Esfuerzos Inducidos
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 INTRODUCCIÓN En el presente trabajo podemos observar que cuando se realiza un
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CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2
INTRODUCCIÓN
En el presente trabajo podemos observar que cuando se realiza una excavación subterránea en un macizo rocoso, los esfuerzos iniciales que existían antes de la excavación se perturban, es decir se inducen nuevos esfuerzos en las inmediaciones de la excavación. Siendo un método que nos permita representar este nuevo campo de esfuerzos es el de las trayectorias de los esfuerzos principales, donde los esfuerzos principales distan notablemente de ser verticales u horizontales en las inmediaciones de la excavación que desvía la trayectoria de los esfuerzos. En la determinación de los esfuerzos radiales, tangenciales, de corte. Así también como de esfuerzos máximos y mínimos, en materiales elásticos, nos apoyamos de la teoría matemática de la elasticidad para realizar nuestros cálculos en un punto dado en coordenadas polares. Donde el valor de K juega un papel importante, y es quien nos da la relación entre el esfuerzo horizontal y esfuerzo vertical. Para nuestro procedimiento tenemos roca Cuarcita de densidad 0.026MN/m3, a una profundidad de 350m, valor de RMR es 55, el esfuerzo de compresión de la roca equivalente es 190MPa y el valor de K=1.35 El cual luego de los cálculos y evaluación de resultados se puede obtener el factor de seguridad (F.S), podemos determinar si dicha excavación e inmediación es estable o inestable. En el cual se le pueda dar el sostenimiento u autosostenimiento adecuado para realizar de manera óptima y segura nuestro trabajo en las labores mineras.
1
Por : Meres Vargas Alí Ivan (20084509A)
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 CÁLCULO DE LOS ESFUERZOS INDUCIDOS La determinación de los esfuerzos inducidos por la excavación circular realizada en el macizo rocoso, se calculó a partir de los siguientes datos: Profundidad de la excavación Densidad de la roca: k Diámetro de la excavación r r
350m 0.026MN/m3 1.35 2a [a, 5a] [0º, 90º] Esfuerzo radial
Esfuerzo tangencial
Esfuerzo de corte
1
Esfuerzo principal
máximo
3
Esfuerzo principal mínimo
Con estos datos calcularemos r , , , 1 , 3 y utilizando las siguientes fórmulas:
r 1/ 2 v (1 k )(1 a 2 / r 2 ) (1 k )(1 4 a 2 / r 2 3a 4 / r 4 ) cos 2
1/ 2 v (1 k )(1 a 2 / r 2 )(1 k )(1 3a 4 / r 4 ) cos 2 1/ 2 v (1 k )(1 2a 2 / r 2 3a 4 / r 4 ) sin 2
1 1/ 2( r ) 1/ 4( r ) 2 2
1/ 2
3 1/ 2( r ) 1/ 4( r ) 2 2
1/ 2
1/ 2 arctan 2 /( r )
Para calcular el factor de seguridad (F.S), vamos a emplear los siguientes datos: Tipo de roca Cuarcita, el cual es una roca Metamórfica no foliada. RMR = 55, entonces GSI = 55 mi = 23, (Se estima utilizando tablas Hoek, 1983, Doruk, 1991 y Hoek et. al.1992) c = 190Mpa Por el criterio de Hoek y Brown determinaremos la resistencia del * material: 1
2
Por : Meres Vargas Alí Ivan (20084509A)
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 m mi e ( RMR 100 ) / 28
s e(GSI 100) / 9
1* 3 c (m 3 / c s ) a a= 1/2
Utilizando las fórmulas mencionadas y con la ayuda de un programa realizado Excel - Macros, se obtuvieron los siguientes resultados: TABLA DE RESULTADOS ESFUERZO RADIAL
TANG.
CORTE
MAX.
MIN.
Inclinación
ESF.
Factor de
de un punto
Resistencia
Seguridad
MAX
ESFUERZO MIN
R
r
r
F. S.
v
v
0
1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80
0,000 3,794 5,651 6,683 7,311 7,721 8,003 8,206 8,356 8,471 8,561 8,632 8,690 8,738 8,778 8,811 8,840 8,864 8,885 8,903 8,919 0,000 3,786 5,645 6,680 7,312 7,725 8,010 8,215 8,367 8,484 8,575 8,648 8,707 8,755 8,796
