Simplified numerical simulation of impact compaction

Vienna Congress on Recent Advances in Earthquake Engineering and Structural Dynamics 2013 (VEESD 2013) C. Adam, R. Heuer, W. Lenhardt & C. Schranz (eds) 28-30 August 2013, Vienna, Austria Paper No. 568

Simplified numerical simulation of impact compaction – limits of application F.J. Falkner1, B. Gruber2, C. Adam1 1 2

Unit of Applied Mechanics, University of Innsbruck, Innsbruck, Austria Pfeifer Planung GmbH, Eppan, Italy

Abstract. This paper discusses modelling strategies of different degree of sophistication for simulating numerically rapid impact compaction. To predict realistically the effects of this highly nonlinear dynamic soil problem, appropriate constitutive models for the soil, and suitable interaction models between different components of the compaction device and the soil must be utilized. As constitutive soil model, the application of the Mohr-Coulomb criterion with a non-associated flow rule and an isotropic hardening law is critically reviewed. The impact is modelled both by a fully nonlinear contact algorithm and a simplified engineering approach based on the theory of perfectly elastic impact. The outcomes of numerical examples leads the conclusion that the more sophisticated contact algorithm leads to a more realistic prediction of the compaction effect, however at the price of increased computational efforts. Keywords: Dynamic soil compaction; Mechanical model; Nonlinear contact algorithm; Numerical simulation

1 INTRODUCTION To prevent buildings from damage due to large or non-uniform settlements, the underground must exhibit homogenously distributed sufficient load-bearing capacity. If, at a specific site, this is not the case, the soil must be compacted and homogenized before building construction. Nowadays, additionally to static compaction devices such as static rollers, several dynamic compaction technologies are available for efficient soil improvement. Depending on the compaction depth to be achieved, it is distinguished between near-surface technologies (e.g. dynamic rollers) and deep compaction techniques (deep vibro-compaction, vibroflotation and deep vibro-replacement, heavy tamping). Based on numerical simulations and experimental measurements in a recently completed research project (Adam et al. 2010, Falkner et al. 2010, Adam et al. 2011) it has been confirmed that the novel rapid impact compactor (RIC) provides middle-deep compaction, and thus, closes the gap between the aforementioned dynamic compaction technologies. In contrast to large-scale field tests, which are generally limited in number, the outcomes of parametric numerical studies have revealed a more global picture of the effect of the RIC on the subsoil. However, realistic numerical modelling of rapid impact compaction requires an appropriate mechanical model. Several effects have to be considered therein, such as nonlinear constitutive soil behaviour, impacts between specific components of the RIC, and wave propagation in the underground. This results in a highly nonlinear mechanical model. The development of efficient, reliable and accurate numerical procedures in the last decades, and the enormous increase of the computing capacity at the same time, permits us to handle and solve such complex mechanical models. In contrast to static analysis, in a dynamic simulation inertia and damping forces are present, resulting in cyclic response. However, constitutive models, mainly developed for ultimate load computations, are often used without any modification for dynamic excitation. The fact that the hysteric constitutive soil behaviour is of minor importance in a static

F.J. Falkner, B. Gruber, C. Adam / VEESD 2013

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analysis but strongly influences the result of dynamic simulations is thereby often ignored. Thus, in the present paper the limitations of the popular Mohr-Coulomb plasticity model simulating dynamic compaction processes are therefore critically reviewed. A more sophisticated constitutive soil model for simulating rapid impact compaction is utilized in Grabe et al. (2011). A further issue that is discussed in more detail is the numerical contact algorithm that models the impact between specific components of the RIC. When using such a contact algorithm, the computation time is drastically increased. Therefore, a comparison to a simplified engineering approach is made, and the differences in the outcomes are shown. The outline of the paper is as follows. In section 2 the components and functionality of the RIC is explained in same detail. Afterwards, the mechanical model for the numerical simulation of soil compaction is described. Special attention is given to the constitutive model, and modelling of the impact. Characteristic results of the numerical simulations are shown in section 4. Final remarks conclude the paper. 2 THE RAPID IMPACT COMPACTOR The RIC is an innovative dynamic compaction device based on the piling hammer technology used to increase the bearing capacity of soils through controlled impacts. The general idea of this method is to drop a falling weight from a relatively low height onto a special foot assembly at a fast rate while the foot remains permanently in contact with the ground (Adam and Paulmichl 2007). At present in the Central European area there are several devices permitting a middle-deep improvement of the ground (Fürpass and Bißmann 2012). The RIC consists mainly of three impact components: impact foot, driving cap, and hammer with the falling weight. The impact foot made of steel has a diameter of 1.5 m. The driving cap connected to the foot allows articulation. Impact foot, driving cap, and falling weight are connected to the so-called hammer rig (see Figure 1). Falling weights of mass 5000, 7000, 9000 or 12000 kg are dropped from a falling height up to 1.2 m at a rate 40 to 60 repetitions per minute. For further details see Adam at al. (2010), Falkner et al. (2010), and Adam et al. (2013).

