HELICAL FOUNDATIONS AND ANCHORS

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

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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

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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.@

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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

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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. D:\HELICAL ANCHORS.DOC

Since about 1975, a number of researchers have studied the 6

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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.

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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 D:\HELICAL ANCHORS.DOC

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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 nonhomogeneities, 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:

n

Qu =

QU i , where

n is the number of helices

i=1

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Qui = qui xA 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

1 qui = cN *c + Di N * + H i N *q 2 A = area of helix Di = diameter of helix H i = Depth from ground surface to helix = effectiveunit weight of soil above helix

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 D:\HELICAL ANCHORS.DOC

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Qu = Q p + Q f + W f

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 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

Φ (deg)

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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

Q p ( max ) = A f ( q q N q

qr

qs

qd

) + W f + Qtu

where: Qp(max)

=

Top plate capacity limit

Af

=

Area of top helix

=

qq

Wf

effective surcharge = H 1

= Effective weight of helical pile alone

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Qtu

=

Tip capacity in uplift (usually neglected)

The Nq term is a bearing capacity factor given by:

Nq= e

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tan

2 _ tan 45 +

2

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qr

= exp - 3.8 tan + (3.07 sin )

log 10 2 I r 1+ sin

1.0

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

Ir =

G E = 2(1+ ) qi tan

qs

qs

qd

E

=

1 qi tan

= 1 + tan

= 1 + tan

= 1+ 2 tan (1 - sin

2

) tan -1

H D1

soil elastic modulus

The cylinder friction capacity, Qf , is computed from the following equation:

H

Qf =

P(z)

v

k u (z)( tan )(z)dz

P(z)

v

k o (z) tan (z)

H1

=

k

H

k o H1

where: P

=

helix perimeter

v

=

effective vertical stress

k

=

coefficient of horizontal earth pressure

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ko

=

effective interface friction angle

=

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:

Q f(reduced)= Q f

r 0

2+ o 3 o = k o tan r

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:

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Qu = Q p + Q f

Uplift capacity of shallow anchors Qu =

k u tan cos

2

2

2

D1 H 1 + 2

3

H 1 tan 3

2 +W in sand: s

The weight of the soil, Ws can be expressed as:

2

Ws=

3

2 1

H 1 ( D ) + D1 + 2 H 1 tan

2

+ ( D1 ) D1 + 2 H 1 tan

2

Das non-dimensionalized these equations into:

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Qp

F q1 =

AH1

2

Ws = 4 + 5.33 H 1 F q2 = AH1 D1

F q1 =

Qp AH1

= 4 k u ( tan ) cos

H1 tan + 8 2 D1 2

H1 D1

2

2

2

0.5 H1 D1

2

tan

2

+ 0.33 tan

2

Similarly: Let

Fq =

H 1 = R2 = 4 R 2 k u ( tan ) cos D1 2 AH1

Qp

+ 4 + 5.33 R 2 tan 2

2

0.5 + 0.33 tan R 2

+ 8R tan

2

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:

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k u = 0.6 + m

H1 D1

The variation of m is given below. Table 2 Variation of m Soil friction angle, Φ

m

(degrees) 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

Qp = Fq AH1=

2

4

F q D1 H 1

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:

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Qf =

2

Da

Da

( H 2n - H 12 ) k u tan

=

average helix diameter.

Therefore the ultimate uplift capacity for a shallow anchor in sand

Qu =

2

4

F q D1 H 1 +

2

D1 + D n ( )( 2 - 2 ) tan H n H 1 ku 2

is:

Figure 5: Variation of ku with H1/D1

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Table 3: Variation of Breakout Factor Fq for Shallow Anchor Condition Fq R =H1/D1

Φ=

Φ=30

Φ=35

Φ=40

Φ=45

25 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

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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

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Qu = Q p + Q f + Q s Q p = bearing capacity of the top helix Q f = frictional resistance of the cone between the helices Q s = shaft friction resistance = 0 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

Qp=

*

resistance

2

F q D1 H 1

4 where F = deep anchor breakout factor * q

Qu =

2

2 D a ( H n 2 - H 1 ) k u max tan

frictional Qf

is

computed using: The two equations can be combined to yield the net

+ where D a = D1 D n 2

ultimate uplift capacity for deep anchors in sand:

Qu =

*

4

2

F q D1 H 1 +

2

D1 + D n 2

( H 2n - H 12 ) k u max tan

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.

