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Solution stability in the dynamic collapse of square aluminium columns
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International Journal of Impact Engineering 34 (2007) 348–359 www.elsevier.com/locate/ijimpeng
Solution stability in the dynamic collapse of square aluminium columns S.A. Meguid, M.S. Attia, J.C. Stranart, W. Wang Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ont., Canada M5S 3G8 Received 27 April 2004; received in revised form 14 September 2005; accepted 18 September 2005 Available online 9 November 2005
Abstract The collapse of empty square aluminium columns is re-examined using dynamic nonlinear elasto–plastic finite element analysis. Earlier work on this topic did not address the influence of FE model reduction on possible numerical anomalies resulting from the use of different processing platforms. Single- and double-sided impact conditions are also considered under both constant and varying initial impact speeds. Specifically, the objective of the study is to investigate the influence of numerical anomalies that are typically overlooked in the literature and the dramatic role they may play in dictating the dynamic mode of collapse. To that end the study investigates the influence of specific symmetric boundary and initial conditions on the resulting mode of collapse and determines the effect of the use of different platforms on the results. The work was carried out using LS-DYNA explicit finite element solver and considers generic geometries that are typically used in industrial applications. The results show clearly that the use of symmetry conditions in FE modelling must be used with proper care and that careful experimental work should be carried out to validate the finite element predictions. r 2005 Elsevier Ltd. All rights reserved. Keywords: Dynamic collapse; Thin-walled; Impact; Crashworthiness; Numerical errors
1. Introduction The Canadian Government and Transport Canada have set a target of a 30% reduction in the average number of road users killed and seriously injured during the 2008–2010 period. Transport Canada’s efforts to improve motor vehicle safety regulations focus on frontal-crash and lateral-impact protection. This, together with a range of environmental concerns and social pressures backed by legislation, has led, and will continue to lead to highly innovative designs, involving lighter materials such as aluminium or magnesium alloys, and cellular materials. Thin-walled extrusions are extensively used to improve vehicle crashworthiness in weak areas such as bumpers, side doors and roof. The ratio of the length to section dimensions of these extrusions is such that it typically promotes failure by buckling. Buckling of thin plates and members has been studied since Euler’s Corresponding author. Tel./fax: +1 416 978 5741.
E-mail address: [email protected] (S.A. Meguid). 0734-743X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2005.09.001
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work on buckling of columns in 1744. Local instability has been found to be the main reason for buckling of thin plates and members. Numerous researchers have investigated the local instability in plates using elastic, elasto–plastic, and plastic hinge analysis methods. For a detailed account of the literature, see the extensive reviews by Jones in [1,2]. The earliest analytical treatment of the quasi-static axial collapse of thin-walled cylindrical columns was conducted by Alexander [3], and Pugsley and Macaulay [4] under the assumptions of perfect folding pattern and rigid plastic material model. They employed an energy balance criterion between the external work done by the axial force with the energy dissipated being into forming plastic hinges at the folds. Abramowicz and Jones [5] incorporated effective crushing distance and varying, as opposed to mean, membrane strains into Alexander’s solution. Their experimental investigations considered both square and circular steel sections for a wide range of lengths and cross-section dimensions using a drop hammer apparatus. Based on the test results and observations, approximate theoretical predictions were developed for the progressive axial crushing of columns using a kinematically admissible plastic analysis. Wierzbicki and Abramowicz [6,7] developed a self-consistent method that describes the crushing behaviour of thin-walled multi-corner structures based on the energy balance postulate. They identified two distinct modes of collapse: (i) quasi-inextensional mode, and (ii) quasi-extensional mode. The former mode is characterised by large rotations (plastic hinge lines) and small axial strains near the column corners, while it is largely governed by membrane stretching in the latter. Similar strategy was followed in [8] where two idealised collapse elements comprising four deformation modes were introduced: one symmetric mode, two asymmetric modes, and an extensional mode of deformation. These asymmetric deformation modes were predicted theoretically and confirmed in the experimental tests. Further enhancements were reported in [9,11] through extending the range of geometrical and material parameters and impact energies beyond those reported in [5]. Different tempering conditions, wall thickness values, and test velocities were examined in [9]. The impact velocity ranged from 8 to 20 m/s using a 56 kg projectile. Quasi-static tests were also carried out to develop a relationship between the dynamic and quasi-static behaviour. The rigorous assessment showed that there is a very good agreement between theoretical predictions and experimental results. Recent work by Jensen et al. [10] examined the dependence of global buckling on the local and global slenderness ratios. Kitagawa et al. [12] analysed axial crushing behaviour of side members in a frontal collision of an automobile. The mechanism of axial collapse was described as being local buckling occurring at the column’s weakest section, along with slight waviness appearing in the column wall. This work was further extended by Han and Park [13], where they studied the crush behaviour of a square column subjected to oblique loads. Their results showed that there is a critical load angle for transition from the axial collapse mode to the bending collapse mode. This critical angle was found to be approximately 71. The value of the mean crush load drops to about 40% of the mean crush load in pure axial collapse beyond the critical load angle. More recently, Reyes et al. [14] compared experimental and LS-DYNA simulations under oblique loadings which induced global buckling. While the force response was captured by the simulations using initial imperfections, the experimentally observed buckling modes were more closely reflected by the simulations which assumed a perfect geometry. In order to adopt these light structures for applications involving safety of the public, a comprehensive understanding of their behaviour under impact loading conditions is necessary. This can be achieved by conducting high fidelity computer models and experimentally validating these models. It is therefore the objective of this study to: (i) carry out nonlinear elasto–plastic finite element analysis of the dynamic collapse of square aluminium columns; (ii) critically assess the influence of enforcing symmetry boundary conditions, component fixation and system initial conditions on the modelling of the columns and the resulting progressive mode of collapse; and (iii) investigate the influence of numerical anomalies that are typically overlooked in the literature and the dramatic role they may play in dictating the dynamic mode of collapse. 2. Finite element modelling 2.1. Geometry and properties of square aluminium columns As a result of earlier published work and range of practical applications, the following dimensions were selected for the aluminium columns of the baseline case: length of tube (L) 300 mm, width (c) 75 mm, and
