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					                               4 ANALYSIS OF LAMINATES
                                      4.1 STATIC ANALYSIS
                    4.1.1 Energy Method of Analysis of Structures
Principle of Total Potential Energy:
      If the deformation of an elastic body can be expressed in terms of admissible functions that satisfy
the kinematic boundary conditions, then the total potential energy (TPE) of the body is stationary. In
other words, the first variation of the TPE with respect to coefficients of the admissible functions is zero.
                                       
Total potential energy,   U  V      ddv   pdu
                                      v0
where       U = Strain energy of the elastic body
            V = Potential energy due to work done by the applied forces
            , u, and p are stress, strain, displacement, and the loading, respectively.

                   
      Principle:         0 ; C i represent the generalized coordinates .
                   c i
Procedure:
Step 1:      Select admissible functions that satisfy the geometric boundary
             conditions of the problem. Define the displacements as a function
             of admissible functions and undetermined constants called
             generalized displacements.                 u  ci  i .

Step 2:      Calculate the total potential energy ( = U + V ) of the body using
             the appropriate strain-displaceme nt and constitutiv e equations.

Step 3:      Minimize the TPE () with respect to the generalized
             displacements. This results in a set of algebraic equations to be
             solved for generalized constants.

Step 4:      Number of equations will be equal to number of constants. Then
             Substitute the constants in the displacement equation. This
             completes the analysis.
Example:    Clamped-clamped rectangular plate subjected to an uniformly
            distributed load, -q.
                                                                          y




                                                                              Material system
Boundary Conditions:                                                                 2
                           Žw
            w = 0; a nd       = 0 @ x = 0 and x = a
                           Žx
                                                                                            1   b
                           Žw
            w = 0; a nd       = 0 @ y = 0 and y = b
                           Žy

Displacement approximation:                                                                         x
                                                                                      a
                       m    n
                                                        2jy
            w(x,y) = •      •    cij 1- cos 2ix 1- cos
                                             a           b
                       i=1 j=1
                                                          2y
One-term approximation: w( x,y) = c 1- cos 2x 1- cos
                                            a              b
Total Potential Energy, : (Specially orthotropic symmetric laminate )
              b
                      a
                                  2  2        2   2          2  2         2       2
(w ) =                   1 D 11 Ž w + 2D 12 Ž w Ž w + D 22 Ž w + 4D 66 Ž w           - qw dx dy
                          2      Žx2         Žx2 Žy2        Žy2         Žx Žy
                  0
          0


Substituting for w in the above e quation, and performing partial differe ntiation                 w.r.to c leads to

                  4ab4 c 3D 11 + 2D 12 + 3D 22 + 4D 66 = qa b
                           a4     a 2b 2   b4     a 2b 2

Example:
For a square (a = b) and isotropic plate {D           11   = D 22 = D, D 12 = D, and D 66 = (1-)D/2}, the expression for c
simplifie s to
                   qa 4
               c=
                  324 D
therefore, the expression for w is
                  4
      w(x,y) = qa    1-cos 2x 1-cos 2y
                            a
              324 D                  b

                                                         qa 4           qa 4
wmax is at the center of the plate,            wmax =         = 0.00128
                                                        84 D            D
Series solution by Evans (1939):
                 qa 4
wmax = 0.00126        ; The approximate solution is about 1% larger than the exact solution.
                  D
    Section 4.2 CLASSICAL METHODS                                                     y
      Analysis of Simply-Supported Laminated Plate                                                SS
            Subjected to an Uniform Load (p)

                                                                                          Material system
                                                                                             2
                                                                                 SS                    1    b   SS




Case 1: Specially Orthotropic Laminate                                                           SS                  x
ŅMaterial axes parallel to the Plate axesÓ - All coupling (Normal-Shear and
Bending-Stretching) terms are ZERO.                                                               a

GDE: D11 w, xxxx  2D12  2D66 w,xxyy  D22 w, yyyy  p                  (1)


Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = -D 11 w,           xx   - D 12 w,   yy   =0

      @ y=0 & b: w = 0 & M     y   = -D 12 w,   xx   - D 22 w,   yy   =0
Selection of Displacement Functions: Because the GDE & BCs are even
derivatives of x and y, we can select a solution in the form:
                      
                                                ny
       w      a mn sin max sin               b                                      (2)
             m1 n1

Loading:

       Different types of loading can be expressed using the Fourier series,
as
                      
                                               n y
        p     pmn sin ma x sin                 b                                     (3)
             m1 n1

The Fourier coefficients are calculated for each type of loading as follows
ab                                            ab      
                  mx    ny                                 mx          n y  mx    ny
  p(x, y) sin    a sin b dxdy                  pmn sin a sin        b sin a sin b dxdy
00                                            0 0 m1 n1

                       ab
                                          mx   ny
        pmn       4
                  ab     p(x, y) sin     a sin b dxdy                                 (4)
                       00

