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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 1
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 1
This Federal Emergency Management Agency (FEMA) CD contains a set of instructional materials for use with FEMA Publication 451, NEHRP Recommended Provisions: Design Examples, in the form of PowerPoint slides with notes. These training materials provide a means for gaining additional knowledge about earthquake engineering as presented in the NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (FEMA 450). Also on the CD is NONLIN, an educational program for dynamic analysis of simple linear and nonlinear structures. The instructional materials can be presented to engineers/architects by a qualified speaker with expertise in the practice of earthquake engineering, can be used by an individual who wishes to enhance his/her understanding of earthquake engineering, or can be applied by engineering academics as the basis for classroom instruction on earthquake-resistant design. The CD contains a series of topic folders. In each folder are pdf files for the PowerPoint presentation, for the notes pages, and for student handouts. Also included is a folder for NONLIN that contains zip files for this program and a file that lists items referenced in the presentation.Any opinions, findings, conclusions, or recommendations expressed in this instructional material publication do not necessarily reflect the views of the Federal Emergency Management Agency. Additionally, neither FEMA nor any of its employees make any warranty, expressed or implied, nor assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, product, or process included in this publication. The opinions expressed herein regarding the requirements of the NEHRP Recommended Provisions, the referenced standards, and the building codes are not to be used for design purposes. Rather the user should consult the jurisdictions building official who has the authority to render interpretation of the code.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 2
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 2
NEHRP Recommended Provisions:Instructional Materials (FEMA 451B)
These instructional materials complement FEMA 451, NEHRP Recommended Provisions: Design Examples
Needed are copies of FEMA 451 and FEMA 450, the 2003 NEHRP Recommended Provisions for New Buildings and Other Structures (Part 1,Provisions, and Part 2, Commentary)
In addition to the Design Examples volume, the training requires copies of FEMA Publication 450, the 2003 NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 3
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 3
FEMA 450 and 451Single copies of both publications are available
at no charge from the FEMA Publications Center at 1-800-480-2520
(order by publication number)
Individual copies of these publications can be obtained at no charge from the FEMA Publications Center, 1-800-480-2520 (order by FEMA Publications number). If multiple copies are needed for presentation of the training materials to a group, e-mail [email protected].
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 4
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 4
Acknowledgments FEMA 451 and 451B were developed for
FEMA by the Building Seismic Safety Council (BSSC) of the National Institute of Building Sciences (NIBS).
The BSSC also manages development and updating of the NEHRP Recommended Provisions.
For information about the BSSC and its member organizations or to download FEMA 451 and 451B, see
http://bssconline.org
This CD was developed by the Building Seismic Safety Council under Contract EMW-1998-CO-0419 between the Federal Emergency Management Agency and the National Institute of Building Sciences. For further information on the Building Seismic Safety Council, see the Councils website www.bssconline.org or contact the Building Seismic Safety Council, 1090 Vermont, Avenue, N.W., Suite 700, Washington, D.C. 20005; phone 202-289-7800; fax 202-289-1092; e-mail [email protected].
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 5
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 5
Acknowledgments
FEMA and the BSSC are grateful to the following individuals for their contribution to these instructional materials:
Finley A. Charney, Ph.D., P.E., Virginia Tech, Blacksburg W. Samuel Easterling, Ph.D., P.E., Virginia Tech James R. Harris, Ph.D., P.E., J. R. Harris and Company,
Denver, Colorado Richard E. Klingner, Ph.D., P.E., University of Texas, Austin James R. Martin, Jr., Ph.D., Virginia Tech Steve Pryor, S.E., Simpson Strong Tie, Inc, Dublin,
California Michael D. Symans, Ph.D., Rensselaer Polytechnic Institute Carin Roberts-Wollmann, Ph.D., P.E., Virginia Tech
Much of this material was originally developed for the Multihazard Building Design Summer Course offered at FEMAs Emergency Management Institute. Managing the development of that course material for the Building Seismic Safety Council (BSSC) was Advanced Structural Concepts, Inc., Blacksburg, Virginia (Finley A. Charney, PhD., PE, President). Further, the course materials were developed in association with the Center for Extreme Load Effects on Structures, Virginia Tech (Finley A. Charney, PhD, PE, Director, and James R. Martin, Jr., Associate Director)
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 6
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 6
Motivation for Earthquake Engineering
Minimize human death and injury Minimize economic loss
Direct (collapse and damage) Indirect (loss of use, business
interruption) Maintain lifelines
Earthquake-resistant design and construction are important in those areas of the nation at risk.Users of this document who are also interested in residential construction are encouraged to consult FEMA Publication 232, Homebuilders Guide to Earthquake-Resistant Design and Construction. This guide provides information on current best practices for earthquake-resistant home design and construction for use by builders, designers, code enforcement personnel, and potential homeowners. It incorporates lessons learned from the 1989 Loma Prieta and 1994 Northridge earthquakes as well as knowledge gained from the FEMA CUREE-Caltech Wood Frame Project. It also introduces and explains the effects of earthquake loads on one- and two-family detached houses and identifies the requirements of the 2003 International Residential Code (IRC) intended to resist these loads.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 7
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 7
Information provided by Property Claims Service
Catastrophic Event Dollar Losses by Year
0
5
10
15
20
25
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
Year
$ B
illio
ns
Catastrophic event is defined as an event that has property loss claims in excess of $5 million.
Average of years 1986 to 1995
Losses Due to All Hazards
Loma Prieta
Northridge
Andrew & Iniki
Natural hazards can be catastrophic to life and property. This slide indicates dollar losses for all natural hazards in the United States for the past several years. The Loma Prieta and Northridge earthquakes were matched in dollar damage by hurricanes Hugo, Andrew and Iniki and all were surpassed by the damage caused by Hurricane Katrina.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 8
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 8
Dollar Losses by Type
Earthquake24.9%
Wind/Hail/Tornado 36.5%
Hurricane/Tropical Storm32.7%
Riot/Civil Disorder1.0%
Explosion/Fire4.5% Other
0.4%
A Significant Portion of Dollar LossDue to Earthquake
Includes Flood
Includes Flood
Earthquakes are a significant hazard but generally cause less dollar damage than wind, rain, and associated flooding. This slide does not break out flood damage, however, it should be emphasized that floods are one of the largest sources of losses due to natural disasters.Nevertheless, mitigation actions to reduce the losses from these natural hazards are cost-effective. In 2006, the National Institute of Building Sciences through its Multihazard Mitigation Council completed a report to the Congress of the United States on behalf of Federal Emergency Management Agency (FEMA) that presents the results of an independent study to assess the future savings from hazard mitigation activities. This study shows that money spent on reducing the risk of natural hazards is a sound investment. On average, a dollar spent by FEMA on hazard mitigation (actions to reduce disaster losses) provides the nation about $4 in future benefits. In addition, FEMA grants to mitigate the effects of floods, hurricanes, tornados, and earthquakes between 1993 and 2003 are expected to save more than 220 lives and prevent almost 4,700 injuries over approximately 50 years. Recent disaster events painfully demonstrate the extent to which catastrophic damage affects all Americans and the federal treasury.
Those interested in reading the report of the study should see http://nibs.org/MMC/mmcactiv5.html
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 9
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 9
Examples of US Earthquake Losses
1906 San Francisco1933 Long Beach1964 Alaska1971 San Fernando Valley1989 Loma Prieta1994 Northridge
These are but a few of the major earthquakes occurring in the United States during the previous century. This presentation emphasizes the Loma Prieta and Northridge earthquakes.The Northridge earthquake, like the 1971 San Fernando Valley earthquake, was a wakeup call to engineers and ultimately resulted in significant changes to building codes. Much of the current emphasis on performance-based engineering is due to the greater than expected damage that occurred as a result of the Northridge earthquake.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 10
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 10
1971 Earthquake in the San Fernando Valley of California
Earth dam located about 20 km from the epicenter. Part of the upstream face lost bearing strength and slipped beneath the water.
This slide emphasizes the fact that damage occurs to nonbuilding structures as well as building structures.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 11
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 11
1971 San Fernando Valley EarthquakeSoft story failure of the Olive View Hospital. The column failure caused a collapse that pinned the ambulances under the rubble, rendering them useless.
