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37 1 Qs = fsu dA = B fdz (21) 0.9 O-cell, good base surface L 0.8 O-cell, base disturbed 0.7 top load test in which Qs = total side resistance (force), fsu = unit side resis- 0.6 tance (stress), A = surface area along the side of the socket, B = socket diameter, and L = socket length. In practice, socket side Qb/Qt 0.5 0.4 resistance capacity is calculated by assuming that a single av- 0.3 erage value of unit side resistance acts along the concreterock 0.2 interface, for each rock layer. This value of f is multiplied by 0.1 the area of the interface to obtain total side resistance Qs, or 0 0 1 2 3 4 5 6 7 8 9 10 Qs = fsu BL (22) L/B Methods for predicting socket side resistance are, therefore, FIGURE 23 Base load transfer interpreted from load tests focused on the parameter fsu. (data from Crapps and Schmertmann 2002). The interaction between a rock mass and drilled shaft that determines side resistance is complex. The principal factors controlling this interaction include: load remained essentially constant. However, the percentage of total load carried by the base after this period was still less Rock material strength; than 10% of the applied load and agreed quite well with pre- Rock mass structure (discontinuities); dictions based on elastic theory. Tang et al. (1994) described Modulus of the concrete relative to modulus of the rock a similar monitoring program on a shaft socketed into karstic mass; dolomite supporting a building on the University of Ten- Shear strength mobilized by dilatancy; nessee campus. Some change in load transfer occurred fol- Confining stress; and lowing the end of construction; however, most of the change Construction-related factors, including roughness of was from side resistance in the overlying soil (decreased) to shaftrock interface. side resistance in the rock socket (increased). Neither case would suggest changes in design of rock sockets to account Geomechanical models that account for these factors (to for the time dependency of load transfer mechanisms. varying degrees) are described in the literature (e.g., Rowe and Pells 1980); however, the methods required to obtain the necessary input parameters normally fall outside the scope of CAPACITY UNDER AXIAL LOADING a typical investigation conducted for the design of highway bridge foundations. More realistically, methods based on the The factored axial resistance of a drilled shaft in compres- strength of intact rock, in some cases with modifications to sion is the sum of the factored side resistance and the fac- account for one or more of the other factors, have been used tored base resistance. The factored resistances are calcu- successfully and are more rational than some of the strictly lated by multiplying appropriate resistance factors by the empirical methods or presumptive values. This approach rep- nominal resistances, which are generally taken as the ulti- resents a practical compromise between oversimplified em- mate values. One approach to the design process depicted pirical methods and more sophisticated numerical methods in Figure 3 of chapter one is to size the foundation initially that might be warranted only on larger projects. The methods to achieve a factored resistance that exceeds the factored are summarized here. loads. The trial design is then analyzed to predict load- displacement response. If necessary, revised trial dimen- sions can then be analyzed until all of the design criteria are Methods Based on Rock Compressive Strength satisfied, including the movement criteria associated with the service limit state. In the case of axial loading, the ultimate A practical approach to evaluating average unit side resis- side and base resistances are required to establish the initial tance is to relate fsu to the strength of the intact rock material. trial design. The rock material strength parameter most often measured is the uniaxial compressive strength (qu). In this approach, val- ues of fsu are determined from full-scale field load tests in Side Resistance which ultimate side resistance (Qs) has been determined. This value is divided by the socket side area (As) to obtain an The ultimate side resistance of a rock socket is the summa- average value of unit side resistance at failure: tion of peak shearing stress acting over the surface of the socket, expressed mathematically by fsu = Qs/As (23)

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38 Early studies relating fsu to qu include those by Rosenberg AASHTO refers to this as the Horvath & Kenney method. A and Journeaux (1976) and Horvath (1978, 1982). Other re- socket that is not specified to be artificially roughened by searchers have continued to expand the available database grooving is considered "smooth" and side resistance is gov- and propose equations relating unit side resistance to rock erned by Eq. 25. If the socket is artificially roughened, Eq. 26 strength on the basis of statistical best-fit analyses. Notable is recommended; however, this requires estimation or mea- studies include those of Williams and Pells (1981), Rowe and surement of roughness as defined by Eq. 27. Armitage (1984, 1987b), Bloomquist and Townsend (1991), McVay et al. (1992), and Kulhawy and Phoon (1993). These Rowe and Armitage (1987b) summarized the available studies, including the proposed equations relating unit