## 3 Results and Discussion
### Stress Transfer Mechanism
#### 3.4.2 Intermediate Flange (\(\sim\)D\({}_{\text{b}}<L_{\text{f}}<L_{\text{s}}\))  
The \(L_{\text{s}}\) value (300 mm) corresponds to \(30D_{\text{b}}\) and \(15D_{\text{b}}\) for \(D_{\text{b}}=10\) and 20 mm, respectively; hence, this section for the intermediate flange length (\(\sim\)\(D_{\text{b}}<L_{\text{f}}<L_{\text{s}}\)) focuses on the stress curves for \(L_{\text{f}}=2D_{\text{b}}\), \(5D_{\text{b}}\), and \(10D_{\text{b}}\) in Figure 4b,c. The schematic of the **B**, **F**, and \(\mathbf{R_{n}}\) waves in this case is shown in Figure 5.  
In the case of a long flange (\(L_{\text{f}}>L_{\text{s}}\)) where only the **F** wave enters the incident bar without the influence of the **B** wave, the magnitude of the **F** wave is amplified according to Equation (4) (The amplitude of the transmitted tensile wave is magnified as the reflected wave from the reference plane is compressive. Particles move to the left direction at both the left and right sides of the reference plane during the passage of the stress wave). If the **F** wave approaches the reference plane within \(\Delta t\) while the **B** wave is being generated, Equations (5) and (6) cannot be applied to the **F** wave; these equations can only be applied to a case where a stress wave reaches a stress-free interface [52; 55]. Figure 5 shows that, once the **F** wave meets the **B** wave at the reference plane (after they are superposed), the plateau height in the superposition period appears to simply be the sum of the **B** and **F** waves; the magnitude of the **F** wave is not amplified (i.e., it is not influenced by the area mismatch). This phenomenon occurs because, according to Equation (4), the generation of the **B** wave (action) and **F** wave (reaction) across the impact plane is governed by the cross-sectional area mismatch during the pulse generation period. In other words, the **B** and **F** waves are simply superimposed while they are being generated under the influence of the area mismatch (Equation (4)). Therefore, when the **F** wave enters the reference plane within the pulse period (\(\Delta t\)), the superposed magnitude is simply the sum of **B** and **F** waves, and no reverberation occurs. In such a case, the **F** wave is also defined in the incident bar, as illustrated in Figures 5 and 6. In the case, where the flange length is intermediate (\(L_{\rm f}\) = 2\(D_{\rm b}\), 5\(D_{\rm b}\), and 10\(D_{\rm b}\) in Figure 4), the first and second wave increases in the stress signal are ascribed to the first-arriving **B** wave and later-arriving **F** wave, respectively. Furthermore, a longer flange increases the travel distance of the **F** wave in the flange structure, delaying the arrival time of the **F** wave. As a result, the second step starts later, and the duration of the superposition decreases with the flange length. All these interpretations are qualitatively consistent with the stress profiles shown in Figure 4.  
In the numerical analysis, the same incident bar length (2000 mm) was considered for all flange lengths. Therefore, the discretisation of the incident bar was identical for each FE model, whereas that of the flange structure varies with different FE models, as they have different flange lengths. This is why the average magnitudes, as well as the peaks and valleys of the first plateau region of the tensile pulses (before the second rises) for different curves are identical, although these are from different FE models with various \(L_{\rm f}\) values. The identical peaks and valleys verify that the **B** wave is generated at the right side of the reference plane and travels alone in the incident bar. After the superposition period, a plateau region which can be solely ascribed to the **F** wave appears in Figure 4. This plateau is the un-superposed latter part of the **F** wave that approaches the reference plane alone (i.e., without the influence of the **B** wave). This latter plateau period of the **F** wave decreases with the flange length, as the arrival of the **F** wave is further delayed.