TOPIC XII: DYNAMICS OF VIBRATION; ATTENUATION, ELASTIC AND INELASTIC
RESPONSE SPECTRA, & EFFECTS OF SOILS ON GROUND MOTION; LIQUEFACTION
Reporters:
Professor:
Abadesco, Laica
Bulquerin, Shenna May
Laurilla, Antonette Mae
Lim, Kessah Marie
BSCE -4B
Engr. Ma Adeline D. Diaz
DYNAMICS OF VIBRATION; ATTENUATION
VIBRATION
 An oscillation of the parts of a fluid or an elastic solid whose equilibrium has been disturbed, or
of an electromagnetic wave.
 Vibration is defined as the oscillating, reciprocating, or other periodic motion of a rigid or elastic
body or medium forced from a position or state of equilibrium.
TYPES OF VIBRATION
• Free Vibration
- Occurs when a mechanical system is set in motion with an initial input and allowed to vibrate freely.
The mechanical system vibrates at one or more of its natural frequencies and damps down to
motionlessness.
Classification of Free Vibrations:
 Longitudinal
 Transverse
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Torsional
• Forced Vibration
- Is when a time-varying disturbance is applied to a mechanical system. The disturbance can be a
periodic and steady-state input, a transit input or a random input.
• Damped Vibration
- Is when an energy of a vibrating system is gradually dissipated by friction and other resistances.
SEISMIC
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Earthquakes and underground explosions can release a lot of energy. That energy ripples away
from its source in a variety of ways.
Some of these waves travel forward and back through the materials. Other waves travel like
ocean waves where they make the material they pass through moves up and down.
Seismic waves are vibration or disturbances that propagates from source, such as explosion or
shockwaves, through the earth until they encounter a reflecting surface and are reflected into a
detector.
ATTENUATION
 When you throw a pebble in a pond, it makes waves on the surface that move out from the place
where the pebble entered the water.
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The waves are largest where they are formed and gradually get smaller as they move away. This
decrease in size, or amplitude, of the waves is called attenuation.
Seismic waves also become attenuated as they move away from the earthquakes source.
SEISMIC ATTENUATION
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describes the energy loss experienced by seismic waves as they propagate. It is controlled by
the temperature, composition, melt content, and volatile content of the rocks through which the
waves travel.
For these reasons, seismic attenuation has the potential to be a valuable source of information
about the Earth’s interior, complementing seismic velocity and allowing more definite
conclusions to be drawn.
Seismic attenuation is an intrinsic property of rocks causing dissipation(waste) of energy as
seismic waves propagate through the subsurface. It results in the decay of amplitude of the
seismic waves.
Attenuation is related to velocity dispersion. The energy of seismic wave is conserved if it
travels through a perfectly elastic medium. Propagating seismic waves loose energy due to.
a. Geometrical spreading (wavefront radiating from a point source is distributed over a
spherical surface of increasing size). Seismic wave amplitudes vary as they travel across
the Earth. As the wavefront moves out from the source, the initial energy released in the
seismic wave is spread over an increasing area and therefore the intensity of the wave
decreases with distance (the case of geometric spreading).
b. Absorption (anelastic attenuation). This is the energy loss due to anelastic processes or
internal friction during wave propagation. This is called intrinsic or anelastic attenuation.
Intrinsic attenuation occurs mostly during shear wave motion associated with lateral
movements of lattice and grain boundaries.
c. Scattering (elastic attenuation). Attenuation is frequency-dependent and is strongly
affected by scattering. Scattering is an important factor caused by the heterogeneity of
the Earth's crust and mantle and availability of hydrocarbon reservoirs. Scattering theory
states that rocks containing oil and gas can cause energy attenuation in seismic waves,
particularly the high frequency waves, passing through them.
ATTENUATION STRUCTURE THROUGH THE EARTH'S SUBSURFACE
Global attenuation model has been obtained from the normal modes and surface waves displays the highest
attenuation in the asthenosphere and inner core and low attenuation in the lithosphere and lower mantle.
Global surface wave attenuation models correspond closely with shear velocity, suggesting that the
temperature is the main controlling factor.
SEISMIC ATTENUATION ANDROCKPROPERTIES
 The attenuation is directly related to the composition of the Earth's layers. Thus, it changes
whenever the changes in the layering composition occur.
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This property of attenuation allows scientists to identify variations in rock properties.
Measurements of seismic attenuation can also provide information of fluid content or zones of
high permeability.
Greater porosity and higher Vp/Vs correspond to higher attenuation.
ELASTIC RESPONSE SPECTRA
Response Spectrum
A response spectrum provides the
maximum response (e.g., acceleration,
velocity, or displacement) of an idealized
SDOF system for different natural
frequencies (or periods). It is used to
estimate the peak response of any structure
with a given natural frequency subjected to
a particular ground motion.
Dampers
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The dampers helps the tower to
balance and reduces the swaying of the
buildings due to heavy wind or
earthquake.
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Dampers are strategically placed
in the building structure to control floor
vibrations and building displacement,
cater for occupancy comfort and
mitigate against major seismic events.
Elastic Response Spectra
Elastic response spectra are derived analytically
by evaluating the Duhamel integral which
provides the total displacement response of
Single Degree of Freedom (SDOF) systems
subjected to earthquake loading. Since
superposition applies (for elastic systems)
convolution integral is valid. The principle of
superposition states that the effect of a number
of simultaneously applied actions is equivalent
superposition of their individual effects
considered one at a time. This design spectrum
is intended for the design of new structures, or
the seismic safety evaluation of existing
structures, to resist future earthquakes.
