portales y brocales cga

Estudios y Diseños - Ruta del Sol

PORTALES FALSOS Y BROCALES - MEMORIA DECÁLCULO ESTRUCTURAL

2361-00-SU-MC-XXX

Los derechos de autor de este documento son de HMV INGENIEROS LTDA-PCA., quien queda exonerada de toda responsabilidad si este documento es alterado o modificado. No se autoriza su empleo o reproducción total o parcial con fines diferentes al contratado.

PORTALES FALSOS Y BROCALES -MEMORIA DE CÁLCULO ESTRUCTURAL

DISEÑOS DE LAS OBRAS NECESARIAS PARA LA CONSTRUCCIÓN DEL PROYECTO

RUTA DEL SOL SECTOR No. 1 VILLETA – GUADUERO – EL KORÁN


INFORME No.2361-00-SU-MC-XXX Revisión 0

CONSORCIO

Mayo 2011

PORTALES FALSOS YBROCALES - MEMORIA DECÁLCULO ESTRUCTURAL

INFORME2361-00-SU-MC-XXX Rev. 0

Bogotá, Mayo 2011

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LISTA DE DISTRIBUCIÓN

Copias de este documento han sido entregadas, según se indica a continuación. Las observaciones que resulten de su revisión y aplicación deben ser informadas a esta oficina para proceder a realizar sus modificaciones:

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ÍNDICE DE MODIFICACIONES

Índice Capítulo Fecha de ObservacionesRevisión Modificado Modificación

0 Revisión General Mayo 2011 Aprobado para construcción



Bogotá, Mayo 2011

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ESTADO DE REVISIÓN Y APROBACIÓN

Tipo de Documento:

INFORME TÉCNICO

TítuloPORTALES FALSOS Y BROCALES - MEMORIA DECÁLCULO ESTRUCTURALDocumento N°:

2361-00-SU-MC-XXX Rev. 0

APROBAC I ÓN

Número de Revisión 0

Elaboró

Nombre Ing. Andrés Cárdenas M X

Firma

Fecha 30 de Mayo de 2011

Revisó

Nombre Ing. Ricardo Rey X

Firma


Nombre Ing. Eduardo Castell X

AprobóCONSORCIO HMV-PCA

Firma


Nombre

AprobaciónCONSORCIO VIAL HELIOS

Firma

Fecha

PORTALES FALSOSY BROCALES -MEMORIA DE

CÁLCULOESTRUCTURAL

INFORME2361-00-SU-MC-XXX - Rev. 0

Bogotá, Mayo de 2011

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2361-00-SU-MC-XXXREVISIÓN 0

TABLA DE CONTENIDO1. INTRODUCCIÓN...........................................................................................................82. OBJETIVO.....................................................................................................................83. CÓDIGOS DE DISEÑO Y ESTÁNDARES.....................................................................84. GEOMETRÍA PORTAL FALSO Y BROCAL..................................................................94.1 GEOMETRÍA EN PLANTA Y ELEVACIÓN..................................................................104.2 MATERIALES..............................................................................................................105. EVALUACIÓN DE CARGAS PORTALES FALSOS....................................................115.1 CASOS DE CARGA.....................................................................................................115.1.1 DC: Carga muerta de componentes estructurales.......................................................115.1.2 Empuje horizontal de terreno (EH) y Empuje vertical de terreno (EV)........................115.2 COMBINACIONES DE CARGA (CCDSP 95)..............................................................126. DISEÑO ESTRUCTURAL PORTAL FALSO................................................................136.1 ESTADO LÍMITE DE RESISTENCIA - DISEÑO A FLEXIÓN PORTAL FALSO..........136.2 ESTADO LÍMITE DE RESISTENCIA - CHEQUEO A FUERZA CORTANTE PORTAL FALSO.....................................................................................................................................156.3 ESTADO LÍMITE DE SERVICIO - CONTROL DE LA FISURACIÓN PORTAL FALSO

166.4 REFUERZO DE RETRACCIÓN Y TEMPERATURA...................................................177. MODELO PORTAL FALSO.........................................................................................177.1 EMPUJE DE SUELOS SOBRE LA ESTRUCTURA....................................................197.2 RESULTADOS DEL MODELO PORTAL FALSO........................................................207.2.1 Momentos actuantes combinación STRENGTH I........................................................207.2.2 Cortantes actuantes combinación. STRENGTH I........................................................217.3 DEFORMACIONES MÁXIMAS PORTAL FALSO........................................................228. EVALUACIÓN DE CARGAS BROCALES...................................................................24


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8.1 CASOS DE CARGA CUANDO EL BROCAL ESTA POR FUERA DE LA EXCAVACIÓN SUBTERRÁNEA.............................................................................................248.1.1 DC: Carga muerta de componentes estructurales.......................................................248.1.2 Empuje horizontal de terreno (EH) y Empuje vertical de terreno (EV) Caso en que el Brocal quede por fuera de la excavación subterránea............................................................248.2 COMBINACIONES DE CARGA (CCDSP 95)..............................................................248.3 CASOS DE CARGA CUANDO EL BROCAL ESTA DENTRO DE LA EXCAVACIÓN 258.4 COMBINACIONES DE CARGA (CCDSP 95)..............................................................259. DISEÑO ESTRUCTURAL BROCAL............................................................................269.1 ESTADO LÍMITE DE RESISTENCIA - DISEÑO A FLEXIÓN BROCAL......................269.2 ESTADO LÍMITE DE RESISTENCIA - CHEQUEO A FUERZA CORTANTE BROCAL

279.3 RESULTADOS DEL MODELO BROCALES................................................................279.3.1 Momentos actuantes combinación STRENGTHI=1,3∗DC+1,3∗EV +1,3∗EH ........289.3.2 Cortantes actuantes combinación STRENGTHI=1,3∗DC+1,3∗EV +1,3∗EH .........289.3.3 Momentos actuantes combinación STRENGTHVII=1,0∗DC+1,0∗EV +1,0∗EH+1,0 EQ..........................................................299.3.4 Cortantes actuantes combinación STRENGTHVII=1,0∗DC+1,0∗EV +1,0∗EH+1,0 EQ..........................................................309.4 DEFORMACIONES MÁXIMAS BROCALES...............................................................31

LISTA DE FIGURAS

Figura 1 Sección Transversal Falso Túnel...............................................................................9Figura 2 Sección Transversal Brocal.....................................................................................10Figura 3 Vista Frontal modelo Falso túnel..............................................................................18Figura 4 Vista Isométrica Falso túnel.....................................................................................18Figura 5 Empuje Vertical de suelos. Portal Falso (kN/m²)......................................................19Figura 6 Empuje Horizontal de suelos. Portal Falso (kN/m²)..................................................19Figura 7 Pesos muertos sobre losa. Portal Falso (kN/m²)......................................................20Figura 8 Momentos actuantes sobre la sección. Portal Falso. Combinación

