Chapter5

5.1 Basic assumptions 5.2 Strain-Displacement Relations 5.3 Stress-Strain Relations 5.4 Stress Resultants 5.5 General Load-Displacement Relations 5.5.1 Laminate stiffness equation 5.5.2 Laminate compliance equation 5.6 Special Class of Laminates Single layer Symmetric laminates Balanced laminates Quasi-isotropic laminate 5.7 Laminate Engineering Properties 5.8 Analysis of Laminated Composite Structures

Laminate and Deformation Parameters y z

t b a

u0 ub

z A B zb y

x

C D

αxzb

A’ B’

αx C’ D’

w

5.1 Basic Assumptions 1. Each layer of the laminate is quasi-homogeneous and orthotropic. 2. The laminate is thin compared to the lateral dimensions and is loaded in its plane. 3. State of stress is plane stress. 4. All displacements are small compared to the laminate thickness. 5. Displacements are continuous throughout the laminate. 6. Straight lines normal to the middle surface remain straight and normal to that surface after deformation. • In-plane displacements vary linearly through the thickness, • Transverse shear strains (γxz & γyz) are negligible. 7.Transverse normal strain εz is negligible compared to the in-plane strains εx and εy. 8. Strain-displacement and stress-strain relations are linear.

5.2 Strain-Displacement Relations Displacements @ mid-plane

u0 = u0 ( x , y)

z

v0 = v0 ( x, y) w = f ( x, y)

u0 ub A B zb

y

C D

From assumption 5 rotation of normal to the mid planes are

∂w ∂x ∂w αy = ∂y αx =

In-plane displacements @ B are:

∂w ∂x ∂w v = v0 − z ∂y

u = u0 − z

From assumption 6, w(x,y,z) = w(x,y)

αxzb

A’ B’

αx C’ D’

w

Using the small deformation linear theory (Assumption 3 & 7), strain-displacement equations can be written as:

∂u ∂u0 ∂ 2w εx = = −z 2 ∂x ∂x ∂x ∂v ∂v0 ∂ 2w = −z 2 εy = ∂y ∂y ∂y

γ xy

∂u ∂v ∂u0 ∂v0 ∂ 2w =γs = + = + − 2z ∂y ∂x ∂y ∂x ∂x∂y

ε z = γ xz = γ yz = 0 Where the mid-plane strains are given by:

ε x0 = ε y0 = 0 γ xy

∂u0 ∂x

∂v0 ∂y

∂u ∂v = γ s0 = 0 + 0 ∂y ∂x

Curvatures are:

∂ 2w κx = − 2 ∂x ∂ 2w κy = − 2 ∂y κ xy

2∂ 2w =κs = − ∂x∂y

Strains at any point (x,y,z) are: 0 κ x  ε x   ε x  ε  =  ε 0  + z κ   y  y  y  κ s  γ s  γ s0 

Layered Material

Isotropic/Orthotropic Material z

z t/2

ε max

t/2

ε max

4 3

x

y

y

x

2 1

-t/2

−ε max

-t/2

−ε max

5.3 Stress-Strain Relations of an kth Layer in a Laminate Qxx σ x  σ  = Q  yx  y  τ s  k  Qsx

Qxy Qyy Qsy

Qxs   Qys  Qss 

k

ε x  ε   y γ s 

z Kth Layer

y

zk Reference plane

x Qxx σ x  σ  = Q  yx  y  τ s  k  Qsx In short:

Qxy Qyy Qsy

{σ }

k x ,y

Qxs   Qys  Qss 

 ε x0  Qxx  0   ε y  + z Qyx γ s0   Qsx   k

= [Q]

k x ,y

Qxy Qyy Qsy

Qxs   Qys  Qss 

k

κ x  κ   y κ s 

{ε }x,y + z[Q] o

k x ,y

{κ } x , y

Isotropic/Single Layered Material z

ε max

t/2

x

y

−ε max

-t/2

Layered Material z

ε max

t/2

Stress variation

4 3

y

x

2 1

-t/2

−ε max

5.4 Force and Moment Resultants y z

N xk

=

Ns

t/2

∫ σ x dz

Mx

Ms

−t / 2

N yk

=

t/2

t/2

∫ σ y dz

=

N sk

=

=

My

Ns

-t/2

x

M yk

=

∫ τ s dz

Ny

∫ σ x zdz

t/2

∫ σ y zdz

−t / 2

Ms

t/2

t/2

−t / 2

Nx

−t / 2 k N xy

M xk

k M xy

=

M sk

=

t/2

∫ τ s z dz

−t / 2

−t / 2

For layered materials:

