Universidad Nacional de San ´ bal de Huamanga Cristobal o Facultad de Ingenier´ıa de Minas, Geolog´ıa y Civil Escuela Profesional de Ingenier´ıa Civil Departamento Acad´ emico de Minas y civil emico
´ lisis Estructural II Analisis a
Trabajo Domiciliario N o3 Docente
:
˜ , Rub´ Ing.Y Ing.YAC ACHAP HAPA A CODE CODENA NA, Ru b´en en Am´eric er icoo
Apellidos y Nombre
Emiliano no Vilchez Alvites , Emilia
Codigo o ´digo
161376 161 37633 33
Ayacucho-Per´ u 2017
An´ alisis alisis Estructura Estructurall II
Resoluci´ on de los Ejercicios Propuestos del Segundo on ´ lisis Matricial de Cap Ca p´ıtu ıtulo lo de dell Li Libr bro o Analisis a de Roberto Aguiar Falconi Estructuras de Estructuras En las siguientes estructuras se ha seleccionado un sistema Q los diagramas de desplazamientos elementales.
para cada ejercicio. Se pide dibujar todos − q para
Ejercicio No 1
Resoluci´ on: on:
†
q 1 = 1 y q i = 0,
Ingenier´ıa Civil
∀i = 1
3
UNSCH
An´alisis Estructural II
Elemento AB: u1 = 0, v 1 = 0, θ1 = 0; u2 = cos β , v 2 = sin β , θ2 = 0
−
Elemento BC:
u1 = 0, v 1 = 0, θ1 = 0; u2 = sin β , v 2 = cos β , θ2 = 0
†
UNSCH
∀i = 2
q 2 = 1 y q i = 0,
4
Ingenier´ıa Civil
An´alisis Estructural II
Elemento AB: u1 = 0, v 1 = 0, θ1 = 0; u2 = cosβ , v 2 = sin β θ2 = 0
−
Elemento BC:
u1 = 0, v 1 = 0, θ1 = 0; u2 = sin β , v 2 = cos β θ2 = 0
−
†
∀i = 2
q 2 = 1 y q i = 0,
Ingenier´ıa Civil
5
UNSCH
An´alisis Estructural II Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u2 = 0, v2 = 0 θ2 = 1rad
Elemento BC:
u1 = 0, v1 = 0, θ1 = 0; u2 = 0, v2 = 0 θ2 = 1rad
Ejercicio No 2
Por el n´ umero de coordenadas generalizadas que muestra el ejercicio, los apoyos debieron ser empotrados, siendo de la siguiente manera:
UNSCH
6
Ingenier´ıa Civil
An´alisis Estructural II
Resoluci´ on:
†
q 1 = 1 y q i = 0,
∀i = 1
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ 2 =
Elemento BC:
u1 = 0, v1 = 0, θ1 = 1; u 2 = 0, v 2 = 0, θ 2 = 0rad
Los otros Elementos
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ 2 = 0rad
†
−1rad
∀i = 2
q 2 = 1 y q i = 0,
Elemento AB y DE:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ 2 = 0rad
Elemento BC:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ 2 =
Ingenier´ıa Civil
7
−1rad UNSCH
An´alisis Estructural II Elemento CD:
u1 = 0, v1 = 0, θ1 = 1rad ; u2 = 0, v 2 = 0, θ 2 = 0
Elemento CF:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
†
−1rad
q 3 = 1 y q i = 0,
∀i = 3
Elemento AB, BC y CF:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ 2 = 0rad
Elemento CD:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
−1rad
Elemento DE:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
−1rad
†
q 4 = 1 y q i = 0,
∀i = 4
Elemento AB,CF,DE:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 =
Elemento BC y CD:
u1 = 1, v1 = 0, θ1 = 0; u 2 = 1, v 2 = 0, θ2 = 0
UNSCH
8
−1, θ = 0 2
Ingenier´ıa Civil
An´alisis Estructural II
Ejercicio No 3
†
∀i = 1
q 1 = 1 y q i = 0,
Elemento AB y EF:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 =
Elemento BC Y DE:
u1 = 0, v1 =