27,755 22,014 18,984 17,191 16,040 15,257 14,698 14,285 13,971 13,727 13,532 13,375 13,246 13,138 13,048 12,972 12,907 12,850 12,801 12,758 12,720 27,658 21,955 18,941 17,155 16,009 15,228 14,671 14,259 13,945 13,701 13,507 13,350 13,221 13,114 13,024
0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,261 0,343 0,366 0,368 0,363 0,355 0,348 0,340 0,334 0,328 0,323 0,318 0,314 0,311
27,76 22,01 18,98 17,19 16,04 15,26 14,70 14,29 13,97 13,73 13,53 13,37 13,25 13,14 13,05 12,97 12,91 12,85 12,80 12,76 12,72 27,66 21,96 18,95 17,17 16,02 15,25 14,69 14,28 13,97 13,72 13,53 13,37 13,24 13,14 13,05
0,000 3,794 5,651 6,683 7,311 7,721 8,003 8,206 8,356 8,471 8,561 8,632 8,690 8,738 8,778 8,811 8,840 8,864 8,885 8,903 8,919 0,000 3,783 5,636 6,667 7,296 7,708 7,991 8,195 8,347 8,463 8,553 8,626 8,684 8,733 8,773
0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,821 1,476 1,999 2,420 2,763 3,046 3,280 3,477 3,644 3,785 3,907 4,012 4,103 4,182
15,596 65,892 80,645 87,961 92,190 94,869 96,680 97,963 98,908 99,625 100,183 100,626 100,983 101,277 101,521 101,726 101,900 102,049 102,177 102,289 102,387 15,596 65,790 80,533 87,856 92,095 94,785 96,606 97,898 98,851 99,574 100,137 100,585 100,947 101,244 101,491
0,562 2,993 4,248 5,117 5,747 6,218 6,578 6,858 7,079 7,258 7,403 7,524 7,624 7,708 7,780 7,842 7,895 7,941 7,982 8,018 8,049 0,564 2,996 4,250 5,117 5,747 6,217 6,576 6,856 7,078 7,256 7,402 7,522 7,622 7,707 7,779
3,050 2,419 2,086 1,889 1,763 1,677 1,615 1,570 1,535 1,508 1,487 1,470 1,456 1,444 1,434 1,425 1,418 1,412 1,407 1,402 1,398 3,039 2,413 2,082 1,887 1,761 1,675 1,614 1,569 1,535 1,508 1,487 1,469 1,455 1,444 1,434
0,000 0,417 0,621 0,734 0,803 0,848 0,879 0,902 0,918 0,931 0,941 0,949 0,955 0,960 0,965 0,968 0,971 0,974 0,976 0,978 0,980 0,000 0,416 0,619 0,733 0,802 0,847 0,878 0,901 0,917 0,930 0,940 0,948 0,954 0,960 0,964
5
3
Por : Meres Vargas Alí Ivan (20084509A)
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40
10
15
20
4
8,830 8,859 8,883 8,905 8,924 8,940 0,000 3,763 5,626 6,672 7,316 7,739 8,032 8,244 8,402 8,523 8,618 8,694 8,755 8,806 8,849 8,884 8,915 8,941 8,964 8,983 9,001 0,000 3,724 5,596 6,660 7,322 7,761 8,067 8,290 8,457 8,586 8,687 8,768 8,835 8,889 8,935 8,974 9,007 9,035 9,060 9,081 9,100 0,000 3,671 5,554
12,947 12,882 12,826 12,777 12,734 12,696 27,371 21,779 18,813 17,051 15,917 15,143 14,590 14,181 13,869 13,626 13,432 13,276 13,148 13,041 12,951 12,875 12,810 12,753 12,704 12,662 12,624 26,902 21,492 18,604 16,880 15,766 15,003 14,457 14,053 13,744 13,503 13,311 13,155 13,028 12,921 12,832 12,756 12,691 12,635 12,586 12,544 12,506 26,265 21,103 18,320
0,308 0,305 0,303 0,301 0,299 0,297 0,000 0,513 0,675 0,721 0,725 0,715 0,700 0,685 0,670 0,657 0,646 0,635 0,627 0,619 0,612 0,606 0,601 0,597 0,592 0,589 0,586 0,000 0,750 0,987 1,054 1,060 1,045 1,023 1,001 0,980 0,961 0,944 0,929 0,916 0,905 0,895 0,886 0,879 0,872 0,866 0,861 0,856 0,000 0,964 1,269
12,97 12,91 12,85 12,80 12,76 12,72 27,37 21,79 18,85 17,10 15,98 15,21 14,66 14,26 13,95 13,71 13,52 13,36 13,24 13,13 13,04 12,96 12,90 12,84 12,80 12,75 12,72 26,90 21,52 18,68 16,99 15,90 15,15 14,62 14,22 13,92 13,68 13,50 13,34 13,22 13,12 13,03 12,95 12,89 12,84 12,79 12,75 12,71 26,26 21,16 18,45
8,807 8,836 8,860 8,882 8,900 8,917 0,000 3,748 5,592 6,623 7,255 7,670 7,958 8,166 8,321 8,439 8,533 8,607 8,668 8,718 8,759 8,794 8,824 8,850 8,872 8,891 8,908 0,000 3,692 5,521 6,553 7,191 7,613 7,907 8,121 8,281 8,405 8,502 8,580 8,643 8,696 8,739 8,776 8,808 8,835 8,858 8,879 8,897 0,000 3,618 5,429
4,252 4,314 4,368 4,416 4,459 4,498 0,000 1,630 2,923 3,954 4,786 5,465 6,025 6,493 6,886 7,220 7,505 7,751 7,963 8,148 8,310 8,452 8,578 8,689 8,789 8,877 8,957 0,000 2,413 4,314 5,827 7,048 8,049 8,880 9,576 10,167 10,670 11,103 11,477 11,802 12,087 12,337 12,557 12,752 12,925 13,080 13,219 13,344 0,000 3,157 5,621
