Static Analysis Of Laminated Composite Circular And Annular Plates

Static Analysis Of Laminated Composite Circular And Annular Plates

Ayisha Powmya.S.E. Department of Built Environment,

Muscat College, Oman

Mattur C. Narasimhan Department of Civil Engineering, National Institute of Technology Karnataka, India

Abstract

Analytical solutions are presented for the static deflection analysis of laminated polar orthotropic circular and annular plates. The analysis is based on the application of the first order shear deformation theory employed. Three linear partial differential equations for axisymmetric deformations are written in terms of displacements u, and w. Chebyshev collocation method is employed for the solution of the evaluation of static deflection problem. Numerical results are presented to show the validity and accuracy of the proposed method. Results of parametric studies conducted to evaluate the effect of parameters like orthotropic ratios, number of layers, lamination sequences and boundary conditions, on the response of laminated polar orthotropic circular and annular plates are also presented.

1. Introduction

   Fiber reinforced laminated composites are being increasingly used in modern engineering applications due to their high specific strength and high specific modulus. The increased application of laminated composites in the primary components in structures like spacecrafts, high speed aircrafts, missiles, gas turbines, etc. are due to the number of advantages they offer in structural, operational, production and/or maintenance aspects.

   The use of advanced composite materials for structural elements brings in the need to develop new analytical and design techniques. With the present level of their application being what it is, it becomes a necessity to develop better mathematical models to predict the mechanical behavior of structural elements made up of such materials, under service loads.

   In the present work, it is proposed to study, the static deflection analysis of laminated polar orthotropic circular and annular plates by Chebyshev collocation

   method. A first order shear deformation theory is used in terms of u, and w. These field variables are expanded in polynomials and then orthogonal point collocation method is used to discretise the governing equations. To demonstrate the convergence of the method, numerical results are presented for clamped and simply supported isotropic and polar orthotropic circular and annular plates. The validity of the analytical solution is confirmed by comparing with data obtained from open literature.

2. Mathematical formulation

   The laminated plate of constant thickness h is composed of polar orthotropic laminae stacking symmetrically or anti-symmetrically about the middle surface of plate. Plate co-ordinates (r, , z) used are as shown in figure 1, where u, v, w denote the displacements of any point of the plate in the corresponding r, , z directions.

   Figure 2.1 Geometry of a n-layered laminate

   In this study, first order shear deformation theory is employed and the general displacement field is assumed in the form

   u (r, , z) = uº(r, ) + z 1(r, ),

   v (r, , z) = vº(r, ) + z 2(r, ),

   w (r, , z) = w(r, ) (1)

   where uº, vº, w denote the displacements of any point

   Nr

   Mr

   N M

   N M

   h/2

   r

   on the middle surface and 1, 2 are the rotations of the normal to the mid-plane about , r axes respectively.

   Strain displacement relations are of the following form

   N r Mr

   (1, z) dz ,

   h/2 r

   in polar co-ordinates

   Qr h/2 rz

   z , z ,

   h/2

   dz

   (6)

   r

   r

   r r

   r

   r

   r

   r

   z ,

   rz

   rz

   rz

   ,

   z z

   (2)

   Q

   z

   where the reference surface strains and curvatures are given by,

   where z is the distance of the lamina from the middle plane.

   Substituting the stress strain relations, we have the

   =

   u

   ,

   =

   1 v

   u ,

   constitutive matrix as

   r r

   r

   Nr

   A11

   A12

   0 B11

   B12 0 0

   0 r

   1 u

   v

   N

   N

   A

   A

   12

   A 22

   0 B12

   B22 0 0

   0

   Nr 0

   0 A 66 0

   0 B66 0

   0

   r =

   r

   v ,

   r

   r

   Mr B11

   B12

   0 D11

   D12 0 0

   0 r

   (7)

   M

   M

   • w

     1 w

     B12

     B22

     0 D12

     D 22 0 0

     0

     = ,

     = ,

     rz 1 r

     z 2

     r

     Mr 0

     0 B66 0

     0 D66 0

     0 r

     Q

     Q

     0 0

     0 0 0

     0 K 2 A

     0

     1

     1 2

     Qr

     44 rz

     r =

     r ,

     =

     r

     1 ,

     0 0

     0 0 0 0

     0 K 2 A55

     z

     = 1 1

     2

     (3)

     where, A =

     n

     (C ) (z z ) ,

     r

     r

     2

     r

     ij

     k1

     ij k k

     k1

     i, j = 1, 2,6,4,5

     According to the shear deformation theory, the

     1 n 2 2

     constitutive equations for the kth layer of a polar orthotropic laminated plate can be written in the

