*Article* **Damping of Dry Sand in Resonant Column-Torsional Simple Shear Device**

**Majd Ahmad \* and Richard Ray**

Structural and Geotechnical Engineering Department, Faculty of Architecture, Civil and Transport Sciences Széchenyi István University, 9026 Gy˝or, Hungary; ray@sze.hu

**\*** Correspondence: ahmad.majd@hallgato.sze.hu

**Abstract:** The damping ratio values of three different Danube sands were measured in the Resonant Column-Torsional Simple Shear device (RC-TOSS). The distinctive configuration of the RC-TOSS device employed in this investigation enabled the performance of both tests using a single sample. This research estimates and compares the damping ratio values measured with three distinct methods (two of which are in the RC test): The Free Vibration Decay (FVD), the Steady-State Vibration (SSV) methods, and the method of calculating the damping ratio from the hysteretic loops generated in the TOSS test. Both dense and loose samples were tested up to a peak-to-peak amplitude shear strain of 1%. The device provides measurements over a wide range of shear strain amplitudes. The results support the employment of the SSV methods at low strains (below 0.005%), while the FVD method gives a better estimate at higher strains (above 0.03%). The two methods and the TOSS results are in agreement with each other between 0.005% and 0.03%. The effect of the number of cycles on the damping ratio was investigated where a significant decrease was observed in the damping ratio with an increasing number of cycles. A parameter is introduced to describe the rate of this decrease, which should be considered during the structural design to reduce maintenance and life-cycle costs and enhance sustainability.

**Keywords:** resonant column test; torsional simple shear test; damping ratio; steady-state vibration; free vibration decay

### **1. Introduction**

When waves propagate through soil during cyclic loading, complex mechanisms contribute to the loss of energy in the material. Fluid flow loss and inelastic friction loss are mainly responsible for energy dissipation in soil [1]. The dominant mechanism depends on the strain level and nonlinear behavior of soil. The damping ratio describes energy dissipation in a system. This parameter plays an important role when modeling dynamic geotechnical and structural problems (e.g., soil response analysis and soil–structure interaction problems). Thus, a substantial effort has been made in the last few decades to introduce experimental dynamic laboratory procedures and in situ methods to study the damping behavior of soils for a wide range of shear strain amplitudes. Resonant Column (RC) and Torsional Simple Shear (TOSS) tests have been developed and improved to measure the damping ratio with reasonable accuracy. The effects of various soil parameters on damping have been further investigated over the years. The most impactful factors are the void ratio, confining pressure, particle shape, and number of loading cycles [2–4].

Damping in materials may take several forms. In soils, geotechnical engineers are mostly concerned with hysteretic and viscous damping, which are proportional to displacement and velocity, respectively. While vibration frequency affects viscous damping, it seems to have no impact on the hysteretic damping. These two forms of damping are results of distinct mechanisms.

Soils have a tendency to dissipate energy even when subjected to low levels of strain [5]. Soils loaded cyclically below the linear strain threshold (*γtl*) exhibit an elastic behavior.

**Citation:** Ahmad, M.; Ray, R. Damping of Dry Sand in Resonant Column-Torsional Simple Shear Device. *Sustainability* **2023**, *15*, 11060. https://doi.org/10.3390/ su151411060

Academic Editors: Jian Zhou, Mahdi Hasanipanah and Danial Jahed Armaghani

Received: 31 May 2023 Revised: 3 July 2023 Accepted: 12 July 2023 Published: 14 July 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Hysteresis loops do not generate at strain amplitudes lower than that threshold. According to Kokusho [6], damping is a consequence of the fluids present in the voids, suggesting that the loss of fluid (viscous damping) is the main mechanism behind the soil's damping at small strain levels (below *γtl*). Conversely, beyond *γtl*, the shear stress–strain curves start to form hysteresis loops when soil is cyclically loaded. This is attributed to the nonlinear behavior of soil at higher strains. A rise in hysteretic damping is observed as the strain amplitude increases, and most of the energy dissipation is a result of inelastic friction. Thus, viscous damping is marginal, and the nature of damping is hysteretic [7,8].

Resonant column testing typically involves two distinct methods for measuring the damping of materials: The Steady-State Vibration (SSV) and the Free Vibration Decay (FVD) [9,10]. Several studies compared these two methods and provided suggestions on their scopes and limitations [11–13]. However, the unique design of the RC-TOSS device used in this study allows for conducting both tests on the same sample. This feature provided an accurate comparison between the three methods of measuring the damping ratio. These three methods are the SSV, FVD, and the method of calculating the damping ratio from the hysteretic loops generated in the TOSS test. This eliminates any potential effect of sample preparation on the results, which facilitates a more precise comparison. Since the damping ratio of soil influences the dynamic response of structures, a better understanding and more accurate estimation of the damping ratio can lead to a reduction in the structural demands and potentially increase the safety and resilience of the design. This can contribute to the long-term sustainability of the structure by preventing or minimizing damage.

