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Collapse of braced tubes under impact loads
Int. J. Impact Engng Vol. 7, No. 2, pp. 125-138, 1988
0734 743X/88$3.00+0.00 © 1988Pergamon Press plc
Printed in Great Britain
COLLAPSE OF BRACED TUBES U N D E R IMPACT LOADS J. R. VEILLETTE and J. F. CARNEY III Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, U.S.A. (Received 21 December 1987)
Summary--The dynamic response of diametrically braced metallic tubes under impact loading conditions is investigated both experimentally and numerically. The effects of strain rate and the initial collapsing mode configuration on the resulting force (acceleration) levelsat the impacted end of the tube are discussed. Differences in energy dissipation characteristics of the braced tubes under quasi-static and impact loading conditions are described as functions of the bracing orientation.
NOTATION a ~i
striker's acceleration accelerationof center of gravity of tube d(t) displacementof striker at time t D experimentalconstant Fc striker's quartz cell force output Fs sledge/tube force m mass of quartz cell and load distribution plate mr mass of striker and distribution plate rh mass of tube p experimentalconstant t time At time increment v(t) striker's velocity at time t Vs initialimpact velocity ~pl equivalent plastic strain rate ~" equivalent stress
INTRODUCTION Metallic tubes which are stiffened across their diameters have recently been employed in impact attenuation devices used in highway safety applications [1, 2]. The diametrical bracing causes the tubes' d y n a m i c response to be a function of the direction of lateral loading. In the application described in [1], thin steel straps (1/8 by 5 in. in cross-section) are inserted into tubes which are 4 ft in diameter. These straps are effective only in tension and are oriented so that they buckle under head-on impact conditions. This behavior allows the crash cushion to exhibit the large collapsing stroke that is required under head-on impact conditions in order to capture the vehicle in a dynamically acceptable manner. U n d e r side impact conditions near the rear of the device, however, the diametrical bracing system is actuated, significantly increasing the stiffness of the tubular cluster. This increased stiffness enables the crash cushion to develop the lateral force required to redirect the errant vehicle away from the rigid hazard. Most of the research performed to date on stiffened tubes subjected to large deformations has been limited to quasi-static loading conditions. In [3], load b o u n d i n g techniques were employed to examine the effect of various orientations of single and symmetric double bracing on the initial collapse loads of tubes. These analytical predictions were c o m p a r e d with experimental data in [4]. This reference also details the post-collapse behavior of the stiffened tubes, and this is sometimes characterized by asymmetric m o d e shapes. Metallic tubes which are employed in crash cushions to dissipate kinetic energy are 125
126
J . R . VFILLF'II't and J. F. CARNkY Ill
subjected to impact loading conditions. Strain rate effects play a significant role in increasing the energy dissipation capacities of such impulsively loaded tubes compared with equivalent tubes loaded quasi-statically. The influence of strain rate effects, however, does not account for the total increase in initial collapse load and the associated energy dissipation which typically occurs under impact loading. In addition to the influence of strain rate, tubes subjected to lateral impact loading conditions exhibit dynamic collapse configurations during the early phases of the impact event which are quite distinct from those of their quasistatic counterparts. This behavior is of great importance in the design of impact attenuation devices since the deceleration levels to which the occupant of an errant vehicle is subjected must remain within allowable limits. In this paper, the impact behavior described above is investigated both experimentally and numerically. The results point out the dramatic differences in quasi-static and impact load-deformation behavior that can occur when two collapse modes differ significantly. The extent of this difference is a function of the orientation of the bracing with respect to the applied loads.
IMPACT
EXPERIMENTS
The experiments were conducted on the specially constructed impact apparatus, described in Fig. 1. The rig includes a 3.40 lb sledge which is fired from a gas gun whose velocity is controlled by setting the gas tank's priming pressure. The launching of the sledge is software controlled through a data acquisition's digital input/output card. Two light beams, set up 5 in. apart and perpendicular to the striker's trajectory, are employed to determine its impact velocity. The exit door is constructed in such a way as to prevent the striker from reentering the guide-rail area after rebounding off the impacted tube and possibly damaging the electronic instrumentation. A high speed camera, operating at 6000 frames s- 1, records the displacement-time history of the events. Force-time histories are obtained by the use of quartz load cells located on the impacting mass and the backup plate. The charges from these load cells are converted in the amplifier to voltages and recorded in digital form on a personal computer at 31.4 kHz through a data acquisition system. The output from the striker's quartz cells must be manipulated in order to obtain the desired sledge/tube force history. Consider the free-body diagram shown in Fig. 2 where Fs is the unknown. The acceleration of every point on the striker assembly is assumed to be the same. Applying Newton's second law of motion:
Fs-Fc=ma,
(1)
since
Fs
a = --, mT
(2)
it follows that Fs =
F~,
(3)
where a = striker's acceleration, Fc = striker's quartz cell force output, Fs = sledge/tube force, m --- mass of the quartz cell load distribution plate (0.0140 slugs), and m t = mass of the striker and distribution plate (0.120 slugs). Substituting the above values of m and tnt into equation (3) yields F~ - 0.883F~.
