A unified formulation for composite quasi-3D elements based on global–local superposition. Part II: Implementation and numerical assessment

Abstract

This two-paper series proposes a unified theory that gives rise to a family of quasi-3D composite elements. The first paper presents the element formulation and its basic capabilities: the ability to capture transverse normal (σ_z) and shear (τ_yz, τ_xz) stresses, suitability for thermoelastic analyses and compliance to both displacements and transverse stress continuity requirements. These capabilities are inherent to the element since a global–local superposition approach is devised that, from inception, guarantees that equilibrium equations, continuity consistency and boundary conditions are fully met. A simple validation analysis was conducted in part I that initially pointed to a very promising direction with high numerical efficiency of the element. This second paper investigates the element numerical performance under different scenarios: use of three- and four-node parent elements, degree of global interpolation functions, adequacy of different local interpolation functions (F₀, F₁, G₀, G₁, H₀, H₁) and consideration of a more practical configuration of a reinforced panel consisting of multiple laminates. Through-the-thickness displacements, strains and stresses are obtained and shown to be of reasonable accuracy. Results are compared against a highly refined mesh of 3D brick elements implemented in a commercial software that provide a benchmark for the elements capabilities.

1 Introduction

Structural components such as plates and shells are largely used in civil, mechanical and aerospace industries. When subjected to transverse loads, they develop through-the-thickness shear and normal strains. These strains are not well described or captured by classical theories commonly employed in the formulation of elements based on pure displacement and rotation interpolation. Unfortunately, most refinements of classical element formulations either are redundant or provide only marginal incremental enhancements, unless transverse stress continuity requirements between layers are imposed. However, in many theories, and consequently in existing higher-order beam and plate finite element formulations, no consideration is made regarding normal stress and strain effects (ε_z ≠ 0) on the bending, buckling and vibration responses when these theories are applied to both monolithic and sandwich laminates. One of the most widespread classical theories is the equivalent single-layer theories [1]. Due to their simplicity, equivalent single-layer theories are popularly employed in the bending, free vibration and buckling analyses of laminated beams and plates. However, these theories are incapable of capturing accurate structural responses in sandwich laminates. Zigzag theories [2, 3], or layerwise displacement field theories, are more accurate, especially in the analysis of sandwich laminates with soft cores. The inherent difficulty with the zigzag theories is, usually, the large number of unknowns that must be treated as degrees of freedom. The newly proposed theory in part I offers a formulation with the same capabilities as the zigzag theory, but with a reduced (and fixed) number of degrees of freedom, independent of the number of layers in the laminate.

The proposed theory can be used in the formulation of new elements applicable to bending, vibration and buckling analyses of both monolithic and sandwich laminates. Initially, elements designed to work under linear regime must be developed, followed by elements able to perform linearized buckling analyses (geometric stiffness matrix) and vibration (mass matrix). It is observed that the new elements are sufficiently accurate and efficient such that they avoid the need to use tridimensional finite element models in situations where the transverse stresses and strains are desired. The avoidance of tridimensional models is of particular relevance in lightweight designs, particularly in aerospace and automotive applications, where thin panels are required. The thinness of such panels naturally leads to tridimensional models with either hundreds of thousands of nodes, if aspect ratio is to be maintained close to 1.0, or to models with highly distorted 3D elements, what compromises accuracy. Moreover, it comes to iterative or nonlinear analyses, and tridimensional models are extremely computer-intensive, requiring hours or even days of simulations. The influence of the temperature profiles (heat conduction problem) on the thermoelastic response of multilayered laminates by the use of more elaborate theories with through-the-thickness capabilities is also assessed. The present paper includes thermoelastic effects in the element formulation, enabling them for analyses where temperature effects are relevant.

One practical example of the application of tridimensional models is crack propagation induced by delamination. Typically, in these cases, a combination of tridimensional modeling, to represent laminae of the laminate, and decohesive elements, to represent interfaces between laminae where cracks originate, is used. A potential application of the new formulation is to replace these complex models by elements with through-the-thickness capabilities that possess features that capture crack opening and propagation. For this purpose, it will be necessary to modify Eqs. (2–4) in part I to allow for discontinuities in the displacement fields u, v and w.

In the accompanying paper part I, a simple four-element validation under uniaxial stress (σ_z) showed promising results. However, no thorough investigations of the elements capabilities were conducted in order to assess its possible limitations, numerical accuracy and stability. Hence, the major objective of this paper part II is to critically assess these aspects and make recommendations toward more practical implementations of the element.

Global and local interpolation functions of increasing order are evaluated (linear, quadratic and cubic). In principle, non-polynomial interpolation functions can be used but the study is restricted to polynomials since they are by far the most commonly used in commercial finite element software. Single and multiple laminate configurations are considered. The ability to model reinforced laminates must not be discounted, disregarded or belittled since reinforced composite panels are one of the most frequently structural components encountered in real applications.

2 Interpolation functions and constitutive relations

In part I, the initial description of the element formulation was given assuming that Lagrange global interpolation functions would be used. This particular approach was adopted simply because it is didactically easier and more intuitive to explain the main features of the element in terms of Lagrange polynomials. Figure 4 in part I shows different “levels” of nodal planes (three levels for the tria and four levels for the quad) and it is more natural to explain the interpolation scheme proposed with the visual aid provided by Fig. 4. However, as reported in part I, a practical weakness of the element is that it cannot represent piecewise variable thickness or drop-off laminates if Lagrange polynomials are chosen as global interpolation functions. The way around this adversity was the use of monomials as global interpolation functions, where the respective coefficients related to each monomial correspond to a global nodal degree of freedom defined on the element mid-plane. Since, theoretically, there is no difference in choosing Lagrange polynomials or monomials (they are linearly dependent on one another), the latter is adopted because the monomials are the device that allow consideration of multiple laminate configurations.

When it comes to the local interpolation functions three possibilities are contemplated: ζ_k, ζ_k² and ζ_k³. The constant coefficient local interpolation function is neglected by virtue of a numerical determent. Notice that the local functions firstly presented in Eqs. (2–4) of part I appear in pairs: F₀/F₁ for u_L^(k), G₀/G₁ for v_L^(k) and H₀/H₁ for w_L^(k). In the case of w_L^(k), if one assumes H₀ = 1 and H₁ is either ζ_k, ζ_k² or ζ_k³, then matrix ${\hat{\mathbf{Q}}}_{33}$ in Eq. (72) of part I becomes singular due to enforcement of the displacement continuity condition in Eq. (9). Evidently this singularity deadly cripples the algorithm since matrix ${\hat{\mathbf{Q}}}_{33}$ must be invertible in order to eliminate the local degrees of freedom w^(k). The same holds with regard to F₀/F₁, matrix ${\hat{\mathbf{Q}}}_{44}$ and u^(k), and also with regard to G₀/G₁, matrix ${\hat{\mathbf{Q}}}_{55}$ and v^(k). More generally, the conditions for these matrices to be invertible are that there must be an even-order term in the local function (1 or ζ_k²) with nonzero first derivative with respect to ζ_k. However, the combination H₀ = 1 and H₁ = ζ_k² is still unacceptable because the term H₁ has derivatives at bottom and top layer surfaces that obey dH₁/dζ_k(− 1) =  − dH₁/dζ_k(+ 1), and since dH₀/dζ_k = 0, it is easy to see that one can make one row of matrix ${\hat{\mathbf{Q}}}_{33}$ identically zero. The conclusion is that the constant coefficient local interpolation function is not viable.

The in-plane interpolation of u_G^(k), v_G^(k) and w_G^(k) are fixed. However, the transverse global interpolation for u_G^(k), v_G^(k) and w_G^(k) and the local interpolation for u_L^(k), v_L^(k) and w_L^(k) can be selected to form a family of elements. Table 1 shows the combinations of global/local interpolations used to build six elements labeled A to F. The global interpolations must be the same for u_G^(k), v_G^(k) and w_G^(k), whereas the local interpolations for u_L^(k), v_L^(k) and w_L^(k) are independent, i.e., they can be different for u_L^(k), v_L^(k) or w_L^(k). Even so, the local interpolations selected for the six types of elements were the same in order to simplify the assessment and reduce the number of combinations. Elements A, C and E contain linear and quadratic monomials that Liu and Li [4] called 1–2 superposition theory (ST). Elements B, D and F contain quadratic and cubic monomials, called 2–3 superposition theory (ST).

