Static analysis of corrugated panels using homogenization models and a cell-based smoothed mindlin plate element (CS-MIN3)

Abstract

Homogenization is a promising approach to capture the behavior of complex structures like corrugated panels. It enables us to replace high-cost shell models with stiffness-equivalent orthotropic plate alternatives. Many homogenization models for corrugated panels of different shapes have been proposed. However, there is a lack of investigations for verifying their accuracy and reliability. In addition, in the recent trend of development of smoothed finite element methods, the cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) has been proposed and successfully applied to many analyses of plate and shell structures. Thus, this paper further extends the CS-MIN3 by integrating itself with homogenization models to give homogenization methods. In these methods, the equivalent extensional, bending, and transverse shear stiffness components which constitute the equivalent orthotropic plate models are represented in explicit analytical expressions. Using the results of ANSYS and ABAQUS shell simulations as references, some numerical examples are conducted to verify the accuracy and reliability of the homogenization methods for static analyses of trapezoidally and sinusoidally corrugated panels.

Keywords

homogenization / corrugated panel / asymptotic analysis / smoothed finite element method (S-FEM) / cell-based smoothed three-node Mindlin plate element (CS-MIN3)

Introduction

Due to the high stiffness-to-weight ratio, corrugated panels and corrugated core panels have been used widely in various applications in civil, mechanical, marine, and aerospace engineering [¹]. Besides, the anisotropic property of the corrugated panels due to different flexure characteristics in two perpendicular directions makes them become promising structures in morphing designing. In practical applications, the overall behavior of corrugated panels is usually taken into consideration [²]. However, the subsequent precise analyses using finite element method (FEM) involve high computational cost. Therefore, they take much time and effort to accomplish parametric studies and optimize structural designs. To overcome such the obstacles, many homogenization methods in which a flat orthotropic plate with equivalent stiffness is derived to replace the original panel have been proposed (see Fig. 1(a)).

There is a rich literature about homogenization techniques for an extensive range of corrugated structures from single-layer sheets of isotropic [²–¹²] or composite material [²,¹¹] to multi-layer structures like corrugated core sandwich plates [¹³–¹⁹] and corrugated laminates [²⁰]. Some of these homogenization models are integrated to parameter analyses and optimization processes [¹²,²¹,²²] along with behavior studies of corrugated structures. Most of the existing homogenization methods belong to engineering approaches in which different boundary conditions and assumptions of strain/stress distribution are used to derive the formulas of equivalent stiffness terms.

Shimansky and Lele [⁴] derived an analytical model for initial transverse stiffness of a sinusoidally corrugated panel and then highlighted the impact of plate thickness and degree of corrugation to this stiffness component. Also working with a sinusoidal profile, Briassoulis [³] reviewed existing classical equivalent models and modified some expressions of extensional and flexural rigidities. Samanta and Mukhopadhyay [⁵] used a similar approach to determine the equivalent extensional rigidities for trapezoidally corrugated sheets. Combining them with the equivalent flexural rigidities derived by McFarland [²³], the authors then qualified the resulting model in buckling, linear static, geometric nonlinear, and free vibration analyses. Liew et al. [⁶–⁹] used a meshless Galerkin method to investigate many mechanical behaviors of stiffened corrugated panels of sinusoidal and trapezoidal profiles. In these studies, besides modifying some formulas of equivalent flexural rigidities, the authors also employed the equivalent transverse shear derivation [²⁴] in their FSDT plate model. Xia et al. [²] formulated generalized expressions to estimate equivalent stiffness terms for thin corrugated laminates of any shape. This generalized model was then extended to cover the transverse shear stiffness by Park et al. [¹¹].

Another approach for homogenization of thin corrugated panels is the asymptotic method proposed by Ye et al. [¹⁰, ²⁵]. The method was based on governing equations of a shell theory in which the variable fields represented in asymptotic expansions were substituted back into the equations. The consequently derived systems of governing differential equations were used to find the relationship between the equivalent plate and the corrugated structure. The method could handle corrugations of any shape as well as provided a complete set of formulas to recover the local fields in the corrugated panels. Although there are many proposal treatments for this kind of structures, a deeper study of their effectiveness and accuracy is necessary.

After the equivalent models are derived, we can replace analyses of corrugated panels by those of equivalent orthotropic plates. The last decades have seen an explosion of studies of various kinds of plate and shell structures. With the emergence of new materials like composite, functionally graded, and piezoelectric ones, many advanced plate and shell structures have been developed and integrated in applications [²⁶,²⁷]. Theories of different levels of complexity have been proposed for modeling and analyzing the behavior of these structures [²⁸,²⁹]. Besides, to obtain more accurate and efficient solutions especially when dealing with the heterogeneity of material as well as the discontinuity of damaged structures, multiscale and phase-field approaches have been taken increasing attention [³⁰–³³]. Among various numerical methods for plate and shell studies [³⁴–⁴⁰], FEM is still the most commonly used due to its simplicity and extensibility for a variety of problems. However, to become applicable to complex analyses of the advanced structures, FEM needs improvements in elemental techniques, accuracy, stability, and computational efficiency, etc. As a result, researching of new FEM with higher performance still takes a particular concern from scientists around the world during last decades.

On another front of the development of numerical methods, Liu and Nguyen-Thoi [⁴¹] have integrated the strain smoothing technique [⁴²] into the FEMs to create a series of smoothed finite element methods (S-FEMs), such as the cell/element-based smoothed FEM (CS-FEM, nCS-FEM) [⁴³,⁴⁴], the node-based smoothed FEM (NS-FEM) [⁴⁵,⁴⁶], the edge-based smoothed FEM (ES-FEM) [⁴⁷], and the face-based smoothed FEM (FS-FEM) [⁴⁸]. These S-FEMs with different properties have been applied to improve the solutions for a wide class of benchmark and practical mechanics problems, especially two-dimensional (2-D) and three-dimensional (3-D) linear elastic mechanics problems. Related to analyses of plate and shell structures, the smoothing techniques have improved the performance of traditional plate elements such as DSG3 [⁴⁹], MITC4 [⁵⁰] and MIN3 [⁵¹] to produce smoothed counterparts such as ES-DSG3 [⁵²,⁵³], NS-DSG3 [⁵⁴], CS-DSG3 [⁵⁵], ES-MIN3 [⁵⁶], CS-MIN3 [⁵⁷], and MISCk [⁵⁸].

