Comparative Experimental Study on the Dynamic and Static Stiffness of Sandy Soils Utilizing Alpan’s Empirical Approach

Abstract

Stiffness parameters are very important and effective in the constitutive models used in finite element analysis. It is not easy or common to obtain these parameters in the laboratory. However, even if the modulus is determined in the small and medium deformation range, there is a need to make transitions in both static and dynamic parameters. In almost all studies, the Alpan approach is used for the relationship between static and dynamic moduli of elasticity. Therefore, a better understanding of this approach is required. In this study, the relationship between static and dynamic stiffness was determined by monotonic triaxial and resonant column tests on five different sand samples with different relative stiffness and grain distributions, and the results were compared with Alpan’s approach. It is not clear which of the initial or maximum modulus of elasticity (E₀), unloading-reloading modulus (E_ur) or secant modulus of elasticity (E₅₀) are used by Alpan for static modulus of elasticity (E_stat). Therefore, the coefficient R_sec = E_stat/E₅₀ was introduced and queried to indicate which E_stat is a multiple of E₅₀. In connection with this, the dynamic modulus of elasticity (E_dyn) was calculated using the small deformation shear modulus (G₀) obtained from resonant column experiments and assuming Poisson’s ratios (ν = 0.2, 0.3, 0.4). It was found that Alpan’s empirical approach achieved a significant degree of agreement for the sands in this study and the studies of other researchers. It was observed that the best agreement between dynamic and static stiffness ratio (E_dyn/E_stat) and static modulus of elasticity (E_stat) for sand specimens in this study was obtained with υ = 0.2 and R_sec = 2. According to the experimental results, it is safe to say that Alpan’s empirical approach is still valid when the values of Poisson’s ratio and E_stat in the very small deformation region are used. Since there are limited studies on E_dyn/E_stat ratio in the literature, it is thought that the findings in this paper will assist engineers and researchers. However, this work would also assist engineers in selecting appropriate stiffness parameters for calibrating constitutive models.

1. Introduction

In recent years, modeling problems with complex geometries and challenging soil conditions has become widely used by engineers and researchers due to significant advances in numerical tools and computational technologies. A major challenge in numerical modeling is the determination of parameters related to the stress–strain behavior of soils affected by both static and dynamic loading conditions. Soils under geotechnical applications such as foundations and embankment structures are subjected to a relatively slow loading rate. The stress–strain behavior of such soils can be described by the static elasticity modulus (E_stat) [1]. On the other hand, the behavior of soils under cyclic and dynamic loading such as earthquakes, winds, explosions, and machine vibrations can be represented and modeled by the dynamic modulus of elasticity (E_dyn), which is considered as a fundamental parameter in geotechnical engineering applications [2].

At very small deformation levels (γ ≤ 10⁻⁵%), where the elastic behavior remains essentially constant, the modulus is defined as the initial shear modulus (G₀) or “dynamic” stiffness, whether obtained by laboratory tests or by measurement or calculation by seismic in situ methods. In contrast, the “static” stiffness is derived from the initial loading curve of oedometer or monotonic triaxial tests. Although the difference between static and dynamic stiffness was initially attributed to the difference in loading rates, later studies have shown that these differences are mainly determined by the deformation levels. The effect of drainage conditions on the stiffness of sandy soils at small to moderate deformations is another important issue that needs to be clarified.

By the 1970s, a comprehensive experimental database of various soil types had been established in the literature, thanks to the more consistent results of monotonic experiments performed under stress or deformation control. However, the acquisition of dynamic parameters was very limited in this period. To overcome this deficiency, ref. [3] proposed an empirical relationship derived from the solution of a differential equation subject to boundary conditions and thus successfully introduced the concept of dynamic-static modulus ratio into the principles of soil mechanics. A rheological model considered by [3] depends on the Kelvin–Voigt rigid body [4,5]. Static elastic modulus (E_stat) is defined as a ratio of constant normal stress, σ₀, to time-dependent axial strains ε(t), as presented in Equation (1).