side data on side resistance of rock sockets, including the data- resistance to rock strength are reviewed briefly. bases used by Williams et al. (1980), Williams and Pells (1981), and Horvath (1982). The suggested correlation for Horvath and Kenney (1979) proposed the following cor- regular clean sockets, defined as roughness classes R1, R2, relation between side resistance and compressive strength: and R3 in Table 17, is given as fsu = b qu (24) fsu = 0.45 qu (28) in which fsu = ultimate unit side resistance, qu = compressive To account for rough sockets, defined as category R4, side strength of the weaker material (rock or concrete), and where resistance is increased and the following is recommended: b ranges from 0.2 (smooth) to 0.3 (rough). Both fsu and qu in Eq. 24 are in units of MPa. Eq. 24 can be expressed in nor- malized form by dividing both unit side resistance and com- fsu = 0.6 qu (29) pressive strength by atmospheric pressure (pa = 0.1013 MPa). This results in the following expression, which is equivalent The correlation suggested by Horvath and Kenney (1979) to Eq. 24 with b = 0.2: as given by Eq. 25 represents a lower bound to the data used by Rowe and Armitage (1987b) fsu q = 0.65 u (25) pa pa Kulhawy and Phoon (1993) incorporated the database compiled by Rowe and Armitage, which included more than A modified relationship was recommended by Horvath et al. 80 load tests from more than 20 sites, and the data reported (1983) to account for shafts with artificially roughened by Bloomquist and Townsend (1991) and McVay et al. (grooved) sockets. The suggested relationship is given by (1992) consisting of 47 load tests to failure from 23 different Florida limestone sites. Linear regression was conducted on fsu = 0.8 [ RF ]0.45 qu (26) two sets of data, one consisting of all data points and the rh Lt other using data that were averaged on a per-site basis. Av- RF = (27) eraging eliminates the bias associated with multiple load rs Ls tests conducted at many of the sites. All stresses are normal- ized by atmospheric pressure, pa (0.1013 MN/m2) and nor- in which RF = roughness factor, rh = average height of as- malized values of one-half of the uniaxial compressive perities, rs = nominal socket radius, Ls = nominal socket length, and Lt = total travel distance along the socket wall strength were plotted against normalized values of average profile. A device (caliper) was used to measure field rough- unit side resistance on a loglog plot as shown in Figure 24 ness for determination of the parameters needed in Eq. 27, as (shown previously as Figure 17). described by Horvath et al. (1993). The following exponential expression provides a best-fit The FHWA Drilled Shaft Manual (O'Neill and Reese 1999) to the available data for rock: and the draft 2006 Interim AASHTO LRFD Bridge Design Specifications have adopted Eqs. 24 to 27 as a recommended fsu qu =C (30) method for selection of design side resistance for shafts in rock. pa 2 pa TABLE 17 SHAFT ROUGHNESS CLASSIFICATION (after Pells et al. 1980) Roughness Class Description R1 Straight, smooth-sided socket; grooves or indentations less than 1 mm deep R2 Grooves 14 mm deep, >2 mm wide, spacing 50200 mm R3 Grooves 410 mm deep, >5 mm wide, spacing 50200 mm R4 Grooves or undulations >10 mm deep, >10 mm wide, spacing 50200 mm

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39 capture all of the mechanisms affecting unit side resistance and because the database incorporates load test results for many different rock types. Use of these empirical correla- tions for design should therefore be conservative (i.e., C = 1) unless a site-specific correlation has been developed that jus- tifies higher values. Research on development of methods that account for these additional factors affecting peak side resistance is summarized here. Methods Based on Additional Rock Mass Parameters Williams et al. (1980) and Rowe and Armitage (1987a) point FIGURE 24 Unit side resistance versus strength (Kulhawy and out that unit side resistance determined strictly by empirical Phoon 1993). correlations with uniaxial compressive strength does not ac- count explicitly for the degree of jointing in the rock mass. Figure 25 shows the potential influence on average side re- where values of the coefficient C = 1 represents a reasonable sistance (denoted by in the figure) of jointing in terms of lower bound, C = 2 represents the mean behavior, and C = 3 the ratio of rock mass modulus to modulus of intact rock. corresponds to an upper bound for artificially roughened Both theoretical and experimental curves show substantially sockets. Kulhawy and Phoon (1993) noted that the expres- reduced side resistance for rock that may have high intact sions and variations are consistent with those reported by strength and stiffness but low mass modulus. Rowe and Armitage (1987b). Values of fsu obtained using Eq. 30 with C = 2 (mean) are identical to the values obtained The draft Interim 2006 AASHTO LRFD Bridge Design using Eq. 28 (Rowe and Armitage) corresponding to the mean Specifications adopt the Horvath and Kenney method de- trend for smooth sockets. The expression of Rowe and Ar- scribed earlier (Eqs. 2527), with the following modification mitage for rough sockets (Eq. 29) corresponds to Eq. 30 with as recommended O'Neill and Reese (1999). Values of unit C = 2.7. side resistance calculated by either Eq. 