HELICAL FOUNDATIONS AND ANCHORS
description
Transcript of HELICAL FOUNDATIONS AND ANCHORS
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC
HELICAL FOUNDATIONS AND TIE BACKS
State of the Art
Richard W. Stephenson
Professor of Civil Engineering
University of Missouri-Rolla
Rolla, Missouri 65409
March 2, 2010
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 2
INTRODUCTION........................................................................................................................... 3
HISTORY ....................................................................................................................................... 3
Modern Usage ..................................................................................................................... 4
HELICAL PILE DESIGN ............................................................................................................... 7
Prototype ............................................................................................................................. 7
Theoretical .......................................................................................................................... 8
Semi -empirical ................................................................................................................... 8
Empirical ............................................................................................................................. 8
UPLIFT CAPACITY OF HELICAL PILES ................................................................................... 9
General ................................................................................................................................ 9
Semi-Empirical Helical Pile Capacity ................................................................................ 9
Individual Plate Capacity Method ........................................................................... 9
Kulhawy Method ................................................................................................... 10
Clemence Method ................................................................................................. 15
Uplift capacity of shallow anchors in sand ............................................... 16
Uplift capacity of deep anchors in sand .................................................... 24
Uplift capacity of helical anchors in clay .................................................. 27
Uplift capacity of shallow helical anchors in clay ........................................................ 27
Uplift capacity of deep helical anchors in clay ............................................................. 30
Empirical Method. ............................................................................................................ 32
BEARING CAPACITY OF HELICAL PILES............................................................................. 33
Bearing Capacity Design of Helical Piles ......................................................................... 33
LATERAL CAPACITY OF HELICAL PILES ............................................................................ 37
Analysis Based on Limiting Equilibrium or Plasticity Theory ............................. 37
Analysis Based on Elastic Theory ......................................................................... 38
Analysis Based on Nonlinear Theory .................................................................... 38
Simplified Method for Nonlinear Analysis of Helical Piles in Clay .................... 39
BIBLIOGRAPHY ......................................................................................................................... 46
EXAMPLE PROBLEMS .............................................................................................................. 48
Uplift Capacity .................................................................................................................. 48
Shallow Anchor in Sand ....................................................................................... 48
Deep Anchor in Sand ............................................................................................ 49
Shallow anchor in Clay ......................................................................................... 51
Deep anchor in Clay .............................................................................................. 52
Anchor in Sand ..................................................................................................... 53
Anchor in Clay ...................................................................................................... 54
Lateral Capacity of a Laterally Loaded Helical Pile in Soft Clay ......................... 56
Lateral Capacity of a Laterally Loaded Helical Pile in Overconsolidated Clay .... 58
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 3
HELICAL FOUNDATIONS AND ANCHORS
STATE OF THE ART
March 2, 2010
R.W. Stephenson, P.E., Ph.D.
INTRODUCTION
Helical piles (helical anchors) are finding increasingly widespread use in the geotechnical
market. These foundations have the advantages of rapid installation and immediate loading
capabilities that offer cost-saving alternatives to reinforced concrete, grouted anchors and driven
piles. The last 12 years have seen the rapid development of rational geotechnical engineering-based
design and analysis procedures that can be used to provide helical pile design solutions
HISTORY
Helical foundations have evolved from early foundations known as Ascrew piles or screw
mandrills.@ The earliest reported screw pile was a timber fitted with an iron screw propeller that
was twisted into the ground(1). The early screw mandrills were twisted into the ground by hand
similar to a wood screw. They were then immediately withdrawn and the hole formed was filled
with a crude form of concrete and served as foundations for small structures. Conventional screw
piles have been in use since the 18th century for support of waterfront and in soft soil conditions for
bridge structures as early as the 19th century.
Power installed foundations were developed in England in the early 1800's by Alexander
Mitchell. In 1833, Mitchell began constructing a series of lighthouses in the English tidal basin
founded on his new Ascrew (1)piles.@
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 4
The first commercially feasible helical anchor was developed in the early 1900's to respond
to a need for rapidly installed guy wire anchors. The anchors were installed and used primarily by
the electrical power industry. The development of reliable truck mounted hydraulic torque drive
devices revolutionized the anchor industry. These advances allowed the installation of helical
anchors to greater depths and in a wider variety of soil conditions than ever before(1).
Modern Usage
Modern helical anchors are earth anchors constructed of helical shaped circular steel plates
welded to a steel shaft (Figure 1). The plates are constructed as a helix with a carefully controlled
pitch. The anchors can have more than one helix located at appropriate spacing on the shaft. The
central shaft is used to transmit torque during installation and to transfer axial loads to the helical
plates. The central shaft also provides a major component of the resistance to lateral loading.
A typical helical anchor installation is depicted in Figure 2. These anchors are turned into the
ground using truck mounted augering equipment. The anchor is rotated into the ground with
sufficient applied downward pressure (crowd) to advance the anchor one pitch distance per
revolution. The anchor is advanced until the appropriate bearing stratum is reached or until the
applied torque value attains a specified value. Extensions are added to the central shaft as needed.
The applied loads may be tensile (uplift), compressive (bearing), shear (lateral), or some
combination.
Helical anchors are rapidly installed in a wide variety of soil formations using a variety of
readily available equipment. They are immediately ready for loading after installation. Large
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 6
multi-helix anchors develop capacities of up to 100,000 lbs. (450 kN).
In the past 20 years, the use of helical anchors has expanded beyond their traditional use in
the electrical power industry. The advantages of rapid installation, immediate loading capability and
resistance to both uplift and bearing loads have resulted in their being used more widely in traditional
geotechnical engineering applications. Reported uses include tiebacks for soil retaining walls,
foundations for lightly loaded structures such as transmission line towers, light poles, tiedowns for
manufactured housing, temporary structures, etc., and for underpinning lightly loaded structures such
as single family dwellings. Because of these uses, there has been an increase in research into the
behavior of helical anchors. Since about 1975, a number of researchers have studied the
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 7
geotechnical principals governing the behavior of helical piles. They have published reports of their
studies of helical anchors under loading and proposed design procedures by which helical pile
performance can be predicted. By far, the majority of this work has been in the anchoring (uplift)
capacity of helical piles(1). However, studies in the lateral and bearing (compression) load
performance are reported as well.