STRENGTH I (kN-m/m).....................................................................................21Figura 9 Cortantes máximos en la sección. Portal Falso. Combinación STRENGTH I

(kN/m)................................................................................................................22Figura 10 Deformaciones máximas verticales. Portal Falso Combinación SERVICE I

(mm)...................................................................................................................23Figura 11 Deformaciones máximas horizontales. Portal Falso Combinación

SERVICE I (mm)................................................................................................23


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Figura 12 Momento actuantes Brocal combinación STRENGTH I (kN-m/m)........................28Figura 13 Cortante máximo V23 (kN), Brocal combinación STRENGTH I (kN/m).................29Figura 14 Momento actuantes Brocal combinación STRENGTH VII (kN-m/m)......................30Figura 15 Cortante máximo V23 (kN), Brocal combinación STRENGTH VII (kN)..................31Figura 16. Deformaciones máximas verticales. Brocal Combinación SERVICE I

(mm)...................................................................................................................32Figura 17 Deformaciones máximas horizontales. Brocal Combinación SERVICE I

(mm)...................................................................................................................32Figura 18 Deformaciones máximas verticales. Brocal Combinación STRENGTH VII

(mm)...................................................................................................................33Figura 19 Deformaciones máximas horizontales. Brocal Combinación

STRENGTHVIII (mm).........................................................................................33

LISTA DE TABLAS

Tabla 1 Radio de esfuerzos por momento en la sección Falso Túnel. Combinación STRENGTH I.....................................................................................................21

Tabla 2 Radio de esfuerzos por cortante en la sección Falso Túnel. Combinación STRENGTH I.....................................................................................................22

Tabla 3 Radio de esfuerzos por momento en la sección de Brocal. Combinación STRENGTH I.....................................................................................................28

Tabla 4 Radio de esfuerzos por Cortante en el Brocal. Combinación STRENGTH I.............29Tabla 5 Radio de esfuerzos por momento en la sección de Brocal. Combinación

STRENGTH VII..................................................................................................30Tabla 6 Radio de esfuerzos por cortante en la sección de Brocal. Combinación

STRENGTH VII..................................................................................................31

LISTA DE APÉNDICES

Apéndice 1: CARACTERÍSTICAS GENERALES DE LA APLICACIÓN SAP2000.................25


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PORTALES FALSOS Y BROCALES - MEMORIA DE CÁLCULO ESTRUCTURAL

INFORME2361-00-SU-MC-XXX Revisión 0

1. INTRODUCCIÓNEl Consorcio HMV INGENIEROS LTDA. – PCA. PROYECTISTAS CIVILES ASOCIADOS LTDA realizará la prestación de Servicios Técnicos para la Elaboración de los Estudios de Detalle y el desarrollo de las Gestiones Ambiental, Predial y Social, exigidos en el Contrato de concesión de Obra pública INCO No. 002 del 14 de Enero de 2010, sus apéndices y demás documentos que lo integran para el Trayecto comprendido entre Villeta y el Korán, Correspondiente al Sector 1 del proyecto denominado Ruta del Sol e incluye la pavimentación del Acceso a Caparrapí y la variante de Guaduas.

El presente documento hace parte de los anexos correspondientes al informe “ESTUDIO GEOTÉCNICO PARA TÚNELES”. En particular contiene la memoria de cálculo estructural de los Portales falsos y los brocales para túneles.

2. OBJETIVOPresentar las hipótesis de diseño, el modelo de estructural, la interpretación de resultados del análisis estructural y el diseño de los elementos estructurales que componen los Portales Falsos a la salida de los Túneles.

3. CÓDIGOS DE DISEÑO Y ESTÁNDARESEl diseño estructural de las obras relacionadas con los Portales Falsos estará regido por el código de puentes local vigente, CCDSP-951.

En los aspectos en que la norma local no sea suficiente se recurrirá al código de puentes de Estados Unidos, AASHTO BDS LRFD 20072.

1 ASOCIACIÓN COLOMBIANA DE INGENIERÍA SÍSMICA (AIS). Código colombiano de diseño sísmico de puentes 1995 (CCDSP-95), Incluyendo Adendo No. 1 de 1996 adoptado mediante Resolución 3600 de 1996 del INVIAS. Bogotá.2 AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORTATION OFFICIALS (AASHTO). Bridge Design Specifications - Load and Resistance Factor Design 2007 SI Units. USA. 2007.


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4. GEOMETRÍA PORTAL FALSO Y BROCALA continuación se presenta la geometría típica del falso túnel y Brocal.

Figura 1 Sección Transversal Falso Túnel


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Figura 2 Sección Transversal Brocal

4.1 GEOMETRÍA EN PLANTA Y ELEVACIÓNLa geometría particular de cada estructura se presenta en planta y sección longitudinal en los planos de dimensiones respectivos.

4.2 MATERIALES


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Se consideró Concreto tipo C de 28 MPa. Se especificó cemento ASTM C1503 tipo I ó ASTM C11574 Tipo GU.

Se consideró como refuerzo principal el constituido por barras corrugadas ASTM A7065 de 420 MPa de límite inferior de fluencia garantizado.

5. EVALUACIÓN DE CARGAS PORTALES FALSOS

5.1 CASOS DE CARGA

5.1.1 DC: Carga muerta de componentes estructurales.Se consideró el peso propio de los materiales estructurales de acuerdo con sus dimensiones definitivas. El peso propio es calculado directamente en el software de análisis estructural.

5.1.2 Empuje horizontal de terreno (EH) y Empuje vertical de terreno (EV).Las cargas de empuje de terreno se calcularon según se muestra a continuación

Empuje de aguasEl lleno exterior del túnel falso se considera drenado. Por tal motivo no se considera presión por ascenso del N.A.F.

Carga viva VehicularEl túnel falso no se diseña paraqué exista la posibilidad de carga viva vehicular encima 7

Factor de interacción suelo estructura (AASHTO 12.11.2.2.1-2)

H fill=5,0m Altura del relleno por encima de la cota clave del túnel

B=11,90m Ancho exterior

3 AMERICAN SOCIETY FOR TESTING OF MATERIALS. ASTM C150/C150M-09 Standard Specification for Portland Cement. USA. 2009.4 AMERICAN SOCIETY FOR TESTING OF MATERIALS (ASTM). ASTM C1157/C1157M-10 Standard Performance Specification for Hydraulic Cement. USA. 2010.5 AMERICAN SOCIETY FOR TESTING OF MATERIALS (ASTM). ASTM A706/A706M-09b Standard Specification for Low-Alloy Steel Deformed and Plain Bars for Concrete Reinforcement. USA. 2009


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F e=(1+ 0,2∗H fill

Bc) Factor de interacción suelo estructura en terraplén

F e=(1+ 0,2∗5,0011,90 )=1,05

F e≤1,15 Cuando se especifican llenos compactados alrededor de la estructura

Cálculo de presiones de terreno (CCSDP A.12.4)