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Table 1 Mechanical properties of aluminium 6061-T4 (after [15]) Engineering strain (%)
Engineering stress (MPa)
0 0.23 0.28 0.97 4.56 7.59 13.44 16.33 20.11
0 138.18 152.83 163.09 196.29 216.80 237.30 242.68 244.14
thickness (h) 2.5 mm; which yields a length-to-width ratio (L/c) of 4 and width-to-thickness ratio (c/h) of 30. Existing work in the literature [11] indicates that with the choice of these geometrical ratios, stable progressive collapse will ensue under both quasi-static and dynamic loading conditions. In addition, columns of longer length (L ¼ 600 mm) were also considered. This would yield a length-to-width ratio of 8, which is near the transition region between folding and global bending failure. Material properties of the aluminium tubes were those of Aluminium alloy 6061 in T4 temper condition. Table 1 illustrates the mechanical properties of this alloy. The T4 condition was specifically chosen in order to ensure sufficient ductility to prevent ductile rupture due to plastic instabilities during loading. 2.2. FE model configuration LS-DYNA explicit FE solver [15] was implemented in the current study. Belytschko–Tsay reduced integration shell element with six degrees of freedom (DOF) per node and five integration points through thickness were used to model the column material. A piecewise-linear plasticity material law was implemented in modelling the column. The striker was modelled using solid elements made of a rigid material model available in LS-DYNA library. Fig. 1 depicts a schematic of the different FE models used in the current study and the associated boundary conditions. They can be summarised as follows: (i) quarter column model (QCM), (ii) half column model (HCM), and (iii) full-column model (FCM). QCM assumes dual symmetry about the y–z and x–y planes and has the advantage of being the most economical in terms of model size and computational resources. The HCM, on the other hand, assumes symmetry about only one plane, while the full column model (FCM) does not employ symmetry conditions and a full model is generated to represent the structure. This is the most expensive model in terms of computational resources. 2.3. Loading and contact conditions Two loading conditions were considered in the current study. In the first, Fig. 2(a), the tube is struck by one rigid platen from one end, while the other end is fully constrained in all DOF. This will be referred to as impact condition I. In the second case, the tube is simultaneously impacted by two rigid platens from both ends, referred to here as impact condition II, Fig. 2(b). It is worth mentioning, however, that the velocity of the striker in these two cases was considered to be constant throughout the impact process. This represents the extreme case of two objects; one of them has a very large mass compared to the other, which yields a conservative estimate of the collapse behaviour of the column. Dynamic simulations at different impact speeds were conducted in order to investigate the different aspects of the collapse process under low and high-velocity impact conditions. Contact surfaces were defined in the different models along two interfaces: (i) tube–striker interface; and (ii) self-contact of the tube walls during collapse. The automatic surface-to-surface contact algorithm available in
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z
z y
y
x x (a)
x
(b)
(c)
Fig. 1. Different FE model configurations: (a) quarter column model (QCM); (b) half column model (HCM), and (c) full column model (FCM).
V V
V (a)
(b)
Fig. 2. Different loading conditions on the column: (a) one-side impact; and (b) two-side impact.
LS-DYNA was used between the tube and the striker, while the single-surface contact algorithm was applied to the tube wall to avoid interpenetration. These contact algorithms consist of placing normal interface springs between all candidate penetrating nodes and the contact surface. The interface stiffness is chosen to be approximately the same order of magnitude as the stiffness of the interface element normal to the interface. 2.4. Triggering mechanism So far, we have limited our discussion to highly idealised finite element models of square aluminium columns and avoided the use of triggers. It is crucially important, however, to realise that these models cannot be realised in real life due to the existence of a number of geometrical and material imperfections Meguid et al. [17] and Hanssen et al. [18] successfully implemented both wedge and sinusoidally shaped geometrical imperfections (triggers) for both empty- and foam-filled aluminium columns under quasi-static crushing. In this study, we implement a wedge-shaped trigger at the top of the columns, which simulates the trigger introduced in the dynamic experiments conducted in [18]. 3. Results and discussion 3.1. Validation of FE models The baseline case for model convergence evaluation is a quarter column model under impact condition I. Both quasi-static and dynamic cases were tested. The striker velocity for the quasi-static case was selected to
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120 Experimental Data Simulation
100
Load [kN]
80 60 40 20 0
(a)
0
50
100
150
Deformation [mm]
(b)
Fig. 3. Comparison of experimental results [9] vs. FE predictions of dynamic collapse of empty columns: (a) final collapse geometry; and (b) instantaneous load–deformation behaviour.
Table 2 Comparison between FE predictions and experimental results [9] of the final axial deformation and mean collapse loads Test
t ¼ 1:8 mm t ¼ 2:0 mm t ¼ 2:5 mm
Simulation
Experiment
ws =we
F smd =F emd
Ws (mm)
F smd ðkNÞ
We (mm)
F smd ðkNÞ
(%)
(%)
133.24 82.98 158.23
21.0 29.7 41.56
129 89 158
21.8 27.9 41.9
103.3 93.2 100.1
96.3 106.5 99.2
be 1 m/s, while a constant velocity of 15 m/s was selected for the dynamic case. It was necessary to implement artificial means to avoid excessively long solution times for the quasi-static case. According to the Courant’s condition [16], increasing the mass of the model by a factor of x2 would increase the minimum solution time step by a factor x. Previous studies [17] illustrated that scaling the mass by a factor of 10 yields desirable computation time and accuracy for quasi-static loading. Sensitivity analyses were conducted and it was found that the solution becomes mesh independent for element size smaller than or equal to 3 mm, which was chosen for the rest of the simulations. Finite element predictions were compared to the experimental results in [9]. These experiments comprised dynamic axial compression of square aluminium sections with dimensions 80 mm 310 mm, at an initial velocity vo ¼ 15:6 m=s. The aluminium alloy used was AA6060 in the T4 temper conditions, with varied wall thickness (1.8, 2.0 and 2.5 mm). Half-column FE models were used in the validation process along with symmetry boundary conditions. The reasons for choosing this model are discussed in the following section. The FE model comprised a total of 5200 shell elements. Fig. 3 depicts a comparison of the final deformed shape and the variation of the instantaneous load between FE predictions and experimental results for a 2.5 mm thick tube. The figure shows reasonable agreement between the experimental results and FE predictions. Very good agreement is observed until the second collapse fold, and the final deformed shape in the two cases is very close. Comparisons were also conducted for 1.8 and 2.0 mm thick tubes. The initial peak load values, subsequent peaks, and plateau regimes are very close in the two cases. Table 2 gives a comparison between the final axial displacements and the corresponding mean loads for FE simulations and tests, where mt is the impactor mass, t is the wall thickness, and wp is permanent axial deformation. Very good correlation can be observed between experimental results and FE predictions.
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Fig. 4. Effect of model configurations on (a) final collapse geometry and; (b).
