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8Design of Laterally Loaded Piles

Manjriker Gunaratne

CONTENTS

8.1 Introduction 327

8.2 Lateral Load Capacity Based on Strength 328

8.2.1 Ultimate Lateral Resistance of Piles 328

8.2.1.1 Piles in Homogeneous Cohesive Soils 328

8.2.1.2 Piles in Cohesionless Soils 335

8.3 Lateral Load Capacity Based on Deflections 339

8.3.1 Linear Elastic Method 339

8.3.1.1 Free-Headed Piles 339

8.3.1.2 Fixed-Headed Piles 341

8.3.2 Nonlinear Methods 343

8.3.2.1 Stiffness Matrix Analysis Method 343

8.3.2.2 Lateral Pressure-Deflection (p-y) Method of Analysis 348

8.3.2.3 Synthesis of p-y Curves Based on Pile Instrumentation 350

8.4 Lateral Load Capacity of Pile Groups 355

8.5 Load and Resistance Factor Design for Laterally Loaded Piles 356

8.6 Effect of Pile Jetting on the Lateral Load Capacity 356

8.7 Effect of Preaugering on the Lateral Load Capacity 360

References 361

8.1 Introduction

Single piles such as sign-posts and lamp-posts and pile groups that support bridge piers andoffshore construction operations are constantly subjected to significant natural lateral loads(such as wind loads and wave actions) (Figure 8.1). Lateral loads can be also introduced onpiles due to artificial causes like ship impacts. Therefore, the lateral load capacity is certainlya significant attribute in the design of piles under certain construction situations.

Unlike in the case of axial load capacity, the lateral load capacity must be determined byconsidering two different failure mechanisms: (1) structural failure of the pile due to yieldingof pile material or shear failure of the confining soil due to yielding of soil, and (2) pile

becoming dysfunctional due to excessive lateral deflections. Although passive failure of theconfining soil is a potential failure mode, such failure occurs only at relatively largedeflections which generally exceed the tolerable movements.

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FIGURE 8.1

Laterally loaded pile.

One realizes that “short” piles embedded in relatively stiffer ground would possibly fail due toyielding of the soil while “long” piles embedded in relatively softer ground would produceexcessive deflections. In view of the above conditions, this chapter is organized to analyzeseparately, the two distinct issues presented above. Hence the discussion will deal with twomain issues: (1) lateral pile capacity from strength considerations, and (2) lateral pile capacitybased on deflection limitations.

On the other hand, piles subjected to both axial and lateral loading must be designed forstructural resistance of the piles as beam-columns.

8.2 Lateral Load Capacity Based on Strength

8.2.1 Ultimate Lateral Resistance of PilesBroms (1964a,b) produced simplified solutions for the ultimate lateral load capacity of pilesby considering both the ultimate strength of the bearing ground and the yield stress of the pilematerial. For simplicity, the Broms (1964a,b) solutions are presented separately for differentsoil types, namely, cohesive soils and cohesionless soils.

8.2.1.1 Piles in Homogeneous Cohesive Soils

When a pile is founded in a predominantly fine-grained soil, the most critical design case isthe case where soil is in an undrained situation. The maximum load that can be applied on thepile depends on the the following factors:

1. Fixity conditions at the top (i.e., free piles or fixed piles). Most single piles can beconsidered as free piles under lateral loading whereas piles clustered in a group by a pilecap must be analyzed as fixed piles.

2. Relative stiffness of the pile compared to the surrounding soil. If the deformationconditions are such that the soil yields before the pile material then the pile is classified as a“short” pile. Similarly, if the pile material yields first, then the pile is considered a “long”pile.

8.2.1.1.1 Unrestrained or Free-Head Piles

Figure 8.2 and Figure 8.3 illustrate the respective failure mechanisms that Broms (1964a,b)assumed for “short” and “long” piles, respectively.

The ultimate lateral resistance Pu can be directly determined from Figure 8.4(a) and (b)based on the geometrical properties and the undrained soil strength. For short piles, Mmax, g,Pu, and f can be determined from Equations (8.1) to (8.4).

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FIGURE 8.2

Deflection, soil reaction, and bending moment distributions for laterally loaded short piles in cohesivesoil. (From Broms, B., 1964a, J. Soil Mech. Found. Div., ASCE, 90(SM3):27–56. Withpermission.)

Since the shear force is zero at the location of maximum moment, from the area of the soilreaction plot (Figure 8.2) one obtains

(8.1)

Similarly, by taking the first moments of Figure 8.2 about the yield point

Mmax=2.25Dg2cu(8.2)

Mmax=Hu(e+1.5D+0.5f)(8.3)

For the total length of the pile,

L=g+1.5D+f(8.4)

8.2.1.1.2 Restrained or Fixed-Head Piles

According to the Broms (1964a) formulations, restrained piles can reach their ultimatecapacity through three separate mechanisms giving rise to (1) short piles, (2) long piles, and

(3) intermediate piles. These failure mechanisms assumed by Broms (1964a) for restrainedpiles are illustrated in Figure 8.5(a)–(c). The assumption that leads to the analytical solutionsis that the moment generated on the pile top can be provided by the pile cap to restrain the pilewith the boundary condition at the top (i.e., no rotation).

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FIGURE 8.3

Failure mechanism for laterally loaded long piles in cohesive soil. (From Broms, B., 1964a, J. SoilMech. Found. Div., ASCE, 90(SM3):27−56. With permission.)

The ultimate lateral load, Pu, of short piles can be directly obtained from Figure 8.4(a). Thereader would notice that this condition is presented through a single curve in Figure 8.4(a) dueto the insignificance of the e parameter. Mmax and KPu can also be determined using thefollowing equations:

Pu=9cuD(L−1.5D)(8.5)

Mmax=Pu(0.5L+0.75D)(8.6)

For long piles, the ultimate lateral load, Pu, can be found from Figure 8.4(b). Then, thefollowing equations can be used to determine/and hence the location of pile yielding:

(8.7)

On the other hand, for “intermediate” piles where yielding occurs at the top (Figure 8.5b), thebasic shear moment and total length consideration in Equations (8.1), (8.4), and (8.8) can beused to obtain Pu:

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FIGURE 8.4

Ultimate lateral resistance of piles in cohesive soils: (a) short piles and (b) long piles. (From Broms,B., 1964a, J. Soil Mech. Found. Div., ASCE, 90(SM3):27–56. With permission.)

