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Static, free vibration, and buckling analysis of plates using strain-based Reissner–Mindlin elements

  • Abderahim BelounarEmail author
  • Sadok Benmebarek
  • Mohamed Nabil Houhou
  • Lamine Belounar
Open Access
Original Research
  • 302 Downloads

Abstract

A quadrilateral and a triangular element based on the strain approach are developed for static, free vibration and buckling analyses of Reissner–Mindlin plates. The four-node triangular element SBTP4 has the three essential external degrees of freedom at each of the three corner nodes and at a mid-side node; whereas the quadrilateral element SBQP has the same degrees of freedom at each of the four corner nodes. Both elements use the same assumed strain functions which are in the linear variation where bending and transverse shear strains are independent and satisfy the compatibility equations. The use of the strain approach allows obtaining elements with higher-order terms for the displacements field. The formulated elements have been proposed to improve the strain-based rectangular plate element SBRP previously published. Several numerical examples demonstrate that the present elements are free of shear locking and provide high-accuracy results compared to the available published numerical and analytical solutions.

Keywords

Strain approach Free vibration Buckling Mindlin plate 

List of symbols

L

Length of plate

k

Shear correction factor

ρ

Material density

ν

Poisson’s ratio

E

Young’s modulus

h

Thickness of plate

β

Angle of the skew plate

D

Flexural rigidity of plate = Eh3/[12(1 − v2)]

G

Shear modulus = E/[2(1 + v)]

λ

Non-dimensional frequency parameter

ω

Angular frequency

λcr

Critical buckling load

αi

Constants in displacement fields

W

Displacement in the z-direction

βx, βy

Rotations about y and x axes, respectively

x, y, z

Co-ordinates system

[Ke]

Element stiffness matrix

[Me]

Element mass matrix

\([K_{\text{g}}^{\text{e}} ]\)

Element geometrical matrix

[K]

Structural stiffness matrix

[M]

Structural mass matrix

[Kg]

Structural geometrical matrix

[C]

Transformation matrix

[P]

Displacement matrix

[Q]

Strain matrix

[G]

Geometrical strain matrix

{F}

Structural nodal force vector

{q}

Structural nodal displacements vector

{qe}

Element nodal displacements vector

Introduction

Analyses of static, buckling and free vibration of plate structures play a large role in structural engineering applications. Considerable research works on analysis of plates are still being conducted (Mackerle 1997, 2002; Leissa 1969, 1987; Liew et al. 1995, 2004).

Designers prefer low-order Reissner–Mindlin plate elements due to their simplicity and efficiency. However, for thin plates, these elements often suffer from the shear locking phenomenon when dealing with thin plates. To overcome shear locking, many research works have been undertaken where the use of the selective reduced integration was first intervened (Zienkiewicz et al. 1971; Hughes et al. 1978; Malkus and Hughes 1978). The formulation procedure used is to divide the strain energy into two parts, one of bending and the other of shear. Then, two different integration rules for these two parts are used. For low-order polynomial elements based on displacement model, such as the four-node classical bilinear element, an exact integration (two Gauss points in each direction) is taken for the bending strain energy; whereas a reduced integration (one Gauss point) is used for the shear strain energy. This selective integration can be provided with a more efficient element but often leads to numerical instability. Considerable investigations have been oriented to develop robust elements using different improved formulations and numerical techniques to avoid shear locking such as mixed formulation, enhanced assumed strain methods, assumed natural strain methods, discrete shear gap method and smoothed finite element method (Lee and Wong 1982; Ayad et al. 1998; Lovadina 1998; César de Sá and Natal Jorge 1999; César de Sá et al. 2002; Cardoso et al. 2008; MacNeal 1982; Bathe and Dvorkin 1985, 1986; Zienkiewicz et al. 1990; Batoz and Katili 1992; Bletzinger et al. 2000; Nguyen-Xuan et al. 2008; Liu and Nguyen-Thoi 2010).

The strain approach has been employed as an alternative to formulate robust plate elements (Belarbi and Charif 1999; Belounar and Guenfoud 2005; Belounar and Guerraiche 2014; Guerraiche et al. 2018; Belounar et al. 2018) to increase the accuracy and stability of the numerical solutions as well as to eliminate shear locking phenomena. The use of the strain approach (Belarbi and Charif 1999; Belounar and Guenfoud 2005; Belounar and Guerraiche 2014; Guerraiche et al. 2018; Belounar et al. 2018; Djoudi and Bahai 2004a, b; Rebiai and Belounar 2013; 2014) has several advantages where it enables to obtain efficient elements with high-order polynomial terms for the displacement functions without the need of including internal nodes. The first developed strain-based Mindlin plate element SBRP (Belounar and Guenfoud 2005) has been adopted for the linear analysis of plates having only rectangular shapes. However, this element suffers from shear locking for very thin plates (Belounar et al. 2018). Then, the formulation of a new three-node strain-based triangular Mindlin plate element SBTMP (Belounar et al. 2018) has been developed for static and free vibration of plate bending. The assumed curvatures and transverse shear strains for the SBRP element (Belounar and Guenfoud 2005) are coupled and contain quadratic terms. The key idea used in this paper is to formulate new elements to overcome shear locking for very thin plates and to improve the accuracy for plates with regular and distorted shapes.

In this paper, a quadrilateral and a triangular strain-based plate element have been formulated for static, free vibration and buckling analyses of plates using Reissner–Mindlin theory. The opportunity is taken to explore the displacements field obtained from the strain-based quadrilateral plate element (SBQP) by applying it to a four-node triangular element strain-based triangular plate with four nodes (SBTP4) having the same degrees of freedom (W, βx, and βy) at each of the three corner nodes and a mid-side node. In the process of formulation, these elements are based on linear variation for the five strain components where bending and transverse shear strains are independent and satisfying the compatibility equations. The numerical study shows that the SBQP and SBTP4 elements pass the patch test, are free of shear locking, and can be found numerically more efficient than the SBRP element (Belounar and Guenfoud 2005).

Formulation of the proposed elements

Derivation of the displacements field

For Reissner–Mindlin plate elements (Fig. 1), the strains in terms of the displacements are given as:
$$\kappa_{x} = \frac{{\partial \beta_{x} }}{\partial x},\quad \kappa_{y} = \frac{{\partial \beta_{y} }}{\partial y},\quad \kappa_{xy} = \left( {\frac{{\partial \beta_{x} }}{\partial y} + \frac{{\partial \beta_{y} }}{\partial x}} \right),\quad \gamma_{xz} = \beta_{x} + \frac{\partial W}{\partial x},\quad \gamma_{yz} = \beta_{y} + \frac{\partial W}{\partial y}.$$
(1a)
Fig. 1

