In this section, the VIV responses of all 9 study cases are presented and discussed, and the Grey relational grade of the 5 parameters (Etension, Ebending, L/Douter, tension force and current velocity) on the amplitude of VIV in cross flow direction for the risers is determined.
3.2. Time History of Displacements and von Mises Stresses for All Study Cases
The time history of displacements in the in-line flow direction (X) and cross flow direction (Y) for all study cases are plotted in
Figure 2 and
Figure 3, respectively. For all the study cases, the results of a typical location, i.e., the middle point of each riser, are presented.
From
Figure 2, the in-line flow direction (X) displacements maintain at a relatively stable value and vibrate slightly after the initial growth for all cases. In case 3, the displacement of the middle point of the riser 1 reached 0.43 m, which is the maximum. The displacement of the middle point of riser 2 in case 5 reached 0.07 m. In cases 6 and 9, the maximum displacements were approximately 0.03 m and 0.02 m, respectively. For all the other cases, the maximum displacements were well below 0.0004 m.
As can be seen in
Figure 3, the cross-flow direction (Y) displacements in all cases demonstrate continuous vibrations. Similarly to the displacements in the in-line flow direction, the amplitude of the middle point of riser 1 in case 3 in the cross flow direction was the maximum (9.01 × 10
−2 m). Then, the amplitude of riser 2 in case 5 was 1.28 × 10
−2 m. For all the other cases, the maximum amplitudes of the middle points of risers were well below 7.1 × 10
−3 m.
The time history of von Mises stress for all study cases is plotted in
Figure 4. For all the study cases, the results of a typical location, i.e., the fixed bottom end of each riser, are shown. Note that, for cases 1 and 7, which are dominated by tension forces, the von Mises stresses are not the maximum at the bottom end.
Figure 4 indicates that the time history of von Mises stresses has the similar change trend to displacement in the cross flow direction for each case. In case 3, the maximum von Mises stress of the bottom end of the riser 1 was 1.3 × 10
8 Pa, followed by riser 3 in case 9 (3.07 × 10
7 Pa). In cases 5 and 2, the maximum von Mises stresses were approximately 2.37 × 10
7 Pa and 2.24 × 10
7 Pa, respectively. For all the other cases, the maximum von Mises stresses were below 8.1 × 10
6 Pa.
3.3. Maximum Displacements and Von Mises Stresses for All Study Cases
Similarly, the time history of displacements in the in-line flow direction (X), the cross flow direction (Y) and von Mises stresses at all locations of risers can be obtained. Based on these data, the maximum displacements and stresses at different locations of risers are presented in
Figure 5,
Figure 6,
Figure 7,
Figure 8 and
Figure 9.
Figure 5a–c shows the maximum displacements in the in-line flow direction (X) for the water depths of 12.5 m (cases 1, 6 and 8), 25 m (cases 2, 4 and 9) and 37.5 m (cases 3, 5 and 7), respectively.
From
Figure 5, deeper water depth results in larger displacement in general. However, cases 4 and 7 do not follow this rule. Also, it is clear that the maximum displacements in the in-line flow direction (X) occurs above the middle point in all the study cases, which indicates the simple support (top end) leads to larger displacement compared with the fixed support (bottom end). According to the procedure of creating
Figure 5, the maximum displacements (X) and the location and time they occur for study cases 1–9 were 2.01 × 10
−4 m (−5 m, 56.4 s), 3.03 × 10
−2 m (−11 m, 14.8 s), 4.44 × 10
−1 m (−16.5 m, 41.2 s), 1.27 × 10
−3 m (−11 m, 55.6 s), 7.18 × 10
−2 m (−16.5 m, 33 s), 3.39 × 10
−3 m (−5.5 m, 36.6 s), 3.23 × 10
−3 m (−16.5 m, 57 s), 5.07 × 10
−4 m (−5 m, 51.4 s) and 2.42 × 10
−2 m (−10 m, 46.6 s), respectively.
Figure 6a–c shows the maximum displacements in the cross flow direction (Y) for the water depths of 12.5 m (cases 1, 6 and 8), 25 m (cases 2, 4 and 9) and 37.5 m (cases 3, 5 and 7), respectively.
As can be seen in
Figure 6, it is also clear that the maximum amplitude in the cross flow direction (Y) occurs above the middle point in all the study cases, which indicates the simple support (top end) leads to larger displacement compared with the fixed support (bottom end). According to the procedure of creating
Figure 6, the maximum amplitudes in the cross flow direction and the location they occur for study cases 1–9 were 7.01 × 10
−5 m (−5 m), 7.62 × 10
−3 m (−10 m), 9.38 × 10
−2 m (−16.5 m), 4.42 × 10
−4 m (−11 m), 1.31 × 10
−2 m (−16.5 m), 1.36 × 10
−3 m (−5 m), 3.72 × 10
−4 m (−15 m), 1.40 × 10
−4 m (−5 m) and 7.00 × 10
−3 m (−10 m), respectively. It has to be noted that the maximum and minimum displacements in the cross flow direction occur at different times and therefore, no time is measured for the maximum amplitude.
Comparing the results in
Figure 5 and
Figure 6, the in-line displacement is much larger than the cross flow displacement. Based on the time when the maximum in-line displacement occurs for all cases,
Figure 7 presents the total displacement distributions of all cases at those specified times. More specifically, the maximum total displacements for cases 1–9 were 2.33 × 10
−4 m (56.4 s), 3.06 × 10
−2 m (14.8 s), 4.44 × 10
−1 m (41.2 s), 1.31 × 10
−3 m (55.6 s), 7.20 × 10
−2 m (33 s), 3.40 × 10
−3 m (36.6 s), 3.26 × 10
−3 m (57 s), 5.13 × 10
−4 m (51.4 s) and 2.42 × 10
−2 m (46.6 s), respectively.