Table 1 Element types with global and local transverse interpolation functions

The choice of the interpolation functions has direct repercussions on the numerical efficiency of the element. Equation (1) shows the expression for the first variation of the element strain energy, where the terms contained therein were defined in the part I of this study. The integrals in Eq. (1) are numerically computed using Gaussian quadrature in the case of quadrilateral elements and similar integration schemes adapted for the case of triangular elements.

$$\delta U_{e} = \delta {\mathbf{q}}_{e}^{T} \int\limits_{A} {\sum\limits_{k = 0}^{n - 1} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {\left\{ {\begin{array}{*{20}c} {{\mathbf{B}}_{u,x}^{(k)T} } \\ {{\mathbf{B}}_{v,y}^{(k)T} } \\ {{\mathbf{B}}_{w,z}^{(k)T} } \\ {{\mathbf{B}}_{v,z}^{(k)T} + {\mathbf{B}}_{w,y}^{(k)T} } \\ {{\mathbf{B}}_{u,z}^{(k)T} + {\mathbf{B}}_{w,x}^{(k)T} } \\ {{\mathbf{B}}_{u,y}^{(k)T} + {\mathbf{B}}_{v,x}^{(k)T} } \\ \end{array} } \right\}^{T} {\mathbf{C}}^{(k)} \left( {\left\{ {\begin{array}{*{20}c} {{\mathbf{B}}_{u,x}^{(k)} } \\ {{\mathbf{B}}_{v,y}^{(k)} } \\ {{\mathbf{B}}_{w,z}^{(k)} } \\ {{\mathbf{B}}_{v,z}^{(k)} + {\mathbf{B}}_{w,y}^{(k)} } \\ {{\mathbf{B}}_{u,z}^{(k)} + {\mathbf{B}}_{w,x}^{(k)} } \\ {{\mathbf{B}}_{u,y}^{(k)} + {\mathbf{B}}_{v,x}^{(k)T} } \\ \end{array} } \right\}{\mathbf{q}}_{e} + \left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ {a_{w,z}^{(k)} } \\ {a_{v,z}^{(k)} } \\ {a_{u,z}^{(k)} } \\ 0 \\ \end{array} } \right\} - \Delta T\left\{ {\begin{array}{*{20}c} {\alpha_{x}^{(k)} } \\ {\alpha_{y}^{(k)} } \\ {\alpha_{z}^{(k)} } \\ 0 \\ 0 \\ {\alpha_{xy}^{(k)} } \\ \end{array} } \right\}} \right)dz} } dA}$$

(1)

Notice in Eq. (1) the integral over the in-plane parent element domain A and the integral along the thickness of any given layer k in the interval [z_k, z_k+1]. The number of Gaussian integration points required to integrate over A is determined by the bending interpolation functions. In the case of the triangular BCIZ element, a complete polynomial of third degree with 10 coefficients is used, requiring the 13-point formula [5]. In the case of the quadrilateral element, a complete polynomial up to the terms of third degree and incomplete in the fourth degree (ξ³η and ξη³) is used, requiring 4-point formula along ξ and 4-point formula along η [6].

An alternative to the numerical integration along [z_k, z_k+1] exits. Interpolation functions and their derivatives appear in matrices B^(k) of Eq. (1). Hence, it is possible to split these matrices into parts of increasing order. For instance, if a global cubic interpolation along z is adopted then B^(k) = B₀^(k) + ζB₁^(k) + ζ ²B₂^(k) + ζ ³B₃^(k). Observe that the B^(k) matrices also depend on the local interpolation functions. Assuming that the 2–3 superposition theory is used (elements types B, D or F), one would have H₀ = ζ_k², H₁ = ζ_k³. Since the relationship ζ_k = (ζh − z_k − z_k+1)/h_k holds, where h is the total laminate thickness and h_k is the thickness of layer k, the local functions H₀ and H₁ can also be split into constant, linear, quadratic and cubic terms in ζ. Substitution in Eq. (1) leads to the appearance of laminate matrices A₀–A₆ of the form

$$({\mathbf{A}}_{0} ,{\mathbf{A}}_{1} ,{\mathbf{A}}_{2} ,{\mathbf{A}}_{3} ,{\mathbf{A}}_{4} ,{\mathbf{A}}_{5} ,{\mathbf{A}}_{6} ) = \sum\limits_{k = 0}^{n - 1} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {(1,\zeta ,\zeta^{2} ,\zeta^{3} ,\zeta^{4} ,\zeta^{5} ,\zeta^{6} ){\mathbf{C}}^{(k)} dz} }$$

(2)

that can be analytically computed, eliminating the need for numerical integration along z.

Since the element implemented has 3D capabilities, the material properties that must be known are: moduli of elasticity in the principal directions E₁^(k), E₂^(k), E₃^(k); shear moduli G₂₃^(k), G₁₃^(k), G₁₂^(k); Poisson coefficients ν₂₃^(k), ν₃₂^(k), ν₁₃^(k), ν₃₁^(k), ν₁₂^(k), ν₂₁^(k); and coefficients of thermal expansion α₁^(k), α₂^(k), α₃^(k). The material stiffness matrix in the principal axes, Q^(k), for layer k is:

$${\mathbf{Q}}^{(k)} = \left[ {\begin{array}{*{20}c} {Q_{11}^{(k)} } & {Q_{12}^{(k)} } & {Q_{13}^{(k)} } & 0 & 0 & 0 \\ {Q_{12}^{(k)} } & {Q_{22}^{(k)} } & {Q_{23}^{(k)} } & 0 & 0 & 0 \\ {Q_{13}^{(k)} } & {Q_{23}^{(k)} } & {Q_{33}^{(k)} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {Q_{44}^{(k)} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {Q_{55}^{(k)} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {Q_{66}^{(k)} } \\ \end{array} } \right]$$

(3)

where Q₁₁^(k) = (1 − ν₂₃^(k)ν₃₂^(k))/(E₂^(k)E₃^(k)μ^(k)), Q₂₂^(k) = (1 − ν₁₃^(k)ν₃₁^(k))/(E₁^(k)E₃^(k)μ^(k)), Q₃₃^(k) = (1 − ν₁₂^(k)ν₂₁^(k))/(E₁^(k)E₂^(k)μ^(k)), Q₁₂^(k) = (ν₁₂^(k) + ν₃₂^(k)ν₁₃^(k))/(E₁^(k)E₃^(k)μ^(k)), Q₁₃^(k) = (ν₁₃^(k) + ν₁₂^(k)ν₂₃^(k))/ (E₁^(k)E₂^(k)μ^(k)), Q₂₃^(k) = (ν₂₃^(k) + ν₂₁^(k)ν₁₃^(k))/(E₂^(k)E₃^(k)μ^(k)), Q₄₄^(k) = G₂₃^(k), Q₅₅^(k) = G₁₃^(k), Q₆₆^(k) = G₁₂^(k) and μ^(k) = (1 − ν₁₂^(k)ν₂₁^(k) − ν₁₃^(k)ν₃₁^(k) − ν₂₃^(k)ν₃₂^(k) − 2ν₂₁^(k)ν₃₂^(k)ν₁₃^(k))/(E₁^(k)E₂^(k)E₃^(k)).

The material stiffness matrix in the local element axes, C^(k), is

$${\mathbf{C}}^{(k)} = \left[ {\begin{array}{*{20}c} {C_{11}^{(k)} } & {C_{12}^{(k)} } & {C_{13}^{(k)} } & 0 & 0 & {C_{16}^{(k)} } \\ {C_{12}^{(k)} } & {C_{22}^{(k)} } & {C_{23}^{(k)} } & 0 & 0 & {C_{26}^{(k)} } \\ {C_{13}^{(k)} } & {C_{23}^{(k)} } & {C_{33}^{(k)} } & 0 & 0 & {C_{36}^{(k)} } \\ 0 & 0 & 0 & {C_{44}^{(k)} } & {C_{45}^{(k)} } & 0 \\ 0 & 0 & 0 & {C_{45}^{(k)} } & {C_{55}^{(k)} } & 0 \\ {C_{16}^{(k)} } & {C_{26}^{(k)} } & {C_{36}^{(k)} } & 0 & 0 & {C_{66}^{(k)} } \\ \end{array} } \right]$$

