Finite element vibration analysis of laminated composite folded plate structures

A nine-noded Lagrangian plate bending finite element that incorporates first-order transverse shear deformation and rotary inertia is used to predict the free and forced vibration response of laminated composite folded plate structures. A 6 × 6 transformation matrix is derived to transform the system element matrices before assembly. The usual five degrees-of-freedom per node is appended with an additional drilling degree of freedom in order to fit the transformation. The present finite element results show good agreement with the available semi-analytical solutions and finite element results. Parametric studies are conducted for free and forced vibration analysis for laminated folded plates, with reference to crank angle, fibre angle and stacking sequence. The natural frequencies and mode shapes, and forced vibration responses furnished here may serve as a benchmark for future investigations.


Introduction
Folded plate structures have a wide range of engineering applications in aircraft fuselages, ship hulls, buildings, bridges and vehicle chassis, among other structures.With the advent of fiber-reinforced laminated composites, the applicability of folded plate structures has increased many folds due to their low weight, high stiffness and high strength properties.Structural properties of the laminated composites can be tailored to realize better performance by controlling the lamination angle and stacking sequence.Literature in the field of dynamic analysis of laminated composite folded plates is scanty and the present paper is meant to address the dynamic behaviour of composite folded plate structures, in view of its tremendous potentials.
Historically, Goldberg and Leve [8] pioneered the exact static analysis of folded plate structures.Irie et al. [10] calculated the natural frequencies of cantilever folded plates by Ritz method.Ohga and Shinematsu [17] applied boundary element-transfer matrix method to solve the bending problem of folded plates.Golley and Grice [9] and Eterovic and Godoy [7] employed finite strip methods.Danial [4] and Danial et al. [5] introduced a concept entitled Spectral Element Method.Liu and Huang [12] had used a finite element-transfer matrix method to analyze one-and two-fold folded plates.Bathe [1] and Zienkeiwicz and Taylor [21] had presented a method of flat shell analysis, which can be directly applied to folded plate structures.However, all these works relate to isotropic folded plate structures only.
For the finite element analysis of laminated composite plates with first and higher order shear deformation theories (FST, HST) several works may be referred to.The works of Reddy [18], Kant et al. [11], Meimaris and Day [16], Bert and Chen [2], Chatterjee and Kulkarni [3], Dong and Chun [6] and Maiti and Sinha [13][14][15] may be cited to name only a few.Recently, Suresh and Malhotra [19] conducted damped free vibration analysis of composite box beams using 4 noded plate finite element with five degrees of freedom (u, v, w, θ x and θ y ) per node.
In the present analysis the FST is adopted to analyze the laminated composite folded plates.A shear correction factor of 5/6 has been assumed, which is derived from the Timoshenko beam concept by applying the energy principle.A comparative study using FST and HST [13][14][15] clearly indicates that the present first order shear deformation theory provides excellent results for moderately thick composite beams, plates and shells for all practical situations.Attention is therefore restricted to application of FST for the development The transformations used to generate the global mass, stiffness and load arrays are presented.The results of free vibration analysis and also transient analysis of cantilever laminated folded plates, subjected to suddenly applied step loading are presented.The transient analysis is done using the Newmark's direct integration scheme [1].Parametric studies are conducted by varying the crank angle, fibre orientation and number of layers for the laminated composite folded plate.The present methodology can easily be applied to the analysis of box beams and closed structures.

Theoretical formulation
In the classical thin plate theory, it is assumed that normals to the mid-plane before deformation remain normal and plane after deformation.This assumption neglects the effect of transverse shear deformation.Here Mindlin's assumptions, as given below, are adopted to incorporate the first order transverse shear effects: i) The deflection of the mid-plane of the plate is small compared to the plate thickness.ii) The transverse normal stress is neglected.iii) Normals to the mid-plane of the plates before deformation remain straight but not necessarily normal after deformation.
The displacements of the plate are fully described by five components: u, v, w, θ x , θ y , where u, v and w are displacements along the x, y and z-directions and θ x and θ y are rotations about y-and x-axes.The positive sign conventions for displacements and stress resultants are illustrated in Fig. 1.

Flat plate finite element formulation for composite plates
The displacements at a node j of a plate element are u j , v j , w j , θ xj and θ yj .The displacements at any point within the element can be expressed as where [I 5 ] is a 5 × 5 identity matrix and N j are Lagrangian interpolation functions [1].For Mindlin plates the following relationship is valid: where θ x and θ y are the total rotations, φ x and φ y are the average shear deformations about the y and x axes, and u 0 , v 0 and w 0 are the mid-plane translations along x, y and z directions.

