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Question 7.7: For a given structure, the overtopping discharge under the p......

For a given structure, the overtopping discharge under the prescribed storm conditions is normally distributed with mean 3 and standard deviation 3 l/s/m-run. Calculate the (deterministic) threshold overtopping rate to ensure a probability of exceedance less than or equal to 0.05.

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Let the threshold be Q_{0}. Then, setting the probability of failure equal to the desired exceedance probability we have

0.05= P_{f} =1-P(Q\lt Q_{0} )=1-\Phi ((Q_{0}-3 )/3)

or

\Phi ((Q_{0}-3 )/3)=0.95.

From the tables in Appendix D, we find that the argument (Q_{0}-3 )/3=1.65.

So Q_{0}=7.95  l/s/m−run.

Appendix D: Normal distribution tables
Table D.1  Standard normal distribution function Φ(z)
Z Φ(z) Z Φ(z) Z Φ(z)
0.00 0.50000 1.00 0.84134 2.00 0.97724
0.02 0.50797 1.02 0.84613 2.02 0.97830
0.04 0.51595 1.04 0.85083 2.04 0.97932
0.06 0.52392 1.06 0.85542 2.06 0.98030
0.08 0.53188 1.08 0.85992 2.08 0.98123
0.10 0.53982 1.10 0.86433 2.10 0.98213
0.12 0.54775 1.12 0.86864 2.12 0.98299
0.14 0.55567 1.14 0.87285 2.14 0.98382
0.16 0.56355 1.16 0.87697 2.16 0.98461
0.18 0.57142 1.18 0.88099 2.18 0.98537
0.20 0.57925 1.20 0.88493 2.20 0.98609
0.22 0.58706 1.22 0.88876 2.22 0.98679
0.24 0.59483 1.24 0.89251 2.24 0.98745
0.26 0.60256 1.26 0.89616 2.26 0.98808
0.28 0.61026 1.28 0.89972 2.28 0.98869
0.30 0.61791 1.30 0.90319 2.30 0.98927
0.32 0.62551 1.32 0.90658 2.32 0.98982
0.34 0.63307 1.34 0.90987 2.34 0.99035
0.36 0.64057 1.36 0.91308 2.36 0.99086
0.38 0.64802 1.38 0.9162 2.38 0.99134
0.40 0.65542 1.40 0.91924 2.40 0.99180
0.42 0.66275 1.42 0.92219 2.42 0.99223
0.44 0.67003 1.44 0.92506 2.44 0.99265
0.46 0.677724 1.46 0.92785 2.46 0.99305
0.48 0.68438 1.48 0.93056 2.48 0.99343
0.50 0.69146 1.50 0.93319 2.50 0.99379
0.52 0.69846 1.52 0.93574 2.52 0.99413
0.54 0.7054 1.54 0.93821 2.54 0.99445
0.56 0.71226 1.56 0.94062 2.56 0.99476
0.58 0.71904 1.58 0.94294 2.58 0.99505
0.60 0.72574 1.60 0.9452 2.60 0.99533
0.62 0.73237 1.62 0.94738 2.62 0.9956
0.64 0.73891 1.64 0.94949 2.64 0.99585
(Continued)

 

