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).