2. A25-ft cantilever beam is shown in the figure at the right. Loads P1-250 lb a
ID: 1767395 • Letter: 2
Question
2. A25-ft cantilever beam is shown in the figure at the right. Loads P1-250 lb and P2-550 lb can be applied at the random positions indicated in the diagram (a, b, and c are random variables). Assume that a and b are independent with the probability mass functions (pmfs) given in the table below. The bending moment, M, induced at the fixed support will depend on the magnitude of the loads and their positions. Compute the pmf of M (lb-ft). Compute its mean, its variance and the most likely bending moment. a(ft) 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 Prob(a) 5.08% 6.29% 7.34% 8.24% 8.86% 10.22% 11.99% 13.41% 14.17% 14.42% b (ft) 1.00 2.00 3.00 4.00 5.00 6.007.008.0 Prob(b) 2.22% 5.23% 7.70% 8.47% 4.41% 10.05% 25.25% 36.67%Explanation / Answer
bV A> 5.08 6.29 7.34 8.24 8.86 10.22 11.99 13.41 14.17 14.42 2.22 0.00112776 0.00139638 0.00162948 0.00182928 0.001967 0.002269 0.002662 0.002977 0.003146 0.003201 5.23 0.00265684 0.00328967 0.00383882 0.00430952 0.004634 0.005345 0.006271 0.007013 0.007411 0.007542 7.7 0.0039116 0.0048433 0.0056518 0.0063448 0.006822 0.007869 0.009232 0.010326 0.010911 0.011103 8.47 0.00430276 0.00532763 0.00621698 0.00697928 0.007504 0.008656 0.010156 0.011358 0.012002 0.012214 4.41 0.00224028 0.00277389 0.00323694 0.00363384 0.003907 0.004507 0.005288 0.005914 0.006249 0.006359 10.05 0.0051054 0.00632145 0.0073767 0.0082812 0.008904 0.010271 0.01205 0.013477 0.014241 0.014492 25.25 0.012827 0.01588225 0.0185335 0.020806 0.022372 0.025806 0.030275 0.03386 0.035779 0.036411 36.67 0.01862836 0.02306543 0.02691578 0.03021608 0.03249 0.037477 0.043967 0.049174 0.051961 0.052878 probability of event=p(a)*p(b) moment=((p1+p2)*a)+(p2*b) p1=250 p2=550 bv a> 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 1 2150 2550 2950 3350 3750 4150 4550 4950 5350 5750 2 2700 3100 3500 3900 4300 4700 5100 5500 5900 6300 3 3250 3650 4050 4450 4850 5250 5650 6050 6450 6850 4 3800 4200 4600 5000 5400 5800 6200 6600 7000 7400 5 4350 4750 5150 5550 5950 6350 6750 7150 7550 7950 6 4900 5300 5700 6100 6500 6900 7300 7700 8100 8500 7 5450 5850 6250 6650 7050 7450 7850 8250 8650 9050 8 6000 6400 6800 7200 7600 8000 8400 8800 9200 9600 probability moment prob*moment 0.001128 2150 2.424684 28617.18169 0.002657 2700 7.173468 53499.77255 0.003912 3250 12.7127 60641.51672 0.004303 3800 16.350488 49371.4715 0.00224 4350 9.745218 18035.93387 0.005105 4900 25.01646 26712.10161 0.012827 5450 69.90715 38718.35488 0.018628 6000 111.77016 26263.78111 0.001396 2550 3.560769 30029.61751 0.00329 3100 10.197977 54959.56491 0.004843 3650 17.678045 60604.63869 0.005328 4200 22.376046 47546.24825 0.002774 4750 13.1759775 16479.24327 0.006321 5300 33.503685 22518.3932 0.015882 5850 92.9111625 28406.94234 0.023065 6400 147.618752 14299.96889 0.001629 2950 4.806966 29258.00752 0.003839 3500 13.43587 52195.68011 0.005652 4050 22.88979 55631.68725 0.006217 4600 28.598108 41619.93121 0.003237 5150 16.670241 13436.32727 0.007377 5700 42.04719 16319.56304 0.018534 6250 115.834375 16285.18973 0.026916 6800 183.027304 4039.160148 0.001829 3350 6.128088 26937.0965 0.00431 3900 16.807128 46572.53336 0.006345 4450 28.23436 47543.31841 0.006979 5000 34.8964 33393.41356 0.003634 5550 20.167812 9742.424791 0.008281 6100 50.51532 9791.729681 0.020806 6650 138.3599 6008.396633 0.030216 7200 217.555776 4.809042602 0.001967 3750 7.37595 23240.36168 0.004634 4300 19.925254 38631.76956 0.006822 4850 33.08767 37272.17222 0.007504 5400 40.523868 23974.69139 0.003907 5950 23.248197 5982.483873 0.008904 6500 57.87795 4207.256995 0.022372 7050 157.719075 422.2498629 0.03249 7600 246.921112 5531.412696 0.002269 4150 9.415686 20931.64531 0.005345 4700 25.121782 33070.31818 0.007869 5250 41.31435 29537.46278 0.008656 5800 50.206772 16662.02867 0.004507 6350 28.619577 3160.37885 0.010271 6900 70.87059 848.2876252 0.025806 7450 192.250975 1779.727637 0.037477 8000 299.81392 24747.54879 0.002662 4550 12.111099 18514.79898 0.006271 5100 31.980927 27322.83192 0.009232 5650 52.162495 21821.00806 0.010156 6200 62.964286 9900.908714 0.005288 6750 35.6912325 1011.54269 0.01205 7300 87.964635 152.8209445 0.030275 7850 237.6567875 13292.41731 0.043967 8400 369.325572 64651.1782 0.002977 4950 14.736249 14902.63063 0.007013 5500 38.573865 19969.09997 0.010326 6050 62.470485 13357.77075 0.011358 6600 74.964582 3918.83433 0.005914 7150 42.2837415 8.265071541 0.013477 7700 103.773285 3541.429429 0.03386 8250 279.3470625 38233.37097 0.049174 8800 432.735336 127879.6497 0.003146 5350 16.829709 10619.95905 0.007411 5900 43.724369 12282.53408 0.010911 6450 70.375305 5932.645337 0.012002 7000 84.01393 421.4245292 0.006249 7550 47.1797235 821.6778319 0.014241 8100 115.350885 11860.73911 0.035779 8650 309.4905125 76540.56708 0.051961 9200 478.044788 210475.9405 0.003201 5750 18.40713 6613.997829 0.007542 6300 47.512458 5938.687352 0.011103 6850 76.05829 1263.879897 0.012214 7400 90.381676 552.1272743 0.006359 7950 50.555799 3698.412065 0.014492 8500 123.18285 24969.30705 0.036411 9050 329.515025 126320.2995 0.052878 9600 507.630144 307788.5489 mean of probability= 7187.384332 2440063.1 variance of probality sum(x*p(x)) sum(((x-mean)^2)*p(x)) most probable
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