Reliability Analysis for the ULS of Shallow Foundations Advanced Foundation Engineering

Transcription

1 Geotechnical Engineering Research Laboratory Dept. of Civil and Environmental Engineering University of Massachusetts Lowell. Reliability Analysis for the ULS of Shallow Foundations The lecture is based on NCHRP Report 651 LRFD DESIGN AND CONSTRUCTION OF SHALLOW FOUNDATIONS FOR HIGHWAY STRUCTURES Samuel G. Paikowsky, Mary C. Canniff, Kerstin Lesny, Aloys Kisse, Shailendra Amatya, and Robert Muganga GeoDynamica Inc. Newton, MA USA Geotechnical Engineering Consultants

2 OUTLINE Background 1. Objectives 2. Method of Approach Databases Database Summary Database Flow Chart Design & Construction Practices Questionnaire BC of Shallow Foundations on Soil 1. Determination of ULS from Case Histories 2. Failure (Ultimate Load) Criteria 3. Uncertainty Evaluation BC of Centric Vertically Loaded Footing on Granular Soils BC of eccentric Vertically Loaded Footing on Granular Soils BC of inclined Loaded Footing on Granular Soils 4. Calibration of Resistance Factors 5. Example 6. Summary and Conclusions BC of Shallow Foundations on Rock 1. Broad Objectives 2. Database UML/GTR RockFound07 3. Rock Classification and Properties 4. Methods of Analyses Selected for Establishing the Uncertainty in B.C. of Foundations on Rock 5. Calibration of resistance factors 6. Summary and Conclusions General Conclusions and Recommendations Summary 2

3 OBJECTIVES NCHRP RESEARCH PROJECT Develop and Calibrate Procedures and Modify AASHTO s Section 10 (Foundations) Specifications for the Strength Limit State Design of Bridge Shallow Foundations. For NCHRP Research Report 651, Google NCHRP 651 3

4 Method of Approach Review Design Cases used in NCHRP Design Cases in Manuals FHWA GEC No. 6, Kimmerling, 2002 FHWA Soils & Foundation Workshop Manual, Cheney and Chassie, 1982 FHWA RD-86/185, Gifford et al., 1987 Examination of Lateral Loads Data on Structures Examination of Load Ranges and statistics of Horizontal and Vertical Loading for the Typical Design Examples and Case Histories Existing AASHTO Specifications and FHWA Manuals AASHTO (2006) FHWA reference manual, Munfakh et al., 2001 FHWA GEC No. 6, Kimmerling, 2002 FHWA Spread Footings of Highway Bridges, Gifford et al., 1987 FHWA Soils & Foundations Workshop Manual, Cheney & Chassie, 1982 NCHRP Questionnaire Determination of DOT Design Methods and Construction Practices of Shallow Foundations Determination of alternative Design Methods Established: AASHTO/FHWA and DOT s Design Methods Complementary and/or Alternative Design Methods Typical Structures under Common Construction Practices Design Cases Load Ranges and their Distributions Available Questionnaires of Foundations Design Methods and Construction Practices NCHRP 24-17, Paikowsky et al NCHRP 12-66, Paikowsky et al., 2005 Univ. of Duisburg-Essen Inst. of S.M. and Found. Eng. Gifu Univ., Japan Japan Geotech. Soc. ISSMFE Figure 41. Flowchart outlining the research plan for Unit I(a) establishing design methods, construction practices, design cases, and loads. 4

5 Method of Approach Literature Identifying Additional Shallow Foundation Load Tests 31 Data Cases Collected in Cornell (Prakoso, 2002) 39 Data Cases Collected in MIT (Zhang and Einstein, 1998) Inst. of S.M & Foundation Eng. UDE Germany Load Testing Program Existing UML/GTR Shallow Foundation Database 329 Load Test Cases Database III Loading of Shallow Foundations on Rock Database II Vertical Inclined & Eccentric Loading of Shallow Foundations on Granular Soils Database I Vertical Centric Loading of Shallow Foundations on Granular Soils Data Solicitation from DOT s across the USA Figure 42. Flowchart outlining the research plan for Unit I(b) establishing databases for shallow foundations load tests. 5

6 Design & Construction Practices Questionnaire Developed and distributed to 161 State Highway Officials, TRB Representatives, and State and FHWA bridge engineers. Obtained responses from 39 states and 1 Canadian Province Previous relevant information was obtained via a questionnaire circulated in 2004 for the research project NCHRP AASHTO LRFD Specifications for the Serviceability in the Design of Bridge Foundation 6

7 Design & Construction Practices - Questionnaire Foundation Alternatives Results on distribution of bridge foundation usage from our previous questionnaires conducted in 1999 and 2004, and the current questionnaire (over the past 3 years, ): shallow foundations driven piles drilled foundations 1999/ %/17% 75%/62% 11%/21% current 17% 59% 24% The use of shallow foundations was not changed overall relative to the last survey (2004). There is a consistent trend, however, in the decrease of the use of driven piles (75%, 62%, and 59% for 1999, 2004, and 2007, respectively) and increase of the use of drilled foundations (11%, 21%, and 24% for 1999, 2004, and 2007, respectively). There is some discrepancy between the total foundation use and the percentage of use specifically addressing piers and abutments. Some of this discrepancy can be attributed to the fact that all foundations include non-bridge structures like buildings, posts and sound barriers. The average use presented above, changes significantly across the country. The presented number that relates to bridge foundations only (with average use of 17.7% for abutments and piers). The use of shallow foundations in the Northeast exceeds by far all other regions of the USA, ranging from 40% in NY, NJ and ME, to 67% in CT. Other heavy users are TN (63%), WA (30%), NV (25%) and ID (20%). In contrast, out of the 39 responding states, six states do not use shallow foundations for bridges at all, and additional eight states use shallow foundations in 5% or less of the highway bridge foundations. 7

8 Design & Construction Practices - Questionnaire Subsurface Conditions for Shallow Foundations Out of all constructed PIERS, 17% were supported by shallow foundations Rock56.3% IGM16.3% Frictional Soil23.9% Cohesive Soil3.4% (cemented soils/ (sand/gravel) (clay/silt) weathered rock) Cohesive Soil breakdown (%): Alabama-3, Arizona-10, Georgia-5, Idaho-10, Illinois-2, Indiana- 20, Michigan-50, Massachusetts-4, Nevada-5, Washington-10 Of those built on cohesive soils, 68% were built without ground improvement measures (geosynthetic, wick drains, etc.) Out of all constructed ABUTMENTS, 19% were supported solely by shallow foundations: Rock55.3% IGM17.3% Frictional Soil 24.4% Cohesive Soil 3.0% (cemented soils/ sand/gravel clay/silt weathered rock) Cohesive Soil breakdown (%): Arizona-5, Georgia-5, Idaho-10, Illinois-10, Michigan-25, Massachusetts-2, Nevada-10, Oregon-1, Vermont-10, Washington-10, CA (Alberta)-10 Of those abutments built on cohesive soils, 50 % were built without ground improvement measures (geosynthetic, wick drains, etc.) Georgia-100, Idaho-100, Michigan-100, Massachusetts-80, Nevada-90, Vermont-50, Washington-5, CA (Alberta)-25 28% have integral bridge abutments supported on shallow foundations (about 25% of all integral abutments), while 68% do not use shallow foundations at all. 8

9 Design & Construction Practices - Questionnaire Subsurface Conditions for Shallow Foundations In summary, 55.8% of the shallow foundations are built on rock (average of piers and abutments) with additional 16.8% on IGM, hence 72.6% of the foundations are build on rock or cemented soils and only 27.4% are built on soils of which 24.2% on granular soils and 3.2% on clay or silt. A further breakdown is presented in Table 1 of Appendix A in the Interim Report. For example, Michigan indicated that 50% of its shallow foundations at the piers location are built on fine grained soils, however, Michigan is using only 5% of its pier foundations on shallow foundations; hence, only 2.5% of the pier foundations are built on clay or silt. Examining all the states this way suggests that the leading state to build bridge foundations on clay is WA (6%) followed by VT (5%), ID (4%), and MI and NV (3.75%) each. Further examination of these facts (in a telephone interview) revealed that WA s use of foundations on silt and clay refers to highly glacial densified soils with SPT N values exceeding 30 for silts and between 40 to 100 for the clays. 9

10 Design & Construction Practices - Questionnaire Subsurface Conditions for Shallow Foundations Twenty-eight states (out of 39) do not build shallow foundations for bridges on cohesive soils at all; hence only 0.8% of all bridge shallow foundations are built on clay or silt including WA, in comparison to 16.9% on rock, 5.4% on IGM and 12.2% on frictional soils. The survey also suggests that only about 60% of the foundations on clay were built without ground improvement measures, hence only about 0.48% of the bridges were actually built on shallow foundations on cohesive soils, practically a marginal number considering the state of these soils as described by WA DOT. Note these numbers do not include the construction of embankments and the B.C. evaluation of embankments and do not consider the issue of -c materials. 10

11 Design & Construction Practices - Questionnaire Foundations on Rock - Implementation About 90% of the states obtain rock cores, evaluate RQD and conduct uniaxial (unconfined) compressive strength tests. About 19% of the states use presumptive values alone, 22% use engineering analyses alone and 59% use both when evaluating B.C. 53% use AASHTO s presumptive values. Other states use or consult the Canadian Foundation Engineering Manual, NY Building Code, NAVFAC, or based their capacity values on local experience (SD, WI, OR, KS, IA, AK). 70% of the responding states would like to see a specific analytical method presented for the evaluation of B.C. of foundations on rock. 25% use Kulhawy and Goodman (1987) analytical method and 33% use Carter and Kulhawy semiempirical design method. Others use: Kulhawy and Goodman (1980) Hoek-Brown, Hoek and Marinos. Two states commented about using GSI (Geotechnical Strength Index) instead of RMR (Rock Mass Rating). 60% evaluate failure by sliding for footings on rock. Seven states do not evaluate sliding because of a requirement to wedge the foundation into the rock 11

