Deep Vibrocompaction at the Natural Frequency of the Soil Response

Deep horizontal vibrocompaction (also referred to as vibroflotation) is a ground improvement technique for the compaction of granular soils. It was developed by the Johann Keller GmbH in the early 1930s and has been widely used ever since (Kirsch and Kirsch 2017). The technique is based on the rearrangement of particles into a dense state by the introduction of mainly horizontal vibration of the vibrator body, resulting in increased soil density. In addition to the reduction of the void volume, changes in the stress state of the soil also result from the increasing horizontal stresses, according to Massarsch and Fellenius (2002) and Massarsch et al. (2020). The benefits of a correctly executed deep horizontal vibrocompaction are increased soil stiffness, a homogenized subsoil, reduced settlements, and a reduction of the liquefaction potential (Kirsch and Kirsch 2017). The main advantages compared to other compaction technologies are the significantly high compaction depth and the homogeneous improvement of soil conditions along the treated depth. Vibroflotation is used for the compaction of granular soils with a comparatively low content of fines, e.g., in land reclamation projects or compaction of loose natural sediments. The practical application of this technique has been extensively discussed in the literature since the 1950s, for example, by D’Appolonia (1953), Thorburn (1975), Wehr and Sondermann (2012), and Kirsch and Kirsch (2017).

The device for deep horizontal vibrocompaction comprises the vibrator body and a variable number of extension tubes to reach the required compaction depth. A flexible coupling connects the vibrator body to the extension tubes. The compaction rig is suspended from a crane or mounted on a specifically designed base machine. An eccentric mass rotating around its vertical axis inside the vibrator body induces the horizontal vibrations of the vibrator. The motor for the eccentric mass is usually located above the mass and is electrically driven (Wehr and Sondermann 2012).

After the vibrator is lowered to the desired compaction depth, the compaction process is performed from bottom to top. During the compaction, the vibrator is either kept at a constant depth for a specified time or the vibrator is withdrawn and then lowered again by typically half the withdrawal height. For the second method, hydraulic pull-down pressure supports the penetration process by activating the dead weight of the compaction rig. This method is called the back-step procedure and is mainly used in slightly cohesive, granular soils.

The compaction success is usually assessed through conventional site investigation methods such as cone penetration tests with (CPTu) or without (CPT) pore-water pressure measurement, standard penetration tests (SPTs), or dynamic probing (DP). CPTu tests have gained worldwide acceptance for their quality control of deep vibrocompaction in recent years. They are a relatively fast and cost-effective method that provides a continuous profile along the depth and more than one reading (Robertson 2006). The application of various site investigation methods and the interpretation of the data is discussed in Robertson (2006), Bo et al. (2012), and Bałachowski and Kurek (2015). Moreover, Covil et al. (1997) and Mitchell and Solymar (1984) point out the influence of aging effects on the results of CPTu tests.

In addition to conventional site investigation, geophysical methods are an alternative way to assess the compaction success of larger areas. However, those methods are rarely used. Application examples are presented by Kim and Park (1999), Karray et al. (2010), and Bitri et al. (2013).

Site investigation methods for assessing the results of the compaction work can only be applied after the compaction work has been completed; therefore, these methods cannot be used to improve the compaction process during the actual compaction work. Moreover, the tests are often time-consuming and are characterized by the random nature of spot-like inspection methods. Therefore, researchers and construction companies have pursued the idea of a vibrator-integrated compaction control to not only evaluate the compaction success, but also control the compaction process. The basic principle of a vibrator-integrated compaction control is the recording and evaluation of process parameters and their correlation with the achieved compaction result.

Poteur (1968, 1971) measured accelerations in the compacted soil in addition to the vibrator motion during large-scale field tests and compared the results to theoretical studies. Morgan and Thomson (1983) determined the amplitude of the vibrator tip for various vibrator types to correlate the amplitude to results from dynamic probing tests. Fellin et al. (2003) presented an analytical model and compared its results to measurements of vibrator motion during compaction of sand fills. Nendza (2006) used a 1:3 scale vibrator model to investigate the influence of various machine and process parameters on the compaction effect. Model tests with a “mini vibrator” were recently also performed by Nagula and Grabe (2020). Theoretical studies by means of analytical, semianalytical, and different numerical approaches have been conducted by Arnold and Herle (2009), Fellin (2000), Nagula and Grabe (2017), Triantafyllidis and Kimmig (2019), and Wotzlaw et al. (2023). In a recent study, Nagy et al. (2021) use an analytical model to investigate the relationship between the actual soil stiffness and the vibrator motion. They present a working hypothesis that allows for an identification of dominant soil mechanical processes leading to the compaction effect in the soil.

