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7/23/2019 Practica de Elementos Finitos
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1.-
Modelo a resolver
Discretizacion
Coordenadas de los nudos
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1.1.
Coordenadas de los Nudos
E = 2500000 Tn/m2
u = 0.15
Esp. = 0.1 m
N x y Elem. Nudos A t1 0 0 1 1 5 7 0.125 0.1
2 0.5 1 2 5 7 0.175 0.1
! 1.5 1 ! 5 " 0.25 0.1
# 2.5 0 # " $ 0.25 0.1
5 0.5 0 5 " # $ 0.25 0.1
" 1.5 0 " 7 2 0.175 0.1
7 0.25 0.5 7 ! 2 0.25 0.1
1 0.5 $ ! 0.25 0.1
$ 2 0.5
25575#5 !!"!1.71 0
D = !!"!1.71 25575#5 0
0 0 10"$57
Formulas a Utilizar
2
2
* 01
* 01
(1 )*0 0
2*(1 )
E E
D E E
E
µ
µ µ
µ
µ
=
− −
+
m x n n x p m x pM x M = M
* * * *T K B D B t A=23 31 12
32 13 21
32 23 13 31 21 12
0 0 01
0 0 0
y y y
B x x x J
x y x y x y
= 1
* 2*2
A J J A= → =
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Elemento 1
N x y A t
1 1 0 0 0.125 0.1
2 5 0.5 0
3 7 0.25 0.5-2 0
0 -1
2 0
-2 0 2 0 0 0 0 -1
% = 0 -1 0 -1 0 2 0 0
-1 -2 -1 2 2 0 0 2
-51150$0 -7"72"!.# -10"$57 11!171!" 2$#117" -$1#!22! -1#0""50
-!!"!1.7 -25575#5 -217!$1! 2$#117" "$05!71 1#0""50 -17$021
%T & D = 51150$0 7"72"!.#! -10"$57 %T&D&%= -$1#!22! 1#0""50 11!171!" -2$#117"
-!!"!1.7 -25575#5 217!$1! -1#0""50 -17$021 -2$#117" "$05!71
0 0 217!$1! -217!$1! -#!#72" -217!$1! #!#72"
7"72"!.#! 51150$0 0 -15!#527 -51150$0 15!#527 -51150$0
1 5 7
1 2 $ 10 1! 1#
1#1#"#.1$ !"7"#.70" -11#2$0.! -175!.12 -2717!.$1 -1$11.5$ 1
!"7"#.70" "!17.1!" 175!.12 -22!7.52 -5#!#7.! -"!$!."2 2
' = %T & D & % & ( & t = -11#2$0 175!.12 1#1#"#.1$ -!"7"#.71 -2717!.$1 1$11.5" $
-175!.12 -22!7.52 -!"7"#.71 "!17.1!" 5#!#7.2" -"!$!."2 10
-2717!.$1 -5#!#7.! -2717!.$1 5#!#7.2" 5#!#7.2" 0 1!
-1$11.5$ -"!$!."2 1$11.5" -"!$!."2 0 12777.2# 1#
% T =
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Elemento 2
N x y A t
1 5 0.5 0 0.1875 0.1
2 8 1 0.53 7 0.25 0.5
0 0
0 -2
1.!!!!!!! 0
0 0 1.!!!!!!! 0 -1.!!!!!! 0 0 0.""""""7
% = 0 -2 0 0.""""""7 0 1.!!!!!!! -1.!!!!!! 0
-2 0 0.""""""7 1.!!!!!!! 1.!!!!!!! -1.!!!!!! 0 1.!!!!!!!
0 0 -217!$1! #!#72" 0 -1##$275 -2$551
-7"72"!.# -51150$0 0 0 102!017$ -102!01 -!#100"0
%T & D = !#100"0 51150$ 72#"!7." %T&D&%= -1##$275 -102!01 502$! 1!071$0
25575#.# 17050!0 1##$275 -2$551 -!#100"0 1!071$0 !0"$05#
-!#100"0 -51150$ 1##$275 -2$551 102!01 -!505"! 15$1!"1
51150$ !#100"0 -1##$275 2$551 -"2011$ -2#171." !#100"
5 8 7
$ 10 15 1" 1! 1#
1521.7!$ 0 -2717!.$1 -5#!#7.! -5#!#7.! 5#!#7.2" $
0 1$115." -1$11.5$ -"!$!."2 1$11.5" -12777.2 10
' = %T & D & % & ( & t = -2717!.$1 -1$11.5$ $#!0$.#"! 2#50$.0# -"71!5.55 -5!2.21 15
-5#!#7.! -"!$!."2 2#50$.0# 575##.757 2$!.022 "!$!."1$ 1"
-5#!#7.! 1$11.5" -"71!5.55 2$!.022 121#!.! -#$01$."1 1!
