8.2 Numerical study in IMPETUS Afea Solver
8.2.2 Material parameter study
It was assumed that the material parameters in the HJC model was one of the reasons why the numerical model overestimated the strength of the concrete plates. Therefore, a param-eter study was conducted in order to improve the model.
The procedure for the parameter study was that a single material parameter would be varied at a time. For all the tested parameters, one greater and one smaller value than the initial would be evaluated. This was done to see what effect an increase, or decrease, of the pa-rameter would have on the maximum load capacity. Note that the papa-rameter efmin was only investigated for decreased values.
The parameters that were investigated in this study was the strength parameters A, B and n, the damage parametersefmin and D1, the compression strength fcand the tensile strength ft. The reason for checking the strength parameters was that their estimated values were disputed in the literature [41, 67, 68], and because the preliminary model overestimated the strength of the material. The damage parameters were investigated because they were nu-merical parameters with little or no basis in physical experiments. The concrete compression strength was checked because in earlier simulations, the cube strength was used. The cube strength gave a higher compression strength than the cylinder strength and since the initial model overestimated the strength, this could have a major influence. Checking the tensile strength was done because a splitting tensile test was known to give a greater value for the tensile strength than a uni-axial tensile test would [69].
CHAPTER 8. NUMERICAL STUDIES
Cohesive strength A
The cohesive strength parameterAis defined as the difference between the complete fractured strength and the undamaged strength. In the original HJC article, it was assumed that the cohesive strength was 0.75 of fc and when correcting for a higher strain rate it was set to 0.79 of fc. From what was understood in the original article, the upper limit for the cohesive strength was 1.0. From the literature, a value of 0.5 was suggested as reasonable [67]. Therefore, a study was conducted to see what effects the parameter had on the load capacity for a range of 0.5-1.0. Table 8.4 lists the maximum investigated load that resulted in both collapse and no collapse. The exact load for which the plate collapsed would lie in between. The results are illustrated in Figure 8.12.
Table 8.4: Effect of parameter A.
A Change No collapse Change Collapse Change [%] load [bar] [%] load [bar] [%]
1 26.6 145 5.5 150 5.3
0.79 - 137.5 - 142.5
-0.5 -36.7 115 -16.4 120 -15.8
Figure 8.12: Effect of parameter A.
CHAPTER 8. NUMERICAL STUDIES Fracture strength B and n
The fracture strength parameter B was set to 1.6 in the original HJC article. The fracture strength parameternwas set to 0.61 based on a good fit with test data provided by Hanchack et al. [70]. Because tri-axial compression tests with varying confinement pressure were needed to estimate these parameters experimentally, the literature was used to find reasonable values for these parameters. Polanco et al. [41] showed that with values close to what was listed in the original article,n=0.75 andB=1.6, the uni-axial compression strength was overestimated by 45%. An attempt to inverse model the parameters was performed by Chen et al. [68].
Here, a domain of B between 1.0-2.0 andnbetween 0.1-1.0 was defined as acceptable. It was also concluded that optimal values for the presented experiments were B=1.38 andn=0.37.
In this parameter study, the impact ofB was therefore checked for 1.0-2.0 and n for 0.3-1.0.
Table 8.5 and 8.6 list the maximum investigated load that resulted in both collapse and no collapse, and the results are illustrated in Figure 8.13 and 8.14.
Table 8.5: Effect of parameterB.
B Change No collapse Change Collapse Change [%] load [bar] [%] load [bar] [%]
2.0 25 145 5.5 150 5.3
1.6 - 137.5 - 142.5
-1.0 -37.5 100 -27.3 105 -26.3
Table 8.6: Effect of parameter n.
n Change No collapse Change Collapse Change [%] load [bar] [%] load [bar] [%]
0.3 -56.5 145 5.5 150 5.3
0.61 - 137.5 - 142.5
-1.0 63.9 120 -12.7 125 -12.3
CHAPTER 8. NUMERICAL STUDIES
Figure 8.13: Effect of parameterB.
Figure 8.14: Effect of parametern.
Damage parameter efmin
efmin is defined as the finite amount of plastic strain to fracture. As this parameter was purely numerical, it was natural to investigate the impact it had on the model. It was known that the damage accumulation in the model was highly dependent on the value of efmin, and the failure of the material could therefore be significantly increased for a lower value of efmin [42].
