Experimental investigation into the axial plastic collapse of steel thin-walled grooved tubes

Int. J. Impact Engng Vol. 4, No. 2, pp. 11%126,1986

0734-743X/86$3.00+0.00 PergamonJournalsLtd.

Printed in Great Britain

E X P E R I M E N T A L I N V E S T I G A T I O N INTO T H E A X I A L PLASTIC C O L L A P S E OF STEEL T H I N - W A L L E D G R O O V E D TUBES A. G. IVIAMALIS*,G. L. VIEGELAHNt,D. E. MANOLAKOS*and W. JOHNSON,§ *Department of Mechanical Engineering, National Technical University of Athens, Greece tDepartment of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, Houghton, Michigan, U.S.A. ~:School of Industrial Engineering, Purdue University, West Lafayette, Indiana, U.S.A. (Received 7 February 1986)

Summary--Experimental investigations were conducted into the quasi-static collapse and energy absorption characteristics of thin-walled steel tubes containing a number of geometrical discontinuities in the form of grooves of constant depth and axial length. The failure modes and load-deflection characteristics are discussed and compared with the corresponding theoretical values. An inextensional collapse mechanism is used to describe a non-symmetrical diamond mode shell folding which takes into account the concept of stationary circumferential and inclinded travelling hinges.

NOTATION A Dg Di Dr L P /~ Pm~, Py tg Y z

cross-sectional area at a grooved region of the shell outside diameter at a grooved region of the shell inside diameter of shell outside diameter at a ringed region of the shell axial length of a groove axial length of a ring axial length of shell axial load mean collapse (post-buckling) load initial peak load yield load wall thickness at grooved regions of the shell wall thickness at ringed regions of the shell mean yield stress of material depth of a groove INTRODUCTION

The plastic collapse of tubular and conical components of circular and non-circular sections under axial compression provide one of the most efficient energy absorbing devices [1]. Experimental studies on the crumpling mechanisms for such devices have been conducted and theoretical methods have been developed using the concepts of stationary and travelling plastic hinges for calculating the buckling load and the amount of energy absorbed [2-5]. In a previous paper [6], the authors have reported on theoretical and experimental investigations into the quasi-static collapse of a thin-walled tube containing a number of geometrical discontinuities in the form of grooves having a constant depth and axial length, a geometry which has potential as an energy-absorbing device. An inextensional collapse mechanism was used for folding the shell in a non-symmetrical diamond mode which takes into account the concept of stationary circumferential and inclined travelling plastic hinges. Experimental confirmation of the theoretical model has been provided by the quasi-static collapse of PVC shells.

§ Permanent address: Ridge Hall, Chapel-en-le-Frith, via Stockport, Derbyshire, SK12 6UD, U.K. 117

A . G . MAMALISet al.

118

In the present paper, predominant concern is with the collapse phenomenon of thinwalled grooved tubes made from low-carbon steel instead of PVC. The failure models and load-deflection characteristics are discussed and compared with the corresponding theoretical predictions. EXPERIMENTAL RESULTS

The axial compression of thin-walled grooved cylindrical tubes was performed between parallel steel platens on a M.T.S. Universal Testing machine. The tests were carried out at a crosshead speed of 10 mm min -1 or an overall compression strain rate of 10 - 3 per second. The test material was low-carbon steel CR 1018. The stress-strain curve from a quasistatic tension test on this material is given in Fig. 1. All the specimens were machined from solid bars to the required size and were tested in the annealed state, the latter being obtained by heating specimens to 900°C for a fixed time and then air-cooling on fire brick. The dimensions of the specimens, similar to those of PVC grooved tubes used in previous work and reported in ref. [6], are presented in Table 1 and refer to the geometrical representation of Fig. 2. The initial lenght L of the specimens and the length and depth of groove were kept constant, whilst the ring length lr varied in the range O<.lr/L<.l, (see Table 1). The end faces were machined square and all specimens were freely and axially compressed in a 'dry' condition.

% E 04 Z J¢

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• 0.2 i-

i

0.02

i

0.04 Naturo[

i

i

i

0.06 tensi|e stroin

0.08

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FIG. 1. Tensile stress-strain curve for annealed steel.

