apuntes de cálculo de estructuras

7/14/2019 Apuntes de Clculo de Estructuras

1/203

Apuntes de Clculo de

Estructuras

Grado en Ingeniera Civil

Escuela Politcnica de Mieres


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5

1.1 .

1.2 . E ,

.

/ :

E , ,

A,

O ,

M

D P

F

P C P

D

A (

) , ,

. C :

E.L. ( )


6/203

6

E.L. S ( )

E

. D , (, ).

L : (S275, S355) (HA25, HA35) (C20, D50)

T : , , ,

T : ,

T : , , ( )

C : , , , ,

S : , , , , .

T : , ( )

P : M F, M G, MM, C, E F, M G

,

, . E E

.

, .


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7

, , . E

.

. E ,

. O

.

S .


8/203

8

. L

. N .

S . A

( , ),

( ) .

E : .

E : , , ( )

E

S .

. S ( )

. S . (F. )

, . S

( )

( ) (

) . E . (F. )

. S

( ) . (F.

)

, . S

(

) ( )

( ) . E .

(F. )


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9

E , (, ,

). S ,

(

).

L , . S ( ). E

( ). E

.

L ( ).

L .

E .


10/203

10

L

: , , . E

. E .

L . E ,

.

E ,

. U

( ).


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11

, , , .,

, . S , ,

.

:

(, ). L ,

O (),

. E

.

,

.

L (N, N)

(N/,

N/). E (N/, N/). E (F, P) (, )


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12

( ). S

( 1.17). S 4

.

: ( 1.17). E

( )

. S 4 .

( 1.17). S

,

. S 4 .

E ,

, .

3

.

U 6 (...) 6

.

3 : , , , . 3 : , , , , , .

L , ,

( ) .

P

(R, R, R) (M , M, M),

.

6 , 6 (3

3 R, R, R, M, M, M) .

3 , 3 (3

R, R, R)


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13

E

. E 2D,

.

2

U 3 (...) 3

.

2 : , ,

1 : .

P

(R, R) (M),

. E .

, 2 (R, R).

, 1 (R R,

).

3 , 3 (2 1 R, R, M).

1 ,

2 (1 R R 1 M).


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14

Apoyo articulado fijo Apoyo articulado mvil Empotramiento

CASOS REALES DE APOYOS EN ESTRUCTURAS DE ACERO

Apoyo articulado fijo Apoyo articulado mvil Empotramiento

CASOS REALES DE APOYOS EN ESTRUCTURAS DE HORMIGN

(

, ) ( , ). L , R M ,

( , = S).


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15

L

( , ) , ,

.

E

.

E

. S

(A, M C)

, .

. L

.

. S

.

( ,

, )

P : GH= R 3=R3

3 , 3 ( M,

V N). S , ,


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16

. E

, (M=0).

P : GH=3 C C(B A1)=

=3CC(BA1)

H ( 1) .

GH=GH+ GH

L , :

GH= B+ R2 =B+R2N

E (

). L ( ) .

L :

GH= R 3=R3

, ,

. GH=0. S

.

GH=R3=33=0 GH=3CC(BA1)=30[(11)+(11)=0 GH=GH+GH=0

O : GH=B+R2N=1+322=0

GH=R3=33=0 GH=3CC(BA1)=30=0 GH=GH+GH=0

E ( ) . S

(M=0).

GH=R3=43=1 GH=3CC(BA1)=30[(21)+(11)=1 GH=GH+GH=0


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17

(GH0). E

.

. S ,

.

GH=R3=43=1 GH=3CC(BA1)=30[(11)=0 GH=GH+GH=1

3 . S , .

GH=R3=63=3 GH=3CC(BA1)=30=0 GH=GH+GH=3

.

.P :

: ,

, H. : . L .


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18

G :

. S .

. N .

: (1+2) 1

2.

Equilibrio en la estructura sin deformar: teora de primer orden

Equilibrio en la estructura deformada: teora de 2 orden. Se tiene en cuenta el desplazamiento Y 1 para obtener losmomentos.


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L :

, . S . T

(N, N, .).

. S

. T (N, N, , .).

(

. S . T (N, N, ,

.).

, .

E ,

. R :

2 (R, R).

1 (R R, ).

3 (2 1 R, R, M).

2 (1 R R 1 M).

L

. :

F = 0

F = 0

M + F ( )=0


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20

D ,

. L 1

,

3 .

E,

.

S 3,

,

. R GH=R3

P . E

:

M =0

E 3 .

P :

1. C .

2. C

. P ,

.

3. L ( ) (

)

: M + F ( )4. L ( ) .

5. L ( ) .

P ( )

. P (N/) L (),

L (N) .

O (, )

( ) M, C A .

q

L L

qL


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21

(V)

(M) .

. E.:

. E.:

, . E.:

. L . E

.

.


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22

( )

. D

.

.

0, . E . E

, .

N,

:

,

:

1

=0

H

(). =0

RB

RH

3P/L

RA


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23

( A

). S

( ).

=0

D R=P/2

I R R=P/2

E .

E .

E .P , .

P , , ,

.

C . D

.

E P/2.

E ( )

E .C .

N .

2

3P/L

P/2

3P/LL/6

P/2L/6L/12

L/2 L/2

5PL/24

P/2

P/2

RB

RH

M

RA


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0 =0

( A

)

=0D R=M/L

I R R=M/L,

.

E .

E M

E .C ( , )

( ).

C . D

.

E .

E ( )C .

N .

3

0 =0

M

M/2

M/2

M/L

M/L M/L

RA

=PL

RB

RH

PL P


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25

( A

)

=0D R=0

I R R=P

E .

E M

E . P .

C

( ).

C . D

.

E . P B .

E ( )

L P.C .

N .

1

=0

=0

( B

, )

=0D M=M

P

PL

P P

RB

RH

M MB


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E .

E M

E M.

E . P M .

C

( ).

C . D

.

N .

N .

2

P =0

=0

( B

)

=0D M=0

D R=P

E .

E M=PL

E . P M=PL.

E .C

( ).

M

M M

RB

RH

P MB=PL PL

P


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27

C . D

.

R,

P. T PL B

0 M.

E R=P.

E ( )

L P.C .

N .

1

+ =0

=0

( B

)

=0

N 3 ( ) ( ). D :

, ( 0) :

M=RL/2=0

L =0

S M=M,

B .

D =0

E .

PL

P P

MB

RB

RHRA

M


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E M

E M=M.

E .

E . P M .

C

( ).

C . D

.

N .

N .

1

+ =0

+ =0

( B

). A, R

B.

=0N 5 ( ) (

, ). D

:

1, ( 0) :

M=RLM =0

MM

M

M

RA

RHB

M

RHA

RBMB

rot2rot1


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2, ( 0) :

M=RL+M=0

D 1 : R=M/L

S : R=M/L. C

, R ( ).

S R 2 : M=M

S M : R=M/L

S R : R=M/L. C ,

R ( ).

E .

E M

E M=M.

E .

E . P M .

C

( ).

C . D

.

E .

E ( )C

( ). E

, M/L.

C .

