Evaluation of lateral capacity for flexible drilled shafts in cohesionless soils

Abstract

This study aimed to evaluate and improve the p–y curve method for use in the lateral analysis of flexible circular drilled shafts in cohesionless soils. Onsite circular drilled shaft lateral load test data from all over the world were collected, and the applicability of O’Neill and Murchison's p–y curve method was evaluated by comparing the measured values with the method's predicted values. Results showed that the current predictions were underestimated after applying the method to a database of flexible circular drilled shafts in cohesionless soils. Then, the difference (ΔQ) between the predicted value (Q_p) and measured value (Q_m) was normalized using the measured value (ΔQ/Q_m) to determine the percent difference, and an average value of − 47% was produced. The difference ratio between the measured and predicted values was proportional to the diameter of the pile. To calibrate the prediction of the current method, this study applied two parameters (α and β) as modifications to reduce ΔQ. The parameters were decided by using the average difference ratio and average absolute difference methods. The reliability and consistency of the prediction of the p–y curve method were enhanced by applying the recommended modifying parameters. Specific design recommendations were also provided based on the proposed modifications on the original p–y curve.

Introduction

Circular drilled shafts, which are cast-in-place bored piles, are commonly used pile foundations in bridge and building structures in urban areas. Compared with driven piles, circular drilled shafts are non-displacement piles and free of noise problems during their installation. However, the bearing behavior of circular drilled shafts may still be influenced by other factors, such as concrete quality, circular drilled hole conditions, and surrounding soil conditions.

Vertical loadings are the most common loads resisted by various types of piles. Many types of vertical loading conditions are considered in constructing pile foundations.^1,2 In recent years, vertical loadings and their subsequent effects on pile foundations have been widely studied around the world.^3–7 With regard to lateral loads, the design criteria limit pile deflections and stresses to within tolerable values. ⁸ The lateral behavior of piles has also been extensively studied,^9–19 and various methods, such as finite element^20,21 and p–y curve,^22,23 are available for analyzing the pile response under lateral loading.

Among these methods, p–y curve is the most commonly used in practice. This method is based on the Winkler model. ²⁴ As presented in Figure 1, the pile is modeled by beam elements, and the soil is modeled by a series of independent nonlinear soil springs. Many p–y curves have been proposed for different soil types; examples include Matlock's method ²⁵ for clay and Cox et al.'s ²² and O’Neill and Murchison's methods ²⁶ for sand. The method proposed by O’Neill and Murchison ²⁶ is widely used for cohesionless soils and has been included in API codes ²⁷ for designing offshore structures. However, their p–y curve is developed from driven steel pipe piles. It only considers the soil parameters and ignores the effect of foundation construction methods. The applicability of this p–y curve to circular drilled shafts needs further investigation.

Figure 1.

Winkler’s elastic foundation model (Winkler, 1867).

The present study aims to investigate the applicability of O’Neill and Murchison's ²⁶ p–y curve to circular drilled shafts in cohesionless soils. Focusing on flexible piles, we apply LPile 2016 software to analyze some large-scale field load tests by using O’Neill and Murchison's ²⁶ p–y curve. A series of statistical analyses is performed to evaluate the differences in lateral predicted and measured capacities at various lateral displacements. On the basis of the evaluation results, an attempt is made to modify the original p–y curve to reduce the global difference, and further suggestions for the use of the modified p–y curve are provided.

Analysis method

This study aims to investigate the applicability of O’Neill and Murchison's p–y curve method to analyzing and predicting the load–displacement behavior of circular drilled shafts subjected to lateral loadings. The study compares the predicted capacities and those measured in field load tests to determine the differences and then modifies the original p–y curve to improve the prediction accuracy. Various in-situ lateral circular drilled shaft load test data are gathered and used to achieve these objectives. The test and soil profile information are tabulated, and the tests are simulated on LPile 2016 software. LPile 2016 is used because it can be employed for the current p–y curve method and its subsequent modifications, thereby providing a reliable analysis.

After collecting sufficient load test results, this study uses the flexible factor K_r suggested by Poulos and Davis ²⁸ and shown in Eq. (1) to characterize the relative stiffness between the pile and soil.

$$K_r={\displaystyle \frac{E_{pile}{I}_{pile}}{E_{soil}{L}^4},}$$

(1)

where E_pile and I_pile are the elastic modulus and moment of inertia of the pile, respectively; E_soil is the soil's elastic modulus; and L is the length of the pile. A pile–soil system with K_r values less than 10⁻² is regarded as having flexible piles. The test data where the piles are classified as flexible are included in the database of this study. The load–displacement curves predicted by LPile 2016 and the load–displacement curves measured from in-situ load tests are compared to assess their relative differences. Some parameters in the p–y curve method are also modified further to reduce the differences between the measured and predicted results.

Improvement of the p-y curve method

Modifications are made on the original p–y curve to decrease the differences between the predicted and measured results of the load tests. Eq. (2) presents O’Neill and Murchison's p–y curve for sand.

$$\mathrm{p}=\mathrm{A}{p}_{u}tanh\left({\displaystyle \frac{kx}{A{p}_u}y}\right),$$

(2)

where p_u is the ultimate soil resistance in force per unit length, A is a factor to account for cyclic or static loading, k is the initial modulus of the subgrade reaction, y is the lateral deflection, and x is the depth of the layer considered. Ultimate soil resistance p_u can be calculated using two equations: one for shallow depths (Eq. [3]) and another one for deep depths (Eq. [4]); whichever of the two provides a smaller p_u value is used.

$$p_{us}=(C_{1}x+C_{2}B){\gamma }^{\mathrm{\prime }}x,$$

(3)

$$p_{ud}=C_{3}B{\gamma }^{\mathrm{\prime }}x,$$

(4)

where C₁, C₂, and C₃ are coefficients that can be determined based on Figure 2; ²⁹ B is the pile diameter; and γ’ is the effective unit weight of the soil. These coefficients are also shown in Eqs. (5)–(7).

$$C_1=tan\omega \left\{K_{P}tan\epsilon +K_0\left[tan{\varphi }^{\mathrm{\prime }}sin\omega \left({\displaystyle \frac{1}{cos\epsilon }+1}\right)-tan\epsilon \right]\right\},$$

(5)

$$C_2=K_P-K_A,$$

(6)

$$C_3=K_P^2(K_P+K_{0}tan{\varphi }^{\mathrm{\prime }})-K_A,$$

(7)

where ϕ’ is the effective friction angle, ε is the angle of the wedge in the horizontal direction equal to ϕ’/2, ω is the angle of the wedge with the ground surface equal to 45° + ϕ’/2, K_p is the coefficient of passive earth pressure equal to $ta{n}^2\left(45+\frac{{\varphi }^{\mathrm{\prime }}}{2}\right)$ , K_o is the coefficient of earth pressure at rest equal to 0.4, and K_a is the coefficient of active earth pressure equal to $ta{n}^2\left(45-\frac{{\varphi }^{\mathrm{\prime }}}{2}\right)$ . Factor A can be determined using Eq. (8) for static loading, and A  =  0.9 for cyclic loading.

$$A=3.0-\left(0.8{\displaystyle \frac{x}{B}}\right)\ge \mathrm{0.9.}$$

(8)

This study adds two parameters, α and β, as shown in Eq. (9) to modify the outermost factor and the factor in the hyperbolic tangent function of Eq. (2), respectively. The ultimate goal is to improve the prediction results of the method, which can be expressed as follows:

$$p=(\alpha )A{p}_{u}tanh\left[(\beta ){\displaystyle \frac{kx}{A{p}_u}y}\right].$$

(9)

Figure 2.

