4. Sismos y Diseno Sismo Resistente

Tema 4

SISMOS Y DISENO SISMO RESISTENTE

Proyecto Estructural - Prof. Michele Casarin 1

INDICE

1. RIESGO SISMICO

2. SISMOLOGIA

3. EFECTOS SISMICOS

4. DINAMICA DE ESTRUCTURAS

5. ESPECTRO DE RESPUESTA Y DISENO

6. SISTEMAS DE VARIOS GRADOS DE LIBERTAD

7. CONCEPTOS DE DISEÑO

8. COVENIN 1756

2Proyecto Estructural - Prof. Michele Casarin

RIESGO SISMICO


RIESGO SISMICO


La historia sismica de nuestro pais revela

que a lo largo del periodo 1530-2002 han

ocurrido mas de 137 eventos sismicos

que han causado algun tipo de dano 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.

-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 SOCIO-

ECONOMICOS 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?

CUAL ES LA SISMICIDAD DE CADA REGION?

SISMOLOGIA


LA TIERRA

SISMOLOGIA


LA TIERRA

02 - Seismology 14/03/2012

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Maria Gabriella Mulas

• The Lithosphere is more firm/ rigid compared to the soft

Mantel.

• The hot Inner core consists of hot soft rock compared

with the cooler rigid rock of the lithosphere. Hence the

inner core drives convection current to the surface

• Tectonic plates are driven by the convective motion of

the material in the earth’s mantle, which in turn is driven

by the heat generated at the earth’s core

FAULTS MOVEMENT

13 Maria Gabriella Mulas

TECTONIC PLATES

14

Earthquake epicenters 1963-2000

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

02 - Seismology 14/03/2012

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TECTONIC PLATES

15

Plate motions can be measured using Very Long Baseline Interferometry

(VLBI) or using the Global Positioning System (GPS)

How fast do the plates move?




GLOBAL DISTRIBUTION OF EARTHQUAKES

16

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TECTONIC PLATES

15

Plate motions can be measured using Very Long Baseline Interferometry

(VLBI) or using the Global Positioning System (GPS)

How fast do the plates move?




GLOBAL DISTRIBUTION OF EARTHQUAKES

16

SISMOLOGIA


ONDAS SISMICAS

02 - Seismology 14/03/2012

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TECTONIC PLATES


TECTONIC PLATES

20

Earthquakes at

divergent and

transform plate

margins have shallow

focuses

Earthquakes at

transform margins

have higher

magnitudes – some of

the highest measured

Most earthquakes occur at lithospheric plate boundaries,

where stresses are produced by plate motion

02 - Seismology 14/03/2012

11


FAULTS AND EARTHQUAKES

21

Faults may range in length from a few millimeters to thousands

of kilometers.

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 dip-

slip faults and described as either normal or reverse, depending

on their motion.

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

as oblique-slip faults.

Usually Tsunamis are created by faults which show dip-slip

and oblique motion.


CAUSES OF EARTHQUAKES

22

Stresses build

up in the crust,

usually due to

lithospheric plate

motions

Rock deform

(strain) as the

result of stress.

The strain is

energy stored in

the rocks.

02 - Seismology 14/03/2012

12


FAULTS AND EARTHQUAKE

23

Relative plate motion at the fault interface is constrained by friction and/or

asperities (areas of interlocking due to protrusions in the fault surfaces). However,

strain energy accumulates in the plates, eventually overcomes any resistance, and

causes slip between the two sides of the fault. This sudden slip, termed elastic

rebound releases large amounts of energy, which constitutes or is the earthquake.


Typically, someone will build a straight reference line such as a road, railroad, pole line,

or fence line across the fault while it is in the pre-rupture stressed state. After the

earthquake, the formerly straight line is distorted into a shape having increasing

displacement near the fault, a process known as elastic rebound.


24

SISMOLOGIA


ONDAS SISMICAS

02 - Seismology 14/03/2012

13



25

When an earthquake fault ruptures, it causes two types of deformation:

static; and dynamic. Static deformation is the permanent displacement

of the ground due to the event. The earthquake cycle progresses from

a fault that is not under stress, to a stressed fault as the plate tectonic

motions driving the fault slowly proceed, to rupture during an

earthquake and a newly-relaxed but deformed state.

Seismic Deformation



26

Like most stories in geology, this one starts

beneath the surface. The continents we live on

are parts of moving plates. Most of the action

takes place where plates meet. Plates may

collide, pull apart, or scrape past each other.

All the stress and strain produced by moving

plates builds up in the Earth’s rocky crust until it

simply can't take it any more. All at once,

CRACK!, the rock breaks and the two rocky

blocks move in opposite directions along a more

or less planar fracture surface called a fault.

The sudden movement generates an earthquake

at a point called the focus. The energy from the

earthquake spreads out as seismic waves in all

directions. The epicenter of the earthquake is the

location where seismic waves reach the surface

directly above the focus.

“hypocenter” is

another name

for the focus

02 - Seismology 14/03/2012

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Maria Gabriella Mulas 27

FAULT AND EARTHQUAKE

Normal fault

We classify faults by how the two rocky blocks on either side of a fault

move relative to each other. The one you see here is a normal fault. A

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

movement. the block farthest to the right that looks kind of like a foot is

the foot wall. The block on the other side of the fault is resting or

hanging on top of the foot wall block and is named hanging wall.

If we hold the foot wall stationary, gravity will normally want to pull the

hanging wall down. Faults that move the way you would expect gravity to

move them normally are called normal faults!

Where the fault has ruptured the Earth surface, that movement along the

fault has produced an elongate fault generated cliff called fault scarp.

foot wall

hanging wall



28

Reverse fault

The fault you see here is a reverse fault. Along a reverse fault one

rocky block is pushed up relative to rock on the other side.

Here’s a way to tell a reverse fault from a normal fault. Take a look at

the side that shows the fault and arrows indicating movement. The

block farthest to the right that is the foot wall. The block on the other

side of the fault is the hanging wall.

If we hold the foot wall stationary, where would the hanging wall go if

we reversed gravity? The hanging wall will slide upwards. When

movement along a fault is the reverse of what you would expect with

normal gravity we call them reverse faults!

FALLA NORMAL

02 - Seismology 14/03/2012

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Normal fault

We classify faults by how the two rocky blocks on either side of a fault

move relative to each other. The one you see here is a normal fault. A

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

movement. the block farthest to the right that looks kind of like a foot is

the foot wall. The block on the other side of the fault is resting or

hanging on top of the foot wall block and is named hanging wall.

If we hold the foot wall stationary, gravity will normally want to pull the

hanging wall down. Faults that move the way you would expect gravity to

move them normally are called normal faults!

Where the fault has ruptured the Earth surface, that movement along the

fault has produced an elongate fault generated cliff called fault scarp.

foot wall

hanging wall



28

Reverse fault

The fault you see here is a reverse fault. Along a reverse fault one

rocky block is pushed up relative to rock on the other side.

Here’s a way to tell a reverse fault from a normal fault. Take a look at

the side that shows the fault and arrows indicating movement. The

block farthest to the right that is the foot wall. The block on the other

side of the fault is the hanging wall.

If we hold the foot wall stationary, where would the hanging wall go if

we reversed gravity? The hanging wall will slide upwards. When

movement along a fault is the reverse of what you would expect with

normal gravity we call them reverse faults!

FALLA REVERSA

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Strike-slip fault

Strike-slip faults have a different type of movement than normal and reverse

faults. You probably noticed that the blocks that move on either side of a

reverse or normal fault slide up or down along a dipping fault surface.

The rocky blocks on either side of strike-slip faults, on the other hand, scrape

along side-by-side. You can see in the illustration that the movement is

horizontal and the rock layers beneath the surface haven't been moved up or

down on either side of the fault.

Take a look where the fault has ruptured the Earth surface. Notice that pure

strike-slip faults do not produce fault scarps.



Real-life

In real-life faulting is not such a

simple picture! Usually faults do not

have purely up-and-down or side-by-

side movement as we described

above. It’s much more common to

have some combination of fault

movements occurring together. For

example, along California’s famous

San Andreas strike-slip fault system,

about 95% of the movement is

strike-slip, but about 5% of the

movement is reverse faulting in

some areas!


FALLA STRIKE-SLIP

02 - Seismology 14/03/2012

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EARTHQUAKES

32

Earthquake focus: center of rupture or slip, seismic waves radiate out

from the focus

Earthquake epicenter – the point on the Earth’s surface over the focus

SISMOLOGIA


ONDAS SISMICAS

1. ONDAS P: 8 KM/S

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DESCRIPTION OF SEISMIC

WAVES


SEISMIC WAVES

40

P-waves – most rapid (8 km/sec)

S-waves – slower (5 km/sec), cannot move through liquids

Surface waves – even slower, move only on surface, most destructive

2. ONDAS S: 5 KM/S, NO SE

MUEVEN EN LIQUIDOS

3. ONDAS SUPERFICIALES: LAS MAS LENTAS, SOLO SE

TRANSMITEN EN LA SUPERFICIE. LAS MAS DESTRUCTIVAS

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SEISMIC WAVES


Surface Waves – seismic waves that tr avel along Ea rth’s surface, most destructive seismic waves

Surface waves travel along the ground and cause the ground and anything resting upon it to move

Body P waves – push-pull waves; they push (compress) and pull (expand) rocks in the direction the waves travel

Body S waves – shake the particles at right angles to their direction of travel

Gases and liquids do not transmit S waves, but do transmit P waves

A seismogram shows all three types of waves: the P waves arrive first, then the S waves, followed by the surface waves last

The waves arrive at different times because they travel at different speeds

SEISMIC WAVES

42

SISMOLOGIA


ONDAS SISMICAS

02 - Seismology 14/03/2012

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BODY WAVES


Velocity equations

density

µ shear modulus (rigidity)

k bulk modulus (rigidity)

because shear modulus (rigidity) for fluid is zero, S waves

cannot propagate through a fluid

consequence of equations is that P waves are 1.7x faster

than S

BODY WAVES

44

sV34 /k

VP

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BODY WAVES


Velocity equations

density

µ shear modulus (rigidity)

k bulk modulus (rigidity)

because shear modulus (rigidity) for fluid is zero, S waves

cannot propagate through a fluid

consequence of equations is that P waves are 1.7x faster

than S

BODY WAVES

44

sV34 /k

VP

03 - Fundamentals of Engineering Seismology 19 March 2012

7

Fundamentals of Engineering Seismology

13P and S wave propagation velocity

Representative values of propagaton velocity of P waves for crustal materials

4.0 - 6.5Crystalline rock

3.0 - 5.0Hard rock, dolomites

2.0 - 3.0Soft rock, dense gravel

0.5 - 2.0 (*)Alluvial material (clay, sand, silt)

(km/ s)Material

(*) lower values are for dry alluvial sediments (above water table)

Representative values of propagaton velocity of S waves for crustal materials

700 – 1500Weathered rock

500 - 1000Soft rock

400 – 800gravel

2500 - 3500Hard rock (crystalline)