t4)
Eight 4-in. diameter mild steel tubular specimens were prepared for testing. All specimens were 2 in. in height and had wall thicknesses of 0.083 in. They were annealed by heating at 900°C for 20 rain and then slowly cooled to room temperature inside a closed oven. High strength wire (0.013 in. diameter) was wound through diametrically opposed holes to form
Collapse of braced tubes under impact loads
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TABLE l . TENSION BRACING ANGLES AND VELOCITIES FOR IMPACT LOADING TESTS
0
(degrees)
0 I0
15 20 25 30 ]5
llflllll/l/llllll/l/lllllll
O0 (unbraced)
b~kup plete
Initial $1.edBe Yel0citq (MPH) 33.3 31.5 32.1 31.5 31.7 32.3 ]3.0 30.8
the tubes' tension bracing; 25 loops were typically used. The various bracing angles and impact velocities are presented in Table 1. Examples of typical force-time histories are presented in Fig. 3 as dashed curves for the 15° bracing case. The impacting sledge/tube output is shown in Fig. 3(a), and the backup plate load cell response is given in Fig. 3(b). Solid curves have been fitted to these virgin data by averaging neighboring peak values. This was done to facilitate the post-processing of the data. The maximum difference in area between the virgin and fitted curves in any of the 16 cases considered (two load cell outputs for eac;~ of eight bracing angle cases) was 6 ~o. A film motion analyser was used to investigate the high speed filmed event. The film analyser is capable of frame-by-frame analysis, has a digitizing screen with an electronic grid, a cursor for co-ordinate data input, and is interfaced with a personal computer. The displacement-time history for the 15 ° bracing case is shown in Fig. 4. Two curves are shown on the figure. The jagged curve was obtained with the film analyser and the smooth curve by integrating the fitted curve to the sledge/tube force history results as described below. The leading edge of the striker has an acceleration of Fs/mT. Assuming rectilinear motion, the sledge's velocity is: v(t)
= v~ - (t F~(t)_dt,
(5)
do mT where vs = initial impact velocity and v(t)= sledge's velocity at time t. Integrating again, the displacement of the sledge at time t, d(t), may be found from the expression d(t)
= fl v(t)dt.
(6)
The differences between the results of the two methods are quite small, demonstrating a reasonable curve fitting procedure to the virgin force data and providing a check on the film analysis results. The force and displacement time histories are coupled to produce the force-displacement
Collapse of braced tubes under impact loads
129
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(b) FIG. 3. 15° bracing virgin (dashed) and fitted (solid) applied load history results: (a) sledge/tube (front), and (b) backup/tube (rear). results for each test. The fitted force history curves were used in compiling these results. This was done because different amplifier filters would result in unique virgin curves, but they would produce approximately the same fitted curves. In addition, the smooth curves allowed the coupling process to be programmed more efficiently. The energy is of main importance, and the virgin data yield essentially the same results. As an example, the coupled results for both the front (sledge side) and rear (backup side) outputs are given in Fig. 5. Recalling Hamilton's principle, and considering the two systems shown in Fig. 6, it follows that the areas under the front and rear force displacement curves should be approximately equal. The maximum difference between the areas was 8 %, producing confidence in the experimental results and evaluation procedures.
130
J.R. VIqLLEIII- and J. F. CARNt~Y lI1
__m
2
i 4
~ 6
i
B
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i I~
EVENT TIME, MSEC
FIG. 4. 15:' bracing motion analyser (jagged) and direct integration (smooth) tube displacement history results.
m .D
S
d
Front
121 W
~E
DISPLACEI'ENT. INCHES
FIG. 5. 15'~ bracing force-displacementresults for front and rear load cells. In the early portion of a typical impact event, the force in the front of the tube is larger than the force in the rear of the tube. This is because the acceleration of the tube's center of gravity during the early phases of the impact event is in the direction of the sledge velocity vector. Later in the event, the acceleration of the tube's center of gravity is in the opposite direction. The film analyser was used to obtain the shapes of the tubes at different levels of deformation. The results are shown in Fig. 7 where the tops of all collapse modes represent the impacting force. Figures 8 and 9 present the sledge and backup plate load-displacement responses, respectively, for the eight double bracing angles considered. ANALYSIS A detailed force-time analysis of the impact event is required, and this can only be done numerically. ABAQUS, a powerful finite element software package [5], was selected for this purpose.
Collapse of braced tubes under impact loads
]
131
Fa(6) ~mTv, l / f i l l
/
+o/
where = mass of the tube = accelaration of the tube's center of gravity
/ F.dx,~/ FBdx FIG. 6. Relationbetweenfrontand rear load histories.
m 0.125 1
0.1875
m
H
B
m
mFr Ii
m mEmlmm JIEimmmm
0.25 0.3125 0.375 0.50 0.6875 FIG. 7. Collapse mode shapes.