(4)

where the lamination angle of layer k is θ_k, s = sinθ_k, c = cosθk, and

$$\begin{gathered} C_{11}^{(k)} = Q_{11}^{(k)} c^{4} + Q_{22}^{(k)} s^{4} + 2c^{2} s^{2} Q_{12}^{(k)} + 4c^{2} s^{2} Q_{66}^{(k)} \hfill \\ C_{12}^{(k)} = (Q_{11}^{(k)} + Q_{22}^{(k)} )c^{2} s^{2} + Q_{12}^{(k)} (c^{4} + s^{4} ) - 4c^{2} s^{2} Q_{66}^{(k)} \hfill \\ C_{13}^{(k)} = Q_{13}^{(k)} c^{2} + Q_{23}^{(k)} s^{2} ,\;C_{23}^{(k)} = Q_{13}^{(k)} s^{2} + Q_{23}^{(k)} c^{2} \hfill \\ C_{16}^{(k)} = (Q_{11}^{(k)} - Q_{12}^{(k)} )c^{3} s + (Q_{12}^{(k)} - Q_{22}^{(k)} )cs^{3} - 2cs(c^{2} - s^{2} )Q_{66}^{(k)} \hfill \\ C_{22}^{(k)} = Q_{11}^{(k)} s^{4} + Q_{22}^{(k)} c^{4} + 2c^{2} s^{2} Q_{12}^{(k)} + 4c^{2} s^{2} Q_{66}^{(k)} \hfill \\ C_{26}^{(k)} = (Q_{11}^{(k)} - Q_{12}^{(k)} )cs^{3} + (Q_{12}^{(k)} - Q_{22}^{(k)} )c^{3} s + 2cs(c^{2} - s^{2} )Q_{66}^{(k)} \hfill \\ C_{33}^{(k)} = Q_{33}^{(k)} ,\;C_{36}^{(k)} = (Q_{13}^{(k)} - Q_{23}^{(k)} )cs \hfill \\ C_{44}^{(k)} = Q_{44}^{(k)} c^{2} + Q_{55}^{(k)} s^{2} ,\;C_{45}^{(k)} = (Q_{55}^{(k)} - Q_{44}^{(k)} )cs,\;C_{55}^{(k)} = Q_{44}^{(k)} s^{2} + Q_{55}^{(k)} c^{2} \hfill \\ C_{66}^{(k)} = (Q_{11}^{(k)} + Q_{22}^{(k)} - 2Q_{12}^{(k)} )c^{2} s^{2} + Q_{66}^{(k)} (c^{2} - s^{2} )^{2} \hfill \\ \end{gathered}$$

(5)

The lamination angle of layer k is θ_k measured with respect to side 1–2 of the element. Figure 4 in part I illustrates the local coordinate system xyz of both tria and quad elements. The x axis is aligned with side 1–2, but it does not have to be. In practical implementation the orientation of the element local coordinate system is determined though the procedure presented in the next section.

3 The plate bending elements

The purpose of this section is to clearly present the implementations of the elements with their in-plane, bending and drilling interpolation functions.

3.1 The triangular plate bending element

The three-node triangle plate bending element to be implemented is shown in Fig. 1 with the mapping from local (ξη) to element coordinates (xy). The traditional area coordinates are ξ, η and ς = 1 − ξ − η. The central node 0 can be subsequently eliminated.

Fig. 1

Three-node element topology and mapping

Since plate bending behavior is to be modeled, nodes 1–3 possess each six degrees of freedom: u, v, w, w_,ξ, w_,η and the drilling θ_z, whereas node 0 possesses only three degrees of freedom: u, v and w. Therefore, there is a total of 21 degrees of freedom. Notice that, in the element coordinate system, the rotational degrees of freedom w_,ξ and w_,η must be computed in terms of the local coordinates x and y, i.e., w_,x and w_,y. This can be accomplished by the chain rule of differentiation:

$$\left\{ {\begin{array}{*{20}c} {w_{,\xi } } \\ {w_{,\eta } } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {x_{,\xi } } & {y_{,\xi } } \\ {x_{,\eta } } & {y_{,\eta } } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {w_{,x} } \\ {w_{,y} } \\ \end{array} } \right\}$$

(6)

and the derivatives of x and y with respect to ξ and η can be obtained using the nodal coordinates (x_i, y_i) and the area coordinates:

$$x = \xi \, x_{1} + \eta \, x_{2} + \varsigma \, x_{3} ,\quad y = \xi \, y_{1} + \eta \, y_{2} + \varsigma \, y_{3}$$

(7)

The interpolation scheme proposed for u and v is quadratic and given by

$$u = \sum\limits_{i = 0}^{3} {\varphi_{i} u_{i} } + \sum\limits_{i = 1}^{3} {P_{ui} \theta_{zi} } ,\quad v = \sum\limits_{i = 0}^{3} {\varphi_{i} v_{i} } + \sum\limits_{i = 1}^{3} {P_{vi} \theta_{zi} }$$

(8)

where

$$\varphi_{1} = \xi - \xi \eta - \xi \varsigma - \eta \varsigma ,\;\varphi_{2} = \eta - \xi \eta - \xi \varsigma - \eta \varsigma ,\;\varphi_{3} = \zeta - \xi \eta - \xi \varsigma - \eta \varsigma ,\;\varphi_{0} = 3(\xi \eta + \xi \varsigma + \eta \varsigma )$$

(9)

and the drilling interpolation functions are those presented in Eq. (36) of part I [7]. The interpolation scheme proposed for w consists in a complete polynomial of third degree with 10 coefficients:

$$w(\xi ,\eta ) = \;a_{0} + a_{1} \xi + a_{2} \eta + a_{3} \xi^{2} + a_{4} \xi \eta + a_{5} \eta^{2} + a_{6} \xi^{3} + a_{7} \xi^{2} \eta + a_{8} \xi \eta^{2} + a_{9} \eta^{3}$$

(10)

The 10 coefficients in Eq. (10) can be determined by imposition of the conditions given in Table 2, leading to the interpolation for w in terms of shape functions N₁-N10 and the nodal degrees of freedom w₁, w_1,ξ, w_1,η, w₂, w_2,ξ, w_2,η, w₃, w_3,ξ, w_3,η, w₀:

$$w = \; N_{1} w_{1} + N_{2}^{**} w_{1,\xi } + N_{3}^{**} w_{1,\eta } + N_{4} w_{2} + N_{5}^{**} w_{2,\xi } + N_{6}^{**} w_{2,\eta } + N_{7} w_{3} + N_{8}^{**} w_{3,\xi } + N_{9}^{**} w_{3,\eta } + N_{10} w_{0}$$

(11)

Table 2 Conditions to be imposed in the determination of the interpolation for w in the triangular element

where

$$N_{1} = 3\xi^{2} - 2\xi^{3} - 7\xi \eta \varsigma ,\;N_{2}^{**} = - \xi^{2} + \xi^{3} + 2\xi \eta \varsigma ,\;N_{3}^{**} = \xi^{2} \eta - \xi \eta \varsigma$$

(12)

$$N_{4} = 3\eta^{2} - 2\eta^{3} - 7\xi \eta \varsigma ,\;N_{5}^{**} = \xi \eta^{2} - 2\xi \eta \varsigma ,\;N_{6}^{**} = - \eta^{2} + \eta^{3} + 2\xi \eta \varsigma$$

(13)

$$N_{7} = 3\varsigma^{2} - 2\varsigma^{3} - 7\xi \eta \varsigma ,\;N_{8}^{**} = \xi \varsigma^{2} - \xi \eta \varsigma ,\;N_{9}^{**} = \eta \varsigma^{2} - \xi \eta \varsigma$$

(14)

$$N_{10} = 27\xi \eta \varsigma$$

(15)

According to Eq. (6), the second and third terms in Eq. (11) are written as

$$N_{2}^{**} w_{1,\xi } + N_{3}^{**} w_{1,\eta } = N_{2}^{**} (x_{1,\xi } w_{1,x} + y_{1,\xi } w_{1,y} ) + N_{3}^{**} (x_{1,\eta } w_{1,x} + y_{1,\eta } w_{1,y} ) = (N_{2}^{**} x_{1,\xi } + N_{3}^{**} x_{1,\eta } )w_{1,x} + (N_{2}^{**} y_{1,\xi } + N_{3}^{**} y_{1,\eta } )w_{1,y} = N_{2}^{*} w_{1,x} + N_{3}^{*} w_{1,y}$$

(16)

where x_1,ξ, y_1,ξ, x_1,η and y_1,η are the terms of the Jacobian matrix. Since the mapping is linear, the Jacobian matrix is constant and x_,ξ = x₁₃, y,_ξ = y₁₃, x_,η = x₂₃ and y_,η = y₂₃, where x_ij = x_i − x_j and y_ij = y_i − y_j. Hence, N₂^* = N₂^**x₁₃ + N₃^**x₂₃ and N₃^* = N₂^**y₁₃ + N₃^**y₂₃. Moreover, rotational degrees of freedom are usually defined as θ_x = w_,y and θ_y =  − w_,x, such that N₂ = N₃^* and N₃ =  − N₂^*.The new interpolation function for w in terms of θ_x and θ_y is

$$w = \;N_{1} w_{1} + N_{2} \theta_{x1} + N_{3} \theta_{y1} + N_{4} w_{2} + N_{5} \theta_{x2} + N_{6} \theta_{y2} + N_{7} w_{3} + N_{8} \theta_{x3} + N_{9} \theta_{y3} + N_{10} w_{0}$$

(17)

Second derivatives of w are required in the computation of strains. These can be evaluated through double application of the chain rule yielding

$$\left\{ {\begin{array}{*{20}c} {w_{,\xi \xi } } \\ {w_{,\xi \eta } } \\ {w_{,\eta \eta } } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {x_{,\xi }^{2} } & {2x_{,\xi } y_{,\xi } } & {y_{,\xi }^{2} } \\ {x_{,\xi } x_{,\eta } } & {x_{,\xi } y_{,\eta } + x_{,\eta } y_{,\xi } } & {y_{,\xi } y_{,\eta } } \\ {x_{,\eta }^{2} } & {2x_{,\eta } y_{,\eta } } & {y_{,\eta }^{2} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {w_{,xx} } \\ {w_{,xy} } \\ {w_{,yy} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {x_{,\xi \xi } } & {y_{,\xi \xi } } \\ {x_{,\xi \eta } } & {y_{,\xi \eta } } \\ {x_{,\eta \eta } } & {y_{,\eta \eta } } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {w_{,x} } \\ {w_{,y} } \\ \end{array} } \right\}$$

