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100 4.1 Overview Chapter 3 presents an analysis of available data that was mostly limited to direct correlations between the loading conditions (e.g., centric, eccentric, and so forth) and the per- formance of the bearing capacity calculation methods. The interpretation of the ï¬ndings in the case of shallow foundations is more complex than the interpretation of the ï¬ndings in the case of deep foundations, as presented, for example, in NCHRP Report 507 (Paikowsky et al., 2004). The reason is that many more parameters can contribute to the trend provided by the data than may be apparent in the ï¬rst evaluation. For example, Section 3.8.2 of this report examined the performance of Carter and Kulhawyâs (1988) equation for the bearing capacity of foundations on rock. The database consisting of tests on shallow foundations and drilled shaft tips suggested large vari- ations between the performances of the two. The natural con- clusion could have been that the load-displacement relations of the tip of a rock socket cannot be applied to the examination of bearing capacity theory. However, further examination of the data suggested that the investigated method (i.e., Carter and Kulhawy) has a bias depending on the rock quality. As the two examined case history databases (i.e., shallow foundations and rock sockets) varied by the rock quality that predominated in each, it was possible to explain the difference in the perform- ance based on rock quality rather than on the type of test. Similarly, the investigation of vertical loading of shallow foun- dations on natural soils as compared to vertical loading of shal- low foundations on controlled soils, presented in Section 3.5, suggested large variations between the two groups. Earlier inter- pretations of the data (e.g., Paikowsky et al., 2008; Paikowsky et al., 2009b; Amatya et al., 2009) naturally followed these ï¬ndings, distinguishing between the groups based on soil place- ment only (i.e., natural versus controlled). Further investigation revealed that part of the reason for variation was the difference in the friction angle of the soils in the investigated groups and the bias of the bearing capacity factor Nγ and its dependence on the magnitude of the internal friction angle. This chapter addresses, therefore, the following issues: 1. Completion of loads and parameters required to carry out the calibration. The distribution functions of the lateral load were previously developed. These are developed to allow for calibrations of sliding resistances. Target reliabil- ity is also established to allow for the calibration of the resistance factors. 2. Investigation and interpretation of the data and ï¬ndings presented in Chapter 3 of this report including sources of uncertainty, size effect, natural versus controlled soil, and the probabilistic approach to missing information. 3. Final determination of recommended resistance factors. 4.2 Uncertainty in Vertical and Lateral Loading 4.2.1 Overview The following discussion presents the chosen characteristics for vertical and lateral loads, dead and live, acting on bridge foundations. Although the subject is beyond the scope of the present research, establishing the lateral load distributions and factors became a necessity for the calibration process and is therefore presented. It is expected that future experimental, analytical, and probabilistic work will enable better analysis and more reliable selection of load distributions. 4.2.2 Vertical Loads NCHRP Report 507 (Paikowsky et al., 2004) established the load distributions and factors used for the ULS and SLS of deep and shallow foundations under vertical loads. These val- ues are based on Table F-1 of NCHRP Report 368, which pro- vides a range for live load (Nowak, 1999). The bias of live load has been taken as the mean of the range provided (1.10â1.20), and the COV is taken as 0.20 instead of 0.18, as presented in NCHRP Report 368. The load factors are from Tables 3.4.1-1 C H A P T E R 4 Interpretations and Appraisal
101 and 3.4.1-2 of LRFD Bridge Design Specifications Section 10: Foundations (AASHTO, 2007). These load factors are listed in Table 49. 4.2.3 Horizontal Loads 4.2.3.1 Horizontal Earth Pressure (Dead Load) The sources of uncertainties in the horizontal earth pres- sures due to soil and surcharge are the variations in soil unit weight and the soil friction angle. Based on the study by Phoon et al. (1995), the ï¬nal report for NCHRP Project 12-55 (DâAppolonia and the University of Michigan 2004) suggests the variation in soil unit weight as the following: ⢠Bias of soil unit weight = 1.00 ⢠COV of 0.10 for in situ (natural) soil conditions ⢠COV of 0.08 for engineered backï¬ll (controlled) ⢠Distribution followed = Normal Also, based on the study by Phoon et al. (1995), the ï¬nal report for NCHRP Project 12-55 (2004) lists the variation in the estimation of the soil friction angle (Ïf) as the following: ⢠Ïf from SPT: Bias = 1.00 to 1.20, COV = 0.15 to 0.20 ⢠Ïf from cone penetration test (CPT) (Kulhawy and Mayne, 1990): Bias = 1.00 to 1.15, COV = 0.10 to 0.15 ⢠Ïf from Lab test: Bias = 1.00 to 1.13, COV = 0.05 to 0.10 ⢠Distribution followed = Lognormal ⢠Reasonable estimate of bias taken as 1.00 At-Rest Earth Pressure Coefficient, K0. Based on the data summarized by Mayne and Kulhawy (1982) for drained and undrained at-rest earth pressure coefï¬cient K0, it was found that the COV of the corresponding transformation, using Jakyâs equation given below (Jaky, 1944), was 0.18 (NCHRP Project 12-55, 2004). K0nc represents K0 for normally consoli- dated cohesionless soil. Table 50 summarizes the variation in K0nc for cohesionless soils, which includes the transformation uncertainty, based on the ï¬nal report for NCHRP Project 12-55 (DâAppolonia and the University of Michigan, 2004). Rankine Active Earth Pressure Coefficient, Ka. The Rankine active earth pressure coefficient is given by the following: The variation of the Rankine active earth pressure coefï¬- cient with the variation in the soil friction angle is presented in Table 51. The coefï¬cients of variation for earth pressure coefï¬cients in Table 51 were obtained by generating 1,000 samples of soil friction angle following lognormal distribu- tion, with COVs of 0.10, 0.15, 0.20, and 0.25, respectively, and limiting maximum soil friction angle to 47°. In Table 51, these COVs are presented under âCOV simâ; âCOV calcâ was obtained using the ï¬rst order approximation in the calculation of COV, as mentioned in the ï¬nal report for NCHRP Project 12-55 (2004). It can be seen that the difference between the estimated COV using the simulation and the first order approximation increases with the increase in the soil friction angle COV. Rankine and Coulomb Passive Earth Pressure Coeffi- cients, Kp. The Rankine passive earth pressure coefï¬cient assumes no friction between the wall and the soil and there- fore results in a conservative estimate of the passive earth pres- sure coefï¬cient, which for frictional material is given by the following: K p f f f = + â = + â ââ â â â 1 1 45 2 1152 sin sin tan ( ) Ï Ï Ï Â° Ka f f f = â + = â â ââ â â â 1 1 45 2 1142 sin sin tan ( ) Ï Ï Ï Â° K nc f0 1 113= â sin ( )Ï Table 49. Load factors and uncertainties in vertical live load and dead load. Load type Load factor1 Bias2 COV2 Live Load (LL) L = 1.75 1.153 0.204 Dead Load (DL) D = 1.25 1.05 0.10 1 Tables 3.4.1-1 and 3.4.1-2 (AASHTO, 2007) 2 Table F-1 of NCHRP Report 368 (Nowak, 1999) 3 Mean of the range 1.10 to 1.20 4 COV of 0.18 rounded to 0.20 Table 50. Ranges of COV of K0nc for ranges of variation in soil friction angle (DâAppolonia and the University of Michigan, 2004). COV of K0nc COV of f 0.05â0.10 0.10â0.15 0.15â0.20 Soil friction angle, f f from Lab Test f from CPT f from SPT 30 0.186â0.202 0.202â0.227 0.227â0.260 35 0.189â0.217 0.217â0.257 0.257â0.303 40 0.195â0.237 0.237â0.295 0.295â0.364
102 Table 51. COV of lateral earth pressure coefficients for different COVs and soil friction angles. f = 2/3 f = 0.5 f = 0.4 f = 0.3 f = 0.2 f = 0.1 f = 0.0 Mean CO V C OV si m C OV calc COV sim COV calc COV sim COV si m C OV sim COV sim COV sim COV sim COV sim 0.10 0.09 0.10 0.10 0.09 0.20 0.17 0.15 0.14 0.12 0.11 0.1 0 0.15 0.14 0.15 0.15 0.13 0.34 0.27 0.24 0.21 0.19 0.17 0.1 5 0.20 0.19 0.21 0.22 0.17 0.64 0.45 0.38 0.33 0.28 0.25 0.2 2 0.25 0.22 0.27 0.27 0.21 1.04 0.61 0.49 0.41 0.35 0.31 0.2 7 0.10 0.12 0.13 0.13 0.11 0.36 0.27 0.23 0.20 0.17 0.15 0.1 3 0.15 0.17 0.19 0.19 0.16 0.70 0.43 0.35 0.30 0.26 0.22 0.1 9 0.20 0.23 0.27 0.26 0.21 1.05 0.63 0.50 0.42 0.35 0.30 0.2 6 0.25 0.27 0.34 0.33 0.25 1.39 0.84 0.67 0.55 0.46 0.39 0.3 3 0.10 0.15 0.16 0.16 0.14 0.58 0.37 0.30 0.25 0.22 0.18 0.1 6 0.15 0.22 0.24 0.24 0.20 0.97 0.59 0.48 0.39 0.33 0.28 0.2 4 0.20 0.28 0.33 0.30 0.25 1.13 0.73 0.59 0.49 0.42 0.35 0.3 0 0.25 0.31 0.43 0.34 0.30 1.19 0.80 0.65 0.55 0.46 0.39 0.3 4 0.10 0.16 0.17 0.17 0.15 0.67 0.42 0.34 0.28 0.24 0.20 0.1 7 0.15 0.22 0.26 0.24 0.21 0.97 0.61 0.49 0.40 0.34 0.28 0.2 4 0.20 0.27 0.36 0.29 0.27 1.07 0.69 0.56 0.47 0.39 0.34 0.2 9 0.25 0.32 0.47 0.33 0.32 1.09 0.75 0.62 0.52 0.44 0.38 0.3 3 0.10 0.17 0.19 0.17 0.16 0.68 0.42 0.34 0.28 0.24 0.20 0.1 7 0.15 0.23 0.30 0.23 0.23 0.84 0.55 0.45 0.37 0.32 0.27 0.2 3 0.20 0.28 0.41 0.27 0.29 0.91 0.63 0.52 0.44 0.37 0.32 0.2 7 0.25 0.33 0.53 0.30 0.35 0.93 0.66 0.55 0.47 0.40 0.35 0.3 0 Notes: f is limited to a max imum of 47deg rees 25 30 35 37 40 * âCOV simâ of earth pressure coefficients calculated from 1000 samples of friction angles assumed to follow lognormal distribut io n Rankine active, Ka Rankine passive, Kp So il friction angle, f Coulomb passive, Kp * COV calc: First order COV of earth pressure coefficients estimated as (DâAppolonia and the University of Michigan, 2004): f f f f f K K K of deviation standard and mean the areand where ) ( ) ( ) ( Table 52. Summary of COVs of earth pressure coefficients. COV K0nc Rankine Ka Rankine Kp30 < f 40 Range Reasonable Range Reasonable Range Reasonable f from Lab Test 0.20â0.22 0.20 0.12â0.17 0.15 0.12â0.17 0.15 f from CPT 0.22â0.26 0.25 0.17â0.23 0.20 0.19â0.23 0.20 f from SPT 0.25â0.33 0.30 0.23â0.28 0.25 0.23â0.28 0.25 The Coulomb passive earth pressure coefï¬cient is used more commonly and is given by the following: where β = angle of wall/interface surface to soil with vertical, δ = friction angle between wall/interface and soil, and α = angle of soil backï¬ll surcharge with the horizontal. Table 51 presents variations in active and passive earth pres- sures for a range of soil friction angles and their COVs. K p f f = â( ) +( ) â +( ) sin sin sin sin sin 2 2 1 β Ï Î² β δ Ï Î´ Ïi i f +( ) +( ) +( ) â¡ â£â¢ ⤠â¦â¥ α β δ β αsin sin ( ) i 2 116 Coulomb passive earth pressure has been presented for β = 90° and α = 0°, i.e., vertical wall and level backï¬ll. Table 52 summarizes the COV results presented in Tables 50 and 51 for lateral earth pressure coefï¬cients; these COVs can be used for at-rest and Rankine active and passive earth pres- sure coefï¬cients. For the Coulomb passive earth pressure coefï¬cient, one can choose a reasonable COV, as has been presented in Table 52, for each ratio of interface friction angle to soil friction angle. For example, for a granular ï¬ll material with the ratio of inter- face friction angle to soil friction angle of about 2â3, when the soil friction angle is estimated from SPT readings, the COV lies in the range of 0.70 to about 1.1. One may choose a rea- sonable COV as 0.85. It should be noted that in Table 51 the maximum conceivable soil friction angle is assumed to be 47°,
hence, there is a drop in the COV calculated for a higher fric- tion angle (40°). The topic of lateral passive earth pressure is complex as it is often associated with the limiting displacement that controls the development of the pressure rather than the theoretical pressures associated with the coefï¬cient. As such, the discus- sion in this section is limited in its scope and addresses solely the current limited needs. With the reasonable estimates of the COVs of soil unit weight and earth pressure coefï¬cients, the lateral pressure due to, for example, active earth pressure can be calculated as (where Ea is active earth pressure and h is height of soil) with a bias of 1.00. This implies that the combined statistics for the mean and standard deviation are the following: Hence When soil friction angles are based on SPT readings, COVs of the horizontal dead load due to at-rest (K0) or active earth pressure (Ka) can be calculated as 0.27 to 0.35. As such, a prac- tical use of a bias of 1.00 and COV of 0.30 is a reasonable rep- resentation of a large range of possibilities for lateral dead load due to earth pressure and can be considered to follow a lognor- mal distribution. Earth Pressure Due to Compaction. A typical distribu- tion of residual earth pressure after compaction of backfill behind an unyielding wall with depth is given in Figure 90. A particular example of granular soil with Ïf of 35°, γ of 125 pcf, and roller load of 500lb/in. compacting a lift thickness of 6 in. when at a distance of 6 in. away from the wall has been presented. It can be seen that the residual earth pressure increases rapidly with depth, with a maximum pressure at around 5 ft below the compacted surface for this example. Table 53 summarizes the variation of the multiplier factor RÏ with the COV of a soil friction angle Ïf of 35°, speciï¬cally for one standard deviation change in Ïf. This range of multi- plier (adjustment) factors was based on the tables of adjust- ment factors by Williams et al. (1987). It is to be noted that these adjustment factors themselves are empirical in nature and are approximate representations of test results with large scatters. From Figure 90, it can be seen that the lateral earth pres- sure after compaction (residual lateral stress) is 800 psf at a COV COV COVEa Ka= +γ2 2 117( ) μ μ μ Ï Ï Ï Î³ γ Ea Ka Ea a Ka h h K = â ( ) + 0 5 0 5 0 52 2 2 2 . . . i i and h i γ( )2 E h Ka a= 0 5. γ i 103 Figure 90. Residual earth pressure after compaction of backfill behind an unyielding wall (based on Clough and Duncan, 1991). 0 400 800 1200 1600 2000 Earth Pressure after Compaction (psf) 35 30 25 20 15 10 5 0 D ep th (f t) At-rest earth pressure (K o = 0 .4) Roller load = 500lb/in Roller 6in from wall Lift thickness = 6in γ = 125lb/ft3 Ï = 35o c = 0 Table 53. Range of multiplier factor R for the estimation of earth pressure due to compaction at a depth of 5 ft of compacted soil for f = 35°, = 125 pcf, roller load = 500 lb/in, distance from wall = 6 in, lift thickness = 6 in, mean of R = 1.00. Ïf = 35 COV Roller Vibrator plates/ rammers 0.10 0.94 â 1.10 0.97 â 1.05 0.15 0.91 â 1.14 0.96 â 1.09 0.20 0.88 â 1.19 0.95 â 1.17 0.25 0.85 â 1.24 0.94 â 1.24 Range of RÏ at 5 ft depth for a variation of 1 s.d. in Ïf depth of 5 ft. When the measured soil friction angle has a COV of 0.20, based on the multiplier factors in Table 53, this residual stress can vary from 704 psf (800 à 0.88) to 952 psf (800 à 1.19). To estimate the uncertainty in the establishment of the resid- ual lateral pressure curve obtained based on a solution proposed by Duncan and Seed (1986) and shown in Figure 90, the bias of the measured lateral earth pressure versus the calculated lateral earth pressure was studied (see Figure 91). The measured earth pressures are from the experimental study by Carder et al. (1977). The residual earth pressures on a concrete retaining wall due to compaction of sand backï¬ll were measured at differ- ent depths. The calculated earth pressures are, as presented by Duncan and Seed (1986), based on an âincremental solution.â