The system response is obtained by specifying time parameters. ABAQUS allows the user to either fix the time increment or let the software choose one through its automatic time stepping process. Herein, all cases were run implementing the automatic time stepping option. This option requires inputting a minimum and maximum time step as well as the overall desired total time. It is possible to satisfy the discretized equilibrium equations at the beginning and end of the time step but overlook important behavior in between. In order to control the accuracy of the results, ABAQUS introduces a half-step residual error parameter. The intermediate time t + At~2 is used to compute the equilibrium residual error. This residual error is then compared to the inputted acceptable half-step residual error. If unacceptable, the time-step is reduced and the process repeated. In the following example, ABAQUS is employed to model the impact experiment for the unbraced tube by first ignoring and then accounting for the effects of strain rate. Figure 10 shows the divisions implemented in describing strain rate independent cases. They consist of 95 cubic beam elements, 30 sledge nodes and 30 backup plate nodes. Thirty gap elements were used to model both the tube to backup plate and tube to sledge interaction. These gap
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node~ t t 2 degreee
teege
a0du
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30 sledge
Ill
mt 4 m
#m-trmt T
l ° 20
nodes ~t.
FIG. 10. ABAQUS discretization impact loading model without strain rate effects.
elements provide a compressive force only if the gap distance is zero. The actual stress-strain behavior, determined from a tensile coupon test, was employed in the ABAQUS run. The rate sensitive model is illustrated in Fig. 11. It differs from Fig. 10 only in that a coarser mesh was used in representing the tube. Here, the solution converged using 69 cubic beam elements, reflecting the fact that including strain rate effects 'stiffens' the tube. ABAQUS accounts for the effects of strain rate by utilizing the empirical relationship [6-8]: ~pl=D t ~ _ l
for
~>~tr o,
(7)
where ~PJ is the equivalent plastic strain rate, 5 is the equivalent stress and D and p are material parameters. For mild steel, D and p are typically taken as 40.4 and 5.0, respectively, and these values were inputted to ABAQUS. Figures 12 and 13 demonstrate the forces which the sledge and backup plate, respectively, exerted on the tube as functions of time. These forces were obtained by keeping track of the gap forces. Both figures show the experimental (light solid line) curves and the numerical results, both with (dark solid line) and without (dashed dark line) strain rate. It is seen that superior results were obtained by incorporating the effects of strain rate on material behavior. Ignoring this effect 'softens' the system and results in an underestimation of its energy dissipation capacity. The sledge's velocity profiles are shown in Fig. 14. Both the strain rate ABAQUS and experimental results predict the sledge's rebound to occur at approximately 9.5 ms.
134
,1 R. VEILLETTE and J. F'. CARNE~ I11
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Im'kup pl~e
/
nodes = ] de~'l nodes st 2 degree.'
N .odest~ mdel ~hettbe
nodes I de~ nodes eL 2 degr'eel
fl~o-t~
pp el~nt,,
FIG. 11. ABAQUS discretization impact loading model including strain rate effects.
DISCUSSION
The eight braced tube experiments involved impact velocities which varied over the narrow range of 30.8-33.3 mph. In view of the weak dependence of flow stress on strain rate (see equation (8) with n = 5), one would expect similar strain rate effects to exist in the eight cases. Figure 15 shows the variation in dimensionless initial collapse load with the double bracing angle under both quasi-static and impact loading conditions. The quasi-static analytical curves and associated experimental values are taken from Refs [3] and [4], respectively. The impact collapse values were taken from Fig. 8. Although quasi-static collapse loads are clearly sensitive functions of the bracing angle, the impact collapse loads are essentially independent of the bracing orientation. This anomalous behavior can be explained by considering the deformation patterns that occur in the early stages of the impact event. Under quasi-static conditions, the early collapse mechanisms are symmetric with respect to the horizontal diameter of the tube. This is not the case under impact loading conditions, as can be seen from Fig. 7. In fact, a localized region of plastic deformation adjacent to the impacted area dominates the early time response of the tube. At later times (and deformations) the impact and quasi-static mode shapes coalesce. Figure 16(a) shows the early stages of the collapse mode for an impact loading. The left figure shows results obtained from ABAQUS and the right figure illustrates an idealized model. It is clear that, at the beginning of the event, neither the bracing nor the majority of the tube experiences any strain. The response is localized, and independent of the bracing orientation. Figure 8 depicts the sledge/tube force-displacement behavior for various bracing angles. As O varies from 0° to 30°, the unstable dynamic post-collapse behavior decreases. This regime is followed by a strain hardening region producing an increase in load-deflection behavior. For ® 7>30° the unstable dynamic post collapse behavior is significantly larger and is also followed by an increase in load deflection response due to strain hardening.
Collapse of braced tubes under impact loads
135
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6 13 EVENT TIME, ~4SEC
FIG. 12. Sledge/tube force history with strain rate ( ~ )
experimental (
10
12
without strain rate ( - - - )
and
).
