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BY 4.0 license Open Access Published by De Gruyter Open Access March 7, 2023

Evaluation of dynamic behavior of varved clays from the Warsaw ice-dammed lake, Poland

  • Krzysztof Nielepkowicz EMAIL logo , Paweł Dobak and Anna Bąkowska
From the journal Open Geosciences


Soil stiffness degradation due to dynamic loading may result in damage to engineering structures. Therefore, dynamic soil properties gained importance in recent years in the areas where vibrations of different origins are transmitted to the ground. The aim of the study was to determine the stiffness properties of the Warsaw varved clay (central Poland), as a measure of its response to dynamic loading. A set of triaxial tests were conducted. Undisturbed samples were subjected to initial deviator stress of 100 kPa, reflecting vertical stress due to construction load, and then to three dynamic stages under axial displacement control conditions. Secant shear modulus G was adopted as a parameter of soil stiffness during dynamic loading. Main methodological result of the study is application of coefficient α (gradient of logarithmic change in stiffness with time) indicating type of soil behavior under dynamic loading. Negative α values (ranging from −0.48 to −0.91) indicated a weakening effect of the applied vibrations (decrease in stiffness), while positive α values (ranging from 0.11 to 0.21) indicated soil strengthening (increase in stiffness). Results of the study may be applied to compare an impact of several factors influencing soil stiffness during dynamic loading

1 Introduction

Dynamic behavior and strength properties of cohesive soils gained importance in recent time due to fast development of transport infrastructure, in particular, roads and railways. Increasing number of roads and railways have to be built in difficult ground conditions, including soft cohesive soils of low stiffness and strength [1,2,3,4]. Thus numerous studies are required: means of mitigating vibrations, techniques for strengthening the ground, and determination of soil mechanical parameters [5,6,7,8,9].

The influence of road or railway traffic on soft cohesive soils can be assessed by combination of both field and laboratory methods. The main objective of field investigations is to collect vibration intensity data (amplitude, acceleration, and frequency values), characteristic for urban transport [10,11]. Collected data can be further applied for laboratory investigations, i.e., shear strength test under cyclic loading.

Shear modulus (G) is a parameter which can be efficiently used for assessment of strength and compressibility of soil under small strain range [12]. There is also a common approach predicting long-term behavior of clayey soils throughout the evaluation of hysteresis loop parameters and calculation of permanent strain [3,13,14]. Some investigations also concern the comparison of applicability of various methods used for predicting stiffness and deformability of soil [15,16,17].

The main objective of this study is to evaluate the behavior of varved clay subjected to road vibration. According to Myślińska [18,19], weakening of varved clays is expected under additional dynamic loading, particularly due to the structural and textural properties of varved clay. We designed a laboratory program of triaxial strength tests to determine the mechanical parameters of varved clay under static and dynamic conditions. It allows us to study shear modulus variation and its dependence on different factors.

Shear modulus can be adopted as a relevant parameter of a long-term behavior of a soft clay, that is more accurate for predicting the magnitude of foundation settlement than commonly used deformation moduli [15]. The applicability of triaxial tests for determining value of shear modulus has more advantages than classical shear strength or deformation tests, and is comparable with other methods like resonant column or bender element test [12]. The tests conducted in this study were executed in the small to medium strain interval (γ = 3.0 × 10−4 to 4.7 × 10−3), which corresponds to the typical geotechnical and engineering situations: retaining walls, foundations, and tunnels. Thus, the results of our investigation can be applied for calculations of bearing capacity of soil subjected not only to static load but also dynamic load, as in case of traffic or the other sources of vibrations.

Triaxial tests were executed under axial displacement control (ADC) method of dynamic loadings, thus our investigations focused on the stiffness degradation (the reduction in stiffness with the increase in the loading cycles), and the factors affecting this process.

2 Methods

2.1 Material origin and properties

Varved clay originates in specific conditions, including existence of ice sheet front, periglacial climate, variation in material supply due to the seasonal conditions, and variation in sedimentation rate. Varved clay occurs in many locations in the northern hemisphere (USA, Poland, Sweden, Estonia, etc.), and the description of their geological properties has been presented by many researchers, e.g., Merta [20], Młynarek and Horvath [21], De Groot and Lutenegger [22], Lu et al. [23], Tankiewicz and Kawa [24], Kalvāns et al. [25], Johnson et al. [26], etc.

