- Email: [email protected]
Axial plastic collapse of thin bi-material tubes as energy dissipating systems
Int. J. Impact Enqn,q Vol, 11, No. 2, pp. 185 196, 1991
0734 743X/91 $3.00+0.00 (" 1991 Pergamon Press plc
Printed in Great Britain
AXIAL PLASTIC COLLAPSE OF THIN BI-MATERIAL T U B E S AS E N E R G Y D I S S I P A T I N G S Y S T E M S A. G.
MAMALIS,
D. E.
MANOLAKOS,
G. A. DEMOSTHENOUSand W.
JOHNSON t
Department of Mechanical Engineering, National Technical University of Athens, Greece; and tRidge Hall, Chapel-en-le-Frith, Derbyshire, U.K. (Received 23 February 1990; and in revised form 20 January 1991 )
Summary The deformation characteristics, crumpling mechanisms and energy absorption efficiency of bi-material circular tubes subjected to axial compression are investigated and analysed both experimentally and theoretically. Theoretical models describing extensible and inextensional type of collapse are proposed and corresponding results are found to be in good agreement with experimental ones.
NOTATION D h Mp n P Pm.x r
t tr W x Y Y" k
diameter of tube height of convolution (lobe) full plastic bending movement per unit width number of lobes per layer (see also Ref. I-9]) mean post-buckling load initial peak load radius of curvature of shell (see also Ref. [9]) wall thickness of shell =tl/t2
plastic work dissipated depth of neutral axis mean yield stress of material equivalent yield stress of bi-material tube = YI/Y2
INTRODUCTION
Crashworthiness studies [1] have caused much attention to be given to analysing the deformation characteristics and determining the energy absorbing efficiency of various thin-walled crashworthy components of different materials, e.g. metals, plastics, composites, etc. [1-3], and for various geometrical shapes--cylindrical tubes, cones, conical frusta, square tubes, square frusta and pyramids, all grooved or ungrooved, etc. [4-6] when subjected to axial compression. Work into the crashworthy characteristics of thin-walled structures has been extended to also cover bi-material components, namely bi-material tubes. We consider bi-material tubes for two reasons: (i) The two materials may each confer on an energy absorbing element their own particular characteristics: for example, if a "core" aluminium tube was used, then covering it with steel might confer an anti-corrosive property or that of supplying a hard surface. Similarly PVC coverings may act so as to supply a generally protective coating. And to our knowledge (ii) bi-material tubes have not been previously reported on. In the present paper combinations of four different materials forming a bi-material thin-walled tube are investigated and their behaviour in axial collapse is analysed both theoretically and experimentally. Well-known theoretical models l-7,9] are recalled and modified appropriately to account for the crumpling behaviour of bi-metalled tubes in the cases of extensible and inextensional modes of collapse, respectively. Theoretical results were found to be in good agreement with experimental ones. 185
1~6
A. (J. MAMALISt't tl[. THEORETICAL
Consider a bimaterial plate consisting of two fully adhering different materials of thicknesses tl and t2, respectively, see Fig. I, and corresponding yield stresses YI and Y2. Examining the full yielding of a cross-section of overall thickness t = t~ + t z and unit width and requiring the forces on a section to total zero, we obtain the position of the plastic neutral axis x from, Y2(t- x)= Y2(x-tl)+
Ylll,
or finally Y2(t+tl)x=
-
Yltl ,
(J)
2Y2
with the restriction that Yttl < Yzt2 • To evaluate the moment per unit width to bend plastically the section Mp, we take moments about the neutral axis and find
Mp= y2 (t- x)2+ y2 (X-tl)2+ yltl(x_t2) 2
2
'
or after simplification
1
2{
2 Y1 • tl-+2 Y1 (tiN} 2 -- (yl~2(tl~2~
Introducing the dimensionless ratios k = Y1/Y2 and t r = tl/t 2 finally we have Mp = (1 + 2ktr + 2kt~ - kZt 2) Y2t~ - 1 + 2 k 6 + 2kt 2 - kZt 2 Y2t 2 4 (1 + tr) 2 4
(2)
Expressing the full plastic bending moment Mp in relation to an equivalent yield stress ~" in the form Mp = YtZ/4 and combining with Eqn (2), gives y. = 1 + 2ktr + 2kt~ - kZt 2"
Y2.
(3)
(1 + tr) 2
Extensible plastic collapse
Consider the well-known crumpling mechanism proposed by Alexander [7,8], as shown in Fig. 2. Taking into account the formation of external or internal convolutions, the following amounts of work dissipated in the various processes which occur--bending in
ove
.....!11 i
Neutral axis
v,
o
FIG. 1. Stress distribution at fully yielded bimetal cross-section.
Axial plastic collapse of thin bi-material tubes as energy dissipating systems
187
" IIllil D
I I I I I I I I Fro. 2. Assumed deformation mode for the extensible collapse.
circular hinges and stretching in the convolutions--can be evaluated: (i) Work dissipated in bending the tube wall,
W 1=2
riD.
+Mp
r
n(D+_2hsinO) dO
dO
}
= 2Mpn(nD + 2h).
(4)
Note, that a "minus" sign would refer to internal convolutions. (ii) Work dissipated in stretching the tube wall to become a convolution,
W2 = ~ 2 fo' YinDit i 2/sin 0 dx i= 1
Di
= 2n(Yltl + YEt2)h 2.
