Journal of Engineering Science and Technology Vol. 10, No.10 (2015) 1350 - 1360 © School of Engineering, Taylor’s University
THE NUMERICAL SIMULATION OF AXIAL CRUMPLING IN GROOVED CIRCULAR PVC TUBES UNDER STATIC COMPRESSION 1,
2
AKRAM S. MAHMOOD *, AYAD A. AL-BADRANY , ARZ Y. RZAYYIG
2
1
2
Civil Engineering Department, Engineering College, Anbar University, Iraq Mechanical Engineering Department, Engineering College, Anbar University, Iraq *Corresponding Author:
[email protected]
Abstract In the simulation works described in this paper the ANSYS finite element package is used to investigate the compressive properties and crushing response of circular grooved PVC tubes subjected to static axial testing. A series of models was created to simulate the static tests performed in the laboratory using PVC tubes with and without grooves featured by the same material combination and external cross-section dimensions, but different grooved length. Simulation works focused on modelling the modes of collapse observed in the series of static compression tests. Satisfactory level of agreement between numerical analysis and experimental results were obtained regarding the main ultimate characteristics of the tested PVC tubes such as peak compressive load and the overall crushing response as the finite element models were refined several times in order to obtain optimum results. Keywords: Finite elements, PVC tubes, Grooved tubes, Static compression.
1. Introduction The most widely used method of design of composite collapsible energy absorbers is by experimental testing in order to quantify mechanical properties and behavior of structures, since the complex failure mechanisms that characterize the overall crushing response of composite structures - such as transverse shear, inter- and intra-laminar cracking, laminate splaying or local wall buckling are difficult to be predicted by analytical methods. However, the high cost of experimental works makes the design by means of numerical methods and especially finite element [1]. 1350
The Numerical Simulation of Axial Crumpling in Grooved Circular PVC . . . . 1351
Nomenclatures Di Dg Dm Dr L Lg Lr t n P Pm Pmax x
Inside diameter of tube with grooves and without grooves, mm Outside diameter of grooved region of the tube, mm Mean diameter of tube, mm Outside diameter at a ringed region of the tube, mm Axial length of tube, mm Axial length of a groove, mm Axial length of a ring, mm Thickness of tube at groove, mm Number of grooves Theoretical of mean crumpling load, kN Experimental of mean crumpling load, kN Experimental initial peak load, kN Depth of a groove, mm
Greek Symbols σ y δ
Yield stress, MPa Reduction in axial length or (deflection), mm
Abbreviations CHS FEA GHT
Circular Hollow Section Finite Element Analysis Grooved Hollow Tubes
The stability of circular hollow section members (tubes) has been studied both analytically and experimentally since the beginning of 20th century by researchers. The buckling behavior of axially compressed cylindrical members was first approximately analyzed by Lorenz in early of (1908). However, experimental test results indicate that cylinders buckled at loads well below those predicted by the early theoretical solutions based on small deflection theory. The establishment of a general theory of structural stability and the influence of imperfections (sensitivity) on the shell stability behavior was studied. Since then, numerous studies on the behaviour of axially compressed circular hollow section (CHS) members have been carried out, in particular, for short cylinders [2]. Concerning the stability behavior of grooved hollow section tubes (GHT), there is a much smaller amount of work. Due to the fact that the radius varies along the longitudinal line, the stability analysis becomes increasingly complex. Axial crumpling of nonmetallic tubes has long been the subject of extensive research [3]. Symmetric and nonsymmetric circular shapes provide perhaps the widest range of all choices for use as absorbing units because of their favorable plastic deformation under axial compression circular tubes have comparatively high energy absorbing capacities and feature a favorable stroke length per unit mass. The cylindrical shells absorbers having symmetric cross-sections may crushing in axi-symmetric and non-symmtric or mixed when subjected axial forces. In fact, it was reported [4 The crumpling of components by splitting or by inversion is reported that the axial crumpling of thin cylinders under quasi-static compression can be classified
