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481 Gu, Q. & Lee, F.-H. (2002). Ge ´otechnique 52, No. 7, 481–493 Ground respon

481 Gu, Q. & Lee, F.-H. (2002). Ge ´otechnique 52, No. 7, 481–493 Ground response to dynamic compaction of dry sand Q. GU and F.-H. LEE The mechanics of dynamic compaction are studied in this paper using two-dimensional ﬁnite element analyses with a large-strain dynamic formulation and a cap model for soil behaviour. Comparison with centrifuge model results shows that stress wave attenuation and improvement effects are realistically predicted. The analyses show that, in the initial blows, stress wave propagation induces transient elasto-plastic K0 compression due to lateral inertia. This preserves the plane wavefront and reduces the attenuation rate of the dynamic stresses with depth. With multiple blows, the effect changes to one of triaxial compression: this sets a limit on the degree of improve- ment that can be achieved in the near ﬁeld. Deeper down, the wavefront adopts a bullet shape, and the attenuation rate rises: this sets a limit on the depth of improvement. Both these phenomena are consistent with the existence of a ‘threshold’ state that has been noted in previous literature on dynamic compaction. The results also show that the depth of improvement is dependent upon the momentum per blow, as well as on the energy per blow. KEYWORDS: compaction; ground improvement; numerical modelling and analysis; sands; stress analysis Nous e ´tudions dans cet expose ´ les me ´canismes de com- pactage dynamique en utilisant des analyses d’e ´le ´ments ﬁnis en deux dimensions avec une formule dynamique de grande de ´formation et un mode `le de comportement du sol. La comparaison avec les re ´sultats du mode `le centri- fuge montre que l’atte ´nuation des ondes de contrainte et les effets d’ame ´lioration sont pre ´vus de manie `re re ´aliste. Les analyses montrent que, lors des frappes initiales, la propagation de l’onde de contrainte entraı ˆne une com- pression e ´lasto-plastique transitoire K0 due a ` une inertie late ´rale. Ceci maintient le front d’onde plat et re ´duit le taux d’atte ´nuation des contraintes dynamiques avec la profondeur. Avec des frappes multiples, l’effet devient un effet de compression triaxiale, ce qui ﬁxe une limite sur le degre ´ d’ame ´lioration qui peut e ˆtre obtenu dans le sol adjacent. Plus profonde ´ment, le front d’onde prend une forme d’obus et le taux d’atte ´nuation augmente, ce qui ﬁxe une limite a ` la profondeur de l’ame ´lioration. Ces deux phe ´nome `nes conﬁrment l’existence d’un e ´tat seuil qui a e ´te ´ signale ´ dans les e ´tudes pre ´ce ´dentes sur le compactage dynamique. Les re ´sultats montrent e ´galement que la profondeur de l’ame ´lioration de ´pend de l’impul- sion de chaque frappe ainsi que de l’e ´nergie de chaque frappe. INTRODUCTION Dynamic compaction (DC), or heavy tamping, is a widely used ground improvement method for compacting dry, un- saturated or well-drained, loose granular material. In this method the soil is compacted by the impulsive stress im- parted by multiple impacts from a tamper. Many empirical studies have been carried out by previous researchers (e.g. Menard, 1974; Leonards et al., 1980; Mayne et al., 1984). Menard & Broise (1976) proposed that the depth of im- provement, d, in metres can be related to the potential energy of the tamper per blow by the relationship d ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ W:H p (1) where W is the weight of the tamper in tonnes and H is the height of drop in metres. Leonards et al. (1980) proposed adding a multiplier of 0·5 to the right-hand side of equation (1) to obtain better agreement with measurements. Mayne et al. (1984) reported that the multiplier can range from 0·3 to 0·8. Lukas (1992) reported that the multiplier can vary from 0·5 to 0·6 for pervious granular soil deposits. The range of suggested multiplier values indicates that other factors, in addition to the energy per blow, may affect the depth of improvement. For example, Oshima & Takada’s (1998) cen- trifuge model test data suggest that impulse per blow may also have an effect. One way of assessing the degree of improvement is via the enforced ground surface settlement. Aziz et al. (1980), Mayne et al. (1984) and Lo et al. (1990) suggested that the enforced ground settlement can be related to the square root of the total drop energy per unit area, termed the applied energy intensity. These experimental studies provided much data for design and construction. However, they shed relatively little light on the mechanism of ground improvement; more insight on the latter has tended to come from analytical or numerical studies. Most of the analytical