Modelos Campos Tensoes_BetI-2014.pdf

BETÃO ESTRUTURAL - ZONAS DE DESCONTINUIDADE Introdução aos modelos de campos de tensões JOÃO F. ALMEIDA MIGUEL S. LOURENÇO Novembro 2014 ESTRUTURAS DE BETÃO I, MESTRADO INTEGRADO EM ENGENHARIA CIVIL, IST 1 RITTER , MÖRSCH : “INTUIÇÃO” (E EQUILÍBRIO) – BASE PARA OS ACTUAIS MODELOS DE CAMPOS DE TENSÕES Identificação das trajectórias da carga EUROCÓDIGO 2 §5.6 Plastic Analysis Estabelecimento de condições de equilíbrio RITTER (1899) MÖRSCH (1909, 1922) Emil Mörsch (1872-1950) “ .... nothing is more practical than a good theory ....” 2 “ESFORÇO TRANSVERSO” EUROCÓDIGO 2 §5.6 Plastic Analysis   VSd - Vccd - VPd  VRd Asw VRd,s = z fywd cot 1  cot  2.5 s fcd VRd,max = cw bw z 1 Ftd = 0.5 VEd cotg  cot  + tan  M V z C  cot g  N z 2 zs M V z  zs T  cot g  N z 2 zs 3 “ESFORÇO TRANSVERSO” ↔ “ Ligação alma – banzos ” Fd Fd Vzd A' vEd = = . hf . x x z Atot Asf hf Fd . fyd  vEd = s cotg  x cotg  vEd   fcd senf cosf 1  cotf  2 (banzos comprimidos) 1  cotf  1.25 (banzos traccionados) 4 “ESFORÇO TRANSVERSO” ↔ “ Torção ” Ak tef,i linha média uk t Asw TEdi fywd cot   VEdi = s 2Ak TRd,máx = 2 cw fcd Ak tef,i sen  cos   Asl fyd TEd = cot  uk 2 Ak T Ed V Ed + 1 T Rd,máx V Rd,máx 5 “ESFORÇO TRANSVERSO” ↔ “Região do apoio” – continuidade entre zonas “B” e “D” z cotg EUROCÓDIGO 2 §5.6 Plastic Analysis q 1 aF  d1 1  cot 1      cot  C 2 z  z 2 Tf  Va cot 1 d V z Va  q0.5  a1  d1  z cot   fsw   Tf T z cot  fsw d1 Va aF d1 cotg “D” “B” “D” Reineck, K.-H., J.; Lourenço, M.; Almeida, J.; Haugerud, S.-A. 2011 : “Gaining experience with strut and tie models for the design of concrete structures”. in Design Examples for strut and tie models, fib Bulletin 61, Lausanne. 6 “ESFORÇO TRANSVERSO” ↔ Transmissão de cargas “próximas” dos apoios Modelo STM Modelo de campos de tensões EUROCÓDIGO 2 §5.6 Plastic Analysis 1 a1  aF  x  cot g 2 F2 1  2a     1   z / 2  a  2z F 3 z  FIP/fib Recommendations 1999 : “Practical Design of Structural Concrete”. FIP-Commission 3 "Practical Design", Sept. 1996. Publ.: SETO, London, 7 Sept. 1999. (Distributed by: fib, Lausanne. Web http://www.fib-international.org) 7 EXERCÍCIO : Dimensionar / Verificar a segurança 1200 kN 1200 kN 3.00 3.00 2.00 3.50 6.25 8.50 6.25 1200 kN "V" [kN] "M" [kN.m] 3600 kN.m EST Ø8//0.10 3Ø16 EST Ø8//0.20 EST Ø8//0.20 3Ø16 EST Ø8//0.10 11Ø25 11Ø25 9Ø20 9Ø20 8 9 3Ø16 EST Ø8//0.10 EST Ø8//0.20 EST Ø8//0.20 EST Ø8//0.10 3Ø16 Em diversas