Influence of Relative Density of Sands on Their Oedometric Moduli

Featured Application

The results of this study will be used to verify the methodology for determining the perpetrator’s mass in forensic science based on a footwear impression left in sands.

Abstract

In laboratory testing, analyzing the influence of various factors on oedometer test results is essential, since the oedometric modulus (E_oed) directly affects e. g. footwear impressions in forensic engineering. This study focused on incremental loading oedometer tests on two types of sand—silty sand (SM) and sand with fine-grained admixture (S-F). For each loading stage, four identically prepared samples were tested at density indices I_D = 0.15, 0.65, and 0.85. The E_oed was calculated using both Slovak Technical Standard (STN) and ISO Standard, which differ in evaluating strain increments at each loading stage. The difference in E_oed between I_D = 0.15 and 0.85 corresponded to about 1.5 to 5.6 times the E_oed value at I_D = 0.15 for SM and about 1.0 to 3.3 times for S-F. Between I_D = 0.65 and 0.85, the difference amounted to about 0.18 to 0.63 times the E_oed value at I_D = 0.65 for SM and about 0.51 to 1.44 times for S-F. The difference in E_oed between STN and ISO ranged from 0.00003 to 0.067 times the STN-based E_oed value for SM and from 0 to 0.017 times for S-F. Results show a very high linear dependence of E_oed on I_D for SM (Pearson R = 0.97–0.99) and high-to-very-high for S-F (Pearson R = 0.83–0.91). Accurate determination of E_oed enables a more reliable and precise assessment of footwear impressions.

1. Introduction

The determination of soil deformation characteristics is among the fundamental tasks of geotechnical engineering, as it enables the quantification of soil behavior under stress and allows for the prediction of structural settlements, stress redistribution, and the overall stability of geotechnical structures. A key parameter is the oedometric modulus E_oed, which represents the relationship between the change in vertical stress and the corresponding vertical strain during one-dimensional compression (consolidation), i.e., without lateral deformation. Accurate determination of this parameter is essential for the design and assessment of foundations, embankments, and earth structures.

The authors in [1] state that buried pipe design requires knowledge of the backfill material to properly design the backfill structure. The strength of the backfill material is described in terms of the soil reaction modulus E′ and the constrained modulus E_oed. As E′ is an empirical parameter, E_oed can be measured in the laboratory through oedometer tests. Therefore, the reliable determination of E_oed is necessary.

The authors in [2] state that there is little research on the standard consolidation test of coarse-grained soil in China, and most scholars focus on isobaric consolidation. However, the standard consolidation test is one of the necessary tests to study the HS model. Therefore they present a large-scaled consolidator with a 300 mm diameter, which is used to measure the relationship between the deformation and pressure of a sandy gravel soil sample by a standard consolidation test. The stress–strain curve under lateral limit conditions is obtained, and the oedometric modulus E_oed can be calculated. Therefore, reliable E_oed values obtained from laboratory tests are important for such modeling.

Because the process of obtaining the compression coefficient (CC) via oedometer test is recognizably time-consuming and experienced, skilled technicians are required to ensure that the test results are reliable; the authors in [3] propose a low-cost, fast, and reliable alternative for estimating this soil parameter utilizing a hybrid metaheuristic optimized neural network. The proposed model was verified at a large-scale real-life urban project where a dataset of the project with 441 samples, which recorded results of the CC and their twelve physical parameters, were used. In this case, the reliability of CC values obtained from oedometer tests is decisive for the applicability of the model.

Time-consuming work and the requirement for experienced, skilled technicians cause the fact that, in laboratory practice, a single sample is rarely tested multiple times but often only once, and the obtained E_oed value is considered the characteristic value used for calculating soil deformation and subsequently for the assessment of the serviceability limit state of geotechnical structures. Even in accordance with [4], ground investigation requirements for category 1 projects are usually minimal, as verifications rely solely on local experience. Therefore, in design practice, engineers use a quadratic relationship between E_oed and I_D, in accordance with [5], or interpolate E_oed values within the given range of the I_D [6]. Settlement, e.g., of a foundation footing, is then calculated as the sum of the compressions of the soil layers beneath the footing, which is directly proportional to the thickness of the soil layer, with the additional stress from structures at the middle of the layer and inversely proportional to E_oed. While this practice is applied in geotechnical engineering, it is unacceptable in forensic engineering.

To our knowledge, there is no study that examines the influence of sand sample homogeneity and sample preparation methods on E_oed values. If sands are used as model sands for investigating the depth of the perpetrator’s footwear impression left in sands, a reliable E_oed value is decisive. This is under the assumption that the perpetrator’s footwear impression left in sands is calculated similarly to the settlement of a foundation footing, for example, a 40% difference in the obtained E_oed values leads to approximately a 40% difference in the estimated mass of the perpetrator, which is unusable. For this reason, there is a need to determine the most accurate E_oed values of model sands possible, considering the effects of sample non-homogeneity, sample preparation methods, and also sand density on the E_oed values of sands.

The transfer of soil particles and the quantification of soil material adhering to footwear have been investigated in forensic studies [7,8]. However, considerably less attention has been paid to the depth of footwear impressions in soils. Authors in [9] present a study focusing on the mechanical response of soils responsible for the formation of plastic (three-dimensional) footwear impressions. Five shoes and their corresponding three-dimensional impressions made in both sand and soil were compared using a grid system and tread descriptors commonly employed in the United Kingdom. However, no analyses of sand or soil deformation parameters were conducted.

The authors in [10] analyzed effects of soil moisture on shoe impression detail and longevity. The shoe impressions are created in a standard topsoil with varying amounts of water and monitored over 48 h. They state that soil moisture has a very large effect on shoeprint detail and longevity. Soil moisture has an inverse relationship with detail longevity, meaning that the more moisture contained within soil, the faster the print will degrade and become useless for identification of class characteristics. However, no information on soil deformation properties and the relation between them and shoe impression is provided.

Even in the Interpol Review Paper of Marks and Impression Evidence 2019–2022 [11], which presents 49 references, there is no study addressing the relationship between the depth of a footwear impression and soil deformation parameters.

The depth of an impression represents a direct mechanical response of the soil to the applied load and, therefore, can provide valuable information about the interaction between the suspect and the ground surface at the time of contact. Despite its potential forensic significance—particularly for estimating loading conditions or even the body weight of an individual—to our knowledge, no research has yet addressed the quantitative relationship between applied stress and impression depth. This gap highlights the need for controlled laboratory experiments aimed at linking measurable soil deformation parameters with the depth of footwear impressions.

While the estimation of loading forces can be relatively reliable, the determination of consistent soil deformation parameters (e.g., the oedometer modulus of sand obtained from incremental loading oedometer tests) is more challenging due to sample inhomogeneities and the random behavior of particles during compaction into the oedometer ring and subsequent loading.

Moreover, different approaches to evaluating vertical strain (i.e., whether the initial sample height or the current height after each loading increment is considered) can also affect the calculated values of the oedometer modulus E_oed. For example, the oedometer modulus can be calculated in accordance with ISO Standard [12] (ISO) as follows:

$$E_{oed}={\displaystyle \frac{\delta {\sigma }_v^{\prime }}{\mathsf{\delta }{\epsilon }_v}},$$

(1)

Using the symbols introduced in this document, the following equation is obtained:

$$E_{oed}={\displaystyle \frac{{\sigma }_{v2}^{\prime }-{\sigma }_{v1}^{\prime }}{\frac{H_i-H_f}{H_i}}},$$

(2)

where σ′_v1 (kPa)—pressure applied to the specimen in the previous load increment, σ′_v2 (kPa)—pressure applied to the specimen in the load increment that is considered, H_i (mm)—initial height, i.e., height of the specimen at the start of the increment, and H_f (mm)—height of the specimen at the end of the increment.

According to the Polish Standard [13] (PN), the primary compression modulus M₀ or secondary modulus M (kPa or MPa) is calculated by the following formula:

$$M_0,M={\displaystyle \frac{∆{\sigma }_i}{\epsilon }}={\displaystyle \frac{∆{\sigma }_i·h_{i-1}}{∆h_i}},$$

(3)

where Δσ_i (kPa, MPa)—pressure increase, Δσ_i = σ_i − σ_i−1; h_i−1 (mm)—specimen height in the oedometer before the pressure increases from σ_i−1 to σ_i; Δh_i (mm)—decrease in sample height in the oedometer ring after pressure increases by Δσ_i; and Δh_i = h_i−1 − h_i.

After arranging Equation (2) according to the PN sign convention, the oedometer modulus according to ISO is calculated using the following formula:

$$E_{oed}={\displaystyle \frac{{\sigma }_{v2}^{\prime }-{\sigma }_{v1}^{\prime }}{\frac{H_i-H_f}{H_i}}}={\displaystyle \frac{∆{\sigma }_i}{\frac{h_{i-1}-h_i}{h_{i-1}}}}={\displaystyle \frac{∆{\sigma }_i·h_{i-1}}{h_{i-1}-h_i}}={\displaystyle \frac{∆{\sigma }_i·h_{i-1}}{∆h_i}},$$

(4)

As we can see, there is no difference between Equations (3) and (4); therefore, the methods for calculating the oedometer modulus according to ISO and PN are the same.