Among these smoothed plate elements, CS-MIN3, which is a combination of the cell-based smoothed technique [⁴³] and the Mindlin plate element, MIN3 [⁵¹], possesses many significant computational advantages. In this method, each triangular element is divided into three sub-triangles on which the MIN3 is employed locally to compute the strains. Afterward, the strain smoothing technique on the entire element is used to smooth the strains on these sub-triangles. The CS-MIN3 is free of shear locking and achieves high accuracy compared to the exact solutions and some others existing elements in the literature [⁵⁷]. Although not as superior as ES-FEM in some cases, the CS-FEM is much simpler in implementation. The CS-FEM is conducted within a target element and does not need any extra information from adjacent elements. With these mentioned advantages, the CS-MIN3 has been applied in different analyses of plate and shell structures. For examples, isotropic plates [⁵⁷,⁵⁹], laminated composite plates [⁶⁰,⁶¹], functional graded plates [⁶²], and recently cracked FGM plates [⁶³]. This paper hence further extends the CS-MIN3 by integrating itself with homogenization models to give homogenization methods for static analyses of corrugated panels of some common shapes.

The layout of the article is as follows. The next section provides a summary of some most cited homogenization models with equivalent stiffness terms given explicitly. Section 3 represents the Galerkin weak form for static analysis and a brief formulation of the CS-MIN3. Some numerical examples to evaluate the reliability and accuracy of these homogenization methods are described in Section 4. Last but not least, the paper ends up with some concluding remarks.

Homogenization methods for corrugated panels

Introduction to homogenization of corrugated structures

In practical applications, depending on the usage and functions, corrugated panel structures have numerous kinds of shape. In this paper, we only consider shell panels of periodic and symmetric corrugations in one direction. Notably, the focus of this work is on trapezoidally and sinusoidally corrugated panels (Figs. 1(b) and 1(c)). The geometry of such structures can be defined by several parameters such as the half-period c, the half-length (of one corrugation unit) l, the half-amplitude f, and the trough angle \alpha. Somewhere in this paper, the period of one corrugation unit ϵ is used instead of the half one c.

The mid-surface of a corrugated panel can be represented analytically using two coordinate systems including a global Cartesian coordinate system xyz and a local coordinate system syn (see Fig. 2(a)). In the xz-plane, the curvilinear coordinate s represents the arch length at a position on the profile curve. The n-axis is the line normal to profile curve in xz-plane. The mid-surface of the corrugated shell panel is obtained by extruding the profile curve in the y-direction. The location of a point belonging to the mid-surface in the global coordinate system is defined as

(1)r(s,y)=x(s)i+yj+z(s)k,

where i, j, and k are unit vectors of x-, y- and z- axes respectively. Then we have dx/ds=\cos \theta and dz/ds=\sin \theta where \theta is the tangential angle at that point. For instance, the trapezoidal and sinusoidal corrugation profiles in Fig. 1(b) and Fig. 1(c) are in turn represented by following equations

(2)s(x)=\{\begin{array}{ll}(x-x_{\mathrm{0}})/\cos \alpha ,& x\in [x_{\mathrm{0}},x_{\mathrm{1}}]\\ (x_{\mathrm{1}}-x_{\mathrm{0}})/\cos \alpha +x-x_{\mathrm{1}}& x\in [x_{\mathrm{1}},x_{\mathrm{2}}]\\ (x_{\mathrm{1}}-x_{\mathrm{0}})/\cos \alpha +x_{\mathrm{2}}-x_{\mathrm{1}}+(x-x_{\mathrm{2}})/\cos \alpha & x\in [x_{\mathrm{2}},x_{\mathrm{3}}]\end{array};z(x)=\{\begin{array}{ll}(x-x_{\mathrm{0}})\mathrm{T}an\alpha ,& x\in [x_{\mathrm{0}},x_{\mathrm{1}}]\\ f& x\in [x_{\mathrm{1}},x_{\mathrm{2}}]\\ -x\mathrm{T}an\alpha & x\in [x_{\mathrm{2}},x_{\mathrm{3}}]\end{array},

where x_{\mathrm{0}}=-c, x_{\mathrm{1}}=-c+f/\mathrm{Tan}\alpha, x_{\mathrm{2}}=-f/\mathrm{Tan}\alpha, x_{\mathrm{3}}=\mathrm{0}, and

(3)s(x)={\displaystyle {\int }_{-c}^x\sqrt{\mathrm{1}+{(dz(\mathrm{T})\mathrm{/d}\mathrm{T})}^{\mathrm{2}}}d\mathrm{T}=}{\displaystyle {\int }_{-c}^x\sqrt{\mathrm{1}+{(f\pi /c)}^{\mathrm{2}}{\cos }^{\mathrm{2}}(\pi \mathrm{T}/c)}d\mathrm{T}};\hspace{1em}z(x)=f\sin (\pi x/c).

Although a shell analysis using FEM can give a precise analysis of corrugated panels, it requires significant computing cost. Homogenization could be a more practical alternative in which orthotropic flat plates with equivalent rigidities are derived. In many engineering approaches, the rigidities are determined through analyzing a primary unit cell called a Representative Volume Element (RVE) [²] (see Fig. 1(a) and Fig. 2(a)). In the local curvilinear coordinate system, the strain energy of the RVE is

(4)U_c={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle {\int }_{\Omega }{\left[\begin{array}{l}N\\ M\\ Q\end{array}\right]}^{\mathrm{T}}{[D]}^{-\mathrm{1}}\left[\begin{array}{l}N\\ M\\ Q\end{array}\right]dsdy},

where \Omega is the mid-surface domain, N={\left[\begin{array}{lll}{N}_s& N_y& N_{sy}\end{array}\right]}^{\mathrm{T}}, M={\left[\begin{array}{lll}{M}_s& M_y& M_{sy}\end{array}\right]}^{\mathrm{T}}, and Q={\left[\begin{array}{ll}{Q}_{sz}& Q_{yz}\end{array}\right]}^{\mathrm{T}} are the stress consultant vectors given by

(5)\left[\begin{array}{l}N\\ M\\ Q\end{array}\right]=\underset{D}{\underbrace{\left[\begin{array}{lll}A& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& D_b& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& D_s\end{array}\right]}}\underset{\epsilon }{\underbrace{\left[\begin{array}{l}{\epsilon }_m\\ \kappa \\ \gamma \end{array}\right]}}=D\mathbf{\epsilon }.