CHAPTER 8. NUMERICAL STUDIES Because of the overestimation of strength in the preliminary study, the upper limit of efmin was set to be the value found in original HJC article [39]. Polanco et al. [41] usedefmin=0.004 to account for the brittle nature of high strength concrete. This seemed to be a reasonable lower limit for this study. Table 8.7 lists the maximum investigated load that resulted in both collapse and no collapse, and the results are illustrated in Figure 8.15.
Table 8.7: Effect of parameterefmin.
efmin Change No collapse Change Collapse Change [%] load [bar] [%] load [bar] [%]
0.004 -60 90 -34.5 95 -33.3
0.005 -50 105 -23.6 110 -22.8
0.007 -30 110 -20.0 115 -19.3
0.01 - 137.5 - 142.5
-Figure 8.15: Effect of parameterefmin.
Damage parameter D1
D1 is described as a numerical damage constant and the damage accumulation is dependent on its value (see Equation (3.4) in Section 3.3.1). Because of the lack of information on
CHAPTER 8. NUMERICAL STUDIES
a reasonable value of the parameter, a range of 0.01-0.2 was simply chosen. It was also a possibility to study the impact of the other damage parameter D2, which was the exponent in the damage accumulation. To limit the scope of the problem, it was chosen to not differ from the original value ofD2, and to keep the expression for the damage accumulation linear.
Table 8.8 lists the maximum investigated load that resulted in both collapse and no collapse, and the results are illustrated in Figure 8.16.
Table 8.8: Effect of parameter D1.
D1 Change No collapse Change Collapse Change [%] load [bar] [%] load [bar] [%]
0.01 -75 115 -16.4 120 -15.8
0.04 - 137.5 - 142.5
-0.20 400 150 9.1 155 8.8
Figure 8.16: Effect of parameter D1. Compression strength fc
As the compression strength was not specified to be cube or cylinder strength in the original article, cube compression strength was chosen as the compression strength for the numerical
CHAPTER 8. NUMERICAL STUDIES model. As the experimental results showed a much lower strength than the numerical model, using the cube strength was a poor choice. Also the strength of concrete is normally based on the cylinder strength. The average cylinder strength found in the material experiments was 54.92 MPa, which was approximately 85% of the cube strength. The effect of changing the concrete strength would therefore have to be investigated. Because more than one material parameter in the HJC model was dependent on the compressive strength, changes were also made to dependent material parameters. The changes are presented in Table 8.9 and the results are shown in Table 8.10 and in Figure 8.17.
Table 8.9: Parameters depending onfc. fc [MPa] G [MPa] Pcrush [MPa] vol,crush
54.92 16239.8 18.4 8.5·10−4 Table 8.10: Effect of parameterfc.
fc Change No collapse Change Collapse Change [%] load [bar] [%] load [bar] [%]
54.92 -15.4 105 -23.6 110 -22.8
64.89 - 137.5 - 142.5
-Figure 8.17: Effect of parameter fc.
CHAPTER 8. NUMERICAL STUDIES
Tensile strength ft
The tensile strength used in the preliminary study was based on the average value found in the material test. Since the plate experienced tensile stress in the shock tube experiments, it was assumed that the tensile strength had a great effect on the response in the numerical model. For this reason, and the fact that the tensile splitting test produces a greater value for the tensile strength than a uniaxial test [69], it was decided to investigate the change by adopting the lowest tensile strength found in the material experiments, ft=3.80 MPa. Table 8.11 lists the maximum investigated load that resulted in both collapse and no collapse, and the results are illustrated in Figure 8.18.
Table 8.11: Effect of parameter ft.
ft Change No collapse Change Collapse Change [%] load [bar] [%] load [bar] [%]
3.80 -8.2 135 -1.8 140 -1.8
4.14 - 137.5 - 142.5
-Figure 8.18: Parameter ft.
CHAPTER 8. NUMERICAL STUDIES
Results
From the material parameter study, it became clear that all of the parameter influenced the collapsing pressure, but to various extents. Within the chosen ranges, a change in efmin caused the largest reduction in capacity. This did not come as a surprise as the damage accumulations is highly sensitive to a change in the parameter efmin. On the contrary, it was surprising that the tensile strength ft of the model did not influence the response more. A change of 8.2% inftonly resulted in a 1.8% reduction in peak pressure. In a problem thought to be highly influenced by tensile forces, this gave an indication that the HJC model might not be the best material model to describe a concrete plate subjected to blast loading. It was also interesting to see that a reduction of 15% of the compressive strength gave approximately 23% reduction in peak pressure capacity. Of the strength parameters, the fracture strength B had the greatest impact, with a change of 37.5% reduced the peak pressure with 27%.