Nx  n N  =  y  k∑ =1  N s 

σ x  σ  dz ∫  y hk −1  τ s  k hk

Mx  n M  =  y  k∑ =1  Ms 

σ x  σ  z dz ∫  y hk −1  τ s  k hk

 N x  n Qxx  N  = Q  yx  y  k∑  N s  = 1 Qsx 

[ N ] x, y

Qxs   Qys  Qss 

Qxy Qyy Qsy

hk n  0 k =  ∑ [Q] x , y ∫ dz  ε k =1 hk −1  

[ ]

 ε x0  h Qxx k  0  ε dz +  y ∫ Qyx γ s0  hk − 1  Qsx  k

x, y

[ ]

[ ]

x, y

Qsy

Qxs   Qys  Qss 

κ x  hk  κ  zdz    y ∫ κ s  hk − 1   k

hk n  k +  ∑ [Q] x , y ∫ zdz [κ ] x , y k =1  hk −1

n  k =  ∑ [Q] x , y (hk − hk −1 ) ε 0 k =1 

= [ A] x , y ε 0

Qxy Qyy

x, y

1 n k +  ∑ [Q] x , y hk2 − hk2−1  2 k =1

(

)[κ ]

x, y

+ [ B] x , y [κ ] x , y

Where: k Ai, j = ∑ [Q]i, j (hk − hk −1)

Extensional stiffness. It relates inplane loads to in-plane strains.

1 n [ ]k ( 2 2 Bi,j = ∑ Q i, j hk − hk −1 ) 2 k =1

Coupling stiffness or in-plane/flexure coupling laminate moduli. It relates in-plane loads to curvatures and moments to in-plane strains.

n

k =1

 M x  n Qxx  M  = Q  yx  y  k∑  Ms  = 1 Qsx 

[ M ] x, y

Qxy Qyy Qsy

Qxs   Qys  Qss 

k

1 n  k =  ∑ [Q] x , y hk2 − hk2−1  ε 0  2 k =1 

)[ ]

(

[ ]

= [ B] x , y ε 0 n

x, y

 ε x0  h Qxx k  0  ε zdz +  y ∫ Qyx γ s0  hk − 1  Qsx  

x, y

Qxy Qyy Qsy

Qxs   Qys  Qss 

1 n k +  ∑ [Q] x , y hk3 − hk3−1  3 k =1

(

 κ x  hk κ  z 2 dz    y ∫  κ s  hk − 1  k

)[κ ]

x, y

+ [ D] x , y [κ ] x , y

Ai, j = ∑ [Q]i, j (hk − hk −1 ) k

k =1

Bi, j =

1 k 2 2 − Q h h [ ] ∑ k k −1 2 k =1 i, j

(

)

(

)

n

1 n k Di, j = ∑ [Q]i, j hk3 − hk3−1 3 k =1

Coupling stiffness or in-plane/flexure coupling laminate moduli. It relates in-plane loads to curvatures and moments to in-plane strains. Bending or flexural stiffness. It relates moments to curvatures.

Final Load-Displacement Equation:

 N x   Axx N  = A  y   yx  N s   Asx

Axy Ayy Asy

Axs  ε x0   Bxx    Ays  ε y0  +  Byx Ass  γ s0   Bsx

 M x   Bxx M  = B  y   yx  M s   Bsx

Bxy Byy Bsy

Bxs  ε x0   Dxx    Bys  ε y0  +  Dyx Bss  γ s0   Dsx

Axs Ays Ass Bxs Bys Bss

Bxx Byx Bsx Dxx Dyx Dsx

 N x   Axx  N   Ayx  y   N s   Asx  M x  =  Bxx     M y   Byx  M s   Bsx 

Axy Ayy Asy Bxy Byy Bsy

Bxy Byy Bsy Dxy Dyy Dsy

Bxy Byy Bsy Dxy Dyy Dsy

Bxs  ε x0  Bys  ε y0    Bss  γ s0  Dxs  κ x    Dys  κ y  Dss  κ s 

Bxs  κ x   Bys  κ y    Bss  κ s  Dxs  κ x   Dys  κ y    Dss  κ s 

OR

o  N   A B  ε   M  =  B D  κ      

Final Displacement-Load Equation:

ε x0  a xx  0  a ε y   yx γ s0   a sx κ  =  c xx  x  κ y   c yx κ   c  s   sx

a xy

a xs

b xx

b xy

a yy a sy c xy

a ys a ss c xs

b yx bsx d xx

b yy bsy d xy

c yy c sy

c ys c ss

d yx d sx

d yy d sy

Where:

[ ] { [ [b] = [ B ] [ D ] [c] = −[ D ] [C ]

[a] = A −1 − [ B*] D *

]} [C ] *

* −1

∗

* −1

[ ]

[d ] = D *

−1

−1

*

b xs   N x  b ys   N y    bss   N s  d xs   M x    d ys   M y  d ss   M s 

[ A ] = inverse of matrix [ A] [ B ] = −[ A ] [ B] [C ] = [ B] [ A ] [ D ] = [ D] − {[ B] [ A ]} [ B] −1

∗

or

[ c ] = [ b ]T

OR

ε o  a b   N    = c d   M    κ  

*

*

−1

−1

−1

5.6 Special Class of Laminates Laminate Staking Sequence:

(angle&thickness/. /. /. /angle N&thickness N)

Constant ply thickness laminate: Ply thickness h (0.005±.0005’) (02/455/902/455/02) Special Laminates: Regular - Ply thickness is constant NOTE: Add more discussion on Symmetric and Balanced laminates

Note: Discuss design considerations

Symmetric Laminates A laminate in which for each layer on one side of a reference plane there is an identical layer on the opposite of the reference plane at equal distance with same thickness, material properties, and orientation. Symmetric definition requires symmetry of both geometry and material properties. Example: (0 2/455/902/455/02) = (02/455/90)s z

Then

1 n k 2 Bij = ∑ Qij (hk − hk2−1 ) 2 k =1

k=n

z

k’

hk'

x

hk

k

Qijk

=

(i, j = x, y, s )

n

∑ Qijk hk tk

k =1

Where

− hk'

k' Qij

=

k=2 k=1

t k = t k' hk =

1 n k = ∑ Qij (hk + hk −1 )(hk − hk −1 ) 2 k =1

1 hk = ( hk + hk − 1 ) 2 t k = hk − hk − 1 Bij = 0, (I,j = x,y,s)

The load-deformation relations become:

and

 N x   Axx N  = A  y   yx  N s   Asx

 M x   Dxx M  = D  y   yx  M s   Dsx

Ayy Asy

Axs   Ays  Ass 

ε xo   o ε y  γ so   

Dxy Dyy Dsy

Dxs   Dys  Dss 

κ x    κ y  κ s   

Axy

Special Cases: a. isotropic layers b. specially orthotropic layers layers c. Angle-Ply layers

Balanced Laminate: A laminate is balanced when it consists of

pairs of layers with identical

thickness and elastic properties but have +θ/- θ orientation of their principal material properties with respect to the laminate reference axes. (152/455/30/-30/-455/-152 )

Qxs (θ ) = m 3 n(Q11 − Q12 − 2Q66 ) + mn 3 (Q12 − Q22 + 2Q66 )

z

where m = cos θ and n = sinθ .

Qis (θ ) = −Qis ( −θ )

k=n

z

−θ θ

K’ k

Pair x

k=2 k=1

For each balanced pair of layers k and k' t k = t k'

θ k = −θ k' n

n

k =1

k =1

Ais = ∑ Qisk (hk − hk −1 ) = ∑ Qisk t k Where I = x, y

Therefore, for each pair Ais = 0 (i=x,y)

Types of Balanced Laminates: Symmetric: [±θ1/±θ2]s Antisymmetric: [θ1/θ2/−θ2/−θ1] Asymmetric: [θ1/θ2/−θ1/−θ2]

Read section 5.8 of text book

b. Antisymmetric Laminate

1 n k 3 Dis = ∑ Qis (hk − hk3−1 ) = 0 3 k =1 (hk3 − hk3−1 ) = (hk3' − hk3' −1 )