3.2. Effect of model configuration Fig. 4(a,b), respectively, show the final deformed geometry and the instantaneous load–deformation curves for the three configurations considered at an impact speed of 15 m/s. It was observed that, for all the models, the first collapse fold was formed at the bottom and was of the extensional type. Nevertheless, subsequent collapse patterns were noticeably different, where the symmetric mode of collapse immediately commenced in the quarter and half models, while the extensional mode of collapse prevailed in the full model case. In fact, symmetry of the geometry and applied load may not be sufficient to implement symmetry boundary conditions in the FE model in buckling analysis. For instance, a global mode of collapse of the Euler-type is realised for certain geometrical aspect ratios and/or above specific threshold impact speed values. Nevertheless, quarter column models are, by definition, incapable of simulating this mode of collapse and are therefore not suited for modelling this case of collapse. Full-column models, on the other hand, develop parasitic numerical errors, which may significantly affect the buckling behaviour of the column. These errors are discussed in detail in Section 3.3. In contrast to these model configurations, half-column models do not suffer from these problems, since symmetry conditions are imposed along one plane only, and global buckling mode is therefore allowed in that plane. Furthermore, the symmetry conditions have suppressed the parasitic modes of deformation. 3.3. Numerical anomalies in untriggered models We have shown in Section 3.2 that the choice of a specific model configuration strongly affects the mode of collapse, and has a relatively moderate effect on the load–deformation response and the associated energy absorption levels. This is in agreement with [14]. In this section, the effect of numerical anomalies upon the
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model response is investigated. These are parasitic numerical errors in the finite element model matrices. The full-column model (FCM) was chosen as the baseline case for investigating the effect of these anomalies, since this model is the least constrained model amongst all considered configurations. This ensures that the effect of these anomalies is not suppressed by displacement constraints. There are two sources for these anomalies. The first is concerned with the computational platforms used in the simulations, while the second is concerned with the numerical precision of the solver. The effect of computational precision upon the solution accuracy has been previously illustrated in [19], where the author considered the solution of a one-dimensional polynomial of the sixth order. The polynomial coefficients have been precisely chosen such that a steep gradient exists in the vicinity of one of the roots, which is the situation commonly encountered in buckling problems. It has been shown that there was an error of approximately 150% in evaluating this specific root when using double-precision computations (32 digits). Single precision computations (16 digits) resulted in higher error values. The authors reproduced the solution in [19] and found that the correct solution was realised only through employing variable-precision computations, commonly used in symbolic analysis, with at least 37-digit computations. A consistent convergence scheme to the incorrect solution was found for up to 36 digits of computations. Appendix A illustrates the details of convergence scheme for this example. This simple example illustrated the crucial importance of scrutinizing the validity of buckling solutions, even if convergence was achieved using numerically accurate doubleprecision computations. For example, two-sided impact simulations (Impact Condition II) were conducted on empty aluminium columns at an initial impact velocity of 15 m/s. The simulations were carried out on: (i) 32-bit AMD-Athlon 1.4 GHz processor with 256 MB of RAM running Windows2000 and LS-DYNA V-960; and (ii) Sun Blade 1000 workstation with 1 GB of RAM and a 64-bit 750 MHz UltraSparc III Processor running SunOS 5.8 (Solaris). Furthermore, LS-DYNA V-960 explicit solver employs a single-precision based calculation system. Fig. 5 illustrates the deformed shape of the aluminium columns at different stages of deformation. It is observed that the collapse behaviour of FCM is almost the same for different processors for the first one-third of the simulation, where the collapse is governed by progressive plastic folding. Nevertheless, in subsequent deformation, the Athlon processor solution changes to bending collapse, while the deformed geometry predicted by the UltraSparc III processor remains stable and shows progressive buckling patterns. This divergence between the two processors is evident in both curves at approx. 30% deformation level. Fig. 6 illustrates the variation of the instantaneous collapse load with the collapse distance. It was found that the difference in energy absorption between the two models reached a maximum of 12% at the end of deformation history. These differences can only be due to the numerical calculations and round off differences between the two machines, which resulted in uncontrolled initiation and growth of spurious buckling modes. It is therefore extremely difficult to judge the soundness of either simulation without proper validation with experimental findings. 3.4. Effect of impact conditions Dynamic axial collapse simulations have been conducted for both impact conditions I and II. The effect of impact speeds was also considered. Although it is established that aluminium is strain rate insensitive, it is anticipated that the impact speed will play an important role in dictating the mode of collapse of the columns because of inertial stiffening. Furthermore, as pointed out in Sections 3.2 and 3.3, it has been established that half-column model possesses the ability to realistically model the buckling process of thin-walled sections and does not suffer from the numerical anomalies problems encountered in full column models. Therefore, the simulations were conducted with this FE model configuration. Fig. 7 illustrates the instantaneous load–deformation curves for different FE model configurations for impact speeds of 1, 10, 15, and 30 m/s, respectively. The collapse behaviour at 1 m/s is characterised by progressive folding of the wall as evidenced by the uniform shape of the instantaneous load–deformation behaviour of the structure for all FE model configurations. In contrast, inertial stiffening significantly affected the collapse behaviour for impact speed 30 m/s such that the collapse pattern is irregular and the folds were steeply developed and collapsed. Furthermore, it has been observed that the folding mode in this case is of the extensional type, compared to symmetric mode for an impact speed of 1 m/s.
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AMD Processor
355
Sun Spark Processor
Deformation 94.5 mm
Deformation 270 mm
Deformation 391.5mm
Deformation 450 mm
Fig. 5. Comparison of collapse geometry at different stages as obtained from: AMD processor and Sun Sparc processor.
160 PC
Unix
Load [kN]
120
80
40
0
0
100
200 300 Deformation [mm]
400
500
Fig. 6. Comparison between the instantaneous load history using different processors for untriggered columns.
Two-sided impact simulations in the current study were mainly concerned with the characteristics of the collapse mode in impact condition II. Attention was therefore devoted to the dynamic collapse at an initial impact speed of 15 m/s. Square aluminium sections having dimensions 600 mm 75 mm 2.5 mm were
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Fig. 7. Instantaneous load–deformation behaviour for different impact speeds.
Striker I
Striker I
Striker II
Striker II (a)
(b)
Fig. 8. Different model configurations under impact condition II.
specifically chosen to reveal the effects of two-sided impact on the mode of the column since the slenderness ratio L=c ¼ 8 lies in the transition region between progressive buckling and global bending modes of collapse. Both full-length and half-length models with symmetry conditions at the mid-length were considered as depicted in Fig. 8. The respective instantaneous load–deformation response for both half- and full-length model configurations are shown in Fig. 9. It is observed that there is a very close match between the response of the column in the two cases. The half-length model exhibited slightly higher response due to the imposed end boundary conditions, which cannot capture buckling modes. Care, however, should be taken for cases comprising high slenderness ratio of the column (L=c), since global buckling becomes increasingly inevitable under these conditions. 3.5. Numerical results of triggered models The situation is quite different in the case where geometric triggers were used. These triggers represent the material and geometrical imperfections, which commonly dictate the initial site and nature of the collapse process. The numerical experiments were conducted on full-length FCM models in impact condition II and triggers were placed at each end of the column in order to maintain geometrical symmetry. Similar to the untriggered case, FE models were run on two different platforms. Fig. 10(a,b) illustrate the instantaneous load–deformation and energy absorption history curves for a trigger amplitude of 0.1 mm, respectively. These figures reveal that the introduction of geometric imperfections dictates the progressive collapse behaviour of the column, and that the numerical anomalies induced a 5% difference in energy absorption of the column in
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160 Half-length Model (with symmetry conditions)
140
Full-length Model
Load [kN]
120 100 80 60 40 20 0
0
100
200
300
400
500
Deformation [mm] Fig. 9. Instantaneous load–deformation behaviour for dual impact of thin-walled square columns.
