 10-m DEM datasets were
comparison of calculated plan curvature values calcu- unavailable.
lated using a 30-m DEM and topographic contours is Plan curvature is a continuous variable as opposed to
presented for a hollow in Fig. 6. a categorical variable, for example, geologic unit. In
Both 10-m and 30-m U. S. Geological Survey DEM order to evaluate the probability of a landslide given
data were used in the analysis. In Kansas, 10-m DEM the plan curvature, the plan curvature values were

Fig. 6. Plan curvature values. This map of a hollow near Leavenworth, Kansas (Fig. 1) shows plan curvature values calculated using the program
written for this study. Positive plan curvature values are concave regions or hollows. Plan curvature values greater than or equal to zero are shown in
bold font. The DEM has 30-m spacing between elevation values.
124 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

zero had very large numbers of grid cells and the

intervals near the positive and negative extremes had a
very few cells. Additionally, landslides in two map areas
in Kansas comprise only 0.5% of the total area. This low
percentage of landslides for the study area leads to high
fluctuations in landslide probabilities in intervals with
few cells (Fig. 7b). In order to increase the number of
grid cells in intervals with few cells and to improve the
probability estimate for each interval, the total range of
values was subdivided into intervals with unequal
ranges of plan curvature. The subdivision that worked
best had a very small interval with a plan curvature value
of zero at the center of the range and each adjacent
interval being twice the range as the interval before. For
these studies, the range centered on a plan curvature of
zero was set to 0.001 (1/m). The ranges for intervals on
either side were 0.002; the next range was 0.004, and the
ranges continued to double away from the zero interval.
Fig. 7b displays a comparison between the results using
equal range intervals and intervals with unequal ranges
of values. Between plan curvature values of − 0.05 and
Fig. 7. Comparison of 10-m and 30-m DEM data. Graph (a) shows the
0.05, the equal and unequal intervals compare favorably
relative distribution of area versus plan curvature in a cumulative form
for the both DEM datasets. Graph (b) shows a comparison of landslide except for the interval containing a plan curvature of
probability between equal and unequal intervals of plan curvature zero. For the regions below − 0.05 and above 0.05, the
values and between the 10-m and 30-m DEM data. The DEM data is probability values for the intervals with equal ranges
for the U. S. Geological Survey Parkville and North Kansas City, display considerably more scatter. This is the region
Kansas and Missouri 7½-minute topographic quadrangles.
where the number of cells in the interval is low (Fig. 7a).
For the remaining analysis, intervals with unequal
subdivided into intervals. The probability was then ranges as defined above will be used.
calculated using Bayes' Theorem: The interval of plan curvature containing zero has a
landslide probability value that is considerably lower
PðlandslidejcurvatureÞ than the two adjacent plan curvature intervals (all data
PðcurvaturejlandslidesÞd PðlandslidesÞ curve in Fig. 8). Part of the reason for this anomaly is
¼ that almost all the low slope areas including valley
PðcurvatureÞ
floors and ridge crests have plan-curvature values near
where P(A|B) is the conditional probability, the zero. This fact combined with the lack of landslides in
probability that A will occur given that B has occurred,
and P(A) is the probability that A will occur (Davis,
1973; Carr, 1995). The probabilities are calculated using
the areal extent of the parameters (landslides and plan
curvature intervals), where P(curvature|landslides) for a
range of curvature values is found by dividing the
number of grid cells with landslides within that
curvature range by the total number of grid cells with
landslides, P(landslides) is the total number of grid cells
with landslides divided by the total number of grid cells,
and P(curvature) is the number of grid cells in each
curvature range divided by the total number of grid
cells.
Fig. 8. Threshold slope comparison. The gray line incorporates all the
A fluctuation in the probability values is observed data from the Atchison East and West, Kansas and Missouri 7½-
when intervals of plan curvature with equal ranges of minute topographic quadrangles. The black line has the grid cells with
values were used (Fig. 7b), because the intervals close to slope gradients less than 4° removed from the data.
 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134 125

