A 4-node thin shell element based on co-rotational for laminated composite structures

Latest revision as of 13:05, 18 December 2018

Abstract

The finite formulation of the 4-node thin shell element based on Co-rotational (CR) and Timoshenko's theory was presented for the analysis of laminated composite structures. Based on the Timoshenko's theory the present thin shell element avoids shear locking behavior, and a bilinear in-plane displacement field is introduced for the coupling of in-plane and bending actions, and such element performs simple formulation and highly efficienct. A highly efficient CR formulation was also established to define the motion of the element with large displacement. The characteristics of the element are more pronounced when shells get thin, each node of the 4-node element has five degrees of freedom. An incremental iterative method based on the Newton Raphson method was used to solve the nonlinear equilibrium equations. A number of numerical examples were given to verify that the formula is computationally efficient and the results showed good agreement.

1. Introduction

Many nonlinear shell element formulations are based on the total Lagrangian (TL) and update Lagrangian (UL), including complex formulations and increasing calculation interval. However, in order to maximum the computational efficiency, a nonlinear finite element method (FEM) is required. A named “CR” approach was introduced by Wempner [1] and Belytschko [2] firstly.

As a new method for improving the computational efficiency of nonlinear element with large displacement problems, the CR has attracted more and more researchers in the last decade. The core of the method is to decompose the analytic process of the element into large displacement and large rotation of the rigid body and local small deformation. When the CR method is used in the process of structural geometric nonlinear analysis, the deformation in the local coordinate system reflects the small strain of the element. However, the translation and rotation of the rigid body reflects the large displacement of the element.

Most of the studies mainly apply the CR method to beam and shell structures. Crisfield et al. [3] based on the CR method to establish a universal model for the analysis of beam, shell and solid structure, the formulas used to build the model are simpler than many of the earlier procedures. Afterwards Hsial et al. [4] and Wen et al. [5] established an axisymmetric thin-walled beam element with second-order accuracy using CR and TL formulation. Tham et al. [6] analyses multi-layered composite plate and shell structures using CR kinematic framework. Cortivo et al. [7] carried out plastic buckling and failure analysis for thin shell structures using CR and ANDES finite element formulations. A triangular shell element formula based on CR method has been obtained by Nour-Omid et al. [8], and then Pacoste [9] and Eriksson et al. [10] developed the instability analysis of the shell structure using the established triangular plate element CR formulations, the difference between them is the parameterization of the rotations.

In order to improve the computational efficiency, Battini [11] introduced three modifications in the CR framework, including local rotations, the number of element local degrees of freedom from 18 to 15 and the parameterization of the global finite rotations. Kim et al. [12] believes that the actual superimposition and the correction factor are not necessary and improve the the transverse shear stiffness, according to it Kim et al. [13] introduced the ANS method and CR formulation with second order accuracy and defines the transverse shear stiffness. To ensure computational efficiency, Kim et al. [14] also improved the 4-node shell element by combining an EAS and ANS. Nowadays, this approach based on beam [15,17,19], rod [18] and shell [16] element is widely used in many fields.

The purpose of the paper is to improve the computational efficiency of the thin shell structures modelling by a high efficiency nonlinear 4-node thin shell element. In the local element formulation, a simple and high efficiency laminated composite element was presented et al. [20], the tangential rotation and the shear strain were obtained by Timoshenko laminated composite beam theory, and a bilinear in-plane displacement field was introduced for the coupling of in-plane and bending actions. In order to consider the movement of the element with large translation and rotation, a highly efficienct CR method was obtained. In the current work, the main idea introduced by Pacoste [9] and Battini [11] was employed. At last, a high efficiency tangent stiffness matrix of 4-node thin shell element was obtained in this paper.

2. Local deformation

In local coordinate, Cen [20] show that tangential rotation and shear strain were obtained by Timoshenko beam theory. Shear strain and rotation fields within the element are then determined by improved rational interpolation, and a bilinear displacement field is introduced for the coupling of in-plane and bending actions. Finally, in order to analyze random laminated composite structures, the 20 degrees of freedom quadrilateral bending element can be used.

2.1 Element shear strain and in-plane strain field of the mid-plane

As shown in Figure1, the local deformational displacements and rotations for random 4-node elements can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \bar{a}_{i} =[\begin{array}{ccccc} {u_{i} } & {v_{i} } & {w_{i} } & {\phi _{xi} } & {\phi _{yi} } \end{array}]^{T} \quad \quad \quad i=1,2,3,4

(1)

Figure 1: Local displacement and rotation at the mid-plane of a laminated composite element.

based on the local deformation of the element mid-plane, displacement and rotation of random point for the elmenet can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l} {u(x,y,z)=u_{0} (x,y)-z\phi _{x} (x,y)} \\ {v(x,y,z)=v_{0} (x,y)-z\phi _{y} (x,y)} \\ {w(x,y,z)=w(x,y)} \end{array}

(2)

The element is shown in Figure 2. Based on Timoshenko beam theory, the tangential rotation and the shear strain along each element side can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \varphi _{s} =\varphi _{si} (1-r)+\varphi _{sj} r+3(1-2\delta )r(1-r)\Gamma

(3)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma _{si} =\delta _{i} \Gamma

(4)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} \Gamma =\frac{2}{d}(w_j-w_i)-{\varphi }_{si}-{\varphi }_{sj}\\ r=s/d \end{array}

(5)

Figure 2: Laminated composite beam element.

