4. Sismos y Diseno Sismo Resistente

SISMOS Y DISENO SISMO RESISTENTE Proyecto Estructural - Prof. Michele Casarin

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Tema 4

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RIESGO SISMICO

2.

SISMOLOGIA

3.

EFECTOS SISMICOS

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DINAMICA DE ESTRUCTURAS

5.

ESPECTRO DE RESPUESTA Y DISENO

6.

SISTEMAS DE VARIOS GRADOS DE LIBERTAD

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CONCEPTOS DE DISEÑO

8.

COVENIN 1756

Proyecto Estructural - Prof. Michele Casarin

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INDICE


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RIESGO SISMICO


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RIESGO SISMICO


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RIESGO SISMICO

RIESGO SISMICO


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La historia sísmica de nuestro país revela que a lo largo del período 1530-2002 han ocurrido más de 137 eventos sísmicos que han causado algún tipo de daño en poblaciones venezolanas

SISMOLOGIA INTRODUCCION -EN LOS ULTIMOS 3 SIGLOS MAS DE 3 MILLONES HAN MUERTO A CAUSA DE SISMOS O DESASTRES CAUSADOS POR SISMOS -70% DE LA TIERRA SE CONSIDERA SISMICAMENTE ACTIVA. 1,000,000,000 PERSONAS VIVEN EN ZONAS CON RIESGO SISMICO -LOS SISMOS PUEDEN CAUSAR PERDIDAS HUMANAS Y PERDIDAS MATERIALES IMPORTANTES. -LOS SISMOS NO PUEDEN PREVENIRSE NI PREDECIR CON PRECISION.


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-NO SON LOS MOVIMIENTOS SISMICOS DIRECTAMENTE LOS QUE CAUSAN PERDIDAS, SINO EL COLAPSO O DANO DE ESTRUCTURAS NO RESISTENTES.

SISMOLOGIA INGENIERIA SISMICA -COOPERACION DE DIFERENTES DISCIPLINAS DE LAS CIENCIAS E INGENIERIAS PARA CONTROLAR LOS RIESGOS SOCIOECONOMICOS DE LOS SISMOS -TRATA DE RESPONDER:

CUAL ES LA RAZON MECANICA POR LA CUAL FALLAN LAS ESTRUCTURAS CON MOVIMIENTOS DEL SUELO? CUALES SON LAS CARACTERISTICAS ESENCIALES QUE LAS ONDAS SISMICAS APLICAN SOBRE ESTRUCTURAS? Y COMO SE PUEDEN EXPRESAR EN FORMA DE ACCIONES DE DISENO?


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CUAL ES LA SISMICIDAD DE CADA REGION?

SISMOLOGIA


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LA TIERRA

TECTONIC PLATES

Seismology

SISMOLOGIA

GLO 14/03/2

How fast do the plates move?

LA TIERRA TECTONIC PLATES

TECTONIC PLATES

GLOBAL DISTRIBUTION OF EARTHQUAKES Plate motions can be measured using Very Long Baseline Interferometry (VLBI) or using the Global Positioning System (GPS)

How fast do the plates move?

Maria Gabriella Mulas

Earthquake e Depth of foc

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Earthquake epicenters 1963-2000 Depth of focus: 70-350 = intermediate (yellow), 0-70 km = shallow (blue) >350Km = deep (red) Maria Gabriella Mulas

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Plate motions can be measured using Very Long Baseline Interferometry (VLBI) or using the Global Positioning System (GPS) 15



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Earthquake epicenters 1963-2000 Depth of focus: 70-350 = intermediate (yellow), 0-70 km = shallow (blue) >350Km = deep (red)

TECTONIC PLATES SISMOLOGIA

Faults may range in length from a few millimeters to thousands of kilometers.

Stresses build up in the crust, usually due to lithospheric plate motions

ONDAS SISMICAS

The fault surface can be horizontal or vertical or some arbitrary angle in between. Faults which move along the direction of the dip plane are dipslip faults and described as either normal or reverse, depending on their motion.

TECTONIC PLATES

Most earthquakes occu where stresses are prod

Rock deform (strain) as the result of stress. The strain is energy stored in the rocks.

Faults which move horizontally are known as strike-slip faults and are classified as either right-lateral or left-lateral. Faults which show both dip-slip and strike-slip motion are known

AND asEARTHQUAKE oblique-slip faults.

FAULTS AND EARTHQU

Usually Tsunamis are created by faults which show dip-slip and oblique motion. Maria Gabriella Mulas

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19 Proyecto Estructural - Prof. Michele Casarin

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Maria

one starts 14/03/2012 foot wall ntsSISMOLOGIA we live on of the action ONDAS SISMICAShanging wall Plates may FAULT AND EARTHQUAKE each other. Normal fault NORMAL We classify faults by how the two rocky blocks on FALLA either side of a fault d by moving move relative to each other. The one you see here is a normal fault. A cky crust until it EARTHQUAKES normal fault drops rock on one side of the fault down relative to the other side. Take a look at the side that shows the fault and arrows indicating at once, Earthquake focus: center of rupture or slip, seismic waves radiate out from the focus movement. the block farthest to the right that looks kind of like a foot is e two rocky Earthquake epicenter – the point on theThe Earth’sblock surface on overthe the focus the foot wall. other side of the fault is resting FAULT ANDorEARTHQUAKE ns along a more hanging on top of the foot wall block and is named hanging wall. If we hold want to pull the Reverse faultthe foot wall stationary, gravity will normally lled a fault. FALLA REVERSA fault hanging wall see down. movefault. the way you awould expect The fault you here is a that reverse Along reverse faultgravity one to “hypocenter” is Faults s an earthquake ault. A move themisnormally called normal rocky block pushed are up relative to rockfaults! on the other side. another name Where the fault has ruptured fault the Earth that movement along atthe nergy he otherfrom the Here’s a way to tell a reverse fromsurface, a normal fault. Take a look for the focus fault has produced an elongate fault generated cliff called fault scarp. cating mic waves in all the side that shows the fault and arrows indicating movement. The foot is Maria Gabriella Mulasfoot wall. The block on the other block farthest to the right that is the 27 arthquake is the r side of the fault is the hanging wall. FALLA STRIKE-SLIP ach the surface l. Strike-slip fault

32 If we hold the foot wall stationary, where would the hanging wall go if Proyecto Estructural - Prof. Michele Casarin Strike-slip faults have a different type we reversed gravity? The hanging wall will slide upwards. When Maria Gabriella Mulas

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ull the

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1. ONDAS P: 8 KM/S

SISMOLOGIA

2. ONDAS S: 5 KM/S, NO SE MUEVEN EN LIQUIDOS

SEISMIC WAVES 3. ONDAS SUPERFICIALES: LAS MAS LENTAS, SOLO SE WAVES SEISMIC TRANSMITEN EN LA SUPERFICIE. LAS MAS DESTRUCTIVAS

ONDAS SISMICAS

P- w aves – most rapid (8 km/sec)

ON OF SEISMIC AVES

S- w aves – slower (5 km/sec), cannot move through liquids Surface w aves – even slower, move only on surface, most destructive

Mulas

39 Maria Gabriella Mulas


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r Surface Waves – seismic w surface, most destructive sei Surface waves travel along t ground and anything resting Body P w aves – push-pull and pull (expand) rocks in th Body S w aves – shake the direction of travel Gases and liquids do not tran P waves A seismogram shows all thre arrive first, then the S waves last The waves arrive at different different speeds 40

