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Grupo de Betão Armado e Pré-Esforçado — IST 19 Esforços e Flechas Elásticas em Lajes Esforços elásticos em lajes vigadas Esforços e flechas elásticas em lajes fungiformes Referências: [10] - Tablas para el cálculo de placas y vigas pared R. Bares, Editorial Gustavo Gili, S.A., Barcelona, 1981. [14] - Béton Armé et Précontraint I/IN, cours du professeur Renaud Favre, 2! Edition, Lausanne 1983. Tipo de y BA x B A B Viga a s % %» Carga + L— + +—— | —++ +— L——+ Ma= My=-PEIP caca) | ME IÉ caÊ(QL-9)] +a+b-+a+ Mináx = (3 E -da”) A 5 lua = EE (28) Wa= Fa LÊ (Sb +68) -a2(2L-0)] + L—+ E a 5 Va= Vo = (a+b) Vas (ab) Vo = ea [É (3b+2a)+8?(2L-2)] 2 M,=- Pb [dele bHL-30)] b 2 RR +. 2, Ene 2É M,= gu 8 IZÉ- Piza (a +b)) v=BB (204b) p Mo =- Do lde + (1:30) +d+-e+ vos LOL ate -ajjár + L+ | y =- PP gcêsablsade(d-e)+ 8L Vo =P (244) at +(2a+b) [a (a+b)]) +b (d-e) Mb(2a-b) Ma(a-2b) M(L-3b? E Ra (L=3b*) — Ma Mb EM A 2 E) 2 Março Meg 5Mar= 1 É E A 2 sq E 280 e va=.6Mab. y óMab v -3Ma(a-2L) y 3Ma(a-2L)| y M. M AR É 1 SRD ap ada +t EE ER —- 20tEI 30tEI —d—h MA O Mi cn M=0 (em todo o vão) Lo 3atEI 3atEI a.- Cof, dil. térm. VatNDEO Nº CRP LE ae SEBIA 9p sodi) sund|t ap sooisgja soIoJs SI — opeSioysa-q1q 2 OpEuLIy OgIog Sp odrus) Grupo de Betão Armado e Pré-Esforçado — IST Esforços elásticos em lajes [10] Er y pi dB ' Pp CE TE a a l I om I a I b | Mae l | l I [ES Pp El x COERERÉ Aa A +—— a =p v=0.15 Y Mxs Mys aç 0.50 0.0991 0.0079 0.1189 0.55 0.0923 0.0103 0.1101 0.60 0.0857 0.0131 0.1015 0.65 0.0792 0.0162 0.0931 0.70 0.0730 0.0194 0.0851 0.75 0.0669 0.0230 0.0777 0.80 0.0611 0.0269 0.0708 0.85 0.0557 0.0307 0.0644 0.90 0.0507 0.0344 0.0584 0.95 0.0462 0.0383 0.0529 1.00 0.0423 0.0423 0.0476 1.10 0.0353 0.0500 0.0390 1.20 0.0293 0.0575 0.0320 1.30 0.0244 0.0644 0.0262 1.40 0.0204 0.0710 0.0216 1.50 0.0173 0.0772 0.0179 1.60 0.0146 0.0826 0.0149 1.70 0.0124 0.0874 0.0124 1.80 0.0107 0.0916 0.0105 1.90 0.0091 0.0954 0.0088 2.00 0.0079 0.0991 0.0074 2 aa | pa x pa x pb x TO Grupo de Betão Armado e Pré-Esfor ado — IST Esforços elásticos em lajes [10] ee b Ee Myvs p | I I [7 que I e H| b | Ma 1 — | | E] | | E botndia | FE Eq EEE E ESSE +— a—— v=0.15 x Mxs Mys Myvs ac 0.50 