In Eq. (5), {[{\epsilon }_m]}^{\mathrm{T}}=\left[\begin{array}{lll}{\epsilon }_s& {\epsilon }_y& {\gamma }_{sy}\end{array}\right], {[\kappa ]}^T=\left[\begin{array}{lll}{\kappa }_s& {\kappa }_y& {\kappa }_{sy}\end{array}\right], and {[\gamma ]}^T=\left[\begin{array}{ll}{\gamma }_{sz}& {\gamma }_{yz}\end{array}\right] are respectively the vectors of membrane strains, curvatures, and transverse shear strains of the shell panel and

(6)A=\left[\begin{array}{lll}{A}_{\mathrm{11}}& A_{\mathrm{12}}& \mathrm{0}\\ A_{\mathrm{12}}& A_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& A_{\mathrm{66}}\end{array}\right];D_b=\left[\begin{array}{lll}{D}_{\mathrm{11}}& D_{\mathrm{12}}& \mathrm{0}\\ D_{\mathrm{12}}& D_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& D_{\mathrm{66}}\end{array}\right];D_s=\left[\begin{array}{ll}{D}_{\mathrm{55}}& \mathrm{0}\\ \mathrm{0}& D_{\mathrm{44}}\end{array}\right],

are the rigidity matrices that consist of extensional, bending, and shearing stiffness components, respectively.

The strain energy of the equivalent plate element is calculated as

(7)U_e={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle {\int }_{\bar{\Omega }}{\left[\begin{array}{l}\bar{N}\\ \bar{M}\\ \bar{Q}\end{array}\right]}^{\mathrm{T}}{\left[\bar{D}\right]}^{-\mathrm{1}}\left[\begin{array}{l}\bar{N}\\ \bar{M}\\ \bar{Q}\end{array}\right]dxdy},

where \mathrm{\Omega }is the mid-plane domain of the equivalent plate element and

(8)\left[\begin{array}{l}\bar{N}\\ \bar{M}\\ \bar{Q}\end{array}\right]=\underset{\_}{\underbrace{\left[\begin{array}{lll}\bar{A}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& {\bar{D}}_b& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\bar{D}}_s\end{array}\right]}}\underset{\_}{\underbrace{\left[\begin{array}{l}{\bar{\epsilon }}_m\\ \bar{\kappa }\\ \bar{\gamma }\end{array}\right]}}=\bar{D}\bar{\epsilon },

are the stress consultant vectors in the plate. All components of the matrices and vectors in Eq. (8) are denoted by adding a bar above their counterparts in Eq. (6). To obtain a homogenization model, we need to identify the extensional, bending, and transverse shear stiffness terms in \bar{A}, {\bar{D}}_b, and {\bar{D}}_s, respectively.

Equivalent stiffness terms of some typical homogenization models

As mentioned in the introduction section, there are two main approaches for homogenization of corrugated panels. In engineering approach, the RVE is constrained in some specific boundary and loading conditions. Equivalent force and energy methods are then applied to derive the equivalent stiffness terms as functions of material and geometric parameters of the original structure (more details in [²]). This approach is highlighted in many studies about trapezoidally and sinusoidally corrugated panels by Samanta et al. [⁵], Peng et al. [⁶–⁹] and Xia et al. [²]. Using thin shell governing equations, the asymptotic approach proposed by Ye et al. [¹⁰] is also applicable for generalized corrugation profiles. Especially, their study finds out that the orthotropic model with no extension-bending coupling is only suitable for symmetric corrugation profiles. For convenience, from this point onwards, we label the four homogenization models as SamantaHM, PengHM, XiaHM, and YeHM, respectively.

Consider a corrugated panel containing n_c corrugation units with half-period c and uniform thickness h. We assume that the panel is made from an isotropic material with Young’s modulus E and Poisson’s ratio v. The equivalent models for the panel when its profile is a trapezoidal, sinusoidal, or general one are respectively summarized in Table 1.

In Table 1, the constant {\alpha }_S and {\alpha }_{\mathrm{T}} are given as follows [⁸]

(9)\begin{array}{ll}{\alpha }_S=\left[\mathrm{1}+{\left({\displaystyle \frac{f}{h}}\right)}^{\mathrm{2}}\mathrm{6}(\mathrm{1}-{\nu }^{\mathrm{2}})\left({\displaystyle \frac{l^{\mathrm{2}}}{c^{\mathrm{2}}}}-{\displaystyle \frac{l}{\mathrm{2}\pi c}}\sin {\displaystyle \frac{\mathrm{2}\pi l}{c}}\right)\right],& {\alpha }_T={\displaystyle {\int }_{\mathrm{0}}^c{z}^{\mathrm{2}}dx}=f^{\mathrm{2}}c-\mathrm{4}{f}^{\mathrm{3}}/\mathrm{3}\mathrm{Tan}\theta \end{array},

where lis the half-length of one corrugation unit. Applying the formulas of equivalent transverse shear terms which were proposed by Semenyuk and Neskhodovskaya [²⁴], Peng et al. [⁷–⁹] derived {\bar{D}}_{\mathrm{44}} and {\bar{D}}_{\mathrm{55}} as proportions of {\bar{D}}_{\mathrm{66}}. The constants k_{\mathrm{1}} and k_{\mathrm{2}} depend on the number of corrugation units n_c, the half-amplitude f, the thickness h, and the total length L as follows

