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英文原文
A plasticity model for the behaviour of footings on sand under combined loading
G.T.HOULSBY1, M.J.CASSIDY2
(1.Departmen of Offshore Engineering, Oxford University, UK; 2.Center for Offshore Foundation Systems, University of Western Australia(formerly at Oxford University))
ABSTRACT：A complete theoretical model is described for the behavior of rigid circular footings on sand, when subjected to combined vertical, horizontal and moment loading. The model, which is expressed in terms of work-hardening plasticity theory, is based on a series of tests specifically designed to allow evaluation of the various components of the theory. The model makes use of the force resultants and the corresponding displacements of the footing, and allows predictions of response to be made for any load or displacement combination. It is verified by comparison with the database of tests. The use of the model is then illustrated by some demonstration calculations for the response of a jack-up unit on sand. This example illustrates the principal purpose of the development, which is to allow a realistic modelling of foundation behaviour to be included as an integral part of a structural analysis.
KEYWORDS: footings/foundations; model tests; numerical modelling and analysis; offshore engineering; plasticity; sands
INTRODUCTION
The purpose of this paper is to describe a theoretical model, based on strain-hardening plasticity theory, which is capable of describing the behaviour of a circular footing on sand when it is subjected to all possible combinations of drained vertical, horizontal and moment loading. The motivation for this work comes principally from the offshore industry, specifically arising from the problem of assessment of jack-up units under extreme loading. The applications are, however, much broader, since the model could be applied to many instances of combined loading of a footing on sand.
Structural engineers carry out detailed analyses of jack-up units, and ask geotechnical engineers to provide them with the values of spring stiffnesses to model the foundations. Geotechnical engineers tend to take the view that such a simplistic view of foundation behaviour is unrealistic. Unfortunately, however, they often describe the complexities and non-linearities of foundation behaviour by a series of ad hoc procedures, which a structural engineer cannot implement within a standard analysis. The purpose of the model described here is to provide a means by which the structural and geotechnical engineers can communicate. Geotechnical engineers must be prepared to re-cast their knowledge of foundation behaviour within a terminology (plasticity theory) that is amenable to numerical analysis. Structural engineers must accept that soil behaviour cannot be described merely by ‘springs’, but can be accommodated if they are prepared to use strain-hardening plasticity theory within their analyses.
The ad hoc procedures for describing foundation behavior under combined loading have their roots in the work on bearing capacity by Meyerhof (1953), and are typified by the procedures described by Brinch Hansen (1970) and Vesic (1973). These methods are adequate for predicting failure under combined loads, but they are unsuitable for numerical analysis, principally because they formulate the problem using a series of factors applied to the bearing capacity formula for vertical loading, modifying it to account for horizontal and moment loading. This renders the analysis unsuitable for direct inclusion in numerical analysis programs. Furthermore the conventional analyses pay no attention to the issue of plastic strains pre-failure, since they treat only the failure problem.
An alternative is to address the problem directly as one of loading within a three-dimensional (V , M , H ) load space, and to explore, for instance, the shape of the yield surface in this space. This approach was pioneered by Roscoe & Schofield (1956), who were also concerned with a problem of soil structure interaction: that of calculating the fully plastic moment resistance of a short pier foundation for a steel framework. The general framework of plotting load paths in (V, M, H) space has been adopted by the offshore industry, but the formulae used to derive the failure surfaces are often based on the shape a nd inclination factor approach (see e.g. Hambly & Nicholson, 1991).
Recently there has been considerable interest in the development of models based on plasticity theory, and on the experimental work necessary to support this approach (e.g. Schotmann, 1989; Nova & Montrasio, 1991: Gottardi & Butterfield, 1993, 1995; Houlsby & Martin, 1992; Martin, 1994). The model described here is intended for the description of drained loading of a circular foundation on dense sand, subjected to an arbitrary combination of vertical, horizontal and moment loads. It is complete in the sense that any load or deformation path can be applied to the footing and the corresponding unknowns (deformations or loads) calculated. The model is based on experimental data by Gottardi & Houlsby (1995) and Gottardi et al. (1999).