According to the Slovak Standard [14] (STN), the oedometer modulus for the stress interval σ_i−1, σ_i is calculated from deformations s_i−1,c, and s_ic at the end of the corresponding loading (unloading) stages by the formula:

$$E_{u}or{E}_f={\displaystyle \frac{h_{or}·\left({\sigma }_i-{\sigma }_{i-1}\right)}{s_{ic}-s_{i-1,c}}},$$

(5)

For reconsolidated testing samples, the original sample height h_or is replaced by the sample height after reconsolidation h_r.

After arranging Equation (2) according to the STN convention, the oedometer modulus according to ISO is calculated by the following formula:

$$E_{oed}={\displaystyle \frac{{\sigma }_{v2}^{\prime }-{\sigma }_{v1}^{\prime }}{\frac{H_i-H_f}{H_i}}}={\displaystyle \frac{{\sigma }_i-{\sigma }_{i-1}}{\frac{\left(h_{or}-s_{i-1,c}\right)-\left(h_{or}-s_{i,c}\right)}{\left(h_{or}-s_{i-1,c}\right)}}}={\displaystyle \frac{\left(h_{or}-s_{i-1,c}\right)·\left({\sigma }_i-{\sigma }_{i-1}\right)}{s_{i,c}-s_{i-1,c}}},$$

(6)

The method for calculating E_oed is not presented in ASTM Standard D2435/D2435M-11 [15].

Comparing Equations (5) and (6), we can see that there is a difference in parts (h_or – s_i−1,c) and h_or. This means that oedometer modulus calculated according to STN will be larger than the oedometer modulus calculated according to ISO and PN (hereinafter referred to simply as ISO).

Footwear impressions in soil can be simulated by applying loads through footwear of various sizes onto model sand placed in a container of sufficient dimensions to prevent boundary effects from the container walls. The largest footwear model requires up to 300 kg of sand, whereas the sand mass compacted in an oedometer sample ring is less than 1 kg. Hence, there is a need to determine the most representative oedometer modulus of the model sand.

In this study, we present incremental loading oedometer tests on two model sands. Four samples were identically prepared by the same person in four oedometers and tested under the same procedure at density indices I_D = 0.15, 0.65, and 0.85. Two methods were used to calculate E_oed according to the STN and ISO standards, which differ in the evaluation of vertical strain increments at each loading step. Standard and robust statistical analyses were performed to identify and exclude outliers. Based on the determined E_oed values for different relative density indices I_D, linear relationships between E_oed and I_D were proposed. The obtained E_oed values will be used for the analysis of footwear impressions in the subsequent experimental research.

2. Materials and Methods

The sand from the Polanka quarry, Czech Republic, and the sand from the Danube River in Bratislava, Slovakia, with a maximum particle diameter of 2 mm, were used in the research. The standard STN EN ISO 17892-1: 2023 [16] was applied to determine the water content, STN EN ISO 17892-4: 2023 [17], to determine the particle size distribution, and STN EN ISO 17892-12: 2019 [18] to determine the liquid limit w_L and plastic limit w_P. The grain-size distribution curves of the sands are shown in Figure 1. The soil classification, in accordance with [5], and their properties are presented in

Table 1. Soil classification and properties.

Soil Parameters	Danube Sand	Polanka Sand
Soil classification [12]	SM	S-F
Water content (%)	2.20	0.38
Plastic limit (%)	20.94	-
Liquid limit (%)	21.99	-
Plasticity index (%)	1.05	-
Mean grain size D50 (mm) Coefficient of Uniformity Cc (-)	0.11	0.36
Coefficient of Uniformity Cu (-)	8.28	3.11
Coefficient of Curvature Cc (-)	2.57	1.23
Specific gravity of the solids (-)	2.73	2.64
Minimum void ratio (-)	1.25	0.74
Maximum void ratio (-)	0.55	0.42

.

Four oedometers (IGHP n. p., Žilina, Slovakia) were used for soil testing. The parameters of the oedometers are listed in

Table A1. Parameters of oedometers and void ratio for various density indices I_D.

Oedometers	Danube Sand				Polanka Sand
	A	B	C	D	A	B	C	D
Height (mm)	30.328	30.188	30.388	30.563	30.328	30.188	30.388	30.563
Diameter (mm)	119.824	120.125	120.150	119.599	119.824	120.125	120.150	119.599
Plunger mass (kPa)	5115	5090	5209	5136	5115	5090	5209	5136
Pressure from plunger (kPa)	4.450	4.406	4.507	4.485	4.450	4.406	4.507	4.485
Apparatus def. at plunger (mm)	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Apparatus def. at 25 kPa (mm)	0.001	0.008	0.009	0.041	0.001	0.008	0.009	0.041
Apparatus def. at 50 kPa (mm)	0.012	0.032	0.038	0.076	0.012	0.032	0.038	0.076
Apparatus def. at 100 kPa (mm)	0.051	0.064	0.108	0.123	0.051	0.064	0.108	0.123
Apparatus def. at 200 kPa (mm)	0.111	0.107	0.199	0.164	0.111	0.107	0.199	0.164
Apparatus def. at 400 kPa (mm)	0.261	0.179	0.277	0.207	0.261	0.179	0.277	0.207
Minimum void ratio (-)	1.25	1.25	1.25	1.25	0.74	0.74	0.74	0.74
Void ratio for ID = 0.15	1.15	1.15	1.15	1.15	0.70	0.70	0.70	0.70
Void ratio for ID = 0.65	0.80	0.80	0.80	0.80	0.54	0.54	0.54	0.54
Void ratio for ID = 0.85	0.66	0.66	0.66	0.66	0.47	0.47	0.47	0.47
Maximum void ratio (-)	0.55	0.55	0.55	0.55	0.42	0.42	0.42	0.42

Note: The dimensions of the oedometer specimen ring were measured using a vernier caliper with a resolution of 0.01 mm. Each diameter was determined as the mean of two measurements taken in perpendicular directions, and the height as the mean of four measurements. Values are reported to three decimal places (rounded to 0.001 mm) to minimize rounding errors in further calculations (e.g., volume determination). The apparatus deformation values represent the best estimates obtained by visual interpolation between dial divisions (0.01 mm resolution) to minimize rounding errors in the E_oed calculation.

. The apparatus has a lever loading system that increases the weight of the load tenfold. For each oedometer, an exactly calculated mass of weights was applied based on its area. Deformations of both the oedometer itself (metal samples) and the soil specimens were measured using an analog dial gauge with a 30 mm measurement range and 0.01 mm resolution.

The masses of the sands compacted into the oedometer specimen rings to achieve density indices of 0.15, 0.65, and 0.85 were calculated from their maximum and minimum dry unit weights and the volumes of the oedometers. Based on the masses of the sands, the sand moisture content, and their particle density, the void ratios were calculated and are presented in Table A1. Sample preparation procedure is presented in Appendix A.2 of Appendix A and in Figure 2.

To eliminate the influence of the deformation of the oedometers themselves on the test results, four metal samples were used. These samples had the same height as the oedometer rings and a diameter 2 mm smaller. Figure 3a shows the four metal samples placed in the oedometers. The oedometers with metal samples were subjected to the same step-loading sequence as that applied to the soil specimens (see Table A1 for apparatus deformation). In the center of the apparatus, the plungers can be seen. Figure 3b shows the oedometers during testing (Danube sand, I_D = 0.15; normal stress 200 kPa).

The oedometer tests were performed using a stepwise loading procedure with the following sequence of loading stages: stress from the plunger weight; 25 kPa; 50 kPa; 100 kPa; 200 kPa; and 400 kPa. In accordance with STN [14], each loading stage lasted 24 h.

The E_oed values were calculated according to ISO (Equation (2)) and STN (Equation (6)). Soil deformation was determined as the difference between the dial gauge reading and the deformation of the oedometer itself for each loading step. The E_oed values were calculated for the following normal effective stress intervals: plunger—25 kPa (hereafter referred to as loading interval No. 1 (LI 1)), 25 kPa–50 kPa (loading interval No. 2 (LI 2)), 50 kPa–100 kPa (loading interval No. 3 (LI 3)), 100 kPa–200 kPa (loading interval No. 4 (LI 4)), and 200 kPa–400 kPa (loading interval No. 5 (LI 5)).

For each sand, four E_oed values obtained from four odometers (A, B, C, and D) for the same loading interval were subjected to standard and robust statistical analyses. In the standard statistical analyses, the mean ($\overline{x}$), range (Range), standard deviation (SD), and coefficient of variation (CV) were calculated according to well-known formulas in statistics.