M/LM/L M/L

M/L M

M

M

M


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E .

E ( )C

( ). E

, M/L.N , ,

(+ , ).

C .

2

+ P=0

+ + 2P=0

( B

). A, R

B.

=0

M/LM/L

M/L

M/L-M/L

-M/L

P

RA

RHB

RBMB

rot2rot1

RHA

2P


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N 5 ( ) (

, ). D

:

1, ( 0) :

M=RL =0, R=0

2, ( 0) :

M=RL+M=0

S R=0 : R=2P C

, R ( ).

S R 2 : M=2PL

S M : R=P/2

S R : R=P/2.

E .

L .

E M=2PL.

E .

E .C

( ).

U .

E ,

:

C . D

.

2PL

2P

P/2

P/2

2P

2PL

PL/4


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32

E .

E ( )

L P.C

( ). E

, P/2 .

C .

E .

E ( )C

2P ( ). E

, , 0.

N , ,

(+ , ).

C .

1

+ =0

=0

2P

P/2

P/2

-P/2-P/2

MB

RB

RH

RA

M


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( B

). =0N 3 ( ) (

). D :

, ( 0) :

M=R2L+M =0

D : R=M/2L C

, R ( ).

S : R=M/2L.

S R : M=M/2. C ,

M ( ).

E .

E M.

E M=M/2.

E . P M .

C

( ).

C . D .

E ( )

E .

L .

T ,

.

C .

N .

M/2L

M/2

M

M/2LM/2L


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2

+ =0

=0

( B

).

=0N 3 ( ) (

). D :

, ( 0) :

M=R2LM =0D : R=M/2L

S : R=M/2L. C

, R ( ).

S R : M=M/2.

E .

E M.

E M=M/2.

E .C

( ).

U

. J ,

:

C . D

.

E ( )

E .

L

.

T .

MB

RB

RHM

RA

M/2L

M/2

M/2

M/2


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C .

N .

a=L/3

M1=M2=PL/2


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1

2 1 1 S GH=R3=33=0

GH=3CC(BA1)=31=3GH=3

2

1 1

1 3 (R, R M)S S GH=R3=63=3

GH=3CC(BA1)=0GH=3

3

E 2

2 21 10 GH=R3=43=1GH=B+R2N=21+4210=5GH=GHGH=51=4


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4

6 1 2 . C 3 .

GH=R3=63=3GH=3CC(BA1)=31[(21)+(21)=1GH=4

GH=GH=1

GH=GH=4


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S . P :

A

C

B

E

S ,

. N .

1. L . D

,

. L , ,

.


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2. H . E

. P

:

C

. P ,

.

P

P

P

P/2

P/2

P/2

P/2

q

L/2L/2

qL/2

q

q

qL/2


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3. L ,

2D .

E :

) Q , ,

.

) Q

. D

.

:

D S A( X).

N

.

S N/

, B L.

C := B/2 [N/

C :P = L= BL/2 [N

L (, , ,

, .) , ,

.

E , ,

,

.

PP P P P

P/2P/2


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S , .

A .


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P ,

. E

,

( ) .

A , . L

.

. G

. E

.

. S

, , . E

.


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S ( ) ,

,

( )

S ( ).

S 2 .

S .

S 2 . A .

C . E 3 .

S . E

. T

.

(1) (2) (3) (4)

(1) (2) (3) (4)

Cordn superior

CerchaViga en celosaCordn inferior

Diagonal

Montante

MontanteCordn su erior

Cordn inferiorDiagonal


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45

C

.

S

. L , ,

( ).

G =N +N 2(N )

+2

0

0

0

E

.

. C

. .

. L .

.

. S

,

.

Generacin internaGeneracin externa Unin por tres barras


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2

( ) .

S :

1. L .

2. L .

E : F=0

E : F=0

E . A,

, ,

. E

.

3. A

. E ,

. A,

.

NINII

R FF

NI RNII

F

Fuerzas sobre el nudo Equilibrio de vectores de fuerza

III

B

F

Nudo con 2 barras alineadas

Equilibrio imposible en el nudo ALneas de accin de las reacciones concurrentes

Equilibrio de momentos imposiblerespecto a B

CELOSAS DE VARIACI N INSTANTNEA

AF

FF

F

Variante Invariante Variacin instantnea


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4. E 0

.) 2 : 0.

) 3 , . L

0 .

L , :

1) O ,

, .

2) E 0.

3) P (2 )

2 (2 ),

.

4) A

.

2

. .

(a)

FF

(b)

0

0

0

F

Eliminacin sucesiva de barras de esfuerzo 0

F

NI=TraccinNII=Compresin

R F


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E .F=0 R+RPP=0F=0 0M=0 R4LP3LPL=0

E , . U .

E , . H ( 4 7) . T .

S ( 1).

C . D

.

L ,

:

0

0 0

0

-P

+P +P

-P 2 -P 2

De modo simplificado:

0 0

0

0 0

0

-P

+P +P

-P 2 -P 20 0

P

Nudo 5

N56

45

Solucin nudo 5

45N52

-P 2

P P

-P 2

P

Nudo 1

N15

N12

45

-P

-P 2

P

Solucin nudo 1

PP

P P

0

0 0

0

1 2 3

4

5 6

7

PP

3L

4L

L

RV1

RV3

RH1

L

1 2 3

4

5 6

7

PP

2L4L

L


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49

E

.

L , :

1) O ,

, .

2) E 0.

3) A 3 (3 ). L 3

. H

3 .

4) P 3 . L , ,

1 2 ( ).

S ,

, .

0

0P

VR

H

C

B

MB=0 N5

MC=0 N2FV=0 N3FH=0 N4

Otra posibilidad de seccin:

Corta 4 barras con esfuerzo no nulo, pero 3 deellas concurren en B y as es posible resolver

planteando inicialmente MB=0

N2

N3

N4 N5

R

P

0

0P

VR

H

V

H

N1

N2

N3

A

B

MA=0 N3

MB=0 N1

FV=0 N2

Considerar barras deesfuerzo 0 y obtener

reacciones

Mtodo de lassecciones


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C

.

L

:

L .

L .

C ,

M

N. A, M/

M.

/

P . S

, C M ? P

R

. O , , ,

,

.

S

MS

VS

MSVS

NcordS

NcordI

Ndiag

h


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51

E

:

C .

C .

L

. C N=V:

=/

E .

S

I. P

, :

A 75% . S 0,75

.

I=0,75I

E , A S. N

I,

S.

I=2[I+A(/2)

2A(/2)


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S .

D

.

L PANDEO. E

.

L , ,

. I .

L 2D

.

S 3 (3 ).S 3 .

S S .S () 6.

E , +3

S (N, R,

), .

RyRx

Rz

VIGA PRATT: diagonales traccionadas.BIEN

VIGA HOWE: diagonales comprimidas.MAL

V


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53

E

:

(

. S . T

(N, N, , .). S ( )

( ).

R .

=0 =0

( A

).

=0

D R=P/2

I R R=P/2

E . D ,

P .

PA

A B

C

D

LL

C


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N

.

D . S

(). D .

E AB AD =(0,5)= 26,565

.