HELICAL PILE DESIGN
The methods available to design helical pile systems and to predict their performance under
load can be divided into four broad categories: prototype (load test), theoretical, semi-empirical and
empirical.
Prototype
In the prototype design method, helical pile capacities are determined by testing a helical pile
identical to the production pile in identical subsurface conditions (5). The results of the prototype
test (load test) are then extrapolated to the rest of the helical piles used at the site. Advantages of this
approach lie in the fact that actual piles are evaluated in their field use conditions. However, this
method requires the a priori selection of helix size and configurations as well as installation depth.
The testing of several helical pile configurations to determine optimum size and spacing is usually
too costly. Consequently, prototype testing is used primarily for proof testing semi-empirical and
empirical designs.
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 8
Theoretical
Theoretical methods utilize soil mechanics theories of the interaction behavior of foundations
and earth materials. The theories use the basic properties of the foundation (strength and
deformability) as well as the basic properties of the soil (strength and compressibility) to create
design procedures that can be applied to different soil structures and different helical pile
configurations. Ideally, the procedures are independent of particular installation equipment and can
be applied to all realistic combinations of helical piles and soil stratigraphies.
Semi -empirical
Unfortunately, because of the complexity of soil stratigraphy and the inability of current soil
mechanics theories to fully describe the actual field performance of a soil, most geotechnical design
procedures are theoretical procedures modified by experience (semi-empirical).
Empirical
Empirical methods are most often developed and used by helical pile manufacturers who
have access to vast quantities of pile behavior data. Empirical methods are based on statistical
correlations of anchor uplift capacity with other, easily measured, parameters such as standard
penetration test (N) values, installation torque, or other indices. The methodology for development
of these correlations and the data on which they are based is usually considered proprietary by the
manufacturers. Results obtained from these methods are highly variable (1)(1)(1)(1).
By far the majority of the research has been directed toward the uplift behavior of helical
piles (helical anchors). This is due primarily to their traditional use as guy line anchors and as tie
downs for transmission towers and tiebacks for retaining structures. Considerably less work has
been carried out on the performance of helical piles under lateral loading. However, significant
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 9
work is available on laterally loaded piles that could possibly be applied to helical piles. Even less
data is reported on the performance of helical piles under bearing (compressive) loading. This is
becoming more important since helical piles are gaining wide use for underpinning and supporting
lightly loaded structures. The following sections will address each of the three design loading
conditions.
UPLIFT CAPACITY OF HELICAL PILES
General
The behavior of any deep foundation is highly complex. Consequently, it is important to
understand the the behavior of helical piles is influenced by the same factors that influence the
behavior of drilled piers and driven piles: i.e., strength and deformation properties of soils, soil non-
homogeneities, groundwater levels, soil plasticity and volume change potential as well as installation
procedures and equipment.
Semi-Empirical Helical Pile Capacity
Individual Plate Capacity Method. One method of computing uplift capacities of helical
piles is the individual plate capacity method. In this method, the uplift capacities are computing
using:
helices of number the is n where,Q=QU
n
=1i
u i
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 10
Qui is the ultimate uplift capacity of the individual helix. Qui can be computed from bearing capacity
theory as:
where:
The first term of equation three is the contribution of soil cohesion to the uplift capacity. The
second term is the contribution of soil friction to the capacity and the third term is the contribution of
soil overburden to the capacity. Nc *, Nγ* and Nq* are bearing capacity factors on cohesion, friction
and surcharge respectively.
For cohesive (clay) soils, Nc* is normally taken as 9.0 for H1/D1 > 3. For H1/D1 3, Nc* is
normally taken as 5.7. Nγ* and Nq* are taken as 0 and 1 respectively.
For helical foundations embedded in cohesionless (sand) soils, c is zero and Nγ* and Nq*
vary as a function of the coefficient of friction (Φ) of the sand. Meyerhof=s values of Nγ* and Nq*
are often used and are presented as Table 1, below.
Kulhawy Method. Kulhawy (1) described a method of analysis of the uplift capacity of
helical anchors by describing their behavior as intermediate between the grouted and spread anchors.
In his model, the upper helix develops a cylindrical shear surface that controls its behavior. The soil
between the helices becomes an effective cylinder if the helices are sufficiently close together. The
shearing resistance along the interface is said to be controlled by the friction angle and state of stress
helix above soilof weightunit effective=
helix to surfaceground from Depth=H
helix of diameter=D
helix of area=A
N H+ND2
1+ cN=q
i
i
*qi
*i
*cui
xAq=Quu ii
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 11
in the disturbed cylinder of soil above the anchor. This disturbance effect can be approximated by
relating the disturbed properties to the in-situ properties in the following equations:
Qu = Ultimate uplift capacity
Qp = Top plate (cone breakout) capacity
Qf = Cylinder friction capacity
Wf = Weight of helical pile (often neglected)
For cohesionless (sand) soils, Kulhawy recommended the following equations:
Table 1 Meyerhof@s
Bearing Capacity Factors
Φ
(deg)
Nc
*
Nq*
Nγ*
0
5.1
1.0
0.0
5
6.5
1.6
0.1
10
8.4
2.5
0.4
15
11.0
3.9
1.1
20
14.8
6.4
2.9
W+Q+Q=Q ffpu
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 12
25 20.7 10.7 6.8
26
22.3
11.8
8.0
28
25.8
14.7
11.2
30
30.1
18.4
15.7
32
35.5
23.2
22.0
34
42.2
29.4
31.1
36
50.6
37.7
44.4
38
61.4
48.9
64.0
40
75.3
64.1
93.6
45
133.9
134.7
262.3
50
266.9
318.5
871.7
where:
Qp(max) = Top plate capacity limit
Af = Area of top helix
H = surchargeeffective = q 1q
W f = Effective weight of helical pile alone
Q + W + ) N q(A=)(Qtufqdqsqrqqfp
max
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 13
Qtu = Tip capacity in uplift (usually neglected)
The Nq term is a bearing capacity factor given by:
2+45e=N
2q
_tantan
1.0 +1
I2)(3.07+3.8-=
r10
qrsin
logsintanexp
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 14
The ζ terms are modification factors for soil rigidity (ζqr), anchor shape (ζqs), and anchor depth (ζqd)
as given below.
with the tan-1
term in radians.