Damage to the Olive View Hospital was particularly disturbing because the structure was relatively new and was designed according to the modern code at the time. The building was a complete loss and had to be demolished. Note that the ambulance canopy in the foreground is a separate structure, and was also a complete loss. Also significant is the fact that the ambulances were trapped in the collapsed canopy and were not available for use.During the 1994 Northridge earthquake, the new Olive View Hospital structure fared rather well, but there were significant losses associated with nonstructural elements and components.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 12
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 12
1989 Earthquake in Loma Prieta, CaliforniaOakland Bay Bridge failure.
Losses of transportation structures are very dramatic and can be among the most costly in terms of loss of life and property and indirect effects. This bridge was out of service for several weeks after the earthquake requiring major rerouting of traffic. The collapse of the Oakland Cyprus Street Viaduct (not shown) was responsible for the loss of 42 lives. There were similar but less catastrophic failures of sections of the Embarcadero Freeway in San Francisco.The Loma Prieta earthquake killed more than 60 people, injured 3,700, and left 12,000 homeless.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 13
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 13
1994 Earthquake in Northridge, California
Bull Creek Canyon Channel Bridge on the Simi Valley freeway near the epicenter to the north. Shear failure of a flared column.
Freeways in the Los Angeles area were not immune to damage during the Northridge earthquake. Ironically, many of the bridges that failed were scheduled for rehabilitation prior to the earthquake.Approximately 60 people were killed by the quake.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 14
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 14
1994 Northridge EarthquakeGavin Canyon Undercrossingon I-5
Another illustration of damage as a result of the Northridge earthquake.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 15
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 15
Examples of Earthquake Losses Outside the United States
1923 Tokyo 1927 China 1985 Chile 1985 Mexico City 1988 Armenia 1993 Hokkaido 1995 Kobe 1999 Turkey, Taiwan 2001 India
Earthquakes occur all over the world and often produce unimaginable destruction. Codes and enforcement in developing countries are often decades behind those of the industrialized world.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 16
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 16
1985 Mexico City EarthquakePino Suarez Towers looking north -- one of the few steel frame buildings to collapse.
The damage in Mexico City was due to an earthquake that occurred more than 350 km away from the city center. The main shock killed 10,000, left 50,000 homeless, and caused $4 billion dollars damage.The vast destruction was attributed in large part to dynamic amplification of seismic waves through the soft soil in Mexico City. The dominant seismic waves had a period of about 2.0 seconds, wreaking havoc on buildings in the 10- to 20-story range.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 17
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 17
1988 Leninakan, Armenia, EarthquakeDamage to a stone bearing wall building. The floor planks
were not tied to the supporting bearing walls.
This is an example of the devastation caused by earthquakes in countries without adequate seismic design building code requirements and/or enforcement. Many (almost complete destruction) precast concrete frame buildings collapsed because of inadequate detailing. This earthquake killed at least 25,000 people, and left 500,000 homeless. Dollar damage was estimated in excess of 13 billion. Overall, 95% of the precast frame structures (5 to 12 stories) in Leninakan collapsed or were damaged beyond repair.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 18
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 18
1995 Kobe, Japan, Earthquake
Distorted train tracks.
The Kobe earthquake killed more than 5,000 people and injured 26,000 others. More than 56,000 buildings were destroyed. Losses were estimated at more than $2 billion. This is more than 10 times the dollar loss for the Northridge earthquake which occurred exactly one year earlier in southern California. This slide was selected to emphasize the point that loss to nonbuilding structures and lifelines can have a significant effect on society. Further, it should be noted that a considerable amount of business and industrial activities that moved from the area after the earthquake never returned.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 19
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 19
Build (Rebuild)
Earthquake!
Learning
Research
Code Development
Typical Cycle
If there is any fortunate aspect of earthquakes, it is that the built environment is an excellent proving ground. Damage occurring during earthquakes is extensively studied and research is performed, ultimately leading to the development of improved building codes. However, it seems that no matter how diligently we react to earthquakes, we are taught new and serious lessons when the next quake strikes. The damage occurring to welded frame structures during the Northridge earthquake is an excellent example.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 20
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 20
The Built Environment(new and existing)
Construction
Architecture
Sociology
Economics
Seismology
Hazard Risk Assessment
Insurance
Government
Research
Education
Geology
EngineeringMaterials
BuildingsBridgesDamsLifelinesOther...
Who Is Involved in Earthquake Hazard Mitigation?
Many disciplines are involved in earthquake hazard mitigation. All groups must work together to provide the level of protection needed by society.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 21
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 21
These Instructional Materials FOCUS on STRUCTURAL ENGINEERING
and New buildings Hazards associated with ground shaking Force-Based approach of 2003 NEHRP
Recommended Provisions (FEMA 450) Examples presented in NEHRP
Recommended Provisions: Design Examples (FEMA 451)
Probabilistic and deterministic based ground motions
New concepts of performance-based engineering
These instructional materials focus almost entirely on new buildings. However, some information is provided for existing buildings, particularly as related to performance-based engineering, and on nonbuilding structures and nonstructural building components.Further, these instructional materials concentrate on ground shaking, which is only one of the many hazards associated with earthquakes (e.g. fault rupture, liquefaction, landslides, flooding, and fire).
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 22
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 22
NEHRP Recommended Provisions (FEMA 450)
IBC and IRC ASCE 7
Published Design Documentsfor New Buildings
1906 San Francisco Earthquake
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 23
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 23
2003 NEHRP Recommended Provisionsfor New Buildings and Other Structures
Uses seismic hazard map (2%-50years) forevaluation purposes
Relies on equal displacement concept toestablish design forces
Utilizes linear elastic static or dynamic analysis
Intended result (obtained somewhat implicitly): Little or no damage for frequent earthquakes Minor nonstructural damage for common earthquakes Life-safety or collapse prevention for rare earthquakes
Deformations checked globally
This slide emphasizes the underlying principles of the NEHRP Recommended Provisions. Performance is evaluated somewhat implicitly, meaning that local deformations in members are not addressed. Before the Northridge earthquake, it was thought that this methodology was sufficient. Many engineers are now moving towards performance-based concepts, particularly in the rehabilitation of existing buildings.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 24
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 24
Other Topics in this SeriesTopic 1 Introduction to CourseTopic 2 Earthquakes Mechanics and EffectsTopic 3 Structural Dynamics of SDOF SystemsTopic 4 Structural Dynamics of MDOF SystemsTopic 5a Seismic Hazard Analysis Topic 5b Ground Motion MapsTopic 6 Inelastic Behavior of Materials and Structures Topic 7 Concepts of Earthquake Engineering [FEMA 451, Ch. 1]Topic 8a Introduction to the NEHRP [FEMA 451, Ch. 2]Topic 8b Overview of Standards used in NEHRP Recommended ProvisionsTopic 9 Seismic Load AnalysisTopic 10 Seismic Design of Structural Steel Structures [FEMA 451, Ch. 5]Topic 11 Seismic Design of Reinforced Concrete Structures [FEMA 451, Ch. 6]Topic 12 Seismic Design of Masonry Structures [FEMA 451, Ch. 9]Topic 13 Seismic Design of Wood Structures [FEMA 451, Ch. 10]Topic 14 Foundation Design [FEMA 451, Ch. 4]Topic 16 Nonstructural Components [FEMA 451, Ch. 13]
Topics 1 through 14 and 16 are the basic topics and include most of the concepts required to understand how earthquake analysis and design procedures are developed (Topics 1-7) and then how they are incorporated into the NEHRP Recommended Provisions and/or ASCE-7. These topics could generally be covered in a four- to five-day course with seven hours of instruction per day. If presented in such a classroom setting, instructors should consider developing a series of group exercises to help illustrate the concepts and to break up a long series of lectures. One of the exercises should use the computer program NONLIN that is included on the FEMA 451B CD.The chapter numbers to the right of some of the topics refer to chapters in FEMA 451, NEHRP Recommended Provisions: Design Examples. In some cases, there is direct correlation between the examples in the slide sets and the FEMA 451 CD. For example, the topics in concrete and steel are related in this manner.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 25
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 25
Other Topics in this SeriesPart 2: Advanced Topics
Topic 15-1 Introduction Topic 15-2 Performance Based EngineeringTopic 15-3 Seismic Hazard AnalysisTopic 15-4 Geotechnical Earthquake EngineeringTopic 15-5a Advanced Analysis, Part 1 of 3 Topic 15-5b Advanced Analysis, Part 2 of 3Topic 15-5c Advanced Analysis, Part 3 of 3Topic 15-6 Passive Energy Systems [FEMA 451, Ch. 6]Topic 15-7 Seismic Isolation [FEMA 451, Ch. 11]Topic 15-8 Nonbuilding Systems [FEMA 451, Ch. 12]
These topics are considered to be advanced topics and would be covered in a separate four-day course. Note that there is considerable overlap between thematerials in Topics 5a and 15-3. As with the previous slide, the chapter numbers to the right of some of the topics refer to chapters in the FEMA 451 volume.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 26
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 26
Chapters in the FEMA 451 Examples CD
Ch. 1 Fundamentals Ch. 2 Guide to the Use of the NEHRP Recommended ProvisionsCh. 3 Structural Analysis (including nonlinear analysis)Ch. 4 Foundation DesignCh. 5 Steel Structures Ch. 6 Reinforced Concrete StructuresCh. 7 Precast Concrete StructuresCh. 8 Composite Steel/Concrete StructuresCh. 9 Masonry Structures Ch. 10 Wood StructuresCh. 11 Seismically Isolated StructuresCh. 12 Nonbuilding StructuresCh. 13 Nonstructural Components
The FEMA 451 CD contains 13 chapters as shown in this slide. The examples are extremely detailed and should be worked into the coursework where possible. Individuals pursuing earthquake engineering knowledge using these presentations for self-study also are strongly encouraged to work through these examples after working through with the presentation information.
