ultimate capacity of pile foundations

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1133ULTIMATE CAPACITY OF

PILE FOUNDATIONS

13.0 CAPACITY OF PILE FOUNDATIONS

In this chapter methods of estimating the ultimate capacity of single piles, both verticaland lateral, and the ultimate capacities, for vertical load, horizontal shear and moment,of pile groups are presented. Estimation of the vertical capacity of piles is limitedmainly to methods based on CPT data. There are also available first principles, orthe so-called geotechnical calculation, approaches which rely heavily on empiricalcorrelations (an interesting review of these methods is given by Burland (2012)).However, the many CPT methods available are also based on empiricism but, with the

wealth of information now available, it can be argued that these have a more robustempirical basis. One limitation on CPT based methods are ground profiles with gravels

rather than sand, silt and clay in these cases empirical methods based on the SPT areneeded.

13.1 Vertical capacity

13.1.1 Preliminary comments

It is usual to distinguish between piles in sands and piles in clays. This suggests thedistinction between short term and long term behaviour, in particular the difference in theresponse of a pile to a long term static load and that to a short term dynamic load such asearthquake or wind. This subdivision between the type of soil in which a pile is embeddedcan be further divided by the method of pile installation: driven or cast in situ.


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Design of Earthquake Resistant Foundations

258

Figure 13.1 Develop m ent w ith he ad set t lem ent o f sha f t an d b ase resistanc e for a

p ile w ith straigh t side s.

The design of a pile foundation requires that the base resistance and the shaft resistancebe evaluated. This can be done on the basis of some estimate of the soil type and soilproperties. The properties being determined either by taking specimens and laboratory

testing or by inference from some in situ test, most commonly a penetrometer test. Analternative is the use of the cone penetration test (CPT) for the direct assessment of pilecapacity in sands and silts; methods developed initially in Holland and Belgium fromabout the middle of the twentieth century.

The ultimate vertical load capacity of a pile is realised when all the shaft resistance and allthe base resistance have been mobilised. Load tests can be done in which the load carriedby the pile base is measured separately from the load carried by the shaft. A typical set ofresults is shown in Figure 13.1.

The important feature in Figure 13.1 is the very stiff response of the shaft resistance in

comparison with the relatively soft response of the base; this understanding is wellestablished having been measured many times. Typically the shaft resistance is mobilisedat vertical displacements of 1% per cent or so of the shaft diameter, whereas considerablylarger displacements, up to 10%, are needed to mobilise the base resistance. The relativecontributions of the base and shaft resistance to the ultimate vertical capacity of a pile inhomogeneous soil depend on the length to diameter ratio. If the pile is end bearing thebase resistance will be more significant. This will also be the case for a pile with anenlarged base, a so-called belled pile.

The ultimate capacity of a pile, Vu, is the sum of these two components less the pileweight:

u su bu= + - WV V V 13.1


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Chapter 13: Ultimate capacity of pile foundations

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where: Vsu is the ultimate shaft resistance,Vbu is the ultimate base resistance,

and W is the pile weight.

When evaluating pile base resistance by geotechnical calculation it is not

uncommon to neglect the weight of the pile and not to include the BNterm in the base bearing strength calculation.

13.1.2 Pile vertical capacity estimated from CPT profiles

The cone penetration test (CPT) was originally developed in Holland, but isnow widely used internationally not only for estimating pile capacities butalso as a site investigation tool. Bustamante and Gianeselli (1982) represent

an approach at correlation between pile capacity and CPT data which is stilloften referred to. Verbrugge(1986) gives a summary of usage in Hollandand Belgium of the CPT for estimating pile capacity in sands and silts.Eslami and Fellenius (1997) evaluate data from a large number of casehistories and conclude that data from the piezometric cone gives the bestestimate of pile capacity. Further examples of the CPT approach aredescribed briefly by Kempfert et al (2003). The Canadian FoundationEngineering Manual (Canadian Geotechnical Society (2007)), presentsmethods for predicting the base and shaft capacities of a range of pile typesin different soil types. These and many other papers indicate how the conepenetration test can be used to estimate pile capacity. In addition, there are

methods based on the Standard Penetration test.

Underlying all these approaches is the admission that the process by whicha pile achieves vertical capacity is complicated and involves several differentmechanisms. CPT probing can be viewed as a model for the installation ofa driven pile. This must be a reasonable starting point for the pile baseresistance, but we will see that it is a little more complex. However, pilecapacity also includes the contribution from the pile shaft; this is dependenton what happens to the soil adjacent to the shaft during the subsequent topile installation. Given these complications an empirical approach linkingCPT resistance to pile capacity is followed, but making some allowance for

the difference in size between the pile and CPT and for the effect ofsubsequent driving on the shaft resistance.

In this section we will concentrate on three recent approaches: CanadianFoundation Engineering Manual (2007), Randolph (2003) and Jardine et al(2005), all of which are based on CPT data.