Relationship between coefficients C₁, C₂, C₃ and the friction angle (Isenhower, et al. 2016).

Parametric study of the modifying factors

The influence of parameter α is investigated, and its sensitivity to the original prediction curve is observed with a small modification. Three values of 1.2, 1.4, and 1.6 are substituted as the trial values of α. As presented in Table 1, the average difference ratio of 1.2 for parameter α is − 0.67, and the total distance value, which is the difference between the lateral displacement results of the maximum and minimum difference ratios denoted as R in Table 1, is 0.13. The average difference ratio of 1.4 for parameter α is − 0.63, and the total distance value, R, is 0.16. The average difference ratio of 1.6 for parameter α is − 0.59, and its R is 0.19. Figure 3 graphically represents the data calculated in Table 1. The higher the values of α are, the closer the differences are between the predicted and measured values.

Figure 3.

Comparison of the influence of the parameter α on the original p-y curve method.

Table 1.

Comparison of the influence of the parameter α on the original p-y curve method.

(α, β)		Difference ratio	Lateral displacement, %B						Mean	R
			0.50	1.00	2.00	3.00	4.00	5.00
(1.0,1.0)	ΔQ/Qm		−0.75	−0.76	−0.74	−0.71	−0.68	−0.66	−0.72	0.10
(1.2,1.0)	ΔQ/Qm		−0.70	−0.72	−0.71	−0.66	−0.62	−0.59	−0.67	0.13
(1.4,1.0)	ΔQ/Qm		−0.66	−0.70	−0.68	−0.62	−0.57	−0.54	−0.63	0.16
(1.6,1.0)	ΔQ/Qm		−0.61	−0.68	−0.65	−0.59	−0.53	−0.49	−0.59	0.19

The average difference ratio exhibits a significant upward trend. The overall distance value shows a small increase, indicating that although the difference ratio is improved overall, the degree of improvement is lower at small displacements than at large displacements.

The influence of parameter β is investigated following the same procedure as that for parameter α. Three values of 1.2, 1.4, and 1.6 are substituted again. As presented in Figure 4, the average difference ratio of 1.2 for parameter α is − 0.69, and the full range value is 0.09. The average difference ratio of 1.4 for parameter α is − 0.66, and the total distance value is 0.09. The average difference ratio of 1.6 for parameter α is − 0.65, and the total distance value is 0.09. The details are presented in Table 2.

Figure 4.

Comparison of the influence of the parameter β on the original p-y curve method.

Table 2.

Comparison of the influence of the parameter β on the original p-y curve method.

(α, β)		Difference ratio	Lateral displacement, %B						Mean	R
			0.50	1.00	2.00	3.00	4.00	5.00
(1.0,1.0)	ΔQ/Qm		−0.75	−0.76	−0.74	−0.71	−0.68	−0.66	−0.72	0.10
(1.0,1.2)	ΔQ/Qm		−0.70	−0.72	−0.72	−0.68	−0.65	−0.63	−0.69	0.09
(1.0,1.4)	ΔQ/Qm		−0.66	−0.71	−0.70	−0.66	−0.63	−0.62	−0.66	0.09
(1.0,1.6)	ΔQ/Qm		−0.62	−0.69	−0.69	−0.65	−0.62	−0.60	−0.65	0.09

The data and comparison chart reveal that for every 0.2 increase in parameter α, the average difference ratio increases by approximately 0.04, and the full range value increases by approximately 0.03. For every 0.2 increase in parameter β, the average difference ratio increases by approximately 0.02, and the full range distance value decreases by approximately 0.005 on the average. Therefore, on the basis of the comparison above, if parameters α and β are substituted with values of the same size and the same interval, parameter α will have a greater influence than parameter β and a larger correction for the original p–y curve method. The average difference ratio is effectively increased, but the increase in parameter α does not help decrease the full range value but causes a certain increase instead. Although parameter β results in a minor increase in the average difference ratio, it reduces the overall distance value. The overall difference ratio and lateral displacement curve can be reduced, and the results can be used to achieve predicted values that are close to that of the measured capacities.

Database

A database of load tests gathered from all over the world is built for this study to assess the accuracy of O’Neill and Murchison's p–y curve method when applied to flexible circular drilled shafts in cohesionless soils. Tables 3 and 4 show the basic soil information, pile material parameters to be used in LPile 2016 software, and test site conditions. A total of 23 sets of load test data are gathered from Kuwait, Romania, the USA, Taiwan, Japan, Portugal, and Singapore. The references for the load test data are presented in Table 5.

Table 3.

Basic data of soil layers for lateral load tests of flexible drilled shafts in cohesionless soils.

