piglet mn

PIGLET

ANALYSIS AND DESIGN OF

PILE GROUPS

M. F. RANDOLPH

OCTOBER 1996

PIGLET

ANALYSIS AND DESIGN OF

PILE GROUPS

M. F. RANDOLPH

OCTOBER 1996

The accuracy of this program has been checked over a period of years, and it is believed that,within the limitations of the analytical model, results obtained with the program are correct.However, the author accepts no responsibility for the relevance of the results to a particularengineering problem.

Technical support in relation to operation of the program, or in respect of engineeringassistance, may be obtained from the author, who reserves the right to make a charge for suchassistance.

Contact Address: Department of Civil and Resource Engineering,The University of Western Australia,Nedlands,Western Australia 6907.

Telephone: +61 8 9380 3075Facsimile: +61 8 9380 1044Email: [email protected]

CONTENTS

Page No.

PART A: GENERAL DESCRIPTION

1. INTRODUCTION 1

2. IDEALISATION OF SOIL PROPERTIES 1

3. RESPONSE OF PILES TO AXIAL LOADING 3

3.1 Solution for single axially loaded piles 3

3.2 Extension of solution to pile groups 5

4. RESPONSE OF PILES TO TORSIONAL LOADING 6

5. RESPONSE OF PILES TO LATERAL LOADING 8

5.1 Deformation of single laterally loaded piles 8

5.2 Interaction between laterally loaded piles 10

6. ANALYSIS OF PILE GROUP 11

6.1 Treatment of raking piles 11

6.2 Allowance for free-standing length of piles 12

7. EXAMPLE APPLICATION 13

PART B: PROGRAM DOCUMENTATION

8. STRUCTURE OF PROGRAM 15

9. PROGRAM INPUT 15

9.1 Interactive data input and editing 15

9.2 Data items 18

10. PROGRAM OUTPUT 22

REFERENCES 23

FIGURE TITLES 25

FIGURES 26

APPENDIX 1 - EXAMPLE OUTPUT

OCTOBER 1996 PIGLET MANUAL M.F. RANDOLPH

1

PART A: GENERAL DESCRIPTION

1. INTRODUCTION

The computer program, PIGLET, analyses the load deformation response of pile groups undergeneral loading conditions. The program is based on a number of approximate, but compact,solutions for the response of single piles to axial, torsional and lateral loading, with dueallowance made for the effects of interaction between piles in the group. In these solutions,the soil is modelled as a linear elastic material, with a stiffness which varies linearly withdepth. No check of the overall stability of the pile group is made within the program; suchcalculations should form a separate part of the design.

The program has developed gradually over the last ten years, with the doctoral research of theauthor (Randolph, 1977) forming the basis for the original version. This manual is the fifthrevision of a report describing the program, originally published in 1980. The current manualis based on the version of PIGLET dated October, 1996.

In order to minimise the amount of computation required, three separate 'scopes' of analysisare identified, depending on the type of loading to be applied to the group. The three casesare:

(a) vertical loading only;(b) vertical and horizontal loading in a single plane;(c) general three-dimensional loading, including torsion.For the latter two cases, the pile group is assumed capped by a rigid pile cap, with the pileseither pinned or built-in to the cap. In the first case, the user may also specify a fully flexiblepile cap.

The pile cap is assumed always to be clear of the ground surface, with no direct transfer ofload to the ground. A non-zero 'free-standing' length of pile may be included between the pilecap and the effective ground surface. The only major geometric limitation on the pile grouplayout is that all the piles are assumed to be of the same length. This limitation arises out ofthe manner in which the axial load deformation response of the pile is calculated.

2. IDEALISATION OF SOIL PROPERTIES

Soil is by nature non-linear in its stress-strain behaviour, even at low stress levels. Thisnon-linearity may be modelled in an approximate fashion for the analysis of single piles bythe use of load transfer methods of analysis, where the soil continuum is replaced by a seriesof springs acting along the length of the pile. Extension of such analysis to pile groups is only


2

possible by adopting a hybrid soil model, combining elastic interactive effects with the loadtransfer analysis of each single pile (O'Neill et al, 1977). This approach is computationallylaborious and is limited by the inconsistency of the approach.

In most applications, it is sufficient to adopt a linear elastic model for the soil for calculatingdeformations and load distributions among piles in a group, under working load conditions.Independent checks should be performed to ensure that the elastic assumption is reasonable atthe load and deformation levels determined. For pile groups of practical size (in terms ofnumber of piles) additional deformation due to interactive effects will generally dominate thatdue to non-linear effects. A possible exception to this is where significant plastic deformationoccurs between pile and soil.

Although the soil has been assumed to deform elastically, less restriction has been imposed onthe relative homogeneity of the soil deposit. It has been assumed that the soil may bemodelled by a material where the stiffness varies linearly with depth. While this does notallow layered soil profiles to be treated rigorously, such deposits may be analysed bychoosing a suitable average stiffness for the strata penetrated by the piles, and adopting alinear variation of stiffness with depth that reflects the general trend present in the actualprofile. In addition, the special case of end-bearing (or partially end-bearing) piles has beencatered for by the inclusion of a facility for specifying a soil of increased stiffness below thelevel of the pile bases.

In summary, the soil is idealised as an elastic material where the stiffness varies as shown inFigure 1. The stiffness is characterised by a shear modulus, G, and Poisson's ratio, , (notingthat the shear modulus is related to the Young's modulus, E, by E = 2(1 + )G ). Theproperties which need to be specified are:

(a) the value of shear modulus at the ground surface, Go.(b) the rate of increase of shear modulus with depth, m = dG/dz;(c) the value of shear modulus at the pile base, Gb.(d) Poisson's ratio for the soil, , assumed constant with depth.Treatment of the axial and lateral response of the piles independently allows additionalfreedom when choosing soil properties. For the majority of piles used in practice,deformation under lateral load occurs only in the upper part of the pile. Because of this, andto allow for the high strains which occur locally near the head of a laterally loaded pile, it isadvantageous to be able to specify different soil properties for the analysis of the lateral loaddeformation response. In particular, it is often advisable to adopt a value of zero for the shearmodulus at ground level, Go, when considering lateral loading. In the program, the same soilproperties are assumed for axial and torsional loading, while different values of shear modulusmay be specified for lateral loading (retaining the same value of Poisson's ratio).


3

3. RESPONSE OF PILES TO AXIAL LOADING

3.1 Solution for single axially loaded piles

An approximate closed form solution for single axially loaded piles has been described indetail for floating piles by Randolph and Wroth (1978a), and extended to end-bearing piles byRandolph and Wroth (1978b). The solution is based on the technique of treating loadtransferred from the pile shaft separately from that at the pile base. The soil is effectivelyconsidered in two layers, divided by an imaginary line drawn at the level of the pile base (seeFigure 2). The upper layer, above the line AB, is considered to be deformed solely by theshear stresses acting down the pile shaft, while the lower layer is deformed by the loadtransmitted to the pile base. Some interaction will occur between the upper and lower layers,which will serve to limit the radial extent of the deformation in the upper layer. To illustratethe method of analysis, the solution for a rigid pile will be developed here.

The load settlement ratio for the pile base is obtained directly from the Boussinesq solution as

PbGbrbwb

=

41

(1)

where P is the load, w the settlement and r the pile radius, the subscript b referring to the pilebase.

