Study of lateral compression of round metallic tubes

Thin-Walled Structures 43 (2005) 895–922 www.elsevier.com/locate/tws

Study of lateral compression of round metallic tubes N.K. Guptaa,*, G.S. Sekhona, P.K. Guptab a

Department of Applied Mechanics, Indian Institute of Technology, Delhi, Hauz khas, New Delhi 110016, India b Civil Engineering Group, Birla Institute of Technology & Science, Pilani 333031, India Received 29 March 2004; accepted 6 December 2004 Available online 23 March 2005

Abstract A detailed experimental and computational investigation of round metallic tubes subjected to quasi-static loading is presented. Experiments were conducted wherein round aluminium and mild steel tubes of different diameter to thickness ratios were subjected to lateral compression in an Instron machine. Their deformation histories and load–compression curves were obtained. The deformation of the tubes has also been studied and analysed with the help of the finite element code FORGE2. Contours of nodal velocity, equivalent strain rate and equivalent strain at different stages of compression are presented and discussed. Experimental and computed results are compared. Basic mechanism of their deformation and the effects of process parameters on deformation behaviour of the tubes are presented and discussed. q 2005 Elsevier Ltd. All rights reserved. Keywords: Lateral compression; Round tubes; FORGE2; Energy absorption

1. Introduction Metallic tubes are frequently used as energy absorbing devices. They deform plastically in several different modes such as inversion and folding. Each mode has an associated energy dissipation capacity [1]. The tubular elements in energy absorbers can be confined * Corresponding author. Tel.: C91 11 659 1178; fax: C91 11 658 1119. E-mail addresses: [email protected] (N.K. Gupta), [email protected] (G.S. Sekhon), [email protected] (P.K. Gupta).

0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2004.12.002

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Notation S~ij K 3~_ij 3_ m K0 a 3 b T tf a p v~f U D t S Sv Sf P q h rp ph k$k m s d/D

deviatoric stress tensor material consistency strain rate tensor effective strain rate strain rate sensitivity index constant term strain hardening parameter effective strain temperature sensitivity term temperature in absolute value shearing stress friction factor sensitivity to sliding velocity relative sliding velocity between tube and platen volume of the deforming tube mean diameter of tube in mm. thickness of tube in mm surface boundary boundary surface on which velocity is prescribed boundary surface on which force is prescribed vertical height at section B1B3 of the deformed tube at any stage of compression width of the deformed tube at middle horizontal section maximum height of deformed tube penalty number hydrostatic pressure Euclidean norm equivalent viscosity effective or equivalent stress compression/average diameter (compression ratio)

in various ways so that when an impact does occur, they deform axially, laterally or in some combination. In the last few decades, several investigations have been carried out to study the response of metallic tubes for their use as energy absorbing elements [2–4]. Earlier studies conducted on lateral collapse of round tubes differ in the mechanisms assumed for the analysis [5–11]. Mutchler [5] examined the behaviour of an aluminium round tube when compressed laterally between two plates. He determined its energy absorbing capacity by using a numerical technique and with the assumption that initially the tube and platen are in a line contact and with progress of deformation, this straight line contact transforms into a rectangular interface. The width of the rectangle was found to increase with increase in

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deformation. The non-contacting portions of the tube were assumed to remain circular. Analytical results were obtained for compression up to 0.5 times the initial diameter of the tube. Runtz and Hodge [6] carried out a limit analysis of the same problem by assuming that plastic hinges are formed at four locations at intersections of horizontal and vertical axis in the tube cross-section. They further assumed that during the compression process, these hinges remain stationary and only rotation about these occurs. The predicted values of the deforming load thus obtained were found to be appreciably lower than the corresponding experimental values. The difference was found to increase with the increase of deformation. Redwood [7] assumed that two plastic hinges are formed at quarter points from the point of application of load [8] and as the deformation proceeds, the hinges move outward and their radius of curvature reduces. The portion under the platen, however, was assumed to remain flat during the compression process. The effect of strain hardening of the tube was taken into account by incorporating strain hardening modulus in their analysis. The analytical results were found to be more reasonable as compared to those predicted by the Runtz and Hodge model. They were, however, still lower than the experimental results. The reason for the discrepancy was attributed to ignoring friction at the platen and tube interface. Reddy and Reid [9–11] proposed that in place of plastic hinges, plastic regions are formed in the vicinity of the quarter points. They divided the deforming arc into two portions, one rigid and the other plastic. They also performed experiments on tubes of different lengths and studied the effect of length on the load– compression curves. The experiments were performed on as-received, annealed and interrupted annealed tubes. In their opinion, when the tubes are tested in the interrupted annealed conditions, the material could be modelled as rigid-perfectly plastic, while in the other states strain hardening should be taken into account. In the present investigation, experiments have been conducted on the lateral compression of round aluminium and mild steel tubes under quasi-static loads. A finite element analysis of the compression process has also been carried out with the help of the FORGE2 code [12]. Results obtained from finite element analysis are compared with the experiments. Contours of nodal velocity, equivalent strain rate and equivalent strain at different stages of compression are presented. On the basis of the developed strain rate and strain during the compression process the mode of deformation has been explained and compared with the analytical models presented in the past. Effects of process parameters on the energy absorbing capacity of tubes and mechanics of their collapse are also discussed. Based on the results obtained, basic mechanism of the tube deformation under lateral loads has been presented. 2. Experimental investigation 2.1. Experimental details Specimens were cut from 3.65 m long commercially available tubes of mild steel and aluminium. The end faces of each specimen were carefully machined on a lathe. The length of the specimens was made equal to 100 mm. The results are presented for unit millimeter length of tube specimens. The cross-sectional dimensions and D/t ratio (i.e. the ratio of mean diameter to wall thickness) of these tubular specimens was also varied.

