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 particular engineering problem.
Technical support in relation to operation of the program, or in respect of engineering assistance, may be obtained from the author, who reserves the right to make a charge for such assistance.
Contact Address:
Department of Civil and Resource Engineering, The University of Western Australia, Nedlands, Western Australia 6907.
Telephone: Facsimile: Email:
+61 8 9380 3075 +61 8 9380 1044
[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
6.
7.
5.2 Interaction between laterally loaded piles
10
ANALYSIS OF PILE GROUP
11
6.1 Treatment of raking piles
11
6.2 Allowance for free-standing length of piles
12
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
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PART A: GENERAL DESCRIPTION 1.
INTRODUCTION
The computer program, PIGLET, analyses the load deformation response of pile groups under general 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 due allowance 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 with depth. No check of the overall stability of the pile group is made within the program; such calculations should form a separate part of the design. The program has developed gradually over the last ten years, with the doctoral research of the author (Randolph, 1977) forming the basis for the original version. This manual is the fifth revision of a report describing the program, originally published in 1980. The current manual is based on the version of PIGLET dated October, 1996. In order to minimise the amount of computation required, three separate 'scopes' of analysis are identified, depending on the type of loading to be applied to the group. The three cases are: (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 piles either pinned or built-in to the cap. In the first case, the user may also specify a fully flexible pile cap. The pile cap is assumed always to be clear of the ground surface, with no direct transfer of load to the ground. A non-zero 'free-standing' length of pile may be included between the pile cap and the effective ground surface. The only major geometric limitation on the pile group layout is that all the piles are assumed to be of the same length. This limitation arises out of the 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. This non-linearity may be modelled in an approximate fashion for the analysis of single piles by the use of load transfer methods of analysis, where the soil continuum is replaced by a series of springs acting along the length of the pile. Extension of such analysis to pile groups is only
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possible by adopting a hybrid soil model, combining elastic interactive effects with the load transfer analysis of each single pile (O'Neill et al, 1977). This approach is computationally laborious 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 calculating deformations 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 at the load and deformation levels determined. For pile groups of practical size (in terms of number of piles) additional deformation due to interactive effects will generally dominate that due to non-linear effects. A possible exception to this is where significant plastic deformation occurs between pile and soil. Although the soil has been assumed to deform elastically, less restriction has been imposed on the relative homogeneity of the soil deposit. It has been assumed that the soil may be modelled by a material where the stiffness varies linearly with depth. While this does not allow layered soil profiles to be treated rigorously, such deposits may be analysed by choosing a suitable average stiffness for the strata penetrated by the piles, and adopting a linear variation of stiffness with depth that reflects the general trend present in the actual profile. In addition, the special case of end-bearing (or partially end-bearing) piles has been catered for by the inclusion of a facility for specifying a soil of increased stiffness below the level of the pile bases. In summary, the soil is idealised as an elastic material where the stiffness varies as shown in Figure 1. The stiffness is characterised by a shear modulus, G, and Poisson's ratio, ν, (noting that the shear modulus is related to the Young's modulus, E, by E = 2(1 + ν)G ). The properties 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, ν, assumed constant with depth.
Treatment of the axial and lateral response of the piles independently allows additional freedom 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, and to allow for the high strains which occur locally near the head of a laterally loaded pile, it is advantageous to be able to specify different soil properties for the analysis of the lateral load deformation response. In particular, it is often advisable to adopt a value of zero for the shear modulus at ground level, Go, when considering lateral loading. In the program, the same soil properties are assumed for axial and torsional loading, while different values of shear modulus may be specified for lateral loading (retaining the same value of Poisson's ratio).
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3.
RESPONSE OF PILES TO AXIAL LOADING
3.1
Solution for single axially loaded piles
M.F. RANDOLPH
An approximate closed form solution for single axially loaded piles has been described in detail for floating piles by Randolph and Wroth (1978a), and extended to end-bearing piles by Randolph and Wroth (1978b). The solution is based on the technique of treating load transferred from the pile shaft separately from that at the pile base. The soil is effectively considered in two layers, divided by an imaginary line drawn at the level of the pile base (see Figure 2). The upper layer, above the line AB, is considered to be deformed solely by the shear stresses acting down the pile shaft, while the lower layer is deformed by the load transmitted 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 illustrate the 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 Pb 4 = G br b wb 1− ν
(1)
where P is the load, w the settlement and r the pile radius, the subscript b referring to the pile base. Turning to the pile shaft, considerations of vertical equilibrium entail that the shear stress, τ, at any depth falls off inversely with the radius, r, as (Cooke, 1974; Baguelin et al, 1975; Frank, 1974) τ r τ= o o r
(2)
where the subscript o denotes conditions at the pile shaft. This equation may be integrated to give the vertical deformation at any radius. In particular, if it is assumed that there is some radius, rm, at which the vertical deformations are effectively zero, then the settlement of the pile shaft may be written τ r ws = ζ o o G
(3)
where ζ = ln(rm/ro). This equation may be combined with equation (1) to give the overall load settlement ratio for a rigid pile of
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Pt 4η 2π = +ρ G r o wt (1 − ν)ξ ζ ro
(4)
where η = rb/ro is the ratio of underream, ξ = G /Gb is the ratio of endbearing, and the subscript t denotes conditions at the top of the pile.
