Resistance of Piles

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Specialists are focusing major attention on methods of evaluating the resistance of piles to hori-

zontal loads. Fedorovskii et al. [1] present results of investigation of characteristic features of the inter-

action between piles and the bed soil, and new methods of analyzing pile foundations. The methods are

approved through laboratory and field experiments.

Based on codification of numerous experimental data on stresses that develop on the lateral sur-

face of piles during their horizontal loading, it is recommended to use

(1)

where ult

(z) is the ultimate reactive pressure, k0(z) is the coefficient of horizontal strength of the soil,

and u(z) is the displacement of the pile at depth z.

It follows from (1) that the support stress or reactive pressure of the soil against the lateral sur-

face of the pile during its horizontal loading is found to be a direct function of its limiting values.

Moreover, this is essentially disregarded in methods specified for analysis of piles subject to horizontal

loads in Construction Rule and Regulation 2.02.03-85 and Building Code 50-102-2003.

The accuracy of the results of the indicated analyses can be evaluated on the basis of archival

data on static pile tests, which were conducted on a construction site at the Western Siberian Metallur-gical Combine in Novokuznetsk [2]. Clayey soils, the characteristics of which are presented in Table 1

(GEE-1 slimy, iron-containing, clayey loam; GEE-2 loess-like (degraded), iron-containing loam)

reside at a depth to 20 m at the plant's construction site.

We tested precast piles with lengths of 10 m (pile 611') and 14 m (pile 758N), and cross section

of 4040 cm, which were fabricated in accordance with GOST 19804.0. The tests were performed in

accordance with GOST 5686. A horizontal load was applied in 10-kN steps to total loads ranging from

0

0

( )( ) ( ),( ) ( ) ( )

ult

ult

k zz u zz k z u z

=+

Soil Mechanics and Foundation Engineering, Vol. 47, No. 6, January, 2011 (Russian Original, No. 6, Nov.-Dec., 2010)

A procedure for the analysis of piles subject to horizontal loads using standard strength

and deformation characteristics of the soils is proposed on the basis of results of experi-

mental investigations.

RESISTANCE OF PILES

TO HORIZONTAL LOADS

B. V. Bakholdin and E. V. TrufanovaN. M. Gersevanov Scientific-Research Institute of Foundationsand Underground Structures, Moscow, Russia.

UDC 624.15

DESIGN

Translated from Osnovaniya, Fundamenty i Mekhanika Gruntov, No. 6, pp. 8-13, November-December, 2010.

0038-0741/11/4706-0215 2011 Springer Science+Business Media, Inc.215


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150 to 170 kN. Results of the tests and analysis of the piles in accordance with Construction Rule and

Regulation 2.02.03-85 and Building Code 50-102-2003 are presented in Figs. 1 and 2. In compiling the

experimental and computed data, their noncorrespondence was established in all cases. Horizontal dis-

placements of the piles based on calculation by a method utilizing the coefficient of subgrade reaction

were found to be 5-20 times higher than the experimental data.

The stresses that develop in the soil at the point where it contacts the pile were measured by

Shakhirev [3]. We used two experimental reinforced-concrete piles with a continuous cross section of

3030 cm and length of 6.0 m with load-cell strain gages established with a spacing of 50 cm on two

opposite sides of the piles. The piles were sunk in clayey soils, the characteristics of which are pre-sented in Table 2.

It follows from Fig. 3 that when the piles are horizontally loaded, basic resistance by the

soil is offered in its upper section, while it is manifested in the lower section only at great depths.

Moreover, the upper zone of the soil increases, and is displaced downward with increasing load.

The pile simultaneously experiences a certain rotation, which, as is indicated below, is one of the

important factors that must be considered in developing a method for the analysis of piles. It is

also established from the experiments that the dependence of pile displacement on load is nonlin-

ear [3].

216

GEENo.

Pilenumber

Soil characteristics

, kN/m3 , deg c, kPa Ip IL e W, %

1 611' 19.6 18 10.0 10 0.98 0.82 28

2 758' 18.9 18 20.0 10 0.70 0.78 25

TABLE 1

170

160

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

Q, kN

10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 u, mm

Fig. 1. Plots showing relationship between head displacement of pile No. 611 'and horizontal load: 1) according to experimental data; 2) BuildingCode 50-102-2003; 3) Construction Rule and regulation 2.02.03-85*;4) proposed procedure.

1 2

3

4


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217

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

Q, kN

10.0 20.0 30.0 40.0 50.0 60.0 70.0 u, mm

Fig. 2. Plots showing relationship between head displacement of pile No. 758 '

and horizontal load: 1) according to experimental data; 2) BuildingCode 50-02-2003; 3) Construction Rule and Regulation 2.02-03-85*;4) proposed procedure.

