Ejercicios Armaduras planas basada método de la rigidez con MATHCAD OK

Gilmer Calderón Quispe Ingenierá Civil UNSCH

Contraseña del área protegida

Por: Gilmer Calderón Quispe

Paea la aradura que se muesta en la gra cálcular las ferzas en las baras La sección tansversal para todas los elementos esrectanglar esrectanglar con una base de 30cm y una altra de 40cm. módulo de elasticidad 2.1E6

N

X

Y

M

Ni

Nj

1

0

0

1

1

2

2

2

0

2

2

3

4

0

3

4

6

0

5

4

3

6

2

3

UNSCH Ayacucho - Perú

A

E

Node

Ux

Uy

Dx

Dy

0.12 2 ⋅ 10 6

1

1

1

0

0

3

0.12 2 ⋅ 10 6

4

1

1

0

0

3

4

0.12 2 ⋅ 10 6

4

5

4

0.12 2 ⋅ 10 6

5

6

5

0.12 2 ⋅ 10 6

NLF

Fx

Fy

6

6

1

6

0.12 2 ⋅ 10

7

2

6

0.12 2 ⋅ 10 6

2

0

-3

8

6

3

0.12 2 ⋅ 10 6

3

0

-3

9

2

5

0.12 2 ⋅ 10 6

10

3

5

0.12 2 ⋅ 10 6

- 1 -


k 1 ＝k 2 ＝k 3 ＝k 5 ‖ k j ≔ ni ← Ni ‖ j ⎡ 120000 0 -120000 0 ⎤ ‖ nj ← Nj ⎢ ‖ 0 0 0 0⎥ j 1 = k ⎢ ⎥ ‖ -120000 0 120000 0 ← xi X ‖ ni 0 0 0 0⎦ ⎣ ‖ ‖ xj ← X nj ‖ ⎡ 20481.238 -30721.857 -20481.238 ‖ yi ← Y ni ⎢ -30721.857 46082.7855 30721.857 ‖ k 4 =⎢ -20481.238 30721.857 20481.238 ‖ yj ← Y nj ‖ 30721.857 -46082.7855 -30721.857 ⎣ 2 2 ‖ ‖ l ← xj - xi + yj - yi ⎡ 20481.238 30721.857 -20481.238 xj - xi ‖ ⎢ 30721.857 46082.7855 -30721.857 ‖λ← l k 6 =⎢ ‖ -20481.238 -30721.857 20481.238 yj - yi ‖ μ ← ―― -30721.857 -46082.7855 30721.857 ⎣ ‖ l ‖ 2 ⎡0 μ ⋅ λ -λ 2 - μ ⋅ λ ⎤ 0 0 0⎤ ‖ A ⋅ E ⎡ λ ⎢ ⎥ 2 2 j j ‖ - μ ⋅ λ - μ ⎥ k 7 = ⎢ 0 80000 0 -80000 ⎥ μ ⋅ λ μ ‖ ―― ⋅ ⎢ 2 0 0 0 0 ⎢ -λ - μ ⋅ λ λ 2 μ ⋅ λ ⎥ ‖ l 2 2 0 -80000 0 80000 ⎣ ⎦ μ ⋅ λ μ ⎦ ⎣ - μ ⋅ λ - μ ‖

⎡0 0 0 80000 k 10 = ⎢ 0 ⎢0 ⎣ 0 -80000

0 0⎤ 0 -80000 ⎥ 0 0⎥ 0 80000 ⎦

‖

F ≔ f r , s ← 0 length N ‖ #n ← ‖ F ← matrix 2 ⋅ #n , 1 , f ‖ for i ∊ 1 , 2‥ length NLF

‖ ‖ ‖ ‖ ‖ ‖ ‖

‖ n ← NLF i

‖ F ‖ 2 ⋅ n - 1 , 1 ← Fxi ‖ F ← Fy i 2⋅n,1 ‖ ‖ return F

30721.857 ⎤ -46082.7855 ⎥ ⎥ -30721.857 46082.7855 ⎦ -30721.857 ⎤ -46082.7855 ⎥ ⎥ 30721.857 46082.7855 ⎦