regiões do elemento, não é aplicável a teoria das peças lineares 3600 kN.m -1926.76 -1670.92 -2275.69 -2275.69 -2651.64 -1536.53 -1495.45 -143.56 -33.77 54 -1545.64 -21.30 -1745.48 22. -102.61 -1562.04 -15.25 -1866.68 -1597.75 -2213.15 -2213.15 -2605.36 -137.52 -1456.92 -1413.97 -35.61 73 -1469.42 -23.23 11. -1681.53 -106.40 -1488.42 -17.17 -1808.81 -1526.69 -134.52 -2151.41 -2151.41 -2559.22 -1379.22 -1334.43 -37.56 15 -1395.24 -0. -25.22 -1619.96 -1416.96 -110.24 -133.95 -1457.89 -19.19 -1753.22 -2090.52 -1303.59 -1257.00 -2090.52 -2513.13 -39.61 -1323.27 -13.05 -27.28 -135.31 -1560.91 -1347.81 -114.12 -1391.50 -1699.98 1533. 68 1532. 64 1531. 55 -21.33 -1230.22 -1181.90 1530. 40 1529. 19 1527. 91 1526. 58 -2030.51 1525. 19 1523. 73 -2030.51 1522. 21 1519. 03 -2467.04 1510. 95 1502. 82 1494. 64 1486. 43 1478. 17 1469. 87 1461. 53 -41.76 1522. 91 1453. 14 1444. 71 -1253.65 1514. 78 1436. 25 -26.94 1506. 60 1427. 74 1473. 43 1481. 78 1490. 10 1498. 37 1465. 05 1419. 19 1410. 60 -138.17 1456. 64 1401. 97 1448. 20 -29.40 1439. 73 1393. 30 -1504.49 -1281.13 1431. 25 1384. 59 -118.06 -1327.69 1422. 74 1373. 72 -1109.33 53 1414. 23 -1159.25 -23.59 -1649.12 1405. 71 1361. 19 -1971.42 1397. 18 1388. 66 1348. 63 1336. 06 -1971.42 -8. -2420.86 1380. 14 1362. 97 1323. 48 1310. 87 1298. 25 1343. 16 1285. 61 1323. 51 -44.00 -41.80 -1186.55 1272. 95 1304. 05 1284. 80 1260. 27 -142.18 1265. 79 1247. 58 1234. 86 -1217.09 -31.59 1222. 12 -1450.81 22 -1039.50 1247. 05 -122.04 -1266.61 -1090.87 1209. 36 1228. 59 5. 1196. 57 -25.99 -1600.67 1183. 77 -1913.29 1210. 46 -1913.29 1170. 95 -2374.52 1192. 67 1156. 40 -57.60 1175. 26 1141. 51 -46.31 -1122.11 1126. 57 1158. 25 -147.04 14 1141. 67 1111. 59 1096. 56 -1155.83 -972.61 18. -33.84 1081. 49 -1025.22 1125. 54 -1399.97 -1208.42 -126.08 1066. 37 -28.54 -1554.66 1109. 89 1051. 21 -1856.16 -1856.16 1036. 01 -2327.96 1094. 76 -74.34 1080. 1020. 76 -1060.48 15 -48.69 29 1005. 46 -152.55 1066. 11 990. 12 -908.86 30. -1097.51 1047. 974. 74 -962.46 -36.17 83 -1352.05 -1153.28 -130.18 1025. 959. 32 -31.24 -1511.08 92 943. 85 -1800.06 -1800.06 1004. -92.00 928. 34 -2281.11 73 -1001.82 78 912. 59 -158.54 -51.12 984. 31 -848.44 892. 47 41. 964. -1042.29 -902.73 74 872. 21 -38.57 -1307.12 -1101.34 946. -134.33 08 851. 81 -34.12 -1469.90 -110.58 831. 26 -1745.03 928. -1745.03 -2233.90 40 810. 55 69 -946.25 -164.87 -791.53 911. -53.57 789. 69 52. 77 -846.15 -990.33 896. 768. 