The loading of a footing clearly results in a complex state of stresses in the soil. In the approach used here the response of the foundation is, however, expressed purely in terms of force resultants (V , M , H ) on the footing. This simplification is very convenient, especially as it allows the model to be coupled directly to a numerical analysis of a structure. It is directly analogous to the use of force resultants (tension, bending moment and shear force) in the analysis of beams and columns. However, it obscures some of the detailed response of the footing—for instance the fact that a real footing probably does not exhibit a truly `elastic' response of the sort employed within the model for certain load combinations. Nevertheless, it proves to be a useful idealisation.
OUTLINE OF THE MODEL
Before giving the detailed mathematical form of the expressions used (see the next section), it is worth describing the model in outline.
The principal concept adopted is that at any penetration of a foundation into the soil, a yield surface in ( V , M , H ) space will be established. Any changes of load within this surface will result only in elastic deformation. Load points that touch the surface can also result in plastic deformation. Although the shape of this surface is assumed constant, the size may vary, with the yield surface expanding as the footing is pushed further into the soil. For simplicity the expansion of the yield surface is taken solely as a function of the plastic component of the vertical deformation.
The model is thus one of the strain-hardening plasticity type. The precise form of the hardening law is specified by a relationship between the size of the yield surface and the plastic vertical deformation.
Within the yield surface, where the deformation is assumed as elastic, the behaviour is specified by a set of elastic constants.
Finally a statement must be made about the flow rule, which determines the ratio between the plastic strains. The simplest type of flow rule is ‘a(chǎn)ssociated flow’, in which the plastic potential is the same as the yield surface. In this model a slight variation is used in that the shape of the yield surface and plastic potential are described by similar mathematical expressions but with different parameter values. It is necessary to introduce these parameters if the modelling of plastic vertical deformations is to be at all reasonable.
There is a striking analogy between the structure of the proposed model and that of constitutive models based on critical-state concepts. In the analogy the vertical load plays the same role as the mean normal stress, p’, the horizontal load or the moment are equivalent to deviator stress, q, and the vertical penetration plays the same role (with a change of sign) as the voids ratio or specific volume. The analogy is pursued in more detail by Houlsby & Martin (1992) and Martin (1994).
DETAILS OF THE MODEL
The model described here is known as Model C (Models A and B were developed by Martin (1994) for footings on clay). The sign conventions and nomenclature used in the following are those suggested by Butterfield et al. (1997) and are shown in Fig. 1. Typical parameter values for Model C are presented in Table 1.
Fig. 1. Sign conventions for load and displacement. Fig. 2. Shape of yield surface
Table 1. Properties used in Model C
Constant
dimension
Explanation
Constrains
Typical value
Notes
R
L
Footing radius
Various
γ
F/L3
Unit weight of soil
20kN/m3
g
Shear modulus factor
400
For equation (2)
kv
Elastic stiffness factor (vertical)
2.65
kh
Elastic stiffness factor (horizontal)
2.3
km
Elastic stiffness factor (moment)
0.46
kc
Elastic stiffness factor (horizontal/moment coupling)
-0.14
h0
Dimension of yield surface (horizontal)
0.116
Maximum value of H /V0 on M=0
m0
Dimension of yield surface (moment)
0.086
Maximum value of M/2RV0 on H=0
α
Eccentricity of yield surface
1.0<α<1.0
-0.2
β1
Curvature factor for yield surface (low stress)
≦1.0
0.9
β1=β2=1 gives parabolic section
β2
Curvature factor for yield surface (high stress)
≦1.0
0.99
β1=β2=1 gives parabolic section
β3
Curvature factor for plastic potential (low stress)
≦1.0
0.55
β4
Curvature factor for plastic potential (high stress)
≦1.0
0.65
αh
Association factor (horizontal)
1.0-2.5
Variation according to equation (9) and αh∞=2.5
αm
Association factor (moment)
1.0-2.15
Variation according to equation (9) and αm∞=2.15
k'
Rate of change in association factors
0.125
f
Initial plastic stiffness factor
0.144
Nγ
Bearing capacity factor (peak)
150-300
δp
Dimensionless plastic penetration at peak
0.0136
Elastic behaviour
The elastic relationship between the increments of load (dV, dM, dH) and the corresponding elastic displacements (dwe, dθe , due ) is
dVdM2RdH=2RGkv000kmkc0kckhdwe2Rdθedue (1)
where R is the radius of the footing, G is a representative shear modulus, and kv, km , kh , kc are dimensionless constants. The values of these constants may be derived using, for instance, finite element analysis of a footing (Bell, 1991; Ngo Tran, 1996), and typical values are given in Table 1. The values of the dimensionless constants depend on the geometry of the footing (e.g. cone angle and depth of embedment) as well as the Poisson's ratio for the sand.