⎡ 20481.238 -30721.857 -20481.238 30721.857 ⎤ -30721.857 46082.7855 30721.857 -46082.7855 k 8 =⎢ ⎥ 20481.238 -30721.857 ⎥ ⎢ -20481.238 30721.857 ⎣ 30721.857 -46082.7855 -30721.857 46082.7855 ⎦ ⎡ 20481.238 30721.857 -20481.238 -30721.857 ⎤ 30721.857 46082.7855 -30721.857 -46082.7855 k 9 =⎢ ⎥ -20481.238 -30721.857 20481.238 30721.857 ⎥ ⎢ ⎣ -30721.857 -46082.7855 30721.857 46082.7855 ⎦

⎡ 0⎤ ⎢ 0⎥ ⎢ ⎥ ⎢ 0⎥ ⎢ -3 ⎥ ⎢ 0⎥ ⎢ -3 ⎥ F = ⎢ ⎥ ⎢ 0⎥ ⎢ 0⎥ ⎢ 0⎥ ⎢ 0⎥ ⎢ ⎥ 0 ⎣ 0⎦

‖

K ≔ f r , s ← 0 length N ‖ #n ← ‖ for i ∊ 1 , 2‥ length M

‖ ‖ ‖ B ← matrix 4 , 2 ⋅ #n , f ‖ ‖ n ← Nii


- 2 -

Gilmer Calderón Quispe Ingenierá Civil UNSCH ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖

‖ m ← Nji ‖ ‖ B1 , 2 ⋅ n - 1 ← 1 ‖ ‖ B2 , 2 ⋅ n ← 1 ‖ ‖ B ←1 ‖ 3,2⋅m-1 ‖ B ←1 ‖ 4,2⋅m T ‖ KG ← KG + B ⋅ k i ⋅ B

⎡ 0.1405 0.0307 0.0307 0.0461 ⎢ 0 ⎢ -0.12 0 0 ⎢ ⎢ 0 0 ⎢ 0 0 K = ⎢ 0 0 ⎢ 0 ⎢ 0 ⎢ 0 0 ⎢ 0 0 ⎢ ⎢ -0.0205 -0.0307 ⎣ -0.0307 -0.0461

-0.12 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2605 0.0307 -0.12 0 0 0 -0.0205 0.0307 0.1261 0 0 0 0 -0.0307 -0.12 0 0.2605 -0.0307 -0.12 0 0 0 0 -0.0307 0.1261 0 0 0 0 0 -0.12 0 0.1405 -0.0307 -0.0205 0 0 0 0 -0.0307 0.0461 0.0307 -0.0205 -0.0307 0 0 -0.0205 0.0307 0.161 -0.0307 -0.0461 0 -0.08 0.0307 -0.0461 0 0 0 -0.0205 0.0307 0 0 -0.12 0 -0.08 0.0307 -0.0461 0 0 0

⎤ 0 0 ⎥ -0.0307 ⎥ -0.0461 ⎥ ⎥ 0 ⎥ 6 -0.08 ⎥ 10 0.0307 ⎥ -0.0461 ⎥ ⎥ 0 ⎥ 0.1722 ⎥ 0 ⎥ 0 …⎦

‖

K_m ≔ for i ∊ 1 , 2‥ length Node

‖ ‖ n ← Node i ‖ ‖ ‖ ‖ if Ux ＝ 0 i ‖ ‖ ‖ ‖ K ← K ‖ ‖ ‖ ‖ else ‖ ‖ ‖ 10 ‖ ‖ C ← max K ⋅ 10 ‖ ‖ ‖ K 2 ⋅ n - 1 , 2 ⋅ n - 1 ← K 2 ⋅ n - 1 , 2 ⋅ n - 1 + C ‖ ‖ ‖ ‖ ‖ if Uy ＝ 0 i ‖ ‖ ‖ ‖ ‖ ‖ K ← K ‖ ‖ else ‖ ‖ ‖ ‖ ‖ C ← max K ⋅ 10 10 ‖ ‖ ‖ K ← K +C 2⋅n,2⋅n 2⋅n,2⋅n ‖ ‖ ‖ ‖ ‖ return K ‖

⎡ 2.6048 ⋅ 10 9 0.0307 ⎢ 2.6048 ⋅ 10 19 ⎢ 0.0307 0 ⎢ -0.12 ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 K_m = ⎢ ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 ⎢ 0 ⎢ 0 ⎢ -0.0205 -0.0307