65 1347. 52 1345. 88 1344. 16 1342. 36 -1052.77 28 1340. 49 -41.05 747. 46 -1265.23 1338. 54 1358. 84 1336. 51 -138.55 1346. 89 1352. 90 1334. 41 1340. 82 -130.07 1332. 22 1328. 49 1334. 68 882. 1329. 95 1326. 31 -37.17 -1431.08 1315. 96 1322. 25 726. 09 1309. 63 1318. 65 1310. 93 -1691.11 00 -1691.11 1303. 27 1303. 16 13 1295. 34 -893.92 1296. 89 -171.47 -738.30 704. 57 -2186.28 1290. 48 869. 1284. 05 1287. 46 1279. 53 1271. 55 63. 1277. 62 -56.04 -792.84 02 1263. 52 1271. 18 682. 89 1255. 43 -941.78 1264. 75 857. 1258. 32 1247. 29 1239. 11 661. 08 1230. 87 41 1251. 92 -1007.71 1222. 58 -43.61 -1226.40 1245. 54 639. -150.47 -142.82 1214. 25 1232. 21 847. 1205. 87 17 -40.43 1218. 57 -1394.56 -688.89 1197. 44 18 -1638.31 25 1205. 14 617. -178.27 -844.97 1186. 92 -1638.31 19 -2138.18 1191. 96 1174. 64 73. -742.90 838. 1162. 32 1179. 05 595. -58.49 61 1149. 97 20 -896.81 1166. 42 1137. 58 1125. 16 573. 1154. 11 831. -966.31 28 -171.78 1112. 71 1142. 14 57 -46.24 -1190.64 -147.17 1100. 22 -643.52 551. 1130. 53 86 1087. 70 -43.91 -185.24 -1360.27 -799.54 55 826. -1586.69 1075. 15 1119. 30 -1586.69 17 -696.44 82. 1062. 58 -2089.55 1108. 48 542. -60.85 822. 1098. 10 1049. 97 66 -855.59 47 1037. 34 536. -194.00 1088. 16 1024. 68 -928.72 -604.74 1078. 1012. 00 26 -1157.92 -48.96 820. 70 -151.57 -192.40 89 1069. 999. 31 -757.78 -47.62 51 73 531. -1328.09 -653.51 -1536.27 986. 60 -1536.27 91. 1061. 83 -2038.76 27 820. 1053. 971. 87 -63.13 -818.29 31 34 956. 43 529. -217.12 -572.34 1045. 97 940. 94 06 -895.07 815. 65 1033. 925. 41 -51.76 -199.77 -1128.17 -156.05 89 527. 76 -719.85 -614.41 99. 909. 83 -51.58 1019. -1297.92 64 -1487.07 30 -1487.07 806. 894. 22 -1974.30 1005. 878. 55 -65.63 29 74 527. -785.09 -545.82 -241.13 47 993. 862. 85 797. 13 28 -865.47 -207.42 847. 09 105. 981. -1101.33 27 -54.64 -160.60 -685.91 -588.00 527. 56 831. 29 -55.82 -1269.62 73 -1439.14 788. 971. -1439.14 08 815. 45 -1910.32 90 17 -68.35 -524.58 -266.04 -756.16 961. 799. 56 76 528. 109. -215.43 781. 783. 62 80 -840.01 953. -567.26 25 -656.18 67 -165.23 -57.61 767. 63 -1077.27 530. -60.34 946. -1243.05 90 774. 751. 59 -1392.50 34 86 -1392.50 1136. 71 -507.95 39 1134. 35 -1846.84 -286.48 1131. 90 1129. 34 735. 50 -71.33 110. 941. 1126. 69 -731.69 40 1123. 93 1121. 07 532. 719. -223.94 1118. 11 1115. 04 1111. 87 36 768. 24 -818.74 937. 1107. 65 -551.52 1100. 39 698. 1137. 84 1139. 23 -630.85 40 1134. 61 33 1093. 07 66 1131. 