Dynamic Analysis of Laminated Composite Plate Structure with Square Cut-Out Under Hygrothermal Load

Dynamic Analysis of Laminated Composite Plate Structure with Square Cut-Out Under Hygrothermal Load

Arun Mukherjee1, Dr. Sreyashi Das (nee Pal)2 and Dr. A. Guha Niyogi3

1PG student, 2Asst. Professor, 3Professor

Department of Civil Engg, Jadavpur University Kolkata-700032, India

Abstract Effect of temperature and moisture on the dynamic properties of epoxy-based laminated composite plates with a square cut-out are investigated. Eight-noded isoparametric plate elements with 6 degrees of freedom per node have been implemented in the present computations. First order transverse shear deformation based on Yang-Norris-Stavsky theory [1] is used along with rotary inertia of the material. The analysis considers material properties of the laminae at elevated temperature and moisture concentrations. Residual stresses due to hygrothermal environment are taken into account. A set of new results with various cut-out ratios and by varying thickness of the plates are presented.

KeywordsFree Vibration, Laminates, Hygrothermal Load, Square cutout

1. INTRODUCTION

   Epoxy-based laminated composite plates with cut-outs are inevitable in naval, aeronautical, mechanical and civil structures. Presence of cut-out changes the stiffness of the structure. Also stress concentration occurs near the opening of the plate. When these structures are subjected to high temperature or moisture, the strength and stiffness of structural composites are strongly affected. The matrix is more susceptible to the elevated temperature and moisture concentrations than the fibre, hence, the deformation is observed to be more in the transverse direction. The rise in hygrothermal stress reduces the elastic modulus of the material and induces initial internal stresses, which may affect the stability as well as the safety of the structures. Thus, knowledge of the dynamic behavior of a structure over a range of temperature and moisture concentrations is essential for the design of structures in a hygrothermal environment.

   Whitney and Ashton [2] considered the effect of environment on the free vibration of laminated composite plates by using Ritz method for symmetric laminates and equilibrium equation of motion for unsymmetric laminates.

   Dhanaraj and Palaninathan [3] showed that the natural frequency decreases linearly with the increase in temperature. Shen and Springer [4, 5] showed that the physical change due to temperature and moisture effects in the resin results in decreasing the tensile properties and reducing the transverse and shear modulus. Sai Ram and P.K. Sinha [6] studied the moisture and temperature effect in free vibration of symmetric and antisymmetric laminates accounting transverse shear deformation with simply supported and clamped boundary condition. Lee et al. [7] have predicted the natural frequencies of composite rectangular plates with cutouts, neglecting shear deformations and rotary inertia. Ashwini kumar and Srivastava [8] studied the free vibration of laminated composites with delaminations around a central cutout. Reddy [9] used the quadratic isoparametric element to study the free vibration of antisymmetric angle-ply laminated plates including transverse shear deformation and rotary inertia based on YNS theory [1]. Ali et al. [10] presented a simplified method for the dynamic analysis of plates with cutouts, based on Rayleigh's principle. A FEM formulation using HSDT is used in the analysis. Marsh et al. [11] studied the moisture absorption and diffusion into epoxy and epoxy-glass composites. Bahrami and Nosier [12] presented the interlaminar hygrothermal normal and shear stress distributions in long laminated composite plates.

   In the present investigation results are presented for symmetric laminates subjected to uniform distribution of temperature and moisture throughout its volume in absence of any external loading. Four layered graphite/epoxy laminates with simply supported and all side clamped boundary conditions are considered. Lamina material properties at elevated temperature and moisture conditions

   [6] are considered. Results showing change in natural frequencies of laminates at elevated temperature and moisture concentrations for various fibre angle orientations, boundary conditions, plate thicknesses and various cut-out ratios are presented. Since the evaluation of shear correction factor from the exact theory of elasticity is difficult in the present case, a commonly used value of 5/6 is assumed [9].

2. MATHEMATICAL FORMULATION

   1. Governing Equations for a Laminate

      A laminated composite plate of uniform thickness, t consisting of a number of thin laminae each of which may

      (C and Co) and (T and To) are the instantaneous and reference moisture concentrations and temperatures respectively.