Quadrilateral and triangular Reissner–Mindlin plate elements

In matrix form, it can be given as
$$\left\{ {\begin{array}{*{20}l} {\kappa_{x} } \hfill \\ {\kappa_{y} } \hfill \\ {\kappa_{xy} } \hfill \\ {\gamma_{xz} } \hfill \\ {\gamma_{yz} } \hfill \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}l} 0 \hfill & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-0pt} {\partial x}}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-0pt} {\partial y}}} \hfill \\ 0 \hfill & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-0pt} {\partial y}}} \hfill & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-0pt} {\partial x}}} \hfill \\ {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-0pt} {\partial x}}} \hfill & 1 \hfill & 0 \hfill \\ {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-0pt} {\partial y}}} \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} W \\ {\beta_{x} } \\ {\beta_{y} } \\ \end{array} } \right\}.$$
(1b)
The five strains, bending (κx, κy and κxy) and transverse shear (γxz and γyz), given in Eq. (1a) cannot be considered independent, for they are in terms of the displacements W, βx and βy and therefore, they must satisfy the compatibility equations (Belounar and Guenfoud 2005) given as:
$$\frac{{\partial^{2} \kappa_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \kappa_{y} }}{{\partial x^{2} }} = \frac{{\partial^{2} \kappa_{xy} }}{\partial x\partial y},\quad \frac{{\partial^{2} \gamma_{xz} }}{\partial x\partial y} - \frac{{\partial^{2} \gamma_{yz} }}{{\partial x^{2} }} + \frac{{\partial \kappa_{xy} }}{\partial x} = 2\frac{{\partial \kappa_{x} }}{\partial y},\quad \frac{{\partial^{2} \gamma_{yz} }}{\partial x\partial y} - \frac{{\partial^{2} \gamma_{xz} }}{{\partial y^{2} }} + \frac{{\partial \kappa_{xy} }}{\partial y} = 2\frac{{\partial \kappa_{y} }}{\partial x}.$$
(2)
The field of displacements due to the three rigid body modes is obtained by having Eq. (1a) equal to zero and the following results are obtained:
$$W = \alpha_{1} - \alpha_{2} x - \alpha_{3} y,\quad \beta_{x} = \alpha_{2} ,\quad \beta_{y} = \alpha_{3} .$$
(3)

The proposed quadrilateral and triangular elements (SBQP and SBTP4) have three degrees of freedom (W, βx and βy) at each of the four nodes. Therefore, the displacements field should contain twelve independent constants and having used three (α1, α2, α3) for the representation of the rigid body modes, the remaining nine constants (α4, α5, …, α12) are to be apportioned among the five assumed strains of the two elements.

The interpolation of the assumed strains field for the present elements (SBQP and SBTP4) is given as:
$$\begin{aligned} & \kappa_{x} = \alpha_{4} + \alpha_{5} y,\,\kappa_{y} = \alpha_{6} + \alpha_{7} x,\quad \kappa_{xy} = \alpha_{8} + (2\alpha_{5} x) + (2\alpha_{7} y), \\ & \gamma_{xz} = \alpha_{9} + \alpha_{10} y,\quad \gamma_{yz} = \alpha_{11} + \alpha_{12} x. \\ \end{aligned}$$
(4)

Assumed bending (κx, κy and κxy) and transverse shear (γxz and γyz) strains given in Eq. (4) of the proposed elements are independent and have only linear terms contrarily for the SBRP element (Belounar and Guenfoud 2005), where bending and transverse shear strains are coupled and quadratic terms are included in the assumed shear strain components.

The bracketed terms of the assumed strains (Eq. 4) are added to have the compatibility equations (Eq. 2) to be satisfied. The strain functions (κx, κy, κxy, γxz, γyz) given by Eq. (4) are substituted into Eq. (1a) and after integration, we obtain:
$$\begin{aligned} & W = - \,\alpha_{4} \frac{{x^{2} }}{2} - \alpha_{5} \frac{{x^{2} y}}{2} - \alpha_{6} \frac{{y^{2} }}{2} - \alpha_{7} \frac{{xy^{2} }}{2} - \alpha_{8} \frac{xy}{2} + \alpha_{9} \frac{x}{2} + \alpha_{10} \frac{xy}{2} + \alpha_{11} \frac{y}{2} + \alpha_{12} \frac{xy}{2} \\ & \beta_{x} = \alpha_{4} x + \alpha_{5} xy + \alpha_{7} \frac{{y^{2} }}{2} + \alpha_{8} \frac{y}{2} + \alpha_{9} \frac{1}{2} + \alpha_{10} \frac{y}{2} - \alpha_{12} \frac{y}{2} \\ & \beta_{y} = \alpha_{5} \frac{{x^{2} }}{2} + \alpha_{6} y + \alpha_{7} xy + \alpha_{8} \frac{x}{2} - \alpha_{10} \frac{x}{2} + \alpha_{11} \frac{1}{2} + \alpha_{12} \frac{x}{2}. \\ \end{aligned}$$
(5a)
The displacement functions obtained from Eq. (5a) are summed to the displacements of rigid body modes given by Eq. (3) to obtain the final displacement shape functions:
$$\begin{aligned} & W = \alpha_{1} - \alpha_{2} x - \alpha_{3} y - \alpha_{4} \frac{{x^{2} }}{2} - \alpha_{5} \frac{{x^{2} y}}{2} - \alpha_{6} \frac{{y^{2} }}{2} - \alpha_{7} \frac{{xy^{2} }}{2} - \alpha_{8} \frac{xy}{2} + \alpha_{9} \frac{x}{2} + \alpha_{10} \frac{xy}{2} + \alpha_{11} \frac{y}{2} + \alpha_{12} \frac{xy}{2} \\ & \beta_{x} = \alpha_{2} + \alpha_{4} x + \alpha_{5} xy + \alpha_{7} \frac{{y^{2} }}{2} + \alpha_{8} \frac{y}{2} + \alpha_{9} \frac{1}{2} + \alpha_{10} \frac{y}{2} - \alpha_{12} \frac{y}{2} \\ & \beta_{y} = \alpha_{3} + \alpha_{5} \frac{{x^{2} }}{2} + \alpha_{6} y + \alpha_{7} xy + \alpha_{8} \frac{x}{2} - \alpha_{10} \frac{x}{2} + \alpha_{11} \frac{1}{2} + \alpha_{12} \frac{x}{2}. \\ \end{aligned}$$