0.0835 0.0088 | -0.0297 | 0.0990 0.55 0.0738 0.0113 | -0.0350 | 0.0872 0.60 0.0647 0.0137 | -0.0400 | 0.0759 0.65 0.0563 0.0166 | -0.0450 | 0.0657 0.70 0.0489 0.0187 | -0.0497 | 0.0565 0.75 0.0423 0.0212 | -0.0540 | 0.0484 0.80 0.0363 0.0233 | -0.0578 | 0.0414 0.8s 0.0313 0.0254 | -0.0612 | 0.0355 0.90 0.0270 0.0274 | -0.0644 | 0.0305 0.95 0.0232 0.0292 | -0.0677 | 0.0262 1.00 0.0201 0.0309 | -0.0699 | 0.0225 1.10 0.0151 0.0335 -0.0741 0.0167 1.20 0.0113 0.0357 -0.0770 0.0126 1.30 0.0088 0.0374 | -0.0793 | 0.0096 1.40 0.0068 0.0386 | -0.0811 | 0.0073 1.50 0.0053 0.0396 | -0.0815 | 0.0057 1.60 0.0042 0.0404 | -0.0825 | 0.0045 1.70 0.0034 0.0410 | -0.0830 | 0.0036 1.80 0.0028 0.0414 | -0.0832 | 0.0029 1.90 0.0023 0.0416 | -0.0833 | 0.0023 2.00 0.0019 0.0417 -0.0833 0.0018 2 2 E pao x pa x pb x pb x Ro 25 Grupo de Betão Armado e Pré-Esforçado — IST Esforços elásticos em lajes [10] Eu y = b a l | fimo] Me ti [oo | b Ms | | ea al. x ER O 4 a——+ v=0.15 Y Mxs ao Mys Mp ac 0.50 0.0570 | -0.1189 | 0.0040 | -0.0205 | 0.0549 0.55 0.0543 | -0.1148 | 0.0054 | -0.0249 | 0.0520 0.60 0.0514 | -0.1104 | 0.0072 | -0.0294 | 0.0490 0.65 0.0483 | -0.1057 | 0.0092 | -0.0341 | 0.0458 0.70 0.0451 | -0.1008 | 0.0114 | -0.0390 | 0.0425 0.75 0.0418 | -0.0957 | 0.0139 | -0.0442 | 0.0393 0.80 0.0385 | -0.0905 | 0.0164 | -0.0496 | 0.0361 0.85 0.0354 | -0.0852 | 0.0191 | -0.0548 | 0.0330 0.90 0.0324 | -0.0798 | 0.0217 | -0.0598 | 0.0301 0.95 0.0295 | -0.0745 | 0.0243 | -0.0648 | 0.0273 1.00 0.0269 | -0.0699 | 0.0269 | -0.0699 | 0.0246 1.10 0.0221 | -0.0608 | 0.0319 | -0.0787 | 0.0201 1.20 0.0182 | -0.0530 | 0.0365 | -0.0869 | 0.0164 1.30 0.0148 | -0.0462 | 0.0406 | -0.0937 | 0.0133 1.40 0.0122 | -0.0405 | 0.0442 | -0.0993 | 0.0108 1.50 0.0100 | -0.0358 | 0.0473 | -0.1041 | 0.0089 1.60 0.0081 | -0.0317 | 0.0499 | -0.1082 | 0.0072 1.70 0.0066 | -0.0282 | 0.0521 | -0.1116 | 0.0059 1.80 0.0055 | -0.0252 | 0.0540 | -0.1143 | 0.0048 1.90 0.0046 | -0.0226 | 0.0556 | -0.1167 | 0.0040 2.00 0.0040 | -0.0205 | 0.0570 | -0.1189 | 0.0034 2 2 2 ir paf x pa x pa x pb x pb x E Grupo de Betão Armado e Pré-Esforçado — IST Esforços elásticos em lajes [10] y=2 y b ! Myvs Mys [| Men CMppM [|] | Myvs e 5 a po E a v=0.15 a Mxs Mxys Mys Myvs ac 