Calculo (Lajes Maciças e Nervuradas)

 

LAJE DE CONCRETO ARMADO COM APOIOS LINEARES Data Projeto Cliente Fernando Musso 10/5/2013 Página Descrição Detalhamento de laje

UNIVERSIDADE FEDERAL DO ESPÍRITO SANTO CENTRO TECNOLÓGI TECNOLÓGICO CO DEPARTAMENTO DE ENGENHARIA CIVIL Laje maciça e nervurada © 2011 Fernando Musso Junior 

NBR 6118:2003

Laje em 2 direções

Laje

Concreto f ck 25 ck MPa Ecs MPa 23800

MACIÇA

NERVURADA  a

Norma

NBR

  m

Aço f yk yk MPa Es MPa

Resumo - Verificações ELU-M - Momento ELU-V - Cortante ELS-W - Fissura ELS-DEF - Flecha

500 210000

Consumo de Materiais Custo (MO+MT+EQ+DI) aço kg 895 8.96 concreto m3 64.0 forma m2 99.9 kg/m3 13.98 k /m2 0.140 m3/m2 aço R$/kg 6.50 450 concr.R$/m 3 65 forma R$/m2 14007 Custo R$

Esforços e Flechas MUSSO

BURKE

CZERNY 

BARES

MARCUS

Status OK! OK! OK! OK!

a e

Ações 3.50 GPP kN/m2 2.00 GEX kN/m2 3.00 Q kN/m2

Ponderação das Ações ELU-M/V ELS-W 1.40 1 0.6 1.40

gG g

   0    0  .    8

Sd / R d 0.90 0.34 0.43 0.86

mm e cm MGk kNm/m MQk kNm/m MSd kNm/m  As cm2/m 2  As ad ado o cm /m 2  A's cm /m  As cm2/nerv f 

M d,lim MSd/M d,lim Status  As,mesa

VGk kN/m VQk kN/m VSd kN/m VRd1 kN/m VSd /VRd1 Status

ELU-V - Força Cortante ae a b 8.05 13.95 8.05 4.39 7.61 4.39 17.42 30.18 17.42 73.73 89.12 73.73 0.24 0.34 0.24 OK! OK! OK!

be 13.95 7.61 30.18 89.12 0.34 OK!

ELS-W - Abertura de Fissura ae a b

be

cm

M kNm/m F Mcr  kNm/m x2 cm I2 cm4 wk mm wlim mm wk /wlim Status

MQP kNm/m

seção Ie,QP cm4 EcsIc kNcm 2 EceIe ceIe EDcsCf  Ie/E mm 0.0 22.86

184..8208 2.84 4045 0.102

1.24% 3.81 58.77 0.90 OK! 4.2

0.51% 1.496 58.77 0.40 OK! e cm

382..8208 4.08 8025 0.130

1.24% 3.81 58.77 0.90 OK!

184..8208 2.84 4045 0.102

382..8208 4.08 8025 0.130

0.34 OK!

0.43 OK!

0.30 0.34 OK!

0.43 OK!

ELS-DEF - Flecha a b QP kN/m2 Mcr  kNm/m 13.10 13.10 f fissurada fissurada 20651 20651 DQP,ela mm 54422667 54422667 Ic,R cm4/m 21157730 21157730 Ic T cm4/m .57 D2 T mm

2.5mm 7 DT,lim

Cf ,lim mm DD T/DT,lim

27.5

32.0

0.86

6.70 12.57 0.02540 10.67 22867 22867 22.9 Status OK!

