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Thin-Walled Structures10 (1990) 247-262

Transverse Collapse o f Th in W alled Sq ua re Tu bes inOp posed Loadings

N . IC G u p t a S . K .S i n h a

D e p a r t m e n t o f A p p l i e d M e c h a n ic s , I n d i a n I n s ti tu t e o f Te c h n o l o g y - - D e l h i, H a u z K h a s ,N e w D e l h i - l l 0 0 16 , I n d i a

(Rece ived 12 Apr i l 1989; acce pted 15 O cto ber 1989)

A B S T R A C T

Th in -wa l l ed squa r e t ub es o f a l u m in ium an d mi ld s t ee l we re sub jec te d t ola te ra l compress ion be tween two oppos ing wedge-shaped a nd f i a t - faced

inden te rs o f na r row wid th in an Ins tron mach ine . Typica l h i sto ries o f the i rdeformat ion load-compress ion response an d the energy absorbed dur ingco l lapse a re d iscussed . A n ana lys i s based on the o rm at ion of p las t ic h inges i sp resen ted to descr ibe the geom et ry changes an d co l lapse behav iour o f thetubes and the resul ts ob ta ined thereby a re com pared wi th the exper iments .

N O TAT I O N

B , HL LoMpPWWA a, etc.

O~AB etc.8tro

W id th an d h e igh t o f t he t ube see F ig . 9a )L e n g t h o f tu b e a n d its o v e r h a n gP l a s ti c m o m e n t p e r u n i t l e n g t hLa te ra l l oadW i d th o f p la t e nEnerg y abso rbe d a t h ing e AB, etc .

An gle of ro ta t ion o f h in ge l ine AB, e tc . see F ig . 9)C o m p r e s s i o n o r re la ti v e i n d e n t e r m o v e m e n tYie ld stress

247Thin-Walled Structures0263-8231/90/ 03-50 1990 Elsevier Sc ien ce Pub l i shers Ltd ,Eng land . P r in t ed in Grea t Br i t a in

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48 N. K. Gupta S . K. Sinh a

1 INTRODUCTION

Thin-walled tubes are increasingly being used in a variety of ways in

energy-absorbing systems, and considerable attention has recently beenpaid to the study of their plasto-mechanics, t-3 Their collapse underlateral compression has been studied extensively in the case of roundtubes 4. 5 and their different combina tions. 6. 7 Such studies in the case oftubes o f square or rectangular cross-section, however, have received lessattention. Bending collapse behaviour of these tubes was studied byKecman, 8 who used a limit analysis techn ique and derived a set ofexpressions for hinge moments and associated angles of rotation. Forthin square rings, Sinha a nd Chitkara 9 determined the plastic collapse

load from the stability analysis of vertical members. To determine thepost-collapse load, they assumed plastic hinges at the mid-point of eacharm of the ring and a constan t coefficient of friction between the identersand the ring surfaces. They later used a hodograph approach to estimatethe collapse load for square and rectangular t tubes subjected totransverse line load.

An a ttempt is made here to study the manne r in which localised plasticmechanisms develop in a square tube when it is subjected to lateralloading between two opposing wedge-shaped or flat-faced platens ofdifferent widths. Experiments were performed 12 on alumi niu m an d mildsteel tubes, and the results obta ined for their history of deformation, post-collapse load-compres sion characteristics and energy absorbed duringcollapse are discussed here. These reveal that the collapse load,deforming shape and the post-collapse behaviour of a tube vary not onlywith the width of inden ter but also with the tube length and the length ofits overhangs. The analysis presented is for a perfectly plastic materialand is based on the assumption that plastic hinges are formed sym-metrically. The energy absorbed and the load-compression characteristicsof the tubes are comp uted and the results thus obt ained compare wellwith the experiments.

2 E ~ E ~ M E N T S

Square tubes of alumin ium a nd mild steel, of dimensions given in Tablel, were commercially obta ined in lengths of about 3-6 m. Specimens ofdifferent lengths varying from 12.5 to 300 mm were cut from these.