TABLE 2

Depth of sampling, m , kN/m3 0, kN/m3 W, % Ip IL e

1 19.3 26.4 26.15 16.85 0.35 0.72

2 19.3 26.5 26.0 13.85 0.21 0.45

5 19.9 26.4 26.2 14.25 0.30 0.70

6 19.2 26.7 33.8 16.95 0.65 0.87

7 18.2 25.4 36.05 18.75 0.52 0.88

a 0

325 kPa

250 kPa

480 kPa

325 kPa

0 25 50 kPa

cm

100

50

0

0

2412

2311

2210

219

19 7

18 6

17 5

43 1

15

15

15

442

45 3

46 4

208

0

375 kPa

300 kPa

500 kPa

375 kPa

0 25 50 kPa

cm

100

50

0

0

2412

2311

2210

21 9

197

18 6

17 5

43 1

17

25

25

20

44 2

45 3

3 46

20 8

b

Fig. 3. Results of stress measurements in area where soil contacts piles [3]:a and b) under horizontal load of 144 and 206 kN, respectively.

1

2

3

4


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It is most convenient to examine laws governing interaction of the "pile-soil" system in a clas-

sical Rankine half space (Fig. 4) with consideration given to the need established in [1] for evaluation

of the resistance of a pile in all steps of its horizontal loading. A pile in a Rankine half space can, in

first approximation, be treated as a plane rod to which a horizontal force Q is applied to its head sec-

tion. The point where this force is applied can be adopted as the origin of coordinates x, z (see Fig.

4).

In the general case, Rankine's solution for the stress state of a medium in a half space bounded

by a slope assumes the following form:

(2)

where is the specific weight of the soil medium, is the surface slope of the medium, and q is the

load distributed over the surface of the soil.

It follows from (2) that for each plane of the slope, it is possible to omit the overlying section

of the soil medium, and treat this plane as the boundary surface. When the surface is horizontal, = 0,solution (2) is simplified, and the paths of the principal stresses, as before, coincide with the slip lines

that parallel the surface of the soil medium. Introducing the parameter c, which takes into account the

cohesion of the clayey soils, we obtain:

(3)

where z

is a coefficient.

1 sin( ) ;

1 sin

zx c

z

z q

= + +

+

2

2

2

2 2 2

cos(1 )(1 );

cos cos

cos(1 ) tan ;

cos cos

sin cos sin sin ,

x

z

xy

z q

z q

+=

+=

= +

218

a

O O

B B

A A

K

z

b

H

hl/2h

l/2h

2/3

(H-h

)

(1-a

)h

ah

C

D

F F

F0

Fmin

Fk

Fk

q

k

zA

zB

zOQ

bQ

Fig. 4. Diagram used for analysis of pile subject to horizontal loadin classical Rankine half space (a) and stress state of soil onpile shaft when functioning in accordance with scheme (b).


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It is accepted to consider that the coefficient z

is a switch: z

= +1 for the active pressure, which

corresponds to the minimum value of the fractional term in (3), and z

= 1 for the development of pas-

sive soil support, i.e., corresponding to the maximum stress state. In truth, the coefficient z

should be

treated as a plasticity index that varies from 1 to 1. Here, it is possible to assume that when a load Q

is applied to the head of a pile at point O (see Fig. 4), it rotates about a certain point A removed from

the by head by a distance z = h. In this connection, the maximum stress state (z=0

= 1) should devel-

op within the pile essentially at point O on plane z = 0. This is explained by the fact that the value of

h will vary during loading, increasing from h

0 to values corresponding to maintenance of limitingequilibrium. The minimum value of the stresses (z=h

= 1) will be maintained at point A, since the soil

at this point is virtually unconsolidated. Under these conditions, the stresses in the section of the pile

from z = 0 to z = h should be considered to vary as z

varies from 1 to 1. The variation in z

over the

depth when z < h can, in first approximation, be assumed to be linear z

= 2z/h 1. At point B, which

corresponds to z = h /2, therefore, it is possible to set z=h/2

= 1 on the assumption of the indicated lin-

earization of the z

values from z. The existence of cohesion c for the soils can be modeled in the cal-

culations by the application of a load q = c/tan over the surface of the soil medium under considera-

tion [4]. When z < h, therefore, the value ofc

in [3] should be determined with consideration given to

this situation

(3')

The stresses on the shaft of the pile xO

,, xA

, and xB

at points O, A, and B, respectively (see

Fig. 4), of the pile shaft can be assumed as:

(4)

Permitting the possibility of linearization and determination of the variation in stresses between

the indicated points, the total resistance F of the soil when a load Q is transmitted onto the pile

(5)

The law describing the stress state of the soil within the surface of a pile and in the near-pile

space also corresponds to experimental data derived from measurement of these stresses; this is appar-

ent from Figs. 3 and 4.

