Cálculo diferencial e integral de una variable 1 Cálculo de volumen.
Cálculo de Lajes.xlsx
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3,5
5m
5,00 m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = ".55
# = 5.00
$ ="55
= %.%&5 cm = '.00 cm(0
E =5.00
= ).()".55
2(.0 5).0 )0.) )2.6 0.")& 0.)%" 0.&)(
M! =q . *!+
= ".( kN.mm!
M# =q . *!+
= ).6 kN.m
m#
N! =q . *!+
= %.) kN.mn!
N# =q . *!+
= 6.5 kN.mn#
,) . q . ! = &." kN/m
,2 . q . ! = (.2 kN/m
," . q . ! = )6.5 kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 1
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,( . q . ! = ".0 kN/m
f = )0"2." = 0.002%"& = 0.2%(cs . $ . "6"'2(.)
16 = = )2".&%5 K3 = 0."(0m!
16 = = 26".0(" K3 = 0.""0m#
16 = = 52.0'" K3 = 0."6'n!
16 = = 6(.'%& K3 = 0."5%n#
As =1" . M!
= ).&%5 cm/m)0 d
As = 1" . M# = 0.%)5 cm/m)0 d
As =1" . N!
= (.60( cm/m 3)245mm a cada 2&cm)0 d
As =1" . N#
= ".5%) cm/m)0 d
-(=
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
364"mm a cada )&cm
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
3)040mm a cada 22cm
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0.)") 2".(5)
v(
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cm
cm
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3,5
5m
5,00 m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = 5.00
# = ".55
$ ="55
= %.%&5 cm = '.00 cm(0
E =".55
= 0.&)5
)05." 55." 26.2 22.5 0.)(" 0.222 0.2(&
M! =q . *!+
= ).5 kN.mm!
M# =q . *!+
= 2.' kN.m
m#
N! =q . *!+
= 6.2 kN.mn!
N# =q . *!+
= &.2 kN.mn#
,) . q . ! = (.6 kN/m
,2 . q . ! = &.2 kN/m
," . q . ! = %.0 kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 2
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f =(062.5
= 0.002(0) = 0.2(0
cs . $ . )6')'%).0
16 = = 2&".&%0 K3 = 0.""0m!
16 = = )(".&%0 K3 = 0.""&m#
16 = = 6%.)20 K3 = 0."5%n!
16 = = 5%.500 K3 = 0."6"n#
As =1" . M!
= 0.&%" cm/m)0 d
As =1" . M#
= ).52( cm/m
)0 d
As =1" . N!
= ".()6 cm/m)0 d
As =1" . N#
= (.0"" cm/m)0 d
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
364"mm a cada 20cm
364"mm a cada 'cm
364"mm a cada &45cm
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)0'.0"
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cm
cm
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3,5
5m
2,30m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = 2."0
# = ".55
$ =2"0
= 5.&50 cm = %.00 cm(0
E =".55
= ).5(2."
2&.6 &5.2 )2.5 )&.5 0.)(( 0."6' 0.25
M! =q . *!+
= ).2 kN.mm!
M# =q . *!+
= 0.5 kN.m
m#
N! =q . *!+
= 2.% kN.mn!
N# =q . *!+
= 2.0 kN.mn#
,) . q . ! = 2.2 kN/m
,2 . q . ! = 5.5 kN/m
," . q . ! = ".& kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 3
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f =)%).'
= 0.000(65 = 0.0(6
cs . $ . "')((2."
16 = = 2(2.%0' K3 = 0."")m!
16 = = 66).56% K3 = 0."25m#
16 = = )0'.'6% K3 = 0."()n!
16 = = )5".'55 K3 = 0.""5n#
As =1" . M!
= 0.&50 cm/m)0 d
As =1" . M#
= 0.2&0 cm/m
)0 d
As =1" . N!