Qisk  N x   Axx  N  A  y   yx  Ns   0 M  = B  x   xx  M y   Byx     M s   Bsx

Axy Ayy 0 Bxy Byy Bsy

=

k' −Qis

0 0 Ass

Bxx Byx Bsx

Bxy Byy Bsy

Bxs Bys Bss

Dxx Dyx 0

Dxy Dyy 0

Bxs   ε xo  Bys   ε yo    Bss  γ so  0  κ x   0  κ y    Dss  κ s 

c. Antisymmetric Cross-ply Laminate (0/90) n

z k = − z k'

Axx = Ayy Axs = Ays = 0

t k = t k'

k Qxx k Qyy k Qxy k Qxs

=

= = =

Bxx = − Byy

k' Qyy

Bxy = Bxs = Bys = Bss = 0 Dxx = Dyy

k' Qxx

k' Qxy

k Qys

=

k' Qxs

 N x   Axx  N  A  y   yx  Ns   0 M  = B  x   xx My   0     Ms   0

=

k' Qys

Axy Axx 0 0 − Bxx 0

Dxs = Dys = 0

=0

0 0 Ass 0 0 0

0

Bxx 0 0

− Bxx 0

Dxx Dyx 0

Dxy Dxx 0

  ε xo   o   ε y   γ so     κ x   κ y    Dss  κ s  0 0 0 0 0

Quasi-Isotropic Laminates y

y

[ A] x , y = [ A] x , y = constant

[a] x , y = [a] x , y = constant x

ϕ x

E x = E x = constant G xy = G xy = constant υ xy = υ xy = constant

η xs = η ys = η xs = η ys = 0 0 / π / 2π / K / n − 1 π   n n  s n

or

 π / 2π / K / π   n n  s

Lay-up is quasi-isotropic for any integer n greater than 2.

Other quasi-isotropic lay-ups:

[0 / ± 45 / 90]s [0 / 60 /− 60]s

5.7 Laminate Engineering Properties

y

 ε xo  axx  o   ε y  = ayx γ so   asx   

Ny Nxy

Nx

Where

Nx

Ny

axs   ays  ass 

Nx  N   y  Ns 

[a] = [ A]−1 For symmetric laminate

{

}

[a] = [ A −1 ] − [ B* ][ D* − 1 ] [C * ]

For general laminate

x

Nxy

axy ayy asy

σ x   N x  1 Average stress: σ y  =  N y     h  τ s   Ns 

 1  o  ε x   Ex Average strain and stress  o   ν xy are related by ε y  = − E x γ so      η xs  E  x

ν yx − Ey 1 Ey η ys Ey

ηsx   Gxy  ηsy  Gxy   1  Gxy 

Nx  N  1  y h  Ns 

Where:

Ex , Ey =

Laminate effective Young’s moduli in the x- and y- directions, respectively

ν xy , ν yx =

Laminate effective Poisson’s ratios

η xs , η ys , ηsx , ηsy = Laminate effective shear coupling coefficients Laminate constants and the [a] are related by: 1 1 1 Ex = , Ey = , Gxy = haxx hayy hass

ν xy = − η xs =

ayx axx

, ν yx = −

axy ayy

, ηsx =

asy ays asx , η ys = , ηsy = axx ayy ass

Further,

ν xy ν yx = Ex Ey

η xs ηsx = Ex Gxy

η ys ηsy = Ey Gxy

Note: if the laminate is non symmetric, use the [a] given by

{

[a] = [ A]−1 − [ B* ][ D* ]

−1

}[C ] *

axs ass

5.8 Analysis Of Laminated Composite Structures Mechanics Equation Classical methods Governing Differential Equations & Boundary Conditions

Energy Formulation

Calculate Displacement Field

Calculate Strain-Stress Field

Apply Failure Criteria

Summary 5.1 Basic assumptions 5.2 Strain-Displacement Relations 5.3 Stress-Strain Relations 5.4 Stress Resultants 5.5 General Load-Displacement Relations 5.5.1 Laminate stiffness equation 5.5.2 Laminate compliance equation 5.6 Special Class of Laminates Single layer Symmetric laminates Balanced laminates Quasi-isotropic laminates 5.7 Laminate Engineering Properties 5.8 Analysis of Laminated Composite Structures