(18)

and, since the Jacobian matrix is constant, x_,ξξ = y_,ξξ = x_,ξη = y_,ξη = x_,ηη = y_,ηη = 0, leading to

$$\left\{ {\begin{array}{*{20}c} {w_{,xx} } \\ {w_{,xy} } \\ {w_{,yy} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {x_{,\xi }^{2} } & {2x_{,\xi } y_{,\xi } } & {y_{,\xi }^{2} } \\ {x_{,\xi } x_{,\eta } } & {x_{,\xi } y_{,\eta } + x_{,\eta } y_{,\xi } } & {y_{,\xi } y_{,\eta } } \\ {x_{,\eta }^{2} } & {2x_{,\eta } y_{,\eta } } & {y_{,\eta }^{2} } \\ \end{array} } \right]^{ - 1} \left\{ {\begin{array}{*{20}c} {w_{,\xi \xi } } \\ {w_{,\xi \eta } } \\ {w_{,\eta \eta } } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {x_{13}^{2} } & {2x_{13} y_{13} } & {y_{13}^{2} } \\ {x_{13} x_{23} } & {x_{13} y_{23} + x_{23} y_{13} } & {y_{13} y_{23} } \\ {x_{23}^{2} } & {2x_{23} y_{23} } & {y_{23}^{2} } \\ \end{array} } \right]^{ - 1} \left\{ {\begin{array}{*{20}c} {w_{,\xi \xi } } \\ {w_{,\xi \eta } } \\ {w_{,\eta \eta } } \\ \end{array} } \right\}$$

(19)

However, when the degree of freedom w₀ is condensed out, the resulting element becomes too stiff under bending since the curvature completeness is lost. In order to overcome this problem, the BCIZ element was implemented. Its idea is to still condense out w₀ but maintaining curvature completeness. Consider the six node quadratic element shown in Fig. 2 whose interpolation functions are

$$\psi_{1} =\; \xi (2\xi - 1),\;\psi_{2} = \eta (2\eta - 1),\;\psi_{3} = \zeta (2\varsigma - 1),\;\psi_{4} = 4\xi \eta ,\;\psi_{5} = 4\eta \varsigma ,\;\psi_{6} = 4\xi \varsigma$$

(20)

Fig. 2

Quadratic interpolation for w

One can express w₀ (the transverse displacement in the center of the element) in terms of the nodal transverse displacements w₁-w₆ through the use of the interpolations functions defined in Eq. (20) evaluated at ξ = η = ς = 1/3:

$$w_{0} = \frac{4}{9}(w_{4} + w_{5} + w_{6} ) - \frac{1}{9}(w_{1} + w_{2} + w_{3} )$$

(21)

Next, the interpolation functions defined in Eq. (17) can be used to compute w₄, w₅ and w₆ in terms of the degrees of freedom w₁, θ_x1, θ_y1, w₂, θ_x2, θ_y2, w₃, θ_x3, θ_y3. w₄ is computed making ξ = η = 1/2 and ς = 0. w₅ is computed making η = ς = 1/2 and ξ = 0. w₆ is computed making ξ = ς = 1/2 and η = 0. Hence,

$$\begin{gathered} w_{4} = \frac{{w_{1} }}{2} + \frac{{y_{21} }}{8}\theta_{x1} - \frac{{x_{21} }}{8}\theta_{y1} + \frac{{w_{2} }}{2} - \frac{{y_{21} }}{8}\theta_{x2} + \frac{{x_{21} }}{8}\theta_{y2} \hfill \\ w_{5} = \frac{{w_{2} }}{2} + \frac{{y_{32} }}{8}\theta_{x2} - \frac{{x_{32} }}{8}\theta_{y2} + \frac{{w_{3} }}{2} - \frac{{y_{32} }}{8}\theta_{x3} + \frac{{x_{32} }}{8}\theta_{y3} \hfill \\ w_{6} = \frac{{w_{1} }}{2} - \frac{{y_{13} }}{8}\theta_{x1} + \frac{{x_{13} }}{8}\theta_{y1} + \frac{{w_{3} }}{2} + \frac{{y_{13} }}{8}\theta_{x3} - \frac{{x_{13} }}{8}\theta_{y3} \hfill \\ \end{gathered}$$

(22)

Substitution of Eq. (22) into (21) gives

$$w_{0} = \frac{1}{3}(w_{1} + w_{2} + w_{3} ) + \frac{{y_{21} - y_{13} }}{18}\theta_{x1} + \frac{{x_{13} - x_{21} }}{18}\theta_{y1} + \frac{{y_{32} - y_{21} }}{18}\theta_{x2} + \frac{{x_{21} - x_{32} }}{18}\theta_{y2} + \frac{{y_{13} - y_{32} }}{18}\theta_{x3} + \frac{{x_{32} - x_{13} }}{18}\theta_{y3}$$

(23)

Equation (23) can now be substituted into Eq. (17) to eliminate w₀ yielding

$$\begin{aligned} w & = (N_{1} + a_{w1} N_{10} )w_{1} + (N_{2} + a_{\theta x1} N_{10} )\theta_{x1} + (N_{3} + a_{\theta y1} N_{10} )\theta_{y1} \\ & \quad { + }(N_{4} + a_{w2} N_{10} )w_{2} + (N_{5} + a_{\theta x2} N_{10} )\theta_{x2} + (N_{6} + a_{\theta y2} N_{10} )\theta_{y2} \\ & \quad { + }(N_{7} + a_{w3} N_{10} )w_{3} + (N_{8} + a_{\theta x3} N_{10} )\theta_{x3} + (N_{9} + a_{\theta y3} N_{10} )\theta_{y3} \\ \end{aligned}$$

(24)

where a_w1 = a_w2 = a_w3 = 1/3, a_θx1 = (y₂₁ − y₁₃)/18, a_θy1 = (x₁₃ − x₂₁)/18, a_θx2 = (y₃₂ − y₂₁)/18, a_θy2 = (x₂₁ − x₃₂)/18, a_θx3 = (y₁₃ − y₃₂)/18, a_θy3 = (x₃₂ − x₁₃)/18. The implementation of the element proceeds normally using the interpolation functions for w defined in Eq. (24).

3.2 The quadrilateral plate bending element

The four-node plate bending element to be implemented is shown in Fig. 3 with the mapping from local (ξη) to element coordinates (xy).

Fig. 3

Four-node element topology and mapping

Since plate bending behavior is to be modeled, nodes 1–4 possess each six degrees of freedom: u, v, w, w_,ξ, w_,η and the drilling θ_z. Therefore, there is a total of 24 degrees of freedom. Notice that, in the element coordinate system, the rotational degrees of freedom w_,ξ and w_,η must be computed in terms of the coordinates x and y, i.e., w_,x and w_,y. This can be accomplished by the chain rule of differentiation as in Eq. (6). The derivatives of x and y with respect to ξ and η can be obtained using the nodal coordinates (x_i, y_i) and the bilinear Lagrange interpolation functions:

$$x = \sum\limits_{i = 1}^{4} {\varphi_{i} x_{i} } ,\;y = \sum\limits_{i = 1}^{4} {\varphi_{i} y_{i} }$$

(25)

The interpolation scheme proposed for u and v is given by

$$u = \sum\limits_{i = 1}^{4} {(\varphi_{i} u_{i} + P_{ui} \theta_{zi} )} ,\;v = \sum\limits_{i = 1}^{4} {(\varphi_{i} v_{i} + P_{vi} \theta_{zi} )}$$

(26)

where

$$\eqalign{ & {\varphi _1} = \frac{1}{4}(1 - \xi )(1 - \eta ),\;{\varphi _2} = \frac{1}{4}(1 + \xi )(1 - \eta ), \cr & {\varphi _3} = \frac{1}{4}(1 + \xi )(1 + \eta ),\;{\varphi _4} = \frac{1}{4}(1 - \xi )(1 + \eta ) \cr}$$