The unstable post-collapse behavior can be explained by comparing the top three hinge locations for the impact collapse mechanism (independent of ®) to the quasi-static collapse mechanism (dependent on 19). Figure 16(b), [-3], shows that as 19 increases from 0° toward 30°, the arc distance between the top three hinges for the quasi-static mechanism decreases. The initial dynamic mechanism therefore approaches the quasi-static mechanism and the unstable response region decreases. For large bracing angles, however, the quasi-static collapse mechanisms progress from a ten-hinge one to an eight-hinge and finally to a fourhinge configuration as 19 increases. This significant difference in character between the quasistatic and impact collapse modes accounts for the large unstable load-deflection responses obtained when 30° ~<19 ~<90° as the initial impact mode transitions into the quasi-static one. It is of interest to note that when 19 ~>45 °, the braced tube's behavior is identical to that of an unbraced tube since the bracing system is ineffective in compression. It is also noteworthy that there is some variation in the initial collapse forces that were measured on the interface of the tube and backup structure, as illustrated in Fig. 9. This variation occurs for 19 >t 35° and is caused by inertia (stress wave) effects which become significant owing to the large difference in the quasi-static and impact collapse modes for these cases. Figures 12-14 illustrate the importance of accounting for strain rate effects in the numerical modeling of impact problems. ABAQUS does an excellent job of predicting the experimental results when strain rate effects are included. Computer runs of this type are quite lengthy, however, and modeling decisions must always be made to strike a balance between accuracy and budgetary considerations. The close agreement of the numerical modeling and experimental results in this case tends to create confidence in the accuracy of the experimentally determined data.
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KVFJ~ TIME, HSEC FIG. 13. Backup/tube force history with strain rate (
), without strain rate ( - - - ) and experimental ).
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FIG. 14. Sledge velocity history with and without strain rate effects.
Collapse of braced tubes under impact loads
137
8.0
5.0 D e x p e r i m e n t a l impact © experimental-quasi-static --analytical predictions
4.0
% ].0
J 2.0
1.0
0.0 0.0
10
20
30
40
50
~qACI~~61.E, DE~qEES FIG. 15. Dimensionless initial collapse loads vs bracing angle for quasi-static and impact loading.
~
__ Independent of bracing angle
\\\\\" \ \ \ \ \ \
(a) I0 hinge
0 ° _< 0_< 30 °
8 hinge
4 hinge
30 ° < 0 < 45 °
0 >_ 45 °
(b) FIG. 16. Collapse modes for: (a) early stages of impact loading, and (b) quasi-static loading.
138
J, R. VEILLETT[:and J. F. CARNI-x, I11 CONCLUSIONS
This e x p e r i m e n t a l a n d n u m e r i c a l i n v e s t i g a t i o n i n t o t h e effect of i m p a c t l o a d i n g c o n d i t i o n s on the d y n a m i c r e s p o n s e of b r a c e d t u b e s has s h o w n t h a t the initial c o l l a p s e b e h a v i o r is largely i n d e p e n d e n t of the b r a c i n g o r i e n t a t i o n . T h i s b e h a v i o r u n d e r i m p a c t has i m p o r t a n t r a m i f i c a t i o n s in the d e s i g n of i m p a c t a t t e n u a t i o n d e v i c e s b e c a u s e c o n s t r a i n t s are usually i m p o s e d o n a c c e p t a b l e d e c e l e r a t i o n levels. A d e s i g n p r o c e d u r e w h i c h a c c o u n t s for strain r a t e effects b u t i g n o r e s t h e l o c a l i z e d n a t u r e of t h e e a r l y t i m e r e s p o n s e c a n result in a n excessively stiff s y s t e m c h a r a c t e r i z e d by u n a c c e p t a b l e o c c u p a n t risk p a r a m e t e r s . Acknowledgemenr - This material is based upon work supported by the National Science Foundation under Grant
No. MSM-8714346.
REFERENCES 1. J. F. CARNEYIII, C. E. DOUGANand M. W. HARGRAVE,The Connecticut Impact Attenuation System. TRB Res. Rec. 1024, 41 50 0985). 2. J. F. CARNEYIII and C. E. DOUGAN,Summary of the results of the crash tests performed on the Connecticut Impact Attenuation System, FHWA-CT-RD-876-1-83-13 (1983). 3. S.R. R ElD, S. L. K. DREWand J. F. CARNEYIII, Energy absorbing capacities of braced metal tubes. Int. J. Mech. Sci, 25, 649-667 (1983). 4. J. F. CARNEYIII and J. R. VEILLETTE,Impact response and energy dissipation characteristics of stiffened metallic tubes. Proc. Int. Conj'. Struct. Impact and Crashworthiness, pp. 564-575 (1984). 5. HIBaITT, KARLSSONand SORENSEN,Inc., Providence, RI, 02906, ABAQUS Manuals: Vol 1, User's Manual; Vol. 2, Theory Manual; Vol. 3, Example Problems Manual; Vol. 4, Systems Manual. 6. G.R. COWPERand P. S. SYMONDS,Strain-hardening and strain-rate effects in the impact loading of cantilever beams. Tech. Report No. 28, Office of Naval Research, Contract Nonr-562-(10) NR-064-406 (1957). 7. M.J. MANJOINE,Influence of rate of strain and temperature on yield stresses of mild steel. J. appl. Mech. 11, A211 A218 (1944). 8. R.J. ASPDENand J. D. CAMPBELL,The effect of loading rate on the elasto-plastic flexure of steel beams. Proc. R. Soc. A290, 266 285 (1966).