E v1=F e∗19 [ kNm3 ]∗(H fill+t top2 ) Empuje de terreno vertical

t top=0,40m Espesor de concreto parte superior de la estructura

E v1=1,05∗19[ kNm3 ]∗(5,0+ 0,402 )=103 ,74[ kNm2 ]

Eh1=10 [ kNm3 ]∗H fill [m ] Empuje de terreno horizontal en la parte superior del

muro (CCDSP95 A.12.4.1.a combinación 1)

Eh1=10 [ kNm3 ]∗5,0 [m ]=50[ kNm2 ] Eh2=10 [ kNm3 ]∗¿ Empuje de terreno horizontal en la parte

inferior del muro (CCDSP95 A.12.4.1.a combinación 1)

Eh2=10 [ kNm3 ]∗(5,0+0,40+8,62+ 0,502 ) [m ]=142,70[ kNm2 ]5.2 COMBINACIONES DE CARGA (CCDSP 95)Se consideraron las siguientes combinaciones de carga:


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STRENGT H I=1,3∗DC+1,3∗EV+1,3∗EH

Para estado límite de resistencia teniendo en cuenta los valores de empuje de terreno de CCDSP95 A.12.4.1.a combinación 1.

STRENGTH II=1,3∗DC+1,3∗EV +1,3∗0,5∗EH

Para estado límite de resistencia teniendo en cuenta los valores de empuje de terreno de CCDSP95 A.12.4.1.a combinación 2.

SERVICEI=DC+EV+EH

Para verificaciones en estado límite de servicio de los resultados de la combinación STRENGTH_I.

SERVICEII=DC+EV +0,5∗EH

Para verificaciones en estado límite de servicio de los resultados de la combinación STRENGTH_II.

6. DISEÑO ESTRUCTURAL PORTAL FALSO

6.1 ESTADO LÍMITE DE RESISTENCIA - DISEÑO A FLEXIÓN PORTAL FALSOLa resistencia a flexión se verificó en las secciones de diseño indicadas anteriormente en muro y losa superior.

Cálculo de parámetros de diseñob=1000 mm Ancho de la sección transversal de muro o losah Altura toral de la seccióndb Diámetro de la barra de refuerzo suministradas Separación de las barras de refuerzo suministradas

d=h−rec−db

2 Altura afectiva de la sección


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ρ=A s

bd Cuantía de refuerzo suministrada

f ' c Resistencia especificada del concreto a los 28 días

f y Resistencia la fluencia del acero de refuerzo

φ f=0,9 Coeficiente de reducción de resistencia a flexión

φ f M n=φ f ρ f y bd2(1−0,59 ρ f y

f ' c ) Momento nominal resistente de la sección

φ f M n=0,9∗0,0061∗420000∗1∗0,312(1−0,59∗0,0061 42000028000 )=209,62kN−m /m

Para la sección de 0,40 m en la bóvedaRefuerzo principal 1N6 c/0,15 m

d=0.40−0.075−0.0192

=0,31

ρ=2,84 /0,15100∗31

=0,0061

f ' c=28,0MPa

φ f M n=0,9∗0,0061∗420000∗1∗0,312(1−0,59∗0,0061 42000028000 )=209,62kN−m /m

Para la sección de 0,50 m en la zapataRefuerzo principal 1N6 c/0,15 m

d=0.50−0.075−0.0192

=0,041

ρ=2,84 /0,15100∗41

=0,0046

f ' c=28,0MPa

φ f M n=0,9∗0,0046∗420000∗1∗0,412(1−0,59∗0,0046 42000028000 )=280,4 kN−m /m


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Para la sección de 0,75 m en la intersección hastial zapataRefuerzo principal 1N6 c/0,15 m

d=0.75−0.075−0.0192

=0,66

ρ=2,84 /0,15100∗66

=0,0029

f ' c=28,0MPa

φ f M n=0,9∗0,0029∗420000∗1∗0,662(1−0,59∗0,0029 42000028000 )=476,3kN−m/m

Verificación de la capacidad de la secciónM u Momento flector último de la sección

M u<φ f Mn

Verificación de la cuantía mínima

f ' cr=0,63√ f 'cMódulo de rotura del concreto

I g=bh3

12Inercia bruta de la sección

M cr=f 'cr I g

h/2Momento de fisuración de la sección

C1=φ f f ybd2 Variable auxilar

C2=C1∗0,59f y

f ' cVariable auxiliar

ρmí n1=(C1−√(C1

2−4C2(1,2M cr)))2C2

Cuantía mínima (cuantía requerida para 1.2*Mcr)


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ρmí n2=(C1−√(C1

2−4C2(1,333M u)) )2C2

Cuantía requerida para 1,330*Mu)

ρmí n=mí n {ρmí n1

ρmí n2 Cuantía mínima de diseño

ρ≥ ρmí n

Verificación de la cuantía máximaϵ u=0,003 Deformación última del concreto

ϵ y=0,002 Deformación de fluencia del acero de refuerzo

β1=0,85−0,05( f 'c−28 )7

0,65≤ β1≤0,85 Coeficiente del modelo de Whitney

ρb=0,85 β1f 'cf y

( ϵu

ϵu+ϵ y) Cuantía balanceada de la sección

ρmáx=0,75 ρb Cuantía máxima de la sección

ρ≤ ρmáx

6.2 ESTADO LÍMITE DE RESISTENCIA - CHEQUEO A FUERZA CORTANTE PORTAL FALSO

La resistencia al cortante se chequeó en las secciones de diseño indicadas anteriormente en muro y losa superior.

Resistencia del concreto al cortante:φc Coeficiente de reducción de resistencia a cortante.

V c Resistencia nominal del concreto a cortante

φcV c=φc0,248√ f c' bd Resistencia de diseño a cortante para las losas vaciadas

monolíticamente con los muros en alcantarillas cajón de una celda


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φcV c=φc√ f 'c6

bd Resistencia de diseño a cortante para los muros.

Para la sección de 0,40 m en la bóveda

φcV c=0,85∗√28

6∗1000∗310=232,4 kN

Para la sección de 0,50 m en la zapata

φcV c=0,85∗√28

6∗1000∗410=307,4 kN

Para la sección de 0,75 m en la intersección hastial zapata

φcV c=0,85∗√28

6∗1000∗660=494,8 kN

Verificación de la resistenciaV u≤φcV c

6.3 ESTADO LÍMITE DE SERVICIO - CONTROL DE LA FISURACIÓN PORTAL FALSOEl control de la fisuración se chequeó en las secciones de diseño indicadas anteriormente en muro y losa superior.