Varved clay of the Warsaw ice-dammed basin (Figure 1) is of heterogeneous provenance, structure, and properties which demonstrate substantial variability in both the vertical and horizontal directions [9,16,17,18,19,20]. The Warsaw ice-dammed basin was formed at the younger Pleistocene and covered most of the Mazovia Lowland. The basin was supplied with the material from the south and east (directions opposite to the glaciated area). This means that the basin was fed by extraglacial rivers from these directions (and not by the proglacial waters). The sediment is a type of extraglacial varved deposits [20].

Figure 1 
                  Location of the sample collection (marked in red) against the extent of Warsaw ice-dammed lake (© OpenStreetMap contributors (Myślińska [27] after Różycki [28], modified).
Figure 1

Location of the sample collection (marked in red) against the extent of Warsaw ice-dammed lake (© OpenStreetMap contributors (Myślińska [27] after Różycki [28], modified).

The specific structure of the varved clay is a result of variation in seasonal conditions at the front of the Vistulian Glaciation ice sheet. Each varve is composed from two alternating layers: the light one (silt) – deposited during the warm period, and the dark one (clay) – deposited during the cold period. Varved clay of the Warsaw ice-dammed lake is also characterized by significant variation in terms of facies. Three types of varves are indicated, each corresponding to different sedimentary zone, depending on the location in the sedimentary basin [20].

  1. Proximal zone – This zone consists of A-type varves (with indistinct gradual boundaries, light layers thicker than dark ones, and total thickness of a varve above 4.0 cm), which is the environment adjacent to the river mouth. Sandy sediments are dominating (with current structures, and without wave ripples), but there are also clayey layers, provided by subglacial streams. This environment is intermediate between fluviatile and basin regime.

  2. Transitional (intermediate) zone – This zone is characterized by the existence of B-type varves (with contrasting sharp boundaries, light and dark layers of equal thickness, and total thickness of a varve in the range of 2.0–4.0 cm), located between the proximal and distal zones. Both current and wave structures can be found on the surface of varves.

  3. Distal zone – This zone consists of C-type varves (with contrasting sharp boundaries, light layers thinner than dark ones, and total thickness of a varve below 2.0 cm), located far from the river mouth. It is characterized by a small amount of sandy-silty sediment and those layers are thinner than in previous zones. Turbidity currents were a main factor of deposition (greater water depths, no wave ripples).

The significant difference between dark and light layers is a proof of the different conditions of the environmental energy. The current and wave activities with bottom suspension currents were the characteristics of the light layers, while quiet sedimentations were the characteristics of the dark ones. The B- and C-type varves are suspected to be a product of suspension currents, while A-type ones are more characteristic of environments close to fluvial [20]. Many of the varves are highly variable in terms of thickness, even within the same type of varve (supernormal varves). Variability does not show any regularity. Changes in thickness are independent between light and dark layers and the clay content is more uniformly dispersed than sandy-silty sediments [20].

In some areas, in the eastern part of the Warsaw ice-dammed lake, the varved clay is covered by a thin, 1–3 m layer of Holocene sand of aeolian origin.

Varved clay is one of the most important units of the subsoil in towns located northeast from Warsaw: Radzymin, Kobyłka, and Zielonka (Figure 1). That is why soil samples for laboratory tests were collected in Kobyłka town (Figure 1). Two parallel drills at a distance of 5 m were executed. The material was collected using Shelby’s thin-walled tubes, from the depth of 1.3–2.6 m. Collected soil corresponds mainly to the A-type type of varves (as described by Merta [20]). Mean values of index properties of silty and clayey layers are presented in Table 1.