The alu min ium tubes were tested in as-received condition, and themild steel tubes were tested in both as-received and ann eal ed conditions.Mild steel tubes were ann eal ed by soaking them at 900C for 30 min

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Transverse collapse o f thin-walled sq uare tubes 249

TABLE 1Specimen Dimensions

Materials Side H ram) Thickness ram)

Alumin ium 12 1.024 1.035 I.i50 i.2

Mild Steel 24 .037 1-0

before allowing them to cool in the furnace for 24 h. The aluminiumtubes were manufactured by extrusion and had sharp square comerswith negligible initial out-of-straightness. The mild steel tubes weremanufactured by cold rolling and by resistance welding. They hadrounded corners and their sides were marginally roun ded outwards. Thetubes were compressed between pairs of identical and opposinginden ters in a 50 ton (500 kN) Instron testing machine (Model 1197) at acrosshead speed of 2 mm min -t. The indenters were made of har deneddie steel and were ground. The wedge angle was 60 or 120 , and thewidths of flat inden ters used in the experiments were 24, 46, 75. 100 or150 mm.

The load -com pres sion curves in the above tests were obtained on theautomatic chart recorder of the machine and the history (shape changes)of the specimen defo rmation was recorded by interrupting the tests atdifferent stages.

3 RESULTS AND DISCUSSION

In the first set of experiments, load -compre ssio n curves were obtained bycompress ing aluminium tubes, each of sides 12, 24, 35 and 50 mm,between 60 wedges. The length o f the tube in each of these experimentswas varied. Typical load-compression curves obtained for differentlengths for 12 and 24-mm tubes are shown in Figs l(a) and l(b)respectively. It is seen that the curves tend to become ind ependent of thespecimen s length as this is increased beyond a particular point. Thedeformation in the case of 12-mm alu min ium tube for all lengths and24-mm tube for the 25-ram length was seen to be generally symmetrical,i.e. the hor izontal plastic hinges were formed at the mid-he ight of thetube. Their load -compression curves, Figs. l(a) and l(b) respectively, are

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2 5 0 N K Gup ta S K S inha

a .

o_J

Fig. 1.

4.00

3.(30

2.003.

,~ 1.00O_J

00

~ ~ L =75 mm

L --50 mmL -- 37.5mmL =25 mm

5 I0COMPRESSION I~ram)

a)

5 0 0 -

t 9 0

3.00

2.00

1.00

00

L - lS0mmL=IO0

N ~ / / ' - L 25

I ~ I ~ 1

5 10 15 20COMPRESSION 6 (ram }

b )

L o a d c o m p r e s s i o n c u r v e s u n d e r6 0 w e d g e s f o r a l u m i n i u m t u b e s o f d i ff e re n tl e n g t h s a n d s i z e :( a) 12 m m a n d ( b ) 2 4 m m .

s m o o t h a n d d r op c o n t i n u o u s l y fr o m t he m a x i m u m l o a d v a lu e u n t il th es ta g e i n t h e t es t w h e n t h e u p p e r a n d t h e l o w e r s i d e s o f t h e tu b e t o u c h e d .F o r s u f f i c ie n t l e n g t h o f t u b e s, h o w e v e r , t h e d e f o r m a t i o n w a s g e n e r a l l yn o n - s y m m e t r i c a l fo r t u b e s iz e s o f 2 4, 3 5 a n d 5 0 m m , a n d t h e l o a d -c o m p r e s s i o n c u r v e s t e n d t o b e o s c i l l a t in g s e e fo r e x a m p l e , F ig . l b ) ) fo r2 4 - m m t u b e o f l e n g t h 5 0 m m o r m o r e . A t t h e f ir st p e a k l o a d , t h e t u b eb u c k l e d a n d a h o r i z o n t a l p la s t ic h i n g e f o r m e d a t a b o u t t w o - th i rd s o f t h eh e i g h t o f t h e tu b e . A s t h e i n d e n t e r n e a r e d t h i s p o s i t i o n , t h e h i n g e b e g a nt o ro ll , a n d t h e f a l l i n g l o a d b e g a n t o r is e o n c e a g a i n . W h e n t h e r o l li n gh i n g e r e a c h e d t h e m i d - h e i g h t o f t h e tu b e, th e l o a d - c o m p r e s s i o n c u r veb e g i n s t o s h o w a n o t h e r f al l. A s t h e te st w a s c o n t i n u e d , t h er e w a s a g a i n ar is e in l o a d w h e n t h e d e f o r m i n g t u b e w a l l s t o u c h e d t h e s i d es o f t h e

