Free vibration characteristics of laminated composite stiffened plates: Experimental and numerical investigation

Journal Pre-proofs Free vibration characteristics of laminated composite stiffened plates: Experimental and Numerical investigation L. Sinha, S.S. Mishra, A.N. Nayak, S.K. Sahu PII: DOI: Reference:

S0263-8223(19)31048-7 https://doi.org/10.1016/j.compstruct.2019.111557 COST 111557

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Composite Structures

Received Date: Revised Date: Accepted Date:

23 March 2019 4 October 2019 9 October 2019

Please cite this article as: Sinha, L., Mishra, S.S., Nayak, A.N., Sahu, S.K., Free vibration characteristics of laminated composite stiffened plates: Experimental and Numerical investigation, Composite Structures (2019), doi: https:// doi.org/10.1016/j.compstruct.2019.111557

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Free vibration characteristics of laminated composite stiffened plates: Experimental and Numerical investigation L. Sinhaa, S. S. Mishraa, A. N. Nayaka*, S. K. Sahub aDepartment

bDepartment

of Civil Engineering, Veer Surendra Sai University of Technology, Burla -768018, Sambalpur, Odisha, India

of Civil Engineering, National Institute of Technology, Rourkela-769008, Odisha, India * Corresponding author. Tel.: +91-9861238403 E-mail address: [email protected]

Abstract This investigation deals with the numerical and experimental study on the free vibration of woven glass fibre laminated composite stiffened plates. Thirty-four stiffened plates fabricated from woven glass fibre and binder (epoxy and hardener) by varying the parameters, such as numbers, types and orientation of stiffeners; depth of stiffener to thickness of plate ratio; and aspect ratio and boundary conditions of plates; are tested in FFT analyser to obtain their natural frequencies. A finite element model is used for vibration of laminated composite stiffened plates for validation of the experimental results. The experimental and numerical results are compared, which shows a very good agreement between them. From this study, it is observed that the above-mentioned parameters significantly influence the fundamental frequency. Since, the experimental investigation on laminated composite stiffened plates is scanty; this study can be considered as a reference for the future work. Key Words: FFT Analyser, Fundamental frequency, Isoparametric element, Laminated Composite, Stiffened Plate, Woven glass fibre fabrics 1. Introduction Stiffened laminated plates are often used in various engineering structures like civil, aerospace and ocean engineering structures because of their high specific stiffness and strength, long fatigue life, excellent thermal characteristics, ease in fabrication, and long durability. Along with the deserved stiffness, elimination or reduction in the vibration 1

response of plate structures can be achieved by changing its natural frequencies by assimilating the stiffeners without increasing the plate thickness altogether to abstain the weight penalty. Aircraft wings and fuselage structures are essentially plates and shells stiffened by longitudinal stringers and transverse or circumferential frames. These structures are usually exposed to dynamic loads during their working period and therefore, the natural frequency and mode shapes which are the dynamic properties, are also of main interest to predict in-service behaviour and resonance frequencies. Hence, a proper study of free vibration analysis is required to exploit these stiffened laminated plates. Extensive research works have been done on isotropic plates with stiffeners by employing various numerical, analytical & semi-analytical methods. Mukherjee and Mukhopadhyay [1] had done a review on dynamic characteristics of stiffened plates. Vibration characteristics of isotropic panels with stiffeners was investigated by various researchers [2-11] employing different models of the finite element method (FEM). Nayak and Bandyopadhyay [12-14] investigated the free and forced vibration behaviour of stiffened conoidal and shallow shells by using FEM. Aksu [15] used finite difference method along with variational method for the free vibration behaviour of stiffened plate. Rayleigh-Ritz method was used by Mizusawa et al. [16] and Bhat [17] for vibration of stiffened plates. Barik and Mukhopadhyay [18] developed a four nodded stiffened isoparametric plate bending element for the modelling of stiffened laminated plates without shear locking phenomena. A semi-analytical finite difference method was used by Mukhopadhyay [19] for the vibration and stability behaviour of rectangular stiffened plates. Xiang and Reddy [20] studied the vibration and buckling analysis of rectangular plates using exact solutions. Olson and Hazell [21] gave a comparative experimental and numerical study of the fundamental frequency of the ribstiffened plates.

2

With the advancement of applications of the laminated composite materials in modern engineering, the attentions of the researchers have been diverted from isotropic stiffened plates to laminated composite ones. Zhang and Yang [22] had done a detail review on analysis of laminated composite plates using FEM. Chattopadhyay et al. [23], Rikards et al. [24] and Prusty and Ray [25] used FEM for the free vibration analysis of stiffened composite plates and shells using first order shear deformation theory. The free vibration of laminated stiffened plates with higher order shear deformation theory was studied by several authors [26-29]. Chandrashekhar and Koli [30] developed a finite element model for the free vibration analysis of laminated stiffened thin and thick plates. Bhar et al. [31] compared the static and vibration analysis of stiffened plates considering first order and higher order shear deformation theory. Nayak and Bandyopadhyay [32] studied free vibration analysis of laminated stiffened shallow shells employing FEM based on the appropriate combination of nine- node isoparametric plate/shell element and three-node isoparametric curved beam element. Chen et al. [33] had done the free vibration analysis of delaminated stiffened composite plates using FEM. Dozio et al. [34] investigated the free vibration characteristics of thin rectangular stiffened plates by combination of analytical & numerical method. Helloty [35] examined the effect of different aspects of stiffener by using FEM with respect to natural frequencies and their modes. Damnjanovic et al. [36] used dynamic stiffness method to study the free vibration analysis of composite stiffened and cracked plates. Qing et al. [37] developed a mathematical model for the free vibration analysis of stiffened composite plates with a semi-analytical solution. Castro and Donadon [38] had investigated the vibration and buckling analysis of deboned composite stiffened panels with semi-analytical approach. It is worth mentioning that the earlier investigators [13, 32, 14, 39, 40, 41, and 42] have used nine node elements with the above first order strain-displacement relationship for the free vibration analysis of isotropic/laminated composite thin plates/shells in their finite element 3