(5b)
The displacement functions of Eq. (5b) and the strain functions of Eq. (4) can be given in matrix form, respectively, as:
$$\{ U\} = [P]\{ \alpha \} = [N]\{ q_{\text{e}} \} ,$$
(6)
$$\{ \varepsilon \} = [Q]\{ \alpha \} = [B]\{ q_{\text{e}} \}$$
(7)
$${\text{Where}}\,[N] = [P][C]^{ - 1}, \quad [B] = [Q][C]^{ - 1}.$$
(8)
And the matrices [P] and [Q] are given as:
$$[P] = \left[ {\begin{array}{*{20}c} 1 & { - x} & { - y} & { - \frac{{x^{2} }}{2}} & { - \frac{{x^{2} y}}{2}} & { - \frac{{y^{2} }}{2}} & { - \frac{{xy^{2} }}{2}} & { - \frac{xy}{2}} & {\frac{x}{2}} & {\frac{xy}{2}} & {\frac{y}{2}} & {\frac{xy}{2}} \\ 0 & 1 & 0 & x & {xy} & 0 & {\frac{{y^{2} }}{2}} & {\frac{y}{2}} & {\frac{1}{2}} & {\frac{y}{2}} & 0 & { - \frac{y}{2}} \\ 0 & 0 & 1 & 0 & {\frac{{x^{2} }}{2}} & y & {xy} & {\frac{x}{2}} & 0 & { - \frac{x}{2}} & {\frac{1}{2}} & {\frac{x}{2}} \\ \end{array} } \right],$$
(9)
$$[Q] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 1 & y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & x & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {(2x)} & 0 & {(2y)} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & y & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & x \\ \end{array} } \right].$$
(10)
And the displacements field, the strains field, and constant parameters vectors are:
$$\{ U\} = \{ W,\beta_{x} ,\beta_{y} \}^{T} ,\quad \{ \varepsilon \} = \{ \kappa_{x} ,\kappa_{y} ,\kappa_{xy} ,\gamma_{xz} ,\gamma_{yz} \}^{T} ,\quad \{ \alpha \} = \{ \alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{12} \}^{T} .$$
(11)
The geometrical strains can be expressed as:
$$\{ \varepsilon^{\text{g}} \} = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial x}} & 0 & 0 \\ {\frac{\partial }{\partial y}} & 0 & 0 \\ 0 & {\frac{\partial }{\partial x}} & 0 \\ 0 & {\frac{\partial }{\partial y}} & 0 \\ 0 & 0 & {\frac{\partial }{\partial x}} \\ 0 & 0 & {\frac{\partial }{\partial y}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} W \\ {\beta_{x} } \\ {\beta_{y} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial x}} & 0 & 0 \\ {\frac{\partial }{\partial y}} & 0 & 0 \\ 0 & {\frac{\partial }{\partial x}} & 0 \\ 0 & {\frac{\partial }{\partial y}} & 0 \\ 0 & 0 & {\frac{\partial }{\partial x}} \\ 0 & 0 & {\frac{\partial }{\partial y}} \\ \end{array} } \right][P]\{ \alpha \} ,$$
(12)
$${\text{where}}\,[G] = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial x}} & 0 & 0 \\ {\frac{\partial }{\partial y}} & 0 & 0 \\ 0 & {\frac{\partial }{\partial x}} & 0 \\ 0 & {\frac{\partial }{\partial y}} & 0 \\ 0 & 0 & {\frac{\partial }{\partial x}} \\ 0 & 0 & {\frac{\partial }{\partial y}} \\ \end{array} } \right][P].$$
We substitute Eq. (6) into Eq. (12), we obtain:
$$\{ \varepsilon^{\text{g}} \} = [G]\{ \alpha \} = [B^{\text{g}} ]\{ q_{\text{e}} \} ,$$
(13)
$${\text{where}}\,[B^{\text{g}} ] = [G][C]^{ - 1} .$$
(14)
And the matrix [G] is given as:
$$[G] = \left[ {\begin{array}{*{20}c} 0 & { - 1} & 0 & { - x} & { - xy} & 0 & { - \frac{{y^{2} }}{2}} & { - \frac{y}{2}} & {\frac{1}{2}} & {\frac{y}{2}} & 0 & {\frac{y}{2}} \\ 0 & 0 & { - 1} & 0 & { - \frac{{x^{2} }}{2}} & { - y} & { - xy} & { - \frac{x}{2}} & 0 & {\frac{x}{2}} & {\frac{1}{2}} & {\frac{x}{2}} \\ 0 & 0 & 0 & 1 & y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x & 0 & y & {\frac{1}{2}} & 0 & {\frac{1}{2}} & 0 & { - \frac{1}{2}} \\ 0 & 0 & 0 & 0 & x & 0 & y & {\frac{1}{2}} & 0 & { - \frac{1}{2}} & 0 & {\frac{1}{2}} \\ 0 & 0 & 0 & 0 & 0 & 1 & x & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right].$$
(15)
The transformation matrix [C] which relates the element nodal displacements ({qe}T = (W1, βx1, βy1, …, W4, βx4, βy4)) to the 12 constants ({α}T = (α1, …, α12)) can be given as:
$$\left\{ {q_{e} } \right\} = \left[ C \right]\left\{ \alpha \right\}$$
(16a)
The constant parameters vector {α} can be derived from Eq. (16a) as follows:
$$\{ \alpha \} = [C]^{ - 1} \{ q_{\text{e}} \} .$$
(16b)
The matrices [N] (Eq. 8), [B] (Eq. 8) and [Bg] (Eq. 14) are obtained, respectively, by substituting Eq. (16b) into Eqs. (6), (7) and (13):
$${\text{Where}}\,\,[C] = \left[ {\begin{array}{*{20}c} {[P_{1} ]} & {[P_{2} ]} & {[P_{3} ]} & {[P_{4} ]} \\ \end{array} } \right]^{T} .$$
(17)
And the matrix [Pi] calculated from Eq. (9) for each of the four element nodes coordinates (xi, yi), (i = 1, 2, 3, 4) to obtain:
$$[P_{i} ] = \left[ {\begin{array}{*{20}c} 1 & { - x_{i} } & { - y_{i} } & { - \frac{{x_{i}^{2} }}{2}} & { - \frac{{x_{i}^{2} y_{i} }}{2}} & { - \frac{{y_{i}^{2} }}{2}} & { - \frac{{x_{i} y_{i}^{2} }}{2}} & { - \frac{{x_{i} y_{i} }}{2}} & {\frac{{x_{i} }}{2}} & {\frac{{x_{i} y_{i} }}{2}} & {\frac{{y_{i} }}{2}} & {\frac{{x_{i} y_{i} }}{2}} \\ 0 & 1 & 0 & {x_{i} } & {x_{i} y_{i} } & 0 & {\frac{{y_{i}^{2} }}{2}} & {\frac{{y_{i} }}{2}} & {\frac{1}{2}} & {\frac{{y_{i} }}{2}} & 0 & { - \frac{{y_{i} }}{2}} \\ 0 & 0 & 1 & 0 & {\frac{{x_{i}^{2} }}{2}} & {y_{i} } & {x_{i} y_{i} } & {\frac{{x_{i} }}{2}} & 0 & { - \frac{{x_{i} }}{2}} & {\frac{1}{2}} & {\frac{{x_{i} }}{2}} \\ \end{array} } \right].$$
(18)