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SEISMIC WAVES

Maria Gabriella Mu Proyecto Estructural - Prof. Michele Casarin

SISMOLOGIA

03 - Fundamentals of Engineering VelocitySeismology equations

BODY WAVES

k 4 / 3 BODY WAVES V

VP

ONDAS SISMICAS

s

density

Velocity equation

µ shear modulus (rigidity) k

bulk modulus (rigidity)

k 4/ 3

V

because shear modulus (rigidity) for fluid is zero, S waves P cannot propagate through a fluid 13 P and S wave propagation velocity consequence of equations is that P waves are 1.7x faster density than S Representative values of propagaton velocity of P waves for crustal materials Maria Gabriella Mulas

(rigidity) µ shear modulus (km/ s)


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Material

Alluvial material (clay, sand, silt) Soft rock, dense gravel Hard rock, dolomites Crystalline rock

k

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0.5 - 2.0 (*) bulk modulus (rigidity) 2.0 - 3.0 3.0 - 5.0 4.0 - 6.5

because shear modulus (rigidity) propagate a fluid Representative values ofcannot propagaton velocity of S waves for crustalthrough materials Materialconsequence (m/ ofs) equations is that Very soft clays (Mexico city) 40 - 80 Normally consolidated clay and silt S 150 - 300 than (*) lower values are for dry alluvial sediments (above water table)

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200 – 400 400 – 800 500 - 1000 700 – 1500 2500 - 3500



Fundamentals of Engineering Seismology

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Medium to dense sand gravel Soft rock Weathered rock Hard rock (crystalline)

SISMOLOGIA

SURFACE WAVES

ONDAS SISMICAS

SURFACE WAVES

Rayleigh waves Rayleigh waves

Love waves

waves Love waves travel faster than Rayleigh waves and thereforeLove arrive earlier Maria Gabriella Mulas

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Love waves travel faster than Rayleigh waves and therefore arrive earlier 45



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a Mulas

SURFACE WAVES

SISMOLOGIA 02 - Seismology DE ONDAS SISMICAS MEDICION

14/03/201

INSTRUMENTS THAT RECORD EARTHQUAKE WAVES

SISMOGRAM

SISMOGRAM

SISMOGRAM

Tells you: SISMOGRAMAS Distance - Time Relations based on the time difference occurred, 1) How far away the earthquake We can determine the distance to an epicenter by finding ves with the ground between p and s –wave arrivals the difference between the arrival of P waves and S waves. and weight tationary, because of the spring, Looking athinge a travel-time graph we can determine how Magnitude of ground motion, based on the amplitude of the S waves 2) far away the epicenter is 49 Travel-time graphs from three or more seismographs can be used to find the exact location of an earthquake epicenter


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Tells you: 1) How far away the earthquake occurred, based on the time difference between p and s –wave arrivals 2) Magnitude of ground motion, based on the amplitude of the S waves

Seismometers: •The paper roll moves with the ground SISMOGRAFOS •The pen remains stationary, because of the spring, hinge and weight Maria Gabriella Mulas


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- Seismology

14/03

SISMOLOGIA MEDICION DE SISMOS RITCHER SCALE MOMENTO DE MAGNITUD MEASUREMENTS OF EARTHQUAKES

MEASUREMENTS OF EARTHQUAKES

ENERGIA SISMICA MEASUREMENTS OF EARTHQUAKES



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70

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MEASUREMENTS OF EARTHQUAKES

SISMOLOGIA

Some Notable Earthquakes

MEDICION DE SISMOS

termed ve recorded ed accelerograms) omplete information ocation otion at the site) quency content,

Indonesia (12/04)

Pakistan (10/05)

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acceleration is , PGA

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SISMOLOGIA PROPAGACION DE ONDAS SEISMIC RISK: determination of ground motions having the required probability of exceedance

-FUENTE (TAMANO Y TIPO) -CAMINO (DISTANCIA Y TIPO DE SUELO)


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-EFECTOS DEL SITIO: TIPO DE SUELO

The case of the Mexico city during the Sept 19th 1985 Michoacán earthquake

SISMOLOGIA

(magnitude=8.2; R ~ 400 km) EL CASO DE CIUDAD DE MEXICO Heavy damage and collapse of 10-14 storey buildings


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Fundamentals of Engineering Seismology

Common types of damage Proyecto Estructural - Prof. Michele Casarin

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strong in shear and continue to support their load EFECTOS SISMICOS and after the earthquakes

EFECTOS SISMICOS

Damage due to soil liquefaction – 1

Mechanism of soil liquefaction

LICUEFACCION

Adapazari, Turkey 1999 Adapazari, Turkey 1999

Izmit, Turkey 1999 Izmit, Turkey 1999

Maria Gabriella Mulas Maria Gabriella Mulas


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Kobe, Japan 1995 Kobe, Japan 1995


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Izmit, Turkey 1999





The weight of the building squeezes the adjacent soil 1964 Nilgata, Japan


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(Courtesy of Prof. Hugo Bachmann)

1964 Nilgata, Japan

http://nisee.berkeley.edu/bertero/html/damage_due_to_liquefaction.html ji-ji_chap8.pdf pag. 7-10 (figs. 8.6-8.18)



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1964 Nilgata, Japan

Sand boils and ground fissures provide Sand boils and ground evidence of liquefaction fissures provide evidence of liquefaction Maria Gabriella Mulas

Maria Gabriella Mulas Maria Gabriella Mulas

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Proyecto Estructural - Prof. Michele Casarin Maria Gabriella Mulas 16

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EFECTOS SISMICOS

hquake Damage - Part I - 1 Landslides

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Landslides - 2

DESLIZAMIENTOS Landslides - 1 inclined mass of soil is suddenly shaken, a When a steeply slip lane can form and the material slides downhill.

Landslides - 2

When a steeply inclined mass of soil is suddenly shaken, a slip lane can form and the material slides downhill.

Structures sitting on the slide move downward Structures below the slide are hitten by falling debris

Before

Before

After


Landslides - 3


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After

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Maria Gabriella Mula

17 Landslide of Turnagain Heights Anchorage, Alaska 1964


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EFECTOS SISMICOS DESLIZAMIENTOS

Landslide of Turnagain Heights Anchorage, Alaska 1964

slides - 3

Landslide of Turnagain Anchorage, Alaska 1964

ji-ji_chap8.pdf photo 8.1-8.5 Maria Gabriella Mulas


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EFECTOS SISMICOS

Ground rupture - 1

Ground rupture - 3

Ground motio

RUPTURA DEL SUELO 1906 Olema, CA

Japan earthquake, March 3,11

Kanto Highway, repaired in one week

Guatem Rails be Maria Gabriella Mulas

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mage - Part I

EFECTOS SISMICOS ARCILLAS DEBILES

The soil pushed against ainst their the piles, breaking their connection with the Piles were superstructure, and dragged by d pushing them away the soil y from the cap beam Maria Gabriella Mulas


Weak clay - Struve Slough Bridge - 2 Weak clay - Struve Slough Bridge - 2

Piles “punctured” the bridge Shear damage was found at the top of the piles

Piles were dragged by the soil

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Piles “punctured” the bridge Shear damage was found at the top of the piles




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Weak clay - Struve Slough Bridge - 1 ve Slough Bridge - 1


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The soft silts and clays were extremely sensitive to the long period (about 2s) ground motion coming from the distant but high-magnitude source. Many EFECTOS SISMICOS medium height buildings (10-14 stories) were damaged or collapsed during ARCILLAS DEBILES the earthquake.

