Enhancement to four-node quadrilateral plate elements by using cell-based smoothed strains and higher-order shear deformation theory for nonlinear analysis of composite structures

Abstract

This paper improves four-node quadrilateral plate elements by using cell-based strain smoothing enhancement and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of composite structures. Small strain-large displacement theory of von Kármán is used in nonlinear formulations of four-node quadrilateral plate elements that have strain components smoothed or averaged over the sub-domains of the elements. From the divergence theory, the displacement gradients in the smoothed strains are transformed from the area integral into the line one. The behavior of composite structures follows the third-order shear deformation theory. The solution of the nonlinear equilibrium equations is obtained by the iterative method of Newton–Raphson with the appropriate convergence criteria. The present numerical results are compared with the other numerical results available in the literature in order to demonstrate the effectiveness of the developed element. These results also contribute a better knowledge and understanding of nonlinear bending behaviors of these composite structures.

Keywords

Composite plates nonlinear bending analysis higher-order shear deformation theory cell-based smoothed strains folded plates

Introduction

In recent years, structures made of composite materials have been using intensively in many engineering applications such as aerospace, marine and civil infrastructure, etc., because they possess many favorable mechanical properties such as high stiffness to weight and low density. This is particularly important in aerospace and submarine structures that require a high stiffness and a considerable amount of weight-saving. For using the composite plates efficiently, it is necessary to develop approximation theories [1 –5] to predict accurately their structural and dynamical behavior. Noor et al. [6 –10] first recommended a three-dimensional (3D) elasticity theory to improve the accuracy of transverse shear stresses. In the 3D elasticity theory, each layer is modeled as a 3D solid, and the accuracy of transverse shear stresses can be significantly improved. However, the computational cost is increased significantly when using the 3D elasticity theory. Many equivalent single layer (ESL) plate theories [11] such as the classical plate theory (CPT), first-order shear deformation theory (FSDT) and higher-order shear deformation plate theory (HSDT) have been then proposed to reduce the 3D models to 2D ones. The CPT relies on the Love–Kirchhoff assumptions ignoring the transverse shear deformation while the FSDT takes into account the effects of shear deformation. The FSDT is simple to implement and can be applied for both thick and thin composite plates. However, the accuracy of solutions using the FSDT is strongly dependent on shear correction factors. In addition, the performance of the FSDT for composite plates has not been satisfactory in predicting the accuracy and smoothed variations of stresses, especially for composite plates with clamped edges, free edges or highly skewed geometry with high stress gradients [12 –21]. The limitations of the FSDT can be overcome by introducing the HSDT. The HSDT has been developed by Reddy [22], Matsunaga [23], Kant and Swaminathan [24], Liu et al. [25], Mantari et al. [26 –30], Singh and Singh [31], Ganapathi and Makhecha [32], Shi [33], and Thai et al. [34]. These models do not need shear correction factors and give more accurate and stable transverse shear stresses.

Nowadays, although many new numerical methods [35 –41] have been developed, the finite element methods (FEM) are still the most efficient and popular ones to successfully analyze plate and shell structures. However, the requirement of C¹-continuous approximation for the displacement fields based on the HSDT theory causes some obstacles when the lower order finite elements are used in the computation. To overcome these shortcomings, Shankara and Iyengar [42] proposed a refined form of HSDT which only requires C⁰ continuity of generalized displacements (C⁰-HSDT). In the C⁰-type HSDT, two additional variables have been included in the displacement field, and hence only the first derivative of transverse displacements is needed. Based on this C⁰-type HSDT, Nayak et al. [43] investigated the C⁰-type four-node and nine-node finite elements for the transient response of orthotropic, layered composite sandwich plates. Ganapathi et al. [44] used C⁰-type eight-node membrane shear-bending element for geometrically nonlinear analysis of laminates. Kant et al. [45, 46] studied a C⁰-type nine-node quadrilateral element for geometrically nonlinear analysis of laminated plates. Phung et al. [47, 48] extend the CS-FEM-MIN3 to geometrically nonlinear analysis of laminated composite plates or functionally graded plates based on the C⁰-type HSDT. Phung et al. [49] also extend the CS-DSG3 method based on the C⁰-type HSDT for analysis of functionally graded plates. Accurate transverse stress evaluation in composite/sandwich thick laminates using a C⁰-HSDT and a novel post-processing technique is proposed by Bhar and Satsangi [50]. Nguyen et al. [51] presented an edge-based MITC3 finite elements to static and vibration analysis of isotropic and functionally graded sandwich plates based on the C⁰-HSDT. Furthermore, C⁰-type global–local HSDT including transverse normal thermal strain for laminated composite plates under thermal loading is given by Zhen and Li [52]. The polygonal element with C⁰-type HSDT for analysis of laminated composite plates is also proposed by Nguyen et al. [53]. The static analysis of functionally graded plates using the C⁰-type HSDT and MITC3 plate elements having strains smoothed on edges is studied by Chau-Dinh et al. [54]. In spite of usually yielding more accurate results than the cell-based smoothing approach, the edge-based smoothing one encounters difficulty when applied to non-planar elements at the folding edges of plates or shell structures. Until now, there are few references using the edge-based smoothing technique for three-node triangular elements with constant stiffness matrices [55 –57] but not for four-node quadrilateral elements.