(5)
Equating the external work We =/5.2h to the internal work Wi = W1 + Wz, yields for the mean load/5
We next minimise/5 with respect to h, i.e. dP =0 dh and find,
h= / ~M,D X/ Yltl + YEt2 and combining the latter with Eqn (2)
h = l- N/l + 2ktr + 2ktZ~ - k2t~ 2 1 + ktr
nx/~2 = l / l + 2kt' + 2ktZ~ - k2tZr ~ . 2 (i+ktr)(1 +tr)
(7)
Since it is not known a priori whether the convolutions will form internally or externally (although mostly the experiments show to be the latter), assuming that the actual value
188
A. (.]. MAMMAS et a/.
of P corresponds to the mean of the values given by Eqn (6), and replacing 11 from Eqn 17) we obtain, P = 2~zx..'zrD\/'Mp~/Ylt 1 +
Y2t2
= 7 z 1 s t ~ S D ° s Y 2 \ / / ( 1 + 2ktr + 2kt~ - k2tr)(l + kt r) = 7zl.stl.SDo.Sy 2 ,~,/(1 + : k t r + 2 k t 2 2 k2t~)(l + kt~)
(8)
(1 + tr)w/1 + t r For similar materials, i.e. Y~ = Y2 = Y or equivalently k = 1, we have from Eqns (2), (7) and (8), respectively, M o = Ft2/4
h = 0.886x/Dt and
(9)
P = 5.6tx/Dt
which, of course, coincide with Alexander's original expressions I-7] when the Tresca yield condition is used. l n e x t e n s i o n a l plastic collapse
For the present theoretical analysis of an inextensional plastic mode of collapse for bi-metallic tubes the model described in Ref. [9] is used for an equivalent tube of the same thickness t with yield stress Y"given by Eqn (3). The following general assumptions are made: (i) the rigid-perfectly plastic material has infinite ductility; (ii) a frictionless compression with both ends of the tubular specimen free to deform; (iii) an unextended middle surface of shell which is not dependent on the compression imposed, that is membrane stretching is neglected; (iv) only the terminal deformation mode (not the history of plastic deformation) is taken into account; (v) the direct and shear stresses are neglected and only the vertical action of the load is considered. Applying the travelling hinge mechanism proposed in Ref. [9] (see also Figs 4 and 7 of [9]) we obtain for the mean post-buckling load P, expressed in terms of Mp, the expression p = 2rcM v D / r - n cot(rt/2n) + 1 + n cosec(~z/2n),
(10)
1 - 2r/h 1
where n the number of lobes per layer, see Notation. r=A___
-1
1+
:-tan(lr/Zn)-I
(11)
and A = 1 + n cosec(rt/2n) - n cot(rt/2n).
112)
In the above analysis, when the Mises yield condition is used, the full plastic bending moment Mp must be multiplied by 2/,¢/3 in Eqn (2) and the lobe height h and the mean load P by the factor x/2/3x/3 in Eqns (7) and (8), respectively (the compressive or tensile yield stress in plane strain bending is 2/x/3 times as great as it is in simple tension). EXPERIMENTAL
The axial compression of thin-walled bi-material cylinders was carried out between the parallel steel platens of a hydraulic testing machine of 100 ton capacity. All tests were
Axial plastic c o l l a p s e o f t h i n b i - m a t e r i a l t u b e s as e n e r g y d i s s i p a t i n g s y s t e m s
189
carried out at a low crosshead speed corresponding to an overall compresssion strain rate of about 10 -3 s -1 The test materials used were PVC, aluminium, copper, mild steel (steel 1) and high strength steel (steel 2) with mean compressive yield stresses predicted from quasi-static compression tests 0.065, 0.20, 0.40, 0.30 and 0.65 kN/mm 2, respectively. All specimens were machined from commercial tubes to the required size, their interfaces (inside surface of the outer tube and the outside surface of the inner one) were roughened to such an extent to allow for an almost fixed tolerance when the inner tube was pressed inside the outer one and therefore to ensure that a very good bonding existed between the bimetals at the interface, approximating that of the assumed bonding in the theoretical model. Attention was also paid to combining the available materials so as to obtain a wide variety of the yield stress ratios of the partner tubes for each case; the ratios ranged between 0.10 and 0.66. Details relating to specimens dimensions are presented in Table 1. The initial axial length of the specimens was kept constant in all cases and equal to 127 mm. The end faces of specimens were machined square and were tested in a "dry" condition. Individual component tubes of bimaterial specimens 1, 2 and 3 were tested and their results superimposed and compared with those of the corresponding bi-material tube. Load deflection curves were obtained using an autographic recorder, see Figs 3-7, and photographs showing the final patterns of the compressed bi-material and component tubes were also taken, see Figs 8 11. Details relating to load and deformation characteristics of the specimens tested were tabulated in Table 1.
300
A
z
~2oo "10 (o
_o O) c
\\ j
-~ 10o
I
"\..f..../
flCl
i
i
0.10
020
i
i
0.30
i
0.40
0.50
Deflection (L) FIG. 3. k o a d ~ t e f l e c t i o n c u r v e s for s p e c i m e n s l - l b ; see T a b l e 1 for details. • sp. l a ; . . . . sp. l b ; . . . . " t o t a l " c u r v e of spec. l a a n d l b .
