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15 20 0.22 18 AASHTO Analysis log normal 0.20 = 0.25 in 16 distribution n = 85 load tests = 0.674 x = 0.916 0.17 14 0.15 Number of Cases Relative Frequency 12 , COV 3.0 normal distribution 0.13 10 2.5 AASHTO 0.10 Bias () 8 = 2.840 COV 2.0 x = 2.265 0.08 6 1.5 0.05 4 1.0 2 0.03 0.5 0 0.00 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.5 1.0 1.5 2.0 2.5 3.0 Bias (Measured Load/Calculated Load) Foundation Settlement (in) (a) (b) Figure 8. (a) Histogram and frequency distributions of measured over calculated loads for a settlement ( = 0.25 in) using AASHTO's analysis method for 85 shallow foundation cases, and (b) variation of the bias () and uncertainty in the ratio between measured to calculated loads for shallow foundations on granular soils under displacements ranging from 0.25 to 3.00 in. able loading conditions is quite complex due to coupled foundation bearing capacity problem. They defined a three- loads and resistances. ULS under combined loading requires term bearing capacity equation by the superposition of the both an attempt to calibrate the existing methodology effects of soil cohesion, soil surcharge, and weight of soil, and an examination of a different approach for design, respectively. For a general case of centric vertical loading of as described in Section 1.5. a rigid strip footing (plain strain problem) on a cohesive- 3. The capacity of shallow foundations on rock under all types frictional soil surface with a uniform surcharge of q, the ulti- of loading is highly dependent on the relative scale of the mate bearing capacity (qu) is given as the following: foundation width to the rock discontinuity spacing and on the nature of the rock and its discontinuities. No established qu = cN c + qN q + (1 2 ) BN (19) bearing capacity theory exists for these cases. The calibra- tion of such cases, both for ULS and SLS (not included in where NCHRP Project 12-66), requires therefore establishing c = soil cohesion; models, using sophisticated analysis methods for evaluating = unit weight of the soil beneath the foundation; both strength and serviceability, and performing a proba- B = footing width; bility evaluation of incomplete data and calibration. q = overburden pressure at the level of the footing base; and Nc, Nq, and N are bearing capacity factors for cohesion, 1.5 Bearing Capacity overburden, and self-weight of soil, respectively. of Shallow Foundations For weightless soil ( = 0), Prandtl (1920) and Reissner 1.5.1 Basic Formulation (1924) developed the following formulas for Nc and Nq : Buismann (1940) and Terzaghi (1943) adopted the solution for metal punching proposed by Prandtl (1920, 1921) to the N c = ( N q - 1) cot f (20)

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16 25 30 0.30 Schmertmann et al. Analysis 0.35 Schmertmann Analysis (1970) log normal (1978) = 0.25 in distribution = 0.068 = 0.25 in log normal n = 81 load tests 25 n = 81 load tests 20 distribution 0.25 x = 0.630 0.30 = -0.324 x = 0.745 20 0.25 0.20 normal distribution Relative Frequency Number of Cases Number of Cases Relative Frequency 15 = 1.296 normal distribution 0.20 15 0.15 x = 0.818 = 0.950 10 0.15 x = 0.734 0.10 10 0.10 5 0.05 5 0.05 0 0.00 0 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Bias (Measured Load/Calculated Load) Bias (Measured Load/Calculated Load) (a) , COV 2.0 Schmertmann 1970 & Schmertmann et al. 1978 Bias () 1970 COV 1978 COV 1970 Bias () Das Bias () 1978 COV Das 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Foundation Settlement (in) (b) Figure 9. (a) Histogram and frequency distributions of measured over calculated loads for a settlement ( = 0.25 in) using Schmertmann (1970) and Schmertmann et al. (1978) analysis methods for 81 shallow foundation case, and (b) variation of the bias () and uncertainty in the ratio between measured to calculated loads for shallow foundations on granular soils under displacements ranging from 0.25 to 3.00 in. f N q = exp ( tan f ) tan 2 45 + (21) 2 Q0 where f = friction angle. L= q The bearing capacity factor Nc is sometimes credited A B E to Caquot and Krisel (1953). These formulas are exact I III closed-form solutions based on Prandtl's assumption of C II rupture surfaces (see Figure 10) in which the downward D movement of the active wedge (I) is resisted by the shear Figure 10. Assumed rupture resistance along the slip surfaces CDE (along the transi- surfaces by Prandtl (1920, 1921).

