Lateral compression of a square or rectangular tube between two parallel narrow width indenters placed non-orthogonally

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E L S E V I E R

Lateral Compression of a Square or Rectangular Tube Between Two Parallel Narrow Width Indenters Placed

Non-Orthogonally

N. K. Gupta & A. Khullar

Applied Mechanics Department, Indian Institute of Technology, Delhi, New Delhi-110016, India

(Received 18 April 1993; accepted 4 January 1994)

A B S T R A C T

Collapse behavior o f aluminum tubes o f square and rectangular cross- sections, when compressed between two identical narrow width indenters placed symmetrically in parallel alignment, is examined. Experiments were performed wherein the angle between the axes o f tube and indenters was varied f r o m 0 ~ to 90Z Load compression curves and deJormation histories o f typical specimens are presented. The collapse o f the tubes was seen to be generally symmetrical, though asymmetries were observed in some tubes.

Considering only the symmetrical mode o f dejormation, an analysis is presentedJor constructing the load-compression curves as well as the shape o[ the d~[brming tube. The analysis considers the energy absorbed in stationary and rolling plastic hinges which are Jormed in the collapsing tube. Computed results thus obtained compare well with the experiments.

N O T A T I O N

Dl Depth at which first horizontal hinge is formed DL Distance of left vertical hinge from center of tube Dr Distance of right vertical hinge from center of tube E~ Energy absorbed in stationary hinges

Er Energy absorbed in rolling hinges H Height of the tube

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L Length o f the tube specimen

LL Length for hinges to form within the specimen ends t Thickness o f the tube

W Width o f the tube

6 Relative vertical displacement o f platens (compression) 61 6 at which left inclined hinges begin to roll

0 Angle between normal to tube axis and indenter axis

I. I N T R O D U C T I O N

An analysis o f the plastic collapse o f a square tube when compressed between two short width indenters was earlier presented by G u p t a and Sinha.~ This analysis considered that the indenters were placed orthogonal to the tube. In this paper we present an improved analysis o f the problem and consider square and rectangular tubes being compressed between two n a r r o w width indenters, which are placed in both orthogonal and non- orthogonal positions relative to the tubes. Experiments were performed on as-received tubes of aluminum, wherein these were compressed between two n a r r o w width indenters in an Instron machine at a crosshead speed o f 2-5mm/min. The angle between the tube and indenter axes was varied from 0 ° to 90 °. L o a d - c o m p r e s s i o n curves and histories o f d e f o r m a t i o n of the tubes were recorded. Based on experimental observations, a c o m p a - tible model which considers the energy a b s o r b e d in rolling and stationary plastic hinges was developed for the analysis. The c o m p u t e d values o f collapse load, location o f hinges, plastic load~zompression curve and history o f d e f o r m a t i o n o f the tube are presented. These results c o m p a r e d well with the experiments.

2. E X P E R I M E N T S

Square and rectangular aluminum tube specimens were compressed between two n a r r o w width indenters which were placed in parallel alignment a b o v e and below the specimen, Fig. I. The clockwise measured angle 0 between the normal to the tube axis and the indenter axis, Fig. 1, was varied from 0 ° (in the orthogonal position) to 90 ° (in the parallel position) in steps o f 15 ° . Specimens were tested in their as-received condition, in an Instron machine o f 5 ton capacity at a crosshead speed o f 2-5 mm/min.

Specimens were 150mm long and were cut from commercially obtained tubes. This was the tube length for all the tests, with the indenters 300 mm in length.

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Collapse behavior of aluminum tubes

Tube

I,oad W i

IMenLel:,i ~H i i Left

i side

z

ot~ y

L : 150mm

.... " ~ W ; ~

' , / / ' t u b e

I . d e . : ~ ( s )

( ~

~ A

/

l, ¸¸ /,'!

Load

t

l I

Right ]

side tl

~ A w

A A

Fig. I. Loading arrangement.