Stiffness matrix of plate element
The stiffness matrix of the plate element assumes the form where Here, {ε} is the strain vector and {δ} is the nodal displacement vector.The strain-displacement matrix [B] is given in [D] is the stiffness matrix given by where and Here, Q ij are the elements of off-axis stress-strain relations.Q k ij relates stresses and strains in the kth layer by the relation and κ is the shear correction factor.Q k ij for the kth layer is expressed as Here, σ 1 , σ 2 , σ 4 , σ 5 , and σ 6 denote σ x , σ y , τ yz , τ zx and τ xy , respectively, and ε 1 , ε 2 , ε 4 , ε 5 , and ε 6 stands for ε x , ε y , γ yz , γ zx and γ xy , respectively.

Mass matrix of plate element
In matrix form the equation of motions for the Mindlin plate may be written as follows: where {F } is the force vector, [ρ] is the inertia matrix, and {A} is the acceleration vector.The mass matrix of the plate element is given by where [N ] are the Lagrangian interpolation functions.

Load vector
The element load vector for forced vibration analysis is given by Ae where q is the transverse load intensity on the element.The integration in every case is carried out over the area of the plate element.Generally, a 3-point Gauss quadrature is adopted to compute the bending stiffness of the elements, whereas a 2-point integration is adopted to calculate the shear stiffness, mass matrix, and element force vector.A 2-point eliminates the shear locking in thin plates.It is also known that a 2point integration for both the mass matrix and force vector is adequate.

Transformations for folded plate
With reference to Fig. 2, the relations between local and global displacements are given as Fig. 2. Local (unprimed) and global (primed) axes system for a typical folded plate element.The least angle contained between positive x and x axes is denoted as (x , x).
Finally, the global stiffness, mass matrices and force vector are expressed as {f However, before applying the transformation, the 45 × 45 stiffness and mass matrices is blown up to 54 × 54 size, to accommodate the nine θ z drilling degrees of freedom per element.The off-diagonal terms corresponding to the θ z terms are set to zero, while a very small positive number is introduced at the corresponding leading diagonal terms.This small number is taken to be 1000 times smaller than the smallest leading diagonal term of the corresponding element matrix before blowing up [1].The load vector is similarly enlarged by incorporating null terms in the θ z positions.
The free vibration analysis involves the solution of and, the method of subspace iteration [1] is adopted to extract the eigenpairs.In the forced vibration analysis, the damping is neglected, and the force term replaces the null vector in the right hand side of Eq. ( 16).Newmark's explicit integration technique [1] is adopted for the transient analysis.