(27)

and the drilling interpolation functions are [8]

$$\begin{gathered} P_{u1} = \frac{1}{16}[l_{41} (1 - \xi )(1 - \eta^{2} )\sin \beta_{41} - l_{12} (1 - \xi^{2} )(1 - \eta )\sin \beta_{12} ] \hfill \\ P_{v1} = \frac{1}{16}[l_{12} (1 - \xi^{2} )(1 - \eta )\cos \beta_{12} - l_{41} (1 - \xi )(1 - \eta^{2} )\cos \beta_{41} ] \hfill \\ P_{u2} = \frac{1}{16}[l_{12} (1 - \xi^{2} )(1 - \eta )\sin \beta_{12} - l_{23} (1 + \xi )(1 - \eta^{2} )\sin \beta_{23} ] \hfill \\ P_{v2} = \frac{1}{16}[l_{23} (1 + \xi )(1 - \eta^{2} )\cos \beta_{23} - l_{12} (1 - \xi^{2} )(1 - \eta )\cos \beta_{12} ] \hfill \\ P_{u3} = \frac{1}{16}[l_{23} (1 + \xi )(1 - \eta^{2} )\sin \beta_{23} - l_{34} (1 - \xi^{2} )(1 + \eta )\sin \beta_{34} ] \hfill \\ P_{v3} = \frac{1}{16}[l_{34} (1 - \xi^{2} )(1 + \eta )\cos \beta_{34} - l_{23} (1 + \xi )(1 - \eta^{2} )\cos \beta_{23} ] \hfill \\ P_{u4} = \frac{1}{16}[l_{34} (1 - \xi^{2} )(1 + \eta )\sin \beta_{34} - l_{41} (1 - \xi )(1 - \eta^{2} )\sin \beta_{41} ] \hfill \\ P_{v4} = \frac{1}{16}[l_{41} (1 - \xi )(1 - \eta^{2} )\cos \beta_{41} - l_{34} (1 - \xi^{2} )(1 + \eta )\cos \beta_{34} ] \hfill \\ \end{gathered}$$

(28)

The interpolation scheme proposed for w consists in a polynomial of fourth degree with 12 coefficients. The polynomial is complete up to term of third degree and incomplete in the fourth degree:

$$w(\xi ,\eta ) = a_{0} + a_{1} \xi + a_{2} \eta + a_{3} \xi^{2} + a_{4} \xi \eta + a_{5} \eta^{2} + a_{6} \xi^{3} + a_{7} \xi^{2} \eta + a_{8} \xi \eta^{2} + a_{9} \eta^{3} + \, a_{10} \xi^{3} \eta + a_{11} \xi \eta^{3}$$

(29)

The 12 coefficients in Eq. (29) can be determined by imposition of the conditions given in Table 3, leading to the interpolation for w in terms of shape functions N₁-N₁₂ and the nodal degrees of freedom w₁, w_1,ξ, w_1,η, …, w₄, w_4,ξ, w_4,η:

$$w = N_{1} w_{1} + N_{2}^{**} w_{1,\xi } + N_{3}^{**} w_{1,\eta } + \cdots + N_{10} w_{4} + N_{11}^{**} w_{4,\xi } + N_{12}^{**} w_{4,\eta }$$

(30)

Table 3 Conditions to be imposed in the determination of the interpolation for w in the quadrilateral element

where

$$\begin{gathered} N_{1} = \frac{1}{8}(2 - 3\xi - 3\eta + 4\xi \eta + \xi^{3} + \eta^{3} - \xi^{3} \eta - \xi \eta^{3} ) \hfill \\ N_{2}^{**} = \frac{1}{8}(1 - \xi - \eta - \xi^{2} + \xi \eta + \xi^{3} + \xi^{2} \eta - \xi^{3} \eta ) \hfill \\ N_{3}^{**} = \frac{1}{8}(1 - \xi - \eta + \xi \eta - \eta^{2} + \xi \eta^{2} + \eta^{3} - \xi \eta^{3} ) \hfill \\ \end{gathered}$$

(31)

$$\begin{gathered} N_{4} = \frac{1}{8}(2 + 3\xi - 3\eta - 4\xi \eta - \xi^{3} + \eta^{3} + \xi^{3} \eta + \xi \eta^{3} ) \hfill \\ N_{5}^{**} = \frac{1}{8}( - 1 - \xi + \eta + \xi^{2} + \xi \eta + \xi^{3} - \xi^{2} \eta - \xi^{3} \eta ) \hfill \\ N_{6}^{**} = \frac{1}{8}(1 + \xi - \eta - \xi \eta - \eta^{2} - \xi \eta^{2} + \eta^{3} + \xi \eta^{3} ) \hfill \\ \end{gathered}$$

(32)

$$\begin{gathered} N_{7} = \frac{1}{8}(2 + 3\xi + 3\eta + 4\xi \eta - \xi^{3} - \eta^{3} - \xi^{3} \eta - \xi \eta^{3} ) \hfill \\ N_{8}^{**} = \frac{1}{8}( - 1 - \xi - \eta + \xi^{2} - \xi \eta + \xi^{3} + \xi^{2} \eta + \xi^{3} \eta ) \hfill \\ N_{9}^{**} = \frac{1}{8}( - 1 - \xi - \eta - \xi \eta + \eta^{2} + \xi \eta^{2} + \eta^{3} + \xi \eta^{3} ) \hfill \\ \end{gathered}$$

(33)

$$\begin{gathered} N_{10} = \frac{1}{8}(2 - 3\xi + 3\eta - 4\xi \eta + \xi^{3} - \eta^{3} + \xi^{3} \eta + \xi \eta^{3} ) \hfill \\ N_{11}^{**} = \frac{1}{8}(1 - \xi + \eta - \xi^{2} - \xi \eta + \xi^{3} - \xi^{2} \eta + \xi^{3} \eta ) \hfill \\ N_{12}^{**} = \frac{1}{8}( - 1 + \xi - \eta + \xi \eta + \eta^{2} - \xi \eta^{2} + \eta^{3} - \xi \eta^{3} ) \hfill \\ \end{gathered}$$

(34)

According to Eq. (6), the second and third terms in Eq. (30) are written as

$$\begin{gathered} N_{2}^{**} w_{1,\xi } + N_{3}^{**} w_{1,\eta } = N_{2}^{**} (x_{1,\xi } w_{1,x} + y_{1,\xi } w_{1,y} ) + N_{3}^{**} (x_{1,\eta } w_{1,x} + y_{1,\eta } w_{1,y} ) \hfill \\ = (N_{2}^{**} x_{1,\xi } + N_{3}^{**} x_{1,\eta } )w_{1,x} + (N_{2}^{**} y_{1,\xi } + N_{3}^{**} y_{1,\eta } )w_{1,y} = N_{2}^{*} w_{1,x} + N_{3}^{*} w_{1,y} \hfill \\ \end{gathered}$$

(35)

where x_1,ξ, y_1,ξ, x_1,η and y_1,η are the terms of the Jacobian matrix evaluated at node 1. For the nodes 2–4, it is also possible to write expressions similar to Eq. (35) leading to the modified interpolation function for w in terms of w_,x and w_,y. Moreover, rotational degrees of freedom are usually defined as θ_x = w_,y and θ_y =  − w_,x, such that N₂ = N₃^* and N₃ =  − N₂^*.The new interpolation function for w in terms of θ_x and θ_y is

$$w = N_{1} w_{1} + N_{2} \theta_{x1} + N_{3} \theta_{y1} + \cdots + N_{10} w_{4} + N_{11} \theta_{x4} + N_{12} \theta_{y4}$$

(36)

Second-order derivatives of w are required in the computation of strains. These can be evaluated through double application of the chain rule yielding Eq. (18). Similarly, third-order derivatives w_,xxx, w_,xxy, w_,xyy and w_,yyy can be evaluated as