11

The l o a d - c o m p r e s s i o n curves o f the tubes were recorded on the machine chart recorder, The histories o f deformation o f the specimens were recorded by repeating experiments on several tubes and interrupting the tests at different stages o f compression.

The stress-strain curves for each tube material were obtained by conducting tension tests on the Instron machine and the curves thus obtained were idealized to be perfectly plastic. Typical experimental and idealized stress-strain curves are shown in Fig. 2.

Tubes o f two rectangular and one square cross-sections were employed

2 C,0 0

1 5 0 0

Z 1 0 0 0 f

5 0 0

× - ~ - ~ Set No, 1

0 . 0 i T , i r I I I r i i r r r i i , i i r i i i i i i i i i i i i r r i i i , i i r . . . . i I I 4 , ~ I , I r I I , I I , I I I , i

0 0 0 0 0 0 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 . 0 2 5 0 . 0 3 0 0 0 5 5

Strain

Fig. 2. Stress versus strain curve for a specimen from set N o . 1.

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TABLE 1

Cross-sectional Geometry of Tube Specimen and Indenter in Different Sets of Tests and Yield Stress of Tube Material

Set Tube specimen, 150 mm long Indenter

1"10.

Height Width Thickness Yield stress Height Width

(mm) (mm) (mm) (N/mm 2) (mm) (ram)

l 23.0 23.0 0.5 147.2 22.5 22.5

2 24-3 36.9 0-7 117.7 22-5 22.5

3 36-9 24.3 0.7 117.7 22-5 22-5

in the present experiments. Respective yield stress values, cross-sectional dimensions o f the tubes and those o f the indenters are given in Table 1.

The experimental l o a d - c o m p r e s s i o n curves for the sets 1, 2 and 3 are shown in Fig. 3. In these curves it can be seen that the magnitude o f the first peak increased with an increase in 0. A second rise is also seen in these curves. The compression at which the second rise begins, reduces with an increase in 0. The load~compression curve for a typical tube of set No. 1, for 0 = 30 °, is given in Fig. 4 to illustrate the different stages o f the collapse. The first peak in the l o a d - c o m p r e s s i o n curve is the collapse load o f the tube. The section which drops after the first peak is associated with the collapse o f the tube occurring with the folding o f stationary plastic hinges. The second rise in the l o a d - c o m p r e s s i o n curve occurs as the two inclined hinges on the left side begin to roll (discussed later). Figure 5 shows p h o t o g r a p h s o f the compressed tubes. The symmetry in the d e f o r m e d shape o f the tubes is clearly seen in the photographs.

3. A N A L Y S I S

F r o m the experimental observations o f the d e f o r m a t i o n geometries, the typical pattern o f hinges assumed to have formed on all faces o f the tube is shown in Fig. 6. The node numbers, considered in the analysis, have been encircled and the hinges have been m a r k e d in R o m a n numbers.

In a tube o f sufficient length and for values o f 0 less than a value say 0L, the collapse mechanism in the deforming tube is restricted to within a certain length o f the tube and it extends to equal distances on either side from the center o f the tube, see Fig. 5. As 0 is increased b e y o n d 0e, and as long as it is less than a value say 0~, plastic hinges cover the entire specimen length. In a n o n - o r t h o g o n a l position of the indenters, the hinges are restricted to within the specimen length towards the right end o f the tube.

W h e n 0 > 0,-, the tube collapses in a manner similar to the collapse o f a

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Collapse behavior of aluminum tubes 13

4 0 0 0 h L

.3000 ~"/',,,

2O00 o

1000

0 0 0

Set No. 1

. . . Theto=00 deg.

+++-++ Theta=15 deg.

o-eo-e¢ Theta=`30 deg.

Theto=45 deg.

4~-@@,0 Theto=60 deg.

~ - ~ e Theta=75 deg.

',,h:,, O-aB-~B ThetQ=90 deg.