For an uniform loading of              qo,
                           ab
                  4q o                       n y      16 qo
        pmn      ab         sin ma x sin    b dxdy   2 mn ,   For odd numbers of m & n.
                           00
Solution:

         Substituting Eqs 2 & 3 in Eq 1 (GDE), we get

                                    pmn
a mn 
               
          4 D11 m   2D12  2D66 m 
                  a
                       4
                                       a         
                                                2 n 2
                                                  b         D22 n 
                                                                  a
                                                                      4
                                                                          
         (5)


                                        16qo sin   mx   sin ny
w                                                 a         b
         m1,3 n1,3 mn
                           6
                               
                               D11m  2 D12  2D66  m 
                                   a
                                     4
                                                       a             
                                                                     2 n 2
                                                                       b       D22 n 
                                                                                     a
                                                                                       4
                                                                                           
         (6)
Case 2: Symmetric Angle Ply Laminate
       Because of symmetry inplane displacements decouple transverse displacement, however bending-twisting
coupling stiffness (D 16 and D 26 ) are non zero.

GDE: D11 w, xxxx  4D16 w, xxxy  2D12  2D66 w, xxyy  4D26 w,xyyy  D22 w, yyyy  p(x,y)                         (7)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = -D 11 w,                       xx   - D 12 w,   yy   - 2D 16 w,   xy   =0

         @ y=0 & b: w = 0 & M              y   = -D 12 w,   xx   - D 22 w,   yy   - 2D 26 w,   xy   =0

Solution:
      Because of non-zero C, it is very difficult to select displacement functions that satisfies both GDE and BCS.
Therefore, total potential energy approach is used to solve such problems. In the TPE method, the displacements
must satisfy only the kinematic boundary conditions. Minimization of the TPE satisfies the GDE and the force
boundary condition integrated over the total boundary.
         ab
           D11 (w, xx )                                                                                                      
                             2
   1
     2                            4D16 w, xx w, xy  2D12 w, xx w, yy  2D66 (w, xy ) 2  4D26 w, yy w,xy  D22 (w, yy ) 2  2 pw dxdy
         00

Assume
                   
                                               ny
         w      a mn sin max sin            b
              m1 n1

Substituting w function and minimizing                  w.r.t amn leads to m*n set of equations. Solution of these equations
will give the coefficients amn .
                  For D 22 / D 11 = 1,        (D 12 + 2D 66 )/ D 11 = 1.5 and
                        D16 / D 11 = D 26 / D 11 = -0.5; m = n = 7
                                                0.00425a 4 p
                                       wmax                   ;
                                                     D11
                  Equivalent Orthotropic sol. (D 16 and D 26 = 0 )
                                            0.00324a 4 p
                         w max EqiOrtho        D11

                                                              0.00452a 4 p
                  Exact solution:          w max Exact    
                                                                  D11




* J. E. Ashton, “An Analogy for Certain anisotropic Plates.”, J. Composite Materials, Vol. 3, 1969, pp. 355-358.
                        Symmetric Angle-ply Laminate (AS4 Carbon/epoxy)
                                                           (q/-q)ns AS4/3501-6 Carbon/Epoxy Laminate
          2.0
                                                                                                                  2.0 2.0
                             n=1
                                                                                                                                   n=1
                             n=2
                                                                                                                                   n=2
          1.6                n=3
                             n=4                                                                                  1.6 1.6          n=3
                                                                                                                                   n=4
                             n=5
                                                                                                                                   n=5
          1.2                n=6
                                                                                                                  1.2 1.2          n=6
                             n=7
D16/D11                                                                                                                            n=7
                             n=8
                             n=9
                                                                                                       D26/D11                     n=8
          0.8                                                                                                                      n=9
                             n=10                                                                                 0.8 0.8
                                                                                                                                   n=10

          0.4
                                                                                                                  0.4 0.4

          0.0
                0   6   12    18    24   30     36    42   48     54   60   66   72     78   84   90              0.0 0.0
                                                                                                                      0 0 6   6
                                                                                                                              12 12 18 24 30 36 42 48 54 60 66 72 78 84 90
                                                                                                                                  18 24 30 36 42 48 54 60 66 72 78 84 90
                                              Theta, θ                                                                                      Theta, θ


          2.0                                               2.0                                                   2.0                           2.0
                                                                                 n=1                                                                         n=1
                                                                                 n=2                                                                         n=2
          1.6                                               1.6                  n=3                              1.6                           1.6          n=3
                                                                                 n=4                                                                         n=4
                                                                                 n=5                                                                         n=5
          1.2                                               1.2                  n=6                              1.2                           1.2          n=6
                                                                                 n=7                   D16/D22                                               n=7
D26/D22                                                                          n=8                                                                         n=8
          0.8                                                                    n=9                                                            0.8          n=9
                                                            0.8                                                   0.8
                                                                                 n=10                                                                        n=10