and numerical studies conducted on DC to date have been based on one-dimensional (1D) models (e.g. Scott & Pearce, 1976; Holeyman, 1985; Smits & Quelerij, 1989; Chow et al., 1992). However, such 1D models cannot explain or capture the lateral spread of the ground improvement effect (e.g. Choa et al., 1979; Harada & Suzuki, 1984; Poran et al., 1992; Oshima & Takada, 1998). Chow et al. (1992) justiﬁed the use of their 1D model by reference to Beine’s (1983) laboratory test data, which showed that, down to a depth of about 1·5 times the tamper’s diameter, there is little or no spreading of vertical stresses. However, since Beine’s data showed only vertical stresses, whether the tamping leads to any permanent build- up of lateral earth pressure around the tamper footprint remains uncertain. Furthermore, Beine (1983) did not indi- cate whether the reported data relate to the initial or sub- sequent blows; ground response is likely to change with successive blows. The deﬁciency of such a 1D model is highlighted in Chow et al. (1994), who proposed a 1D wave model for the area directly beneath the footprint but had to resort to an empirical relation established from ﬁeld data for lateral spreading. Poran & Rodriguez (1991, 1992) used a two-dimensional (2D) ﬁnite element (FE) model to study the behaviour of dry sand under DC, and presented predictions of plastic volumetric strains below and beyond the range of tamper. However, their analyses were unable to model the Manuscript received 11 April 2001; revised manuscript accepted 9 April 2002. Discussion on this paper closes 1 March 2003, for further details see p. ii. Centre for Protective Technology, Department of Civil Engineer- ing, National University of Singapore. tailing-off of tamper settlement with successive tamping. Poran & Rodriguez (1991, 1992) attributed this discrepancy to the non-associated ﬂow rule of their cap model, which has an angle of dilation of zero. This would have suppressed any shear-induced dilatancy that would have moderated the build-up of tamper settlement. Moreover, no results were given on the stress changes undergone by the soil during successive blows, so that ground improvement mechanisms remain unclear. The objective of this paper is to examine ground response during repeated tamping blows at a single point, and some key factors that can affect the degree and extent of compac- tion. The important considerations in the modelling of this problem are ﬁrst discussed. A reality check on the model prediction of stress wave propagation is then made by comparing computed and measured dynamic stress wave attenuation patterns at several depths below the ground surface at the point of impact. A further check on the prediction of ground improvement effect is then conducted by comparing model prediction with measurements of rela- tive density, Dr, from Oshima & Takada’s (1998) centrifuge model DC tests. The effects of energy and momentum on the depth of improvement are also examined using this comparative exercise. By examining the stress paths of several points beneath the footprint, the stress changes and progressive densiﬁcation of the ground arising from succes- sive impacts during DC are then clariﬁed. Finally, the effects of momentum and tamper base area on depth and degree of improvement are examined. MODELLING CONSIDERATIONS Large deformation The study was conducted using the dynamic FE program CRISDYN (Goh, 1995), which has an Updated Lagrangian large-strain formulation for dynamic problems incorporated to reﬂect the large strains in the soil during impact (Poran & Rodriguez, 1991). The detailed formulation of the Updated Lagrangian method has been presented by Bathe (1996) for the static problem. Its adaptation to the dynamic problem involves essentially incorporating the necessary inertial ef- fects (e.g. Goh et al., 1998). As the adaptation involved is relatively minor, it will not be repeated herein. Compared with Goh et al.’s (1998) formulation for inertial effects, the only difference is that the material density is conﬁgured as a state variable for each integration point, rather than as a material property, in order to allow it to be changed in each time step. This ensures that the mass of each element is kept relatively constant even with the occurrence of large volu- metric strain. Following Poran & Rodriguez (1992), the impulse from the tamper was modelled by prescribing an initial velocity and acceleration for the tamper elements. Constitutive model The choice of constitutive model was based on the consideration that the ground improvement effect is caused by transient effective stress increase induced uploads/Voyage/ ground-response-to-dynamic-compaction-of-dry-sand 1 .pdf