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Transverse collapse of thin walled squa re tubes 25

w e d g e s. Ty p i c a l d e f o r m a t i o n s ta g es f o r a 2 4 - m m a l u m i n i u m t u b e a r es h o w n i n F i g . 2 (i )- (v i ). I t i s s e e n f r o m F i g . 2( vi ) t h a t t h e p o r t i o n o f t h e t u b ed i r e c t ly u n d e r t h e i n d e n t e r d e f o r m e d t h e m o s t . A s e c t i o n i n th i s p o r t i o na p p e a r s t o h a v e u n d e r g o n e m a x i m u m e n l a r g e m e n t o f t h e h o r iz o n t a ld i m e n s i o n a n d m a x i m u m s h o r t e n i n g o f t h e v e rt ic a l d i m e n s i o n . T h isp l a st ic d e f o r m a t i o n g r a d u a l l y d i m i n i s h e d a w a y f ro m t h e l o a d e d z o n e s ot h a t b e y o n d a c e r t a i n d i s t a n c e t h e t u b e r e m a i n e d t o t a l l y u n d e f o r m e d .S i m i l a r b e h a v i o u r w a s a ls o r e v e a l e d b y t h e st ee l t u be s . A s a n e x a m p l e o ft h e n a t u r e o f t h e i r l o a d - c o m p r e s s i o n b e h a v i o u r , r e s u lt s f o r 3 7 - ra m s t ee lt u b e s o f d i f f e r e n t l e n g t h s t e s te d i n a s - r e c e i v e d c o n d i t i o n a r e s h o w n i nF i g. 3. I n g e n e r a l , t h e r a ti o o f t h e p e a k l o a d t o m e a n p o s t - c o l la p s e l o a d i nt h e c a s e o f s te e l t u b e s w a s l o w e r t h a n f o r t h e a l u m i n i u m t u b e s, w h i c h w a sp o s s i b l y d u e t o t h e i n i t i a l o u t - o f - s t r a i g h t n e s s o f th e l a tt e r.

T h e s e c o n d s e t o f e x p e r i m e n t s c o n s i s t e d o f c o m p r e s s i n g t h e 2 4-.3 5 - a n d 5 0 - m m s i z e t u b e s b e t w e e n t w o i d e n t i c a l f i a t - f a c e d n a r r o ww i d t h p l at e n s. I n t e re s ti n g ly, t h e m o d e o f d e f o r m a t i o n i n s o m e o f t he s et es ts w a s o b s e r v e d t o b e s y m m e t r i c a l , w h e r e a s i n o t h e r s it w a s f o u n d t o ben o n - s y m m e t r i c a l , a l t h o u g h t h e t e s t c o n d i t i o n s w e r e i d e n t i c a l . T h i sp h e n o m e n o n r e q u ir e s f u r t h e r in v e s ti g at io n ; h o w e v e r, v a r ia t i o n s i nt h i c k n es s a n d m a t e r i a l p r o p e rt i es o f t he t u b e m a y h a v e c o n t r i b u t e d t os u c h v a r i a t i o n i n b e h a v i o u r. I n F i g s 4 ( a ) a n d 4 ( b ) , t y p i c a l l o a d -c o m p r e s s i o n c u r ve s a re p r e s e n t e d f o r t h e 2 4 -m m a l u m i n i u m t ub e s ,c o m p r e s s e d b e t w e e n 2 4 - m m w i d e p l a te n s . F i g u r e 4 (a ) sh o w s c u r ve s f ord i f f e r e n t t u b e l e n g t h s a n d f o r a l l t h o s e c a s e s w h e r e t h e d e f o r m a t i o nm o d e s w e r e o b s e r v e d t o b e s y m m e t r i c a l : th e s e c u rv e s a r e s m o o t h , d r o pc o n t i n u o u s l y a n d a r e s i m i l a r i n n a t u r e t o t h o s e o f 1 2 -m m a l u m i n i u mt u be s c o m p r e s s e d b e t w e e n w e d g e - s h a p e d i n d e n t e r s (S e e F ig . l ta )) . T h el o a d - c o m p r e s s i o n c u r v e s f or 2 4 - m m t u b es c o m p r e s s e d b e t w e e n fla ti n d e n t e r s w h e r e th e d e f o r m a t i o n w a s n o n - s y m m e t r i c a l a re p r e s e n t e d inF i g . 4 (b ). I n b o t h c a s e s, h o w e v e r , th e l o a d - c o m p r e s s i o n c ur ~ e s d e p e n do n t h e l e n g t h o f t h e t u b e u n t i l it is g r e a t e r t h a n a p a r t i c u l a r v a l u e . I nc o m p r e s s i o n u n d e r w e d g e s o r u n d e r n a r r o w w i d th p l at en s , w h e n e v e r t h el e n g t h o f t h e t u b e t e s t e d w a s r e d u c e d , its d e f o r m a t i o n t e n d e d t o b e i n am o d e w h i c h w a s a t ra n s i t io n f r o m t h e a b o v e l o c a l i z e d m o d e to th e r in gm o d e .

