Optimium sandwich design

Optimum sandwich design

by

M. P. Nieuwenhuizen

Presented to the fifth

_{European Aeronautical Congress}

Venice. Sept. 12 th-15 th 1962

ROYAL NETHERLANDS AIRCRAFT FACTORIES F OK K E R S CH I PH QL - A M ST E R DA M _{- N E T H E R LANDS}

Optimum sandwich design

by

M. P. Nliuwsnhuizsn

SUMMARY

The buckling load for flat compressed sandwich panels with several edge conditions is given !n a simple form.

In addition, for two cases, namely, plates with the unloaded edges free to translate and

rotate and plates with the unloaded edges simply supported, a method is given by means

of which the optimum designed construction can be determined as a function of the load

factor.

The -Instability stress for conventional stiffened panels decreases much more rapidly with the panel length than that for sandwich panels. This means that In a sandwich panel. wing structure the rib spacing can be much larger than that in a wing with conventional stiffe-ned panèls.

However, due to the increase In rib spacing the Influence of lateral loads, such_{as air}

suction and tank pressurization becomes increasingly important.

In this report a method is given by means of which an estimation can be made of the

maximum allowable axial load on a panel in cases where lateral loads are acting.

In addition an optimum design theory for minimum weight for those combined loads is given.

CONTENTS

L INTRODUCTION

GENERAL INSTABILITy FORMULAS

OPTIMUM CONSTRUCTIONS

3.1 GENERAL

2 COLUMN

3.3 OPTIMUM DESIGNED SANDWICH. PANELS WITH ALL EDGES

SIMPLY SUPPORTED

SANDWICH PANELS WITH SIMPLY SUPPORTED EDGES UNDER LATERAL AND AXIAL LOADINGS

NOM ENCLATURE

Panel width

end fixity coefficient

Yc

k buckling coefficient

L half wave length

m number of half wave lengths In X direction

n number of half wave lengths In Y direction

q uniformly distributea lateral load

2tf

t

core height

tf thickness of one face

t

_tc+2tt

U deflection

U, maximum defleàtion

V shear deflection

E.tt.t bending stiffness

2(1-vi per unit width

E Young's modulus

Et tangent modulus

2EEt

Er

E+Et

core shear modulus

L panel length L'us

load per unit width In X-direction

R. l_..

StGc

W weight per unit surface area

P/w efficiency criterium 'L loading parameter

face material density

core density

G Y4k1 stress

G bending stress

Poisson's ratio

OPTIMUM SANDWICH DESIGN

1. INTRODUCTION

Sandwiches have been very popular as food but _{as a construction method it has nevér} enjoyed such a popularity.

There were, and still are, a lot of difficulties and unknown facts _{concerning sandwich}

constructions.

Even the stress analysis of sandwich construätions is different from _{the more common}

constructions.

This lecture deals only with compression members of sandwich construction and the

results will be shown in an easily usable form.

Although the treatment is mainly theoretical the results have been _{experimentally}

verified at the Fokker factory in Holland and found to be sufficiently reliable. In order to make the lecture brief the mathematical analysis will be summarised.

2. GENERAL INSTABILITY FORMULAS

On graph i the general buckling formula is given for a sandwich panel loaded in

compression and all edges simply supported.

Explanation of the nomenclature: L panel length

L ha:Lfwave length

b panel width

m number of half waves in X-direction n number of half waves in Y-direction

E(t'_t çtf t

B=12(1,

2(1v bending stiffness

core height

tf

thickness of one face

t= t+2t

sandwich thickness

S tG

G shear modulus of the core

The general formula wifi be extrapolated to extremes.

CASEI

_b=o.

The minimum instability load occurs for m i i. e. the case When only one buckle

occurs in the loading (or X) direction.

This yields

+st,

But for this case there is no Influence of the support of the unloaded edges and

therefore it is reasonable to use this solution for a sandwich column.

CASEn

C=oo

It can bé shown that the in i n i mu m instability load occurs In the case when only one buckle occurs in Y-direction.

Thé result can be compared with that for columns of case I and it Is found that the Instability loads are equal when the following two conditions are fulfilled.

column

b.

. panel

C . (column length) b panel width.

The factor

(xa_B'

is small and neglected ( see case 11 ). b" S J

CASE ill L is finite

For a panel with finite length the buckling load is different from case U, because the buckle pattern is influénced by the support of the edges I. e. the ratio

b

The difference is very important for values 1

Comparison with the buckling load for columns in case I shows that the instability

loads are equal if:

S (for column) S (1+L'/b') The instability load can also be written as:

a

r

in which

1= b

3. OPTIMUM CONSTRUCTIONS

3.1. General

The buckling formulas given in graph 1 will be used to derive the optimum

sand-wich construction, that is, a construction with the ratio P1w a maximum.