,

.~....~. A-~-z,~ Single s p e c i m e n

£i

2.0 4 0 6 0 8 0 10.0 12.0 140 16.0 18.0 20.0

C o m p r e s s i o n ( r a m )

"t Set No 2 ond 5 . . . Theto=00 deg. (Set 2)

3 0 0 0 t ; ' ~H~-~+Theto=15 deg. (Set 2~

~ 0~6~)e There=50 deg. (Set 2)

.lq,, o ~ . ~ T h e t a = 4 5 deg, (Set 2)

J\l@ 4>,~3~Theto=60 deg, (Set 2)

' ', s ~ - ~ Theto=00 deg. Set .3

~'.~\', u u ~ : z Theto= 15 deg, tSet .3t

Z ' 2 0 0 0 ~,, (%. ~-~a-t~ Theto=50 deg. t'Set `3t

~.~,,"~.\ vv-~v--v Thet(:]=45 deg ISet .31

7,,~.,'x'. oeeo-o Theto=60 deg ISet `3)

~ ~ - ~ . ~ . : . ~ ..--~-:-= - ~ - . ~ - - . , ~ . a ) - - ~ ~ - ~ ~ _ _ _ _ 4

o o 5 0 10.0 1~0 2 0 0 2s.o ~00

C o m p r e s s i o n ( m m )

Fig. 3. (a and b) Experimental load-compression curves for different values of 0.

single tube compressed between two platens with plastic hinges extending over the entire specimen length on each face. The analysis presented is for 0 < 0L and symmetrical mode o f deformation (Fig. 6). The undeformed ends o f the tube behave as rigid bodies.

In the case where 0 < 0L, the collapse occurs when the load-compres- sion curve begins to drop with increasing compression and plastic hinges are seen to have formed on the tube. These hinges as idealized for the analysis are shown in Fig. 6. At collapse (Fig. 4) four sets o f plastic hinges are formed on the tube (Figs 5 and 6). They are:

(1) A horizontal hinge, number III, under the indenter at mid-height o f the tube.

(2) Two vertical hinges (slightly convex outward) numbered I and V, at

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2 0 0 0

1500

1000 •

5 0 0 •

Set No 1

~)~,,,,~- Collapse

'9,

"0. 1ncllined h i n g e s on the left begin to roll t 6 !

" " - - - . . . . o

0-ec-ee T h e t o = 5 0 deg.

[ ~ 1 w [ ~ I ~ F ~ 1 ~ I ~ 1 I ~ [ ~ ' ~ I ~ [ I I ~ j F ~ I ~ l r l ~ T ~ I I l 1 w r ~ I 1 1 ~ F ~ 1 r [ ~ l ~ 1 ~ r r 1 ~ w I ~ F [ l ~ l ~ 1 ~ [ [ r [ ~ I

0.0 2.0 4.0 6.0 8 0 10.0 1 2 0 1 4 0 16.0 18.0 2 0 . 0

Compression (mm)

Fig. 4. Experimental load~compression curve illustrating different stages of the tube collapse.

a certain distance from the edges of the indenters on both the vertical faces.

(3) F o u r inclined hinges between the two ends o f the horizontal hinge at mid-height of the tube and top and bottom o f the two vertical hinges. The two inclined hinges (II and IV) in the upper half of the tube are curved and concave upward and ones in the lower half of the tube are concave d o w n w a r d (Fig. 5).

(4) Stationary hinges on the top and b o t t o m surfaces and corner edges o f the tube (Fig. 6). These are n u m b e r e d as VI, VII, VIII, XI and XII.

In the beginning, collapse o f the tubes progresses with the rotation of all plastic hinges. At some compression 6t, the faces of the indenters come into contact with the o u t w a r d deforming portions o f the vertical face, along the inclined hinges on the left side (II). F r o m this point onwards the inclined plastic hinges are forced to roll by the indenters because these are in contact. The rolling hinges (II) take up positions, such that the horizontal hinge (III), at mid-height of the tube, becomes longer and the rolling inclined hinges become shorter as shown later in Fig. 9. This rolling of inclined hinges occurs for non-orthogonal positions o f the indenters only.