12 Design & Construction Practices - Questionnaire Foundations on Rock - Implementation 70% of the states do not analyze lateral displacement as they use limiting measures (key way, dowling, etc.) as described above. NY specifies geologic inspection during construction to ensure rock quality and key way or dowelling is ordered if necessary. 75% of the responding states limit the eccentricity of footings on rock. Most of the states follow AASHTO recommendations for e/b ⅜, some use e/b ¼ based on the FHWA Soils and Foundations Manual that also meets the AASHTO standards specification. WY, SD, and Alberta use e/b 1 6 with Alberta specifying that either eccentricity is maintained within limits or an effective foundation size is used in which the dimensions are reduced by twice the eccentricity (e.g. B = B 2e). 70% of the states do not analyze settlement of footings on rock as it is not being seen as an issue of importance and the settlement is limited to 0.5in. 28% use AASHTO procedures for broken/jointed rock with NV also using Kulhawy (1987) and the Army EM Note questionnaire did not address differences between competent/hard rock and soft rock/igms 12

13 Design & Construction Practices - Questionnaire Foundations on Soil - Implementation All states follow either AASHTO s LRFD or ASD guidelines, only a small number of responders use presumptive values. 58% use the theoretical general B.C. equation. 53% of the responders find it reasonable to omit the load inclination factors and 63% limit the eccentricity of the footing mostly with e/b 1 6 to ¼ (standard specifications e/b = 1 6, LRFD specifications e/b = ¼). MA responded that load inclination factors must be used in the final design of the footing. PA commented that when inclination factors were considered together with factored loads, it resulted in an increased footing size; hence, unfactored loads are used. 45% do not decrease the soil s strength parameters considering punching shear, while 23% do so. Seven states commented that punching shear is not a viable option as foundations are not built in loose soil conditions or alternatively settlement criteria prevails especially under such conditions. 58% use the AASHTO procedures presented for footings on a slope. NV, ID and MI commented that the charts are not clear and needs to be improved. WA, and NC commented on the use of Meyerhoff s method, also presented by the Navy Design Manual (NAVFAC), essentially identical to the AASHTO presentation. OR commented that the provided foundations on slope analysis results with a reasonable approach (somehow conservative) while PA commented that experience shows that sometimes this analysis results with a drastically larger footing. 13

14 Design & Construction Practices - Questionnaire Foundations on Soil - Implementation 30% of the responding states do not use the AASHTO procedures for footings on a layered soil, while 38% of the responders do use these procedures. Eighteen states commented about the procedures; ID, MI, VT and WI commented that they calculate the B.C. for the layer with the lower strength. IA and OR commented that under such conditions alternative foundation solutions are examined. Only 28% (with 40% responding with No) of the responders use the semi-empirical procedures described in section for evaluation of bearing capacity. The majority of the states that commented about the procedure expressed the opinion that the method is used for a rough evaluation only as an initial estimation and/or in comparison to other methods. Oregon commented that the SPT method usually yields higher capacity and settlement controls the design. Nineteen states responded when asked for comments about the currently existing resistance factors being all about the same value. Some states stated that they have not enough experience with LRFD to judge the resistance factors values. NC and NH suggested combining all resistance factors to be 0.45, while OR, PA, VT, and WA commented that the resistance factors are in line with the factor of safety range (2.5 to 3.0) used in the ASD methodology and hence result with similar design as that obtained using ASD. 70% evaluate failure by sliding with about half (33%) use the full foundation area and 30% use the effective foundation area. 14

15 Design & Construction Practices - Questionnaire Foundations on Soil - Implementation Only 13% consider passive resistance for the lateral resistance of the shallow foundations and all utilize a limited value due to a limited displacement. Many responding states expressed concern with a long term reliance on a passive resistance. WA commented that it is rarely used to meet sliding criterion of extreme events and MN commented it is used in front of shear keys only. Traditionally no safety margin is provided to settlement analysis though it typically controls the size of shallow foundations. When asked about it, 35% answered the issue should not be of concern and 25% answered it should. From those responded, some recognized that it needs to be researched (CT, MI, TN) while others hold the notion that a safety margin on B.C. already addresses the issue (HI, ME, NJ, NC, WA) or that settlement calculations are conservative to begin with (NH, NC). Only two states stated that they conduct plate load tests, one of which (CT) referred to tests from over 20 years ago and the other to three recent tests (MA). When asked to comment on any related subject, 13 states responded. A major concern expressed by MI was written by a bridge designer referring to the difficulties in using effective width for bearing capacity calculations as it requires iterations for each load case for service and strength. More so, the division of responsibilities between the geotechnical section (providing allowable pressure) and structural section (examining iteratively final design) is a source for problems. The engineer proposes to have allowable contact stresses for service and strength based on gross footing width and eccentricity limited to B/6. (The issue of allowable to ULS is not so clear and the engineer was contacted). 15

16 DATABASES UML-GTR ShalFound07 Database 499 cases built in ACCESS platform, currently being updated to 549 cases. Out of it, 415 cases are suitable for ULS. UML-GTR RockFound07 Capacity Database 122 Cases of load tests to failure including 61 rock sockets, 33 shallow foundations on rock surface, 28 shallow foundations below surface 16

17 Assembled Databases UML-GTR ShalFound07 (predominantly granular soils) 499 test cases UML-GTR RockFound07 (rocks) 122 test cases Database I centric vertical loading Database II eccentric and inclined loadings and their combinations Database III centric vertical loading 17

18 BC Shallow Foundations on Soil - OUTLINE 1. BC of Shallow foundations BC Factors BC modification Factors 2. Determination of ULS from case histories ULS and Modes of Failure Overview Modes of Failure 3. Failure (Ultimate Load) Criteria Minimum slope criteria (Vesić, 1963) Limited settlement criterion of 0.1B (Vesić, 1975) Log-log plot of load-settlement curve (DeBeer, 1967) Two-slope criterion Selection of failure criteria (representative values and minimum slope) Examples in soil & rock 4. Uncertainty Evaluation BC of Centric Vertically Loaded Footing on Granular Soils Database overview Calculated BC missing soil parameters and equations used for BC calculations 5. Calibration 6. Summary and Conclusions 18

19 Bearing Capacity of Shallow Foundation General Formulation Centric vertical loading of a rigid footing Buismann (1940) and Terzaghi (1943) adopted solution for metal punching proposed by Prandtl (1920, 1921) and proposed the Ultimate Bearing Capacity q u c N c q N q 1 2 B N Q 0 L = q A I B III E C II D 19

20 Bearing Capacity Factors N c and N q These factors have exact solutions and were given by Prandtl (1920) and Reissner (1924) for weightless soils proposal for N c is credited to Caquot and Kerisel (1953) N 1cot f c N q N q exp 2 f tan tan f

21 Bearing Capacity Factor N No closed form solution present and proposals from different authors exist Formulas based on Empirical Relations Meyerhof (1963): N 1 tan1. N q 4 N N q Brinch Hansen (1970): f f tan Muhs (1971) and Eurocode 7 (2005): N 2 N q 1 tan f Ingra and Baecher (1983) for square footings: exp Formulas based on Analytical Derivation Vesic (1973): N N f N q 1 tan f 2 For a complete list, refer to NCHRP Report

22 Different Proposed NFactors 1000 Bearing capacity factor, N Vesic (1973) Meyerhof (1963) Brinch Hansen (1970) Chen (1975) Ingra & Baecher (1983) EC7 (2005) Michalowsky (1997) Bolton & Lau (1993) Hansen (1968) Zhu et al (2001) Gudehus (1981) Steenfelt (1977) Friction angle, f (deg) B.C. factor N vs. based on empirical, analytical and numerical derivations 22

23 General Bearing Capacity Equation Based on modifications by Meyerhoff (1953, 1963), Brinch Hansen (1961, 1970) and Vesić (1973, 1975): q cn s d i qn s d i u c c c c q q q q 1 B' N s d i 2 where s, d, and i are modification factors for footing shape, footing embedment depth and load inclination, respectively, effective width B = B - 2e e = load eccentricity 23

24 Eccentricity The effect of eccentric loading on the bearing capacity is usually accounted for via Meyerhof s (1953) effective area consideration. The bearing capacity is calculated for the footings effective dimensions given by: L' B' L 2e L B 2e B with e B = M L /V and e L = M B /V (35) where M, M B and M L = the moments loading in L and B directions, respectively V = the total vertical load e L and e B = load eccentricities along footing length L and footing width B, respectively. qu e Meyerhof (1953): 1 2 (36) q B u, centric 2 q u e exp 12 Giraudet (1965): q (37) u, centric B qu e 1.9 Ticof (1977): 1 q B (38) u, centric

25 Eccentricity qu e Bowles (1996): 1 e for (39) q B B u, centric qu e e Paolucci and Pecker (1997): 1 for 0.3 (40) q 0. B B u, centric Ingra and Baecher (1983): qu e e (41) q B B u, centric qu e Gottardi and Butterfield (1993): 1 q 0.36B (42) u, centric 2 qu e Perau (1995, 1997): q B (43) u, centric 25

26 Eccentricity: Reduction factors for Foundations Under Vertical-Eccentric Loading Test were carried on footings with different length to width ratios Meyerhof (1953) Ticof (1977) Ingra & Baecher (1983) Perau (1995) Experimental results (n=61) It can be seen that the Meyerhof s proposal is closest to the lower boundary of the test results. q u / q u,centric Load eccentricity to footing width ratio, e/b 26

27 Shape Factors Table 4 Shape factors proposed by different authors Reference De Beer (1961) as modified by Vesic (1973) EC 7 (2005) and DIN 4017 (2006) Footing base shape Rectangle Circle and Square Rectangle Circle and Square s c s q s B Nq B 1 1 tan f B L N L L 1 N N c c q 1 tan f sq Nq 1 N 1 q sq Nq 1 N 1 q B 1 sin f L 1 sin f B L 0.7 Meyerhof (1963) Rectangle B K p L 1 ; for K ( B/ L); for 10 f f p K p B 10.1 K p; L 2 f tan 45 2 Perau (1995, 1997) Rectangle 11.6tan B/ L B 1 L f B L Zhu and Michalowski (2005) Rectangle 2 1 (0.6sin f 0.25) B / for 30 ; 1 (1.3sin 0.5)( L/ B) f exp( L/ B) f for 30 f L 27