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into seven different categories, based on experimental results: (a) sequential concertina; (b) sequential diamond; (c) Euler; (d) Concertina and diamond; (e) Simultaneous concertina; (f) Simultaneous diamond; and (g) tilling of tube axis [5]. Several experimental, analytical and numerical studies have appeared in recent years, which help in understanding of the plastic mechanics of the collapse phenomenon. These experimental studies [6-8], have shown the effect of geometrical shapes on the mode of deformation and mean crushing load. Finite element simulations are used widely in the automotive industry to predict the response of energy dissipating structures. Such simulations are used in the development process to compare different concepts and to improve the behaviour. In order to make correct decisions the predictability of the models is crucial. Thin-walled structures are used as energy dissipating structures in cars. Normally, these structures are modelled by use of plane stress shell elements. In several investigations, a reasonable agreement between shell-based finite element simulations and experimental results has been found. Peixinho et al. [9]obtained good agreement between numerical simulations and experimental results for axial crushing of thin-walled sections made of high-strength steels. Good correlation was also found between experiments and simulations of thin-walled high-strength steel sections subjected to axial crushing by Tarigopula et al. [10]. Yamashita et al. [11] showed that computations and experiments agreed well for axial crushing of cylindrical structures with various polygonal cross-sections. In some of the simulations the mean crushing force was overestimated, owing to material fracture in the experiments. Also Fyllingen et al. [12] studied the influence of element type and formulation is investigated in finite element analyses of aluminium profiles subjected to axial crushing. It is shown that solid elementbased simulations give predictions in better agreement with experiments. On other side, energy absorption and mean crushing load of thin-walled steel tubes containing annular grooves are theoretically studied by Hosseinipour [13]. The grooves are introduced in the tube to force the plastic deformation to occur at predetermined intervals along the tube. In this paper, some experimental and numerical results are reported for static axial compression of circular tubes with and without grooves was made from PVC. The load-deflection curved of specimens was proposed in this study. The manner in which the tubes collapse is compared with results of ANSYS package using solid elements. The aim is to assess the accuracy, robustness and efficiency of the different element types and formulations.
2. Experimental Work and Theoretical Models 2.1. Experimental work Circular tubes with and without grooves of PVC with the same sizes were subjected to axial compression in an Instron machine. The speed of testing was (5mm/min) and load compression curves in the test were obtained on the automatic recorder of the machine. The initial value of yield stress was estimated to be 69 MPa. Details relating to the geometrical dimensions are presented in Table 1 which refers to the geometrical shape as shown in Fig. 1. For grooved specimens the Journal of Engineering Science and Technology
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length of grooves Lg, and depth of grooves, x values were kept constant and equal to (3.125mm) and (0.75mm), respectively. While the ungrooved length (Lr) was varied in the range (0< Lr/L≤1). All specimens were tested at room temperature. The values of mean crumpling load (P m) and initial peak load (Pmax) were deduced from the load-deflection curves as shown in Table 1. The value of mean load equals the plastic work (area under the curve) divided on deflection.
Dr (mm)
n Number grooves
Ratio
Dg (mm)
125 125 125 125 125
1 0.176 0.144 0.1 0.05 0.025
Inside Diameter (Di) (mm)
125
Outside diameter
Groove Length (Lg) (mm)
1 2 3 4 5 6
Axial length (L) (mm)
Trial No.
Table 1. The geometry and dimensions of grooved PVC circular tubes.
3.125
41.5 41.5 41.5 41.1 41.1 21.1
47.5 47.4 47.41 47.42 47.6
49 49 49 49 49 49
0 5 6 8 14 20
3.125 3.125 3.125 3.125
P
Dr x
Dg
x
Lr Lg
Di
L
P Fig. 1. Section of a Grooved Tube. Journal of Engineering Science and Technology
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2.2. Theoretical model 2.2.1. Mathematical model Based on experimental results in Table 2 and literatures survey[14-16 ]. The value of average crumpling force under static conditions can be estimated as follows:
0.0299L f P q y Dm t L 0.2742
(1)
where q = 5.10 material constant.