Development of the full solution, which takes account of compression of the pile is given in detail by Randolph and Wroth (1978a). Effectively, equation (3) is taken to act at each point down the length of the pile, much as in a linear load transfer analysis. The final expression for the load settlement ratio is 4η 2π tanh(µ ) +ρ ζ µ (1 − ν)ξ Pt ro = µ tanh( ) 4η G r o wt 1+ πλ(1 − ν)ξ µ ro
(5)
where, summarising the various dimensionless parameters: η ξ ρ λ ζ µ
= = = = = =
rb/ro G /Gb G /G Ep/G ln(rm/ro) 2 / ζλ ( /ro)
(ratio of underream for underreamed piles) (ratio of end-bearing for end-bearing piles) (variation of soil modulus with depth) (pile-soil stiffness ratio) (measure of radius of influence of pile) (measure of pile compressibility).
It should be noted that Ep is the Young's modulus of a solid pile with equivalent cross-sectional rigidity to the actual pile. Thus Ep = (EA)p/(πro2), where (EA)p is the actual cross-sectional rigidity of the pile. A suitable expression for the maximum radius of influence, rm, is rm = {0.25 + ξ[2.5ρ(1 - ν) - 0.25]}
(6)
Figure 3 shows the variation of the load settlement ratio with slenderness ratio /ro for η = ξ = 1, ν = 0.3. It has been found that these values are in reasonably good agreement with those computed using charts from Poulos and Davis (1980), in spite of the simplifying assumptions adopted in the analytical solution given above, and making allowance for the possible scope for error when using the various multiplicative factors taken from the charts in Poulos and Davis. For long compressible piles, the results from Poulos and Davis, which are based on boundary 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 long piles, leading to numerical inaccuracies.
From Figure 3, it may be seen that there are combinations of slenderness ratio, /ro, and
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stiffness ratio, λ, beyond which the load settlement ratio becomes independent of the pile length. It can be shown that insignificant load is transmitted to the pile base for such long piles. This limiting behaviour is the converse of a stiff rigid pile, and corresponds to the case where the pile starts behaving as if it were infinitely long, with no load reaching the lower region. The two limits may be quantified. Piles may be taken as essentially rigid where /ro is less than 0.5(Ep/G )0⋅5. Equation (5) then reduces to equation (4). At the other extreme, for piles where /ro is greater than about 3(Ep/G )0⋅5, tanh(µ ) approaches unity and equation (5) reduces approximately (exactly for ρ = 1) to
Pt = πρ 2λ / ζ G r o wt
(7)
As expected, the load settlement ratio is now independent of the length of the pile (since no load reaches the lower end). The modulus G should be interpreted as the soil shear modulus at the bottom of the active part of the pile, that is, at a depth that corresponds to z/ro = 3(Ep/G )0⋅5, 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 suitable interaction factors to account for the proximity of other piles. However, such an approach ignores an important facet of group behaviour, which is the transfer of a higher percentage of load to the bases of piles within a group than for isolated piles. This phenomenon can be modelled by considering separately the interaction of the displacement fields around the pile shaft, from the corresponding interaction at the level of the pile bases. Interaction between neighbouring piles may be modelled (Cooke, 1974) by the superposition of the displacement fields of each pile. Thus the settlement of one pile may be thought of as made up of the sum of the settlement due to its own loading (without the presence of the neighbouring piles) and the settlements due to the displacement fields of each of the other piles . At the level of the pile bases, equation (1) gives the settlement of pile (i) due to its own loading, while the settlement due to a neighbouring pile (j), at spacing sij, may be approximated by
(wb )ij =
2r b 1− ν (P b )j (wb )j = ( ) 2πGb sij πs ij
(8)
In a similar manner, equation (3) is assumed to give the settlement of pile (i) due to its own loading, while the settlement due to a neighbouring pile (j) at spacing sij is approximated by
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τ r j (ws )ij = ln (r m / sij)( o o ) ρG
(9)
Equations (8) and (9) enable flexibility matrices to be formed relating the settlements at the pile bases to the corresponding loads, and likewise the settlements at each pile mid-depth to the average shear stresses down each pile shaft. For very stiff piles, the settlement is uniform down the pile length, and these matrices may be inverted, for a given distribution of pile head settlement, to give {Pb} and {τo} and hence the overall load settlement behaviour of the pile group. For compressible piles, the solution is more complex, as the values of wt, ws and wb will be different for any particular pile, due to compression of the pile. However, the compression of the pile may be calculated for a given load in the pile. Thus, after some matrix algebra described in detail by Randolph and Wroth (1979), it is possible to arrive at an overall flexibility matrix relating the pile head settlements {wt} to the total loads {Pt} taken by each pile. Comparison of the approximate method of analysis presented here with the results of more rigorous boundary element analysis shows good agreement. Figure 4(a) shows the values of Pt/(G rowt) for the three different pile positions in a 3 x 3 group of rigid piles embedded in a homogeneous soil, for a range of slenderness ratios, /ro. The boundary element analyses, obtained using the program, PGROUP (Banerjee and Driscoll, 1978) generally yield values of load settlement ratio which are some 10 % lower than the approximate analysis. For compressible piles, with a stiffness ratio of λ = 1000, the agreement is also good, as shown in Figure 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 analytical solution 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 pile base. At the pile base, the torque, T, may be related to the angle of twist, φ, using the established solution for the torsion of a rigid punch: Tb G br 3b φb
