MATRIZ DE OPERACIONALIZACIÓN DE VARIABLES CUESTIONARIO SOBRE HISTORIETA

G. Buscarnera, R. Nova & M. Vecchiotti Politecnico di Milano, Milan, Italy

C. Tamagnini & D. Salciarini University of Perugia, Perugia, Italy

ABSTRACT: In this paper, the soil-foundation-structure interaction processes due to wind loading of a wind turbine founded on a circular raft are modelled by means of the FE method. The cyclic/dynamic response of the soil-foundation system is assumed to be fully described by a set of uncoupled viscous dampers and a nonlinear, inelastic macroelement which links the external loads and moment acting on the foundation with its displacements and rotation. The constitutive equations of the macroelement is formulated within the framework of the theory of strain hardening elastoplasticity. To account for cyclic loading effects, it is assumed that irreversible displacement increments may occur even for loading paths contained within a conveniently defined state boundary surface. The corresponding flow rule and hardening law can be formulated by means of a suitable mapping rule, establishing a correspondence between the current loading state and a uniquely defined image state on the state boundary surface. By means of the macroelement approach, it is possible to couple in a simple and efficient way the dynamic behaviour of the structure to the foundation movements. Settlement and rotation of the structure can then be predicted for any convenient number of cycles.

1 INTRODUCTION

The analysis of the behaviour of structures subject to cyclic or transient loading requires the modelling of the interaction between the foundation and the soil underneath. This can be done in various ways. For instance, the soil volume which undergoes significant deformations under the applied loads can be described as an inelastic continuum and the resulting continuum model can be analyzed by means of the finite ele-ment method. Due to the strong nonlinearity of soil behaviour, the use of such an approach in connec-tion with any realistic inelastic soil model would be extremely time-consuming.

A possible alternative approach consists of lump-ing the soil-foundation system compliance into a finite (small) number of discrete deformable elements – typically elastic springs and viscous dashpots – which act on the superstructure as uncoupled deformable constraints placed at the foundation level. The springs are typically used to reproduce the soil stiffness under applied loads and moments, while the viscous dash-pots are responsible for modelling the damping effects (geometrical dissipation and mechanical damping) which occurs in the subsoil.

This latter method is clearly much more cost-effective, but has several important limitations. The assumed linear behaviour of the springs implies fully reversible response under closed cyclic loading paths.

Moreover, the uncoupled character of the equations describing the load-displacement response of the

deformable constraints makes it impossible to repro-duce a number of important features of the observed response of shallow foundations. In particular, no ver-tical displacement is predicted under the cyclic action of a rocking moment and a horizontal force, in sharp contrast with the available experimental observations, see e.g., di Prisco et al. (2002).

The goal of this paper is to show that it is pos-sible to overcome this difficulty by adopting a third, different approach which, while retaining the simplic-ity and efficiency of the deformable constraints, can effectively reproduce the nonlinear, irreversible and hysteretic response of shallow foundations subject to non-proportional, cyclic/dynamic loading conditions.

In this case, a set of suitable inelastic constitutive equations – a so-called “macroelement” as originally defined by Nova & Montrasio (1991) – linking the load increments applied to the foundation to the corre-sponding work-conjugated displacement increments, are derived within the framework of the theory of strain-hardening elastoplasticity. The macroelement which reproduces the soil/foundation system at the global level can then be easily coupled to any FE struc-tural model to analyze quite complex soil-structure interaction (SSI) problems.

It is worth noting that other approaches can be used to define the inelastic constitutive equations of the macroelement. For example, Salciarini & Tamagnini (2009) have developed an inelastic macroelement within the framework of the theory of hypoplasticity, which is characterized by a very small (pseudo)elastic

domain and a continuous dependence of the tan-gent stiffness from the loading direction (incremental non-linearity).

In order to demonstrate the effectiveness of the macroelement approach in the analysis of complex SSI problems, the elastoplastic macroelement devel-oped by Nova & Montrasio (1991) has been used to simulate the cyclic/dynamic response of a wind tur-bine founded on a circular raft foundation, subject to time-dependent wind forces. This particular type of structure – characterized by relatively low verti-cal loads, and high horizontal loads and moments – represents an interesting benchmark to investigate the influence of possible coupling effects between the different degrees of freedom of the system.

The outline of the paper is as follows. In Sect. 2, the details of the macroelement developed by Nova &

Montrasio (1991) are given. The modification of the basic isotropic-hardening model to account for cyclic/dynamic loading effects is discussed in Sect. 3.

