These tests are the quasi-static tests from Sindre Hellem Sølvernes’ master thesis in 2015 [16].
The test is also studied in the paper by Gruben et al. in 2017 [17]. Two types of indenters are used, and denoted indenter A and indenter B. The numerical model in this chapter was recieved from David Morin and is the model Sindre made during his work. The model is still described here.
5.1 Geometry
5.1.1 Test specimen
The test specimen consists of a 1250 mm wide and 1375 long plate with 3 mm thickness and six stringers are welded to it with intermittent welds. The throat size is 3 mm and length of the weld is 15 mm. These are placed with a center-to-center distance of 45 mm. The stringers have a height of 65 mm and a thickness of 3 mm. The stringers are an L-shape and the width of the bottom is 15 mm (thickness not included). The geometry of the test specimen is shown in the figure below.
Figure 5.1: (a) Stiffened steel plate seen from the bottom, (b) cross section of test specimen, (c) stiffener cross-section, and (d) length and center-to-center distance of the fillet welds. [17]
5.1.2 Test rig frame
This specimen is set between two frame parts that are bolted together by using 8 M16 bolts.
Teflon sheets are used between the frame and test specimen to reduce friction. Shim plates with a thickness of 8 mm are also used to give extra support for the test specimen. It is also 50 mm cutouts for the stiffeners in the bottom frame. The frame with the test specimen installed, is shown in Figure 5.2
Figure 5.2: (a) Specimen clamped between bottom and top frame, (b) cross-section in the longitudinal direction, (c) lower support frame side, (d) profile in the width direction, (e) cross section in longitudinal direction, and (f) cross section in width direction. [17]
5.1.3 Indenters
The geometry for indenter A and B, are shown in Figure 5.3. Indenter A has a cylindrical shape with rounded corners with 25 mm radius, and has a length of 350 mm. Indenter B has a hemispherical shape with a radius of 50 mm.
(a) Indenter A
(b) Indenter B
Figure 5.3: Geometry of the indenters [16]
5.2 Material
The material used for the test specimen is steel grade DOMEX 355 MC E. The material model for the test specimen is the same as used in the previous tests, and described in section 3.1 on page 10, but the material parameters are of course different. These are shown in the following table.
Table 5.1: Material parameters for the stiffened steel plates σ0[MPa] E [GPa] K[MPa] n εplateau ν ρ[tons/m3]
404 210 772 0.173 0.024 0.33 7.85
The material used in the frame is elastic-perfectly plastic with a yield limit of 355 MPa. The material used for the bolts is just elastic with the same Young’s modulus.
5.3 Numerical model
5.3.1 Mesh
This test is run with three different mesh sizes for the test specimen: 3 mm (le/te= 1), 12 mm (le/te = 4) and 30 mm (le/te = 10). The mesh with 3 mm elements is shown in Figure 5.4 and has just an area around the indenter that are set to 3 mm to reduce computational time. S4R elements in Abaqus are used in the plate. The indenters are modelled as discrete rigid with R3D elements with a mesh size of 6 mm. The frame is also modelled using S4R elements, and has a 15 mm mesh. The bolts has a mesh size of 10 mm. A symmetry plane across the middle of the plate is also utilized in the simulations.
(a) close-up of the mesh close to the indenter (b) Seen from below
(c) Whole model with 3 mm elements Figure 5.4: 3 mm mesh of the stiffened steel plates
5.3.2 Contact
A general contact formulation is also used in these tests. The friction coefficient is also here set to 0.3 in the tangential direction. Small stabilizing forces of 0.01 kN is added for the shim plates to push the plates towards the frame in order to avoid that the parts are set in motion by propagating stress waves. A friction coefficient of 0.04 is adopted between the teflon sheets and the plate. This coefficient between teflon and steel is taken from Engineers Handbook [18].
5.4 Results
5.4.1 Indenter A
(a) First integration point without mesh scaling (b) First integration point with mesh scaling
(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 5.5: Results - Indenter A
The results shown for the middle integration point is of course similar to what Sindre got in his simulations using the BWH criterion [16]. Using the first integration point instead, leads to more conservative results and could be good for design where high safety factors are used.
When using the mesh scaling technique, Figure 5.5d (middle integration point) shows a bit too strong scaling, while Figure 5.5b (first integration point) actually shows a small, but a bit too weak scaling, however, it is still more conservative compared to using the middle integration point.
The plate does not fracture for the 30mm mesh without the mesh scaling but when using it, the test specimen fractures first in tip of the stringers below the indenter. The force continues to increase due to this, and fails in the plate where the curve gets a rapid drop in force. All the elements below the indenter fails fast when reaching this point, and drops to zero. This happens much sooner when using the first integration points, as seen on the figure.