T A

B. E .

A

B

C

D /2/2

PP

P+1,118P+1,118P

B N

N

A

/2

N

N

A

B

C

D /2/2


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R .

E

.

D . S

(). D .

D . S

(). D .

N

N

P

PAN

N

P

AB

C D

L


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R , .

=0 =0

( A

).

=0

D R=P/2

I R R=P/2

E . D ,

P .

A B C

D E

L

LL

A

B

C D

+P

P+P2

P


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57

N

.

D . S

(). D .

T A

D B. P B 3

, ,

. P ,

.

P N=P/2N=N

P DE D. P

, A D

. N B.

B

45

N

N

/2

45

N

A

/2

45

N

N

A B C

D E /2/2


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P :

T D:

T B:

E :

E A:

A B C

D E /2/2

2

A

D/2

N

N

N

A B C

D E /2/2

1

A B C

D E /2/2

P/2 P/2

+P

+P/2 +P/2P/2 P/2

D

45

N

/2

45

/2


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E , , .

L 2L

:

V 0 PL/2. P

M

.

E

. U N D. P :

E ( ) . U

N B. P :

PL/2

P/2

P/2

A B C

D E /2/2

P/2 P/2

+P

+P/2 +P/2P/2 P/2

A

/2

45

N

N


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E .

D

, :

E , , .

:

L L

P20

+P+P +P

L L

A B C

D E /2/2

P/2 P/2

+P

+P/2 +P/2P/2 P/2

/2

/2

N

N

/2

/2

N

N

-(Barra BE)+(Barra EC)

+(Barra AD)-(Barra DB)


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: 67, 34 28

:

: =102

:

A


62/203


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63

3.1 .

1. : .

2. : H,

.

3. :

.

4. :

. E , .

5. :

.

6. :

. S 2 5.

3.2 . P

:

: , ( ) .

: .

: H, .


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64

E .

(X, Y, Z).

.

S (, , ) .

.

A (, , )

:

S , 6

( ).

. C H. L L H

L. E


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65

( M Y) P . E

.

.

E

(, ) . Y

,

. P ,

PTV.

()S :

. C

.

. T

.

(A )


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66

. T=T. S

.

P :

1. U .2. U .

S , PTV (PDV PFV)

U ( ) .

S ( ),

. T , .

U ( ) .

S .

.

E (F, R, .)

(, , .) (M) () E ,

.

E ,

PTV .

E ,

PTV. E , F,

.


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67

.

E (N, V, M, M ) .

E ,

.

E . E . L

( ).

L :

C (EA, GA, EI, GK) E

, G , A , A

, I K .

E :

1. E

.

+++= Lt

LLLi dMdMdVduNT


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68

2. E

.

:

S ,

() .

E PFV .

:

S , ()

.

E PDV

.

...

... ( ) I D D

H D F E.

... (. ) I D . E

H D G E.

C , PTV .

D :

1. E .

2. L

. E

.

3. S , ,

.

4. E PFV ,

.

T i = N duL + V d + M d + M t d

L

L

L


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69

D ( )

L P .

P

. C ,

, , ,

.

T ,

.

Sistema real de desplazamientos y deformaciones Sistema virtual de fuerzas externas e internasA , P F V,

.

T = ()

T = +

N ()


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D :

N .

3 3

1. 3 3BB

L L

e iT TPL PL PL

M VM dx V dx EI GA EIEI GA

=

= + = +

P

, . S ,

.

A,

M M , ,

L/3 M M. S PL L

PL/3. E (EI), ,

.


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71

S ,

.

D

:

1. S

2. E

3. E ,

. L ( )

.

D (

) .

D A

PDV.

P

,

R. E,

( ) .

C , (

) , . E ,

. L

.

A , P D V,

.

T =R + C

BA

P

L /2 L /2

C


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72

T =0. Y .

N (R)

P , , ,

P

( R).

D C

PDV.

P

. E,

.

E M . S , M . C ,

.

T = C

T = M

N (M)

A A A

e i

e

i

T T

PT R P R P 0 R2 2 2T 0

=

= + + = =

=

BA

P

L/2 L/2

C


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73

C ,

, ,

. A (

) .


74/203


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75

C ,

(GH>0). P ( )

P T V (P.D.V.

P.F.V.)

C , P D V

M (

) F F.

E M F,

. P

.

P M F

. L , 3 :

, (R, R, M). S

,

( ).

GH=R3=(3+)3=

N .

GH=3CC(BA1)=0

1 2 i k n......

pP

M

...

A . E

, . A,

, .E .


76/203

76

.

S ,

.

1. E . Y

C 1:

GH=R3=(3+)3=

GH=3CC(BA1)=0

GH=

S

GH=B+R3N

2. E

. E ,

. A,

( ).

3. A ,

. E .

D =

+

X.

E (

). E . D :

01

m

i i ik k

k

X =

= + =

D:

= .


77/203

77

= .

X=

= ()

( )S PFV M C U.

P (

)

.

E, PFV : 1=T 0.

1.L L L

L L L

e i

o o o

io i i i i mo

muelles

o o oio i i i i mo

muelles

T T

M N VM dx N dx V dx F

EI EA GA

M N VM dx N dx V dx FEI EA GA

=

= + + +

= + + +

P : 0

( ) ( ). E

.

A, . E

( ) ( ).

1 2 i k n......

pP

M...

io

1 2 i k n...... ...

1


78/203

78

P

.

1 2 i k...... ...

ik

Xk

= 1

1 2 i k...... ...

1

E, PFV : 1=T .

1. L L L

L L L

e i

k k k

ik i i i i mk muelles

k k kik i i i i mk

muelles

T T

M N V

M dx N dx V dx FEI EA GA

M N VM dx N dx V dx F

EI EA GA

=

= + + +

= + + +

P :

( ) (

). E

.

A, . E

( )

( ).

C

.

C

:


79/203

79

1 1 1

1

1

1

0

. . .

. . .

0

n

o k k

k

n

i io i k k

k

n

n no n k k

k

X

X

X

=

=

=

= + =

= + =

= + =

E . P

.

C ,

. E :

C +1 : X (0) , .

O .

L +1 .

+=

+=

+=

=

=

=

n

iio

n

iio

n

iio

1i

1i

1i

XVVV

XNNN

XMMM


80/203

80

ECAD: E ,

.

DA: = 6E/5

AE= E/65

E =

AE = A= .

L EI ( E I).

E EI .

L , AE( ).

E AE

, . L

GA= .

L 1:

E F ( ). P

, . E

A .

GH=R3=53=2

P ,

B . A

2 , AB .

P

Sm

AE EI

EI

EI

A

DE

x

y

F

B

EI

4m

2m

2m 1m

2m3m


81/203

81

GH=3CC(BA1)= (21) =1

E GH=21=1. P

.

L

: . D

,

. E AB

A . P 4 (3 F 1 A).

GH=43=1

P ,

. C

,

. A, AB A, ( )

, X.

P , A

X .

O

X AB. L .

S .

P

X1

Sm

AE EI

EI

EI

A

DE

F

CB

EI

X X

A B


82/203

82

A:

D A = A

+ A .