101,698 101,875 102,026 102,156 102,270 102,369 15,596 65,489 80,207 87,550 91,820 94,543 96,393 97,712 98,687 99,429 100,009 100,470 100,844 101,151 101,407 101,622 101,805 101,962 102,097 102,216 102,319 15,596 65,001 79,685 87,067 91,393 94,171 96,070 97,430 98,441 99,214 99,818 100,301 100,693 101,016 101,285 101,512 101,705 101,870 102,014 102,138 102,248 15,596 64,347 79,002
7,841 7,894 7,940 7,981 8,017 8,048 0,570 3,005 4,256 5,120 5,747 6,215 6,574 6,853 7,074 7,253 7,398 7,519 7,619 7,704 7,776 7,838 7,892 7,938 7,979 8,015 8,046 0,580 3,020 4,266 5,125 5,749 6,215 6,572 6,851 7,072 7,250 7,396 7,517 7,617 7,702 7,775 7,837 7,890 7,937 7,978 8,014 8,045 0,594 3,042 4,283
1,425 1,418 1,412 1,407 1,402 1,398 3,008 2,395 2,071 1,879 1,756 1,672 1,611 1,567 1,533 1,506 1,485 1,468 1,454 1,443 1,433 1,425 1,418 1,411 1,406 1,401 1,397 2,956 2,365 2,053 1,867 1,747 1,665 1,606 1,563 1,530 1,504 1,483 1,466 1,453 1,441 1,432 1,423 1,416 1,410 1,405 1,401 1,397 2,886 2,325 2,027
Por : Meres Vargas Alí Ivan (20084509A)
0,968 0,971 0,974 0,976 0,978 0,980 0,000 0,412 0,614 0,728 0,797 0,843 0,875 0,897 0,914 0,927 0,938 0,946 0,952 0,958 0,963 0,966 0,970 0,973 0,975 0,977 0,979 0,000 0,406 0,607 0,720 0,790 0,837 0,869 0,892 0,910 0,924 0,934 0,943 0,950 0,956 0,960 0,964 0,968 0,971 0,973 0,976 0,978 0,000 0,398 0,597
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20
25
30
5
6,644 7,330 7,791 8,115 8,353 8,533 8,672 8,782 8,870 8,942 9,002 9,052 9,095 9,131 9,163 9,190 9,213 9,234 0,000 3,606 5,503 6,623 7,340 7,827 8,175 8,431 8,626 8,777 8,898 8,995 9,075 9,141 9,197 9,244 9,285 9,320 9,350 9,377 9,400 0,000 3,531 5,444 6,599 7,352 7,870 8,243 8,521 8,733 8,900 9,033 9,140
16,648 15,561 14,814 14,278 13,879 13,574 13,336 13,146 12,992 12,865 12,759 12,670 12,595 12,530 12,474 12,426 12,383 12,346 25,480 20,622 17,971 16,361 15,309 14,581 14,056 13,665 13,365 13,130 12,942 12,790 12,664 12,559 12,471 12,396 12,332 12,277 12,228 12,186 12,149 24,570 20,066 17,566 16,030 15,016 14,311 13,800 13,417 13,123 12,891 12,706 12,556
1,355 1,363 1,344 1,316 1,287 1,259 1,235 1,213 1,194 1,178 1,163 1,151 1,140 1,130 1,121 1,114 1,107 1,101 0,000 1,149 1,512 1,615 1,624 1,601 1,568 1,533 1,501 1,472 1,446 1,423 1,404 1,386 1,371 1,358 1,346 1,336 1,327 1,319 1,312 0,000 1,299 1,709 1,825 1,836 1,810 1,772 1,733 1,697 1,664 1,635 1,609
16,83 15,78 15,06 14,55 14,16 13,87 13,64 13,46 13,31 13,19 13,09 13,01 12,93 12,87 12,82 12,77 12,73 12,70 25,48 20,70 18,15 16,62 15,63 14,94 14,45 14,08 13,80 13,58 13,41 13,26 13,15 13,05 12,97 12,90 12,84 12,79 12,75 12,71 12,67 24,57 20,17 17,80 16,37 15,43 14,79 14,32 13,97 13,70 13,49 13,33 13,19
6,463 7,110 7,542 7,846 8,068 8,236 8,365 8,467 8,549 8,616 8,671 8,718 8,757 8,790 8,819 8,844 8,865 8,884 0,000 3,529 5,322 6,362 7,022 7,467 7,783 8,015 8,190 8,326 8,434 8,521 8,591 8,650 8,699 8,740 8,775 8,806 8,832 8,855 8,875 0,000 3,429 5,208 6,258 6,934 7,396 7,726 7,969 8,154 8,297 8,411 8,502
7,577 9,162 10,468 11,560 12,484 13,272 13,950 14,537 15,047 15,493 15,885 16,230 16,536 16,808 17,050 17,267 17,462 17,638 0,000 3,847 6,817 9,172 11,090 12,684 14,031 15,182 16,173 17,033 17,783 18,439 19,016 19,525 19,976 20,377 20,734 21,054 21,341 21,599 21,831 0,000 4,466 7,875 10,580 12,801 14,669 16,268 17,650 18,853 19,907 20,832 21,648
86,448 90,856 93,711 95,677 97,095 98,152 98,964 99,600 100,110 100,524 100,866 101,151 101,391 101,596 101,772 101,924 102,056 102,173 15,596 63,556 78,200 85,743 90,262 93,218 95,269 96,755 97,867 98,723 99,396 99,935 100,373 100,734 101,035 101,289 101,505 101,691 101,851 101,991 102,113 15,596 62,662 77,332 85,012 89,675 92,753 94,902 96,465 97,638 98,540 99,250 99,816
