G Mylonakis

HG Poulos

JM Roesset

MF Randolph

RF Scott

WD Guo

X Karatzia

Approximate closed-form solutions are derived for the settlement of axially loaded piles embedded in inhomogeneous soil by employing an energy method based on the Winkler model of soil reaction. Simple, linear shape functions for the attenuation of pile displacement with depth result in easy-to-use formulas. Two alternative methods are discussed for selecting the slope of the linear function. The proposed methodology is compared with rigorous (yet complicated) closed form solutions and an alternative approximation using a representative homogeneous soil

Crispin, Jamie

Su, Huize

Round, Katherine

Mylonakis, George

Explore Bristol Research

                          Crispin, J., Su, H., Round, K., & Mylonakis, G. (2018). Energy Formulationfor Axial Pile Head Stiffness in Inhomogeneous Soil. In Sustainability Issuesfor the Deep Foundations: Proceedings of the 2nd GeoMEast InternationalCongress and Exhibition on Sustainable Civil Infrastructures, Egypt 2018 –The Official International Congress of the Soil-Structure Interaction Groupin Egypt (SSIGE) (pp. 181-190). Springer, Cham.https://doi.org/10.1007/978-3-030-01902-0_15Peer reviewed versionLink to published version (if available):10.1007/978-3-030-01902-0_15Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia Springer at https://doi.org/10.1007/978-3-030-01902-0_15 . Please refer to any applicable terms of use of thepublisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms    Page 1                                           Energy Formulation for Axial Pile Head Stiffness in Inhomogeneous Soil  Jamie Crispin1, Huize Su2, Katherine Round2 and George Mylonakis3, PhD  1PhD Student, University of Bristol, Bristol, UK; j.crispin@bristol.ac.uk 2Student, University of Bristol, Bristol, UK;  3Chair in Geotechnics and Soil Structure Interaction, University of Bristol, Bristol, UK  ABSTRACT: Approximate closed-form solutions are derived for the settlement of axially loaded piles embedded in inhomogeneous soil by employing an energy method based on the Winkler model of soil reaction. Simple, linear shape functions for the attenuation of pile displacement with depth result in easy-to-use formulas. Two alternative methods are discussed for selecting the slope of the linear function. The proposed methodology is compared with rigorous (yet complicated) closed form solutions and an alternative approximation using a representative homogeneous soil.  INTRODUCTION  The Winkler model for axially loaded piles is a popular and cost-effective approach as it allows the development of simple, closed form solutions for predicting pile settlement without resorting to numerical analysis (Randolph and Wroth 1978; Poulos and Davis 1980; Scott 1981; Mylonakis and Gazetas 1998; Guo 2012; Crispin et al. 2018). Comparison of these solutions to more rigorous continuum analyses allows the Winkler spring stiffness to be determined by matching the settlement at the pile head between the two models (Mylonakis 2001; Syngros 2004; Guo 2012; Anoyatis 2013). In addition, the model can be extended to inhomogeneous soils in a straightforward manner by employing depth-dependent springs. For soil stiffness varying according to a power-law function of depth, closed-form solutions for pile head stiffness are available in the literature (Scott 1981; Guo 2012; Crispin et al. 2018). However, these solutions are complex, expressed in terms of special functions of mathematical physics, and are limited to a handful of inhomogeneous soil profiles.  