A Review of Refined Shear Deformation Theories of ...

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A Review of Refined Shear Deformation Theories of Isotropic and Anisotropic Laminated Plates Y. M. Ghugal and R. P. Shimpi Journal of Reinforced Plastics and Composites 2002; 21; 775 DOI: 10.1177/073168402128988481 The online version of this article can be found at: http://jrp.sagepub.com/cgi/content/abstract/21/9/775

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A Review of Refined Shear Deformation Theories of Isotropic and Anisotropic Laminated Plates Y. M. GHUGAL AND R. P. SHIMPI* Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai - 400 076, India ABSTRACT: A review of displacement and stress based refined theories for isotropic and anisotropic laminated plates is presented. Various equivalent single layer and layerwise theories for laminated plates are discussed together with their merits and demerits. Exact elasticity solutions for the plate problems are cited, wherever available. Various critical issues related to plate theories are presented, based on the literature reviewed. KEY WORDS: refined plate theory, shear deformation, isotropic thick plates, laminated thick plates, equivalent single layer theories, layerwise theories, elasticity solutions of plates, review of plate theories.

INTRODUCTION HE EXTENSIVE USE of fibrous composite materials in aircraft, automotive, shipbuilding and other industries has stimulated interest in the accurate prediction of the structural behaviour of laminated plates. Most of the advanced composites in use to date have low ratio of transverse shear modulus to the high inplane tensile modulus and, therefore, the transverse shear deformation plays important role in the structural analysis of composite structures. The most attractive properties of composite materials are the high strength-to-weight and high stiffness-to-weight ratios and because of many other superior physical properties of unidirectional fiber reinforced polymer matrix composites, they are excellent material for high-performance structures. The mechanics of such composite materials is widely covered in the monographs and books by Broutman and Krock [1,2], Ashton et al. [3], Calcote [4], Ashton and Whitney [5], Jones [6], Christensen [7], Tsai and Hahn [8], Agarwal and Broutman [9], Hull [10] and Daniel and Ishai [11].

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*Author to whom correspondence should be addressed.

Journal of REINFORCED PLASTICS AND COMPOSITES, Vol. 21, No. 9/2002 0731-6844/02/09 0775–39 $10.00/39 DOI:10.1106/073168402025748 © 2002 Sage Publications


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A detailed review of the literature related to the structural mechanics of composite beams, plates anf shells is presented by Bert and Francis [12], and Noor [13]; while the structural analysis of structures composed of composite material is discussed by Vinson and Chou [14], Vinson and Sierkowsky [15], Whitney [16], and Reddy [17]. The extensive use of advanced composite materials in the various high-performance structures led to the development of refined theories for analysis of such structures in order to address the correct structural behavior. The objective of this paper is to present a comprehensive review of refined theories for shear deformable isotropic and anisotropic laminated plates based on the various methods of development. Since the development of refined structural theories for laminated plates (made up from advanced fiber reinforced composite materials) have their origins in the refined theories of isotropic plates, detailed review of the theories for homogeneous, isotropic plates is also presented. DEVELOPMENT OF REFINED SHEAR DEFORMATION THEORIES Thick plate analysis is basically a three-dimensional problem. The two-dimensional theories can be derived by making suitable assumptions concerning the kinematics of deformation or the state of stress through the thickness of the plate. There are many methods available for reducing the three-dimensional equations of the theory of elasticity to two-dimensional equations in the theory of plates. The principal aspect of this reduction problem is given by Gol’denveizer [18]. These methods are grouped into five categories: 1. The method of hypotheses. (a) The semi-inverse method. (b) The method of trigonometric functions. (c) The method of full range Fourier series expansion. 2. The method of expansion. (a) The method of initial functions (b) The method of symbolic integrations 3. The asymptotic method. 4. The method of successive approximations. 5. The mixed method. In all the above methods, reduction is carried out with the use of thickness coordinate except for the asymptotic expansion in which, generally, small parameter like thickness to length ratio is used in the expansion. In the method of expansion, generally, Taylor series, MacLaurin series, Legendre polynomials and Bernstein polynomials are used.


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A Review of Refined Shear Deformation Theories

The merits and demerits of these methods relative to each other are discussed by Gol’denveizer [18], Kil’chevskiy [19], Donnell [20] and Noor and Burton [21]. THEORIES FOR ISOTROPIC PLATES Classical Plate Theory Kirchhoff [22,23] developed the well-known classical plate theory (CPT). It is based on the Kirchhoff hypothesis that straight lines normal to the undeformed midplane remain straight and normal to the deformed midplane and do not undergo thickness stretching (i.e., they are inextensible in thickness direction). In accordance with the kinematic assumptions made in the CPT all the transverse shear and transverse normal strains are zero. The displacement field of the theory is u = uo ( x, y) - z

¶w ¶x

v = vo ( x, y) - z

¶w ¶y

w = w( x, y)

where o - x - y - z is the rectangular cartesian co-ordinate system; o - x - y is the undeformed midplane; u, v, w are displacements in x, y, z directions respectively; uo, vo and w are the unknown functions of position (x,y). The boundary value problem based on this theory can be formulated in terms of transverse displacement using principle of virtual work. Kirchhoff [22,23] resolved the famous controversy concerning the nature and number of proper boundary conditions. The theory reached maturity with the addition of Kirchhoff’s boundary conditions. The classical plate bending theory permits the satisfaction of two boundary conditions per edge. An improved version of this theory is presented by Volokh [24]. Kirchhoff theory is widely used for static bending, vibrations and stability of thin plates in the area of solid structural mechanics. Various applications of this theory are presented by Love [25], Timoshenko and Krieger [26], Timoshenko and Gere [27], Dym and Shames [28], Szilard [29], Ugural [30] and many others. A brief survey of flexure of elastic plates is presented by Strel’bitskaya and Matoshko [31]. A review devoted to analysis of the classical theory of thin isotropic plates is given by Vasil’ev [32]. Since the transverse shear deformation is neglected in Kirchhoff theory, it can not be applied to thick plates wherein shear deformation effects are more significant. Thus, its suitability is limited to only thin plates.


