modelos campos tensoes_beti-2014
DESCRIPTION
Modelos Campos Tensoes_BetITRANSCRIPT
1
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
2
EUROCÓDIGO 2 §5.6 Plastic Analysis
RITTER (1899)
RITTER , MÖRSCH : “INTUIÇÃO” (E EQUILÍBRIO) – BASE PARA OS ACTUAIS MODELOS DE CAMPOS DE TENSÕES
Identificação das trajectórias da carga
Estabelecimento de condições de equilíbrio
MÖRSCH (1909, 1922)
Emil Mörsch (1872-1950)
“ .... nothing is more practical than a good theory ....”
3
EUROCÓDIGO 2 §5.6 Plastic Analysis
VSd - Vccd - VPd VRd
VRd,s = Asw
s z fywd cot
VRd,max = cw bw z 1 fcd
cot + tan Ftd = 0.5 VEd cotg
1 cot 2.5
sz
zNgcot
2
V
z
MC
s
s
z
zzNgcot
2
V
z
MT
“ESFORÇO TRANSVERSO”
4
vEd = Fd
hf . x
Fd
x =
Vzd
z .
A'
Atot
Asf
s . fyd vEd
hf
cotg =
Fd
x cotg
1 cotf 2 (banzos comprimidos)
1 cotf 1.25 (banzos traccionados)
vEd fcd senf cosf
“ESFORÇO TRANSVERSO” ↔ “ Ligação alma – banzos ”
5
Ak
linha média
uk
tef,i
A tsw
s fywd cot VEdi =
TEdi
2Ak
Asl fyd
uk =
TEd
2 Ak cot
TRd,máx = 2 cw fcd Ak tef,i sen cos
TEd
TRd,máx +
VEd
VRd,máx 1
“ESFORÇO TRANSVERSO” ↔ “ Torção ”
6
EUROCÓDIGO 2 §5.6 Plastic Analysis
Va
T
C
z
q
dd1
aF d1 cotg
z cotg
Tf
fsw
V
cot
2
1
z
d
z
a
2
1cot 1F
1
1af cotVT
cotz
cotzda5.0qVf 11asw
“ESFORÇO TRANSVERSO” ↔ “Região do apoio” – continuidade entre zonas “B” e “D”
“B” “D”
“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.
EUROCÓDIGO 2 §5.6 Plastic Analysis
z2a2/z1z
a2
3
1
F
F2
Modelo STM Modelo de campos de tensões
gcotxa2
11a F
“ESFORÇO TRANSVERSO” ↔ Transmissão de cargas “próximas” dos apoios
7 FIP/fib Recommendations 1999 : “Practical Design of Structural Concrete”. FIP-Commission 3 "Practical Design", Sept. 1996. Publ.: SETO, London, Sept. 1999. (Distributed by: fib, Lausanne. Web http://www.fib-international.org) 7
8
1200 kN 1200 kN
1200 kN
"V" [kN]
"M" [kN.m]
3600 kN.m
3.5
0
6.25
2.0
0
8.50
3.00
6.25
3.00
11Ø25
9Ø20
3Ø16 EST Ø8//0.20EST Ø8//0.10
11Ø25
9Ø20
3Ø16EST Ø8//0.20 EST Ø8//0.10
EXERCÍCIO : Dimensionar / Verificar a segurança
9
ORIENTAÇÃO DO MODELO SEGUNDO AS TRAJECTÓRIAS ELÁSTICAS
EXERCÍCIO : Dimensionar / Verificar a segurança
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540.89535.87530.82
525.75520.35
514.64509.04
503.54498.17
492.93487.83
482.89478.12
473.53
469.14
464.94
460.96
457.19
453.66
450.36
447.30
444.50
440.33
435.34
430.61
426.15
421.79
417.17
412.87
408.88
405.22
401.89
398.89
396.21
393.87
391.86
390.18
388.84
387.83
387.15
386.80
386.78
386.43
383.86
381.28
378.71
376.13
373.55
370.97
368.39
365.80
363.22
360.63
356.52
351.87
346.90
341.62
336.03
330.14
323.94
317.4
3
310.6
3
303.5
4
296.1
7
288.5
2
280.6
0
272.4
4
263.5
1
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7
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6
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2
64.4
2
60.8
6
57.8
0
54.7
2
51.6
2
48.5
2
45.4
3
42.3
8
39.3
836
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33.6
030
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28.0
825
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22.6
120
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18.2
016
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3307.513307.503307.483307.463307.443307.423307.403307.383307.363307.343307.323307.293307.273307.253307.243307.243307.253307.263307.273307.293307.303307.333307.353307.373307.403307.433307.473307.513307.543307.593307.633307.683307.633307.553307.453307.333307.203307.063306.903306.733306.543306.333306.123305.883305.643305.373305.103304.803304.503304.183303.963303.733303.503303.283303.063302.843302.623302.403302.183301.963301.743301.533301.313301.103300.893300.673300.463300.183299.733299.253298.723298.163297.553296.913296.223295.493294