Furthermore, to evaluate the accuracy and predictability of each method, a statistical analysis was ... (1999), Brinch Hansen (1961), and Meyerhof et al.
5iYGEC Paris 2013
Comparative Study between the Methods Used for Estimating Ultimate Lateral Load of Piles in Sandy Soil Etude comparative entre les méthodes d'estimation de charge latérale ultime des pieux dans un sol sableux Mahmoud F. AWAD-ALLAH1, Noriyuki YASUFUKU2 1,2 Departement of Civil and Structural Engineering, University of Kyushu, Japan. ABSTRACT – In this paper, a comparison study based on results of experimental lateral pile load tests was carried out between a new proposed method and widely used methods for predicting the ultimate lateral resistance of piles in sandy soil. The proposed technique agrees well with those obtained from the centrifuge tests with average error of -1.30%, while other methods of design gave significantly higher average errors. Furthermore, to evaluate the accuracy and predictability of each method, a statistical analysis was performed using four independent statistical criteria, namely: (1) the best-fit line with the corresponding 2 coefficient of correlation, r , (2) the cumulative probability concept, (3) the 20% accuracy level derived from Log Normal distribution, and (4) the arithmetic mean (µ) and coefficient of variation (COV). 1. Introduction Numerous methods have been published in the literature for predicting the ultimate lateral resistance to piles in cohesionless soils (Brinch Hansen 1961; Broms 1964; Meyerhof et al. 1981; Petrasovits and Award 1972; etc.). Basically, the main difference between those methods is the assumed distribution pattern of lateral earth pressure in front of pile during loading; thus, each method gives different value for ultimate lateral load. A key element in the design of laterally loaded piles is the determination of the ultimate lateral earth pressure that can be exerted by the soil against the pile movement. After full mobilization of lateral soil pressure due to lateral loading, some methods assume that pile rotates at the pile base (e.g. Broms 1964, and Awad-Allah et al. 2011). However, other methods by Petrasovits and Award (1972), Prasad and Chari (1999), Brinch Hansen (1961), and Meyerhof et al. (1981) consider the point of pile rotation at a certain depth below the ground level. Consequently, this paper introduces a comparison study between a proposed method (Awad et al. 2011) and the other widely used method for determining the ultimate lateral pile capacity. Because the proposed method considers the ultimate resistance of the soil (not of the pile), it is applicable to both flexible and rigid piles. To evaluate the accuracy of the method of Awad-Allah et al. 2011, it is first used to calculate the ultimate lateral resistance of model test piles in a centrifuge (using 22 laboratory lateral pile load tests). In order to unify the comparison study, those experimental tests were selected to achieve maximum horizontal displacements within range of 15% to 20% of pile diameter at pile head. Therefore, the obtained ultimate lateral loads were measured at this criterion.
2. Methods for predicting ultimate lateral resistance Four methods have been selected to be used for comparison in this paper; three of them have been widely used in the literature and practice, while the fourth one was proposed by the authors in 2011. The selected methods are: Broms 1964, Petrasovits and Award 1972, Prasad and Chari 1999, and Awad-Allah et al. 2011. Broms’s method (Broms 1964) was selected mainly due to its popularity in practice. In this method, it is assumed that a lateral concentrated load acts at pile base and therefore the pile rotates according to the pile base. The ultimate lateral soil resistance pu at fully mobilized passive state is given as follows:
pu 3K p v'
(1) Petrasovits and Award (1972) developed a method in which the effect of active earth pressure was considered in the estimation of lateral soil pressure, as given in the following formula:
pu (3.7 K p K a ) v' (2) Prasad and Chari (1999) proposed Eq. 3 for predicting ultimate soil resistance for laterally loaded pile in cohesionless soil. This method is adopted because it was developed relatively recently.
pu 10(1.3 tan 0.3) v' (3) 2 ° Where: Kp= tan (45 +φ/2) = passive earth pressure 2 ° coefficient; Ka= 1/tan (45 +φ/2) = 1/Kp = active earth pressure coefficient; φ= internal friction angle of soil; and σv = vertical effective stress.
5iYGEC Paris 2013 3. The proposed technique to calculate ultimate lateral load
EI
Awad-Allah et al. (2011) have introduced new technique to calculate the ultimate lateral pile capacity which considers the effect of both of active earth pressure and side shear resistance of soil during lateral movement of pile. In this method, it is assumed that the distribution of the soil reaction is as shown in Fig. 1. The basic idea is that the pile is deflected towards the right side by the applied lateral force. Thus, passive earth pressure is created in front of pile while active earth pressure is created behind the pile shaft. Accordingly, soil resistance to the lateral movement of the pile can be expressed in two components: (1) the frontal normal reaction (qmax) and (2) the side friction reaction (max), as shown in Fig. 2. Therefore, ultimate lateral resistance that can be exerted by the soil against the pile lateral movement, pu, can be expressed as given in Eq. 4.