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First-order Shear Deformation Theories First-order shear deformation theories (FSDTs) can be considered as improvements over the classical Kirchhoff thin plate theory. This is achieved by including a gross transverse shear deformation in the kinematic assumptions. The transverse shear strain is assumed to be constant across the thickness. Inclusion of average shear deformation allows the normality restrictions of the classical plate theory to be relaxed somewhat. DISPLACEMENT-BASED FIRST-ORDER SHEAR DEFORMATION THEORY The displacement field of the theory is u = uo ( x, y) + zf x ( x, y) v = vo ( x, y) + zf y ( x, y) w = w( x, y)

where uo, vo and w are the unknown functions of co-ordinates x and y; fx and fy are the rotations of a transverse normal about the y-axis and x-axis, respectively. A treatise by Panc [33] provides an exhaustive coverage of refined static theories for homogeneous isotropic plates. For the static analysis of thick homogeneous isotropic plate, Bolle [34], Hencky [35], Uflyand [36] and Mindlin [37] extended the work of Timoshenko [38,39]. Timoshenko was the first to examine both the effects of shear deformation and rotatory inertia in the analysis of beams. The Mindlin’s theory for thick plate has been extended to laminated anisotropic plates by Yang, et al. [40], Whitney and Pagano [41]. These displacement based theories are well known as first order shear deformation theories (FSDTs) in the literature. However, these theories suffer from the drawback of use of problem dependent shear correction factors. STRESS-BASED FIRST-ORDER SHEAR DEFORMATION THEORY Reissner [42,43] was the first to provide a consistent stress-based plate theory which incorporates the effect of shear deformation. The derivation given by Reissner results in displacement field of the first order shear deformation theory. This theory is further discussed by Donnell et al. [44]. Green [45] has pointed out that Reissner’s equations can be obtained directly from the three dimensional elasticity equations without recourse to variational considerations. Similar theory for isotropic thick plate is also developed by Schafer [46]. Frederick [47] applied Reissner’s theory to the bending of thick circular plates on an elastic foundation. Medwadowsky and Pister have extended Reissner’s the-



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ory to strong cylindrical bending of elastic plates [48]. Salerno and Goldberg [49] reformulated the Reissner theory in terms of a fourth- and second-order governing equations and further discussed by Volterra [50] and it is generalised by Gol’denveizer [51]. Medovikov [52] improved the theory of Reissner [53] of flexure of plates in respect of more exact boundary conditions and Berdichevskii [54] formulated it in terms of stress functions. Carley and Langhaar [55] applied it to a plate subjected uniformly distributed bending load and solution obtained is compared with experimental results. Goldenveizer et al. [56] obtained the Timoshenko-Reissner type theories of plates by asymptotic method. Speare and Kemp [57] formulated the Reissner’s [42,43,53] theory explicitly in terms of transverse displacement as the single variable, including the effects of both transverse shear stress and transverse direct stress. A single sixth order governing differential equation is obtained for the plate deflection. Medwadowsky [58] has extended Reissner’s theory to orthotropic plates. Donnell [20,59] type stress based theories are extended to laminated composites by Armanios [60,61]. Pryor et al., [62] developed a finite element formulation based upon the Reissner’s theory. Similar formulations for Reissner and Mindlin’s plate theories are given by Alliney and Carnicer [63], Chen and Pan [64]. In these theories, the transverse shear strain is assumed to be constant through the thickness, and thus shear correction factor to correct the strain energy of shear deformation is required. These factors are problem dependent. Second-Order Shear Deformation Theories The second order shear deformation theories by Naghdi [65], Pister and Westmann [66], Whitney and Sun [67], Nelson and Lorch [68] give marginally improved results vis-a-vis FSDT, but suffer from the same drawbacks of the FSDT. The displacement field of these theories can typically be described as u = uo ( x, y) + zf x ( x, y) + z2 y x ( x, y) v = vo ( x, y) + zf y ( x, y) + z2 y y ( x, y) w = w( x, y) + zf z ( x, y) + z2 y z ( x, y)

where uo, vo, wo, fx, fy, fz, yx, yy, yz are unknown functions of co-ordinates x and y. Higher-Order Shear Deformation Theories The limitations of classical plate theory and first order shear deformation theories forced the development of higher order and equivalent shear deformation the-