Based on the Equation (4), element shear strain of every edge can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} {\gamma }_{s1}=-\frac{{\delta }_1}{d_1}[2(w_2-w_3)+(c_1{\varphi }_{x2}-b_1{\varphi }_{y2})+(c_1{\varphi }_{x3}-b_1{\varphi }_{y3})]\\ {\gamma }_{s2}=-\frac{{\delta }_2}{d_2}[2(w_3-w_4)+(c_2{\varphi }_{x3}-b_2{\varphi }_{y3})+(c_2{\varphi }_{x4}-b_2{\varphi }_{y4})]\\ {\gamma }_{s3}=-\frac{{\delta }_3}{d_3}[2(w_4-w_1)+(c_3{\varphi }_{x4}-b_3{\varphi }_{y4})+(c_3{\varphi }_{x1}-b_3{\varphi }_{y1})]\\ {\gamma }_{s4}=-\frac{{\delta }_4}{d_4}[2(w_1-w_2)+(c_4{\varphi }_{x1}-b_4{\varphi }_{y1})+(c_4{\varphi }_{x2}-b_4{\varphi }_{y2})] \end{array}

(6)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{i} =\frac{6\lambda _{i} }{1+12\lambda _{i} } ,\quad \quad \lambda _{i} =\frac{D_{di} }{C_{di} d_{i}^{2} } \quad \quad \quad (i=1,2,3,4)

(7)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d_{i} (i=1,2,3,4)

is the length of element edge,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): b_{i} (i=1,2,3,4)
is the y orientation distance between two adjacent nodes, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_{i} (i=1,2,3,4)
is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x
orientation distance between two adjacent nodes.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): D_{di}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_{di}

 are the shear and bending stiffness.

According to the element shear strain of every edge Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma _{si} (i=1,\; 2,\; 3,\; 4) , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma _{s}^{*}

can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma _{s}^{*} =\left[\begin{array}{cccc} {d_{1} \gamma _{s1} } & {d_{2} \gamma _{s2} } & {d_{3} \gamma _{s3} } & {d_{4} \gamma _{s4} } \end{array}\right]

(8)

Based on the geometrical relationship of two adjacent edges, shear strain on the crossing point 1 can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{\begin{array}{c} {\gamma _{x1} } \\ {\gamma _{y1} } \end{array}\right\}=\frac{1}{b_{3} c_{4} -b_{4} c_{3} } \left[\begin{array}{cc} {b_{3} } & {-b_{4} } \\ {c_{3} } & {-c_{4} } \end{array}\right]\left\{\begin{array}{c} {\gamma _{s4}^{*} } \\ {\gamma _{s3}^{*} } \end{array}\right\}

(9)

Shear strain of the other nodes can be written similarly. So shear strain field can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma =\left\{\begin{array}{c} {\gamma _{xz} } \\ {\gamma _{yz} } \end{array}\right\}=\left[\begin{array}{c} {\gamma _{x1} N_{1}^{0} +\gamma _{x2} N_{2}^{0} +\gamma _{x3} N_{3}^{0} +\gamma _{x4} N_{4}^{0} } \\ {\gamma _{y1} N_{1}^{0} +\gamma _{y2} N_{2}^{0} +\gamma _{y3} N_{3}^{0} +\gamma _{y4} N_{4}^{0} } \end{array}\right]=B_{s} \bar{a}

(10)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} N_1^0=\frac{1}{4}\left(1-\xi \right)\left(1-\eta \right),\mbox{ }\mbox{ }N_2^0=\frac{1}{4}\left(1+\xi \right)\left(1-\eta \right),\\ N_3^0=\frac{1}{4}\left(1+\xi \right)\left(1+\eta \right),\mbox{ }\mbox{ }N_4^0=\frac{1}{4}\left(1-\xi \right)\left(1-\eta \right) \end{array}

(11)

2.2 Rotation and in-plane displacement field

The element normal and tangential of the first edge is shown in Figure 1, each edge of the element is assumed to be linearly distributed.

Based on the Equation (3), the normal and tangential rotations of the first edge can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} {\varphi }_{n1}=-\frac{1}{2d_1}[b_1({\varphi }_{x2}+{\varphi }_{x3})+c_1({\varphi }_{y1}+{\varphi }_{y3})]\\ {\varphi }_{s1}=\frac{3}{2d_1}(1-2{\delta }_1)(w_3-w_2)-\frac{1}{4d_1}(1-6{\delta }_1)\cdot [c_1({\varphi }_{x2}+{\varphi }_{x3})-b_1({\varphi }_{y2}\_{\varphi }_{y3})] \end{array}

(12)

Normal and tangential rotations of the other edges can be obtained similarly.

In global coordinate, Normal and tangential rotations of the midpoint on the first edge are represented as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\left\{\begin{array}{c} {\varphi }_{x1}\\ {\varphi }_{y1} \end{array}\right\}}_{first\mbox{ }edge}=\frac{1}{d_1}\left[\begin{array}{cc} -b_1 & c_1\\ -c_1 & -b_1 \end{array}\right]\left\{\begin{array}{c} {\varphi }_{n1}\\ {\varphi }_{s1} \end{array}\right\}

(13)

Element rotation field can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \phi _{x} =\sum _{i=1}^{8}N_{i} \phi _{xi} ,\quad \quad \phi _{y} =\sum _{i=1}^{8}N_{i} \phi _{yi}

(14)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} N_1=-\frac{1}{4}\left(1-\xi \right)\left(1-\eta \right)\left(1+\xi +\eta \right),\mbox{ }\mbox{ }N_2=-\frac{1}{4}\left(1+\xi \right)\left(1-\eta \right)\left(1-\xi +\eta \right),\\ N_3=-\frac{1}{4}\left(1+\xi \right)\left(1+\eta \right)\left(1-\xi -\eta \right),\mbox{ }\mbox{ }N_4=-\frac{1}{4}\left(1-\xi \right)\left(1+\eta \right)\left(1+\xi -\eta \right),\\ N_5=\frac{1}{2}\left(1-{\eta }^2\right)\left(1+\xi \right),\mbox{ }\mbox{ }N_6=\frac{1}{2}\left(1-{\xi }^2\right)\left(1+\eta \right),\\ N_7=\frac{1}{2}\left(1-{\eta }^2\right)\left(1-\xi \right),\mbox{ }\mbox{ }N_8=\frac{1}{2}\left(1-{\xi }^2\right)\left(1-\eta \right) \end{array}

(15)

The bilinear in-plane displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u^{0}

and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v^{0}
can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u^{0} =\sum _{i=1}^{4}N_{i}^{0} u_{i} ,\quad \quad v^{0} =\sum _{i=1}^{4}N_{i}^{0} v_{i}

(16)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_{i}^{0} (i=1,2,3,4) are the quadrilateral element shape functions.