This paper investigates an energy method based on a corresponding solution for laterally loaded piles by Karatzia and Mylonakis (2012, 2016). The proposed method applies the theory of virtual work to the Winkler model via an approximate shape function describing the attenuation of pile displacement with depth, to provide an easier to apply yet still realistic solution. By relating the Winkler modulus to soil properties using either the concentric cylinder model (Randolph and Wroth 1978) or by matching the results to the aforementioned continuum analyses, this solution can be applied to design problems.  METHOD  Eq. (1) shows the familiar governing equation for an axially loaded pile attached to a bed of Winkler springs with depth dependent modulus, k(z) (Scott 1981).  𝐸𝑝𝐴 𝑤′′(𝑧) − 𝑘(𝑧)𝑤(𝑧) = 0        (1)      Page 2                                           where Ep and A denote the pile Young’s modulus and cross-sectional area, respectively, and w(z) is the pile settlement at depth z. By considering the work done by a small virtual displacement profile, w*(z), over the pile length, L, Eq. (1) can be expressed in an integral form as:  ∫ 𝐸𝑝𝐴 𝑤′′(𝑧)𝑤∗(𝑧)𝑑𝑧𝐿0− ∫ 𝑘(𝑧)𝑤(𝑧)𝑤∗(𝑧)𝑑𝑧𝐿0= 0    (2)  Following the related solution for laterally loaded piles by Karatzia and Mylonakis (2012, 2016), a dimensionless shape function, 𝜙(𝑧), is used to describe both the real and virtual displacements by normalising with the corresponding value at the pile head. The pile head stiffness, K0, can then be expressed in terms of this shape function in the following virtual work equation:  𝐾0 = 𝐸𝑝𝐴 ∫ [𝜙′(𝑧)]2𝑑𝑧𝐿0+ ∫ 𝑘(𝑧)[𝜙(𝑧)]2𝑑𝑧𝐿0+ 𝛽2𝐾𝑏    (3)  where 𝛽 = 𝜙(𝐿) is the fraction of the pile head settlement propagating to the pile base, and Kb is the stiffness of the lumped spring used to model the pile base reaction. Note that the first term expresses the energy stored in the pile body, the second term the energy stored in the distributed Winkler springs that represent the stiffness (mainly through shearing) of the surrounding soil, and the final term the energy stored in the base spring that represents the soil below the pile. The sum of the three energies equals the work done by the force at the pile head under a unit virtual displacement.  Unlike in the lateral case where most piles of practical dimensions may be treated as infinitely long beams, the pile base conditions have a significant effect on the axial response. Therefore, the shape functions derived from the homogeneous soil case are relatively complex and do not lend themselves to the simple formulations desired. As an alternative, this paper investigates the benefits from using a simple linear shape function given in Eq. (4) for predicting pile head stiffness.  𝜙(𝑧) =𝑤(𝑧)𝑤(0)=𝑤∗(𝑧)𝑤∗(0)= 1 − (1 − 𝛽)𝑧𝐿       (4)  Homogeneous Soil In the case of homogeneous soil, k(z)=k, and the normalised pile head stiffness approximated using Eqs. (3) and (4) is given by:  𝐾0𝐸𝑝𝐴𝜆=1𝜆𝐿(1 − 𝛽)2 +13𝜆𝐿(𝛽2 + 𝛽 + 1) + 𝛽2Ω     (5)  where λ is a load transfer parameter (units of Length-1) controlling the attenuation of pile displacement with depth and Ω is a dimensionless base stiffness coefficient:  𝜆 = √𝑘𝐸𝑝𝐴  ,  Ω =𝐾𝑏𝐸𝑝𝐴𝜆              (6a,b)      Page 3                                           The coefficient Ω can take any value between 0, representing a fully-floating pile, and ∞, representing an end-bearing pile. Using Eq. (6a) and 𝑘 ≈ 0.6𝐸𝑠 (Roesset and Angelides 1980), λL can be approximated by (Mylonakis 1995):  𝜆𝐿 ≈ 0.85 (𝐸𝑝𝐸𝑠)−12(𝐿𝑑)         (7)  where Es is the Young’s modulus of the soil and d is the pile diameter.  