Low_ICOSSAR-2001-Paper.pdf

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Stochastic nonlinear p-y analysis of laterally loaded piles

B.K. Low & C.I. TehSchool of Civil & Structural Engineering, Nanyang Technological University, Singapore 639798

Wilson H. TangSchool of Civil Engineering, Hong Kong University of Science & Technology, Hong Kong, China

Keywords: laterally loaded pile, nonlinear p-y curves, optimization, reliability, stochastic.

ABSTRACT: The deflection and bending moment of laterally loaded single piles are studied usingnonlinear and strain-softening p-y curves. The soil-pile interaction problem is solved using an al-ternative rigorous numerical procedure based on constrained optimization. The numerical proce-dure is then extended to reliability analysis in which the Hasofer-Lind index is computed based on

the alternative perspective of an expanding ellipsoid in the original space of the variables. The soilresistance is modeled stochastically to reflect spatial variation. Multicriteria reliability-based designof a laterally loaded pile is also illustrated. The advantages of the proposed method are its simplic-ity and transparency, and its implementation in the ubiquitous spreadsheet platform.

1 INTRODUCTION

The analysis of a laterally loaded pile based on the p-y curve concept is commonly done using thefinite element or the finite difference method. The problem is akin to the classical problem of the

beam on elastic foundation, except that in a laterally loaded pile the springs which model the resis-tance offered by the soil medium (Fig. 1) typically follow a nonlinear resistance-deflection behav-ior (nonlinear p-y curves). The nonlinear p-y curves are also likely to vary with depth. Hence thedeterministic soil-structure interaction analysis is intricate, and traditionally requires iterative nu-merical methods. Extending the deterministic analysis to a reliability analysis is hitherto a compli-cated task, by virtue of the implicit, numerical, and iterative nature of the performance function,and the need to account for the spatial correlation of the soil spring properties.

The objectives of the paper are twofold: (a) to present an alternative method of performing non-

Figure 1. (a) Laterally loaded pile in soil. (b) Soil spring idealization. (c) A segment of length dz.

pile

PH

soil

pile

PH

soil

z

y

dz

Q i+1

M i

M i+1

∆ zi+1

Q i

This 8-page paper appeared in the CDRom Proceedings of the Eight International Conference onStructural Safety and Reliability (ICOSSAR), Newport Beach, California, 17-22 June 2001.



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linear p-y analysis of laterally loaded piles which is as accurate as the finite element or finite dif-ference approach; (b) to demonstrate that the proposed deterministic numerical approach is readilyextended to a stochastic nonlinear p-y analysis of laterally loaded pile in which the soil spring

properties are modeled as one dimensional random field.The deterministic numerical procedure presented in the first part of the paper is based on con-

strained optimization in spreadsheet with a minimal amount of programming. It extends Low &Teh (1999)’s linear elastic analysis into the nonlinear and strain softening realm of the p-y curves.Similarly, the second part of the paper extends Low & Teh (1999)’s probabilistic analysis of later-ally loaded pile to a stochastic nonlinear analysis which accounts for spatial auto-correlation of thesoil spring properties. The stochastic procedure presented herein further illustrates the capability ofthe ellipsoid optimization approach for reliability analysis described in Low & Tang (1997a). The

procedure deals with cross-correlation and auto-correlation directly in the original space of thevariables, without the need for orthogonal transformation of the correlation matrix. The numericalderivatives and optimization search are performed automatically without user-intervention.

The deterministic and stochastic numerical procedures are presented one after the other. Theconstrained optimization approach used for the former also affords opportunities for the latter. Theresult is that the stochastic analysis is performed with relative clarity and transparency. Had con-

ventional deterministic and probabilistic computation methods been used, the task would have beenmore complicated.

2 DETERMINISTIC NUMERICAL PROCEDURE USING CONSTRAINED OPTIMIZATION

The pile is modeled as a beam and discretized into finite segments. Strain compatibility and equi-librium among the segments and between the segments and the soil springs are formulated. The

proposed numerical procedure was applied in Low & Teh (1999) to a pile in soil with uniform andlinear p-y relationship, and its accuracy verified. The flexibility of the method is enhanced here byallowing non-uniform discretization of the pile, choosing the pile head response as the optimizationcriteria and pile toe deflections as changing variables. The refined procedure is explained below byapplying it to a case with nonlinear, strain-softening p-y curves.

This problem is described in Tomlinson (1994, Example 8.2). A steel tubular pile having anoutside diameter d of 1.3 m and a wall thickness of 0.03 m forms part of a pile group in a breastingdolphin. The flexural rigidity E pI p of the pile is 4829082 kNm 2. The pile is embedded 23 m in thestiff overconsolidated clay with undrained shear strength c u = 150 kN/m 2, and protrudes 26 mabove the sea bed.