### **2. Testing Device, Materials, and Methods**

Hollow, cylindrical soil samples were tested using a combined Resonant Column-Torsional Simple Shear device (RC-TOSS). The samples have an outer diameter of 6 cm, an inner diameter of 4 cm, and a height of 14 cm. A drive head applies torsional loading on the top of the sample (the free end), while the bottom of the specimen is fixed, as shown in Figure 1. The drive system consists of a set of two magnets inserted inside four coils. Two proximitors are fixed on the measurement post to measure the gap between the sensors and targets attached to the specimen. The difference between the two gaps gives the rotation from which the shear strain is found at each loading step. Torque is calculated from the known current passing through the coils during the test. The RC test requires displacement measurements with very high accuracy at very low amplitude (*γ* = 10−4%). An accelerometer mounted on the drive head provides such accurate measurements. A multimeter and an oscilloscope read and record the response curve, resonance frequency, and decaying response of the sample. The device is capable of conducting RC and TOSS tests on the same sample. A detailed description of the device and the methods of calibration are present in [3,14].

Three different types of Danube sands were selected for testing. Samples A and B retained a very low percentage of fines, while the fine content for sample C was 21.11%. The soil comprised fluvial sediments of the Danube River, extracted from a location near the channel of the river. The particles were sub-angular to rounded. The properties of the samples are shown in Table 1.

A vacuum applied a confining pressure of 97 KPa on the tested samples. The preparation of dense specimens was performed via the pluviation method. The soil was poured slowly from a height of 50 cm to achieve the thinnest thread of soil possible. On the other hand, to prepare samples in a loose state, the dry filling with tamping method was applied. Dry soil was poured through a glass funnel with a 14 cm spout into the mold. Pouring started with the spout touching the base ring, and then the spout was constantly and carefully raised and moved around to touch the surface of the soil. The testing plan appears in Table 2.

**Figure 1.** Cross-section of the RC-TOSS testing device. **Figure 1.** Cross-section of the RC-TOSS testing device.


**e** 



**Dr**

**Void Ratio Relative Density Maximum Shear Stress** 

**Gmax**

#### *2.1. Damping in the Torsional Simple Shear Device* damping in most analyses for the sake of mathematical convenience. "The equivalent vis-

*2.1. Damping in the Torsional Simple Shear Device* 

*Sustainability* **2023**, *15*, x FOR PEER REVIEW 4 of 14

Around load reversal points throughout cyclic torsional loading, the soil recovers its stiffness due to the reengagement and interlocking of the previously slipped contacts between particles in the opposite direction. The shear stress–strain path forms a hysteresis loop as demonstrated in Figure 2a. The gradient of the straight line that joins the endpoints of the hysteresis loop indicates the typical level of shear stiffness exhibited by the soil, which is also referred to as the secant shear modulus (*Gsec*). Although soil damping is recognized to be hysteretic in nature, it is commonly substituted with equivalent viscous damping in most analyses for the sake of mathematical convenience. "The equivalent viscous damping is determined in such a manner as to yield the same dissipation of energy per cycle as that produced by the actual damping mechanism" [15]. The damping ratio (*D*) represents the ratio of the energy absorbed in one cycle of vibration to the potential energy at maximum displacement in that cycle [16]. The following equation calculates the equivalent damping ratio *D*: cous damping is determined in such a manner as to yield the same dissipation of energy per cycle as that produced by the actual damping mechanism" [15]. The damping ratio (*D*) represents the ratio of the energy absorbed in one cycle of vibration to the potential energy at maximum displacement in that cycle [16]. The following equation calculates the equivalent damping ratio *D*: = 4 ∗ ் (1) where *AL* is the area of the loop (shadowed in Figure 2a) and *AT* is the area of the triangle bounded by a straight line defining the secant modulus at the point of maximum strain (Figure 2a). Subroutines in Visual Basic for Applications (VBA) in Excel were created to calculate the area between the loop and line that connects its two ends. The subroutine divides this area into very small slices that can be considered trapezoids (Figure 2b). The sum of the

Around load reversal points throughout cyclic torsional loading, the soil recovers its stiffness due to the reengagement and interlocking of the previously slipped contacts between particles in the opposite direction. The shear stress–strain path forms a hysteresis loop as demonstrated in Figure 2a. The gradient of the straight line that joins the endpoints of the hysteresis loop indicates the typical level of shear stiffness exhibited by the soil, which is also referred to as the secant shear modulus (*Gsec*). Although soil damping is recognized to be hysteretic in nature, it is commonly substituted with equivalent viscous

$$D = \frac{A\_L}{4\pi \ast A\_T} \tag{1}$$

where *A<sup>L</sup>* is the area of the loop (shadowed in Figure 2a) and *A<sup>T</sup>* is the area of the triangle bounded by a straight line defining the secant modulus at the point of maximum strain (Figure 2a). accelerometer measurements in the RC test. Therefore, TOSS tests in this study allow for damping ratio measurements at strain amplitudes above 0.02%.