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17 tion zone [II] and passive wedge [III]) and the overburden Muhs and Weiss (1969) and Muhs (1971) adapted by pressure, q. Eurocode 7 (2005) and DIN 4017 (2006): N = 2 ( N q - 1) tan f (23) 1.5.2 The Factor N 1.5.2.1 N Formulations Brinch Hansen (1970): A closed-form analytical solution for the bearing capacity N = 1.5 ( N q - 1) tan f (24) problem including the effects of the unit weight of the soil beneath the footing via the factor N is not possible. Different Steenfelt (1977): solutions for N were developed based on empirical relations, analytical derivations, or numerical analyses. Some of these N = ( 0.08705 + 0.3231 sin ( 2 f ) - 0.04836 sin 2 ( 2 f )) solutions are listed below and are presented in Figure 11 for comparison. N q exp tan f - 1 (25) 2 1.5.2.2 Formulas Based on Empirical Relations Gudehus (1981): Formulas based on empirical relations are the following: ( N = exp 5.19 ( tan f ) 1.5 ) -1 (26) Meyerhof (1963): Ingra and Baecher (1983) for footings with L/B = 6: N = ( N q - 1) tan (1.4 f ) (22) N = exp ( -1.646 + 0.173 f ) (27) 1000 Ingra and Baecher (1983) for square footings: N = exp ( -2.046 + 0.173 f ) (28) 100 1.5.2.3 Formulas Based on Analytical Derivations Formulas based on analytical derivations are the following: Vesic (1973): Bearing capacity factor, N 10 N = 2 ( N q + 1) tan f (29) Chen (1975): Vesic (1973) N = 2 ( N q + 1) tan ( 45 + f 2 ) (30) Meyerhof (1963) 1 Brinch Hansen (1970) Chen (1975) Michalowski (1997) for a rough footing base: Ingra & Baecher (1983) L/B=6 N = exp ( 0.66 + 5.11 tan f ) tan f (31) Ingra & Baecher (1983) L/B=1 0.1 EC7 (2005) Zhu et al. (2001): Michalowski (1997) Bolton & Lau (1993) N = ( 2 N q + 1) tan (1.07 f ) (32) Zhu et al. (2001) Gudehus (1981) Steenfelt (1977) 1.5.2.4 Formulas Based on Numerical Analyses 0.01 0 5 10 15 20 25 30 35 40 45 50 There is one formula based on numerical analyses: Friction angle, f (deg) Bolton and Lau (1993): Figure 11. Bearing capacity factor N versus friction angle (f ) according to different proposals. N = ( N q - 1) tan (1.5 f ) (33)

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18 1.5.3 General Bearing Capacity Formulation 1 Meyerhof (1953) The basic equation by Terzaghi has been modified to ac- Giraudet (1965) count for the effects of the shape of the footing, load inclina- Ticof (1977) 0.8 Ingra & Baecher (1983) tion, load eccentricity, and shear strength of the embedment Bowles (1996) depth on the ultimate bearing capacity. Some of these modi- Paolucci & Pecker (1997) fications were incorporated originally by Meyerhof (1953) Gottardi & Butterfield (1993) 0.6 Perau (1995) and then further enhanced by Meyerhoff (1963), Brinch qu / qu, centric Hansen (1961, 1970), and Vesic (1973, 1975) to give what is known as the General Bearing Capacity Equation: 0.4 qu = cN c sc dc ic + qN q sq dq iq + (1 2 ) B N s d i (34) where 0.2 si = shape factors, ii = load inclination factors, di = depth factors, and 0 B = is the effective (i.e., functional) width of the footing 0 0.1 0.2 0.3 0.4 0.5 considering the effect of load eccentricity (see Equa- Load eccentricity to footing width ratio, e/B tion 35). Figure 12. Reduction factors for shallow foundations under vertical-eccentric load. Various approaches for the calculation of these factors including evaluation and critical review are presented in the Meyerhof (1953): following sections. 2 qu e = 1- 2 (36) 1.5.4 Eccentricity qu , centric B The effect of eccentric loading on the bearing capacity is Giraudet (1965): usually accounted for via Meyerhof 's (1953) effective area consideration. Bearing capacity is calculated for the footings' qu e 2 effective dimensions by the following: = exp -12 (37) qu , centric B L = L - 2 i eL B = B - 2 i eB with e B = M L V and e L = M B V (35) Ticof (1977): 2 qu e where = 1 - 1.9 (38) M, MB and ML = the moments loading in L and B direc- qu , centric B tions, respectively; V = the total vertical load; and Bowles (1996): eL and eB = load eccentricities along footing length (L) and footing width (B), respectively. qu e e = 1- for 0 < < 0.3 (39) qu , centric B B In contrast, other approaches describe the decrease in the bearing