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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
Proyecto Estructural - Prof. Michele Casarin 3
RIESGO SISMICO
Proyecto Estructural - Prof. Michele Casarin 4
RIESGO SISMICO
Proyecto Estructural - Prof. Michele Casarin 5
RIESGO SISMICO
Proyecto Estructural - Prof. Michele Casarin 6
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
Proyecto Estructural - Prof. Michele Casarin 7
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
Proyecto Estructural - Prof. Michele Casarin 8
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
Proyecto Estructural - Prof. Michele Casarin 9
LA TIERRA
SISMOLOGIA
Proyecto Estructural - Prof. Michele Casarin 10
LA TIERRA
02 - Seismology 14/03/2012
7
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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Maria Gabriella Mulas
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?
Maria Gabriella Mulas
Earthquake epicenters 1963-2000
Depth of focus: 70-350 = intermediate (yellow), 0-70 km = shallow (blue) >350Km = deep (red)
GLOBAL DISTRIBUTION OF EARTHQUAKES
16
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
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?
Maria Gabriella Mulas
Earthquake epicenters 1963-2000
Depth of focus: 70-350 = intermediate (yellow), 0-70 km = shallow (blue) >350Km = deep (red)
GLOBAL DISTRIBUTION OF EARTHQUAKES
16
SISMOLOGIA
Proyecto Estructural - Prof. Michele Casarin 11
ONDAS SISMICAS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
TECTONIC PLATES
19 Maria Gabriella Mulas
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
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Maria Gabriella Mulas
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.
Maria Gabriella Mulas
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
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Maria Gabriella Mulas
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.
Maria Gabriella Mulas
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.
FAULTS AND EARTHQUAKE
24
SISMOLOGIA
Proyecto Estructural - Prof. Michele Casarin 12
ONDAS SISMICAS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
FAULTS AND EARTHQUAKE
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
Maria Gabriella Mulas
FAULTS AND EARTHQUAKE
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
Maria Gabriella Mulas
FAULT AND EARTHQUAKE
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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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
Maria Gabriella Mulas
FAULT AND EARTHQUAKE
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
02 - Seismology 14/03/2012
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Maria Gabriella Mulas 29
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.
FAULT AND EARTHQUAKE
Maria Gabriella Mulas 30
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!
FAULT AND EARTHQUAKE
FALLA STRIKE-SLIP
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
FAULT AND EARTHQUAKE
31 Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 13
ONDAS SISMICAS
1. ONDAS P: 8 KM/S
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
DESCRIPTION OF SEISMIC
WAVES
39 Maria Gabriella Mulas
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
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
SEISMIC WAVES
41 Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 14
ONDAS SISMICAS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
BODY WAVES
43 Maria Gabriella Mulas
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
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
BODY WAVES
43 Maria Gabriella Mulas
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
Fundamentals of Engineering Seismology
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
Proyecto Estructural - Prof. Michele Casarin 15
ONDAS SISMICAS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
SURFACE WAVES
45
SU
RF
AC
E W
AV
ES
Maria Gabriella Mulas
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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Maria Gabriella Mulas
SURFACE WAVES
45
SU
RF
AC
E W
AV
ES
Maria Gabriella Mulas
SURFACE WAVES
46
Love waves travel faster than Rayleigh waves and therefore arrive earlier
Love waves
Rayleigh waves
SISMOLOGIA
Proyecto Estructural - Prof. Michele Casarin 16
MEDICION DE ONDAS SISMICAS
SISMOGRAFOS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
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
25
Maria Gabriella Mulas
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
SISMOGRAMAS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
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
53 Maria Gabriella Mulas
Distance - Time Relations
SISMOGRAM
54
SISMOLOGIA
Proyecto Estructural - Prof. Michele Casarin 17
MEDICION DE SISMOS
RITCHER SCALE
MOMENTO DE MAGNITUD
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
MEASUREMENTS OF EARTHQUAKES
69 Maria Gabriella Mulas
MEASUREMENTS OF EARTHQUAKES
70
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
MEASUREMENTS OF EARTHQUAKES
69 Maria Gabriella Mulas
MEASUREMENTS OF EARTHQUAKES
70
ENERGIA SISMICA
SISMOLOGIA
Proyecto Estructural - Prof. Michele Casarin 18
MEDICION DE SISMOS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
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
MEASUREMENTS OF EARTHQUAKES
75 Maria Gabriella Mulas
Some Notable Earthquakes
76
Indonesia (12/04)
Pakistan
(10/05)
SISMOLOGIA
Proyecto Estructural - Prof. Michele Casarin 19
PROPAGACION DE ONDAS
02 - Seismology 14/03/2012
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Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 20
EL CASO DE CIUDAD DE MEXICO
03 - Fundamentals of Engineering Seismology 19 March 2012
15
Fundamentals of Engineering Seismology
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
Fundamentals of Engineering Seismology
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
Proyecto Estructural - Prof. Michele Casarin 21
Earthquake Damage - Part I 4/22/2012
2
Maria Gabriella Mulas
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
Maria Gabriella Mulas 7
Structural damage - 2
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.
Maria Gabriella Mulas 8
Structural damage - 3
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
Proyecto Estructural - Prof. Michele Casarin 22
LICUEFACCION
Earthquake Damage - Part I 4/22/2012
3
Maria Gabriella Mulas
DAMAGE AS A RESULT OF
SOIL PROBLEMS
9 Maria Gabriella Mulas 10
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
Maria Gabriella Mulas 11
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
Maria Gabriella Mulas 12
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)
Earthquake Damage - Part I 4/22/2012
4
Maria Gabriella Mulas 13
Damage due to soil liquefaction – 1
Izmit, Turkey 1999
Maria Gabriella Mulas 14
Damage due to soil liquefaction – 2
Adapazari, Turkey 1999
Kobe, Japan 1995
Maria Gabriella Mulas 15
1964 Nilgata, Japan
Damage due to soil liquefaction – 3
Maria Gabriella Mulas 16
Damage due to soil liquefaction – 4
Sand boils and ground
fissures provide
evidence of liquefaction
Earthquake Damage - Part I 4/22/2012
4
Maria Gabriella Mulas 13
Damage due to soil liquefaction – 1
Izmit, Turkey 1999
Maria Gabriella Mulas 14
Damage due to soil liquefaction – 2
Adapazari, Turkey 1999
Kobe, Japan 1995
Maria Gabriella Mulas 15
1964 Nilgata, Japan
Damage due to soil liquefaction – 3
Maria Gabriella Mulas 16
Damage due to soil liquefaction – 4
Sand boils and ground
fissures provide
evidence of liquefaction
Earthquake Damage - Part I 4/22/2012
4
Maria Gabriella Mulas 13
Damage due to soil liquefaction – 1
Izmit, Turkey 1999
Maria Gabriella Mulas 14
Damage due to soil liquefaction – 2
Adapazari, Turkey 1999
Kobe, Japan 1995
Maria Gabriella Mulas 15
1964 Nilgata, Japan
Damage due to soil liquefaction – 3
Maria Gabriella Mulas 16
Damage due to soil liquefaction – 4
Sand boils and ground
fissures provide
evidence of liquefaction
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 23
DESLIZAMIENTOS
Earthquake Damage - Part I 4/22/2012
5
Maria Gabriella Mulas
Landslides - 1
When a steeply inclined mass of soil is suddenly shaken, a
slip lane can form and the material slides downhill.