Table 1
Landslide data
Recent landslides Older landsides Map
2 2 2 area2
Number Average area (m ) Total area (km ) Number Average area (m ) Total area (km )
(km2)
Atchison East and West, KS 145 880 0.13 85 11,600 0.98 202.35
Easton East and West, KS 550 530 0.29 880 9900 8.71 299.19
Leavenworth, KS 226 880 0.20 430 16,300 7.03 171.42
McLouth and Jarbalo, KS 553 600 0.33 554 7700 4.3 299.60
Parkville and North Kansas City, KS 65 330 0.02 95 1900 0.18 63.69
Potter and Oak Mills, KS 329 750 0.25 604 11,600 7.00 279.10
Marietta, OH (Pomeroy, 1984) 708 2500 1.78 – – – 111.85
Wheeling, WV (Lessing et al., 1976) 290 2300 0.67 971 10,900 10.60 122.39
Wheeling, WV and OH (Davies and 78 14,100 1.10 314 122,300 38.42 147.41
Ohlmacher, 1978)

regions of low slope lowers the overall probability for Marietta, Ohio, study area includes only recent earth-
the interval including a plan curvature value of zero. flow features. The Wheeling, West Virginia study area
One solution is to use a minimum slope-gradient thresh- was mapped as part of two different landslide studies.
old value. A dataset where grid cells with the slope Lessing et al. (1976) produced a map of West Virginia
gradient below a minimum threshold of 4° were re- portion of the Wheeling quadrangle as part of a project
moved has an increase in the probability of landslides to map landslides in 28 7½-minute quadrangles in the
for the interval of plan curvature containing zero state. Davies and Ohlmacher (1978) mapped the entire
(Fig. 8). Even with a 4° slope gradient a portion of the Wheeling quadrangle as part of a study mapping land-
anomaly remains. A low slope threshold may have the slide features in the Appalachian Plateau from Pennsyl-
unwanted effect of removing some low slope grid cells vania to northern Alabama. The total area mapped as
from along hillsides and within landslides. The use of recent and older landslide is smaller; the individual
low slope thresholds should be evaluated based on the recent and older landslides are smaller (Table 1); and the
conditions in the study area under consideration. All area mapped as susceptible to landslides is larger on the
the data points will be used in the analysis presented in landslide map of Wheeling by Lessing et al. (1976)
this paper. relative to the landslide map by Davies and Ohlmacher
A comparison of the landslide probability results
between the 10-m and 30-m DEM datasets was con-
ducted for a portion of the Parkville and North Kansas
City 7½-minute quadrangles. The results of the compa-
rison indicate a shift in the probabilities between the 10-
m and the 30-m DEM datasets (Fig. 7b). The 30-m DEM
dataset had lower landslide probabilities for the region
of plan curvature values shown. However, the relative
distribution of landslide probabilities with respect to the
intervals of plan curvature remains essentially the same.

3.3. Statistics results

Several landslide maps were used in this study,

including 5 published and 1 unpublished landslide-
inventory studies in the Kansas portion of the Kansas
City metropolitan area (Ohlmacher, 2000a, 2003, 2004a,
b, 2005), along with landslide-inventory studies from
Marietta, Ohio (Pomeroy, 1984), and Wheeling, West
Virginia (Lessing et al., 1976; Davies and Ohlmacher,
1978). Table 1 contains numbers and areal extent of Fig. 9. Plots of landslide probability versus plan curvature for
landslides from the landslide-inventory studies. The landslide-inventory datasets from northeastern Kansas.
126 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