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The range of D/t values are from 7.2 to 37.4 for aluminium tubes and from 10.4 to 36.8 for mild steel tubes (see Table 1). These specimens were tested after annealing. The annealing of the aluminium and mild steel specimens was done by soaking them at 360 8C for 30 min and 910 8C for 10 min, respectively, and allowing them to cool in the furnace gradually for 24 h. A universal testing machine (Instron model 1197) of 50 t capacity was employed for experimentation. Specimens were centrally positioned on the bottom platen of the machine. The upper platen was moved at a constant downward velocity of 10 mm/min. The compression process was continued until there was hardly any gap between the top and bottom halves of the tube at its middle. The load–compression curves were recorded with the automatic chart recorder of the machine. The deformed profiles of the specimens were measured and traced at different stages of compression with the help of a profile projector. 2.2. Experimental results and discussions It was observed that mode of deformation of the tubular specimens of a given material is unaffected by their D/t ratio. Therefore, deformed profiles at different stages of compression for only one tube specimen each from aluminium (Sp. No. A503, Table 1) and mild steel (Sp. No. S504, Table 1) are presented in Fig. 1(a) and (b). The corresponding experimental load–compression curves for these specimens are presented in Fig. 2. Labels 1, 2, 3, etc. on these curves indicate the stages of deformation at which deformed profiles are presented in Fig. 1(a) and (b), respectively. It is interesting to study the effect of diameter D, thickness t and D/t ratio on the energy absorbing capacity of round tubes subjected to lateral compression. Variation of specific Table 1 Material properties of annealed round tubes subjected to lateral compression Sp. No.

Average diameter (D) (mm)

Thickness (t) (mm)

D/t

Aluminium A251 A252 A253 A383 A381 A503 A502 A501 A802

24.50 23.38 21.93 34.73 37.04 47.66 48.92 48.70 79.20

0.84 1.62 3.03 3.13 1.12 3.44 1.68 1.30 2.18

Mild steel S504 S253 S503 S301 S451

44.80 24.41 47.90 30.02 42.74

3.34 2.34 2.80 1.34 1.16

Material properties K0 (MPa)

a

m

29.17 14.43 7.239 11.09 33.07 13.85 29.12 37.46 36.33

159.7 140.3 159.1 161.3 159.4 153.3 145.1 161.3 125.2

1.243 1.416 1.456 1.653 1.453 1.451 1.452 1.652 1.253

0.0251 0.0819 0.0259 0.0271 0.0259 0.0244 0.0284 0.0272 0.0215

14.24 10.43 17.10 22.40 36.84

288.2 256.3 269.5 319.5 279.4

2.35 2.34 2.39 3.39 2.59

0.13510 0.01431 0.01610 0.01700 0.01510

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Fig. 1. True deformed profiles at different stages of lateral compression. (a) Sp. No. A503, (b) Sp No. S504.

1.00 0.90

A503

0.80

6

S504

Load (kN/mm)

0.70 0.60 0.50

5

0.40

4

0.30 0.20

3 2

1

0.10 0.00 0.0

1 0.2

3

2 0.4

0.6

4

5

6

0.8

1.0

Compression/Average diameter Fig. 2. Experimental load–compression curves during lateral compression of aluminium specimen A503 and mild steel specimen S504 (Labels 1, 2, 3, etc. show sequences of profiles in Fig. 1(a) and (b).

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Specific Energy (Joules/mm3)

0.016 0.014

0. 1

0. 2

0. 3

0. 4

0.012

0. 5

0. 6

0. 7

0. 8

0.01 0.008 0.006 0.004 0.002 0 5

10

15

20

25

30

35

40

D/t Ratio Fig. 3. Specific energy–compressionion curves for specimens of different D/t ratios made of aluminium.

energy absorption with D/t ratio for different compression ratios for aluminium specimens is depicted in Fig. 3. It is clear from this figure that the energy absorbing capacity of tubes increases with decrease in D/t ratio throughout the compression process. By observing Fig. 3, it is clear that the specific energy varies exponentially with D/t ratio throughout the compression process. The relationship may be written as Specific energy Z ð0:0106e4:0633d=D ÞðD=tÞK1:22 where the d/D is the compression ratio and specific energy is in J/mm3. 0.25 Aluminium specimens of diameter 50 mm

Load (kN/mm)

0.20

0.15

D/t = 13.85 D/t = 29.05 D/t = 37.46

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

Compression/Average diameter Fig. 4. Load–compression curves for specimen of different thicknesses but same diameter.

1.0

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901

Specific Energy (Joules/mm3)

0.008 0.007

Aluminium specimens of diameter 50 mm

0.006

D/t = 13.85

0.005

D/t = 29.05 D/t = 37.46

0.004 0.003 0.002 0.001 0.000 0.0

0.2

0.4

0.6

0.8

1.0

Compression/Average diameter Fig. 5. Specific energy–compression curves for specimen of different thicknesses but same diameter.

Figs. 4 and 5 show load–compression and specific energy–compression curves for specimens of equal diameter but different thicknesses. It is observed from these figures that the energy absorbing capacity increases with increase in tube thickness. It is also found from inspection of load–compression curves that the constant slope portion in the middle of curve decreases in range with increase in tube thickness. The effect of thickness on mean collapse load is presented in Fig. 6. It is found that the mean collapse load increases with increase in tube thickness. The rate of increase of mean collapse load with tube thickness is greater for tubes of larger diameter and vice-versa.

Mean Collpase load (kN/mm)

0.240 Experimental (25 mm diameter)

0.210

Computational (25 mm diameter)

0.180

Experimental (50 mm diameter)

0.150

Computational (50 mm diameter)

0.120 0.090 0.060 0.030 0.000 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

Average diameter/thickness Fig. 6. Variation of mean collapse load with average diameter/thickness ratio for specimens of same diameter but different thicknesses.

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Specific Energy (Joules/mm3)

0.020 Aluminium specimens of 3 mm thickness

0.016

D/t = 7.23

0.012

D/t = 13.85 D/t = 11.09

0.008

0.004

0.000 0.0

0.2

0.4

0.6

0.8

Compression/Average diameter Fig. 7. Specific energy–compressionion curves for specimens of different diameters but same thickness.

Fig. 7 shows the specific energy–compression curves for aluminium specimens of equal thickness but of different diameters. It is observed that the energy absorbing capacity decreases with increase in tube diameter. Variation of mean collapse load (defined as ratio of the area under the load compression curve and the total compression of the tube) with tube diameter shows that the mean collapse load tends to decrease with increase in tube diameter (Fig. 8). However, the mean collapse load becomes more or less constant beyond a certain diameter of the tube, depending upon its thickness.

