# Correction of Young's Modulus Calculation in the Flexural Mode of Resonant Column Test
## 2 Analysis Method of Flexural Test
### Corrected Calibration Formula of Flexural Test  
Owing to the complex cross-section of the EDD, the product of its mass and the centroid of mass \(\overline{m}_{d}\) and the sum of the moments of inertia \(\overline{I}_{T1}\) cannot be directly obtained. A calibration block with a known mass added to the calibration rod is required to indirectly estimate \(\overline{m}_{d}\) and \(\overline{I}_{T1}\). After the flexural resonant frequency is measured under this condition (\(\omega_{2}\)), substituting the mass property and resonant frequency into Equation (13) yields:  
\[{\omega_{2}}^{2}=\frac{3 EI_{b}}{\left\{\frac{33}{140}m_{T}+\overline{m}_{a}+ \overline{m}_{c}+\overline{m}_{d}+\overline{m}_{am}+\overline{I}_{T2}\right\} L^{3}}=\frac{3 EI_{b}}{\left\{\frac{33}{140}m_{T}+\overline{m}_{a}+\overline{m}_{c}+ \overline{m}_{d}+\overline{m}_{am}+\overline{I}_{T1}+l_{am}\right\}L^{3}} \tag{15}\]  
where \(\overline{m}_{am}\) is the product of the mass of the calibration block and its centroid height; \(\overline{I}_{T2}\) is the total rotational inertia of the top plate, top cap, EDD, and calibration block; and \(l_{am}\) is the rotational inertia of the calibration block. The calibration block is aligned with an angle of \(45^{\circ}\) in respect to the vibration direction to avoid the interruption with the wire and measuring instruments. Hence, the moment of inertia of the calibration block \(l_{am}\) is calculated as follows:  
\[I_{am}=\frac{I_{y}+I_{z}}{2}+\frac{I_{y}-I_{z}}{2}\cos 2\beta-I_{yz}\sin 2\beta \tag{16}\]  
where \(I_{y}\) and \(I_{z}\) are the rotational inertia of the calibration block parallel and perpendicular to the vibration direction, respectively, and \(\beta\) is the angle with respect to the vibration direction. The product of the mass of the EDD and its centroid and the summation of the moment of inertia (\(\overline{I}_{T1}\)) can be obtained by combining Equations (14) and (15):  
\[\overline{m}_{d}+\overline{I}_{T1}=\frac{{\omega_{2}}^{2}(\overline{m}_{am}+ I_{am})-\left({\omega_{1}}^{2}-{\omega_{2}}^{2}\right)\left(\frac{33}{140}m_{T}+ \overline{m}_{a}+\overline{m}_{c}\right)}{{\omega_{1}}^{2}-{\omega_{2}}^{2}} =\overline{m}_{d}+\overline{I}_{d}+\overline{I}_{d}+\overline{I}_{c} \tag{17}\]  
where \(\overline{I}_{d}\) is the product of the rotational inertia of EDD, and constants \(\left(\frac{3}{\pi L}\right)^{2}\), and \(\overline{I}_{a}\) and \(\overline{I}_{c}\) are the products of the rotational inertia of the top plate and top cap and constant \(\left(\frac{3}{\pi L}\right)^{2}\), respectively, which can be calculated according to Equation (18):  
\[I=\frac{1}{4}mr^{2} \tag{18}\]  
where \(r\) is the radius of the top plate/cap. The mass of the EDD can be measured, but the calculation for the centroid and moment of inertia is complicated. In addition, \(\overline{m}_{d}\) and \(\overline{I}_{d}\) are also functions of mass. Therefore, the centroid function and the rotational inertia function are integrated into an equal centroid for subsequent calculations as expressed in the following formula:  
\[\overline{m}_{d}+\overline{I}_{d}=m_{d}\times f(h)+m_{d}\times f(r)=m_{d} \times(f(h)+f(r))=m_{d}\times h_{d} \tag{19}\]  
where \(m_{d}\) is the mass of the EDD, \(f(h)\) is the centroid function of the EDD, \(f(r)\) is the rotational inertia function of the EDD, and \(h_{d}\) is the equivalent centroid position of the EDD accounting for the mass inertia and rotational inertia. When \(h_{d}\) is determined by Equation (17), it is replaced with (14) to calculate the \(E\) of the calibration rod or specimen:  
\[E=\frac{{\omega_{1}}^{2}L^{3}\Big{(}\frac{33}{140}m_{T}+\overline{m}_{a}+ \overline{m}_{c}+\overline{I}_{a}+\overline{I}_{c}+m_{d}\times h_{d}\Big{)}}{3I _{b}} \tag{20}\]  
The equivalent centroid position of the EDD is obtained by using the above-mentioned calibration procedure and formula. Hence, the \(E\) of the soil from the flexural resonant frequency can be calculated using the computed value of \(h_{d}\).