=
16 3
(10)
Down the pile shaft, it may be shown that the angle of twist is related to the interfacial shear stress, τo, by (Randolph, 1981b) 6
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φ=
M.F. RANDOLPH
τo 2G
(11)
For rigid piles, the above two equations may be combined to give an overall torsional stiffness of Tt 16 η3 = + 4 πρ 3 3ξ ro G r oφ t
(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 pile shaft. The situation is then similar to that for most laterally loaded piles, with the pile length no longer affecting the performance of the pile. For piles of intermediate length, the torsional stiffness may be written 16η3 tanh(µ ) + 4πρ µ Tt 3ξ ro = 3 3 G r oφ t 32 η G tanh(µ ) 1+ µ 3πρ ξG p 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 µ is now given by µ = 8G / Gp ( / r o ). The similarity of the above expression with that for axially
loaded piles (equation (5)) is evident. The torsional stiffness Tt/(Gro3φt) for homogeneous soil conditions is plotted against the stiffness ratio Gp/G for various pile slenderness ratios, /ro in Figure 5. The transition from flexible behaviour (where the pile length does not effect the stiffness), for /ro ≥ (Gp/G)0.5, to rigid behaviour for /ro ≤ 0.125(Gp/G)0.5, may be clearly seen. The limiting form of equation (13) for long piles is
Tt
= πρ 2Gp / G G r 3oφ t
(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 noted by Poulos (1975) are of benefit. Firstly, he showed from a series of model tests, that values of shear modulus for the soil, deduced from axial load tests, gave good predictions of the response of a pile under torsional loading. Thus, in choosing soil properties as input to
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PIGLET, the same shear modulus profile may be adopted for both axial and torsional response of the pile group. The second observation made by Poulos (1975) was that there was no evidence of an interaction effect between neighbouring piles under torsional loading. This finding conforms with what might be anticipated intuitively, and enables the torsional response of piles within a group to be estimated directly from the equations given above, with no additional factors to allow 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 torsionally loaded piles. Even for soil idealised as an elastic continuum, no simple closed form solution is forthcoming. The solution which has been adopted in the program is one developed by Randolph (1981a) by curve fitting the results of finite element analyses of laterally loaded piles embedded in elastic 'soil'. It was found that, for piles which behave flexibly under lateral load, simple power law relationships could be developed giving the lateral deflection, u, and the rotation, θ, of the pile at the soil surface, in terms of the pile stiffness and the soil properties. The relationships are similar in form to those arising from considering the soil as a Winkler 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. The term 'flexible' is taken to refer to piles where the load deformation characteristics would not be altered by increasing the length of the pile. Thus piles that are longer than their critical length behave as 'flexible' piles. The large majority of piles used in practice fall into this category. Since the solution is, by its nature, approximate, a further simplification has been introduced concerning the soil properties - the shear modulus, G, and Poisson's ratio, ν. Randolph (1977) showed that the effect of Poisson's ratio could be allowed for to sufficient accuracy by considering a single elastic property given by G* = G(1 + 3ν/4)
(15)
The solution detailed below is in terms of the single parameter G* rather than the true elastic parameters G and ν. The critical length of the pile is determined as
c
(
= 2r o E p / Gc
8
)2 / 7
(16)
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where Ep is the equivalent Young's modulus of the pile, given by Ep = (EI)p/(πro4/4)
(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 a stiffness varying linearly with depth as G = Go + mz
(18)
the parameter Gc is given by Gc = Go* + 0.5m* c = (1 + 3/4ν)(Go + 0.5m c)
(19)
The evaluation of the critical length from equations (16) and (19) requires some iteration except in the extreme cases of a homogeneous soil (where Gc = Go*) or a soil where the modulus 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, θ, at the soil surface may be evaluated as
(E p / Gc) u=
1/ 7
ρc Gc
(E p / Gc )
1/ 7
θ=
ρc Gc
H M 0.27 0.30 + ( c / 2) ( c / 2 )2 H + 0.80 ρ c 0.30 2 / 2 ( ) c
(
M 3 c / 2)
(20)
where H and M are, respectively, the lateral load and bending moment acting at the soil surface. The factor ρc gives the degree of homogeneity for the soil in a similar manner to the factor ρ in the analysis of axially loaded piles. It is conveniently defined as the ratio of the value of G* at a depth of c/4 to the value of G* at a depth of c/2 (see Figure 6). Thus
ρc =
Go* + m * c / 4 Go* + m* c / 4 = Gc Go* + m* c / 2
(21)
It should be noted that ρc varies from unity for a homogeneous soil down to 0.5 for a soil where the stiffness is proportional to depth. In equations (20), the product ρcGc is merely the value 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 soil stiffness at a depth of one quarter of the active, or critical, length of pile.