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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

Deep Anchor Breakout Factor, Fq*

uplift capacity.

100

10 20

30

40

50

Soil Friction Angle (deg)

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Uplift capacity of helical anchors in clay. Failure of helical piles in clay soils is

s u = cu

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.

Qu = Q p + Q f

For shallow anchors: where: Qp = bearing capacity of the top helix

Q p = A1 c F c + W s = A1 ( cu F c + )

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: D:\HELICAL ANCHORS.DOC

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9 8 7 6

Fc

5 4 3 2 1 0 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

Rcr =

H1 D1

= 0.107 cu + 2.5 7 cr

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:

Qf =

D1 + D n cu ( H n H 1 ) 2

Combining:

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Qu =

D1 + D n ( H n - H 1 ) cu 2

2

4

D1 ( c u F c + H 1 ) +

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

Qu = Q p + Q f + Q s

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.

Qp =

Qf =

4

( D12 )(9 cu + H 1 )

D1 + D n cu ( H n H 1 ) 2

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:

Qs = Ds H 1 ca

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Qu =

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4

( D12 )(9 cu + H 1 ) +

D1 + D n ( H n - H 1 ) cu + D s H 1 c a 2

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All ultimate uplift capacities should be divided by an appropriate factor of safety to set the

Qallow=

Qu FS

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

Qu = K t xT 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.

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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

qult = cN c + q( N q - 1) capacity equation: where:

c

=

soil cohesion

q

=

overburden pressure = γHi

γ

=

effective unit weight

Hi

=

depth to helix

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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)

n

Qult =

Ai [ ci N c i + qi ( N q i - 1)]

i

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Nc* and Nq*

1000

H/D

100

7 4

Nc

1

7 4 1

10

Nq 1 0

5

10

15

20

25

30

35

40

45

50

Soil Friction Angle (deg)

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Nqi=

=

Bearing capacity factor on overburden for helix i(Figure 11)

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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.

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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

4

d y EI 4 = -ky dx

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

4

EI

2

d y d y +Q 2 + Es y = 0 4 dx dx

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:

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March 2, 2010 (7:49PM)

4

EI

2

d y d y + Q 2 + E s y + w(x)= 0 4 dx dx

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 D:\HELICAL ANCHORS.DOC

39

March 2, 2010 (7:49PM)

required parameters are the ratio of length to diameter and the elastic modulus of the pile shaft. If

k h = nh z

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.

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March 2, 2010 (7:49PM)

Table 4: Soil Parameters for Figure 13 Description

N

Cu

Φ

(blows)

(kPa)

(deg)

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

Very Loose

2

0

28

Loose

7

0

29

Medium

20

0

33

Dense

40

0

39

Very Dense

50+

0

43

Clays

Sands

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March 2, 2010 (7:49PM)

200

FF

C

ME

I

SA

ND

100

CL

A

Y

D

SO

FT

RY VE

50

LO

OS

ES

AN

Axial Load (kN)

150

DI

UM

M LO ED O S IU M ES C AN LA D Y&

I ST

178 kN BRACKET STRENGTH LIMIT

Y LA

0 0.0

1.0

V

Y ER

2.0

SO

FT

CL

A

Y

3.0

4.0

5.0

6.0

Shaft Length (m)

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March 2, 2010 (7:49PM)

Table 5: Range of Coefficient of Subgrade Reaction Soil Type

kh (kN/m3)

nh (kN/m4)

Over-consolidated stiff

15700-31400

Terzaghi (1955)