TABLE

Specimen No. 8 9 10 I1 12 13 14

Inside diameter, Outside diameter Di Dr Dg (mm) (mm) (mm) 41.0 41.0 41.1 41.1 41.0 41.1 40.9

44.3 44.3 44.3 44.3 44.3 44.3 42.8

-42.7 42.7 42.9 42.8 42.8 --

1

Axial Ring P/Pv length length Number Length Buckling load (kN) of of last Initial Mean P L lr Rings (mm) peak Ext Theor. Ext. Theor. (mm) (mm) 127 127 127 127 127 127 127

127.0 25.4 19.1 12.7 6.35 3.18 0.0

1 4 5 7 13 19

Y=0.250 kN mm -2 is the initial yield stress at 3% strain (Fig. 1). Py ='it y(D2g - D2i)/4. /g=3.175 mm. z=0.75 mm.

00 12.7 15.5 15.4 2.8 5.3 0.0

33.1 22.2 21.8 21.4 19.1 17.8 13.3

21.0 9.1 8.8 11.4 10.6 10.3 5.9

19.1 9.6 8.8 11.2 10.1 9.5 8.3

0.380 0.326 0.334 0.384 0.359 0.368 0.189

0.346 0.344 0.334 0.378 0.341 0.339 0.267

Axial plastic collapse of steel thin-walled grooved tubes

119

TABLe. 1. (cont.) Specimen No. 8

9 10 11 12 13 14

Buckling modes for the various levels of folds a b c d e f g h A 2D 2D 2D 2D A-2D A

A 2D 2D 2D 2D 3D A

A : Axisymmetric ring. 2D : 2-diamond (Ellipse). 3D : 3-diamond (Triangle).

A 2D 2D 2D 2D 3D A

Remarks:

A A 2D 2D 2D 2D 3D 3D A A A A A

All All All All All

patterns patterns patterns patterns patterns

: axisymmetric ring; Do/Di : 56/35 : Ellipse; 59/56,56/,64/56,56/ : Ellipse; 51/55,55/,55/52,52/ : Ellipse; 65/,61/,53/,57 : Ellipse;

All patterns : axisymmetric ring; Do/Di : 50/37

Do : outside diameter of axisymmetric ring. Di : inside diameter of axisymmetric rings. Note : Dimensions are in mm.

~

P

Dr

0~

FIo. 2. A schematic diagram of grooved tubes in axial loading.

T h e test results are s u m m a r i z e d in T a b l e 1. P h o t o g r a p h s of buckling m o d e s which w e r e t a k e n during the various crumpling stages, along with terminal t o p and b o t t o m views of the b u c k l e d s p e c i m e n s , are s h o w n in Figs 3-7. T h e variation of buckling load with deflection, i.e. shortening of axial length of the shell, was followed using an a u t o g r a p h i c r e c o r d e r (see Figs 8a and b). T h e values of the initial p e a k load Pmax and the m e a n post-buckling load P ( o b t a i n e d by carefully m e a s u r i n g the area u n d e r the load--deflection curve for the post-buckling region and dividing it by the c o r r e s p o n d i n g deflection) are t a b u l a t e d also in T a b l e 1. Initially, the shell b e h a v e s elastically, and the testing m a c h i n e load rises at a steady rate. T h e elastic region is c o m m o n and yielding begins with the s a m e slope for all specimens e x a m i n e d , as described a b o v e (see Fig. 8). T h e rate of load increase t h e n falls; the load, h o w e v e r , rises to a m a x i m u m value, but as the instability begins, it falls off rapidly until a fold is d e v e l o p e d , which extends to its m a x i m u m degree.

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1

2

3

4

5

100

50 0 mm AL

6

7

8

10

9

11 I

I

20ram FIG. 3. Buckling modes for spec. 9, see Table 1 for details, 1-9: views of progressive collapse; 10: a top view; 11: a bottom view.