17 Maria Gabriella Mulas
Landslides - 2
18
Structures sitting on the
slide move downward
Structures below the
slide are hitten by falling
debris
Before
After
Maria Gabriella Mulas 19
Landslides - 3
ji-ji_chap8.pdf photo 8.1-8.5
Maria Gabriella Mulas
Landslide of Turnagain Heights
Anchorage, Alaska 1964
20
Earthquake Damage - Part I 4/22/2012
5
Maria Gabriella Mulas
Landslides - 1
When a steeply inclined mass of soil is suddenly shaken, a
slip lane can form and the material slides downhill.
17 Maria Gabriella Mulas
Landslides - 2
18
Structures sitting on the
slide move downward
Structures below the
slide are hitten by falling
debris
Before
After
Maria Gabriella Mulas 19
Landslides - 3
ji-ji_chap8.pdf photo 8.1-8.5
Maria Gabriella Mulas
Landslide of Turnagain Heights
Anchorage, Alaska 1964
20
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 24
DESLIZAMIENTOS
Earthquake Damage - Part I 4/22/2012
5
Maria Gabriella Mulas
Landslides - 1
When a steeply inclined mass of soil is suddenly shaken, a
slip lane can form and the material slides downhill.
17 Maria Gabriella Mulas
Landslides - 2
18
Structures sitting on the
slide move downward
Structures below the
slide are hitten by falling
debris
Before
After
Maria Gabriella Mulas 19
Landslides - 3
ji-ji_chap8.pdf photo 8.1-8.5
Maria Gabriella Mulas
Landslide of Turnagain Heights
Anchorage, Alaska 1964
20
Earthquake Damage - Part I 4/22/2012
5
Maria Gabriella Mulas
Landslides - 1
When a steeply inclined mass of soil is suddenly shaken, a
slip lane can form and the material slides downhill.
17 Maria Gabriella Mulas
Landslides - 2
18
Structures sitting on the
slide move downward
Structures below the
slide are hitten by falling
debris
Before
After
Maria Gabriella Mulas 19
Landslides - 3
ji-ji_chap8.pdf photo 8.1-8.5
Maria Gabriella Mulas
Landslide of Turnagain Heights
Anchorage, Alaska 1964
20
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 25
RUPTURA DEL SUELO
Earthquake Damage - Part I 4/22/2012
6
Maria Gabriella Mulas 21
1906 Olema, CA
Ground rupture - 1
Maria Gabriella Mulas 22
Ground rupture - 2
Maria Gabriella Mulas 23
Ground rupture - 3
Japan earthquake, March 3,11
Kanto Highway, repaired in
one week
Maria Gabriella Mulas
Ground motion
24
Guatemala earthquake, 1976
Rails bent in Gualan
Earthquake Damage - Part I 4/22/2012
6
Maria Gabriella Mulas 21
1906 Olema, CA
Ground rupture - 1
Maria Gabriella Mulas 22
Ground rupture - 2
Maria Gabriella Mulas 23
Ground rupture - 3
Japan earthquake, March 3,11
Kanto Highway, repaired in
one week
Maria Gabriella Mulas
Ground motion
24
Guatemala earthquake, 1976
Rails bent in Gualan
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 26
ARCILLAS DEBILES
Earthquake Damage - Part I 4/22/2012
8
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Weak clay - Struve Slough Bridge - 2
30
Piles “punctured”
the bridge
Shear damage was
found at the top of
the piles
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Weak clay – Cypress Viaduct - 2
32
Earthquake Damage - Part I 4/22/2012
8
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Weak clay - Struve Slough Bridge - 2
30
Piles “punctured”
the bridge
Shear damage was
found at the top of
the piles
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Weak clay – Cypress Viaduct - 2
32
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 27
ARCILLAS DEBILES
Earthquake Damage - Part I 4/22/2012
9
Maria Gabriella Mulas
Weak clay – Cypress Viaduct - 3
33
Bent reinforcement bar in
failed support column
Inadequate confinement
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Weak clay – Mexico City 1985 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.
35 Maria Gabriella Mulas
Weak clay – Mexico City 1985 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
Proyecto Estructural - Prof. Michele Casarin 28
TSUNAMI
Earthquake Damage - Part I 4/22/2012
10
Maria Gabriella Mulas 37
Tsunami
•very long wavelength, deep wavebase
•speeds up to 800 km/hour, 20 meters high
Maria Gabriella Mulas 38
Tsunami – wave propagation times
Maria Gabriella Mulas 39
Tsunami - 1964 Alaska Earthquake
Maria Gabriella Mulas
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
Earthquake Damage - Part I 4/22/2012
10
Maria Gabriella Mulas 37
Tsunami
•very long wavelength, deep wavebase
•speeds up to 800 km/hour, 20 meters high
Maria Gabriella Mulas 38
Tsunami – wave propagation times
Maria Gabriella Mulas 39
Tsunami - 1964 Alaska Earthquake
Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 29
TSUNAMI
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 30
FALLAS EN CONEXIONES CON LAS FUNDACIONES
Earthquake Damage - Part II 4/22/2012
2
Maria Gabriella Mulas 7
Structural damage - 2
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.
Maria Gabriella Mulas 8
Structural damage - 3
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
Maria Gabriella Mulas
DAMAGE AS A RESULT OF
STRUCTURAL PROBLEMS
9
Maria Gabriella Mulas 10
Ground motion
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Foundation failure - 2
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.
Earthquake Damage - Part II 4/22/2012
2
Maria Gabriella Mulas 7
Structural damage - 2
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.
Maria Gabriella Mulas 8
Structural damage - 3
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
Maria Gabriella Mulas
DAMAGE AS A RESULT OF
STRUCTURAL PROBLEMS
9
Maria Gabriella Mulas 10
Ground motion
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Foundation failure - 2
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.
