Hydroelastic responses and interactions of mega floating fuel storage modules 4

CHAPTER FOUR Verification of 8-node NC-QS Mindlin Plate Element 4.1 Overview In the previous chapter, we have introduced a modified 8-node non-conforming quadratic-serendipity (NC-QS) Mindlin plate element for modelling the floating structure as an equivalent solid plate in the hydroelastic analysis In this chapter, we shall verify the validity, convergence and accuracy of this modified plate element in predicting the hydroelastic deflection and stress resultants of VLFS Comparison studies will be made with existing numerical results obtained by Yago and Endo (1996) on a 300m × 60m × 2m VLFS as well as results of Wang and Wang (2006) for a 180m × 80m × 15m FFSF Note that Yago and Endo (1996) used the Kirchhoff plate element in modelling the VLFS whereas Wang and Wang (2006) employed the 4-node Mindlin plate element (Q4 element in short) in modelling the FFSF The dimensions, material properties and operating sea states of the VLFS (Yago and Endo, 1996) and the FFSF (Wang and Wang, 2006) are given in Table 4.1 57 Verification of 8-node NC-QS Mindlin Plate Element Table 4.1 Dimensions, material properties and operating sea states of VLFS (Yago and Endo, 1996) and FFSF (Wang and Wang, 2006) Input Data Length of structure 2L2 Width of structure 2L1 Depth of structure h Draft d VLFS (Yago and Endo, 1996) FFSF (Wang and Wang, 2006) 60m 80m 300m 2m 0m (zero draft assumption) 0.13 0.2 11.9GPa Density ρ 256.25kg/m3 Water depth H Wave direction Wavelength λ /L 15m 0.5m Young’s modulus E Poisson’s ratio ν 180m ∞ Head sea 0.4, 0.5, 0.6, 1.0 24GPa 2400kg/m3 18m Head sea 0.4, 0.6, 0.8, 1.0 We first perform a free vibration test to assess the rate of convergence of the NC-QS element We then compare the hydroelastic deflection modeled using the NC-QS element with their counterparts predicted by the Kirchhoff element and the Q4 element It is shown herein that a significant improvement in the stress resultants is achieved by the NC-QS element over the corresponding results furnished by the Q4 element The hydroelastic deflection presented hereafter is for unit wave amplitude unless otherwise stated For convenience, the numerical model that employs the Kirchhoff plate element, Q4 element and NC-QS element will, respectively, be known as Kirchhoff plate model, Q4 plate model and NC-QS plate model for brevity 58 Verification of 8-node NC-QS Mindlin Plate Element 4.2 Free Vibration Test We first perform a free vibration analysis of a flat, isotropic rectangular plate of length L2 = 4m , width L1 = 1m and two different plate thicknesses h = 0.1(2L1 ) and h = 0.01(2L1 ) in order to study the convergence of the NC-QS element and assess its performance against the Q4 element These plate dimensions were used by Wang et al (2001) The 1st and 2nd bending vibration frequencies of the Q4 element and NC-QS element are compared with the results obtained by Wang et al (2001) in Table 4.2 Note that Wang et al (2001) used the Mindlin plate model and various methods for the vibration analysis They recommended the use of Method C which is the Ritz method plus a penalty functional to satisfy the natural boundary conditions at the free edges The adopted Ritz functions used for approximating the transverse deflection and shear rotations are mathematically complete two-dimensional polynomial functions of degree p By examining the natural frequencies of the plate with h / (2 L1 ) = 0.1 and h / (2 L1 ) = 0.01 , it can be seen that both Q4 element and NC-QS element frequencies converge to the result of Ritz Method C as the number of elements increases The natural frequencies shown in bold indicate that the natural frequencies of the Mindlin plate element have converged to the results of Method C The NC-QS element is found to provide faster convergence as compared to the Q4 element One of the major drawback of the Q4 element is that it requires a large number of degree of freedoms (DOFs) in order to achieve convergence when h / (2 L1 ) gets smaller, i.e when h / (2 L1 ) = 0.01 On the other hand, the NC-QS element converges to the 59 Verification of 8-node NC-QS Mindlin Plate Element solution by Method C with lesser DOFs required when the ratio of h / (2 L1 ) = 0.01 The results obtained imply that no spurious modes are present in the NC-QS element; whereas the Q4 element contains one zero energy spurious mode The occurrence of spurious modes due to rank deficiency in its stiffness matrix is a result of using too low an order of integration Although NC-QS element also uses the reduced integration method, the additional non-conforming modes aid in overcoming the rank