13.1.2.1 Ca na dia n Found at ion Engine er ing Ma nua l (2007)

In this method there is a pair of simple equations linking pile baseresistance and shaft resistance with the CPT qc. Note that this method is

intended to cover both bored and driven piles. The equations are:


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Table 13.1 Fa c tor kc fo r p i le ver tica l c ap ac i ty for eq uat ion 13.2( f r om Canad ianFou nd a t ion Eng inee ring M a nua l (2007))

Table 13.2 Factor for pi le ver t ica l capaci ty for equat ion 13.3 ( from Ca nad ianFou nd a t ion Eng ine er ing Ma nua l (2007))

qc

qs (MPa)


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t c caq k q 13.2

cs

qq

13.3

where: qc is the cone penetration resistance corrected if necessary forpore pressure effects

qt is the unit base resistanceqs is the unit shaft resistance

qca is the average CPT resistance at the depth of the pile baseaveraged over a distance of 1.5 pile diameters above andbelow the pile tip

kc is a bearing capacity factor based on soil type and pile typegiven in Table 13.1

is a friction coefficient given in Table 13.2.

To estimate the vertical capacity of the pile one needs, first, to find the unitbase resistance, qt, this is then multiplied by the pile base area to give Vbu.

The shaft resistance is evaluated with depth along the pile shaft and thetotal shaft resistance, Vsu, obtained by integration of the shaft shear stressdistribution.

Belled piles

Belled piles are handled by evaluating the end resistance in the downwarddirection using the base area, and for estimates of uplift resistance the areaof the annulus defined by the diameters of the pile shaft and the enlargedbase.

13.1.2.2 Randolph (2003) driven piles in sand

The Rankine lecture of Professor Randolph reviewed recent research andpresented a procedure for estimating the capacity of driven piles in sand.

There are two things to consider the base resistance and shaft resistance.Randolph suggests that the base resistance be taken as 0.4 qca (averaged,over a depth of about 1 to 2 pile diameters above and below the pile tip(similar to the method given in the Canadian Foundation EngineeringManual). The reason for the use of 0.4qc is that the vertical capacity of thepile base will be regarded, from a design point of view, as having beendeveloped when the downward displacement is about 10% of the pile base

diameter. At this displacement the full penetration resistance of the soil willnot have been mobilised. This means that our idea of pile base failure is notactually the same as the CPT penetration resistance, despite the commentsmade above that a penetrometer is often thought of as a model of the pile.

Understanding of the mechanism of pile shaft resistance for driven piles isan important breakthrough which has occurred in the last decade or so.Consider some point at a depth z beneath the ground surface adjacent to apile shaft, Figure 13.2. During the driving process, when the pile tip is at adepth z, we can think of the conditions there being similar to those whenthe CPT cone is at that depth. Subsequently the pile shaft is driven some


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Figure 13.2 Definitions of p a ram ete rs for estim a ting sha ft resistanc e o f

dr iven p i les. (af te r Ja rd ine et a l (2005)) .

distance past this point. This means that soil immediately adjacent to thepile shaft at depth z continues to experience very severe shearing as the pile

penetrates further, the effect of which is accentuated by the cyclic nature ofthe pile driving process. The exact mechanism is a little complex but it is aninteraction between the lateral confining effect of the soil remote from thepile shaft (depends on the small strain soil modulus) and the volumedecrease of the soil adjacent to the pile shaft induced by repeated cyclicshearing during the driving. Measurements with specially instrumented pileshafts have confirmed this mechanism, which has been called frictionfatigue (White and Lehane (2004)).

Randolphs (2003) equation for estimation of the pile shaft shear stress is:

2

h/ Rs vo cv min max mintan K K K e 13.4

where: cv is the constant volume angle of shearing resistance of the soiladjacent to the pile shaft (determined by a ring shear test)

Kmax is typically 1 to 2% of qc/voKmin lies in the range 0.2 to 0.4.

is about 0.05 for typical pile diameters (this is related tothe pile shaft roughness).

Using equation 13.4 we can find the distribution of shear stress along the

pile shaft, integration then gives the pile shaft capacity.


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13.1.2.3 Jard ine et al (2005) estima tes of ver tica l pi le c ap ac i ty o f

d rive n piles.

A series of very well executed full scale, thoroughly instrumented, pile load

tests was the main impetus behind the development of the ICP (ImperialCollege Pile) method, Jardine et al (2005), for the design of driven piles.The approach developed has many similarities with the approach outlinedby Randolph in his 2003 Rankine lecture. The method covers driven pilesin silica sand and clay.

As with Randolph there are two basic mechanisms; one for the baseresistance and the other for the shaft resistance. The base resistance is afraction of the cone tip penetration resistance, justified with arguments thatare similar to but not identical to those of Randolph. The shaft resistancerecognises the effect of repeated shearing of the material adjacent to the

pile shaft leading to a progressive reduction of the shear stress between thepile shaft and the surrounding soil after the base of the pile passes a givenposition.

Driven piles in clay

The base capacity is given by (from Table 6 in Jardine et al (2005)):

0 8b caq . q Undrained loading 13.5

1 3b caq . q Drained loading 13.6

where: qca is the average CPT resistance near the pile tip defined as forequation 13.2.

Several steps are required to obtain the shaft shear stress (from Table 5 inJardine et al (2005)):

ff rc f

c

Ktan

K 13.7

where: Kf/Kc is known as the loading factor (= 0.8 regardless ofwhether the direction of loading or drainageconditions)

rc is the long term effective radial stress acting on thepile shaft

f is the angle of shearing resistance between the pileshaft and the adjacent clay (somewherebetween the peak and residual values).

0 42 0 20

10

2 2 0 016 0 870

rc c vo

. .c vy

vy t

K

and

K . . YSR . I YSR (h/R)

with

I log S

13.8


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Figure13.3 Va ria t ions w ith tim e o f the ra d ial effec t ive stress a g a inst the

sha fts of d rive n p iles in four cla y soil profi les. (a fter Jardine et a l(2005)).

and: St is the sensitivity of the clay

YSR is the yield stress ratio (=vy/vo) a ratio similar to theconventional overconsolidation ratio, but with the expectation thatthe yield stress is determined more accurately than is possible in theconventional oedometer.

