KSCE Journal of Civil Engineering (2015) 19(1):74-80 Copyright ⓒ2015 Korean Society of Civil Engineers DOI 10.1007/s12205-014-0020-6
Geotechnical Engineering
pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205
TECHNICAL NOTE
A Variational Solution for Nonlinear Response of Laterally Loaded Piles with Elasto-plastic Winkler Spring Model Fayun Liang*, Hao Zhang**, and Ke Yang*** Received January 9, 2013/Revised November 17, 2013/Accepted January 19, 2014/Published Online July 7, 2014
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Abstract Piles are frequently used to support lateral loads. Nonlinear p-y analysis is one of the most widely used methods to design laterally loaded piles. However, nonlinear p-y analysis requires special computer programs to perform the analysis. In this paper, an alternative solution based on a variational approach is presented to capture the behaviors of laterally loaded piles using Winkler beams on elastoplastic foundation. The displacements of piles are represented by finite series, and no discretisation of the pile shaft is required. Therefore, the proposed nonlinear solution can be easily implemented using the general computational software such as MATLAB. To facilitate the use of the numerical solution, methods for determining the parameters of the model are provided. The validity of the proposed method is confirmed through comparisons with existing theoretical solutions. Case studies are also presented to show the application of the present method to laterally loaded piles in field. The simplicity and the relative ease of the solution makes the proposed method a good alternative approach to analyze the nonlinear response of laterally loaded piles. Keywords: laterally loaded piles, elasto-plastic winkler spring model, nonlinear response, displacement, bending moment, variational approach ··································································································································································································································
1. Introduction Piles are widely used to support laterally loaded structures, such as bridges, high-rise buildings, and wind turbines. Several analytical methods have been developed for analyzing piles under lateral loads, including the elastic subgrade reaction approach by Reese and Matlock (1956), the elastic continuum method by Poulos (1971a; 1971b), Randolph (1981) and Chen (2008), and the p-y curve method by Matlock (1970) and Reese and Welch (1975). Among them, the p-y curve method is one of the most widely used methods to design laterally loaded piles. These aforementioned methods, however, require complex computer programs to perform full numerical analysis (Shen and Teh, 2004). In an attempt to provide a relatively easy and efficient alternative to analyze the nonlinear response of piles under lateral loads, a variational approach was employed in this work to analyze laterally loaded piles. The studies of pile based on a variational method started decades ago. Shen et al. (1997) proposed a variational method to solve the problem of vertically loaded pile groups, and this method was adopted in the investigation of laterally loaded pile groups embedded in homogeneous soil (Shen and Teh, 2002), as well as of a laterally loaded single pile in a one-layer soil profile with stiffness
increasing with depth (Shen and Teh, 2004). Then, Yang and Liang (2006) further developed the variational approach for laterally loaded piles in layered soil profiles with varying soil stiffness. Recently, Salgado et al. (2013) presented a variational elastic solution for axially loaded piles in multilayered soils that accounted for both vertical and radial soil displacements. However, these methods assumed the linear responses of the laterally loaded piles in real soil and were majorly for pile foundations under small lateral working loads. In this paper, in order to take the nonlinear relationship between soil pressure and pile deflection into account, a variational approach using Winkler beams on elasto-plastic foundation is employed as an alternative approach to solve the problem of a laterally loaded pile. This solution is very efficient since discretisation of the pile shaft is no longer required, and only a limited number of the terms in the assumed finite series (represent the displacements of the piles) can give solutions with an agreeable accuracy. In addition, the determinations of the parameters of the elasto-plastic Winkler spring model are provided and the computational software MATLAB is applied to deduce and perform the numerical calculations. The proposed method is validated against a computer program LPILE based on the p-y method and an elasto-plastic solution presented by Yokoyama (1977). Case studies are also presented to show the
*Professor, Dept. of Geotechnical Engineering, Tongji University, Shanghai 200092, China (Corresponding Author, E-mail:
[email protected]) **Ph.D. Student, Dept. of Geotechnical Engineering, Tongji University, Shanghai 200092, China (E-mail:
[email protected]) ***Geotechnical Engineer, CH2M HILL, Chantilly, VA 20151, USA (E-mail:
[email protected]) − 74 −
A Variational Solution for Nonlinear Response of Laterally Loaded Piles with Elasto-plastic Winkler Spring Model
application of the present method to laterally loaded piles in field.
surface; K can be assumed to linearly increasing with depth, and nh is the constant of subgrade reaction, following a relationship, K = K0 + nh · z.