31400-62800

Terzaghi (1955

>62800

Terzaghi (1955

Reference

clay Over-consolidated very-stiff clay Over-consolidated hard clay Normally consolidated soft clay

Normally consolidated organic clay

160-3260

Reese and Matlock(1965)

270-540

Davisson and Prakash (1962)

108-270

Peck and Davisson (1962)

108-810

Davisson (1970)

54

Davisson (1970)

27-108

Wilson and Hilts (1967)

Peat

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March 2, 2010 (7:49PM)

Table 6: Formulas for Normalizing Load and Moment Factors Formula

Soil Condi-

Pile-head

Loading

Normalizing Load

Normalizing moment

Number

tion

Condition

Condition

factor

factor

(3)

(4)

(1)

(6)

(2) 1

Constant kh

(5) Free-head

Lateral Force

Pc = 2EIλ3u*

Mcmax =0.3224Pc/λ

Po 2

Constant kh

Free-head

Moment Mo

Mc = 2EIλ2u*

Mcmax = Mc

3

Constant kh

Fixed-head

Lateral Force

Pcf = 4EIλ3u*

Mcmax =0.5Pf c/λ

Pc = (1/249)EIλ3u*

Mcmax =0.7714 Pf

Po 4

Linear kh

Free-head

Lateral Force Po

c

/λ

5

Linear kh

Free-head

Moment Mo

Mc =(1/1.619)EIλ2u*

Mcmax = Mc

6

Linear kh

Fixed-head

Lateral Force

Pcf =(1/.9279)4EIλ3u*

Mcmax =0.9271Pf

Po

Note: For constant kh:

D:\HELICAL ANCHORS.DOC

c

/λ

= 4 k h d/4 E p I p For linear k : = 5 nh d/ E p I p h

44

March 2, 2010 (7:49PM)

Table 7: Regression Equations for Maximum Deflection and Moment Equation

Soil

Pile-head

Loading

Regression equation of

Regression equation of

Number

Condition

Condition

Condition

load-max. deflection

load-max. Moment

(1)

(2)

(3)

(4)

(5)

(6)

1

Constant kh

Free-head

Lateral

u/ u Po = * c P 0.32u/ u + 0.66

*

Force Po

2

Constant kh

Free-head

3

Constant kh

Fixedhead

4

Linear kh

Free-head

Moment Mo

u/ u Mo = c 0.15u/ u* + 0.87 M

Lateral

u/ u Po = * c P f 0.37u/ u + 0.67

*

*

Force Po

Lateral

*

u/ u Po = * c P f 0.15u/ u + 0.82

Force Po

5

Linear kh

Free-head

6

Linear kh

Fixedhead

D:\HELICAL ANCHORS.DOC

Po = 1.10 M max c c P M max M o = 1.00 M max c c M M max Po = 1.07 M max c c P M max

Po = 1.111 M max c c P M max

Moment Mo

u/ u Mo = c 0.075u/ u* + 0.91 M

M o = 1.00 M max c c M M max

Lateral

u/ u* Po = * c P f 0.19u/ u + 0.83

Po = 1.09 M max c c P M max

*

Force Po

45

0.52

1.0

0.54

0.70

1.0

0.72

March 2, 2010 (7:49PM)

BIBLIOGRAPHY

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March 2, 2010 (7:49PM)

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March 2, 2010 (7:49PM)

EXAMPLE PROBLEMS Uplift Capacity Shallow Anchor in Sand

Qu =

D1 + D n ( )( 2 - 2 ) tan H n H 1 ku 2

2

4

F q D1 H 1 +

2

Given the situation shown in Figure EX-1. Using equation 31:

36 R = H1 = = 3.6 D1 10

Interpolating from Table 3, Fq = 30.06. ku = 1.3 (Figure 5). 2

Qu =

4

30.06x105

10 3+ 12 2

10 +7.5 (105)( 8 2 - 32 )1.3x tan 35 2x12

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:

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March 2, 2010 (7:49PM)