14

cm

eq = d= f  = bn = en = w= c=

14.0 11.0 5.0 13.0 65 100.0 2.0

cm cm cm cm cm cm cm

a

x ELS-DEF 1 0.4

x x x x x a = apoio simples

be 12.5 9 24.32 13.27 52.62 13.31 13.64 0.00

0.51% 1.496 58.77 0.40 OK! f  mm

r lx

h=

a e

vão menor (a) = 8.00 m

Q

ELU-M - Momento Fletor  ae a b 8.0 12.5 8.0 9 9 9 10.76 24.32 10.76 5.87 13.27 5.87 23.27 52.62 23.27 5.22 5.22 13.31 5.59 13.64 5.59 0.00 0.00 0.00

  =    )    b    (   r   o    i   a   m   o    ã   v

e

x

e

x

x x x x e = engaste

x x x x    0    0 x    8      9 x   c    8 88 88Ø1 Ø12. 2.5c 5c99-20 200 0 x    Ø    8 x    8 88Ø8c9-800 x x    0 Armação    0    2 x      9   c x    5  .    2 x    1    Ø x    8    8 x x x x x x x x x x x    0    0  .    0

   7    2  .    3    2

23.27

0.00

Momentos Fletores - MSd (kNm/m)    2    6  .    2    5

x

x x x x x x x x x x x x x x x

x

x

x

x

x

x

G = 8.05 Q = 4.39 12.44

x

x

x x x x x x x 52.62 x x x x x x x x x x

x x x x x DT = 27.45 x    4    6    5 Dlim = 32.00    5    9    1 x    5    9    4  .  .  .    0  .    2    1    3    6  .  .    3 DCf = 0.00    2    1    7 x    8    4    1 x   =   =       =       = Reações e Flecha    G    Q    G    Q x (kN/m) (mm) x x 21.56 x G = 13.95 x Q = 7.61 x x x x x x x x x x x x

 

Referência: Flechas e Momentos (poisson = 0,0) - MARCUS - adaptado por MUSSO; Reações - NBR 6118 (item (it em 14.7.6.1) - MUSSO pa pa4/(Ecsh3) multiplicar por  pa2 a Tipo 1 f m mam maem mbm mbem r a r aaee r b 1.0 0.04557 0.03646 0 0.03646 0 0.25000 0 0.25000 1.1 0.05485 0.04388 0 0.03626 0 0.27273 0 0.25000 1.2 0.06426 0.05141 0 0.03570 0 0.29167 0 0.25000 1.3 0.07346 0.05877 0 0.03478 0 0.30769 0 0.25000 b 1.4 0.08215 0.06572 0 0.03353 0 0.32143 0 0.25000 b 1 a 1.5 0.09012 0.07210 0 0.03204 0 0.33333 0 0.25000 1.6 0.09728 0.07782 0 0.03040 0 0.34375 0 0.25000 1.7 0.10361 0.08289 0 0.02868 0 0.35294 0 0.25000 a2 0.15625 0.12500 0 0.02364 0 0.50000 0 0.25000 a Tipo 2 f m mam maem mbm mbem r a r aaee r b 1.0 0.03401 0.02721 0 0.03341 0.08929 0.18301 0 0.23205 1.1 0.04303 0.03442 0 0.03446 0.09539 0.20131 0 0.24052 1.2 0.05225 0.04180 0 0.03492 0.09839 0.21962 0 0.24631 1.3 0.06141 0.04913 0 0.03496 0.09860 0.23792 0 0.24942 b 1.4 0.07027 0.05622 0 0.03465 0.09658 0.25607 0 0.25000 2 b 1 . 5 0 . 0 7 8 6 6 0 . 0 6 2 9 3 0 0 . 0 3 4 0 6 0 . 0 9 2 9 8 0 . 2 7 2 3 3 0 0.25000 a 1.6 0.08645 0.06916 0 0.03323 0.08836 0.28656 0 0.25000 1.7 0.09357 0.07485 0 0.03220 0.08322 0.29911 0 0.25000 a2 0.15625 0.12500 0 0.02838 0.06757 0.50000 0 0.25000 a f m mam maem mbm mbem r a r aaee r b Tipo 3 1.0 0.02242 0.01794 0 0.02668 0.06944 0.14434 0 0 1.1 0.02987 0.02390 0 0.02886 0.07799 0.15877 0 0 1.2 0.03803 0.03043 0 0.03042 0.08482 0.17321 0 0 1.3 0.04662 0.03730 0 0.03143 0.08963 0.18764 0 0 1.4 0.05535 0.04428 0 0.03196 0.09237 0.20207 0 0 b 1.5 0.06396 0.05117 0 0.03212 0.09317 0.21651 0 0 b 3 a 1.6 0.07227 0.05781 0 0.03196 0.09232 0.23094 0 0 1.7 0.08011 0.06409 0 0.03154 0.09019 0.24537 0 0 1.8 0.08740 0.06992 0 0.03091 0.08711 0.25944 0 0 a2 0.15625 0.12500 0 0.02918 0.07937 0.50000 0 0 a Tipo 4 f m mam maem mbm mbem r a r aaee r b 1.0 0.03401 0.03341 0.08929 0.02721 0 0.23205 0.40192 0.18301 1.1 0.03847 0.03842 0.09818 0.02543 0 0.24423 0.42302 0.18301 1.2 0.04223 0.04286 0.10479 0.02346 0 0.25438 0.44060 0.18301 1.3 0.04534 0.04667 0.10964 0.02146 0 0.26297 0.45547 0.18301 b