2. Definition of the Problem 3. Procedures of Variational Solution For laterally loaded piles, the soil pressure and deflection has a nonlinear relationship that can be simulated by using an elastoplastic Winkler spring model proposed by Madhav et al. (1971). The model is shown in Fig. 1, in which K = modulus of subgrade reaction; py = yield soil resistance per unit length of pile; and u* = yield displacement of the soil. For a laterally loaded pile, yielding is likely to occur near the top of the pile at relatively low loads in real soils (Broms 1964b; Yokoyama, 1977; Poulos and Davis, 1980) and the yield zone propagates downward as the applied loads increase (Guo, 2009). Therefore, the soil-pile system is divided into two domains (Hsiung 2003; Guo 2006) as shown in Fig. 2, including yielding zone and elastic zone. In Fig. 2, Ht represents the applied lateral load at the top of a pile; Pz is the soil resistance per unit pile length at depth z; L and D are the length and diameter of a pile, respectively; and Ls is the thickness of the yield zone of soil, L0 is the distance between the loading point and the top of the soil
3.1 Potential Energy The problem defined in Fig. 2 will be solved using the variational approach. The total potential energy πp of a pile-soil system depicted in Fig. 2 is defined by Eq. (1): π p = Up + ∫
L0 + LS L0
1 --- u*Ku* + ( yz – u* )py dz 2
1 L + --- ∫ L + L yz Kyz dz – Ht yt 2 0
(1)
S
where, 2
2
dy 1 L Up = --- ∫ 0EP IP ⎛ --------2-z⎞ dz ⎝ dz ⎠ 2
(2)
In Eq. (2), Ep = Young’s modulus of the pile Ip = Moment of inertia of the pile section Up = Elastic strain energy of the pile Yz = Deflection of the pile at depth z The second and third terms on the right side of Eq. (1) are the energy consumed by the soil resistance of the yield zone and elastic zone, respectively. The fourth term is the input work performed by the lateral load (Ht) acting at the pile head. And yt = pile deflection at the pile head. 3.2 Deflection Series A finite series successfully used by Shen and Teh (2002) and Yang and Liang (2006) is employed in this paper to represent the pile deflections under applied lateral loads. The approximate horizontal deflections of a pile (yz) due to a lateral load at the pile head can be expressed by Eq. (3): T
Fig. 1. Elasto-plastic Winkler Spring Model
yz = { Z h } { δ h }
(3)
where, T
⎧ z πz 2πz nπz ⎫ { Zh } = ⎨ 1, --- , sin -----, sin ---------, …, sin --------- ⎬ L L L ⎭ ⎩ L { δh } = { ah, bh, βh1, βh2, …, βhn }
Fig. 2.
Notations for the Soil–pile System: (a) Pile Profile and (b) Soil Profile
Vol. 19, No. 1 / January 2015
T
(4) (5)
In the above equations, ah = constants representing the rigid body movement of a pile under lateral loads; bh = constants capturing the deflections due to the rotation of a pile; βhi = constants capturing the nonlinear mode of the deflected pile; and n = number of terms used in the trigonometric function. These expressions for capturing pile deflections are validated in the subsequent sections, where the results from the computer program LPILE and an elasto-plastic solution by Yokoyama (1977) are compared with the proposed solutions.
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Fayun Liang, Hao Zhang, and Ke Yang
3.3 Minimization of Potential Energy According to the principle of minimum potential energy, πp should be an extremum with respect to the admissible deflection field characterized by Eq. (3) which is expressed as: ∂π --------p = 0 (i = 1, 2, …, n+2) ∂δi
(6)
where δ i denotes the undetermined coefficients in Eq. (5) for the lateral load. Using Eq. (1), the potential energy can then be reduced to: L ∂Up L + L ∂yz --------- + ∫ ------- py dz + ∫ L ∂δi L ∂δi 0
∂y ∂y Kyz -------z dz = Ht -------t ∂δi ∂δi
S
0 + LS
0
2
∂y T K --------z- { Zh } dz { δh } ∂δhi
L +L ∂y = Ht --------ht- –∫ L py { Zh }dz ∂δhi 0
S
(8)
0
where, δhi = constants in the vectors { δh } ; and yht = deflections of the pile at the pile head under the applied lateral load. Eq. (8) can be expressed in matrix forms as follows: ( [ KpH ] + [KsH ]) { δh } = { H }
(9)
4
[K sH ] =
O 4
L
T ∫ L + L {Zh }K{ Zh } dz 0
S
T
L0 + LS
{ H } = { H, 0, …, 0 } – ∫ L
0
py { Zh }dz
(14)
Table 1. Values of nh (MN/m3) for Sand (Liang, 2002)
(10) n
4 1/2
Es D Es D - ---------K = ------------------------2 ( 1 – υ )D ref Ep Ip
(12)
0 1
4.1 Modulus of Subgrade Reaction For soils, on the basis of field test data of piles under lateral loads, the Carter (1984) equation which is revised from the Vesic (1961) equation can be used to determine the modulus of subgrade reaction K as follows:
(11)
0 4
The behavior analysis of laterally loaded pile using the proposed method requires some key parameters to be determined, such as the modulus of subgrade reaction K, the yield soil resistance per unit length of pile py and the yield displacement u*. Several methods are employed in this paper to determine these parameters.