-0.12 0 0 0 0 0 0 0 0 0 0 0 0.2605 0.0307 -0.12 0 0 0 0.0307 0.1261 0 0 0 0 -0.12 0 0.2605 -0.0307 -0.12 0 0 0 -0.0307 0.1261 0 0 29 0 0 -0.12 0 2.6048 ⋅ 10 -0.0307 0 0 0 0 -0.0307 2.6048 ⋅ 10 39 -0.0205 -0.0307 0 0 -0.0205 0.0307 -0.0307 -0.0461 0 -0.08 0.0307 -0.0461 0 0 -0.0205 0.0307 0 0

- 3 -

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 10 6 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Gilmer Calderón Quispe Ingenierá Civil UNSCH ⎣ -0.0307

-0.0461

0

‖ ‖ ‖ ‖ ‖ n ← Node i ‖ ‖ ‖ ‖ if Dx ＝ 0 ∧ Dy ＝ 0 i i ‖ ‖ ‖ ‖ ‖ F ← F ‖ ‖ 2⋅n-1,1 2⋅n-1,1 ‖ ‖ ‖ F ← F n ‖ ‖ 2⋅n,1 2⋅ ,1 ‖ ‖ ‖ ‖ ‖ else ‖ ‖ ‖ C ← max K ⋅ 10 10 ‖ ‖ ‖ ‖ ‖ F 2 ⋅ n - 1 , 1 ← F 2 ⋅ n - 1 , 1 + Dxi ⋅ C ‖ ‖ ‖ ← F + Dy ⋅ C ‖ F ‖ i 2⋅n,1 ‖ ‖ ‖ 2⋅n,1 ‖ ‖ return F

F_m ≔ for i ∊ 1 , 2‥ length Node

‖

Δ ≔ f r , s ← 0 ‖ D ← matrix 2 ⋅ length N , 1 , f ‖ for i ∊ 1 , 2 ‥ 2 ⋅ length N

‖ ‖ ‖ Di , 1 ← concat “U” , num2str i ‖ ‖ ‖ R ← K_m -1 ⋅ F_m ‖ D R

‖ P i ≔ f r , s ← 0 length N ‖ #n ← ‖ B ← matrix 4 , 2 ⋅ #n , f ‖ n ← Ni i ‖ ‖ m ← Nj ‖ i ‖ B ‖ 1,2⋅n-1←1 ‖ B ‖ 2,2⋅n←1 ‖ ‖ B3 , 2 ⋅ m - 1 ← 1 ‖ ‖ B4 , 2 ⋅ m ← 1 ‖ ‖ R ← K_m -1 ⋅ F_m ‖ φ ← B ⋅ R ‖ ni ← Ni ‖ i ‖ ‖ nj ← Nji ‖ ‖ xi ← X ni ‖ xj ← X


-0.08

0.0307 -0.0461

0

0

…⎦

⎡ 0⎤ ⎢ 0⎥ ⎢ ⎥ ⎢ 0⎥ ⎢ -3 ⎥ ⎢ 0⎥ ⎢ -3 ⎥ F_m = ⎢ ⎥ ⎢ 0⎥ ⎢ 0⎥ ⎢ 0⎥ ⎢ 0⎥ ⎢ ⎥ 0 ⎣ 0⎦

⎡ ⎡ “U1” ⎤ ⎢ ⎢ “U2” ⎥ ⎢⎢ ⎥ ⎢ ⎢ “U3” ⎥ ⎢ ⎢ “U4” ⎥ ⎢ ⎢ “U5” ⎥ ⎢ ⎢ “U6” ⎥ Δ=⎢⎢ ⎥ ⎢ ⎢ “U7” ⎥ ⎢ ⎢ “U8” ⎥ ⎢ ⎢ “U9” ⎥ ⎢ ⎢ “U10” ⎥ ⎢⎢ ⎥ “U11” ⎣ ⎣ “U12” ⎦

⎡ 0 ⎤⎤ ⎢ 0 ⎥⎥ ⎢ ⎥⎥ ⎢ 0.000001587 ⎥ ⎥ ⎢ -0.000099033 ⎥ ⎥ ⎢ -0.000001587 ⎥ ⎥ ⎢ -0.000099033 ⎥ ⎥ ⎢ ⎥⎥ ⎢ 0 ⎥⎥ ⎢ 0 ⎥⎥ ⎢ -0.000010713 ⎥ ⎥ ⎢ -0.000072242 ⎥ ⎥ ⎢ ⎥⎥ 0.000010713 ⎣ -0.000072242 ⎦ ⎦