In the robust statistical analyses, the following parameters were calculated. First, the values were sorted as

$$Eoe{d}_1\le Eoe{d}_2\le Eoe{d}_3\le Eoe{d}_4$$

(7)

Median (Mdn): the mean of the two middle values:

$$\mathrm{Mdn}={\displaystyle \frac{Eoe{d}_2+Eoe{d}_3}{2}}$$

(8)

First quartile (Q1):

$$Q_1={\displaystyle \frac{Eoe{d}_1+Eoe{d}_2}{2}}\left(\mathrm{average}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{lowest}\mathrm{values}\right)$$

(9)

Third quartile (Q3):

$$Q_3={\displaystyle \frac{Eoe{d}_3+Eoe{d}_4}{2}}\left(\mathrm{average}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{highest}\mathrm{values}\right)$$

(10)

Interquartile range (IQR):

$$\mathrm{IQR}=Q_3-Q_1$$

(11)

Lower fence (LF):

$$\mathrm{LF}=Q_1-1.5\cdot \mathrm{IQR}$$

(12)

Upper fence (UF):

$$\mathrm{UF}=Q_3+1.5\cdot \mathrm{IQR}$$

(13)

Median Absolute Deviation (MAD):

$$\mathrm{MAD}=\mathrm{median}(\mid Eoe{d}_i-\mathrm{Mdn}\mid )$$

(14)

Individual Modified Z-scores (MiA, MiB, MiC, MiD): for the Eoed values from odometers A, B, C, and D, respectively, according to [19]:

$$M_i={\displaystyle \frac{0.6745(Eoe{d}_i-\mathrm{Mdn})}{\mathrm{MAD}}}$$

(15)

Values of $M_i>3.5$ indicate outlying Eoed values [19].

3. Results

The measured deformations of specimens (including deformations of the oedometers) are presented in

Table A2. Measured deformations of specimens for various density index I_D (including deformations of the oedometers).

Measured Deformations (mm)	Danube Sand				Polanka Sand
	A	B	C	D	A	B	C	D
Density index ID = 0.15
After plunger load	0.089	0.143	0.163	0.054	0.001	0.001	0.001	0.001
After load 25 kPa	0.557	0.678	0.684	0.430	0.134	0.217	0.212	0.221
After load 50 kPa	0.891	1.146	1.118	0.902	0.285	0.343	0.372	0.368
After load 100 kPa	1.272	1.591	1.604	1.418	0.440	0.449	0.590	0.532
After load 200 kPa	1.684	2.182	2.198	2.012	0.600	0.562	0.802	0.713
After load 400 kPa	2.122	2.718	2.803	2.680	0.771	0.732	1.022	0.918
Density index ID = 0.65
After plunger load	0.003	0.001	0.021	0.015	0.001	0.001	0.000	0.000
After load 25 kPa	0.068	0.197	0.188	0.152	0.119	0.162	0.149	0.188
After load 50 kPa	0.190	0.349	0.324	0.292	0.221	0.278	0.272	0.302
After load 100 kPa	0.389	0.538	0.532	0.470	0.362	0.382	0.398	0.431
After load 200 kPa	0.639	0.722	0.850	0.748	0.517	0.490	0.518	0.553
After load 400 kPa	0.958	0.975	1.288	1.203	0.683	0.614	0.652	0.691
Density index ID = 0.85
After plunger load	0.001	0.002	0.001	0.001	0.001	0.001	0.001	0.001
After load 25 kPa	0.041	0.125	0.073	0.132	0.090	0.082	0.076	0.154
After load 50 kPa	0.109	0.229	0.158	0.263	0.166	0.163	0.141	0.266
After load 100 kPa	0.241	0.352	0.319	0.432	0.279	0.213	0.231	0.388
After load 200 kPa	0.421	0.535	0.576	0.694	0.399	0.281	0.325	0.509
After load 400 kPa	0.708	0.761	0.967	1.108	0.558	0.379	0.432	0.639

Note: The soil specimen deformation values represent the best estimates obtained by visual interpolation between dial divisions (0.01 mm resolution) to minimize rounding errors in the E_oed calculation.

. The void ratios and the normalized void ratios at the end of the loading stage are presented in

Table A3. Void ratio e after loading (Danube sand, Polanka sand).

Voi Ratio e (-)	Danube Sand				Polanka Sand
	A	B	C	D	A	B	C	D
Density index ID = 0.15
No load	1.152	1.152	1.152	1.152	0.699	0.699	0.699	0.699
After plunger load	1.146	1.142	1.141	1.148	0.699	0.699	0.699	0.699
After load 25 kPa	1.113	1.104	1.104	1.125	0.691	0.687	0.687	0.689
After load 50 kPa	1.090	1.073	1.076	1.094	0.683	0.681	0.680	0.682
After load 100 kPa	1.066	1.043	1.046	1.061	0.677	0.677	0.672	0.676
After load 200 kPa	1.041	1.004	1.011	1.022	0.671	0.673	0.665	0.668
After load 400 kPa	1.020	0.971	0.973	0.978	0.670	0.667	0.657	0.659
Density index ID = 0.65
No load	0.798	0.798	0.798	0.798	0.536	0.536	0.536	0.536
After plunger load	0.798	0.798	0.797	0.797	0.536	0.536	0.536	0.536
After load 25 kPa	0.794	0.787	0.787	0.791	0.530	0.528	0.529	0.529
After load 50 kPa	0.787	0.779	0.781	0.785	0.526	0.524	0.524	0.525
After load 100 kPa	0.778	0.770	0.773	0.777	0.520	0.520	0.522	0.521
After load 200 kPa	0.766	0.761	0.759	0.763	0.516	0.517	0.520	0.517
After load 400 kPa	0.756	0.750	0.738	0.739	0.515	0.514	0.517	0.512
Density index ID = 0.85
No load	0.656	0.656	0.656	0.656	0.471	0.471	0.471	0.471
After plunger load	0.656	0.656	0.656	0.656	0.471	0.471	0.471	0.471
After load 25 kPa	0.654	0.650	0.652	0.651	0.467	0.468	0.468	0.466
After load 50 kPa	0.651	0.645	0.649	0.646	0.464	0.465	0.466	0.462
After load 100 kPa	0.646	0.640	0.644	0.639	0.460	0.464	0.465	0.459
After load 200 kPa	0.639	0.633	0.635	0.627	0.457	0.463	0.465	0.455
After load 400 kPa	0.632	0.624	0.618	0.607	0.457	0.462	0.464	0.451

and

Table A4. Normalized void ratio e/e₀ (Danube sand, Polanka sand).

Voi Ratio e (-)	Danube Sand				Polanka Sand
	A	B	C	D	A	B	C	D
Density index ID = 0.15
No load	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
After plunger load	0.995	0.991	0.990	0.997	1.000	1.000	1.000	1.000
After load 25 kPa	0.966	0.959	0.959	0.976	0.989	0.983	0.984	0.986
After load 50 kPa	0.946	0.931	0.934	0.950	0.978	0.975	0.973	0.977
After load 100 kPa	0.925	0.906	0.908	0.921	0.969	0.969	0.961	0.967
After load 200 kPa	0.903	0.872	0.877	0.887	0.961	0.963	0.952	0.956
After load 400 kPa	0.885	0.843	0.845	0.849	0.959	0.955	0.940	0.943
Density index ID = 0.65
No load	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
After plunger load	1.000	1.000	0.998	0.999	1.000	1.000	1.000	1.000
After load 25 kPa	0.995	0.986	0.987	0.992	0.989	0.985	0.987	0.986
After load 50 kPa	0.987	0.976	0.979	0.984	0.980	0.977	0.978	0.979
After load 100 kPa	0.975	0.965	0.969	0.974	0.971	0.970	0.973	0.971
After load 200 kPa	0.961	0.954	0.952	0.957	0.962	0.964	0.970	0.964
After load 400 kPa	0.948	0.941	0.925	0.927	0.960	0.959	0.965	0.955
Density index ID = 0.85
No load	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
After plunger load	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
After load 25 kPa	0.997	0.990	0.995	0.992	0.991	0.992	0.993	0.988
After load 50 kPa	0.992	0.984	0.990	0.985	0.984	0.986	0.989	0.981
After load 100 kPa	0.984	0.976	0.982	0.974	0.977	0.985	0.987	0.973
After load 200 kPa	0.974	0.964	0.969	0.956	0.970	0.982	0.987	0.965
After load 400 kPa	0.963	0.951	0.943	0.926	0.969	0.979	0.984	0.956

, respectively. The relationships between normalized void ratio e/e₀ and vertical effective stress σ′ are shown in Figure 4 (Danube sand) and Figure 5 (Polanka sand).

As can be seen in Table A3, Danube sand has a large initial void ratio, so the E_oed values are expected to be low. Polanka sand has a smaller initial void ratio, resulting in higher expected E_oed values. This fact can also be seen in Figure 4 and Figure 5, where the normalized void ratios of Polanka sand are larger than those of Danube sand. The random variation in the curves from different oedometers indicates that there are no systematic errors. The closer agreement of the curves at I_D = 0.65 for both sands suggests that the scatter of E_oed values among different oedometers is smaller at this density. This behavior is likely caused by non-uniform compaction within the samples: samples at I_D = 0.15 are very easy to compact, whereas samples at I_D = 0.85 are much more difficult to compact.