(10)\{\begin{array}{l}{k}_{\mathrm{1}}={\displaystyle \frac{\mathrm{2}}{\pi }}\sqrt{\mathrm{1}+{\gamma }^{\mathrm{2}}{n}_c^{\mathrm{2}}}\bar{E}(k,\pi /\mathrm{2})\\ k_{\mathrm{2}}={\displaystyle \frac{\mathrm{2}\sqrt{\mathrm{1}+{\gamma }^{\mathrm{2}}{n}_c^{\mathrm{2}}}}{\mathrm{3}\pi {\epsilon }_{\mathrm{0}}^{\mathrm{2}}{n}_c^{\mathrm{2}}}}\left[\left(\mathrm{1}+{\displaystyle \frac{\mathrm{3}{\epsilon }_{\mathrm{0}}^{\mathrm{2}}{n}_c^{\mathrm{2}}}{\mathrm{1}+{\gamma }^{\mathrm{2}}{n}_c^{\mathrm{2}}}}\right)\bar{F}\left(k,{\displaystyle \frac{\pi }{\mathrm{2}}}\right)+({\gamma }^{\mathrm{2}}{n}_c^{\mathrm{2}}-\mathrm{1})\bar{E}\left(k,{\displaystyle \frac{\pi }{\mathrm{2}}}\right)\right]\end{array},

where

(11)\begin{array}{lll}\{\begin{array}{l}\gamma ={\displaystyle \frac{\mathrm{2}\pi f}{L}}\\ {\epsilon }_{\mathrm{0}}^{\mathrm{2}}=\mathrm{4}{\pi }^{\mathrm{2}}{h}^{\mathrm{2}}/\mathrm{12}/L^{\mathrm{2}}\\ k=\gamma n_c/\sqrt{\mathrm{1}+{\gamma }^{\mathrm{2}}{n}_c^{\mathrm{2}}}\end{array}& \mathrm{and}& \{\begin{array}{l}\bar{F}(k,\varphi )={\displaystyle {\int }_{\mathrm{0}}^{\varphi }{\displaystyle \frac{d\theta }{\sqrt{\mathrm{1}-k^{\mathrm{2}}{\sin }^{\mathrm{2}}\theta }}}}\\ \bar{E}(k,\varphi )={\displaystyle {\int }_{\mathrm{0}}^{\varphi }\sqrt{\mathrm{1}-k^{\mathrm{2}}{\sin }^{\mathrm{2}}\theta }d\theta }\end{array}\end{array}.

As in [⁹], the shear correction factor \kappa =\mathrm{5}/\mathrm{6} is imposed to the equivalent transverse shear stiffness terms.

Other geometrically dependent constants are (see details in [²,¹⁰])

(12)\begin{array}{l}{I}_{\mathrm{1}}={\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}l}{\left({\displaystyle \frac{dx}{ds}}\right)}^{\mathrm{2}}ds}\\ I_{\mathrm{2}}={\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}l}{z}^{\mathrm{2}}ds}\end{array};\hspace{1em}\begin{array}{ll}{C}_{\mathrm{1}}=\langle \sqrt{a}\phi \rangle , C_{\mathrm{2}}=\langle \sqrt{a}\rangle , C_{\mathrm{3}}={\displaystyle \frac{C_{\mathrm{1}}}{C_{\mathrm{2}}}},& C_{\mathrm{6}}=-\mathrm{12}{C}_{\mathrm{4}}{\left({\displaystyle \frac{\epsilon }{h}}\right)}^{\mathrm{2}}-C_{\mathrm{5}},C_{\mathrm{7}}=\mathrm{1}/\langle {\displaystyle \frac{\sqrt{a}}{{\psi }_{\mathrm{1}}}}\rangle \\ C_{\mathrm{4}}=\langle \varphi A\rangle ,C_{\mathrm{5}}=\langle \mathrm{1}/\sqrt{a}\rangle ,& C_{\mathrm{8}}=\langle \sqrt{a}{\phi }^{\mathrm{2}}\rangle , C_{\mathrm{9}}=\langle \sqrt{a}{\displaystyle \frac{h^{\mathrm{2}}}{\mathrm{3}}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{a}}}{\displaystyle \frac{{\psi }_{\mathrm{2}}}{{\psi }_{\mathrm{1}}}}\rangle \end{array},

where the operator \langle ·\rangle is defined by \langle f\rangle ={\displaystyle {\int }_{-\mathrm{1}/\mathrm{2}}^{+\mathrm{1}/\mathrm{2}}f(X)dX;X=x/\epsilon } and

(13)\begin{array}{l}\phi (X)=z(x(X))/\epsilon \\ \varphi =dz(x)/dx=d\phi (X)/dX\\ a=\mathrm{1}+{\varphi }^{\mathrm{2}}\end{array}\hspace{1em}\hspace{1em}\begin{array}{l}A(X)=-{\displaystyle {\int }_{\mathrm{0}}^X\sqrt{a}\phi (Y)dY}+C_{\mathrm{3}}{\displaystyle {\int }_{\mathrm{0}}^X\sqrt{a}dY}\\ {\psi }_{\mathrm{1}}=\mathrm{1}+{{\varphi }^{\prime }}^{\mathrm{2}}{h}^{\mathrm{2}}/\mathrm{48}{\epsilon }^{\mathrm{2}}{a}^{\mathrm{3}}\\ {\psi }_{\mathrm{2}}={{\varphi }^{\prime }}^{\mathrm{2}}{h}^{\mathrm{4}}/{\mathrm{12}}^{\mathrm{2}}{\epsilon }^{\mathrm{2}}{a}^{\mathrm{2}}\end{array}.

The particular values of these constants for trapezoidal or sinusoidal profiles are offered in the Appendix.

Finite element method for analysis of equivalent orthotropic plate

In this paper, we consider the panels which are subjected to distributed loads in simply supported or clamped boundary conditions along all boundary edges. The homogenization models represented in section 2 are now used to approximate the static behavior of the original corrugated panels. Particularly, the first-order shear deformation theory (FSDT) is applied in the derivation of the Galerkin weak form while the static problems are then solved using the cell-based smoothed MIN3 (CS-MIN3).

Galerkin weak form for static analyses of equivalent orthotropic plates

The displacements in x-, y-, and z-directions of the plate (as in Fig. 2(b)) are respectively given as follows

(14)\begin{array}{l}U(x,y,z)=u(x,y)+z{\beta }_x(x,y)\\ V(x,y,z)=v(x,y)+z{\beta }_y(x,y)\\ W(x,y,z)=w(x,y)\end{array}.