formulations and obtained satisfactory results. Moreover, Le [43] authored a book on vibration of shells and rods which contributes significantly for the formulation of elastic/ piezoelectric shells and rods including low and high frequency. Le [44] reported the formulation of anisotropic piezoelectric shells by using variational asymptotic method. Le and Yi [45] have presented an asymptotically exact theory for the formulation of elasticpiezoceramic sandwich shells using an asymptotically exact two dimensional theory. Similarly, Le [46] reported an asymptotically exact two dimensional theory of functionally graded piezoelectric shells using the variational asymptotic method. Since plate is a special case of shell having very high value of radii i.e. equal to infinity, the above formulation will be immensely helpful for formulating the thin laminated plate in order to get most accurate results. On the other hand, very limited experimental investigation has been conducted on the vibration study of plates without/ with stiffeners. In order to get the actual characteristics of a structure and for the validation of the numerical and analytical results; recently a few experimental investigations on the free vibration of laminated composite bare plates [42, 4750] and shells [51] are studied. Moreover, very few works were carried out on the experimental investigation of free vibration of stiffened plates. Thinh et al. [52] investigated the vibration characteristics of laminated stiffened plates consisted of 6 longitudinal and 9 transverse stiffeners in different boundary conditions experimentally. Thinh et al. [53] investigated the free vibration and bending behaviour of laminated stiffened composite plates made of glass fibre/polyester with different shapes of stiffeners, i.e. rectangular, Tee shaped and hat shaped by experimentally and using FEM with four sides fixed, two sides fixed and two sides’ free boundary conditions. From the review of literature, it is found that there is a vast scope for carrying out experiments for the measurements of natural frequency of laminated stiffened composite 4

plates. Moreover, the previous experimental works considered only the effects of shape of stiffener and boundary conditions. An effort has, therefore, been made to investigate the vibration characteristics of stiffened composite plates experimentally considering the effects of number, types and orientation of stiffeners; depth of stiffener to plate thickness ratio; and aspect ratio and boundary conditions of plates. For the validation of these experimental results, computer code is developed by using the standard finite element model available in the literature. 2. Finite Element Formulation

In this formulation, the stiffened plate surface is discretized into various elements. Every element of the stiffened plate is further considered as the stiffened plate element, which is the appropriate combination of the plate and the beam element if there is the presence of stiffeners in that element. The effects of element stiffness and mass matrices of stiffeners are being lumped at the nodes of the plate element considering proper transformations for eccentricity and the location of the beam elements; thereby it is possible to place the beam element at any convenient position along x- or y-axes with reference to the overall geometry of plate element and nodal configurations. The stiffeners oriented along x-direction are termed as the x-directional stiffeners and those along y-direction as the y-directional stiffeners. The element stiffness matrix and the mass matrix of the stiffened plate element are obtained by combining the nine-node isoparametric plate element and three-node isoparametric beam element. A typical stiffened plate element is shown in Fig. 1. First order shear deformation theory of plates is used in the formulation. 2.1 Plate element The plate on the xy- plane is rectangular of sides a and b with uniform thickness h. The plate thickness is assumed to compose of a composite laminate which, in turn, may consist of a number of thin layers along with fibre orientations (Fig. 2). The Principal material directions 5

are indicated by 1 and 2. The plate element has 5 degrees of freedom (dof) i.e. u, v, w, α and β. Here, along x, y and z axes the displacements are u, v and w, respectively, and along x and y axes, the rotations are α and β, respectively. The isoparametric thin nine-node plate element is considered here as per Nayak and Bandopadhyay [13]. According to the modified Sanders’ first-approximation-theory, the strains {  } are given as follows:



 y  xy  xz  yz T   x0  y0  xy0  xz0  yz0   z k x T

x

ky

k xy

k xz

k yz



T

(1)

Where the first and second vectors of the right hand side represent the mid-surface strains and the curvatures, respectively, and are related to DOFs as





0 x

k x



0 y

ky

0 xy

k xy



k xz

0 xz



k yz



0 T yz



T

 u   x

    x

 y

v y

u v w   y x x

   y x

 0 0 

w    y 

T

T

(2) (3)

The force, moment and shear resultants {F} are obtained from the stresses as follows:

x   Nx    N   y  y   xy   N xy    h2    x z  Mx  F        dz  M y   h  y z  M xy  2  xy z        xz   Q xz     Q yz    yz   

(4)

The normal stresses are represented by x and y along x and y directions and are the shear stresses are represented by xy, xz and yz in yz, xy and xz planes. On simplifying the above equation (4), the following expression for {F} is obtained as follows:

6

 N x   A11 N    y   A12  N xy   A16     M x   B11    M y   B12 M xy   B16     Q xz   0    Q yz   0

A12

A16

B11

B12

B16

A22

A26

B12

B22

A26

A66

B16

B26

B26 B66

B12

B16

D11

D12

D16

B22

B26

D12

D22

B26

B66

D16

D26

D26 D66

0

0

0

0

0

0

0

0

0 0

0 0 0 0 0 0 k 44 Q44 k 45 Q45

0  x    0    y0  0   xy0    0  k x    0  k y   0  k xy    k 45 Q45   xz0   k 55 Q55   yz0 

0

(5)

Where, k44, k45 and k55 are the shear correction factors to account for parabolic variation of the transverse shear strains and each of these is approximately taken as 5/6. Aij, Bij, Dij and Sij are usual extensional, bending-extensional coupling, bending and shear stiffnesses [32] and given as follows: Aij   Qij k  z k  z k 1 ; i, j  1,2,6 nl

(6)

k 1

Bij 





1 nl  Qij  z k2  z k21 ; i, j  1,2,6 2 k 1 k



(7) (8)



1 nl Dij   Qij k z k3  z k31 ; i, j  1,2,6 3 k 1

  z nl

S ij 

Qijk

k

(9)

 z k 1 ; i, j  4,5

k 1

Where, nl is the number of layers of lamina. zk and zk-1 are the distance obtained from the mid-plane of laminate to the bottom of the kth and (k-1) th lamina, respectively and (Qij) k is the element corresponding to the ith row and jth column of the off-axis elastic constant matrix for the kth lamina. The Eq. (5) can be written as:

F   D 

(10)

Where, [D] is the elasticity matrix.