As the indenter moves down, both ends of the two vertical hinges (I, V) move inwards towards the indenter accompanied with the rotation of hinges on the top and b o t t o m faces (XI, XII) (Figs 5 and 6). For non- orthogonal positions o f the indenter, the u n d e f o r m e d ends o f the tubes move such that they no longer lie along the tube axis. This is due to the

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Collapse behavior of aluminum tubes 15

Fig. 5. Deformed shapes of tubes of set No. 1 for varying O; (a) side view (b) end view (c) top view.

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- - - - - -- J'J~J'J~ / W

- _ . . . . / n d e n t e r H ~

/ ) c - ,, : \ ;, ":r2 " - Q I ~ )

,, - -~'q, a PLastic

,' h i n g e W[o ~

/" ^! ⁾

/ , Z '

: Z" v! _ } / ~ + v i i : F - VlII / .... (~.~

left (.2~[ ... II X/'(4S. ) : ~ 5', R i g h t l s i d e i ( ' : 3 ":;~s,~2-:-:-i<t" ]\ ~ : s i d e H

/

I!15).~. ",l~w. : J , ' ' _ _ _ - ---- ] ' . 6 Y Nodeaode '~u(~a~

\ " " ~ 2 \ ' \

\ \ \

Fig. 6. Showing the hinges at collapse: Roman numbers indicate hinges and encircled numbers indicate the nodes.

• /

/ / .

x' q ' / " C , ,''>" 0

/ ," ~, /

H Wi/cos@ Wtan@ H

Fig. 7. Deformed shape of tube in plan.

l

^w

fact that the vertical hinges on the left and the right side are at different distances from the center o f the tube, see Figs 5 and 7.

Collapse o f the tube occurs when under the indenter, four connected panels on the four faces o f the tube become isolated from each other as well as from the undeformed end portions o f the tube by the formation o f plastic hinges I, V, VI, VII, VIII and XII at their edges. Plastic hinges II, II1, IV, IX, X and XI are also formed within the panels on the sides of the tube. Their folding and rolling permits the panels on the sides to deform.

This is similar to the collapse o f plates under a combination o f in-plane and edge loading.

The values o f limiting angles, 0L and 0s, defined earlier are dependent on the dimensions o f the tube and the indenter.

When the overhang h > HI2 (Fig. 8) the extent o f the collapse mechan- ism formed is within the tube length and the corresponding angle, 0e, for this sufficient overhang is given as:

OL = t a n - I ( L I / W ) - sin I ( W i / ( W 2 +

LI2) ½)

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Collapse behavior of aluminum tubes 17

I .... + ~;/',~ I w

h H W i ,/cos~'~ WtanG H h

(~)

A B i

I

]Indenter] ]

~, _ D 1 ~.,~ Dr . .~

L e .

4 L / 2 _ I,,'E

(b)

Fig. 8. (a) T o p face o f t h e t u b e . (b) L o c a t i o n o f t h e vertical h i n g e s .

where

LI = L - 3 H t a n ( f l ) (2)

Correspondingly the m i n i m u m length o f the tube required, for a given 0, such that h > HI2 is:

LL = Wi/cos (0) + W t a n (0) + 3 H t a n (fi) (3) where L is the length o f the specimen and Wi and W are the indenter and tube widths. The angle fi is equal to 45 °.