25 or Eq. 26 are mod- ified to account for rock mass behavior in terms of RQD, Kulhawy et al. (2005) recently reexamined the data avail- modulus ratio (EM/ER), and joint condition using the factor able and attempted to evaluate them in a more consistent as defined in Table 18. In Table 18, fdes is the reduced unit manner. Only data showing load-displacement curves to fail- ure were incorporated into the analysis, so that the "inter- preted failure load" could be established in a consistent man- ner. Based on the updated analysis, the authors recommend the use of Eq. 30 with C = 1 for predicting side resistance of normal rock sockets for drilled shafts. The authors also note the importance of using compressive strength values (qu) obtained from laboratory uniaxial compression tests, not from point load tests. In summary, Eq. 30 with C = 1 provides a conservative estimate of design ultimate side resistance, based on the most up-to-date analysis of the available data. Use of C values greater than 1 for design should be verified by previous ex- perience or load testing. Load test results that exhibit values of C in the range of 2.7 to 3 demonstrate the potential in- crease in side resistance that is possible if the sidewalls of the socket are roughened. These upper-bound values of fsu should only be considered when they can be validated by field load testing. The methods described rely on empirical relationships between side resistance and a single parameter, uniaxial com- pressive strength, to represent the rock mass. There is signif- FIGURE 25 Effect of rock mass modulus on average unit side icant scatter in the database because the relationships do not resistance (Rowe and Armitage 1987a).

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40 TABLE 18 in which c = construction method reduction factor, as de- SIDE RESISTANCE fined in Table 19; n = ratio of rock mass modulus to uniaxial REDUCTION BASED ON MODULUS REDUCTION compressive strength of intact rock (EM/qu); v = Poisson's (O'Neill and Reese 1999) ratio; r = mean roughness height; and ds = socket diameter. EM/ER = fdes/fsu Implementation of the SRC in design requires an estimate 1.0 1.0 of socket roughness in terms of r. As noted by the authors, 0.5 0.8 reliable measurements of roughness are not undertaken in 0.3 0.7 routine design. However, the SRC factor incorporates many 0.1 0.55 0.05 0.45 of the significant parameters that influence side resistance, including rock mass modulus, Poisson's ratio, and intact rock strength, and provides a framework for taking into account socket roughness and construction effects. side resistance recommended for design. The modulus re- duction ratio (EM/ER) is given in Table 13, in chapter two, The SRC method represents the type of approach that based on RQD. However, application of the -factor may be holds promise for improved methods for selecting design side questionable because the RQD and rock mass modulus were resistance. Although more detailed guidance is required for not accounted for explicitly in the original correlation analy- determination of socket roughness and construction effects, sis by Horvath and Kenney (1979). Because the load test data improvements in reliability of design equations are possible included sites with RQD less than 100 and modulus ratio val- only if the relevant factors controlling side resistance are in- ues less than one, it would appear that these factors affected corporated properly. Advancement of the SRC or other robust the load test results and are therefore already incorporated methods can be facilitated by promoting the awareness of into the resulting correlation equations. engineers involved in field load testing of the importance of collecting appropriate data on rock mass characteristics. Docu- Interface roughness is identified by all researchers as hav- mentation of RMS and modulus along with careful observation ing a significant effect on peak side resistance. Pells et al. and documentation of construction procedures would allow (1980) proposed the roughness classification that assigns a these methods to be evaluated against load test results. The key rock socket to one of the categories R1 through R4 as defined parameter that is currently missing from the database is socket in Table 17. The criteria are based on observations of sock- roughness. O'Neill et al. (1996) point out that roughness can be ets drilled in Sydney sandstone and the classification report- quantified approximately by making electronic or mechanical edly forms the basis for current practice in that city (Seidel caliper logs of the borehole, and that such borehole calipers are and Collingwood 2001). The Rowe and Armitage (1987b) available commercially. Seidel and Collingwood (2001) de- correlation equations were developed by distinguishing be- scribe a device called the Socket-Pro that is operated remotely tween roughness classes R1R3 (Eq. 28) and roughness class and records sidewall roughness to depths of 60 m. R4 (Eq. 29). Horvath et al. (1983) proposed the roughness factor (RF) defined in Eqs. 26 and 27 as presented earlier. Despite these efforts, selection of side resistance for rock- Geomaterial-Specific Correlations socket design in the United States is done mostly without considering interface roughness explicitly. Correlations between unit side resistance and intact rock strength that are based on a