⎡ -0.19038 ⎤ ⎢ 0 ⎥ P 1 = ⎢ ⎥ 0.19038 ⎣ 0 ⎦

⎡ 0.38077 ⎤ ⎡ -0.19038 ⎤ ⎢ 0 ⎥ ⎢ 0 ⎥ P 2 = ⎢ ⎥ P 3 = ⎢ ⎥ -0.38077 0.19038 ⎣ 0 ⎦ ⎣ 0 ⎦

⎡ 3.60555 ⎤ ⎢ 0 ⎥ P 4 = ⎢ ⎥ ⎢ -3.60555 ⎥ ⎣ 0 ⎦

⎡ 2.57115 ⎤ ⎡ 3.60555 ⎤ ⎢ 0 ⎥ ⎢ 0 ⎥ P 5 = ⎢ ⎥ P 6 = ⎢ ⎥ ⎢ -2.57115 ⎥ ⎢ -3.60555 ⎥ ⎣ 0 ⎦ ⎣ 0 ⎦

⎡ -2.14327 ⎤ ⎢ 0 ⎥ P 7 = ⎢ ⎥ 2.14327 ⎣ 0 ⎦

⎡ -1.02966 ⎤ ⎡ -1.02966 ⎤ ⎢ 0 ⎥ ⎢ 0 ⎥ P 8 = ⎢ ⎥ P 9 = ⎢ ⎥ 1.02966 1.02966 ⎣ 0 ⎦ ⎣ 0 ⎦

⎡ -2.14327 ⎤ ⎢ 0 ⎥ P 10 = ⎢ ⎥ 2.14327

- 4 -

Gilmer Calderón Quispe Ingenierá Civil UNSCH n

‖ ‖ yi ← Y ni ‖ ‖ yj ← Y nj ‖ 2 2 ‖ ‖ l ← xj - xi + yj - yi ‖ xj - xi ‖ λ ← ―― l ‖ ‖ μ ← yj - yi ‖ l ‖ ⎡ λ μ 0 0⎤ ‖ ‖ ← - μ λ 0 0 ‖ T ⎢ 0 0 λ μ ⎥ ⎢ ⎥ ‖ 0 0 μ λ ⎣ ⎦ ‖ ⎡ 1 0 -1 0 ⎤ ‖ A ⋅ E i i ‖ 0 0 0 0 ‖ kl ← ―― ⋅ ⎢ -1 0 1 0 ⎥ l ⎢ ⎥ ‖ ⎣ 0 0 0 0⎦ ‖ ‖ p ← kl ⋅ T ⋅ φ ‖ return p


- 5 -


Calcule las reacciones en los sopores y la ferza en cada uno delos elementos que componen la aradura que se muesta en la gra mediante el método directo de la rigidez. Comprebe el equilibrio en las juntas para vericar que los resultados obtenidos son corectos.  es constante.


- 6 -


N

X

Y

M

Ni

Nj

A

E

Node

Ux

Uy

Dx

Dy

1

0

0

1

1

2

1

1

1

0

1

0

0

2

5

0

2

2

3

1

1

4

1

1

0

0

3

9

0

3

3

4

1

1

7

0

1

0

0

4

12.5

-5

4

4

5

1

1

5

16

0

5

5

6

1

1

6

21

0

6

6

7

1

1

7

25

0

7

8

7

1

1

NLF

Fx

Fy

8

21

5

8

9

8

1

1

9

16

5

9

10

9

1

1

8

0

-3

10

12.5

5

10

11

10

1

1

9

0

-2

11

9

5

11

12

11

1

1

10

0

-1

12

5

5

12

13

12

1

1

11

0

-6

13

0

5

13

1

13

1

1

12

0

-3

14

1

12

1

1

13

3.6

-2

15

2

12

1

1

16

12

3

1

1

17

3

11

1

1

18

3

10

1

1

19

10

5

1

1

20

5

9

1

1

21

5

8

1

1

22

6

8

1

1

⎡ 0.2 0 -0.2 0 ⎤ ⎢ 0 0 0 0⎥ k 1 =⎢ ⎥ -0.2 0 0.2 0 ⎣ 0 0 0 0⎦

⎡ 0.25 0 -0.25 0 ⎤ ⎢ 0 0 0 0⎥ k 2 =⎢ ⎥ -0.25 0 0.25 0 0 0 0⎦ ⎣ 0

⎡ 0.0539 0.077 -0.0539 -0.077 ⎤ 0.077 0.11 -0.077 -0.11 k 4 =⎢ ⎥ -0.0539 -0.077 0.0539 0.077 0.077 0.11 ⎦ ⎣ -0.077 -0.11