The use of this yield stress ratio means that the application of this approachto residual clays is not clear as these materials do appear to have a yield

stress or preconsolidation pressure (cf sections 4.3.2 and 4.4.5).

After driving, the clay adjacent to the pile shaft consolidates. Some data forthis is shown in Figure 13.3. London clay is a highly overconsoldatedmaterial and is apparent from Figure 13.3 that the long term lateral effectivestress against the pile shaft is less than the in situ value for clays with anundrained shear strength of about 100 kPa or more. On the other hand forthe soft Bothkennar clay, low yield stress ratio, the lateral effective stressagainst the pile shaft is greater than the in situ lateral effective stress beforethe pile is installed. It is of note that the consolidation process comes tocompletion in a few days for all four clays.


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Driven piles in sand

The base capacity is given by (from Table 3 in Jardine et al (2005)):

1 0 5b ca CPTq q . log D/ D 13.9

where: D is the pile shaft diameter and DCPT is the diameter ofthe cone penetrometer (0.036 m).

A lower limit of 0.30qca is suggested for piles with D > 0.9 m.

Several steps are required to obtain the shaft shear stress (from Table 2 inJardine et al (2005)):

0 13 0 38

0 029

2

f rf cv

rf rc rd

. .

rc c vo a

rd

tan

with :

and :

. q /P h/R

and :

G r/R

13.10a

where: r is the radial expansion that must occur to allow the pileshaft to slip past a given level, about equal to the

average peak to trough roughness of the pile surfacePa is atmospheric pressureG is the shear modulus of the sand remote from the pile

shaft (independent of the direction of loadingor drainage conditions)

cv is the constant volume angle of shearing resistance ofthe sand adjacent to the pile shaft. It should bemeasured in interface shear tests.

G is obtained from the correlation of Baldi et al (1989):

12

0 5

6

0 0203

0 00125

1 216 10

c

.

c a vo

G q A B C

where :

q P

A .

B .

C . x

13.10b

For piles subject to tension loading:

0 8f rc rd cv . tan 13.10c


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Figure 13.4 Increa se w ith t im e of the shaf t ca pa c i ty for dr iven pi les in sand . (af ter

Jardine et a l (2005)).

After driving the shaft capacity of piles in sand has been observed to increase. Somedata on this are shown in Figure 13.4. It is apparent that over a period of about threeyears there is a three-fold increase in capacity.

13.1.3 Effect of cyclic vertical loading on pile vertical capacity

When designing pile foundations for earthquake resistance cyclic axial loading needs toconsidered. Under gravity loading the pile is expected to sustain some vertical load. Anearthquake superimposes on this a cyclic vertical load which increases and decreases theactions applied to the pile. Intuitively one would expect that this cyclic loading willdecrease ultimate vertical load capacity of the pile; laboratory and field testing confirm thatthis is the case. This effect is important not only for earthquake loading but also forfoundations of offshore structures and wind turbines, which may be subject to manythousands of cycles of loading. A typical earthquake can be expected to apply up to tensof cycles so one might expect that cyclic degradation would appear not to be so severe;however, it has been observed that the maximum rate of degradation of shaft capacity

occurs in the first few tens of cycles. Experimental and computational work on this hasbeen reported by, among others, Lee and Poulos (1993), Poulos (1981, 1988 and 1989),

Turner and Kulhawy (1990), and Abel-Raman and Achmus (2011).

In summary the effect of the cyclic loading leads to a decrease in pile capacity, a decreasein pile stiffness, and an accumulation of pile displacement. The effects are more severe

when the amplitude of the cyclic load is a larger fraction of the static pile capacity, and asthe static load applied to the pile increases. Poulos (1988) gives a diagrammaticrepresentation of these effects, which is shown here in Figure 13.5a. Notice that thePoulos diagram also includes the case where the pile is subject to static tensile load. Thestraight line boundaries give conditions which lead to failure under monotonic loading,

that is without cycling.


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The mechanism by which the cyclic loading decreases the pile capacity revolves aroundthe manner in which the interface between the pile shaft and surrounding soil responds tocyclic shear stress. Figure 3.31 gives some insight into the mechanism and shows howcyclic shearing at constant normal stress leads a reduction in volume. However, adjacent

to a pile shaft the boundary conditions are better modelled as constant normal stiffness, soduring cyclic shearing there is a reduction in the normal stress at the pile shaft interfaceand hence a reduction in frictional resistance. Many investigations, for example Ooi andCarter (1987) and Fakharian and Evgin (1997), have made of cyclic soil-shaft interfacefriction using constant normal stiffness laboratory test rigs. This information shows howsensitive the normal stress at the interface is to cyclic shear stress and normal stiffness.One surprising conclusion is that, as the relative density of the soil in which the pile isinstalled increases, the rate of reduction in normal effective stress is more severe. Thismeans that the mechanism which leads to the cyclic degradation in pile shaft capacity isthe same as that used by Randolph and Jardine et al for estimating the static shaft capacityof a driven pile in cohesionless soil.