(**a**) Dynamic properties from hysteretic loops (**b**) Integration element for half a loop

**Figure 2.** Hysteretic loops due to cyclic loading in TOSS test. **Figure 2.** Hysteretic loops due to cyclic loading in TOSS test.

*2.2. Damping in the Resonant Column Device*  2.2.1. The Free Vibration Decay Method (FVD) The resonant column test starts by applying a torsional oscillation on the top of the specimen with an increasing amplitude sine wave. For each tested strain amplitude, the Subroutines in Visual Basic for Applications (VBA) in Excel were created to calculate the area between the loop and line that connects its two ends. The subroutine divides this area into very small slices that can be considered trapezoids (Figure 2b). The sum of the areas multiplied by two denotes *A<sup>L</sup>* in Equation (1).

frequency progressively increases to obtain the dynamic response of the specimen. The frequency that corresponds to the maximum response amplitude is the resonance frequency of the sample. The dynamic shear modulus is a function of the fundamental fre-The proximitors incorporated in the TOSS test have low accuracy compared to the accelerometer measurements in the RC test. Therefore, TOSS tests in this study allow for damping ratio measurements at strain amplitudes above 0.02%.

#### quency based on wave propagation and elasticity theories. Once the operator finds the *2.2. Damping in the Resonant Column Device*

#### resonance frequency, they cut the power and track the decaying response on a storage 2.2.1. The Free Vibration Decay Method (FVD)

The resonant column test starts by applying a torsional oscillation on the top of the specimen with an increasing amplitude sine wave. For each tested strain amplitude, the frequency progressively increases to obtain the dynamic response of the specimen. The frequency that corresponds to the maximum response amplitude is the resonance frequency of the sample. The dynamic shear modulus is a function of the fundamental frequency based on wave propagation and elasticity theories. Once the operator finds the resonance frequency, they cut the power and track the decaying response on a storage oscilloscope. A multimeter records the decaying curve on an Excel sheet where damping is determined from the logarithmic decrement equation [5]:

$$\delta = \frac{1}{N} \ln \frac{Z\_1}{Z\_{1+N}} = \frac{2\pi D}{\sqrt{1 - D^2}}\tag{2}$$

where *δ* is the logarithmic decrement, *N* is the number of cycles, *Z*<sup>1</sup> is the first amplitude, *Z*1+*<sup>N</sup>* is the amplitude after *N* cycles (Figure 3a), and *D* is the damping ratio. higher number of cycles is needed in the SSV method to plot the frequency-response curve [19].

**Figure 3.** Damping in Resonant Column test. **Figure 3.** Damping in Resonant Column test.

**3. Results and Discussions**  Every test starts with very small strain RC measurements to obtain the maximum For small values of the damping ratio, as found in soil, <sup>√</sup> <sup>1</sup> − *<sup>D</sup>*<sup>2</sup> ∼= 1, the damping expression becomes:

$$D = \frac{\delta}{2\pi} = \frac{1}{2N\pi} \ln \frac{Z\_1}{Z\_{1+N}} \tag{3}$$

no effect of the cyclic loading on the dynamic behavior of soil [3,20,21]. Next, cyclic TOSS tests load the sample for two cycles at progressively higher stress levels (5–10–15–20–25– 30–35–40–45–50 KPa). RC tests are continued after reaching the maximum shear strain that can be measured using the proximitors (just below 1% peak-to-peak strain). The six RC tests provided a total of 115 damping measurements. This allowed for comparison with 54 data points from the TOSS test. Typical response curves for the SSV method appear in Figure 4 for ascending strain levels. At higher strain amplitudes (above 0.1%), dynamic instability prevented precise response data from being recorded by the multimeter. For the FVD method, however, very small strain levels inhibited accurate damping measurements, and relatively accurate decay-response curves required several attempts. During the disconnection from drive Ray [14] suggests using a small number of cycles (*N*) when driving at high amplitude. A larger *N* would cause a reduction in strain amplitude by a factor of about 3 over the measurement interval. ASTM D4015 [17] recommends the use of less than 10 cycles. Mog and Anbazhagan [13] conducted resonant column tests using the GCTS device to investigate the effect of the number of successive cycles (*N*) used in measuring the damping ratio. They observed an increase in the damping ratio when increasing the number of cycles up to 10 cycles. Nonetheless, beyond 10 cycles, the damping ratio starts to decrease for a higher number of cycles in the measurements (i.e., for 20, 30, and 50 cycles). Damping ratio measurements determined by this method scatter significantly. Therefore, 2 or 3 successive cycles in the decay response curve should determine the damping ratio in the RC test.