This paper presents a novel numerical procedure based on the modification of smoothed quadrilateral element [58 –69] associated with the C⁰-type HSDT for geometrically nonlinear analyses of composite plates. The higher-order shear deformation plate theory is involved in the formulation in order to avoid using the shear correction factors and to improve the accuracy of transverse shear stresses. In the present method, the membrane and bending strains are smoothed over sub-quadrilateral domains of elements. As a result, the membrane and bending stiffness matrices are integrated along the boundary of the smoothing domains instead of over the element surfaces. And the shear stiffness matrix is based on reduced-integration technique to remove the shear-locking phenomenon. Compared with the conventional FEM, the present approach requires more computational time for the gradient matrices of the membrane and bending strains when more than one smoothing domain are employed. However, the present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to show the performance of the proposed method and results obtained are compared to other published methods in the literature.

The paper is outlined as follows. First, a brief review of C⁰-type HSDT and a weak form of plate model are described in the C⁰-type higher-order shear deformation theory and weak form for plate model section. The Formulations of cell-based smoothed four-node quadrilateral plate element (SRIQ28) section presents a formulation of smoothed reduced-integration element SRIQ28 with 28 degrees of freedom for composite plates. Several numerical examples are provided in the Numerical results section. Finally, some concluding remarks are drawn.

C⁰-type higher-order shear deformation theory and weak form for plate model

Let Ω be the domain in R² occupied by the mid-plane of the plate. The displacements of an arbitrary point in the plate are expressed as [42]

\begin{array}{l} u (x, y, z) = u_{0} + (z - \frac{4 z^{3}}{3 h^{2}}) β_{y} - \frac{4 z^{3}}{3 h^{2}} ϕ_{x} \\ v (x, y, z) = v_{0} - (z - \frac{4 z^{3}}{3 h^{2}}) β_{x} - \frac{4 z^{3}}{3 h^{2}} ϕ_{y} - \frac{h}{2} \leq z \leq \frac{h}{2} \\ w (x, y, z) = w_{0} \end{array}

(1)

where

u_{0}, v_{0}

and

w_{0}

are axial and transverse displacements at the mid-surface of the plates, respectively;

ϕ_{x}, ϕ_{y}, β_{x}, β_{y}

are rotations due to the bending and shear effects. It can be seen that the present theory is composed of seven unknowns: three axial and transverse displacements, four rotations with respect to the

y

- and

x -

axis as shown in Figure 1.

For large deformation analysis, the in-plane vector of Green–Lagrangian strain in a plate element is

ε = {\begin{cases} ε_{x} \\ ε_{y} \\ ε_{x y} \end{cases}} = {\begin{cases} u_{, x} + \frac{1}{2} w_{, x}^{2} \\ v_{, y} + \frac{1}{2} w_{, y}^{2} \\ u_{, y} + v_{, x} + w_{, x} w_{, y} \end{cases}}

(2)

where the comma indicates partial differentiation with respect to the coordinate subscript that follows. Substituting equation (1) into equation (2) and considering the von Karman’s large deflection assumption, the in-plane strain vector can be rewritten as

ε = ε_{0} + z κ_{1} - \frac{4}{3 h^{2}} z^{3} κ_{2}

(3)

in which

ε_{0} = {\begin{matrix} u_{0, x} \\ v_{0, y} \\ u_{0, y} + v_{0, x} \end{matrix}} + {\begin{cases} \frac{1}{2} w_{0, x}^{2} \\ \frac{1}{2} w_{0, y}^{2} \\ w_{0, x} w_{0, y} \end{cases}} = ε_{0}^{L} + ε_{0}^{N L}

(4)

κ_{1} = [\begin{matrix} β_{y, x} \\ - β_{x, y} \\ β_{y, y} - β_{x, x} \end{matrix}]

(5)

κ_{2} = [\begin{matrix} β_{y, x} + ϕ_{x, x} \\ - β_{x, y} + ϕ_{y, y} \\ β_{y, y} - β_{x, x} + ϕ_{x, y} + ϕ_{y, x} \end{matrix}]

(6)

ε_{0}^{N L} = \frac{1}{2} [\begin{matrix} w_{0, x} & 0 \\ 0 & w_{0, y} \\ w_{0, y} & w_{0, x} \end{matrix}] {\begin{matrix} w_{0, x} \\ w_{0, y} \end{matrix}} = \frac{1}{2} Ψ θ

(7)

The transverse shear strain vector is given as

γ = ε_{s} + z^{2} κ_{s}

(8)

in which

ε_{s} = [\begin{matrix} w_{0, x} + β_{y} \\ w_{0, y} - β_{x} \end{matrix}]

(9)

κ_{s} = - \frac{4}{h^{2}} [\begin{matrix} ϕ_{x} + β_{y} \\ ϕ_{y} - β_{x} \end{matrix}]