Shaft No.	Test site	Shaft length	Shaft diameter	Soil profile		GWTa	γt or γ'b	φ'c	kd	Dre	SPT Ni	Esf
		L (m)	B (m)	Depth (m)	Description	(m)	(kN/m3)	(°)	(kN/m3)			(kN/m2)
F1	Kuwait	5.0	0.30	0–3	Medium dense reddish brown dry silty sand	_g	18.7	30	12973	35	10	4629
				3–4			17.3	31	17678	40	13
				4–5			18.2	32	22610	44	16
F2	Kuwait	5.0	0.33	0–3	Medium dense reddish brown dry silty sand	_g	18.7	30	12973	35	10	4629
				3–4			17.3	31	17678	40	13
				4–5			18.2	32	22610	44	16
F3	Kuwait	5.0	0.35	0–3	Medium dense reddish brown dry silty sand	_g	18.7	30	12973	35	10	4629
				3–4			17.3	31	17678	40	13
				4–5			18.2	32	22610	44	16
F4	Kuwait	5.0	0.39	0–3	Medium dense reddish brown dry silty sand	_g	18.7	30	12973	35	10	4629
				3–4			17.3	31	17678	40	13
				4–5			18.2	32	22610	44	16
F5	Nanzi, Kaohsiung, Taiwan	38.0	1.50	0–1.8	SM	1.8	18.5	30	12973	32	9	6723
				1.8–3.6			8.3	30	9406	32	9
				3.6–38			8.5	34	20422	53	22
F6	Bedok, Singapore	25.9	1.00	0–10	Very loose sand	0.5	8.3	29	6640	22	6	4334
				10–21	Medium dense silty sand		8.3	30	9406	32	9
				21–25.9	Dense silty sand		8.5	34	20422	56	24
F7	Portugal	23.5	1.00	0–15.8	Sand	0	8.5	30	9406	41	14	5655
				15.823.5	Muddy sand		8.5	35	23300	50	20
F8	Portugal	39.5	1.20	0–15.8	Sand	0	8.5	30	9406	41	14	6029
				15.8–39.5	Muddy sand		8.5	35	23300	50	20
F9	Japan	29.5	1.55	0–15	NR	NRh	16.0	28	4242	11	3	3471
				15–17			19.0	32	22610	43	15
				17–29.5			18.7	31	17678	38	12
F10	Tai Tam, Taoyuan, Taiwan	16.0	1.20	0–16	SM	_g	8.5	40	39783	74	39	10190
F11	Taiwan	33.9	1.58	0–3	SM	3.0	18.6	28	4242	15	4	7920
				3–10			8.5	33	17621	50	20
				10–16			8.5	36	26281	65	30
				16–27			8.5	34	20422	55	23
				27–33.9			8.5	40	39783	81	46
F12	Yunlin, Taiwan	29.5	0.50	0–6	SM	4.68	20.5	35	38767	56	24	8422
				6–13.7			8.5	35	23300	56	24
				13.7–22.7			8.5	33	17621	59	26
				22.7–29.5			9.1	32	14869	79	44
F13	Houston, Texas, USA	28.9	0.91	0–1.05	Fine sand	1.05	8.5	33	27769	52	21	6773
				1.05–28.9			8.5	33	17621	52	21
F14	Houston, Texas, USA	38.7	1.07	0–1.06	Fine sand	1.05	8.5	33	27769	52	21	6773
				1.05–38.7			8.5	33	17621	52	21
F15	Houston, Texas, USA	38.5	1.07	0–1.07	Fine sand	1.05	8.5	33	27769	52	21	6773
				1.05–38.5			8.5	33	17621	52	21
F16	Houston, Texas, USA	38.5	1.07	0–1.08	Fine sand	1.05	8.5	33	27769	52	21	6773
				1.05–38.6			8.5	33	17621	52	21
F17	Mustang Island, Texas, USA	15.0	0.50	0–15	Sand	NRh	10.4	35	38767	61	27	7994
F18	Neapolitan area	14.0	0.60	0–4	Pozzolana sand	1.5	8.5	33	27769	50	20	8917
				4–8	Pozzolana sand		8.5	32	14869	43	15
				8–12	Volcanic sand		9.1	38	32662	74	39
				12–14	Volcanic sand		9.5	47	73657	99	75
F19–22	South Surra, Kuwait	5.0	0.30	0–0.6	Medium dense cemented silty sand	0.6	17.8	35	38767	40	13	9698
				0.6–3			8.5	35	23300	56	24
				3–3.5			8.5	43	52324	52	21
				3.5–4.5	Medium dense to very dense silty sand with cemented lumps		9.5	43	52324	95	60
				4.5–5			9.5	43	52324	99	90
F23	Romania	33.0	0.88	0–10	Silty sand	0	8.0	29	6640	25	7	4828
				10–33	Loose/medium dense fine sand		8.5	32	14869	43	15

a – GWT: ground water table; b – γ_t or γ': total unit weight (for GWT exist), effective unit weight(for GWT not exist); c-ϕ': effective friction angle; d – k: Initial modulus of subgrade reaction (Isenhower, et al. 2016); e – Dr: Relative density; f – E_s : elastic modulus for soil ( ${\mathrm{E}}_{\mathrm{s}}=9.08\times {\mathrm{N}}^{0.66}\times {\mathrm{P}}_{\mathrm{a}}$ ) where P_a is the atmospheric pressure and N is the SPT-N value (Ohya et al.,1982): g – __: water table below shaft tip; h-NR: not report (Regarded as lower than pile tip). i – SPT-N: original SPT-N.

Table 4.

The material parameters of the pile body in the lateral load test of the flexible drilled shafts in cohesionless soils.

Shaft No.	Test site	Shaft length	Shaft diameter	e a	Ec b	Eγ c	fY d	fc' e	Rebar size	Amount of rebar	ρf	cg	Ich	Kri
		L (m)	B (m)	(m)	(MPa)	(MPa)	(MPa)	(MPa)			(%)	(mm)	(m4)
F1	Kuwait	5.0	0.30	0.30	23243	200000	413.7	24 .50	#7	6	3.29	25	0.00040	3.19E-03
F2	Kuwait	5.0	0.33	0.30	23243	200000	413.7	24 .50	#7	6	2.72	25	0.00058	4 .68E-03
F3	Kuwait	5.0	0.35	0.30	23243	200000	413.7	24 .50	#7	6	2.41	25	0.00074	5.92E-03
F4	Kuwait	5.0	0.39	0.30	23243	200000	413.7	24 .50	#7	6	1.94	25	0.00114	9.12E-03
F5	Nanzi, Kaohsiung, Taiwan	38.0	1.50	0.30	24351	200000	413.7	27 .44	#10	12	0.56	76	0.24850	4.50E-04
F6	Bedok, Singapore	25.9	1.00	0.50*	24351	200000	413.7	27 .44	#8	15	0.97	76	0.04909	6.13E-04
F7	Portugal	23.5	1.00	0.50*	29000	200000	413.7	34 .30	#8	16	1.04	76	0.04909	8.25E-02
F8	Portugal	39.5	1.20	0.50*	31500	200000	413.7	34 .30	#8	16	0.72	76	0.10179	2.18E-04
F9	Japan	29.5	1.55	0.50*	24351	200000	413.7	27 .44	#10	24	1.04	76	0.28187	2.61E-03
F10	Tai Tam, Taoyuan, Taiwan	16.0	1.20	0.50*	24351	200000	413.7	27 .44	#10	8	0.58	76	0.1018	3.75E-03
F11	Taiwan	33.9	1.58	1.00	24623	200000	413.7	27 .47	#10	26	1.09	75	0.3059	7.20E-02
F12	Yunlin, Taiwan	29.5	0.50	0.50*	27225	200000	413.7	34 .30	#8	8	2.08	76	0.0031	1.31E-05
F13	Houston, Texas, USA	28.9	0.91	0.09	24351	200000	413.7	27 .44	#9	14	1.38	76	0.0343	1.78E-04
F14	Houston, Texas, USA	38.7	1.07	0.09	24351	200000	413.7	27 .44	#9	14	1.01	76	0.0636	1.02E-04
F15	Houston, Texas, USA	38.5	1.07	0.21	24351	200000	413.7	27 .44	#9	14	1.01	76	0.0636	1.04E-04
F16	Houston, Texas, USA	38.5	1.07	0.25	24351	200000	413.7	27 .44	#9	14	1.01	76	0.0636	1.04E-04
F17	Mustang Island, Texas, USA	15.0	0.50	0.50*	24351	200000	413.7	27 .44	#8	8	2.08	76	0.0031	1.85E-04
F18	Neapolitan area	14 .0	0.60	0.35	24351	200000	413.7	27 .44	#8	10	1.80	76	0.0064	4.52E-04
F19–22	South Surra, Kuwait	5.0	0.30	0.30	24351	200000	413.7	27 .44	#8	6	4 .30	25	0.0004	1.60E-03
F23	Romania	33.0	0.88	0.80	25205	200000	413.7	29.4	#8	8	0.67	76	0.02944	1.30E-04

a – e: load eccentricity; b – E_c: young's modulus for concrete; c-E_y : elastic modulus for reinforcing bars; d – f_y : yield stress for steel; e – f_c' : compressive strength for concrete; f – ρ: percentage of steel; g – c: concrete cover to edge of bars; h - I_c: moment of inertia for piles ( ${\mathrm{I}}_{\mathrm{c}}=\frac{1}{64}\pi ({\mathrm{b}}^4)$ ) where b is the diameter of the shaft; i - K_r: the factor of flexibility for piles.