Elemento BE:
u1 = 1, v1 = 0, θ1 = 0; u 2 = 1, v 2 = 0, θ2 = 0
Ingenier´ıa Civil
−1, θ
1
−1, θ = 0 2
= 0; u 2 = 0, v2 = 0, θ2 = 0
9
UNSCH
An´alisis Estructural II u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento CD:
†
q 2 = 1 y q i = 0,
∀i = 2
Elemento AB, BC y CD:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento BE y EF:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
Elemento DE:
u1 = 0, v1 = 0, θ1 =
†
UNSCH
−1rad; u
2
−1rad
= 0, v 2 = 0, θ2 = 0
∀i = 3
q 3 = 1 y q i = 0,
10
Ingenier´ıa Civil
An´alisis Estructural II Elemento AB, BE, y EF:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento BC y ED:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 =
Elemento CD:
u1 = 1, v1 = 0, θ1 = 0; u 2 = 1, v 2 = 0, θ2 = 0
−1, θ = 0rad 2
Ejercicio No 4
†
q 1 = 1 y q i = 0,
Elemento AB:
Ingenier´ıa Civil
∀i = 1
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 =
11
−1, θ = 0 2
UNSCH
An´alisis Estructural II u1 = 0, v1 =
Elemento CD:
−1, θ
1
= 0; u 2 = 0, v2 = 0, θ2 = 0
Elemento BC:
u1 = cos α, v1 = sin α, θ1 = 0; u2 = sin α, v2 = cos α θ2 = 0
†
∀i = 2
q 2 = 1 y q i = 0,
Elemento AB y BC:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento CD:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
UNSCH
12
−1rad
Ingenier´ıa Civil
An´alisis Estructural II
Ejercicio No 5
†
q 1 = 1 y q i = 0,
∀i = 1
Elemento AB:
u1 = 0, v1 = 0, θ1 = 1rad ; u2 = 0, v 2 = 0, θ2 = 0
Elemento BC:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Ingenier´ıa Civil
13
UNSCH
An´alisis Estructural II
†
∀i = 2
q 2 = 1 y q i = 0,
Elemento AB:
u1 = 0, v 1 = 0, θ1 = 0; u2 = cos α, v 2 = sin α θ2 = 0
−
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento BC:
†
UNSCH
∀i = 3
q 3 = 1 y q i = 0,
14
Ingenier´ıa Civil
An´alisis Estructural II
Elemento AB:
u1 = 0, v 1 = 0, θ1 = 0; u2 = sin α, v2 = cos α θ2 = 0
u1 = 0, v1 = 1, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento BC:
†
∀i = 4
q 4 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
Elemento BC:
u1 = 0, v1 = 0, θ1 = 1rad ; u2 = 0, v 2 = 0, θ2 = 0
Ingenier´ıa Civil
15
−1rad
UNSCH
An´alisis Estructural II
†
∀i = 5
q 5 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento BC:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
−1rad
Ejercicio No 6
UNSCH
16
Ingenier´ıa Civil
An´alisis Estructural II
†
∀i = 1
q 1 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 1rad ; u2 = 0, v 2 = 0, θ2 = 0
Elemento BC:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
†
∀i = 2
q 2 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
Elemento BC:
u1 = 0, v1 = 0, θ1 = 1rad ; u2 = 0, v 2 = 0, θ2 = 0
Ingenier´ıa Civil
17
−1rad
UNSCH
An´alisis Estructural II
†
∀i = 3
q 3 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento BC:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
−1rad
Ejercicio No 7
†
UNSCH
∀i = 1
q 1 = 1 y q i = 0,
18
Ingenier´ıa Civil
An´alisis Estructural II
Elemento AB:
u1 = 0, v 1 = 0, θ1 = 0; u2 = cos α, v 2 = sin α θ2 = 0
−
Elemento BC:
u1 = cos α, v1 = sin α, θ1 = 0; u2 = 0, v 2 = 0 θ2 = 0
†
