Safety and Security Engineering

83

Finite element analysis of hexagonal tube structures under axial loading M. R. Said1, R. Ahmad2 & A. Alias3 1

Faculty of Mechanical Engineering, Kolej Universiti Teknikal Kebangsaan Malaysia, Melaka, Malaysia 2 Universiti Sains Malaysia, Malaysia 3 Universiti Teknologi Malaysia, Malaysia

Abstract Elastic-perfectly plastic material is assumed in the model using the non-linear facilities of the finite element code ABAQUS in hexagonal tube structures. A four-noded quadrilateral (S4R) of shell elements is used in the analysis. The load-displacement curves are examined and compared with experiment. It has been found both experimentally and throughout the finite element method that the form of the two load-displacement curves are approximately the same throughout the compression. The FE mean load is in close agreement with experiment. However, the FE plastic fold length is found to have overestimated the experimental result by 10%. The strain history is also examined and it is found that the form of the predicted curves are in agreement with the experiments at locations away from the end. Keywords: finite element analysis, hexagonal tube, axial compression, quasi-static.

1

Introduction

Energy absorption systems made of thin-walled structures have been given great attention by researchers in the past, particularly over the last few decades. The initiation of buckling of axially compressed cylindrical tubes has been a subject of intense investigation by many famous researchers. Mallock [1] was the first to report experiments on the crushing of thin circular tubes, observing progressive axisymmetric folding in the thicker tubes and non-axisymmetical folding in the thinner ones. Alexander [2] presented the first of the analyses for the prediction

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

84 Safety and Security Engineering of a mean crushing load, Fm for a tube subjected to axial compression progressively folding. By equating the work done by the applied (mean) load and the work dissipated by plastic deformation, he obtained an expression for the mean load, Fm, minimising from which the geometry of the plastic half fold length, defined by H, was obtained as 1.347 rt , where t is thickness and r is the radius of the tubes. A discussion of the assumptions made by Alexander [2] and the effects of these have been studied by several authors. These have been summarised by Jones [3]. In all these models and analyses, the tube wall moves out of its initial position while in reality there is outward as well as inward movement. This anomaly was first addressed by Reddy and Reid [4] albeit by an over prescribed hinge (four) mechanism. Wierzbicki et al [5] presented a more realistic three hinge mechanism representing a concentina collapse mode. The mechanism allows for radially inward as well outward folding. An eccentricity factor, e was introduced to define the part of the plastic fold, H that falls outside of the initial position. However, the mean load, Fm deduced was independent of the eccentricity factor, e, which was indeterminate. Singace et al [6] further modified Wierzbicki’s analysis enabling the determination of the eccentricity factor, e and the value obtained agreed well with the experimental results. However, non-circular tube structures, particularly of square, rectangular and hexagonal shape, seem to be more popular than circular tubes in the automobile industry. Efforts have been made to improve the capability of energy absorption of sheet metal tubes, particularly of empty, wood-filled and foam-filled rectangular tubes under static and dynamic crushing. Reid and Reddy [7], Reid et al. [8] and Reddy and Al-Hassani [9] have shown that filling the tubes with foam or wood increases the energy absorbed by inducing a larger overall plastic deformation of the shell wall. Said [10] presented an experimental measurement of elastic half wavelength and plastic fold length during the crushing of rectangular tubes. It is noted that the plastic fold length depends on the elastic wavelength, which can also be found from the deformation pattern. Abramowicz and Wierzbicki [11] developed the theory for predicting crush behaviour for multicorner columns with an arbitrary corner angle. This was applied to find the crushing strength of a hexagonal tube under axial compression to produce a 0.4

simple expression for the mean crushing as Fm/Mo= 80.92 (b/t) . The most recent study was done by Seitzberger et al [12] on an hexagonal aluminium foam-filled tube subjected to quasi-static axial loading. Surprisingly, there are not many Finite Element Analyses (FEA) of tubes, particularly in hexagonal, crushed under axial loadings. Kormi et al. [13] used the ABAQUS FE package to investigate the crash response of a thin cylindrical shell under the action of a static and dynamic load, which induced a combination of both compression and twisting responses. It has been illustrated by Reid and Yang [14] that FE investigations of gross plastic deformations of axially compressed tubes can yield extremely useful information. In this paper, the finite element code has been used to compare the experimental results of the deforming mode, plastic wave-length and the strain histories of hexagonal tube under axial compression. The FE results are in good agreement with experiments [15].