⎡ 0.2 0 k 5 =⎢ -0.2 ⎣ 0

⎡ 0.0609 -0.0762 -0.0609 0.0762 ⎤ -0.0762 0.0952 0.0762 -0.0952 k 7 =⎢ ⎥ ⎢ -0.0609 0.0762 0.0609 -0.0762 ⎥ ⎣ 0.0762 -0.0952 -0.0762 0.0952 ⎦

k 10 =

⎡ 0.2857 0 -0.2857 0 ⎤ 0 0 0 0


k 11 =

⎡ 0.0539 -0.077 -0.0539 0.077 ⎤ ⎢ -0.077 0.11 0.077 -0.11 ⎥ k 3 =⎢ ⎥ -0.0539 0.077 0.0539 -0.077 0.11 ⎦ ⎣ 0.077 -0.11 -0.077 0 -0.2 0 ⎤ 0 0 0 ⎥ 0 0.2 0 0 0 0⎦

⎡ 0.2 0 k 8 =⎢ -0.2 ⎢ ⎣ 0

⎡ 0.25 0 k 6 =⎢ -0.25 ⎣ 0

0 -0.25 0 ⎤ 0 0 0 ⎥ 0 0.25 0 0 0 0⎦

⎡ 0.2857 0 -0.2857 0 ⎤ 0 -0.2 0 ⎤ 0 0 0 0 0 0 0 ⎥ k 9 =⎢ ⎥ 0 0.2 0 ⎥ -0.2857 0 0.2857 0⎥ ⎢ 0 0 0⎦ 0 0 0⎦ ⎣ 0

⎡ 0.25 0 -0.25 0 ⎤ 0 0 0 0

k 12 =

⎡ 0.2 0 -0.2 0 ⎤ 0 0 0 0

- 7 -

Gilmer Calderón Quispe Ingenierá Civil UNSCH -0.2857 0 0 ⎣ 0 ⎡0 0 0 0.2 k 13 = ⎢ ⎢0 0 ⎣ 0 -0.2

0.2857 0 0 0⎦

0 0 ⎤ 0 -0.2 ⎥ 0 0 ⎥ 0 0.2 ⎦

-0.25 0 0 ⎣ 0

-0.2 0 ⎣ 0 0

0.2 0 0 0⎦

⎡ 0.0707 0.0707 -0.0707 -0.0707 ⎤ ⎡0 0 0.0707 0.0707 -0.0707 -0.0707 k 15 = 0 0.2 k 14 = ⎢ ⎥ ⎢ ⎢ -0.0707 -0.0707 0.0707 0.0707 ⎥ ⎢0 0 ⎣ -0.0707 -0.0707 0.0707 0.0707 ⎦ ⎣ 0 -0.2

0 0 ⎤ 0 -0.2 ⎥ 0 0 ⎥ 0 0.2 ⎦

⎡ 0.0609 -0.0762 -0.0609 0.0762 ⎤ -0.0762 0.0952 0.0762 -0.0952 k 16 = ⎢ ⎥ ⎢ -0.0609 0.0762 0.0609 -0.0762 ⎥ ⎣ 0.0762 -0.0952 -0.0762 0.0952 ⎦

⎡0 0 0 0.2 k 17 = ⎢ ⎢0 0 ⎣ 0 -0.2

⎡ 0.0539 0.077 -0.0539 -0.077 ⎤ 0.077 0.11 -0.077 -0.11 k 18 = ⎢ ⎥ -0.0539 -0.077 0.0539 0.077 ⎥ ⎢ 0.077 0.11 ⎦ ⎣ -0.077 -0.11

⎡ 0.2707 ⎢ 0.0707

0.0707 -0.2 6.7919 ⋅ 109 0


0 0 -0.25 0 0.4187 -0.0762 -0.0539 0.077 0 0 0 0 0 0 0 0 0 0 -0.0539 -0.077 0 0 -0.0609 0.0762 0 0