G = soil shear modulus
E = soil elastic modulus
The cylinder friction capacity, Qf , is computed from the following equation:
where:
P = helix perimeter
v = effective vertical stress
k = coefficient of horizontal earth pressure
tantan q
1
)+2(1
E=
q
G=I
ii
r
tan+1=qs
tan+1=qs
D
H)-(12+1=
1
1-2
qd tansintan
(z)(z)kP(z)k
k=
)(z)dz(z)(kP(z)=Q
ov
H
Ho
uv
H
H
f
1
1
tan
tan
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 15
= effective interface friction angle
ko = Coefficient of earth pressure at rest
= Effective stress soil friction angle
__/__ = 0.9
k/ko = 5/6
The friction capacity of the helical pile system is reduced due to disturbance caused by pile
installation. Kulhawy accounted for this by using a reduced uplift capacity according to the
following equation:
Clemence Method. A significant series of studies on helical anchor uplift capacity was
done by Clemence (1), and later summarized in Mitsch and Clemence (1) and Mooney, Adamczak,
and Clemence (1). They extended the work of previous researchers with extensive full scale field
tests, scale model laboratory tests, and theoretical analysis. These researchers suggested that helical
pile uplift capacity could be divided into two broad categories: shallow anchors and deep anchors.
They stated that the uplift capacity is provided by:
0
r
ff(reduced) Q=Q
tan k=
3
+2=
oo
o
r
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 16
Uplift capacity of shallow anchors
in sand:
The weight of the soil, Ws can be expressed as:
Das non-dimensionalized these equations into:
Q+Q=Qfpu
W+3
2H
+2
HD
2 k = Q s
312
112uu
tan
costan
2H2+D)D(+
2H2+D+)D(H
3=W 11111
2
211s tantan
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 18
Similarly:
Let
Combining:
Fq is called the breakout factor by Das. To determine Fq the value of ku must be determined. Mitsch
and Clemence(11) showed that this value varies with the soil friction angle, Φ. Their values can be
expressed as:
HA
Q=F
1
p
q1
20.33+
D
H
0.5
D
H
2)(k4=
HA
Q=F
1
11
1
2
2u
1
p
q1tancostan
2D
H8+
2D
H5.33+4=
HA
W=F
1
1
2
2
1
1
2
1
sq2
tantan
R=D
H
1
1
2 8R+
2R5.33+4+
20.33+
R
0.5
2)(kR4=
HA
Q=F
22
2u
2
1
p
q
tantan
tancostan
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 19
The variation of m is given below.
Table 2 Variation of m
Soil friction angle, Φ
(degrees)
m
25
0.033
30
0.075
35
0.180
40
0.250
45
0.289
The magnitude of ku increases with H1/D1 up to a maximum value and remains constant after
that. This maximum value is attained at (H1/D1)cr = Rcr . The variation of ku with H1/D1 and Φ are
plotted in Figure 4. Substituting the appropriate value of ku and R into the previous equation, the
variation of the breakout factor is shown in Figure 5 and Table 3. Now,
The frictional resistance that occurs at the interface of the cylinder is given as:
D
H m+0.6=k
1
1u
HDF4
=HAF=Q 121q1qp
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 20
Da = average helix diameter.
Therefore the ultimate uplift capacity for a shallow anchor in sand
is:
tank)H-H(D2
= Q u21
2naf
tank)H-H)((2
D+D
2+HDF
4=Q u
21
2n
n11
21qu
Figure 5: Variation of ku with H1/D1
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 21
Table 3: Variation of Breakout Factor Fq for Shallow
Anchor Condition
Fq
R =H1/D1
Φ =
25
Φ=30
Φ=35
Φ=40
Φ=45
0.5
5.27
5.54
5.87
6.23
6.61
1.0
6.74
7.38
8.25
9.18
10.17
1.5
8.41
9.54
11.16
12.91
14.77
2.0
10.27
12.01
14.64
17.49
20.53
2.5
12.33
14.82
18.72
22.99
27.54
3.0
14.6
17.97
23.44
29.46
35.94
3.5
21.48
28.84
36.99
45.74
4.0
25.35
34.95
45.64
57.13
4.5
41.81
55.44
70.18
5.0
49.46
66.56
85
5.5
78.97
101.68
6.0
92.76
120.34
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 22
6.5 108.01 141.06
7.0
124.78
163.98
7.5
189.14
8.0
216.69
8.5
246.73
9.0
279.34
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 24
Uplift capacity of deep anchors in sand:
The magnitude of the Fq = Fq* is determined by setting R = Rcr and ku = ku(max) in equation 25.
Fq* has been plotted in Figure 8.
The frictional
resistance Qf is
computed using:
The two equations can
be combined to yield the net
ultimate uplift capacity for
deep anchors in sand:
If the helices are
placed too close to each
other, the average net ultimate uplift capacity of each anchor may decrease due to the overlapping
and interference of the individual failure zones.
0=resistance friction shaft=Q
helices the between cone the of resistance frictional=Q
helix top the ofcapacity bearing=Q
Q+Q+Q=Q
s
f
p
sfpu
factor breakout anchor deep =F where
HDF4
= Q
*q
121
*qp
2
D+D=D where
k)H-2H( D 2
= Q
n1a
u 21nau
tanmax
tanmaxk)H-H(2
D+D
2+HDF
4 = Q u
21
2n
n11
21
*qu
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 26
It is recommended that the optimum spacing of the helices be about 3D1 apart. A factor of safety of
2.5 or more should be applied to the ultimate uplift capacity to determine the allowable or working
uplift capacity.