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FEMA 451B HandoutsFEMA 451B Notes Introduction 1 - 27
Instructional Material Complementing FEMA 451, Design Examples Introduction 1 - 27
Structural engineering:The art of using materials that
have properties which can only be estimatedto build real structures that
can only be approximately analyzedto withstand forces that
are not accurately knownso that our responsibility to the
public safety is satisfied.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 1
Earthquakes Mechanics and Effects
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 2
Earthquakes: Cause and Effect
Why earthquakes occur How earthquakes are measured Earthquake effects Mitigation strategy Earthquake time histories
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 3
Seismic Activity: 1961-1967
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 4
Plate Boundaries
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 5
Plate Tectonics: Driving Mechanism
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 6
Plate Tectonics: Details in Subduction Zone
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 7
Seismicity of North America
Pacific Plate
North AmericanPlate
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 8
Seismicity of Alaska
Pacific Plate
North American Plate
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 9
Faults and Fault Rupture
Fault plane
Hypocenter(focus)
Rupture surface
Epicenter
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 10
Types of Faults
Strike slip(left lateral)
Strike slip(right lateral)
Normal Reverse (thrust)
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 11
New fence
Time = 0 Years
Elastic Rebound Theory
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 12
Old fence
New road
Time = 40 Years
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 13
Old fence
Time = 41 Years
New road
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 14
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 15
Seismic Wave Forms(Body Waves)
Compression wave(P wave)
Shear wave(S wave)
Direction of
propagation
Direction of
propagation
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 16
Love wave Rayleigh wave
Seismic Wave Forms(Surface Waves)
Direction of
propagation
Direction of
propagation
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 17
Love wavesP waves S waves
Arrival of Seismic Waves
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 18
Relationship Between Reservoir Leveland Seismic Activity at Koyna Dam, India
Inflow
Reservoir level
Earthquake frequency
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 19
Effects of Seismic Waves
Fault rupture Ground shaking Landslides Liquefaction Tsunamis Seiches
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 20
Surface Fault Rupture, 1971 Earthquake in San Fernando, California
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 21
If a saturated sand is subjected to groundvibrations, it tends to compact and decrease in volume.
If drainage is unable to occur, the tendency todecrease in volume results in an increase inpore pressure.
If the pore water pressure builds up to the point atwhich it is equal to the overburden pressure, theeffective stress becomes zero, the sand loses itsstrength completely, and liquefaction occurs.
Cause of Liquefaction
Seed and Idriss (1971)
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 22
Liquefaction Damage, Niigata, Japan, 1964
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 23
Liquefaction and Lateral Spreading, 1993 Earthquake in Kobe, Japan
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 24
Landslide on Coastal Bluff,1989 Earthquake in Loma Prieta, California
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 25
Cause of Tsunamis
Tsunamis are created by a sudden vertical movement of the sea floor.
These movements usually occur insubduction zones.
Tsunamis move at great speeds, often 600to 800 km/hr.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 26
Tsunami Damage, Seward, Alaska, 1964
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 27
Result of Ground Shaking, 1994 Earthquake in Northridge, California
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 28
Earthquake effect Strategy Fault rupture AvoidTsunami/seiche AvoidLandslide AvoidLiquefaction Avoid/resistGround shaking Resist
Mitigation Strategies
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 29
Measuring Earthquakes
INTENSITY Subjective Used where instruments are not available Very useful in historical seismicity
MAGNITUDE Measured with seismometers Direct measure of energy released Possible confusion due to different measures
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 30
Modified Mercalli IntensityDeveloped by G. Mercalli in 1902 (after a previous
version of M. S. De Rossi in the 1880s)
Subjective measure of human reaction and damage
Modified by Wood and Neuman to fitCalifornia construction conditions
Intensity range I (lowest) to XII (most severe)
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 31
Modified Mercalli Intensity
Not felt except by a few under especiallyfavorable circumstances.
Felt only by a few persons at rest, especially onupper floors if buildings. Suspended objects may swing.
Felt quite noticeably indoors, especially onupper floors of buildings. Standing automobiles mayrock slightly. Vibration like passing truck.
I.
II.
III.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 32
Modified Mercalli IntensityDuring the day, felt indoors by many, outdoors byfew. At night, some awakened. Dishes, windows,doors disturbed; walls make creaking sound. Sensationlike heavy truck striking building. Standing automobilesrocked noticeably. [0.015 to 0.02g]
Felt by nearly everyone, many awakened. Somedishes and windows broken. Cracked plaster.Unstable objects overturned. Disturbance of trees, polesand other tall objects. [0.03 to 0.04g]
Felt by all. Many frightened and run outdoors.Some heavy furniture moved. Fallen plaster anddamaged chimneys. Damage slight. [0.06 to 0.07g]
IV.
V.
VI.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 33
Modified Mercalli IntensityEverybody runs outdoors. Damage negligible inbuildings of good design and construction, slight tomoderate in well built ordinary structures, considerablein poorly built or badly designed structures. Noticedby persons driving cars. [0.10 to 0. 15g]
Damage slight in specially designed structures,considerable in ordinary construction, great inpoorly built structures. Fall of chimneys, stacks,monuments. Sand and mud ejected is smallamounts. Changes in well water. Persons drivingcars disturbed. [0.25 to 0.30g]
VII.
VIII.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 34
Modified Mercalli IntensityDamage considerable in specially designedstructures, well designed frame structures thrownout of plumb, damage great in substantial buildingswith partial collapse. Buildings shifted off foundations.Ground cracked conspicuously. Underground pipesbroken. [0.50 to 0.55g]
Some well built wooden structures destroyed. Mostmasonry and frame structures destroyed withfoundations badly cracked. Rails bent. Landslidesconsiderable from river banks and steep slopes.Shifted sand and mud. Water splashed over banks.[More than 0.60g]
IX.
X.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 35
Modified Mercalli Intensity
Few, if any, (masonry) structures left standing. Bridges destroyed. Broad fissures in ground. Underground pipelines completely out of service.Earth slumps and land slips in soft ground.Rails bent greatly.
Damage total. Waves seen on ground surface. Linesof sight and level distorted. Objects thrown into air.
XI.
XII.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 36
Isoseismal Map for the Giles County, Virginia,Earthquake of May 31, 1897.
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 37
Isoseismal MapFor New MadridEarthquake ofDecember 16, 1811
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 38
Isoseismal Mapfor 1886 CharlestonEarthquake
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 39
Isoseismal Map for February 9, 1971,San Fernando Earthquake
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 40
Comparison of Isosiesmal Intensity for Four Earthquakes
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 41
Comparisons of Various Intensity Scales
MMI = Modified MercalliRF = Rossi-ForelJMA = Japan Meteorological AgencyMSK =Medvedez-Spoonheur-Karnik
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Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 42
Instrumental Seismicity
ML = Log [Maxumum Wave Amplitude (in mm/1000)]
Recorded Wood-Anderson seismograph
100 km from epicenter
Magnitude (Richter, 1935)
Also called local magnitude
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 43
M = Log A +f(d,h) +CS + CR
A is wave amplitude
F(d,h) accounts for focal distance and depth
CS and CR, are station and regional corrections
Magnitude (in general)
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 44
MS Surface-wave magnitude (Rayleigh waves)
mb Body-wave magnitude (P waves)
MB Body-wave magnitude (P and other waves)
mbLg (Higher order Love and Rayleigh waves)
MJMA (Japanese, long period)
Other Wave-Based Magnitudes
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 45
Moment Magnitude
Seismic moment = MO = AD
Where: = modulus of rigidityA = fault rupture areaD = fault dislocation or slip
Moment magnitude = MW = (Log MO-16.05)/1.5
[Units = force times distance]
(Units = dyne-cm)
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 46
Moment Magnitude vs Other Magnitude Scales(Magnitude Saturation)
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 47
0
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10 11 12
Intensity
Mag
nitu
de
Richter (Local)MbLg
Approximate RelationshipBetween Magnitude and Intensity
0 0.67 1LM I= +
66.149.0 0 += ImbLg
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 48
Seismic Energy ReleaseLog E = 1.5 MS + 11.8
1E+12
1E+14
1E+16
1E+18
1E+20
1E+22
1E+24
1E+26
1E+28
0 2 4 6 8 10
Magnitude, Ms
Ener
gy, E
rgs
..
