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33 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, core-stones 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 conditions--such 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 chimney-like 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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34 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) Prandtl-type 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 Bell-Terzaghi 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 cone-shaped 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 (1-II) 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 (3-II) is equal to the major principal stress of Prism I (1-I) such that the bearing capacity is qult JcNcr (73) Prism I Prism II - B qq + 2 tan I-I = 3-II I-II = qo Figure 33. Mohr Circle analysis of bearing capacity based on straight-line failure planes and prismatic zones of triaxial compression and shear (based on Sowers, 1979).

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36 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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37 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 bearing-pressure 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 lower-bound 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 qu-core (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 qu-core = 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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38 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 High-very 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 Medium-high Not applicable (60) condition(1) (2) Sedimentary rocks: cemented shale, siltstone, sandstone, limestone without 1,0004,000 Medium-high Not applicable cavities, thoroughly cemented (2080) in conglomerates, all in sound Rocks condition(1) (2) 5001,000 Compaction shale and other (1020) argillaceous Low-medium 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 in-situ, 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 Mohr-Coulomb ternal friction (f ) and unconfined compressive strength (qu) failure envelopes described in Figure 37 or from Equation 79, (Mohr-Coulomb 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 open-jointed 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 (a-c), Brown, 1980, see also Section 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 Mohr-Coulomb 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