.723293.913293.063292.173291.233290.263289.243288.193287.093288.183290.563293.043295.633298.323301.113304.003306.993310.083313.273316.563319.943323.423326.993330.663334.423338.263342.203334.453326.923320.573315.493311.773309.513308.813309.763312.483317.063323.603332.223343.013356.083371.533389.473409.993422.803394.013364.603334.643304.183273.293242.043210.513178.793146.983115.213083.603052.333021.582991.602962.692935.212909.632883.112849.602812.362771.592727.472680.192629.912576.822521.062462.822402.242339.512274.772208.192139.952070.221999.151926.931856.681810.861764.951718.941672.831626.621580.291533.851487.301440.621393.811346.881299.811252.601205.251157.761110.111062.321015.08991.25967.44943.65919.88896.13872.40848.69824.99801.31777.64753.98730.33706.69683.05659.41635.77612.13588.48575.91563.90551.85539.74527.57515.34503.05490.70478.28465.80453.24440.62427.92415.15402.30389.36376.35363.25356.20349.89343.45336.88330.17323.33316.37309.27302.05294.71287.25279.67271.97264.16256.23248.19240.04231.77226.04221.07216.09211.11
206.15201.21
196.32191.47
186.68181.96
177.31172.75
168.27163.89
159.60
155.41
151.32
147.33
143.44
140.49
137.81
135.31
132.99
130.87
128.94
127.22
125.70
124.39
123.29
122.41
120.75
118.70
117.28
116.49
116.36
116.87
118.05
119.90
122.37
122.11
122.16
122.53
123.25
124.34
125.79
127.65
129.91
132.59
135.71
139.29
143.34
147.86
152.8
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882. 00869. 02
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847. 25
838. 61
831. 57
826. 17
822. 47
820. 51
820. 31
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806. 29
797. 27
788. 90
781. 25
774. 39
768. 40
764. 04
763. 10
763. 83
766. 31
770. 63
776. 86
785. 08
795. 34
807. 68
822. 13
838. 70
857. 40
878. 21
901. 10
926. 06
948. 69
973. 56
1002. 0
2
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9
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4
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7
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0
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1
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4
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5
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0
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3
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8
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6
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2
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2987. 89
1533.681532.641531.551530.401529.191527.911526.581525.191523.731522.211519.031510.951502.821494.641486.431478.171469.871461.531453.141444.711436.251427.741419.191410.601401.971393.301384.591373.721361.191348.631336.061323.481310.871298.251285.611272.951260.271247.581234.861222.121209.361196.571183.771170.951156.401141.511126.571111.591096.561081.491066.371051.211036.011020.761005.46990.12974.74959.32943.85928.34912.59892.47872.21851.81831.26810.55789.69
768.65747.46726.09
704.57682.89
661.08639.17
617.19595.20
573.28551.55
542.66
536.26
531.83
529.06
527.64
527.28
527.73
528.80
530.34
532.24
535.12
539.44
542.82
545.27
546.81
547.47
547.25
546.20
544.34
519.11
493.88
471.78
452.41
435.40
420.44
407.24395.58
385.24376.06
367.91360.68
354.26348.60343.61339.27335.52328.56311.62295.48280.16265.68252.07239.37227.64216.93207.34199.01192.12186.94180.35172.61164.68156.51148.04139.21129.98
120.30110.13
99.49
88.45
77.23
66.26
1347.521345.881344.161342.361340.491338.541336.511334.411332.221329.951326.311318.651310.931303.161295.341287.461279.531271.551263.521255.431247.291239.111230.871222.581214.251205.871197.441186.921174.641162.321149.971137.581125.161112.711100.221087.701075.151062.581049.971037.341024.681012.00999.31986.60971.87956.43940.94925.41909.83894.22
