Álgebra Matricial y Métodos Matriciales Para Sistemas Hiperestáticos

LGEBRA MATRIC MA TRICIAL IAL – MÉTODOS MATRICIALES PARA SISTEMAS HIPERESTÁTICO S INGENIERÍA SÍSMICA

ALUMNO:

MEDINA VILLANUEVA, JEISON THAILOR

DOCENTE: RODRIGUEZ PLASENCIA, EDWIN

ÁLGEBRA MATRICIAL – MÉTODOS MATRICIALES PARA SISTEMAS HIPERESTÁTICOS

INTRODUCCIÓN El pr!"# $"%&r' ( $")!#$*+$-" ! r.r + l&! #'+! “Álgebra Matriial! / “M"t#$#% Matriiale% &ara Si%te'a% Hi&ere%t(ti#%!, Hi&ere%t(ti#%!, l&! 01l! !r1" ( *r+" +/0(+ p+r+ !&l0$&"+r pr&2l'+! rl+$&"+(&! +l A"1l$!$! E!#r0#0r+l, / !&" ( 2+! p+r+ 0r!&! '1! +)+"3+(&!, &'& I"*"$r4+ S4!'$+5 El &26#$)& pr$"$p+l ! #"r &"&$'$"#& !&2r l "#"($'$"#& !&2r !&2r  '+#r '+#r$ $! ! ( ($% ($%rr"# "#!! -r( -r(" "!, !, &"& &"& rr !0! !0! #$p& #$p&!! / l+! l+! &pr &pr+ +$& $&" "!! 70 70 &" &" ll+ ll+!! ! r+l r+l$3 $3+" +"55 A!4 #+'2 #+'2$8 $8" " +pr +pr"( "(r r l&! l&! '8#&(&! '+#r$$+l! 70 "&! +/0(+" " l+ !&l0$-" + pr&2l'+! ( !$!#'+! 9$pr!#1#$&!, &'& !&" l M8#&(& ( l;$2$l$(+(! / l M8#&(& ( l+ R$*$(35 L+ pr&2l'1#$+ +"# !#&! #'+!, ! (2$(& 70 "& ! 0"#+ &" l&! &"&$'$"#&! pr)$&! / "!+r$&! p+r+ p&(r (!+rr&ll+r pr&2l'+! &'pl6&! (&"( ! )+ "!+r$& #"r &"&$'$"#& !&2r '+#r$! & !&2r !&l0$-" ( !$!#'+! 9$pr!#1#$&!< "& ! p0( ll*+r + (+rl (+rl!! !&l0 !&l0$$-" " (2$ (2$(& (& +l p&& p&& &"& &"&$' $'$ $"# "#& & & +pr +pr"($ "($3+ 3+6 6 70 70 #"'&! !&2r !#&! #'+!< p&r l& 70 !# $"%&r' 20!+ '&!#r+r ( '+"r+ !"$ll+ / pr$!+ l (!+rr&ll& ( +(+ #'+ p+r+ #"r 0" '6&r "#"($'$"#&5 E" l C+p4#0l& C+p4#0l& I ! '0!#r+ l& rl+$&"+( rl+$&"+(& & +l Ál*2r+ M+#r$$+l, M+#r$$+l, (&"( (&"( ! ;p&" l&! #$p&! ( '+#r$!, l+! &pr+$&"! 70 ! p0(+" (+r &" 8!#+! / !0! pr&p$(+(!5 E" l C+p4#0l& II ! (+ + &"&r !&2r l&! '8#&(&! p+r+ !&l0$&"+r !$!#'+! 9$pr!#1#$&!, ;p&"$"(& l M8#&(& ( l;$2$l$(+( / l M8#&(& ( l+ R$*$(35 E" l C+p4#0l& III ! '0!#r+ 6'pl&! !"$ll&! / ( %1$l &'pr"!$-" p+r+ %+)&rr l '6&r "#"($'$"#& ( +(+ #'+5

=>P1*$"+


OB)ETI*OS OB)ETI*O GENERAL+ ,

I")!#$*+r l& rl+$&"+(& +l Ál*2r+ M+#r$$+l / !&2r l&! M8#&(&! M+#r$$+l! p+r+ S$!#'+! H$pr!#1#$&!5