0734 743X/88$3.00+0.00 © 1988Pergamon Press plc
Printed in Great Britain
COLLAPSE OF BRACED TUBES U N D E R IMPACT LOADS J. R. VEILLETTE and J. F. CARNEY III Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, U.S.A. (Received 21 December 1987)
Summary--The dynamic response of diametrically braced metallic tubes under impact loading conditions is investigated both experimentally and numerically. The effects of strain rate and the initial collapsing mode configuration on the resulting force (acceleration) levelsat the impacted end of the tube are discussed. Differences in energy dissipation characteristics of the braced tubes under quasi-static and impact loading conditions are described as functions of the bracing orientation.
NOTATION a ~i
striker's acceleration accelerationof center of gravity of tube d(t) displacementof striker at time t D experimentalconstant Fc striker's quartz cell force output Fs sledge/tube force m mass of quartz cell and load distribution plate mr mass of striker and distribution plate rh mass of tube p experimentalconstant t time At time increment v(t) striker's velocity at time t Vs initialimpact velocity ~pl equivalent plastic strain rate ~" equivalent stress
INTRODUCTION Metallic tubes which are stiffened across their diameters have recently been employed in impact attenuation devices used in highway safety applications [1, 2]. The diametrical bracing causes the tubes' d y n a m i c response to be a function of the direction of lateral loading. In the application described in [1], thin steel straps (1/8 by 5 in. in cross-section) are inserted into tubes which are 4 ft in diameter. These straps are effective only in tension and are oriented so that they buckle under head-on impact conditions. This behavior allows the crash cushion to exhibit the large collapsing stroke that is required under head-on impact conditions in order to capture the vehicle in a dynamically acceptable manner. U n d e r side impact conditions near the rear of the device, however, the diametrical bracing system is actuated, significantly increasing the stiffness of the tubular cluster. This increased stiffness enables the crash cushion to develop the lateral force required to redirect the errant vehicle away from the rigid hazard. Most of the research performed to date on stiffened tubes subjected to large deformations has been limited to quasi-static loading conditions. In [3], load b o u n d i n g techniques were employed to examine the effect of various orientations of single and symmetric double bracing on the initial collapse loads of tubes. These analytical predictions were c o m p a r e d with experimental data in [4]. This reference also details the post-collapse behavior of the stiffened tubes, and this is sometimes characterized by asymmetric m o d e shapes. Metallic tubes which are employed in crash cushions to dissipate kinetic energy are 125
126
J . R . VFILLF'II't and J. F. CARNkY Ill
subjected to impact loading conditions. Strain rate effects play a significant role in increasing the energy dissipation capacities of such impulsively loaded tubes compared with equivalent tubes loaded quasi-statically. The influence of strain rate effects, however, does not account for the total increase in initial collapse load and the associated energy dissipation which typically occurs under impact loading. In addition to the influence of strain rate, tubes subjected to lateral impact loading conditions exhibit dynamic collapse configurations during the early phases of the impact event which are quite distinct from those of their quasistatic counterparts. This behavior is of great importance in the design of impact attenuation devices since the deceleration levels to which the occupant of an errant vehicle is subjected must remain within allowable limits. In this paper, the impact behavior described above is investigated both experimentally and numerically. The results point out the dramatic differences in quasi-static and impact load-deformation behavior that can occur when two collapse modes differ significantly. The extent of this difference is a function of the orientation of the bracing with respect to the applied loads.
IMPACT
EXPERIMENTS
The experiments were conducted on the specially constructed impact apparatus, described in Fig. 1. The rig includes a 3.40 lb sledge which is fired from a gas gun whose velocity is controlled by setting the gas tank's priming pressure. The launching of the sledge is software controlled through a data acquisition's digital input/output card. Two light beams, set up 5 in. apart and perpendicular to the striker's trajectory, are employed to determine its impact velocity. The exit door is constructed in such a way as to prevent the striker from reentering the guide-rail area after rebounding off the impacted tube and possibly damaging the electronic instrumentation. A high speed camera, operating at 6000 frames s- 1, records the displacement-time history of the events. Force-time histories are obtained by the use of quartz load cells located on the impacting mass and the backup plate. The charges from these load cells are converted in the amplifier to voltages and recorded in digital form on a personal computer at 31.4 kHz through a data acquisition system. The output from the striker's quartz cells must be manipulated in order to obtain the desired sledge/tube force history. Consider the free-body diagram shown in Fig. 2 where Fs is the unknown. The acceleration of every point on the striker assembly is assumed to be the same. Applying Newton's second law of motion:
Fs-Fc=ma,
(1)
since
Fs
a = --, mT
(2)
it follows that Fs =
F~,
(3)
where a = striker's acceleration, Fc = striker's quartz cell force output, Fs = sledge/tube force, m --- mass of the quartz cell load distribution plate (0.0140 slugs), and m t = mass of the striker and distribution plate (0.120 slugs). Substituting the above values of m and tnt into equation (3) yields F~ - 0.883F~.
t4)
Eight 4-in. diameter mild steel tubular specimens were prepared for testing. All specimens were 2 in. in height and had wall thicknesses of 0.083 in. They were annealed by heating at 900°C for 20 rain and then slowly cooled to room temperature inside a closed oven. High strength wire (0.013 in. diameter) was wound through diametrically opposed holes to form
Collapse of braced tubes under impact loads
,
127
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J i
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fi °,-i
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128
R. V|:ILL[ IIE and J. F. C,XR'~[Y Ill
,I.
