Closure to “Lateral Pipe-Soil Interaction in Sand with Reference to Scale Effect” by P. J. Guo and D. F. E. Stolle

The writers thank the discusser for his interest in the paper. Dr. Peek raised some concerns about the scale effect in pipe–soil interaction, particularly the effect of particle sizes in centrifuge tests, which is referred to as the “true scale effect” in the discussion. It should be noted, however, the scale effect is used in the original paper to describe the effect of pipe size (i.e., size effect) as well as the differences between a full-scale model and a reduced-scale model (i.e., model effect), especially under $1g$ conditions.

It is well established that granular materials are pressure sensitive, even for perfectly rigid grains interacting by frictional forces with an intergrain friction coefficient that is independent of the magnitude of the normal force. For example, dense sand tends to dilate more under shear and yield a higher friction angle. An increase in confining pressure, however, tends to restrain dilation, which results in a lower peak friction angle. Moreover, the failure mode and the strain required to fully mobilize soil strength vary with confining stresses. Similar phenomena have been observed in ensembles of steel balls and glass beads without particle crushing during deformation. Consequently, the writers believe that stress-level dependency is not induced by particle crushing, at least not in the normal stress range encountered in the soil–pipe interaction problems discussed in the paper. Of course, the writers agree that particle crushing may contribute to the stress-level dependency of granular soils under high confining pressures.

The differences between a full-scale test and a reduced-scale model test under $1g$ conditions, which is referred to as “model effect” in the original paper, are attributed to the effect of stress level on soil behavior. When the reduced-scale model test is performed in the centrifuge, the model effect vanishes (as shown in Fig. 8 of the original paper). The “true scale effect”, which describes the differences between a full-scale test and a reduced-scale test in the centrifuge in the discussion, is not discussed in the original paper. The writers agree that the continuum FEM cannot simulate the “true scale effect” induced by variation of particle sizes. This is because even though the classical continuum mechanics deals with stresses at each point, the theory, when applied to granular soils, is only applicable to representative volumes of soil mass. Given that the thickness of shear zone developed in sand is approximately 5 to 15 times of particle size (Scarpelie and Wood 1982; Viggiani et al. 2001; Nemat-Nasser and Okada 2001), the dimension of a representative volume element at failure might be approximately in the order of 100 times of particle size. When the dimension of a reduced-scale model approaches this limit, the model test will change its continuum feature to discrete, yielding very much different results. Numerous studies have been conducted to investigate the influence of particle size on centrifuge model tests (Tatsuoka et al. 1991; Palmer 2003; Stuit 1995).

It should be noted that an aspect of the original paper is to demonstrate that the pressure sensitivity of soil and the comparatively low stress level in reduced-scale model tests under $1g$ conditions yield different soil properties from those encountered in-situ, which results in the scale effect and the size effect. The discussion suggests that the size effect is owing to changes in the cohesion $c$ and the elastic modulus $E$ by the same factor. This issue, however, is not considered in the paper.

In closure, the writers agree that one can demonstrate that the same results might be obtained by changing either soil parameters or model sizes in numerical modeling, particularly when the soil properties are assumed to be independent of stress level. In physical modeling, however, the scale effect cannot be considered as identical to the effect owing to variation in the cohesion $c$ and the elastic modulus $E$ by a factor.

Erratum

The following correction should be made to the original paper:

In Fig. 4 on page 341, H series modeling should reflect the size effect, while D series modeling reflects the effect of burial depth.