Let u^{\mathrm{T}}=\left[\begin{array}{lllll}u& v& w& {\beta }_x& {\beta }_y\end{array}\right] be the field vector containing, in that order, the displacements in x-, y-, and z- directions of the mid-plane and the rotation angles about y- and x- axes of its normal. Using the small deformation theory, the membrane, bending, and transverse shear strains of the equivalent orthotropic plate are

(15)\begin{array}{lll}{\epsilon }_m=L^{m}u;& \kappa =L^{b}u;& \gamma =L^{s}u\end{array},

where the partial differential operators are

(16)\begin{array}{lll}{L}^m=\left[\begin{array}{lllll}\partial /\partial x& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \partial /\partial y& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \partial /\partial y& \partial /\partial x& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right];& L^b=\left[\begin{array}{lllll}\mathrm{0}& \mathrm{0}& \mathrm{0}& \partial /\partial x& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \partial /\partial y\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \partial /\partial y& \partial /\partial x\end{array}\right];& L^s=\left[\begin{array}{lllll}\mathrm{0}& \mathrm{0}& \partial /\partial x& \mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \partial /\partial y& \mathrm{0}& \mathrm{1}\end{array}\right]\end{array}.

When the plate is subjected to a distributed load b={\left[\begin{array}{lllll}\mathrm{0}& \mathrm{0}& p(x,y)& \mathrm{0}& \mathrm{0}\end{array}\right]}^{\mathrm{T}}, the Galerkin weak form of the corresponding static equilibrium equation can be written as

(17){\displaystyle {\int }_{\mathrm{\Omega }}\delta {\epsilon }_m^{\mathrm{T}}\overline{A}{\epsilon }_m}\mathrm{d}\mathrm{\Omega }+{\displaystyle {\int }_{\mathrm{\Omega }}\delta {\kappa }^{\mathrm{T}}{\overline{D}}_b\kappa }\mathrm{d}\mathrm{\Omega }+{\displaystyle {\int }_{\mathrm{\Omega }}\delta {\gamma }^{\mathrm{T}}{\overline{D}}_s\gamma }\mathrm{d}\mathrm{\Omega }={\displaystyle {\int }_{\mathrm{\Omega }}\delta u^{\mathrm{T}}b}\mathrm{d}\mathrm{\Omega },

where the material matrices \bar{A}, {\bar{D}}_b, and {\bar{D}}_s are substituted from those of a homogenization model. We see that the membrane and bending deformations are separated in the governing equation. As a result, in this paper, the membrane deformation is approximated linearly using the constant strain triangular element (CST) and the bending deformation is approximated using the cell-based smoothed MIN3 (CS-MIN3). For short, we still call the resulting flat shell element CS-MIN3.

A brief introduction to formulation of the CS-MIN3 method

In the formulation of the CS-MIN3, each triangular element {\mathrm{\Omega }}_e is divided into three sub-triangles {\Delta }_{\mathrm{1}}, {\Delta }_{\mathrm{2}}, and {\Delta }_{\mathrm{3}} by connecting the central point O of the element to three field nodes \mathrm{1}, 2 and 3 (see Fig. 2c). Then, in each sub-triangle, the MIN3 is used to estimate the strain fields. The formulation of MIN3 on the whole element (\mathrm{1},2,3) can be described shortly as follows (more details in [⁵⁷]).

Firstly, the rotation angles {\beta }_x and \beta y_{} are approximated linearly and the deflection w is approximated quadratically. We have

(18)u={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{F}_i{d}_i},

in which d_i and [F_i] are nodal displacement vector and shape function matrix corresponding to the node i

(19)\begin{array}{ll}{[d_i]}^{\mathrm{T}}=\left[\begin{array}{lllll}{u}_i& v_i& w_i& {\beta }_{x i}& {\beta }_{y i}\end{array}\right];& F_i=\left[\begin{array}{lllll}{N}_i& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& N_i& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& N_i& H_i& L_i\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& N_i& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& N_i\end{array}\right]\end{array}.

The shape functions N_i, H_i, and L_i are given as in [⁵¹]. As a result, the strain fields of a MIN3 element are

(20)\begin{array}{lll}\begin{array}{l}{\epsilon }_m=L^{m}u={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{L}^m{F}_i{d}_i^e}={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{B}_i^m{d}_i^e}=B^m{d}^e\\ \kappa =L^{b}u={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{L}^b{F}_i{d}_i^e}={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{B}_i^b{d}_i^e}=B^b{d}^e\\ \gamma =L_S^{s}u={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{L}^s{F}_i{d}_i^e}={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{B}_i^s{d}_i^e}=B^s{d}^e\end{array}& \mathrm{where}& \begin{array}{l}{d}^e={\left[\begin{array}{lll}{d}_{\mathrm{1}}^e& d_{\mathrm{2}}^e& d_{\mathrm{3}}^e\end{array}\right]}^{\mathrm{T}}\\ B^m=\left[\begin{array}{lll}{B}_{\mathrm{1}}^m& B_{\mathrm{2}}^m& B_{\mathrm{3}}^m\end{array}\right]\\ B^b=\left[\begin{array}{lll}{B}_{\mathrm{1}}^b& B_{\mathrm{2}}^b& B_{\mathrm{3}}^b\end{array}\right]\\ B^s=\left[\begin{array}{lll}{B}_{\mathrm{1}}^s& B_{\mathrm{2}}^s& B_{\mathrm{3}}^s\end{array}\right]\end{array}\end{array}.

We see that the approximated membrane and bending strains of MIN3 are constant, while the approximated transverse shear strain is linear. From the Galerkin weak form of static analysis, the stiffness matrix of a MIN3 element can be derived as follows

(21)K_e^{MIN\mathrm{3}}=A_e{(B^m)}^{\mathrm{T}}\overline{A}{B}^m+A_e{(B^b)}^{\mathrm{T}}{\overline{D}}_b{B}^b+{\displaystyle {\int }_{{\mathrm{\Omega }}_e}{(B^s)}^{\mathrm{T}}{\overline{D}}_s{B}^s}\mathrm{d}\mathrm{\Omega },

where A_e is the area of the entire triangular element.