20 30 40 50
Soil Friction Angle (deg)
10
100
Deep
An
ch
or
Bre
ak
ou
t F
acto
r, F
q*
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 27
Uplift capacity of helical anchors in clay. Failure of helical piles in clay soils is
normally analyzed using the Φ= 0 condition. The soil shear strength is then characterized as:
Uplift capacity of shallow helical anchors in clay. For shallow anchors (H1/D1 7), the
failure surface at ultimate load extends from the top helix to the ground surface (Figure 9 ). If the
H1/D1 ratio is relatively large then the failure zone will not extend to the ground surface and the deep
anchor situation controls.
For shallow anchors:
where: Qp = bearing capacity of the top helix
Qf = bearing due to friction along enclosed cylinder between helices.
Where A1 = area of the top helix
Fc = breakout factor
γ = unit weight of soil above top helix
H1 = distance between the ground surface and the top helix
Fc is related to the bearing capacity factor Nc in that it increases with depth of embedment up to a
maximum of 9 at the critical Rcr = (H1/D1)cr value that depends on the undrained cohesion, cu
(kN/m2) as in:
c=s uu
Q+Q=Qfpu
)+Fc(A=W+ F c A = Q cu1sc1p
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 29
The variation of the breakout factor Fc is plotted as a function of (H1/D1)/(H1/D1)cr in Figure 10.
The frictional resistance of the cylinder of soil between the helices can be computed from:
Combining:
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(H1/D1)/(H1/D1)cr
0
1
2
3
4
5
6
7
8
9
Fc
72.5+c0.107=D
H=R u
1
1
cr
cr
)H H( c 2
D+D = Q 1nu
n1
f
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 30
Uplift
capacity of deep
helical anchors in
clay. For the deep
anchor condition (H1/D1)> (H1/D1)cr deep anchor criteria holds (Figure 11). The capacity for this
case is given below.
Where Qs = resistance due to adhesion at the interface of the clay and the anchor shaft located
above the top helix.
Where ca is the adhesion and varies from about 0.3cu for stiff clays to about cu for soft clays and Ds is
the shaft diameter. Combining:
c)H-H(2
D+D+ )H+ Fc( D
4 = Q u1n
n11cu
21u
Q+Q+Q=Qsfpu
)H+c)(9D(4
=Q 1u21p
)H H( c 2
D+D = Q 1nu
n1
f
cHD=Q a1ss
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 32
All ultimate uplift capacities should be divided by an appropriate factor of safety to set the
allowable(working) factor of safety, i.e.,
Empirical Method.
Empirical methods are most often developed and used by anchor manufacturers who have
access to vast quantities of anchor behavior data. These methods are based on statistical correlations
of anchor uplift capacity with other, easily measured, parameters such as standard penetration test
(N) values, installation torque, or other indices. The methodology for development of these
correlations and the data on which they are based are usually considered proprietary by the
manufacturers. Results obtained from these methods are highly variable.
The most widely used correlation is with installation torque. In this method, the total anchor
capacity is computed from the installation torque as:
where: Kt is the empirical factor relating installation torque and uplift capacity and T is the average
installation torque. Currently, Kt values are reported between 3 feet-1
for large (8 inch) extension
shafts to around 10 feet-1
for all small (3 inch) shafts. 10 feet-1
is most widely used in the industry.
FS
Q=Q u
allow
xTK=Q tu
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 33
BEARING CAPACITY OF HELICAL PILES
Although helical piles have been used as tower foundations for many years, the design
loading for these foundations is not bearing (compression) but uplift. It is only relatively recently
that helical piles have been used in primarily bearing conditions. In particular, these foundations are
being used in the retrofit or underpinning of distressed lightly loaded structures.
There are several advantages of helical piles for foundation underpinning(1). Of particular
importance is the general relationship between installation torque and helical pile capacity. It is
possible to develop site-specific Kt values from preliminary field load testing and use the results as
quality control values for the production piles. Other advantages include the ease of extending pile
length by adding on extension shafts, the lack of influence of water table or caving soils, ability to
install in low-overhead, low noise or other restricted areas. Helical anchor shafts are relatively small
in diameter and by that develop low lateral stresses and low drag along their lengths. This makes
them particularly applicable in expansive soil conditions.
Bearing Capacity Design of Helical Piles
The bearing capacity of helical piles in compression is based upon the general bearing
capacity equation:
where: c = soil cohesion
q = overburden pressure = γHi
γ = effective unit weight
Hi = depth to helix
1)-Nq(+cN=q qcult
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 34
Nc= and Nq= are bearing capacity factors for circular plates at varying H/D values.
Although there are some minor differences in these values depending upon the particular theory
adopted, in general Nc= and Nq= are taken from Figure 11(1).
The bearing capacity of a multi-helix system is the sum of the individual capacities of the
individual helices if they are spaced appropriately far apart, i.e., three times the plate diameter or
greater.