0.50 0.0405 | -0.0833 | 0.0024 | -0.0143 | 0.0296 0.55 0.0394 | -0.0817 | 0.0033 | -0.0172 | 0.0286 0.60 0.0378 | -0.0794 | 0.0046 | -0.0206 | 0.0275 0.65 0.0360 | -0.0767 | 0.0061 | -0.0242 | 0.0261 0.70 0.0339 | -0.0737 | 0.0079 | -0.0280 | 0.0246 0.75 0.0315 | -0.0704 | 0.0098 | -0.0320 | 0.0231 0.80 0.0293 | -0.0668 | 0.0103 | -0.0360 | 0.0214 0.85 0.0269 | -0.0631 | 0.0139 | -0.0400 | 0.0196 0.90 0.0247 | -0.0593 | 0.0160 | -0.0440 | 0.0180 0.95 0.0224 | -0.0554 | 0.0181 | -0.0480 | 0.0164 1.00 0.0202 | -0.0515 | 0.0202 | -0.0515 | 0.0149 1.10 0.0164 | -0.0449 | 0.0242 | -0.0585 | 0.0121 1.20 0.0131 | -0.0388 | 0.0287 | -0.0643 | 0.0098 1.30 0.0105 | -0.0336 | 0.0306 | -0.0690 | 0.0078 1.40 0.0084 | -0.0291 | 0.0332 | -0.0728 | 0.0063 1.50 0.0066 | -0.0254 | 0.0353 | -0.0757 | 0.0051 1.60 0.0053 | -0.0223 | 0.0369 | -0.0779 | 0.0041 1.70 0.0042 | -0.0198 | 0.0383 | -0.0797 | 0.0033 1.80 0.0035 | -0.0176 | 0.0392 | -0.0812 | 0.0027 1.90 0.0028 | -0.0158 | 0.0399 | -0.0824 | 0.0022 2.00 0.0024 | -0.0143 0.0405 | -0.0833 | 0.0018 E) x pa? | x pa?) xpb? | x pb? | x Pã Eh? Grupo de Betão Armado e Pré-Esforçado — IST Esforços elásticos em lajes [10] y=2 E b eee l l M. dis fama 1 2 b | re pe I | BEER ER CERERa) === +—— àa———+ v=0.15 al a? u Mxs Mys Myas c c 0.30 0.1133 0.0143 0.0256 2157 4.1259 0.40 | 0.0994 | 0.0254 | 0.0415 | 1.1222 | 2.0659 0.50 | 0.0832 | 0.0341 | 0.0575 | 0.6529 | 1.1539 0.60 0.0683 0.0441 0.0724 0.4063 0.6940 0.70 0.0559 0.0535 0.0848 0.2652 0.4356 0.80 0.0459 0.0611 0.0922 0.1775 0.2771 0.90 | 0.0373 | 0.0699 | 0.1037 | 0.1271 | 0.1937 1.00 0.0305 0.0764 0.1108 0.0913 0.1358 1.20 0.0213 0.0883 0.1201 0.0504 0.0706 1.50 0.0128 0.0992 0.1264 0.0235 0.0305 2.00 0.0063 0.1114 0.1316 0.0083 0.0100 4 4 x pa? | x pb? | xpb? | x Pa |, Pa Eh? Eh? 29 Grupo de Betão Armado e Pré-Esforçado — IST Esforços elásticos em lajes [10] ag y dá b | Mys Mya ] | E | H I Mys H polo ome Hd | Mas |] m | ad |Mra s T] fo e o pg ff +— a————+ v=0.15 1 2 x Mxs Mys Myas Myvs Myva de ac 0.30 0.1044 0.0128 0.0230 -0.0377 | -0.0693 1.3464 2.3163 0.40 0.0725 0.0184 0.0307 | -0.0472 | -0.0782 | 0.5588 0.9711 0.50 0.0520 0.0232 0.0361 | -0.0557 | -0.0846 | 0.2758 0.4559 0.60 0.0394 0.0267 0.0403 -0.0612 | -0.0852 | 0.1511 0.2316 0.70 0.0298 0.0299 0.0426 -0.0662 | -0.0854 | 0.0903 0.1289 0.80 0.0223 0.0319 0.0438 | -0.0707 | -0.0853 | 0.0578 0.0773 0.90 0.0168 0.0349 0.0442 | -0.0744 | -0.0850 | 0.0386 0.0490 1.00 0.0126 0.0362 0.0445 -0.0771 | -0.0848 0.0266 0.0325 1.20 0.0078 0.0386 0.0447 | -0.0805 | -0.0846 | 0.0138 0.0159 1.50 0.0041 0.0408 0.0449 -0.0828 | -0.0845 0.0059 0.0066 2.00 0.0018 0.0417 0.0450 | -0.0833 | -0.0845 0.0019 0.0021 h) 7] 2 2 2 pa” paí x pai x pb x pb x pb x pb x EEE x Ra 31 32 Grupo de Betão Armado e Pré-Esforçado — IST Esforços elásticos em lajes [10] y=2 b 4Myvs Myva p T E My | Ip BS 146 2NMyas VU] db Myys Mas É: 4Myes Myva x 1] i É CEREERR. 4 a—— v=0.15 al a? Y Mxs | Mxvs | Mys | Myas | Myvs | Myva c c 0.30 -0.0489 | -0.3833 | 0.0019 0.0078 | -0.0131 | -0.0333 | 0.3764 0.9700 0.40 -0.0025 | -0.2783 | 0.0068 0.0173 | -0.0242 | -0.0545 | 0.2565 0.5909 0.50 0.0132 | -0.2004 | 0.0120 0.0268 | -0.0335 | -0.0709 | 0.1613 0.3433 0.60 0.0192 | -0.1476 | 0.0177 0.0333 | -0.0416 | -0.0798 | 0.1031 0.1981 0.70 0.0190 | -0.1106 | 0.0217 0.0384 | -0.0493 | -0.0837 | 0.0674 0.1211 0.80 0.0173 | -0.0865 | 0.0247 0.0413 | -0.0561 | -0.0848 | 0.0452 0.0753 0.90 0.0154 | -0.0691 | 0.0270 0.0426 | -0.0616 | -0.0850 | 0.0315 0.0484 1.00 0.0125 | -0.0559 | 0.0288 0.0435 | -0.0664 | -0.0851 | 0.0226 0.0324 1.20 | 0.0086 | -0.0387 | 0.0327 | 0.0443 | -0.0734 | -0.0848 | 0.0124 | 0.0159 1.50 0.0047 | -0.0248 | 0.0370 0.0449 | -0.0793 | -0.0846 | 0.0055 0.0066 2.00 0.0021 -0.0139 | 0.0406 0.0450 | -0.0830 | -0.0845 | 0.0018 0.0021 4 4 x pa? | xpa? | xpb? | xpb2 | xpb?|xpb?|xBa |,Pa Eh? Eh? Grupo de Betão Armado e Pré-Esforçado — IST Esforços elásticos em lajes [10] Y= a y b ) My AM p Mys ! 2 Mas Mas ra d E ia ã EEE, 4 as. v=0.0 al a? DÊ Mxs Mxvs Mys Myas | Myvs | Myva c c 0.30 -0.0482 | -0.4168 | 0.0032 | 0.0072 | -0.0133 | -0.0593 | 0.3562 | 1.0526 0.35 | -0.0292 | -0.3720 | 0.0045 | 0.0116 | -0.0182 | -0.0608 | 0.3062 | 0.8642 0.40 | -0.0137 | -0.3335 | 0.0076 | 0.0163 | -0.0233 | -0.0802 | 0.2664 | 0.7248 0.45 | -0.0024 | -0.2981 | 0.0100 | 0.0214 | -0.0285 | -0.0885 | 0.2282 | 0.5753 0.50 0.0064 | -0.2678 | 0.0126 | 0.0265 | -0.0338 | -0.0958 | 0.1958 | 0.4562 0.55 0.0133 | -0.2382 | 