200 – 400Medium to dense sand

150 - 300Normally consolidated clay and silt

40 - 80Very soft clays (Mexico city)

(m/ s)Material


14P- and S- wave propagation velocity

2)21(

)1(222

Ratio between P and S wave propagation velocity

For = 0.25, = 3

typically used in engineering

seismology

SISMOLOGIA


ONDAS SISMICAS

02 - Seismology 14/03/2012

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SURFACE WAVES

45

SU

RF

AC

E W

AV

ES


SURFACE WAVES

46

Love waves travel faster than Rayleigh waves and therefore arrive earlier

Love waves

Rayleigh waves

02 - Seismology 14/03/2012

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SURFACE WAVES

45

SU

RF

AC

E W

AV

ES


SURFACE WAVES

46

Love waves travel faster than Rayleigh waves and therefore arrive earlier

Love waves

Rayleigh waves

SISMOLOGIA


MEDICION DE ONDAS SISMICAS

SISMOGRAFOS

02 - Seismology 14/03/2012

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INSTRUMENTS THAT RECORD

EARTHQUAKE WAVES

49

Seismometers:

•The paper roll moves with the ground

•The pen remains stationary, because of the spring, hinge and weight Maria Gabriella Mulas

SISMOGRAM

50

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

02 - Seismology 14/03/2012

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INSTRUMENTS THAT RECORD

EARTHQUAKE WAVES

49

Seismometers:

•The paper roll moves with the ground

•The pen remains stationary, because of the spring, hinge and weight


SISMOGRAM

50

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

SISMOGRAMAS

02 - Seismology 14/03/2012

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We can determine the distance to an epicenter by finding the difference between the arrival of P waves and S waves. Looking at a travel-time graph we can determine how far away the epicenter is

Travel-time graphs from three or more seismographs can be used to find the exact location of an earthquake epicenter

SISMOGRAM


Distance - Time Relations

SISMOGRAM

54

SISMOLOGIA


MEDICION DE SISMOS

RITCHER SCALE

MOMENTO DE MAGNITUD

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MEASUREMENTS OF EARTHQUAKES



70

02 - Seismology 14/03/2012

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70

ENERGIA SISMICA

SISMOLOGIA


MEDICION DE SISMOS

02 - Seismology 14/03/2012

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Typically, the ground motion records, termed seismographs or time histories, have recorded acceleration (these records are termed accelerograms)

Time histories theoretically contain complete information about the motion at the instrumental location

Time histories (i.e., the earthquake motion at the site) can differ dramatically in duration, frequency content, and amplitude.

The maximum amplitude of recorded acceleration is termed the peak ground acceleration, PGA



Some Notable Earthquakes

76

Indonesia (12/04)

Pakistan

(10/05)

SISMOLOGIA


PROPAGACION DE ONDAS

02 - Seismology 14/03/2012

42


SEISMIC RISK: determination of ground motions

having the required probability of exceedance

83

-FUENTE (TAMANO Y TIPO)

-CAMINO (DISTANCIA Y TIPO DE

SUELO)

-EFECTOS DEL SITIO: TIPO DE

SUELO

SISMOLOGIA


EL CASO DE CIUDAD DE MEXICO

03 - Fundamentals of Engineering Seismology 19 March 2012

15


29

The case of the Mexico city during the Sept

19th 1985 Michoacánearthquake (magnitude=8.2; R ~ 400 km)

Effects of geological irregularities on earthquake

ground motion

Heavy damage and collapse of 10-14 storey buildings


30

The case of the Mexico city during the Sept

19th 1985 Michoacánearthquake

VS profile at SCT and CDAO stations

Effects of geological irregularities on earthquake

ground motion

EFECTOS SISMICOS


Earthquake Damage - Part I 4/22/2012

2


Size of the earthquake - 2

Engineers to not design structures on the base of magnitude, but on the

peak ground acceleration and displacement at the site.

Attenuation curves relate the peak ground acceleration to the magnitude

of the earthquake with the distance from the fault rupture.

Seismic hazard map: the contour lines provide the peak acceleration

based on attenuation curves.

5 Maria Gabriella Mulas 6

Structural damage - 1

Structural damage does not usually occur for M < 5.0

Most damage is caused by strong shaking and/or failure

of the surrounding soil

Damage can result also from surface ruptures, failure of

nearby lifelines or failure of more vulnerable structures.

These are usually secondary effects but can become

predominant in some cases (1999, JiJi Earthquake in

Taywan)

See report ji-ji_chap9.pdf page 11 and 12



Damage State Functionality Repairs Required Expected outage

None (pre-yield) (1) No loss None None

Minor/slight (2) Slight loss Inspect, adjust, patch < 3 days

Moderate (3) Some loss Repair components < 3 weeks

Major/extensive (4) Considerable loss Rebuild components < 3 months

Complete/collapse (5) Total loss Rebuild structure > 3 months

Damage can mean anything from minor cracks to total

collapse: we need to specify categories of damage

Levels of damage can be adopted in design to guarantee a

level of performance.



Most engineered structures are designed to prevent

collapse only: structures must have sufficient ductility to

survive an earthquake.

This means that elements will yield and deform, but they

will be strong in shear and continue to support their load

during and after the earthquakes

Common types of damage

EFECTOS SISMICOS


LICUEFACCION


3


DAMAGE AS A RESULT OF

SOIL PROBLEMS


Soil liquefaction - 1

When loose saturated sands, silts or gravel are shaken,

the material consolidates, reducing the porosity and

increasing water pressure.

The ground settles, often unevenly, tilting and toppling

structures that were formerly supported by the soil.

The buildings – with little damage - fell as the liquefied

soil lost its ability to support them; collapse can take

place hours after earthquake


Soil liquefaction - 2

• Structures supported on liquefied soil topple

• Structures that retain liquefied soil are pushed forward

• Structures buried in liquefied soil (as tunnel or culverts)

float to the surface

• A culvert is a device used to channel water. It may be used to allow

water to pass underneath a road, railway, or embankment for

example


Mechanism of soil liquefaction

http://nisee.berkeley.edu/bertero/html/damage_due_to_liquefaction.html

ji-ji_chap8.pdf pag. 7-10 (figs. 8.6-8.18)

The weight of the building squeezes the adjacent soil (Courtesy of Prof. Hugo Bachmann)


4


Damage due to soil liquefaction – 1

Izmit, Turkey 1999



Adapazari, Turkey 1999

Kobe, Japan 1995


1964 Nilgata, Japan




Sand boils and ground

fissures provide

evidence of liquefaction


4



Izmit, Turkey 1999




Kobe, Japan 1995


1964 Nilgata, Japan





fissures provide



4



Izmit, Turkey 1999




Kobe, Japan 1995


1964 Nilgata, Japan





fissures provide


EFECTOS SISMICOS


DESLIZAMIENTOS


5


Landslides - 1

When a steeply inclined mass of soil is suddenly shaken, a

slip lane can form and the material slides downhill.


Landslides - 2

18

Structures sitting on the

slide move downward

Structures below the

slide are hitten by falling

debris

Before

After


Landslides - 3

ji-ji_chap8.pdf photo 8.1-8.5


Landslide of Turnagain Heights

Anchorage, Alaska 1964

20


5


Landslides - 1




Landslides - 2

18


slide move downward



debris

Before

After


Landslides - 3





20

EFECTOS SISMICOS


DESLIZAMIENTOS


5


Landslides - 1




Landslides - 2

18


slide move downward



debris

Before

After


Landslides - 3





20


5


Landslides - 1




Landslides - 2

18


slide move downward



debris

Before

After


Landslides - 3





20

EFECTOS SISMICOS


RUPTURA DEL SUELO


6


1906 Olema, CA

Ground rupture - 1


Ground rupture - 2


Ground rupture - 3

Japan earthquake, March 3,11

Kanto Highway, repaired in

one week


Ground motion

24

Guatemala earthquake, 1976

Rails bent in Gualan


6


1906 Olema, CA

Ground rupture - 1


Ground rupture - 2


Ground rupture - 3

Japan earthquake, March 3,11

Kanto Highway, repaired in

one week


Ground motion

24

Guatemala earthquake, 1976

Rails bent in Gualan

EFECTOS SISMICOS


ARCILLAS DEBILES


8


Weak clay - Struve Slough Bridge - 1

29

The soil pushed against

the piles, breaking their

connection with the

superstructure, and

pushing them away

from the cap beam

Piles were

dragged by

the soil



30

Piles “punctured”

the bridge

Shear damage was

found at the top of

the piles


Weak clay – Cypress Viaduct - 1

31

During the Loma Prieta

1999 earthquake the

upper deck of Cypress

Viaduct collapsed in two

regions

The collapse was the

result of the weak pin

connections at the base of

the columns of the upper

frame

The soft bay mud was

sensitive to long period

motion and caused large

motions that overstressed

the pinned connection



32


8



29

The soil pushed against

the piles, breaking their

connection with the

superstructure, and

pushing them away

from the cap beam

Piles were

dragged by

the soil



30

Piles “punctured”

the bridge

Shear damage was

found at the top of

the piles



31

During the Loma Prieta

1999 earthquake the

upper deck of Cypress

Viaduct collapsed in two

regions

The collapse was the

result of the weak pin

connections at the base of

the columns of the upper

frame

The soft bay mud was

sensitive to long period

motion and caused large

motions that overstressed

the pinned connection



32

EFECTOS SISMICOS


ARCILLAS DEBILES


9



33

Bent reinforcement bar in

failed support column

Inadequate confinement


Weak clay – Mexico City 1985 earthquake

Mexico City was located 350 Km from the epicenter of the magnitude 8.1

earthquake, but the city is underlain by an old lake bed composed by

soft silts and clays.

34



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

medium height buildings (10-14 stories) were damaged or collapsed during

the earthquake.



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

medium height buildings (10-14 stories) were damaged or collapsed during

the earthquake.

36

EFECTOS SISMICOS


TSUNAMI


10


Tsunami

•very long wavelength, deep wavebase

•speeds up to 800 km/hour, 20 meters high


Tsunami – wave propagation times


Tsunami - 1964 Alaska Earthquake


Japan earthquake of March 11, 2011

40

http://www.corriere.it/esteri/11_marzo_18/Il-disastro-del-

Giappone_d8540410-5159-11e0-b0a4-77b20470b36e.shtml?fr=correlati


10


Tsunami

•very long wavelength, deep wavebase

•speeds up to 800 km/hour, 20 meters high


Tsunami – wave propagation times


Tsunami - 1964 Alaska Earthquake


Japan earthquake of March 11, 2011

40

http://www.corriere.it/esteri/11_marzo_18/Il-disastro-del-

Giappone_d8540410-5159-11e0-b0a4-77b20470b36e.shtml?fr=correlati

EFECTOS SISMICOS


TSUNAMI

EFECTOS SISMICOS


FALLAS EN CONEXIONES CON LAS FUNDACIONES

Earthquake Damage - Part II 4/22/2012

2
























STRUCTURAL PROBLEMS

9


Ground motion


Foundation failure - 1

11

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

When the foundation is too small, it can

become unstable and rock over



12

Overturning due to

foundation failure

We have seen pile damage due to weak clay on the Struve

Slough bridge (1989 Loma Prieta).