Ai = individual plate area
ci = cohesion of soil at and beneath helix I
qi = γiHi = overburden pressure at helix i
Nci= = Bearing capacity factor on cohesion for helix i(Figure 11)
1)]-N(q+NcA=Q iqiicii
n
i
ult[
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 35
0 5 10 15 20 25 30 35 40 45 50
Soil Friction Angle (deg)
1
10
100
1000
Nc*
an
d N
q*
Nc
Nq
14
7
H/D
7 4 1
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 36
Nqi= = Bearing capacity factor on overburden for helix i(Figure 11)
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 37
LATERAL CAPACITY OF HELICAL PILES
Lateral loads and moments may be transferred to helical pile foundations by the supported
structures due to a variety of reasons. The load can come from wind loading, line breakage for tower
structures, axial load eccentricities in underpinning foundations and from other sources. Since the
extension shafts used with these piles have diameters less than about 2.0 inches and may be as long
as 50 feet or more, the slenderness ratios roughly 100 to 200 are typical. These values of slenderness
ratios make buckling of the shaft a matter of concern. On the other hand, the applied loads are
generally a small fraction of the shaft=s compressive yield load and the connection hardware
between the shaft and the supported foundation provides significant restraint against rotation. The
available approaches for the analysis of laterally loaded vertical piles can be broadly grouped under
the following categories:
A. Analysis based on limiting equilibrium or plasticity theory.
B. Analysis based on elastic theory.
C. Analysis based on nonlinear theory.
Analysis Based on Limiting Equilibrium or Plasticity Theory. The theories of Brinch
Hansen(1) and Meyerhof and Ranjan(1) were developed for rigid piles assuming that the limiting or
maximum soil resistance is acting against the pile (Figure 12) when it is subjected to the ultimate
lateral load. The pile is assumed to deflect sufficiently to develop full soil resistance along the length
considered. This is not true for small deflections. Because of the diameter of the extension shafts
used in helical pile foundations, these anchors behave as flexible piles rather than rigid piles so that
this technique is not appropriate.
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 38
Analysis Based on Elastic Theory. Methods based on elastic theory commonly assume that
the soil behaves as a series of closely spaced independent elastic springs (Winkler=s assumption).
Using the beam-on-elastic-foundation approach, basic equations have been developed by various
investigators (Reese and Matlock(1), Davisson and Gill(1) for different variations of modulus of
subgrade reaction (k). The governing equation is:
Non-dimensional coefficients have been given for the solution of the pile problem to obtain
deflection, moment and shear, etc., along the pile lengths (Stephenson and Puri(1).
Analysis Based on Nonlinear Theory. In general, because of the inherent complexities of
determining soil-structure interaction using nonlinear theories, most of the solutions are computer
based. The most widely used computer program is a finite difference model LPILEPLUS
developed by
Lyman Reese and his colleagues at the University of Texas(1). The computer model uses the
following equation:
where: y = lateral displacement
x = distance along the axis
EI = the bending stiffness of the pile, and
Es = the secant modulus of the soil response curve.
If a distributed lateral load w acts along some portion of the shaft length, the final equation becomes:
-ky=dx
ydEI
4
4
0=y E+dx
ydQ+
dx
ydEI s2
2
4
4
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 39
The advantages of the computer solutions are:
The bending stiffness EI of the pile can be varied along its length.
The soil secant modulus can vary from point to point along the pile=s length and as a
function of deflection (nonlinearly).
The effect of the axial load on deflection and bending moment can be considered.
The effect of bending on compression loaded helical anchors was studied by Hoyt, et. Al.
(20). They used LPILEPLUS
to model three full-scale loading tests. The modeling showed that
buckling is a practical concern only in the softest soils, and this agrees with past analyses and
experience on other types of piles (Sowers and Sowers(1). They presented a figure (Figure 13) that
can be used to decide whether a particular application is clearly stable (well to the right and below
the applicable boundary line), clearly unstable (well to the left of or above the boundary), or
questionable (close to the boundary). Their criteria for soil classification is given in Table 4. They
suggested that load tests could be used to resolve questionable applications.
Simplified Method for Nonlinear Analysis of Helical Piles in Clay. Hsiung and Chen(1)
have presented a simplified method for the analysis and design of long piles under lateral loads
(moment) in uniform clays. The method is based on the concept of the coefficient of subgrade
reaction with consideration of the soil properties in the elastoplastic range. To use their method four
parameters are needed, two for the soil behavior and two for the pile. For the soil, the coefficient of
subgrade reaction (kh) and the yield displacement of the soil (u*) are required. For the pile, the two
0=w(x)+y E+dx
ydQ+
dx
ydEI s2
2
4
4
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 40
required parameters are the ratio of length to diameter and the elastic modulus of the pile shaft. If
the coefficient of subgrade reaction varies with depth, then:
The range of the analytical parameters are determined as follows:
1. The coefficient of subgrade reaction, kh has been compiled by Poulos and Davis(1) and are
shown in Table 5. kh is taken in the range of 4,000-26,000 kN/m3 for an overconsolidated
clay. A value of nh in the range of 200-1300 kN/m4 is used.
2. Yielding displacement of soil, u* is taken in the range of 12-25 mm according to Bowles(1).
3. Ratio of length to diameter (L/d).
4. Elastic modulus of the pile shaft, Ep.
The results of this study showed that both the load-maximum deflection and load-maximum
moment relationships may be expressed by normalized curves based on regression analysis. The
equations used for normalizing the Load and Moment Factors are given in Table 6. Table 7 gives the
regression equations for maximum deflection and moment.