5,137 5,757 6,221 6,577 6,855 7,076 7,254 7,400 7,520 7,621 7,705 7,778 7,840 7,893 7,940 7,980 8,016 8,048 0,612 3,070 4,308 5,158 5,776 6,239 6,594 6,871 7,092 7,269 7,414 7,534 7,634 7,718 7,790 7,851 7,904 7,950 7,990 8,026 8,057 0,635 3,107 4,344 5,193 5,810 6,273 6,629 6,906 7,126 7,303 7,447 7,565
1,849 1,734 1,655 1,599 1,556 1,524 1,499 1,479 1,463 1,450 1,438 1,429 1,421 1,414 1,409 1,404 1,399 1,395 2,800 2,275 1,995 1,827 1,717 1,642 1,588 1,547 1,517 1,492 1,473 1,458 1,445 1,434 1,425 1,418 1,411 1,406 1,401 1,397 1,393 2,700 2,216 1,956 1,799 1,696 1,625 1,573 1,535 1,506 1,483 1,465 1,450
Por : Meres Vargas Alí Ivan (20084509A)
0,710 0,781 0,829 0,862 0,887 0,905 0,919 0,930 0,939 0,947 0,953 0,958 0,962 0,966 0,969 0,972 0,974 0,976 0,000 0,388 0,585 0,699 0,772 0,821 0,855 0,881 0,900 0,915 0,927 0,936 0,944 0,951 0,956 0,960 0,964 0,968 0,971 0,973 0,975 0,000 0,377 0,572 0,688 0,762 0,813 0,849 0,876 0,896 0,912 0,924 0,934
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00
35
40
6
9,229 9,303 9,365 9,418 9,463 9,502 9,536 9,566 9,592 0,000 3,447 5,379 6,573 7,364 7,917 8,319 8,620 8,853 9,035 9,182 9,301 9,399 9,481 9,550 9,609 9,660 9,704 9,742 9,775 9,805 0,000 3,359 5,309 6,545 7,378 7,968 8,400 8,727 8,980 9,180 9,341 9,472 9,580 9,671 9,748 9,814 9,870 9,919 9,961 9,998 10,031
12,432 12,328 12,241 12,166 12,103 12,047 11,999 11,957 11,920 23,564 19,451 17,118 15,663 14,693 14,012 13,516 13,143 12,855 12,628 12,445 12,297 12,174 12,072 11,985 11,912 11,849 11,794 11,746 11,704 11,667 22,491 18,794 16,640 15,272 14,348 13,694 13,214 12,850 12,569 12,346 12,167 12,021 11,900 11,799 11,713 11,641 11,578 11,524 11,476 11,435 11,398
1,587 1,567 1,550 1,535 1,522 1,511 1,500 1,491 1,483 0,000 1,410 1,855 1,981 1,993 1,964 1,923 1,881 1,841 1,805 1,774 1,746 1,722 1,701 1,682 1,666 1,652 1,639 1,628 1,618 1,609 0,000 1,478 1,944 2,076 2,088 2,058 2,016 1,971 1,929 1,892 1,859 1,830 1,804 1,782 1,763 1,746 1,731 1,718 1,706 1,696 1,686
13,08 12,99 12,92 12,85 12,80 12,75 12,71 12,67 12,64 23,56 19,57 17,40 16,08 15,20 14,59 14,15 13,82 13,57 13,38 13,22 13,10 13,00 12,91 12,84 12,79 12,74 12,69 12,66 12,62 12,60 22,49 18,93 16,96 15,74 14,93 14,36 13,95 13,64 13,41 13,23 13,09 12,98 12,89 12,81 12,75 12,70 12,65 12,62 12,59 12,56 12,53
8,576 8,637 8,688 8,731 8,768 8,800 8,827 8,850 8,871 0,000 3,324 5,093 6,160 6,858 7,339 7,685 7,941 8,135 8,285 8,403 8,498 8,575 8,639 8,691 8,736 8,773 8,805 8,832 8,856 8,877 0,000 3,219 4,985 6,076 6,800 7,304 7,667 7,936 8,139 8,296 8,419 8,517 8,595 8,660 8,712 8,757 8,794 8,825 8,852 8,875 8,895
22,369 23,009 23,577 24,084 24,537 24,943 25,308 25,637 25,934 0,000 4,996 8,768 11,772 14,268 16,400 18,253 19,876 21,307 22,572 23,692 24,685 25,567 26,352 27,052 27,677 28,237 28,739 29,191 29,599 29,967 0,000 5,419 9,469 12,719 15,465 17,856 19,971 21,855 23,536 25,037 26,375 27,569 28,634 29,584 30,432 31,191 31,870 32,480 33,028 33,522 33,969
100,277 100,656 100,971 101,236 101,461 101,654 101,821 101,965 102,092 15,596 61,706 76,453 84,318 89,156 92,377 94,636 96,281 97,515 98,463 99,205 99,796 100,274 100,665 100,990 101,261 101,491 101,687 101,855 102,001 102,128 15,596 60,733 75,623 83,722 88,767 92,148 94,524 96,253 97,545 98,532 99,301 99,909 100,397 100,793 101,120 101,391 101,618 101,811 101,975 102,116 102,239
7,664 7,747 7,817 7,877 7,928 7,973 8,012 8,046 8,076 0,662 3,153 4,393 5,245 5,866 6,331 6,688 6,965 7,185 7,360 7,502 7,618 7,715 7,795 7,862 7,920 7,969 8,011 8,048 8,080 8,108 0,693 3,207 4,458 5,319 5,947 6,418 6,778 7,056 7,274 7,448 7,587 7,699 7,792 7,868 7,931 7,985 8,030 8,069 8,103 8,132 8,157