b

4 a

a2

0..0 02 27 77 27 3 0 0.02821 0.02858 0.03125 f m 0.01345 0.01604 0.01834 0.02033 0.02201 0.02341 0.02456 0.02551 0.02630 0.02695 0.02749 0.03125

0..0 03 36 55 72 5 0 0.03715 0.03768 0.04167 mam 0.01794 0.02138 0.02445 0.02710 0.02934 0.03121 0.03275 0.03402 0.03506 0.03593 0.03665 0.04167

0..0 07 79 85 64 3 0 0.08025 0.08081 0.08333 maem 0.04167 0.04951 0.05622 0.06172 0.06612 0.06959 0.07230 0.07442 0.07609 0.07739 0.07843 0.08333

0..0 00 19 06 64 0 0 0.00879 0.00804 0.00804 mbm 0.01794 0.01767 0.01698 0.01604 0.01497 0.01387 0.01279 0.01177 0.01082 0.00995 0.00916 0.00916

0..0 01 28 04 41 0 0 0.01667 0.01515 0.01515 mbem 0.04167 0.04092 0.03904 0.03652 0.03374 0.03093 0.02824 0.02575 0.02348 0.02144 0.01961 0.01961

0 0 0 0 0 r a 0 0 0 0 0 0 0 0 0 0 0 0

0..3 39 80 44 06 2 0 0.39623 0.40142 0.50000 r aaee 0.25000 0.27273 0.29167 0.30769 0.32143 0.33333 0.34375 0.35294 0.36111 0.36842 0.37500 0.50000

0..1 14 44 43 34 4 0 0.14434 0.14434 0.14434 r b 0 0 0 0 0 0 0 0 0 0 0 0

0..2 25 50 00 00 0 0 0.25000 0.25000 0.25000 r bbee 0.25000 0.25000 0.25000 0.25000 0.25000 0.25000 0.25000 0.25000 0.25000 0.25000 0.25000 0.25000

0..9 95 44 35 54 2 0 0.96305 0.96970

0..0 04 55 64 46 8 0 0.03695 0.03030

a

0.50000 0.59417 0.67465 0.74067 0.79346 0.83505 0.86761 0.89307 0.91303 0.92873 0.94118 a

0.33333 0.42265 0.50903 0.58815 0.65763 0.71681 0.76618 0.80680 0.83997 0.86695 0.88889 a

0.83333 0.87981 0.91203 0.93456 0.95051 0.96200 0.97039 0.97661 0.98130 0.98489 0.98765 a

a

0.50000 0.59417 0.67465 0.74067 0.79346 0.83505 0.86761 0.89307 0.91303 0.92873 0.94118

b

0.50000 0.40583 0.32535 0.25933 0.20654 0.16495 0.13239 0.10693 0.08697 0.07127 0.05882 b

