1

Reese’s Sand Model

Compiled by Jinchi Lu and Liangcai He

1 Introduction

Some of the early soil models for predicting the behavior of the sand around a pile subjected to

lateral loading were given by Terzaghi (1955), Parker and Reese (1971), and Reese, Cox and

Koop (1974). The model reported here is the one proposed by Reese, Cox and Koop (1974).

The Reese, Cox and Koop (1974) sand model is represented by a family of p-y curves of the soil at depths

of interest (Figure 1). Each p-y curve in Figure 1 stands for the lateral response of soil at a specific depth.

In the so-called p-y curve, p is the lateral soil resistance per unit length of pile and y is the displacement of

the soil. This model is based on a series of field testing of two laterally loaded pipe piles in sand in 1966

(Cox, Reese, and Grubbs, 1974).

Figure 1 Illustration of the Reese, Cox and Koop (1974) sand model.

2

2 Practical application

Reese, Cox and Koop (1974) noted the soil model is good for driven pile in sand above or below

water table under both static and cyclic loads. However, caution shall be exercised when

applying the model to other piles that greatly differ from the test piles (tests are briefly described

in the following section), for example a pile subjected to high frequency lateral loads due to

earthquake.

3 Tests performed

The tests on which this model is based were carried out at a test site on Mustang Island near

Corpus Christi, Texas. The test site (Figure 2) was 40 ft wide by 60 ft long.

Figure 2 Illustration of test site.

3

Two soil borings were made in the test sites as shown in Figure 2. Standard penetration tests and

laboratory experiments were also performed. The soil in the site consisted of silty fine sand and

fine sand. The soil profile of the test site is shown in Figure 3. The top 40 ft of the site was

medium dense sand overlain by 10 ft soft clay. Below the soft clay is 20 ft dense sand. The

original ground surface was 5 ½ ft above water table. Before installation of the piles, the top 5 ½

ft layer was removed. The water table was thereafter maintained above the new ground surface

during loading.

Two piles made from 24 in diameter pipe with a wall thickness 3/8 in were tested. The pile

material was A-53 grade B seamless steel. Each pile consisted of 3 sections. The bottom is 38 ft

un-instrumented section. The middle is a 32 ft section instrumented with strain gages. The top is

10 ft section without strain gages. Displacement transducer and load cell were installed at the

pile top. The test piles were installed by driving with a hammer.

Figure 3 Test site soil profile.

10’ soft clay

40’ Medium Dense Sand

20’ Dense Sand

Test Pile 1 φ24” Steel Pipe

32’ instrumented

38’ uninstrumented

10’ uninstrumented

Test Pile 1 φ24” Steel Pipe

4

Before pile installation, the test piles were calibrated and pile EI constants (E is Young’s

modulus and I is inertial of moment) were obtained. During installation of the piles, the bottom

section was driven into the sand first, the sections above it were then welded on it and driving

continued until the pile was driven to the desired depth.

Both piles were tested under static and cyclic loadings. On pile 1, the static tests were performed

first, and the cyclic tests were conducted one month after the static tests. But on pile 2 the cyclic

tests were performed first, the static tests were performed shortly after the cyclic tests.

During static tests, loads were first applied in increments of 2,500 lbs to 15,000 lbs and then in

increments of 1,250 lbs from 15,000 lbs to a maximum of 60,000 lbs. The outputs of instruments

were recorded at each load level. Data were also taken as loads were decreased in increments of

10,000 lbs back to 0.

During cyclic tests, the same peak 1,000 lbs to 3,000 lbs was applied first in both forward and

backward directions. Subsequently the peaks applied in forward direction were between 10,000

and 60,000 lbs, and the peak in the backward direction was about 1/3 of the forward direction.

The cycle time is about 16 ~ 20 seconds. At each load level, several cycles were needed in order

to obtain a stable/repeated pile deflection output.

4 Determination of p-y curve from test results

According to Euler-Bernoulli beam theory the strain ε at a location is given by

ε=κ*d (1)

where d is the location relative to neutral axis and κ is curvature.

The moment of a section is given by

M = EI*κ (2)

Using equation (1), the measured strains at side A and B at a certain cross section can be expressed as:

εa=κ*da, and εb=κ*db,

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which yields:

κ= (εa - εb)/(da - db) (3)

Equation (3) can be used to obtain the curvature of a pile at a certain cross section since (da - db)

is the known distance from A to B. This equation has the advantage of eliminating the effects of

bending in the other direction when analyzing concrete piles. Once curve is calculated, M can be

found by employing Equation (2).

