Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019
the TSDT developed by Reddy (Reddy, 1984) is the most commonly used one. The transverse shear deformation effect is considered in TSDT. It also satisfies the zero- traction boundary conditions on the top and bottom surfaces of a plate. Thus, a shear correction factor is not needed. Though the equations of motion for Reddy’s TSDT and Levinson’s theory (Levinson, 1980) are different both these theories use the same displacement field. The displacement field of Reddy’s third-order shear deformation theory may be expressed as,
= (, ) + [(, ) − 2
4 3 (
4 3 (
ℎ) ((, ) + )]
= (, ) + [(, ) − 2
ℎ) ((, ) + = (, )
)]
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.
Murthy (Murthy, 1981) developed a higher-order shear deformation theory and formulated it for unsymmetric laminates, symmetric laminates and classical orthotropy. In such similar attempts, a few TSDTs were formulated by Ambartsumian (Ambartsumian, 1960) (Ambartsumian, 1969), Librescu (Librescu, 1967), Shirakawa (Shirakawa, 1983) and Bhimaraddi & Stevens (Bhimaraddi & Stevens, 1984) among others. In 1984, Reddy (Reddy, 1984) reviewed all TSDTs proposed until then and established an equivalence among them. Phan & Reddy (Phan & Reddy, 1985) proposed a higher-order shear deformation theory that accounted for parabolic distribution of the transverse shear stresses. Pandya & Kant (Pandya & Kant, 1988) incorporated a linear variation of transverse normal strains and parabolic variation of transverse shear strains through plate thickness. A nine-node Lagrangian parabolic isoparametric plate bending element was used by them for the finite element analysis. Murty & Vellaichamy (Murty & Vellaichamy, 1988) developed a higher-order shear deformation theory with provision for cubic variation of in-plane displacements and parabolic variation of the normal displacement. Using the principle of virtual displacements Ren-Huai & Ling-Hui (Ren-Huai & Ling-Hui, 1991) developed a TSDT that accounted for parabolic variation of transverse shear strains through the thickness. Singh & Rao (Singh & Rao, 1996) developed a four node rectangular element with fourteen degrees of freedom and it used it in conjunction with a TSDT to study the effect of various parameters such as lay-up, side to thickness ratio, aspect ratio, type of loadings, boundary conditions on stability characteristics of laminated plates. Vuksanovic (Vuksanovic, 2000) developed a TSDT that could take a parabolic distribution of shear strains across the plate thickness and cubic variation for in-plane displacements.
A-360 (3)
Idlbi et al. (Idlbi, et al., 1997) in 1997 made a comparative study of CLPT, FSDT, TSDT and TSDPT (sine type), through which they concluded TSDPT to better than the others especially when interlayer continuity requirements are included. Much later Carrera (Carrera, 2007) compared three different TSDT models – one having five displacement variables, and the other two having three displacement variables. While the second TSDT was reduced from five displacement variables to three by enforcing homogeneous transverse stress conditions, the third was done so by considering non-homogeneous transverse stress conditions. He concluded that the use of non-homogeneous transverse stress conditions led to superiority of the third model over the second one. However, in general, the original model (first one) still had better estimations the other two.
2.3 (b) Trigonometric shear deformation theory (TgSDT)
As the name suggests, trigonometric functions are used to describe the shear deformation plate theories called trigonometric shear deformation theory (TgSDT). TgSDT is richer than polynomial functions, simple, more accurate and the stress-free surface boundary conditions can be guaranteed a priori (Mantari, et al., 2012). TgSDT was realized by Levy (Levy, 1877) using sinusoidal functions in the displacement field. The displacement field of the Levy’s TgSDT are as follows,
= ∑ 2+1
∑ (2+1) ℎ
=0
=0
= ∑ 2+1
∑ (2+1) ℎ
=0
=0
= ∑ 2
=0
(, ) + (, )
(, ) + (, )
(, )
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.
Other such TgSDTs have been developed for plate and shells using sine, hyperbolic sine and cosine functions. The trigonometric functions describe the warping through the thickness of the plate during rotation due to transverse shear. Kil’chevskiy (Kil'chevskiy, 1965) in his book discussed TgSDTs in detail. He solved several static and dynamic problems on shells which were used as benchmark problems till much later. However, he neglected the dissipative forces in the analysis. Stein and Jegley (Stein & Jegley, 1987) used a TgSDT and described the displacement fields using algebraic and trigonometric terms. To find the displacements and stresses they used both potential and complementary energymethods. Jegley (Jegley, 1988) in a technical report for NASA, USA studied the effects of transverse shear deformation and anisotropy on natural vibration frequencies of laminated cylinders by using a TgSDT. He also reported that the
©2019: The Royal Institution of Naval Architects (4)
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