878.55862.85847.09831.29
815.45799.56
783.62767.63
751.59735.50
719.36698.66
677.85659.33
641.85625.51
610.39
596.56584.09
573.01
563.37
555.16
548.38
543.01
539.01
536.34
534.93
534.74
535.70
538.38
541.67
543.95
545.21
545.49
544.80
543.17
540.63
537.22
532.96
527.91
522.12
515.63
508.50
500.79
492.57
483.91
474.87
465.55
455.13
438.80
422.90
407.43
392.37
377.74
363.53
349.73
336.35323.38
310.82298.66
286.91275.56
264.62254.07
243.91234.15
224.57213.32
202.17191.16
180.30169.64
159.20149.05
139.22129.77120.77112.27104.3597.9193.8189.8386.0082.3278.8075.4672.3069.3466.6064.1261.9060.00
1358. 84 1352. 90 1346. 89 1340. 82 1334. 68 1328. 49 1322. 25 1315. 96 1309. 63 1303. 27 1296. 89 1290. 48 1284. 05 1277. 62 1271. 18 1264. 75 1258. 32 1251. 92 1245. 54 1232. 21 1218. 57 1205. 14 1191. 96 1179. 05 1166. 42 1154. 11 1142. 14 1130. 53 1119. 30 1108. 48 1098. 10 1088. 16 1078. 70 1069. 731061. 27 1053. 34 1045. 97
1033. 76 1019. 301005. 74
993. 13981. 56
971. 08961. 76
953. 67946. 86
941. 40937. 33
934. 87
934. 46
935. 40
937. 69
941. 30
946. 23
952. 46
959. 95
961. 17
954. 76
949. 52
945. 50
942. 81
941. 53
941. 74
943. 54
947. 02
952. 26
959. 34
968. 33
979. 30
992. 30
1007. 37
1024. 5
5
1043. 8
3
1065. 2
5
1088. 7
8
1114. 4
3
1142. 1
7
1171. 9
8
1204. 9
2
1241. 9
2
1283. 1
9
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0
1375. 7
2
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8
1495. 5
2
1566. 9
9
1646. 2
4
1733. 2
3
1827. 9
6
1930. 3
4
2040. 2
9
2157. 69
2282. 40
2414. 28
2553. 15
2698. 85
2851. 20
3010. 04
3175. 20
3346. 52
1136.711134.351131.901129.341126.691123.931121.071118.111115.041111.871107.651100.391093.071085.671078.691071.741064.741057.671050.551043.371036.141028.851021.501014.101006.65999.14991.58982.28971.22960.15949.09938.04927.01915.99905.00
894.05883.14
872.28861.48
850.75840.10
829.54819.08
808.73796.76
783.00769.38
755.91742.60
729.48716.56
703.86691.42
679.25
667.38655.84
644.67
633.89
623.55
613.67
604.29
595.44
589.47
585.88
583.37
581.91
581.48
582.04
583.56
586.01
589.35
593.56
598.61
598.03
596.67
594.89
592.71
590.11
587.08
583.63
579.75
575.44
565.00
550.01
535.38
521.09
507.12
493.48
480.14
467.10
454.34
441.86
429.63
417.67
405.94
394.46
383.20
372.17
361.35
350.74
340.34
330.14
319.73
308.56
297.63
286.96
276.53
266.34
256.40
246.70
237.24
228.01
219.03
210.27
201.75
193.46
185.40
177.56
169.95
162.56
155.40
148.41
138.55
129.18
120.29
111.88
103.96
96.53
89.59
83.15
77.20
71.76
66.82
62.40
58.49
55.11
52.06
47.31
42.89
38.80
35.0431.61
28.5025.73
23.2921.18
19.4117.99
16.92
1139. 23 1137. 84 1134. 61 1131. 30 1127. 92 1124. 47 1120. 96 1117. 39 1114. 02 1110. 94 1107. 90 1104. 91 1101. 98 1099. 12 1096. 33 1093. 64 1091. 04 1088. 56 1086. 18 1082. 23 1076. 94 1071. 88 1067. 04 1062. 45 1058. 12 1054. 06 1050. 29 1046. 83 1043. 68 1040. 86 1038. 39 1036. 271034. 53 1033. 16 1032. 19
1031. 63 1031. 491031. 01
1026. 951023. 88
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1020. 871022. 00
1024. 231027. 57
1032. 031037. 61
1044. 311052. 14
1061. 101071. 18
1082. 371094. 68
1108. 101122. 62
1137. 02
1134. 351133. 00
1133. 05
1134. 60
1137. 73
1142. 54
1149. 09
1157. 49
1167. 78
1180. 05
1194. 34
1210. 69
1229. 15
1246. 94
1265. 60
1287. 19
1311. 71
1339. 17
1369. 60
1403. 01
1445. 80
1503. 99
1569. 40
1642. 00
1721. 72
1808. 53
1902. 33
2003. 05
2110. 61
2224. 91
2345. 85
2473. 33
2607. 242747. 48
2893. 93
3046. 50
3205. 073369. 55
3539. 833715. 81
994.84991.92988.94985.88982.75979.53976.24972.87969.42965.89962.28957.02949.92942.76935.54928.26920.93913.54906.10898.62891.09883.52875.91868.27860.60852.92