Turning to the pile shaft, considerations of vertical equilibrium entail that the shear stress, ,at any depth falls off inversely with the radius, r, as (Cooke, 1974; Baguelin et al, 1975;Frank, 1974)

=oro

r(2)

where the subscript o denotes conditions at the pile shaft. This equation may be integrated togive the vertical deformation at any radius. In particular, if it is assumed that there is someradius, rm, at which the vertical deformations are effectively zero, then the settlement of thepile shaft may be written

ws = oroG (3)

where = ln(rm/ro).This equation may be combined with equation (1) to give the overall load settlement ratio fora rigid pile of


4

P tG rowt

=

41 ( ) +

2pi

ro(4)

where = rb/ro is the ratio of underream, = G /Gb is the ratio of endbearing, and thesubscript t denotes conditions at the top of the pile.

Development of the full solution, which takes account of compression of the pile is given indetail by Randolph and Wroth (1978a). Effectively, equation (3) is taken to act at each pointdown the length of the pile, much as in a linear load transfer analysis. The final expressionfor the load settlement ratio is

P tG

rowt=

41 ( ) +

2pi

tanh ( )

ro

1+ 4pi 1 ( )

tanh ( )

ro

(5)

where, summarising the various dimensionless parameters:

= rb/ro (ratio of underream for underreamed piles) = G

/Gb (ratio of end-bearing for end-bearing piles) = G /G (variation of soil modulus with depth) = Ep/G (pile-soil stiffness ratio) = ln(rm/ro) (measure of radius of influence of pile) = 2 / ( /ro) (measure of pile compressibility).

It should be noted that Ep is the Young's modulus of a solid pile with equivalentcross-sectional rigidity to the actual pile. Thus Ep = (EA)p/(piro2), where (EA)p is the actualcross-sectional rigidity of the pile. A suitable expression for the maximum radius ofinfluence, rm, is

rm = {0.25 + [2.5(1 - ) - 0.25]} (6)

Figure 3 shows the variation of the load settlement ratio with slenderness ratio /ro for = =1, = 0.3. It has been found that these values are in reasonably good agreement with thosecomputed using charts from Poulos and Davis (1980), in spite of the simplifying assumptionsadopted in the analytical solution given above, and making allowance for the possible scopefor error when using the various multiplicative factors taken from the charts in Poulos andDavis. For long compressible piles, the results from Poulos and Davis, which are based onboundary element analysis, give higher values of pile stiffness than obtained using equation(5). The higher values may be partly due to relatively coarse discretisation of the very longpiles, leading to numerical inaccuracies.

From Figure 3, it may be seen that there are combinations of slenderness ratio, /ro, and


5

stiffness ratio, , beyond which the load settlement ratio becomes independent of the pilelength. It can be shown that insignificant load is transmitted to the pile base for such longpiles. This limiting behaviour is the converse of a stiff rigid pile, and corresponds to the casewhere the pile starts behaving as if it were infinitely long, with no load reaching the lowerregion.

The two limits may be quantified. Piles may be taken as essentially rigid where

/ro is lessthan 0.5(Ep/G )05. Equation (5) then reduces to equation (4). At the other extreme, for pileswhere

/ro is greater than about 3(Ep/G )05, tanh( ) approaches unity and equation (5)reduces approximately (exactly for = 1) to

P tG

rowt= pi 2 / (7)

As expected, the load settlement ratio is now independent of the length of the pile (since noload reaches the lower end). The modulus G

should be interpreted as the soil shear modulusat the bottom of the active part of the pile, that is, at a depth that corresponds to z/ro =3(Ep/G )05, rather than at z = .

3.2 Extension of solution to pile groups

One possible approach for analysing pile groups is to use equation (5), together with suitableinteraction factors to account for the proximity of other piles. However, such an approachignores an important facet of group behaviour, which is the transfer of a higher percentage ofload to the bases of piles within a group than for isolated piles. This phenomenon can bemodelled by considering separately the interaction of the displacement fields around the pileshaft, from the corresponding interaction at the level of the pile bases.

Interaction between neighbouring piles may be modelled (Cooke, 1974) by the superpositionof the displacement fields of each pile. Thus the settlement of one pile may be thought of asmade up of the sum of the settlement due to its own loading (without the presence of theneighbouring piles) and the settlements due to the displacement fields of each of the otherpiles . At the level of the pile bases, equation (1) gives the settlement of pile (i) due to its ownloading, while the settlement due to a neighbouring pile (j), at spacing sij, may beapproximated by

wb( )ij = 2rbpisij wb( )j =1 ( )2piGb

Pb( )jsij

(8)

In a similar manner, equation (3) is assumed to give the settlement of pile (i) due to its ownloading, while the settlement due to a neighbouring pile (j) at spacing sij is approximated by


6

ws( )ij = ln rm / sij( ) oro( )jG (9)

Equations (8) and (9) enable flexibility matrices to be formed relating the settlements at thepile bases to the corresponding loads, and likewise the settlements at each pile mid-depth tothe average shear stresses down each pile shaft. For very stiff piles, the settlement is uniformdown the pile length, and these matrices may be inverted, for a given distribution of pile headsettlement, to give {Pb} and {o} and hence the overall load settlement behaviour of the pilegroup.

For compressible piles, the solution is more complex, as the values of wt, ws and wb will bedifferent for any particular pile, due to compression of the pile. However, the compression ofthe pile may be calculated for a given load in the pile. Thus, after some matrix algebradescribed in detail by Randolph and Wroth (1979), it is possible to arrive at an overallflexibility matrix relating the pile head settlements {wt} to the total loads {Pt} taken by eachpile.

Comparison of the approximate method of analysis presented here with the results of morerigorous boundary element analysis shows good agreement. Figure 4(a) shows the values ofPt/(G rowt) for the three different pile positions in a 3 x 3 group of rigid piles embedded in ahomogeneous soil, for a range of slenderness ratios, /ro. The boundary element analyses,obtained using the program, PGROUP (Banerjee and Driscoll, 1978) generally yield values ofload settlement ratio which are some 10 % lower than the approximate analysis. Forcompressible piles, with a stiffness ratio of = 1000, the agreement is also good, as shown inFigure 4(b).

4. RESPONSE OF PILES TO TORSIONAL LOADING

The next type of loading to be considered is that of torsion about the pile axis. An analyticalsolution for the torsional response of piles has been presented by Randolph (1981b).Development of the solution follows the same lines as for the case of axially loaded piles,with the load transfer down the pile shaft being considered separately from that at the pilebase.

At the pile base, the torque, T, may be related to the angle of twist, , using the establishedsolution for the torsion of a rigid punch:

TbGbrb

3b=

163

(10)

Down the pile shaft, it may be shown that the angle of twist is related to the interfacial shearstress, o, by (Randolph, 1981b)


7

= o2G

(11)

For rigid piles, the above two equations may be combined to give an overall torsional stiffnessof

TtG ro

3t=

163

3 + 4pi

ro(12)

where the parameters are as defined previously for axial loading.

In practice, few piles will behave as rigid piles under torsional loading. Usually,deformations induced by torsion reduce to negligible magnitude at some level down the pileshaft. The situation is then similar to that for most laterally loaded piles, with the pile lengthno longer affecting the performance of the pile. For piles of intermediate length, the torsionalstiffness may be written

TtG ro

3t=

163

3 + 4pitanh ( )

ro

1 + 323pi

3G Gp

tanh ( )

ro

(13)

where Gp is the shear modulus of a solid pile of the same torsional rigidity as the actual pile.The remaining parameters are the same as in equation (5), except that the quantity is nowgiven by = 8G / Gp

/ ro( ). The similarity of the above expression with that for axiallyloaded piles (equation (5)) is evident.The torsional stiffness Tt/(Gro3t) for homogeneous soil conditions is plotted against thestiffness ratio Gp/G for various pile slenderness ratios,

/ro in Figure 5. The transition fromflexible behaviour (where the pile length does not effect the stiffness), for /ro (Gp/G)0.5, torigid behaviour for /ro 0.125(Gp/G)0.5, may be clearly seen. The limiting form of equation(13) for long piles is

TtG ro

3t= pi 2Gp / G (14)

where G is interpreted as the shear modulus at a depth of z = ro(Gp/G )0.5.In applying these solutions to the torsional response of piles within a group, two results notedby Poulos (1975) are of benefit. Firstly, he showed from a series of model tests, that values ofshear modulus for the soil, deduced from axial load tests, gave good predictions of theresponse of a pile under torsional loading. Thus, in choosing soil properties as input to


8

PIGLET, the same shear modulus profile may be adopted for both axial and torsionalresponse of the pile group.