A simple method to estimate β, referred to here as exact β, is to use the exact solution for homogeneous soil (Crispin et al. 2018):  𝛽 =𝑤(𝐿)𝑤(0)= [Ω sinh(𝜆𝐿) + cosh (𝜆𝐿)]−1      (8)  However, in light of the approximate shape function in Eq. (4), this choice would result in equilibrium of external forces on the pile not being satisfied. Considering equilibrium of vertical forces on the pile yields the following expression for pile head stiffness:  𝐾0 = ∫ 𝑘(𝑧)𝜑(𝑧)𝑑𝑧𝐿0+ 𝛽𝐾𝑏        (9)  For the homogeneous soil case, using the shape function in Eq. (4), Eq. (9) evaluates to:  𝐾0𝐸𝑝𝐴𝜆=12(1 + 𝛽)𝜆𝐿 + 𝛽 Ω        (10)  This expression can be used with the exact β in Eq. (8). Alternatively, an improved equation for β that obeys the equilibrium condition, referred to here as matched β, can be obtained by equating Eq. (10) with Eq. (5) and solving for β:  𝛽 =6−(𝜆𝐿)26[1+Ω𝜆𝐿]+2(𝜆𝐿)2         (11)  Figure 1a compares the proposed linear shape functions using both the matched β and exact β against the exact shape function for an example pile in homogeneous soil. It is evident that the matched β results in a slightly closer fit to the exact shape function. Figure 1b shows the variation of β with dimensionless length, λL, when Ω=0. Both estimates are very similar for λL<1.5, which covers a wide range of pile dimensions. However, the matched β behaves unexpectedly at larger values, predicting upward displacement of the pile base, which is clearly inadmissible.  In Figure 2, Eq. (5) and Eq.(10) with both the exact β and matched β are compared to the closed form solution to Eq. (1) for homogeneous soil (Mylonakis and Gazetas 1998):  𝐾0𝐸𝑝𝐴𝜆=Ω+tanh (𝜆𝐿)1+Ωtanh (𝜆𝐿)         (12)      Page 4                                           Both approximate solutions match Eq. (12) well when λL<1 (i.e. L/d<40 for Ep/Es=103), with the matched β providing the closest results.  Inhomogeneous Soil For homogeneous soils, the utility of the proposed approximate solution is limited by the simplicity of the exact solution. However, in inhomogeneous soils exact solutions have only been found for a limited number of soil stiffness variations with depth. The resulting formulae use special functions, such as those of the Airy or the Bessel type, that are unfamiliar to many practicing engineers. A simpler solution to derive and implement is desired. Extension of the proposed energy solution to inhomogeneous soil profiles only requires integration of the second term in Eq. (3) using a depth varying spring stiffness, k(z), rather than solving the full governing differential equation. This paper investigates Winkler modulus varying according to the power-law function of depth given in Eq. (13). Closed form solutions to similar stiffness variations have been provided by Scott (1981), Guo (2012) and Crispin et al. (2018).  𝑘(𝑧) = 𝑘𝐿 [𝑎 + (1 − 𝑎)𝑧𝐿]𝑛  ,  𝑎 = (𝑘0𝑘𝐿)1𝑛            (13a,b)  where k0 and kL are the Winkler modulus at the ground surface and pile base, respectively, 𝑎 is an inhomogeneity parameter accounting for non-zero surface stiffness and n is an inhomogeneity exponent. Substituting this variation in Winkler modulus into Eq. (3) and performing the integration yields:  𝐾0𝐸𝑝𝐴𝜆𝐿=(1−𝛽)2𝜆𝐿𝐿+𝜆𝐿𝐿(1−𝑎)3[(1−𝑎𝑛+1)(1−𝑎𝛽)2(𝑛+1)−2(1−𝑎𝑛+2)(1−𝑎𝛽)(1−𝛽)(𝑛+2)+(1−𝑎𝑛+3)(1−𝛽)2(𝑛+3)] + 𝛽2𝛺𝐿  (14)  where λL and ΩL are the values of λ and Ω considering the soil properties at the pile base:  𝜆𝐿 = √𝑘𝐿𝐸𝑝𝐴  ,  Ω𝐿 =𝐾𝑏𝐸𝑝𝐴𝜆𝐿              (15a,b)  An alternative approximate approach to account for soil inhomogeneity is to use the exact solution for homogeneous soil in Eq. (12) with the average Winkler modulus over the pile length, kav. In this case:  𝑘𝑎𝑣𝑘𝐿=1−𝑎𝑛(𝑛+1)(1−𝑎) ,  𝜆 = 𝜆𝐿√𝑘𝑎𝑣𝑘𝐿 ,  Ω = Ω𝐿√𝑘𝐿𝑘𝑎𝑣        (16a,b,c)  This is a generalisation of the approximation for linear stiffness variations proposed by Randolph and Wroth (1978).  