Table 1

Average values of index parameters of the varved clay

Parameters Silty layer Clayey layer
Clay content (%) 25 48
Silt content (%) 74 51
Sand content (%) 1 1
Particle density (Mg/m3) 2.69 2.76
Water content (%) 27.3 34.3
Plastic limit (%) 19.8 25.9
Liquid limit (%) 35.9 59.8

Clay content shows significant variation which is noticeable macroscopically. This may cause unusual differentiation of mechanical properties, as the horizontal shear strength along the clayey layer may be lower than the vertical shear strength of varved clay. Thus, it is of particular importance to apply adequate methods for predicting the behavior of the varved clay under different loading conditions [9].

In respect of the history of geological loads, Zawrzykraj [17] proved that varved clay of the Warsaw ice-dammed lake are normally consolidated soils, and have not been covered by either the ice sheet or the large amount of younger deposits. But these deposits can demonstrate behavior comparable to some overconsolidated soils because of the post-sedimentary chemical cementation process, occurring in conditions of lowering of water table.

2.2 Test procedure

The aim of the investigation was to study the dynamic behavior of the varved clay in conditions corresponding to in situ conditions of typical construction in Kobyłka: subsoil loaded by the construction of unit stress of 100 kPa at the foundation level and additionally subjected to intermittent vibrations due to traffic. The dynamic behavior of the samples was compared to their static behavior, thus the laboratory test program included two types of triaxial CU (consolidated undrained) tests: under static conditions (monotonic loading) and under dynamic conditions (cyclic loading). All tests were conducted in GDS Ltd dynamic triaxial testing apparatus under the CU conditions (CU tests) according to the following procedure:

  • The samples were oriented in the direction corresponding to their orientation in natural conditions. The specimens were trimmed to an average dimension of D = 35.0–38.4 mm and H/D = 1.83–2.65

  • Prior shearing, each test consisted of the following stages: initial loading under effective stress, three saturation steps with three B-checks, consolidation stage, initial deviator stress increase, three dynamic stages, and final static shearing

  • Initial effective stress applied on each sample was equal to 50 kPa, in case of preventing swelling

  • After that, the back-pressure saturation was undertaken, average B-value for all the samples was 0.93

  • Next stage was isotropic consolidation under effective confining pressure, respectively, σ c = 100, 200, or 400 kPa

  • After consolidation, samples were subjected to an increase in the initial deviator stress to 100 kPa to reflect the vertical stress that is applied to the soil due to the construction load

  • Then, axial loading was applied under undrained conditions

  • For samples tested under static conditions, monotonic loading with constant rate of strain of 0.01 mm/min were executed

  • For samples tested under dynamic conditions, three consecutive stages of ADC tests were performed (Table 2)

  1. Dynamic stage I – high amplitude vibration: displacement amplitude A = 0.1 mm, frequency f = 5 Hz, and number of cycles N = 1,000

  2. Dynamic stage II – low amplitude vibration: A = 0.01 mm, f = 5 Hz, and N = 10,000

  3. Dynamic stage III – high amplitude vibration: A = 0.1 mm, f = 5 Hz, and N = 1,000

Table 2

Parameters of vibration applied in the dynamic stages

Test parameters Dynamic stage I Dynamic stage II Dynamic stage III
Amplitude A (mm) 0.1 0.01 0.1
Frequency f (Hz) 5 5 5
Number of cycles N (–) 103 104 103
Acceleration a (mm/s2) 98.7 9.9 98.7

Vibration parameters were designed with regards to correspond to the different traffic conditions in site, and the technical limitations of the triaxial apparatus.

After each dynamic stage, samples were subjected to mid-phase loadings, so that each dynamic stage started from 100 kPa of initial deviator stress. General conditions of dynamic phases were presented on the schematic graphs (Figure 2).

Figure 2 
                  Scheme of cyclic loadings.
Figure 2

Scheme of cyclic loadings.

Since no shear failure was observed during dynamic loading, after the dynamic stage III, the monotonic shearing was executed with a constant rate of strain of 0.01 mm/min to determine the post-dynamic shear strength.

In total, 24 tests were conducted. Under dynamic conditions, 17 samples were tested: 4 samples consolidated under σ c = 100 kPa, 7 samples under σ c = 200 kPa, and 6 samples under σ c = 400 kPa. Under static conditions 7 samples were tested: 2 samples under σ c = 100 kPa, 2 samples under σ c = 200 kPa, and 3 samples under σ c = 400 kPa.