Figure 1. Rapid impact compactor (left), impact foot with driving cap (centre top), compaction spots (centre bottom), and compaction flow (right).


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The RIC provides middle-deep compaction of gravels, sands, silts, industrial by-products, tailings material, and landfills up to a depth of 4 to 7 m efficiently and economically, and is thus an ideal amendment between near-surface and deep compaction technologies. 3 DESCRIPTION OF THE MECHANICAL MODEL Subsequently, specific properties of the mechanical model utilized for predicting the effect of rapid impact compaction are discussed with emphasis on the dynamic characteristics of the problem. Numerical modelling is based on the commercial finite element software ABAQUS (ABAQUS 2010), and therefore, underlies the capabilities of this software. 3.1 Discretization of the subsoil The subsoil is considered as a homogeneous half-space with infinite horizontal surface and infinite soil depth. However, geometry and loading conditions of RIC induced soil compaction allows a significant reduction of the geometric dimension of the mechanical model, dividing the half-space into a near-field and a far-field. The near-field comprises the soil domain that undergoes actual compaction. Bilinear finite elements are used to discretize this soil domain and the components of the compaction device. In contrast, bilinear infinite elements model the far-field to permit unconstrained propagation of the soil waves into the infinite half-space. Figure 2 shows a sketch of the numerical model. The infinite elements facilitate viscous dampers at the interface between the near-field and the far-field, and consequently, reflection of the waves at this interface is avoided. For details it is referred to Lysmer and Kuhlemeyer (1969).

Figure 2. Discretization of the compaction device and the subsoil divided into a near-field and a far-field.

Material damping in the near-field is considered by means of Rayleigh damping (Chopra 2011). From the assumption that viscous damping is 5% at two selected frequencies, the mass and stiffness proportional damping parameters are determined. The first frequency corresponds to the fundamental frequency of the near-field. Lysmer (1965) developed an equivalent single-degree-of-freedom (SDOF) system for vibrating foundations resting on a half-space. The natural frequency of this SDOF system is used as second frequency when computing the damping coefficients. In spite of its simple numerical implementation it should be kept into mind that the Rayleigh model damps modes lower than the


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fundamental frequency much higher than physically justified. Numerical damping within the implicit Hilbert-Hughes-Taylor time integration scheme provides additional damping of higher modes (Wriggers 2008). 3.2 Constitutive soil model In general, soil compaction goes along with permanent soil deformation, as e.g., it can be seen in Figure 1 (centre bottom). It is therefore obvious to describe the soil subjected to dynamic compaction by means of an elastic-plastic constitutive law. Several phenomenological plasticity and hypoplasticity models (such as Kolymbas (2012)) have been developed for non-cohesive soils. Almost all of these models have in common that they have been derived considering static loading only. In particular, currently the Mohr-Coulomb model is predominantly used for geotechnical problems in engineering practice, due to its simplicity and physical clearness. Thus, in an effort to keep computational soil modelling readily comprehensible, in several previous studies (Falkner et al. 2010, Adam et al. 2011) the Mohr-Coulomb model has also been applied for predicting rapid impact compaction of homogenous isotropic soil, although this problem is dynamic by nature. This implies that soil compaction is directly related to yielding, and subsequently to the predicted equivalent plastic strains. However, based on this model soil compaction can be predicted qualitatively only. Subsequently, this approach is critically reviewed. Alternatively, in Pistrol et al. (2012) a modified Drucker-Prager/cap soil model is assessed simulation of dynamic roller compaction. For a plasticity model (i) the yield function, (ii) evolution equations for plastic strains and internal variables, and (iii) hardening and/or softening functions must be defined properly. In the discussed approach, compaction is initiated, if a stress state on the Mohr-Coulomb failure (yield) surface is attained. In Figure 3(a) a meridional section of the failure surface fn is depicted, where p is the hydrostatic pressure of the current stress state (positive in compression), and s is a deviatoric stress measure. Parameters c denotes the cohesion, and ϕ refers to the friction angle. Application of an associated flow rule for plastic strain evolution would overestimate the volumetric straining and consequently the hydrostatic stress state. Therefore, a non-associated flow rule governs the evolution

Figure 3. Mohr-Coulomb model: (a) failure surface, (b) flow potential, (c) soil volume before failure, (d) soil volume after failure.


