## 3 Results and Discussion
### Maximum Displacements and Von Mises Stresses of All Three Risers  
Similarly, the time history of displacements in in-line flow direction (X), cross flow direction (Y) and von Mises stresses for all locations of risers can be acquired. According to these data, the maximum displacements and stresses at different locations of risers are obtained and presented in Figures 8-12. Figure 8a shows the maximum Ax/D for a velocity of 0.36 m/s, Figure 8b presents the maximum Ax/D for velocity of a 1.22 m/s, and Figure 8c illustrates the maximum Ax/D for a velocity of 2.13 m/s. From Figure 8, for all cases riser 3 had the maximum Ax/D and riser 1 had the minimum Ax/D. Also it could be found that the maximum Ax/D occurred above the middle point of each riser, which indicated the simple support lead to larger displacement compared with the fixed support. Based on Figure 8 and the procedure of creating it, the maximum Ax/D for risers 1, 2 and 3 for 0.36 m/s were 0.0023 at \(-\)11 m (60.6 s), 0.0039 at \(-\)11 m (74.6 s) and 0.0085 at \(-\)11 m (62.6 s), respectively. The maximum Ax/D for risers 1, 2 and 3 for 1.22 m/s were 0.0263 at \(-\)11 m (71.0 s), 0.0455 at \(-\)11 m (28.8 s) and 0.100 at \(-\)11 m (68.0 s), respectively. The maximum Ax/D for risers 1, 2 and 3 for 2.13 m/s were 0.0805 at \(-\)11 m (46.6 s), 0.140 at \(-\)11 m (39.8 s) and 0.290 at \(-\)11 m (54.4 s), respectively. Figure 9a shows the maximum Ay/D for velocity of 0.36 m/s, Figure 9b presents the maximum Ay/D for velocity of 1.22 m/s, and Figure 9c illustrates the maximum Ay/D for velocity of 2.13 m/s. Figure 8: Maximum displacements/outside diameter (Ax/D) in in-line flow direction (X) of all three risers. From Figure 9, for all cases riser 3 had the maximum ranges of Ay/D and riser 1 had the minimum ranges of Ay/D. Also it could be found that the maximum ranges of Ay/D occurred above the middle point of each riser, which indicated the simple support lead to larger vibrations compared with the fixed support. Based on Figure 9 and the procedure of creating it, the maximum ranges of Ay/D for risers 1, 2 and 3 for 0.36 m/s were 0.00048 at \(-\)11 m, 0.0019 at \(-\)11 m and 0.003 at \(-\)11 m, respectively. The maximum ranges of Ay/D for risers 1, 2 and 3 for 1.22 m/s were 0.0063 at \(-\)11 m, 0.018 at \(-\)11 m and 0.0282 at \(-\)9 m, respectively. The maximum ranges of Ay/D for risers 1, 2 and 3 for 2.13 m/s were 0.0233 at \(-\)10 m, 0.044 at \(-\)11 m and 0.076 at \(-\)11 m, respectively. Here it is noted that the maximum Ay/D and the minimum Ay/D occurred at different times, and therefore no time was measured for the maximum range of Ay/D. From Figures 8 and 9, the in-line displacement was much larger than the cross flow displacement. Based on the time when the maximum Ax/D occur for all cases, Figure 10 presents the total displacement distributions of riser 3 at those specified times. Here we have to note, the displacement distributions of risers 1 and 2 were similar to those of riser 3, and the only difference was the specific maximum displacement value for each riser. More specifically, the maximum displacements for riser 1 were 0.00071 m at 60.6 s, 0.0079 m at 71 s and 0.024 m at 46.6 s for flow velocities of 0.36 m/s, 1.22 m/s and 2.13 m/s respectively, the maximum displacements for riser 2 were 0.0013 m at 74.6 s, 0.015 m at 28.8 s and 0.046 m at 39.8 s for flow velocities of 0.36 m/s, 1.22 m/s and 2.13 m/s respectively, and the maximum displacements for riser 3 were 0.0027 m at 62.6 s, 0.031 m at 68 s and 0.090 m at 54.4 s for flow velocities of 0.36 m/s, 1.22 m/s and 2.13 m/s respectively. Figure 9: Maximum displacements cross flow direction (Y) of all three risers. Figure 11 shows the maximum von Mises stresses of all three risers for the velocity of 0.36 m/s, 1.22 m/s, and 2.13 m/s, respectively. Figure 11: Maximum von Mises stresses of all three risers. Figure 10: Maximum displacement of the riser 3: (**a**) at 62.6 s with 0.36 m/s, (**b**) at 68.0 s with 1.22 m/s and (**c**) at 54.4 s with 2.13 m/s.