NUMERICAL ANALYSIS OF NSF USING FEM
6.2 NSF ON END-BEARING SINGLE PILES
6.2.6 Degree of Mobilization for End-bearing Piles
Both centrifuge model tests and the numerical analysis presented above have clearly demonstrated the partial mobilization of NSF around neutral point which will lead to overestimation of maximum downdrag forces if the popular β method is to be used to calculate dragloads in piles. Poulos and Davis (1980) proposed a “correction factor” for cases in which full slip does not occur around the neutral point. Based on simplified boundary element method, they concluded that a decrease in pile-soil stiffness ratio or pile length-diameter ratio tends to decrease the correction factor, but reckoned that the effects should be generally small. This is not in line with the postulation by Matyas and Santamarina (1994) that the relative stiffness between the soil and the pile would be expected to substantially alter the partially mobilized zone.
So far, no conclusive guidelines in this aspect are available. For example, Singapore Code of Practice for Foundation (CP4: 2003) introduces a factor, η, to account for the degree of mobilization when calculating NSF and proposes that the value is “typically 0.67, although 1.0 may be used in specific cases”. A better understanding and estimation of the mobilization factor will directly lead to a better estimation of maximum dragload on a pile which is important in the design of piles with NSF.
Both centrifuge model tests and the numerical analyses have demonstrated that the length of the partial mobilization zone above the neutral point was observed to be
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dependent on the intensity of surcharge loads. For example, in the present specific condition, the partial mobilization zone ranges from about 3 m at the water drawdown stage to less than 1 m at the surcharge stage. As such, the mobilization factor in each case will be different. On the other hand, the relative stiffness of the pile and the surrounding soil may affect the results substantially as proposed by Matyas and Santamrina (1994). What is more, the effect of the pile length-diameter ratio (hereinafter referred to as pile slenderness ratio) may deserve a careful investigation as well, although Poulos and Davis (1980) suggested that the effects may be minor.
To gain a better understanding on the mobilization factor under various pile-soil conditions, an extensive parametric study was conducted using the axisymmetric FEM analysis which has been calibrated against centrifuge model test as presented in the preceding sections. Before presenting the numerical results, some symbols and terminologies used are explained as follows:
(1) η -- degree of mobilization of NSF due to the partial mobilization of NSF around neutral point, which is defined as,
(6.24)
where Qn,mob is the actual mobilized maximum dragload at the neutral point. Qn,β is the calculated maximum dragload at the neutral point based on β method which is given as:
(6.25)
, , n mob
n
Q Q β η=
, 0Zn '
n v
Q β =∫ βσ dz
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in which Zn is the depth of the neutral point and β assumes a typical value of 0.24 for clay based on the present model tests.
(2) K – pile-soil stiffness ratio, the same as that used by Poulos and Davis (1980), which is defined as
(6.26)
where Ep and Es are the Young’s modulus of the pile and soil, respectively. RA is the pile section ratio which is unity for solid piles and, in case of hollow pile section, is equal to:
(6.27)
in which d is the pile outer diameter and Ap is the solid area of the pile cross section.
(3) L/d – pile length-diameter ratio, namely pile slenderness ratio, where L and d are the pile length and pile diameter, respectively.
The modulus of concrete pile is typically in the range of 2.0E+7 to 5.0E+7 kPa depending on the grade of concrete. Steel piles typically have modulus approximately one order higher than concrete. However, they are typically of hollow sections with a relatively smaller section ratio RA as defined in Eq. (6.27). On the other hand, NSF problem typically involves soft soils which normally possess relatively low stiffness,
s A p
E R K = E
4 / d2
RA Ap
=π
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say, not exceeding 20,000 kPa. In view of the above data, the pile-soil stiffness ratio K as defined by Eq. (6.26) is likely to be in the following range:
(6.28)
A stocky end-bearing pile with length L = 12 m and diameter d = 1 m, resulting in a slenderness ratio of 12, and a relatively much slender pile with L = 48 m and d = 1 m, resulting in a L/d ratio of 48, are used to illustrate the variation of degree of mobilization under various pile and soil conditions. In each case, the pile-soil stiffness ratio was varied from 1,000 to 40,000, while the surcharge was increased from 10 kPa in steps to 70 kPa. For clarity of presentation, only the analysis results of surcharge 10 kPa and 55 kPa are presented in Fig. 6.7. It should be noted that the depth z has been normalized by the clay depth L, while the dragload along the pile shaft has been normalized by the maximum dragload at the neutral point, Qn,β calculated by the β method. For comparison purpose, the normalized β curves are also plotted in Fig. 6.7.
It can be seen that in all cases, the normalized dragload around the neutral point (namely the pile toe in the present case) is less than the corresponding β curve. For a stocky pile shown in Fig. 6.7(a), the normalized dragload on the pile is not much affected by the pile-soil stiffness ratio K, regardless of magnitudes of surcharge intensity. This is understandable as an end-bearing stocky pile normally has small pile settlement under dragload and the relatively larger soil settlement tend to play a dominate effect and cause a large degree of mobilization of NSF on the pile. On the other hand, for a slender pile shown in Fig. 6.7(b) with a relatively small magnitude of surcharge loading of 10 kPa, the relative pile-soil settlement tends to increase as pile- soil stiffness ratio K increases, resulting in an increase of degree of mobilization of
000 , 40 000
,
1 ≤K ≤
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NSF. With a much larger surcharge magnitude of 55 kPa, the much larger soil settlement dominates the relative pile-soil settlement resulting in a much larger degree of mobilization of NSF with the effect of K substantially diminished as shown in Fig.
6.7(b).
The results of all the analyses on the degree of mobilization of NSF for various pile slenderness ratio L/d, pile-soil stiffness ratio K and intensity of surcharge loading q are summarized in Fig. 6.8. The general findings are presented as follows:
(a) Under the same pile slenderness ratio L/d, the degree of mobilization increases with surcharge intensity, but tends to taper off after the surcharge intensity is beyond 40 kPa. On the other hand, increase in pile-soil stiffness ratio K tends to increase the degree of mobilization as well, but with minor degree of influence for stocky pile (see for example Fig. 6.8(a)) and substantial degree of influence for slender pile (see for example Fig. 6.8(d)).
(b) Under the same pile-soil stiffness ratio K, the degree of mobilization tends to decrease with increase in pile slenderness ratio, but increases with the increase in surcharge loadings.
(c) Under the same surcharge loading, the degree of mobilization generally increases with pile-soil stiffness K. This is more obvious for slender piles than for stocky piles (compare, for example, Fig. 6.8(d) with Fig. 6.8(a)). A very large magnitude of surcharge will dominate the degree of mobilization with the influence of pile-soil stiffness ratio greatly diminishes.
It can be seen from Fig. 6.8 that the degree of mobilization can vary within very large ranges from as low as about 35% to as high as about 95% depending on the pile-
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soil conditions and surcharge intensity. From a practical point of view, Fig. 6.8 provides a useful guide for deriving the specific degree of mobilization, η, and hence more appropriate evaluation of maximum dragload on end-bearing piles subjected to NSF. It thus supplement qualitatively to some Foundation Codes like Singapore CP4 when it comes to the selection of an appropriate degree of mobilization to be adopted in practical applications.