     Bij =

     (Cij ) k (z k zk1 ) ,

     2

     2

     k1

     following form in polar co-ordinates.

     i, j = 1, 2,6

     D = 1 n (C

     ) (z 3 z3 ) ,

     3

     3

     r

     C11

     C12

     0 r

     ij

     k1

     ij k k

     k1

     C12

     C22

     0 ,

     where i, j = 1, 2,6 (8)

     r

     0

     0 C66 r

     Aij are the extensional stiffnesses, Bij are the bending-

     rz

     C44

     0 rz

     extension coupling stiffnesses, Dij are the bending

     0

     C

     (4)

     stiffnesses and K2 is the Shear correction factor

     z

     55 z

     introduced to account for non-uniform distribution of the transverse shear strains through the thickness of the

     where C11 =

     E

     Er

     1 r

     , C12 =

     r E

     1 r

     = Er ,

     1 r

     plate, which is taken as ²/12.

     n is the total number of layers in the laminate.

     The stress resultants and stress couples defined in (6)

     C22 =

     1 r

     and (7) must satisfy the following equilibrium equations (Ravichandran, 1989)

     Nr 1 Nr (N r N ) 0 ,

     C44 = Grz , C55 = Gz, C66 = Gr (5)

     Where E and E are Youngs moduli of elasticity in r

     r r r

     Nr N 2 Nr 0

     and d

     r

     irec

     tions.

     r and

     are Poissons ratios in r and

     r r r

     directions. Gr, Gz and Grz are the shear moduli in the respective planes.

     The stress resultants acting on a laminate are obtained as:

     Qr 1 Q Qr q(r, ) 0 ,

     r r r

     Mr 1 Mr (M r M ) Q

     r r r r

     Mr

     1 M

     Mr

     1 2 U

     1 U U

     2

     Q

     (9)

     b11 (a b)2 2 ( a (1 )b) (a b) b22 ( a (1 )b) 2

     r r r

     1 2

     1

     d11 (a b)2 2 ( a (1 )b) (a b) d 22 ( a (1 )b) 2

     If the analysis is now restricted to only axisymmetric case, the deformations are symmetrical about z-axis,

     a 44 W

     (12)

     thus the stresses and strains are independent of and

     h 2

     r

     z

     0 , vº = 0, 2 = 0 and also (

     ) = 0

     2.1 Polynomial series solution by collocation at

     The governing equations for axi-symmetric case then can be written in terms of mid-plane displacements as,

     Chebyshev zeroes

     The Chebyshev polynomials Tn (x) are a class of

     2 u u u 2

     orthogonal polynomials, which are defined as,

     A11

     A22 B11 1 1 B22 1 0

     r 2

     r r r 2 r 2

     r r r 2

     Tn (x) cosn

     K2A

     1

     • 1

     2w

     w

     q(r) 0

     Where cos x

     , 1 x 1

     (13)

     44 r

     2 u

     r

     u

     r2

     u

     r r

     2

     These provide a solution of the second order differential equation,

     B11

     B22 D11 1 1

     2 " ' 2

     r2

     r r r2 r2

     r r

     (1 x

     ) Tn x Tn n

     Tn 0

     (14)

     1 2

     w

     (10)

     where ( ) '

     d ( )

     D22 r 2

     K A44 1

     r

     dx

     In many applications, it is advantageous to re-define the

     r

     r

     For convenience, the following dimensionless

     polynomials in the range

     0 1 . The shifted

     parameters are introduced

     Chebyshev polynomial

     T* ()

     of degree r in the range

     u (a b) w

     1 (a b)

     0 1

     is derived from the polynomial

     Tr (x)

     U ,

     h 2

     W ,

     h

     ,

     h

     {1 x 1} using a linear transformation,

     r b ,

     a b

     q (a b) 4

     p ,

     ET h 4

     2

     K A

     K A

     a44 44 ,

     ET h

     1 x 1

     2

     i.e., Tr () cosrt

     where cost (2 1)