          0.4                                               0.4                                                   0.4                           0.4


          0.0                                                0.0                                                   0.0                          0.0
                0   6   12   18     24   30    36    42    48 54 60 66 72 78 84 90
                                                                 0 6 12 18 24 30 36                     42   48   54 600    6
                                                                                                                            66 12 18 24 30 36
                                                                                                                               72 78 84 90      42 048   54 60 66 72 78 84 90
                                                                                                                                                         6 12 18 24 30 36 42 48   54   6
                                                    Theta, θ                                                                              Theta, θ
Case 3: Anti-Symmetric Cros s- Ply Laminate

Prope rties:

       A11 = A 22 and D 11 = D 22 , and B 22 = -B 11

       Other non zero terms are: A       12 ,   A 66 , D 12 , and D 66 .

       Normal-stretching and bending-twisting terms are zero.

GDE:
       A11 u, xx  A66 u, yy  A12  A66 v,xy  B11 w,xxx  0                        (8)

       A12  A66 u, xy  A66 v, xx    A11 v, yy  B11 w ,yyy  0                    (9)

       D11 (w,xxxx  w,yyyy )  2D12  2 D66 w,xxyy  B11 (u,xxx  v,yyy )  p       (10)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = B 11 u, x -D11 w,                 xx   - D 12 w, yy = 0

                     v = 0 & Nx = A 11 u, x + A 12 v, y -B11 w, xx = 0


       @ y=0 & b: w = 0 & M        y   = -B 11 v, y -D12 w, xx - D 22 w, yy = 0

                     u = 0 & Ny = A 12 u, x + A 22 v, y +B11 w, yy = 0
Displacement Functions:

                              
                                                ny
                   u       Amn cos ma x sin    b
                          m1 n1
                            
                                                ny
                   v       Bmn sin ma x cos    b
                          m1 n1
                            
                   w        Cmn sin ma x sin ny
                                        
                                                 b
                          m1 n1


                                        y
Loading:            p  qo sin a sin
                                x
                                        b
     Anti-Symmetric Cross-Ply Laminate

        0.25
                           (0/90)n Laminate
                           AS4/3501-6 Carbon/Epoxy
             0.2


        0.15
B22/(hA11)
             0.1


        0.05


              0
                   0   1    2     3    4       5     6   7   8   9   10
                                           n
Case 4: Anti-Symmetric Angle-Ply Laminate

Prope rties:

       Non Zero Extensional Stiffness: A         11 ,   A 22 , A 12 and A 66.

       Non Zero Flexural Stiffness:           D11, D22 , D 12 , and D 66 .

       Bending-Extensional Coupling Stiffness:                 B16 and B 26

GDE:
       A11 u, xx  A66 u, yy  A12  A66 v,xy  3B16 w, xxy  B26 w, yyy  0                                    (11)

       A12  A66 u, xy  A66 v, xx    A22 v,yy  B16 w, xxx  3B26 w,xyy  0                                   (12)

       D11 w ,xxxx  2D12  2D66 w, xxyy  D22 w,yyyy  B16 (3u, xxy  v, xxx )  B26 (u, yyy  3v, xyy )  p   (13)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = B 16 (u, y +v, x ) - D 11 w, xx - D 12 w, yy = 0

                     u = 0 & Nxy = A 66 (u, y +v, x ) - B 16 w, xx - B 26 w, yy = 0


       @ y=0 & b: w = 0 & M        y   = B 26 (u, y +v, x ) - D 12 w, xx - D 22 w, yy = 0

                     v = 0 & N xy = A 66 (u, y +v, x ) - B 16 w, xx - B 26 w, yy = 0
                                              T11  A11 (  )  A66 (  )
                                                          m 2            n 2
                                                            a             b
     Displacement Function                    T12 ( A12  A66 )(  )(  )
                                                                 m     n
                                                                a     b
                              mx cos n y
         u     Amn sin
                                              T13  [3B16 (  ) 2  B26 (  ) 2 ](  )
                               a        b                     m            n       n
              m1 n1
                
                                                               a            b       b
                            m x        ny
         v        Bmn cos a sin            T22  A66 (  ) 2  A22 (  ) 2
                                                         m             n
                                         b
              m1 n1                                      a            b
                
