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For simplicity, it is assumed here that the load polygon in and capacity are important to quantify for the design of shal
Figure 30 expands evenly. low foundations on both soil and rock. The research herein
Finally, the foundation stability is verified, if it can be shown addresses, however, only the bearing capacity (i.e., the ULS of
that shallow foundations).
F (. . . , R ,i , Ld ) < 0 (71) 1.7.2 Failure Mechanisms of Foundations
Ld Ld
on Rock
where L d is one design load combination of the set of all design Failure of foundations on rock may occur as the result of
load combinations, Ld, which need to be checked. If the inequal one of several mechanisms, as shown in Figure 31 (Franklin
ity (Equation 71) is fulfilled, all design load combinations are and Dusseault, 1989). The failure modes are described by the
located inside the design failure surface. Canadian Foundation Engineering Manual (Canadian Geo
technical Society, 2006) in the following way:
1.7 Bearing Capacity of Shallow 1. Bearing capacity failures occur when soil foundations are
Foundations on Rock overloaded (see Figures 31a and b). Such failures, although
uncommon, may occur beneath heavily loaded footings on
1.7.1 Overview
weak clay shales.
The bearing capacity of foundations founded on rock 2. Consolidation failures, common in weathered rocks, occur
masses depends mostly on the ratio of joint spacing to foun where the footing is placed within the weathered profile
dation width, as well as intact and rock mass qualities like (see Figures 31c and e). In this case, unweathered rock
joint orientation, joint condition (open or closed), rock type, corestones are pushed downward under the footing load
and intact and mass rock strengths. Failure modes may con because of a combination of low shear strength along clay
sist of a combination of modes, some of which include bear coated lateral joints and voids or compressible fillings in
ing capacity failure. Limited review of the bearing capacity of the horizontal joints.
foundations on rock, as well as the relationships among bear 3. A punching failure (see Figure 31d) may occur where the
ing capacity mechanisms, unconfined compressive strength foundation rock comprises a porous rock type, such as
(qu), and other rock parameters is presented. Emphasis is shale, tuff, and porous limestone (chalk). The mechanism
placed on classifications and parameters already specified by includes elastic distortion of the solid framework between
AASHTO and methods of analysis utilized in this study for the voids and the crushing of the rock where it is locally
bearing capacity calibrations. highly stressed (Sowers and Sowers, 1970). Following such
Loads on foundation elements are limited by the structural a failure, the grains are in much closer contact. Continued
strength, the ultimate (geotechnical) limit state (strength), leaching and weathering will weaken these rock types, result
and the load associated with the serviceability limit state. The ing in further consolidation with time.
relationships among these limits when applied to founda 4. Slope failure may be induced by foundation loading of the
tions on rock are often vastly different than when they are ground surface adjacent to a depression or slope (see Fig
applied to shallow foundations on soil. For typical concrete ure 31f). In this case, the stress induced by the foundation
strengths in use today, the strength of the concrete member is sufficient to overcome the strength of the slope material.
is significantly less than the bearing capacity of many rock 5. Subsidence of the ground surface may result from collapse
masses. The structural design of the foundation element will of strata undercut by subsurface voids. Such voids may be
dictate, therefore, the minimum element size and, conse natural or induced by mining. Natural voids can be formed
quently, the maximum contact stress on the rock. In other by solution weathering of gypsum or rock salt and are com
loading conditionssuch as intensely loaded pile tips, con monly encountered in limestone terrain (see Figure 31g).
centrated loads of steel supports in tunnels, or the bearing When weathering is focused along intersecting vertical
capacity of highly fractured or softer homogeneous rocks (such joints, a chimneylike opening called a pipe is formed,
as shale and sandstone)the foundation's geotechnical limit which may extend from the base of the soil overburden
state (bearing capacity) can be critical. While settlement (i.e., to a depth of many tens of meters. When pipes are covered
serviceability) is often the limit that controls the design load by granular soils, the finer silt and sand components can
of shallow foundations on soil, for many rocks the load re wash downward into the pipes, leaving a coarse sand and
quired to develop common acceptable settlement limits well gravel arch of limited stability, which may subsequently
exceeds the bearing capacity values. As such, both settlement collapse (see Figure 31h).