0 0

⎡0 0 ⎢ 0 0.2 k 22 = ⎢ ⎢0 0 ⎣ 0 -0.2

0 0 0 0 0 0 0 0 0 0 0 0 -0.0762 -0.0539 0.077 0.5152 0.077 -0.11 0.077 0.1078 0 -0.11 0 0.2199 0 -0.0539 -0.077 0 -0.077 -0.11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.077 0 0 -0.11 0 0 0 0 0 -0.2 0 0 0.0762 0 0 -0.0952 0 0 0 0 0 0 0 0

0 0

⎡0 0 0 0 ⎤ 0 -0.2 0 0.2 ⎥ k 20 = ⎢ 0 0 ⎥ ⎢0 0 0 0.2 ⎦ ⎣ 0 -0.2

0 0 ⎤ 0 -0.2 ⎥ 0 0 ⎥ 0 0.2 ⎦

⎡ 0.0539 -0.077 -0.0539 0.077 ⎤ -0.077 0.11 0.077 -0.11 k 19 = ⎢ ⎥ -0.0539 0.077 0.0539 -0.077 ⎥ ⎢ 0.11 ⎦ ⎣ 0.077 -0.11 -0.077

⎡ 0.0707 0.0707 -0.0707 -0.0707 ⎤ ⎢ 0.0707 0.0707 -0.0707 -0.0707 ⎥ k 21 = ⎢ ⎥ ⎢ -0.0707 -0.0707 0.0707 0.0707 ⎥ ⎣ -0.0707 -0.0707 0.0707 0.0707 ⎦

⎡ 0.2707 0.0707 -0.2 0 0.0707 0.2707 0 0 ⎢ 0 0.45 0 ⎢ -0.2 0 0 0.2 ⎢ 0 ⎢ 0 0 -0.25 0 ⎢ 0 0 0 0 ⎢ 0 0 0 ⎢ 0 0 0 0 ⎢ 0 ⎢ 0 0 0 0 ⎢ 0 0 0 0 ⎢ 0 0 0 ⎢ 0 0 0 0 ⎢ 0 ⎢ 0 0 0 0 K = ⎢ 0 0 0 0 ⎢ 0 0 0 ⎢ 0 0 0 0 ⎢ 0 ⎢ 0 0 0 0 ⎢ 0 0 0 0 ⎢ 0 0 0 ⎢ 0 ⎢ 0 0 0 0 ⎢ 0 0 0 0 ⎢ 0 0 0 0 ⎢ 0 ⎢ -0.0707 -0.0707 0 ⎢ -0.0707 -0.0707 0 -0.2 ⎢ 0 0 0 0 -0.2 0 0 ⎣ 0

0.25 0 0 0⎦

0 0

0 0

0 0 0 0 0 0 -0.0539 -0.077 0.3785 0.0707 -0.2 0 0 0 -0.0707 -0.0707 0 0 -0.0539 0.077 0 0 0 0 0 0

0 0

0 0 ⎤ 0 -0.2 ⎥ ⎥ 0 0 ⎥ 0 0.2 ⎦

0 0 0 0 0 0 0 0 0 0 0 0 -0.077 0 -0.11 0 0.0707 -0.2 0.4906 0 0 0.45 0 0 0 -0.25 0 0 -0.0707 0 -0.0707 0 0 0 -0.2 0 0.077 0 -0.11 0 0 0 0 0 0 0 0 0 0 0 0 0 …

0 0

⎤

⎡ 0 ⎤ 0 ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ F = ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ -3 ⎥ ⎢ 0 ⎥ ⎢ -2 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ -1 ⎥ ⎢ 0 ⎥ ⎢ -6 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ -3 ⎥ ⎢ 3.6 ⎥ ⎣ -2 ⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥

- 8 -

⎡ 0 ⎤ ⎢ 0 ⎥

Gilmer Calderón Quispe Ingenierá Civil UNSCH ⎢- . ⎢ 0 0 ⎢ 0 0 ⎢ 0 ⎢ 0 0 ⎢ 0 ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 K_m = ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 ⎢ 0 0 ⎢ ⎢ -0.0707 -0.0707 ⎢ -0.0707 -0.0707 ⎢ 0 ⎢ 0 0 -0.2 ⎣