The error introduced by both approximate solutions, Eq. (14) with the matched β in Eq. (11) and Eqs. (12) and (16), is compared in Figure 3 for λL=1 (i.e. L/d≈40 for Ep/Es=103) and Ω = 0 (when the error in both approximations peaks). The exact solution to Eq. (1) when k(z) is described by Eq. (13) is given by (Crispin et al. 2018):     Page 5                                            𝐾0𝐸𝑝𝐴𝜆𝐿= 𝑎𝑛2[𝐼𝜈−1(𝜒0)𝐼1−𝜈(𝜒𝐿)−𝐼1−𝜈(𝜒0)𝐼𝜈−1(𝜒𝐿)]+𝛺𝐿[𝐼𝜈−1(𝜒0)𝐼−𝜈(𝜒𝐿)−𝐼1−𝜈(𝜒0)𝐼+𝜈(𝜒𝐿)][𝐼−𝜈(𝜒0)𝐼𝜈−1(𝜒𝐿)−𝐼+𝜈(𝜒0)𝐼1−𝜈(𝜒𝐿)]+𝛺𝐿[𝐼−𝜈(𝜒0)𝐼+𝜈(𝜒𝐿)−𝐼+𝜈(𝜒0)𝐼−𝜈(𝜒𝐿)]  (17)  where Iν(χ) is the modified Bessel functions of the first kind, of order ν and argument χ:  𝜈 =1𝑛+2  ,  𝜒𝐿 =2𝜆𝐿𝐿(1−𝑎)(𝑛+2)  , 𝜒0 = 𝜒𝐿𝑎𝑛+22          (18a,b,c)  The performance of this solution has been evaluated against a database of pile field test results in London Clay (Voyagaki et al. 2018; Crispin et al. 2018) by employing the concentric cylinder model and magical radius proposed in Randolph and Wroth (1978).  The average homogeneous solution performs better the closer the stiffness variation is to a uniform distribution (a=1 or n=0), while the accuracy of the proposed solution is almost entirely independent of the inhomogeneity function. For a>0.3, the average homogeneous solution provides a better approximation of the pile head stiffness. However, for a=0 and n=1, the error in the average homogeneous solution is approximately double that for the proposed solution. Similar results have been observed for different pile lengths and base conditions with only the magnitude of the errors varying. Consequently, Eqs. (14) and (9) have been used to derive a new expression for pile head stiffness when a=0 incorporating an inhomogeneous matched β. The general expression is given in Eq. (19) and simplified expressions for different soil stiffness variations and base conditions are provided in Table 1.  𝐾0𝐸𝑝𝐴𝜆𝐿=(𝑛+2)𝜆𝐿𝐿[(𝑛+2)(𝑛+3)+2Ω𝐿𝜆𝐿𝐿]+(𝜆𝐿𝐿)3+(𝑛+1)(𝑛+2)2(𝑛+3)Ω𝐿(𝑛+1)(𝑛+2)2(𝑛+3)[1+Ω𝐿𝜆𝐿𝐿]+(𝑛+1)(𝑛+2)2(𝜆𝐿𝐿)2   (19)  Figure 4 shows the variation of predicted pile head stiffness with length for the proposed expression, the equivalent homogeneous soil approximation using Eqs. (12) and (16) and the exact solution in Eq. (17) when Ω = 0. The variation in error with pile length of Eq. (19) is compared to the average homogeneous solution when Ω = 0 in Figure 5. The proposed expression has approximately half the error for n=1 and it is less than 12% for λL<1.5, covering most likely pile dimensions (i.e. L/d<60 for Ep/Es=103).  CONCLUSIONS  An approximate solution has been developed using the Winkler model and virtual work to predict the response of axially loaded piles in inhomogeneous soils. The performance of this solution has been evaluated against an exact closed-form solution and an approximation considering an equivalent homogeneous soil and the same Winkler model. The main findings of this study are: • The proposed energy solution, in conjunction with an elementary shape function, approximates pile head stiffness well for piles of most practical     Page 6                                           dimensions. The best results are obtained when equilibrium of the pile is considered to establish the slope of the shape function (matched β). • The homogeneous approximation performs better the closer the soil stiffness profile is to homogeneous. The accuracy of the energy solution is relatively independent of soil inhomogeneity. • A simple expression for pile head stiffness, derived using the energy method, is provided in Eq. (19) for inhomogeneous soil with vanishing surface stiffness. This expression has half the error when compared to the homogeneous approximation for soil stiffness varying linearly with depth. Further investigation into this method using more complex shape functions could yield accurate expressions for pile head stiffness using well known elementary functions.  ACKNOWLEDGMENTS  The authors would like to thank Dr. Elia Voyagaki for her assistance in preparing the figures. The first author would also like to thank the Engineering and Physical Sciences Research Council for their support (grant number: EP/N509619/1).  