Table 5.

References for lateral load test of flexible drilled shafts in cohesionless soils.

Shaft No.	References
F1–4	Ismael, N.F. (2010). "Behavior of Step Tapered Bored Piles in Sand under Static Lateral Loading", Journal of Geotechnical and Geoenvironmental Engineering, Vol. 136, No. 5, 669–676.
F5	Fu, C.F. and Huang, R.L. (1991). "Discussion on the Estimation Method of Horizontal Site Reaction Coefficient Based on the Results of Field Lateral Pile Load Test", The 4th Conference on Current Researches in Geotechnical Engineering in Taiwan, 811–818.
F6	Goh, A.T.C. and Lam, T.S.K. (1988). "Lateral Load Tests on Some Bored Piles in Singapore", 5th Australia - New Zealand Conference on Geomechanics, 509–513.
F7–8	Portugal, J.C. and Pinto, P.S. (1993). "Analysis and Design of Piles under Lateral Loads", Deep Foundations on Bored and Auger Piles, 309–312.
F9	Nakai, S. and Kishida, H. (1982). "Nonlinear Analysis of a Laterally Loaded Pile", Proceedings 4th International Conference on Numerical Methods in Geomechanics, Edmonton, 835–842.
F10	Diagnostic Engineering Consultants, Ltd (2001). "Report on Static Load Test of Bored Piles for Datan Power Plant Project: GT1", Taiwan.
F11	Hsueh, C.K., Lin, S.S. and Chern, S.G. (2004). "Lateral Performance of Drilled Shaft Considering Nonlinear Soil and Structure Material Behavior'', Journal of Marine Science and Technology, Vol. 12, No. 1, 62–70.
F12	Wang, L.H. and Chen K.H. (1998). "Study of Behavior of Mechanics for Laterally Loaded Piles", Feng Chia University.
F13–16	Little, R.L. and Briaud, J.L. (1988). "Full Scale Cyclic Lateral Load Tests on Six Single Piles in Sand", Geotechnical Division Civil Engineering Department Texas A&M University College Station, Texas.
F17	Drosos, V.A., Gerolymos, N. and Gazetas, G. (2006). "Non-Linear Analysis of Laterally Loaded Piles: Calibration of a New Method", Conference:10th International Conference on Piling and Deep Foundations.
F18	Fenelli, G.B. and Galateri, C. (1981). "Soil Modulus for Laterally Loaded Bored Piles in Pozzolana", 10th International Conference on Soil Mechanics and Foundation Engineering Stockholm, 703–708.
F19–22	Ismael, N.F. (1990). "Behavior of Laterally Loaded Bored Piles in Cemented Sands", Journal of Geotechnical Engineering, Vol. 116, No. 11, 1678–1699.
F23	Botea, E., Abramescu, T. and Manoliu, I. (2010). "Large Diameter Piles under Arial and Lateral Loads", 8th International Conference on Soil Mechanics and Foundation Engineering (Moscow).

According to the gathered data, the soil types range from muddy sand and extremely loose to dense and silty sand, with some pozzolana and volcanic sand. The shaft lengths range from 5 m to 38.7 m, and the shaft diameters range from 0.30 m to 1.58 m. Groundwater tables are presented in Table 5 together with the original N values of the standard penetration test (SPT-N) that range from 4 to greater than 50. Notably, the k values (initial modulus of subgrade reaction) from Table 3 are based on the method of Isenhower et al.. ²⁹ In addition, to simplify the computation of the moment of inertia, this study assumes that all of the circular drilled shafts are homogenous cylindrical in shape. Additionally, to simplify the calculations and because the data used in this study are secondhand data, cohesion is assumed to be equal to zero for all the data. To calculate the elastic modulus of soil (E_s), the relation of SPT-N and E_s is utilized. ³⁰

Analysis of measured and predicted results

After the results of the gathered lateral load test data are compiled, LPile 2016 software is employed to analyze and predict the lateral load–displacement curve by using O’Neill and Murchison's p–y curve method. Table 6 presents the predicted and measured loads with their corresponding lateral displacements and the relative differences of the measured and predicted loads ΔQ (ΔQ  =  Q_p – Q_m) with respect to the same amount of displacements. The table also presents the normalized differences between the predicted and measured loads and measured load ΔQ/Q_m.

Table 6.

Lateral prediction of flexible drilled shafts in cohesionless soil layer and results of load displacement measurement.