25
140 PC Unix
120
PC
Unix
20 Energy [kJ]
Load [kN]
100 80 60
15 10
40 5 20 0 (a)
0
100
200 300 Deformation [mm]
400
0
500 (b)
0
100
200 300 Deformation [mm]
400
500
(c) Fig. 10. Dynamic collapse of empty columns for an FCM model using different platforms: (a) instantaneous load; (b) energy absorption; and (c) final deformed geometry.
this case. This is further depicted in Fig. 10(c), which illustrates the final deformed geometries of the column using different platforms. It is envisaged, however, that the position of the trigger as well as its amplitude and frequency will play a major role in dictating the collapse. An insight into the determination of realistic triggers should be obtained from the practical situation considered, the associated experimental data, and the collapse history of the component. Furthermore, numerical simulations of the collapse of half-column models were conducted in order to investigate the sensitivity of HCM to numerical anomalies associated with the computational precision of
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different platforms. Simulations were performed under both Windows and UNIX environments similar to the full-column model cases. It was observed that there is an excellent match between the two platforms, which indicates that the role played by numerical anomalies is almost insignificant and that HCM are reasonably robust for finite element analyses. 4. Conclusions Three different FE models were developed to simulate the dynamic collapse of square aluminium columns. The results of these different models, which are typically used in the literature, show that care must be taken when applying symmetry boundary conditions and ignoring the use of triggers. It is further shown that the numerical errors in untriggered models could lead to erroneous collapse modes and that appropriate checks must be made to overcome this situation. The results show that stiffening due to inertial effects, present in the high-speed collapse of dynamically loaded columns, manifest themselves in the higher crippling loads and greater energy absorption. It is worth noting that the analysis of the full column model, though expensive in both preparation and computer time, is sensitive to very small numerical errors. These errors could lead to fictitious modes of collapse. It is therefore important that careful assessment be made of the potential mode of collapse prior to modelling so that an appropriate selection of model configuration can be made for the prediction of the successive and/or global collapse modes. Furthermore, it has been shown that half column model has successfully overcome these problems since it is, by definition, able to model global buckling modes of collapse and is sufficiently constrained along one plane of symmetry, which eliminated the spurious modes of deformation encountered in the untriggered full model configuration. Acknowledgements The second author is partially funded by Ontario Graduate Scholarship for Science and Technology (OGSST), which he gratefully acknowledges. This work is also supported by the Auto21 NCE. Appendix A The example cited in [19] consisted of the following two-dimensional function: x f ðx; yÞ ¼ 333:75y6 þ x2 ð11x2 y2 y6 121y4 2Þ þ 5:5y8 þ 2y It is required to evaluate the value of f ðx; yÞ for the integer set (x*, y*) ¼ (77617, 33096). Clearly, these values were specifically chosen such that the gradient of f ðx; yÞ is very steep in the neighbourhood of x and y . We have investigated the computational precision of the solution using variable precision arithmetic (VPA) computations in MAPLE 7.0 software. In order to avoid any intervention from built-in algorithm, the function was defined using basic arithmetic operations, i.e. x3 ¼ x x x. In fact, using built-in functions would yield totally different results from those reported herein. It should also be noted that the results are a function of the version of Maple and the platform. Table A1 depicts selected values of f ðx ; y Þ for different
Table A1 Computational precision of f ðx ; y Þ using variable-precision arithmetic No. of digits
f ðx ; y Þ
No. of digits
f ðx ; y Þ
No. of digits
f ðx ; y Þ
1 3 5 7 10 12
2.00E+36 1.17 1.00E+32 1.172604 1.17260394 1.17260394
15 18 21 25 27 30
1.00E+22 1.17260394 1.17260394 3E+12 1.17260394 10000001.17
32 34 37 40 70 100
200001.1726 998.827396 0.82739606 0.82739606 0.82739606 0.82739606
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values of computational precision. It is first observed that the convergence scheme is severely oscillatory, although it stabilises with increasing the computational precision. Furthermore, the italic values in Table A1 refer to an erroneous value of f ðx ; y Þ ¼ 1:17260394, which persisted with increasing the number of digits up to 27 digits. Under some conditions, this value was reported when using 31, 32, 33 and 34 significant digits. Nevertheless, the correct value of f ðx ; y Þ, which is equal to 0.82739606 was not obtained until the computations comprised 37 digits, at which the solution was stabilised. This rather simple example clearly shows that the correct evaluation of the function could not be performed using even double-precision computations, and variable precision arithmetic, which is not commonly used in standard FE packages, was required to evaluate the function. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Jones N. Recent studies on the dynamic plastic behavior of structures—an update. Appl Mech Rev 1996;49(10):S112–7. Jones N. Recent studies on the dynamic plastic behavior of structures. Appl Mech Rev 1989;42(4):95–115. Alexander JM. An approximate analysis of thin cylindrical shells under axial loading. Quart J Mech Appl Math 1960;13:10–5. Pugsley A, Macaulay M. The large scale crumpling of thin cylindrical columns. Quart J Mech Appl Math 1960;13:1–9. Abramowicz W, Jones N. Dynamic progressive buckling of circular and square tubes. Int J Impact Eng 1986;4:243–70. Wierzbicki T, Abramowicz W. On the crushing mechanics of thin-walled structures. J Appl Mech 1983;50:727–39. Abramowicz W, Wierzbicki T. Axial crushing of multicorner sheet metal columns. J Appl Mech 1989;56:113–20. Abramowicz W, Jones N. Dynamic axial crushing of square tubes. Int J Impact Eng 1984;2:179–208. Langseth M, Hopperstad OS. Static and dynamic axial crushing of square thin-walled aluminium columns. Int J Impact Eng 1996;18(7):949–68. Jensen Ø, Langseth M, Hopperstad OS. Experimental investigations on the behaviour of short to long square aluminium tubes subjected to axial loading. Int J Impact Eng 2004;30:973–1003. Abramowicz W, Jones N. Transition from initial global bending to progressive buckling of tubes loaded statically and dynamically. Int J Impact Eng 1997;19:415–37. Kitagawa Y, Hagiwara I, Tsuda M. Dynamic analysis of thin-walled columns with arbitrary section geometry subjected to axial crushing. JSME-Int-J-Ser-1 1992;35(2):189–93. Han DC, Park SH. Collapse behavior of square thin-walled columns subjected to oblique loads. Thin-walled Struct 1999;35:167–84. Reyes A, Langseth M, Hopperstad OS. Square aluminum tubes subjected to oblique loading. Int J Impact Eng 2003;28:1077–106. Hallquist JO. LS-DYNA3D User’s Manual, Version: LS-DYNA 960. LSTC, Livermore Software Technology Corporation, 2001. Bathe KJ. Finite element procedures. NJ: Prentice-Hall; 1996. Meguid SA, Stranart JC, Heyerman J. On the layered micromechanical three dimensional finite element modelling of foam-filled columns. Finite Elem Anal Des 2004;40:1035–57. Hanssen AG, Langseth M, Hopperstad OS. Static and dynamic crushing of square aluminium columns with aluminium foam filler. Int J Impact Eng 2000;24:374–83. Rump SM. Algorithms for verified inclusions. Proc. Reliability in Computing: The Role of Interval Methods in Scientific Computing. New York: Academic Press; 1988. pp. 109–26.