According to (4) and (5), we have

(6)

when c = 0.

In the general case, the depth zF

at which the resultant F is applied from the coordinate ori-

gin will depend on h, and can be adopted as zF

= ah , where a is a coefficient that varies from 1/2

to 2/3.

21 sin

1 ,4 1 sin

hF

= + +

( 2 ).2 2 2 4

xO xB xB xAxO xB xA

h hF

+ + = + = + +

1 sin; ;

1 sin xO c xA c

h

= = +

+

1 sin1 when 1.

tan 1 sin

zc z

z

c

+

219


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The exact value of a can be determined, proceeding from the principle of the displacement of

gravitating load platforms in accordance with (5). The largest of these values can be adopted for safety-

margin analyses.

For known F and a, the maximum bending moment Mmax

in the pile when loaded horizontally

Mmax

= ahQ. (7)

The displacement U of the pile head under a load Q can be determined with respect to the dis-placement of point B (see Fig. 4) due to the bending stresses acting on plane z = h /2 as compared with

their minimum value

(8)

which had previously been specified, where xB

is the stress at point B according to (4), x,min

is the pre-

viously effective stress corresponding to the minimum stress state in plane z = h /2, as determined from

(3) when z = h /2 and z

= 1, b is a dimension of the zone of propagation of the bending stress state in

plane z = h /2, and E is the compression modulus of the soil.The dimension b of the zone can be determined from the equilibrium condition of prism OBCD

(see Fig. 4)

(9)

where F0 is the resultant supporting force of the soil against face OB of prism OBCD, Fx,min is the resul-tant force of the minimal stress state on face CD of prism OBCD.

In sandy soils:

(10)

The displacement U of the pile head in conformity with (8), (9), and (10)

(11)

in sandy soils.

Using (5-7) and (11), it is therefore possible to evaluate both the maximum bending moment in

the piles when acted upon by a horizontal load Q, and also the displacements of their head section. For

this purpose, it is necessary to know the depth of the point at which the pile is fixed in the soil, and

also the support FK

of the soil in the lower section of the pile.

For absolutely rigid piles, the maximum possible support FK

of the soil can be determined by the

overall soil prism. The actual soil support in each specific case will be smaller, however, since it is real-

ized by the soil only when necessary.

221 sin

2 18 tan 1 sin

B

hU U

E

= = +

2

0 ,min

1 1 sin; .

2 2 2 1 sinxB x

h hF F

= = +

0 ,min

0 ,min

tan2

or ,

tan2

x

x

hF b c F

F Fb

hc

= + +

=+

,min,

2

xB x

BU bE

=

220


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For actual piles, soil support will be realized not in the region of the lower end, but over the

entire length of section KA of the pile (see Fig. 4). In evaluating the specific-pressure distribution of the

support along section KA, it is possible to state that its value will increase with depth from zero at point

A. If the increase is assumed to be linear, therefore, the resultant support FK

will be situated at a dis-

tance (H h)/3 from the lower end of the pile (where H is the depth to which the pile is embedded) on

the condition that the horizontal stress K

around the end of the pile will not exceed the maximum value

determined from (3).

Basic calculations associated with evaluation of the resistance of horizontally loaded piles can bereduced to solution (6) in conjunction with equations whereby the forces in the rotating section where

the pile rotates about point A are in equilibrium, and all forces are projected onto the x axis

(12)

Q F+ FK

= 0 . (13)

Solving Eqs. (6), (12), and (13) simultaneously for h, we have

(14)

for sandy soils when c = 0 and a 2/3.

The roots of this equation are readily determined from the Kardan formula with use of comput-

er programs, or by the simple method of determining Q based on assigned values of h.

For clayey soils (c 0), Eq. (5) should be used in lieu of (6) without causing principal difficul-

ties, but will make the calculations more cumbersome, although the procedure employed here for the

calculations is retained in its entirety.

Calculations based on the proposed procedure as applies to Figs. 1 and 2 suggest good conver-gence between computed and experimental data.

An approximated accounting of three-dimensional deformations of a soil medium as apples to cir-

cular piles can be achieved with use of Berezantsev's development [4] in which he demonstrates that in

the case of three-dimensional deformation, limiting equilibrium of a point of a soil medium with coordi-

nates coinciding with principal stresses 1,

2, and

3when

1>

2>

3sets in under the condition

(15)

where and c are the angle of internal friction and cohesion of the soil, respectively.

This condition is mandatory for the three-dimensional problem in any coordinate system.Incomplete limiting equilibrium is theoretically defined by (15), since in addition to this condi-

tion, it is possible to propose yet another two relationships between principal stresses; this is uncharac-

teristic when only a horizontal force acts on the pile.