1,438 1,428 1,419 1,412 1,406 1,401 1,397 1,393 1,389 2,589 2,151 1,913 1,767 1,670 1,603 1,555 1,519 1,492 1,470 1,453 1,439 1,428 1,419 1,411 1,405 1,400 1,395 1,391 1,387 1,384 2,472 2,081 1,864 1,730 1,640 1,578 1,533 1,499 1,474 1,454 1,438 1,426 1,416 1,408 1,401 1,395 1,391 1,386 1,383 1,380 1,377
Por : Meres Vargas Alí Ivan (20084509A)
0,942 0,949 0,955 0,959 0,964 0,967 0,970 0,973 0,975 0,000 0,365 0,560 0,677 0,754 0,807 0,844 0,873 0,894 0,910 0,923 0,934 0,942 0,949 0,955 0,960 0,964 0,968 0,971 0,973 0,975 0,000 0,354 0,548 0,668 0,747 0,803 0,843 0,872 0,894 0,912 0,925 0,936 0,945 0,952 0,957 0,962 0,966 0,970 0,973 0,975 0,977
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 45
1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60
50
55
7
0,000 3,267 5,237 6,516 7,392 8,019 8,483 8,836 9,111 9,329 9,504 9,648 9,768 9,867 9,952 10,024 10,086 10,140 10,187 10,228 10,265 0,000 3,176 5,165 6,487 7,407 8,071 8,567 8,946 9,242 9,478 9,668 9,825 9,955 10,064 10,156 10,235 10,303 10,362 10,413 10,459 10,498 0,000 3,087 5,096 6,459 7,420 8,122 8,648 9,052 9,369
21,385 18,118 16,148 14,869 13,993 13,366 12,902 12,549 12,274 12,056 11,881 11,737 11,617 11,518 11,433 11,361 11,299 11,245 11,198 11,157 11,120 20,279 17,441 15,655 14,466 13,637 13,037 12,590 12,247 11,980 11,766 11,594 11,452 11,335 11,236 11,152 11,081 11,019 10,966 10,919 10,878 10,842 19,206 16,785 15,178 14,075 13,292 12,719 12,287 11,955 11,694
0,000 1,500 1,974 2,108 2,120 2,090 2,047 2,001 1,959 1,921 1,887 1,858 1,832 1,810 1,790 1,773 1,758 1,744 1,732 1,722 1,712 0,000 1,478 1,944 2,076 2,088 2,058 2,016 1,971 1,929 1,892 1,859 1,830 1,804 1,782 1,763 1,746 1,731 1,718 1,706 1,696 1,686 0,000 1,410 1,855 1,981 1,993 1,964 1,923 1,881 1,841
21,39 18,27 16,49 15,37 14,62 14,09 13,70 13,42 13,21 13,05 12,92 12,82 12,74 12,68 12,63 12,59 12,55 12,52 12,50 12,48 12,46 20,28 17,59 16,00 14,97 14,27 13,78 13,43 13,17 12,98 12,83 12,72 12,64 12,58 12,53 12,49 12,45 12,43 12,41 12,39 12,38 12,37 19,21 16,93 15,51 14,56 13,90 13,44 13,12 12,88 12,71
0,000 3,117 4,891 6,014 6,770 7,299 7,681 7,963 8,175 8,337 8,462 8,561 8,640 8,704 8,755 8,798 8,833 8,863 8,888 8,909 8,928 0,000 3,024 4,817 5,979 6,771 7,329 7,731 8,025 8,245 8,411 8,538 8,636 8,713 8,774 8,822 8,861 8,893 8,920 8,942 8,960 8,975 0,000 2,943 4,765 5,974 6,808 7,397 7,820 8,128 8,354
0,000 5,712 9,946 13,388 16,361 19,011 21,406 23,577 25,542 27,313 28,906 30,332 31,607 32,747 33,764 34,673 35,487 36,215 36,869 37,457 37,988 0,000 5,852 10,168 13,743 16,917 19,829 22,528 25,026 27,322 29,415 31,309 33,012 34,536 35,896 37,109 38,189 39,153 40,012 40,781 41,470 42,089 0,000 5,816 10,100 13,739 17,081 20,256 23,291 26,170 28,865
15,596 59,785 74,894 83,278 88,558 92,113 94,612 96,423 97,768 98,786 99,572 100,186 100,673 101,065 101,383 101,644 101,861 102,042 102,195 102,325 102,437 15,596 58,906 74,312 83,028 88,569 92,310 94,933 96,823 98,212 99,251 100,041 100,648 101,122 101,495 101,793 102,033 102,229 102,389 102,522 102,634 102,727 15,596 58,132 73,907 82,994 88,818 92,756 95,507 97,471 98,896
0,729 3,273 4,541 5,418 6,059 6,539 6,904 7,184 7,401 7,571 7,705 7,813 7,899 7,969 8,027 8,075 8,115 8,149 8,178 8,202 8,223 0,769 3,348 4,643 5,545 6,206 6,699 7,071 7,353 7,569 7,734 7,862 7,962 8,041 8,103 8,152 8,193 8,225 8,252 8,274 8,292 8,308 0,812 3,434 4,766 5,700 6,388 6,900 7,282 7,568 7,782
2,350 2,007 1,813 1,689 1,606 1,548 1,506 1,475 1,452 1,434 1,420 1,409 1,401 1,394 1,388 1,383 1,379 1,376 1,373 1,371 1,369 2,228 1,933 1,759 1,645 1,568 1,514 1,475 1,447 1,426 1,410 1,398 1,389 1,382 1,376 1,372 1,369 1,366 1,364 1,362 1,360 1,359 2,111 1,860 1,704 1,600 1,528 1,477 1,441 1,415 1,397
Por : Meres Vargas Alí Ivan (20084509A)