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of the plastic strains. In Figure 3(b) the underlying smooth flow potential g proposed by Menetrey and Willam (1995) is depicted in the p-s plane. According to this potential the volumetric plastic strain increases during yielding (compaction), see Figure 3(b). However, this is the root of one major drawback of the Mohr-Coulomb model implemented in ABAQUS, i.e. the physically meaningless prediction of an increase of the soil volume during yielding (compare with Figure 3(c) and 3(d)), which is supposed to be related to soil compaction. Even if the dilatation angle ψ is set to zero, the applied plastic potential cannot describe a volume decrease observed in reality. Furthermore, the Mohr-Coulomb criterion cannot predict compaction in a condition of pure hydrostatic pressure. Adding a cap failure surface to the failure surface, combined with an appropriate plastic flow potential, could eliminate this deficiency of the original Mohr-Coulomb model. 3.3 Impact modelling RIC induced soil compaction is a result of the impact between the falling weight and the driving cap. Since the components of the subsystem driving cap-impact foot-soil are in loose contact, the impact induced forces are transmitted impulsively through each component. The magnitude of the energy transfer into the subsoil is essential for its compaction. Therefore, a suitable mechanical model must be chosen to model impact loading appropriately. In the present study two different mechanical impact models for predicting soil compaction are evaluated. In the first model, which represents a simplified engineering approach, the driving cap together with impact foot and falling weight are considered as rigid bodies. Assuming that no mechanical energy is dissipated during the impact (perfectly elastic impact) the velocity of the impact foot after the impact can be computed according to Ziegler (1998). (1) In this equation and denotes the mass of the falling weight and the impact foot, respectively. is the velocity of the falling weight immediately before the impact, with h = 1.2 m denoting the falling height and g the acceleration of gravity. When deriving Eq. (1) standard assumptions of the perfectly elastic impact, such as no position change of the rigid bodies during the impact, are applied (Ziegler 1998). Constant pressure , (2) applied at t = 0 to the top surface of the driving cap with area , and removed at impact time according to a Heaviside function, models impact induced change of momentum. In all numerical simulations, impact time is assumed to be 10-3 s. In this simplified numerical procedure only normal stresses can be transmitted between impact foot and soil. Therefore, the impact foot can slide in horizontal direction without transmitting shear stresses. Within the more elaborated second approach, a nonlinear contact algorithm models load transmission between (i) falling weight-driving cap, (ii) driving cap-impact hammer, and (iii) impact hammer-soil. The commercial finite element software ABAQUS (ABAQUS 2010) provides several possibilities for modelling contact between deformable bodies. In the present study, the method of Lagrange multipliers is used (Bathe 2002, Wriggers 2008). Thereby, an opening function measures the distance between nodes of defined contact surfaces. If during the analysis the distance becomes negative, i.e. body penetration is initiated, a contact pressure is applied in opposite direction to the two contact surfaces to set the distance function equal to zero. This contact pressure corresponds to the Lagrange


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multiplier. It should be recognized that the total number of unknowns increases, compared to the simplified engineering approach (Gruber 2012). For more details it is referred to ABAQUS (2010). 4 APPLICATION AND ASSESSMENT In the following selected results of a comprehensive numerical study of rapid impact compaction (Gruber 2012) of homogeneous silty fine sand are presented. In all example problems the MohrCoulomb plasticity model is utilized to describe the constitutive behaviour of the soil subjected to impact loading. In Table 1 the utilized soil parameters of silty fine sand for the Mohr-Coulomb plasticity model are listed. An isotropic hardening law for the cohesion c as shown in Figure 4 is applied to consider strength increase of the soil. For the compaction equipment standard constitutive parameters of elastic steel are used. The mass of the falling weight and the impact foot including the driving cap is 9000 kg and 4000 kg, respectively. Table 1. Soil properties of silty fine sand for the Mohr-Coulomb plasticity model. Youngs modulus E [MN/m²]

Poisson ratio ν [-]

Density ρ [kg/m³]

Cohesion c [kN/m²]

Friction angle ρ Dilatancy angle ψ [°] [°]

10

0.30

2000

2

26

8

Figure 4. Isotropic hardening law for the cohesion c in dependence of the equivalent plastic strain PEEQ.