⎡ ⎡ “U1” ⎤ “U2” ⎢⎢ ⎥ ⎢ ⎢ “U3” ⎥ ⎢ ⎢ “U4” ⎥ ⎢ ⎢ “U5” ⎥ ⎢ ⎢ “U6” ⎥ ⎢⎢ ⎥ ⎢ ⎢ “U7” ⎥ ⎢ ⎢ “U8” ⎥ ⎢ ⎢ “U9” ⎥ ⎢ ⎢ “U10” ⎥ ⎢⎢ ⎥ ⎢ ⎢ “U11” ⎥ ⎢ ⎢ “U12” ⎥ ⎢ ⎢ “U13” ⎥ Δ=⎢⎢ “U14” ⎥ ⎢⎢ ⎥ ⎢ ⎢ “U15” ⎥ ⎢ ⎢ “U16” ⎥ ⎢ ⎢ “U17” ⎥ ⎢ ⎢ “U18” ⎥ ⎢⎢ ⎥ ⎢ ⎢ “U19” ⎥ ⎢ ⎢ “U20” ⎥ ⎢ ⎢ “U21” ⎥ ⎢ ⎢ “U22” ⎥ ⎢⎢ ⎥ ⎢ ⎢ “U23” ⎥ ⎢ ⎢ “U24” ⎥ ⎢ ⎢ “U25” ⎥ ⎣ ⎣ “U26” ⎦

⎡ 3.6364 ⎤ ⎢ 0 ⎥ P 1 = ⎢ ⎥ -3.6364

. 0 0.2 -0.25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.2 0 0 0 0

- . 0 0.4187 -0.0762 -0.0539 0.077 0 0 0 0 0 0 0 0 0 0 -0.0539 -0.077 0 0 -0.0609 0.0762 0 0

0 0 0 -0.0762 -0.0539 0.077 0.5152 0.077 -0.11 0.077 6.7919 ⋅ 10 19 0 -0.11 0 6.7919 ⋅ 10 29 0 -0.0539 -0.077 0 -0.077 -0.11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.077 0 0 -0.11 0 0 0 0 0 -0.2 0 0 0.0762 0 0 -0.0952 0 0 0 0 0 0 0 0

⎥ ⎢ 0 ⎥ ⎥ 0 ⎢ 0 ⎥ ⎥ ⎢ 0 ⎥ 0 ⎥ ⎢ 0 ⎥ 0 ⎥ ⎢ ⎥ -0.0539 ⎥ ⎢ 0 ⎥ ⎥ ⎢ 0 ⎥ -0.077 ⎥ ⎢ 0 ⎥ 0.3785 ⎥ ⎢ 0 ⎥ 0.0707 ⎥ ⎢ ⎥ ⎥ -0.2 ⎢ 0 ⎥ ⎥ ⎢ 0 ⎥ 0 ⎥ ⎢ 0 ⎥ 0 F_m ⎥ =⎢ 0 ⎥ 0 ⎥ ⎢ ⎥ ⎥ -0.0707 ⎢ 0 ⎥ ⎥ ⎢ -3 ⎥ -0.0707 ⎥ ⎢ 0 ⎥ 0 ⎥ ⎢ -2 ⎥ 0 ⎥ ⎢ ⎥ ⎥ -0.0539 ⎢ 0 ⎥ ⎥ ⎢ -1 ⎥ 0.077 ⎥ ⎢ 0 ⎥ 0 ⎥ ⎢ ⎥ 0 ⎥ ⎢ -6 ⎥ ⎥ 0 ⎢ 0 ⎥ ⎥ ⎢ -3 ⎥ 0 ⎥ ⎢ 3.6 ⎥ 0 ⎥ ⎣ -2 ⎦ 0 … ⎦

