04. Doble Integracion

RESISTENCIA DE MATERIALES II (CIV -203)  -203) 

1. M ÉTODO ÉTODO DOBLE INTEGRACION 

a. Reacciones en la viga De la estática.

3 ⋅ M o

∑ M  A ＝ 0

 R B L + 2 ⋅ M o + M o ＝ 0

 R B ≔ -

∑ F V ＝ 0

 R A - R B ＝ 0

3 ⋅ M o  R A ≔ ―― L

L

b. Ecuaciones de Momeno !leco" #a"a cada "amo TRAMO AD x ≤

Corte I-I

⎛ 3 ⋅ M o ⎞ ⎟x ⎝ L ⎠

L

3

 M  AD ≔ ⎜

Corte IIII-II

L

TRAMO DE

⎛ 3 ⋅ M o ⎞ ⎟ x - M o ⎝ L ⎠

3

2 L 3


 M  DE ≔ ⎜

Corte III-II III-IIII TRAMO TRAMO EB

⎛ 3 ⋅ M o ⎞ ⎟ x - 3  M o ⎝ L ⎠

 M  EB ≔ ⎜

c. M $odo $odo Doble Ineg"aci%n

Corte I-I

TRAMO AD x ≤

L

 E ⋅ I ⋅

3

d2 dx

2

2

y ＝ M  AD

⎛ 3 ⋅ M o ⎞ ⎟x ⎝ L ⎠ 1 ⎛ 3 ⋅ M o x ＝ ⎜ ⋅  EI  ⎝ L 1 ⎛ 3 ⋅ M o x ＝ ⎜ ⋅  EI  ⎝ L

Ecuación de Momento Flector :

 M  AD x ≔ ⎜

Ecuación de Deformacion Anular:

θ AD

Ecuacion de Defle!ion:

y AD

Corte IIII-II

2 TRAMO D E
 E ⋅ I ⋅

d2 dx

⎞ + C1⎟ 2 ⎠

x2

⎞ + C1 ⋅ x + C2⎟ 6 ⎠

x3

2

2

y ＝ M  DE

⎛ 3 ⋅ M o ⎞ ⎟ x - M o ⎝ L ⎠ ⎞ 1 ⎛ 3 ⋅ M o x 2 x ＝ ⎜ ⋅ - M o ⋅ x + C3⎟ 2  EI  ⎝ L ⎠

Ecuación de Momento Flector :

 M  DE x ≔ ⎜

Ecuación de Deformacion Anular:

θ DE

Ecuacion de Defle!ion:

y DE x ＝

2 Corte III-III III-III TRAMO TRAMO EB L
 E ⋅ I ⋅

Ecuación de Momento Flector :

 M  EB x ≔

Ecuación de Deformacion Anular:

θ EB x ＝

Ecuacion de Defle!ion:

y EB

⎞ x2 1 ⎛ 3 ⋅ M o x 3 ⎜ ⋅ - M o ⋅ + C3 ⋅ x + C4⎟ 6 2  EI  ⎝ L ⎠

d2 d x2

2

y ＝ M  EB

⎛ 3 ⋅ M o ⎞ ⎝

L

⎠

2 L
x - 3  M o

⎞ 1 ⎛ 3 ⋅ M o x 2 ⎜ ⋅ - 3  M o ⋅ x + C5⎟ 2  EI  ⎝ L ⎠ 3 2 ⎞ x 1 ⎛ 3 ⋅ M o x ⎜ ⋅ - 3  M o ⋅ + C5 ⋅ x + C6⎟ x ＝ 6 2  EI  ⎝ L ⎠

RESISTENCIA DE MATERIALES II (CIV -203) 

"or Condiciones de A#o$o x≔0

y A ≔ 0

⎛ 3 ⋅ M o x 3 ⎞ 0＝⎜ ⋅ + C1 ⋅ x + C2⎟ → 0 ＝ C2 6 ⎝ L ⎠ x≔

C2 ≔ 0

y B ≔ 0

2 ⎛ 3 ⋅ M o x 3 ⎞ solve , C6 x 0＝⎜ ⋅ - 3  M o ⋅ + C5 ⋅ x + C6⎟ ―― →  M o ⋅ L 2 - C5 ⋅ L 6 2 ⎝ L ⎠

2

C6 ≔  M o ⋅ L - C5 ⋅ L

"or Continuidad x≔

θ I ＝ θ II 

3

y I ＝ y II 

⎛ 3 ⋅ M o x 2 ⎞ ⎛ 3 ⋅ M o x 2 ⎞ solve , C3  M o ⋅ L ⎜―― ⋅ ― + C1⎟ ＝ ⎜―― ⋅ ― - M o ⋅ x + C3⎟ ―― → C1 + ―― 2 2 3 ⎝ L ⎠ ⎝ L ⎠