     (15)

     (16)

     1. Aij ,

        T

        T

     2. Bij ,

     T

     T

     d Dij

     (11)

     The shifted polynomials satisfy the recurrence relations

     E

     E

     T

     T

     ij h

     ij E h 2

     ij E h 3

     *

     T

     T

     r1

     ()

     *

     • T

     • T

     r1

     () 2 (2 1) T* ()

     (17)

     r

     r

     and the orthogonality conditions

     where i, j = 1, 2,6

     Here, ET is the reference Youngs modulus. In case of

     1 T () T ()

     0

     = /2

     for

     for

     m n

     m n 0

     (18)

     laminated composites with layers of same material, ET is taken to be the Youngs modulus in the direction

     0 (1 )

     for

     m n d

     m n d

     m n 0

     transverse to fiber direction.

     Thus the governing differential equations of the plate can now be expressed in dimensionless form as,

     1. ., the shifted Chebyshev polynomials are orthogonal with respect to the weighing function

        1

        1 2 U 1 U U

        W()

        1/2

        (19)

        a11 (a b)2 2 ( a (1 )b) (a b) a 22 ( a (1 )b) 2

        {(1 )}

        1 2

        1

        Any continuous function g() in the interval 0 1 ,

        b11 (a b)2 2 ( a (1 )b) (a b) b22 ( a (1 )b) 2 0

        1

        1 2 W

        1 W

        can be represented by a infinite series of the form

        a 44 (a b) ( a (1 )b) (a b) 2

        ( a (1 )b)

        g * *

        h 2

        g()

        T () grTr () 2 r1

        (20)

        p 0

        (a b)3

        where gi

        (i=0,1,2,.) are the coefficients to be

        determined so as to obtain a best possible fit. The series in eqn.(20) is a fast converging one and good

        approximation is obtained by taking a finite number of

        N1 (n 1)(n 2) n3

        (n 1) n2

        terms in the above series. i.e.,

        d11

        n1

        n

        (a b)2

        n

        ( a (1 )b) (a b)

        g * N *

        N1

        • d

     n1

     n

     g()

     T () gr Tr ()

     N

     N

     2

     2

     r1

     (21)

     22

     n1

     ( a (1 )b)2

     gr r0

     T* ()

     a44

     p

     N1

     n

     n1

     n1 (n 1)Wnn2 0

     (26)

     r

     r

     Where, the plus sign indicates that the first term of the series is to be halved.

     For a known function g(), using the orthogonality conditions (eqn.18), the coefficients gr can be calculated as

     There are 3(N+1) constants to be determined. The stipulation of the three boundary conditions at each edge provide six equations. 3(N-1) additional equations are obtained by forcing the satisfaction of each of the

     2 1 g() T* ()

     , 0 r N

     (22)

     three differential equations at the (N-1) zeroes of

     gr

     r d

     T () ,

     0 1

     – the (N-1)th degree shifted

     0

     (1 )

     (N1)

     r

     r

     The shifted Chebyshev polynomial T*() of degree r

     Chebyshev Polynomial.

     The Nth degree Chebyshev Polynomial T

     has N

     has r zeroes in the range 0 1 , which are used as

     N

     zeroes at

     the collocation points in the present study.

     1 (2 i 1)

     The (N-1) collocation points are taken at the zeroes of the Chebyshev polynomial. For static loads, the radial displacement, deflection and rotation are expanded as

     i 2 1 cos 2N

     i = 1, 2, N (27)

     polynomials in . Thus forcing the satisfaction of each of the three

     U(), (), W()

     Un , n , Wnn1

     (23)

     differential equations at the (N-1) zeroes of

     T

     T

     (N1)

     () ,

     N1

     N1

     n1

     0 1

     along with the stipulation of the three

     Substituting eqn. (23) in eqn. (12), the following equations are obtained.

     boundary conditions at each edge results in a set of simultaneous equations of the form,

     [L11] {U} [L12 ] {} [L13] {W} {0}

     N1 (n 1)(n 2)Unn3 (n 1)Unn2

     [L } {U} [L ] {} [L ] {W} {p}

     11

     11

     1. n1

        N1

        (a b)2

        Unn1

        ( a (1 )b) (a b)