Earthquake Damage - Part II 4/22/2012
3
Maria Gabriella Mulas 13
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
Maria Gabriella Mulas 14
Foundation connection – timber structure
Hold-down
Maria Gabriella Mulas 15
Foundation connection
San Fernando,
California,
Earthquake February
1971.
Collapsed highway
overpass,
INTERSTATE 5 and
14 interchange.
The interchange was
built on consolidated
sand
Maria Gabriella Mulas 16
Foundation connection
San Fernando,
California,
Earthquake February
1971.
Collapsed highway
overpass,
INTERSTATE 5 and
14 interchange.
Maria Gabriella Mulas 17
Foundation connection
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
Maria Gabriella Mulas 18
Foundation connection (special structures)
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 31
FALLAS EN CONEXIONES CON LAS FUNDACIONES
Earthquake Damage - Part II 4/22/2012
3
Maria Gabriella Mulas 13
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
Maria Gabriella Mulas 14
Foundation connection – timber structure
Hold-down
Maria Gabriella Mulas 15
Foundation connection
San Fernando,
California,
Earthquake February
1971.
Collapsed highway
overpass,
INTERSTATE 5 and
14 interchange.
The interchange was
built on consolidated
sand
Maria Gabriella Mulas 16
Foundation connection
San Fernando,
California,
Earthquake February
1971.
Collapsed highway
overpass,
INTERSTATE 5 and
14 interchange.
Maria Gabriella Mulas 17
Foundation connection
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
Maria Gabriella Mulas 18
Foundation connection (special structures)
Earthquake Damage - Part II 4/22/2012
3
Maria Gabriella Mulas 13
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
Maria Gabriella Mulas 14
Foundation connection – timber structure
Hold-down
Maria Gabriella Mulas 15
Foundation connection
San Fernando,
California,
Earthquake February
1971.
Collapsed highway
overpass,
INTERSTATE 5 and
14 interchange.
The interchange was
built on consolidated
sand
Maria Gabriella Mulas 16
Foundation connection
San Fernando,
California,
Earthquake February
1971.
Collapsed highway
overpass,
INTERSTATE 5 and
14 interchange.
Maria Gabriella Mulas 17
Foundation connection
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
Maria Gabriella Mulas 18
Foundation connection (special structures)
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 32
FALLAS POR ENTREPISO DEBIL “SOFT STORY”
Earthquake Damage - Part II 4/22/2012
4
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story
20 Maria Gabriella Mulas 21
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
Maria Gabriella Mulas 22
Soft Story, L’Aquila 2009
Same situation
also in this side
Maria Gabriella Mulas 23
Soft Story, L’Aquila 2009
Same situation
also in this side!
Maria Gabriella Mulas 24
Soft Story, L’Aquila 2009
Intermediate soft-story in a 3-story house
Earthquake Damage - Part II 4/22/2012
4
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story
20 Maria Gabriella Mulas 21
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
Maria Gabriella Mulas 22
Soft Story, L’Aquila 2009
Same situation
also in this side
Maria Gabriella Mulas 23
Soft Story, L’Aquila 2009
Same situation
also in this side!
Maria Gabriella Mulas 24
Soft Story, L’Aquila 2009
Intermediate soft-story in a 3-story house
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 33
FALLAS POR ENTREPISO DEBIL “SOFT STORY”
Earthquake Damage - Part II 4/22/2012
4
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story
20 Maria Gabriella Mulas 21
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
Maria Gabriella Mulas 22
Soft Story, L’Aquila 2009
Same situation
also in this side
Maria Gabriella Mulas 23
Soft Story, L’Aquila 2009
Same situation
also in this side!
Maria Gabriella Mulas 24
Soft Story, L’Aquila 2009
Intermediate soft-story in a 3-story house
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 34
FALLAS POR ENTREPISO DEBIL “SOFT STORY”
Earthquake Damage - Part II 4/22/2012
5
Maria Gabriella Mulas 25
Soft Story, L’Aquila 2009
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story at mid level
27
10-story SRC
building with 3rd
floor collapse
Maria Gabriella Mulas
Soft story at mid level
28
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
Soft story at mid level
29
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 35
FALLAS POR ENTREPISO DEBIL “SOFT STORY”
Earthquake Damage - Part II 4/22/2012
5
Maria Gabriella Mulas 25
Soft Story, L’Aquila 2009
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story at mid level
27
10-story SRC
building with 3rd
floor collapse
Maria Gabriella Mulas
Soft story at mid level
28
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
Soft story at mid level
29
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 36
FALLAS POR ENTREPISO DEBIL “SOFT STORY”
Earthquake Damage - Part II 4/22/2012
5
Maria Gabriella Mulas 25
Soft Story, L’Aquila 2009
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story at mid level
27
10-story SRC
building with 3rd
floor collapse
Maria Gabriella Mulas
Soft story at mid level
28
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
Soft story at mid level
29
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
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.
Earthquake Damage - Part II 4/22/2012
5
Maria Gabriella Mulas 25
Soft Story, L’Aquila 2009
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story at mid level
27
10-story SRC
building with 3rd
floor collapse
Maria Gabriella Mulas
Soft story at mid level
28
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
Soft story at mid level
29
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 37
FALLAS MOMENTOS TORSIONALES
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
Earthquake Damage - Part II 4/22/2012
5
Maria Gabriella Mulas 25
Soft Story, L’Aquila 2009
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Soft story at mid level
27
10-story SRC
building with 3rd
floor collapse
Maria Gabriella Mulas
Soft story at mid level
28
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
Soft story at mid level
29
Mid story collapse,
Kobe earthquake
Maria Gabriella Mulas
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
Proyecto Estructural - Prof. Michele Casarin 38
FALLAS MOMENTOS TORSIONALES
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 39
FALLAS MOMENTOS TORSIONALES
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 40
FALLAS MOMENTOS TORSIONALES
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 41
FALLAS POR CORTE
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
Earthquake Damage - Part II 4/22/2012
6
Maria Gabriella Mulas
Torsional moments
31 Maria Gabriella Mulas
Torsional moments
32
Torsional failure of the
top of the column
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Pounding - 2
34 Maria Gabriella Mulas
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
35 Maria Gabriella Mulas
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.
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 42
FALLAS POR CORTE
Earthquake Damage - Part II 4/22/2012
7
Maria Gabriella Mulas
Shear failure – Mt McKinley Apartments
Great Alaska Earthquake, 1964
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.
