Collapse Load Analysis of Square and Rectangular Tubes ... · 346 N. K. Gupta, Atul Khullar r t T W...

ELSEVIER

Thin-Walled Structures 21 (1995) 345 358 ~) 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0263-8231/95/$9.50

Collapse Load Analysis of Square and Rectangular Tubes Subjected to Transverse In-Plane Loading

N. K. Gupta & Atul Khullar

Department of Applied Mechanics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-1 10016, India

(Received 7 August 1991; revised version received 2 April 1993; accepted 7 July 1993)

a0

ah

A c

E

f H I

J k Mo P P1 P2

A B S T R A C T

Experiments to determine the collapse load of aluminum and mild steel tubes, of square and rectangular cross-sections, between parallel rigid platens were carried out in an Instron machine. A two stage analysis for the collapse of these tubes was carried out by considering the out-of-straightness of arms, corner radius, Jriction between the platens and the specimen and stability of the vertical arms. Results thus obtained compare well with experiments.

N O T A T I O N

Ampli tude of sinusoidal initial out-of-straightness of the vertical arms Initial out-of-straightness of horizontal arms at mid-span Cross-sectional area of the tube walls Corner radius taken as eccentricity for loading Tangent modulus of elasticity Extreme fiber distance from mid-plane Average height of the tube specimens Second moment of area of the tube walls ( T/ Elfl)½ (P/E1) ' Magnitude of bending moment at the corners Load on each vertical arm of the tube Hal f the load acting at mid-span of horizontal arm Load acting at an eccentricity e from the corners

345

346 N. K. Gupta, Atul Khullar

r

t

T W

o- 0

0-ma x

Radius of gyration of the tube walls Average thickness of the tube specimens Tension in the horizontal arms Average width of the tube specimens

Stiffness ratio of the horizontal arms Modified stiffness ratio for tension if:I/F) Yield stress Maximum extreme fiber stress Coefficient of friction between platens and specimen

1 INTRODUCTION

A comprehensive study of round tubes under lateral loading has been conducted in the past.~ 3 Post collapse behavior of single square tubes and their cross layered systems have also been studied for different loading conditions by Gupta and Sinha. 4-6 Bending collapse of square and rectangular tubes has been dealt with by Kecman. 7 This paper is concerned with the analysis of the collapse load of square and rectangular tubes subjected to lateral in-plane loading between two rigid platens, see Fig. 1.

Sinha and Chitkara 8 have presented the collapse load analysis of square tubes considering stability and the out-of-straightness of the vertical arms. We present here a two stage analysis describing the collapse of these tubes by considering the out-of-straightness of both the horizontal and vertical arms, corner radius, friction between platens and the tube and stability of the vertical arms. The first stage corresponds to the formation of hinges at the mid-height of the two vertical arms. Whereas, the second stage hinges are formed at the four corners, as shown later. Results obtained are compared with the experiments which were conducted on an Instron machine.

2 EXPERIMENTAL

Tubes of square and rectangular cross-sections and made of aluminum and mild steel were commercially obtained and cut to make specimens of 50 mm length. Other dimensions of the tubes (Fig. 2) and their determined yield stress o-0, from tensile tests in an lnstron machine using the large strain extrapolation definition, are given in Table 1. Aluminum tubes were annealed by soaking them at 300°C for 30 min and allowing these to cool in the furnace.

Annealed and as-received specimens of aluminum and as-received

Collapse load analysis of square and rectangular tubes 347

L o c ~ d

/ , / / ' - . ~7 / ., / - / / P r o t ~ q

T ~ ~k~p

:i: / / - / / /

/ ' , \

Fig. 1. L oad i ng a r r angemen t .

specimens of steel were compressed between two rigid parallel platens in an Instron machine of 50 T capacity at a crosshead speed of 2 mm/min. Their load versus compression (relative platen displacement) curve was recorded on the machine chart recorder.

In the tests it was seen that on commencement of loading, the horizontal arms were first flattened. At the end of this stage, the load begins to act at the corners, see Fig. 2. On increasing compression the vertical arms kept on deforming outwards while the horizontal arms bent inwards. At some stage hinges were formed at or near the mid-height o f the vertical arms. After the collapse, the entire horizontal arm continued to deform inwards in a near circular shape. Thus at collapse hinges were effectively formed at the four corners and at mid-height of the vertical sides, as shown in Fig. 3.

Experiments revealed that prediction of the behavior of these tubes would require consideration of various geometrical and structural factors, which include corner radius, out-of-straightness of horizontal arms and effect of friction under the loads, in addition to the out-of-straightness of vertical arms and their stability.