capacity with the increase in the eccentricity of the Paolucci and Pecker (1997): load using reduction factors. These factors indicate the ratio 1.8 of the average ultimate bearing capacity under eccentric load- qu e e = 1- < 0.3 0.5 B for (40) ing, qu , to that under the centric vertical loading, qu,centric. The qu , centric B formulas are mostly based on small-scale model tests on cohe- sionless soils without embedment, i.e., embedment depth of Ingra and Baecher (1983): the foundation (Df) = 0 and c = 0. Some approaches are spec- 2 ified below, and their evaluations are presented in Figure 12. qu e e = 1 - 3.5 + 3.03 (41) The approaches are the following: qu , centric B B

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19 Gottardi and Butterfield (1993): 1 Meyerhof (1953) qu e Ticof (1977) = 1- (42) Ingra & Baecher (1983) qu , centric 0.36 B 0.8 Perau (1995) Experimental results Perau (1995, 1997): (n=61) qu e = 1 - 2.5 (43) 0.6 qu / qu, centric qu , centric B Figure 12 presents the ratio of eccentric to centric load capac- 0.4 ity versus the ratio of load eccentricity to the smaller foot- ing width (B) of a strip footing. From the figure, it can be seen that the influence of load eccentricity in the approaches of Meyerhof (1953), Ticof (1977), and Ingra and Baecher (1983) 0.2 is very similar. The curve according to Bowles (1996) shows a different progression, beyond an eccentricity of e/B = 0.1. Here, the decrease of the bearing capacity is less pronounced as com- 0 pared to the three aforementioned approaches. In contrast, 0 0.1 0.2 0.3 0.4 0.5 the approach by Giraudet (1965) shows a completely different Load eccentricity to footing width ratio, e/B progression and a much smaller reduction of bearing capacity Figure 13. Reduction factors for shallow foundations for smaller load eccentricities. One cannot derive conclusions under vertical-eccentric load compared with test results regarding the validity of the different approaches based on this from different authors as presented by Perau (1995). figure alone. For example, it seems that Meyerhof's (1953) The experimental results presented are from Ramelot approach leads to a greater bearing capacity; however, this is and Vanderperre (1950) as cited by Derken (1969) not entirely so. The change in the shape factors because of for B/L = 1; Meyerhof (1953) for B/L = 1, 1/6, and 6; the change in the footing size, as effective width and effective Schultze (1952) for B/L = 2; Das (1981) for B/L = 1/3; length, must be considered as well. Giraudet (1965) for B/L = 1/3.5; and Eastwood (1955) for B/L = 1/1.8, 1/2.25, and 1/3. Figure 13 shows some of the reviewed approaches together with experimental results cited by Perau (1995). It can be seen that the three selected equations (Meyerhof, 1953; Ticof, 1977; width (L/B) ratios under centric vertical loading and without and Ingra and Baecher, 1983) represent a lower boundary of embedment have been modeled and analyzed. the experimental results. Figures 14 and 15 present the numerical values of the afore- mentioned shape factors s and sq, respectively, versus the 1.5.5 Shape Factors foundation width to length ratios, B/L. Due to the fact that the bearing capacity of Equation 19 was developed for strip The effect of a foundation shape other than a strip footing footings assuming plain strain conditions, the values of the (plain strain condition) has to be considered with foundation shape factors approach unity for long footings (as B/L 0). shape factors. A footing is theoretically defined as a strip foot- Practically, the value of s is within the range of 1 0.05 for ing for the length to width ratios of L/B > 10. Practically, foun- L/B 6.7 (B/L 0.15), and the value of sq is within the same dations possessing the ratio of L/B > 5 already behave as strip range for L/B 10.0 (B/L 0.10) for most cases. footings (Vesic, 1975). Due to the difficulties in obtaining For footings with dimension ratios close to unity (approach- mathematical solutions that consider the effect of a founda- ing equidimension), the deviations of the shape factors from tion shape, semi-empirical approaches have been formulated. the unity proposed by different authors show that