Maria Gabriella Mulas 38
Shear failure
Maria Gabriella Mulas 39
Shear failure (short column)
Maria Gabriella Mulas 40
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 41
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 42
Shear damage (L’Aquila 2009)
Earthquake Damage - Part II 4/22/2012
7
Maria Gabriella Mulas
Shear failure – Mt McKinley Apartments
Great Alaska Earthquake, 1964
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.
Maria Gabriella Mulas 38
Shear failure
Maria Gabriella Mulas 39
Shear failure (short column)
Maria Gabriella Mulas 40
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 41
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 42
Shear damage (L’Aquila 2009)
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 43
FALLAS POR CORTE (COLUMNA CORTA)
Earthquake Damage - Part II 4/22/2012
7
Maria Gabriella Mulas
Shear failure – Mt McKinley Apartments
Great Alaska Earthquake, 1964
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.
Maria Gabriella Mulas 38
Shear failure
Maria Gabriella Mulas 39
Shear failure (short column)
Maria Gabriella Mulas 40
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 41
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 42
Shear damage (L’Aquila 2009)
Earthquake Damage - Part II 4/22/2012
7
Maria Gabriella Mulas
Shear failure – Mt McKinley Apartments
Great Alaska Earthquake, 1964
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.
Maria Gabriella Mulas 38
Shear failure
Maria Gabriella Mulas 39
Shear failure (short column)
Maria Gabriella Mulas 40
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 41
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 42
Shear damage (L’Aquila 2009)
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 44
FALLAS POR CORTE
Earthquake Damage - Part II 4/22/2012
7
Maria Gabriella Mulas
Shear failure – Mt McKinley Apartments
Great Alaska Earthquake, 1964
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.
Maria Gabriella Mulas 38
Shear failure
Maria Gabriella Mulas 39
Shear failure (short column)
Maria Gabriella Mulas 40
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 41
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 42
Shear damage (L’Aquila 2009)
Earthquake Damage - Part II 4/22/2012
7
Maria Gabriella Mulas
Shear failure – Mt McKinley Apartments
Great Alaska Earthquake, 1964
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.
Maria Gabriella Mulas 38
Shear failure
Maria Gabriella Mulas 39
Shear failure (short column)
Maria Gabriella Mulas 40
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 41
Shear failure (short column)
L’Aquila 2009
Maria Gabriella Mulas 42
Shear damage (L’Aquila 2009)
Earthquake Damage - Part II 4/22/2012
8
Maria Gabriella Mulas 43
Shear damage (L’Aquila 2009)
Maria Gabriella Mulas 44
Flexural failure of columns
Maria Gabriella Mulas 45
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.
Maria Gabriella Mulas 46
Flexural failure
Kobe earthquake, flexural failure
Hanshin expressway
Maria Gabriella Mulas 47
Plastic hinge at the base of a column
Maria Gabriella Mulas 48
Column failure
Insufficient transversal reinforcement results in lack of
confinement for columns
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 45
FALLAS POR FLEXION
Earthquake Damage - Part II 4/22/2012
8
Maria Gabriella Mulas 43
Shear damage (L’Aquila 2009)
Maria Gabriella Mulas 44
Flexural failure of columns
Maria Gabriella Mulas 45
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.
Maria Gabriella Mulas 46
Flexural failure
Kobe earthquake, flexural failure
Hanshin expressway
Maria Gabriella Mulas 47
Plastic hinge at the base of a column
Maria Gabriella Mulas 48
Column failure
Insufficient transversal reinforcement results in lack of
confinement for columns
Earthquake Damage - Part II 4/22/2012
8
Maria Gabriella Mulas 43
Shear damage (L’Aquila 2009)
Maria Gabriella Mulas 44
Flexural failure of columns
Maria Gabriella Mulas 45
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.
Maria Gabriella Mulas 46
Flexural failure
Kobe earthquake, flexural failure
Hanshin expressway
Maria Gabriella Mulas 47
Plastic hinge at the base of a column
Maria Gabriella Mulas 48
Column failure
Insufficient transversal reinforcement results in lack of
confinement for columns
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 46
FALLAS POR FLEXION
Earthquake Damage - Part II 4/22/2012
8
Maria Gabriella Mulas 43
Shear damage (L’Aquila 2009)
Maria Gabriella Mulas 44
Flexural failure of columns
Maria Gabriella Mulas 45
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.
Maria Gabriella Mulas 46
Flexural failure
Kobe earthquake, flexural failure
Hanshin expressway
Maria Gabriella Mulas 47
Plastic hinge at the base of a column
Maria Gabriella Mulas 48
Column failure
Insufficient transversal reinforcement results in lack of
confinement for columns
Earthquake Damage - Part II 4/22/2012
8
Maria Gabriella Mulas 43
Shear damage (L’Aquila 2009)
Maria Gabriella Mulas 44
Flexural failure of columns
Maria Gabriella Mulas 45
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.
Maria Gabriella Mulas 46
Flexural failure
Kobe earthquake, flexural failure
Hanshin expressway
Maria Gabriella Mulas 47
Plastic hinge at the base of a column
Maria Gabriella Mulas 48
Column failure
Insufficient transversal reinforcement results in lack of
confinement for columns
EFECTOS SISMICOS
Proyecto Estructural - Prof. Michele Casarin 47
FALLAS EN NODOS
Earthquake Damage - Part II 4/22/2012
9
Maria Gabriella Mulas 49
Failure of beam-column node
Maria Gabriella Mulas 50
Test on a beam-column node
Earthquake Damage - Part II 4/22/2012
9
Maria Gabriella Mulas 49
Failure of beam-column node
Maria Gabriella Mulas 50
Test on a beam-column node
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 48
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.
Element of Mechanics of Particles
KINETICS OF PARTICLES
ENERGY AND MOMENTUM
METHOD
42
Element of Mechanics of Particles
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.
Element of Mechanics of Particles
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.
Elements of Mechanics of Particles
12
Element of Mechanics of Particles
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.
Element of Mechanics of Particles
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.
Element of Mechanics of Particles
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
Element of Mechanics of Particles
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,
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 49
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.
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 50
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.
Elements of Mechanics of Particles
13
Element of Mechanics of Particles
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
Element of Mechanics of Particles
MECHANICAL VIBRATIONS
OF A PARTICLE
50
Element of Mechanics of Particles
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.
Element of Mechanics of Particles
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
Elements of Mechanics of Particles
13
Element of Mechanics of Particles
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
Element of Mechanics of Particles
MECHANICAL VIBRATIONS
OF A PARTICLE
50
Element of Mechanics of Particles
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.
