nota de aula de flexão exercicio
DESCRIPTION
Exercícios de Flexão em barras de aço.TRANSCRIPT
![Page 2: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/2.jpg)
RESUMO• Pré-Dimensionamento;
• Dimensionamento
FLA FLM
FLT
- a p
- p < a r
- a> r
- m pm
- pm < m rm
- m> rm
- lt plt
- plt < lt rlt
- lt> rlt
Cálcula-se o Mn
![Page 3: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/3.jpg)
RESUMO
• Flecha
• Cisalhamento
EIql4
3845 360
Lmáx
![Page 4: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/4.jpg)
Exercicio 01: Dimensionar uma viga em perfil I de abas inclinadas para um vão de 3,00m.
• Dados: Md = 19 kNm Vd = 25 kN
q = 12 kN/m Aço A-36
fy = 250 MPa fu = 400 MPa
![Page 5: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/5.jpg)
a) Pré-Dimensionamento
• Adotando p bMn=bMpl= bZx fy= Md
• Para perfil I – Z 1,12 Wx Wx = Z / 1,12 logo Wx = 75,40 cm3
3
336
10*250*9,010*19
44,84
44,8410*44,846
3
cmZ
cmmZyb f
Md
![Page 6: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/6.jpg)
Perfil a ser adotado• De acordo com a Tabela E.4 para Wx = 75cm3
![Page 7: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/7.jpg)
• Perfil I – 127 x 14,8 kg/m
A = 18,80 cm2 tf = 0.83cm bf = 7.62cm
tw =0,533cm d = 12,7cm h = 11,04 cm
Ix = 511 cm4 Iy = 50,2 cm4
Wx = 80,40 cm3 Wy = 13,20 cm3
rx = 5,21cm ry =1,63cm
![Page 8: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/8.jpg)
Dimensionamento
1. Verificação da Flambagem Local da Alma
• Mna = Z.fy
83,23533,07,12
wth
a
44,1015,35,3 6
9
10*25010*210
yfE
p
a p – a viga é compacta quanto a alma.
2
2/22 2
.2 hw
tdffii
httbdAZ f
![Page 9: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/9.jpg)
• Mna = Z.fy= 91,31x10-6.250x10+6 =22828,43 Nm
• bMn= 0,9 x 22828,43 = 20545,58Nm > 19000 Nm
• Pela FLA está Ok!
322/04,11
283,0
27,12 31,91
2
04,11533,083,062,72 cmxZ
![Page 10: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/10.jpg)
Dimensionamento
2. Verificação da Flambagem Local da Mesa
• Mna = Z.fy
60,483,0*262,7
2 ft
bm
01,1138,038,0 6
9
10*25010*210
yfE
pm
m pm – a viga é compacta quanto a mesa.
2
2/22 2
.2 hw
tdffii
httbdAZ f
![Page 11: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/11.jpg)
• Mna = Z.fy= 91,31x10-6.250x10+6 =22828,43 Nm
• bMn= 0,9 x 22828,43 = 20545,58Nm > 19000 Nm
• Pela FLM está Ok!
322/04,11
283,0
27,12 31,91
2
04,11533,083,062,72 cmxZ
![Page 12: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/12.jpg)
Dimensionamento
3. Verificação da Flambagem Local com Torção
Perfil com dois eixos de simetria
20004,18463,1300
y
b
rL
LT
71,5075,175,1 6
9
10*25010*210
yfE
pLT
LT > pLT – a viga não possui elementos compactos.
![Page 13: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/13.jpg)
Dimensionamento
De acordo com a tabela 8.1
MPafr
WfrfyMr
EG
AIGE
LT
I
tdA
I
tdA
GE
gt
t
fg
t
fg
115
)(
415,6
385,022
2 )()(
42
1
LTCMrC
LT MrrbLT
b21
221 4707,0 11
![Page 14: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/14.jpg)
Dimensionamento
NmMr
WfrfyMr
Nm
LT
LT
10854
10*4,80*10*)115250()(
11,4911415,6
57,3301539
)10*8,18(*)10*46,3(*)10*210(*385,0
66
10*46,3
10*)83,07,12(10*8,182
1
4891
8
224
43533,0
383,0
3 46,3*04,11*62,72333
cmI btt
![Page 15: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/15.jpg)
1 2 MrLT Cb LT pLT
3301539,57 4911,11 10854Nm 1 184,04 50,71
54,2151085411
11
22
21
221
)57,3301539(*111,4911*4
1085457,3301539*1*707,0
4707,0
LT
LTCMrC
LT
r
MrrbLT
b
pLT< LT rLT Viga com elementos semicompactos
pLTrLT
pLTLTLTplnLT MrMplMM
)(
MrLT = 10854 Nm e Mpl = 22828,43 Nm - Cálculado Anteriormente
![Page 16: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/16.jpg)
NmM
MrMplMM
nLT
pLTrLT
pLTLTLTplnLT
39.1314271,5054,215
71,5004,184)1085443,22828(43,22828
)(
Mpl MrLT rLT LT pLT
22828,43 10854Nm 215,54 184,04 50,71
Para o FLT - 0,9*13142,39=11828,15 < 19000 = Md Não Ok! Redimensionar.