                                              T23  [ B16 (  ) 2 3B26 (  ) 2 ](  )
                                       ny                 m              n        m
         w      Cmn sin ma x sin      b                    a             b        a
              m1 n1
                                              T33  D11 (  ) 4  2( D12  2 D66 )(  ) 2 (  ) 2  D22 (  ) 4
                                                         m                         m       n             n
                                                           a                        a       b             b
      Loading:
                         x         y
           p  po Sin         Sin                     T12T23 T22T13
                         a          b         Amn                  p0
                                                           D
                                              Bmn  12 13  11 23 p0
                                                    T T T T
                                                          D
                                                                2
                                            Cmn 
                                                    T11T22 T12
                                                                  p0
                                                         D
                                              where
                                                   T11 T12 T13
                                              D  T12 T22 T23
                                                  T13 T23 T33
        Anti-Symmetric Angle-Ply Laminate
                         (q/q)n AS4/3501-6 Laminate
       0.2       (q/q)n Laminate              n=
                 AS4/350-6                      1
      0.15
                 Carbon/epoxy

B16/(hA11)
       0.1                                     n=
                                                2                      n = 10
                                               n=
      0.05                                     n3=
                                                4
         0
             0    10   20    30      40   50         60   70           80        90
                                    q, Degrees
                                                          0.8

                                                                            (q/-q)n Laminate
                                                                            AS4/350-6
                                                          0.6
                                                                            Carbon/epoxy                  n=1
                                                     B26/(hA11)
                                                          0.4
                                                                                                                          n = 10
                                                                                                          n=2
                                                          0.2                                              n=3
                                                                                                           n=4

                                                               0
                                                                   0        10        20   30   40   50   60    70   80   90

                                                                                                 q Degrees
                       VARIATION OF WMAX WITH q
                Anti-Symmetric Angle-Ply laminate (a/b=1)
                10
                                                             n=1


                 8
wmax104/p0b4




                 6
                                                              2
                                                              3
                 4                                         Orthotropic


                 2         Material AS4/3501-6 Carbon/Epoxy Composite
                                   (q/-q)n laminate

                 0
                     0        15                      30                 45
                                          q, deg




                                                                              Anil Bhragava
           Variation of WMAX With a/b Ratio
          Anti-Symmetric Angle-Ply laminate
100
           Material AS4/3501-6 Carbon/Epoxy Composite
                   (30/-30)n laminate
                                                          n=1

 75

                                                           2
                                                           3
 50                                                     Orthotropic




 25



  0
      0                      5                    10                  15
                                        a/b
4.3 Buckling Analysis of Laminated Plates                                             y
                                                                                          Ny
4.3.1 Buckling Equations of Equilibrium.                                                       N xy


                                                                                 Nx               Nx

        N x  N yx
                     0
        x      y
        N xy  N y
                     0
         x     y                                                                                     x
                                                                     N xy
        2 Mx      2 M xy  2 M y       2w     2w        2w                           Ny
               2                  N x 2  N y 2  2N xy       0 (1)
        x 2       xy      y 2       x      y         xy


Where N x , N y , and N xy are the edge loads.
If the prebuckling state of the laminate is not flat we can re-write the above
equation in a variation of the prebuckled state.
          4.3.2 Buckling of S-S laminated Plate Subjected toN x Loading.
                                                 y

                                                                  SS

                                                 SS
                                        Nx               Material system SS   Nx
                                                            2
                                                                       1 b


                                                                  SS
                                                                               x
                                                                   a


Case (1): Symmetric Specially Orthotropic laminate
GDE: D11w ,xxxx  2D12  2D66 w,xxyy  D22 w , yyyy  N x w, xx  0              (2)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = -D 11 w,            xx   - D 12 w,   yy   =0

      @ y=0 & b: w = 0 & M      y   = -D 12 w,   xx   - D 22 w,   yy   =0

Selection of Displacement Functions: Because the GDE & Bcs are even derivatives of x and y, we can select a
solution in the form:
                
                                               ny
       w     a mn sin max sin               b                                                    (3)
            m1 n1

Where m and n are number of buckled half-waves in x- and y-directions respectively. Substituting for ŌwÕ in the
GDE, we get for a non-trivial solution

       N x mn   2    
                         D        11 a 
                                     m 2     2D12  2D66 n   D22 n 
                                                             b
                                                                  2
                                                                        b       
                                                                             4 a 2
                                                                               m                     (4)

Buckling load is a function of                (i)     Elastic properties of the material.
                                              (ii)    Geometry (b and aspect ratio, a/b)
                                              (iii)   Number of half-waves in x- and y-directions.

Minimum buckling load occurs @ n = 1.