Element of Mechanics of Particles
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
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 51
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
Maria Gabriella Mulas, Paolo Martinelli
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
15 Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 2
16
Pergola at the Macuto Sheraton Hotel damaged by the
earthquake of July 29, 1967 (Venezuela, Caracas)
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
Maria Gabriella Mulas, Paolo Martinelli
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
15 Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 2
16
Pergola at the Macuto Sheraton Hotel damaged by the
earthquake of July 29, 1967 (Venezuela, Caracas)
Dynamics of Structures: an introduction 15/04/2012
5
Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 3
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.
17 Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 4
18
Real case: two simple structural models, one of aluminum
and the other of plexiglass
Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 5
Real case: the amplitude
of oscillations will decay
with time
The energy dissipating
mechanism, called
damping, must be
included in the structural
modeling.
19 Maria Gabriella Mulas, Paolo Martinelli
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
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 52
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.
Dynamics of Structures: an introduction 15/04/2012
5
Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 3
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.
17 Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 4
18
Real case: two simple structural models, one of aluminum
and the other of plexiglass
Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 5
Real case: the amplitude
of oscillations will decay
with time
The energy dissipating
mechanism, called
damping, must be
included in the structural
modeling.
19 Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
3
Maria Gabriella Mulas, Paolo Martinelli
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
9 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Linear modeling of dissipating effects
Viscous damper
zcfD
Equations of motion
Qkqqcqm
11 Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
5
Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 3
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.
17 Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 4
18
Real case: two simple structural models, one of aluminum
and the other of plexiglass
Maria Gabriella Mulas, Paolo Martinelli
SIMPLE STRUCTURES - 5
Real case: the amplitude
of oscillations will decay
with time
The energy dissipating
mechanism, called
damping, must be
included in the structural
modeling.
19 Maria Gabriella Mulas, Paolo Martinelli
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
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 53
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)
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 54
ESTRUCTURAS 1 GRADO DE LIBERTAD
Dynamics of Structures: an introduction 15/04/2012
6
Maria Gabriella Mulas, Paolo Martinelli
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.
21 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
23 Maria Gabriella Mulas, Paolo Martinelli
Force-displacement relation: linearly elastic
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
Dynamics of Structures: an introduction 15/04/2012
6
Maria Gabriella Mulas, Paolo Martinelli
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.
21 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
23 Maria Gabriella Mulas, Paolo Martinelli
Force-displacement relation: linearly elastic
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
Dynamics of Structures: an introduction 15/04/2012
6
Maria Gabriella Mulas, Paolo Martinelli
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.
21 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
23 Maria Gabriella Mulas, Paolo Martinelli
Force-displacement relation: linearly elastic
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
Dynamics of Structures: an introduction 15/04/2012
6
Maria Gabriella Mulas, Paolo Martinelli
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.
21 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
23 Maria Gabriella Mulas, Paolo Martinelli
Force-displacement relation: linearly elastic
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
Dynamics of Structures: an introduction 15/04/2012
7
Maria Gabriella Mulas, Paolo Martinelli
Force-displacement relation: linearly elastic
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.
25 Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
7
Maria Gabriella Mulas, Paolo Martinelli
Force-displacement relation: linearly elastic
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.
25 Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
8
Maria Gabriella Mulas, Paolo Martinelli
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
29 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: external force
D’Alembert’s principle
The idealized one-story frame is subjected to an external
dynamic force p(t) in the direction of u
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
Maria Gabriella Mulas, Paolo Martinelli
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
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 55
ESTRUCTURAS 1 GRADO DE LIBERTAD
LA ESTRUCTURA IDEALIZADA DE UN 1 PISO SUJETA A UNA FUERZA DINAMICA P(t) EN LA DIRECCION DE u:
2nda LEY DE NEWTON:
Dynamics of Structures: an introduction 15/04/2012
8
Maria Gabriella Mulas, Paolo Martinelli
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
29 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: external force
D’Alembert’s principle
The idealized one-story frame is subjected to an external
dynamic force p(t) in the direction of u
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
Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
8
Maria Gabriella Mulas, Paolo Martinelli
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
29 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: external force
D’Alembert’s principle
The idealized one-story frame is subjected to an external
dynamic force p(t) in the direction of u
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
Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
8
Maria Gabriella Mulas, Paolo Martinelli
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
29 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: external force
D’Alembert’s principle
The idealized one-story frame is subjected to an external
dynamic force p(t) in the direction of u
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
Maria Gabriella Mulas, Paolo Martinelli
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
ELASTICO:
INELASTICO:
Dynamics of Structures: an introduction 15/04/2012
8
Maria Gabriella Mulas, Paolo Martinelli
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
29 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: external force
D’Alembert’s principle
The idealized one-story frame is subjected to an external
dynamic force p(t) in the direction of u
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
Maria Gabriella Mulas, Paolo Martinelli
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
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 56
ESTRUCTURAS 1 GRADO DE LIBERTAD
Dynamics of Structures: an introduction 15/04/2012
8
Maria Gabriella Mulas, Paolo Martinelli
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
29 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: external force
D’Alembert’s principle
The idealized one-story frame is subjected to an external
dynamic force p(t) in the direction of u
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
Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
8
Maria Gabriella Mulas, Paolo Martinelli
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
29 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: external force
D’Alembert’s principle
The idealized one-story frame is subjected to an external
dynamic force p(t) in the direction of u
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
Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
9
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: earthquake excitation
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.
Maria Gabriella Mulas, Paolo Martinelli
PROBLEM STATEMENT AND
SOLUTION METHODS
36
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 57
ESTRUCTURAS 1 GRADO DE LIBERTAD
Dynamics of Structures: an introduction 15/04/2012
9
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: earthquake excitation
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.
Maria Gabriella Mulas, Paolo Martinelli
PROBLEM STATEMENT AND
SOLUTION METHODS
36
Dynamics of Structures: an introduction 15/04/2012
9
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: earthquake excitation
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.
Maria Gabriella Mulas, Paolo Martinelli
PROBLEM STATEMENT AND
SOLUTION METHODS
36
Dynamics of Structures: an introduction 15/04/2012
9
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
Equation of motion: earthquake excitation
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.
Maria Gabriella Mulas, Paolo Martinelli
PROBLEM STATEMENT AND
SOLUTION METHODS
36
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 58
ESTRUCTURAS 1 GRADO DE LIBERTAD
METODO DE SOLUCION:
-Se conoce la masa, rigidez y coeficiente de amortigumiento. Se conoce la excitacion externa ya sea en forma de una fuerza
dinamica P(t) o en forma de desplazamiento del suelo Ug(t). Las condiciones de inicio son U=0
-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.
