Críticas a los axiomas de la Utilidad Esperada

Based on plasticity theories, the ultimate end-bearing resistance, qb, for piles in granular soils may be expressed in terms of vertical effective stress, σv', and the bearing capacity factor, N_q as :

qb = Nq σv' [6.3]

Nq is generally related to the angle of shearing resistance, φ'. Values of Nq factor quoted in the literature vary considerably. N_q can be determined based on the bearing capacity factor in Table 3.1. Davies & Chan (1981) suggested the values presented by Brinch Hansen (1970), while both Poulos & Davis (1980) and Fleming et al (1992) recommended the use of factors derived by Berezantzev et al (1961), which is also supported by Vesic (1967). Poulos & Davis (1980) further suggested that for the determination of Nq, the value of φ' should be reduced by 3° to allow for possible loosening effect of installation. For general design purposes, it is suggested that the Nq values based on Poulos & Davis (1980) as presented in Figure 6.2 may be used.

The calculated ultimate end-bearing resistance should conservatively be limited to 10 MPa, unless higher values have been justified by loading tests. It is prudent to apply an upper limit on the q_b value because the angle of shearing resistance and hence the

end-bearing resistance may be reduced due to suppressed dilation and possible crushing of soil grains at high pressure.

The ultimate shaft resistance (τs) for piles in granular soils may be expressed in terms of effective stresses as follows :

τs = c' + Ks σv' tan δs [6.4]

τs = β σv' (where c' is taken as zero) [6.5]

where Ks = coefficient of horizontal pressure which depends on the relative density and state of the soil, method of pile installation, and material, length and shape of the pile

σv' = mean vertical effective stress

δs = angle of interface friction along pile/soil interface β = shaft resistance coefficient

The angle of interface friction is primarily a function of the nature of pile material and the state of the ground, and it can be reasonably determined in a shear box test (Lehane, 1992). For bored piles in granular soils, δs can be taken as equal to the friction angle of the shearing resistance, φ'. Ks may be related to the coefficient of earth pressure and the ratio Ks/Ko varies between 0.67 and 1 (Kulhawy, 1984). The determination of Ko is notoriously difficult as it is a function of stress history and not a fundamental soil property. In the case of

Figure 6.2 – Relationship between Nq and φ' (Poulos & Davis, 1980)

For driven piles, φ' = φ'1 + 40 2 For bored piles, φ' = φ'₁ – 3 where φ'1 is the angle of shearing resistance prior to installation.

10 100 1000

25 30 35 40 45

Angle of Shearing Resistance, φ' (°) Be

ari ng Ca pa cit y Fa cto r, Bearing Capac

ity Factor, Nq

saprolites, the Ko value may be lower than that given by the conventional formula Ko = 1 - sin φ' due to possible effects of bonding (Vaughan & Kwan, 1984). This is supported by deduction from field measurements in Hong Kong as reported by Endicott (1982) and Howat (1985).

It should be noted that the Ks value is a function of the method of pile construction.

In view of the uncertainties associated with assessing Ko and the effects of construction method, it may be more reasonable to consider the combined effect as reflected by the β values deduced from loading tests on piles in saprolites. It must be noted that in relating τs to σv' with the use of the β factor, it is assumed that there is no cohesion component (c').

Although there may be some cohesion for undisturbed saprolites, the effect of construction on c' of the soil at the interface with the pile is difficult to evaluate and may be variable. The β values back analysed from pile loading tests would have included any contribution from c' in the measured τs.

So (1991) postulated that the shaft resistance of a pile in a bonded soil such as dense saprolites may be dominated by the increase in horizontal stresses due to its tendency to dilate during shearing. This may explain isolated loading test results (e.g. Holt et al, 1982;

Sweeney & Ho, 1982) which indicated a continual increase in shaft resistance at large relative displacement of up to about 4% of pile diameter (viz. 39 mm). Based on cavity expansion theory, So (1991) suggested that the dilation and hence the shaft resistance in a small-diameter pile will be greater than that in a large-diameter pile. At present, this remains a conceptual model and has not been sufficiently validated by loading test results. However, it is possible that this dilation effect compensates the small insitu stresses in the saprolites such that pile capacity is broadly similar to that in a sedimentary granular deposit. On the other hand, Nicola & Randolph (1993) and Lehane & Jardine (1994) discussed the effect of pile stiffness on the mobilisation of shaft resistance.

Table 6.3 summarises the range of β values interpreted from the pile loading tests conducted in saprolites in Hong Kong. These values are comparable to those suggested by Meyerhof (1976) for bored piles in granular soils (Figure 6.3). These values may be used for bored piles in granular soils.

Available instrumented loading test data from large-diameter bored piles in saprolites (Appendix A) indicate that substantial shaft resistance is mobilised at a relative pile-soil movement of about 1% pile diameter (about 10 to 15 mm), in many cases. Based on the available loading test results in Hong Kong, it is suggested that the calculated average ultimate shaft resistance should be limited to 150 kPa for granitic saprolites unless a higher value can be justified by site-specific loading tests. Plumbridge et al (2000a) reported the results of loading tests on shaft-grouted bored piles and barrettes for the West Rail project.

The maximum shaft resistance measured was 220 kPa. For preliminary design of piles in saprolites, the typical values given in Tables 6.3 may be used to calculate the shaft resistance using the effective stress method. It should be noted that values of β in Table 6.3, are based on back analysis of field test data. Therefore, the effective stress method is essentially a semi-empirical design approach.

Table 6.3 – Typical Values of Shaft Resistance Coefficient, β, in Saprolites and Sand

Type of Piles Type of Soils Shaft Resistance Coefficient, β

Saprolites 0.1 – 0.4

Driven small displacement piles

Loose to medium dense sand⁽¹⁾ 0.1 – 0.5

Saprolites 0.8 – 1.2

Driven large displacement piles

Loose to medium dense sand⁽¹⁾ 0.2 – 1.5

Saprolites 0.1 – 0.6

Bored piles &

barrettes

Loose to medium dense sand⁽¹⁾ 0.2 – 0.6 Shaft-grouted bored

piles & barrettes

Saprolites 0.2 – 1.2

Notes : (1) Only limited data is available for mobilised shaft resistance measured in loose to medium dense sand.

(2) Refer to Appendix A for details.

0 0.1 0.2 0.3 0.4 0.5

30 32 34 36 38 40

Angle of Shearing Resistance, φ' (°)

Shaft Resistance Coefficient, β

Figure 6.3 – Relationship between β and φ' for Bored Piles in Granular Soils (Figure adopted from Poulos & Davis (1980) based on interpretation of results given by Meyerhof (1976))

It should be cautioned that data also exist in Hong Kong for large-diameter bored piles showing very low shaft resistance in dense to very dense granitic saprolites, although it is possible that these were a result of problems associated with pile construction. In view of the possible adverse effects of construction, the assumptions concerning design parameters, construction method and workmanship should be verified by load testing of instrumented piles when friction bored piles are proposed, until sufficient local experience has been built up.

The behaviour of piles in colluvium may be greatly affected by the presence of boulders (e.g. Chung & Hui, 1990). However, a lower bound estimate may be made based on the properties of the matrix material and using the effective stress method for design.