#### coils, an electric pulse often produces a strong transient. This pulse creates noise that distorts the readings of the most critical first few cycles, since damping is calculated from the 2.2.2. The Steady-State Vibration Method (SSV)

first three cycles before strain levels decrease substantially. This noise did not affect the response at strain amplitudes higher than 0.005%, and the peaks could be spotted easily for the first few cycles. Figure 5 compares the damping ratio measurements of the three methods. Initially, the SSV method (also termed the half-power bandwidth method) was created to calculate the modal damping ratio (ξ) of a structure by measuring the width of peaks in its frequency response function. However, it could be used to measure the damping ratio of soil during the RC test. In this technique, the width of the frequency-response curve near the resonance determines the logarithmic decrement (*δ*). The half-power bandwidth (∆ω) is the width of the peak where the magnitude of the frequency corresponds to 1/ √ 2 ∗ *Pmax* [18]. Equation (4) determines *δ* (GCTS-CATS, 2007):

$$\delta = \frac{\pi (f\_2^2 - f\_1^2)}{2f\_r^2} \ast \sqrt{\frac{P^2}{P\_{\max}^2 - P^2}} \frac{\sqrt{1 - 2D^2}}{1 - D^2} \tag{4}$$

where *f* <sup>1</sup> and *f* <sup>2</sup> are frequencies below and above the resonance where the strain amplitude is *P*, *Pmax* is the maximum amplitude (or resonant amplitude), and *f<sup>r</sup>* is the resonant frequency (Figure 3b).

Since the damping in soils is small and the amplitude *P* is *Pmax*/ √ 2, Equation (4) becomes:

$$
\delta \cong \frac{\pi (f\_2 - f\_1)}{f\_r} \tag{5}
$$

Then, the damping ratio can be expressed as:

$$D \cong \frac{(f\_2 - f\_1)}{2f\_r} \tag{6}$$

According to Mog and Anbazhagan [13], at strain levels below 0.005%, there can be a variation of up to 15% in the damping ratio when measured using the SSV method and FVD method with two consecutive cycles. However, the variation can be even greater (up to 50%) when a higher number of successive cycles (3, 7, and 10 cycles) are used. These results agree with Senetakis et al. [11] where a scatter of 15% was also reported. The ambient noise and the number of applied cycles during the RC test justify this scatter. A much higher number of cycles is needed in the SSV method to plot the frequency-response curve [19].

#### **3. Results and Discussions**

Every test starts with very small strain RC measurements to obtain the maximum shear modulus (*Gmax*) and minimum damping ratio (*Dmin*). The RC tests continue with ascending strain amplitudes up to the volumetric-threshold shearing strain (*γtv*), which is around 0.01% for sand. Below *γtv*, the behavior is nonlinear but still elastic and there is no effect of the cyclic loading on the dynamic behavior of soil [3,20,21]. Next, cyclic TOSS tests load the sample for two cycles at progressively higher stress levels (5–10–15–20–25–30–35–40–45–50 KPa). RC tests are continued after reaching the maximum shear strain that can be measured using the proximitors (just below 1% peak-to-peak strain). The six RC tests provided a total of 115 damping measurements. This allowed for comparison with 54 data points from the TOSS test.

Typical response curves for the SSV method appear in Figure 4 for ascending strain levels. At higher strain amplitudes (above 0.1%), dynamic instability prevented precise response data from being recorded by the multimeter. For the FVD method, however, very small strain levels inhibited accurate damping measurements, and relatively accurate decay-response curves required several attempts. During the disconnection from drive coils, an electric pulse often produces a strong transient. This pulse creates noise that distorts the readings of the most critical first few cycles, since damping is calculated from the first three cycles before strain levels decrease substantially. This noise did not affect the response at strain amplitudes higher than 0.005%, and the peaks could be spotted easily for the first few cycles. Figure 5 compares the damping ratio measurements of the three methods. *Sustainability* **2023**, *15*, x FOR PEER REVIEW 7 of 14

 (**a**) Dense sample A (**b**) Loose sample A

 (**c**) Dense sample B (**d**) Loose sample B

**Figure 4.** Response curves in the RC test for Sample C. **Figure 4.** Response curves in the RC test for Sample C.

 (**a**) Loose sample, test #5 (**b**) Dense sample, test #6

**Figure 4.** Response curves in the RC test for Sample C.

**Figure 5.** Damping results of the six tests with the three methods. **Figure 5.** Damping results of the six tests with the three methods.