Element matrices

The standard weak form for free vibration and buckling can, respectively, be expressed as:
$$\int\limits_{{S_{\text{e}} }} {\delta \{ \varepsilon \} }^{T} [D]\{ \varepsilon \} {\text{d}}S + \int\limits_{{S_{\text{e}} }} {\delta \{ U\} }^{T} [T]\{ \ddot{U}\} {\text{d}}S = 0,$$
(19)
$$\int\limits_{{S_{\text{e}} }} {\delta \{ \varepsilon \} }^{T} [D]\{ \varepsilon \} {\text{d}}S + \int\limits_{{S_{e} }} {\delta \{ \varepsilon^{\text{g}} \} }^{T} [\tau ]\{ \varepsilon^{\text{g}} \} {\text{d}}S = 0.$$
(20)
By substituting Eqs. (6), (7) and (13) into Eqs. (19) and (20), we obtain:
$$\delta \{ q_{\text{e}} \}^{T} \left( {\int\limits_{{S_{\text{e}} }} {[B]^{T} [D][B]} dS} \right)\{ q_{\text{e}} \} + \delta \{ q_{\text{e}} \}^{T} \left( {\int\limits_{{S_{\text{e}} }} {[N]^{T} [T][N]} {\text{d}}S} \right)\{ \ddot{q}_{\text{e}} \} = 0,$$
(21)
$$\delta \{ q_{\text{e}} \}^{T} \left( {\int\limits_{{S_{\text{e}} }} {[B]^{T} [D][B]} {\text{d}}S} \right)\{ q_{\text{e}} \} + \delta \{ q_{\text{e}} \}^{T} \left( {\int\limits_{{S_{\text{e}} }} {[B^{\text{g}} ]^{T} [\tau ][B^{\text{g}} ]} {\text{d}}S} \right)\{ q_{\text{e}} \} = 0.$$
(22)
Where the element stiffness, mass and geometrical stiffness matrices ([Ke], [Me], \([K_{\text{g}}^{\text{e}} ]\)), are, respectively, as:
$$\begin{aligned} & [K^{\text{e}} ] = \int_{{S_{\text{e}} }} {\left[ B \right]^{T} [D][B]} {\text{d}}S \\ & [K^{\text{e}} ] = [C]^{ - T} \underbrace {{\left( {\int {[Q]^{T} [D][Q]} \det (J){\text{d}}\xi {\text{d}}\eta } \right)}}_{{[K_{0} ]}}[C]^{ - 1} = [C]^{ - T} [K_{0} ][C]^{ - 1} , \\ \end{aligned}$$
(23)
$$\begin{aligned} \,[M^{\text{e}} ] = \int_{{S_{\text{e}} }} {[N]^{T} [T][N]} {\text{d}}S \\ & [M^{\text{e}} ] = \left[ C \right]^{ - T} \underbrace {{\left( {\int {[P]^{T} [T][P]} \det (J){\text{d}}\xi {\text{d}}\eta } \right)}}_{{[M_{0} ]}}[C]^{ - 1} = [C]^{ - T} [M_{0} ][C]^{ - 1} , \\ \end{aligned}$$
(24)
$$\begin{aligned} & \left[ {K_{\text{g}}^{\text{e}} } \right] = \int_{{S_{e} }} {\left[ {B^{\text{g}} } \right]^{T} [\tau ][B^{\text{g}} ]} {\text{d}}S \\ & \left[ {K_{\text{g}}^{\text{e}} } \right] = [C]^{ - T} \underbrace {{\left( {\int {[G]^{T} [\tau ][G]} \det (J){\text{d}}\xi {\text{d}}\eta } \right)}}_{{[K_{g0} ]}}[C]^{ - 1} = [C]^{ - T} [K_{g0} ][C]^{ - 1} . \\ \end{aligned}$$
(25)
The stress–strain relationship is given by:
$$\{ \sigma \} = [D]\{ \varepsilon \} ,$$
(26)
where \(\{ \sigma \} = \{ M_{x} ,M_{y} ,M_{xy} ,T_{x} ,T_{y} \}^{T} ,\) \(\{ \varepsilon \} = \{ \kappa_{x} ,\kappa_{y} ,\kappa_{xy} ,\gamma_{xz} ,\gamma_{yz} \}^{T} .\)
where [D], [D]b, [D]s are, respectively, rigidity, bending rigidity, shear rigidity matrices and [T] is the matrix containing the mass material density:
$$\begin{aligned} & [D] = \left[ {\begin{array}{*{20}c} {[D]_{\text{b}} } & 0 \\ 0 & {[D]_{\text{s}} } \\ \end{array} } \right],\quad [D]_{\text{b}} = \frac{{Eh^{3} }}{{12(1 - \upsilon^{2} )}}\left[ {\begin{array}{*{20}c} 1 & \upsilon & 0 \\ \upsilon & 1 & 0 \\ 0 & 0 & {\frac{(1 - \upsilon )}{2}} \\ \end{array} } \right] \\ & [D]_{\text{s}} = khG\left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right], \\ \end{aligned}$$
(27)
$$[T] = \rho \left[ {\begin{array}{*{20}c} h & 0 & 0 \\ 0 & {\frac{{h^{3} }}{12}} & 0 \\ 0 & 0 & {\frac{{h^{3} }}{12}} \\ \end{array} } \right],$$
(28)
$$[\sigma_{0} ] = \left[ {\begin{array}{*{20}c} {\sigma_{x}^{0} } & {\sigma_{xy}^{0} } \\ {\sigma_{xy}^{0} } & {\sigma_{y}^{0} } \\ \end{array} } \right],\quad [\tau ] = \left[ {\begin{array}{*{20}c} {h[\sigma_{0} ]} & 0 & 0 \\ 0 & {\frac{{h^{3} }}{12}[\sigma_{0} ]} & 0 \\ 0 & 0 & {\frac{{h^{3} }}{12}[\sigma_{0} ]} \\ \end{array} } \right],$$
(29)
where \(\sigma_{x}^{0}\), \(\sigma_{y}^{0}\) and \(\sigma_{xy}^{0}\) are the in-plane stresses.

The matrices [K0], [M0] and [Kg0] given in Eqs. (23), (24) and (25) are numerically computed with exact Gauss and Hamer rule integration, respectively, for quadrilateral and triangular elements (SBQP and SBTP4). The element stiffness, mass and geometrical matrices ([Ke], [Me] and \([K_{\text{g}}^{\text{e}} ]\)) can then be obtained. These are assembled to obtain the structural stiffness, mass and geometrical matrices ([K], [M] and [Kg]).

For static analysis, we use
$$[K]\{ q\} = \{ F\} .$$
(30)
For free vibration, we use
$$([K] - \omega^{2} [M])\{ q\} = 0.$$
(31)
For the buckling analysis, we use
$$([K] - \lambda_{\text{cr}} [K_{\text{g}} ])\{ q\} = 0.$$
(32)

Numerical validation

To validate the accuracy and efficiency of the formulated quadrilateral and triangular elements (SBQP and SBTP4), several numerical examples have been investigated for static, free vibration and buckling analysis of isotropic plates where the patch test of rigid body modes and the mechanic patch test are first carried out. The obtained results of the SBQP and SBTP4 elements are compared with other numerical and analytical solutions available in the literature.

Patch test of rigid body modes

To verify that both SBQP and SBTP4 elements pass the patch test of rigid body modes, the eigenvalues of the stiffness matrix for a single element are computed for various shapes and different aspect ratio. The only three zero eigenvalues obtained correspond to the three rigid displacement modes for a plate.

Mechanic patch test

In this patch test, a rectangular plate of (L = 2a = 40) length and (2b = 20) width simply supported at the three corner 1,2 and 3 (W1 = W2 = W3 = 0) is considered where the plate is modeled by several elements as shown in Fig. 2 (Batoz and Dhatt 1990) for various side–thickness L/h ratio (10,100 and 1000). The plate boundaries are subjected to solicitations that produce the state of constant moments (or stresses). For the case of Mn = 1 applied on all sides (Fig. 2), the obtained results are Mx = My = 1 everywhere in the plate (Table 1). Whereas for the case of Mns = 1 applied on all sides (Fig. 2), the obtained results at any points of the plate are Mxy = 1 (Table 1). The results given in Table 1 confirm that both SBQP and SBTP4 elements fulfill the mechanic patch test.
Fig. 2

Quadrilateral and triangular meshes for the patch test (E = 1000, ν = 0.3)

Table 1

Results of mechanic patch test

Elements

Applied load

Moments in the plate

L/h

10

100

1000

SBQP

Mn = 1

Mx = My

1

1

1

Mns = 1

M xy

1

1

1

SBTP4

Mn = 1

Mx = My

1

1

1

Mns = 1

M xy

1

1

1

Square plates

A classical benchmark is first studied of square plate bending problem (Fig. 3) with different boundary conditions and various thickness–side (h/L) ratios subjected to a uniform load (q = 1), where the shear locking free test and convergence investigation of central deflection are considered in this study.
Fig. 3

Square plate with a mesh of N × N elements (L = 10, E = 10.92, ν = 0.3, k = 5/6)

Shear locking free test is considered for a clamped square plate with several values of ratios (L/h = 10–1,000,000) using a mesh of 12 × 12. The central deflection results of the plate illustrated in Table 2 and Fig. 4, confirm that the new formulated elements (SBQP and SBTP4) are able to solve the shear locking problem when the plate thickness becomes gradually small. However, it is observed that the SBRP element (Belounar and Guenfoud 2005) exhibits from shear locking phenomena for (L/h > 100).
Table 2

Deflections at the center [(WD/qL4)100] of a clamped square plate with different aspect ratios

L/h

10

100

1000

10,000

100,000

1,000,000

Taylor and Auricchio (1993)

SBQP

0.1490

0.1254

0.1252

0.1252

0.1252

0.1252

 

SBTP4

0.1508

0.1256

0.1252

0.1252

0.1252

0.1252

0.1267

SBRP

0.1453

0.1035

0.0074

7.90 × 10−5

7.90 × 10−7

7.90 × 10−9

 
Fig. 4

Shear locking test (Wc/WRef) of a clamped square plate

Now, convergence tests of a square plate are investigated with three cases of boundary conditions [clamped, soft simply supported SS1 (W = 0), and hard simply supported SS2 (W = βs = 0)]. Various values of h/L ratios of 0.1, 0.01, and 0.001 are considered for thick, thin and very thin plates, respectively. The obtained results of the vertical displacement at the center of the plate are presented in Tables 3, 4 and 5 and Figs. 5, 6 and 7, which show that:
Table 3