Shaft	Shaft length	Shaft diameter	Q (kN)	Lateral displacement, (%B)
No.	L (m)	B (m)		0.5	1.0	2.0	3.0	4.0	5.0
F1	5	0.30	ρa (m)	0.002	0.003	0.006	0.009	0.012	0.015
			Qmb (kN)	70	93.5	114.4	127.6	137.9	148.3
			Qpc (kN)	6.5	11.1	16.9	22.4	27.2	31.4
			ΔQ d	−63.5	−82.4	−97.5	−105.2	−110.8	−116.9
			ΔQ/Qm	−0.91	−0.88	−0.85	−0.83	−0.80	−0.79
F2	5	0.33	ρa (m)	0.002	0.003	0.007	0.010	0.013	0.017
			Qmb (kN)	67.1	91.9	122.5	134.4	146.3	158.2
			Qpc (kN)	8.6	13.4	21	28	33.7	38.9
			ΔQ d	−58.4	−78.5	−101.5	−106.4	−112.6	−119.3
			ΔQ/Qm	−0.87	−0.86	−0.83	−0.79	−0.77	−0.75
F3	5	0.35	ρa (m)	0.002	0.004	0.007	0.011	0.014	0.018
			Qmb (kN)	90.6	112.5	141.2	159.5	173.8	184.4
			Qpc (kN)	10	14.8	23.9	31.9	38.6	44.1
			ΔQ d	−80.6	−97.7	−117.3	−127.6	−135.2	−140.3
			ΔQ/Qm	−0.89	−0.87	−0.83	−0.80	−0.78	−0.76
F4	5	0.39	ρa (m)	0.002	0.004	0.008	0.012	0.016	0.02
			Qmb (kN)	133.9	159.9	185.5	205.7	218.7	231.7
			Qpc (kN)	12.1	20.2	30.6	40.7	49.1	55.1
			ΔQ d	−121.8	−139.6	−154.9	−165	−169.6	−176.6
			ΔQ/Qm	−0.91	−0.87	−0.84	−0.80	−0.78	−0.76
F5	38	1.50	ρa (m)	0.008	0.015	0.03	0.045	0.06	0.075
			Qmb (kN)	537.9	856.5	1023.9	1130.7	1192.8	1233.6
			Qpc (kN)	375.4	489.8	737.8	933.2	1031.1	1076.8
			ΔQ d	−162.5	−366.7	−286.1	−197.5	−161.8	−156.8
			ΔQ/Qm	−0.30	−0.43	−0.28	−0.18	−0.14	−0.13
F6	25.9	1.00	ρa (m)	0.005	0.01	0.02	0.03	0.04	0.05
			Qmb (kN)	150.4	225.4	300.8	323.4	345.1	357.4
			Qpc (kN)	106.3	144.4	199.7	253.5	299.1	332.8
			ΔQ d	−44.1	−81	−101.1	−69.9	−46	−24.6
			ΔQ/Qm	−0.29	−0.36	−0.34	−0.22	−0.13	−0.07
F7	23.5	1.00	ρa (m)	0.005	0.01	0.02	0.03	0.04	0.05
			Qmb (kN)	160.8	234.7	304.9	338.6	358.4	371.5
			Qpc (kN)	135.1	170.4	232.6	292	339.3	376.7
			ΔQ d	−25.7	−64.3	−72.2	−46.6	−19.1	5.2
			ΔQ/Qm	−0.16	−0.27	−0.24	−0.14	−0.05	0.01
F8	39.5	1.20	ρa (m)	0.006	0.012	0.024	0.036	0.048	0.06
			Qmb (kN)	239.4	371.5	501.0	566.9	606.8	633.6
			Qpc (kN)	204.2	250.7	340.8	426.9	491.6	532.3
			ΔQ d	−35.2	−120.9	−160.2	−140.0	−115.2	−101.3
			ΔQ/Qm	−0.15	−0.33	−0.32	−0.25	−0.19	−0.16
F9	29.5	1.55	ρa (m)	0.008	0.015	0.031	0.046	0.062	0.077
			Qmb (kN)	443.1	587.6	828.2	966.1	1104.0	1212.3
			Qpc (kN)	270.5	419.9	598.2	821.0	1038.0	1230.3
			ΔQ d	−172.7	−167.8	−230.0	−145.1	−66.0	18.0
			ΔQ/Qm	−0.39	−0.29	−0.28	−0.15	−0.06	0.02
F10	16	1.20	ρa (m)	0.006	0.012	0.024	0.036	0.048	0.060
			Qmb (kN)	650.7	857.1	1016.9	1084.3	1121.5	1145.0
			Qpc (kN)	281.0	365.5	519.1	588.0	608.7	616.5
			ΔQ d	−369.7	−491.7	−497.8	−496.4	−512.8	−528.6
			ΔQ/Qm	−0.57	−0.57	−0.49	−0.46	−0.46	−0.46
F11	33.9	1.58	ρa (m)	0.008	0.016	0.032	0.047	0.063	0.079
			Qmb (kN)	512.9	767.7	1159.3	1451.9	1665.0	1862.5
			Qpc (kN)	359.2	487.9	820.1	1171.7	1445.2	1624.7
			ΔQ d	−153.7	−279.8	−339.2	−280.2	−219.8	−237.8
			ΔQ/Qm	−0.30	−0.37	−0.29	−0.19	−0.13	−0.13
F12	29.5	0.50	ρa (m)	0.003	0.005	0.010	0.015	0.020	0.025
			Qmb (kN)	48.1	74.5	117.1	152.8	216.7	257.0
			Qpc (kN)	33.0	50.4	79.6	102.0	119.1	133.2
			ΔQ d	−15.1	−24.1	−37.5	−50.8	−97.6	−123.8
			ΔQ/Qm	−0.31	−0.32	−0.32	−0.33	−0.45	−0.48
F13	28.85	0.91	ρa (m)	0.005	0.009	0.018	0.027	0.037	0.046
			Qmb (kN)	140.5	237.7	368.5	478.2	575.3	662.4
			Qpc (kN)	154.0	191.1	265.1	327.1	375.5	415.0
			ΔQ d	13.6	−46.6	−103.5	−151.1	−199.8	−247.3
			ΔQ/Qm	0.10	−0.20	−0.28	−0.32	−0.35	−0.37
F14	38.6744	1.07	ρa (m)	0.005	0.011	0.021	0.032	0.043	0.053
			Qmb (kN)	240.1	401.5	637.8	806.1	904.6	976.2
			Qpc (kN)	192.1	257.5	366.7	449.5	511.5	559.8
			ΔQ d	−48.1	−144.1	−271.1	−356.6	−393.2	−416.4
			ΔQ/Qm	−0.20	−0.36	−0.43	−0.44	−0.44	−0.43
F15	38.4944	1.07	ρa (m)	0.005	0.011	0.021	0.032	0.043	0.053
			Qmb (kN)	217.2	352.2	543.4	683.4	771.1	835.1
			Qpc (kN)	181.3	248.4	350.3	433.8	498.8	539.4
			ΔQ d	−35.9	−103.8	−193.1	−249.6	−272.3	−295.7
			ΔQ/Qm	−0.17	−0.30	−0.36	−0.37	−0.35	−0.35
F16	38.5444	1.07	ρa (m)	0.005	0.011	0.021	0.032	0.043	0.053
			Qmb (kN)	261.4	426.7	616.6	724.0	793.1	841.2
			Qpc (kN)	193.0	245.4	348.6	428.4	492.9	537.3
			ΔQ d	−68.4	−181.3	−268.0	−295.6	−300.2	−304.0
			ΔQ/Qm	−0.26	−0.43	−0.44	−0.41	−0.38	−0.36
F17	15	0.50	ρa (m)	0.003	0.005	0.010	0.015	0.020	0.025
			Qmb (kN)	27.8	55.6	106.3	137.5	168.8	200.0
			Qpc (kN)	36.0	45.8	62.5	77.9	89.8	101.3
			ΔQ d	8.3	−9.8	−43.7	−59.6	−79.0	−98.7
			ΔQ/Qm	0.30	−0.18	−0.41	−0.43	−0.47	−0.49
F18	14	0.60	ρa (m)	0.003	0.006	0.012	0.018	0.024	0.030
			Qmb (kN)	221.9	310.2	418.8	449.4	466.5	477.3
			Qpc (kN)	48.9	60.8	83.2	101.7	116.7	130.4
			ΔQ d	−173.0	−249.4	−335.7	−347.8	−349.8	−347.0
			ΔQ/Qm	−0.78	−0.80	−0.80	−0.77	−0.75	−0.73
F19	5	0.30	ρa (m)	0.002	0.003	0.006	0.009	0.012	0.015
			Qmb (kN)	44.1	68.2	99.7	114.6	127.5	135.6
			Qpc (kN)	10.5	15.9	24.2	31.2	36.5	41.3
			ΔQ d	−33.6	−52.3	−75.5	−83.5	−91.0	−94.3
			ΔQ/Qm	−0.76	−0.77	−0.76	−0.73	−0.71	−0.70
F20	5	0.30	ρa (m)	0.002	0.003	0.006	0.009	0.012	0.015
			Qmb (kN)	50.0	68.1	91.3	105.9	116.2	125.9
			Qpc (kN)	10.5	15.9	24.2	31.2	36.5	41.3
			ΔQ d	−39.5	−52.2	−67.2	−74.7	−79.7	−84.6
			ΔQ/Qm	−0.79	−0.77	−0.74	−0.71	−0.69	−0.67
F21	5	0.30	ρa (m)	0.002	0.003	0.006	0.009	0.012	0.015
			Qmb (kN)	55.0	74.5	103.3	125.1	134.4	143.8
			Qpc (kN)	10.5	15.9	24.2	31.2	36.5	41.3
			ΔQ d	−44.5	−58.6	−79.1	−93.9	−98.0	−102.4
			ΔQ/Qm	−0.81	−0.79	−0.77	−0.75	−0.73	−0.71
F22	5	0.30	ρa (m)	0.002	0.003	0.006	0.009	0.012	0.015
			Qmb (kN)	57.9	76.8	98.7	120.6	135.6	148.9
			Qpc (kN)	10.5	15.9	24.2	31.2	36.5	41.3
			ΔQ d	−47.4	−60.9	−74.5	−89.4	−99.1	−107.5
			ΔQ/Qm	−0.82	−0.79	−0.76	−0.74	−0.73	−0.72
F23	33	0.88	ρa (m)	0.004	0.009	0.018	0.026	0.035	0.044
			Qmb (kN)	58.7	88.6	126.3	155.0	165.2	173.5
			Qpc (kN)	49.4	71.8	116.5	147.1	171.9	191.4
			ΔQ d	−9.3	−16.9	−9.8	−7.9	6.7	17.9
			ΔQ/Qm	−0.16	−0.19	−0.08	−0.05	0.04	0.10