The constrained optimization approach is shown in Fig. 2, for the case where a cyclic force of421 kN is applied at 26 m above the seabed. Only the embedded portion of the pile requires soil-structure interaction analysis. The deflection of the pile head at 26 m above the seabed can be in-ferred from statics once the deflection and rotation of the pile at seabed level are known. The Mat-lock p-y curves for clays, as summarized in Eqs. 6.66-6.70 of Tomlinson (1994), have been used.In this case the Matlock p-y curves start from p = 0 at y = 0, and increases as cubical parabolas withthe deflection y to reach a maximum p value at y = 0.0975 m. For y between 0.0975 m and 0.4875m, the p-y curve decreases linearly, and remains constant beyond y = 0.4875 m. Hence the p-ycurves are nonlinear, exhibits strain-softening, and varies with depth, as shown in Fig. 3.

The numerical method proposed in this paper is as follows. The 23 m embedded length L of the pile is divided into 15 segments of varying length ∆ z. The segmental lengths increase in geometric progression by the common ratio r ∆ z, which is computed using the built-in GoalSeek routine in Mi-crosoft Excel. The depth coordinates in Fig. 2 under the column “z i” are:

101 z z z ∆+= (1a)

( )211 −−∆− −+= ii zii z zr z z , for i > 2 (1b)



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Figure 2. Numerical procedure for nonlinear p-y analysis of a steel tubular pile in a breasting dolphin.

Figure 3. Matlock’s nonlinear p-y curves for the case in Fig. 2

-1500

-1000

-500

0

500

1000

1500

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

y (m )

p ( k N / m

)

γ ' = 11.8 kN/m 3

c u = 150 kN/m 2

J = 0.25

B = 1.3 m

ε = 0.01

z = 22 m

z = 0 m

z = 0 m

z = 22 m

12 m

d L P H Mz0 ∆z1 r ∆z yhead Equilibrium maxdiff

1.3 23.00 421 10946 1 1.0584 0.994 0.0E+00 0.0E+00 5.138E-07

io z i EpIp yi yi' ksec d p i Q i Mi yprev0 0.00 4829082 0.06024 -0.01625 5964.7 -359.30 421.00 10946.00 0.060241 1.00 4829082 0.04512 -0.01396 7885.1 -355.78 63.46 11187.94 0.045122 2.06 4829082 0.03164 -0.01152 10865.2 -343.81 -306.77 11058.05 0.031643 3.18 4829082 0.02017 -0.00902 15919.6 -321.15 -679.25 10503.37 0.020174 4.36 4829082 0.01101 -0.00657 25825.6 -284.27 -1038.18 9480.84 0.011015 5.62 4829082 0.00431 -0.00430 52158.3 -224.91 -1357.70 7969.60 0.004316 6.95 4829082 0.00006 -0.00237 692419.0 -39.86 -1533.56 6022.07 0.000067 8.35 4829082 -0.00205 -0.00093 99705.8 204.25 -1418.00 3906.94 -0.002058 9.84 4829082 -0.00254 -0.00001 93119.4 236.10 -1090.35 2034.68 -0.002549 11.42 4829082 -0.00203 0.00042 115967.3 235.92 -718.62 610.07 -0.00203

10 13.08 4829082 -0.00116 0.00048 180890.8 210.29 -346.68 -271.99 -0.0011611 14.85 4829082 -0.00041 0.00032 388667.0 159.49 -20.43 -582.70 -0.00041

12 16.72 4829082 -0.00002 0.00013 1067795.8 25.05 151.89 -420.86 -0.0000213 18.69 4829082 0.00005 0.00001 1143757.9 -62.77 114.61 -128.86 0.0000514 20.79 4829082 0.00002 -0.00002 1224158.7 -29.63 17.95 -2.28 0.0000215 23.00 4829082 -0.00001 -0.00002 1276299.6 13.42 0.00 0.00 -0.00001

Deflection y (m)

0

5

10

15

20

25

-0.05 0.00 0.05 0.10Moment M (kNm)

0

5

10

15

20

25

-20000 0 20000Soil reaction (kN/m)

0

5

10

15

20

25

-500 0 500

z = 0

Stiff claycu = 150 kPa

z = 23 m

water

y = 60 mm

y = 1 m

PH = 421 kN

steel pipe pile,d=1.3m

e = 26 mseabed seabed

Boxed cells contain equations

seabed



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For uniform discretization, r ∆ z = 1. Other configurations and finer discretization can be used.The third column shows the flexural rigidity E pI p of the pile at the nodal points. The fourth col-

umn (y i) computes the lateral pile deflection y, as follows:

( ) ( ) 12

1112

2

211 *1 ++

+++++++ ′⎟⎟ ⎠

⎞⎜⎜⎝ ⎛

∆∆

+∗∆−−∗∆∆+= ii

i

iiiiiii y z

z z y y z z y y (2)

in which iii z z z −=∆ ++ 11 , and 122 +++ −=∆ iii z z z . The above equation has been derived byquadratic curve fitting over the nodes i to i+2 , based on the computed deflection values y i+1 and y i+2 at z i+1 and z i+2 , respectively, and the gradient y ’i+1 at z i+1 .

(The last two deflection values in column y i, at pile toe and at z = 20.79 m, respectively, wereinitially zero. The results shown in Fig. 2 were obtained after constrained optimization, asdescribed later.)

The fifth column, labeled y i′, gives the slope dy/dz of the pile at various z values, as follows:

( )ii

i p pi p p

ii z z I E

M I E

M y y −

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ +

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −′=′ +

+

+ 1

1

1 *5.0 (3)

The above equation is an expression of the first moment area theorem, which relates the changeof slope between any two points on a beam to the bending moment diagram. At the pile toe, thevalue of y′ is calculated as ( yn – y n-1) / ( zn − zn-1 ), where n is the last nodal number, 15 in this case.

A short function named “ksecd”, which stands for (secant k)*d, is created in the Visual Basic programming environment (VBA) of Microsoft Excel. The code is shown in Fig. 4. This function isused in column “ksecd” of Fig. 2, and reads its parameter y from column “yprev”. A program “SubIterate_ksecd” is also created (Fig. 4) to iteratively update the secant modulus of the p-y curves, intandem with the updated pile deflections of the column labeled “yprev”.

Figure 4. User-created VBA function and procedure for nonlinear p-y analysis of pile

The column labelled p i computes the soil reaction (kN/m) as:

ihi yd k p *−= (4)

where the negative sign is due to soil reaction acting in the opposite direction to pile deflection.The shear force Q and the bending moment M are:

( ) ( )iiiiii z z p pQQ −×+×−= +++ 111 5.0 (5)

Function ksecd (gammaz, cu, J, z, b, gammaB, epsilon, y) pu = (3 * cu + gammaz + J * z / b * cu) * b

If pu > 9 * cu * b Then pu = 9 * cu * byc = 2.5 * epsilon * bprest = 0.72 * puxr = 6 * b * cu / (gammaB + J * cu)If z < xr Then prest = prest * z / xry = Abs(y)If y < 10 ^ (-4) Then y = 10 ^ (-4)If y <= 3 * yc Then p = 0.5 * pu * (y / yc) ^ (1 / 3)If y > 3 * yc And y < 15 * yc Then

tem = (15 * yc - y) / (12 * yc) * (0.72 * pu - prest)p = (prest + tem)

End IfIf y >= 15 * yc Then p = prestksecd = p / yEnd Function

Sub Iterate_ksecd () Application.ScreenUpdating = Falsedel = 0.000001 'convergence criterionFor i = 1 To 200

'Ranges "yprev" & "yi" are predefined column names.If i = 1 Then Range("yprev").Value = 0.0001If i > 1 Then Range("yprev").Value = Range("yi").ValueSolverSolve TrueIf Range("maxdiff") < del Then Exit For

Next i Application.ScreenUpdating = TrueEnd Sub

Matlock p-y model





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ues. These average responses alone cannot be used to judge, at the design stage, whether the pilewill perform satisfactorily with respect to some specified permissible pile head deflection and tol-erable bending moment. A more rational approach would need to take into account not just themean values of the parameters but also their uncertainty. The partial factors in Eurocode 7 and theload and resistance factor design (LRFD) method are attempts in this direction.