0.66667 0.57735 0.49097 0.41185 0.34237 0.28319 0.23382 0.19320 0.16003 0.13305 0.11111 b

0.16667 0.12019 0.08797 0.06544 0.04949 0.03800 0.02961 0.02339 0.01870 0.01511 0.01235 b

b

0.50000 0.40583 0.32535 0.25933 0.20654 0.16495 0.13239 0.10693 0.08697 0.07127 0.05882

 

aa

ab

8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0

8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0

aa

ab

8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0

14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222

aa

8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0

ab

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

a

0.58333 0.59079 0.60958 0.63478 0.66265 0.69072 0.71757 0.74248 0.76517 0.78561 0.80392 a

0.76190 0.74563 0.73762 0.73706 0.74244 0.75207 0.76437 0.77808 0.79227 0.80632 0.81982 a

0.86111 0.84401 0.83036 0.82073 0.81527 0.81366 0.81535 0.81963 0.82578 0.83317 0.84127

aa

ab

14.222 14.222 14.222 14.222

8.0 8.0 8.0 8.0

0.66518 0.69573 0.72712 0.75671

14.222 14.222 14.222 14.222 14.222 14.222

8.0 8.0 8.0 8.0 8.0 8.0

70 86 39 42 0 0.8 0.82743 0.84522 0.86063 0.87402 0.88567

aa

ab

14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222

14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222

aa

ab

14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

a

a

0.76563 0.76982 0.78039 0.79456 0.81024 0.82603 0.84114 0.85515 0.86791 0.87941 0.88971 a

0.84375 0.83627 0.83430 0.83687 0.84272 0.85066 0.85971 0.86914 0.87848 0.88743 0.89583

aa

ab

24.0 24.0

8.0 8.0

0.76852 0.79802

2 24 4..0 0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

8 8..0 0 8.0 8.0 8.0 8.0 8.0 8.0 8.0

0 0..8 82 44 60 37 9 0.86529 0.88124 0.89471 0.90613 0.91587 0.92422 0.93141

aa

ab

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222 14.222

aa

ab

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

a

a

0.81481 0.82887 0.84458 0.86012 0.87460 0.88764 0.89918 0.90931 0.91816 0.92590 0.93266 a

0.86111 0.86360 0.86986 0.87826 0.88755 0.89691 0.90586 0.91416 0.92172 0.92854 0.93464

 

UNIVERSIDADE FEDERAL DO ESPÍRITO SANTO DISCIPLINA: ESTRUTURAS DE CONCRETO VERIFICAÇÃO DO ESTADO LIMITE ÚLTIMO DE RESISTÊNCIA À MOMENTO FLETOR (ELU-M) - MOMENTO POSITIVO norma NB R NB R E C2 E C2 direção 2 2 2 2 posição a b a b PROPRIEDADES DO CONCRETO f ck 25 25 ck gc 1.40 1.40 f cd 17.857 17.857 cd h 0.85 0.85 l 0.80 0.80 f cctm 2.565 2.565 tm PROPRIEDADES DO AÇO f yk 500 500 yk gs 1.15 1.15 f yd 434.78 434.78 yd Es 210000 210000 eyd 0.00207 0.00207 es' 0.00159 0.00159 ss'd 334.09 334.09

25 1.50 16.667 0.85 0.80 2.565

25 1.50 16.667 0.85 0.80 2.565

500 1.15 434.78 200000 0.00217 0.00138 275.76

500 1.15 434.78 200000 0.00217 0.00138 275.76

CENTRO TECNOLÓGICO PROFESSOR: FERNANDO MUSSO JUNIOR

MPa MPa

MPa

NBR: 1,4; EC2: 1,5 f ck ck/gc 0.85 0.80 2/3 0,3f ck ck

MPa MPa MPa

MPa

1.15 f yk yk/gs NB NBR: R: 210 210000 00;; E EC2 C2:: 2 20 0000 000 0 f yydd/Es 3,5‰(xlim - d')/xlim SE(es'