845.22837.52
828.21817.91
807.68797.51
787.43777.44
767.56757.79
748.15738.65
729.31720.14
711.16702.38
693.81
685.48
677.39
669.11
658.25
647.75
637.66
627.99
618.78
610.05
601.83
594.25
587.29
580.84
574.89
569.45
564.50
560.06
556.10
552.63
549.64
547.11
545.05
544.09
542.95
541.40
539.44
537.06
534.26
531.02
527.35
523.24
518.69
513.70
508.26
502.37
496.03
489.25
482.01
474.30
466.07
457.87
449.69
441.54
431.19
419.53
408.10
396.92
385.97
375.27
364.81
354.59
344.61
334.87
325.37
316.12
307.10
298.33
289.79
281.50
273.45
265.64
258.07
250.75
243.67
236.82
228.20
219.65
211.33
203.24
195.39
187.76
180.37
173.21
166.28
159.58
153.12
146.88
140.88
135.10
129.56
124.25
119.17
114.33
109.72
105.34
101.20
96.92
90.09
83.37
76.74
70.20
62.59
55.34
48.43
41.89
35.71
29.91
24.49
19.47
14.85
10.68
6.97
4.49
3.15
1.91
0.76
-0.30
-1.26
-2.12
-2.89
-3.56
-4.14
-4.62
-5.01
-5.29
-5.48
1011. 05 1012. 79 1012. 25 1011. 67 1011. 07 1010. 46 1009. 84 1009. 23 1008. 64 1008. 08 1007. 55 1007. 07 1006. 65 1006. 30 1006. 02 1005. 83 1005. 74 1005. 75 1005. 88 1005. 87 1004. 48 1003. 32 1002. 40 1001. 73 1001. 33 1001. 21 1001. 39 1001. 88 1002. 69 1003. 85 1005. 35 1007. 21 1009. 461012. 08
1015. 111018. 55 1022. 41
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1079. 891090. 89
1102. 911115. 93
1129. 971145. 03
1161. 101178. 19
1196. 281215. 39
1235. 511256. 64
1262. 931264. 07
1262. 79
1259. 411259. 25
1262. 261268. 44
1277. 73
1290. 13
1305. 59
1324. 09
1345. 62
1370. 14
1397. 641428. 10
1461. 511497. 83
1537. 041579. 11
1624. 011698. 10
1781. 611871. 45
1967. 602070. 01
2178. 642293. 44
2414. 362541. 32
2674. 27
2813. 152957. 88
3108. 403264. 64
3426. 533594. 01
3767. 013945. 46
4129. 30
2449.872449.842449.812449.772449.732449.692449.652449.612449.562449.522449.472449.412449.362449.302449.282449.272449.282449.292449.312449.342449.382449.432449.482449.542449.612449.692449.772449.862449.962450.072450.192450.312450.392450.432450.452450.432450.392450.322450.222450.092449.932449.752449.532449.292449.012448.712448.382448.022447.632447.262447.202447.172447.162447.172447.212447.282447.362447.482447.612447.782447.962448.172448.412448.672448.952449.262449.592449.932450.222450.412450.502450.472450.352450.122449.782449.342448.792448.142447.392446.532445.562444.492443.322442.052440.672441.412443.292445.472447.962450.762453.882457.322461.072465.152469.562474.292479.352484.742490.472496.542502.942509.682516.762496.412476.552457.472439.212421.782405.222389.552374.812361.042348.272336.552325.932316.442308.142301.102295.362290.992284.112265.202245.112223.822203.492186.472167.342146.122122.812097.392069.852040.122008.161973.861937.141897.911856.121811.931770.251731.311691.941653.531614.68
1575.161535.05
1494.461453.47
1412.151370.60
1328.861287.02
1245.131203.26
1161.481119.86
1078.45
1037.33987.82
931.94
877.38
824.26
772.72
722.90
674.96
629.05
585.35
543.99
505.14
468.90
435.36
404.58
376.56
351.30
328.71
308.72
291.19
276.0
0
241.6
5
205.3
7
171.6
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266.34
256.40
246.70
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228.01
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201.75
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177.56
169.95
162.56
155.40
148.41
138.55
129.18
120.29
111.88
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83.15
77.20
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58.49
55.11
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-954.00
-933.35
-918.90
-904.33
-889.62
-874.76
-859.71
-844.46
-828.99
-813.26
-797.26
-780.97
-764.35
-747.39
-730.06
-712.33
-694.18
-675.59
-660.13
-648.83