Numerical results and discussion
The finite element formulation described in the earlier section has been used to compare the present results with the published ones and also to generate nu-merical results to study the effects of crank angle, fibre angle and number of plies used, on the nondimensional frequencies and responses of composite folded plates.The definitions for non-dimensionalised frequencies are given along with Tables 1 and 2.
Example 1.The results of free vibration analysis of isotropic single-fold and double-fold cantilever folded plates (Fig. 3) are presented in Table 1 and compared with those of Liu et al. [12] and Irie et al. [10].The geometry of the folded plates is defined in Fig. 3.Here elastic modulus E, Poisson's ratio ν, and density ρ are taken as 10.92 × 10 9 N/m 2 , 0.30, and 1000 kg/m 3 , respectively.The length of the cantilever is taken as 1.5 m for single fold folded plates and 2.0 m for twofold folded plates.The present analysis is carried out with multiple layers of isotropic material.The results are found to be in good agreement with previous works on isotropic folded plates.
bending mode.It may be noted that the ridgeline remains nearly undeformed for these modes and is instrumental in imparting additional stiffness compared to flat plates.This, in turn, has resulted in higher nondimensional fundamental frequencies to folded plates compared to flat plate, as seen in Table 2.
Table 3 provides natural frequencies for 4-layered symmetric and anti-symmetric angle-ply and cross ply laminates.Here it is observed that anti-symmetric angle-ply with lower values of fibre angle produces higher fundamental frequency.It may also be noted from Table 4 that the fundamental frequencies have an  increasing trend with the increase in the number of layers, while the fibre angles and thickness are kept invariant.Anti-symmetric plies have shown higher fundamental frequencies.
Tables 5, 6 and 7 represent the first three natural frequencies for two-fold folded plates (refer Fig. 3).6 × 3 mesh are used.For β = 120 • , the mode shapes of a two-fold composite folded plate are shown in Fig. 5(b).The first mode is a torsion mode.The second mode is a symmetric bending mode and the third is a symmetric mode too, though complex in shape.However this trend is not repeated for β = 90 • , as shown in Fig. 5(a).Thus, Fig. 5 shows that the mode shapes for two-fold folded plates depend on the crank angles β heavily, as also on the fibre angles.This is unlike single fold folded plates, where only fibre angles and number of plies are of importance, and the values of first and the third frequencies are nearly invariant for a given fibre angle and number of plies irrespective of the crank angle α.From Table 5 it is observed that for the type [θ/−θ/θ], there is a decreasing trend of first frequencies with increase in the value of θ, but the case of 2-fold plate with a crank of 90 • now shows a reverse trend.It may also be noted that for isotropic and composite single-fold plates and two-fold isotropic folded plates the first frequency is found to remain practically unchanged for same material properties and fibre angles (refer Tables 1, 2, 3, and 4, and [10,12]).This is also not generally true for 2-fold composite folded plates and the crank angle is seen to affect the fundamental frequencies considerably.From Table 6 it is seen that anti-symmetric plies produce higher fundamental frequencies.Table 7 reveals that an increase in number of layers increases the fundamental frequencies.It may also be noted here that for single-fold folded plates the [30 • /−30 • ] 3 laminations produce highest fundamental frequencies whereas the 0 • or (0 • /90 • ) 2 lamination produce the highest fundamental frequencies for two-fold folded plates.Finally, Table 8 shows a comparative study of results obtained by applying 2-point and 3-point integration for mass matrices.The table reveals that there is no perceptible difference of results, and a 2-point integration may be employed for computational efficiency.
Example 3. To check the accuracy of the code developed for forced vibration, the central deflection of a square, clamped, cross-ply laminate has been obtained using this code.The data used are L = 25 cm, 4 .A 4 × 4 mesh is used.The central deflection due to a suddenly applied step loading q = 10 N/cm 2 , that is imposed at time t = 0 and is not withdrawn, is presented in Fig. 6.Damping  has not been accounted in this analysis.The time step for the explicit integration scheme is taken as 5 microseconds.The results conform satisfactorily to those of Kant [11].
Example 4. The comparative study of mid-point tip deflection of two-fold cantilever folded plate (Fig. 3) with crank angle, β = 120 • is applied for two ply The results are plotted in Fig. 7.A time step of 5 and 10 µs has produced same responses.Here the guideline of Tsui and Tong [20] has been followed to fix the

Conclusion
This paper is concerned with the dynamic analysis of laminated composite cantilever folded plate structures.For the numerical simulation of the problem finite element technique has been used.Various folded plate configurations have been considered in the present study.The results obtained from the present formulation for isotropic folded plate structure are compared with the semi-analytical solutions and finite element results available in the open literature and a good agreement is observed.A set of new results for laminated composite cantilever folded plate structures with various lamination schemes, crank angles, and stacking sequence are presented.Certain differences in free vibration modes of isotropic and composite folded plates have been highlighted.The first order shear deformation theory is found to work well for folded plates where the five regular degrees of freedom is appended with a sixth drilling degree of freedom as suggested in Zienkiewicz [21] and Bathe [1].

Example 5 .Example 6 .
In Fig.8the effect of number of plies is illustrated for single-fold E-glass-Epoxy composite, cantilever folded plates with crank angle α = 120 • .The data used are the same as those in Example 4. The two-ply and four-ply laminates show slight change in response in the second cycle, but the four-ply symmetric and anti-symmetric laminates have produced practi-cally identical results.The six-ply anti-symmetric laminate has been observed to behave identically with the four-ply laminates.In Fig.9the effect of the crank angle is illustrated for two fold E-glass-Epoxy composite, cantilever folded plates with crank angles 90 • , 120 • , and 150 • .The data used are the same as those in Example 4. The 90 • cranked plate shows stiffest behaviour and quickly approaches to attain the state of static de-

Table 1
Results of free vibration analysis of isotropic folded plate

Table 2
Non-dimensional natural frequencies for single fold composite cantilever folded plates

Table 6
Non-dimensional natural frequencies for single fold composite cantilever folded plates