Mesh sensitivity study

A mesh sensitivity study was performed to see how the element size could influence the capacity of the benchmark model. All results in this section are compared with the benchmark model with five elements over the thickness.

Procedure of mesh study

The procedure in the mesh sensitivity study was to load the concrete plate to failure so that the minimum pressure needed for the plate to collapse could be found. This was done by applying a blast load approximated by a general Friedlander equation with a varying

CHAPTER 6. PRELIMINARY STUDIES peak pressure. The approach made it easy to observe the impact the element size had on the collapsing pressure. However, it demanded that each model with different element sizes needed several analyses to find the collapsing load. For each model, it was initially chosen the same peak pressure that collapsed the initial model. From the response of the tested pressure, the next peak pressure would either be reduced if the plate collapsed, or raised if the plate withstood the tested pressure. To reduce the number of simulation for each mesh, a limit of 5 bar was chosen as an acceptable difference between the pressure collapsing the plate and the pressure the plate withstood. This approach was chosen, as opposed to using a gradually increased load, because the loading would be in the dynamic domain. A blast load would therefore give an earlier collapse.

(a) Typical fluctuation (b) Collapse vs no collapse

Figure 6.15: Displacement of top central element.

Figure 6.16 illustrates the different levels of damage D in the simulations presented for the simulations in IMPETUS.

Figure 6.16: Damage levels in the HJC model

CHAPTER 6. PRELIMINARY STUDIES

(a) t=0.1 ms (b) t=0.5 ms

(c) t=1.0 ms (d) t=2.0 ms

Figure 6.17: Collapse for 5 elements over the thickness and P_r = 142.5 bar.

It should be noted that each analysis was stopped when it was clear that the concrete plate would collapse. A collapse could be detected by examining the maximum displacement of the upper central element. If the displacement of the element fluctuated after it reached its first maximum, it was assumed that the plate would not collapse. This is shown in Figure 6.15a. However, if the displacement continued to grow (see Figure 6.15b) and the beginning of a failure mechanism could be spotted (see Figure 6.17), the plate was assumed to collapse.

The procedure was chosen to save computational costs, and seemed reasonable since the maximum deflection and damage occurred in the very start of the loading. The maximum run time for analyses that did not collapse was set to 5 ms.

Results of the mesh study

Table 6.12 shows the results from the mesh study. It has been differentiated between no collapse and collapse, and the actual collapsing pressure for the plate would therefore lie between the two. The models with different element size, were compared with the benchmark model with five elements over the thickness.

CHAPTER 6. PRELIMINARY STUDIES Table 6.12: Results from the mesh study.

# Elements over No collapse Difference Collapse Difference the thickness load [bar] [%] load [bar] [%]

2 152.5 10.9 155 10.5

3 145 5.5 150 5.3

4 147.5 7.0 152.5 7.0

5 137.5 0 142.5 0

6 135 -1.8 140 -1.8

8 107.5 -21.8 112.5 -21.1

10 100 -27.3 105 -26.3

12 95 -30.9 100 -29.8

Figure 6.18: Results from the mesh study

Figure 6.18 shows that a finer mesh led to a lower collapse load and that the model had a clear mesh dependency. Due to restrictions of computational power, further mesh refinement was not studied and no clear mesh convergence was found.

CHAPTER 6. PRELIMINARY STUDIES

(a) 2 elements (b) 3 elements (c) 4 elements (d) 5 elements

(e) 6 elements (f) 8 elements (g) 10 elements (h) 12 elements

Figure 6.19: Failure mechanism for different meshes

Figure 6.19 illustrates the failure mechanism for the models with different element sizes. It is evident that the element size affected the crack propagation and that a too coarse mesh failed to capture the failure well. From Figure 6.18, it can be seen that a mesh with three elements had a lower collapse load than a mesh with 4 elements, even though a finer mesh would sug-gest a lower collapse load. This can be explained by the location of element boundaries. For the models with a sufficiently fine mesh, a crack can be seen developing through the thickness with an angle of about 30 degrees. If the crack developed sufficiently, collapse would occur.

The node splitting algorithm struggled to represent this crack with few elements over the thickness.

The results show that the benchmark model was mesh sensitive. To capture the failure mechanisms, a mesh with 8 elements over the thickness would be used for further prelimi-nary studies, as it seemed to be a good compromise between accuracy and computational time. Table 6.13 shows the CPU time for a run-time of 1.0 ms for the different models with collapse.

CHAPTER 6. PRELIMINARY STUDIES Table 6.13: CPU time for different meshes.

# Elements over thickness Number of elements CPU time [s]

2 128 20

From the mesh study, it was determined to further investigate the model with 8 elements over the thickness. The main objective of the preliminary study was to find the maximum load that the concrete plates would withstand without collapsing. This section looks into other features that could be described as damage, and the following properties were investigated:

1. At what load spalling would occur

2. At what load node-splitting would appear 3. The damage accumulation

To save computational time, the analyses were only run for 2.0 ms as the different features appeared within this time.

Results

A selection of the load cases is shown in Table 6.14. The different cases were selected to show the different failure features of the model. It should be noted that the loads listed in Table 6.14 were not the exact loads for which the features first occurred, but shows an upper limit of 5 bar. This limit was used because a more accurate approach would have required too