very careful Various relations proposed for shape factors, si, are listed in consideration is required in the choice of the shape factors. Table 4. For eccentrically loaded footings, the effective foot- The values suggested by Meyerhof (1963) for s (see Figure 14) ing dimensions B and L have to be used to compute the are always greater than unity and increase with the decrease shape factors (e.g., AASHTO, 2007; EC 7, 2005). in the width to length ratio (B/L). In contrast, the values cal- The presented shape factors in Table 4 are empirical except culated with other equations decrease below unity as the ratio for the expressions by Zhu and Michalowski (2005) that have increases. The reason for this is that Meyerhof's (1963) val- been derived from numerical simulations. For example, to ues of N for a strip footing (B/L 0) are smaller than those determine the shape factor, s, footings with different length to for a circle (B/L = 1), and the bearing capacities for the footing

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20 Table 4. Shape factors proposed by different authors. Footing Reference base shape sc sq s modified by Vesic De Beer (1970) as B Nq B B Rectangle 1 1 tan f 1 0.4 L Nc L L (1973) Circle and Nq 1 1 tan f 0.6 Square Nc sq N q 1 DIN 4017 (2006) EC 7 (2005) and B B Rectangle 1 sin f 1 0.3 Nq 1 L L Circle and sq N q 1 1 sin f 0.7 Square Nq 1 Meyerhof (1963) B 1 ; for f 0 1 0.1 Kp; B L Rectangle 1 0.1 Kp 1 0.1K p ( B / L ); L Kp tan 2 45 f for f 10 2 Perau (1995, 1997) 1 1.6 tan f 1 B /L Rectangle Not applicable 2 B B 1 1 L L Zhu and Michalowski 1 (0.6sin 2 f 0.25) B / L for f 30 ; (2005) Rectangle Not applicable Not applicable 1 (1.3sin 2 f 0.5)( L / B )1.5 exp( L / B ) for f 30 with width to length ratios between B/L 0 and B/L = 1 are angle. The depth factors proposed by Brinch Hansen are linearly interpolated values. Hence, a consistent set of equa- greater than those proposed by Meyerhof. The depth factors tions for the bearing capacity factors and their modifications listed in AASHTO (2007) are also shown in Figure 16. These by the same author are recommended for use in the bearing values lie between the expressions proposed by Meyerhof capacity calculation. In summary, the foundation shape (vary- and Brinch Hansen. ing between strip to equidimensional footing) and hence, the shape factor have an important influence on the ultimate bear- 1.5.7 Load Inclination Factors ing capacity. An inclination in the applied load always results in a reduced bearing capacity, often of a considerable magnitude (Brinch 1.5.6 Depth Factors Hansen, 1970). Meyerhof (1953) suggested that the vertical If the foundation is placed with a certain embedment depth, component of the bearing capacity under a load inclined at an Df, below the ground surface, the bearing capacity is affected angle to the vertical is obtained using the following inclina- in two ways: one, by the overburden pressure, q = Df, and tion factors: two, via the shear strength of the soil above the base level. ic = iq = (1 - 90 ) 2 Table 5 presents typically used expressions of the depth (44) factors. Figure 16 presents the values of the depth factor dq i = (1 - f ) 2 versus the friction angle for the different expressions pro- (45) vided in Table 5. In contrast to the factors proposed by Meyerhof (1963), the depth factor dq according to Brinch These expressions were modified by Meyerhof and Koumoto Hansen (1970) decreases with the increase in the soil friction (1987) and presented for cases of footings on a sand surface,

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21 1.2 1 0.8 Shape factor, s 0.6 Vesic (1973) Din 4017 (2006) Perau (1995) 0.4 Meyerhof (1963) f =25 Meyerhof (1963) f =37 Zhu and Michalowski (2005) f =25 Zhu and Michalowski (2005) f =37 0.2 0 0.2 0.4 0.6 0.8 1 Footing side ratio, B/L Figure 14. Shape factor s proposed by different authors versus footing side ratio, B/L. 2 Vesic (1973) f =25 Vesic (1973) f =37 DIN 4017 (2006) f =25 1.8 DIN 4017 (2006) f =37 Perau (1995) f =25 Perau (1995) f =37 1.6 Shape factor, sq 1.4 1.2 1 0 0.2 0.4 0.6 0.8 1 Footing side ratio, B/L Figure 15. Shape factor sq proposed by different authors versus footing side ratio, B/L.