        21

        [L31] {U} [L

        22

        32 ] {} [L

        23

        33] {W} {0}

        (28)

        a 22

        n1 ( a (1 )b) 2

        Where { U}T

        {U , U

        ,……………..U }

        N 1 (n 1)(n 2)

        n 3

        (n 1)

        n 2

        1 2 n1

     2. n n

     {}T {1 , 2 ,……………… n1}

     11 n 1

     N 1

     (a b)2

     n 1

     ( a (1 )b) (a b)

     { W}T {W1, W2 ,………..Wn1}

     (29)

     • b

     n 0

     (24)

     It can be seen that [L13] = [0] and [L21] = [0].

     22 n 1 ( a (1 )b) 2

     (n 1) n2

     n1

     These algebraic equations are then solved using LU

     n n

     Decomposition method. (William H. Press et al, 2002)

     a N1

     (a b)

     ( a (1 )b)

     The coefficients are then substituted in eqn. (23) to get

     n1 n n

     n1 n n

     44 (n 1)(n 2)W n3 (n 1)W n2

     radial displacement, transverse deflection and rotation at any point inside the plate.

     (a b)

     p

     p 0

     (a b)3

     ( a (1 )b)

     (25)

     N1 (n 1)(n 2)Unn3

     (n 1)Unn2

3. Convergence and comparison studies

   b11

   2

   n1

   N1

   (a b)

   Unn1

   ( a (1 )b) (a b)

   In order to validate the procedure implemented, static analysis has been conducted on isotropic plates. The

   b22

   n1 ( a (1 )b) 2

   convergence analysis of the Chebyshev collocation method for static deflection of circular and annular

   plates is first carried out. Table 3.1 shows the results of analysis of an isotropic circular plate subjected to axisymmetric pressure q, with increasing number of terms used in each of the series in eqn. (23). It can be observed that sufficiently accurate, converged solution is obtained taking 10 12 terms used in each of the series.

   Isotropic circular and annular plates subjected to various types of pressure loads are now analyzed. Table 3.2 and Table 3.3 shows the results of static deflection and rotation of the transverse normal in radial direction, obtained using the present solution. It can be observed that the present solution provides results comparing reasonably well with the results obtained by CPT [Warren C. Young (1989)]. The deviations between the results can be attributed in fact to effect of shear deformations as the plates considered herein are moderately thick plates. (a/h = 10)

   Table 3.1 Static deflection of isotropic circular plates – convergence study

   a = 100 mm, h=10 mm, = 0.3, q = 100 N/mm2, E = 2.10 kN/mm2

   Boundary condition: clamped edges

   2 w

   W 10

   h

   Number of terms in Chebyshev series N	Maximum deflection W
   4	0.8438
   6	0.8437
   8	0.8438
   10	0.8437
   12	0.8437

   Table 3.2 Static solution for isotropic circular plates

   a/h = 10 , = 0.3

   Type of load		Boundary Condition			Present (FSDT)	Warren C. Young (1989) (CPT)
   Uniformly Distributed Load		Clamped		Kymax	0.01622	0.01563
   		Simply Supported		Kymax Kmax	0.06430 0.09615	0.06370 0.09615
   Uniformly Varying Load		Clamped		Kymax	0.00693	0.00667
   		Simply Supported		Kymax Kmax	0.03257 0.05128	0.03231 0.05128
   Parabolically Varying Load		Clamped		Kymax	0.00362	0.00347
   		Simply Supported		Kymax Kmax	0.01964 0.03205	0.01949 0.03205
   Wmax = Kymax	q a 4 D		q a3 max = Kmax D

   Table 3.3 Static solution for isotropic annular plates

   a/h = 10 , b/a = 0.1, = 0.3

   Type of load	Boundary Condition		Present (FSDT)	Reference (CPT)*
   Uniformly Distributed Load	C C	Kymax	0.0020	0.0018
   	Sa C	Kymax	0.0025	0.0025
   		Kb	0.0100	0.0135
   	Ca S	Kymax	0.0044	0.0040
   		Ka	0.0140	0.0147
   	S S	Kymax Kb	0.0061 0.0196	0.0060 0.0264
   		Ka	0.0176	0.0198
   Uniformly Varying Load	C C	Kymax	0.0010	0.0009
   	Sa C	Kymax	0.0013	0.0013
   		Kb	0.0043	0.0059
   	Ca S	Kymax	0.0026	0.0024
   		Ka	0.0087	0.0093