where Dref = 1.0 m; D = diameter of a pile; Ep = elasticity modulus of a pile; Ip = second moment of inertia of a pile section; Es = modulus of soil; and υ = Poisson’s ratio of the soil. The linear relationship between K and depth is shown in Fig. 2(b). The alternative empirical correlation for the modulus of subgrade reaction is described below for different kinds of soil. For sand, assuming the modulus of subgrade reaction vary linearly with depth, the constant of subgrade reaction nh is determined from Table 1 developed in Liang (2002), which has been successfully adopted by Yang and Liang (2006) to numerically solve the problem of modeling laterally loaded piles in layered soils. For clay, the modulus of subgrade reaction K can be assumed to be constant with depth except for the surface portion of the soil. The K values are adopted from Terzaghi (1955) and Poulos and Davis (1980), respectively; the former
where [K pH ] and [ KsH ] are the matrices reflecting the pile and elastic zone soil stiffness under a lateral load, respectively; { H } is the vector representing the lateral load applied at the pile head, excluding the effect of yield zone soil. The details of these aforementioned matrices are given by:
Ep Ip π [K pH ] = -------------3 2L
(13)
4. Methods for Determining Input Parameters
3.4 Numerical Solution Substituting Eqs. (2) and (3) into Eq. (7) gives:
0 + LS
2
dy M = EpIp --------2-z dz
(7)
Equation (7) is the governing variational equation of a laterally loaded single pile in an elasto-plastic soil mass.
∂y ∂ ⎛ --------2-z⎞ 2 T ⎝ L L ∂z ⎠ ∂ { Zh } ------------------dz + ∫L ∫ 0EPIP ----------------2 ∂δhi ∂z
deflection of a pile) at the ground surface is compared with the specified yield displacement (u*). If the displacement exceeds u*, increase Ls by a length of 0.01L; calculate yz again and compare the displacement at the place of Ls. (c) The procedure is then repeated until soil displacements below Ls become less than u*. Then, yz and the thickness of the yield zone can be obtained. The moment of the pile at any depth can be calculated using Eq. (13).
The numerical solutions are obtained in three steps. (a) For a specified load, the initial value of the thickness of the yield zone (Ls) is first assumed to be zero. With the matrices [KpH ] , [K sH ] and {H} obtained according to Eqs. (10) ~ (12), the vectors { δh } and the deflections (yz) of a pile under a lateral load can successively be calculated using Eq. (9) and Eq. (3). (b) The soil displacement (where the soil displacement is equal to the − 76 −
SPT N 2-4 4 - 10 10 - 20 20 - 30 30 - 50 50 - 60
Above water 5.4 - 6.8 6.8 - 16.3 16.3 - 24.4 24.4 - 43.4 43.4 - 65.1 65.1 - 70.6
Below water 4.1 - 5.4 5.4 - 10.9 10.9 - 16.3 16.3 - 24.4 24.4 - 35.3 35.3 - 40.7
KSCE Journal of Civil Engineering
A Variational Solution for Nonlinear Response of Laterally Loaded Piles with Elasto-plastic Winkler Spring Model
ones recommended as 7.16, 14.3, and 28.6 MPa for stiff, very stiff, and hard clay respectively, while the later ranged between 80cu and 320cu. Davisson (1970) suggested a more conservative value of K = 67cu (cu = undrained shear strength). 4.2 Yield Soil Resistance and Yield Displacement For soils, the yield soil resistance per pile length py at depth z (see Fig. 1) can respectively be determined using Eqs. (15), (16) and (18) given by Broms (1964a and 1964b). Accordingly, the corresponding yield displacement u* (see Fig. 1) can then be obtained based on Eq. (17) for sand and Eq. (19) for clay, respectively (Madhav et al., 1971 and Guo, 2006). For sand: py = 3γzKpD
(15)
2 o φ K p = tan ⎛ 45 + ---⎞ ⎝ 2⎠
(16)
u* = py /K
(17)
For clay, py = 9cuD
(18)
u* = py /K
(19)
where, γ = the unit weight of the soil; Kp = the coefficient of passive earth pressure calculated by the Rankine earth pressure theory; φ = measured angle of internal friction; D = the pile diameter; and cu = the undrained shear strength.