      The stiffness coefficients of the laminate are defined as

      2 y

      1 (A , B , D ) = In

      fzk

      [Q

      ] {1, z, z2} dz,

      ij ij ij

      b x

      k=1

      zk-1

      ij k

      for i, j =1,2,6

      and (4)

      a (A ) = In

      fzk

      [Q

      ] dz,

      Fig. 1: Arbitrarily oriented lamina, Axis 3 coincides with z axis

      ij k=1

      zk-1

      ij k

      for i, j = 4,5

      be arbitrarily oriented at an angle (Fig. 1) with reference to the x-axis of the co-ordinate system is considered. The constitutive equations for the plate, when it is subjected to uniform temperature and moisture concentration is given by,

      {F}=[D]{}-{FN}, (1)

      where,

      {F} = Mechanical force resultant

      = {Nx, Ny, Nxy, Mx, My, Mxy, Qx, Qy}T,

      k

      {FN} = Hygrothermal force resultant

      is a shear correction factor, taken as 5/6, to take account for the non uniform distribution of the transverse shear strain across the thickness of the laminate.

      k

      [Qij ] is defined as,

      ij k

      [Q ] = [R1]-1[Qij] [R1]

      k

      for i, j=1,2,6

      and

      ={NxN, NyN, NxyN, MxN, MyN, MxyN, QxN, QyN, 0, 0}T,

      [Q

      ij]

      = [R2]-1[Qij ] [R2]

      k

      {} = Strain vector= {-x,

      -y, -xy, x, y, xy, x, y}T

      where,

      for i, j=4,5

      A11

      r

      A12

      A16

      B11

      B12

      B16

      0 0 l

      [R1] =

      cos20 sin20 2sin 0 cos 0 sin20 cos20 2sin 0 cos 0 ,

      A12 A22 A26

      I

      A16 A26 A66

      B12 B22 B26

      B16 B26 B66

      0 0 I

      0 0

      sin 0 cos 0 sin 0 cos 0 cos2

      0 sin20

      IB B B D D D 0 0 I

      [D]=I 11 12 16

      11 12 16

      I (2)

      I 0	0	0	0	0	0	A44	A45I
      l 0	0	0	0	0	0	A45	A55J

      IB12 B22 B26 IB16 B26 B66

      D12 D22 D26 D16 D26 D66

      0 0 I

      0 0 I

      [R2]

      cos 0 sin 0

      = [sin 0 cos 0 ],

      Q11 Q12 0

      [Q ]

      = Q12 Q22 0

      where, the hygrothermal force and moment resultants are

      ij k

      0 0 Q66

      {NN

      x NNy NNxy

      T n

      } =I

      k=1

      fzk

      zk-1

      [Qij ] [e]k

      dz,

      k

      [Qij ]

      = [Q44

      for i, j=1,2,6

      0 ], i, j = 4,5

      (i, j = 1, 2 and 6)

      and (3)

      k 0 Q55

      in which, Q11 = E1/(1 1221),

      {MN

      MN MN

      }T = In

      fzk

      z [Q ] [e] dz,

      Q = E /(1 ),

      x y xy

      k=1

      zk-1

      ij k k

      12 12 2

      12 21

      (i, j = 1, 2 and 6).

      k k

      where {e}k {ex ey exy}T = Hygrothermal strain = [R]{H1, H2}T (C-C0) + [R]{1, 2}T (T-T0)

      cos20 sin20

      in which [R] = sin20 cos20 ,

      sin 20 sin 20

      Q22 = E2/(1 1221), Q44 = G13, Q55 = G23.

      Where, E1 and E2 are Youngs moduli along the 1 and 2 axes, respectively and ij is Poissons ratio of transverse strain in the j direction to axial strain in i direction when stressed along i-direction only.