Cálculo de la tensión máxima en la fibra extrema a tracción (AASTHO 5.7.3.4-1)

dc=rec+db

2 Parámetro de cálculo de distribución del refuerzo

γe=1 Factor de exposición

βs=1+dc

0,7 (h−dc ) Parámetro de cálculo de distribución del refuerzo


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f smáx=123000 [MPa∙mm]γ e

βs (s+2dc ) Tensión máxima permitida en el acero a tracción

Calculo de la tensión de la fibra extrema a tracción en estado límite de servicio (AASHTO C12.11.3-1)

e=M s

P s+d−h

2Parámetro de cálculo de la sección elástica fisurada

j=0,74+0,1 ed≤0,9 Parámetro de cálculo de la sección elástica fisurada

k= 1

1− j de

Parámetro de cálculo de la sección elástica fisurada

Ps Fuerza axial en el elemento en estado límite de servicio (Ps<0 = compresión)Ms Momento flector en el elemento en estado límite de

servicio

f s=M s+Ps(d−h

2 )A s∑ ¿ jkd ¿

Tensión calculada en el acero de refuerzo a tracción

Verificación del control de la fisuraciónf s≤ f smáx

6.4 REFUERZO DE RETRACCIÓN Y TEMPERATURA

A s ret/ cara=3,6cm2

cara


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7. MODELO PORTAL FALSOEl modelo está compuesto por una malla tridimensional de elementos finitos apoyados sobre resortes en su base, simulando un apoyo elástico. (Modulo de reacción vertical adoptado para el apoyo k = 0,06 N/mm³). Para considerar la restricción lateral que ejerce el suelo a ser empujado desde el interior del túnel se considero también una restricción lineal de resortes con igual valor de k que para la base del falso túnel. El modelo fue elaborado en SAP2000 y corrido con no linealidad para evitar que los resortes tuvieran la capacidad de tomar tensión (solo se considera que el suelo es capaz de soportar compresión).


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Figura 3 Vista Frontal modelo Falso túnel

Figura 4 Vista Isométrica Falso túnel


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7.1 EMPUJE DE SUELOS SOBRE LA ESTRUCTURA

Figura 5 Empuje Vertical de suelos. Portal Falso (kN/m²)


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Figura 6 Empuje Horizontal de suelos. Portal Falso (kN/m²)

Figura 7 Pesos muertos sobre losa. Portal Falso (kN/m²)


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7.2 RESULTADOS DEL MODELO PORTAL FALSO

7.2.1 Momentos actuantes combinación STRENGTH I

Figura 8 Momentos actuantes sobre la sección. Portal Falso. Combinación STRENGTH I (kN-m/m)

Tabla 1 Radio de esfuerzos por momento en la sección Falso Túnel. Combinación STRENGTH I

Actuante Resistente Radio de esfuerzosPunto Mua (kN-m/m) Mur (kN-m/m) Mua / Mur

Bóveda Superior -55 209,62 0,26

Transición Bóveda Hastial -160 209,62 0,76

Esquina hastial zapata 360 476,3 0,76

Zapata -240 280,4 0,86


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7.2.2 Cortantes actuantes combinación. STRENGTH I

Figura 9 Cortantes máximos en la sección. Portal Falso. Combinación STRENGTH I (kN/m)

Tabla 2 Radio de esfuerzos por cortante en la sección Falso Túnel. Combinación STRENGTH I

Actuante Resistente Radio de esfuerzosPunto Vua (kN/m) Vur (kN/m) Vua / Vur

Bóveda Superior 17 232 0,07

Transición Bóveda Hastial 75 232 0,32

Esquina hastial zapata 450 494 0,91

Zapata 300 307 0,98

7.3 DEFORMACIONES MÁXIMAS PORTAL FALSO

Para la combinación SERVICEI=DC++EV+EH se tienes las siguientes deformaciones


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Figura 10 Deformaciones máximas verticales. Portal Falso Combinación SERVICE I (mm)


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Figura 11 Deformaciones máximas horizontales. Portal Falso Combinación SERVICE I (mm)

8. EVALUACIÓN DE CARGAS BROCALES

8.1 CASOS DE CARGA CUANDO EL BROCAL ESTA POR FUERA DE LA EXCAVACIÓN SUBTERRÁNEA

8.1.1 DC: Carga muerta de componentes estructurales.Se consideró el peso propio de los materiales estructurales de acuerdo con sus dimensiones definitivas. El peso propio es calculado directamente en el software de análisis estructural.

8.1.2 Empuje horizontal de terreno (EH) y Empuje vertical de terreno (EV) Caso en que el Brocal quede por fuera de la excavación subterránea.Las cargas de empuje de terreno se calcularon según se muestra a continuación

Factor de interacción suelo estructura (AASHTO 12.11.2.2.1-2)H fill=5,0m Altura del relleno por encima de la cota clave del Brocal

en el caso que se encentre por fuera de la excavación subterránea


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B=11,90m Ancho exterior

F e=(1+ 0,2∗H fill

Bc) Factor de interacción suelo estructura en terraplén

F e=(1+ 0,2∗5,0011,90 )=1,05

Cálculo de presiones de terreno en el caso de encontrarse por fuera de la excavación subterránea (CCSDP A.12.4)

E v1=1,05∗19[ kNm3 ]∗(5,0+ 0,402 )=103 ,74[ kNm2 ] Eh1=10 [ kNm3 ]∗5,0 [m ]=50[ kNm2 ] Eh2=10 [ kNm3 ]∗(5,0+0,40+8,62+ 0,502 ) [m ]=142,70[ kNm2 ]8.2 COMBINACIONES DE CARGA (CCDSP 95)Combinaciones de cargas ultimas para diseño

STRENGT H I=1,3∗DC+1,3∗EV+1,3∗EH

STRENGTH II=1,3∗DC+1,3∗EV +1,3∗0,5∗EH

Combinaciones de carga admisibles para la verificación de condiciones de servicio

SERVICEI=DC+(¿+ℑ)+EV +EH

SERVICEII=DC+(¿+ℑ)+EV +0,5∗EH

8.3 CASOS DE CARGA CUANDO EL BROCAL ESTA DENTRO DE LA EXCAVACIÓN Empuje Vertical


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Para el caso de presión vertical se considera una presión correspondiente a 10,0 m de presión de roca = 10,0 X 22,0 kN/m³ = 220,0 kN/m²

Empuje Horizontal Para el caso de presión horizontal de suelos se considera

Eh1=0,50∗22[ kNm3 ]∗(10,0+ 0,302 )=111,65 [ kNm2 ] Eh2=0,50∗22 [ kNm3 ]∗(10,0+0,30+8,62+ 0,502 ) [m ]=210,90 [ kNm2 ]

Empuje por sismoPara considerar los efectos de sismo transversalmente en la sección se considera un incremento de 0,15 g a nivel de roca en los empujes de un lado de la sección.

