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Pile foundation modelling for inelastic earthquake analyses of large structures
Pile foundation modelling for inelastic earthquake analyses of large structures Stavros A. A n a g n o s t o p o u l o s
Institute of Engineering Seismology and Earthquake Engineering, Hapsa 1, Thessaloniki, Greece (Received May 1982)
Modelling of pile foundations for nonlinear earthquake response analyses of steel template offshore structures is examined, related problems are discussed and practical solutions are recommended. A parametric investigation is then used as the basis of assessing the effects of uncertainty in some of the parameters on the response of the foundation and superstructu re. Key words: earthquakes, pile foundations, earthquake response analysis
Introduction Pile foundations are generally used to support structures on soft soil with low bearing capacity. Offshore softs generally fall into this category, and for this reason a major class of pile-supported structures are the Fixed steel platforms used for developing offshore oil and gas. Following the criteria in API-RF2A, ~offshore structures in regions of high seismicity should be designed for two levels of earthquake shaking: the strength and the ductility level earthquake. The latter represents an extreme and rare event, for which the structure should have sufficient ductility to avoid collapse by dissipating seismic energy through inelastic action. Analysis requirements for the ductility level earthquake call for realistic representations of the structure-foundation system, which should adequately model the interaction with the surrounding soil masses and preserve the basic characteristics of the structural response. This paper examines current analytical models that are used to simulate pile foundations of steel offshore structures for inelastic earthquake response analyses of the complete structure-foundation system. Problems and possible inconsistencies in selecting the parameters required to describe the hysteretic force-deformation and energy dissipation characteristics at the soil-pile interface are discussed and practical solutions are recommended. Because some of these parameters cannot be measured directly but must be derived from other soil properties, which may be either inherently uncertain or load-dependent, there is usually substantial uncertainty associated with mathematical idealizations of pile foundations in nonlinear seismic response analyses. The effects of such uncertainties on the response of the superstructure and foundation are
investigated, and recommendations are then made as to which parameters are important and require careful consideration. It is not the purpose of this paper to recommend new analytical procedures for studying the behaviour of piles isolated from the structure. Rather, its objective is to provide the designer-analyst with information necessary to correctly model the soil-pile interface, under the constraints of existing computer programs for extreme earthquake response analysis of the complete structurefoundation system. General considerations The most rehable method for evaluating the survivability of structures under extreme earthquake shaking is to carry out inelastic dynamic analyses of detailed structurefoundation models, using several earthquake records selected on the basis of site-specific seismicity studies. For systems with a large number of degrees of freedom, however, solutions of this type can be extremely expensive. Thus, economic considerations are a primary factor that will dictate solution strategies, modelling details and the degree of sophistication to be sought. To make such analyses feasible it will be almost imperative to use first a simple, inexpensive model for most of the preliminary investigations (for example, to screen out candidate input motions, and to investigate various simplifying modelling assumptions), and to use then a detailed model to check survivability of the structure with a minimum of computer runs. Typically, discrete-element formulations have been used for inelastic earthquake studies, in which the piles are
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Eng. Struct., 1983, Vol. 5, July 215
Pile foundation modelling: S A Anagnostopoulos divided into segments by introducing mass nodes along their axes. 2-4 These nodes are supported by near-field soil elements (three-component spring-dashpots) that simulate the force-deformation characteristics of the surrounding soil and energy dissipation due to material and radiation (geometric) damping. The far ends of these elements can either have prescribed motion or be connected to free-field elements. Free-field soil elements may be used to account for local soil conditions at the site. 4 Alternatively, if the computer program has the capability to accept independent support excitations, it will be more economical to carry out the free-field amplification studies separately and then use the resulting motions at the elevations of the near-field elements as input. The third, and simplest, alternative will be to have the same motion prescribed at all the far ends of the near-field soils. In this case, the free-field motion at the mudline or at some other appropriate horizon should be used. This horizon can be the centroid of lateral and axial soil support reactions for horizontal and vertical motions, respectively. Typically, the lateral soil reaction is centred 7-15 m below the mudline, while the vertical soil reaction is centred in the bottom third of the pile. It should be realized, of course, that the approach to be followed depends on the philosophy and procedures used to arrive at the input motions selected for the analysis. For example, if a real surface recording from a site very similar to the site of the structure has been selected as input, it may not be very appropriate to include free-field soil elements in the analysis (owing to the danger of accounting twice for local soil effects). For the near-field discretization, the soil can be divided into layers and nodal points may be selected along the piles at the middle of each layer. To minimize the number of degrees of freedom, the thickness of the layers should be variable, increasing gradually towards the tip of the pile. In three-dimensional models, consideration should be given to the fictitious increase in soil strength along directions of pile movement other than the x and y directions used to specify the nonlinear force-deformation curves of the near-field elements.
Force-deformation characteristics Lateral response Force-deformation information on the lateral response of piles can be taken from the design guidelines in APIRP2A, which give load-deflection (p-y) curves for soft clays and sands as functions of soil properties and depth. These curves, an example of which is sketched in Figure 1, have been developed on the basis of field and laboratory tests for monotonic and cyclic loads, s' 6 Available algorithms 3,4 can simulate the degradation associated with cyclic loading as well as the formation and closing of gaps, which has been observed in field experiments for certain types of clay. Gap formation takes place in a surface zone of essentially unconfined response, where the soil is moulded as the pile moves back and forth owing to load reversals. Recently, certain modifications to Matlock's p-y curves for soft clays have been proposed, based on the analysis of experimental information from several pile tests. 7 These modifications result in 'stiffer' p-y curves with greater ultimate resistance and lead to better agreement with a wider set of experimental results. For other types of soil, such as stiff clays, silts etc., there are no standard, well-established p-y curves. The analyst must
216 Eng. Struct., 1983, Vol. 5, July
I
Z
1.0
Gap
O5
i
L
m~O
l/
@
/
Z >~Zcr
10 b . . . . . . . .
0
3
8
Y/Yc //
Yc
-0.5
-0,5
-10
Figure I
Lateral load-deflection curves for soft clays
l O12"-p°r AP'-RP2A
I/ 0
,o L °°r AP'-RP2A
J / 10
Z/Z c
l
o'i 0
10
Z/Zc
Skin friction
End b e a r i n g Figure 2 Axial load-deflection curves for clays and sands (a), skin friction; (b), end bearing
consult the literature (for example, references 8 and 9) and then use a great deal of judgement. The p-y curves in API-RP2A are based on statically applied loadings and thus proper consideration should be given to the strain-rate effects associated with seismic excitations. These effects seem to increase the stiffness and strength substantially] ° For the purpose of the analyses considered here, an increase in soil strength and stiffness of the order of 30 per cent appears to be a reasonable choice.