Figure 13.5b gives some data of cyclic shaft capacity for driven piles in clay. Theamplitude of the cyclic load is denoted by Pc normalised with respect to the static loadcapacity of the pile, Qc. Looking at the data for 10 cycles it is apparent that thedegradation of pile axial capacity during cyclic loading will be a significant consideration inthe design of earthquake resistant pile foundations. The mean load Po, that is the staticload carried by the pile, is likely to be one third or less of the static capacity Qc. FromFigure 15.5.b we can see that failure would occur in 10 cycles if Pc/Qo is about 0.5. Poulosemphasises that the rate of degradation is very sensitive to the degradation properties forthe soil in which the pile is embedded for clays, sands and compressible sands. Theconclusion from these comments is that the cyclic axial capacity of piles requires careful

consideration when designing pile foundations to resist earthquake loads as the capacitiesobtained from section 13.1.2 are for static, not cyclic, loading.

Figure 13.5 (a) The Poulos stability diagram for cyclically loaded piles, (b) Cyclic loaddata for driven piles in clay (after Poulos (1988)).

a b


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Figure 13.6 Pile axial load cyclic data on 76 mm diameter model drilled shafts in drysand (after Turner and Kulhawy (1990)).

Figure 13.6 gives some data from Turner and Kulhawy (1990). This indicates theaccumulation of permanent displacement for the cyclic loading of laboratory models ofdrilled shafts in sand; again emphasising that this is an important effect even when the

amplitude of the cyclic force is small in relation to the static capacity.

13.2 Lateral capacity

13.2.1 Background

Now we need to evaluate pile head lateral load and moment combinations which willmobilise the ultimate resistance of the pile-soil system.

There are two possibilities for consideration which depend on the length of the pile.Firstly, in the case of a relatively long pile at some depth the ultimate moment of the pilesection will be reached and a plastic hinge formed. When this happens any further attemptto increase the lateral load will simply cause unlimited rotation at the plastic hinge. Thusthe ultimate capacity of a long pile is limited by the moment capacity of the pile section.For a reinforced concrete pile this is affected by the axial load carried by the pile.

Secondly, in the case of a short pile the depth of local failure along the pile shaft reachesthe pile depth before the ultimate moment capacity of the pile section is reached. In thiscase the ultimate lateral capacity of the system is determined by the soil properties. Todistinguish the two cases herein the second is sometimes referred to as a pole rather than apile.


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Figure 13.7 Ultima te p ressure d ist ribut ion ag ainst a lateral ly load ed pi le in c ohe sive

soi l af ter Brom s; (a) free hea d a nd (b ) fixed h ea d p i le.

Figure13.8 Ult imate pressure distr ibution against a lateral ly loaded pile in

c ohe sionless soi l af ter Brom s; (a ) free hea d a nd (b) f ixed he ad .

The above discussion is in terms of a horizontal shear being applied at the top of the pile.No mention has been made of any constraint so this applies to a free head pile. Thepossibility that the pile head is restrained against rotation must also be considered. Headfixity increases the ultimate lateral capacity of a pile as a negative fixing moment equal tothe yield moment of the pile section must be mobilised to generate the ultimate capacity.

In the case of a short pole shear failure of the ground adjacent occurs before the plasticmoment of the pile section is reached. The transition between a pole and a pile occurs forthe length at which the fixing moment reaches the plastic moment of the section at thesame load as the capacity of the soil is reached. For a long pile the ultimate capacity of thesystem is not reached until the head moment as well as the maximum moment along thepile shaft reach the yield moment of the pile section.

13.2.2 The Broms and Budhu and Davies methods

The best known approach to estimating the ultimate lateral capacity of a pile is that ofBroms (1968a and b). Broms distinguishes between piles in cohesive soils and those in

cohesionless soils. In each case he proposes a convenient simple method of estimatingthe maximum lateral pressure that the soil can mobilise. The approach is intended to

Hu

plastic

hingesfc

"elastic"soil

bending momensoil reaction

9suD

eo

MyieldMyield

b

Hu

plastic

hinge

e

fc

Myield

M =HueD

soil atfailure

"elastic"soil

bending momentsoil reaction

9suD

eo

a

Hu

plastic

hinge

e

fs

Myield

M =HueD

soil atfailure

"elastic"soil

z


3KpD v

a

Hu

plastic

hingesfs

"elastic"soil


3KpD v

Myield

Myield

b

g not e for these two diagrams


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account for three dimensional effects. The capacity of a given pile can be assessed fromthe charts given in the Broms papers. However Budhu and Davies (1986 and 1987) andBudhu and Davies (1988) have developed simple equations which cover these cases.

Piles in cohesive soilsBroms' proposed pressure distribution for a pile in cohesive soil is given in Figure 13.7 forthe free head and fixed head cases. Brom's recognizes that the actual distribution of lateralpressure against the pile shaft will be complex when all the soil pressure is mobilised, hesimplifies this by assuming that there will be zero pressure for distance of 1.5 diametersfrom the ground surface below which it is 9suD; the notation for this distance is eo (Figure13.7). Broms gives a chart for estimating the lateral pile capacity based on the soil reactiondistribution given in Figure 13.7.

Davies and Budhu (1986) also give expressions which give the ultimate lateral capacity ofpiles. We discuss only those for long piles here. The ultimate lateral capacity of a free head

pile embedded in saturated clay is given by:

2 0.52cu u f)= (2 + 100 - 10f nsH D 13.11

where: f is e/D, with e = M/H + eo + g.

andy

c 3u

10M=n

s D13.12

The position of the yield moment (and the length of pile shaft over which failure occurs)is given by:

uoc

u

H e= +f

9 Ds

13.13

For the fixed head case the ultimate lateral capacity will be given by:

2 0.5uu c= 2sH D n 13.14

The above four equations are based on the same assumptions as Broms with theexception the eo for Budhu and Davies is 0.6 m rather than 1.5D for Broms.