7

By implementing the methodology of Nayak and Bandopadhyay [13], the element stiffness

 

matrix K pe



and mass matrix M pe



of unstiffened plate element obtained based on

minimum potential energy are as follows: 1 1

K     B DB J dd

(11)

T

pe

1 1

1 1

(12)

M   N mN J dd T

pe

11

Where,  and  are the local natural co-ordinates, B is the local derivatives of shape functions, [N] and [m] are the matrices representing shape function and inertia respectively, J is the Jacobian matrix. 2 x 2 reduced numerical integration with Gaussian quadrature is

applied to obtain the element stiffness and mass matrices in order to avoid shear locking. 2.2 Beam Element A beam of rectangular section having width, bst and depth, dst and made of laminated composite material is considered for the stiffeners. The stiffeners are oriented along x- or yaxes. The x- stiffener and y- stiffener elements have four dof, i.e. usx, wsx, αsx and βsx and vsy, wsy, αsy and βsy, respectively. The strain-displacement equations of the x-/y-direction stiffeners can be derived from the generalized displacement fields of the respective stiffener. The constitutive relation for the x-stiffeners is,

F   D    D B   sx

sx

sx

(13)

sx

sx

sxi

Where the force resultants of x-stiffeners are given as follows:

F   N sx

sx x

M

sx x

T

sx x

Q xzsx



(14)

T

 

and Bsx  is the local derivative of the shape function, D sx is the elasticity matrix and  sxi  is the nodal displacement of the x-stiffener elements. 8

 

The force resultants F sx of x-stiffeners are expressed as:

 N xsx   A b  sx   B b M x    sx    sx  Tx   B16 bst  Q xzsx    0 sx 11 st sx 11 st

sx 11 st sx 11 st

B b D b



sx 16 st sx 16 st

B b D b

 u sx    x 0   sx    0    x     0 sx   x    K 55sx A44sx   wsx   sx  x 



1 sx Q44  Q66sx d st bst3 6 0

D16sx bst 0

(15)

Where, nlx

(16)

Aijsx   (Qijsx ) k z k  z k 1 ; i, j  1,2,6 k 1





(17)

1 nlx sx (Qij ) k z k3  z k31 ; i, j  1,2,6  3 k 1





(18)

nlx

(19)

Bijsx 

1 nlx sx (Qij ) k z k2  z k21 ; i, j  1,2,6  2 k 1

Dijsx  S

sx ij

  (Qijsx ) k  z k  z k 1 ; i, j  4,5 k 1

Where, nlx is the number of layers of lamina and Qijsx is the off-axis elastic constants of a lamina of the x-direction stiffeners. Aijsx , Bijsx , Dijsx and Sijsx are usual extensional, extensionalbending coupling, bending and shear stiffnesses [32]. N xsx , M xsx Txsx and Qxzsx are usual axial force, moment, torsion and shear resultants for x- stiffener element. Similarly, the force-strain relation can also be calculated for the y- stiffeners. The stiffness and mass matrix of x-and ystiffeners are found by using the procedure used by Nayak and Bandyopadhyay [13]. The effect of the eccentricity, esx of the x-directional stiffener is considered and the nodal dof

 sxi  of the x- stiffener are transformed to the plate dof  i p  as follows:  u sx  1  w  0  sx      sx  0   sx  0

0 0 0 0

0 esx 1 0 0 1 0 0

u p  0  p  v 0  p   w  , i  1,2,3 0  p    1 i  p    i

(20)

9

The equation (20) can be compacted in the form of:

 sxi   T sx  i p 

(21)

The element stiffness matrix [Ksxe] and mass matrix [Msxe] of the x-stiffener elements are derived by using minimum potential energy principle and are presented below:

K sxe    T

T

1

sx

 B  D B T  J d sx T

sx

sx

(22)

sx

1

M sxe     T sx  N sx T m sx N sx T sx  J d 1

(23)

T

1

N  , m  and T  are the shape function matrix, inertia matrix and the transformation sx

sx

sx

matrix of the x-stiffener respectively. The transformation matrix takes care of the arbitrary position of the x-directional stiffener in the plate element without increasing the degrees of freedom as given by Palani et al. [4]. Similarly, the element matrices of the stiffener element in y-direction can also be obtained. The element matrices are obtained in local natural co-ordinate form applying 2x2 reduced integration of Gaussian quadrature to avoid shear locking. Finally, the element stiffness matrix [Ke] and element mass matrix [Me] of the stiffened plate element are obtained by adding the respective matrices of plate element, the x direction stiffener element and y direction stiffener element and are given as follows: [Ke]= [Kpe] + [Ksxe] + [Ksye]

(24)

[Me]= [Mpe] + [Msxe] + [Msye]

(25)

2.3 Solution procedure The structural stiffness matrix K and mass matrix M is obtained by assembling the element stiffness and mass matrices. The equation of free vibration of stiffened composite plate without damping is presented in the eigen value form: 10

K    M d   0

(22

)

(26)

2

Where,  represents the natural frequency and d represents the mode shape. A subspace iteration technique is used to solve Eq. 26 to compute the natural frequency. Then, in presence or absence of the generalized displacements u, v, w,  and  at various nodes of the discretized structure the boundary conditions are applied. The zero displacement boundary conditions are incorporated by deleting the corresponding terms from global matrices and vectors. 3. Experimental Programme The experimental investigation is performed to obtain the natural frequencies of laminated glass fibre reinforced polymer (GFRP) stiffened plates. The composite plates are fabricated with GFRP woven fabrics with epoxy resin. In this section, the details of fabrication of samples, determination of the material properties and free vibration testing procedure are presented. 3.1 Fabrication of samples The materials used for casting of the laminated composite plates are glass fibre woven fabric, Epoxy, Hardener, polyvinyl alcohol. Under room temperature, plates are fabricated by using hand layup technique. The fabrication consists of glass fibre and epoxy with the ratio of weight of fibre and epoxy as 50:50 with 00/900 orientation of fibre. The laminates contain 8 layers of each epoxy and woven roving glass fibre identically. For fabrication of laminates woven roving glass fibbers were cut into required dimension. The binding material was prepared by adding 92% epoxy resin with 8% hardeners. By using contact moulding, the combination of plies is done with required sequence. A plastic sheet is kept on the rigid surface for mould releasing purpose and for releasing agent silicon spray or polyvinyl alcohol is applied. A brush is used to apply the gel coat (epoxy and hardener) to the plastic sheet and 11

then the fibre is placed, and then lamination starts. By the help of a steel roller, the entrapped air is removed. The outer layer of the laminated plate with epoxy and hardener coat (gel coat) provides a smooth external surface which protects it from direct contact to the external exposure conditions. This method is repeated up to obtaining eight stacking plies. Lastly another plastic sheet is placed over the laminate. Thereafter flat plywood is kept over it. A load is applied by keeping the concrete cubes over it for three days at room temperature for better bonding between the layers of GFRP fabrics. The mould is released from the sheet after proper curing and cut into required shape with the help of diamond cutter. The stages of fabrication of composite plates are given in Fig. 3. The typical laminated composite stiffened plates with varying number and orientations are shown in Fig. 4. 3.2 Determination of material properties The material properties of the GFRP laminate are obtained by conducting tensile coupon tests as per ASTM specifications [58]. For testing purpose, specimens are cut in longitudinal direction and 45° inclined to this direction, as specified in ASTM specifications. The dimensions of the coupon specimen are furnished in Table 1. The specimens are cut by size according to ASTM standard and the thickness of specimen was measured with the help of screw gauge. The coupon is tested in the universal testing machine INSTRON 3382 for measuring the Young’s modulus as per ASTM specifications [58] and is shown in Fig. 5. The applied loading rate is 0.2 mm/minute. The typical stress-strain curve of the [0/90]2s coupon specimens considered is shown in Fig.6. The test results of GFRP coupon specimens are presented in Table 2. The proportion of lateral strain to longitudinal stain is taken for the calculation of Poisson’s ratio directly by using bidirectional strain gauge. Shear modulus is calculated according to Jones [60] formula as given below:

G12 

1

2 4 1 1    12 E 45 E11 E 22 E11

(27) 12

Where, E11, E22 = Elastic moduli of laminated plate and stiffener along 1- and 2-directions E45 = Elastic modulus in 45-degree orientation ν12, ν13, ν23= Poisson’s ratios of laminated plate and stiffener G12, G13, G23= Shear moduli of laminated plate

 = density of laminated plate and stiffener The material properties of the specimen tested in universal testing machine (INSTRON 3382) are presented in Table 3. 3.3 Free vibration test The free vibration test of the GFRP laminated stiffened plates are carried out with the help of FFT analyser, accelerometer, impact hammer, laptop and pulse software. The mechanical assembly for the vibration test is shown in Fig. 7. First, the stiffened laminated composite plate has been fixed in the fabricated frame as per the specific boundary condition, such as clamped (CCCC), simply supported (SSSS) and cantilever (CFFF). The typical set ups for the above boundary conditions are presented in Fig. 8. The FFT analyzer is connected with the transducers, cables, laptop and modal hammer. Pulse lab shop is the platform used in the computer to search the information from the FFT Analyzer and generate the required Frequency response. For frequency response function (FRF), the modal hammer is struck five times at a solitary point on the specimen. The function of accelerometer, which is attached to the plate by the help of bees wax, is to receive the response when the excitation has been made on it. Then, it sends the response to the FFT analyser. FFT Analyser is utilized to get a time varying signal from the accelerometer and convert it into a frequency domain signal. FFT analyzer is associated with a computer platform in which the pulse software and is used to obtain the average output from the analyzer. Various response curves are plotted on the 13

screen of the laptop and then the FRFs are specifically estimated. A usual FRF (pulse report) measurement from FFT analyser is carried out which is shown in Figs 9 (a), (b). Fig. 9 (a), shows the frequencies of various modes of vibrations which are measured from the corresponding peaks of FRF and Fig. 9(b), gives an idea about the exactness of the measurement by the coherence of nearly equal to 1. Therefore, after obtaining the coherence of nearly equal to 1, the frequencies of various laminated stiffened plates are recorded. 4. Results and discussions In this section, the fundamental frequencies obtained from the experimental and numerical investigations for the GFRP laminated stiffened plates with varying numbers of stiffeners, stiffeners types and orientations; depth of stiffener to plate thickness ratio; aspect ratio and boundary conditions of plate are presented to study their behaviour. The fundamental frequency of the stiffened plate is expressed in the form of non-dimensional fundamental frequency (ωnd), which is given as follows. ωnd =b2 ( / E22 h2) ½

(28)

Moreover, the accuracy of the present FEM code is established by performing the convergence studies and validation of the present results with the previous results reported in the literature. Thereafter, parametric study is presented to understand the effects of various parameters on the frequencies of vibration of stiffened plates. 4.1 Convergence studies The convergence study of the FEM formulation is done for the laminated composite plate stiffened with 1 x 1 orthogonal stiffeners with respect to mesh size. The variation of ωnd versus mesh size of the problems has been presented in Fig. 10 for both CCCC and SSSS boundary conditions. From the Fig. 10, it is shown that the value of frequency converges at mesh sizes of 6 x 6 and 8 x 8 for SSSS and CCCC boundary conditions, respectively. Hence, 14

these mesh sizes are considered for obtaining the ωnd values of the stiffened plate for further study. 4.2 Validation of present formulation For the validation of the computer code developed for the free vibration of laminated composite stiffened plates, the following examples are considered to compare the present results to those obtained by the earlier investigators. Non-dimensional fundamental frequencies [ωnd =b2 ( / E22h2) ½] of symmetric SSSS cross-ply square laminated bare plate obtained from the present code and the existing literature [41, 54-57] are indicate in Table 4. It is evident from Table 4 that the frequencies of vibration match closely with those of the previous studies. Thereafter, the natural frequency of laminated composite stiffened plates with central x-direction stiffener having clamped boundary condition are obtained by the present study and compared with the earlier investigation [59] as indicated in Table 5. The present results match closely with the results of earlier investigators validating the present formulation. 4.3 Parametric study In this section, a set of typical laminated stiffened plates is considered for both experimental and numerical investigations for parametric study with respect to number, type and orientation of stiffeners; aspect ratio; and stiffener depth to plate thickness ratio; and boundary conditions of the plates. Numbers of stiffeners are considered as 3. The dimensions of the laminated stiffened plates are given below. Length (a) = width (b) =235mm, thickness (h) =2.8mm, a/b=0.5,1,1.5 and 2. bst/h =4 and dst/h=2, 4, 6, 8 for section 4.3.4 and for the other sections dst/h=6. Where, bst and dst are the stiffener width and depth respectively. In this study, b, h and bst are kept constant. The other parameters are varying for the above parametric study. 15

