Earthquake Engineering Analysis

EARTHQUAKE
ENGINEERING
ANALYSIS
Shieh-Kung Huang
黃謝恭
1

Shieh-Kung
Huang
Copyright © 2016 by Pearson Education, Inc. All rights reserved.
2021 HAITI EARTHQUAKE
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在我們開始前，
要知道地震是真的離我們很近。
讓我們回到約半年之前…

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Courtesy of C. S. Prentice et al.

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CAUSES AND EFFECTS OF NATURAL HAZARDS
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Human death toll caused by major natural hazards
Earthquake financial losses
Courtesy of Estrada and Lee, 2008
Chapter 1.1 Introduction of Earthquake

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INTRODUCTION OF SEISMOLOGY
Chapter Outline
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CHAPTER 1
1.1 Introduction of Earthquake
1.2 Tectonic Plates
1.3 Effects of Earthquakes
1.4 Earthquakes in Taiwan
1.5 Earthquake Waves
1.7 Earthquake-related Application

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NATURAL HAZARDS
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CAUSES AND EFFECTS OF NATURAL HAZARDS
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Correlation between typical hazard events and social and economic consequences

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EARTHQUAKE MYTHS
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中國古老的傳說中，地震是一種稱為鰲魚的動物所引起的；日本的神話裡，地震是鯰魚在興風作浪；
希臘有海神波賽頓引發地震的神話故事；北美有些原住民部落相信大地是由幾隻烏龜駝著的，每當烏龜彼
此爭吵，大地就會顫動；過去中美洲的住民則以為抬著大地的神會偶爾將大地抖一抖。
國家地震工程研究中心“安全耐震的家認識地震工程”

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GLOBAL DISTRIBUTION OF EARTHQUAKES
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From: http://ds.iris.edu/seismon/index.phtml

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GLOBAL DISTRIBUTION OF EARTHQUAKES
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Courtesy of internet geography https://www.internetgeography.net/igcse-geography/the-natural-environment-igcse-geography/

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EARTHQUAKES AND TECTONIC PLATES
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Courtesy of Advanced Subsidary and Advanced GCE Geography
https://geoplatetectonics.weebly.com/earthquakes-and-volcanoes.html
Courtesy of USGS https://pubs.usgs.gov/gip/dynamic/fire.html
世界上最常發生地震的地區，大致與板塊的接合線相符，這些地區被
歸納為三大地震帶：環太平洋地震帶、歐亞地震帶、中洋脊地震帶。
根據統計，全世界超過80%的地震發生在環太平洋地震帶，10%～15%
發生在歐亞地震帶，中洋脊地震帶只有大約5%。
Chapter 1.2 Tectonic Plates

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WORLDWIDE TECTONIC PLATES
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Courtesy of EarthHow https://earthhow.com/7-major-tectonic-plates/

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7 MAJOR TECTONIC PLATES
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Courtesy of EarthHow https://earthhow.com/7-major-tectonic-plates/

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RING OF FIRE
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Convergent boundaries or
subduction zones are where two
plates collide into each other.
These are the most common type
of tectonic plate along the Pacific
Ring of Fire.
The Ring of Fire tectonic
plates collide and sink into the
ocean floor at zones of
subduction. This causes the most
active and violent areas of
earthquakes on the planet.
Courtesy of Wikiwand https://www.wikiwand.com/en/Ring_of_Fire

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EARTH CUTAWAY SCHEMATIC
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地球的構造由內而外可
大致分為：地核、地函、
地殼三個部分。地核位於地
球深度2,900公里以下至地
心處，又分為內核和外核。
地殼就是我們所站著的土地，
平均厚度只有35公里。如果把地
球以蘋果來比喻，地殼的厚度相
當於蘋果皮，不過這個「蘋果皮」
並非完整連續，而是像拼圖一樣，
由一塊一塊的板塊拼湊而成的。
地函是由固態岩石及部分融熔的岩漿所
構成。上部地函的堅硬部分與地殼合稱岩石
圈，厚度約100公里；岩石圈下方有一層部
分融熔的岩漿稱為軟流圈。

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MANTLE CONVECTION
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地殼下方的軟流圈，因為地球內部的高溫產
生旺盛的對流作用，經常有熔岩自中洋脊湧出。
冷卻後的熔岩形成新的岩塊，使中洋脊兩側的板
塊不斷往外擴張，因而對相鄰的板塊造成推擠。
板塊運動的速度緩慢而難以察覺，但隨著板
塊的推擠和變形，持續累積的能量可能在瞬間爆
發，使板塊之間相互錯動而引發地震，因此板塊
交界處的地震發生頻率高。

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TECTONIC PLATES MOVEMENT
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Courtesy of internet geography https://www.internetgeography.net/igcse-
geography/the-natural-environment-igcse-geography/

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TECTONIC PLATES MOVEMENT
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Constructive (divergent) plate boundaries Destructive (convergent) plate boundaries
Conservative (passive) plate boundaries

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EARTHQUAKES IN THE 20TH & 21ST CENTURY
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Chapter 1.3 Effects of Earthquakes

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EARTHQUAKES IN THE 20TH & 21ST CENTURY
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• 2021/7/28 Chignik (USA) Earthquake
• 2017/9/7 Chiapas (Mexico) Earthquake
• 2015/9/16 Illapel (Chile) Earthquake
• 2012/4/11 Indian Ocean Earthquakes
• 2011/3/11 Tōhoku Earthquake 東日本地震
• 2010/2/27 Chile Earthquake 智利地震
• 2010/1/12 Heidi Earthquake 海地地震
• 2008/5/12 Wenchuan (China) Earthquake 汶川地震
• 2004/12/26 Indian Ocean (Sumatra, Indonesia) Earthquake 印度尼西亞地震
• 2001/1/26 Gujarat (India) Earthquake
• 1999/9/21 Chi-chi (Taiwan) Earthquake 集集地震
• 1999/8/17 Kocaeli (Turkey) Earthquake 土耳其地震
• 1995/1/19 Kobe (Japan) Earthquake 日本神戶地震
• 1994/1/17 Northridge (USA) Earthquake 美國北嶺地震
• 1989/10/17 Loma Prieta (USA) Earthquake 美國舊金山地震
• 1985/9/19 Mexico Earthquake 墨西哥地震

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EFFECTS OF EARTHQUAKES
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Earthquakes exact a heavy toll
on all aspects of exposed societal
systems. They can have several
direct and indirect effects as shown
in the following figure.

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EFFECTS OF EARTHQUAKES – LOMA PRIETA EQ
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The San Francisco-Oakland Bay bridge collapsed during Loma
Prieta earthquake on Oct. 17, 1989.

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EFFECTS OF EARTHQUAKES – KOBE EQ
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The Hanshin expressway lies on its side after being toppled by the Kobe earthquake on Jan. 17, 1995.

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EFFECTS OF EARTHQUAKES – KOBE EQ
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Upper 4 floors of 8-storied high kobe city office building in Chuo-ku, which collapsed due to failure
of 6th floor were demolished and lower floors are retrofitted for further use.

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EFFECTS OF EARTHQUAKES – CHI-CHI EQ
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The collapsed Wuchang temple in Jiji after the Chi-chi earthquake on Sep. 21, 1999.
921 Earthquake Museum of Taiwan

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EFFECTS OF EARTHQUAKES – CHI-CHI EQ
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The fault rupture passed through the northern part of the Shih-
Kang dam with an up-thrust of about 8 m, destroying
spillways and gates. Due to its length of 700 m, the dam
behaved as a flexible structure, practically following the
imposed deformation during Chi-chi earthquake(photos
adapted from Hwang 2000).
921 Earthquake Museum of Taiwan
Other photos can be found at
https://www.cna.com.tw/project/20190916-
921earthquake/imgcomparison.html

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EFFECTS OF EARTHQUAKES – INDIAN OCEAN EQ
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A village near the coast of Sumatra lies in ruins after the
Tsunami that struck South East Asia. U.S. Navy photo by
Photographer's Mate 2nd Class Philip A. McDaniel.
Courtesy of Wikipedia https://en.wikipedia.org/wiki/2004_Indian_Ocean_earthquake_and_tsunami
Indian Ocean earthquake on Dec. 26, 2004.

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Courtesy of GEOGRAPHY MYP/GCSE/DP https://www.jkgeography.com/causes-of-tsunami.html Courtesy of USGS https://pubs.usgs.gov/
Courtesy of Ahmed Ismail and Zain Hajee https://ysjournal.com/the-science-behind-tsunamis/

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Most tsunami are caused by large earthquakes on the sea floor when slabs of rock move past each other
suddenly, causing the overlying water to move. The resulting waves move away from the source of the
earthquake event.

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Most tsunami are caused by large earthquakes on the sea floor when slabs of rock move past each other
suddenly, causing the overlying water to move. The resulting waves move away from the source of the
earthquake event.

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EFFECTS OF EARTHQUAKES – TŌHOKU EQ
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Courtesy of NOAA https://www.ngdc.noaa.gov/hazard/11mar2011.html
Fukushima Daiichi nuclear disaster
and the ground radioactive map
during Tōhoku earthquake on Mar.
11, 2011.