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FIGURE 8.5

Failure mechanisms for laterally loaded restrained piles in cohesive soils: (a) short piles, (b)intermediate piles and (c) long piles. (From Broms, B., 1964a, J. Soil Mech. Found. Div.,ASCE, 90(SM3):27–56. With permission.)

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(8.1)

My=2.25cuDg2–9cuDf(1.5D+0.5f)(8.8)

L=g+1.5D+f(8.4)

Example 8.1Estimate the ultimate lateral load that can be applied on the steel H pile (HP 250×62)

shown in Figure 8.6 assuming that the pile cap can provide the moment required at the piletop to keep it from rotating. The yield strength of steel is 300 MPa. The CPT test results (qc)for the site are also plotted in Figure 8.6(a). The Atterberg limits for the clay are: LL=60 andPL=25 and the saturated unit weight of clay is 17.5 kN/m3.

FIGURE 8.6

(a) Illustration for Example 8.1.

(b) HP section.

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From steel section tables and Figure 8.6(b)

Sxx=0.711×10−3m3, d=256My=Sxx σy=(0.711) (10−3) (300) MN m=213.3 kN m.

From the qc profile in Figure 8.6(a), qc can be expressed as

qc=4.7+0.04z MPa

From Robertson and Campanella (1983)

From Bowles (1996)

where PI is the plasticity index of the soil.One obtains the following su profile for PI=35:

Su=(1/13.16)[(4.7+0.04z)+0.001{(9.8z)(l−0.5)−(17.5–9.8)z}]=0.357+0.0028z MPa

su ranges along the length of the pile from 357 to 385 kPa showing the linear trend with depththat is typical for clays. Due to its relatively narrow range, it can be reasonably averagedalong the pile depth to be about 371 kPa

cu=371

Assume that the ground conditions and the pile stiffness are such that it behaves as a short pile.Then from Figure 8.4(a) or Equation (8.5), for an embedment length of 10 m/0.256 m= 39,

Pu/cuD2 can be extrapolated as Pu/cuD2=337But cuD2=24.314 kN, and hence Pu=8.22 MN.Thus, if the pile does not yield, it can take 8.22 MN before the soil fails.In order to check the maximum moment in the pile, Equation (8.6) can be applied.

Mmax=Pu(0.5L+0.75D)=8.22(0.5×10+0.75×0.256) MNm=42.68 MNm

But My=213.3 kN m. Hence the pile would yield long before the clay, and the pile has to bereanalyzed as a long pile.

From Figure 8.4(b),

Hence, the ultimate lateral load that can be applied on the given pile is about 600 kN.

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8.2.1.2 Piles in Cohesionless Soils

Based on a number of assumptions, Broms (1964b) formulated analytical methodologies todetermine the ultimate lateral load capacity of a pile in cohesionless soils as well. The mostsignificant assumptions were: (1) negligible active earth pressure on the back of the pile dueto forward movement of the pile bottom, and (2) tripling of passive earth pressure along thetop front of the pile. Hence

(8.9)

where is the effective vertical overburden pressure and is theangle of internal friction (effective stress).

8.2.1.2.1 Free-Head Piles

By following terminology similar to that in the case of cohesive soils, the failure mechanismsof short and long piles are illustrated in Figure 8.7 and Figure 8.8, respectively.

The ultimate lateral load for short piles can be estimated from Figure 8.9(a) or thefollowing equation.

(8.10)

Then, the location of the maximum moment (f in Figure 8.7) can be determined by thefollowing equation.

(8.11)

Finally, the maximum moment can be estimated by Equation (8.12)

(8.12)

FIGURE 8.7

Failure mechanism for laterally loaded short pile in cohesionless soil. (From Broms, B., 1964b, J. SoilMech. Found. Div., ASCE, 90(SM3):123–156. With permission.)

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FIGURE 8.8

Failure mechanism for laterally loaded long piles in cohesionless soil (From Broms, B., 1964b, J. SoilMech. Found. Div., ASCE, 90(SM3):123–156. With permission.)

If the Mmax value computed from Equation (8.12) is larger than Myield for the pile material,then obviously the pile behaves as a long pile and the actual ultimate lateral load Pu can becomputed from Equations (8.11) and (8.12) by setting Mmax=Myield.

On the other hand, Figure 8.9(b) enables one to determine the ultimate lateral load for longpiles directly.

8.2.1.2.2 Restrained or Fixed-Head Piles

For restrained short piles, consideration of horizontal equilibrium in Figure 8.10(a) yields

Pu=1.5γL2DKP(8.13)

Hence Pu can be found either from Equation (8.13) or Figure 8.9(a). Also, from Figure 8.10(a)it follows that

(8.14)

If Mmax computed from Equation (8.14) is larger that Myield for the pile material, then thefailure mechanism in Figure 8.10(b) applies. For this case, the following expression can bewritten for the moment about the pile bottom from which the ultimate lateral load can becomputed:

(8.15)

The above solution only applies if the moment Mmax at a depth of f computed

(8.11)

is less than Myield for the pile material.

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FIGURE 8.9

Ultimate lateral resistance of piles in cohesionless soils: (a) short piles, (b) long piles. (From Broms,B., 1964b, J. Soil Mech. Found. Div., ASCE, 90(SM3):123–156. With permission.)

Finally, if the above Mmax is larger than Myield, then the failure mechanism in Figure 8.10(c)applies. Thus, the ultimate lateral load can be computed from the following equation or itsnondimensional form in Figure 8.9(b).

