Estimating Young Moduli in Sands from the Normalized N 60 Blow Count

. This paper presents statistical correlations between the static Young modulus E s and the normalized SPT penetration resistance N 60 (corresponding to 60% of the theoretical energy of the SPT test) for preliminary estimation of settlements in sedimentary sands. The correlations have been established through the statistical interpretation of many experimental studies available in the literature, based on power regression analyses. Equations and charts are presented to estimate the mean value E s and related statistical limits of E s as a function of N 60 . The obtained correlations have been used to compare the calculated settlements of a rigid raft bearing on a multi-layered sand deposit (comprising distinct relative densities) with the measured settlements in eight points underneath the raft.


Introduction
The use of seismic tests to obtain correlations between the shear wave velocity V s and N SPT is widely used in foundation engineering. However, most of the published correlations do not provide the statistical inferences used to support the published results. Also, many of these correlations do not take into consideration normalized N 60 blow counts, causing a large scatter of the data compiled from different countries. Hanumantharao & Ramana (2008) compiled several correlations between V s and N SPT (uncorrected) worldwide for different soil types. It is well known that the low amplitude shear modulus (G max ) is directly related to the shear wave velocity and to the density of the soil through which the wave travels (Richart et al., 1970). Thus, G max is estimated by measuring the shear wave velocity and the soil density to predict the stress-strain behavior of soils under low amplitude dynamic excitation. Unfortunately the use of seismic tests is not yet widespread in Brazil, while the SPT is still the most used technique for site investigation there and in many other countries, what supports the development of several correlations between V s and N SPT worldwide. Anbazhagan & Sitharam (2010) developed recent relationships between V s and N SPT and pointed out that better correlations could be obtained by correlating N SPT and G max also to the overburden stress.
The most common relationship between V s to N SPT is of the type given by Eq. 1, although Eq. 2 is also commonly used. In both equations A, B and C are regression constants and N may represent either N SPT or N 60 .
The correlations compiled by Hanumantharao & Ramana (2008) for sands have been used to form the database of the present paper, along with further correlations by Ohsaki & Iwasaki (1973), Jafari et al. (2002), Athanasopoulos (1995), Wride & Robertson (1997), Wride et al. (2000) and Robertson et al. (2000). In this paper, a power regression analysis (Eq. 1) is used to relate statistically V s to N 60 . It is worth noting that the velocity of the shear wave can be determined reliably by seismic field tests such as crosshole, downhole, uphole, seismic piezocone, seismic refraction, among others. The velocity V s and all parameters related to V s are limited to small strains, of the order of 10 -4 % ( Barros, 1997). Hanumantharao & Ramana (2008) mention that the measured shear wave velocities obtained from different field methods differ only by the order of 10 to 15%. This may be regarded as negligible when compared to the uncertainties to estimate dynamic moduli and make feasible the use of correlations between V s and N 60 , even when V s is determined by different field techniques. Giachetti et al. (2006) and Moura (2007) reported a difference of only 6.7% in the average shear wave velocity determined by downhole and crosshole tests in a fine clayey sand from the region of Bauru (State of São Paulo, Brazil).
Starting from published correlations between V s and N SPT , regression equations relating G max and E max (small strain dynamic Young modulus) to V s are readily obtained by well known equations of the classical theory of elasticity. Further correlations between E max and the static Young modulus E s , according to Buzdugan (1972), enable the final correlations between the static Young modulus E s and the corrected N 60 blow count for sands, as proposed in this paper. Hanumantharao & Ramana (2008) compiled several correlations between V s and N SPT for clayey and sandy soils. In the present paper, however, emphasis is given only to correlations in sands, according to Table 1. Wride et al. (2000) and Robertson et al. (2000) summarized the Canlex research project (Canadian Liquefaction Experiment) whose main objective was the study of soil liquefaction in saturated sandy soil. From the Canlex database some correlations between shear wave velocity V s and N 60 have been established and presented in the research report from different sites, also included in Table 1. The fourth column in Table 1 refers to the range of N SPT corresponding to the sands investigated by each author. The fifth column refers to the range of N 60 for the correlations established in the Canlex research project.