Eh1=111 ,65[ kNm2 ]∗1.15=128,4 (Incluye empuje de tierras y sismo)

Eh2=210 ,90 [ kNm2 ]∗1,15=242 ,5 (Incluye empuje de tierras y sismo)

8.4 COMBINACIONES DE CARGA (CCDSP 95)STRENGT HVII=1,0∗DC+1,0∗EV +1,0∗EH+1,0 EQ

9. DISEÑO ESTRUCTURAL BROCAL

9.1 ESTADO LÍMITE DE RESISTENCIA - DISEÑO A FLEXIÓN BROCAL


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Sección de bóveda t = 0,30Refuerzo principal 1N6 C/0,10

d=0.30−0.075−0.0192

=0,21

ρ=2,84 /0,10100∗21

=0,0135

f ' c=28,0MPa

φ f M n=0,9∗0,0135∗420000∗1∗0,212(1−0,59∗0,0135 42000028000 )=198,2 kN−m/m

Sección zapata t = 0,50Refuerzo principal 1N6 C/0,10

d=0.50−0.075−0.0192

=0,41

ρ=2,84 /0,10100∗41

=0,0069

f ' c=28,0MPa

φ f M n=0,9∗0,0069∗420000∗1∗0,412(1−0,59∗0,0069 42000028000 )=411,6−m/m

Para la sección de 0,75 m en la intersección hastial zapataRefuerzo principal 1N6 C/0,10

d=0.75−0.075−0.0192

=0,66

ρ=2,84 /0,10100∗66

=0,0043

f ' c=28,0MPa


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φ f M n=0,9∗0,0043∗420000∗1∗0,662(1−0,59∗0,0043 42000028000 )=681,1 kN−m /m

9.2 ESTADO LÍMITE DE RESISTENCIA - CHEQUEO A FUERZA CORTANTE BROCAL

Para la sección de 0,30 m en la bóveda

φcV c=0,85∗√28

6∗1000∗210=157,4 kN

Para la sección de 0,50 m en la zapata

φcV c=0,85∗√28

6∗1000∗410=307,4 kN

Para la sección de 0,75 m en la intersección hastial zapata

φcV c=0,85∗√28

6∗1000∗660=494,8 kN

9.3 RESULTADOS DEL MODELO BROCALES


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9.3.1 Momentos actuantes combinación STRENGT H I=1,3∗DC+1,3∗EV+1,3∗EH

Figura 12 Momento actuantes Brocal combinación STRENGTH I (kN-m/m)

Tabla 3 Radio de esfuerzos por momento en la sección de Brocal. Combinación STRENGTH I


Bóveda Superior 56,0 198,2 0,28

Transición Bóveda Hastial 150 198,2 0,75


Zapata 400 411,6 0,97

9.3.2 Cortantes actuantes combinación STRENGT H I=1,3∗DC+1,3∗EV+1,3∗EH


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Figura 13 Cortante máximo V23 (kN), Brocal combinación STRENGTH I (kN/m)

Tabla 4 Radio de esfuerzos por Cortante en el Brocal. Combinación STRENGTH I


Bóveda Superior 122 157,4 0,77

Transición Bóveda Hastial 66,7 157,4 0,42


Zapata 250 307,4 0,81

9.3.3 Momentos actuantes combinación STRENGT HVII=1,0∗DC+1,0∗EV +1,0∗EH+1,0 EQ


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Figura 14 Momento actuantes Brocal combinación STRENGTH VII (kN-m/m)

Tabla 5 Radio de esfuerzos por momento en la sección de Brocal. Combinación STRENGTH VII





Zapata 400 411,6 0,97

9.3.4 Cortantes actuantes combinación STRENGT HVII=1,0∗DC+1,0∗EV +1,0∗EH+1,0 EQ


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Figura 15 Cortante máximo V23 (kN), Brocal combinación STRENGTH VII (kN)

Tabla 6 Radio de esfuerzos por cortante en la sección de Brocal. Combinación STRENGTH VII





Zapata 250 307,4 0,81

9.4 DEFORMACIONES MÁXIMAS BROCALES


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Figura 16. Deformaciones máximas verticales. Brocal Combinación SERVICE I (mm)

Figura 17 Deformaciones máximas horizontales. Brocal Combinación SERVICE I (mm)


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Figura 18 Deformaciones máximas verticales. Brocal Combinación STRENGTH VII (mm)


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Figura 19 Deformaciones máximas horizontales. Brocal Combinación STRENGTHVIII (mm)


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Apéndice 1: CARACTERÍSTICAS GENERALES DE LA APLICACIÓN SAP20006

FEATURES

SAP2000, ETABS, SAFE, and CSiBridge are software packages from Computers and Structures, Inc. for structural analysis and design. Each package is a fully integrated system for modeling, analyzing, designing, and optimizing structures of a particular type:

SAP2000 for general structures, including stadiums, towers, industrial plants, off shore structures, piping systems, buildings, dams, soils, ma chine parts and many others

ETABS for building structures SAFE for floor slabs and base mats CSiBridge for bridge structures

At the heart of each of these software packages is a common analysis engine, referred to throughout this manual as SAPfire. This engine is the latest and most powerful version of the well-known SAP series of structural analysis programs.

OBJECTS AND ELEMENTS

The physical structural members in a SAP2000 model are represented by objects. Using the graphical user interface, you “draw” the geometry of an object, then “assign” properties and loads to the object to completely define the model of the physical member.

The following object types are available, listed in order of geometrical dimension: Point objects, of two types:

o Joint objects: These are automatically created at the corners or ends of all other types of objects below, and they can be explicitly added to model supports or other localized behavior.

o Grounded (one-joint) support objects: Used to model special support behavior such as isolators, dampers, gaps, multilinear springs, and more.

Line objects, of several typeso Frame objects: Used to model beams, columns, braces, and trusses; they may be straight or curvedo Cable objects: Used to model flexible cableso Ten don objects: Used to prestressing tendons in side other objectso Connecting (two-joint) link objects: Used to model special member behavior such as isolators, dampers, gaps, multilinear springs, and more.

Unlike frame, cable, and tendon objects, connecting link objects can have zero length. Area objects: Used to model walls, floors, and other thin-walled members, as well as two-dimensional sol ids (plane stress, plane strain, and axisymmetric

solids). Solid objects: Used to model three-dimensional solids.

STATIC AND DYNAMIC ANALYSIS

Many different types of analysis are available using program SAP2000. These include: Linear static analysis Modal analysis for vibration modes, using eigen vectors or Ritz vectors Response- spectrum analysis for seismic response Other types of linear and non linear, static and dynamic analysis

These different types of analyses can all be defined in the same model, and the results combined for output.

Loads

Loads represent actions upon the structure, such as force, pressure, support displacement, thermal effects, ground acceleration, and others. You may define named Load Patterns containing any mixture of loads on the objects. The program automatically computes built-in ground acceleration loads.

In order to calculate any response of the structure due to the Load Patterns, you must define and run Load Cases which specify how the Load Patterns are to be applied (e.g., statically, dynamically, etc.) and how the structure is to be analyzed(e.g., linearly, nonlinearly, etc.) The same Load Pattern can be applied differently in different Load Cases. By default, the program creates a linear static Load Case corresponding to each Load Pattern that you define.