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Comparison of results calculated from equations (20) with existing solutions obtained by boundary element analyses shows good agreement over a wide range of pile-soil stiffness ratios. 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 similar treatment of the interaction between laterally loaded piles as was possible for axially loaded piles. 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 interaction factors, f, may be estimated to sufficient accuracy from the expression
(
α f = 0.6ρc E p / Gc
) (1+ cos2 β)rso 1/ 7
(22)
where s is the spacing between the axes of the piles and β is the angle which the direction of loading 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 head piles subjected to force loading (zero moment at the soil surface). In that case, it is found that the coefficient 0.6 in equation (22) should be replaced by 0.4 to give a reasonable fit to factors computed by Poulos' program DEFPIG (Poulos, 1980). In addition, at very close spacings, the 1/s variation of α 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 α is replaced by a parabolic variation wherever α is calculated to be greater than 1/3. To summarise, the interaction factor αuH, giving the increase in deflection for free head piles subjected to lateral load H, is calculated from
(
α = 0.4ρc E p / G c
) (1 + cos2 β)rso 1 /7
(23)
where α uH = α
for α ≤ 0.333
and α uH = 1 −
2 27α
for α > 0.333
Randolph (1981a) has compared values of αuH calculated from these expressions with values obtained from Poulos' program DEFPIG.
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The other interaction factors, αuM (deflection due to moment loading), αθH (rotation due to force loading) and αθM (rotation due to moment loading) may be estimated to sufficient accuracy by αuM = αθH ≈ αuH2 and
(24) αθM ≈ αuH
3
Poulos and Randolph (1983) have compared tabulated values of interaction factors obtained from this approach and from the boundary element program DEFPIG. In general, the agreement is reasonably good, with a tendency for interaction factors given by the present approach to decay more rapidly with increasing pile spacing, than shown by the DEFPIG results.
6.
ANALYSIS OF PILE GROUP
The separate solutions for axial, torsional and lateral response of piles must be combined in order to analyse a pile group under general loading conditions. Before outlining how this is achieved, two important practical features of pile groups must be catered for - namely the presence of raking piles, and the possibility of a free-standing length of pile between pile cap and 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 of interaction between neighbouring piles.
The treatment of the first of these effects in the analysis is straightforward. The solutions outlined in the previous sections are used to calculate the stiffness matrices in terms of local pile axes (i.e. in terms of axial, torsional and lateral loads and deflections). When the overall group stiffness matrix is formed, the coordinate axes of each pile are transformed to global axes (vertical and horizontal). It should be noted that the bending moments induced in the piles by horizontal loading are relatively sensitive to the angle of rake of the piles. The use of raking piles instead of vertical piles for a particular foundation may well enable economies to be 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, as must the danger of using raking piles in circumstances where large vertical movements of 11
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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 to spread the foundation load over a greater volume of soil. The ability of the program to cope with piles raking in any direction (rather than in any one particular vertical plane) is an important one, since it enables the true spacing between pile centres to be calculated at a given depth. Consider, for example, a square 2 x 2 group of piles with /ro = 40 and a pile spacing at ground level of s = 6ro. If the piles rake diagonally outwards at 1 in 8, the true average spacing down the shafts of the piles at adjacent corners is 11ro. If the analysis is restricted to piles raking in one direction only, the spacing between adjacent corner piles normal 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. For two piles which rake away from each other (see Figure 8), axial and lateral loading on pile A may be assumed to cause interactive displacement of pile B in mode (i) (Figure 8(b) - where the induced movements are parallel and normal to pile A) or in mode (ii) (Figure 8(c) - where the induced movements are parallel and normal to pile B). Poulos (1979) has discussed the merits of either choice in the analysis of pile groups with raking piles. He points out that the assumption of interaction in mode (ii) conforms with the reciprocal theorem of Betti, while that 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 present analysis. Adoption of this mode of interaction between piles has the additional advantage of enabling the axial load deformation behaviour of the piles to be considered independently from the lateral load deformation behaviour, before combining the two to obtain the overall deformation 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 should be ignored in the analysis of the load deformation characteristics of the pile group. In effect the pile cap is considered suspended above the top of the soil strata in which the piles are founded (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 the free-standing section of pile as a simple cantilever. New flexibility matrices are formed relating the deformations and loads at the underside of the pile cap before combining these to give the required load deformation characteristics of the complete group. To allow for the situation where the upper part of a pile is cased as it passes through softer soil, it is possible in the program to specify different pile properties in the free-standing section than in the main part of the pile.