a – ρ: lateral displacement; b – Q_m : measured load; c – Q_p :predicted load; d – ΔQ: Q_p – Q_m.

According to Chen and Lee, ³¹ for circular drilled shafts in cohesionless soils, a 5% pile diameter lateral displacement range can represent the overall trend of most load–displacement curves. Thus, the present study focuses on the lateral displacement range of 5% pile diameter for the comparison of pile capacity.

To assess the relative differences between the measured and original predicted results, the study presents the predicted Q_p (vertical axis) and measured Q_m (horizontal axis) values in Figure 5 in accordance with their respective lateral displacements. The figure shows a comparison of the difference between the prediction and measurement of the database when the displacement is 0.5%, 1%, 2%, 3%, 4%, and 5% of the pile diameters (a–f, respectively). In these figures, the slope of the regression line is less than 1 when the predicted values are lower than the measured values. When the predicted values are larger than the measured values, the slope is greater than 1. After the lateral displacements based on pile diameter are analyzed as 0.5%B, 1%B, 2%B, 3%B, 4%B, and 5%B, the slopes of the regression lines are computed as 0.61, 0.57, 0.63, 0.70, 0.75, and 0.77, respectively. The distribution of these slopes indicates that the values predicted by the original p–y curve method are all underestimated compared with the measured values, and the degree of underestimation is inversely proportional to the increase in displacement. The smaller the displacement is, the more underestimated the prediction is. Table 7 presents the statistical results of the comparison between the original prediction and the measured values from the load test results.

Figure 5.

Difference between the original predicted and measured values of flexible drilled shafts in cohesionless soils

Table 7.

Statistics of the difference between the predicted and measured values.

Data	Statistics	Lateral displacement, (%B)						Overall mean	Rb
		0.50	1.00	2.00	3.00	4.00	5.00
ΔQa/Qm	n	23	23	23	23	23	23	-	0.09
	Mean	−0.45	−0.52	−0.51	−0.47	−0.45	−0.43	−0.47
	SD	0.36	0.26	0.25	0.27	0.28	0.30	0.29

a

– ΔQ = Q_p – Q_m; ^b – R: range.

For the selected tests, the results predicted by the original p–y curve method are lower than the actual measurement results. The overall difference ratio is − 0.47, and this can be further improved. When the lateral displacement reaches 1%B, the difference ratio provides the lowest estimate, and its average underestimation is − 0.52, which is the average difference ratios of all piles in the database. However, when the lateral displacement reaches 5%B, the difference ratio is less underestimated at − 0.43. The overall fluctuation is small, and the difference in the full range value (the maximum value minus the minimum value) is only 0.09, which is denoted by R.

The prediction capacities of the original p–y curve method underpredict the measured capacity. This result may be attributed to the various factors that could affect the overall prediction. One of such factors is the fact that the original p–y curve method is derived from the results of driven piles that are installed using a very different method from that used for the installation of circular drilled shafts. Applying it to circular drilled shafts could produce a low predicted capacity, which calls for a calibration of the current method to suit circular drilled shafts. Additionally, this study assumes that cohesion is equal to zero, which may reduce the predicted capacity when the method is used. Moreover, the soil layers are simplified into just one type which is the soil type that is dominant throughout the entire shaft. Another factor is the assumption that the circular drilled shafts in this study are smooth cylindrical piles, which is not the case on site because drilling and casting concrete produce irregularities in the soil–shaft interface, thus increasing the actual capacity results. ³²

After reorganizing the above-mentioned data, Table 8 and Figure 6 present the ratio of the difference between pile diameter and displacement. Notably, the closer the difference ratio is to zero, the closer the predicted values are to the measured values. The results suggest that the difference ratios of flexible circular drilled shafts are generally negative, which represent the general prediction results that are under-predicted. As the pile diameter of the flexible circular drilled shafts increases, the difference ratio also increases, which means that the prediction method predicts father away from the measured values. This phenomenon indicates that when the p–y curve method is used for flexible circular drilled shafts, the difference ratio is proportional to the pile diameter.

Figure 6.

Relationship between the pile diameter and the difference ratio ΔQ/Q_m.

Table 8.

Relationship between the pile diameter and the difference ratio ΔQ/Q_m.

Shaft	Shaft diameter,	ΔQ/Qm
No.	B (m)	0.5%B	1.0%B	2.0%B	3.0%B	4.0%B	5.0%B
F1	0.30	−0.91	−0.88	−0.85	−0.83	−0.80	−0.79
F19	0.30	−0.76	−0.77	−0.76	−0.73	−0.71	−0.70
F20	0.30	−0.79	−0.77	−0.74	−0.71	−0.69	−0.67
F21	0.30	−0.81	−0.79	−0.77	−0.75	−0.73	−0.71
F22	0.30	−0.82	−0.79	−0.76	−0.74	−0.73	−0.72
F2	0.33	−0.87	−0.86	−0.83	−0.79	−0.77	−0.75
F3	0.35	−0.89	−0.87	−0.83	−0.80	−0.78	−0.76
F4	0.39	−0.91	−0.87	−0.84	−0.80	−0.78	−0.76
F12	0.50	−0.31	−0.32	−0.32	−0.33	−0.45	−0.48
F17	0.50	0.30	−0.18	−0.41	−0.43	−0.46	−0.49
F18	0.60	−0.78	−0.80	−0.80	−0.77	−0.75	−0.73
F23	0.88	0.37	0.15	0.13	0.22	0.39	0.46
F13	0.91	0.10	−0.20	−0.28	−0.32	−0.35	−0.37
F6	1.00	−0.29	−0.36	−0.34	−0.22	−0.13	−0.07
F7	1.00	−0.16	−0.27	−0.24	−0.14	−0.05	0.01
F14	1.07	−0.20	−0.36	−0.43	−0.44	−0.44	−0.43
F15	1.07	−0.17	−0.30	−0.36	−0.37	−0.35	−0.35
F16	1.07	−0.26	−0.43	−0.44	−0.41	−0.39	−0.36
F8	1.20	−0.15	−0.33	−0.32	−0.25	−0.19	−0.16
F10	1.20	−0.57	−0.57	−0.49	−0.46	−0.46	−0.46
F5	1.50	−0.30	−0.43	−0.28	−0.18	−0.14	−0.13
F9	1.55	−0.39	−0.29	−0.28	−0.15	−0.06	0.02
F10	1.58	−0.30	−0.37	−0.29	−0.19	−0.13	−0.13