0.0154 | 0.0319 | -0.0395 | -0.1022 | 0.1670 | 0.3712 0.60 0.0158 | -0.2137 | 0.0182 | 0.0376 | -0.0455 | -0.1076 | 0.1422 | 0.3002 0.65 0.0178 | -0.1885 | 0.0211 | 0.0424 | -0.0515 | -0.1124 | 0.1226 | 0.2447 0.70 0.0180 | -0.1697 | 0.0239 | 0.0463 | -0.0572 | -0.1165 | 0.1048 | 0.2016 0.75 0.0178 | -0.1508 | 0.025 0.0502 | -0.0632 | -0.1198 | 0.0903 | 0.1678 0.80 0.0174 | -0.1355 | 0.0296 | 0.0528 | -0.0688 | -0.1227 | 0.0780 | 0.1408 0.85 0.0168 | -0.1212 | 0.0321 | 0.0550 | -0.0741 | -0.1258 | 0.0675 | 0.1172 0.90 0.0158 | -0.1084 | 0.0345 | 0.0568 | -0.0793 | -0.1280 | 0.0582 | 0.1001 0.95 0.0148 | -0.0975 | 0.0370 | 0.0583 | -0.0838 | -0.1303 | 0.0510 | 0.0841 1.00 0.0137 | -0.0878 | 0.0394 | 0.0597 | -0.0882 | -0.1323 | 0.0447 | 0.0707 1.10 0.0116 | -0.0730 | 0.0435 | 0.0616 | -0.0958 | -0.1357 | 0.0326 | 0.0521 1.20 0.0093 | -0.0619 | 0.0474 | 0.0629 | -0.1018 | -0.1378 | 0.0235 | 0.0378 1.30 0.0076 | -0.0528 | 0.0506 | 0.0639 | -0.1061 | -0.1390 | 0.0205 | 0.0276 1.40 0.0060 | -0.0456 | 0.0530 | 0.0641 | -0.1098 | -0.1401 | 0.0182 | 0.0213 1.50 0.0048 | -0.0402 | 0.0549 | 0.0642 | -0.1124 | -0.1403 | 0.0133 | 0.0175 1.73 0.0023 | -0.0300 | 0.0582 | 0.0638 | -0.1158 | -0.1418 | 0.0071 | 0.0088 2.00 0.0012 | -0.0224 | 0.0602 | 0.0630 | -0.1168 | -0.1435 | 0.0040 | 0.0057 q q x pa? | x pa? | x pb? |x pb? |x pb? |x pb? |x Pã |, Pa E à E Eh? Eh? Grupo de Betão Armado e Pré-Esforçado — IST Flechas elásticas em alguns tipos de lajes [14] 4 ac=k 5. PL P - carga uniformemente distribuida 384 D L - vão h - espessura da laje Eh? E - módulo de elasticidade “Hald=v)? v - coeficiente de Poisson Tipo de laje k Laje simplesmente 5 1.000 apoiada Laje bi-encastrada 0.200 Laje quadrada simplesmente apoiada nos quatro bordos aa nos quatro bordos Laje quadrada apoiada nos cantos 2.250 É CERTA RS - Laje quadrada encastrada 0.098 É CE Painel quadrado interior de É É 0.446 laje fungiforme 35 Grupo de Betão Armado e Pré-Esforçado — IST Flechas e esforços elásticos de lajes fungiformes [14] Geometria Painel | ! Painel i de canto + + E + 1 de bordo 1 Jr=cem a[M a7 38 Grupo de Betão Armado e Pré-Esforçado — IST Flechas e esforços elásticos de lajes fungiformes [14] Tabela 1 - Painel de bordo simplesmente apoiado