       N x m  
                 b
                     2
                     2   
                         D       11   mb 2  2D12  2D66   D22 bm 2 
                                       a
                                                                      a                              (5)

Minimization of the above                Eq 5 w.r.to aspect ratio (a/b), we get

       N x min  2b  D11 D22  D12  2D66 
                             2
                         2                                                                           (6)

for    a / b  m 4 D11 / D22 . Notice that Eq 6 is independent of length (a) and m.
Example:     For D 11 /D22 = 10 and              (D 12 + 2D66)/D 22 = 1

             N x min        2  2 D22
                                  b2
                                            10 1

Is otropic Plate:


      Nx Iso     2 D
                    b2     
                           
                           mb 2
                           a       2  bm 
                                           a    2
                                                    
                   Buckling of Simp ly Sup ported laminated p lates
                        (n=m=1 for Carbon/Epoxy, Boron/Epoxy
                             & Graphite/Epoxy Laminate)
          60



                                    Gr/Epoxy (E 1/E2 = 45.9)
          50



                                               C/Epoxy (E 1/E2 = 13.8)
          40
N xb 2
 2 D22
     2
 Nx b  30                                               B/Epoxy (E 1/E2 = 10)
______
  2
_ D22

          20




          10




          0
               0    1               2                   3                       4   5

                                    Aspect ratio
                                   AspectRatio (a/b)(a/b)
                   Buckling of Simply Supported laminated plates under in-plane load
                                         Carbon/Epoxy Laminate
          70


                             (m = 1)n=2            (m = 2)n=2

          60


                                                                                          M in. Buckling Load
          50                                                                                     n=2
                                                                    (m = 3)n=2



N x b22   40
 Nx b
______
  2
    D22
 _2 D22   30
                                          (m = 1)n=1
                                                                         (m = 2)n=1
          20                                                                                           (m = 3)n=1



          10




           0
               0      1      2            3        4            5          6          7         8         9         10

                                                                                                    M in. Buckling Load
                                                       Aspect Ratio (a/b)                                  n=1
Case 2: Symmetric Angle Ply Laminate
       Because of symmetry inplane displacements decouple transverse displacement, however bending-twisting
coupling stiffness (D 16 and D 26 ) are non zero.

GDE: D11w ,xxxx  4D16 w, xxxy  2D12  2D66 w,xxyy  4 D26 w,xyyy  D22 w, yyyy  N x w ,xx  0    (7)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = -D 11 w,            xx   - D 12 w,   yy   - 2D 16 w,   xy   =0

      @ y=0 & b: w = 0 & M      y   = -D 12 w,   xx   - D 22 w,   yy   - 2D 26 w,   xy   =0

Solution:    Energy method is used to obtain the solution.


Assumed displacements:
                
                                    ny
      w      a mn sin max sin    b
            m1 n1
Case 3: Anti-Symmetric Cross- Ply Laminate
Prope rties:

       A11 = A 22 and D 11 = D 22 , and B 22 = -B 11

       Other non zero terms are: A       12 ,   A 66 , D 12 , and D 66 .

       Normal-stretching and bending-twisting terms are zero.

GDE:
       A11 u, xx  A66 u, yy  A12  A66 v,xy  B11 w,xxx  0                                   (8)

       A12  A66 u, xy  A66 v, xx    A11 v, yy  B11 w ,yyy  0                               (9)

       D11 (w, xxxx  w, yyyy )  2D12  2D66 w, xxyy  B11 (u, xxx  u, yyy )  N x w,xx  0   (10)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = B 11 u, x - D 11 w, xx - D 12 w, yy = 0

                     v = 0 & Nx = A 11 u, x + A 12 v, y -B11 w, xx = 0


       @ y=0 & b: w = 0 & M        y   = -B 11 v, y -D12 w, xx - D 22 w, yy = 0

                     u = 0 & Ny = A 12 u, x + A 22 v, y + B 11 w, yy = 0
Displacement Functions:

                        u  u cos ma x sin ny
                                   
                                            b
                                         ny
                        v  v sin ma x cos b                            (11)
                        w  w sin ma x sin n y
                                   
                                             b

Buckling solution:

        N x mn  m  T33 
                     a       2
                                    2T12 T23T13 T22 T13 T11T23
                                             T11T22 T12
                                                        2

                                                        2
                                                                2

                                                                       (12)

Where
        T11  A11 m   A66 nb 
                         2             2
                    a

                            a 
        T12  A12  A66 m  nb 
                                 


        T13  B11 m 
                             3
                     a
                                                                           (13)
        T22  A11 nb   A66 m 
                        2               2
                                a

        T23  B11 nb 
                        3


        T33  D11   
                    
                    m 4
                     a      nb 4  2D12  2D66 ma 2 nb 2
                                       
Case 4: Anti-Symmetric Angle-Ply Laminate
Prope rties:

       Non Zero Extensional Stiffness: A         11 ,   A 22 , A 12 and A 66.

       Non Zero Flexural Stiffness:           D11, D22 , D 12 , and D 66 .