Axial response Soil resistance to the axial movement of a pile is due to skin friction and end bearing. The force/deformation (f-z) characteristics for both types of action are nonlinear and depend on several factors such as soil properties, pile diameter, rate and history of loading etc. Experimental results suggest that these force-deformation characteristics may be adequately represented by an elastic, perfectly plastic relationship u-14 as shown in Figure 2. Ultimate unit skin friction and end bearing capacities for different types of soil are recommended in API-RP2A. What is not so well established is the idealized elastic stiffness or the yield displacement at which full capacity is mobilized. An expression for the yield displacement at which maximum skin friction is reached has been given in reference 15, based on linear soil behaviour. More recently, theoreticalf-z curves have also been derived in reference 16. For the present work, an analysis along the lines of
Pile foundation modelling: S. A. Anagnostopoulos the area of the triangle (012). This leads to the following expression for z e :
references 15 and 16 was carried out, using a hyperbolic stress-strain curve for the soil, while allowing the possibility for the maximum skin friction to be less than the shear strength of the soil. This analysis is based on the assumptions that the soil around the pile deforms in pure shear and that soil deformations are negligible beyond a radial distance of 10ro from the pile axis, where ro is the pile radius)s With these assumptions, the axial displacement of a pile segment is:
zc =
2cro Go
F(fu/C )
(5)
in which F ( f u / c ) is a somewhat complicated function of fu/c. For engineering purposes, this function can be approximated in the interval 0 ~< fu/C ~< 1 by:
F(fu/C ) ~ 1.87(fu/c) ''s 10r o tQ
and so :
3' dr
z = ~
(1)
*/
z e ~ 3.7 - Go
ro
in which 3' is the shear strain in the soil. The hyperbolic stress-strain relationship for the soil is given by: r
3' -
C
-
(2)
-
¢
--+3' Go in which 7. is the shear stress, c is the shear strength, and Go is the shear modulus at zero strain. Using simple statics, we can express the shear stress r at a distance r from the pile axis in terms of the shear stress 7.o at the soil-pile interface: ro
7. = 7.0 -
(3) r
Making use of equations (2) and (3) and carrying out the integration, we obtain: 7.0 10---z = 7.oro In c Go 7"0
(6)
This expression gives values for z e that are in the range of the experimental data 1~ and should be viewed only as an estimate of a parameter that can vary considerably. It will be seen later, however, that its effect on the response of the superstructure or foundation is not significant. Even less important is the effect of the yield displacement of the elastoplastic end-bearing element, because for long piles, such as those used for offshore structures, practically all the resistance is due to skin friction. In reference 11, a value of yield displacement in the range of 0.04d to 0.06d (d = pile diameter) is recommended for end bearing in either clay or sand. As in the lateral response, strain-rate effects appear to increase the ultimate axial pile resistance) ° The same factor (1.3) used for lateral response may also be used for axial response. Damping
(4)
1----
¢
I f f n is the unit skin friction capacity, then the yield displacement z e of the elastoplastic axial support may be determined by equating the shaded area in Figure 3 with
There are two main sources of damping in the soil: material and radiation damping. Material damping is predominantly of a hysteretic nature and is explicitly accounted for by the nonlinear force-deformation relationships of the near field elements (except at low strains at which the nonlinear soil reactions have been linearized). Radiation damping is due to energy dissipation by waves propagating in the half-space. This type of damping is simulated by the viscous dashpots associated with the lateral and axial nearfield soil elements. Rough estimates of the dashpot constants can be obtained by making use of available solutions for piles embedded in an elastic soil medium. Under strong earthquake motions, however, the soil in the vicinity of the pile becomes highly nonlinear and thus the validity of elastic solutions is at best questionable. Unfortunately, there does not seem to be any alternative at present for the problem at hand. Only for an isolated, single pile has an attempt been made to explore the effects of soil nonlinearity on pile dynamics using finite elements, a transmitting boundary and an iteratively linear procedure. 17 Under these limitations, estimates of the dashpot constants can be derived as follows.
Lateral dashpo ts The complex, horizontal reaction per unit length of pile is expressed by: as
O Figure 3
Zc
fh = G ( S ~ + i s ~ ) u ( z , t) z
Determination of yield displacement for elastoplastic skin friction spring
(7)
where San , 2h are functions of dimensionless frequency ao( = coro~/(p/G)) and Poisson s ratio, G is the shear modulus of the soil medium, u(z, t) is the lateral pile
Eng. Struct., 1983, Vol. 5, July 217
Pile foundation modelling: S. A. Anagnostopoulos displacement (harmonic) at depth z from the mudline, i = x / - 1 , co is the excitation frequency, ro is the pile radius, and p is the soil density. The imaginary part of this reaction represents the radiation damping force fhd, which can be written as:
End-bearing dashpots Using the complex reaction at the tip of the pile, 21 we can obtain as before the following equation for the dashpot constant Cb of the end-bearing element: Cb
GS2h OU(Z, t) fhd --
(8)
60 Ot because, for harmonic motion, au(z, t)/Ot = i~ou(z, t). It follows then that the dashpot constant Ch corresponding to a pile segment of length L is:
(16)
ao
For Poisson's ratio of 0.5, S2b/a o can be approximated by: zl S2b
~-- 7.414--2.986ao+4.324aZ--l.782a~
(17)
t/o
GS2hL Ch - - -
(9)
or, in terms of the frequency parameter ao: Ch = --S2h VsproL
: --S2b Vspr2o
(10)
In the frequency range of practical interest, say for 0 ~< ao ~ 1.5, this ratio varies from 7.41 to 6.65. It is therefore recommended that for practical applications the dashpot constant Cb of the end-bearing element be estimated from:
ao
where Vs = x/(G/p) is the shear-wave velocity of the soil medium. In the range 0 ~< ao ~< 1.5, which covers the frequency range of interest for earthquake excitations, and for Poisson's ratio of 0.4, the ratio S2h/a o c a n be approximated by :~9 S2h 56.55 - - -~ 0.96 + ao 4.68 + ao
(11)
This ratio will vary between 10.11 and 13.04 for ao in the range 0 ~< ao ~< 1.5. For typical soil properties and pile sizes, and for the lower periods that dominate translational structural response, the ratio S2h/ao is very close to 13. Therefore, the frequency independent dashpot constant for lateral motion may be computed, for practical purposes, by:
Ch ~- 13VsproL
(12)
Skin-friction dashpots The foregoing derivation can also be applied to axial pile motion, leading to the following equation for the constant Cv of the axial skin-friction dashpots: S2v Cv = VsProL
(13)
Cb ~- 7Vspr~
(18)
The shear wave velocity in equations (12), (15) and (18) should reflect free-field conditions during the earthquake; that is, it should correspond to a shear modulus G compatible with the strains induced in the free-field by the motion considered. For clays, however, typical values of G associated with shear wave propagation owing to seismic excitations are of the order of 300c to 1000c (reference 22), while the values of G implied by the forcedeformation curves, i.e. G = c/3eso, are of the order of 33c to 100c (for eso = 0.003 to 0.01). As a result, the dashpots will be unrealistically 'stiff' compared with the p-y curves. The resulting inconsistency suggests a parameter uncertainty inherent in the spring-dashpot idealization. For cases where the spring stiffness is reduced with increasing deformations, the inconsistency becomes even greater. This latter problem can be alleviated by making the dashpot constants proportional to the tangent stiffness of the springs, that is by using C' = C(St/So ) instead of C, where St and So are the tangent and initial spring stiffness, respectively. The foregoing discussion suggests that damping is perhaps the most uncertain of all the foundation parameters, and thus it will be prudent to bound the solution by examining a range of damping values.
ao
Mass
For any value of Poisson's ratio, S2v/ao can be approximated in the range 0 ~< ao ~< 2 by: 2° S2v 0.7022 - - ~- 6.059 + ao 0.01616+a0
(14)
It can be seen that in this case the ratio S2v/ao and consequently the dashpot constant Cv is more frequencydependent than the horizontal dashpot constant Ch (in the frequency ranges of interest). In practical problems, the pile radius is generally greater than 0.60 m, the fundamental structural period for vertical motion usually less than about 0.5 s and the shear wave velocity in alluvium deposits under earthquake-induced strains may be of the order of 300 m/s or less. This establishes a rough upper limit for the ratio S2v/ao equal to 23, while a lower limit is 6.4 for ao = 2 (from the validity range of equation (14)). For practical applications requiring frequency-independent dashpots, it is then suggested that an average value be used, that is:
Cv ~- 15VsproL 218 Eng. Struct., 1983, Vol. 5, July
(15)
The mass to be assigned to nodes along a pile should include the mass of the pile, the mass of soil inside the pile and some added (or virtual) mass representing the portion of surrounding soils accelerated by the pile. The added mass is not easy to compute. Moreover, it is not a constant quantity, but varies with the induced deformations. Rough estimates may be obtained by assuming a strain field around the pile and integrating the squares of the corresponding displacements over the volume of the assumed strain field. In this manner, the added mass m a per unit length of pile may be expressed as: ma
: CmPnr2° ~- P l f u2dA
(19)
A
where Cm is the added mass coefficient, p the soil density, ro the pile radius, u the displacement field in the soil (normalized to unity at the pile surface) and A the near field area contributing to the added mass.
