Problema 1

INGENIERÍA CIVIL

EJERCICIO N°1

Hallar los desplazamientos y fuerzas internas finales en los extremos de los

elementos mediante un análisis P-Delta. En el nudo B existe una rotula.

Considerar elementos son perfiles W18x40, E = 2.1 x 106 kg/cm2.

SOLUCIÓN

1. ANALISIS DE PRIMER ORDEN

1.1. MATRIZ DE RIGIDEZ DE LOS ELEMENTOS

ANÁLISIS ESTRUCTURAL II

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ELEMENTO 1


M1

E A1 CX2

L1

3 E I2 CY2

L13

E A1L1

3 E I2

L13

CX CY

1E A1 CX

2L1

3 E I2 CY2

L13

1E A1L1

3 E I2

L13

CX CY

3 E I2 CY

L12

0

E A1L1

3 E I2

L13

CX CY

E A1 CY2

L1

3 E I2 CX2

L13

1E A1L1

3 E I2

L13

CX CY

1E A1 CY

2L1

3 E I2 CX2

L13

3 E I2 CX

L12

0

1E A1 CX

2L1

3 E I2 CY2

L13

1E A1L1

3 E I2

L13

CX CY

E A1 CX2

L1

3 E I2 CY2

L13

E A1L1

3 E I2

L13

CX CY

3 E I2 CY

L12

0

1E A1L1

3 E I2

L13

CX CY

1E A1 CY

2L1

3 E I2 CX2

L13

E A1L1

3 E I2

L13

CX CY

E A1 CY2

L1

3 E I2 CX2

L13

3 E I2 CX

L12

0

3 E I2 CY

L12

3 E I2 CX

L12

3 E I2 CY

L12

3 E I2 CX

L12

3 E I2L1

0

0

0

0

0

0

0

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ELEMENTO 2


KE1

11592.855389

15285.961958

11592.855389

15285.961958

513.53568

15285.961958

20509.666531

15285.961958

20509.666531

385.15176

11592.855389

15285.961958

11592.855389

15285.961958

513.53568

15285.961958

20509.666531

15285.961958

20509.666531

385.15176

513.53568

385.15176

513.53568

385.15176

3209.598

M1

E A1 CX2

L1

3 E I2 CY2

L13

E A1L1

3 E I2

L13

CX CY

1E A1 CX

2L1

3 E I2 CY2

L13

1E A1L1

3 E I2

L13

CX CY

3 E I2 CY

L12

0

E A1L1

3 E I2

L13

CX CY

E A1 CY2

L1

3 E I2 CX2

L13

1E A1L1

3 E I2

L13

CX CY

1E A1 CY

2L1

3 E I2 CX2

L13

3 E I2 CX

L12

0

1E A1 CX

2L1

3 E I2 CY2

L13

1E A1L1

3 E I2

L13

CX CY

E A1 CX2

L1

3 E I2 CY2

L13

E A1L1

3 E I2

L13

CX CY

3 E I2 CY

L12

0

1E A1L1

3 E I2

L13

CX CY

1E A1 CY

2L1

3 E I2 CX2

L13

