## 1. Introduction

As it is well known, engineering theories for plates and shells simplify the three-dimensional (3D) elasticity problem by introducing the kinematic hypothesis, which leads to simpler mathematical problems. Therefore, such simplified theories have limitations that are strictly related to the initial hypotheses. The present work is based on the so-called Reissner–Mindlin theory or First-order Shear Deformation Theory, which is used to study moderately thick plates [

1,

2]. The term “moderately thick” refers to the fact that the plate is not “thin” as in the Classical Laminated Plate Theory (CLPT) or Kirchhoff-Love Theory and not “thick” as in the classical 3D theory of elasticity. Once the physical problem is mathematically well-posed, it is generally solved via numerical methods due to the complexity of finding analytical or semi-analytical solutions for general configurations. The present work aims to show a peculiar behavior in the solution of such problems by comparing the results obtained using strong and weak form finite element methods when the plates are in free vibrations. In particular, the authors compare the results obtained with two- and three-dimensional theories as a function of the plate thickness and material configuration, such as isotropic and laminated composite.

The free vibration problem of plates with regular shape has been developed since the XIX century. The study on vibrations of composite laminates started in the early 60s and is nowadays still a topic of great interest among researchers. In particular, plates of arbitrary shape and made of composite materials are investigated in the following article.

Rectangular plates were the first structures to be studied, because for some configurations an analytical or semi-analytical solution could be found. In 1970, Srinivas et al. [

3] studied thick homogeneous and laminated plates. In 1978, Nelson [

4] presented the vibration problem of rectangular plates and bars with a solution that satisfied the Mindlin equations. The same topic was presented by Ali and Atwal [

5]. In 1992, Lee and Lim [

6] studied squared isotropic and orthotropic plates with in-plane forces. Farsa et al. in 1993 [

7,

8] used the Generalized Differential Quadrature (GDQ) method for the vibration of rectangular orthotropic and anisotropic laminated plates. Coupled rectangular plates were investigated in 1996 by Bardell et al. [

9]. The GDQ method was considered in 1998 by Wang et al. [

10]. In 1999, Huang and Sakiyama [

11] studied rectangular plates with holes of different shapes. In 2003, Karami and Malekzadeh [

12] proposed a new version of the GDQ method in which multiple boundary conditions were implemented in the weighting coefficients for the derivative approximation in square and rectangular plates. In the same year, Liew et al. [

13,

14] used the Ritz method and the FSDT for studying laminated rectangular plates with central holes. In 2004, Huang and Li [

15] studied the flexural strength of plates with anti-symmetrical lamination schemes. Seok et al. [

16] presented the out-of-plane motions of cantilevered rectangular plates. In 2006, Singh and Tanveer [

17] investigated multi-connected rectangular plates in several configurations. In 2007, Shu et al. [

18] used the finite difference method based on the least squares, for the free vibration problems of thin isotropic plates of general shape. Among all, square plates with semi-circular cut-outs were considered. In 2008, Houmat [

19] studied the free vibration problem of plates with curvilinear edges and square plates with holes. In the same year, Secgin and Sarigul [

20] used the Discrete Singular Convolution (DSC) method for the free vibration of square plates. Quadrilateral plates were studied with a Fringing meshfree interpolation technique by Bui et al. [

21]. In 2011 and 2012, Dozio and Carrera [

22,

23] presented the free vibration problem of quadrilateral plates with various thicknesses and different plate theories. A modified version of the Differential Quadrature (DQ) method was proposed by Eftekhari and Jafari in 2013 [

24] for rectangular plates with several boundary conditions. Quadrilateral plates were studied using the DQ method also by Nassar et al. [

25]. In 2014, Kurtaran [

26] investigated the effect of element shape on the free vibrations of plates made of Functionally Graded Materials (FGMs) [

27,

28,

29,

30,

31,

32,

33,

34,

35]. Finally, Kumar et al. [

36] have recently proposed laboratory tests and theoretical analyses for the free vibrations of composite laminated plates made of glass/epoxy.

When the plate is not of regular shape, numerical or complex analytical approaches should be considered. However, due to the geometric distortion, the accuracy of a methodology might be lower than others according to the applied distortion. The free vibration problem of skew plates was first studied by Claasen [

37] in 1963. Nair and Durvasula [

38] in 1973 and Mizusawa et al. [

39] in 1979 also studied the same problem. In 1980, Kanaka Raju and Hinton [

40] and in 1988, Gorman [

41] studied simply-supported and fully clamped skew plates. The finite element method was used by Bardell in 1992 [

42] for the free vibration problem of skew plates and analogous studies were conducted by Liew and Wang [

43] in 1993. Several geometries, with skew plates with different skew angles, were studied using the DQ method by Bert and Malik in 1996 [

44] and Hosokawa et al. [

45] in the same year. Han and Dickinson [

46] proposed the study of skew plates with symmetric lamination schemes. Wang [

47,

48] published a similar study using the First-order Shear Deformation Theory (FSDT). A shear-deformable finite element was used for the free vibration problem of skew plates by Sheikh et al. [