![Page 17: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/17.jpg)
• Perfil I – 152,4 x 18,5 kg/m
A = 23,60 cm2 tf = 0,92cm bf = 8,46 cm
tw =0,584cm d = 15,24cm h = 13,40 cm
Ix = 919 cm4 Iy = 75,7 cm4
Wx = 120,6 cm3 Wy = 17,90 cm3
rx = 6,24cm ry =1,79cm
![Page 18: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/18.jpg)
Redimensionamento
1. Verificação da Flambagem Local da Alma
• Mna = Z.fy
94,22584,04,13
wth
a
44,1015,35,3 6
9
10*25010*210
yfE
p
a p – a viga é compacta quanto a alma.
2
2/22 2
.2 hw
tdffii
httbdAZ f
![Page 19: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/19.jpg)
• Mna = Z.fy= 144,11x10-6x250x10+6 =36028,91Nm
• bMn= 0,9 x 36028,91= 34426,03Nm > 19000 Nm
• Pela FLA está Ok!
322/4,13
292,0
224,15 11,144
2
4,13584,092,046,82 cmxZ
![Page 20: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/20.jpg)
Dimensionamento
2. Verificação da Flambagem Local da Mesa
• Mna = Z.fy
60,492,0*246,8
2 ft
bm
01,1138,038,0 6
9
10*25010*210
yfE
pm
m pm – a viga é compacta quanto a mesa.
2
2/22 2
.2 hw
tdffii
httbdAZ f
![Page 21: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/21.jpg)
• Mna = Z.fy= 144,11x10-6x250x10+6 =36028,91Nm
• bMn= 0,9 x 36028,91= 34426,03Nm > 19000 Nm
• Pela FLM está Ok!
322/4,13
292,0
224,15 11,144
2
4,13584,092,046,82 cmxZ
![Page 22: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/22.jpg)
Dimensionamento
3. Verificação da Flambagem Local com Torção - Redimensionamento
Perfil com dois eixos de simetria
20060,16779,1300
y
b
rL
LT
71,5075,175,1 6
9
10*25010*210
yfE
pLT
LT > pLT – a viga não possui elementos compactos.
![Page 23: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/23.jpg)
Dimensionamento
De acordo com a tabela 8.1
MPafr
WfrfyMr
EG
AIGE
LT
I
tdA
I
tdA
GE
gt
t
fg
t
fg
115
)(
415,6
385,022
2 )()(
42
1
24707,021
221 11 LTCMr
CLT Mrr
bLT
b
![Page 24: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/24.jpg)
Dimensionamento
NmMr
WfrfyMr
Nm
LT
LT
16281
10*6,120*10*)115250()(
52,6063415,6
05,4499772
)10*60,23(*)10*12,5(*)10*210(*385,0
66
10*12,5
10*)92,024,15(10*6,232
1
4891
8
224
43584,0
392,0
3 12,5*40,13*46,82333
cmI btt
![Page 25: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/25.jpg)
1 2 MrLT Cb LT pLT
4499772,05 6063.52 16281Nm 1 167,60 50,71
68,2471628111
11
2
)05,4499772(*152,6063*4
1628105,4499772*1*707,0
24707,0
22
21
221
LT
LTCMrC
LT
r
MrrbLT
b
pLT< LT rLT Viga com elementos semicompactos
pLTrLT
pLTLTLTplnLT MrMplMM
)(
MrLT = 16281 Nm e Mpl = 31056 Nm - Cálculado Anteriormente
![Page 26: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/26.jpg)
NmM
MrMplMM
nLT
pLTrLT
pLTLTLTplnLT
04,2220171,5068,247
71,5060,167)6,1613431056(31056
)(
Mpl MrLT rLT LT pLT
31056 16134,6Nm 247,68 167,60 50,71
Para o FLT - 0,9*22201,04=19980 > 19000 = Md Ok!