= = + = 0

= 1 ( A). E 0 A.

= 1 .

= 1 1.

= 1

( 0)

( 1).

0 1 , E ,


83/203

83

1

. E

2/5 1/5 .

( 1 0) .

GA= AB EA= .

C , . D

,

:

D , ( 2):

E , AB

0 .

E , 0

0, 0 0.

:

D

( 1).


84/203

84

P ( 1 1)

.

D , :

E AB , EA, :

E :

P :

P A, 0:

= + X= 0

S , .

L A P/9 (

).

S 0 1

.

M , M = M+ MX


85/203

85

E , = +

E , = +

D .

:

E . E

.

. C

.

:

. (

)

E

:

.

( ) .

.

E

:

C A, :

>


86/203

86

E ,

.

E , ,

.

( 0).

Sistema hiperesttico Sistema base sin incgnitas=S0

E 0 =

. E ,

,

, , .

Sistema hiperesttico Sistema base sin incgnitas=S0


87/203

87

0

= . E ,

, , ,

. (EA) .

0

=()/ . E ,

, ( )

()/ , ,

. (EI) .

: .

Caso de carga simtrica Caso de carga antisimtrica

, .

A A

A A

, ,

:


88/203

88

L . P

, :

( ).

E .

E /

, V .

P

. E

.

A .

Carga simtrica Carga antisimtrica


89/203

89

.

E

( ).

.

,

:

.

.

E

/ ,

.

. E

. A

.


90/203

90

,

,

, ,

. I

.

E (

) .

.

C ,

, :

1. (

)

2. E ( ).

3.

.

4. C (

0) ( 1, 2).

5. .

6.

.

. E ,

3, .


91/203

91

E C 2 2.5. S ,

.

C AE. S

. T 1

Sistema hiperesttico (sobra una barra para serisosttico)

Sistema base (sistema isosttico con incgnitahiperesttica X1)

GH=B+R2N=6+324=1 GH=R3=33=0 GH=GHGH=1

D ,

: X.

L C X . L ,

.

S , .

1 .

1 1 X.

Sistema 0 (sistema base sin X1) Sistema 1 (con carga unidad en la incgnita X1)

E 1 0, . E

1 1 11 1 0o X = + =

L

( ). A,

.

i

barrasn

1i

1ii

barrasn

1i

ioo LAE

NNyL

AE

NN 1i111i1

=

=

=

===

P1

P2A

B

P1

P

X1 X

1

1

1

A

B

P1

P2

1

1

A

B

1

1

X1= 1

X1

= 1

A

B


92/203

92

E X

X 1 X

.

P D

,

D .

Sistema real de deformaciones (sistema total) Sistema virtual de fuerzas (carga unidad)

E ,

.

11

1.

n barrasin barras

i iii ii

i

VD

VD

TVE TVIN

N LNN L AE

AE

==

==

=

==

E ,

D .

DA: = 6E/5

AE= E/65

E =

AE = A= .

P 1 P 2

B

V D D

C A

1

B

A

D

C

P

Sm

AE EI

EI

EI

A

DE

x

y

F

B

EI4m

2m

2m 1m

2m3m


93/203

93

A, AB.

.

.

A

.

AB.

,

.

.

(AE E

).

E =6/9

6/9

.

E . E

.

AB. C ,

AB.

E .

.

2P/EI

2P/9EI16P/9EI

6P/9EI

P/9AE P/9AE

1t

2tm

2tm


94/203

94

( )

1muelleVD m

L L AB

TVE TVI

M NM dx M N dx

EI AE

=

= + +

D EF,

:

A, DE:

E , ()

.

T AB

.

E :

A , ( ).

P :

E ,

.

E .P

.

(16P/9+6P/9)/20=(16P/9)/=1620/22=165/11

D .

E , . A B

=(N/AE)L=P3/9AE , AB .


95/203

95

E : =/.

.

E

, :

C .


96/203

96


97/203

97

4 1

2 . C

3

.

G=3=43=1

G=3CC(BA1)=31(21)+(21)=1G=2

.

. , ,

().

, , :

C , , .

C , .

E 1,

.

D

,

:

.

E :

=++=0

=++=0E

=. (D )


98/203

98

E

2

1

8

5

G=3=33=0G=B+2=8+325=1

G=10=1

.

E

. D

.

, :

. .

D

,

: 1

1.

E :

=+ 1 0

.

2

1

3

1

G=3=63=3

G=3CC(BA1)=30(21)=1G=31=2

.

A ,

. A, :


99/203

99

C , , .

C , .

L , .

: D ,

: 2 2.E :

=+X+X=0=+X+X=0Q 2 0 .

XX

X

T


100/203

100


101/203

101

,

, , , ,

.

, 0,10 , 1,2.10

.

. .

C AE GA

. E

.

(1 ) (2

). .

G=3=43=1

,

.

G=1

.

,

.

.


102/203

102

: E :

=+X=0T ) ) , .

P . E , 1 X

( 0) ( 1).

L .

0: E 0 , .

S .

E . A , .

1:

E 1.S M, V N.1 O R=3/2, R=1/2, R=02 O

E 0 1, :

3/2

1

1/2

L/2

1

=T=1,21050

)T=50C

X

)T=50C )T=20C T=20C


103/203

103

310 1 0

31 3,610

2

TVE TVI

N dx T L m

=

= = =

C , ,

, . E

.

E 1 1, :

31

11 1

1 1 1 11

3 2 2 2 3 2 2 8

TVE TVI

M L L L L L LM dx L

EI EI EI EI

=

= = + =

D ,

, . P

, L/2

L.

D +X=0 :

101 2

11

12

T EIX

L

= =

A , EI, . L 1

.

L 0 1

X. = = 0+ 1

M , M = M+ MX. T M .

E , V = V+ VX. T V .

E , N = N+ NXT N .

D X

.


104/203

104

P

=T

.

1 B:

E , B.S M, V N.

1 O R=1, R=1, R=12 O

D :

E M M

E N .

N ( ).

N ( ).

1 6 11 1

3B B B

TVE TVI

M TEIM dx N dx L L T L T L

EI L EI

=

= + = + =

C ,

B.

11

1 1

1

L

L

1

18TEI/L

12TEI/L

6TEI/L

6TEI/L


105/203

105

1 D:

E , D.S M, V N.1 O R=1/L, R=1/L, R=02 O

D , . S

.

1 6 1 11 1

3D D B

TVE TVI

M TEIM dx N dx L T L T

EI L EI L

=

= + = + =

0: E 0 , .S .S ,

. (TT)/2=0

E . A , . , .

1:

E 1.E )

=(TT)/

)T=20C T=20C

1/L

1/L

1

1


106/203

106

E 0 1, :

10 1 01 0

TVE TVI

M dx

=

= =

C , ,

, . E

. A M

, .

E 1 1. E ):

31

11 1

1 1 1 11

3 2 2 2 3 2 2 8

TVE TVI

M L L L L L LM dx L

EI EI EI EI

=

= = + =

D +X=0 :

1 0X =

E , ), .