The length of pile shaft required for this solution to be valid is:

0.5ceff c n= 0.4DL 13.15

The above equations give essentially the same prediction for the ultimate lateral capacity asthose of Broms (1964a).

_____________________________________________________

Example 13.1 Consider a reinforced concrete pile 0.75 m in diameter and 20 m inlength. Estimate the ultimate lateral resistance of a pile assuming that the ratio of theapplied ground level pile head actions, horizontal shear to applied moment, is 2.3, that theyield moment of the pile section is 1575 kNm, and that the undrained shear strength of

the soil is 50 kPa.


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e = 1/2.3 = 0.43.

To this we add the 0.6 m that Budhu and Davies recommend.

Thus e = 1.03 m and f = 1.03/0.75 = 1.37.

nc = 10x1575/50x0.753 = 746.7

The effective length of the pile shaft for the valid application of equation 13.9 is:

Lc = 0.4x0.75x746.70.5 = 8.2 m, thus length OK.

Hu = 50x0.752(41.0 - 10x1.37) = 768 kN ((2x746.7 + 100x1.372)0.5 = 41.0 )

The depth to the maximum moment is:

fc = 768/9x50x0.75 + 0.6 = 2.88 m._____________________________________________________

Budhu and Davies (1988) also give a set of equations for ultimate lateral capacity of pilesembedded in normally consolidated clay which has a linear increase in undrained shearstrength with depth, a case not covered by Broms. The ultimate lateral capacity of a freehead pile embedded in normally consolidated saturated clay is:

3 0.67 0.75 -0.25l lu

( )exp= 0.5c -2D n f nH 13.16

where: c is the rate of increase in undrained shear strength with depth (kPa/m).

The ratio nl is defined by:

yl 4

M10=n

cD13.17


ul 2H=f 9cD 13.18

For the fixed head case the ultimate lateral capacity is given by:

3 0.67lu = 0.8c D nH 13.19

Piles in cohesionless soils

Broms' proposed pressure distribution for a pile in cohesionless soil is given in Figure 13.8for both the free head and fixed head cases. Brom's recognizes that the actual distributionof lateral pressure against the pile shaft will be complex and of a three dimensional nature

when all the soil pressure is mobilised, he simplifies this by assuming that the soil reactionalong the pile shaft is controlled by 3Kp as shown in the diagram. Broms gives a chart for


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estimating the lateral pile capacity based on the soil reaction distribution given in Figure13.8.

The Budhu and Davies (1987) expressions for the ultimate lateral capacity of a free head

pile embedded in cohesionless soil are:

3 0.67 0.75 -0.25u p s s= 0.35 exp(-1.6 )H K D n f n 13.20

where: Kp is the coefficient of passive earth pressure = (1 + sin)/(1 - sin)and f is M/HD.

The ratio ns is defined by:

ys 4

p

M10=n

K D13.21


us

p

H2=f

3 DK13.22

where: is the unit weight of the sand chosen to give the effective verticalstresses ( for a saturated sand).

For the fixed head case the ultimate lateral capacity will be given by:

3 0.67p cu = K D n0.56H 13.23

0.33eff s = 0.8DnL 13.24

The length of pile shaft required for this solution to be valid is:

The above equations give essentially the same predictions of pile lateral capacity as thoseof Broms (1964b).

____________________________________________________Example 13.2 Consider a reinforced concrete pile 0.75 m in diameter and 20 m inlength. Assume that the ratio of the applied pile head actions, horizontal shear to appliedmoment, is 2.3, that the yield moment of the pile section is 1575 kNm, and that the pile isembedded in saturated sand with an angle of shearing resistance of 35 degrees.

e = 1/2.3 = 0.43 and f = 0.43/0.75 = 0.57.

Kp = (1 + sin35)/(1 - sin35) = 3.69

ns = 10x1575/3.69x10x0.754 = 1349

The effective length of the pile shaft for the valid application of equation 13.18 is:


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Lc = 0.8x0.75x13490.33 = 6.5 m, thus length OK.

(-1.6x0.570.75x1349-0.25) = -0.17 exp(-0.17) = 0.84

Hu = 0.35x13490.67x3.69x10x0.753x0.84 = 572 kN

The depth to the maximum moment is:

fc = (2x572/3x3.69x10x0.75)0.5 = 3.71 m._____________________________________________________

13.3 Pile group capacity

13.3.1 Vertical capacity of a pile group

There are two cases: a free standing group, which has a cap having no interaction with theunderlying soil, and the pile raft in which the group cap is in contact with the underlyingmaterial. The pile raft case is not considered herein.

There are two modes of failure for a group. When there are a large number of closelyspaced piles the soil is so heavily reinforced that it fails as a block. With fewer more

widely spaced piles the capacity of the group is determined by the individual pilecapacities. The concept of group efficiency arises; Poulos and Davis (1980) discuss anumber of ways in which the efficiency is defined. Their preferred definition requires the

evaluation of the ultimate capacity of the block of soil reinforced by the pile groupassuming that it acts as a unit:

G G G GB u c a= + 2( + )LV s N cB L B L 13.25

where: VB is the ultimate capacity of the block,LG is the length (between pile extremities) of the pile group,BG is the width (between pile extremities) of the pile group,L is the length of the piles,Nc is the deep foundation bearing strength factor (about 9.0 rather than 5.14

for a shallow foundation),

and ca is the adhesion along the block boundary.