$${\mathrm{E}}_{\mathrm{stat}}={\displaystyle \frac{{\mathsf{\sigma }}_0}{\mathsf{\epsilon }}}={\displaystyle \frac{\mathrm{E}}{1-\exp \left(-\frac{\mathrm{t}}{{\mathrm{T}}_{\mathrm{ret}}}\right)}}$$

(1)

where T_ret, denotes the retardation time of the force applied in rheologic model; E represents the modulus of elasticity for the springs-within model; and t refers to the duration of the static test. In this context, the value of E is employed in the static elasticity formula. T_ret may be interpreted as the characteristic duration over which the stress induced by dynamic or static loading is transferred to the soil skeleton. For dynamic loading states, the Kelvin–Voigt rigid body exhibits dynamic elastic modulus (E_dyn) influenced by the periods (T) of cyclic loading, given by Equation (2).

$${\mathrm{E}}_{\mathrm{dyn}}={\displaystyle \frac{{\mathsf{\sigma }}_0}{{\mathsf{\epsilon }}_0}}=\mathrm{E}{\left[1+\left(2{\mathsf{\pi }\mathrm{T}}_{\mathrm{ret}}/{\mathrm{T}}^2\right)\right]}^{1/2}$$

(2)

where $\mathrm{T} = 2\mathsf{\pi }/\mathsf{\omega }$ denotes the vibration period of dynamic stress. Alpan’s concept is still widely accepted and applied by researchers and engineers today. This relationship allows a direct transition between E_dyn/E_stat and E_stat. However, in Alpan’s work, it is unclear whether the E_stat is equal to E₀, E_ur or E₅₀. This relation needs to be clarified.

Similar to the [3] concept, the German Geotechnical Society (DGGT) proposed a correlation between modulus ratio and static modulus retrieved from oedometer test results [6]. The relationship between dynamic and static modulus is explained by comparing the measured constraint modulus (M_dyn for dynamic constraint modulus, M_stat for static constraint modulus). Following [3], ref. [7] proposed a correlation illustrating the relationship between static and dynamic modulus in the “Recommendations of the Soil Dynamics Working Committee” of the DGGT. The correlation between dynamic and static moduli is expressed in terms of the oedometer modulus for one-dimensional compression (zero lateral deformation), with M_stat = M_oedo. Ref. [7] further contributed to the DGGT findings by expanding the E_S range from approximately 1 MPa to 300 MPa, while the dynamic to static ratio increases significantly according to DGGT (Figure 1). It can be observed that Alpan’s proposal exhibits limitations in estimating the E_dyn/E_stat when compared to the curves presented by both the [6,7].

This paper presents the results of a comprehensive experimental study aimed at assessing the small to medium strain stiffness parameters of the different types of sands. The experimental outcomes of five different types of sandy soils are evaluated and analyzed in conjunction with the Alpan concept. To achieve this, the index and classification properties of the sandy specimens were determined. Subsequently, the relative density of the specimens was defined in accordance with traditional density classifications, ranging from medium dense to dense. The specimens were subjected to both monotonic compression in static tests (MTX) and high-frequency vibration in resonant column tests (RCT). While there are uncertainties regarding the applicability of the empirical correlations derived from these relationships (Figure 1) in terms of soil type, index properties and stress conditions, the experimental program carried out covers sands with different firmness (loose and tight) and uniformity under different environmental stresses.

2. Materials and Test Methods

2.1. Materials

Clayey sand and silty sand specimens (S1 to S4) collected from the city center of Çanakkale consisted of alluvial sands deposited by the Sarıçay Stream during the prehistoric period. These sandy specimens were obtained as part of a geotechnical research conducted within a research project funded by TÜBİTAK [9], aiming to investigate basin edge effects in Çanakkale. The sampling site is located at 40°09′19″ N, 26°24′47″ E. The depth from which the specimens were retrieved ranged between 3 and 14 m, primarily determined through standard penetration tests (SPTs). Additionally, geophysical measurements using the multichannel analysis of surface waves (MASW) method were conducted to estimate the in situ relative density and to define the confining stress levels to be applied during laboratory testing.