C3 ≔ C1 +

2 ⎛ 3 ⋅ M o x 3 ⎞ ⎛ 3 ⋅ M o x 3 ⎞ solve , C4  M o ⋅ L 2 x ⎜―― ⋅ ― + C1 ⋅ x + C2⎟ ＝ ⎜―― ⋅ ― - M o ⋅ ― + C3 ⋅ x + C4⎟ ―― → -―― 6 6 2 18 ⎝ L ⎠ ⎝ L ⎠

C4 ≔ -

x≔

2 3

θ II ＝ θ III 

 M o ⋅ L

3

 M o ⋅ L 2

18

y II ＝ y III 

⎛ 3 ⋅ M o x 2 ⎞ ⎛ 3 ⋅ M o x 2 ⎞ solve , C1 5 ⋅ M o ⋅ L ⎜―― ⋅ ― -  M o ⋅ x + C3⎟ ＝ ⎜―― ⋅ ― - 3  M o ⋅ x + C5⎟ ―― → C5 - ――― 2 2 3 ⎝ L ⎠ ⎝ L ⎠

C1 ≔ C5 -

5 ⋅ M o ⋅ L 3

RE%O&'IE(DO E& %I%TEMA TE(EMO%: C1 ≔ -

 M o ⋅ L

C2 ≔ 0

6

C3 ≔

 M o ⋅ L

C4 ≔ -

6

 M o ⋅ L 2

18

C5 ≔

3 ⋅ M o ⋅ L 2

C6 ≔ -

 M o ⋅ L 2

2

Reemplazando tenemos las ecuaciones finales de cada tramo:

Corte I-I

TRAMO AD x ≤

L

3

Ecuación de Deformacion Anular:

1 ⎛ 3 ⋅ M o x 2  M o ⋅ L ⎞ θ AD x ＝ ― ―― ⋅ ― - ―― 2 6 ⎠  EI  ⎝ L

Ecuacion de Defle!ion:

y AD x ＝

Corte II-II

TRAMO D E

L

3


2 L 3

 M o ⋅ L ⎞ 1 ⎛ 3 ⋅ M o x 2 ⎜ ⋅ - M o ⋅ x + ⎟  EI  ⎝ L 2 6 ⎠ 2  M o ⋅ L  M o ⋅ L 2 ⎞ 1 ⎛ 3 ⋅ M o x 3 x ⎜ ⎟ x ＝ ⋅ - M o ⋅ + ⋅x6 2 6 18 ⎠  EI  ⎝ L

Ecuación de Deformacion Anular:

θ DE x ＝

Ecuacion de Defle!ion:

y DE

Corte III-III TRAMO EB

2 L
1 ⎛ 3 ⋅ M o x 3  M o ⋅ L ⎞ ⎜ ⋅ ⋅ x⎟ 6 6  EI  ⎝ L ⎠

3 ⋅ M o ⋅ L ⎞ 1 ⎛ 3 ⋅ M o x 2 ⎜ ⋅ - 3  M o ⋅ x + ⎟ 2 2  EI  ⎝ L ⎠ 2 3 2 ⎛ 3 ⋅ M o ⋅ L  M o ⋅ L ⎞ x 1 3 ⋅ M o x ⎜ ⎟ ⋅ - 3  M o ⋅ + ⋅xx ＝  EI  ⎝ L 6 2 2 2 ⎠

Ecuación de Deformacion Anular:

θ EB x ＝

Ecuacion de Defle!ion:

y EB

RESISTENCIA DE MATERIALES II (CIV -203) 

%O&)CIO(: x≔0 θ A x ＝

x≔

y D

1 ⎛ 3 ⋅ M o x 2  M o ⋅ L ⎞ ⎜ ⋅ ⎟  EI  ⎝ L 2 6 ⎠

 M o ⋅ L θ A 0 ＝ -―― 6 ⋅ EI 

θ ＝ θ B

θ B x ＝

x≔

θ ＝ θ A

3

3 ⋅ M o ⋅ L ⎞ 1 ⎛ 3 ⋅ M o x 2 ⎜ ⋅ - 3  M o ⋅ x + ⎟ 2 2  EI  ⎝ L ⎠

θ B L ＝ 0

y ＝ y D

1 ⎛ 3 ⋅ M o x 3  M o ⋅ L ⎞ ⎜ ⋅ ⋅ x⎟ x ＝  EI  ⎝ L 6 6 ⎠

y D

⎛L⎞ ⎝3⎠

 M o ⋅ L

2

＝ -―― 27 ⋅ EI