Modification of the P-Y curve method

After determining that an underestimation occurs when using the original p–y curve method, the model is adjusted to improve the accuracy of the predictions of the method by using parameters α and β in Eq. (9). Parameter α is adjusted first, followed by parameter β, to slightly adjust the results and initially reduce the difference ratio greatly. Two considerations are adopted in modifying the p–y curve method. The first consideration is to adjust parameters α and β so that the average difference ratio and standard deviation are close to the actual value. The second consideration is to assess the average absolute difference between 1 (indicating that Q_p and Q_m are equal) and the results of the normalization of Q_p to Q_m (Q_p/Q_m) and adjust α and β to achieve average absolute differences that are as close to 0 as possible in all displacement ranges.

Modification of parameters α and β based on average difference ratio

Three values of 2.0, 3.0, and 4.0 are used for parameter α. The results of the modifications using the three parameters are shown in Figure 7. The figure presents the relationship of displacement (X axis) to the normalized difference of the predicted to the measured value (Y axis) and how the values are affected by the amount of parameter α being introduced to the equation.

Figure 7 illustrates that parameter α greatly increases the predicted capacity of the p–y curve model until the predicted values become close to the measured ones. The detailed results are presented in Table 9. The average difference ratios of 1.0, 2.0, 3.0, and 4.0 for parameter α are − 0.47, − 0.25, − 0.10, and 0.01, respectively, and the full range values are 0.09, 0.15, 0.15, and 0.17 with standard deviations of 0.29, 0.39, 0.45, and 0.51, respectively.

Figure 7.

Prediction and measurement statistics of flexible drilled shafts improved by parameter α.

Table 9.

The prediction and measurement statistics of flexible drilled shafts by varying the parameter α.

Modifying parameters		ΔQ/Qm	Lateral displacement, %B						Overall	R
α	β		0.5	1.0	2.0	3.0	4.0	5.0	mean
1.0	1.0	Meana	−0.45	−0.52	−0.51	−0.47	−0.45	−0.43	−0.47	0.09
		SDb	0.36	0.26	0.25	0.27	0.28	0.30	0.29
2.0	1.0	Meana	−0.28	−0.34	−0.29	−0.22	−0.20	−0.19	−0.25	0.15
		SDb	0.46	0.37	0.36	0.38	0.39	0.38	0.39
3.0	1.0	Meana	−0.16	−0.19	−0.11	−0.05	−0.04	−0.05	−0.10	0.15
		SDb	0.53	0.43	0.44	0.45	0.45	0.43	0.45
4.0	1.0	Meana	−0.07	−0.09	0.03	0.08	0.07	0.05	0.01	0.17
		SDb	0.59	0.50	0.51	0.51	0.49	0.46	0.51

a

- Mean based on 23 load tests; ^b - Standard deviation based on 23 load tests.

With the increase in parameter α, its average difference ratio, total range value, and standard deviation exhibit an increasing trend. The influence of parameter α on the prediction results decreases with the increase in displacement, which is based on the pile diameter. This result makes the prediction results highly dispersed, and the standard deviation is increased, which is a phenomenon that occurs when parameter α is used to modify the p–y curve method.

The result of parameter α  =  4.0 is the closest to the actual value. Continuous adjustment of the parameter only makes the difference ratio deviate farther from the measured value and does not meet the convergence conditions set in the study because the difference ratio at 5% pile diameter displacement is positive. Therefore, this value is not considered for use in this study. However, when parameter α is 3.0, the result is the closest to the actual value, and the negative value still has room for some improvement. Therefore, parameter α  =  3.0 is used in the modification and analysis to determine parameter β.

The values of parameters β are adjusted to 1.5, 1.6, and 1.7. These values are chosen based on the parametric study section of this paper. Figure 8 and Table 10 show the effects of each of the tested β parameters on the results by using parameter α  =  3.0 and analyzing the effects of the three values of parameter β.

Figure 8.

The predicted and measured statistics of the flexible drilled shafts improved by (3.0, 1.5), (3.0, 1.6), and (3.0, 1.7).

Table 10.

The prediction and measurement statistics of flexible drilled shafts using different sets of α and β.

Modifying parameters		ΔQ/Qm	Lateral displacement, B%						Overall	R
α	β		0.5	1.0	2.0	3.0	4.0	5.0	mean
3.0	1.5	Meana	−0.03	−0.08	0.00	0.03	0.01	−0.01	−0.01	0.11
		SDb	0.63	0.52	0.53	0.52	0.50	0.47	0.53
3.0	1.6	Meana	−0.01	−0.06	0.01	0.04	0.02	−0.01	0.00	0.10
		SDb	0.64	0.53	0.54	0.53	0.50	0.47	0.53
3.0	1.7	Meana	0.00	−0.05	0.03	0.05	0.02	−0.01	0.01	0.09
		SDb	0.65	0.53	0.55	0.53	0.50	0.47	0.54

a

- Mean based on 23 load tests; ^b - Standard deviation based on 23 load tests.

Figure 8 and Table 10 reveal that the average difference ratios of the parameters (α, β) of (3.0, 1.5), (3.0, 1.6), and (3.0, 1.7) are − 0.01, 0.00, and 0.01, respectively. The full range values are 0.11, 0.10, and 0.09, and the standard deviations are 0.53, 0.53, and 0.54. With the increase in parameter β, the average difference ratio and standard deviation still exhibit an increasing trend, but the full range value shows a decreasing trend.

According to the results, the parameters (3.0, 1.6) and (3.0, 1.7) meet the following requirements: (1) the average difference ratio must be within  ±  0.01, (2) the maximum difference ratio must be less than 0.1, (3) the minimum difference ratio must be greater than 0.1, and (4) the full range value must be less than or equal to 0.1. However, comparison of the average difference ratio and standard deviation indicates that the parameters (3.0, 1.6) are closer to the actual value and standard deviation than the parameters (3.0, 1.7). This study suggests the use of parameter α  =  3.0 and parameter β  =  1.6 as flexible circular drilled shafts in cohesionless soils because the difference is small. The modified p–y curve model is expressed as Eq. (10).

$$p=3.0A{p}_{u}tanh\left(1.6{\displaystyle \frac{kx}{A{p}_u}y}\right)$$

(10)

Modification of α and β based on average absolute difference

The goal of the modification of parameters α and β is to reduce the differences between the measured and predicted values in all displacement ranges and decrease the average absolute differences. To achieve this goal, α and β are substituted with various positive numbers until the average absolute difference between 1 (indicating that Q_p and Q_m are equal) and the results of the normalization of Q_p to Q_m (Q_p/Q_m) becomes close to 0 in all loading stages to achieve prediction values that are the closest to the measured values. This condition indicates that at an average absolute difference close to 0, the substituted constant parameters provide modifications that make the p–y curve method have the closest mean predicted values in all loading conditions and displacement ranges on the basis of the percent of the diameter.