4.3.1 Effect of numbers of Stiffeners The experimental and numerical non-dimensional fundamental frequencies (ωnd) for laminated composite stiffened plates are obtained for CCCC boundary conditions with four different aspect ratios (a/b), i.e., 0.5, 1, 1.5 and 2 and are shown in Figs. 11 and 12 for varying number of stiffeners i.e. x direction stiffeners and orthogonal stiffeners, respectively. From both the Figs.11and 12, it is shown that, the experimental and FEM results are in good agreement. From Figs.11 and 12, it is seen that ωnd obtained from both the experimental and numerical investigations increase with the increase in number of x-direction and orthogonal stiffeners. It is worth mentioning that the percentage increase in the value of ωnd is higher when the number of x-direction stiffeners and orthogonal stiffeners increase from 1 to 2 in comparison to the increase in stiffeners from 2 to 3. This may be due to the fact that, as the number of stiffeners increases, the stiffness of plate will increase and therefore the frequency increases. The frequency of a structure is influenced by stiffness and mass of that structure in an interactive manner. When a stiffener is added to a plate, the stiffness and mass both increases but the increase in mass is independent of the position of the stiffener where as the increase in stiffness is very much influenced by the stiffener position. Therefore, the frequency of the plate increases or decreases depending on the increase in stiffness K or mass [M] matrices due to the addition of stiffeners in an interactive manner, as given in Eq. (26). 4.3.2 Effect of Types of Stiffeners The variations of experimental and numerical values of ωnd of laminated composite plates with a/b ratio of 0.5, 1, 1.5 and 2 for different types of x-direction central stiffeners (eccentric top and bottom, concentric) are shown in Fig.13. It is shown that for rectangular plates, the eccentric top stiffener shows highest values of ωnd followed by eccentric bottom and concentric stiffeners. It may be due to the fact that, the first and second moments of the area 16

of the stiffeners of plates stiffened with eccentric stiffeners about the centroidal axis of the laminated plate are more than those with concentric stiffeners. Therefore, the laminated stiffened plates with eccentric type of stiffeners are showing better performance in increasing the values of ωnd in comparison to that of the concentric ones. But for the square plate, the eccentric top and bottom types of stiffener shows same value of ωnd. The variation of the experimental values of ωnd of concentric and eccentric (top and bottom) types of stiffeners of laminated composite stiffened plate with aspect ratio of 1.5 having two different orientations (along x or y directions) with the three modes are shown in Fig. 14. It is shown that the eccentric top x- and y-direction stiffeners show the maximum improvement in the value of ωnd followed by eccentric bottom and concentric types x- and y-direction of stiffeners. Hence, for all the orientation of the stiffeners, the eccentric stiffeners show superior performance in comparison to concentric stiffeners. 4.3.3 Effect of orientations of Stiffeners The behaviour of the eccentrically stiffened plates along x- and orthogonal directions, are investigated by considering examples of plates with stiffeners for different aspect ratio (a/b) of 0.5, 1, 1.5 and 2 are considered. The variation of the frequencies of first three modes (experimental) with aspect ratio for two numbers of x-direction stiffeners and orthogonal stiffeners are shown in Fig. 15. It is observed that for the aspect ratio of 0.5, 1.5 and 2 i.e. rectangular plates, the plates stiffened with cross stiffeners show higher frequencies of vibration in comparison to the plates with same number of x-direction stiffeners. i.e. 1x1 stiffened plates have higher value of ωnd than stiffened plates with two numbers of xdirection stiffeners. In the case of the plates with aspect ratio 1.5 and 2, the x-direction stiffeners are treated as long span stiffeners. For plate with aspect ratio 0.5, the x-direction stiffeners are considered as short span stiffeners. Higher frequencies are observed for plates

17

with aspect ratios 1, i.e. for eccentrically stiffened square plate along with x-direction stiffeners in comparison to orthogonal direction stiffeners. 4.3.4 Effect of depth of stiffener to thickness of plate (dst/h) ratio The experimental and numerical values of ωnd for laminated composite stiffened plates (a/b=1) are obtained for various values of (dst/h) ratio i.e.2, 4, 6 and 8 with increasing number of eccentric stiffeners in x- direction and orthogonal direction and are shown in Figs. 16 and 17, respectively. It is seen that as the (dst/h) ratio increases, the experimental and numerical values of ωnd of the laminated composite stiffened square plate increases for both x- / y- and orthogonal direction stiffeners. From Fig. 16, the percentage increase in the value of ωnd is 46.3%, 13.4% and 8.35% with the increase in dst/h ratio from 2 to 4, 4 to 6 and 6 to 8 respectively for single stiffener. For two stiffeners, the percentage increase in the value of ωnd is 56 %, 23 % and 24 % when the dst/h ratio increases from 2 to 4, 4 to 6 and 6 to 8, respectively. For 3 stiffeners, the percentage increase in the value of ωnd is 53 %, 23 % and 27 % when the dst/h ratio increases from 2 to 4, 4 to 6 and 6 to 8, respectively. It can be shown from Fig.17 that the percentage increase in the value of ωnd is significant when the dst/h ratio increases from 2 to 4 but the increase rate of ωnd value decreases with the increase in dst/h ratio from 4 to 6 and 6 to 8 in case of orthogonal stiffeners. Since the plate thickness remains constant, the value of thickness of stiffener increases with increasing dst/h ratio. As the stiffness of the plate with stiffeners increase, the frequencies also increase. It is found that ωnd of the stiffened plate increases at a faster rate with the increase of dst/h ratio at the initial stage and thereafter, the rate of increase of ωnd decreases with the increase in the value of dst/h. It is worthy to note that these findings will be used to decide the stiffener depth for a particular number and a specific plate thickness to obtain the required ωnd. Similarly, for definite values of both dst and h, the optimum number of stiffeners can be determined to obtain the required value of ωnd. 18

4.3.5 Effect of aspect ratio of plate The variations of the experimental and numerical values of ωnd for the laminated plates with increase in number of x-direction stiffeners and orthogonal stiffeners for various aspect ratios (0.5, 1.0, 1.5 and 2.0) are shown in Figs. 11 and 12, respectively. Similarly, the variations of the experimental and numerical values of ωnd with aspect ratio for different types of stiffeners are shown in Fig. 13. The variation of the experimental values of ωnd with aspect ratio for stiffeners with different orientations (x- and orthogonal) are shown in Fig. 15. The variations of the experimental and numerical values of ωnd with aspect ratio for different boundary conditions (CCCC, SSSS, and CFFF) are shown in Fig. 18. It is found that the values of ωnd decrease with increase in aspect ratio. By comparing aspect ratios, the laminated composite stiffened plates with aspect ratio of 0.5 are most efficient followed by aspect ratios 1.0, 1.5 and 2.0 by the provision of x-direction and orthogonal stiffeners. Therefore, the short and wide plates (aspect ratio=0.5) with stiffeners in both directions show the good performance in comparison to long and narrow (aspect ratio=1.5 and 2.0) plates stiffened with corresponding stiffeners. 4.3.6 Effect of different boundary conditions The experimental and numerical values of ωnd for laminated composite stiffened plates with aspect ratios (0.5,1.0,1.5 and 2.0) and dst/h ratios (2,4,6 and 8) are obtained for CCCC, SSSS, CFFF boundary conditions and are shown in Figs. 18 and 19, respectively. It is seen that the both the experimental and numerical values of ωnd of the laminated composite stiffened plates at CCCC boundary conditions show higher values as compared to SSSS and CFFF boundary conditions. CFFF condition shows less value as compared to other two conditions as expected. Therefore, the laminated composite stiffened plates in CCCC boundary conditions are preferred over SSSS and CFFF boundary conditions.