The second observation made by Poulos (1975) was that there was no evidence of aninteraction effect between neighbouring piles under torsional loading. This finding conformswith what might be anticipated intuitively, and enables the torsional response of piles within agroup to be estimated directly from the equations given above, with no additional factors toallow for effects of interaction.

5. RESPONSE OF PILES TO LATERAL LOADING

5.1 Deformation of single laterally loaded piles

The analysis of laterally loaded piles is much more complex than that for axially or torsionallyloaded piles. Even for soil idealised as an elastic continuum, no simple closed form solutionis forthcoming. The solution which has been adopted in the program is one developed byRandolph (1981a) by curve fitting the results of finite element analyses of laterally loadedpiles embedded in elastic 'soil'. It was found that, for piles which behave flexibly underlateral load, simple power law relationships could be developed giving the lateral deflection,u, and the rotation, , of the pile at the soil surface, in terms of the pile stiffness and the soilproperties. The relationships are similar in form to those arising from considering the soil as aWinkler material characterised by a coefficient of subgrade reaction (e.g. Reese and Matlock,1956; Matlock and Reese, 1960). As in the latter type of analysis, the concept of a 'critical'length of pile is used, this depth being the depth to which the pile deforms appreciably. Theterm 'flexible' is taken to refer to piles where the load deformation characteristics would notbe altered by increasing the length of the pile. Thus piles that are longer than their criticallength behave as 'flexible' piles. The large majority of piles used in practice fall into thiscategory.

Since the solution is, by its nature, approximate, a further simplification has been introducedconcerning the soil properties - the shear modulus, G, and Poisson's ratio, . Randolph (1977)showed that the effect of Poisson's ratio could be allowed for to sufficient accuracy byconsidering a single elastic property given by

G* = G(1 + 3/4) (15)

The solution detailed below is in terms of the single parameter G* rather than the true elasticparameters G and .

The critical length of the pile is determined as

c = 2ro Ep / Gc( )2/7 (16)


9

where Ep is the equivalent Young's modulus of the pile, given by

Ep = (EI)p/(piro4/4) (17)

(EI)p being the flexural rigidity of the pile. The quantity Gc in equation (16) is the value of G*at a depth of half the critical pile length. For a soil idealised as an elastic material, with astiffness varying linearly with depth as

G = Go + mz (18)

the parameter Gc is given by

Gc = Go* + 0.5m* c = (1 + 3/4)(Go + 0.5m c) (19)

The evaluation of the critical length from equations (16) and (19) requires some iterationexcept in the extreme cases of a homogeneous soil (where Gc = Go*) or a soil where themodulus is proportional to depth (Go* = 0, then c = 2ro(Ep/m*ro)2/9 ).For piles which are longer than their critical length, the lateral deflection, u, and rotation, , atthe soil surface may be evaluated as

u =Ep / Gc( )1/7

cGc0.27 H

c / 2( )+ 0.30M

c / 2( )2

=Ep / Gc( )1/7

cGc0.30 H

c / 2( )2+ 0.80 c

M

c / 2( )3

(20)

where H and M are, respectively, the lateral load and bending moment acting at the soilsurface. The factor c gives the degree of homogeneity for the soil in a similar manner to thefactor in the analysis of axially loaded piles. It is conveniently defined as the ratio of thevalue of G* at a depth of c/4 to the value of G* at a depth of c/2 (see Figure 6). Thus

c =Go* + m* c / 4Go* + m

*

c / 2=

Go* + m* c / 4Gc

(21)

It should be noted that c varies from unity for a homogeneous soil down to 0.5 for a soilwhere the stiffness is proportional to depth. In equations (20), the product cGc is merely thevalue of G* at a depth of c/4. Thus for piles of a given critical length (i.e. stiffness ratio,Ep/Gc), the deformation under given loading conditions is inversely proportional to the soilstiffness at a depth of one quarter of the active, or critical, length of pile.


10

Comparison of results calculated from equations (20) with existing solutions obtained byboundary element analyses shows good agreement over a wide range of pile-soil stiffnessratios. Detailed results from such comparisons have been reported by Randolph (1981a).

5.2 Interaction between laterally loaded piles

The complexity of the displacement field around a laterally loaded pile precludes a similartreatment of the interaction between laterally loaded piles as was possible for axially loadedpiles. However, for piles that are loaded laterally with the pile head restrained against rotation(so-called fixed head or socketed piles), Randolph (1981a) has shown that the interactionfactors, f, may be estimated to sufficient accuracy from the expression

f = 0.6c E p / Gc( )1/7 1+ cos2 ( )ros

(22)

where s is the spacing between the axes of the piles and is the angle which the direction ofloading makes to a line passing through the pile axes (see Figure 7).The same form of expression may be used for interaction of deflection between two free headpiles subjected to force loading (zero moment at the soil surface). In that case, it is found thatthe coefficient 0.6 in equation (22) should be replaced by 0.4 to give a reasonable fit to factorscomputed by Poulos' program DEFPIG (Poulos, 1980). In addition, at very close spacings,the 1/s variation of can lead to unrealistically high interaction factors. In order to avoid this,and to allow to tend to unity as s tends to zero, the hyperbolic variation of is replaced by aparabolic variation wherever is calculated to be greater than 1/3. To summarise, theinteraction factor uH, giving the increase in deflection for free head piles subjected to lateralload H, is calculated from

= 0.4c Ep / Gc( )1/7 1 + cos2 ( )ros

(23)

where

uH = for 0.333

and

uH =1 2

27for > 0.333

Randolph (1981a) has compared values of uH calculated from these expressions with valuesobtained from Poulos' program DEFPIG.


11

The other interaction factors, uM (deflection due to moment loading), H (rotation due toforce loading) and M (rotation due to moment loading) may be estimated to sufficientaccuracy by

uM = H uH2

and (24)M uH3

Poulos and Randolph (1983) have compared tabulated values of interaction factors obtainedfrom this approach and from the boundary element program DEFPIG. In general, theagreement is reasonably good, with a tendency for interaction factors given by the presentapproach to decay more rapidly with increasing pile spacing, than shown by the DEFPIGresults.

6. ANALYSIS OF PILE GROUP

The separate solutions for axial, torsional and lateral response of piles must be combined inorder to analyse a pile group under general loading conditions. Before outlining how this isachieved, two important practical features of pile groups must be catered for - namely thepresence of raking piles, and the possibility of a free-standing length of pile between pile capand bearing strata.

6.1 Treatment of raking piles

The main reasons for using raking piles instead of vertical piles are:

(a) to transfer a portion of the horizontal load at the pile cap into axial load down the pile;(b) to increase the average spacing between piles, thus transferring the load from the

foundation over a greater volume of soil and, in effect, decreasing the amount ofinteraction between neighbouring piles.