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Japanese port city of Sendai in Miyagi
Before After
Courtesy of Kaushik Patowary
https://www.amusingplanet.com/2011
/03/before-and-after-satellite-photos-
of.html

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Sendai Airport
Before After
of.html

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Iwaki City (磐城市)
Before After
of.html

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Natori City (名取市)
Before After
of.html

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of.html
Fukushima nuclear power plant
Before After

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EFFECTS OF EARTHQUAKES – WORLDWIDE DEATHS
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Courtesy of Angelier,1986

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PLATE STRUCTURE OF TAIWAN
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三個板塊交於臺灣的運動造成複雜的褶皺地形和斷層，直到今天，菲律賓海板塊仍然以平均每年7~11
公分的速度向歐亞板塊推擠，使中央山脈海拔高度持續上升，碰撞前緣的海岸山脈，每年以大約2到3公分
的速度長高。
Courtesy of Angelier,1986
Chapter 1.4 Earthquakes in Taiwan

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PLATE STRUCTURE OF TAIWAN
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From 中研院GPS監測成果統合資料

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MOUNTAIN BUILDING IN TAIWAN
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臺灣島的形成，是由於菲律賓海板塊與歐亞板塊互相推擠，
使海底的沉積岩隆起而露出海面。這種因為板塊推擠使岩層產生
隆起、褶皺和斷層的現象，稱為造山運動。
臺灣在地形上以花東縱谷為界，左右分屬不同的板塊，縱谷
以東的海岸山脈屬於菲律賓海板塊，以西的中央山脈及西部山麓
平原屬於歐亞板塊。
Courtesy of 台灣的大地構造
http://homepage.ntu.edu.tw/~tengls/geo-
info_tectonic.htm

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ORIGIN OF FAULT
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板塊的推擠效應就如同推土機一樣，使原本連續的內陸地層發生斷裂，引發內陸地震。

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FAULT AND EARTHQUAKE
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• Earthquake
shaking or trembling of the earth that accompanies rock movements extending anywhere from the
crust to 680 km below the Earth’s surface. It is the release of stored elastic energy caused by sudden
fracture and movement of rocks inside the Earth. Part of the energy released produces seismic waves,
like P, S, and surface waves, that travel outward in all directions from the point of initial rupture. These
waves shake the ground as they pass by. An earthquake is felt if the shaking is strong enough to
cause ground accelerations exceeding approximately 1.0 centimeter/second squared.
• Fault
a fracture or zone of fractures in rock along which the two sides have been displaced relative to
each other. If the main sense of movement on the fault plane is up (compressional; reverse) or down
(extensional; normal), it is called a dip-slip fault. Where the main sense of slip is horizontal the fault is
known as a strike-slip fault. Oblique-slip faults have both strike and dip slip.
• Fault Plane
The plane along which the break or shear of a fault occurs. It is a plane of differential movement,
that can be vertical as in a strike slip fault or inclined like a subduction zone fault.
• Fault zone
Since faults do not usually consist of a single, clean fracture, the term fault zone is used when
referring to the zone of complex deformation that is associated with the fault plane.

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• Dip-slip
Dip-slip faults can be again classified into the types “reverse” and “normal”. A normal fault occurs
when the crusts extended. Alternatively such a fault can be called an extensional fault. The hanging
wall moves downward, relative to the footwall. A downthrown block between two normal faults dipping
towards each other is called a graben. An up throw block between two normal faults dipping away
from each other is called a horst. Low-angle normal faults with regional tectonic significance may be
designated detachment faults.
• Epicenter
the point on the Earth’s surface directly above the focus of an earthquake.
Courtesy of Lindeburg and Baradar, 2001
• Focus (Hypocenter)
the point on the fault at which the first movement or
break occurred directly.
• Asperity
literally “roughness”. It is an area on a fault that is
stuck or locked. A type of surface roughness appearing
along the interface of 2 faults. Physics the elastically
compressed region of contact between two surfaces
caused by the normal force.
• Compression
fractional decrease of volume due to pressure.

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• Dip Angle
The angle between the fault plane and horizontal ground surface is the dip angle d. It is measured
downwards from the horizontal surface and it takes values between 0 and 90°.
• Strike Angle
The strike angle f is the clockwise angle relative to North and it varies between 0 and 360°. It
shows the direction of fault strike that is defined as the line of intersection of the fault plane and the
ground surface. The strike of a fault is defined such that the hanging wall is always on the right and
footwall block is on the left.
Courtesy of Sucuoğlu and Akkar, 2014
• Slip Angle
The slip (rake) angle l shows the direction of relative
motion of hanging wall with respect to footwall. It is
measured relative to fault strike and it varies between
±180°.

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3 TYPES OF FAULTS
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Normal Fault
The upper block moves
downward relative to the lower
block. (dip-slip)
Reverse/Thrust Fault
The upper block moves upward
relative to the lower block. (dip-
slip)
Strike-slip Fault
Both blocks slide horizontally
across one another. (Strike-slip)
Courtesy of EarthHow https://earthhow.com/types-of-faults/
• Dip-slip: Incline split with vertical movement (upwards or downwards)
• Strike-slip: Straight split with horizontal movement (right or left lateral)
• Creep: If movement is slow at the fault, it’s called “creep”. By definition, “creep” means the fault is
always absent of sudden movements that could create an earthquake.
• Earthquake: But if the two plates have a sudden jerky movement, this generates enough force to
produce an “earthquake”. At this point, elastic waves shoot outwards which is the force one would feel
from an earthquake.

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OTHER TYPES OF FAULTS
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Courtesy of Lindeburg and Baradar, 2001

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ACTIVE FAULTS IN TAIWAN
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斷層在地表破裂處所形成的軌跡，稱為斷層線。根據中央地質調查所
的研究，臺灣共有42條活動斷層，其中的12條斷層在過去一萬年之內曾經
發生錯動，歸納為第一類活動斷層；11條斷層在過去十萬年內曾經發生錯
動，歸納為第二類活動斷層；其餘19 條斷層的活動性尚待詳加調查，因此
列為存疑性活動斷層。
Courtesy of Duruo Huang and Wenqi Du, 2017

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EARTHQUAKES IN TAIWAN
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Eurasian Plate
Philippine Sea Plate
Date
(UTC+8)
Area
Affected
ML Dead
Houses
Destroyed
1916/08/28 Central Taiwan 6.8 16 614
1920/06/05 Hualien 8.3 5 273
1927/08/25 Tainan 6.5 11 214
1935/04/21
Hsinchu,
Taichung
7.1 3,276 17,907
1935/07/17
Hsinchu,
Taichung
6.2 44 1,734
1941/12/17 Chiayi 7.1 360 4,520
1946/12/05 Tainan 6.1 74 1,954
1959/08/15 Pingtung 7.1 16 1,214
1964/01/18 Chiayi, Tainan 6.3 106 10,924
1999/09/21 Island-wide 7.3 2,415 51,711
2006/12/26
Pingtung,
Kaohsiung
7.0 2 3
2016/2/6
Kaohsiung,
Tainan
6.6 117 many

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HISTORICAL REMAINS FROM TAIWAN EARTHQUAKE
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• 南投縣名間鄉傾斜的電塔
車籠埔斷層穿切過台三線旁的電塔，造成電塔傾斜、電纜線斷裂。
傾斜的電塔已保存為921地震震災紀念塔。
• 南投縣名間鄉彎曲的鐵軌
車籠埔斷層穿切過台三線旁的集集支線鐵道，造成鐵軌彎曲。

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SEISMIC WAVES
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• Seismic waves can be divided into two types of waves which are propagated differently during
earthquakes:
− Body waves travel through the interior of the Earth. They create ray paths refracted by the varying
density and Young's modulus (stiffness) of the Earth’s interior. The density and modulus, in turn,
vary according to temperature, composition, and phase.
− Surface waves are elastic waves which propagate along the surface of the earth and whose
energy decays exponentially with depth. They contain most of the long period energy (periods
greater than 20 sec) generated by earthquakes and recorded at teleseismic distances.
Example of vertical component record
for the October 14, 1996 Solomon
Islands earthquake at BDSN station
ORV showing the arrivals of multiply
reflected body wave phases forming a
higher-mode Rayleigh wave train in
front of the fundamental mode (R1).
Courtesy of Harsh K. Gupta https://link.springer.com/referenceworkentry/10.1007/978-90-481-8702-7_143
Example of vertical component record showing
earth-circling Rayleigh wave trains (marked
R1...R8) following the M8.8 Maule Chile
earthquake of February 27, 2010.
Chapter 1.5 Earthquake Waves

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Focus
Fault
Epicenter
Fault Scarp
(Fault Cliff)
Seismic Waves

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SEISMIC WAVES – BODY WAVES
64
• Two types of waves may be identified in the earthquake motions that are propagated deep within the
earth:
− P waves, in which the material particles move along the path of the wave propagation inducing an
alternation between tension and compression deformations. The P or Primary wave designation
refers to the fact that these normal stress waves travel most rapidly through the rock and therefore
are the first to arrive at any given point.
− S waves, in which the material particles move in a direction perpendicular to the wave propagation
path, thus inducing shear deformations. The S or Secondary wave designation refers
correspondingly to the fact that these shear stress waves travel more slowly and therefore arrive
after the P waves.
Courtesy of Clough and Penzien, 1995

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SEISMIC WAVES – SURFACE WAVES
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• When the vibratory wave energy is propagating near the surface of the earth rather than deep in the
interior, two other types of waves known as Rayleigh and Love waves can be identified.
− The Rayleigh surface waves are tension-compression waves similar to the P waves except that
their amplitude diminishes with distance below the surface of the ground.
− Similarly the Love waves are the counterpart of the S body waves; they are shear waves that
diminish rapidly with distance below the surface. Following figure illustrates the nature of these
four types of elastic earthquake waves.