zn=k hh
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 41
Table 4: Soil Parameters for Figure 13
Description
N
(blows)
Cu
(kPa)
Φ
(deg)
Clays
Very Soft
1
10
0
Soft
3
19
0
Medium
6
38
0
Stiff
12
72
0
Very Stiff
24
143
0
Hard
32+
287
0
Sands
Very Loose
2
0
28
Loose
7
0
29
Medium
20
0
33
Dense
40
0
39
Very Dense
50+
0
43
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 42
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Shaft Length (m)
0
50
100
150
200
Ax
ial
Lo
ad
(k
N)
STIFF C
LAY
MEDIIU
M SAND
MEDIU
M CLAY &
LOOSE SAND
SOFT C
LAY
VE
RY
LO
OS
E S
AN
D
VER
Y S
OFT C
LAY
178 kN BRACKET
STRENGTH LIMIT
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 43
Table 5: Range of Coefficient of Subgrade Reaction
Soil Type
kh (kN/m
3)
nh (kN/m
4)
Reference
Over-consolidated stiff
clay
15700-31400
Terzaghi (1955)
Over-consolidated
very-stiff clay
31400-62800
Terzaghi (1955
Over-consolidated
hard clay
>62800
Terzaghi (1955
Normally consolidated
soft clay
160-3260
Reese and Matlock(1965)
270-540
Davisson and Prakash (1962)
Normally consolidated
organic clay
108-270
Peck and Davisson (1962)
108-810
Davisson (1970)
Peat
54
Davisson (1970)
27-108
Wilson and Hilts (1967)
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 44
Table 6: Formulas for Normalizing Load and Moment Factors
Formula
Number
(1)
Soil Condi-
tion
(2)
Pile-head
Condition
(3)
Loading
Condition
(4)
Normalizing Load
factor
(5)
Normalizing moment
factor
(6)
1
Constant kh
Free-head
Lateral Force
Po
P
c = 2EIλ
3u
*
Mcmax
=0.3224P
c/λ
2
Constant kh
Free-head
Moment Mo
M
c = 2EIλ
2u
*
Mcmax = M
c
3
Constant kh
Fixed-head
Lateral Force
Po
P
cf = 4EIλ
3u
*
Mcmax =0.5Pf
c/λ
4
Linear kh
Free-head
Lateral Force
Po
P
c = (1/249)EIλ
3u
*
Mcmax =0.7714 Pf
c/λ
5
Linear kh
Free-head
Moment Mo
Mc =(1/1.619)EIλ
2u
*
Mcmax = M
c
6
Linear kh
Fixed-head
Lateral Force
Po
P
cf =(1/.9279)4EIλ
3u
*
Mcmax =0.9271Pf
c/λ
Note: For constant kh: 4
pph IEd/4k=For linear kh:
5pph IEd/n=
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 45
Table 7: Regression Equations for Maximum Deflection and Moment
Equation
Number
(1)
Soil
Condition
(2)
Pile-head
Condition
(3)
Loading
Condition
(4)
Regression equation of
load-max. deflection
(5)
Regression equation of
load-max. Moment
(6)
1
Constant kh
Free-head
Lateral
Force Po
0.66+u0.32u/
uu/=
P
P*
*
c
o
M
M1.10 =
P
Pc
0.52
c
o
max
max
2
Constant kh
Free-head
Moment Mo
0.87+u0.15u/
uu/=
M
M*
*
c
o
M
M1.00 =
M
Mc
1.0
c
o
max
max
3
Constant kh
Fixed-
head
Lateral
Force Po
0.67+u0.37u/
uu/=
P
P*
*
cf
o
M
M1.07 =
P
Pc
0.54
c
o
max
max
4
Linear kh
Free-head
Lateral
Force Po
0.82+u0.15u/
uu/=
P
P*
*
cf
o
M
M1.111 =
P
Pc
0.70
c
o
max
max
5
Linear kh
Free-head
Moment Mo
0.91+u0.075u/
uu/=
M
M*
*
c
o
M
M1.00 =
M
Mc
1.0
c
o
max
max
6
Linear kh
Fixed-
head
Lateral
Force Po
0.83+u0.19u/
uu/=
P
P*
*
cf
o
M
M1.09 =
P
Pc
0.72
c
o
max
max
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 48
EXAMPLE PROBLEMS
Uplift Capacity
Shallow Anchor in Sand
Given the situation shown in Figure EX-1. Using equation 31:
Interpolating from Table 3, Fq = 30.06.
ku = 1.3 (Figure 5).
Qu = 5164+6021 = 11,185 lbs
FS = 2.5
Qallow = 11,185 = 4474 lbs = 4.5kips
If the water surface were at the ground surface, then:
tank)H-H)((2
D+D
2+HDF
4=Q u
21
2n
n11
21qu
3.6=10
36=
D
H=R
1
1
35)1.3x3-8(105)(2x12
7.5+10
2+3
12
1030.06x105
4=Q 22
2
utan
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 49
Qu = 5902 lbs
FS = 2.5
Qallow = 5902/2.5 = 2361 lbs = 2.4 kips
Deep Anchor in Sand
Given the situation shown in Figure EX-2. Using equation 31:
For Φ = 35 , m = 0.18 (Table 2)
pcf 55.4=62.4-117.8=-==watersat
35)1.3x3-8(55.4)(2x12
7.5+10
2+3
12
1030.06x55.4
4=Q 22
2
utan
7.2=10
72=
D
H=R
1
1
D
Hm+0.6=5) (Figure 1.5=k
1
1
cr
umax
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 50
Fq* = 50 (Figure 8)
Qu = 17181+10737 = 27918 lbs
FS = 2.5
Qallow = 27918/2.5 = 11,167 lbs = 11.1 kips
If the water surface were at
the ground surface, then:
Qu = 14730 lbs
FS = 2.5
Qallow = 14730/2.5 = 5892 lbs = 5.9 kips
R < 5=0.18
0.9=
D
H
D
H+0.6=1.5
1
1
cr
1
1
cr
39) (equation k)H-H)((2
D+D
2+HDF
4=Q u
21
2n
n11
21qu
tan
35)1.5x6-11(105)(2x12
7.5+10
2+6
12
1050x105
4=Q 22
2
utan
35)1.5x6-11(55.4)(2x12
7.5+10
2+6
12
1050x55.4
4=Q 22
2
utan
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 51
Shallow anchor in Clay
Given the situation shown in Figure EX-3. Using
equation 38
Fc = 9
D1 = 12 in = 30.5 cm
Qu = 33.6 + 66.3 = 99.9 kN
)+Fc(A=W+ F c A = Q cu1sc1p
72.5+c0.107=D
H=R u
1
1
cr
cr
7.7=2.5+0.107(49)=
D
H=RR
1
1
cr
cr
kN 33.6=19.5)+ (49x9100
30.5
4 = Q
2
p
)H H( c 2
D+D = Q 1nu
n1
f
kN 66.3=(3x0.305)] [(8x0.305) 49.5 2x100
10)2.54+(12 = Q
f
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 52
Deep anchor in Clay
Assume ca = 0.9c = 0.9(48.0) = 43.2 kN/m2.