(8.16)

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FIGURE 8.10

Failure mechanisms for restrained piles in cohesionless soils: (a) short piles, (b) intermediate piles,and (c) long piles. (From Broms, B., 1964b, J. Soil Mech. Found. Div., ASCE, 90(SM3):123–156. With permission.)

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8.3 Lateral Load Capacity Based on Deflections

The maximum permissible ground line deflection must be compared with the lateraldeflection of a laterally loaded pile to fulfill one important criterion of the design procedure.A number of commonly adopted methods to determine the lateral deflection are discussed inthe ensuing sections.

8.3.1 Linear Elastic MethodA laterally loaded pile can be idealized as an infinitely long cylinder laterally deforming in aninfinite elastic medium (Pyke and Beikae, 1984) with the horizontal deformation governed bythe following equation:

P=khy(8.17)

But, from distributed load vs. moment relations,

(8.18)

where B is the width of pile and EPI is the pile stiffness.Then the equation governing the lateral deformation can be expressed by combining (8.17)

and (8.18) as

(8.19)

The characteristic coefficient of the solution to y is defined by

(8.20)

1/βis also known as the nondimensional length, where kh is the coefficient of horizontalsubgrade reaction.

Broms (1964a,b) showed that a laterally loaded pile behaves as an infinitely stiff memberwhen the coefficient βis less than 2. Further, when βL≥4, it was shown to behave as aninfinitely long member in which failure occurs when the maximum bending moment exceedsthe yield resistance of the pile section.

For the simple situation where kh can be assumed constant along the pile depth, Hetenyi(1946) derived the following closed-form solutions:

8.3.1.1 Free-Headed Piles

8.3.1.1.1 Case (1): Lateral Deformation due to Load H

The following expressions can be used in conjunction with Figure 8.11, for a pile of width d.Horizontal displacement

(8.21a)

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FIGURE 8.11

Aid for using Table 8.1 for lateral load.

Slope

(8.21b)

Moment

(8.21c)

Shear force

V=−HKVH(8.21d)

The influence factors KΔH, KθH, KMH, and KVH are given in Table 8.1.

8.3.1.1.2 Case (2): Lateral Deformation due to Moment M

The following expressions can be used with Figure 8.12.Horizontal displacement

(8.22a)

Slope

(8.22b)

Moment

M=M0KMM(8.22c)

Shear force

V=−2M0/βKVM(8.22d)

The influence factors KΔM, KθM, KMM, and KVM are also given in Table 8.1.

