# Correction of Young's Modulus Calculation in the Flexural Mode of Resonant Column Test
## 3 Calibration and Verification of Proposed Formula  
The GDS resonant column apparatus was utilized to perform the flexural test. The aluminum calibration rod provided by the GDS was used as a dummy specimen to verify the proposed calibration formula. Three aluminum calibration rods with different diameters (10, 12.5, and 15 mm) but similar height (140 mm) (Figure 4) were used to obtain an equivalent centroid position of the EDD. Three input voltages (0.02, 0.05, and 0.1 V) were applied for the flexural resonant test. The test results must be the same because the aluminum rod performs linearly under these voltages. For each voltage, the flexural resonant frequency was measured with and without the calibration block (0.134 kg) (Figure 4) as described in Section 2.1. In addition, the torsional mode test was performed to measure \(G\) and thus estimate \(\
u\). According to the information provided by the GDS, the properties of aluminum rods were as follows: shear modulus of 26 GPa, Young's modulus of 69 GPa, and Poisson's ratio of 0.33. Test and analysis results of the flexural mode test were reported with and without rotational inertial correction and are shown in Table 1, where f1 is the measured resonant frequency with calibration rod, f2 is the flexural resonant frequency of the calibration rod with the attached calibration block, h\({}_{d}\) is the calculated equivalent centroid position, and \(E\) is the calculated Young's modulus from the individual centroid positions obtained from each pair of tests. The individual centroid heights obtained from three input voltages were further averaged, and the final \(E\) was calculated based on this value. Table 1 shows that h\({}_{d}\) is higher with the rotational inertia correction compared with that without the inertia  
Figure 4: Aluminum calibration rods and calibration block used in this study. correction. As a result, the calculated \(E\) accounting for the rotational inertia effect was higher than that without the inertia correction, especially for the rod with a large diameter (i.e., large bending stiffness). Although the change of \(E\) was not significant with and without the inertia correction, the calculated \(E\) accounting for the rotational inertia effect was closer to the actual value (69 GPa). This observation agrees with [12] that used a different approach to calculate \(E\) and obtained a higher \(E\) compared with that calculated according to [1]. However, regardless of the correction for the inertial effect of rotation, the modulus of the aluminum calculated from the resonant frequency was lower than the actual value. Similar observations were reported by [PERSON] and [PERSON] [13], who stated that the \(E\) of the same calibration rod obtained from the resonant column test is approximately 63-65 GPa. This finding confirms the reliability of the current measurements. The measured flexural resonant frequency slightly differed for the rods with the same diameter under different voltages. The frequency difference was only 0.1-0.2 Hz, but this minor variation caused a substantial difference in the \(E\) estimation. The difference of \(E\) was 8-10% when the rotational inertia effect was not considered. By contrast, the difference of estimated \(E\) was reduced to 5-8% after the rotational inertia correction, indicating that this step can reduce the sensitivity of \(E\) estimation. The equivalent centroid height of the EDD determined from the three tests of different rod diameters was further averaged as the final adopted value in Table 2.  
Table 2 shows the comparison of the torsional mode test results and the estimated \(G\). Only the maximum and minimum resonant frequencies of the test with the same rod size from Table 1 were compared. The resonant frequency in torsional mode was more stable than that in the flexural mode. As a result, only one \(G\) value was obtained under different input voltages.