REFERENCES  Anoyatis G. (2013) Contribution to kinematic and inertial analysis of piles by analytical and experimental methods. PhD Thesis. University of Patras, Patras, Greece.  Crispin J.J., Leahy C.P. and Mylonakis G. (2018) Winkler model for axially-loaded piles in inhomogeneous soil. under review.  Guo W.D. (2012). Theory and practice of pile foundations. CRC Press.  Karatzia X. and Mylonakis G. (2012) Horizontal response of piles in inhomogeneous soil: simple analysis. In Proceedings 2nd International Conference on performance-based design in earthquake geotechnical engineering. (M. Maugeri ed). Taormina, Italy. 1312-1327.  Karatzia X. and Mylonakis G. (2016) Horizontal stiffness and damping of piles in inhomogeneous soil. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 143(4), 04016113. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001621  Mylonakis G (1995) Contributions to static and seismic analysis of piles and pile-supported bridge piers. PhD thesis, State University of New York at Buffalo. New York, NY, USA.  Mylonakis G. (2001) Winkler modulus for axially loaded piles. Géotechnique, 51(5): 455-461.  Mylonakis, G. and Gazetas, G. (1998). Settlement and additional internal forces of     Page 7                                           grouped piles in layered soil. Géotechnique, 48(1), 55-72.  Poulos HG and Davis EH (1980) Pile foundation analysis and design. Wiley. New York, NY, USA.  Randolph M.F. and Wroth C.P. (1978). Analysis of deformation of vertically loaded piles. Journal of the Geotechnical Engineering Division, ASCE. 104(12), 1465-1488.  Roesset J.M. and Angelides D. (1980) Dynamic stiffness of piles. Numerical methods in offshore piling. London. Institution of Civil Engineers: 75-82.  Scott R.F. (1981). Foundation analysis. Prentice Hall.  Syngros K. (2004) Seismic Response of Piles and Pile-Supported Bridge Piers Evaluated through Case Histories, PhD Thesis, The City College of the City University of New York, New York, NY, USA.  Voyagaki E., Crispin J.J., Gilder C., Nowak P., O’Riordan N., Patel D. and Vardanega P.J. (2018) Analytical approaches to predict pile settlement in London Clay. these proceedings.  TABLES  Table 1. Summary of proposed pile head stiffness expressions (Eq. 19) for different soil stiffness variations and base conditions  𝐾0𝐸𝑝𝐴𝜆𝐿 General case 0 < 𝐿 < ∞, 0 ≤ Ω𝐿 < ∞ Perfectly floating pile, ΩL = 0 Perfectly end-bearing pile, ΩL → ∞ n=0 4𝜆𝐿(3 + Ω𝐿𝜆𝐿𝐿) + (𝜆𝐿𝐿)3 + 12Ω𝐿12(1 + Ω𝐿𝜆𝐿𝐿) + 4(𝜆𝐿𝐿)2 12𝜆𝐿𝐿 + (𝜆𝐿𝐿)312 + 4(𝜆𝐿𝐿)2 (𝜆𝐿𝐿)2 + 33𝜆𝐿𝐿 𝑛 = 1/2 10𝜆𝐿𝐿(35 + 8Ω𝐿𝜆𝐿𝐿) + 16(𝜆𝐿𝐿)3 + 525Ω𝐿525(1 + Ω𝐿𝜆𝐿𝐿) + 150(𝜆𝐿𝐿)2 350𝜆𝐿𝐿 + 16(𝜆𝐿𝐿)3525 + 150(𝜆𝐿𝐿)2 16(𝜆𝐿𝐿)2 + 105105𝜆𝐿𝐿 𝑛 = 1 6𝜆𝐿𝐿(6 + Ω𝐿𝜆𝐿𝐿) + (𝜆𝐿𝐿)3 + 72Ω𝐿72(1 + Ω𝐿𝜆𝐿𝐿) + 18(𝜆𝐿𝐿)2 36𝜆𝐿𝐿 + (𝜆𝐿𝐿)372 + 18(𝜆𝐿𝐿)2 (𝜆𝐿𝐿)2 + 1212𝜆𝐿𝐿        Page 8                                           FIGURES   a) b)   FIG. 1. a) Exact and approximate shape functions for example pile. b) Variation of exact β and matched β with pile length, Ω=0.  FIG. 2. Variation of exact and approximate pile head stiffness in homogeneous soil with pile length.     Page 9                                             FIG. 3. Error in pile head stiffness estimation for different soil stiffness profiles.  FIG. 4. Variation of exact and approximate pile head stiffness in inhomogeneous soil (described by Eq. 13) with pile length, a=0.      Page 10                                            FIG. 5. Error in head stiffness estimation with pile length, a=0.  

Energy Formulation for Axial Pile Head Stiffness in Inhomogeneous Soil

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Energy Formulation for Axial Pile Head Stiffness in Inhomogeneous Soil

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