Regression Equations Relating E max to N 60
Selection of appropriate correction factors are required to convert N SPT into N 60 to normalize the correlations in Table 1 according to the actual energy delivered during the SPT test in each country. Several authors have proposed correction factors to account for the actual energy transmitted to the rods in the SPT test (De Mello, 1971;Kovacs et al., 1977;Palacios, 1977;Schmertmann & Palacios, 1979;Belincanta, 1985;Danziger et al., 2008, among others). The ISSMFE (1989) established 60% of the theoretical potential energy as the international reference. Therefore, N SPT should be converted to N 60 by the expression: In Eq. 3 E represents the actual energy delivered to the rods in the SPT test and E 60 refers to 60% of the theoreti-cal potential energy of the SPT hammer. If the energy E is measured, the above expression should be used. Otherwise, an estimated value for E based on past experience is required. Therefore, correcting N SPT is essential to compare correlations from different countries. Décourt (1989) pointed out that the energy just below the anvil (E) can be obtained as: where e 1 , e 2 , e 3 , are the efficiencies (or correction factors). The efficiency factor e 1 relates the kinetic energy just before the impact, being mainly dependent on the way the hammer is lifted and released. Values of e 1 suggested by Décourt (1989) are shown in Fig. 1.
The factor e 2 is the ratio between the energy just below the anvil and the kinetic energy just before the impact and it is dependent on the anvil mass (Skempton, 1986). Figure 2 summarizes the main results (Décourt, 1989). 90 Soils and Rocks, São Paulo, 35(1): 89-98, January-April, 2012.
The efficiency factor e 3 is due to the rod length. According to Schmertmann & Palacios (1979) the driving energy would only be fully transmitted to the rods if they had a minimum critical length. This would occur because in most cases the first compression wave pulse is reflected in the lower end of the sampler as a tension wave. Therefore the tension wave induces a separation between the hammer and the rods, preventing further transfer of energy. Recent researches, however, have shown that the subsequent (secondary) impacts in the same blow contribute to full energy transmission, indicating that the energy just below the anvil is independent of the length of the rod stem, and thus the factor e 3 should be taken as unity. Odebrecht (2003) and Odebrecht et al. (2005) have shown that the potential energy resulting from the penetration of the sampler should be added to the nominal potential energy, which is significant mostly in case of small rod lengths in soft clays and loose sands. Aoki & Cintra (2000) pointed out that the energy producing the sampler penetration (which is associated to the N value) is the one that reaches the sampler, and not the one below the anvil. Thus, the corresponding energy loss over the rods should also be taken into account. Thus another factor e 4 that quantifies the energy loss over the rods should also be included in Eq. 4. Unfortunately, very few data is available about this factor (e.g., Cavalcante, 2002;Odebrecht, 2003;Johnsen & Jagello, 2007).
It turns out that the only reliable way to quantify the SPT energy losses is by proper measurement of the actual energy delivered to the sampler. In this paper, N 60 values are assigned according to average correction factors C reported by Décourt et al. (1989), as shown in Table 2. For example, the value C = 1.05 (USA) in Table 2 refers to the average value for 0.75 (donut/rope-cathead); 1.00 (safety/hope/cathead) and 1.40 (safety/free fall). The correction factor C is given by: For the Brazilian SPT, Décourt (Table 2) proposed: Further research based on actual energy measurements on the SPT hammers mostly used in Brazil (e.g., Belincanta, 1985Belincanta, , 1998Cavalcante, 2002;Odebrecht, 2003) has indicated: Equations 7 and 8 are used in Session 3 to enable correlations for practical application in Brazil. Hanumantharao & Ramana (2008) pointed out that most correlations may produce acceptable V s predictions for N SPT values up to 40. According to the Authors experience in settlement predictions based on N SPT , the correlations above require good engineering judgment for N SPT > 30 and should be avoided for N SPT above 50 or below 4.
To relate E max to N 60 , the following equations of the classical theory of elasticity are used: From Eqs. 9 and 10: In the equations above n is the Poisson ratio, r the soil unit mass, g the soil unit weight and g the acceleration of gravity. Equation 11 enables to relate E max to N 60 by assigning a power regression curve to V s (Eq. 1). Therefore, each pair N 60 x E max (for each regression equation in Table 1) is plotted in Fig. 3 for all integer N SPT and N 60 values according to the ranges given by the two last column of Table 1. N SPT values are corrected to N 60 according to Table 2. The points in Fig. 3 have been plotted by assigning n = 0.3, g = 18 kN/m 3 and g = 9.8 m/s 2 in Eq. 11. According to Conde de Freitas (2010)    impact on Eqs. 9, 10 and 11. Therefore, a constant value g = 18 kN/m 3 was assigned throughout this work. Attention should be pointed out that Eqs. 9 and 10 have been derived from the classical theory of elasticity that assumes a constant modulus for soil mass. Such a simplification is only justified in the proposed correlations as long as its use has been conceived as preliminary settlement estimations.