The calculated E_oed of Danube sand according to the STN (excluding deformations of the oedometers) are presented in

Table 2. Oedometric moduli E_oed for various density index I_D according to the STN, Danube sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Danube Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	1.327	2.340	4.421	8.590	20.999	9.686	6.810	9.449	15.915	35.785	15.894	13.262	16.257	25.198	44.143
B	1.174	1.700	3.655	5.509	13.012	3.292	5.896	9.614	21.410	33.357	5.381	9.434	16.587	21.563	39.205
C	1.127	1.876	3.652	6.041	11.532	3.943	7.100	11.010	13.387	16.882	9.888	13.566	16.696	18.306	19.417
D	1.870	1.748	3.258	5.527	9.780	6.526	7.277	11.665	12.896	14.836	6.962	7.959	12.526 (*)	13.829	16.476
Standard descriptive statistics
$\overline{x}$ (MPa)	1.397	1.916	3.747	6.417	13.831	5.862	6.771	10.435	15.902	25.215	9.531	11.055	15.517	19.724	29.810
Range	0.696	0.641	1.162	3.082	11.218	6.394	1.381	2.216	8.514	20.948	10.513	5.607	4.171	11.369	27.667
SD	0.322	0.292	0.487	1.470	4.958	2.907	0.614	1.079	3.903	10.881	4.635	2.792	2.003	4.834	13.899
CV (%)	23.04	15.26	12.99	22.91	35.85	49.59	9.07	10.34	24.54	43.15	48.63	25.26	12.91	24.51	46.62
Robust descriptive statistics
MiA	0.488	4.049 (*)	2.611	7.109 (*)	3.642 (*)	1.856	−0.419	−0.746	0.565	0.777	2.236	0.625	−0.506	1.030	0.880
MiB	−0.861	−0.861	0.004	−0.697	0.309	−0.810	−3.062	−0.603	3.020	0.600	−0.911	−0.625	0.506	0.319	0.587
MiC	−0.488	0.488	−0.004	0.652	−0.309	−0.539	0.419	0.603	−0.565	−0.600	0.438	0.724	0.843	−0.319	−0.587
MiD	5.272 (*)	−0.488	−1.345	−0.652	−1.040	0.539	0.930	1.170	−0.784	−0.749	−0.438	−1.106	−11.960 (*)	−1.195	−0.762
Eoed	1.272	1.812	3.654	5.784	12.272	5.235	6.955	10.312	14.651	25.119	8.425	11.348	16.422	19.934	29.311

Notes: Numbers with an asterisk are larger than 3.5; corresponding E_oed are outliers and will be excluded from the evaluation of E_oed. Bold, underlined, italic number: the lowest CV. Bold, underlined number: the highest CV.

and

Table A5. Oedometric moduli E_oed for various density index I_D according to the STN, Danube sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C, and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Danube Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	1.327	2.340	4.421	8.590	20.999	9.686	6.810	9.449	15.915	35.785	15.894	13.262	16.257	25.198	44.143
B	1.174	1.700	3.655	5.509	13.012	3.292	5.896	9.614	21.410	33.357	5.381	9.434	16.587	21.563	39.205
C	1.127	1.876	3.652	6.041	11.532	3.943	7.100	11.010	13.387	16.882	9.888	13.566	16.696	18.306	19.417
D	1.870	1.748	3.258	5.527	9.780	6.526	7.277	11.665	12.896	14.836	6.962	7.959	12.526	13.829	16.476
Standard descriptive statistics
$\overline{x}$ (MPa)	1.397	1.916	3.747	6.417	13.831	5.862	6.771	10.435	15.902	25.215	9.531	11.055	15.517	19.724	29.810
Range	0.696	0.641	1.162	3.082	11.218	6.394	1.381	2.216	8.514	20.948	10.513	5.607	4.171	11.369	27.667
SD	0.322	0.292	0.487	1.470	4.958	2.907	0.614	1.079	3.903	10.881	4.635	2.792	2.003	4.834	13.899
CV (%)	23.04	15.26	12.99	22.91	35.85	49.59	9.07	10.34	24.54	43.15	48.63	25.26	12.91	24.51	46.62
Robust descriptive statistics
Mdn	1.272	1.812	3.654	5.784	12.272	5.235	6.955	10.312	14.651	25.119	8.425	11.348	16.422	19.934	29.311
Q1	1.195	1.724	3.455	5.518	10.656	3.617	6.353	9.532	13.141	15.859	6.171	8.696	14.391	16.068	17.946
Q3	1.599	2.108	4.038	7.316	17.005	8.106	7.188	11.388	18.662	34.571	12.891	13.414	16.642	23.381	41.674
IQR	0.403	0.384	0.582	1.798	6.349	4.489	0.835	1.806	5.521	18.712	6.720	4.718	2.250	7.313	23.728
LF	0.591	1.148	2.582	2.821	1.133	−3.116	5.100	6.823	4.859	−12.208	−3.908	1.620	11.016	5.098	−17.645
UF	2.204	2.684	4.911	10.013	26.529	14.839	8.441	14.046	26.944	62.638	22.971	20.049	20.017	34.350	77.266
MAD	0.077	0.088	0.198	0.266	1.616	1.617	0.233	0.780	1.509	9.260	2.253	2.066	0.220	3.446	11.365
MiA	0.488	4.049	2.611	7.109	3.642	1.856	−0.419	−0.746	0.565	0.777	2.236	0.625	−0.506	1.030	0.880
MiB	−0.861	−0.861	0.004	−0.697	0.309	−0.810	−3.062	−0.603	3.020	0.600	−0.911	−0.625	0.506	0.319	0.587
MiC	−0.488	0.488	−0.004	0.652	−0.309	−0.539	0.419	0.603	−0.565	−0.600	0.438	0.724	0.843	−0.319	−0.587
MiD	5.272	−0.488	−1.345	−0.652	−1.040	0.539	0.930	1.170	−0.784	−0.749	−0.438	−1.106	−11.960	−1.195	−0.762
Eoed	1.272	1.812	3.654	5.784	12.272	5.235	6.955	10.312	14.651	25.119	8.425	11.348	16.422	19.934	29.311

. The calculated E_oed of Danube sand according to the ISO (excluding deformations of the oedometers) are presented in

Table 3. Oedometric moduli E_oed for various density index I_D according to the ISO, Danube sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C, and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Danube Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	1.323	2.297	4.292	8.243	19.906	9.685	6.795	9.394	15.737	35.160	15.894	13.245	16.205	25.040	43.691
B	1.169	1.662	3.520	5.230	12.118	3.292	5.859	9.513	21.074	32.677	5.381	9.397	16.479	21.357	38.649
C	1.210	1.834	3.523	5.744	10.774	3.940	7.058	10.906	13.200	16.520	9.888	13.537	16.630	18.179	19.176
D	1.867	1.726	3.170	5.293	9.189	6.523	7.250	11.583	12.749	14.553	6.961	7.935	12.449 (*)	13.690	16.190
Standard descriptive statistics
$\overline{x}$ (MPa)	1.392	1.880	3.626	6.127	12.997	5.860	6.741	10.349	15.690	24.728	9.531	11.029	15.441	19.566	29.427
Range	0.698	0.635	1.122	3.013	10.717	6.393	1.391	2.189	8.324	20.607	10.513	5.602	4.181	11.350	27.500
SD	0.323	0.287	0.474	1.429	4.759	2.907	0.617	1.071	3.823	10.691	4.635	2.795	2.002	4.818	13.769
CV (%)	23.21	15.28	13.07	23.32	36.62	49.61	9.15	10.35	24.36	43.24	48.63	25.34	12.97	24.62	46.79
Robust descriptive statistics
MiA	0.494	4.055 (*)	2.953	7.156 (*)	3.897 (*)	1.859	−0.389	−0.728	0.573	0.786	2.236	0.627	−0.434	1.037	0.888
MiB	−0.855	−0.926	−0.005	−0.757	0.310	−0.810	−3.163	−0.621	2.983	0.601	−0.911	−0.627	0.434	0.312	0.585
MiC	−0.494	0.423	0.005	0.592	−0.310	−0.539	0.389	0.621	−0.573	−0.601	0.438	0.722	0.915	−0.312	−0.585
MiD	5.233 (*)	−0.423	−1.344	−0.592	−1.039	0.539	0.960	1.225	−0.776	−0.748	−0.438	−1.103	−12.336 (*)	−1.195	−0.764
Eoed	1.267	1.780	3.521	5.518	11.446	5.232	6.927	10.210	14.468	24.599	8.425	11.321	16.342	19.768	28.913

Notes: Numbers with an asterisk are larger than 3.5; corresponding E_oed are outliers and will be excluded from the evaluation of E_oed. Bold, underlined number: the highest CV.