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EXAMPLE APPLICATION
As an example of the application of PIGLET, model tests on a group of six piles embedded in sand (Davisson and Salley, 1970) have been analysed. Output from the analysis is given in Appendix A. A group of six tubular aluminium piles of external diameter 12.7 mm (0.5 in) and wall thickness 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 10 shows the pile layout and applied loads. Equivalent Young's modulus for the piles may be calculated 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, in order to determine the axial stiffness. Davisson and Salley (1970) report an average stiffness of 0.82 kN/mm (4860 lbf/in) with a standard deviation of 0.15 kN/mm (840 lbf/in). It is reasonable to assume that the shear modulus of the sand is proportional to the effective stress level (and thus to depth below the sand surface). With this assumption, and adopting a value for Poisson's ratio of 0.25, equation (5) may be used to deduce the shear modulus profile necessary to yield the above value of axial stiffness for the piles. This process leads to an expression 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 piles under 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 the tops of the six piles. These results compare favourably with those obtained from the program PIGLET, the error in the predicted bending moments and in the largest axial loads being generally less than 10%. The computed lateral deflection of 0.28 mm (0.011 in) is some 1.2 standard 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 taken from 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 this discrepancy may be attributed to the assumption of a homogeneous value of shear modulus for the soil in the PGROUP analysis. This assumption is likely to be a less good approximation for the sand than taking a shear modulus which is proportional to depth, and in this case leads to significant under-prediction of the bending moments induced in the piles.
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TABLE 1 COMPARISON WITH MODEL TEST RESULTS OF DAVISSON AND SALLEY (1970)
LOADS AT HEAD OF EACH PILE Pile No.
Axial Load (N) Meas'd
PIGLET PGROUP
Lateral Shear Load (N) Meas'd
PIGLET PGROUP
Bending Moment (Nm) 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
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PART B: PROGRAM DOCUMENTATION 8.
STRUCTURE OF PROGRAM
The program PIGLET has been structured so that the complexity, or 'scope', of analysis may be 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 output are 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 flexible cap, 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 running the program, different 'configurations' (of pile layout, or soil parameters) may be considered sequentially, merely changing the relevant data items between each analysis. For each configuration, a number of different loading cases may be considered. A flow chart for the program is shown in Figure 11. The only major branch point occurs for the case of vertical loading only (NSCOPE = 1), where the extra option of a fully flexible pile cap entails a different approach than for a rigid pile cap. As supplied, the maximum problem size is set to 300 piles, which requires about 1.5 Mb of RAM to run. Alternative versions can be supplied that can analyse larger groups, at the cost of greater memory requirements. The program will run in either a DOS or Windows environment.
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 switch between the various screens in order to expedite modification of any given set of data. When the program is run, the user has a choice of editing a previously saved datafile, or inputing new data interactively (starting with default values that are generally zero). After the data have been entered, there is a facility to save the data in an unformated datafile.
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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 the highlighted option. Limited help screens are available, by pressing the 'F1' key at any stage. The first help screen lists the various editing keys, with a brief description of the usual function of each key. A secondary 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 10 character field. Real items of data may be input as integers (without a decimal point), or as a real number containing a decimal point. With either form, an exponential scaling factor may be 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 to another, press 'Enter' or use the 'Up' or 'Down' arrows. For screens where data are entered in columns (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 as a 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 numeric input 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, additional notes are provided on the following pages to guide new users of the program. As in any engineering problem, a consistent set of units must be used in the input data. No system of units is assumed by the program. Essentially, the user must decide what unit of force (F) and what unit of length (L) are to be adopted. Data are then input in appropriate units according to the type of data. Thus values of modulus should be in units of F/L2, bending moments in FL, 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 when searching the current directory for alternative datafiles (or output files). When entering a 'general specification' of a file, an appropriate extension will be assumed, provided no extension (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 of data that may have been carried out.
Space bar
For numeric fields, the space bar clears all the remaining characters in the data field (to the right of, and including, the cursor). For loading options (load or deflection control, rigid or flexible cap) the space bar is used to toggle between the different options.
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Ctrl PgUp
Moves to the previous screen (adopting the data values displayed on the current screen).
Ctrl PgDn
Moves to the subsequent screen (adopting the data values displayed on the current 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 next screen 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 below current row
F1
Help screen (press F1 again for any secondary help screen, or Esc to revert to data entry).
