01. Ejemplo 1 (Doble Integracion)

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

1. M ÉTODO ÉTODO DOBLE INTEGRACION De la estática. ∑ M B ＝ 0

RC L - M 2 + M 1 ＝ 0

M 2 - M 1 RC ≔ ―― L

∑ F V ＝ 0

R B + RC ＝ 0

R B ≔ -⎜

⎛ M 2 - M 1 ⎞ ⎟ L ⎝ ⎠

b. Ecuaciones de Momento Flecto !aa cada tamo Corte I-I

TRAMO AB x ≤ L

M AB ≔ - M 1

Cort Cortee II-I II-III

TRAM TRAMO O BC L < x ≤ 2 L

Corte Corte III-II III-IIII TRAMO TRAMO CD 2 L < x ≤ 3 L

⎛ M 2 - M 1 ⎞ ⎟⋅ x-L L ⎝ ⎠

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

M BC ≔ - M 1 - ⎜

M CD ≔ - M 1 - ⎜

c. M "todo "todo Doble Inte#aci$n Corte I-I

TRAMO TRAMO AB x ≤ L

E ⋅ I ⋅

d2 dx

2

2

y ＝ M AB

Ecuación de Momento Flector :

M AB x ≔ - M 1

Ecuación de Deformacion Anular:

θ AB x ＝

Ecuacion de Defle!ion:

⎞ x 1 ⎛ y AB x ＝ ― ⎜- M 1 ⋅ ― + C1 ⋅ x + C2⎟ EI ⎝ 2 ⎠

Cort Cortee II-I II-III TRAM TRAMO O BC L < x ≤ 2 L

E ⋅ I ⋅


M BC x ≔ - M 1 - ⎜

1 EI

- M 1 ⋅ x + C1 2

d2 dx

2

2

y ＝ M BC

⎛ M 2 - M 1 ⎞ ⎟⋅ x-L L ⎝ ⎠

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


θ BC


y BC

Corte III-III III-III TRAMO TRAMO CD 2 L < x ≤ 3 L

E ⋅ I ⋅


M CD x ≔ - M 1 - ⎜


θCD x ＝


yCD x ＝

d2

2

d x2

y ＝ M CD

1 EI

1 EI

⎛ M 2 - M 1 ⎞ ⎛ M 2 - M 1 ⎞ ⎟ x-L +⎜ ⎟ ⋅ x - 2 L L L ⎝ ⎠ ⎝ ⎠ 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜- M ⋅ x - ⎜ M 2 - M 1 ⎟ x - L + ⎜ M 2 - M 1 ⎟ ⋅ x - 2 L + C ⎟ 1 5 L L 2 2 ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ 3 3 ⎛ ⎞ 2 ⎛ ⎞ ⎛ ⎞ ⎜- M ⋅ x - ⎜ M 2 - M 1 ⎟ x - L + ⎜ M 2 - M 1 ⎟ ⋅ x - 2 L + C ⋅ x + C ⎟ 1 5 6 L L 2 ⎝ 6 6 ⎝ ⎠ ⎝ ⎠ ⎠

RESISTENCIA DE MATERIALES II (CIV -203)

EI ≔ 2500000

"or condicion del #ro$lema:

L ≔ 2.5

"or Condiciones de A#o%o x≔0

0.018 ＝ x ≔ 2.5

0＝

y A ≔ 0.018 2 ⎞ solve , C2 x 1 ⎛ + C1 ⋅ x + C2⎟ ―― → 45000.0 ⎜- M 1 ⋅ EI ⎝ 2 ⎠

C2 ≔ 45000.0

y B ≔ 0

⎞ solve , C1 x2 1 ⎛ + C1 ⋅ x + C2⎟ ―― → 1.25 ⋅ M 1 - 18000.0 ⎜- M 1 ⋅ EI ⎝ 2 ⎠

C1 ≔ 1.25 ⋅ M 1 18000.0

"or Continuidad x ≔ 2.5

θ I ＝ θ II

y I ＝ y II

⎛ ⎛ M 2 - M 1 ⎞ x - L 1 ⎜ - M 1 ⋅ x + C1 ＝ - M 1 ⋅ x - ⎜ ⎟⋅ EI EI ⎝ L 2 ⎝ ⎠ 1

2 ⎞ 1 x 1 ⎛ - M 1 ⋅ + C1 ⋅ x + C2 ＝ EI ⎝ 2 ⎠ EI

x≔5

2

⎞ solve , C 3 + C3⎟ ―― → 1.25 ⋅ M 1 - 18000.0 ⎠

⎛ 2 ⎜- M ⋅ x - ⎛ M 2 - M 1 ⎞ ⋅ x - L 1 L 2 ⎝ 6 ⎝ ⎠

θ II ＝ θ III

3

C3 ≔ 1.25 ⋅ M 1 18000.0

⎞ solve , C 4 + C3 ⋅ x + C4⎟ ―― → 45000.0 ⎠

C4 ≔ 45000.0

y II ＝ y III

⎛ ⎛ M 2 - M 1 ⎞ x - L 1 ― - M 1 ⋅ x - ―― ⋅ ―― EI ⎝ L 2 ⎝ ⎠

2

⎞

⎛ ⎛ M 2 - M 1 ⎞ x - L 1 + C3 ＝ ― - M 1 ⋅ x - ―― ―― L 2 ⎠ EI ⎝ ⎝ ⎠

2

2 ⎞ solve , C ⎛ M 2 - M 1 ⎞ x - 2 L 5 + ―― ⋅ ――― + C5 ―― → 1.25 ⋅ M 1 - 18000.0 L 2 ⎝ ⎠ ⎠

C5 ≔ 1.25 ⋅ M 1 18000.0 ⎛ ⎛ M 2 - M 1 ⎞ x - L 1 ⎜ x2 ⎟⋅ - M 1 ⋅ -⎜ EI ⎝ 2 L 6 ⎝ ⎠

3

⎞ 1 + C3 ⋅ x + C4⎟ ＝ ⎠ EI

⎛ 2 ⎛ ⎞ ⎜- M ⋅ x - ⎜ M 2 - M 1 ⎟ x - L 1 2 L 6 ⎝ ⎝ ⎠

3

⎛ M 2 - M 1 ⎞ x - 2 L ⎟⋅ +⎜ L 6 ⎝ ⎠

3

⎞ solve , C 6 + C5 ⋅ x + C6⎟ ―― → 45000.0 ⎠

C6 ≔ 45000 x ≔ 7.5

y D ≔ 0.012

⎛ ⎛ M 2 - M 1 ⎞ x - L x2 1 ⎜ 0.012 ＝ - M 1 ⋅ -⎜ ⎟ EI ⎝ L 2 ⎝ 6 ⎠

x≔5

yc ≔ 0

⎛ 2 ⎛ M 2 - M 1 ⎞ x - L x 1 ⎜ 0＝ - M 1 ⋅ ⋅ EI ⎝ L 2 ⎝ 6 ⎠

3

⎛ M 2 - M 1 ⎞ x - 2 L +⎜ ⎟⋅ L 6 ⎝ ⎠

3

⎞ float , 3 + C5 ⋅ x + C6⎟ ―― → 0.012 ＝ -0.00000458 ⋅ M 1 + -0.00000292 ⋅ M 2 - 0.036 ⎠

0.048 ＝ -0.00000458 ⋅ M 1 - 0.00000292 ⋅ M 2 3

⎞ float , 3 + C3 ⋅ x + C4⎟ ―― → 0 ＝ -0.00000208 ⋅ M 1 + - 4.17 ⋅ 10 -7 ⋅ M 2 - 0.018 ⎠

-0.00000208 ⋅ M 1 - 4.17 ⋅ 10-7 ⋅ M 2 ＝ 0.018

RESISTENCIA DE MATERIALES II (CIV -203)

RE&O'(IE)DO E' &I&TEMA TE)EMO&: -1

⎡ -0.00000458 -0.00000292 ⎤ ⎡ 0.048 ⎤ ⎡ -7816.0499935154452487 ⎤ ⋅ → ⎣ -0.00000208 - 4.17 ⋅ 10-7 ⎦ ⎣ 0.018 ⎦ ⎣ -4178.9352841435824523 ⎦

M 1 ≔ -7816.050 M 2 ≔ -4178.935

01. Ejemplo 1 (Doble Integracion)

Recommend Documents