As long as the foundation is embedded in good material, it

usually has ample strength and ductility to survive large

earthquakes.


2
























STRUCTURAL PROBLEMS

9


Ground motion



11

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

When the foundation is too small, it can

become unstable and rock over



12

Overturning due to

foundation failure

We have seen pile damage due to weak clay on the Struve

Slough bridge (1989 Loma Prieta).

As long as the foundation is embedded in good material, it

usually has ample strength and ductility to survive large

earthquakes.


3


Foundation connection

Houses need to be

anchored to the

foundation with hold-

downs connected to the

stud walls and anchor

bolts connected to the

sill plates.

Otherwise, the house

will fall off its foundation

Timber structure


Foundation connection – timber structure

Hold-down



San Fernando,

California,

Earthquake February

1971.

Collapsed highway

overpass,

INTERSTATE 5 and

14 interchange.

The interchange was

built on consolidated

sand



San Fernando,

California,

Earthquake February

1971.

Collapsed highway

overpass,

INTERSTATE 5 and

14 interchange.



The major cause of damage to electrical transformers, storage bins,

and a variety of other structures is the lack of secure connection to

the foundation

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


Foundation connection (special structures)

EFECTOS SISMICOS


FALLAS EN CONEXIONES CON LAS FUNDACIONES


3



Houses need to be

anchored to the





sill plates.



Timber structure



Hold-down



San Fernando,

California,

Earthquake February

1971.

Collapsed highway

overpass,

INTERSTATE 5 and

14 interchange.

The interchange was


sand



San Fernando,

California,

Earthquake February

1971.

Collapsed highway

overpass,

INTERSTATE 5 and

14 interchange.





the foundation

Pull-out of column


foundation





footings







3



Houses need to be

anchored to the





sill plates.



Timber structure



Hold-down



San Fernando,

California,

Earthquake February

1971.

Collapsed highway

overpass,

INTERSTATE 5 and

14 interchange.

The interchange was


sand



San Fernando,

California,

Earthquake February

1971.

Collapsed highway

overpass,

INTERSTATE 5 and

14 interchange.





the foundation

Pull-out of column


foundation





footings






EFECTOS SISMICOS


FALLAS POR ENTREPISO DEBIL “SOFT STORY”


4


Soft story

19

Loma Prieta earthquake damage in San Francisco. The soft

first story is due to construction of garages in the first story

and resultant reduction in shear strength


Soft story


Soft Story, L’Aquila 2009

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



Same situation

also in this side



Same situation

also in this side!



Intermediate soft-story in a 3-story house


4


Soft story

19





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



Same situation

also in this side



Same situation

also in this side!




EFECTOS SISMICOS




4


Soft story

19





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



Same situation

also in this side



Same situation

also in this side!




EFECTOS SISMICOS




5




Soft story at mid level

26

During the Kobe earthquake, many tall buildings had damage

at the midstory, often due to designing the upper floors for a

reduced seismic load

Most buildings in Japan are either built of RC or of SRC (steel

and reinforced concrete). However, the design practice was to

discontinue either the RC or the SRC above a certain floor



27

10-story SRC

building with 3rd

floor collapse



28

Mid story collapse,

Kobe earthquake



29

Mid story collapse,

Kobe earthquake


Torsional moments

Curved, skewed and eccentrically supported structure often experience

torsion during earthquakes

30

Nine-story building in Kobe, Japan

Shear walls on 3 sides, a moment resisting frame on the 4th

The 1995 earthquake caused a torsional moment on the

building. The first-story column on the east side failed, the

building leaned to east and eventually collapsed.

EFECTOS SISMICOS




5





26









27

10-story SRC

building with 3rd

floor collapse



28

Mid story collapse,

Kobe earthquake



29

Mid story collapse,

Kobe earthquake


Torsional moments



30






EFECTOS SISMICOS




5





26









27

10-story SRC

building with 3rd

floor collapse



28

Mid story collapse,

Kobe earthquake



29

Mid story collapse,

Kobe earthquake


Torsional moments



30







5





26









27

10-story SRC

building with 3rd

floor collapse



28

Mid story collapse,

Kobe earthquake



29

Mid story collapse,

Kobe earthquake


Torsional moments



30






EFECTOS SISMICOS


FALLAS MOMENTOS TORSIONALES


6


Torsional moments


Torsional moments

32

Torsional failure of the

top of the column


Pounding - 1

Collisions between adjacent structures due to insufficient separation

gaps have been witnessed in almost every major earthquake since

the 1960’s.

Earthquake-induced pounding between inadequately separated

structures may cause considerable damage or even lead to a

structure’s total collapse.

33


Pounding - 2


Shear

Most building structures use shear walls or moment resisting

frames to resist lateral forces during earthquakes.

Damage to these systems may vary from minor cracks to

complete collapse.

Shear damage is often related to insufficient transverse

reinforcement


Shear failure – Mt McKinley Apartments

Great Alaska Earthquake, 1964

36

Mt. Mc Kinley Apartments: 14-story RC building

composed of narrow exterior shear walls and spandrel

beams, as well as exterior and interior columns and a

central Tower.


5





26









27

10-story SRC

building with 3rd

floor collapse



28

Mid story collapse,

Kobe earthquake



29

Mid story collapse,

Kobe earthquake


Torsional moments



30






EFECTOS SISMICOS




6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.

EFECTOS SISMICOS




6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.


6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.


6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.


6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.

EFECTOS SISMICOS


FALLAS POR CORTE


6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.


6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.


6


Torsional moments


Torsional moments

32


top of the column


Pounding - 1



the 1960’s.




33


Pounding - 2


Shear




complete collapse.


reinforcement




36




central Tower.

EFECTOS SISMICOS


FALLAS POR CORTE


7




37

The spandrel beam between

the wall had large X cracks

associated with shear

damage as the building

moved back and forth

A wide shear crack

split the wall in two,

directly below a

horizontal beam.


Shear failure


Shear failure (short column)



L’Aquila 2009



L’Aquila 2009


Shear damage (L’Aquila 2009)


7




37






A wide shear crack


directly below a

horizontal beam.


Shear failure





L’Aquila 2009



L’Aquila 2009



EFECTOS SISMICOS


FALLAS POR CORTE (COLUMNA CORTA)


7




37






A wide shear crack


directly below a

horizontal beam.


Shear failure





L’Aquila 2009



L’Aquila 2009




7




37






A wide shear crack


directly below a

horizontal beam.


Shear failure





L’Aquila 2009



L’Aquila 2009



EFECTOS SISMICOS


FALLAS POR CORTE


7




37






A wide shear crack


directly below a

horizontal beam.


Shear failure





L’Aquila 2009



L’Aquila 2009




7




37






A wide shear crack


directly below a

horizontal beam.


Shear failure





L’Aquila 2009



L’Aquila 2009




8




Flexural failure of columns


Flexural failure

Insufficient transverse reinforcement results in lack of confinement

for columns. This allows longitudinal reinforcement to buckle and

the concrete to fall off from the column.


Flexural failure

Kobe earthquake, flexural failure

Hanshin expressway


Plastic hinge at the base of a column


Column failure

Insufficient transversal reinforcement results in lack of

confinement for columns

EFECTOS SISMICOS


FALLAS POR FLEXION


8






Flexural failure





Flexural failure


Hanshin expressway




Column failure




8






Flexural failure





Flexural failure


Hanshin expressway




Column failure



EFECTOS SISMICOS


FALLAS EN NODOS


9


Failure of beam-column node


Test on a beam-column node


9


Failure of beam-column node


Test on a beam-column node

DINAMICA DE ESTRUCTURAS


MECANICA

-LAS LEYES 1 Y 3 SON SUFICIENTE PARA ESTUDIAR CUERPOS ESTATICOS O EN MOVIMIENTOS

SIN ACELERACION

-CUANDO UN CUERPO SE ACELERA SE REQUIERE LA 2nda LEY DE NEWTON, PARA RELACIONAR EL

MOVIMIENTO CON LAS FUERZAS ACTUANTES.

-2nda LEY DE NEWTON: F= M x A

Elements of Mechanics of Particles

11

Element of Mechanics of Particles

Dynamic Equilibrium – D’Alembert’s

Principle

12 - 41

• Alternate expression of Ne wt on ’s second law,

ectorinertial vam

amF

0

• With the inclusion of the inertial vector, the system

of forces acting on the particle is equivalent to

zero. The particle is in dynamic equilibrium.

• Methods developed for particles in static

equilibrium may be applied, e.g., coplanar forces

may be represented with a closed vector polygon.

• Inertia vectors are often called inertial forces as

they measure the resistance that particles offer to

changes in motion, i.e., changes in speed or

direction.

• Inertial forces may be conceptually useful but are

not like the contact and gravitational forces found

in statics.


KINETICS OF PARTICLES

ENERGY AND MOMENTUM

METHOD

42


Introduction

13 - 43

• Previously, problems dealing with the motion of particles were

solved through the fundamental equation of motion,

Current chapter introduces two additional methods of analysis.

.amF

• Method of work and energy: directly relates force, mass,

velocity and displacement.

• Method of impulse and momentum: directly relates force,

mass, velocity, and time.


Work of a Force

13 - 44

• Differential vector is the particle displacement. rd

• Work of the for ce is

dzFdyFdxF

dsF

rdFdU

zyx

cos

• Work is a scalar quantity, i.e., it has magnitude and

sign but not direction.

force. length • Dimensions of work are Units are

J 1.356lb1ftm 1N 1 J 1 joule

-LEYES DE TRABAJO. RESORTES.


12


Work of a Force

13 - 45

• Work of a force du ring a finite displacement,

2

1

2

1

2

1

2

1

cos

21

A

Azyx

s

st

s

s

A

A

dzFdyFdxF

dsFdsF

rdFU

• Work is represented by the area under the

curve of Ft plotted against s.


Work of a Force

13 - 46

• Work of a constant force in rectilinear motion,

xFU cos21

• Work of the force of gr avity,

yWyyW

dyWU

dyW

dzFdyFdxFdU

y

y

zyx

12

21

2

1

• Work of the weight is equal to product of

weight W and vertical displacement y.

• Work of the weight is positive when y < 0,

i.e., when the weight moves down.


Work of a Force

13 - 47

• Magnitude of the force exerted by a spring is

proportional to deflection,

lb/in.or N/mconstant spring k

kxF

• Work of the force exerted by spring,

222

1212

121

2

1

kxkxdxkxU

dxkxdxFdU

x

x

• Work of the force exerted by spring is positive

when x2 < x1, i.e., when the spring is returning to

its undeformed position.

• Work of the force exe rted by the spring is equal to

negative of area under curve of F plotted against x,

xFFU 2121

21


Work of a Force

13 - 48

Forces which do not do work (ds = 0 or cos 0 :

• weight of a body when its center of gravity moves

horizontally.