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SEISMIC WAVES – SURFACE WAVES
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From: https://scweb.cwb.gov.tw/zh-tw/guidance/faqdetail/21

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PROPAGATION OF SEISMIC WAVES
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SEISMIC WAVES TRAVELLING ACROSS EARTH
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Courtesy of Physical Geology, University of Saskatchewan
https://openpress.usask.ca/physicalgeology/chapter/3-2-
understanding-earth-through-seismology-2/
• The relative arrival times of the P and S waves can be interpreted in
terms of the distance of the observatory from the focus if the
properties of the materials through which the waves travel are known;
also this relative delay time provides evidence regarding reflection
and refraction of the earthquake waves from the boundaries between
concentric layers of rock having different moduli of elasticity and
densities.

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TRAVELLING TIME AND DIRECTION OF EARTHQUAKE
69
Courtesy of IRIS https://www.iris.edu/hq/inclass/video/travel_time_curves_described

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ENERGY AND FREQUENCY OF SEISMIC WAVES
70
• Seismic waves can be divided into two types of waves which are propagated differently during
earthquakes:
− Body waves travel through the interior of the Earth. They create ray paths refracted by the varying
density and Young's modulus (stiffness) of the Earth’s interior. The density and modulus, in turn,
vary according to temperature, composition, and phase.
− Surface waves are elastic waves which propagate along the surface of the earth and whose
energy decays exponentially with depth. They contain most of the long period energy (periods
greater than 20 sec) generated by earthquakes and recorded at teleseismic distances.

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SEISMIC INSTRUMENTATION IN TAIWAN
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• TSMIP (Taiwan Strong Motion Instrumentation Program, Central Weather Bureau):
started in 1992, TSMIP connects over 800 strong motion stations (including borehole stations) that
consist of 110 real-time stations and 60 building arrays.
• TREIRS (Taiwan Rapid Earthquake Information Release System, Central Weather Bureau):
has been operational since 1996, TREIRS routinely determines the location and magnitude of
earthquakes in the Taiwan region within one minute of occurrence.
• BATS (Broadband Array in Taiwan for Seismology, Institute of Earth Sciences, Academia
Sinica):
initiated in 1992 and has produced high-quality data in sufficient quantity since early 1996, the number
of permanent broadband stations is 28, including 2 located in the South China Sea.
• NCREE (National Center for Research on Earthquake Engineering) network:
entered operation in early 2012, the NCREE network has produced data for the research on
earthquake engineering with 33 permanent broadband stations.
臺灣在1897年引進第一部地震觀測儀—格雷．米爾恩地震儀(Gray-
Milne)，臺灣從此進入科學的地震觀測時代。一百多年來，隨著地震觀測
儀的功能提升、地震觀測網的架設以及近年來電腦與資訊系統的應用，臺
灣地震觀測的即時性與精確度有了長足的進步。

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SEISMIC INSTRUMENTATION OF CWB
72
• Seismic instrumentation of Central Weather
Bureau (CWB)
• Taiwan Strong Motion Instrumentation Program
(TSMIP)
− 650 free-field stations
− 57 sets of strong motion arrays in structures
− 200-250 samples/sec
− 16/24-bits resolution
• Taiwan Rapid Earthquake Information Release
System (TREIRS)
− 75 rapid stations
− 50 samples/sec
− 16-bits resolution
− An integrated seismic early warning system

Shieh-Kung
Huang
SEISMIC INSTRUMENTATION OF CWB
73
• 57 sets of strong motion arrays in structures

Shieh-Kung
Huang
SEISMIC INSTRUMENTATION OF IES AND NCREE
74
• Broadband Array in Taiwan for Seismology (BATS)
Network
• NCREE (National Center for Research on
Earthquake Engineering) Network
• Other High-tech Fabs
Network Manufacturer Model Output Bandwidth
BATS
Kinemetrics
Nanometrics
STS-2
Trillium 240
Vel.
Vel.
0.0083-50 Hz
1/240-35 Hz
NCREE Guralp CMG-6TD Vel. 1/30-100 Hz
Fab B Kinemetrics Episensor ES-T Accel. DC-200 Hz
Fab C Tokyo Sokushin AS-2000 Accel. DC-200 Hz
Kinemetrics STS-2 Kinemetrics Trillium 240 Guralp CMG-6TD Kinemetrics CMG-6TD

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Huang
RECORDINGS OF SEISMIC INSTRUMENTATION
75
下圖為某次地震時，某地震觀測站的地震記錄。該地震觀測站共有東西向、南北向、垂直向三部地震
儀，記錄不同震動方向的地震加速度，從圖中可以明顯看出P波和S波抵達的時間，大約相差3秒。

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76
因為地震觀測站的密度高，中央氣象局得以
在1999年集集地震時，記錄車籠埔斷層沿線的震
動情形，進而推估斷層的破裂行為以及斷層對於
建築物造成的危害效應。由於其他國家很少在斷
層附近設置密集的地震監測站，因此這份完整的
近斷層活動記錄，也就成為各國地震工程學家進
行研究時的珍貴資料。

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77

Shieh-Kung
Huang
STRONG MOTION ARRAY
78
由100多個地震觀測站組成，觀測站內部設有強震
儀、弱震儀和數據機，地震時可以即時將地震資料透過
網路線，傳輸到中央氣象局的資料處理中心，迅速計算
出地震規模、震央、各地震度等資訊，以發布地震速報
並作為強震時的救災決策參考。
即時地震監測網將地震資料以網路線送往花蓮、南
部、臺北的資料處理中心，構成地震速報系統
Chapter 1.6 Earthquake Characteristics

Shieh-Kung
Huang
CWB EARTHQUAKE REPORT
79
每當臺灣發生有感地震，中央氣象局會在地震發生3分鐘內，於官方網站上發布如同以下畫面的地震報
告，下面這張圖是發生於1999年9月21日的集集地震報告。

Shieh-Kung
Huang
CWB EARTHQUAKE REPORT
80
每當臺灣發生有感地震，中央氣象局會在地震發生3分鐘內，於官方網站上發布如同以下畫面的地震報
告，下面這張圖是發生於1999年9月21日的集集地震報告。

Shieh-Kung
Huang
LOCATION OF EARTHQUAKES
81

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Huang
MAGNITUDE AND INTENSITY OF EARTHQUAKE
82
• 規模（Magnitude）：以釋放的能量來表示
Based on energy released
• 震度（Intensity）：以破壞程度及結構物振動大小來表示
Based on structural damage severity

Shieh-Kung
Huang
EARTHQUAKE MAGNITUDE
83
地震規模是指地震所釋放的能量，臺灣所採用的計算方式為芮氏規模，在敘述時以「規模5.0」、「規
模7.3」的方式來表示，數字的後面不加「級」字。人類歷史上曾發生規模最大的地震，根據美國地質調查
所觀測的記錄，發生於1960年5月22日南美洲的智利，規模9.5。

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Huang
RICHTER MAGNITUDE SCALE
84
The Richter scale – also called the Richter magnitude scale or Richter's
magnitude scale – is a measure of the strength of earthquakes, developed by
Charles Francis Richter and presented in his landmark 1935 paper, where he
called it the "magnitude scale". This was later revised and renamed the local
magnitude scale, denoted as ML or ML.
The Richter magnitude of an earthquake is determined from the logarithm of
the amplitude of waves recorded by seismographs (adjustments are included to
compensate for the variation in the distance between the various seismographs
and the epicenter of the earthquake). The original formula is:
where A is the maximum excursion of the Wood–Anderson seismograph, the
empirical function A0 depends only on the epicentral distance of the station, d. In
practice, readings from all observing stations are averaged after adjustment with
station-specific corrections to obtain the ML value.
10 10 0 10
0
log log ( ) log
( )
L
A
M A A
A
d
d
 
= − =  
 

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RICHTER MAGNITUDE SCALE
85
The computation of ML can also be done from the
nomogram given in the following that requires P- and S-wave
arrival times and the maximum amplitude readings on a
Wood-Anderson seismograph.