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

Safety and Security Engineering

85

However, there have not been many studies on the finite element analysis (FEA) to confirm the results of the crushing strength. The present study is to compare the FE results with experiment [15]. This includes the mode of deformation, plastic folding, strain histories and the load-displacement characteristics.

2

Finite element modelling

The finite element code ABAQUS/Explicit [16] has been employed to model hexagonal tube under quasi-static loading. The tube is made of mild steel. Only one type of material model, i.e. elastic-perfectly plastic was used in the analysis for the as-received material. The model curve is shown in Figure 1 along with the nominal stress-plastic strain curve. One (longitudinal) half of a tube was modelled and the length of the tube considered was 170 mm. A four-noded quadrilateral (S4R) of shell elements was used in the analysis. With these elements, the transverse shear strains are calculable at the reduced integration points as well as the membrane and bending strains. The element size was 10 mm by 5.3125 mm with a thickness of 1.87 mm. Hence, there were 384 elements with 429 nodes in the 170 mm long specimen. However, the thickness does not appear in the model as a plane strain of the element is used. Hence, the chamfered edges could not be modelled in this case. The base of the tube was constrained completely while a rigid body element type R3D4 was used to model the platen that compressed the tube axially from the top. The initial contact between the platen and the tube was modelled and later with other parts of the tube as it folds. The coefficient of friction for all contacts was taken as 0.3. 500

Vy= 435 MPa, E=212 GPa, X=0.27

Nominal stress (MPa)

400

300 For as-received material.

200

elastic-perfectly plastic nominal stress-strain curve

100

0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

strain, H

Figure 1:

3

Material properties used in ABAQUS analysis model.

Result of FEA and comparison with the experiment

3.1 The load-displacement characteristics and the mode of deformation Figure 2 shows the comparison between experimental and the predicted loaddisplacement traces. The agreement shows that the combination of the

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

86 Safety and Security Engineering coefficient of friction of 0.3 and the assumed elastic-perfectly plastic material model is satisfactory. The experimental load-displacement curve is for an unchamfered specimen [13]. However, the first peak load produced by FEA is higher by 8% than that in the experiment. This could be the effect of the built in lower end of the tube in the FE model. The predicted loads at the second, third, and fourth peaks gradually increase as also observed in experiment. The second and third valley loads overestimate the experiment by about 20% but the first valley underestimates it by about the same amount. Secondary fluctuations appear in every fold in the FEA but only in the first fold in the experiment. In general, the form of the two load-displacement curves is approximately the same throughout the compression, of 100 mm. The mean load up to displacement of 95 mm is approximately 82 kN in the FE analysis, which is close to the experimental mean load of 84 kN. The deforming mesh shown in Figure 3a exhibits a concentina mode. It starts to crush at the top simply supported end, instead of the built in bottom end. The cross-sectional views of deforming models at mid face are also shown in Figure 3b at various degrees of displacement. Points 0 to 6 in Figure 3b are associated with point those on loaddisplacement curve shown in Figure 2. The plastic fold length, λp was found to be approximately 36 mm, which overestimates the experimental result by 10%. The final deformed cross-sectional configurations (experiment and FE analysis) are shown in Figure 4. 220 0

200

ABAQUS model: Elastic-perfectly plastic

180

experiment ABAQUS

Axial load, F ( kN )

160

6

4

2

140

Fm=82 kN

120 100 80 60 3

40 20

5

Fm=84 kN

1

0 0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

displacement,G ( mm )

Figure 2:

Comparison between displacement curves.