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ories to avoid the use of shear correction factors, to include correct cross sectional warping and to get the realistic variation of the transverse shear strains and stresses through the thickness of plate. Refined shear deformation theories based on the power series expansion for displacements with respect to the thickness coordinate, and truncating the series at required order of thickness coordinate are called the higher-order shear deformation theories. This type of series expansion was initially proposed by Basset [69]. Donnell [59], Lee and Donnell [70] obtained the solution of thick plates under normal and tangential loads respectively, applied on both the surfaces by expanding the three dimensional stresses and displacements in the form of infinite series in thickness coordinate with first term representing classical thin plate theory results. Teregulov [71] presented a general method of formulating refined theories of plates and shells which is based on the expansion of displacements, stresses, and strains in terms of thickness coordinate and use of generalized variational principle of the nonlinear elasticity. The expansion is truncated at the cubic power of thickness coordinate. Such an approach is also used by Preusser [72]. Kil’chevskiy [19] used the method of expansion based on power series in terms of thickness coordinate for plates and shells to represent the higher order displacement field. He also used the Fourier series (sine and cosine) in thickness coordinate in the displacement field for the analysis of plates and shells. Poniatovskii [73,74] developed the refined plate theories without assumptions about the nature of the deformations of the transverse linear elements; by expanding the stresses in a series of Legendre polynomials in the thickness coordinate and using the Castigliano principle. The higher order theories based on series expansions are developed by Donnell [59], Reissner [75], Provan and Koeller [76], Lo et al. [77–79] and are modified by Levinson [80], Murthy [81], Reddy [82], Blocki [83] to get the parabolic shear stress distribution through the thickness of plate and to satisfy the shear stress free surface conditions on the top and bottom surfaces of the plate to avoid the need of shear correction factors. Lo et al. type theory has been discussed by Christensen [7]. The displacement field of the third order shear deformation theory is u = uo ( x, y) + zf x ( x, y) + z2 y x ( x, y) + z3 x x ( x, y) v = vo ( x, y) + zf y ( x, y) + z2 y y ( x, y) + z3 x y ( x, y) w = w( x, y) + zf z ( x, y) + z2 y z ( x, y)

where uo, vo, wo, fx, fy, fz, yx, yy, yz, xx, xy, are the unknown functions of co-ordinates x and y. Krenk [84] constructed a hierarchy of plate theories in systematic manner utilizing expansions of orthogonal polynomials and, as a special



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case, also developed the third order parabolic shear deformation theory. Krenk’s work is very similar in concept to that of Lo et al. [78,79]. Kelkel [85] developed a third order refined plate theory using orthogonal polynomials. Lisitsyn [86] obtained approximate analytical solutions of the three-dimensional problem of the theory of elasticity, for plates rigidly clamped along the edges using Legendre polynomials, i.e., orthogonal power polynomials. See also the work of Lisitsyn and Krivenko [87]. Kant [88] presented a third-order refined shear deformation theory for analysis of thick elastic plates which gives rise to a twelfth order governing differential equation system. Shvabyuk [89] developed the generalised parabolic shear deformation theory for thick isotropic plates. Lee et al. [90] used Reddy’s [82] higher order shear deformation theory for the nonlinear analysis of thick isotropic plate. Hanna and Leissa [91] developed a refined theory for thick plate based on the Levinson’s theory, containing four displacement variables. The theory allows the quadratic variation of transverse displacement. There is no critical evaluation of variationally consistent and inconsistent refined shear deformation theories (see Levinson [92]). REFINED THEORIES BUILT UPON CLASSICAL PLATE THEORY Third order theories built upon classical plate theories and satisfying the requirement of shear stress free conditions at the top and bottom surfaces of the plate are developed by Panc [33], Kromm [93], Vlasov [94], Narasimhamurthy [95,96], Panc [97], Kaczkowsky [98], Reissner [99,100], Krishna Murty [101–103], Lewinski [104,105], Krishna Murty and Vellaichamy [106], Soldatos [107], Savithri and Varadan [108], and are revieved and generalised by Reddy [109] and Jemielita [110,111]. In these theories through the thickness shear stress distribution is parabolic. The typical displacement field of such theories is u = uo ( x, y) - z

é 4 æ zö2 ù ¶w + z ê1 - ç ÷ ú f x ( x, y) è ø ¶x ëê 3 h ûú

v = vo ( x, y) - z

é 4 æ zö2 ù ¶w + z ê1 - ç ÷ ú f y ( x, y) ¶y êë 3 è h ø úû w = w( x, y)

where uo, vo and w are the uknown functions; and fx and fy are the rotations of a transverse normal about the y-axis and x-axis, respectively. Refined theories built upon classical plate theory, upto fifth order in the thick-


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ness coordinate are examined by Bhimaraddi and Stevens [112,113]. Nair et al. [114,115] presented parabolic shear deformation theory with linear variation of inplane displacement and quadratic variation of transverse displacement in thickness coordinate. THEORIES BASED ON SEMI-INVERSE METHOD Kromm [93], Ambartsumyan [116–119], Panc [97] and Reissner [120–123] basically used the semi-inverse method to develop the refined shear deformation theories. It is based on assuming the distribution of the transverse shear stresses and using the constitutive relations to derive expressions for the inplane displacements which are nonlinear in the thickness coordinate. Reissner [100] improved earlier refined plate theory (Reissner [99]) by including the effect of transverse normal strain. This semi-inverse approach is extended for isotropic thick plates by Gevorkyan [124], Voyiadjis and Baluch [125], Baluch et al. [126], Voyiadjis et al. [127], Voyiadjis and Pecquet [128], Voyiadjis and Kattan [129,130], and generalised by Muhammad et al. [131] for bending of thick plates. Jemielita [111] presented the direct and variational methods in forming theories of plates. In all these refined theories transverse shear stress distribution is parabolic and satisfies the zero shear stress conditions at the top and bottom surfaces of the plate. THEORIES BASED ON REISSNER’S MIXED VARIATIONAL PRINCIPLE Reissner [132,133] originally developed the mixed variational principle and further modified it to obtain the variationally consistent refined shear deformation theories (see Reissner [134–142], Pagano [143]) Reissner [99] obtained a twelfth-order thick plate theory for plate transverse bending including the effect of transverse shear deformation. Reissner [121] derived another twelfth-order thick plate theory by using a mixed variational theorem and obtained equilibrium equations and boundary conditions. The twelfth order plate theory is identical for the isotropic case to Lo et al.’s [77] third order displacement based theory with six displacement variables. Chen and Archer [144], Lewinski [145] obtained the twelfth-order theory for moderately thick plates using the Lo et al. [77] type third order displacement field. Based on a mixed variational theorem, He [146] derived the twelfth-order theory for thick isotropic plate for the three components of displacements and transverse normal stress in plate problems. His displacement field is built upon the classical plate theory. Prusakov [147] developed the three versions of the twelfth order bending theory using the Legendre polynomials in terms of the transverse coordinate and the Hu-Washizu, Reissner and a combination of Lagrange and Reissner variational principles.