3 Results

3.1 Shear modulus parameter

Modulus of deformation was calculated on the basis of the following equation (Figure 3):

(1) E N = dev N , max dev N , min ε N , max ε N , min ,

where E N is the modulus of deformation at the Nth cycle; dev N,max and dev N,min are the maximum and minimum deviator stress at the Nth cycle; and ε N,max and ε N,min are the maximum and minimum axial strains at the Nth cycle, respectively.

Figure 3 
                  Shear modulus on the basis of hysteresis loop of Nth cycle.
Figure 3

Shear modulus on the basis of hysteresis loop of Nth cycle.

Then, shear modulus (G) was calculated from the following equations [29]:

(2) Δ γ = ( 1 + ν ) · Δ ε ,

(3) G N = E N 2 · ( 1 + ν ) ,

where ν is the Poisson coefficient; Δε is the axial strain amplitude; Δγ is the shear strain amplitude; and G N is the shear modulus at the Nth cycle.

The Poisson coefficient equal to 0.5 was taken into calculations, as the soil samples were fully saturated prior to triaxial loading.

Characteristic loops for 1, 100, 1,000, and 10,000 cycles of dynamic stages were selected to determine the values of G (Table 3). The results of G values (Table 3) were analyzed with regards to vibration amplitude A, number of cycles N, and effective confining pressure (effective stress at consolidation) σ c .

Table 3

Statistical variability of modulus G

Number of dynamic stage/shear strain range Effective confining pressure Number of samples Number of dynamic cycles Shear modulus
Average value Standard deviation Variability coefficient Median Difference
σ c N N G av σ n–1 ν G med G maxG min
(kPa) (–) (–) (MPa) (MPa) (–) (MPa) (MPa)
Dynamic stage I/3.0 × 10−3 to 4.7 × 10−3 100 4 1 14.4 1.9 0.13 14.2 4.4
100 10.8 1.6 0.15 10.6 3.7
1,000 9.7 1.5 0.15 9.5 3.4
200 7 1 20.0 2.5 0.12 19.3 6.0
100 16.0 1.6 0.10 16.0 4.1
1,000 14.6 1.4 0.09 14.5 3.4
400 6 1 26.5 3.1 0.12 26.4 6.6
100 21.8 4.2 0.19 22.5 10.9
1,000 20.3 4.0 0.20 21.5 10.7
Dynamic stage II/3.0 × 10−4 to 4.7 × 10−4 100 4 1 18.8 2.5 0.13 18.0 5.7
1,000 20.0 2.9 0.15 18.9 6.3
10,000 20.9 3.1 0.15 19.9 7.0
200 7 1 27.1 4.6 0.17 25.3 10.9
1,000 27.9 4.4 0.16 25.4 10.7
10,000 28.3 4.2 0.15 26.2 10.8
400 6 1 36.7 3.2 0.09 38.5 7.7
1,000 37.2 3.6 0.10 37.6 11.0
10,000 37.9 3.9 0.10 38.7 10.8
Dynamic stage III/3.0 × 10−3 to 4.7 × 10−3 100 4 1 13.3 2.2 0.17 12.7 5.1
100 10.9 1.9 0.17 10.4 4.4
1,000 10.0 1.7 0.17 9.6 3.9
200 7 1 19.5 2.3 0.12 19.6 6.5
100 16.4 1.9 0.12 15.9 5.1
1,000 15.1 1.6 0.11 14.6 4.3
400 6 1 27.2 2.9 0.11 27.8 6.5
100 23.3 2.9 0.13 23.9 6.1
1,000 21.4 3.1 0.15 22.3 7.3

3.2 Vibration amplitude

The results can indicate two qualitatively different ways of soil response to dynamic loading applied at the same frequency of vibrations but with different amplitudes:

  • In the dynamic stages I and III, when the amplitude was higher (A = 0.1 mm) after 1,000 cycles, G av values decreases c.a. 25%;

  • While in the dynamic stage II, when the amplitude was lower (A = 0.01 mm), an increase in G av value by several percentage was registered.