Let the central point of the element {\mathrm{\Omega }}_e be the node 0. Using the above formulas while replacing the triangle (\mathrm{1},\mathrm{2},\mathrm{3}) by the sub-triangles (\mathrm{1},\mathrm{2},\mathrm{0}), (\mathrm{0},\mathrm{2},\mathrm{3}) and (\mathrm{1},\mathrm{0},\mathrm{3}), we can derive the MIN3 approximation of strain fields in each sub-element

(22)\begin{array}{ll}\begin{array}{l}{\epsilon }_{{\Delta }_i}^m=B_{{\Delta }_i}^m{d}_{{\Delta }_i}^e\\ {\kappa }_{{\Delta }_i}=B_{{\Delta }_i}^b{d}_{{\Delta }_i}^e\\ {\gamma }_{{\Delta }_i}=B_{{\Delta }_i}^s{d}_{{\Delta }_i}^e\end{array}, i=\mathrm{1},\mathrm{2},\mathrm{3};& \begin{array}{lll}{d}_{{\Delta }_{\mathrm{1}}}^e=\left[\begin{array}{l}{d}_{\mathrm{1}}^e\\ d_{\mathrm{2}}^e\\ d_{\mathrm{0}}^e\end{array}\right],& d_{{\Delta }_{\mathrm{2}}}^e=\left[\begin{array}{l}{d}_{\mathrm{0}}^e\\ d_{\mathrm{2}}^e\\ d_{\mathrm{3}}^e\end{array}\right],& d_{{\Delta }_{\mathrm{3}}}^e=\left[\begin{array}{l}{d}_{\mathrm{1}}^e\\ d_{\mathrm{0}}^e\\ d_{\mathrm{3}}^e\end{array}\right]\end{array}\end{array}.

The cell-based strain smoothing technique on the whole element is then applied to derived the smoothed membrane {\tilde{\epsilon }}^m, smoothed curvature

\tilde{\kappa }

, and smoothed transverse shear strains \tilde{\gamma }. Using the constant smoothing function {\Phi }_e(x)=\{\begin{array}{l}\mathrm{1}/A_e\\ \mathrm{0}\end{array}\begin{array}{l}\mathrm{if} x\in {\Omega }_e\\ \mathrm{if} x\notin {\Omega }_e\end{array}, we have

(23)\begin{array}{l}{\tilde{\epsilon }}^m={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{\displaystyle {\int }_{{\Delta }_i}{\epsilon }_{{\Delta }_i}^m{\Phi }_e(x)d\Omega }}={\displaystyle \frac{\mathrm{1}}{A_e}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{A}_{{\Delta }_i}{\epsilon }_{{\Delta }_i}^m}={\displaystyle \frac{\mathrm{1}}{A_e}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{A}_{{\Delta }_i}{B}_{{\Delta }_i}^m{d}_{{\Delta }_i}^e}\\ \tilde{\kappa }={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{\displaystyle {\int }_{{\Delta }_i}{\kappa }_{{\Delta }_i}{\Phi }_e(x)d\Omega }={\displaystyle \frac{\mathrm{1}}{A_e}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{A}_{{\Delta }_i}{\kappa }_{{\Delta }_i}}={\displaystyle \frac{\mathrm{1}}{A_e}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{A}_{{\Delta }_i}{B}_{{\Delta }_i}^b{d}_{{\Delta }_i}^e}}\\ \tilde{\gamma }={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{\displaystyle {\int }_{{\Delta }_i}{\gamma }_{{\Delta }_i}{\Phi }_e(x)d\Omega }={\displaystyle \frac{\mathrm{1}}{A_e}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{\displaystyle {\int }_{{\Delta }_i}{\gamma }_{{\Delta }_i}(x)d\Omega }}={\displaystyle \frac{\mathrm{1}}{A_e}}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}{\displaystyle {\int }_{{\Delta }_i}{B}_{{\Delta }_i}^s(x)d\Omega d_{{\Delta }_i}^e}}}\end{array},

where A_{{\Delta }_i} is the area of the sub-triangle i.

The displacement vector d_{\mathrm{0}}^e at the center point \mathrm{0} is simply the average of the three displacement vectors d_{\mathrm{1}}^e, d_{\mathrm{2}}^e, and d_{\mathrm{3}}^e, i.e.,

(24)d_{\mathrm{0}}^e=(d_{\mathrm{1}}^e+d_{\mathrm{2}}^e+d_{\mathrm{3}}^e)/\mathrm{3.}

By substituting Eq. (24) into Eq. (23) we can represent the smoothed strains in terms of nodal displacement vectors d^e (as given in Eq. (20)) by

(25)\begin{array}{lll}{\tilde{\epsilon }}^m={\tilde{B}}_e^m{d}^e,& \tilde{\kappa }={\tilde{B}}_e^b{d}^e,& \tilde{\gamma }={\tilde{B}}_e^s{d}^e\end{array}.

As a result, the smoothed elemental stiffness matrix of the CS-MIN3 are derived as follows

(26){\tilde{K}}_e^{CS-MIN\mathrm{3}}=A_e{\left({\tilde{B}}_e^m\right)}^{\mathrm{T}}\bar{A}{\tilde{B}}_e^m+A_e{\left({\tilde{B}}_e^b\right)}^{\mathrm{T}}{\bar{D}}_b{\tilde{B}}_e^b+A_e{\left({\tilde{B}}_e^s\right)}^{\mathrm{T}}{\bar{D}}_s{\tilde{B}}_e^s.

Numerical examples and discussion

In this section, several numerical examples are conducted to verify the accuracy and reliability of some typical homogenization models in static analyses of corrugated panels. First, a validation study will clarify the computational ability of the CS-MIN3 in the simulation of orthotropic plates. Second, the CS-MIN3 will be integrated with the homogenization models (SamantaHM, PengHM, XiaHM, and YeHM) to give four homogenization methods, labeled by CS-MIN3-SamantaHM, CS-MIN3-PengHM, CS-MIN3-XiaHM, and CS-MIN3-YeHM, for the static analyses of panels of trapezoidal and sinusoidal profiles. Here, the formulas for equivalent transverse shear terms in PengHM are added to the other three methods to make them compatible with the FSDT plate theory. The analysis results are then compared with those of benchmark software including ANSYS Workbench 16.0 and ABAQUS/CAE 6.14-1. Consistently, in all numerical examples, equivalent orthotropic plates will be analyzed using CS-MIN3 with a constructive triangular mesh of 21\times 21 nodes while the original shell simulations will be conducted using ANSYS and ABAQUS with constructive rectangular meshes (see Fig. 3). The element types used in the software are SHELL181 and S4R, respectively.