0,000 0,343 0,537 0,661 0,744 0,802 0,844 0,875 0,898 0,916 0,930 0,941 0,949 0,956 0,962 0,967 0,971 0,974 0,977 0,979 0,981 0,000 0,332 0,529 0,657 0,744 0,805 0,850 0,882 0,906 0,924 0,938 0,949 0,957 0,964 0,969 0,974 0,977 0,980 0,983 0,985 0,986 0,000 0,323 0,524 0,657 0,748 0,813 0,859 0,893 0,918
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40
60
65
8
9,622 9,827 9,996 10,136 10,254 10,354 10,439 10,513 10,577 10,633 10,682 10,725 0,000 3,004 5,030 6,432 7,433 8,169 8,723 9,151 9,488 9,758 9,976 10,156 10,306 10,432 10,539 10,631 10,710 10,778 10,838 10,891 10,937 0,000 2,928 4,971 6,409 7,445 8,211 8,792 9,242 9,596 9,880 10,111 10,301 10,460 10,593 10,707 10,804 10,888 10,961
11,485 11,316 11,176 11,061 10,963 10,880 10,810 10,749 10,696 10,650 10,609 10,573 18,200 16,170 14,730 13,709 12,969 12,420 12,003 11,681 11,426 11,221 11,055 10,918 10,803 10,707 10,625 10,555 10,495 10,442 10,396 10,356 10,320 17,290 15,613 14,325 13,377 12,676 12,150 11,747 11,433 11,183 10,983 10,819 10,684 10,571 10,476 10,395 10,325 10,265 10,213
1,805 1,774 1,746 1,722 1,701 1,682 1,666 1,652 1,639 1,628 1,618 1,609 0,000 1,299 1,709 1,825 1,836 1,810 1,772 1,733 1,697 1,664 1,635 1,609 1,587 1,567 1,550 1,535 1,522 1,511 1,500 1,491 1,483 0,000 1,149 1,512 1,615 1,624 1,601 1,568 1,533 1,501 1,472 1,446 1,423 1,404 1,386 1,371 1,358 1,346 1,336
12,58 12,49 12,43 12,38 12,35 12,32 12,30 12,29 12,28 12,27 12,26 12,26 18,20 16,30 15,02 14,14 13,52 13,09 12,78 12,56 12,41 12,31 12,24 12,19 12,16 12,14 12,13 12,13 12,13 12,13 12,13 12,14 12,14 17,29 15,72 14,56 13,73 13,14 12,72 12,42 12,22 12,09 12,00 11,95 11,93 11,92 11,92 11,93 11,94 11,96 11,97
8,522 8,648 8,743 8,816 8,871 8,914 8,948 8,975 8,996 9,013 9,027 9,038 0,000 2,877 4,738 6,000 6,879 7,502 7,949 8,270 8,503 8,672 8,794 8,884 8,949 8,996 9,031 9,057 9,076 9,090 9,101 9,108 9,114 0,000 2,825 4,733 6,053 6,981 7,643 8,115 8,453 8,692 8,860 8,976 9,056 9,111 9,147 9,171 9,185 9,194 9,199
31,352 33,618 35,659 37,485 39,110 40,550 41,826 42,957 43,960 44,851 -44,356 -43,648 0,000 5,583 9,708 13,322 16,781 20,208 23,611 26,943 30,137 33,128 35,871 38,345 40,548 42,494 44,205 -44,295 -42,980 -41,826 -40,810 -39,914 -39,121 0,000 5,136 8,958 12,431 15,919 19,556 23,350 27,226 31,063 34,730 38,122 41,175 43,869 -43,783 -41,753 -40,002 -38,491 -37,186
99,944 100,723 101,308 101,752 102,093 102,356 102,562 102,724 102,853 102,957 103,040 103,108 15,596 57,487 73,690 83,178 89,303 93,449 96,333 98,372 99,826 100,870 101,623 102,168 102,565 102,854 103,067 103,223 103,339 103,424 103,488 103,534 103,568 15,596 56,984 73,651 83,554 89,993 94,362 97,393 99,512 100,993 102,024 102,734 103,220 103,547 103,766 103,908 103,998 104,052 104,081
7,942 8,061 8,151 8,218 8,269 8,308 8,338 8,361 8,378 8,392 8,402 8,410 0,857 3,528 4,905 5,882 6,604 7,141 7,539 7,831 8,044 8,196 8,305 8,381 8,434 8,470 8,495 8,511 8,521 8,526 8,529 8,530 8,529 0,902 3,626 5,057 6,084 6,849 7,419 7,839 8,142 8,355 8,500 8,594 8,653 8,687 8,704 8,709 8,707 8,701 8,692
1,383 1,373 1,366 1,361 1,357 1,354 1,352 1,350 1,349 1,348 1,348 1,347 2,000 1,791 1,651 1,554 1,486 1,438 1,404 1,380 1,364 1,352 1,345 1,340 1,336 1,334 1,333 1,333 1,333 1,333 1,333 1,334 1,334 1,900 1,727 1,600 1,509 1,444 1,398 1,365 1,343 1,328 1,319 1,314 1,311 1,310 1,310 1,311 1,312 1,314 1,316
Por : Meres Vargas Alí Ivan (20084509A)
0,937 0,950 0,961 0,969 0,975 0,980 0,983 0,986 0,989 0,990 0,992 0,993 0,000 0,316 0,521 0,659 0,756 0,824 0,873 0,909 0,934 0,953 0,966 0,976 0,983 0,989 0,992 0,995 0,997 0,999 1,000 1,001 1,002 0,000 0,310 0,520 0,665 0,767 0,840 0,892 0,929 0,955 0,974 0,986 0,995 1,001 1,005 1,008 1,009 1,010 1,011
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80 2,00