The experimental p-y curve can then be computed by the following two equations with appropriate boundary conditions

dxy ∫ ∫= κ (4)

)(2

2

xMdxdp = (5)

Soil displacement y expressed by Equation (4) can normally be obtained by numerical

integration using trapezoidal rule with appropriate accuracy. However, caution should be excised

when solving Equation (5). Error from experiments will be seriously amplified if directly

differentiating experimental moment using numerical technique. To avoid this problem,

polynomial curve fitting the experimental moments up to the order of 6 may be required.

5 Construction of p-y curve

With their field tests and some rudimentary theoretical studies as a basis, Reese, Cox, and Koop

(1974) proposed a procedure for predicting the family of p-y curves based on the soil properties

and pile dimensions. Procedures were proposed for both static loading and cyclic loading.

Basically, the p-y curve consists of four segments (Figure 4). An initial segment with slope k0 is

intersecting with a parabola at point k. The parabola is tangent to the third segment at point m.

The third segment is a straight line with slope kmu. The last segment representing the ultimate soil

resistance is a horizontal line intersecting with the third segment at point u. The key steps of the

p-y curve construction are to determine points k, m, and u and to find the parameters to express

the parabola. The procedures are are explained below in seven steps.

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Step 1. Obtain soil internal friction angle φ, effective unit weight γ’, and pile diameter b

Note: Pile diameter b can be in any length unit.

Figure 4 Construction of p-y curve.

Step 2. Compute following parameters

α = φ /2

β = 45 + φ /2

K0 = 0.4

Ka = tan2(45- φ /2)

Step 3. Compute ultimate soil resistance pu per unit length of pile at depth z of interest by taking

the smaller of the values given by the following two equations.

−−+

+−

+−=

bKzK

zbKzp

a

st

)tansin(tantan

)tantan()tan(

tancos)tan(

sintan'

0

0

αβφβ

αβφβ

βαφβ

βφγ (6)

βφγβγ 40

8 tantan')1(tan' zbKzbKp asd +−= (7)

Note: equation (6) was derived from the assumed wedge failure mode shown in Figure 5(a),

while equation (7) was derived from the assumed block failure mode shown in Figure 5(b). Since

the two failure modes are very rudimentary, the ultimate resistance of sand around a laterally

loaded pile does not yield to the completely rational analysis. Correction factors A and B (Figure

6) from a considerable amount of empiricism are involved as shown in the next step.

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(a) Wedge failure mode (b) Block failure mode

Figure 5 Two types of soil failure mode

Step 4. Determine the point u and point m

yu = 3b/80, pu = Apu

ym = b/60, pm = Bpu

where b is pile diameter in any length unit, A is a correction factor from empiricism that can be

found from empirical curves in Figure 6(a), B is another empirical factor that can be obtained

from Figure 6(b). Note that for static case, use As and Bs, for cyclic case, use Ac and Bc in Figure

6(a) and Figure 6(b).

(a) empirical curves for factor A (a) empirical curves for factor B

Figure 6 Empirical correction factors A and B

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Step 5. Find the initial slope of the p-y curve:

k0 = kz

where z is depth and k is sub-grade modulus for sand. A rough value of k can be found from

Table 1.

Table 1 Reese, Cox, and Koop (1974) recommended k values for sand (static and cyclic loading)

Relative Density Loose Medium Dense

k for Sand above water table

MN/m3

(lb/in.3)

6.8

(25.0)

24.4

(90.0)

61.0

(225.0)

k for Submerged Sand

MN/m3

(lb/in.3)

5.4

(20.0)

16.3

(60.0)

34

(125.0)

Step 6. Establish the parabolic section of the p-y curve in the form: nCyp /1=

mmu

m

ykp

n = where kmu is the slope of the line between points u and m

nm

m

yp

C /1=

Step 7. Determine the point k:

1−

=

nn

k kzCy , pk= kzyk

6 Example

The example shown here is to use the above steps to reproduce the curves given by Reese and

Impe (2001). The soil and pile parameters are:

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Submerged medium dense sand: γ’ = 9.8 kN/m3, φ = 35o, k = 16300 kN/m3. Pile diameter =

0.61m. The curves in Figure 7 match those given by Reese and Impe (2001).

Figure 7 Reese p-y curve examples.

7 Parametric Study

Although soil effective unit weight, sub-grade modulus, and internal friction angle are correlated,

a parametric study was still conducted to examine the effects of each single parameter on p-y

curves. The effects of pile diameter were also investigated.