(1978). Although results from both mapping projects are observed in Kansas and Marietta (Figs. 9 and 10a). In
shown, it is beyond the scope of this study to evaluate general, two conclusions can be drawn from the stat-
the quality of the two Wheeling landslide-inventory istical analysis: 1) hillsides with planar plan curvature
maps. have the highest susceptibility to landslides, and 2)
Plots of the landslide (earth flow and earth slide) the hillsides with concave curvature are slightly more
probability with respect to plan curvature are shown in susceptible to landslides than hillsides with convex
Figs. 7–10. The Kansas results show that hillsides with curvature.
planar plan curvature have the highest probability of
landslides and that areas with concave curvature 4. Discussion
(hollows) have slightly higher probabilities than areas
with convex curvature (Figs. 7–9). Differences in the What causes the decrease in susceptibility as the plan
maximum landslide probability for each individual area curvature becomes more concave? Carson and Kirkby
may represent the effects of other factors such as (1972) observed that plan curvature affects the down-
changes in material properties and the distribution of slope movement of soil in two ways. Where plan cur-
slope gradients on landslide susceptibility. An analo- vature is concave, water flow is concentrated into the
gous pattern of landslide probability with respect to plan hollow, which will increase the moisture content of the
curvature is observed in the recent landslide data for soil and the amount of time the soil will remain saturated.
Marietta and Wheeling (Fig. 10a). However, when both This, in turn, will increase erosion and decrease stability
the recent and older landslides are analyzed for the of the soil. However, soil being transported downslope
Wheeling map by Lessing et al. (1976), the maximum by various mechanisms including slope wash, creep, and
landslide probability is shifted into the region of con- landsliding will also converge in areas with concave plan
cave slopes (Fig. 10b). For larger plan curvatures values curvature causing the soil to accumulate in the hollow
(more concave), the probability of landslides decreases and stabilizing the slope. Carson and Kirkby noted that
as in all of the other study areas. The cause of the shift in the rate of ground-surface lowering is directly related to
the maximum probability of landslides in the dataset by the downslope sediment flux and inversely related to the
Lessing et al. (1976) was not investigated in this study. curvature (concavity positive). Thus, the rate of ground-
The landslide-probability results from the landslide- surface lowering should be less in hollows.
inventory map of Wheeling, West Virginia by Davies The convergence of ground water in areas with con-
and Ohlmacher (1978) (Fig. 10b) are similar to those cave plan curvature is one important factor. Anderson
and Burt (1978a,b) installed a series of piezometers in a
hollow and measured the distribution of pore-water
pressures. Their results show that ground-water flow
does converge within an area of concave plan curvature.
A combination modeling and field study for an isolated
hollow showed that areas of convergent topography
(concave plan curvature) tend to remain saturated be-
tween storms due to convergence of ground-water flow
(Jackson and Cundy, 1992). O'Loughlin (1986) mod-
eled a complex topography with several hollows by
examining the upslope catchment area associated with
adjacent segments of contours. In hollows, the catch-
ment area is larger than on noses because water flow is
converging in the hollow. The results showed that areas
with the highest plan curvature saturate first and as the
duration of the storm increases, areas with lower plan
curvature values begin to saturate. Areas that become
saturated first and remain saturated should have higher
pore-water pressures and should be more susceptible to
landslides. The ground-water model of O'Loughlin was
Fig. 10. Plots of landslide probability versus plan curvature for
landslide datasets from (a) Marietta, OH and Wheeling, WV using only
combined with a simple one-dimensional slope stability
the recent landslide data and (b) Wheeling, WV using both recent and model in order to determine areas susceptible to debris
older landslide data. flows (Montgomery and Dietrich, 1994). The model
 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134 127