The mean of the bias was found to be 1.005 and the bias COV was 0.215. Based on the results obtained in Table 53, it can be con- cluded that for the compaction case presented in Figure 90, the worst-case calculated COV of multiplier factor RÏ approaches the COV of the friction angle. Incorporating the effect of the result obtained in Figure 91, the combined COV for the esti- mation of residual lateral earth pressure due to compaction is approximately using the COVs of Ïf = 0.20, RÏ = 0.20 and residual earth pressure estimator = 0.215, respectively. Using COVs of Ïf and RÏ as 0.15 and 0.09 results in a combined COV of 0.27. The range of COV is thus 0.27 to 0.35. Hence, it may be said that a bias of 1.00 and COV of 0.30 would provide a reasonable estimate of the residual earth pressure due to compaction. 4.2.3.2 Lateral Pressure from Live Loads In order to assess the horizontal lateral pressure due to live load, uncertainties in different components of the live load must be assessed (A.S. Nowak, personal communication, 2006). ACI 318 (Szerszen and Nowak, 2003) lists the following: Wind (50-year maximum) bias = 0.78 COV = 0.37 Snow bias = 0.82 COV = 0.26 Earthquake bias = 0.66 COV = 0.56 In 1983, the Ontario Ministry of Transport used the fol- lowing for the assessment of lateral forces for the Toronto subway (OHBDC, 1979, 1983, 1993): 0 35 0 20 0 20 0 2152 2 2. . . .= + +( ) Temperature bias = 1.00 COV = 0.25 Shrinkage and creep bias = 0.90 COV = 0.20 Wind (75-year maximum) bias = 0.85 COV = 0.25 Braking force (railways) bias = 1.00 COV = 0.10 There are no exact measurements available, but wind load is similar to other forces and a limited parametric study seems to be reasonable. Experts (A.S. Nowak, personal communi- cation) suggest that a bias of 1.00 and COV of 0.15 should be used for the lateral pressure due to live loads. 4.2.3.3 Summary of Horizontal Loads Assuming that lateral loading due to dead load (LFD: lateral force due to dead load) is mostly due to soil and surcharge, possibly compacted, the following load distribution and load factors (load factors from AASHTO, 2007, Table 3.4.1-2) have been chosen for at-rest and active earth pressures: λLFD = bias of lateral loading due to dead load = 1.00, COVLFD = 0.30 and is assumed to follow lognormal distribution with the following distribution in soil unit weight γ (assumed to follow normal distribution): λγ = bias of soil weight = 1.00, COVγ = 0.10 for in-situ (natural) soil conditions, COVγ = 0.08 for engineered backfill (controlled soil condition) Load factor for at-rest earth pressure, γEH0 = 1.35, and load factor for active earth pressure, γEHa = 1.50. Assuming the lateral loading due to live load (LFL: lateral force due to live load) is mostly shear loads from wind, tem- perature variation, and creep and shrinkage transferred via the bearing pads, the following distributions and load factors have been chosen: λLFL = 1.00, COVLFL = 0.15 and assumed to follow lognor- mal distribution Load factor for lateral live load, γLFL = 1.00 (assumed) 4.3 Calibration Methodology 4.3.1 Overview of Calibration Procedures Probability-based limit state designs are presently carried out using methods categorized into three levels (Thoft- Christensen and Baker, 1982): ⢠Level 3 includes methods of reliability analysis utiliz- ing full probabilistic descriptions of design variables and the true nature of the failure domains (limit states) to calculate the exact failure probability, for example, using MCS techniques. Safety is expressed in terms of failure probability. 104 Figure 91. Measured versus calculated residual earth pressures. Measured earth pressures at Transport and Road Research Laboratory experimental concrete retaining wall by Carder et al. (1977) and calculated earth pressure using the incremental solution proposed by Duncan and Seed (1986) (bias = measured/calculated). 0 5 10 15 20 0 5 10 15 20 measured lateral pressure (kPa) ca lc ul at ed la te ra l p re ss ur e (k Pa ) n = 12 bias mean = 1.005 bias COV = 0.215
⢠Level 2 involves a simpliï¬cation of Level 3 methods by expressing the uncertainties of the design variables in terms of mean, standard deviation, and/or COV and may involve either approximate iterative procedures (e.g., FOSM, FORM and SORM analyses) or more accurate techniques like MCS to evaluate the limit states. Safety is expressed in terms of a reliability index. ⢠Level 1 is more of a limit state design than a reliability analy- sis. Partial safety factors are applied to the predeï¬ned nom- inal values of the design variables (namely the loads and resistance(s) in LRFD); however, the partial safety factors are derived using Level 2 or Level 3 methods. Safety is measured in terms of safety factors. Regardless of the probabilistic design levels described above, the following steps are involved in the LRFD calibration process: 1. Establish the limit state equation to be evaluated. 2. Deï¬ne the statistical parameters of the basic random vari- ables or the related distribution functions. 3. Select a target failure probability or reliability value. 4. Determine load and resistance factors consistent with the target value using reliability theory. More applicable to an AASHTO LRFD geotechnical application is a variation in which structural selected load factors are utilized to deter- mine resistance factors for a given target value. Chapter 1 of this report reviewed the limit state equations to be evaluated, and Chapter 2 developed their evaluation to establish the statistical parameters to be used. The statisti- cal parameters to be used are further investigated in the following sections of this chapter to finally establish the parameters to be used in the calibration. The load charac- teristics were developed and presented in Section 4.2. The following section outlines the selected target reliability and develops the resistance factors based on the methodology presented in Sections 1.3.5 and 1.4. 4.3.2 Target Reliability 4.3.2.1 Methods of Establishing Target Reliability As has been pointed out in NCHRP Report 507 (Paikowsky et al., 2004), in general, two methods are used to generate target reliability levels: (1) basing them on the reliability lev- els implicit in current WSD codes and (2) using cost-beneï¬t analysis with optimum reliability proposed on the basis of minimum total cost, which includes the cost of economic losses and consequences due to failure. In establishing a target reliability level using the ï¬rst method, the reliability levels implied in the current design practice are calculated. The target level is usually taken as the mean of the reliability levels of representative designs. Such target reliabil- ity can be thought of as related to the acceptable risks in cur- rent practice and hence an acceptable starting point for code revision. The second method is based on the concept that safety measures are associated with cost; therefore, âsafety essentially is a matter not only of risk and consensus about acceptable risks, but also of costâ (Schneider, 2000). Even though attempts have been made to determine the cost of failure (Kanda and Shah, 1997), it is hard to assign the cost of failure, especially when it incorporates human injury or loss of life. 4.3.2.2 Target Reliability Based on Current WSD It has been found that the reliability levels of foundations designed using WSD factors of safety can vary considerably (e.g., Phoon and Kulhawy, 2002; Honjo and Amatya, 2005). Hence, the recommendation of a target reliability index based on the reliability levels implied in the current WSD practice requires some judgment. A literature survey shows that very few authors have dealt with the determination of the target reliability of shallow foun- dations. Phoon and Kulhawy (2002) calculated the reliability indexes for different COVs in the operative horizontal stress coefï¬cient of soil. Taking the soil property variability into account, it was shown that reliability indexes lie in an approx- imate range of 2.6 to 3.7, with an average of 3.15. Designs for square footings with embedment depth ratios (ratio of embed- ment depth to footing width) of 1 and 3 and 50-year return period wind loads of 50% and 33.33% of the uplift capacity of the footings were evaluated. A target level of 3.2 was decided for ULS. However, this target level is speciï¬c only for footings subject to uplift loads. In NCHRP Report 343 (Barker et al., 1991), which forms a basis for the resistance factor in the current edition of AASHTO LRFD Bridge Design Speciï¬cations, it was found that the reliability indexes obtained using âRational Theoryâ varied from 1.3 to 4.5 for the bearing capacity of footings on sand and from 2.7 to 5.7 for footings on clay (Allen, 2005). They concluded that a target reliability of 3.5 should be used for footings (for the reference, the resistance component was taken equal to the factor of safety times the summation of the effect of load combination and the reliability indexes calcu- lated for a ratio of dead load to live load of 3). A target level of 3.5 was used for the code calibration for foundations in the National Building Code of Canada (NRC, 1995). Becker (1996) mentions that this target reliability was the average of the range of 3.0 to 4.0 obtained using a semi- analytical approach to ï¬t WSD for the typical load combina- tions in Canadian structural design, with ductile behavior and normal consequence of failure. This range of reliability level matches with the range obtained from an updated 105
database included in the ï¬nal report for NCHRP Project 20-7/ Task 186 (Kulicki et al., 2007)âfor a majority (about 120) of the 124 bridges analyzed, the reliability index for superstructures was between 3 and 4. A target reliability level of 3.5 is taken in the current AASHTO LRFD Bridge Design Specifications (1994) (for the structural system) for the most common load combination, dead load and maximum 75-year live load (Strength I). Further, a range of 2.5 to 3.0 for drilled shafts and 2.0 to 2.5 for a redundant foundation system such as a pile group of more than four piles was suggested by Barker et al. (1991). Paikowsky et al. (2004) suggested a target reliability of 3.0 for a nonredundant deep foundation system (system with four or less piles) and, along with the study by Zhang et al. (2001), suggested 2.33 for a redundant deep foundation system. 4.3.2.3 Recommended Target Reliability General Considerations. It would be logical and conven- ient to assign at the present stage a target level for foundations equal to that assigned for superstructures. In order to fulï¬ll one of the main goals of the LRFD, the reliability level of the foun- dation system should be comparable to that of the structural system. However, the actual resulting reliability level of the combined system of super- and sub-structures (including soil- structure interaction) is unknown, even though a target level equal to that obtained for the superstructure is assigned for substructures. It may also be of interest to note that due consideration should be given to applying structural safety concepts to geo- technical designs (Phoon and Kulhawy, 2002) for two reasons. First, it is unrealistic to assign a single âtypicalâ variation (of COV) to each soil parameter, even those obtained from direct measurements taking into consideration the inherent soil vari- ability, measurement errors, and transformation uncertain- ties. Usually, a range has to be provided even for datasets of satisfactory quality, taking into consideration important details like soil type, number of samples per site, distribu- tion of depositions and measurement techniques. Second, it is important to consider the vital role of the geotechnical engineer in appreciating and recognizing the complexities of soil behavior and the inherent limitation of âsimplisticâ empirical geotechnical models used in the prediction of such behavior. Current Study Calibrations. For the present calibration of resistance factors for shallow foundations, a target relia- bility range of 3.0 (pf = 0.135%) to 3.5 (pf = 0.023%) will be examined. This range encompasses the nonredundant target reliability used for deep foundations (β = 3.0) to the target reli- ability assigned in the current LRFD Bridge Design Speciï¬ca- tions for shallow foundations. There are two major reasons at this stage for leaving the target reliability as a range: (1) using the different resistance factors obtained from the target relia- bility range allows evaluation of the associated range of equiv- alent factors of safety and hence identiï¬cation of suitability to WSD and (2) shallow foundation design includes two distinct groups of foundations for which the controlling limit state is different. By and large, shallow foundations on soil are con- trolled by the SLS, and, therefore, the target reliability of the ULS and the associated resistance factor are of secondary prac- tical importance and must be evaluated against the service- ability limits. In contrast, for foundations built on rock, the ULS is by and large the controlling criterion as either struc- tural or geotechnical failure will take place before the limit settlement will be mobilized. As such, the chosen target reli- ability actually controls the safety of the structure. An addi- tional aspect affecting the aforementioned discussion is the fact that the uncertainty in the determination of capac- ity for foundations on rock is of higher complexity (as it is subjected to discontinuities that control the rock strength), and, hence, a possible logical outcome of the proposed range is the use of two different target reliabilities: one for shallow foundations on soil and the other for shallow foundations on rock. Examined Target Reliability Range. Resistance factors for three target reliabilitiesâ3.0 (pf = 0.135%), 3.25 (pf = 0.058%), and 3.5 (pf = 0.023%)âare examined as a ï¬rst stage in the pres- ent study for the uncertainty established by the databases and selected methods of analysis. Figure 92 illustrates the range of resistance factors calculated based on a typical range of bias and a wide range in the uncertainty of the resistance using load characteristics from NCHRP Report 507âs calibration for the three examined target reliabilities. Considering âtypicalâ val- ues of resistance with a lognormal distribution, with a bias of 1.5, and a COV of 0.3, the resistance factors for the target reli- abilities of 3.00, 3.25, and 3.50 are 0.64, 0.58, and 0.53, respec- tively. The three resistance factors roughly translate into a cost difference of 20% between the higher and the lower resistance factor (assuming, for simplicity, direct relations among load, size, and cost). 4.3.3 Load Conditions, Distributions, Ratios, and Factors The loading conditions are taken as those presented in Table 49 and Section 4.2.3.3. The actual load transferred from the superstructure to the foundations is, by and large, unknown because very little long-term research has been focused on the subject. The load uncertainties are taken, therefore, as those used for superstructure analysis. The LRFD Bridge Design Spec- ifications (AASHTO, 2007) provide four load combinations for the standard strength limit state (dead, live, vehicular, and wind loads) and two for the extreme limit states (earthquake 106