28 Shape Factor: s The value of s is within the range of 1±0.05 for L/B

29 Shape Factor: s q The value of s q is within the range of 1±0.05 for L/B

30 Shape Factor: s c For soil with f =20 and c=5kpa (0.1ksf) The value of s c is within the range of 1±0.05 for L/B 10.0 side ratio factor s c [-] EC 7 (2005) de Beer (1965) Meyerhof (1963) B/L [-] 30

31 Depth Factors Table 5 Depth factors proposed by different authors Author d c d q d Meyerhof (1963) d c 10.2 K p Df B d D f 10.1 K for 10 B 1 for 0 q p f f d d q Brinch Hansen (1970) and Vesic (1973) d c d d q q N 1 d C 1 d N q tan q 1 q f D f / B 1: 2 d 12tan 1sin D / B D q f f f f / B 1: 2 d 12tan 1sin arctan D / B q f f f 1 where K p 2 tan (45 f / 2) 31

32 Depth Factor: d q 32

33 Load Inclination Factors An inclination in the applied load always results in a reduced bearing capacity, often of a considerable magnitude (Brinch Hansen, 1970). Meyerhof (1953) suggested that the vertical component of the bearing capacity under a load inclined at an angle to the vertical, is obtained using the following inclination factors. i i 1 /90 (44) c q 2 2 i (1 / f ) (45) These expressions were modified by Meyerhof and Koumoto (1987), and presented for the cases of footings on the surface of sand, when embedment ratio (D f / B) is unity, and for footings on the clay surface as given below. Assuming that a footing with a perfectly rough base on the sand surface starts to slide when the load inclination angle to the vertical is approximately equal to the soil s friction angle, the following expression was proposed: sin i cos 1 for Df / B 0, c 0 sin f (46) 33

34 Load Inclination Factors For a particular case of footings with embedment ratio equal to 1 in a soil with friction angle greater than 30, the inclination factor was expressed as: For footings on the surface of clay: i cos 1sin for c 0 c i cos 1sin for 30, D / B 1, c0 a cos 10.81sin for c c undrained shear strength of the clay a n f f (47) (48) where c a = adhesion between the clay and the base of the footing Muhs and Weiss (1969) suggested, based on DEGEBO (Deutsche Forschungsgesellschaft für Bodenmechanik) tests with large scale models of shallow footings on sands, that there is a distinct difference in the load inclination effects when the inclination is in the direction of the longer side L and when in the direction of the shorter side B. Thus, the direction of load inclination as well as the ratio B/L affect on the inclination factor. Brinch Hansen (1970) incorporated the inclination effects as: i q 1 0.5H 5 (49) 0.7H (50) i 1 V A' ccot f V A' ccotf 5 34

35 Load Inclination Factors Vesic (1975) proposed the factors in the following forms: i q 1 H n 1 (51) (52) V A' ccotf V A' ccotf n 2 LB / 2 BL / cos sin 1 LB / 1 BL / 2 2 i H n1 (53) where H and V are the horizontal and vertical components of the applied inclined load P (Figure 17), is the projected direction of the load in the plane of the footing, measured from the side of length L in degrees; L and B as defined in Equation 35, A is the effective area of the footing, and c is soil cohesion. P V Figure 17 Inclined load without eccentricity, and the projected direction H L B 35

36 Load Inclination Factor: i q Plots valid for horizontal component of load normal to the footing length c=0, f =35, and D f /B=0 36

37 Load Inclination Factor: i Plots valid for horizontal component of load normal to the footing length c=0, f =35, and D f /B=0 37

38 Load Inclination Factors The inclination factor i c results from Caquot s theorem of corresponding stress states (De Beer and Ladanyi 1961 and Vesić 1970 as cited by Vesić 1975) are: 1 iq 1 iq ic iq iq N tan N 1 for f 0 (54a) where i q is given by Equation 51. nh for f 0 1 A cn Reduction coefficients for the case of a load inclination related to the case of a centrically and vertically loaded footings can be found in the references of Figure 20. These expressions were determined based on model foundation test results on sand without embedment; and as such, are valid for the case of D f =0, c =0. Ticof (1977) qu H (55) q V u, centric c f q Ingra and Baecher (1983) qu H H (56) q V V u, centric Gottardi and Butterfield (1993) qu H 1 (57) q 0.48V 2 u, centric i c c 2 (54b) 38

39 Reduction Factor for Load Inclination Effects of load inclination on ultimate bearing capacity Based on model tests on sands: D f =0, c=0 inclined vs. vertical-centric 39

40 Load Inclination Factor: i c Plotted for square footings, with base area of 1m 2 (10.75ft 2 ) on soil with f =20 and c=5kpa (0.1ksf) (V=Accot f ) Load inclination factor, i c Vesic (1975) Meyerhof (1963) Load ratio, H/V 40

41 Determination of ULS from Case Histories Ultimate Limit State (ULS) consists of: Exceeding load carrying capacity of the ground supporting the foundation Sliding, uplift and/or overturning Three principle modes of shear failure under foundation: General shear failure Local shear failure Punching shear failure 41

42 General Shear Failure Characterized by well-defined failure pattern of a continuous slip surface Load-displacement curve shows a prominent peak (Vesic, 1975) 42

43 General Shear Failure Load test of a 3inch footing under centric vertical loading Slip surfaces developed under and on the sides of the footing developed after general shear failure (Selig and McKee, 1961) One sided rupture failure surface from a vertical, eccentric loading (Jumkins, 1956) 43

44 Local Shear Failure Characterized by failure pattern clearly visible only immediately below footing Load-displacement curve does not show a clear peak (Vesic, 1975) 44

45 Punching Shear Failure Failure pattern is not easy to observe Compression of the soil immediately below footing occurs; no movement of soils on the sides Jerks and sudden movements in the vertical dir. (Vesic, 1975) 45

46 Modes of Failure and Relative Density Generally speaking, a general shear failure takes place if the soil is incompressible and punching shear failure if compressible Failure mode also depends on embedment ratio and loading type (Vesic, 1963 modified by De Beer, 1970) Failure Load criteria for Local Shear and Punching Shear failures 46

47 Failure (Ultimate Load) Criteria Minimum Slope Failure Load Criterion (Vesic, 1963) Limited Settlement Criterion of 0.1B (Vesic, 1975) Log-log Plot of Load-settlement Curve (De Beer, 1967) Two-slope Criterion Recommended Criterion Failure Interpretation Examples in Soils and Rocks 47

48 Minimum Slope Failure Criteria, Vesic (1963) Ultimate Load is the point where the slope of the curve first reaches a steady, minimum value or zero 48

49 10% Width Settlement Criterion, Vesic (1975) Cases in which minimum slope on the curve cannot be established with certainty Conservative estimates and may be problematic for larger foundations 49

50 Log-log plot of Load-Settlement, De Beer (1967) Ultimate Load defined as the change in load-settlement curve as the point of break of the curve (Circled Dots) Found to be very conservative compared to Minimum Slope 50

51 Two Slope Criterion Ultimate Load taken as the intersection of the two asymptotes to the curve at the beginning and the end of loading test Sometimes a range of loads is possible; take mean value Relative settlement, Se / B (%) range of failure 1 Base Pressure (kpa) 10 D r = 25% 51

52 Recommended Failure Criterion Minimum Slope Failure Load Criterion, Vesic (1963) Failure load interpreted were for 196 cases using each of the proposed methods Representative Failure Load defined as the mean value of all the failure loads interpreted using each criterion Mean of the ratio was 0.98 and Failure Loads for most cases could be interpreted using Minimum Slope criterion the criterion was chosen as the standard BC interpretation method 52

53 Interpreted Failure Load Example FOTID #35 (TAMU, Riverside Campus) Reported by Briaud and Gibbens (1994) in Geotechnical Special Publication No. 41 (ASCE) 39in x 39in square footing with 28in embedment Soil information Ground level 11.5ft: silty fine sand 11.5ft 23ft: med dense silty sand GWT at 16.0ft SPT-N counts as shown Average soil unit weight = 118pcf Average relative density = 50.75% 53

54 Interpreted Failure Load Example 0 load intensity (tsf) load intensity (tsf) settlement, Se (in) tsf relative settlement, Se / B (%) Interpreted Failure load using Min Slope criterion = 13.94tsf

55 Uncertainty Evaluation Granular Soils BC of centric vertically loaded footings Database Overview Calculated Bearing Capacity Soil Parameters Equations used for BC calculations 55

56 Database Overview UML-GTR ShalFound07 Foundation type Plate load tests B 1m Small footings 1 < B 3m Large footings 3 < B 6m Rafts & Mats B > 6m Predominant Soil Type Country Total Sand Gravel Cohesive Mix Others Germany Others Total Note: Mixed are cases with alternating layers of sand or gravel and clay or silt Others are cases with either unknown soil types or with other granular materials like loamy Scoria 1m 3.3ft 56

57 Database Overview UML-GTR ShalFound07: Database I (pcf) (in parentheses) Frequency Number of data 238 Minimum 28.8 Maximum Mean Std dev 4.35 Frequency Number of data 238 Minimum 9.93 (63.26) Maximum (146.02) Mean (107.67) Std dev 2.37 (15.10) Internal friction angle, (deg) Soil unit weight, (kn/m3) 57

58 Database Overview Controlled Soil Conditions (pcf) (in parentheses) Frequency Number of data 185 Minimum 28.8 Maximum Mean Std dev 4.42 Frequency Number of data 185 Minimum 9.93 (63.27) Maximum (117.0) Mean (103.70) Std dev 2.18 (13.87) Internal friction angle, (deg) Soil unit weight, (kn/m3) 58