Pile foundation modelling: S. A. Anagnostopoulos
Lateral motion For lateral motion, the displacement field resulting from the Mindlin solution for a horizontal force P acting parallel to the x axis within a linear, isotropic half-space, was assumed.: To further simplify the computations, displacement components perpendicular to the direction of loading were neglected. For Poisson's ratio equal to 0.5, Mindlin's equation takes the form (in cylindrical coordinates):
Table2
Values of added mass coefficient for axial motion
X
0
0.5
0.67
0.8
0.95
6 8 10
3.89 5.80 7.90
3.08 4.52 6.09
2.65 3.87 5.21
2.21 3.21 4.32
1.35 1.96 2.65
2, = radius of integration/pile radius; # = "rmax/C
3P II 1
1
s o.` + s
o,z) =
[
+ r~ cos~0 ~ -
+
1 .s
+
2bz~ SI~.s] 1
(20) 6bz~t
(20)
where u x is the horizontal displacement of the half-space, E is Young's modulus for the half-space, b is the depth of the force P from the free surface, and $1 = r: + (z-- b) ~ and S: = r ~ + (z + b) 2. Added mass coefficients have been obtained from equations (19) and (20) for various depths (b = z = kro) and circular areas of integration (r = Xro), and these are summarized in Table 1. The rather wide range of values should not cause any great concern, because the dynamic response is quite insensitive to changes in the mass assoCiated with nodes along the piles.: It should also be noted that as the near field soils become softer owing to cyclic pile motions, the added mass coefficient decreases, becoming equal to 1.0 in the extreme case that the soil fluidizes. For practical applications, a constant value of Cm ~- 3.0 appears to be a reasonable choice.
.......
WATERLEVEL
E --- ~¢.~._,
~/'~S':q
--- ~-so~-sol a-so-
~,-so~ ~-so-~ ~-so-
F
,
\
,,.
~'6~........
, . . . . . . . . . . . . . . .
so-~ so-~ .so%
rh
rh
1
X--#
~-so
in which ro is the pile radius, r is the distance from the pile axis, and X and # are parameters defining the various displacement distributions around the pile. X determines the zone of influence (radial distance = Xro) and # is a parameter in the interval 0 ~ ~ < 1 to simulate soil softening effects as they would be materialized by a hyperbolic stress/strain with ro/C = #. Results are summarized in Table 2 for different values of X and #. It can be seen that as the soft around the pile softens, the added mass coefficient decreases. For practical applications, a value of Cm ~ 4.0 may be considered as reasonable. Again, it should
Tab/e 1 motion
Values of added mass coefficient for horizontal pile
k ~.
2
5
10
20
50
100
6 8 10 20
3.25 3.94 4.49 6.30
2.86 3.51 4.07 5.98
2.53 3.07 3.54 5.23
2.34 2.79 3.16 4.51
2.22 2.60 2.91 3.95
2.17 2.54 2.83 3.76
h = radius of integration/pile radius k = depth of pile segment/pile radius
~--so-
r
"1
SO=
o
Axial motion For axial motion, several displacement fields have also been used'to establish a range of values for the added mass coefficient. These displacement fields are described by: =
.OOL,.E
H ...... ,,,H,,,.,
~-so-~
s~-~
l
Figure 4 Platform model
be kept in mind that the dynamic response of the structure and its foundation is not sensitive to even large variations in the mass associated with nodes along the piles. Parameter s t u d y
A limRed parameter study aimed at investigating the effects of some of the uncertainties discussed earlier was carried out, by performing several analyses of the platform model shown in Figure 4. For these analyses, an augmented version of the DRAIN-2D computer program w a s u s e d . 23 The soil prof'fle from which near-field and free-field dement properties were computed is shown in Figure 5. Two different earthquakes were employed: the 1966 Park field-Array No. 2 and a deconvoluted version of the 1971 Pacoima Dam recording. These are both near-field recordings. The strongest horizontal component of each of these recordings had been used earlier in separate free-field amplification studies to excite the soil profile in Figure 5 and produce site-appropriate motions at the discretization levels along the piles. For these studies, the soil stress/ strain curves had been idealized by Ramberg-Osgood
Eng. Struct., 1983, Vol. 5, July
219
Pile foundation modelling: S. A. Anagnostopoulos Properties Depth (m)
Layer
Soil t y p e
Thickness (m)
q5
C (kPa)
es0
Mudline
Pile m o d e l
0.0
--10.0
-20.0
-
1
Sand (loose)
3.0
30 °
-
-
2
Clay
4.4
-
57.4
0.018
3 . . . . .
Silt
3.8 . . . . . .
36°------- 57.4-----0.018
4
Sand ( m e d i u m )
4.3
36 °
-
-
5
Sand ( m e d i u m )
7.3
37 °
-
--
6
Sand ( m e d i u m )
7.7
36 °
--
7
Clay
20.7
-
191.0
0.018
8
Clay
17.1
-
191.0
0.018
9
Clay
15.7
-
191.0
0.018
. . . . . .
3 0 . 0
-40.0
--50.0
-60.0
-70.0
-80.0
A, Figure 5
Soil p r o f i l e
relationships. The horizontal motion of the soil at 2.5 m below the mudhne and the as-recorded (unmodified) vertical component were used as global excitations (that is, at all support points) for all the cases analysed, except one. For this, the motions obtained at each discretization level were used as multiple support excitations. Spring-dashpot properties were computed as described earlier, with dashpot constants based on shear moduti estimated for compatibility with the near-field force-deformation curves (for example, Gso = c/3eso for clays). In all but one of the analyses, the dashpots were initial-stiffness proportional. The results are summarized in Table 3, which also contains brief descriptions of the cases examined. These are in reference to a standard model symbolized as (A) and listed in the first line. For example, the second line in the table is for a variation of the standard model having the strength and stiffness of the lateral springs 50% higher, the third line is for a variation having half the strength and stiffness of the standard lateral springs, and so on. In each case, only the properties stated in the table were different from corresponding properties of the standard model. For the nineth and tenth cases, the shear wave velocities used to determine dashpot constants were based on the free-field shear modulus variation corresponding to two-thirds the maximum free-field shear strains (average for the two earthquakes). This gave substantially higher dashpot constants for all axial dashpots (three to five times) as well for the lateral dashpots in the clay zones (three to eight times), while it reduced the lateral dashpots in the middle sand layers by up to 50%.