For specimen 9 (with/~/L=0.20) after the first ellipse formed in groove 3, a progressive collapse was developed in groove 4 with the formation of an ellipse orthogonal to that of groove 3 (see 5 in Fig. 3); and further, in groove 2 parallel to the one of groove 4 (see 6 in Fig. 3). Note that in the final stages of deformation, the top circular end of the specimen finally distorted to essentially an ellipse, also losing contact with the upper steel platen along its major axis (see 6--8 in Fig. 3). Worth mentioning is the fact that the deformed levels of ellipse in rings 3 and 4 did not remain horizontal but moved downwards until the ends touched the bottom platen (see 5 in Fig. 3). This resulted in an increase of buckling load shown in these stages of deformation as indicated in Fig. 8(a). Enveloping ring 1 when the initial elliptical collapse mode in groove 2 moved upwards (see 4 in Fig. 4) and the formation of an ellipse along the circumference of groove 1 at ~r/2 rotation to the previous one (4 and 5 in Fig. 4) are eha~eteristie stages of deformation after initiation of collapse for specimen 10 with a ratio/,/L=0.15. The final collapse stages are dearly indicated in Fig. 4. Note that the top end of the collapsed grooved specimen is elliptical. The subsequent stages of deformation for specimen 11 (UL=0.10), specimen 12

Axial plastic collapse of steel thin-walled groovedtubes

1

2

3

4

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5

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7

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20rnm Fro. 4. Bucklingmodesfor spec. 10, see Table 1 for details, 1-9: viewsof progressivecollapse; 10:a top view; 11: a bottom view. (UL=0.05) and specimen 13 (/JL=0.025), showing the progressive collapse of grooves in an elliptical (specimens 11 and 12) or triangular (specimen 13) manner with the levelby-level rotations of ~r/2 or ~r/3 radians, respectively; the top and bottom views of distorted specimens are shown in Figs 5-7. For comparison purposes, two tubes with no geometrical discontinuities, of the same initial length and with the inside diameter of the grooved specimens but with different wall thickness corresponding to lr/L=l (specimen 8) (i.e. with an outside diameter Dr, see Fig. 2) and to l/L=O (specimen 14) (i.e. with an outside diameter Dg, see Fig. 2) were collapsed in an axisymmetric extensible buckling mode. They showed the well known initial elastic behaviour and the typical oscillatory load--deflection curve of a post-buckling region. DISCUSSION AND CONCLUSIONS For almost all the cases of the grooved tubes examined some common deformation characteristics are apparent in Figs 3-8; compare these with similar observations made during the axial collapse of PVC grooved tubes of similar geometries [6].

A.G. MAMALISet

122

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6

7

8

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&L

10

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20ram Flo. 5. Bucklingmodesfor spec. 11, see Table 1 for details, 1-9: viewsof progressivecollapse; 10: a top view; 11: a bottom view.

The initial elastic behaviour of the shell with a rise of load at a steady rate with the same slope, the appearance of the peak load and the rapid falling off of the load are all evident in the present work. At the peak load the material yields and a buckle (a circumferential plastic hinge line) into a two-lobe type of failure forms in a groove, i.e. at the thinnest section of the tube. It develops at a distance between 1/2 and 4/5 of the axial length from the top end of the tube adjacent to the platen until the cross-section had become ellipti,'ad, (see 2 in Figs 3-7). After the initial peak load corresponding to the formation of the elliptical fold described above, the load increased only slightly, showing the characteristic features of column-buckling, (see Fig. 8 for specimens 9-13). Inclined plastic hinge lines at each side of the groove, where deformation has been initiated, were developed leading to the plastic collapse of the adjacent ring-sections of the tube with the formation of a two-lobe (elliptical) type of failure (specimens 9-12) or a three-lobe (triangular) type (specimen 13). In the two-lobe type of failure a level-by-level rotation of ~r/2 radians appears, whilst in three-lobe type this rotation is ~r/3 radians.

Axial plastic collapse of steel thin-walled grooved tubes

1

2

3

123

4

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6

7

8

9

AL

10

11

FIG. 6. Buckling modes for spec. 12, see Table 1 for details, 1-9: views of progressive collapse; 10: a top view; 11: a bottom view.