OB)ETI*OS ESPEC-.ICOS+ , D."$r +(+ &"p#& rl+$&"+(& +l Ál*2r+ M+#r$$+l, (."$"(& ,

'+#r$!, #$p&! / &pr+$&"! rl+$&"+(&! + 8!#+!5 E;pl$+r l&! M8#&(&! ( R$*$(3 / ( l;$2$l$(+(, l&! 01l! "&! +/0(+r1" " l+ !&l0$-" ( S$!#'+! H$pr!#1#$&!5

?>P1*$"+


CAP-TULO I ÁLGEBRA MATRICIAL /0 MATRI1 /0/0 De23ii43 U"+ '+#r$3 ! 0"+ &r("+$-" ( l'"#&! ( %&r'+ r#+"*0l+r5 P&r l& 0+l ! +.r'+ 70 0"+ '+#r$3 ( &r(" @m;n” , ! 0"+ &r("+$-" ( l'"#&! " @ m” .l+! / “n” &l0'"+!, ( l+ !$*0$"# %&r'+: •

D&"(:  P+r+ "&'2r+r 0"+ '+#r$3 ! 0!+ 0"+ l#r+

'+/!0l+5  U" l'"#& *"8r$& ( l+ '+#r$3 A ! (!$*"+ '($+"# aij , (&"( l&! !024"($! $"($$+" l l0*+r (&"( ! 02$+ l l'"#& ("#r& ( l+ '+#r$3: l pr$'r !024"($ i ! r.r + l+ .l+ / l !*0"(& !024"($ &l0'"+5

j

! r.r + l+

/050 Matri6 Re3gl43 # *et#r .ila E! 0"+ '+#r$3 ( 0"+ !&l+ .l+ / ( “n” &l0'"+!5 [ A ] xn= [ a 1

a12 a13 a14

11

[ A ] x =[ 4

E65:

1

6

9

a1 n ]

…

71

5

8

]

/070 Matri6 C#l8'3a # *et#r C#l8'3a E! 0"+ '+#r$3 ( 0"+ !&l+ &l0'"+ / ( “m” .l+!5

[] a 11

[ B ]mx

1=

a21 a31 ⋮

am 1

E65:

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[] 2 7

[ B ] x = 5

1

8 3 9

/090 Matri6 Tra3%&8e%ta L+ '+#r$3 #r+"!p0!#+ ( &#r+ '+#r$3 ! &2#$" '($+"# l $"#r+'2$& .l+! / &l0'"+!, ! ($r< 0+"(& l+! .l+! ! &")$r#" " &l0'"+!, / l+! &l0'"+! " .l+!5 E65:

[ ] 1

4

8

[ A ] x = 2

6

5

7

3

9

3

[ A ]

3

T

3 x 3 =

[ ] 1

2

7

4

6

3

8

5

9

50 TIPOS DE MATRICES 50/0 Matri6 C8a$ra$a E! 0+"(& l "'r& ( .l+! ! $*0+l +l "'r& ( &l0'"+!5 E65:

[ ]

[ A ] x = 2 2

2

6

4 1

5050 Matri6 Diag#3al E! 0+"(& l&! ('1! l'"#&! 70 !#1 %0r+ ( l+ ($+*&"+l pr$"$p+l, !&" "0l&! & r&!5

[ E ] mxn=

[ ] a11

0

0 0

0

0

a22

0 0

0

0

0

0

0

0 ⋱

0

0

0

0 0

am =n

a33

0

0

E65:

>P1*$"+


[ ] 3

0

0

[ E ] x = 0

6

0

0

0

2

3

3

5070 Matri6 I$e3ti$a$ # U3itaria E! 0+"(& l&! l'"#&! ( l+ ($+*&"+l pr$"$p+l ( l+ '+#r$3 !#1 &"%&r'+(& p&r "'r&! 0"& =, / l&! ('1! !&" "0l&! & r&!5 [ I ] =

[ ] 1

0

0

0

1

0

0

0

1

5090 Matri6 E%alar E! 0+"(& #&(&! l&! l'"#&! ( l+ ($+*&"+l pr$"$p+l !&" $(8"#$&! & $*0+l!5 E65: [ D ] x 3

[ ]