-637.78
-626.97
-616.43
-606.16
-596.20
-586.54
-577.22
-568.24
-559.36
-550.49
-541.87
-533.50
-525.39
-517.54
-509.96
-502.63
-495.11
-487.68
-480.57
-473.80
-467.36
-461.27
-455.54
-450.17
-445.16
-440.52
-436.25
-432.36
-428.85
-425.71
-422.94
-420.54
-418.50
-416.83
-415.51
-414.53
-414.18
-414.04
-414.08
-414.3
0
-414.1
1
-411.7
8
-409.7
9
-408.1
3
-406.7
9
-405.7
8
-405.0
8
-404.6
8
-404.5
9
-404.7
9
-405.2
7
-406.0
2
-407.0
4
-408.3
2
-409.8
3
-411.5
9
-413.5
7
-415.7
7
-418.1
8
-420.7
8
-423.1
7
-424.8
5
-426
.67
-427
.88
-429
.29
-430
.90
-432
.69
-434
.64
-436
.76
-439
.03
-441
.44
-444
.00
-446
.70
-449
.53
-452
.49
-455
.59
-458
.83
-462
.20
-465
.72
-469
.37
-473
.17
-482
.07
-492
.50
-503
.54
-515
.14
-527
.25
-539
.79
-552
.73
-566
.02-5
79.61
-593
.47-6
07.57
-621
.88-6
36.37
-651
.02-6
65.82
-680
.74-6
95.78
-689
.77-6
83.18
-677
.36-6
72.69
-669
.53-6
68.21
-669
.02-6
72.18
-677
.86-6
86.19
-697
.25
-711
.05
-727
.59
-746
.84
-768
.74
-793
.23
-818
.74
-840
.01
-865
.47
-895
.07
-928
.72
-966
.31
-100
7.71
-105
2.77
-110
1.34
-115
3.28
-120
8.42
-126
6.61
-132
7.69
-139
1.50
-145
7.89
-152
6.69
-159
7.75
-167
0.92
-749.52
-737.78
-726.02
-714.21
-702.30
-690.27
-678.08
-664.34
-650.44
-636.70
-623.12
-609.71
-596.47
-583.41
-570.54
-557.87
-545.41
-533.18
-521.19
-509.45
-497.99
-486.81
-475.95
-465.43
-455.98
-447.38
-438.90
-430.54
-422.32
-414.22
-406.25
-398.39
-390.66
-383.05
-375.55
-368.17
-360.89
-353.72
-346.65
-339.67
-332.93
-326.63
-321.53
-318.33
-315.38
-312.68
-310.24
-308.07
-306.17
-304.55
-303.20
-302.14
-301.36
-300.87
-300.66
-300.7
4
-301.1
0
-301.7
4
-302.6
6
-303.8
5
-305.3
1
-307.0
4
-309.0
3
-311.2
7
-311.1
8
-310.8
8
-310.5
6
-308.8
7
-307.6
8
-307.0
1
-306.8
6
-307.2
4
-308.1
5
-309.5
9
-311
.56
-314
.04
-317
.03
-320
.50
-324
.45
-328
.85
-333
.68
-338
.93
-344
.57
-350
.58
-356
.94
-363
.63
-368
.36
-370
.47
-373
.39
-377
.13
-381
.65
-386
.86
-392
.71
-399
.13
-406
.09
-413
.52
-421
.38
-429
.64
-438
.26
-447
.20
-456
.45
-465
.97
-475
.75
-485
.44
-489
.40
-492
.83
-495
.90-4
98.72
-501
.44-5
04.17
-507
.01-5
10.06
-513
.40-5
17.09
-521
.20-5
25.76
-530
.83-5
36.42
-542
.56-5
49.26
-550
.30-5
38.69
-527
.96-5
18.20
-509
.53
-502
.07
-495
.96
-491
.34
-488
.35
-487
.13
-487
.82
-490
.52
-495
.32
-502
.28
-511
.43
-522
.78
-536
.30
-551
.52
-567
.26
-588
.00
-614
.41
-653
.51
-696
.44
-742
.90
-792
.84
-846
.15
-902
.73
-962
.46
-102
5.22
-109
0.87
-115
9.25
-123
0.22
-130
3.59
-137
9.22
-145
6.92
-153
6.53
1200 kN 1200 kN
1200 kN
"V" [kN]
"M" [kN.m]
3600 kN.m
3.5
0
6.25
2.0
0
8.50
3.00
6.25
3.00
11Ø25
9Ø20
3Ø16 EST Ø8//0.20EST Ø8//0.10
11Ø25
9Ø20
3Ø16EST Ø8//0.20 EST Ø8//0.10
Em diversas regiões do elemento, não é aplicável a teoria das peças lineares
10
ZONAS DE DESCONTINUIDADE ↔ Identificação das regiões “D”
D
D B
D
D
D
B
B
B D
D
D
D
B D
D
D
B
D
B
B
B
D
D
D B D B
B
D
F
h
z=h
B B
D
B
D
B
Schlaich, J.; Schäfer, K; Jennewein, M. (1987): Toward a consistent design for structural concrete. PCI-Journ. V.32
ZONAS “B” (Bernoulli) – de CONTINUIDADE ZONAS “D” (discontinuity) - de DESCONTINUIDADE
Necessidade de :
• CONSISTÊNCIA
• CONTINUIDADE
na modelação e dimensionamento de zonas “B” e “D”
11
EUROCÓDIGO 2 §5.6 Plastic Analysis 5.6.4 Análise com modelos de escoras e tirantes
(1) Os modelos de escoras e tirantes poderão ser utilizados para as verificações em relação ao
ELU das regiões de continuidade (estado fendilhado de vigas e lajes, ver 6.1 a 6.4) e para as
verificações em relação ao ELU e para a definição das disposições construtivas das regiões
de descontinuidade (ver 6.5). Em geral, estas estendem-se até uma distância h (altura da
secção do elemento) da descontinuidade.