19

5. Conclusions On the basis of the above experimental and numerical investigation on the vibration characteristics of laminated stiffened plates, the conclusions obtained are as follows; 1. The results obtained from the experimental investigation and numerical model developed using FEM are in good agreement for all parameters considered. 2. Non-dimensional fundamental frequencies of the stiffened composite plates increase with the increase in number of stiffeners in any orientation as well as with increase in dst/h ratio. 3. The enhancement in the frequency is maximum for short and wide (a/b = 0.5) plates, followed by square (a/b = 1.0) plates and then the long and narrow (a/b = 1.5, 2.0) plates for clamped, simply supported and cantilever boundary conditions. As expected, the clamped stiffened plates give higher frequency in comparison to the simply supported and cantilever ones due to restraint at the edges. 4. For rectangular plates, the plate stiffened with orthogonal stiffeners shows higher fundamental frequency compared to the plate with same number of x-direction stiffeners. But in the case of square plate, the x-direction stiffeners show higher fundamental frequency than the orthogonal stiffeners. 5. The eccentric stiffeners show superior performance in comparison to the concentric stiffeners for all aspect ratios and orientations. 6. The present parametric study reveals the free vibration behaviour of laminated composite stiffened plates, which will be helpful in the primary selection of the number, types and orientation of stiffeners, boundary conditions (clamped, simply supported and cantilever) and aspect ratio and depth of stiffener to thickness of plate ratio.

20

The findings will build the confidence in the designers using FEM codes for designing the laminated stiffened plates. The experimental results will also serve as benchmark for future investigators for dynamic analysis of woven fibre composite stiffened plates. Data Availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. References 1. Mukherjee A, Mukhopadhyay M. A review of dynamic behaviour of stiffened plates 1986. 2. Mukherjee A, Mukhopadhyay M. Finite element free vibration of eccentrically stiffened plates. Comput Struct 1988; 30:1303–17. 3. Koko TS, Olson MD. Vibration analysis of stiffened plates by super elements. J Sound Vib 1992; 158:149–67. 4. Palani GS, Iyer NR, Rao TA. An efficient finite element model for static and vibration analysis of eccentrically stiffened plates/shells. Comput Struct 1992; 43:651–61. 5. Harik IE, Guo M. Finite element analysis of eccentrically stiffened plates in free vibration. Comput Struct 1993; 49:1007–15. 6. Chen CJ, Liu W, Chern SM. Vibration analysis of stiffened plates. Comput Struct 1994; 50:471–80. 7. Holopainen TP. Finite element free vibration analysis of eccentrically stiffened plates. Comput Struct 1995; 56:993–1007. 8. Hamedani SJ, Khedmati MR, Azkat S. Vibration analysis of stiffened plates using finite element method. Lat Am J Solids Struct 2012; 9:1–20.

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9. Yadav D, Sharma A, Shivhare V. Free vibration analysis of isotropic plate with stiffeners using finite element method. Eng Solid Mech 2015; 3: 167–76. 10. Siddiqui H, Shivhare V. Free vibration analysis of eccentric and concentric isotropic stiffened plate using ANSYS. Eng Solid Mech 2015; 3:223–34. 11. Nayak AN, Satpathy L, Tripathy PK. Free vibration characteristics of stiffened plates. Int J Adv Struct Eng 2018:1–15. 12. Nayak AN, Bandyopadhyay JN. Free vibration analysis and design aids of stiffened conoidal shells. J Eng Mech 2002; 128:419–27. 13. Nayak AN, Bandyopadhyay JN. On the free vibration of stiffened shallow shells. J Sound Vib 2002; 255:357–82. 14. Nayak AN, Bandyopadhyay JN. Dynamic response analysis of stiffened conoidal shells. J Sound Vib 2006; 3:1288–97. 15. Aksu G. Free vibration analysis of stiffened plates by including the effect of in plane inertia. J Appl Mech 1982; 49:206–12. 16. Mizusawa T, Kajita T, Naruoka M. Vibration of stiffened skew plates by using Bspline functions. Comput Struct 1979; 10:821–6. 17. Bhat RB. Vibrations of panels with non-uniformly spaced stiffeners. J Sound Vib 1982; 84:449–52. 18. Barik M, Mukhopadhyay M. A new stiffened plate element for the analysis of arbitrary plates. Thin-Walled Struct 2002; 40:625–39. 19. Mukhopadhyay M. Vibration and stability analysis of stiffened plates by semianalytic finite difference method, part I: consideration of bending displacements only. J Sound Vib 1989; 130:27–39. 20. Xiang Y, Reddy JN. Buckling and vibration of stepped, symmetric cross-ply laminated rectangular plates. Int J Struct Stab Dyn 2001; 1:385–408.

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21. Olson MD, Hazell CR. Vibration studies on some integral rib-stiffened plates. J Sound Vib 1977; 50:43–61. 22. Zhang YX, Yang CH. Recent developments in finite element analysis for laminated composite plates. Compos Struct 2009; 88:147–57. 23. Chattopadhyay B, Sinha PK, Mukhopadhyay M. Finite element free vibration analysis of eccentrically stiffened composite plates. J Reinf Plast Compos 1992; 11:1003–34. 24. Rikards R, Chate A, Ozolinsh O. Analysis for buckling and vibrations of composite stiffened shells and plates. Compos Struct 2001; 51:361–70. 25. Prusty BG, Ray C. Free vibration analysis of composite hat-stiffened panels by method of finite elements. J Reinf Plast Compos 2004; 23:533–47. 26. Iyengar NGR, Shankara CA. Free vibration studies including transverse shear on a class of laminated plates. Struct Vib Acoust 1989:145–51. 27. Ghosh AK, Biswal KC. Free-vibration analysis of stiffened laminated plates using higher-order shear deformation theory. Finite Elem Anal Des 1996; 22:143–61. 28. Hernandes JA, Almeida SFM, Nabarrete A. Stiffening effects on the free vibration behavior of composite plates with PZT actuators. Compos Struct 2000; 49:55–63. 29. Sadek EA, Tawfik SA. A finite element model for the analysis of stiffened laminated plates. Comput Struct 2000; 75:369–83. 30. Chandrashekhara K, Kolli M. Free vibration of eccentrically stiffened laminated plates. J Reinf Plast Compos 1997; 16:884–902. 31. Bhar A, Phoenix SS, Satsangi SK. Finite element analysis of laminated composite stiffened plates using FSDT and HSDT: A comparative perspective. Compos Struct 2010; 92:312–21. 32. Nayak AN, Bandyopadhyay JN. Free vibration analysis of laminated stiffened shells. J Eng Mech 2005; 131:100–5.