The horizontal load on a circular pile can be calculated in a cylindrical coordinate system z, r,

u. Figure 5 shows the computational diagram of the forces acting on the pile, and the stress state of the

soil medium in the near-pile space.

According to this diagram, two slip surfaces, which run at angles += +(/4 /2) in the direc-

tion of the larger principal stress 1

exist at each point of the soil medium. Consequently, two families

1 3 1 21 tan ,cos 2 2

c

+ =

3 2

2 1 1 sin; 1 ;

2 4 1 sin

2 2 0

h HF Q

H h

h Hh Qh QH

+ = = + + + + =

2(1 ) ( ) 0;

3KQh Fh a F H h =

221


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of slip lines exist in the near-pile soil medium. If the angle formed by the direction of 1

and the r axis

is designated by , and the angle that is made by the same axis r and a slip line of the first family by, the stress components that develop in the soil medium could be expressed in terms of principal stress-

es 1,

2, and

3:

(16)

Solving these relationships simultaneously with (15) as applies to a granular medium when c = 0,we obtain:

r

= (1 + sin cos2);

z

= (1 sin cos2) ;

rz

= sin sin2 ;

= (1

3)/2sin . (17)

Disregarding the roughness of the lateral surface of the pile and considering that no forces, with

the exception of the weight of the soil itself, exist in the zone of soil support when the pile is horizon-

1 2 1 2

1 31 2

1 2

cos2 ;2 2

cos 2 ;2 2

sin 2 .2

r

z

ru

+ = +

+=

=

222

Fig. 5. Computational diagram of circular pile subjected to horizontalload with indication of active forces and stress state of soil mediumin near-pile space in cylindrical coordinate system z, r, u.


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tally loaded, it can be approximately assumed that z

= z. Considering this, Eq. (17) can be represented

in the following form:

(18)

Since a horizontal load acts on the pile, it is possible to assume that the directions of active

stress 1

are also horizontal in the vicinity of the lateral surface of the pile, and, consequently, = 0,

cos2= 1, and sin2= 0. The stress state of the soil on the lateral surface of the pile will therefore cor-

respond to the condition:

(19)

It is logical that at a certain distance from the axis of the pile, which is equal to r = r0

+ b0,

complete attenuation of horizontal displacement will be observed due to the soil's shear resistance. In

that case, only downward displacement of the soil is theoretically possible under its own weight, for

example, as a result of incomplete consolidation, which usually occurs due to minor additional pres-

sures. At a distance r = b, consequently, = 90o, cos2 = 1, and sin2 = 0, since the active forces

are vertical:

(20)

Intermediate r

values corresponding to values of from 0 to 90o

will exist in the section from

r = r0

to r = r0

+ b0.

The result obtained is expected, since the stresses on the lateral surface of a circular pile may

attain maximum limiting values in its upper displaced zone, and correspond to the minimum stress state

at a certain distance from the pile. Considering the stress state of the soil support F on the lateral sur-

face of the active zone of the pile when its cross section is circular,

(21)

where D = 2r0

is the diameter of the pile.

Comparison of (21) with (6) for solution of the problem under consideration in the plane state-

ment suggests their qualitative agreement. When used in calculations of plane solutions, however, this

comparison implies the need for the introduction of appropriate correction factors.

The relationships presented make it possible to account for nonlinearity of deformation phenom-

ena, and are based on use of actual standard strength and deformation characteristics , c, and E of

soils, and not on empirical coefficients as specified in regulatory documents.

02/ 4

0 0 0

1 sin 1 sincos ,

1 sin 2 1 sin

rh h DF h z dz dr d

+ += =

1 sin;

1 sin

; 0.

r

z rz

z

z

=

+= =

1 sin;

1 sin

; 0.

r

z rz

z

z

+=

= =

1 sin cos 2; ;

1 sin cos 2

sin sin 2.

1 sin cos 2

r z

rz

z z z

z z

+= = =

= =

223


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REFERENCES

1. V. G. Fedovskii, S. N. Levachev, S. V. Kurillo, and Yu. M. Kolesnikov, Piles in HydrotechnicalConstruction. Educational Textbook [in Russian], Izdatel'stvo ASV, Moscow (2003).

2. B. V. Bakholdin and E. V. Trufanova, Certain relative comparisons between analysis of horizontallyloaded piles and experimental data, Proceedings of the International Scientific-Technical Conference"Problems of Soil Mechanics and Foundation Engineering Under Complex Soil Conditions [in Russian],Vol. 1, Ufa (2006).

3. V. B. Shakhirev, Performance of horizontally loaded piles: author's abstract of dissertation for

Candidate of Technical Sciences, Moscow (1966).4. V. G. Berezantsev, Axisymmetric Problem of the Theory of Limiting Equilibrium of a Granular Medium

[in Russian], Gostekhizdat, Moscow (1952).

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