The equation of dynamic equilibrium for linear elastic structural system with mass (m), stiffness
(k), and damping (c) is as follows:
Where the term ü is the ground acceleration, Thus, expresses the equilibrium of inertial (mü),
damping (cú) and elastic (ku) forces and the earthquake loading –müg.
Natural Time Period (T)
Natural Time period is the time need to finish one complete cycle of vibration. It
depends on:
Mass of the structure (m)
Stiffness of the structure (k)
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The higher the mass (m), the higher the natural time period (t)
 The higher the stiffness (k), the lower the natural time period (t)
Fast Vibration
Fast Vibration
Slow Vibration
Slow Vibration
Displacement/Deformation Response Spectra
The family of curves corresponding to Sd values is called
displacement response spectrum.
Velocity Response Spectra
The family of curves corresponding to Sd values is
called displacement response spectrum.
Acceleration Response Spectra
The Family of curves corresponding to a values is called
acceleration response specrum.
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Relationship Between the 3 Spectra:
Elastic spectra are useful tools for structural design and assessment. They, however, do not
account for inelasticity stiffness reduction and strength degradation experienced by structures
during severe earthquakes. Structural systems are not designed to resist earthquake forces in
their elastic range, but for very few cases because of the economy of the construction.
INELASTIC SPECTRA
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Inelastic response spectra describe how structures
behave during strong earthquakes when they go
beyond their elastic limit and experience permanent
deformation.
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Unlike elastic response spectra, which assume
buildings return to their original shape, inelastic
response spectra account for damage, energy
dissipation, and nonlinear behavior.
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Engineers use these spectra to design earthquakeresistant structures by predicting how much
deformation is acceptable before failure.
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Elastic vs. Inelastic Response Spectra
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Elastic Response: Structure returns to its original shape
after deformation.
Inelastic Response: Structure undergoes permanent
deformation when stressed beyond its yield point.
The yield point is the transition between elastic and
inelastic behavior.
Factors Affecting Inelastic Response
1. Material Properties
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Strength, ductility, and yield stress determine
how a material behaves under stress.
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Brittle materials (e.g., concrete) fail suddenly, while ductile materials (e.g.,
steel) absorb more energy
2.
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Loading Conditions
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Earthquake magnitude, duration, and
frequency influence inelastic response.
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Stronger & longer earthquakes cause
more permanent deformation.
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3.
Structural Design
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Buildings with reinforced structures and
damping systems withstand inelastic
deformations better.
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Proper bracing and flexible designs
improve earthquake resilience.
4. Soil-Structure Interaction
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Soft soil amplifies ground shaking,
increasing inelastic deformation.
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Hard rock foundations reduce
excessive movement.
Force-Deformation Relationship
When structures enter the inelastic range, they undergo
plastic deformation, allowing them to absorb and dissipate
seismic energy.
Key Concepts:
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Elastic Region: The initial phase where structures return to their original shape after
deformation.
Yield Point: The threshold beyond which permanent deformation begins.
Plastic Deformation: The inelastic phase where energy dissipation occurs.
Hysteresis Loop: A graphical representation of cyclic loading, showing how structures lose
and absorb energy over repeated cycles.
Practical Example:
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A steel frame under cyclic loading will deform plastically, forming a stable hysteresis loop,
indicating its energy dissipation capacity.
Brittle structures (e.g., unreinforced concrete) exhibit a sharp drop in load capacity after
yielding, leading to sudden failure.
Importance of Inelastic Response Spectra in Earthquake Engineering
Engineers use inelastic response spectra to design earthquake-resistant structures.
Structures should withstand strong earthquakes without sudden collapse.
Helps in defining seismic codes and building standards.
EFFECTS OF SOILS ON GROUND MOTION; SOIL LIQUEFACTION
Soil Liquefaction
also called earthquake liquefaction, is a phenomenon in which the strength and stiffness of a
soil is reduced by earthquake shaking or other rapid loading.
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Effects of Soil Liquefaction
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1. Loss of bearing strength - In soil liquefaction, the bearing capacity of soil is being reduced.
The combination of liquefaction and earthquake forces causes the buildings to lose their
equilibrium.
2. Lateral spreading - When surface material extends or spreads on gentle slopes. Lateral
spreading causes damage to foundations of buildings, pipelines, railway lines and cause
shaking at pile due to increased lateral loads.
3. Sand boiling - When liquefaction happens under a tightly compacted crust, the water
pressure beneath the surface causes the water to escape in the form of a bubble. These
result in the formation of boiling water.
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4. Flow failures - are the most catastrophic ground failures caused by liquefaction. These
failures commonly displace large masses of soil laterally. Flows develop in loose saturated
sands or silts on relatively steep slopes.
5. Settlement - The downward movement or the sinking of a structure's foundation. It is
mostly caused by changes in the underlying soil.
Examples of Soil Liquefaction due to Earthquake
The 1994 Niigata earthquake caused widespread liquefaction in Niigata,
Japan which destroyed many buildings.
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Lateral spreading due to magnitude 7.2 Bohol Earthquake last October
2013.
On July 27, 2022, a magnitude 7.3 earthquake shook several parts of
Luzon. Its epicenter was recorded in Abra and its intensity was felt in
Metro Manila. The tremors are said to have caused mysterious pits.
According to PHILVOCS, these pits are called 'sand boils.'
A landslide blocking a road on the Halsema Highway near to
Bontoc last July 2022 due to magnitude 7.0 earthquake in Abra
Province, a northern part of Philippines.
Row houses in San Francisco, California tilted by soil liquefaction
after the San Francisco earthquake of 1906.
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