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TABLE 8.1

Influence Factors for the Linear Solution

βL Z/L K(ΔH) K(θH) K(MH) K(VH) K(ΔM) K(θM) K(MM) K(VM)2.0 0 1.1376 1.1341 0 1 −1.0762 1.0762 1 0

2.0 0.125 0.8586 1.0828 0.1848 0.5015 −0.6579 0.8314 0.9397 0.2214

2.0 0.25 0.6015 0.9673 0.262 0.1377 −0.2982 0.6133 0.7959 0.3387

2.0 0.375 0.3764 0.8333 0.2637 −0.1054 −0.0376 0.4366 0.6138 0.3788

2.0 0.5 0.1838 0.7115 0.218 −0.2442 0.1463 0.3068 0.4262 0.3639

2.0 0.625 0.0182 0.6192 0.1491 −0.2937 0.2767 0.222 0.2564 0.3101

2.0 0.75 −0.1288 0.5628 0.0776 −0.2654 0.3747 0.1757 0.1208 0.2282

2.0 0.875 −0.2659 0.5389 0.0222 −0.1665 0.4572 0.1578 0.0318 0.1241

2.0 1 −0.3999 0.5351 0 0 0.5351 0.1551 0 0

3.0 0.125 0.6459 0.8919 0.2508 0.3829 −0.3854 0.6433 0.8913 0.2514

3.0 0.25 0.3515 0.6698 0.3184 0.0141 −0.0184 0.3493 0.6684 0.3202

3.0 0.375 0.1444 0.4394 0.285 −0.1664 0.1607 0.1429 0.436 0.2887

3.0 0.5 0.0164 0.2528 0.2091 −0.2223 0.2162 0.0168 0.2458 0.215

3.0 0.625 −0.0529 0.1271 0.1272 −0.2057 0.2011 −0.0489 0.1148 0.1353

3.0 0.75 −0.0861 0.0584 0.0594 −0.1519 0.1524 −0.0763 0.0396 0.0684

3.0 0.875 −0.1021 0.0321 0.0154 −0.0807 0.0916 −0.0839 0.0069 0.0225

3.0 1 −0.113 0.0282 0 0 0.0282 −0.0847 0 0

4.0 0 1.0008 1.0015 0 −0.0000 0.0282 −0.0847 0.0000 0

4.0 0.1250 0.5323 0.8247 0.2907 0.2411 −0.2409 0.5344 0.8229 0.2910

4.0 0.2500 0.1979 0.5101 0.3093 −0.1108 0.1136 0.2010 0.5082 0.3090

4.0 0.3750 0.0140 0.2403 0.2226 −0.2055 0.2118 0.0178 0.2397 0.2200

4.0 0.5000 −0.0590 0.0682 0.1243 −0.1758 0.1858 −0.0558 0.0720 0.1176

4.0 0.6250 −0.0687 −0.0176 0.0529 −0.1084 0.1200 −0.0696 −0.0043 0.0406

4.0 0.7500 −0.0505 −0.0488 0.0147 −0.0475 0.0538 −0.0616 −0.0206 −0.0025

4.0 0.8750 −0.0239 −0.0552 0.0014 −0.0101 −0.0033 −0.0535 −0.0096 −0.0148

4.0 1.0000 0.0038 −0.0555 −0 0.0000 −0.0555 −0.0517 −0.0000 −0

5.0 0 1.0003 1.0003 0 1.0000 −1.0003 1.0002 1.0000 0

5.0 0.1250 0.4342 0.7476 0.3131 0.1206 −0.1210 0.4343 0.7472 0.3133

5.0 0.2500 0.0901 0.3628 0.2716 −0.1817 0.1818 0.0907 0.3620 0.2720

5.0 0.3750 −0.0466 0.1013 0.1461 −0.1919 0.1930 −0.0455 0.1002 0.1461

5.0 0.5000 −0.0671 −0.0157 0.0494 −0.1133 0.1163 −0.0654 −0.0161 0.0482

5.0 0.6250 −0.0456 −0.0435 0.0026 −0.0412 0.0461 −0.0444 −0.0409 −0.0012

5.0 0.7500 −0.0197 −0.0369 −0.0088 −0.0008 0.0055 −0.0221 −0.0276 −0.0159

5.0 0.8750 0.0002 −0.0279 −0.0044 0.0108 −0.0139 −0.0110 −0.0086 −0.0125

5.0 1.0000 0.0167 −0.0259 −0 0.0000 −0.0259 −0.0091 −0.0000 −0

8.3.1.2 Fixed-Headed Piles

Due to the elastic nature of the solution, lateral deformation of the fixed-headed piles can behandled by superimposing the deformations caused by: (1) the known deforming lateral forceand the unknown restraining pile head moment, or (2) the known deforming moment and theunknown restraining pile head moment. Then, by setting the pile head rotation to zero (forfixed end conditions), the unknown restraining moment and hence the resultant solution canbe determined.

Example 8.2The 300mm wide steel pile shown in Figure 8.13 is one member of a group held together

by a pile cap that exerts a lateral load of 8 kN on the given pile and a certain magnitude of amoment required to restrain the rotation at the top. It is given that the coefficient of

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FIGURE 8.12

Aid for using Table 8.1 for moment.

FIGURE 8.13

Illustration for Example 8.2.

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horizontal subgrade modulus is 1000 kN/m3 and invariant with the depth. Determine thelateral deflection and the restraining moment at the top. Assume that the second moment ofarea (I) of the steel section is 2.2×10−6 m4 and the elastic modulus of steel to be 2.0× 106kPa.

But L=3.75 m, therefore, βL=7Then, determine the lateral displacement and the slope due to a force 8 kN (Equation 8.21)

If the restraining moment needed at the top is M, then the lateral displacement and the slopedue to M are evaluated as follows (Equation 8.22):

For restrained rotation at the top,0.056M+0.219=0; M=−3.93 kN mThen ΔM=0.108mHence, the total lateral displacement is ΔM+ΔH=0.216 m.

8.3.2 Nonlinear MethodsSeveral nonlinear numerical methods have become popular nowadays due to the availabilityof superior computational capabilities. Of them the most widely used ones are the stiffnessmatrix method of analysis and the lateral force-deflection (p−y) approach.

8.3.2.1 Stiffness Matrix Analysis Method

This method is also known as the finite element method due to the similarity in the basicformulation of the conventional finite element method and the stiffness matrix analysismethod. First, the pile is discretized into a number of one-dimensional (beam) elements.Figure 8.14 shows a typical discretization of a pile in preparation for load-deflection analysis.The following notation applies to Figure 8.14:

1, 2,…,N (in bold)—node numberPi (i even)—internal lateral forces on pile elements concentrated (lumped) at the

nodes

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FIGURE 8.14

Stiffness matrix method of analyzing laterally loaded piles.

Pi (i odd)—internal moments on pile elements concentrated (lumped) at the nodesXi (i even)—nodal deflection of each pile elementXi (i odd)—nodal rotation of each pile elementKj—lateral soil resistance represented by an equivalent spring stiffness (kN/m)Based on slope-deflection relations in structural analysis, the following stiffness relation

can be written for a free pile element (i.e., 1,2):

(8.23)

where EI is the stiffness of the pile and L is the length of each pile element.If the pile is assumed to be a beam on an elastic foundation, then the modulus of lateral

subgrade reaction kh at any depth can be related to the lateral pile deflection at that depth bythe following expression:

p=khy(8.24)

Hence the spring stiffness Kj can be expressed conveniently in terms of the modulus of lateralsubgrade reaction kh as follows:

For buried nodes:

Kj=LBkh(8.25)

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For surface node:

Kj=0.5LBkh(8.26)

where B is the pile width (or the diameter).

8.3.2.1.1 Estimation of the Modulus of Horizontal (Lateral) Subgrade Reaction kh

Bowles (1996) suggests the use of the following relation to evaluate kh (at different nodes)corresponding to different depths.

kh=Ah+BhZn

(8.27)

where Ah and Bh are evaluated using the bearing capacity expressions as follows:

Ah=Fw1CmC(cNc+0.5γBNγ)(8.28a)

Bh=Fw2CmCγNq(8.28b)

where Z is the depth of the evaluated location.The following values are suggested by Bowles (1996) for the above constants:

when using units of kN/m3, Cm=1.5–2.0, n=0.4–0.6, and Fw1, Fw2=1.0 for square andHP piles and in cohesive soils. Fw1 =1.3–1.7; Fw2=2.0−4.4 for round piles.

Thus, Equations (8.25)–(8.28) can be used to evaluate kh and hence Kj at each relevant node.This is illustrated in the following example.

Example 8.3Evaluate the equivalent spring stiffness at each node of the 300 mm×300 mm square pile

shown in Figure 8.15. Assume that the overburden corrected average SPT(N) value and theunit weight of the sand layer is 15 and 16.5 kN/m3, respectively.

SolutionFor N=15, from Equation (3.23), Φ≈34°. Also from Table 3.1, Nc=42, Nq=29, and Nγ=29.From Equations (8.28)

Ah=Fw1CmC(cNc+0.5γBNγ)=(1.0)(1.5)(40)(0.5)(16.5)(0.3)(29)=4306.5Bh=Fw2CmCγNq=1.0(1.5)(40)(16.5)(29)=28,710

Applying Equation (8.27) kh=As+BhZn−4307+28710Z0.5 kN/m3.Figure 8.15 also shows the kh distribution with the depth and the equivalent spring stiffness

corresponding to each node.It must be noted that Equations (8.25) and (8.26) have been applied to determine the K

values.The element stiffness matrices given by expressions such as Equation (8.23) can be

assembled to produce the global stiffness matrix [K] using basic principles of structural

analysis. During the assembling process, the spring stiffness Kj of each underground node canbe added to the corresponding diagonal element of [K]

[P]=[K][X](8.29)

Then, knowing the global force vector one can solve Equation (8.29) to obtain the globaldeflection vector.