pu qmax max (4) Where: η= shape factor to account for the nonuniform distribution of earth pressure in front of the pile= 0.8; and ζ= shape factor to account for the non-uniform distribution of lateral shear drag= 1. The ultimate frontal earth pressure (qmax) can be given by Eq. 5. Side friction reaction (max) can be computed the same as the ultimate vertical shear resistance of piles estimated with Eq. 6.
qmax 3( K p K a ) v' (5)
max 2K tan v' (6) The differential equation of laterally loaded piles in cohesionless soil can be given by Eq. 7, and the general solution of the forth order differential equation is obtained by using MATLAB (Eq.8).
d4y pu z 0 dz 4
y
(7)
5
pu z C C 1 z 3 2 z 2 C3 z C4 (8) 120 EI 6 2
Figure 2. Proposed distribution of front earth pressure and side shear resistance around pile subjected to lateral load (After Awad-Allah et al. 2011). Where: EI = Bending (flexural) stiffness of pile, y = lateral deflection of the pile, z = depth below ground level; and C1, C2, C3, and C4 are four unknown constants of integration that can be calculated from the boundary conditions of the problem (Fig. 1). Hence, by solving the differential equation using three boundary conditions, and by making some mathematical manipulations, it now follows that the ultimate lateral pile capacities Hu can be calculated as given in Eq. 9.
H u Pu * B
pu L2 e 6(1 ) L
(9)
Where: Pu = ultimate lateral load per unit length of pile (kN/m); pu = ultimate lateral soil resistance 2 (kN/m ); e= lateral load eccentricity; L= pile embedded length; and B = pile diameter (m). The expressions developed in the preceding sections [Eqs. (1), (2), (3), and (4)] were used to predict the ultimate lateral loads of piles in cohesionless soil. Those expressions were applied into Eq. 9 so that different values for predicted ultimate lateral load, (Hu)p, can be estimated. Afterward, those values were compared to the corresponding measured ultimate pile capacities measured from model test in a centrifuge (Hu)m. 4. Analysis of the results 4.1 Percentage of average error
Figure 1. Distribution of ultimate frontal soil force against pile shaft (After Awad-Allah et al. 2011).
Twenty-two laboratory lateral pile load tests have been utilized in this work. Percentage of average error obtained from each method of design
5iYGEC Paris 2013 can be calculated using Eq. 10. It can be seen that Prasad and Chari (1999), and Awad-Allah et al. (2011) yield the lowest average error percentages of -0.93% and -1.30%, respectively, compared to other two methods. On the other hand, the highest error percentages are 45.46% and 53.90% obtained from the methods of Broms (1964), and Petrasovits and Award (1972), respectively, which basically neglected the effect of side shear resistance between pile shaft and soil medium during pile lateral movement.
Error%
(Q )
(Qu )m x100 (Qu )m
u p
(10)
4.2 Statistical analysis An evaluation scheme using four criteria was considered in order to ranking the methods, as follows: 4.2.1. Best Fit Line Criterion (R1) Linear best fit using regression analysis was performed for each method and the corresponding 2 coefficient of determination (r ) was obtained; this is used to test the strength of best fit equation. Practically, the method which yields close value to 2 (1) for best fit equation as well as (r ) is close to (1) is considered the most predictable method. Fig.3 shows best fit line analysis for the measured versus the predicted ultimate loads (trend line of data), and Table 1 gives best fit equation together with the associated coefficient of 2 determination, r , for each method. It is obvious that Awad-Allah et al. (2011) method gave trend line o that almost coincides with the inclined line of 45 , and it has best fit equation of (Hu)p = 1.05 (Hu)m with 2 r = 0.94. This indicates that this method has an excellent predictability and high correlation strength; consequently, it can be used with high reliability and confidence. Table 1. Best fit calculations for assessment of ultimate lateral load. Method of Ranking 2 Best fit equation r design (R1) Awad-Allah (Hu)p = 1.05 (Hu)m 0.94 1 et al. (2011) Prasad and (Hu)p = 1.29 (Hu)m 0.89 3 Chari (1999) Broms (Hu)p = 1.32 (Hu)m 0.94 2 (1964) Petrasovits 4 and Award (Hu)p = 1.62 (Hu)m 0.94 (1972) 4.2.2. Cumulative Probability Criterion (R2) The cumulative probability concept is utilized to help in quantifying the accuracy of the investigated methods in prediction the ultimate lateral capacity
of piles. The method which gives P50 value closer to (1) in conjunction with lower (P90-P50) range is considered the best. The procedures of this criterion are: sort the ratio (Hu)p/(Hu)m for each method in an ascending order. The smallest value is given i = 1 and the largest is given number i = n, where n is the number of case studies considered in the analysis. The cumulative probability value, CPi, for each value of (Hu)p/(Hu)m is given, as follows:
CPi
i n 1
(11)
Table 2 summaries the results and ranking of each method of design. It is clear that, Awad-Allah et al. (2011) method is ranked as number 1, because it gives P50 value that approaches to 1 and at the same time gives low value of (P90-P50). It is obvious that the ranking of other methods are as follows: Broms (1964) is in the second position; Prasad and Chari (1999) is in the third order; and in Petrasovits and Award (1972) is in the fourth (last) position with the lowest P50 value as well as the highest value of (P90-P50).
