Free Vibration Analysis of Symmetrically Laminated Folded Plate Structures Using an Element-Free Galerkin Method

Abstract

An element-free Galerkin method for the solution of free vibration of symmetrically laminated folded plate structures is introduced. Employing the mature meshfree folded plate model proposed by the author, a folded laminated plate is simulated as a composite structure of symmetric laminates that lie in different planes. Based on the first-order shear deformation theory (FSDT) and the moving least-squares (MLS) approximation, the stiffness and mass matrices of the laminates are derived and supposed to obtain the stiffness and mass matrices of the entire folded laminated plate. The equation governing the free vibration behaviors of the folded laminated plate is thus established. Because of the meshfree characteristics of the proposed method, no mesh is involved to determine the stiffness and mass matrices of the laminates. Therefore, the troublesome remeshing can be avoided completely from the study of such problems as the large deformation of folded laminated plates. The calculation of several numerical examples shows that the solutions given by the proposed method are very close to those given by ANSYS, using shell elements, which proves the validity of the proposed method.

1. Introduction

Because of high strength/weight ratio, easy forming, and low cost, folded plate structures have been widely used in many engineering branches, such as roofs, corrugated-cores, and cooling towers. They have much higher load carrying capacity compared to flat plates. Before the invention of fiber-reinforced material, folded plates were often made of medal or timber. The application of fiber-reinforced material to folded plate structures was a remarkable advance in engineering, which combined the advantages of fiber-reinforced material and folded plate structure directly and made the structure even lighter and stiffer.

The study of isotropic folded plates had a quite long history, and a variety of methods had emerged. In early days, researchers were short of powerful numerical tools and tried to analyze the structures with various approximations. The beam method and the theory that ignores relative joint displacement were introduced [1]. Although the methods were weak in dealing with generalized folded plate problems, they were simple and fulfilled the demand of fast and easy computation in engineering. Therefore, they are still used in some design environments, where accurate analysis is not the first concern. Researchers such as Gaafar [2], Yitzhaki [3], Yitzhaki and Reiss [4], and Whitney et al. [5] were the first to consider the relative joint displacement of the structures in their methods, which led to more precise analysis results. Goldberg and Leve [6] used the two-dimensional theory of elasticity and the two-way slab theory to analyze folded plates. In their method, both the simultaneous bending and the membrane action of a folded plate were taken into account, and the degree of freedom (DOF) of each point along the joint of the folded plate was chosen to be four (three components of translation and one rotation). Niyogi et al. [7] considered this method as the first to give an exact static solution for folded plates. Yitzhaki and Reiss [4] took the moments along the joints of folded plates as unknown and applied the slope deflection method to the analysis of the folded plates. Bar-Yoseph and Hersckovitz [8] proposed an approximated method for folded plates based on Vlasov’s theory of thin-wall beams. Their method considered a folded plate as a monolithic structure composed of longitudinal beams, which can give good results for long folded plates. Bandyopadhyay and Laad [9] compared two classical methods for folded plates and studied the suitability of these methods for the preliminary analysis of folded plate structures. Lai et al. [10] gave an equation of the middle surface of a simply supported cross V-shaped folded plate roof by using the inclined coordinate system and generalized functions, sign function and step function, and carried out a nonlinear analysis for the folded plate.

The development of computation techniques and computers has aroused research interest in numerical methods for folded plates. A number of methods, such as the finite strip methods (Cheung [11], Golley and Grice [12], Eterovic and Godoy [13]), the combined boundary element-transfer matrix method (Ohga et al. [14]), and the finite element methods (FEM) (Liu and Huang [15], Perry et al. [16], Niyogi et al. [7], and Duan and Miyamoto [17]), have been introduced to solve folded plate problems. Among these methods, the FEMs are the most successful. They are very versatile as they can deal with the problems with complicated geometry, boundary conditions, or loadings easily. However, FEM also has disadvantages. Their solution of a problem is based on the meshes that discretize the problem domain, and any dramatic change of the problem domain will lead to remeshing of the domain, which results in programming complexity, diminished accuracy, and long computation time. Regarding the disadvantage, some researchers proposed the element-free, meshfree, or meshless methods [18–22]. As alternatives to FEMs, the meshfree methods construct their approximated solution of a problem completely in terms of a set of ordered or scattered points that discretize the problem domain; that is, their solution relies on the points other than meshes. No element is required. Without the limit of meshes, the meshfree methods are more applicable than the FEMs and avoid the aforementioned difficulties caused by remeshing in the FEMs.