S : M, V N . S

CD.

P

=(TT)/

.

1 B:

E

, B.Y ).


107/203

107

D

E M .

N ( , )

.

2( ) ( )11

2 2i s i s

B B

TVE TVI

T T T T LM dx L L

h h

=

= = =

1 D:

E , D.Y ).

N , (

) . S MD .

1 0D D

TVE TVI

M dx

=

= =

C ,

D. P

=0 ).

1/L

1/L

1

1


108/203


109/203

109

,

, ,

1. ,

.

2. ( ,

).

3. .

2,1.10/

, 1210

, 1627010

, 0,30 1,2.105

3330

3(1)31(21)+(21)1

1

.

. , .

+0.

.

0


110/203

110

E 0 , , .U AC AB.

=T=1,21025=310

=(TT)/=1,210

[12,5(12,5)/0,3=10

1

E 1.R:

R=0R=R=0

D .D AB BC,

. S

. T

1 0,

0 . A, 1 0.

3 410 1 0 1 0

1 810 17 1310 30 0,077

2M dx N dx m = + = + =

21 111 1 1

1 1 1 2 8817 130 2,1310

3AC

M NM dx N dx m

EI AE EI AE

= + = + =

D =+X=0.

X=0,077/2,1310=3,61 N

:

M , M = M+ MX.

E , V = V+ VX.

1 1

1 1

T


111/203

111

E , N = N+ NX.

E

. H .

A .

S : A.

R:R=1

N , . E . A 0.

N ,

.

38 8

1 1 1 2 28,9817 810 17 0

3 2,110 1627010 2

0,00864

A A A A

MM M dx N dx

EI

m

= + + = + + =

=

B .

S : B.

1


112/203

112

R:R=1/30R=1/30

N , . E . A 0.

N ,

.

31 0 0,510 17 0 0,0425

2

B B B B

MM dx M dx N dx rad

EI

= + + = + =

E .

A.P

. E 2, :

A, ,

.

S : A.

R:R=1

N ,

.

4

4 8

1 13,630 1310 30 0,00857

1210 2,110A A AN

N N dx mAE

= + = = L .

C =M/EI+

,

. E ,

.

E .

1


113/203

113

,

1. ,

.

2. ,

.

3. .

, 510

,

3532

3(1)0

2

,

.

.

,


114/203

114

D , , . P X=X.A, X= X=X. A B F .E :

=

+

X=0

0: P 0 . A .

1:

E

1 2 . P . R .

D . N , .

T ,

(EA=). A ( ).

010 1

1 1 2 2202

2

MM dx

EI EI

= =

20 N

10 N10 N

XX


115/203

115

111 1

1 1 2 222

3

MM dx

EI EI

= =

D +X=0

X=/=15 N

:

M , M = M+ MX.

E , V = V+ VX.

E , N = N+ NX.

O M N V.

B F , , =.

P , . P .

S : B.

R:R=1M=2S , .

N ,

M.

55

1 1 1 1 140 140 2202 (20 10)22 9,3310

3 2 3 3510B BM

M dx mEI EI EI EI

= = + = = =

59,3310F

m =

.

C E , , =. P ,

. P .

S : C.


116/203

116

R:M=1S , .

N ,

M.

55

1 1 10 10 (20 10)12 210

2 510D DM

M dx radEI EI EI

= = = = =

5210E

rad =

.

P

. A,

.

N

EA=.


117/203

117

, AB

BC, .C:

. D (,

).

. D C

D: AE=3E/2, =73/(8E)

:

.

E , , E, .

E :

=+X=0

X=/M

L LL/2

1 1

1 1

1

0

T

X X

X X

B

(=1)

T

E, I, L

A

B

C

D

A, E, , L

E, I, L

T


118/203

118

:

:

X=/=L

D :

=M+MX

=V+VX

=N+NX

D

:

=

L

qLL/2

qL2

T

N

+1

N

1

V

L

V

*/EI + * /AE2

*/EI + *


119/203

119

D C:

M

L

1

S

C*/EI

L


120/203

120

E

:

D . B D E

DA:

E=1,310

; =10 ; =260 ; =

/2E

=9,3610

= =

=6,9210

P .

L,E,IL,E,I

H

A

S=EI/L

M=PL/2

P

L/2L/2L/2

R EI

S=4EI/L

S=EI/L

P/2

L, E, I

R EI

S=3EI/2L


121/203

121

:

D D A B

DA:

E=10000 ; =4 ; =4 ; =50

C;

=10C; AE=8000 ; AE==1,8310

A B

DA: AE=48E/

L/2,E,I L/2,E,I

2,A,E,A,E

=30=30

2QQ

T

L,E,I

L,A,E,


122/203


123/203

123

C C 1, :

1. ( )

2. (, , )

3. .

E

() . D

:

) C:

) P : ( )

.

) E: ( )

.

) P :

.


124/203

124

L

. E M M

I. P

.

E ,

.

E : E, C L

:

M F ( ). R

. M R ( ). P .

L . L

. E

, ,

. P ,

.

A (

) (

). E ,


125/203

125

A

,

().

S

( G.L.).

. L

.

L . P

, .

C ,

.

C =

( GL ) ( 1 GL 0 )


126/203

126

S . E ,

(,).

S ,

. E

3

(, , , , , ) (F, F, F, M, M, M). S

,

.

P 2

(, , ).

E

(F, F, M)

.

u (A) F (A)

AA

x

A A

y

A A

Fu

v F

M

= =

E

(, ) .

E

(F, F) .

u F x

y

Fu

Fv

= =

E

(, , ).

E

(F, M, M).

u Fz

x x

y y

w F

M

M

= =


127/203

127

E 3

(, , )

.

E 3

(F, F, F) .

u Fx

y

z

u Fv F

w F

= =

( )

P

3 3

(, , , , , ).

E 3

3 (F, F, F, M, M,

M) .

u F

x

y

z

x x

y y

z z

u Fv F

w F

M

M

M

= =

S

, . L

. L

(,

).


128/203

128

L

, .

E (

), ( )

.

1 1

2 2

3 3

4 4

5 5

6 6

P

NV

M

N

V

M

= =

L

.

E .1 1

2 2

P N

N

= =

L

,

.

E ,

.

1 1

2 2

3 3

4 4

5 5

6 6

F

V

M

T

V

M

T

= =

( )

3

4

5

6

7

8

9

10

11

12

= =

1 1

2 2

3

4

5

6

7

8

9

10

11

12

N

V

VT

M

M

N

V

V

T

M

M

P


129/203

129

E , ()

. E

.

E A

,

.

E B

.

S

. E , .

P (

). P P

, H

.

A, : (

) .

P ,

.

L , ,

. E

(= ).

= M ( )

= M ( )

C , ,

.

L :

P ,

1 ...


130/203

130

P

( ).

C

3 : .

S 0

:

P :

... () 1 (=1) 2 (=1) 3 (=1)

1 (N) EA

L

2 () 312EI

L

2

6EI

L

3 (M) 26EI

L

4EI

L

1 .