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22 Table 5. Depth factors proposed by different authors. Author dc dq d Df Meyerhof Df dq 1 0.1 K p for f 10 (1963) dc 1 0.2 K p B d dq B 1 for 0 f 1 dq Df / B 1: Brinch dc dq Hansen N C tan f dq 1 2 tan 1 sin 2 Df / B f f (1970) and 1 dq 1 Vesic dq Df / B 1: (1973) Nq 1 2 dq 1 2 tan f 1 sin f arc tan D f / B where K p tan 2 (45 f / 2) when the embedment ratio (Df /B) is unity, and for footings For footings on the surface of clay: on a clay surface, as shown in Equations 46 through 48. Assum- ing that a footing with a perfectly rough base on a sand surface i = cos (1 - sin ) for ca = 0 starts to slide when the load inclination angle to the vertical = cos (1 - 0.81 sin ) for ca = cn = undrained shear is approximately equal to the soil's friction angle, the follow- strength of the clay (48) ing expression was proposed: where ca = adhesion between the clay and the base of the sin footing. i = cos 1 - for D f B = 0, c = 0 (46) sin f Muhs and Weiss (1969) suggested, based on DEGEBO (Deutsche Forschungsgesellschaft fr Bodenmechanik) tests For the case of footings with an embedment ratio equal to with large-scale models of shallow footings on sands, that 1 in a soil with a friction angle greater than 30, the inclina- there is a distinct difference between load inclination effects tion factor was expressed as the following: when the inclination is in the direction of the longer side, L, and when the inclination is in the direction of the shorter i = cos (1 - sin ) for f > 30, D f B = 1, c = 0 (47) side, B. Thus, the direction of load inclination as well as the 1.4 Brinch Hansen (1970) Df /B=1 1.3 Brinch Hansen (1970) Df /B=2 Brinch Hansen (1970) Df /B=4 Depth factor, dq Meyerhof (1963) Df /B=1 1.2 Meyerhof (1963) Df /B=2 Meyerhof (1963) Df /B=4 AASHTO (2007) Df /B=1 1.1 AASHTO (2007) Df /B=2 AASHTO (2007) Df /B=4 1 28 30 32 34 36 38 40 42 44 Soil friction angle, f (deg) Figure 16. Depth factor dq proposed by different sources versus soil friction angle, f.

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23 ratio B/L affect the inclination factor. Brinch Hansen (1970) 1 incorporated the inclination effects as Brinch Hansen (1970) Meyerhof (1953) f =35 0.8 Vesic (1975) L/B =1 (=90) 5 0.5 H Vesic (1975) L/B =5 (=90) Load inclination factor, iq iq = 1 - (49) (V + Ac cot f ) 0.6 5 0.7 H i = 1 - (50) 0.4 (V + Ac cot f ) 0.2 Vesic (1975) proposed the factors in the following forms: n 0 H iq = 1 - (51) (V + Ac cot f ) 0 0.2 0.4 0.6 0.8 1 Load ratio, H/V n+1 Figure 18. Load inclination factor iq versus load H ratio, H/V, for c = 0, f = 35, and any Df /B. i = 1 - (52) (V + Ac cot f ) (2 + L B) 2 (2 + B L ) 2 The inclination factor ic results from Caquot's theorem of n= cos + sin (53) (1 + L B ) (1 + B L ) corresponding stress states (De Beer and Ladanyi 1961 and Vesic 1973 as cited by Vesic 1975): where 1 - iq 1 - iq H and V = the horizontal and vertical components of the ic = iq - = iq - for f > 0 (54a) N c tan f Nq - 1 applied inclined load, P (see Figure 17); = the projected direction of the load in the plane of nH ic = 1 - for f = 0 (54b) the footing, measured from the side of length L A c N c in degrees; A = the effective area of the footing; where iq is given by Equation 51. c = soil cohesion; and L and B are as defined in Equation 35. 1 Figures 18 and 19 are graphical presentations of Equa- Brinch Hansen (1970) tions 49 through 53 for load inclination factors iq and i , Meyerhof (1953) f =35 respectively. 0.8 Meyerhof & Koumoto (1987) Df /B=0, f =35 Load inclination factor, i Vesic (1975) L/B =1 (=90) 0.6 Vesic (1975) L/B =5 (=90) V P 0.4 0.2 H L 0 B 0 0.2 0.4 0.6 0.8 1 Load ratio, H/V Figure 17. Inclined load without eccentricity and Figure 19. Load inclination factor i versus load the projected direction, . ratio, H/V, for c = 0, f = 35, and Df /B = 0.