5. Validations with Existing Solutions The above calculations have been coded into MATLAB. It is found that 10 × 10 stiffness matrices [KpH] and [KsH] yield sufficiently accurate results. The accuracy of the proposed solution is verified with the computer program LPILE based on the p-y method and an elasto-plastic solution by Yokoyama (1977) (a semi-analytical method treating the pile as an infinite Winkler beam on elasto-plastic foundation with a constant modulus of subgrade reaction). One free-head pile example by Yokoyama (1977) is evaluated. The following input parameters were used: D = 0.5 m, L =12 m, L0 = 0 m, EpIp = 102000 kN·m2, K = 10000 kN/m2 (proposed in the example by Yokoyama), γ = 10 kN/m3, φ = 30o, Ht = 200 kN, and the yield soil resistance per pile length py and yield displacement u* at depth z below the ground surface were determined using Eqs. (15) ~ (17). The pile deflections predicted by Yokoyama (1977), by LPILE and by the proposed solution are presented in Fig. 3(a). Compared with LPILE, the pile displacement from this study is much closer to that from the solution of Yokoyama (1977). Meanwhile, the pile displacements obtained from both our analysis and the solution of Yokoyama (1977) are slightly larger than that of the LPILE. Moreover, it can be seen in Fig. 3(b) that the maximum bending moment of the pile from our analysis is slightly smaller than those from the LPILE and the solution of Yokoyama (1977). Vol. 19, No. 1 / January 2015
Fig. 3. Comparison between: (a) The Displacement, (b) Bending Moment Profiles
6. Case Studies 6.1 Case History 1 The proposed solution for laterally loaded piles in an elastoplastic soil is validated against field test results of several fully instrumented lateral load tests. This case study is based on a lateral load test reported by Cox et al. (1974). The test was performed on a free-head steel tube pile embedded in sand at Mustang Island, Texas, with a diameter of 0.61 m and an embedment length of 21 m. The information pertaining to the steel pipe pile and the soil are briefly summarized as follows: According to the results from Standard Penetration Tests (SPT) given by Cox et al. (1974), an average SPT N value of the sand within 10D (where D is the pile diameter) below ground surface was reported as 18 blows per 30 cm. The stiffness of the sand is assumed to increase linearly with depth, and the modulus of subgrade reaction K0 is assumed to be
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Fayun Liang, Hao Zhang, and Ke Yang
Fig. 4. Comparison in: (a) Pile Deflections at Ground Surface, (b) Maximum Bending Moment (Cox et al., 1974)
Fig. 5. Comparison in: (a) Pile Head Deflections, (b) Maximum Bending Moment (Matlock, 1970)
zero. The constant of subgrade reaction for the sand layer (nh) is estimated to be 15 MN/m3 using Table 1. Other required input parameters are given by Cox et al. (1974) and Reese et al. (1974) as follows: EpIp = 163 000 kN·m2; γ = 10.4 kN·m3; and φ = 39o, and the yield soil resistance per pile length py and yield displacement u* at depth z below the ground surface were obtained using Eqs. (15) ~ (17). The distance between the loading point and the top of the soil surface was 0.305 m. Comparisons between the measured and predicted about pile deflections at ground surface and the maximum bending moment in Figs. 4(a) and 4(b) show that the predicted deflections and the maximum bending moment tend to be smaller than the measured values when the load exceeds 100 kN. Nevertheless, the difference between the predicted and the measured is within 10%.