According to Brumund and Leonards (1972), Seed and Silver (1972), and Youd (1972), the densification process in the compaction of granular soils primarily depends on the shear strain amplitude. The shear strain amplitude was also identified by Arnold and Herle (2009) as the most important factor for densification by vibroflotation. Massarsch (2023) notes that in addition to the shear strain amplitude, the number of vibration cycles also has a significant influence on the compaction process. Both the shear strain amplitude and the number of vibration cycles are influenced by machine and process parameters, e.g., the excitation frequency and the holding times determine the number of vibration cycles. The shear strain amplitude in the soil caused by deep horizontal vibrocompaction is also dependent on the excitation frequency and also, for example, on the applied tip load and the jetted water.

The term “deep vibratory compaction” is sometimes used as a collective term for horizontal and vertical vibratory compaction methods in larger depths. The interaction system of compaction equipment and soil and its resonance characteristics differ greatly, depending on the dominant direction of excitation. A comprehensive overview on deep vertical vibratory compaction is given by Massarsch (2023).

This paper focuses exclusively on deep horizontal vibrocompaction (vibroflotation). Theoretical considerations are made using a simple single-degree-of-freedom (SDOF) model to develop the novel concept of deep vibrocompaction at the natural frequency of the soil response. The presented concept has the potential for the development of an “online” control of the compaction process and the optimization of compaction by adjusting the parameters of the process during the compaction work.

In contrast to the state-of-the-art procedure of aiming for a compaction at the natural frequency of the interaction system, the authors mentioned a new concept in Kopf et al. (2023). This different and novel approach for an optimized compaction is presented in this paper.

The dynamically excited vibrator causes a response of the soil in the form of a resulting soil contact force, denoted by $F_{\mathrm{b}}$. The soil contact force comprises the entire response of the soil system, including the soil stiffness, radiation damping (both of which might be frequency-dependent and/or nonlinear), as well as a vibrating soil mass. The dynamic properties of the soil are unknown and no attempt is made to describe them, e.g., in the form of a spring stiffness or a dashpot coefficient. It is proposed to calculate the resulting soil contact force $F_{\mathrm{b}}$ and its direction from measurements and directly measure the displacement amplitude $A$ of the vibrator and its direction. It is suggested that an angle of ${\phi }_{b,\mathrm{ef}}=90°$ between the directions of the soil contact force $F_{\mathrm{b}}$ and the displacement $A$ results in the best possible compaction, since it is a compaction at the natural frequency of the soil response. The amplitudes and directions of $F_{\mathrm{b}}$ and $A$ will change of course as the soil is compacted.

The applicability of the concept requires fulfillment of the following conditions:

Given suitable soil conditions, conditions 1, 2, and 4 are considered fulfilled for deep horizontal vibrocompaction. In order to fulfill condition 4, it is assumed that the vibrator performs a rigid-body motion along a cone of rotation, with the tip of the cone located at the flexible coupling. The vibrator movement is therefore described with only an SDOF, the rotation angle of the vibrator along the cone around the vertical axis. The admissibility of this approach is not necessarily obvious, but it is shown in the following that this simplification is permissible at least for the S-series vibrator under investigation in almost all registered motion states.

The formulated prerequisites for the application of the concept limit confirm its transferability to other forms of vibratory compaction. Vibratory roller compaction and vibratory plate compaction, for example, are characterized by the challenging contact conditions between compaction equipment and soil. With the exception of very soft soils, vibratory rollers and plate compactors will periodically or aperiodically loose contact with the soil. Therefore, continuous contact (condition 1) cannot be guaranteed, and the compaction energy is transmitted to the soil in a series of impacts instead of a harmonic loading (condition 2). Deep vertical vibratory compaction is similar to vibroflotation in some aspects, and the resonance behavior of the vibrator-probe-ground system has already been investigated by Massarsch and Fellenius (2005) for this vibratory compaction method. However, the concept of vibrocompaction at the natural frequency of the soil response is not applicable to vertically oscillating probes, as the resulting soil contact force does not rotate around the vertical axis of the probe (condition 4).

