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UNIVERSIFACULTAD DE IN
Integrantes:1.- 2.-
.- !.-
Se"estre: S#$t%"& - Seg'n(&
E)ERCICIO: VIGA CONTINUAMETODO: RIGIDECES
1 1 2
0.35 0.45
* 5.50
E + 2.1E * a /+*G + 0.*E * /3*4+ 1.2*
.5*E * 6N7"
Y
1.- Ge&"etr8a (e9 r& 9e"a:
2.- Dat&s:6N7" 2
6N7" 2
6N7" 2
; R+
.- E
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1 1 2
0.35 0.45
* 5.50
j= 4n= 3
NUDO1 0.002 5.50
12.50! 21.00
=ARRA NI NF1 1 22 2 3
3 4
1 1 2
*
!.- N'"era>%?n (e N'(&s @ =arras:
5.- C&&r(ena(as (e 9&s N'(&s:
%
.- In>%(en>%as (e 9as =arras:
.- Des$9aBa"%ent&s (e 9as =arras: +*
( 1
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=ARRA1 0 0 02 0
0
=ARRAY L%
9% "%" " "
1 5.50 0 5.50 1 0 0.2 7.00 0 7.00 1 0 0.
8.50 0 8.50 1 0 0.
=ARRA 1: ara /+*
6 + EI7L4 22 4
=ARRA 2: ara /+*
6 + EI7L4 6/L 2
6/L 6/L2 6/L 4
=ARRA : ara /+*
6 + EI7L-6/L -6/L
-6/L 4 2-6/L 2 4
ara /+*NUDO 2 NUDO NUDO !
107142 14260.714 33275 0
14261 13438.931 -10663.16 -24924
V1 H1 V2 H2d 1
d 1 d 2 d 3d 2 d 3 d 4
0.- Cara>ter8st%>as Ge&"#tr%>as @ E9 st%>as (e >a(a =arra:
J.- Matr%B (e R%g%(eB (e >a(a =arra:J.1.- E $res%&nes Genera9es:
H1 H2
H1 V2 H2
12/L 2
V1 H1 H212/L 2
1*.- Matr%B (e R%g%(eB T&ta9: 6
(1
(2
( (!
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33275 -10663.16 207785.29 70617.6
0 -24923.88 70617.647 141235
ara /+*NUDO 2 NUDO NUDO !
1E-005 -1.9E-005 -2.2E-006 -2E-006
-2E-005 0.0001408 2.28E-006 2E-005
-2E-006 2.28E-006 6.20E-006 -3E-006
-2E-006 2.37E-005 -2.7E-006 1E-005
=ARRA 1:-12 .*! T-"12 .*! T-"
V+ 1 .5* T V+VK+ 1 .5* T VK+
-12 .*! 12 .*! -2*!.1
1 .5* 1 .5* 1 5.*
78.13
+-476.0496.88
-301.04
11.- Matr%B (e R%g%(eB In ersa: 6 -1
( 1 ( 2 ( ( !
6 -1 +
12.- Matr%B (e Cargas E ter%&res: 12.1.- F'erBas @ M&"ent&s (e E"$&tra"%ent& er e>t&:
MF+ M F+MKF+ MKF+
12.2.- Ve>t&res (e Cargas :
- MF
F F
MF
M
V F
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0.1
1 1 2
*
00.01073
( +-0.07549
( +0.00015
-0.01553
V% + 6% (%M%
=ARRA 1: ara /+*
6 +91.72
561.57
=ARRA 2: ara /+*
6 +-561.5722.60
-505.22
=ARRA : ara /+*
6 +153.06505.22
0.00
1 .- Res&9'>%?n (e9S%ste"a (e E>'a>%&nes: ( + 6 -1
( 1( 2(( !