The collapse load o f a tube, compressed between two short width indenters placed symmetrically on the top and the bottom, was found to be same as the collapse o f a tube o f length Le, when compressed between two platens. 2 As shown in Fig. 8, Le is the b o t t o m width o f a trapezium A B C D u n d e r the indenter and can be written as,

Le = Wi/cos (0) + 2 H t a n (fl) (4)

The horizontal hinge, III, under the indenter (Fig. 6) is f o r m e d at mid- height o f the tube and thus its depth DI from the top is,

Dl = HI2 (5)

Length o f this hinge is taken equal to the tube thickness. The inclined hinges, II, IV, and the horizontal hinge, III, are f o r m e d simultaneously. The inclined hinges are curved such that no distinct horizontal portion o f the hinges is clearly visible u n d e r the indenter at small compressions after the elastic collapse. At large compressions (Fig. 5) it is nearly Wi/cos 0 long.

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In the experiments, the location o f the vertical hinges, I and V, was observed as shown in Figs 6 and 8. The vertical hinges are stationary during compression o f the tube. When the indenter is placed obliquely over the tube, the deformation model of the tube during its compression (Figs 6 and 8(a)) is best described by considering an increased effective width o f the indenter. This increase in the width of the indenter is considered such that edge between nodes 10 and 8 (or 7 and 9) is rotated outwards about node 10 (or node 7) by an angle ~. The value o f • was lneasured in all the deformed tubes and it was found to be nearly equal to 0 3 f o r 0 ~ < 0 L a n d 0 / 2 f o r G > 0 > 0 L .

DL and DR, the distances o f the left vertical hinge I and the right vertical hinge V from the tube center, are given by:

for 0 ~<

0L,

DE -- ( W t a n (0)/2 + Wi/(2 cos (0))) + H tan (fi) (6)

DR = DE -- Wtan(20/3) (7)

for 0~ > 0 > 0L,,

DE = L/2 (8)

and for computing ^{D R}the indenter edge is rotated out by an angle 0/2 about node l0 (or node 7). For 0 > 0L, node l0 m a y lie between nodes 12 and 13.

The four inclined hinges, II, IV (and two symmetrical), extend from the ends o f the first horizontal hinge, III, and the top and b o t t o m o f the two vertical hinges, I, V, see Figs 6 and 9.

- : - - ~i~deate!;~ - ~{, 5 I

Ca/

L - - ~]uctent el~

• S i

(b)

Fig. 9. P o s i t i o n o f i n d e n t e r a n d l o c a t i o n o f h i n g e s at (a) c o m m e n c e m e n t o f r o l l i n g o f left i n c l i n e d h i n g e s a n d (b) at s o m e c o m p r e s s i o n a f t e r r o l l i n g h a s b e g u n .

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Co~lapse behavior o f a l u m i n u m tubes 19

14 w I~

I / i \'*,

D L . . . . 1-~-/~ • C

Fig. 10. Deformed cross-section of the tube, under the indenter.

The coordinates of each of the 15 nodes (Fig. 6) define the deforming geometry of the tube. C o m p u t a t i o n s to locate the coordinates of these nodes are carried out by first determining the coordinates of points in contact with the indenters (7, 8, 9 and 10). The coordinates of node 4 are then determined from simple geometric considerations as shown in Fig. 10. The z coordinate remains unchanged due to symmetrical loading, the x and y coordinates of other nodes (5, 11, 12, 13, etc.) are thereafter determined by solving simultaneous equations for distances of these nodes from two other nodes, whose coordinates are already known, such that these distances remain constant during the tubes compression. For exam- ple, the coordinates of node 5 are determined by using the known coordinates of nodes 4 and 8. The distances are taken to remain constant as deformations, if any, in the panels bounded by plastic hinges are ignored.

Having c o m p u t e d the geometry of the deformed tube for any 6, orien- tation of the rigid panels, defined by the coordinates of any three nodes on it, is then determined. The change in angle between the normals to any two adjacent panels gives the change in angle of the c o m m o n stationary hinge. The work done at any rotating stationary hinge is determined as:

Ea = M p L h Oh (9)

where Mp is the plastic moment. Lh and Oh are the length of the hinge and the rotation of the hinge at a given compression.