The presented concept proposes to calculate the resulting soil contact force $F_{\mathrm{b}}$ and its direction from measurements and directly measure the displacement amplitude $A$ of the vibrator and its direction. It is suggested to strive for an angle of ${\phi }_{\mathrm{b},\mathrm{ef}}=90°$ between the directions of the soil contact force $F_{\mathrm{b}}$ and the displacement $A$ to optimize the densification process.

It has not yet been possible to apply the proposed concept in specially designed experimental studies. The concept is therefore tested using selected measurement data from the experimental field tests that were carried out in 2019 before the methodology presented was developed.

The amplitude $A$ and the lead angle $\phi$ were recorded during the field tests and are evaluated for a variation of the excitation frequency. In addition to the standard excitation frequency of $f=25\mathrm{Hz}$, a reduced frequency of $f=18\mathrm{Hz}$ and an increased frequency of $f=25\mathrm{Hz}$ were tested. Three compaction tests were performed for each of the frequency variants (Fig. 4).

The amplitude ratio $A/A_{\infty }$ is plotted over the lead angle $\phi$ in Fig. 12 for seven compaction steps at depths of –10 m to –5 m. The results are shown for both horizontal vibrator directions ($x$ = transverse to wing direction and $y$ = in wing direction) and each of the three excitation frequencies. The results in Fig. 12 confirm that a lead angle of $\phi =90°$ (left edge of the diagram), as suggested in the literature, is far from being achieved in practice. Detailed representations are given in Fig. 13 for the $x$-direction (transverse to wing direction), and in Fig. 14 for the $y$-direction (in wing direction), respectively. The geometry of the vibrator in its setup utilized for the tests with wings in the $y$-direction causes different bedding conditions in the two horizontal directions and thus a direction-dependent motion behavior of the vibrator.

The vibrator motion only shows a small variability transverse to the wing direction ($x$) in case of the high excitation frequency of $f=28\mathrm{Hz}$ (Figs. 12 and 13). The high amplitude of the excitation force $F_{\mathrm{e}}$ dominates the vibrator motion and the influence of the soil is minimal. In contrast, the amplitude ratios of the low excitation frequency of $f=18\mathrm{Hz}$ show higher variability. They cover a large range of the trailing angle ${\phi }_{\mathrm{b},\mathrm{ef}}$ of the soil contact force and even get fairly close to the natural frequency of the soil response at ${\phi }_{\mathrm{b},\mathrm{ef}}=90°$.

The increased excitation frequency of $f=28\mathrm{Hz}$ leads to a significantly larger amplitude of the excitation force of $F_{\mathrm{e}}\approx 900\mathrm{kN}$ (Table 1). However, only about a third of it is utilized as soil contact force (Fig. 13). The amplitude of the excitation force is significantly smaller for the reduced frequency of $f=18\mathrm{Hz}$ ($F_{\mathrm{e}}\approx 400\mathrm{kN}$, see Table 1), but it is used much more efficiently and about half of it is utilized as soil contact force. An increased flow rate of the jetted water at the cone ($200\mathrm{L}/\min$ instead of $150\mathrm{L}/\min$) was required to operate the vibrator at the increased frequency of $f=28\mathrm{Hz}$ without exceeding the power and temperature limit of the vibrator.

The compaction success was monitored with CPTu tests. The tests were carried out 18 to 21 days after treatment at the centers of the triangular grid of the compaction tests with the same vibration frequencies (Fig. 4). A full set of CPTu results for the three locations after treatment in comparison with the results obtained prior to treatment is given in Fig. 15. With exception of the near-surface area, cone resistance and sleeve friction were low in the sand fill but increased after treatment by vibrocompaction. The degree of improvement is approximately the same below the groundwater table for the cone resistance and the sleeve friction, whereby the mean values after treatment are around five times higher than before compaction. Compaction with an excitation frequency of $f=18\mathrm{Hz}$ tended to lead to the highest absolute values of cone resistance, while pronounced sleeve friction values were recorded after treatment with $f=28\mathrm{Hz}$. With only one CPTu test per parameter set, the differences in the CPTu results after compaction cannot be attributed exclusively to the different excitation frequencies, as these also depend on the local initial conditions. However, it is noteworthy that compaction with a reduced excitation frequency leads to densification results below the groundwater table that are at least on par with the results after treatment with higher frequencies.