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FIGURE 8.15

Illustration for Example 8.3.

Example 8.4The 300mm wide steel pile shown in Figure 8.16 is one member of a group held together

by a pile cap that exerts a lateral load of 8 kN on the given pile and a moment of certainmagnitude required to restrain the rotation at the top. It is given that the coefficient ofhorizontal subgrade modulus is 1000 kN/m3 and invariant with the depth. Determine therelevant force and deflection vectors assuming that the total number of nodes is 6. Alsoillustrate the solution procedure to obtain the lateral deflection of the pile and the momentrequired at the cap. Assume that the second moment of area (I) of the steel section is 2.2×10−6

m4 and the elastic modulus of steel is 2.0×106 kPa.

SolutionThe equivalent spring stiffness has been computed as in Example 8.3 and indicated in Figure8.16. As shown in Figure 8.16, the only external forces applied on the pile are the onesapplied by the pile cap and the soil reactions at the bottom that assure fixity. It is also notedthat the spring associated with the bottom-most node has been added to the unknown force P12.

Hence, the external force vector is given by the following equation:

[P]=[M1 8 0 0 0 0 0 0 0 0 M11 P12]T

(8.30)

On the other hand, the deflection vector is given by the following equation

[X]=[0 A θ2 Δ2 θ3 Δ3 θ4 Δ4 θ5 Δ5 0 0](8.31)

in which it is assumed that the rotation at the top is restrained due to the pile cap (i.e., X1=θ1=0) and the translation as well as the rotation at the bottom are retrained by the groundfixity (i.e., X12=Δ6=0 and X11= 0 6=0). The required lateral deflection is Δ.

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FIGURE 8.16

Illustration for Example 8.4.

The stiffness matrices for the first four elements and the fifth element are expressed by thefollowing matrices:

Hence, the assembled and modified (for springs) global stiffness matrix would be

(8.32)

If [K] in Equation (8.32) is rearranged and partitioned so that

[P]1=[K]11[X]1+[K]12[X]2(8.33a)

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and

[P]2=[K]21[X]1+[K]22[X]2(8.33b)

where

[P]1=[M1 M11 P12](8.34a)

[P]2=[8 0 0 0 0 0 0 0 0]T

(8.34b)

(8.34c)

[X]2=[Δ θ2 Δ2 θ3 Δ3 θ4 Δ4 θ5 Δ5](8.34d)

and[K]11, [K]12, [K21] , and [K]22 are the corresponding 3×3, 3×9, 9×3, and 9×9 par of [K] as

illustrated below:

(8.35)

From Equation (8.33b) [X]2 can be expressed as

(8.36)

Substituting the above result and Equation (8.34c) in Equation (8.33a),

(8.37)

Hence, the unknown external forces can be determined from Equation (8.37). Accordingly,from Equation (8.36), Δ=0.304 m and M1=16 kN m. Then, by substitution in Equation (8.36)the unknown deflections [X]2 can be determined.

Finally, the moments and the shear forces along the pile length can be determined bysubstituting the nodal deflections in the individual element equations such as Equation (8.23).

8.3.2.2 Lateral Pressure-Deflection (p-y) Method of Analysis

The following form of Equation (8.18) is employed in the p-y curve approach (Figure 8.17)developed by Reese (1977):

(8.38)

where p'=soil reaction per unit pile length.

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FIGURE 8.17

Set of p−y curves. (Reese, L.C., 1977, J. Geotech. Eng., ASCE, 103(GT4):283–305. With permission.)

It is observed that the difference between Equations (8.38) and (8.19) is that Equation (8.38)accounts for the shear and moment effects induced by the axial force P(z) due to the finitecurvature of the pile produced by lateral loading (Figure 8.18). Hence, the shear force and thedistributed soil reaction on the pile at any depth can be expressed as

(8.39)

(8.40)

The finite difference (FD) form of the above equation is given as (Reese, 1977):

(8.41)

where

Rm=EmIm(8.42)

is the stiffness of the mth mode, ym is the lateral deflection at the mth node, h is the finitedifference step size (nodal distance along the pile), Pz is the axial force at the mth node (depth

z). The parameter km defined in Equation (8.43) can be evaluated for each node m bypredicting the p−y curve corresponding to the depth, of that node, z.

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FIGURE 8.18

Representation of the deflected pile. (From Reese, L.C., 1977, J. Geotech. Eng., ASCE,103(GT4):283–305. With permission.)

p'=kmy(8.43)

Finally, using the following boundary conditions:

1. shear and moment are zero at the bottom of the pile,2. lateral load and the moment (or the slope or the rotational restraint) at the pile top are

known,

the FD algorithm in Equation (8.41) can be solved and the lateral deflection, pile rotation, andmoment and shear along the pile can be numerically determined at any location. According toReese (1977), the p−y methodology implies that the behavior of the soil at any depth isindependent of its behavior at other locations, which is strictly not true. However,experiments seem to indicate that the above implication is justified under practicalcircumstances.

8.3.2.3 Synthesis of p−y Curves Based on Pile Instrumentation

Strain gauge readings obtained along the length of a laterally tested pile can be employed todevelop the lateral load transfer curves (p−y curves) at a finite number of points along the pile(Hameed, 1998). The values of p (horizontally distributed load intensity) and y (lateraldeflection) at any pile location at a given lateral loading stage can be determined using thefollowing analytical procedure. From the simple beam theory,

(8.44)

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where y is the lateral deflection, z is the vertical coordinate along the pile, h is the distancefrom the neutral axis of the pile cross-section to the strain gauge location, and εis the straingauge reading at z.