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33. Chen H, Wang M, Bai R. The effect of nonlinear contact upon natural frequency of delaminated stiffened composite plate. Compos Struct 2006; 76:28–33. 34. Dozio L, Ricciardi M. Free vibration analysis of ribbed plates by a combined analytical–numerical method. J Sound Vib 2009; 319:681–97. 35. El-Helloty A. Free Vibration Analysis of Stiffened Laminated Composite Plates. Int J Comput Appl 2016; 156:975–8887. 36. Damnjanović E, Marjanović M, Nefovska-Danilović M. Free vibration analysis of stiffened and cracked laminated composite plate assemblies using shear-deformable dynamic stiffness elements. Compos Struct 2017; 180:723–40. 37. Qing G, Qiu J, Liu Y. Free vibration analysis of stiffened laminated plates. Int J Solids Struct 2006; 43:1357–71. 38. Castro SGP, Donadon M V. Assembly of semi-analytical models to address linear buckling and vibration of stiffened composite panels with debonding defect. Compos Struct 2017; 160:232–47. 39. Pandit MK, Haldar S, Mukhopadhyay M. Free vibration analysis of laminated composite rectangular plate using finite element method. J Reinf Plast Compos 2007; 26:69–80. 40. Kalita K, Haldar S. Free vibration analysis of rectangular plates with central cutout. Cogent Eng 2016; 3:1163781. 41. Kalita K, Ramachandran M, Raichurkar P, Mokal SD, Haldar S. Free vibration analysis of laminated composites by a nine node isoparametric plate bending element. Adv Compos Lett 2016; 25:096369351602500501 42. Mandal A, Ray C, Haldar S. Free vibration analysis of laminated composite skew plates with cut-out. Arch Appl Mech 2017; 87:1511–23. 43. Le KC. Elastic rods. Vib. Shells Rods, Springer; 1999, p. 311–48.

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44. Le KC. The theory of piezoelectric shells. J Appl Math Mech 1986; 50:98–105. 45. Le KC, Yi J-H. An asymptotically exact theory of smart sandwich shells. Int J Eng Sci 2016; 106:179–98. 46. Le KC. An asymptotically exact theory of functionally graded piezoelectric shells. Int J Eng Sci 2017; 112:42–62. 47. Rath MK, Sahu SK. Vibration of woven fibre laminated composite plates in hygrothermal environment. J Vib Control 2012; 18:1957–70. 48. Mohanty J, Sahu SK, Parhi PK. Numerical and experimental study on free vibration of delaminated woven fibre glass/epoxy composite plates. Int J Struct Stab Dyn 2012; 12:377–94. 49. Panda HS, Sahu SK, Parhi PK. Hygrothermal effects on free vibration of delaminated woven fibre composite plates–numerical and experimental results. Compos Struct 2013; 96:502–13. 50. Prasad E V, Sahu SK. Vibration Analysis of Woven Fibre Metal Laminated Plates— Experimental and Numerical Studies. Int J Struct Stab Dyn 2018; 18:1850144. 51. Biswal M, Sahu SK, Asha A V. Vibration of composite cylindrical shallow shells subjected to hygrothermal loading-experimental and numerical results. Compos Part B Eng 2016; 98:108–19. 52. Thinh TI, Khoa NN. Free vibration analysis of stiffened laminated plates using a new stiffened element. Tech Mech 2008; 28:227–36. 53. Thinh TI, Quoc TH. Finite element modeling and experimental study on bending and vibration of laminated stiffened glass fibre/polyester composite plates. Comput Mater Sci 2010; 49: S383–9. 54. Xiang S, Chen YT. An Analytical Solution of Natural Frequency for Symmetric laminated composite plates. Adv. Mater. Res., vol. 785, Trans Tech Publ; 2013, p.

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253–6. 55. Ferreira AJM, Roque CMC, Jorge RMN. Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions. Comput Methods Appl Mech Eng 2005; 194:4265–78. 56. Liew KM, Huang YQ, Reddy JN. Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method. Comput Methods Appl Mech Eng 2003; 192:2203–22. 57. Reddy JN. Mechanics of composite plates–theory and analysis. Chem Rubber Company, Boca Raton, FL 1997 58. ASTM Standard: D3039/D3039M-08: standard test method for tensile properties of Polymer matrix composite materials; 2008 59. Ahmadian MT, Zangeneh MS. Application of super elements to free vibration analysis of laminated stiffened plates. J Sound Vib 2003; 5:1243–52. 60. Jones RM. Mechanics of composite materials. New York: McGraw-Hill; 1975.

26

Figures

Fig.1. Typical laminated composite stiffened plate

(a)

(a)

(b)

Fig.2. A typical laminated plate (a) Details of lamination (b) Fibre orientation of a lamina

27

(a)

(b)

(e)

(c)

(f)

(d)

(g)

Fig.3. Stages of fabrication of stiffened composite plates: (a)Cutting of glass fibre(b)Application of releasing agent on mould releasing sheet (c)Application of primary gel coat (d)Placing of fibre (e)Removal of entrapped air by a steel roller (f)Placing of mould releasing sheet on top of plate(g) Placing of concrete cubes for compression purpose

(a)

(d)

(b)

(e)

28

(c)

(f)

Fig.4.Typical GFRP laminated plate stiffened with different stiffeners (a) x-direction central stiffener (b) y-direction central stiffener (c) three numbers of x-directional stiffeners (d) 1 x 1 orthogonal stiffener (e) 2 x 2 orthogonal stiffeners (f) 3 x 3 orthogonal stiffeners

Fig.5.Experimental set up for the testing of coupons in the universal testing machine (INSTRON 3382)

Fig.6. The typical stress-strain curve of the coupon [0/90]2s specimens obtained from the INSTRON 3382