Before the analysis is given, some assumptions will be made. Their effect on the

calculated buckling load is negligible.

The assumptions are:

1. _{The core cannot take axial forces and therefore the axial load must be taken by}

the faces

P=2tfG

2., _{The general formüla for the bending stiffness per unit width is:}

Ef (t'-t1)

- 12(1-v )

The faces of the sandwich are in general so thin that their individual stiffnesses àan

be neglected.

Moreover even 4 times the individual bending stiffness of the faces is negligible. This yields a simplified form for the bending stiffness:

B E.tct.tf

2(1-vI

3, Since the core isnot able to take axial loads the shear force is:

Qtc(tc+tf)

Now the energy due to shearing is

;a

2G 2S

Since the shear energy stored in the faces can be. neglected it f011ows that:

SD

tc and a good approximation is

StGc

The 3 expressions mentioned above will be introduced in the buckling formula. Firstly the column will be considered.

3.2. Column

(see graph 2)

Having introduced the values for the bending stiffness and core-shear stiffness the buckling formula becomes:

t E(1-r)r.t

PG.2tfDr.tD,

_E(i)t'

The weight of the sandwich consists of the following: Weight of the facings

Weight of the core Weight of the bond

Theweight of the bond is in general small _{compared with the combined weights of the}

faces and the core and is therefore not considered.

The analysis is simplified by thì procedúre withoutmaking a fault of any significance.

The weight per unit surface area is

W=rf.2tf+7t[Çr+rc(1_r)]

t

It is known that:

=i.t.

_{V(1_l)(1I1_}

Now the equations for the structural _{parameter fand the efficiency parameter}

can be written as:

p 2Gr

?

4i_ï1_r)(1_

Gr

W

_['rtr+;l-r)]

The construction can be considered an optimum. design if, for a certain value of.

f, the value of

is a maximúm or if f

. constant then d =0

From the formulas for f and

_{It. is clear that those quantities are}

functions of ₆ and

r.

When the stress exceeds the proportional limit _{Young's modulus for the face material}

is no longer a constant but varies with the stress _Et

_f(oi

Many tests proved that the use of the tangent _{modulus gives reliable results for panels}

loadéd above the proportional limit. The use of the reduced modulus _{2 E.Et}

Er

gives only slightly different results. _E+Et

To describe the stress-strain curve for _{a chosen material the formula of Ramberg}

and 9sgood has been used:

E

,6fl

-.+B.

)

"JI

The optimum construction exists _{when :}

IP

t,=

di-

=0

Mathematically speaking this means:

d(-)=

d6+

I ¡ P% or. +

r' Lí.

J6

r

dc5 d = d 6+ a- d r =0 or +

!W

r _ldG From these formulas follows:

P p and E =0 constant =0 = constant

(i.)

ut

_¡w

¡r

¡r

The physical meaning of this formula is _{schematically given on graph 3.}

From the formula it follows that for the _{considered - value:}

R R&(1+1+1

r= -

_{2f-1 4 (2f-1) [(R+f) .-+f-1]} In which 9 and

f=i

Yc

Sandwich constructions have most advantage _{when used as compression members,}

and, like all óther compressed structures, the materials with a high specific strength/

weight ratio are most suitable. This must be taken _{into account when selecting the}

faces

Such a material is the American specified _{7075-T6 material, identica], with the}

Itallafl P-AZ 5,8 and French AZ 5 GU.

The tangent modulus as a function of the _{stress for this material ia shown on graph}

4. This is based on a line Stress and it must _{be noted that the tangent modulus for}

10

Forthe core the aluminium-magnesium type is selected. The foil material is hydro-nalium 5052 or AG3. This type of core is stronger than the older 3003 (Al Mn) foil-core.

The American Hexel company produces at the moment core of 5056 material, which Is claimed to be 15% stronger.

To calculate the optimum construction the procedure is as follows

1. Take a value of G

( E and R are known for the material)

?. Calculate r.

f and

can be calculated using and

r.

The results of the calculations are given in graphs 5 to 8 inclusive and to get an idea

about the structurai efficiency some figures are also given for a coñventional Z

-stiffened riveted panel of optimum design.

In graph 5 the buckling stress of sandwich columns is plotted against the_structuräl

index

From this graph it can be seen that the stress changes only slightly in _{a wide range}

of .. values.