220
Eng. S t r u c t . , 1 9 8 3 , V o l . 5, J u l y
The response variables selected for comparison are the maximum horizontal structural displacement, maximum rotational ductility factor for legs and piles, and the number of brace failures. Although these four variables alone can hardly give a complete picture of the response, they do indicate the severity of inelastic action and allow a quantitative comparison between the various cases. It must be noted, though, that for members carrying large axial loads, as do legs and piles of offshore platforms, the ductility factor alone is not a good index of potential failure, since the plastic rotation capacity of tubular members is substantially reduced in the presence of axial loads. However, all the cases examined here showed insignificant variations of maximum axial forces in legs and piles, and thus it was decided that the ductility factor could be an adequate index for purposes of comparison. In the lower part of the table, the member numbers of the struts that failed are listed for the various cases. As a general observation it can be noted that both earthquakes produced the same qualitative trends. This can be explained by the fact that the two motions have rather similar characteristics, especially since they are both filtered through the same soil profile. Capacity (or strength) changes in either lateral or axial springs appear to cause noticeable changes in superstructure response, particularly in the legs. The pile response is also affected, but to a lesser degree. On the other hand, changes in yield displacements (or stiffness) of either lateral or axial springs, while keeping capacities the same, have very little effect on either the structural or the pile responses. This suggests that it suffices
Pile foundation modelling: S. A. Anagnostopoulos Table 3
Results
(a) Summary Parkfield quake
Case 1 2 3 4 5 6 7 8 9 10 11 12 13
Standard properties: (A) strength and stiffness: 1.5 x (A) Lateral springs strength and stiffness: 0.5 x (A) yield displacements: 1.5 x (A) yield displacements: 0.5 x (A) strength and stiffness: 1.5 x (A) Axial springs strength and stiffness: 0.5 x (A) yield displacements: 1.5 x (A) yield displacements: 0.5 x (A) Lateral deshpots from free field Axial deshpots from free field Dashpotstangent Multiple support excitation
Pacoima Dam quake
~ (cm)
Legs /~r
Struts N.F.
Piles ~ur
86.3 99.3 73.5 86.0 86.5 87.2 75.7 88.4 82.7 87.7 85.1 82.3 91.2"
3.36 5.12 1.63 3.23 3.53 3.53 EL 3.47 3.05 4.22 3.29 2.27 2.70
7 4 5 7 7 7 0 7 7 7 7 5 5
3.04 2.72 2.11 2.95 3.04 3.14 2.23 2.95 3.09 2.08 3.00 3.64 3.02
6 (cm) 92.0 96.0 85.3 91.9 92.7 94.2 75.7 92.4 90.3 92.4 90.7 90.2 116.8"
Legs /~r
Struts N.F.
Piles /~r
3.95 4.76 2.59 3.87 4.13 4.10 1.01 4.04 3.82 5.00 3.87 3.18 3.40
6 5 5 6 7 7 0 6 6 5 6 5 5
2.80 2.63 2.76 2.77 2.87 2.80 1.96 2.72 2.65 1.81 2.77 3.49 2.50
(b) Details of failed struts Case no.
Parkfield quake
Pacoima Dam quake
1 2 3 4 5 6 7 8 9 10 11 12 13
9, 12, 13, 14, 15, 16, 30 12, 13, 16,30 13, 14, 15, 16, 30 Same as case 1 Same as case 1 Same as case 1 No failures Same as case 1 Same as case 1 Same as case 1 Same as case 1 9, 12, 14, 16, 30 9, 12, 14, 16, 30
10, 11, 13, 14, 11, 13, 14, 15, 10, 11, 13, 15, Same as case 1 10, 11, 13, 14, 10, 11, 13.14, No failures Same as case 1 Same as case 1 10, 11, 13, 15, Same as case 1 10, 11,13, 15, 10, 11, 13, 15,
5 = maximum structural displacement; #r = ductility factor = 1 + (Max. pl. hinge E L = members elastic. * Absolute displacement
to use only rough estimates of yield displacements for all support springs. Increases in the values of lateral dashpot constants, as a result of using shear wave velocities compatible with freefield strains, reduce, as might be expected, plastic deformations in the piles and force more inelastic action into the jacket legs. The observed differences are similar, if not greater, in size to the differences caused by varying the strength and stiffness of the p - y curves. On the other hand, changes in axial dashpot constants have practically no effect on any of the response variables examined, because the lateral motion is the dominant component of excitation. Use of tangent stiffness-proportional dashpots implies a decrease in damping and a subsequent increase in pile response. This produced a softer support for the jacket and resulted in lower superstructure response, following a 'soft story' type of behaviour. For the last case examined, the horizontal motions at the nine discretization levels were applied at the ends of the near-field elements as independent support excitations, along with the unmodified vertical component. The result was a rather small reduction in the overall response, in agreement with intuition, which suggests that phase differences among the input motions at the various levels would have a beneficial effect. This should not, however, be considered as a general trend. For a different structure, soil
rotation)/(MpL/6EI);
15, 30 30 30 15, 16, 30 15, 16, 30
30 30 30
N.F. = number of members failed;
profile and earthquake excitations, opposite results may be obtained.
Conclusions The results of the present study are strictly applicable to the structure and motions considered herein. In a qualitative sense, however, they indicate certain trends and thus may prove useful in suggesting which foundation parameters may require more reliable data. In summary, it was found that the structural as well as pile response is much more sensitive to variations in strengths (that is, lateral bearing or skin friction capacities) than yield displacements (stiffness). Changes in lateral dashpot constants caused noticeable changes in the response, but similar changes in axial dashpots had practically no effect on it. Also, the difference between the responses resulting from the use of initial or tangent stiffness-proportional dashpots suggests a need for checking both possibilities, since neither idealization can simulate the real phenomenon correctly. In addition, there is an apparent need for parametric variation of the dashpot constants, because of incompatibility between free.field shear moduli and elastic stiffness of the p - y curves for clays. Finally, the use of multiple support excitations to reflect motion modification with depth should be considered, but only in
Eng. St r uct . , 1983, V o l . 5, J u l y
221
Pile foundation modelling: S. A. Anagnostopoulos r e l a t i o n to t h e p h i l o s o p h y e m p l o y e d in selecting i n p u t motions.