E A1L1

3 E I2

L13

CX CY

E A1 CY2

L1

3 E I2 CX2

L13

3 E I2 CX

L12

0

3 E I2 CY

L12

3 E I2 CX

L12

3 E I2 CY

L12

3 E I2 CX

L12

3 E I2L1

0

0

0

0

0

0

0

INGENIERÍA CIVIL


INGENIERÍA CIVIL

ELEMENTO 3


KE2

31974.138

0

31974.138

0

0

0

128.38392

0

128.38392

641.9196

31974.138

0

31974.138

0

0

0

128.38392

0

128.38392

641.9196

0

641.9196

0

641.9196

3209.598

INGENIERÍA CIVIL


INGENIERÍA CIVIL

1.2. MATRIZ DEL SISTEMA

1.3. FUERZA EN LOS NUDOS

1.4. FUERZA DE EMPOTRAMIENTO PERFECTO


4

5

6

KE3

1002.999375

0

2005.99875

1002.999375

0

2005.99875

0

39967.6725

0

0

39967.6725

0

2005.99875

0

5349.33

2005.99875

0

2674.665

1002.999375

0

2005.99875

1002.999375

0

2005.99875

0

39967.6725

0

0

39967.6725

0

2005.99875

0

2674.665

2005.99875

0

5349.33

KE

43566.993389

15285.961958

31974.138

0

0

15285.961958

20638.050451

0

128.38392

641.9196

31974.138

0

32977.137375

0

2005.99875

0

128.38392

0

40096.05642

641.9196

0

641.9196

2005.99875

641.9196

8558.928

Fn

0

0

10

0

0

M1

E A1 CX2

L1

12 E I2 CY2

L13

E A1L1

12 E I2

L13

CX CY

6 E I2 CY

L12

1E A1 CX

2L1

12 E I2 CY2

L13

1E A1L1

12 E I2

L13

CX CY

6 E I2 CY

L12

E A1L1

12 E I2

L13

CX CY

E A1 CY2

L1

12 E I2 CX2

L13

6 E I2 CX

L12

1E A1L1

12 E I2

L13

CX CY

1E A1 CY

2L1

12 E I2 CX2

L13

6 E I2 CX

L12

6 E I2 CY

L12

6 E I2 CX

L12

4 E I2L1

6 E I2 CY

L12

6 E I2 CX

L12

2 E I2L1

1E A1 CX

2L1

12 E I2 CY2

L13

1E A1L1

12 E I2

L13

CX CY

6 E I2 CY

L12

E A1 CX2

L1

12 E I2 CY2

L13

E A1L1

12 E I2

L13

CX CY

6 E I2 CY

L12

1E A1L1

12 E I2

L13

CX CY

1E A1 CY

2L1

12 E I2 CX2

L13

6 E I2 CX

L12

E A1L1

12 E I2

L13

CX CY

E A1 CY2

L1

12 E I2 CX2

L13

6 E I2 CX

L12

6 E I2 CY

L12

6 E I2 CX

L12

2 E I2L1

6 E I2 CY

L12

6 E I2 CX

L12

4 E I2L1

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1.5. MATRIZ DE FUERZA EN EL SISTEMA