Table D.1 (Continued)  Standard normal distribution function Φ(z)
Z Φ(z) Z Φ(z) Z Φ(z)
0.66 0.74537 1.66 0.95154 2.66 0.99609
0.68 0.75174 1.68 0.95352 2.68 0.99631
0.70 0.75803 1.70 0.95543 2.70 0.99653
0.72 0.76423 1.72 0.95728 2.72 0.99673
0.74 0.77035 1.74 0.95907 2.74 0.99692
0.76 0.77637 1.76 0.96079 2.76 0.99710
0.78 0.7823 1.78 0.96246 2.78 0.99728
0.80 0.78814 1.80 0.96406 2.80 0.99744
0.82 0.79389 1.82 0.96562 2.82 0.99759
0.84 0.79954 1.84 0.96711 2.84 0.99774
0.86 0.8051 1.86 0.96855 2.86 0.99788
0.88 0.81057 1.88 0.96994 2.88 0.99801
0.90 0.81593 1.90 0.97128 2.90 0.99813
0.92 0.82121 1.92 0.97257 2.92 0.99824
0.94 0.82639 1.94 0.97381 2.94 0.99834
0.96 0.83147 1.96 0.97500 2.96 0.99846
0.98 0.83645 1.98 0.97614 2.98 0.99855
1.00 0.84134 2.00 0.97724 3.00 0.99865
Table D.2  Standard normal distribution function Φ(z) for larger values of z
Z Φ(z)
3.00 0.99865010
3.05 0.99885579
3.10 0.99903239
3.15 0.99918364
3.20 0.99931286
3.25 0.99942297
3.30 0.99951657
3.35 0.99959594
3.40 0.99966307
3.45 0.99971970
3.50 0.99976737
3.55 0.99980738
3.60 0.99984089
3.65 0.99986887
3.70 0.99989220
3.75 0.99991158
3.80 0.99992765
3.85 0.99994094
3.90 0.99995190
3.95 0.99996092
4.00 0.99996832
4.05 0.99997439
4.10 0.99997934
4.15 0.99998337
4.20 0.99998665
4.25 0.99998931
4.30 0.99999146
4.35 0.99999319
4.40 0.99999458
4.45 0.99999570
4.50 0.99999660
4.55 0.99999731
4.60 0.99999788
4.65 0.99999834
4.70 0.99999869
4.75 0.99999898
4.80 0.99999920
4.85 0.99999938
4.90 0.99999952
4.95 0.99999962
5.00 0.9999997133
480   Appendix D
Table D.3  Standard normal density function values, Φ(z)
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.3989 0.3989 0.3989 0.3988 0.3986 0.3984 0.3982 0.398 0.3977 0.3973
0.1 0.397 0.3965 0.3961 0.3956 0.3951 0.3945 0.3939 0.3932 0.3925 0.3918
0.2 0.3910 0.3902 0.3984 0.3885 0.3876 0.3867 0.3857 0.3847 0.3836 0.3825
0.3 0.3814 0.3802 0.3790 0.3778 0.3765 0.3752 0.3739 0.3725 0.3712 0.3697
0.4 0.3683 0.3668 0.3653 0.3637 0.3621 0.3605 0.3589 0.3572 0.3555 0.3538
0.5 0.3521 0.3503 0.3485 0.3467 0.3448 0.3429 0.3410 0.3361 0.3372 0.3352
0.6 0.3332 0.3312 0.3292 0.3271 0.3251 0.323 0.3209 0.3187 0.3166 0.3144
0.7 0.3123 0.3101 0.3079 0.3056 0.3034 0.3011 0.2989 0.2966 0.2943 0.2920
0.8 0.2897 0.2874 0.2850 0.2827 0.2803 0.2780 0.2756 0.2732 0.2709 0.2685
0.9 0.2661 0.2637 0.2613 0.2589 0.2565 0.2541 0.2516 0.2492 0.2468 0.2444
1.0 0.2420 0.2396 0.2371 0.2347 0.2323 0.2299 0.2275 0.2251 0.2227 0.2203
1.1 0.2179 0.2155 0.2131 0.2107 0.2083 0.2059 0.2036 0.2012 0.1989 0.1965
1.2 0.1942 0.1919 0.1895 0.1872 0.1849 0.1826 0.1804 0.1781 0.1758 0.1736
1.3 0.1714 0.1691 0.1669 0.1647 0.1626 0.1604 0.1582 0.1561 0.1539 0.1518
1.4 0.1497 0.1476 0.1456 0.1435 0.1415 0.1394 0.1374 0.1354 0.1334 0.1315
1.5 0.1295 0.1276 0.1257 0.1238 0.1219 0.1200 0.1182 0.1163 0.1145 0.1127
1.6 0.1109 0.1092 0.1074 0.1057 0.1040 0.1023 0.1006 0.0989 0.0973 0.0957
1.7 0.094 0.0925 0.0909 0.0893 0.0878 0.0863 0.0848 0.0833 0.0818 0.0804
1.8 0.079 0.0775 0.0761 0.0748 0.0734 0.0721 0.0707 0.0694 0.0681 0.0669
1.9 0.0656 0.0644 0.0632 0.0620 0.0608 0.0596 0.0584 0.0573 0.0562 0.0551
2.0 0.0540 0.0529 0.0519 0.0508 0.0498 0.0488 0.0478 0.0468 0.0459 0.0449
2.1 0.0440 0.0431 0.0422 0.0413 0.0404 0.0396 0.0387 0.0379 0.0371 0.0363
2.2 0.0355 0.0347 0.0339 0.0332 0.0325 0.0317 0.0310 0.0303 0.0297 0.0290
2.3 0.0283 0.0277 0.0270 0.0264 0.0258 0.0252 0.0246 0.0241 0.0235 0.0229
2.4 0.0224 0.0219 0.0213 0.0208 0.0203 0.0198 0.0194 0.0189 0.0184 0.0180
2.5 0.0175 0.0171 0.0167 0.0163 0.0158 0.0154 0.0151 0.0147 0.0143 0.0139
2.6 0.0136 0.0132 0.0129 0.0126 0.0122 0.0119 0.0116 0.0113 0.0110 0.0107
2.7 0.0104 0.0101 0.0099 0.0096 0.0093 0.0091 0.0088 0.0086 0.0084 0.0081
2.8 0.0079 0.0077 0.0075 0.0073 0.0071 0.0069 0.0067 0.0065 0.0063 0.0061
2.9 0.0060 0.0058 0.0056 0.0055 0.0053 0.0051 0.0050 0.0048 0.0047 0.0046
3.0 0.0044 0.0043 0.0042 0.0040 0.0039 0.0038 0.0037 0.0036 0.0035 0.0034
3.1 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 0.0025 0.0025
3.2 0.0024 0.0023 0.0022 0.0022 0.0021 0.0020 0.0020 0.0019 0.0018 0.0018
3.3 0.0017 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 0.0013 0.0013
3.4 0.0012 0.0012 0.0012 0.0011 0.0011 0.0010 0.0010 0.0010 0.0009 0.0009
3.5 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 0.0007 0.0007 0.0006
3.6 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0004
3.7 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 0.0003 0.0003 0.0003
3.8 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002
3.9 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 0.0001
4.0 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

 

Appendix D   481
Table D.4  Complement of the standard normal distribution function, 1 – Φ(z)
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
0.3 0.3821 0.3738 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.17110 0.1685 0.1660 0.1635 0.1611
1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
1.2 0.11510 0.11310 0.11120 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.025 0.0244 0.0239 0.0233
2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.002 0.0019
2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
3.6 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

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