Tsunami

Tsun

EFECTOS SISMICOS Earthquake Damage - Part I TSUNAMI

•very long wavelength, deep wavebase •speeds up to 800 km/hour, 20 meters high Tsunami – wave propagation times

Tsunami

•very long wavelength, deep wavebase •speeds up to 800 km/hour, 20 meters high

Tsunami - 1964 Alaska Earthquake Maria Gabriella Mulas


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Japan earthquake of March 11, 2011


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EFECTOS SISMICOS


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TSUNAMI

Ground motion

Foundatio

Foundation failure - 1 EFECTOS SISMICOS

Foundation failure - 2

FALLAS EN CONEXIONES CON LAS FUNDACIONES

Earthquake Damage - Part II

The connection to the foundation or to an adjacent member is more likely to be damaged during the eq.than the foundation itself. Materials that cannot resist lateral forces should be avoided

Ove foun

Foundation connection

When the foundation is too small, it can become unstable and rock over

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Timber structure

Other will fa

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Hous We have seen pile damage due to weak ancho Slough bridge (1989 Loma Prieta). found down As long as the foundation is embedded in stud w usually has ample strength and ductility boltst When the foun sill pla earthquakes. become unsta

Foundation connection

Foundation c

EFECTOS SISMICOS

FALLAS EN of CONEXIONES CON LAS FUNDACIONES The major cause damage to electrical transformers, storage bins, ction Foundation connection (special structures) and a variety of other structures is the lack of secure connection to the foundation

age to electrical transformers, storage bins, uctures is the lack of secure connection to




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Pull-out of column reinforcement from the foundation The longitudinal rebars did not have sufficient development length to transfer the force to the footings Insufficient confinement reinforcements in the footings and columns

Pull-out of column reinforcement from the foundation The longitudinal rebars did not have sufficient development length to transfer the force to the footings Insufficient confinement reinforcements in the footings and columns

EFECTOS SISMICOS Soft story

e in San Francisco. The soft ma Prieta earthquake damage in San Francisco. The soft of garages in the first story story is due to construction of garages in the first story strength

resultant reduction in shear strength


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t story Soft story FALLAS POR ENTREPISO DEBIL “SOFT STORY”

EFECTOS SISMICOS Soft Story, L’Aquila 2009

FALLAS POR ENTREPISO DEBIL “SOFT STORY” Soft-story collapse No damage on vertical walls The bottom column is totally detached from the upper beam Transverse reinforcement is absent, longitudinal reinforcement is insufficient

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Story, L’Aquila 2009 EFECTOSSoft SISMICOS

So

FALLAS POR ENTREPISO DEBIL “SOFT STORY”



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Durin at th redu Most and disco

Soft story at mid level EFECTOS SISMICOS



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10-story SRC building with 3rd floor collapse

Soft story at mid leve


Mid story collapse, Kobe earthquake



M Ko

Maria

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EFECTOS SISMICOS Soft story at mid level

Soft story at mid level

Torsional moments EFECTOS SISMICOS

FALLAS MOMENTOS TORSIONALES

Torsional mom

Torsional moments

Curved, skewed and eccentrically supported stru torsion during earthquakes


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Nine-story building in Kobe, Japan Shear walls on 3 sides, a moment resisting The 1995 earthquake caused a torsional m Proyecto Estructural - Prof. Michele building. The 31first-story column onCasarin the east

Torsional moments EFECTOS SISMICOS

P


C

Torsional failure of the top of the column


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E



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Earthquake-induced pounding between inadequately separated structures may cause considerable damage or even lead to a EFECTOS SISMICOS FALLAS MOMENTOS TORSIONALES structure’s total collapse.

unding -2 EFECTOS Pounding - 2 SISMICOS

Shear


Most b frames MostDamag building frames to res comple Damage to t Shear complete col reinforc

Shear damag reinforcemen


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Maria Gabriella M

lure MtMcKinley McKinleyApartments Apartments re – –Mt Shear failure – Mt McKinley Apartments ska Earthquake, 1964 kaEFECTOS Earthquake, 1964 SISMICOS Great Alaska Earthquake, 1964


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FALLAS POR CORTE

horizontal beam.

EFECTOS SISMICOS

aria Gabriella Mulas

Shear failure

CORTE

A wide shear crack split the wall in two, directly below a horizontal beam.

The spandrel beam between the wall had large X cracks associated with shear damage as the building moved back and forth

The spandrel beam between the wall had large X cracks associated with shear damage as the building moved back and forth

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McKinley Apartments FALLAS POR quake, 1964

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SISMICOS earEFECTOS failure (short column)

Shear failure (short column)

FALLAS POR CORTE (COLUMNA CORTA)


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L’A

EFECTOS SISMICOS FALLAS POR CORTE

Shear damage (L’Aquila 2009)

Shear damage (L’Aquila 2009)

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Flexural failureSISMICOS of columns EFECTOS

Flexural fa

FALLAS POR FLEXION

ure of columns

Flexural failure

Insufficient transverse reinforcement results in lack of confinem Insufficient tran for columns. This allows longitudinal reinforcement to buckle an the concrete to fall off from the column. for columns. Th 44


the concrete to


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EFECTOS SISMICOS

astic hinge at the base of a column

umn

FALLAS POR FLEXION

Column failure

Column failure

Insufficient transversal reinforce confinement for columns

Maria Gabriella 47 Insufficient transversal reinforcement results in Mulas lack of confinement for columns

Proyecto Estructural - Prof. Michele Casarin Maria Gabriella Mulas

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EFECTOS SISMICOS


Test on a beam-column node

Test on a beam-column node

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Failure of beam-column node FALLAS EN NODOS


nt of Mechanics of Particles

MECANICA

13 - 45

-LAS LEYES 1 Y 3 SON SUFICIENTE PARA ESTUDIAR CUERPOS ESTATICOS O EN MOVIMIENTOS brium – D’Alembert’s SIN ACELERACION

-2nda LEY DE NEWTON: F= Mex A of Nwt • Alternate expression

F

ma

on ’s second law,

0

ma inertial vector Magnitude of the force a spring • •With the inclusion of theexerted inertialby vector, the is system -LEYES DE TRABAJO. RESORTES. deflection, ofproportional forces actingtoon the particle is equivalent to zero. FThekx particle is in dynamic equilibrium. k developed spring constant N/m or in lb/in. • Methods for particles static Proyecto Estructural - Prof. Michele Casarin equilibrium may be applied, e.g., coplanar forces

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e

-CUANDO UN CUERPO SE ACELERA SE REQUIERE LA 2nda LEY DE NEWTON, PARA RELACIONAR EL MOVIMIENTO CON LAS FUERZAS ACTUANTES.

DINAMICA DE ESTRUCTURAS VIBRACIONES DE PARTICULAS -SON MOVIMIENTOS DE UNA PARTICULA O CUERPO EN EL CUAL OSCILA CON RESPECTO A UN PUNTO DE EQUILIBRIO. -EL PERIODO DE VIBRACION (T=s) EL TIEMPO REQUERIDO PARA QUE UN SISTEMA COMPLETE UN CICLO COMPLETO DE MOVIMIENTO -LA FRECUENCIA (f=hertz=1/s) ES EL NUMERO DE CICLOS POR UNIDAD DE TIEMPO -LA AMPLITUD ES EL DESPLAZAMIENTO MAXIMO DEL CUERPO DESDE EL PUNTO DE EQUILIBRIO

-SE CONSIDERA UNA VIBRACION LIBRE CUANDO EL MOVIMIENTO ES MANTENIDO POR LAS FUERZAS INERCIALES. CUANDO UNA FUERZA HARMONICA ES APLICADA SE LE LLAMA VIBRACION FORZADA.