Hence, the lateral deflection (y) can be expressed as

(8.45)

Similarly, by using Equations (8.19) and (8.44), the distributed soil load (p) can be expressedas

(8.46)

Therefore, it can be seen that both p and y values can be found from a mathematicallyapproximated (fitted) εcurve to measured flexural strains. This is usually achieved either byfitting a cubic spline function between successive strain data points (Li and Byrne, 1992) orby fitting a higher-order polynomial to all of the strain data points (Ting, 1987). The fittingprocedure can be illustrated as follows.

Example 8.5For the model pile shown in Figure 8.19, assuming that the measured longitudinal strains

are given by Figure 8.20, illustrate the fitting procedure that can be used to generate the p−ycurves at specific depths.

SolutionThe distance z is measured from the pile tip which is located 1.0m below the ground surface.In order to closely trace all of the strain data, the following polynomial with five coefficients(ai) can be considered:

y=a1z6+a2z7+a3Z8+a4Z9+a5Z10

(8.47)

FIGURE 8.19

Illustration for Example 8.5.

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FIGURE 8.20

The measured strains in Example 8.5 and the polynomial fit. (From Hameed, R.A., 1998, lateral LoadBehavior of Jetted and Preformed Piles, Ph.D. dissertation, University of South Florida,Tampa, FL. With permission.)

It can be seen that the terms up to z5 have been discarded from Equation (8.47) since the piledeflection and all of its derivatives up to the fifth derivative are generally considered zero atthe pile tip (z=0) (Ting, 1987). This is because the deflection, slope, moment, shear, and thelateral pressure due to the applied lateral load are negligible at the pile tip. Thus, bycombining Equations (8.44) and (8.47), the strain at any location within the embedded part ofthe pile can be expressed by the following function with five unknown coefficients ai, i=1–5:

(8.48)

Then four pairs of strain gauge readings and the known soil pressure (p=0) at the soil surface(z=z0=1.0 m) can be used to determine the unknown ai(i=1–5). Furthermore, a third-degreepolynomial was employed for approximating the deflection (y) of the free portion of the pile(above the ground level). This ensures that the p=0 condition is satisfied all over the freeportion since the fourth derivative of this polynomial (p in Equation 8.18) automatically dropsout. Consequently, the deflection above the soil surface can be given by the followingfunction with four unknown coefficients bi, i= 0–3:

y=b0+b1(z−z0)+b2(z−z0)2+b3(z−z0)3

(8.49)

Three of the above constants (bi, i=0–3) were determined by matching the deflection, slopeand moment of the free pile portion with the corresponding values of the embedded portion asdetermined by Equations (8.47) and (8.48), at the soil surface (z=z0). The fourth bi constantcan be determined by setting the moment at the lateral loading level to zero.

The distributions of deflections and soil pressure computed using the above methodologyare illustrated in Figure 8.21 and Figure 8.22, respectively. In this case, the model pile is

assumed to be embedded in an unsaturated soil bed of unit weight 16.2 kN/m3. Figure 8.23shows the analytical predictions of the lateral load behavior of the piles at specified depth.

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FIGURE 8.21

Computed lateral displacement vs. depth. (From Hameed, R.A., 1998, Lateral Load Behavior of Jettedand Preformed Piles, Ph.D. dissertation, University of South Florida, Tampa, FL. Withpermission.)

FIGURE 8.22

Computed soil pressure vs. depth. (From Hameed, R.A., 1998, Lateral Load Behavior of Jetted andPreformed Piles, Ph.D. dissertation, University of South Florida, Tampa, FL. Withpermission.)

FIGURE 8.23

Analytically predicted load-displacement behavior. (From Hameed, R.A., 1998, Lateral LoadBehavior of Jetted and Preformed Piles, Ph.D. dissertation, University of South Florida,Tampa, FL With permission.)

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In order to eliminate the depth dependency, p−y curves can be normalized using a soilparameter which depicts the mean normal stress level. p/EsB is a suitable normalizedparameter for this purpose since Emax (elastic modulus at very low strains) used to computep/EsB shows a strong mean normal stress dependence(Li and Byrne, 1992). Hameed et al.(2000) determined Emax from the measured coefficient of horizontal subgrade reaction, Kmax,using the following expressions (Glick, 1948; Bowles, 1996):

(8.50)

(8.51)

where K's and Es have the same units (kPa) and K's is the horizontal subgrade modulus, Lp isthe pile length, B is the pile width, and υs, is Poisson’s ratio.

Kmax at each depth can be obtained from the initial stiffness of the experimentallydetermined p−y curves (Figure 8.23) (Hameed, 1998). Similarly, the ultimate soil pressures(pu) can be obtained from p−y curves at each depth by fitting the experimentally developedp−y curve with a hyperbolic function of the form p=y/(a+by) (Kondner, 1963; Georgiadis etal., 1991). The pu value for each fitted curve is expressed by the curve parameter, 1/b, sincepu=1/b when y→∞. The variations of Kmax and pu with depth are shown in Figure 8.24 andFigure 8.25, respectively.

Based on the foregoing discussion p−y curve can be expressed as (Hameed, 1998):

(8.52)

On the other hand, two other popular mathematical formats for p−y curves have beenprovided by Reese et al. (1974) and Murchison and O’Neill (1984). These are illustrated inFigure 8.26 and Figure 8.27, respectively.

FIGURE 8.24

Variation of Kmax with depth. (From Hameed, R.A., Gunaratne, M, Putcha, S., Kuo, C, and Johnson, S.,2000, ASTM Geotech. Testing J., 23(3). With permission.)

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FIGURE 8.25

Variation of pu with the depth. (From Hameed, R.A., Gunaratne, M., Putcha, S., Kuo, C., and Johnson,S., 2000, ASTM Geotech. Testing J., 23(3). With permission.)

