MEMORIAL DE CALCULO

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ROTEIRO ENGENHEIRO ESTRUTURAL
PASSO 1 - O CAMINHO DA CARGA
� Pré-dimensionamento das lajes
≔n 1 Numero de engastes
≔lx 410 Menor vão ou o com mais engastes
≔ly 450
≔l =⋅ly 0.7 315
≔D =―――――
⋅(( -2.5 ⋅0.1 n)) l
100
7.56
≔H =+D 2.5 10.06 laje com 10 cm
� Definição das cargas
Cargas (kN/m²)
≔Sc 2 Carga para escritórios
≔Cp 1 Revestimento
≔PP =⋅0.10 25 2.5 PP laje
≔q =++Sc Cp PP 5.5
� Momentos e reações das lajes
1ª Pergunta: Tem vizinhos? 
2ª Pergunta: A laje está no mesmo nivel
3ª Pergunta: Mais de 2/3 da laje é continua e está no mesmo nivel? 
Laje 01
≔lx 1.00 ≔ly 4.10
=―
ly
lx
4.1 Acima de 2 laje biapoiada
≔Mx =――
⋅q lx2
8
0.688 ≔Rx =――
⋅q lx
2
2.75
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Laje 02 (2 engaste) tabela 3
≔lx 3.13 ≔ly 4.1
=―
ly
lx
1.31 ≔kx 0.746 ≔mx 23.93 ≔nx 10.72 ≔my 41.06 ≔ny 18.39
momento fletor positivo e negativo em X
≔Mx =――
⋅q lx2
mx
2.252 ≔Xx =―――
⋅-q lx2
nx
-5.026
momento fletor positivo e negativo em y
≔My =――
⋅q lx2
my
1.312 ≔Xy =―――
⋅-q lx2
ny
-2.93
Reações de apoio em X e Y
≔Ry =――
⋅q lx
3
5.738 ――
kN
m
≔Rx =⋅⋅q lx ――――
-3
⎛
⎜
⎝
―
lx
ly
⎞
⎟
⎠
2
6
6.935 ――
kN
m
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Laje 03 (2 engastes) tabela 3
≔lx 4.1 ≔ly 4.5 lx = 4.1 pois é o menor vão
=―
ly
lx
1.098 ≔kx 0.585 ≔mx 31.59 ≔nx 13.67 ≔my 37.53 ≔ny 16.24
momento fletor positivo e negativo em X
≔Mx =――
⋅q lx2
mx
2.927 ≔Xx =―――
⋅-q lx2
nx
-6.763
momento fletor positivo e negativo em y
≔My =――
⋅q lx2
my
2.463 ≔Xy =―――
⋅-q lx2
ny
-5.693
Reações de apoio em X e Y
≔Ry =――
⋅q lx
3
7.517 ――
kN
m
≔Rx =⋅⋅q lx ――――
-3
⎛
⎜
⎝
―
lx
ly
⎞
⎟
⎠
2
6
8.155 ――
kN
m
Laje 04 (3 engastes) tabela 5
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Laje 04 (3 engastes) tabela 5
≔lx 1.37 ≔ly 1.67 lx = 4.1 pois é o menor vão
=―
ly
lx
1.219 ≔kx 0.816 ≔mx 34.7 ≔nx 14.71 ≔my 59.53 ≔ny 29.19
momento fletor positivo e negativo em X
≔Mx =――
⋅q lx2
mx
0.297 ≔Xx =―――
⋅-q lx2
nx
-0.702
momento fletor positivo e negativo em y
≔My =――
⋅q lx2
my
0.173 ≔Xy =―――
⋅-q lx2
ny
-0.354
Reações de apoio em X e Y
≔Ry =――
⋅q lx
3
2.512 ――
kN
m
≔Rx =⋅⋅q lx ――――
-3
⎛
⎜
⎝
―
lx
ly
⎞
⎟
⎠
2
6
2.922 ――
kN
m
Laje 05 (4 engastes) tabela 6
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Laje 05 (4 engastes) tabela 6
≔lx 1.37 ≔ly 2.43
=―
ly
lx
1.774 ≔kx 0.907 ≔mx 28.76 ≔nx 13.22 ≔my 90.09 ≔ny 41.42
momento fletor positivo e negativo em X
≔Mx =――
⋅q lx2
mx
0.359 ≔Xx =―――
⋅-q lx2
nx
-0.781
momento fletor positivo e negativo em y
≔My =――
⋅q lx2
my
0.115 ≔Xy =―――
⋅-q lx2
ny
-0.249
Reações de apoio em X e Y
≔Ry =――
⋅q lx
3
2.512 ――
kN
m
≔Rx =⋅⋅q lx ――――
-3
⎛
⎜
⎝
―
lx
ly
⎞
⎟
⎠
2
6
3.368 ――
kN
m
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Laje 06 (3 engastes)