A similar expression could be developed for an effective stress analysis.

The group efficiency is defined as:

ult . capacity of group

=sum of ult . caps. of individual piles

13.26

Poulos and Davis suggest the following formula for obtaining the ultimate capacity of thegroup:


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2 2 22G 1 B

1 1 1= +

V V Vn13.27

where: VG is the ultimate capacity of the group,

V1 is the ultimate capacity of the individual piles,and n is the number of piles in the group.

The group efficiency is then obtained from:

221

2 2B

1 Vn= 1 +

V13.28

Note that using this approach of Poulos and Davis that the concept of group efficiency is

not needed. They introduce it as other more traditional methods need to evaluate beforethe group capacity can be evaluated.

Model tests of pile groups in sand show that the group efficiency may be greater thanunity, Poulos and Davis (1980, p. 36-37).

13.3.2 Combined load capacity of a pile group

We have considered the ultimate vertical capacity of a pile group now we need to look atthe capacity of a group when subject of vertical, shear and moment loading. The challengefor this is to find a way in which the various mechanisms can contribute to the overallgroup capacity without overlooking some mechanism that gives a smaller capacity thanthe one chosen. The complication is that the vertical capacity of the piles contributes to

both the vertical and moment capacity of the group and that the piles also contribute tothe shear capacity of the group. Thus we need to find some way of partitioning the waythe capacity of the piles contributes to the various mechanisms.

Figure 13.9 Free b od y dia gram for a 2 x 2 pi le free -stand ing free-he ad group .

M

V

H

2(H/4)2(H/4)

[2(V/4) - 2(M/2s)] [2(V/4) +2(M/2s)]

s


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In Figure 13.9 free body diagrams show how the various applied loads might be splitbetween the various mechanisms. The free body diagram for the free-standing free-headgroup is clear enough. For the fixed-head free-standing case the diagram is very similarexcept that the yield moment, My, of the pile section is mobilised at the top of each pile,

this in effect means that the moment applied to the pile group is: M - 4M y(in the generalcase the factor 4 is replaced by n, where n is the number of piles in the group). In both ofthese mechanisms the pile shafts contribute to the moment resistance, the vertical loadresistance as well as the shear capacity of the mechanism. Thus we need to place somerestriction on the way the soil along the pile shaft contributes to these variousmechanisms. It is proposed that no element of soil can contribute resistance to more than onemechanism. In the case of the horizontal shear this means that a certain length of the pileshaft should be dedicated to providing horizontal resistance and so it makes nocontribution to the vertical and moment resistance. Referring to section 13.2.2 we haveexpressions (equations 13.7 and 13.18) for the length of pile shaft that is required todevelop the ultimate lateral pile capacity.

Note from equations 13.5, 13.10 and 13.14 that the ultimate lateral capacity of the pile is afunction of the yield moment of the pile section. This means that there will be somepossibility for adjusting the relative contributions of the various mechanisms by adjustingthe pile section.

Moment capacity of pile groups

Firstly we need to consider the vertical capacity of the individual piles. We will assumethat they are embedded in a cohesive soil and that a short term total stress assessment isrequired. The ultimate downward capacity is denoted as VD and the uplift capacity as Vu.

These are given, using the equations 13.1 to 13.4, by:

bD a u= LC + 9V c s A 13.29

where: Ab is the area of the pile base.

b sU a u= LC + 9 ( - )V c s A A 13.30

where: As is the cross-sectional area of the pile shaft.

As the length of the piles increases these two capacities will tend to converge if the soil hasuniform properties. On the other hand if the pile base is founded on a much stronger

layer the two vertical capacities will be substantially different.

For the 2x2 pile group as illustrated in Figure 13.9 we need to consider two limiting cases -the applied moment acting parallel to the side of the group and across the diagonal of thegroup. We will consider the moment parallel case first. From Figure 13.10 we note thatthere are three possible capacity mechanisms available for the group when the moment isapplied about an axis parallel to the side of the group. We will look at each of these inturn.


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Figure13.10 Mom ent resistanc e m ec ha nism s for a 2x2 free -he ad free -stan ding pi le

g roup .

Mechanism 1. Moment and force equilibrium give:

1 22

2

cap p D

L D

sM sV V

VR V


2

2

cap p D U

D U

M s V V

V V V

Note that this mechanism is available only for one value of V.


3 22

2

cap p u

R D

sM sV V

VR V

These moment relationships are plotted in Figure 13.11, with a little manipulation it isfound that Mcap(1p) = Mcap(3p) when V = 18suAs.

From Figure 13.10 there are three additional capacity mechanisms when the moment isapplied about a diagonal axis. As above we examine each of these mechanisms in turn.


4 42

3

cap d D

L D

sM V V

R V V

MV

MV

MV

1

2

3

s

s

RLx2VDx2

VUx2 VDx2

VUx2 RRx2

Pointof rotation

MV

s

s

s/ 2

4

5

6

RL VDx2 VD

VU

VU VUx2 RR

RCx2 VD


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277

Figure 13.11 Com bined ver tical load and m om ent c ap ac i ty for a free-stand ing free-

head 2x2 pi le g roup.


