First Quartile Min Max Third Quartile Median Mean 0.17 0.12 P1M1T1 Test Series One, Model One Tests
Bed Slope Side Slope D50
0.5 0.4 0.038 m
Model 1
Test Conditions
5-117 It is important to understand the scatter of the calculated MN results. The scatter informs about the variation of the MN in the local failure zones. The box-whisker diagrams in Figure 53 display the deviation of the calculated MN values for each test in Test series one. From
Appendix E each test had a large dataset of calculated MN values. There were more than 17
MN results obtained at each failure region for each test. Therefore, the variations illustrated in all the seven tests in Figure 53 can be deemed reliable and representative of the local failure variations of the MN values. Appendix F shows the standard deviations of the MN values for each test to express the variations within the local failure regions of the riprap.
The small variations in the MN values are shown by the compressed nature of the box. A compressed box illustrates the small deviations in the scatter of the MN values from the mean MN value. Test P1M3T2 has the smallest deviations in the MN values. The minimum and maximum MN values of test P1M3T2 are closer to the box. On the other hand, test P1M3T1 showed the largest variation in the calculated MN values.
It is apparent from Figure 52 and Figure 53 that the MN in the failure regions varies. The MN values range from a minimum of 0.105 to a maximum of 0.509 in Test series one. This is a large discrepancy in the MN values. The large variation in the obtained MN values does not facilitate the ease of determining the exact MN value to define the hydraulic incipient failure conditions of the riprap.
When a large value of the MN value is chosen to define the incipient failure condition of riprap, there is an advantage of specifying low-cost riprap median size. However, there is also a risk of specifying an undersized riprap median stone size. Moreover, if a very small MN is chosen, then a large stone size may be chosen, which can have cost implications on the overall riprap protection project. However, the advantage is that a safe riprap median stone size may be specified.
Considering the above advantages and disadvantages of choosing a MN value to define the incipient motion condition, the MN values of 0.12 and 0.17 were initially considered to be the safe region to define the MN for riprap incipient motion conditions. However, it was not clear whether to choose the 0.12 or the 0.17 since the seven critical MN tests had MN within the 0.12-0.17 MN envelope. To guide the decision, an exceedance probability statistical analysis on the critical tests with the MN values lying in the 0.12-0.17 envelop was executed.
In Figure 53 the tests P1M1T2, P1M1T3 and P1M2T3 were the three critical tests with the minimum MN values between the 0.12 and the 0.17 MN envelope.
5-118 Therefore, an exceedance probability analysis was performed on the MN datasets of the three tests. The MN of the three critical tests P1M1T2, P1M1T3 and P1M2T3 were listed and sorted with the lowest MN value at the bottom and the highest MN for each test. Thereafter, the 5% percentile was calculated using the Microsoft Excel internal mathematical function. The resulting MN value was assumed to be the 5% of non-exceedance and 95% probability of exceedance value. Table 16 below provides a summary of the results of the analysis.
Table 16: Summary of MN values with 95% probability of exceedance P1M1T2 (MN) P1M1T3 (MN) P1M2T3 (MN) 0.362 0.369 0.264 0.359 0.360 0.263 0.357 0.326 0.255 0.352 0.310 0.254 0.324 0.280 0.252 0.303 0.251 0.246 0.299 0.240 0.237 0.251 0.235 0.237 0.251 0.230 0.236 0.250 0.225 0.233 0.243 0.208 0.218 0.240 0.202 0.217 0.231 0.192 0.208 0.229 0.191 0.190 0.218 0.134 0.188 0.217 0.123 0.179 0.208 0.105 0.172 0.200 0.161 0.184 0.134 0.124 0.125 0.181 0.119 0.134 5% Percentile
From Table 16, it was observed that the critical MN was 0.119 obtained from test P1M1T3. Test P1M2T3 obtained a MN with a 95% probability of exceedance of 0.134 which lies between the 0.12-0.17 envelop. However, the 95% exceedance probability MN of test P1M1T2 was 0.181 which is greater than the 0.17 upper limit MN of the envelope. Therefore, the critical MN value obtained with a 95% probability of exceedance for Test series one was determined to be 0.119. The critical MN value of 0.119 was very close to the lower limit MN value of 0.12 that was determined by Rooseboom (1992).
5-119
5.2 Test Series Two MN Analysis
Similar tests to those of Test series one was conducted to perform Test series two riprap incipient failure tests. However, there were few differences in the Test series two tests performed compared to Test series one. One of the major differences was the 0.075 m median stone size tested in Test series two tests. Secondly, in testing Test series two there were more tests performed than in Test series one. About 15 tests were performed for Test series two and five tests in each designed steep bed slope were tested.
The MN in each test were calculated using the dimensionless MN criteria as defined by
Equation 57. The relevant correction factors were applied to account for the steep bed and
steep side bank slopes. However, only Equation 58 was used to calculate the steep bed correction factor. The side slope correction factor was not applied in the calculation of the Test series two MN values. The reason for this was because the water depth inducing the incipient failure of riprap did not encroach the side slopes of the riprap. Therefore, it seemed logical to assume that the side slope correction factor did not have an influence on the final critical MN values.
Table 17 provides a summary of all the important input parameters that were required in order
to calculate the MN values of Test series two physical laboratory tests.
Table 17: Hydraulic input parameters to determine MN values 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
5-120
Table 17 shows that only the median stone size and the settling velocities changed in the Test
series two MN input parameters. The average local water depths and average bed slopes in the failure region varied for each test depending on the failure conditions. The average local water depths and average local slopes measured for Test series two tests are summarised in Appendix
D.
The median stone size of 0.075 m (D50) for Test series two tests were used for the riprap protection. The assumed rock density of 2700 kg/m3 (ρr)was chosen based on the hornfels rock material type. The riprap rock settling velocity was based on the experimentally observed and tested results summarised in Appendix B. The settling velocity results from the laboratory settling velocity tests were based on the 0.075 m median stone size for Test series two tests. The rock angle of repose was 40° (ɸr).
The density of the water (ρw)was assumed to be 1000 kg/m3 and the kinematic viscosity (ѵ) was assumed to be 1.13E-06 m2/s at 15 °C. The reason for the two assumptions was explained in section 5.2.
By substituting the average local failure water depths and average local slopes summarised in
Appendix D into Equation 57 and the input variables in Table 17, the MN and Re* for Test series two tests were successfully calculated. A summary of all the Test series two MN and Re* can be obtained from Appendix E.
Test series two MN and Re* results from Appendix E were plotted onto the Liu diagram as shown in Figure 54. Two characteristic curves which were developed by Rooseboom (1992) and Armitage (2002) were plotted onto Figure 54 to form an envelope of critical incipient motion MN.
In Figure 54 most of the measured MN results plot above the 0.17 upper limit. About ten MN points plot below the 0.17 upper limit. From the MN of 0.17, the measured MN reached up to a maximum of 0.433. The lowest MN was 0.091 read from Figure 55.
Figure 54 and Figure 55 do not show the actual variations in the MN for each test in Test
series two. However, all the points are plotted as a scatter. The scatter of the MN results in
Figure 54 and Figure 55 points can be perceived as a representation of the different
possibilities defining the MN values in the riprap incipient failure regions. Therefore, the box- whisker diagram in Figure 56 was produced to display the variation in the MN values for each test in Test series two.
5-121 Figure 54: Test series two (incipient failure of D50=0.075 m riprap dumped on 0.5, 0.4 and 0.333 steep bed slopes with 0.4 steep side bank slope) MN
results plotted onto the Liu diagram.
0.00 0.50 1.00 1.50 2.00 2.50 1 10 100 1000 10000 100000