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MOMENT MAGNITUDE SCALE
86
The moment magnitude scale (MMS; denoted explicitly with Mw or Mw, and
generally implied with use of a single M for magnitude) is a measure of an
earthquake's magnitude ("size" or strength) based on its seismic moment. It was
defined in a 1979 paper by Thomas C. Hanks and Hiroo Kanamori. Similar to the
Richter magnitude scale (ML), it uses a logarithmic scale; small earthquakes have
approximately the same magnitudes on both scales.
where M0 is the seismic moment, which is a measure of the fault slip and area
involved in the earthquake. Its value is the torque of each of the two force couples
that form the earthquake's equivalent double-couple.
10 0
2
log 10.7
3
w
M M
= −

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COMPARISON BETWEEN MAGNITUDE SCALES
87
Courtesy of Robert W. Day, 2012
• Surface Wave Magnitude Scale, Ms
• Japan Meteorological Agency Magnitude Scale, MJMA
• Long-period Body Wave Magnitude Scale, mB
• Short-period Body Wave Magnitude Scale, mb
Courtesy of Robin K. McGuire, 2004

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COMPARISON BETWEEN MAGNITUDE SCALES
88

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RELATIONSHIP BETWEEN MAGNITUDE AND ENERGY
89
Once the Richter magnitude, ML, is known, an approximate relation can be used to calculate the
energy, E, radiated. The magnitude is empirically related to the amount of earthquake energy released by
the formula
By this formula, the energy increases by a factor of 32 for each unit increase of magnitude. More
important to engineers, however, is the empirical observation that earthquakes of magnitude less than 5
are not expected to cause structural damage, whereas for magnitudes greater than 5, potentially
damaging ground motions will be produced.
10
log 11.8 1.5 L
E M
= +

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EARTHQUAKE INTENSITY
90
震度指的是，地震時人們對於地面震動的感受程
度，或物品因震動遭受破壞的程度。中央氣象局利用
地震觀測站所記錄的最大加速度，計算出各地區的最
大震度，表達方式為數字後加「級」，如：「臺中市
6 級」、「臺北市4級」。
在一場地震當中，設在不同地區的地震觀測站所
記錄到的最大震度不盡相同，如果將最大震度相同的
區域彼此連結，可以繪製成右頁的等震度圖，圖中的
震度約略以震央為中心，向外遞減，顯示地震能量擴
散的情形。
Intensity of Chi-chi Earthquake

Shieh-Kung
Huang
EARTHQUAKE INTENSITY – JAPAN
91

Shieh-Kung
Huang
EARTHQUAKE INTENSITY – TAIWAN (OLD SCALE)
92
Courtesy of 中央氣象局數位科普網
https://edu.cwb.gov.tw/PopularScience/index.php

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Huang
EARTHQUAKE INTENSITY – TAIWAN (NEW SCALE)
93
Courtesy of 中央氣象局數位科普網 https://edu.cwb.gov.tw/PopularScience/index.php

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Huang
EARTHQUAKE INTENSITY – USA (MMI)
94
The modified Mercalli intensity scale (MM or MMI), developed from Giuseppe
Mercalli's Mercalli intensity scale of 1902, is a seismic intensity scale used for
measuring the intensity of shaking produced by an earthquake.
It measures the effects of an earthquake at a given location, distinguished
from the earthquake's inherent force or strength as measured by seismic
magnitude scales (such as the "Mw" magnitude usually reported for an
earthquake). While shaking is caused by the seismic energy released by an
earthquake, earthquakes differ in how much of their energy is radiated as seismic
waves. Deeper earthquakes also have less interaction with the surface, and their
energy is spread out across a larger volume.

Shieh-Kung
Huang
EARTHQUAKE INTENSITY – COMPARISON
95
Courtesy of 中央氣象局數位科普網 https://edu.cwb.gov.tw/PopularScience/index.php

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PEAK GROUND MOTION
96
For engineering purposes, the time variation of
ground acceleration is the most useful way of defining
the shaking of the ground during an earthquake.
Actually, the ground acceleration governs the
response of structures to earthquake excitation.
Courtesy of Chopra, 2020

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Huang
PEAK GROUND MOTION
97
North–south component of horizontal ground acceleration recorded at the Imperial Valley Irrigation
District substation, El Centro, California, during the Imperial Valley earthquake of May 18, 1940. The
ground velocity and ground displacement were computed by integrating the ground acceleration.
Peak Ground Acceleration (PGA)
Peak Ground Velocity (PGV)
Peak Ground Displacement (PGD)

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PEAK GROUND MOTION
98
• Ground Motion Attenuation
3 4
1 2
1 2 3 4
ln( )
lnPGA ln( )
PGA b D b
b b M
b b M b D b
e e e +
= + + +

=

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Huang
ARIAS INTENSITY
99
The Arias Intensity (IA) is a measure of the strength of a ground motion. It
determines the intensity of shaking by measuring the acceleration of transient
seismic waves. It was proposed by Chilean engineer Arturo Arias in 1970, and has
been found to be a fairly reliable parameter to describe earthquake shaking
necessary to trigger landslides.
where T is the duration of signal above threshold. Theoretically the integral should
be infinite. The Arias Intensity could also alternatively be defined as the sum of all
the squared acceleration values from seismic strong motion records.
2
0
( )
2
T
A
I a t dt
g

= 

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ARIAS INTENSITY
100
A plot that portrays the buildup of this energy with time for a strong motion record is known as a
Husid plot (Husid 1969). It plotted normalized Arias intensity in accordance with the following equation.
The normalized Arias intensity ranges from 0 to 1, as illustrated in Figure 2.3. Husid used the interval from
0 to 95% of Arias intensity as the duration of strong shaking of earthquake record (1969). Trifunace and
Brady suggested that the interval between 5% to 95% of the Arias intensity was a more appropriate
choice for the significant duration (1975). The Trifunace and Brady definition of duration is illustrated on
the Husid plot in Figure 2.4.
2
0
2
0
( )
( )
( )
t
T
a t dt
h t
a t dt
=



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EMPIRICAL MODELS FOR SIGNIFICANT DURATION
101

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RUPTURE LENGTH
102
• Length of Active Fault (Rupture Length)
The following equation correlates the Richter magnitude, ML, with the approximate total rupture
length, L (in kilometers), involved in an earthquake. Such correlations are very site-dependent, and
event then, there is considerable scatter in such data. This equation should be considered only
representative of the general (approximate) form of the correlation.
Courtesy of Gudmundsson et al., 2013
• Length of Fault Slip
The following equation (as derived by King
and Knopoff in 1968) correlates the Richter
magnitude, ML, and the rupture length, L (in
meters), which the approximate length of
vertical or horizontal fault slip displacement, D
(in meters). As the above equation, this
correlation should be considered representative
of the general relationship.
10
log 1.02 5.77
L
L M
= −
6 2
10
log (10 ) 1.9 2.65
L
LD M
 = −

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RUPTURE LENGTH
103

Shieh-Kung
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RUPTURE AREA
104
The following figure shows the relationship between rupture area and magnitude. Larger rupture
areas indicate large-magnitude earthquakes. The rupture area of small magnitude events (i.e.,
magnitudes less than 6) can be represented by a circle and such seismic sources are referred to as point-
source in seismology. The rupture area tends to become rectangular (i.e., extended source) for larger
magnitudes. For such cases the rupture geometry is characterized by the width (W) and length (L) of the
rupture area. There are many empirical models in the literature that relate the magnitude of earthquakes
with the rupture dimensions (e.g., Wells and Coppersmith 1994).

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Huang
APPLICATIONS OF SEISMIC INSTRUMENTATION
105
Chapter 1.7 Earthquake-related Application

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106
疾駛中的高鐵列車，若遇上地震強烈晃動，極可能引發列車出軌，造成慘重傷亡。假設2010年3月4日
台灣發生內陸地震，震央在高雄甲仙鄉，地震規模6.4，其中有一列車，剛駛離板橋站不久，有另一部列車，
在苗栗縣與新竹縣交界上行駛(如下左圖)。
最先收到地震訊號的是高鐵台南歸仁站，地震波傳遞我們以S波來考慮，S波平均行進速度假設為每秒4
公里，那麼S波由甲仙傳至歸仁需8.7秒；S波由甲仙傳至新竹縣與苗栗縣縣界需43.5秒；由甲仙傳至高鐵板
橋站需55.1秒(如下右圖)。另外，列車從接收地震警訊到啟動煞車時間假定為1秒整。
• 55.1-8.7-1=45.4 秒 (監測系統為板橋處的列車多爭取的煞車時間)
• 43.5-8.7-1=33.8 秒 (監測系統為竹苗處的列車多爭取的煞車時間)
由此例可知，離震央越遠，能爭取的煞車時間越長；離震央越近，
能爭取的煞車時間越短。至於列車是否能及時停駛，還與列車當時
速度大小有關。至少列車趕緊減速，必定能減少意外傷亡的機會與
程度。

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107
• Earthquake Early Warning (EEW)
The idea of an earthquake early warning system was
proposed more than 150 years ago and published as a letter to
the editor the San Francisco Daily Chronicle in 1868 by J.D.
Cooper, a medical doctor in the Bay Area.
• P-wave: showed as pink
− also called primary or compression wave
− fast (6~7 km/s)
− small and nondestructive amplitude
− brings the information of the earthquake
• S-wave: showed as red
− also called secondary or shear wave
− slow (3~4 km/s)
− the major destructive energy
• Regional (Front-detection, Network-based) EEW System:
Seismometers installed in the earthquake source area
provide early warnings and send back to distant areas before
the seismic waves will occur.
• On-site EEW System:
Earthquake information is determined from the initial portion of P-wave recorded by onsite
seismometers and seismic intensity of following S-wave is predicted for emergency response.