. - - sp. 1 ; - -
'+I A
•
loo
i
0.20
OA.O
0.80
Deflection (L) FIG. 4. L o a d - d e f l e c t i o n c u r v e s for s p e c i m e n s 2 - 2 b ; see T a b l e 1 for details. - - . . . . sp. 2a; . . - - sp. 2 b ; . . . . " t o t a l " c u r v e o f spec. 2 a a n d 2b.
sp. 2; - - -
TABLE 1. Buckling loads (kN)
o~
Mean yield stress (kN/mm 2)
70.3 75.4 70.3
0.20 0.30
40.5 45.7 40.5
0.065 0.65
70.3 75.0 70.3
0.20 0.40
Diameter (mm) Spec. no.
Material (ext/int)
1 la lb
Inside
Yield stress ratio
Wall thickness (mm)
Outside DO
Interm. Dm
AI-St I AI Stl
80.0 80.0 75.4
75.4
2 2a 2b
PVC-St2 PVC St2
50.0 50.0 45.7
45.7
3 3a 3b
A1-Cu AI Cu
79.0 79.0 75.0
75.0
4
AI-PVC
78.4
72.2
66.5
0.33
5.95
3.10
2.85
5
Stl-PVC
89.3
82.2
75.1
0.22
7.10
3.55
3.55
* Refers to "total" Ioad~teflection curve.
(k)
l
0.66
4.85
t~
t2
Wall thickness ratio lr
294.3 89.3 166.8
177.6
153.2
13.7
Mixed (A-2D) Mixed (A-2D) Mixed (A-2D)
210.0 18.6 175.6
147.2
136.4
-7.9
Concertina Diamond (3D) Concertina
235.5 68.7 166.8
143.2
146.1
2.0
Concertina Concertina Concertina
1.09
152.1
98.1
110.2
10.9
Mixed (A-2Dt
1.00
340.4
202.4
210.3
3.8
Mixed (A-2D)
2.55 0.83 2.15 2.60 0.50
0.85
4.35 2.00 2.35
% Dif.
Deformation mode
Theor.
0.90
4.75
Mean post-buckling P Exper.
2.30 0.10
Initial peak Omax
164.8"
140.3"
126.6"
Axial plastic collapse of thin bi-material tubes as energy dissipating systems
191
300
"o 2OO eo _go
tr .
',,
I
"~
.
k ~' o 100
\
X'~ ~J/ •
0
,'
"., ."
•
,~/f~-,l,
~\
".~, #'~,...j
"\ ...,..., f \ . . )
,
/ ' T " C L'. "k
0.20
"~, " ~ . . ~ .
0.6 0
0.40 Deflection (L)
F I G . 5. Load~leflection curves for specimens 3-3b; see Table 1 for details. - - . - - . sp. 3 a ; . . . . sp. 3 b ; - - - "total" curve of spec. 3a and 3b.
sp. 3;
200 A
z
0 --
100
o
o
i
020
0'.40
o16o
Deflection (L) F I G . 6. Load-deflection curve for specimen 4; see Table 1 for details.
400 Z "10
_8 o~ 200
c m
0 ,,'n
0
' 0.20
0140
0160
Deflection (L) F I G . 7. Load~teflection curve for specimen 5; see Table 1 for details.
DISCUSSION
AND
CONCLUSIONS
Deformation characteristics By comparison of the results from monomaterial and bi-material tubes see specimens 1-5 in Table 1 and Figs 8711, the following remarks may be made. --Concertina and/or diamond type modes of collapse were observed for the specimens tested; the nature of their deformation mode depends on the yield stress ratio k and, as would be expected mainly on the collapse mode of the harder component when the thickness ratio is around 0.5; see details in Table 1. - - F o r materials of highly different strength characteristics (Y~ << II1) the deformation mode of the harder material determines the crushing characteristics of the bi-material tube, e.g. the PVC-St2 tube (spec. 2) and the steel component follow the concertina mode, but the
192
A.G.
MAMAt.IS el a[.
sp. 1
St
AI.
2cm
4cm
I
I
I
(a)
I
(b)
sp. l a
sp. l b
2 cm
!
(c)
•
(d)
FIG. 8. (a) Side view of spec. 1; (b) Enlarged view of (a); (c) Side view of spec. la; (d) Side view of spec. lb.
PVC tube, individually compressed, develops a 3D-inextensional mode of collapse; see Fig. 9 and Table 1 for details. - - F o r the same specimens as above during a concertina type of collapse the softer material is trapped and extruded severely in the convolutions of the harder component, diminishing in this manner its thickness; full adhesion at the internal edge of convolution is maintained whilst adhesion is almost totally lost at the outer edge, see Fig. 9 for the PVC-St2 specimen. The separation which is shown in Figs 8(a) and (b) and Fig. 9(d), for example, clearly identifies a region of very severe plastic bending deformation to which careful attention would need to be given when developing a bi-material tube product. A study of this separation proces is clearly a matter necessitating an investigation in its own right. - - F o r medium yield stress ratios k, adhesion is lost at both ends of a convolution, probably due to the slight relative sliding of the component materials at the beginning of convolution formation, but it is also maintained at the extended area of the contact surface; see spec. 3 in Fig. 10.
Axial plastic collapse of thin bi-material tubes as energy dissipating systems
sp. 2
i
St
PVC
1 cm
193
5mm
I
I
(a)
I
(b)
sp. 2o sp. 2b
1 cm !