52

2

cap d D U

D U

C

sM V V

V V VR


6 42

3

cap d U

R U

sM V V

R V V

The moment relationships for mechanisms 4, 5 and 6 are plotted in Figure 13.11 alongwith those for mechanisms 1, 2 and 3. If we ask what value of V gives Mcap(4d) = Mcap(6d)we get V = 18suAs, which is the same value as that obtained for the equality of mechanism1 and 3. Figure 13.9 shows that the diagonal moment case Mcap(5d) gives the smallestmoment capacity for values of V up to about 50% of the maximum vertical capacity ofthe pile group, 4VD, and for larger values of V it is Mcap(1p) that is critical. In practice thismeans that Mcap(5d) will control design as one would not expect to have a 2x2 pile groupunder static conditions carrying a vertical load as great as 50% of the ultimate verticalcapacity of the group.

V

4VD4VU

4

6

3

5

1

2

vertical

failure

uplift failure

2VD

18suAs

[4VD - 1.4(VD +VU)]

M/s


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All these moment capacity relations are plotted in Figure 13.11. It is apparent that thediagonal case Mcap5d gives the critical moment capacity for the likely range of verticalloads.

Similar calculations can be done for the ultimate moment capacity of a 3x3 group. It isfound that once again the diagonal mechanism is critical but that the range of verticalloads for which it is critical is increased. If the dimensions of the pile cap are the samethen the pile spacing for a 3x3 group will be about half that of a 2x2 group. This wouldgive the 3x3 group about 40 to 50% more ultimate moment capacity than the 2x2 groupfor the same size of pile. On the other hand the efficiency of each pile in a 3x3 group willbe smaller, so the difference between the two will be less.

Free-standing fixed head groups

The free-standing fixed-head pile group is very similar to the above case except that themoment equilibrated by asymmetric axial loading of the piles is M - nMy, n is the number

of piles in the group.Lateral capacity of pile groups

The lateral load capacity of a group is developed, as explained at the beginning of section13.3, by assigning the length of the pile shaft given in equations 13.12 and 13.16 toproviding lateral resistance only. This in effect reduces VD and VU for each pile.

_____________________________________________________

Example 13.3 A simple "building" is shown below. Investigate the ultimate capacity ofthe pile group foundation. Take the yield moment of the pile shafts to be 1575 kNm.

Assume that the shaft shear stress assessed from CPT test data is 75% of the undrained

shear strength of the clay. Use 25 kN/m3 for the unit weight of concrete.

The weight of the structure (assuming the roof loading is equivalent to that of a floor) is:

W = 6x8x8.52 = 3468 kN.

The seismic mass is 247.5 tonnes. The spectral acceleration was 0.45g so the base shear is1093 kN. (These details come from a related example in Chapter 20.)

The base moment is generated by the concentrated mass acting at 0.7 of the buildingheight. Thus:

M = 0.7x(5x3.5)x1093 = 13389 kNm.

We will evaluate the ultimate lateral capacity of the pile group first. For a fixed head pilewe have from equation 13.12:

nc = 10x1575/50x0.753 = 746.7

From equation 13.14 the ultimate lateral capacity is:

Hu = 2x50x0.752x(746.5)0.5 = 1537 kN


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Details of the building supported by a 2x2 pile group foundation (more details givenin example 20.3)

The length of pile shaft required to generate this, from equation 13.13, is:

fc = 1537/9x50x0.75 + 0.6 = 5.15 m

Note from example 15.2 that a lateral force of about 900 kN generated the yield momentin the pile shaft. Thus we will restrict our ultimate lateral capacity to 900 kN, rather than1537 kN, but we will dedicate 5.15 m of each pile shaft to generating this with theconsequence that the length of pile shaft available for axial loading is 14.85 m.

This gives an ultimate lateral capacity for the pile group of 3600 kN (assuming no groupaction between the piles as the pile spacing, 7.5 m, is rather greater than the length of pileshaft required to generate the lateral capacity). Thus the lateral capacity of the group iseasily able to accommodate the applied base shear of 1093 kN.

The weight of each pile is (length 20 m from example 18.3)

5 floors @ 3.5m

2x2 fixed-head free-standing pile group

piles 20 mlong by0.75 mdiameter

pile spacing 10 diameters

20 m

0.75 m

8.5 m

loading 8 kN/m2

clay: su =50 kPa, Es =500su

Ep =25 GPa

s =10D =7.5 m

s=10D

2 3

1

4

=0o

=45o

=90o

8.5 m


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258

Wp = 25x20x3.1416x0.752/4 = 221 kN

The downward vertical capacity of each pile is:

Vud = 50x0.75x14.85x3.1416x0.75 + (9x50 + 18x20)x3.1416x0.752

/4 221= 1448 kN

The upward vertical capacity of each pile is:

Vud = 50x0.75x14.85x3.1416x0.75 + 221 = 1533 kN

The moment capacity of the group for the diagonal mechanism is Mcap5d is:

Mu = 7.5x(1448 + 1533)/1.414 = 15811 kNm.

This moment capacity is generated by axial force increments in two diagonally oppositepiles. Under the applied moment the force increments are:

F = 13389/1.414x7.5 = 1262.5 kN.

Assuming that each pile in the group continues to carry the one quarter of the weight ofthe building the forces in the diagonally opposite piles are:

F1 = 3468/4 + 1262.5 = 2129.5 kN

F2 = 3468/4 - 1262.5 = -395.5 kN.