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108
In the past two decade, progress has been made towards implementation of earthquake early
warning in Japan, Taiwan, Mexico, Southern California, Italy, and Romania.
• Mexico: Seismic Alert System (SAS)
• Japan: Urgent Earthquake Detection & Alarm System (UrEDAS)
• Turkey: Istanbul Earthquake Rapid Response & Early Warning (IERREW) System
Courtesy of Richard M. Allen, et al., 2009.

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109
• P-alert
https://palert.earth.sinica.edu.tw/index.php
• 複合式地震速報
氣象局（強震即時警報）＋國家地震工程研究中心（現地型地震預警）
https://pwaver.com/

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INTRODUCTION OF SEISMIC HAZARD ANALYSIS
Chapter Outline
110
CHAPTER 2
2.1 Introduction of Seismic Hazard Analysis
2.2 Seismicity Recurrence
2.3 Attenuation of Ground Motion
2.4 Exceedance Probability for Seismic Hazard

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FRAMEWORK OF EARTHQUAKE ENGINEERING
111
Chapter 2.1 Introduction of Seismic Hazard Analysis

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EARTHQUAKE LOSS ESTIMATIONS
112
The fundamental components of earthquake loss assessment are (i) hazard, (ii) inventory and (iii)
vulnerability or fragility, as depicted in the following figure.

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SEISMIC HAZARD ANALYSIS
113
Also known as
• Seismic Hazard and Risk Analysis
• Seismic Hazard Assessment
• Seismic Hazard and Risk Assessment
estimates the level of a ground-motion intensity
parameter (e.g., peak ground acceleration, PGA, peak
ground velocity, PGV, and spectral acceleration, Sa, at
different vibration periods, etc.) that would be produced
by future earthquakes.
The seismic hazard analysis can be simply divided
into the two main components as the deterministic and
probabilistic seismic hazard analysis (abbreviated as
DSHA and PSHA, respectively).

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DETERMINISTIC SEISMIC HAZARD ANALYSIS (DSHA)
114

Shieh-Kung
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PROBABILISTIC SEISMIC HAZARD ANALYSIS (PSHA)
115

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SEISMIC ZONING SCHEME
116
Chapter 2.2 Seismicity Recurrence

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SEISMIC ZONING SCHEME
117
S01~S13 D01~S05

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SEISMICITY RECURRENCE
118
The frequency of earthquakes generated by seismic sources defines the earthquake recurrence
model that is used in PSHA. For a certain magnitude m, the earthquake recurrence model gives the mean
yearly number of earthquakes exceeding m. The pioneering study by Gutenberg and Richter (1944)
proposed the simplest, nevertheless very useful, earthquake recurrence relationship as
10
log m
v a bm
= −
The earthquake recurrence model of
Gutenberg and Richter (1944) was
derived by compiling the earthquake
catalog in Southern California and sorting
them by the total number of earthquakes
exceeding different magnitudes (M). The
total number of earthquakes exceeding
each magnitude is normalized by the total
time span covered by the earthquake
catalog to describe the mean annual rate
of exceedance vm, of an earthquake of
magnitude m.

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TRUNCATED SEISMICITY RECURRENCE
119
The observed saturation towards larger magnitudes can be addressed by using the truncated
Gutenberg-Richter earthquake recurrence model (McGuire and Arabasz 1990). This model caps the
annual frequency of earthquakes for a given mmax. Of course, there are more complicated earthquake
recurrence models in the literature (e.g., characteristic earthquake recurrence model; Youngs and
Coppersmith 1985).
As indicated in the previous paragraph,
truncated Gutenberg-Richter requires mmax
information to describe the maximum
earthquake size that can be generated by
the considered seismic source. mmax is
determined either from the compiled
catalog information (e.g., Mueller 2010) or
from the empirical expressions that
estimate mmax by using fault rupture
dimensions (e.g., Wells and Coppersmith
1994; Leonard 2010).
min
2 2
10 1 min 2 min min max
max
log ( ) ( )
0
m
a m m
v a b m m b m m m m m
m m



= − − − −  

 


Shieh-Kung
Huang
As far as the physics of earthquakes are of concern
(elastic rebound theory), given a region, the occurrence of
earthquakes are not independent of each other. To this end,
being stationary in time and having no memory on the
occurrence of earthquakes, the Poisson process does not
fully represent the actual earthquake mechanism. However,
earthquakes can be assumed as randomly occurring,
independent events in time when the foreshocks and
aftershocks are removed from earthquake catalogs. (This is,
in fact, the major reason behind the removal of foreshocks
and aftershocks from earthquake catalogs as indicated
previously).
FREQUENCY OF OCCURRENCE
120
In some textbook, the seismicity recurrence (also known as frequency of occurrence) is equivalently
derived by the earthquake number
10
10
10 10
10 10 10
log
log
log log
log log log and
m
m
m
m
n a b m
n
N a b m
N
v N a b m
v a N b m a a N b b
 
= −
 
 = −
 
 + = −
   
 = − −  = − =

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PROBABILITYDENSITYFUNCTIONOFSEISMICITYRECURRENCE
121
Based on the Poisson process, the conditional probability can be described by the following
expression
The numerator in the above expression is the rate of earthquakes between mmin and m m. The
denominator is rate of earthquakes having magnitudes greater than mmin. The final form of cumulative
distribution function becomes
The corresponding probability density function is
, min
min
min , min
Rate of eqrthquakes with
( )
Rate of eqrthquakes with
m m
M
m
v v
m M m
F m
m M v
−
 
= =

min
min
min
( )
min
10 10
( )
10
1 10
a bm a bm
M a bm
b m m
F m
m m
− −
−
− −
−
=
 
= − 
 
min
min
( )
( )
min
( ) ( )
1 10
ln(10)10
M M
b m m
b m m
d
f m F m
dm
d
dm
b m m
− −
− −
=
 
= −
 
= 
10
Hint: ln(10)10
x
x
d
dx
=

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122
Again, in some textbook, the probability density function of seismicity recurrence is equivalently
derived by the earthquake number
where n is the number of earthquakes of magnitude m or greater per unit of time. Often, a lower-bound
mmin other than magnitude 0 is used; the equivalent expression is
The cumulative distribution function for earthquake magnitude m is
The corresponding probability density function is
10
log 10 10
a b m
m m
n a b m n  
−
 
= −  =
min
( )
10 10 b m m
a
m
n 
 − −
=
min
min
( )
( )
min
( ) ( )
1 10
ln(10)10
M M
b m m
b m m
d
f m F m
dm
d
dm
b m m

− −

− −
=
 
= −
 

= 
min
min
min min min
min min
min
( ) ( )
( )
( )
min
( )
10 10 10 10
10 10
1 10
m m
M
m
b m m b m m
a a
b m m
a
b m m
n n
F m
n
m m
 
 
− − − −

 − −

− −
−
=
−
=
= − 
10
Hint: ln(10)10
x
x
d
dx
=

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123

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TRUNCATED MAGNITUDE
124
Explanatory diagram for graphical representation of the energy release method of calculating upper
bound magnitude and mean annul energy release

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TRUNCATED MAGNITUDE OF TAIWAN
125

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SEISMICITY RECURRENCE OF TAIWAN
126

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ATTENUATION OF GROUND MOTION
127
The ground-motion intensity parameters (e.g., PGA, PGV, spectral ordinates such as Sa at different
vibration periods etc.) at a specific location are correlated the source, path and site effects. These effects
are mainly described by independent variables such as magnitude, source-to-site distance, site class and
style-of-faulting.
However, the most obvious piece of information to be gained from an earthquake record is the PGA.
Partly because it is so easy to obtain, and partly because earthquake forces are proportional to
acceleration. PGV and PGD also have their uses, with growing interest in velocity in recent years. The
characteristics of ground motion vary with the size of the event at source and with the distance from the
source. Transitionally, the peak ground motions y have been described as a function of magnitude M and
epicentral distance D (as well as focal distance R) from the source, in the general form
The coefficients b1 to b4 vary depending on the data set to which the equation is fitted, and have been
modified for different regions as more data have become available.
1 2 3 4
ln ln( )
y b b M b D b
= + + +
Chapter 2.3 Attenuation of Ground Motion

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ATTENUATION OF GROUND MOTION
128

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EXCEEDANCE PROBABILITY OF A THRESHOLD LEVEL
129
2
ln
ln
ln
ln
1 1
( ) exp
2
2
y
y
y
y
y
P Y y dy


 

 
 
−
 
 = −  
 
 
 
 