E

1

=

M

M

1


131/203

131

E

. =

A A 1:

A,

,

:

3 2

2

0 0

12 60

6 40

EA

LN uEI EI

V vL L

MEI EI

L L

=

E 3 ,

3 33.

11 1 1 11 1

1

1

1

i j n

i ii ij ini i

j ji jj jnj j

n ni nj nnn n

k k k k P

k k k k P

K K

k k k k P

k k k k P

= =

L L L

M M M MM M

L L L

M M M MM M

L K L

M M M MM M

L L L

,

.

= ,

.

E :

= +...+ +... + +...+

, .

1


132/203

132

P . L

...

.

1 0 1 00 0 0 0

k1 0 1 0

0 0 0 0

EA

L

=

S 66 3 ...

.

E

.

I

EA

L

sim0 12EIL

3

0 6EI

L24EI

L

EAL

0 0 EA

L

0 12EI

L36EI

L2 0

12EI

L3

0 6EI

L

2

2EI

L

0 6EI

L

2

4EI

L

K =

K=

12EI

L3

6EI

L24EI

L

sim

12EI

L36EI

L212EI

L3

6EIL

2 2EIL 6EIL

2 4EIL


133/203

133

E , 6

. C 2 , 2 .

K=

12EI

sim

0

4EI

L

0 GJt

L

0

0

0

12EI

L3

6EI

L2

00

6EI

L2

2EI

L

0

6EI

L2

4EI

L

L3

6EI

L2

GJtL

GJtL

12EI

L3

x

y

z

2

6

8

12

1185

2

14

6 34 10

5

3

1

9

11

7

912 10

7

K =

EA

L

0 12EIz

L3

0 0 12EIy

L3

0 0 0GJt

L

0 0- 6EIy

L2

04EIy

L

0

6EI

L2 0 0 0

4EIzL

EA

L

0 12EIz

L3

0 0 12EIy

L3

0 0 0GJt

L

0 0

-

6EIy

L2

04EIy

L

0 6EI

0 0 0 4EIz

L

EA

L

0

0

0

0

0

-

L

2

12EIz

L3

0

0

0

6EI

L

2

12EIy

L3

0GJt

L

- 6EIy

L2

0

0 0

-

-

z

0

0

00 0 0

0 0- 6EI

L2

z

06EIy

L2 0

0

0

0

2EIy

L

2EIz

L0

sim

z

-

z


134/203

134

E .

( ),

,

.

, .

.

:

= +

= +

, :

'

'

cos

cosx x

y y

P Psen

senP P

=

, :

1'

'

cos

cosx x

y y

P Psen

senP P

=

D , ,

:

cos

cos

senR

sen

=

=

z

Py'

Px

Px'

x'

x

y

y'

Py

ny'

nx'

P

Pysen

Pycos

Px sen

Px cos


135/203

135

S

( 66) , :

i i

j j

i i

j j

T

T

T

T

R 0P' P'

P' P0 R

R 0''

' 0 R

T

T

P L P

y

L

= =

= =

E

.

T

cos 0 0 0 0

cos 0 0 0 0

0 0 1 0 0 0L

0 0 0 cos 0

0 0 0 cos 0

0 0 0 0 0 1

sen

sen

sen

sen

=

L :

( )P ' ' ' ' 'T T Tk L P k L P L k L = = =

C P=, :

' Tk L k L=

S L:

cos 0 0 0 0

cos 0 0 0 0

0 0 1 0 0 0L

0 0 0 cos 0

0 0 0 cos 0

0 0 0 0 0 1

sen

sen

sen

sen

=

A,

L L.


136/203

136

P . D

66 (3 .. ).

S ,

, 3 ... .

P

P

:

P=

P

F . E .

F=

E , , ,

.


137/203

137

E

1, 2 3.1 2 3

1

2

3

a a a a

b b

c c

d d

u

u

u

u

= = =

==

=

E , :

i i i B u

( 1) ( ) ( 1)d d n n

=

=

=

= .B= ( )

=

=

.

L .

1 2 3 0a a a a

F P P P = E ,

:

i i iF

( 1) ( ) ( 1)

C P

n n d d

=

=

F=

P=

.

C= (

)

L

. P=

11 1aa ab

a a

b b

ba bb

k kP

Pk k

=

() 1


138/203

138

22 2aa ac

a a

c c

ca cc

k kP

Pk k

=

() 2

33 3aa ad

a a

d d

da dd

k k

PP

k k

=

() 3

D :

1 2 3 0a a a a

F P P P = =

. E

.

( ){ { {

ab ac adaa

1 1 2 2 3 3 1 2 3 1 2 3a aa a ab b aa a ac c aa a ad d aa aa aa a ab b ac c ad d

K K KK

F k u k u k u k u k u k u k k k u k u k u k u= + + + + + = + + + + +1442443

D ,

.

D ,

.

1 1

i iii ij

j jji jj

n n

F u

. .. .

. .. .

. .. .

F u. . . K . . . K . . .

. .. .

.. .. .

. .. .F u. . . K . . . K . . .

. .. .

. .. .

. .. .

F u

F K u

= =

Fila"i"

Fila"j"

Col "i" Col "j"


139/203

139

9 .. (3 ) .

2.

.

,

:

cos 0 0 0 0

cos 0 0 0 0

0 0 1 0 0 0L

0 0 0 cos 0

0 0 0 cos 0

0 0 0 0 0 1

sen

sen

sen

sen

=

3.

...

. .

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9


140/203

140

E .

E M.

L ""

.

S , ( =

).

E

. L

, ...

.

L ...

. L

...

.

P ... :

L=

R= . E .

L 1, 2, 3, 16 17

.

E

.


141/203

141

:

L L

R R

L LLL LR

RL RRR R

F uF u K KF ,

K KF u F uu

= = =

:

E .

E (EACCE) F.

LL L LRF K u K uL R= + RL L RRF K u K uR R= +

:

LL LF K uL=

F

.

, , , ,

.

: R RL LF K u=

.

C

.

.

(, , ). A,

:

T TP' L P L k = =

, , :

T' ' ' k' L P k = =

,

.


142/203

142

E

( )

. S , (

.. , )

.

A,

,

.

E ,

.

E ()

()

.

E P

. P

. E

(P) .

C :

P=

P L:


143/203

143

P=LP

C

REACCIONES POR

APOOS EN CADA NO DE LOS G.D.L. RESRINGIDOS ( ).

(

)

' ' ' 'empP k P= +

E , :

P

. H

.

) ) ) ) ( )nbci

ij *iji i i i i itotal eq externa

j 1

F F F F C Pexterna eq

=

= + = +

P

.

) ) ) )ij ij ij ij ij ijtotal emp emp empT

P' P' P' ' ' P' ' L P'ij ij ij ijk k = + = + = +

E ,

. E

, 0,

P P

.


144/203

144

L

( ) ( ). C

(= L

).

P== L

L , ,

, .

P=L P

= LP

O, :

E

.

( )nbci

ij ij oiji total i externa i eq i externa i i eq

j 1

F ) F ) F ) F ) C P P=

= + = + +

Tij ij ij ij ij oijtotal emp empP' ) k' L P' ) P' )

= + +

C

.