$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {w_{,\xi \xi \xi } } \\ {w_{,\xi \xi \eta } } \\ {w_{,\xi \eta \eta } } \\ {w_{,\eta \eta \eta } } \\ \end{array} } \right\} & = \left[ {\begin{array}{*{20}c} {x_{,\xi }^{3} } & {3x_{,\xi }^{2} y_{,\xi } } & {3x_{,\xi } y_{,\xi }^{2} } & {y_{,\xi }^{3} } \\ {x_{,\xi }^{2} x_{,\eta } } & {x_{,\xi }^{2} y_{,\eta } + 2x_{,\xi } x_{,\eta } y_{,\xi } } & {2x_{,\xi } y_{,\xi } y_{\eta } + x_{,\eta } y_{,\xi }^{2} } & {y_{,\xi }^{2} y_{,\eta } } \\ {x_{,\xi } x_{,\eta }^{2} } & {x_{,\eta }^{2} y_{,\xi } + 2x_{,\xi } x_{,\eta } y_{,\eta } } & {x_{,\xi } y_{,\eta }^{2} + 2x_{,\eta } y_{,\xi } y_{,\eta } } & {y_{,\xi } y_{,\eta }^{2} } \\ {x_{,\eta }^{3} } & {3x_{,\eta }^{2} y_{,\eta } } & {3x_{,\eta } y_{,\eta }^{2} } & {y_{,\eta }^{3} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {w_{,xxx} } \\ {w_{,xxy} } \\ {w_{,xyy} } \\ {w_{,yyy} } \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {3x_{,\xi } x_{,\xi \xi } } & {3(x_{,\xi \xi } y_{,\xi } + x_{,\xi } y_{,\xi \xi } )} & {3y_{,\xi } y_{,\xi \xi } } \\ {2x_{,\xi } x_{,\xi \eta } + x_{,\xi \xi } x_{,\eta } } & {2(x_{,\xi \eta } y_{,\xi } + x_{,\xi } y_{,\xi \eta } ) + x_{,\xi \xi } y_{,\eta } + x_{,\eta } y_{,\xi \xi } } & {2y_{,\xi } y_{,\xi \eta } + y_{,\xi \xi } y_{,\eta } } \\ {2x_{,\eta } x_{,\xi \eta } + x_{,\xi } x_{,\eta \eta } } & {2(x_{,\xi \eta } y_{,\eta } + x_{,\eta } y_{,\xi \eta } ) + x_{,\eta \eta } y_{,\xi } + x_{,\xi } y_{,\eta \eta } } & {2y_{,\eta } y_{,\xi \eta } + y_{,\xi } y_{,\eta \eta } } \\ {3x_{,\eta } x_{,\eta \eta } } & {3(x_{,\eta } y_{,\eta \eta } + x_{,\eta \eta } y_{,\eta } )} & {3y_{,\eta } y_{,\eta \eta } } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {w_{,xx} } \\ {w_{,xy} } \\ {w_{,yy} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {x_{,\xi \xi \xi } } & {y_{,\xi \xi \xi } } \\ {x_{,\xi \xi \eta } } & {y_{,\xi \xi \eta } } \\ {x_{,\xi \eta \eta } } & {y_{,\xi \eta \eta } } \\ {x_{,\eta \eta \eta } } & {y_{,\eta \eta \eta } } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {w_{,x} } \\ {w_{,y} } \\ \end{array} } \right\} \\ \end{aligned}$$

(37)

4 A technique to minimize Jacobian variation in quadrilateral elements

Evaluation of the Jacobian of a mapping transformation is necessary in the most common finite element numerical procedures. If the Jacobian of the transformation is constant, then it is expected that the numerical integration, usually carried out using the Gaussian integration rule, will be more accurate because fractions of polynomials are avoided in the computation of strains. This document describes a procedure to make the Jacobian of a four-node quadrilateral element as constant as possible simply by selecting an appropriate local reference system. Since transformation of local to global reference systems is almost always inevitable, the procedure presented does not add significant burden to the numerical processing cost.

The topology of the four-node quadrilateral element is shown in Fig. 4. Originally, the local coordinate system assigned to this element is xy and the nodal coordinates in this original system are given by x_i, y_i, for i = 1, 2, 3, 4. A new rotated local coordinate system can be defined as $\overline{x}\overline{y}$ in which the nodal coordinates are $\overline{x}_{i}$, $\overline{y}_{i}$, for i = 1, 2, 3, 4. The relationship between the two local reference systems is

$$\overline{x} = (x - x_{0} )\cos \beta + (y - y_{0} )\sin \beta ,\;\overline{y} = (y - y_{0} )\cos \beta - (x - x_{0} )\sin \beta$$

(38)

Fig. 4

Four-node element topology and mapping

where x₀, y₀ and β are parameters to be determined.

In the computation of strains, it is necessary to evaluate derivatives of displacements with respect to x and y, what can be accomplished by the chain rule of differentiation:

$$\left\{ {\begin{array}{*{20}c} {\overline{u}_{,\xi } } \\ {\overline{u}_{,\eta } } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\overline{x}_{,\xi } } & {\overline{y}_{,\xi } } \\ {\overline{x}_{,\eta } } & {\overline{y}_{,\eta } } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\overline{u}_{,x} } \\ {\overline{u}_{,y} } \\ \end{array} } \right\}$$

(39)

where $\overline{u}$ is the displacement along $\overline{x}$. In order to compute the elements of the 2 × 2 Jacobian matrix in Eq. (39), the required interpolation functions and their derivatives are

$$\varphi_{1} = \frac{1}{4}(1 - \xi )(1 - \eta ),\;\varphi_{2} = \frac{1}{4}(1 + \xi )(1 - \eta ),\;\varphi_{3} = \frac{1}{4}(1 + \xi )(1 + \eta ),\;\varphi_{4} = \frac{1}{4}(1 - \xi )(1 + \eta )$$

(40)

$$\varphi_{1,\xi } = \frac{1}{4}(\eta - 1),\;\varphi_{2,\xi } = \frac{1}{4}(1 - \eta ),\;\varphi_{3,\xi } = \frac{1}{4}(1 + \eta ),\;\varphi_{4,\xi } = \frac{ - 1}{4}(1 + \eta )$$

(41)

$$\varphi_{1,\eta } = \frac{1}{4}(\xi - 1),\;\varphi_{2,\eta } = \frac{ - 1}{4}(1 + \xi ),\;\varphi_{3,\eta } = \frac{1}{4}(1 + \xi ),\;\varphi_{4,\eta } = \frac{1}{4}(1 - \xi )$$

(42)

The elements of the 2 × 2 Jacobian matrix are

$$\overline{x}_{,\xi } = \frac{1}{4}[(\eta - 1)\overline{x}_{1} + (1 - \eta )\overline{x}_{2} + (1 + \eta )\overline{x}_{3} - (1 + \eta )\overline{x}_{4} ] = \frac{1}{4}[( - \overline{x}_{1} + \overline{x}_{2} + \overline{x}_{3} - \overline{x}_{4} ) + \eta (\overline{x}_{1} - \overline{x}_{2} + \overline{x}_{3} - \overline{x}_{4} )]$$

(43)

$$\overline{x}_{,\eta } = \frac{1}{4}[(\xi - 1)\overline{x}_{1} - (1 + \xi )\overline{x}_{2} + (1 + \xi )\overline{x}_{3} + (1 - \xi )\overline{x}_{4} ] = \frac{1}{4}[( - \overline{x}_{1} - \overline{x}_{2} + \overline{x}_{3} + \overline{x}_{4} ) + \xi (\overline{x}_{1} - \overline{x}_{2} + \overline{x}_{3} - \overline{x}_{4} )]$$

(44)

$$\overline{y}_{,\xi } = \frac{1}{4}[(\eta - 1)\overline{y}_{1} + (1 - \eta )\overline{y}_{2} + (1 + \eta )\overline{y}_{3} - (1 + \eta )\overline{y}_{4} ] = \frac{1}{4}[( - \overline{y}_{1} + \overline{y}_{2} + \overline{y}_{3} - \overline{y}_{4} ) + \eta (\overline{y}_{1} - \overline{y}_{2} + \overline{y}_{3} - \overline{y}_{4} )]$$

(45)

$$\overline{y}_{,\eta } = \frac{1}{4}[(\xi - 1)\overline{y}_{1} - (1 + \xi )\overline{y}_{2} + (1 + \xi )\overline{y}_{3} + (1 - \xi )\overline{y}_{4} ] = \frac{1}{4}[( - \overline{y}_{1} - \overline{y}_{2} + \overline{y}_{3} + \overline{y}_{4} ) + \xi (\overline{y}_{1} - \overline{y}_{2} + \overline{y}_{3} - \overline{y}_{4} )]$$

(46)

Examination of Eqs. (43–46) permits to conclude that, in order for the Jacobian to be constant, the terms multiplied by both ξ and η must be zero, i.e., $\overline{x}_{1} - \overline{x}_{2} + \overline{x}_{3} - \overline{x}_{4} = 0$ and $\overline{y}_{1} - \overline{y}_{2} + \overline{y}_{3} - \overline{y}_{4} = 0$. However, both conditions cannot be rigorously met simultaneously. If one intuitively defines the quadratic error f as

$$f = (\overline{x}_{1} - \overline{x}_{2} + \overline{x}_{3} - \overline{x}_{4} )^{2} + (\overline{y}_{1} - \overline{y}_{2} + \overline{y}_{3} - \overline{y}_{4} )^{2}$$

(47)

and substitutes Eq. (38) into Eq. (47), the result is

$$f = (x_{1} - x_{2} + x_{3} - x_{4} )^{2} + (y_{1} - y_{2} + y_{3} - y_{4} )^{2}$$

(48)

or, the classical quadratic error is independent of x₀, y₀ and ϕ. Clearly, this is not a viable path. Another possibility is to recognize that a perfectly rectangular element, whose sides are parallel to the $\overline{x}$ and $\overline{y}$ axes, satisfies the four relations $\overline{x}_{1} = \overline{x}_{4}$, $\overline{x}_{2} = \overline{x}_{3}$, $\overline{y}_{1} = \overline{y}_{2}$ and $\overline{y}_{3} = \overline{y}_{4}$, leading to a constant Jacobian matrix. Thus, a new quadratic error g is now defined as

$$g = (\overline{x}_{1} - \overline{x}_{4} )^{2} + (\overline{x}_{2} - \overline{x}_{3} )^{2} + (\overline{y}_{1} - \overline{y}_{2} )^{2} + (\overline{y}_{3} - \overline{y}_{4} )^{2}$$