ABAQUS

and

experimental

load-

The energy histories in the crushing analysis are shown in Figure 5, which include the total strain energy, energy dissipated, energy due to friction and external work. It shows that the energy dissipating by friction is very small and does not influence the energy absorbed. The other energies gradually increase with small fluctuate in the ns, which indicate the formation of folds. The total energy is the sum of dissipated energy, energy due to friction and a small amount of elastic strain energy.

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

Safety and Security Engineering

Figure 3:

87

The ABAQUS deforming mesh of a tube under quasi-static axial compression (a) half-model isometric view (b) Cross-sectional view (only one side) at various levels of displacement (point 0 – 6 referred to Figure 2).

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

88 Safety and Security Engineering

Figure 4:

A comparison deformed specimens, experiment and ABAQUS, at displacement of 95 mm.

12 10

external work (allwk) total strain energy (allie) energy dissipated (allpd) energy due to friction(allfd)

Energy ( kNm)

8 6 4 2 0 -2 0

20

40

60

80

100

120

displacement (mm)

Figure 5:

Graph of energy-displacement curve from FEA for a single hexagonal tube under quasi-static compression.

The predicted load-displacement curves for the single hexagonal tube, with length of 200 mm under axial compression were also analysed. The form of the two load-displacement curves is the same, and the agreement is also good. 3.2 Strain histories. Said [13] has discussed the observed strain history for a specimen length, L of 200 mm under quasi-static axial compression. Typical measured strain histories at two different locations (15 mm and 60 mm away from the top end and midface) are compared with the corresponding strains computed by the FEA in Figure 6. The curves shown are up to the displacement of 35 mm and beyond this displacement, the comparison between FEA and experimental results is not appropriate due to the saturation of the strain gauge. WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

Safety and Security Engineering

89

Axial strain(%)

0 a)

-1 -2

circumferential strain, (%)

-3 3.0

b)

2.5 2.0 1.5 1.0 0.5 0.0

0.0

c)

circumferential strain, (%) Axial strain(%)

-0.5 -1.0 -1.5 -2.0 -2.5

Figure 6:

0.5

d)

0.4 0.3 0.2 0.1 0.0

0

5

10

15

20

25

displacement (mm)

30

35

Strain-displacement curves, FEA (solid lines) and experiments (dashed lines) (a) and (b) axial and circumferential strains at distance of 15 mm from top end (c) and (d) axial and circumferential strains at a distance of 16 mm from the top end.

200 200 180 180 160 160

120 120

Load (kN)

Load (kN)

140 140

100 100 80 80 60 60 40 40 20 20 00

-2.5 -2.5

Figure 7:

-2.0 -2.0

-1.5 -1.0 -0.5 -1.5 -1.0 -0.5 axial strain strain (%) axial (%) (a) (a)

0.0 0.0 0.0 0.0

0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 -0.5 -0.5 circumferential strain circumferential strain(%) (%) (b) (b)

-0.4 -0.4

-0.3 -0.3 -0.2 -0.2 -0.1 -0.1 axialstrain strain (%) axial (%) (c) (c)

0.00 0.0 0.00 0.0

0.05 0.05 0.10 0.10 0.15 0.15 0.20 0.20 0.25 0.25 circumferential strain circumferential strain(%) (%) (d) (d)

Typical load-strain curves, FEA (solid lines) and experiment (dotted lines) at mid-face (a) (b) axial and circumferential strain at 15 mm, and (c) and (d) Axial and circumferential strain at 60 mm.

The axial and circumferential strains 15 mm away from the top end are shown in Figure 6a and Figure 6b, respectively. The deviation between predicted and measured axial strains gradually increases and while the strain gauge output

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

90 Safety and Security Engineering saturates a displacement of about 5 mm, the predicted strains increased monotonically. For the case of circumferential strain curves, the deviation fluctuates, between 40% (at δ = 5 mm), 10% (δ = 15 mm), 30% (δ = 20 mm) and then (after δ = 23 mm) gradually increases. The measured strains saturate and even show a decrease after δ = 12 mm. The axial strains at 60 mm from the end (Figure 6c) are in good agreement. However, the circumferential strains (Figure 6d), the predicted strain curve are overestimated by about 50%, but the form of variation is the same. The loads versus strain variations are also plotted in Figures 7.It can be seen in all cases that the form of the predicted curves are in agreement with the experiments at locations away from the end.