A = 1 2( )

2

T T +

C = 1 2( )T T

h

D

.

P ()

. E ()

().


145/203

145

P ,

AB,

(). L B :

B

1 2( )

2

T Tu L

+=

B .

1 2( )T TLh

=

B A.

21 2( )

2

T Tv L

h

= ( )

D ,

.

.

1 2' #emp

21 2

1 2

0

00 00

( )0P k'' k' k' 2

( )

2( )

T TL

uT T

v Lh

T TL

h

+ = = =

P

:

E

(3 6).

1 2

1 2

' #emp

1 2

1 2

( )

20

( )

P( )

20

( )

T TEA

T TEI

h

T TEA

T TEI

h

+

= +


146/203

146

L.

P=L P

= LP

E

. S .. .

) ) ( )nbci

ij ij oij #iji i i i itotal externa

j 1

F F C P P Peq

=

= + + +

) ) ) )Tij ij ij ij ij oij #ij

total emp emp empP' k' L P' P' P' = + + +

S ,

C ,

L.

P=L P

= LP

E

. S .. .

) ) ( )nbci

ij ij oij #ij &iji i i i i itotal externa

j 1

F F C P P P Peq

=

= + + + +

) ) ) ) )Tij ij ij ij ij oij #ij &ij

total emp emp emp empP' k' L P' P' P' P'= + + + +

P

. E

. S L

D .

P'emp =k' ' donde ' =

0

0

0

0

0

& & &


147/203

147

L

.

E ,

0.

0D

u' u'

u' v'

' '

= =

P

:

LL LL D L

D DL DD D

F K K u

F K u F K K u

= =

S F,

, :

= L

F= L F

C ,

F:

' 'T T

L LL LD L

D DL DD D

F K K L u

F L K L K L u

=

S , ,

. A,

3 , , ,

.


148/203

148

E ,

. A:

1

2

3

0 0

siendo 0 0

0 0

ap

ii

k

K k

k

=

ap api ii iy F = K u

,

( ).

E , ,

. E

:


149/203

149

( )

:

1 0 1 0

0 0 0 0k

1 0 1 0

0 0 0 0

EA

L

=


150/203


151/203

151

,

.

: 210, 1,210

, 1 , 2,8610

, 2,7810

L ,

, . E

, (3 4)

.

P AB ,

. E ,

.

P

.

0 1 0 1 0 0 96

0 0 0 0 0 0 02' '

(2 ) 1 0 1 0 (2 ) 962

0 0 0 0 0 0 0

empAB AB

EAP k

T L T LL

= = =


152/203

152

.

.

44 60.

1

2

3

4

1 30 0

2 24896

3 1 0 0 0 48 32 2 '96 481 3

0 002 2 48 3

3 10 0

2 2

eqAB empAB empABP P L P

= = = =

(6).

,

.

11

22

3

4

5

56

6

48

48 3

48

48 3 100

R

R

F

R

R

+

=

1 1

2 2

43 33

44 44

5 5

6 6

0 0

0 0

2,8610

2,7810

0 0

0 0

uu

u

= =

.

.

= + = : =

+

60

1

2

4

3


153/203

153

1 1' 1'

2 2' 2'

4 3 3' 3'

4 4 4' 4'

1/ 2 3 / 2 0 01 0 1 0 0 96 115,5

0 0 0 0 3 / 2 1/ 2 0 0 0 0 02'

1 0 1 0 2,8610 96 115,52 0 0 1/ 2 3 / 20 0 0 0 2,7810 0 00 0 3 / 2 1/ 2

AB

EAP

L

= + =

BC

BC .

4 3 3' 3'

4 4 4' 4'

5 5' 5'

6 6' 6'

1 0 1 0 57, 22,8610

0 0 0 0 02,7810'

1 0 1 0 57,20

0 0 0 0 00

BC

EAP

L

=

=

=

=

= =

D .

57 2

115 5

4

3

6

5

57,2

1

2

4

3

115 5


154/203

154

, , :

) .

)

.

) .

:

10

5

510

210/

1,2 10

10

10

D . A

.

A AB BC,

:

1 0 1 0

0 0 0 0' '

1 0 1 00 0 0 0

AB BC

EAk k

L

= =

1 0 1 0

0 0 0 0'

1 0 1 02

0 0 0 0

AC

EAk

L

=

L

:

L

:

P AB: 0. L

.

P BC: 90

P AC: 45

C ,

.

2

14

3

6

5

A B

C


155/203

155

2 2

2 2

2 2

2 2

cos cos cos cos

cos cos ' /

cos cos cos cos

cos cos

T

sen sen

sen sen sen senk L k L AE L

sen sen

sen sen sen sen

= =

AB

:

1

2

3

4

1 0 1 0

0 0 0 0

1 0 1 0

0 0 0 0

AB

EAk

L

=

BC (90) CD (45) :

3

4

5

6

0 0 0 0

0 1 0 1

0 0 0 0

0 1 0 1

BC

EAk

L

=

11 1 1 12 2 2 2

21 1 1 12 2 2 2

51 1 1 12 2 2 2

61 1 1 12 2 2 2

2AC

EAk

L

=

. A,

( ).

5 61 2 3 4

1

2

3

4

5

6

1 1 1 11 1 0

2 2 2 2 2 2 2 21 1 1 1

0 02 2 2 2 2 2 2 2

1 0 1 0 0 0

0 0 0 1 0 11 1 1 1

0 02 2 2 2 2 2 2 21 1 1 1

0 1 12 2 2 2 2 2 2 2

m

EAL

mEA

L

S

EA SKL

+ +

= +

+

A/.


156/203

156

3 5 61 2 4

1

2

3

4

5

6

1,35 0,35 01 0,35 0,35

0,35 0,39 0 0 0,35 0,35

5000 1 0 1,04 0 0 0

0 0 0 1 0 1

0,35 0,35 0 0 0,35 0,35

0,35 0,35 0 1 0,35 1,35

K

=

D (2,3 4) (1,5 6) .

.

D .

10C AC.

1'

2'

5'

6'

00 1 0 1 0 6

00 0 0 0 0 0' '

1 0 1 0 62 ( 2 )0 0 0 0 0 00

empAC AC

AC

EAP k

TL L T L

= = =

(

45) .

1

2

5

6

1 1 0 0 6 11 1 0 0 0 12

' 3 20 0 1 1 6 12

0 0 1 1 0 1

eqAC empAC empAC P P L P

= = = =

(6).

,

.

1

12

3

4

5

56

6

3 2

10 3 2

10

5

3 2

3 2

R

F

R

R

+ =

+

1

22

33

44

5

6

0

0

0

u

uu

u

=

C , (1, 5 6)

0, .


157/203

157

(2, 3 4) :

LL L LRF K u K uL R= + , LL LF K uL=

0 .

2 3 42 222

3 333

4 444

LL L

10 3 20,39 0 0

F K u 10 50000 1,04 0

50 0 1

L

u

u

u

+ = =

10+3

10=50001,04

5=5000

(1, 5 6) :

RL L RRF K u K uR R= + , RL LF K uR=

0 .