U-- Speeiaen Striker r
"
~uartz cells
- - Otstributi0nplate FIG. 2. Striker d i s t r i b u t i o n plate.
TABLE l . TENSION BRACING ANGLES AND VELOCITIES FOR IMPACT LOADING TESTS
0
(degrees)
0 I0
15 20 25 30 ]5
llflllll/l/llllll/l/lllllll
O0 (unbraced)
b~kup plete
Initial $1.edBe Yel0citq (MPH) 33.3 31.5 32.1 31.5 31.7 32.3 ]3.0 30.8
the tubes' tension bracing; 25 loops were typically used. The various bracing angles and impact velocities are presented in Table 1. Examples of typical force-time histories are presented in Fig. 3 as dashed curves for the 15° bracing case. The impacting sledge/tube output is shown in Fig. 3(a), and the backup plate load cell response is given in Fig. 3(b). Solid curves have been fitted to these virgin data by averaging neighboring peak values. This was done to facilitate the post-processing of the data. The maximum difference in area between the virgin and fitted curves in any of the 16 cases considered (two load cell outputs for eac;~ of eight bracing angle cases) was 6 ~o. A film motion analyser was used to investigate the high speed filmed event. The film analyser is capable of frame-by-frame analysis, has a digitizing screen with an electronic grid, a cursor for co-ordinate data input, and is interfaced with a personal computer. The displacement-time history for the 15 ° bracing case is shown in Fig. 4. Two curves are shown on the figure. The jagged curve was obtained with the film analyser and the smooth curve by integrating the fitted curve to the sledge/tube force history results as described below. The leading edge of the striker has an acceleration of Fs/mT. Assuming rectilinear motion, the sledge's velocity is: v(t)
= v~ - (t F~(t)_dt,
(5)
do mT where vs = initial impact velocity and v(t)= sledge's velocity at time t. Integrating again, the displacement of the sledge at time t, d(t), may be found from the expression d(t)
= fl v(t)dt.
(6)
The differences between the results of the two methods are quite small, demonstrating a reasonable curve fitting procedure to the virgin force data and providing a check on the film analysis results. The force and displacement time histories are coupled to produce the force-displacement
Collapse of braced tubes under impact loads
129
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(b) FIG. 3. 15° bracing virgin (dashed) and fitted (solid) applied load history results: (a) sledge/tube (front), and (b) backup/tube (rear). results for each test. The fitted force history curves were used in compiling these results. This was done because different amplifier filters would result in unique virgin curves, but they would produce approximately the same fitted curves. In addition, the smooth curves allowed the coupling process to be programmed more efficiently. The energy is of main importance, and the virgin data yield essentially the same results. As an example, the coupled results for both the front (sledge side) and rear (backup side) outputs are given in Fig. 5. Recalling Hamilton's principle, and considering the two systems shown in Fig. 6, it follows that the areas under the front and rear force displacement curves should be approximately equal. The maximum difference between the areas was 8 %, producing confidence in the experimental results and evaluation procedures.
130
J.R. VIqLLEIII- and J. F. CARNt~Y lI1
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2
i 4
~ 6
i
B
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i I~
EVENT TIME, MSEC
FIG. 4. 15:' bracing motion analyser (jagged) and direct integration (smooth) tube displacement history results.
m .D
S
d
Front
121 W
~E
DISPLACEI'ENT. INCHES
FIG. 5. 15'~ bracing force-displacementresults for front and rear load cells. In the early portion of a typical impact event, the force in the front of the tube is larger than the force in the rear of the tube. This is because the acceleration of the tube's center of gravity during the early phases of the impact event is in the direction of the sledge velocity vector. Later in the event, the acceleration of the tube's center of gravity is in the opposite direction. The film analyser was used to obtain the shapes of the tubes at different levels of deformation. The results are shown in Fig. 7 where the tops of all collapse modes represent the impacting force. Figures 8 and 9 present the sledge and backup plate load-displacement responses, respectively, for the eight double bracing angles considered. ANALYSIS A detailed force-time analysis of the impact event is required, and this can only be done numerically. ABAQUS, a powerful finite element software package [5], was selected for this purpose.
Collapse of braced tubes under impact loads
]
131
Fa(6) ~mTv, l / f i l l
/
+o/
where = mass of the tube = accelaration of the tube's center of gravity
/ F.dx,~/ FBdx FIG. 6. Relationbetweenfrontand rear load histories.
m 0.125 1
0.1875
m
H
B
m
mFr Ii
m mEmlmm JIEimmmm
0.25 0.3125 0.375 0.50 0.6875 FIG. 7. Collapse mode shapes.