59 Database Overview Natural soil conditions (pcf) (in parentheses) Frequency Number of data 53 Minimum Maximum Mean Std dev 3.56 Frequency Number of data 53 Minimum (106.51) Maximum (146.01) Mean (121.44) Std dev 1.64 (10.47) Internal friction angle, (deg) Soil unit weight, (kn/m3) 59

60 UML-GTR ShalFound07: Database I Cases in/on granular soils German tests (pcf) (in parentheses) Frequency Number of data 153 Minimum Maximum Mean Std dev 3.07 Frequency Number of data 153 Minimum 9.93 (63.26) Maximum (117.04) Mean (104.80) Std dev 2.30 (14.63) Internal friction angle, (deg) Soil unit weight, (kn/m3) 60

61 UML-GTR ShalFound07: Database I Cases in/on granular soils Non-German tests (pcf) (in parentheses) Frequency Number of data 85 Minimum 28.8 Maximum 46.0 Mean Std dev 4.91 Frequency Number of data 85 Minimum (84.04) Maximum (146.02) Mean (112.84) Std dev 2.29 (14.59) Internal friction angle, (deg) Soil unit weight, (kn/m3) 61

62 Database Overview Footing sizes 100 L (ft) Number of data 238 Footing width, B (m) B (ft) Width in m (ft) Minimum 0.04 (0.13) Maximum 38.1 (125.0) Mean 1.25 (4.10) Median 0.09 (0.30) Std dev 3.40 (11.15) Length in m (ft) Minimum (0.177) Maximum 57.7 (189.3) Mean 2.62 (8.56) Median 0.15 (0.49) Std dev 6.80 (22.34) Footing length, L (m) 62

63 UML-GTR ShalFound07: Database I Footing sizes Controlled soil conditions 10 L (ft) Number of data 185 Footing width, B (m) B (ft) Width in m (ft) Minimum 0.04 (0.13) Maximum 1.00 (3.28) Mean 0.18 (0.60) Median 0.09 (0.30) Std dev 0.22 (0.73) Length in m (ft) Minimum (0.177) Maximum 2.0 (6.56) Mean 0.33 (1.07) Median 0.09 (0.30) Std dev 0.46 (1.50) Footing length, L (m) 63

64 UML-GTR ShalFound07: Database I Footing sizes Natural soil conditions 100 L (ft) Number of data 53 Footing width, B (m) B (ft) Width in m (ft) Minimum 0.55 (1.80) Maximum 38.1 (125.0) Mean 4.96 (16.27) Median 3.81 (12.50) Std dev 5.86 (19.23) Length in m (ft) Minimum 0.65 (2.13) Maximum 57.7 (189.3) Mean (34.74) Median 5.64 (18.50) Std dev (36.88) Footing length, L (m) 64

65 UML-GTR ShalFound07: Database I Cases in/on granular soils German tests L (ft) Number of data Footing width, B (m) B (ft) Width in m (ft) Minimum 0.04 (0.13) Maximum 1.0 (3.28) Mean 0.16 (0.53) Median 0.09 (0.30) Std dev 0.20 (0.66) Length in m (ft) Minimum 0.09 (0.30) Maximum 2.0 (6.56) Mean 0.31(1.01) Median 0.09 (0.30) Std dev 0.48 (1.59) Footing length, L (m) 65

66 UML-GTR ShalFound07: Database I Cases in/on granular soils Non-German tests 100 L (ft) Number of data 85 Footing width, B (m) B (ft) Width in m (ft) Minimum (0.17) Maximum 38.1 (125.0) Mean 3.20 (10.50) Median 2.01 (7.0) Std dev 5.15 (16.90) Length in m (ft) Minimum (0.18) Maximum 57.7 (189.3) Mean 6.76 (22.18) Median 2.01 (7.0) Std dev (33.3) Footing length, L (m) 66

67 In which: Equations used for BC calculation The bearing capacity equation specified in AASHTO (2008) with minimal necessary adjustment has been used to calculate the bearing capacity of a footing of length L and width B and supported by a soil with cohesion c, average friction angle f and average unit weights 1 and 2 above and below the footing base, respectively. The format presented in equation (95) is based on the general bearing capacity formulation used by Vesić (1975) as presented in section equation (34). The numbering in the parenthesis represents the proposed numbering for the modified AASHTO specifications. q cn γ D N 05. γ BN n cm 1 f qm 2 γm N N N N c scd ci N s d i qm q q q q N s d ( a-1) (95) ( a-2) (96) ( a-3) (97) γm γ γ γ γ ( a-4) (98) where: c = cohesion, taken as undrained shear strength c u in total stress analysis or as cohesion c in effective stress analysis (ksf) N c = cohesion term bearing capacity factor as specified in Tables 25 and 26 (dim.) N q = surcharge (embedment) term bearing capacity factor as specified in Tables 25 and 26 (dim.) N = unit weight (footing width) term bearing capacity factor as specified in Tables 25 and 26 (dim.) = moist or submerged unit weight of soil above the bearing depth of the footing (kcf) = moist or submerged unit weight of soil below the bearing depth of the footing (kcf) D f = footing embedment depth (ft.) B = footing width (ft.), equal to the physical footing width B in case of centric loading or effective footing width B in case of eccentric loading s c, s, s q = footing shape correction factors as specified in Table 27 (di.) d c,d,, d q = depth correction factors to account for the shearing resistance along the failure surface passing through the soil above the bearing elevation as specified in Table 28 (dim.) ic,i,i q = load inclination factors as specified in Table 29 (dim.). cm i c 67

68 Equations used for BC calculation The effective vertical stress calculated at the base of the footing ( 0 ) should be used or alternatively, an average weighted soil unit weight ( 1,ave ) should be used above the base. Below the base an average soil unit weight ( 2,ave ) should be used within a zone of 1.5B. The highest anticipated groundwater level should be used in design. In Tables 27 to 29 B and L are either the physical footing dimensions in case of centric loading or have to be substituted by the effective footing dimensions B and L in case of eccentric loading. In Table 29 H and V are the unfactored horizontal and vertical loads, in (kips), respectively. The angle is the projected direction of load in the plane of the footing, measured from the side of the footing length L (deg.). Figure 17 (AASHTO Figure a-1) shows the conventions for determining. The parameter n is defined according to equation (99). D f i D i n 2 L/ B 2 B/ L 1 L/ B 1 B/ L 2 2 cos sin ( a-5) (99) 68

69 Equations used for BC calculation The depth correction factor should be used only when the soils above the footing bearing elevation are competent and there is no danger for their removal over the foundation s lifetime, otherwise, the depth correction factor should be taken as 1.0, or D f should be reduced to include the competent, secured depth only. The depth correction factors presented in Table 28 refers when applicable to the effective foundation width B. Some design practices use the physical footing width (B) for evaluating the depth factors under eccentric loading as well. The calibration presented in this study was conducted using B. The use of B in the depth factors expressions, results with a more conservative evaluation as discussed by Paikowsky et al. (2009a). Table 25 Bearing capacity factors N c (Prandtl, 1921), N q (Reissner, 1924), and N (Vesic, 1975) (AASHTO Table a-1) Factor Friction Angle Cohesion Term (N c ) Unit Weight Term (N ) Surcharge Term (N q ) Bearing Capacity Factors N c, N, N q f = f f > 0 (N q -1)cot f 2(N q + 1)tan exp tan tan 45 f f 2 69

70 Equations used for BC calculation Table 26 Bearing capacity factors N c (Prandtl, 1921), N q (Reissner, 1924), and N (Vesic, 1975) (AASHTO Table a-2) f N c N q N f N c N q N

71 Equations used for BC calculation Table 27 Shape correction factors s c, s, s q. (Vesić, 1975) (AASHTO Table a-3) Factor Friction Angle Cohesion Term (s c ) Unit Weight Term (s ) Surcharge Term (s q ) Shape Factors f = L s c, s, s q B N q f > 0 1 B B L N B c L 1 tan L f Table 28 Depth correction factors d c, d, dq. (Brinch Hansen, 1970) (AASHTO Table a-4) Factor Depth Correcti on Factors d c, d, d q Friction Angle f = 0 Cohesion Term (d c ) for D f B: D 0.4 B 1 f Unit Weight Term (d ) Surcharge Term (d q ) for D f > B: D 0.4arctan B 1 f 1 d q f > 0 d q 1.0 N 1 q for D f B: 2 Df 1 2 tan f 1 sin f B for D f > B: 2 Df 1 2 tan 1 sin arctan f f B 71

72 Equations used for BC calculation Table 29 Load inclination factors i c, i, i q. (Vesić, 1975) (AASHTO Table a-5) Factor Friction Angle Cohesion Term (i c ) Unit Weight Term (i ) Surcharge Term (i q ) Load Inclination Factors i c, i, i q f = f > 0 N c L B c H n 1 1 N i 1 i q q q 1 n f cot L B c V H 1 n f cot L B c V H 1 72

73 Soil parameters Estimation based on SPT-N blow counts Soil friction angle Peck, Hanson and Thornburn (PHT) as modified by Kulhawy and Mayne (1990): f exp N 1 N 60 corrected using Liao and Whitman s correction (1996) 60 Soil unit weight Paikowsky et at (1995): ( N1) 99 (pcf) for 146pcf 73

74 Estimation of f from SPT-N PHT and Hatanaka and Uchida Comparison Bias using Hatanaka and Uchida (1996) n = 15 Mean bias Bias using Peck, Hanson and Thornburn as mentioned in Kulhawy and Mayne (1990) 74

75 Bias of Estimated BC Cases with Vertical Centric Loading Vertical Centric Loading n = 173; mean bias = 1.59, COV = Natural soil conditions ( f from SPT-N counts) n = 14; no. of sites = 8 mean = 1.00 COV = Controlled soil conditions (D r 35%) n = 159; no. of sites = 7 mean = 1.64 COV = B > 1.0m n =6 no. of sites = 3 mean = 1.01 COV = < B 1.0m n =8 no. of sites = 7 mean = 0.99 COV = B 0.1m n = 138 no. of sites = 5 mean = 1.67 COV = < B 1.0m n =21 no. of sites = 3 mean = 1.48 COV = Figure 60 Summary of bias (measured over calculated BC) for vertical centric loading cases (Database I); 0.1m = 3.94in; 1m = 3.28ft. 75