Three constants, namely, 2.0, 3.0, and 4.0, are used to modify parameter α. The parameters are tested by not modifying parameter β to test the effects on the absolute difference. Table 11 shows that among the three values of parameter α, α  =  2.0 provides the mean absolute difference value that is the closest to 0 at 0.10 compared.

Table 11.

Mean absolute difference between predicted and measured lateral capacities.

(α, β)	Mean %Ba						Average
	0.5	1.0	2.0	3.0	4.0	5.0	|1-%B|
(1.0, 1.0)	0.61	0.57	0.63	0.70	0.75	0.77	0.33
(2.0, 1.0)	0.79	0.76	0.90	0.99	1.03	1.03	0.10
(3.0, 1.0)	0.92	0.64	1.10	1.18	1.20	1.17	0.18
(4.0, 1.0)	1.00	1.05	1.26	1.32	1.32	1.28	0.21
(2.0, 1.1)	0.81	0.78	0.93	1.02	1.05	1.04	0.10
(2.0, 1.2)	0.83	0.80	0.95	1.04	1.06	1.05	0.09
(2.0, 1.3)	0.85	0.82	0.97	1.05	1.08	1.06	0.09
(2.0, 1.4)	0.86	0.84	0.99	1.07	1.09	1.07	0.09
(2.0, 1.5)	0.88	0.86	1.01	1.09	1.10	1.07	0.09
(2.0, 1.6)	0.90	0.88	1.03	1.10	1.11	1.08	0.09

Note: ^a – Mean Q_m/Q_p of the 23 data at set displacement ranges in terms of percentage diameter.

After concluding that α  =  2.0 is suitable to be substituted for the equation, parameter β is then adjusted to fine-tune the results. In this study, parameter β is analyzed using various values, as shown in Table 11. The values are tested until the mean absolute value changes from exhibiting a downward trend to exhibiting an upward trend. At β  =  1.4, the predicted values are close to the measured values; they provide the mean absolute difference results that are the closest to 0 and then increase again at β  =  1.5.

According to the results, the parameters (2.0, 1.4) meet the requirements of the average difference between 1 (indicating that Q_p and Q_m are equal) and the results of the normalization of Q_p to Q_m (Q_p/Q_m). Therefore, this study suggests the use of parameter α  =  2.0 and parameter β  =  1.4 in the p–y curve method to predict the lateral load for flexible circular drilled shafts in cohesionless soils with an average absolute difference of 0.09. The results also indicate that the modifications are usable for all displacement ranges. The modified p–y curve model is expressed as Eq. (11).

$$p=2.0A{p}_{u}tanh\left(1.4{\displaystyle \frac{kx}{A{p}_u}y}\right).$$

(11)

With the use of Eqs. (10) and (11) to predict the lateral capacity of flexible circular drilled shafts in cohesionless soil conditions via the p–y curve method, the predicted Q_p (vertical axis) and the measured Q_m (horizontal axis) values are shown in Figure 9 in accordance with their lateral displacements. The figure presents a comparison of the difference between the prediction and measurement of the database when the displacements are 0.5%, 1%, 2%, 3%, 4%, and 5% of the pile diameters in (a), (b), (c), (d), (e), and (f), respectively, with the (α, β) parameters used. Under all lateral displacement conditions, the results of the new models are closer to the measured values compared with those of the original p–y curve method. At displacement values of 0.5% and 1.0% of the diameter, the predicted values are the closest to the measured values.

Figure 9.

Difference between the predicted and measured values of flexible drilled shafts in cohesionless soils with modifying factors (α, β).

To properly assess the capabilities of the two recommended equations in predicting the capacity, Figure 10 presents a test sample from the database (F14) displaying the measured load–displacement curve together with the curves predicted using the original and modified p–y curve method. Utilizing the modified equations provides a closer prediction compared with the original p–y curve method. Moreover, using Eq. (10) provides a higher estimate of the capacity, whereas using Eq. (11) provides a lower estimate in comparison with the measured value. After averaging all the data, this study recommends that the two equations be utilized for their respective strengths.

Figure 10.

Sample load-displacement curve comparison.

The complete analysis indicates that the modified equation of Eq. (10) should be utilized if the field engineer or designer aims to obtain a good average difference ratio, which produces the lowest value among all the modifications. However, if the designer aims to achieve consistent predictions through different displacement ranges based on the average absolute difference, then Eq. (11) is recommended for use by field engineers and designers.

Conclusions and recommendations

This study analyzed the current prediction method for determining the capacity of circular drilled shafts in cohesionless soils under lateral loading conditions. Various flexible circular drilled shaft lateral load test results from around the world were used to assess the differences in the measured capacities from load tests and capacities predicted using the prediction model of the p–y curve method. The following conclusions and design recommendations were obtained:

When flexible circular drilled shaft capacities were predicted in the cohesionless soil layer by using the original p–y curve method, the predicted values of the lateral loads were underestimated compared with the measured values, and this underestimation was roughly proportional to the pile diameter. The difference ratio between the average lateral load difference ratio was approximately − 47%, and the overall distance value, R, was 0.09.

When the pile diameter increased, the difference ratio also increased. The closer the difference ratio was to a positive number, the closer the predicted result was to the actual value. This phenomenon appears to show that when the p–y curve method of sand is used for flexible piles, the difference ratio is proportional to the pile diameter.

The average lateral load difference ratio converged from − 7.2% by increasing the original parameter (α, β)  =  (1.0, 1.0) to the improved parameter (α, β)  =  (3.0, 1.6), thus producing load difference ratios of − 0.2%.

The improved parameter (α, β)  =  (2.0, 1.4) produced the lowest average absolute difference of 0.087 compared with the other parameters, indicating that the modifications are applicable to all displacement ranges.

When the flexible circular drilled shaft was predicted in cohesionless soils by using the original p–y curve method, the standard deviation was 0.29. After gradually increasing the values of parameters α and β, the p–y curve increased with the increase in pile diameter. This condition increased the standard deviation to 0.53 when the improved parameters (α, β) were (3.0, 1.6).

The improved formulas for the p–y curve of flexible circular drilled shafts in cohesionless soils are shown in Eqs. (10) and (11). These modified equations are recommended to be utilized by designers and field engineers for their corresponding strengths.

$$p=3.0A{p}_{u}tanh\left(1.6{\displaystyle \frac{kx}{A{p}_u}y}\right),$$

(10)

$$p=2.0A{p}_{u}tanh\left(1.4{\displaystyle \frac{kx}{A{p}_u}y}\right).$$

(11)

Although the parameters α and β used in this study can reduce the difference ratio and the total distance value, respectively, they do not play an effective role in reducing the standard deviation. Further studies can be performed to develop other methods or parameters for reducing the difference ratio, full range, and standard deviations.

This study used 23 sets of in-situ pile load test data for flexible circular drilled shafts. Additional data can be collected in the future to improve the reliability of the modified p–y curve equations.