S i m i l a r b e h a v i o u r w a s r e v e a l e d b y t h e s t e e l t u b e s w h e n c o m p r e s s e db e t w e e n 2 4 - m m f i a t - f a c e d i n d e n t e r s . F i g u r e s 5 ( a ) a n d 5 ( b ) s h o w t h et y p ic a l l o a d - c o m p r e s s i o n c u r v e s fo r t h e 3 7 - m m t u b es m a d e o f m i l d ste eli n a s - r e c e i v e d a n d a n n e a l e d c o n d i t i o n s r e s p ec t iv e l y. T h e i n i ti a l o u t -o f -s t r a i g h t n e s s o f st ee l t u b e s a g a i n p o s s i b l y r e s u l ts i n t h e l o w e r p e a k t om e a n l o a d r at io t h a n f or a l u m i n i u m t ub es .

F i g u r e 6 s h o w s l o a d - c o m p r e s s i o n c u rv e s fo r 2 4 - m m a l u m i n i u m t ub e s

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252 , ~ K. Gup ta . S K Sin ha

i ) i i )

i i i ) i v )

v ) v i )F i g . 2 . S t a g e s o f d e f o r m a t i o n o f a t y p i c a l 2 4 m m a l u m i n i u m r o b e o f l e n g t h 150 m m .

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Transverse collapse of thin walled square tubes 253

L=150mmo o

s oo

~ 2 so

5 10 15 20

COMPRESSION 6 ram)

F i g 3. Load-compression curves fo r 37-ram steel tubes in as-received condit ion , under60 wedges.

each of 150-mm length, obtained in tests with different inden ter widths.The variation of peak load with variation in in dente r width is shown inthe inset. The variation of peak load vs the length of tube, when ot hercondi tions are same, is shown in Figs 7 a) an d 7 b) for 24-mmaluminium and 37-mm steel tubes respectively when compressedbetween 60 wedges and 24-mm fiat indenters.

The peak load from the above results can be written in the form

P = po W q for0

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;0

IO0

~0

LO0

0i

4 oo

i

3 oc

,.

kO

~o

X\

/~-

c

=o

~

=85

,

\ .~

_.

~-

._

~oo

\~--J~

~--c

~o

PO

ZOC

,0C

10

i

A

t

t

I

~

z

_|

J

J

l

|

)

5

10

1

1

0

5

1

1

COt4

RESSION G

m m

)

~

COMPRESSION 6

{m m.