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THEORIES BASED ON SPECIAL REDUCTION TECHNIQUES OF 3-D ELASTICITY EQUATIONS Cheng [148] presented a method for the solution of three dimensional elasticity equations, and with the method deduced directly a refined theory of plates from the exact three dimensional elasticity theory without employing ad hoc assumptions. However, transverse loading is not considered in the development. Cheng used Boussinesq-Galerkin solution of Naviers’s equations to derive the refined theory. Barret and Ellis [149] extended the Cheng’s work for the isotropic plate subjected to a transverse loading and Wang [150,151] extended it for transversely isotropic i.e., orthotropic plates. Wang and Shi [152] derived a new thick plate theory using Papkovich-Neuber solution of three dimensional elasticity. Bisegna and Sacco [153] proposed a general procedure to deduce wide class of plate theories from the three-dimensional linear theory of elasticity by imposing suitable constraints on the strain and stress field and using the standard and powerful Lagrange multipliers. They rationally deduced FSDT and Lo et al. type higher order shear deformation theories from 3-D elasticity theory. Piltner [154] presented a solution of thick isotropic plate using method of complex variables and complex functions. Piltner [155] presented the generalized Trefftz method for the solution of isotropic and anisotropic thick elastic plates. THEORIES BASED ON METHOD OF INITIAL FUNCTIONS Vlasov [156,157], Vlasov and Leont’ev [158] initially developed refined plate theories using the method of initial functions in which the displacements and transverse stresses are expanded in a Taylor series or Maclaurin series in the thickness coordinate. It is an approach which asymptotically converges to that of three dimensional linear elasticity. This method is used by Iyengar et al. [159] for thick isotropic plates. Babadzhanyan et al. [160] used the method of initial functions to develop the refined shear shear deformation theory of plate bending for isotropic plate to satisfy three natural boundary conditions. Bahar [161] adopted a state space approach and Rao and Das [162] adopted a matrix approach to the method of initial functions. Iyengar and Pandya [163,164], Sun and Archer [165], Vlasov [166] applied it to orthotropic laminated plates. THEORIES BASED ON LURE’S METHOD OF SYMBOLIC INTEGRATION OF EQUILIBRIUM EQUATIONS Lur’e [167] suggested a method for the symbolic integration of equations of the theory of elasticity. It is a general power series method of solution of elasticity equations. Aksentian and Vorovich [168] used this method for the bending analysis of isotropic, homogeneous thin plate and Lekhnitskii [169], Nigul [170], Prokopov [171], Vorovich and Malkina [172], Gruzdev and Prokopov [173,174], Ustinov [175], Igarashi and Takizawa [176], Igarashi [177] applied it for the bend-


784


ing analysis of isotropic thick plates. Igarashi et al. [178,179] employed it for the bending analysis of an anisotropic thick plate. THEORIES BASED ON ASYMPTOTIC METHOD OF INTEGRATION OF EQUILIBRIUM EQUATIONS One of the effective methods of constructing two-dimensional equations of the theory of plates and shells on the basis of three-dimensional equations of the theory of elasticity is the method of asymptotic integration developed by Gol’denveizer [18,51,56,180]. In this method the small, positive, dimensionaless parameter such as ratio of thickness to length i.e., the slenderness parameter, is generally used to reduce the three-dimensional elasticity equations to recursive sets of two-dimensional equations, i.e., the boundary value problems, governing the interior and edge zone responses of the plate. The boundary value problems in this method have an iterative character. This method is used by Friedrichs and Dressler [181], Kolos [182], Souchet [183], Shamrovskii [184], Niordson [185], Zimmermann [186], Gregory and Wan [187], Kaplunov and Nol’de [188] for the analysis of isotropic plates, and Gusein-Zade [189], Zakharov [190], Volokh [191], Savoia et al. [192], Hodges et al. [193], Sutyrin and Hodges [194], Sutyrin [195], Tarn [196] used it for the analysis of laminated plates. Trigonometric Shear Deformation Theory There exists another class of refined shear deformation theories, wherein use of trigonometric function is made to take into account shear deformation effects. Levy [197] developed a refined theory for thick isotropic plate for the first time using sinusoidal functions in the displacement field. The displacement field of the theory is as follows: u= v=