Acknowledgment This paper is based on experience gained by the author when working for Shell Development Co. The free-field motion analyses were performed by Earle Doyle of Shell Development Co. References 1 2 3 4 5 6 7 8 9
222
'API recommended practice for planning, designing and constructing fixed offshore platforms', API-RP2A, 1 l t h edn, American Petroleum Institute, Washington DC, 1980 Penzien, J. 'Soft-pile foundation interaction' in Earthquake Engineering (ed. Wiegel, P. L.), Prentice-Hall, Englewood Cliffs, NJ, 1970, pp. 349-381 Marshall, P. W. et al. ' Inelastic dynamic analysis of tubular offshore structures', Proc. Of]shore Technology Conf., OTC 2908, III, 1977, pp. 235-246 Arnold, P. et al. 'SPSS - a study of soil-prie-structure systems in severe earthquakes', Proc. Offshore Technology Conf., OTC 2749, I, 1977, pp. 189-202 Matlock, H. 'Correlations for design of laterally loaded piles in soft clay', Proc. Offshore Technology Conf., OTC 1204, 1970, pp. 577-594 Reece, L. C. et al. 'Analysis of laterally loaded piles in sand', Proc. Offshore Technology Conf., OTC 2080, 1974, pp. 473483 Stevens, J. B. and Audibert, J. M. E. 'Re-examination of ply curve formulations', Proc. Offshore Technology Conf., OTC 3402, 1979, pp. 397-403 Reece, L. C. et al. ' Field testing and analysis of laterally loaded pries in stiff clay', Proc. Offshore Technology Con]., OTC 2312, 1975, pp. 671-690 Baguelin, F. and Jezequel, J. 'Etude experimentale du comportement des pieux sollicites hofizontalement', Bulletin de Liaison de Laboratoire des Ponts et Chaussees, No. 62, 1972, pp. 129-169
Eng. Struct., 1983, Vol. 5, July
10 11 12 13 14 15 16 17 18
19 20 21 22 23
Bea, R. G. et al. 'Dynamic response of laterally and axially loaded piles', Proc. Offshore Technology Conf., OTC 3749, 1980, pp. 129-139 Vijayvergiya, V. N. ' Load-movement characteristics of piles', paper presented at PORTS 77 Conference, 1977 Bea, R. G. and Doyle, E. H. 'Parameters affecting axial capacity of piles in clays', Proc. Offshore Technology (¥mf, OTC 2307, 1975, pp. 611-623 Coyle, H. M. and Reece, L. C. 'Load transfer for axially loaded pries in clay', J. Soil Mech. Found. Div., ASCE, 1966, 92 (SM2), 1 Coyle, H. M. and Sulalman, J. H. 'Skin friction for steel pries in sand', J. SoilMech. Found. Div., ASCE, 1967, 93 (SM6), 261 Holmquist, D. V. and Maltock, H. 'Resistance-displacement relationships for axially-loaded piles in soft clay', Proc. Offshore Technology Conf., OTC 2474, 1976, pp. 553-569 Kraft, L. M. et al. 'Theoretical t/z curves', J. Geot. Engr. Div., ASCE, 1981, 107 ( G T l l ) , 1543 Angelides, D. and Roesset, J. M. 'Nonlinear lateral dynamic stiffness of piles',J. Geot. Engr. Div., ASCE, 1981, 107 (GT11), 1443 Baranov, V. A. 'On the calculation of excited vibrations of an embedded foundation' (in Russian), Voprosy Dynamiki i Prochnocti, No. 14, Polytechnic Inst. of Riga, 1967, pp. 195-209 Beredugo, Y. O. and Novak, M. 'Coupled horizontal and rocking v~ration of embedded footings', Can. Geot. J., 1972, 9,477 Novak, M. and Beredugo, Y. O. 'Vertical vibration of embedded footings', J. Soil Mech. Found. Div., ASCE, 1972, 98 (SM12), 1291 Novak, M. 'Vertical vibration of floating piles', ,L Eng. Mech. Div., ASCE, 1977, 103 (EM1), 153 Seed, H. B. and Idriss, I. M. 'Soft moduli and damping factors for dynamic response analyses', Report EERC 70-10, Earthquake Engng Res. Center, Univ. of California, Berkeley, 1970 Canaan, A. E. and Powel, G. H. 'General purpose computer program for inelastic dynamic response of plane structures', Report EERC 73-6, Earthquake Engng Res. Center, Univ. of California, Berkeley, 1973
Institute of Engineering Seismology and Earthquake Engineering, Hapsa 1, Thessaloniki, Greece (Received May 1982)
Modelling of pile foundations for nonlinear earthquake response analyses of steel template offshore structures is examined, related problems are discussed and practical solutions are recommended. A parametric investigation is then used as the basis of assessing the effects of uncertainty in some of the parameters on the response of the foundation and superstructu re. Key words: earthquakes, pile foundations, earthquake response analysis
Introduction Pile foundations are generally used to support structures on soft soil with low bearing capacity. Offshore softs generally fall into this category, and for this reason a major class of pile-supported structures are the Fixed steel platforms used for developing offshore oil and gas. Following the criteria in API-RF2A, ~offshore structures in regions of high seismicity should be designed for two levels of earthquake shaking: the strength and the ductility level earthquake. The latter represents an extreme and rare event, for which the structure should have sufficient ductility to avoid collapse by dissipating seismic energy through inelastic action. Analysis requirements for the ductility level earthquake call for realistic representations of the structure-foundation system, which should adequately model the interaction with the surrounding soil masses and preserve the basic characteristics of the structural response. This paper examines current analytical models that are used to simulate pile foundations of steel offshore structures for inelastic earthquake response analyses of the complete structure-foundation system. Problems and possible inconsistencies in selecting the parameters required to describe the hysteretic force-deformation and energy dissipation characteristics at the soil-pile interface are discussed and practical solutions are recommended. Because some of these parameters cannot be measured directly but must be derived from other soil properties, which may be either inherently uncertain or load-dependent, there is usually substantial uncertainty associated with mathematical idealizations of pile foundations in nonlinear seismic response analyses. The effects of such uncertainties on the response of the superstructure and foundation are
investigated, and recommendations are then made as to which parameters are important and require careful consideration. It is not the purpose of this paper to recommend new analytical procedures for studying the behaviour of piles isolated from the structure. Rather, its objective is to provide the designer-analyst with information necessary to correctly model the soil-pile interface, under the constraints of existing computer programs for extreme earthquake response analysis of the complete structurefoundation system. General considerations The most rehable method for evaluating the survivability of structures under extreme earthquake shaking is to carry out inelastic dynamic analyses of detailed structurefoundation models, using several earthquake records selected on the basis of site-specific seismicity studies. For systems with a large number of degrees of freedom, however, solutions of this type can be extremely expensive. Thus, economic considerations are a primary factor that will dictate solution strategies, modelling details and the degree of sophistication to be sought. To make such analyses feasible it will be almost imperative to use first a simple, inexpensive model for most of the preliminary investigations (for example, to screen out candidate input motions, and to investigate various simplifying modelling assumptions), and to use then a detailed model to check survivability of the structure with a minimum of computer runs. Typically, discrete-element formulations have been used for inelastic earthquake studies, in which the piles are
0141-0296/83/03215-08/$3.00 © 1983 Butterworth & Co. (Publishers) Ltd
Eng. Struct., 1983, Vol. 5, July 215
Pile foundation modelling: S A Anagnostopoulos divided into segments by introducing mass nodes along their axes. 2-4 These nodes are supported by near-field soil elements (three-component spring-dashpots) that simulate the force-deformation characteristics of the surrounding soil and energy dissipation due to material and radiation (geometric) damping. The far ends of these elements can either have prescribed motion or be connected to free-field elements. Free-field soil elements may be used to account for local soil conditions at the site. 4 Alternatively, if the computer program has the capability to accept independent support excitations, it will be more economical to carry out the free-field amplification studies separately and then use the resulting motions at the elevations of the near-field elements as input. The third, and simplest, alternative will be to have the same motion prescribed at all the far ends of the near-field soils. In this case, the free-field motion at the mudline or at some other appropriate horizon should be used. This horizon can be the centroid of lateral and axial soil support reactions for horizontal and vertical motions, respectively. Typically, the lateral soil reaction is centred 7-15 m below the mudline, while the vertical soil reaction is centred in the bottom third of the pile. It should be realized, of course, that the approach to be followed depends on the philosophy and procedures used to arrive at the input motions selected for the analysis. For example, if a real surface recording from a site very similar to the site of the structure has been selected as input, it may not be very appropriate to include free-field soil elements in the analysis (owing to the danger of accounting twice for local soil effects). For the near-field discretization, the soil can be divided into layers and nodal points may be selected along the piles at the middle of each layer. To minimize the number of degrees of freedom, the thickness of the layers should be variable, increasing gradually towards the tip of the pile. In three-dimensional models, consideration should be given to the fictitious increase in soil strength along directions of pile movement other than the x and y directions used to specify the nonlinear force-deformation curves of the near-field elements.
Force-deformation characteristics Lateral response Force-deformation information on the lateral response of piles can be taken from the design guidelines in APIRP2A, which give load-deflection (p-y) curves for soft clays and sands as functions of soil properties and depth. These curves, an example of which is sketched in Figure 1, have been developed on the basis of field and laboratory tests for monotonic and cyclic loads, s' 6 Available algorithms 3,4 can simulate the degradation associated with cyclic loading as well as the formation and closing of gaps, which has been observed in field experiments for certain types of clay. Gap formation takes place in a surface zone of essentially unconfined response, where the soil is moulded as the pile moves back and forth owing to load reversals. Recently, certain modifications to Matlock's p-y curves for soft clays have been proposed, based on the analysis of experimental information from several pile tests. 7 These modifications result in 'stiffer' p-y curves with greater ultimate resistance and lead to better agreement with a wider set of experimental results. For other types of soil, such as stiff clays, silts etc., there are no standard, well-established p-y curves. The analyst must
216 Eng. Struct., 1983, Vol. 5, July
I
Z
1.0
Gap
O5
i
L
m~O
l/
@
/
Z >~Zcr
10 b . . . . . . . .