Q]

(b)

;g

4

Ioad-c

mprc

on

curves

fo

2

am Muminium lubcs c

mpe

under

2

mm wid

pae

when dcorma hm wa

asymme

c

or h

n

ymme

c

g

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Transverse collapse of thin walled square tubes 255

.o o / l ~

0 5 10 15 20COMPRESSION 6 (mm)

a)

I ~

~ L -- 17 5m mX

Q- 5.00 1 5 0c ~ - - 1 2 5< ----100o ~ 7 5J ~ 5 0

_ . - - - - - 2 5i l

5 10 15 20

C O M PR E SS IO N 6 ( m m )

b )

Fig. 5. Loa d-c omp res si on curves for 37-mm steel tubes, compressed un de r 24 mm wideplatens, in a) as-received or b) ann eal ed condition.

affec ted leng th i s app rec ia b ly red uced see F ig . 8 ). I t m ay be seen tha t them ax im um a ffec ted leng th fo r l onge r t ubes , symm et r i ca l l y p l aced ab ou ta n i n d e n t e r o f w i d t h W e x t e n d s o n e i th e r s id e o f t h e l o a d e d p o r t io n b ynea r ly H on ly.

I t i s seen f rom the exp er im ents tha t the co l lapse o f a tube occurs in twoways : t he ho r i zon t a l h inge i s f o rmed e i t he r a t t he m id -he igh t , w hen t hede fo rma t ion is s a id t o be sy mm et r i ca l , o r a t abou t two- th i rd s o f its he igh t ,when t he de fo rma t ion i s non - symmet r i ca l . I n t he fo l l owing ana ly s i s ,howeve r, we have cons ide red on ly t he fo rmer ca se fo r a s i t ua t i on o fsu ff ic i en t ove rhang .

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256 N K. Gupta S K Sinha

=lS0mm ~ .200

l O 0 0 [ / ~ m m o sO 100 Is o

05 10

COMPRESSION 8 ram )

F i g 6 Load-compression curves for 24-mm alum inium tub es of 150-mm lengthobtained in compression under flat platen of different widths.

a) t,ao mm BARS

0 50 100 150LENGTH OF TUBE m m )

15-00 r- 24ra m BAR Sb) I

6 o WEDGES

TO O0 ~ ~

5.00 J / / - - - - ANNEALED

; / /

25 50 75 100 125 150 175

LENGTH OF TU BE m m )

F i g. 7 . Va r i a t i o n o f p e a k l o a d w i t h t h e l e n g t h o f atube w h e n c o m p r e s s e d b e t w e e nwedges or 24-mm bars: a) for 24-mm alu min ium an d b) for 37-mm steel tubes.

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Transverse collapse of thin walled square tubes 257

1 2 5

l t0C }

Etom 7 ~. ii

iLI.O,r

- S OZU J. . /

0

U .

< 0 I t | I I t I0 2 5 5 0 7 S 1 0 0 1 2 S 1 5 0 1 7 S

O R I G I N A L L E N G T H O F T U B E ( m m ) -. ..- -. ,, ,, ,

F i g . 8 . V a r i a t i o n i n a f f e c t e d l e n g t h o f t h e t u b e w i t h i ts o r i g i n a l l e n g t h f o r 2 4 r a m t u b e oi) alum inium and ii) steel.

4 A N A LY S I S

On the ba s i s o f t he obse rva t i ons d i s cus sed above , a m ech an i sm o f shapecha nge s fo r the sym m et r ica l mo de of co l lapse i s p rese n ted be low. I t isa s sum ed tha t t he ene rgy is abso rbed i n ro t a ti on o f t he concen t r a t edp l a s ti c h inges , a s show n schem a t i ca l l y i n F ig. 9 , an d t he m a te r ia l i n pa r tso the r t han t he se h inge l i ne s r ema ins r i g id .

Fo r a g iven re la t ive inde n te r m ove m ent 5 a se t o f express ions fo r theangles o f ro ta t ion of the h inge l ines i s de r ived as fo l lows .