29

Fig.7. Mechanical assembly used for vibration test (a) Display Unit (b) Impact hammer (2302-5) (c) Accelerometer (4507) (d) FFT Analyser (Bruel and Kajer-3560)

(a)

(b)

(c)

Fig.8. Fixation of typical GFRP laminated stiffened plate in (a) Clamped (b) Simply supported (c) Cantilever boundary conditions

30

(a)

(b) Fig.9. FRF measurement obtained from FFT analyser (a) Frequency–Response (b) Coherence of test specimen

31

Fig.10.Convergence study of the Clamped and simply supported stiffened laminated composite plate element with stiffeners in orthogonal orientation [a/b =1, bst/h = 4, dst/h = 6, ν12= ν13=ν23=0.25, E11 = E22=16.07 × 109 N/mm2, G12 = G13= G23= 2.814 × 109 N/mm2, ρ = 1664 kg/m3, b = 0.235m, ωnd =b2 ( / E22 h2) ½]

Fig.11.Variations of ωnd of laminated composite stiffened plates with numbers of x-direction stiffeners

32

Fig.12.Variations of ωnd of laminated composite stiffened plates with numbers of stiffeners in orthogonal direction

Fig.13.Variations ofωnd of laminated composite stiffened plates with aspect ratio for different types (eccentric top, eccentric bottom and concentric) of x-direction central stiffener

33

6.0

Eccentric top(x direction stiffener) Eccentric bottom(x direction stiffener) Concentric (x direction stiffener) Eccentric top(y direction stiffener) Eccentric bottom(y direction stiffener) Concentric (y direction stiffener)

Non Dimensional fundamental frequency

5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1

2

3

Mode Number

Fig.14.Variations of ωnd of laminated composite stiffened plate (a/b=1.5) with respect to modes with various types of stiffeners are in x and y direction

Fig.15.Variations of ωnd of laminated composite stiffened plates obtained from experimentation with respect to aspect ratio for two numbers of x direction stiffeners and orthogonal stiffeners

34

Fig.16.Variations of ωnd of square stiffened plates with dst/h ratio, for varying numbers of x / y- direction stiffeners

Fig.17.Variations of ωnd of square stiffened plates with dst/h ratio for varying number of orthogonal stiffeners

35

Fig.18.Variations of experimental and numerical values of ωnd of stiffened plates with aspect ratio for CCCC, SSSS and CFFF boundary conditions

Fig.19.Variations of experimental and numerical values of ωnd of square stiffened plates with varying dst/h ratio for different boundary conditions

36

Tables Table1 Cross-section of the coupon specimen for tensile test Specimen Ply orientation Ply layer Overall Length(mm) Width (mm) Thickness(mm) 1

0/90

8

250

25

2.8

2

+45/-45

8

250

25

2.8

Table 2 Test results of GFRP Coupons Specimen Sp. Max Stress No. Load at s (KN) Max Load (MPa)

(0/90)2s

(45/-45)2s

1

18.51

263.62

% strain at Max Load (%) 2.30

Load at Break (KN)

Stress 0.2% Yield (MPa)

Strain at Break (%)

Extensi on at Break (mm)

Young's Modulu s (GPa)

18.38

14.62

2.31

3.46

15.04

2

21.38

304.56

2.48

21.38

16.21

2.48

3.72

16.78

3

21.32

303.65

2.54

21.29

15.83

2.55

3.83

16.40

Mean

20.40

290.61

2.44

20.35

15.55

2.45

3.67

16.07

1

8.09

110.09

7.99

6.432

46.04

18.64

36.533

9.371

2

8.85

119.55

7.75

6.681

50.04

11.51

23.027

9.664

3

4.66

62.96

5.81

2.484

20.08

9.31

18.627

7.710

Mean

7.2

97.53

7.18 3

5.199

38.72

13.153

26.062

8.915

Table 3 Material properties of glass/epoxy laminated composite E11 (GPa)

E22 (GPa)

E45 (GPa)

G12 = G13 = G23 (GPa)

ν12, ν13, ν23

ρ (kg/m3)

16.07

16.07

8.915

2.814

0.25

1664

37

Table 4 Comparison of the non-dimensional fundamental frequency (ωnd) of simply supported symmetric cross-ply (0o/90o/0o) square plates Author

Methods

a/h 10

100

Xiang [54]

Analytical

14.776

18.830

Ferreira [55]

Radial basis functions method

14.804

18.355

Reddy [57]

Exact

14.767

18.891

Liew et al [56]

Moving least squares differential quadrature method

14.767

18.769

Kalita et al [41]

FEM

14.772

18.83

Present

FEM

14.767

18.831

Note: a/b = 1, b = 10.0m, E11/ E22=40, G12 = G13 =0.6 E22, G23 =0.5 E22, ν12=0.25, ρ =1 ωnd =b2 ( / E22 h2) ½ Table 5 Comparison of the natural frequencies (Hz) of clamped laminated composite plate with xdirection stiffener Method

Mode

Ahmadian and Zangeneh [59] Super-element ANSYS [00] [00]

Present

Ahmadian and Zangeneh [59] Present

FEM (N9) [00]

Superelement [900]

ANSYS [900]

FEM (N9) [900]

1

1357.3

1324.1

1297.6

891.4

888.5

854.4

2

1661.3

1629.7

1553.7

1349.3

1329.4

1353.7

3

2099.0

2017.4

2062.7

2054.4

2017.4

1947.6

4

2360.1

2303.0

2228.1

2060.9

2027.6

2064.7

5

3518.1

3232.6

3271.9

2308.6

2240.4

2249.4

6

3566.5

3263.4

3442.3

3138.2

3052.3

2946.6

Note: Lamination: [00/900/00/900/00] for plate and [00] and [900] for stiffener, E11=120 GPa, E22=7.9 GPa, G12=5.5GPa, v12= 0.33, ρ=1580 kg/m3, Plate: a= 0.2m, b= 0.2m; t= 0.003m: Stiffener: hs = 0.01 m, bs = 0.006m Conflict of Interest and Authorship Conformation Form 38

Please check the following as appropriate:

o

All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

o

This manuscript has not been submitted to, nor is under review at another journal or other publishing venue.

o

The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:

Author’s name

Affiliation

Mrs. Leena Sinha

VSS University of Technology, Burla, Odisha, India

Ms. Supra Swagatika Mishra

VSS University of Technology, Burla, Odisha, India

Prof. Amar Nath Nayak VSS University of Technology, Burla, Odisha, India Prof. Shishir Kunar SahuVSS University of Technology, Burla, Odisha, India

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