• reaction at a roller moving along its track, and

• reaction at frictionless surface when body in contact

moves along surface,

• reaction at frictionless pin supporting rotating body,



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.

-CUANDO NO SE CONSIDERA EL AMORTIGUAMIENTO DEL SISTEMA, SE LE LLAMA SISTEMA NO

AMORTIGUADO. TODAS LAS VIBRACIONES SON AMORTIGUADAS HASTA CIERTO PUNTO.



VIBRACIONES DE PARTICULAS

-SI UNA PARTICULA ES DESPLAZADA DE SU PUNTO DE EQUILIBRIO Y SOLTADA SIN VELOCIDAD, LA

PARTICULA ENTRARA EN UN MOVIMIENTO HARMONICO SIMPLE.


13


Particle Kinetic Energy: Principle of Work &

Energy

13 - 49

dvmvdsF

ds

dvmv

dt

ds

ds

dvm

dt

dvmmaF

t

tt

• Consider a particle of mass m acted upon by force F

• Integrating from A1 to A2 ,

energykineticmvTTTU

mvmvdvvmdsFv

v

s

st

221

1221

212

1222

12

1

2

1

• The work of the force is equal to the change in

kinetic energy of the particle.

F

• Units of work and kinetic energy are the same:

JmNms

mkg

s

mkg

2

22

21 mvT


MECHANICAL VIBRATIONS

OF A PARTICLE

50


Introduction

51

• Mechanical vibration is the motion of a particle or body which

oscillates about a position of equilibrium. Most vibrations in

machines and structures are undesirable due to increased stresses

and energy losses.

• Time interval required for a system to complete a full cycle of the

motion is the period of the vibration.

• Number of cycles per unit time defines the frequency of the vibrations.

• Maximum displacement of the system from the equilibrium position is

the amplitude of the vibration.

• When the motion is maintained by the restoring forces only, the

vibration is described as free vibration. When a periodic force is applied

to the system, the motion is described as forced vibration.

• When the frictional dissipation of energy is neglected, the motion

is said to be undamped. Actually, all vibrations are damped to

some degree.


Free Vibrations of Particles. Simple

Harmonic Motion

52

• If a particle is displaced through a distance xm from its

equilibrium position and released with no velocity, the

particle will undergo simple harmonic motion ,

0kxxm

kxxkWFma st

• General solution is the sum of two particular solutions,

tCtC

tm

kCt

m

kCx

nn cossin

cossin

21

21

• x is a periodic function and n is the natural circular

frequency of the motion.

• C1 and C2 are determined by the initial conditions:

tCtCx nn cossin 21 02 xC

nvC 01tCtCxv nnnn sincos 21


13


Particle Kinetic Energy: Principle of Work &

Energy

13 - 49

dvmvdsF

ds

dvmv

dt

ds

ds

dvm

dt

dvmmaF

t

tt

• Consider a particle of mass m acted upon by force F

• Integrating from A1 to A2 ,

energykineticmvTTTU

mvmvdvvmdsFv

v

s

st

2

21

1221

212

1222

12

1

2

1

• The work of the force is equal to the change in

kinetic energy of the particle.

F

• Units of work and kinetic energy are the same:

JmNms

mkg

s

mkg

2

22

21 mvT


MECHANICAL VIBRATIONS

OF A PARTICLE

50


Introduction

51

• Mechanical vibration is the motion of a particle or body which

oscillates about a position of equilibrium. Most vibrations in

machines and structures are undesirable due to increased stresses

and energy losses.

• Time interval required for a system to complete a full cycle of the

motion is the period of the vibration.

• Number of cycles per unit time defines the frequency of the vibrations.

• Maximum displacement of the system from the equilibrium position is

the amplitude of the vibration.

• When the motion is maintained by the restoring forces only, the

vibration is described as free vibration. When a periodic force is applied

to the system, the motion is described as forced vibration.

• When the frictional dissipation of energy is neglected, the motion

is said to be undamped. Actually, all vibrations are damped to

some degree.



Harmonic Motion

52

• If a particle is displaced through a distance xm from its

equilibrium position and released with no velocity, the

particle will undergo simple harmonic motion ,

0kxxm

kxxkWFma st

• General solution is the sum of two particular solutions,

tCtC

tm

kCt

m

kCx

nn cossin

cossin

21

21

• x is a periodic function and n is the natural circular

frequency of the motion.

• C1 and C2 are determined by the initial conditions:

tCtCx nn cossin 21 02 xC

nvC 01tCtCxv nnnn sincos 21



ESTRUCTURAS SIMPLES

-SI LA ESTRUCTURA SIMPLE ES DESPLAZADA Y SOLTADA, EMPEZARA A OSCILAR O VIBRAR CON

RESPECTO A SU POSICION INICIAL (VIBRACION LIBRE)

Dynamics of Structures: an introduction 15/04/2012

4

Maria Gabriella Mulas, Paolo Martinelli

SINGLE DEGREE OF FREEDOM

SYSTEMS: EQUATION OF

MOTION

13 Maria Gabriella Mulas, Paolo Martinelli

ACKNOWLEDGEMENTS

The figures in the following slides come from:

Anil K. Chopra

Dynamics of Structures. Theory and Application to

Earthquake Engineering. 3rd edition

Pearson/ Prentice Hall 2007

14


SIMPLE STRUCTURES - 1

“Simple” because can b

e

idealized as a lumped mass m supported by a

massless structure with stiffness k in the lateral direction

Lateral motion is “small”: the structure deform within the elastic range



16

Pergola at the Macuto Sheraton Hotel damaged by the

earthquake of July 29, 1967 (Venezuela, Caracas)


4


SINGLE DEGREE OF FREEDOM

SYSTEMS: EQUATION OF

MOTION


ACKNOWLEDGEMENTS

The figures in the following slides come from:

Anil K. Chopra

Dynamics of Structures. Theory and Application to

Earthquake Engineering. 3rd edition


14



“Simple” because can b

e

idealized as a lumped mass m supported by a

massless structure with stiffness k in the lateral direction

Lateral motion is “small”: the structure deform within the elastic range



16

Pergola at the Macuto Sheraton Hotel damaged by the

earthquake of July 29, 1967 (Venezuela, Caracas)


5



If the idealized system is displaced and then released, it will

start to oscillate (or vibrate) back and forth about its initial

equilibrium position: FREE VIBRATION

In an ideal case, the structure will oscillate indefinitely,

without any energy dissipation: the kinetic energy will convert

in potential energy and viceversa.



18

Real case: two simple structural models, one of aluminum

and the other of plexiglass



Real case: the amplitude

of oscillations will decay

with time

The energy dissipating

mechanism, called

damping, must be

included in the structural

modeling.


Single-degree-of-freedom system – 1

Idealization of a 1-story structure: a mass m lumped at

the roof level, a massless frame providing stiffness, a

viscous damper that dissipates energy.

Two types of dynamic loading:

• external force in the lateral direction

• ground motion imposed at the base (earthquake)

20



ESTRUCTURAS SIMPLES

-LA AMPLITUD DE LOS DESPLAZAMIENTOS DISMINUYE CON EL TIEMPO GRACIAS AL

AMORTIGUAMIENTO, QUE ES UN MECANISMO DE DISIPACION DE ENERGIA QUE DEBE SER

INCLUIDO EN LOS CALCULOS.


5











18







with time


mechanism, called

damping, must be


modeling.









20


3


Discretization with lumped mass (and

forces) method

Hypothesis:

the “external” forces

(dynamic + inertia +

damping) are lumped forces

or are applied to rigid

bodies.

)(),...,(),(),( 21 tqtqtqtxv n

Degrees of freedom

(in a finite number)

DOFs


Lumped mass method:

derivation of equations of motion

The equations of motions are the

conditions of equilibrium of

rigid bodies that are subjected to

dynamic forces, to restoring

elastic forces, to inertia forces

(and to damping forces).

0)(1

n

i

jjij ij qmqktFQkqqm

Matrix form

10


Linear modeling of dissipating effects

Viscous damper

zcfD

Equations of motion

Qkqqcqm


SDOF linear system

v(t) displacement

m mass

k stiffness

f(t) dynamic external force

c viscous damping coefficient

Hp. Bending deformation only

Equation of motion

3h

EJ24k

)(tfkvvcvm

ktfmtfvv2v 2

1

2

11 /)(/)(

.m

k2

1km2

c

2

1

m

cv

1

Damping

ratio

12


5











18







with time


mechanism, called

damping, must be


modeling.









20



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.

-EXISTEN DOS TIPOS DE CARGAS DINAMICAS:

1) FUERZA LATERAL EXTERNA

2) DESPLAZAMIENTO DEL SUELO EN LA BASE (SISMO)





6


Single-degree-of-freedom system - 2

Degrees of freedom for dynamic analysis is the number of

independent parameters required to define the displaced

position of all the masses relative to their original position.

More DOFs are typically necessary to define the stiffness

properties of a structure compared to the DOFs necessary

for the dynamic analysis.


Force-displacement relation

A static external

force is applied

on roof,

balanced by an

internal

restoring force

The internal restoring

force depends on the

relative displacement u.

The fs-u relation can be

either linear or non

linear

22


Force-displacement relation: linearly elastic

systems - 1

For a linear system the relation between the lateral

force fs and the resulting displacement u is linear:

fs = ku

The resisting force is a single-valued function of u: the

system is elastic

k is the stiffness of the system: it represents the force

that must be applied to obtain an unit displacement



systems - 2

The lateral stiffness of this frame depends on both the

columns and beam stiffness. It will vary between the two

extreme values of infinite and null beam stiffness

= Ib / 4Ic

L= 2h

24


6











A static external

force is applied

on roof,

balanced by an

internal

restoring force






linear

22



systems - 1



fs = ku


system is elastic





systems - 2




= Ib / 4Ic

L= 2h

24


6











A static external

force is applied

on roof,

balanced by an

internal

restoring force






linear

22



systems - 1



fs = ku


system is elastic





systems - 2




= Ib / 4Ic

L= 2h

24


6











A static external

force is applied

on roof,

balanced by an

internal

restoring force






linear

22



systems - 1



fs = ku


system is elastic





systems - 2




= Ib / 4Ic

L= 2h

24


7



systems - 3

The stiffness matrix is computed with the standard method

of displacement method, by imposing an unit value to one

coordinate (dof) while the remaining are equal to zero.

The stiffness coefficients are the forces that are necessary

to maintain the system in equilibrium. They can be thought

as the reactions of additional constraints, inserted to

impose the desired values to the dofs.


Force-displacement relation: inelastic systems

For inelastic systems, the restoring force is no longer a

single valued function of the displacement/deformation:

We are interested in studying the dynamic response of

inelastic systems because almost all the structures are

designed with the expectation that they will evolve in the

nonlinear range (cracking, yielding, damage etc) during

the intense ground shaking caused by an earthquake.

u,uff ss

26

Maria Gabriella Mulas, Paolo Martinelli 27

Inelastic force-deformation relation: panel zone of

a steel welded beam-to-column connection

No strength degradation; no stiffness degradation

Stable cycles, large amount of dissipated energy Maria Gabriella Mulas, Paolo Martinelli

Damping force - 1

In damping, the energy of the vibrating system is

dissipated by various mechanisms: thermal 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.