(2) Poderão também efectuar-se verificações em relação ao estado limite de utilização utilizando
modelos de escoras e tirantes, por exemplo, a verificação das tensões do aço e o controlo da
largura das fendas, se forem asseguradas as condições de compatibilidade aproximada dos
modelos de escoras e tirantes ...... .
(3) Os modelos de escoras e tirantes são constituídos por escoras que representam campos de
tensão de compressão, por tirantes que representam as armaduras e por nós. As forças nos
elementos de um modelo de escoras e tirantes deverão ser determinadas mantendo, no estado
limite último, o equilíbrio com as cargas aplicadas. Os elementos dos modelos de escoras e
tirantes deverão ser dimensionados de acordo com as regras indicadas em 6.5.
(4) Os tirantes de um modelo de escoras e tirantes deverão coincidir, em posição e direcção,
com as armaduras correspondentes.
(5) Os modelos de escoras e tirantes podem ser definidos a partir das isostáticas de tensão e
das distribuições de tensões obtidas pela teoria elástica linear ou a partir do método da
trajectória de cargas.
Todos os modelos de escoras e tirantes poderão ser optimizados por critérios de energia.
NP EN1992-1-1, 2009 – EC2 - ESTRUTURAS DE BETÃO
12
MODELOS DE CAMPOS DE TENSÕES – EQUILÍBRIO EXTERIOR
Começar por estabelecer o equilíbrio das forças aplicadas na fronteira da região (Equílibrio Exterior), que poderão ser forças exteriores ou esforços internos em secções adjacentes, obtidos da análise corrente das zonas “B” .
150 250
171 229
113
50
287
350
200200
200 200
142 258
100 300
0.21L 0.58L 0.21L 0.21L 0.58L 0.21L
0.21L 0.58L 0.21L 0.21L 0.58L 0.21L
0
0
0
M
F
F
y
x
F1
F2
F1F2
B
D
D D
D
B
V
T
C
M
M
N=V
T'C'
N
p
p
B
13
F1
F2
F1F2
90 kN
330 kN
0.40
0.28
0.3
2
1 1
T
T 90330
MODELOS DE CAMPOS DE TENSÕES - EQUILÍBRIO EXTERIOR ↔ “TRAJECTÓRIAS DAS FORÇAS”
- Identificar as forças na fronteira da região em estudo ; - “Encaminhar” as cargas através da região; - Introduzir escoras ou tirantes (resultantes de campos de tensões), de forma a equilibrar os “nós” do modelo.
MODELOS DE CAMPOS DE TENSÕES - EQUILÍBRIO EXTERIOR ↔ “TRAJECTÓRIAS DAS FORÇAS”
- Identificar as forças na fronteira da região em estudo ; - “Encaminhar” as cargas através da região; - Introduzir escoras ou tirantes (resultantes de campos de tensões), de forma a equilibrar os “nós” do modelo.
R = q x (l/2)
F = q x (l/2)
C
T
C
C
T
R
C
14
15
90 kN
330 kN
0.40
90 kN
330 kN
0.40
90 kN
330 kN
0.40
0.28
0.3
2
1 1
T
T 90330
“TRAJECTÓRIA DAS FORÇAS” MODELO VERIF. DA SEGURANÇA / PORMENORIZAÇÃO
MCT - “TRAJECTÓRIAS DAS FORÇAS” ↔ MODELO ↔ VERIF. SEGURANÇA / PORMENORIZAÇÃO
16
M z
M
T'
C'
T
C
x
x'
z'
z
C'-C
C
T
C
C'-C
T
C'
{
T'
{
M
T'
M
C
T
C'
z'
z
M
T'
M
C
T
C'
z'
z
Node region
Stress Field Resultant
Prismatic Shape Strut
Tie
Fan shape strut
EQUILÍBRIO EXTERIOR “TRAJECTÓRIAS DAS FORÇAS” MODELO
VERIFICAÇÃO DA SEGURANÇA / PORMENORIZAÇÃO
ORIENTAÇÃO DO MODELO SEGUNDO AS TRAJECTÓRIAS ELÁSTICAS 1200 kN 1200 kN
3.5
0
6.25
2.0
0
8.50
3.00
6.25
3.00
D
B
D
1200 kN
2.0
0
D
1200 kN
C
T
M=3600kN.m
-18MPa
18MPa
Elastic
-27.7MPa
411MPa
Cracked
0.5
34
-17.6MPa
Plastic
0.4
30
2271kN
2271kN
1.8
0
M=3600kN.m M=3600kN.m M=3600kN.m
As
2.0
0
0.30
EXERCÍCIO : Dimensionar / Verificar a segurança
Escolher o tipo de distribuição de tensões na fronteira, de forma coerente com o adoptado na zona “B” adjacente
Equilíbrio exterior
17
18
EXERCÍCIO : Dimensionar / Verificar a segurança
z2a2/z1z
a2
3
1
F
F2
1200 kN
1200 kN
2271
2271
0.4
300.400x0.30
0.400x0.30
37
8
-184
-420 -1147
-420
983 1167
11
67
2271
-2271
-1650
-1607
-1167
Modelo
19
ORIENTAÇÃO DO MODELO SEGUNDO AS TRAJECTÓRIAS ELÁSTICAS
11Ø25
9Ø20
3Ø16 EST Ø8//0.20EST Ø10//0.20
12ESTØ12
1.00
2.40
11Ø25
9Ø20
3Ø16 EST Ø8//0.20EST Ø8//0.10
1200 kN
1200 kN
1200 kN
1200 kN
EXERCÍCIO : Dimensionar / Verificar a segurança
20
EUROCÓDIGO 2 §5.6 Plastic Analysis
“TRAJECTÓRIAS DAS FORÇAS” ↔ MODELO
21
“ORIENTAÇÃO” DO MODELO SEGUNDO AS TRAJECTÓRIAS ELÁSTICAS
Schlaich, J.; Schäfer, K; Jennewein, M. (1987): Toward a consistent design for structural concrete. PCI-Journ. V.32
5.6.4 Análise com modelos de escoras e tirantes (NP EN1992-1-1, 2009 – EC2) (5) Os modelos de escoras e tirantes podem ser definidos a partir das isostáticas de tensão e das
distribuições de tensões obtidas pela teoria elástica linear ..................
