7.5 Lateral Capacity and Deflection — Individual Foundation
7.5.3 Lateral Capacity and Deflection — p–y Method
One of the most commonly used methods for analyzing laterally loaded piles is the p–y method, in which soil reactions to the lateral deflections of a pile are treated as localized nonlinear springs based on the Winkler’s assumption. The pile is modeled as an elastic beam that is supported on a deformable subgrade.
The p–y method is versatile and can be used to solve problems including different soil types, layered soils, nonlinear soil behavior; different pile materials, cross sections; and different pile head connection conditions.
Analytical Model and Basic Equation
An analytical model for pile under lateral loading with p–y curves is shown on Figure 7.9. The basic equation for the beam-on-a-deformable-subgrade problem can be expressed as
(7.30) where
= lateral deflection at point along the pile
= bending stiffness or flexural rigidity of the pile
= axial force in beam column
= soil reaction per unit length, and ; where is the secant modulus of soil reaction
= lateral distributed loads
The following relationships are also used in developing boundary conditions:
(7.31)
(7.32)
(7.33)
L¢ 1 5. B
¢ = - L L 1 5. B
L0 L0=(H+23L) / (2H+L)
L0¢ 1 5. B
¢ = - L0 L0 1 5. B
Pu
Pu
9cuB L( –1.5B) 1.5g¢BL2Kp
ÓÌ
= ẽ for cohesive soil
for cohesionless soil
EId y
dx P d y dx p q
x 4
4 2
2 0
- + + =
y x
EI Px
p p= -E ys Es
q
M EId y
= - dx44
Q dM
dx P dy
xdx
= - +
q = dy dx
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Deep Foundations 7-29
where is the bending moment, is the shear force in the beam column, is the rotation of the pile.
The p–y method is a valuable tool in analyzing laterally loaded piles. Reasonable results are usually obtained. A computer program is usually required because of the complexity and iteration needed to solve the above equations using the finite-difference method or other methods. It should be noted that Winkler’s assumption ignores the global effect of a continuum. Normally, if soil behaves like a continuum, the deflection at one point will affect the deflections at other points under loading.
There is no explicit expression in the p–y method since localized springs are assumed. Although p–y curves are developed directly from results of load tests and the influence of global interaction is included implicitly, there are cases where unexpected outcomes resulted. For example, excessively large shear forces will be predicted for large piles in rock by using the p–y method approach, where the effects of the continuum and the shear stiffness of the surrounding rock are ignored. The accuracy of the p–y method depends on the number of tests and the variety of tested parameters, such as geometry and stiffness of pile, layers of soil, strength and stiffness of soil, and loading conditions. One should be careful to extrapolate p–y curves to conditions where tests were not yet performed in similar situations.
Generation of p–y Curves
A p–y curve, or the lateral soil resistance p expressed as a function of lateral soil movement y, is based on backcalculations from test results of laterally loaded piles. The empirical formulations of p–y curves are different for different types of soil. p–y curves also depend on the diameter of the pile, the strength and stiffness of the soil, the confining overburden pressures, and the loading conditions. The effects of layered soil, battered piles, piles on a slope, and closely spaced piles are also usually considered. Formulation for soft clay, sand, and rock is provided in the following.
p–y Curves for Soft Clay
Matlock [35] proposed a method to calculate p–y curves for soft clays as shown on Figure 7.10. The lateral soil resistance p is expressed as
(7.34) FIGURE 7.9 Analytical model for pile under lateral loading with p–y curves.
M Q q
p 0.5 y y50
--- Ë ¯ Ê ˆ1 3§ pu
pu
ÓÔ ÌÔ ẽ
= y<yp = 8y50
y≥yp 1681_MASTER.book Page 29 Sunday, January 12, 2003 12:36 PM
© 2003 by Taylor & Francis Group, LLC
7-30 Bridge Engineering: Substructure Design
in which
= ultimate lateral soil resistance corresponding to ultimate shear stress of soil
= lateral movement of soil corresponding to 50% of ultimate lateral soil resistance
= lateral movement of soil
The ultimate lateral soil resistance is calculated as
(7.35)
where is the effective unit weight, is the depth from ground surface, is the undrained shear strength of the clay, and is a constant frequently taken as 0.5.
