application of laminate analysis to composite...
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Application of Laminate Analysis to Composite Structures
• Composite sandwich structures • Composite grid structures
Composite Sandwich Structure
Geometry of symmetric composite sandwich structure for laminate analysis
[ ] ( )
[ ] [ ] [ ][ ] ( ) [ ] ccijffij
cf
cfij
cccijf
ccfij
kkkkijij
hQtQ
ht
hQ
hhQt
hhQ
zzQA
+=
=
−
++
−−+
−−−=
=−= ∑=
−
2
222222
3
11
Laminate stiffnesses for symmetric sandwich structures (from J. R. Vinson, The Behavior of Sandwich Structures of
Isotropic and Composite Materials, CRC Press, 1999)
Extensional stiffnesses
Flexural stiffnesses
[ ] ( ) [ ]
[ ] [ ]
−
++
−−
+
+
−−−
−=−= ∑
=−
3333
333
1
31
3
2231
2231
2231
31
cf
cfij
cccij
fcc
fijk
kkkijij
ht
hQ
hhQ
thh
QzzQD
Some corrections may be needed when using Classical Lamination Theory (CLT)
for sandwich structures
• CLT neglects transverse shear strains and , which may not be negligible in composite sandwich structures due to low shear modulus of foam or honeycomb core
• Sandwich structures have several possible failure modes due to the core or core/skin interactions that are not present in composite laminates
xzγ yzγ
Failure modes of a sandwich beam in 3-point bending
Failure modes of a sandwich beam in three-point bending. (From Steeves, C.A. and Fleck, N.A. 2004. International Journal of Mechanical Sciences, 46, 585–608. With permission.)
Geometry of a sandwich beam in 3-point bending
Geometry of a sandwich beam in three-point bending. (From Steeves, C.A. and Fleck, N.A. 2004. International Journal of Mechanical Sciences, 46, 585–608. With permission.)
Failure loads for sandwich beams in 3-point bending
LdfbtfP
σ4=
For microbuckling of face sheet (from D. Zenkert, An Introduction to Sandwich Construction, 1995)
where = face sheet axial compressive strength b = beam width = face sheet thickness d = c +
fσ
ft
ft
For core shear failure (from D. Zenkert, An Introduction to Sandwich Construction, 1995)
bdcP τ2=
where = core shear strength cτ
For face sheet wrinkling (from D. Zenkert, An Introduction to Sandwich Construction, 1995)
32
cGcEfELdbtP f=
where = face sheet material Young’s modulus = core material Young’s modulus = core material shear modulus
fE
cEcG
For face sheet indentation (from C. A. Steeves and N. A. Fleck, Int. Journal of Mechanical Sciences, 2004
= LdfEc
fbtP 3
2σπ
where L = beam span length = core compressive strength cσ
Composite grid structures
Orthogrid Isogrid
Why Composite Grid Structures?
• Excellent damage tolerance (redundant load paths) • Open structure easily inspected for defects or damage• Unidirectional composite ribs offer maximum strength
and stiffness• Several high volume manufacturing processes can be
used (e.g., pultrusion, thermoplastic stamping)
Twintex™ E-glass/polypropylene composite isogrid
Advanced Composite Grid Structure in Launch Vehicle Fairing (Source: Boeing Company)
Large (3 m x 3 m) interlocked composite grid structure fabricated from pultruded carbon/epoxy ribs and rib caps (Source: S. W. Tsai, Stanford University, 2001)
1
2
x
y
Nx
Mx
Mxy
Nx
Mx
Mxy
θ
Family of parallel ribs for modeling of grid structure as a laminated plate having equivalent stiffnesses (From H. J. Chen and S. W. Tsai, Journal of Composite Materials, 1996)
dAEN xx
xε
=
[ ]
=2233
3422
3224
nmmnnmmnnnm
nmnmm
dAE
A x
=
662616
262221
161211
AAAAAAAAA
Force per unit length parallel to family of ribs
where A = cross-sectional area of ribs = Young’s modulus along rib direction = strain along rib direction
xExε
Transforming both force per unit length and strain from local coordinate x,y to global coordinates 1,2, the extensional stiffnesses for an equivalent flat laminate plate are;
where m = cos θ, n = sin θ Similarly, the global flexural stiffness matrix is
−−= 4
43
12136.3
316
16 hw
hwhwJ
Where = Young’s modulus and shear modulus of rib h = rib height I, J = moment of inertia and torsional constant of rib
G,xE
Using superposition of each parallel family of ribs, taking into account the orientation of each family of ribs, the total grid stiffnesses are; For the orthogrid with two families of ribs at θ = 0o and 90o
[ ]
=
6600
00
00
Ad
AEd
AE
A x
x
orthogrid
[ ]
=
200
0000
1
GJIE
IE
dD x
x
orthogrid
For the isogrid with three parallel families of ribs at θ = 0o, 60o and –60o
[ ]
=
100031013
43
dAEA x
isogrid
[ ]
+++++
=τ
ττττ
100031013
43
dIED x
isogrid
where
IEGJ
x
=τ 3
121 whI =
−−= 4
43
12136.3
316
16 hw
hwhwJ
For more details on analysis of composite grid structures, see the following papers;
Chen, H. J., and Tsai, S. W., “Analysis and optimum design of composite grid structures,” Journal of Composite Materials, 30(4), 503-534 (1996). Huybrechts, S., and Tsai, S. W., “Analysis and behavior of grid structures,” Composites Science and Technology, 56(9), 1001-1015 (1996).