2.3 Element in-plane strain and curvature field of the mid-plane

Element in-plane strain can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \varepsilon ^{0} =B_{e} \bar{a}

(17)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l} B_e=[\begin{array}{cccc} B_{e1} & B_{e2} & B_{e3} & B_{e4} \end{array}]\\ B_{ei}=\left[\begin{array}{ccccc} \frac{\partial N_i^0}{\partial x} & 0 & 0 & 0 & 0\\ 0 & \frac{\partial N_i^0}{\partial y} & 0 & 0 & 0\\ \frac{\partial N_i^0}{\partial y} & \frac{\partial N_i^0}{\partial x} & 0 & 0 & 0 \end{array}\right]\\ \overline\mathit{\boldsymbol{a}}=\left[\begin{array}{cccc} \overline\mathit{\boldsymbol{a}}_1 & \overline\mathit{\boldsymbol{a}}_2 & \overline\mathit{\boldsymbol{a}}_3 & \overline\mathit{\boldsymbol{a}}_4 \end{array}\right] \end{array}\qquad (i=1,2,3,4)

(18)

According to section 2.2, element curvature field can be given

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \kappa =B_{b} \bar{a}

(19)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_{b}

is bendding strain matrix.

Based on the constitutive equation of laminated composite plate, the relationship between strain and displacement in mid-plane can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{\varepsilon }}_p=\left\{\begin{array}{c} \mathit{\boldsymbol{\varepsilon }}^0\\ \mathit{\boldsymbol{\kappa}} \end{array}\right\}=\left\{\begin{array}{c} B_e\\ B_b \end{array}\right\}\overline\mathit{\boldsymbol{a}}=B_p\overline\mathit{\boldsymbol{a}}

(20)

2.4 Element stiffness matrix

The stiffness matrix of the 4-node element is derived from the principle of minimum potential energy, element shear strain and in-plane strain field:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{l}^{e} =\int _{-1}^{1}\int _{-1}^{1}B_{P}^{T} C_{P} B_{P} \left|J\right|d\xi d\eta + \int _{-1}^{1}\int _{-1}^{1}B_{S}^{T} C_{S} B_{S} \left|J\right|d\xi d\eta

(21)

where shear strain matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_{S}

 and in-plane strain matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_{P} 
of the mid-plane can be found in Equations (10) and (20), respectively,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_p
is stress-strain matrix of laminated composite plate,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_s
is shear stiffness, and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left|J\right| 
is the Jacobian determinant. The element stiffness matrix was reduced to a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 20\times 20 
matrix, which was calculated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3\times 3 
Gauss integral.

3. Co-rotational formulation

CR approach is more efficient than TL and UL to analyze thin wall structure models. Based on CR description, a lot of plate and shell elements including 3 node and 4 node were established and confirmed by many numerical illustrations and tests. According to CR description, element kinematics from initial undeformed to final deformed attitude can be divided into two sections [7]. The first section is rigid rotation removing deformational part.and translation. The second step is a local deformation and rotation under local coordinate system. Element deformation displacement and rotation were established conveniently in section 2.

3.1 Parameterization of the rigid rotations and coordinate systems

In order to represent large three-dimensional rotations, a parameterization approach of the rigid rotations is introduced [9]. Orthogonal matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R

can be obtained by three separate parameter variables. Firstly, the rotational vector can be defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{\Psi}= \boldsymbol{e}\psi

(22)

According to the definition above, as shown in Figure 3, rotation can be obtained by an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Psi

 defined by the unit vector  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): e

. So the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Psi

is obtained by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \psi =\sqrt{{\Psi }_1^2+{\Psi }_2^2+{\Psi }_3^2}

(23)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\Psi }_i

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i=1,2,3

) are the components of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Psi .

Figure 3: Rotational vector.

In terms of the definition above, the orthogonal matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R

can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R =\boldsymbol I_3+\frac{sin\psi }{\psi }\Psi +\frac{1}{2}{\left(\frac{sin(\psi /2)}{\psi /2}\right)}^2{\Psi }^2

(24)

Based on the conception of the rigid rotation, the global rotation matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_i^g

at point  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i
can be defined by  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_r
.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_i^g=\boldsymbol I+\frac{sin\psi }{\psi }{\Psi }_i^g+\frac{1}{2}{\left(\frac{sin(\psi /2)}{\psi /2}\right)}^2{\Psi }_i^g{}^2

,   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \psi =\Vert {\Psi }_i^g\Vert

(25)

Figure 4: Element kinematics and coordinate systems.

As shown in Figure 4, we define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol e_i

as the vectors of the local frame in the current deformed configuration. Then, the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_r
[9] defining rotations can be obtained as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_r=\left[\begin{array}{ccc} \boldsymbol e_1 & \boldsymbol e_2 & \boldsymbol e_3 \end{array}\right]

(26)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} \boldsymbol e_1=\frac{\left(\boldsymbol r_j^g+\boldsymbol u_j^g-\boldsymbol r_i^g-\boldsymbol u_i^g\right)}{\Vert \boldsymbol r_j^g+\boldsymbol u_j^g-\boldsymbol r_i^g-\boldsymbol u_i^g\Vert }\\ \boldsymbol e_3=\frac{\boldsymbol x_j\times \boldsymbol x_k}{\Vert \boldsymbol x_j\times \boldsymbol x_k\Vert }\\ \boldsymbol e_2=\boldsymbol e_1\times \boldsymbol e_3\\ \boldsymbol x_j=\boldsymbol r_j^g+\boldsymbol u_j^g-\boldsymbol r_i^g-\boldsymbol u_i^g\\ \boldsymbol x_k=\boldsymbol r_k^g+\boldsymbol u_k^g-\boldsymbol r_i^g-\boldsymbol u_i^g \end{array}

(27)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol u_i^g

and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol u_j^g
denote displacement from the initial to current coordinate, respectively.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol r_i^g
and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol r_j^g
denote displacement vector of initial nodes  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i
and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): j
in the global system, respectively.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol x_j
and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol x_k
denote displacement vector along the orientations  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): j
and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k
of the line connecting two nodes, respectively.

