7/28/2019 CHP6.pdf

1/20

CHP6. MECHANICAL BEHAVIOR

OF THIN LAMINATED PLATES

7/28/2019 CHP6.pdf

2/20

LAMINATE WITH MIDPLANE SYMMETRY

Membrane behavior

We consider in the following a laminate with midplanesymmetry. The total thickness of the laminate is denoted as h.

It consists of n plies. Ply number k has a thickness denoted as

ek. Plane x,y is the plane of symmetry

Loading

7/28/2019 CHP6.pdf

3/20

Displacement Field

The elastic displacement at each point of the laminate is

assumed to be two dimensional, in thex,y plane of the laminate.It has the components: uo, vo. The strains can be written as:

7/28/2019 CHP6.pdf

4/20

In an analogous manner ;

with

and

with

Therefore

7/28/2019 CHP6.pdf

5/20

Remarks

Aijare independentof the stacking order of the plies. ( ????? )

Normal stress resultants Nxor Nycreate angular distortions (A13, A23).This coupling will disappear if the laminate is balanced. (A13 = A23 = 0

???)

It is possible to substitute the stress resultants Nx, Ny, Txy with the globalaverage

stresses

7/28/2019 CHP6.pdf

6/20

Calculating the term without knowing the thickness h (but using

percentage of certain direction of plies p (in %)

7/28/2019 CHP6.pdf

7/20

Inversion

Determination ofapparent moduli Ex, Ey,

Gxy, nxy, nyx of LAMINATE related tomembrane behavior

7/28/2019 CHP6.pdf

8/20

Practical determination of a laminate subject to

membrane loading

Given: The stress resultants are given and denoted as: Nx, Ny, Txy.

Estimate the ply proportions in the four orientations.

Assume that plies are identical (same material and same thickness).

Problem :

The apparent elastic moduli of the laminate and the associated

coupling coefficients, in order to estimate strains under loading

(deformation)

The minimum thickness for the laminate in order to avoid rupture of

one of the plies in the laminate

7/28/2019 CHP6.pdf

9/20

Principle and procedure of calculation

Given :

Nx, Ny Txy

Fiber and matrix material

Assume : all plies have the same material.

1st step

Estimate the ply proportions in the four orientations.

7/28/2019 CHP6.pdf

10/20

Principle and procedure of calculation

2nd step

Calculate the term of (1/h)*[A]

OR

7/28/2019 CHP6.pdf

11/20

3rd step

Determine Apparent Moduli of laminate

Principle and procedure of calculation

Inversion of

matrix

Determine : Ex, Ey, Gxy, xy, yx, OF LAMINATE !!!!

h (unknown)

Determine : h ox, h oy, h xy

7/28/2019 CHP6.pdf

12/20

4th step

Determine h using Tsai-Hill criteria ( failure condition for

each ply ( or direction)

Principle and procedure of calculation

In x y coordinate ( laminate one)

Introducing h for each ply or direction

7/28/2019 CHP6.pdf

13/20

Principle and procedure of calculation

( in l, t coordinate )

Apply Tsai Hill for each ply or direction

Found the biggest h thickness of laminate

7/28/2019 CHP6.pdf

14/20

Flexure Behavior

7/28/2019 CHP6.pdf

15/20

Flexion Loading (having mid-plane symmetrie)

7/28/2019 CHP6.pdf

16/20

and Are in opposite sign (due to mid-plan

symmetri)

OR

WITH

7/28/2019 CHP6.pdf

17/20

Using the same calculation forMx and Mxy we obtain

Cij depends on stacking sequence of theplies

7/28/2019 CHP6.pdf

18/20

7/28/2019 CHP6.pdf

19/20

Laminate without midplane symmetry (coupled

Membrane-Flexure Behavior)

(for ply k) (for laminate, not equal to zero (no

midplane sym)

Similarly for Membrane loadings

7/28/2019 CHP6.pdf

20/20

Inversion of matrix determine strains and rotations deformation