30 -170.07 1127. 92 -60.74 -1055.65 1124. 47 1120. 96 1085. 67 1078. 69 -65.58 92 677. 1117. 39 535. 1114. 02 934. 1071. 74 85 -1215.29 -1345.13 764. 1110. 94 1064. 74 -1345.13 -493.69 87 1107. 90 12 -296.21 102. 1104. 91 1057. 67 659. -1780.33 04 1050. 55 -76.98 1101. 98 -229.62 -707.33 934. 1043. 37 33 1099. 12 46 1036. 14 1096. 33 763. 641. 539. 1093. 64 1028. 85 -793.23 -536.30 14 1021. 50 85 44 -604.47 1091. 04 10 935. -178.56 1014. 10 -67.55 1088. 56 -1025.65 625. 79. 40 -73.58 1006. 65 51 1086. 18 999. 14 542. -1294.69 -1181.05 -479.55 -301.83 1082. 23 -1294.69 763. 937. 610. -233.31 991. 58 82 -85.55 1076. 94 -1710.13 -682.34 69 39 57 83 982. 28 1071. 88 55. 971. 22 -522.78 1067. 04 596. 941. 545. -768.74 -587.21 766. 30 960. 15 56 1062. 45 27 -187.05 -74.55 949. 09 -997.31 1058. 12 31 -81.93 43 946. 938. 04 584. -307.92 1054. 06 -237.53 -467.54 09 -1245.40 546. -1147.91 -1245.40 32. 23 1050. 29 927. 01 -94.62 770. 81 -1640.35 -660.17 952. 915. 99 573. 1046. 83 63 905. 00 01 -511.43 46 1043. 68 -746.84 -575.20 1040. 894. 05 547. 69 563. 776. 959. 86 -195.53 -81.75 47 -90.63 -970.61 9. 1038. 37 -242.29 883. 14 -314.43 -457.68 86 95 39 -1197.37 1036. -1197.37 -1115.81 27 872. 28 -104.16 555. 547. -640.94 961. -1571.06 1034. 861. 48 16 25 785. 17 53 -502.28 -12.74 1033. -565.42 08 16 850. 75 -727.59 954. 1032. 548. 546. 840. -251.43 -204.00 -89.16 10 38 -99.64 76 19 -321.29 -945.51 20 -449.99 829. 795. 1031. 54 -1150.72 63 543. -1150.72 -1084.70 949. -114.13 34 1031. 819. 544. -624.79 -34.98 01 52 49 -1502.36 08 34 -495.32 1031. 808. -557.87 807. 945. 01 73 539. -262.48 -711.05 50 1026. 68 796. 01 519. -328.47 95 -212.47 -96.78 -108.98 76 -921.94 -444.45 11 -57.14 1023. 942. 994. 84 991. 92 783. -1105.57 988. 94 88 985. 88 1012. 25 1012. 79 1011. 05 -124.52 536. -1105.57 1011. 67 982. 75 00 -1054.48 81 1010. 46 1011. 07 979. 53 822. 1021. 493. -611.83 1009. 84 976. 24 34 1009. 23 -1434.34 972. 87 769. 83 1008. 64 969. 42 88 -490.52 1008. 08 38 13 -273.45 941. 1020. 1007. 55 965. 89 -552.54 1007. 07 82 962. 28 755. 534. -697.25 -79.36 53 1006. 65 957. 02 471. -335.91 91 1020. 1006. 30 949. 92 -118.61 -220.92 -104.61 93 78 -441.01 838. 87 1006. 02 942. 76 -899.85 742. 1005. 83 941. 935. 54 -1062.08 60 -135.28 70 1022. 1005. 74 -1062.08 928. 26 534. 452. -1025.09 74 1005. 75 -602.21 00 920. 93 729. 41 -284.33 -1367.11 -101.77 1005. 88 48 74 -487.82 1024. 913. 54 1005. 87 857. 943. 23 -549.38 906. 