4.1 Prediction of soil compaction Figure 5 shows at certain time instances (as indicated) the distribution of the hydrostatic pressure in the soil after the first impact of the falling weight. The results of these studies are based on a mechanical model that utilizes the more accurate contact formulation implemented in ABAQUS (2010), as previously described. The radius and the depth of the considered cylindrical soil segment are 15 m and 8 m, respectively. Immediately after the impact, close to the impact foot the soil surface is subject to tensile stresses (grey coloured area in Figure 5). These stresses are related to the Rayleigh waves (Studer et al. 2007), and thus, in the field the soil surface is radially cracked. Since the horizontal deformability of the soil is constrained, the hydrostatic pressure increases enormously below the compaction foot immediately after the impact. Subsequently, shear waves transport the impact energy to a large extent into the half-space, and simultaneously, plastic deformations develop in the subsoil. In the considered mechanical model it is assumed that soil compaction is associated to plastic deformations. The last subfigure of Figure 5, which shows the hydrostatic stress 2 s after the


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Figure 5. Development of the hydrostatic pressure in the soil after the first impact [N/m²].

impact where the soil is almost at rest, an increased hydrostatic pressure up to a depth of 3 m as a result of the first impact. In the Mohr-Coulomb plasticity model the only internal variable is the so-called equivalent plastic strain (PEEQ). This quantity memorizes the plastic loading regime, and therefore, gives information about soil compaction. Figure 6 depicts the distribution of the logarithmic scaled equivalent plastic strain of a vertical section through the centre of the half-space at two different time instances. Thereby, a threshold of 2% of the equivalent plastic strain separates the compacted space from the non-compacted subsoil. According to the compaction domain shown to the right, 2 s after the first impact a compaction depth of almost 4 m is predicted. It is noted that the underlying Mohr-Coulomb plasticity model underpredicts the magnitude of compaction, because the hydrostatic pressure under the compaction foot is very large, see Figure 5. However, for pure hydrostatic stress states the MohrCoulomb model cannot predict plastic deformation, as explained before. Furthermore, the volumetric plastic strain is larger than zero, and hence, soil displacement is predicted rather than compaction.

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6m


8m

Figure 6. Equivalent plastic strains in the soil after the first impact.

Figure 7. Kinetic energy of the mechanical model for the two different impact models.

4.2 Assessment of impact modelling strategies Next, the effect of the two different impact modelling strategies, as described before, on compaction prediction is discussed. Figure 7 shows the kinetic energy evolution of the considered mechanical models with respect to time. It is readily observed that immediately after the impact the kinetic energy based on the contact model is about 15% higher in comparison to the corresponding outcome of the simplified approach based on the theory of perfectly elastic impact. This can be partly led back to the assumption of the theory of perfectly elastic impact that the mass of the falling weight is not present after it strikes the driving cap. In contrast, in the contact model the falling weight remains after the impact on the driving cap, and hence, more energy is transferred into the soil. A further consequence of this larger energy input into the half-space is a different prediction of the compaction domain, as shown in Figure 8. In this figure for both models the distribution of the PEEQ at time t = 2.0 s after the first impact is given. It can be seen that the predicted compaction depth based


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6m

on the contact model is 4.0 m, while the perfectly elastic impact approach estimates this quantity with 3.5 m only. However, the horizontal extension of the PEEQ is smaller when using the simplified model. According to field tests the predicted compaction depth of 4 m is more realistic. Note that a simplified geometry of the impact foot, however with same mass and same contact area of the actual foot, was used when applying the theory of perfectly elastic impact.

8m

Contact formulation

Perfectly elastic impact

Figure 8. Comparison of soil compaction based on different impact modelling strategies.

5 CONCLUSIONS In the present study a mechanical model for the numerical prediction of rapid impact soil compaction was presented. Two crucial issues, i.e. modelling of the impact problem and the soil behaviour, were discussed in some detail. For the impact problem advantages and disadvantages of a simplified engineering approach and a more sophisticated contact formulation were set in contrast. The latter formulation requires fewer assumptions when establishing the mechanical impact model. Numerical simulations revealed that the input energy is considerably larger when using the contact model, resulting in a deeper and more realistic compaction effect. Therefore, for further numerical studies of rapid impact compaction this impact model should be the preferred preference. However, increased computational efforts must be accepted. In the present study the Mohr-Coulomb plasticity model was used to describe the constitutive behaviour of soil under impact loading. However, as major drawback of this simple to apply and wellaccepted constitutive model, failure (compaction) of soil in a condition of pure hydrostatic pressure cannot be predicted. Hence, a cap surface in the direction of the hydrostatic axis should be added to the failure surface. Furthermore, the plastic potential must be adopted to predict appropriately volumetric plastic strains leading to a volume decrease of the soil, which is a main characteristic of a compaction process. It can be concluded that there exists a need for appropriate constitutive models that describe sufficiently accurate the hysteric behaviour of soil subjected dynamic loads. Further (theoretical) research should be directed in the development of (simple) material models for simulating dynamic compaction.


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