Dynamics of Structures: an introduction 15/04/2012
11
Maria Gabriella Mulas, Paolo Martinelli
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
41 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
Dynamics of Structures: an introduction 15/04/2012
11
Maria Gabriella Mulas, Paolo Martinelli
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
41 Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
Maria Gabriella Mulas, Paolo Martinelli
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
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 59
VIBRACION LIBRE DE ESTRUCTURAS SIN AMORTIGUAMIENTO
Vibration of SDOF systems 15/04/2012
3
Maria Gabriella Mulas
Free Vibrations of Particles. Simple
Harmonic Motion
Free vibrations of a 1-story undamped frame
9 Maria Gabriella Mulas
Free Vibrations of Particles. Simple
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
Maria Gabriella Mulas
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.
Maria Gabriella Mulas
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
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 60
VIBRACION LIBRE DE ESTRUCTURAS CON AMORTIGUAMIENTO
Vibration of SDOF systems 15/04/2012
4
Maria Gabriella Mulas
Sample Problem 1
13
mkN6mkN4 21 kk • Springs in series:
- determine the spring constant for equivalent spring
- apply the approximate relations for the harmonic
motion of a spring-mass system
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
Maria Gabriella Mulas
FREE VIBRATION
DAMPED SYSTEMS
14
Maria Gabriella Mulas
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.
Maria Gabriella Mulas
Damped Free Vibrations - 2
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
- nonvibratory motion
• Light damping: c < cc
tCtCex ddtmc
cossin 212
2
1c
ndc
cdamped frequency
Vibration of SDOF systems 15/04/2012
4
Maria Gabriella Mulas
Sample Problem 1
13
mkN6mkN4 21 kk • Springs in series:
- determine the spring constant for equivalent spring
- apply the approximate relations for the harmonic
motion of a spring-mass system
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
Maria Gabriella Mulas
FREE VIBRATION
DAMPED SYSTEMS
14
Maria Gabriella Mulas
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.
Maria Gabriella Mulas
Damped Free Vibrations - 2
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
- nonvibratory motion
• Critical damping: c = cc tnetCCx 21 - double roots
- nonvibratory motion
• Light damping: c < cc
tCtCex ddtmc cossin 21
2
2
1c
ndc
cdamped frequency
Vibration of SDOF systems 15/04/2012
5
Maria Gabriella Mulas
Damped Free Vibrations - 3
17
• Light damping: c < cc
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
Maria Gabriella Mulas
Damped free vibrations - 4
18
cc
c
Critical damping is the smallest value of damping that inhibits
oscillations completely
Structures are usually underdamped
Maria Gabriella Mulas
Damped free vibrations - 5
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
Maria Gabriella Mulas
Damped free vibrations - 6
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.
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 61
VIBRACION LIBRE DE ESTRUCTURAS CON AMORTIGUAMIENTO
Vibration of SDOF systems 15/04/2012
5
Maria Gabriella Mulas
Damped Free Vibrations - 3
17
• Light damping: c < cc
tCtCex ddtmc cossin 21
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
Note:in this figure x0 is to be read xm
Maria Gabriella Mulas
Damped free vibrations - 4
18
cc
c
Critical damping is the smallest value of damping that inhibits
oscillations completely
Structures are usually underdamped
Maria Gabriella Mulas
Damped free vibrations - 5
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
Maria Gabriella Mulas
Damped free vibrations - 6
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
Vibration of SDOF systems 15/04/2012
5
Maria Gabriella Mulas
Damped Free Vibrations - 3
17
• Light damping: c < cc
tCtCex ddtmc cossin 21
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
Note:in this figure x0 is to be read xm
Maria Gabriella Mulas
Damped free vibrations - 4
18
cc
c
Critical damping is the smallest value of damping that inhibits
oscillations completely
Structures are usually underdamped
Maria Gabriella Mulas
Damped free vibrations - 5
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
Maria Gabriella Mulas
Damped free vibrations - 6
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
Vibration of SDOF systems 15/04/2012
5
Maria Gabriella Mulas
Damped Free Vibrations - 3
17
• Light damping: c < cc
tCtCex ddtmc cossin 21
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
Note:in this figure x0 is to be read xm
Maria Gabriella Mulas
Damped free vibrations - 4
18
cc
c
Critical damping is the smallest value of damping that inhibits
oscillations completely
Structures are usually underdamped
Maria Gabriella Mulas
Damped free vibrations - 5
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
Maria Gabriella Mulas
Damped free vibrations - 6
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
Misma masa y rigidez
Vibration of SDOF systems 15/04/2012
6
Maria Gabriella Mulas
Damped free vibrations - 7
Effect of damping on period
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Real dampers - 1
24
Vibracion forzada de un sistema ocurre cuando es sometida a fuerzas periodicas o desplazamientos periodicos
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 62
VIBRACION FORZADA DE ESTRUCTURAS SIN AMORTIGUAMIENTO
Vibration of SDOF systems 15/04/2012
7
Maria Gabriella Mulas
Real dampers - 2
25 Maria Gabriella Mulas
FORCED VIBRATIONS OF
UNDAMPED SYSTEMS
HARMONIC FORCE
26
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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,
Vibration of SDOF systems 15/04/2012
8
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Harmonic vibrations of undamped systems
k/p0u
00u
2.0/
0n
n
30
Harmonic force
System response for
k
pust
00
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Harmonic Vibrations:
Displacement Response Factor
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”
Vibration of SDOF systems 15/04/2012
8
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Harmonic vibrations of undamped systems
k/p0u
00u
2.0/
0n
n
30
Harmonic force
System response for
k
pust
00
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Harmonic Vibrations:
Displacement Response Factor
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”
Vibration of SDOF systems 15/04/2012
8
Maria Gabriella Mulas
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
n
nnn tsin
/k
ptsin
/k
putcosutu
20
20
1
1
1
100
tsin/k
ptsinBtcosAtu
nnn 2
0
1
1
Particular solution
Complementary solution
Maria Gabriella Mulas
Harmonic vibrations of undamped systems
k/p0u
00u
2.0/
0n
n
30
Harmonic force
System response for
k
pust
00
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Harmonic Vibrations:
Displacement Response Factor
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”
Vibracion forzada de un sistema ocurre cuando es sometida a fuerzas periodicas o desplazamientos periodicos
DINAMICA DE ESTRUCTURAS
Proyecto Estructural - Prof. Michele Casarin 63
VIBRACION FORZADA DE ESTRUCTURAS CON AMORTIGUAMIENTO
Vibration of SDOF systems 15/04/2012
10
Maria Gabriella Mulas
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)
Maria Gabriella Mulas
FORCED VIBRATIONS OF
DAMPED SYSTEMS
HARMONIC FORCE
38
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Damped Forced Harmonic Vibrations – 2
General solution
40
Vibration of SDOF systems 15/04/2012
10
Maria Gabriella Mulas
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)
Maria Gabriella Mulas
FORCED VIBRATIONS OF
DAMPED SYSTEMS
HARMONIC FORCE
38
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Damped Forced Harmonic Vibrations – 2
General solution
40
Vibration of SDOF systems 15/04/2012
11
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Damped Forced Harmonic Vibrations – 4
Response for n
42
000
050
1
uu
.