When conducting parametric study, the p-y curves at 2 m depth were plotted using the values of

the basic parameters as below:

Soil properties: γ’ = 9.8 kN/m3, k = 16300 kN/m3, φ = 35o, Pile diameter = 1.0m.

kN/m

m

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7.1 Effects of subgrade modulus k

Figure 8 shows the effects of subgrade modulus k on p-y curve. It can be seen that as subgrade

modulus increases from 5,000 to 60,000 kN/m3, p-y curves at 2 m depth vary only a small

amount. Therefore, soil subgrade modulus k has no significant effects on p-y curve.

Figure 8 Effects of soil subgrade modulus k on p-y curve

7.2 Effects of soil friction angle

Figure 9 shows the effects of soil internal friction angle φ on p-y curve. It can be seen that as soil

internal friction angle φ increases from 250 to 400, p-y curves at 2 m depth vary a lot. Therefore,

soil internal friction angle φ has significant effects on p-y curve.

k = 5,000 kN/m3, 16,300 kN/m3, 35,000 kN/m3, 60,000 kN/m3,

Z = 2m

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Figure 9 Effects of soil internal friction angle φ on p-y curve

7.3 Effects of soil unit weight

Figure 10 shows the effects of soil unit weight γ on p-y curve. It can be seen that as soil unit

weight γ increases from 10 to 20 kN/m3, p-y curves at 2 m depth vary a lot. Therefore, soil unit

weight γ has significant effects on p-y curve.

Figure 10 Effects of soil unit weight γ on p-y curve

φ = 35o

φ = 30o

φ = 25o

Z = 2m

γ’ = 20 kN/m3

γ’ = 18 kN/m3

γ’ = 16 kN/m3

γ’ = 10 kN/m3

φ = 40o

Z = 2m

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7.4 Effects of soil pile diameter

Figure 11 shows the effects of pile diameter b on p-y curve. It can be seen that as pile diameter b

increases from 1.0 to 2.0 m, the slope of the initial segment of the p-y curves the same depth

keep the same. The slope increases as depth goes deeper.

It also can be seen that the ultimate soil resistance increases as pile diameter increases. This is

due to the fact that more soil was involved to resist pile deflection as pile diameter increases.

The interesting thing is that the middle part of the p-y curves at 12 m depth (Figure 11 (a))

displays opposite diameter effects from those at 2 m depth (Figure 11 (b)). At 2 m depth (Figure

11 (a)) soil resistance in the middle part of the p-y curves increases as increasing pile diameter,

while at 12 m depth (Figure 11 (b)) soil resistance decreases as increasing pile diameter. This

may be due to the different soil failure modes at different depth. At 2 m depth soil fails following

the wedge failure mode (Figure 5(a)), while at 12 m depth soil fails following the block failure

mode (Figure 5(b)).

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(a)

(b)

Figure 11 Effects of soil diameter on p-y curve

kN/m

Z = 2m

m

m

kN/m

Z = 12m

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8 FB-Pier Runs

The model is a single pile subjected a lateral load at the pile head (Figure 11). The soil layer is

uniform sand with water table at the pile head. The pile has a diameter of 0.5m and a height of

10m. The objective of the FB-Pier runs performed is to study the effects of the soil properties

(internal friction angle φ, subgrade modulus K and unit weight γ) on the behavior of Reese sand

model.

(a) (b)

Figure 12 Single pile model used in FB-Pier runs

Note that the above mentioned soil properties are dependent with each other in reality. One may

not change some of them and keep the other constant. However, in order to study the behavior of

the Reese sand model in FB-Pier program, we did the following runs (changing one and keeping

the other two constant).

8.1 Effect of the internal friction angle

Figure 12 shows the load-deflection curve and the moment distribution of the pile for different

internal friction angles (32, 36, 40 degrees), while keeping the subgrade modulus K = 37.5

MN/m3, and unit weight 18.7 KN/m3.

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Figure 13 Load-deflection curve (left) and moment distribution (right) of the pile for different friction angle

8.2 Effect of the subgrade modulus

Figure 13 shows the load-deflection curve and the moment distribution of the pile for different

subgrade moduli (3.75, 37.5 and 61 MN/m3), while keeping the internal friction angle 36

degrees, and the unit weight 18.7 KN/m3.

It is noted that the curves for the subgrade modulus = 37.5 and 61 MN/m3 are almost the same.