The compression at which the two inclined hinges on the left begin to roll and the load--compression curve begins to rise is 6i. This is the compression at which the indenters come into contact with the inclined hinges, II (and a symmetrical one), on the left (Fig. 9). The computation o f 6[ requires deter- mination of the distance between (the straight lines formed by) the left b o t t o m edge of the top indenter and the inclined hinge II.

Knowing the initial and final positions of the ends of the rolling hinges, the area traversed by these is determined. The accuracy of this calculation is increased by taking the s u m m a t i o n of the area traversed during small increments (t) of compression between the initial and the final positions.

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The work done by a rolling hinge 3 is determined from eqn (10):

Er = 2Mp/Rr (1o)

The average rolling radius Rr, determined from experiments is 6-3 times the tube thickness.

The total energy, Et, absorbed in all hinges is c o m p u t e d over two stages o f compression: (1) when 5 < 5~ and all hinges are stationary at their initial collapse locations; (2) when 5 > 5~ and the inclined hinge II dissi- pates energy by rolling.

The total energy, Et, is thus given by,

El -- EEl, + EE~ (11)

The load at any 5 is given by the slope of the cumulative energy compression curve at the compression 5. This load is determined by computing the total work done by all the hinges at two values o f compression, at a small interval on either side of 5, and then determining the slope linearly between these two points. In computations this gap between the two values o f compression, on either side o f 5, was kept equal to 14/1000.

4. R E S U L T S A N D D I S C U S S I O N S

The c o m p u t e d load-compression curves are shown for typical specimens o f set No. 1 in Fig. 11, and their energy~zompression curves are shown in Fig. 12. These show very good agreement with the experiments.

The c o m p u t e d collapse load, based on the predicted equivalent length o f a tube compressed between two platens, is c o m p a r e d with the experiments in Fig. 13. The c o m p u t e d values show very good agreement with the experiments.

In Fig. 14 the c o m p u t e d and the observed location o f the horizontal hinge under the indenter on both the front and back faces of the tubes are compared. The predicted and experimentally observed location o f the vertical hinges on the left and right sides o f the tube are also given in Fig. 14, for both front and back faces o f the tube. This was done to show the a s y m m e t r y due to the variations observed in the experiments.

The c o m p u t e d results are in good agreement with the experimental values.

The compression at which a load~zompression curve in an experiment began to rise, on c o m m e n c e m e n t o f rolling of inclined hinges, II, shows good agreement with the c o m p u t e d values, for all values of 0 (Fig. 15).

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2000 -

1500

2

~D 1000 o o

5 0 0

Set No 1 t

: ', ... Computed, 00 d e g

i0",i t*q bq Computed, 15 deg.

~.,,,,.,9. 00000 Computed, .50 deg.

¢ o o c o Computed, 45 deg 7,.",,', . . . ~ Experiment, 00 deg.

t ', "S~,. ~.--~-+ Experiment 15 deg. / ., ,.,.-. 0 - ~ - 0 0 Experiment, 50 deg /

~'.,.£-£... o-s,~s.~> Experiment, 45 de 9 /

", "- - - \ . . _ - + . . . . ~"

0 0.0 2 0 4 0 6 0 8.0 1 0 0 1 2 0 1 4 0 1 6 0 18.0 2 0 0

C o m p r e s s i o n ( r a m )

Fig. 11. Experimental and c o m p u t e d l o a d ~ c o m p r e s s i o n curves.

20000

f _ 15000 E E z

10000

5000

0 0 0

Set No 1

+ _ 0

~ d , 00 deg.

~ ~ U-'''''~ . . . . Computed, 15 deg.

~ U - "-'~'~- 00000 Computed, .50 de9.

c o . c o Computed, 45 deg

.~.~o~¢/~_..~.~ . . . Experiment, 00 deg.

~ . ~ . ~ +-+H+ Exper!ment, 15 deg.

0-$0~0$ Experiment, 30 deg.