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-CUANDO NO SE CONSIDERA EL AMORTIGUAMIENTO DEL SISTEMA, SE LE LLAMA SISTEMA NO AMORTIGUADO. TODAS LAS VIBRACIONES SON AMORTIGUADAS HASTA CIERTO PUNTO.

monic Motion

DINAMICA DE ESTRUCTURAS VIBRACIONES DE PARTICULAS Free Vibrations of Particles. Simple • If aDEparticle is displaced through a dista Motion -SI UNA Harmonic PARTICULA ES DESPLAZADA SU PUNTO DE EQUILIBRIO Y SOLTADA SIN VELOCIDAD, LA

equilibrium position and released with • If a particle is displaced through a distance particle will undergo simple harmonic equilibrium position and released with no ma particle F W k st simple x harmonic kx mot will undergo

PARTICULA ENTRARA EN UN MOVIMIENTO HARMONICO SIMPLE.

ma

F

mx kx 0

mx kx

W

k st

x

kx

0


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• General solution is the oftwo twoparticu par • General solution is thesum sum of k kk x C1 sink t C2 cos t x C1 sin tm C2 cos m t

SIMPLE STRUCTURES - 3

DINAMICA DE ESTRUCTURAS SIMPLE STRUCTURES e- 1

S

STRUCTURES - 2 ESTRUCTURAS SIMPLE SIMPLES

“Simple” because can b idealized as a lumped masssystem m supported by a If the idealized is displaced and then released,P -SI LA structure ESTRUCTURA SIMPLE ES DESPLAZADA Y SOLTADA, EMPEZARA A OSCILAR O VIBRAR CON e massless withPergola stiffness k in the lateral direction at the Macuto Sheraton Hotel damaged by the and forth about its init start to oscillate (or vibrate) back RESPECTO A SU POSICION INICIAL (VIBRACION LIBRE) earthquake of July deform 29, 1967 (Venezuela, Caracas) Lateral motion is “small”: the structure within the elastic range equilibrium position: FREE VIBRATION

Maria Gabriella Mulas, Paolo Martinelli

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16 In an ideal case, the structure15 will oscillate indefinitely, Proyecto Estructural - Prof. Michele Casarin


without any energy dissipation: the kinetic energy will convert Maria Gabriella Mulas, 9 in potential energy and viceversa.

DINAMICA DE ESTRUCTURAS Maria Gabriella Mulas, Paolo Martinelli

Paolo Martinelli

Maria Gabriella M

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ESTRUCTURAS SIMPLES

-LA AMPLITUD DE LOS DESPLAZAMIENTOS DISMINUYE CON EL TIEMPO GRACIAS AL SDOF system SIMPLE STRUCTURES 5 UN MECANISMO AMORTIGUAMIENTO, QUE -ES DE linear DISIPACION DE ENERGIA QUE DEBE Single-degree-of-fr SER INCLUIDO EN LOS CALCULOS.

v(t) m k f(t) c

Real case: the amplitude of oscillations will decay with time The energy dissipating mechanism, called damping, must be included in the structural modeling.

Idealization of a 1

Hp. Bending deformat the roof level, a ma

viscous damper th 24 EJ of dynam kTwo types 3 h force in t • external • ground motion im

mv cv kv

f (t )


2

2

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Equation of motion

displacement mass stiffness dynamic external viscous damping c


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DINAMICA DE ESTRUCTURAS ESTRUCTURAS 1 GRADO DE LIBERTAD -IDEALIZACION DE UNA ESTRUCTURA DE UN PISO: LA MASA M ES CONCENTRADA EN EL TECHO, SOBRE UN PORTICO SIN MASA PERO QUE TIENE RIGIDEZ, JUNTO UN AMORTIGUADOR VISCOSO QUE DISIPA ENERGIA.

Single-degree-of-freedom system – 1

1)

FUERZA LATERAL EXTERNA

2)

DESPLAZAMIENTO DEL SUELO EN LA BASE (SISMO)


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-EXISTEN DOS TIPOS DE CARGAS DINAMICAS:

Force-displacement relation: linearly elastic systems - 3

Force-displacement relation: inelastic systems The internal restoring For a linear system the relation betwe force depends on the Force-displacement relation he stiffness matrix is computed with the standard method relative displacement For inelastic the restoring force is no longer a u. systems, force f and the resulting displacement displacement method, by imposing an unit value to one single valued function of the relation: displacement/deformation: s Force-displacement linearly elastic ESTRUCTURAS 1toGRADO DE LIBERTAD oordinate (dof) while the remaining are equal zero. The fs-u relation systems can be - 3 he stiffness coefficients are the forces that are necessary fs f s u ,u fs = ku maintain the system in equilibrium. They can be thought either linear or non s the reactions of additional constraints, inserted to linear We are interested in studying the dynamic response of mpose the desired values to the dofs. The resisting force is a single-valued fu inelastic systems because is almost all the structures are the standard method The stiffness matrix computed with s is the number of T designed with the expectation that they will evolve in the Aelastic static external of displacement method, imposing system is Maria Gabriella Mulas, Paolo Martinelli nonlinear range (cracking, yielding,by damage etc) during an unit value to one 22 e the displaced the intense ground caused an earthquake.are equal to zero. f coordinate (dof)shaking while the by remaining force is applied original position. is the stiffness of forces the system: it repre r The k stiffness coefficients are the that are necessary on roof, to maintain system equilibrium.to They can bean thought ine the stiffness that the must beinapplied obtain unit Td as the reactions ofbalanced additional constraints, by aninserted to DOFs necessary e orce-displacement relation: linearly elastic impose the desired values to the dofs.


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ystems -2 nelastic force-deformation relation: panel zone of

a steel welded beam-to-column connection

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e lateral stiffness of this frame depends on both the lumns and beam stiffness. It will vary between the two stic treme values of infinite and null beam stiffness


Damping force - 1

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internal restoring force

In damping, the energy of the vibrating system is Mariathermal Gabriella Mulas, Paolo Martinelli dissipated by various mechanisms: effects, internal friction of the material, friction at steel connections, opening and closing of micro-cracks in reinforced concrete and so on. The damping in actual structures is idealized by a linear viscous damper: the damper coefficient is selected to reproduce the actual energy dissipation. Maria Gabriella Maria Mulas, Paolo Martinelli Mulas, Paolo Martinelli We only consider linearGabriella viscous damper:

li

The internal restoring force depends on the relative displacement u.linearly e Proyecto Estructural -relation: Prof. Michele Casarin Force-displacement 54


Newton’s law

ized one-story frame is subjected to an external idealized one-story frame is subjected to an external in the direction of u DINAMICA DE ESTRUCTURAS ed one-story frame is subjected mic force in the direction force p(t) p(t) in the direction of uof uto an external 1 GRADO one-story frame is subjected to an external rceESTRUCTURAS p(t) inThe theidealized direction of DE u LIBERTAD dynamic force p(t) in the direction of u

ton’s second law states that: second law states that: econd law states that: 2ndaNewton’s LEY DE NEWTON: second law states that: aw states equation can that: be rewritten as: ation can be rewritten on can be The rewritten as:as: equation can be rewritten as: e elastic range we obtain: be rewritten as: stic range we obtain: In the elastic range we obtain: ELASTICO: ic range we obtain: e inelastic range : INELASTICO: In: the lastic range : inelastic range : stic werange obtain:


55

LA ESTRUCTURA IDEALIZADA DE UN 1 PISO SUJETA A UNA FUERZA DINAMICA P(t) EN LA DIRECCION DE u:

pure components, the stiffness, damping and mass components. The external force applied to the complete amics of Structures: an DE introduction DINAMICA ESTRUCTURAS system is distributed among the three components, and ESTRUCTURAS 1 GRADO DE LIBERTAD the sum fs + fD + fI must equal the applied force p(t) Mass – spring – damper system

D’Alembert’s principle is based on the notion of a fictitious “inertia force”, equal to the product of the times its acceleration, and acting in a direction opp to the acceleration. With inertia force included, the Equati system is in equilibrium at each instant

fI

mu

This is the classical SDF system analyzed in textbooks on mechanical vibration and elementary physics. We will refer f D and cu forced (harmonic) f I muvibrations f s ku to study free to this system Maria Gabriella Mulas, Paolo Martinelli

pt

fI

fD

fS

0


32

fI

Newton’s law

D’Alembert’s Principle Proyecto Estructural - Prof. Michele Casarin

56

8

Newton’s law


e classical SDF system analyzed in textbooks on al vibration and elementary physics. We will refer tem to study free and forced (harmonic) vibrations

Equation of motion: earthquake excitation ESTRUCTURAS 1 GRADO Maria GabriellaDE Mulas, LIBERTAD Paolo Martinelli

ut t

ut

33

fI

ug t

ug ground displacement u relative displacement Equation of motion: earthquake excitation D’Alembert’s Newton’s law t u absolute displacement Principle

fI

fD

t

33

fI

mu t

mu cu ku

mu g t

fS

0

n of motion: earthquake excitation

mu cu

f s u ,u


fD

fI

mu t t

mu cu ku

mu g t

fS

0

mu cu

f s u ,u

ut

ug t

ug ground displacement u relative displacement ut absolute displacement

mu t

ug t

mu g t


mu t

ug t

34

mu g t 34

PROBLEM STATEMENT AND SOLUTION METHODS

The system undergoing base motion can be analyzed a Proyectoas Estructural - Prof. Michele Casarin

57


ut t

Principle

ntroduction DINAMICA DE ESTRUCTURAS Maria Gabriella Mulas, Paolo Martinelli

41

ESTRUCTURAS 1 GRADO DE LIBERTAD METODO DE SOLUCION:

-Se conoce la masa, rigidez y coeficiente de amortigumiento. Se conoce la excitacion externa ya sea en forma de una fuerza dinamica P(t) o en forma de desplazamiento del suelo Ug(t). Las condiciones de inicio son U=0

Duhamel’s integral

-Se requiere la respuesta de la estructura, ya sea en forma de desplazamientos, velocidades, aceleraciones o fuerzas internas.

f solution of the differential equation

-Luego que se calculo el desplazamiento de respuesta U(t) de la estructura, se calculan los esfuerzos internos para cada In this approach the external force is represented a los instante de tiempo utilizando ANALISIS ESTATICOS: 1) Se le puede aplicar la deformacion a la estructura as y hallar esfuerzos internos. 2) Se le puede aplicar la fuerza estatica equivalente P, lashort cual aplicada en dado momento debe resultar en sequence of infinitesimally impulses. la misma derformacion U calculado anteriormente. 3) En sistemas inelasticos se deben hacer calculos paso a paso The response of the system to an applied force p(t) at incrementales.

completely the problemImplicit we must assign the initial in this result are the “at rest” initial conditions.


58

ion-Para ofresolver motion of a SDOF system subjected toresponses t, isgrado obtained addingdethe all the la fuerza la ecuacion diferencial de time segundo se utilizaby la integral Duhamel, donde seto representa externa como una secuencia de cortos differential impulsos infinitos. impulses up to that time: orce is a second order equation:

DINAMICA ESTRUCTURAS undamped frame of a 1-story Free vibrations DE


59

VIBRACION LIBRE DE ESTRUCTURAS SIN AMORTIGUAMIENTO

am x m

2 n

DINAMICA DE ESTRUCTURAS 2 Damped - 1s2 0.040 m 6.93Free rad s Vibrations am 1.920 m

VIBRACION LIBRE DE ESTRUCTURAS CON AMORTIGUAMIENTO Damped free vibrations 4 degree by • All vibrations are damped to- some

briella Mulas

ns - 1


13due to dry friction, fluid friction, or forces internal friction.

• With viscous damping due to fluid friction, F

ma :

W

k

st

mx cx kx

x

cx 0

mx

c cc

Damped Free Vibrations - 2

60

• Substituting x = e t and dividing through by e t • All vibrations are damped to some degree by • Characteri yields the characteristic equation, forces due to dry friction, fluid friction, or 2 m 2 c internal friction. c c k m 2 c k 0 2m 2m m Normalmente en estructuras el amortiguamiento es UNDERDAMPEDcc 2m n • With viscous damping due to fluid friction, damping is thedamping smallest value of damping that inhibits • Heavy dam the critical coefficient such that W k st x Critical cx m•x Define F ma : completely x C1e 1t 2 mx cx kx oscillations 0 cc k k 0 cc 2m 2m n t t 2 m m m Structures are usually underdamped • Substituting x = e and dividing through by e • Critical da Proyecto Estructural - Prof. Michele Casarin yields the characteristic equation,Maria Gabriella Mulas

Critical damping is the smallest value of damping that inhibits oscillations completely Damped free vibrations - 5

DINAMICA DE ESTRUCTURAS Structures are usually underdamped on period Effect of damping 2 VIBRACION LIBRE DE ESTRUCTURAS CON AMORTIGUAMIENTO Vibration of SDOF systems

c

El periodo con namortiguamiento Tdn es1 1 d cc mayoramplitude que Tn. 18

xm

v0

n x0

2

T

2 Damped freenxvibrations -7 0 damped perio Effect d D of damping on period

T

1

Damped free vibrations - 6 Effect of damping on amplitude decay

d

Td

n

c cc

1

Tn

= xm

2 n

1

2

c

damped frequency – lower than cc natural frequency

2

natural period

For d typica Maria Gabriella Mu engin the va such signif the na vibra

damped period – longer than

2 Misma y rigidez 1 masa natural period Free vibration due to an initial displacement applied to four


61


2

c

Forced Vibrations: harmonic excitation DINAMICA DE ESTRUCTURAS Harmonic vibrations of undamped systems

VIBRACION FORZADA DE ESTRUCTURAS SIN AMORTIGUAMIENTO Harmonic vibrations of undamped systems

tems

Vibracion forzada de un sistema ocurre cuando es sometida a fuerzas periodicas o desplazamientos periodicos Forced vibrations - Occur when a system is subjected to a periodic force or a periodic Vibration of SDOF systems displacement of a support. forced frequency f

Particular solution

solution

Harmonic vibrations of undamped systems u p (notations of Chopra’s textbook) ust 0st 0 0 k mu ku p 0 sin t

mentary solution sin t

0 nt

1

at t

p 01 k 1

1

/ n 2/

sinF2 tsinmat :

u u0

0

up t

p0 k 1

uc t

A cos nt B sin nt

1 / n 2

n

P sine steadystat

steadystat e m

mx kx 29

ft

W

Pm sin 29

k

st

x

mx

W

ut

k

A cos nt

st

m ux t ukx 0 cos knt

ft

x

sin t

u0

System response System response for for /

Complementary solution 1

t

sin t

n

/0.2n

0.2

u0

u0 0

u0

u 0n p0 / k n p0 / k

0

2x / m n

p0 t 1 2 sin nt p0 k f1Maria /Gabriella k 1 Mulas n

transient

Harmon

Particular solution

n

p0 sin k 1f m

m sin

p0 k

u0

B sin nt

n


u

Harmonic force


1 / n 2

sin t

30

steadystat e

27


29

62

ticular solution

Harmonic force

30

Vibration of SDOF systems

DINAMICA DE ESTRUCTURAS Damped Forced HarmonicDE Vibrations –1 VIBRACION FORZADA ESTRUCTURAS CON AMORTIGUAMIENTO Damped Forced Harmonic Vibrations – 2 Problem statement and ocurre steady state response Vibracion forzada de un sistema cuando essolution sometida a fuerzas periodicas o desplazamientos periodicos General Damped Forced Harmonic Vibrations - 3 Steady state