The local deformation of point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

can be expressed by  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\overline{a}}_i
, as shown in section 2. According to the define of rigid frame in the local coordinate,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\overline{a}}_i
can be redefined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\overline{a}}_i=\boldsymbol R_r^T(\boldsymbol r_i^g+\boldsymbol u_i^g-\boldsymbol r_c^g-\boldsymbol u_c^g)-\boldsymbol r_c^1

(28)

As shown in Figure 4, noting that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_r\boldsymbol {\overline{R}}_i=\boldsymbol R_i^g\boldsymbol R_0

, the local rotations are defined by the matrices   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\overline{R}}_i
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(i=1,2,3,4\right)

as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\overline{R}}_i=\boldsymbol R_r^T \boldsymbol R_i^g \boldsymbol R_0

,   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(i=1,2,3,4\right).

(29)

The differentiation can be given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \boldsymbol {\overline{R}}_i=\delta {\tilde{\overline{\varphi }}}_i \boldsymbol {\overline{R}}_i,\delta \boldsymbol R_i^g=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\tilde{\varphi }}_i^g \boldsymbol R_i^g,\delta \boldsymbol R_r= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\tilde{\varphi }}_r^g \boldsymbol R_r.

(30)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\tilde{\overline{\varphi }}}_i

,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\tilde{\varphi }}_i^g
,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\tilde{\varphi }}_r^g
and the corresponding associated vectors  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\overline{\varphi }}_i
,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\varphi }_i^g
,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\varphi }_r^g
denote spatial angular variations, which superimposed onto orthogonal matrices  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\overline{R}}_i
,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_i^g
and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_r
.

Using Equations (28), (29) and (30), the differentiation of Equations (28) can be given as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} \delta \boldsymbol {\overline{a}}_i=\boldsymbol R_r^T\delta \boldsymbol u_i^g-\boldsymbol R_r^T\delta {\tilde{\varphi }}_r^g\left(\boldsymbol r_i^g+\boldsymbol u_i^g-\boldsymbol r_c^g-\boldsymbol u_c^g\right)\\ =\delta \boldsymbol u_i^e-\delta {\tilde{\varphi }}_r^e\left(\boldsymbol {\overline{a}}_i+\boldsymbol r_c^1\right)=\delta \boldsymbol u_i^e+\boldsymbol s_i\delta {\tilde{\varphi }}_r^e \end{array}

(31)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol s_i =\boldsymbol {\overline{a}}_i+\boldsymbol r_c^1

(32)

The differentiation of Equation (30) can be given as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{ll} \delta\boldsymbol {\overline{R}} & = \boldsymbol R_{r}^{T} \delta \boldsymbol R_{i}^{g} \boldsymbol R_{0} +\delta \boldsymbol R_{r}^{T} \boldsymbol R_{i}^{g} \boldsymbol R_{0} \\ & =\delta \boldsymbol\tilde{\phi }_{i}^{e} \boldsymbol R_{r}^{T} \boldsymbol R_{i}^{g} \boldsymbol R_{0} -\delta \boldsymbol\tilde{\phi }_{r}^{e} \boldsymbol R_{r}^{T} \boldsymbol R_{i}^{g} \boldsymbol R_{0} =\left(\delta \boldsymbol\tilde{\phi }_{i}^{e} -\delta \boldsymbol\tilde{\phi }_{r}^{e} \right)\boldsymbol\bar{R}_{i} \end{array}

(33)

According to Equations (30) and (33), the following expression is obtained as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \boldsymbol\bar{\phi }_{i} = \delta \boldsymbol{\phi }_{i}^{e}- \boldsymbol{\phi }_{r}^{e}

(34)

3.2 Transformation matrix

The transformation matrix is needed to convert the internal force and tangent stiffness matrix. In the global coordinate, the 4-node element displacement is defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \boldsymbol a_g={\left[\begin{array}{cccccccc} \delta \boldsymbol u_1^g{}^T & \delta {\psi }_1^g{}^T & \delta \boldsymbol u_2^g{}^T & \delta {\psi }_2^g{}^T & \delta \boldsymbol u_3^g{}^T & \delta {\psi }_3^g{}^T & \delta \boldsymbol u_4^g{}^{} & \delta {\psi }_4^g{}^T \end{array}\right]}^T

(35)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol u_i^g\left(i=1,2,3,4\right)

are the global displacement vectors of the nodes,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\varphi }_i^g\left(i=1,\mbox{ }2,\mbox{ }3,\mbox{ }4\right)
denotes the spatial angular variations, defined in Equation (30).

Referring to Equations (23), (25) and (30), the transformation matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol T_m

was defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\varphi }_i^g=\boldsymbol T_m({\psi }_i^g)\delta {\psi }_i^g

(36)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol T_m({\psi }_i^g)=\left(\boldsymbol R_i^g+\boldsymbol I_3\right)/\sqrt{1-{\psi }_1^2-{\psi }_2^2-{\psi }_3^2}

(37)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\psi }_i\,\left(i=1,2,3\right)

are the components of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \psi 
,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol I_3
is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":):  3\times 3
unit diagonal matrix.