Mean Collpase load (kN/mm)

0.240 3 mm thickness (Experimental)

0.210

3 mm thickness (Computational)

0.180

2 mm thickness (Experimental)

0.150

2 mm thickness (Computational)

0.120 0.090 0.060 0.030 0.000 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

Average diameter/thickness Fig. 8. Variation of mean collapse load with D/t ratio for specimens of same thickness but different diameters.

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3. Computational study 3.1. Governing equations Finite element formulations for non-linear problems of plasticity are classified into solid formulations and flow formulations [13]. In the flow formulation which is employed here, the elastic components of strain are neglected as small compared to their plastic counterparts. An updated Lagrangian reference system is employed wherein the velocities are considered as the basic unknowns and the incompressibility condition is incorporated using a penalty function. The overall deformation is analysed in terms of a large number of deformation steps. Linearized relationship between the stress and strain rate is assumed to exist during each step and a quasi-steady state is assumed for each incremental solution. The computational procedure is linked to a re-zoning procedure. The computational time is not excessive. The details of the formulation and solution technique can be found in Ref. [14]. Each deformation step is treated as a boundary value problem. At the beginning of a given step, the problem domain U (i.e. the volume occupied by the deforming tube), the state of inhomogeneity and the values of material parameters are supposed to be given or determined already. The velocity vector v~ is prescribed on a part of surface Sv together with traction on the remainder of surface Sf. Solution to the incremental problem at any given time provides the velocity and stress distributions that satisfy the governing equations in the body as well as boundary conditions on the surface. The material is assumed as homogeneous, isotropic, incompressible and rigid viscoplastic. The constitutive relation for such a material is given by the Norton–Hoff law [15] as follows pffiffiffi S~ ij Z 2Kð 33_ÞmK1 3~_ij (1) where 3_ ¼



2~ ~ 3_ $3_ 3 ij ij

1=2

;

3~_ij ¼ 1=2ðvi;j þ vj;i Þ

where S~ ij , 3~_ij , K and m represents the components of the deviatoric stress tensor, strain rate tensor, material consistency and strain rate sensitivity index, respectively. The vi is the component of velocity in the direction i at any point of the problem domain. The incompressibility condition is written as below div v~ Z 0 over the problem domain U

(2)

where v~ is the velocity vector at any point of the domain. The material consistency K depends upon the thermo-mechanical condition of the material. For most metals, the behaviour of K can be approximated by means of the following multiplicative law K Z K0 ð1 C a3Þeb=T

(3)

where K0 is a constant, a is the strain hardening parameter, b is the temperature sensitivity term and T is the absolute temperature. The values of the parameters K0, a, b and m can be found by conducting uniaxial tensile tests at different strain rates and temperatures.

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By suitable choice of these parameters, Eqs. (1) and (3) can approximate the mechanical behaviour of most of the metals at different temperatures and strain rate ranges. Using above equations, the constitutive equation for uniaxial case gets the form as follows s Z K0t ð1 C a3Þ3_m

(4) pffiffiffi mC1 b=T where s is the equivalent stress for uniaxial case and K0t Z K0 ð 3Þ e . The friction between the workpiece and the tool is modelled with a viscoplastic law tf Z KaK0 ð1 C a3Þ3_m v~f kv~f kpK1

(5)

where k$k indicates the norm of a vector, v~f is the sliding velocity between tube and platen, a is the friction factor and p is a material parameter whose value is often taken equal to m. Application of the virtual work rate principle to the deforming tube gets to the following functional in terms of the velocity field v ð ð ð pffiffiffi K aK d ~ Z ð 33_ÞmC1 dU K f~ $v~ dSf C kv~ kpC1 dSv~ JðvÞ ðm C 1Þ ðp C 1Þ f Sf

U

Sv~

ð

1 C rp Kð3~_ii Þ2 dU 2

(6)

U

where rp is a penalty number which is assigned a sufficiently large value (y106–107). For ~ must attain a minimum value. In order to get the minimum, the deforming equilibrium JðvÞ body divided into finite elements and the velocity field discretized in terms of the nodal speeds by using shape functions as below v~ Z Nk v~k

(7)

where k is the kth element node, v~ is a velocity component, v~k its value at a node, and N a shape function. For carrying out interaction Gaussian quadrature is used. So-called reduced integration is employed to avoid overstiff response. Let N denote the total number of (global) nodes of the finite element mesh and 2N the degree of freedom of the system (or the finite element mesh). The minimization of Eq. (6) amounts to the problem of finding a 2N-dimensional vector of velocities v~ in the following way vJ dv~ Z 0 vv~

(8)

where dv~ represents a virtual nodal velocity vector. For solving Eq. (8) Newton–Raphson method is employed. In the present case, application of this method leads to iterative solution of the following equations ~ v~i Z K~gi H$D

(9)

v~iC1Z~vC i li$Dv~i

(10)

where g~ i is the gradient vector, H~ the Hessian matrix of the function Jðv~i Þ, li is a descent parameter, Dv~i is the steepest descent direction, and the subscript i denotes the ith iteration. Suitable convergence tests have to be incorporated in algorithm for terminating iterations.

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After getting the solution in terms of nodal velocities, deforming tube configuration is updated by changing the nodal coordinates with the help of the following equation ~ t C Dtt Þ Z Xðt ~ t Þ C vðt ~ t ÞDtt Xðt

(11)

where X~ denotes the vector (of size 2N!1) of nodal coordinates and tt represents time. The problem is solved by dividing the total deformation process into a large number of increments. Eqs. (9)–(11) are solved at each time step, starting from ttZ0. The deforming tube geometry is updated at each time increment which is made sufficiently small for the sake of accuracy. Finite difference procedure [16] is used to obtain continuous distribution of strain and stress in the problem domain. Having completed deviatoric stresses S~ij , the hydrostatic pressure ph is obtained by first expressing the equations of equilibrium in the form ph;i C S~ ij;j Z 0

(12)

whenever the mesh is updated, a check is also made to detect whether one or more elements of the mesh have got overly distorted. If any such case is discovered, a new more favourable mesh is automatically generated. This is done to avoid numerical problems. The remeshing of six nodded isoparametric triangular elements is achieved through a Declaunary–Voronoi type algorithm [17]. Values of 3 at the nodes of the newly created mesh are found by interpolation of the corresponding values at the nodes of the older, distorted mesh. 3.2. Computational model and its features

VERTICAL AXIS OF SYMMETRY

Lateral compression of a metallic round tube has been visualized as a plane strain problem and it involves symmetry of deformation about two mutually perpendicular axes. Therefore, only one upper right quarter portion of the tube cross-section has been taken as the problem domain (see Fig. 9). Six nodded isoparametric triangular elements have been

O (0,0)

Y TOP PLATEN

Y

X Z HORIZONTAL AXIS OF SYMMETRY

X

Fig. 9. Proposed finite element model used for computational study.