Th e re la tive ro ta t ion o f y ie ld l ine BC i s

aBc = si n -I 5/2/-/) 1)

The ene rgy abso rbed i n BC and t h r ee more i den t i ca l y i e ld l ine s B i C ian d two iden t i ca l li ne s be low) becom es

WBc = 4M pB sin -n8121-0 2)

wh ere B i s the l en g th o f the y ie ld line BC an d Mp i s the fu l ly p las t i cm o m e n t p e r u n i t le n g th , g i v e n b y

m p = O o t 2/ 4 3)

S imi l a r ly, ro t a t i on o f h inge l i ne D E , Dn En an d two o the r i den t i ca l l ine sa n d c o r r e s p o n d i n g e n e rg y a b s o r b e d a r e g i v en a s

aOE = sin-I cS/2/-/) 4)

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258 N. K. Gupta S K . S i n h a

D C C1

AH _ ~ _ w j _ H _t

B

a ) U N O E F O R M E D

C ) D E F O R M E D

/ 8 1 2

E * ~ E L J1 j \ ~ L _

A i A'.t - '

b ) SECTIONA B C s 2

L 8 ~ s / 2

d) SECTION A B C

e ) ELEVATION OFE B B 1 J A I A f ) S E C T IO N A S A T d )

Fig . 9 . Col lapse m odel o f the tube show ing p las tic h inges for the tube .

a n d

W DE = 4 M p B s i n - I ( 8 / 2 H ) ( 5)

T h e s e e q u a t i o n s f o r h i n g e l in e E F a re

T h e r o t a t i o n o f h i n g e l i n e s A B , B E , E A ( a n d t h e i r c o u n t e r p a r t s ) a r ec o m p u t e d b y a l l o w i n g s o m e i n - p l a n e d e f o r m a t i o n o f t he t r i an g le A B Eu s i n g t h e g e o m e t r y o f d e f o r m a t i o n s h o w n i n F ig . 9 :

A A ' = [ (A B ) 2 - (A 'B)2 ] '~ = [ (H I2 ) 2 -H/2 - c5/2 )21 /Z

I f E ' is t h e p r o j e c t i o n o f E o n t h e A B C p l a n e ( se e F ig s . 9 (e ) a n d 9 (f )) t h e n

E E ' = [ (E B ) 2 - (E B ' )2 ] '/2

o r

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f

a n d

Transverse collapse of thin wailed square tubes

= [ (AA ) 2 + (A E )2] ~

259

a A B ---~

w h e r e

E P = A E s in Z E A P

T h e e n e r g y a b s o r b e d i n A B a n d s e v e n m o r e i d e n t i c a l y i e ld l in e st o g e th e r b e c o m e s

WAS = 8 M p (H / 2 )a A B = 4 M p H a A a (9)

T h i s h o w e v e r , n e g l e c t s t h e w o r k d o n e d u e t o s t r e t c h i n g .R o t a t i o n o f y i e l d li n e E A , i.e . t h e a n g l e b e t w e e n t h e t w o t ri a n g l e s F E A

a n d E A B , m a y b e d e t e r m i n e d b y c o n s t r u c t in g a t ri a n gl e G J K s u c h t h a t Gis t h e m i d - p o i n t o n E F, G J is n o r m a l t o t h e c u r r e n t y i e l d l i ne E A a n d J Kis n o r m a l t o E A i n t h e p l a n e o f t r i a n g l e E B A , K b e i n g a p o i n t o n E B . T h ea n g l e o f ro t a t io n o f E A t h e n is

~EA = ~ -- Z G J K

a n d c a n b e e v a l u a t e d b y so l v in g th e t ri a n g l e G J K . T h e e n e r g y a b s o r b e di n E A a n d s e v e n m o r e i d e n t i c a l y ie l d l in e s t o g e th e r b e c o m e s

WEA = 8M pEA aEA =8 9 4 4 M p H a E A (10)

To c a l c u l a te t h e a n g l e o f ro t a t i o n o f y i e l d l i ne B E , a n o r m a l A B f r o m A t oE B e x t e n d e d i s d r a w n a n d t h e a n g l e A B A t h e n g iv e s t h e r o ta t i o n o f E B .H e n c e

a E a = t a n - I ( A A / A B ) ( l l )

a n d t h e e n e r g y a b s o r b e d i n E B a n d s e v en o t h e r i d e n t i c a l y i e ld l in e st o g e th e r b e c o m e s