(c)
I
(d)
FIG. 9. (a) Side view of spec. 2; (b) Enlarged view of (a); (c) Side view of spec. 2a; (d) Side view of spec. 2b.
--Less de-adhesion/separation of components is observed with diamond type of collapse; compare specimens 1-3 in Figs 8-10. Load-deflection curves
For specimens l-3,.a "total" load-deflection curve is introduced as obtained by the superposition of the load-deflection curves for component tubes, individually compressed, see Figs 3-5. This "total" curve, although not valid in the plastic region, seems to give useful conclusions from the comparison of the combined collapse of the individual components with that of the bimaterial tube from the energy absorption capability point of view. The toad-deflection curves obtained are characterized by the elastic region of a steady-state increase of load until reaching a peak value, after which the post-buckling
194
A. (;. MAMALISet al.
sp. 3
1 cm I
5 turn I
k
((:I)
J
(b)
sp. 3a
sp. 3b
lcm I
(c)
,J
(d)
FIG. 10. (a) Side view of spec. 3; (b) Enlarged view of (a); (c) Side view of spec. 3a; (d) Side view of spec. 3b.
region ensues consisting of consequent troughs and peaks of the same order of maximum load as in the case of the concertina mode or of decreased maximum load as in the diamond type of folding; see Figs 3 and 4, respectively. Comparing the bi-material load-compression curves with individual and "total" ones and taking also into consideration deformation modes, the following general remarks may be drawn: - - I n general, the initial peak values Pmax are greater than in the case of the superimposed ones, demonstrating higher initial strength with the bi-material tubes, see Figs 3-5 and Table 1. - - T h e inclination of the elastic load-deflection line is closer to that of the relatively harder material and appears to coincide with the superimposed curve, see Figs 3-5. - - T h e initial post-buckling region (first lobe) is characterized by a load drop of lower rate of decrease than that of the "total" curve; it is significantly influenced by the crushing
Axial plastic collapse of thin bi-rnaterial'tubes as energy dissipating systems
195
PVC ;t5
20m,m
(a)
AI.
._,,,,.,
20mm I
I
(b) FIG. 11. (a) Side view of spec. 5; (b) Side view of spec. 4. behaviour of the harder material in cases where its component materials are of highly different strength, see Fig. 4 for the PVC-St2 specimen. - - T h e fluctuations of the curve in the post-buckling region follow, in general, the variations of the corresponding curve of the harder material, see Figs 3-5. - - W h e n compressing materials of very different strength in a concertina mode, the post-buckling peaks show a step-wise load increase, this may be due to separate buckling of the component tubes at different times and due to the loss of adhesion, see Fig. 4. This fact is not apparent in diamond folding, see Figs 3, 6 and 7. - - T h e mean post-buckling load P, obtained by measuring the area under the load-deflection curve and dividing by total compression, is greater in bi-material tubes than in the superimposed ones, showing in this manner a higher energy absorbing capacity when compressing bi-material tubes as opposed to individual ones. --Theoretical and experimental values of P are found to be in good agreement with predicted values with a deviation of about _+ 10% (see also Table 1) and with the best agreement applying to the concertina deformation mode, see Table 1.
196
A. C]. MAMALISC/ Ol.
It is n o t e d h o w e v e r , t h a t the effect of factors like the e v a l u a t i o n of b e n d i n g radii of the v a r i o u s lobes with t h i c k n e s s for different thicknesses a n d m a t e r i a l s of the l a m i n a e of the shell and the a m o u n t of s e p a r a t i o n b e t w e e n the l a m i n a e , o n the p r o p o s e d e n e r g y a b s o r b i n g m e c h a n i s m needs to be f u r t h e r i n v e s t i g a t e d . T h e r e f o r e , an e x t e n s i v e e x p e r i m e n t a l w o r k on the c r a s h w o r t h y b e h a v i o u r of b i - m a t e r i a l t u b e s u n d e r static a n d e l e v a t e d s t r a i n - r a t e l o a d i n g is u n d e r way.
REFERENCES 1. W. JOHNSONand A. G. MAMALIS,Crashworthiness of ~2,hicles. 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 (19831. 3. A. G. MAMALIS,D. E. MANOLAKOS,G. L. VIEGELAHN,N. M. VAXEVANID[Sand W. JOHNSON, On the inextensional axial collapse of thin PVC conical shells. Int. J. Mech. Sci. 28, 323 (1986L 4. A. G. MAMALIS,D. E. MANOLAKOSand G. L. VIEGELAHN,The axial crushing of thin PVC tubes and frusta of square cross-section. Int. J. Impact Engng 8, (1989). 5. A.G. MAMALIS,D. E. MANOLAKOS,G. L. VIEGELAHN,N. M. VAXEVANII)ISand W. JOHNSON,The inextensional collapse of grooved thin-walled cylinders of PVC under axial loading. Int. J. Impact En,qn~l 4, 41 (1986). 6. A. G. MAMALIS,D. E. MANOLAKOS,G. L. VIEGELAHNand W. JOHNSON,Energy absorption and deformation modes of thin PVC tubes internally grooved when subjected to axial plastic collapse. Proc. C of the Institution of Mechanical Engineers (1989). 7. J. M. ALEXANDER,An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q. J. Mech. Appl. Math. 13, I0 (1960). 8. W. JOHNSON, Impact Strength of Materials. Van Nostrand, New York {1972). 9. W. JOHNSON, P. D. SODENand S. T. S. AL-HASSANI,lnextensional collapse of thin-walled tubes under axial compression. J. Strain Anal. 12, 4 (1977).