A more rigorous and direct approach is to evaluate the reliability index as defined by Hasoferand Lind (1974). The classical reliability analysis procedure (FORM) based on transformed spaceis widely known (e.g., Ang & Tang, 1984, Haldar & Mahadevan, 1999). Alternatively, a practicaland transparent procedure that achieves the same result was presented in Low & Tang (1997a),

based on the perspective of an expanding ellipsoid tangent to the limit state surface in the originalspace of the random variables. Concepts of coordinate transformation are not required. Correlationis accounted for by setting up the quadratic form directly. Nonnormals are dealt with using estab-lished relationships between nonnormal distribution and its equivalent normal distribution. Iterativesearching is performed automatically using constrained optimization in spreadsheet. An advantageof the method is that partial derivatives are not part of the user-input; the method may therefore beattractive for cases with complicated or non-explicit performance functions. This intuitive approachis extended below for stochastic nonlinear p-y analysis and reliability-based design of a laterally

loaded pile.To illustrate, reliability analysis will be performed on the laterally loaded pile in Fig. 2. The 23m embedded length of the pile is discretized into 30 unequal segments, with ∆z1=0.3 m and r ∆z

=1.0822. The random variables are the lateral load P H at pile head and the undrained shear strengthcu at 31 nodal points along the embedded pile length below the seabed. The P H is assumed to benormally distributed, with mean value 421 kN and a coefficient of variation of 25%. Since themean undrained shear strength c u typically exhibits an increasing trend with depth, it will be as-sumed here that c u = 150 + 2z, kPa. The standard deviations of the 31 c u random variables are equalto 30% of their respective mean values. The following established negative exponential model isadopted to model the spatial variation of the c u values:

( ) ( )[ ]

e j Depthi Depth

ij δ ρ

−−=

(10)

An autocorrelation distance δ = 2 m is used. A 32 × 32 correlation matrix is set up. The first en-try is the coefficient of P H. The other entries are the correlation coefficients of the c u values at the31 nodes along the pile length. A vector (x i – m i)/σi of 32 components is also set up, where m i andσi are the mean and standard deviation of a random variable x i. The following quadratic form of theHasofer-Lind reliability index is then entered using the matrix functions of the spreadsheet:

[ ] ⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡ −= −

∈i

ii

T

i

ii

F x

m x R

m x

σ σ β 1min (11)

in which [R ]−1 is the inverse of the correlation matrix and F the failure region. The built-in optimi-

zation routine (Solver) in Microsoft Excel is invoked to predefine the optimization setting as fol-lows: “ to minimize ” the quadratic form (a 32-dimensional ellipsoid in original space), “ by chang-ing ” the 32 random variables (x i) and the last two nodal deflections (y n and y n-1, at and adjacent tothe pile toe respectively), “ subject to the constraints ” that the ellipsoid be tangent to the limit statesurface, that the equilibrium equations be satisfied, and that the 32 random variables be greater thanzero. The bottom two displacements (y n and y n-1) are reset to zero, and the x i values are assignedtheir respective mean values. The program code “Sub Iterate_ksecd” shown in Fig. 4 is then used toinvoke Solver iteratively to obtain the most probable failure point x* (i.e. the “design point”) atwhere the ellipsoid touches the limit state surface. The procedure does not require orthogonal trans-formation of the correlation matrix, and has been applied in Low & Tang (1997b) to reliabilityanalysis of reinforced embankments on soft ground.



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The β index obtained was 1.514 with respect to the limit state surface defined by max(columnM i) – 15490 = 0, where 15490 kNm is the bending moment of the pile that will cause yieldingstress σy ( = 417 MPa) at the outer edge of the annular steel cross section. A spreadsheet’s built-infunction Max(.) is conveniently used to return the largest bending moment in the column M i. Thisis because the M i column is computed numerically and iteratively (Eqs. 2 to 6), and it is not knownin advance where along the pile the maximum moment will occur.

At the most probable failure point (where the ellipsoid touches the limit state surface) the valueof the lateral load at pile head is P H = 580.3 kN, i.e., at 1.513 σPH from the mean value of P H, whilethe 31 autocorrelated c u values deviates only very slightly from their mean values. This is not sur-

prising given the e = 26 m cantilever length above the seabed; in fact the maximum bending mo-ment occurs at a depth of only 1.36 m below the seabed, or 27.36 m from the pile head. Hence, forthe case in hand, pile yielding caused by bending moment is sensitive to the applied load at pilehead, and not sensitive to the uncertainty of the shear strength below seabed. However, separate re-liability analysis for cases where the lateral load acts on pile head near ground surface (with zerocantilever length) indicates that the response is sensitive to both the lateral load at pile head and thesoil shearing resistance within the first few meters of the ground surface. The different sensitivitiesfrom case to case is automatically reflected in reliability analysis aiming at target index value (next

section), but will be difficult to consider in codes based on partial factors.

4 ILLUSTRATIVE EXAMPLE OF MULTI-CRITERIA RELIABILITY-BASED DESIGN OF ALATERALLY LOADED PILE IN A BREASTING DOLPHIN

The steel tubular pile that forms part of a pile group in a breasting dolphin (Fig. 2) has been exam-ined probabilistically in section 3 based on the single performance function of bending failuremode. The reliability index is only 1.514. In design, a reliability index of 3.0 is often stipulated.Further, deflection criterion also need to be considered. In the following illustrative design exam-

ple, it is assumed that 1.4 m is the maximum tolerable pile head deflection, which means a secanttilt angle of about 1.4 in 26, or about 3 degrees, with respect to seabed.