Central deflection [(WD/qL4)100] for clamped square plates with uniform load

h/L

Elements

4 × 4

8 × 8

10 × 10

12 × 12

16 × 16

Taylor and Auricchio (1993)

0.001

SBQP

0.1149

0.1235

0.1246

0.1252

0.1258

0.1265

SBTP4

0.1150

0.1235

0.1246

0.1252

0.1258

SBRP

2.77 × 10−5

0.0011

0.0032

0.0074

0.0234

0.01

SBQP

0.1151

0.1237

0.1248

0.1254

0.1260

0.1267

SBTP4

0.1153

0.1239

0.1250

0.1256

0.1261

SBRP

0.0027

0.0558

0.0860

0.1035

0.1179

0.1

SBQP

0.1372

0.1473

0.1484

0.1490

0.1497

0.1499

SBTP4

0.1446

0.1507

0.1509

0.1508

0.1507

SBRP

0.0903

0.1384

0.1429

0.1453

0.1476

Table 4

Central deflection [(WD/qL4)100] for SS1 square plates with a uniform load

h/L

Elements

4 × 4

8 × 8

10 × 10

12 × 12

16 × 16

Taylor and Auricchio (1993)

0.001

SBQP

0.3858

0.4014

0.4032

0.4041

0.4050

0.4062

SBTP4

0.3859

0.4014

0.4032

0.4041

0.4051

SBRP

8.43 × 10−4

0.0152

0.0363

0.0697

0.1624

0.01

SBQP

0.3861

0.4019

0.4037

0.4048

0.4058

0.4062

SBTP4

0.3864

0.4021

0.4040

0.4050

0.4061

SBRP

0.0673

0.3115

0.3589

0.3802

0.3962

0.1

SBQP

0.4228

0.4450

0.4493

0.4522

0.4556

0.4617

SBTP4

0.4277

0.4487

0.4523

0.4545

0.4572

SBRP

0.3587

0.4311

0.4407

0.4463

0.4524

Table 5

Central deflection [(WD/qL4)100] for SS2 square plates with a uniform load

h/L

Elements

4 × 4

8 × 8

10 × 10

12 × 12

16 × 16

Taylor and Auricchio (1993)

0.001

SBQP

0.3858

0.4014

0.4032

0.4041

0.4050

0.4062

SBTP4

0.3859

0.4014

0.4032

0.4041

0.4050

SBRP

6.23 × 10−4

0.0147

0.0357

0.0691

0.1619

0.01

SBQP

0.3860

0.4016

0.4034

0.4043

0.4052

0.4064

SBTP4

0.3862

0.4017

0.4034

0.4044

0.4053

SBRP

0.0523

0.3081

0.3572

0.3789

0.3952

0.1

SBQP

0.4079

0.4227

0.4244

0.4253

0.4261

0.4273

SBTP4

0.4110

0.4240

0.4253

0.4260

0.4266

SBRP

0.3260

0.4048

0.4131

0.4175

0.4218

Fig. 5

Central deflection [(WD/qL4)100] for clamped square plates

Fig. 6

Central deflection [(WD/qL4)100] for SS1 square plates

Fig. 7

Central deflection [(WD/qL4)100] for SS2 square plates

  • Faster convergence towards analytical solutions (Taylor and Auricchio 1993) is obtained using only a small number of elements for all cases of ratios (h/L = 0.1, 0.01, and 0.001) and boundary conditions.

  • The SBQP and SBTP4 elements have similar behaviors for thin and very thin plates (h/L = 0.01, 0.001); whereas, for thick plates (h/L = 0.1), the SBTP4 element is a little better than the SBQP element.

  • Both proposed elements are free from shear locking phenomena where they are able to provide excellent results for thin and very thin plates (h/L = 0.01, 0.001).

  • Slow convergence to analytical solutions (Taylor and Auricchio 1993) is obtained using the SBRP element (Belounar and Guenfoud 2005) for thick and thin plates (h/L = 0.1, 0.01) and suffers from shear locking for very thin plates (h/L = 0.001).

Skew plates

To show the performance of the present elements to the sensitivity of mesh distortion, two examples of thin skew plates subjected to a uniform load (q = 1) are considered which are known in the literature as severe tests and studied by many researchers (Razzaque 1973; Morley 1963). The first is concerned with Razzaque’s skew plate (Razzaque 1973) (β = 60°) with simply supported on two sides and free on the other sides (Fig. 8). The results of the vertical displacement at the center of the plate using uniform meshes N = 2, 4, 8, 12 and 16 are given in Table 6 and Fig. 9 for (h/L = 0.001). The obtained results for both elements (SBQP and SBTP4) are in quite a good agreement with the reference solution given by Razzaque (1973). But it can be seen that the SBTP4 element is a little better than the SBQP and MITC4 (Nguyen-Xuan et al. 2008) elements.
Fig. 8

Skew plates (a Razzaque, b Morley) with N × N meshes (L = 100, E = 10.92, ν = 0.3, k = 5/6)

Table 6

Convergence of central displacement (Wc) for the Razzaque’s skew plate

Elements

Wc = Wc (D/qL4) × 102

2 × 2

4 × 4

6 × 6

8 × 8

12 × 12

16 × 16

SBQP

0.4835

0.7180

0.7596

0.7643

0.7732

0.7807

SBTP4

0.5312

0.7124

0.7518

0.7670

0.7792

0.7841

MITC4 (Nguyen-Xuan et al. 2008)

0.3856

0.6723

0.7357

0.7592

0.7765

0.7827

Razzaque (1973)

  

0.7945

   
Fig. 9

Central displacement (Wc/WRef) for the Razzaque’s skew plate

The second example treated by Morley (β = 30°) (Morley 1963) is simply supported (W = 0) on all sides (Fig. 8). Using meshes of N = 4, 8, 16 and 32, the obtained vertical displacement at the center of the plate are presented in Table 7 and Fig. 10 for h/L = 0.01 and 0.001. It can be observed that for h/L = 0.01, the results of the SBTP4 and SBQP elements are in good agreement with the reference solution (Morley 1963); whereas, for h/L = 0.001, the SBTP4 element is more efficient than the SBQP and MITC4 (Chen and Cheung 2000) elements.
Table 7

Convergence of central displacement (Wc) for the Morley’s skew plate

Mesh

Wc = Wc (D/qL4) × 103

L/h = 0.01

L/h = 0.001

SBQP

SBTP4

MITC4 (Chen and Cheung 2000)

SBQP

SBTP4

MITC4 (Chen and Cheung 2000)

4 × 4

0.231

0.372

0.359

0.143

0.369

0.358

8 × 8

0.323

0.388

0.357

0.206

0.324

0.343

16 × 16

0.380

0.411

0.383

0.280

0.324

0.343

32 × 32

0.405

0.419

0.404

0.339

0.366

0.359

Morley (1963)

0.408

0.408

Fig. 10

Central displacement (Wc/WRef) for the Morley’s skew plate

Free vibration of square plates

Convergence tests of the formulated quadrilateral and triangular elements are first undertaken for simply supported (W = βs = 0) and clamped plates with two thickness–side ratios (h/L = 0.005 and 0.1) (Fig. 3). The results of the first six non-dimensional frequencies (λ = (ω2ρL4h/D)1/4) using the SBQP and SBTP4 elements with four regular meshes (N = 4, 8, 16 and 22) are presented in Tables 8, 9, 10 and 11 and Figs. 11 and 12 together with the four-node mixed interpolation of tensorial component MITC4 (Nguyen-Thoi et al. 2012), the discrete shear gap triangle DSG3 (Nguyen-Thoi et al. 2012) and the edge-based smoothed discrete shear gap triangular ES-DSG (Nguyen-Thoi et al. 2012) elements. It can be demonstrated that:
Table 8

Six first nondimensional frequency parameters (λ) of a SSSS thin square plate (h/L = 0.005)