Using Equations (31), (32) and (34), the matrix can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \overline{\boldsymbol a}=\boldsymbol{PE}^T\delta \boldsymbol a_g

(38)

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 5\times 6

blocks  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":):  \boldsymbol P
is the matrix removing deformational part:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} \boldsymbol P_{ij}=\left[\begin{array}{cc} \frac{\partial {\overline{\boldsymbol u}}_i^e}{\partial \boldsymbol u_j^e} & \frac{\partial {\overline{\boldsymbol u}}_i^e}{\partial {\varphi }_j^e}\\ \frac{\partial {\varphi }_i^e}{\partial \boldsymbol u_j^e} & \frac{\partial {\varphi }_i^e}{\partial {\varphi }_j^e} \end{array}\right]=\boldsymbol I_{56}{\delta }_{\mbox{ij}}-\boldsymbol A_i\boldsymbol G_j^T\\ \boldsymbol E=\boldsymbol{diag}(\boldsymbol R_r,\boldsymbol R_r,\boldsymbol R_r,\boldsymbol R_r,\boldsymbol R_r,\boldsymbol R_r,\boldsymbol R_r,\boldsymbol R_r) \end{array}

(39)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol I_{56}

is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":):  5\times 6
unit diagonal matrix.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol R_r
can be found in section 3.1.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta }_{ij}
,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol A_i
,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol G_j
and more detail process of the element kinematics are defined by local and global displacement and rotation vector [9,11].

According to the rotate transformation relation obtained by the change of variables from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\phi }_i^g

to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {\psi }_i^g
.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol B_{m} =\boldsymbol{diag}(\boldsymbol I_{3} ,\boldsymbol T_{m} (\boldsymbol \psi _{1}^{g} ),\boldsymbol I_{3} ,\boldsymbol T_{m} (\boldsymbol \psi _{2}^{g} ),\boldsymbol I_{3} ,\boldsymbol T_{m} (\boldsymbol \psi _{3}^{g} ),\boldsymbol I_{3} ,\boldsymbol T_{m} (\boldsymbol \psi _{4}^{g} ))

(40)

Using Equations (38) and (40), the final transformation matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): T

is obtained as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol T=\boldsymbol {PE}^T\boldsymbol B_m

(41)

3.3 Element internal force and tangent stiffness matrix

According to the transformation matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol T

in section 3.2, the element global internal force can be defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol F =\left[\begin{array}{cccccccc} {\boldsymbol F_{1}^{T} } & {\boldsymbol M_{1}^{T} } & {\boldsymbol F_{2}^{T} } & {\boldsymbol M_{2}^{T} } & {\boldsymbol F_{3}^{T} } & {\boldsymbol M_{3}^{T} } & {\boldsymbol F_{4}^{T} } & {\boldsymbol M_{4}^{T} } \end{array}\right]=\boldsymbol T^{T} \boldsymbol f_{l}

(42)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol F_i

and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol M_i
denote the internal force vector of the node  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i
for translation and rotation, respectively. The internal force of local deformation  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol f_l
is defined and expressed in [20].

The element local stiffness matrix and established transformation matrix by CR can be found in sections 2.3 and 3.2, Using Equations (21) and (41), the element global tangent stiffness matrix is defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol K_T=\boldsymbol T^T \boldsymbol K_l^e \boldsymbol T + \boldsymbol K_{\sigma }

(43)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} \boldsymbol K_{\sigma }=\boldsymbol{diag}(0_3,\boldsymbol K_{\sigma 1,}0_3,\boldsymbol K_{\sigma 2,}0_3,\boldsymbol K_{\sigma 3,}0_3,\boldsymbol K_{\sigma 4}),\\ \boldsymbol K_{\sigma i}=\partial \boldsymbol T_m^T (\boldsymbol M_i)/\partial {\psi }_i(i=1,2,3,4) \end{array}

(44)

In Equation (43), the element tangent stiffness matrix is reduced from a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 24\times 24

matrix to a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 20\times 20
matrix. In Equation (44), the  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0_3
denotes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3\times 3
zero matrix, Equation (44) is explained in detail in [11].

4. Numerical experiments

The 4-node thin shell element is used to carry out linear and geometrically nonlinear analysis. This part is mainly to assess the accuracy and efficiency of the element by numerical experiments. All of the numerical results are compared with references and standard 4-node (QUAD 4) thin shell element, which is established and extensively used based on standard updated Lagrangian formulation (UL). The standard 4-node thin shell element tangent stiffness matrix is still a 24×24 block. Generally speaking, the results from the standard finite element with small grid can be used as the standard of structure design. So the numerical results from the present element can be compared with the results of the QUAD 4, for which the model can be divided into grids small enough. At the same time, the convergence feature of the present method was also verified by analytical solution for simply supported orthotropic symmetry plates.

4.1 Linear analysis

4.1.1 Simply supported laminated composite plates

All edges are provided with a simple supported restraint and a uniform load applied to the surface. The plate with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4\times 4

meshes is used in the analysis. The geometry and material properties of T700S/5405 can be obtained as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l} a=b=100mm\quad ,\quad a/h=100\quad ,\quad E_{1} =131100N/mm^{2} \quad ,\quad \\ E_{2} =8580N/mm^{2}\quad ,\quad G_{12} =G_{13} =G_{23} =5000N/mm^{2} \quad ,\quad \quad v_{12} =0.36 \end{array}

The center deflections are presented in the non-dimensional form using the equation [14]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \bar{w}=w\left(\frac{E_{2} h^{3} }{qa^{4} } \right)\times 10^{3}

(45)

and are compared with standard element and analytical solution in Table 1. The ply stacking sequence and ply angle influence the stiffness and strength of laminated composite plate in aircraft structural design. Hence, the laminated composite plate is more designable than isotropy plate.

Table 1. Maximum displacements in Non-dimensional form.