≔lx 1 ≔ly 3.29 Menor 0,50 ou maior que 2 biapoiada 
=―
ly
lx
3.29
momento fletor positivo e negativo em X
≔Mx =――
⋅q lx2
8
0.688
Reações de apoio em X 
≔Rx =――
⋅q lx
2
2.75 ――
kN
m
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Laje 07 (2 engastes) - Tabela 4
≔lx 3.29 ≔ly 3.92
=―
ly
lx
1.191 ≔kx 0.912 ≔mx 31.93 ≔nx 13.16 ≔my 70.6
momento fletor positivo e negativo em X
≔Mx =――
⋅q lx2
mx
1.864 ≔Xx =―――
⋅-q lx2
nx
-4.524
momento fletor positivo e negativo em y
≔My =――
⋅q lx2
my
0.843
Reações de apoio em X e Y
≔Ry =――
⋅q lx
3
6.032 ――
kN
m
≔Rx =⋅⋅q lx ――――
-3
⎛
⎜
⎝
―
lx
ly
⎞
⎟
⎠
2
6
6.923 ――
kN
m
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Equilibrio de momentos
L2 e L3
� Equilibrio de momentos negativo é o maior valor entre a média e 80% do 
maior momento
≔Mmedia =――――
+2.93 6.76
2
4.845
≔M80 =⋅6.76 0.8 5.408
Ajuste do momento positivo apenas no vão cujo o momento negativo foi reduzido
≔Majust =+2.93 ―――――
(( -6.76 5.41))
2
3.605
L2 e L4 / L5
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L2 e L4 / L5
� Equilibrio de momentos negativo é o maior valor entre a média e 80% do 
maior momento
≔Mmedia =―――
+0.70 5
2
2.85
≔M80 =⋅5 0.8 4
Ajuste do momento positivo apenas no vão cujo o momento negativo foi reduzido
≔Majust =+2.25 ―――
(( -5 4))
2
2.75
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Dimensionamento das lajes
≔Asmin =⋅⋅10 100 0.0015 1.5 cm² Adotando 6.3 mm 
≔n =――
1.5
0.31
4.839 =――
100
n
20.667 6,3 mm c/20
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� L2 e L3 (horizontal)
≔bw 100
≔d 7.5
≔Md1 =⋅131 1.4 183.4 ≔Md2 =⋅541 1.4 757.4 ≔Md3 =⋅361 1.4 505.4
≔Kc1 =―――
⋅bw d2
Md1
30.671 ≔Ks 0.023
≔As =⋅Ks ――
Md1
d
0.562 =Asmin 1.5
≔Kc2 =―――
⋅bw d2
Md2
7.427 ≔Ks 0.024
≔As =⋅Ks ――
Md2
d
2.424 ≔n =――
As
0.31
7.818 =――
100
n
12.79 6,3 mm c/13
≔Kc3 =―――
⋅bw d2
Md3
11.13 ≔Ks 0.024
≔As =⋅Ks ――
Md3
d
1.617 ≔n =――
As
0.31
5.217 =――
100
n
19.168 6,3 mm c/20
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� L2 e L4 / L5 (vertical)
≔bw 100
≔d 7.5
≔Md1 =⋅400 1.4 560 ≔Md2 =⋅275 1.4 385
≔Kc1 =―――
⋅bw d2
Md1
10.045 ≔Ks 0.024
≔As =⋅Ks ――
Md1
d
1.792 ≔n =――
As
0.31
5.781 =――
100
n
17.299 6,3 mm c/19
≔Kc2 =―――
⋅bw d2
Md2
14.61 ≔Ks 0.023
≔As =⋅Ks ――
Md2
d
1.181 =Asmin 1.5
� L3 (vertical)
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� L3 (vertical)
≔bw 100
≔d 7.5
≔Md1 =⋅570 1.4 798 ≔Md2 =⋅246 1.4 344.4
≔Kc1 =―――
⋅bw d2
Md1
7.049 ≔Ks 0.024
≔As =⋅Ks ――
Md1
d
2.554 ≔n =――
As
0.31
8.237 =――
100
n
12.14 6,3 mm c/12
≔Kc2 =―――
⋅bw d2
Md2
16.333 ≔Ks 0.023
≔As =⋅Ks ――
Md2
d
1.056 =Asmin 1.5
L6 Armadura minima
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L7
≔bw 100
≔d 7.5
≔Md1 =⋅452 1.4 632.8 ≔Md2 =⋅186 1.4 260.4
≔Kc1 =―――
⋅bw d2
Md1
8.889 ≔Ks 0.024
≔As =⋅Ks ――
Md1
d
2.025 ≔n =――
As
0.31
6.532 =――
100
n