∀i = 2
q 2 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 =
Elemento BC:
u1 = 0, v1 = 0, θ1 = 1rad ; u2 = 0, v 2 = 0, θ2 = 0rad
Ingenier´ıa Civil
19
−1rad
UNSCH
An´alisis Estructural II
Ejercicio No 8
†
q 1 = 1 y q i = 0,
∀i = 1
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 =
Elemento CD:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
UNSCH
20
−1, θ = 0 2
Ingenier´ıa Civil
An´alisis Estructural II
Elemento BC:
u1 = cos α, v 1 = sin α, θ1 = 0; u2 = 0, v 2 = 0 θ2 = 0
−
†
q 2 = 1 y q i = 0,
∀i = 2
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 1, v 2 = 0, θ2 = 0
Elemento CD:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0
Elemento BC:
u1 = cos α, v1 = sin α, θ1 = 0; u2 = 0, v 2 = 0 θ2 = 0
Ingenier´ıa Civil
21
UNSCH
An´alisis Estructural II
†
∀i = 3
q 3 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 1, v 2 = 0, θ2 =
−0,2rad
Elemento CD:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 1, v 2 = 0, θ2 = 0
Elemento BC:
u1 = 0, v 1 = 0, θ1 = 0,2rad; u2 =
L(tan β
√ 2 − 1) , v
2
=
θ2 = 0
UNSCH
22
L(tan β
√ 2 − 1)
Ingenier´ıa Civil
An´alisis Estructural II
†
∀i = 4
q 4 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 = 0, θ2 = 0rad
Elemento CD:
u1 = 0, v1 = 0, θ1 = 0; u 2 = 0, v 2 =
−1, θ = 0 2
Elemento BC:
u1 = 0, v 1 = 0, θ1 =; u2 =
√ 2, v = −√ 2 2
θ2 = 0
Ingenier´ıa Civil
23
UNSCH
An´alisis Estructural II
Ejercicio No 9
†
∀i = 1
q 1 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0; u2 = 0, v 2 =
−1,
Elemento BC:
u1 = 1, v1 = 0; u2 = 0, v 2 = 0,
Elemento AC, AF, CF, CD, u1 = 1, v1 = 0; u2 = 0, v 2 = 0, CE, DE, EF:
UNSCH
24
Ingenier´ıa Civil
An´alisis Estructural II
†
∀i = 2
q 2 = 1 y q i = 0,
Elemento AB:
u1 = 0, v1 = 0; u2 = 1, v 2 = 0,
Elemento BC:
u1 = 0, v1 = 1; u2 = 0, v 2 = 0,
Elemento AC, AF, CF, CD, u1 = 1, v1 = 0; u2 = 0, v 2 = 0, CE, DE, EF:
†
∀i = 3
q 3 = 1 y q i = 0,
Ingenier´ıa Civil
25
UNSCH
An´alisis Estructural II Elemento AB, AF, ED, EF: u1 = 0, v1 = 0; u2 = 0, v 2 = 0, Elemento BC:
u1 = 0, v1 = 0; u2 = 1, v 2 = 0,
Elemento CD:
u1 = 1, v1 = 0; u2 = 0, v 2 = 0,
Elemento AC:
u1 = 0, v 1 = 0; u2 = cos α, v 2 = sin α
−
Elemento CE:
u1 = sin α, v 1 = cos α; u2 = 0, v 2 = 0
UNSCH
26
Ingenier´ıa Civil
An´alisis Estructural II
†
∀i = 4
q 4 = 1 y q i = 0,
Elemento AB, AF, ED, EF: u1 = 0, v1 = 0; u2 = 0, v 2 = 0, Elemento BC:
u1 = 0, v1 = 0; u2 = 0, v 2 = 1,
Elemento CD:
u1 = 0, v1 = 1; u2 = 0, v 2 = 0,
Elemento AC:
u1 = 0, v 1 = 0; u2 = cos α, v 2 = sin α
Ingenier´ıa Civil
27
UNSCH
An´alisis Estructural II
Elemento CE:
u1 =
†
− cos α, v
= sin α; u2 = 0, v 2 = 0 1
∀i = 5
q 5 = 1 y q i = 0,
Elemento CD:
u1 = 0, v1 = 0; u2 = 1, v 2 = 0,
Elemento DE:
u1 = 0, v1 = 0; u2 = 0, v 2 =
−1,
Elemento AB, AF, AC, BC, u1 = 0, v1 = 0; u2 = 0, v 2 = 0, CF, CE, EF:
UNSCH
28
Ingenier´ıa Civil
An´alisis Estructural II
†
∀i = 6
q 6 = 1 y q i = 0,
Elemento CD:
u1 = 0, v1 = 0; u2 = 0, v 2 = 1,
Elemento DE:
u1 = 0, v1 = 0; u2 = 1, v 2 = 0,
Elemento AB, AF, AC, BC, u1 = 0, v1 = 0; u2 = 0, v 2 = 0, CF, CE, EF:
Ingenier´ıa Civil
29
UNSCH