N

N

n=0

n=0

N

N

n=0

n=0

å z2n +1un (x, y) + å sin å z2n +1vn (x, y) + å sin w=

(2n + 1)pz f x ( x, y) h (2n + 1)pz f y ( x, y) h

N

å z2n wn (x, y)

n=0

where un, vn and wn are the unknown functions; fx and fy are the rotations of a transverse normal about the y-axis and x-axis, respectively. Stein [198,199], Stein et al. [200], also proposed such theories and applied



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them to isotropic plates in the modified form. Kil’chevskiy [19], Donnell [20], Panc [33], Green [45], Kromm [93], Vorovich and Malkina [172], Voin [201] also provided solution of thick plates using sine functions in thickness coordinate. The use of sine functions can also be found in the work of Cheng [148], Matveev et al. [202], Pecastaings [203], Ladeveze and Pecastaings [204] and Chen and Rajagopal [205]. The theories containing trigonometric functions involving thickness coordinate in the displacement fields are designated as trigonometric shear deformation theories (TSDTs).

LAMINATE THEORIES The various theories originally developed for homogeneous isotropic thin, moderately thick and very thick plates are extended to the laminated anisotropic plates and also new refined shear deformation theories are developed for laminated plates for the accurate analysis of anisotropic laminated composites. These theories are basically divided into two catagories: 1. Equivalent single layer theories; and 2. Discrete layer theories or layerwise theories. Equivalent Single Layer Theories The historical account of analysis of anisotropic plates utilising classical plate theory and shear deformable plate theories is given by Vinson and Chou [14]. The theories adopting the equivalent single layer approach, are developed by expanding the displacement field in a power series expansion through the thickness. Reviews of these theories have been given by Bert and Francis [12], Noor and Burton [21], Grigolyuk and Kogan [206], Reissner [207], Reddy and Chandrashekhara [208], Reddy [109,209–214], Reddy and Robbins [215], Kapania and Raciti [216], Nemish and Khoma [217,218], Mallikarjuna and Kant [219], Noor, Burton and Bert [220], Liu and Li [221], Liew et al. [222], and Varadan and Bhaskar [223]. It is apparent from the reviews that many of the higher order shear deformation theories available in the literature can be viewed as special cases of the third order shear deformation theory of Lo et al. [77]. CLASSICAL LAMINATE PLATE THEORY (CLPT) The classical laminate plate theory is based on the Kirchhoff hypothesis that straight lines normal to the undeformed midplane remain straight and normal to the deformed midplane and do not undergo stretching in thickness direction. These assumptions imply the vanishing of the transverse shear and transverse normal strains.


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The classical laminate theory has been used in the stress analysis of composite plates. However, it is only accurate for thin composite laminates. The laminate plate theories based on the Kirchhoff’s [22,23] hypothesis have been developed by Ashton and Whitney [5], Ambartsumyan [119], Reissner and Stavsky [224], Stavsky [225], Dong, et al. [226], Lekhnitskii [227], Arkhipov [228], Whitney and Leissa [229], Whitney [230–232], Tamurov and Grud’eva [233] and are summarised by Whitney [16] and Reddy [17]. Konieczny and Wozniak [234] proposed a simple method of formulation of 2D-theories for composite thin plates made up of an arbitrary inhomogeneous linear elastic material. Wang et al. [235] used the finite strip method based on the CLPT to determine bending solutions of orthotropic plates. Pagano [236–238] showed the inadequacy of the classical laminate plate theory (CLPT) for the analysis of thick laminates as the theory is based on a linear displacement across the entire laminate with shear deformation neglected.

FIRST-ORDER SHEAR DEFORMATION LAMINATE THEORIES (FSDLTS) The laminated plate versions of the Mindlin and Reissner plate theories are due to Yang et al. [40], Whitney and Pagano [41], Reissner [239], Qi and Knight [240], Knight and Qi [241,242], Wang and Chou [243], Sun and Whitney [244], Chow [245,246]. Pryor and Barker [247] developed a finite element formulation based upon the FSDLT for cross-ply symmetric and unsymmetric laminated plates. Hinton [248], Reddy and Chao [249], Owen and Li [250], Rolfes and Rohwer [251] developed finite elements based on FSDLT for composite laminated plates. Ha [252] developed the finite element model for sandwich plates based on FSDLT. Byun and Kapania [253] used FSDLT and Chebysev and a class of orthogonal polynomials to predict interlaminar stresses in laminated plates. Dobyns [254] employed it for static and dynamic analysis of orthotropic plates. Turvey [255] presented the exact (closed-form) and approximate analyses for the flexure of angle-ply and cross-ply laminated rectangular plates based on FSDLT. Kabir [256] presented analytical solution to shear flexible rectangular plates with arbitrary laminations based on FSDLT. In FSDLT, however, shear correction factor is required to account for strain energy of shear deformation. These factors depend upon the constituent ply properties, ply layup, fiber orientation, boundary conditions, and the particular application. The procedure to find these problem dependent factors is presented by Whitney [257], Chatterjee and Kulkarni [258]. Vlachoutsis [259] presented a study on shear correction factors with typical applications using energy principles. It is shown that these factors for multilayered composite plates are different from those of homogeneous plates.