Meshing

Elements

Mode sequence number

1

2

3

4

5

6

4 × 4

SBQP

4.4004

7.1140

7.1140

8.6298

10.7342

10.7342

SBTP4

4.4001

7.1126

7.1139

8.6298

10.7336

10.7336

MITC4 (Nguyen-Thoi et al. 2012)

4.6009

8.0734

8.0734

10.305

15.0109

15.0109

DSG3 (Nguyen-Thoi et al. 2012)

5.5626

8.8148

11.8281

13.4126

18.1948

19.2897

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.9168

8.1996

9.4593

11.5035

14.2016

15.0164

8 × 8

SBQP

4.4315

7.0383

7.0383

8.7997

10.1044

10.1044

SBTP4

4.4313

7.0377

7.0379

8.7988

10.1029

10.1029

MITC4 (Nguyen-Thoi et al. 2012)

4.4812

7.2519

7.2519

9.2004

10.7796

10.7796

DSG3 (Nguyen-Thoi et al. 2012)

4.7327

7.4926

8.2237

10.2755

11.6968

12.4915

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.5376

7.2981

7.4659

9.6486

10.8937

11.0280

16 × 16

SBQP

4.4398

7.0271

7.0271

8.8618

9.9727

9.9727

SBTP4

4.4398

7.0267

7.0267

8.8612

9.9714

9.9714

MITC4 (Nguyen-Thoi et al. 2012)

4.4522

7.0792

7.0792

8.9611

10.1285

10.1285

DSG3 (Nguyen-Thoi et al. 2012)

4.5131

7.1502

7.3169

9.3628

10.3772

10.4461

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.4641

7.0870

7.1193

9.0582

10.1444

10.1489

22 × 22

SBQP

4.4412

7.0256

7.0256

8.8722

9.9535

9.9535

SBTP4

4.4411

7.0252

7.0253

8.8717

9.9522

9.9522

MITC4 (Nguyen-Thoi et al. 2012)

4.4477

7.0531

7.0531

8.9247

10.0349

10.0349

DSG3 (Nguyen-Thoi et al. 2012)

4.4781

7.0905

7.1718

9.1455

10.1643

10.1814

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.4537

7.0565

7.0729

8.9731

10.0410

10.0422

 

Exact (Abbassian et al. 1987)

4.4430

7.0250

7.0250

8.8860

9.93500

9.93500

Table 9

Six first nondimensional frequency parameters (λ) of a SSSS thick square plate (h/L = 0.1)

Meshing

Elements

Mode sequence number

1

2

3

4

5

6

4 × 4

SBQP

4.3296

6.8538

6.8538

8.2153

10.0531

10.0531

SBTP4

4.3212

6.7855

6.8014

8.1724

9.8150

9.8166

MITC4 (Nguyen-Thoi et al. 2012)

4.5146

7.6192

7.6192

9.4471

12.2574

12.2574

DSG3 (Nguyen-Thoi et al. 2012)

4.9970

8.1490

9.4311

11.354

14.1290

14.9353

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.7376

7.6580

8.4524

10.1882

12.1227

12.7533

8 × 8

SBQP

4.3566

6.7647

6.7647

8.3052

9.3981

9.3981

SBTP4

4.3522

6.7315

6.7409

8.2739

9.2985

9.2986

MITC4 (Nguyen-Thoi et al. 2012)

4.4025

6.9402

6.9402

8.6082

9.8582

9.8582

DSG3 (Nguyen-Thoi et al. 2012)

4.4891

7.0697

7.2530

9.1263

10.2195

10.3361

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.4433

6.9495

7.0727

8.8487

9.8575

9.9221

16 × 16

SBQP

4.3639

6.7488

6.7488

8.3412

9.2637

9.2637

SBTP4

4.3625

6.7384

6.7415

8.3305

9.2333

9.2333

MITC4 (Nguyen-Thoi et al. 2012)

4.3753

6.7918

6.7918

8.4166

9.3728

9.3728

DSG3 (Nguyen-Thoi et al. 2012)

4.3943

6.8227

6.8587

8.5447

9.4557

9.4616

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.3846

6.7922

6.8196

8.4744

9.3666

9.3698

22 × 22

SBQP

4.3650

6.7466

6.7466

8.3473

9.2437

9.2437

SBTP4

4.3643

6.7408

6.7425

8.3413

9.2270

9.2271

MITC4 (Nguyen-Thoi et al. 2012)

4.3711

6.7692

6.7692

8.3872

9.3009

9.3009

DSG3 (Nguyen-Thoi et al. 2012)

4.3809

6.7854

6.8037

8.4543

9.3441

9.3457

ES-DSG3 (Nguyen-Thoi et al. 2012)

4.3759

6.7692

6.7834

8.4173

9.2968

9.2976

 

Exact (Abbassian et al. 1987)

4.3700

6.7400

6.7400

8.3500

9.2200

9.2200

Table 10

Six first nondimensional frequency parameters (λ) of a CCCC thin square plate (h/L = 0.005)

Meshing

Elements

Mode sequence number

1

2

3

4

5

6

4 × 4

SBQP

6.2197

9.6440

9.6440

11.0163

47.3130

47.3130

SBTP4

6.2185

9.6418

9.6428

11.0195

32.7089

33.3128

MITC4 (Nguyen-Thoi et al. 2012)

6.5638

11.523

11.523

13.9510

62.6050

62.6054

DSG3 (Nguyen-Thoi et al. 2012)

8.4197

12.772

14.965

17.2580

21.3890

21.7600

ES-DSG3 (Nguyen-Thoi et al. 2012)

6.9741

10.193

11.476

13.0550

15.4040

15.9360

8 × 8

SBQP

6.0465

8.7528

8.7528

10.484

12.0573

12.0881

SBTP4

6.0458

8.7509

8.7511

10.4819

12.0533

12.0842

MITC4 (Nguyen-Thoi et al. 2012)

6.1235

9.0602

9.0602

11.019

12.998

13.0263

DSG3 (Nguyen-Thoi et al. 2012)

6.7161

9.7867

10.567

12.998

14.531

15.3143

ES-DSG3 (Nguyen-Thoi et al. 2012)

6.1982

9.0117

9.2894

11.562

12.795

13.0357

16 × 16

SBQP

6.0097

8.6083

8.6083

10.4173

11.5973

11.6256

SBTP4

6.0091

8.6068

8.6069

10.4152

11.5938

11.6222

MITC4 (Nguyen-Thoi et al. 2012)

6.0285

8.6801

8.6801

10.5443

11.7989

11.8266

DSG3 (Nguyen-Thoi et al. 2012)

6.1786

8.8759

9.0680

11.2450

12.2180

12.2992

ES-DSG3 (Nguyen-Thoi et al. 2012)

6.0355

8.6535

8.7081

10.6580

11.7430

11.7720

22 × 22

SBQP

6.0041

8.5875

8.5875

10.4084

11.5342

11.5620

SBTP4

6.0036

8.5861

8.5862

10.4065

11.5309

11.5588

MITC4 (Nguyen-Thoi et al. 2012)

6.0140

8.6252

8.6252

10.4750

11.6390

11.6661

DSG3 (Nguyen-Thoi et al. 2012)

6.0889

8.7239

8.8202

10.8567

11.8519

11.8845

ES-DSG3 (Nguyen-Thoi et al. 2012)

6.0158

8.6075

8.6353

10.5252

11.6032

11.6293

 

Exact (Abbassian et al. 1987)

5.9990

8.568

8.568

10.4070

11.4720

11.4980

Table 11

Six first nondimensional frequency parameters (λ) of a CCCC thick square plate (h/L = 0.1)