Figure 3. Correlation between measured and predicted ultimate load for each method. 4.2.3. 20% Accuracy Level Criterion (R3) The 20% accuracy level denotes that the predicted lateral pile capacity (Hu)p lies within the range between 0.8 and 1.2 the measured capacity (Hu)m. Using the Log Normal probability function, the probability of predictability of the pile capacity at different accuracy levels can be determined. At a specified accuracy level (i.e., 20% of accuracy), the higher the probability is, the better the accuracy of this method is. Table 3 shows the results of analysis and the ranking of methods. It is obviously that, at 20% accuracy the method of Petrasovits and Award (1972) has the highest probability of accuracy that equals 50.00% (R3=1), and therefore it is ranked
5iYGEC Paris 2013 number in the first order. Whereas, the method of Petrasovits and Award (1972) comes in the last order with the lowest value of probability of accuracy 22.73% (R3=4). The methods of AwadAllah et al. (2011) and Broms (1964) are ranked in the second and third order, respectively. Table 2. Cumulative probability results of methods used in this study. Method of Ranking P50 P90 P90 – P50 design (R2) Awad-Allah 0.94 1.93 0.99 1 et al. (2011) Prasad and 0.80 1.67 0.86 2 Chari (1999) Broms 1.26 2.02 0.76 3 (1964) Petrasovits 1.46 3.00 1.54 4 and Award (1972) Table 3. Results of 20% accuracy range of prediction ultimate bearing capacity. Probability at Method of 20% accuracy Ranking (R3) design range (%) Awad-Allah et 50.00 2 al. (2011) Prasad and 59.09 1 Chari (1999) Broms (1964) 36.36 3 Petrasovits and 22.73 4 Award (1972) 4.2.4. Statistical Parameters Criterion (R4) The precision of each method can be evaluated by measuring the scatter of results around the mean value, µ, for the ratio [(H u)p/(Hu)m], and by calculating a parameter defined as, COV, which is equal to standard deviation, s, divided by µ. The most accurate method gives, µ = 1, and COV = 0, respectively. This case is ideal, however, in reality the method is better when, µ is nearly (1), and COV is asymptotic to zero. Table 4 summaries the results of the statistical parameters (µ, s, and COV) for each method used in this comparison study. It can be seen that the most accurate method, based on this criterion, is Awad-Allah et al. (2011) which considers each of side shear and frontal earth pressure resistance between pile shaft and soil medium. 4.2.5. Overall ranking index (RI) An overall rank index, RI, is defined as the sum of ranking values obtained from the four criteria (RI=R1+R2+R3+R4). The lower the ranking index is, the better the performance of the method is, i.e. in accuracy and predictability. It can be seen that the proposed method (AwadAllah et al. 2011) for prediction of ultimate lateral capacity came in the first order with the lowest ranking index (RI=5). The method of Prasad and
Chari (1999) had the second order with total ranking index (RI=8). On the other hand, methods of Broms (1964) and Petrasovits and Award (1972) indicated the highest ranking indexes of RI=11 and 16, respectively. Therefore, they were ranked in the third and fourth order, respectively. Table 4. Statistical parameters for assessment of ultimate lateral load. Ranking Method µ s COV (R4) Awad-Allah et 1.00 0.47 0.47 1 al. (2011) Prasad and 0.99 0.46 0.47 2 Chari (1999) Broms (1964) 1.46 1.21 0.83 3 Petrasovits 1.54 0.72 0.47 4 and Award (1972) 5. Summary and conclusion This paper establishes an evaluation scheme to assess the accuracy and predictability of the most widely used methods for estimating the ultimate lateral pile capacity in sandy soils. The conclusions of this study can be summarized as follows: 1) The proposed method introduced by AwadAllah et al. (2011) yields remarkably satisfactory results since the accuracy and the predictability of this technique have been estimated versus well-known design methods. 2) Active earth pressure and side shear resistance, which are ignored in most of the design methods, have a significant part in determination of lateral pile capacity.
6. References Awad-Allah. M. F., Yasufuku, N. and Omine, K. (2011). Proposed analytical solution for estimating of ultimate lateral capacity of piles in sandy soil. International Journal of GeoEngineering, Vol. 3, No. 4, pp. 29-39. Brinch Hansen, J. (1961). The ultimate resistance of rigid piles against transversal forces. Bulletin No. 12, Danish Geotechnical Institute, Copenhagen, pp. 5–9. Broms, B. B. (1964). Lateral resistance piles on cohesionless soils. J. of Soil Mechanics and Foundation Engineering, ASCE, (SM3), pp. 123–156. Petrasovits, G. and Award, A. (1972). Ultimate lateral resistance of a rigid pile in cohesionless soil. Proceedings of 5th European Conference on Soil Mechanics and Foundation Engineering, Madrid, Vol.3, pp. 407-412. Prasad, Y. V. S. N. and Chari, T. R. (1999). Lateral capacity of model rigid piles in cohesionless soils. Soils Found., Vol. 39, No. (2), pp. 21–29.