0734 743X/91 $3.00+0.00 (" 1991 Pergamon Press plc
Printed in Great Britain
AXIAL PLASTIC COLLAPSE OF THIN BI-MATERIAL T U B E S AS E N E R G Y D I S S I P A T I N G S Y S T E M S A. G.
MAMALIS,
D. E.
MANOLAKOS,
G. A. DEMOSTHENOUSand W.
JOHNSON t
Department of Mechanical Engineering, National Technical University of Athens, Greece; and tRidge Hall, Chapel-en-le-Frith, Derbyshire, U.K. (Received 23 February 1990; and in revised form 20 January 1991 )
Summary The deformation characteristics, crumpling mechanisms and energy absorption efficiency of bi-material circular tubes subjected to axial compression are investigated and analysed both experimentally and theoretically. Theoretical models describing extensible and inextensional type of collapse are proposed and corresponding results are found to be in good agreement with experimental ones.
NOTATION D h Mp n P Pm.x r
t tr W x Y Y" k
diameter of tube height of convolution (lobe) full plastic bending movement per unit width number of lobes per layer (see also Ref. I-9]) mean post-buckling load initial peak load radius of curvature of shell (see also Ref. [9]) wall thickness of shell =tl/t2
plastic work dissipated depth of neutral axis mean yield stress of material equivalent yield stress of bi-material tube = YI/Y2
INTRODUCTION
Crashworthiness studies [1] have caused much attention to be given to analysing the deformation characteristics and determining the energy absorbing efficiency of various thin-walled crashworthy components of different materials, e.g. metals, plastics, composites, etc. [1-3], and for various geometrical shapes--cylindrical tubes, cones, conical frusta, square tubes, square frusta and pyramids, all grooved or ungrooved, etc. [4-6] when subjected to axial compression. Work into the crashworthy characteristics of thin-walled structures has been extended to also cover bi-material components, namely bi-material tubes. We consider bi-material tubes for two reasons: (i) The two materials may each confer on an energy absorbing element their own particular characteristics: for example, if a "core" aluminium tube was used, then covering it with steel might confer an anti-corrosive property or that of supplying a hard surface. Similarly PVC coverings may act so as to supply a generally protective coating. And to our knowledge (ii) bi-material tubes have not been previously reported on. In the present paper combinations of four different materials forming a bi-material thin-walled tube are investigated and their behaviour in axial collapse is analysed both theoretically and experimentally. Well-known theoretical models l-7,9] are recalled and modified appropriately to account for the crumpling behaviour of bi-metalled tubes in the cases of extensible and inextensional modes of collapse, respectively. Theoretical results were found to be in good agreement with experimental ones. 185
1~6
A. (J. MAMALISt't tl[. THEORETICAL
Consider a bimaterial plate consisting of two fully adhering different materials of thicknesses tl and t2, respectively, see Fig. I, and corresponding yield stresses YI and Y2. Examining the full yielding of a cross-section of overall thickness t = t~ + t z and unit width and requiring the forces on a section to total zero, we obtain the position of the plastic neutral axis x from, Y2(t- x)= Y2(x-tl)+
Ylll,
or finally Y2(t+tl)x=
-
Yltl ,
(J)
2Y2
with the restriction that Yttl < Yzt2 • To evaluate the moment per unit width to bend plastically the section Mp, we take moments about the neutral axis and find
Mp= y2 (t- x)2+ y2 (X-tl)2+ yltl(x_t2) 2
2
'
or after simplification
1
2{
2 Y1 • tl-+2 Y1 (tiN} 2 -- (yl~2(tl~2~
Introducing the dimensionless ratios k = Y1/Y2 and t r = tl/t 2 finally we have Mp = (1 + 2ktr + 2kt~ - kZt 2) Y2t~ - 1 + 2 k 6 + 2kt 2 - kZt 2 Y2t 2 4 (1 + tr) 2 4
(2)
Expressing the full plastic bending moment Mp in relation to an equivalent yield stress ~" in the form Mp = YtZ/4 and combining with Eqn (2), gives y. = 1 + 2ktr + 2kt~ - kZt 2"
Y2.
(3)
(1 + tr) 2
Extensible plastic collapse
Consider the well-known crumpling mechanism proposed by Alexander [7,8], as shown in Fig. 2. Taking into account the formation of external or internal convolutions, the following amounts of work dissipated in the various processes which occur--bending in
ove
.....!11 i
Neutral axis
v,
o
FIG. 1. Stress distribution at fully yielded bimetal cross-section.
Axial plastic collapse of thin bi-material tubes as energy dissipating systems
187
" IIllil D
I I I I I I I I Fro. 2. Assumed deformation mode for the extensible collapse.
circular hinges and stretching in the convolutions--can be evaluated: (i) Work dissipated in bending the tube wall,
W 1=2
riD.
+Mp
r
n(D+_2hsinO) dO
dO
}
= 2Mpn(nD + 2h).
(4)
Note, that a "minus" sign would refer to internal convolutions. (ii) Work dissipated in stretching the tube wall to become a convolution,
W2 = ~ 2 fo' YinDit i 2/sin 0 dx i= 1
Di
= 2n(Yltl + YEt2)h 2.