Thus for a certain value of P the distance L , which may be the rib spacing in a

wing, may be increased çonsiderably without significant change in the buckling_stress.

This is typical for sandwich constructions and Is in fact their main advantage.

For most wings the value of

'

will fall between 5 and 25 and in this region there

Is a variation of the column buckling stress of only 2 kg/mm

In other words the rib spacing may be increased 500% and this causes a decrease of tI'e buckling stress of only 2 kg/mm2.

This offers us the possibility to reduce the number of wing ribs.

Moreover, from the same graph It follows that for a lighter core thé buckling stress

Is higher than that for a heavier core. However it must be noted that the _{sandwich is}

thicker for the lighter core than fora heavier one. This can be seen in graph 6.

But a thicker sandwich may have several disadvantages e. g. in a wing structure. The effective height is decreasing, thus the axial load is Increasing. The tank content is decreasing.

3. The allowable lateral loadings are decreasing with decrease of the core

density.

In general the stabilisation of the faces necessitates a core with sufficient strength and stiffness..

A comparison is made on graph 7 between sandwich columns and ordinary

Z-stiff-ened panels.

It must be notéd that the buckling stress of Z-stiffened panels depends largely on

the loading parameter f.

The allowable buckling Stress of Z-stiffened panels decreases rapidly with increase

of the rib spacing L . Many more ribs are needed than for sandwich constructions.

However the stress is not a good parameter for the comparison of the constructions, since in a sandwich construction there is also core present and for this reason, for

the comparison, the structural efficiency criterion

.,. must be

used.

This is given in graph 8.

It ca be seen that for high values of f the Z-stiffèned panels become more

effi-cient than sandwich columns. The same can be stated of panels stiffened with oher types of stringers.

Z-stiffened páneis are more efficient than sandwich columns for .f 24 ,which

means for small rib distances.

In the foregoing analysis a column loaded In compression was considered.

However for connections and sealing of the sandwich construction, edge members

and local reinforcements must be introduced, which are in general heavier than for ordthary stiffened panél8. It is still the authors opinion that sandwiches can be

.12

3.3. Optimum designed sandwich panels with all edges simply

supported.

. . .

For an ordinary stiffened panel the influence of the support of the unloaded edges on. the buckling load is small or negligible, since the bending stiffness of those panels, in planes parallel to the stiffeners, is very low.

However this is not true for sandwich panels since the panel has .about the same ben-ding rigidity in all directions.

Support of the unloaded edges has a favourable influence on the buckling stress of sandwich columns.

However due to the support of the unloaded edges the optimum construction also

differs from that of columns.

This will be analyzed in the following manner:

(see also graph 9 ).

For a panel of infinite length the buckling load was found to be:

¿n'B

ba

Again P and P

w

=

and the efficiency index is

[i+Vi+.:5B"]

2r

n +

n/E(1(1 r67Gr)

6(1-vi

G.(1lVa) Gr w

rf+; (1-r)

are functions of ₃ and r

For an optimum design it is required that the value of certain value of

b

Mathematically this can be expressed as

P p

¡(-f)

-¡r

is a maximum for a / L baS The buckling, load for a column was

n'B

F=

+ s The column and panel buckling stress are equal if:

Hence;

rG

[R

-

(R+f)r] u";-

--

r(f-1)(1-r)

The procedure to calculate the optimum construction for every value of the structural Index .. is the same as that for columns.

The values of f and Gc are chosen.

Calculation procedure

G is assumed

For the chosen face material

R ja known

r is calculated

_!. and

..

can be calculated from the values of G. and

r.

In graph 10 the results are given of some caicülations.

The face material is 7075-T6 and the core is of the Al Mg type with a density of 45 kg/rn3. From graph 10 it follows that the stress is dependent on the ratio - to a very Small extent.

On graph 11 sandwich columns and panels are compared.

The loading parameters for columns and panels are respectively and Only the ratio _r .ÇL differs for the chosen loading parameters.

It follows that for a panel the total sandwich thickness for an optimum design is less than for a column.

4. SAÑDWICH PANELS WITH SIMPLY SUPPORTED EDGES UNDER LATERAL AND

AXIAL LOADINGS.

Since thé buckling stress is only slightly dependent upon the load factor. it was

possible to have large rib spacings for sandwich constructions.

However an increase in the rib-spacing means that lateral loadings cause higher stress and may become severe.