76 Bias of Estimated BC Cases with Vertical Centric Loading Number of observations Vertical-centric loading n = 173 mean = 1.59 COV = lognormal distribution normal distribution Frequency Interpreted bearing capacity, qu,meas using Minimum Slope criterion (Vesic, 1963) (ksf) Vertical-centric loading Data (n = 173) Data best fit line No bias line Bias, q u,meas / q u,calc Calcualted bearing capacity, q u,calc (Vesic, 1975 and modified AASHTO) (ksf) Figure 61. (a) Histogram and probability density functions of the bias and (b) relationship between measured and calculated bearing capacity for all cases of vertical centrically loaded shallow foundations. 76

77 Bias of Estimated BC Cases with Vertical Centric Loading Number of observations Controlled soil conditions n = 159 mean = 1.64 COV = lognormal distribution normal distribution Bias, q u,meas / q u,calc Frequency Interpreted bearing capacity, qu,meas using Minimum Slope criterion (Vesic, 1963) (ksf) Calcualted bearing capacity, q u,calc (Vesic, 1975 and modified AASHTO) (ksf) Figure 62. (a) Histogram and probability density functions of the bias and (b) relationship between measured and calculated bearing capacity for vertical centrically loaded shallow foundations on controlled soil conditions Controlled soil conditions Data (n = 159) Data best fit line No bias line 77

78 Bias of Estimated BC Cases with Vertical Centric Loading Number of observations Natural soil conditions n = 14 mean = 1.00 COV = normal distribution lognormal distribution Frequency Interpreted bearing capacity, qu,meas using Minimum Slope criterion (Vesic, 1963) (ksf) Natural soil conditions Data (n = 14) Data best fit line No bias line Bias, q u,meas / q u,calc Calcualted bearing capacity, q u,calc (Vesic, 1975 and modified AASHTO) (ksf) Figure 63. (a) Histogram and probability density functions of the bias and (b) relationship between measured and calculated bearing capacity for vertical centrically loaded shallow foundations on natural soil conditions. 78

79 Bias versus Footing Width 3.8 Footing width, B (ft.) Natural Soil Condition (n =14) Controlled Soil Condition (n =158) 1s.d. (x) number of cases in each interval 2.6 Bias, (34) (90) (4) (5) (5) (2) (12) (3) (3) Footing width, B (m) Figure 99. Variation of the bias in bearing resistance versus footing size for cases under vertical-centric loadings: controlled and natural soil conditions. 79

80 Bias versus Footing Width B(ft) 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) Bias (5) (90) (2) (3) (4) 1.5 (4) (5) (1) (34) (1) (17) (3) (1) (2) Footing width, B (m) Figure 100. Variation of the bias in bearing resistance versus footing size for cases under vertical-centric loadings: f 43 and f <

81 Uncertainty in N 1000 q u / (0.5 B s ) N from load tests; n = 125 N (Vesic, 1973) N = exp(0.39 f ) (R 2 = 0.666) friction angle, f (deg) Comparison of bearing capacity factor calculated based on test results; N =q u / (0.5 Bs ) from 125 tests carried out in controlled soil conditions (tests by Perau, 1995) and N proposed by Vesic (1973) in the range of soil friction angle of 42 and 46 81

82 Uncertainty in N N = [q u / (0.5 B s )] / NVesic load test data; n = 125 = exp(0.205 f 8.655) (R 2 = 0.351) Friction Angle, f (deg) Figure 93. The ratio ( N ) between the back-calculated B.C. factor N based on experimental data to that proposed by Vesić versus soil friction angle. 82

83 Uncertainty in N Data BC bias (n = 131) Bearing Capacity (BC) bias, N bias, 2 Bias Friction Angle, f (deg) Figure 94. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the B.C. factor N ( N ) versus the soil s friction angle for footings under vertical-centric loadings. 83

84 Uncertainty in B.C Mean bias,bc (n = 172) 1 s.d. (x) no. of cases in each interval BC = 0.308exp( f ) (R 2 =0.200) % confidence interval Bias (2) (4) (2) (3) (12) (30) (90) (2) 1.0 (3) (4) (4) (14) 0.5 (2) Friction angle f (deg) Figure 103. Bearing resistance bias vs. average soil friction angle (taken f 0.5) including 95% confidence interval for all cases under vertical-centric loading. 84

85 Uncertainty in B.C % confidence interval for 2.5 Resistance factor based on database (x) no. of cases in each interval Recommended f for Controlled soil conditions Bias (2) (4) (3) Recommended f for Natural soil conditions (2) (3) (4) (12) (2) (30) (4) (14) (90) (2) Resistance factor, n = Friction angle f (deg) Figure 104. Recommended resistance factors for soil friction angles (taken f 0.5) 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. 85

86 Uncertainty in B.C. Table 58 Resistance factors for vertical-centric loading cases based on the bias vs. f best-fit line of equation (121) and the COV of natural vs. controlled soil conditions Soil friction angle f (deg) Mean bias (Equation 121) Resistance factor ( T = 3) Soil Condition Natural Controlled (COV = 0.35) (COV = 0.25) MCS Reco MCS Reco

87 5. Calibration of Resistance Factors Vertical Load Distributions Using the same distributions used for strength limit state for piles (NCHRP 507) and service limit state of foundations (NCHRP 12-66): Based on Nowak (1999) NCHRP 368 Dead Load D = 1.25 QD = 1.05 COV QD = 0.1 (as recommended by Nowak) Live Load L = 1.75 QL = 1.15 COV QL = 0.2 Table F1 by Nowak QL = 1.1 to 1.2, COV QL = 0.18 (Selected in consultation with Billal Ayyub) Dead to Live Load Ratio = 2.0 (see discussion in NCHRP 507) 87

88 Recommended Resistance Factors for Vertical- Centric Loading Table 59 Recommended resistance factors for Vertical-Centric loading cases Soil friction angle f (deg) Recommended resistance factor ( T = 3) Soil Conditions Natural Controlled

89 Final Resistance Factors Controlled Conditions Table 66 Recommended resistance factors for shallow foundations on granular soils placed under controlled conditions Soil friction angle f Vertical-centric or -eccentric Loading conditions Inclined-centric Inclined-eccentric Positive Negative Notes: 1) f 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 T =

90 Final Resistance Factors Natural Conditions Table 67 Recommended resistance factors for shallow foundations on natural deposited granular soil conditions Loading conditions Soil friction angle f Vertical-centric or -eccentric Inclined-centric Inclined-eccentric Positive Negative Notes: 1) f 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 T =

91 Intermediate Conclusions and Summary It was found that for the footings of larger sizes (B>3m (9.9ft)), the load tests were not carried out to the failure load Biases for the tests in Natural Soil Condition and Controlled Soil Conditions were analyzed separately For the footing sizes in similar ranges (0.1m < B 1.0m), the scatter of bias was larger for footings on/in natural soil conditions The majority of the relevant data refers to small size foundations (B 3.3ft (1.0m)) on controlled compacted material. Many of the highway shallow foundations on soils are built on compacted materials and hence, the statistical data of the uncertainty can be used for that purpose There appears to be a trend of increase in bias with the footing size within the range of footing sizes available for testing (which seems to conform with the observation made by Vesic (1969)) 91

92 ULS of Inclined Loading x 2 F 1 M 2 D M 1 F 2 M 3 F 3 b 2 x 3 b 3 x 1 g f (a) Loading convention F 1 F 1 F F 1 1,const. = const. F 3 arctan e = const M 2 increasing F 3 (b) Radial load path (c) Step-like load path Figure 64. Loading convention and load paths used during tests. 92

93 ULS of Inclined Loading 0 Vertical load, F 1, F 10 (kn) Horizontalload, F 3 (kn) MoD2.1 = 3 0 Vertical displacement, u 1 (mm) MoD2.1 = 3 MoD2.2 = 8 MoD2.3 = 14 MoA2.1 = 0F F 1 (kips) u 1 (in) Horizontal displacement, u 3 (mm) MoD2.2 = 8 MoD2.3 = F 3 (kips) u 3 (in) Figure 65. Load displacements curves for model tests conducted by Montrasio (1994) with varying load inclination: (a) vertical load vs. vertical displacement and (b) horizontal load vs. horizontal displacement. 93

94 Bias of Estimated BC Cases with Vertical-Eccentric Loading (using B) Table 32 Summary of the statistics for biases of the test results for vertical-eccentric loading when using effective foundation width B Tests No. of cases Minimum slope criterion Two slope criterion Mean Std. Dev. COV Mean Std. Dev. COV DEGEBO radial load path Montrasio/Gottardi radial load path Perau radial load path All cases 17 (15) (41) Radial Load Path Gradual increase of loads keeping the eccentricity constant 94

95 Bias of Estimated BC Cases with Vertical-Eccentric Loading (using B) Table 33 Summary of the statistics for biases of the test results for vertical-eccentric loading when using foundation width B Tests No. of cases Minimum slope criterion Two slope criterion Mean Std. Dev. COV Mean Std. Dev. COV DEGEBO radial load path Montrasio/Gottardi radial load path Perau radial load path All cases 17 (15) (41) Radial Load Path Gradual increase of loads keeping the eccentricity constant 95

96 Bias of Estimated BC Cases with Vertical-Eccentric Loading (using B) Number of observations Vertical-eccentric loading n = 43 mean = 1.83 COV = lognormal distribution normal distribution Frequency Interpreted bearing capacity, qu,meas using Minimum Slope criterion (Vesic, 1963) (ksf) Vertical-eccentric loading Data (n = 43) Data best fit line No bias line Bias, q u,meas / q u,calc Calcualted bearing capacity, q u,calc (Vesic, 1975 and modified AASHTO) (ksf) Figure 66. (a) Histogram and probability density functions of the bias and (b) relationship between measured and calculated bearing capacity for all cases of vertical eccentrically loaded shallow foundations