70
75
80
9
11,024 11,080 11,130 0,000 2,863 4,920 6,388 7,455 8,248 8,851 9,319 9,689 9,986 10,227 10,427 10,593 10,733 10,852 10,953 11,041 11,118 11,185 11,243 11,295 0,000 2,811 4,879 6,371 7,463 8,278 8,899 9,382 9,764 10,071 10,322 10,528 10,700 10,846 10,969 11,075 11,166 11,245 11,315 11,376 11,430 0,000 2,772 4,848 6,359 7,469 8,300
10,167 10,127 10,092 16,505 15,133 13,975 13,091 12,424 11,917 11,526 11,219 10,974 10,777 10,615 10,482 10,370 10,276 10,196 10,127 10,067 10,015 9,970 9,930 9,894 15,868 14,743 13,692 12,859 12,219 11,728 11,346 11,045 10,805 10,610 10,450 10,318 10,207 10,114 10,034 9,965 9,906 9,855 9,809 9,770 9,734 15,399 14,456 13,483 12,688 12,069 11,589
1,327 1,319 1,312 0,000 0,964 1,269 1,355 1,363 1,344 1,316 1,287 1,259 1,235 1,213 1,194 1,178 1,163 1,151 1,140 1,130 1,121 1,114 1,107 1,101 0,000 0,750 0,987 1,054 1,060 1,045 1,023 1,001 0,980 0,961 0,944 0,929 0,916 0,905 0,895 0,886 0,879 0,872 0,866 0,861 0,856 0,000 0,513 0,675 0,721 0,725 0,715
11,99 12,01 12,02 16,51 15,21 14,15 13,35 12,77 12,36 12,06 11,87 11,75 11,68 11,65 11,65 11,66 11,69 11,72 11,75 11,78 11,82 11,85 11,87 11,90 15,87 14,79 13,80 13,03 12,44 12,02 11,72 11,51 11,39 11,34 11,33 11,36 11,40 11,46 11,51 11,57 11,62 11,67 11,71 11,75 11,79 15,40 14,48 13,54 12,77 12,18 11,74
9,201 9,201 9,200 0,000 2,788 4,746 6,125 7,106 7,809 8,313 8,670 8,918 9,085 9,193 9,260 9,298 9,319 9,327 9,328 9,324 9,317 9,309 9,300 9,290 0,000 2,764 4,770 6,204 7,237 7,986 8,528 8,913 9,175 9,343 9,440 9,488 9,505 9,504 9,492 9,474 9,455 9,435 9,415 9,395 9,377 0,000 2,749 4,796 6,278 7,357 8,151
-36,054 -35,068 -34,207 0,000 4,467 7,827 11,005 14,374 18,109 22,267 26,783 31,482 36,118 40,456 44,337 -42,301 -39,446 -37,044 -35,030 -33,336 -31,907 -30,693 -29,655 -28,763 0,000 3,583 6,312 8,999 12,014 15,605 19,956 25,141 31,017 37,172 43,051 -41,774 -37,469 -33,992 -31,211 -28,983 -27,184 -25,716 -24,505 -23,494 -22,641 0,000 2,510 4,444 6,417 8,751 11,749
104,093 104,094 104,086 15,596 56,620 73,753 84,066 90,825 95,436 98,636 100,857 102,377 103,390 104,042 104,444 104,677 104,798 104,848 104,852 104,829 104,788 104,739 104,684 104,628 15,596 56,380 73,941 84,633 91,703 96,572 99,978 102,345 103,938 104,945 105,523 105,809 105,910 105,901 105,830 105,727 105,611 105,490 105,371 105,257 105,149 15,596 56,238 74,148 85,151 92,497 97,621
8,682 8,670 8,659 0,945 3,723 5,212 6,295 7,111 7,724 8,176 8,498 8,716 8,853 8,931 8,966 8,974 8,965 8,946 8,922 8,895 8,868 8,842 8,817 8,793 0,983 3,812 5,358 6,497 7,369 8,034 8,532 8,888 9,123 9,256 9,312 9,316 9,288 9,244 9,194 9,141 9,091 9,043 8,999 8,958 8,921 1,013 3,884 5,478 6,669 7,594 8,317
1,318 1,319 1,321 1,814 1,671 1,555 1,468 1,404 1,358 1,326 1,304 1,291 1,283 1,280 1,280 1,282 1,285 1,288 1,291 1,295 1,298 1,302 1,305 1,308 1,744 1,625 1,517 1,431 1,368 1,321 1,288 1,265 1,252 1,246 1,245 1,248 1,253 1,259 1,265 1,271 1,277 1,282 1,287 1,291 1,295 1,692 1,591 1,487 1,403 1,338 1,290
Por : Meres Vargas Alí Ivan (20084509A)
1,011 1,011 1,011 0,000 0,306 0,522 0,673 0,781 0,858 0,913 0,953 0,980 0,998 1,010 1,018 1,022 1,024 1,025 1,025 1,025 1,024 1,023 1,022 1,021 0,000 0,304 0,524 0,682 0,795 0,878 0,937 0,979 1,008 1,027 1,037 1,043 1,045 1,044 1,043 1,041 1,039 1,037 1,035 1,032 1,030 0,000 0,302 0,527 0,690 0,808 0,896
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2
85
90
10
2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 4,20 4,40 4,60 4,80 5,00 1,00 1,20 1,40 1,60 1,80
8,935 9,429 9,820 10,135 10,391 10,603 10,780 10,929 11,055 11,164 11,258 11,339 11,411 11,474 11,529 0,000 2,748 4,830 6,351 7,473
11,214 10,917 10,680 10,487 10,329 10,197 10,087 9,994 9,915 9,847 9,788 9,736 9,691 9,652 9,617 15,112 14,281 13,355 12,583 11,976