3 A N A L Y S I S

In this analysis we consider that a square or a rectangular tube collapses in symmetrical mode. Further it is assumed that there is sinusoidal out-of- straightness of its vertical arms and it has circular shaped corners.

348 N. K. Gupta, A tul Khullar

TABLE 1 Dimens ions and Material Properties of Specimens

Set Dimensions (mm) ~(, ( N/mm') ao ah e no. (mm) (ram) (mm)

Width Height Thickness Annealed As received

Aluminum specimens A1 22-4 22.5 0.44 97.6 98.1 0.060 0,123 0.000 A2 23-4 36-1 0-72 113.6 118.7 0.050 0,080 0.000 A3 36-1 23-6 0.69 113.6 118.7 0.055 0,270 0.000 A4 36.7 36.7 1.59 178.5 200-2 0.115 0,070 0.000 A5 36.4 60.4 0-95 153.6 201-1 0.025 0,020 0-000 A6 60.3 36-4 0.95 153-6 201.1 0.045 0,085 0-000 A7 46.8 46.7 1.27 179.6 247.3 0-015 0,035 0.000

Steel specimens S 1 44-0 44.1 1-43 577-4 0.845 0,465 3.500 $2 35.1 36-0 1-12 519-9 0.410 0.500 2.375 $3 23-9 24-0 1-19 523-6 0.005 0-440 3-375 $4 17.8 18-7 1.04 508-3 0.005 0.005 2-188

I i t

J

i ,

4

i

F ~ : l k' >

Fig. 2. Idealized geometry of the tube and loading on it.

It is assumed that the hinges are first formed at mid-height of the vertical arms when the maximum combined stress in the extreme fiber, am~x, due to direct compression and bending, equals the yield stress a0, and then at the corners, see Fig. 3. What follows formation of these hinges is referred to as the first and second stage, respectively.

Due to assumed symmetry, we consider that the idealized structure


P~_ P2

i " hlNC4@ - - / [

'1 F r -~ f i ~ t o g @ ~ k~ir 9 c - 7 / ~ J

F=, Fs L

Fig. 3. Collapsed tube shape showing location of hinges.

consists o f half the vertical arm and the horizontal arm is taken as a rotational spring at the end.

After the formation of the first set of hinges at mid-height of the vertical arms, a second set o f hinges at the corners will either be formed simultaneously, because of the increased instability of the altered structural arrangement, or may require an additional force if the first set of hinges were formed at a load much smaller than the stability load of the tubes because of large initial imperfections.

Friction between platens and the specimen causes tension in the horizontal arms when the load begins to act at the corners. Tension increases the end rotational stiffness of the horizontal arms as discussed below.

3.1 Tension in the horizontal arms

Friction at the corner load points results in tension in the deforming horizontal arms, which increases the effective stiffness of the arms.

The configuration assumed for the horizontal arm of the tube along with the loading on it is shown in Fig. 4.

Here, span moment, M, is given by

M = Mo - Ty (1)

and, flexural equilibrium gives,

d2y Elfl ~xSx z + M = 0 (2)

3 5 0 N. K. Gupta, Atul Khullar

~ " , I~ '~: L i .

Ve

1

• /

Fig. 4. Horizontal arm in tension with end moments.

Substi tut ing a7/0 = 2FfiflM° and ./2 =ETfiT in eqn (2), we get

d2y dx 2 ./.2y q_ ~/0 = 0

Solution of eqn (3) is given as,

?o0 y = Cle jx q- C2 e-ix + - - j2

Applying boundary condit ions y(x = 0, W) = 0 eqn (4) gives,

(3)

(4)

f/io [[(1-e-JW)eiX +(eJW-1)e-JXl ] Y = 7 ~-;~-~ U~) + l (5)

and thus,

dy Mo [ [ ( 1 - e - J W ) e J ' - ( e m ' - l)e Jx]l dx .-] -~-7~" --~7-~)

Equat ion (6) can also be written as,

dY(~c = 0) - MoW dx " 2EIfl

(6)

(7)

where,

= fl/F

and

F = [ 2 ( 2 - e ~w - e jw) 1 L 7WTe-~--eta) I

(8)

(9)

Thus, for end m o m e n t loading, the effect of friction can be accounted for by dividing the stiffness ratio fl of the horizontal arm by F.


3.2 First stage hinge formation

Considering the initial out-of-straightness of magnitude ah at mid-span of the horizontal arms, corner eccentricity e of loading and sinusoidal out-of- straightness of the vertical arms of amplitude a0 (Fig. 2) an expression for the load at which the first set of hinges are formed at mid-height of the vertical arms is developed by assuming an idealized structure as shown in Fig. 5.