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NEXT GENERATION ATTENUATION (NGA)
130
In 2003, five developer teams were selected to participate in a Pacific Earthquake Engineering
Research Center (PEER) project to empirically develop Next Generation Attenuation (NGA) empirical
ground motion models (EGMMs). Finally, The general functional form of the our EGMM is given by the
equation
where fi are functions of magnitude (M), source-to-site distance (R), style of faulting (F), hanging-wall
effects (HW), shallow site conditions (S), and sediment depth (D).
1 2 3 4 5 6
ln ( ) ( ) ( ) ( ) ( ) ( ) T
y f M f R f F f HW f S f D 
= + + + + + +

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PROBABILISTIC SEISMIC HAZARD ANALYSIS
131
Given a seismic source, the computation of mean annual rate for ground-motion intensity parameter
Y exceeding a threshold level y can be described by the integral given as
In this expression, vm(M > mmin) is the mean annual exceedance rate of earthquakes with magnitudes
greater than mmin for the considered seismic source. The probability density functions of earthquake
recurrence and source-to-site distance are described by fM(m) and fR(r | m), respectively.
This equation also accounts for the existence of multiple (ns) seismic sources that have the potential
of affecting the project site.
max max
min
min ,
0
( ) ( ) ( ) ( | ) ( )
m r
m R M
m r
m
P Y y v M m P Y y f r m f m drdm
 =  
 
max max
min
min ,
0
1
( ) ( ) ( ) ( | ) ( )
s
n
m r
m i R M
m r
m
i
=
 =  
  
Chapter 2.4 Exceedance Probability for Seismic Hazard

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132
max max
min
min ,
0
( ) ( ) ( ) ( | ) ( )
m r
m R M
m r
m
 =  
 

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133
The mean annual rate for ground-motion intensity parameter Y exceeding a threshold level y is
described as
In the conditional probability point of view, the equations stand
So, the annual probability of exceedance can be re-written as
This actually indicates that the probability is a multiply of
1) the earthquake recurrence rate
2) the exceedance probability of a threshold level,
which is much more making sense for us.
max max
min
min ,
0
( ) ( ) ( ) ( | ) ( )
m r
m R M
m r
m
 =  
 
1 2
1 2 1 2 1 2 2
2
( )
( | ) or ( ) ( | ) ( )
( )
P E E
P E E P E E P E E P E
P E
= =
max max
min
min ,
0
min ,
min
( ) ( ) ( ) ( | ) ( )
( ) ( ) ( , ) ( , )
( ) ( , , ) ( , , )
m r
m R M
m r
m
m m r
m
v M m P Y y f r m d r m
v M m P Y y r m d Y y r m
 =  
=  
=   
 



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PROBABILITY OF EXCEEDANCE
134
For any given site on the map, the computer calculates the ground motion effect (peak acceleration)
at the site for all the earthquake locations and magnitudes believed possible in the vicinity of the site.
Each of these magnitude-location pairs is believed to happen at some average probability per year. Small
ground motions are relatively likely, large ground motions are very unlikely. Beginning with the largest
ground motions and proceeding to smaller, we add up probabilities until we arrive at a total probability
corresponding to a given probability, P, in a particular period of time, t.
The probability P comes from ground motions larger than the ground motion at which we stopped
adding. The corresponding ground motion (peak acceleration) is said to have a P probability of
exceedance (PE) in t years. The map contours the ground motions corresponding to this probability at all
the sites in a grid. Thus the maps are not actually probability maps, but rather ground motion hazard maps
at a given level of probability.
A typical seismic hazard map may have the title, "Ground motions having 90 percent probability of
not being exceeded in 50 years." The 90 percent is a "non-exceedance probability"; the 50 years is an
"exposure time." An equivalent alternative title for the same map would be, "Ground motions having 10
percent probability of being exceeded in 50 years."
From USGS Earthquake Hazards 201

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RETURN PERIOD
135
Let assume probability P = 0.10, 0.05, or 0.02, respectively. The approximate annual probability of
exceedance is the ratio, P*/50, where P* = P(1+0.5P) is an approximation to the value -ln(NPE, non-
exceedance probability). The inverse of the annual probability of exceedance is known as the "return
period (RP)," which is the average number of years it takes to get an exceedance.
A typical seismic hazard map may have the title, "Ground motions having 90 percent probability of
not being exceeded in 50 years." The 90 percent is a "non-exceedance probability"; the 50 years is an
"exposure time." An equivalent alternative title for the same map would be, "Ground motions having 10
percent probability of being exceeded in 50 years." A typical shorthand to describe these ground motions
is to say that they are 475-year return-period ground motions. This means the same as saying that these
ground motions have an annual probability of occurrence of 1/475 per year.

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RETURN PERIOD
136
A 10% probability of exceedance in 50 years corresponds to a 475-year return period, and the
question sometimes arises, "What is special about the 475-year return period?" This period is derived by
assuming a Poisson process for ground motion occurrences, wherein the probability of an event, P, is
related to the annual frequency of exceedance of the ground motion g and exposure time t through
Rearranging this gives
Substituting a probability 0.1 and an exposure time of 50 years gives g = 0.002107 per year, which is 1/475
years.
The same result can be obtained from the binominal distribution, which represents the Poisson
process in discrete form. If it is assumed only the exceedances of ground motion in successive years are
independent and that the probability of non-exceedance in any year is 1 - g, then a 90% probability of non-
exceedance in 50 years is expressed as
which gives g = 0.002105 per year for 50 years, which again is 1/475 years. The slight difference from the
Poisson result arises because of the discrete representation of time with the binomial distribution.
1 exp( )
P t
g
= − −
ln(1 ) ln(NPE) (1 0.5 )
P P P
t t t
g
− +
= − = − 
NPE (NPE in one years)
(1 ) 0.9
ln(NPE)
ln(1 )
t
t
t
g
g
=
= − =
 − =

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USGS National Seismic Hazard map, peak ground acceleration expressed in % g (gravity) for a 2%
probability of exceedance in 50 years.
SEISMIC HAZARD MAP
137

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Huang
Seismic hazard map for horizontal peak ground acceleration in % g (gravity) for 10% and 2% probability
of exceedance in 50 years.
SEISMIC HAZARD MAP
138

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SPECTRAL ACCELERATION
139
PGA is what is experienced by a particle on the ground, and Sa is approximately what is experienced
by a building, as modeled by a particle mass on a massless vertical rod having the same natural period of
vibration as the building.
PGA is a good index to hazard for short buildings, up to about 7 stories. To be a good index, means
that if you plot some measure of demand placed on a building, like inter story displacement or base shear,
against PGA, for a number of different buildings for a number of different earthquakes, you will get a
strong correlation.
PGA is a natural simple design parameter since it can be related to a force and for simple design one
can design a building to resist a certain horizontal force. PGV, peak ground velocity, is a good index to
hazard to taller buildings. However, it is not clear how to relate velocity to force in order to design a taller
building.
Sa would also be a good index to hazard to buildings, but ought to be more closely related to the
building behavior than peak ground motion parameters. Design might also be easier, but the relation to
design force is likely to be more complicated than with PGA, because the value of the period comes into
the picture.
PGA, PGV, or Sa are only approximately related to building demand/design because the building is
not a simple oscillator, but has overtones of vibration, each of which imparts maximum demand to
different parts of the structure, each part of which may have its own weaknesses. Duration also plays a
role in damage, and some argue that duration-related damage is not well-represented by response
parameters.

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SPECTRAL ACCELERATION
140
On the other hand, some authors have shown that non-linear response of a certain structure is only
weakly dependent on the magnitude and distance of the causative earthquake, so that non-linear
response is related to linear response (Sa) by a simple scalar (multiplying factor). This is not so for peak
ground parameters, and this fact argues that Sa ought to be significantly better as an index to
demand/design than peak ground motion parameters.
There is no particular significance to the relative size of PGA and Sa. On the average, these roughly
correlate, with a factor that depends on period. While PGA may reflect what a person might feel standing
on the ground in an earthquake, I don't believe it is correct to state that Sa reflects what one might "feel" if
one is in a building. In taller buildings, short period ground motions are felt only weakly, and long-period
motions tend not to be felt as forces, but rather disorientation and dizziness.