To fit a regression power curve through the 357 points in Fig. 3a, a transformed linear regression analysis is carried out by plotting log(N 60 ) x log(E max ), as in Fig. 3b (Conde de Freitas, 2010). The desired regression power curve in Fig. 3a is readily determined by mapping back the linear regression curve (Fig. 3b) into the original plot (N 60 x E max ), producing: For comparison, an alternative relationship between N SPT (uncorrected) and E max is readily obtained from Eq. 13 below, widely used in the oil industry in Brazil for design of foundations for machinery (Petrobras, 2008;Machado, 2010): From Eqs. 9 and 13: Equations 12 and 14 are compared in Session 3 to estimate the static Young modulus E s , the main objective of this paper. Experimental data by Machado (2010) has confirmed the adequacy of Eq. 13 to estimate the shear modulus G max for design of foundations for machinery.
Statistical limits for the points in Fig. 3a have been determined in Fig. 4 (Conde de Freitas, 2010) according to Neter et al. (1982); Pacheco & Lima (1996), for n standard deviations about the mean regression curve (n ranging from 0.5 to 2).

Static Young Moduli
Equations 12 and 14 are applicable to engineering problems related to small strains, as in dynamic analyses of foundations for machinery. For static problems, however, a reduction factor is to be applied to the dynamic modulus E max to estimate the corresponding static Young modulus E s . Laboratory tests on reconstituted samples in sands indicate that the dynamic shear modulus may be reduced more than tenfold for shear deformations of 10 -3 % to 1% (Barros, 1997;Moura, 2007). According to Kulhawy & Mayne (1990); Moura (2007), the shear modulus for static loads is about 5 to 10% G max . A similar reduction is also reported by Sitharam et al. (2004), Fig. 5. Therefore it is important to keep in mind that estimates of E s (or G s ) should be made according to the range of shear deformation expected for each problem under consideration (Silveira et al., 2006). 92 Soils and Rocks, São Paulo, 35 (1) Buzdugan (1972) summarized the range of variation of dynamic (E max ) and static (E s ) Young moduli (Table 3) for foundation design. An average ratio E max /E s = 3 is inferred from Table 3 to estimate E s from Eqs. 12 and 14 in sands. This ratio is recommended only to well-designed foundations whose applied pressure is far from failure, as expected in large dimension foundations like rafts, tanks and silos. The ratio E max /E s is expected to increase for smaller safety factors (larger shearing strains) and therefore good engineering judgment and experience are required for practical use of the correlations proposed herein. Furthermore, the ratio E max /E s increases for clayey soils (Buzdugan, 1972). Therefore, the ratio E max /E s = 3 as well the correlations presented in this paper should also be restricted to sedimentary pure sands with negligible amount of fines, considering that even small percentages of fines are likely to produce higher E max /E s ratios.
The regression curves in Fig. 4 (N 60 x E max ) are converted into N 60 x E s in Fig. 6 for E max /E s = 3. The corresponding mean power curve in Fig. 6 (determined from Eq. 12) is given by: For comparison, the static modulus E s estimated from Eq. 14 for E max /E s = 3 is: For practical applications using the Brazilian SPT, curve (a) in Fig. 6 is obtained from Eq. 16 for C = 1.20 (Eq. 7), as: Alternatively, curve (b) in Fig. 6 is obtained from Eq. 16 for C = 1.37 (Eq. 7), as: Fig. 6 is very close to the curve corresponding to 1.0 standard deviations above the mean curve, whereas curve (b) nearly coincides with the curve corresponding to 0.5 standard deviations above the mean curve. The good agreement between curves (a), (b) and the mean curve given by Eq. 15 seems to indicate that the regression curves in Fig. 6 provide a simple and reasonable statistical procedure to estimate E s , in addition to providing a simple way to account for the uncertainty in the predictions (varia-Soils and Rocks, São Paulo, 35 (1)  It should be pointed out that the Young modulus E s increases with depth in nearly normally consolidated sand deposits and this can be easily accounted for in FE modeling. In the present study, however, the variation of E s with depth was modeled by Eq. 19 after subdividing the sand deposit into several thin layers represented by individual average N 60 values which account indirectly for the overconsolidation condition.