WEB = 8Mr, HaEB (12)

T h e e n e rg y a b s o r b e d i n t h e h i n g e l i n e s b e l o w t h e p l a t e n s , i.e . B B 1 , A A 1 ,L L 1 , e t c . i s g i v e n b y

WpL 8MpWcos - (1 - 8 / 1 t ) (13)T h e t o ta l v a l u e o f t h e e n e r g y a b s o r b e d d u e t o a r e l at iv e i n d e n t e rm o v e m e n t 8 th u s b e c o m e s

L E A P = L E A A - / B A A

I f E P is t h e n o r m a l f r o m p o i n t E t o t h e li n e A B e x t e n d e d , t h e n r o t a ti o n o fh i n g e l i n e A B i s g i v e n b y

rr/2 - L E P E = r t/ 2 - t a n -~ ( E E / E P ) (8 )

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260 N K Gup ta S K S inh a

1 S ' 0 0 1 ~ E X P E R I M E N TA L- - - T H E O R E T I C A L

10.0C

~ ~ 2 4 m mz RECEIV

o , ' I ~ ~ 2 4 r a m

5.00

/ S T E E L0 , , , I ~ , i J ~ , , I

0 5 10 15 20COMPRESSION ( ram)

a )

150.00 ~

EXPERIMENTALE THEOR E T I C A ~ / , ~

W

>

h i

0 ~ - - ' ' ~ ' , , I I I I i I I f r ~ I i l r , 1

0 5 10 15 20D E F O R M AT I O N 6 ( r a m )

b )

F ig . 10 . T h e o r e t i c a l a n d e x p e r im e n t a l c u rv e s fo r a l u m i n i u m a n d s te e l i n a n n e a l e d a n da s - r e c e i v e d c o n d i t i o n ; ( a )]oad-compression ( b ) e n e r g y - c o m p r e s s io n .

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Transve r se co l lapse o f th in wa i l ed squa re tubes 261

W 8 = WBc ~W E l W EF ] WAB ~ [ YEA ~ WEB + W pL 14)

Th e load P6 a t an y ~5 i s t hen ca l cu l a t ed by w r i ti ng

t 8 = [ W 8 + A 8 - W S I / A 8 (15)

w h e r e A 8 is a n i n c r e m e n t o f t h e i n d e n t e r m o v e m e n t .T h e o r e t i c a l l o a d - c o m p r e s s i o n a n d e n e rg y - c o m p r e s s i o n c u r v e s a r e

p re sen t ed fo r some typ i ca l c a se s o f squa re a lu m in iu m an d s t eel t ubesin Figs . 10(a) a n d 10(b) respect ively, w i th the exp er im en tal resul ts .O sc i l l a t ion in the exp er im enta l c urve fo r as- rece ived s tee l i s due to non-symmet r i ca l de fo rma t ion .

5 C O N C L U S I O N S

The se exp e r imen t s r evea l t ha t t he na tu r e an d ex t en t o f de fo rm a t ion o f asqua re t ube u nde r l a te r a l com pres s ion depe nds on i ts he igh t , l eng th , t heo v e r h a n g a n d t h e w i d th o f t h e i n d e n te r. F o r1 2 m m tubes , co l lapse wasa lways found t o be sym me t r ica l , w i th a ho r i zon t a l h ing e fo rmed a t t hetube ' s m id -he igh t . The l oad -compres s ion cu rve i n t h i s c a se i s con -t inuo us ly fa l l ing . Fo r com pres s ion un de r wedges , the co l lapse o f 24-, 35-an d 50 -mm tubes was gene ra l l y s een t o be non - sy mm et r i ca l , w i th t heho r i zo n t a l h inge fo rm ing a t abou t two- th i rd s o f t he he igh t o f t he t ubea n d p r o d u c i n g o s c i ll a ti n g l o a d - c o m p r e s s i o n g r a p h s . F o r c o m p r e s s i o nof 24-, 37- o r 50-m m tubes wi th f l a t- faced ind en te rs , how ever, thede fo rma t ion i n d i f f e r en t expe r imen t s was obse rved t o be e i t he rsym m et r i ca l o r non - sym me t r i ca l . I f t he l eng th o f tube i s equa l t o t hep l a t en w id th , t he tube de fo rm s i n a r ing mode . W he n tube l eng th i s morethan t he p l a t en w id th , such t ha t t he ove rh ang i s inc r ea sed f rom ze ro , thede fo rm a t ion ex t ends t o the t ube ' s f u ll l eng th un t i l t he ove rhang is abou t1 . 5 H - 2 H . W ith fu r the r inc r ea se i n leng th o f tub e , how eve r, the de fo rm edleng th on e i t he r s i de f rom the edge o f t he l oaded zone becom es nea r ly H .Th i s s i t ua t i on i s ana ly sed by con s ide r ing t he co l lapse t o be symm et r ica l .Th e r e su lt s t hus ob t a ined com pare we l l w i th the expe r imen t s .