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Rigid Plastic
(a) (b)
Rigid Porous
(c) (d)
(e) (f)
Cavity
(g) (h)
Figure 31. Mechanisms of foundation failure from Franklin and Dusseault (1989),
adapted from Sowers (1979): (a) Prandtltype shearing in weak rock, (b) shearing
with superimposed brittle crust, (c) compression of weathered joints, (d) compres
sion and punching of porous rock underlying a rigid crust, (e) breaking of pinnacles
from a weathered rock surface, (f) slope failure caused by superimposed loading,
(g) collapse of a shallow cave, and (h) sinkhole caused by soil erosion into solution
cavities (Canadian Geotechnical Society, 2006).
1.7.3 Bearing Capacity Failure Mechanisms shows three simple possible analyses associated with the
ratio of foundation width to joint spacing and the joint
Out of the various aforementioned possible failures of conditions.
foundations on rock, this research is focused on those as
sociated with bearing capacity mechanisms. The mecha 1. Closed Spaced Open Joints: Figure 32a illustrates the
nism of potential failure in jointed rocks depends mostly on condition where the joint spacing, s, is a fraction of B, and
the size of the loaded area relative to the joint spacing, joint the joints are open. The foundation is supported by un
opening, and the location of the load. Figure 32 (a through c) confined rock columns; hence, the ultimate bearing
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35
Compression
zones
B S S B B
Comp.
zone
Split
S
Compression zone
(a) Close, open joints, S B: Splitting
Unconfined Compressions Compression Zones (after Bell) (after Meyerhof, Bishoni)
B B
Rigid H Rigid H
Shear
Weak compressible
Weak compressible
(d) Thick rigid layer over weak (e) Thin rigid layer over weak
compressible layer: Flexure Failure compressible layer: Punching Failure
Figure 32. Bearing capacity failure modes of rock (based on Sowers, 1979).
capacity approaches the sum of the unconfined compres the major principal stress of Prism I and is expressed in
sive strengths of each of the rock prisms. Because all rock Equation 72:
columns do not have the same rigidity, some will fail be
f
fore others reach their ultimate strength; hence, the total qult = 2c tan 45 + (72)
2
capacity is somewhat less than the sum of the prism
strengths. where c is cohesion, and f is friction angle of the rock mass.
2. Closed Spaced Joints in Contact: The BellTerzaghi analy 3. Wide Joints: If the joint spacing is much greater than the
sis is shown in Figure 32b. When s > B (see Figure 32c), the proposed
closed so that pressure can be transmitted across them failure mechanism is a coneshaped zone forming below
without movement, the rock mass is essentially treated the foundation that splits the block of rock formed by
as a continuum, and the bearing capacity can be evalu the joints. Equation 73 can be used to approximate the
ated in the way shown in Figure 33 in which the major bearing capacity assuming that the load is centered on
principal stress of Prism II (1II) is equal to the embed the joint block and little pressure is transmitted across
ment confining stresses qo, and the minor principal the joints:
stress of Prism II (3II) is equal to the major principal
stress of Prism I (1I) such that the bearing capacity is qult JcNcr (73)
Prism I Prism II

B
qq + 2
tan II = 3II III = qo
Figure 33. Mohr Circle analysis of bearing capacity based on
straightline failure planes and prismatic zones of triaxial
compression and shear (based on Sowers, 1979).