An appropriate value of G is one of the most difficult parameters to establish for the model. Recognising that the mobilised shear stiffness is strongly dependent on the shear strain, the value has to be a compromise one that is representative of typical strains in the soil. It has been determined here by fitting of overall curves to experimental data. The shear modulus also depends on stress level, and is typically proportional to approximately the square root of the mean effective stress. It is convenient therefore to estimate the shear modulus through use of a formula such as
GPa=gVAPa (2)
where Pa is atmospheric pressure, V is a representative vertical load on the foundation, A =π r2 is the plan area of the foundation, and g is a dimensionless constant. A typical value of g is approximately 400 for medium dense sand, but would be expected to depend mildly on the relative density. Note that equation (2) represents a different scaling relationship than was used in Cassidy (1999), and is suggested on the basis of more recent work.
Yield surface
The yield surface is most conveniently expressed in dimensionless terms, using the variables v =V/V0, m=M/2RV0, h=H/V0, where V0is the parameter that defines the size of the yield surface. The chosen form of the surface that fits the observed behaviour of footings well is that used by Martin(1994):
f=hh02+mm02-2αhh0mm0-β12v2β11-v2β2=0 (3)
where the factor
β12=β1+β3β1+β2β1β1β2β22
is introduced so that h0 and m0 have simple physical interpretations. This surface may seem unnecessarily complicated, and it is perhaps useful to consider a simplified form in which a = 0 and β1=β2=1:
f=hh02+mm02-16v21-v2=0 (4)
It is straightforward to show that this is a ‘rugby ball’ shaped surface that is elliptical in section on planes at constant V , and parabolic on any section including the V -axis: see Fig. 2. Although there is some theoretical justification for this choice of shape (particularly in the (V , M ) plane), it is largely chosen empirically. The size of the surface is determined by the point on the surface at maximum V value, which is given by (V ,M ,H )=(V0, 0, 0). The shape of the surface is determined by the two parameters h0 and m0, which determine the ratios of H/V and M/2RV at the widest section of the surface, which occurs at V = V0/2.
The factor a in equation (3) allows the ellipse to become eccentric (that is, the principal axes are no longer aligned with the H - and M -axes). This is necessary for accurate modeling of the experimental data, and accounts for the fact that if, for instance, the footing is subjected to a horizontal load from left to right, a clockwise moment will produce a different response from an anticlockwise moment. The factors β1 and β2 are introduced following Nova & Montrasio (1991). They have two advantages: (a) the position of the maximum size of the elliptical section can be moved from V =V0/2 to V=β2 V0/(β1+ β2), thus fitting experimental data better; and (b) by choosing β1﹤1 and β2﹤1 the sharp points on the surface at V=0 and V=V0 can be eliminated, which has advantages in the numerical implementation of the model. If β1 =β2 =0.5 , the yield surface becomes an ellipsoid. The factor β12 in equation (3) is simply so that h0 and m0 retain their original meanings.
Strain hardening
The form of the strain-hardening expression can be determined from a vertical load-penetration curve, since for pure vertical loading V0=V. Typical load-penetration curves are shown in Fig.3, showing a peak in the load-penetration curve for the dense sand tested by Gottardi & Houlsby (1995). An expression that fitts the data well, and which is shown in Fig.3, is
V0=kwp1+kwpmV0m-2wpwpm+wpwpm2 (5)
where k is an initial plastic stiffness, wp is the plastic component of the vertical penetration, V0m is the peak value of V0,and wpm is the value of wp at this peak. No special significance is attached to this particular form of the fit to the vertical load-penetration response, and alternative expressions that fitted other experimental data could also be appropriate.