(10)

The total strain can be written as

\hat{ε} = {\hat{ε}}^{L} + {\hat{ε}}^{N L} = {[\begin{matrix} ε_{0}^{L} & κ_{1} & κ_{2} & ε_{s} & κ_{s} \end{matrix}]}^{T} + {[\begin{matrix} ε_{0}^{N L} & 0 & 0 & 0 & 0 \end{matrix}]}^{T}

(11)

The composite plate is usually made of several orthotropic layers in which the stress–strain relation for the kth orthotropic lamina with the arbitrary fiber orientation maps to the reference is

{\begin{cases} σ_{x} \\ σ_{y} \\ σ_{x y} \\ τ_{x z} \\ τ_{y z} \end{cases}}^{(k)} = {[\begin{matrix} {\bar{Q}}_{11} & {\bar{Q}}_{12} & {\bar{Q}}_{16} & 0 & 0 \\ {\bar{Q}}_{21} & {\bar{Q}}_{22} & {\bar{Q}}_{26} & 0 & 0 \\ {\bar{Q}}_{61} & {\bar{Q}}_{62} & {\bar{Q}}_{66} & 0 & 0 \\ 0 & 0 & 0 & {\bar{Q}}_{55} & {\bar{Q}}_{54} \\ 0 & 0 & 0 & {\bar{Q}}_{45} & {\bar{Q}}_{44} \end{matrix}]}^{(k)} {\begin{cases} ε_{x} \\ ε_{y} \\ ε_{x y} \\ γ_{x z} \\ γ_{y z} \end{cases}}^{(k)}

(12)

where

{\bar{Q}}_{i j} (i, j = 1, 2, 4, 5, 6)

are the material constants of kth layer in global coordinates system. Under weak form, the normal forces, bending moments, higher-order moments, shear forces and higher-order shear forces can then be computed through the following relations

{\begin{cases} \tilde{N} \\ \tilde{M} \\ \tilde{P} \\ \tilde{Q} \\ \tilde{R} \end{cases}} = [\begin{matrix} A & B & c_{1} E & 0 & 0 \\ B & D & c_{1} F & 0 & 0 \\ c_{1} E & c_{1} F & c_{1}^{2} H & 0 & 0 \\ 0 & 0 & 0 & c_{2} \hat{A} & c_{2} \hat{B} \\ 0 & 0 & 0 & c_{2} \hat{B} & c_{2}^{2} \hat{D} \end{matrix}] {\begin{cases} ε_{0} \\ κ_{1} \\ κ_{2} \\ ε_{s} \\ κ_{s} \end{cases}}

(13)

with

(A, B, D, E, F, H) = \int_{- h / 2}^{h / 2} (1, z, z^{2}, z^{3}, z^{4}, z^{6}) {\bar{Q}}_{i j} d z i, j = 1, 2, 6

(14)

(\hat{A}, \hat{B}, \hat{D}) = \int_{- h / 2}^{h / 2} (1, z^{2}, z^{4}) {\bar{Q}}_{i j} d z i, j = 4, 5

(15)

Formulations of cell-based smoothed four-node quadrilateral plate element (SRIQ28)

Discretize the bounded domain $Ω$ of plates into N_c finite elements such that $Ω= \cup_{c = 1}^{N_{c}} Ω_{c}$ and $Ω_{i} \cap Ω_{j} = \emptyset$ with $i \neq j$ . The finite element displacements are approximated according to the associated node displacements as

u = \sum_{i = 1}^{N_{n}} [\begin{matrix} N_{i} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & N_{i} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & N_{i} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & N_{i} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & N_{i} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & N_{i} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & N_{i} \end{matrix}] q_{i}

(16)

where N_n is the total number of nodes of the mesh,

N_{i} = \frac{1}{4} (1 + ξ_{i} ξ) (1 + η_{i} η)

is the shape function of the four-node serendipity element,

q_{i}^{T} = [\begin{matrix} u_{i} & v_{i} \end{matrix}

\begin{matrix} w_{i} & β_{x i} & β_{y i} & ϕ_{x i} & ϕ_{y i} \end{matrix}]

is the displacement vector of the nodal degrees of freedom of u associated to the ith node, respectively. The membrane, bending and shear strains can be then expressed in the matrix forms as

\begin{array}{l} ε_{0}^{L} = \sum_{i} B_{0 i}^{L} q_{i} κ_{1} = \sum_{i = 1} B_{b 1 i} q_{i} κ_{2} = \sum_{i = 1} B_{b 2 i} q_{i} \\ ε_{0}^{N L} = \sum_{i = 1} B_{0 i}^{N L} q_{i} ε_{s} = \sum_{i = 1} B_{s 1 i} q_{i} κ_{s} = \sum_{i = 1} B_{s 2 i} q_{i} \end{array}