global database (e.g., Figure 24, Seidel and Haberfield (1994) developed a theoretical model Eq. 30) exhibit scatter and uncertainty because the results of interface roughness that accounts for the behavior and char- reflect the variations in interface shear strength of different acteristics of socket interfaces under CNS conditions. Rough- rock types, interface roughness, and other factors that control ness is modeled using a quasi-probabilistic approach that side resistance. For this reason, selection of design side re- involves fractal geometry to predict the distribution and char- sistance values based on such correlations should be consid- acteristics of asperities. Results of the interface model and ered as first-order estimates and the philosophy underlying laboratory CNS testing are incorporated into the computer pro- their use for design is that a lower-bound, conservative rela- gram ROCKET that predicts the axial load-displacement curve, tionship should be used (e.g., C = 1 in Eq. 30). Alternatively, including post-peak behavior. Extending this work, Seidel and correlations have been developed for specific geomaterials. Collingwood (2001) proposed a nondimensional parameter Correlations identified by the literature review and the survey defined as the shaft resistance coefficient (SRC) to account for are summarized here. the factors that influence side resistance, as follows: Florida Limestone Limestone formations in Florida are n r characterized by highly variable strength profiles, the pres- SRC = c (31) ence of cavities that may be filled with soil, and interbedding 1 + ds of limestone with sand and marine clay layers (Crapps 1986). fsu = (SRC) qu (32) Locally, geotechnical engineers distinguish between "lime-

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41 TABLE 19 CONSTRUCTION METHOD REDUCTION FACTORS, c (Seidel and Collingwood 2001) Construction Method c Construction without drilling fluid Best practice construction and high level of construction control (e.g., socket 1.0 sidewalls free of smear and remolded rock) Poor construction practice or low-quality construction control (e.g., smear or 0.30.9 remolded rock present on rock sidewalls) Construction under bentonite slurry Best practice construction and high level of construction control 0.70.9 Poor construction practice or low level of construction control 0.30.6 Construction under polymer slurry Best practice construction and high level of construction control 0.91.0 Poor construction practice or low level of construction control 0.8 rock" and "limestone"; the former defined informally as used for design should be limited to the design strength of the material with qu less than approximately 13.8 MN/m2 (2,000 shaft concrete. psi). McVay et al. (1992) conducted a study of design meth- ods used to predict unit side resistance of drilled shafts in Side resistance values in Florida limestone have also been Florida limestone. Based on a parametric finite-element evaluated using a small-scale field pullout test devised by study and a database of 14 case histories consisting of full- Schmertmann (1977) for the Florida DOT and shown scale load tests and field pullout tests, the following expres- schematically in Figure 26. A grout plug is placed into a 140- sion was found to provide a reasonable estimate of ultimate mm-diameter cored hole at the bottom of a 165-mm-diameter unit side resistance: hole drilled to the test depth in rock. Overburden soils are supported by a 200-mm-diameter casing. The grout plug is 1 reinforced with a wire cage and a threaded high-strength steel fsu = qu qt (33) 2 bar extends from the bottom of the plug to the ground sur- face. A center hole jack is used to apply a pullout force to the In which qu = uniaxial compressive strength and qt = split ten- bar. The grout plug is typically 610 mm (2 ft) in length, but sile strength. To account for the effect of material strength other lengths are also used. The average unit side resistance variability on side resistance, the authors recommend a min- is taken as the measured pullout force divided by the sidewall imum of 10 (preferably more) core samples be tested in un- interface area of the plug (Eq. 23). Results of pullout tests confined compression and splitting tensile tests. The mean were included in the database of McVay et al. (1992) that values of qu and qt are used in Eq. 33. The standard error forms the basis of Eq. 33, and McVay et al. recommend the of the mean from the laboratory strength tests can be used to test as an alternative method for estimating side resistance for estimate the expected variation from the mean side resis- design. tance, for a specified confidence level. Cohesionless IGM The FHWA Drilled Shaft Manual According to Lai (1998), design practice by the Florida (O'Neill and Reese 1999) recommends a procedure for cal- DOT is based on a modified version of the McVay et al. culating unit side resistance specifically for cohesionless relationship in which spatial variations in rock quality are IGM. These are granular materials exhibiting SPT N60-values incorporated by multiplying the unit side resistance, accord- between 50 and 100. The method follows the general ing to Eq. 33, by the average percent recovery (REC) of rock approach for calculating side resistance of drilled shafts in core expressed as a decimal, or: granular soils, given by fsu = v ' K