2.4 Scale effects

In physical modelling studies, it is seldom possible to replicate precisely all details of the prototype and some approximations have to be made. It is important to recognise that model studies are not perfect and to inquire into the nature of any shortcomings, often referred to as scale effects, and to evaluate their magnitude. The influence of the non-uniform acceleration field created in centrifuge models is an example of a scale effect and was considered in section 2.2.1. The following are some examples of scale effects; others may be relevant in any particular centrifuge study and it is the responsibility of the centrifuge worker to establish for any particular project the extent to which model test results can be extrapolated to a prototype scale.

A good technique for checking for scale effects is the ‘modelling of models’.

It is particularly useful when no prototype is available for verifying the model test results. Centrifuge models of different scale are tested at appropriate accelerations such that they then correspond to the same prototype. The models should predict the same behaviour and thus provide a useful internal check on the modelling procedure. However, it should be noted that the scale range of the models is usually limited. For example, it might be possible to model an event at 40 g and 120 g on the same centrifuge. This is only a range of scale of 3 compared to the scale of 40 scale needed to extrapolate from the larger model to the prototype. Thus, while ‘modelling of models’ provides a valuable internal check on the modelling procedure, it does not in itself provide a guarantee that model data can be extrapolated successfully to the prototype scale.

2.4.1 Particle size effects

The most common question asked of centrifuge workers is how can centrifuge modelling be justified if the soil particles are not reduced in size by a factor of N.

In increasing the model scale to the prototype in the mind’s eye, it might appear sensible to also increase the particle size. Thus a fine sand used in a 1:100 scale model might be thought of as representing a gravel. But by the same argument, a clay would then be thought of as representing a fine sand. This argument is clearly flawed since a clay has very different stress-strain characteristics to a fine sand. There could be a problem if an attempt was made to model at high acceleration and hence at very small scale an event in a prototype soil consisting mainly of a coarse soil (gravel). In that case, the soil grain size would be significant when compared to model dimensions and it is unlikely that the model would mobilise the same stress-strain curve in the soil as would be the case in the prototype. Local effects of the soil grains would influence the behaviour rather than the soil appearing like a continuum as would be the case in the prototype.

It is therefore sensible to develop simple guidelines on the critical ratio between a major dimension in the model to the average grain diameter to avoid problems of particle size effects. This was the approach adopted by Ovesen (1979, 1985) who investigated the performance of circular foundations on sand by undertaking a series of experiments using different sized models at different accelerations such that they corresponded to the same prototype. The data were generally internally consistent which validated the centrifuge technique, though it was noted that there was some deviation from the common behaviour when the ratio of foundation diameter to grain size was less than about 15. Thus the particle size scale effect could be quantified to some extent. However, this approach may be too simplistic and in some cases it may be necessary to consider the ratio of particle size to shear band width (Tatsuoka et al., 1991). The important point is to recognise that in some circumstances particle size effects may be important and the model test series should include sufficient relevant investigation to assess its significance in the problem being studied.

2.4.2

Rotational acceleration field

While a centrifuge is an extremely convenient method of generating an artificial high gravitational acceleration field, problems are created by the rotation about a fixed axis. The inertial radial acceleration is proportional to the radius which leads to a variation with depth in the model; this was discussed in section 2.2.1.

Also, this acceleration is directed towards the centre of rotation and hence in the horizontal plane, there is a change in its direction relative to vertical across the width of the model. There is therefore a lateral component of acceleration, the effect of which needs to be recognised. For a model having a half width of 200mm and an effective radius of 1.6m, this lateral acceleration has a maximum value of 2/16 or 0.125 times the ‘vertical’ acceleration. This can be quite significant if there is a major area of activity near a side wall of the model container. For some centrifuges, the major vertical plane lies in the horizontal plane of rotation and if the radius of the centrifuge is relatively small, some centrifuge workers have adopted the practice of shaping models to take account of the radial nature of the acceleration field. Alternatively, it is good practice to ensure that major events occur in the central region of the model where the error due to the radial nature of the acceleration field is small.

Another problem caused by generating the acceleration field by rotation is the Coriolis acceleration which is developed when there is movement of the model in the plane of rotation. This might be the horizontal movement of base shaking earthquake simulation on models whose major vertical plane lies parallel to the plane of rotation. In trying to avoid this, many centrifuges now arrange for the major vertical plane of the model to be perpendicular to the plane of rotation.

However, there may be vertical velocities in the plane of rotation and the effect of Coriolis accelerations needs to be assessed. The following gives guidelines for

CENTRIFUGES IN MODELLING: PRINCIPLES AND SCALE EFFECTS 29

the range of velocities in a model for which Coriolis effects could be considered negligible. The Coriolis acceleration a_c is related to the angular velocity, , of the centrifuge and the velocity, v, of a mass within the model as:

(2.25) The inertial acceleration, a, of the model is:

(2.26) where V is the velocity of the model in centrifuge flight. It is generally assumed that Coriolis effects would be negligible if the ratio a_c/a was less than 10% which implies v < 0.05V (see section 7.2). This gives an upper limit on v for relatively slow events.

At the other extreme, for example, the high velocity of soil ejected during blast simulation, it was argued that the radius of curvature, r_c, of the path followed by a moving mass in the model should not be less than the effective radius of the centrifuge. The Coriolis acceleration can then be written as:

i.e.

(2.27) Since then for Thus it is concluded that the range of velocities within a model which would not lead to significant Coriolis effects is given by:

(2.28) Further discussion of Coriolis effects is presented in chapter 7.

2.4.3 Construction effects

Geotechnical engineering is often concerned with the effects of construction and this can pose many difficulties for the centrifuge worker. It is very difficult to excavate or build during centrifuge flight. The soil is very heavy and any model equipment needs to be small, lightweight and very strong and usually requires skilful design. Though there are many difficulties, new techniques and devices are being developed and these are often reported at speciality centrifuge conferences.

In modelling construction or installation processes, the first thoughts need to be directed towards defining the essential details which have to be modelled and, importantly, those details which are of secondary importance and can be taken into account in some approximate way. Even if approximations are made, centrifuge data will still be useful as these can be taken into account in any back-analysis, so verifying the analysis for later application to the prototype event. An example is modelling embankment construction on soft clay where the foundation

behaviour is the main feature under investigation rather than the actual embankment. Many centrifuge centres have developed hoppers which can be used to build up an embankment made of dry sand during centrifuge flight rather than use the proposed embankment material or construction procedure.

Nevertheless, it is still possible to study the foundation strata under this loading and any changes in behaviour when, for example, ground improvement techniques are used.

Craig (1983) considered construction effects in the context of modelling pile foundations. If the performance of the piles under lateral loading is being investigated, then it would be reasonable to adopt the easy approach of installing the piles prior to starting the centrifuge. Although the stress distribution due to installation is not correctly modelled, this is a minor effect on the overall performance. However, if the performance of the piles under axial loading is to be studied it is essential to install the piles during centrifuge flight since the load capacity is critically dependent on the lateral stresses developed during installation.

2.5