(17)

where

\begin{array}{l} B_{0 i}^{L} = [\begin{matrix} N_{i, x} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & N_{i, y} & 0 & 0 & 0 & 0 & 0 \\ N_{i, y} & N_{i, x} & 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ B_{b 1 i} = [\begin{matrix} 0 & 0 & 0 & 0 & N_{i, x} & 0 & 0 \\ 0 & 0 & 0 & - N_{i, y} & 0 & 0 & 0 \\ 0 & 0 & 0 & - N_{i, x} & N_{i, y} & 0 & 0 \end{matrix}] \\ B_{b 2 i} = [\begin{matrix} 0 & 0 & 0 & 0 & N_{i, x} & N_{i, x} & 0 \\ 0 & 0 & 0 & - N_{i, y} & 0 & 0 & N_{i, y} \\ 0 & 0 & 0 & - N_{i, x} & N_{i, y} & N_{i, y} & N_{i, x} \end{matrix}] \\ B_{0 i}^{N L} = Ψ G_{i} = [\begin{matrix} N_{i, x} & 0 \\ 0 & N_{i, y} \\ N_{i, y} & N_{i, x} \end{matrix}] [\begin{matrix} 0 & 0 & N_{i, x} & 0 & 0 & 0 & 0 \\ 0 & 0 & N_{i, y} & 0 & 0 & 0 & 0 \end{matrix}] \\ B_{s 1 i} = [\begin{matrix} 0 & 0 & N_{i, x} & 0 & N_{i} & 0 & 0 \\ 0 & 0 & N_{i, y} & - N_{i} & 0 & 0 & 0 \end{matrix}] \\ B_{s 2 i} = [\begin{matrix} 0 & 0 & 0 & 0 & N_{i} & N_{i} & 0 \\ 0 & 0 & 0 & - N_{i} & 0 & 0 & N_{i} \end{matrix}] \end{array}

(18)

For smoothing strategy, as shown in Figure 2, a quadrilateral element domain $Ω_{C}$ is further divided into nc smoothing cells. The generalized strain field is smoothed by a weighted average of the original generalized strains using the strain smoothing operation for each smoothing cell as follows

\begin{array}{l} {\tilde{ε}}_{0}^{L} = \frac{1}{A_{c}} \int_{Ω_{c}} ε_{0}^{L} (x) d Ω {\tilde{ε}}_{0}^{N L} = \frac{1}{A_{c}} \int_{Ω_{c}} ε_{0}^{N L} (x) d Ω \\ {\tilde{κ}}_{1} = \frac{1}{A_{c}} \int_{Ω_{c}} κ_{1} (x) d Ω {\tilde{κ}}_{2} = \frac{1}{A_{c}} \int_{Ω_{c}} κ_{2} (x) d Ω \end{array}

(19)

where

{\tilde{ε}}_{0}^{L}

{\tilde{ε}}_{0}^{N L}

{\tilde{κ}}_{1}

{\tilde{κ}}_{2}

are the smoothed strains and A_c is the area of the smoothing cell

Ω_{C}

Introducing the approximation of the linear membrane strain by the quadrilateral finite element and applying the divergence theorem, the smoothed membrane strain can be obtained as

{\tilde{ε}}_{0}^{L} = \frac{1}{A_{c}} \int_{Γ_{c}} n (x) u (x) d Γ = \frac{1}{A_{c}} \int_{Γ_{c}} \sum_{i = 1}^{4} n (x) N_{i} (x) q_{i} d Γ = \sum_{i = 1}^{4} {\tilde{B}}_{0 i}^{L} q_{i}

(20)

where

{\tilde{B}}_{0 i} (x_{C}) = \frac{1}{A_{C}} \int_{Γ_{C}} [\begin{matrix} N_{i} n_{x} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & N_{i} n_{y} & 0 & 0 & 0 & 0 & 0 \\ N_{i} n_{y} & N_{i} n_{x} & 0 & 0 & 0 & 0 & 0 \end{matrix}] d Γ

(21)

in which n_x and n_y are the components of the outward unit vector n normal to the boundary

Γ_{C}

. Applying Gauss integration along four segments of the boundary

Γ_{C}

of the smoothing domain

Ω_{C}

, the above equation can be rewritten in algebraic form as

{\tilde{B}}_{0 i} (x_{C}) = \frac{1}{A_{C}} \sum_{b = 1}^{4} [\begin{matrix} \sum_{n = 1}^{n G} w_{n} N_{i} (x_{b n}) n_{x} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \sum_{n = 1}^{n G} w_{n} N_{i} (x_{b n}) n_{y} & 0 & 0 & 0 & 0 & 0 \\ \sum_{n = 1}^{n G} w_{n} N_{i} (x_{b n}) n_{y} & \sum_{n = 1}^{n G} w_{n} N_{i} (x_{b n}) n_{x} & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(22)

where

n G

is the number of Gauss integration points,

x_{b n}

is the Gauss point and

w_{n}

is the corresponding weighting coefficients. Equation (23), which relates to the in-plane translations (approximated by bilinear shape functions), is evaluated by one Gauss point

(n G = 1)

. In a similar way, the smoothed nonlinear membrane strain over the element domain