The lateral movement of soil corresponding to 50% of ultimate lateral soil resistance is calculated as
(7.36) where is the strain of soil corresponding to half of the maximum deviator stress. Table 7.12 shows the representative values of .
p–y Curves for Sands
Reese et al. [53] proposed a method for developing p–y curves for sandy materials. As shown on Figure 7.11, a typical p–y curve usually consists of the following four segments:
FIGURE 7.10 Characteristic shape of p–y curve for soft clay. [After Matlock, (1970)35]
Segment Curve type Range of y Range of p p–y curve
1 Linear 0 to 0 to
2 Parabolic to to
3 Linear to to
4 Linear
pu y50 y
pu
pu
3 g¢x ---c Jx
B---
+ +
Ë ¯
Ê ˆcB
Ó9cB ÔÌ Ợ
= x<xr (6B) g¢B
---c +J
Ë ¯
Ê ˆ
= § x≥xr
¢
g x c
J
y50
y50=2 5. e50B e50
e50
yk pk p=( )kx y yk ym pk pm
p p y
m y
m n
= Ê
ËÁ ˆ
¯˜ ym yu pm pu
p p p p
y y y y
m
u m
u m
= + - m
- ( - )
≥yu pu p=pu
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Deep Foundations 7-31
where , , , and can be determined directly from soil parameters. The parabolic form of Segment 2, and the intersection with Segment 1 ( and ) can be determined based on ,
, , and as shown below.
Segment 1 starts with a straight line with an initial slope of , where is the depth from the ground surface to the point where the p–y curve is calculated. is a parameter to be determined based on relative density and is different whether above or below water table. Representative values of are shown in Table 7.13.
TABLE 7.12 Representative Values of
Consistency of Clay Undrained Shear Strength, psf
Soft 0–400 0.020
Medium stiff 400–1000 0.010
Stiff 1000–2000 0.007
Very stiff 2000–4000 0.005
Hard 4000–8000 0.004
1 psf = 0.048 kPa.
FIGURE 7.11 Characteristic shape of p–y curves for sand. [After Reese, et al. (1974)53]
TABLE 7.13 Friction Angle and Consistency Friction Angle and Consistency Relative to
Water Table
29°–30° 30°–36° 36°–40°
(Loose) (Medium Dense) (Dense)
Above 20 pci 60 pci 125 pci
Below 25 pci 90 pci 225 pci
1 pci = 272 kPa/m.
ym yu pm pu
yk pk ym
yu pm pu
kx x
k k
e50
e50
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© 2003 by Taylor & Francis Group, LLC
7-32 Bridge Engineering: Substructure Design Segment 2 is parabolic and starts from end of Segment 1 at
and , the power of the parabolic
Segments 3 and 4 are straight lines. , , , and are expressed as
(7.37)
(7.38) (7.39) (7.40) where is the diameter of a pile; and are coefficients that can be determined from Figures 7.12 and 7.13, depending on either static or cyclic loading conditions; is equal to the minimum of and , as
(7.41)
FIGURE 7.12 Variation of with depth for sand. [After Reese, et al. (1974).53]
y p y
k kx
m m
n n
= ẩẻÍ
ù ûú
( )-
/ ( )
1 1
pk=( )kx yk
n y p
p p y y
m m
u m
u m
= -
- Ê ËÁ
ˆ
¯˜
ym yu pm pu
y b
m= 60
y b
u=3 80 pm=B ps s pu=A ps s
b As Bs
ps pst psd
p x
K x b x
K x K b
st
o
o a
= - +
- +
+ - -
È
ẻ ÍÍ Í
ù û úú g ú
j b
b j a b
b j b a
b f j a
tan sin tan( )cos
tan
tan( ) ( tan tan ) tan (tan tan tan )
As 1681_MASTER.book Page 32 Sunday, January 12, 2003 12:36 PM
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Deep Foundations 7-33
(7.42) (7.43) in which is the friction angle of soil; is taken as ; is equal to ; is the coefficient of the earth pressure at rest and is usually assumed to be 0.4; and is the coefficient of the active earth pressure and equals to .