We only consider linear viscous damper:

28


7



systems - 3

The stiffness matrix is computed with the standard method

of displacement method, by imposing an unit value to one

coordinate (dof) while the remaining are equal to zero.

The stiffness coefficients are the forces that are necessary

to maintain the system in equilibrium. They can be thought

as the reactions of additional constraints, inserted to

impose the desired values to the dofs.


Force-displacement relation: inelastic systems

For inelastic systems, the restoring force is no longer a

single valued function of the displacement/deformation:

We are interested in studying the dynamic response of

inelastic systems because almost all the structures are

designed with the expectation that they will evolve in the

nonlinear range (cracking, yielding, damage etc) during

the intense ground shaking caused by an earthquake.

u,uff ss

26

Maria Gabriella Mulas, Paolo Martinelli 27

Inelastic force-deformation relation: panel zone of

a steel welded beam-to-column connection


Stable cycles, large amount of dissipated energy Maria Gabriella Mulas, Paolo Martinelli

Damping force - 1

In damping, the energy of the vibrating system is

dissipated by various mechanisms: thermal 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.

We only consider linear viscous damper:

28


8


Damping force - 2

The linear viscous damper models the energy dissipation

at deformation amplitudes within the elastic limit of the

structure.

Additional energy is dissipated by the inelastic behavior of

materials: this phenomenon is known as hysteretic

damping, and is best modeled through the proper

modeling of the hysteretic relations of materials and/or

components in the inelastic range.

ucfD


Equation of motion: external force

Newton’s law

The idealized one-story frame is subjected to an external

dynamic force p(t) in the direction of u

Newton’s second law states that:

The equation can be rewritten as:

In the elastic range we obtain:

In the inelastic range :

30



D’Alembert’s principle



D’Alembert’s principle is based on the notion of a

fictitious “inertia force”, equal to the product of the mass

times its acceleration, and acting in a direction opposite

to the acceleration. With inertia force included, the

system is in equilibrium at each instant

31

umf I 0SDI ffftp


Stiffness, mass and damping components

Under the action of the external force p(t) the state of the

system is described by

We can visualize the system as the combination of three

pure components, the stiffness, damping and mass

components. The external force applied to the complete

system is distributed among the three components, and

the sum fs + fD + fI must equal the applied force p(t)

tuandtu,tu

kufs ucfD umf I

32




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

2nda LEY DE NEWTON:


8


Damping force - 2



structure.






ucfD



Newton’s law







30











31

umf I 0SDI ffftp










tuandtu,tu

kufs ucfD umf I

32


8


Damping force - 2



structure.






ucfD



Newton’s law







30











31

umf I 0SDI ffftp










tuandtu,tu

kufs ucfD umf I

32


8


Damping force - 2



structure.






ucfD



Newton’s law







30











31

umf I 0SDI ffftp










tuandtu,tu

kufs ucfD umf I

32

ELASTICO:

INELASTICO:


8


Damping force - 2



structure.






ucfD



Newton’s law







30











31

umf I 0SDI ffftp










tuandtu,tu

kufs ucfD umf I

32





8


Damping force - 2



structure.






ucfD



Newton’s law







30











31

umf I 0SDI ffftp










tuandtu,tu

kufs ucfD umf I

32


8


Damping force - 2



structure.






ucfD



Newton’s law







30











31

umf I 0SDI ffftp










tuandtu,tu

kufs ucfD umf I

32


9


Mass – spring – damper system

33

This is the classical SDF system analyzed in textbooks on

mechanical vibration and elementary physics. We will refer

to this system to study free and forced (harmonic) vibrations

Newton’s law D’Alembert’s

Principle


Equation of motion: earthquake excitation

34

0SDI fff tutumtumf gt

I

tumkuucum g

tututu gt

ug ground displacement

u relative displacement

ut absolute displacement

tumu,ufucum gs



35

The system undergoing base motion can be analyzed as a

stationary base system, subjected to an effective force due

to the ground excitation.

The force is mass proportional.


PROBLEM STATEMENT AND

SOLUTION METHODS

36





9



33





Principle



34


I

tumkuucum g

tututu gt




tumu,ufucum gs



35







SOLUTION METHODS

36


9



33





Principle



34


I

tumkuucum g

tututu gt




tumu,ufucum gs



35







SOLUTION METHODS

36


9



33





Principle



34


I

tumkuucum g

tututu gt




tumu,ufucum gs



35







SOLUTION METHODS

36




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

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

internas.

-Luego que se calculo el desplazamiento de respuesta U(t) de la estructura, se calculan los esfuerzos internos para cada

instante de tiempo utilizando ANALISIS ESTATICOS: 1) Se le puede aplicar la deformacion a la estructura y hallar los

esfuerzos internos. 2) Se le puede aplicar la fuerza estatica equivalente P, la cual aplicada en dado momento debe resultar en

la misma derformacion U calculado anteriormente. 3) En sistemas inelasticos se deben hacer calculos paso a paso

incrementales.

-Para resolver la ecuacion diferencial de segundo grado se utiliza la integral de Duhamel, donde se representa la fuerza

externa como una secuencia de cortos impulsos infinitos.


11


Methods of solution of the differential equation

The equation of motion of a SDOF system subjected to

external force is a second order differential equation:

To define completely the problem we must assign the initial

conditions, that is displacement and velocity at time t=0.

Typically, the structure is at rest before the onset of the

dynamic excitation


Classical solution

The complete solution of the differential equation is the sum

of the complementary solution uc(t) and the particular

solution up(t)

Since the differential equation is of second order, two

constants of integration are involved

They appear in the complementary solution and are

evaluated from the initial conditions

42

tututu cp


Duhamel’s integral

43

In this approach the external force is represented as a

sequence of infinitesimally short impulses.

The response of the system to an applied force p(t) at

time t, is obtained by adding the responses to all the

impulses up to that time:

Implicit in this result are the “at rest” initial conditions.

We will use this method to compute the response to

earthquake excitation


Other methods not used in this course

Transform methods: based on Laplace and Fourier

transforms, they provide a powerful tool to solve differential

equations (we solve in the so called frequency domain,

opposed to time domain of the previous methods)

Numerical methods represent the practical approach to solve

differential equations of motion for non linear systems

They are based on an incremental form of the motion

equations: we do not find the function u(t) and its derivatives,

but only a discrete series of values of displacement, velocity

and acceleration.

44


11


Methods of solution of the differential equation

The equation of motion of a SDOF system subjected to

external force is a second order differential equation:

To define completely the problem we must assign the initial

conditions, that is displacement and velocity at time t=0.

Typically, the structure is at rest before the onset of the

dynamic excitation


Classical solution

The complete solution of the differential equation is the sum

of the complementary solution uc(t) and the particular

solution up(t)

Since the differential equation is of second order, two

constants of integration are involved

They appear in the complementary solution and are

evaluated from the initial conditions

42

tututu cp


Duhamel’s integral

43

In this approach the external force is represented as a

sequence of infinitesimally short impulses.

The response of the system to an applied force p(t) at

time t, is obtained by adding the responses to all the

impulses up to that time:

Implicit in this result are the “at rest” initial conditions.

We will use this method to compute the response to

earthquake excitation


Other methods not used in this course

Transform methods: based on Laplace and Fourier

transforms, they provide a powerful tool to solve differential

equations (we solve in the so called frequency domain,

opposed to time domain of the previous methods)

Numerical methods represent the practical approach to solve

differential equations of motion for non linear systems

They are based on an incremental form of the motion

equations: we do not find the function u(t) and its derivatives,

but only a discrete series of values of displacement, velocity

and acceleration.

44



VIBRACION LIBRE DE ESTRUCTURAS SIN AMORTIGUAMIENTO

Vibration of SDOF systems 15/04/2012

3



Harmonic Motion

Free vibrations of a 1-story undamped frame



Harmonic Motion

10

txx nm sin

• Velocity-time and acceleration-time curves can be

represented by sine curves of the same period as the

displacement-time curve but different phase angles.

2sin

cos

tx

tx

xv

nnm

nnm

tx

tx

xa

nnm

nnm

sin

sin

2

2


Sample Problem 1

11

A 50-kg block moves between vertical

guides as shown. The block is pulled

40mm down from its equilibrium

position and released.

For each spring arrangement, determine

a) the period of the vibration, b) the

maximum velocity of the block, and c)

the maximum acceleration of the block.

SOLUTION:

• For each spring arrangement, determine

the spring constant for a single

equivalent spring.

• Apply the approximate relations for the

harmonic motion of a spring-mass

system.


Sample Problem 1

12

mkN6mkN4 21 kk SOLUTION:

• Springs in parallel:

- determine the spring constant for equivalent spring

mN10mkN104

21

21

kkP

k

kkP

- apply the approximate relations for the harmonic

motion of a spring-mass system

nn

n srad.kg

N/m

m

k

2

141450

104

s 444.0n

srad 4.141m 040.0

nmm xv

sm566.0mv

2sm00.8ma2

2

srad 4.141m 040.0

nmm axa



VIBRACION LIBRE DE ESTRUCTURAS CON AMORTIGUAMIENTO


4


Sample Problem 1

13

mkN6mkN4 21 kk • Springs in series:




nn

n srad.kg

400N/m

m

k

2

93650

2

s 907.0n

srad .936m 040.0

nmm xv

sm277.0mv

2sm920.1ma

2

2

srad .936m 040.0

nmm xa

mkN4.2

111

21

21

21

21

21

kk

kkk

kkPk

k

P

k

P


FREE VIBRATION

DAMPED SYSTEMS

14


Damped Free Vibrations - 1

15

• With viscous damping due to fluid friction,

:maF

0kxxcxm

xmxcxkW st

• Substituting x = e t and dividing through by e t

yields the characteristic equation,

m

k

m

c

m

ckcm

22

220

• Define the critical damping coefficient such that

ncc m

m

kmc

m

k

m

c220

2

2

• All vibrations are damped to some degree by

forces due to dry friction, fluid friction, or

internal friction.



16

• Characteristic equation,

m

k

m

c

m

ckcm

22

220

nc mc 2 critical damping coefficient

• Heavy damping: c > cc tt eCeCx 21

21 - negative roots

- nonvibratory motion

• Critical damping: c = cc tnetCCx 21 - double roots


• Light damping: c < cc

tCtCex ddtmc

cossin 212

2

1c

ndc

cdamped frequency


4


Sample Problem 1

13

mkN6mkN4 21 kk • Springs in series:




nn

n srad.kg

400N/m

m

k

2

93650

2

s 907.0n

srad .936m 040.0

nmm xv

sm277.0mv

2sm920.1ma2

2

srad .936m 040.0

nmm xa

mkN4.2

111

21

21

21

21

21

kk

kkk

kkPk

k

P

k

P


FREE VIBRATION

DAMPED SYSTEMS

14



15

• With viscous damping due to fluid friction,

:maF

0kxxcxm

xmxcxkW st

• Substituting x = e t and dividing through by e t

yields the characteristic equation,

m

k

m

c

m

ckcm

22

220

• Define the critical damping coefficient such that

ncc m

m

kmc

m

k

m

c220

2

2

• All vibrations are damped to some degree by

forces due to dry friction, fluid friction, or

internal friction.