and collision loads). The load combination for Strength I (Z) was therefore applied in its primary form, as shown in the fol- lowing limit state: where R = the strength or resistance of the footing, D = dead load, and LL = vehicular live loads. The probabilistic char- acteristics of the random variables D and LL are provided in Table 49 for vertical loads and in Section 4.2.3.3 for lat- eral loads. For the strength or resistance (R), the probabilis- tic characteristics are developed in Chapters 3 and 4, based on the databases for the various methods and conditions of analysis. Paikowsky et al. (2004) examined the inï¬uence of the ratio of dead load to live load, demonstrating very little sensitivity of the resistance factors to that ratio, with overall decrease of the resistance factors with the increase in the ratio of dead load to live load. Large ratios of dead load to live load represent condi- tions of bridge construction typically associated with very long bridge spans. The relatively small inï¬uence of the ratio of dead load to live load on the resistance factor led Paikowsky et al. (2004) to use a typical ratio of 2.0, knowing that the obtained factors are by and large applicable for long span bridges, being on the conservative side. This ratio was adopted, therefore, for the present study calibrations as well. Discussion of the ratio of dead load to live load for lateral loads is presented later in this chapter. Z R D LL= â â ( )118 4.4 Examination of the Factor N as a Source of Uncertainty in Bearing Capacity Analysis 4.4.1 Overview Section 3.5 examined the uncertainty in the bearing capac- ity of footings in/on granular soils subjected to vertical-centric loading. This load type pertains to 173 case histories of data- base UML-GTR ShalFound07. A summary of the bias is pre- sented as a ï¬ow chart in Figure 60 and histograms and relations between measured and calculated capacities in Figures 61 to 65. The analysis of the data indicated the following: 1. Overall, the mean bias (measured over predicted capacity) was greater than 1 (mλ = 1.59 for n = 173) pointing out a systematic capacity underprediction. 2. The mean bias (mλ) of the footings on natural soil condi- tions was 1.0, and the mean bias (mλ) of the footings on controlled soil conditions was 1.64. 3. Previous ï¬ndings suggested resistance factors based on the separation between natural and controlled soils, using the above ï¬ndings (Paikowsky et al., 2008; Amatya et al., 2009; Paikowsky et al., 2009b). A clear variation exists between the cases of the foundations on natural soils and the cases of the foundations on controlled soils by a factor of 1.6. The source of this large variation in the 107 Figure 92. Calculated resistance factors as a function of the bias and COV of the resistance for the chosen vertical loading distributions and ratios under the range of the examined target reliabilities. 0 2 31 Bias (λ) 0 0.4 0.8 1.2 1.6 2 R es is ta nc e Fa ct or (Ï ) FOSM λQL = 1.15 COVQL = 0.2 QD/QL = 2.0 γD = 1.25 CO V = 0.1 =β β β 3.0 0 = 3.2 5 = 3.5 0 COV = 0.3 COV = 0. 6 COV = 0.9 β = 3.00, 3.25, 3.50 COVQD = 0.1 λQD = 1.05 γL = 1.75
bias was further investigated, especially other parameters that could affect this variation and could be the source for the large bias in the prediction. Section 1.5.2 discusses the fact that no closed-form analytical solution exists for the bearing capacity problem formulation once the soil weight effect beneath the foundation is considered. The factor Nγ has been, therefore, evaluated by many researchers with varying results, as demon- strated in Figure 11. The investigation of the factor Nγ using the robust database assembled for this study is presented in the following section in view of the aforementioned bias ï¬ndings. 4.4.2 The Uncertainty in the Bearing Capacity Factor N For foundations tested on the surface of granular soils, the bearing capacity (Equation 19) becomes a function of the term γ Nγ only, as the cohesion and embedment terms are zeroed. The bearing capacity factor Nγ can then be back-calculated and the obtained factor (termed NγExp) can be evaluated against that proposed by Vesic´ (1973) (termed NγVesic´) and used in this study (see Equation 29 and Table 26). The bias of the term Nγ can be deï¬ned as the following: One hundred and twenty ï¬ve relevant cases were investigated in which the foundation was tested on the ground surface, and the groundwater was below the zone of the foundation inï¬u- ence. Figure 93 presents the scatter and exponential ï¬t of the λ γ Ïγ γ γ γ N u q f N N q Bs N = = ( ) +( ) Exp Vesic 0 5 2 1 11 . tan ( 9) bias in Nγ obtained for soils with friction angles between 42° and 46°. The data points representing the bias in Nγ presented in Figure 93 suggest a clear trend in which the bias Nγ increased as the soilâs internal friction increased beyond about Ïf ⥠43°. The best ï¬t line of the bias λNγ versus internal friction Ïf, as expressed in Figure 93, can be used to develop an expression for a modiï¬ed bearing capacity factor Nγ that would better match the experimental data: The large scatter of the data results in a coefï¬cient of deter- mination (R2) of 0.351 for Equation 120. 4.4.3 Re-examination of the Uncertainty in Bearing Capacity of Footings in/on Granular Soils Accounting for the Bias in the Factor N The effect of the bias in Nγ established in Section 4.4.2 is examined in this section by comparing the bias of the calcu- lated bearing capacity under different loading conditions to the bias established for Nγ. Figures 94 to 98 describe the bias of the calculated bearing capacity for soil friction angles between 42.5° and 46.0° (for which Equation 120 is valid) for different loading conditions. For the case of vertical-centric loading (Figure 94), the bias of the bearing capacity calculation over- laps that of Nγ, suggesting that the bias observed for the inves- tigated cases can be mostly attributed to the bias in Nγ. This N Nf f γ Î³Ï Ï Exp Vesic for ° = â( ) ⤠exp . . . 0 205 8 655 42 5 ⤠46 120° ( ) 108 Figure 93. The ratio (N) of the back-calculated bearing capacity factor N (based on experimental data) and the bearing capacity factor proposed by Vesicâ (1973) versus soil friction angle. 42 43 44 45 46 Friction Angle, f (deg) 0 0.5 1 1.5 2 2.5 3 N = [q u / (0 . 5 B s )] / N V es ic load test data; n = 125 = exp(0.205 f 8.655) (R2 = 0.351) Figure 94. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the bearing capacity factor N (N) versus the soil friction angle for footings under vertical-centric loadings. 43 44 45 46 Friction Angle, Ïf (deg) 0 0.5 1 1.5 2 2.5 3 Bi as Data Bearing Capacity bias (n = 131) Bearing Capacity (BC) bias, N bias,
conclusion is subjected, however, to the fact that most of the cases are related to surface loading, hence, used for establish- ing the bias in Nγ. For the cases related to vertical-eccentric and inclined-centric loading (Figures 95 and 96), the data suggests that the trends are similar, and, hence, the bias in Nγ may be a significant contributor to the bias in the bearing capacity calculations. The biases do not overlap because the cases involved in eccentric and inclined loading are highly sensitive to many other factors that affect the bearing capacity. The cases involved in inclined-eccentric loading (Figures 97 and 98) have a small number of data cases and the bearing capacity is highly sensitive to the loading conditions. Overall, the data presented in Figures 94 to 98 suggest that the bias in the bearing capacity factor Nγ is a major contributor to the uncertainties in the bear- ing capacity estimation regardless of the load combinations acting on the footing. 109 Figure 95. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the bearing capacity factor N (N) versus the soil friction angle for footings under vertical-eccentric loadings. 43 44 45 46 Friction Angle, Ïf (deg) 0 0.5 1 1.5 2 2.5 3 Bi as , λ Data Bearing Capacity bias (n = 26) Bearing Capacity (BC) bias, λ Nγ bias, λNγ Figure 96. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the bearing capacity factor N (N) versus the soil friction angle for footings under inclined-centric loadings. 43 44 45 46 Friction Angle, f (deg) 0 0.5 1 1.5 2 2.5 3 Bi as , Data Bearing Capacity bias (n = 29) Bearing Capacity (BC) bias, N bias, Figure 97. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the bearing capacity factor N (N) versus the soil friction angle for footings under inclined-eccentric, positive moment loadings. 44 44.5 45 45.5 46 Friction Angle, f (deg) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Bi as , Data Bearing Capacity bias (n = 8) Bearing Capacity (BC) bias, bias, Figure 98. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the bearing capacity factor N (N) versus the soil friction angle for footings under inclined-eccentric, negative moment loadings. 44 44.5 45 45.5 46 Friction Angle, f (deg) 0 1 2 3 4 5 6 7 8 Bi a s, Data Bearing Capacity bias (n = 7) Bearing Capacity (BC) bias, bias,
4.5 Examination of Footing Size Effect on the Uncertainty in Bearing Capacity Analysis Figure 99 presents the ratio of measured to calculated bear- ing capacity (the bias λ) versus footing width for vertical- centric loaded footings on/in natural and controlled soils. Overall, no easily identiï¬able trend appears in Figure 99 other than a general trend of some increase in the bias with the increase in footing size for natural soils, subjected to the pre- sented scatter. Figure 100 shows the mean bias of the bearing resistance ver- sus the footing size for all the cases in controlled and natural soil conditions combined. The 95% conï¬dence interval of the mean bias versus the footing size is also presented for friction angles less than and greater than 43° (the reason for making Ïf = 43° the separator is related to the uncertainty in the factor Nγ presented in Section 4.4). The following observations related to the database on which Figure 100 was based can be made: smaller footings were tested on soils with larger friction angles, Ïf ⥠43°, and larger footings were tested on soils with smaller friction angles, Ïf < 43°. Overall, it can be concluded that what can be perceived as a reduction in the bias with an increase in the foundation size seems to be more associated with the bias in Nγ associated with the internal friction angle. Other conclusions are difï¬cult to derive due to the small number of cases associated with large footings (i.e., 1 to 3 cases for footings greater than 1 m) as compared to 135 cases in the small footing category. 4.6 In-Depth Re-Examination of the Uncertainty in Bearing Capacity of Footings in/on Granular Soils Under Vertical-Centric Loading 4.6.1 Identification of Outliers and Fit of Distributions for Calibrations 4.6.1.1 Overview The bearing capacity of footings in granular soils is highly controlled by the bearing capacity factor Nγ, in particular for foundations on or near the surface. The factor Nγ is very sen- sitive to the magnitude of the soilâs internal friction angle Ïf as expressed by Equation 29, presented in Table 26, and illus- trated in Figure 11. Section 4.3 investigated the source of the bias underlying the bearing capacity analysis, demonstrating that the bias increases with the increase in the internal friction angle (when exceeding 42.5°) and is closely associated to the bias in the expression of Nγ as illustrated in Figures 94 to 96. The varying bias with the soilâs internal friction angle sug- gests that the development of the resistance factors should follow this trend, unless a correction to the methodology is developed and the expression of Nγ is modiï¬ed. The latter, although it may have some advantages, is problematic for several reasons, including the need to change an established methodology and modiï¬cations of an expression based on a database that, while extensive, may be modiï¬ed in the future. As the resistance factors should be developed considering the 110 0.1 1 10 0.1 1 10 0.2 0.6 1 1.4 1.8 2.2 2.6 3 3.4 3.8 Bi as , 0.01 Footing width, B (m) (5) (34) (4) (90) (5) (2) (12) (3) (3) Natural Soil Condition (n =14) Controlled Soil Condition (n =158) 1s.d. (x) number of cases in each interval Footing width, B (ft) Figure 99. Variation of the bias in bearing resistance versus footing size for cases under vertical-centric loadings: controlled and natural soil conditions.
