Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019 2.
LITERATURE REVIEW ON EQUIVALENT SINGLE LAYER THEORIES
The static and dynamic behavior of composite plates and shells can be simulated using either equivalent single layer theories or three-dimensional elasticity theories. Using suitable assumptions, equivalent single layer theories are derived from three-dimensional elasticity theories (Reddy, 2004). In general, the equivalent single layer theories account for shear deformation using certain assumptions. Equivalent Single Layer theories (ESL) can be further classified Classical Laminate Plate Theory (CLPT), First- Order Shear Deformation Theory (FSDT) and Higher Order Shear Deformation Theories (HSDT). In the context of his paper, three-dimensional theories are not discussed. Readers may look at the excellent works of Jin (Jin et al., 2015a, 2015b, 2015c), (Jin et al., 2014a, 2014b) and Su (Su, et al., 2015a, 2015b) (Su, et al., 2014a, 2014b, 2014c, 2014d) (Su, et al., 2016a, 2016b, 2016c) and (Ye et al.,2014), (Ye et al.,2015), (Ye and Jin, 2016), (Ye et al.,2016a, 2016b), (Ye et al.,2017) on three-dimensional vibrational analysis.
2.1
CLASSICAL LAMINATE PLATE THEORY (CLPT)
Classical laminate plate theory (CLPT) is the simplest of the equivalent single layer theories. CLPT which is based on Kirchhoff–Love hypothesis assumes that the straight lines remain straight and perpendicular to the midplane after deformation. Due to this shear and normal strains vanishes which in turn leads to neglecting the transverse shear and normal deformation effects (Kirchoff, 1850). Thus, the applicability of CLPT is limited to thin plates/shells which leads to erroneous solutions for thick and moderately thick plates and shells where the shear and normal deformation effects are considerable. Further, CLPT violates stress-free boundary conditions at top, bottom surfaces. It underpredicts the deflections in plates and shells and overpredicts Eigenfrequencies and buckling loads (Cosentino & Weaver, 2010). However, CLPT gives relatively good results for symmetric and balanced laminates under the effect of pure bending or pure tension (Khandan, et al., 2012). The displacement fields of CLPT may be expressed as,
= (, ) −
= (, ) −
= (, ) Where , , are displacements in , , directions
respectively., , are unknown functions of position(, ).
CLPT despite its shortcoming has been popular among researchers due to its simple form and computational inexpensive nature. Since 3D plate or shells are idealized as
(1)
2D plate or shells there is a significant reduction in the total number of variables which in turn saves a lot of computational costs. CLPT was initially propounded by Kirchhoff (Kirchhoff, 1850) and was later extended by Love (Love, 2013), Timoshenko and Goodier (Timoshenko & Goodier, 1971) and Volokh (Volokh, 1994). Volokh (Volokh, 1994) tried to enhance the classical form of CLPT by assuming the shear forces as statically equivalent to “rotated” bending and twisting moments instead of defining it as an integral over the plate thickness of the transversal shear stresses. Timoshenko and Krieger (Timoshenko & Woinowsky-Krieger, 1959), Timoshenko and Gere (Timoshenko & Gere, 1961), Dym and Shames (Dym, et al., 1973), Szilard (Szilard & Nash, 1974), Ugural (Ugural & Ugural, 1999), Ashton and Whitney (Ashton & Whitney, 1970), Ambartsumyan (Ambartsumian, 1970), Lekhnitskii (Lekhnitskii, 1968), Arkhipov (Arkhipov, 1968) and Tamurov and Grud’eva (Tamurov & Grud'eva, 1974) also made significant contributions that helped in making the theory more popular.
Reissner and Stavsky (Reissner & Stavsky, 1961) were the first researchers to apply the CLPT to heterogeneous aeolotropic elastic plates. Stavsky (Sky, 1961) use CLPT to study multilayer aeolotropic plate subject to in-plane forces and transverse loading. Dong et al. (Dong, 1962) formulated the CLPT for analysing electrostatic extension and flexure of laminated plates and shells having small thickness.
By using CLPT and including Von Karman nonlinear terms Whitney and Leissa (Whitney & Leissa, 1969) formulated the governing equations of laminated plates. They also included the inertia effect and thermal stresses. Whitney (Whitney, 1969a) further used the CLPT to study bending of simply supported rectangular plates. He also successfully modelled the effect of transverse shear deformation to predict flexural vibration frequencies and buckling loads. He then extended the theory to study anti- symmetric cross-ply and angle-ply laminates under transverse loading (Whitney, 1969b). Whitney (Whitney, 1969c) also showed the effect of bending-extensional coupling in cylindrical bending of laminated plates.
Konieczny and Wozniak (Konieczny & Wozniak, 1994) used CLPT to study composites plates of arbitrary inhomogeneous linear-elastic material. Wang et al. (Wang, et al., 1997) used CLPT to strip element method is presented to determine bending solutions of orthotropic plates. CLPT has been extensively reviewed by Vasil’ev (Vasil'Ev, 1992) for isotropic plates and by Vinson and Chou (Vinson & Chou, 1975) for anisotropic plates. The limitations of CLPT have been shown by a few researchers, notably Pagano (Pagano, 1969) (Pagano, 1970a, 1970b).
By comparing the CLPT results with the theory of elasticity solutions. Pagano (Pagano, 1969) highlighted that at low span-to-depth ratios CLPT leads to poor approximation but convergences towards an exact
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©2019: The Royal Institution of Naval Architects
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