CLCULO DAS REAES DE APOIO EM UMA VIGA

BIENGASTADA CARGA POLINOMIAL

L

E I

Figura 1 - Viga biengastada submetida carga polinomial

q(x) = ai xi

n

i=0

ai

N2 V2 M2

Figura 2 - Sistema Principal

y(x)

(x)

(x)

Figura 3 Deformaes no sistema principal

y2(L) = 0

2(L) = 0

2(L) = 0

V2

N2

M2;

q(x);

ij i j

Fx = 0

Fy = 0

Mp = 0

d(x)

dx=

M(x)

EI

d2y(x)

dx2=

M(x)

EI

d(x)

dx=

p(x)

EA

EI(x)= M(x)dx

EIy(x) = (x)dx

EA(x) = p(x)dx

1j(0) = 0

y1j(0) = 0

1j(0) = 0

L

Figura 4 - Carregamento V2 e respectiva deformada

Fy = 0 V11 + V2 = 0 V11 = V2

Fx = 0 N11 = 0

Mp = 0 M11 + V2L = 0 M11 = V2L

Ms = 0 M11 V11x + M(x) = 0 M(x) = V11x M11

M(x) = V2x + V2L

M(x) = V2(L x)

EIi1(x) = M(x)dx

EIi1(x) = V2(L x)dx

EIi1(x) = V2 (xL 1

2x2) + 1

EIyi1(x) = i1(x)dx

EIyi1(x) = V2 (xL 1

2x2) + 1dx

EIyi1(x) = V2 (1

2Lx2

1

6x3) + 1x + 2

[V2 (Lx 1

2x2) + 1]

0= 0

V2 (L 0 1

202) + 1 = 0

=

[V2 (1

2Lx2

1

6x3) + 2]

0= 0

V2 (1

2L 02

1

603) + 2 = 0

=

EIi1(x) = V2 (xL 1

2x2) EIyi1(x) = V2 (

1

2Lx2

1

6x3)

EI21 = [V2 (Lx 1

2x2)]

L

EI21 = V2 (L L 1

2L2)

EI21 =1

2V2L

2

EIy21 = [V2 (1

2Lx2

1

6x3)]

L

EIy21 = V2 (1

2L L2

1

6L3)

EIy21 =1

3V2L

3

Mp = 0 M12 + M2 = 0 M12 = M2

Ms = 0 M12 + M(x) = 0 M(x) = M12

M(x) = M2

EIi2(x) = M(x)dx

EIi2(x) = M2dx

EIi2(x) = M2x + 1

EIyi2(x) = i2(x)dx

EIyi2(x) = M2x + 1dx

EIyi2(x) =1

2M2x

2 + 1x + 2

[M2x + 1]0 = 0

M2 0 + 1 = 0

=

[1

2M2x

2 + 2]0

= 0

1

2M2 0

2 + 2 = 0

=

EIi2(x) = M2x EIyi2(x) =1

2M2x

2

EI22 = [M2x]L

EI22 = M2L

EIy22 = [1

2M2x

2]L

EIy22 =1

2M2L

2

Figura 6 Carregamento N2 e respectiva deformada

Fx = 0 N13 + N2 = 0 N13 = N2

EAi3(x) = p(x)dx

EAi3(x) = N2dx

EAi3(x) = N2x + 1

[N2x + 1 = 0

N2 0 + 1 = 0

=

EAi3 = N2x

EA23 = [N2x]L

EI23 = N2L

Figura 7 - Carregamento q(x) e respectiva deformada

Fy = 0 V14 q(x)dx

L

x=0

= 0 V14 = q(x)dxL

0

V14 = aixi

n

i=0

dxL

0

= ai xi

L

0

dx

n

i=0

V14 = [ai

i + 1xi+1

n

i=0

]

0

L

= ai

i + 1(Li+1 0i+1)

n

i=0

=

+ +

=

Mp = 0 M14 q(x)xdx

L

x=0

= 0 M14 = q(x)xdxL

0

M14 = x aixidx

n

i=0

= aixi+1dx

n

i=0

L

0

L

0

= [ai

i + 2 xi+2

n

i=0

]

0

L

M14 = ai

i + 2(Li+2 0i+2)

n

i=0

=

+ +

=

Mxs = 0 M(xs) + M14 V14xs + q(x)(xs x)dx

x

x=0

= 0

M(xs) = M14 + V14xs q(x)(xs x)dxxs

0

=

ai

i + 2Li+2

n

i=0

+ xs ai

i + 1Li+1

n

i=0

(xs x) aixi

n

i=0

dxxs

0

M(xs) = (ai

i + 2Li+2 +

a1i + 1

Li+1xs ai(xixs x

i+1)dxxs

0)

n

i=0

M(xs) = (ai

i + 2Li+2 +

aii + 1

Li+1xs (ai

i + 1xsx

i+1 ai

i + 2xi+2)

0

xs)

n

i=0

M(xs) = (ai

i + 2Li+2 +

aii + 1

Li+1xs aii!

(i + 2)!xs

i+2)

n

i=0

EIi4(x) = M(x)dx = ai [1

i + 2Li+2 +

1

i + 1Li+1x

i!

(i + 2)!xi+2]

n

i=0

dx

EIi4(x) = ai [1

i + 2Li+2 +

1

i + 1Li+1x

i!

(i + 2)!xi+2] dx

n

i=0

() = [

+ + +

( + )+

!