International Journal of Impact Engineering 34 (2007) 348–359 www.elsevier.com/locate/ijimpeng
Solution stability in the dynamic collapse of square aluminium columns S.A. Meguid, M.S. Attia, J.C. Stranart, W. Wang Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ont., Canada M5S 3G8 Received 27 April 2004; received in revised form 14 September 2005; accepted 18 September 2005 Available online 9 November 2005
Abstract The collapse of empty square aluminium columns is re-examined using dynamic nonlinear elasto–plastic finite element analysis. Earlier work on this topic did not address the influence of FE model reduction on possible numerical anomalies resulting from the use of different processing platforms. Single- and double-sided impact conditions are also considered under both constant and varying initial impact speeds. Specifically, the objective of the study is to investigate the influence of numerical anomalies that are typically overlooked in the literature and the dramatic role they may play in dictating the dynamic mode of collapse. To that end the study investigates the influence of specific symmetric boundary and initial conditions on the resulting mode of collapse and determines the effect of the use of different platforms on the results. The work was carried out using LS-DYNA explicit finite element solver and considers generic geometries that are typically used in industrial applications. The results show clearly that the use of symmetry conditions in FE modelling must be used with proper care and that careful experimental work should be carried out to validate the finite element predictions. r 2005 Elsevier Ltd. All rights reserved. Keywords: Dynamic collapse; Thin-walled; Impact; Crashworthiness; Numerical errors
1. Introduction The Canadian Government and Transport Canada have set a target of a 30% reduction in the average number of road users killed and seriously injured during the 2008–2010 period. Transport Canada’s efforts to improve motor vehicle safety regulations focus on frontal-crash and lateral-impact protection. This, together with a range of environmental concerns and social pressures backed by legislation, has led, and will continue to lead to highly innovative designs, involving lighter materials such as aluminium or magnesium alloys, and cellular materials. Thin-walled extrusions are extensively used to improve vehicle crashworthiness in weak areas such as bumpers, side doors and roof. The ratio of the length to section dimensions of these extrusions is such that it typically promotes failure by buckling. Buckling of thin plates and members has been studied since Euler’s Corresponding author. Tel./fax: +1 416 978 5741.
E-mail address: [email protected] (S.A. Meguid). 0734-743X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2005.09.001
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work on buckling of columns in 1744. Local instability has been found to be the main reason for buckling of thin plates and members. Numerous researchers have investigated the local instability in plates using elastic, elasto–plastic, and plastic hinge analysis methods. For a detailed account of the literature, see the extensive reviews by Jones in [1,2]. The earliest analytical treatment of the quasi-static axial collapse of thin-walled cylindrical columns was conducted by Alexander [3], and Pugsley and Macaulay [4] under the assumptions of perfect folding pattern and rigid plastic material model. They employed an energy balance criterion between the external work done by the axial force with the energy dissipated being into forming plastic hinges at the folds. Abramowicz and Jones [5] incorporated effective crushing distance and varying, as opposed to mean, membrane strains into Alexander’s solution. Their experimental investigations considered both square and circular steel sections for a wide range of lengths and cross-section dimensions using a drop hammer apparatus. Based on the test results and observations, approximate theoretical predictions were developed for the progressive axial crushing of columns using a kinematically admissible plastic analysis. Wierzbicki and Abramowicz [6,7] developed a self-consistent method that describes the crushing behaviour of thin-walled multi-corner structures based on the energy balance postulate. They identified two distinct modes of collapse: (i) quasi-inextensional mode, and (ii) quasi-extensional mode. The former mode is characterised by large rotations (plastic hinge lines) and small axial strains near the column corners, while it is largely governed by membrane stretching in the latter. Similar strategy was followed in [8] where two idealised collapse elements comprising four deformation modes were introduced: one symmetric mode, two asymmetric modes, and an extensional mode of deformation. These asymmetric deformation modes were predicted theoretically and confirmed in the experimental tests. Further enhancements were reported in [9,11] through extending the range of geometrical and material parameters and impact energies beyond those reported in [5]. Different tempering conditions, wall thickness values, and test velocities were examined in [9]. The impact velocity ranged from 8 to 20 m/s using a 56 kg projectile. Quasi-static tests were also carried out to develop a relationship between the dynamic and quasi-static behaviour. The rigorous assessment showed that there is a very good agreement between theoretical predictions and experimental results. Recent work by Jensen et al. [10] examined the dependence of global buckling on the local and global slenderness ratios. Kitagawa et al. [12] analysed axial crushing behaviour of side members in a frontal collision of an automobile. The mechanism of axial collapse was described as being local buckling occurring at the column’s weakest section, along with slight waviness appearing in the column wall. This work was further extended by Han and Park [13], where they studied the crush behaviour of a square column subjected to oblique loads. Their results showed that there is a critical load angle for transition from the axial collapse mode to the bending collapse mode. This critical angle was found to be approximately 71. The value of the mean crush load drops to about 40% of the mean crush load in pure axial collapse beyond the critical load angle. More recently, Reyes et al. [14] compared experimental and LS-DYNA simulations under oblique loadings which induced global buckling. While the force response was captured by the simulations using initial imperfections, the experimentally observed buckling modes were more closely reflected by the simulations which assumed a perfect geometry. In order to adopt these light structures for applications involving safety of the public, a comprehensive understanding of their behaviour under impact loading conditions is necessary. This can be achieved by conducting high fidelity computer models and experimentally validating these models. It is therefore the objective of this study to: (i) carry out nonlinear elasto–plastic finite element analysis of the dynamic collapse of square aluminium columns; (ii) critically assess the influence of enforcing symmetry boundary conditions, component fixation and system initial conditions on the modelling of the columns and the resulting progressive mode of collapse; and (iii) investigate the influence of numerical anomalies that are typically overlooked in the literature and the dramatic role they may play in dictating the dynamic mode of collapse. 2. Finite element modelling 2.1. Geometry and properties of square aluminium columns As a result of earlier published work and range of practical applications, the following dimensions were selected for the aluminium columns of the baseline case: length of tube (L) 300 mm, width (c) 75 mm, and
