First Quartile Min Max Third Quartile Median Mean 0.17 0.12
P2M1T1 Test Series Two, Model One Tests
Bed Slope Side Slope D50
0.5 0.4 0.075 m
Model 1 Test Conditions
P2M2T1 Test Series Two, Model Two Tests
Bed Slope Side Slope D50
0.4 0.4 0.075 m
Model 2 Test Conditions
P2M3T1 Test Series Two, Model Three Tests
Bed Slope Side Slope D50
0.333 0.4 0.075 m
Model 3 Test Conditions Stellenbosch University https://scholar.sun.ac.za
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Figure 56 shows the 15 successful test results of the MN analysis performed at the laboratory.
The main data required to plot the box whiskers in Figure 56 were the basic statistics shown in the legend of the graph in Figure 55, on the right-hand side. A summary of the basic statistics of the MN values used for plotting the box-whisker diagram of Test series two can be found in
Appendix F.
Tests P2M1T3 and P2M1T4 were considered to produce unreliable MN box-whisker plots. This was because of the insufficient water depth and slope data measured in the failure region of the two tests. Only six and eight MN values could be obtained from the failure region of test P2M1T3 and P2M1T4, respectively. There were 20 possible measurements of water depth and slope for each test. Therefore, since the two tests only produced insufficient measurements for the analysis, no conclusions can be drawn based on the two tests. However, the tests have been included as a reference. The two tests are enclosed with a blue cloud in Figure 56.
Test P2M1T1 and P2M1T2 show the largest deviation of MN results. The standard deviation calculations summarised in Appendix F show that P2M1T2 has the largest standard deviation MN value from the mean MN value of 0.08. The P2M1T1 has the second largest standard deviation from the mean MN value of 0.079. The vertically wide boxes show a clear physical illustration of how the two tests MN results significantly vary compared to the rest of the tests in Test series two.
There were two MN test results that showed the smallest deviations from the mean value of the MN in Test Series two. The smallest value of the MN standard deviation of 0.013 was obtained from test P2M2T3. The second smallest value of the MN standard deviation was 0.021 obtained from test P2M1T5. The small deviations in the MN values are shown by the vertically squeezed nature of the boxes, illustrating the closeness of the MN values measured at the laboratory relative to the mean MN value.
Most of the box-whisker plots in Figure 56 are positioned above the 0.17 MN upper limit value. The whole box plot including the minimum values on the box plots plotted above the 0.17 MN upper limit value except the two tests with a blue cloud.
Tests P2M1T1, P2M2T2, P2M2T4 and P2M3T4 are the only four reliable tests with the minimum MN values that are less than the 0.17 upper limit MN value. The four previously mentioned tests of Test series two were analysed by means of the probability of exceedance. The probability of exceedance analysis assisted in defining the critical MN value that has a 95% probability of exceedance with respect to the observations made at the laboratory.
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Table 18 shows a summary of the sorted MN and the calculated 95% probability of exceedance
MN values at the bottom of the table.
Table 18: Summary of MN with 95% probability of exceedance P1M1T1 (MN) P2M2T2 (MN) P2M2T4 (MN) P2M3T4 (MN) 0.366 0.245 0.287 0.357 0.358 0.237 0.276 0.332 0.356 0.234 0.274 0.331 0.347 0.229 0.270 0.266 0.343 0.226 0.265 0.259 0.286 0.223 0.258 0.258 0.28 0.223 0.251 0.255 0.28 0.206 0.251 0.241 0.273 0.199 0.245 0.230 0.215 0.199 0.242 0.228 0.212 0.197 0.240 0.227 0.209 0.197 0.227 0.217 0.198 0.195 0.218 0.200 0.191 0.194 0.210 0.196 0.184 0.19 0.205 0.163 0.149 0.168 0.197 0.158 0.142 0.144 0.146 0.155 0.122 0.131 0.122 0.153 0.091 0.142 0.139 0.127 0.142 0.152 5% Percentile
The Microsoft Excel function was used to calculate the 5% percentile MN value for the tests, which also represented the 95% probability of exceedance value of the observed MN values. The four tests in Table 18 all had 18 or more MN values measured for each test. Therefore, these were reliable tests to perform the probability of exceedance analysis.