The SSV method generally provides higher damping values than the FVD method, especially at higher strain levels. At the same time, there is a closer agreement between the two methods at medium strain amplitudes (0.005–0.03%). At higher strain amplitudes, the frequency-response curve becomes asymmetric, compromising the accuracy of the damping ratio calculations in the SSV method and causing an overestimation of their values. Such behavior also occurs when comparing with the TOSS tests, as damping values ob-The SSV method generally provides higher damping values than the FVD method, especially at higher strain levels. At the same time, there is a closer agreement between the two methods at medium strain amplitudes (0.005–0.03%). At higher strain amplitudes, the frequency-response curve becomes asymmetric, compromising the accuracy of the damping ratio calculations in the SSV method and causing an overestimation of their

tained from the TOSS tests agree more with the FVD method in the RC test at the strain

become difficult at strain levels below 0.02%. However, the two tests are in agreement and their combined response can generate damping-ratio curves for a wide range of strains. Multiple effects and uncertainties cause the scatter seen in the damping measurements. The sources of variability include the decreasing strain amplitude with time (FVD), response curve asymmetry (SSV), and the number of cycles and stiffening behavior in the TOSS test. Such effects cause difficulties when comparing the damping ratio obtained from different methods. As a result, they produce a scatter that may exceed 40% in the

Due to the nature of the TOSS test measurements and insufficient accuracy of the

The best fit for all the damping ratio data points from the six tests using the three methods appears in Figure 7, along with a comparison with two earlier studies [22,23] on

data when comparing the SSV and FVD methods, as seen in Figure 6.

levels where the SSV method is questionable.

dry sand.

values. Such behavior also occurs when comparing with the TOSS tests, as damping values obtained from the TOSS tests agree more with the FVD method in the RC test at the strain levels where the SSV method is questionable.

Due to the nature of the TOSS test measurements and insufficient accuracy of the proximitors at very small strain amplitudes, comparisons between the RC and TOSS tests become difficult at strain levels below 0.02%. However, the two tests are in agreement and their combined response can generate damping-ratio curves for a wide range of strains.

Multiple effects and uncertainties cause the scatter seen in the damping measurements. The sources of variability include the decreasing strain amplitude with time (FVD), response curve asymmetry (SSV), and the number of cycles and stiffening behavior in the TOSS test. Such effects cause difficulties when comparing the damping ratio obtained from different methods. As a result, they produce a scatter that may exceed 40% in the data when comparing the SSV and FVD methods, as seen in Figure 6. *Sustainability* **2023**, *15*, x FOR PEER REVIEW 9 of 14

**Figure 6.** Comparison of damping between SSV and FVD for all samples. **Figure 6.** Comparison of damping between SSV and FVD for all samples. 2

40 45 50 TOSS SSV The best fit for all the damping ratio data points from the six tests using the three methods appears in Figure 7, along with a comparison with two earlier studies [22,23] on dry sand. **Figure 6.** Comparison of damping between SSV and FVD for all samples. 0 0 5 10 15 **Damping-FVD (%)**

> The modified hyperbolic model based on the Hardin–Drnevich model provides an excellent presentation of the modulus-reduction curve. This model employs two curve-

> To estimate the damping ratio with correlations, it is common practice to relate it with the normalized shear modulus using a second-degree polynomial of (*G/Gmax*)

> > <sup>=</sup> <sup>1</sup> 1 + <sup>ቀ</sup> ቁ

The reference shear strain in this model is defined as the strain amplitude when the shear modulus reduces to one-half of *Gmax* [23]. The fit for all the samples is shown in

<sup>=</sup> <sup>1</sup> 1 + <sup>ቀ</sup> ቁ

After finding the correlation for the shear-modulus-degradation curve, the following

The reference shear strain in this model is defined as the strain amplitude when the shear modulus reduces to one-half of *Gmax* [23]. The fit for all the samples is shown in

After finding the correlation for the shear-modulus-degradation curve, the following

+ ଶ <sup>൬</sup> ௫

+ ଶ <sup>൬</sup> ௫

(7)

40%

(7)

൰+ଷ (8)

൰+ଷ (8)

 ௫

equation models the damping ratio increase with increasing strain amplitude:

 ௫

௫ ൰ ଶ

௫ ൰ ଶ

equation models the damping ratio increase with increasing strain amplitude:

where *C*1*, C*2*,* and *C*<sup>3</sup> are curve-fitting constants. The constants are found using the least square method, by minimizing the summation of the squared errors between the equation

where *C*1*, C*2*,* and *C*<sup>3</sup> are curve-fitting constants. The constants are found using the least square method, by minimizing the summation of the squared errors between the equation

=ଵ <sup>൬</sup>

=ଵ <sup>൬</sup>

with the normalized shear modulus using a second-degree polynomial of (*G/Gmax*) [11,24,25]. **Figure 7.** A fit of damping data points for each method. **Figure 7.** A fit of damping data points for each method.