   	S S	Kymax	0.0035	0.0034
   		Kb Ka	0.0114 0.0108	0.0137 0.0119
   Parabolically Varying Load	C C	Kymax	0.0006	0.0005
   	Sa C	Kymax	0.0007	0.0007
   		Kb	0.0025	0.0031
   	Ca S	Kymax Ka	0.0017 0.0060	0.0016 0.0064
   	S S	Kymax	0.0022	0.0022
   		Kb	0.0074	0.0083
   		Ka	0.0073	0.0080

   Wmax = Kymax

   q a 4 D

   , = K

   q a3 D

   * Warren C. Young (1989)

   a Boundary condition at inner edge of the plate

4. Parametric study

   In order to bring out the effect of shear deformation in case of plates undergoing axisymmetric deformations, different shear moduli are considered.

   0.5 E

   assumed to be appropriate even in the case of composite plates and the same is followed here. It can be seen that, the increase in orthotropy ratio decreases the displacements for all the lamination schemes considered herein. Also, in case of annular plates, among the different lamination sequences, the least deflection is observed when the laminate is made up of

   (1) Grz

   0.75 E

   (4) Grz

   1. 0.375 E

      laminae in which fibers are arranged in 0º only. Surprisingly, in case of circular plates with simply supported boundary condition, the least deflection is

      1. Grz (5) Grz

         1

         0.625 E

         1

         0.25 E

         observed when the laminate is made up of laminae in which fibers are arranged in 90º only. In case of unsymmetric and anti-symmetric laminates, since the

      2. Grz

      1

      (6) Grz

      1

      coupling rigidity [Bij] exists, radial dsplacements are

      Table 4.1 shows the results of analysis of an isotropic plate undergoing axisymmetric deformations. For very large value of shear modulus, the results of the present work are comparable with the results given by Warren

      C. Young (1989). It can also be observed from this table that the results with different values of shear moduli considered (in the narrow range) are very close to each other and to the value due to CPT, which neglects shear deformation.

      Table 4.2 and Table 4.3 show the results of a study conducted to know the effect of orthotropy ratio on the transverse deflection and radial displacement of laminated polar orthotropic circular and annular plates respectively. The need to select proper shear correction factor K2 in the analysis makes the first order shear deformation theory of the present work somewhat

      also observed.

      Table 4.1 Effect of shear deformation on the deflection of plate

      a/h = 10, Boundary condition: clamped edges

      G (1 ) E	Kymax
      	Present (FSDT)	(CPT)*
      0.75 0.625 0.5 0.375 0.25	0.01562 0.01602 0.01610 0.01622 0.01642 0.01682	0.01563

      q a 4

      empirical. There are different recommendations for the selection of shear correction factors. Quite often the value of 2/12 recommended for isotropic plates is

      Wmax = Kymax

      D

      * Warren C. Young (1989)

      Table 4.2 Effect of orthotropy ratio on displacements of laminated polar orthotropic circular plates

      a = 100 mm, h = 10 mm, r = 0.31, Grz/E = 0.253, q = 100 N/mm2 ,

      BC	Displacement	Er/E	5	10	15	30
      		Lamination scheme
      		90º	0.0582	0.0408	0.0327	0.0223
      		90º/0º/90º	0.0559	0.0378	0.0297	0.0195
      		90º/0º/0º/90º	0.0516	0.0333	0.0256	0.0166
      	W	0º/90º	0.0469	0.0304	0.0234	0.0153
      Clamped		90º/0º/90º/0º	0.0438	0.0271	0.0206	0.0136
      		0º	0.0428	0.0250	0.0191	0.0125
      		0º/90º/0º	0.0417	0.0248	0.0190	0.0125
      		0º/90º/90º/0º	0.0403	0.0232	0.0186	0.0123
      	U	0º/90º	0.0172	0.0056	0.0040	0.0027
      		90º/0º/90º/0º	0.0026	0.0018	0.0014	0.0008
      		0º	0.6379	0.4349	0.4204	0.4090
      		0º/90º/90º/0º	0.2270	0.0494	1.2854	0.3627
      		0º/90º/0º	0.1846	0.4104	0.3639	0.2728
      	W	0º/90º	0.1933	0.1820	0.1328	0.0796
      Simply		90º/0º/90º/0º	0.1812	0.1063	0.0759	0.0425
      Supported		90º/0º/0º/90º	0.1459	0.0835	0.0597	0.0340
      		90º/0º/90º	0.1418	0.0815	0.0585	0.0336
      		90º	0.1404	0.0810	0.0584	0.0337
      	U	0º/90º	0.0914	0.0498	0.0334	0.0193
      		90º/0º/90º/0º	0.0098	0.0074	0.0057	0.0033