This may indicate that a slight strengthening of soil is possible at lower vibration amplitudes, while higher vibration amplitudes cause weakening of the structure, which is expressed by the decrease in G av due to dynamic loading.

3.3 Number of cycles

The changes in G av values during dynamic loading are not linear. The dependence of G av value on the number of cycles was approximated by a logarithmic function (Figure 4a–c).

(4) G av N = α ln N + G av N = 1 ,

where G av is the average value of shear modulus, N is the number of Nth cycle, α is the gradient of logarithmic curve, and G avN=1 is the G av value at 1st cycle of a particular dynamic stage.

Figure 4 
                   Shear modulus changes during: (a) the dynamic stage I, (b) the dynamic stage II, and (c) the dynamic stage III.
Figure 4

Shear modulus changes during: (a) the dynamic stage I, (b) the dynamic stage II, and (c) the dynamic stage III.

The value of α coefficient is an indicator of the trend of G av changes during the dynamic loading: negative values (α < 0) correspond to the decrease in G av which indicate soil weakening, while positive values (α > 0) correspond to the increase in G av which indicates soil strengthening.

Comparison of α values allows for a quantitative evaluation of G av changes. The most significant reduction in G av value (from −0.70 to −0.91) was observed in the dynamic stage I – in the samples where no dynamic loads were applied so far. The soil weakened after dynamic stage I, and the soil was then subjected to dynamic stage II with vibrations of significantly lower amplitude but much higher number of cycles. This resulted in an increase in G-value, which is expressed in positive α values (from 0.11 to 0.21). This slight strengthening influenced the quantitative changes in soil response when dynamic loads of high amplitude were applied again (dynamic stage III). Although a further reduction in G av value during the dynamic stage III was observed, the quantitative indicators of this reduction were slightly smaller (α values range from −0.48 to −0.84) compared to those observed in the dynamic stage I.

The changes in G av values were analyzed in the two intervals: during the initial cycles and during the subsequent cycles of dynamical loading. It was assumed that for the dynamic stages I and III, the initial cycles are cycles from N = 1 to N = 100, and for the dynamic stage II the initial cycles are from N = 1 to N = 1,000. The subsequent cycles are, respectively, from N = 100 to N = 1,000, and from N = 1,000 to N = 10,000. The initially rapid changes in G av values during the initial cycles and smaller changes during the subsequent cycles of each dynamic phase (Figure 4a–c) are indicated not only by the parameters of the logarithmic model G av = f(N), but also by comparisons of values of absolute changes in ΔG avN gradients.

In the dynamic stage I, ΔG avN determined in the initial cycles decreased 27–30 times faster compared to the smaller ΔG avN changes noted during the subsequent cycles.

In the dynamic stage II, the increase in ΔG avN determined in the initial cycles was only 7–19 times greater compared to much smaller increase during the subsequent cycles.

In the dynamic stage III, in the initial cycles, decreases in ΔG avN values were only 3–19 times higher than during the subsequent cycles.

Thus, varying dynamic loading conditions in subsequent dynamic stages result in less intense changes in the mechanical soil parameters.

3.4 Effective confining stress

Soil response to dynamic loading is also significantly dependent on effective confining stress at consolidation σ c (Figure 5a and b). In a particular dynamic phase, G av values increase with the increase in the effective confining pressure σ c . This relationship is well represented by the following equation:

(5) G av = β · ln ( σ c ) + b ,

where G av is the average value of shear modulus, σ c is the consolidation effective stress, β is the gradient of logarithmic curve, and b is the intercept of logarithmic curve

Figure 5 
                     av variation as a function of consolidation effective stress: (a) in the dynamic stages I and III and (b) in the dynamic stage II.
Figure 5

G av variation as a function of consolidation effective stress: (a) in the dynamic stages I and III and (b) in the dynamic stage II.

Coefficient of determination (R 2) ranged from 0.994 to 0.999. G av values for samples consolidated under σ c = 200 kPa and σ c = 400 kPa were, respectively, 44% and almost 100% higher than for samples consolidated under σ c = 100 kPa.