For flexural deformation, equilibrium is given by,

d2y EI~x2+ M - - 0 (10)

Span moment M = P(y + ao s inQcx/H))- M0, where M0 is the restraint moment due to the horizontal arm, and for,

P = Pl + P2, k2 -- P 37/0 M0 - ~ , =-E---~ and c~ = ~

where P1 is half the load acting at mid-span of the horizontal arm and P2 is the load that acts on it at a distance e fuom the corner after the horizontal arm has been flattened (Fig. 2). Equilibrium eqn (10) then becomes,

d2Y + k2y -k2ao sin(~x) + ~/0 (1 1) - - z

d x 2

The solution of eqn (11) is given by,

r~ ( P ] + P~ 2 )

M o

P

Fig. 5. Idealized structure for the computation of first stage load.


y = Ct cos(kx) + C2 sin(kx) + ao k 2 sin(c~x) Mo

+ (0~ 2 -- k 2) k 2 (12)

F o r the b o u n d a r y condi t ions , which are;

y (x = 0) = 0 and dy/dx (x H/2) = 0

eqn (12) gives,

y = k ~ 2 1 - cos(kx) - tan (kill2) sin(kx) -~ k 2 ao sin(ex)

( ~ 2 - k 2 ) (13)

and thus,

dY - M° [sin(kx)- tan (kH/2)cos(kx)J dx k

ao ~k 2 cos(~x) + (14) (0~ 2 - - k 2)

dy/dx at x = 0 is compa t ib le with end ro t a t ion 0 o f the hor izon ta l member , whose stiffness ra t io is fi and the express ion for the same is,

MoW PIW 2 P 2 e ( W - e ) 9 - A + + ( 1 5 )

2EIfi 8EI/3 2EI/3

where/3 is the modi f ied stiffness ra t io as per eqn (8). On equa t ing eqns (14) and (15) we get,

[ Pao~ p, W2 P 2 e ( W - e ) ]

_ k2) 8/3 (16)

Since yo=aos in (c~x) , Y t = Y + Y o and ) ' t m a x = y t ( x = H / 2 ) and on combin ing these with eqn (13), we get,

3( 2 Mo ao (17) Yt ma x = ~ - [ 1 - sec(kH/2)] + k 2 )

F r o m overall equi l ibr ium we have,

Mm~x = M(x = H/2) = PYt max - Mo -

9

Pao ~- 0{2 - - k 2

M0 sec (kH/2) (18)

and thus for c om b ined axial and bending stresses at x = H/2, we get,

P Mmax(f/r 2) am~,x -- + (19)

A A


w h e r e f i s the distance of the extreme fiber from mid-plane, A is the cross- sectional area and r the radius of gyration.

The collapse load of the full tube is 2P, corresponding to the condition, Oma x = O" 0 . The maximum value of P1 is determined by satisfying the equation,

M0 W 2 P1 W 3 ah -- - - + - - (20)

8EIfl 24EIfl

When the load exceeds a certain value, the horizontal arms separate from the platens. This is taken to occur when P2 satisfies the condition,

MoW 2 PzeW 2 Pz e3 ah -- - - + - - (21)

8EIfl 8EIfi 6EIfl

At any load greater than the maximum value of P~ and less than the value of P2 at which separation occurs, the magnitudes of P~ and P2 are determined by equating the relative vertical deformat ion at mid-span of the horizontal arm to ah.

The expression for buckling load of a perfectly straight sided rectangular tube is given by Timoshenko. 9 This equation, when modified by considering the effect of friction, is written in the form,

fl tan(kH/2) 1 -+ = 0 (22)

(W/H) (kill2)

the value of (kill2) lies between re/2 and rc in this case. The expression in the denominator of eqn (16) reduces to this expres-

sion and thus eqn (22) gives the expression for maximum load capacity of a slender sided tube.

Using eqns (13) and (16) which account for the stability and, considering out-of-straightness of the horizontal arms ah and axial shortening, we get the value of relative vertical displacement of the platens at any load until it reaches the first hinge load. The computat ions were carried out by dividing the vertical arm into a number of nodes. Adding axial shortening and relative displacement at mid-span of the horizontal arm to the chan- ged chord length of the bent arc (vertical arm) we get the elastic compression corresponding to a load, less than the first stage load. The value o f / , is 0.5 in the computations. 8

3.3 Second stage hinge formation

If the first stage hinges are formed at a load close to the buckling load then the second set o f hinges at the corners will form simultaneously as deformations


at that stage increase with no increase in load. In this case the second stage load capacity is given by the buckling load of the altered tube structure. This occurs when the geometric imperfections are negligibly small. Whereas, if the first set of hinges are formed at a load much smaller than the buckling load of the tube then, depending on the stiffness and geometry of the altered structural arrangement, second stage hinge may form either by the buckling of the tube or it may sustain a higher load with further compression before the second set of hinges are formed. After this the structure becomes incapable of sustaining higher loads with increasing compression.