Bui et al. [23], Bui and Nguyen [24], and Somireddy and Rajagopal [25] have introduced the meshfree methods for vibration analysis of laminated plates. However, few studies on folded laminated plates have been found. There are only Niyongi et al. [7] and Lee et al.’s [26] work on vibration and the author’s work on bending with a meshfree method, which is also the motive for this paper.

The objective of this paper is to introduce an element-free Galerkin method based on the first-order shear deformation theory (FSDT) [27, 28] for the free vibration analysis of folded laminated plates. A symmetrical folded laminated plate is regarded as a composite structure composed of symmetric laminates. The analysis process includes (a) deriving the stiffness and mass matrices of the symmetric laminates that make up a folded plate by the element-free Galerkin method; (b) considering the laminates as super elements and superposing their stiffness and mass matrices to obtain the global stiffness and mass matrices of the folded plate. Some numerical examples are used to demonstrate the convergence and accuracy of the proposed method. The calculated results are compared with the results from the finite element analytical software ANSYS. The proposed method may be used as a potential meshfree tool for the analysis of laminated shell structures.

2. Moving Least-Squares Approximation

In the moving least-squares approximation (MLS) [18], a function in a domain can be approximated by in the subdomain and where are the monomial basis functions, are the corresponding coefficients, is a factor that measures the domain of influence (or the support) of the nodes, and is the number of basis functions. In this paper, the quadratic basis () are used for the laminates. The unknown coefficients are obtained by the minimization of a weighted discrete norm where or is the weight function that is associated with node , outside , is the number of nodes in that make the weight function , and are the nodal parameters. The minimization of in (2) with respect to leads to a set of linear equations where The coefficients are then derived from (4): By substituting (7) into (1), the approximation is expressed in a standard form as where the shape function is given by From (6), we obtain and thus (9) can be rewritten as

3. Meshless Model of a Laminate

The first step in our analysis is to obtain the stiffness and mass matrices of the laminates that make up a folded plate. The meshless model for a laminate in the local coordinate, as shown in Figure 1, is prescribed with a set of nodes. The DOF of every node is , where , , and are the nodal translations of the laminate in the -direction, -direction, and -direction, respectively. and are the rotation about the -axis and the -axis, respectively. The laminate is assumed to have layers, and the thickness of each layer is (). Therefore, the thickness of th layer is .

Figure 1 

Meshfree model of a laminate.

3.1. Displacement Approximation

Based on the FSDT and the MLS approximation, the displacements of the laminate can be approximated by where are the nodal parameters of the th node of the laminate, is the number of nodes of the laminate, and and are independent of . The shape functions are obtained from (9), and the cubic spline function is used as the weight function. Equation (12) can be written in a matrix form as The strains of the laminate are defined as where “” refers to the derivatives of , and “” refers to the derivatives of .

3.2. Governing Equation

In free vibration, the strain energy and kinetic energy of the laminate are, respectively, where () are the material stiffness that are defined in [28], is the shear correction factor, and is the thickness of the laminate. is the density of the material.

According to Hamilton’s principle, The substitution of (14) to (17) into (19) gives where If , we obtain For the symmetric laminates, .

4. Formulation for Folded Laminated Plates

In the paper, a folded laminated plate is regarded as a composite structure composed of laminates. We have obtained the stiffness and mass matrices of a single laminate. Therefore, the next step is to take each laminate of the composite structure as a super element, to superpose their stiffness and mass matrices by applying the displacement compatibility conditions along the joints between the laminates, and to give the governing equation of the entire folded laminated plate (Figure 2).

Figure 2 

A meshfree model of a folded laminated plate that is made up of two super elements (laminates). The domain of influence of node is indicated by the dashed line.

