# Correction of Young's Modulus Calculation in the Flexural Mode of Resonant Column Test
## 3 Calibration and Verification of Proposed Formula
After the equivalent centroid positions were averaged (i.e., one h\({}_{d}\) was used for calculation), the difference in the calculated \(E\) was greatly reduced to approximately 2%. Under the same frequency difference, the difference of \(\
u\) was \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Test Conditions and Results**} & \multicolumn{6}{c}{**Without Inertia Correction**} & \multicolumn{6}{c}{**With Inertia Correction**} \\ \hline
**Diameter** & \multicolumn{2}{c}{**Input**} & \multicolumn{2}{c}{**Test Conditions and Results**} & \multicolumn{2}{c}{\(\overline{\text{h}_{d}}\) = 2.87 mm**} & \multicolumn{2}{c}{\(\overline{\text{h}_{d}}\) = 3.01 mm**} \\ \cline{2-13}
**Diameter** & \multicolumn{2}{c}{**Input Voltage (V)**} & \multicolumn{2}{c}{**Torsional**} & \multicolumn{2}{c}{**Flexural**} & \multirow{2}{*}{_G_ (GPa)**} & \multirow{2}{*}{_E (GPa)**} & \multirow{2}{*}{_v (-)} & \multirow{2}{*}{_E (GPa)_} & \multirow{2}{*}{_v (-)} & \multirow{2}{*}{_E (GPa)_} & \multirow{2}{*}{_v (-)} & \multirow{2}{*}{_E (GPa)_} & \multirow{2}{*}{_v (-)} \\ \cline{2-13}
**(mm)** & \multicolumn{2}{c}{**(Volt)**} & \multicolumn{2}{c}{**Frequency (Hz)**} & & & & & & & & \\ \hline \multirow{3}{*}{**10**} & 0.1 & 34.5 & 17.2 & \multirow{3}{*}{26.69} & 66.29 & 0.24 & 67.71 & 0.27 \\ \cline{2-13}  & 0.02 & 34.5 & 17 & & 64.75 & 0.21 & 66.14 & 0.24 \\ \hline \multirow{3}{*}{**12.5**} & 0.02 & 53.9 & 25.7 & \multirow{3}{*}{26.72} & 60.69 & 0.14 & 62.00 & 0.16 \\ \cline{2-13}  & 0.1 & 53.9 & 25.5 & & 59.76 & 0.12 & 61.04 & 0.14 \\ \cline{2-13}  & 0.02 & 76.6 & 37.1 & & 61.03 & 0.18 & 62.40 & 0.20 \\ \cline{2-13}  & 0.1 & 76.6 & 36.7 & & 59.72 & 0.15 & 61.06 & 0.18 \\ \hline \hline \end{tabular} \({}^{*}\)\(\overline{\text{h}_{d}}\) is obtained by averaging h\({}_{d}\) of different diameters in Table 1.  
\end{table} 
Table 2: Shear modulus, Young’s modulus, and Poisson’s ratio of calibration rod from the torsional and flexural resonant column tests. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Test Conditions and Results**} & \multicolumn{6}{c}{**Without Inertia Correction**} & \multicolumn{6}{c}{**With Inertia Correction**} \\ \hline
**Diameter** & \multicolumn{2}{c}{**Input**} & \multicolumn{2}{c}{**Test Conditions and Results**} & \multicolumn{6}{c}{\(\overline{\text{h}_{d}}\) = 2.87 mm**} & \multicolumn{2}{c}{\(\overline{\text{h}_{d}}\) = 3.01 mm**} \\ \cline{2-13}
**(mm)** & \multicolumn{2}{c}{**Input Voltage (V)**} & \multicolumn{2}{c}{**Torsional**} & \multicolumn{2}{c}{**Flexural**} & \multirow{2}{*}{_G_ (GPa)**} & \multirow{2}{*}{_E (GPa)_} & \multirow{2}{*}{_v (-)} & \multirow{2}{*}{_E (GPa)_} & \multirow{2}{*}{_v (-)} & \multirow{2}{*}{_E (GPa)_} & \multirow{2}{*}{_v (-)} \\ \cline{2-13}  & **(Voll)** & **Frequency (Hz)** & & & & & & & & \\ \hline \multirow{3}{*}{**10**} & 0.1 & 34.5 & 17.2 & \multirow{3}{*}{26.69} & 66.29 & 0.24 & 67.71 & 0.27 \\ \cline{2-13}  & 0.02 & 34.5 & 17 & & 64.75 & 0.21 & 66.14 & 0.24 \\ \hline \multirow{3}{*}{**12.5**} & 0.02 & 53.9 & 25.7 & \multirow{3}{*}{26.72} & 60.69 & 0.14 & 62.00 & 0.16 \\ \cline{2-13}  & 0.1 & 53.9 & 25.5 & & 59.76 & 0.12 & 61.04 & 0.14 \\ \cline{2-13}  & 0.02 & 76.6 & 37.1 & & 61.03 & 0.18 & 62.40 & 0.20 \\ \cline{2-13}  & 0.1 & 76.6 & 36.7 & & 59.72 & 0.15 & 61.06 & 0.18 \\ \hline \hline \end{tabular} \({}^{*}\)\(\overline{\text{h}_{d}}\) is obtained by averaging h\({}_{d}\) of different diameters in Table 1.  
\end{table} 
Table 1: Flexural resonant test results of the calibration rods. approximately 13-17% without inertia correction and 12-16% with inertia correction. This finding proves that rotational inertia correction can improve the modulus estimation from the resonant test. An additional test was performed on a soil specimen to evaluate the proposed calibration formula. The soil specimen (7 cm in diameter and 15 cm in height) was made of dry Ottawa C778 sand with a density of 1.67 g/cm\({}^{2}\).