For the 3mm mesh, the curve continues in a pretty straight line after the first element fails.
When looking at how the fracture develops (see Figure 5.6), we can see that the fracture develops in a straight line which allows the membrane stresses to continue to be carried in the transverse direction.
(a) Right before fracture
(b) At 200 mm indentation
Figure 5.6: Plastic equivalent strain before fracture and for 200 mm indentation for 3 mm mesh with BWH using first integration point
The fracture for the 12 mm mesh seems to develop in arbitrary directions after the stringer fails first. The plastic equivalent strain for the 12 mm mesh with the first integration point and mesh scaling right before the fast drop in force is shown in the following figure:
Figure 5.7: Plastic equivalent stress at 150 mm indentation for 12 mm mesh first integration point with mesh scaling
After this, the elements above the indenter fails fast and the drop in force is observed.
5.4.2 Indenter B
The results for indenter B are shown in Figure 5.8. It is also observed here that when using the middle integration point, the mesh scaling is too strong. For the first integration point, the biggest mesh size (30 mm) with mesh scaling is too weak, but instability occurs at the same spot as the middle integration point. The first integration point also gives more conservative results here, especially for the 3 mm and 12 mm mesh. The sudden drop in force is because of the shape of the indenter, which penetrates the plate and goes through pretty fast after first fracture occurs.
(a) First integration point without mesh scaling (b) First integration point with mesh scaling
(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 5.8: Results - Indenter B
6 Conclusion
The results show more conservative or pretty much the same results for all cases studied, except for the 2-HP case explained in section 4.4.5, when using the first integration point instead of the middle integration point that is intended for BWH. This could be interesting for design where large safety factors are used, but more tests and simulations are needed on this topic.
The mesh scaling technique used in this study shows overall pretty good results for most cases.
However, it seems to give too conservative results for for some cases. The reason for this is prob-ably that the scaling is used for the whole test specimen in all the cases instead of only at places where it is needed, as explained in 2.2.3.
References
[1] M. Storheim. Structural response in ship-platform and ship-ice collisions [doctoral thesis].
Norwegian University of Science and Technology, 2016.
[2] S. Ehlers et al. Simulating the collision response of ship side structures: A failure criteria benchmark study. International Shipbuilding Progress, 55:127–144, 2008.
[3] H.S. Alsos et al. On the resistance to penetration of stiffened plates, part ii: Numerical analysis. International Journal of Impact Engineering, 36:875–887, 2009.
[4] Abaqus 2016. Dassault syst`emes, 2015.
[5] O.S. Hopperstad, T. Børvik. Materials Mechanics - Part I. 2015.
[6] O.S. Hopperstad, T. Børvik. Impact Mechanics – Part 1: Modelling of plasticity and failure with explicit finite element methods. 2017.
[7] Engineers edge. Von Mises criterion. https://www.engineersedge.com/material_
science/von_mises.htm, Accessed april 2018.
[8] Z. Marciniak et al. Mechanics of sheet metal forming. Butterworth-Heinemann, 2002.
[9] R. Hill. On discontinous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids, 1:19–30, 1952.
[10] J.D. Bressan, J.A. Williams. The use of a shear instability criterion to predict local necking in sheet metal deformation. International Journal of Mechanical Sciences, 25:155–168, 1983.
[11] H.S. Alsos et al. Analytical and numerical analysis of sheet metal instability using a stress based criterion. International Journal of Solids and Structures, 45:2042–2055, 2008.
[12] J. Broekhuijsen. Ductile failure and energy absorption of y-shape test section [master thesis].
The Netherlands: Delft University of Technology, 2003.
[13] M. Storheim et al. A damage-based failure model for coarsely meshed shell structures. In-ternational Journal of Impact Engineering, 83:59–75, 2015.
[14] H.S. Alsos et al. On the resistance to penetration of stiffened plates, part i – experiments.
International Journal of Impact Engineering, 36:799–807, 2009.
[15] T. Wang et al. Finite element analysis of welded beam-to-column joints in aluminium alloy enaw 6082 t6. Finite Elements in Analysis and Design, 44:1–16, 2007.
[16] S. Sølvernes. Impact behaviour of stiffened steel plates [master thesis]. Norwegian University of Science and Technology, 2015.
[17] G. Gruben et al. Low-velocity impact behaviour and failure of stiffened steel plates. Marine Structures, 54:73–91, 2017.
[18] Engineers Handbook. Coefficient of friction.http://www.engineershandbook.com/Tables/
frictioncoefficients.htm, Accessed june 2018.