G av values are thus conditioned by the following:

  • the value of effective confining pressure σ c

  • parameters of dynamic loading in successive stages (dynamic stages I, II, and III)

  • the number of cycles for which G av values are determined

The characteristics of the variation in G av values determined as a function of σ c are illustrated in Figure 5a and b. In all analyzed cycles, G av/ σ c gradient in the range 100–200 kPa was higher and decreased in the range 200–400 kPa. This confirms the reduced response to dynamic loadings for samples consolidated at higher effective pressure.

Increase in G av value as a function of σ c is observed at both conditions when higher amplitude of vibration leads to structural weakening of the soil material, and at lower amplitude of vibration resulting in soil strengthening. G av values in the dynamic stages I and III, where the structural weakening was observed, were smaller than G av values obtained in the dynamic stage II where the soil strengthening was observed. The increase in G av = f( σ c ) in the dynamic stage II is characterized by β values ranging from 12.3 to 12.9 (Figure 5b), whereas in the dynamic stage I, by significantly smaller β values (from 7.6 to 8.7). In the dynamic stage III, the effects of the previous varying loading conditions most probably got accumulated and β values were slightly higher (from 8.2 to 10.1) than in the dynamic stage I (Figure 5a). These quantitative comparisons show how the loading history may indicate the dependence of G av values on stress in the soil mass.

The comparison of factors influencing G av changes takes into account dynamic stages for samples consolidated under particular effective confining pressure (Figure 6). The ratio of ΔG av/G avi−1, where ΔG av = G avi-G avi−1, was assumed as an indicator of the changes. ΔG av/G avi−1 ratio was analyzed in the two periods of each dynamic stage: the initial cycles and the subsequent cycles. The most significant changes in ΔG av/G avi−1 ratio were observed during the initial cycles, and thus when a given dynamic loading is applied to soil straight after the “deviatoric loading.” During subsequent cycles, much smaller changes were observed (several percent smaller than during the initial cycles). This indicates that the trend of rapidly decreasing ΔG av/G avi−1 change is continued and confirms that it is reasonable to analyze them for a larger number of cycles.

Figure 6 
                  Change in average shear modulus in relation to effective confining pressure at each dynamic stage.
Figure 6

Change in average shear modulus in relation to effective confining pressure at each dynamic stage.

The use of vibration with different amplitudes is also reflected in the changes in ΔG avi/G avi−1 ratio. During the dynamic stage I, the initial cycles cause the most significant changes in ΔG av/G avi−1 proportional to the effective pressure at consolidation σ c . During the dynamic stage II, ΔG av/G avi−1 ratio decreases (up to few %) at higher values of effective pressure at consolidation. The least significant changes in ΔG av/G avi−1 ratio are observed during the dynamic stage III. Also, in the dynamic stage III, ΔG av/G avi−1 changes are similar in the samples consolidated at 200 and 400 kPa, thus it may indicate the unification of material characteristics of soil due to dynamic loading and application of higher values of effective pressure at consolidation σ c .

4 Discussion

Dynamic loads of amplitude A = 0.1 mm and frequency f = 5 Hz were assumed to correspond to the conditions of intensive road traffic. In these conditions, a rapid decrease in the G av modulus was observed during the initial cycles, which tends to stabilize during subsequent cycles.

In the conditions corresponding to normal road traffic, the loads of amplitude A = 0.01 mm and f = 5 Hz were applied. In these conditions, an increase in G av value was observed. This indicates a dynamically conditioned slight strengthening of the soil under dynamic loading, most probably due to the rearrangement of soil particles, advantageous for strengthening of soil structure and, at the same time, not enough to cause significant increase in pore water pressure. This trend is typical of the behavior of soils under low strain conditions [12]. Also, initial increase in the deviator stress under drained conditions before the dynamic stage II was an important factor responsible for observable strengthening.