bias change with the soilâs internal friction angle, Ïf, it is also reasonable to pursue the identification of data outliers for subsets based on the magnitude of Ïf. 4.6.1.2 Outliers and Examination of Fit of Distributions for Ïf = 45° ± 0.5 The largest dataset in the UML-GTR ShalFound07 database is for footings tested under vertical-centric loadings. Subsets of data are formed for each identiï¬able internal friction angle, Ïf (± 0.5°). The largest subset is for Ïf = 45 ± 0.5° (90 cases), the mean and COV of the bias for which are found to be 1.81 and 0.203, respectively. Figure 101 presents a comparison of a standard normal quantile of the bias data to predicted quan- tiles of the theoretical normal and lognormal distributions. At least one possible outlier, a footing with a bias of 3.51, can be observed for both the normal and the lognormal distributions. Removal of this data point can result in a better fit of the dataset to the normal distribution, which is further quantiï¬ed by the goodness-of-ï¬t test. In this sense, the outliers identiï¬ed here imply that their removal improves the dataset so it better ï¬ts a theoretical distribution. The Ï-squared goodness-of-ï¬t (GOF) tests have been car- ried out to test the ï¬t of the theoretical normal and lognormal distributions to follow the bearing resistance bias for n = 90 cases, along with the datasets after the removal of some identi- ï¬able outliers. Table 54 lists in detail a number of trials and the corresponding Ï-squared values obtained from the GOF tests. If the Ï-squared values obtained for an assumed distribution are greater than the acceptable Ï-squared values of a certain signiï¬cance level (usually of 1% or 5%), the distribution is rejected. For n = 90, the Ï-squared value for the lognormal dis- tribution is 63.0 and the Ï-squared value for the normal distri- bution is 228.9, both of which are greater than the Ï-squared values of 21.66 at the 1% signiï¬cance level and 16.92 at the 5% signiï¬cance level, respectively. Hence, both distributions do not ï¬t the data well and are rejected by the Ï-squared GOF test. The smaller Ï-squared value for the lognormal distribution (in comparison to the Ï-squared value of the normal distribution) for this dataset suggests, however, that the lognormal distribu- tion provides a better ï¬t. It can be seen from the trials outlined in Table 54 that the removal of outliers from either or both the higher and the lower tails of the bias distribution does not result in an accept- able Ï-squared value for either the normal or the lognormal distribution. Hence, the removal of outliers from the distribu- tion tails does not render normal or lognormal distribution acceptable, while a comparatively better ï¬t ï¬uctuates between normal and lognormal distribution, based on the Ï-squared GOF test. Hence, all the available data for the cases in/on soil with Ïf = 45° have been used for the resistance factor calibra- tion without the identiï¬cation and removal of outliers and assumed to follow lognormal distribution. In Figure 101, there are four footings with a bias smaller than 1.0, the smallest being λ = 0.82, for which the assumed lognormal distribution overpredicts the bias in the lower tail region, which is more critical than the higher tail region (because bias less than 1.0 means the calculated resistance was more than the actual resistance). This circumstance is exam- ined in Section 4.6.2.4 following the resistance factor calibra- tion in order to ensure that the resistance factor developed for Ïf = 45° results in acceptable risk in design. 111 0.01 0.1 1 Footing width, B (m) 0.5 1 1.5 2 2.5 3 3.5 Bi as 0.1 1 10B (ft) (5) (34) (4) (90) (5) (2) (1) (17) (1) (3) (4) (3) (1) (2) Mean bias BC (n = 172) s.d. (x) no. of cases in each interval 95% confidence interval for f (n = 135) 95% confidence interval for f (n = 37) Figure 100. Variation of the bias in bearing resistance versus footing size for cases under vertical-centric loadings: f >â 43° and f < 43°. -2 0 2 St an da rd n o rm al q ua nt ile 0.5 1 1.5 2 2.5 3 3.5 4 Bias λ Vertical-centric loading f = 45 Total data (n = 90) Normal distribution Lognormal distribution λ = 1.81 COVλ = 0.203 Figure 101. Standard normal quantile of bias data (measured over predicted bearing capacity) for f = 45 +â 0.5° and predicted quantiles of normal and lognormal distributions.
4.6.1.3 Outliers and DFs for Internal Friction Angles Other than 45° Procedures similar to those described in Section 4.6.1.2 have been performed for the data subsets of Ïf other than 45°. For Ïf = 44° (n = 30, mλ = 1.40 and COV = 0.250), both the normal and lognormal distributions are accepted by the Ï-squared GOF test for the 1% signiï¬cance level. The log- normal distribution provides a better ï¬t, with a Ï-squared value of 13.74 versus 17.82 for the normal distribution. For Ïf = 43°, 42°, 38°, 36°, and 32°, although the normal dis- tributions provide better ï¬ts than the lognormal distributions, lognormal distributions have been considered. This is done because lognormal distribution is naturally expected to better represent the dataset of a ratio (i.e., bias) restricted by values greater than zero or due to similar behavior, small dataset, and so forth as further detailed. For Ïf = 43° (n = 14, mλ = 1.34, and COV = 0.283), the Ï-squared value is 18.53 for normal versus 22.69 for lognormal. For Ïf = 42° (n = 4, mλ = 1.60, and COV = 0.416), the Ï-squared value is 12.20 for normal versus 12.74 for lognormal. For Ïf = 38° (n = 12, mλ = 1.26, and COV = 0.215), the Ï-squared value is 16.75 for normal versus 74.62 for log- normal. The minimum bias of 0.55, which is overpredicted by the lognormal distribution for this dataset, will be examined following the resistance factor calibration. For Ïf = 36° (n = 4, mλ = 1.20, and COV = 0.233), the Ï-squared value is 19.78 for normal versus 21.61 for lognormal, and, for Ïf = 32° (n = 4, mλ = 1.25, and COV = 0.347), the Ï-squared value is 10.77 for a normal distribution versus 11.15 for lognormal. For Ïf = 35° (n = 4), the mean bias is found to be 2.00 and the bias COV is 0.528, which is exceptionally high compared to the COVs for the datasets of the closer-in-magnitude friction angles, which is around 0.2. Though the GOF test shows that both normal and lognormal distributions are acceptable, with lognormal being a better fit, the case with the highest bias, λ = 3.57, has been considered an outlier. The comparison of the standard normal quantiles of the dataset and the theo- retical distributions is shown in Figure 102(a). The result- ing dataset after the removal of this case has a mean of 1.47 and COV of 0.088 (examination of the database shows that the remaining three cases are from the same site, hence explaining the very small COV). Comparison of the stan- dard normal quantiles of the filtered dataset and the theo- retical distributions is shown in Figure 102(b). Lognormal distribution is considered for this dataset also. Hence, only one outlier was removed from the total dataset, resulting in 172 cases used for the resistance factor calibration for vertical- centric loading. 4.6.2 The Statistics of the Bias as a Function of the Soilâs Internal Friction Angle and Resulting Resistance Factors 4.6.2.1 In-Depth Examination of Subsets Based on Internal Friction Angle Tables 55 through 57 present the biases evaluated for the bearing capacity estimation according to the soilâs friction angles. The corresponding resistance factors have been obtained for a target reliability index βT of 3.0 (exceedance probability of 0.135%). Table 55 presents the cases in con- trolled soil conditions while Table 56 shows the cases in nat- ural soil conditions. Table 57 presents all the cases in the 112 Table 54. -squared values for the fitted lognormal and normal distributions for vertical-centric loading cases on/in soil with an internal friction angle (f) of 45°. Ï-squared values n Lognormal distribution Normal distribution Comments 90 63.0 228.9 Mean = 1.81, COV = 0.203; all data for Ïf = 45° 89 515.0 60.3 Mean = 1.79, COV = 0.179; highest bias (3.51) removed (data beyond 2s.d.) 89 60.3 428.0 Mean = 1.822, COV = 0.195; case with 3rd lowest bias (0.87) removed; this case is on the lower bias tail and the farthest from theoretical lognormal quantile 88 57.9 724.0 Mean = 1.83, COV = 0.186; 2 cases with 2nd and 4th lowest biases (0.85 and 0.87) removed; in lower bias tail and farthest two from theoretical lognormal quantile 87 805.0 43.6 Mean = 1.83, COV = 0.185; 2nd and 4th lowest bias cases (0.85 and 0.87) and the case with the highest bias (3.51) removed 87 62.5 927.0 Mean = 0.81, COV = 0.161; 2nd and 4th lowest bias cases (0.85 and 0.87) and the case with the 2nd highest bias (2.37) removed 87 57.5 1,418.0 Mean = 1.84, COV = 0.177; 2nd, 3rd and 4th lowest bias cases (0.85, 0.85 and 0.87) removed Note: Acceptable Ï-squared value for significance level of 1% is 21.666 and for significance level of 5% is 16.919.