15.309 6,3 mm c/17
≔Kc2 =―――
⋅bw d2
Md2
21.601 ≔Ks 0.023
≔As =⋅Ks ――
Md2
d
0.799 =Asmin 1.5
Levantamento de carga das vigas
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Levantamento de carga das vigas
≔Alv 4.5
≔PPV19 =⋅⋅0.19 0.40 25 1.9 ――
kN
m²
≔PPV14 =⋅⋅0.14 0.40 25 1.4 ――
kN
m²
� Dimensionamento das vigas
V1
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Vsk1 18.8≔Md1 =⋅1920 1.4 ⋅2.688 103
Trecho 1
≔Kc =―――
⋅bw d2
Md1
9.677 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14 cm²
≔As =―――
⋅Ks Md1
d
1.744 cm² ≔n =――
As
1.25
1.395 2x 12,5mm
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≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
é coeficiente de redução do concreto que leva αv2
em conta microfissuração da biela de compressão
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
VRd2 é o esforço cortante resistente de calculo - Ruptura da biela precisa ser maior 
que a cortante atuante na viga
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
fctd - Resistencia de calculo do concreto à tração - Valor tabelado para 
concreto de 30 Mpa é 1450 kN/m²
≔fctd 1450 ――
kN
m²
Vc - Parcela do esforço cortante resistido pelo proprio concreto
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
Vsw - Parcela do esforço cotante que precisa ser resistido por uma armadura 
≔Vsw1 =-⋅Vsk1 1.4 Vc -34.841
≔fyd =――
50
1.15
43.478 ――
kN
cm²
Asw - Área de aço transversal 
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
-2.406 cm²
≔Asw.min =⋅0.12 bw 2.28
V2
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V2
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅870 1.4 ⋅1.218 103 ≔Vsk1 18.8
P1 P2
Trecho 1
≔Kc =―――
⋅bw d2
Md1
21.356 ≔Ks 0.023
≔As =―――
⋅Ks Md1
d
0.757 cm² ≔Asmin =⋅⋅bw h 0.0015 1.14 cm²
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
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≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161
≔Vsw1 =-⋅Vsk1 1.4 Vc -34.841
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
-2.406 cm² ≔Asw.min =⋅0.12 bw 2.28
V4
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅1990 1.4 ⋅2.786 103 ≔Vsk1 25.3
≔Md2 =⋅3150 1.4 ⋅4.41 103 ≔Vsk3 40.7
≔Md3 =⋅1560 1.4 ⋅2.184 103 ≔Vsk3 20.8
P5 P3 V22/P14
Trecho 1
≔Kc =―――
⋅bw d2
Md1
9.336 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14 cm²
≔As =―――
⋅Ks Md1
d
1.807 cm² ≔n =――
As
1.25
1.446 2x 12,5mm
Trecho 2
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Trecho 2
≔Kc =―――
⋅bw d2
Md2
5.898 ≔Ks 0.024
≔As =―――
⋅Ks Md2
d
2.861 cm² 1x 12,5mm + 2x 10mm
Trecho 3
≔Kc =―――
⋅bw d2
Md3
11.91 ≔Ks 0.024
≔As =―――
⋅Ks Md3
d
1.417 cm² 2x 10mm
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
≔Vsw1 =-⋅Vsk3 1.4 Vc -32.041
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
-2.213 cm²
≔Asw.min =⋅0.12 bw 2.28