0
3
8
Y/Yc //
Yc
-0.5
-0,5
-10
Figure I
Lateral load-deflection curves for soft clays
l O12"-p°r AP'-RP2A
I/ 0
,o L °°r AP'-RP2A
J / 10
Z/Z c
l
o'i 0
10
Z/Zc
Skin friction
End b e a r i n g Figure 2 Axial load-deflection curves for clays and sands (a), skin friction; (b), end bearing
consult the literature (for example, references 8 and 9) and then use a great deal of judgement. The p-y curves in API-RP2A are based on statically applied loadings and thus proper consideration should be given to the strain-rate effects associated with seismic excitations. These effects seem to increase the stiffness and strength substantially] ° For the purpose of the analyses considered here, an increase in soil strength and stiffness of the order of 30 per cent appears to be a reasonable choice.
Axial response Soil resistance to the axial movement of a pile is due to skin friction and end bearing. The force/deformation (f-z) characteristics for both types of action are nonlinear and depend on several factors such as soil properties, pile diameter, rate and history of loading etc. Experimental results suggest that these force-deformation characteristics may be adequately represented by an elastic, perfectly plastic relationship u-14 as shown in Figure 2. Ultimate unit skin friction and end bearing capacities for different types of soil are recommended in API-RP2A. What is not so well established is the idealized elastic stiffness or the yield displacement at which full capacity is mobilized. An expression for the yield displacement at which maximum skin friction is reached has been given in reference 15, based on linear soil behaviour. More recently, theoreticalf-z curves have also been derived in reference 16. For the present work, an analysis along the lines of
Pile foundation modelling: S. A. Anagnostopoulos the area of the triangle (012). This leads to the following expression for z e :
references 15 and 16 was carried out, using a hyperbolic stress-strain curve for the soil, while allowing the possibility for the maximum skin friction to be less than the shear strength of the soil. This analysis is based on the assumptions that the soil around the pile deforms in pure shear and that soil deformations are negligible beyond a radial distance of 10ro from the pile axis, where ro is the pile radius)s With these assumptions, the axial displacement of a pile segment is:
zc =
2cro Go
F(fu/C )
(5)
in which F ( f u / c ) is a somewhat complicated function of fu/c. For engineering purposes, this function can be approximated in the interval 0 ~< fu/C ~< 1 by:
F(fu/C ) ~ 1.87(fu/c) ''s 10r o tQ
and so :
3' dr
z = ~
(1)
*/
z e ~ 3.7 - Go
ro
in which 3' is the shear strain in the soil. The hyperbolic stress-strain relationship for the soil is given by: r
3' -
C
-
(2)
-
¢
--+3' Go in which 7. is the shear stress, c is the shear strength, and Go is the shear modulus at zero strain. Using simple statics, we can express the shear stress r at a distance r from the pile axis in terms of the shear stress 7.o at the soil-pile interface: ro
7. = 7.0 -
(3) r
Making use of equations (2) and (3) and carrying out the integration, we obtain: 7.0 10---z = 7.oro In c Go 7"0
(6)
This expression gives values for z e that are in the range of the experimental data 1~ and should be viewed only as an estimate of a parameter that can vary considerably. It will be seen later, however, that its effect on the response of the superstructure or foundation is not significant. Even less important is the effect of the yield displacement of the elastoplastic end-bearing element, because for long piles, such as those used for offshore structures, practically all the resistance is due to skin friction. In reference 11, a value of yield displacement in the range of 0.04d to 0.06d (d = pile diameter) is recommended for end bearing in either clay or sand. As in the lateral response, strain-rate effects appear to increase the ultimate axial pile resistance) ° The same factor (1.3) used for lateral response may also be used for axial response. Damping
(4)
1----
¢
I f f n is the unit skin friction capacity, then the yield displacement z e of the elastoplastic axial support may be determined by equating the shaded area in Figure 3 with
There are two main sources of damping in the soil: material and radiation damping. Material damping is predominantly of a hysteretic nature and is explicitly accounted for by the nonlinear force-deformation relationships of the near field elements (except at low strains at which the nonlinear soil reactions have been linearized). Radiation damping is due to energy dissipation by waves propagating in the half-space. This type of damping is simulated by the viscous dashpots associated with the lateral and axial nearfield soil elements. Rough estimates of the dashpot constants can be obtained by making use of available solutions for piles embedded in an elastic soil medium. Under strong earthquake motions, however, the soil in the vicinity of the pile becomes highly nonlinear and thus the validity of elastic solutions is at best questionable. Unfortunately, there does not seem to be any alternative at present for the problem at hand. Only for an isolated, single pile has an attempt been made to explore the effects of soil nonlinearity on pile dynamics using finite elements, a transmitting boundary and an iteratively linear procedure. 17 Under these limitations, estimates of the dashpot constants can be derived as follows.
Lateral dashpo ts The complex, horizontal reaction per unit length of pile is expressed by: as
O Figure 3
Zc
fh = G ( S ~ + i s ~ ) u ( z , t) z
Determination of yield displacement for elastoplastic skin friction spring
(7)
where San , 2h are functions of dimensionless frequency ao( = coro~/(p/G)) and Poisson s ratio, G is the shear modulus of the soil medium, u(z, t) is the lateral pile
Eng. Struct., 1983, Vol. 5, July 217
Pile foundation modelling: S. A. Anagnostopoulos displacement (harmonic) at depth z from the mudline, i = x / - 1 , co is the excitation frequency, ro is the pile radius, and p is the soil density. The imaginary part of this reaction represents the radiation damping force fhd, which can be written as:
End-bearing dashpots Using the complex reaction at the tip of the pile, 21 we can obtain as before the following equation for the dashpot constant Cb of the end-bearing element: Cb
GS2h OU(Z, t) fhd --
(8)
60 Ot because, for harmonic motion, au(z, t)/Ot = i~ou(z, t). It follows then that the dashpot constant Ch corresponding to a pile segment of length L is:
(16)
ao
For Poisson's ratio of 0.5, S2b/a o can be approximated by: zl S2b
~-- 7.414--2.986ao+4.324aZ--l.782a~
(17)
t/o
GS2hL Ch - - -
(9)
or, in terms of the frequency parameter ao: Ch = --S2h VsproL
: --S2b Vspr2o
(10)
In the frequency range of practical interest, say for 0 ~< ao ~ 1.5, this ratio varies from 7.41 to 6.65. It is therefore recommended that for practical applications the dashpot constant Cb of the end-bearing element be estimated from:
ao
where Vs = x/(G/p) is the shear-wave velocity of the soil medium. In the range 0 ~< ao ~< 1.5, which covers the frequency range of interest for earthquake excitations, and for Poisson's ratio of 0.4, the ratio S2h/a o c a n be approximated by :~9 S2h 56.55 - - -~ 0.96 + ao 4.68 + ao
(11)
This ratio will vary between 10.11 and 13.04 for ao in the range 0 ~< ao ~< 1.5. For typical soil properties and pile sizes, and for the lower periods that dominate translational structural response, the ratio S2h/ao is very close to 13. Therefore, the frequency independent dashpot constant for lateral motion may be computed, for practical purposes, by:
Ch ~- 13VsproL
(12)
Skin-friction dashpots The foregoing derivation can also be applied to axial pile motion, leading to the following equation for the constant Cv of the axial skin-friction dashpots: S2v Cv = VsProL
(13)
Cb ~- 7Vspr~
(18)
The shear wave velocity in equations (12), (15) and (18) should reflect free-field conditions during the earthquake; that is, it should correspond to a shear modulus G compatible with the strains induced in the free-field by the motion considered. For clays, however, typical values of G associated with shear wave propagation owing to seismic excitations are of the order of 300c to 1000c (reference 22), while the values of G implied by the forcedeformation curves, i.e. G = c/3eso, are of the order of 33c to 100c (for eso = 0.003 to 0.01). As a result, the dashpots will be unrealistically 'stiff' compared with the p-y curves. The resulting inconsistency suggests a parameter uncertainty inherent in the spring-dashpot idealization. For cases where the spring stiffness is reduced with increasing deformations, the inconsistency becomes even greater. This latter problem can be alleviated by making the dashpot constants proportional to the tangent stiffness of the springs, that is by using C' = C(St/So ) instead of C, where St and So are the tangent and initial spring stiffness, respectively. The foregoing discussion suggests that damping is perhaps the most uncertain of all the foundation parameters, and thus it will be prudent to bound the solution by examining a range of damping values.