8.4 Lateral Load Capacity of Pile Groups

In most actual foundation applications, since piles installed as a cluster invoke group action, itis important for the foundation designer to be knowledgeable of the response of a group ofpiles to lateral loads. Ruesta and Townsend (1997) performed a field test involving an isolatedsingle pile and a large-scale test group of 16 prestressed concrete piles spaced three (3)diameters apart to study how the lateral load characteristics of pile groups relate to those ofindividual piles in the group. Of the many in situ testing methods used to predict the p−ycurves, SPT and pressuremeter test predictions were corroborated by the strain gage andinclinometer readings. Ruesta and Townsend (1997) concluded that an overall average pmultiplier of 0.55 was needed for the individual p−y curves to predict the overall lateralresponse of the pile group.

FIGURE 8.26

Analytically predicted load-displacement behavior. (From Reese, L.C., Cox, W.R., and Koop, F.D.,1974, Analysis of laterally loaded piles in sand, Proceedings of the 6th OffshoreTechnology Conference, Houston, TX, paper OTC 2080, pp. 473–483. With permission.)

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FIGURE 8.27

Analytically predicted load-displacement behavior. (From Murchison, J.M. and O’Neill, M.W., 1984,Evaluation of p–y relationship in cohesionless soils, in Analysis and Design of PileFoundation, ASCE, New York, pp. 174–191. With permission.)

8.5 Load and Resistance Factor Design for Laterally Loaded Piles

Based on FHWA (1998) recommendations, design of laterally loaded piles involvesdetermining the maximum lateral ground line deflections at the “service limit state” and themaximum moment at the “strength limit state” for an individual pile considering theinstallation method for the selected pile section and comparing it with the tolerabledeformation and the maximum factored axial resistance of the pile, respectively, in order notto exceed both limits.

FHWA (1998) recommends that the allowable stress design (ASD) methods used toestimate the lateral resistance of a single pile or pile group can also be used for load andresistance factor design (LFRD) with the pile or the pile group subjected to the factored lateralloads, axial loads and moments, and the resulting factored axial and bending stresses arecompared with the factored axial and bending capacities of the pile.

8.6 Effect of Pile Jetting on the Lateral Load Capacity

Water jetting can be utilized as an effective aid to impact pile driving when hard strata areencountered above the designated pile tip elevation. During jetting, the immediateneighborhood of the pile is first liquefied due to high pore pressure induced by the water jetand subsequently densified with its dissipation. In addition, the percolating water also createsa filtration zone further away from the pile. Hence, jetting invariably causes substantialdisturbance to the surrounding soil, which results in a notable change in the lateral loadbehavior. Tsinker (1988) and Hameed et al. (2000) investigated the lateral load performanceof driven and jetted-driven model piles installed under the same in situ soil conditions, bycomparing the normalized p−y curves of driven piles to those of jetted-driven piles (Figure8.28). They also explored the effect of jet water pressure, soil unit weight, and groundwaterconditions on the p−y characteristics. Based on the above study, Hameed et al. (2000)developed approximate guidelines for predicting the lateral load behavior of jetted piles basedon that of piles impact driven under similar soil conditions.

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FIGURE 8.28

Comparison of p–y curves of driven (UD1) and jetted (UJ2) piles. Hameed, R.A., 1998, Lateral LoadBehavior of Jetted and Preformed Piles, Ph.D. dissertation, University of South Florida,Tampa, FL. With permission.)

In the Hameed et al. (2000) study, Kmax ratios (Kjet/Kdriven) and pu ratios (pu,jet/Pu,driven) obtainedfrom the model testing program were plotted against the nondimensional jetting pressure(π3=P0/k2γ) (k=permeability coefficient of the foundation soil) and are shown in Figure 8.29and Figure 8.30, respectively. Each data point represents the mean of five ratio values. The K-ratio and pu-ratio can be related to nondimensional jetting pressure by Equations (8.53a) and(8.53b). The foundation soil was a sand contaminated by bentonite clay.

FIGURE 8.29

Effect of pile jetting on Kmax. (From Hameed, R.A., Gunaratne, M., Putcha, S., Kuo, C, and Johnson,S., 2000, ASTM Geotech. Testing J., 23(3). With permission.)

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FIGURE 8.30

Effect of pile jetting on pu. (From Hameed, R.A., Gunaratne, M., Putcha, S., Kuo, C., and Johnson, S.,2000, ASTM Geotech. Testing J., 23(3). With permission.)

(8.53a)

(8.53b)

where, α1, α2, β1, and β2 are soil type dependent parameters which can be determined by therespective intercepts and slopes. Equations (8.53) and (8.54) produce on a log-log scale. Thefitted values are shown in Table 8.2(a) and (b).

TABLE. 8.2

Parameters for Equations (8.53a) and (8.53b)

Constant γ=16.2 kN/m3

UnsaturatedSaturated γ=14.8 kN/m3

UnsaturatedSaturated

(a) Equation(8.53a)

α1 165.32 748.82 110.8 237.42

β1 −0.323 −0.323 −0.323 −0.323

(b) Equation(8.53b)

α2 3509.67 797.02

β2 −0.4 −0.4

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The values for β1 andβ2 seem to be independent of the unit weight of the foundationmedium and the groundwater condition. On the other hand, the values of α1 andα2 seem toincrease with the foundation medium unit weight. Hence, one can assume the variation of α1and α2 to be linear proportional to the unit weight of the foundation medium.

Example 8.6If a field p-y curve of a driven pile in a clayey sand similar to the soil tested by Hameed et

al. (2000) is available based on either (1) experimental data, (2) Reese et al. (1974) method(Figure 8.25), or (3) Murchison and O’Neill’s (1984) method (Figure 8.26), and if oneneglects the possible errors due to scale effects, then one can generate the p-y characteristicsfor a pile to be jetted in the same soil type using the following procedure.

SolutionIn order to illustrate this, assume that a p-y curve based on Murchison and O’Neill’s (1984)method is available for a driven pile in a clayey sand site (with k=1.592×10−3 cm/sec andγd=15.76 kN/m3 above the groundwater table) and that the relevant Murchison and O’Neillparameters at a 3D depth are Ad=1, pu,d=900.00 kPa, and Kmax,d=30000.00 kN/m3 (Figure8.27). The subscript “d” indicates a driven pile. Using these values, the corresponding p-ycurve can be plotted in Figure 8.30.