5#!#7.2" -12777 -5!2.21 "!$!."1$ -#$01$."1 121#!.! 1#
% T =
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Elemento 3
N x y A t
1 5 0.5 0 0.25 0.12 6 1.5 0
3 8 1 0.5
-1 0
0 -1
1 0
-1 0 1 0 0 0 0 -1
% = 0 -1 0 -1 0 2 0 0
-1 -1 -1 1 2 0 0 2
-25575#5 -!!"!1.7 -10"$57 !"##501 1#705 -1#705 -70!!25
-!!"!1.7 -25575#5 -10"$57 1#705 !"##501 70!!2#.1 1#705
%T & D = 25575#5 !!"!1.71 -10"$57 %T&D&%= -1#705 70!!2#.1 !"##501 -1#705
-!!"!1.7 -25575#5 10"$57 -70!!2#. 1#705 -1#705 !"##501
0 0 217!$1! -217!$1! -217!$1! -217!$1! 217!$1!
7"72"!.#! 51150$0 0 -7"72"!.# -51150$0 7"72"!.#! -51150$0
5 6 8
$ 10 11 12 15 1"
$1112.5!2 !"7"#.70" -!"7"#.71 -175!.12 -5#!#7.! -1$11.5$ $
!"7"#.70" $1112.5!2 175!.12 !"7"#.70" -5#!#7.! -12777.2 10
' = %T & D & % & ( & t = -!"7"#.71 175!.12 $1112.5!2 -!"7"#.71 -5#!#7.! 1$11.5" 11
-175!.12 !"7"#.70" -!"7"#.71 $1112.5!2 5#!#7.2" -12777.2 12
-5#!#7.! -5#!#7.! -5#!#7.! 5#!#7.2" 10"$5."5 0 15
% T =
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-1$11.5$ -12777 1$11.5" -12777.2 0 25575#.# 1"
Elemento 4
N x y A t1 6 1.5 0 0.25 0.1
2 9 2 0.5
3 8 1 0.5
0 0
0 -2
1 0
0 0 1 0 -1 0 0 1
% = 0 -2 0 1 0 1 -1 0
-2 0 1 1 1 -1 0 1
0 0 -217!$1! #!#72" 0 -217!$1! -217!$1!
-7"72"!.# -51150$0 0 0 102!017$ -7"72"!.# -51150$0
%T & D = 25575#5 !!"!1.71 10"$57 %T&D&%= -217!$1! -7"72"!.# !"##501 1#705
!!"!1.71 25575#5 10"$57 -217!$1! -51150$0 1#705 !"##501
-25575#5 -!!"!1.7 10"$57 -217!$1! 7"72"!.#! -1#705 70!!2#.1
!!"!1.71 25575#5 -10"$57 217!$1! -51150$0 -70!!2#. 1#705
6 9 8
11 12 17 1 15 1"
10"$5."5 0 -5#!#7.! -5#!#7.! -5#!#7.! 5#!#7.2" 11
0 25575#.# -1$11.5$ -12777.2 1$11.5" -12777.2 12
' = %T & D & % & ( & t = -5#!#7.! -1$11.5$ $1112.5!2 !"7"#.70" -!"7"#.71 -175!.12 17
-5#!#7.! -12777 !"7"#.70" $1112.5!2 175!.12 !"7"#.70" 1
% T =
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-5#!#7.! 1$11.5" -!"7"#.71 175!.12 $1112.5!2 -!"7"#.71 15
5#!#7.2" -12777 -175!.12 !"7"#.70" -!"7"#.71 $1112.5!2 1"
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)*ara las multiplicacion de
matrices+
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Elemento 5
N x y A t
1 6 1.5 0 0.25 0.1
2 4 2.5 0
3 9 2 0.5
-1
-2
-1
2 -1 0 1 0 0 0
2 % = 0 -1 0 -1 0 2
0 -1 -1 -1 1 2 0
-217!$1! -15!#527 -25575#5 -!!"!2 -10"$57 !"##501 1#705
-#!#72" -51150$0 -!!"!2 -25575#5 -10"$57 1#705 !"##501
-217!$1! 15!#527 %T & D = 25575#5 !!"!1.7 -10"$57 %T&D&%= -1#705 70!!2#.