Note that the wavelength of buckling is fully restricted by the geometrical discontinuities of the grooved specimens. Thus, for the tubes with four or five rings (specimens 9 and 10, respectively) the lobes are formed after each ring, for the tube with seven rings (specimen 11) the wavelength consists of two adjacent rings and for specimens with more rings (specimens 12 and 13) their structural behaviour approaches that of the non-grooved specimens in inextensional buckling, with wavelengths almost equal. For the grooved specimens which show characteristic features of column-buckling the load then increased only slightly. The post-buckling phase is extended with successive rises and falls in load level as the various plastic regions form and develop with a clear oscillatory character. Secondary peaks of the buckling load at the final stages of deformation are due to the touching of the bottom platen by deformed ellipses moving downwards. The maximum load (initial peak) decreases with increase in the number of grooves (see Table 1 and Fig. 8). Values of the mean post-buckling load P for the grooved specimens 9-13, obtained from the theoretical model outlined in ref. [6] and for the tubes without grooves (specimens 8 and 14) using Alexander's analysis from ref. [7], are given in Table 1. It will

124

A . G . MAMALISet al.

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0 mm

6

7

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&k

10

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11

FIo. 7. Buckling modes for spec. 13, see Table I for details, 1-9: views of progressive collapse; 10: a top view; 11: a bottom view.

be convenient to compare the mean load/5 with the end yield load Pv that would cause simple yielding of the groove (thinnest section of the shell where plastic collapse is initiated) in direct compression. If Y is the initial uniaxial yield stress of the material (see Fig. 1), then P y = ~ Y' ( D2g- D2i)/4 .

(1)

Theoretical and experimental values of P/Py from Table 1 are plotted against the ratio lr/L in Fig. 9. These results for steel fit the equation /~ =0.04-~- +0.330. PY

(2)

The icorresponding equation deriveddn :ref. :[@for ,PVC grooved cylinders is

±

L Pv =0.03 L +0.356

which is similar to equation (2), as shown in Fig. 9.

(3)

Axial plastic collapse of steel thin-walled grooved tubes

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FIG. 8(a). Load-deflection curves for specimens 8--11. -----; spec. 10: - -

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; spec. 8: - - . - - ; .; spec. 11.

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(b) Load-deflection curves for specimens ]2-14. - - , ;

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: experimental; • : theoretical.

126

A . G . MAMALISet al.

Acknowledgements--Professor W. Johnson wishes to acknowledge the support of the Leverhulme Trust and the kindness of Professor Norman Jones for reading and commenting on the manuscript. REFERENCES 1. W. JOHNSON and A. G. MAMALIS, Crashworthiness of Vehicles. Mechanical Engineering Publications, London (1978). 2. A. G. MAMALISand W. JOHNSON,The quasi-static crumpling of thin-walled circular cylinders and frusta under axial compression. Int. J. Mech. Sci. 25, 713 (1983). 3. A. G. MAMALIS,W. JOHNSONand G. L. VIEGELAHN,The crumpling of steel thin-walled tubes and frusta under axial compression at elevated strain-rates: some experimental results. Int. J. Mech. Sci. 26, 537 (1984). 4. A. G. MAMALIS,D. E. MANOLAKOS,G. L. V1EGELAHN,N. M. VAXEVANIDISand W. JOHNSON, On the unextensional axial collapse of thin PVC conical shells. Int. J. Mech. Sci. 28, 323-335 (1986). 5. A. G. MAMALIS, D. E. MANOLAKOS, S. SAIGAL, G. L. VIEGELAHN, and W. JOHNSON, Extensible plastic collapse of thin-wall frusta as energy absorbers. Int. J. Mech. Sci. 28, 219-229 (1986). 6. A. G. MAMALIS,D. E. MANOLAKOS,G. L. VIEGLAHN,N. i . VAXEVAN1DISand W. JomqSON, The intextensional collapse of grooved thin-walled cylinders of PVC under axial loading. Int. J. Impact. Engng. 4, 41-56 (1986). 7. J. M. ALEXANOER,An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q. J. appl. Math. 13, 10-15 (1960).