The system response is obtained by specifying time parameters. ABAQUS allows the user to either fix the time increment or let the software choose one through its automatic time stepping process. Herein, all cases were run implementing the automatic time stepping option. This option requires inputting a minimum and maximum time step as well as the overall desired total time. It is possible to satisfy the discretized equilibrium equations at the beginning and end of the time step but overlook important behavior in between. In order to control the accuracy of the results, ABAQUS introduces a half-step residual error parameter. The intermediate time t + At~2 is used to compute the equilibrium residual error. This residual error is then compared to the inputted acceptable half-step residual error. If unacceptable, the time-step is reduced and the process repeated. In the following example, ABAQUS is employed to model the impact experiment for the unbraced tube by first ignoring and then accounting for the effects of strain rate. Figure 10 shows the divisions implemented in describing strain rate independent cases. They consist of 95 cubic beam elements, 30 sledge nodes and 30 backup plate nodes. Thirty gap elements were used to model both the tube to backup plate and tube to sledge interaction. These gap
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133
Collapse of braced tubes under impact loads utot
l°~zo~
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mt
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T-I
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prate nodes
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nodes et
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teege
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Ill
mt 4 m
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l ° 20
nodes ~t.
FIG. 10. ABAQUS discretization impact loading model without strain rate effects.
elements provide a compressive force only if the gap distance is zero. The actual stress-strain behavior, determined from a tensile coupon test, was employed in the ABAQUS run. The rate sensitive model is illustrated in Fig. 11. It differs from Fig. 10 only in that a coarser mesh was used in representing the tube. Here, the solution converged using 69 cubic beam elements, reflecting the fact that including strain rate effects 'stiffens' the tube. ABAQUS accounts for the effects of strain rate by utilizing the empirical relationship [6-8]: ~pl=D t ~ _ l
for
~>~tr o,
(7)
where ~PJ is the equivalent plastic strain rate, 5 is the equivalent stress and D and p are material parameters. For mild steel, D and p are typically taken as 40.4 and 5.0, respectively, and these values were inputted to ABAQUS. Figures 12 and 13 demonstrate the forces which the sledge and backup plate, respectively, exerted on the tube as functions of time. These forces were obtained by keeping track of the gap forces. Both figures show the experimental (light solid line) curves and the numerical results, both with (dark solid line) and without (dashed dark line) strain rate. It is seen that superior results were obtained by incorporating the effects of strain rate on material behavior. Ignoring this effect 'softens' the system and results in an underestimation of its energy dissipation capacity. The sledge's velocity profiles are shown in Fig. 14. Both the strain rate ABAQUS and experimental results predict the sledge's rebound to occur at approximately 9.5 ms.
134
,1 R. VEILLETTE and J. F'. CARNE~ I11
I=dtuff-t0-tul~g~ elts~ts
Im'kup pl~e
/
nodes = ] de~'l nodes st 2 degree.'
N .odest~ mdel ~hettbe
nodes I de~ nodes eL 2 degr'eel
fl~o-t~
pp el~nt,,
FIG. 11. ABAQUS discretization impact loading model including strain rate effects.
DISCUSSION
The eight braced tube experiments involved impact velocities which varied over the narrow range of 30.8-33.3 mph. In view of the weak dependence of flow stress on strain rate (see equation (8) with n = 5), one would expect similar strain rate effects to exist in the eight cases. Figure 15 shows the variation in dimensionless initial collapse load with the double bracing angle under both quasi-static and impact loading conditions. The quasi-static analytical curves and associated experimental values are taken from Refs [3] and [4], respectively. The impact collapse values were taken from Fig. 8. Although quasi-static collapse loads are clearly sensitive functions of the bracing angle, the impact collapse loads are essentially independent of the bracing orientation. This anomalous behavior can be explained by considering the deformation patterns that occur in the early stages of the impact event. Under quasi-static conditions, the early collapse mechanisms are symmetric with respect to the horizontal diameter of the tube. This is not the case under impact loading conditions, as can be seen from Fig. 7. In fact, a localized region of plastic deformation adjacent to the impacted area dominates the early time response of the tube. At later times (and deformations) the impact and quasi-static mode shapes coalesce. Figure 16(a) shows the early stages of the collapse mode for an impact loading. The left figure shows results obtained from ABAQUS and the right figure illustrates an idealized model. It is clear that, at the beginning of the event, neither the bracing nor the majority of the tube experiences any strain. The response is localized, and independent of the bracing orientation. Figure 8 depicts the sledge/tube force-displacement behavior for various bracing angles. As O varies from 0° to 30°, the unstable dynamic post-collapse behavior decreases. This regime is followed by a strain hardening region producing an increase in load-deflection behavior. For ® 7>30° the unstable dynamic post collapse behavior is significantly larger and is also followed by an increase in load deflection response due to strain hardening.
Collapse of braced tubes under impact loads
135
I
I
!11 tl i I11 '( 2
! 4
6 13 EVENT TIME, ~4SEC
FIG. 12. Sledge/tube force history with strain rate ( ~ )
experimental (
10
12
without strain rate ( - - - )
and
).