4

Conclusion

The comparison between experiments and computer modelling of hexagonal tubes under axial compression was investigated. The FE results suggest that the coefficient of friction of 0.3 in the modelling and elastic-perfectly plastic material model is satisfactory. The mean load was 82 kN in the FE analysis, which was close to the experimental mean load of 84 kN. The plastic fold length,

λp in FE was also close to experiment. The mode of deformation of axial compression of the hexagonal tube was compared with FEA. FEA gave excellent results for the case of axial compression of single hexagonal tubes which deform in a concertina mode. The FE strain curves were also compared with experiments. It can be noticed that the form of the predicted curves was in agreement with the experiments at locations away from the end.

References [1] [2] [3] [4] [5] [6]

A. Mallock, “ Note on the instability of tubes subjected to end pressure, and on the folds in a flexible material.” Proc. of the Royal Society of London, vol. LXXXI, Series A., 1908. J.M. Alexander, “An approximate analysis of the collapse of thin cylindrical shells under axial loading”, Quart. J.Mech. Appl. Math, vol. 13, 1960, pp 10-14 N. Jones, Structural Impact, Cambridge University Press, U.K., 1989. T.Y. Reddy, and S.R. Reid, “Plastic folding of cylinders under axial compression” Presented at Euro Mech. 1st European Solid Mech Conference at Munich, 1990 T.Wierzbicki, S.U. Bhat, W. Abramowicz, and D. Brodkin, “Alexander revisited-a two folding elements models of progressive crushing of tubes” Int. J. Solids Structures vol. 29, 1992, pp 3269-3288. A.A. Singace, H. Elsobky, and T.Y. Reddy, “On the eccentricity factor in the progressive crushing of tubes” Int. J. Solids Structures vol. 32, 1995, pp 3589-3602.

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)

Safety and Security Engineering

[7] [8] [9] [10] [11] [12]

[13] [14] [15] [16]

91

S.R. Reid, and T.Y. Reddy, “Axial crushing of foam-filled tapered sheet metal tubes”. Int. J. Mech.Sci.,Vol.28,No.10, 1986, pp 643-656. S.R. Reid, T.Y. Reddy, and M.D. Gray, “Static and dynamics axial crushing of foam-filled sheet metal tubes” Int. J. Mech. Sci.,28, 1986, pp295-322 T.Y. Reddy, and S.T.S. Al-Hassani, “Axial crushing of wood-filled square metal tubes”. Int. J.Mech.Sci. Vol.35, No.3/4, 1993, pp 231-246. M.R. Said, “Axial compression of empty and filled rectangular tubes”. MSc. (UMIST) Dissertation, 1988 W. Abramowicz, and T. Wierzbicki, “Axial crushing of multicorners sheet metal columns” J. App. Mech., Trans ASME., 56, 1989,pp 113-120. M. Seitzberger,. F.G. Rammerstorfer, R. Gradinger, H.P. Degischer, M. Blaimschein, C. Walch, “Experimental studies on the quasi-static axial crushing of steel columns filled with aluminium foam” International Journal of Solids and Structures, 37,2000, pp 4125-4147. K. Kormi, D.C. Webb, and W. Johnson “The crash response of circular tubes under general applied loading”. Int. J. Impact Engng, Vol. 13, No.2, 1993 pp 243-257 J.D. Reid, and K.H. Yang “Crashworthiness and occupant protection in transportation systems”. ASME Technical Publishing, 1993, New York. M. R. Said,. “Energy absorption in certain cellular structures under uniaxial and biaxial loading” PhD. Thesis, UMIST (2000). Hibbit, Karlsson and Sorensen Inc., ABAQUS User’s Manual, ABAQUS Theory Manual and ABAQUS Examples Manual, 1996

WIT Transactions on The Built Environment, Vol 82, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line)