Based on the first-order shear deformation theory, a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth, folded and multi-shell laminated composite structures. The two smaller components of the mid-surface normal vector of shell at a node are defined as nodal rotational variables in the co-rotational local coordinate system. In the global coordinate system, two smaller components of one vector, together with the smallest or second smallest component of another vector, of an orthogonal triad at a node on a non-smooth intersection of plates and/or shells are defined as rotational variables, whereas the two smaller components of the mid-surface normal vector at a node on the smooth part of the plate or shell (away from non-smooth intersections) are defined as rotational variables. All these vectorial rotational variables can be updated in an additive manner during an incremental solution procedure, and thus improve the computational efficiency in the nonlinear solution of these composite shell structures. Due to the commutativity of all nodal variables in calculating of the second derivatives of the local nodal variables with respect to global nodal variables, and the second derivatives of the strain energy functional with respect to local nodal variables, symmetric tangent stiffness matrices in local and global coordinate systems are obtained. To overcome shear locking, the assumed transverse shear strains obtained from the line-integration approach are employed. The reliability and computational accuracy of the present 3-node triangular shell finite element are verified through modeling two patch tests, several smooth and non-smooth laminated composite shells undergoing large displacements and large rotations.

Laminated composite shell structures are extensively used in pressure vessels, aircraft and spacecraft, automotive and other industries due to their high strength- and stiffness-to-mass ratios, excellent damage tolerance, superior fatigue response characteristics, and good damping behaviors under dynamic loads. By choosing an appropriate combination of reinforcement and matrix material, manufacturers can produce properties that exactly fit the requirements of a particular structure design [

Various computational formulations have been proposed for modeling composite shells and plates, and can be broadly classified into three categories: (1) single-director theories with anisotropic constitutive relations and (2) multi-director theories, which include multi-layer formulations, within which each layer has a single director with its own anisotropic constitutive relation, and (3) 3-D continuum theories. Examples of single-director theories include the classical laminated plate/shell theory (CLPT), the first-order shear deformation laminated plate/shell theory (FSDT), and the higher-order shear deformation laminated plate/shell theories (HSDT). Examples of multi-director theories include the layer-wise theory (LWT) and the zig-zag theory (ZZT). Examples of 3-D continuum theories include solid-shell formulations with a single layer or a multilayer structure.

In the single-director category, the CLPT is restricted to thin shell structures, as the effects of transverse shear strains and thickness strains are ignored. Based on the CLPT, Madenci et al. [

Also in the single-director category, under the FSDT, the transverse shear strains are assumed to remain constant through the thickness, and the shell normal does not need to remain perpendicular to the mid-surface after deformation, while the inextensibility of transverse shell normal is assumed. Based on this theory, Peng et al. [

The HSDT provides a more accurate description of the transverse shear stress distributions by introducing more independent displacement parameters. Within the HSDT, Chen et al. [_{1} weak-continuity. Tran et al. [_{0}-type higher-order shear deformation theory for static, free vibration and buckling analyses of laminated composite plates. Jin et al. [_{0}-type 3-node triangular plate element with linear interpolation functions for the analysis of multi-layered composite plates based on the mixed global-local higher-order theory.

In the multi-director category, the LWT assumes a layer-wise deformation pattern, and it can predict the interlaminar stresses accurately. Liu et al. [

The ZZT describes a piecewise continuous displacement field in the plate thickness direction and fulfills interlaminar continuity of transverse stresses at each layer interface. Carrera [

The 3D continuum-based theory accounts for fully 3D constitutive behaviors, so the interlaminar stress of composite laminates can be effectively captured. Houmat [

Despite these developments, numerical formulations based on the theories of HSDT, LWT, ZZT and 3D continuum-based theory often lead to high computational costs, which is a major concern in their practical applications. In the present study, a 3-node co-rotational triangular composite shell finite element is developed based on FSDT, where vectorial rotational variables are employed as rotational variables, two smaller components of one vector, together with the smallest or second smallest component of another vector, of an orthogonal triad initially oriented along the global coordinate system axes at each node on a non-smooth intersection of plates and/or shells are defined as vectorial rotational variables, while two smaller components of the mid-surface normal vector of shell at other nodes are defined as vectorial rotational variables. The resulting element tangent stiffness matrices are symmetric owing to the commutativity of nodal variables in calculating the second derivatives of strain energy with respect to local nodal variables and the second derivatives of the local nodal variables with respect to global nodal variables. Using such vectorial rotational variables, triangular and quadrilateral shell elements have been developed for large displacement and large rotation analyses of smooth shell structures [

There have been continuous efforts in developing triangular shell finite elements with high computational accuracy and convergence [

The outline of the paper is as follows. Section 2 describes the co-rotational framework and the kinematics of the 3-node triangular composite shell element. Section 3 presents the composite shell finite element formulation in the co-rotational local co-ordinate system. Section 4 gives the transformation relationship between the local and global responses. Several numerical examples are analyzed in Section 5 to verify the numerical accuracy of the present finite element. Conclusions are presented in Section 6.