97 Bias of Estimated BC Cases with Vertical-Eccentric Loading (using B) Table 33 Summary of the statistics for biases of the test results for verticaleccentric loading when using foundation width B Tests DEGEBO radial load path Montrasio/Gottar di radial load path Perau radial load path All cases No. of cases Minimum slope criterion Two slope criterion Mean Std. Dev. COV Mean Std. Dev. COV 17 (15) (41) number of cases for Two slope criterion 97

98 Bias of Estimated BC Cases with Vertical-Eccentric Loading 3.0 n = 43 (4) (7) (2) (11) Bias 1.5 (6) (4) 1.0 (9) 0.5 Mean bias, BC 1 s.d. (x) no. of cases in each interval BC = exp( f ) (R 2 =0.01) 95% confidence interval Friction angle f (deg) 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. 98

99 Bias of Estimated BC Cases with Vertical-Eccentric Loading Bias, Vertical-eccentric loading (n = 43) B 4.0in (0.1m) B = 1.65ft (0.5m) B = 3.3ft (1.0m) Load eccentricity ratio, e/b Figure 106. Bearing resistance bias vs. load eccentricity ratio e/b for vertical-eccentric loading. 99

100 Bias of Estimated BC Cases with Inclined-Centric Loading (using B) Tests DEGEBO/ Montrasio/Gottardi radial load path Perau/Gottardi step-like load path All cases No. of cases 1 number of cases for Two slope criterion Minimum slope criterion Two slope criterion Mean Std. Dev. COV Mean Std. Dev. COV 26 (24) (37)

101 Bias of Estimated BC Cases with Inclined-Centric Loading Number of observations Inclined-centric loading n = 39 mean = 1.43 COV = lognormal distribution normal distribution Frequency Interpreted bearing capacity, qu,meas using Minimum Slope criterion (Vesic, 1963) (ksf) Inclined-centric loading Data (n = 39) Data best fit line No bias line Bias, q u,meas / q u,calc Calcualted bearing capacity, q u,calc (Vesic, 1975 and modified AASHTO) (ksf) Figure 67. (a) Histogram and probability density functions of the bias and (b) relationship between measured and calculated bearing capacity for all cases of inclined centric loaded shallow foundations. 101

102 Bias of Estimated BC Cases with Inclined-Eccentric Loading (using B) Table 35 Summary of the statistics for biases of the test results for inclinedeccentric loading when using effective foundation width B Tests DEGEBO/Gottardi radial load path Step-like load path Montrasio/ Gottardi Perau positive eccentricity Perau negative eccentricity All step-like load cases No. of cases Minimum slope criterion Two slope criterion Mean Std. Dev. COV Mean Std. Dev. COV All cases

103 Bias of Estimated BC Cases with Inclined-Eccentric Loading Number of observations Inclined-eccentric loading n = 29 mean = 2.43 COV = lognormal distribution normal distribution Bias, q u,meas / q u,calc Frequency Interpreted bearing capacity, qu,meas using Minimum Slope criterion (Vesic, 1963) (ksf) Calcualted bearing capacity, q u,calc (Vesic, 1975 and modified AASHTO) (ksf) Figure 68. (a) Histogram and probability density functions of the bias and (b) relationship between measured and calculated bearing capacity for all cases of inclined eccentrically loaded shallow foundations Inclined-eccentric loading Data (n = 29) Data best fit line No bias line 103

104 Bias of Estimated BC Cases with Inclined-Eccentric Loading (using B) Table 36 Summary of the statistics for biases of the test results for inclinedeccentric loading when using foundation width B Tests DEGEBO/Gottardi radial load path Montrasio/ Gottardi No. of cases Minimum slope criterion Two slope criterion Mean Std. Dev. COV Mean Std. Dev. COV Step-like load path Perau positive eccentricity Perau negative eccentricity All step-like load cases All cases

105 Loading Directions for Inclined-Eccentric Loadings F 1 e 3 b Moment acting 3 in direction opposite to the lateral 3 loading negative b e 3 F 1 F 3 F 3 eccentricity Moment acting 3in the same direction as the lateral 3 loading positive eccentricity (a) along footing width b b F F M 2 1 F M F F 1 e 2 b b e 2 F 1 F 2 Moment acting 2 in direction opposite to the lateral 2 loading negative F 2 eccentricity Moment acting 2in the same direction as the lateral 2 loading positive eccentricity b b F 1 F 1 M 3 M 3 (b) along footing length + - F + 2 F + 2 Figure 69. Loading directions for the case of inclined-eccentric loadings: (a) along footing width and (b) along footing length 105

106 Bias of Estimated BC Cases with Inclined-Eccentric Loading Number of observations Inclined-eccentric loading Negative eccentricity n = 7 mean = 3.43 COV = lognormal distribution normal distribution Bias, q u,meas / q u,calc Frequency Interpreted bearing capacity, qu,meas using Minimum Slope criterion (Vesic, 1963) (ksf) Calcualted bearing capacity, q u,calc (Vesic, 1975 and modified AASHTO) (ksf) Figure 71. (a) Histogram and probability density functions of the bias and (b) relationship between measured and calculated bearing capacity for all cases of inclined eccentrically loaded shallow foundations under negative eccentricity Inclined-eccentric loading Negative eccentricity Data (n = 7) Data best fit line No bias line 106

107 5. Calibration of Resistance Factors Vertical Load Distributions Using the same distributions used for strength limit state for piles (NCHRP 507) and service limit state of foundations (NCHRP 12-66): Based on Nowak (1999) NCHRP 368 Dead Load D = 1.25 QD = 1.05 COV QD = 0.1 (as recommended by Nowak) Live Load L = 1.75 QL = 1.15 COV QL = 0.2 Table F1 by Nowak QL = 1.1 to 1.2, COV QL = 0.18 (Selected in consultation with Billal Ayyub) Dead to Live Load Ratio = 2.0 (see discussion in NCHRP 507) 107

108 Final Resistance Factors Natural Conditions Table 67 Recommended resistance factors for shallow foundations on natural deposited granular soil conditions Loading conditions Soil friction angle f Vertical-centric or -eccentric Inclined-centric Inclined-eccentric Positive Negative Notes: 1) f 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 T =

109 Conclusion and Summary It was found that for the footings of larger sizes (B>3m (9.9ft)), the loading tests were not carried to the failure load Biases for the tests in Natural Soil Condition and Controlled Soil Conditions were analyzed separately For the footing sizes in similar ranges (0.1m < B 1.0m), the scatter of bias was more for footings on/in natural soil conditions The majority of the relevant data refers to small size foundations (B 3.3ft (1.0m)) on controlled compacted material. Many of the highway shallow foundations on soils are built on compacted materials and hence, the statistical data of the uncertainty can be used for that purpose There appears to be a trend of increase in bias with the footing size within the range of footing sizes available for testing (which seems to conform with the observation made by Vesic (1969)) 109

110 Conceptual Design Influence of Serviceability s based on Serviceability Limit States Developed as a part of Project NCHRP Bias = measured load / calculated load for a given settlement For reliability index = 1.28 (p f = 10%), and load factors taken as unity Bias of LL = 1.15, COV QL = 0.2 Bias of DL = 1.05, COV QD = 0.1 Method AASHTO Range of Settlement (inch) Resistance Factor Efficiency Factor / 0.00 < < <

111 Conceptual Design Granular Soils Example 2 NCHRP Report 651: Known Load and Settlement Central Pier of a bridge in Billerica (Rangeway Rd over Rte.3) (B ) Design (factored) load is kips for ultimate limit state and 2750kips for service limit state (unfactored) Allowable settlement 1.5inches 111

112 Conceptual Design Granular Soils Subsurface condition Footing rests on gravel borrow of unit wt 120.0pcf (18.85kN/m 3 ) and the soil friction angle considered to be 38, which replaces approx. 3ft of loose granular fill overlaying 5.5ft of coarse sand and gravel underlain by a rock layer GWT present at foundation level Length of the required foundation = 52.4ft (fixed) 112

113 Loading Convention and Notations x 2 F 1 D M 1 M 2 F 3 F 2 M 3 L x 3 B x 1 The vertical centric loading is F1; F2 and F3 are horizontal loadings along the transverse (x2-direction or z-direction) and longitudinal (x3-direction or y-direction) directions of the bridge, respectively. M3 is the moment about the longitudinal direction (x3- or y-axis) due to transverse loading and M2 is the moment about the z-axis (transverse direction) due to longitudinal loading. The load eccentricity across the footing width is eb = M2/F1 and across the footing length is el = M3/F1. The resultant load inclination is given by. 2 2 F F / 2 3 F1 113

114 Conceptual Design Granular Soils Unfactored Resistances (ksf) Figure H-5. Variation of unfactored bearing resistance for Strength-I and Service-I limit states with effective footing width for Example 2 (NCHRP Report 651) Note: The settlement calculations are done for B and transformed to B Unfactored resistance (limit states) (ksf) Effective footing width, B (m) Effective footing width, B (ft) Strength LS C2 load combination (vertical eccentric) Strength LS C7 load combination (2-way load inlination and 2-way eccentricity) Service LS AASHTO (2007) Service LS Schmertmann (1978) Service LS Hough (1959) Service I loading Unfactored resistance (limit states) (MPa) 114

115 Conceptual Design Granular Soils Unfactored Resistances (kips) Strength LS C2 load combination (vertical eccentric) Strength LS C7 load combination (2-way load inlination and 2-way eccentricity) Service LS AASHTO (2007) Service LS Schmertmann (1978) Service LS Hough (1959) Service I loading Figure H-5 cont. Variation of unfactored bearing resistance for Strength-I and Service-I limit states with effective footing width for Example 2 (NCHRP Report 651) Unfactored resistance (limit states) (kips) Effective footing width, B (m) Unfactored resistance (limit states) (MN) Effective footing width, B (ft) 115