1!.- Ca9>'9& (e F'erBas @ M&"ent&s F%na9es:
VM
M1M2
M1V2M
V1M1M2
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arJ1. 2 5 1.5 -5 1.5
*.** *.** 2 .!
ar05. 0 55 .2 -55 .2
*.** *.** 2 .2
ara /+*
-2 !.21 ;N
15.- Ca9>'9& (e F'erBa en e9 Res&rte:
F R+ 6 RP( 2 F R+F R+ F R+
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AD CENTRAL DEL ECUADOR GENIERIA, CIENCIAS FISICAS Y MATEMATICA
SCUELA DE INGENIERIA CIVIL
ESTRUCTURAS III
5*6n7"
2 !
0.40 0.55 0.50 0.70
7.00 8.50
+*
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5*6n7"
2 !0
0.40 0.55 0.50 0.70
7.00 8.50
2 !0
( 2( !
(
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/+* /3*% I%
/% /%EI7L EI7LQ1 /+++++++++++
158 0.0027 1.20 0.021 10148 9938.4220 0.0055 1.20 0.019 16638 16320.1350 0.0143 1.20 0.021 35309 34570.3
=ARRA 1: ara /3*
6 + EI7LQ1 /+++++++++++4 + 2 -2 - 4 +
=ARRA 2: ara /3*
6 + EI7LQ1 /+++++++++++4 + 6/L 2 -6/L 6/L2 - 6/L 4 +
=ARRA : ara /3*
6 + EI7LQ1 /+++++++++++-6/L -6/L
-6/L 4 + 2 --6/L 2 - 4 +
ara /3*NUDO 2 NUDO NUDO !
105561 13989 32323 0
13989 13239 -10414 -24403
2 " !
H1 H2
H1 V2 H2
12/L 2
V1 H1 H212/L 2
(1
(2
( (!
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32323 -10414 204618 68402
0 -24403 68402 139020
ara /3*NUDO 2 NUDO NUDO !
1.3E-005 -2E-005 -2E-006 -2E-006
-2E-005 0.00014 2E-006 2E-005
-2E-006 2E-006 6E-006 -3E-006
-2E-006 2E-005 -3E-006 1E-005
=ARRA 2: =ARRA :-2*!.1 T-" - *1.*! T-"2*!.1 T-" *1.*! T-"1 5.** T V+ 212.5* T1 5.** T VK+ 212.5* T
2*!.1 - *1.*! *
* 1 5.** 212.5* 212.5*
( 1 ( 2 ( ( !
6 -1 +
MF+MKF+
simétricacargaser por
0y0 para 12
2
≠== β β pL
2
pL=
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-! .*!
- *1.*!J .00
2 !0
00.01077
-0.07589
0.00010
-0.01554
=ARRA 1: ara /3*
6 +85.68
556.26
=ARRA 2: ara /+*
6 +-556.26
23.73
-502.66
=ARRA : ara /+*
6 +153.36502.66
0.00
( 1( 2(( !
M1M2
M1V2M
V1M1M2
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a /+*-5*5.22 5*5.22 *.
* 22. * 15 .* 2 1.J!
-2 !.21a /3*
-5*2. 5*2. *.
2 . 15 . 2 1. !
-2 5.
ara /3*
-2 5. ;N
F R+
F R+
6 RP( 2
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=ARRA 1: ara /+*
*
6 +40592.05 20296.0220296.02 40592.05
=ARRA 2: ara /+*
6 +66550 14260.71 33275
14260.71 4074.49 14260.7133275 14260.71 66550
=ARRA : ara /+*
6 +5864.441 -24923.88 -24923.88-24923.88 141235.3 70617.65-24923.88 70617.65 141235.3
J.2.- Va9&res Ca9>'9a(&s:
( 1
( 1 ( 2 (
( 2 ( ( !
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.*!
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*
*
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=ARRA 1: ara /3*
*
6 +39963.3363 19667.313619667.3136 39963.3363
=ARRA 2: ara /3*
6 +65598 13989 3232313989 3997 1398932323 13989 65598
=ARRA : ara /3*
6 +5741.77783 -24402.5558 -24402.5558-24402.5558 139019.686 68402.0385-24402.5558 68402.0385 139019.686
( 1
( 1 ( 2 (
( 2 ( ( !