       Bending-Extensional Coupling Stiffness:                 B16 and B 26


GDE:
       A11 u, xx  A66 u, yy  A12  A66 v,xy  3B16 w, xxy  B26 w, yyy  0                                     (14)

       A12  A66 u, xy  A66 v, xx    A22 v,yy  B16 w, xxx  3B26 w,xyy  0                                    (15)

       D11 w ,xxxx  2D12  2D66 w, xxyy  D22 w,yyyy  B16 (3u, xxy  v, xxx )  B26 (u, yyy  3v, xyy )  N x w,xx  0 (16)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = B 16 (u, y +v, x ) - D 11 w, xx - D 12 w, yy = 0

                     u = 0 & Nxy = A 66 (u, y +v, x ) - B 16 w, xx - B 26 w, yy = 0


       @ y=0 & b: w = 0 & M        y   = B 26 (u, y +v, x ) - D 12 w, xx - D 22 w, yy = 0

                     v = 0 & Nxy = A 66 (u, y +v, x ) - B 16 w, xx - B 26 w, yy = 0
Displacement Functions:

                                        ny
                       u  u sin ma x cos b
                                              ny
                       v  v cos ma x sin       b                            (17)

                       w  w sin        mx sin ny
                                         a       b

Buck ling Solution:


        Nx mn     a 2
                                                                      
                                                           2       2
                                        2T12 T23T13 T22 T13 T11T23
                      m     T 33                         2                 (18)
                                                T11T22 T12

Where

        T11  A11 m   A66 nb 
                        2                 2
                    a

                          a 
        T12  A  A66 m  nb 
                12
                               


                
        T13   3B16 m   B26 n 
                       a
                              2
                                   b
                                                 2
                                                     
                                                     n
                                                       b   
                                                                                        (19)
                    2
                              
                                      2
        T22    A22 nb        A66 m
                                   a

        T23  16 m    3B26 n   a 
                                        
                         2            2 m
                B    a             b

        T33  D11 m   2 D12  2D66 m                          D22 nb 
                         4                                     2 n 2              4
                    a                      a                      b
                         Equation   Edge                                                                                Comments (Assumptions)
          Loads          No.        Conditio
                                                                           Buckling Loads
                                    ns
                                                                                                                                  Ny
                                                                                                                             
                         2.2.2-14   All                                                                                 1.
Biaxial Compression                                                                                                             Nx
                                                                                                                             m,n  1,2,...,
      Y                             sides
                                                                  4                        2                2.Equation to be minimized with
          Ny                        S-S                    D11m4 b   2D12  2D66 m2 n2 b   D22 n4       respect to
                                                         
                                                         2
                                                                   a                        a                 3.a/b finite
                                               Nx ,c r  2                                                    
                                                        b                      
                                                                               2 b
                                                                                      2
                                                                                                                      4. Reference 2.2-20
Nx                  Nx
                                                                            m    n 2
                                                                                                                
                                                                              a                           min
                                                                                                                 ,

               Ny   X




                                                                                                                              m,n  1,2,...,
                         2.2.2-15   All                                                                                 1. Equation to be minimized with
Uniaxial Compression
                                                         2                                     n4 b 2 
                                                                                                                        respect to
                                    sides                         b 2                                  
                                               Nx ,c r  2  D11m2    2D12  2D66n 2  D22 2              2.a/b finite
                                    S-S                 b       a                                    
                                                                                                  m a  min        3. Reference 2.2-20
                                                                                                              ,


      Y


                         2.2.2-16   All                                                                                 1.a/b=infinite

                                                                      2

                                                                           D                                 
 Nx                 Nx                                                                                                  2. Reference 2.2-21
                                    sides
                                    S-S          Nx ,c r  2                        D22  D12  2D66
                                                                   b2          11
                    X




                         2.2.2-17   All                2  2 b 2
                                                                                     1 a 2        
                                                                                                                      1.a/b=finite

                                    sides      Nx,cr  2 D11m    2.67D12  5.33D22 2    D66           2.Minimized
                                                      b     a                  m b 
                                                                                                        
                                                                                                                    3. Reference 2.2-21
                                    fixed

                         2.2.2-18   All                                                                                 1.a/b=infinite
                                                              2
                                    sides
                                    fixed
                                                  Nx ,c r 
                                                              b2
                                                                   4.6     D11 D22  2.67D12  5.33D66                2. Reference 2.2-21
                                2.2.2-19   Three                                                                                                    1.a/b=finite
                                                                            D                                                                       2.Minimize Ks with respect
                                           sides S-S        N x ,cr     K s 11                                                                     to
       Uniaxial                            and one                          b2                                                                         m,n  1,2,...,

     compression                           side free    Where Ks is found from the solution of the transcendental equation                          3. Reference 2.2-19

                                                               2 mb 2  2 mb 2 
                                                                      
                                                                                               tan  
                                                              
                                                                    a  
                                                                                      a  x y
                                                                   mb 2  2 mb 2        
                                                                 
                                                               2
                                                                                    x y tan 
                                                                    a  
                                                                                      a        
           Y

               fre
                                                                                                     
     b         e
Nx       s.s          s.s                               With                                                                                    1
         .      s.s   .
                                                                                                                             1              2
                .a          X
                                                                 D 2 mb 2  D 2 mb 2 N b2  D 2  D mb 
                                                            mb  3                                                                    
                                                                             11              x  11   3           
                                                                  D   a   D   a    D D   D  a 
                                                             a  22               22         11  22   22       
                                                                