31 1000
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 49
Seismic Energy Release
1E+12
1E+14
1E+16
1E+18
1E+20
1E+22
1E+24
1E+26
1E+28
0 2 4 6 8 10
Magnitude, Ms
Ener
gy, E
rgs
..
Nuclear bomb
1964 Alaska earthquake
1906 San Francisco earthquake
1972 San Fernando earthquake
Atomic bomb
1978 Santa Barbara earthquake
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 50
Ground Motion Accelerograms
Sources: NONLIN (more than 100 records) Internet (e.g., National Strong Motion Data Center) USGS CD ROM
Uses: Evaluation of earthquake characteristics Development of response spectra Time history analysis
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 51
Sample Ground Motion Records
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 52
Ground Motion Characteristics
Acceleration, velocity, displacement Effective peak acceleration and velocity Fourier amplitude spectra Duration (bracketed duration) Incremental velocity (killer pulse) Response spectra Other (see, for example, Naiem and Anderson 2002)
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 53
Corrected vs Uncorrected Motions
Corrections made primarily:
To remove instrument response To account for base line shift
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 54
-500
-250
0
250
500
0 1 2 3 4 5 6 7 8 9 10
0
50
100
150
200
0 1 2 3 4 5 6 7 8 9 10
0
200400
600
800
0 1 2 3 4 5 6 7 8 9 10
Time, sec
Acceleration, in/sec2
Velocity, in/sec
Displacement, in
Base Line Correction for Simple Ground Motion
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 55
-600
-400
-200
0
200
400
600
0 5 10 15 20 25 30 35 40 45
-600
-400
-200
0
200
400
600
0 5 10 15 20 25 30 35 40 45
-600
-400
-200
0
200
400
600
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5
Tim e (s e c )
Horizontal acceleration (E-W), cm/sec2
Vertical acceleration (E-W), cm/sec2
Horizontal acceleration (N-S), cm/sec2
Typical Earthquake Accelerogram Set
Time, Seconds Loma Prieta Earthquake
-463 cm/sec2
-500 cm/sec2
-391 cm/sec2
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 56
-600
-400
-200
0
200
400
600
0 5 10 15 20 25 30 35 40 45
Bracketed duration
0.05g
Definition of Bracketed Duration
Time, Seconds
Acceleration, cm/sec2
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 57
Definition of Incremental Velocity
Time, Seconds
-400
-300
-200
-100
0
100
200
300
400
8 9 10 11 12
-6 0 0
-4 0 0
-2 0 0
0
2 0 0
4 0 0
6 0 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5
Acceleration, cm/sec2
Time, Seconds
Acceleration, cm/sec2
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 58
Concept of Fourier Amplitude Spectra
-6 0 0
-4 0 0
-2 0 0
0
2 0 0
4 0 0
6 0 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5
Acceleration, cm/sec2
N points at timestep dt
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 10 20 30
Frequency (Hz)
N/2 points at frequency df
Normalized Fourier Coefficient
)2cos()2sin()2cos()(2/
100
2/
1
2/
1000 j
N
jj
N
j
N
jjjg jfAajfbjfaatv
== =
++=++&&
22jjj baA +=Ndtdff /10 ==
=
j
jj a
barctan
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 59
Concept of Fourier Amplitude Spectra
-50-40-30-20-10
01020304050
0.00 0.10 0 .20 0 .30 0 .40 0.50 0.60 0 .70 0 .80 0 .90 1.00
T im e, S econds
Am
plitu
de
0
2
4
6
8
1 0
1 2
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
F r e q u e n c y , H z .
Four
ier A
mpl
itude
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 60
Ground Motion Frequency Content (1)
-600
-400
-200
0
200
400
600
0 10 20 30 40 50
-600
-400
-200
0
200
400
600
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15 20 25 30
Frequency (Hz)
Four
ier A
mpl
itude
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15 20 25 30
Frequency (Hz)Fo
urie
r Am
plitu
de
Horizontal acceleration (E-W), cm/sec2
Vertical acceleration (E-W), cm/sec2
Time, Seconds
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 61
-6 0 0
-4 0 0
-2 0 0
0
2 0 0
4 0 0
6 0 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 10 20 30
Frequency (Hz)
Four
ier A
mpl
itude
- 4 0
-3 0
-2 0
-1 0
0
1 0
2 0
3 0
4 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 10 20 30
Frequency (Hz)
Four
ier A
mpl
itude
-1 5
-1 0
-5
0
5
1 0
1 5
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 50.0
0.2
0.4
0.6
0.8
1.0
1.2
0 10 20 30
Frequency (Hz)
Four
ier A
mpl
itude
`
Horizontal acceleration, cm/sec2
Horizontal velocity, cm/sec
Horizontal displacement, cm
Ground Motion Frequency Content (2)
Time, Seconds
-463 cm/sec2
-30.7 cm/sec
11.0 cm
-
Instructional Material Complementing FEMA 451, Design Examples Earthquake Mechanics 2 - 62
-0.40
-0.20
0.00
0.20
0.40
0.00 1.00 2.00 3.00 4.00 5.00 6.00
TIME, SECONDS
GR
OU
ND
AC
C, g
T=2.0 Seconds
T=0.6 Seconds
El Centro Earthquake Record
Maximum Displacement Response Spectrum
Development of an Elastic DisplacementResponse Spectrum
-4.00
-2.00
0.00
2.00
4.00
0.00 1.00 2.00 3.00 4.00 5.00 6.00
DIS
PLA
CEM
ENT,
in.
-8.00
-4.00
0.00
4.00
8.00
0.00 1.00 2.00 3.00 4.00 5.00 6.00
DIS
PLA
CEM
ENT,
In.
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10
PERIOD, Seconds
DIS
PLA
CEM
ENT,
inch
es
-
SDOF Dynamics 3 - 1Instructional Material Complementing FEMA 451, Design Examples
Structural Dynamics ofLinear Elastic Single-Degree-of-Freedom
(SDOF) Systems
-
SDOF Dynamics 3 - 2Instructional Material Complementing FEMA 451, Design Examples
Structural Dynamics
Equations of motion for SDOF structures Structural frequency and period of vibration Behavior under dynamic load Dynamic magnification and resonance Effect of damping on behavior Linear elastic response spectra
-
SDOF Dynamics 3 - 3Instructional Material Complementing FEMA 451, Design Examples
Importance in Relation to ASCE 7-05 Ground motion maps provide ground
accelerations in terms of response spectrumcoordinates.
Equivalent lateral force procedure gives base shear in terms of design spectrum and period of vibration.
Response spectrum is based on 5% critical damping in system.
Modal superposition analysis uses design response spectrum as basic ground motion input.
-
SDOF Dynamics 3 - 4Instructional Material Complementing FEMA 451, Design Examples
Idealized SDOF Structure
Mass
Stiffness
Damping
F t u t( ), ( )
t
F(t)
t
u(t)
-
SDOF Dynamics 3 - 5Instructional Material Complementing FEMA 451, Design Examples
F t( )f tI ( )
f tD ( )0 5. ( )f tS0 5. ( )f tS
F t f t f t f tI D S( ) ( ) ( ) ( ) = 0
f t f t f t F tI D S( ) ( ) ( ) ( )+ + =
Equation of Dynamic Equilibrium
-
SDOF Dynamics 3 - 6Instructional Material Complementing FEMA 451, Design Examples
-40
0
40
0.00 0.20 0.40 0.60 0.80 1.00
-0.50
0.00
0.50
0.00 0.20 0.40 0.60 0.80 1.00
-15.00
0.00
15.00
0.00 0.20 0.40 0.60 0.80 1.00
-400.00
0.00
400.00
0.00 0.20 0.40 0.60 0.80 1.00
Acceleration, in/sec2
Velocity, in/sec
Displacement, in
Applied Force, kips
Observed Response of Linear SDOF
Time, sec
-
SDOF Dynamics 3 - 7Instructional Material Complementing FEMA 451, Design Examples
Observed Response of Linear SDOF(Development of Equilibrium Equation)
-30.00
-15.00
0.00
15.00
30.00
-0.60 -0.30 0.00 0.30 0.60
Displacement, inches
-4.00
-2.00
0.00
2.00
4.00
-20.00 -10.00 0.00 10.00 20.00
Velocity, In/sec
-50.00
-25.00
0.00
25.00
50.00
-500 -250 0 250 500
Acceleration, in/sec2
Spring Force, kips Damping Force, Kips Inertial Force, kips
Slope = k= 50 kip/in
Slope = c= 0.254 kip-sec/in
Slope = m= 0.130 kip-sec2/in
f t k u tS ( ) ( )= f t c u tD ( ) &( )= f t m u tI ( ) &&( )=
-
SDOF Dynamics 3 - 8Instructional Material Complementing FEMA 451, Design Examples
F t( )f tI ( )
f tD ( )0 5. ( )f tS0 5. ( )f tS
m u t c u t k u t F t&&( ) & ( ) ( ) ( )+ + =
Equation of Dynamic Equilibrium
f t f t f t F tI D S( ) ( ) ( ) ( )+ + =
-
SDOF Dynamics 3 - 9Instructional Material Complementing FEMA 451, Design Examples
Mass
Includes all dead weight of structure May include some live load Has units of force/acceleration
Inte
rnal
For
ce
Acceleration
1.0M
Properties of Structural Mass
-
SDOF Dynamics 3 - 10Instructional Material Complementing FEMA 451, Design Examples
Damping
In absence of dampers, is called inherent damping Usually represented by linear viscous dashpot Has units of force/velocity
Dam
ping
For
ce
Velocity
1.0C
Properties of Structural Damping
-
SDOF Dynamics 3 - 11Instructional Material Complementing FEMA 451, Design Examples
Damping
Dam
ping
For
ce
Displacement
Properties of Structural Damping (2)
Damping vs displacement response iselliptical for linear viscous damper.