The treatment of the first of these effects in the analysis is straightforward. The solutionsoutlined in the previous sections are used to calculate the stiffness matrices in terms of localpile axes (i.e. in terms of axial, torsional and lateral loads and deflections). When the overallgroup stiffness matrix is formed, the coordinate axes of each pile are transformed to globalaxes (vertical and horizontal). It should be noted that the bending moments induced in thepiles by horizontal loading are relatively sensitive to the angle of rake of the piles. The use ofraking piles instead of vertical piles for a particular foundation may well enable economies tobe made in the choice of pile section. To balance this benefit of raking piles, the difficulties(and possible inaccuracies in positioning) in installing such piles must be borne in mind, asmust the danger of using raking piles in circumstances where large vertical movements of


12

piles or soil are possible. Fleming et al (1985) discuss this point in more detail.Some discussion concerning reason (b) above, is appropriate. Raking piles may be used tospread the foundation load over a greater volume of soil. The ability of the program to copewith piles raking in any direction (rather than in any one particular vertical plane) is animportant one, since it enables the true spacing between pile centres to be calculated at agiven depth. Consider, for example, a square 2 x 2 group of piles with

/ro = 40 and a pilespacing at ground level of s = 6ro. If the piles rake diagonally outwards at 1 in 8, the trueaverage spacing down the shafts of the piles at adjacent corners is 11ro. If the analysis isrestricted to piles raking in one direction only, the spacing between adjacent corner pilesnormal to this direction would be constant at the surface spacing of 6ro.

It is also necessary to consider the mode in which interaction is assumed to take place. Fortwo piles which rake away from each other (see Figure 8), axial and lateral loading on pile Amay be assumed to cause interactive displacement of pile B in mode (i) (Figure 8(b) - wherethe induced movements are parallel and normal to pile A) or in mode (ii) (Figure 8(c) - wherethe induced movements are parallel and normal to pile B). Poulos (1979) has discussed themerits of either choice in the analysis of pile groups with raking piles. He points out that theassumption of interaction in mode (ii) conforms with the reciprocal theorem of Betti, whilethat in mode (i) does not. Clearly both modes are idealisations of the real situation. However,in order to satisfy the reciprocal theorem, the second mode has been adopted in the presentanalysis. Adoption of this mode of interaction between piles has the additional advantage ofenabling the axial load deformation behaviour of the piles to be considered independentlyfrom the lateral load deformation behaviour, before combining the two to obtain the overalldeformation characteristics of the foundation.

6.2 Allowance for free-standing length of piles

In many situations, the soil immediately below the pile cap may be relatively soft and shouldbe ignored in the analysis of the load deformation characteristics of the pile group. In effectthe pile cap is considered suspended above the top of the soil strata in which the piles arefounded (see Figure 9). The resulting free-standing length of pile must be taken into account.This is achieved by modifying the axial, torsional and lateral flexibility matrices of the piles(which relate deformations and loads at the top of the bearing stratum), treating thefree-standing section of pile as a simple cantilever. New flexibility matrices are formedrelating the deformations and loads at the underside of the pile cap before combining these togive the required load deformation characteristics of the complete group. To allow for thesituation where the upper part of a pile is cased as it passes through softer soil, it is possible inthe program to specify different pile properties in the free-standing section than in the mainpart of the pile.


13

7. EXAMPLE APPLICATION

As an example of the application of PIGLET, model tests on a group of six piles embedded insand (Davisson and Salley, 1970) have been analysed. Output from the analysis is given inAppendix A.

A group of six tubular aluminium piles of external diameter 12.7 mm (0.5 in) and wallthickness 0.8 mm (0.03 in), embedded in a tank of dry fine sand to a depth of 0.533 m (21 in),were loaded through a pile cap suspended just above the level of the sand surface. Figure 10shows the pile layout and applied loads. Equivalent Young's modulus for the piles may becalculated as

Axial loading: Ep = Eal[1 - (ri/ro)2] = 16,300 MPa (2.37 x 106 psi)Lateral loading: Ep = Eal[1 - (ri/ro)4] = 28,900 MPa (4.19 x 106 psi).Each of the six piles in the group was load tested axially, prior to forming the pile cap, inorder to determine the axial stiffness. Davisson and Salley (1970) report an average stiffnessof 0.82 kN/mm (4860 lbf/in) with a standard deviation of 0.15 kN/mm (840 lbf/in). It isreasonable to assume that the shear modulus of the sand is proportional to the effective stresslevel (and thus to depth below the sand surface). With this assumption, and adopting a valuefor Poisson's ratio of 0.25, equation (5) may be used to deduce the shear modulus profilenecessary to yield the above value of axial stiffness for the piles. This process leads to anexpression for the shear modulus, G, of

G = 4.2z MPa (15.3z psi) [z in metres (in)].

The above variation of shear modulus has been used to analyse the complete group of pilesunder the loading shown in Figure 10. The output for the analysis is given in Appendix A.

Table 1 summarises measured values of axial load, lateral load and bending moment at thetops of the six piles. These results compare favourably with those obtained from the programPIGLET, the error in the predicted bending moments and in the largest axial loads beinggenerally less than 10%. The computed lateral deflection of 0.28 mm (0.011 in) is some 1.2standard deviations larger than the measured deflection of 0.23 mm (0.09 in).Also shown in Table 1 are loads obtained from the program PGROUP (the values are takenfrom the PGROUP users' manual). Although predictions of axial and lateral loads are good,there is considerable discrepancy in the values of bending moments. Much of thisdiscrepancy may be attributed to the assumption of a homogeneous value of shear modulusfor the soil in the PGROUP analysis. This assumption is likely to be a less goodapproximation for the sand than taking a shear modulus which is proportional to depth, and inthis case leads to significant under-prediction of the bending moments induced in the piles.


14

TABLE 1

COMPARISON WITH MODEL TEST RESULTSOF DAVISSON AND SALLEY (1970)

LOADS AT HEAD OF EACH PILE

Pile No. Axial Load (N) Lateral Shear Load (N) Bending Moment (Nm)Meas'd PIGLET PGROUP Meas'd PIGLET PGROUP Meas'd PIGLET PGROUP

1 80.1 73.5 69.0 22.2 14.5 17.4 1.23 1.16 0.79

2 56.9 73.5 69.0 23.1 14.5 17.4 1.27 1.16 0.79

3 28.9 34.5 31.6 16.9 14.8 16.0 1.20 1.19 0.77

4 24.0 34.5 31.6 16.0 14.8 16.0 1.18 1.19 0.77

5 12.2 5.8 14.0 23.1 19.4 20.5 1.49 1.51 0.96

6 13.6 5.8 14.0 16.9 19.4 20.5 1.22 1.51 0.96


15

PART B: PROGRAM DOCUMENTATION

8. STRUCTURE OF PROGRAM

The program PIGLET has been structured so that the complexity, or 'scope', of analysis maybe chosen by the user. The user may choose between three alternatives:

(a) Analysis for vertical loading only (piles are assumed to be vertical).(b) Analysis for vertical and horizontal loading in one plane only (piles are assumed to be

raked only in the plane of loading).(c) Full analysis of pile group under vertical, horizontal and torsional loading (piles may be

raked in any direction).The advantages of this choice are that the amount of input data, computer effort and outputare all determined by the scope of the analysis, being a minimum for (a) and a maximum for(c). In addition, for vertical loading only, the program allows specification of a fully flexiblecap, as opposed to the rigid cap assumed in the other options.

The program is fully interactive in terms of data input and running. In the course of runningthe program, different 'configurations' (of pile layout, or soil parameters) may be consideredsequentially, merely changing the relevant data items between each analysis. For eachconfiguration, a number of different loading cases may be considered. A flow chart for theprogram is shown in Figure 11. The only major branch point occurs for the case of verticalloading only (NSCOPE = 1), where the extra option of a fully flexible pile cap entails adifferent approach than for a rigid pile cap.

As supplied, the maximum problem size is set to 300 piles, which requires about 1.5 Mb ofRAM to run. Alternative versions can be supplied that can analyse larger groups, at the costof greater memory requirements. The program will run in either a DOS or Windowsenvironment.