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PIGLET MANUAL
M.F. RANDOLPH
Data Items
Problem Title and Scope The title may consist of up to 78 alphanumeric characters. The 'scope' of the problem has been discussed in Section 8 of the manual. Essentially it defines the complexity of the applied loading, with options of (a) vertical loading only, (b) vertical and horizontal loading in one plane, 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 standard version. As discussed earlier, a version catering for 500 or greater number of piles can be supplied, provided sufficient computer memory is available. The length of each pile is assumed the same, with an overall value equal to the sum of the embedded portion and a free-standing length (which may be zero). The Young's modulus of the pile is that of a pile having equivalent cross-sectional rigidity (for axial loading) or bending rigidity (for lateral loading) as the real pile. Thus, for a pile of radius ro, the value of Young's modulus for axial loading is Ep = (EA)p/(πro2) while for lateral loading, Ep = (EI)p/(πro4/4) In order to allow for the possibility of a change in pile cross-section at ground level, different values of Young's modulus may be specified for the free-standing lengths of pile. For torsional loading, the torsional rigidity of the pile is obtained from the bending rigidity, taking Poisson's ratio for the pile material as 0.3. For non-circular piles, it is important that the radius of the idealised pile is chosen realistically. It is suggested that the cross-sectional area of the idealised pile should be chosen so as to equal the gross (enclosed) area of the actual pile. For H section piles, the gross area should be taken as that of the encompassing rectangle. For lateral loading, there is a choice between whether the piles are to be assumed fixed into the 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. In addition, where lateral loading is involved, angles of rake must be specified in radians, either in 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
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co-ordinates (and angles of rake where appropriate) are to be edited. However, it is possible to 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 using the '>' edit key to copy the new radius to every other pile in the group. Note that it is possible to add or delete piles on this screen (overriding the number of piles set on the previous screen). The program assumes a right-handed set of coordinate axes (x, y, z), with the z axis pointing vertically 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 shows this 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 for lateral loading (the profile for torsional loading is assumed the same as for axial loading). As discussed in Section 2 of the manual, the shear modulus profile is assumed to increase linearly with 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 sudden increase in modulus at the base of the pile (for end-bearing piles) may be input. If this value is set to less than the value that would be calculated from the linear variation of shear modulus, then the program corrects it to that value (thus the program does not permit an abrupt 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 is chosen so as to reflect the true average shear modulus over the depth of penetration of the piles, and also the trend of variation of soil modulus with depth. Since piles deflect under lateral loading only in the upper ten diameters or so, it is possible to specify different values of 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 significantly stiffer stratum of soil. While such piles are not strictly 'end-bearing' piles, the stiffer stratum of 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 modulus below the pile bases, Gb, is chosen by means of the expression (Lee, 1991)
1 1 G z = 1 − e1 − h / = + 1 − Gb Gh G h G 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
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M.F. RANDOLPH
chosen by inspection of the available soil data. For cohesive soil, it is common practice to correlate shear modulus with the shear strength su. At working load levels, the axial deformation of piles may be estimated reasonably well by taking shear modulus values in the range 200 ≤ G/su ≤ 400 Under lateral loading, the high strains which occur in the soil close to the pile give rise to lower 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 penetration tests, 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 with depth in sand (below the water table) may be expressed as G = mz, with m in the range 1 MPa/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 of soft rocks such as chalk can be extremely high, installation of bored or driven piles tends to break up the block structure of the rock. The relevant shear modulus is then that associated with large strains (see Wakeling, 1970; Randolph and Wroth, 1978b). Further guidance on the 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 specified explicitly (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 loads or 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').
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M.F. RANDOLPH
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 are assumed 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) must be specified for each pile. This may be accomplished by specifying a uniform load (or deflection) for each pile - using the 'U' option - and then changing individual values on required piles. For lateral loading, in addition to loads or deflections specified for the pile cap, a mixed mode of 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 fixing moments 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, and moments are taken as positive in the sense of rotating the x axis towards the z axis (for loading in the x:z plane) and rotating the y axis towards the z axis (for loading in the y:z plane). This sign convention differs from the usual right-handed axis rule, as indicated in Figure 13. Profiles of Bending Moment and Lateral Deflection For analyses which involve lateral loading, profiles of bending moments and lateral deflection relative to the immediately surrounding soil may be output for specified piles. A choice is given as to whether (a) no profiles, (b) profiles of bending moment only, or (c) profiles of bending moment and lateral deflection, are required. For three-dimensional loading, separate choices 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 profiles are to be calculated (using the space bar as a toggle, and the edit keys to move from pile to pile). It should be emphasised that, since the free field soil deflections (due to interaction between piles) are not included in the relative lateral deflection profile, the deflection output for the pile head will not correspond with the total lateral deflection for the pile head (except for analyses with only one pile in the group).
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10. PROGRAM OUTPUT Output from the program consists of lines of up to 120 characters, with up to 60 lines per page. 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 each pile, and (optionally) profiles of bending moment and lateral deflection down specified piles.
(5)
Where more than a single load case is analysed, summary tables of output are included where the loads and deflections at the head of each pile are summarised for each load case.
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 output may 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 first instance, and this file is subsequently printed where required. The file may be edited using any standard text editor. The sign convention for lateral loads and moments for each pile follows that for specifying the applied loads, with lateral load being taken as positive in the direction of the positive x and y axes, and moments taken as positive in the sense of rotating the x axis towards the z axis (for loading in the x:z plane) and rotating the y axis towards the z axis (for loading in the y:z plane).
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REFERENCES 1.
Baguelin F., Bustamante M., Frank R. and Jezequel J.F. (1975), 'La capacite portante des 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 loaded piles 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 pile groups 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 piles and pile groups', CIRIA Technical Note 108.
5.
Cooke R.W. (1974), 'Settlement of friction pile foundations', Proc. Conf. on Tall Buildings, Kuala Lumpur, 7-19.
6.
Davisson M.T. and Salley J.R. (1970), 'Model study of laterally loaded piles', J. of Soil Mech. 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), 'Piling Engineering', 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 Curie University).
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 pile groups with non-linear soil response and pile-soil pile interaction', Proc. 9th Offshore Technology Conf., Vol 2, 245-256.
12.
Poulos H.G. (1971), 'Behaviour of laterally loaded piles, I - Single piles, II - Pile groups', J. Soil Mech and Found. Engng Div., ASCE, Vol 97, No SM5.
13.
Poulos H.G. (1973), 'Load-deflection prediction for laterally loaded piles', Australian Geomechanics Journal, Vol G3, No 1.
14.