ao
Mass
For any value of Poisson's ratio, S2v/ao can be approximated in the range 0 ~< ao ~< 2 by: 2° S2v 0.7022 - - ~- 6.059 + ao 0.01616+a0
(14)
It can be seen that in this case the ratio S2v/ao and consequently the dashpot constant Cv is more frequencydependent than the horizontal dashpot constant Ch (in the frequency ranges of interest). In practical problems, the pile radius is generally greater than 0.60 m, the fundamental structural period for vertical motion usually less than about 0.5 s and the shear wave velocity in alluvium deposits under earthquake-induced strains may be of the order of 300 m/s or less. This establishes a rough upper limit for the ratio S2v/ao equal to 23, while a lower limit is 6.4 for ao = 2 (from the validity range of equation (14)). For practical applications requiring frequency-independent dashpots, it is then suggested that an average value be used, that is:
Cv ~- 15VsproL 218 Eng. Struct., 1983, Vol. 5, July
(15)
The mass to be assigned to nodes along a pile should include the mass of the pile, the mass of soil inside the pile and some added (or virtual) mass representing the portion of surrounding soils accelerated by the pile. The added mass is not easy to compute. Moreover, it is not a constant quantity, but varies with the induced deformations. Rough estimates may be obtained by assuming a strain field around the pile and integrating the squares of the corresponding displacements over the volume of the assumed strain field. In this manner, the added mass m a per unit length of pile may be expressed as: ma
: CmPnr2° ~- P l f u2dA
(19)
A
where Cm is the added mass coefficient, p the soil density, ro the pile radius, u the displacement field in the soil (normalized to unity at the pile surface) and A the near field area contributing to the added mass.
Pile foundation modelling: S. A. Anagnostopoulos
Lateral motion For lateral motion, the displacement field resulting from the Mindlin solution for a horizontal force P acting parallel to the x axis within a linear, isotropic half-space, was assumed.: To further simplify the computations, displacement components perpendicular to the direction of loading were neglected. For Poisson's ratio equal to 0.5, Mindlin's equation takes the form (in cylindrical coordinates):
Table2
Values of added mass coefficient for axial motion
X
0
0.5
0.67
0.8
0.95
6 8 10
3.89 5.80 7.90
3.08 4.52 6.09
2.65 3.87 5.21
2.21 3.21 4.32
1.35 1.96 2.65
2, = radius of integration/pile radius; # = "rmax/C
3P II 1
1
s o.` + s
o,z) =
[
+ r~ cos~0 ~ -
+
1 .s
+
2bz~ SI~.s] 1
(20) 6bz~t
(20)
where u x is the horizontal displacement of the half-space, E is Young's modulus for the half-space, b is the depth of the force P from the free surface, and $1 = r: + (z-- b) ~ and S: = r ~ + (z + b) 2. Added mass coefficients have been obtained from equations (19) and (20) for various depths (b = z = kro) and circular areas of integration (r = Xro), and these are summarized in Table 1. The rather wide range of values should not cause any great concern, because the dynamic response is quite insensitive to changes in the mass assoCiated with nodes along the piles.: It should also be noted that as the near field soils become softer owing to cyclic pile motions, the added mass coefficient decreases, becoming equal to 1.0 in the extreme case that the soil fluidizes. For practical applications, a constant value of Cm ~- 3.0 appears to be a reasonable choice.
.......
WATERLEVEL
E --- ~¢.~._,
~/'~S':q
--- ~-so~-sol a-so-
~,-so~ ~-so-~ ~-so-
F
,
\
,,.
~'6~........
, . . . . . . . . . . . . . . .
so-~ so-~ .so%
rh
rh
1
X--#
~-so
in which ro is the pile radius, r is the distance from the pile axis, and X and # are parameters defining the various displacement distributions around the pile. X determines the zone of influence (radial distance = Xro) and # is a parameter in the interval 0 ~ ~ < 1 to simulate soil softening effects as they would be materialized by a hyperbolic stress/strain with ro/C = #. Results are summarized in Table 2 for different values of X and #. It can be seen that as the soft around the pile softens, the added mass coefficient decreases. For practical applications, a value of Cm ~ 4.0 may be considered as reasonable. Again, it should
Tab/e 1 motion
Values of added mass coefficient for horizontal pile
k ~.
2
5
10
20
50
100
6 8 10 20
3.25 3.94 4.49 6.30
2.86 3.51 4.07 5.98
2.53 3.07 3.54 5.23
2.34 2.79 3.16 4.51
2.22 2.60 2.91 3.95
2.17 2.54 2.83 3.76
h = radius of integration/pile radius k = depth of pile segment/pile radius
~--so-
r
"1
SO=
o
Axial motion For axial motion, several displacement fields have also been used'to establish a range of values for the added mass coefficient. These displacement fields are described by: =
.OOL,.E
H ...... ,,,H,,,.,
~-so-~
s~-~
l
Figure 4 Platform model
be kept in mind that the dynamic response of the structure and its foundation is not sensitive to even large variations in the mass associated with nodes along the piles. Parameter s t u d y
A limRed parameter study aimed at investigating the effects of some of the uncertainties discussed earlier was carried out, by performing several analyses of the platform model shown in Figure 4. For these analyses, an augmented version of the DRAIN-2D computer program w a s u s e d . 23 The soil prof'fle from which near-field and free-field dement properties were computed is shown in Figure 5. Two different earthquakes were employed: the 1966 Park field-Array No. 2 and a deconvoluted version of the 1971 Pacoima Dam recording. These are both near-field recordings. The strongest horizontal component of each of these recordings had been used earlier in separate free-field amplification studies to excite the soil profile in Figure 5 and produce site-appropriate motions at the discretization levels along the piles. For these studies, the soil stress/ strain curves had been idealized by Ramberg-Osgood
Eng. Struct., 1983, Vol. 5, July
219
Pile foundation modelling: S. A. Anagnostopoulos Properties Depth (m)
Layer
Soil t y p e
Thickness (m)
q5
C (kPa)
es0
Mudline
Pile m o d e l
0.0
--10.0
-20.0
-
1
Sand (loose)
3.0
30 °
-
-
2
Clay
4.4
-
57.4
0.018
3 . . . . .
Silt
3.8 . . . . . .
36°------- 57.4-----0.018
4
Sand ( m e d i u m )
4.3
36 °
-
-
5
Sand ( m e d i u m )
7.3
37 °
-
--
6
Sand ( m e d i u m )
7.7
36 °
--
7
Clay
20.7
-
191.0
0.018
8
Clay
17.1
-
191.0
0.018
9
Clay
15.7
-
191.0
0.018
. . . . . .