*3.1. Damping Ratio Correlation* 

FVD

fitting parameters:

fitting parameters:

[11,24,25].

Figure 8a for = 0.1 and = 0.974.

Figure 8a for = 0.1 and = 0.974.

#### *3.1. Damping Ratio Correlation*

samples, the equation becomes:

To estimate the damping ratio with correlations, it is common practice to relate it with the normalized shear modulus using a second-degree polynomial of (*G/Gmax*) [11,24,25].

The modified hyperbolic model based on the Hardin–Drnevich model provides an excellent presentation of the modulus-reduction curve. This model employs two curvefitting parameters: *Sustainability* **2023**, *15*, x FOR PEER REVIEW 10 of 14

$$\frac{G}{G\_{\text{max}}} = \frac{1}{1 + \left(\frac{\gamma}{\gamma\_r}\right)^d} \tag{7}$$

The reference shear strain *γ<sup>r</sup>* in this model is defined as the strain amplitude when the shear modulus reduces to one-half of *Gmax* [23]. The fit for all the samples is shown in Figure 8a for *γ<sup>r</sup>* = 0.1 and *a* = 0.974. = 8.5 ൬ ௫ ൰ − 40 ൬ ௫ ൰ + 33 (9) Figure 8b demonstrates how the equation fits the RC-TOSS damping measurements.

ଶ

**Figure 8.** Shear modulus and damping ratio correlations fit with test data. **Figure 8.** Shear modulus and damping ratio correlations fit with test data.

*3.2. Effect of Torsional Cyclic Loading on Damping*  After exceeding the volumetric-shear-strain threshold (௧௩), the cyclic loading causes an increase in soil stiffness [26–28]. Figure 9 of the shear-stress–strain curves for 100 cycles After finding the correlation for the shear-modulus-degradation curve, the following equation models the damping ratio increase with increasing strain amplitude:

$$D = \mathcal{C}\_1 \left(\frac{G}{G\_{\text{max}}}\right)^2 + \mathcal{C}\_2 \left(\frac{G}{G\_{\text{max}}}\right) + \mathcal{C}\_3 \tag{8}$$

high number of cycles on the dynamic properties of soil. No agreement has been found regarding the effect of high-strain-amplitude cyclic loading. Thus, in this study, cyclic TOSS tests were conducted with increasing stress levels ranging between 10 KPa to 60 KPa with a maximum peak-to-peak strain amplitude of 1%. Each test applies 100 cycles where the damping ratio is calculated for all cycles. where *C*1, *C*2, and *C*<sup>3</sup> are curve-fitting constants. The constants are found using the least square method, by minimizing the summation of the squared errors between the equation and the lab measurements. For the damping ratio measurement in this study for all tested samples, the equation becomes:

$$D = 8.5 \left(\frac{G}{G\_{\text{max}}}\right)^2 - 40 \left(\frac{G}{G\_{\text{max}}}\right) + 33\tag{9}$$

Figure 8b demonstrates how the equation fits the RC-TOSS damping measurements.

#### *3.2. Effect of Torsional Cyclic Loading on Damping*

**Figure 9.** The stiffening behavior of the sample during cyclic torsional loading. The decrease in the damping ratio with the increasing number of cycles (N) at differ-After exceeding the volumetric-shear-strain threshold (*γtv*), the cyclic loading causes an increase in soil stiffness [26–28]. Figure 9 of the shear-stress–strain curves for 100 cycles shows how the hysteresis cycles decrease in size. Therefore, the damping ratio decreases with the increasing number of cycles. Several authors have detected this effect [6,29,30]. However, most studies have focused on the effect of low-strain cyclic vibration with a high number of cycles on the dynamic properties of soil. No agreement has been found regarding the effect of high-strain-amplitude cyclic loading. Thus, in this study, cyclic TOSS tests were conducted with increasing stress levels ranging between 10 KPa to 60 KPa with a maximum peak-to-peak strain amplitude of 1%. Each test applies 100 cycles where the damping ratio is calculated for all cycles.

ent stress levels is shown in Figure 10 for sample C at two different states; (a) is a loose

0

0.2

0.4

**G/Gmax [-]**

0.6

0.8

1

**Figure 9.** The stiffening behavior of the sample during cyclic torsional loading. **Figure 9.** The stiffening behavior of the sample during cyclic torsional loading. sample and (b) is a dense sample. It is difficult to estimate the rate of change for each stress