All the 5% percentile MN values calculated above fell below Armitage’s 0.17 upper limit MN value. Moreover, all the 5% percentile MN values of the four tests in Table 18 fall above the 0.12 lower limit MN value. Therefore, all these values fall within the envelope of 0.12-0.17 MN values. The highest critical MN value with a 95% probability of exceedance was 0.152, obtained from test P2M3T4. The lowest critical MN value with a 95% probability of exceedance was 0.127. The lowest MN value of 0.127 was also very close to the lower limit of 0.12 that was obtained by Rooseboom (1992).
5-126 Test series one and Test series two tests were very similar in terms of the testing procedure followed and the behaviour of riprap failure. Therefore, similar results were expected for the critical MN value. However, Test series one tested a smaller 0.038 m median stone size while Test series two tested a larger stone size of 0.075 m, approximately two times the Test series one median stone size. So, the two critical MN value s of 0.119 and 0.127 obtained from the probability of exceedance analysis were satisfactory results. As anticipated, very similar results were obtained. The percentage difference between the two MN values for the critical incipient failure conditions for Test series one and Test series two was 6.5%.
5.3 Test Series Three MN Analysis
The main observation made in Test series one and Test series two tests was that the riprap failed in the steep bed area and did not fail on the side bank area. Therefore, in Test series one and two the side bank incipient motion conditions could not be studied. Consequently, for Test series three it was decided that the angular riprap dumped in the steep bed area must be glued with an adhesive. The glueing of the riprap over the bed area allowed the testing of the ten tests for Test series three. In Test series three, five tests were performed over the steep bed slopes of 0.5 and the remaining five tests were performed on 0.333 steep slopes.
To complete the MN value calculations based on the observed water depths and slopes for the ten tests in Test series three, the hydraulic input parameters in Table 19 were applied into
Equation 57. The hydraulic input parameters in Table 19 are the same as those used in Test
series two of the tests. However, the only difference was the location of the vicinity of the measured average local water depths and slopes in the incipient failure regions of the angular riprap. The incipient failure region was in the steep bed area for Test series one and two, while the local failure regions on Test series three were at or near the toe of the side bank. The observed average local hydraulic water depths and average local bed slopes at incipient failure regions for Test series three were summarised in Appendix D.
5-127 Table 19: Hydraulic input parameters to determine MN for Test series two tests
Input Parameter Value Unit
D50 0.075 m
ρr 2700 kg/m3
ρw 1000 kg/m3
vss 0.8352 m/s
ɸr 40 °
αAngle (side slope) 21.77 °
αSlope (side slope) 0.4
θAngle (bed slope) Varies °
θSlope (bed slope) Varies
g 9.81 m/s2
So Varies
Dw Varies m
ѵ 1.13E-06 m2/s at 15 °C
The MN values were calculated by applying the water and riprap hydraulic input properties in
Table 19, plus the average local water depths and average local slopes from Appendix D into Equation 57.
Each of the calculated MN values for Test series three were summarised in Appendix E. The calculated MN values and Re* from Appendix E were then plotted onto the Liu diagram shown in Figure 57. Only two data points were less than the 0.17 upper limit MN value. Most of the points plotted in abundance from an approximate MN value of 0.22 and upwards. Higher MN values for Test series three were expected, due to the high flow rates and water depths that were instigated incipient failure of riprap dumped on the 0.4 steep side bank slope. Therefore, the general overall high MN values displayed in the Liu diagram was anticipated.
5-128 Figure 57: Test series three MN results plotted onto the Liu diagram.
Figure 58 displays a closer look at the Liu diagram for Test series three. A distinction between the tests performed on the 0.333 steep bed slope
and the 0.5 steep bed slope was shown with the two differently coloured dots. The physical hydraulic model with the 0.5 steep bed slope produced MN values that were generally lower than the MN values of the models tested on the 0.333 steep bed slopes.
0.00 0.50 1.00 1.50 2.00 2.50 1 10 100 1000 10000 100000 MN
Particle Reynolds Number