The co-rotational framework for the 3-node shell finite element is depicted in

where

In the deformed configuration, the vectors _{12} and _{13} are defined as

where _{i}

In the global coordinate system, we define the following vector that consists of all the global DOFs (degrees of freedom) for each element:

where

where

On the other hand, if Node

The remaining components of the vectors

Since the norm of a unit vector is identical to 1, defining the vectorial rotational variables as above can avoid ill-conditioning in updating the mid-surface normal vector at a node on the smooth part of the plate or shell (away from non-smooth intersections) or orientation vectors of an orthogonal triad at a node on a non-smooth intersection of plates or shells by properly controlling the size of loading step in a nonlinear incremental solution procedure.

There are 15 degrees of freedom per element in the local coordinate system

where

The relationship between the local and global translational displacements can be expressed as follows:

where,

At any node of smooth shells or any node away from non-smooth shell intersections, the relationship between the shell directors expressed in the local and global coordinate systems can be described as

At any node on intersections of non-smooth shells, the following relationships between the shell directors in the local and global coordinate systems hold:

where

For convenience,

The finite element shape functions are expressed in the natural coordinate system as follows:

The displacement at any point of the element is calculated as follows:

In the initial configuration, the shell director at Node

where

To ensure the uniqueness of the shell director at any node shared by multiple adjacent elements in smooth shell regions, the following averaging procedure is adopted:

The Green–Lagrange strain of the nonlinear shallow shell theory is adopted. For convenience, the strain vector is divided into membrane strain vector

where,

The potential energy of a 3-node triangular composite shell finite element is defined as

where

where, _{1} and E_{2} are the elastic moduli, G_{12}, G_{23} and G_{31} are the shear moduli [

By enforcing the stationarity condition to the potential energy, we have

which yields the element internal force vector:

where

By taking the first-order derivative of the internal force vector with respect to the local nodal variables, a symmetric element tangent stiffness matrix is obtained

For elastic laminated shell elements, the integration along the shell thickness direction is decoupled from the integration on the mid-surface, so

where

_{ci} is the elastic matrix of the

where

where

To alleviate shear locking phenomenon, the assumed transverse shear strain vector and its first-order derivatives with respect to local nodal variables are employed. The modified line integration approach [

where

The internal force

where

The element tangent stiffness

where

the sub-matrices in

To verify the reliability and computational accuracy of present 3-node co-rotational triangular composite shell element, two patch tests, and several smooth and non-smooth composite shell problems are solved, and the solutions are compared to numerical results from literatures [

Patch tests for the membrane behavior and the transverse out-of-plane bending behavior of plate and shell elements were suggested by MacNeal et al. [

In the membrane patch test, the displacements

To construct a constant stress state of the plate under out-of-plane bending, the displacements

For a linear problem, the theoretical solution for the stresses at the top and bottom surfaces of the plate is

In the present triangular shell element formulation, vectorial rotational variables are defined. These can be calculated from the prescribed rotations,

The rectangular plate is meshed into triangular elements (

Laminated cylindrical shells subjected to a point load at the central point B are studied. The shell geometry is shown in

We consider two groups of lamination schemes consisting of 12 layers and 48 layers, respectively. The first group includes four 12-layer lamination shells arranged as (0°_{4}/90°_{4}/0°_{4}) and (90°_{4}/0°_{4}/90°_{4}), respectively. Here, 0° denotes the circumferential direction of the cylinder, and 90° denotes the cylinder's axial direction. For instance, (90°_{4}/0°_{4}/…) indicates that there are 4 laminas at 90 degrees followed by 4 laminas at 0 degrees, etc.). The second group consists of three 48-layer laminations arranged as (0°_{6}/90°_{6}/0°_{6}/90°_{6})_{s}, (45°_{2}/−45°_{2}/0°_{2}/90°_{2})_{3s,} and (45°_{2}/−45°_{2})_{6s}, respectively, where the subscript “s” indicates that the 48 laminas are arranged symmetrically with respect to the shell mid-surface, and the number “3” or “6” before the letter “s” denote that 3 or 6 groups of laminae arranged as those in the parentheses. The material properties of these laminated cylindrical shells are given in