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=ARRA 1: ara /+*
6 ( +217.7662
435.53231
=ARRA 2: ara /+*
6 ( +-357.4073
-152.398925-709.385165
=ARRA : ara /+*
6 ( +-59.4375
806.260165-301.041667
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M +-126.04
6 ( +211.7256
126.04 430.219506
V +-204.17
6 ( +-352.0945
175.00 -151.273881204.17 -706.822662
V +212.50
6 ( +-59.1360
-301.04 803.697662301.04 -301.041667
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=ARRA 1: ara /3*
V +-126.04126.04
=ARRA 2: ara /3*
V +-204.17175.00204.17
=ARRA : ara /3*
V +212.50-301.04301.04
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! .- Estabilidad y determinación est"tica e#terna de la estructura
= 6
C*%"# + nN" L# #"%#d$"#
.D.I. = 3L# #"%#d$"# !# ' n $ *n: $"# + n ;!
$ .- Estabilidad interna de la armadura
.2.- N = 3 - 12 15
% .- Estabilidad y determinación est"tica total de la armadura
N! = N < N" 21! = 3 21
C*%"# + nN! = !
& .- 'oordenadas de los nudos
NUDO1 0.00 0.00 0.002 5.00 0.00 9.00
10.00 0.00 0.00! 2.50 4.00 2.005 5.00 4.00 6.00
7.50 4.00 2.005.00 8.00 4.00
( .- )ncidencias de las barras
NUDO DIAMETRO
N" > .
?A@ 0
.1.- CB $ * >+ $#' d '# '$'# , + #% n! d '# #"%#d$"# + *n !"+#n $'#
N! > . !
% @% B%
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Q""1 1 4 25 490.872 1 5 25 490.87
2 5 25 490.87! 2 6 25 490.875 3 6 25 490.87
3 4 25 490.874 5 25 490.87
0 5 6 25 490.87J 6 4 25 490.87
1* 4 7 25 490.8711 5 7 25 490.8712 6 7 25 490.87
1* .- +espla amientos de los nudos y en cada una de las barras
NUDO INICIAL=ARRA
1 0 0 0 d12 0 0 0 d4
0 0 0 d4! 0 0 0 d75 0 0 0 d7
0 0 0 d1d1 d2 d3 d4
0 d4 d5 d6 d7J d7 d8 d9 d1
1* d1 d2 d3 d1011 d4 d5 d6 d1012 d7 d8 d9 d10
11 .- '"lculo de longitudes, cosenos directores y rigide a#ial en cada
=ARRA@ B L
Q" Q" Q" Q"1 2.5 4 2 5.122 5 4 6 8.77
0 4 -3 5.00! 2.5 4 -7 8.445 -2.5 4 2 5.12
-7.5 4 2 8.732.5 0 4 4.72
0 2.5 0 -4 4.72
INICIAL Qn % FINAL Qn
% @% B%
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J -5 0 0 5.001* 2.5 4 2 5.1211 0 4 -2 4.4712 -2.5 4 2 5.12
12 .- atri de rigide para cada barra de la armadura
=ARRA 14.79 7.66 3.83 -4.797.66 12.26 6.13 -7.663.83 6.13 3.07 -3.83
-4.79 -7.66 -3.83 4.79-7.66 -12.26 -6.13 7.66-3.83 -6.13 -3.07 3.83
=ARRA 0.00 0.00 0.00 0.000.00 13.19 -9.90 0.000.00 -9.90 7.42 0.000.00 0.00 0.00 0.000.00 -13.19 9.90 0.000.00 9.90 -7.42 0.00
=ARRA 54.79 -7.66 -3.83 -4.79
-7.66 12.26 6.13 7.66-3.83 6.13 3.07 3.83-4.79 7.66 3.83 4.797.66 -12.26 -6.13 -7.663.83 -6.13 -3.07 -3.83