#!#72" -51150$0 -!!"!2 -25575#5 10"$57 -70!!25 1#705
#!#72" 0 0 0 217!$1! -217!$1! -217!$1!
0 102!017$ 7"72"!.# 51150$0 0 -7"72"! -51150$0
6 4 9
11 12 7 17
1$1112.5! !"7"#.7 -!"7"5 -175!.12 -5#!#7.!
!"7"#.71 $1112.5 175!.1 !"7"#.71 -5#!#7.!
5 ' = %T & D & % & ( & -!"7"#.71 175!.1 $1112.5 -!"7"#.71 -5#!#7.!
-175!.12 !"7"#.7 -!"7"5 $1112.5! 5#!#7.!
7-5#!#7.! -5#!# -5#!# 5#!#7.! 10"$5.7
-1$11.5$ -12777 1$11." -12777 0
% T =
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Elemento 6
N x y A t
1 7 0.25 0.5 0.1875 0.1
2 8 1 0.5
3 2 0.5 1
-2
0
0.""""""7
1.!!!!!!! -1.!!!!! 0 1.!!!!!! 0 0 0
1.!!!!!!! % = 0 -1.!!!!!! 0 -0."""""7 0 2
-1.!!!!!! -1.!!!!! -1.!!!!!! -0."""""7 1.!!!!!! 2 0
-2$551 2$551 -!#100"0 -51150$ -1##$275 "#7$11! 2"1#!7$
102!01 -"2011$ -51150$ -!#100"0 -1##$275 2"1#!7$ "#7$11!
-!505"! -2#171." %T & D = !#100"0 51150$ -72#"! %T&D&%= -!505"! 2#171."
15$1!"1 !#100" -25575# -17050!0 1##$275 -15$1!"1 !#100"
"#7$11! -2"1#!7$ 0 0 217!$1! -2$551 -2$551
-2"1#!7$ "#7$11! 7"72"!.# 51150$0 0 -102!01 -"2011$
7 8 2
1! 1# 15 1" !
5121#!.# #$01$." -"71!" -2$! -5#!#7.!
#$01$."1 121#! 5!2.22 "!$!."2 -5#!#7.!
8 ' = %T & D & % & ( & -"71!5.55 5!2.22 $#!0$.5 -2#50$. -2717!.$1
-2$! "!$!." -2#510 575##.7" 5#!#7.!
7-5#!#7.! -5#!# -2717# 5#!#7.! 1521.7#
-1$11.5$ -12777 1$11." -"!$!."2 0
% T =
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Elemento 7
N x y A t
1 8 1 0.5 0.25 0.1
2 3 1.5 1
3 2 0.5 1
-1
-1
-1
1 0 0 1 0 -1 0
2 % = 0 -2 0 1 0 1
0 -2 0 1 1 1 -1
-217!$1! -7"72"!.# 0 0 -217!$1! #!#72" 0
-217!$1! -51150$0 -7"72"! -51150$0 0 0 102!017$
-217!$1! 7"72"!.#! %T & D = 25575#5 !!"!1.7 10"$57 %T&D&%= -217!$1! -7"72"!
217!$1! -51150$0 !!"!1.7 25575#5 10"$57 -217!$1! -51150$0
#!#72" 0 -25575#5 -!!"!2 10"$57 -217!$1! 7"72"!.#
0 102!017$ !!"!1.7 25575#5 -10"$57 217!$1! -51150$0
8 3 2
15 1" 5 " !
510"$5.7 0 -5#!# -5#!#7.! -5#!#7.!
0 25575# -1$12 -12777 1$11.5$
6 ' = %T & D & % & ( & -5#!#7.! -1$12 $1112.5 !"7"#.71 -!"7"#.71
-5#!#7.! -12777 !"7"#.7 $1112.5! 175!.12
-5#!#7.! 1$11." -!"7"5 175!.12 $1112.5!