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SECOND-ORDER SHEAR DEFORMATION LAMINATE THEORIES (SSDLTS) Nelson and Lorch [68], Whitney and Sun [67] presented a second-order refined structural theory for the static and dynamic analysis of laminated orthotropic plates. Engblom and Ochoa [260] used SSDLT to develop finite element formulation for the static analysis of cross-ply laminated plates. Kwon and Akin [261] developed a parabolic shear deformation theory without shear correction factor and developed the mixed finite element formulation. Fares [262] presented the second-order shear deformation theory for free vibration analysis of anisotropic plates. Tessler [263] developed a new two-dimensional laminate plate theory for the linear elastostatic analysis of thick composite plates (inplane displacements are linear and transverse displacement is quadratic in thickness coordinate). Sadek [264] developed a serendipity finite element for the analysis of laminated plates based on SSDLT. Most of the SSDLTs suffers from the use of shear correction factors as required in FSDLT. HIGHER-ORDER SHEAR DEFORMATION LAMINATE THEORIES Various refined plate models for symmetric and unsymmetric thick laminates can be derived from the theory of Lo, Christensen and Wu [77,78] to capture the structural behaviour of laminate subjected to transverse loads. Lo et al. type theory is more suitable for C0 type finite element formulations. The finite element formulations based on the higher-order shear deformation theories (HSDT) suitable for symmetric and unsymmetric laminated composites are developed by Kant and Pandya [265], Pandya and Kant [266–269], Mallikarjuna and Kant [270], Kant and Mallikarjuna [271], Chomkwah and Avula [272], Kant and Manjunatha [273], Kant and Kommineni [274], Chang [275], Chang and Huang [276], Paul and Rao [277,278], Mohan et al. [279], Maiti and Sinha [280,281], Sadek [265]. Reddy [212] presented the review of finite element models of the continuum based theories and two-dimensional plate theories used in the analysis of composite laminates. Rohwer [282] presented a comparative study of seven different refined theories with the advantages and disadvantages of the respective higher order theories. Basar [283] presented comparative study of CLPT, FSDT, HSDT, layerwise FSDT and exact theory for composite laminates. Tseng and Wang [284] developed a finite strip formulation based on the eleven variables theory of Lo et al. [77]. Soldatos and Watson [285] proposed a method for improving the stress analysis performance of one- and two-dimensional theories for laminated plates. The maximum number of independent variables varies upto twelve in the Lo et al. type theory depending upon the choice of displacement functions for the symmetric and unsymmetric laminates.


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PARABOLIC SHEAR DEFORMATION THEORIES The third order parabolic shear deformation theories for the bending analysis of thick laminated plates are developed by Murthy [81], Krishna Murty [103], Krishna Murty and Vellaichamy [106], Bhimaraddi and Stevens [112], Nair et al. [115], Ambartsumyan [116,117,119], Librescu [286], Shirakawa [287], Reddy [82,288,289], Reddy and Phan [290], Liu and He [291]. Reddy [209,292] developed the mixed variational formulation for CLT, FSDT and HSDT (3rd order) shear deformation theories of laminated composite plates. The displacement field of Reddy’s third order shear deformation theory is as 2 é ¶w ö ù 4æ zö æ u = uo ( x, y) + z êf x ( x, y) - ç ÷ ç f x ( x, y) + ÷ ú ¶x ø ûú 3 è hø è ëê 2 é ¶w ö ù 4æ zö æ v = vo ( x, y) + z êf y ( x, y) - ç ÷ ç f y ( x, y) + ÷ ú ¶y ø úû 3 è hø è êë

w = w( x, y)

where uo, vo and w are the unknown functions; fx and fy are the rotations of a transverse normal about the y-axis and x-axis, respectively. Doong et al. [293] developed a new seven variables, parabolic shear deformation theory (PSDT) from the Lo et al. [77] theory. Rychter [294] presented a family of accurate plate theories for anisotropic and composite plates. Sun and Shi [295] presented a third order shear deformation theory based on the classical laminate plate theory. Work of Savithri and Varadan [296] also belongs to this class of theories. Carvelli and Savoia [297] investigated the accuracies of laminated plate theories for the analysis of angle-ply multilayered plates using classical laminate plate theory (CLPT), first order shear deformation theory (FSDT) and Reddy’s [82] higher order shear deformation theory (HSDT). The analytical and finite element solutions using parabolic shear deformation theories are presented by many research workers (e.g., Reddy [82], Phan and Reddy [290,298], Khdeir et al. [299], Khdeir and Reddy [300], Shankara and Iyengar [301], Shu and Sun [302]). Bose and Reddy [303,304] presented analytical and finite element solutions using the unified third order plate theory. Osternik and Barg [305], Peshtmaldzhyan [306], Rasteryaev and Prusakov [307], Andreev and Nemirovskii [308], Piskunov et al. [309], Rehfield and Valisetty [310], Valisetty and Rehfield [311], Rasskazov et al. [312], Ren [313–315], Voyiadjis and Baluch [316] developed the refined parabolic shear deformation theory based on the semi-inverse method for thick composite plates. Jalali and Taheri [317] used the semi-inverse method to develop an analytical semi-exact analysis of laminates in conjection with Laplace transformation.