The decrease in the damping ratio with the increasing number of cycles (N) at different stress levels is shown in Figure 10 for sample C at two different states; (a) is a loose The decrease in the damping ratio with the increasing number of cycles (N) at different stress levels is shown in Figure 10 for sample C at two different states; (a) is a loose sample and (b) is a dense sample. It is difficult to estimate the rate of change for each stress level due to the sequence of the TOSS test and cyclic loading that the sample has experienced at lower stress levels. However, the higher the stress level, the higher the effect of (*N*) on damping up to a specific limit. Beyond this, the stress level seems to have a minor influence on the rate of decrease in the damping ratio. Specimens with a higher void ratio or lower relative density exhibit a more significant reduction in damping ratio. For instance, at stress levels of 20 KPa and 30 KPa, the damping ratio decreased by 50% and 54% after 100 cycles for a dense sample, while a loose sample decreased by 61% and 64% (Figure 11). *D<sup>N</sup>* is the damping ratio calculated from a cyclic TOSS test at the Nth cycle, and *D<sup>1</sup>* is the damping ratio of the first cycle (maximum damping ratio at that stress level). Results show that log(*DN/D*1) is linearly proportional to log(*N*), and the slope of log(*DN/D*1) − log(*N*) plot represents a parameter (*r*) that describes the rate of decrease in damping ratio with the number of cycles. level due to the sequence of the TOSS test and cyclic loading that the sample has experienced at lower stress levels. However, the higher the stress level, the higher the effect of (*N*) on damping up to a specific limit. Beyond this, the stress level seems to have a minor influence on the rate of decrease in the damping ratio. Specimens with a higher void ratio or lower relative density exhibit a more significant reduction in damping ratio. For instance, at stress levels of 20 KPa and 30 KPa, the damping ratio decreased by 50% and 54% after 100 cycles for a dense sample, while a loose sample decreased by 61% and 64% (Figure 11). *DN* is the damping ratio calculated from a cyclic TOSS test at the Nth cycle, and *D1* is the damping ratio of the first cycle (maximum damping ratio at that stress level). Results show that log(*DN/D*1) is linearly proportional to log(*N*), and the slope of log(*DN/D*1) − log(*N*) plot represents a parameter (*r*) that describes the rate of decrease in damping ratio with the number of cycles. = (/ଵ) () (10) The values of the parameter (*r*) also vary in a manner similar to damping ratio meas-

and the lab measurements. For the damping ratio measurement in this study for all tested

− 40 ൬

Figure 8b demonstrates how the equation fits the RC-TOSS damping measurements.

After exceeding the volumetric-shear-strain threshold (௧௩), the cyclic loading causes an increase in soil stiffness [26–28]. Figure 9 of the shear-stress–strain curves for 100 cycles shows how the hysteresis cycles decrease in size. Therefore, the damping ratio decreases with the increasing number of cycles. Several authors have detected this effect [6,29,30]. However, most studies have focused on the effect of low-strain cyclic vibration with a high number of cycles on the dynamic properties of soil. No agreement has been found regarding the effect of high-strain-amplitude cyclic loading. Thus, in this study, cyclic TOSS tests were conducted with increasing stress levels ranging between 10 KPa to 60 KPa with a maximum peak-to-peak strain amplitude of 1%. Each test applies 100 cycles where

௫

Correlation TOSS SSV FVD

0.001 0.01 0.1 1

**Shear Strain, (%)**

൰ + 33 (9)

= 8.5 ൬

 (**a**) Shear modulus (**b**) Damping ratio **Figure 8.** Shear modulus and damping ratio correlations fit with test data.

*3.2. Effect of Torsional Cyclic Loading on Damping* 

the damping ratio is calculated for all cycles.

௫ ൰ ଶ

0

10

**Damping (%)**

20

30

40

samples, the equation becomes:

H-D model Sample A, Dense Sample A, Loose Sample B, Dense Sample B, Loose Sample C, Dense Sample C, Loose

0.0001 0.001 0.01 0.1 1 10

**, p-p [%]**

menter en vyzzen: 
$$r = \frac{\log(D/D\_1)}{\log(N)}\tag{10}$$

above 0.1%.

**Figure 10.** The decrease in damping with increasing number of cycles in TOSS test. **Figure 10.** The decrease in damping with increasing number of cycles in TOSS test.

**Figure 11.** The rate of decrease in damping during cyclic loading for Sample C. **Figure 11.** The rate of decrease in damping during cyclic loading for Sample C.

**4. Conclusions**  Several methods are in use nowadays to measure the damping ratio in practical applications. The scatter in the measurements is typically present in the literature due to uncertainties in the strain amplitudes, frequency-response curve symmetry, and the effect The values of the parameter (*r*) also vary in a manner similar to damping ratio measurements, and it is difficult to find a pattern in the change in (*r*) with the increasing strain amplitude. Generally, (*r*) takes values between (−0.15 and −0.24) for strain amplitudes above 0.1%.