AREA =ENERGYDISSIPATED
-
SDOF Dynamics 3 - 12Instructional Material Complementing FEMA 451, Design Examples
Includes all structural members May include some seismically nonstructural members Requires careful mathematical modelling Has units of force/displacement
Sprin
g Fo
rce
Displacement
1.0K
Properties of Structural StiffnessS
tiffn
ess
-
SDOF Dynamics 3 - 13Instructional Material Complementing FEMA 451, Design Examples
Is almost always nonlinear in real seismic response Nonlinearity is implicitly handled by codes Explicit modelling of nonlinear effects is possible
Sprin
g Fo
rce
Displacement
Properties of Structural Stiffness (2)S
tiffn
ess
AREA =ENERGYDISSIPATED
-
SDOF Dynamics 3 - 14Instructional Material Complementing FEMA 451, Design Examples
Undamped Free Vibration
)cos()sin()( 00 tututu
+=&
m u t k u t&&( ) ( )+ = 0Equation of motion:
0u&Initial conditions:
0uA&
= B u= 0Solution: =km
Assume: u t A t B t( ) sin( ) cos( )= + 0u
-
SDOF Dynamics 3 - 15Instructional Material Complementing FEMA 451, Design Examples
= km
f = 2
Tf
= =1 2
Period of Vibration(sec/cycle)
Cyclic Frequency(cycles/sec, Hertz)
Circular Frequency (radians/sec)
Undamped Free Vibration (2)
-3-2-10123
0.0 0.5 1.0 1.5 2.0
Time, seconds
Dis
plac
emen
t, in
ches
T = 0.5 sec
u0
&u01.0
-
SDOF Dynamics 3 - 16Instructional Material Complementing FEMA 451, Design Examples
Approximate Periods of Vibration(ASCE 7-05)
xnta hCT =
NTa
1.0=
Ct = 0.028, x = 0.8 for steel moment framesCt = 0.016, x = 0.9 for concrete moment framesCt = 0.030, x = 0.75 for eccentrically braced framesCt = 0.020, x = 0.75 for all other systems
Note: This applies ONLY to building structures!
For moment frames < 12 stories in height, minimumstory height of 10 feet. N = number of stories.
-
SDOF Dynamics 3 - 17Instructional Material Complementing FEMA 451, Design Examples
Empirical Data for Determinationof Approximate Period for Steel Moment Frames
8.0028.0 na hT =
-
SDOF Dynamics 3 - 18Instructional Material Complementing FEMA 451, Design Examples
Periods of Vibration of Common Structures
20-story moment resisting frame T = 1.9 sec10-story moment resisting frame T = 1.1 sec1-story moment resisting frame T = 0.15 sec
20-story braced frame T = 1.3 sec10-story braced frame T = 0.8 sec1-story braced frame T = 0.1 sec
Gravity dam T = 0.2 secSuspension bridge T = 20 sec
-
SDOF Dynamics 3 - 19Instructional Material Complementing FEMA 451, Design Examples
SD1 Cu> 0.40g 1.4
0.30g 1.40.20g 1.50.15g 1.6< 0.1g 1.7
computedua TCTT =
Adjustment Factor on Approximate Period(Table 12.8-1 of ASCE 7-05)
Applicable ONLY if Tcomputed comes from a properlysubstantiated analysis.
-
SDOF Dynamics 3 - 20Instructional Material Complementing FEMA 451, Design Examples
If you do not have a more accurate period (from a computer analysis), you must use T = Ta.
If you have a more accurate period from a computeranalysis (call this Tc), then:
if Tc > CuTa use T = CuTa
if Ta < Tc < TuCa use T = Tc
if Tc < Ta use T = Ta
Which Period of Vibration to Usein ELF Analysis?
-
SDOF Dynamics 3 - 21Instructional Material Complementing FEMA 451, Design Examples
Damped Free Vibration
u t e u t u u tt DD
D( ) cos( )&
sin( )= + +
0 0 0
m u t c u t k u t&&( ) &( ) ( )+ + = 0Equation of motion:u u0 0&Initial conditions:
Solution:
Assume: u t e st( ) =
= =c
mccc2
D = 12
-
SDOF Dynamics 3 - 22Instructional Material Complementing FEMA 451, Design Examples
= =c
mccc2
Damping in Structures
cc is the critical damping constant.
Time, sec
Displacement, in
is expressed as a ratio (0.0 < < 1.0) in computations.
Sometimes is expressed as a% (0 < < 100%).
Response of Critically Damped System, =1.0 or 100% critical
-
SDOF Dynamics 3 - 23Instructional Material Complementing FEMA 451, Design Examples
-30.00
-15.00
0.00
15.00
30.00
-0.60 -0.30 0.00 0.30 0.60
Displacement, inches
-4.00
-2.00
0.00
2.00
4.00
-20.00 -10.00 0.00 10.00 20.00
Velocity, In/sec
-50.00
-25.00
0.00
25.00
50.00
-500 -250 0 250 500
Acceleration, in/sec2
Spring Force, kips Damping Force, Kips Inertial Force, kips
True damping in structures is NOT viscous. However, for lowdamping values, viscous damping allows for linear equations and vastly simplifies the solution.
Damping in Structures
-
SDOF Dynamics 3 - 24Instructional Material Complementing FEMA 451, Design Examples
Damped Free Vibration (2)
-3-2-10123
0.0 0.5 1.0 1.5 2.0
Time, seconds
Dis
plac
emen
t, in
ches
0% Damping10% Damping20% Damping
-
SDOF Dynamics 3 - 25Instructional Material Complementing FEMA 451, Design Examples
Damping in Structures (2)Welded steel frame = 0.010Bolted steel frame = 0.020
Uncracked prestressed concrete = 0.015Uncracked reinforced concrete = 0.020Cracked reinforced concrete = 0.035
Glued plywood shear wall = 0.100Nailed plywood shear wall = 0.150
Damaged steel structure = 0.050Damaged concrete structure = 0.075
Structure with added damping = 0.250
-
SDOF Dynamics 3 - 26Instructional Material Complementing FEMA 451, Design Examples
Inherent damping
Added damping
is a structural (material) propertyindependent of mass and stiffness
critical%0.7to5.0=Inherent
is a structural property dependent onmass and stiffness anddamping constant C of device
critical%30to10=Added
Damping in Structures (3)
C
-
SDOF Dynamics 3 - 27Instructional Material Complementing FEMA 451, Design Examples
ln uu
1
22
21
=
u uu
1 2
22
For alldamping values
For very lowdamping values
Measuring Damping from Free Vibration Test
-1
-0.5
0
0.5
1
0.00 0.50 1.00 1.50 2.00 2.50 3.00Time, Seconds
Ampl
itude
u e t0
u1u2 u3
-
SDOF Dynamics 3 - 28Instructional Material Complementing FEMA 451, Design Examples
Undamped Harmonic Loading
)sin()()( 0 tptuktum =+&&Equation of motion:
-150-100
-500
50100150
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, Seconds
Forc
e, K
ips
po=100 kips= 0.25 sec
= frequency of the forcing function
2
=T
T
-
SDOF Dynamics 3 - 29Instructional Material Complementing FEMA 451, Design Examples
Solution:
Particular solution:
Complimentary solution:
u t C t( ) s in ( )=
u t A t B t( ) sin( ) cos( )= +
= )sin()sin(
)/(11)( 2
0 ttkptu
Undamped Harmonic Loading (2)
m u t k u t p t&&( ) ( ) s in ( )+ = 0 Equation of motion:
Assume system is initially at rest:
-
SDOF Dynamics 3 - 30Instructional Material Complementing FEMA 451, Design Examples
Define
=
( )u t pk
t t( ) sin( ) sin( )=
0 21
1
Static displacementSteady state
response(at loading frequency)
Transient response(at structures frequency)
Loading frequency
Structures natural frequency
Undamped Harmonic Loading
Dynamic magnifier
-
SDOF Dynamics 3 - 31Instructional Material Complementing FEMA 451, Design Examples
-10-505
10
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-10-505
10
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-10-505
10
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, seconds
spac
ee
t,-200-100
0100200
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
= 2 rad / sec = 4 rad / sec uS = 5 0. .in = 0 5.
