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47 Ls d = 1 + 0.4 B (65) Qc where sv = vertical spacing between discontinuities, td = aperture (thickness) of discontinuities, B = socket diameter, and Ls = depth of socket (rock) embedment. A method to calculate ultimate unit base resistance from Ec,c Er,r PMT is also given by CGS as follows: L qult = K b ( p1 - po ) + v (66) where p1 = limit pressure determined from PMT tests averaged over a depth of two diameters above and below socket base elevation, po = at-rest total horizontal stress measured at base elevation, Eb, b v = total vertical stress at base elevation; and B Kb = socket depth factor given as follows: FIGURE 34 Axially loaded rock socket, elastic analysis. H/D 0 1 2 3 5 7 Kb 0.8 2.8 3.6 4.2 4.9 5.2 shaft is subjected to a vertical compressive force Qc as- sumed to be uniformly distributed over the cross-sectional The two CGS methods described earlier are adopted in the area of the shaft resulting in an average axial stress b = draft 20006 Interim AASHTO LRFD Bridge Design Specifi- 4Q/(B2). cations (2006). Early solutions to the problem of a single compressible AXIAL LOAD-DISPLACEMENT BEHAVIOR pile in an elastic continuum were used primarily to study the response of deep foundations in soil (e.g., Mattes and Analysis of the load-displacement behavior of a drilled shaft Poulos 1969; Butterfield and Banerjee 1971; Randolph and is an essential step in a rational design. Design of most sock- Wroth 1978). In most cases, the solutions were not directly ets is governed by the requirement to limit settlement to a applicable to rock sockets because they did not cover the specified allowable value. The problem of predicting verti- typical ranges of modulus ratio (Ec/Er) or embedment ratio cal displacement at the top of a rock socket has been studied (L/B) of rock sockets, but they did provide the basic through theoretical and numerical analyses along with lim- methodology for analysis of the problem. Osterberg and ited results from full-scale field load testing. Methods that Gill (1973) used an elastic finite-element formulation to an- appear to have the most application to design of highway alyze rock sockets with D/B ranging from zero to 4 and the bridge foundations are summarized in this section. modulus ratios ranging from 0.25 to 4. Their analysis also considered differences between the modulus of the rock be- The basic problem is depicted in Figure 34 and involves neath the base (Eb) and that along the shaft (Er). Results predicting the relationship between an axial compression showed the influence of these parameters on load transfer, load (Qc) applied to the top of a socketed shaft and the in particular the relative portion of load carried in side re- resulting axial displacement at the top of the socket (wc). The sistance and transmitted to the base, but did not provide a concrete shaft is modeled as an elastic cylindrical inclusion method for predicting load-displacement behavior for de- embedded within an elastic rock mass. The cylinder of depth sign. Pells and Turner (1979) and Donald et al. (1980) con- L and diameter B has Young's modulus Ec and Poisson's ducted finite-element analyses assuming elastic and elasto- ratio vc. The rock mass surrounding the cylinder is homoge- plastic behaviors. Their numerical results were used to neous with Young's modulus Er and Poisson's ratio vc, determine values of the dimensionless influence factor (I) whereas the rock mass beneath the base of the shaft has that can be used to predict elastic deformation using the Young's modulus Eb and Poisson's ratio vb. (Note: some general equation authors use Er to denote modulus of rock in elasticity solu- tions; elsewhere in this report, Er denotes modulus of intact Qc wc = I (67) rock and EM is the rock mass modulus of deformation.) The Er B

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48 Values of the influence factor were presented in the form of charts for a range of modulus and embedment ratios com- mon for rock sockets. Graphs are also provided showing the ratio of applied load transferred to the base (Qb/Qc). These studies provided the first practical methods for predicting the load-displacement response of rock sockets. Their principal limitation lies in the assumption of a full bond between the shaft and the rock; that is, no slip. Observations from load tests; for example, Horvath et al. (1983), show that peak side resistance may be reached at displacements on the order of 5 mm. Rupture of the interface bond begins at this point, re- sulting in relative displacement (slip) between the shaft and surrounding rock. Under service load conditions, most rock sockets will undergo displacements that reach or exceed the full slip condition and should be designed accordingly. Analyses that account for both fully bonded conditions and full slip conditions provide a more realistic model of load- displacement response. Rowe and Pells (1980) conducted a theoretical study based on finite-element analyses of rock-socketed shafts that accounts for the possibility of slip at the shaftrock interface. The analysis treats the shaft and rock as elastic materials, but provides for plastic failure within the rock or the concrete FIGURE 35 Design chart for shaft displacement and base load shaft and for slip at the cohesive-frictional and dilatant rock transfer, complete socket (Rowe and Armitage 1987a). shaft interface. At small loads, the shaft, rock, and interface are linearly elastic and the shaft is fully bonded to the rock. Armitage solutions represent a standard against which Slip is assumed to occur when the mobilized