along the lateral boundaries and that faults are observed

between the mudflow and the undisturbed soil outside
the landslide (Hutchinson, 1970). Slope stability ana-
lysis that includes the end effects of the landslide indi-
cate that incorporating forces acting on vertical sides of
the landslide increases the factor of safety (Baligh and
Azzouz, 1975; Hutchinson and Del Prete, 1985; Gens
et al., 1988; Hungr et al., 1989). However, the lateral
forces acting on vertical surfaces within landslides —
for example, columns used in three-dimensional anal-
ysis of landslides, appear to be insignificant in slope
stability analysis (Hungr, 1987).
The role of lateral forces with respect to relative
susceptibility to landsliding is evaluated in this paper by
examining the forces acting on a column with a unit
surface area (Fig. 11). The column represents a unit
slope element and is not an element of a discretized
landslide in a slope stability analysis. Slope stability
methods that include the end effects exist to determine
the factor of safety and potential failure surface for
individual landslides (Baligh and Azzouz, 1975; Hungr,
Fig. 11. Schematic drawing of a hollow and the trapezoidal column 1987; Gens et al., 1988; Chen et al., 2001; Xie et al.,
used as part of the factor of safety analysis. 2003). The analysis presented below is developed solely
as a measure of the effect of lateral forces associated
with plan curvature on landslide susceptibility.
calculates the minimum steady state rainfall required for The column has a trapezoidal shape in plan view and
a region of the slope to fail. The results indicate that is very small compared to the area of the hillside with
steep channel areas in hollows will fail first. These concave plan curvature. For this analysis, the column is
studies indicate that within a hollow the areas with the assumed to be at rest. The potential direction of motion
highest plan curvature (most concave) should be the of the column is perpendicular to the parallel ends of the
most susceptible. As the plan curvature decreases, the trapezoid and from the longer end toward the shorter end
susceptibility to landslides should also decrease. (Fig. 12). For the purpose of this analysis, the parallel
The above discussion only incorporates the effects ends of the trapezoid can be thought of as contours;
associated with the convergence of water in the hollow. however, it must be remembered that contours in nature
In a manner analogous to the convergence of water, the
material of the landslide is attempting to converge as it
moves downslope in a hollow. The convergence of
material will lead to lateral forces on the sides of the
particles (out-of-the-plane of landslide motion). As
adjacent soil grains in the soil mantle are attempting to
move in the direction of steepest decent, the grains will
converge in areas of concave plan curvature and diverge
in areas of convex plan curvature. Additionally, in the
study areas the earth flows and earth slides are occurring
in fine-grained soils containing clay minerals and
cohesion is also adding lateral forces to the analysis.
Lateral forces have a role in slope-stability analyses
where end effects are included. Hutchinson (1961) in-
cluded friction acting on the lateral boundaries of land-
Fig. 12. Comparison of an arch to the motion of the trapezoidal
slides in a list of forces resisting slope failure. Field columns of soil within a landslide. For an arch, the blocks are being
evidence from studies of mudflows in Kent, England driven downward by gravity; where as, the plan curvature of the
indicates that displacement is confined to a narrow zone hollow causes the trapezoidal columns to converge.
128 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