16

• Characteristic equation,

m

k

m

c

m

ckcm

22

220

nc mc 2 critical damping coefficient

• Heavy damping: c > cc tt

eCeCx 2121 - negative roots


• Critical damping: c = cc tnetCCx 21 - double roots



tCtCex ddtmc cossin 21

2

2

1c

ndc

cdamped frequency


5



17


tCtCex ddtmc

cossin 212

2

1c

ndc

cdamped frequency

2

0

2

D

0n02

2

2

1m xxv

CCx

tsinexx dtmc

m2

02

D

0n01 xC

xvC

1

21

C

Ctan

Note:in this figure x0 is to be read xm


Damped free vibrations - 4

18

cc

c

Critical damping is the smallest value of damping that inhibits

oscillations completely

Structures are usually underdamped



Effect of damping on period

19

2

0

2

D

0n0m x

xvx

amplitude

= xm

2

n

2

c

nd 1c

c1 damped frequency – lower than

natural frequency

2

nd

1

TT damped period – longer than

natural period



Effect of damping on amplitude decay

20

cc

c

Free vibration due to an initial displacement applied to four

SDOF systems having the same mass and stiffness but different

damping ratios

The rate of decay increases with damping

Normalmente en estructuras el amortiguamiento es UNDERDAMPED

El periodo con amortiguamiento Td es

mayor que Tn.



VIBRACION LIBRE DE ESTRUCTURAS CON AMORTIGUAMIENTO


5



17



2

2

1c

ndc

cdamped frequency

2

0

2

D

0n02

2

2

1m xxv

CCx

tsinexx dtmc

m2

02

D

0n01 xC

xvC

1

21

C

Ctan




18

cc

c







19

2

0

2

D

0n0m x

xvx

amplitude

= xm

2

n

2

c

nd 1c


natural frequency

2

nd

1


natural period




20

cc

c



damping ratios



5



17



2

2

1c

ndc

cdamped frequency

2

0

2

D

0n02

2

2

1m xxv

CCx

tsinexx dtmc

m2

02

D

0n01 xC

xvC

1

21

C

Ctan




18

cc

c







19

2

0

2

D

0n0m x

xvx

amplitude

= xm

2

n

2

c

nd 1c


natural frequency

2

nd

1


natural period




20

cc

c



damping ratios



5



17



2

2

1c

ndc

cdamped frequency

2

0

2

D

0n02

2

2

1m xxv

CCx

tsinexx dtmc

m2

02

D

0n01 xC

xvC

1

21

C

Ctan




18

cc

c







19

2

0

2

D

0n0m x

xvx

amplitude

= xm

2

n

2

c

nd 1c


natural frequency

2

nd

1


natural period




20

cc

c



damping ratios


Misma masa y rigidez


6




21

For damping values

typical of most

engineering structures

the value of damping is

such that there is no

significant change on

the natural period of

vibration

cc

c2

nd 1


Decay of motion

22

2Dn

D 1

2expTexp

Ttu

tu

Ratio of two successive peaks:

21i

i

1

2exp

u

u

2

1

2

u

uln

21i

i

is the logarithmic decrement


Decay of motion

23

Logarithmic decrement over j cycles:

21

1

2

1

1

21

1

j

j

u

uln

j

jexpjexpu

u

Number of cycles elapsed for a 50%

reduction in amplitude:

11.0j %50

ji

i

u

uln

j2

1Experimental determination of damping


Real dampers - 1

24

Vibracion forzada de un sistema ocurre cuando es sometida a fuerzas periodicas o desplazamientos periodicos



VIBRACION FORZADA DE ESTRUCTURAS SIN AMORTIGUAMIENTO


7


Real dampers - 2


FORCED VIBRATIONS OF

UNDAMPED SYSTEMS

HARMONIC FORCE

26


Forced Vibrations: harmonic excitation

27

:maF

xmxkWtP stfm sin

tPkxxm fm sin

xmtxkW fmst sin

tkkxxm fm sin

Forced vibrations - Occur

when a system is subjected to

a periodic force or a periodic

displacement of a support.

f forced frequency


Forced Harmonic Vibrations

28

txtCtC

xxx

fmnn

particulararycomplement

sincossin 21

22211 nf

m

nf

m

f

mm

kP

mk

Px

tkkxxm fm sin

tPkxxm fm sin

At f = n, forcing input is in

resonance with the system.

tPtkxtxm fmfmfmf sinsinsin2

Substituting particular solution into governing equation,


8


Harmonic vibrations of undamped systems

(notations of Chopra’s textbook)

29

000

0

uuuutat

tsinpkuum

nn

p tsin/k

ptu

20

1

1

tsinBtcosAtu nnc

esteadystat

n

transient

nnn

n tsin/k

ptsin

/k

putcosutu

20

20

1

1

1

100

tsin/k

ptsinBtcosAtu

nnn 2

0

1

1

Particular solution

Complementary solution



k/p0u

00u

2.0/

0n

n

30

Harmonic force

System response for

k

pust

00


Harmonic Vibrations:

Displacement Response Factor

31

For / n<1 or < n

u(t) and p(t) have the same algebraic sign:

the displacement is said to be in phase with

the applied force

For / n>1 or > n

u(t) and p(t) have opposite algebraic sign:

the displacement is said to be out of phase

relative to the applied force

k

putsin

k

ptu

tsin/

utsin/k

ptu

stst

n

st

n

p

00

0

2020

1

1

1

1




32

n

n

nstd

dstp

u

uR

tRututu

180

0

/1

1

sinsin

20

0

00

2n

Three cases:

Force “slowly varying”

Resonance

Force “rapidly varying”


8




29

000

0

uuuutat

tsinpkuum

nn

p tsin/k

ptu

20

1

1

tsinBtcosAtu nnc

esteadystat

n

transient

nnn

n tsin/k

ptsin

/k

putcosutu

20

20

1

1

1

100

tsin/k

ptsinBtcosAtu

nnn 2

0

1

1

Particular solution




k/p0u

00u

2.0/

0n

n

30

Harmonic force

System response for

k

pust

00




31

For / n<1 or < n



the applied force

For / n>1 or > n




k

putsin

k

ptu

tsin/

utsin/k

ptu

stst

n

st

n

p

00

0

2020

1

1

1

1




32

n

n

nstd

dstp

u

uR

tRututu

180

0

/1

1

sinsin

20

0

00

2n

Three cases:


Resonance



8




29

000

0

uuuutat

tsinpkuum

nn

p tsin/k

ptu

20

1

1

tsinBtcosAtu nnc

esteadystat

n

transient

n

nnn tsin

/k

ptsin

/k

putcosutu

20

20

1

1

1

100

tsin/k

ptsinBtcosAtu

nnn 2

0

1

1

Particular solution




k/p0u

00u

2.0/

0n

n

30

Harmonic force

System response for

k

pust

00




31

For / n<1 or < n



the applied force

For / n>1 or > n




k

putsin

k

ptu

tsin/

utsin/k

ptu

stst

n

st

n

p

00

0

2020

1

1

1

1




32

n

n

nstd

dstp

u

uR

tRututu

180

0

/1

1

sinsin

20

0

00

2n

Three cases:


Resonance


Vibracion forzada de un sistema ocurre cuando es sometida a fuerzas periodicas o desplazamientos periodicos



VIBRACION FORZADA DE ESTRUCTURAS CON AMORTIGUAMIENTO


10


Sample Problem 2

37

W = 350 lb

k = 4(350 lb/in) rad/s 5.57n

• Evaluate the magnitude of the periodic force due to the

motor unbalance. Determine the vibration amplitude

from the frequency ratio at 1200 rpm.

ftslb001941.0sft2.32

1

oz 16

lb 1oz 1

rad/s 125.7 rpm 1200

2

2m

f

lb 33.157.125001941.02

126

2mrmaP nm

in 001352.0

5.577.1251

300033.15

122

nf

mm

kPx

xm = 0.001352 in. (out of phase)



DAMPED SYSTEMS

HARMONIC FORCE

38


Damped Forced Harmonic Vibrations – 1

Problem statement and steady state response

39

2

222

1

2tan

21

1

nf

nfc

nfcnf

m

m

m

cc

cc

x

kP

xmagnification

factor

phase difference between forcing and steady

state response

tPkxxcxm fm sin particulararycomplement xxx



General solution

40


10


Sample Problem 2

37

W = 350 lb

k = 4(350 lb/in) rad/s 5.57n

• Evaluate the magnitude of the periodic force due to the

motor unbalance. Determine the vibration amplitude

from the frequency ratio at 1200 rpm.

ftslb001941.0sft2.32

1

oz 16

lb 1oz 1

rad/s 125.7 rpm 1200

2

2m

f

lb 33.157.125001941.02

126

2mrmaP nm

in 001352.0

5.577.1251

300033.15

122

nf

mm

kPx

xm = 0.001352 in. (out of phase)



DAMPED SYSTEMS

HARMONIC FORCE

38



Problem statement and steady state response

39

2

222

1

2tan

21

1

nf

nfc

nfcnf

m

m

m

cc

cc

x

kP

xmagnification

factor

phase difference between forcing and steady

state response

tPkxxcxm fm sin particulararycomplement xxx



General solution

40


11


Damped Forced Harmonic Vibrations - 3

Steady state

41

050

20

.

./ n

k/pu

u

n 00

00

Due to damping, the transient vanishes and the steady state

is the response after a sufficient time

The response is an harmonic function having the same

circular frequency of the forcing term



Response for n

42

000

050

1

uu

.

/ n

nDFor lightly damped systems:

The response is now bounded!