The index properties of the soils were determined in accordance with the relevant ASTM standards [10,11] and are summarized in

Table 1. Index properties of the tested granular materials.

Sand No	Uniformity Coeff. Cu [-]	Coff. Of Curvature, CC [-]	Average Diameter, D50 [mm]	Specific Gravity, Gs [-]	Fine Content FC [%]	Maximum Void Ratio, emax [-]	Minimum Void Ratio emin [-]	Unified Soil Class. System (USCS)
S1	27.0	6.0	0.52	2.65	16	0.960	0.590	SC
S2	6.20	2.9	0.38	2.64	11	0.830	0.500	SW
S3	13.6	3.4	0.12	2.67	28	1.020	0.600	SC
S4	25.0	1.5	0.08	2.70	45	1.230	0.670	SC
S5	6.00	1.28	0.63	2.64	0	0.674	0.415	SW

. Granulometry curves obtained from sieve analysis are also presented in Figure 2 and accompanying legend photographs are attached. All sand samples were SW or SC according to the unified soil classification system. A commercially standardized clean sand sample (S5), as defined by EN 196-1 [12], was examined to broaden the range of sand types considered. Sample S2 also contained 11% fine content (FC). On the other hand, sand S3 and sand S4 had 28% and 45% FC, respectively. For these sands, two different relative density values were determined, corresponding to medium (D_R = 45%) and dense conditions (D_R = 70–90%).

2.2. Test Methods

Specimens with a diameter of 50 mm and a height of 100 mm, as presented Figure 3 and Figure 4, were prepared using the dry deposition method to achieve the target relative density specified in

Table 2. Initial conditions and experimental testing program.

Sand No	Type of Experiment	Initial Void Ratio e0 (-)	Relative Density DR (%)	Confining Stress σ3 (kPa)	Initial Shear Modulus G0 (MPa)	Initial Static Modulus E0 (MPa)	Secant Static Modulus E50 (MPa)
S1	RCT, MTX	0.794	45	50	55.1	13.4	20.0
	RCT, MTX			100	71.3	32.7	22.0
	RCT, MTX			200	111.9	58.1	30.0
	RCT, MTX	0.701	70	50	60.9	35.2	25.3
	RCT, MTX			100	78.1	57.2	33.0
	RCT, MTX			200	122.4	96.8	48.0
S2	RCT, MTX	0.682	45	50	69.6	19.0	27.0
	RCT, MTX			100	93.6	29.3	35.0
	RCT, MTX			200	143.8	59.2	50.7
	RCT, MTX			400	214.1	147.6	82.9
	RCT, MTX	0.599	70	100	96.7	41.6	40.0
	RCT, MTX			200	148.8	77.9	55.3
	RCT, MTX			400	216.5	200.0	100.0
S3	RCT, MTX	0.831	45	100	83.04	20.0	17.2
	RCT, MTX			200	136.1	53.9	24.5
	RCT, MTX			400	203.4	64.1	32.0
	RCT, MTX	0.726	70	100	84.4	36.4	20.0
	RCT, MTX			200	139.4	50.0	30.3
	RCT, MTX			400	206.8	51.0	37.0
S4	RCT, MTX	0.978	45	100	91.8	38.3	12.0
	RCT, MTX			200	111.7	35.0	16.0
	RCT, MTX			400	176.1	76.8	21.0
	RCT, MTX	0.838	70	100	92.5	43.2	15.7
	RCT, MTX			200	131.9	59.4	20.8
	RCT, MTX			400	203.4	78.1	26.0
S5	RCT, MTX	0.56	45	50	70.0	47.4	40.2
	RCT, MTX			100	85.0	68.6	53.2
	RCT, MTX			150	118.0	90.0	63.7
	RCT, MTX			200	136.1	111.2	83.0
	RCT, MTX			300	157.9	128.2	102.8
	RCT, MTX	0.44	90	50	83.6	54.7	44.0
	RCT, MTX			100	117.7	80.0	56.3
	RCT, MTX			150	149.9	107.0	68.0
	RCT, MTX			200	169.5	121.3	98.5