10 435. 716. 1004. 48 -686.19 54 898. 62 40 -343.58 56 40 1027. 535. 1003. 32 -128.52 -229.37 -112.65 -439.60 57 891. 09 420. 26 -124.52 1002. 40 703. -879.14 947. 883. 52 70 -146.37 86 44 66. -1020.36 1032. 1001. 73 -1020.36 878. 02 875. 91 -996.40 03 -295.12 -598.48 1001. 33 868. 27 691. 538. 407. -1300.81 23 21 1037. 42 24 -487.13 1001. 21 952. 61 860. 60 38 77. -548.32 1001. 39 679. 395. -147.77 -351.45 -677.86 26 58 1044. 852. 92 -138.69 901. 25 45 1001. 88 -120.90 -440.11 -237.82 541. 88. 31 385. 1002. -859.68 845. 22 -157.77 24 69 667. 10 959. -980.58 67 -305.81 1052. -980.58 1003. 34 85 837. 52 38 376. 49 -968.31 99. -598.29 14 06 1005. 655. -1235.57 828. 21 -171.69 -488.35 1061. 35 926. 1007. 84 543. 367. 91 13 -549.26 110. 968. -359.50 10 21 817. 91 95 -672.18 -149.10 06 1009. 360. 68 30 33 644. 1071. -442.40 -129.36 46 -246.25 807. 68 120. 67 -169.43 -316.40 18 1012. 354. 26 545. -841.35 797. 98 -942.86 08 51 -942.86 129. 948. 979. -196.43 1082. 633. 21 348. 60 1015. 787. -940.69 -600.16 30 37 11 69 43 89 343. 61 139. 21 -1171.56 -491.34 1018. 777. 339. 27 545. -552.09 1094. 55 148. 04 -367.70 44 623. -159.74 992. 68 1022. 335. 52 -669.02 41 767. 55 49 156. 51 -138.04 -326.88 328. 56 -446.32 973. -181.32 30 -254.68 56 164. 68 1108. 311. 62 -222.17 1026. -823.98 -907.36 613. 295. 48 56 69 172. 61 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-418.99 696. 492. -775.56 99 -648.98 05 72 -825.41 39 -538.69 57 92 980. 02 689. -357.51 1235. -602.02 618. 589. -353.33 1194. 60 -194.17 -397.26 1318. 987. 83 51 78 483. -689.77 35 -747.45 1241. -305.09 -496.07 34 682. -747.45 58 1256. 06 91 996. 59 610. -728.44 71 92 64 -268.54 675. -360.76 1210. -238.68 1004. 593. 474. -427.76 71 05 -746.33 1262. -776.51 -662.46 79 56 87 -375.18 69 -550.30 93 1013. 668. -406.92 1382. 1283. 56 601. 32 -204.07 -614.78 29 465. -368.42 1229. 1264. 1023. 83 598. -730.68 659. 19 -730.68 -695.78 55 -313.12 -506.45 07 15 85 80 61 1032. -280.22 594. -249.47 -712.66 1262. 1327. 98 651. 455. -430.30 25 -310.90 1246. -368.74 81 79 13 1449. 1041. -725.53 -748.65 598. -410.00 -660.61 20 51 -549.26 30 94 1259. 1051. 643. 587. 03 438. -214.50 95 -607.68 41 44 80 -719.94 29 -719.94 1265. 1061. -316.50 -680.74 -501.59 -214.07 636. -289.28 1375. 1259. 78 23 596. 422. -258.79 60 580. 1518. 25 -698.92 1071. 90 -349.49 67 -425.98 -411.00 89 72 53 628. 84 -735.14 -712.67 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-680.46 -525.76 1503. 