A formula that models post-peak work softening as well as pre-peak performance was essential. However, equation (5) unrealistically implies V0→0 as wp→∞. Therefore it can be used only for a limited range of penetrations. It is assumed that for most properly designed foundations on dense sand, loading post-peak would not be expected; however, for a complete model capable of fitting post-peak behaviour more realistically, equation (5) can be altered to
V0=kwp+fp1-fpwpwpm2V0m1+kwpmV0m-2wpwpm+11-fpwpwpm2 (6)
where fp is a dimensionless constant that describes the limiting magnitude of vertical load as a proportion of V0m (that is , V0→fpV0m as wp→∞). It is possible to use the same parametric values of k, V0m and wpm as in equation (5). For realistic footing designs in which it was not required to describe softening, a much simpler equation than equation (6) could be used. The precise form of this equation is not in fact central to the model; all that is required is a convenient expression that fits observed data and defines V0 as a function of wp.
Fig. 3. Theoretical fit of the vertical load tests
Plastic potential
In the ( M/2R, H ) plane an associated flow rule is found to model the ratios between the plastic displacements well, but this is not the case in the (V , M/2R) or (V , H ) planes, for which an associated flow rule is found to predict unrealistically large vertical displacements. A plastic potential different from the yield surface must therefore be specified. A convenient expression is, however, very similar to that used for the yield surface:
g=h'h02+m'm02-2αh'h0m'm0-αv2β34v'2β31-v'2β4=0 (7)
Where
β34=β3+β4β3+β4β3β3β4β42
and αv is an association parameter (associated flow is given by αv=1.0). Note that the condition g=0 is used to define a dummy parameter V0’ which gives the intersection of the plastic potential with the V-axis. The primed parameters are defined by v’=V / V0’, m’=M/2RV0’ and h’=H/ V0’. Factors β3 and β4 have been introduced, which can be chosen independently from β1 and β2. The association parameter αv allows for variation of the vertical displacement magnitude, with values greater than 1.0 resulting in the increase of the vertical displacements. It also controls the position of the ‘parallel point’ as defined by Tan (1990), which is the point on the yield locus at which the footing could rotate (or move sideways) at constant vertical load and with no further vertical deformation. Accurate prediction of this point is important as it describes the transition between settlement and heave of the footing and where sliding failures will occur. In the analogy with critical-state models, this point plays the same role as the critical state. When associated flow is used (αv=1, β3=β1, β4=β2) the parallel point occurs at v=β2/(β1+β2): that is, the largest constant vertical load section of the yield surface. As αv is decreased, the position of the parallel point moves to a lower value of vertical load, but the exact expression for the value of v becomes very complex. The modelling of realistic vertical displacements and of the position of the parallel point are linked, and with only one parameter it is difficult to model both adequately.
Increasing h0 or m0 with two association factors, rather than scaling the vertical component, enables the plastic potential's shape to change in the radial plane. This consequently changes radial plastic displacements. This method has the advantage of more flexibility in modelling subtle differences between horizontal and moment loading results. Using two association factors the plastic potential may be defined as
g=h'αhh02+m'αmm02-2αh'm'αhαmh0m0-β34v'2β31-v'2β4=0 (8)
If αh and αm are constant and equal, equation (7) is equivalent to equation (8) for the same value of αv.
In fact it was found that experimental data can be fitted well only if the αh and αm factors are themselves taken as variable. The values of αh and αm that best fit both the radial displacement and constant V tests of Gottardi & Houlsby (1995) were found to be hyperbolic functions of plastic displacement histories:
αh=k'+αh∞upwpk'+2Rθpwp (9)
αm=k'+αm∞2Rθpwpk'+2Rθpwp (10)
where k’ determines the rate of change of the association factors. For no previous radial displacements, αh and αm equate to 1 and associated flow is assumed. The rates at which αh and αm vary in Model C are depicted in Fig.4. With the plastic potential defined as in equation (6), the following values were evaluated: β3=0.55 ; β4=0.65; αh∞=2.5; αm∞=2.15; k’=0.125. Further details of the development of the plastic potential in equation (8) and comparisons between the theory and experimental data can be found in Cassidy (1999).
Partially drained behaviour
The model described above is based on data from tests on dry sand, and thus describes fully drained behaviour. For realistic loading times of large offshore foundations, partially drained behaviour is expected, and the above model would need to be modified to take into account the transient pore pressures beneath the foundation. Both Mangal (1999) and Byrne (2000) have carried out model tests equivalent to those used here, but on saturated sand and at loading rates where partially drained behaviour occurs. They record that