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Table 1 Mechanical properties of aluminium 6061-T4 (after [15]) Engineering strain (%)
Engineering stress (MPa)
0 0.23 0.28 0.97 4.56 7.59 13.44 16.33 20.11
0 138.18 152.83 163.09 196.29 216.80 237.30 242.68 244.14
thickness (h) 2.5 mm; which yields a length-to-width ratio (L/c) of 4 and width-to-thickness ratio (c/h) of 30. Existing work in the literature [11] indicates that with the choice of these geometrical ratios, stable progressive collapse will ensue under both quasi-static and dynamic loading conditions. In addition, columns of longer length (L ¼ 600 mm) were also considered. This would yield a length-to-width ratio of 8, which is near the transition region between folding and global bending failure. Material properties of the aluminium tubes were those of Aluminium alloy 6061 in T4 temper condition. Table 1 illustrates the mechanical properties of this alloy. The T4 condition was specifically chosen in order to ensure sufficient ductility to prevent ductile rupture due to plastic instabilities during loading. 2.2. FE model configuration LS-DYNA explicit FE solver [15] was implemented in the current study. Belytschko–Tsay reduced integration shell element with six degrees of freedom (DOF) per node and five integration points through thickness were used to model the column material. A piecewise-linear plasticity material law was implemented in modelling the column. The striker was modelled using solid elements made of a rigid material model available in LS-DYNA library. Fig. 1 depicts a schematic of the different FE models used in the current study and the associated boundary conditions. They can be summarised as follows: (i) quarter column model (QCM), (ii) half column model (HCM), and (iii) full-column model (FCM). QCM assumes dual symmetry about the y–z and x–y planes and has the advantage of being the most economical in terms of model size and computational resources. The HCM, on the other hand, assumes symmetry about only one plane, while the full column model (FCM) does not employ symmetry conditions and a full model is generated to represent the structure. This is the most expensive model in terms of computational resources. 2.3. Loading and contact conditions Two loading conditions were considered in the current study. In the first, Fig. 2(a), the tube is struck by one rigid platen from one end, while the other end is fully constrained in all DOF. This will be referred to as impact condition I. In the second case, the tube is simultaneously impacted by two rigid platens from both ends, referred to here as impact condition II, Fig. 2(b). It is worth mentioning, however, that the velocity of the striker in these two cases was considered to be constant throughout the impact process. This represents the extreme case of two objects; one of them has a very large mass compared to the other, which yields a conservative estimate of the collapse behaviour of the column. Dynamic simulations at different impact speeds were conducted in order to investigate the different aspects of the collapse process under low and high-velocity impact conditions. Contact surfaces were defined in the different models along two interfaces: (i) tube–striker interface; and (ii) self-contact of the tube walls during collapse. The automatic surface-to-surface contact algorithm available in
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z
z y
y
x x (a)
x
(b)
(c)
Fig. 1. Different FE model configurations: (a) quarter column model (QCM); (b) half column model (HCM), and (c) full column model (FCM).
V V
V (a)
(b)
Fig. 2. Different loading conditions on the column: (a) one-side impact; and (b) two-side impact.
LS-DYNA was used between the tube and the striker, while the single-surface contact algorithm was applied to the tube wall to avoid interpenetration. These contact algorithms consist of placing normal interface springs between all candidate penetrating nodes and the contact surface. The interface stiffness is chosen to be approximately the same order of magnitude as the stiffness of the interface element normal to the interface. 2.4. Triggering mechanism So far, we have limited our discussion to highly idealised finite element models of square aluminium columns and avoided the use of triggers. It is crucially important, however, to realise that these models cannot be realised in real life due to the existence of a number of geometrical and material imperfections Meguid et al. [17] and Hanssen et al. [18] successfully implemented both wedge and sinusoidally shaped geometrical imperfections (triggers) for both empty- and foam-filled aluminium columns under quasi-static crushing. In this study, we implement a wedge-shaped trigger at the top of the columns, which simulates the trigger introduced in the dynamic experiments conducted in [18]. 3. Results and discussion 3.1. Validation of FE models The baseline case for model convergence evaluation is a quarter column model under impact condition I. Both quasi-static and dynamic cases were tested. The striker velocity for the quasi-static case was selected to
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120 Experimental Data Simulation
100
Load [kN]
80 60 40 20 0
(a)
0
50
100
150
Deformation [mm]
(b)
Fig. 3. Comparison of experimental results [9] vs. FE predictions of dynamic collapse of empty columns: (a) final collapse geometry; and (b) instantaneous load–deformation behaviour.
Table 2 Comparison between FE predictions and experimental results [9] of the final axial deformation and mean collapse loads Test
t ¼ 1:8 mm t ¼ 2:0 mm t ¼ 2:5 mm
Simulation
Experiment
ws =we
F smd =F emd
Ws (mm)
F smd ðkNÞ
We (mm)
F smd ðkNÞ
(%)
(%)
133.24 82.98 158.23
21.0 29.7 41.56
129 89 158
21.8 27.9 41.9
103.3 93.2 100.1
96.3 106.5 99.2
be 1 m/s, while a constant velocity of 15 m/s was selected for the dynamic case. It was necessary to implement artificial means to avoid excessively long solution times for the quasi-static case. According to the Courant’s condition [16], increasing the mass of the model by a factor of x2 would increase the minimum solution time step by a factor x. Previous studies [17] illustrated that scaling the mass by a factor of 10 yields desirable computation time and accuracy for quasi-static loading. Sensitivity analyses were conducted and it was found that the solution becomes mesh independent for element size smaller than or equal to 3 mm, which was chosen for the rest of the simulations. Finite element predictions were compared to the experimental results in [9]. These experiments comprised dynamic axial compression of square aluminium sections with dimensions 80 mm 310 mm, at an initial velocity vo ¼ 15:6 m=s. The aluminium alloy used was AA6060 in the T4 temper conditions, with varied wall thickness (1.8, 2.0 and 2.5 mm). Half-column FE models were used in the validation process along with symmetry boundary conditions. The reasons for choosing this model are discussed in the following section. The FE model comprised a total of 5200 shell elements. Fig. 3 depicts a comparison of the final deformed shape and the variation of the instantaneous load between FE predictions and experimental results for a 2.5 mm thick tube. The figure shows reasonable agreement between the experimental results and FE predictions. Very good agreement is observed until the second collapse fold, and the final deformed shape in the two cases is very close. Comparisons were also conducted for 1.8 and 2.0 mm thick tubes. The initial peak load values, subsequent peaks, and plateau regimes are very close in the two cases. Table 2 gives a comparison between the final axial displacements and the corresponding mean loads for FE simulations and tests, where mt is the impactor mass, t is the wall thickness, and wp is permanent axial deformation. Very good correlation can be observed between experimental results and FE predictions.
