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RESISTENCIA DE MATERIALES II (CIV-203)

1. MÉTODO DOBLE INTEGRACION De la estática. ∑ MB ＝ 0

RC ((L)) - M2 + M1 ＝ 0

M2 - M1 RC ≔ ――― L

∑ FV ＝ 0

RB + RC ＝ 0

⎛ M2 - M1 ⎞ RB ≔ -⎜―――⎟ L ⎝ ⎠

b. Ecuaciones de Momento Flector para cada tramo Corte I-I

TRAMO AB x ≤ L

MAB ≔ -M1

Corte II-II TRAMO BC L < x ≤ 2 L

Corte III-III TRAMO CD 2 L < x ≤ 3 L

⎛ M2 - M1 ⎞ MBC ≔ -M1 - ⎜―――⎟ ⋅ ((x - L)) L ⎝ ⎠

⎛ M2 - M1 ⎞ ⎛ M2 - M1 ⎞ MCD ≔ -M1 - ⎜―――⎟ ((x - L)) + ⎜―――⎟ ⋅ ((x - 2 L)) L L ⎝ ⎠ ⎝ ⎠

c. Método Doble Integración

Corte I-I

TRAMO AB x ≤ L

d2 2 E ⋅ I ⋅ ―― y ＝ MAB d x2

Ecuación de Momento Flector :

MAB ((x)) ≔ -M1

Ecuación de Deformacion Angular:

1 ⎛⎝-M1 ⋅ x + C1⎞⎠ θAB ((x)) ＝ ―― EI

Ecuacion de Deflexion:

⎞ 1 ⎛ x2 yAB ((x)) ＝ ―― + C1 ⋅ x + C2⎟ ⎜-M1 ⋅ ―― EI ⎝ 2 ⎠

Corte II-II TRAMO BC L < x ≤ 2 L

d2 2 E ⋅ I ⋅ ―― y ＝ MBC d x2

Ecuación de Momento Flector : Ecuación de Deformacion Angular: Ecuacion de Deflexion:

Corte III-III TRAMO CD 2 L < x ≤ 3 L Ecuación de Momento Flector : Ecuación de Deformacion Angular: Ecuacion de Deflexion:

⎛ M2 - M1 ⎞ MBC ((x)) ≔ -M1 - ⎜―――⎟ ⋅ ((x - L)) L ⎝ ⎠ 2 ⎛ ⎞ ⎛ M2 - M1 ⎞ ((x - L)) 1 ⎜ θBC ((x)) ＝ ―― -M1 ⋅ x - ⎜―――⎟ ⋅ ―――+ C3⎟ ⎟⎠ EI ⎜⎝ L 2 ⎝ ⎠ 3 ⎛ ⎞ 2 ⎛ M2 - M1 ⎞ ((x - L)) 1 ⎜ x yBC ((x)) ＝ ―― -M1 ⋅ ―― - ⎜―――⎟ ⋅ ―――+ C3 ⋅ x + C4⎟ ⎟⎠ EI ⎜⎝ 2 L 6 ⎝ ⎠ d2 2 E ⋅ I ⋅ ―― y ＝ MCD d x2 ⎛ M2 - M1 ⎞ ⎛ M2 - M1 ⎞ MCD ((x)) ≔ -M1 - ⎜―――⎟ ((x - L)) + ⎜―――⎟ ⋅ ((x - 2 L)) L L ⎝ ⎠ ⎝ ⎠ 2 2 ⎛ ⎞ ⎛ M2 - M1 ⎞ ((x - L)) ⎛ M2 - M1 ⎞ ((x - 2 L)) 1 ⎜ ( ) θCD (x) ＝ ―― -M1 ⋅ x - ⎜―――⎟ ―――+ ⎜―――⎟ ⋅ ―――― + C5⎟ ⎟⎠ EI ⎜⎝ L 2 L 2 ⎝ ⎠ ⎝ ⎠ 3 3 ⎛ ⎞ ⎛ M2 - M1 ⎞ ((x - 2 L)) 1 ⎜ x 2 ⎛ M2 - M1 ⎞ ((x - L)) yCD ((x)) ＝ ―― -M1 ⋅ ―― - ⎜―――⎟ ―――+ ⎜―――⎟ ⋅ ―――― + C5 ⋅ x + C6⎟ ⎟⎠ EI ⎜⎝ 2 L 6 L 6 ⎝ ⎠ ⎝ ⎠

RESISTENCIA DE MATERIALES II (CIV-203)

Por condicion del problema: Por Condiciones de Apoyo x≔0

EI ≔ 2500000

L ≔ 2.5

yA ≔ 0.018 m

⎞ solve , C2 1 ⎛ x2 0.018 ＝ ―― → 45000.0 + C1 ⋅ x + C2⎟ ――― ⎜-M1 ⋅ ―― EI ⎝ 2 ⎠

x ≔ 2.5

C2 ≔ 45000.0

yB ≔ 0 m

⎞ solve , C1 1 ⎛ x2 0 ＝ ―― → 1.25 ⋅ M1 - 18000.0 + C1 ⋅ x + C2⎟ ――― ⎜-M1 ⋅ ―― EI ⎝ 2 ⎠