3=

3

0

0

0

3

0

0

0

3

50:0 Matri6 Si'"tria C0+"(& l+ '+#r$3 &r$*$"+l ! $*0+l + !0 '+#r$3 #r+"!p0!#+5 E65: [ C ] x = 3

3

[ C ] x T

3

[ ] 3

7

8

7

4

2

8

2

6

3=

[ ] 3

7

8

7

4

2

8

2

6

T D&"(: [ C ] =[ C ]

50;0 Matri6 Tria3g8lar S8&eri#r E! 0+"(& #&(&! l&! l'"#&! p&r (2+6& ( l+ ($+*&"+l pr$"$p+l ( l+ '+#r$3, !&" r&!5 E65: F>P1*$"+


[ F ] x = 3

3

[ ] 3

7

8

0

4

2

0

0

6

50<0 Matri6 Tria3g8lar I3=eri#r E! 0+"(& #&(&! l&! l'"#&! p&r "$'+ ( l+ ($+*&"+l pr$"$p+l ( l+ '+#r$3, !&" r&!5 E65: [ F ] x 3

3=

[ ] 3

0

0

7

4

0

8

2

6

70 OPERACIONES CON MATRICES 70/0 A$ii43 > S8%trai43 $e Matrie% L+ !0'+ / r!#+ ( '+#r$! ! (+ !$'pr / 0+"(& +'2+! '+#r$! #$"" l '$!'& &r(", ! ($r l "'r& ( .l+! / &l0'"+!5  l+ !0'+" & r!#+" l&! #8r'$"&! 70 &0p+" l&! '$!'&! l0*+r! " l+! '+#r$!5 7050 M8lti&liai43 $e Matrie% P+r+ 70 ! p0(+ r+l$3+r l+ '0l#$pl$+$-" "#r (&! '+#r$!, l "'r& ( &l0'"+! ( l+ pr$'r+ '+#r$3 (2 !r $*0+l +l "'r& ( .l+! ( l+ !*0"(+ '+#r$35 L+ '0l#$pl$+$-" ! r+l$3+ '0l#$pl$+"(& +(+ .l+ ( l+ pr$'r+ '+#r$3 p&r +(+ &l0'"+ ( l+ !*0"(+ '+#r$3< / p+r+ll+'"# ! r+l$3+ 0"+ +($$-"5 [ A ] x 2

[

2=

x 11 x 21

x 12 x 22

]

/

[ B ] x 2

3=

[

y 11 y 21

y 12 y 22

y 13 y 23

]

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( x ∗ y + x ∗ y ) 11

11

12

21

x

(¿ ¿ 11∗ y + x ∗ y ) 12

12

22

x

(¿ ¿ 11∗ y + x ∗ y ) ¿ ¿( x ∗ y + x ∗ y ) 13

12

23

11

22

21

21

x

¿

(¿ ¿ 21∗ y + x ∗ y ) x ¿(¿ ¿ 21∗ y + x ∗ y ) [ C ] x = [ A ]∗[ B ]=¿ 12

22

22

2

13

22

23

3

7050/0 M8lti&liai43 $e Matri6 &#r 83 E%alar P+r+ '0l#$pl$+r 0" !+l+r k p&r 0"+ '+#r$3, !$'pl'"# ! '0l#$pl$+ l !+l+r p&r +(+ l'"#& ( l+ '+#r$35 [ A ] x 2

[

2=

x 11 x 21

x 12 x 22

]

Escalar =k

E"#&"!:

[

k ∗[ A ] 2 x 2=

x 11∗k x 21∗k

]

x 12∗k x 22∗k

7070 Deter'i3a3te% •

U"+ '+#r$3 ! ( pr$'r &r(" 0+"(& "$+'"# #$"

0"

!&l&

l'"#&

/

(."$'&!

l+

(#r'$"+"# ( A &'& •

S$ l+ '+#r$3 A ! 0"+ '+#r$3 0+(r+(+ ( !*0"(&

&r(" , !0 (#r'$"+"# ! r+l$3+ ( l+ !$*0$"# '+"r+:

>P1*$"+


7090 I3?er%a $e 83a Matri6 M"t#$# $e Ga8%% – )#r$a3 C&"!$!# " &l&+r 60"#& + l+ '+#r$3 ( &r$*" A, l+ '+#r$3 I("#$(+( I / 9+r &pr+$&"! p&r .l+!, 70 +%#+r+" + +'2+! '+#r$!, &" l &26#& ( #r+"!%&r'+r l+ '+#r$3 A " l+ '+#r$3 I("#$(+(5

90 PROPIEDADES DEL ÁLGEBRA MATRICIAL 90/0 Le> A%#iati?a &ara la M8lti&liai43 $e Matrie% S+ [ A ] mxn , [ B ] nxp y [ C ] pxq < "#&"! [ A ]∗( [ B ]∗[ C ] ) =( [ A ]∗[ B ] )∗[ C ] !#1 (."$(&5 9050 Le>e% Di%trib8ti?a% &ara la M8lti&liai43 $e Matrie% S$ l+ '0l#$pl$+$-" ( l+! '+#r$! ! (.", "#&"! ! p0( +.r'+r: S+ [ A ] mxn , [ B ] nxp y [ C ] pxq < "#&"! [ A ]∗( [ B ] + [ C ] )=( [ A ]∗[ B ] ) +( [ A ]∗[ C ] )

( [ A ] + [ B ] )∗[ C ]=( [ A ]∗[ C ] ) +( [ B ]∗[ C ] ) 9070 Le> $e la Matri6 I$e3ti$a$ L+ '0l#$pl$+$-" ( 0+l70$r '+#r$3 p&r l+ '+#r$3 I("#$(+(, l r!0l#+(& ! $*0+l + l+ '$!'+ '+#r$3 '0l#$pl$+(+5 [ A ]∗[ I ] =[ A ] 9090 El &r#$8t# $e 'atrie% e3 ge3eral 3# e% #3'8tati?#0

CAP-TULO II >P1*$"+


MÉTODOS MATRICIALES PARA SISTEMAS HIPERESTÁTICOS /0 SISTEMAS HIPERESTÁTICOS /0/0 De23ii43 E! 0+"(& l+! 0+$&"! ( l+ !#1#$+ & ( 70$l$2r$& "& !&" !0.$"#! p+r+ 9+ll+r l+! r+$&"! & %0r3+! $"#r"+! 70 p&! l !$!#'+5 E! ($r ;$!#" '1! %0r3+! +#0+"#! 70 0+$&"! ( 70$l$2r$&, p&r l+ 0+l #$" '1! (  *r+(&! ( l$2r#+(5

/050 Ti&#%

/050/0 Si%te'a% E@teri#r'e3te Hi&ere%t(ti#% C0+"(& l+! 0+$&"! ( 70$l$2r$& r!0l#+" $"!0.$"#! p+r+ 9+ll+r l+! r+$&"!5 •

/05050 Si%te'a% I3teri#r'e3te Hi&ere%t(ti#% •

C0+"(& l+! 0+$&"! ( 70$l$2r$& r!0l#+" $"!0.$"#! p+r+ 9+ll+r l&! !%0r3& $"#r"&!5

50 MÉTODOS MATRICIALES PARA SOLUCIÓN DE SISTEMAS HIPERESTÁTICOS 50/0 M"t#$# $e la% .le@ibili$a$e% 50/0/0 De23ii43 C&"!$!# " (%&r'+$&"! p+r+ +l0l+r %0r3+! / (!pl+3+'$"#&! " 0"+ !#r0#0r+5 A !0 )3 !0p&" l+ !0prp&!$$-" ( (!pl+3+'$"#&! " #8r'$"&! ( !#r0#0r+! !#1#$+'"# (#r'$"+(+!5 •

50/050 M"t#$# $e De%arr#ll# Al !0p&"r l&! (!pl+3+'$"#&! " !#r0#0r+! !#1#$+'"# (#r'$"+(+!, ! l+! p0( &"!$(r+r + !#+! &'& !#r0#0r+! pr$'+r$+!5 P+r+ &2#"r !#+ !#r0#0r+ pr$'+r$+ ! 9+ l+ !0pr!$-" ( +p&/&! & ! #r+"!%&r'+ l +p&/& " &#r& '1! !$'pl5 Pr& l+! &"($$&"! 70 (2" #"r 8!#+! (2" !r !#+2$l$(+( & $!&!#+#$$(+(5 •