Poulos H.G. (1975), 'Torsional response of piles', J. Geot. Engng Div., ASCE, Vol 101, No GT10.
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M.F. RANDOLPH
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 pile groups', 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. of Geot. 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 loaded piles', 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 the analysis 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 pile groups', Geotechnique, Vol 29, No 4.
26.
Reese L.C. and Matlock H. (1956), 'Non-dimensional solutions for laterally loaded piles', Proc. 8th Texas Conf. on Soil Mech.
27.
Wakeling T.R.M. (1970), 'A comparison of the results of standard site investigation methods against the results of a detailed geotechnical investigation in Middle Chalk at Mundford, Norfolk', Proc. Conf. on In Situ Investigations in Soils and Rocks, British Geotechnical Society, London.
28.
Wroth C.P., Randolph M.F., Houlsby G.T. and Fahey M. (1979), 'A review of the engineering properties of soils with particular reference to the shear modulus', Cambridge University Engineering Department Research Report, CUED/D - Soils TR 75.
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M.F. RANDOLPH
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 in homogeneous soil
Figure 7
Plan view of two piles subjected to lateral loading
Figure 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
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PIGLET MANUAL
FIGURES
26
M.F. RANDOLPH
OCTOBER 1996
PIGLET MANUAL
M.F. RANDOLPH
Shear modulus, G
Solid cylindrical pile Radius: ro Equivalent modulus, Ep
Gavg = ρG
G
Depth, z
Figure 1 Assumed variation of soil shear modulus with depth
27
Gb ≥ G
OCTOBER 1996
PIGLET MANUAL
M.F. RANDOLPH
Pt Pt = Ps + Pb
Shaft response
A
B
Ps
A
B Pb
A'
B' Base response
Figure 2 Uncoupling of effects due to pile shaft and base
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PIGLET MANUAL
M.F. RANDOLPH
60 Pt G ro w t 50
3000
40 1000
30 300
20 100
10
30
λ = 10
0 1
10 100 Pile slenderness ratio, /ro
1000
(a) ρ = 0.5 80 Pt 70 G ro w t 60 50 40 30 20 10 0
3000
1000
300 100 30
λ = 10
1
10 100 Pile slenderness ratio, /ro
1000
(b) ρ = 0.75
Pt G ro w t
100
3000
80 60
1000
40
300 100
20 λ = 10
0 1
30
10 100 Pile slenderness ratio, /ro
(c) ρ = 1 Figure 3 Load settlement ratios for axially loaded piles 29
1000
OCTOBER 1996
PIGLET MANUAL
M.F. RANDOLPH
Figure 4 Comparison of load-settlement ratios for piles in a 3 x 3 pile group in homogeneous soil 30
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M.F. RANDOLPH
10000
Tt G r o3 φt
/ro = 200 100 50 25
1000
10
100
10
1 1
10
100
1000 10000 100000 1E+06 λ = Gp/G
Figure 5 Torsional stiffness factors for piles in homogeneous soil
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M.F. RANDOLPH
Modified shear modulus, G*
Gc
G* = G(1 + 3ν/4)
Solid cylindrical pile Radius: ro Equivalent modulus, Ep Ep = 2 ro G c
c
c/2
Gz =
2/7
c
Depth, z
Figure 6 Notation for analysis of laterally loaded pile
32
c /4
= ρcGc
OCTOBER 1996
PIGLET MANUAL
M.F. RANDOLPH
s β
Figure 7 Plan view of two piles subjected to lateral loading
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OCTOBER 1996
PIGLET MANUAL
uA
αwA
wA
Pile A
M.F. RANDOLPH
αuA
Pile B
(a) Interactive displacements of Pile B parallel to Pile A
αuA uA
αwA
wA
Pile A
Pile B
(a) Interactive displacements of Pile B axial and lateral
Figure 8 Choice of modes for interaction between pairs of non-parallel piles
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Pile cap
Depth of free-standing section of piles Level of bearing strata Penetration of piles into bearing strata
Piles
Figure 9 Allowance for free-standing length of piles
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M.F. RANDOLPH
0.222 kN 0.138 kN 76 mm
fine, dry sand 3
piles 0.533 m long
127 mm Plan View
1
Figure 10 Model pile test arrangement (Davisson and Salley, 1970)
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INDATA Read (new) data
Start
VSTIF Axial load-deformation response of pile group VERTLD Flexibility and stiffness of group, response to load cases (vertical loading only)
1
NSCOPE 2, 3 HSTIF Lateral load-deformation response (x:z plane)
2
NSCOPE 3 HSTIF Lateral load-deformation response (y:z plane) TSTIF Torsional load-deformation response of pile group FORMGS Form terms in overall group stiffness matrix GENLD Flexibility and stiffness of group; response to load cases
Yes
No
Modify data?