9. PROGRAM INPUT

9.1 Interactive data input and editing

Data input into the program is by means of interactive screen input, with the ability to switchbetween the various screens in order to expedite modification of any given set of data. Whenthe program is run, the user has a choice of editing a previously saved datafile, or inputingnew data interactively (starting with default values that are generally zero). After the datahave been entered, there is a facility to save the data in an unformated datafile.


16

For screens that present options (such as choosing the scope of the analysis), the 'Up' and'Down' arrows may be used to change among the options, while pressing 'Enter' accepts thehighlighted option.

Limited help screens are available, by pressing the 'F1' key at any stage. The first help screenlists the various editing keys, with a brief description of the usual function of each key. Asecondary help screen is available when inputing particular data (such as pile properties),giving guidance on the requested items of data.

Data values are input in free format (within the datafields shown on each screen), over a 10character field. Real items of data may be input as integers (without a decimal point), or as areal number containing a decimal point. With either form, an exponential scaling factor maybe included. For example, a pile length of 40 m could be input as '40', '40.', '4E+1', '0.4E+2'or '4.E+1' (or several other, more convoluted, ways!). To move from one item of data toanother, press 'Enter' or use the 'Up' or 'Down' arrows. For screens where data are entered incolumns (such as the pile group geometry, or the loading details), the 'Tab' and 'Shift (back)Tab' keys may be used to move horizontally through the data field.

The space bar is used in two ways. Where there is a choice of options, the space bar is used asa tab between different options (for example, to tab between specifying 'Load', 'Deflection' or'Fixed Head' modes of specifying the loading applied to the pile cap). For ordinary numericinput of a data field, the space bar is used to clear the remaining characters in the field.

The description of each item of data is intended to be self-explanatory. However, additionalnotes are provided on the following pages to guide new users of the program. As in anyengineering problem, a consistent set of units must be used in the input data. No system ofunits is assumed by the program. Essentially, the user must decide what unit of force (F) andwhat unit of length (L) are to be adopted. Data are then input in appropriate units accordingto the type of data. Thus values of modulus should be in units of F/L2, bending moments inFL, and so forth.

The program assumes filename extensions of .DAT for datafiles and .OUT for output files.Other extensions may be used, although they will not be recognised by the program whensearching the current directory for alternative datafiles (or output files). When entering a'general specification' of a file, an appropriate extension will be assumed, provided noextension (and no '.') is provided in the name.The principal actions of the editing keys are described below:

Esc Generally returns the screen to its original status, reversing any editing ofdata that may have been carried out.

Space bar For numeric fields, the space bar clears all the remaining characters in thedata field (to the right of, and including, the cursor).For loading options (load or deflection control, rigid or flexible cap) thespace bar is used to toggle between the different options.


17

Ctrl PgUp Moves to the previous screen (adopting the data values displayed on thecurrent screen).

Ctrl PgDn Moves to the subsequent screen (adopting the data values displayed on thecurrent screen).

Home Moves to the top (left) of the data items.End Moves to the bottom (right) of the data items.Tab Tabs right through columns of data.

Shift Tab Tabs left through columns of data.

Enter Accepts the current data item, and moves to the next item (or to the nextscreen if on the last data item on the current screen).

Up Arrow Moves up the column of data fields.

Down Arrow Moves down the column of data fields.

Left Arrow Moves left within the data field.

Right Arrow Moves right within the data field.

PgUp Scrolls up pile numbers by (up to) 10 rows.PgDn Scrolls down pile numbers by (up to) 10 rows.Ins Toggles between insert or overwrite mode (denoted by size of cursor on

most screen types).Del Deletes character at cursor position.

Backspace Deletes character to left of cursor and moves cursor back one space.

' = ' Ditto function (provides same value as in row immediately above).' < ' Multiple ditto function for every row above current row

' > ' Multiple ditto function for every row below current row

F1 Help screen (press F1 again for any secondary help screen, or Esc to revertto data entry).


18

9.2 Data Items

Problem Title and Scope

The title may consist of up to 78 alphanumeric characters. The 'scope' of the problem hasbeen discussed in Section 8 of the manual. Essentially it defines the complexity of the appliedloading, with options of (a) vertical loading only, (b) vertical and horizontal loading in oneplane, or (c) full three-dimensional loading including torsion.Pile Parameters

The maximum number of piles that the program can analyse is set to 300 in the standardversion. As discussed earlier, a version catering for 500 or greater number of piles can besupplied, provided sufficient computer memory is available. The length of each pile isassumed the same, with an overall value equal to the sum of the embedded portion and afree-standing length (which may be zero). The Young's modulus of the pile is that of a pilehaving equivalent cross-sectional rigidity (for axial loading) or bending rigidity (for lateralloading) as the real pile. Thus, for a pile of radius ro, the value of Young's modulus for axialloading is

Ep = (EA)p/(piro2)

while for lateral loading,

Ep = (EI)p/(piro4/4)

In order to allow for the possibility of a change in pile cross-section at ground level, differentvalues of Young's modulus may be specified for the free-standing lengths of pile. Fortorsional loading, the torsional rigidity of the pile is obtained from the bending rigidity, takingPoisson's ratio for the pile material as 0.3.

For non-circular piles, it is important that the radius of the idealised pile is chosenrealistically. It is suggested that the cross-sectional area of the idealised pile should be chosenso as to equal the gross (enclosed) area of the actual pile. For H section piles, the gross areashould be taken as that of the encompassing rectangle.

For lateral loading, there is a choice between whether the piles are to be assumed fixed intothe pile cap or pinned to the pile cap (zero moment at pile cap level).Pile Group Geometry

For each pile, values of shaft radius, base radius and (x, y) co-ordinates must be input. Inaddition, where lateral loading is involved, angles of rake must be specified in radians, eitherin the x:z plane (where loading is restricted to one plane only), or in both x:z and y:z planes.The program initially assumes that the pile radii are identical for each pile, and that only the


19

co-ordinates (and angles of rake where appropriate) are to be edited. However, it is possibleto change individual values of radius (using the Tab or Shift Tab keys to access the data field),or to alter the pile radii throughout the group by changing the radius of the first pile and usingthe '>' edit key to copy the new radius to every other pile in the group. Note that it is possibleto add or delete piles on this screen (overriding the number of piles set on the previousscreen).The program assumes a right-handed set of coordinate axes (x, y, z), with the z axis pointingvertically downwards. Angles of rake should be input in radians measured from the z axis,positive values indicating a pile lying between the x and z (or y and z) axes. Figure 12 showsthis sign convention. The maximum angle of rake that is permitted is 1 radian.

Soil Parameters

The value of Poisson's ratio for the soil is assumed the same for all types of loading - axial,lateral or torsional. Different profiles of shear modulus may be specified for axial and forlateral loading (the profile for torsional loading is assumed the same as for axial loading). Asdiscussed in Section 2 of the manual, the shear modulus profile is assumed to increase linearlywith depth. The user specifies the value at the ground surface (which must be non-negative)and the gradient with depth (also non-negative). In addition, for vertical loading a suddenincrease in modulus at the base of the pile (for end-bearing piles) may be input. If this valueis set to less than the value that would be calculated from the linear variation of shearmodulus, then the program corrects it to that value (thus the program does not permit anabrupt decrease in the value of shear modulus at the pile base).For irregular soil profiles, it is important that the linear variation of soil modulus with depth ischosen so as to reflect the true average shear modulus over the depth of penetration of thepiles, and also the trend of variation of soil modulus with depth. Since piles deflect underlateral loading only in the upper ten diameters or so, it is possible to specify different valuesof soil modulus for lateral loading than for axial (and torsional) loading.In many instances, piles are installed so that they finish at some depth above a significantlystiffer stratum of soil. While such piles are not strictly 'end-bearing' piles, the stiffer stratumof soil will reduce the overall settlement of the group. For a stratum with shear modulus Gh,at a depth h (greater than the pile length

) it is recommended that the value of shear modulusbelow the pile bases, Gb, is chosen by means of the expression (Lee, 1991)

1 11

1 1

G GGG

e

Gb hz

h

h

z

= +

=

=

/(25)

For values of h greater than 4

, the presence of the stiffer stratum of soil may, conservatively,be ignored.