K>P1*$"+


5050 M"t#$# $e la% Rigi$ee% 5050/0 De23ii43 E" !# '8#&(& l+! $"-*"$#+! !&" l&! (!pl+3+'$"#&! ( l&! "0(&!, ! ($r< " l&! +p&/&!, )&l+($3&! & " l&! p0"#&! ( 0"$-" ( (&! & '1! l'"#&!5 L&! (!pl+3+'$"#&! ( l&! "0(&! rpr!"#+" l&! *r+(&! ( l$2r#+( ( l+ !#r0#0r+5 E!# '8#&(& 9+ 0!& ( l+! %-r'0l+! ( 'p&#r+'$"#&, #+l! &'&: •

•

•

505050 505050 505050 505050 505050 505050 505050 505050 505050 505050 505050 505050 505050 505050

M"t#$#

$e De%arr#ll# El pr$'r p+!& ! r!#r$"*$r l&! (!pl+3+'$"#&! ( l&! "&(&! (!&"&$(&!, ! ($r< rpr!"#+rl&! " 0"+ $(+l$3+$-" ( 'p&#r+'$"#&5 E" !# '8#&(& ! 0!+" %0r3+! & '&'"#&! pr&(0$(&! p&r l&! (!pl+3+'$"#&! 0"$#+r$&!5 E!#+! %0r3+! & '&'"#&! pr&(0$(&! ! ll+'+" •

•

rigideces.

= > P 1 * $ " +


CAP-TULO III APLICACIONES /0 OPERACIONES CON MATRICES0 /0/0 ADICIÓN  SUSTRACCIÓN DE MATRICES Sea la% 'atrie%+ [ A ] x 3

3=

[ ] 3

1

8

0

2

4

7

3

6

/ [ B ] x 3

[

3=

2

6

0

5

4

3

−1

0

8

]

<

Hallar A  B > A , B SOLUCIÓN •

AB [ A ] + [ B ]=

[ A ] + [ B ]=

•

[ ][

[

3

1

8

0

2

4

7

3

6

5

7

8

5

6

7

6

3

14

2

6

0

5

4

3

−1

0

8

+

]

]

AB

[ ][ 3

1

8

[ A ] −[ B ] = 0

2

4

7

3

6

[

1

[ A ] −[ B ] = −5 8

2

6

0

5

4

3

−1

0

8

−

−5 −2

8

3

−2

1

]

]

== > P 1 * $ " +


/050 MULTIPLICACIÓN DE MATRICES Sea la% 'atrie%+

[ ]

[ A ] x = 2

2

1 3

2

[ B ] x =

/

4

2

3

[

5

6

7

0

8

9

]

Hallar C  A  B SOLUCIÓN

[

( 1∗5 +2∗0 ) ( 1∗6 +2∗8) ( 1∗7 + 2∗9 ) [ C ] x =[ A ]∗ [ B ] = ( 3∗5 + 4∗0 ) ( 3∗6 + 4∗8 ) ( 3∗7 + 4∗9) 2

3

[ C ] x = 2

3

[

5

22

25

15

50

57

]

]

/050/0 MULTIPLICACIÓN DE UN ESCALAR POR UNA MATRI1 Sea la 'atri6+

[ ]

[ A ] x = 1 2

2

3

2 4

Escalar =3

Hallar 7A SOLUCIÓN E"#&"!:

[

∗ [ A ] x = 1∗3 3∗ 3

3

2

2

∗ 4∗3 2 3

] 3

[

[ A ] =

3

6

9

12

]

=? > P 1 * $ " +


/070 DETERMINANTE DE UNA MATRI1 Sea la 'atri6+ [ A ] x 2

[ ]

2=

5

2

3

4

Hallar la Deter'i3a3te $e [ A ] SOLUCIÓN Det . [ A ]= (5∗ 4 )− ( 3∗2 ) =14

Sea la 'atri6+

[

[ B ] x = 3

3

3

2

1

0

2

−5

−2

1

4

]