0,700 0,685 0,670 0,657 0,646 0,635 0,627 0,619 0,612 0,606 0,601 0,597 0,592 0,589 0,586 0,000 0,261 0,343 0,366 0,368
11,41 11,18 11,05 10,99 11,01 11,07 11,15 11,24 11,32 11,40 11,47 11,54 11,60 11,65 11,69 15,11 14,29 13,37 12,60 12,01
8,737 9,162 9,454 9,631 9,714 9,733 9,718 9,686 9,649 9,610 9,573 9,539 9,507 9,478 9,451 0,000 2,742 4,816 6,330 7,443
15,781 21,305 28,662 37,498 -43,613 -36,150 -30,541 -26,477 -23,518 -21,316 -19,639 -18,330 -17,287 -16,440 -15,742 0,000 1,293 2,299 3,350 4,643
101,270 103,856 105,604 106,653 107,143 107,258 107,166 106,980 106,759 106,532 106,314 106,109 105,921 105,748 105,591 15,596 56,167 74,305 85,517 93,059
8,874 9,286 9,561 9,704 9,735 9,691 9,612 9,520 9,429 9,344 9,267 9,197 9,135 9,079 9,029 1,032 3,932 5,558 6,785 7,751
1,254 1,229 1,214 1,208 1,209 1,216 1,225 1,235 1,244 1,253 1,261 1,268 1,274 1,280 1,285 1,661 1,570 1,469 1,385 1,319
0,960 1,007 1,039 1,058 1,067 1,070 1,068 1,064 1,060 1,056 1,052 1,048 1,045 1,042 1,039 0,000 0,301 0,529 0,696 0,818
2,00
8,313
11,503
0,363
11,54
8,273
2,20
8,956
11,133
0,355
11,19
8,900
6,410
98,385
8,523
1,269
0,909
9,044
102,267
9,140
1,230
0,978
2,40
9,457
10,839
0,348
10,92
9,375
13,354
105,133
9,626
1,200
1,030
2,60
9,854
10,603
0,340
10,73
9,723
21,127
107,195
9,986
1,180
1,068
2,80
10,173
10,411
0,334
10,65
9,938
35,178
108,459
10,187
1,170
1,092
3,00
10,434
10,254
0,328
10,68
10,004
-37,337
108,843
10,188
1,174
1,099
3,20
10,649
10,124
0,323
10,80
9,970
-25,424
108,645
10,058
1,187
1,096
3,40
10,828
10,014
0,318
10,94
9,904
-19,001
108,261
9,898
1,202
1,088
3,60
10,980
9,921
0,314
11,07
9,835
-15,351
107,855
9,747
1,216
1,081
3,80
11,108
9,842
0,311
11,18
9,770
-13,074
107,473
9,612
1,229
1,074
4,00
11,219
9,774
0,308
11,28
9,711
-11,541
107,128
9,496
1,240
1,067
4,20
11,314
9,715
0,305
11,37
9,659
-10,448
106,820
9,395
1,249
1,061
4,40
11,397
9,664
0,303
11,45
9,613
-9,633
106,546
9,307
1,258
1,056
4,60
11,470
9,619
0,301
11,52
9,571
-9,005
106,302
9,230
1,266
1,052
4,80
11,533
9,579
0,299
11,58
9,535
-8,508
106,085
9,163
1,272
1,048
5,00
11,590
9,544
0,297
11,63
9,502
-8,106
105,891
9,103
1,278
1,044
1,00
0,000
15,015
0,000
15,02
0,000
0,000
15,596
1,039
1,650
0,000
1,20
2,740
14,221
0,000
14,22
2,740
0,000
56,146
3,948
1,563
0,301
1,40
4,823
13,312
0,000
13,31
4,823
0,000
74,364
5,586
1,463
0,530
1,60
6,349
12,548
0,000
12,55
6,349
0,000
85,650
6,826
1,379
0,698
1,80
7,474
11,945
0,000
11,95
7,474
0,000
93,263
7,808
1,313
0,821
2,00
8,318
11,475
0,000
11,47
8,318
0,000
98,670
8,599
1,261
0,914
2,20
8,964
11,105
0,000
11,11
8,964
0,000
102,656
9,244
1,220
0,985
2,40
9,467
10,812
0,000
10,81
9,467
0,000
105,682
9,774
1,188
1,040
2,60
9,866
10,577
0,000
10,58
9,866
0,000
108,034
10,214
1,162
1,084
2,80
10,186
10,386
0,000
10,39
10,186
0,000
109,899
10,581
1,141
1,119
Por : Meres Vargas Alí Ivan (20084509A)
CALCULO DE LOS ESFUERZOS INDUCIDOS 201 2 3,00
10,448
10,229
0,000
10,45
10,229
0,000
110,145
10,542
1,148
1,124
3,20
10,664
10,099
0,000
10,66
10,099
0,000
109,391
10,258
1,172
1,110
3,40
10,845
9,989
0,000
10,84
9,989
0,000
108,756
10,028
1,192
1,098
3,60
10,997
9,897
0,000
11,00
9,897
0,000
108,215
9,841
1,208
1,088
3,80
11,126
9,818
0,000
11,13
9,818
0,000
107,753
9,685
1,223
1,079
4,00
11,237
9,750
0,000
11,24
9,750
0,000
107,354
9,553
1,235
1,071
4,20
11,333
9,691
0,000
11,33
9,691
0,000
107,008
9,442
1,245
1,065
4,40
11,416
9,640
0,000
11,42
9,640
0,000
106,706
9,347
1,255
1,059
4,60
11,489
9,595
0,000
11,49
9,595
0,000
106,440
9,264
1,263
1,054
4,80
11,553
9,555
0,000
11,55
9,555
0,000
106,206
9,193
1,270
1,050
5,00
11,610
9,520
0,000
11,61
9,520
0,000
105,998
9,130
1,276
1,046
.
11
Por : Meres Vargas Alí Ivan (20084509A)