buckling_of_laminated_composite.pdf

7/29/2019 buckling_of_laminated_composite.pdf

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Mca Composite Training Page 1/20

Mcanique des Composites - Mechanical behaviour of composites materials

Buckling of laminated composite construction

Buckling of straight beams, plate-Euler theory for beams, composite plates buckling

-Influence of shear modulus

-Curved beam

-Stability of plates strengthened by longitudinal ribs

Face wrinkling of sandwich structures

-Influence of core material, skin stiffness


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Buckling: definition

Eulers formula for isotropic column under compression

-The maximum axial load that a structural component (a.k.a., column) can support when it is on the verge of buckling iscalled the critical load, Pcr.

-Any additional load greater than Pcr will cause the column to buckle and therefore deflect laterally.

Buckling is a geometric instability and is related to material stiffness, column length, and the cross-sectional dimensions of the column. Strength does not play a role in buckling.

Structure wants to move from one previous state of equilibrium to another one which at Pcr corre-sponds to less energy.

P Pcr

stable

U=NRJ>UoUo

Uo U=NRJ


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Overview: Post-buckling

If P/Pc (buckling) happens, it does not necessarily mean critical failure

-Structure is in a post-buckling state,

-When load returns to zero, the structure could return to its initial state or enter another state.

Figure 1 - Example (thin plate with stiffeners)

Post-buckling is the ability of a structure to carry loads well in excess of the initial buckling load.

-Post-buckling is often associated with a change in load sharing, because buckled plate has a lower equivalent modulusthan straight initial plate.

before bucklingpost-buckling

F

U

axial displacement

buc

kli

ng


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Buckling and flow

Buckling needs compression stresses:

Buckling is heavily dependant upon boundary conditions: free edge, simply supported, clamped

-in real structures, boundary conditions are a mix of several ideal conditions,

-use a coefficient K for semi-clamped (mean value of simply supported and clamped),

-choose most pessimistic conditions.

simple compressivein plane flow

double compressivein plane flow

in plane shear flow

in plane torsion


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Buckling and composites

Buckling is not specific to composite materials, but some factors could promote buckling

-thin plate,

-very different stiffnesses along different directions,

-coupling could have a softening effect,

-observation of an unperfect plane due to residual stresses,

-sandwich skins are a special case.

Other structures as well as intact plane should be analysed using buckling or stability analysis

-parts with holes

-damaged structures (delamination, loss of mechanical properties)

-parts with initial imperfections (curvature)

-thickness variations, curved structure

-environmental effects

Eulers load with axial modulus and transverse shear modulus

(1)

PCr

1

L2

kBC

2E1I

----------------------2

GS------+

-----------------------------------= where L= beam length

kBC Eulers coefficient depending on boundary conditions.(=1 simply supported ends, =4 for

both clamped ends,= 2 for one end simply support and the other clamped, =1/4 for one endclamped and the other free.


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Buckling and hygrothermal analysis

When subjected to an increase of temperature or moisture expansion, laminates want to expand.

-If the structure is restrained, compressive flow would occur.

-Note that in the general case of anisotropic materials, a decrease in hygrothermal conditions could also create compres-sive flow in the part (negative thermal expansion coefficient, coupling).

Example: buried composite pipe with end restrained:


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Governing equations for bending, buckling and vibration of laminatedplates

Kirchhoff assumptions

3 coupled governing differential equations in U,V and w:

-derivation order 4 and 3 for normal displacement w

-derivation order 3 and 2 for in-plane displacements U and V

x

Nxy

Nxy+ 0=

x

Nxyy

Ny+ 0=

x

x

Mx y

y

My 2 x

y

Mxy( )+ + p=

equilibrium differential equations

0

A B

B D

1N

M=

x0 x

U=

y0 y

V=

xy0

x

V

y

U+=

x

x2

2

w=

y

y2

2

w=

xy

2x

y

w( )=

CLT mid-plane strainand curvature


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Buckling equations for laminated plates

A plate buckles when the in-plane compressive load get so large that the originally flat equilibriumstate is no longer stable:

-the plate deflects into a non-flat (wavy) configuration.

-if there is a membrane-bending coupling B, the plate under in-plane compression will always bend. There is prebucklingbending deformation which softens the plate because of buckling. This effect is very difficult to get using analysis.

Closed form solution of laminates is only valid for orthotropic plates (symmetrical and balanced):

-D16=D26=0, A16=A26=0 and B=0

P=0

P




Linear buckling calculation for plate: principle

Governing equations from a membrane pre-buckled state

-where denotes a variation of force, moment or displacement in pre-buckled state (membrane behaviour)-N load is applied in a plane

-membrane state: w= w, no membrane-bending coupling B=0 so M = D simplified equation

-w is a function of trigonometric series

-buckling solution is an eigenvalue problem

x

Nx y

Nxy+ 0= x

Nxy y

Ny+ 0=

x

x Mx

y

y My 2

x

y Mxy( ) Nx

x2

2

w Nyy2

2

w 2Nxyx

y

w( )+ + + + + 0=

D11

x4

4

w2 D

122D

66+( )

x2

2

y2

2

wD

22y4

4

w+ + Nx

x2

2

wNy

y2

2

w2Nxy

x

yw

( )+ +=




Techniques for solving buckling problems

Exact solution with many restrictions and assumptions

Finite element calculation or finite differences

Rayleigh Ritz and Galerkin method

-Often variable separation assumption in x and y coordinate

-Put a reliable form for the displacement w which satisfies boundary conditions

-Minimise energy




Simply supported laminated plates under in-plane load

Uniaxial flow with all sides simply supported

solution :