      The non-linear strains can be expressed as,

      xnl = [u-2,x + v-2,x + w 2,x + 2z(u-,xy,x v-,xx,x) + z2(2y,x + 2x,x)]/2,

      ynl = [u-2,y + v-2,y + w 2,y + 2z(u-,yy,y v-,yx,y) + z2(2x,y + 2y,y)]/2,

      1. Element Mass Matrix

         The elemental mass matrix [6] is defined as

         e -a/2 -b/2

         [M ] = fa/2 fb/2 [N]T[p][N]dxdy

      2. Element Load Vectors

         (10)

         xynl = [u-,xu-,y + v-,xv-,y + w ,xw ,y + z(u-,yy,x + u-,xy,y)

         z(v-,yx,x + v,xx,y) + z2(y,xy,y + x,xx,y)],

         The element load vector [6] due to hygrothermal forces and moment is given by

         xznl

         = [u-,x,y + v-,xx + z(yy,x + xx,x)],

         {PN} = f[B]T{FN}dxdy (11)

         e

         yznl = [u-,y,y + v-,yx + z(yy,y + xx,y) (5)

         where, u, v and w are x, y and z component of structural displacement respectively and x and y are total rotations of a plate element along x and y directions.

   2. Finite Element Formulation

   Eight-noded isoparametric plate elements with 6 degrees of freedom per node have been implemented in the present computations. The stiffness matrix, the initial stress stiffness matrix, the mass matrix and the nodal load vectors of the element are derived by using the principle of minimum potential energy [6]. The element displacements are expressed in terms of their nodal values by using the element shape functions (Ni((, 1))and are given by

   1. Solution Process

   The stiffness matrix, the geometric stiffness matrix, the mass matrix and the load vectors of the element, given by equations (8-11), are calculated by expressing the integral in local natural co-ordinates, and of the element and then performing Gaussian integration. Generally, a 3 point Gauss quadrature is adopted to compute the bending stiffness of the elements, whereas a 2 point integration is applied to calculate the shear stiffness and mass matrix. The purpose of reduced integration is to reduce the shear stiffness of the element. Then the element matrices are expressed accordingly to global co-ordinates from local co-ordinates.

   The initial displacements (i) are then evaluated from equilibrium equation,

   i=1

   uO = I8 Ni((, 1)u0 ,

   i

   V = I8 N ((, 1)V ,

   [K]{i} = {P} + {PN} (12)

   After obtaining the initial stress resultants {Nxi, Nyi, Nxyi,

   O i=1 i 0i

   w = I8 N ((, 1)w , (6)

   Mxi, Myi, Mxyi, Qxi, Qyi} from (1), the natural frequencies

   i=1 i i

   i=1

   i

   x = I8 Ni((, 1)0x ,

   i=1 i yi

   y = I8 N ((, 1)0

   in which uoi, voi, wi, xi, yi are the displacements at a node i.

   1. Element Stiffness Matrix

      The linear strain matrix {} is obtained by expression

      {} = [B]{e} (7)

      The element stiffness matrix is given by

      of vibration of the plate are determined from the general

      equation of vibration,

      l[K] + [K] n2[M]l = 0

      This generalized eigenvalue problem is solved by subspace iteration method.

3. NUMERICAL RESULTS AND DISCUSSION The finite element formulation described in the earlier

   section has been used to generate numerical results to study the effects of cutout under hygrothermal environment of a laminated composite plate. Parametric studies have been

   e a/2 -b/2

   [K ] = fa/2 fb/2 [B]T[D][B]dx dy

   (8)

   provided for variation in cut-out ratio, stack up sequence, plate thicknesses, boundary conditions and various

   Where [B] is the linear strain displacement matrix and

   [D] is the elasticity matrix [6].

   1. Element Geometric Stiffness Matrix

      Element stiffness matrix [6] is given by

      Ge -a/2 -b/2

      [Kr ] = fa/2 fb/2 [G]T {Sr}[G]dx dy (9)

      temperature and moisture concentrations.

      1. Mesh Convergence Study

         A mesh convergence study has been carried out to obtain minimum number of elements required to achieve the non-dimensional natural frequencies as accurately as possible optimizing the computation time. For this purpose, a square laminated composite plate having a square cutout (a/s=0.4) is taken and analyzed under simply supported (SSSS) and clamped (CCCC) boundary conditions. The

         side-to-thickness ratio (=s/t) is taken as 75. Three mesh sizes containing 144 (=9×9 mesh), 200 (=10×10 mesh) and

         220 (=11×11 mesh) elements are considered for the convergence study. Details of a typical mesh having 200 elements are shown in Fig. 2.