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For continuous strip foundations: where
JcNcr s = the spacing between a pair of vertical open dis
qult = (74) continuities,
L
2.2 + 0.18 f = the friction angle of intact rock, and
B N = the bearing capacity factor given by:
where
B and L = width and length of the footing, respectively; f
N = tan 2 45 + (77)
J = a correction factor dependent upon the thick 2
ness of the foundation rock below the footing
and the width of the footing; and 4. Thick and Thin Rigid Rock Layer over Weak Compress
Ncr = bearing capacity factor. ible Layer: As shown in Figures 31d, 32d, and 32e, depend
ing on the ratio H/B and S/B and on the flexural strength
Based on laboratory test results and the Ncr solution by of the rock stratum, two forms of failure occur when the
Bishoni (1968), J is estimated by the following: rock formation consists of an extensive hard seam under
H H lain by a weak compressible stratum. If the H/B ratio is large
5 J = 0.12 + 0.4 (75a) and the flexural strength is small, the rock failure occurs
B B
by flexure (see Figure 32d). If the H/B ratio is small, punch
H
>5 J =1 (75b) ing is more likely (see Figure 32e). The same analysis can
B also be used for designs with hard rock layers over voids.
where H is the average spacing between a pair of horizon Bearing capacity calculations for flexural or punching fail
tal discontinuities. ure are proposed by Lo and Hefny (2001) and by ASCE
Values of Ncr derived from models for splitting failure (Zhang and Einstein, 1998; Bishoni, 1968; Kulhawy, 1978).
depend on the s/B ratio and f , which will be discussed later.
The values for square footings are 85% of the circular.
1.7.4 The Canadian Foundation
Graphical solutions for the bearing capacity factor (Ncr)
Engineering Manual
and correction factor (J) by Bishoni (1968) are provided in
Figures 34a and 34b, respectively. The bearing capacity fac The bearing capacity methods for foundations on rock
tor (Ncr) is given by Goodman (1980): proposed by the Canadian Foundation Engineering Manual
(Canadian Geotechnical Society, 2006) are described to be
s 1
(cot f )
2
2N suitable for all ranges of rock quality, noting that the design
N cr = 1
1 + N B N bearing pressure is generally for SLSs not exceeding 25 mm
1
(1 in.) settlement. The Canadian Foundation Engineering
 N ( cot
t f ) + 2N 2 (76) Manual (Canadian Geotechnical Society, 2006) considers a
150 S
q0 = cNcr J
For square foundations B
shape correction = 0.85
H
100 = 50 1
45
Ncr 40
35 J
50 30 0.5
25
20
0 0
0 5 10 15 20 0 2 4 6 8 10 12
S/B H/B
(a) bearing capacity factors for circular foundation on (b) correction factor J for rock layer
jointed rock, with S/B > 1 and H/B > 8. thickness, H
Figure 34. Bearing capacity factors for rock splitting (based on
Bishoni, 1968).
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Table 6. Coefficients of discontinuity soil or rock debris), and for a foundation width greater than
spacing, Ksp (Canadian Geotechnical 300 mm (1 ft). For sedimentary rocks, the strata must be hor
Society, 2006). izontal or nearly so.
The bearingpressure coefficient, Ksp, as given in Figure 35,
Discontinuity spacing
Ksp takes into account the size effect and the presence of discon
Description Distance m (ft)
Moderately close 0.3 to 1 (1 to 3) 0.1 tinuities and includes a nominal safety factor of 3 against the
Wide 1 to 3 (3 to 10) 0.25 lowerbound bearing capacity of the rock foundation. The
Very wide > 3 (> 10) 0.4
factor of safety against general bearing failure (ULSs) may
be up to ten times higher. For a more detailed explanation,
rock to be sound when the spacing of discontinuities is in the Canadian Foundation Engineering Manual (Canadian Geo
excess of 0.3 m (1 ft). When the rock is sound, the strength technical Society, 2006) refers to Ladanyi et al. (1974) and
of the rock foundation is commonly in excess of the design Franklin and Gruspier (1983) who discuss a special case of
requirements provided the discontinuities are closed and are foundations on shale. It is often useful to estimate a bearing
favorably oriented with respect to the applied forces, i.e., the pressure for preliminary design on the basis of the material de
rock surface is perpendicular to the foundation, the load has no scription. Such values must be verified or treated with cau
tangential component, and the rock mass has no open discon tion for final design. Table 7 presents presumed preliminary
tinuities. Under such conditions, the design bearing pressure design bearing pressure for different types of soils and rocks.