Loading (kips)
Steady stateresponse (in.)
Transientresponse (in.)
Total response(in.)
-
SDOF Dynamics 3 - 32Instructional Material Complementing FEMA 451, Design Examples
Steady stateresponse (in.)
Transientresponse (in.)
Total response(in.)
Loading (kips)
= 4 rad / sec 4 rad / sec uS = 5 0. .in = 0 99.
-500
-250
0
250
500
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-150-100
-500
50100150
0.00 0 .25 0 .50 0 .75 1.00 1 .25 1 .50 1 .75 2 .00
-500
-250
0
250
500
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-80
-40
0
40
80
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00Time, seconds
p,
-
SDOF Dynamics 3 - 33Instructional Material Complementing FEMA 451, Design Examples
-80
-40
0
40
80
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00Time, seconds
Dis
plac
emen
t, in
.
2 uS
Undamped Resonant Response Curve
Linear envelope
-
SDOF Dynamics 3 - 34Instructional Material Complementing FEMA 451, Design Examples
Steady stateresponse (in.)
Transientresponse (in.)
Total response (in.)
Loading (kips)
= 4 rad / sec 4 rad / sec uS = 5 0. .in = 1 01.
-150-100
-500
50100150
0 .00 0.25 0 .50 0 .75 1 .00 1.25 1 .50 1 .75 2.00
-500
-250
0
250
500
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-500
-250
0
250
500
0 .0 0 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50 1 .75 2 .00
-80
-40
0
40
80
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
T im e, seconds
p,
-
SDOF Dynamics 3 - 35Instructional Material Complementing FEMA 451, Design Examples
Steady state response (in.)
Transient response (in.)
Total response (in.)
Loading (kips)
=8 rad / sec = 4 rad / sec uS = 5 0. .in = 2 0.
-150-100
-500
50100150
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-6
-3
0
3
6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-6
-3
0
3
6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-6
-3
0
3
6
0.00 0.25 0.50 0 .75 1 .00 1 .25 1.50 1.75 2.00
T im e, seconds
p,
-
SDOF Dynamics 3 - 36Instructional Material Complementing FEMA 451, Design Examples
-12.00
-8.00
-4.00
0.00
4.00
8.00
12.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Frequency Ratio
Mag
nific
atio
n Fa
ctor
1/(1
- 2 ) In phase
180 degrees out of phase
Resonance
Response Ratio: Steady State to Static(Signs Retained)
-
SDOF Dynamics 3 - 37Instructional Material Complementing FEMA 451, Design Examples
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Frequency Ratio
Mag
nific
atio
n Fa
ctor
1/(1
- 2 )
Response Ratio: Steady State to Static(Absolute Values)
Resonance
Slowlyloaded
1.00
Rapidlyloaded
-
SDOF Dynamics 3 - 38Instructional Material Complementing FEMA 451, Design Examples
Damped Harmonic Loading
m u t cu t k u t p t&&( ) &( ) ( ) sin( )+ + = 0 Equation of motion:
-150-100-50
050
100150
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, Seconds
Forc
e, K
ips
po=100 kips
sec25.02 ==T
-
SDOF Dynamics 3 - 39Instructional Material Complementing FEMA 451, Design Examples
Solution:
Assume system is initially at rest
Particular solution:
Complimentary solution:
u t C t D t( ) sin( ) cos( )= +
[ ]u t e A t B tt D D( ) sin( ) cos( )= +
Damped Harmonic LoadingEquation of motion:
m u t cu t k u t p t&&( ) &( ) ( ) sin( )+ + = 0
D = 12
=c
m2
[ ]u t e A t B tt D D( ) sin( ) cos( )= + + +C t D tsin( ) cos( )
-
SDOF Dynamics 3 - 40Instructional Material Complementing FEMA 451, Design Examples
Transient response at structures frequency(eventually damps out)
Steady state response,at loading frequency
D pk
o=
+2
1 22 2 2
( ) ( )
Damped Harmonic Loading
C pk
o=
+1
1 2
2
2 2 2
( ) ( )
)cos()sin( tDtC +
+[ ]u t e A t B tt D D( ) sin( ) cos( )= +
-
SDOF Dynamics 3 - 41Instructional Material Complementing FEMA 451, Design Examples
-50
-40
-30
-20
-10
0
10
20
30
40
50
0.00 1.00 2.00 3.00 4.00 5.00
Time, Seconds
Dis
plac
emen
t Am
plitu
de, I
nche
sBETA=1 (Resonance)Beta=0.5Beta=2.0
Damped Harmonic Loading (5% Damping)
-
SDOF Dynamics 3 - 42Instructional Material Complementing FEMA 451, Design Examples
Static21
-50
-40
-30
-20
-10
0
10
20
30
40
50
0.00 1.00 2.00 3.00 4.00 5.00
Time, Seconds
Dis
plac
emen
t Am
plitu
de, I
nche
sDamped Harmonic Loading (5% Damping)
-
SDOF Dynamics 3 - 43Instructional Material Complementing FEMA 451, Design Examples
Harmonic Loading at ResonanceEffects of Damping
-200
-150
-100
-50
0
50
100
150
200
0.00 1.00 2.00 3.00 4.00 5.00
Time, Seconds
Dis
plac
emen
t Am
plitu
de, I
nche
s
0% Damping %5 Damping
-
SDOF Dynamics 3 - 44Instructional Material Complementing FEMA 451, Design Examples
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Frequency Ratio,
Dyn
amic
Res
pons
e A
mpl
ifier
0.0% Damping5.0 % Damping10.0% Damping25.0 % Damping
RD = +
11 22 2 2( ) ( )
Resonance
Slowlyloaded Rapidly
loaded
-
SDOF Dynamics 3 - 45Instructional Material Complementing FEMA 451, Design Examples
Summary Regarding Viscous Dampingin Harmonically Loaded Systems
For systems loaded at a frequency near their natural frequency, the dynamic response exceeds the static response. This is referred to as dynamic amplification.
An undamped system, loaded at resonance, will have an unbounded increase in displacement over time.
-
SDOF Dynamics 3 - 46Instructional Material Complementing FEMA 451, Design Examples
Summary Regarding Viscous Dampingin Harmonically Loaded Systems
Damping is an effective means for dissipating energy in the system. Unlike strain energy, which is recoverable, dissipated energy is not recoverable.
A damped system, loaded at resonance, will have a limited displacement over time with the limit being (1/2) times the static displacement.
Damping is most effective for systems loaded at or near resonance.
-
SDOF Dynamics 3 - 47Instructional Material Complementing FEMA 451, Design Examples
LOADING YIELDING
UNLOADING UNLOADED
F F
F
u
F
u
u u
EnergyStored
EnergyDissipated
EnergyRecovered
TotalEnergyDissipated
CONCEPT of ENERGY STOREDand Energy DISSIPATED
12
1
3
2
4
3
-
SDOF Dynamics 3 - 48Instructional Material Complementing FEMA 451, Design Examples
Time, T
F(t)
General Dynamic Loading
-
SDOF Dynamics 3 - 49Instructional Material Complementing FEMA 451, Design Examples
General Dynamic Loading Solution Techniques
Fourier transform Duhamel integration Piecewise exact Newmark techniques
All techniques are carried out numerically.