For situations where no values of shear modulus are available for the soil, values of G must be


20

chosen by inspection of the available soil data. For cohesive soil, it is common practice tocorrelate shear modulus with the shear strength su. At working load levels, the axialdeformation of piles may be estimated reasonably well by taking shear modulus values in therange

200 G/su 400

Under lateral loading, the high strains which occur in the soil close to the pile give rise tolower secant modulus. It is suggested that G should be chosen in the range

100 G/su 200

for the lateral load deformation behaviour of the pile group.

For non-cohesive soil, or where the only data available are results of standard penetrationtests, it is suggested that the simple (but conservative) guideline of G = N MPa be adopted(see Randolph, 1981b). A less conservative correlation has been proposed by Wroth et al(1979), who suggest

G/pa 40N0.77

where pa is atmospheric pressure (100 kPa). In general, the variation of shear modulus withdepth in sand (below the water table) may be expressed as G = mz, with m in the range 1MPa/m (loose virgin sand) up to 5 MPa/m (dense sand).In soft rocks, the effects of pile installation must be allowed for. While the in situ modulus ofsoft rocks such as chalk can be extremely high, installation of bored or driven piles tends tobreak up the block structure of the rock. The relevant shear modulus is then that associatedwith large strains (see Wakeling, 1970; Randolph and Wroth, 1978b). Further guidance onthe choice of shear modulus may be found in Wroth et al (1979).Load Cases

Up to 20 separate load cases may be specified for each analysis. Loading may be specifiedexplicitly (as forces and moments) or may be given as imposed deformations of the pile cap.The pile cap is assumed rigid accept for the case of vertical loading only, when arbitrary loadsor deflections may be specified at the head of each pile.

The user is asked to specify, for each load case, whether the loading is specified:

(a) in terms of loads applied to the pile cap ('Load');or

(b) in terms of deflections applied to the pile cap ('Deflection').


21

For vertically loaded pile groups, the user may choose between:

(a) rigid pile cap ('Rigid Cap');or

(b) flexible pile cap ('Flex. Cap').Where a rigid pile cap is specified (or assumed for problems involving horizontal loading),the user must then specify the loads (or deflections) acting on the pile cap. All loads areassumed to act at pile cap level (z = 0), through the origin (x = y = 0).For vertical loading applied through a flexible pile cap, individual loads (or deflections) mustbe specified for each pile. This may be accomplished by specifying a uniform load (ordeflection) for each pile - using the 'U' option - and then changing individual values onrequired piles.

For lateral loading, in addition to loads or deflections specified for the pile cap, a mixed modeof loading is allowed, where loads are applied to a pile cap that is prevented from rotating(so-called 'Fixed Head'). Vertical and horizontal loads (and torque) are specified, the fixingmoments to provide zero rotation being calculated by the program.

It must be emphasised that all loads are assumed to act through the origin x = y = z = 0.Horizontal loads are taken as positive in the direction of the positive x and y axes, andmoments are taken as positive in the sense of rotating the x axis towards the z axis (forloading in the x:z plane) and rotating the y axis towards the z axis (for loading in the y:zplane). This sign convention differs from the usual right-handed axis rule, as indicated inFigure 13.

Profiles of Bending Moment and Lateral Deflection

For analyses which involve lateral loading, profiles of bending moments and lateral deflectionrelative to the immediately surrounding soil may be output for specified piles. A choice isgiven as to whether (a) no profiles, (b) profiles of bending moment only, or (c) profiles ofbending moment and lateral deflection, are required. For three-dimensional loading, separatechoices are given for the x:z plane and the y:z plane.

Having established what profiles are required, the user can specify for which piles the profilesare to be calculated (using the space bar as a toggle, and the edit keys to move from pile topile). It should be emphasised that, since the free field soil deflections (due to interactionbetween piles) are not included in the relative lateral deflection profile, the deflection outputfor the pile head will not correspond with the total lateral deflection for the pile head (exceptfor analyses with only one pile in the group).


22

10. PROGRAM OUTPUT

Output from the program consists of lines of up to 120 characters, with up to 60 lines perpage. The form of the output is reasonably self-explanatory. It consist of four main sections:

(1) Front page and two further pages reflecting the input data.(2) Response of pile group to unit deformations of the pile cap, giving loads and moments

at the head of each pile, in local coordinates.

(3) Overall stiffness and flexibility matrices for the group.(4) Response of the pile group to load cases specified by the user. This section includes

loads and resulting deformations of the pile cap, loads and moments at the head of eachpile, and (optionally) profiles of bending moment and lateral deflection down specifiedpiles.

(5) Where more than a single load case is analysed, summary tables of output are includedwhere the loads and deflections at the head of each pile are summarised for each loadcase.

The user has the option of full output (as given above), or slim-line output, where sections (2)and (3) above are omitted. The overall quantity of output is not large, but slim-line outputmay be preferred when using a slow printer - for example, attached to a microcomputer.

It is recommended that output from the program is directed to a computer file in the firstinstance, and this file is subsequently printed where required. The file may be edited usingany standard text editor.

The sign convention for lateral loads and moments for each pile follows that for specifyingthe applied loads, with lateral load being taken as positive in the direction of the positive xand y axes, and moments taken as positive in the sense of rotating the x axis towards the zaxis (for loading in the x:z plane) and rotating the y axis towards the z axis (for loading in they:z plane).


23

REFERENCES

1. Baguelin F., Bustamante M., Frank R. and Jezequel J.F. (1975), 'La capacite portantedes pieux', Annales de l'Institut Technique du Batiment et des Travaux Publics, Suppl.330, Serie SF116, pp 1-22.

2. Banerjee P.K. and Davies T.G. (1978), 'The behaviour of axially and laterally loadedpiles embedded in non-homogeneous soils', Geotechnique, Vol 28, No 3, 309-326.

3. Banerjee P.K., Driscoll R.M.C. and Davies T. (1978), 'Program for the analysis of pilegroups of any geometry subjected to horizontal and vertical loads and moments,PGROUP, (3.0)', HECB/B/7, Department of Transport, HECB, London.

4. Butterfield R. and Douglas R.A. (1981), 'Flexibility coefficients for the design of pilesand pile groups', CIRIA Technical Note 108.

5. Cooke R.W. (1974), 'Settlement of friction pile foundations', Proc. Conf. on TallBuildings, Kuala Lumpur, 7-19.

6. Davisson M.T. and Salley J.R. (1970), 'Model study of laterally loaded piles', J. of SoilMech. and Found. Engg Div., ASCE, Vol 96, No SM5.

7. Fleming W.G.K., Weltman A.J., Randolph M.F. and Elson W.K. (1985), 'PilingEngineering', Surrey University Press, Glasgow.

8. Frank R. (1974), 'Etude theorique du comportement des pieux sous charge verticale;introduction de la dilatance', Dr-Eng. Thesis, University Paris VI (Pierre et Marie CurieUniversity).

9. Lee C.Y. (1991), 'Discrete layer analysis of axially loaded piles and pile groups',Computers and Geotechnics, Vol. 11, 295-313.

10. Matlock H. and Reese L.C. (1960), 'Generalised solutions for laterally loaded piles', J.Soil Mech. and Found. Engng Div., ASCE, Vol 86, No SM5.