=

=

2

10 3+ Qu = 30.06x55.4 4 12 2

sat

-

water

= 117.8 - 62.4 = 55.4 pcf

10 +7.5 (55.4)( 8 2 - 32 )1.3x tan 35 2x12

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: 72 R = H 1 = = 7.2 D1 10

k umax = 1.5 (Figure 5) = 0.6 + m

H1 D1

cr

For Φ = 35 , m = 0.18 (Table 2)

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March 2, 2010 (7:49PM)

1.5 = 0.6 + H 1 D1 H1 D1

= cr

cr

0.9 = 5< R 0.18

Fq* = 50 (Figure 8) Qu =

2

4

F q D1 H 1 +

2

D1 + D n ( )( 2 - 2 ) tan H n H 1 ku 2

2

10 6+ Qu = 50x105 4 12 2

(equation 39)

10 +7.5 (105)( 112 - 6 2 )1.5x tan 35 2x12

Qu = 17181+10737 = 27918 lbs FS = 2.5 Qallow = 27918/2.5 = 11,167 lbs = 11.1 kips If the water surface were at 2

10 6+ Qu = 50x55.4 4 12 2

10 +7.5 (55.4)( 112 - 6 2 )1.5x tan 35 2x12

the ground surface, then:

Qu = 14730 lbs FS = 2.5 Qallow = 14730/2.5 = 5892 lbs = 5.9 kips

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March 2, 2010 (7:49PM)

Shallow anchor in Clay Given the situation shown in Figure EX-3. Using equation 38 Q p = A1 c F c + W s = A1 ( cu F c + )

H 1 1 0.107 + 2.5 7 cu Rcr =Rcr = H = R = 0.107(49) + 2.5 = 7.7 D1 D cr 1 cr

Fc = 9 D1 = 12 in = 30.5 cm 2

30.5 (49x9 + 19.5) = 33.6 kN Qp= 4 100

Qu = 33.6 + 66.3 = 99.9 kN

Qf =

D1 + D n cu ( H n H 1 ) 2

Qf =

(12 + 10)2.54 49.5 [(8x0.305) (3x0.305)] = 66.3 kN 2x100

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March 2, 2010 (7:49PM)

( D12 )(9 cu + H 1 ) 4 D1 + D n ( H n - H 1 ) cu 2 + Ds H 1 ca

Qu = +

Deep anchor in Clay

Assume ca = 0.9c = 0.9(48.0) = 43.2 kN/m2.

Qu =

4

+

[(1x.305 )2 )[9x48.0+ 19.5x(6x.305)] (12 + 10).305 (11 - 6)(.305)48.0 12x2x100

2x.305 6x.305x43.2 12 Qu = 34.17 + 0.64 + 12.63 = 47.4 kN +

FS = 3.0 Qall = Qu/FS = 235.3/3 = 78 kN

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March 2, 2010 (7:49PM)

Bearing Capacity of a Helical Pile in Com-pression Anchor in Sand (Figure EX-5)

qult = cN c + q( N q - 1) (equation 49)

c=0 γ = 105 pcf

n

Qult =

Ai [ ci N c i + qi ( N q i - 1)]

i

Φ = 35 deg (10/12 )2 = 0.545 sf 4 2 2 D 2 = (10/12 ) = 0.545 sf 4 4 2 2 D3 = (12/12 ) = 0.545 sf 4 4 10 H 1 = 3.6 2 N q 1 = 77 D H 1= 1= 3 2 D4 = (10/12 ) = 0.307 sf 12 D1 4 4 10 H 2 = 5.6 N q 2 = 90 D2 = H 2 = 4.67 12 D2 10 H 3 = 7.6( max = 7) N q 3 = 110 D3 = H 3 = 6.34 12 D3 7.5 H 4 = 12.8( max = 7) N q 4 = 110 D4 = H 4 = 8.0 12 D4 2

A1 = A2 = A3 = A4 =

D1 = 4

q1 =

H 1 = (105)(3) = 315 psf q 2 = 2 H 2 = (105)(3 + 5/3) = 490 psf q3 = D:\HELICAL ANCHORS.DOC