         Development Agency, Bangalore, India). The plate has side-to-thickness ratio (=s/t) of 75. First five Non-dimensionalized frequencies are obtained with a square (a/s = 0.4) cutout. The results are compared with those of Ashwini Kumar and R.P. Shrivastava [8] as shown in Table

         2. From the table it is clear that the two results are in good agreement.

         Table 2: First Five Non-dimensional Frequency

         MODE	Ashwini kumar et al. [8]	Present
         1	14.86	14.86
         2	26.68	28.02
         3	29.91	28.02
         4	49.76	50.54
         5	54.10	53.45

         Fig. 2: Discretization details for a square plate with square cutout

         Natural frequencies obtained for different meshes under different boundary conditions are tabulated in Table 1.

         s2

         wi = wr t (

         where,

         p

         )

         /E2

         Table 1: Non-Dimensional Frequency for Different Mesh Sizes

         MODE NO.	SSSS			CCCC
         	9 x 9	10×10	11 x 11	9 x 9	10×10	11 x 11
         1	14.95	14.86	14.86	14.95	14.86	14.86
         2	28.06	28.02	28.02	28.06	28.02	28.02
         3	28.06	28.02	28.02	28.06	28.02	28.02

         It can be observed that the fundamental frequency converges for a mesh containing 200 elements, or more. Hence, for subsequent studies, a mesh containing 200 (10×10 mesh) elements is taken.

      2. Validation Study

         Two validation studies are made to test the accuracy of the computer program and the results are compared with those available in the literature. The cases are as follows:

         1. Validation for laminated composite plate with square cutout

         2. Validation for laminated composite plate under hygro-thermal load

            1. Laminated Composite Plate With A Square Cutout

               A square laminated composite plate, with all edges simply supported, is considered [8]. The lay-up sequence is [(45/02)3(90/02/90)2]s. Each lamina has material properties as follows: E1 = 13.00 x 1010 N/m2; E2 = 1.00 1010 N/m2; G12 = G31 = 0.5 x 1010 N/m2; G23 = 0.33 x 1010 N/m2;

               12 = 0.35; = 1500kg/m3 (available from Aeronautical

               wi =Non-dimensional frequency wr = Natural frequency in rad/sec s = Length of the square plate

               t = Thickness of the plate

               = Density of the plate

            2. Laminated Composite Plate under Hyro-thermal Load

            A simply supported square plate, with side to thickness ratio, s/t = 100, has been analyzed under increased temperature of 325K and moisture concentration = 0.1% and the first four non-dimensional frequencies are compared in Table 3, with those obtained by K. S. Sai Ram and P. K. Sinha [6]. The layup sequence of the laminate is taken as (0/90/90/0). Graphite-epoxy material with E1=130 GPa, E2=9.5GPa, G12=6GPa, G13= G12, G23= 0.5G12, 12=0.3, 1=

            0.0, 2= 0.44 and density = 1600kg/m3are used. Material properties are assumed to be constant for this purpose, i.e., they do not vary with change in hygro-thermal conditions. From Table 3, it is evident that the program used, are giving quite accurate results.

            Table 3: First Four Non-dimensional Frequencies under Uniform Thermal and Hygral loading

            MODE NO.	Temperature=325K		Moisture Conc.=0.1%
            	Sai Ram et al.	Present	Sai Ram et al.	Present
            1	8.088	8.096	9.429	9.438
            2	19.196	19.297	20.679	20.783
            3	39.324	39.489	40.068	40.236
            4	45.431	46.165	46.752	47.505

      3. Case Studies

      1) Study of Dynamic Behavior of Laminated Composite Plate for Varying cutout size

      • A drastic increase in frequency is observed at a/s=0.8 for the plate clamped on all four edges. Thus for moderate and large cutouts the stiffness of the plate increases rapidly with the cut out area for clamped boundary

        Various sizes of square cut-outs are

        considered to

        condition.

        examine the effect of cut-out size on the natural frequency of the laminate. The laminate (1.0m x 1.0m) has a varying side to thickness ratio (=s/t) of 25 and40. The layup sequence of the laminate is (0/90/90/0). The material properties for Graphite-epoxy material are given as follows: E1=130 GPa, E2= 9GPa, G12=6GPa, G13= G12, G23= 0.5G12,

        12=0.3, 1= 0.0, 2= 0.44 and density = 1600kg/m3 are used.