These lateral loadings may be air-suction, tank pressurisation or the crushing effect

Not only the. allowable buckling stress but also the optimum construction is influencec

by the lateral loadings. ( see graph 12 ).

The differential equation for a plate. on which lateral loads only are acting is

+2

+i2.L

q

xuI

- B

For a solid plate the deflectión düe to shear is so small that it can be left out of the analysis.

But in general this is not true for sandwiches and the shearing deflection must be

taken into. account.

i i

B,u

u

For combined lateral and axll loads the differential equation is

(1±)

+(2±.)

u +ì-!_J_( _pìL)

S

x'

S hy&

JyN B q

The deflection must now be split into: deflection due to lateral loads.

u'

= u sin. .sìn.

!1.

extra deflection due to axial loads.

- -

.nx .iy

u = u

_51T

Therefore the differential equation for the sandwich under lateral and axial loads can

alsO be written, as : . ,'

14

(u'+ i) +( 2- j.)

(u'+ ü) +

From this the allowable axial buckling load can be derived as I

P=_t6

(-+-)

ad

This can also be written as :

in which is the buckling load of the panel without lateral loading. The former formula was developed by Professor Vianello.

In the following figure the physical meaning is given schematically.

-

-- eLastic

[

,/"

" infLuence of pLasticity.

u0

The axial stress can be split into axial Btress caused by the compression and the axial stress caused by bending.

This results In varying axial stresses across the panel (all edges simply supported) A safe calculation of the allowable stress is made using the tangent modulus of the

face material at the centre (that is at the station with the n'rImum stress).

Furthermore it io assurnedthat the deflection shape has not been influenced by the plasticity.

Ttus If the deflection follows a sine curve then this remains a sine curve. In the plastic field of the face material.

The ma,rirnum stress is

. ri

mi.

q see graph 13 )

G.

E(tc+tf) 3E' lT' 2(1-ev') (u

-

u)('

+ 15

Fôr ä uniformly distributed load this is

U0 E(tc+tf+'V)

max= _{2(1_v&)} 16g Z B(_i.+-i L' b'. u0.2tf..

'V is Poisson's ratio and will be considered constant.

Assuming that the configuration and the lateral load

knowns in the equation are

'max E(6'may.) and u0

p f(u0) can now be calculated as follows

max. is assumed

E is known for the chosen material and _6'rnax

U is solved

When u. is known h can be calculated

16

P=2tf (6ria,r6b)

After some trials the maximum value of P can be calculated. But this is hot

necessarily the optimum design.for the considered load.

The panel under the considered combined loadings is optimum if for a maximum

allowable value of .. the panel has the lowest weight.

For à constant value, q of the uniformly distributed lateral load the relation between def1ection and panel dimensions ( incorporating the thicknesses) is linear.

This means that an increase in the dimensions with a constant fáctor causes a deflec -tion increase with the same fáctor.

Moreover this is found to be true for panels under cOmbined loadings. For constant

values of the lateral load q, and the axial stress 6 the same relation exists

between the dimensions and deflections for similar panels. For á given panel with known values of:

lateral loading q

facing/sandwich thickness ratio r

b L

T

and

T

the value of _f(u.) can be calculated, but the result can also be used for

un-etrriI1r panels by relating the deflection to the panel _dimensions.

Therefore a new parameter .

is introduced.

bb

Thé Inverse of the slenderness ratio is called _O.

For a panel with dimensions such that

4.

i and an uniformly distributed.

lateral loading q = 0,25kg/cm the optimum design is calculated.

The panel is of optimum design if, for a given value of the panel has the

lowest weight.

To arrive at the optimum cònstruction the following calculation procedure has been

performed :.

i.

_{e and}

I are assumed,

For the assumed constant values of _O

_{and r}

the stress

G" la

varied, thus also the value óf

.

wÑ Q

,

and 6'theueof

canbe calculated.

The maximum value of is important.

For (

-.

) max is

( ) max calculated.

In graph 14 the values of r

,,

r

and (5 are plotted against

Thróugh the points representing the values for certain e - r combinations

a line can be drawn.

Those lines are not given on graph 14, because they are flat.

On graph 14 is given the envelope of all those lines, which given also the optimum construction.

On the same graph the curves are given for a panel without lateral loads and it is found that for the considered case the allowable stress is considerably decreased. It will be clear that the possibility to increase the rib distance, since the instability load Is less sensitive to rib distance than for ordinary stiffened panels, cannot be used to achieve maximum advantage since the lateral loadings are playing an im-portant rôle and lower the allowable axial loads.