o tan ' (35) REC(%) 1 ( fsu )design = qu qt (34) 100% 2 in which fsu = ultimate unit side resistance, v' = vertical effec- tive stress, Ko = in situ coefficient of lateral earth pressure, and Lai (1998) also recommends using larger diameter double- ' = effective stress friction angle of the IGM. The modifica- tube core barrels (61 mm to 101.6 mm inner diameter) for ob- tions to account for cohesionless IGM behavior are incorpo- taining samples of sufficient quality for laboratory strength rated into empirical correlations with the N-value as follows: tests. Analysis of the laboratory strength data involves dis- carding all data points above or below one standard deviation 'p = 0.2 N 60 pa (36) about the mean, then using the mean of the remaining values as input to Eq. 34. Crapps (2001) recommends using RQD in 'p OCR = (37) place of REC in Eq. 34 and points out that values of qu and qt v'

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42 nut plate steel bearing centerhole plate jack timber 35 mm dia 200-mm dia threaded bar casing top of rock drilled hole, 165 mm dia 0.61 m grout plug 0.76 m cored FIGURE 27 Factor for cohesive IGM (O'Neill et al. 1996). w/ #9 wire cage hole, 140 mm dia different value of interface friction applies, then the parame- FIGURE 26 Small-scale pullout test used in Florida limestone ter can be adjusted by (after Crapps 1986). tan rc = 26 (41) 0.34 tan 30 o N ' = tan -1 60 (38) The method is based on work reported by Hassan et al. 12.2 + 20.3 'v (1997) in which detailed modeling and field testing were pa conducted to study side resistance of shafts in clayshales of Texas. O'Neill and Reese (1999) provide additional equa- K o = (1 - sin ')OCR sin ' (39) tions for modifying side resistance for roughness, the pres- ence of soft seams, and other factors, and the reader is where p ' = preconsolidation stress, v ' = average vertical advised to consult their work for these additional details. effective stress over the layer, N60 = SPT N-value corre- sponding to 60% hammer efficiency, and OCR = overconsoli- dation ratio. O'Neill et al. (1996) reported good agreement Correlations with In Situ Tests with results of load tests on shafts in residual micaceous sands The survey responses indicate that several states use mea- in the Piedmont province (Harris and Mayne 1994) and gran- surements from field penetration tests to estimate unit side ular glacial till in the northeastern United States. resistance in weak rock. As an example, consider the Texas cone penetration test (TCPT) described in chapter two. The Soft Argillaceous Rock Side resistance in weak shales and Texas DOT Geotechnical Manual, which is accessible on- claystones can be approximated for design using the relation- line (2005), presents graphs for estimating design values of ships given previously for rock. Alternatively, a procedure allowable unit side and base resistances as a function of the for evaluating unit side resistance specifically in argillaceous penetration resistance (millimeters of penetration per 100 (containing clay) cohesive IGMs is presented in the FHWA blows), for materials that exhibit TCPT blowcounts of Drilled Shaft Manual (O'Neill and Reese 1999). The ulti- greater than 100 blows per 300 mm. For materials exhibiting mate unit side resistance is given by fewer than 100 blowcounts, separate graphs are provided for allowable values of unit side and base resistances. The Mis- fsu = qu (40) souri DOT also reports using the TCPT to correlate allow- able side and base resistances in weak rock. where qu = compressive strength of intact rock and = em- pirical factor given in Figure 27. In Figure 27, n = fluid pres- sure exerted by the concrete at the time of the pour and p = Base Resistance atmospheric pressure in the same units as n. As indicated in Figure 27, the method is based on an assumed value of Load transmitted to the base of a rock-socketed shaft, ex- interface friction angle rc = 30 degrees. If it is known that a pressed as a percentage of the axial compression load applied

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43 at the head, can vary over a wide range at typical working Rock Mass Conditions Failure loads. Several authors suggest a typical range of 10% to 20% Joint Dip Angle, from Joint Bearing Capacity horizontal Spacing Illustration Mode Equation No. of the head load (Williams et al. 1980; Carter and Kulhawy B (a) Brittle Rock: Local shear failure caused 1988), and some authors suggest that base resistance should by localized brittle fracture INTACT / MASSIVE Eq. 43 be neglected entirely for rock-socket design (Amir 1986). Elasticity solutions show that base load transfer depends on N/A S>>B (b) Ductile Rock: General shear failure along the embedment ratio (L/B) and the modulus ratio (Ec/Er). The well-defined shear surfaces Eq. 43 ratio of base load to applied load (Qb/Qc) decreases with in- creasing L/B (see Figure 21) and increases with increasing (c) Open Joints: Compression failure of modular ratio. As discussed previously, there is ample evi- individual rock columns Eq. 44 STEEPLY DIPPING JOINTS dence that base resistance should not be discounted in most s SB failure; near vertical joints Eqs. 5354 The ultimate base resistance of a rock-socketed drilled s (f) General shear failure shaft, Qb, is the product of the limiting normal stress, or bear- S < B or JOINTED S > B if with potential