(49)

Substitution of Eq. (38) into Eq. (49) yields

$$\eqalign{ g & = (x_{14}^2 + x_{23}^2 + y_{12}^2 + y_{34}^2)\frac{{1 + \cos 2\beta }}{2} \cr & \quad + (x_{21}^2 + x_{43}^2 + y_{14}^2 + y_{23}^2)\frac{{1 - \cos 2\beta }}{2} \cr & \quad + 2({x_{14}}{y_{14}} + {x_{23}}{y_{23}} + {x_{21}}{y_{12}} + {x_{43}}{y_{34}})\frac{{\sin 2\beta }}{2} \cr}$$

(50)

where x_ij = x_i − x_j and y_ij = y_i − y_j. Notice that Eq. (50) is independent of x₀ and y₀. Enforcing ∂g/∂β = 0 leads to

$$\tan 2\beta = \frac{{x_{13} y_{24} + x_{24} y_{13} }}{{x_{13} x_{24} - y_{13} y_{24} }}$$

(51)

A good property of Eq. (51) is that the denominator x₁₃x₂₄ − y₁₃y₂₄ is never zero, provided the quadrilateral element is not degenerated. Numerically, this is an advantage since the function at an can be used without further complications. As far as the parameters x₀ and y₀ are concerned, they can be arbitrarily selected as the centroid of the element, i.e., x₀ = (x₁ + x₂ + x₃ + x₄)/4 and y₀ = (y₁ + y₂ + y₃ + y₄)/4.

A geometrical interpretation for Eq. (51) can be given recognizing that the angle β₁₃, formed between diagonal 13 and the x axis, satisfies tanβ₁₃ = y₁₃/x₁₃, and that the angle β₂₄, formed between diagonal 24 and the x axis, satisfies tanβ₂₄ = y₂₄/x₂₄. Hence, Eq. (51) can be rewritten in the form

$$\tan 2\beta = \frac{{\frac{{y_{13} }}{{x_{13} }} + \frac{{y_{24} }}{{x_{24} }}}}{{1 - \frac{{y_{13} }}{{x_{13} }}\frac{{y_{24} }}{{x_{24} }}}} = \frac{{\tan \beta_{13} + \tan \beta_{24} }}{{1 - \tan \beta_{13} \tan \beta_{24} }} = \tan (\beta_{13} + \beta_{24} )$$

(52)

leading to the conclusion that β = (β₁₃ + β₂₄)/2, or that angle β is the angle formed between the bisector of the angle formed by the diagonals 13 and 24 and the x axis. This is precisely the orientation of the element coordinate system implemented in the Nastran CQUAD4 element.

As an example, the element with nodal coordinates x₁ = 0, y₁ = 0, x₂ = 1, y₂ = 0, x₃ = 1, y₃ = 3, x₄ =  − 1, y₄ = 1 is investigated. Straight use of Eq. (51) or (52) reveals that β = 22.5°. The quadratic error function g in Eq. (50) is computed for different values of β and plotted in Fig. 5, where a substantial variation of the error can be observed.

Fig. 5

Error variation in a four-node quadrilateral element

5 Numerical assessment

Table 1 presents the six possibilities of element types that were implemented. In practice, it was observed that numerical results have shown to be mostly insensitive to the choice of element type. Elements C–F perform a little better compared to elements A–B in terms of convergence, i.e., they require fewer elements in the mesh in order to achieve accurate results. However, their quadratic and cubic global interpolation functions demand higher computational processing in order to numerically evaluate integrals along thickness. The conclusion is that the relatively better accuracy of elements C–F is offset by the numerical performance of elements A–B.

Regarding the local interpolation functions in Table 1, the linear/quadratic pair and the quadratic/cubic pair perform equally, at least in the simulations carried out in this work. Again, since cubic polynomials entail the necessity to use more integration points in the quadrature formulae, the pair linear/quadratic local functions is the preferred choice.

In light of these preliminary conclusions, element A (the simplest), both tria and quad topologies, are employed in the simulations. The mechanical properties used are those of a typical unidirectional ply of a UD [9]: E₁ = 150.0 GPa, E₂ = E₃ = 10.0 GPa, G₁₂ = G₁₃ = 5.0 GPa, G₂₃ = 3.378 GPa, ν₁₂ = ν₁₃ = 0.30, ν₂₃ = 0.48; α₁ = 1.39 × 10⁻⁷ K⁻¹, α₂ = α₃ = 9.0 × 10⁻⁶ K⁻¹. In the 0° layer, the principal directions (1, 2, 3) are aligned with the global axes (x, y, z).

5.1 A single laminate plate

In the first case, a square plate with edge lengths a = b = 1 m is investigated in detail. The laminate is chosen in the cross-ply [0°/90°], where the two layers have equal thicknesses. Thick and thin laminates are considered. In the thick laminate case, the total thickness is h = 0.25 m, and in thin laminate case, the total thickness is h = 0.05 m. A measure of the relative thinness of the laminates is given by the ratio s = a/h. Hence, the thick laminate has s = 4 and the thin laminate has s = 20. The boundary conditions assumed are illustrated in Fig. 6 and they are: u(0, y, 0) = w(0, y, 0) = 0, u(a, y, 0) = w(a, y, 0) = 0, v(x, 0, 0) = w(x, 0, 0) = 0 and v(x, b, 0) = w(x, b, 0) = 0. The plate is loaded on the top surface (z =  + h/2) with p₀ = σ_z = 100 MPa, while the bottom surface (z =  − h/2) is free.

Fig. 6

Boundary conditions of single laminate plate

Three models of the plate were considered: a 33 × 33 mesh of the new quad elements, a 2 × (33 × 33) mesh of the new tria elements and a 100 × 100 × 20 mesh of traditional eight-node brick elements. Both thick (s = 4) and thin (s = 20) laminate plates were modeled using the same meshes that can be seen in Fig. 7. The meshes shown in Fig. 7 were selected following a convergence analysis. A series of coarser meshes were tried before reaching the 33 × 33 mesh utilized. In the case of the tridimensional model, the problem lies in the imposition of the boundary conditions on the bottom and top surfaces. If too few 3D elements are used in the thickness direction, the enforced conditions σ_z(− h/2) = 0, σ_z(+ h/2) = p₀, τ_xz(± h/2) = 0 and τ_yz(± h/2) = 0 are not accurately satisfied. Hence, it was determined that 20 elements must be used in the z direction. The number of 3D elements in the x and y directions (100 × 100) follows from consideration of the aspect ratio.

Fig. 7

Quad, tria and brick element meshes

The results are presented for displacement, strain and stress distributions along thickness. Distributions of w, ε_x, ε_y, ε_z, σ_x, σ_y, σ_z are shown at x = a/2, y = b/2, whereas distributions of u, v, γ_yz, γ_xz, γ_xy, τ_yz, τ_xz, τ_xy are shown at x = a/4, y = b/4. All results are normalized according to Eq. (53).

$$\begin{gathered} u^{*} = \frac{{100E_{2} }}{{p_{0} hs^{3} }}w,\;v^{*} = \frac{{100E_{2} }}{{p_{0} hs^{3} }}w,\;w^{*} = \frac{{100E_{2} }}{{p_{0} hs^{4} }}w \hfill \\ \varepsilon_{x}^{*} = \frac{1}{{E_{1} }}\varepsilon_{x} ,\;\varepsilon_{y}^{*} = \frac{1}{{E_{1} }}\varepsilon_{y} ,\;\varepsilon_{z}^{*} = \frac{{E_{2} }}{{p_{0} s^{2} }}\varepsilon_{z} \hfill \\ \gamma_{yz}^{*} = \frac{{E_{3} }}{{p_{0} s}}\gamma_{yz} ,\;\gamma_{xz}^{*} = \frac{{E_{3} }}{{p_{0} s}}\gamma_{xz} ,\;\gamma_{xy}^{*} = \frac{{E_{3} }}{{p_{0} s^{2} }}\gamma_{xy} \hfill \\ \sigma_{x}^{*} = \frac{1}{{p_{0} s^{2} }}\sigma_{x} ,\;\sigma_{y}^{*} = \frac{1}{{p_{0} s^{2} }}\sigma_{y} ,\;\sigma_{z}^{*} = \frac{1}{{p_{0} }}\sigma_{z} \hfill \\ \tau_{yz}^{*} = \frac{1}{{p_{0} s}}\tau_{yz} ,\;\tau_{xz}^{*} = \frac{1}{{p_{0} s}}\tau_{xz} ,\;\tau_{xy}^{*} = \frac{1}{{p_{0} s^{2} }}\tau_{xy} \hfill \\ \end{gathered}$$

(53)

The results for the thick laminate (s = 4) are grouped in Figs. 8, 9 and 10, whereas the results for the thin laminate (s = 20) are grouped in Figs. 11, 12 and 13. Figures 8 and 11 show the normalized displacements u^*, v^* and w^*. Figures 9 and 12 show the normalized strains ε_x^*, ε_y^*, ε_z^*, γ_yz^*, γ_xz^*, γ_xy^*. Figures 10 and 13 show the normalized stresses σ_x^*, σ_y^*, σ_z^*, τ_yz^*, τ_xz^*, τ_xy^*. In Figs. 8, 9, 10, 11, 12 and 13, the z coordinate is normalized according to z^* = z/h such that − 0.5 ≤ z^* ≤ 0.5.