        The cut out ratio (= a/s) is varied from 0.2 to 0.8 with increment of 0.2. External temperature is taken as 300K and

      • For second and third modes, the frequency decreases for increasing cutout ratio up to 0.6 for simply supported boundary conditions. After that, the frequency increases considerably. Similar phenomenon can be observed for clamped boundary condition. In this case (CCCC) the increase in natural frequency is much more compared to SSSS condition. It is also observed for the all the cases

        the moisture concentration is taken as 0.5%. Two different types of boundary conditions, namely simply supported

        above, that the second and third modes are showing

        nearly similar frequencies. This is due to the similar

        (SSSS) and clamped (CCCC), are taken in the analysis.

        mode shapes occurring owing to the structure symmetry.

        Results obtained are shown in Table 4. To visualize the effect of cut out size a plot of the fundamental frequencies with various cut out sizes is presented in Fig 3.

      • It is observed that for smaller cut outs with s/t = 40, a/s =

        0.2 to 0.4, the fundamental frequency decreases slightly, for simply supported edge condition. This indicates that stiffness of the plate decreases marginally when a smaller cut out is present. With the increase in cut out size, a/s =0.4 to 0.8, the fundamental frequency

        2500

        Frequency (Hz)

        2000

        1500

        1000

        500

        C=0.5%,

        s/t=40, S SSS

        C=0.5%,

        s/t=25, S SSS

        C=0.5%,

        s/t=40, C CCC

        decreases so rapidly that instability occurs and no 0

        frequency could be found out. For s/t=25, the

        fundamental frequency increases with increase in cutout

        0 0.2 0.4

        0.6 0.8 1

        C=0.5%,

        s/t=25, C CCC

        ratios. Here the decrease in mass is much compared to

        Cutout Ratio

        the decrease in stiffness. Hence there increase in frequency.

        is an overall

        Fig. 3: Graphical plot for fundamental frequency (Hz) vs.

        varying cutout size boundary conditions

        for different s/t ratios and

        Table 4: First Three Natural Frequencies (Hz)for varying cutout size, boundary condition and s/t ratios

        		Mode No.	cutout ratio
        			Simply supported (SSSS)				Clamped (CCCC)
        			0.2	0.4	0.6	0.8	0.2	0.4	0.6	0.8
        T = 300K, C = 0.5%	s/t = 40	1	85.90	74.30	–	–	180	214.4	389.1	1425.0
        		2	227.80	208.80	–	–	342.9	324.4	440.6	1454.8
        		3	227.90	209.50	–	–	343.0	325.0	444.3	1457.9
        	s/t = 25	1	157.9	159.5	168.2	200.9	293.8	358.8	636.8	2093.4
        		2	371.0	338.9	308.4	351.1	533.5	508.2	697.9	2114.4
        		3	371.0	339.1	309.9	362.4	533.6	508.4	698.9	2116.5

        2) Study of Dynamic Behavior of Laminated Composite Plate Structure under Thermal Load

        Here the effect of increased thermal load on the dynamic behaviour of laminated composite plate (1.0m x 1.0m) with

        First three natural frequencies (Hz) are shown in Table 6. The fundamental frequency vs. temperature is plotted in Fig. 4.

        a square cutout (a/s=0.4), for different side to thickness ratio, has been analyzed. The layup sequence of the laminate is taken as (0/90/90/0). The temperature is varied from 300K to 400K. The material used in the analysis is graphite /epoxy

      • From the figure, it is

        temperature, generally reducing its stiffness.

        observed that with increase in

        the structure turns soft thus

        with the following properties as given in Table 5.