for both monotonic triaxial (MTX) and resonant column (RC) tests (Figure 4). Tests were carried out in dry conditions; therefore, no porewater pressure (PWP) was measured. To prevent the dispersion of the sandy specimens, a vacuum pressure ranging from 10 to 30 kPa was applied to the samples via the pipe at the top cap prior to the application of confining stress (refer to Figure 3a,b). Subsequently, the samples were isotropically consolidated at the mean principal stress indicated in Table 2.

The RC tests employ a pneumatic stress-controlled system, which utilizes an electric motor to generate shear strain (torsion) at the specimen’s top cap. The RC test is based on the theory of the wave propagation of prismatic bars. The testing apparatus not only automatically applies the necessary confining pressure (σ₃) via the control panel depicted in Figure 4b but also independently measures and records shear stress (τ) and strain (γ) at the specimen’s top cap, as shown in Figure 4c. The resonant frequencies of the excited torque range from 60 to 160 Hz for fine sands (S1 to S4) and from 80 to 180 Hz for clean sand.

In the MTX tests, the hydraulic system was employed to apply σ₃ isotropically, as illustrated in Figure 3c. To ensure the acquisition of precise stress–strain curves during the tests, a minimum rate of 0.05 mm/min was selected at the shearing stage, owing to the data-reading capabilities of the S-type load cell in the apparatus. The tests were conducted until an axial strain of 15% was achieved (Figure 3d). Table 2 presents the initial conditions and testing program for both types of tests, as well as the measured modulus values obtained from each test.

3. Results and Discussion

Within the scope of this study, using the stress–strain relationship obtained from monotonic triaxial tests as a result of the laboratory studies presented in detail, above and whole test curves of resonant column and monotonic triaxial tests were generated, as given in Figure 5. It should be noted that in the monotonic triaxial experiments conducted in this study, E_ur could not be measured. Instead, the values of E₀ and E₅₀ were determined by evaluating the stress–strain curves given in Figure 6a. E_dyn is derived from G₀ obtained from resonant column tests considering the Poisson’s ratio range (υ = 0.2, 0.3 and 0.4) by using E_dyn = 2G₀(1 + υ) relation.

Figure 5 shows that the shear modulus (G₀) values obtained from resonant column tests for five different sand samples (S1–S5) increase with increasing confining stress. The specimens appear to be confined within a narrow range under similar stress conditions; however, the S5 specimen consistently exhibits higher G₀ values compared to the others. This difference can be attributed to the fact that S5 is clean sand, since the grain-to-grain contact is more direct and the stiffness properties are generally higher in sands without fine grains.

The initial elastic modulus (E₀) and secant elastic modulus (E₅₀) obtained from monotonic triaxial compression tests were evaluated for five different sand specimens at varying relative densities (D_R = 0.45 and 0.70). As shown in Figure 5, both modulus values increase with increasing confining stress. In specimens with high relative density (D_R ≥ 0.70), E₀ and E₅₀ values are significantly higher, especially specimen S5, which exhibited higher modulus of elasticity values compared to other sands under the same confining stress.