Also assume that one is interested in synthesizing a p-y curve at a depth of 3D for a fieldjetting pressure of 861.88kPa (125 psi). The equivalent nondimensional jetting pressurescorresponding to the above soil properties must be determined by the nondimensional jettingpressure, (P0/k2γ) (Table 8.3). The constants α1, α2, β1, and β2 can be obtained by linearinterpolation based on the values given in Table 8.2(a) and (b). Table 8.3 shows theinterpolated values for a 15.76 kN/m3 unit weight. It has been assumed that the range ofvalues shown in Table 8.2 is generally valid for any combination of unit weight andpermeability for soils similar to the one tested by Hameed et al. (2000).

Using Equations (8.53a) and (8.53b), and Table 8.3, the K-ratio and pu-ratio can bedetermined as 0.14 and 0.43, respectively. Thus, the corresponding p-y parameters at a 3Ddepth, for the pile to be jetted at 861.88 kPa are, Aj=1.0, Kmax,j=4200.00 kN/m3 , and pu,j =387kPa. The corresponding p-y curve is also plotted in Figure 8.31. This example shows how onecan use Equation (8.53) to generate p-y curves for a pile to be jetted at any desired pressure inthe field. It must be noted that the same procedure can be extended to p-y curves for drivenpiles also available in terms of the Reese et al. (1974) method or experimental data.

TABLE 8.3

Interpolated Parameters for Use in Equations (8.53a) and (8.53b)

Unit weight (kN/m3) P0/k2γ α1 α2 β1 β2

15.76 2.11×109 147.90 2322.96 −0.323 −0.4

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FIGURE 8.31

Synthesized p–y curve for the jetted pile. (From Hameed, R.A., Gunaratne, M., Putcha, S., Kuo, C,and Johnson, S., 2000, ASTM Geotech. Testing J., 23(3). With per-mission.)

8.7 Effect of Preaugering on the Lateral Load Capacity

Hameed et al. (1998) developed similar relationships between the p–y curve parameters ofpreaugered piles and driven piles as shown in the following equations

(8.54a)

(8.54b)

where α3, α4, β3, and β4 are soil type dependent constants, which can be evaluated from Table8.4.

TABLE. 8.4

Parameters for Equations (8.54a and 8.54b)

γ=16.2 kN/m3 γ=14.8 kN/m3

Constant Unsaturated Saturated Unsaturated Saturated(a) Equation (8.54a)

α3 0.14 0.69 0.38 0.69

β3 −1.17 −1.17 −1.17 −1.17

(b) Equation (8.54b)

α4 0.64 0.39

β4 −0.68 −0.68

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References

Bowles, J.E., 1996, Foundation Analysis and Design, 5th edn, McGraw-Hill, New York.Broms, B., 1964a, Lateral resistance of pile in cohesive soil, Journal of Soil Mechanics Foundation

Division, ASCE, 90(SM3):27–56.Broms, B., 1964b, Lateral resistance of pile in cohesionless soil, Journal of Soil Mechanics

Foundation Division, ASCE, 90(SM3): 123–156.Federal Highway Administration, 1998, Load and Resistance Factor Design (LRFD) fo r Highw ay

Bri Substructures, Federal Highway Administration, Washington, DC.Georgiadis, S.M., Anagnostopoulos, C., and Saflekou, S., 1991, Centrifugal testing of laterally loaded

piles in sand, Canadian Geotechnical Journal, 27:208–216.Glick, F.H., 1948, Influence of soft ground in the design of long piles, 2nd ICSMFE, Vol. 4, pp. 84–88.Hameed, R.A., 1998, Lateral Load Behavior of Jetted and Preformed Piles, Ph.D. Dissertation,

University of South Florida, USA.Hameed, R.A., Gunaratne, M., Putcha, S., Kuo, C., and Johnson, S., 2000, Laterally loaded behavior

of jetted piles, ASTM Geotechnical Testing Journal, 23(3), 358–368.Kondner, R.L., 1963, Hyperbolic stress-strain response: cohesive soils, Journal of Soil Mechanics

Foundation Division, ASCE, 89(1):115–143.Li, γ. and Byrne, P.M., 1992, Lateral pile response to monotonic pile head loading, Canadian

Geotechnical Journal, 29:955–970.Murchison, J.M. and O’Neill, M.W., 1984, Evaluation of p-y relationship in cohesionless soils, In

Analysis and Design of Pile Foundation, ASCE, New York, pp. 174–191.Pyke, R. and Beikae, M., 1984, A new solution for the resistance of single piles to lateral loading,

Laterally Loaded Deep Foundation: Analysis and Performance, ASTM STP 835, pp. 3–20.Reese, L.C., 1977, Laterally loaded piles: program documentation, Journal of Geotechnical

Engineering, ASCE, 103(GT4):283–305.Reese, L.C, Cox, W.R., and Koop, F.D., 1974, Analysis of laterally loaded piles in sand, Proceedings

of the 6th Offshore Technology Conference, Houston, TX, paper OTC 2080, pp. 473−483Robertson, P.K. and Campanella, R., 1983, Interpretation of cone penetration tests, Part I. Sand,

Canadian Geotechnical Journal, 20(4):718–733.Ruesta, P.F. and Townsend, F.C., 1997, Evaluation of laterally loaded pile group at Roosevelt bridge,

Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(12):1 153–1161.Ting, J.M., 1987, Full-scale dynamic lateral pile response, Journal of Geotechnical Engineering,

ASCE, 113(1):30−45.Tsinker, G.P., 1988, Pile jetting, Journal of Geotechnical Engineering, ASCE, 114(3):326–334.Hetenyi, M., 1946, Beams on Elastic Foundations, University of Michigan Press, Ann Arbor,

Michigan.

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