      w

      w

      W U

      h

      u a p

      Table 4.3 Effect of orthotropy ratio on displacements of laminated polar orthotropic annular plates

      a = 100 mm, h = 10 mm, b = 10 mm, r = 0.31, Grz/E = 0.253, q = 100 N/mm2

      BC	Displacement	Er/E	5	10	15	30
      		Lamination scheme
      		90º	1.2031	1.0107	0.8956	0.7150
      		90º/0º/90º	1.1303	0.9063	0.7823	0.6038
      		90º/0º/0º/90º	0.9981	0.7600	0.6437	0.4938
      	W	0º/90º	0.8386	0.6513	0.5634	0.4489
      C C		90º/0º/90º/0º	0.7408	0.5472	0.4704	0.3831
      		0º/90º/90º/0º	0.5934	0.4470	0.3941	0.3375
      		0º/90º/0º	0.5746	0.4354	0.3857	0.3327
      		0º	0.5673	0.4309	0.3825	0.3309
      	U	0º/90º	0.2743	0.2283	0.1888	0.1236
      		90º/0º/90º/0º	0.1108	0.0796	0.0610	0.0360
      		90º	3.9604	1.7014	1.4370	1.0204
      		90º/0º/90º	2.9180	1.6621	1.2933	0.9317
      		90º/0º/0º/90º	2.3716	1.5949	1.2860	0.8963
      		0º/90º	1.7369	1.1620	0.9406	0.6936
      		90º/0º/90º/0º	1.7061	1.1210	0.9014	0.6576
      	W	0º/90º/90º/0º	1.3923	0.9238	0.7560	0.5806
      S – S		0º/90º/0º	1.3455	0.8932	0.7346	0.5722
      		0º	1.3255	0.8814	0.7265	0.5692
      	U	0º/90º	0.3120	0.3015	0.2637	0.2036
      		90º/0º/90º/0º	0.1333	0.1162	0.0954	0.0664

      W 10

   2. w h

   U 10

5. Conclusions

   Studies have been conducted to understand the static deflection characteristics of composite circular and annular plates by the Chebyshev collocation technique using first order shear deformation theory. Convergence tests were conducted for the Chebyshev collocation technique and it can be seen that there is excellent convergence even when we take four or six terms in the series for the problems considered. Further, numerical results have been worked out for isotropic and polar orthotropic laminated circular and annular plates with different combinations of clamped and simply supported boundary conditions.

   From the results it can be seen that, the increase in orthotropy ratio decreases the displacements for all the lamination schemes considered. Also, in case of annular plates, among the different lamination sequences, the least deflection is observed when the laminate is made up of laminae in which fibers are arranged in 0º only. Surprisingly, in case of circular plates with simply supported boundary condition, the least deflection is observed when the laminate is made up of laminae in which fibers are arranged in 90º only. In case of unsymmetric and anti-symmetric laminates, since the coupling rigidity [Bij] exists, radial displacements are also observed.

6. References

1. Ravichandran, V., Some studies on the analysis of circular multilayer plates, M.Tech. Dissertation, Department of Applied Mechanics, IIT Madras, India,1989.

2. Warren C. Young, ROARKS FORMULAS for stress and strain, Mc-Graw Hill Book Company, 6th edition, 1989.

3. William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, U.S.A, 2002.

4. Fox, L., and Parker, I.B., Chebyshev Polynomials in Numerical Analysis, Oxford University Press, 1968.

5. Robert M. Jones, Mechanics of composite materials, Hemisphere Publishing Co., New York, 1975.

6. Antia, H.M., Numerical methods for scientists and engineers, Hindustan book agency, Newdelhi, India, 2002.