( + )!+]

=

+

EIyi4(x) = i4(x)dx

EIyi4(x) = ai [1

i + 2Li+2x +

1

2(i + 1)Li+1x2

i!

(i + 3)!xi+3]

n

i=0

+ 1dx

() = [

( + )+ +

( + )+

!

( + )!+] + +

=

[ ai [1

i + 2Li+2x +

1

2(i + 1)Li+1x2

i!

(i + 3)!xi+3]

n

i=0

+ 1]

0

= 0

ai [1

i + 2Li+2 0 +

1

2(i + 1)Li+1 02

i!

(i + 3)! 0i+3]

n

i=0

+ 1 = 0

=

[ ai [1

2(i + 2)Li+2x2 +

1

6(i + 1)Li+1x3

i!

(i + 4)!xi+4] + 2

n

i=0

]

0

= 0

ai [1

2(i + 2)Li+2 02 +

1

6(i + 1)Li+1 03

i!

(i + 4)! 0i+4] + 2 = 0

n

i=0

=

EIi4(x) = ai [1

i + 2Li+2x +

1

2(i + 1)Li+1x2

i!

(i + 3)!xi+3]

n

i=0

EIyi4(x) = ai [1

2(i + 2)Li+2x2 +

1

6(i + 1)Li+1x3

i!

(i + 4)!xi+4]

n

i=0

EI24 = [ ai [1

i + 2Li+2x +

1

2(i + 1)Li+1x2

i!

(i + 3)!xi+3]

n

i=0

]

L

EI24 = ai [1

i + 2Li+2 L +

1

2(i + 1)Li+1 L2

i!

(i + 3)!Li+3]

n

i=0

EI24 = ai [1

i + 2Li+3 +

1

2(i + 1)Li+3

i!

(i + 3)!Li+3]

n

i=0

EI24 = aiLi+3 [

1

i + 2+

1

2(i + 1)

i!

(i + 3)!]

n

i=0

EI24 = aiL

i+3

2

n

i=0

1

i + 3

EIy24 = [ ai [1

2(i + 2)Li+2x2 +

1

6(i + 1)Li+1x3

i!

(i + 4)!xi+4]

n

i=0

]

L

EIy24 = ai [1

2(i + 2)Li+2L2 +

1

6(i + 1)Li+1L3

i!

(i + 4)!Li+4]

n

i=0

EIy24 = ai [1

2(i + 2)Li+4 +

1

6(i + 1)Li+4

i!

(i + 4)!Li+4]

n

i=0

EIy24 = aiLi+4 [

1

2(i + 2)+

1

6(i + 1)

i!

(i + 4)!]

n

i=0

=

+

( + )!

( + )! ( + )

=

2 = y2 = 2 =

0

2j = y2j = 2j = 0

2j = N2L = 0; N2 = 0

y2j = 2

3V2L

3 +1

2M2L

2 aiL

i+4

6

(i + 2)!

(i + 4)! (2i + 9)

n

i=0

2j =1

2V2L

2 + M2L aiL

i+3

2

n

i=0

1

i + 3

1

3V2L

3 +1

2M2L

2 = A

1

2V2L

2 + M2L = B

(2

3

1

2) V2L

3 + (1 1)M2L2 = BL 2A

1

6V2L

3 = BL 2A = aiL

i+4

2

n

i=0

1

i + 3+

aiLi+4

3

(i + 2)!

(i + 4)! (2i + 9)

n

i=0

1

6V2L

3 =1

6 aiL

i+4(i + 2)!

(i + 4)!(i + 6)

n

i=0

V2 = aiLi+1

(i + 2)!

(i + 4)!(i + 6)

n

i=0

V2 M2

M2 = 1

LB

1

2V2L

M2 = aiL

i+2

2

n

i=0

1

i + 3

1

2 aiL

i+2(i + 2)!

(i + 4)!(i + 6)

n

i=0

M2 = a1Li+2

(i + 2)!

(i + 4)!

n

i=0

N2, V2 M2

N1 = N1i

V1 = V1i

M1 = M1i

N1 = N13 = 0

V1 = V2 q(x)dxL

0

V1 = aiLi+1

(i + 2)!

(i + 4)!(i + 6)

n

i=0

a1Li+1

1

(i + 1)

n

i=0

V1 = aiLi+1 [

(i + 2)!

(i + 4)!(i + 6)

1

(i + 1)]

n

i=0

V1 = 6 aiLi+1

n

i=0

i!

(i + 4)!(i + 2)

M1 = M14 V2L M2

M1 = q(x)xdxL

0

V2L M2

M1 = aiL

i+2

i + 2

n

i=0

aiLi+2

(i + 2)!

(i + 4)!(i + 6)

n

i=0

+ a1Li+2

(i + 2)!

(i + 4)!

n

i=0

M1 = aiLi+2 [

1

i + 2

(i + 2)!

(i + 4)!(i + 6) +

(i + 2)!

(i + 4)!]

n

i=0

M1 = 2 aiLi+2

n

i=0

(i + 2)!

(i + 4)!

N1 = 0 N2 = 0

V1 = 6 aiLi+1

n

i=0

i!

(i + 4)!(i + 2) V2 = aiL

i+1(i + 2)!

(i + 4)!(i + 6)

n

i=0

M1 = 2 aiLi+2

n

i=0

(i + 2)!

(i + 4)!M2 = a1L

i+2(i + 2)!

(i + 4)!

n

i=0