= ).&05 cm/m)0 d
As =1" . N#
= ).)'& cm/m)0 d
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
364"mm a cada )%cm
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
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"5.')5
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cm
cm
cm
cm
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3,5
5m
4,00m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = (.00
# = ".55
$ ="55
= %.%&5 cm = '.00 cm(0
E =".55
= 0.%'(
5(." 55.( )%.) )%.& 0.)(( 0.2%) 0.25
M! =q . *!+
= ).' kN.mm!
M# =q . *!+
= ).' kN.m
m#
N! =q . *!+
= 5.& kN.mn!
N# =q . *!+
= 5.6 kN.mn#
,) . q . ! = ".& kN/m
,2 . q . ! = &." kN/m
," . q . ! = 6.5 kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 4
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f =)66(.0
= 0.00)6'5 = 0.)&0
cs . $ . '%)("5.'
16 = = 220.5'( K3 = 0."")m!
16 = = 225.06" K3 = 0."")m#
16 = = &".5") K3 = 0."5"n!
16 = = &5.'6' K3 = 0."52n#
As =1" . M!
= 0.'&5 cm/m)0 d
As =1" . M#
= 0.'56 cm/m
)0 d
As =1" . N!
= ".)20 cm/m)0 d
As =1" . N#
= ".0)2 cm/m)0 d
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
3)040mm a cada 25cm
3)040mm a cada 26cm
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6".2("
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cm
cm
cm
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3,5
5m
4,30m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = (."0
# = ".55
$ ="55
= %.%&5 cm = %.00 cm(0
E =".55
= 0.%"(."
6)." 5(." )'.5 )'.2 0.)(( 0.26% 0.25
M! =q . *!+
= 2.0 kN.mm!
M# =q . *!+
= 2.2 kN.m
m#
N! =q . *!+
= 6.2 kN.mn!
N# =q . *!+
= 6." kN.mn#
,) . q . ! = (.0 kN/m
,2 . q . ! = &.5 kN/m
," . q . ! = &.0 kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 5
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f =2222.2
= 0.002%'5 = 0.2%'
cs . $ . &6&625.'
16 = = )5(.2%' K3 = 0.""5m!
16 = = )"6.6&) K3 = 0.""%m#
16 = = ('.0%) K3 = 0."&(n!
16 = = (%."25 K3 = 0."&(n#
As =1" . M!
= ).)'( cm/m)0 d
As =1" . M#
= )."60 cm/m
)0 d
As =1" . N!
= (.)') cm/m)0 d
As =1" . N#
= (.25& cm/m)0 d
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
364"mm a cada 22cm
3)0400mm a cada )'cm
3)0400mm a cada )%cm
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&0.("
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cm
cm
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3,5
5m
2,20m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = 2.20
# = ".55
$ =220
= 5.500 cm = %.00 cm(0
E =".55
= ).6)2.2
20.' 55.0 '.2 )2." 0.")& 0.)%" 0.("6
M! =q . *!+
= ).5 kN.mm!
M# =q . *!+
= 0.6 kN.m
m#
N! =q . *!+
= ".( kN.mn!
N# =q . *!+
= 2.6 kN.mn#
,) . q . ! = (.5 kN/m
,2 . q . ! = 2.6 kN/m
," . q . ! = 6.2 kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 6
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,( . q . ! = ".6 kN/m
f =)52."
= 0.0006&% = 0.06%cs . $ . 22(&(0.)
16 = = 200.'62 K3 = 0."""m!
16 = = 52%.%(6 K3 = 0."26m#
16 = = %%.(62 K3 = 0."(&n!
16 = = ))%.26' K3 = 0."(0n#
As =1" . M!
= 0.')) cm/m)0 d
As =1" . M#
= 0.""' cm/m)0 d
As =1" . N!