In this study, special emphasis was placed on defining the static stiffness of the tested soils in accordance with the conceptual framework proposed by [3]. Therefore, it becomes important which of the modules E₀, E_ur and E₅₀ in Figure 6a is E_stat. To understand this,

$${\mathrm{R}}_{\sec }={\displaystyle \frac{{\mathrm{E}}_{\mathrm{stat}}}{{\mathrm{E}}_{50}}}$$

(3)

relation was defined. Here, R_sec is the ratio between static elastic modulus (E_stat) and the secant modulus (E₅₀). R_sec will be 1.0 when E_stat is assumed as E₅₀. On the other hand, [13] suggests R_sec = 3.0 by using the E_stat = E_ur for HS and HSsmall constitutive models. Ref. [3] probably used the notation Ei as the static modulus of elasticity and defined it as shown in Figure 6b. This issue was discussed by [7,14] as to which of the static moduli of elasticity (E₀, E_ur, E₅₀) defined in Figure 6a is E_stat. As a result, they stated that the relationship E_stat = E_ur = 3E₅₀ is more appropriate for E_stat. In other words, considering υ = 0.2, they obtained R_sec = 3.

Figure 7 illustrates the relationships between elastic moduli and their respective ratios, as calculated for five distinct sand specimens under varying assumptions of Poisson’s ratio (υ = 0.2, 0.3, 0.4). The [3] correlation posits that an increase in the static elastic modulus corresponds to a decrease in the E_dyn/E_stat ratio. This hypothesis is predicated on the notion that in loose and low-stiffness sands, the dynamic elastic modulus is substantially greater than the static modulus; however, this disparity diminishes as soil stiffness increases. The experimental data acquired in this study predominantly exceed the Alpan correlation curve, particularly at lower E_stat values. The correlation between E_dyn/E₀ and E₀ demonstrates a closer alignment with the Alpan trend, whereas the E_dyn/E₅₀ and E₅₀ relationship exhibits greater variability. The results reveal that Poisson’s ratio significantly affects E_dyn and hence E_dyn/E_stat ratio.

The data obtained in this study were analyzed in comparison with the experimental results presented by [8], who conducted experiments on four different sands (Sands 1, 2, 3 and 4) with varying grain distributions, utilizing the [3] correlation (Figure 8). Ref. [8] measured Poisson’s ratio (υ) during the tests about 0.2. Therefore, comparisons are made for υ = 0.2 and compared for different R_sec ratios (R_sec = 1, 2, 3, and 4). A high level of agreement was observed between the sands tested in this study (S1–S5) and the data reported by [8] in terms of E_dyn/R_secE₅₀ ratios. This consistency supports the validity and reliability of the experimental methods and analytical approaches employed in both studies. However, the curve proposed by [3] underestimates the experimentally obtained E_dyn/R_secE₅₀ (R_sec = 1) values by approximately 1.5 to 2.5 times (Figure 8a). Nonetheless, when Figure 8b,c are considered, the assumption of R_sec = 2–3 indicates that the experimental data are in good agreement with the curve proposed by Alpan. However, for higher R_sec values (R_sec = 4), this relationship deteriorates slightly (Figure 8d). Furthermore, the distribution observed among the specimens varies according to the fines content of the sands. The clean sand specimens (S5) exhibit lower E_dyn/E_stat ratios, indicating a more rigid internal structure and a smaller discrepancy between dynamic and static stiffness. Conversely, specimens with higher fines content (S1–S4) display a broader range of variation. Consequently, the position of the data relative to the Alpan curve also varies with fines content: as the fines content decreases, the data tend to align more closely with the sand region defined by [3].

In order check the reliability of the experimental results, statistical error analyses were conducted on the case discussed in Figure 8. In relation to this, Figure 9 presents the variation in the differences (residuals) between the stiffness ratios predicted according to the [3] equation and the experimental data with respect to the R_sec parameter. As R_sec increases, the model tends to converge to the measured values; especially for R_sec = 2, the residuals decrease to the lowest level. This shows that the assumption that E_stat equals twice the secant elastic modulus (E₅₀) definition is the most compatible with the Alpan approach. On the other hand, a wider distribution of residuals in both the positive and negative directions is observed for R_sec = 1 and R_sec = 4, which proves that the model tends to deviate in these extreme cases.