Ω_{C}

can be written as

{\tilde{ε}}_{0}^{N L} = \sum_{i = 1}^{4} {\tilde{B}}_{0 i}^{N L} q_{i}

(23)

where

{\tilde{B}}_{0 i}^{N L}

is the smoothed nonlinear gradient matrix in the smoothing cell given as

{\tilde{B}}_{0 i}^{N L} = \tilde{Ψ} {\tilde{G}}_{i}

(24)

in which

{\tilde{G}}_{i} = \frac{1}{A_{c}} \sum_{g = 1}^{4} [\begin{matrix} 0 & 0 & N_{i} (x_{g}^{G}) n_{x} & 0 & 0 & 0 & 0 \\ 0 & 0 & N_{i} (x_{g}^{G}) n_{y} & 0 & 0 & 0 & 0 \end{matrix}] l_{g}^{c},

(25)

\tilde{Ψ} = \sum_{i = 1}^{4} [\begin{matrix} \frac{1}{A_{c}} \sum_{j = 1}^{4} N_{i} (x_{j}^{G}) n_{x} l_{j}^{c} w_{i} & 0 \\ 0 & \frac{1}{A_{c}} \sum_{j = 1}^{4} N_{i} (x_{j}^{G}) n_{y} l_{j}^{c} w_{i} \\ \frac{1}{A_{c}} \sum_{j = 1}^{4} N_{i} (x_{j}^{G}) n_{y} l_{j}^{c} w_{i} & \frac{1}{A_{c}} \sum_{j = 1}^{4} N_{i} (x_{j}^{G}) n_{x} l_{j}^{c} w_{i} \end{matrix}]

(26)

and

w_{i}

is the deflection at the node

i

of the element.

The smoothed bending strain over the element domain $Ω_{C}$ is expressed as

{\tilde{κ}}_{1} = \sum_{i = 1}^{4} {\tilde{B}}_{b 1 i} q_{i} {\tilde{κ}}_{2} = \sum_{i = 1}^{4} {\tilde{B}}_{b 2 i} q_{i}

(27)

where

\begin{array}{l} {\tilde{B}}_{b 1 i} = \frac{1}{A_{c}} \sum_{b = 1}^{4} (\begin{matrix} 0 & 0 & 0 & 0 & N_{i} (x_{b}^{G}) n_{x} & 0 & 0 \\ 0 & 0 & 0 & - N_{i} (x_{b}^{G}) n_{y} & 0 & 0 & 0 \\ 0 & 0 & 0 & - N_{i} (x_{b}^{G}) n_{x} & N_{i} (x_{b}^{G}) n_{y} & 0 & 0 \end{matrix}) l_{b} \\ {\tilde{B}}_{b 2 i} = \frac{1}{A_{c}} \sum_{b = 1}^{4} (\begin{matrix} 0 & 0 & 0 & 0 & N_{i} (x_{b}^{G}) n_{x} & N_{i} (x_{b}^{G}) n_{x} & 0 \\ 0 & 0 & 0 & - N_{i} (x_{b}^{G}) n_{y} & 0 & 0 & N_{i} (x_{b}^{G}) n_{y} \\ 0 & 0 & 0 & - N_{i} (x_{b}^{G}) n_{x} & N_{i} (x_{b}^{G}) n_{y} & N_{i} (x_{b}^{G}) n_{y} & N_{i} (x_{b}^{G}) n_{x} \end{matrix}) l_{b} \end{array}

(28)

The reduced integration strategy is for dealing with shear locking. The solution of plate problems by independent specification of slopes and middle surface displacements is attractive due to its simplicity and its ability to reproduce shear deformation. Unfortunately, elements of this type become much too stiff as the thickness is reduced. With SRIQ28 element, one way to obtain a suitable improved flexibility is simply to use the different smoothing schemes for the individual components of the integrand of the stiffness matrix. The use of nc >1 for the bending and membrane parts and nc =1 for the shear part leads to a formulation which is free from shear locking without hurting convergence properties. From this ideal, the shear strain is expressed as

{\tilde{ε}}_{s} = \sum_{i = 1}^{4} {\tilde{B}}_{s 1 i} q_{i} {\tilde{κ}}_{s} = \sum_{i = 1}^{4} {\tilde{B}}_{s 2 i} q_{i}

(29)

where

\begin{array}{l} {\tilde{B}}_{s 1 i} = (\begin{matrix} 0 & 0 & \frac{1}{A_{c}} \int_{Γ_{c}} N_{i} (x) n_{x} d Γ & 0 & N_{i} (x) & 0 & 0 \\ 0 & 0 & \frac{1}{A_{c}} \int_{Γ_{c}} N_{i} (x) n_{y} d Γ & - N_{i} (x) & 0 & 0 & 0 \end{matrix}) \\ {\tilde{B}}_{s 2 i} = (\begin{matrix} 0 & 0 & 0 & 0 & N_{i} (x) & N_{i} (x) & 0 \\ 0 & 0 & 0 & - N_{i} (x) & 0 & 0 & N_{i} (x) \end{matrix}) \end{array}