⎡ -199.078391 ⎤ ⎤ 0 ⎢ ⎥⎥ ⎢ -217.260209 ⎥ ⎥ ⎢ -155.663251 ⎥ ⎥ ⎢ -231.805664 ⎥ ⎥ ⎢ -277.827152 ⎥ ⎥ ⎢ ⎥⎥ 0 ⎢ ⎥⎥ 0 ⎢ ⎥⎥ ⎢ 307.745801 ⎥ ⎥ ⎢ -284.216702 ⎥ ⎥ ⎢ ⎥⎥ ⎢ 301.200346 ⎥ ⎥ ⎢ -140.677065 ⎥ ⎥ ⎢ 295.963982 ⎥ ⎥ ⎢ ⎥⎥ 0 ⎢ ⎥⎥ ⎢ 98.63808 ⎥ ⎥ ⎢ -140.677065 ⎥ ⎥ ⎢ 68.910807 ⎥ ⎥ ⎢ -294.216702 ⎥ ⎥ ⎢ ⎥⎥ ⎢ 48.101717 ⎥ ⎥ ⎢ -474.411882 ⎥ ⎥ ⎢ 29.392626 ⎥ ⎥ ⎢ -307.827152 ⎥ ⎥ ⎢ ⎥⎥ 8.010807 ⎥ ⎥ ⎢ ⎢ -155.663251 ⎥ ⎥ ⎢ 26.010807 ⎥ ⎥ ⎣ -10 ⎦⎦

⎡ 3.6364 ⎤ ⎢ 0 ⎥ P 2 = ⎢ ⎥ -3.6364


⎡ 15.5118 ⎤ ⎢ 0 ⎥ P 3 = ⎢ ⎥ -15.5118

⎡ 9.2342 ⎤ ⎢ 0 ⎥ P 4 = ⎢ ⎥ -9.2342

⎡ 1.3091 ⎤ ⎢ 0 ⎥ P 5 = ⎢ ⎥ -1.3091

- 9 -


⎡ 1.3091 ⎤ ⎢ 0 ⎥ P 6 = ⎢ ⎥ -1.3091 ⎣ 0 ⎦

⎡ -2.0956 ⎤ ⎢ 0 ⎥ P 7 = ⎢ ⎥ 2.0956 ⎣ 0 ⎦

⎡ -5.3455 ⎤ ⎡ 3.6 ⎤ 0 0 P 11 = ⎢ ⎥ P 12 = ⎢ ⎥ ⎢ 5.3455 ⎥ ⎢ -3.6 ⎥ ⎣ 0 ⎦ ⎣ 0 ⎦

⎡ -5.9455 ⎤ ⎢ 0 ⎥ P 8 = ⎢ ⎥ 5.9455 ⎣ 0 ⎦

⎡ 2⎤ 0 P 13 = ⎢ ⎥ ⎢ -2 ⎥ ⎣ 0⎦

⎡ 8.4987 ⎤ ⎡ 6⎤ ⎢ 0 ⎥ ⎢ 0⎥ P 16 = ⎢ ⎥ P 17 = ⎢ ⎥ -8.4987 -6 ⎣ 0 ⎦ ⎣ 0⎦

⎡ 6.5568 ⎤ ⎢ 0 ⎥ P 21 = ⎢ ⎥ ⎢ -6.5568 ⎥ ⎣ 0 ⎦


⎡ -5.9455 ⎤ ⎢ 0 ⎥ P 9 = ⎢ ⎥ 5.9455 ⎣ 0 ⎦

⎡ 0.0872 ⎤ ⎢ 0 ⎥ P 18 = ⎢ ⎥ -0.0872 ⎣ 0 ⎦

⎡ -5.3455 ⎤ ⎢ 0 ⎥ P 10 = ⎢ ⎥ 5.3455 ⎣ 0 ⎦

⎡ -5.1426 ⎤ 0 P 14 = ⎢ ⎥ ⎢ 5.1426 ⎥ ⎣ 0 ⎦

⎡ 1.1335 ⎤ ⎢ 0 ⎥ P 19 = ⎢ ⎥ -1.1335 ⎣ 0 ⎦

⎡0⎤ 0 P 15 = ⎢ ⎥ ⎢0⎥ ⎣0⎦

⎡ 2⎤ ⎢ 0⎥ P 20 = ⎢ ⎥ -2 ⎣ 0⎦

⎡0⎤ ⎢0⎥ P 22 = ⎢ ⎥ ⎢0⎥ ⎣0⎦

- 10 -


[1] "RESOLUCIÓN DE ARMADURAS EN 2D CON EL MÉTODO MATRICIAL DE LA RIGIDEZ " David Oriz Soto [2] INTRODUCTION TO FINITE ELEMENTS IN ENGINEERING 3ra edicion Tirupaty R. Chandrupatla Ashok D. Belegundo


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Ejercicios Armaduras planas basada método de la rigidez con MATHCAD OK

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