2.2.2. Finite element modelling (ANSYS) In ANSYS version 12.0 there are basically many types of eight-node hexahedron solid elements. One of most recommended in elastic analysis is Solid45 element, where that constant stress solid element. This element had three degrees of freedom per node resulting in a total of 24 degrees of freedom. The fully integrated S/R element uses an eight-point Gauss integration rule. Fully integrated solid elements have the tendency of locking up in constant volume bending modes as the Poisson’s ratio approaches 0.5. In order to avoid locking, an average pressure is used over the element in ANSYS. Solid 45 element was used to valid the solution conditions of this study. Buckling and limit load analysis Limit load investigations of thin-walled structures are usually started with a linear buckling analysis. The results are buckling modes and load factors. Load factors are estimates for an upper limit of the ultimate load; buckling modes show, how the structure will buckle. As it is well known that the structures are most sensitive against imperfections in the shape of the lower buckling modes, they also give an idea of a conservative imperfection and can be applied to the ideal model as geometric (stress-free) modifications which is a simple function in the case of ANSYS. Buckling and prestressed modal analysis taking into account the current state of loading after a nonlinear static solution is possible to get proper information about the structural behavior. Both can also be used to investigate whether a convergence problem is due to a numerical or due to a physical instability. Loading control and path-following methods If imperfections are superimposed on a perfect shell structure, the bifurcation problem usually changes to a non-linear stress problem or a snap-through problem depending on the post-critical behavior. Knowing this behavior is essential for safety considerations. However, in a standard force-controlled analysis approaching the critical load will end up in non-convergence of the solution process; displacement control often is not possible and is only helpful if a characteristic displacement is chosen which is often difficult. Therefore, arc-length methods are preferred as implemented in ANSYS which allow controlling the load level together with the length of the displacement increment. This permits to compute the postcritical load-deflection path although force-type loads are applied.
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3. Results and Discussion 3.1. Crumpling load and mechanism of deformation The results of experimental work and finite element analysis (ANSYS) are presented as shown in Figs. 2 to 7 and Table 2. The figures of samples show the fluctuations in the load values with reduction in axial length. Good agreement and almost conformance can clearly be distinguished for the initial peak load and progress of fold. However, the experimental displacements are showed to be somewhat shifted ratio to the predicted values. Also the values of initial peak load and average load in Table 2 gave fair agreement with data of software ANSYS.
Trial No.
Table 2.The variation of values of mean load and initial peak load on grooved PVC circular tubes. n Number grooves
Experimental of mean load (Pm) (kN.mm)
Exp. Initial peak load (Pmax) (kN)
FEA Initial peak Load (kN)
Difference (FEExp/Exp) (%)
1 2 3 4 5 6
0 5 6 8 14 20
18.6 17.6 17.3 17 16.8 16.58
36.1 29.4 31 30 29.8 30.2
36.8 30.1 31.5 30.9 30.5 31.06
+1.9 +2.4 +1.6 +3.0 +2.3 +2.8
40
35
Crumpling load (kN)
30
25
Exp data FEA
20
15
10
5
0 0
10
20
30
40
50
60
70
80
Reduction in axial length (mm)
Fig. 2. Crumpling load verse reduction in axial length of sample (1).
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35
30
Crumpling Load (kN)
25
20 Exp. Dtat FEA 15
10
5
0 0
10
20
30
40
50
60
70
80
Reduction in axial Length (mm)
Fig. 3. Crumpling load verse reduction in axial length of sample (2). 35
30
Crumpling Load (kN)
25
20 Exp. data FEA 15
10
5
0 0
10
20
30
40
50
60
70
80
Reduction in axial length (mm)
Fig. 4. Crumpling load verse reduction in axial length of sample(3). 35
30
Crumpling Load (kN)
25
20 Exp Data FEA 15
10
5
0 0
10
20
30
40
50
60
70
80
Reduction in axial length (mm)
Fig. 5. Crumpling load verse reduction in axial length of sample (4). Journal of Engineering Science and Technology
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35
30
Crumpling Load(kN)
25
20 Exp. data FEA 15
10
5
0 0
10
20
30
40
50
60
70
80
Reduction in axial Length (mm)
Fig. 6. Crumpling load verse reduction in axial length of sample (5). 35
30
Crumpling Load (kN)
25
20 Exp. data FEA 15
10
5
0 0
10
20
30
40
50
60
70
80
Reduction in axial Length (mm)
Fig. 7. Crumpling load verse reduction in axial length of sample (6). The mode of deformation by plastic buckling of cylindrical shells can be show in Fig. 8, this figure refer to one shape of deformations is nonsymmetric mode and the process deformities of lobe takes place with progress the fold generally. This mechanism consists of a set of three stationary plastic hinges separating two outward moving portions which undergo hoop stretching. In Fig. 9 shows the finite element mode of deformation for a same sample in Fig. 8, good agreement was issued for this mode. In the present model analysis in order to get a frictionless postbuckling response of the deflected circular tube, Fig. 10 shows the buckling stages of the grooved and specimen deformations.
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Fig. 8. The stage deformation of single lobe for tube with grooved.
Fig. 9. The stage deformation by FEA (ANSYS model).