(30)

in which N_i is the shape function value at node i of the element. However, calculation of the shear part still uses Gauss integration to calculate –N_i of

_{\tilde{B}}^{s 1 i}

and

_{\tilde{B}}^{s 2 i}

, because the integration of –N_i in the domain

Ω_{C}

cannot be transformed to a line integration on the boundary

Γ_{C}

. Finally, the element tangent stiffness matrix is modified as

{\tilde{K}}_{T} = {\tilde{K}}_{L} + {\tilde{K}}_{N L} + {\tilde{K}}_{g}

(31)

where

{\tilde{K}}_{L} = \sum_{i = 1}^{n c} {\tilde{B}}_{L i}^{T} D^{*} {\tilde{B}}_{L i} A_{i}

(32)

{\tilde{K}}_{N L} = \sum_{i = 1}^{n c} {\tilde{B}}_{N L i}^{T} D^{*} {\tilde{B}}_{N L i} A_{i}

(33)

{\tilde{K}}_{g} = \sum_{i = 1}^{n c} {\tilde{G}}_{i}^{T} \hat{N} {\tilde{G}}_{i} A_{i}

(34)

\begin{array}{l} {\tilde{B}}_{L} = [\begin{matrix} {\tilde{B}}_{0}^{L} \\ {\tilde{B}}_{b 1} \\ {\tilde{B}}_{b 2} \\ {\bar{B}}_{s 1} \\ {\bar{B}}_{s 2} \end{matrix}], {\tilde{B}}_{N L} = [\begin{matrix} {\tilde{B}}_{0}^{N L} \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}], \hat{N} = [\begin{matrix} N_{x} & N_{x y} \\ N_{x y} & N_{y} \end{matrix}] \\ D^{*} = [\begin{matrix} A & B & c_{1} E & 0 & 0 \\ B & D & c_{1} F & 0 & 0 \\ c_{1} E & c_{1} F & c_{1}^{2} H & 0 & 0 \\ 0 & 0 & 0 & c_{2} \hat{A} & c_{2} \hat{B} \\ 0 & 0 & 0 & c_{2} \hat{B} & c_{2}^{2} \hat{D} \end{matrix}] \end{array}

(35)

in which the number of smoothing cells nc =1 or 2 for shear part or membrane and bending parts in this study, respectively. The internal forces at the time t computed from the stress state in the structures can be rewritten as

\tilde{F} t = \int_{Ω} {({\tilde{B}}_{L} + {\tilde{B}}_{N L})}^{t} σ^{*} d Ω

(36)

in which the stress resultant after the

i^{t h}

iteration is

σ t_{i + 1}^{*} = σ t i_{*} + Δ t σ^{*}

(37)

Finally, the nonlinear equations can be rewritten as

\tilde{K} t_{T} Δ q = P t + Δ t - F t

(38)

Numerical results

In this section, we will test and assess the SRIQ28 element through numerical examples. In all examples, the Newton–Raphson method and automatic incremental algorithm is used to solve the nonlinear finite element equations. Especially, the procedure will be extended with bubble drilling rotation β_z for analysis of folded plate problems. The convergence tolerance of displacement is taken to be 0.001.

Isotropic plate

In this example, the geometrically nonlinear analysis of an isotropic square plate using the SRIQ28 based on the C⁰-type HSDT is presented. The plate of the length a = 10 and the thickness h = 1 is simply supported on the boundary as shown in Figure 3(a). The isotropic material has the Young’s modulus E = 7.8 × 10⁶ and the Poisson’s ration ν = 0.3. The plate is subjected to a uniform load q with load parameter $\bar{q} = q a^{4} / E h^{4}$ increased from 0 to 250. The non-dimensional central deflection $\bar{w} = w (a / 2, a / 2, 0) / h$ of plate subjected to the increment of the applied load by the SRIQ28 is plotted in Figure 3(b). As compared with the Reddy’s solution [70], the results of SRIQ28 are more accurate than those given by the cell-based smoothed three-node element (CS-FEM-MIN3) [48].

Two-layer [0°/90°] square plate

Consider a two-layer [0°/90°] square laminated composite plate with the length a and the thickness h, in which a/h = 50 or a/h = 100. The thickness of each lamina is h/2. The plate is subjected to a uniform load q as demonstrated in Figure 4(a). The material parameters are E₁/E₂ = 40, ν₁₂ = 0.25, G₁₂/E₂ = 0.6, G₂₃/E₂ = 0.5, G₁₃/G₁₂ = 1. The non-dimensional central deflections $\bar{w} = w (a / 2, a / 2, 0) / h$ of the plate corresponding to the load parameter $\bar{q} = q a^{4} / E_{2} h^{4}$ given by the present element SRIQ28, the CS-FEM-MIN3 [47] and the ANSYS commercial package [47] are illustrated in Figure 4(b) for a/h = 50 and Figure 4(c) for a/h = 100. It can be seen that the present results excellently agree with those of the references for the both cases of the length-to-thickness ratios.