2 3 41 3 21 1

5 3 35 5

6 3 466

3 2 2,95100,35 1 0

F K u 3 2 5000 1,92100,35 0 0

103 2 0,35 0 1

R RL L

R

R

R

= + =

:

==200(2,9510

)=0,59 ( )

= =200(1,9210

)=0,38 ( )

= +

AB

AB .


158/203

158

1 1' 1'

3 2 2' 2'

3 3 3' 3'

3 4 4' 4'

1 0 1 0 0 9,6

0 0 0 0 2,9510 0'

1 0 1 0 1,9210 9,6

0 0 0 0 10 0

AB

EAP

L

=

=

=

=

= =

BC . 90.

3 3 3'

3 4 4'

5 5'

6 6'

1 0 1 0 0 1 0 0 51,9210

0 0 0 0 1 0 0 0 010'

1 0 1 0 0 0 0 1 50

0 0 0 0 0 0 1 0 00

BC

EAP

L

= =

AC 45

( ).

1 1' 1'

3 2 2' 2'

5 5' 5'

6 6' 6'

1 0 1 0 1 1 0 0 0 6 13,3

0 0 0 0 1 1 0 0 2,9510 0 02'

1 0 1 0 0 0 1 1 0 6 13,322

0 0 0 0 0 0 1 1 0 0 0

AC

EAP

L

= + =

. C , .

2

1

4

3

9,6

13,3

5


159/203

159

, ,

) . .

) ,

.

) .

) .

10

10/

210/

10

120

/

10

. ( =

) . 2

.

,

.

3 2 3 2

1 3 2 4 3 5 4 6

1 32 23

2 4

3 2 3 2 3 5

4 6

2 2

12 6 12 6

6 4 6 2 1, 2 6 1, 2 6

' ' 10 6 40 6 2012 6 12 6

1, 2 6 1,2 6

6 20 6 406 2 6 4

AB BC

EI EI EI EI

L L L L

EI EI EI EI

L L L Lk k

EI EI EI EI

L L L LEI EI EI EI

L L L L

= = =

. ,

( ).

2

1

4

3

6

5


160/203

160

1 2 3 4 5 61

2

3 3

4

5

6

1,6 6 1, 2 6 0 0

6 40 6 20 0 0

10 1,2 6 2, 4 0 1, 2 6

6 20 0 80 6 20

0 0 1, 2 6 1,2 60 0 6 20 6 42

K

=

.

.

.

,

.

3' 32

4' 4

5' 5

6' 6

2

2 5

25312'

52 253

12

empBC empBC

qL

qL

P P

qL

qL

=

=

=

=

= = =

3

4

5

6

5

253

525

3

eq BC empBC P P

= =

(6).

,

.

1

2

33

4253

55

6253

10

0

5

120

5

RF

R

=

11

22

3

44

5

66

0

0

u

u

uu

u

=

, (3 5)

0, .


161/203

161

(1, 2, 4 6)

:


0 .

1 2 4 61 1 1

12 3 2 2

2

4 4 43353 4

25 6 6 63 4

LL L

10 1,6 6 6 0

F K u 0 10 6 40 20 0

6 20 80 20

0 0 20 42

L

u

u

u

u

= =

1

LL LKLu F

=

(3 5) :


0 .

1

1 2 4 63 23 3 3

5 45 5

6

0,028

5 0,0026F K u 10 1, 2 6 0 6

5 0,00320 0 6 6

0,0013

R RL L

R

R

= =

:

==410(0,028)=11,2 ( )

= =210(0,0013)=2,6

= +


162/203

162

AB

.

1 1' 1'

2 2' 2 '3

3 3' 3'

4 4' 4 '

1, 2 6 1, 2 6 0,028 1, 2

6 40 6 20 0,0026 0

' 10 1, 2 6 1, 2 6 0 1, 2

6 20 6 40 0,0032 12

ABP

=

=

=

=

= =

BC

,

.

3 3' 3' 3'

4 4' 4 ' 4'3

5 5' 5 ' 5 '

6 6' 6 ' 6 '

1, 2 6 1, 2 6 0 5 16, 4

6 40 6 20 0, 0032 8,33 110,3' 10

1, 2 6 1, 2 6 0 5 6, 46 20 6 40 0,0013 8,33 3,6

BC

P

=

=

=

=

= + =

. ,

B 120 , .

12

1 21 2

3 6

6 4

110 3

16 4

3,612

16 4

1 2

AB

B C

110 3

6 4


163/203

163

,

, .

) .

) .

)

.

0

0

0/

000

,0

0

6 (2 ):

:

:

/=210/

2/2=210/

,

3 :

5

1 0 1 0

0 0 0 0k' 210

1 0 1 0

0 0 0 0

=

.


164/203

164

3

45BC

5

6

1 0 1 0

0 0 0 0k 210

1 0 1 0

0 0 0 0

=

, 90 (=1,

=0, =0 )

1

25AB

3

4

0 0 0 0

0 1 0 1k k' 210

0 0 0 0

0 1 0 1

TL L

= =

, 45 (=1/2,

=1/2, =1/2 )

1

25

AC

5

6

1 1 1 12 2 2 2

1 1 1 12 2 2 2k k' 2101 1 1 1

2 2 2 21 1 1 1

2 2 2 2

TL L

= =

66.

.

1 2 3 4 5 61

2

5 3

4

5

6

0,5 0,5 0 0 0,5 0,5

0,5 1,5 0 1 0,5 0,5

210 0 0 1 0 1 0

0 1 0 1 0 00,5 0,5 1 0 1,5 0,5

0,5 0,5 0 0 0,5 0,5

K

=

.

.


165/203

165

3'

4'

5'

6'

0 1 0 1 0 0 240

0 0 0 0 0 0 0 0' '

1 0 1 0 240

0 0 0 0 0 0 0 0

empBC BC

EA T

EAP k

TL TL EA T L

= = = =

>

(3 6 )

==:

,

( ) :

3 3'

4 4'

5 5'

6 6'

240

0'

240

0

empBC empBC P P

=

=

=

=

= =

3

4

5

6

240

0

240

0

eq BC empBC P P

= =

(6).

,

. .

1 1

2 2

3 3

4

5 5

6 6

240

1000

240

R

R

RF

R

R

=

+

4

0

0

0

0

0

uu

=

(4) :


0 .

5

41000 210 u =


166/203

166

D :

34 510Lu u m

= =

(1, 2, 3, 5, 6)

:


0 .

:

==10000(0,005238)=52,38

[ ]

41 1

12 225

3 343

5 55

6 66

00

10001240 210 0,005 0

0240 0

00

0

R

R

R

R

R

= = +

=0

=1000

=0+240=240

=0240=240

=0

BC = + ,

.

3' 3' 3' 3'

4' 4' 4 ' 4'5BC BC

5' 5' 5' 5'

6' 6 ' 6' 6'

1 0 1 0 0 240 240

0 0 0 0 0,005 0 0P' =k' ' ' 210 1 0 1 0 0 240 240

0 0 0 0 0 0 0

BC empBC

kN

PkN

apuntes de cálculo de estructuras

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