=ARRA 6.14 0.00 9.82 -6.140.00 0.00 0.00 0.009.82 0.00 15.71 -9.82
-6.14 0.00 -9.82 6.140.00 0.00 0.00 0.00
-9.82 0.00 -15.71 9.82
=ARRA J20.62 0.00 0.00 -20.62
0.00 0.00 0.00 0.00
6 G +
6 G +
6 G +
6 G +
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0.00 0.00 0.00 0.00-20.62 0.00 0.00 20.62
0.00 0.00 0.00 0.000.00 0.00 0.00 0.00
=ARRA 110.00 0.00 0.00 0.000.00 18.44 -9.22 0.000.00 -9.22 4.61 0.000.00 0.00 0.00 0.000.00 -18.44 9.22 0.000.00 9.22 -4.61 0.00
13 .- atri de rigide total para la armadura
NUDO !(1 (2
NUDO !(1 45.04 10.68(2 10.68 27.00( 15.16 13.50
NUDO 5(! -6.14 0.00(5 0.00 0.00
( -9.82 0.00
NUDO ( -20.62 0.00(0 0.00 0.00(J 0.00 0.00
NUDO (1* -4.79 -7.66(11 -7.66 -12.26(12 -3.83 -6.13
0.0776 0.00820.0082 0.1116
-0.0482 -0.09090.0046 -0.0463
-0.0019 -0.0009-0.0025 -0.02500.0776 0.00810.0270 0.00260.0431 0.00710.0261 0.05620.0078 0.02390.0169 0.0246
6 G +
6 G +
6 T +
6 T-1
+
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1! .- Matr%B (e >argas
"#d* d '+ "!#d % = 12!#d* d #" # n = 1
0 1 NNUDO 40 2 N
0 3 N10 4 N
NUDO 50 5 N + 0 6 N
0 7 NNUDO 60 8 N
0 9 N0 10 N
NUDO 7-15 11 N0 12 N
1! .- esolución del sistema de ecuaciones
)$ "F# +n! "n# < )$ "F# ;! "n# = 0 ?G@H?d@ < ? @ =
-0.0719d1 %%NUDO 4-0.8212d2 %%
1.1630 d3 %%2.1365 d4 %%
NUDO 5-0.9832d5 %%( + -0.2173d6 %%
-0.0404d7 %%
NUDO 6-0.1787d8 %%-0.9827d9 %%0.2881 d10 %%
NUDO 7-1.0259d11 %%0.5109 d12 %%
1$ .- '"lculo de fuer as internas en cada barra
=ARRA 9Q;N7"" Q" Q"
?d@ = ?G@-1H?- @
A%PE%7L% Q - % Q@ - @%
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1 20.120 -0.072 0.488 -0.8212 11.747 2.137 0.570 -0.983
20.617 2.137 0.000 -0.983! 12.212 -0.040 0.296 -0.1795 20.120 -0.040 -0.488 -0.179
11.805 -0.072 -0.859 -0.82121.854 2.208 0.530 -0.162
0 21.854 -2.177 0.530 0.805J 20.617 -0.032 -1.000 -0.643
1* 20.120 0.360 0.488 -0.20511 23.050 -1.848 0.000 -0.04312 20.120 0.329 -0.488 -0.847
1% .- '"lculo de fuer as de reacción y comprobación del e/uilibrio est"
FUER A
F1 -4.47 0.49 0.78 0.39F2 7.29 0.57 0.46 0.68F -13.53 0.00 0.80 -0.60F! 8.77 0.30 0.47 -0.83F5 -10.13 -0.49 0.78 0.39F -0.57 -0.86 0.46 0.23
R; =
9 + >&s " + >&s n + >&s
REACCIONES +
2 3
15 kN
R 2 R 5 R 8
R 1
R 3
R 4
R 6
R 7
R 9
10 kN
1 2 3
z
4 5 6
7
1 2 34
5
6
7 8
9
1011
12
y
x1
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R =RF =
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!,!+ #% n! +nd ! "%+n#d#
"n#
COORDENADASNUDO INICIAL
" .