Figure 11 Flow chart for PIGLET
37
Stop
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M.F. RANDOLPH
x
x y
Plan view y
z
Negative rake
Elevation - x:z plane
z Negative rake
Positive rake
Positive rake
Elevation - y:z plane
Figure 12 Sign convention for pile rake
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PIGLET MANUAL
M.F. RANDOLPH
x
x
Px
Px
Mx
My to z Mx to z
Tx to y
Mz
My
Py
Py y
Pz
y
Pz
Note: Mx to z = -My
z
z
(a) Conventional right-hand notation
(b) PIGLET notation
Figure 13 Sign convention for loading
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PIGLET MANUAL
APPENDIX 1 - EXAMPLE OUTPUT
40
M.F. RANDOLPH
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PIGLET MANUAL
Output from Pile Group Analysis Program
-
PIGLET
M.F. RANDOLPH
Version dated October, 1996
Analysis of group of six model piles in sand - Davisson and Salley (1970)
Page
Example output for program manual
******************************************************************************* ******************************************************************************* **
**
**
**
**
PPPPPPP
IIIIII
GGGGGG
**
PP
PP
II
GG
**
PP
PP
II
GG
**
PPPPPPP
II
GG
**
PP
II
GG
**
PP
II
GG
**
PP
IIIIII
TTTTTTTT
**
EE
TT
**
LL
EE
TT
**
LL
EEEEE
TT
**
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41
0
OCTOBER 1996
PIGLET MANUAL
Output from Pile Group Analysis Program
-
PIGLET
M.F. RANDOLPH
Version dated October, 1996
Analysis of group of six model piles in sand - Davisson and Salley (1970)
Pile group analysis for
Page
Example output for program manual
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
42
1
OCTOBER 1996
PIGLET MANUAL
Output from Pile Group Analysis Program
-
PIGLET
M.F. RANDOLPH
Version dated October, 1996
Analysis of group of six model piles in sand - Davisson and Salley (1970)
-
G(0) = dG/dz = G(b) =
0.000E+00 4.300E+03 2.292E+03
Nu = 0.250
Lateral load-deformation - G(0) = dG/dz =
0.000E+00 4.300E+03
Nu = 0.250
Parameters for axial load-deformation behaviour are:
Shear modulus at level of pile bases is G(L) = Rho = G(L/2)/G(L) =
Xi = G(L)/G(b) =
2.292E+03
5.000E-01
Shear modulus below pile bases is G(b) =
2.292E+03
1.000E+00
Poissons ratio is nu =
0.250
Rm = (0.25+xi*(2.5*rho*(1-nu)-0.25))*L + Pile stiffness ratio is Epa/G(L) =
7.839E-02 =
5.781E-01
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) = Rhoc=G(Lc/4)/G(Lc/2) =
Critical depth is Lc =
6.809E+02
5.000E-01
Critical slenderness ratio is Sc =
4.200E+01
2.667E-01
Lateral flexibilities (isolated, at mudline) of first pile are: u/H =
2.725E-02
th/H or u/M = th/M =
2.271E-01
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
43
2
Example output for program manual
Soil properties are:
Axial load-deformation
Page
OCTOBER 1996
PIGLET MANUAL
Output from Pile Group Analysis Program
-
PIGLET
Version dated October, 1996
Analysis of group of six model piles in sand - Davisson and Salley (1970)
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
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)
1.9988E+00
For unit horizontal deflection of pile cap - in x direction:
Pile
Axial
Lateral
Moments
no,
loads
loads (x)
(x to z)
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)
M.F. RANDOLPH
For unit rotation of pile cap - x towards z about y axis:
Pile
Axial
Lateral
Moments
no,
loads
loads (x)
(x to z)
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
44
Page
3
Example output for program manual
OCTOBER 1996
PIGLET MANUAL
Output from Pile Group Analysis Program
-
PIGLET
M.F. RANDOLPH
Version dated October, 1996
Analysis of group of six model piles in sand - Davisson and Salley (1970)
Overall group stiffness matrix is :
Total
Total
Total
vertical
horizontal
moment
load
load (x)
(x to z)
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
Unit vertical deflection
Overall group flexibility matrix is :
Unit vertical load
Unit horizontal load (x)
Unit moment (x to z)
Vertical
Horizontal
Rotation (x
deflection
deflection
to z about
x = y = 0.0
(x dir.)
y axis)
3.9183E-04
-2.2756E-11
2.3463E-10
-2.2756E-11
2.1913E-03
-3.6407E-03
2.3463E-10
-3.6407E-03
2.9293E-02
45
Page
4
Example output for program manual
OCTOBER 1996
PIGLET MANUAL
Output from Pile Group Analysis Program
-
PIGLET
Version dated October, 1996
Analysis of group of six model piles in sand - Davisson and Salley (1970)
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 to z)
8.6986E-05
(x)
2.8164E-04
M.F. RANDOLPH
-3.3544E-04
Pile
Axial
Lateral
Moments
no,
loads
loads (x)
(x to z)
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
46
Page
5
Example output for program manual
OCTOBER 1996
PIGLET MANUAL
Output from Pile Group Analysis Program
-
PIGLET
M.F. RANDOLPH
Version dated October, 1996
Analysis of group of six model piles in sand - Davisson and Salley (1970)
Load case no.
1 out of
Page
Example output for program manual
1
Profiles of bending moments in the (x,z) plane and (optionally) lateral deflections (relative to soil) in the x direction for specified
Pile number
Depth
3 piles
1
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
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
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
Pile number
Depth
Pile number
Depth
3
5
47
6