Effect of damping

43

nDFor lightly damped systems:



Maximum deformation and phase lag

44

k

putsin

k

puR

k

pu ststD

00

000

tsinRk

ptsinutu d

00

0< <

COMPONENTES: 2 HORIZONTALES, 1 VERTICAL

ESPECTRO DE RESPUESTA


DESCRIPCION DE LA EXCITACION SISMICA

Response spectrum of linear systems 22/04/2012

1

Earthquake Response of Linear Systems Maria Gabriella Mulas

Course “Buildings in Seismic A

r

eas”

Instructor: Maria Gabriella Mulas


Acknowledgments

The figures in this powerpoint come from:

Anil K. Chopra

Dynamics of Structures. Theory and Application to Earthquake

Engineering. 3rd edition


2


DESCRIPTION OF

EARTHQUAKE EXCITATION


Recorded Ground Motions (horizontal component)

To define earthquakes – ground

shaking: time variation of

ground acceleration

3 components: 2 horizontal, 1

vertical

Strong-motion accelerographs

Frequency range of recording

without excessive distorsion:

0-15 Hz for analog instruments

up to 30 Hz for digital ones

First record in 1933, Long Beach

earthquake

4


2


Recorded PGA, Loma Prieta Earthquake of

October 17, 1989

5

Values recorded

at many different

locations


Horizontal ground acceleration (Parkfield Station)

Ground motion is

presumed to be

known and

independent of the

structural response

RIGID SOIL:

NO SOIL-

STRUCTURE

INTERACTION

Time interval at

which numerical

values are defined:

0.01-0.02 s

6


EQUATION OF MOTION AND

RESPONSE PARAMETERS


Equation of motion for earthquake

excitation - 1

ut= total displ. u = relative displ. ug = ground displ.

ut = u + ug

kufucfumf sD

t

I

0sDI ffftumkuucum

kuucum

g

t 0

The relative displacement u(t) of the system is the same that we would

obtain by applying to the stationary base system the effective load:

tumtp geff

8

SISMO EL CENTRO: COMPONENTE N-S



ECUACION DE MOVIMIENTO Y PARAMETROS DE RESPUESTA

EL DESPLAZAMIENTO U(t) DEL SISTEMA SERIA EL MISMO QUE OBTUVIERAMOS SI

APLICARAMOS UNA FUERZA EFFECTIVA:


2



October 17, 1989

5

Values recorded

at many different

locations



Ground motion is

presumed to be

known and

independent of the

structural response

RIGID SOIL:

NO SOIL-

STRUCTURE

INTERACTION

Time interval at

which numerical

values are defined:

0.01-0.02 s

6



RESPONSE PARAMETERS



excitation - 1


ut = u + ug

kufucfumf sD

t

I

0sDI ffftumkuucum

kuucum

g

t 0



tumtp geff

8


2



October 17, 1989

5

Values recorded

at many different

locations



Ground motion is

presumed to be

known and

independent of the

structural response

RIGID SOIL:

NO SOIL-

STRUCTURE

INTERACTION

Time interval at

which numerical

values are defined:

0.01-0.02 s

6



RESPONSE PARAMETERS



excitation - 1


ut = u + ug

kufucfumf sD

t

I

0sDI ffftumkuucum

kuucum

g

t0



tumtp geff

8


2



October 17, 1989

5

Values recorded

at many different

locations



Ground motion is

presumed to be

known and

independent of the

structural response

RIGID SOIL:

NO SOIL-

STRUCTURE

INTERACTION

Time interval at

which numerical

values are defined:

0.01-0.02 s

6



RESPONSE PARAMETERS



excitation - 1


ut = u + ug

kufucfumf sD

t

I

0sDI ffftumkuucum

kuucum

g

t 0



tumtp geff

8




-GRAFICAS: HISTORIA DE RESPUESTA DE

DEFORMACION DE SDOF PARA SISMO DE EL

CENTRO.

-DOS SISTEMAS CON EL MISMO T Y

AMORTIGUAMIENTO, TENDRAN LA MISMA

RESPUESTA

-PARAMETROS DE RESPUESTA DE INTERES:

DEFORMACIONES RELATIVAS (PARA CALCULO

DE ESFUERZOS INTERNOS), DEFORMACIONES

TOTALES, ACELERACION TOTAL.

-ENTRE MAYOR EL Tn, MAYOR SERA LA

DEFORMACION MAXIMA.


3



excitation - 2

Two systems with the same Tn and ς will have the same response

Numerical methods are necessary to determine structural response!

Response parameters of interest are:

• Deformations or relative displacements u to which internal forces

are related

• Total displacements (to avoid pounding)

• Total acceleration if the structure is supporting sensitive

equipments (for example, industrial plants and nuclear reactor

vessel).

tuuuu gnn

22 ,T,tu)t(u n


Deformation response history of SDF systems

to El Centro ground motion

Time required to

complete a cycle

when subjected to

this earthquake

ground motion is

close to the natural

period.

The longer the

period, the greater

the peak

deformation

Same ς, different Tn Same Tn , different ς

10


Pseudo acceleration response of SDF

system to ElCentro Ground motion

Internal forces can be evaluated by

static analysis of the structure at

each instant t. Preferred approach:

Static equivalent force fs is the

force that, applied statically to the

system, would produce the same

deformation u(t)

2

2

2

2

2

n

n

n

ns

s

T

tutA

tmAtumf

tkuf

A(t) is the pseudo-acceleration


Application of pseudo-acceleration concept

12

thVthftM

tmAtfV

bsb

sb

The base shear and overturning moment depend on the pseudo-

acceleration.

Base shear balances the static equivalent force.

Base overturning moment balances its moment with respect to the

foundation and is provided by axial loads in columns.

fs(t)

Mb(t) Vb(t)

Static analysis of the

structure subjected to the

static equivalent force fs


3



excitation - 2





are related




vessel).

tuuuu gnn

22 ,T,tu)t(u n




Time required to

complete a cycle

when subjected to

this earthquake

ground motion is


period.

The longer the

period, the greater

the peak

deformation


10










deformation u(t)

2

2

2

2

2

n

n

n

ns

s

T

tutA

tmAtumf

tkuf




12

thVthftM

tmAtfV

bsb

sb


acceleration.




fs(t)

Mb(t) Vb(t)







-Fs ES LA FUERZA ESTATICA EQUIVALENTE

A(t) SE LE LLAMA PSEUDO ACELERACION


3



excitation - 2





are related




vessel).

tuuuu gnn

22 ,T,tu)t(u n




Time required to

complete a cycle

when subjected to

this earthquake

ground motion is


period.

The longer the

period, the greater

the peak

deformation


10










deformation u(t)

2

2

2

2

2

n

n

n

ns

s

T

tutA

tmAtumf

tkuf




12

thVthftM

tmAtfV

bsb

sb


acceleration.




fs(t)

Mb(t) Vb(t)





3



excitation - 2





are related




vessel).

tuuuu gnn

22 ,T,tu)t(u n




Time required to

complete a cycle

when subjected to

this earthquake

ground motion is


period.

The longer the

period, the greater

the peak

deformation


10










deformation u(t)

2

2

2

2

2

n

n

n

ns

s

T

tutA

tmAtumf

tkuf




12

thVthftM

tmAtfV

bsb

sb


acceleration.




fs(t)

Mb(t) Vb(t)







-EL CORTE BASAL Y EL MOMENTO DE VOLCAMIENTO

SON DEPENDIENTES DE LA PSEUDO ACELERACION.

-EL CORTANTE BASAL EQUILIBRIA LA FUERZA

ESTATICA EQUIVALENTE


3



excitation - 2





are related




vessel).

tuuuu gnn

22 ,T,tu)t(u n




Time required to

complete a cycle

when subjected to

this earthquake

ground motion is


period.

The longer the

period, the greater

the peak

deformation


10










deformation u(t)

2

2

2

2

2

n

n

n

ns

s

T

tutA

tmAtumf

tkuf




12

thVthftM

tmAtfV

bsb

sb


acceleration.




fs(t)

Mb(t) Vb(t)





3



excitation - 2





are related




vessel).

tuuuu gnn

22 ,T,tu)t(u n




Time required to

complete a cycle

when subjected to

this earthquake

ground motion is


period.

The longer the

period, the greater

the peak

deformation


10










deformation u(t)

2

2

2

2

2

n

n

n

ns

s

T

tutA

tmAtumf

tkuf




12

thVthftM

tmAtfV

bsb

sb


acceleration.




fs(t)

Mb(t) Vb(t)






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.

-ESTOS VALORES SON

OBTENIDOS DE UN FINITO DE

SISTEMAS DE UN GRADO DE

LIBERTAD CON UN FACTOR DE

AMORTIGUAMIENTO DEFINIDO.


4


RESPONSE SPECTRUM

CONCEPT


Response Spectrum Concept

14

A plot of the peak value of a response quantity as a function

of the natural period Tn or related quantities as fn or ωn is

called response spectrum for that quantity.

Practical mean to characterize a ground motion:

,,max,

,,max,

,,max,

0

0

0

nt

n

nt

n

nt

n

TtuTu

TtuTu

TtuTu Deformation R. S.

Relative velocity R.S.

Acceleration R.S.

By definition, the peak response is positive; the sign is dropped

because it is usually irrelevant for design.

Each plot is derived for a SDF having a fixed damping ratio ς.


Deformation Response Spectrum for El Centro

Ground Motion


Deformation, pseudo-velocity and pseudo-

acceleration response spectra ( = 2%)

Pseudo-velocity:

2222

2

22220

0

0

mVVkkDkuE

uVDT

DV

ns

nn

Pseudo-acceleration:

Wg

AA

g

WmAfV

uDT

DA

sb

t

n

n

00

0

2

2 2

A/g base shear coefficient

16


4


RESPONSE SPECTRUM

CONCEPT



14





,,max,

,,max,

,,max,

0

0

0

nt

n

nt

n

nt

n

TtuTu

TtuTu



Acceleration R.S.






Ground Motion




Pseudo-velocity:

2222

2

22220

0

0

mVVkkDkuE

uVDT

DV

ns

nn


Wg

AA

g

WmAfV

uDT

DA

sb

t

n

n

00

0

2

2 2


16




PASOS

-DEFINIR ACELERACION DEL SUELO EN FUNCION DEL

TIEMPO

-SELECCIONAR T Y AMORTIGUAMIENTO

-CALCULO DE DESPLAZAMIENTOS Y DESPLAZAMIENTO

MAXIMO

-DERIVAR HASTA OBTENER ACELERACION

-REPETIR PARA COMBINACIONES DESEADASX


5


Combined D-V-A response spectrum ( = 2%)

log-log scale

DVA

n

n

n

nT

2


Response spectrum, El Centro ground motion

( = 0, 2, 5,10 and 20 %)

Spectrum covers

a wide range of

periods and

damping values

18


Pseudo acceleration response spectrum, linear scale

Deformation response spectrum, log scale

Linear scale

From this plot:

Base shear coefficient

(lateral force)

19

From this plot:

Peak deformation

= 0, 2, 5,10 and 20 %


Construction of a response spectrum

Necessary steps:

1. Numerically define

2. Select Tn and ς

3. Compute u(t)

4. Determine the peak value u0

5. Spectral ordinates are D= u0 V = ωnD A = (ωn)2 D

6. Repeat steps 2 to 5 for a range of Tn and ς covering all the possible

engineering systems of interest

7. Plot the results obtained (three separate spectra or one combined

spectrum)

A large computational effort!

20

tug


4


RESPONSE SPECTRUM

CONCEPT



14





,,max,

,,max,

,,max,

0

0

0

nt

n

nt

n

nt

n

TtuTu

TtuTu



Acceleration R.S.