Mox Tu m, dy (apoio) Miy =[m, dx (apoio) É fu Moy=[ m,dx (apoio) RE RAS O is +—in+in+— ke My=/m, dx (vão) PAUSE A L q - carga uniformemente distribuida Q - ângulo da tangente à deformada com a horizontal fegho | LL | mis mar fasso caem =iiay-emaged May M5y méy 1.5 0.1 0.051 -0.343 0.028 0.001 -0.026 -0.360 0.070 0.079 0.076 0.082 0.2 0.045 -0.293 0.027 0.001 -0.026 -0.312 0.068 0.076 0.075 0.080 0.3 | 0.039 -0.278 0.026 0.001 -0.025 -0.297 0.066 0.072 0.073 0.078 0.4 0.035 -0.269 0.025 0.002 -0.023 -0.288 0.065 0.069 0.072 0.076 2.0 0.1 0.049 -0.405 0.028 0.001 -0.026 -0.431 0.070 0.078 0.076 0.082 0.2 0.041 -0.327 0.026 0.001 -0.025 -0.355 0.067 0.073 0.074 0.078 03 0.033 -0.304 0.024 0.002 -0.022 -0.332 0.064 0.068 0.071 0.075 0.4 0.026 -0.2922 0.023 0.003 -0.019 -0.320 0.061 0.063 0.068 0.071 2s 01 0.048 -0.439 0.027 0.001 -0.026 -0.469 0.069 0.077 0.076 0.081 0.2 |0.039 -0.345 0.026 0.002 -0.024 -0.379 0.066 0.072 0.073 0.077 0.3 0.030 -0.317 0.024 0.003 -0.020 -0.351 0.062 0.065 0.070 0.073 0.4 0.021 -0.303 0.021 0.004 -0.015 -0.338 0.059 0.059 0.067 0.069 Laje sem capitel| 0.054 -0.224 0.028 0.000 -0.026 -0.230 0.072 0.081 0.078 0.084 GE? qr? qe RqULE EAGBDAAGIES | QDO go q Lei uê hº/h LYL Max Miy M2y May a| a3 as 9y 98y Ts 01 |-0.0343 -0.0550 -0.0360 0.0743[0.0445 0.0940 00852] 0.314 0.307 0.2 |-0.0544 -0.0382 -0.0581 0.0717 | 0.0355 0.0878 0.0794| 0.300 0.293 0.3 |-0.0689 -0.0271 -0.0746 0.0690 | 0.0293 0.0830 0.0754| 0.288 0.281 0.4 |-0.0793 -0.0194 -0.0871 0.066 | 0.0256 0.0797 0.0724 | 0.280 0.272 20 | 01 |-0.0405 -0.0496 -0.0431 0.0735 | 0.0404 0.0913 0.0827] 0.308 0.301 02 |-0.0630 -0.0317 -0.0686 0.0697 | 0.0280 0.0827 0.0748| 0.288 0.281 0.3 |-0.0788 -0.0209 -0.0875 0.0656 | 0.0197 0.0758 0.0689] 0.271 0.265 0.4 |-0.0904 -0.0139 -0.1022 0.0619 | 0.0148 0.0708 0.0646 | 0.258 0.252 2.5 0.1 |-0.0439 -0.0466 -0.0469 0.0731 | 0.0384 0.0901 0.0815] 0.306 0.299 02 |.0.0679 -0.0276 -0.0474 0.0687|0.0245 0.0802 0.0725| 0.282 0.275 0.3 |-0.0844 -0.0168 -0.0950 0.0640 | 0.0152 0.0725 0.0658 | 0.263 0.257 0.4 |-0.0967 -0.0102 -0.1108 0.0593]0.0101 0.0665 0.610 [0.247 0.241 Laje sem capitel -0.0740 -0.0740 0.0761 | 0.0538 €.0998 0.0907 | 0.328 0.320 3 EE E DT al? ql? ql ql [ca EM e JEM EM 