49] in 2002. In 2003, Karami and Malekzadeh [

50,

51] implemented a DQ technique for quadrilateral plates with straight edges of skew and trapezoidal shape. The FSDT was considered by Liew et al. [

52] for investigating the skew plate strength. Garg et al. [

53] studied skew plates with laminated and sandwich configurations. In 2008, several studies were published on skew plates by Civalek [

54], Das et al. [

55], Nallim and Oller [

56], and Zhou and Zheng [

57]. Ashour [

58] focused his research on skew plates symmetrically clamped and Gurses et al. [

59] presented skew plates studied with FSDT. In 2012, Rao and Babu [

60] presented thin skew plates symmetrically laminated with holes. Skew plates, made of composite materials, were studied by Dozio and Carrera [

22,

23] and also by Srinivasa et al. [

61]. The DQ method was used for studying skew plates with free edges by Wang and Wu [

62]. Other studies were presented by Kurtaran [

26], Zhang and Xiao [

63], Zhang [

64] and Wang et al. [

65]. Finally, Mohazzab and Dozio [

66] presented a study on skew plates in 2015.

Circular, annular, and sectorial plates have been thoroughly studied in the last 40 years. First studies were conducted by Ramakrishnan and Kunukasseril [

67] in 1976. Irie et al. [

68,

69] published two works on orthotropic circular plates with simply-supported edges. Clamped circular plates were presented by Maruyama and Ichinomiya [

70]. Srinivasan and Thiruvenkatachari [

71,

72] treated the problem of annular isotropic plates in free vibrations for the first time. An analytical solution for sectorial plates was presented by Harik and Molaghasemi [

73] in 1989. Sectorial plates were analyzed by Misuzawa and Kajita [

74], whereas annular plates were considered by Misuzawa and Takami [

75]. In the same topic, other researchers focused their studies using different techniques such as Liew and Lam [

76] and Xiang et al. [

77]. In 1995, McGee et al. [

78,

79] presented two papers on annular plates with several boundary conditions, as well as Bert and Malik in 1996 [

44]. Circular Mindlin plates (with and without discontinuities) were investigated by Liew et al. and Liu et al. [

80,

81,

82,

83,

84] using the DQ method. In 2002, Zhong [

85] applied the triangular DQ method to sectorial plates using a curvilinear mapping for triangular elements. In 2003, Civalek and Catal [

86] used the HDQ method for linear static and free vibration problems of annular and circular plates. Between 2003 and 2004 Liew et al. [

14,

52,

87] published several papers on sectorial and annular plates with the DQ method. The buckling and vibration of sectorial plates was investigated by Sharma et al. [

88]. Similarly, Nie and Zhong [

89] and Shu et al. [

18] presented other studies on circular plates. In 2008, Dong [

90] presented a three-dimensional analysis of annular plates in free vibrations. Several papers about sectorial and annular plates were presented by Civalek [

91,

92], Xing and Liu [

93], and Zhou et al. [

94]. Civalek and Ozturk [

95] studied the free vibrations of sectorial plates using a quadrilateral finite element with curvilinear edges. Finally, plates with curvilinear edges were studied by Kurtaran [

26].

The most difficult geometries to treat in numerical analysis are the triangular and trapezoidal shapes, due to their geometry, which can lead to strong shape distortion. Triangular isotropic and orthotropic plates were studied for the first time by Lam et al. [

96] in 1990 and by Dubliner [

97] in 1991. Triangular isosceles plates were investigated by Kitipornchai et al. [

98] and partially supported plates were presented by Mirza and Alizadeh [

99], whereas cantilever plates of triangular and trapezoidal shape were illustrated by Qatu [

100]. In 1996, Abrate [

101] and Karunasena et al. [

102] studied simply-supported triangular plates. Singh and Saxena [

103], Karunasena and Kitipornchai [

104], and Singh and Hassan [

105] investigated triangular plates with variable thicknesses. In 2000, Sakiyama and Huang [

106] dedicated their studies to triangular plates of variable thickness and in the same year Zhong [

107] presented isosceles plates with the DQ method. Sheikh et al. [

108] showed a shear-deformable finite element for laminated composite plates of several shapes. Trapezoidal plates were presented by Karami and Malekzadeh [

50] and Karami et al. [

51]. The previously cited Shu et al. [

18] also studied trapezoidal plates with the finite differences method based on the least squares. The same for Civalek [

54] in 2008, who also presented trapezoidal plates. Plates of trapezoidal shape were presented by Nallim and Oller [

56] and Civalek and Gurses [

109]. Thick trapezoidal plates were deeply studied by El-Sayad and Ghazy [

110], Quintana and Nallim [

111], and Rango et al. [

112,

113].