	Analytical solution	standard element (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 20\times 20 )	Present element (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4\times 4 )
[45/-45]_s	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 7.05\times 10^{-4}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 7.3\times 10^{-4}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 7.12\times 10^{-4}
[90/0]_s	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 9.54\times 10^{-4}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 9.98\times 10^{-4}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 9.67\times 10^{-4}

4.1.2 Simply-supported laminated composite plates

The plate with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4\times 4

meshes is used in the analysis. Aspect ratio (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a/h =100

) is used to verify present element. The relevant parameters are as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} a=b=100mm\quad ,\quad E_1/E_2=25\quad ,\quad E_2=1.0\times {10}^6N/mm^2\\ G_{12}=G_{13}=0.5\times {10}^6N/mm^2\quad ,\quad G_{23}=0.2\times {10}^6N/mm^2\quad ,\quad v_{12}=0.25 \end{array}

Maximum displacements are obtained using the equation (45), and are compared with the strandard element and the references by Kim et al. [13-14] and Pucha et al. [21] in Table 2. The results show that present element can be well applied to the analysis of the actual engineering structure.

Table 2. Maximum displacements in Non-dimensional form.

Solution method	Mesh	5/-5	15/-15	30/-30	45/-45
Present	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4\times 4	7.083	9.34	10.77	10.25
Strandard element (UL)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 40\times 40	7.145	9.82	10.36	10.17
Kim et al. [13]	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4\times 4	7.015	9.505	10.52	10.12
Kim et al. (EAS-ANS) [14]	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4\times 4	7.025	9.468	10.56	10.16
Pucha et al. [21]		7.1298	9.1077	9.1718	9.0793

4.1.3 Simply supported laminated spherical shell

A laminated spherical shell shown in Figure 5 is analyzed. The surface of the structures is applied with a uniform load. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 8\times 8

meshes are used to analyze the spherical shell structure with lay-ups of (0/90/0/90/0/90/0/90/0). The relevant parameters are used as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} h/a=0.001\quad , \quad E_1/E_2=40\quad , \quad E_2=5.1713\times {10}^9N/mm^2\\ G_{12}/E_2=0.6\quad , \quad G_{23}/E_2=0.6\quad , \quad v_{12}=0.25\quad , \quad a=b \end{array}

Figure 5: Geometry proprieties of laminated spherical shell.

The center deflection of the laminated spherical shell structure is compared with the non-dimensional form (as shown in Equation (45)) of the standard element and the references by Kim et al. [13,14] and Noor et al. [22]. The results are tabulated in Table 3.

Table 3. Non-dimensional center deflection of laminated spherical shell.

Solution method	Mesh	Non-dimensional center deflection
Present	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 8\times 8	5.874E-05
Standard element (UL)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 40\times 40	5.92E-05
Kim et al. [13]	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 8\times 8	5.844E-05
Kim et al. (EAS-ANS) [14]	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 8\times 8	5.890E-05
Noor et al. [22]	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 8\times 8	5.916E-05

4.2 Nonlinear analysis

4.2.1 Laminated composite plate

A multi-layer laminated composite plate is used to nonlinear analyze. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 8\times8

meshes and lay-ups of  [45/-45/0/0/45/-45/90/90] are used in this analysis. The length of the plate is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a=b=254mm

. The total thickness is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h=2.114mm . All edges are clamped. The surface of the structures is applied with a uniform load. The material properties are used as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} E_1=13.1\times {10}^4N/mm^2\quad ,\quad E_2=E_3=1.303\times {10}^4N/mm^2\quad ,\quad v_{12}=v_{23}=v_{13}=0.38\\ G_{12}=G_{13}=0.641\times {10}^4N/mm^2\quad ,\quad G_{23}=0.4721\times {10}^4N/mm^2 \end{array}

Maximum displacements of the plate is compared with that of the standard element with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 40\times 40

meshes and the references by Kim et al. [14] and Lee et al. [23]. The results are shown in Figure 6, normalized deflection: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{w}=w/h

.

Figure 6: Load-deflection curve of laminated composite plate.

4.2.2 Pinched cantilever cylinder

In this example, nonlinear analysis of pinched cantilever cylinder is carried out. The cylinder is depicted in Figure 7, which clamped at B face and applied two opposite loads at A face. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 16\times 16

regular meshes are used to analyze the structure.

Figure 7: Pinched cantilever cylinder.

It is clear that this example reflects the nonextensional deformations for the element, this phenomenon will not occur in actual engineering. By the large deformation analysis, the vertical displacement at A of the cantilever cylinder is compared with that of the standard element (UL) with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 50\times 50

and the references by Battini [11] and Norachan et al. [24] as shown in Figure 8. The results analyzed by present element showed a satisfied precision and efficiency.

Figura 8: Load-displacement curve of cantilever cylinder.

4.2.3 Double-curved thin shell structure

Double-curved thin shell structure can be seen everywhere in the actual project. The numerical example is to use present element to carry out the nonlinear analysis of the double-curved structure, which is always used in many engineering structures and shows significant different nonlinear. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 16\times 10

meshes and lay-ups of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): [45/90/0/-45]_{S}
are used in this analysis. The length of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): b=2000 mm
and width of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a=1250 mm
are shown in Figure 9, and the radius of the two sides in the direction of width are  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r_1 = 318mm

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r_2 = 504mm

and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r'_1 = 1015mm

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r'_2 = 772mm , respectively, the length of the section with radius as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r_1

in the direction of width is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c=645mm

, the length of the section with radius as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r'_1

in the direction of width is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d=860mm

, the total thickness of the laminates is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h=3.2mm . The material properties of T700/QY9511 are used as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l} {E_{1} =1.334\times 10^{5} N/mm,\quad \quad E_{2} =1.008\times 10^{4} N/mm} \\ {G_{12} =G_{13} =G_{23} =4.9\times 10^{3} N/mm,\quad \quad v_{12} =0.31} \end{array}

Figure 9: Geometry proprieties of double-curved thin shell structure.