113 -2 -1 0 1 2 St an da rd n or m al q ua nt ile 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Bias Vertical-centric loading f = 35 Total data (n = 4) Identified outlier (n = 1) Normal distribution Lognormal distribution = 2.00 COV = 0.528 -1.5 -1 -0.5 0 0.5 1 1.5 St an da rd n or m al q ua nt ile 1 1.5 2 2.5 Bias Vertical-centric loading f = 35 Filtered data (n = 3) Normal distribution Lognormal distribution = 1.47 COV = 0.088 (a) (b) Figure 102. Standard normal quantile of bias data for f = 35 +â 0.5° and predicted quantiles of normal and lognormal distributions (a) for all data and (b) with the outlier removed. Table 55. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled soil conditions for vertical-centric loading. Bias Resistance factor ( T = 3) Friction angle f ( 0.5 deg) n Mean COV MCS Preliminary 46 2 1.81 0.071 1.655 1.00 45 90 1.81 0.203 1.194 1.00 44 30 1.40 0.250 0.807 0.80 43 14 1.34 0.283 0.700 0.70 42 4 1.60 0.416 0.700 0.70 39 1 1.02 -- -- -- 38 11 1.32 0.122 1.081 1.00 36 3 1.34 0.079 1.206 1.00 35 3 1.47 0.088 1.300 1.00 43 to 46 136 1.67 0.247 0.971 0.95 38 3 22 1.38 0.225 0.855 0.85 all angles 158 1.63 0.252 0.934 0.90 Table 56. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in natural soil conditions for vertical-centric loading. Bias Resistance factor ( T = 3) Friction angle f n Mean COV MCS Preliminary 33 2.5 (all angles) 14 1.00 0.329 0.457 0.45 database, both controlled and natural soil conditions, under vertical-centric loadings. All the cases in the controlled soil conditions are in soils with relative densities above 35%. Graphical presentation of the bias in bearing resistance esti- mation versus soil friction angle is shown in Figure 103. The error bars represent one standard deviation of the mean bias for each friction angle, taken as a range of Ïf ± 0.5°, and the number in parentheses represents the number of cases in each of the friction anglesâ subsets. 4.6.2.2 Factor Development Based on Data Trend The bias in bearing resistance estimation for the cases under vertical-centric loading, both in/on controlled and natural soil
conditions, can be expressed by the best ï¬t line in Figure 103 and in Equation 121, for which the coefï¬cient of determina- tion is 0.200. This line shows that the bearing resistance bias (λBC) increases with an increase in the soil friction angle: The details provided in Tables 55 and 56 indicate that the data available for controlled soil conditions relate to soils λ ÏBC f= ( )0 308 0 0372 121. exp . ( ) with higher friction angles compared to that for natural soil conditions. The bias expressed by Equation 121 has been used to develop resistance factors for the whole range of soil friction angles for both controlled and natural soil conditions. Based on Tables 55 and 56, the COVs of the bias for all the controlled and natural soil condition cases are 0.252 and 0.329, respectively. Hence, COVλ of 0.25 and 0.35 may be taken to represent the COVs of the biases for the controlled soil and nat- 114 Table 57. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled and natural soil conditions combined, for vertical-centric loading. Bias Resistance factor ( T = 3) Friction angle f ( 0.5 deg) n Mean COV MCS Preliminary 46 2 1.81 0.071 1.655 1.00 45 90 1.81 0.203 1.194 1.00 44 30 1.40 0.250 0.807 0.80 43 14 1.34 0.283 0.700 0.70 42 4 1.60 0.416 0.700 0.70 39 2 0.83 0.330 0.378 0.35 38 12 1.26 0.215 0.804 0.80 36 4 1.20 0.233 0.727 0.70 35 3 1.47 0.088 1.300 1.00 34 2 1.09 0.135 0.865 0.85 33 3 1.03 0.126 0.836 0.80 32 4 1.25 0.347 0.542 0.50 30.5 2 0.98 0.423 0.339 0.30 43 to 46 136 1.67 0.247 0.971 0.95 36 ± 3 36 1.23 0.296 0.619 0.60 all angles 172 1.58 0.278 0.838 0.80 30 32 34 36 38 40 42 44 46 Friction angle f (deg) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Bi a s (2) (90) (30) (14)(4) (2) (12) (4) (3) (2) (3) (4) (2) Mean bias, BC (n = 172) 1 s.d. (x) no. of cases in each interval BC = 0.308 à exp(0.0372Ïf) (R2=0.200) 95% confidence interval Figure 103. Bearing resistance bias versus average soil friction angle (taken f +â 0.5°) including 95% confidence interval for all cases under vertical-centric loading.
ural soil conditions, respectively. Table 58 presents the resis- tance factors calculated using these statistics for friction angles ranging from 30° to â¥45°, on foundations in/on natural and controlled soil conditions. Figure 104 presents the recommended resistance factors for controlled and natural soil conditions detailed in Table 58. Fig- ure 104 also presents a comparison of the recommended resis- tance factors to those obtained in Table 57 (based on the database) and the 95% confidence interval of the bearing resistance bias. It can be observed that the recommended resistance factors follow the trend in the bearing resistance bias with the soil friction angle. The cases for which a small resistance factor was developed based on a very small sub- set (two cases each) could be justifiably overruled in the context of the established trend and the large datasets sup- porting it. 4.6.2.3 Recommended Resistance Factors The recommended resistance factors for vertical-centric loading cases are presented for different friction angles in Table 59 based on the values calculated and recommended in Table 58. The values in Table 59 are applicable for soils with relative densities greater than 35%. Further consider- ation is necessary for soils with friction angles less than 30° combined with relative densities less than 35%. For these soils, which are in a very loose state, it is recom- mended either to consider ground improvement to a depth of at least twice the footing width (subjected to a settlement criterion), ground replacement, or an alternative founda- tion type. 4.6.2.4 Examination of the Recommended Resistance Factors A rough estimate of the equivalent factor of safety (FS) for a resistance factor of Ï, developed using a ratio of dead load to live load of 2.0; dead-load, load factor of 1.25; and live-load, 115 Table 58. Resistance factors for vertical-centric loading cases based on the bias versus f best fit line of Equation 121 and the COV of natural versus controlled soil conditions. Resistance factor ( T = 3) Soil Conditions Natural Controlled (COV = 0.35) (COV = 0.25) Soil friction angle f (deg) Mean bias (Equation 121) MCS Rec.* MCS Rec. 30 0.94 0.403 0.40 0.542 0.50 35 1.13 0.485 0.45 0.652 0.60 37 1.22 0.524 0.50 0.703 0.70 38 1.27 0.545 0.50 0.732 0.70 40 1.36 0.584 0.55 0.784 0.75 45 1.64 0.704 0.65 0.946 0.80 *Rec. = Recommended 30 32 34 36 38 40 42 44 46 Friction angle Ïf (deg) 0.0 0.2 0.4 0.6 0.8 1.0 R es ist an ce fa ct or , Ï 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Bi as λ (2) (90) (30) (14)(4) (2) (12) (4) (3) (2)(3) (4) (2) 95% confidence interval for λ Resistance factor based on database (x) no. of cases in each interval Recommended f for Controlled soil conditions Recommended f for Natural soil conditions n = 172 Figure 104. Recommended resistance factors for soil friction angles (taken f +â0.5°) between 30° and 46°, with comparisons to 95% confidence interval and resistance factors obtained for the cases in the database; the bubble size represents the number of data cases in each subset.
load factor of 1.75 was presented by Paikowsky et al. (2004) and expressed by the following equation: The highest recommended resistance factor in Table 59 is Ï = 0.80 for Ïf ⥠45°, developed assuming the data follow a lognormal distribution. A rough estimate of the equiva- lent factor of safety for this resistance factor is given by Equa- tion 122 as 1.77. A safe design requires that the condition in Equation 123 is met: where qcalc is calculated bearing capacity and qmeas is measured bearing capacity. The minimum allowable bias for the given FS is, therefore, the reciprocal of the FS, i.e., the minimum bias for which the design will be safe is 1/FS = 0.565. This bias is much smaller than the smallest bias of the dataset, λ = 0.82, for which the standard normal quantile is seen to be over- predicted by the assumed lognormal distribution (see Fig- ure 101). A bias of 0.82, therefore, results in a safe design, and all the footing cases in the database are safe upon the applica- tion of the developed resistance factor. It can, therefore, be concluded that the methodology of utilizing the trend and the assumption of the lognormal distribution for the bias is acceptable for resistance factor calibration and is justiï¬ed by the outcome. 4.7 In-Depth Re-Examination of the Uncertainty in Bearing Capacity of Footings in/on Granular Soils Under Vertical-Eccentric Loading 4.7.1 Examination of the Bias for Controlling Parameters The investigation presented in Section 4.4.3 and Figure 95 suggested that the bias in the bearing capacity factor Nγ can be associated with the general trend of the bias for the bearing q FS qcalc meas⤠( )123 FS â1 4167 122. ( )Ï capacity analysis of footings under vertical-eccentric loading. The relations shown in Figure 95 are not similar to those in Figure 94 (investigating footings under vertical-centric load- ing); hence, additional evaluation is required for cases not under vertical-centric loading. The variation of the bearing capacity bias with the soilâs fric- tion angle is presented in Figure 105 for cases under vertical- eccentric loading (each error bar represents one standard deviation). It can be observed that for Ïf = 35 ± 0.5° the mean bias of the seven cases is higher than for the other soil friction angles with a relatively lower COV. These seven cases are related to a single site and compiled from the DEGEBO litera- ture. Hence, for the determination of the best ï¬t line of the bias versus the friction angle, these seven cases were excluded. The trend in Figure 105 suggests a possible decrease in the bias with the increase in the friction angle, which is contrary to the trend established for the case of vertical-centric loading (see Fig- ure 94) or the trend seen in Figure 95 for the soilâs friction angles in the range of 43.5° to 46.0°. The data in Figure 105 suggest that no clear, unique correlation exists between the bias and the soilâs internal friction angle, and, even upon the exclusion of the aforementioned seven cases, the coefï¬cient of determi- nation (R2) is 0.01, essentially indicating that a correlation does not exist. The data in Figure 105 may indicate, therefore, that either for the eccentric loading and/or the available data for such cases, factors other than the soilâs friction angle contribute signiï¬cantly to the bias. Figure 106 presents the relationship between the bias of vertical-eccentric loading of foundations and the magnitude of the eccentricity normalized by the foundationâs width, i.e., e/B. Forty-three cases have been tested with load eccentricity 116 Table 59. Recommended resistance factors for vertical-centric loading cases. Recommended resistance factor ( T = 3) Soil conditions Soil friction angle f (deg) Natural Controlled 30â34 0.40 0.50 35â36 0.45 0.60 37â39 0.50 0.70 40â44 0.55 0.75 45 0.65 0.80 30 32 34 36 38 40 42 44 46 Friction angle f (deg) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Bi as (7) (6) (4) (2) (9) (4) (11) Mean bias, BC 1 s.d. (x) no. of cases in each interval BC = 2.592´exp(-0.01124 f) (R2=0.01) 95% confidence interval n = 43 Figure 105. Bearing resistance bias versus soil friction angle for cases under vertical-eccentric loadings (seven cases for f = 35° [all from a single site] have been ignored for obtaining the best fit line).
117 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Load eccentricity ratio, e/B 0.5 1 1.5 2 2.5 3 3.5 B ia s, Vertical-eccentric loading (n = 43) B ⤠4.0 in (0.1m) B = 1.65 ft (0.5m) B = 3.3 ft (1.0m) Figure 106. Bearing resistance bias versus load eccentricity ratio e/B for vertical-eccentric loading. 34 35 36 37 38 39 40 41 42 43 44 45 46 Soil friction angle, f (deg) 0.5 1 1.5 2 2.5 3 3.5 B ia s, Vertical-eccentric loading e/B = 1/6 (Total n = 21) B = 1.65 ft (0.5m) B = 3.3 ft (1.0m) B ⤠4.0 in (0.1m) Figure 107. Change in bearing resistance bias with soil friction angle for tests with a load eccentricity ratio of e/B = 1/6. ratios ranging from 0.025 to 0.333 (1/40 to 1/3), with a major- ity of them having an eccentricity ratio of 1/6. It can be seen that while the larger foundations mostly have higher biases, there appears to be no correlation between the bearing resistance bias and the load eccentricity ratio. The large scatter that appears for the small foundations may be related to the physical difï¬- culties of conducting such tests where eccentric loads need to be applied to a small footing. A closer examination of the relationship between the bias and the magnitude of the eccentricity is presented in Figure 107 for a given eccentricity ratio of e/B = 1/6 versus friction angle Ïf. Cases with various footing widths are available for this eccentricity ratio only (see Figure 106), while tests with other load eccentricity ratios mostly utilize footings of widths less than or equal to 4 in (â0.1 m). While a best ï¬t line for these data would show a decrease in the bias with an increase in Ïf, the data related to small footings only are relevant to higher friction angles. Figure 107 thus emphasizes that the effect of the footing size on the bearing resistance bias when testing eccen- trically loaded foundations is more signiï¬cant compared to the effect of the soilâs friction angle. Hence, calibrating resistance factors using this dataset, based on Ïf, cannot be justiï¬ed, as has been done for the vertical-centric loading cases. 4.7.2 Identification of Outliers The data presented in Figures 105 and 107 lead to the con- clusion that in the absence of a clear underlying factor to explain the bias, resistance factors may be developed for both natural and controlled soil conditions and a range of Ïf and then compared to the resistance factors developed for vertical- centric loading.