/ n
nDFor lightly damped systems:
The response is now bounded!
Maria Gabriella Mulas
Damped Forced Harmonic Vibrations – 5
Effect of damping
43
nDFor lightly damped systems:
Maria Gabriella Mulas
Damped Forced Harmonic Vibrations – 6
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
Proyecto Estructural - Prof. Michele Casarin 64
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
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
Pearson/ Prentice Hall 2007
2
Maria Gabriella Mulas
DESCRIPTION OF
EARTHQUAKE EXCITATION
3 Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
2
Maria Gabriella Mulas
Recorded PGA, Loma Prieta Earthquake of
October 17, 1989
5
Values recorded
at many different
locations
Maria Gabriella Mulas
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
Maria Gabriella Mulas
EQUATION OF MOTION AND
RESPONSE PARAMETERS
7 Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 65
ECUACION DE MOVIMIENTO Y PARAMETROS DE RESPUESTA
EL DESPLAZAMIENTO U(t) DEL SISTEMA SERIA EL MISMO QUE OBTUVIERAMOS SI
APLICARAMOS UNA FUERZA EFFECTIVA:
Response spectrum of linear systems 22/04/2012
2
Maria Gabriella Mulas
Recorded PGA, Loma Prieta Earthquake of
October 17, 1989
5
Values recorded
at many different
locations
Maria Gabriella Mulas
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
Maria Gabriella Mulas
EQUATION OF MOTION AND
RESPONSE PARAMETERS
7 Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
2
Maria Gabriella Mulas
Recorded PGA, Loma Prieta Earthquake of
October 17, 1989
5
Values recorded
at many different
locations
Maria Gabriella Mulas
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
Maria Gabriella Mulas
EQUATION OF MOTION AND
RESPONSE PARAMETERS
7 Maria Gabriella Mulas
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
t0
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
Response spectrum of linear systems 22/04/2012
2
Maria Gabriella Mulas
Recorded PGA, Loma Prieta Earthquake of
October 17, 1989
5
Values recorded
at many different
locations
Maria Gabriella Mulas
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
Maria Gabriella Mulas
EQUATION OF MOTION AND
RESPONSE PARAMETERS
7 Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 66
ECUACION DE MOVIMIENTO Y PARAMETROS DE RESPUESTA
-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.
Response spectrum of linear systems 22/04/2012
3
Maria Gabriella Mulas
Equation of motion for earthquake
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
9 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
11 Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
3
Maria Gabriella Mulas
Equation of motion for earthquake
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
9 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
11 Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 67
ECUACION DE MOVIMIENTO Y PARAMETROS DE RESPUESTA
-Fs ES LA FUERZA ESTATICA EQUIVALENTE
A(t) SE LE LLAMA PSEUDO ACELERACION
Response spectrum of linear systems 22/04/2012
3
Maria Gabriella Mulas
Equation of motion for earthquake
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
9 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
11 Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
3
Maria Gabriella Mulas
Equation of motion for earthquake
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
9 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
11 Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 68
ECUACION DE MOVIMIENTO Y PARAMETROS DE RESPUESTA
-EL CORTE BASAL Y EL MOMENTO DE VOLCAMIENTO
SON DEPENDIENTES DE LA PSEUDO ACELERACION.
-EL CORTANTE BASAL EQUILIBRIA LA FUERZA
ESTATICA EQUIVALENTE
Response spectrum of linear systems 22/04/2012
3
Maria Gabriella Mulas
Equation of motion for earthquake
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
9 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
11 Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
3
Maria Gabriella Mulas
Equation of motion for earthquake
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
9 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
11 Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 69
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.
Response spectrum of linear systems 22/04/2012
4
Maria Gabriella Mulas
RESPONSE SPECTRUM
CONCEPT
13 Maria Gabriella Mulas
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.
Maria Gabriella Mulas
Deformation Response Spectrum for El Centro
Ground Motion
15 Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
4
Maria Gabriella Mulas
RESPONSE SPECTRUM
CONCEPT
13 Maria Gabriella Mulas
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.
Maria Gabriella Mulas
Deformation Response Spectrum for El Centro
Ground Motion
15 Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 70
CONCEPTO DE ESPECTRO DE RESPUESTA
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
Response spectrum of linear systems 22/04/2012
5
Maria Gabriella Mulas
Combined D-V-A response spectrum ( = 2%)
log-log scale
DVA
n
n
n
nT
2
17 Maria Gabriella Mulas
Response spectrum, El Centro ground motion
( = 0, 2, 5,10 and 20 %)
Spectrum covers
a wide range of
periods and
damping values
18
Maria Gabriella Mulas
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 %
Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
4
Maria Gabriella Mulas
RESPONSE SPECTRUM
CONCEPT
13 Maria Gabriella Mulas
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.
Maria Gabriella Mulas
Deformation Response Spectrum for El Centro
Ground Motion
15 Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 71
CONCEPTO DE ESPECTRO DE RESPUESTA
Response spectrum of linear systems 22/04/2012
7
Maria Gabriella Mulas
RESPONSE SPECTRUM
CHARACTERISTICS
25 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
7
Maria Gabriella Mulas
RESPONSE SPECTRUM
CHARACTERISTICS
25 Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 72
CONCEPTO DE ESPECTRO DE RESPUESTA
Response spectrum of linear systems 22/04/2012
8
Maria Gabriella Mulas
Spectrum on normalized scale, 5% damping
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
Response spectrum of linear systems 22/04/2012
8
Maria Gabriella Mulas
Spectrum on normalized scale, 5% damping
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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 DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 73
ESPECTRO ELASTICO DE DISENO
Response spectrum of linear systems 22/04/2012
10
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Elastic design spectrum
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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!