Nevertheless, as pointed out by the author in [20], due to a lack of Kronecker delta properties in the meshfree shape functions given by (9), and that of (20) are nodal parameters other than actual nodal displacements, the stiffness and mass matrices cannot be directly superposed. The full transformation method that was first introduced by Chen et al. [19] to enforce the essential boundary conditions is extended by the author in the paper to modify the stiffness and mass matrices before a superposition. After the modification, the essential boundary conditions can be implemented as those in FEMs.

4.1. Modification of Stiffness and Mass Matrices

From (1), the actual displacement of the nodes, , can be approximated by where and are the nodal parameters. Therefore, the nodal parameters can be expressed in terms of the actual displacement where .

For the folded laminated plate, we accordingly have where are the actual nodal displacement (translation) in the direction and are the corresponding nodal parameters. , , , , , , , and have similar definition. Equation (26) can be written as where are the actual nodal displacement of all nodes. is a modification matrix which combines five . Substituting (27) into (20) and premultiplying both sides of the equation with , we obtain Assuming that and , we have the modified governing equation.

4.2. The Governing Equation of the Folded Laminated Plate

Because the stiffness and mass matrices of a laminate in (20) were established in a local coordinate attached to the laminate (Figure 1), the matrices in (29) need to be transformed to the global coordinates before the superposition. The stiffness and mass matrices and the nodal displacement in the global coordinates are where is the coordinate transformation matrix derived in [20] (note: a drilling degree of freedom has been added to , and and must be expanded accordingly by inserting some zero elements). If there are (no. 1 to ) coincident nodes along the joint between super elements 1 and 2 (Figure 2), we have where   are the actual displacement of the nodes of super element 1 along the joint between super element 1 and super element 2 and  are the actual displacement of the nodes of super element 2 along the joint between super element 1 and super element 2. After necessary elementary transformation, the governing equation of super elements 1 and 2 can be written in block forms where and are the actual displacement of the nodes of super elements 1 and 2 that are not along the joint, respectively. Equation (32) are supposed to give the equation governing the dynamic behaviors of the entire structure Taking we have The solution of the corresponding eigenvalue problem gives us the free vibration frequencies of the folded laminated plate.

5. Results and Discussion

In order to show the convergence and accuracy of the proposed method, several numerical examples are calculated with the method and the finite element software ANSYS. For all the laminates in the examples, the plies are assumed to have the same thickness and material properties:  Pa,  Pa,  Pa,  Pa, , and  kg/m³. Unless otherwise specified, for each example a total of two cases of symmetric laminates, cross-ply or angle-ply, which make up the folded plates, are studied: (0°/90°/90°/0°) and (−45°/45°/−45°/45°/−45°…)₁₀. In ANSYS, the folded laminated plates in the examples are all modeled as shells, and the linear layered structural shell element SHELL99 [29] is used to discretise the folded plates.

5.1. Validation Studies

To carry out the validation studies, a cantilever square laminate with a lamination scheme of (−45°/45°/45°/−45°) is considered (Figure 3). The width of the plate is 1.8 m, and the thickness is 0.018 m. The solution from ANSYS (5000 elements to discretise the laminate) is taken to be the exact solution. The validation studies consist of a convergence study and a study on the effect of the size of support and the completeness order of the basis functions on the convergence of solutions.

Figure 3 

A cantilevered laminate.

Firstly, we choose a certain meshless scheme (11 × 11 nodes for the laminate) and let the scaling factor and the completeness order of the basis function vary. defines the size of the support of nodes. In this paper, a rectangular support is used and where , are the lengths of the support in the and directions, respectively, and , are the distances between two neighbouring nodes in the - and -directions, respectively. The dimensionless fundamental frequency of the laminate as calculated by the proposed method under different values of and is shown in Figure 4 and is compared with the solution that is given by ANSYS. The dimensionless frequency is defined as where is the width of the laminate and is the vibration frequency. From Figure 4, it can be observed that for a certain meshless scheme (in this case 11 × 11 nodes), all of the solutions for different completeness orders () of the basis functions converge when the support size () is larger than 5. Higher completeness orders () need a larger support size to make the solution converge (the solution under converges at while the solution under converges at ).