shear stress approximate methods can be compared and verified. reaches the interface strength, assumed to be governed by a MohrCoulomb failure criterion: Rowe and Armitage (1987a,b) developed design charts for two contact conditions at the base of the socket: (1) a "com- = cpeak + n tanpeak (68) plete socket," for which full contact is assumed between the base of the concrete shaft and the underlying rock; and (2) a where cpeak = peak interface adhesion, n = interface normal shear socket, for which a void is assumed to exist beneath the stress, and peak = peak interface friction angle. Once slip oc- base. These conditions are intended to model the socket curs, it is assumed that c and degrade linearly with relative arrangements and methods of loading used in field load test- displacement between the two sides of the ruptured interface ing. When clean base conditions during construction can be from the peak values to residual values (cresidual, residual) at a verified and instrumentation is provided for measuring base relative displacement r. Roughness of the interface is mod- load, a complete socket is assumed. Frequently, however, base eled in terms of a dilatancy angle and a maximum dilation, resistance is eliminated by casting the socket above the base of and strain softening of the interface is considered. Modeling the drilled socket, in which case the test shaft is modeled as a of the interface in this way provides a good mechanistic rep- shear socket. The boundary condition at the base of a shear resentation of the load-displacement behavior of a rock- socket under axial compression is one of zero stress. The charts socketed shaft, as described in the beginning of this chapter given in Rowe and Armitage (1987a,b) provide a rigorous (Figure 19). method for analyzing rock-socket load-displacement behavior. From these studies, Rowe and Armitage (1987a,b) pre- Closed Form Solutions pared design charts that enable construction of a theoretical load-displacement curve in terms of (1) the influence factor An approximate method given by Kulhawy and Carter I used to calculate axial displacement by Eq. 67 and (2) the (1992b) provides simple, closed-form expressions that are at- ratio of load Qb/Qt transmitted to the socket base, where Qb = tractive for design purposes and yield results that compare base load and Qt = total load applied to the top of the shaft. well with those of Rowe and Armitage. For axial compres- Figure 35 is an example of the chart solution for a complete sion loading, the two cases of complete socket and shear socket. The charts offer a straightforward means of calculat- socket are treated. Solutions were derived for two portions of ing load-displacement curves and have been used by practi- the load-displacement curve depicted in Figure 19; the initial tioners for the design of bridge foundations. The Rowe and linear elastic response (OA) and the full slip condition (re-

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49 gion beyond point B). The closed-form expressions cannot 2. For the full slip portion of the load-displacement curve. predict the load-displacement response between the occur- (a) Shear socket: rence of first slip and full slip of the shaft (AB). However, the nonlinear finite-element results indicate that the progression Q wc = F1 c - F2 B (78) of slip along the socket takes place over a relatively small Er B interval of displacement. Comparisons of the bilinear curve given by the closed-form expressions with results of Rowe in which F1 = a1(2BC2 - 1BC1) 4a3 (79) and Armitage (1987b) indicated that this simplification is c F2 = a2 (80) reasonably accurate for the range of rock-socket conditions Er encountered in practice. C1,2 = exp[2,1L]/(exp[2L] - exp[1L]) (81) The closed-form expressions for approximating the load- - ( 2 + 4 ) 1 2 displacement curves for complete and shear socket are given 1,2 = (82) here. For a full description of the assumptions and deriva- 2 tions the reader is referred to Carter and Kulhawy (1988) and E B2 = a1 c (83) Kulhawy and Carter (1992b). Er 4 E 1. For the linearly elastic portion of the load-displace- = a3 c B (84) ment curve. Er (a) Shear socket (zero stress at the base): a1 = (1 + vr) + a2 (85) Er Bwc 1 2 Er cosh [L ] E 1 = a2 = (1 - c ) r + (1 + r ) (86) Ec (69) 2Qc B inh [L ] si Ec 2 tan tan c Er a3 = in which wc = downward vertical displacement at Ec (87) 2 tan the butt (top) of the shaft and where is defined by: (b) Complete socket: 2 2L 2 ( L )2 = (70) Q wc = F3 c - F4 B B Er B (88) = ln [5(1 - vr)L/B] (71) = Ec/Gr (72) in which Gr = Er / [2(1 + vr] (73) F3 = a1(1BC3 - 2BC4) 4a3 (89) where Gr = elastic shear modulus of rock mass. - 2 c F4 = 1 - a1 1 B a2 (90) D4 - D3 Er (b) Complete socket: D3,4 C 3 ,4 = (91) D4 - D3 4 1 2 L tanh [L ] 1+ 1 - b B L D3,4 = (1 - 2 Er b ) + 4 a3 + a1 2,1 B exp [ 2,1 L ] Gr Bwc (92) = (74) Eb 2Qc 4 1 2 2 L tanh [L ] + 1 - b B L The magnitude of load transferred to the base of where the shaft (Qb) is given by = Gr /Gb (75) Qb B 2 c = P3 + P4 (93) Gb = Eb/ [2(1 + vb] (76) Qc Qc The magnitude of load transferred to the base of in which the shaft (Qb) is given by P3 = a1(1 - 2) B exp[(1 + 2)L]/(D4 D3) (94) 4 1 1 P4 = a2(exp[2D] - exp[1L])/(D4 D3) (95) 1 - b cosh [L ] Qb = (77) Qc 4 1 2 2 L tanh [L ] The solutions given previously (Eqs. 6995) are easily + 1 - b B L implemented by spreadsheet, thus providing designers with