1
4
7
5
2
6
3
1
2
3
4
5
6
7
11
8
1210
9
z
x
2.50
10.00
2.00
9.00
2.00
2.00
3.00
2.50 2.50 2.50
10 kN
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-1.00 0.00 0.00 20.62 20.620.49 0.78 0.39 20.12 4.790.00 0.89 -0.45 23.05 0.00
-0.49 0.78 0.39 20.12 4.79
-7.66 -3.83 3.81-12.26 -6.13 3.05
-6.13 -3.07 4.587.66 3.83 -3.81
12.26 6.13 -3.056.13 3.07 -4.58
0.00 0.00 1.07-13.19 9.90 1.71
9.90 -7.42 -3.000.00 0.00 -1.07
13.19 -9.90 -1.71-9.90 7.42 3.00
7.66 3.83 8.71-12.26 -6.13 -4.64
-6.13 -3.07 -2.32-7.66 -3.83 -8.71
12.26 6.13 4.646.13 3.07 2.32
0.00 -9.82 6.140.00 0.00 0.000.00 -15.71 -9.820.00 9.82 -6.140.00 0.00 0.000.00 15.71 9.82
0.00 0.00 4.790.00 0.00 7.66
6 G +
6 G +
6 G +
6 G +
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0.00 0.00 3.830.00 0.00 -4.790.00 0.00 -7.660.00 0.00 -3.83
0.00 0.00 4.79-18.44 9.22 -7.66
9.22 -4.61 -3.830.00 0.00 -4.79
18.44 -9.22 7.66-9.22 4.61 3.83
NUDO 5( (! (5 ( (
15.16 -6.14 0.00 -9.82 -20.6213.50 0.00 0.00 0.00 0.0022.47 -9.82 0.00 -15.71 0.00-9.82 16.09 3.05 4.58 -6.140.00 3.05 34.08 -15.45 0.00
-15.71 4.58 -15.45 48.95 9.820.00 -6.14 0.00 9.82 37.410.00 0.00 0.00 0.00 -13.620.00 9.82 0.00 -15.71 -20.49
-3.83 0.00 0.00 0.00 -4.79-6.13 0.00 -18.44 9.22 7.66-3.07 0.00 9.22 -4.61 3.83
MATRI INVERSA
-0.0482 0.0046 -0.0019 -0.0025 0.0776-0.0909 -0.0463 -0.0009 -0.0250 0.00810.2016 0.0907 0.0027 0.0511 -0.04800.0907 0.1781 -0.0418 -0.0177 0.00740.0027 -0.0418 0.0831 0.0276 -0.00310.0511 -0.0177 0.0276 0.0606 -0.0041
-0.0480 0.0074 -0.0031 -0.0041 0.1261-0.0162 0.0092 -0.0038 -0.0051 0.0438-0.0319 -0.0705 0.0294 0.0392 0.0701-0.0145 0.0261 -0.0109 -0.0145 0.0261-0.0170 -0.0237 0.0377 0.0027 0.00760.0117 0.0186 -0.0633 0.0107 0.0173
6 G +
6 G +
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0.781 1.163 0.390 -4.470 C*%&0.456 -0.217 0.684 7.291 T"#0.800 -0.217 -0.600 -13.529 C*%&0.474 -0.983 -0.829 8.772 T"#0.781 -0.983 0.390 -10.128 C*%&0.458 1.163 0.229 -0.567 C*%&0.000 -1.380 0.848 0.000 T"#0.000 -0.765 -0.848 -11.029 C*%&0.000 2.146 0.000 0.650 T"#0.781 -0.652 0.390 -4.803 C*%&0.894 0.728 -0.447 -8.385 C*%&0.781 1.494 0.390 -4.803 C*%&
tico total de la armadura
REACCIONES EN LOSR1 R2 R R! R5
-2.18 -3.49 -1.754.15 3.32 4.99
0.00 -10.822.60 4.16
-1.J *.1 - .2! -2. * .