The CPTu results above the groundwater table cannot be directly attributed to the different excitation frequencies and must be viewed with caution. In order not to exceed the power and temperature limit of the vibrator, which is harmful to the unit, the motor was automatically regulated back above the groundwater table, whereby the frequency was reduced below the initial setting and the vibrator had to be pulled faster.

In Table 1, additional parameters of the compaction tests are listed. The compaction with the reduced frequency of $f=18\mathrm{Hz}$ required less than half the electrical energy, a shorter period of time, and less jetted water, but still achieved compaction results on par with the results after compaction with an increased frequency of $f=28\mathrm{Hz}$ (Fig. 15). The differences in electrical energy demand and required compaction time are less pronounced when comparing the results for compaction with $f=18\mathrm{Hz}$ to the standard, but are still in favor of the reduced excitation frequency. It is assumed that the increased electrical energy demand required for compaction at high excitation frequencies, but which does not lead to an improvement in compaction success, merely results in increased machine wear.

The good compaction results after treatment with a reduced excitation frequency despite the significantly reduced amplitude of the excitation force are attributed to the large variability of the trailing angle ${\phi }_{\mathrm{b},\mathrm{ef}}$ and the compaction closer to the natural frequency of the soil response. For practical application in vibrocompaction projects, it is therefore recommended to adapt the process parameters of the vibrator to the conditions found on site and not to rely solely on the theoretical amplitude of the excitation force $F_{\mathrm{e}}$.

The state of research suggests that the best possible compaction results in deep horizontal vibrocompaction are achieved when the vibrator operates at the natural frequency of the vibrator-soil interaction system (lead angle $\phi =90°$). In the scope of a previous research project, a vibrator was specially equipped with sensors to allow a lead angle controlled compaction. The compaction results were not satisfactory and the state of research concept proved to be unsustainable in practical application. This finding led to the research presented in this paper and the development of the concept of compaction at the natural frequency of the soil response.

The presented concept suggests to calculate the resulting soil contact force $F_{\mathrm{b}}$ and its direction from measurements and directly measure the displacement amplitude $A$ of the vibrator and its direction. It is proposed that an angle of ${\phi }_{\mathrm{b},\mathrm{ef}}=90°$ between the directions of the soil contact force and the displacement amplitude results in an optimized compaction. The applicability of the concept requires continuous contact between the vibrator and the soil, a harmonic loading of the soil, the admissibility of describing the vibrator movement with a SDOF, and the soil contact force to rotate around the vertical axis of the vibrator.

It was shown on the example of measurement data that the simplified approach of describing the vibrator movement with only a SDOF is permissible at least for the vibrator under the investigated conditions. A procedure for the determination of the unknown soil contact force $F_{\mathrm{b}}$ and the trailing angle ${\phi }_{\mathrm{b},\mathrm{ef}}$ was shown by means of the force-displacement relation in the excitation plane. Compaction at the natural frequency of the soil response is given for a trailing angle of ${\phi }_{\mathrm{b},\mathrm{ef}}=90°$. The introduced equations and the diagram in Fig. 11 allow a quick determination of the soil contact force and the trailing angle based on measurements of the vibration amplitude and the lead angle during compaction.

The method for determining the soil contact force $F_{\mathrm{b}}$ and the trailing angle ${\phi }_{\mathrm{b},\mathrm{ef}}$ was applied to measurement data from compaction tests as part of a land reclamation project in Southeast Asia, which were carried out in 2019 before the concept was developed. Tests with three different excitation frequencies were investigated. CPTu tests were conducted at selected points of the triangular compaction grid before treatment and at the center points of the grid after compaction to monitor compaction success.

The main findings of the research for the practical application of deep horizontal vibrocompaction are as follows:

All data, models, and code generated or used during the study appear in the published article.