![Page 27: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/27.jpg)
4. Verificação da força Cortante
• k= 5,34 - sem enrijecedores transversais
• Como a < p Vn = Vpl = 0,6 Awfy = 0,6dtwfy
Vn = 0,6.(15,24 – 2*0.92)x0.584*10-4x250*106
Vn = 117384 N• Vn = 0.9x117384 = 105645,6 > 25000N ok!
94,22584,04,13
wth
a
46,7108,108,1 6
9
10*25010*210*34,5
yfkE
pv
![Page 28: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/28.jpg)
5. Verificação da Flecha
mmmm
m
mm
xEIqL
L
33,86,6
0066,0
33,8
max
10*91910*2103*10*12
3845
3845
3603000
360max
89
434
![Page 29: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/29.jpg)
Questão 02: Para o perfil VS 550 x 64 (Tabela F1), determine a máxima carga uniformemente distribuida para um vão não travado de uma viga biapoiada de 2,50m.
• Dados fy = 250 MPa ; fu = 400 MPa = 1,4
VS 550 x 64 Tabela F1
A = 81 cm2
Ix = 42556 cm4
Wx = 1547 cm3
rt = 6,5 cm
It = 19 cm4
ry = 5,53 cm
![Page 30: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/30.jpg)
Dimensionamento
1. Verificação da Flambagem Local da Alma
• Mna = Z.fy
28.843,6531
wth
a
44,1015,35,3 6
9
10*25010*210
yfE
p
a p – a viga é compacta quanto a alma.
2
2/22 2
.2 hw
tdffii
httbdAZ f
![Page 31: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/31.jpg)
• Mna = Z.fy= 1727,78x10-6.250x10+6 =431,75 kNm
• bMn= 0,9 x 431,75 = 388,56kNm
322/531
25,9
2550 1727776
2
5313,65,92502 mmxZ
![Page 32: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/32.jpg)
Dimensionamento
2. Verificação da Flambagem Local da Mesa
15,135.9*2250
2 ft
bm
01,1138,038,0 6
9
10*25010*210
yfE
pm
m > pm – a viga não é compacta quanto a mesa.
16,2460,060,0 6
9
10*)115250(10*210
)( frfy
Er m
pm < m < rm – a viga é semicompacta quanto a mesa.
![Page 33: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/33.jpg)
• Mrm = (fy – fr)Wx = (250 –115)*106x1547*10-6
• Mrm = 208845 Nm = 208,845 kNm
kNmM
M
MrMplMM
nm
nm
pmrm
pmmplnm
64,393
88,1016,24
88,1015,13)84,20875,431(75,431
)(
![Page 34: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/34.jpg)
Dimensionamento
3. Verificação da Flambagem Local com Torção
Perfil com dois eixos de simetria
20020,4553,5250
y
b
rL
LT
71,5075,175,1 6
9
10*25010*210
yfE
pLT
LT < pLT – a viga com elementos compactos.
Mna = Z.fy= 1727,78x10-6.250x10+6 =431,75 kNm
![Page 35: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/35.jpg)
Dimensionamento
• Portanto: Mn = 0,9x393,64 = 354,27 kNm
• Tem-se: Md = M Mn então
M 354,27/1,4 253,05 kNm.
Como o Momento máximo na viga Bi-apoiada:
mkNL
MqM qL /9,323
5,2
05.253*88228
2
![Page 36: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/36.jpg)
![Page 37: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/37.jpg)
![Page 38: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/38.jpg)
![Page 39: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/39.jpg)
![Page 40: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/40.jpg)
![Page 41: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/41.jpg)
![Page 42: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/42.jpg)
![Page 43: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/43.jpg)
![Page 44: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/44.jpg)
![Page 45: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/45.jpg)
![Page 46: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/46.jpg)
![Page 47: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/47.jpg)
![Page 48: Nota de Aula de Flexão Exercicio](https://reader038.vdocuments.pub/reader038/viewer/2022102619/563dbb17550346aa9aaa33ff/html5/thumbnails/48.jpg)
3 33