Validation example

The following example is taken from [⁶⁴] where a simply supported orthotropic plate of different length-to-width ratios (L/W) and different thickness-to-length ratios (h/L) is subjected to a uniform pressure q_{\mathrm{0}}. The material of the plate is given by

(27)\begin{array}{l}{E}_{\mathrm{2}}/E_{\mathrm{1}}=\mathrm{0.52500},\hspace{0.17em}{\mu }_{\mathrm{12}}=\mathrm{0.44046},\hspace{0.17em}{\mu }_{\mathrm{21}}=\mathrm{0.23124}\\ G_{\mathrm{12}}/E_{\mathrm{1}}=\mathrm{0.26293},G_{\mathrm{13}}/E_{\mathrm{1}}=\mathrm{0.15991},\hspace{0.17em}{G}_{\mathrm{23}}/E_{\mathrm{1}}=\mathrm{0.26681},\end{array}

and the non-dimensional central deflection {\bar{w}}_c of the plate is derived using the following formula [⁶⁴]

(28){\bar{w}}_c=w_c{E}_{\mathrm{1}}/h{q}_{\mathrm{0}}(\mathrm{1}-{\nu }_{\mathrm{12}}{\nu }_{\mathrm{21}}),

where w_c is the actual central deflection.

Considering fifteen study cases listed in Table 2 while taking E_{\mathrm{1}}=\mathrm{1}, L=\mathrm{1}, and q_{\mathrm{0}}=\mathrm{1}, we compute the non-dimensional central deflections of the plate in simply supported (SSSS) and clamped (CCCC) boundary conditions. The results by five different FEMs including DSG3, MIN3, CS-DSG3, CS-MIN3, and ES-DSG3 are represented in Table 3 and Table 4 along with those by ABAQUS plate simulation and the exact 3D analysis [⁶⁴]. All five FEMs use a regular triangular mesh of \mathrm{21}\times \mathrm{21} nodes while ABAQUS uses a regular rectangular mesh of \mathrm{21}\times \mathrm{41} nodes for the cases L/W=\mathrm{0.5}, \mathrm{21}\times \mathrm{21} nodes for the cases L/W=\mathrm{1}, and \mathrm{41}\times \mathrm{21} nodes for the cases L/W=\mathrm{2}.

In the fifteen study cases, all FEMs give well-agreed results compared to those of ABAQUS while there are small discrepancies compared to the exact solutions of 3D analysis [⁶⁴]. We can explain this by the simplification and approximation of the FSDT that has been used in this study. Taking ABAQUS results as references, the mean absolute percentage errors (MAPE) listed at the bottom of each table show that all five elements give acceptable accuracy in many cases of plate shape and thickness. With maximum relative errors only 1.63% and 1.12%, CS-MIN3 and ES-DSG3 show outstanding performances. They also have a good convergences as shown in Fig. 4 where the cases 7 and 9 in SSSS boundary condition are demonstrated.

Static analyses of a trapezoidally corrugated panel

We now study the static behavior of a square trapezoidally corrugated panel using four homogenization methods including CS-MIN3-SamantaHM, CS-MIN3-PengHM, CS-MIN3-XiaHM, and CS-MIN3-YeHM. The material and geometry parameters of the panel are given as follows

(29)\begin{array}{l}E=\mathrm{21}\hspace{0.17em}\mathrm{GPa,}\hspace{0.17em}\nu =\mathrm{0.3},\hspace{0.17em}h=\mathrm{0.00635}\hspace{0.17em}m,\\ L=W=\mathrm{0.9144}\hspace{0.17em}\mathrm{m,}\hspace{0.17em}f=\mathrm{0.0127}\hspace{0.17em}\mathrm{m,}\hspace{0.17em}\alpha =\mathrm{45}°\hspace{0.17em}\mathrm{and}\hspace{0.17em}{n}_c=\mathrm{9.}\end{array}

This specimen of the trapezoidal profile was first examined in Samanta and Mukhopadhyay [⁵] and then cited in Xia et al. [²] and Ye [⁶⁵]. The effect of the number of corrugation units (n_c), half-amplitude (f), and trough angle (\alpha) to the accuracy of four homogenization methods will be in turn figured out.

The equivalent stiffness terms computed by different homogenization methods are listed in Table 5 along with those computed by VAPAS [⁶⁵], a 3D elasticity numerical code for equivalent plate modeling of panels with microstructure. As mentioned, in this table, all transverse shear stiffness terms are derived using the proposed strategy in PengHM. We see that all stiffness terms of YeHM are well-agreed with VAPAS results which are chosen as references. While the only significant deviation to the reference values in XiaHM’s results occurs at the bending term {\bar{D}}_{\mathrm{22}} (7.36%), many equivalent extensional and bending terms of SamantaHM and PengHM are much different to those of VAPAS.

Effect of the number of corrugation units

To validate the quality of these homogenization models, the panel with three different degrees of corrugation, n_c=\mathrm{3},\mathrm{9}, and 15 , is in turn subjected to a uniformly distributed load of 100 Pa in simply supported (SSSS) and clamped (CCCC) boundary conditions. Simulations of the original panels which are conducted in ANSYS and ABAQUS are taken as references. The larger the n_c is, the finer mesh for shell modelling is required. For example, the mesh sizes used in ANSYS when n_c=\mathrm{3},\mathrm{9}, and \mathrm{15} are 2107, 5785, and 9471 nodes, respectively. All homogenization methods use a much coarser mesh with only 441 nodes.

The deflections of the equivalent plates and the original panels along x- and y-central lines are depicted in Figs. 5 and 6. Table 6 shows the relative errors of central deflections between four homogenization methods and the ANSYS shell simulations. The error is defined as follows

(30)\mathrm{error}={\displaystyle \frac{w_c^{\mathrm{HM}}-w_c^{\mathrm{ANSYS}}}{w_c^{\mathrm{ANSYS}}}}\times \mathrm{100}\%.

where w_c^{\mathrm{HM}}and w_c^{\mathrm{ANSYS}} are the central deflections computed by a homogenization method and ANSYS shell simulation, respectively.

We see that when the number of corrugation units n_c is small, all equivalent models cannot produce completely the deformation shapes of the original shell models. This limitation is improved when n_c is increased. Among four methods, the CS-MIN3-YeHM gives most agreed central deflections to the references, especially when the number of corrugation units is large. For example, in the case of the simply supported panel with n_c=\mathrm{15}, the absolute error of CS-MIN3-YeHM is only 0.34% while those of other methods are larger than 8%. In general, we can observe that the CS-MIN3-PengHM is softer while CS-MIN3-XiaHM is stiffer than the CS-MIN3-YeHM. The deflection of the whole panel (n_c=\mathrm{9}) and its YeHM equivalent plate are depicted in Fig. 7. We can see that this homogenization model can produce a well-matched approximation for the deformation of the original panel.