Four layer [0°/90°/90°/0°] square plate

A four-layer [0°/90°/90°/0°] square laminated composite plate of the length a = 12 and the thickness h = 0.096 in Figure 5(a) is clamped on the four edges. The thickness of each lamina is h/4. The material properties of each lamina are E₁ = 1.8282 × 10⁶, E₂ = 1.8315 × 10⁶, ν₁₂ = 0.23949, G₁₂ = G₁₃ = G₂₃ = 3.125 × 10⁵. The plate is applied a uniform load q with the load parameter $\bar{q} = q a^{4} / E_{2} h^{4}$ increasing from 0 to 2. Figure 5(b) shows the curves of the central deflection and the load parameter given by the present element SRIQ28 and such references as the four-node quadrilateral element RDKQ [46], cell-based smoothed three-node element CS-FEM-MIN3 [47], experiment solution [47] and ANSYS commercial package [47]. Figure 5(b) also points out that when compared with the solution of the ANSYS commercial package, the results of the SRIQ28 are better than those provided by the RDKQ or CS-FEM-MIN3.

Four-layer [0°/90°/90°/0°] skew plate

A four-layered [0°/90°/90°/0°] clamped skew plate with the lengths a, b and the thickness h is subjected to uniform load q as shown in Figure 6. The thickness of each layer is h/4. The b-long edges skew an angle α in respect of the x-axis. The material properties used in this analysis are E₁/E₂ = 10, ν₁₂ = 0.22, G₁₂/E₂ = 0.33, G₂₃/E₂ = 0.2, G₁₃/G₁₂ = 1. For comparison with the analytical solution proposed by Upadhyay and Shukla [71], the normalized displacement at the plate center and the load parameter are defined as $\bar{w} = w (a / 2, a / 2, 0) / h$ and $\bar{q} = q a^{4} / E_{2} h^{4}$ , respectively.

We first consider the problem with the length-to-width ratio a/b = 1, the length-to-thickness ratio a/h = 10 or 100, and the angle of skew α = 0°, 30° or 60°. The normalized deflection–load parameter curves for the thick (a/h = 10) and thin (a/h = 100) plates given by the SRIQ28 element are, respectively, plotted in Figure 7(a) and (b). For the thin plate, an excellent agreement with the analytical solution [71] is achieved.

Figure 7(c) shows the results for the plate with a/h = 20 and α = 0°, 15°, 30°, 45° or 60°. By changing the length-to-width ratio a/b from 1 to 2, we obtain the load–displacement curve in Figure 7(d). The effect of modular ratio E₁/E₂ on the normalized deflection at the plate center in the case of a/h = 20 and a/b = 1 for different skew angles α = 0°, 30° or 60° is shown in Figure 7(e) for E₁/E₂ = 1 and in Figure 7(f) for E₁/E₂ = 1.

Figure 7 shows that the present approach can give reasonable results with the analytical solution [71] for a variety of shapes and the length-to-thickness ratios. However, when the skew angle α increases, the shapes of the elements in the mesh become thin and long rhomboids greatly different from the square shape of the mapping elements in the natural coordinates. This causes errors in the numerical integration of the stiffness matrices based on Gaussian quadrature points. As a result, there are some errors between the present result and the analytical solution in Figure 7(c) to (e) for the large angle of skew.

Five-layer [0°/90°/0°/90°/0°] or [45°/−45°/45°/−45°/45°] trapezoidal plate

The next performance is tested for nonlinear bending behavior of thin (a/h = 200) five-layer symmetric or unsymmetric clamped trapezoidal plate under uniform load as shown in Figures 8 and 9. The material properties used in this analysis are E₁/E₂ = 25, ν₁₂ = 0.25, G₁₂/E₂ = 0.5, G₂₃/E₂ = 0.2, G₁₃/G₁₂ = 1.

With symmetric trapezoidal plate, the parameters are given as follows: a/b = 1, α₁ = α₂ and c/a = 0.5 or 0.7 for two cases. With unsymmetric trapezoidal plate, we have a/b = 1.2, c/b = 1 and angle α₂ = 0.

The normalized displacement and load are defined as: $\bar{w} = minimum (w) / h$ , $\bar{q} = q a^{4} / E_{2} h^{4}$ . The solutions of the present method are presented in Figure 10(a) and (b). These results are compared with results of the Element-free Galerkin method proposed by Watts et al. [72]. It is again found that the obtained solutions are in good agreement with other existing ones for both cases of cross-ply and angle-ply.

Two layer [θ°/−θ°] circular plate

The large deformation analysis of a two-layered clamped circular plate under uniform pressure q is considered in this section as Figure 11. With θ = 0°, the geometry data and material properties are: radius R = 100, thickness h = 2, Young’s modulus E = 10⁷, Poisson’s ratio $ν = 0.3$ . Owing to symmetry, a quadrant of the plate is discretized and analyzed with 27 elements. The computed normalized central deflections w_c/h of the present method, together with the analytic solution of Schoop [73], and with MISQ20-FSDT [58] are displayed in Figure 12. It is interesting to note that, the present method shows good results when compared with exact solutions.