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used to discretize the domain. The contact between the platen and tube surface has been assumed as sliding unilateral. The following constitutive relation models mechanical behaviour of the tube material s Z K0t ð1 C a3Þ3_m pffiffiffi where K0t Z K0 ð 3ÞmC1 eb=T , and s is the effective stress, K0 is a constant, a is the strain hardening parameter, m is strain rate sensitivity index, b is the temperature sensitivity term and T is the absolute temperature. If the compression process is performed at a more or less constant temperature then the value of eb/T remains same at all points of the deforming body. To determine the above material parameters namely K0, a and m uniaxial tensile tests were conducted at three different strain rates. Special tensile test specimens were prepared by cutting the same tubes in their axial direction as were used for carrying out the compression tests. A universal testing machine (MTS model no. 810 of 250 kN capacity) was employed to perform the tests. Load–deformation curves were recorded. The true stress versus true strain curves were calculated from the recorded load–compression curves. Fig. 10 shows a typical true stress versus true strain curves used to calculate the material properties of tube specimens. Table 1 presents these parameters for different specimens. The total deformation of the specimen is divided into a large number of small steps or increments. The value of the incremental strain in each increment was varied from 0.2 to 1% of the current tube height. The number of elements actually used in the finite element mesh for discretization varied between 500 and 1100. CPU time required for completion of different simulations was approximately varied between 120 and 400 s on a SUN workstation. The number of remeshings required to simulate the complete compression process was varied between 3 and 5. The computer memory required to store the results of these simulations was varied from 4 to 9 MB. A convergence study was carried out for

300

True Stress (MPa)

250 200

0.1 mm/min

150

2 mm/min 100

20 mm/min Idealized

50 0 0.000

0.020

0.040

0.060

True Strain Fig. 10. Typical true stress and true strain graph of uniaxial tension test for Sp. No. A503.

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optimizing the element size. In the present computational study, simulations were continued until the compression of the tube section was equal to that achieved on the actual specimens during experiments. 3.3. Verification of computational model The proposed finite element model was validated by comparing the predicted load– compression curves and the deformed shapes at different stages of the compression process with those found from the experiments. To compare the experimental and computed deformed profiles only one quarter portion of the tube has been taken and on this two key parameters P and q have been designated in Fig. 11. Computed deformed profiles at various stages of compression for tube specimen A503 and S504 are presented in Fig. 12(a) and (b), respectively. The corresponding comparison of experimental and computed load–compression curves are shown in Fig. 13. Labels 1, 2, 3, etc. on these figures indicate the different stages of the compression process. Comparison of the characteristic dimensions (namely height at the vertical axis of symmetry P and width at the horizontal axis of symmetry q) as shown in Fig. 11 of computed and experimental deformed profiles at different stages of compression for the above specimens are presented in Table 2. The two sets of deformed profiles are found to match with each other fairly closely. Experimental and computational findings of the lateral compression of tubes of different D/t values were also compared. Load–compression curves for certain tube specimens of aluminium and mild steel are presented in Fig. 14. It can be observed that they match very well. Fig. 15 depicts the computed deformed profiles of these specimens. The characteristic dimensions of the final deformed profiles of tubes of different D/t values are presented in Table 3. These dimensions match fairly well. The mean collapse load was calculated from both the experimental and computational load–compression curves. The results are presented in Table 4. The predicted values of the mean collapse load are found to be somewhat greater than those obtained from experiments. Fig. 16 shows the computed and measured variation of the mean collapse load with D/t value for the specimens made from aluminium and mild steel. The variation is non-linear and the value of the mean collapse load is found to decrease with increase in D/t value.

Fig. 11. Quarter cross-section of typically deformed profile of tube showing zones and points of interest.

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Fig. 12. Computed deformed mesh at different stages of compression. (a) Aluminium specimen A503, (b) mild steel specimen S504.

4. Analysis of typical computational results To discuss the compression process, 12 key point designated as A1, A2, A3, G1, G2, G3, H1, H2, H3, B1, B2, B3 along with the two fibres H1H3 and G1G3 at angles of 30 and 608

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1.00 0.90 Aluminium Experimental

0.80

Aluminium Computational

0.70 Load (kN/mm)

5

Mild steel Computational

0.60

Mild steel Experimental

0.50 0.40 0.30

4

3

0.20

5

2

1

4

0.10 0.0

3

2

1

0.00 0.2

0.4

0.6

0.8

1.0

Compression/Average diameter Fig. 13. Computed and experimental load–compressionion curves during lateral compression of aluminium specimen A503 and mild steel specimen S504, (Labels 1, 2, 3, etc. show sequences of profiles in Fig. 12(a) and (b).

from the vertical axis of symmetry are selected on the quarter portion of the tube crosssection bounded by the two planes of symmetry namely the horizontal plane OA3 and the vertical plane OB3 as shown in Figs. 9 and 11. Three zones of interest have also been demarcated within the above portion of the tube cross-section. These can be described as Table 2 Dimensions of the deformed profiles of aluminium tube A503 and mild steel tube S504 at different stages of compression Compression of tube (mm)

Maximum height h (mm)

Vertical height at section B1B3 (P) (mm)

Width at middle horizontal section (q) (mm)

Exp.

Comp.

Exp.

Comp.