Other plates of complex shape are elliptic plates, pentagonal, heptagonal, and plates with holes. The free vibrations of elliptic plates were presented by Singh and Chakraverty [

114] in 1992, followed by Bert and Malik [

44] in 1996. Not-homogeneous elliptic plates were presented by Chakraverty et al. [

115] in 2005 and the DQ method was used by Xing and Liu [

93], and Civalek and Ozturk [

95]. Elliptic laminated plates were presented by Bui et al. [

21] and Kurtaran [

26].

Pentagonal and heptagonal plates were not investigated much in the past. Some studies by Ghazi et al. [

116] and Xing and Liu [

93] must be mentioned.

Complex analysis is carried out for plates with holes due to the strong geometric discontinuity, rather than the geometric distortion. Plates with holes were presented by Lim and Liew in 1995 [

117]. Plates with holes were also considered by Houmat [

19] and Liu et al. [

118]. In 2011, Bui et al. [

21] studied laminated plates in free vibrations with openings. Moreover, laminated pierced plates were presented by Rao and Babu [

60] and recently by Rango et al. [

112,

113].

Other studies [

119,

120,

121,

122,

123,

124,

125,

126,

127,

128,

129,

130,

131,

132,

133,

134,

135,

136,

137,

138,

139,

140,

141,

142,

143,

144,

145,

146,

147,

148,

149,

150,

151,

152,

153,

154,

155,

156,

157,

158,

159,

160,

161,

162,

163,

164,

165,

166,

167,

168,

169,

170,

171,

172,

173,

174] regarding plates made of composite materials and higher order theories are noted. They are important because they represent the background of the present work. The problem of plates can be treated with innovation. Other methods, both numerical and analytical, the interested reader can find in the list of papers given.

## 2. First-Order Laminated Plate Theory

The so-called First-order Shear Deformation Theory (FSDT) or Reissner–Mindlin theory is the most common and well-known theory for studying plates and shell structures. It is not the purpose of the present work to extensively present such theory but to compare the numerical results obtained using FSDT with different approaches, in particular strong and weak formulations. Therefore, only main equations are reported in the following. According to the given hypothesis of FSDT, the displacement field considers five kinematic parameters as shortly given below

where the three-dimensional displacements

$U,V,W$ are indicated with capital letters, whereas the kinematic parameters

${u}_{x},{u}_{y},w,{\beta}_{x},{\beta}_{y}$ (three in-plane displacements and two rotations) with small letters. The graphical representation of the displacement field (1) is given in

Figure 1.

The present theory considers eight strain characteristics on the plate middle surface instead of the three-dimensional strain tensor. Such quantities can be divided into three groups, the in-plane characteristics ${\epsilon}_{x}^{0},{\epsilon}_{y}^{0},{\gamma}_{xy}^{0}$, the curvatures ${k}_{x},{k}_{y},{k}_{xy}$ and the shear strains ${\gamma}_{xz},{\gamma}_{yz}$. The latter by definition are constant through the thickness, therefore, the present theory needs a shear correction factor for a correct approximation of the shear stresses.

Equilibrium equations are of the same number as the degree of freedom of the model and they are a function of eight stress resultants which correspond to the eight strain characteristics of the kinematic field. The present theory considers also the rotary inertia as indicated in the equilibrium equations given below.

where

${N}_{x},{N}_{y},{N}_{xy}$ are the in-plane forces,

${M}_{x},{M}_{y},{M}_{xy}$ represent bending moment and torque and

${T}_{x},{T}_{y}$ are the shear forces which are constant through the thickness by hypothesis. The inertia terms

${I}_{i}$ are defined as

It is recalled that, the rotary inertias are significant when the thickness of the plate is relevant when compared to its in-plane dimensions. A graphical representation of the stress resultants is given in

Figure 2.

The theory of laminated composite materials considers strain and stress components that vary ply by ply according to the following general constitutive law

It is remarked that, Equation (4) considers a general orientation of the orthotropic layers (indicated by the superscript

${}^{(k)}$ within the stacking sequence. The stiffness constants

${\overline{Q}}_{ij}^{(k)}$ are expressed by classical relationships that can be found in [

174,

175].

The constitutive equations for FSDT theory can be summarized by the following system

where the two vectors have the same meaning as aforementioned.

${A}_{ij},{B}_{ij},{D}_{ij}$ for

$i,j=1,2,6$ are the membrane, coupling and bending stiffnesses. The shear correction factor is indicated as

$\kappa $ and it is taken as

$\kappa =5/6$.

The present study focuses on the free vibration behavior of plates as a function of the relationships given by Equation (5). For instance, when a plate made with a single isotropic ply is considered ${B}_{ij}=0$, thus, the in-plane behavior is un-coupled with the bending part, moreover ${A}_{16}={A}_{26}={D}_{16}={D}_{26}=0$. Analogously for cross-ply symmetric laminates. On the contrary, for a general lamination scheme made of orthotropic plies all the coefficients of the constitutive law are not null. In this regard, it is noted that, numerical issues occur when some of these combinations are considered as it will be discussed in the sections below.