factors remain essentially at 1.0, with the exception of four cases related to Ïf = 41°, for which a large scatter had been observed (see Figure 105). In addition, the amount of data available for some of the Ïf subsets is comparatively small. It has also been concluded in Section 4.5 that using the available data, the effect of footing size on the bias cannot be isolated from the effect of the soil friction angle. All these conditions lead to the issue of whether it is practical and appropriate to use the dataset for vertical-eccentric loading conditions alone for the resistance factor calibration of this loading situation. Since vertical-centric loading is the simplest loading mode, the uncertainties involved in estimating the resistance of footings under vertical-eccentric loading are assumed to be not less than those involved in the case of footings under vertical-centric loading. This assumption is based on the fol- lowing: (1) when the source of the lateral load is not perma- nent, the foundation supports vertical-centric loading only, and (2) very often the magnitudes of the lateral loads (and hence eccentricity) are not known at the bridge foundation design stage (see Section 3.1, in particular, Section 3.1.7). This means that the resistance factors for vertical-eccentric loading conditions have to be either equal to or less than the ones rec- ommended for the vertical-centric loading in Table 59. 4.7.4 Examination of the Recommended Resistance Factors for Vertical-Eccentric Loading The bias mean for vertical-eccentric loading is slightly higher than the bias mean for vertical-centric loading (1.83 ver- sus 1.58); hence, the same resistance factors used for vertical- centric loadings are recommended for vertical-eccentric loadings. Based on Equations 122 and 123, the minimum allowable bias for the highest resistance factor of 0.80 is 0.565. The bear- ing resistance biases of all the cases under vertical-eccentric loading in the database (the minimum being λ = 0.80) are thus 118 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Bias λ -3 -2 -1 0 1 2 3 St an da rd n or m al q ua nt ile Vertical-eccentric loading Total data (n = 43) Normal distribution Lognormal distribution λ = 1.83 COVλ = 0.351 Figure 108. Standard normal quantile of the bias for all vertical-eccentric loading cases and the predicted quantiles of theoretical distributions. Table 60. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled soil conditions for vertical-eccentric loading. Bias Resistance factor ( T = 3) Friction angle f ( 0.5 deg) n Mean CO V MCS Preliminary 46 11 1.80 0.227 1.109 1.00 45 4 1.53 0.199 1.021 1.00 44 9 1.27 0.182 0.889 0.85 43 2 1.88 0.238 1.122 1.00 41 4 2.06 0.604 0.426 0.40 40 6 1.77 0.203 1.168 1.00 35 7 2.69 0.148 2.063 1.00 43 to 46 26 1.58 0.257 0.892 0.85 40 to 46 36 1.67 0.325 0.772 0.75 all angles 43 1.83 0.351 0.783 0.75 Figure 108 presents the standard normal quantile of the dataset with the theoretical predictions of normal and log- normal distributions. The presented relations visually suggest a good match between the lognormal distribution and the data. The Ï-squared GOF tests verify that the data follow the lognormal distribution better than the normal distribu- tion (accepted both at the 1% and 5% significance levels), with Ï-squared values of 8.34 for lognormal distribution ver- sus 11.74 for normal distribution. As the data follow the log- normal distribution, no outliers are identiï¬ed. 4.7.3 The Statistics of the Bias as a Function of the Soilâs Internal Friction Angle and Resulting Resistance Factors The bias in the bearing resistance estimation for footings under vertical-eccentric loadings evaluated for subsets of each Ïf (±0.5)° are presented in Table 60. The associated resistance
safe upon the application of the recommended factors. A dif- ferent approach may be taken, assuming the eccentric loads are permanent, hence, allowing for resistance factors higher than those applied for vertical-centric loading. This condition is examined via the effective width (Bâ²) versus the actual width of footings under vertical-centric loading, i.e., both founda- tion sizes are examined (B based on Ï for vertical-centric and Bâ² based on Ï for vertical-eccentric) and the larger foundation size prevails. Such examination allows review of the recom- mended resistance factors for vertical-centric versus vertical- eccentric conditions. A limited examination of this issue follows. In Figure 105, the mean bias of vertical-eccentric loading for friction angles between 40° and 46° is 1.60. Assuming the mean bias to remain a constant at 1.60 for all friction angles and the COV of the bias of the bearing resistance to be related to natural and controlled soil conditions, i.e., 0.35 and 0.30, respectively, the obtained resistance factors are as follows: Natural soil conditions, for all Ïf : Ï = 0.65 (Ï obtained from MCS = 0.687) Controlled soil conditions, for all Ïf : Ï = 0.75 (Ï obtained from MCS = 0.796) Taking these two separate databases, one for vertical-centric and the other for vertical-eccentric, two sets of resistance fac- tors, one for controlled soil conditions and one for natural soil conditions can be obtained, as presented in Table 61. Table 61 demonstrates that the recommended resistance factors based on the extensive data available from vertical-centric load tests, although they may be conservative, will be also safe when applied for footings designated to be subjected to load-eccentricity. This is validated when compared to the resistance factors developed based on vertical-eccentric load tests (under the aforementioned assumptions) shown in Table 61 as well. An additional examination of the effect of the eccentric- ity ratio (ratio of load eccentricity to footing width) on the designed footing was carried out. A strip foundation on the surface of soil with a unit weight of 124.7pcf (19.6 kN/m3) was analyzed, hence eliminating the effects of the founda- tionâs shape and depth. Bearing resistances of the strip foot- ing under an eccentric load with a given eccentricity ratio, altered from 1/4 to 1/100, were estimated using the bearing capacity equation and expressed as bearing resistance versus the effective footing width Bâ² (B â 2e), (Meyerhof, 1953). Because the effective footing width is used, the effect of eccentricity is âeliminatedâ and the vertical-eccentric load case is essentially transferred to the vertical-centric case, i.e., the resulting effective footing width is the same regardless of the load eccentricity ratio. For example, for a required fac- tored load of 369 ton/ft (1,000 kN/m), the required effective footing width (Bâ²) using the recommended Ï of 0.60 for a Ïf of 35° has been found to be about 6.25 ft (1.90 m) for eccen- tricity ratios of 1/4 as well as 1/100. In Figures 109(a) and 109(b), the bearing resistance versus the effective footing width plots have been presented for e/B = 1/4 and 1/100, respectively, for a frictional soil with an internal friction angle (Ïf) of 35°. It should be noted that the design (physical) footing width in both cases is different as B = Bâ² + 2e and hence depends on the magnitude of the eccentricity. Based on the examination above, it can be said that the recommended resistance fac- tors using the vertical-centric test data results in an accept- able design for vertical-eccentric loading conditions and that separate sets of resistance factors are not required. The test results in the UML-GTR ShalFound07 database for vertical-eccentric loadings did not enable evaluation of the performance of Meyerhofâs effective width model (1953), i.e., the uncertainty in defining Bâ² = B â 2e or the ability of the eccentricity ratio to exceed the limiting compression contact value of 1/6. Some discussion of the subject using other sources follows. 119 Table 61. Comparison of the recommended resistance factors based on vertical-centric loading to those obtained based on Figure 105 for vertical-eccentric loading. Resistance factor ( T = 3) Controlled soil conditions Natural soil conditions Soil friction angle f Recommended Vertical-centric and vertical- eccentric Vertical- eccentric based on Figure 105 Recommended Vertical-centric and vertical- eccentric Vertical- eccentric based on Figure 105 0.50 0.40 0.60 0.45 0.70 0.50 0.75 0.55 0.80 30°â34° 35°â36° 37°â39° 40°â44° ⥠45° 0.75 0.65 0.65
The limiting eccentricity value of e/B = 1/6 is developed from a theory assuming a linear stress distribution under a rigid footing subjected to eccentric loading (the combination of cen- tric load and a moment similar to the stress distribution in a beam). As such, when the eccentricity ratio is 1/6, the founda- tion is subjected to compression with one edge under no (zero) stress. When the eccentricity ratio exceeds 1/6, the foundation is expected to be subjected to âtension,â hence the contact area between the foundation and the soil decreases. It is well under- stood that the load distribution under the foundation depends on the relative stiffness of the foundation/soil system and, hence, is not necessarily linear. Expected load distributions under vertical-centric loading proposed by Terzaghi and Peck (1948) were veriï¬ed experimentally by Paikowsky et al. (2000) using tactile sensor technology and demonstrating concave stress load distribution across a rigid footing in granular soil. The effect of eccentricity (not presented in Paikowsky et al., 2000) was measured as one side stress concentration support- ing the one-sided extensive slip surfaces developing under an eccentrically loaded foundation as illustrated in Figure F-3 (Appendix F) by Jumikis (1956). A theoretical study was presented by Michalowski and You (1998) examining Meyerhof âs aforementioned effective width rule (1953) in calculations of the bearing capacity of shallow foundations. Michalowski and You developed a limit analysis solution for eccentrically loaded strip footings and assessed the effective width rule. The isometric slip lines developed by Michalowski and You invoking the kinematic approach of limit analysis resemble qualitatively the above described exper- imental observations. Michalowski and You concluded that for smooth footings, realistic footing models, and cohesive soils, Meyerhof âs effective width rule is a reasonable account of eccentricity in bearing capacity calculations. It is only for sig- niï¬cant bonding at the soil interface (i.e., no separation or per- fect adhesion) and for large eccentricities (e.g., e/B greater than 0.25) that the effective width rule signiï¬cantly underestimates bearing capacity (for clays). For cohesive-frictional soil, this underestimation decreases with an increase in the internal fric- tion angle, becoming more and more âaccurateâ with limited eccentricity. The examination and discussion presented in Sections 4.6 and 4.7 lead to the following recommendations: 1. The use of resistance factors developed and recommended for vertical-centric loading (see Table 59) could and should be extended to be used with vertical-eccentric loading. 2. The rule of effective foundation size (Bâ² = B â 2e) pro- posed by Meyerhof (1953) is not overly conservative and results in realistic bearing capacity predictions for the foundation-soil conditions expected to be encountered in bridge construction (rough surface foundations on granular soils). 120 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Effective footing width, B' (m) 0 500 1000 1500 Be ar in g re sis ta nc e p er u ni t l en gt h, Q u (k N/ m ) f = 35 Eccentricity ratio e/B = 1/4 Unfactored Bearing Capacity Factored - Natural, = 0.45 (centric) Factored - Natural, = 0.65 (eccentric) Factored - Controlled, = 0.60 (centric) Factored - Controlled, = 0.75 (eccentric) Required vertical-eccentric load (a) (b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Effective footing width, B' (m) 0 500 1000 1500 B ea ri n g re sis ta nc e pe r un it le ng th , Q u (k N /m ) f = 35 Eccentricity ratio e/B = 1/100 Unfactored Bearing Capacity Factored - Natural, = 0.45 (centric) Factored - Natural, = 0.65 (eccentric) Factored - Controlled, = 0.60 (centric) Factored - Controlled, = 0.75 (eccentric) Required vertical-eccentric load Figure 109. Comparison of the required effective footing widths for different eccentricity ratios: (a) e/B = 1/4 and (b) e/B = 1/100 for a strip footing resting on a soil with internal friction angle (f) = 35°.
3. The independence of the calculated effective foundation size (Bâ²) from the magnitude of the eccentricity and the aforementioned recommendations/observations provide a solution for the design problems presented by various DOTs (see Section 3.1.7), in which the eccentricity is unknown at the early design stage. The solution justiï¬es the calculated foundation size during early design to be referred to as the effective foundation that can then be modiï¬ed by twice the eccentricity at the ï¬nal design stage. 4. In light of the presented material, there is no clear evi- dence allowing an increase in the foundation eccentricity ratio for permanent loading beyond e/B = 1/6. 5. For combined loading (permanent and variable), an argu- ment can be made that the eccentricity ratio can be increased to e/B = 1/3 for which half of the foundation is under âtensionâ conditions. Some performance-based design codes (e.g., DIN 1054) allow that limit. As no clear data exists to support such an increase, it is recom- mended that until further research is carried out in the area, the eccentricity of the combined loading will be limited to e/B ⤠1/4, as allowed in the AASHTO standard specifications (4.4.8) or recommended in Section 8.4.3.1 of FHWA-NHI-06-089 Soils and Foundation Manual. (FHWA, 2006). 4.8 In-Depth Re-Examination of the Uncertainty in Bearing Capacity of Footings in/on Granular Soils Under Inclined-Centric Loading 4.8.1 Examination of the Bias for Controlling Parameters In the case of footings under inclined-centric loadings, an additional factor involved is the load inclination to the verti- cal, when compared to the case of footings under vertical- centric loadings. Figure 110 examines the variations in the bias versus the angle of load inclination (to the vertical), according to footing sizes. The scatter shows that there is no clear trend of the bias associated with either the load inclination angle or the footing size. All the larger footings (B ⥠1.65 ft) were tested under inclined loads with θ = 0° (inclination along the footing length, see Figure 17), while the smaller footings were subjected to inclined loads with θ = 90° (inclination along the footing width). Although it appears that the bias increases with an increase in the load inclination for θ = 0° while for θ = 90° the bias decreases with an increase in the inclination angle, it is difficult to isolate the effect of the footing size, except in the vicinity of load inclination of 10°. For the tests with inclination angles around 10° carried out on different footing sizes, it can be observed that the orientation switched between θ = 0° and 90° has no effect on the bias, which suggests that no corre- lation exists with the orientation of the load. This obser- vation should be qualified, however, by the fact that the dataset for loading orientations between 0° and 90° is not sufficiently large to make a general statement. The resis- tance factors can thus be further examined in relation to the soilâs friction angle. The total number of data points available for inclined- centric loading is 39 (bias mean = 1.43 and COV = 0.295), while the soil friction angles ranged from 46 (±0.5°) to 38 (±0.5°). As a result, the identiï¬cation of outliers based on the data subset for each Ïf (±0.5°) may not be practical because of the small data subsets. The standard normal quantiles of the data and those predicted by the developed normal and log- normal distributions are presented in Figure 111. A visual observation clearly shows that the data fits the normal dis- tribution, while for the data to follow the lognormal distri- bution, some outliers in the lower tail region (especially 121 0 5 10 15 20 25 30 Load inclination to the vertical (deg) 0 0.5 1 1.5 2 2.5 B ia s, Inclined-centric loading (Total n = 39) B ⤠4 in (0.1m); θ = 90° B = 1.65 ft (0.5m); θ = 0° B = 3.3 ft (1.0m); θ = 0° Figure 110. Bias versus load inclination for footings under inclined- centric loading.