Aluminium specimen A503 12.0 39.1 26.0 25.1 30.1 21.0 34.2 16.9 37.4 13.7

36.94 23.53 19.23 12.64 9.784

36.81 23.32 18.41 12.38 9.057

61.54 68.88 70.72 72.34 73.62

62.15 68.84 70.83 72.02 73.63

Mild steel tube S504 9.8 38.34 18.1 30.04 23.8 24.34 28.8 19.34 33.8 14.34 36.9 11.24

38.25 29.40 23.29 17.82 12.05 7.262

38.20 29.50 23.30 17.88 11.74 8.062

55.46 60.48 63.74 66.04 68.32 69.68

56.12 61.10 64.18 66.55 68.46 69.64

Exp., experimental; Comp., computational.

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(a) 0.03

(b) 0.05

Sp. No. A251

Sp. No. A502 Computational

0.04

Experimental

Load (kN/mm)

Load (kN/mm)

Computational

0.02

0.01

Experimental

0.03 0.02 0.01

0.00 0.0

0.2

0.4

0.6

0.8

0.00 0.0

1.0

Compression/Average diameter (c) 0.30

(d) 1.00

Sp. No. S301

0.4

0.6

0.8

1.0

Sp. No. S253 Computational

0.80

Experimental

Load (kN/mm)

Load (kN/mm)

Computational

0.20

0.2

Compression/Average diameter

0.10

Experimental

0.60 0.40 0.20

0.00 0.0

0.2

0.4

0.6

0.8

Compression/Average diameter

1.0

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Compression/Average diameter

Fig. 14. Comparison of experimental and computed load–compression curves for different specimens of aluminium and mild steel during lateral compression.

Zone I: Region B1B3C 0 C adjoining the end B1B3 on the vertical axis of symmetry. Zone II: A1A3E 0 E adjoining the end A1A3 on the horizontal axis of symmetry. Zone III: Region C 0 D 0 E 0 EDC locked between Zones I and II undergoes to elastic deformation.

Fig. 15. Computed deformed meshes of different specimens of aluminium and mild steel after compression.

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Table 3 Dimensions of the deformed shapes of aluminium and mild steel tubes at final stages of compression Sp. code no.

Compression of tube (mm)

Maximum height (h) (mm)

Vertical height at section B1B3 (P) (mm)

Width at middle horizontal section (q) (mm)

Exp.

Comp.

Exp.

Comp. 37.26 35.52 34.46 56.48 52.47 73.56 74.54 73.63 121.24

Aluminium A251 A252 A253 A381 A383 A501 A502 A503 A802

20.2 17.9 15.3 30.8 23.2 38.6 39.9 37.4 66.7

5.14 7.10 9.86 7.36 14.66 11.40 10.70 13.70 14.68

1.88 3.46 7.26 3.06 12.36 8.64 5.16 9.78 5.08

1.804 3.32 8.88 2.35 12.4 5.8 4.354 9.057 4.18

37.56 36.66 34.68 57.38 53.08 74.9 75.82 73.6 122.26

Mild steel S504 S253 S503 S301 S451

36.9 20.40 39.60 25.40 36.40

11.24 6.35 11.10 5.96 7.50

7.26 4.66 7.16 3.58 3.18

8.06 5.18 6.66 3.23 2.38

69.68 38.54 74.16 46.48 66.02

69.64 38.265 74.04 46.20 65.38

Exp., experimental; Comp., computational.

The proposed simulation model was used to obtain detailed computational results of lateral compression of each specimen, however, computational results of a typical specimen (Sp. No. A503, Table 1) are presented below. Input data for this typical case is: average diameter of tube (D)Z47.66 mm, thickness of tube (t)Z3.44 mm, K0Z153 MPa, Table 4 Mean collapse load of annealed round tubes subjected to lateral compression Sp. code no.

Average diameter (D) (mm)

Thickness (t) (mm)

D/t

Aluminium A251 A252 A253 A383 A381 A503 A502 A501 A802

24.50 23.38 21.93 34.73 37.04 47.66 48.92 48.70 79.20

0.84 1.62 3.03 3.13 1.12 3.44 1.68 1.30 2.18

Mild steel S504 S253 S503 S301 S451

44.80 24.41 47.90 30.02 42.74

3.34 2.34 2.80 1.34 1.16

Mean collapse load (kN/mm) Experimental

Computational

29.17 14.43 7.239 11.095 33.07 13.85 29.12 37.46 36.33

0.1239 0.0350 0.2092 0.11891 0.01158 0.08964 0.01811 0.01124 0.02007

0.1182 0.02963 0.2298 0.11479 0.01274 0.10104 0.019055 0.01228 0.0219

14.24 10.43 17.10 22.40 36.84

0.2433 0.2016 0.1402 0.0787 0.02207

0.2363 0.2219 0.13182 0.07933 0.0291

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0.300 Mean Collpase load (kN/mm)

0.270

Experimental Aluminium

0.240 Computational Aluminium

0.210 0.180

Experimental Mild Steel

0.150 0.120

Computational Mild Steel

0.090 0.060 0.030 0.000 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

Average diameter/thickness Fig. 16. Variation of mean collapse load with diameter/thickness ratio.

mZ0.0244, aZ1.451, aZ0.45. The problem domain was discretized into 1086 elements and 2460 nodes. Strain increment in each step of compression was 1%. The total displacement of 37.4 mm of top platen was completed in 153 steps (increments). 4.1. Development of equivalent strain rate and equivalent strain To clearly understand the deformation process, the history of deformation of 12 key points designated as A1, A2, A3, G1, G2, G3, H1, H2, H3, B1, B2, B3 and two planes A1A3 and B1B3 (see Fig. 11) are collected. The points A2, G2, H2, B2 are located in the middle of the planes A1A3, G1G3, H1H3 and B1B3, respectively. Fig. 17 presents the variation of equivalent strain rate ð3_Þ at all these 12 points during the deformation process. It is very clear from this figure that 3_ of the points B1 and B2 increase up to 0.18 compression ratio (8.6 mm compression) and then reduces and becomes nearly zero. The equivalent strain rate 3_ at B1 is higher in comparison with that of B3. This may be due to the nearness of the point B3 with the platen. Equivalent strain rate 3_ at A1, A2 and A3 increases throughout the compression process. If one compares the magnitudes of 3_ of these points one can see that the magnitude at location A1 is highest while at A3 is lowest and at A2 is intermediate. This clearly indicates that the tube is under compression and bending. The magnitude of 3_ at locations G1, G2, and G3 remains almost zero till the compression process reached to 0.76 compression ratio (36.6 mm compression) and there after suddenly increases. The variation shows that the region around layer G1G3 remains undeformed till 0.76 compression ratio and after that deformation starts in the nearby region. The position of the layer G1G3 at this stage remains just left of the tube–platen contact. The magnitudes of equivalent strain rate ð3_Þ at H1 and H3 remain almost zero up to 17.56 mm compression, there after it suddenly start increasing and reaches to a magnitude of 0.00528 and 0.00412, respectively. As further compression proceeds those values