3 0 . 0
-40.0
--50.0
-60.0
-70.0
-80.0
A, Figure 5
Soil p r o f i l e
relationships. The horizontal motion of the soil at 2.5 m below the mudhne and the as-recorded (unmodified) vertical component were used as global excitations (that is, at all support points) for all the cases analysed, except one. For this, the motions obtained at each discretization level were used as multiple support excitations. Spring-dashpot properties were computed as described earlier, with dashpot constants based on shear moduti estimated for compatibility with the near-field force-deformation curves (for example, Gso = c/3eso for clays). In all but one of the analyses, the dashpots were initial-stiffness proportional. The results are summarized in Table 3, which also contains brief descriptions of the cases examined. These are in reference to a standard model symbolized as (A) and listed in the first line. For example, the second line in the table is for a variation of the standard model having the strength and stiffness of the lateral springs 50% higher, the third line is for a variation having half the strength and stiffness of the standard lateral springs, and so on. In each case, only the properties stated in the table were different from corresponding properties of the standard model. For the nineth and tenth cases, the shear wave velocities used to determine dashpot constants were based on the free-field shear modulus variation corresponding to two-thirds the maximum free-field shear strains (average for the two earthquakes). This gave substantially higher dashpot constants for all axial dashpots (three to five times) as well for the lateral dashpots in the clay zones (three to eight times), while it reduced the lateral dashpots in the middle sand layers by up to 50%.
220
Eng. S t r u c t . , 1 9 8 3 , V o l . 5, J u l y
The response variables selected for comparison are the maximum horizontal structural displacement, maximum rotational ductility factor for legs and piles, and the number of brace failures. Although these four variables alone can hardly give a complete picture of the response, they do indicate the severity of inelastic action and allow a quantitative comparison between the various cases. It must be noted, though, that for members carrying large axial loads, as do legs and piles of offshore platforms, the ductility factor alone is not a good index of potential failure, since the plastic rotation capacity of tubular members is substantially reduced in the presence of axial loads. However, all the cases examined here showed insignificant variations of maximum axial forces in legs and piles, and thus it was decided that the ductility factor could be an adequate index for purposes of comparison. In the lower part of the table, the member numbers of the struts that failed are listed for the various cases. As a general observation it can be noted that both earthquakes produced the same qualitative trends. This can be explained by the fact that the two motions have rather similar characteristics, especially since they are both filtered through the same soil profile. Capacity (or strength) changes in either lateral or axial springs appear to cause noticeable changes in superstructure response, particularly in the legs. The pile response is also affected, but to a lesser degree. On the other hand, changes in yield displacements (or stiffness) of either lateral or axial springs, while keeping capacities the same, have very little effect on either the structural or the pile responses. This suggests that it suffices
Pile foundation modelling: S. A. Anagnostopoulos Table 3
Results
(a) Summary Parkfield quake
Case 1 2 3 4 5 6 7 8 9 10 11 12 13
Standard properties: (A) strength and stiffness: 1.5 x (A) Lateral springs strength and stiffness: 0.5 x (A) yield displacements: 1.5 x (A) yield displacements: 0.5 x (A) strength and stiffness: 1.5 x (A) Axial springs strength and stiffness: 0.5 x (A) yield displacements: 1.5 x (A) yield displacements: 0.5 x (A) Lateral deshpots from free field Axial deshpots from free field Dashpotstangent Multiple support excitation
Pacoima Dam quake
~ (cm)
Legs /~r
Struts N.F.
Piles ~ur
86.3 99.3 73.5 86.0 86.5 87.2 75.7 88.4 82.7 87.7 85.1 82.3 91.2"
3.36 5.12 1.63 3.23 3.53 3.53 EL 3.47 3.05 4.22 3.29 2.27 2.70
7 4 5 7 7 7 0 7 7 7 7 5 5
3.04 2.72 2.11 2.95 3.04 3.14 2.23 2.95 3.09 2.08 3.00 3.64 3.02
6 (cm) 92.0 96.0 85.3 91.9 92.7 94.2 75.7 92.4 90.3 92.4 90.7 90.2 116.8"
Legs /~r
Struts N.F.
Piles /~r
3.95 4.76 2.59 3.87 4.13 4.10 1.01 4.04 3.82 5.00 3.87 3.18 3.40
6 5 5 6 7 7 0 6 6 5 6 5 5
2.80 2.63 2.76 2.77 2.87 2.80 1.96 2.72 2.65 1.81 2.77 3.49 2.50
(b) Details of failed struts Case no.
Parkfield quake
Pacoima Dam quake
1 2 3 4 5 6 7 8 9 10 11 12 13
9, 12, 13, 14, 15, 16, 30 12, 13, 16,30 13, 14, 15, 16, 30 Same as case 1 Same as case 1 Same as case 1 No failures Same as case 1 Same as case 1 Same as case 1 Same as case 1 9, 12, 14, 16, 30 9, 12, 14, 16, 30
10, 11, 13, 14, 11, 13, 14, 15, 10, 11, 13, 15, Same as case 1 10, 11, 13, 14, 10, 11, 13.14, No failures Same as case 1 Same as case 1 10, 11, 13, 15, Same as case 1 10, 11,13, 15, 10, 11, 13, 15,
5 = maximum structural displacement; #r = ductility factor = 1 + (Max. pl. hinge E L = members elastic. * Absolute displacement
to use only rough estimates of yield displacements for all support springs. Increases in the values of lateral dashpot constants, as a result of using shear wave velocities compatible with freefield strains, reduce, as might be expected, plastic deformations in the piles and force more inelastic action into the jacket legs. The observed differences are similar, if not greater, in size to the differences caused by varying the strength and stiffness of the p - y curves. On the other hand, changes in axial dashpot constants have practically no effect on any of the response variables examined, because the lateral motion is the dominant component of excitation. Use of tangent stiffness-proportional dashpots implies a decrease in damping and a subsequent increase in pile response. This produced a softer support for the jacket and resulted in lower superstructure response, following a 'soft story' type of behaviour. For the last case examined, the horizontal motions at the nine discretization levels were applied at the ends of the near-field elements as independent support excitations, along with the unmodified vertical component. The result was a rather small reduction in the overall response, in agreement with intuition, which suggests that phase differences among the input motions at the various levels would have a beneficial effect. This should not, however, be considered as a general trend. For a different structure, soil
rotation)/(MpL/6EI);
15, 30 30 30 15, 16, 30 15, 16, 30
30 30 30
N.F. = number of members failed;
profile and earthquake excitations, opposite results may be obtained.
Conclusions The results of the present study are strictly applicable to the structure and motions considered herein. In a qualitative sense, however, they indicate certain trends and thus may prove useful in suggesting which foundation parameters may require more reliable data. In summary, it was found that the structural as well as pile response is much more sensitive to variations in strengths (that is, lateral bearing or skin friction capacities) than yield displacements (stiffness). Changes in lateral dashpot constants caused noticeable changes in the response, but similar changes in axial dashpots had practically no effect on it. Also, the difference between the responses resulting from the use of initial or tangent stiffness-proportional dashpots suggests a need for checking both possibilities, since neither idealization can simulate the real phenomenon correctly. In addition, there is an apparent need for parametric variation of the dashpot constants, because of incompatibility between free.field shear moduli and elastic stiffness of the p - y curves for clays. Finally, the use of multiple support excitations to reflect motion modification with depth should be considered, but only in
Eng. St r uct . , 1983, V o l . 5, J u l y
221
Pile foundation modelling: S. A. Anagnostopoulos r e l a t i o n to t h e p h i l o s o p h y e m p l o y e d in selecting i n p u t motions.
Acknowledgment This paper is based on experience gained by the author when working for Shell Development Co. The free-field motion analyses were performed by Earle Doyle of Shell Development Co. References 1 2 3 4 5 6 7 8 9
222
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