Settlement Analysis of a Raft on Multilayered Sand Deposit
The following is an application of Eq. 19 to predict the settlements of an instrumented nearly rigid raft bearing on a sedimentary sand deposit investigated by Lopes et al. (1994), and Lopes (2000). Those authors describe the measured settlement of a raft foundation during a period up to 3 years after completion of the structure. The raft supports the building housing a diesel power generator and applies to the soil a net uniform pressure Dp = 123 kN/m 2 (building + generator). The raft dimensions (16.6 m x 27.0 m) and the instrumentation points are shown in Fig. 7 (Lopes et al., 1994).
The sand deposit has been subdivided into six layers whose deformation parameters have been assigned in Table 4 according to the normalized N 60 blow count obtained from Fig. 8.
The soil profile is subdivided into six moderately inclined sand layers interpreted from the SPT boring logs, as depicted in Fig. 9 (input geometry to the finite element program Plaxis 3-D Foundation). The raft bears on top of the sand deposit above which was placed a 4-m high fill. The water table is assumed to coincide with the base of the raft (4 m below the fill surface). The Young modulus for each sand layer is estimated by Eq. 19 taking the average normalized N 60 blow counts in Table 4. Other input parameters to the numerical analysis are listed in Table 5.
In Table 5, g t is the total (saturated) unit weight, g h the natural (non saturated) unit weight, n the Poisson ratio, c the cohesion and f the friction angle. Figure 10 shows the displacement contours at a longitudinal cross section passing through the center of the raft. The measured and the calculated displacements at the instrumentation points are shown in     The thicknesses h i , the oedometric moduli E 0i and the settlement r i for each sand layer are shown in Table 7. The estimated total settlement is r = 22.87 mm. This is higher than the average measured settlement of 15.06 mm due to the assumption of infinite loading and also because Eq. 22 does not consider the confinement provided by the 4-m high fill and the stiffness of the raft (both accounted for in the FEM analysis).
To account for those effects, the following equation from the theory of elasticity modified by Barata (1962Barata ( , 1984Barata ( , 1986 is used: The following values are plugged in Eq. 23 (Barata, 1962(Barata, , 1984(Barata, , 1986; l = 0.96 (Mindlin's coefficient); c D = 1.09 (shape factor, rigid foundation); Dp = 123 kN/m 2 ; B = 16.6 m (foundation width); n = 0.3 (Poisson ratio); I = 0.63 (influence factor accounting for a rigid boundary about 12 m below the raft); E s = 61500 kN/m 2 (weighted mean static Young modulus, according to Table 4). The settlement estimated by Eq. 23 is r = 19.9 mm. This is about 33% higher than the average FEM predictions (which coincided with the average measured settlement) and 15% lower than the estimated 1-D settlement.
Considering that Burland and Burbidge chart provides an upper bound for the expected settlements, the results shown in Fig. 11 seem to indicate the adequacy of the estimated Young moduli according to Eq. 19 to calculate the raft settlements.

Conclusions
This paper presents correlations between the static Young modulus E s and the normalized penetration resistance N 60 for pure sands, aiming at preliminary settlement predictions. The proposed correlations can also be extended to preliminary estimates of the oedometric modulus E o in pure sands.
The correlations in this paper are limited to the statistical interpretation of several results published in the literature. Therefore, practical application of equations and charts presented in this paper should be supported further by load tests and other in-situ tests. Eq. 19 is generally suited to applied pressures sufficiently far from failure, as expected in well-designed rafts and foundations for tanks and silos where reasonably low static shearing strains result. For higher shearing strains, it is recommended to select E s values below the mean trend equation in Fig. 6. In contrast, E s values above the mean trend curve should be used with increased caution and only when supported by load tests.
The proposed correlations have been used to predict satisfactorily the settlements of a nearly rigid raft supporting the structure of a diesel power generator on sedimentary sand. Good predictions have been obtained by 3-D FEM analysis taking E s values from Eq. 19. Reasonable predictions have also been achieved by one-dimensional calculations using the oedometric modulus E o (Eqs. 21 and 22) and the elasticity equation modified by Barata (Eq. 23). The results coming from the three models plotted satisfactorily on Burland and Burbidge chart, indicating that the proposed correlations are useful for preliminary estimation of settlements in sedimentary sands.