The unstable post-collapse behavior can be explained by comparing the top three hinge locations for the impact collapse mechanism (independent of ®) to the quasi-static collapse mechanism (dependent on 19). Figure 16(b), [-3], shows that as 19 increases from 0° toward 30°, the arc distance between the top three hinges for the quasi-static mechanism decreases. The initial dynamic mechanism therefore approaches the quasi-static mechanism and the unstable response region decreases. For large bracing angles, however, the quasi-static collapse mechanisms progress from a ten-hinge one to an eight-hinge and finally to a fourhinge configuration as 19 increases. This significant difference in character between the quasistatic and impact collapse modes accounts for the large unstable load-deflection responses obtained when 30° ~<19 ~<90° as the initial impact mode transitions into the quasi-static one. It is of interest to note that when 19 ~>45 °, the braced tube's behavior is identical to that of an unbraced tube since the bracing system is ineffective in compression. It is also noteworthy that there is some variation in the initial collapse forces that were measured on the interface of the tube and backup structure, as illustrated in Fig. 9. This variation occurs for 19 >t 35° and is caused by inertia (stress wave) effects which become significant owing to the large difference in the quasi-static and impact collapse modes for these cases. Figures 12-14 illustrate the importance of accounting for strain rate effects in the numerical modeling of impact problems. ABAQUS does an excellent job of predicting the experimental results when strain rate effects are included. Computer runs of this type are quite lengthy, however, and modeling decisions must always be made to strike a balance between accuracy and budgetary considerations. The close agreement of the numerical modeling and experimental results in this case tends to create confidence in the accuracy of the experimentally determined data.
136
.I R. V~i[t~]lrt:and J. F. Car~l-~ I11 i i i i i i I i
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), without strain rate ( - - - ) and experimental ).
(
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TINIE, NISEC
FIG. 14. Sledge velocity history with and without strain rate effects.
Collapse of braced tubes under impact loads
137
8.0
5.0 D e x p e r i m e n t a l impact © experimental-quasi-static --analytical predictions
4.0
% ].0
J 2.0
1.0
0.0 0.0
10
20
30
40
50
~qACI~~61.E, DE~qEES FIG. 15. Dimensionless initial collapse loads vs bracing angle for quasi-static and impact loading.
~
__ Independent of bracing angle
\\\\\" \ \ \ \ \ \
(a) I0 hinge
0 ° _< 0_< 30 °
8 hinge
4 hinge
30 ° < 0 < 45 °
0 >_ 45 °
(b) FIG. 16. Collapse modes for: (a) early stages of impact loading, and (b) quasi-static loading.
138
J, R. VEILLETT[:and J. F. CARNI-x, I11 CONCLUSIONS
This e x p e r i m e n t a l a n d n u m e r i c a l i n v e s t i g a t i o n i n t o t h e effect of i m p a c t l o a d i n g c o n d i t i o n s on the d y n a m i c r e s p o n s e of b r a c e d t u b e s has s h o w n t h a t the initial c o l l a p s e b e h a v i o r is largely i n d e p e n d e n t of the b r a c i n g o r i e n t a t i o n . T h i s b e h a v i o r u n d e r i m p a c t has i m p o r t a n t r a m i f i c a t i o n s in the d e s i g n of i m p a c t a t t e n u a t i o n d e v i c e s b e c a u s e c o n s t r a i n t s are usually i m p o s e d o n a c c e p t a b l e d e c e l e r a t i o n levels. A d e s i g n p r o c e d u r e w h i c h a c c o u n t s for strain r a t e effects b u t i g n o r e s t h e l o c a l i z e d n a t u r e of t h e e a r l y t i m e r e s p o n s e c a n result in a n excessively stiff s y s t e m c h a r a c t e r i z e d by u n a c c e p t a b l e o c c u p a n t risk p a r a m e t e r s . Acknowledgemenr - This material is based upon work supported by the National Science Foundation under Grant
No. MSM-8714346.
REFERENCES 1. J. F. CARNEYIII, C. E. DOUGANand M. W. HARGRAVE,The Connecticut Impact Attenuation System. TRB Res. Rec. 1024, 41 50 0985). 2. J. F. CARNEYIII and C. E. DOUGAN,Summary of the results of the crash tests performed on the Connecticut Impact Attenuation System, FHWA-CT-RD-876-1-83-13 (1983). 3. S.R. R ElD, S. L. K. DREWand J. F. CARNEYIII, Energy absorbing capacities of braced metal tubes. Int. J. Mech. Sci, 25, 649-667 (1983). 4. J. F. CARNEYIII and J. R. VEILLETTE,Impact response and energy dissipation characteristics of stiffened metallic tubes. Proc. Int. Conj'. Struct. Impact and Crashworthiness, pp. 564-575 (1984). 5. HIBaITT, KARLSSONand SORENSEN,Inc., Providence, RI, 02906, ABAQUS Manuals: Vol 1, User's Manual; Vol. 2, Theory Manual; Vol. 3, Example Problems Manual; Vol. 4, Systems Manual. 6. G.R. COWPERand P. S. SYMONDS,Strain-hardening and strain-rate effects in the impact loading of cantilever beams. Tech. Report No. 28, Office of Naval Research, Contract Nonr-562-(10) NR-064-406 (1957). 7. M.J. MANJOINE,Influence of rate of strain and temperature on yield stresses of mild steel. J. appl. Mech. 11, A211 A218 (1944). 8. R.J. ASPDENand J. D. CAMPBELL,The effect of loading rate on the elasto-plastic flexure of steel beams. Proc. R. Soc. A290, 266 285 (1966).