FS = 3.0
Qall = Qu/FS = 235.3/3 = 78 kN
cHD+
c)H-H(2
D+D+
)H+c)(9D(4
=Q
a1s
u1nn1
1u21u
kN 47.4=12.63+0.64+34.17=Q
26x.305x43.12
2x.305+
.06)(.305)48-(1112x2x100
10).305+(12+
05)]19.5x(6x.3+)[9x48.0)[(1x.3054
=Q
u
2
u
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 53
Bearing Capacity of a Helical Pile in Com-pression
Anchor in Sand (Figure EX-5)
c = 0
γ = 105 pcf
Φ = 35 deg
49) (equation 1)-Nq(+cN=q qcult
1)]-N(q+NcA=Q iqiicii
n
i
ult[
sf0.307=4
)(10/12=
4
D=A
sf0.545=4
)(12/12=
4
D=A
sf0.545=4
)(10/12=
4
D=A
sf0.545=4
)(10/12=
4
D=A
224
4
223
3
222
2
221
1
110=4N 7)=12.8(=D
H 8.0=H
12
7.5=D
110=3N 7)=7.6(=D
H 6.34=H
12
10=D
90=2N 5.6=D
H 4.67=H
12
10=D
77=1N 3.6=D
H 3=H
12
10=D
q
4
444
q
3
333
q
2
222
q
1
111
max
max
psf 840=5)+(105)(3=H=q
psf 665=10/3)+(105)(3=H=q
psf 490=5/3)+(105)(3=H=q
psf 315=(105)(3)=H=q
444
333
222
111
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 54
Anchor in Clay (Figure EX-6)
c = 1000 psf
γ = 124 pcf
Φ = 0 deg
Nq= = 1 (Figure 11)
kips 35.2=3
105=
FS
Q=Q
kips 105 = lbs 105,487=28,367+39,867+24,035+13,219=
0)](110)(0.307)(84+5)(110)(0.545)(66+0)(90)(0.545)(49+15)(77)[(0.545)(3=
NqA=]Nq+NcA=Q
ult
allow
qiii
4
1
iqiicii
n
i
ult[
49) (equation 1)-Nq(+cN=q *q
*cult
1)]-N(q+NcA=Q *qii
*ciii
n
i
ult[
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 55
16=4N 7)=9.6(=D
H 8.0=H
12
10=D
15=3N 6.34=D
H 6.34=H
12
12=D
14=2N 4.67=D
H 4.67=H
12
12=D
10=1N 3.0=D
H 3=H
12
12=D
c
4
444
c
3
333
c
2
222
c
1
111
max
sf0.545=4
)(10/12=
4
D=A
sf0.785=4
)(12/12=
4
D=A
sf0.785=4
)(12/12=
4
D=A
sf0.785=4
)(12/12=
4
D=A
224
4
223
3
222
2
221
1
kips 13.1=3
39.3=
FS
Q=Q
kips 39.3=
lbs 39,335=8720+11,775+10,990+7850=
)(16)0.545(1000+
)(15)0.785(1000+)(14)0.785(1000+
)(10)0.785(1000=Q
ult
allow
ult
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 56
Lateral Capacity of a laterally Loaded Helical Pile in Medium Clay
Assume the situation is as shown in Figure EX-7. Also assume:
Load = 40 kN
c = 1000 psf = 48 kPa
Shaft Properties:
L = 8 ft = 2.4 m
2 inch square outside dimension
wall thickness = 05 inch
Check to see if buckling will be a concern.
From Table 4, soil is a medium clay.
Using Figure 13, situation plots significantly below is medium clay line, therefore, buckling is not a
concern.
Lateral Capacity of a Laterally Loaded Helical
Pile in Soft Clay
Assume the situation is as shown in Figure EX-7.
Also assume:
Load = 40 kN
c = 400 psf = 19 kPa
Shaft Properties:
L = 8 ft = 2.4 m
2 inch square outside dimension
wall thickness = 0.5 inch
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 57
Check to see if buckling will be a concern.
From Table 4, soil is a soft clay.
Using Figure 13, situation plots near the soft clay line, therefore, buckling is a concern.
Helical shaft properties:
Ep = 30 106 psi = 20.7 10
7 kPa
I = π(do4 -di
4)/64 = 4.8 10
-7 m
4
From Table 5 assume nh = 500 kN/m4
Assume u* = 15 mm = yield deflection
Using equation 6 from Table 6
Using equation 6 from Table 7
mm 69=m -0.069=u
0.83+u0.19u/
uu/=
P
P*
*
cf
o
Determine the maximum moment.
Using Equation 6 from Table 6
6) (Table 0.519=)10)(4.8x107x0.032/(20.*500=IE/d*n= 5 -775ppoh
kN 0.331=uIE0.9279
1=P
*3ppcf
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 58
Using Equation 6 from Table 7
m-kN 359=M
M
M1.09 =
P
Pcf
0.72
cf
o
max
max
Lateral Capacity of a Laterally Loaded Helical Pile in Overconsolidated Clay
Assume the situation is as shown in Figure EX-7. Also assume:
Load = 40 kN
c = 2000 psf = 96 kPa
Shaft Properties:
L = 8 ft = 2.4 m
2 inch square outside dimension
wall thickness = 0.5 inch
Check to see if buckling will be a concern.
From Table 4, soil is a stiff clay.
Using Figure 14, situation plots to the right of the
stiff clay line, therefore, buckling is not a concern.
Helical shaft properties:
Ep = 30 106 psi = 20.7 10
7 kPa
I = π(do4 -di
4)/64 = 4.8 10
-7 m
4
From Table 5 assume kh = 20000 kN/m3
m-kN 0.519=P
0.9271=M
cfc
max
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 59
Assume u* = 15 mm = yield deflection
Using equation 2 from Table 6
Using equation 2 from Table 7
mm 3.064=u
0.87+u0.15u/
uu/=
M
M*
*
c
o
Determine the maximum moment.