579. 14 61 63 -297.55 66 1990. 574. 1228. -569.35 99 88 75 82 -306.81 1827. 1397. -331.60 286. -621.88 -489.96 30 1569. 64 48 549. 91 1244. 185. -414.80 569. 96 -646.30 64 -684.43 1428. 96 575. 275. -335.40 17 40 56 2109. -423.97 -664.02 1259. -267.64 44 10 -671.50 -671.50 -596.25 1930. -307.59 48 547. 264. 1642. -521.20 53 563. 66 62 1461. 34 1271. -302.64 11 -560.35 00 51 65 651. 57 646. 07 640. 55 634. 98 629. 37 623. 73 618. 05 612. 32 607. 48 74 565. 254. -335.75 219. 657. 03 602. 99 -607.57 -488.97 2236. 1285. 667. 84 662. 45 598. 45 593. 88 00 07 -413.43 2040. 1721. 1497. 673. 19 589. 26 558. -344.79 -633.84 545. -428.19 -672.67 88 48 678. 51 584. 59 243. 683. 80 57 83 91 72 579. 89 -652.66 1302. -278.37 29 689. 06 -662.93 05 -662.93 575. 14 -317.76 98 694. 29 550. 234. 03 -583.44 1808. 1537. 570. 36 699. 49 553. 15 -517.09 2372. 2157. 04 1321. 704. 66 565. 54 01 -300.37 -551.51 70 53 710. 46 544. 224. 245. 14 560. 68 -340.05 57 16 718. 83 69 -593.47 1579. 1344. 555. 78 -354.23 -432.96 -488.71 1902. 727. 25 09 535. 213. -412.59 -661.30 549. 11 -621.71 2515. 2282. 96 735. 73 550. 85 38 32 -289.13 33 1368. -654.74 -328.06 12 -641.66 744. 28 545. 89 -654.74 202. "M" [kN.m] 1624. 542. 77 45 752. 90 17 -570.55 16 40 2003. 01 540. 89 1396. 95 521. 191. -513.40 761. 60 543. -299.75 16 262. 05 1698. 535. 87 57 770. 37 -438.29 09 -542.78 2414. 59 -363.74 2665. -344.49 10 1426. 779. 23 -579.61 530. 82 180. 30 -489.12 2110. 541. -412.25 -650.34 62 28 33 788. 18 -609.95 1781. 525. 75 -338.48 538. 40 -646.95 -299.91 507. 169. 64 -646.95 61 1459. -631.06 797. 22 79 520. 2553. 61 2823. 70 35 23 12 159. 20 806. 36 -557.54 15 2224. 1871. 1494. 514. -444.19 -510.06 815. 59 539. 149. 05 273. -300.53 64 -373.31 28 533. 91 45 65 824. 93 493. 139. 22 -534.10 2698. 1532. 509. 44 -349.07 04 33 -566.02 1967. 834. 37 2345. 48 129. 77 -639.81 -490.16 2987. -412.41 -349.03 17 -639.54 85 60 503. -639.54 -598.57 843. 92 120. 77 -310.72 85 99 1573. 54 537. 89 528. -620.87 2070. 853. 59 112. 27 -450.66 2851. 2473. 21 498. 06 480. 104. 35 01 863. 37 17 88 -382.94 -544.35 278. 1616. 14 97. 91 -302.53 93. 81 -507.01 20 33 2178. 76 873. 26 492. 89. 83 -525.41 2607. 1661. 890. 14 93 524. 534. 86. 00 82. 32 -353.79 3010. 78. 80 64 -359.70 -629.71 -552.73 467. 75. 46 -632.54 77 487. -491.73 -413.04 87 72. 30 -632.54 907. 94 26 69. 34 -321.54 -587.62 66. 60 64. 12 61. 90 60. 00 24 1709. 83 04 2293.