1

H 3 = (105)(3 + 10/3) = 665 psf q4 = 4 H 4 = (105)(3 + 5) = 840 psf 53 3

March 2, 2010 (7:49PM)

n

Qult =

A [c N i

i

4 c i + qi N q i ] =

i

Ai qi N qi 1

= [(0.545)(315)(77)+ (0.545)(490)(90)+ (0.545)(665)(110)+ (0.307)(840)](110) = 13,219 + 24,035 + 39,867 + 28,367 = 105,487 lbs = 105 kips Q allow=

Qult 105 = = 35.2 kips FS 3

qult = cN*c + q( N*q - 1) (equation 49)

n

Ai [ ci N ci + qi ( N qi - 1)] *

Qult = i

*

Anchor in Clay (Figure EX-6) c = 1000 psf

γ = 124 pcf Φ = 0 deg Nq= = 1 (Figure 11)

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March 2, 2010 (7:49PM)

12 H 1 = 3.0 N c 1 = 10 H1= 3 12 D1 2 12 D12 H 2)= (12/12 = 4.67 N sf D2 = H 2 c 2 = 14 = = =4.67 0.785 A112 D 2 4 4 2 12 D 22 H 3 )= (12/12 =2 = H 3 = 6.34 N csf3 = 15 D3 A =6.34 0.785 12 4 = 4D3 2 2 10 H(12/12 4 ) max = 7) N c 4 = 16 3 = 9.6( D4 = A3 =H 4 =D8.0 = = 0.785 sf 12 D4 4 4 D1 =

2

A4 =

D4 = 4

(10/12 )2 = 0.545 sf 4

Qult = 0.785(1000)(10) + 0.785(1000)(14) + 0.785(1000)(15) + 0.545(1000)(16) = 7850 + 10,990 + 11,775 + 8720 = 39,335 lbs = 39.3 kips Q allow =

D:\HELICAL ANCHORS.DOC

Qult 39.3 = = 13.1 kips FS 3

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March 2, 2010 (7:49PM)

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 D:\HELICAL ANCHORS.DOC

56

March 2, 2010 (7:49PM)

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: 106 psi = 20.7

Ep

=

30

I

=

π(do4 -di4)/64 = 4.8

107 kPa 10-7 m4

From Table 5 assume nh = 500 kN/m4

= 5 nh * d o / E p I p = 5 500* 0.032/(20.7x 107 )(4.8x 10-7 ) = 0.519 (Table 6)

Assume u* = 15 mm = yield deflection

Pcf =

1 Ep I p 0.9279

3

* u = 0.331 kN

Using equation 6 from Table 6

Using equation 6 from Table 7 u/ u* Po = * c P f 0.19u/ u + 0.83 u = -0.069 m = 69 mm

Determine the maximum moment. Using Equation 6 from Table 6 D:\HELICAL ANCHORS.DOC

57

March 2, 2010 (7:49PM)

c

M

c max

= 0.9271

Pf

= 0.519 kN - m

Using Equation 6 from Table 7 Po = 1.09 M max c c Pf Mf

0.72

M max = 359 kN - m

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: 106 psi = 20.7

Ep

=

30

I

=

π(do4 -di4)/64 = 4.8

107 kPa 10-7 m4

From Table 5 assume kh = 20000 kN/m3 D:\HELICAL ANCHORS.DOC

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March 2, 2010 (7:49PM)

= 4 k h * d o /4 E p I p = 4 20000* 0.057/4(20.7x 107 )(4.8x 10-7 ) = 1.304 m-1 (Table 6)

Assume u* = 15 mm = yield deflection

Mc = 2 Ep I p

2

* u = 5.1 kN - m

Using equation 2 from Table 6

Using equation 2 from Table 7 u/ u* Mo = c 0.15u/ u* + 0.87 M u = 3.064 mm

Determine the maximum moment. Using Equation 2 from Table 6

c c M max = M = 5.1 kN - m

Since Mcmax