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Stepanenko [318] proposed third order refined theory including transverse shear and transverse normal effects for isotropic and orthotropic plates. Mukoed [319] presented the geometrically nonlinear analysis of orthotropic plates using first-order (Reissner-Mindlin type) and third-order (Ambartsumyan type) theories. In all the equivalent single layer (ESL) theories, transverse shear stresses obtained by using the constitutive relations are discontinuous at interfaces between layers which is contradictory against the equilibrium conditions. Thus, many authors (e.g., Lo et al. [79]) suggested the use of three-dimensional equilibrium equations to obtain the transverse shear and transverse normal stresses through the thickness of the laminate. QUASI-THREE DIMENSIONAL THEORIES Ulyashina [320], Jakobsen and Jensen [321] developed the quasi-three dimensional theories based on the simplifying assumptions regarding the distribution of the stress or strain state in the laminate or in the individual layers without making a priori assumptions about the distribution of the different response quantities in the thickness direction.

Trigonometric Shear Deformation Theory The refined shear deformation theories for laminated plates using sine, hyperbolic sine and cosine functions (i.e., trigonometric functions) to describe the warping through the thickness of the plate during rotation due to transverse shear or in other words, to take into account transverse shear deformation effects, are developed by Kil’chevskiy [19], Bhimaraddi and Stevens [113], Stein [198], Stein et al. [200], Stein and Jegley [322], Jegley [323], Stein and Bains [324], Touratier [325–327], Soldatos [328], Beakou and Touratier [329], Becker [330,331], Muller and Touratier [332], Idlbi et al. [333], Touratier and Faye [334], and Shimpi and Ghugal [335]. Refined shear deformation theories for laminated plate (e.g., Shimpi and Ghugal [335]) which utilise the constitutive relations to obtain meaningful transverse shear stresses, which are single valued at layer interfaces, are rare in the literature.

Layerwise Theories or Discrete Layer Theories To overcome the drawbacks of equivalent single layer theories, layerwise theories have been proposed. These theories are divided into two catagories depending on the number of unknowns in the kinematic model.


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LAYER DEPENDENT THEORIES The theories, wherein the number of unknowns in the model is dependent on the number of layers, are developed by Srinivas [336], Seide [337], Toledano and Murakami [338], Cho et al. [339], Lee and Liu [340], Reddy [341], Reddy et al. [342], Barbero et al. [343], and Robbins and Reddy [344], Zhu and Lam [345]. These theories are often prohibitively expensive for multilayer laminates. Reddy [341], Robbins and Reddy [344], Soldatos [346], Gaudenzi [347], Gaudenzi et al. [348] and Negishi and Hirashima [349] proposed a generalized laminate plate theory that could account for any degree of approximation of the distribution of the inplane and transverse displacement through a proper selection of variables and functions. Many well-known plate theories can be derived as special cases of this theory. Eventhough these theories are the most accurate ones, they are computationally inefficient. Zhu and Lam [345] used the Rayleigh-Ritz solution for local stresses in composite laminates. He used the cubic spline functions in the displacement field. LAYER INDEPENDENT THEORIES Discrete layer theories in which the number of unknowns in the model does not depend on the number of layers in the laminate are called as layer independent theories. These theories are further subdivided into two classes as 1. First-Order Discrete Layer Theories: These theories are also called as first-order zig-zag or layerwise theories. In these theories a piecewise linear displacement function is superimposed over a linear displacement field. The theories of Mau [350], Chou and Carleone [351], Durocher and Solecki [352], Di Sciuva [353,354], Murakami [355], He et al. [356], Schmidt and Librescu [357], Bisegna and Sacco [358] belong to this catagory. Most of the authors of these theories used the zig-zag functios or the Heaviside unit function to get the kinky, zigzag distribution of inplane displacements through the thikness of laminate since this type of distribution is given by exact elasticity solutions obtained by Pagano [236,237] for thick laminated plates. However, due to the low order of their assumed displacement field, the transverse shear stresses are constant through the thickness in these theories. These theories are counterpart of the single layer first-order shear deformation theories. 2. Refined or Higher-order Discrete Layer Theories: Liu and Li [221] showed that the zig-zag form of inplane displacement distribution through the thickness cannot be represented by simply increasing the order of the polynomial series as is usually done in equivalent single layer theories. Hence, a new class of discrete layer theories has been developed by superimposing a piecewise cubic varying displacement field over a linear displacement field. Such theories are developed by Whitney [230], Ren [313–315], Toledano and Murakami [359], Di Sciuva [360,361], Di Sciuva and Icardi [362], Bhaskar and Varadan



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[363], Savithri and Varadan [364–366], Reddy et al. [367], Cho and Parmerter [368,369], Cho and Kim [370], Liu et al. [371], Icardi [372], Lee et al. [373], Lee et al. [374], Lee and Cao [375], He and Zhang [376,377]. Moazzami and Sandhu [378] proposed a second order layerwise laminated plate theory. Layerwise theories, using Lo et al. theory, are developed for the analysis of laminated plates by Wu and Kuo [379], Wu and Hsu [380]. Layerwise theories based on the semi-inverse method are developed by Ambartsumyan [119], Osternik and Barg [305], Prisyazhnyuk and Piskunov [381] for the multilayered anisotropic plates. Karama et al. [382] developed a layerwise trigonometric shear deformation theory using zig-zag function. These layerwise theories are highly cumbersome and computationally more demanding because of more number of unknown variables. Furthermore, in these theories post-processing analysis based on equilibrium equations is required to obtain all the transverse stresses, and this is a tedious task. There are very few theories that use constitutive relations to obtain transverse shear stresses. 3. Other Higher-order Layerwise Theories: Biot [383], Khoroshun [384,385], Khoroshun and Patlashenko [386], Patlashenko [387] developed the discrete layer theories by dividing the plate into thin layers and assuming a uniform state of stress and/or strain within each sublayer. The resulting displacements are nonlinear in the thickness coordinate. EXACT ELASTICITY SOLUTIONS The accuracy of any refined shear deformation theory is always verified with respect to the exact elasticity solution. The available elasticity solutions for isotropic and anisotropic laminated plates are presented here. 1. Isotropic Plates: Srinivas, et al. [388,389] developed a three dimensional, linear, small deformation theory of elasticity solutions by the direct method for the flexure and the free vibration of simply supported homogeneous, isotropic, thick rectangular plates under arbitrary loading. The solutions are in series form. Exact solutions, based on couple-stresses plane strain theory of elasticity, are presented by Hsu [390] for the normal, shear and couple-stresses, displacements and rotations of an infinite plate under cylindrical bending. Little [391] provided the exact elasticity solution for flexure of simply supported rectangular isotropic plate. Levinson [392] presented an exact three dimensional solution for the problem of a transversely loaded, simply supported rectangular plate of arbitrary thickness within the linear theory of elastostatics. The solution, obtained in a semi-inverse fashion, satisfies all the boundary conditions of the problem in a