% T =
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5#!#7.! -12777 -175! !"7"#.71 -!"7"#.71
Elemento 8
N x y A t
1 8 1 0.5 0.25 0.1
2 9 2 0.5
3 3 1.5 1
-2
0
1
1 -1 0 1 0 0 0
1 % = 0 -1 0 -1 0 2
-1 -1 -1 -1 1 2 0
-217!$1! 217!$1! -25575#5 -!!"!2 -10"$57 !"##501 1#705
7"72"!.#! -51150$0 -!!"!2 -25575#5 -10"$57 1#705 !"##501
-1#705 -70!!2#. %T & D = 25575#5 !!"!1.7 -10"$57 %T&D&%= -1#705 70!!2#.
70!!2#.1 1#705 -!!"!2 -25575#5 10"$57 -70!!25 1#705
!"##501 -1#705 0 0 217!$1! -217!$1! -217!$1!
-1#705 !"##501 7"72"!.# 51150$0 0 -7"72"! -51150$0
8 9 3
15 1" 17 1 5
6$1112.5! !"7"#.7 -!"7"5 -175!.12 -5#!#7.!
!"7"#.71 $1112.5 175!.1 !"7"#.71 -5#!#7.!
9 ' = %T & D & % & ( & -!"7"#.71 175!.1 $1112.5 -!"7"#.71 -5#!#7.!
-175!.12 !"7"#.7 -!"7"5 $1112.5! 5#!#7.!
% T =
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8-5#!#7.! -5#!# -5#!# 5#!#7.! 10"$5.7
-1$11.5$ -12777 1$11." -12777 0
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-1 0 -1
0 -1 -1
1 0 -1
0 -1 1
0 0 2
0 2 0
-1#705 -70!!25 -217!$1! -7"72"!.#
70!!2#. 1#705 -217!$1! -51150$0
!"##501 -1#705 -217!$1! 7"72"!.#!
-1#705 !"##501 217!$1! -51150$0
-217!$1! 217!$1! #!#72" 0
7"72"!.# -51150$0 0 102!017$
1
-1$11.5$ 116
-12777 12
1$11.5$ 74
-12777
0 179
25575#.5 1
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-1.!!!!!! 0 -1.!!!!!!
0 -1.!!!!!! -1.!!!!!!
1.!!!!!! 0 -0."""""7
0 -0."""""7 1.!!!!!!
0 0 2
0 2 0
-!505"! -15$1!"1 -2$551 -102!01
2#171." !#100" -2$551 -"2011$
502$! -1!071$0 -1##$275 102!01
-1!071$0 !0"$05# 2$551 -!#100"0
-1##$275 2$551 #!#72" 0
102!01 -!#100"0 0 102!017$
#
-1$11.5$ 1!7
-12777 1#
1$11.5$ 158
-"!$!."2 1"
0 !2
1$115.$ #
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0 0 -2
0 -2 0
1 0 1
0 1 1
-1 0 1
0 1 -1
-217!$1! -217!$1! -217!$1! 217!$1!
-7"72"! -51150$0 7"72"!.# -51150$0
!"##501 1#705 -1#705 -70!!2#.
1#705 !"##501 70!!2#. 1#705
-1#705 70!!2#. !"##501 -1#705
-70!!25 1#705 -1#705 !"##501
#
5#!#7.! 158
-12777 1"
-175!.12 53
!"7"#.71 "
-!"7"#.71 !
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$1112.5! #
-1 0 -1
0 -1 -1
1 0 -1
0 -1 1
0 0 2
0 2 0
-1#705 -70!!25 -217!$1! -7"72"!.#
70!!2#. 1#705 -217!$1! -51150$0
!"##501 -1#705 -217!$1! 7"72"!.#!
-1#705 !"##501 217!$1! -51150$0
-217!$1! 217!$1! #!#72" 0
7"72"!.# -51150$0 0 102!017$
"
-1$11.5$ 158
-12777 1"
1$11.5$ 179
-12777 1
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0 53
25575#.5 "
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Elemento 1 Elemento 5
1 5 7 6 4 9
1 2 $ 10 1! 1# 11 12 7 17 1
1#1#"#.1$# !"7"#.705$ -11#2$0.2 -175!.12 -2717!.$1! -1$11.5" 11
$1112.5!2 !"7"#.705$ -!"7"#.705$ -175!.1202 -5#!#7.2"1 -1$11.557
!"7"#.705$ "!17.1!55 175!.1202 -22!7.517 -5#!#7.2" -"!$!."1$ 2 !"7"#.705$ $1112.5!2 175!.1202 !"7"#.705$ -5#!#7.2"1 -12777.2!