116 Conceptual Design Granular Soils Factored Resistances (ksf) 25 Effective footing width, B (m) g ( ) Strength LS, C2 load Strength LS, C7 load Strength LS AASHTO, 2007) Service LS AASHTO (2007) Service LS Schmertmann (1978) Service LS Hough (1959) Service I loading Strength I, C2 loading Strength I, C7 loading Factored ultimate limit state and unfactored service limit state (ksf) Factored ultimate limit state and unfactored service limit state (MPa) Effective footing width, B (ft) 0 Figure H-6 Variation of factored bearing resistance for Strength-I and unfactored resistance for Service-I limit state with effective footing width for Example 2 (NCHRP Report 651) 116

117 Factored ultimate limit state and unfactored service limit state (kips) Conceptual Design Granular Soils Factored Resistances (kips) Effective footing width, B (m) Effective footing width, B (ft) Factored ultimate limit state and unfactored service limit state (MN) g ( ) Strength LS, C2 load Strength LS, C7 load Strength LS AASHTO, 2007) Service LS AASHTO (2007) Service LS Schmertmann (1978) Service LS Hough (1959) Service I loading Strength I, C2 loading Strength I, C7 loading Strength Limit State loading 3688kips (phi=0.45) B = 8.9ft (B =7.9ft+2x0.5) Service Limit state of 2750kips B=4.5ft (B =3.5+2x0.5) Figure H-6 cont. Variation of factored bearing resistance for Strength-I and unfactored resistance for Service-I limit state with effective footing width for Example 2 (NCHRP Report 651) 117

118 Intermediate Conclusions The Strength Limit State governs the footing dimensions in this design example with a requirement for B=8.9ft vs. B=4.5ft for the service limit state The bridge was designed with B=13.1ft most likely due to the differences in design procedures (especially settlement) 118

119 BC Shallow Foundations on Rock - OUTLINE 1. Broad Objectives 2. Database UML/GTR RockFound07 3. Rock Classification and Properties 4. Methods of Analyses Selected for Establishing the Uncertainty in B.C. of Foundations on Rock 5. Calibration evaluation of resistance factors 6. Summary and Conclusions 119

120 1. Broad Objectives Examining the methods for B.C. and displacement evaluation of shallow foundations on rock. Establishing the uncertainty of the methods in order to develop the resistance factors. In contrast to shallow foundations on soil, the design of shallow foundations on hard rock is by and large controlled by the B.C. and not by settlement. Both however are investigated 120

121 2. DATABASE UML/GTR RockFound 07 Comprised of 122 foundation case histories of load tests in/on rock and IGM s. The database has 61 footings cases (28 cases D>0, 33 cases D=0) and 61 rock socket cases for which the base behavior (load and displacement) under loading was monitored. 89 of the 122 cases were used for the uncertainty determination of the settlement of foundations on rock. 121

122 2. Database UML/GTR RockFound07 80 Cases Sites Rock Types Number Non-Embedded Shallow Foundations Square - 4 Circular < B < 23 B avg = 2.76 ft Embedded Shallow Foundations Circular < B < 3 B avg = 1.18 ft Rock Sockets Circular < B < 9 B avg = 2.59 ft Distribution of Case Histories used in B.C. Analysis 0 122

123 2. Database UML/GTR RockFound Cases Sites Rock Types Number Non-Embedded Shallow Foundations Square - 4 Circular < B < 23 B avg = 3.08 ft Embedded Shallow Foundations Circular < B < 3 B avg = 0.93 ft Rock Sockets Circular < B < 9 B avg = 2.58 ft Distribution of Case Histories used in Settlement Analysis 0 123

124 2. Database UML/GTR RockFound07 No. of Observations Mean = 118 ksf COV = log-normal distribution 122 Rock Socket and Footings cases range of concrete strength >> >> >> Unconfined Compressive Strength, q u (ksf) AASHTO table : Geomechanical Classification of Rock Masses: Relative rating (for RMR) Lowest ksf Highest 15 >4320ksf One definition for IGM q u < 20ksf Distribution of the Unconfined Compressive Strength (q u ) for all 122 Case Histories in Database UML/GTR RockFound Frequency 124

125 2. Database UML/GTR RockFound07 No. of Observations Mean = 2.57 (ft) COV = log-normal distribution 122 Rock Socket and Footings cases m B = 3.93ft D=0 m B = 1.18ft D>0 m B = 2.47ft RockS Frequency 14 normal distribution B (ft) Distribution of the Foundation Width (B) for all 122 Case Histories in Database UML/GTR RockFound07 125

126 2. Database UML/GTR RockFound Rock Socket and Footings cases Mean = COV = m RMR = 65 All m RMR = 65 D=0 No. of Observations log-normal distribution normal distribution Frequency m RMR = 44 D>0 m RMR Note = 74 RockS RMR > 85 v. good rock RMR Distribution of RockMass Rating (RMR) for all 122 Case Histories in Database UML/GTR RockFound07 126

127 2. Database UML/GTR RockFound Rock Socket and Footings cases 0.49 m ql2 = 580ksf All m ql2 = 1647ksf D=0 No. of Observations Mean = ksf COV = 4.26 log-normal distribution Frequency m ql2 = 51ksf D>0 m ql2 = 244ksf RockS >> Interpreted Foundation Capacity, (q L2 ) (ksf) 0 Distribution of ql2 (ksf) for all 122 Case Histories in Database UML/GTR RockFound07 127

128 3. Rock Classification and Properties Rock is a natural aggregate of minerals that cannot be readily broken by hand and that will not disintegrate on a first wetting and drying cycle. A rockmass comprises blocks of intact rock that are separated by discontinuities such as cleavage, bedding planes, joints and faults. These naturally formed discontinuities create weakness zones within the rockmass, thereby reducing the material strength. 128

129 3. Rock Classification and Properties Rock is classified with respect to its geological origin or lithology as follows: Igneous rocks, such as granite, diorite and basalt, which are formed by the solidification of molten material, either by intrusion of magma at depth in the earth's crust, or by extrusion of lava at the earth's surface. Sedimentary rocks, such as sandstone, limestone and shale, which are formed by lithification of sedimentary soils. Metamorphic rocks, such as quartzite, schist, marble and gneiss, which were originally igneous or sedimentary rocks, and which have been altered physically and sometimes chemically or mineralogically, by the application of intense heat and/or pressure at some time in their geological history. 129

130 3. Rock Classification and Properties The strength and stiffness properties of rockmasses are required in the design of foundations in or on rock. These properties are functions of the properties of the intact rock and the discontinuities. The two most commonly used rockmass classification systems in Civil engineering are: 1. Rockmass Rating (RMR), Bieniawski, (1974) with several modifications up to used in tunneling and foundations, adopted by the International society for Rock mechanics (ISRM) and the South African Council of Scientific and Industrial Research (CSIR). 2. Q-system, Barton et al., 1974 used in tunneling and adopted by the Norwegian Geotechnical Institute index (NGI-index) In this study, the RMR classification system was adopted because it is most commonly used, it was favored by the available rock property data of the case histories and GSI noted by two states is based on the RMR-system. 130

131 3. Rock Classification and Properties The RMR-system uses the following 6 parameters, whose ratings are added to obtain a total RMR-value: i. Unconfined compressive strength of intact rock material (q u ) ii. Rock quality designation (RQD) iii. Joint or discontinuity spacing (s) iv. Joint condition v. Ground water condition vi. Joint orientation. See AASHTO Tables &2 for the above parameters and relative ratings (5 first and table 2 for vi) Hoek et al. (1995) introduced the GSI-system as a means of estimating the strength and deformation properties of jointed rockmasses. For RMR > 18 the GSI = RMR (Bieniawski, 1976). 131

132 4. Failure Modes of Foundations on Rock B.C. Failure Modes of Rock (Sowers, 1979) 132

133 4. Methods of Analyses Selected for Establishing the Uncertainty in B.C. of Foundations on Rock The ratio of the measured and interpreted capacity (q L2 ) to the calculated B.C. (q ult ) (the bias) was used to asses the uncertainty of the different design methods. The calculated B.C. (q ult ) was determined in 5 ways, namely: (a) following the semi empirical method by Carter and Kulhawy (1988) (b) following the analytical method by Goodman (1989) (c) following the Hoek and Brown (1980) failure criterion (d) utilizing the N c* and q u based on the relationship developed by Zhang and Einstein s (1998) and examined in this study (e) following relationships between measured or interpreted B.C. (q L2 ) and q u developed in this study as a function of rockmass quality utilizing AASHTO (2007) RMR ranges Only (a) and (b) are presented in NCHRP research report 651 The margin of safety of the AASHTO (2007) presumptive values was also examined. 133

134 Hirany and Kulhawy (1988) Failure criterion Hirany and Kulhawy (1988) proposed the L1-L2 method for interpreting the "failure" load or "ultimate" capacity of foundations from loaddisplacement curves. The unique peak or asymptote value in the curves is taken as the measured or interpreted capacity (QL2=qL2). For 79 cases ql2 could be evaluated, 43 cases are based on reported failure load. 134

135 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Using the Hoek-Brown strength criterion, Carter and Kulhawy (1988) developed the curved strength envelope represented by Equation 1 for B.C. evaluation of jointed rockmasses: 2 mq 0. 5 sq 1 3 u 3 u (1) in which s 1 = major principal effective stress s 3 = minor principal effective stress q u = unconfined compression strength of the intact rock s and m = empirically determined strength parameters for the rockmass, which are to some degree analogous to c and of the Mohr-Coulomb failure criterion 135

136 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Using the limit-equilibrium approach, Carter and Kulhawy (1988) developed a lower bound to the B.C. for strip and circular footings on jointed rock masses presented below. q ult m s qu (2) 136