Load Patterns

You can define as many named Load Patterns as you like. Typically you would have separate Load Patterns for dead load, live load, wind load, snow load, thermal load, and so on. Loads that need to vary in dependently, either for de sign purposes or because of how they are applied to the structure, should be defined as separate Load Patterns. After defining a Load Pattern name, you must assign specific load values to the objects as part of that Load Pattern. Each Load Pattern may include:

Self-Weight Loads on Frame and/or Shell elements Concentrated and Distributed Span Loads on Frame elements Uniform Loads on Shell elements Force and/or Ground Displacement Loads on Joints Other types of loads described in the CSI Analysis Reference Manual

Each object can be subjected to multiple Load Patterns.

Acceleration Loads

6 COMPUTERS AND STRUCTURES, INC. SAP2000® - Basic Analysis Reference Manual - Linear and Nonlinear Static and Dynamic Analysis and Design of Three-Dimensional Structures. Berkeley, California, USA . CSI, 2009

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The program automatically computes three Acceleration Loads that act on the structure due to unit translational accelerations in each of the three global directions. They are determined by d’Alembert’s principal, and are denoted mx, my, and mz. These loads are used for applying ground accelerations in response-spectrum analyses, and are used as starting load vectors for Ritz-vector analysis. These loads are computed for each joint and element and summed over the whole structure. The Acceleration Loads for the joints are simply equal to the negative of the joint translational masses in the joint local coordinate system. These loads are transformed to the global coordinate system. The Acceleration Loads for all elements are the same in each direction and are equal to the negative of the element mass. No coordinate transformations are necessary. The Acceleration Loads can be transformed into any coordinate system. In the global coordinate system, the Acceleration Loads along the positive X, Y, and Z axes are denoted UX, UY, and UZ, respectively. In a local coordinate system defined for a response-spectrum analysis, the Acceleration Loads along the positive local 1, 2, and 3 axes are denoted U1, U2, and U3, respectively.

Load Cases

Each different analysis performed is called a Load Case. You assign a label to each Load Case as part of its definition. These labels can be used to create additional combinations and to control output. The basic types of Load Cases are:

Linear static analysis Modal analysis Response-spectrum analysis

You may define any number of each different type of Load Case for a single model. Other types of Load Cases are also available. By default, the program creates a linear static Load Case for each Load Pattern that you define, as well as a single modal Load Case for the first few eigen-modes of the structure. Linear combinations and envelopes of the various Load Cases are available through the SAP2000 graphical interface.

Linear Static Analysis

The static analysis of a structure involves the solution of the system of linear equations represented by:

where K is the stiffness matrix, r is the vector of applied loads, and u is the vector of resulting displacements. See Bathe and Wilson (1976). For each linear static Load Case that you define, you may specify a linear combination of one or more Load Patterns and/or Acceleration Loads to be applied in vector r .Most commonly, however, you will want to solve a single Load Pattern in each linear static Load Case, and combine the results later.

Modal Analysis

You may define as many modal Load Cases as you wish, although for most purposes one case is enough. For each modal Load Case, you may choose either eigen vector or Ritz-vector analysis.

Eigenvector Analysis

Eigenvector analysis determines the undamped free-vibration mode shapes and frequencies of the system. These natural Modes provide an excellent insight into the behavior of the structure. They can also be used as the basis for response-spectrum analyses, although Ritz vectors are recommended for this purpose. Eigen vector analysis involves the solution of the generalized eigen value problem:

where K is the stiffness matrix, M is the diagonal mass matrix, W2 is the diagonal matrix of eigen values, and F is the matrix of corresponding eigen vectors (mode shapes). Each eigen value-eigen vector pair is called a natural Vibration Mode of the structure. The Modes are identified by numbers from 1 to n in the order in which the modes are found by the program. The eigen value is the square of the circular frequency, w, for that Mode. The cyclic frequency, f, and period, T, of the Mode are related to w by:

You may specify the number of Modes, n, to be found. The program will seek the n lowest- frequency (longest- period) Modes. The number of Modes actually found, n, is limited by:

The number of Modes requested, n The number of mass degrees of freedom in the model

A mass degree of freedom is any active degree of freedom that possesses translational mass or rotational mass moment of inertia. The mass may have been assigned directly to the joint or may come from connected elements. Only the Modes that are actually found will be available for any subsequent response-spectrum analysis processing.

Ritz-vector Analysis

Research has indicated that the natural free-vibration mode shapes are not the best basis for a mode-superposition analysis of structures subjected to dynamic loads. It has been demonstrated (Wilson, Yuan, and Dickens, 1982) that dynamic analyses based on a special set of load-dependent Ritz vectors yield more accurate results than the use of the same number of natural mode shapes. The reason the Ritz vectors yield excellent results is that they are generated by taking into account the spatial distribution of the dynamic loading, whereas the direct use of the natural mode shapes neglects this very important information. The spatial distribution of the dynamic load vector serves as a starting load vector to initiate the procedure. The first Ritz vector is the static Displacement vector corresponding to the starting load vector. The remaining vectors are generated from are ocurrence relationship in which the mass matrix is multiplied by the previously obtained Ritz vector and used as the load vector for the next static solution. Each static solution is called a generation cycle. When the dynamic load is made up of several in dependent spatial distributions, each of these may serve as a starting load vector to generate a set of Ritz vectors. Each generation cycle creates as many Ritz vectors as there are starting load vectors. If a generated Ritz vector is redundant or does not excite any mass degrees of freedom, it is discarded and the corresponding starting load vector is re moved from all subsequent generation cycles. For seismic analysis, including response-spectrum analysis, you should use the three acceleration loads as the starting load vectors. This produces better response spectrum results than using the same number of eigen Modes. Standard eigen solution techniques are used to orthogonalize the set of generated Ritz vectors, resulting in a final set of Ritz-vector Modes. Each Ritz-vector Mode consists of a mode shape and frequency. The full set of Ritz-vector Modes can be used as a basis to represent the dynamic Displacement of the structure. Once the stiffness matrix is triangularized it is only necessary to statically solve for one load vector for each Ritz vector required. This results in an extremely efficient algorithm. Also, the method automatically includes the advantages of the proven numerical techniques of static condensation, Guyan reduction, and static correction due to higher-mode truncation.

When a sufficient number of Ritz-vector Modes have been found, some of them may closely approximate natural mode shapes and frequencies. In general, however, Ritz- vector Modes do not represent the intrinsic characteristics of the structure in the same way the natural eigen-Modes do. The Ritz-vector Modes are biased by the starting load vectors. You may specify the total number of Modes, n, to be found. The total number of Modes actually found, n, is limited by:

The number of Modes re quested, n The number of mass degrees of freedom present in the model


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The number of natural free- vibration modes that are excited by the starting load vectors (some additional natural modes may creep in due to numerical noise)

A mass degree of freedom is any active degree of freedom that possesses translational mass or rotational mass moment of inertia. The mass may have been assigned directly to the joint or may come from connected elements. Only the Modes that are actually found will be available for any subsequent response-spectrum analysis processing.