~ < x ~ , ,ExFe:im,ent, 4 5 deg.,

2.0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 C o m p r e s s i o n ( r a m )

Fig. 12. Experimental and computed energy~ompression curves.

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

Square and rectangular aluminum tubes in their as-received state were compressed between a pair of parallel, orthogonally and non-orthogonally placed, short width indenters. The collapse behavior of the tubes was studied by recording the load-compression curves and observing the deforming geometry of the tubes by interrupting and terminating the tests at various stages of compression and taking photographs of the tube specimens.

Based on the experimental observations an analysis for the symmetrical mode of collapse has been presented for the computation of limiting

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4 () C I

~ >

_j J

- - , } 0 0 ~ > - - . ' - o

Z ~ ~ - f _ 4> "

o

i 1 Set 1, C o m p u t e d

(D 10 0 ~ Set 2, E x p e r i m e n t

o . s o o c . Set 2, C o m p u t e d

@,S,,I~,@ Set 5, E x p e r i m e n t

~,~--,~-@ Set 5, C o m p u t e d

i i , r l ~ l l ~ i i i i i i i i i , I I I T ~ - I i i l ~ [ 1 1 p l l l l l l l l , , l l l l l l l l r ~ i , l ~ l l l q ~ l : , 1 1 1 , 1

0 0 10.0 20.0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 80.0

T h e t a ( d e g . )

Fig. 13. Experimental and computed first elastic collapse load.

4 0 . 0 L

1) 5 0 0

;£

:Z 2(}' 0

Z 7

1:7 i 1 : : )

¢, 0 [, (h

5,et No 1

- - ' ~ : ; - i [{2. - % "

~ C o m p u t e d h i n g e ) .~

0@0~,¢ "Front side ,(Fred to

0 9 . ~ ) Back side left 9

0 0 0 0 0 C o m p u t e d h i n q e }

@@@'@@ F r o q t side ,',F n d to

@@¢~",~C~ Beck side r, I t :

@@@@@ C o m £ , u t e d hln( r ,

\

. . . , , , i , I , t r T ' ' F t ~ - - - ~ ~ I r r t r , ' I I t T I T T ' T T ~ T ~ 1 T v ~ , , , , , i ~ , , , , , , , , 7

• 0 ' 3 C ' -I ~ l , 0 4 ( "} ~c-: 0 ( 4 : ) 7 0 0 8 0 0

T h e t a ' , d c . ~ . ,

Fig. 14. Experimental and computed depth of first horizontal hinge, and distance of the left and right vertical hinges from respective ends of the tube.

loading geometry, collapse load, location o f hinges at collapse, deforming geometry o f the tube and the load~zompression and energy-compression curves. The results o f the analysis compared well with those o f the experiments.

1.

REFERENCES

Gupta, N. K. & Sinha, S. K., Transverse collapse of thin walled square tubes in opposed loadings. Thin-Walled Structures, 10 (1990) 247-62.

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~ 2 o o E

1 5 . 0

1 0 0

~ " ' - . .

~ 5 . 0 ",~"

8 Q

O 0 . . . O 0

/

/ / -

/ f

\, . /

\ /

\ / ~ . . . . S e t 1, F i r s t r i s e

\ / × ~ ~ S e t 1 , S e c o n d rise

\ / . . . S e t 1, C o m p u t e d

\ /

1 0 0 2o 0 s o o 4o.0 50.0 6 0 o 7 0 0 80.o

T h e t a ( d e g )

Fig. 15. Experimental and computed compressions at which the load~zompression curves begin to rise.

2. Gupta, N. K. & Khullar, A., Collapse load analysis of square and rectan- gular tubes subjected to transverse in-plane loading. Thin-Walled Structures, 21 (1995) 345 58.

3. Meng, Q., AI-Hasani, S. T. S. & Soden, P. D., Axial crushing of square tubes.

Int. J. Mech. Sci., 25 (1983) 747-73.