Figure 10 illustrates the residual behavior of the Alpan correlation across various Poisson’s ratios (υ). It is evident that for υ = 0.4, the residual distribution is most constrained, yielding predictions that are closest to the model. In contrast, for υ = 0.2 and υ = 0.3, the residual values are higher and exhibit greater variability, suggesting that the model’s accuracy improves with an increase in Poisson’s ratio within the Alpan equation. Systematic deviations are particularly noticeable for υ = 0.2, which may indicate a structural inadequacy in the model. Furthermore, υ = 0.2 was experimentally measured in the study by [8], and the residuals from these data are also depicted in Figure 10.

Table 3. Results of the error and regression analysis.

Case Definition	MAE	MSE	RMSE	R2	Slope	Intercept
Rsec = 3, ν = 0.2	3.3274	12.316	3.5095	0.63	0.70	−1.37
Rsec = 3, ν = 0.3	3.0612	10.702	3.2714	0.59	0.76	−1.49
Rsec = 3, ν = 0.4	2.8161	9.2618	3.0433	0.59	0.82	−1.60
ν = 0.2, Rsec = 1	3.3212	21.687	4.6569	0.58	2.10	−4.12
ν = 0.2, Rsec = 2	2.0445	5.245	2.2902	0.66	1.05	−2.06
ν = 0.2, Rsec = 3	3.3274	12.316	3.5095	0.63	0.70	−1.37
ν = 0.2, Rsec = 4	4.1259	18.205	4.2668	0.60	0.53	−1.03

contains model performance metrics for different combinations of R_sec and Poisson’s ratio. When the mean absolute error (MAE), mean squared error (MSE) and root mean squared error (RMSE) values are analyzed, the lowest error is obtained for υ = 0.2 and R_sec = 2 (MAE = 2.04, RMSE = 2.29). This indicates that this parameter combination provides the highest agreement with experimental data. The coefficient of determination (R² = 0.66) for the same combination also supports this agreement. Conversely, at extreme values such as R_sec = 1 and R_sec = 4, error metrics increase, and model accuracy decreases.

4. Concluding Remarks

This paper presents the findings from a series of monotonic triaxial, and resonant column tests conducted on sandy specimens with varying characteristics. The primary objective of the study is to determine the dynamic and static stiffness of sandy soils and their interrelationship. Based on the results of this study, the following conclusions can be drawn:

• This study examines sands classified as fine sand “SC” (S1 and S3) and well graded “SW” (S2, S4 and S5) according to the Unified Soil Classification System (USCS). These soils were tested with varying fine content (FC = 0–45%) and different levels of relative density (D_R = 45% and DR = 70–90%). Therefore, the results obtained are valid for sandy soil with similar properties to the tested specimens.
• The measured E_dyn/E_stat values are generally underestimated by the Alpan approach. When Poisson’s ratio is assumed to be small, the Alpan curve is approximated more closely. The results reveal that Poisson’s ratio significantly affects E_dyn and hence the E_dyn/E_stat ratio.
• The ratio between dynamic and static modulus as proposed by Alpan, the static elastic modulus (E_stat) is not explicitly defined. The coefficient R_sec = E_stat/E₅₀ was introduced that E_stat is a multiple of E₅₀. Test results revealed the best agreement between dynamic and static stiffness ratio (E_dyn/E_stat) and static modulus of elasticity (E_stat) for R_sec = 2 (when v is assumed 0.2). Statistical error analyses support this finding.
• According to the experimental results, it is safe to say that Alpan’s empirical approach is still valid when the values of Poisson’s ratio and E_stat in the small deformation region are used. Ref. [3] diagram was found to be practical and appropriate when both static and dynamic tests were not performed on a soil.
• Since there are limited studies on E_dyn/E_stat ratio in the literature, it is thought that the findings in this paper will contribute to engineers and researchers. However, this work would also assist engineers in selecting appropriate stiffness parameters for calibrating constitutive models.
• It is observed that E_dyn/E_stat increases as the fine content increases. This and the updating of the mathematical expression originally proposed by [3] through new index properties (such as C_u and D₅₀), confining stress and relative density are the subject of ongoing research.