#### of the number of cycles. Luckily, the impact of this scatter is marginal in dynamic prob-**4. Conclusions**

lems, such as site response analysis. Previous studies have not provided the opportunity to compare the measurement methods for damping ratios in both RC and TOSS tests. However, this study benefits from the exceptional design of the device, which enables the execution of both tests on a single sample. This allows for a direct comparison of the results without the interference of external factors, disturbances, or sample preparation. Steady-State Vibration (SSV), Free Vibration Decay (FVD), and damping in the Torsional Simple Shear device have been investigated and compared for three types of Danube sands with different relative densities. Based on the results, it is recommended to use the SSV method at very low strain amplitudes (below 0.005%) where the response curve is symmetrical. For this strain range, the FVD measurements are unreliable due to the noise created by the device when turning off the drive power. At medium strain levels (0.005%– 0.03%), the two methods and the TOSS results agree with each other. However, at higher strain amplitudes (above 0.03%), it is advised to employ the FVD method to measure the damping ratio, due to the reduced accuracy of the SSV method. The SSV-response curves are not symmetrical anymore, which compromises the use of the half-bandwidth method, causing an overestimation in the damping ratio values. Achieving a more precise estimation of the damping ratio of soil leads to improving the performance and safety of structures under dynamic loads, which can contribute to the longevity of the design, reducing the overall environmental footprint associated with maintenance. The effect of the shear stiffening behavior during cyclic loading in the TOSS test was investigated. After exceeding the volumetric shear strain threshold, a considerable decrease in the damping ratio was observed. This trend is more evident in loose samples where damping can decrease by up to 64% after 100 cycles compared to a 54% decrease in Several methods are in use nowadays to measure the damping ratio in practical applications. The scatter in the measurements is typically present in the literature due to uncertainties in the strain amplitudes, frequency-response curve symmetry, and the effect of the number of cycles. Luckily, the impact of this scatter is marginal in dynamic problems, such as site response analysis. Previous studies have not provided the opportunity to compare the measurement methods for damping ratios in both RC and TOSS tests. However, this study benefits from the exceptional design of the device, which enables the execution of both tests on a single sample. This allows for a direct comparison of the results without the interference of external factors, disturbances, or sample preparation. Steady-State Vibration (SSV), Free Vibration Decay (FVD), and damping in the Torsional Simple Shear device have been investigated and compared for three types of Danube sands with different relative densities. Based on the results, it is recommended to use the SSV method at very low strain amplitudes (below 0.005%) where the response curve is symmetrical. For this strain range, the FVD measurements are unreliable due to the noise created by the device when turning off the drive power. At medium strain levels (0.005%–0.03%), the two methods and the TOSS results agree with each other. However, at higher strain amplitudes (above 0.03%), it is advised to employ the FVD method to measure the damping ratio, due to the reduced accuracy of the SSV method. The SSV-response curves are not symmetrical anymore, which compromises the use of the half-bandwidth method, causing an overestimation in the damping ratio values. Achieving a more precise estimation of the damping ratio of soil leads to improving the performance and safety of structures under dynamic loads, which can contribute to the longevity of the design, reducing the overall environmental footprint associated with maintenance.

dense samples for the same stress level of the applied cycles. Considering this decrease during the design phase can improve the analysis of the dynamic response of structures to reach a more resilient design of the infrastructure when subjected to dynamic loading, this can reduce the need for repairs or reconstruction after seismic events, thereby minimizing resource consumption and waste generation, contributing to long-term sustainability. The effect of the shear stiffening behavior during cyclic loading in the TOSS test was investigated. After exceeding the volumetric shear strain threshold, a considerable decrease in the damping ratio was observed. This trend is more evident in loose samples where damping can decrease by up to 64% after 100 cycles compared to a 54% decrease in dense samples for the same stress level of the applied cycles. Considering this decrease during the design phase can improve the analysis of the dynamic response of structures to reach a more resilient design of the infrastructure when subjected to dynamic loading, this can

reduce the need for repairs or reconstruction after seismic events, thereby minimizing resource consumption and waste generation, contributing to long-term sustainability.

**Author Contributions:** Conceptualization, M.A. and R.R.; methodology, M.A. and R.R.; software, R.R.; investigation, M.A.; resources, R.R.; writing—original draft preparation, M.A.; writing—review and editing, R.R.; visualization, M.A.; supervision, R.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is funded by the Stipendium Hungaricum Scholarship, grant number 192800.

**Data Availability Statement:** The data presented in this study can be provided by the author upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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