Figure 4 

Variation of dimensionless fundamental frequency of the laminate under different and .

Secondly, we vary the meshless scheme and obtain the variations of the dimensionless fundamental frequency under certain completeness order of the basis functions (), which are shown in Figures 5, 6, 7, and 8, respectively. The solution that is given by ANSYS is also in the figures for comparison. Figures 5 to 8 indicate that for certain , the solution converges when the number of nodes increases. For an , the solutions for larger support sizes () converge before those for smaller support sizes do.

Figure 5 

Variation of dimensionless fundamental frequency of the laminate, .

Figure 6 

Variation of dimensionless fundamental frequency of the laminate, .

Figure 7 

Variation of dimensionless fundamental frequency of the laminate, .

Figure 8 

Variation of dimensionless fundamental frequency of the laminate, .

From the studies, we find that when the order of basis functions and the support size for the laminate, the solutions are precise enough with a relatively lower computational cost. Therefore, all of the following examples are calculated with , .

5.2. A Folded Plate That Is Made Up of Two Laminates

A clamped laminated folded plate that is made up of two identical square laminates is studied (Figure 9). The width of each laminate is  m, and the thickness  m. The dimensionless frequencies of the first five vibration modes of the folded plate that are obtained by the proposed method (11 × 11 nodes for each laminate) are listed in Tables 1 and 2 alongside the results that are given by ANSYS (3200 elements to discretise the folded plate) for comparison, and the first five mode shapes of vibration of the folded plate are also plotted graphically in Figures 10 and 11. Different crank angles of the folded plate, and , are considered. The results of the two methods are very close.

Table 1 

Dimensionless free vibration frequencies of the onefold laminated plate with the lamination scheme (0°/90°/90°/0°).

Table 2 

Dimensionless free vibration frequencies of the onefold laminated plate with the lamination scheme (−45°/45°/−45°/45°/−45°…)₁₀.

Figure 9 

The onefold laminated plate with two sides fixed.

Figure 10 

First five vibration mode shapes of the clamped folded plate ().

Figure 11 

First five vibration mode shapes of the clamped folded plate ().

When the folded plate is clamped at one side, which makes it a cantilevered folded plate (Figure 12), the dimensionless frequencies of the first five vibration modes of the structure are listed in Tables 3 and 4, and the five vibration mode shapes are shown in Figures 13 and 14.

Table 3 

Dimensionless free vibration frequencies of the cantilevered onefold laminated plate with the lamination scheme (0°/90°/90°/0°).

Table 4 

Dimensionless free vibration frequencies of the cantilevered onefold laminated plate with the lamination scheme (−45°/45°/−45°/45°/−45°…)₁₀.

Figure 12 

The cantilevered onefold laminated plate.

Figure 13 

First five vibration mode shapes of the cantilevered folded plate ().

Figure 14 

First five vibration mode shapes of the cantilevered folded plate ().

5.3. A Folded Plate That Is Made Up of Three Laminates

A folded plate that is made up of three identical square laminates and clamped at one side is studied (Figure 15, ). The width of each laminate is  m, and the thickness  m. The dimensionless frequencies of the first five vibration modes of the folded plate, which are calculated with the proposed method and ANSYS, are listed in Tables 5 and 6, and the first five mode shapes of vibration of the folded plate are also plotted graphically in Figures 16 and 17. In ANSYS, 4800 elements are used to discretise the structure. The agreement between the two sets of results is good.

Table 5 

Dimensionless free vibration frequencies of the cantilevered twofold laminated plate with the lamination scheme (0°/90°/90°/0°).

Table 6 

Dimensionless free vibration frequencies of the cantilevered twofold laminated plate with the lamination scheme (−45°/45°/−45°/45°/−45°…)₁₀.

Figure 15 

A cantilevered folded plate that is made up of three identical square laminates.

Figure 16 

First five vibration mode shapes of the cantilevered folded plate () that is made up of three identical square laminates.