(5)
Equating the external work We =/5.2h to the internal work Wi = W1 + Wz, yields for the mean load/5
We next minimise/5 with respect to h, i.e. dP =0 dh and find,
h= / ~M,D X/ Yltl + YEt2 and combining the latter with Eqn (2)
h = l- N/l + 2ktr + 2ktZ~ - k2t~ 2 1 + ktr
nx/~2 = l / l + 2kt' + 2ktZ~ - k2tZr ~ . 2 (i+ktr)(1 +tr)
(7)
Since it is not known a priori whether the convolutions will form internally or externally (although mostly the experiments show to be the latter), assuming that the actual value
188
A. (.]. MAMMAS et a/.
of P corresponds to the mean of the values given by Eqn (6), and replacing 11 from Eqn 17) we obtain, P = 2~zx..'zrD\/'Mp~/Ylt 1 +
Y2t2
= 7 z 1 s t ~ S D ° s Y 2 \ / / ( 1 + 2ktr + 2kt~ - k2tr)(l + kt r) = 7zl.stl.SDo.Sy 2 ,~,/(1 + : k t r + 2 k t 2 2 k2t~)(l + kt~)
(8)
(1 + tr)w/1 + t r For similar materials, i.e. Y~ = Y2 = Y or equivalently k = 1, we have from Eqns (2), (7) and (8), respectively, M o = Ft2/4
h = 0.886x/Dt and
(9)
P = 5.6tx/Dt
which, of course, coincide with Alexander's original expressions I-7] when the Tresca yield condition is used. l n e x t e n s i o n a l plastic collapse
For the present theoretical analysis of an inextensional plastic mode of collapse for bi-metallic tubes the model described in Ref. [9] is used for an equivalent tube of the same thickness t with yield stress Y"given by Eqn (3). The following general assumptions are made: (i) the rigid-perfectly plastic material has infinite ductility; (ii) a frictionless compression with both ends of the tubular specimen free to deform; (iii) an unextended middle surface of shell which is not dependent on the compression imposed, that is membrane stretching is neglected; (iv) only the terminal deformation mode (not the history of plastic deformation) is taken into account; (v) the direct and shear stresses are neglected and only the vertical action of the load is considered. Applying the travelling hinge mechanism proposed in Ref. [9] (see also Figs 4 and 7 of [9]) we obtain for the mean post-buckling load P, expressed in terms of Mp, the expression p = 2rcM v D / r - n cot(rt/2n) + 1 + n cosec(~z/2n),
(10)
1 - 2r/h 1
where n the number of lobes per layer, see Notation. r=A___
-1
1+
:-tan(lr/Zn)-I
(11)
and A = 1 + n cosec(rt/2n) - n cot(rt/2n).
112)
In the above analysis, when the Mises yield condition is used, the full plastic bending moment Mp must be multiplied by 2/,¢/3 in Eqn (2) and the lobe height h and the mean load P by the factor x/2/3x/3 in Eqns (7) and (8), respectively (the compressive or tensile yield stress in plane strain bending is 2/x/3 times as great as it is in simple tension). EXPERIMENTAL
The axial compression of thin-walled bi-material cylinders was carried out between the parallel steel platens of a hydraulic testing machine of 100 ton capacity. All tests were
Axial plastic c o l l a p s e o f t h i n b i - m a t e r i a l t u b e s as e n e r g y d i s s i p a t i n g s y s t e m s
189
carried out at a low crosshead speed corresponding to an overall compresssion strain rate of about 10 -3 s -1 The test materials used were PVC, aluminium, copper, mild steel (steel 1) and high strength steel (steel 2) with mean compressive yield stresses predicted from quasi-static compression tests 0.065, 0.20, 0.40, 0.30 and 0.65 kN/mm 2, respectively. All specimens were machined from commercial tubes to the required size, their interfaces (inside surface of the outer tube and the outside surface of the inner one) were roughened to such an extent to allow for an almost fixed tolerance when the inner tube was pressed inside the outer one and therefore to ensure that a very good bonding existed between the bimetals at the interface, approximating that of the assumed bonding in the theoretical model. Attention was also paid to combining the available materials so as to obtain a wide variety of the yield stress ratios of the partner tubes for each case; the ratios ranged between 0.10 and 0.66. Details relating to specimens dimensions are presented in Table 1. The initial axial length of the specimens was kept constant in all cases and equal to 127 mm. The end faces of specimens were machined square and were tested in a "dry" condition. Individual component tubes of bimaterial specimens 1, 2 and 3 were tested and their results superimposed and compared with those of the corresponding bi-material tube. Load deflection curves were obtained using an autographic recorder, see Figs 3-7, and photographs showing the final patterns of the compressed bi-material and component tubes were also taken, see Figs 8 11. Details relating to load and deformation characteristics of the specimens tested were tabulated in Table 1.
300
A
z
~2oo "10 (o
_o O) c
\\ j
-~ 10o
I
"\..f..../
flCl
i
i
0.10
020
i
i
0.30
i
0.40
0.50
Deflection (L) FIG. 3. k o a d ~ t e f l e c t i o n c u r v e s for s p e c i m e n s l - l b ; see T a b l e 1 for details. • sp. l a ; . . . . sp. l b ; . . . . " t o t a l " c u r v e of spec. l a a n d l b .