        Table 5: Degraded Elastic Moduli for varying Temperature

        Elastic Moduli (GPa)	Temp (K)
        	300	325	350	375	400
        E1	130	130	130	130	130
        E2	9.5	8.5	8.0	7.5	7.0
        G12	6.0	6.0	5.5	5.0	4.75

        Simply supported (SSSS) and clamped (CCCC) boundary conditions are taken in the analysis. The layup sequence is taken as (0/90/90/0).

      • The decrease in natural frequency is nearly nonlinear for

        thinner plates (s/t=40).

      • Clamped boundary condition imparts higher siffnesses to the structure than simply supported conditions.

      • First three mode shapes for clamped case are plotted in Fig. 5.

      • From the figure it is seen that owing to the similarity in mode shapes for the second and third modes, the second and third modes are showing nearly same frequency of vibration.

        Table 6: First Three Natural Frequency (Hz) for varying Temperature

        	Mode No.	Temp (K)
        		SSSS					CCCC
        		300	325	350	375	400	300	325	350	375	400
        s/t = 40	1	116.4	107.0	96.1	85.2	75.7	241.5	232.6	223.8	215.4	208.4
        	2	230.5	223.8	215.5	207.6	201.7	343.7	336.2	327.2	318.4	311.9
        	3	230.5	223.8	215.7	207.9	202.2	343.7	336.2	327.4	318.8	312.4
        s/t = 25	1	182.6	176.3	168.0	160.1	154.1	375.8	367.9	359.3	350.9	344.4
        	2	353.3	347.7	338.6	329.5	323.4	521.7	514.5	503.3	491.8	484.0
        	3	353.3	347.8	338.7	329.6	323.5	521.7	514.5	503.3	491.9	484.1

        400

        350

        Frequency (Hz)

        300

        250

        200

        150

        100

        50

        0

        s/t =40, SSSS

        s/t = 25, SSSS

        s/t = 40, CCCC

        s/t = 25, CCCC

        300 325

        350 375 400

        Temperature (K)

        425

        Fig. 4: Graphical plot for fundamental frequency vs. varying temperature for various boundary conditions and side to thickness ratio

        Mode 1

        Mode 2

        Mode 3

        300K 350K 400K

        Fig. 5. First three mode shapes of vibration of (0/90/90/0) laminate for increase in temperature

   2. Study of Dynamic Behavior of Laminated Composite Plate Structure under Hygral Load

      Here the effect of increased hygral load on the dynamic behaviour of laminated composite plate of same material and geometry, as previous case has been analysed. The moisture concentration is varied from 0 to 1.5%. Graphite-epoxy material with the following material properties (Table 7) are used in the analysis.

      Table 7: Degraded Elastic Moduli for various Moisture Concentrations

      Elastic Moduli (GPa)	Moisture Concentration, C (%)
      	0.00	0.25	0.50	0.75	1.00	1.25	1.50
      E1	130	130	130	130	130	130	130
      E2	9.5	9.25	9.0	8.75	8.5	8.5	8.5
      G12	6.0	6.0	6.0	6.0	6.0	6.0	6.0

      First three natural frequencies (Hz) for different cases are shown in Table 8 and a plot of fundamental frequency and changing moisture concentration is presented in Fig. 6.

      • From the figure, it is observed that fundamental frequency decreases with increase in moisture concentration. This is due to reduction in stiffness with increasing hygral load.

      • The decrease in natural frequency is nearly nonlinear for thinner plates (s/t=40).

      • Instability occurs at 1.0% moisture concentration for the thinner plate at simply supported boundary condition.

      • Clamped boundary condition imparts higher stiffnesses to the structure than simply supported conditions.

      • From the figure, it is also observed that, the less stiff a structure is, chances of instability is more with increase in moisture concentration.

      • The mode shapes are plotted in Fig. 7. Here also similar frequency of vibration for second and third modes is observed as in Table 8 due to similarity in mode shapes.