= 2.)5& cm/m)0 d
As =1" . N#
= ).5%) cm/m)0 d
-(=
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
364"0mm a cada )(cm
364"0mm a cada )'cm
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0.252 20.62
v(
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cm
cm
cm
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4,7
5m
5,00m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = (.&5
# = 5.00
$ =(&5
= )).%&5 cm = )2.00 cm(0
E =5.00
= ).05(.&5
(0.2 (0.2 )(." )(." 0.")& 0.)%" 0.")&
M! =q . *!+
= ".6 kN.mm!
M# =q . *!+
= ".6 kN.m
m#
N! =q . *!+
= )0." kN.mn!
N# =q . *!+
= )0." kN.mn#
,) . q . ! = '.% kN/m
,2 . q . ! = 5.& kN/m
," . q . ! = '.% kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 7
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,( . q . ! = 5.& kN/m
f =""0%.'
= 0.00226& = 0.22&cs . $ . )(5'&22.)
16 = = 2(&."%5 K3 = 0.""0m!
16 = = 2(&."%5 K3 = 0.""0m#
16 = = %%.000 K3 = 0."(&n!
16 = = %%.000 K3 = 0."(&n#
As =1" . M!
= ).26& cm/m)0 d
As =1" . M#
= ).26& cm/m)0 d
As =1" . N!
= ".&(6 cm/m)0 d
As =1" . N#
= ".&(6 cm/m)0 d
-(=
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
364"0mm a cada 2(cm
364"0mm a cada 2(cm
3)0400mm a cada 2)cm
3)0400mm a cada 2)cm
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0.)%" "'.6%"
v(
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cm
-
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4,7
5m
3,00m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = ".00
# = (.&5
$ ="00
= &.500 cm = %.00 cm(0
E =(.&5
= ).5%"
26.6 &%.& )2." )&.5 0.)(( 0."&& 0.25
M! =q . *!+
= 2.2 kN.mm!
M# =q . *!+
= 0.& kN.m
m#
N! =q . *!+
= (.% kN.mn!
N# =q . *!+
= "." kN.mn#
,) . q . ! = 2.% kN/m
,2 . q . ! = &.( kN/m
," . q . ! = (.' kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 8
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f =526.5
= 0.00)"'" = 0.)"'
cs . $ . "&&'&).0
16 = = )"&.5(& K3 = 0.""&m!
16 = = (06.'5" K3 = 0."2&m#
16 = = 6".60" K3 = 0."5'n!
16 = = '0.(') K3 = 0."(&n#
As =1" . M!
= )."(% cm/m)0 d
As =1" . M#
= 0.((2 cm/m
)0 d
As =1" . N!
= ".)0( cm/m)0 d
As =1" . N#
= 2.)0' cm/m)0 d
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
364"0mm a cada 22cm
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
3%400mm a cada )6cm
364"0mm a cada )(cm
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"(.6&'
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cm
cm
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4,7
5m
3,30m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = "."0
# = (.&5
$ =""0
= %.250 cm = '.00 cm(0
E =(.&5
= ).(("."
2'.0 &).' )2.% )&.5 0.)(( 0."5' 0.25
M! =q . *!+
= 2.( kN.mm!
M# =q . *!+
= ).0 kN.m
m#
N! =q . *!+
= 5.5 kN.mn!
N# =q . *!+
= (.0 kN.mn#
,) . q . ! = ".) kN/m
,2 . q . ! = &.& kN/m
," . q . ! = 5.( kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 9
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f =&&0.%
= 0.00)""6 = 0.)"(
cs . $ . 5&&0%6.)
16 = = )&".0'5 K3 = 0.""(m!
16 = = (2'.)55 K3 = 0."2&m#
16 = = &6.(00 K3 = 0."52n!
16 = = )0(.(5( K3 = 0."("n#
As =1" . M!
= ).25( cm/m)0 d
As =1" . M#
= 0.('5 cm/m
)0 d
As =1" . N!