Analysis using mean parametric values of the previous section results in a pile head deflectionof 0.986 m. This average deflection does not provide information on the reliability of not exceedingthe 1.4 m tolerable limit, because the uncertainties of the random variables have not been reflectedin estimates based on mean values.

To illustrate multi-criteria reliability based design, suppose it is desired to select external diame-ter d and steel wall thickness t of the tubular pile so as to achieve a reliability index of 3.0 with re-spect to both the pile head deflection limit state and pile bending moment limit state. The mean andcovariance structure of P H and c u profile are as in the previous section. Pile embedment length is 23m. Note that the external diameter d and wall thickness t affect Eqs. 2 to 9 and both limit state func-tions, by affecting the moment of inertia I , the Matlock p-y curves (Fig. 3, in which B = d), and theyield moment M y = 2 σyI/d, where σy is the yield stress (417 MPa) of high-tensile alloy steel.

Figure 5 shows the curves of reliability index β = 3 as a function of the thickness t and external

Figure 5. Combinations of pile external diameter and wall thickness for reliability index of 3.

1.35

1.4

1.45

1.5

30 32 34 36 38 40

St e e l w a l l t h i ck n e s s ( m m )

E x

t e r n a

l d i a m e

t e r

( m

)

β = 3, (bend ing)

β = 3, (deflection)a

b

c



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diameter d of the tubular pile, for the deflection performance criterion and pile bending failuremode. The configuration at location a (i.e. 32 mm annular wall thickness and external diameter1.42 m) yields a reliability index of 3 with respect to both the deflection and the bending momentmodes. Configurations along line ac will have a reliability index of 3 for bending moment mode,and greater than 3 for deflection mode, whereas configurations along line ab means a reliability in-dex of 3 for deflection mode, and greater than 3 for bending moment mode.

5 SUMMARY AND CONCLUSIONS

An alternative practical numerical method based on constrained optimization has been presentedfor nonlinear strain-softening p-y analysis of laterally loaded single piles. Only a minimal amountof programming in the spreadsheet environment is required. The deterministic numerical procedurewas then extended into reliability analysis, in which the soil springs were modeled stochastically toreflect spatial variation of soil resistance. An intuitive expanding ellipsoid perspective in the origi-nal space of the random variables (Low and Tang, 1997) greatly simplifies the computation of thereliability index as defined by Hasofer-Lind, such that the transition from the numerical soil-pile

interaction deterministic analysis to a stochastic nonlinear and strain-sftening p-y analysis requireslittle additional effort, despite the highly implicit nature of the performance function. The proce-dure for cross-correlated and autocorrelated variables is the same as for uncorrelated variables, anddoes not involve orthogonal transformation of the correlation matrix.

Multicriteria reliability-based design was illustrated for a steel tubular pile that forms part of a pile group in a breasting dolphin. Both ultimate limit state and serviceability limit state were con-sidered. Advantages of reliability-based design over design based on partial factors were discussed.

REFERENCES

Ang, H. S., and Tang, W. H. (1984). Probability concepts in engineering planning and design, vol.2 Decision, risk, and reliability. John Wiley, New York.

Haldar, A., and Mahadevan, S. (1999). Probability, reliability and statistical methods in engineering design.John Wiley, New York.

Hasofer, A. M., and Lind, N. C. (1974). “Exact and invariant second-moment code format.” J. of Engrg.Mech., ASCE, New York, 100(1), 111-121.

Low, B. K., and Tang, W. H. (1997a). “Efficient reliability evaluation using spreadsheet.” J. of Engrg. Mech. , ASCE, New York, 123(7), 749-752.

Low, B. K., and Tang, W.H. (1997b). “Reliability analysis of reinforced embankments on soft ground.” Ca-nadian Geotechnical Journal , Ottawa, Canada, 34(5), 672-685.

Low, B. K., and Teh, C. I. (1999). “Probabilistic Analysis of Pile Deflection Under Lateral Loads.” Interna-tional Conference on Applications of Statistics and Probability (ICASP8) , A. A. Balkema, Rotterdam,The Netherlands, Vol. 1, pp. 407-414.

Tomlinson, M. J. (1994). Pile design and construction practice, 4th Ed., E & FN Spon, London.

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Low_ICOSSAR-2001-Paper.pdf