may be estimated from the following approximate relation:
1.7.5 Goodman (1989)
qa = Ksp × qucore (78)
The considered mode of failure is shown in Figures 36a
where through 36c, in which a laterally expanding zone of crushed
qa = design bearing pressure; rock under a strip footing induces radial cracking of the rock
qucore = average unconfined compressive strength of rock on either side (Goodman, 1989). The strength of the crushed
(as determined from ASTM D2938); and rock under the footing is described by the lower failure enve
Ksp = an empirical coefficient, which includes a factor of lope (curve for Region A) in Figure 37, while the strength of
safety of 3 (in terms of WSD) and ranges from 0.1 the less fractured neighboring rock is being described by the
to 0.4 (see Table 6 and Figure 35). upper curve in the same figure (curve for Region B). The largest
horizontal confining pressure that can be mobilized to support
The factors influencing the magnitude of the coefficient are the rock beneath the footing (Region A in Figure 37) is ph, de
shown graphically in Figure 35. The relationship given in Fig termined as the unconfined compressive strength of the adja
ure 35 is valid for a rock mass with spacing of discontinuities cent rock (Region B of Figure 37). This pressure determines the
greater than 300 mm (1 ft), aperture of discontinuities less lower limit of Mohr's circle tangent to the strength envelope of
than 5 mm (0.2 in.) (or less than 25 mm [1 in.] if filled with the crushed rock under the footing. Triaxial compression tests
0.5
=0
0.4 /c
1 3+ c
0.00 K sp = B
. 0 0 2 10 1+300
0 c
Value of Ksp
0.3 c = spacing of discontinuities
5
0.00 = aperture of discontinuities
B = footing width
0.2
0.010
Valid for 0.05 < c/B < 2.0
0.020
0 < /c < 0.02
0.1
0
0 0.4 0.8 1.2 1.6 2
Ratio c/B
Figure 35. Bearing pressure coefficient (Ksp) (based on Canadian
Geotechnical Society, 2006).
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Table 7. Presumed preliminary design bearing pressure (Canadian Geotechnical
Society, 2006).
Types and Preliminary design
Strength of
Group conditions bearing pressure (5) Remarks
rock material
of rocks kPa (ksf)
These values are
Massive igneous and based on the
metamorphic rocks (granite, 10,000 assumption that the
Highvery high
diorite, basalt, gneiss) in (200) foundations are
sound condition (2) carried down to
unweathered rock.
Foliated metamorphic rocks
3,000
(slate, schist) in sound Mediumhigh Not applicable
(60)
condition(1) (2)
Sedimentary rocks: cemented
shale, siltstone, sandstone,
limestone without 1,0004,000
Mediumhigh Not applicable
cavities, thoroughly cemented (2080)
in conglomerates, all in sound
Rocks condition(1) (2)
5001,000
Compaction shale and other (1020)
argillaceous Lowmedium Not applicable
rocks in sound condition (2)(4) 1,000
(20)
Broken rocks of any kind with
moderately close spacing of
discontinuities (0.3 m [11.8
in]) or greater), except Not applicable (See note 3) Not applicable
argillaceous rocks (shale),
limestone, sandstone, shale
with closely spaced bedding
Heavily shattered or
Not applicable (See note 3) Not applicable
weathered rocks
Notes:
1. The above values for sedimentary or foliated rocks apply where the strata or the foliation are level or nearly so, and,
then, only if the area has ample lateral support. Tilted strata and their relation to nearby slopes or excavations should
be assessed by a person knowledgeable in this field of work.