The normal displacement at A of the double-curved thin shell structure is compared with that of the standard 4-node thin shell element with 80×40 meshes in Figure 10, displacements of double-curved thin shell structure are obtained by load of 1.2 KN using standard element and present element, respectively, as shown in Figure 11. The relative error between present element with big grid and standard element with small grid is 4%, and the relative error between big grid and small grid models using standard element is 56.7%. The results showed that the present element is acceptable for the double-curved structure.

Figure 10: Load-displacement curve of double-curved thin shell structure.

style="padding:10x

(a) using standard element (80×40)

(b) using standard element (16×10)

(c) using present element (16×10)

Figure 11: Displacements of laminated thin cylinder by 1.2 KN.

5. Conclusions

This paper mainly presented a highly efficient nonlinear computer modelling approach for laminated composite thin shell structures. The analysis method mainly includes 3 points:

In the local coordinate, a simple and highly efficienct 4-node thin shell element formulation was presented, and the number of local degrees of freedom was reduced,

A 4-node thin shell element CR formulation was established considering the element with large displacement,

Finally, a highly efficienct element tangent stiffness matrix reduced from a 24×24 matrix to a 20×20 matrix was established to analyze laminated composite thin shell structures.

The test results presented in Section 4 show that all of the numerical results are the same, but the element presented in this paper is more efficient than the standard element. Especially, large deformation analysis of pinched cantilever cylinder (Example 4.2.2) and double-curved thin shell (Example 4.2.3) can be used to detect the results of geometric nonlinear analysis for other engineering structures. It is indicated that the present element is more effective since it can save much computation time, which is more applicable for engineering computation. The present work can be applied in designing and analyzing composite thin shell structure.

Acknowledgements

The work presented in this article was supported by the National Natural Science Foundation of China (Grant Nos. 51175424 & 51475369), the Basic Research Foundation of Northwestern Polytechnical University (Grant No. JC201238), and the Aeronautical Science Foundation of China (Grant No. 2016ZD53036).

References

[1] G.A. Wempner, Finite elements, finite rotation and small strains of flexible shells, International Journal of Solids and Structures, 5 (1969) 117-153.

[2] T. Belytschko, L. Schwer, Non-linear transient finite element analysis with convected co-ordinates, International Journal of Numerical Methods and Engineering, 7(9) (1973) 255-271.

[3] M.A. Crisfield, G.F. Moita, A unified co-rotational framework for solids, shells and beams, Solids Structures, 33 (1996) 2969-2992.

[4] K.M. Hsial, Y.L. Wen, A co-rotational formulation for thin-walled beams with monosymmetric open section, Computer Methods in Applied Mechanics and Engineering, 190 (2000) 1163-1185.

[5] Y.L. Wen, K.M. Hsial, Co-rotational formulation for geometric nonlinear analysis of doubly symmetric thin-walled beams, Computer Methods in Applied Mechanics and Engineering, 190 (2001) 6023-6052.

[6] C.L. Tham, Z. Zhang, A. Masud, An elasto-plastic damage model cast in a co-rotational kinematic framework for large deformation analysis of laminated composite shells, Computer Methods in Applied Mechanics and Engineering, 194 (2005) 2641-2660.

[7] N.D. Cortivo, C.A. Felippa, H. Bavestrello, W.T.M. Silva, Plastic buckling and collapse of thin shell structures, using layered plastic modeling and co-rotational ANDES finite elements, Computer Methods in Applied Mechanics and Engineering, 198 (2009) 785-798.

[8] B. Nour-Omid, C.C. Rankin, Finite rotation analysis and consistent linearization using projectors, Computer Methods in Applied Mechanics and Engineering, 93 (1991) 353-384.

[9] C. Pacoste, Co-rotational flat facet triangular elements for shell instability analysis, Computer Methods in Applied Mechanics and Engineering, 156 (1998) 75-110.

[10] A. Eriksson, C. Pacoste, Element formulation and numerical techniques for stability problems in shells, Computer Methods in Applied Mechanics and Engineering, 191 (2002) 3775-3810.

[11] J.M. Battini, A modified corotational framework for triangular shell elements, Computer Methods in Applied Mechanics and Engineering, 196 (2007) 1905-1914.

[12] K,D. Kim, G.R. Lomboy, S.C. Han, A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells, Comput. Mech. 30(4) (2003) 330-42.

[13] K.D. Kim, C.S. Lee, S.C. Han, A 4-node co-rotational ANS shell element for laminated composite structures, Composite Structures, 80 (2007) 234-252.

[14] K.D. Kim, S.C. Han, S. Suthasupradit, Geometrically non-linear analysis of laminated composite structures using a 4-node co-rotational shell element with enhanced strains, International Journal of Non-Linear Mechanics, 42 (2007) 864-881.

[15] R. Alsafadie, M. Hjiaj, J.M. Battini, Corotational mixed finite element formulation for thin-walled beams with generic cross-section, Comput. Methods Appl. Mech. Engrg. 199 (2010) 3197-3212.

[16] F.S. Almeida, A.M. Awruch, Corotational nonlinear dynamic analysis of laminated composite shells, Finite Elements in Analysis and Design, 47 (2011) 1131-1145.

[17] T.N. Le, J.M. Battini, M.A. Hjiaj, Consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures, Comput. Methods Appl. Mech. Engrg. 269 (2014) 538-565.

[18] S. Faroughi, H.H. Khodaparast, M.I. Friswell, Non-linear dynamic analysis of tensegrity structures using a co-rotational method, International Journal of Non-Linear Mechanics, 69 (2015) 55-65.

[19] W. Wang, X.P. Zhu, Z. Zhou, J.B. Duan, A method for nonlinear aeroelasticity trim and stability analysis of very flexible aircraft based on co-rotational theory, Journal of Fluids and Structures, 62 (2016) 209-229.

[20] S. Cen, Y.Q. Long, Z.H. Yao, A new element based on the first-order shear deformation theory for the analysis of laminated composite plates, Engineering Mechanics, 1(19) (2002) 1-8.