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0.035

Equivalent strain rate

0.03 0.025 0.02

B1

B2

B3

A1

A2

A3

G1

G2

G3

H1

H2

H3

0.015 0.01 0.005 0 0

0.1

0.2 0.3 0.4 0.5 Compression/Average diameter

0.6

0.7

0.8

Fig. 17. Variation of equivalent strain rate 3_ of the points A1, A2, A3, G1, G2, G3, H1, H2, H3, B1, B2, B3 during the compression process for Sp. No. A503.

reduce and attain almost zero value at 30.2 mm tube compression. Magnitude of 3_ at H2 remains zero throughout the compression process. Variation of equivalent strain ð3Þ at all these 12 points during the deformation process is shown in Fig. 18. The magnitude of 3 at points A1, A3, and A2 increases continuously throughout the compression process. Magnitudes of 3 at points A1, A3, and A2 are in the decreasing order at every instant of compression process. Variation in magnitudes is due to the presence of increasing axial force and increasing bending moment at the tube layer A1A3.

Equivalent strain

1 0.9

B1

B2

B3

0.8

A1

A2

A3

0.7

G1

G2

G3

0.6

H1

H2

H3

0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2 0.3 0.4 0.5 Compression/Average diameter

0.6

0.7

0.8

Fig. 18. Variation of equivalent strain 3 of the points A1, A2, A3, G1, G2, G3, H1, H2, H3, B1, B2, B3 during the compression process for Sp. No. A503.

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Equivalent strain 3 at the points B1 and B2 increases uniformly up to 0.18 compression ratio (8.6 mm compression) and attains magnitude of 0.16 and 0.14, respectively. In the remaining compression process these values remain unchanged. This is because the deformation in the surrounding region of B1B3 layer occurs only up to 8.6 mm compression. Variation of equivalent strain 3 at locations H1 and H3 shows that it remains almost zero up to 0.4 compression ratio, there after it suddenly start increasing uniformly and reaches a magnitude of 0.05 and 0.04, respectively, at 0.6 compression ratio. Magnitude of 3 at H2 remains zero throughout the compression process. This clearly indicates that the deformation in the region around layer H1H3 occurs only for a short period (between 0.4 and 0.6 compression ratio) and the section is under almost pure bending. Equivalent strain of points G1, G2, and G3 remains zero up to 0.76 compression ratio (36.6 mm compression). Sudden increase in 3 occurs after this compression ratio. The variation shows that the region around layer G1G3 remains undeformed till 0.76 compression ratio and after that deformation starts in the nearby region. 4.2. Development of plastic zones It is clear now that the plastic regions form at two zones, which are marked as I and II (Fig. 11). To understand clearly the development of plastic regions in these zones the variation of the equivalent strain, equivalent stress, normal stress and shear stress at two planes B1B3 in Zone I and A1A3 in Zone II are plotted in Figs. 19 and 20 at six intermediate stages of compression. Planes B1B3 and A1A3 move downward and outward during the lateral compression of tube on the two axes of symmetries OY and OX, respectively. Therefore, the positions (coordinates) of the points on the planes B1B3 and A1A3 change continuously throughout the compression process. The current coordinates of points on these planes are taken on X-axis and stress and strain components on Y axis. Variation of equivalent strain, equivalent stress, and normal stress at section B1B3 (Fig. 19) show that their magnitudes increase only up to 8.6 mm (increment No. 24) of compression. In the later part of compression process their magnitude do not increase significantly. In the region around section B1B3 actual deformation occurs only up to 8.6 mm compression while after that this region undergoes rigid body displacement. Due to this the strains and stresses increase only up to 8.6 mm compression and after that they remain more or less unchanged. Fig. 20 shows that the magnitudes of equivalent strain, equivalent stress and normal stress increase throughout the compression process. Major increase in magnitudes occurs after 8.6 mm (increment No. 24) of compression. It shows that up to 8.6 mm compression major part of external workdone is dissipated in creating the deformation in the region around the section B1B3 while only a little part of it is dissipated in the region around section A1A3. After 8.6 mm compression the major part of external work done is utilized in creating deformation in the region around the section A1A3. The shear stress remains absent on the planes A1A3 and B1B3 throughout the compression process. In the initial stage of compression, the normal stress distribution on plane A1A3 is almost symmetrical about the zero stress point whereas in advance stages of compression it turns out unsymmetrical. This indicates that in the initial stages moment is dominant over the axial force and in the advance stages vice-versa. From Figs. 19 and 20, it can be concluded that

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0.16

(a)

0.14 Equivalent Strain

0.12 0.1 Increment No. 2 Increment No. 24 Increment No.32 Increment No. 100 Increment No. 130 Increment No. 152

0.08 0.06 0.04 0.02 0 0

4

8 12 16 20 Y coordinate of width at B1B3

24

28

Equivalent Stress (Mpa)

(b) 300

200 Increment No. 2 Increment No. 24 Increment No.32 Increment No. 100 Increment No. 130 Increment No. 152

100

0 0

(c)

4

8 12 16 20 Y coordinate of width at B1B3

28

24

28

400 300

Normal Stress (Mpa)

24

Increment No. 2 Increment No. 24 Increment No.32 Increment No. 100 Increment No. 130 Increment No. 152

200 100 0 –100 –200 –300 –400 0

4

8

12

16

20

Y coordinate of width at B1B3

Shear Stress (Mpa)

(d) 4

0

–4

Increment No. 2

Increment No. 24

Increment No.32

Increment No. 100

Increment No. 130

Increment No. 152

–8 0

4

8

12

16

20

24

28

Y coordinate of width at B1B3

Fig. 19. Variation of (a) equivalent strain, (b) equivalent stress, (c) normal stress, and (d) shear stress at section B1B3 for Sp. No. A503.