= 2.''5 cm/m)0 d
As =1" . N#
= 2.)"( cm/m)0 d
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
364"0mm a cada 25cm
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
3%400mm a cada )6cm
3%400mm a cada 2"cm
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"&.)%&
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cm
cm
-
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4,7
5m
5,00m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = 5.00
# = (.&5
$ =(&5
= )).%&5 cm = )2.00 cm(0
E =(.&5
= 0.'55
(%.5 55.& )&.) )%.( 0.)(( 0.2'" 0.25
M! =q . *!+
= ".( kN.mm!
M# =q . *!+
= 2.' kN.m
m#
N! =q . *!+
= '.5 kN.mn!
N# =q . *!+
= %.% kN.mn#
,) . q . ! = (.& kN/m
,2 . q . ! = '.5 kN/m
," . q . ! = %.) kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 10
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f =(062.5
= 0.00)'25 = 0.)'"
cs . $ . 2))0"6&.6
16 = = 26'."62 K3 = 0.""0m!
16 = = "0'."(' K3 = 0."2'm#
16 = = '(.'&) K3 = 0."(6n!
16 = = )02.)') K3 = 0."((n#
As =1" . M!
= ).)6( cm/m)0 d
As =1" . M#
= ).0)0 cm/m
)0 d
As =1" . N!
= ".(6) cm/m)0 d
As =1" . N#
= ".)'% cm/m)0 d
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
3)0400mm a cada 2"cm
3)0400mm a cada 25cm
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5&."&)
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cm
cm
cm
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4,7
5m
1,50m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = ).50
# = (.&5
$ =)50
= ".&50 cm = %.00 cm(0
E =(.&5
= ".)&).5
M =q . *!+
= 0.6 kN.m2(
N =q . *!+
= 7).2 kN.m)2
16 = = ('6.()0 K3 = 0."26M
16 = = 2(%.205 K3 = 0.""0N
As =1" . M
= 0."6) cm/m)0 d
Espaamento mnimo = 2 x h ou 20cm = (6,3mm a cada 16cm - 1,96cm/m
As = 1" . N = 0.&") cm/m)0 d
)05. bw . d
)05. bw . d
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
*Asmnimo
+ = )420cm/m 7 364"mm a cada 26
!s"
= 0,2 x 1,96 = 0,392 ou 0,90cm/m (o meno# = !s"
= 0,392 ($,0mm a cada 30cm
LAJE 11
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cm
cm
-
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4,7
5m
5,00m
fck = 20 Mpa
CA = 50
q = 6.5 kN/m
Agressividade = Urbano
! = (.&5
# = 5.00
$ =(&5
= )).%&5 cm = )2.00 cm(0
E =5.00
= ).05(.&5
(0.2 (0.2 )(." )(." 0.")& 0.)%" 0.")&
M! =q . *!+
= ".6 kN.mm!
M# =q . *!+
= ".6 kN.m
m#
N! =q . *!+
= )0." kN.mn!
N# =q . *!+
= )0." kN.mn#
,) . q . ! = '.% kN/m
,2 . q . ! = 5.& kN/m
," . q . ! = '.% kN/m
m! m# n! n# v) v2 v"
-)=
-2=
-"=
LAJE 12
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,( . q . ! = 5.& kN/m
f = ""0%.' = 0.00226& = 0.22&cs . $ . )(5'&22.)
16 = = 2(&."%5 K3 = 0.""0m!
16 = = 2(&."%5 K3 = 0.""0m#
16 = = %%.000 K3 = 0."(&n!
16 = = %%.000 K3 = 0."(&n#
As =1" . M!
= ).26& cm/m)0 d
As = 1" . M# = ).26& cm/m)0 d
As =1" . N!
= ".&(6 cm/m)0 d
As =1" . N#
= ".&(6 cm/m)0 d
-(=
q . *!+(
)05. bw . d
)05. bw . d
)05. bw . d
)05. b . d
364"0mm a cada 2(cm
364"0mm a cada 2(cm
3)0400mm a cada 2)cm
3)0400mm a cada 2)cm
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0.)%" "'.6%"
v(
-
7/23/2019 Clculo de Lajes.xlsx
49/49
cm