for failure failure along joints; moderately ing capacity, qult, at the base and the cross-sectional area of 20 < < 70 we dge can dipping joint sets. Eqs. 4552 de ve lop the shaft base (Ab): along joints (g) Rigid layer over weak H compressible layer: rigid Failure is initiated by Qb = qult Ab = qult [1/4 B ] 2 (42) LAYERED Limiting va lue of H weak, compressible tensile failure caused by flexure of rigid upper layer N/A w/re to B is 0 < < 20 dependent upon (h) Thin rigid layer over Analytical solutions for bearing capacity of rock are based material H rigid weak compressible layer: Failure is by punching properties N/A on the general bearing capacity equation developed for soil, weak, compressible shear through upper layer with appropriate modifications to account for rock mass FRACTURED (g) General shear failure characteristics such as spacing and orientation of discontinu- with irregular failure surface through fractured N/A S << B rock mass; two or more Eq. 57 ities, condition of the discontinuities, and strength of the rock closely spaced joint sets mass. Typical failure modes for foundations bearing on rock are shown in Figure 28. The failure modes depicted were in- FIGURE 28 Bearing capacity failure modes in rock (after Rock tended to address shallow foundations bearing on rock (Sow- Foundations 1994). ers 1976); however, the general concepts should be applica- ble to bearing capacity of deep foundations. The cases shown can be placed into four categories: massive, jointed, layered, is either intact or tightly jointed (no compressible or gouge- and fractured rock. filled seams) and there are no solution cavities or voids below the base of the pier. O'Neill and Reese (1999) recommend limiting base resistance to 2qu if the embedment into rock is Massive Rock less than one diameter. In rock with high compressive strength, the designer also must determine the structural ca- For this case, the ultimate bearing capacity will be limited to pacity of the shaft, which may govern the allowable normal the bearing stress that causes fracturing in the rock. An intact stress at the base. rock mass can be defined, for purposes of bearing capacity analysis, as one for which the effects of discontinuities are insignificant. Practically, if joint spacing is more than four Jointed Rock Mass to five times the shaft diameter, the rock is massive. If the base is embedded in rock to a depth of at least one diame- When discontinuities are vertical or nearly vertical ( > 70), ter, the failure mode is expected to be by punching shear and open joints are present with a spacing less than the socket (Figure 28, mode a). In this case, Rowe and Armitage diameter (S < B, Figure 28, mode c), failure can occur (theo- (1987b) stated that rock fracturing can be expected to occur retically) by unconfined compression of the poorly con- when the bearing stress is approximately 2.7 times the rock strained columns (Sowers 1979). Bearing capacity can be uniaxial compressive strength. For design, the following is estimated from recommended: qult = qu = 2c tan (45 + 1/2 ) (44) qult = 2.5 qu (43) where qu = uniaxial compressive strength and c and are Other conditions that must be verified are that the rock to MohrCoulomb strength properties of the rock mass. If the a depth of at least one diameter below the base of the socket nearly vertical joints are closed (Figure 28, mode d), a gen-

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44 eral wedge failure mode may develop and the bearing ca- pacity can be approximated using Bell's solution for plane strain conditions: B qult = cN c sc + N s + DN q sq (45) 2 in which B = socket diameter; = effective unit weight of the rock mass; D = foundation depth; Nc, N, and Nq are bearing capacity factors; and sc, s, and sq are shape factors to account for the circular cross section. The bearing capacity factors and shape factors are given by: Nc = 2 N ( N + 1) (46) N = N ( N 2 - 1) (47) Nq = N 2 (48) N = tan 2 45 + (49) 2 FIGURE 29 Correction factor for discontinuity spacing Nq (Kulhawy and Carter 1992a). sc = 1 + (50) Nc s = 0.6 (51) velop along the discontinuity planes. Eq. 45 can be used, sq = 1 + tan (52) but with strength parameters representative of the joints. Rock Foundations (1994) recommends neglecting the first In these equations (4652), the values of c and are RMS term in Eq. 45 based on the assumption that the cohesion properties, which may be difficult to determine accurately for component of strength along the joint surfaces is highly rock mass beneath the base of drilled shafts. uncertain. If joint spacing S is greater than the socket diameter (Figure 28, mode e), failure occurs by splitting, leading Layered Rocks eventually to general shear failure. This problem has been evaluated by Bishnoi (1968) and developed further by Sedimentary rock formations often consist of alternating Kulhawy and Goodman (1980). The solution can be hard and soft layers. For example, soft layers of shale in- expressed by: terbedded with hard layers of sandstone. Assuming the base of the shaft is bearing on the more rigid layer, which is un- qult = J c Ncr (53) derlain by a soft layer, failure can occur either by flexure if the rigid layer is relatively thick or by punching shear if the in which J = a correction factor that depends on the ratio of rigid layer is thin (Figure 28, modes g and h). Both modes are horizontal