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EARTHQUAKE RESPONSE SPECTRUM
141

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142

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Huang
143

Shieh-Kung
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144

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UNIFORM HAZARD RESPONSE SPECTRUM (UHRS)
145
Traditional
Method
UHRS

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UNIFORM HAZARD RESPONSE SPECTRUM (UHRS)
146

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COMPARISON OF DESIGN SPECTRUM AND UHRS
147
(a) Design spectrum for Patna for 5 % damping from 2 % and 10 % probability of exceedance in 50 years
and IS 1893 (2002) at centre of the city.
(b) Design spectrum for Patna for 5 % damping from 2 % and 10 % probability of exceedance in 50 years
and IS 1893 (2002) at the northeastern part of the city.
Courtesy of Panjamani Anbazhagan et al., 2019

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REVIEW OF STRUCTURAL DYNAMICS
Chapter Outline
148
CHAPTER 3
3.1 Introduction of Structural Dynamics
3.2 Single-degree-of-freedom Systems
3.3 Response of Free Vibration
3.4 Earthquake Response of Linear Systems
3.5 Response Spectrum
3.6 Muliti-degree-of-freedom Systems
3.7 Free Vibration of MDOF Systems
3.8 Damping in Structures
3.9 Force Vibration of MDOF Systems

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3.1 INTRODUCTION OF STRUCTURAL DYNAMICS
149
• Structural Dynamics
Determination of responses of
structures under the effect of dynamic
loading
• Responses
Responses are usually included the
displacement, velocity, and acceleration.
• Dynamic Loading
Dynamic loading is a loading whose
magnitude, direction, sense and point of
application changes in time.
Chapter 3 Review of Structural Dynamics

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3.1 INTRODUCTION OF STRUCTURAL DYNAMICS
150
• (Modeling) Assumption
− Discrete vs. Continuous
− Lumped vs. Distributed
• Dimension
− Structural member
− Finite element
• (Analysis) Domain
− Time
− Frequency

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3.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS
151
• Simple Structures
We begin our study of structural dynamics with simple
structures; these structures simple because they can be idealized
as a concentrated or lumped mass m supported by a massless
structure with stiffness k in the lateral direction.
• Degrees of Freedom
The number of independent displacements required to define
the displaced positions of all the masses relative to their original
position is called the number of degrees of freedom (DOFs) for
dynamic analysis. Thus we call this simple structure a single-
degree-of-freedom (SDOF) system.
0
mu ku
+ =

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152
• Damping
The process by which vibration steadily
diminishes in amplitude is called damping. It is
usually represented in a highly idealized manner.
This idealization is therefore called equivalent
viscous damping.
• Damping in Real Structures
− Opening and closing of microcracks
− Friction in connections
− Friction between structure and non-structure
elements
Mathematical description of these components
is almost impossible, so the modelling of damping
in real structures is usually assumed to be
equivalent viscous damping.

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153
• SDOF system
The system considered is shown schematically and It consists of a mass m concentrated at the
roof level, a massless frame that provides stiffness to the system, and a viscous damper (also known
as a dashpot) that dissipates vibrational energy of the system. The beam and columns are assumed to
be inextensible axially.
where the constant c is the viscous damping coefficient, which is a measure of the energy dissipated
in a complete cycle.
0
mu cu ku
+ + =

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154
• Force–Displacement Relation
The internal force resisting the displacement u is equal and opposite to the external force fS. It is
desired to determine the relationship between the force fS and the relative displacement u associated
with deformations in the structure during oscillatory motion. This force–displacement relation would be
linear at small deformations but would become nonlinear at larger deformations.

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155
• Linear Elastic System:
− Elastic material
− First-order analysis
• Inelastic System:
− Plastic material
− Higher-order analysis
S
f k u
= 
( , )
S
f f u u
=

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156
• Equation of Motion
The following figure is the free-body diagram at time t with the mass replaced by its inertia force.
The forces acting on the mass at some instant of time are balanced according to D’Alember’s principle
of dynamic equilibrium. These include the external force p, the elastic (or inelastic) resisting force fS,
the damping resisting force fD, and the inertial force fI.
or
and or ( , )
S D D S
D S S
p f f mu mu f f p
f cu f ku f f u u
− − = + + =
 = = =

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157
• Mass–Spring–Damper System
We have introduced the SDOF system by idealizing a one-story structure, an approach that
should appeal to structural engineering students. However, the classic SDOF system is the mass–
spring–damper system of the following figure.
or
and or ( , )
D S
D S S
mu cu ku p mu f f p
f cu f ku f f u u
+ + = + + =
 = = =

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158
• Solution of A Linear SDOF System
The equation of motion for a linear SDF system subjected to external force is the second-order
differential equation derived earlier.
The initial displacement and initial velocity at time zero must be specified to define the
problem completely. Typically, the structure is at rest before the onset of dynamic excitation, so that
the initial velocity and displacement are zero. A brief review of four methods of solution is given in the
following.
− Classical Solution
Complete solution of the linear differential equation of motion consists of the sum of the
complementary solution and the particular solution.
− Duhamel’s Integral
Another well-known approach to the solution of linear differential equations, such as the
equation of motion of an SDOF system, is based on representing the applied force as a sequence
of infinitesimally short impulses.
Duhamel’s integral provides an alternative method to the classical solution if the applied force p(t)
is defined analytically by a simple function that permits analytical evaluation of the integral.
( ) ( ) ( ) ( )
mu t cu t ku t p t
+ + =
(0)
u
(0)
u
0
0
( ) (1 cos ) when 0, (0) , and ( ) 0
n
p
u t t c p p p t
k

= − = = =
 
0
1
( ) ( )sin ( )
t
n
n
u t p t d
m
   

= −


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159
− Frequency-Domain Method
The Laplace and Fourier transforms provide powerful tools for the solution of linear differential
equations, in particular the equation of motion for a linear SDOF system. Because the two
transform methods are similar in concept, here we mention only the use of Fourier transform,
which leads to the frequency-domain method of dynamic analysis.
− Other Numerical Methods
The preceding three dynamic analysis methods are restricted to linear systems and cannot
consider the inelastic behavior of structures anticipated during earthquakes if the ground shaking
is intense. The only practical approach for such systems involves numerical time-stepping
methods, for example, Newmark-beta method, Runge-Kutta method, or state-space method
(which are presented latter). These methods are also useful for evaluating the response of linear
systems to excitation—applied force p(t) or ground motion—which is too complicated to be defined
analytically and is described only numerically.
1
( ) ( ) ( )
2
i t
u t H P e d

  


−
= 

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160
Schematic of Duhamel’s Integral and Frequency-Domain Method
Input Signal, u(t)
System
(i.e. Filter)
Output Signal, y(t)
0 10 20 30 40
-4
-2
0
2
4
Time
Signal
0 10 20 30 40
-10
-5
0
5
10
Time
Signal
0 5 10 15 20
10
0
10
1
10
2
10
3
Frequency (Hz)
Magnitude
0 5 10 15 20
10
-2
10
-1
10
0
10
1
Frequency (Hz)
Magnitude
0 5 10 15 20
10
-2
10
0
10
2
10
4
Frequency (Hz)
Magnitude
0
( ) ( ) ( )
t
y t h u t d
  
= −

h(t) is Transfer Function

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3.3 RESPONSE OF FREE VIBRATION
161
• Undamped Free Vibration
Free vibration is initiated by disturbing the system from its static equilibrium (or undeformed, u(0)
=0) position by imparting the mass some displacement and velocity at time zero.
The time required for the undamped system to complete one cycle of free vibration is the natural
period of vibration of the system, which we denote as Tn, in units of seconds. It is related to the natural
circular frequency of vibration, ωn, in units of radians per second:
(0)
( ) (0)cos sin
n n
n
u
u t u t t
 

= +
(0)
u
(0)
u

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162
A system executes several cycles in 1 sec. This natural cyclic frequency of vibration is denoted by
The units of fn are hertz (Hz) [cycles per second (cps)]; fn is obviously related to ωn through
The term natural frequency of vibration applies to both ωn and fn.
By solving the dynamic equilibrium, we can further find the natural circular frequency of vibration
is related to mass and stiffness.
1
n
n
f
T
=
2
n
n
f


=
n
k
m
 =
Tn
n
fn

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163
• Viscously Damped Free Vibration
Setting p(t)=0 in dynamic equilibrium gives the differential equation governing free vibration of
SDOF systems with damping:
where ζ is the damping ratio or fraction of critical damping as:
The damping coefficient ccr is called the critical damping coefficient because it is the smallest value of
c that inhibits oscillation completely.
2
( ) ( ) ( ) 0 ( ) ( ) ( ) 0
( ) 2 ( ) ( ) 0
n n
c k
mu t cu t ku t u t u t u t
m m
u t u t u t
 
+ + =  + + =
 + + =
cr
cr
2
and 2 2
2
n
n n
c c k
c m km
m c
 
 
= = = = =

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164
• Underdamped Free Vibration
The time

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165
• Comparison between Underdamped and Damped Free Vibration
The time required for the undamped system to complete one cycle of free vibration is changed
because the natural circular frequency of vibration, ωn, is affected by the damping.
This is the natural frequency of damped vibration. The natural period of damped vibration or the
natural frequency of damped vibration, is related to the one without damping by
2
1 where
D n n
k
m
   
= − =
2
2
1
2
or
2
1
n D
D D
D n
T
T f
f

 
 

−
= = = =
−
(0) (0)
( ) (0)cos sin
nt n
D D
D
u u
u t e u t t
 
 

−  
+
= +
 
 

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166
• Attenuation of Motion
Ratio between displacement at an arbitrary time, t, and the one after a period, TD, is independent
of time
and
Hence, the natural logarithm of the above ratio is called logarithmic decrement.
(0) (0) ( )
( ) (0)cos sin
( )
n n D
t T
n
D D
D D
u u u t
u t e u t t e
u t T
 