V6
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V6
≔bw 14
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅2280 1.4 ⋅3.192 103 ≔Vsk1 18.8
V22 V23
Trecho 1
≔Kc =―――
⋅bw d2
Md1
6.004 ≔Ks 0.023
≔As =―――
⋅Ks Md1
d
1.984 cm²
≔Asmin =⋅⋅bw h 0.0015 0.84 cm²
2x 10mm + 1x 8mm
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
263.736
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
45.066 kN
≔Vsw1 =-⋅Vsk1 1.4 Vc -18.746
≔fyd =――
50
1.15
43.478 ――
kN
cm²
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≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
-1.295 cm²
≔Asw.min =⋅0.12 bw 1.68
V7
≔bw 14
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅3760 1.4 ⋅5.264 103 ≔Vsk1 34.5
V19 V21
Trecho 1
≔Kc =―――
⋅bw d2
Md1
3.641 ≔Ks 0.025 ≔Asmin =⋅⋅bw h 0.0015 0.84 cm²
≔As =―――
⋅Ks Md1
d
3.557 cm² 3x 12.5mm 
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
263.736
――
kN
m²
≔fctd 1450
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≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
263.736
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
45.066 kN
≔Vsw1 =-⋅Vsk1 1.4 Vc 3.234
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
0.223 cm²
≔Asw.min =⋅0.12 bw 1.68
V8
≔bw 14
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅2120 1.4 ⋅2.968 103 ≔Vsk1 25.7
V22 V23
Trecho 1
≔Kc =―――
⋅bw d2
Md1
6.458 ≔Ks 0.025 ≔Asmin =⋅⋅bw h 0.0015 0.84 cm²
≔As =―――
⋅Ks Md1
d
2.005 cm² 2x 10mm + 1x 8mm
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≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
263.736
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
45.066 kN
≔Vsw1 =-⋅Vsk1 1.4 Vc -9.086
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
-0.628 cm²
≔Asw.min =⋅0.12 bw 1.68
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V10
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅2640 1.4 ⋅3.696 103 ≔Vsk1 26
≔Md2 =⋅3170 1.4 ⋅4.438 103 ≔Vsk2 38.8
≔Md3 =⋅1400 1.4 ⋅1.96 103 ≔Vsk3 19.1
P7 P8 P9
Trecho 1
≔Kc =―――
⋅bw d2
Md1
7.038 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md1
d
2.397 cm² 3x 10mm 
≔Kc =―――
⋅bw d2
Md2
5.861 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md2
d
2.879 cm² 2x 12.5mm + 1x 10mm
≔Kc =―――
⋅bw d2
Md3
13.271 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md3
d
1.271 cm² 2x 10mm 
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
≔As =―――
⋅Ks Md3
d
1.271 cm² 2x 10mm 
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
≔Vsw1 =-⋅Vsk2 1.4 Vc -6.841
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
-0.473 cm²
≔Asw.min =⋅0.12 bw 2.28
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
V14
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅2640 1.4 ⋅3.696 103 ≔Vsk1 26
V22 V23
Trecho 1
≔Kc =―――
⋅bw d2
Md1
7.038 ≔Ks 0.024
≔As =―――
⋅Ks Md1
d
2.397 cm²