deficiency in its stiffness matrix Bathe (1996) claimed that elements with spurious modes will introduce large errors or lead to an unstable solution Hence, this type of elements with spurious modes is unsuitable for engineering applications Table 4.2 Comparison of vibration frequencies (rad/s) furnished by Q4 element, NC-QS element and Ritz Method C (Wang et al., 2001) 0.1 Method DOF’s 1st mode 2nd mode Ritz Method C h/ L1 p=20 1323 7803 19683 1600 7563 23103 p=10 p=15 p=20 1323 7803 19683 43923 1600 7563 23103 44943 1.334 1.338 1.334 1.333 1.334 1.334 1.334 1.338 1.338 1.338 1.343 1.339 1.338 1.338 1.338 1.338 1.338 1.338 3.682 3.719 3.675 3.669 3.125 3.681 3.682 3.711 3.711 3.711 3.764 3.719 3.713 3.712 3.710 3.710 3.710 3.710 Q4 element NC-QS element Ritz Method C 0.01 Q4 element NC-QS element 60 Verification of 8-node NC-QS Mindlin Plate Element 4.3 Hydroelastic Analysis After accessing the convergence of the NC-QS element, we will now perform hydroelastic analyses on the VLFS treated by Yago and Endo (1996) and the FFSF studied by Wang and Wang (2006) We first check the hydroelastic deflections computed from our numerical model against their counterparts obtained by Yago and Endo (1996) and Wang and Wang (2006) We then compute the stress resultants of the FFSF and compare them with those given by Wang and Wang (2006) The comparison studies on the validity and accuracy of the present numerical model in predicting the deflections and stress resultants are given in the subsequent sections 4.3.1 Verification of Numerical Model using Yago and Endo’s (1996) VLFS Hydroelastic analyses on Yago and Endo’s (1996) VLFS defined in Table 4.1 were carried out for different wavelengths λ/L = 0.4, 0.5, 0.6 and 1.0 Note that Yago and Endo (1996) employed the Kirchhoff plate element in modelling the VLFS as a solid plate The water depth is assumed to be infinite and the regular waves propagate in a head sea (i.e incident wave angle θ = π / ) Figure 4.1 shows the comparison of the hydroelastic deflections computed using our NC-QS plate model with those obtained by Yago and Endo’s Kirchhoff plate model The experimental results are also presented The hydroelastic deflections w/A shown in Fig 4.1 are taken along the centerline of the VLFS Good agreement is obtained between the present numerical results and Yago and Endo’s numerical and experimental results; thereby establishing the correctness of the NC-QS plate model in performing hydroelastic analysis 61 Verification of 8-node NC-QS Mindlin Plate Element 1.2 w/A (m/m) 1.2 0.8 0.8 0.4 0.4 0.0 (a) λ /L=0.4 0.0 1.2 (b) λ /L=0.5 w/A (m/m) 1.2 0.8 0.8 0.4 0.4 0.0 (c) λ /L=0.6 -0.5 -0.25 0.0 y/L (m) -0.25 0.5 (d) λ /L =1.0 -0.5 -0.25 -0.25 y/L (m) x/L (m) 0.5 Fig 4.1 Numerical and experimental results for Yago and Endo’s (1996) VLFS at wavelength (a) λ/L = 0.4 (b) λ/L = 0.5 (c) λ/L = 0.6 (d) λ/L = 1.0 Water depth H = ∞ (  ) Present numerical NC-QS plate model (­­­­) Yago & Endo’s numerical Kirchhoff plate model ( ) Yago & Endo’s experimental results 4.3.2 Verification of Numerical Model using Wang and Wang’s (2006) FFSF To be certain that the proposed numerical model is capable of providing accurate results for the hydroelastic analyses, we perform hydroelastic analyses on FFSF that was treated by Wang and Wang (2006) and compare the results 62 Verification of 8-node NC-QS Mindlin Plate Element 4.3.2.1 Hydroelastic Deflection We shall compare the hydroelastic deflections of the FFSF obtained from the NC-QS plate model with Wang and Wang’s (2006) Q4 plate model results The dimensions and material properties of Wang and Wang’s (2006) FFSF are given in Table 4.1 For the sea state, four wavelengths λ/L = 0.4, 0.6, 0.8 and 1.0 are considered The water depth H is assumed to be 18m and the regular waves propagate in a head sea The hydroelastic deflections w/A of the FFSF obtained from the NC-QS plate model are compared with the Q4 plate model results in Fig 4.2 It can be seen that the present numerical results agree very well with their counterparts predicted by the Q4 plate model The rigid body responses obtained from the software package WAMIT (Lee, 2006) are also included in Fig 4.2 It can be seen that rigid body responses of FFSF are obviously inaccurate since the structure is rather deformable under wave action Therefore, hydroelastic analysis is essential for such a flexible structure in order to obtain the correct dynamic response of the structure 4.3.2.2 Stress Resultants Next we compute the stress resultants of the FFSF under wave action The bending moments and shear forces at the free edges should not violate the free-edge boundary conditions as given in Eqs (2.12a-c) Comparisons of the stress resultants between the Q4 plate model and the NC-QS plate model are presented in Figs 4.3 and 4.4 The bending moment M yy and shear forces Q y are taken along the longitudinal line, i.e line parallel to the y-axis in Fig 2.1 Both Q4 and NC-QS 63 Verification of 8-node NC-QS Mindlin Plate Element elements are able to provide bending moments that satisfy the natural boundary conditions (i.e bending moment vanish along y / L = -0.5 and y / L = 0.5) Referring to Fig 4.4, the shear forces of the Q4 element are erroneous since they not vanish at the free edges because of the occurrence of shear locking phenomenon and spurious modes On the other hand, the shear forces predicted by the NC-QS plate model vanish at the free edges of the plate This shows that the addition of non-conforming modes (see Eqs B4b-c in Appendix B) and the selective integration method (see Eqs C1 and C2 of Appendix C) as suggested by Kim and Choi (1992) and Bathe (1996) aid significantly in the correct prediction of the stress resultants in the floating structures 64 1.2 0.4 0.3 w/A (m/m) w/A (m/m) 1.0 0.2 0.1 0.8 0.6 0.4 0.2 0.0 0.0 (b) λ/L = 0.6 0.0 0.2 0.4 (a) λ /L = 0.2 0.0 0.2 0.4 0.6 y/L (m/m) 0.8 1.0 y/L (m/m) 1.2 w/A (m/m) 1.6 1.2 1.0 2.0 1.6 0.8 0.8 0.4 0.0 (c) λ/L = 0.6 0.0 0.2 0.4 0.8 0.4 (a) λ/L = 1.0 0.6 y/L (m/m) 0.8 1.0 0.0 0.0 0.2 0.4 0.6 y/L (m/m) 0.8 1.0 65 Fig 4.2 Comparison of maximum deflection w furnished by NC-QS element and Q4 element for wavelength (a) λ/L = 0.4 (b) λ/L = 0.6 (c) λ/L = 0.8 (d) λ/L = 1.0 Water depth H = 18m (  ) present NC-QS plate model (­­­­) Wang and Wang’s (2006) Q4 plate model ( ) WAMIT (rigid body motion) Verification of 8-node NC-QS Mindlin Plate Element 2.0 w/A (m/m) 0.6 Verification of 8-node NC-QS Mindlin Plate Element λ/L Q4 Element NC-QS Element (a) 0.4 (b) 0.6 (c) 0.8 (d) 1.0 Fig 4.3 Comparison of bending moment Myy furnished by NC-QS element and Q4 element for wavelength (a) λ/L = 0.4 (b) λ/L = 0.6 (c) λ/L = 0.8 (d) λ/L = 1.0 66 Verification of 8-node NC-QS Mindlin Plate Element λ/L Q4 Element NC-QS Element (a) 0.4 (b) 0.6 (c) 0.8 (d) 1.0 Fig 4.4 Comparison of shear forces Qy furnished by NC-QS element and Q4 element for wavelength (a) λ/L = 0.4 (b) λ/L = 0.6 (c) λ/L = 0.8 (d) λ/L = 1.0 67 Verification of 8-node NC-QS Mindlin Plate Element 4.4 Concluding Remarks We have established the validity, convergence and accuracy of the modified 8-node NC-QS Mindlin plate element for use in predicting the deflection and stress resultants on the floating plate In the development of the modified element, non-conforming modes are added to the conventional basis functions of the 8-node element so as to remove spurious modes Moreover, in the computation of the stiffness matrix, the selective integration method was used to avoid shear locking phenomenon In the convergence tests, the results showed that the NC-QS element produced faster convergence when compared to existing Q4 element, especially when the thickness of the plate gets smaller The exceptionally good agreement between the deflection predicted using the present numerical model with that of Yago and Endo’s (1996) and Wang and Wang’s (2006) provides the confidence of employing the proposed NC-QS plate model in modelling the mega floating fuel storage modules for the hydroelastic analysis Unlike the Kirchhoff plate model, the NC-QS plate model is also proven to be applicable for modelling both thin and thick solid plate for hydroelastic analysis Stress resultants predicted by the NC-QS plate model are found to be more accurate than the Q4 plate model, especially the shear forces In sum, the NC-QS plate model provides analysts with a robust tool for the hydroelastic analysis of VLFS It takes into consideration the effects of transverse shear deformation and rotary inertia and it does not suffer from shear locking phenomenon and spurious modes In the subsequent chapters, we will adopt the 68 Verification of 8-node NC-QS Mindlin Plate Element NC-QS plate model in performing the hydroelastic analysis on the FFSF 69 ... Method DOF’s 1st mode 2nd mode Ritz Method C h/ L1 p=20 1323 7803 19683 1600 7563 23103 p=10 p=15 p=20 1323 7803 19683 43 923 1600 7563 23103 44 943 1.3 34 1.338 1.3 34 1.333 1.3 34 1.3 34 1.3 34 1.338... and Endo’s (1996) and Wang and Wang’s (2006) provides the confidence of employing the proposed NC-QS plate model in modelling the mega floating fuel storage modules for the hydroelastic analysis... -0.5 and y / L = 0.5) Referring to Fig 4. 4, the shear forces of the Q4 element are erroneous since they not vanish at the free edges because of the occurrence of shear locking phenomenon and spurious