Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019


Example 2: Composite square plate with central square cut-out


A simply supported, cross-ply (0/90), square laminate with thickness ratio (h/a=0.01) having square cut-outs at the center is considered (Figure. 1b). The study is made for different cut-out sizes where the edges of the cut-out are taken parallel to the edges of the plate. A mesh converge is carried out (not shown here) and in case of composite laminates, convergence is seen at 20x20. Thus, henceforth in this study, the same mesh is used unless otherwise stated. The present results are reported in Table 3 along with those of Sheikh et al. (Sheikh, et al., 2004). Material properties are considered as E1=25E2, G12= G13= 0.5E2, G23=0.2E2 and ν12=0.25.


From the above two examples, it is seen that the current finite element formulation is capable of producing highly accurate results. Thus, the same formulation is used for


analysis of composite skew plate having a skew cut-out at the center of the plate under different situations.


5.2 NUMERICAL RESULTS


Example 3: Perforated composite skew plates with and without rotary inertia.


A simply supported skew-symmetric cross-ply (0/90/0), having a skew cut-out (0.2a × 0.2b) at the plate center is considered (Figure. 1b). The plate is analysed with different thickness ratios (h/a=0.01, 0.1 and 0.2). Both types of mass lumping (MLORI and MLWRI) schemes are used. From Table 4, it is seen that for thin plate there is no effect of rotary inertia. As thickness increases, the effect of rotary inertia also increases. Percentage change of results of both the lumping schemes is also been presented in Table 4. Since the lumping scheme MLWRI is useful for both thick and thin plates, the subsequent examples have been studied for mass lumping scheme MLWRI.


Table 2. Frequencies = 2√ℎ ⁄ of a simply supported isotropic skew plate (h/a=0.01) Skew angle (α)


Source 30° 45° Present (16 × 16)


Liew and Lam (Liew & Lam, 1990) Present (16 × 16)


Liew and Lam (Liew & Lam, 1990) 2 3


24.89 25.07 34.77 34.94


52.59 52.90 66.20 66.42


First five natural frequencies 1


71.62 72.34


100.09 100.87


4


83.71 84.78


106.83 107.78


5


122.56 -


140.46 -


Table 3. Frequencies = 2√ 2⁄ /ℎ of a square laminate with square cut-out at plate center (h/a=0.01, a=b) Cut-out size


Source 0.2a × 0.2a 0.4a × 0.4b 0.6a × 0.6b Present (20 × 20)


Sheikh et al. (Sheikh, et al., 2004) Present (20 × 20)


Sheikh et al. (Sheikh, et al., 2004) Present (20 × 20)


Sheikh et al. (Sheikh, et al., 2004)


First five natural frequencies 1


2


9.11 9.12 9.09 9.09


11.14 11.11


25.41 25.50 20.41 20.30 18.51 18.54


3


25.41 25.51 20.43 20.30 18.51 18.55


4


38.00 38.04 35.48 35.46 32.71 32.94


5


53.99 54.03 44.60 44.28 34.34 34.27


Table 4. Frequencies = 2√ 2⁄ /ℎ of a simply supported, cross-ply (0/90/0) skew laminate having skew cut-out (0.2a × 0.2b) at the plate center (a=b, α = 30°)


Mass lumping MLORI*


MLWRI& % variation


MLORI (20 × 20) MLWRI (20 × 20) % variation


MLORI (20 × 20) MLWRI (20 × 20) % variation


h/a 0.01 0.1 0.2 1


15.45 15.45 0


12.70 12.58 0.95 9.60 9.43 1.8


2


First five natural frequencies 3


27.88 27.88 0


21.45 21.23 1.036 12.65 12.65 0


*Mass lumping without rotary inertia / &Mass lumping with rotary inertia ©2019: The Royal Institution of Naval Architects A-367


50.31 50.29 0


25.31 25.31 0


14,78 14.55 1.58


4


50.71 50.70 0


27.45 27.22 0.845 16.55 16.02 3.31


5


65.91 65.88 0


35.18 34.69 1.41


18.41 18.04 2.051


Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |  Page 8  |  Page 9  |  Page 10  |  Page 11  |  Page 12  |  Page 13  |  Page 14  |  Page 15  |  Page 16  |  Page 17  |  Page 18  |  Page 19  |  Page 20  |  Page 21  |  Page 22  |  Page 23  |  Page 24  |  Page 25  |  Page 26  |  Page 27  |  Page 28  |  Page 29  |  Page 30  |  Page 31  |  Page 32  |  Page 33  |  Page 34  |  Page 35  |  Page 36  |  Page 37  |  Page 38  |  Page 39  |  Page 40  |  Page 41  |  Page 42  |  Page 43  |  Page 44  |  Page 45  |  Page 46  |  Page 47  |  Page 48  |  Page 49  |  Page 50  |  Page 51  |  Page 52  |  Page 53  |  Page 54  |  Page 55  |  Page 56  |  Page 57  |  Page 58  |  Page 59  |  Page 60  |  Page 61  |  Page 62  |  Page 63  |  Page 64  |  Page 65  |  Page 66  |  Page 67  |  Page 68  |  Page 69  |  Page 70  |  Page 71  |  Page 72  |  Page 73  |  Page 74  |  Page 75  |  Page 76  |  Page 77  |  Page 78  |  Page 79  |  Page 80  |  Page 81  |  Page 82  |  Page 83  |  Page 84  |  Page 85  |  Page 86  |  Page 87  |  Page 88  |  Page 89  |  Page 90  |  Page 91  |  Page 92  |  Page 93  |  Page 94  |  Page 95  |  Page 96  |  Page 97  |  Page 98  |  Page 99  |  Page 100  |  Page 101  |  Page 102  |  Page 103  |  Page 104  |  Page 105  |  Page 106  |  Page 107  |  Page 108  |  Page 109  |  Page 110  |  Page 111  |  Page 112  |  Page 113  |  Page 114  |  Page 115  |  Page 116  |  Page 117  |  Page 118  |  Page 119  |  Page 120  |  Page 121  |  Page 122  |  Page 123  |  Page 124  |  Page 125  |  Page 126  |  Page 127  |  Page 128  |  Page 129  |  Page 130  |  Page 131  |  Page 132  |  Page 133  |  Page 134  |  Page 135  |  Page 136  |  Page 137  |  Page 138  |  Page 139  |  Page 140  |  Page 141  |  Page 142  |  Page 143  |  Page 144  |  Page 145  |  Page 146  |  Page 147  |  Page 148  |  Page 149  |  Page 150  |  Page 151  |  Page 152  |  Page 153  |  Page 154  |  Page 155  |  Page 156  |  Page 157  |  Page 158  |  Page 159  |  Page 160  |  Page 161  |  Page 162  |  Page 163  |  Page 164  |  Page 165  |  Page 166