≔Asmin =⋅⋅bw h 0.0015 1.14
3x 10mm
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
≔Vsw1 =-⋅Vsk1 1.4 Vc -24.761
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
-1.71 cm²
≔Asw.min =⋅0.12 bw 2.28
V19
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅4410 1.4 ⋅6.174 103 ≔Vsk1 32.7
≔Md2 =⋅1980 1.4 ⋅2.772 103 ≔Vsk2 52
≔Md3 =⋅2460 1.4 ⋅3.444 103
P5 P7 P13
Trecho 1
≔Kc =―――
⋅bw d2
Md1
4.213 ≔Ks 0.025 ≔Asmin =⋅⋅bw h 0.0015 1.14
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
≔Kc =―――
⋅bw d2
Md1
4.213
≔As =―――
⋅Ks Md1
d
4.172 cm² 2x 16mm 
≔Kc =―――
⋅bw d2
Md2
9.383 ≔Ks 0.024
≔As =―――
⋅Ks Md2
d
1.798 cm²
≔Asmin =⋅⋅bw h 0.0015 1.14
2x 10mm + 1x 6.3mm
≔Kc =―――
⋅bw d2
Md3
7.553 ≔Ks 0.024
≔As =―――
⋅Ks Md3
d
2.234 cm²
≔Asmin =⋅⋅bw h 0.0015 1.14
2x 12.5mm + 1x 10mm
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
≔Vsw1 =-⋅Vsk2 1.4 Vc 11.639
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
0.804 cm²
≔Asw.min =⋅0.096 bw 1.824 5 mm c/ 15mm
V21
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
V21
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅2460 1.4 ⋅3.444 103 ≔Vsk1 46.5
≔Md2 =⋅2870 1.4 ⋅4.018 103 ≔Vsk2 75
≔Md3 =⋅6480 1.4 ⋅9.072 103
P3 P8 P17
Trecho 1
≔Kc =―――
⋅bw d2
Md1
7.553 ≔Ks 0.025 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md1
d
2.327 cm² 2x 16mm 
≔Kc =―――
⋅bw d2
Md2
6.474 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md2
d
2.606 cm² 2x 10mm + 1x 6.3mm
≔Kc =―――
⋅bw d2
Md3
2.867 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md3
d
5.885 3x 16mm cm²
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
cm²≔As =―――
⋅Ks Md3
d
5.885 3x 16mm 
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
≔Vsw1 =-⋅Vsk2 1.4 Vc 43.839
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
3.028 cm²
≔Asw.min =⋅0.096 bw 1.824
5 mm c/ 12mm
V22
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
V22
≔bw 19
≔h 40 ≔fck 30
≔d =-40 3 37
≔Md1 =⋅6420 1.4 ⋅8.988 103 ≔Vsk1 51.3
≔Md2 =⋅8930 1.4 ⋅1.25 104 ≔Vsk2 88.6
≔Md3 =⋅3130 1.4 ⋅4.382 103
P1 P9 P14
Trecho 1
≔Kc =―――
⋅bw d2
Md1
2.894 ≔Ks 0.025 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md1
d
6.073 cm² 3x 16mm 
≔Kc =―――
⋅bw d2
Md2
2.081 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md2
d
8.109 cm² 4x 16mm + 1x 10
≔Kc =―――⋅bw d2
Md3
5.936 ≔Ks 0.024 ≔Asmin =⋅⋅bw h 0.0015 1.14
≔As =―――
⋅Ks Md3
d
2.842 cm² 2x 12.5mm + 1x 8mm
Created with PTC M
athcad Express. See www.m
athcad.com
 for m
ore inform
ation.
≔αv2 =
⎛
⎜
⎝
-1
⎛
⎜
⎝
――
fck
250
⎞
⎟
⎠
⎞
⎟
⎠
0.88
≔fcd =――――
⋅fck 1000
1.4
21428.571 ――
kN
m²
≔VRd2 =⋅⋅⋅⋅0.27 αv2 fcd ――
bw
100
――
d
100
357.927
≔fctd 1450 ――
kN
m²
≔Vc =⋅⋅⋅0.6 fctd ――
bw
100
――
d
100
61.161 kN
≔Vsw1 =-⋅Vsk2 1.4 Vc 62.879
≔fyd =――
50
1.15
43.478 ――
kN
cm²
≔Asw =―――――
Vsw1
⋅⋅0.9 ――
d
100
fyd
4.343 cm² 6,3mm c/ 14mm