The second set of hinges are considered to form at positions directly under the corner loads rather than at the corners because a sustained mechanism will be the one in which there is no reduction in stresses at hinge locations on increasing compression.

In determining the second hinge load we have imposed the geometry, including the remaining out-of-straightness of the horizontal arm, when the first stage hinges are formed and a bending moment at the location of first hinge whose magnitude is a0 t2/6. In a pure bending case a moment of this magnitude will result in the formation of a hinge in the walls of the tube. We then determine the load at which the stresses (combined or bending only) equal a0. In the case where the equations indicate no positive value of this load for the imposed geometry and moment, then the second stage load capacity is taken as its buckling load and the corresponding compression is taken to be the same as that for the first stage. It is assumed that until the second stage hinges are formed the material properties of the horizontal arms remain unchanged.

The idealized structure is shown in Fig. 6, where al, H1 are the current out-of-straightness and half height of the tube for the altered geometry when the first stage hinges have just formed, and M~ = a0 t2/6.

Equilibrium of flexural forces gives,

d2 . E I d x 2 + M = 0 (23)

and

M = P ( y + al x /H1) -- Mo

where M0 is the restraint moment due to the horizontal arm. = al and ~;/o Mo get, Putting k 2 = ~, ~/j ~ = ~- we

d2y t- k2y = -k2al x + ]l~/0 dx 2

(24)

(25)

and the solution for (25) is given as,


P

Second s±c~ge

(H1)H/2 x ~ hinge

~ First stage

~ ~ hinge

P Fig. 6. Idealized structure for the computation of second stage load.

y = C1 cos(kx) + C2 sin(kx) - ~l 1 X -~- )~Io/k 2 (26)

Considering y (x = O) = O, y ( x = Hi) = 8 and the overall equilibrium, i.e. Mo = P(al + 8 ) - M~, where M1 is the moment applied and is repre- sentative of the moment that is present when the first stage hinges are formed. From this we get,

/~lf I

[ (al + 8) (1 - cos(kill)) ( + [ s--~n (kH-]-) ~ ~,al

and

J~fl al X

(27)

[ ( dxd--Y-Y = k sin(kx) al + 6 -

[ (al + 8) (1 - cos(kill)) /1~/1 -k k [ s-~n(k-Hl) ~ ( a l - l - 8 - - I c - f f - ) ] c ° s ( k x ) - g t l

(28)

As in the case of first stage hinges, equating d y / d x ( x --- 0) to 0, given in eqn (15), we get,


[kl2W 2 k22e(W-e) ~I1m ~lltan(kH1/2)]

(a, + 6) = [- ~ 4 2~ + ~' 2/~ (29) [ k k2w] tan(kill) 2~

here, k l 2 = P1/EI a n d k2 2 = P2/EI. We want the value of P for a corresponding positive M0 which will

satisfy the condition O-ma x = O" 0 . Where,

O'ma x = P/A + (Mo f/r2)/A when e = 0 and

a m a x = ( ( M 0 ÷ P 2 e ) f / r 2 ) / A w h e n e > 0

(30)

(31)

Therefore, if P is less than the first stage load then the second hinge will form simultaneously with the formation of the first set of hinges.

The expression in the denominator of the expression for (a~ + 6) (eqn (29)) is equivalent to the buckling load of a column (of height HI2) supported by a horizontal member, which is given by,

(kH/2) (W/H) tan(kH/2) -1 -~ /~ = 0 (32)

the value of (kill2) lies between 0 and ~/2. The expression has been derived in a manner similar to that by Timosh-

enko 9 for a cantilever column with fixed base by replacing the fixed support by an elastic rotational spring at the end and accounting for the effect of friction. In an actual situation the structural transition from that of the first stage to that of the second stage will be gradual, with reducing bending stiffness at mid-height of the vertical arm and increasing moment.

Computed load and displacement values for first and second stage hinges are presented, along with the experimental results, in Table 2. The collapse load for a given tube is the one which is the larger of the first stage and second stage loads. Except for two steel specimens of the $3 and $4 series, the second hinge load for all other tubes was smaller than the first hinge load. Thus collapse occurred with the formation of the first set of hinges itself in the cases, wherein the out-of-straightness of arms and corner radii were small.