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pointwise manner and is in the form of double Fourier sine series. Similarly, an exact elasticity soluton for free vibration of a simply supported rectangular plate is also presented by Levinson [393]. 2. Laminated Plates: Srinivas, Rao and Rao [388,389] developed exact elasticity solutions for the flexure and free vibration of simply supported homogeneous, isotropic, thick rectangular plates and also extended them to three-ply laminated plates of isotropic materials. However, the number of the simultaneous equations increases with number of layers. Srinivas and Rao [394,395] presented a unified exact analysis for the statics and dynamics of a class of thick laminates. A three dimensional, linear, small deformation theory of elasticity solution is developed for the flexure, free vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Similar solutions for the same class of problems (i.e., static and eigenvalue) are given by Wittrick [396]. Hussainy and Srinivas [397] analysed rectangular composite plates with various fiber orientations using exact elasticity solution. Pagano [236–238], Pagano and Wang [398], Pagano and Hatfield [399], developed exact elasticity solutions for cross-ply laminated plates subjected to cylindrical and bidirectional bending. Kulkarni and Pagano [400] extended these solutions to the vibration of composite laminates. However, the nature of exact solution is governed by the specific parameter which depends on the combination of the material, geometric and loading properties. Piskunov et al. [401] expanded the class of known three dimensional solutions for laminated plates made of orthotropic materials by providing the more general exact solution for laminated plates with an arbitrary number of layers arranged asymmetrically and subjected to normal and tangential loads. No restrictions are imposed on the ratios of the physicomechanical and geometric parametres. Kaprielian et al. [402] presented an exact elasticity solution for laminated plates based on a generalization of Mitchell’s exact plane stress theory. There is no restriction on the number of laminae. A three layered symmetric isotropic laminated plate is analysed. Fan and Ye [403,404] provided exact elasticity solutions for the statics and dynamics of laminated thick plates with orthotropic layers by overcoming the disadvantage (i.e., the number of simultaneous equations increases with the number of layers) associated with exact solutions given by Srinivas et al. [388,389,394,395]. Noor and Burton [405], Savoia and Reddy [407] developed the three dimensional elasticity solutions for cross-ply and antisymmetric angle-ply rectangular plates. Solutions developed are in series form. Pan [406] has given an exact solution for transversely isotropic, simply supported and layered rectangular plates.



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Wu and Wardenier [408] performed an exact elasticity solution for simply supported thick orthotropic plates subjected to arbitrary loading. Restrictions on the specific parameter governing the nature of the exact solution given by Pagano [236,237] are removed and this parameter could be any real number. Barker et al. [409], Wanthal and Yang [410], Kong and Cheung [411], Miravete and Duenas [412] presented the three dimensional finite element solutions for laminated composites including transverse shear and transverse normal stress effects. Three layered symmetric cross-ply laminated plates are analyzed. CONCLUDING REMARKS Refined shear deformation theories for isotropic and anisotropic plates are reviewed in this paper. Following are the crucial issues related with the refined theories where further research is required. 1. Many refined theories are developed for symmetric cross-ply laminated plates either subjected to cylindrical bending or subjected to bidirectional bending. However, research work on unsymmetric laminated plates is still in rudimentary stage. 2. If the unsymmetry in the lay-up is not properly accounted for in the theory, refined theories even with more than five displacement variables appeared to be insufficient and inefficient while dealing with unsymmetric laminates. 3. Refined theories based on trigonometric functions representing the shear deformation effects are not fully explored and need critical evaluation. 4. Higher order theories with more than five displacement variables are highly cumbersome and computationally more demanding. In all the equivalent single layer theories, transverse shear stresses obtained are double valued at layer interfaces when obtained using constitutive relations. This necessiates post-processing using equilibrium equations to obtain meaningful, single valued shear stresses. 5. Refined shear deformation theories for laminated plate which utilise the constitutive relations to obtain meaningful transverse shear stresses, which are single valued at layer interfaces, are rare in the literature. 6. There is no critical evaluation of variationally consistent and inconsistent refined shear deformation theories. REFERENCES 1. Broutman, L. J., and Krock, R. H. (Eds.), 1967, Modern Composite Materials, Addison-Wesley Pub. Co., New York. 2. Broutman, L. J., and Krock, R. H. (Eds.), 1974, Composite Materials, Vol. 1–7, Academic Press, New York.


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