                                                                                                                                            
                                                                                                                                              
                                                                                                                                                1
                                                                                                                             1               
                                                                                                                                                2
                                                                                                                           2  
                                                                  
                                                            mb  D3  mb   D11  m b   Nx b 2  D11   D3  mb 
                                                                            2                   2
                                                                                     2                  2
                                                                                                                                               
                                                                                                           
                                                                    
                                                             a  D22   a   D22   a   D11  D22   D22  a 
                                                                     
                                                                                                             
                                                                 
                                                                                                                                             
                                                                                                                                               
                                                                              4Gx y    x y y x 
                                                                                    1
                                                       D3  D12  2D66 ,                              xy
                                                                                      Ey

                                2.2.2-20   Three                                                                                                    M=1 is minimum
                                                                                 m2 2 E x                                    
                                                                      3 Gx y
                                                                                                                                                    Reference 2.2-36
                                           sides S-S
                                                          Nx ,c r  h                                                          
                                           one side
                                           free
                                                                         b2 12a 2 (1  x y y x
                                                                                                                             
                                                                                                                                



                                                                                   h 3Gx y                       a
                                                               Nx ,c r                      2           ,         
                                                                                         b                       b
4.4 Free Vibration Analysis of Laminated
                  Plates
4.4.1 Equations of Equilibrium of a Transversely Vibrating Laminate

                N x  N yx
                           0
                x     y
                N xy  N y
                            0
                 x     y
                2 Mx     2 Mxy  2 My
                    2  2 xy       2  h w  0
                                             ÝÝ                       (1)
                 x                y

Where    w t 2 , acceleration
            2
      ÝÝ
       4.4.2 Vibrations of a S-S laminated Plate.

Case (1): Symmetric Specially Orthotropic laminate
GDE: D11w ,xxxx  2D12  2D66 w,xxyy  D22 w , yyyy  w,tt  0               (2)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = -D 11 w, xx - D 12 w, yy = 0

       @ y=0 & b: w = 0 & M             y   = -D 12 w, xx - D 22 w, yy = 0

Selection of Displacement Functions: Free vibrations of an elastic
continuum is harmonic, hence we can express the time variation of
displacement in terms of sin, cos or ei t functions. Using the principle of
separation of variable, we can write the m & n th mode of vibration in the
form of
                   it                 ny
       w  amn e         sin ma x sin    b                                      (3)

Where  is the natural frequency of vibration, expressed in terms of
radians/sec. Substituting for ŌwÕ in the GDE, for non-trivial solution, we
get
        4

            
 2   D11 m   2D12  2D66 m 
              a
                    4
                                   a                
                                                   2 n 2
                                                     b      D22 n 
                                                                  b
                                                                    4
                                                                               (4)

 is function of         (i)     Elastic properties of the material.
                         (ii)    Geometry (a and a/b)
                         (iii)   Number of half-waves in x- and y-directions.
The fundamental or the lowest frequency is when m=n=1.

                                             D11 b 4  2 D12 2D66  b 2  1
                                              22 a 
  Fundamental   2
                                     D22
                                                                       a  
                             2
                                                                                                          (5)
                   b                         
                                              D                D22             

Example:       For D 11 /D22 = 10, (D 12 + 2D 66 )/D 22 = 1,
               and a/b=1 (Square Plate)
                                                                                                                    Specially Orthotropic          Isotropic
              2       D22               4              2 2         4
            b2                10m  2m n  n
                                                                                                                                        Mode
                                                                                                                         Nodal line         1st    Nodal line
For isotropic material:                       
                                                 b2
                                                    2
                                                          D
                                                             m    2
                                                                        n   2
                                                                                 
                                                                                                                         Nodal line                Nodal line
                                                                                                                                            2nd
                     Specially orthotropic                                           Isotropic
 Mode              m       n       b 2                   D22
                                                                        m            n
                                                                                            b
                                                                                                     2    
                                                                                                                D
   Ist             1             1               3.606                  1            1            2.0
                                                                                                                                             3rd
  2nd              1             2               5.831                  1            2            5.0
  3rd              1             3              10.440                  2            1            5.0
  4th              2             1              13.000                  2            2            8.0

                                                                                                                                            4th
Case 2: Symmetric Angle Ply Laminate
      Because of symmetry, the inplane and transverse displacements GDE will decouple into 3 independent
Equations. However bending-twisting coupling stiffness (D   16 and D 26 ) are non zero.