discontinuity spacing to socket diameter (H/B) as controlled fundamentally by the tensile strength of the intact shown in Figure 29, c = rock mass cohesion, and Ncr = a bear- rock, which can be approximated as being on the order of ing capacity factor given by 5% to 10% of the uniaxial compressive strength. According to Sowers (1979) neither case has been studied adequately and no analytical solution proposed. The failure modes de- 2N2 S 1 N cr = ( cot ) 1- - N ( cot ) + 2 N (54) picted in Figure 28 merely suggest possible methods for 1 + N B N analysis. where N is given by Eq. 49. If the actual RMS properties are not evaluated, Kulhawy and Carter (1992a) suggest that rock Fractured Rock Mass mass cohesion in Eq. 53 can be approximated as 0.1qu, where qu = uniaxial compressive strength of intact rock. Rock mass A rational approach for calculating ultimate bearing capac- cohesion can also be estimated from the HoekBrown ity of rock masses that include significant discontinuities strength properties using Eq. 17 given in chapter two. (Figure 28, mode g) is to apply Eq. 45 with appropriate RMS properties, c' and '. However, determination of c' and For the case of moderately dipping joint sets (Figure 28, ' for highly fractured rock mass is not straightforward mode f, 20 < < 70), the failure surface is likely to de- because the failure envelope is nonlinear and there is no stan-

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45 dard test method for direct measurement. One possible ap- where H ' = horizontal stress in Zone 2. To satisfy equilibrium, proach is to employ the HoekBrown strength criterion the horizontal stress given by Eq. 56 is set equal to 3 ' in described in chapter two. The criterion is attractive because Zone 1. Substituting 3 ' = qu s0.5 into Eq. 55 and considering that (1) it captures the nonlinearity in the strength envelope that 1 ' = qult yields is observed in jointed rock masses and (2) the required para- qult = qu s a + ( mb s a + s ) a meters can be estimated empirically using correlations to (57) GSI and RMR, also described in chapter two. To use this approach, it is necessary to relate the HoekBrown strength The assumption of zero vertical stress at the bearing ele- parameters (mb, s, and a) to MohrCoulomb strength pa- vation may be overly conservative for many rock sockets. A rameters (c' and '); for example, using Eqs. 16 and 17 in similar derivation can be carried out with the overburden chapter two. stress taken into account, resulting in the following. Let ( 'v ,b ) a Alternatively, several authors (Carter and Kulhawy 1988; Wyllie 1999) have shown that a conservative, lower-bound A = 'v ,b + qu mb + s (58) qu estimate of bearing capacity can be made directly in terms of HoekBrown strength parameters by assuming a failure where 'v,b = vertical effective stress at the socket bearing mode approximated by active and passive wedges; that is, elevation, which is also the minor principal stress in Zone 2. the Bell solution for plane strain. The failure mass beneath Then the foundation is idealized as consisting of two zones, as shown in Figure 30. The active zone (Zone 1) is subjected a A to a major principal stress (1 ') coinciding at failure with the qult = A + qu mb + s (59) qu ultimate bearing capacity (qult) and a minor principal stress (3 ') that satisfies equilibrium with the horizontal stress in A limitation of Eqs. 5759 is that they are based on the the adjacent passive failure zone (Zone 2). In Zone 2, the assumption of plane strain conditions, corresponding to minor principal stress is vertical and conservatively as- a strip footing. Kulhawy and Carter (1992a) noted that for sumed to be zero, whereas the major principal stress, acting a circular foundation the horizontal stress between the two in the horizontal direction, is the ultimate strength accord- assumed failure zones may be greater than for the plane ing to the HoekBrown criterion. From chapter two, the strain case, resulting in higher bearing capacity. The analy- strength criterion is given by sis is therefore conservative for the case of drilled shafts. a ' Eqs. 5759 require determination of a single rock strength '1 = '3 + qu mb 3 + s (55) qu property (qu) along with an approximation of the HoekBrown strength parameters. In chapter two, the HoekBrown strength where 1 ' and 3 ' = major and minor principal effective parameters are correlated to GSI by Eqs. 1215. This allows a stresses, respectively; qu = uniaxial compressive strength of correlation to be made between the GSI of a rock mass; the intact rock; and mb, s, and a are empirically determined value of the coefficient mi for intact rock as given in Table 11 strength parameters for the rock mass. For Zone 2, setting the (chapter two), and the bearing capacity ratio qult/qu by Eq. 57. vertical stress 3' = 0 and solving Eq. 55 for 1 ' yields The resulting relationship is shown graphically in Figure 31. The bearing capacity ratio is limited by an upper-bound value '1 = ' H = qu s a (56) of 2.5, corresponding to the recommendation of Rowe and Armitage (Eq. 43). Qult maximum qult/qu = 2.5 2.5 m i = 33 25 20 15 2.0 qult 10 qult/qu 1.5 4 qusa 1.0 0.5 Zone 1 Zone 2 (active Zone 2 0.0 10 20 30 40 50 60 70 80 90 100 (passive wedge) wedge) (passive wedge) Geological Strength Index (GSI) FIGURE 30 Bearing capacity analysis. FIGURE 31 Bearing capacity ratio versus GSI.

