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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Testing Device, Materials, and Methods

^{−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].

#### 2.1. Damping in the Torsional Simple Shear Device

_{L}is the area of the loop (shadowed in Figure 2a) and A

_{T}is the area of the triangle bounded by a straight line defining the secant modulus at the point of maximum strain (Figure 2a).

_{L}in Equation (1).

#### 2.2. Damping in the Resonant Column Device

#### 2.2.1. The Free Vibration Decay Method (FVD)

_{1}is the first amplitude, Z

_{1+N}is the amplitude after N cycles (Figure 3a), and D is the damping ratio.

#### 2.2.2. The Steady-State Vibration Method (SSV)

_{1}and f

_{2}are frequencies below and above the resonance where the strain amplitude is P, P

_{max}is the maximum amplitude (or resonant amplitude), and f

_{r}is the resonant frequency (Figure 3b).

## 3. Results and Discussions

_{max}) and minimum damping ratio (D

_{min}). The RC tests continue with ascending strain amplitudes up to the volumetric-threshold shearing strain (${\gamma}_{tv}$), which is around 0.01% for sand. Below ${\gamma}_{tv}$, 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.

#### 3.1. Damping Ratio Correlation

_{max}) [11,24,25].

_{max}[23]. The fit for all the samples is shown in Figure 8a for ${\gamma}_{r}$ = 0.1 and a = 0.974.

_{1}, C

_{2}, and C

_{3}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:

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

_{N}is the damping ratio calculated from a cyclic TOSS test at the N

^{th}cycle, and D

_{1}is the damping ratio of the first cycle (maximum damping ratio at that stress level). Results show that log(D

_{N}/D

_{1}) is linearly proportional to log(N), and the slope of log(D

_{N}/D

_{1}) − log(N) plot represents a parameter (r) that describes the rate of decrease in damping ratio with the number of cycles.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Sample ID | Mean Particle Diameter | Effective Particle Diameter | Uniformity Coefficient | Fines Content | Max Void Ratio | Min Void Ratio | Liquid Limit for Fines | Plastic Limit for Fines | Plastic Index for Fines |
---|---|---|---|---|---|---|---|---|---|

d_{50}[mm] | d_{10}[mm] | C_{u}[-] | FC [%] | e_{max}[-] | e_{min}[-] | w_{l}[%] | w_{p}[%] | I_{p}[%] | |

A | 0.211 | 0.109 | 2.06 | 7.56 | 0.81 | 0.52 | - | - | - |

B | 0.243 | 0.130 | 2.18 | 5.69 | 0.79 | 0.516 | - | - | - |

C | 0.107 | 0.013 | 9.85 | 21.11 | 0.9 | 0.524 | 30.4 | 19.7 | 10.7 |

Test # | Sample ID | Void Ratio | Relative Density | Maximum Shear Stress |
---|---|---|---|---|

e [-] | D_{r}[-] | G_{max}[KPa] | ||

1 | A | 0.77 | 0.14 | 84,500 |

2 | A | 0.57 | 0.83 | 115,000 |

3 | B | 0.73 | 0.22 | 79,700 |

4 | B | 0.58 | 0.77 | 101,500 |

5 | C | 0.85 | 0.13 | 69,300 |

6 | C | 0.63 | 0.72 | 88,000 |

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**MDPI and ACS Style**

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

**AMA Style**

Ahmad M, Ray R.
Damping of Dry Sand in Resonant Column-Torsional Simple Shear Device. *Sustainability*. 2023; 15(14):11060.
https://doi.org/10.3390/su151411060

**Chicago/Turabian Style**

Ahmad, Majd, and Richard Ray.
2023. "Damping of Dry Sand in Resonant Column-Torsional Simple Shear Device" *Sustainability* 15, no. 14: 11060.
https://doi.org/10.3390/su151411060