Technical University of Łódź

Department of Strength of Material and Structures

M.Kotelko, Z. Kołakowski, R.J. Mania

LOAD-BEARING CAPACITY OF THIN-WALLED THREE-LAYERED STRUCTURES –

RECENT ACHIEVEMENTS IN THEORETICAL ANALYSES

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MAIN TOPICS-RECENT ADVANCES IN RESEARCH

• - the buckling and post-buckling analyses of multi-layered structures based on the asymptotic approach leading to the lower-bound threshold criterion,

• - equivalent approach used in the analysis of three-layered sandwich structures with honeycomb core (simplified equivalent models, which enable to avoid a complexity of a real sandwich structure and to determine the load-bearing capacity approximately),

• -   kinematical method allowing to avoid extremely complex analyses of the multi-layered structure’s post-buckling behavoiur in the elasto-plastic range and finally leads to the upper-bound estimation of ultimate loads,

• -    finite element analyses (FEA) used to investigate both buckling and ultimate loads.

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Examples of three-layered structures

c)

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BUCKLING AND POST-BUCKLING ANALYSIS – ASYMPTOTIC METHOD

Geometrical relationships

,2

),(

),(

,,,,

2,

2,2

1,

2,

2,2

1,

yixixiyiixyixy

yiyiyiiy

xixixiix

wwvu

uwv

vwu

Equilibrium equations

ix i xx iy i yy ixy i xyw w w , , , ., ,

N N N u

N N N v

N w N w N w N w M M

ix x ixy y iy i y y

ixy x iy y ix i x x

ix i x x iy i y y ixy i x y ixy i y x ix xx iy yy ixy xy

, , , ,

, , , ,

, , , , , , , , , , ,

,

,

.

0

0

2 0+ M

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...

...

)()()0(

)()()0(

jk

ikjj

iji

jkikj

jiji

NNN

UUU

N

U

Displacements and sectional force fields

Honeycomb sandwich panel under uniform compression

a

b

t f

h c

E ,G ,ff f E ,Gc c

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Equivalent single plate models

Equivalent rigidity method Equivalent weight method

=

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)1(12 2

3

eq

eqeqeq

tED

)1(12

)2[(2

33

f

cfcf hthE

f

eq

feq

f

eq

feq

ffcceq

Gt

tG

Et

tE

tthht

2

2

463 22

f

cavcffeq

eq

htt

WW

2

FE model

r

tGG

r

tGG cxzcyz

00 ;576.0

Ex = Ey = 0 for 25.02

0 ff

cc

tE

tEh

r

EtE c

z 3.1

2 0

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t0 - thickness of the cell foil2r – size of the hexagonal cellcoefficient depending on structural parameters of sandwich core

fac ings ortho trop ichom ogen ized core

FE models - continuation

The FEM has been applied to the buckling and post-buckling analysis of multi-layered channel-section members subject to compression [3] and also of sandwich three-layered plates with honeycomb core under compression [2]. In both cases the FE model was built from shell elements of six degrees of freedom at each node. ANSYS ver.7.1 software package has been applied.

The eight node non-linear layered SHELL91 or SHELL99 element were used. These elements allow to account for up to 100 layers of different thickness and material properties (either isotropic or orthotropic).

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FEM results – deformation at collapse

Shortennig of loaded edges 33.3 mm

Kinematical method

dVP pij

Vij ),(

Principle of virtual velocities

where - generalised displacement, - rate of change of the generalised displacement, P - generalised force, - vector of kinematic parameters of the plastic mechanism, - vector of geometrical parameters of the plastic mechanism, - rate of change of the plastic train tensor.

ijp

Energy of plastic deformation dissipated at stationary yiled-lines

kk

pkp mlW ~)(

n

i

i

jijiipp ttttmm

1

1

100

0 ])2([~

Plastic moment capacity for the multi-layered wall:

Exemplary results: Honeycomb sandwich plate under uniform compression

Square plate 500x500 [mm], hc = 3, tf =1.5 [mm], aluminium facings and honeycomb core made from aluminium foil (Ef =71 070 MPa, f0 =268 MPa), cav = 54 kg/m3

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a

b

t f

h c

E ,G ,ff f E ,Gc c

Three-layered girder under pure bending

a

bt

d

c

a

bt

d

c

h

t

0

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g

h

Sandwich honeycomb box-section

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 2 4 6 8

M [k

Nm

]*e3

1

12

2

500

500

h=31

Structural behaviour of the sandwich honeycomb girder1-sandwich girder 2-aluminium girder of equivalent weight

aluminium facings and honeycomb core made from aluminium foil

tf =3 mm, hc =25 mm,

Ef =71 070 MPa,

f0=268 MPa, f = 2.7 g/cm3,

cav = 54.4.kg/m3

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a

b

t f

h c

E ,G ,ff f E ,Gc c

Composite material box-section

100

100

h=1.25

I.Case: steel/alum./steel

E MPa

Y

MPa

II.Case: steel/comp./steel

E  

Y

MPa

Outer layer 200 000 384 Outer layer 200 000 MPa

1400

Centre layer 70 000 123 Centre layer E1 = 55e9 Pa

E2=46.5e9PaY1=602

Y2= 598  

 

Material data:

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Load-bearing capacity estimation

0

1

2

3

4

0 0,2 0,4 0,6 0,8 1g/h

M/M

cr

1 2 3

0

2

4

6

8

0 0,2 0,4 0,6 0,8 1g/h

M/M

cr1 2 3

1 – I. Lower Bound L-B-C, 2 – II. Lower Bound L-B-C, 3 – Upper Bound L-B-C 

I. Case (steel/alum./steel)

II. Case (steel/comp./steel)

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Three-layered girder: Lightness factor =M/W

0

4

8

12

0 0,2 0,4 0,6 0,8 1g/h

M/W

[N

m/k

g]

*e3 2 1

1 – I.Case (steel/alum./steel), 2 – II. Case (steel/comp./steel)

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GENERAL REMARKS

• The method consisting in compiling the post-buckling asymptotic analisis and kinematic approach leading to Upper Bound L-B-C estimation occures to be efficient, particularly at the initial stage of structural design, since one obtaines a quick response using algorithms much simpler than those related to Finite Element analysis. The latter becomes very complicated and time consuming in the considered case. At a more advanced stage of research the presented analyses can be also carried out simultaneuosly with FE analysis in order to verify purely numerical results.

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CONCLUSIONS – HONEYCOMB SANDWICH PLATES

•The equivalent weight method, in the case of sandwich plates with honeycomb core, gives strongly underestimated results except the lowest values of hc/tf ratios. Thus, basically this seems to be inadequate in a wide

range of sandwich panels parameters. However, the model may be useful for the assessment of load-capacity to weight ratio of a real sandwich plate in comparison with a plate of an equivalent weight (i.e. in optimisation procedures).

•The equivalent rigidity method provides more realistic estimation of ultimate loads, although overestimated for relatively low hc/tf ratios. The

method is applicable for relatively high hc/tf ratios. For the range of

elastic and geometrical parameters analysed within this study the limit ratio is about 4.

•A very approximate upper-bound estimation of ultimate loads derived from the kinematical approach provides one with reasonably realistic results. The agreement of ultimate loads obtained in the way described and experimental results is satisfactory.

CONCLUSION – THREE-LAYERED GIRDERS UNDER PURE BENDING

•The lower- and upper bound estimation of the L-B-C are close that indicates the compilation of the post-buckling asymptotic analysis with the kinematical method to give a satisfactory approximation of both L-B-C and the structural behaviour in the whole range of loading, up to and beyond the ultimate load.

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