The stiffness of the soil was assumed to be constant with depth. The modulus of subgrade reaction, K, is estimated to be 200 cu (an average value) using Poulos and Davis’s (1980) suggestion, and the yield soil resistance per pile length py and yield displacement u* at depth z below the ground surface are given by Eqs. (18) and (19). Other required input parameters are: EpIp = 31 280 kN·m2; and the distance between the loading point and the ground surface was 0.305 m. Comparisons between the measured and predicted deflections at the pile head and the maximum bending moment are shown in Figs. 5(a) and 5(b). The predicted deflections are somewhat over-predicted for the load level below 40kN and when the load exceeds 40kN, the predicted deflections and the maximum bending moment tend to be smaller than the measured values.
6.2 Case History 2 Another case study is based on a lateral load test reported by Matlock (1970). The lateral load test was performed on a freehead pile with a diameter of 0.324 m and a total embedment length of 12.8 m. The pile was embedded in clay at Sabine, Texas. The shaft and soil information are presented as follows: the average undrained shear strength cu was reported as 14.4 kPa.
6.3 Case History 3 In addition, two more field lateral load tests are also used to validate the proposed solutions. This case study is based on a lateral load test reported by Brown et al. (1988). The test was performed on a free-head steel pipe pile with a diameter of 0.273 m and an embedment length of 13.115 m. This field lateral load tests were performed at the Foundation Test Facility on the campus of the University of Houston. The information pertaining to the steel pipe pile and the soil
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A Variational Solution for Nonlinear Response of Laterally Loaded Piles with Elasto-plastic Winkler Spring Model
Fig. 6. Comparison in: (a) Pile Deflections at Ground Surface, (b) Maximum Bending Moment (Brown et al., 1988)
Fig. 7. Comparison in: (a) Pile Deflections at Ground Surface, (b) Maximum Bending Moment (Kishida and Nakai, 1977)
are briefly summarized as follows: According to the results from Standard Penetration Tests (SPT) given by Brown et al. (1988), the average SPT N value of the sand within 10D (where D is the pile diameter) below ground surface was reported as 40 blows per 30 cm. With a zero value of K0, the stiffness of the sand is assumed to increase linearly with depth. The constant of subgrade reaction for the sand layer (nh) is estimated to be 30 MN/m3 using Table 1. Other required input parameters are given by Brown et al. (1988) as follows: EpIp = 18824 kN·m2; γ = 15.8 kN/m3; and φ = 38.5o, and the yield soil resistance per pile length py and yield displacement u* at depth z are obtained using Eqs. (15) ~ (17). The distance between the loading point and the top of the soil surface was 0.305 m. The predicted pile deflections at ground surface and the maximum bending moment along pile shaft are compared with the measured results in Figs. 6(a) and 6(b). The predicted deflections are underestimated for the load level below 70kN and when the load exceeds 70 kN, the predicted deflections and the maximum bending moment tend to be greater than the measured values.
length of 17.5 m. The pile was driven into a silt layer with a uniform undrained shear strength. The shaft and soil information are presented as follows: the average undrained shear strength cu was reported as 18.8 kPa. The stiffness of the soil is assumed to be constant with depth. The modulus of subgrade reaction, K, is estimated to be 200 cu (an average value) using Poulos and Davis’s (1980) suggestion, and the yield soil resistance per pile length py and yield displacement u* at depth z below the ground surface are given by Eqs. (18) and (19). Other required input parameters are: EpIp = 298200 kN·m2; and the distance between the loading point and the ground surface was 0.1 m. Comparisons between the measured and predicted pile deflections at ground surface and the maximum bending moment are shown in Figs. 7(a) and 7(b). The comparisons show that the computed results agree reasonably well with the measured field results.
6.4 Case History 4 Kishida and Nakai (1977) reported a field lateral load test on a free-head pile with a diameter of 0.61 m and a total embedment Vol. 19, No. 1 / January 2015
7. Conclusions A variational solution based on elasto-plastic Winkler spring model is proposed as an alternative solution to analyze laterally loaded piles in yielding soils. The proposed method is validated against the existing available solutions and published field test
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Fayun Liang, Hao Zhang, and Ke Yang
results. The proposed method is more versatile than the existing solutions because of its relative ease of use and its capability to consider the nonlinearity of soil properties. Case studies of field lateral load test results show the applicability of the proposed solution and the recommended methods to determine the modulus of subgrade reaction, the yield soil resistance and yield displacement. Therefore, the proposed method has the potential to be used as an alternative approach in engineering practices for the analysis of laterally loaded piles.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 41172246), and National Key Basic Research Program of China (973 Program, Grant No. 2013CB036304). Financial support from these organizations is gratefully acknowledged.
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