and

Table A6. Oedometric moduli E_oed for various density index I_D according to the ISO, Danube sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C, and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Danube Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	1.323	2.297	4.292	8.243	19.906	9.685	6.795	9.394	15.737	35.160	15.894	13.245	16.205	25.040	43.691
B	1.169	1.662	3.520	5.230	12.118	3.292	5.859	9.513	21.074	32.677	5.381	9.397	16.479	21.357	38.649
C	1.210	1.834	3.523	5.744	10.774	3.940	7.058	10.906	13.200	16.520	9.888	13.537	16.630	18.179	19.176
D	1.867	1.726	3.170	5.293	9.189	6.523	7.250	11.583	12.749	14.553	6.961	7.935	12.449	13.690	16.190
Standard descriptive statistics
$\overline{x}$ (MPa)	1.392	1.880	3.626	6.127	12.997	5.860	6.741	10.349	15.690	24.728	9.531	11.029	15.441	19.566	29.427
Range	0.698	0.635	1.122	3.013	10.717	6.393	1.391	2.189	8.324	20.607	10.513	5.602	4.181	11.350	27.500
SD	0.323	0.287	0.474	1.429	4.759	2.907	0.617	1.071	3.823	10.691	4.635	2.795	2.002	4.818	13.769
CV (%)	23.21	15.28	13.07	23.32	36.62	49.61	9.15	10.35	24.36	43.24	48.63	25.34	12.97	24.62	46.79
Robust descriptive statistics
Mdn	1.267	1.780	3.521	5.518	11.446	5.232	6.927	10.210	14.468	24.599	8.425	11.321	16.342	19.768	28.913
Q1	1.189	1.694	3.345	5.261	9.981	3.616	6.327	9.453	12.975	15.537	6.171	8.666	14.327	15.934	17.683
Q3	1.595	2.066	3.907	6.994	16.012	8.104	7.154	11.245	18.405	33.919	12.891	13.391	16.555	23.199	41.170
IQR	0.406	0.372	0.562	1.732	6.031	4.488	0.827	1.791	5.431	18.382	6.720	4.725	2.228	7.264	23.487
LF	0.581	1.137	2.502	2.663	0.935	−3.116	5.087	6.767	4.828	−12.036	−3.908	1.579	10.986	5.037	−17.547
UF	2.204	2.623	4.751	9.592	25.058	14.836	8.395	13.931	26.551	61.491	22.970	20.478	19.896	34.095	76.400
MAD	0.077	0.086	0.176	0.257	1.464	1.616	0.228	0.756	1.494	9.062	2.253	2.070	0.213	3.431	11.230
MiA	0.494	4.055	2.953	7.156	3.897	1.859	−0.389	−0.728	0.573	0.786	2.236	0.627	−0.434	1.037	0.888
MiB	−0.855	−0.926	−0.005	−0.757	0.310	−0.810	−3.163	−0.621	2.983	0.601	−0.911	−0.627	0.434	0.312	0.585
MiC	−0.494	0.423	0.005	0.592	−0.310	−0.539	0.389	0.621	−0.573	−0.601	0.438	0.722	0.915	−0.312	−0.585
MiD	5.233	−0.423	−1.344	−0.592	−1.039	0.539	0.960	1.225	−0.776	−0.748	−0.438	−1.103	−12.336	−1.195	−0.764
Eoed	1.267	1.780	3.521	5.518	11.446	5.232	6.927	10.210	14.468	24.599	8.425	11.321	16.342	19.768	28.913

. Similarly, the calculated E_oed of Polanka sand according to the STN and ISO (excluding deformations of the oedometers) are presented in

Table 4. Oedometric moduli E_oed for various density index I_D according to the STN, Polanka sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Polanka Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	4.696	5.400	13.034	30.238	287.981	5.298	8.307	14.823	31.829	377.975	7.044	11.630	20.431	50.397	671.956
B	2.975	7.399	20.397	43.126	61.608	4.045	8.203	20.964	46.586	115.663	8.477	13.240	83.856	120.752	232.215
C	3.084	5.799	10.266	25.114	42.799	4.450	8.082 (*)	27.132	104.784	108.527	9.439	21.102	75.969	1012.917	209.569
D	3.500	6.822	13.061	21.831	37.732	4.262	9.672	18.636	37.732	64.343	5.594	9.923	20.375	38.204	70.260
Standard descriptive statistics
$\overline{x}$ (MPa)	3.564	6.355	14.190	30.077	107.530	4.514	8.566	20.389	55.233	166.627	7.639	13.974	50.158	305.567	296.000
Range	1.721	1.999	10.131	21.295	250.249	1.253	1.590	12.309	72.955	313.632	3.844	11.179	63.480	974.713	601.696
SD	0.788	0.918	4.341	9.362	120.738	0.548	0.743	5.159	33.586	142.715	1.681	4.942	34.508	472.968	260.892
CV (%)	22.11	14.45	30.59	31.13	112.28	12.15	8.67	25.30	60.81	85.65	22.01	35.36	68.80	154.78	88.06
Robust descriptive statistics
MiA	3.608 (*)	−0.864	−0.007	0.411	13.321 (*)	3.140	0.311	−1.093	−0.944	6.989 (*)	−0.404	−0.327	−0.674	−0.575	3.757 (*)
MiB	−0.814	1.032	3.547 (*)	2.479	0.531	−1.037	−0.311	0.256	0.405	0.094	0.404	0.327	0.865	0.575	0.094
MiC	−0.535	−0.485	−1.342	−0.411	−0.531	0.312	−1.038	1.610	5.725 (*)	−0.094	0.945	3.525 (*)	0.674	15.155 (*)	−0.094
MiD	0.535	0.485	0.007	−0.938	−0.818	−0.312	8.480 (*)	−0.256	−0.405	−1.255	−1.221	−1.022	−0.675	−0.774	−1.255
Eoed	3.292	6.311	13.047	27.676	52.204	4.356	8.255	19.800	42.159	112.095	7.761	12.435	48.200	85.574	220.892

Notes: Numbers with an asterisk are larger than 3.5; corresponding E_oed are outliers and will be excluded from the evaluation of E_oed.

and

Table A7. Oedometric moduli E_oed for various density index I_D according to the STN, Polanka sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C, and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Polanka Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	4.696	5.400	13.034	30.238	287.981	5.298	8.307	14.823	31.829	377.975	7.044	11.630	20.431	50.397	671.956
B	2.975	7.399	20.397	43.126	61.608	4.045	8.203	20.964	46.586	115.663	8.477	13.240	83.856	120.752	232.215
C	3.084	5.799	10.266	25.114	42.799	4.450	8.082	27.132	104.784	108.527	9.439	21.102	75.969	1012.917	209.569
D	3.500	6.822	13.061	21.831	37.732	4.262	9.672	18.636	37.732	64.343	5.594	9.923	20.375	38.204	70.260
Standard descriptive statistics
$\overline{x}$ (MPa)	3.564	6.355	14.190	30.077	107.530	4.514	8.566	20.389	55.233	166.627	7.639	13.974	50.158	305.567	296.000
Range	1.721	1.999	10.131	21.295	250.249	1.253	1.590	12.309	72.955	313.632	3.844	11.179	63.480	974.713	601.696
SD	0.788	0.918	4.341	9.362	120.738	0.548	0.743	5.159	33.586	142.715	1.681	4.942	34.508	472.968	260.892
CV (%)	22.11	14.45	30.59	31.13	112.28	12.15	8.67	25.30	60.81	85.65	22.01	35.36	68.80	154.78	88.06
Robust descriptive statistics
Mdn	3.292	6.311	13.047	27.676	52.204	4.356	8.255	19.800	42.159	112.095	7.761	12.435	48.200	85.574	220.892
Q1	3.030	5.599	11.650	23.472	40.266	4.153	8.143	16.729	34.781	86.435	6.319	10.777	20.403	44.300	139.914
Q3	4.098	7.111	16.729	36.682	174.795	4.874	8.989	24.048	75.685	246.819	8.958	17.171	79.912	566.834	452.085
IQR	1.069	1.511	5.079	13.210	134.529	0.720	0.847	7.319	40.905	160.384	2.639	6.395	59.509	522.534	312.171
LF	1.427	3.333	4.031	3.658	−161.528	3.073	6.872	5.751	−26.576	−154.141	2.631	1.184	−68.860	−739.501	−328.342
UF	5.701	9.377	24.438	56.496	376.588	5.954	10.260	35.026	137.042	487.395	12.916	26.764	169.176	1350.636	920.342
MAD	0.262	0.711	1.398	4.204	11.938	0.202	0.113	3.071	7.378	25.660	1.197	1.659	27.797	41.274	80.978
MiA	3.608	−0.864	−0.007	0.411	13.321	3.140	0.311	−1.093	−0.944	6.989	−0.404	−0.327	−0.674	−0.575	3.757
MiB	−0.814	1.032	3.547	2.479	0.531	−1.037	−0.311	0.256	0.405	0.094	0.404	0.327	0.865	0.575	0.094
MiC	−0.535	−0.485	−1.342	−0.411	−0.531	0.312	−1.038	1.610	5.725	−0.094	0.945	3.525	0.674	15.155	−0.094
MiD	0.535	0.485	0.007	−0.938	−0.818	−0.312	8.480	−0.256	−0.405	−1.255	−1.221	−1.022	−0.675	−0.774	−1.255
Eoed	3.292	6.311	13.047	27.676	52.204	4.356	8.255	19.800	42.159	112.095	7.761	12.435	48.200	85.574	220.892