. - - sp. 1 ; - -
'+I A
•
loo
i
0.20
OA.O
0.80
Deflection (L) FIG. 4. L o a d - d e f l e c t i o n c u r v e s for s p e c i m e n s 2 - 2 b ; see T a b l e 1 for details. - - . . . . sp. 2a; . . - - sp. 2 b ; . . . . " t o t a l " c u r v e o f spec. 2 a a n d 2b.
sp. 2; - - -
TABLE 1. Buckling loads (kN)
o~
Mean yield stress (kN/mm 2)
70.3 75.4 70.3
0.20 0.30
40.5 45.7 40.5
0.065 0.65
70.3 75.0 70.3
0.20 0.40
Diameter (mm) Spec. no.
Material (ext/int)
1 la lb
Inside
Yield stress ratio
Wall thickness (mm)
Outside DO
Interm. Dm
AI-St I AI Stl
80.0 80.0 75.4
75.4
2 2a 2b
PVC-St2 PVC St2
50.0 50.0 45.7
45.7
3 3a 3b
A1-Cu AI Cu
79.0 79.0 75.0
75.0
4
AI-PVC
78.4
72.2
66.5
0.33
5.95
3.10
2.85
5
Stl-PVC
89.3
82.2
75.1
0.22
7.10
3.55
3.55
* Refers to "total" Ioad~teflection curve.
(k)
l
0.66
4.85
t~
t2
Wall thickness ratio lr
294.3 89.3 166.8
177.6
153.2
13.7
Mixed (A-2D) Mixed (A-2D) Mixed (A-2D)
210.0 18.6 175.6
147.2
136.4
-7.9
Concertina Diamond (3D) Concertina
235.5 68.7 166.8
143.2
146.1
2.0
Concertina Concertina Concertina
1.09
152.1
98.1
110.2
10.9
Mixed (A-2Dt
1.00
340.4
202.4
210.3
3.8
Mixed (A-2D)
2.55 0.83 2.15 2.60 0.50
0.85
4.35 2.00 2.35
% Dif.
Deformation mode
Theor.
0.90
4.75
Mean post-buckling P Exper.
2.30 0.10
Initial peak Omax
164.8"
140.3"
126.6"
Axial plastic collapse of thin bi-material tubes as energy dissipating systems
191
300
"o 2OO eo _go
tr .
',,
I
"~
.
k ~' o 100
\
X'~ ~J/ •
0
,'
"., ."
•
,~/f~-,l,
~\
".~, #'~,...j
"\ ...,..., f \ . . )
,
/ ' T " C L'. "k
0.20
"~, " ~ . . ~ .
0.6 0
0.40 Deflection (L)
F I G . 5. Load~leflection curves for specimens 3-3b; see Table 1 for details. - - . - - . sp. 3 a ; . . . . sp. 3 b ; - - - "total" curve of spec. 3a and 3b.
sp. 3;
200 A
z
0 --
100
o
o
i
020
0'.40
o16o
Deflection (L) F I G . 6. Load-deflection curve for specimen 4; see Table 1 for details.
400 Z "10
_8 o~ 200
c m
0 ,,'n
0
' 0.20
0140
0160
Deflection (L) F I G . 7. Load~teflection curve for specimen 5; see Table 1 for details.
DISCUSSION
AND
CONCLUSIONS
Deformation characteristics By comparison of the results from monomaterial and bi-material tubes see specimens 1-5 in Table 1 and Figs 8711, the following remarks may be made. --Concertina and/or diamond type modes of collapse were observed for the specimens tested; the nature of their deformation mode depends on the yield stress ratio k and, as would be expected mainly on the collapse mode of the harder component when the thickness ratio is around 0.5; see details in Table 1. - - F o r materials of highly different strength characteristics (Y~ << II1) the deformation mode of the harder material determines the crushing characteristics of the bi-material tube, e.g. the PVC-St2 tube (spec. 2) and the steel component follow the concertina mode, but the
192
A.G.
MAMAt.IS el a[.
sp. 1
St
AI.
2cm
4cm
I
I
I
(a)
I
(b)
sp. l a
sp. l b
2 cm
!
(c)
•
(d)
FIG. 8. (a) Side view of spec. 1; (b) Enlarged view of (a); (c) Side view of spec. la; (d) Side view of spec. lb.
PVC tube, individually compressed, develops a 3D-inextensional mode of collapse; see Fig. 9 and Table 1 for details. - - F o r the same specimens as above during a concertina type of collapse the softer material is trapped and extruded severely in the convolutions of the harder component, diminishing in this manner its thickness; full adhesion at the internal edge of convolution is maintained whilst adhesion is almost totally lost at the outer edge, see Fig. 9 for the PVC-St2 specimen. The separation which is shown in Figs 8(a) and (b) and Fig. 9(d), for example, clearly identifies a region of very severe plastic bending deformation to which careful attention would need to be given when developing a bi-material tube product. A study of this separation proces is clearly a matter necessitating an investigation in its own right. - - F o r medium yield stress ratios k, adhesion is lost at both ends of a convolution, probably due to the slight relative sliding of the component materials at the beginning of convolution formation, but it is also maintained at the extended area of the contact surface; see spec. 3 in Fig. 10.
Axial plastic collapse of thin bi-material tubes as energy dissipating systems
sp. 2
i
St
PVC
1 cm
193
5mm
I
I
(a)
I
(b)
sp. 2o sp. 2b
1 cm !
(c)
I
(d)
FIG. 9. (a) Side view of spec. 2; (b) Enlarged view of (a); (c) Side view of spec. 2a; (d) Side view of spec. 2b.