/

0.2

n

0.05

u0 u0

cx kx

and steady

xm

Pm sin

ft

/k

Due to damping, the transient vanishes and the steady state x ary axsufficient particular time is thecomplement response after The response is an harmonic function having the same 1 circular frequency of the forcing term magnification Proyecto Estructural - Prof. Michele Casarin

x

xm 39

n p0

2


40

63

nification ctor mx

0

Horizontal ground acceleration (Parkfield Station)

ESPECTRO DE RESPUESTA Recorded Ground Motions (horizontal component)

DESCRIPCION DE LA EXCITACION SISMICA SISMO EL CENTRO: COMPONENTE N-S To define earthquakes – ground shaking: time variation of ground acceleration

ues recorded many different ations

3 components: 2 horizontal, 1 vertical Strong-motion accelerographs Frequency range of recording without excessive distorsion:

Ground presum known indepen structur RIGID NO SO STRUC INTER

0-15 Hz for analog instruments up to 30 Hz for digital ones

Time in which n values a 0.01-0.0

First record in 1933, Long Beach earthquake

5



64

Maria Gabriella Mulas COMPONENTES: 2 HORIZONTALES, 1 VERTICAL

ND t S ESPECTRO DE RESPUESTA f mu f

cu

f

ku

I MOVIMIENTO Y PARAMETROS D s ECUACION DE RESPUESTA uation of motion forDE earthquake

ut= total displ. u = relative displ.

citation - 1

fI

fD

mu

fs 0

mut

fI

t

ug = ground displ.

cu ku 0

ut = u + ug fD

cu

fs

ku

cu mu ku cu ku 0 mu f f fmu 0 g mu cu ku mu t relative displacement u(t) of the system is the same thattha we w ative displacement u(t)The of the system is the same obtain by applying to the stationary base system the effective load t mu t by applying to the stationary basepsystem the effectiv t

I

D

s

g

ut= total displ. u = relative displ. ug = ground displ. EL DESPLAZAMIENTO U(t) DEL SISTEMA SERIA EL MISMO QUE OBTUVIERAMOS SI ut =FUERZA u + ug EFFECTIVA: APLICARAMOS UNA eff g

fI

fD

fs

0

fD

cu

mu t

fs

ku

cu ku 0

peff t

mu cu ku mu g t relative displacement u(t) of the system is the same that we would in by applying to the stationary base system the effective load:

peff t

mug tMaria


mug t

Gabriella Mulas Proyecto Estructural - Prof. Michele Casarin

65

f I 7 mut

Deformation response history of SDF systems to El Centro ground motion

ESPECTRO DE RESPUESTA ECUACION DE TMOVIMIENTO PARAMETROS DE RESPUESTA Same ς, different Same T , Y different ς n

n

-GRAFICAS: DE RESPUESTA DE Time required HISTORIA to DEFORMACION complete a cycle DE SDOF PARA SISMO DE EL CENTRO. when subjected to this earthquake -DOS SISTEMAS CON EL MISMO T Y ground motion is AMORTIGUAMIENTO, TENDRAN LA MISMA close to the natural RESPUESTA period. -PARAMETROS The longer the DE RESPUESTA DE INTERES: period, the greater DEFORMACIONES RELATIVAS (PARA CALCULO the peak DE ESFUERZOS INTERNOS), DEFORMACIONES deformationACELERACION TOTAL. TOTALES,

systems

-ENTRE MAYOR EL Tn, MAYOR SERA LA Deformation response history of SDF systems DEFORMACION MAXIMA. to El Centro ground motion Maria Gabriella Mulas

2 n

u

ug t

u( t ) u t ,Tn ,

10

Same ς, different Tn

Same Tn , different ς

66

f motion for earthquake -2

Time require Proyecto Estructural - Prof. Michele Casarin complete a c

force that, applied statically to the ESPECTRO DE RESPUESTA system, would Application produce the same se of SDF of pseudo-acceleration con motion ECUACION DE MOVIMIENTO Y PARAMETROS deformation DE u(t)RESPUESTA

by

fs

:

fs

At

e

2 n

ut

fs(t)

mA t 2 n

Static structure static eq

ut

2 Vb(t)

2 n

2 Tn

Mb(t)

Vb Mb t

fs t

mA t

hf s t

hVb t

A(t) is the pseudo-acceleration

-Fs ES LA FUERZA ESTATICA EQUIVALENTE The base shear and overturning moment depend o Maria Gabriella Mulas acceleration. A(t) SE LE LLAMA PSEUDO ACELERACION Base shear balances the static equivalent force. Base overturning moment balances its moment w foundation and is provided by axial loads in colum 11



67

as

m

ku t

of Application pseudo-acceleration concept of pseudo-acceleration concept

ESPECTRO DE RESPUESTA

ECUACION DE MOVIMIENTO Y PARAMETROS DE RESPUESTA

fs(t)

fs(t)

Static analysis Static of the analysis of the structure subjected to BASAL thesubjected to theDE VOLCAMIENTO -ELstructure CORTE Y EL MOMENTO SON DEPENDIENTES DE LA PSEUDO static equivalent force fs ACELERACION. static equivalent force fs

Vb Mb t

Mb(t)

fs t

mAVbt

fs t

b t t hf s t hf s t MhV b

mA t hVb t

The base shear and overturning moment depend on the pseudoacceleration. Proyecto Estructural - Prof. Michele Casarin

nd overturning moment depend on the pseudo-

68

Vb(t) Mb(t)

-EL CORTANTE BASAL EQUILIBRIA LA FUERZA ESTATICA EQUIVALENTE

Deformation Response Spectrum for El Centro ESPECTRO DE RESPUESTA Ground Motion

CONCEPTO DE ESPECTRO DE RESPUESTA -UN PLOTEO DE LOS VALORES PICO DE RESPUESTA EN FUNCION DE LOS PERIODOS NATURALES DE LA ESTRUCTURA ES LLAMADO ESPECTRO DE RESPUESTA.

trum of linear systems

Response Spectrum Concept


Practical mean to characterize a ground mot

u0 Tn ,

max u t , Tn ,

Deformation R. S

u0 Tn ,

max u t , Tn ,

Relative velocity

u0 Tn ,

max u t , Tn ,

Acceleration R.S

t

t

t

By definition, the peak response is positive; the because it is usually irrelevant for design. Proyecto Estructural - Prof. Michele Casarin 15 Each plot is derived for a SDF having a fixed d

69

RESPONSE SPECTRUM CONCEPT

-ESTOS VALORES SON OBTENIDOS DE UN FINITO DE SISTEMAS UN value GRADO DE quantity a A plot ofDE the peak of a response of the natural Tn or related DE quantities as LIBERTAD CONperiod UN FACTOR called response spectrum for that quantity. AMORTIGUAMIENTO DEFINIDO.