and

Table 5. Oedometric moduli E_oed for various density index I_D according to the ISO, Polanka sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Polanka Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	4.696	5.376	12.916	29.849	283.324	5.298	8.275	14.720	31.502	372.900	7.044	11.596	20.327	50.017	665.556
B	2.975	7.348	20.187	42.576	60.680	4.045	8.161	20.793	46.096	114.196	8.477	13.208	83.492	120.156	230.877
C	3.084	5.760	10.153	24.715	41.950	4.450	8.045 (*)	26.923	103.784	107.387	9.438	21.056	75.711	1008.817	208.700
D	3.500	6.782	12.936	21.539	37.054	4.262	9.625	18.498	37.352	63.524	5.594	9.886	20.249	37.873	69.467
Standard descriptive statistics
$\overline{x}$ (MPa)	3.564	6.317	14.048	29.670	105.752	4.514	8.527	20.234	54.684	164.502	7.638	13.936	49.945	304.215	293.650
Range	1.721	1.972	10.034	21.037	246.269	1.253	1.581	12.203	72.282	309.376	3.844	11.170	63.243	970.944	596.089
SD	0.788	0.908	4.296	9.260	118.818	0.548	0.739	5.114	33.279	140.735	1.681	4.936	34.392	471.132	258.024
CV (%)	22.11	14.38	30.58	31.21	112.36	12.15	8.66	25.28	60.86	85.55	22.01	35.42	68.86	154.87	87.87
Robust descriptive statistics
MiA	3.608 (*)	−0.859	−0.005	0.417	13.248 (*)	3.138	0.332	−1.094	−0.945	6.978 (*)	−0.404	−0.327	−0.674	−0.575	3.726 (*)
MiB	−0.814	1.033	3.519 (*)	2.483	0.535	−1.037	−0.332	0.255	0.404	0.091	0.404	0.327	0.863	0.575	0.093
MiC	−0.535	−0.490	−1.344	−0.417	−0.535	0.312	−1.017	1.616	5.737 (*)	−0.091	0.945	3.515 (*)	0.674	15.144 (*)	−0.093
MiD	0.535	0.490	0.005	−0.932	−0.814	−0.312	8.247 (*)	−0.255	−0.404	−1.258	−1.221	−1.022	−0.675	−0.774	−1.256
Eoed	3.292	6.271	12.926	27.282	51.315	4.356	8.218	19.646	41.724	110.792	7.761	12.402	48.019	85.086	219.788

Notes: Numbers with an asterisk are larger than 3.5; corresponding E_oed are outliers and will be excluded from the evaluation of E_oed. bold, underlined, italic number: the lowest CV; bold, underlined number: the highest CV.

and

Table A8. Oedometric moduli E_oed for various density index I_D according to the ISO, Polanka sand.

Oed. and Param.	Oedometric Moduli Eoed Obtained from Oedometers A, B, C, and D and Their Descriptive Statistics (Standard and Robust Measures) for 5 Intervals of Loading (No. 1 to 5), Polanka Sand
	Density Index ID = 0.15					Density Index ID = 0.65					Density Index ID = 0.85
	1	2	3	4	5	1	2	3	4	5	1	2	3	4	5
Oedometric moduli obtained from oedometers
A	4.696	5.376	12.916	29.849	283.324	5.298	8.275	14.720	31.502	372.900	7.044	11.596	20.327	50.017	665.556
B	2.975	7.348	20.187	42.576	60.680	4.045	8.161	20.793	46.096	114.196	8.477	13.208	83.492	120.156	230.877
C	3.084	5.760	10.153	24.715	41.950	4.450	8.045	26.923	103.784	107.387	9.438	21.056	75.711	1008.817	208.700
D	3.500	6.782	12.936	21.539	37.054	4.262	9.625	18.498	37.352	63.524	5.594	9.886	20.249	37.873	69.467
Standard descriptive statistics
$\overline{x}$ (MPa)	3.564	6.317	14.048	29.670	105.752	4.514	8.527	20.234	54.684	164.502	7.638	13.936	49.945	304.215	293.650
Range	1.721	1.972	10.034	21.037	246.269	1.253	1.581	12.203	72.282	309.376	3.844	11.170	63.243	970.944	596.089
SD	0.788	0.908	4.296	9.260	118.818	0.548	0.739	5.114	33.279	140.735	1.681	4.936	34.392	471.132	258.024
CV (%)	22.11	14.38	30.58	31.21	112.36	12.15	8.66	25.28	60.86	85.55	22.01	35.42	68.86	154.87	87.87
Robust descriptive statistics
Mdn	3.292	6.271	12.926	27.282	51.315	4.356	8.218	19.646	41.724	110.792	7.761	12.402	48.019	85.086	219.788
Q1	3.029	5.568	11.535	23.127	39.502	4.153	8.103	16.609	34.427	85.456	6.319	10.741	20.288	43.945	139.083
Q3	4.098	7.065	16.562	36.212	172.002	4.874	8.950	23.858	74.940	243.548	8.958	17.132	79.601	594.486	448.216
IQR	1.069	1.497	5.027	13.085	132.500	0.720	0.847	7.249	40.513	158.092	2.639	6.391	59.314	520.542	309.133
LF	1.427	3.323	3.994	3.499	−159.247	3.073	6.832	5.736	−26.343	−151.682	2.361	1.155	−68.683	−736.868	−324.616
UF	5.701	9.310	24.102	55.840	370.751	5.954	10.221	34.731	135.710	480.686	12.916	26.718	168.572	1345.299	911.916
MAD	0.262	0.703	1.392	4.155	11.813	0.202	0.115	3.036	7.297	25.336	1.197	1.661	27.731	41.142	80.705
MiA	3.608	−0.859	−0.005	0.417	13.248	3.138	0.332	−1.094	−0.945	6.978	−0.404	−0.327	−0.674	−0.575	3.726
MiB	−0.814	1.033	3.519	2.483	0.535	−1.037	−0.332	0.255	0.404	0.091	0.404	0.327	0.863	0.575	0.093
MiC	−0.535	−0.490	−1.344	−0.417	−0.535	0.312	−1.017	1.616	5.737	−0.091	0.945	3.515	0.674	15.144	−0.093
MiD	0.535	0.490	0.005	−0.932	−0.814	−0.312	8.247	−0.255	−0.404	−1.258	−1.221	−1.022	−0.675	−0.774	−1.256
Eoed	3.292	6.271	12.926	27.282	51.315	4.356	8.218	19.646	41.724	110.792	7.761	12.402	48.019	85.086	219.788

, respectively.

As shown in Table 2, Table 3, Table 4 and Table 5, both SD and CV vary with I_D and the load intervals. For the Danube sand, the lowest CV value (9.07%) was recorded at I_D = 0.65, load interval No. 2 (STN, see Table 2, bold, underlined, italic number), whereas the highest CV value (49.61%) was recorded at I_D = 0.65, load interval No. 1 (ISO, see Table 3, bold, underlined number). For the Polanka sand, the lowest CV value (8.66%) occurred at I_D = 0.65, load interval No. 2 (ISO, see Table 5, bold, underlined, italic number), while the highest CV value (154.87%) occurred at I_D = 0.85, load interval No. 4 (ISO, see Table 5, bold, underlined number). Analysis of all CV values in Table 2, Table 3, Table 4 and Table 5 indicates that their variation is random, and there is no extreme case in which the minimum CV for a certain I_D and load interval exceeds the maximum CV for another I_D and load interval. Therefore, it can be concluded that no systematic error is present, and the variation in CV values exhibits no systematic dependence on I_D or load interval.

Due to the high CV value, a robust statistical analysis was performed to identify and exclude outliers. It was found that all measured E_oed values did not exceed the LF and UF; even the value E_oed = 1008.817 MPa, which occurred at I_D = 0.85, load interval No. 4 (ISO, see Table 5, bold, underlined, italic number), did not exceed the upper fence UF = 1345.299 MPa (ISO, see Table A8, bold, underlined number).

In addition, another robust statistical analysis using the Individual Modified Z-score method was applied to further identify and exclude outliers. Individual Modified Z-scores (MiA, MiB, MiC, MiD) for the E_oed values obtained from oedometers A, B, C, and D, respectively, were calculated according to Equation (15) and are presented in Table 2, Table 3, Table 4 and Table 5. Individual Modified Z-scores greater than 3.5 (number with an asterisk) indicate outliers.