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Equivalent Strain

(a)

1 Increment No. 2 Increment No.24 Increment No.32 Increment No. 100 Increment No. 130 Increment No. 152

0.8 0.6 0.4 0.2 0 22

27

32

37

X coordinate of width at A1A3

Equivalent Stress (Mpa)

(b) 600

Increment No. 2 Increment No. 24 Increment No.32 Increment No. 100 Increment No. 130 Increment No. 152

500 400 300 200 100 0 22

27

32

37

X coordinate of width at A1A3

Normal Stress (Mpa)

(c) 900

Increment No. 2 Increment No.32 Increment No. 130

600

Increment No. 24 Increment No. 100 Increment No. 152

300 0 –300 –600 22

27 32 X coordinate of width at A1A3

37

Shear Stress (Mpa)

(d) 200

0

–200

Increment No. 2

Increment No. 24

Increment No.32

Increment No. 100

Increment No. 130

Increment No. 152

–400 22

27 32 X coordinate of width at A1A3

37

Fig. 20. Variation of (a) equivalent strain, (b) equivalent stress, (c) normal stress, and (d) shear stress at section A1A3 for Sp. No. A503.

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up to 8.6 mm compression (increment No. 24) the Zone I reaches in the plastic state and afterwards the plastic region develops in Zone II. 4.3. Distribution of contours Although the analysis provides results for each increment of compression, contour distributions of nodal velocity, equivalent strain rate and equivalent strain have been plotted for only three typical stages of the compression process, corresponding to platen displacement of (I) 8.6, (II) 34.2 and (III) 37.4 mm, respectively. 4.3.1. Velocity distribution During the compression process the platen moves downward at a uniform speed of 10 mm/s. It imparts velocities of varying magnitudes and directions at different material points. Fig. 21(a) and (b) depict the distribution of velocity vectors in magnitude and direction at different points in the tube. It is observed that during the initial stage of compression, the dominant movement is downward in Zone I. This is accompanied by a smaller movement of section A1A3 in the outward direction. It indicates that before 8.6 mm compression the deformation mainly concentrates around the Zone I. With progress of compression, the difference between the above two velocities tends to increase. The minimum value of nodal velocity occurs at a point on the outer periphery in the D 0 E 0 tube profile portion. The lower value contours are concentric about the minimum velocity point, which indicates that the tube rotates about this point as centre of rotation. Largest magnitude of velocity occurs always on the B1B3 plane. This is due to the rigid body motion of the Zone I. 4.3.2. Equivalent strain rate pffiffiffiffiffiffiffi The equivalent strain rate 3_ Z 2=3ð_32xx C 3_2xy C 3_2zz C 2_32xy Þ1=2 , since due to the zero volume change 3_xx C 3_yy C 3_zz Z 0, and 3_zz y0 duepto ffiffiffi the plane strain condition. Therefore, 3_xx ZK_3yy . So the equivalent strain rate 3_ Z 2= 3ð_32xx C 3_2xy Þ1=2 means the variation of equivalent strain rate can depict qualitatively variation of strain rate or other way change in strain between two consecutive increments during the compression process if shear strain rate is negligible. Fig. 21(c) depicts the variation of the equivalent strain rate. At each stage of the compression process, two zones of intense activity can be seen, one just left of the current point (or area) of contact between the tube and the platen and the other adjoining the horizontal axis of symmetry (referred to as Zone II). The location of the former zone changes with time because of the changing location of the contact area. During the initial stage of compression, the highest strain rate occurs in Zone I whilst in the following stages, it occurs in Zone II. Smallest rates are found in Zone III. The distribution confirms the development of plastic regions in Zones I and II. 4.3.3. Equivalent strain The variation of equivalent strain is shown in Fig. 21(d). It is clear from this figure that in the initial stage of compression, the major deformation occurs in Zone I. But later on, most of the deformation in tube takes place in Zone II. The maximum value of equivalent

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Fig. 21. Predicted variation of (a) magnitude of nodal velocity, (b) direction of nodal velocity, (c) equivalent strain rate 3_, and (d) equivalent strain 3, at three stages of compression (Sp. No. A503).

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strain in Zone I is only 0.155 whereas it is as high as 0.9005 in Zone II. The strain in Zone III is relatively very small.

5. Discussion pffiffiffi The equivalent strain and strain component relationship can be written as 3 Z 1= 3 ! ð432xx C 232xypÞ1=2 ffiffiffi since 3_xx ZK_3yy so 3xx ZK3yy . If the magnitude of shear strain is very small then 3 Z 2= 33xx . Means equivalent strain shows the qualitative variation of 3xx in the tube. The maximum value of 3xx or 3yy under the hypothesis of a linear distribution of strain through the thickness for complete flattening of the tube is equal to thickness/outer diameterZ 3.44/51.1Z0.067 (corresponding to the change in curvature 1/R). Since the every crosssection of the tube is compressed in such a way that the change in curvature is more than 1/R, so strain at the extreme fibres will also be higher than 0:067ð3Z 0:077Þ. The shear stress on two planes A1A3 and B1B3 is negligible (see Figs. 19(d) and 20(d)). The maximum strain (3xx) at points B1 and B3 are 0:129ð3 Z 0:15Þ corresponding to a curvature change of 2/R. This strain occurs at 0.18 compression ratio (8.6 mm compression). The maximum strain at points H1 and H3 are 0:073ð3 Z 0:083Þ corresponding to a curvature change of 1.2/R. This section acquires this change in curvature at 0.7 compression ratio. The strain at points A1 and A3 continue to increase monotonically and acquires a value of 0:7794ð3 Z 0:9Þ at compression ratio 0.79. After measuring the final deformed shape it was found that the change in curvature of points B1 and B3 was 2/R, H1 and H3 was greater than 1/R. Therefore, the final maximum strain of these points should not be equal. These final strains with their corresponding compression ratio at different locations are in good agreement with the experimental findings of the Reddy and Reid (see Fig. 13, Ref. [11]). The following points can be concluded on the basis of the above discussion: 1. Equivalent strain at both the extreme fibres of the different layers (except Zone I) of the tube is not equal in magnitude throughout the compression process. This indicates that the tube is subjected to bending and compression. 2. The highest value of the equivalent strain at extreme fibres of the different layers is always more than the value of the strain (t/D) obtained by the hypothesis of a linear balanced distribution of strain through the thickness. 3. Tube is deformed under the development of two plastic zones (around B1B3 and A1A3) and the tube between these plastic zones remains almost undeformed. Out of these two zones, one zone (around B1B3) continuously spreads with the progress of compression while other around A1A3 remains concentrated at its position.