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50 a simple analytical tool for assessing the likely ranges of be- mobilized at the strength limit state. As stated by O'Neill and havior for trial designs. A spreadsheet solution provides the Reese (1999), "the issue of whether ultimate side resistance opportunity to easily evaluate the effects of various input pa- should be added directly to the ultimate base resistance to ob- rameters on load-displacement response. When combined tain an ultimate value of resistance is a matter of engineering with appropriate judgment and experience, this approach judgment." Responses to Question 20 of the survey (Appen- represents a reasonable analysis of rock-socketed drilled dix A) show a wide range in the way that side and base shafts. The method of Carter and Kulhawy presented herein resistances are combined for design of rock sockets. Several is also adopted in the FHWA Drilled Shaft Manual (O'Neill states indicated that they follow the guidelines given by and Reese 1999) for analysis of load-displacement response O'Neill and Reese in the FHWA Drilled Shaft Manual of single drilled shafts in rock (see Appendix C of the Man- (1999). Three possible approaches are described here. The ual). Reese and O'Neill also present methods for predicting first applies to the case where load testing or laboratory shear load-displacement response of shafts in cohesive IGMs and strength tests prove that the rock is ductile. In this case, the cohesionless IGMs. The equations are not reproduced here, ultimate values of side and base resistance are added directly. but are given as closed-form expressions that can be imple- If no field or lab tests are conducted, a "fully reduced fric- mented easily using a spreadsheet. tional shearing resistance at the interface" is used to compute side resistance and this value is added to the ultimate base re- sistance. The fully reduced strength is taken as the residual Other Methods shear strength of the rock = 'htanrc, where 'h = horizontal effective stress normal to the interface and rc = residual A computer program that models the axial load-displacement angle of interface friction between rock and concrete. A behavior of a rock-socketed shaft, based on the methods de- value of 25 degrees is recommended in the absence of mea- scribed by Seidel and Haberfield (1994) and described surements and 'h is assumed to be equal to the vertical briefly earlier, has been developed. The program ROCKET effective stress 'v. A second approach is recommended for requires the following input parameters: cases in which base resistance is neglected in design. In this case, the ultimate side resistance is recommended for design Drained shear strength parameters of the intact rock, at the strength limit state, unless "progressive side shear fail- Rock mass modulus and Poisson's ratio, ure could occur," in which case the resistance should be Foundation diameter, reduced "according to the judgment of the geotechnical Initial normal stress, and engineer." Mean socket asperity length and mean socket asperity angle. Several states indicate in their response to Question 20 that load testing, especially using the Osterberg load cell, is Although some of these parameters are determined on a one of the ways in which the issue is addressed. Load tests routine basis for the design of rock-socketed foundations, that provide independent measurements of side and base asperity characteristics are not typically evaluated. Drained resistance as a function of displacement and that are carried triaxial tests are also not considered routine by most trans- to large displacements provide the best available data for es- portation agencies in the United States. However, this ap- tablishing resistance values. When testing is not conducted, proach is promising because it provides a theoretical basis the major uncertainty (i.e., judgment) is associated with the for predicting rock-socket behavior that encompasses more question of whether or not side shear behavior will exhibit of the important parameters than the empirical approaches significant strain softening. Research is needed to provide now available to predict side resistance. guidance on what conditions are likely to produce a "brittle" response of the side resistance. Most load test data in which Combining Side and Base Resistances side resistance is measured independently do not exhibit a se- vere decrease in side resistance with large displacement. A A fundamental aspect of drilled shaft response to axial com- study with the objective of identifying the factors that con- pression loading is that side and base resistances are mobilized trol stressstrain behavior at the shaftrock interface at large at different downward displacements. Side resistance typi- displacement is needed. A large amount of data have been cally reaches a maximum at relatively small displacement, produced in recent years from load tests using the Osterberg in the range of 5 to 10 mm. Beyond this level, side resis- cell (O-cell) and such data would be useful for a study of tance may remain constant or decrease, depending on the post-peak interface behavior. The results would be most stressstrain properties of the shaft-rock interface. Ductile useful to practitioners if guidance could be provided on spe- behavior describes side resistance that remains constant or cific rock types, drilling methods, construction practices, decreases slightly with increasing displacement. If the inter- and ranges of confining stress that are likely to cause strain face is brittle, side resistance may decrease rapidly and softening at the interface. These factors could then be used significantly with further downward displacement. One as indicators of cases for which further field or laboratory question facing the designer is how much side resistance is testing is warranted.