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Fig. 4. Effect of model configurations on (a) final collapse geometry and; (b).
3.2. Effect of model configuration Fig. 4(a,b), respectively, show the final deformed geometry and the instantaneous load–deformation curves for the three configurations considered at an impact speed of 15 m/s. It was observed that, for all the models, the first collapse fold was formed at the bottom and was of the extensional type. Nevertheless, subsequent collapse patterns were noticeably different, where the symmetric mode of collapse immediately commenced in the quarter and half models, while the extensional mode of collapse prevailed in the full model case. In fact, symmetry of the geometry and applied load may not be sufficient to implement symmetry boundary conditions in the FE model in buckling analysis. For instance, a global mode of collapse of the Euler-type is realised for certain geometrical aspect ratios and/or above specific threshold impact speed values. Nevertheless, quarter column models are, by definition, incapable of simulating this mode of collapse and are therefore not suited for modelling this case of collapse. Full-column models, on the other hand, develop parasitic numerical errors, which may significantly affect the buckling behaviour of the column. These errors are discussed in detail in Section 3.3. In contrast to these model configurations, half-column models do not suffer from these problems, since symmetry conditions are imposed along one plane only, and global buckling mode is therefore allowed in that plane. Furthermore, the symmetry conditions have suppressed the parasitic modes of deformation. 3.3. Numerical anomalies in untriggered models We have shown in Section 3.2 that the choice of a specific model configuration strongly affects the mode of collapse, and has a relatively moderate effect on the load–deformation response and the associated energy absorption levels. This is in agreement with [14]. In this section, the effect of numerical anomalies upon the
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model response is investigated. These are parasitic numerical errors in the finite element model matrices. The full-column model (FCM) was chosen as the baseline case for investigating the effect of these anomalies, since this model is the least constrained model amongst all considered configurations. This ensures that the effect of these anomalies is not suppressed by displacement constraints. There are two sources for these anomalies. The first is concerned with the computational platforms used in the simulations, while the second is concerned with the numerical precision of the solver. The effect of computational precision upon the solution accuracy has been previously illustrated in [19], where the author considered the solution of a one-dimensional polynomial of the sixth order. The polynomial coefficients have been precisely chosen such that a steep gradient exists in the vicinity of one of the roots, which is the situation commonly encountered in buckling problems. It has been shown that there was an error of approximately 150% in evaluating this specific root when using double-precision computations (32 digits). Single precision computations (16 digits) resulted in higher error values. The authors reproduced the solution in [19] and found that the correct solution was realised only through employing variable-precision computations, commonly used in symbolic analysis, with at least 37-digit computations. A consistent convergence scheme to the incorrect solution was found for up to 36 digits of computations. Appendix A illustrates the details of convergence scheme for this example. This simple example illustrated the crucial importance of scrutinizing the validity of buckling solutions, even if convergence was achieved using numerically accurate doubleprecision computations. For example, two-sided impact simulations (Impact Condition II) were conducted on empty aluminium columns at an initial impact velocity of 15 m/s. The simulations were carried out on: (i) 32-bit AMD-Athlon 1.4 GHz processor with 256 MB of RAM running Windows2000 and LS-DYNA V-960; and (ii) Sun Blade 1000 workstation with 1 GB of RAM and a 64-bit 750 MHz UltraSparc III Processor running SunOS 5.8 (Solaris). Furthermore, LS-DYNA V-960 explicit solver employs a single-precision based calculation system. Fig. 5 illustrates the deformed shape of the aluminium columns at different stages of deformation. It is observed that the collapse behaviour of FCM is almost the same for different processors for the first one-third of the simulation, where the collapse is governed by progressive plastic folding. Nevertheless, in subsequent deformation, the Athlon processor solution changes to bending collapse, while the deformed geometry predicted by the UltraSparc III processor remains stable and shows progressive buckling patterns. This divergence between the two processors is evident in both curves at approx. 30% deformation level. Fig. 6 illustrates the variation of the instantaneous collapse load with the collapse distance. It was found that the difference in energy absorption between the two models reached a maximum of 12% at the end of deformation history. These differences can only be due to the numerical calculations and round off differences between the two machines, which resulted in uncontrolled initiation and growth of spurious buckling modes. It is therefore extremely difficult to judge the soundness of either simulation without proper validation with experimental findings. 3.4. Effect of impact conditions Dynamic axial collapse simulations have been conducted for both impact conditions I and II. The effect of impact speeds was also considered. Although it is established that aluminium is strain rate insensitive, it is anticipated that the impact speed will play an important role in dictating the mode of collapse of the columns because of inertial stiffening. Furthermore, as pointed out in Sections 3.2 and 3.3, it has been established that half-column model possesses the ability to realistically model the buckling process of thin-walled sections and does not suffer from the numerical anomalies problems encountered in full column models. Therefore, the simulations were conducted with this FE model configuration. Fig. 7 illustrates the instantaneous load–deformation curves for different FE model configurations for impact speeds of 1, 10, 15, and 30 m/s, respectively. The collapse behaviour at 1 m/s is characterised by progressive folding of the wall as evidenced by the uniform shape of the instantaneous load–deformation behaviour of the structure for all FE model configurations. In contrast, inertial stiffening significantly affected the collapse behaviour for impact speed 30 m/s such that the collapse pattern is irregular and the folds were steeply developed and collapsed. Furthermore, it has been observed that the folding mode in this case is of the extensional type, compared to symmetric mode for an impact speed of 1 m/s.
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AMD Processor
355
Sun Spark Processor
Deformation 94.5 mm
Deformation 270 mm
Deformation 391.5mm
Deformation 450 mm
Fig. 5. Comparison of collapse geometry at different stages as obtained from: AMD processor and Sun Sparc processor.