These Individual Modified Z-scores represent, in most cases, the maximum of the four E_oed values for a given load interval; however, in two cases, they also represent the minimum E_oed values for a given load interval (Table 2: 12.526 MPa (with an asterisk); Table 3: 12.449 MPa (with an asterisk), both at I_D = 0.85, loading interval No. 3; Table 4: 8.082 MPa (with an asterisk); Table 5: 8.045 MPa (with an asterisk), both at I_D = 0.65, loading interval No. 2).

Overall, the analysis of all LF/UF and Individual Modified Z-score values in Table 2, Table 3, Table 4 and Table 5 indicates that their variation is random and shows no systematic dependence on I_D or load interval. No systematic error was detected.

Unlike the arithmetic mean, which is highly sensitive to outliers, the median provides a robust estimate of central tendency. Given the presence of outliers in the tested datasets (identified using the Modified Z-score) and the small sample size, the median was chosen as the most consistent estimate of the characteristic value of the E_oed. For four values obtained from four odometers, the median equals the mean of the two middle values after removing the maximum and minimum. To ensure methodological consistency when comparing results across different density indices, the same robust measure of central tendency (median) was applied to all datasets. This prevents comparisons of statistics with inherently different properties. The characteristic E_oed values are reported in the last row of Table 2, Table 3, Table 4 and Table 5.

Differences in oedometric moduli E_oed of Danube sand (according to STN and ISO) caused by different density index I_D are presented in

Table 6. Differences in oedometric moduli E_oed caused by different density index I_D (STN, ISO, Danube sand).

Differences in Oedometric Moduli Eoed Between Various ID (STN, ISO, Danube Sand)
Standard	STN						ISO
ID	ID = 0.15 and ID = 0.65		ID = 0.15 and ID = 0.85		ID = 0.65 and ID = 0.85		ID = 0.15 and ID = 0.65		ID = 0.15 and ID = 0.85		ID = 0.65 and ID = 0.85
No. of Load interval	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)
1	3.963	311.51	7.153	562.31	3.190	60.94	3.965	312.98	7.158	565.02	3.193	61.03
2	5.143	283.81	9.536	526.23	4.393	63.16	5.147	289.11	9.541	535.95	4.394	63.44
3	6.658	182.24	12.768	349.48	6.110	59.25	6.689	189.95	12.821	364.10	6.132	60.06
4	8.867	153.29	14.150	244.64	5.284	36.06	8.950	162.19	14.250	258.23	5.300	36.63
5	12.847	104.68	17.039	138.84	4.192	16.68	13.153	114.92	17.467	152.61	4.314	17.54

. Similarly, differences in oedometric moduli E_oed of Polanka sand (according to STN and ISO) caused by different density index I_D are presented in

Table 7. Differences in oedometric moduli E_oed caused by different density index I_D (STN, ISO, Polanka sand).

Differences in Oedometric Moduli Eoed Between Various ID (STN, ISO, Polanka Sand)
Standard	STN						ISO
ID	ID = 0.15 and ID = 0.65		ID = 0.15 and ID = 0.85		ID = 0.65 and ID = 0.85		ID = 0.15 and ID = 0.65		ID = 0.15 and ID = 0.85		ID = 0.65 and ID = 0.85
No. of Load Interval	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)
1	1.064	32.32	4.469	135.74	3.405	78.17	1.064	32.32	4.469	135.74	3.405	78.16
2	1.945	30.81	6.125	97.05	4.180	50.63	1.947	31.05	6.131	97.76	4.184	50.91
3	6.753	51.75	35.153	269.42	28.400	143.43	6.719	51.98	35.093	271.49	28.374	144.43
4	14.483	52.33	57.899	209.20	43.415	102.98	14.442	52.93	57.804	211.88	43.363	103.93
5	59.891	114.73	168.688	323.13	108.797	97.06	59.477	115.91	168.474	328.31	108.997	98.38

.

As shown in Table 6 and Table 7, the difference in E_oed between I_D = 0.15 and I_D = 0.85 ranged from 138.8 to 562.3% (Danube sand, SM, STN) and 152.6 to 565.0% (Danube sand, SM, ISO); 97.0 to 323.1% (Polanka sand, S-F, STN) and 97.7 to 328.3% (Polanka sand, S-F, ISO), which are very high. The difference in E_oed between I_D = 0.65 and I_D = 0.85 ranged from 16.6 to 63.1% (Danube sand, SM, STN) and 17.5 to 63.4% (Danube sand, SM, ISO); 50.6 to 143.4% (Polanka sand, S-F, STN) and 50.9 to 144.4% (Polanka sand, S-F, ISO), which are much smaller. For Danube sand, the lowest difference occurs for loading interval 5 (normal stress 200 kPa–400 kPa), whereas for Polanka sand, the lowest difference occurs for loading interval 2 (normal stress 25 kPa–50 kPa), which indicates different behavior of the sands under the same loading conditions.

Differences in oedometric moduli E_oed between STN and ISO for Danube sand and Polanka sand are presented in

Table 8. Differences in oedometric moduli E_oed between STN and ISO (Danube sand, Polanka sand).

Differences in Oedometric Moduli Eoed Between Various STN and ISO (STN-ISO)
Sand	Danube Sand						Polanka Sand
ID	ID = 0.15		ID = 0.65		ID = 0.85		ID = 0.15		ID = 0.65		ID = 0.85
No. of Load Interval	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)	Diff. (MPa)	Diff. (%)
1	0.005	0.41	0.003	0.06	0.0003	0.003	0.0002	0.003	0.0000	0.000	0.0003	0.003
2	0.032	1.76	0.028	0.41	0.027	0.24	0.039	0.63	0.037	0.45	0.033	0.27
3	0.132	3.62	0.102	0.99	0.080	0.49	0.121	0.93	0.154	0.78	0.181	0.38
4	0.266	4.60	0.182	1.24	0.166	0.83	0.394	1.42	0.435	1.03	0.488	0.57
5	0.827	6.73	0.521	2.07	0.398	1.36	0.889	1.70	1.303	1.16	1.104	0.50

. As can be seen, the E_oed values according to STN are higher than those according to ISO, which is consistent with Equations (2) and (6). However, the differences are small, with the maximum difference of 6.7% occurring for I_D = 0.15, loading interval No. 5 (Danube sand). For Polanka sand, the maximum difference is 1.7%, also occurring for I_D = 0.15, loading interval No. 5. In general, it can be observed that the differences are larger under higher loads (larger loading interval number). An exception is Polanka sand at I_D = 0.85, where the percentage difference (0.50%) for loading interval No. 5 is smaller than for loading interval No. 4 (0.57%) due to the high value of E_oed for loading interval No. 5.

Based on the obtained E_oed values for different I_D, a linear relationship between E_oed and I_D was proposed in the form E_oed = a·I_D + b. The values of a and b, as well as the Pearson’s correlation coefficient R and the coefficient of determination R², are given in

Table 9. Linear relationship between the oedometer modulus E_oed and the relative density index I_D.

Loading Interval	Danube Sand				Polanka Sand
	a	b	R	R2	a	b	R	R2
1	9.784 (9.209)	−0.406 (0)	0.984 (0.995)	0.968 (0.991)	5.565	2.075	0.860	0.739
2	12.988 (12.328)	−0.467 (0)	0.981 (0.995)	0.962 (0.989)	7.823	4.661	0.900	0.811
3	17.366	0.472	0.976	0.953	43.077	3.171	0.834	0.695
4	19.885	2.314	0.995	0.991	72.252	11.626	0.866	0.750
5	25.213	7.785	0.999	0.998	217.267	7.801	0.917	0.840

Note: Parameters in parentheses were obtained from the constrained linear regression (forcing the line to pass through the origin (0, 0)).

. Based on the R values for Polanka sand and the interpretation of correlation strength [20], it can be stated that the linear dependence of E_oed on I_D, with R ranging from 0.834 to 0.917, is high (R = 0.70–0.90) to very high (R = 0.90–1.00).

For Danube sand (loading intervals No. 3–5), the Pearson’s correlation coefficient R ranges from 0.995 to 0.999, defining the linear dependence of E_oed on I_D as very high. Regarding loading interval No. 1, the initial unconstrained linear regression between the oedometer modulus E_oed and the relative density index I_D yielded the model E_oed = 9.784·I_D − 0.406 (MPa), with a very high linear dependence (R = 0.984). The negative intercept (−0.406) suggests a nonphysical negative modulus as I_D approaches zero. In such a case, according to some geotechnical laboratory practices, constrained regressions are applied for parameters that are assumed to be zero (e.g., forcing zero cohesion in direct shear tests accordance with [21]). By analogy, the regression in this study was recalculated by constraining the line to pass through the origin (0, 0), resulting in E_oed = 9.209·I_D (MPa). This constrained model demonstrates an even higher linear dependence (R = 0.995) compared to the initial unconstrained fit (R = 0.984). Although E_oed does not necessarily equal zero when I_D = 0, this approach provides a physically acceptable and simplified correlation that is valid for I_D ≥ 0.15, corresponding to the tested range of relative densities (see Table 9, Danube sand, loading intervals No. 1). Similarly, for loading interval No. 2, the regression was recalculated by constraining the line to pass through the origin (0, 0), resulting in E_oed = 12.328·I_D (MPa) with a Pearson’s correlation coefficient R = 0.995, indicating a very high linear dependence of E_oed on I_D. The unconstrained linear relationship between the oedometer modulus E_oed and the relative density index I_D for the Danube sand and the Polanka sand is presented in Figure 6.