.

R E F E R E N C E S

Ezra, A. A. Fay, R. J., An a ssessm ent of energy abs orb ing devices forprospective use in aircraft impact situations. InDynamic Response ofStructures, ed. G. He,,, ,an and N. Perrone, Pergamon Press, Oxford, 1972,pp. 225--46.

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262 N. K. Gupta. S. h: Sinha

2. Joh nso n , W. Re id , S . R ., M eta l l i c ene rgy d i ss ipa t ing sys tem,A p p l i e dM e c h a n i c s R e v i e w s31(3) (1978) 277-88.

3. R e id , S . R ., La t e r a l l y com pr e s sed m e ta l t ubes as im pac t ene rgy abso rbe r s . I nS t r u c t u r a l C r a s h w o r t h i n e ssBut te rw or th , Lo nd on , 1983, pp . 1 -43.

4 . Re id , S . R . Re ddy , T. Y., Effec t s o f s t ra in ha rd en in g on the l a te ra lc o m p re s s ion o f t ubes be t w ee n r i g id p l a te s ,In t . J . So l . S t ruc t .14(3) (1978)213-25.

5. Reddy , T. Y. Re id , S . R ., P he no m e n a a s soc i a t ed w i t h t he c ru sh in g o f me t a ltubes be tween r ig id p la tes ,In t . J . So l . S t ruc t .16(6) (1980) 545-62.

6 . John s on , W., Re id , S . R . Red dy, T. Y., T h e com pre s s ion o f c ro s sed l aye r s o fth in tubes ,In t . J . Mech . Sc i . .19(7) (1977) 423-37.

7. R edd y, T. Y. Re id , S . T., La t e r a l com pre s s io n o f t ub es an d t ub e - sys t e mswi th s ide cons t ra in t s ,In t . J . Mech . Sc i .21(3) (1979) 187-99.

8. K e c m a n , D ., B e n d i n g c o l l a p s e o f r e c t a n g u l a r a n d s q u a r e s e c ti o nt ub es In t . J .

Mech . Sc i . .25(9/10) (1983) 623-36.9 . S inha . D. K. Ch i tkara , N . R ., P las t i c co l lapse o f squ are r ings ,Int . J . Sol .

S t ruc t . . 18(9) (1982) 819-26.10. S inha . D. K. Ch i tkara , N . R ., A s im pl i f i ed so lu t io n fo r p las t i c co l lapse

loads o f squa re t ube s sub j ec t ed t o oppos ed t r an sve r se l i ne l oad s ,A c t aM e c h a n i c a . 44(3--4)(1982) 177-86.

11. S inha , D . K . Ch i t ka r a , N . R ., D e t e r m ina t i on o f p l a s ti c co l l ap se l oa ds o fr e c t ang u l a r t ub es ,A c t a M e c h a n i c a .51(3--4) (1984) 199-215.

12. S i n h a . S . K ., La t e r a l com pr e s s ion o f squa re t ubes a nd t he i r c ro s s l aye r edsys te m s , P hD the si s, I nd i an In s t i tu t e o f Te chn o l og y - - De lh i , 1988.