The characteristics of the wind turbine considered in this study and the results of preliminary quasi-static, monotonic simulations are presented in Sect. 4, while the results of a series of dynamic simulations are dis-cussed in Sect. 5. Finally, Sect. 6 summarizes the main conclusions.

2 THE ELASTOPLASTIC MACROELEMENT MODEL OF NOVA & MONTRASIO 2.1 Constitutive equations

In the macroelement model developed by Nova &

Montrasio (1991), the mechanical response of a cir-cular foundation of diameter D resting on a homoge-neous soil layer is described globally by means of the following constitutive equation in rate-form:

relating the increment of the generalized load vector:

and the increment of the generalized displacement vector:

In the above equations, V , H and M are the verti-cal load, the horizontal load and the rocking moment, respectively, while u, v and θ are the vertical displace-ment, the horizontal displacement and the rotation of the foundation (seeFig. 1).

The (3× 3) matrix K in eq. (1) is the generalized tangent stiffness matrix of the foundation-soil system.

In order to reproduce the non-linear and irreversible response of the foundation, K is assumed to depend on the current generalised load Q, on the loading direction

Figure 1. Components of generalized load and displace-ment.

Figure 2. Representation of the elastic domain, of the yield surface and of the bounding surface in the H :V space.

(in displacement space) η, and on the previous load-ing history of the system, globally accounted for by a set of suitable internal variables. In the simplest case considered by Nova & Montrasio (1991), the internal variables reduce to a single scalar quantity Vc, which controls the size of the elastic domain, i.e., the convex locus in load space within which the response of the macroelement is assumed to be linear elastic.

Within the framework of the theory of plasticity, the elastic domain is defined as follows:

where f is the yield function and the curve f = 0 is the yield surface (see Fig. 2). In the Nova &

Montrasio (1991) model, the yield surface is given by the following expression:

where ψ, µ and β are model constants.

If a certain load state such as point A in Fig. 2 lies within the elastic domain, any (infinitesimal) load-ing path will produce recoverable displacements. The same will occur if the current state lies on the yield surface (point B inFig. 2) and the load increment points towards the interior of the elastic domain (load increment dQ2inFig. 2).

On the contrary, when the load increment points outwards, as for dQ1in Fig. 2, the loading process is

plastic and the generalised displacement increment is partly irreversible.

Given the plastic potential function:

the plastic displacement increment is given by the flow rule:

where ≥ 0 is the plastic multiplier. In eq. (6):

λMand λHare material constants, and Vgis a dummy variable. When λM= λH= 1 the plastic flow is asso-ciative.

In the Nova & Montrasio (1991) model, the evolu-tion equaevolu-tion for the internal variable is provided by the following hardening law:

where α, γ, R0and VMare model constants, and:

is the hardening function for Vc.

From eq. (8) it is clear that VMrepresents the max-imum size of the yield surface, i.e., the size of the failure locus (Fig. 2).

From the consistency condition, df= 0, the follow-ing expression is obtained for the plastic multiplier:

in which:

is a strictly positive scalar quantity and (−∂f /∂Vc)hcis the so-called plastic modulus. Substituting the expres-sion (9) in the flow rule and taking into account the elastic constitutive equation, the following expres-sions are obtained for the elastoplastic tangent stiffness matrix of eq. (1):

where K^eis the elastic stiffness matrix.

2.2 Calibration of model constants

The model is fully characterized by the elastic stiff-ness matrix K^e and 9 material constants controlling the shape of the yield locus, of the plastic potential, and the amount of hardening. It is shown in Montrasio and Nova (1997) that many of them vary in a rela-tively narrow range and their actual values play only a secondary role on the overall response of the foun-dation. In fact, only the variables R0, VM, µ, ψ are relevant in this respect. Furthermore these constants can be easily related to traditional soil constants, such as the elastic modulus of the soil or the friction angle at the soil-foundation interface.

For instance, consider a rigid circular foundation resting on a deep, uniform sand layer modeled as a linear elastic material. The settlement increment
s induced by a centered vertical load increment
V can be derived from the theory of elasticity as follows:

The parameter R0 can be easily determined from eq. (12), given the elastic properties of the soil and the foundation geometry.

The value of VM– representing the bearing capacity of the foundation under a perfectly centered vertical load – can be determined from the classic bearing capacity formulas. If the soil is cohesionless and the depth of the foundation is small with respect to its diameter the contribution of the cohesion and the lateral surcharge vanish, and:

where, according to Poulos et al. (2001), the bearing capacity factor Nγis given by:

and the shape factor sγcan be assumed equal to:

The constant µ is identified with the tangent of the friction angle ˆφat the foundation-soil interface.