11. O'Neill M.W., Ghazzaly O.I. and Ha H.B. (1977), 'Analysis of three-dimensional pilegroups with non-linear soil response and pile-soil pile interaction', Proc. 9th OffshoreTechnology Conf., Vol 2, 245-256.

12. Poulos H.G. (1971), 'Behaviour of laterally loaded piles, I - Single piles, II - Pilegroups', J. Soil Mech and Found. Engng Div., ASCE, Vol 97, No SM5.

13. Poulos H.G. (1973), 'Load-deflection prediction for laterally loaded piles', AustralianGeomechanics Journal, Vol G3, No 1.

14. Poulos H.G. (1975), 'Torsional response of piles', J. Geot. Engng Div., ASCE, Vol 101,No GT10.


24

15. Poulos H.G. (1979), 'An approach for the analysis of offshore pile groups', Proc. Conf.on Numerical Methods in Offshore Piling, ICE, London, 119-126.

16. Poulos H.G. (1979), 'Settlement of single piles in non-homogeneous soil', J. Geot.Engng Div., ASCE, Vol 105, No GT5.

17. Poulos H.G. (1980), 'Users' guide to prgram DEFPIG - deformation analysis of pilegroups', School of Civil Engineering, University of Sydney.

18. Poulos H.G. and Davis E.H. (1980), 'Pile foundation analysis and design', John Wiley &Sons, New York.

19. Poulos H.G. and Randolph M.F. (1983), 'Pile group analysis: a study of two methods', J.of Geot. Eng., ASCE, Vol 109, No 3, 355-372.

20. Randolph M.F. (1977), 'A theoretical study of the performance of piles', PhD Thesis,University of Cambridge.

21. Randolph M.F. (1981), 'Analysis of the behaviour of piles subjected to torsion', J. ofGeot. Engng Div., ASCE, Vol 107, No GT8, pp 1095-1111.

22. Randolph M.F. (1981), 'The response of flexible piles to lateral loading', Geotechnique,Vol 31, No 2, pp 247-259.

23. Randolph M.F. and Wroth C.P. (1978), 'Analysis of deformation of vertically loadedpiles', J. of the Geot. Eng. Div., ASCE, Vol 104, No GT12, 1465-1488.

24. Randolph M.F. and Wroth C.P. (1978), 'A simple approach to pile design and theanalysis of pile tests', Proc. Symp. on Behaviour of Deep Foundations, ASTM STP 470,484-499.

25. Randolph M.F. and Wroth C.P. (1979), 'An analysis of the vertical deformation of pilegroups', Geotechnique, Vol 29, No 4.

26. Reese L.C. and Matlock H. (1956), 'Non-dimensional solutions for laterally loadedpiles', Proc. 8th Texas Conf. on Soil Mech.

27. Wakeling T.R.M. (1970), 'A comparison of the results of standard site investigationmethods against the results of a detailed geotechnical investigation in Middle Chalk atMundford, Norfolk', Proc. Conf. on In Situ Investigations in Soils and Rocks, BritishGeotechnical Society, London.

28. Wroth C.P., Randolph M.F., Houlsby G.T. and Fahey M. (1979), 'A review of theengineering properties of soils with particular reference to the shear modulus',Cambridge University Engineering Department Research Report, CUED/D - SoilsTR 75.


25

FIGURE TITLES

Figure 1 Assumed variation of soil shear modulus with depth

Figure 2 Uncoupling of effects due to pile shaft and base

Figure 3 Load settlement ratios for compressible piles

Figure 4 Notation for analysis of laterally loaded piles

Figure 5 Torsional stiffness factor for piles in homogeneous soil

Figure 6 Comparison of load settlement ratios for piles in a 3 x 3 pile group inhomogeneous soil

Figure 7 Plan view of two piles subjected to lateral loadingFigure 8 Choice of modes for interaction between pairs of non-parallel piles

Figure 9 Allowance for free-standing length of piles

Figure 10 Model pile test arrangement (Davisson and Salley, 1970)Figure 11 Flow chart for PIGLET

Figure 12 Sign convention for pile rake

Figure 13 Sign convention for loading


26

FIGURES


27

Figure 1 Assumed variation of soil shear modulus with depth

Shear modulus, G

Depth, z

G

Gavg = G

Solid cylindrical pileRadius: roEquivalent modulus, Ep

G Gb


28

Figure 2 Uncoupling of effects due to pile shaft and base

B

A B

A' B'

PtPt = Ps + Pb

APb

PsShaft response

Base response


29

0

10

20

30

40

50

60

1 10 100 1000Pile slenderness ratio, /ro

= 10 30100

300

3000

1000

to

t

wrGP

(a) = 0.5

01020304050607080


= 10 30100

300

3000

1000

to

t

wrGP

(b) = 0.75

0

20

40

60

80

100


= 10 30100

300

3000

1000

to

t

wrGP

(c) = 1Figure 3 Load settlement ratios for axially loaded piles


30

Figure 4 Comparison of load-settlement ratios for piles in a 3 x 3 pile group inhomogeneous soil


31

1

10

100

1000

10000

1 10 100 1000 10000 100000 1E+06 = Gp/G

t3

o

t

G rT

/ro = 200

100502510

Figure 5 Torsional stiffness factors for piles in homogeneous soil


32

Figure 6 Notation for analysis of laterally loaded pile

cSolid cylindrical pileRadius: roEquivalent modulus, Ep

Modified shearmodulus, G*

Depth, z

Gc

Gz = /4 = cGc

c/2c

G* = G(1 + 3/4)

c

o

p

cr

EG2

2 7=

/


33

Figure 7 Plan view of two piles subjected to lateral loading

s


34

(a) Interactive displacements of Pile B parallel to Pile A

Figure 8 Choice of modes for interaction between pairs of non-parallel piles

Pile A Pile B

(a) Interactive displacements of Pile B axial and lateral

Pile A Pile B

wA

uAwA

uA

wA

uAwA

uA


35

Figure 9 Allowance for free-standing length of piles

Pile cap

Piles

Level of bearing strata

Depth of free-standingsection of piles

Penetration of piles intobearing strata


36

Figure 10 Model pile test arrangement (Davisson and Salley, 1970)

3

1

fine, dry sand

0.222 kN0.138 kN

piles 0.533 mlong

76 mm

127 mm

Plan View


37

Figure 11 Flow chart for PIGLET

VSTIFAxial load-deformationresponse of pile group

HSTIFLateral load-deformation

response (x:z plane)

HSTIFLateral load-deformation

response (y:z plane)

TSTIFTorsional load-deformation

response of pile group

FORMGSForm terms in overall group

stiffness matrix

GENLDFlexibility and stiffness of

group; response to load cases

VERTLDFlexibility and stiffness of

group, response to load cases(vertical loading only)

INDATARead (new) data

NSCOPE

NSCOPE

Modify data?