-10.00 K R; = 0.00
9
x
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15.00 K R = 0.000.00 K RF = 0.00
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NUDO FINAL
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0.00 0.00 0.00 0.00 0.0012.26 3.07 7.66 3.83 6.1318.44 4.61 0.00 0.00 -9.2212.26 3.07 -7.66 -3.83 6.13
=ARRA 23.05 4.58 -3.81 -3.05 -4.582.44 3.66 -3.05 -2.44 -3.663.66 5.49 -4.58 -3.66 -5.49
-3.05 -4.58 3.81 3.05 4.58-2.44 -3.66 3.05 2.44 3.66-3.66 -5.49 4.58 3.66 5.49
=ARRA !1.71 -3.00 -1.07 -1.71 3.002.74 -4.80 -1.71 -2.74 4.80
-4.80 8.40 3.00 4.80 -8.40-1.71 3.00 1.07 1.71 -3.00-2.74 4.80 1.71 2.74 -4.804.80 -8.40 -3.00 -4.80 8.40
=ARRA -4.64 -2.32 -8.71 4.64 2.322.48 1.24 4.64 -2.48 -1.241.24 0.62 2.32 -1.24 -0.624.64 2.32 8.71 -4.64 -2.32
-2.48 -1.24 -4.64 2.48 1.24-1.24 -0.62 -2.32 1.24 0.62
=ARRA 00.00 -9.82 -6.14 0.00 9.820.00 0.00 0.00 0.00 0.000.00 15.71 9.82 0.00 -15.710.00 9.82 6.14 0.00 -9.820.00 0.00 0.00 0.00 0.000.00 -15.71 -9.82 0.00 15.71
=ARRA 1*7.66 3.83 -4.79 -7.66 -3.83
12.26 6.13 -7.66 -12.26 -6.13
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6.13 3.07 -3.83 -6.13 -3.07-7.66 -3.83 4.79 7.66 3.83
-12.26 -6.13 7.66 12.26 6.13-6.13 -3.07 3.83 6.13 3.07
=ARRA 12-7.66 -3.83 -4.79 7.66 3.83
12.26 6.13 7.66 -12.26 -6.136.13 3.07 3.83 -6.13 -3.077.66 3.83 4.79 -7.66 -3.83
-12.26 -6.13 -7.66 12.26 6.13-6.13 -3.07 -3.83 6.13 3.07
NUDO NUDO (0 (J (1* (11 (12
0.00 0.00 -4.79 -7.66 -3.830.00 0.00 -7.66 -12.26 -6.130.00 0.00 -3.83 -6.13 -3.070.00 9.82 0.00 0.00 0.000.00 0.00 0.00 -18.44 9.22
0.00 -15.71 0.00 9.22 -4.61-13.62 -20.49 -4.79 7.66 3.8327.27 7.46 7.66 -12.26 -6.13
7.46 30.25 3.83 -6.13 -3.077.66 3.83 9.58 0.00 0.00
-12.26 -6.13 0.00 42.97 3.04-6.13 -3.07 0.00 3.04 10.74
0.0270 0.0431 0.0261 0.0078 0.01690.0026 0.0071 0.0562 0.0239 0.0246
-0.0162 -0.0319 -0.0145 -0.0170 0.01170.0092 -0.0705 0.0261 -0.0237 0.0186
-0.0038 0.0294 -0.0109 0.0377 -0.0633-0.0051 0.0392 -0.0145 0.0027 0.01070.0438 0.0701 0.0261 0.0076 0.01730.0825 0.0165 -0.0417 0.0180 0.03870.0165 0.1271 -0.0145 0.0185 0.0174
-0.0417 -0.0145 0.2087 -0.0018 0.00360.0180 0.0185 -0.0018 0.0526 -0.02170.0387 0.0174 0.0036 -0.0217 0.2024
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8/17/2019 7.- Viga Continua
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" + n+ n
" + n+ n
" + n" + n
+ n" + n
+ n" + n" + n" + n
OYOSR R R0 RJ
8.12-7.27
4.94 -7.91 -3.950.49 -0.26 -0.13
-*.0! -5.! 0.1 !.*0
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C$%&' '# *nd+ + n
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81/86
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