To enhance the informational value of the visual presentation, error bars have been added to the E_oed values (determined according to ISO), indicating the standard deviation (see Figure 7 and Figure 8).

As can be seen in Figure 7 and Figure 8, SD values are smaller at lower I_D (lower E_oed) and also at lower vertical effective stresses (loading intervals LI 1–3). Larger SD values occur at higher I_D (higher E_oed) and higher vertical effective stresses (loading intervals LI 4–5). Extreme values are also observed (see Figure 8, I_D = 0.85, loading intervals LI 4 and LI 5). Such outliers were successfully detected using the Modified Z-score method and were excluded from the E_oed evaluation.

To examine the behavior of sands under loading, the relationship between m_v and vertical effective stress σ′_v is presented in Figure 9. The relationship between log (E_oed) and vertical effective stress σ′_v is presented in Figure 10. The power function is defined as:

$$m_v=C_0{\left({\sigma }_v^{\prime }\right)}^n,$$

(16)

where C₀ and n correspond to the fitting parameters of the power function. Values of C₀ and n are presented in

Table 10. Parameters of the power function for various I_D.

	Danube Sand			Polanka Sand
	ID = 0.15	ID = 0.65	ID = 0.85	ID = 0.15	ID = 0.65	ID = 0.85
C0	113.9	12.074	4.8135	78.469	108.11	79.036
n	−0.798	−0.553	−0.436	−1.005	−11.168	−1.243
R	0.994	0.994	0.996	1.000	0.998	0.991
R2	0.988	0.988	0.993	0.999	0.996	0.981

. Figure 9 indicates that the soil compressibility decreases, demonstrating strain-hardening behavior resulting from incremental loading.

The graphical analysis (Figure 10) confirms a change in material behavior. For all tested densities, we observe that at low stresses (approximately up to 100 kPa), the increase in the compressibility modulus E_oed is rapid, followed by a gradual decrease in the rate of modulus increase at higher stresses. This point of slope change is interpreted as the probable boundary of particle-bond yielding, with its precise location being slightly influenced by the relative density I_D.

4. Discussion

To the best of our knowledge, there are no indicative (reference) CV values for oedometer moduli obtained from laboratory testing. Authors in [22] state that because of the natural heterogeneity of soils, the estimates of c and tanφ obtained through different samples drawn from the same deposit present a certain degree of variability, and this has stimulated the formulation of statistical analysis of the shear strength parameters. Based on results presented in the literature review, it emerges that the coefficient of variation of tanφ is generally considered to be in the range 10% to 20%, while that of c seems to be between 20% and 60%.

The maximum CV value for Danube sand, considering all I_D values and loading intervals, is 49.59% (E_oed obtained in accordance with STN) and 49.61% (E_oed obtained in accordance with ISO), see Table 2 and Table 3 (bold, underlined number), which are therefore lower than the CV of cohesion c (60%). For Polanka sand, 60% of the CV values (E_oed from 9 loading intervals out of 15, obtained in accordance with STN and ISO, see Table 4 and Table 5) are lower than 60%. We assume that the high CV values are caused not only by soil heterogeneity but also by human factors (even though the tests were prepared and conducted by the same person following the same procedure, differences in compaction forces cannot be entirely ruled out). Although it is difficult to compare the CV of E_oed with the CV of soil cohesion c, the mentioned values may serve as reference values for further research.

Authors in [23] present the results of oedometer tests on Osorio sand, which is classified as non-plastic, uniform fine sand. The specific gravity of the solids is 2.63 (Polanka sand: 2.64). Mineralogical analysis showed that sand particles are predominantly quartz (Polanka sand particles contain 91.6% SiO₂ [24]). The grain size is purely fine sand (Polanka sand is classified as sand with fine-grained admixture S-F, with sub-angular to rounded particles) with a mean diameter of 0.16 mm (Polanka sand: 0.36 mm), with uniformity and curvature coefficients of 1.9 and 1.2, respectively (Polanka sand: 3.11 and 1.23). The minimum and maximum void ratios are 0.6 and 0.9, respectively (Polanka sand: 0.42 and 0.74). The authors suggest that the relationship between the oedometer modulus E_oed and the relative density index I_D for Osorio sand is E_oed = 41.18·I_D + 29.40 MPa. Among the ten interval-specific experimental regressions obtained in this study (see Table 9), the regression for Polanka sand, loading interval No. 4, E_oed = 72.255·I_D + 11.626 MPa, provides the closest approximation to the literature correlation. Notably, this regression intersects the literature correlation at approximately I_D ≈ 0.572 (E_oed ≈ 52.9 MPa), providing a point of reference, even though the tested Polanka sand differs from those in the literature.

A limitation of this study is that particle shape, mineral composition, particle-crushing potential, and other characteristics of the sands that influence compression behavior were not sufficiently analyzed (are available only for Polanka sand, see the previous paragraph). Therefore, it will be necessary in further research to perform tests to determine these parameters and relate them to the obtained E_oed values.

For the incremental loading tests, separate linear regressions were derived for each loading interval. This approach allows a more detailed representation of the behavior of the tested sands under varying stress conditions, capturing changes in stiffness with each incremental load. The resulting interval-specific regression models reliably describe the behavior of the sands over the investigated range of relative densities. The regressions are considered valid for I_D between 0.15 and 0.85, corresponding to the experimental range. Due to the very high linear dependence, it is reasonable to interpolate within this range with confidence, while extrapolation outside this interval is not supported by the data.

Regarding the difference in E_oed caused by different approaches to evaluating vertical strain increments at each loading step (STN vs. ISO), authors in [25] present results of incremental loading oedometer tests on firm CI (clay of intermediate plasticity). For the loading interval 100–200 kPa, the difference in E_oed resulting from the two approaches is 0.266 MPa (6.803 MPa vs. 6.537 MPa), corresponding to 3.91%. This value of 3.91% is the closest to that (4.60%) observed for Danube sand at I_D = 0.15 for the same loading interval (see Table 8, loading interval No. 4).

The oedometer tests in this study were limited to one-way loading, focusing only on compression. Unloading and reloading cycles were not considered, so the elastic rebound could not be assessed. To address this limitation, further tests including unloading–reloading cycles will be conducted in future research to capture the full compressive and elastic behavior of sands.

Outliers were successfully detected using the Modified Z-score method and were excluded from the E_oed evaluation. However, the small sample size (n = 4 per condition) limits the generalizability of the results. Although the median was chosen as the most consistent estimate of the characteristic E_oed value, the reliability of the mean of the two middle values after removing the maximum and minimum may not be very high. Therefore, it is useful in future research to increase the number of tests. Although performing repeated oedometer tests is more time-consuming, it is methodologically valuable; for instance, five repetitions on the same oedometer are planned for future work. After obtaining more data, it is necessary to consider whether a relationship between E_oed and I_D other than linear exists. However, since there are few data points in this study, only a linear relationship can be considered. Due to the very high linear dependence of E_oed on I_D for SM (Pearson R = 0.97–0.99) and the high-to-very-high dependence for S-F (Pearson R = 0.83–0.91), the applicability of this relationship in this case is justified.

5. Conclusions

The results of the oedometer tests show that the E_oed values for the same loading interval, obtained from four different oedometers, are not identical, even though the samples were prepared and tested in the same manner by the same operator and considering the deformation of the oedometers themselves. It is assumed that the differences were caused not only by the inhomogeneity of the samples but also by the preparation process, which cannot be perfectly identical for each tested sample.

The IQR method for outlier exclusion does not remove outliers when only four values are available, even though laboratory experience clearly indicates that some values are outliers. Such outliers were successfully detected using the Modified Z-score method.

The relative density significantly affects the E_oed of the examined sands. The difference in E_oed between I_D = 0.15 and I_D = 0.85 reached up to 565.0% for Danube sand (SM) and 328.3% for Polanka sand (S-F).

The results show a very high linear dependence of E_oed on I_D for SM (Pearson R = 0.97–0.99) and a high-to-very-high dependence for S-F (Pearson R = 0.83–0.91).

The difference in E_oed between STN and ISO ranged From 0.003 to 6.7% for SM and 0.0 to 1.7% for S-F. A difference of 6.7% should not be considered negligible, as it may affect the values of footwear impressions, thereby influencing the reliability of estimating a perpetrator’s weight in forensic engineering.

The obtained E_oed values will be applied in subsequent experimental analyses of footwear impressions.