Modal Analysis Results

Each modal Load Case results in a set of modes. Each mode consists of a mode shape (normalized deflected shape) and a set of modal properties. These are available for display and printing from the SAP2000 graphical interface. This information is the same regardless of whether you use eigen vector or Ritz-vector analysis, and is described in the following subtopics.

Periods and Frequencies

The following time properties are given for each Mode: Period, T, in units of time Cyclic frequency, f, in units of cycles per time; this is the inverse of T Circular frequency, w, in units of radians per time; w = 2 p f Eigen value, w2, in units of radians per time squared

Modal Stiffness and Mass

For each mode shape, only the relative deflection values are important. The overall scaling is arbitrary. In SAP2000, the modes shapes are each normalized, or scaled, with respect to the mass matrix such that:

This quantity is called the modal mass. Similarly, the modal stiffness is defined as:

Regardless of how the modes are scaled, the ratio of modal stiffness to modal mass always gives the modal eigenvalue:

Participation Factors

The modal participation factors are the dot products of the three Acceleration Loads with the modes shapes. The participation factors for Mode n corresponding to Acceleration Loads in the global X, Y, and Z directions are given by:

where jn is the mode shape and mx, my, and, mz are the unit Acceleration Loads. These factors are the generalized loads acting on the Mode due to each of the Acceleration Loads. They are referred to the global coordinate system. These participation factors indicate how strongly each mode is excited by the respective acceleration loads.

Participating Mass Ratios

The participating mass ratio for a Mode provides a relative measure of how important the Mode is for computing the response to the Acceleration Loads in each of the three global directions. Thus it is useful for determining the accuracy of response- spectrum analyses. The participating mass ratios for Mode n corresponding to Acceleration Loads in the global X, Y, and Z directions are given by:

where fxn, fyn, and fzn are the participation factors defined in the previous subtopic; and Mx, My, and Mz are the total unrestrained masses acting in the X, Y, and Z directions. The participating mass ratios are expressed as percentages. The cumulative sums of the participating mass ratios for all Modes up to Mode n are printed with the individual values for Mode n. This provides a simple measure of how many Modes are required to achieve a given level of accuracy for ground acceleration loading. If all eigen Modes of the structure are present, the participating mass ratio for each of the three Acceleration Loads should generally be 100%. However, this may not be the case in the presence of certain types of Constraints where symmetry conditions prevent some of the mass from responding to translational Accelerations.

Response-Spectrum Analysis

The dynamic equilibrium equations associated with the response of a structure to ground motion are given by:

where K is the stiffness matrix; C is the proportional damping matrix; M is the diagonal mass matrix; u, u& , and &u& are the relative displacements, velocities, and accelerations with respect to the ground; mx, my, and mz are the unit Acceleration Loads; and u&&gx , u&&gy , and u&&gz are the components of uniform ground


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Acceleration. Response-spectrum analysis seeks the likely maximum response to these equations rather than the full time history. The earth quake ground acceleration in each direction is given as a digitized response-spectrum curve of pseudo-spectral acceleration response versus period of the structure. Even though accelerations may be specified in three directions, only a single, positive result is produced for each response quantity. The response quantities included displacements, forces, and stresses. Each computed result represents a statistical measure of the likely maximum magnitude for that response quantity. The actual response can be expected to vary within a range from this positive value to its negative. No correspondence between two different response quantities is available. No information is available as to when this extreme value occurs during the seismic loading, or as to what the values of other response quantities are at that time. Response-spectrum analysis is per formed using mode superposition (Wilson and Button, 1982). Modes may have been computed using eigenvector analysis or Ritz-vector analysis. Ritz vectors are recommended since they give more accurate results for the same number of Modes. You must define a Modal Load Case that computes the modes, and then refer to that Modal Load Case in the definition of the Reponse-Spectrum Case. Response-spectrum can consider high-frequency rigid response if requested and if appropriate modes have been computed. When eigen modes are used, you should request that static correction vectors be computed. This information is automatically available in Ritz modes generated for ground acceleration. In either case, you must be sure to have sufficient dynamical modes below the rigid frequency of the ground motion. Any number of response- spectrum analyses can be defined in a single model. You assign a unique label to each response-spectrum Load Case. Each case can differ in the acceleration spectra applied, the modal Load Case used to generate the modes, and in the way that results are combined. The following subtopics describe in more detail the parameters that you use to define each case.

Local Coordinate System

Each response-spectrum case has its own response- spectrum local coordinate system used to define the directions of ground acceleration loading. The axes of this local system are denoted 1, 2, and 3. By default these correspond to the global X, Y, and Z directions, respectively. You may change the orientation of the local coordinate system by specifying a coordinate angle, ang (the de fault is zero). The local 3 axis is always the same as the vertical global Z axis. The local 1 and 2 axes coincide with the X and Y axes if angle ang is zero. Otherwise, ang is the angle in the horizontal plane from the global X axis to the local 1 axis, measured counter clock wise when viewed from above. This is illustrated in Figure 22 (page 77). Response-Spectrum Functions. A Response- spectrum Function is a series of digitized pairs of structural- period and corresponding pseudo- spectral acceleration values. You may define any number of Functions, assigning each one a unique label. You may scale the acceleration values whenever the Function is used. Specify the pairs of period and acceleration values as:

t0, f0, t1, f1, t2, f2, ..., tn, fn where n + 1 is the number of pairs of values given. All values for the period and acceleration must be zero or positive. These pairs must be specified in order of increasing period.

Response-Spectrum Curve

The response- spectrum curve for a given direction is defined by digitized points of pseudo-spectral acceleration response versus period of the structure. The shape of the curve is given by specifying the name of a Response- spectrum Function.

Figure 22 Definition of Response Spectrum Local Coordinate System

If no Function is specified, a constant function of unit acceleration value for all structural periods is assumed. The pseudo spectral acceleration response of the Function may be scaled by the factor sf. After scaling, the acceleration values must be in consistent units. See Figure 23 (page 78). The response-spectrum curve chosen should reflect the damping that is present in the structure being modeled. Note that the damping is inherent in the response spectrum curve itself. It is not affected by the damping ratio, damp, used for the CQC or GMC method of modal combination, although normally these two damping values should be the same. If the response-spectrum curve is not defined over a period range large enough to cover the modes calculated in the modal Load Case, the curve is extended to larger and smaller periods using a constant acceleration equal to the value at the nearest defined period.

Figure 23 Digitized Response-Spectrum Curve

Modal Combination

For a given direction of acceleration, the maximum displacements, forces, and stresses are computed throughout the structure for each of the Vibration Modes. These modal values for a given response quantity are combined to produce a single, positive result for the given direction of acceleration using one of the following methods. The method of modal combination may be affected by the modal damping ratio, damp, specified for the response spectrum load case. Given response-spectrum curves will be adjusted, if necessary, to reflect this damping value.