Effect of corrugation amplitude

Remaining n_c=\mathrm{9}, we replace the half-amplitude f of the above panel by f/{\mathrm{2}}^{r_f} where the amplitude modification coefficient r_f=-\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{..},\mathrm{7} and then compute the corresponding central deflections. The results of four homogenization methods and two shell simulations are listed in Tables 7 and 8. We see that when the amplitude tends to zero, all homogenization methods and shell analyses approach the solution of the flat plate, except the CS-MIN3-SamantaHM in SSSS boundary condition. Figure 8 shows the relative errors between homogenization methods and ANSYS shell analysis. We can observe that the CS-MIN3-YeHM give the closet results (relative error less than 5%) to those of ANSYS shell simulation while it seems to be bounded by the CS-MIN3-XiaHM and the CS-MIN3-YeHM. Notably, the CS-MIN3-PengHM and CS-MIN3-XiaHM show poor performances when the corrugation amplitude is high.

Effect of trough angle

With the same static problem where n_c=\mathrm{9} and f unchanged, Table 9 represents the central deflections of the trapezoidally corrugated panels corresponding to five different values of trough angle \alpha =\mathrm{30}°,\mathrm{45}°,\mathrm{60}°,\mathrm{75}°,\mathrm{90}°. From the relative errors shown in Fig. 9, we see that only CS-MIN3-YeHM produces good accuracy in all cases of trough angle, especially in SSSS boundary condition. Meanwhile, the CS-MIN3-PengHM is likely too soft and SamantaHM and XiaHM too stiff to ANSYS solutions.

Static analysis of a sinusoidally corrugated panel

Consider a square sinusoidally corrugated panel that has following material and geometric properties

(31)\begin{array}{l}E=\mathrm{30}\hspace{0.17em}\mathrm{GPa,}\hspace{0.17em}\nu =\mathrm{0.2},\hspace{0.17em}h=\mathrm{0.005}\hspace{0.17em}m,\\ L=W=\mathrm{7.04}\hspace{0.17em}\mathrm{m,}\hspace{0.17em}f=\mathrm{0.11}\hspace{0.17em}m\hspace{0.17em},\hspace{0.17em}\mathrm{and}\hspace{0.17em}{n}_c=\mathrm{11.}\end{array}

This panel was first analyzed in Ye et al. [⁶⁵] where a uniformly distributed load of 50 Pa is applied in simplify supported (SSSS) boundary conditions.

The equivalent stiffness terms for this panel are represented in Table 10. The VAPAS results [⁶⁵] are also included as references. It can be seen that both XiaHM and YeHM produce terms close to those of VAPAS, while PengHM does not.

Effect of the number of corrugation units

The accuracy of the homogenization methods in static analysis of the panel under simply supported (SSSS) and clamped (CCCC) boundary conditions will be verified by comparing their deflections along two central lines with those of ANSYS and ABAQUS shell simulations. We consider three different values of the number of corrugation units, n_c=\mathrm{4}, 11, and 18. To capture the curve profiles of the corresponding panels, three constructive rectangular meshes of 6035, 22479, and 44795 nodes are used in ANSYS shell modeling, respectively. Again we only use a constructive triangular mesh of 441 nodes for all equivalent plates.

Figures. 10 and 11 represent the deflections of the panels along two central lines while the relative errors between homogenization methods and ANSYS shell simulation are listed in Table 11. It can be seen that when the number of corrugation units is small, for example n_c=\mathrm{4}, all three homogenization models cannot produce exactly the deformation shapes in ANSYS and ABAQUS results where local deformations of the panel are dominated. When n_c is increased, the CS-MIN3-XiaHM and the CS-MIN3-YeHM obtain better approaches to ANSYS solution while the CS-MIN3-PengHM becomes too soft. It can be noticed that the results of CS-MIN3-YeHM are bounded by those of CS-MIN3-PengHM and CS-MIN3-XiaHM. The contour plots for the deflection of the whole panel (n_c=\mathrm{18}) and its YeHM equivalent plate are depicted in Fig. 12. We can observe that the CS-MIN3-YeHM produces a well-matched deformation shape.

Effect of thickness

Keeping n_c=\mathrm{11}, the thickness h of the above panel is changed to {\mathrm{2}}^{r_h}h where the thickness modification coefficient r_h=-\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{...},\mathrm{7}. The central deflections of the resulting panels under SSSS and CCCC boundary conditions are listed in Tables 12 and 13, respectively. Figure 13 shows the relative errors of central deflections between homogenization methods and ANSYS shell analysis. Among three homogenization methods, CS-MIN3-YeHM gives the closest and stable results to those of ANSYS. Again, we can observe that in most cases of thickness, the CS-MIN3-YeHM result is bounded by those of the CS-MIN3-PengHM and CS-MIN3-XiaHM.

Conclusions

In this paper, some typical homogenization models for corrugated panels are reviewed and represented by explicit formulas. The CS-MIN3 based on the first-order shear deformable theory (FSDT) is integrated with three homogenization models (by Samanta et al., Peng et al., Xia et al. and Ye et al.) to give the so-called CS-MIN3-SamantaHM, CS-MIN3-PengMH, CS-MIN3-XiaMH and CS-MIN3-YeMH for static analysis of corrugated panels of trapezoidal and sinusoidal profiles. The accuracy of these homogenization methods is evaluated through several numerical examples. In conclusion, we can withdraw the following points:

(1) The CS-MIN3 shows a good performance in static analyses of orthotropic plates and therefore is a suitable method for static analyses of the homogenization models of corrugated panels.

(2) Among four homogenization methods for trapezoidally and sinusoidally corrugated panels, the CS-MIN3-YeMH yields good agreement and stable results with ANSYS shell simulations. In many cases, it is observed that CS-MIN3-PengHM is softer while CS-MIN3-XiaHM is stiffer than CS-MIN3-YeHM.

(3) The geometric parameters such as the number of corrugation units, half-amplitude, trough angle, and panel thickness have influences on the accuracy of homogenization models.

(4) Although losing a certain amount of accuracy, homogenization methods can give a prediction of static deformation of corrugated panels while using a much coarser mesh. This advantage should enable more efficient optimization and reliability analyses in designing of corrugated structures.

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