Using Equation 2 from Table 6
Since Mcmax <Mo, part of soil is yielding.
Using Equation 2 from Table 7
m-kN 1.146=M
M
M1.00 =
M
Mc
1.0
c
o
max
max
max
1. Wilson, Guthlac, AThe Bearing Capacity of Screw Piles and Screwcrete Cylinders,@ J. Inst
of Civil Engineers, London, Vol 34, pp 4-93, 1950.
6) (Table m 1.304=)10)(4.8x10.7x0.057/4(20*20000=IE/4d*k= -14 -774ppoh
m-kN 5.1=uIE2=M*2
ppc
m-kN 5.1=M=Mcc
max
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 60
1. Stephenson, R.W., ADesign and Applications of Helical Earth Anchors,@ (1988),
unpublished notes of a seminar for geotechnical engineering graduate students, University of
Missouri-Rolla, Oct. 20, 1988.
1. Clemence, S.P., Thorsten, R.E., and Edwards, B., AHelical Anchors: Overview of
Application and Design@ (1990), Foundation Drilling, Dec./Jan. 1990, P.P. 8-12.
1. Das, Braja M. Earth Anchors, Developments in Geotechnical Engineering, Series No. 50,
Elsevier, NY 1990.
1. Udwari, J.J., Rodgers, T.E., and Singh, H., "A Rational Approach to the Design of High
Capacity ;Multi-helix Screw Anchors," Proceedings, Seventh IEEE/PES Transmission and
Distribution Conference and Exposition, April 1-6, 1979, pp. 606-609.
1. Lutenegger, A.J., Smith, B.L., and Kabir, M.G., "Use of In Situ Tests to Predict Uplift
Performance of Multihelix Anchors," Special Topics in Foundations, ASCE, pp. 93-108.
1. A.B. Chance Co. Encyclopedia of Anchoring, The A.B. chance Co., 1977.
1. Hoyt, R.M., and S.P. Clemence (1989), "Uplift Capacity of Helical Anchors in Sand,"
Proceedings of the XII International Conference on Soil Mechanics and Foundation
Engineering, Rio De Janeiro, Aug, 1989, pp 1019-1022.
1. Kulhawy, F.H., "Uplift Behavior of Shallow Soil Anchors - An Overview", Uplift Behavior
of Anchor Foundations in Soil, ASCE, New York, pp. 1-25, 1985.
1. Clemence, Samuel P., "The Uplift and Bearing Capacity of Helix Anchors in Soil," Contract
Report TT112-1, 3 Volumes, Niagara Mohawk Power Corporation, Syracuse, New York
(1984).
1. Mitsch, M.P., and Clemence, S.P., AThe Uplift Capacity of Helical Anchors in Sand,@
Uplift Behavior of Anchor Foundations in Soil, ASCE, New York, pp. 26-47 (1985).
1. Mooney, J.S., Adamczak, S., and Clemence, S.P., "Uplift Capacity of Helical Anchors in
Clay and Silt," Uplift Behavior of Anchor Foundations in Soil, ASCE, pp. 48-72 (1985).
1. Carville, Chester, A., P.E., and Walton, Robert W. , AFoundation Repair Using Helical
Screw Anchors,@ Foundation Upgrading and Repair for Infrastructure Improvement,@
Geotechnical Special Publication, N. 50, American Society of Civil Engineering, New York,
1995.
March 2, 2010 (7:49PM) D:\HELICAL ANCHORS.DOC 61
1. Meyerhof, G.G., ABearing Capacity and Settlement of Pile foundations,@ Journal of the
Geotechnical Engineering Division, American Society of
Civil Engineers, Vol. 102, No. GT3, pp. 197-228, 1976.
1. Hansen, Brinch, J., AThe Ultimate Resistance of Rigid Foundation Piles Against Transveral
Forces,@ Bulletin-12, Danish Geotechnical Institute, Copenhagen, 1951.
1. Meyerhof, G.G. and Ranjan, Gopal, AThe Bearing Capacity of Rigid Piles Under Inclined
Load in Sand, Part I: Vertical Piles,@ Canadian Geotechnical Journal, Vol. 9, 1972,.
1. Reese, L.C., and Matlock, H., ANon-dimensional Solution for Laterally Loaded Piles with
Soil Modulus Assumed Proportional to Depth,@ Proceedings of the Eighth Texas
Conference on Soil Mechanics and Foundation Engineering, Austin TX, 1965.
1. Davisson, M.T. and Gill, H.L., Laterally Loaded Piles in layered Soil System,@, Journal of
the Soil Mechanics and Foundation Division, Vol. 89, No. SM3, American Society of Civil
Engineers, 1963.
1. Stephenson, Richard W., and Puri, Vijay, AUltimate Capacity of A.B. Chance Helical
Anchors,@ Unpublished report, prepared for the A.B. Chance Co., Centralia MO, 1981.
1. Hoyt, Robert, Seider, Gary, Reese, Lymon C., and Wang, Shin-Tower, ABuckling of Helical
Anchors Used for Underpinning,@ Foundation Upgrading and Repair for Infrastructure
Improvement,@ Geotechnical Special Publication, N. 50, American Society of Civil
Engineering, New York, 1995.
1. Sowers, G.B. and Sowers, G.F., Introductory Soil Mechanics and Foundations, Third Edition,
Macmillan Publishing Co., New York, NY, 1970.
1. Hsiuing, Yun-mei and Chen, Ya-ling, ASimplified Method for Analyzing laterally Loaded
single Piles in Clay,@ Journal of Geotechnical and Geoenvironmental Engineering, Vol.
123, No. 11, November, 1997.
1. Poulos, H.G., and Davis, E.HH., Pile Foundaton Analysis and Design, John Wiley & sons,
Inc., New York, 1971.
1. Bowles, J.E., Foundation Analysis and Design, McGraw-Hill, New York, 1988.