Figure 8a–c shows the maximum von Mises stresses for the water depths of 12.5 m (cases 1, 6 and 8), 25 m (cases 2, 4 and 9) and 37.5 m (cases 3, 5 and 7), respectively.
It is noted that only the failure of steel risers (cases 7, 8 and 9) can be verified by the von Mises stresses. Based on the American Bureau of Shipping standard [
37], 67% of the yield stress of X80 steel (371 MPa) is the allowable stress. Therefore, the steel risers studied in this paper would not fail for all cases 7 (6.6 MPa), 8 (3.5 MPa) and 9 (30.7 MPa). In contrast, von Mises stresses of FRP composite risers 1 and 2 cannot be utilized to verify the failure because the global stresses cannot represent the stress distributions in each composite laminae. The global stress distributions of composite risers 1 and 2 in
Figure 8 are only used to determine the locations and time of the maximum stresses and these data can be used to verify the stress failure in each composite layer in
Section 3.4.
For all study cases, the maximum von Mises stresses occur at bottom end of all risers, except in case 1 (−2.5 m) and case 7 (top end). In the middle parts above the middle point are the locations where second maximum von Mises stresses happen. In the point which is approximately 25% of the riser length to the bottom end, the minimum von Mises stresses occur. This is because the fixed support at the bottom leads to maximum stress of the bending situation, and the middle parts have the largest deformation which also results in large stresses. In case 1, the tension force and vertical deformation together lead to maximum stresses at approximately −2.5 m, while in case 7, the tension force is dominant and the maximum stresses occur at the top end. For the part which is approximately 25% of the riser length to the bottom end, a comparatively small deformation happens and also relatively far from the fixed support, therefore, the minimum von Mises stresses occur.
Based on the time when the maximum von Mises stresses occur for all cases,
Figure 9 presents the von Mises stress distributions of all cases at those specified times. More specifically, the maximum von Mises stresses for cases 1–9 were 2.68 MPa at −2.5 m (56.4 s), 23.67 MPa at bottom end (15 s), 135.21 MPa (42.2 s), 3.00 MPa at bottom end (30.6 s), 23.37 MPa at bottom end (33 s), 8.16 MPa at bottom end (37 s), 6.02 MPa at top end (35 s), 3.36 MPa at bottom end (26.4 s) and 23.37 MPa at bottom end (46.6 s), respectively.
3.6. Grey Relational Analysis of Multiple Parameters
The parameters considered in this paper that affect the VIV amplitude in cross flow direction of the risers are E
tension, E
bending, L/D
outer, tension force and current velocity, listed in
Table 6. It is noted that, for a composite laminate, there can be a significant difference between its effective moduli in tension and bending (E
tension and E
bending). In this paper, E
tension is calculated based on the 3D effective properties of the composite tube using the 3D laminate property theory [
40,
41]. E
bending is evaluated using static analyses of the FEA models of the selected lay-ups with Solid186 (layered brick) under bending situations. More specifically, the FEA model for calculating E
bending is a cantilever pipe (30 m) under a transverse force of 1000 N and therefore, the
, where,
P is the transverse force,
L is the pipe length,
I is the moment of inertia, and
is tip displacement of the pipe.
In this paper, the Grey relational analysis (Equations (5)–(12)) is utilized to determine the Grey relational grade of the 5 parameters on the amplitude of VIV in cross flow direction for the riser.
where,
X0 is original (reference) data and
X1 to
X5 are the comparative series. More specifically,
X0 is the VIV amplitude in cross flow direction;
X1 is the E
tension;
X2 is the E
bending;
X3 is the L/D
outer;
X4 is the tension force;
X5 is the current velocity, and 1–9 in brackets are the study cases.
The detailed values for
X0 to
X5 are presented below:
For X0 to X5, the units are different and their values have a very large range. Hence, these sequences are normalized to comparability sequences using a process of Grey relational generation (Equation (6)). Here, is the mean value of Xi(n), n (1,9).
From Equation (6),
Y0 to
Y5 are:
Equation (7) is utilized to calculate the absolute difference between the original data and the comparative series.
From Equation (7), Δ
1 to Δ
5 are:
The maximum and minimum value of each data set from Equation (7) are obtained using Equations (8) and (9).
Equation (10) is utilized to calculate the Grey relational coefficient which is used to determine how close each parameter sequence is to its reference sequence. More specifically, the larger the Grey relational coefficient, the closer the variable sequence is to its reference sequence.
where
. The identification coefficient
is used to adjust the distinction between normalized reference series and comparative series. From the former study, the rank of the Grey relational grade cannot change with the change of
value and
only has an effect on the magnitude of the relational coefficient [
42]. In this paper,
, which leads to medium distinguishing effect and stability, is utilized [
43].
From Equation (11),
Z01 to
Z05 are:
Finally, Equation (12) is used to calculate the Grey relational grade (GRG), which is defined as the numerical measure of the relevancy between the reference sequence (
X0) and the comparability sequence (
X1 to
X5) and indicates the degree of similarity between the comparability sequence and the reference sequence [
44]. The higher the GRG of the comparability sequence of a parameter, the more similar this comparability sequence is to the reference sequence, i.e., the parameter from this comparability sequence affects reference sequence (VIV amplitude in crossflow direction) more significantly.
From Equation (12), .
According to the GRG value for each parameter, the flow velocity has the largest effect, then the L/Douter, followed by the tension force, and after that, the Etension, and Ebending.