        Table 8: First Three Natural Frequency (Hz) for varying Moisture Concentrations

        	Mode No.	Moisture Concentration, C (%)
        		SSSS							CCCC
        		0.0	0.25	0.50	0.75	1.00	1.25	1.50	0.0	0.25	0.50	0.75	1.00	1.25	1.50
        s/t = 40	1	116.4	97.6	74.3	39.2	–	–	–	241.5	228.4	214.4	199.3	182.8	162.3	136.6
        	2	230.5	219.7	208.8	198	–	–	–	343.7	334	324.4	314.9	305.6	295.8	285.5
        	3	230.5	219.8	209.5	199.6	–	–	–	343.7	334.1	325	316.3	308.1	299.9	291.7
        s/t =25	1	182.6	171.2	159.5	147.3	134.7	119.1	100.8	375.8	367.3	358.8	350.3	341.8	333.3	324.3
        	2	353.3	346	338.9	332.1	325.5	318.7	311.7	521.7	514.9	508.2	501.8	495.6	489.8	484
        	3	353.3	346	339.1	332.5	326.2	319.8	313.4	521.7	514.9	508.4	502.2	496.2	490.9	485.5

        400

        350

        Frequency (Hz)

        300

        250

        200

        150

        100

        50

        0

        s/t =40, SSSS s/t = 25, SSSS

        s/t =

        40, CCCC

        s/t =

        25, CCCC

        0.00 0.25 0.50 0.75 1.00 1.25 1.50

        Moisture Concentration (%)

        1.75

        Fig. 6: Graphical plot for fundamental frequency vs. varying moisture concentration for various boundary conditions and side to thickness ratio

        0.0%

        0.5%

        1.0%

        Mode 1

        Mode 2

        Mode 3

        Fig. 7. First three mode shapes of vibration of (0/90/90/0) laminate with increase in moisture concentration

   3. Study of Dynamic Behavior of Laminated Composite Plate Structure with a cutout under increasing hygrothermal load for various Layup Sequences

   Here the effect of hygrothermal load on the dynamic behaviour of simply supported laminated composite plate (1.0m x 1.0m, s/t=25 and s/t=40) with a square cutout with

   • From the figures, it can be seen that (45/-45)S is the stiffest, whereas (0/90)S have the least stiffnesses.

   • Also, for increasing moisture concentration, symmetric cross ply shows instability much earlier than others.

     (a/s=0.4), for different symmetric fibre angle, has been analyzed.

   • The rate of drop of natural frequency is less for

     (60/-60)S.

     The temperature is varied from 300K to

     400K and the

   • The rate of degradation of material is more pronounced

   moisture concentration is varied from 0.0% to 1.5%. The

   fundamental frequencies (Hz) are plotted in Fig. 8 and Fig. 9.

   due to increased moisture effects than due to thermal changes.

   250

   s/t=40,(0/90)s

   200 s/t=40,(45/-45)s

   Frequency (Hz)

   150 s/t=40,(60/-60)s

   100

   s/t=25,(0/90)s

   50 s/t=25,(45/-45)s

   0

   300 325

   350 375 400 425

   Temperature (K)

   s/t=25,(60/-60)s

   Fig. 8: Graphical plot for fundamental frequency vs. varying temperature for various layup sequences

   250

   s/t=40,(0/90)s

   200 s/t=40,(45/-45)s

   Frequency (Hz)

   150

   s/t=40,(60/-60)s

   100

   50

   0

   0.00 0.25 0.50

   0.75 1.00 1.25 1.50 1.75

   s/t=25,(0/90)s s/t=25,(45/-45)s s/t=25,(60/-60)s

   Moisture Concentration (%)

   Fig. 9: Graphical plot for fundamental frequency vs. varying moisture concentration for various layup sequences

4. CONCLUSION

   This paper is concerned with the dynamic analysis of laminated composite plate structures with a square cutout under hygrothermal load. For the numerical simulation of the problem finite element technique has been used. Various cutout sizes, plate thicknesses, hygral and thermal load, lay up sequences and boundary configurations have been considered in the present study. A set of new results are presented. It has been shown that boundary condition plays an important role in providing stiffness of a plate structure with a cutout. This is also observed that higher moisture concentration and higher temperature turns the structure soft and there is reduction in stiffnesses as well as natural frequency. Less stiff plate has higher chances of instability under higher hygral load. Also instability of a structure largely depends on the layup sequence of the laminated composite structure.

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