[21] N.S. Pucha, J.N. Reddy, A mixed shear flexible finite element for the analysis of laminated plates, Comput. Meth. Appl. Mech. Eng. 44(2) (1984) 213-27.

[22] N A.K. oor, M.D. Mathers, Anisotropy and shear deformation in laminated composite plates, AIAA J, 14 (1975) 282-5.

[23] K.D. Lee, K W. anok-Nukulchai, A nine-node assumed strain finite element for large deformation analysis of laminated shells, Int. J. Num. Meth. Eng. 42 (1998) 777-798.

[24] P. Norachan, S. Suthasupradit, K.D. Kim, A co-rotational 8-node degenerated thin-walled element with assumed natural strain and enhanced assumed strain, Finite Elements in Analysis and Design, 50 (2012) 70-85.

@@ Line 39: / Line 39: @@
 | [[Image:draft_Aparicio Nogué_939719944-image2.png|534px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 1:''' Local displacement and rotation at the mid-plane of a laminated composite element.
+| colspan="1" | '''Figure 1:''' Local displacement and rotation at the mid-plane of a laminated composite element.
 |}
@@ Line 96: / Line 96: @@
 | [[Image:draft_Aparicio Nogué_939719944-image7.png|480px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 2:''' Laminated composite beam element.
+| colspan="1" | '''Figure 2:''' Laminated composite beam element.
 |}
@@ Line 295: / Line 295: @@
 & \frac{\partial N_i^0}{\partial y} & 0 & 0 & 0\\
 \frac{\partial N_i^0}{\partial y} & \frac{\partial N_i^0}{\partial x} & 0 & 0 & 0
-\end{array}\right]\mbox{ }(i=1,2,3,4)\\
+\end{array}\right]\\
 \overline\mathit{\boldsymbol{a}}=\left[\begin{array}{cccc}
 \overline\mathit{\boldsymbol{a}}_1 & \overline\mathit{\boldsymbol{a}}_2 & \overline\mathit{\boldsymbol{a}}_3 & \overline\mathit{\boldsymbol{a}}_4
 \end{array}\right]
-\end{array}</math>
+\end{array}\qquad (i=1,2,3,4)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (18)
@@ Line 325: / Line 325: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\mathit{\boldsymbol{\epsilon }}_p=\left\{\begin{array}{c}
+| <math>\mathit{\boldsymbol{\varepsilon }}_p=\left\{\begin{array}{c}
-\mathit{\boldsymbol{\epsilon }}^0\\
+\mathit{\boldsymbol{\varepsilon }}^0\\
-\overline\mathit{\boldsymbol{\kappa}}
+\mathit{\boldsymbol{\kappa}}
 \end{array}\right\}=\left\{\begin{array}{c}
 B_e\\
@@ Line 389: / Line 389: @@
 | [[Image:draft_Aparicio Nogué_939719944-image49.png|342px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 3:''' Rotational vector.
+| colspan="1" | '''Figure 3:''' Rotational vector.
 |}
@@ Line 421: / Line 421: @@
 | [[Image:draft_Aparicio Nogué_939719944-image57.png|600px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 4:''' Element kinematics and coordinate systems.
+| colspan="1" | '''Figure 4:''' Element kinematics and coordinate systems.
 |}
@@ Line 839: / Line 839: @@
 | [[Image:draft_Aparicio Nogué_939719944-image137.png|318px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 5:''' Geometry proprieties of laminated spherical shell.
+| colspan="1" | '''Figure 5:''' Geometry proprieties of laminated spherical shell.
 |}
@@ Line 900: / Line 900: @@
 | style="padding:10px;"|[[Image:draft_Aparicio Nogué_939719944-image143.png|416px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 6:''' Load-deflection curve of laminated composite plate.
+| colspan="1" | '''Figure 6:''' Load-deflection curve of laminated composite plate.
 |}
@@ Line 912: / Line 912: @@
 | style="padding:10px;|[[Image:draft_Aparicio Nogué_939719944-image144.png|400px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 7:''' Pinched cantilever cylinder.
+| colspan="1" | '''Figure 7:''' Pinched cantilever cylinder.
 |}
@@ Line 944: / Line 944: @@
 | [[Image:draft_Aparicio Nogué_939719944-image157.png|400px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 9:''' Geometry proprieties of double-curved thin shell structure.
+| colspan="1" | '''Figure 9:''' Geometry proprieties of double-curved thin shell structure.
 |}
@@ Line 954: / Line 954: @@
 | [[Image:draft_Aparicio Nogué_939719944-image158.png|404px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 10:''' Load-displacement curve of double-curved thin shell structure.
+| colspan="1" | '''Figure 10:''' Load-displacement curve of double-curved thin shell structure.
 |}
@@ Line 972: / Line 972: @@
 | (c) using present element (16&#x00d7;10)
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figura 11:''' Displacements of laminated thin cylinder by 1.2 KN.
+| colspan="1" | '''Figure 11:''' Displacements of laminated thin cylinder by 1.2 KN.
 |}

Latest revision as of 13:05, 18 December 2018

Abstract

1. Introduction

2. Local deformation

2.1 Element shear strain and in-plane strain field of the mid-plane

2.2 Rotation and in-plane displacement field

2.3 Element in-plane strain and curvature field of the mid-plane

2.4 Element stiffness matrix

3. Co-rotational formulation

3.1 Parameterization of the rigid rotations and coordinate systems

3.2 Transformation matrix

3.3 Element internal force and tangent stiffness matrix

4. Numerical experiments

4.1 Linear analysis

4.1.1 Simply supported laminated composite plates

4.1.2 Simply-supported laminated composite plates

4.1.3 Simply supported laminated spherical shell

4.2 Nonlinear analysis

4.2.1 Laminated composite plate

4.2.2 Pinched cantilever cylinder

4.2.3 Double-curved thin shell structure

5. Conclusions

Acknowledgements

References

Document information

Document Score

Share this document

Keywords

claim authorship