5.1. Effect of friction between tube–platen interface Effect of friction at the interface of platen and tube has been studied by performing the simulation study for four different values of friction factor a, which are 0.45, 0.35, 0.20,

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0.40 friction factor = 0.05 friction factor = 0.20

Load(kN/mm)

0.30

friction factor = 0.35 friction factor = 0.45

0.20

0.10

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Compression/Average diameter Fig. 22. Effect of friction factor on load–compressionion curve (Sp. No. A503).

and 0.05. Fig. 22 shows the load–compression curve during lateral compression of specimen A503 for different values of friction factor a. It can be concluded from this figure that, the friction presents between tube–platen interface does not affect the energy absorbing capacity of round tubes subjected to lateral compression.

6. Mode of collapse To understand the pattern of deformation occurring during lateral compression of tubes, the compression process is divided into three stages, referred to as the initial, intermediate, and final. The progress of deformation during lateral compression of round tubes is typified by the collapse of specimen number A503 (Fig. 1(a)). Initial tube profile is convex over its whole periphery. Therefore, there is only a line contact between the tube and the platen, creating stress concentration at and around B1B3 (Fig. 11). In the initial stage of compression, therefore, deformation is mainly restricted to Zone I (or the site of the first plastic hinge). However, the area of contact gradually builds up and the portion B3C 0 of the boundary flattens out. The load required to continue compression tends to increase (Fig. 2). Increase of contact area and strain hardening of material in Zone I lead to gradual decrease of rate of deformation in Zone I. The site of dominant deformation next shifts to Zone II. This marks the beginning of the intermediate stage of compression, and formation of the second plastic hinge around A1A3. However, the required load is higher as compared to that in stage I. This is because for a given load the magnitude of bending moment at A1A3 is lower than that at B1B3. With increasing downward movement of the platen, the character of platen tube contact undergoes marked change. The first plastic hinge moves outward from B1B3 to CC 0 and still further away. As this happens, there is loss of contact over the boundary B3C 0 –D 0 . If the platen load was to remain unchanged, the bending moment in Zone II would actually decrease. To compensate the decrease in the so-called

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‘moment arm’ the platen load increases. In the final stage of the compression process, point B1 comes quite close to the horizontal axis OA3 and the curvature in Zone II starts increasing at a high rate. This necessitates increase of bending moment in Zone II, and the corresponding load also increases sharply.

7. Conclusions Lateral collapse of aluminium and mild steel round tubes of different D/t ratios was investigated experimentally and computationally. load–compression curves and deformed shapes of the specimens at different stages of compression were recorded. The predicted curves and deformed shapes at different stages of compression are found to match well with those found from experiments. These tubes were found to collapse due to formation of two plastic zones along with the elastic deformation of tube in between. The load–compression curves of different tube specimens of different D/t values have similar kind of variation. These curves can be divided into two parts. First part, which have constant slope while the second part of load–compression curves has increasing slope at every instant of compression. Among the specimens of different D/t values specimens of lower D/t values have lower range of second part of load–compression curve with high load values and therefore their energy absorbing capacity is more as compared to the tubes of higher D/t values. It is seen that specimens of lower diameters have higher energy absorbing capacity and mean collapse load among the specimens of equal thickness but different diameters. The mean collapse load increases with decrease in diameter for the specimens of equal thickness but different diameters. The energy absorbing capacity and mean collapse load increases with increase in thickness for the specimens of equal diameter and different thicknesses. The friction between tube–platen interfaces has negligible effect on the load– compression and energy–compression curves during the compression.

References [1] Ezra AA, Fay RJ. An assessment of energy absorbing devices for prospective use in aircraft impact situations. In: Hormone G, Perron N, editors. Dynamic response of structures. Proceeding of a symposium held at Stanford University, California; June 28 and 29, 1972. p. 225–46. [2] Jones N, Wierzbicki T. Structural crashworthiness. London: Butterworth & Co.; 1983. [3] Gupta NK. Plasticity and impact mechanics. NewAge International (P) Limited, New Delhi, India; 1996. [4] Johnson W, Reid SR. Metallic energy dissipating systems. Appl Mech Rev 1978;31:277–88. [5] Mutchler LD. Energy absorption of aluminium tubing. J Appl Mech 1960;27:740–3. [6] DeRuntz JA, Hodge PG. Crushing of a tube between rigid plates. J Appl Mech 1963;30:391–5. [7] Redwood RG. Discussion of (ref. 6) crushing of a tube between rigid plates. J Appl Mech 1964;31:357–8. [8] Burton RH, Craig JM. An investigation into the energy absorbing properties of metal tubes loaded in the transverse direction. BSc (Engg) report, University of Bristol, Bristol, England; 1963. [9] Reid SR, Reddy TY. Effect of strain hardening on the lateral compression of tubes between rigid plates. Int J Solids Struct 1978;14:213–25. [10] Reddy TY, Reid SR. Lateral compression of tubes and tube systems with side constraints. Int J Mech Sci 1979;21:187–99.

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[11] Reddy TY, Reid SR. Phenomena associated with the crushing of metal tubes between rigid plates. Int J Solids Struct 1980;16:545–62. [12] FORGE2. Finite element analysis code for metal forming problems version 2.5, cemef, sofia Antipolis, France; 1996. [13] Pittman JFT. Numerical analysis of forming processes. Chichester: Wiley; 1984. [14] Gupta PK. An investigation into large deformation behaviour of metallic tubes. PhD thesis, Indian Institute of Technology Delhi, India; 2000. [15] Hoff NJ. Approximate analysis of structures in presence of moderately large creep deformations. Q Appl Math 1954;12. [16] Liszka T, Orkisz J. The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput Struct 1980;11:83–95. [17] Bellet M, et al. editors. Plasticity and metal forming. Seminar proceedings Sophia antipolies; 1990.