1.6. DESPLAZAMIENTOS DEL SISTEMA

1.7. VECTOR DE FUERZAS EN LOS ELEMENTOS

ELEMENTO 1


1

23

4

5

1

2

3

54

En eje global En eje local

f1 KE1 u1 fo1 T1 f1

Fo

0

9.375

0

15.625

15.625

FE Fn Fo

FE

0

9.375

10

15.625

15.625

u KE( )1FE

u

0.005642

0.003632

0.005952

0.000331

0.002923

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ELEMENTO 2


f1

9.894

11.76

9.894

11.76

4.296

fo1

15.344915

0.859297

15.344915

0.859297

4.296484

N1 fo10 0 15.34491539272632207007

En eje local En eje global

f2 KE2 u2 Fo fo2 T2 f2

N2 fo20 0 9.8943867009968007842018

fo2

9.894387

11.760354

9.894387

13.239646

3.698229

f2

9.894

11.76

9.894

13.24

3.698

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ELEMENTO 3

2. ANALISIS DE SEGUNDO ORDEN (i=1)

ELEMENTO 1



f3 KE3 u3 fo3 T3 f3

fo3

13.239646

0.105613

4.120682

13.239646

0.105613

3.698229

f3

0.106

13.24

4.121

0.106

13.24

3.698

N3 fo30 0 13.2396457848396980006443

M1

E A1 CX2

L1

1 3 E I2 CY2

L13

E A1L1

3 1 E I2

L13

CX CY

1E A1 CX

2L1

3 1 E I2 CY2

L13

1E A1L1

31 E I2

L13

CX CY

3 2 E I2 CY

L12

0

E A1L1

3 1 E I2

L13

CX CY

E A1 CY2

L1

3 1 E I2 CX2

L13

1E A1L1

3 1 E I2

L13

CX CY

1E A1 CY

2L1

3 1 E I2 CX2

L13

3 2 E I2 CX

L12

0

1E A1 CX

2L1

3 1 E I2 CY2

L13

1E A1L1

3 1 E I2

L13

CX CY

E A1 CX2

L1

3 1 E I2 CY2

L13

E A1L1

3 1 E I2

L13

CX CY

3 2 E I2 CY

L12

0

1E A1L1

3 1 E I2

L13

CX CY

1E A1 CY

2L1

3 1 E I2 CX2

L13

E A1L1

3 1 E I2

L13

CX CY

E A1 CY2

L1

3 1 E I2 CX2

L13

3 2 E I2 CX

L12

0

3 2 E I2 CY

L12

3 2 E I2 CX

L12

3 2 E I2 CY

L12

3 2 E I2 CX

L12

3 2 E I2L1

0

0

0

0

0

0

0

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ELEMENTO 2


00

1

2

0

0

0

KE1

11590.497602

15287.730298

11590.497602

15287.730298

511.075447

15287.730298

20508.340276

15287.730298

20508.340276

383.306585

11590.497602

15287.730298

11590.497602

15287.730298

511.075447

15287.730298

20508.340276

15287.730298

20508.340276

383.306585

511.075447

383.306585

511.075447

383.306585

3194.221543

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ELEMENTO 3


KE2

31974.138

0

31974.138

0

0

0

126.0087432247552047399777

0

126.0087432247552047399777

639.9381028247728244840897

31974.138

0

31974.138

0

0

0

126.0087432247552047399777

0

126.0087432247552047399777

639.9381028247728244840897

0

639.9381028247728244840897

0

639.9381028247728244840897

3199.690514123864122420449

45

1

2

3

4

5

M1

E A1 CX2

L1

1 3 E I2 CY2

L13

E A1L1

3 1 E I2

L13

CX CY

1E A1 CX

2L1

3 1 E I2 CY2

L13

1E A1L1

31 E I2

L13

CX CY

3 2 E I2 CY

L12

0

E A1L1

3 1 E I2

L13

CX CY

E A1 CY2

L1

3 1 E I2 CX2

L13

1E A1L1

3 1 E I2

L13

CX CY

1E A1 CY

2L1

3 1 E I2 CX2

L13

3 2 E I2 CX

L12

0

1E A1 CX

2L1

3 1 E I2 CY2

L13

1E A1L1

3 1 E I2

L13

CX CY

E A1 CX2

L1

3 1 E I2 CY2

L13

E A1L1

3 1 E I2

L13

CX CY

3 2 E I2 CY

L12

0

1E A1L1

3 1 E I2

L13

CX CY

1E A1 CY

2L1

3 1 E I2 CX2

L13

E A1L1

3 1 E I2

L13

CX CY

E A1 CY2

L1

3 1 E I2 CX2

L13

3 2 E I2 CX

L12

0

3 2 E I2 CY

L12

3 2 E I2 CX

L12

3 2 E I2 CY

L12

3 2 E I2 CX

L12

3 2 E I2L1

0

0

0

0

0

0

0

M1

E A1 CX2

L1

12 1 E I2 CY2