This should not happen in reality since only the subgrade modulus is changed here.

Figure 14 Load-deflection curve (left) and moment distribution (right) of the pile for different

subgrade moduli

0

20

40

60

80

100

120

140

0.00 0.02 0.04 0.06 0.08 0.10

Pile Head Displacement (m)

Load

(KN

)

Φ = 32°Φ = 36°Φ = 40°

0123456789

10

-20 0 20 40 60 80

Moment (KN-m)

Dep

th (m

)

Φ = 32°Φ = 36°Φ = 40°

0

20

40

60

80

100

120

140

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Pile Head Displacement (m)

Load

(KN

)

k = 3.75MN/m3k = 37.5MN/m3k = 61MN/m3

0

1

2

3

4

5

6

7

8

9

10

-10 0 10 20 30 40 50 60

Moment (KN-m)

Dep

th (m

)

k = 3.75MN/m3k = 37.5MN/m3k = 61MN/m3

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8.3 Effect of unit weight

Figure 14 shows the load-deflection curve and the moment distribution of the pile for different

unit weights (16, 18.7 and 21 KN/m3), while keeping the internal friction angle 36 degree and the

subgrade modulus K = 37.5 MN/m3.

Figure 15 Load-deflection curve (left) and moment distribution (right) of the pile for different unit weights

8.4 Runs with loose, medium and dense sand

Figure 15 shows the load-deflection curve and the moment distribution of the pile for realistic

sand types (loose, medium and dense). The soil properties for the above sand types are shown in

the following table:

Loose sand Medium sand Dense sand Internal friction angle φ (°) 29.5 33 38

Subgrade modulus K (MN/m3) 6.8 24.4 61

Unit weight γ (KN/m3) 16 18.7 21

0

20

40

60

80

100

120

140

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Pile Head Displacement (m)

Load

(KN

)

γ = 16 KN/m3γ = 18.7 KN/m3γ = 21 KN/m3

0

1

2

3

4

5

6

7

8

9

10

-10 0 10 20 30 40 50 60 70

Moment (KN-m)

Dep

th (m

)γ = 16 KN/m3γ = 18.7 KN/m3γ = 21 KN/m3

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Figure 16 Load-deflection curve (left) and moment distribution (right) of the pile for loose, medium and

dense sand

9 Summary

The Reese sand model is summarized as follows:

• The model was originally developed for the design of laterally loaded piles in sand under

both static and cyclic loads in offshore structures. The tests were performed on driven

piles in sand below water table only. Only two piles were tested. The two piles had the

same diameter of 24 in. The cyclic loads applied in the tests were at very low frequency

(about 0.05 Hz).

• Caution shall be exercised when applying the model to other piles that greatly differ from

the test piles, for example a pile subjected to high frequency lateral loads due to

earthquake.

• This model is good for driven piles in sand below or above water table.

• The initial linear portion of Reese p-y curve is independent of pile diameter.

• The nonlinear portion will have larger soil resistance when pile diameter increases at

shallow depth. The soil resistance at nonlinear portion will decrease at certain depth as

pile diameter increases.

• The effects of each parameter are uncertain when considered alone. Soil friction angle,

subgrade modulus, and unit weight are correlated with each other.

0

20

40

60

80

100

120

140

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Pile Head Displacement (m)

Load

(KN

)

LooseMediumDense

0

1

2

3

4

5

6

7

8

9

10

-20 0 20 40 60 80

Moment (KN-m)

Dep

th (m

)

LooseMediumDense

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10 References

Cox, William R., Lymon C. Reese, and Berry R. Grubbs (1974). “Field Testing of Laterally

Loaded Piles in Sand”, Proceedings of Offshore Technology Conference, 6200 North Central

Expressway Dalla, Texas, paper number OTC 2079.

Parker, Frazier and Lymon C. Reese (1971), “Lateral Pile-Soil Interaction Curves for Sand,”

Proceedings, The International Symposium on the Engineering Properties of Sea-Floor Soils and

Their Geophysical Identification, University of Washington, Seattle, Washington, July 25, 1971.

Reese, Lymon C., William R. Cox, and Francis D. Koop (1974). “Analysis of Laterally Loaded

Piles in Sand”, Proceedings of Offshore Technology Conference, 6200 North Central

Expressway Dalla, Texas, paper number OTC 2080.

Terzaghi, Karl (1955), “Evaluation of Coefficients of Subgrade Reaction,” Geotechnic, Vol.5.5,

December, 1955, pp. 297-326.