4 CONCLUSIONS

The collapse of square and rectangular tubes is analyzed by considering out-of-straightness of both the vertical and horizontal arms and the corner


TABLE 2 Experimental Collapse Load and Corresponding Compression Along with Computed

Loads and Compressions for Unit Specimen Length

Set Experimental no. (at collapse) Computed values, # = 0.5

Load Compression First hinge Second hinge (N) (mm)

Load Compression Buckling Load Compression Buckling load load

As received spectmens, per A1 14.0 0.450 0.120 15.8 2.6 0.120 2.6 A2 28.1 0.420 0.250 30.2 5.9 0.250 5-9 A3 31.0 0.690 0.209 60.4 7-8 0.209 7.8 A4 323.4 0.840 0.237 403.4 65.7 0.237 65.7 A5 61.8 0-420 0.206 62.4 12.5 0.206 12.5 A6 113-1 0.400 0.221 142.6 17.9 0.221 17.9 A7 186.0 0.590 0.169 209.6 34.2 0.169 34.2 S1 183-4 0.860 1.056 977.5 125.1 1.056 159.4 $2 166-7 0-746 1.072 699-1 101-0 1.072 115.4 $3 220.7 0-860 0.894 1904-0 175.5 0.934 311.2 $4 264.4 0.262 0.012 2104.0 199.9 0-032 351.2

unit length 14-8 28.5 30.3

294.9 59.5

127.1 201.8 137.9 114.8 84.9

152.2

Annealed specimens, per unit length A1 13.7 0.280 14-4 0.292 16.4 2.7 0-292 2.7 A2 24.4 0.240 22.7 0.282 26.3 5.1 0-282 5.1 A3 42.1 1.000 46.1 0.585 54.5 7-1 0.585 7.1 A4 243.9 0.650 260.9 0.292 296.6 48.3 0.292 48.3 A5 54-6 0.300 56.6 0.113 58.1 11.6 0.113 11-6 A6 108.4 0.410 121.4 0.152 143.5 18.0 0-152 18-0 A7 119.9 0-300 148.7 0.151 152.5 24.8 0.151 24.8

radius . A two s tage s tabi l i ty analys is o f the ver t ical a r m s h o w e d tha t w h e n one or all o f the geome t r i c imper fec t ions are large the tubes co l lapse at loads m u c h smal le r t h a n the buck l ing load and exhibi t large bend ing d e f o r m a t i o n s a t col lapse. Whereas , when their g e o m e t r i c shapes are near ly true, co l lapse o f the tube occurs by buckl ing . T h e t rans i t ion f r o m the buck l ing to the bend ing m o d e occurs w h e n the effect ive eccentr ic i ty o f l oad ing due to a0 and e resul ts in large bend i n g stresses at mid -he igh t o f the ver t ical a r m s as c o m p a r e d to direct c o m p r e s s i v e stress.

The m a g n i t u d e s o f a0, ah a n d e for which the second stage hinges are f o r m e d by bend ing r a the r t han buck l ing also depends on the s lenderness o f the tube walls. Thus , the values o f geome t r i c imper fec t ions , for which the second s tage load exceeds the first s tage load, will be d i f ferent for tubes wi th d i f ferent g e o m e t r y .


R E F E R E N C E S

1. Reddy, T. Y. & Reid, S. R., Phenomena associated with the crushing of metal tubes between rigid plates. Int. J. Sol. Struct., 16(15) (1980) 545-62.

2. Reid, S. R., Laterally compressed metal tubes as impact energy absorbers. In Structural Crashworthiness. Butterworth, London, 1983, pp. 1-43.

3. Reid, S. R. & Reddy, T. Y., Effects of strain hardening on the lateral compression of tubes between rigid plates. Int. J. Sol. Struct., 14(12) (1978) 213-25.

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

5. Gupta, N. K. & Sinha, S. K., Collapse of a laterally compressed square tube resting on a flat base. Int. J. Sol. Struct., 26(5/6) (1990) 601 15.

6. Gupta, N. K. & Sinha, S. K., Lateral compression of cross layers of square section tubes. Int. J. Mech. Sei., 32(7) (1990) 565-80.

7. Kecman, D., Bending collapse of rectangular and square section tubes. Int. J. Mech. Sci., 25(9/10) (1983) 623-36.

8. Sinha, D. K. & Chitkara, N. R., Plastic collapse of square rings. Int. J. Sol. Struct., 18(18) (1982) 819-26.

9. Timoshenko, S. P. & Gere, J. M., Elastic buckling of bars and frames. In Theory o f Elastic Stability. McGraw Hill, Singapore, 1988, pp. 46~158.

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