are rarely parallel. Adjacent to the slanted sides of the along the base of the column. The third and fourth terms
trapezoid are additional trapezoidal columns. These in numerator of Eq. (2) and the second term in the
columns are rotated relative to the central column, and numerator of Eq. (3) are the lateral resisting forces on
the direction of movement of the adjacent trapezoidal the slanted sides of the trapezoidal column. In a
column relative to the central column leads to conver- cohesive soil on a hillside with planar plan curvature
gence of the columns. An equivalent model could be (α = 0), a lateral resisting force will still exist as is shown
developed using square columns and accounting for the by the third term in Eq. (2). The solid line in Fig. 13
convergence between adjacent columns. The downslope shows the percentage increase in factor of safety with
driving force for the column was determined by using increasing convergence angle for a case of a cohesion-
the resultant force in the downslope direction parallel to less soil with an angle of internal friction ϕ of 27° and
the basal failure surface and is based on the weight of the with a slope gradient θ of 27°. The water forces U and V
column. The stress analysis used to determine the lateral were ignored in Fig. 13 because the determination of the
forces is somewhat analogous to that for an arch except water forces will require an understanding of the dis-
in the case of the arch, the driving forces for each block tribution of hydraulic heads and pore-water pressures
are parallel (gravity); where as, the downslope driving along with details of the direction of ground-water
forces for the trapezoidal columns converge (Fig. 12). A migration. The soil density is 20 kN/m3, and the thick-
factor of safety equation for the trapezoidal column was ness of the landslide is 2 m. Changes in the density of
developed using an approach similar to the infinite- the landslide material and the thickness of the landslide
slope approach and is: have no effect on the results for a cohesionless soil. Over
the range of convergence angles considered the per-
ðCab=coshÞ þ ðW −U Þcoshtan/ þ 2ðCdb=cosaÞ þ 2ðW −V Þsinhsinatan/
FS ¼ centage increase in factor of safety is approximately a
W sinh
ð2Þ straight line, because the increase in factor of safety is
controlled by the sine of the convergence angle. For
where C is the cohesion, a is the average width (per- convergence angles greater than those used in Fig. 13,
pendicular to motion) of the trapezoid, b is the hori- the rate of percentage increase in the factor of safety
zontal length of the trapezoid between the parallel ends, should slow and the line will no longer be straight.
d is the vertical thickness of the column, W is the weight The effect of cohesion was evaluated by setting a
of the column, U is the water force acting along the cohesion value equal to 3 kPa, which is the maximum
base, V is the water force acting along the side, θ is the value for the residual cohesion in the Appalachian
slope gradient, and ϕ is the angle of internal friction. Plateau. The dashed line in Fig. 13 is for this cohesive
The convergence angle α is the angle between a slanted soil and indicates that the change in factor of safety is not
side of the trapezoid and a side of an equivalent rect- as great as in the cohesionless soil. However, two issues
angle. The derivation of this equation is presented in the render the percentage change in factor of safety (Fig. 13)
Appendix. An approximate relationship between the slightly misleading. The magnitude of the factor of safety
convergence angle and the plan curvature κ is given by: for planar plan curvature (α = 0) for the cohesive soil is
about 130% higher than an equivalent cohesionless soil
j ¼ 2tana
because for the cohesive soil, cohesion acts on the sides
As was stated earlier, the residual cohesion in soils of the trapezoidal blocks. Also, the magnitude of the
from landslides in the Appalachian Plateau is very close change in factor of safety (FSα = 25 − FSα = 0) is roughly
to zero. Without cohesion, Eq. (2) simplifies to: the same for the two soils: 0.43 for the cohesionless soil
and 0.55 for the cohesive soil. The slight increase in
ðW −U Þcoshtan/ þ 2ðW −V Þsinhsinatan/ magnitude for the cohesive soil is because as the con-
FS ¼ ð3Þ vergence angle increases the surface area of the sides of
W sinh
the trapezoidal column increases and the resistance due
If the convergence angle α is zero (planar plan cur- to cohesion increases along with the frictional resistance.
vature), Eq. (3) simplifies to the standard factor of safety Thus, the addition of cohesion does increase the factor
equation for a cohesionless soil. of safety as the convergence angle increases, and the
Eqs. (2) and (3) indicate that as the convergence magnitude of the increase in factor of safety is larger for
angle (plan concavity of a hillside) increases, the lateral the cohesive soil.
resisting forces and the factor of safety increase. The How do these results relate to debris flows? This
first two terms in the numerator of Eq. (2) and the first study lacks data from regions with debris flows. Debris
term in the numerator of Eq. (3) are the resisting forces flows can initiate in areas with hollows and in areas with
 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134 129