L13

E A1L1

12 1 E I2

L13

CX CY

6 2 E I2 CY

L12

1E A1 CX

2L1

12 1 E I2 CY2

L13

1E A1L1

12 1 E I2

L13

CX CY

6 2 E I2 CY

L12

E A1L1

12 1 E I2

L13

CX CY

E A1 CY2

L1

12 1 E I2 CX2

L13

6 2 E I2 CX

L12

1E A1L1

12 1 E I2

L13

CX CY

1E A1 CY

2L1

12 1 E I2 CX2

L13

6 2 E I2 CX

L12

6 2 E I2 CY

L12

6 2 E I2 CX

L12

4 3 E I2L1

6 2 E I2 CY

L12

6 2 E I2 CX

L12

2 4 E I2L1

1E A1 CX

2L1

121 E I2 CY2

L13

1E A1L1

12 1 E I2

L13

CX CY

6 2 E I2 CY

L12

E A1 CX2

L1

12 1 E I2 CY2

L13

E A1L1

12 1 E I2

L13

CX CY

6 2 E I2 CY

L12

1E A1L1

12 1 E I2

L13

CX CY

1E A1 CY

2L1

12 1 E I2 CX2

L13

6 2 E I2 CX

L12

E A1L1

121 E I2

L13

CX CY

E A1 CY2

L1

121 E I2 CX2

L13

6 2 E I2 CX

L12

6 2 E I2 CY

L12

6 2 E I2 CX

L12

24 E I2L1

6 2 E I2 CY

L12

6 2 E I2 CX

L12

4 3 E I2L1

INGENIERÍA CIVIL


INGENIERÍA CIVIL

2.1. MATRIZ DEL SISTEMA





1

23

4

5

KE3

999.02729393432097302

0

2004.6744107610618713

999.02729393432097302

0

2004.6744107610618713

0

39967.6725

0

0

39967.6725

0

2004.6744107610618713

0

5342.2651907767272888

2004.6744107610618713

0

2676.4324522675209185

999.02729393432097302

0

2004.6744107610618713

999.02729393432097302

0

2004.6744107610618713

0

39967.6725

0

0

39967.6725

0

2004.6744107610618713

0

2676.4324522675209185

2004.6744107610618713

0

5342.2651907767272888

KE

43564.635602

15287.730298

31974.138

0

0

15287.730298

20634.34902

0

126.008743

639.938103

31974.138

0

32973.165294

0

2004.674411

0

126.008743

0

40093.681243

639.938103

0

639.938103

2004.674411

639.938103

8541.955705

Fn

0

0

10

0

0

Fo

0

9.375

0

15.625

15.625

FE Fn Fo

FE

0

9.375

10

15.625

15.625

INGENIERÍA CIVIL

2.5. DEZPLAZAMIENTO DEL SISTEMA

2.6. FUERZAS INTERNAS EN LOS ELEMENTOS

ELEMENTO 1


1

2

3

4

5


f1 KE1 u1 fo1 T1 f1

f1

9.88

11.764

9.88

11.764

4.333

fo1

15.339058

0.845819

15.339058

0.845819

4.333176

N11 fo10 0 15.33905845138967619015881673

1N1

N11 0.000382 OK

u KE( )1FE

u

0.005714

0.003686

0.006023

0.000331

0.002942

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ELEMENTO 2



f2 KE2 u2 Fo fo2 T2 f2

f2

9.88

11.764

9.88

13.236

3.641

fo2

9.88009

11.763755

9.88009

13.236245

3.641476

N22 fo20 0 N2 9.8943867009968007842018

1N2

N22 0.00145

NO CUMPLE

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ELEMENTO 3

3. SEGUNDA ITERACION

ELEMENTO 1


f3 KE3 u3 fo3 T3 f3

f3

0.12

13.236

4.201

0.12

13.236

3.641

fo3

13.236245

0.11991

4.200859

13.236245

0.11991

3.641476

N33 fo30 0 13.236244525530037460859387175

1N3

N33 0.00026 OK


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ELEMENTO 2

ELEMENTO 3


11

3

( )3cos ( )

sin ( ) ( ) cos ( )

1 0.97131545179998249836204205174

21

3

( )2sin ( )

sin ( ) ( ) cos ( )

2 0.99521105874420055338039453191

KE1

11590.498503

15287.729623

11590.498503

15287.729623

511.076388

15287.729623

20508.340783

15287.729623

20508.340783

383.307291

11590.498503

15287.729623

11590.498503

15287.729623

511.076388

15287.729623

20508.340783

15287.729623

20508.340783

383.307291

511.076388

383.307291

511.076388

383.307291

3194.227424

11

3

( )3cos ( )

sin ( ) ( ) cos ( )