 

−  
+
= +  =
 
+
 
2 2
2 2
1 1
2
1
( ) 2
where and
( ) 1
n D
T n i
n D
D n i
u t T u
e e T T e
u t T u
 
 
 
 
− −
+
= = = =  =
+ −
2
2
1
2
ln 2 where 1 1
1
i
i
u
u

d d  

+
= =  = − 
−

Shieh-Kung
Huang
3.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS
167
• Earthquakes in Taiwan
Annually, there is more than 4000 earthquakes, including over 200 sensible earthquakes.
Date
(UTC+8)
Area
Affected
ML Dead
Houses
Destroyed
1916/08/28 Central Taiwan 6.8 16 614
1920/06/05 Hualien 8.3 5 273
1927/08/25 Tainan 6.5 11 214
1935/04/21
Hsinchu,
Taichung
7.1 3,276 17,907
1935/07/17
Hsinchu,
Taichung
6.2 44 1,734
1941/12/17 Chiayi 7.1 360 4,520
1946/12/05 Tainan 6.1 74 1,954
1959/08/15 Pingtung 7.1 16 1,214
1964/01/18 Chiayi, Tainan 6.3 106 10,924
1999/09/21 Island-wide 7.3 2,415 51,711

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168
• Earthquake Excitation
For engineering purposes, the time
variation of ground acceleration is the
most useful way of defining the shaking of
the ground during an earthquake.
Actually, the ground acceleration
governs the response of structures to
earthquake excitation.

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169
North–south component of horizontal ground acceleration recorded at the Imperial Valley
Irrigation District substation, El Centro, California, during the Imperial Valley earthquake of May 18,
1940. The ground velocity and ground displacement were computed by integrating the ground
acceleration.

Shieh-Kung
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170
• Earthquake–induced Force
In earthquake-prone regions, the principal problem of structural dynamics that concerns structural
engineers is the response of structures subjected to earthquake-induced motion of the base of the
structure.
where ug(t) is the displacement of the ground
ut(t) is the total (or absolute) displacement (of the mass)
The concept of dynamic equilibrium is used. From the free-body diagram including the inertia
force fI, the equation of dynamic equilibrium is
( ) ( ) ( )
t
g
u t u t u t
= +
0 and ( ) ( ) ( )
( ) ( ) ( ) ( ) or ( ) ( ) ( ( ), ( )) ( )
t
I D S I g
g g
f f f f mu t mu t mu t
mu t cu t ku t mu t mu t cu t f u t u t mu t
+ + = = = +
 + + = − + + = −

Shieh-Kung
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171
The ground motion can therefore be replaced by the effective earthquake force (indicated by the
subscript “eff”):
eff ( ) ( )
g
p t mu t
= −
Courtesy of Wikiwand https://www.wikiwand.com/en/Seismic_base_isolation

Shieh-Kung
Huang
172
The above equation governs the motion (or the response) of a linear SDOF system subjected to
ground acceleration.
Dividing this equation by mass m gives
When the responses are evaluated, please know the responses are:
− Absolute responses
− Relative responses (to ground)
− Relative responses (to other points)
eff
( ) ( ) ( ) ( ) ( )
g
mu t cu t ku t p t mu t
+ + = = −
2 2
( ) ( ) ( ) ( )
( ) 2 ( ) ( ) ( ) or ( ) 2 ( ) ( ) 0
g
t
n n g n n
c k
u t u t u t u t
m m
u t u t u t u t u t u t u t
   
+ + = −
 + + = − + + =

Shieh-Kung
Huang
173
• Response History
The following figure shows the deformation response of SODF systems to El Centro ground
motion.

Shieh-Kung
Huang
3.5 RESPONSE SPECTRUM
174
• Concept of Response Spectrum
A plot of the peak value of a response quantity as a function of the natural vibration period Tn of
the system, or a related parameter such as circular frequency ωn or cyclic frequency fn, is called the
response spectrum for that quantity.

Shieh-Kung
Huang
175
• Concept of Response Spectrum
A plot of the peak value of a response quantity as a function of the natural vibration period Tn of
the system, or a related parameter such as circular frequency ωn or cyclic frequency fn, is called the
response spectrum for that quantity.
A variety of response spectra can be defined depending on the response quantity that is plotted.
Consider the following peak responses:
The deformation response spectrum is a plot of deformation against Tn for fixed ζ . A similar plot for
velocity is the relative velocity response spectrum, and for total acceleration is the acceleration
response spectrum.
For engineering purposes, the relative velocity response spectrum is replaced by the pseudo-
velocity response spectrum and the acceleration response spectrum is replaced by the pseudo-
acceleration response spectrum.
0
0
0
( , ) max ( , , )
( , ) max ( , , )
( , ) max ( , , )
n n
t
n n
t
t t
n n
t
u T u t T
u T u t T
u T u t T
 
 
 




Shieh-Kung
Huang
176
• Spectral Responses
Considering the peak responses, the spectral displacement, Sd, spectral velocity, Sv, and spectral
acceleration, Sa, can be defined as
And, the relative velocity response spectrum is replaced by the pseudo response spectrums and the
can be defined as
if and only if ζ is small.
0
0
0
( , ) max ( , , )
( , ) max ( , , )
( , ) max ( , , )
d n n
t
v n n
t
t t
a n n
t
S u T u t T
S u T u t T
S u T u t T
 
 
 
 
 
 
2
max ( , , )
max ( , , ) named as
max ( , , ) named as
d n
t
v n n d
t
t
a n n d
t
S u t T
S u t T S PSV
S u t T S PSA

 
 

 
 

Shieh-Kung
Huang
177
The procedure to determine the deformation response spectrum.
0
0
0
( , ) max ( , , )
( , ) max ( , , )
( , ) max ( , , )
n n
t
n n
t
t t
n n
t
u T u t T
u T u t T
u T u t T
 
 
 




Shieh-Kung
Huang
178
The response spectrum for El Centro ground motion with various damping ratios.

Shieh-Kung
Huang
179
The mean spectra with probability distributions for the construction of elastic design spectrum.

Shieh-Kung
Huang
3.6 MULITI-DEGREE-OF-FREEDOM SYSTEMS
180
• Simple System: Two-story Shear Building
We first formulate the equations of motion for the simplest possible muliti-degree-of-freedom (MDOF)
system, a highly idealized two-story frame subjected to external forces p1(t) and p2(t). In this idealization
the beams and floor systems are rigid (infinitely stiff) in flexure, and several factors are neglected: axial
deformation of the beams and columns, and the effect of axial force on the stiffness of the columns. This
shear-frame or shear-building idealization, although unrealistic, is convenient for illustrating how the
equations of motion for an MDF system are developed.
Similar with Chapter 1.1, we can develop the dynamic equilibrium as:
1
1 1 1 1
2
2 2 2 2
or
0
0
j Sj Dj j j j j Dj Sj j
S
D
S
D
D S
p f f m u m u f f p
f
m u f p
f
m u f p
− − = + + =
 
       
 + + =
 
       
       
 
 + + =
mu f f p

Shieh-Kung
Huang
181
This matrix equation represents two ordinary differential equations governing the displacements
u1 and u2 of the two-story frame subjected to external dynamic forces p1(t) and p2(t). Each equation
contains both unknowns u1 and u2. The two equations are therefore coupled and in their present form
must be solved simultaneously.
1 1 2 2 1
2 2 2 2
1 1 2 2 1
2 2 2 2
or
or
S
S
S
D
D
D
f k k k u
f k k u
f c c c u
f c c u
+ −
     
= =
     
−
   
 
+ −
     
= =
     
−
     
 + + =
f ku
f cu
mu cu ku p

Shieh-Kung
Huang
182
• Mass–Spring–Damper System
We have introduced the linear two-DOF system by idealizing a two-story frame—an approach
that should appeal to structural engineering students. However, the classic two-DOF system, shown in
the following figure, consists of two masses connected by linear springs and linear viscous dampers
subjected to external forces p1(t) and p2(t).
1 1 1 2 2 1 1 2 2 1 1
2 2 2 2 2 2 2 2 2
0
or
0
m u c c c u k k k u p
m u c c u k k u p
+ − + −
             
+ + = + + =
             
− −
             
mu cu ku p

Shieh-Kung
Huang
183
• General Formulation of N-story Shear Building
Although the shear-frame or shear-building idealization is unrealistic in some manners, it is still
convenient and, most importantly, useful for studying the fundamental structural control of an MDOF
system. the dynamic equilibrium is the same as:
− Inertia Forces
− Damping Forces
I D S
+ + =
f f f p
1 1
2 2
3 3
0 0 0
0 0 0
0 0 0
0 0 0
I
n n
m u
m u
m u
m u
   
   
   
   
= =
   
   
   
   
f mu
m1
m2
m3
mn-1
mn
…
m1
m2
1 2 2 1
2 2 3 3 2
3 3 4 3
0 0
0
0 0
0 0 0
D
n n
c c c u
c c c c u
c c c u
c u
+ −
   
   
− + −
   
   
= = − +
   
   
   
   
f cu

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