Meshing

Elements

Mode sequence number

1

2

3

4

5

6

4 × 4

SBQP

5.9066

8.6852

8.6852

9.9470

13.0167

13.0351

SBTP4

5.8216

8.4187

8.4313

9.7144

12.0144

12.1514

MITC4 (Nguyen-Thoi et al. 2012)

6.1612

9.5753

9.5753

11.254

14.089

14.1377

DSG3 (Nguyen-Thoi et al. 2012)

6.8748

9.8938

11.085

12.636

15.103

15.6402

ES-DSG3 (Nguyen-Thoi et al. 2012)

6.2662

8.7952

9.6625

10.911

12.610

13.1360

8 × 8

SBQP

5.7477

8.0260

8.0260

9.4133

10.5267

10.5889

SBTP4

5.7098

7.9273

7.9340

9.3085

10.3191

10.3775

MITC4 (Nguyen-Thoi et al. 2012)

5.8079

8.2257

8.2257

9.7310

10.992

11.0457

DSG3 (Nguyen-Thoi et al. 2012)

5.9547

8.3618

8.6293

10.299

11.342

11.5397

ES-DSG3 (Nguyen-Thoi et al. 2012)

5.8068

8.0861

8.2701

9.8397

10.760

10.8960

16 × 16

SBQP

5.7140

7.9117

7.9117

9.3446

10.2143

10.2656

SBTP4

5.7022

7.8816

7.8843

9.3110

10.1541

10.2045

MITC4 (Nguyen-Thoi et al. 2012)

5.7288

7.9601

7.9601

9.4230

10.326

10.3752

DSG3 (Nguyen-Thoi et al. 2012)

5.7616

7.9935

8.0525

9.5772

10.415

10.4697

ES-DSG3 (Nguyen-Thoi et al. 2012)

5.7250

7.9211

7.9627

9.4499

10.263

10.3126

22 × 22

SBQP

5.7088

7.8949

7.8949

9.3351

10.1697

10.2195

SBTP4

5.7023

7.8784

7.8800

9.3165

10.1370

10.1863

MITC4 (Nguyen-Thoi et al. 2012)

5.7166

7.9204

7.9204

9.3764

10.2280

10.2771

DSG3 (Nguyen-Thoi et al. 2012)

5.7337

7.9381

7.9686

9.4589

10.2760

10.3246

ES-DSG3 (Nguyen-Thoi et al. 2012)

5.7141

7.8990

7.9206

9.3896

10.1935

10.2411

 

Exact (Abbassian et al. 1987)

5.7100

7.8800

7.8800

9.3300

10.1300

10.1800

Fig. 11

Six first frequencies of a simply supported square plate with a 4 × 4 mesh

Fig. 12

Six first frequencies of a clamped square plate with a 22 × 22 mesh

  • Both elements (SBQP and SBTP4) agree well with analytical solutions (Abbassian et al. 1987) and other elements (MITC4, DSG3, and ES-DSG) (Nguyen-Thoi et al. 2012).

  • Figures 11 and 12 show that the SBQP and SBTP4 elements produce more accurate results than those given by other elements (MITC4, DSG3, and ES-DSG) (Nguyen-Thoi et al. 2012) when few elements are employed (4 × 4 mesh).

Having verified the convergence rate of the formulated elements, thin square plates (h/L = 0.005) with five different kinds of boundary conditions (SSSF, SFSF, CCCF, CFCF, and CFSF) for a 22 × 22 mesh are considered. The results of the four non-dimensional frequencies (λ = ωL2(ρh/D)1/2) are presented in Table 12 and the first four mode shapes of SSSF and CFCF plates are plotted in Figs. 13 and 14. For all cases of boundary condition, the following can be concluded:
Table 12

Four first nondimensional frequency parameters (λ) of a thin square plate (h/L = 0.005)

Boundary conditions

Elements

Mode sequence number

1

2

3

4

SSSF

SBQP

11.6920

27.7371

41.3354

59.0370

SBTP4

11.6914

27.7350

41.3290

59.0289

MITC4 (Nguyen-Thoi et al. 2012)

11.7085

27.8259

41.5907

59.4952

DSG3 (Nguyen-Thoi et al. 2012)

11.7553

28.2580

41.8252

61.1274

ES-DSG3 (Nguyen-Thoi et al. 2012)

11.6817

27.8143

41.3866

59.5521

Exact (Leissa 1969)

11.6850

27.7560

41.1970

59.0660

SFSF

SBQP

9.6426

16.1396

36.6991

39.1045

SBTP4

9.6422

16.1381

36.6952

39.0984

MITC4 (Nguyen-Thoi et al. 2012)

9.6560

16.1594

36.8250

39.3439

DSG3 (Nguyen-Thoi et al. 2012)

9.6608

16.3096

37.5011

39.4050

ES-DSG3 (Nguyen-Thoi et al. 2012)

9.6402

16.1214

36.8606

39.1664

Exact (Leissa 1969)

9.6310

16.1350

36.7260

38.9450

CCCF

SBQP

24.0205

40.0559

63.8154

76.9320

SBTP4

24.0158

40.0475

63.7906

76.9100

MITC4 (Nguyen-Thoi et al. 2012)

24.0559

40.1776

64.2683

77.5923

DSG3 (Nguyen-Thoi et al. 2012)

24.2149

41.4350

64.6795

80.2128

ES-DSG3 (Nguyen-Thoi et al. 2012)

23.8927

40.1428

63.4463

77.6415

Exact (Leissa 1969)

24.0200

40.0390

63.4930

76.7610

CFCF

SBQP

22.2733

26.5042

43.6303

61.7962

SBTP4

22.2691

26.4981

43.6205

61.7722

MITC4 (Nguyen-Thoi et al. 2012)

22.3107

26.5333

43.7558

62.2403

DSG3 (Nguyen-Thoi et al. 2012)

22.3132

27.0330

45.4552

62.2851

ES-DSG3 (Nguyen-Thoi et al. 2012)

22.1684

26.4128

43.8441

61.4711

Exact (Leissa 1969)

22.2720

26.5290

43.6640

64.4660

 

SBQP

15.2347

20.6194

39.7292

49.7881

 

SBTP4

15.2331

20.6163

39.7230

49.7750

CFSF

MITC4 (Nguyen-Thoi et al. 2012)

15.2590

20.6440

39.8569

50.1204

 

DSG3 (Nguyen-Thoi et al. 2012)

15.2635

20.9362

40.9260

50.1777

 

ES-DSG3 (Nguyen-Thoi et al. 2012)

15.2002

20.5789

39.9116

49.7129

 

Exact (Leissa 1969)

15.2850

20.6730

39.8820

49.5000

Fig. 13

First four mode shapes of SSSF square plate using the SBQP element

Fig. 14

First four mode shapes of CFCF square plate using the SBTP4 element

  • The present results are very close to analytical solutions (Leissa 1969) and are more accurate than those of the MITC4, DSG3 and ES-DSG elements (Nguyen-Thoi et al. 2012).

  • The two elements (SBQP and SBTP4) have similar behavior, are shear locking free and their accuracy is insensitive to boundary conditions.