The value of m (number of half wavelength in x direction) depends upon aspect ratio:

w Amn

mxa--

nyb--

sinsin= NxNx

a

bm is the number of buckled halfwavelength in x direction and n iny direction

smallest value for n=1x =

2

b2

D11 D22 k

k =D11

D22

m

2

+2 D12 + 2 D66

D11 D22+

D22

D11

m

2 where is the aspect ratio =a/b

m m - 1D11

D22

4

< < m m + 1D11

D22

4 allows us to find m value




Figure 2 - critical load

It is usual to take a conservative approach using least possible value of the series which is:

(2)

where kn is a series which converges to 1 (1.025 for m=4)

l'quation (2) is also valid for clamped loaded edges

Uniaxial flow with various boundary conditions

-see Handbook of thin plate buckling and post-buckling or Mil Hdbk 17-3

Uniaxial loading: long plate with all sides fixed

simplified complete formula

= a/b = a/b

NxNx

D11=751 D22=751

D12=96 D66 =61

D11=7510 D22=751

D12=96 D66 =61

x = 2

22 D66 + D 12 + D22 D11

b2




long plate only a/b>4

Uniaxial loading : three sides simply supported and one unloaded edge freelong plate only a/b>4

Biaxial loading : plate with all sides simply supported

minimise

where N is the ratio Ny/Nx

xmil =

25.33 D66 + 2.67 D12 + 4.6 D22 D11

b2

xmil =12 D66

b2

+

2D11

a2

xmil =

2

b2

D11 m4 b

a

4

+ 2 D12 + 2 D66 m2

n2 b

a

4

+ D22 n4

m2 b

a

2

+ N n2




Buckling and laminate stacking sequence

Stability is strongly affected by laminate stacking sequence.

General rules that define the best laminate stacking sequence for buckling do not exist.

Figure 3 - example: evolution of buckling load for an antisymmetrical angle ply laminate

formula for buckling of simply supported long plate subjected to compressive flow

angle +-

Nx=2 2

b2

Qpm11 Qpm22 + Qpm12 + 2 Qpm66h

3

12




Curved structures

Special case of the ends of a cylinder under external pressure

-dished end is equivalent to a square plate under pressure

-curved ends are very sensitive to buckling when sujected to external pressure

-internal pressure creates positive membrane flow in the curved end

critical external pressure rf. 1for a long cylinder: (diameter D and thickness h)

(3)

valid if ratio K=L/D is greater than (L = length between stiffeners):

(4)

1. NF T 57-900 3-10.2 and 3

PC

2E2

1 1221---------------------------

h

D----

3=

KC

2.5

2-------

E1

E2

------4 D

2h------=

K KC

>


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Damaged structures

Damaged plane structure due to manufacturing, in service manipulation or impact

-can create delamination for monolithic laminate or skin debonding for sandwich structures

-when delamination dimension (length or width) reaches a critical value-> local buckling of the laminate

-Analytical formula (laminate buckling), FEM calculation, more difficult to calculate delamination development

delamination in a monolithic laminate

development of the delamination

critical value of the defect


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Buckling and FEM

Linear buckling

-provides critical factor kc, if loads are multiplied by kc then buckling would occur

-kc = reserve factor

Non-linear buckling

-perform a non-linear analysis to take into account material non-linearity or geometric non-linearities

-global stiffness matrix will be recalculated for each step.

Material properties and FEM

-Buckling is very sensitive to material properties,

-use mean values of modulus, knockdown coefficient p2/22

Interlaminar shear: influence on buckling

-For thick plate for example, transverse shear displacement can be significant.

-CLT fail, use FEM with 3D model


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Local buckling in sandwich structures

When a sandwich is loaded with flexion, one skin is in a compressive state of stress

Skin buckling is called wrinkling. It depends upon the properties of the skin and of the core.

-local skin buckling can crush the core, debond the skins,...

Another form of local buckling is dimpling (or intercell buckling)

-critical compressive stress for wrinkling by Howard G. Allen is:

-With Young modulus Ef = 12394 Mpa and Ec = 139 Mpa

There is a criticial buckling stress of about 320 Mpa.

Other formulas for wrinkling

crit 1 = B 1 Ef

1

3 Ec

2

3

Ef: Young modulus of the skin

Ec Young modulus of the core normal to the skin or z thickness direction

B1 is a function of Poissons ratio of the core

B 1 = 3 12 3 - c2

1 + c2 -

1

3 c = 0.95


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NASA1

where

-ef: thickness of the skin 0.95mm

-ec: thickness of the core Nida 38.5mm

We find m = 191 MPa

NASA CR1457

where

-Gc: shear modulus of the core 25 MPa

We find m = 190 MPa

formula for Intracell buckling

HEXCEL

where

-l = 1

-s: diameter of the cell 9mm

We find ib = 276 MPa

m = 0.82 Ef Ecef

ec

m = 0.5 G c E c E f 3

ib = 2Ef

ef

s

2


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Finite element model for skin buckling

top skin is put into compression

Figure 1 First buckling mode

We find a compressive stress of 376 Mpa.

buckling_of_laminated_composite.pdf

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