2. Sound rock conditions allow minor cracks at spacing not closer than 1 m (39.37 in).
3. To be assessed by examination insitu, including test loading if necessary.
4. These rocks are apt to swell on release of stress, and on exposure to water they are apt to soften and swell.
5. The above values are preliminary estimates only and may need to be adjusted upwards or downwards in a specific case.
No consideration has been made for the depth of embedment of the foundation. Reference should be made to other
parts of the Manual when using this table.
on broken rock can define the latter strength envelope, and to the joint spacing (s), the rock foundation can be compared
thus the bearing capacity can be found (Goodman, 1989). to a column whose strength under axial load should be approx
Examination of Figure 37 leads to the conclusion that the imately equal to the unconfined compressive strength (qu). If
bearing capacity of a homogeneous, discontinuous rock mass the footing contacts a smaller proportion of the joint block, the
cannot be less than the unconfined compressive strength of bearing capacity increases toward the maximum value consis
the rock mass around the footing, and this can be taken as tent with the bearing capacity of homogeneous, discontinuous
the lower bound. If the rock mass has a constant angle of in rock, obtained with the construction of the MohrCoulomb
ternal friction (f ) and unconfined compressive strength (qu) failure envelopes described in Figure 37 or from Equation 79,
(MohrCoulomb material), the mechanism described in Fig which takes into account the friction angle (f ) of the homo
ure 37 establishes the bearing capacity as geneous discontinuous rock. This problem was studied by
Bishoni (1968), who assumed that some load is transferred
qult = qu ( N + 1) (79) laterally across joints. Modifying this boundary condition
for an openjointed rock mass in which lateral stress trans
where N is calculated using Equation (77). fer is zero, yields
Figure 38 depicts a footing resting on a portion of a single
( N 1)
joint block created by orthogonal vertical joints each spaced 1 S N
distance s. Such a condition might arise, for example, in weath qult = qu N  1 (80)
N  1 B
ered granite (Goodman, 1989). If the footing width (B) is equal
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39
B
s
Cracking Crushing
(a) (b)
Figure 38. Footing on rock
with open, vertical joints
(based on Goodman, 1989).
Wedging Punching
Comparing the results of Goodman's (1989) computa
(c) (d) tions with Equations 79 and 80 shows that open joints reduce
the bearing capacity only when the ratio S/B is in the range
from 1 to 5. The bearing capacity of footings on rock with open
joints increases with increasing f for any of the S/B ratios rang
ing from 1 to 5.
Shear
(e) 1.7.6 Carter and Kulhawy (1988)
Figure 36. Modes of failure of a footing on rock Carter and Kulhawy (1988) suggested that the Hoek and
including development of failure through crack Brown strength criterion for jointed rock masses (Hoek and
propagation and crushing beneath the footing (ac), Brown, 1980, see also Section 1.8.2.4) can be used in the eval
punching through collapse of voids (d), and shear uation of bearing capacity. The curved strength envelope for
failure (e) (based on Goodman, 1989). jointed rock mass can be expressed as
1 = 3 + (mqu 3 + squ )0.5
2 (81)
Strength of Rock Mass where
(region B) 1 = major principal effective stress,
3 = minor principal effective stress,
qu = uniaxial compressive strength of the intact rock.
s and m = empirically determined strength parameters for
the rock mass, which are to some degree anal
Strength of Rock Mass ogous to c and f of the MohrCoulomb failure
(region A) criterion.
Carter and Kulhawy (1988) suggested that an analysis of the
Ph qf
bearing capacity of a rock mass obeying this criterion can be
made using the same approximate technique as used in the
qf
Strip footing Bell (1915) solution. The details of this approach are described
in Figure 39. A lower bound to the failure load was calculated
by finding a stress field that satisfies both equilibrium and the
B Ph B
failure criterion. For a strip footing, the rock mass beneath
the foundation may be divided into two zones with homoge
neous stress conditions at failure throughout each, as shown
A
in Figure 39. The vertical stress in Zone I is assumed to be
Figure 37. Analysis of bearing capacity on rock zero, while the horizontal stress is equal to the uniaxial com
(based on Goodman, 1989). pressive strength of the rock mass, given by Equation 81 as
s0.5qu. For equilibrium, continuity of the horizontal stress