Furthermore, with θ = 15°, 30° and 45°, the material properties are E₁/E₂ = 10, ν₁₂ = 0.22, G₁₂/E₂ = 0.33, G₂₃/E₂ = 0.2, G₁₃/G₁₂ = 1. The load–central deflection curves are shown in Figure 13.

Single-fold plate

The geometric non-linear behavior of the one-layer [0°] clamped folder plate in Figure 14(a) is analyzed. The folded plate is still subjected to a uniformly distributed load. The load is applied vertically to the flat plates. Young’s modulus and the Poisson ratio of the material of the plates are E = 3 × 10⁹ and $ν = 0.3$ , respectively. The load–central deflection curve of the flat plate A as obtained by the proposed method is shown in Figure 15(a), and is compared to the results that are obtained by a large deflection analysis using ANSYS and by the spline strip kernel particle method of Liew et al. [74]. In ANSYS, the folded plate is discretized by 5000 identical rectangular Shell181 meshes (2500 meshes for each panel). But in the proposed method, the folded plate is only discretized by 450 elements as shown in Figure 14(b). Very good agreement between the three sets of results is obtained.

In the next example, four-layered folded plates [0°/θ°/θ°/0°] with θ = 15°, 30°, 45°, 60° and 90° are considered. The material properties of each ply are E₁/E₂ = 25, ν₁₂ = 0.25, G₁₂/E₂ = 0.5, G₂₃/E₂ = 0.2, G₁₃/G₁₂ = 1. These results of deflection are plotted in Figure 15(b) and (c) for ratio a/h = 10 and 20. Especially, for a four-layered folded plate [0°/45°/45°/0°], the load–central deflection curves are shown in Figure 15(d) with ratio a/h = 10, 20, 30 and 40.

We now change the folded angle of the plates to 60° and make the 60°-folded plates to be cantilevered as shown in Figure 16(a).

The deflections along x = 0.75 m and y = 0.5 m of laminate A, computed by the proposed method using 300 elements as Figure 16(b), are shown in Figure 17(a), compared with the deflections calculated by ANSYS using 5581 nodes and by Liew et al. [75]. The results of the three methods are very close. The 60°-folded plates of the four-layered [0°/θ°/θ°/0°] composite plates with θ = 15°, 30°, 45° and 60° are considered here. Once again, the deflections along x = 0.75 m and y = 0.5 m of laminate A with ratio h/a = 0.06 and 0.1 are plotted in Figure 17(b) to (e). By changing the number of layer and the angle of ply, the deflections along y = 0.5 m are shown in Figure 17(f) with the ratio h/a = 0.1.

Cantilevered folded plate

Finally, the geometric non-linear behavior of a one-layered [0°] cantilevered folded plate that is made up of four identical square flat plates is shown as Figure 18(a). The flat plates are joined with each other vertically. Young’s modulus and the Poisson ratio of the material of the folded plate are E = 3 × 10⁹ and $ν = 0.3$ , respectively. The folded plate is subjected to a uniformly distributed load that is applied vertically to each flat plate. The load–central deflection curve of the flat plate A as obtained by the proposed method is shown in Figure 19(a), along with the results from a large deflection analysis using ANSYS and the method of Liew [74] for comparison. In the ANSYS analysis, 14.400 identical rectangular Shell63 meshes (3600 meshes for each panel) are employed to discretize the folded plate. But in the proposed method, the folded plate is only discretized by 400 elements as shown in Figure 18(b). Very good agreement between the three sets of results is obtained.

The results of central deflection of the flat plate A are also plotted in Figure 19(b) with ratio a/h = 5, 10, 15 and 20, respectively.

Conclusions

In this paper, the SRIQ28 element is further developed and successfully applied to geometrically nonlinear analysis of composite plates and folded structures in the framework of the C⁰-HSDT. Numerical examples have been carried out and the present element is found to yield satisfactory results in comparison with other available numerical results using finite element as well as meshfree methods. It is observed that the present approach remains accurate for nonlinear analysis of both moderately thin and thick structures. In addition, the formulation and implementation of the present element for geometrically nonlinear analysis of the multi-layered structures based on the HSDT are simple. The success of the present flat element provides a further demonstration of efficient flat quadrilateral elements for nonlinear analysis.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, or/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02–2017.304.

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Enhancement to four-node quadrilateral plate elements by using cell-based smoothed strains and higher-order shear deformation theory for nonlinear analysis of composite structures

Abstract

Keywords

Introduction

C0-type higher-order shear deformation theory and weak form for plate model

Formulations of cell-based smoothed four-node quadrilateral plate element (SRIQ28)

Numerical results

Isotropic plate

Two-layer [0°/90°] square plate

Four layer [0°/90°/90°/0°] square plate

Four-layer [0°/90°/90°/0°] skew plate

Five-layer [0°/90°/0°/90°/0°] or [45°/−45°/45°/−45°/45°] trapezoidal plate

Two layer [θ°/−θ°] circular plate

Single-fold plate

Cantilevered folded plate

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

References

C⁰-type higher-order shear deformation theory and weak form for plate model