137 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Summary of the statistics for the Ratio of Measured to Calculated B.C. using Carter and Kulhawy s (1988) Method Cases n No. of Sites Mean of Bias m Standard deviation COV All Foundations All rock sockets All footings Sub-categorization showed that the more detailed rock measurements are available, the lower the uncertainty. e.g. 39 Rock Socket cases with measured discontinuity spacing had a COV =

138 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Table 38 Summary of the statistics for the ratio of measured (q L2 ) to calculated bearing capacity (q ult ) of rock sockets and footings on rock using Carter and Kulhawy (1988) method Cases n No. of Sites m COV All rock sockets All rock sockets on fractured rock All rock sockets on non-fractured rock Rock sockets on non-fractured rock with measured discontinuity spacing (s') Rock sockets on non-fractured rock with s' based on AASHTO (2007) All footings All footings on fractured rock All footings on non-fractured rock Footings on non-fractured rock with measured discontinuity spacing (s') Footings on non-fractured rock with s' based on AASHTO (2007) n = number of case histories m = mean of biases = standard deviation COV = coefficient of variation Calculated capacity based on equation (82a) 138

139 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Interpreted Foundation Capacity q L2 (ksf) q L2 = q ult ) (n = 119; R 2 = 0.921) q L2 = q ult ) (Revised) (n = 119; R 2 = 0.917) q L2 = q ult 58 Footings cases 61 Rock Socket cases 119 All cases with revised equation Carter and Kulhawy (1988) Bearing Capacity q ult (ksf) Relationship between Carter and Kulhawy (1988) calculated bearing capacity (q ult ) using two variations (equations 82a and 82b) and the interpreted bearing capacity (q L2 ). 139

140 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Table 39 Summary of the statistics for the ratio of measured (q L2 ) to calculated bearing capacity (q ult ) using Carter and Kulhawy (1988) method categorized by the rock quality and foundation type Foundation type All Rock Sockets Footings Cases n No. of Sites m COV RMR > <RMR< <RMR< <RMR< RMR > <RMR< <RMR< <RMR< RMR > <RMR< <RMR< <RMR< n = number of case histories; m = mean of biases; = standard deviation; COV = coefficient of variation; Calculated capacity based on equation (82a) 140

141 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Rock-sockets and Footing cases Carter and Kulhawy (1988) mean = 8.00 COV = Foundation cases on Fractured Rocks Carter and Kulhawy (1988) mean = 4.05 COV = Number of observations 12 8 lognormal distribution normal distribution Frequency Number of observations 3 2 lognormal distribution normal distribution Frequency Bias, q u,meas / q u,calc Figure 76. Distribution of the ratio of the interpreted bearing capacity (q L2 ) to the bearing capacity (q ult ) calculated using Carter and Kulhawy s (1988) method (equation 82a) for the rock sockets and footings in database UML-GTR RockFound Bias, q u,meas / q u,calc Figure 77. Distribution of the ratio of the interpreted bearing capacity (q L2 ) to the bearing capacity (q ult ) calculated using Carter and Kulhawy s (1988) method (equation 82a) for foundations on fractured rock in database UML-GTR RockFound

142 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock 4 = 3.42 COV = Standard normal quantile B.C. of All Cases in rocks using Carter and Kulhawy (1988) Total data (n = 119) Normal distribution Lognormal distribution Bias Figure 117. Comparison of the unfiltered bias for BC calculated using Carter and Kulhawy (1988) method for total cases in/on rocks in the database and the theoretical normal and lognormal distributions. 142

143 4a Carter and Kulhawy (1988) - B.C. of Foundations on Rock Table 69 Calibrated resistance factors for different datasets of resistance bias obtained using Carter and Kulhawy s (1988) method Dataset No. of cases Bias Resistance factor ( T = 3) Mean COV MCS Recommended All cases RMR RMR < RMR < RMR <

144 4b Goodman (1989) - B.C. of Foundations on Rock Goodman (1989) considered the mode of failure presented in a through c, in which a laterally expanding zone of crushed rock under a strip footing induces radial cracking of the rock on either side. 144

145 4b Goodman (1989) - B.C. of Foundations on Rock Strength of crushed rock under footing lower envelope. Strength of the less fractured neighboring rock upper envelope. P h in the figure is equal to q u of the adjacent rock (Zone B) which is the largest confining stress that can be mobilized to support the rock under the footing (Zone A). The figure suggests that B.C. of a homogeneous discontinuous rockmass can not be less than the q u of the rockmass around the footings and this can be taken as the lower bound. 145

146 4b Goodman (1989) - B.C. of Foundations on Rock The lower bound is represented by the following Equation: in which q N ult tan q u N (3) (4) Goodman (1989) developed the B.C. Equation 5 for footings resting on orthogonal vertical joints each spaced distance s in which lateral stress transfer is nil. q ult q u N 1 N 1 S B N 1 N 1 (5) 146

147 4b Goodman (1989) - B.C. of Foundations on Rock Summary of the statistics for the Ratio of Measured to Calculated B.C. using Goodman s (1989) Method Cases n No. of Sites Mean of Bias m Standard Deviation COV All Foundations All rock sockets All footings Sub-categorization suggests that if more details of rock measurements are available, the uncertainty is reduced Rock Socket cases with measured discontinuity spacing had a COV l = Rock Socket cases with measured discontinuity spacing and friction angle had a COV l =

148 4b Goodman (1989) - B.C. of Foundations on Rock Table 40 Summary of the statistics for the ratio of measured (q L2 ) to calculated bearing capacity (q ult ) of rock sockets and footings on rock using Goodman (1989) method Cases n No. of Sites m COV All Measured discontinuity spacing (s') and friction angle ( f ) Measured discontinuity spacing (s') Measured friction angle ( f ) Fractured Fractured with measured friction angle ( f ) Non-fractured Non-fractured with measured s' and measured f Non-fractured with measured discontinuity spacing (s') Non-fractured with measured friction angle ( f ) Spacing s' and f, both based on AASHTO (2007) Discontinuity spacing (s') based on AASHTO (2007) Friction angle ( f ) based on AASHTO (2007) n = number of case histories m = mean of biases = standard deviation COV = coefficient of variation 148

149 4b Goodman (1989) - B.C. of Foundations on Rock Interpreted Foundation Capacity q L2 (ksf) q L2 = 2.16 q ult ) (n = 119; R 2 = 0.897) q L2 = q ult 1 58 Footings cases 61 Rock Socket cases Goodman (1989) Bearing Capacity q ult (ksf) Figure 78. Relationship between Goodman s (1989) calculated bearing capacity (q ult ) and the interpreted bearing capacity (q L2 ). 149

150 4b Goodman (1989) - B.C. of Foundations on Rock Rock-sockets and Footing cases Goodman (1989) mean = 1.35 COV = Foundation cases on Fractured Rocks Goodman (1989) mean = 1.24 COV = Number of observations lognormal distribution normal distribution Frequency Number of observations lognormal distribution normal distribution Frequency Bias, q u,meas / q u,calc Bias, q u,meas / q u,calc 0 Figure 79. Distribution of the ratio of the interpreted bearing capacity (q L2 ) to the bearing capacity (q ult ) calculated using Goodman s (1989) method for the rock sockets and footings in database UML-GTR RockFound07. Figure 80. Distribution of the ratio of the interpreted bearing capacity (q L2 ) to the bearing capacity (q ult ) calculated using Goodman s (1989) method for foundations on fractured rock in database UML-GTR RockFound07 150

151 4b Goodman (1989) - B.C. of Foundations on Rock 4 = 1.35 COV = Standard normal quantile B.C. using Goodman (1989) All data Total data (n = 119) Normal distribution Lognormal distribution Bias Figure 113. Comparison of the unfiltered bias for BC calculated using Goodman (1989) method for all data and the theoretical normal and lognormal distributions. 151

152 4b Goodman (1989) - B.C. of Foundations on Rock Table 42 Summary of the statistics for the ratio of measured (q L2 ) to calculated bearing capacity (q ult ) using Goodman (1989) method categorized by the rock quality Foundation type All Rock Sockets Footings Cases n No. of Sites m COV RMR > <RMR< <RMR< <RMR< RMR > <RMR< <RMR< <RMR< RMR > <RMR< <RMR< <RMR< n = number of case histories; m = mean of biases; = standard deviation; COV = coefficient of variation 152

153 4b Goodman (1989) - B.C. of Foundations on Rock Table 68 Calibrated resistance factors for different datasets of resistance bias obtained using Goodman s (1989) method Dataset No. of cases Bias Resistance factor ( T = 3) Mean COV MCS Recommended All data Measured friction angle f Measured spacing s Measured friction angle f and s

154 5. Calibration of Resistance Factors Outline of Major Points Both Monte Carlo (MC) simulation and the First Order Second Moment (FOSM) methods were used for the resistance factors calculations. The resistance factors are mostly in the range of 0.1 to 0.5. The resistance factors of Carter and Kulhawy (1988) are greater than one under three categories due to extremely high bias. Although theoretically there is no restriction for the resistance factor magnitude, practically it often leads to misconception as to the economics of a method, as will be further explained. The tables include the number of case histories and sites, mean bias and COV for each examined method of analysis and its application procedure or subcategory. For each of the three examined target reliabilities, the following is presented: 1. Rounded resistance factor based on the values initially evaluated. 2. Efficiency factor, a measure evaluating the relative efficiency of each design method with the higher value representing a more effective method. Such measure is required as often design engineers evaluate the economic value of a design method by the absolute value of the factor of safety or resistance factor (e.g. lower F.S. or higher, representing a more efficient method). A discussion and presentation of the concept are presented by Paikowsky et al. (2004). 154

155 5. Calibration of Resistance Factors Table 70 Recommended resistance factors for foundations in/on rock based on T = 3.0 (p f = 0.135%) Method of Analysis Equation Application Efficiency Factor / (%) Carter and Kulhawy (1988) Goodman (1989) q q m s ult For fractured rocks: For non-fractured rocks: u qult qu N 1 s qult qu N N 1 B 1 ( N 1) N 1 All RMR RMR < RMR < RMR < All Measured f Measured s Measured s and f

156