C1 ≔ 1.25 ⋅ M1 - 18000.0

Por Continuidad x ≔ 2.5

θI ＝ θII

yI ＝ yII

2 ⎛ ⎞ solve , C ⎛ M2 - M1 ⎞ ((x - L)) 3 1 1 ⎜ ⎛⎝-M1 ⋅ x + C1⎞⎠ ＝ ―― -M1 ⋅ x - ⎜―――⎟ ⋅ ―――+ C3⎟ ――― → 1.25 ⋅ M1 - 18000.0 ―― ⎟⎠ EI EI ⎜⎝ L 2 ⎝ ⎠

C3 ≔ 1.25 ⋅ M1 - 18000.0

3 ⎛ ⎞ solve , C ⎞ 4 1 ⎛ x2 1 ⎜ x 2 ⎛ M2 - M1 ⎞ ((x - L)) + C1 ⋅ x + C2⎟ ＝ ―― -M1 ⋅ ―― - ⎜―――⎟ ⋅ ―――+ C3 ⋅ x + C4⎟ ――― → 45000.0 ―― ⎜-M1 ⋅ ―― ⎟⎠ EI ⎝ 2 2 L 6 ⎠ EI ⎜⎝ ⎝ ⎠

C4 ≔ 45000.0

x≔5

θII ＝ θIII

yII ＝ yIII

2 2 2 ⎛ ⎞ ⎛ ⎞ solve , C ⎛ M2 - M1 ⎞ ((x - L)) ⎛ M2 - M1 ⎞ ((x - L)) ⎛ M2 - M1 ⎞ ((x - 2 L)) 5 1 ⎜ 1 ⎜ -M1 ⋅ x - ⎜―――⎟ ⋅ ―――+ C3⎟ ＝ ―― -M1 ⋅ x - ⎜―――⎟ ―――+ ⎜―――⎟ ⋅ ―――― + C5⎟ ――― → 1.25 ⋅ M1 - 18000.0 ―― ⎟⎠ EI ⎜⎝ ⎟⎠ EI ⎜⎝ L 2 L 2 L 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

C5 ≔ 1.25 ⋅ M1 - 18000.0 3 3 3 ⎛ ⎛ ⎞ ⎞ ⎛ M2 - M1 ⎞ ((x - L)) ⎛ M2 - M1 ⎞ ((x - L)) ⎛ M2 - M1 ⎞ ((x - 2 L)) solve , C6 1 ⎜ x2 1 ⎜ x2 -M1 ⋅ ―― - ⎜―――⎟ ⋅ ――― + C3 ⋅ x + C4⎟ ＝ ―― -M1 ⋅ ―― - ⎜―――⎟ ――― + ⎜―――⎟ ⋅ ―――― + C5 ⋅ x + C6⎟ ―――→ 45000.0 ―― ⎟⎠ EI ⎜⎝ ⎟⎠ EI ⎜⎝ 2 2 L 6 L 6 L 6 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

C6 ≔ 45000 x ≔ 7.5

yD ≔ 0.012 m

3 3 ⎛ ⎞ ⎛ M2 - M1 ⎞ ((x - L)) ⎛ M2 - M1 ⎞ ((x - 2 L)) float , 3 1 ⎜ x2 0.012 ＝ ―― -M1 ⋅ ―― - ⎜―――⎟ ―――+ ⎜―――⎟ ⋅ ―――― + C5 ⋅ x + C6⎟ ――― → 0.012 ＝ -0.00000458 ⋅ M1 + -0.00000292 ⋅ M2 - 0.036 ⎟ EI ⎜⎝ 2 L L 6 6 ⎝ ⎠ ⎝ ⎠ ⎠

x≔5

yc ≔ 0 m

0.048 ＝ -0.00000458 ⋅ M1 - 0.00000292 ⋅ M2

3 ⎛ ⎞ ⎛ M2 - M1 ⎞ ((x - L)) float , 3 1 ⎜ x2 0 ＝ ―― -M1 ⋅ ―― - ⎜―――⎟ ⋅ ―――+ C3 ⋅ x + C4⎟ ――― → 0 ＝ -0.00000208 ⋅ M1 + -⎛⎝4.17 ⋅ 10 -7⎞⎠ ⋅ M2 - 0.018 ⎟⎠ EI ⎜⎝ 2 L 6 ⎝ ⎠

-0.00000208 ⋅ M1 - ⎛⎝4.17 ⋅ 10 -7⎞⎠ ⋅ M2 ＝ 0.018

RESISTENCIA DE MATERIALES II (CIV-203)

RESOLVIENDO EL SISTEMA TENEMOS:

⎡ -0.00000458 -0.00000292 ⎤ ⎢ -7 ⎥ ⎣ -0.00000208 -⎛⎝4.17 ⋅ 10 ⎞⎠ ⎦

-1

⎡ -7816.0499935154452487 ⎤ ⎡ 0.048 ⎤ ⋅⎢ → ―⎢ ⎥ 0.018 ⎣ ⎦ ⎣ -4178.9352841435824523 ⎥⎦

M1 ≔ -7816.050 M2 ≔ -4178.935