Young's moduli | Shear moduli | Poisson's ratio | |
---|---|---|---|

12 layers | _{1} = 3.3 GPa, _{2} = 1.1 GPa |
_{12} = _{13} = _{23} = 0.66 GPa |
_{12} = _{21} = 0.25 |

48 layers | _{1} = 150 GPa, _{2} = 10 GPa |
_{12} = _{13} = 6 Gpa, _{23} = 4 GPa |
_{12} = _{21} = 0.25 |

Due to symmetry, only a quarter of these laminated cylindrical shells are studied by using 8

A laminated channel section cantilever is subjected to a concentrated load at the upper corner point, as shown in _{ref} with P_{ref} = 100 lb. Two cross-ply lamination schemes are considered, including (0°/90°/0°) and (90°/0°/90°), where the 0° and 90° directions are respectively along the t_{1}-axis and the t_{2}-axis of the local material coordinate system depicted in _{1} = 10^{7} lb/in^{2}, _{2} = 4 × 10^{5} lb/in^{2}, _{12} = _{13} = 2 × 10^{5} lb/in^{2}, _{23} = 8 × 10^{4} lb/in^{2}, _{12} = _{21} = 0.333.

The laminated channel section cantilever is modeled using respectively (4 + 4 + 6) × 72 × 2 and (6 + 6 + 9) × 108 × 2 CR3T element meshes, where the web is discretized respectively by 6 × 72 × 2 and 9 × 108 × 2 elements, and the upper and the lower flanges are discretized by 4 × 72 × 2 and 6 × 108 × 2 elements, respectively. The load-deflection curves at the upper corner point are presented in

The deformed shapes of the channel section cantilever with different lamination schemes are presented in

A stiffened doubly curved cylindrical panel is subjected to a lateral force at the midpoint of the free curved edge (_{0}_{ref}_{ref}. Two lamination schemes are considered as (0°/90°/90°/0°) and (90°/0°/0°/90°), where the 0° and 90° directions are respectively along the t_{1}-axis and t_{2}-axis of local material coordinate system depicted in _{1}^{5} MPa, _{2}^{3} MPa, _{12}_{13}^{3} MPa, _{23}^{3} MPa, _{12} = _{21} = 0.25.

The stiffened doubly curved cylindrical panel with lamination scheme (0°/90°/90°/0°) is modeled using (30 + 30 + 14)

The deformed shapes of the stiffened doubly curved cylindrical panel are presented in _{A} = 0.2167 m for lamination scheme (0°/90°/90°/0°) and W_{A} = 0.1692 m for lamination scheme (90°/0°/0°/90°) in

A laminated cantilever sickle shell is subjected to a lateral force at the free end, as shown in _{1}^{7}, _{2}^{7}, _{12}_{13}^{7}, _{23}^{7}, _{12} = _{21} = 0.3.

The cantilever sickle shell is modeled using (40 + 40)

The deformed shapes of the sickle shell at different levels of the lateral tip load are presented in

A 3-node co-rotational triangular composite shell element for large deformation analysis of smooth, folded and multi-shell laminated composite structures is proposed. Different from other existing elements using traditional rotational variables, vectorial rotational variables are employed in the present element under a co-rotational framework. All nodal variables are additive in the nonlinear solution procedure, and the global tangent stiffness matrix is symmetric, which enhances the computational efficiency and saves computer storage resource. To overcome shear locking phenomenon, the conforming transverse shear strains are replaced with assumed transverse shear strains by using the line integration method. The computational performance of the developed finite element formulation is demonstrated through solving several smooth and non-smooth laminated composite shell structural problems.

^{0}-type higher-order shear deformation theory for analysis of laminated composite plates

^{0}triangular elements based on the refined zigzag theory for multilayer composite and sandwich plates

^{0}three-node triangular element based on preprocessing approach for thick sandwich plates

^{0}ANS element for geometrically non-linear structural analysis

_{3}shell finite element enriched by interpolation covers

The first-order derivatives of membrane strains with respect to local nodal variables:

where

The first-order derivatives of shear strains with respect to local nodal variables:

The first-order derivatives of bending strain with respect to local nodal variables:

Sub-matrices of the transformation matrix

where,

Case 1: If Node

In

where

Case 2: If Node

In

In

where

The first-order derivatives of the transformation matrix

In

Case 1: If Node

In

Case 2: If Node

where