Typically, ˆφ= φcv= 3φ/4.

According to Montrasio & Nova (1997), the con-stant ψ can be deduced from the following empirical relation:

where d is the depth of the foundation base.

Figure 3. Bounding surface and cone-shaped elastic domain. Point I is the image point of the current loading state P.

3 MODIFICATION FOR CYCLIC LOADING CONDITIONS

Macroelement models developed within the frame-work of strain-hardening elastoplasticity exhibit important limitations when the cyclic/dynamic response of shallow foundations is of concern. In fact, in isotropic hardening macroelement models, the irre-versible displacements accumulated during the first loading cycle determine an expansion of the elas-tic domain, so that the subsequent loading cycles are fully contained within the elastic domain. As a result, the foundation response is fully reversible, and the model is unable to capture such relevant phenomena as hysteresis or ratcheting.

Various strategies can be pursued to improve the model performance for cyclic loading conditions.

Here, the so-called Bounding Surface (BS) approach is adopted (di Prisco et al. 2002), following the strat-egy proposed by Dafalias & Herrmann (1982) for continuum plasticity.

In this approach, plastic deformations can occur even for loading states within the BS. The plastic modulus controlling the magnitude of the plastic dis-placement rates is assumed to be a function of the distance of the current state from the BS, determined by suitably defining an image point on the BS by means of a non-invertible mapping rule. A small elas-tic region is maintained in the vicinity of the current load state, similar to the concept of elastic ‘bubble’

introduced by Al Tabbaa and Wood (1989).

In particular, it is postulated that the elastic domain is delimited by a small cone closed at the end by a spherical cap, as shown in Fig. 3. The cone is entirely contained within the BS, whose size Vcis determined by the plastic deformations accumulated during the monotonic loading stage.

In fact, the BS coincides with the yield locus of the standard Nova & Montrasio (1991) model, so that when the current state is on the BS (e.g., during virgin monotonic loading) the response of the two models are identical.

The elastic cone is defined by the position of the centre of the spherical ‘cap’ (point A inFig. 3) and by its radius, which is set to a small fixed quantity. The

Figure 4. Definition of memory surface and of the distances δ1and δ2.

angle at the cone apex, at the origin of the load space, is determined by enforcing a C1continuity condition at the transition between the cone and the spherical cap.

Assume now that a point P on the yield surface represents the current loading state and that the stress increment dQ is such that plastic loading occurs. Then, the plastic deformation increment is determined by the following modified flow rule:

In Eq. (17), the plastic multiplier  as well as the gradient to the plastic potential ∂g/∂Q are calculated at the image point I, enforcing the consistency condi-tion at Q= Q on the bounding surface. As shown in Fig. 4, the image point I is determined by the inter-section of the bounding surface with the straight line AP. The matrix  is a diagonal matrix which operates on the plastic flow vector as a weighting function: the smaller the components of , the smaller the magni-tude of plastic displacement increment. In particular, each component of  is evaluated as:

where α= V , H or M. In eq. (18), ζαand ξαare mate-rial constants; the variable δ is a scalar measure of the distance of the current state to the BS, and ρkis a scalar memory variable.

The distance δ is evaluated by introducing the con-cept of memory surface, as shown in Fig. 4. The domain within the bounding surface is divided into two regions by a cone with vertex at the origin and opening angle ω equal to the maximum load obliquity reached upon reloading:

Figure 5. A typical medium size wind turbine.

The distance δ is then computed as:

where the quantities δ1and δ2are defined in Fig. 4, and χis a model constant. According to eq. (17), the mag-nitude of the components αdecrease with increasing distance of the current state from the BS. Therefore, the plastic flow magnitude decreases as δ increases.

The memory variable ρkappearing in eq. (18) is a function of the loading history and is updated formally as Vc/VM in eq. (8). For the details of the evolution equations of the yield locus and the memory surface, the reader is referred to di Prisco et al. (2002).

Finally, the elastic stiffness matrix is defined as follows:

where the coefficients K_V^e, K_H^e and K_M^e are computed as:

according to Gazetas (1991).

4 SSI ANALYSIS OF A WIND TURBINE:

QUASISTATIC CONDITIONS

In this section, the macroelement model illustrated in the previous sections is used to analyze the quasi-static behaviour of a medium size (850 kW) wind turbine similar to the one shown in Fig. 5.