Free vibration of parallelogram plates

A cantilever parallelogram plate of skew angle = 60° with two h/L ratios (0.001 and 0.2) is studied (Fig. 15) using 22 × 22 mesh. The computed six non-dimensional frequencies (λ = ωL2/π2(ρh/D)1/2) and the mode shapes are illustrated in Table 13 and Fig. 16, respectively. These results are compared with other numerical (DSG3, ES-DSG3, and MITC4) (Nguyen-Thoi et al. 2012) and analytical solutions (Karunasena et al. 1996). It can be seen that the SBQP and SBTP4 elements have a good accuracy compared to exact solutions (Karunasena et al. 1996) and are good competitors to ES-DSG3 and MITC4 (Nguyen-Thoi et al. 2012) and better than DSG3 (Nguyen-Thoi et al. 2012).
Fig. 15

Cantilever skew plate with a mesh of N × N elements

Table 13

Frequency parameters (λ) of cantilever skew plates (CFFF)

Mode

h/L

SBQP

SBTP4

DSG3 (Nguyen-Thoi et al. 2012)

ES-DSG3 (Nguyen-Thoi et al. 2012)

MITC4 (Nguyen-Thoi et al. 2012)

Exact (Karunasena et al. 1996)

1

0.001

0.3990

0.3988

0.4019

0.3981

0.3984

0.3980

2

0.9594

0.9568

0.9949

0.9532

0.9552

0.9540

3

2.5871

2.5776

2.6392

2.5692

2.5776

2.5640

4

2.6392

2.6351

2.8569

2.6508

2.6395

2.6270

5

4.2143

4.2046

4.3554

4.2030

4.2163

4.1890

6

5.1612

5.1475

6.0079

5.2283

5.1728

5.1310

1

0.2

0.3781

0.3778

0.3783

0.3772

0.3777

0.3770

2

0.8188

0.8183

0.8187

0.8129

0.8190

0.8170

3

1.9890

1.9869

1.9738

1.9573

1.9911

1.9810

4

2.1695

2.1670

2.1982

2.1786

2.1748

2.1660

5

3.1150

3.1097

3.1374

3.0999

3.1224

3.1040

6

3.7649

3.7585

3.8689

3.8050

3.7835

3.7600

Fig. 16

Mode shapes of a cantilever skew plate with h/L = 0.2

Buckling of square plates subjected to uniaxial compression

Square plates subjected to uniaxial compression (Fig. 17) with h/L of 0.01 is analyzed for both simply supported (SSSS) and clamped (CCCC). The buckling load factor is defined as Kh = λcrL2/(π2/D). The results of the buckling load factor for the SBQP and SBTP4 elements using 4 × 4, 8 × 8, 12 × 12, 16 × 16 and 20 × 20 meshes are presented in Table 14 and Fig. 18. For all cases of boundary condition, the two elements (SBQP and SBTP4) have similar results and converge to analytical solutions (Timoshenko and Gere 1970). In addition, these elements have excellent accuracy compared to other elements (DSG3 and ES-DSG3) (Nguyen-Xuan et al. 2010a, b).
Fig. 17

Square plate subjected to axial compression

Table 14

Convergence of uniaxial buckling load factor (Kh) of square plates with (h/L = 0.01)

Plates type

Elements

4 × 4

8 × 8

12 × 12

16 × 16

20 × 20

Timoshenko and Gere (1970)

SSSS

SBQP

3.8452

3.9568

3.9790

3.9869

3.9905

4.00

SBTP4

3.8434

3.9558

3.9782

3.9862

3.9899

DSG3 (Nguyen-Xuan et al. 2010a, b)

7.5891

4.8013

4.3200

4.1590

4.0889

ES-DSG3 Nguyen-Xuan et al. (2010a, b)

4.7023

4.1060

4.0368

4.0170

4.0089

CCCC

SBQP

11.1243

10.3089

10.1625

10.1120

10.0887

10.07

SBTP4

11.1082

10.2955

10.1498

10.0999

10.0774

DSG3 (Nguyen-Xuan et al. 2010a, b)

31.8770

14.7592

11.9823

11.0446

10.6282

ES-DSG3 (Nguyen-Xuan et al. 2010a, b)

14.7104

11.0428

10.3881

10.2106

10.1410

Fig. 18

Convergence of uniaxial buckling load factor (Kh/Kexact) of square plates with h/L = 0.01

The results of the buckling load factor (Kh) and the relative error using 20 × 20 mesh are presented in Table 15. Numerical results of the SBQP and SBTP4 elements are in good agreement with analytical solutions (Timoshenko and Gere 1970) and other numerical solutions (Nguyen-Xuan et al. 2010a, b; Tham and Szeto 1990; Vrcelj and Bradford 2008; Liew and Chen 2004).
Table 15

Uniaxial buckling load factor (Kh) of square plates with (h/L = 0.01)

Plates type

SBQP

SBTP4

DSG3 (Nguyen-Xuan et al. 2010a, b)

ES-DSG3 (Nguyen-Xuan et al. 2010a, b)

Liew and Chen (2004)

Ansys (Liew and Chen 2004)

Timoshenko and Gere (1970)

Tham and Szeto (1990)

Vrcelj and Bradford (2008)

SSSS

3.9905 (− 0.24%)

3.9862 (− 0.34%)

4.0889 (2.22%)

4.0089 (0.22%)

3.9700 (− 0.75%)

4.0634 (1.85%)

4.00 (0.0%)

4.00 (0.0%)

4.0006 (0.02%)

CCCC

10.0887 (0.18%)

10.0774 (0.07%)

10.6282 (5.54%)

10.1410 (0.70%)

10.1501 (0.8%)

10.1889 (1.18%)

10.07 (0.0%)

10.08 (0.1%)

10.0871 (0.17%)

Buckling of square plates subjected to biaxial compression

Square plate subjected to biaxial compression (Fig. 19) with three essential boundary conditions (SSSS, CCCC, SCSC) is considered for h/L = 0.01 using a mesh of 16 × 16. The buckling load factor results (Kh = λcrL2/(π2/D)) of the proposed elements are presented in Table 16 with analytical (Timoshenko and Gere 1970) and other numerical solutions (Nguyen-Xuan et al. 2010a, b; Tham and Szeto 1990; Vrcelj and Bradford 2008). It can be seen that both elements (SBQP and SBTP4) provide results which agree well with analytical solutions (Timoshenko and Gere 1970) and other solutions (Nguyen-Xuan et al. 2010a, b; Tham and Szeto 1990; Vrcelj and Bradford 2008) for all cases of boundary condition.
Fig. 19

Square plate subjected to biaxial compression

Table 16

Biaxial buckling load factor (Kh) of square plates with (h/L = 0.01)

Plates type

SBQP

SBTP4

DSG3 (Nguyen-Xuan et al. 2010a, b)

ES-DSG3 (Nguyen-Xuan et al. 2010a, b)

Timoshenko and Gere (1970)

Tham and Szeto (1990)

Vrcelj and Bradford (2008)

SSSS

1.9934

1.9931

2.0549

2.0023

2.00

2.00

2.0008

CCCC

5.3039

5.2991

5.6419

5.3200

5.31

5.61

5.3260

SCSC

3.8331

3.8279

4.0108

3.8332

3.83

3.83

3.8419

Conclusion

A simple and efficient quadrilateral and triangular strain-based finite elements have been presented for static, free vibration and buckling analyses of Reissner–Mindlin plates. The four-node strain-based triangular element SBTP4 has the three engineering external degrees of freedom at each of the three corner nodes and one mid-edge point, while the quadrilateral element SBQP has the same engineering degrees of freedom at each of the four corner nodes. These developed elements passed successfully both patch and benchmark tests for plate bending problems. Numerical results show that the SBQP and SBTP4 elements are shear locking free, stable and superior to the original strain-based rectangular plate element (SBRP) (Belounar and Guenfoud 2005) which suffers from shear locking when the plate thickness becomes progressively very thin and has less rate of convergence to analytical solutions for thick and thin plates. The obtained results using both strain-based elements (SBQP and SBTP4) show that a rapid convergence to analytical solutions can be achieved with relatively coarse meshes compared with other robust elements based on different methods. In perspective, these elements can be superposed with membrane robust elements to construct shell elements for the analysis of complex shell structures.

Notes

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Authors and Affiliations

  • Abderahim Belounar
    • 1
    Email author
  • Sadok Benmebarek
    • 1
  • Mohamed Nabil Houhou
    • 1
  • Lamine Belounar
    • 1
  1. 1.NMISSI Laboratory, Faculty of Science and TechnologyBiskra UniversityBiskraAlgeria

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