includes plan curvature requires knowledge of the local

relationship between plan curvature and landslides.
This, in turn, requires quality landslide-inventory data
and quality digital elevation data. From the quality data,
the local relationship between plan curvature and land-
slide susceptibility can be developed. Knowledge of the
relationship between plan curvature and landslides is
very important in statistical methods like multiple
logistic regression where plan curvature can be treated
as a continuous variable. It is not as critical in statistical
methods where plan curvature is treated as a categorical
Fig. 13. Plot showing the percentage increase in factor of safety with variable and subdivided into intervals. For the statistical
increasing convergence angle. analysis presented above which categorized the plan-
curvature values, the optimum results were obtained by
using unequal ranges of plan-curvature values.
planar plan curvature. Some of the 1976 Dorset Landsliding, slope wash, gullying, subsidence, tec-
Mountain debris flows in Vermont (Fig. 1) initiated in tonics, creep, and other processes shape the local land-
regions of the mountainside with planar plan curvature form geometry and play a role in the relationship
(Fig. 14). The mobilized landslide material moved into a between plan curvature and landslide susceptibility.
stream channel and continued down the mountain in the The region surrounding the Ohio River in Ohio and
stream channel. Insufficient data exists at Dorset West Virginia is dominated by a landform consisting of
Mountain to do a statistical analysis on plan curvature.
In the Blue Ridge Province of Virginia where debris
flows dominate, Wieczorek et al. (1997) also observed
higher landslide susceptibility on hillsides with planar
plan curvature. A study of debris slides in Idaho re-
vealed no relationship between plan curvature and
landslide susceptibility (Gritzner et al., 2001).
The preceding results from this study with respect to
cohesionless soils suggest that susceptibility to debris
flows that begin as landslides should be highest in areas
of planar plan curvature. However, other factors that
affect the initiation of debris flows may influence the
lateral resisting forces. Iverson et al. (1997) indicate
that mobilization of debris flows involve widespread
Coulomb failure, liquefaction of the material, and
conversion of translational energy to vibrational energy.
They further state that the factors leading to liquefaction
include an upward component of ground-water flow and
a densification of the material with a resulting increase
in pore-water pressure. The process of densification and
increase of pore-water pressure might lead to a reduction
in the lateral resisting stresses by reducing the effective
stress along the sides of the trapezoidal column. This
reduction of resisting forces on the sides will decrease
the factor of safety when compared to landslides without
liquefaction. Thus, the relationship between plan cur-
vature and debris flows may be different from the
relationship for earth flows and earth slides.
This analysis indicates that the relationship between
plan curvature and landslide type is complex. A valid Fig. 14. Debris flow on Dorset Mountain in Vermont. This 1976 debris
landslide-susceptibility or landslide-hazard analysis that flow started in a region of relatively planar plan curvature.
130 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

alternating hollows and noses. The same is true for landslide mapping in the Appalachian Plateau by
northeastern Kansas, especially in the vicinity of the City William E. Davies was supported by the U. S. Geo-
of Leavenworth. Some hollows are very broad (Fig. 5d). logical Survey. I would also like to thank John Davis for
Contours in these areas have a complex quasi-sinusoidal helpful comments on an early draft of this manuscript.
shape (Fig. 4). Earth flows and earth slides are the Bill Haneberg and an anonymous reviewer provided
dominant slope process in these regions. Areas with insightful comments that improved this paper.
planar plan curvature are the most susceptible to land-
slides. Regions of the hillside that have developed slight Appendix A
concavity because of landslide events will be more
susceptible than adjacent regions with slight convexity. The following section presents the derivation of the
The hollows will increase in size, and a hollow and nose equations used in the analysis of plan curvature and
terrain will develop. Thus, areas with planar plan cur- landslides. The analysis is based on the infinite slope
vature should not remain planar. Understandably, slope model for slope stability. A very small block of material
wash, gullying, and creep are active in both study areas. is isolated for analysis from the entire slope. For this
The contribution of karst in the study areas is very minor analysis the block is shaped like a parallelogram in the
because the thicknesses of the limestone and other karst- cross section (side view in Fig. A1) and the top of the
prone units are too thin relative to the local relief to create block is a trapezoid (map view in Fig. A1). A static's
sinkhole features of this magnitude. When compared approach is used to create the equation for the factor of
to the plate boundaries along the west coast of North safety. The depth of the block is d and extends from the
America, the contribution of tectonics in this region is ground surface to the slide plane. The slide plane is
relatively minor. The hills in these study areas are most parallel to the ground surface. The average width of the
likely being shaped by landslides, slope wash, and creep, block normal to the downslope direction is a (map view
and this landform of hollows and noses may be the result. in Fig. A1). The horizontal thickness of the block in the
Thus, a landslide hollow may represent an end-member downslope direction is b (side view in Fig. A1), and b is
landform in terrain dominated by earth flows and earth very small as compared to the length of the slope in the
slides.