GDE: D11 w ,xxxx  4D16 w,xxxy  2D12  2D66 w, xxyy  4D26 w, xyyy  D22 w,yyyy  w,tt  0        (7)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = -D 11 w,             xx   - D 12 w,     yy   - 2D 16 w,   xy   =0

      @ y=0 & b: w = 0 & M       y   = -D 12 w,   xx   - D 22 w,     yy   - 2D 26 w,   xy   =0

Solution:    Energy method is used to obtain the solution.

              =U+ V+ T
                                                                          2
Where T is the kinetic energy of the laminate,          T   1
                                                             2    w dV
                                                                    Ý
                                                                 V

Assumed displacements:

                          ny it
       w  amn sin ma x sin b e
Case 3: Anti-Symmetric Cross- Ply Laminate
Prope rties:

       A11 = A 22 and D 11 = D 22 , and B 22 = -B 11

       Other non zero terms are: A       12 ,   A 66 , D 12 , and D 66 .

       Normal-shear and bending-twisting terms are zero.

GDE:
       A11 u, xx  A66 u, yy  A12  A66 v,xy  B11 w,xxx  0                                (8)

       A12  A66 u, xy  A66 v, xx    A11 v, yy  B11 w ,yyy  0                            (9)

       D11 (w, xxxx  w, yyyy )  2D12  2D66 w, xxyy  B11 (u, xxx  u, yyy )  w,tt  0   (10)


Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = B 11 u, x - D 11 w, xx - D 12 w, yy = 0

                     v = 0 & Nx = A 11 u, x + A 12 v, y -B11 w, xx = 0


       @ y=0 & b: w = 0 & M        y   = -B 11 v, y -D12 w, xx - D 22 w, yy = 0

                     u = 0 & Ny = A 12 u, x + A 22 v, y + B 11 w, yy = 0
Displacement Functions:
       u  u cos mx sin ny e it
                    a           b
                          ny
        v  v sin ma x cos b e it
                                              (11)

        w  w sin ma x sin ny e it
                   
                            b

Natural Frequency:
               4
                 2T T T T T 2 T T 2
       2   T33  12 23 T13 T 22 13 11 23
                            11 22 T
                                     2
                                    12
                                               
                                               (12)


Where
        T11  A11 m   A66 b 
                        2     n      2
                   a

                           a 
        T12  A12  A66 m b 
                              n


        T13   B11m 
                            3
                    a
                                                           (13)
        T22      
                A11 n
                    b
                      2
                            
                             A66 m
                                  a
                                    2


        T23  B11 n 
                      3
                   b

        T33  D11 a   n   2D12  2D66 m  n 
                   m b 
                        4    4                    2    2
                                               a    b
Case 4: Anti-Symmetric Angle-Ply Laminate
Prope rties:
               Non Zero Extensional Stiffness: A 11 , A 22 , A 12 and A 66.
               Non Zero Flexural Stiffness:     D11, D22 , D 12 , and D 66 .
               Bending-Extensional Coupling Stiffness:          B16 and B 26

GDE:
        A11 u, xx  A66 u, yy  A12  A66 v,xy  3B16 w, xxy  B26 w, yyy  0                                    (14)
        A12  A66 u, xy  A66 v, xx  A22 v,yy  B16 w, xxx  3B26 w,xyy  0                                     (15)
D11 w ,xxxx  2D12  2D66 w, xxyy  D22 w,yyyy  B16 (3u, xxy  v, xxx )  B26 (u, yyy  3v, xyy )  w,tt  0   (16)

Boundary Conditions :
     @ x=0 & a: w = 0 & Mx = B 16 (u, y +v, x ) - D 11 w, xx - D 12 w, yy = 0
                 u = 0 & Nxy = A 66 (u, y +v, x ) - B 16 w, xx - B 26 w, yy = 0

       @ y=0 & b: w = 0 & M y = B 26 (u, y +v, x ) - D 12 w, xx - D 22 w, yy = 0
                  v = 0 & Nxy = A 66 (u, y +v, x ) - B 16 w, xx - B 26 w, yy = 0

Displacement Functions:
                                                      ny
                                    u  u sin ma x cos b e it
                                               

                                                      ny it
                                    v  v cos ma x sin  b e                                                        (17)

                                    w  w sin   mx sin n y e i t
                                                 a        b
Natural Frequency:


                                                          
                                                 2       2
         2      4             2T12T23T13 T22 T13 T11T23
                      T33 
                                      T11T22 T2
                                                                   (18)
                                                12


Where
   T11  A11 m   A66 b 
                     2   n           2
              a

                    a 
   T12  A  A66 m b 
           12
                       n


             
   T13   3B16 m   B26 b 
                 a
                            2
                            n             2
                                           
                                             n
                                              b
                                                                          (19)
   T22        
                 n 2
             A22 b          A66 m
                                a
                                  2


   T23  16 m   3B26 b  m 
                               a
                  2        n 2
           B a

   T33  D11 m   2D12  2D66 m                       D22 n 
                     4                               2 n 2            4
              a                    a                   b            b

				
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