Thus one pile would be subject to a tensile force at the head. For the other a verticaldownward force of 2130 kN is carried. However this downward force is greater than thedownward capacity of 1448 kN estimated above for the reduced pile shaft length (even

with the full 20 m of pile shaft active the downward capacity is still less than the forcegenerated). Thus the pile group does not have capacity matching or exceeding the appliedactions. As there will be cyclic axial loading of the piles the realizable shaft adhesion islikely to be less than the 0.75su assumed. This suggests that the free-standing pile groupfoundation might exhibit the plunging failure mode inferred by Zeevaert (1991) to haveoccurred during the 1985 Mexico City earthquake. As discussed in section 2.1.2Zeevaert's suggestion for avoiding this problem is to have a pile-raft foundation andmobilise the soil bearing capacity beneath the structure.

As an alternative to the pile-raft solution the pile shafts could be belled to provide greaterbase area and hence greater vertical capacity.

_____________________________________________________


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References

Abdel-Raman, K. and Achmus, M. (2011) Behaviour of foundation piles for offshorewind energy plants under axial cyclic loading. Proceedings of the 2011 SIMULIACustomer Conference.

Broms, B. B. (1964a) "Lateral resistance of piles in cohesive soils", Proc. ASCE, Jnl. SoilMech. and Found. Div., Vol. 90 SM3, pp. 27-63.

Broms, B. B. (1964b) "Lateral resistance of piles in cohesionless soils", Proc. ASCE, Jnl.Soil Mech. and Found. Div., Vol. 90 SM3, pp. 123-156.

Budhu, M. and Davies, T. G. (1987) "Nonlinear analysis of laterally loaded piles incohesionless soils", Canadian Geotechnical Journal, Vol. 24, pp. 289-296.

Budhu, M. and Davies, T. G. (1988) "Analysis of laterally loaded piles in soft clays", Jnl.Geotech. Eng., Proc. ASCE, Vol. 114 No. 1, pp. 21-39.

Burland, J. B. (2012) Behaviour of single piles under vertical loads. Chapter 22, ICEManual of Geotechnical Engineering, pp. 231-245. ICE.

Bustamante, M. and Gianeselli, L. (1982) Pile bearing capacity by means of staticpenetrometer CPT, Proceedings of the Second European Symposium onPenetration Testing, Amsterdam, pp. 493-499.

Canadian Geotechnical Society (2007) Canadian Foundation Engineering Manual, 4th edition,pp. 269-272. Bitech, Richmond, British Columbia.

Davies, T. G. and Budhu, M. (1986) "Nonlinear analysis of laterally loaded piles inheavily overconsolidated clays", Geotechnique Vol. 36 No. 4, pp. 527-538.

Eslami, A. and Fellenius, B. H. (1997) Pile capacity by direct CPT and CPTu methodsapplied to 102 case histories, Canadian Geotechnical Journal, Vol. 34, pp. 886 904.

Fakharian, K. and Evgin, E. (1997) Cyclic simple shear behaviour of sand-steel interfacesunder constant normal stiffness condition. Journal of Geotechnical andGeoenvironmental Engineering, 123 (12), pp. 1096-1105.

Jardine, R., Chow, F., OVery, R and Standing, J (2005) ICP design methods for driven piles insands and clay, Thomas Telford, London.

Kempfert, H-G., Eigenbrod, K. D. and Smoltczyk, U. (2003) Pile foundations in:Geotechnical Engineering Handbook, Vol. 3, pp. 84 227. Ernst & Sohn, Berlin.

Lee, C. Y. and Poulos, H. P. (1993) Cyclic analysis of axially loaded piles in calcareoussoils, Canadian Geotechnical Journal, 30, pp. 82-95.

Ooi, L. H. and Carter, J. P. (1987) A constant normal stiffness direct shear device forstatic and cyclic loading, Geotechnical Testing Journal, 10(1), pp. 3-12.

Poulos, H. G. and Davis, E. H. (1980) Pile Foundation Analysis and Design, Wiley.Poulos, H. P. (1981) Cyclic axial response of a single pile. Journal of Geotechnical

Engineering, 107 (GT1), pp. 41-58.Poulos, H. G. (1982) "Single pile response to cyclic lateral load", Jnl. Geotech. Eng., Vol.

108 GT 3, pp. 355-377.Poulos, H. G. (1988) "Some recent developments in pile design and determination of

design parameters", Proc. NZ Geomechanics Society Symposium on PileFoundations for Engineering Structures, Hamilton, pp. 1-17.

Poulos, H. G. (1988) "Cyclic stability diagram for axially loaded piles", Jnl. Geotech.Eng., Vol. 114 No. 8, pp. 877-897.

Poulos, H. G. (1989) "Cyclic axial loading analysis of piles in sand", Jnl. Geotech. Eng.,Vol. 115 No. 6, pp. 836-852.

Randolph, M. F. (2003) Science and empiricism in pile foundation design,Geotechnique, Vol. 53, No. 10, pp. 847 877.


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Turner, J. P. and Kulhawy, F. H. (1990) "Drained uplift capacity of drilled shafts underrepeated axial loading", Jnl. Geotech. Eng., Vol. 116 No. 3, pp. 470-491.

Verbrugge, J. C. (1986), "Pile foundations design using CPT results", StructuralEngineering Practice, Vol. 3, pp. 93-112.

White, D. J. and Lehane, B. M. (2004) Friction fatigue on displacement piles in sand,Geotechnique, Vol. 54, No. 10, pp. 645 658.

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