5. Conclusion

The statistical analysis presented above indicates that

hillsides with planar plan curvature are the most sus-
ceptible to earth flows and earth slides. Hillsides with
concave plan curvature (hollows) are slightly more
susceptible to landslides than hillsides with convex plan
curvature (noses). Although conventional wisdom
suggests that the convergence of ground and surface
water in hollows should increase landslide susceptibil-
ity, the resisting forces between soil particles that results
from convergence of material increases the stability of
hollow. The type of landslide along with other geo-
morphic processes may play a role in the nature of the
local relationship between plan curvature and landslide
susceptibility. The local relationship is important to the
treatment of plan curvature in statistical landslide sus-
ceptibility and hazard analysis. A landform composed
of hollows and noses is a relatively stable landform in
regions dominated by earth flows and earth slides.

Acknowledgments

The Kansas landslide mapping and research program

is supported by the Kansas Geological Survey. The Fig. A1. Sketch of trapezoidal column showing stresses.
 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134 131

downslope direction. The weight W of the block is given

by:

W ¼ gs abd

where γs is the unit weight of the soil. In order to

simplify the mathematics the coordinate system will be
rotated so that the X and Y axes will be parallel to the
slide plane and the Z axis will be normal to the slide
plane (Fig. A1). The dip of the slide plane is given by the
angle θ. The weight is then resolved into a force normal
to the X–Y plane N and a force parallel to the X axis WX.
These force components are given by the following
equations:

N ¼ W cosh

and

WX ¼ W sinh:

The material upslope of the block applies a force on

the upslope end of the block Ru and a force Rd in the
opposite direction is applied to the downslope end of the Fig. A2. Water forces acting on the trapezoidal column.
block by the material downslope of the block. Cohesion
and friction along the base and sides create tractions approaches 0. Here, the magnitudes of the forces on the
parallel to these surfaces, Trb and Trs respectively. The upslope and downslope ends are equal or:
tractions along the sides are in the X–Y plane but are not
in the direction of the X axis (map view in Fig. A1). The Ru iRd :
convergence angle α is defined as the angle between the
X-axis direction and the side of the block. Thus, the This assumption will apply to blocks with converg-
traction on the side of the in the X-axis direction TrsX is ing sides as long as the convergence angle α is small. If
given by: the convergence is greater that 45°, then the assumption
may not be valid because the area of the downslope end
TrsX ¼ Trs cosa: will be considerably smaller than the upslope end.
Applying the assumption to Eq. (1) and assuming the
The weight of the block as the block attempts to
block is at equilibrium yields:
move down the slide plane applies a normal force to the
sides of the block WXN (Fig. A1) which is given by: X
FX ¼ WX −Trb −2TrsX ¼ 0 ð2Þ
WX N ¼ WX sina ¼ W sinhsina:
Moving the tractions to the right side of Eq. (2) and
Now, summing the forces and tractions in the X dividing both sides by WX yields the factor of safety
direction yields: equation at equilibrium:
X Trb þ 2TrsX
FX ¼ WX −Trb −2TrsX þ Ru −Rd : ð1Þ FS ¼ 1 ¼ : ð3Þ
WX
The constant 2 in front of the side traction variables
Using the Mohr–Coulomb failure criteria and
accounts for the two sides of the block. Eq. (1) cannot be
removing the equilibrium restriction yields the factor
simplified by taking moments about any point within the
of safety equation:
block. In order to simplify Eq. (1), the standard as-
sumption of the infinite slope model will be applied.
This model assumes that the thickness of the block b is ðCab =coshÞ þ N tan/ þ ðCdb =coshÞ þ 2WX N tan/
FS ¼
much smaller than the length or that the value of b WX
132 G.C. Ohlmacher / Engineering Geology 91 (2007) 117–134

or the analysis presented in this paper, the block is assumed

to be at rest and in order for it to move, it needs to move
ðCab=coshÞ þ W coshtan/ þ 2ðCdb=cosaÞ þ 2W sinhsinatan/
FS ¼ material laterally out of its path downslope.
W sinh
ð4aÞ
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