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


solution as the span-to-depth ratio increases. He also showed the limitations of the theory for sandwich plate (Pagano, 1970a) and unidirectional and angle-ply composites (Pagano, 1970b).


2.2


FIRST-ORDER SHEAR DEFORMATION THEORY (FSDT)


Due to the inherent flaws of CLPT, the first-order shear deformation theory (FSDT) was propounded by Mindlin (Mindlin, 1951). By considering a linear variation of in- plane displacements through the thickness, FSDT accounts for the shear deformation effect. The displacement fields of FSDT may be expressed as,


= (, ) +(, ) = (, ) +(, ) = (, )


(2)


Where , , are displacements in , , directions respectively;


, , are unknown functions of


position(, ); and are the rotations of a transverse normal about the -axis and -axis, respectively.


However, FSDT requires a shear correction factor. Thus, the predictions of FSDT are largely dependent on the considered shear correction factor which accounts for the strain energy of shear deformation. The shear correction factor depends on geometry, loading and boundary conditions and thus may be difficult to determine. In fact, the accurate estimation of the shear correction factor for FSDT has been a research concern by itself.


Bolle (Bolle, 1947), Hencky (Hencky, 1947), Uflyand (Uflyand, 1948), Yang et al. (Yang, et al., 1966), Whitney and Pagano (Whitney & Pagano, 1970), Qi and Knight (Qi & Knight Jr, 1996), Knight and Qi (Knight & Qi, 1997a, 1997b), Wang and Chou (Wang & Chou, 1972), Sun and Whitney (Sun & Whitney, 1973), Chow (Chow, 1971) (Chow, 1975) initiated further investigations on FSDTs.


Using energy principles Whitney (Whitney, 1973), Chatterjee and Kulkarni (Chatterjee & Kulkarni, 1979), Vlachoutsis (Vlachoutsis, 1992) presented a study on shear correction factors. They also established that multi- layered composite plates and homogeneous plates require separate values of shear correction factors. Gruttmann and Wagner (Gruttmann & Wagner, 2017) also detailed shear correction factors for layered plates and shells. FSDT is suitable for thin and moderately thick plates/shells. For thick plates, it deviates slightly from the exact solution.


It is worth mentioning here that Reissner (Reissner, 1947) (Reissner, 1945) also developed a theory that considers the shear deformation effect. However, Thai and Kim (Thai & Kim, 2015) have pointed out in a recent review that the Reissner theory is not similar to the Mindlin one. Wang et al. (Wang, et al., 2001) derived the bending relations between Mindlin and Reissner quantities to establish the


©2019: The Royal Institution of Naval Architects


differences between the two theories. The displacement variation across the thickness may or may not be linear in case of Reissner theory since it considers a linear bending stress distribution and a parabolic shear stress distribution (Wang, et al., 2001). Thai and Kim (Thai & Kim, 2015) argue that it is erroneous to refer to the Reissner theory as the FSDT since FSDT essentially implies a linear variation of the displacements through the thickness. Moreover, the normal stress is not included in the Mindlin theory (Panc, 1975).


Bhaskar and Varadan (Bhaskar & Varadan, 1993) used the combination of Navier’s approach and a Laplace transform technique to solve the equations of equilibrium. Onsy et al. (Roufaeil & Tran-Cong, 2002) presented a finite strip solution for laminated plates. Pryor and Barker (Barker & Pryor Jr, 1971) developed a finite element formulation based upon the FSDT for cross-ply symmetric and unsymmetric laminated plates. Ha (Ha, 1990) developed the finite element model for sandwich plates based on FSDT. Byun and Kapania (Byun & Kapania, 1992) used FSDT to predict interlaminar stresses in laminated plates. Dobyns (Dobyns, 1981) employed FSDT for analysis of orthotropic plates. Turvey (Turvey, 1977) presented the analyses for laminated rectangular plates using FSDT. Kabir (Kabir, 1996) presented an analytical solution to shear flexible rectangular plates with arbitrary laminations based on FSDT. Some recent applications of FSDT may be found at (Alavi & Eipakchi, 2018) (Civalek, 2017) (Pandit, et al., 2007) (Yu, et al., 2015) (Yu, et al., 2016) (Zhang, et al., 2015a) (Kalita & Haldar, 2015, 2016, 2017, 2018) (Kalita, et al., 2018a, 2018b) (Kalita, et al., 2016a, 2016b, 2016c) (Kalita, et al., 2015).


2.3


HIGHER ORDER SHEAR DEFORMATION THEORIES (HSDT)


Since accurate estimation of the shear correction factor is essential for correct prediction by FSDT, higher-order shear deformation theories (HSDT) were introduced. In HSDT, the displacement components are expanded in a power series of the thickness coordinate. In general, by including more and more terms in the expansion series, the desired accuracy may be achieved. Higher-order variations of the in-plane displacements or both in-plane and transverse displacements through the thickness are considered in higher-order shear deformation theories. Thus, in HSDT the effects of shear deformation or both shear and normal deformations are accounted for. HSDT is realized by either considering polynomial shape functions or non-polynomial shape functions.


2.3 (a) Third-order shear deformation theory (TSDT)


It was Vlasov (Vlasov, 1957), who initially developed a third-order displacement field that could satisfy the stress- free boundary conditions at the top and bottom surfaces of a plate. Jemielita (Jemielita, 1975), Krishna Murty (Krishna Murty, 1987) and Schmidt (Schmidt, 1977) were some of the first researchers to propose TSDT. However,


A-359


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