Although the following vibration of higher amplitude (A = 0.1 mm) caused once again reduction in G av values, its absolute values are from 3.0 to 5.4% higher in comparison with the corresponding data from the dynamic stage I. The lower amplitude vibration in combination with the increase in the deviatoric static loading to 100 kPa between the dynamic stages (Figure 2) result in soil strengthening. This may be explained as a consequence of the reorientation of soil particles during dynamic conditions, and an increase in the density of the soil due to particle compaction during deviatoric static loading.

G av values show an increasing tendency to stabilize at higher effective confining pressures (Figure 6). This may be interpreted as an effect of the earlier strengthening of soil strength at the stage of isotropic consolidation when the excess pore water pressure was dissipated and the soil particles were precompacted. Also, short stages of mid-phase under drained loading conditions right before the dynamic stage II and the dynamic stage III led to the dissipation of excess pore water pressure generated in the previous dynamic stage and, in consequence, increasing of the effective stress and strengthening of the soil structure increased. This type of behavior of Warsaw varved clay during the dynamic stages I and III may be compared with the other clayey soils which show rapid decrease in shear modulus during the first dozens of cycles and its stability after large number of cycles [3,30].

Comparing G av values dependence on consolidation effective stress (Figure 5), we can indicate a few important factors. Geological loading history and diagenetic processes, including actual or apparent overconsolidation of the soil [17] may indicate the dependence of G av on the stress in the soil mass. The rate of G av increase as a function of consolidation effective stress represented by β coefficient also depends on the strain level. For the dynamic stage II (strain range of: 3.0 × 10−4 to 4.7 × 10−4), G av increase is distinctly higher (β ranging from 12.3 to 12.9) than for the dynamic stage I and the dynamic stage III (strain range: 3.0 × 10−3 to 4.7 × 10−3 and β ranging from 7.6 to 10.1). This indicates the important role of nonlinearity of G av in the function of consolidation effective stress.

5 Conclusion

Test results lead to the following methodical conclusions:

  1. G modulus is a convenient parameter for evaluation of stiffness changes during the different dynamic loading conditions;

  2. Investigations of soils subjected to dynamic loads should be focused on evaluation of factors influencing the changes in stiffness, such as different amplitudes of vibrations, number of loading cycles, increase in pore water pressure, and confining pressure;

  3. The quantitative change in G av in logarithmic dependence on the number of cycles during dynamic loading allows the introduction of α coefficient, which differentiates the soil behavior. The threshold value of the amplitude A at α = 0 is an important indicator of the changes in the vibration influence on the behavior of soils of a given origin. Negative α values indicate a weakening effect of the applied vibrations (stiffness decreasing), while positive α values indicate a soil strengthening (stiffness increasing);

  4. The quantitative change in G av in logarithmic dependence on the effective confining pressure can be characterized by the β coefficient. This parameter may be applied for the assessment of impact of the confining stress on soil reaction to dynamic loading;

  5. During dynamic loading, ΔG av/G avi−1 ratio has a decreasing tendency, which may indicate achieving reversible quasi-elastic behavior of the soil;

  6. Application of similar test program on the soils of different genesis may give a comparative, new understanding of the behavior of soils.


We would like to thank Dr eng. Kamil Kiełbasiński and the Geological-Drilling Company PAWLAK for its technical assistance during the soil sampling. The authors also wish to express their greatest gratitude to Reviewers for their insightful suggestions and valuable comments on the manuscript.

  1. Funding information: This study was supported by funds of the University of Warsaw – Faculty of Geology: 501-D113-01-1130302 (Department of Engineering Geology and Geomechanics) and 501-D113-01-1130103 (Laboratory of Applied Geology).

  2. Author Contributions: conceptualization: K.N.; methodology: K.N. and A.B.; validation: A.B. and P.D.; formal analysis: K.N. and P.D.; investigation: K.N.; resources: K.N.; writing – original draft preparation: K.N. and P.D.; writing – review and editing: A.B. and P.D.; visualization: K.N., A.B., and P.D.; supervision: P.D. All authors have read and agreed to the published version of the manuscript.

  3. Conflict of interest: The authors declare no conflict of interest.


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Received: 2022-10-29
Revised: 2023-01-12
Accepted: 2023-01-24
Published Online: 2023-03-07

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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