Response spectrum of linear systems 22/04/2012
10
Maria Gabriella Mulas
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
Maria Gabriella Mulas
Elastic design spectrum
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
Maria Gabriella Mulas
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
Maria Gabriella Mulas
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 DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 74
ESPECTRO ELASTICO DE DISENO
Response spectrum of linear systems 22/04/2012
12
Maria Gabriella Mulas
Pseudo-velocity design spectrum
48 Maria Gabriella Mulas
Pseudo-acceleration design spectrum
49
Log scale
Linear scale
Maria Gabriella Mulas
Deformation design spectrum
50 Maria Gabriella Mulas
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 DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 75
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
Maria Gabriella Mulas 2
Acknowledgments
The figures in this powerpointcome from:
Anil K. Chopra
Dynamics of Structures. Theory and Application to Earthquake
Engineering. 3rd edition
Pearson/ Prentice Hall 2007
Color pictures have been downloaded from the web
Maria Gabriella Mulas 3
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
Maria Gabriella Mulas
Damage not economically repairable
Imperial County Services Building after the Imperial
Valley, California earthquake of Oct. 15, 1979
Inelastic Design Spectrum 15/05/2012
2
Maria Gabriella Mulas
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
Maria Gabriella Mulas
DESCRIPTION OF POSSIBLE
NON LINEAR BEHAVIORS
6
Maria Gabriella Mulas
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
Maria Gabriella Mulas 8
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
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 76
ESPECTRO INELASTICO DE DISENO
SISTEMA ELASTICO SIN AMORTIGUAMIENTO SISTEMA ELASTOPLASTICO SIN AMORTIGUAMIENTO
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 77
ESPECTRO INELASTICO DE DISENO
ESPECTRO DE RESPUESTA
Proyecto Estructural - Prof. Michele Casarin 78
ESPECTRO INELASTICO DE DISENO
Inelastic Design Spectrum 15/05/2012
11
Maria Gabriella Mulas
Theory of Ductility Factor
41 Maria Gabriella Mulas 42
Construction of inelastic design spectrum
From the elastic design spectrum, the inelastic design spectrum is obtained by
applying the equations in slide 40
Maria Gabriella Mulas 43
Inelastic pseudoacceleration design
spectrum (84.1th percentile) linear scale
This format of
the inelastic
design
spectrum is
contained in
seismic codes
Maria Gabriella Mulas 44
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
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
80
Proyecto Estructural - Prof. Michele Casarin
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
81
Proyecto Estructural - Prof. Michele Casarin
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.
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
82
Proyecto Estructural - Prof. Michele Casarin
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.
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
83
Proyecto Estructural - Prof. Michele Casarin
VIBRACION LIBRE DE SISTEMAS MGL SIN AMORTIGUAMIENTO
El período natural de vibración es el tiempo requerido para
completer un ciclo de movimiento harmónico en uno de los
modos naturales de vibración. Su inversa es llamada
frecuencia natural.
Las N raices de Wn son conocidas
como los valores eigen (eigenvalues),
de los cuales obtenemos Tn. El primer
período T1 es llamado fundamental.
Luego de conocer Wn podemos
calcular la forma modal Φn
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
84
Proyecto Estructural - Prof. Michele Casarin
VIBRACION LIBRE DE SISTEMAS MGL SIN AMORTIGUAMIENTO
La solución del problema EIGEN no da valores de
amplitud de Φ sino su forma. Los N vectores de Φn son
los modos naturales de vibración, cada uno asociado
con un T. Son llamados naturales porque son
propiedades del sistema, que dependen solo de la
rigidez y de la masa.
Una matriz 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.
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
85
Proyecto Estructural - Prof. Michele Casarin
VIBRACION LIBRE DE SISTEMAS MGL CON AMORTIGUAMIENTO
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
86
Proyecto Estructural - Prof. Michele Casarin
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.
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
87
Proyecto Estructural - Prof. Michele Casarin
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.
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
88
Proyecto Estructural - Prof. Michele Casarin
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.
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
89
Proyecto Estructural - Prof. Michele Casarin
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.
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
90
Proyecto Estructural - Prof. Michele Casarin
COMBINACION DE RESPUESTAS MODALES
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
91
Proyecto Estructural - Prof. Michele Casarin
COMBINACION DE RESPUESTAS MODALES
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
92
Proyecto Estructural - Prof. Michele Casarin
EJEMPLO
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
93
Proyecto Estructural - Prof. Michele Casarin
EJEMPLO
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
94
Proyecto Estructural - Prof. Michele Casarin
EJEMPLO
SISTEMAS DE VARIOS GRADOS DE LIBERTAD
95
Proyecto Estructural - Prof. Michele Casarin
EJEMPLO
CONCEPTOS DE DISEÑO SÍSMICO
96
Proyecto Estructural - Prof. Michele Casarin
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.
CONCEPTOS DE DISEÑO SÍSMICO
97
Proyecto Estructural - Prof. Michele Casarin
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.
CONCEPTOS DE DISEÑO SÍSMICO
98
Proyecto Estructural - Prof. Michele Casarin
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
CONCEPTOS DE DISEÑO SÍSMICO
99
Proyecto Estructural - Prof. Michele Casarin
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
CONCEPTOS DE DISEÑO SÍSMICO
100
Proyecto Estructural - Prof. Michele Casarin
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.
CONCEPTOS DE DISEÑO SÍSMICO
101
Proyecto Estructural - Prof. Michele Casarin
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.
CONCEPTOS DE DISEÑO SÍSMICO
102
Proyecto Estructural - Prof. Michele Casarin
ESTADOS LÍMITES DE DISEÑO SÍSMICO
CONCEPTOS DE DISEÑO SÍSMICO
103
Proyecto Estructural - Prof. Michele Casarin
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
CONCEPTOS DE DISEÑO SÍSMICO
104
Proyecto Estructural - Prof. Michele Casarin
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
CONCEPTOS DE DISEÑO SÍSMICO
105
Proyecto Estructural - Prof. Michele Casarin
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.
CONCEPTOS DE DISEÑO SÍSMICO
106
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN PLANTA
CONCEPTOS DE DISEÑO SÍSMICO
107
Proyecto Estructural - Prof. Michele Casarin
TORSION
CONCEPTOS DE DISEÑO SÍSMICO
108
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN PLANTA
CONCEPTOS DE DISEÑO SÍSMICO
109
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN PLANTA
CONCEPTOS DE DISEÑO SÍSMICO
110
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN PLANTA
CONCEPTOS DE DISEÑO SÍSMICO
111
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
112
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
113
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
114
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
115
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
116
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
117
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
118
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN
CONCEPTOS DE DISEÑO SÍSMICO
119
Proyecto Estructural - Prof. Michele Casarin
REGULARIDAD EN ELEVACIÓN