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46 Alternatively, the bearing capacity ratio can be related ap- 1000 Ultimate Unit Base Resistance, proximately to the rock mass description based on RMR qmax = 2.5 (Table 9) using an earlier correlation given by Hoek and Brown (1988). The resulting HoekBrown strength parame- 100 ters (m, a, and s) are substituted into Eq. 57 to obtain the bear- qult (MPa) ing capacity ratio as a function of RMR. This relationship qult = a (qu)0.5 is shown graphically in Figure 32. Both figures are for the 10 case of zero overburden stress at the bearing elevation. To ac- count for the depth of embedment and resulting surcharge a = 6.6 4.8 stress, Eqs. 58 and 59 can be used. 3.0 1 0.1 1 10 100 Method Based on Field Load Tests Uniaxial Compressive Strength qu (MPa) Zhang and Einstein (1998) compiled and analyzed a database FIGURE 33 Unit base resistance versus intact rock strength of 39 load tests to derive an empirical relationship between (derived from Zhang and Einstein 1998). ultimate unit base resistance (qult) and uniaxial compressive strength of intact rock (qu). Reported values of uniaxial com- pressive strength ranged from 0.52 MPa to 55 MPa, although most were in the range of relatively low strength. The authors with due consideration of the limitations associated with pre- relied on the interpretation methods of the original references dicting a rock mass behavior on the basis of a single strength to determine ultimate base capacity and acknowledge that parameter for intact rock. Rock mass discontinuities are not some uncertainties and variabilities are likely to be incorpo- accounted for explicitly, yet they clearly must affect bearing rated into the database as a result. The results are shown capacity. By taking this empirical approach, however, rock on a loglog plot in Figure 33. The linear relationship mass behavior is accounted for implicitly because the load recommended by Rowe and Armitage (1987a) is shown for tests on which the method is based were affected by the char- comparison. Based on statistical analysis of the data, the fol- acteristics of the rock masses. Additional limitations to the lowing recommendations are given by the authors: approach given by Zhang and Einstein are noted in a discus- sion of their paper by Kulhawy and Prakoso (1999). Lower bound: qult = 3.0 qu (60) None of the analytical bearing capacity models described Upper bound: qult = 6.6 qu (61) above by Eqs. 44 through 59 and depicted in Figure 28 have been evaluated and verified against results of full-scale field Mean: qult = 4.8 qu (62) load tests on rock-socketed drilled shafts. The primary rea- son for this is a lack of load test data accompanied by suffi- Eqs. 6062 provide a reasonably good fit to the available cient information on rock mass properties needed to apply data and can be used for estimating ultimate base resistance, the models. 3.0 maximum qult/qu = 2.5 Canadian Geotechnical Society Method 2.5 The Canadian Foundation Engineering Manual [Canadian 2.0 Geotechnical Society (CGS) 1985] presents a method to es- E timate ultimate unit base resistance of piles or shafts bearing qult/qu 1.5 D C B on rock. The CGS method is described as being applicable to 1.0 A sedimentary rocks with primarily horizontal discontinuities, 0.5 where discontinuity spacing is at least 0.3 m (1 ft) and dis- continuity aperture does not exceed 6 mm (0.25 in.). The 0.0 method is given by the following: POOR GOOD VERY POOR VERY GOOD FAIR INTACT qult = 3qu K sp d (63) A. Carbonate rocks with well-developed cleavage: dolomite, limestone, marble. B. Lithified argillaceous rocks: mudstone, siltstone, shale, and slate. in which C. Arenaceous rocks with strong crystals and poorly developed crystal cleavage: sandstone and quartzite. D. Fine-grained polyminerallic igneous crystalline rocks: andesite, dolerite, diabase, and rhyolite. sv E. Coarse-grained polyminerallic igneous and metamorphic crystalline rocks: amphibolite, gabbro, 3+ gneiss, quartz, norite, quartz-diorite. K sp = B (64) td FIGURE 32 Bearing capacity ratio as a function of rock type 10 1 + 300 and RMR classification (see Table 9). sv