bias gradually increases with an increase in the soil friction angle. The resistance factor is calibrated using the mean obtained by the best fit line. 4.8.2 The Statistics of the Bias as a Function of the Soilâs Internal Friction Angle and Resulting Resistance Factors The statistics of the bearing resistance bias for the cases under inclined-centric loadings are presented in Table 62 for subsets of each Ïf (±0.5°), while the best ï¬t line obtained from the regression analysis of the biases available for 38° < Ï â¤ 46° in Figure 112, is provided by Equation 124. λ Ï= +1 25 0 0041 124. . ( )f 122 -3 -2 -1 0 1 2 3 St an da rd n o rm a l q ua nt ile 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Bias λ Inclined-centric loading Total data (n = 39) Normal distribution Lognormal distribution λ = 1.43 COVλ = 0.295 Figure 111. Standard normal quantile of bias data for all data for inclined-centric loading and predicted quantiles of theoretical distributions. 30 32 34 36 38 40 42 44 46 Friction angle f (deg) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Bi as (1) (3) (6) (4) (4) (11) (10) Mean bias, BC 1 S.D. (x) no. of cases in each interval BC = 1.25 + 0.0041 f 95% confidence interval n = 39 Figure 112. Variation of the bias in bearing resistance versus soil friction angle for cases under inclined- centric loadings. Table 62. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled soil conditions for inclined-centric loading. Bias Resistance factor ( T = 3) Friction angle f ( 0.5 deg) n Mean COV MCS Preliminary 46 10 1.81 0.104 1.555 1.00 45 11 1.08 0.376 0.442 0.45 44 4 1.17 0.347 0.520 0.50 43 4 1.43 0.166 1.055 1.00 40 6 1.64 0.217 1.050 1.00 39 3 1.42 0.151 1.088 1.00 38 1 1.14 -- -- -- 43 to 46 29 1.39 0.322 0.665 0.65 all angles 39 1.43 0.295 0.737 0.70 with biases of less than 1.0) need to be removed. However, lognormal distribution has been assumed to be followed by the resistance bias without removing the outliers because the lower tail region (where the resistance bias is less than 1.0) is a critical region for determination of the resistance factors as it is associated with the area of concern in which the loading may exceed the resistance. It should be noted that in such a case, the use of a lognormal distribution would result in a more conservative resistance factor eval- uation than otherwise. Other practices, such as âfittingâ the distribution to the tail (ignoring the bulk of the data), should be discouraged and are not perceived as mathemat- ically or otherwise justifiable. Further examination of the variation of bias with the mag- nitude of the soilâs friction angle is presented in Figure 112 for cases under inclined-centric loading (each error bar repre- sents 1 standard deviation). The best fit line suggests that the
The COV of the bias (COVλ) obtained for the data is used as a reference value; thus, a COVλ of 0.35 is adopted for controlled soil conditions (even though a maximum COVλ of 0.376 was obtained for Ïf = 45°), and a COVλ of 0.40 is adopted for natu- ral soil conditions. Table 63 presents the resistance factors for inclined-centric loading cases for Ïf ranging from 38° to 46° using Equation 124 to obtain the bias for each soil friction angle and COVλ values of 0.35 and 0.40, assumed based on the uncertainty evaluation. The minimum bias for the highest resistance factor obtained using the equivalent factor of safety relationship in Equa- tion 122 is 0.423 (0.60/1.4167). The minimum biases of the data are 0.37 and 0.57 (both with Ïf = 45 ± 0.5°), which means that the resistance factor needs to be reduced further. The required resistance factor for λ = 0.37 is approximately 0.52 (= 0.37 à 1.4167), which can be taken as 0.50. Hence, the resis- tance factors for both controlled soil conditions and natural soil conditions are rounded off to a much lower number than resistance factors obtained from the MCS. 4.9 In-Depth Re-Examination of the Uncertainty in Bearing Capacity of Footings in/on Granular Soils Under Inclined-Eccentric Loading 4.9.1 Extent of Database The number of reliable data points for the inclined-eccentric loading cases for which the positive and negative loading eccentricities could be clearly distinguished are 15 in total. Eight were tested under a positive loading eccentricity, and seven were tested under a negative loading eccentricity. The resistance factors obtained using the bias statistics for these cases have been used here for guidance only. 4.9.2 Inclined-Eccentric, Positive Loading Eccentricity Condition Table 64 summarizes the bias statistics for the eight footing cases under inclined-eccentric, positive (or reversible) loading eccentricity. The resistance factor obtained based on the bias statistics was 0.65, but as could be observed in all other cases of loading, the recommended resistance factor may be taken as low as 0.50. 4.9.3 Inclined-Eccentric, Negative Loading Eccentricity Condition Table 65 summarizes the bias statistics for the seven footing cases under inclined-eccentric, negative loading eccentricity. The preliminary resistance factor obtained based on the bias statistics was 1.00 for the available cases of soil friction angle, but 123 Table 63. Recommended resistance factors for inclined-centric loading cases. Resistance factor ( T = 3) Soil conditions Natural Controlled (COV = 0.40) (COV = 0.35) Soil friction angle f (deg) Mean bias (from Eq. 5) MCS Rec* MCS Rec 38 1.41 0.522 0.45 0.605 0.45 42 1.42 0.526 0.45 0.610 0.50 45 1.43 0.530 0.50 0.614 0.50 46 1.44 0.533 0.50 0.618 0.55 *Rec = recommended. Table 64. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled soil conditions for inclined-eccentric, positive (or reversible) loading eccentricity. Bias Resistance factor ( T = 3) Friction angle f ( 0.25 deg) n Mean COV MCS Preliminary 45.0 5 2.52 0.505 0.687 0.65 44.5 3 1.55 0.158 1.158 1.00 all angles 8 2.16 0.506 0.587 0.55 Table 65. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled soil conditions for inclined-eccentric, negative loading eccentricity. Bias Resistance factor ( T = 3) Friction angle f ( 0.25 deg) n Mean COV MCS Preliminary 45.0 4 3.78 0.640 2.043 1.00 44.5 3 2.96 0.187 0.703 0.70 all angles 7 3.43 0.523 0.887 0.85
as could be observed in all other cases of loading, the recom- mended resistance factor may be conservatively reduced to 0.80. 4.10 Summary of Recommended Resistance Factors for Footings in/on Granular Soils Tables 66 and 67 present the resistance factors recom- mended for use in the design of shallow foundations in/on granular soils (controlled soil conditions and natural soil con- ditions, respectively) with soil friction angles (Ïf) in the range of 30° to 45° and relative density (DR) ⥠35%. The resistance factors for controlled soil conditions are to be used when the foundations are placed in/on compacted engineering ï¬lls extending to a depth of no less than two (2.0) times the foun- dation width below the foundation base. The internal friction angle in such cases is to be determined by laboratory testing. Use of the resistance factors for natural soil conditions is rec- ommended when the foundations are placed on/in the in situ soil, and the soilâs internal friction angle is assumed to be eval- uated from correlations with Standard Penetration Testing. 4.11 Goodmanâs (1989) Semi-Empirical Bearing Capacity Method for Footings in/on Rock 4.11.1 Identification of Outliers The Ï-squared GOF tests have been carried out on the datasets containing all the cases and subsets: (1) cases with measured friction angle, (2) cases with measured rock discon- tinuity spacing sâ², and (3) cases with both friction angle and sâ² measured. Figure 113 presents the standard normal quantile of the unï¬ltered bias data for all cases with the theoretical normal and lognormal distributions based on the calculated mean and standard deviation. The Ï-squared values of the normal and lognormal distributions are found to be 121.28 and 18.79, respectively. The match observed in Figure 113 and the GOF test results indicate that the lognormal distribution is the matching underlying distribution for the data with an accep- tance level of the GOF test at 1% (for which the acceptable highest Ï-squared value is 21.67). These results also mean that no outliers need to be identiï¬ed for the dataset of all cases. Figures 114 and 115 present the standard normal quantiles of the unï¬ltered bias data for the cases with measured rock friction angle and measured rock discontinuity spacing, respec- tively, along with the relations predicted from the theoretical 124 Table 66. Recommended resistance factors for shallow foundations on granular soils placed under controlled conditions. Loading conditions Inclined-eccentric Soil friction angle f Vertical-centric or -eccentric Inclined-centric Positive Negative 0.50 0.60 0.40 0.40 0.70 0.70 0.45 0.45 0.75 0.75 0.50 0.80 0.55 0.50 0.80 30°â34° 35°â36° 37°â39° 40°â44° ⥠45° Notes: (1) Ïf determined by laboratory testing. (2) Compacted controlled fill or improved ground are assumed to extend below the base of the footing to a distance to at least two (2.0) times the width of the foundation (B). If the fill is less than 2B thick, but overlays a material equal or better in strength than the fill itself, then the recommendation stands. If not, then the strength of the weaker material within a distance of 2B below the footing prevails. (3) The resistance factors were evaluated for a target reliability (βT) = 3.0. Table 67. Recommended resistance factors for shallow foundations on natural deposited granular soil conditions. Loading conditions Inclined-eccentric Soil friction angle f Vertical-centric or -eccentric Inclined-centric Positive Negative 0.40 0.65 0.45 0.35 0.50 0.40 0.70 0.55 0.45 0.40 0.65 0.50 0.45 0.75 Notes: (1) Ïf determined from Standard Penetration Test results. (2) Granular material is assumed to extend below the base of the footing at least two (2.0) times the width of the foundation. (3) The resistance factors were evaluated for a target reliability (βT) = 3.0. 30°â34° 35°â36° 37°â39° 40°â44° ⥠45° -4 -2 0 2 4 St an da rd n or m al q ua nt ile 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Bias λ Bearing Capacity using Goodman (1989) All data Total data (n = 119) Normal distribution Lognormal distribution λ = 1.35 COVλ = 0.535 Figure 113. Comparison of the unfiltered bias for bearing capacity calculated using the Goodman (1989) method for all data and the theoretical normal and lognormal distributions.
normal and lognormal distributions. For the dataset of cases with measured friction angle presented in Figure 114, the Ï-squared value for the normal distribution is found to be 64.35 while that for the lognormal distribution is 15.60, which is accepted with a significance level of 5%. For the dataset of cases with measured rock discontinuity spacing presented in Figure 115, the Ï-squared value for the normal distribution is found to be 113.92 while that for the lognor- mal distribution is 11.99, which is also accepted with a sig- nificance level of 5%. Figure 116 examines the standard normal quantile for the resistance bias dataset of cases with both friction angle and dis- continuity spacing measured along with the predicted rela- tions for the theoretical normal and lognormal distributions. The Ï-squared value from the GOF tests obtained for the nor- mal distribution is 66.27 while that for the lognormal distri- bution is 11.77. Based on the data and analyses of Figures 113 to 116, it can be concluded that the bias associated with Goodmanâs (1989) analysis of shallow foundations on rock as an entire set and its subsets match the lognormal distribution, and no outliers exist for the examined datasets. 4.11.2 Calibration of Resistance Factors Table 68 shows the resistance factors (Ï) obtained from the MCS using one million samples for each dataset considered. As can be expected, the uncertainties in the estimated bearing resistance decrease with the increase in the available reliable information, thereby increasing the conï¬dence of the estimated resistances, and thus resulting in higher resistance factors. When all data are used, without differentiating between data for which the rock properties information is available from the ï¬eld and testing and data for which rock properties information is esti- mated by the outlined procedure, the recommended resistance factor is 0.30. The resistance factor can be increased to 0.45 when the relevant rock properties, i.e., rock friction angle and rock discontinuity spacing, are measured values. Figures 113 and 114 indicate that the assumed lognormal distribution overpredicts the bias in the lower tail regions of the 125 Bearing Capacity using Goodman (1989) All cases with measured friction angle Total data (n = 98) Normal distribution Lognormal distribution -4 -2 0 2 4 St an da rd n or m al q ua nt ile 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Bias λ λ = 1.41 COVλ = 0.541 Figure 114. Comparison of the unfiltered bias for bearing capacity calculated using the Goodman (1989) method for all data on rocks with measured friction angles and the theoretical normal and lognormal distributions. All cases with measured discontinuity spacing Total data (n = 83) Normal distribution Lognormal distribution -4 -2 0 2 4 St an da rd n or m al q ua nt ile 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Bias λ Bearing Capacity using Goodman (1989) λ = 1.43 COVλ = 0.461 Figure 115. Comparison of the unfiltered bias for bearing capacity calculated using the Goodman (1989) method for all data on rocks with measured discontinuity spacing sâ and the theoretical normal and lognormal distributions. All cases with measured discontinuity spacing and measured friction angle Total data (n = 67) Normal distribution Lognormal distribution -4 -2 0 2 4 St an da rd n or m al q ua nt ile 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Bias λ Bearing Capacity using Goodman (1989) λ = 1.51 COVλ = 0.459 Figure 116. Comparison of the unfiltered bias for bearing capacity calculated using the Goodman (1989) method for all data on rocks with measured discontinuity spacing and friction angle and the theoretical normal and lognormal distributions.