Fig. 8

Normalized displacement distributions along thickness for s = 4. w^* is plotted at x = a/2 and y = b/2. u^* and v^* are plotted at x = a/4 and y = b/4

Fig. 9

Normalized strain distributions along thickness for s = 4. ε_x^*, ε_y^*, ε_z^* are plotted at x = a/2 and y = b/2. γ_yz^*, γ_xz^*, γ_xy^* are plotted at x = a/4 and y = b/4

Fig. 10

Normalized stress distributions along thickness for s = 4. σ_x^*, σ_y^*, σ_z^* are plotted at x = a/2 and y = b/2. τ_yz^*, τ_xz^*, τ_xy^* are plotted at x = a/4 and y = b/4

Fig. 11

Normalized displacement distributions along thickness for s = 20. w^* is plotted at x = a/2 and y = b/2. u^* and v^* are plotted at x = a/4 and y = b/4

Fig. 12

Normalized strain distributions along thickness for s = 20. ε_x^*, ε_y^*, ε_z^* are plotted at x = a/2 and y = b/2. γ_yz^*, γ_xz^*, γ_xy^* are plotted at x = a/4 and y = b/4

Fig. 13

Normalized stress distributions along thickness for s = 20. σ_x^*, σ_y^*, σ_z^* are plotted at x = a/2 and y = b/2. τ_yz^*, τ_xz^*, τ_xy^* are plotted at x = a/4 and y = b/4

Judging by Figs. 8 and 11, it is fair to say that the displacements are in good agreement with the fine traditional brick element FE mesh results. Notice that the horizontal axes in the w* distributions are shifted, i.e., 3.6 ≤ w* ≤ 4.4 for s = 4 and 0.42 ≤ w* ≤ 0.46 for s = 20. In Fig. 11, u* and v* are seen to be almost perfectly linear, what is expected as the laminate gets thinner. The result is more conservative since the traditional FE model in Nastran delivers a smaller w*. Similar trends regarding the distribution of the w displacement were observed in a previous paper [10], where a possible explanation for such behavior is related to the choice of the local interpolation functions.

The ε_x^*, ε_y^*, ε_z^*, γ_xz^* strain results for s = 4 and s = 20 are acceptable. The trends, magnitudes and discontinuity jumps are all well captured by the new element, both its quad and tria versions. One very positive aspect of the new element is that the transverse normal strain ε_z^* computed is quite accurate. However, when it comes to γ_xz^*, γ_xy^* strains the results do not compare favorably against the traditional FE solution; particularly, γ_xz^* does not behave well, even presenting a discontinuity for the tria mesh case at z^* = 0. A procedure to enhance accuracy of γ_xz^*, γ_xy^*, τ_yz^*, τ_xz^* can be achieved through post-processing of the results. The force equilibrium equations along x and y can be integrated through the thickness to yield Eq. (54). Once these improved transverse shear stresses are available, one can solve Eq. (4) to determine γ_yz, γ_xz as in τ_yz = C₄₄^(k)γ_yz + C₄₅^(k)γ_xz and τ_xz = C₄₅^(k)γ_yz + C₅₅^(k)γ_xz.

$$\tau_{xz} = - \int\limits_{ - h/2}^{z} {(\sigma_{x,x} + \tau_{xy,y} )dz} ,\;\tau_{yz} = - \int\limits_{ - h/2}^{z} {(\sigma_{y,y} + \tau_{xy,x} )dz}$$

(54)

The tendencies of the strain results are carried on to the stresses. It is noticed that σ_x^*, σ_y^*, σ_z^*, τ_xy^*stress results are good, but τ_yz^*, τ_xz^* are not; particularly, τ_xz^* does not behave well. The good σ_z^* behavior is remarkable: It varies from 0 at the bottom surface up to 1 at the top surface, as expected. It is worth mentioning that the τ_xz^* obtained by Nastran is not zero at bottom (z^* =  − 0.5) and top (z* = 0.5) surfaces. This goes to show that even the fine mesh used by the commercial software is still incapable of predicting the correct answer τ_xz^* = 0. This is a quite alarming conclusion since, usually, solid FE models applied to laminate structures use only one or two solid elements per layer.

The traditional FE brick element mesh has 642,663 degrees of freedom, whereas the meshes used in partnership with the quad and tria elements have only 13,872 degrees of freedom, or almost 47 times less. This reflects in huge savings in the computational processing. Actually, it was observed that the factor of 47 times reflects directly in the processing time taken by the FE numerical solver, even recalling that the quadrature rule of numerical integration was used over each layer. The FE brick element mesh resulted in numerical models that required as much as 25 × more processing time than the quad and tria meshes.

5.2 Stiffened panel

A more representative and practical composite structure is illustrated in Fig. 14 along with the dimensions chosen. A circular cylindrical panel with radius 5 m has two equally spaced reinforcements. One edge of the panel is clamped (no displacements and no rotations). The opposite edge is subject to a shear loading N_s = 10 kN/m that represents torsion.

Fig. 14

Circular cylindrical stiffened composite panel

The panel has multiple laminates whose material properties are the same as those used in the previous square plate example. Representing a more realistic application, each layer thickness is 0.15 mm. The layer orientations are defined according to y axis of Fig. 14, i.e., 0° ply angle means that the fibers are parallel to the y axis. The panel skin laminate is the [0°/90°]_6S with a total of 24 layers. The reinforcement base laminate is the [90°/0°]₃ laminate with 6 layers in total. The reinforcement web is the [0°/90°]_5S with a total of 20 layers. Notice that in the region where the stiffener base attaches to the panel skin the resulting laminate has 30 layers: [[0°/90°]_6S, [90°/0°]₃].

Two models for this panel are simulated. The first model is composed of traditional four-node plate laminate elements based on Mindlin formulation. Sixty elements are used in the hoop direction (x axis), 40 elements are used in the longitudinal direction (y axis) and 4 elements are used along the stiffener height (z axis). Four elements are used to model the base of each stiffener. In the second model, the same mesh is used, but the new element formulated is employed.

Figures 15 and 16 show the results is terms of u displacement along x axis. The negative displacements (blue hue) observed in the Mindlin mesh on the region near the free tip of the stiffeners were not observed in the new element model. However, the displacement distribution over most of the reinforced panel agrees very well. The greatest advantage of the model built with the newly proposed element is its ability to assess through-the-thickness effects. Evidently, the Mindlin elements cannot predict neither ε_z nor σ_z and may provide inaccurate results in terms of γ_yz, γ_xz, τ_yz, τ_xz. Figure 17 shows a magnified view of the skin/stiffener attachment region. The laminate representing the connection skin/stiffener base with a total of 30 layers is clearly seen. Additionally, the transverse normal stress σ_z distribution is captured. This information is invaluable if studies toward stiffeners detachment are to be done.

Fig. 15

Mindlin element mesh used to model the circular cylindrical stiffened composite panel. Result for the u displacement distribution

Fig. 16

Mesh of new elements used to model the circular cylindrical stiffened composite panel. Result for the u displacement distribution

Fig. 17

Detail of the attachment region where stiffener and skin connect

6 Conclusions

Two versions of a new element are proposed (quad and tria). Their numerical implementations were described and performance assessment conducted. The element results compare well against a commercial FE package that used traditional eight-node brick elements to build a 3D model. Displacement, strain and stress magnitude, tendencies and discontinuities were reasonably well predicted. The quad element is superior to the tria, what is expected. It is good practice to use as many quads as possible and recur to a few trias only to avoid severe mesh distortions.

One important feature of the new element is that it can take advantage of previously existing meshes originally created to work with traditional quads and trias formulated under Mindlin hypothesis and build on them through-the-thickness capabilities. This is because the new element requires only that the parent tria or quad nodal positions are available, and from there, the element matrices and vectors can be computed without the need to create specially purposed 3D meshes.

Another great feature of the element is its ability to model multiple laminate structures such as the reinforced panel considered. In the simulations performed, the Mindlin element mesh presented in Fig. 15 was taken as the starting point for the model created using the new elements and depicted in Fig. 16. Figure 17 shows that the through-the-thickness capabilities of the new element allow one to compute, for instance, σ_z, τ_yz and τ_xz stresses and 3D displacements, which can subsequently be used in conjunction with composite failure criteria [11] in the investigation of challenging phenomena involving crack propagation [12].