1 0.98152615934230165479999593701

21

3

( )2sin ( )

sin ( ) ( ) cos ( )

2 0.99691763534597798788599600945

KE2

31974.138

0

31974.138

0

0

0

126.012175918909308265710294377

0

126.012175918909308265710294377

639.940969714236051592583403988

31974.138

0

31974.138

0

0

0

126.012175918909308265710294377

0

126.012175918909308265710294377

639.940969714236051592583403988

0

639.940969714236051592583403988

0

639.940969714236051592583403988

3199.70484857118025796291701994

INGENIERÍA CIVIL

3.1. MATRIZ DE LA ESTRUCTURA




11

6sx1 1 cx1( )

N33 1 L12

2 E I2

0.996041 31

4sx1 0.99868

21

6sx1 1 cx1( )[ ] 0.99934 4

1

2sx1 cx1 1.000661

KE3

999.0283144093194061

0

2004.6747510814039457

999.0283144093194061

0

2004.6747510814039457

0

39967.6725

0

0

39967.6725

0

2004.6747510814039457

0

5342.2670066702373982

2004.6747510814039457

0

2676.4319976553781974

999.0283144093194061

0

2004.6747510814039457

999.0283144093194061

0

2004.6747510814039457

0

39967.6725

0

0

39967.6725

0

2004.6747510814039457

0

2676.4319976553781974

2004.6747510814039457

0

5342.2670066702373982

KE

43564.636503

15287.729623

31974.138

0

0

15287.729623

20634.352959

0

126.012176

639.94097

31974.138

0

32973.166314

0

2004.674751

0

126.012176

0

40093.684676

639.94097

0

639.94097

2004.674751

639.94097

8541.971855

Fn

0

0

10

0

0

Fo

0

9.375

0

15.625

15.625

INGENIERÍA CIVIL


3.5. DEZPLAZAMIENTO DEL SISTEMA

3.6. VECTOR DE FUERZAS INTERNAS EN CADA ELEMENTO

ELEMENTO 1


1

23

4

5

1

2

3

4

5

En eje local

f1 KE1 u1

FE Fn Fo

FE

0

9.375

10

15.625

15.625

u KE( )1FE

u

0.005714

0.003686

0.006023

0.000331

0.002942

INGENIERÍA CIVIL

ELEMENTO 2


En eje global

fo1 T1 f1

f1

9.88

11.764

9.88

11.764

4.333

fo1

15.339071

0.845821

15.339071

0.845821

4.333149

N111 fo10 0 15.3390712958647681622446995342508

1N11

N111 0.000001

OK

En eje local

En eje global

f2 KE2 u2 Fofo2 T2 f2

f2

9.88

11.764

9.88

13.236

3.641

fo2

9.8801

11.763764

9.8801

13.236236

3.64149

N222 fo20 0 N2 9.8943867009968007842018

OK

INGENIERÍA CIVIL

ELEMENTO 3

4. COMPARACION CON EL PROGRAMA SAP 2000


1N22

N222 0


f3 KE3 u3 fo3 T3 f3

fo3

13.236236

0.1199

4.200813

13.236236

0.1199

3.64149

f3

0.12

13.236

4.201

0.12

13.236

3.641

N333 fo30 0

1N33

N333 0

INGENIERÍA CIVIL

5. SIN P-DELTA

A. DEFORMACION


INGENIERÍA CIVIL

B. FUERZA AXIAL


INGENIERÍA CIVIL

C. FUERZA CORTANTE

D. MOMENTO FLECTOR

5.1. CON P-DELTA

A. DEFORMACION


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B. FUERZA AXIAL


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C. DIAGRAMA DE FUERZA CORTNATE

D. DIAGRAMA DE MOMENTOS


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Problema 1

Documents

Transcript of Problema 1