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