Start

Stop

1

2, 3

2

3

NoYes


38

x

y

y

z

z

x

Plan view

Elevation - x:z plane

Elevation - y:z plane

Negativerake

Negativerake

Positiverake

Positiverake

Figure 12 Sign convention for pile rake


39

x

y

z

Px

Py

Pz

MyMz

Mx

x

y

z

Px

Py

Pz

Mx to zTx to y

My to z

(a) Conventional right-hand notation (b) PIGLET notation

Figure 13 Sign convention for loading

Note: Mx to z = -My


40

APPENDIX 1 - EXAMPLE OUTPUT


41

Output from Pile Group Analysis Program - PIGLET Version dated October, 1996 Page 0

Analysis of group of six model piles in sand - Davisson and Salley (1970) Example output for program manual

*******************************************************************************

*******************************************************************************

** **

** **

** PPPPPPP IIIIII GGGGGG LL EEEEEEEE TTTTTTTT **

** PP PP II GG GG LL EE TT **

** PP PP II GG LL EE TT **

** PPPPPPP II GG GG LL EEEEE TT **

** PP II GG GG LL EE TT **

** PP II GG GG LL EE TT **

** PP IIIIII GGGGGG LLLLLLLL EEEEEEEE TT **

** **

** **

*******************************************************************************

*******************************************************************************


42



Pile group analysis for 6 piles under vertical and horizontal loading in one plane only

Pile details are

Embedded length = 5.330E-01

Freestanding length = 3.000E-03

Equivalent Youngs modulus of embedded section of piles:

Axial : 1.630E+07 , Lateral : 2.890E+07

Equivalent Youngs modulus of freestanding section of piles:

Axial : 1.630E+07 , Lateral : 2.890E+07

Piles are assumed to be fixed into the pile cap

Pile layout details are:

Pile no. Radius Base radius X co-ord Y co-ord Rake psi(x) Rake psi(y)

1 6.350E-03 6.350E-03 1.270E-01 3.800E-02 3.330E-01 0.000E+00

2 6.350E-03 6.350E-03 1.270E-01 -3.800E-02 3.330E-01 0.000E+00

3 6.350E-03 6.350E-03 0.000E+00 3.800E-02 0.000E+00 0.000E+00

4 6.350E-03 6.350E-03 0.000E+00 -3.800E-02 0.000E+00 0.000E+00

5 6.350E-03 6.350E-03 -1.270E-01 3.800E-02 -3.330E-01 0.000E+00

6 6.350E-03 6.350E-03 -1.270E-01 -3.800E-02 -3.330E-01 0.000E+00


43



Soil properties are:

Axial load-deformation - G(0) = 0.000E+00 dG/dz = 4.300E+03 G(b) = 2.292E+03 Nu = 0.250

Lateral load-deformation - G(0) = 0.000E+00 dG/dz = 4.300E+03 Nu = 0.250

Parameters for axial load-deformation behaviour are:

Shear modulus at level of pile bases is G(L) = 2.292E+03 Rho = G(L/2)/G(L) = 5.000E-01 Shear modulus below pile bases is G(b) = 2.292E+03 Xi = G(L)/G(b) = 1.000E+00 Poissons ratio is nu = 0.250

Rm = (0.25+xi*(2.5*rho*(1-nu)-0.25))*L + 7.839E-02 = 5.781E-01 Pile stiffness ratio is Epa/G(L) = 7.112E+03 Axial flexibility of pile no. 1 (isolated, at mudline) = 1.205E-03

Parameters for lateral load-deformation behaviour are:

Gc = (G(0)+(Lc/2)*Gm*(1.+0.75nu) = 6.809E+02 Rhoc=G(Lc/4)/G(Lc/2) = 5.000E-01 Critical slenderness ratio is Sc = 4.200E+01

Critical depth is Lc = 2.667E-01

Lateral flexibilities (isolated, at mudline) of first pile are: u/H = 2.725E-02 th/H or u/M = 2.271E-01 th/M = 3.211E+00

The following pages of output give the load deformation behaviour of the pile group under

vertical and horizontal loading in one plane


44



1) For unit vertical deflection of pile cap:

Pile Axial Lateral Moments

no, loads loads (x) (x to z)

1 4.5565E+02 -2.7696E+01 1.9988E+00

2 4.5565E+02 -2.7696E+01 1.9988E+00

3 3.9672E+02 1.5977E-06 -1.2664E-07

4 3.9672E+02 1.7555E-06 -1.1469E-07

5 4.5565E+02 2.7696E+01 -1.9988E+00

6 4.5565E+02 2.7696E+01 -1.9988E+00

2) For unit horizontal deflection of pile cap - in x direction:



1 2.1356E+02 5.1790E+01 -3.7772E+00

2 2.1356E+02 5.1790E+01 -3.7772E+00

3 5.3510E-06 5.0051E+01 -3.6705E+00

4 8.0212E-06 5.0051E+01 -3.6705E+00

5 -2.1356E+02 5.1790E+01 -3.7772E+00

6 -2.1356E+02 5.1790E+01 -3.7772E+00

3) For unit rotation of pile cap - x towards z about y axis:



1 7.8413E+01 -7.0744E+00 8.1712E-01

2 7.8413E+01 -7.0744E+00 8.1712E-01

3 1.4083E-06 -2.1535E+00 4.6609E-01

4 5.9677E-06 -2.1535E+00 4.6609E-01

5 -7.8413E+01 -7.0743E+00 8.1712E-01

6 -7.8413E+01 -7.0743E+00 8.1712E-01


45



Overall group stiffness matrix is :

Total Total Total

vertical horizontal moment

load load (x) (x to z)

Unit vertical deflection 2.5521E+03 -9.3999E-06 -2.1610E-05

Unit horizontal movement (x) -9.3999E-06 5.7511E+02 7.1477E+01

Unit rotation (x to z) -2.1610E-05 7.1477E+01 4.3021E+01

Overall group flexibility matrix is :

Vertical Horizontal Rotation (x deflection deflection to z about

x = y = 0.0 (x dir.) y axis)

Unit vertical load 3.9183E-04 -2.2756E-11 2.3463E-10

Unit horizontal load (x) -2.2756E-11 2.1913E-03 -3.6407E-03

Unit moment (x to z) 2.3463E-10 -3.6407E-03 2.9293E-02


46



Load case no. 1 out of 1

Pile loads and deformations

Vertical Horizontal Moment

load load (x) (x to z)

2.2200E-01 1.3800E-01 5.7000E-03

Vertical Horizontal Rotation

deflection defn (x) (x to z)

8.6986E-05 2.8164E-04 -3.3544E-04



1 7.3479E-02 1.4550E-02 -1.1641E-03

2 7.3479E-02 1.4550E-02 -1.1641E-03

3 3.4509E-02 1.4819E-02 -1.1901E-03

4 3.4509E-02 1.4819E-02 -1.1901E-03

5 5.7918E-03 1.9369E-02 -1.5118E-03

6 5.7918E-03 1.9369E-02 -1.5118E-03


47



Load case no. 1 out of 1

Profiles of bending moments in the (x,z) plane and (optionally) lateral deflections (relative to soil) in the x direction for specified 3 piles

Pile number 1

Depth 0.00E+00 3.33E-02 6.67E-02 1.00E-01 1.33E-01 1.67E-01 2.00E-01 2.33E-01

Moment -1.12E-03 -6.46E-04 -2.59E-04 5.75E-06 1.41E-04 1.84E-04 1.89E-04 1.31E-04

Defn u 1.42E-04 1.38E-04 1.17E-04 9.01E-05 6.45E-05 4.64E-05 2.54E-05 7.59E-06

Pile number 3

Depth 0.00E+00 3.33E-02 6.67E-02 1.00E-01 1.33E-01 1.67E-01 2.00E-01 2.33E-01

Moment -1.15E-03 -6.63E-04 -2.68E-04 2.84E-06 1.41E-04 1.86E-04 1.92E-04 1.34E-04

Defn u 1.44E-04 1.40E-04 1.19E-04 9.17E-05 6.56E-05 4.73E-05 2.59E-05 7.75E-06

Pile number 5

Depth 0.00E+00 3.33E-02 6.67E-02 1.00E-01 1.33E-01 1.67E-01 2.00E-01 2.33E-01

Moment -1.45E-03 -8.25E-04 -3.14E-04 3.25E-05 2.05E-04 2.56E-04 2.56E-04 1.76E-04

Defn u 1.98E-04 1.89E-04 1.59E-04 1.21E-04 8.58E-05 6.14E-05 3.35E-05 9.99E-06

piglet mn

Documents