Hallar la Deter'i3a3te $e [ B ] SOLUCIÓN

Det . [ B ] =3

[

3

2

0

2

−2

1

1

][

−5 −2 4

3

2

0

2

−2

1

1

][

−5 + 1 4

3

2

1

0

2

−5

−2

1

4

]

E"#&"! &2#"'&!: Det . [ B ] =3

[ ] [− ] [− ] 2 1

−5 −2 4

0

2

−5 + 1 4

0

2

2

1

= > P 1 * $ " +

ÁLGEBRA MATRICIAL – MÉTODOS MATRICIALES PARA SISTEMAS HIPERESTÁTICOS Det . [ B ] =3∗( 8 + 5 ) −2∗( 0 −10 ) + 1∗(1 + 4 )=63

50 MÉTODOS MATRICIALES PARA SISTEMAS HIPERESTÁTICAS ?5=5 MÉTODO DE LA RIGIDE1 •

H+ll+r l+ M+#r$3 ( R$*$(3 [ K ] ( l+ !$*0$"# !#r0#0r+5

SOLUCIÓN GRADOS DE LIBERTAD DE LA ESTRUCTURA

= > P 1 * $ " +


a) Primer Grado de Libertad

K 11=

K 21=

K 31=

12 EI

l

3

+

12 EI

l

3

=

24 EI

l

3

−6 EI l

2

−6 EI l

2

2 S*0"(& Gr+(& ( L$2r#+(

=F > P 1 * $ " +


K 12=

K 22=

−6 EI l

2

4 EI

l

+

K 32=

4 EI

l

=

8 EI

l

2 EI

l

 Trr Gr+(& ( L$2r#+(

K 13=

K 23=

K 33=

−6 EI l

2

2 EI

l 8 EI

l

Ensamblando la matriz de rigidez

[ k ]

= > P 1 * $ " +


[ k ] =

[

24 EI

l

3

−6 EI l

2

−6 EI l

2

−6 EI −6 EI l 8 EI l

2

l 2 EI l

2

2 EI

8 EI

l

l

] CONCLUSIONES

•

S l&*r- ;pl$+r #&(& l& rl+$&"+(& +l 1l*2r+ M+#r$$+l, $"l0/"(& l&! #$p&! ( '+#r$!, l+! &pr+$&"! 70 ! p0(" 9+r &" ll+!5 S ;pl$- l&! '8#&(&! p+r+ !$!#'+! 9$pr!#1#$&!, l&! 01l! !&" M8#&(& ( l;$2$l$(+(! / M8#&(& ( R$*$(35

•

S '&!#r+r&" +l*0"&! 6r$$&! ( +pl$+$-" p+r+ %+$l$#+r 0"+ '6&r &'pr"!$-" ( l&! #'+! ;p0!#&!5

= > P 1 * $ " +


BIBLIOGRA.-A •

J&!8 L0$! C+'2+, Ap0"#! ( A"1l$!$! E!#r0#0r+l I, Pr$'r+

•

E($$-", =K?5 J+$' Br+)& 2r!, M+#r$! / D#r'$"+"#!, Pr$'r+ E($$-"5 E370$l Ur$l, El'"#&! ( Ál*2r+ M+#r$$+l5

•

LINOGRA.-A ÁLGEBRA MATRICIAL Ftt&+0$it8t#r0#'$eter'i3a3te%$eter'i3a3t e0Ft'l Ftt&+re8r%#%ti0e$8ai#30e%$e%arte%eb'ate riale%$i$ati#%'atrie%'atrie%#&erai#3e%I0F t' Ftt&+eb8r%#080lae%%#3te3tattaF'e3tea e5:Ja,/,5/,5J/5MaterialK5J$el K5JC8r%#7$=;:,=9 P 1 * $ " +


K5JINTRODUCCIONK5JALK5JALGEBRA K5JMATRICIAL0&$= MÉTODOS MATRICIALES Ftt&+$a$83083a?0e$8bit%trea'/J/
=K > P 1 * $ " +

Álgebra Matricial y Métodos Matriciales Para Sistemas Hiperestáticos

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