-
SDOF Dynamics 3 - 50Instructional Material Complementing FEMA 451, Design Examples
dt
( ) odFF Fdt
= +
dF
dt
Piecewise Exact Method
Fo
-
SDOF Dynamics 3 - 51Instructional Material Complementing FEMA 451, Design Examples
Initial conditions 00, =ou 00, =ou&
Determine exact solution for 1st time step
)(1 uu = )(1 uu && = )(1 uu &&&& =
)(1, uuo = )(1,0 uu && =Establish new initial conditions
Obtain exact solution for next time step
)(2 uu = )(2 uu && = )(2 uu &&&& =
LOOP
Piecewise Exact Method
-
SDOF Dynamics 3 - 52Instructional Material Complementing FEMA 451, Design Examples
Piecewise Exact Method
Advantages:
Exact if load increment is linear Very computationally efficient
Disadvantages:
Not generally applicable for inelastic behavior
Note: NONLIN uses the piecewise exact method forresponse spectrum calculations.
-
SDOF Dynamics 3 - 53Instructional Material Complementing FEMA 451, Design Examples
Newmark Techniques
Proposed by Nathan Newmark General method that encompasses a family of different
integration schemes Derived by:
Development of incremental equations of motion Assuming acceleration response over short time step
-
SDOF Dynamics 3 - 54Instructional Material Complementing FEMA 451, Design Examples
Newmark MethodAdvantages:
Works for inelastic response
Disadvantages:
Potential numerical error
Note: NONLIN uses the Newmark method forgeneral response history calculations
-
SDOF Dynamics 3 - 55Instructional Material Complementing FEMA 451, Design Examples
-0.40
-0.20
0.00
0.20
0.40
0.00 1.00 2.00 3.00 4.00 5.00 6.00
TIME, SECONDS
GR
OU
ND
AC
C, g
Development of Effective Earthquake Force
-
SDOF Dynamics 3 - 56Instructional Material Complementing FEMA 451, Design Examples
Earthquake Ground Motion, 1940 El Centro
Many ground motions now are available via the Internet.
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60
Time (sec)
Gro
und
Acce
lera
tion
(g's
)
-30
-20
-10
0
10
20
30
40
0 10 20 30 40 50 60
Time (sec)
Gro
und
Velo
city
(cm
/sec
)
-15
-10
-5
0
5
10
15
0 10 20 30 40 50 60
Time (sec)
Gro
und
Disp
lace
men
t (cm
)
-
SDOF Dynamics 3 - 57Instructional Material Complementing FEMA 451, Design Examples
m u t u t c u t k u tg r r r[&& ( ) && ( )] & ( ) ( )+ + + = 0
mu t c u t k u t mu tr r r g&& ( ) & ( ) ( ) && ( )+ + =
Development of Effective Earthquake Force
-0.40
-0.20
0.00
0.20
0.40
0.00 1.00 2.00 3.00 4.00 5.00 6.00
TIME, SECONDS
GR
OU
ND
AC
C, g
Ground Acceleration Response History
gu&&tu&&
ru&&
-
SDOF Dynamics 3 - 58Instructional Material Complementing FEMA 451, Design Examples
)()()()( tumtuktuctum grrr &&&&& =++
)()()()( tutumktu
mctu grrr &&&&& =++
2=mc 2=
mk
Divide through by m:
Make substitutions:
Simplified form of Equation of Motion:
)()()(2)( 2 tutututu grrr &&&&& =++ Simplified form:
-
SDOF Dynamics 3 - 59Instructional Material Complementing FEMA 451, Design Examples
)()()(2)( 2 tutututu grrr &&&&& =++
Ground motion acceleration history
Structural frequency
Damping ratio
For a given ground motion, the response history ur(t) is function of the structures frequency and damping ratio .
-
SDOF Dynamics 3 - 60Instructional Material Complementing FEMA 451, Design Examples
Change in ground motion or structural parameters and requires re-calculation of structural response
Response to Ground Motion (1940 El Centro)
-6
-4
-2
0
2
4
6
0 10 20 30 40 50 60
Time (sec)
Stru
ctur
al D
ispl
acem
ent (
in)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60
Time (sec)
Gro
und
Acce
lera
tion
(g's)
Excitation applied to structure with given and
Peak displacement
Computed response
SOLVER
-
SDOF Dynamics 3 - 61Instructional Material Complementing FEMA 451, Design Examples
0
4
8
12
16
0 2 4 6 8 10
PERIOD, Seconds
DIS
PLA
CEM
ENT,
inch
es
The Elastic Displacement Response SpectrumAn elastic displacement response spectrum is a plotof the peak computed relative displacement, ur, for anelastic structure with a constant damping , a varyingfundamental frequency (or period T = 2/ ), respondingto a given ground motion.
5% damped response spectrum for structureresponding to 1940 El Centro ground motion
-
SDOF Dynamics 3 - 62Instructional Material Complementing FEMA 451, Design Examples
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0 1 2 3 4 5 6 7 8 9 10 11 12
Time, Seconds
Dis
plac
emen
t, In
ches
= 0.05T = 0.10 secUmax= 0.0543 in.
0.00
2.00
4.00
6.00
8.00
10.00
0.00 0.50 1.00 1.50 2.00
Period, Seconds
Dis
plac
emen
t, In
ches
Computation of Response Spectrum for El Centro Ground Motion
Elastic response spectrum
Computed response
-
SDOF Dynamics 3 - 63Instructional Material Complementing FEMA 451, Design Examples
= 0.05T = 0.20 secUmax = 0.254 in.
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
0 1 2 3 4 5 6 7 8 9 10 11 12
Time, Seconds
Dis
plac
emen
t, In
ches
0.00
2.00
4.00
6.00
8.00
10.00
0.00 0.50 1.00 1.50 2.00
Period, Seconds
Dis
plac
emen
t, In
ches
Computation of Response Spectrumfor El Centro Ground Motion
Elastic response spectrum
Computed response
-
SDOF Dynamics 3 - 64Instructional Material Complementing FEMA 451, Design Examples
= 0.05T = 0.30 secUmax = 0.622 in.
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
0 1 2 3 4 5 6 7 8 9 10 11 12
Time, Seconds
Dis
plac
emen
t, In
ches
0.00
2.00
4.00
6.00
8.00
10.00
0.00 0.50 1.00 1.50 2.00
Period, Seconds
Dis
plac
emen
t, In
ches
Computation of Response Spectrumfor El Centro Ground Motion
Elastic response spectrum
Computed response
-
SDOF Dynamics 3 - 65Instructional Material Complementing FEMA 451, Design Examples
= 0.05T = 0.40 secUmax = 0.956 in.
-1.20
-0.90
-0.60
-0.30
0.00
0.30
0.60
0.90
1.20
0 1 2 3 4 5 6 7 8 9 10 11 12
Time, Seconds
Dis
plac
emen
t, In
ches
0.00
2.00
4.00
6.00
8.00
10.00
0.00 0.50 1.00 1.50 2.00
Period, Seconds
Dis
plac
emen
t, In
ches
Computation of Response Spectrumfor El Centro Ground Motion
Elastic response spectrum
Computed response
-
SDOF Dynamics 3 - 66Instructional Material Complementing FEMA 451, Design Examples
= 0.05T = 0.50 secUmax = 2.02 in.
-2.40
-1.80
-1.20
-0.60
0.00
0.60
1.20
1.80
2.40
0 1 2 3 4 5 6 7 8 9 10 11 12
Time, Seconds
Dis
plac
emen
t, In
ches
0.00
2.00
4.00
6.00
8.00
10.00
0.00 0.50 1.00 1.50 2.00
Period, Seconds
Dis
plac
emen
t, In
ches
Computation of Response Spectrumfor El Centro Ground Motion
Elastic response spectrum
Computed response
-
SDOF Dynamics 3 - 67Instructional Material Complementing FEMA 451, Design Examples
= 0.05T = 0.60 secUmax= -3.00 in.
Computation of Response Spectrumfor El Centro Ground Motion
-3.20-2.40-1.60-0.800.000.801.602.403.20
0 1 2 3 4 5 6 7 8 9 10 11 12
Time, Seconds
Dis
plac
emen
t, In
ches
0.00
2.00
4.00
6.00
8.00
10.00
0.00 0.50 1.00 1.50 2.00
Period, Seconds
Dis
plac
emen
t, In
ches
Elastic response spectrum
Computed response
-
SDOF Dynamics 3 - 68Instructional Material Complementing FEMA 451, Design Examples
Complete 5% Damped Elastic DisplacementResponse Spectrum for El Centro
Ground Motion
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Period, Seconds
Dis
plac
emen
t, In
ches
-
SDOF Dynamics 3 - 69Instructional Material Complementing FEMA 451, Design Examples
Development of PseudovelocityResponse Spectrum
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
0.0 1.0 2.0 3.0 4.0Period, Seconds
Pseu
dove
loci
ty, i
n/se
c
DTPSV )(
5% damping
-
SDOF Dynamics 3 - 70Instructional Material Complementing FEMA 451, Design Examples
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
0.0 1.0 2.0 3.0 4.0Period, Seconds
Pseu
doac
cele
ratio
n, in
/sec
2
DTPSA 2)(
Development of PseudoaccelerationResponse Spectrum
5% damping
-
SDOF Dynamics 3 - 71Instructional Material Complementing FEMA 451, Design Examples
The pseudoacceleration response spectrum represents the total acceleration of the system, not the relative acceleration. It is nearly identical to the true total acceleration response spectrum for lightly damped structures.