Pseudo-velocity: ESPECTRO DEresponse RESPUESTA Pseudo acceleration spectrum, linear scale CONCEPTO2DE ESPECTRO DE RESPUESTA Deformation response spectrum, log scale V D D V u ku02

Es0

kD 2

2

0

Tn

2

2

kV

mV 2

n

2

Pseudo-acceleration: A Vb 0

2 n

f s0

2 Tn

D mA

2

PASOS

Linear scale -DEFINIR ACELERACION DEL SUELO EN FUNCION DEL From this plot: TIEMPO -SELECCIONAR Y AMORTIGUAMIENTO Base shear Tcoefficient -CALCULO DESPLAZAMIENTOS Y DESPLAZAMIENTO (lateralDE force) MAXIMO

2

D

W A g

u0t

-DERIVAR HASTA OBTENER ACELERACION -REPETIR PARA COMBINACIONES DESEADASX

= 0, 2, 5,10 and 20 %

A W g

A/g base shear coefficient

From this plot: Peak deformation Proyecto Estructural - Prof. Michele Casarin

70

n

Spectrum on normalized scale, 5% damping ESPECTRO DE Response spectra for = 0,RESPUESTA 2, 5 and 10% Spectru

d 10%

Plotted on normalized scales CONCEPTO DE ESPECTRO DE RESPUESTA

27Gabriella Mulas Maria

The adoption of normalized scales shows more directly the relation between the response spectrum and the ground motion parameters

Th is ve R. tec cu be re sp an

27



71

The adoption of normalized scales shows more directly the relation between the response spectrum and the ground motion parameters

Long period system (very flexible) ESPECTRO DE RESPUESTA Short period system (very stiff – rigid) T > Tf =15s

CONCEPTO DE ESPECTRO DE RESPUESTAn Tn < Ta = 0.035s

Da Ai Ma stat gro

The mass moves rigidly with the ground A approaches the ground acceleration and D is very small Neglecting damping: 2 n

u 2 n

u

At u

t 0

u

ug

At ug

ut t

ut

u

A ug0

ut


30 Proyecto Estructural - Prof. Michele Casarin Maria Gabriella Mulas

72

29

0

ESPECTRO DE RESPUESTA Elastic design spectrum Mean spectrum over 10 ground motions

Elasti

ESPECTRO ELASTICO DE DISENO At the same site, response spectrum due to different earthquake can differ; all of them are jagged. It is not possible to predict a jagged response spectrum for a future ground motion!

he same peak

The needs arise for design spectra smooth or composed of straight lines: they are necessary for design of new structures, or the seismic safety evaluation of existing structures

h of them Vi and Ai provide probability each period Tn


42

Recommended period values in the plot! 40

Mean spectru The design statistical ana recorded response spec Factors If noneadop ha the motions scales are over 10 recor Connecting a The impo values gives • The spec ma response • The dis connecting al one• standard The fau gives the mea • The geo standard dev site Mean spectru • The thro loc idealized lines (dashed figure)


73

l analysis of the er I of ground motions me history and the cceleration

Comparison between design spectrum and ESPECTRO DE RESPUESTA response spectrum

ESPECTRO ELASTICO DE DISENO


74

The jagged response spectrum is a description of a particular ground motion The smooth design spectrum is a specification of the level of seismic design force, or deformation, as a function of period and damping and is an average representation of many ground motions Differences are expected!

Damage not economically repairable ESPECTRO DE RESPUESTA

rum

ESPECTRO INELASTICO DE DISENO

tures are designed for base smaller than the elastic : will deform beyond the c range when subjected to und motion represented by 4g design spectrum

Inelastic Design Spectrum

Collapse

MAGE will occur

essful design will control ge to keep it acceptable

Im p e r ia lC o u n ty S e r v ic e s B u ild in g a fte rth e Im p e r ia l V a lle y ,C a lifo r n ia e a r th q u a k e o fO c t.1 5 ,1 9 7 9

irable damage: frequent quakes 3


O liv e V ie w H o s p ita l,P s y c h ia tr ic D a y C a r e C e n te ra fte rS a n Proyecto Estructural Prof. Michele Casarin F e r n a n d o e a r th q u a k e o fF e b 9 ,1 9 7 1

75

ollapse: strong earthquakes

ESPECTRO DE RESPUESTA ESPECTRO INELASTICO DE DISENO

SISTEMA ELASTOPLASTICO SIN AMORTIGUAMIENTO


76

SISTEMA ELASTICO SIN AMORTIGUAMIENTO

ESPECTRO DE RESPUESTA


77


pseudoacceleration design ESPECTROInelastic DE RESPUESTA

spectrum (84.1th percentile) linear scale



78

This format of the inelastic design spectrum is contained in seismic codes

SISTEMAS DE VARIOS GRADOS DE LIBERTAD ESTRUCTURA 2 NIVELES IDEALIZADA SOMETIDA A P1(t) y P2(t) -Losas son infinitamente rigidas. -Columnas y vigas infinitamente rigidas axialmente -Masas concentradas en cada nivel


79

-Amortiguamiento lineal viscoso respresenta la disipacion de energia.


80



Tenemos como product un Sistema de dos ecuaciones diferenciales que gobiernan los desplazamientos U1(t) y U2(t) para el Sistema de dos niveles sometido a P1(t) y P2(t).


81

Ambas ecuaciones contienen las dos incognitas, por lo tanto deben ser resueltas simultaneamente.

SISTEMAS DE VARIOS GRADOS DE LIBERTAD VIBRACION LIBRE DE SISTEMAS MGL SIN AMORTIGUAMIENTO Vibración es iniciada por curva A El movimiento no es harmónico


82

Se origina un movimiento harmónico gracias a la correcta proporción constant de U. Estas dos formas deformadas son modos naturales de vibración.

SISTEMAS DE VARIOS GRADOS DE LIBERTAD VIBRACION LIBRE DE SISTEMAS MGL SIN AMORTIGUAMIENTO El período natural de vibración es el tiempo requerido para completer un ciclo de movimiento harmónico en uno de los modos naturales de vibración. Su inversa es llamada frecuencia natural.


83

Las N raices de Wn son conocidas como los valores eigen (eigenvalues), de los cuales obtenemos Tn. El primer período T1 es llamado fundamental. Luego de conocer Wn podemos calcular la forma modal Φn

SISTEMAS DE VARIOS GRADOS DE LIBERTAD VIBRACION LIBRE DE SISTEMAS MGL SIN AMORTIGUAMIENTO La solución del problema EIGEN no da valores de amplitud de Φ sino su forma. Los N vectores de Φn son los modos naturales de vibración, cada uno asociado con un T. Son llamados naturales porque son propiedades del sistema, que dependen solo de la rigidez y de la masa.

Una matriz diagonal puede ser construida con los N valores eigen.


84

Una matriz cuadrada con todos los valores de Φ, donde cada columna es el modo natural.



85

VIBRACION LIBRE DE SISTEMAS MGL CON AMORTIGUAMIENTO

SISTEMAS DE VARIOS GRADOS DE LIBERTAD VIBRACION FORZADA CON AMORTIGUAMIENTO

-Determinar propiedades de la estructura: matrices M, K y evaluar matrices de amortiguamiento -Determinar frecuencias naturales Wn y modos de vibración Φn

-Calcular respuesta en cada modo, primero el desplazamiento del nodo U(t) y luego la fuerza asociada en el elemento.


86

-Combinar las contribuciones en las respuestas de cada modo para obtener la respuesta total.