data for all cases as well as for the cases with measured Ïf, respectively. The minimum bias observed for both of these datasets is 0.19, and the second lowest is 0.29 (for both, the rock discontinuity spacing sâ² is based on AASHTO [2007]). A rough estimate of the equivalent factor of safety for a given calibrated resistance factor is given by Equation 122, while the equivalent minimum allowable bias for which the design will be safe for the given resistance factor is given by the reciprocal of the equivalent factor of safety (Equation 123). Thus, the minimum allowable biases for the recommended resistance factors are the following: (1) 0.21 for Ï = 0.30, (2) 0.25 for Ï = 0.35, (3) 0.28 for Ï = 0.40, and (4) 0.32 for Ï = 0.45, respectively. Except for the single case of the minimum bias of 0.19 (which can be taken as a marginal case), the results imply safe design when Ï = 0.30 is taken, i.e., all the data result in safe design on the application of the recommended resistance factors. 4.12 Carter and Kulhawyâs (1988) Semi-Empirical Bearing Capacity Method for Footings in/on Rock 4.12.1 Identification of Outliers The information and analyses presented in Section 3.8.2 suggest that the bearing resistance bias obtained using the Carter and Kulhawy (1988) method depends on the type of foundation, i.e., a rock socket drilled into rock or a shallow foundation in/on the rock. It is also observed that a systematic variation exists in the bearing resistance bias with the rock quality. When examining both factors, the data suggested (Sec- tion 3.8.2.2) that the bias variation attributed to the foundation type is actually controlled by the bias relation to the rock qual- ity within the independent databases for each of the founda- tion types. As such, GOF tests have been carried out on the datasets categorized according to the rock RMR and the resis- tance factors developed for each of these subgroups. Comparisons of the standard normal quantiles of the data- sets for (1) the total cases in/on rocks, (2) the cases in/on rocks with RMR ⥠85, and (3) the cases in/on rocks with 65 ⤠RMR < 85 are presented in Figures 117, 118 and 119, respectively. Except in the case of Figure 119, it can be observed that the lognormal distribution ï¬ts the data better than the normal distribution. The Ï-squared GOF tests have been carried out for all the data subsets, classiï¬ed according to rock RMR values, to check the suitability of the assumption that the datasets can be mod- eled by lognormal distributions. The Ï-squared values obtained for the normal distribution (N) and the lognormal distribution (LN), respectively are the following: (1) 481.64 for N and 16.22 for LN for the total cases for rocks (n = 119); (2) 15.87 for N and 15.61 for LN for RMR ⥠85 (n = 23); (3) 18.97 for N and 31.82 for LN for 65 ⤠RMR < 85 (n = 57); (4) 11.58 for N and 9.12 for LN for 44 ⤠RMR < 65 (n = 17); and (5) 13.34 for N and 10.43 for LN for 3 ⤠RMR < 44 (n = 22). The Ï-squared values at the 1% and 5% signiï¬cance levels are 21.66 and 16.92, respectively; hence, the GOF tests show that a majority of the data subsets follow lognormal distributions and that no outliers need to be identiï¬ed. 4.12.2 Calibration of Resistance Factors Based on the datasets, for a majority of which the GOF tests show that lognormal distributions can be assumed to model the bias distribution, the resistance factors have been calibrated using MCS using one million samples. These factors are pre- sented in Table 69. If no RMR information is available, the rec- 126 Table 68. Calibrated resistance factors for different datasets of resistance bias obtained using Goodmanâs (1989) method. Bias Resistance factor ( T = 3)Dataset No. of cases Mean COV MCS Recommended All data 119 1.35 0.535 0.336 0.30 Measured friction angle, f 98 1.41 0.541 0.346 0.35 Measured spacing, s 83 1.43 0.461 0.437 0.40 Measured friction angle, f, and 67 1.51 0.459 0.464 0.45 s Total data (n = 119) Normal distribution Lognormal distribution -4 -2 0 2 4 St an da rd n or m al q ua nt ile 0 5 10 15 20 25 30 35 45 5540 50 Bias λ Bearing Capacity of all cases in rocks using Carter and Kulhawy (1988) λ = 3.42 COVλ = 0.554 Figure 117. Comparison of the unfiltered bias for bearing capacity calculated using the Carter and Kulhawy (1988) method for total cases in/on rocks in the database and the theoretical normal and lognormal distributions.
ommended Ï is 0.35. When the rock has RMR ⥠85 the recom- mended Ï is 0.50. For rocks with RMR lower than 85, Ï = 1.00. 4.13 Summary of Recommended Resistance Factors for Shallow Foundations in/on Rock Table 70 summarizes (based on the information presented in Tables 68 and 69) the recommended resistance factors to be used in evaluation of the bearing capacity of shallow founda- tions on rock. The resistance factors for both examined meth- ods are presented along with the efï¬ciency factors providing a measure for the relative efï¬ciency of the methods. Goodmanâs (1989) method performed exceptionally well consistently, regardless of rock quality. Improvement in the methodâs performance with an increase in knowledge trans- lates into an increase in the resistance factor and the associated method efï¬ciency. The performance of the Carter and Kulhawy (1988) method has a âbuilt-inâ safety that increases as the rock quality decreases. As such, the methodâs bias changes with the rock quality (expressed via RMR), and a calibration was required following the rock quality designation. The relatively higher resistance factors are a byproduct of the large bias of the method and, hence, do not represent efï¬cient design as expressed by the low efï¬ciency factor of the methodâs application compared to Goodmanâs (1989) method. 4.14 Sliding Friction Resistance 4.14.1 Parametric Study Evaluating the Resistance Factor as a Function of the Ratio of Dead to Live Load The probabilistic characteristics of the parameter contribut- ing directly to the sliding friction resistance, the friction coefï¬- cient ratio ( fc), have been presented in Section 3.9 and summarized in Table 48. The uncertainties in the friction coef- ï¬cient ratio ( fc) follow one-to-one transformation to the slid- ing resistance, i.e., the mean of sliding resistance = vertical load à (mean of fc à tan Ïf) and the standard deviation (s.d.) of sliding resistance = vertical load à (s.d. of fc à tan Ïf). Hence, 127 -3 -2 -1 0 1 2 3 St an da rd n or m al q ua nt ile 0 1 2 3 4 5 6 7 8 Bias λ RMR ⥠85 Total data (n = 23) Normal distribution Lognormal distribution λ = 2.93 COVλ = 0.651 Bearing Capacity of all cases in rocks using Carter and Kulhawy (1988) Figure 118. Comparison of the unfiltered bias for bearing capacity calculated using the Carter and Kulhawy (1988) method for all cases in rocks with RMR >â 85 and the theoretical normal and lognormal distributions. 0 1 2 3 4 5 6 7 98 Bias λ -4 -2 0 2 4 St an da rd n or m al q ua nt ile Bearing Capacity of all cases in rocks using Carter and Kulhawy (1988) 65 ⤠RMR < 85 Total data (n = 57) Normal distribution Lognormal distribution λ = 3.78 COVλ = 0.463 Figure 119. Comparison of the unfiltered bias for bearing capacity calculated using the Carter and Kulhawy (1988) method for all cases in rocks with 65 <â RMR < 85 and the theoretical normal and lognormal distributions. Table 69. Calibrated resistance factors for different datasets of resistance bias obtained using Carter and Kulhawyâs (1988) method. Bias Resistance factor ( T = 3) Dataset No. of cases Mean COV MCS Recommended All cases 119 8.00 1.240 0.372 0.35 RMR 85 23 2.93 0.651 0.535 0.50 65 RMR < 85 57 3.78 0.463 1.149 1.00 44 RMR < 65 17 8.83 0.651 1.612 1.00 3 RMR < 44 22 23.62 0.574 5.295 1.00
128 Table 70. Recommended resistance factors for foundations in/on rock based on T = 3.0 (pf = 0.135%). Method of analysis Equation Application Efficiency factor / (% ) All 0.35 4.4 RMR 85 0.50 17.1 65 RMR < 85 26.5 44 RMR < 65 11.3 Carter and Kulhawy (1988) ul t u q q m s 3 RMR < 44 1.00 4.2 All 0.30 22.2 Measured f 0.35 24.8 Measured s 0.40 28.0 Good ma n (1989) For fractured rocks: 1 ult u q q N For non-fractured rocks: ( 1 ) 1 1 1 N N ult u s q q N N B Measured s and f 0.45 29.8 Table 71. Resistance factors obtained from MCS simulations for footings, either cast in place or prefabricated, in soils with various friction angles, along with the effect of ratios of lateral dead load to lateral live load. (a) Cast-in-place footings Resistance factor from MCS ( MCS) At-rest earth pressure Active earth pressure fobtained from LFD/LFL = 2 LFD/LFL = 4 LFD/LFL = 5 LFD/LFL = 7 LFD/LFL = 2 LFD/LFL = 4 LFD/LFL = 5 LFD/LFL = 7 SPT 0.469 0.455 0.452 0.447 0.507 0.498 0.496 0.492 CPT 0.516 0.499 0.494 0.488 0.558 0.545 0.542 0.537 Lab test 0.558 0.535 0.530 0.523 0.603 0.585 0.581 0.576 (b) Prefabricated footings Resistance factor from MCS ( MCS) At-rest earth pressure Active earth pressure fobtained from LFD/LFL = 2 LFD/LFL = 4 LFD/LFL = 5 LFD/LFL = 7 LFD/LFL = 2 LFD/LFL = 4 LFD/LFL = 5 LFD/LFL = 7 SPT 0.195 0.193 0.193 0.191 0.211 0.212 0.211 0.211 CPT 0.217 0.213 0.212 0.210 0.234 0.233 0.232 0.232 Lab test 0.239 0.234 0.232 0.230 0.258 0.256 0.255 0.253 Table 72. Recommended resistance factors for sliding resistance () for soil friction angles based on different tests and lateral pressure due to at-rest or active earth pressure for cast-in-place and prefabricated footings. Resistance factor for sliding friction ( ) ( T = 3) At-rest earth pressure Active earth pressure fobtained from Cast in- place 1 Prefabricated 2 Cast in- place 1 Prefabricated 2 SPT 0.40 0.45 CPT 0.45 0.50 Lab test3 0.50 0.20 0.55 0.20 1 tan s = 0.91 tan f ; 2 tan s = 0.53 tan f , 3 Any laboratory shear strength measurement of f
the form of the limit state function for sliding resistance is essentially the same as that for the bearing resistance (see Equa- tion 118), which can be expressed as where ZÏ is the load combination for sliding, RÏ is sliding resis- tance of a footing, LFD is lateral load due to dead load, and LFL is lateral load due to live load. A summary of the uncertainties in the lateral loads and the load factors as recommended in AASHTO (2007) are presented in Section 4.2.3.3. Analogous to the calibration of resistance factors for the bearing resistance, the inï¬uence of the ratio of lateral dead load to the lateral live load has been studied and presented here. Z RÏ Ï= â âLFD LFL ( )125 Based on the loadings for the design example bridges consid- ered in the current research study, it is found that the ratios of LFD to LFL range from 4 to 7. As a result, the resistance factors for sliding resistance have been calibrated for LFD to LFL ratios varying from 2 to 7 and the corresponding results are presented in Table 71 for cast-in-place and prefabricated footings. 4.14.2 Resistance Factors The calculated resistance factors presented in Table 71 suggest that the ratio of LFD to LFL does not have a pronounced effect on the magnitude of the resistance factors. As a result, selected resistance factors are recommended for use for sliding resistance of footings on granular materials as presented in Table 72. 129