Simulation model validation, single tube

The simulation model of a single AISI 304 tube was validated against the experimental quasi-static tests done by DiPaolo and Tom (2009). In their experiment, 305 mm long 50×50 AISI 304 tubes with measured average thickness of 1.50 mm were used in axial crush tests. Crush initiators, machined grooves on the full width of the tube, were used on opposite sides. Another set of initiators were used below the first pair on adjacent sides. Test were performed in three temperatures: -46 °C, 22 °C and 93 °C. Mechanical properties were tested in uniaxial tension tests. For AISI 304 the following values were obtained in room temperature: 0.2% proof stress 449 MPa, ultimate strength 706 MPa and ultimate elongation 0.50. Grooved end caps, that restrict the movement in transverse direction to the compression, were used in the experimental axial crush tests. [64] In sim- ulation of this work, fixed support was used at the bottom edges and all degrees of free- dom expect y-translation were restricted at the top edges. 200 mm movement downwards in y-direction was set to the top of the tube. Local imperfections were obtained from a linear buckling analysis. First and second buckling modes were tested, and the second buckling mode gave better agreement with the experimental test: both folding shape and

force–displacement curve were closer to the experimental data. The amplitude of the im- perfection was b/200, where b is the width of the tube. The rounding of the corners or the crush initiators were not modelled. Geometry and constraints are shown in Figure 8.

Nonlinear material model was used in the simulation. The material model was constructed using the two-stage Ramberg–Osgood material model presented in Design Manual for Structural Stainless Steel [3]. 0.2% proof strength, ultimate strength and ultimate elonga- tion values were taken from the experimental data of [64], whereas the Yong’s modulus E and Ramberg–Osgood coefficient n were obtained from the design manual. True stress–

logarithmic strain curve was determined from the engineering stress–engineering strain curve. Multilinear isotropic hardening was chosen in ANSYS to model the plasticity. The true stress–logarithmic strain curve was converted to multilinear true stress–logarithmic plastic strain curve with 10 data points. The material modeling is shown in detail in ap- pendix 1. Effects of cold working on the strength of the corners were not considered. The model was meshed with linear SHELL181 shell elements. According to ANSYS Mechan- ical manual, SHELL181 is suitable for large strain nonlinear applications [114]. Different mesh sizes were tested, and 2.0 mm element size was chosen for suitable results and ef- ficiency. Frictional contacts with 0.2 friction coefficient and thickness effect were used, and self-contact of bodies were considered. The experiments in [64] were quasi-static with speed of 2.5 mm/min, but the simulation was carried out in 1.0 s analysis time for

Figure 8. On the left geometry and constraints of the simulation model. Constraints were scoped to the edges at the top and bottom of the tube. On the right two first modes from linear buckling analysis (LBA).

Displacement (0,0,0) → (0,–200,0) Rotation (0,0,0) 305 mm

Displacement (0,0,0) Rotation (0,0,0)

LBA1 LBA2

shorter calculation times. Displacement of 200 mm in that time equals to 200 mm/s. Us- ing linear buckling mode 2 imperfection with b/200 amplitude, the ratio of maximum kinetic energy to total energy was only 0.09% and thus the inertial effects are negligible.

Mass scaling of elements up to 1000 times the original mass was applied in the analysis settings to shorten the calculation time.

Figure 9 shows the folding behavior of AISI 304 tubes in experiments [64] and in FEA validation model. From the figure can be seen that the folding behavior of both tubes, with 1^st and 2^nd LBA mode imperfections, corresponded well to the experimental results but the shape of the tube with 2^nd LBA mode imperfection was closer to the experimental tests. The difference can be seen at the width of the fold (relative to the width of the tube, seen on the front face in the figure) which was smaller for the 2^nd LBA mode. Below the lowest completed folds on the sides new forming folds also behaved more accurately with the 2^nd LBA imperfection model. One remarkable difference between the simulation model and experiments was the behavior of the top of the tube: there was a short part of undeformed tube in experiments, but in the simulation the top part of the tube experienced severe plastic deformation and was folded so that at the end of the analysis the top edges of the tube were at the level of the first folds in vertical direction.

Figure 9. Axial crush test results for AISI 304 tubes in different temperatures from [64]

at the top and simulation results (plastic strain, mm/mm) with first and second buckling mode imperfections at the bottom. 2^nd LBA mode imperfection gives better correspond- ence with the experimental results.

Mode 1 Mode 2

-46 °C 22 °C 93 °C

Figure 10 presents the force–displacement curves for simulation and axial crush experi- ments. Both 1^st and 2^nd LBA mode imperfection models with b/200 amplitude are shown.

The curve with 2^nd LBA mode imperfection (black line) had very good correlation with the experimental curve. The number of force peaks and the displacement which they oc- cur at were quite accurately predicted at the range of 0…170 mm crush distance, which was the experimental data range. The magnitudes of the force peaks were also close to the experimental values. With the 1^st LBA mode imperfection the displacements between the force peaks were shorter than in experimental tests, but with the 2^nd LBA imperfection the peak distance was close to the experimental data. The peak force in experiments was 107.8 kN [64], whereas the first force peak in simulation using 2^nd linear buckling mode as imperfection was 132.1 kN, so the simulation model overestimated the initial peak force by 22.5%. The error of peak force with 1^st linear buckling mode was 23.8%. It should be noted, that the crush initiators were not modelled and thus it was expected that Figure 10. At the top force–displacement curve from simulation with 1^st and 2^nd LBA mode imperfections, b/200 imperfection amplitude. Experimental curves in different temperatures below [64]. S6-6 is the room temperature curve.

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140 160 180 200

Force (kN)

Displacement (mm)

Mode 1 amp=b/200 Mode 2 amp=b/200

-46 °C 22 °C 93 °C

the initial force peak was overestimated in simulation. Other load peaks of 1^st and 2^nd LBA mode imperfection models and tests had a quite good correspondence at around 50–

70 kN loads, depending on the peak, though the places and shapes of the peaks were not as good relative to the tests with 1^st LBA mode imperfection than with the 2^nd LBA mode imperfection.

Table 4. Mean crushing forces (Fmean) and initial peak forces (Fpeak) for experiments in [64] and simulation models with 1^st and 2^nd LBA mode imperfections. The errors of the simulation models are also presented.

In the experiments the initial and secondary folding phase mean loads were 46.6 kN and 46.5 kN, respectively [64]. In the simulation, the mean load for the whole displacement range were 52.7 kN and 51.8 kN for 1^st and 2^nd LBA mode imperfection models, respec- tively. These values mean 13.1% and 11.1% overprediction errors for the mean load.

Mean and peak loads with corresponding errors are gathered together in Table 4. Based on the folding behavior presented in Figure 9, axial load–displacement curve presented in Figure 10 and force values given in Table 4, the simulation model with b/200 imper- fection amplitude and 2^nd LBA mode imperfection shape can give reasonable predictions for this quasi-static axial crush of a single AISI 304 (EN 1.4301) square tube with around 10% error in mean crushing force.