Ground Motion




Pseudo-velocity:

2222

2

22220

0

0

mVVkkDkuE

uVDT

DV

ns

nn


Wg

AA

g

WmAfV

uDT

DA

sb

t

n

n

00

0

2

2 2


16





7


RESPONSE SPECTRUM

CHARACTERISTICS


Response spectrum and ground motion (El

Centro 1940 , = 0, 2, 5,10 and 20 %)

26

Dashed lines

represent the

peak values of

ground

acceleration,

velocity and

displacement

Response

spectrum

depicts the

SDOF system

response to the

ground motion


Response spectra for = 0, 2, 5 and 10%

Plotted on normalized scales

27

The adoption

of normalized

scales shows

more directly

the relation

between the

response

spectrum and

the ground

motion

parameters


Spectrum on normalized scale, 5% damping

28

The dashed line

is the idealized

version of the

R.S. : formal

techniques of

curve fitting can

be used to

replace the real

spectrum with

an idealized one


7


RESPONSE SPECTRUM

CHARACTERISTICS


Response spectrum and ground motion (El

Centro 1940 , = 0, 2, 5,10 and 20 %)

26

Dashed lines

represent the

peak values of

ground

acceleration,

velocity and

displacement

Response

spectrum

depicts the

SDOF system

response to the

ground motion


Response spectra for = 0, 2, 5 and 10%

Plotted on normalized scales

27

The adoption

of normalized

scales shows

more directly

the relation

between the

response

spectrum and

the ground

motion

parameters



28

The dashed line

is the idealized

version of the

R.S. : formal

techniques of

curve fitting can

be used to

replace the real

spectrum with

an idealized one





8



Spectral regions

29

Goal:

Study properties of the

R.S. over various

ranges of the natural

period of the system

Dashed line: idealized

version of the R.S.

Tn < Ta

Tn > Tf

Other periods

30

31

32


Short period system (very stiff – rigid)

Tn < Ta = 0.035s

30

The mass moves rigidly

with the ground

A approaches the ground

acceleration and D is

very small

Neglecting damping:

00

2

2

g

t

t

g

n

gn

uAu

uuutA

tAu

uuu

29


Long period system (very flexible)

Tn > Tf =15s

31

D approaches ug0 and

A is very small

Mass remains

stationary while the

ground moves below

0

00

gg

t

uDtutu

tAtu

29


Intermediate period systems - 1

Short-period systems Ta= 0.035s < Tn < Tc= 0.50s

A exceeds - amplification depends on Tn and ς

Between Tb = 0.125s and Tc = 0.5 s A may be assumed as constant

at a value equal to amplified by a factor depending on ς

This region is called acceleration sensitive region because the

structural response is mostly related to the ground acceleration

Long-period systems Td= 3.0 s < Tn < Tf= 15.0 s

D exceeds - amplification depends on Tn and ς

Between Td= 3s and Te =15 s D may be assumed as constant at a

value equal to amplified by a factor depending on ς

This region is called displacement sensitive region because the

structural response is mostly related to the ground displacement

32

0gu

0gu

0gu

0gu


8



Spectral regions

29

Goal:

Study properties of the

R.S. over various

ranges of the natural

period of the system

Dashed line: idealized

version of the R.S.

Tn < Ta

Tn > Tf

Other periods

30

31

32


Short period system (very stiff – rigid)

Tn < Ta = 0.035s

30

The mass moves rigidly

with the ground

A approaches the ground

acceleration and D is

very small

Neglecting damping:

00

2

2

g

t

t

g

n

gn

uAu

uuutA

tAu

uuu

29


Long period system (very flexible)

Tn > Tf =15s

31

D approaches ug0 and

A is very small

Mass remains

stationary while the

ground moves below

0

00

gg

t

uDtutu

tAtu

29


Intermediate period systems - 1

Short-period systems Ta= 0.035s < Tn < Tc= 0.50s

A exceeds - amplification depends on Tn and ς

Between Tb = 0.125s and Tc = 0.5 s A may be assumed as constant

at a value equal to amplified by a factor depending on ς

This region is called acceleration sensitive region because the

structural response is mostly related to the ground acceleration

Long-period systems Td= 3.0 s < Tn < Tf= 15.0 s

D exceeds - amplification depends on Tn and ς

Between Td= 3s and Te =15 s D may be assumed as constant at a

value equal to amplified by a factor depending on ς

This region is called displacement sensitive region because the

structural response is mostly related to the ground displacement

32

0gu

0gu

0gu

0gu



ESPECTRO ELASTICO DE DISENO


10


Elastic design spectrum

40

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!

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



41

The design spectrum should be representative of ground motions

recorded at the site during past earthquakes.

If none has been recorded, the DS should be based on ground

motions recorded at other sites under similar conditions

The important factors to be matched are:

• The magnitude of the earthquake

• The distance of site from causative fault

• The fault mechanism

• The geology of the travel path of seismic waves from source to

site

• The local conditions at the site


Mean design spectrum - 1

42

Design spectrum must be based on statistical analysis of the

response spectra for an ensemble of a number I of ground motions

Each accelerogram is characterized by the time history and the

peak values of displacement, velocity and acceleration

Accelerograms are scaled (up or down) to the same peak

acceleration.

The response spectrum is computed for each of them

For each period Tn we have I values of Di, Vi and Ai

Statistical analysis of spectral ordinates will provide probability

distribution, mean and standard deviation at each period Tn

ig

ig

ig

ig uuuandtu 000


Mean spectrum over 10 ground motions

43

Mean spectrum based on

statistical analysis of the

response spectra.

Factors adopted to normalize

the scales are the mean values

over 10 records.

Connecting all the mean

values gives the mean

response spectrum;

connecting all the mean plus

one standard deviation values

gives the mean plus one

standard deviation spectrum .

Mean spectrum is easily

idealized through straight

lines (dashed lines in the

figure) Recommended period values in the plot!


10



40

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!

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



41

The design spectrum should be representative of ground motions

recorded at the site during past earthquakes.

If none has been recorded, the DS should be based on ground

motions recorded at other sites under similar conditions

The important factors to be matched are:

• The magnitude of the earthquake

• The distance of site from causative fault

• The fault mechanism

• The geology of the travel path of seismic waves from source to

site

• The local conditions at the site


Mean design spectrum - 1

42

Design spectrum must be based on statistical analysis of the

response spectra for an ensemble of a number I of ground motions

Each accelerogram is characterized by the time history and the

peak values of displacement, velocity and acceleration

Accelerograms are scaled (up or down) to the same peak

acceleration.

The response spectrum is computed for each of them

For each period Tn we have I values of Di, Vi and Ai

Statistical analysis of spectral ordinates will provide probability

distribution, mean and standard deviation at each period Tn

ig

ig

ig

ig uuuandtu 000


Mean spectrum over 10 ground motions

43

Mean spectrum based on

statistical analysis of the

response spectra.

Factors adopted to normalize

the scales are the mean values

over 10 records.

Connecting all the mean

values gives the mean

response spectrum;

connecting all the mean plus

one standard deviation values

gives the mean plus one

standard deviation spectrum .

Mean spectrum is easily

idealized through straight

lines (dashed lines in the

figure) Recommended period values in the plot!



ESPECTRO ELASTICO DE DISENO


12


Pseudo-velocity design spectrum


Pseudo-acceleration design spectrum

49

Log scale

Linear scale


Deformation design spectrum


Comparison between design spectrum and

response spectrum

51

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!



ESPECTRO INELASTICO DE DISENO

Inelastic Design Spectrum 15/05/2012

1

Inelastic Design SpectrumMaria Gabriella Mulas

Course “Buildings in Seismic Areas”

Instructor: Maria Gabriella Mulas


Acknowledgments

The figures in this powerpointcome from:

Anil K. Chopra

Dynamics of Structures. Theory and Application to Earthquake

Engineering. 3rd edition


Color pictures have been downloaded from the web


Elastic vs. Building Code Design Spectrum

Structures are designed for base

shear smaller than the elastic

value: will deform beyond the

elastic range when subjected to

a ground motion represented by

the 0.4g design spectrum

DAMAGE will occur

Successful design will control

damage to keep it acceptable

No collapse: strong earthquakes

Repairable damage: frequent

earthquakes


Damage not economically repairable

Imperial County Services Building after the Imperial

Valley, California earthquake of Oct. 15, 1979


2


Collapse

Olive View Hospital, Psychiatric Day Care Center after San

Fernando earthquake of Feb 9, 1971

The hospital had open only one month before the earthquake


DESCRIPTION OF POSSIBLE

NON LINEAR BEHAVIORS

6


Force-deformation behavior

7

Since the 1960’s thousands of laboratory tests have been

conducted to determine the force-deformation behavior for

earthquake condition of:

•Structural members

•Assemblage of members

•Scaled model of structures

•Small full-scale structures

Response depends on both structural material and the

structural systems.

Common feature: force-deformation relationship shows

hysteresis loops under cyclic deformation due to inelastic

behavior


Force deformation relation: panel zone of a steel

welded beam-to-column connection


Stable cycles, large amount of dissipated energy




SISTEMA ELASTICO SIN AMORTIGUAMIENTO SISTEMA ELASTOPLASTICO SIN AMORTIGUAMIENTO





11


Theory of Ductility Factor


Construction of inelastic design spectrum

From the elastic design spectrum, the inelastic design spectrum is obtained by

applying the equations in slide 40


Inelastic pseudoacceleration design

spectrum (84.1th percentile) linear scale

This format of

the inelastic

design

spectrum is

contained in

seismic codes


Inelastic deformation design spectrum

For Tn>Tc

peak deformation

of inelastic system

is independent on

ductility and equal

to the peak

deformation of the

corresponding

elastic system

Equal displacement rule

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

-Amortiguamiento lineal viscoso respresenta la disipacion

de energia.

79

Proyecto Estructural - Prof. Michele Casarin


80



81


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).

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

simultaneamente.


82


VIBRACION LIBRE DE SISTEMAS MGL SIN AMORTIGUAMIENTO

Vibración es iniciada por curva A

El movimiento no es harmónico

Se origina un movimiento harmónico gracias a la correcta proporción constant de U. Estas dos

formas deformadas son modos naturales de vibración.


83



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.

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


84



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 cuadrada con todos los

valores de Φ, donde cada columna es

el modo natural.

Una matriz diagonal puede ser

construida con los N valores eigen.


85


VIBRACION LIBRE DE SISTEMAS MGL CON AMORTIGUAMIENTO


86


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.

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

respuesta total.


87


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.

-Combinar las contribuciones en las

respuestas de cada modo para obtener

la respuesta total.


88


HISTORIA DE RESPUESTA

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.


89


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.

-CQC (complete quadratic combination). Excelentes resultados para estructuras

con frencuencias naturales cercanas.


90




91




92


EJEMPLO


93


EJEMPLO


94


EJEMPLO


95


EJEMPLO

CONCEPTOS DE DISEÑO SÍSMICO

96


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.

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

gravitacionales.


97


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.

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

fallas.


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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

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


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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

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


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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

-Irregularidades en planta y elevación.


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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

Pérdidas humanas deben prevenirse inclusive para los sismos mas Fuertes. Ocurren daños

irreparables pero nunca el colapso.


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ESTADOS LÍMITES DE DISEÑO SÍSMICO


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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.

-Ductilidad requiere atención en el detallado


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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

-Momentos adicionales son transmitidos al resto de la

estructura


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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.

-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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