--Less de-adhesion/separation of components is observed with diamond type of collapse; compare specimens 1-3 in Figs 8-10. Load-deflection curves
For specimens l-3,.a "total" load-deflection curve is introduced as obtained by the superposition of the load-deflection curves for component tubes, individually compressed, see Figs 3-5. This "total" curve, although not valid in the plastic region, seems to give useful conclusions from the comparison of the combined collapse of the individual components with that of the bimaterial tube from the energy absorption capability point of view. The toad-deflection curves obtained are characterized by the elastic region of a steady-state increase of load until reaching a peak value, after which the post-buckling
194
A. (;. MAMALISet al.
sp. 3
1 cm I
5 turn I
k
((:I)
J
(b)
sp. 3a
sp. 3b
lcm I
(c)
,J
(d)
FIG. 10. (a) Side view of spec. 3; (b) Enlarged view of (a); (c) Side view of spec. 3a; (d) Side view of spec. 3b.
region ensues consisting of consequent troughs and peaks of the same order of maximum load as in the case of the concertina mode or of decreased maximum load as in the diamond type of folding; see Figs 3 and 4, respectively. Comparing the bi-material load-compression curves with individual and "total" ones and taking also into consideration deformation modes, the following general remarks may be drawn: - - I n general, the initial peak values Pmax are greater than in the case of the superimposed ones, demonstrating higher initial strength with the bi-material tubes, see Figs 3-5 and Table 1. - - T h e inclination of the elastic load-deflection line is closer to that of the relatively harder material and appears to coincide with the superimposed curve, see Figs 3-5. - - T h e initial post-buckling region (first lobe) is characterized by a load drop of lower rate of decrease than that of the "total" curve; it is significantly influenced by the crushing
Axial plastic collapse of thin bi-rnaterial'tubes as energy dissipating systems
195
PVC ;t5
20m,m
(a)
AI.
._,,,,.,
20mm I
I
(b) FIG. 11. (a) Side view of spec. 5; (b) Side view of spec. 4. behaviour of the harder material in cases where its component materials are of highly different strength, see Fig. 4 for the PVC-St2 specimen. - - T h e fluctuations of the curve in the post-buckling region follow, in general, the variations of the corresponding curve of the harder material, see Figs 3-5. - - W h e n compressing materials of very different strength in a concertina mode, the post-buckling peaks show a step-wise load increase, this may be due to separate buckling of the component tubes at different times and due to the loss of adhesion, see Fig. 4. This fact is not apparent in diamond folding, see Figs 3, 6 and 7. - - T h e mean post-buckling load P, obtained by measuring the area under the load-deflection curve and dividing by total compression, is greater in bi-material tubes than in the superimposed ones, showing in this manner a higher energy absorbing capacity when compressing bi-material tubes as opposed to individual ones. --Theoretical and experimental values of P are found to be in good agreement with predicted values with a deviation of about _+ 10% (see also Table 1) and with the best agreement applying to the concertina deformation mode, see Table 1.
196
A. C]. MAMALISC/ Ol.
It is n o t e d h o w e v e r , t h a t the effect of factors like the e v a l u a t i o n of b e n d i n g radii of the v a r i o u s lobes with t h i c k n e s s for different thicknesses a n d m a t e r i a l s of the l a m i n a e of the shell and the a m o u n t of s e p a r a t i o n b e t w e e n the l a m i n a e , o n the p r o p o s e d e n e r g y a b s o r b i n g m e c h a n i s m needs to be f u r t h e r i n v e s t i g a t e d . T h e r e f o r e , an e x t e n s i v e e x p e r i m e n t a l w o r k on the c r a s h w o r t h y b e h a v i o u r of b i - m a t e r i a l t u b e s u n d e r static a n d e l e v a t e d s t r a i n - r a t e l o a d i n g is u n d e r way.
REFERENCES 1. W. JOHNSONand A. G. MAMALIS,Crashworthiness of ~2,hicles. 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 (19831. 3. A. G. MAMALIS,D. E. MANOLAKOS,G. L. VIEGELAHN,N. M. VAXEVANID[Sand W. JOHNSON, On the inextensional axial collapse of thin PVC conical shells. Int. J. Mech. Sci. 28, 323 (1986L 4. A. G. MAMALIS,D. E. MANOLAKOSand G. L. VIEGELAHN,The axial crushing of thin PVC tubes and frusta of square cross-section. Int. J. Impact Engng 8, (1989). 5. A.G. MAMALIS,D. E. MANOLAKOS,G. L. VIEGELAHN,N. M. VAXEVANII)ISand W. JOHNSON,The inextensional collapse of grooved thin-walled cylinders of PVC under axial loading. Int. J. Impact En,qn~l 4, 41 (1986). 6. A. G. MAMALIS,D. E. MANOLAKOS,G. L. VIEGELAHNand W. JOHNSON,Energy absorption and deformation modes of thin PVC tubes internally grooved when subjected to axial plastic collapse. Proc. C of the Institution of Mechanical Engineers (1989). 7. J. M. ALEXANDER,An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q. J. Mech. Appl. Math. 13, I0 (1960). 8. W. JOHNSON, Impact Strength of Materials. Van Nostrand, New York {1972). 9. W. JOHNSON, P. D. SODENand S. T. S. AL-HASSANI,lnextensional collapse of thin-walled tubes under axial compression. J. Strain Anal. 12, 4 (1977).