Figure 17 

First five vibration mode shapes of the cantilevered folded plate () that is made up of three identical square laminates.

When the crank angle and the laminates are assumed to be connected with one another, we obtain a tub structure with three folds (Figure 18). The dimensionless frequencies of the first five vibration modes of the structure are listed in Tables 7 and 8, and the first five mode shapes are shown in Figure 19.

Table 7 

Dimensionless free vibration frequencies of the cantilevered tub structure with the lamination scheme (0°/90°/90°/0°).

Table 8 

Dimensionless free vibration frequencies of the cantilevered tub structure with the lamination scheme (−45°/45°/−45°/45°/−45°…)₁₀.

Figure 18 

A tub structure with three folds.

Figure 19 

First five vibration mode shapes of the tub structure.

5.4. A Laminated Shell

If the three laminates in Section 5.3 are joined with each other vertically, we obtain a laminated shell (or half of a box structure) (Figure 20). The structure is pinned at points A, B, and C. All of the DOFs, except the rotations of these points, are set to zero.

Figure 20 

A laminated shell structure.

Three lamination schemes are considered. Case  1: laminate 1 is taken to be (45°/−45°/−45°/45°) and laminates 2 and 3 to be (−45°/45°/45°/−45°), as is demonstrated in Figure 21(a); Case  2: laminate 1 is taken to be (−45°/45°/45°/−45°) and laminates 2 and 3 to be (45°/−45°/−45°/45°), as is demonstrated in Figure 21(b); and Case  3: laminate 1 is taken to be (90°/0°/0°/90°) and laminates 2 and 3 to be (0°/90°/90°/0°), as is shown in Figure 21(c).

(a)

(b)

(c)

Figure 21 

Lamination schemes of the laminated shell.

The dimensionless frequencies of the first five vibration modes of the structures are computed by both the proposed method and ANSYS and listed in Tables 9, 10, and 11, and the first five mode shapes are shown in Figures 22, 23, and 24. In ANSYS, 4800 elements are used to discretise the structure. The agreement of the two sets of results is good.

Table 9 

Dimensionless free vibration frequencies of the laminated shell (Figure 21(a)).

Table 10 

Dimensionless free vibration frequencies of the laminated shell (Figure 21(b)).

Table 11 

Dimensionless free vibration frequencies of the laminated shell (Figure 21(c)).

Figure 22 

First five vibration mode shapes of the laminated shell (Case  1).

Figure 23 

First five vibration mode shapes of the laminated shell (Case  2).

Figure 24 

First five vibration mode shapes of the laminated shell (Case  3).

6. Conclusions

An element-free Galerkin method that is based on the FSDT is proposed for the free vibration analysis of folded symmetrically laminated plate structures. A folded laminated plate is considered to be a composite structure of flat symmetrical laminates. The global stiffness and mass matrices of the folded plate are formed by superposing the stiffness and mass matrices of the laminates that are derived with the meshfree method. In order to ensure the success of the superposition, a treatment initially developed for the enforcement of essential boundary conditions is extended to modify the stiffness and mass matrices before the superposition, which has overcome the difficulties that the EFG handles displacement compatibility and improved the applicability of EFG to composite structures. The proposed method does not rely on meshes; therefore, mesh disorder due to the large deformation of problem domain is avoided. The convergence and accuracy of the proposed method are demonstrated by a comparison of the solutions of several examples with those that are given by ANSYS. Good agreement between the two sets of results is observed. The proposed method used a relatively small number of nodes to obtain the calculated results close to the solutions given by ANSYS with a large number of nodes, and the linear analysis by the proposed meshless method in the paper can be the basis for future nonlinear analysis. The treatment introduced with the proposed method has provided a clue for EFG to be applied to composite structures.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work that is described in this paper has been supported by the grants awarded by the National Natural Science Foundation of China (Projects nos. 11102044, 51168003, and 11562001), the Systematic Project of Guangxi Key Laboratory of Disaster Prevention and Structural Safety (Project no. 2012ZDX07), and Key Project of Guangxi Science and Technology Lab Center (Fund no. LGZX201101).