22
EUROCÓDIGO 2 §5.6 Plastic Analysis
Modelos baseados em soluções elásticas : “Viga parede” c/ 3 apoios
FIP/fib Recommendations 1999 : “Practical Design of Structural Concrete”. FIP-Commission 3 "Practical Design", Sept. 1996. Publ.: SETO, London, Sept. 1999. (Distributed by: fib, Lausanne. Web http://www.fib-international.org)
EUROCÓDIGO 2 §5.6 Plastic Analysis
23
Modelos baseados em soluções elásticas : “Viga parede” contínua (tramo interior)
FIP/fib Recommendations 1999 : “Practical Design of Structural Concrete”. FIP-Commission 3 "Practical Design", Sept. 1996. Publ.: SETO, London, Sept. 1999. (Distributed by: fib, Lausanne. Web http://www.fib-international.org)
24
Definir a zona “D” e estabelecer o equilíbrio exterior. Identificar o “caminho das cargas” ↔ seguir a trajectória das forças através da região.
Elaborar modelos simples. O refinamento do modelo deve ser efectuado gradualmente, se necessário.
“Orientar” o modelo com base nas trajectórias de tensões elásticas.
Desvios moderados em relação à solução elástica não comprometem o comportamento da estrutura. As trajectórias de tracções principais não coincidem em geral com as disposições de armaduras adoptadas no projecto. O modelo deve reproduzir, com proximidade, os campos de compressões principais, sendo os campos traccionados definidos de forma coerente com a disposição das armaduras principais adoptadas.
A compreensão e o exercício de aplicação do método permitem tirar partido da tipificação e repetitividade dos modelos.
Cargas concentradas têm ângulos de desvio elevados (em geral, maiores que 60º). Não devem assim utilizar-se ângulos entre as escoras e tirantes principais “pequenos” (inferiores a ≈ 40º/45º).
Dispôr os tirantes de forma a corresponderem a distribuições de armaduras correntes e exequíveis.
As escoras representam campos de compressões, com largura proporcionada ao valor da resultante.
MCT/STM - CONSTRUÇÃO DO MODELO
EUROCÓDIGO 2 §5.6 Plastic Analysis
25 Almeida, J., Lourenço, M., 2010 : “Abutment Shear Wall of Viaduct 1 on Algarve Highway”, fib Congress, Washington.
- As trajectórias de tracções principais não coincidem em geral com as disposições de armaduras adoptadas no projecto. - O modelo deve reproduzir, com proximidade, os campos de compressões principais, sendo os campos traccionados definidos de forma coerente com a disposição das armaduras principais adoptadas. - Explorar a tipificação e repetitividade dos modelos
EXEMPLO : Definição do modelo a partir das distribuições de tensões elásticas
EUROCÓDIGO 2 §5.6 Plastic Analysis
26
TRAJECTÓRIAS ELÁSTICAS
PORMENORIZAÇÃO DE ARMADURAS
MODELOS
Trajectórias elásticas ↔ Campos de compressões ↔ Modelo global da região ↔ Pormenorização
EUROCÓDIGO 2 §5.6 Plastic Analysis
27 Almeida, J., Lourenço, M., 2010 : “Abutment Shear Wall of Viaduct 1 on Algarve Highway”, fib Congress, Washington.
DEFINIÇÃO DO MODELO (da modelação global às regiões locais)
Verificação dos “Nós” ↔ Modelos locais ↔ Pormenorização
28
No essencial, • Verificação das regiões dos “NÓS” (concentrados), zonas de intersecção de escoras e/ou tirantes:
Controlo de tensões no betão (valores limites dependentes do estado de tensão multi-axial no nó); Pormenorização detalhada / condições de amarração das armaduras.
