Estruturas - Diagramas de Esforços

DIAGRAMA DAS FORÇAS CORTANTES E DO MOMENTO FLETOR

A

L = 8 m

B

q = 15kN/m


A

L = 8 m

B

F = 15 kN/m x 8m = 120 kN

120kN

4m


120kN

4mL = 8 m

BA

∑FR = 0

VA + VB = 120kN

∑MA = 0

(-120 . 4)+(VB . 8) = 0

VB = 60 kN

VA + VB = 120kN

VA + 60 = 120

VA = 60 kN

V VA B


120kN

8 m4 m

Distância = 60 kN / 15kN/m = 4 m

60 – 120 = -6060

4 mDQ


8 m

4 m

4 m

60

120kN

60

DQ


60

MMÁX = q . L2

8

MMÁX = 15 . 82

8

MMÁX = 120kNm

DQ

DMF

4 m

120kN

4 m

8 m

60

MMÁX

60


120kN

4 m

60

4 m

60

8 m

MMÁX = 120kNm

DQ

DMF


DQ

DMF

120kN

4 m

4 m

60

60

8 m

MMÁX = 120kNm


50kN

4 m

A B


200kNm50kN

4 m

A B

∑FR = 0

50 - VA = 0

MA = (-50 . 4)

VA = 50kN

VA = 50kN

MA = 200kNm


200kNm50kN

4 m

A B

VA = 50kN


200kNm50kN

4 m

A B

50 50

50-50=0

0DQ


200kNm 50kN10kN

4 m

DQ

A B

DMF

50

-200

50

0

0


50kN

4 m

A B

q = 30kN/m

FR = 30kN/m . 4m

FR = 120kN

FR = 120kN


50kN

4 m

A B

120kN

2 m

∑FV = 0

120 + 50 - VA = 0

VA = 170kN

MA = (-120.2)+(-50.4)

VA = 170kN

MA = 440kNm

440 kNm


50kN

4 m

A B

120kN

2 m

170

120kN + 50kN = 170kN

DQ

440kNm


50kN

4 m

A B

120kN

2 m

170 170kN-120kN = 50kN

50

DQ

440kNm


50kN

4 m

A B

120kN

2 m

170 50kN-50kN = 0

50

DQ

440kNm

0


50kN

4 mA B

120kN

2 m

17050

0

0

DQ

DMF

440kNm

-440


50kN

4 mA

B

120kN

2 m

170 50

0

0

DQ

DMF

440kNm

-440


30kN 50kN 10kN

3 m 3 m2 m2 m

A B

∑MA = 0

(-30.2)+(-50.5)+(-10.7)+(VB.10)= 0

VB = 38kN

VBVA

∑FV = 0 => -VA + 30 + 50 + 10 - VB = 0

VA + VB = 90kN

VA + 38 = 90

VA = 52kN


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m

52

DQ

VA = 52 VB = 38


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m

52

DQ

22

52-30

VA = 52 VB = 38


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m

52

DQ

22

52-3022-50

-28

VA = 52 VB = 38


30kN 50kN 10kN

2 m

A

3 m 2 m

52

DQ

22

52-3022-50

-28-28-10

-38

3 m

B

VA = 52 VB = 38


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m

52

DQ

22

52-3022-50

-28-28-10

-38

-38+38

0

VA = 52 VB = 38


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m

DMF

S1 S2S3

VA = 52 VB = 38


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m

DMF

S1 S2S3

S1 Esq.

MS1 = (-52 . 2)

MS1 = 104 kNm

104

VA = 52 VB = 38

2 m


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m

DMF

S1

104

S2S3

S2 Esq.

MS2 = (-52 . 5) + (30 . 3 )

MS2 = 170 kNm

170

VA = 52 VB = 38

30kN

5 m


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 mVB = 38VA = 52

DMF

S3 Esq.

MS3 =(-52 . 7)+(30 . 5)+(50 . 2)

MS3 = 114 kNm

S1

104

S2S3

17030kN 50kN

7 m

114


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 mVB = 38VA = 52

DMF

MVB = VB . 3 = 38 . 3

S1

104

S2S3

170

114

VB

3 m

-114+114=0

0

MVB = 114 kNm


30kN 50kN 10kN

B

3 m2 m

A

3 m 2 m VB = 38VA = 52

DMF

S1

104

S2 S3

170114

DQ

0

5222

-28

-38

0


A B

q1 = 10 . 1

10kN

q2 = 10 . 7

70kN

q2 = 10 . 7

20kN20kN


A B

∑MA = 0

(20.1)+(10.0,5)+(-70.3,5)+(VB.7)(-20.8)= 0

20 + 5 - 245 - 160 + 7VB = 0

VB = 380/7 => VB = 54,28kN

VA = 65,72kN

∑FV = 0

VA + VB -20 -10 -70 -20 = 0

VA + VB = 120kN

VB = 54,28kN

10kN 70kN 20kN

VA + VB = 120kN

VA + 54,28 = 120kN

VA = 65,72kN

20kN


A B

10kN 70kN 20kN20kN

DQ

-20

Iniciou pela esquerda

as forças para cima são

positivas

65,72kN 54,28kN

-20-10=-30

-30


A B

10kN 70kN 20kN20kN

DQ

-20

-30

-30+65,72=35,72

35,72

65,72kN 54,28kN


A B

10kN 70kN 20kN20kN

DQ

-20

-30

d= FV / q

d = 35,72 / 10

d= 3,572m

3,57m

35,72

65,72kN 54,28kN


A B

10kN 70kN 20kN20kN

DQ

-20

-30 35,72 –70 = –34,28

-34,28

3,57m

35,72

65,72kN 54,28kN


A B

10kN 70kN 20kN20kN

DQ

-20

-30-34,28 +54,28 = 20

20

-34,28

3,57m

35,72

65,72kN 54,28kN


A B

10kN 70kN 20kN20kN

DQ

-20

-30

20 – 20 = 0

0

20

-34,28

3,57m

35,7265,72kN 54,28kN


A B

10kN 70kN 20kN20kN

DQ

-20-30

0

20

A1= (20.1)+(10.1/2)

A1= 25A2= (20 . 2 / 2)

A2= 20

3,57m

-34,28

35,72

65,72kN 54,28kN

Calcular a área:

Calcular a área:


A B

10kN 70kN 20kN20kN

DMF

25 20

65,72 54,28

Sabe-se que os pontos são para cima por se tratar de viga em balanço e sua ferragem ser negativa


A B

10kN 70kN 20kN20kN

DMF

25 20

65,72 54,28


A B

10kN 70kN 20kN20kN

DQ

-20-30

0

20 A2= (20 . 2 / 2)

A2= 203,57m

-34,28

35,72

65,72 54,28

A1= (20.1)+(10.1/2)

A1= 25

A3= (35,72 . 3 / 2)

A3= 63,76

MMAX= 63,76 - 25

MMAX= 38,76kNm

A4= (34,28 . 3,428 / 2)

A4= 58,76

MMAX= 58,76 - 20

MMAX= 38,76kNmPode calcular por qualquer

lado, o resultado é o mesmo.


A B

10kN 70kN 20kN20kN

DMF

25 20

MMAX = 38,76

65,72 54,28


A B

10kN 70kN 20kN20kN

DMF25 20

MMAX = 38,76

65,72 54,28

0

0DQ

-30

203,57m

-34,28

35,72

-20

Estruturas - Diagramas de Esforços

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