160 PC
Unix
Load [kN]
120
80
40
0
0
100
200 300 Deformation [mm]
400
500
Fig. 6. Comparison between the instantaneous load history using different processors for untriggered columns.
Two-sided impact simulations in the current study were mainly concerned with the characteristics of the collapse mode in impact condition II. Attention was therefore devoted to the dynamic collapse at an initial impact speed of 15 m/s. Square aluminium sections having dimensions 600 mm 75 mm 2.5 mm were
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Fig. 7. Instantaneous load–deformation behaviour for different impact speeds.
Striker I
Striker I
Striker II
Striker II (a)
(b)
Fig. 8. Different model configurations under impact condition II.
specifically chosen to reveal the effects of two-sided impact on the mode of the column since the slenderness ratio L=c ¼ 8 lies in the transition region between progressive buckling and global bending modes of collapse. Both full-length and half-length models with symmetry conditions at the mid-length were considered as depicted in Fig. 8. The respective instantaneous load–deformation response for both half- and full-length model configurations are shown in Fig. 9. It is observed that there is a very close match between the response of the column in the two cases. The half-length model exhibited slightly higher response due to the imposed end boundary conditions, which cannot capture buckling modes. Care, however, should be taken for cases comprising high slenderness ratio of the column (L=c), since global buckling becomes increasingly inevitable under these conditions. 3.5. Numerical results of triggered models The situation is quite different in the case where geometric triggers were used. These triggers represent the material and geometrical imperfections, which commonly dictate the initial site and nature of the collapse process. The numerical experiments were conducted on full-length FCM models in impact condition II and triggers were placed at each end of the column in order to maintain geometrical symmetry. Similar to the untriggered case, FE models were run on two different platforms. Fig. 10(a,b) illustrate the instantaneous load–deformation and energy absorption history curves for a trigger amplitude of 0.1 mm, respectively. These figures reveal that the introduction of geometric imperfections dictates the progressive collapse behaviour of the column, and that the numerical anomalies induced a 5% difference in energy absorption of the column in
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160 Half-length Model (with symmetry conditions)
140
Full-length Model
Load [kN]
120 100 80 60 40 20 0
0
100
200
300
400
500
Deformation [mm] Fig. 9. Instantaneous load–deformation behaviour for dual impact of thin-walled square columns.
25
140 PC Unix
120
PC
Unix
20 Energy [kJ]
Load [kN]
100 80 60
15 10
40 5 20 0 (a)
0
100
200 300 Deformation [mm]
400
0
500 (b)
0
100
200 300 Deformation [mm]
400
500
(c) Fig. 10. Dynamic collapse of empty columns for an FCM model using different platforms: (a) instantaneous load; (b) energy absorption; and (c) final deformed geometry.
this case. This is further depicted in Fig. 10(c), which illustrates the final deformed geometries of the column using different platforms. It is envisaged, however, that the position of the trigger as well as its amplitude and frequency will play a major role in dictating the collapse. An insight into the determination of realistic triggers should be obtained from the practical situation considered, the associated experimental data, and the collapse history of the component. Furthermore, numerical simulations of the collapse of half-column models were conducted in order to investigate the sensitivity of HCM to numerical anomalies associated with the computational precision of
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different platforms. Simulations were performed under both Windows and UNIX environments similar to the full-column model cases. It was observed that there is an excellent match between the two platforms, which indicates that the role played by numerical anomalies is almost insignificant and that HCM are reasonably robust for finite element analyses. 4. Conclusions Three different FE models were developed to simulate the dynamic collapse of square aluminium columns. The results of these different models, which are typically used in the literature, show that care must be taken when applying symmetry boundary conditions and ignoring the use of triggers. It is further shown that the numerical errors in untriggered models could lead to erroneous collapse modes and that appropriate checks must be made to overcome this situation. The results show that stiffening due to inertial effects, present in the high-speed collapse of dynamically loaded columns, manifest themselves in the higher crippling loads and greater energy absorption. It is worth noting that the analysis of the full column model, though expensive in both preparation and computer time, is sensitive to very small numerical errors. These errors could lead to fictitious modes of collapse. It is therefore important that careful assessment be made of the potential mode of collapse prior to modelling so that an appropriate selection of model configuration can be made for the prediction of the successive and/or global collapse modes. Furthermore, it has been shown that half column model has successfully overcome these problems since it is, by definition, able to model global buckling modes of collapse and is sufficiently constrained along one plane of symmetry, which eliminated the spurious modes of deformation encountered in the untriggered full model configuration. Acknowledgements The second author is partially funded by Ontario Graduate Scholarship for Science and Technology (OGSST), which he gratefully acknowledges. This work is also supported by the Auto21 NCE. Appendix A The example cited in [19] consisted of the following two-dimensional function: x f ðx; yÞ ¼ 333:75y6 þ x2 ð11x2 y2 y6 121y4 2Þ þ 5:5y8 þ 2y It is required to evaluate the value of f ðx; yÞ for the integer set (x*, y*) ¼ (77617, 33096). Clearly, these values were specifically chosen such that the gradient of f ðx; yÞ is very steep in the neighbourhood of x and y . We have investigated the computational precision of the solution using variable precision arithmetic (VPA) computations in MAPLE 7.0 software. In order to avoid any intervention from built-in algorithm, the function was defined using basic arithmetic operations, i.e. x3 ¼ x x x. In fact, using built-in functions would yield totally different results from those reported herein. It should also be noted that the results are a function of the version of Maple and the platform. Table A1 depicts selected values of f ðx ; y Þ for different
Table A1 Computational precision of f ðx ; y Þ using variable-precision arithmetic No. of digits
f ðx ; y Þ
No. of digits
f ðx ; y Þ
No. of digits
f ðx ; y Þ
1 3 5 7 10 12
2.00E+36 1.17 1.00E+32 1.172604 1.17260394 1.17260394
15 18 21 25 27 30
1.00E+22 1.17260394 1.17260394 3E+12 1.17260394 10000001.17
32 34 37 40 70 100
200001.1726 998.827396 0.82739606 0.82739606 0.82739606 0.82739606
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values of computational precision. It is first observed that the convergence scheme is severely oscillatory, although it stabilises with increasing the computational precision. Furthermore, the italic values in Table A1 refer to an erroneous value of f ðx ; y Þ ¼ 1:17260394, which persisted with increasing the number of digits up to 27 digits. Under some conditions, this value was reported when using 31, 32, 33 and 34 significant digits. Nevertheless, the correct value of f ðx ; y Þ, which is equal to 0.82739606 was not obtained until the computations comprised 37 digits, at which the solution was stabilised. This rather simple example clearly shows that the correct evaluation of the function could not be performed using even double-precision computations, and variable precision arithmetic, which is not commonly used in standard FE packages, was required to evaluate the function. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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