• Dimensionamento e disposição adequada das áreas de armadura, para os esforços obtidos do modelo. • Adopção de armaduras mínimas, distribuídas nas 2 direcções, em ambas as faces.
VERIFICAÇÃO DA SEGURANÇA DE ZONAS (D)
29
EUROCÓDIGO 2 §5.6 Plastic Analysis
Escora prismática Escora em leque Escora em “garrafa”
• Adopção de armaduras mínimas, distribuídas nas 2 direcções, em ambas as faces • Reforço eventual da armadura transversal distribuída, em situações de transmissão de “carga concentrada” ↔ “carga concentrada”
Os “NÓS” como “intercecções dos Campos de Tensões”
30 fib Bulletin 52, 2010 : “Structural Concrete – Textbook on behaviour, design and performance”, vol. 2, fib, Lausanne
Ex. de condições de amarração das armaduras nos nós
NÓS “CCT”
cotause 112
Rd,max 0.85
1 - fck
250 fcd
cosusenaa 12
ba
C
1
11c
ba
C
2
22c
u
VERIFICAÇÃO DA SEGURANÇA DE ZONAS “D” : “NÓS”
As tensões c2 dependem de “u”, altura de distribuição das armaduras no tirante
31
EUROCÓDIGO 2 §5.6 Plastic Analysis
C2T2
T1T3
C2
a2
T1 T2
T2/nT2/nT2/nT2/nT2/n
sw
NÓS “CTT”
M
T'
M
C
T
C'
z'
z
Node region
Stress Field Resultant
Prismatic Shape Strut
Tie
Fan shape strut
32
[Nota: o valor de 0.75 acima indicado (EN1992-1), será provavelmente reduzido para ≈ 0.60 , em futura revisão da norma]
EUROCÓDIGO 2 §5.6 Plastic Analysis
NÓS “CCC”
1
2
3
II I1,II 1,I=2,I=3,I=02,II3,II
C3
C2
C1
1
3
a3
a2
a1
I
II
33
EUROCÓDIGO 2 §5.6 Plastic Analysis
34
(GERAIS):
fib Model Code 2010, Bulletin 65, 66 (2012): Design of concrete structures – Final Draft, fib, Lausanne, 2012
NP EN1992-1-1 2010 : “Projecto de estruturas de betão – Parte 1-1: regras gerais e regras para edifícios”, IPQ, Lisboa.
Reineck, K.-H., Almeida, J., Appleton, J., Corres, H., Friedrich, T., Ganz, H., Kalny, M., Miehlbradt, M., Westerberg, B. (FIP WG on Recommendations for Practical Design of Structural Concrete) 1999 : “FIP/fib Recommendations for Practical Design of Structural Concrete”, SETO, London.
Muttoni, A., Schwartz, J., Thürlimann, B. 1998 : “Design of Concrete Structures With Stress Fields”, Birkhäuser, Basel.
Almeida, J., Lourenço, M. 2011 : “Modelos de Campos de Tensão – Zonas D”, Apresentação preparada para a disciplina Betão Estrutural, Mestrado em Engenharia de Estruturas, Instituto Superior Técnico, Lisboa
Almeida, J., Lourenço, M., 2010 : “Abutment Shear Wall of Viaduct 1 on Algarve Highway”, fib Congress, Washington.
Lourenço, M.; Almeida, J., Bóia, R. 2011 : “Continuous Deep Beam”. in Design examples for strut and tie models, fib Bulletin 61, Lausanne, pp.45-52
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, pp.197-216.
Schlaich, J.; Schäfer, K; Jennewein, M. 1987: “Toward a consistent design for structural concrete”. PCI-Journ. V.32
(AVALIAÇÃO E SELECÇÃO DE MODELOS):
Almeida, J. 2012 : “Modelos de Campos de Tensões para Betão Estrutural – dos Fundamentos à Aplicação em Projecto”, lição de síntese apresentada no âmbito de provas de agregação, IST, Julho 2012.
Lourenço, M.; Almeida, J. 2013 : “Adaptive Stress Field Models. Formulation and Validation”, ACI, Structural Journal, 110-S08, Farmington Hills, USA, January-February 2013, pp. 71-81.
Lourenço, M.; Almeida, J. 2013 : “Adaptive Stress Field Models. Assessment of Design Models”, ACI, Structural Journal, 110-S09, Farmington Hills, USA, January-February 2013, pp. 83-93.
REFERÊNCIAS :
“Betão Estrutural - Zonas de Descontinuidade : Introdução aos modelos de campos de tensões” , MIEC, Estruturas de Betão I, IST, Nov. 2014
João F. Almeida ([email protected])