Case 1

b=oo

Cse 2

L=oo

ÇOse 3

L

is finite

B

E

(t_

t

E tctf t

12(1-v') 2(1-v1

L

m*7t fl17t*

-b1

f.B\2.

1+-r-+

_b2S)

*B1b

_ti2

ba

_Lt.

bJ

kB

-

b1

+

s (1+L'/b

mrr'

fl17t1

L'

+

b'

(tc

+tf)

4

.GtG

'C

P =G.2tf =G.r.t =

w=rf.2tftC=[rfrrC(1_r)]

t

(1r)(1

)

2Gr

OPTIMUM

SANDWICHCOLUMNS.

Graph 2

[rfr+;.(i_r)]

* E (1

-

r) r. t

4(1vi L

3ttE(1-.r) r.t

+

_4(1'viL1.tG

rG)

Gr

P P

(T)

_d6+

_&drO

-P

dG+

P

p

OPTIMUM

SANDWICHCOLUMNS.

Graph 3 A _-

-pl'

-I,

pLasticity

infLuence

in which R

P

w

otimum

-t- =

constant.

projectiòn

of the optima.

R

R 1+)+1

r

2f-1

-

4(2f-1)[(R+f)ff-i]'

Et. 1OT( Kg,*m1)

Etf()

MATERIAL

:

7075....T6

3000

3500

¿000

Graph 4 4400

6

(kg/i)

1

¿200

3600

3000

2/.00

1800

1200

600

BUCKLJNGSTRESSES OF SANDW1CH COLUMNS

Graph S

CORE DENSITY

30 Kq/m5

CORE DENStTY ¿5 Kg/ma

. . CORE DENSITY WABENRÄUMGEWICHT DENSIT NIDA 10

P/L

1.00 1000

&tf

r

014

012

010

₀₀₈

0,06

0,04

002

0,1 CORE DENSITY l.5

Kq/mJ\

CORE DENSITY 3OKg/m

I

10

P/L

_Kg/cm

100 Graph 6 1000

G

(Kg,/cn

¿200

3600:

3000

2400

1800

1200

600

Graph 7

SANDWICH

core

"-Z STIFFENED

PANELS.

0,1 1 10

P/L..

100 1000

Kg/cm*

P/W

.IOT'(cm)

t

lì.

1,2

l-o

0.8

06.

Oil.

SANDWICH

CORE

'45 Kg/rn

-z

STIFFENED PANELS.

loo

0,1

i

lo

P/L

Kg,cmt

Graph 8 1000

OPTIMUM

SANDWICH

PANELS.

Graph 9

PANEL

_QE

INFINITE,

LEÑGTH.

4nB

J

I

_¡

\1+

bS

COLUMN,:

xB

D..

L

I-

-.I

S

for

b=1

[14.

b F? P

w

rfr+Ç (.1-r)

OPTIMUM FOR:

=

[_

(R+f)r]'/1_ "'=

oc

r(f1.)(1_r)

0.5

Optimum designed sandwich

panels with

simply supported

edges.

Graph 10

lo

G(Kg/mrv

¿o

1,5

P'

4' 1.0 0.5

4100

14m)

Comparison

sandwich colUmns

and

panels.

G Kg1tm'

3600

3100

-/

I I

_I

I

_/

o

10 Graph 11 100 .15 0.10

SANDWICHES

UNDER

LATERAL AND

AXIAL

_LOADS.

ONLY LATERAL.LOADS

.

.3u

₂ U. .q

+

=

B

COMBINED

LATERAL AND AXIAL

LOADS.

.sI

I

(1-.--)

o ø

U_dU=.J_(q_.pdU

S

. S

xy"

B

: (u'+)+(2-f).(u+ü)+

- i

r

(u+u)=

_B

[q'-P--(u+u)

RESULT.

u0-

u

û0

(Vianetto)

inftuence

of, plasticity.

-*

u0

q

OPTIMUM SANDWICH PANELS UNDER AXIAL

AND LATERAL

LOADINGS.

G'=

6b

+.G-=

n which

P2k

b

rb

eer

s.

and

9=

-j;-(G-.:. b

¿e J*E(1.t)(

-2(1v')

u.,tB(4T

)"_.

,tL

i:'

L*L

S

L& h&.

j

9tE(1r)r(4+1)

TL1+

4(1-v)Gc

-Graph 13

¿E(1

,v)r(_+1)a

16g

4(1-vI

4

P/

'w

Optimum

designed

sandwich

paneLs

¡n

compression

with

and

without

Latergi

Loads.

q = 025 k9/cm

L/b= 1