SISTEMAS DE VARIOS GRADOS DE LIBERTAD VIBRACION FORZADA CON AMORTIGUAMIENTO -Determinar propiedades de la estructura: matrices M, K y evaluar matrices de amortiguamiento -Determinar frecuencias naturales Wn y modos de vibración Φn

-Combinar las contribuciones en las respuestas de cada modo para obtener la respuesta total. Proyecto Estructural - Prof. Michele Casarin

87

-Calcular respuesta en cada modo, primero el desplazamiento del nodo U(t) y luego la fuerza asociada en el elemento.

SISTEMAS DE VARIOS GRADOS DE LIBERTAD HISTORIA DE RESPUESTA


88

SE PUEDEN OBTENER LOS VALORES PICO DIRECTAMENTE DEL ESPECTRO DE RESPUESTA, SIN TENER QUE REALIZAR UN CALCULO DE HISTORIA DE RESPUESTA PRECISO. ESTOS VALORES NO SERAN EXACTOS PERO ES SUFICIENTE PARA CALCULOS ESTRUCTURALES.

SISTEMAS DE VARIOS GRADOS DE LIBERTAD COMBINACION DE RESPUESTAS MODALES Como combinar los valores picos de cada modo para obtener el valor de respuesta total? No sabemos el momento en que ocurre la respuesta máxima en cada modo. Existen varios métodos:

-Suma de los valores absolutos picos de la respuesta. Muy conservador. -SRSS (square root sum of squares). Excelentes resultados para estructuras con frecuencias naturales bien separadas.


89

-CQC (complete quadratic combination). Excelentes resultados para estructuras con frencuencias naturales cercanas.



90

COMBINACION DE RESPUESTAS MODALES



91

COMBINACION DE RESPUESTAS MODALES



92

EJEMPLO



93

EJEMPLO



94

EJEMPLO



95

EJEMPLO

CONCEPTOS DE DISEÑO SÍSMICO FILOSOFÍA DE DISEÑO -Se diseña con sismos de 100 a 500 años de período de retorno, muy altos para resistir en el rango elástico. -Estructuras son diseñadas para resistencias de un 15-25% de la respuesta elástica. -Esperamos que las estructuras resistan gracias a deformaciones grandes inelásticas y disipación de energía gracias al comportamiento inelástico de materiales.


96

-Probabilidad annual de falla elástica: 1 a 3% por fuerzas sísmicas, 0,01% por cargas gravitacionales.

CONCEPTOS DE DISEÑO SÍSMICO DEL RANGO ELÁSTICO AL INELÁSTICO -De observaciones en campo se determinó que las fallas no eran necesariamente por falta de Resistencia -Si la Resistencia estructural se podía mantener sin mucha degradación, la estructura puede resistir el sismo y muchas veces ser reparada. (ductilidad) -Las filosofìas de diseño pasaron de la Resistencia a grandes cargas laterales, a la evasiòn de estas, dando paso para el diseño inelástico.


97

-No todos los mecanismos inelásticos son aceptados: unos disipan energía y otros ocasionan fallas.

CONCEPTOS DE DISEÑO SÍSMICO FILOSOFÍA DE DISEÑO POR CAPACIDAD CIERTAS FORMAS ESTRUCTURAS TIENEN MAS DUCTILIDAD: -Regularidad en planta y elevación

-Ubicación de puntos de plastificación (rótulas plásticas) -Con la selección adecuada de configuración estructural, Resistencia para mecanismos inelasticos no deseados es amplificada. Ej: Resistencia a corte de vigas concreto armado.

PRINCIPIOS BÁSICOS: -Selección de configuración estructural para una respuesta inelástica -Selección y detallado adecuado de puntos de deformación inelástica


98

-Diferencia de resistencias adecuadas para evitar fallas en lugares y formas indeseadas

CONCEPTOS DE DISEÑO SÍSMICO FILOSOFÍA DE DISEÑO POR CAPACIDAD CIERTAS FORMAS ESTRUCTURAS TIENEN MAS DUCTILIDAD: -Regularidad en planta y elevación

-Ubicación de puntos de plastificación (rótulas plásticas) -Con la selección adecuada de configuración estructural, Resistencia para mecanismos inelasticos no deseados es amplificada. Ej: Resistencia a corte de vigas concreto armado.

PRINCIPIOS BÁSICOS: -Selección de configuración estructural para una respuesta inelástica -Selección y detallado adecuado de puntos de deformación inelástica


99

-Diferencia de resistencias adecuadas para evitar fallas en lugares y formas indeseadas

CONCEPTOS DE DISEÑO SÍSMICO CAUSAS COMUNES DE FALLAS -Entrepiso débil -Entrepiso blando -Poco confinamiento en columnas de Concreto armado. -Ignorar aporte de rigidez de elementos no estructurales.

-Fallas a flexion o corte de elementos principales resistentes a sismo -Mal detallado de nodos y conexiones viga-columna


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-Irregularidades en planta y elevación.

CONCEPTOS DE DISEÑO SÍSMICO ESTADOS LÍMITES DE DISEÑO SÍSMICO -ESTADO LÍMITE DE SERVICIO Diseño para sismos frecuentes (período de retorno 50 años). Protección de edificios importantes como hospitales, estaciones de bombero, etc. Se limita el daño para que no afecte el funcionamiento del edificio. Se resiste en rango elástico. -ESTADO LÍMITE DE CONTROL DE DAÑO

Representa el límite entre daños reparables y no reparables. Probabilidad baja de ocurrencia en vida útil del edificio. Se espera fluencia del acero, grietas y desconchamiento del concreto. -ESTADO LÍMITE DE SUPERVIVENCIA


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Pérdidas humanas deben prevenirse inclusive para los sismos mas Fuertes. Ocurren daños irreparables pero nunca el colapso.

CONCEPTOS DE DISEÑO SÍSMICO


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ESTADOS LÍMITES DE DISEÑO SÍSMICO

CONCEPTOS DE DISEÑO SÍSMICO RIGIDEZ, RESISTENCIA Y DUCTILIDAD -Importante chequear derives -Relaciona las cargas con las deformaciones -Dependiente de E, G y geometría. -Resistencia Sy determinada por diseñador.

-El límite de ductilidad corresponde a determinada degradación de Resistencia. -Falla frágil: agotamiento de Resistencia sin ninguna advertencia -Falla dúctil: no implica colapso structural.


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-Ductilidad requiere atención en el detallado

CONCEPTOS DE DISEÑO SÍSMICO RÓTULAS PLÁSTICAS -Cuando se alcanza el momento plástico, la sección no tiene mas rigidez de “reserva” para incrementar el momento flexionante -Alcanzamos la rótula plástica, ya que se comporta y gira como una rótula o rodillo


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-Momentos adicionales son transmitidos al resto de la estructura

CONCEPTOS DE DISEÑO SÍSMICO CONFIGURACION ESTRUCTURAL -Edificio no debería ser muy pesado -La estructura debería ser sencilla y simétrica, en planta y elevación

-Debería tener una distribución uniforme de peso, rigidez, resistenca y ductilidad. -La estructura debería tener la mayor cantidad de lineas resistentes posibles. -Estructura redundante e hiperestática. -Elementos no estructurales deberían estar unidos o separados adecuadamentes. Entre mas rígida o mas resistenta la estructura, menos influyen los elementos no estructurales.


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-Simetría, simplicidad, redundancia y regularidad.



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REGULARIDAD EN PLANTA



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TORSION



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REGULARIDAD EN ELEVACIÓN



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4. Sismos y Diseno Sismo Resistente

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