Fig. 10. Buckling stages of the grooved PVC tubes (five grooves sample).
4. Conclusions The following conclusions have been derived: The values of mean crumpling load decrease with ratio (Lr/L). The value of initial peak load decreases when using grooves but this value stays constant approximately with increasing the number of grooves. The values of mean crumpling load for circular tube with grooves is a function of mechanical properties, geometrical dimensions and ratio (Lr/L).
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The experimental results gave good agreement with mathematical model and Finite element model. That is issued the ability of numerical model to analyze the buckling load and load deformations behaviour of grooved PVC tubes. The patterns of buckling modes are easily illustrated in present FEA model. Also the grooved thickness is represented the focus of buckling folds, because of the decreasing in wall thickness at this point.
References 1. 2. 3.
4.
5.
6.
7.
8. 9.
10.
11.
12.
13.
Farley, G.; and Jones, R. (1992). Crushing characteristics of continuous fibrereinforced composite tubes. Journal of Composite Materials; 26, 37–50. Aljawi, A.A.; and Ahmed, M.H. (1999). Finite Element Analysis of the Axial Compression of Tube. Alexandia Enginerring Journal, 38(6), A445-A452. Aljawi, A.A. (2000). Finite Element and Experimental Analysis of Axially Compressed Plastic Tubes. Belgium Society of Mechanical and environmental Engineering, 45(1), 3-10. Karagiozova, D.; Alves, M.; and Jones, N. (2000). Inertia Effects in Axisymmetrically Deformed cylindrical shells under Axial Impact. Int.J. Impact Engineering.,24 1083-1115. Lu, G.; Ong, L.S.; Wang, B.; and Ng, H.W. (1994). An Experimental Study on Tearing Energy in Spliting Square Metal Tubes. Int. J.Mech.Sci.,36, 10871097. Mamalis, A.; Manolakos, D.; Loannidis, M.; Kostazos, P.; and Dimitriou, C. (2003). Finite Element Simulation of the Axial Collapse of Metallic ThinWalled Tubes with Octagonal Cross-Section, Thin-Walled Structures, 41, 891-900. Liu, Y.C.; and Day, M.L. (2007). Simplified Modeling of Thin-Walled Tubes with Octagonal Cross-section Axial Crushing. WCECS 2007 conference,San Francisco , USA, 24-26. Liu, Y.C.; and Day M.L. (2006). Simplified Modeling of Thin-Walled Box Section Beam. International Joural of Crashworthiness, 11(3), 263-272. Peixinho, N.; Jones, N.; and Pinho, A. (2003). Experimental and numerical study in axial crushing of thin walled sections made of high-strength steels. Journal of Physique IV France, 110, 717–22. Tarigopula, V.; Langseth, M.; Hopperstad, O.S.; and Clausen, A.H. (2006). Axial crushing of thin-walled high-strength steel sections. International Journal of Impact Engineering, 32, 847–82. Yamashita, M.; Gotoh, M.; and Sawairi, Y. (2003). Axial crush of hollow cylindrical structures with various polygonal cross-setcions: Numerical simulation and experiment. Journal of Material Processing Technology, 140, 59–64. Fyllingen, Ø.; Hopperstad, O.S.; Hanssen, A.G.; and Langseth, M. (2010). Modelling of tubes subjected to axial crushing. Thin-Walled Structures, 48,134–142. Hosseinipour, S.J. (2003). Mathematical model for thin-walled grooved tubes under axial compression. Materials and Design, 24, 463–469.
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14. Mamalis, A.G.; Manolakos, D.E.; Viegelahn, G.L.; Vaxevanidis, N.M.; and Johnson, W. (1986). On the Inextensional Axial Collapse of Thin PVC Conical Shells, International Journal of Mechanical Sciences, 28, 323-335. 15. Alghamdi, A.A.; Aljawi, A.A.; Abu-Mansour, T.M.; and Mazi, R.A. (2000). Axial Crushing of Frusta Between Two Oarallel Plates, in Structural Failure and Plasticity. IMPLAST 2000, ZhaoX.L and Grzebieta, R. H.(Eds), Pergamon, New York, USA., 545-550. 16. Mamalis, A.G.; Manolakos, D.E.; Viegelahn, G.L.; Vaxevanidis, N.M. and Johnson, W. (1986). The Inextentional Collapse of Grooved Thin-walled Cylinders of PVC under Axial Loading. International Journal of Impact Engineering, 4(1), 41-56.
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