The weight of the turbine and the rotor is 320 kN, and these are mounted on a 30 m high conical steel tower. The external tower diameter varies from a max-imum of 3.5 m at the base to a minmax-imum of 2.0 m at the top. The moment of inertia of its section varies from about 0.56 m⁴at the base to about 0.13 m⁴at the top.

Table 1. Model constants adopted in the simulations.

ψ µ λM λH β

0.30 0.60 0.286 0.80 0.95

Rur(GPa) α γ ζV ζH= ζM

1.26 2.83 1.71 4.00 20.00

ξV ξH= ξM χ

50.00 100.00 0.01

The weight of the tower is 406 kN. The foundation is a circular concrete raft, with diameter D ranging from 12 to 14 m, and a thickness equal to 0.1D. It is worth not-ing that the weight of the foundation is larger than the entire weight of the structure (rotor+turbine+tower).

The foundation rests on a coarse-grained soil layer, with a friction angle of 40^◦, shear modulus of 40 MPa and Poisson’s ratio equal to 0.30. The unit weight of the soil is assumed equal to 20 kN/m³.

Starting from the given soil properties and foun-dation sizes, the macroelements have been calibrated following the guidelines provided by Nova & Montra-sio (1991), MontraMontra-sio & Nova (1997) and di Prisco et al. (2002). Table 1 shows the values of the model constants adopted in the simulations.

4.1 Monotonic loading conditions

To demonstrate the capability of the macroelement to reproduce the wind turbine behavior under monotonic loading conditions, a 3D FE model of the 12 m circular foundation raft has been developed, using the FE code Abaqus (Simulia, 2010). The foundation soil has been modeled as an elastic-perfectly plastic material, with Mohr-Coulomb yield condition. The soil-foundation contact has been modeled as a unilateral frictional con-tact, to simulate the possible opening of the contact at high load eccentricities.

The foundation has been subjected to the follow-ing loadfollow-ing program: 1) application of the vertical load corresponding to the weight of the structure and the foundation; 2) application of a horizontal load of 400 kN at the turbine hub, simulating the wind action on the rotor. For the chosen soil properties and foun-dation diameter, this value of the horizontal load is relatively close to the horizontal collapse load.

The macroelement predictions for loading stage (2), in terms of horizontal load-displacement curve, are compared to the 3D FE simulation in Fig. 6. The results show that, at least for load values within the range of working load levels (i.e., H≤ 200 kN), the macroele-ment prediction matches reasonably well the FE results. However, the macroelement approach allows for a dramatic increase in computational efficiency, as the 3D FE analysis required several hours of CPU time to complete, while only fractions of seconds are needed for the macroelement simulation.

This means that the macroelement approach can be effectively used in the simulation of soil-structure interaction problems characterized by complex load-ing conditions with large number of cycles, whereas the FE approach would be practically inapplicable.

Figure 6. Horizontal load-displacement curves for wind loading under monotonic, quasi-static conditions: compar-ison between macroelement and 3D FE predictions.

4.2 Cyclic loading conditions

To assess the effects of cyclic wind loading, a second series of simulations has been performed under quasi-static conditions, considering two different founda-tions with a diameter of 12 m and 14 m, respectively.

In this case, the wind load – a horizontal force applied at the turbine hub – is assumed to vary with time according to the following relation:

with H0= 400 kN.

The results of the simulations are shown in Figs. 7 and 8 in terms of horizontal displacements and rota-tions, respectively, as a function of the number of cycles.

It is interesting to note that, in both cases, both horizontal displacements and rotations reach a station-ary condition characterized by constant maximum and minimum values very rapidly, in just a few cycles.

As expected, the stationary values of displacements and rotations depend on the foundation size, the larger movements being associated with the smaller foundation.

As far as vertical displacements are concerned, the results in Fig. 9 (computed for foundation diameters ranging from 13 to 16 m) show that foundation set-tlements increase steadily with the number of cycles, reaching values which, after 1000 cycles, are about one order of magnitude larger than observed horizon-tal movements. This effect is only slightly influenced by the foundation diameter.

As far as vertical displacements are concerned, the results in Fig. 9 (computed for foundation diameters ranging from 13 to 16 m) show that foundation set-tlements increase steadily with the number of cycles, reaching values which, after 1000 cycles, are about one order of magnitude larger than observed horizon-tal movements. This effect is only slightly influenced by the foundation diameter.