Periodic and Rigid Response

For all modal combination methods except Absolute Sum, there are two parts to the modal response for a given direction of loading: periodic and rigid. The distinction here is a property of the loading, not of the structure. Two frequencies are defined, f1 and f2, which define the rigid-response content of the ground motion, where f1 £ f2. For


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structural modes with frequencies less than f1 (longer periods), the response is fully periodic. For structural modes with frequencies above f2 (shorter periods), the response is fully rigid. Between frequencies f1 and f2, the amount of periodic and rigid response is interpolated, as described by Gupta (1990). Frequencies f1 and f2 are proper ties of the seismic input, not of the structure. Gupta defines f1 as:

where SAmax is the maximum spectral acceleration and SVmax is the maximum spectral velocity for the ground motion considered. The de fault value for f1 is unity. Gupta defines f2 as:

where f r is the rigid frequency of the seismic input, i.e., that frequency above which the spectral acceleration is essentially constant and equal to the value at zero period (infinite frequency). Others have defined f2 as:

The following rules apply when specifying f1 and f2: If f2 = 0, no rigid response is calculated and all response is periodic, regard less of the value specified for f1. Otherwise, the following condition must be satisfied: 0 £ f1 £ f2. Specifying f1 = 0 is the same as specifying f1 = f2.

For any given response quantity (displacement, stress, forces, etc.), the periodic response, Rp , is computed by one of the modal combination methods described below. The rigid response, Rr , is always computed as an algebraic (fully correlated) sum of the response from each mode with frequency above f2, and an interpolated portion of the response from each mode between f1 and f2. The total response, R, is computed by one of the following two methods:

SRSS, as recommended by Gupta (1990) and NRC (2006), which assumes that these two parts are statistically independent:

Absolute Sum, for compatibility with older methods:

Please note that the choice of using the SRSS or Absolute Sum for combining periodic and rigid response is independent of the modal combination or the directional combination methods described below.

CQC Method

The Complete Quadratic Combination technique for calculating the periodic response is described by Wilson, DerKiureghian, and Bayo (1981). This is the default method of modal combination. The CQC method takes into account the statistical coupling between closely-spaced Modes caused by modal damping. Increasing the modal damping increases the coupling between closely-spaced modes. If the damping is zero for all Modes, this method de generates to the SRSS method.

GMC Method

The General Modal Combination technique for calculating the periodic response is the complete modal combination procedure described by Equation 3.31 in Gupta (1990). The GMC method takes into account the statistical coupling between closely-spaced Modes similarly to the CQC method, but uses the Rosenblueth correlation coefficient with the time duration of the strong earth quake motion set to infinity. The result is essentially identical to the CQC method. Increasing the modal damping increases the coupling between closely-spaced modes. If the damping is zero for all Modes, this method degenerates to the SRSS method.

SRSS Method

This method for calculating the periodic response combines the modal results by taking the square root of the sum of their squares. This method does not take into account any coupling of the modes, but rather assumes that the response of the modes are all statistically in de pendent.

Absolute Sum Method

This method combines the modal results by taking the sum of their absolute values. Essentially all modes are assumed to be fully correlated. This method is usually over-conservative. The distinction between periodic and rigid response is not considered for this method. All modes are treated equally.

NRC Ten-Percent Method

This technique for calculating the periodic response is the Ten-Per cent method of the U.S. Nuclear Regulatory Commission Regulatory Guide 1.92. The Ten-Percent method assumes full, positive coupling between all modes whose frequencies differ from each other by 10% or less of the smaller of the two frequencies. Modal damping does not affect the coupling.

NRC Double-Sum Method

This technique for calculating the periodic response is the Double-Sum method of the U.S. Nuclear Regulatory Commission Regulatory Guide 1.92. The Double-Sum method assumes a positive coupling between all modes, with correlation coefficients that depend upon damping in a fashion similar to the CQC and GMC methods, and that also depend upon the duration of the earthquake. You specify this duration as parameter td as part of the Load Cases definition.

Directional Combination

For each displacement, force, or stress quantity in the structure, modal combination produces a single, positive result for each direction of Acceleration. These directional values for a given response quantity are combined to produce a single, positive result. Two methods are available for combining the directional response, SRSS and Absolute Sum.

SRSS Method

This method com bines the response for different directions of loading by taking the square root of the sum of their squares:


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where R1, R2 , and R3 are the modal-combination values for each direction. This method is invariant with respect to coordinate system, i.e., the results do not depend upon your choice of coordinate system when the given response-spectrum curves are the same in each direction. This is the recommended method for directional combination, and is the default.

Absolute Sum Method

This method combines the response for different directions of loading by taking the sum of their absolute values. A scale factor, dirf, is available for reducing the interaction between the different directions.

Specify dirf = 1 for a simple absolute sum:

This method is usually over-conservative. Specify 0 < dirf < 1 to combine the directional results by the scaled absolute sum method. Here, the directional results are combined by taking the maximum, over all directions, of the sum of the absolute values of the response in one direction plus dirf times the response in the other directions.

For example, if dirf = 0.3, the spectral response, R, for a given displacement, force, or stress would be:

where:

and R1, R2 , and R3 are the modal- combination values for each direction. The results obtained by this method will vary depending upon the coordinate system you choose. Results obtained using dirf = 0.3 are comparable to the SRSS method (for equal input spectra in each direction), but may be as much as 8% unconservative or 4% over-conservative, depending upon the coordinate system. Larger values of dirf tend to produce more conservative results.

Response-Spectrum Analysis Results

Certain information about each response- spectrum Load Case is available for printing from the SAP2000 graphical interface. This information is described in the following subtopics.

Damping and Accelerations

The modal damping and the ground Accelerations acting in each direction are given for every Mode. The damping value printed for each Mode is just the specified damping ratio, damp, specified for the load case (unless advanced damping is defined for the model, in which case it will be larger.) The Accelerations printed for each Mode are the actual values as interpolated at the modal period from the response- spectrum curves after scaling by the specified value of sf. The Accelerations are always referred to the local axes of the response spectrum analysis. They are identified in the output as U1, U2, and U3.

Modal Amplitudes

The response-spectrum modal amplitudes give the multipliers of the mode shapes that con tribute to the displaced shape of the structure for each direction of Acceleration Load. For a given Mode and a given direction of acceleration, this is the product of the modal participation factor and the response-spectrum acceleration, divided by the eigen value, w2, of the Mode. This amplitude, multiplied by any modal response quantity (displacement, force, stress, etc.), gives the contribution of that mode to the value of the same response quantity reported for the response-spectrum load case. The acceleration directions are always referred to the local axes of the response spectrum analysis. They are identified in the output as U1, U2, and U3.


ESTRUCTURAL


Bogotá, Mayo 2011

Página 43

portales y brocales cga

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