-11#2$0.2 175!.1202 1#1#"#.1$# -!"7"#.70" -2717!.$1! 1$11.557 $5
-!"7"#.705$ 175!.1202 $1112.5!2 -!"7"#.705$ -5#!#7.2"1 1$11.557
-175!.12 -22!7.517 -!"7"#.70" "!17.1!55 5#!#7.2"1 -"!$!."1$ 10 -175!.1202 !"7"#.705$ -!"7"#.705$ $1112.5!2 5#!#7.2"1 -12777.2!-2717!.$1! -5#!#7.2" -2717!.$1! 5#!#7.2"1 5#!#7.2"1 0 1!7
-5#!#7.2"1 -5#!#7.2"1 -5#!#7.2"1 5#!#7.2"1 10"$5."52 0
-1$11.5" -"!$!."1$ 1$11.557 -"!$!."1$ 0 12777.2! 1# -1$11.557 -12777.2! 1$11.557 -12777.2! 0 25575#.#7"
Elemento 2 Elemento 6
5 8 7 7 8 2
$ 10 15 1" 1! 1# 1! 1# 15 1" ! #
1521.7!$1 0 -2717!.$1! -5#!#7.2" -5#!#7.2" 5#!#7.2"1 $5
121#!.!7" #$01$."07 -"71!5.5#$$ -2$!.0222 -5#!#7.2"1 -1$11.557
0 1$115.57 -1$11.5" -"!$!."1$ 1$11.557 -12777.2# 10 #$01$."07 121#!.!7" 5!2.212# "!$!."1$ -5#!#7.2"1 -12777.2!
-2717!.$1! -1$11.5" $#!0$.#"2$ 2#50$.0!$ -"71!5.55 -5!2.212 15
8
-"71!5.5#$$ 5!2.212# $#!0$.#"2$ -2#50$.0!$ -2717!.$1! 1$11.557
-5#!#7.2" -"!$!."1$ 2#50$.0!$ 575##.757 2$!.0222 "!$!."1$ 1" -2$!.0222 "!$!."1$ -2#50$.0!$ 575##.757 5#!#7.2"1 -"!$!."1$
-5#!#7.2" 1$11.557 -"71!5.55 2$!.0222 121#!.!7" -#$01$."0 1!7
-5#!#7.2"1 -5#!#7.2"1 -2717!.$1! 5#!#7.2"1 1521.7!$1 0
5#!#7.2"1 -12777.2# -5!2.212 "!$!."1$ -#$01$."0 121#!.!7" 1# -1$11.557 -12777.2! 1$11.557 -"!$!."1$ 0 1$115.57
Elemento 3 Elemento 7
5 6 8 8 3 2
$ 10 11 12 15 1" 15 1" 5 " ! #
$1112.5!2 !"7"#.705$ -!"7"#.70" -175!.12 -5#!#7.2" -1$11.5" $5
10"$5."52 0 -5#!#7.2"1 -5#!#7.2"1 -5#!#7.2"1 5#!#7.2"1
!"7"#.705$ $1112.5!2 175!.1202 !"7"#.705$ -5#!#7.2" -12777.2# 10 0 25575#.#7" -1$11.557 -12777.2! 1$11.557 -12777.2!
-!"7"#.70" 175!.1202 $1112.5!2 -!"7"#.70" -5#!#7.2" 1$11.557 11 6 -5#!#7.2"1 -1$11.557 $1112.5!2 !"7"#.705$ -!"7"#.705$ -175!.1202-175!.12 !"7"#.705$ -!"7"#.70" $1112.5!2 5#!#7.2"1 -12777.2# 12 -5#!#7.2"1 -12777.2! !"7"#.705$ $1112.5!2 175!.1202 !"7"#.705$
-5#!#7.2" -5#!#7.2" -5#!#7.2" 5#!#7.2"1 10"$5."52 0 158
-5#!#7.2"1 1$11.557 -!"7"#.705$ 175!.1202 $1112.5!2 -!"7"#.705$
-1$11.5" -12777.2# 1$11.557 -12777.2# 0 25575#.#7" 1" 5#!#7.2"1 -12777.2! -175!.1202 !"7"#.705$ -!"7"#.705$ $1112.5!2
Elemento 4 Elemento 8
6 9 8 8 9 3
11 12 17 1 15 1" 15 1" 17 1 5 "
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