Note About the Pseudoacceleration Response Spectrum
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
0.0 1.0 2.0 3.0 4.0Period, Seconds
Pseu
doac
cele
ratio
n, in
/sec
2
5% damping
Peak groundacceleration
-
SDOF Dynamics 3 - 72Instructional Material Complementing FEMA 451, Design Examples
m u t u t c u t k u tg r r r[&& ( ) && ( )] & ( ) ( )+ + + = 0
mu t c u t k u t mu tr r r g&& ( ) & ( ) ( ) && ( )+ + =
PSA is TOTAL Acceleration!
-0.40
-0.20
0.00
0.20
0.40
0.00 1.00 2.00 3.00 4.00 5.00 6.00
TIME, SECONDS
GR
OU
ND
AC
C, g
Ground Acceleration Response History
gu&&tu&&
ru&&
-
SDOF Dynamics 3 - 73Instructional Material Complementing FEMA 451, Design Examples
Difference Between Pseudo-Accelerationand Total Acceleration
(System with 5% Damping)
0.00
50.00
100.00
150.00
200.00
250.00
300.00
350.00
0.1 1 10Period (sec)
Acce
lera
tion
(in/s
ec2 )
Total Acceleration Pseudo-Acceleration
-
SDOF Dynamics 3 - 74Instructional Material Complementing FEMA 451, Design Examples
Difference Between Pseudovelocityand Relative Velocity
(System with 5% Damping)
0
5
10
15
20
25
30
35
40
0.1 1 10Period (sec)
Velo
city
(in/
sec)
Relative Velocity Pseudo-Velocity
-
SDOF Dynamics 3 - 75Instructional Material Complementing FEMA 451, Design Examples
Displacement Response Spectrafor Different Damping Values
0.00
5.00
10.00
15.00
20.00
25.00
0.0 1.0 2.0 3.0 4.0 5.0Period, Seconds
Dis
plac
emen
t, In
ches
0%5%10%20%
Damping
-
SDOF Dynamics 3 - 76Instructional Material Complementing FEMA 451, Design Examples
Pseudoacceleration Response Spectrafor Different Damping Values
0.00
1.00
2.00
3.00
4.00
0.0 1.0 2.0 3.0 4.0 5.0Period, Seconds
Pseu
doac
cele
ratio
n, g
0%5%10%20%
Damping
Peak groundacceleration
-
SDOF Dynamics 3 - 77Instructional Material Complementing FEMA 451, Design Examples
Damping Is Effective in Reducing the Response for (Almost) Any Given Period
of Vibration
An earthquake record can be considered to be the combination of a large number of harmonic components.
Any SDOF structure will be in near resonance with oneof these harmonic components.
Damping is most effective at or near resonance.
Hence, a response spectrum will show reductions due todamping at all period ranges (except T = 0).
-
SDOF Dynamics 3 - 78Instructional Material Complementing FEMA 451, Design Examples
Damping Is Effective in Reducing the Response for Any Given Period of
Vibration
Time (sec)
Am
plitu
de
Example of an artificially generated wave to resemble a real time ground motion accelerogram.
Generated wave obtained by combining five different harmonic signals, each having equal amplitude of 1.0.
-4.00-2.000.002.004.00
0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0
`
-
SDOF Dynamics 3 - 79Instructional Material Complementing FEMA 451, Design Examples
The Artificial Wave Is the Sum of Five Harmonics
T = 5.0 s
T = 4.0 s
T = 3.0 s
Time (sec)
Am
plitu
de
-1-0.5
00.5
1
0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0
`
-1-0.5
00.5
1
0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0
`
-1-0.5
00.5
1
0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0
`
-
SDOF Dynamics 3 - 80Instructional Material Complementing FEMA 451, Design Examples
T = 2.0 s
T = 1.0 s
Time (sec)
Am
plitu
de
Summation
-1-0.5
00.5
1
0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0
`
-1-0.5
00.5
1
0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0
`
-4.00-2.000.002.004.00
0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0
`
The Artificial Wave Is the Sum of Five Harmonics
-
SDOF Dynamics 3 - 81Instructional Material Complementing FEMA 451, Design Examples
FFT curve for the combined wave
Frequency (Hz)
Four
ier a
mpl
itude
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Frequency Ratio,
Dyn
amic
Res
pons
e A
mpl
ifier
0.0% Damping5.0 % Damping10.0% Damping25.0 % Damping
Damping Reduces the Responseat Each Resonant Frequency
-
SDOF Dynamics 3 - 82Instructional Material Complementing FEMA 451, Design Examples
Use of an Elastic Response SpectrumExample StructureK = 500 k/inW = 2,000 kM = 2000/386.4 = 5.18 k-sec2/in = (K/M)0.5 =9.82 rad/secT = 2/ = 0.64 sec5% critical damping
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Period, Seconds
Dis
plac
emen
t, In
ches
At T = 0.64 sec, displacement = 3.03 in.
-
SDOF Dynamics 3 - 83Instructional Material Complementing FEMA 451, Design Examples
Use of an Elastic Response SpectrumExample StructureK = 500 k/inW = 2,000 kM = 2000/386.4 = 5.18 k-sec2/in = (K/M)0.5 =9.82 rad/secT = 2/ = 0.64 sec5% critical damping
At T = 0.64 sec, pseudoacceleration = 301 in./sec2
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Period, Seconds
Pseu
doac
cele
ratio
n, in
/sec
2
Base shear = M x PSA = 5.18(301) = 1559 kips
-
SDOF Dynamics 3 - 84Instructional Material Complementing FEMA 451, Design Examples
Response Spectrum, ADRS Space
0.00
0.20
0.40
0.60
0.80
1.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Displacement, inches
Pseu
doac
cele
ratio
n, g
Diagonal lines representperiod values, T
T = 0.64s
-
SDOF Dynamics 3 - 85Instructional Material Complementing FEMA 451, Design Examples
0.1
1
10
100
0.1 1 10 100 1000
Circular Frequency , Radiand per Second
PSEU
DO
VELO
CIT
Y, in
/sec
1.0
D=10.0
0.01
.01
0.1
0.1 0.001
1.0
Line of increasingdisplacement
Line of constantdisplacement
Four-Way Log Plot of Response Spectrum
PSVD =
Circular Frequency (radians/sec)
-
SDOF Dynamics 3 - 86Instructional Material Complementing FEMA 451, Design Examples
0.1
1
10
100
0.1 1 10 100 1000
Circular Frequency , Radiand per Second
PSEU
DO
VELO
CIT
Y, in
/sec
100
PSA=1000
10000
100
100000
10 1000
10000Line of increasingacceleration
Line of constantacceleration
Four-Way Log Plot of Response Spectrum
PSA PSV=
Circular Frequency (radians/sec)
-
SDOF Dynamics 3 - 87Instructional Material Complementing FEMA 451, Design Examples
0.1
1
10
100
0.1 1 10 100 1000
Circular Frequency , Radiand per Second
PSEU
DO
VELO
CIT
Y, in
/sec
Four-Way Log Plot of Response Spectrum
10000
1000
100
10
10.1
ACCE
LERA
TION,
in/se
c2
100
10
1.0
0.1
0.01
0.001
DISPLACEMENT, in
Circular Frequency (radians/sec)
-
SDOF Dynamics 3 - 88Instructional Material Complementing FEMA 451, Design Examples
0.10
1.00
10.00
100.00
0.01 0.10 1.00 10.00
PERIOD, Seconds
PSEU
DO
VELO
CIT
Y, in
/sec
1.0
10.0
0.1
0.01
Acceleration, g
0.001
10.0
0.101.0
0.01
0.001
Disp
lacem
ent, i
n.
Four-Way Log Plot of Response SpectrumPlotted vs Period
-
SDOF Dynamics 3 - 89Instructional Material Complementing FEMA 451, Design Examples
Development of an ElasticResponse Spectrum
0.10
1.00
10.00
100.00
0.01 0.10 1.00 10.00
PERIOD, Seconds
PSEU
DO
VELO
CIT
Y, in
/sec
1.0
10.0
0.1
0.01
Acceleration, g
0.001
10.0
0.10
1.0
0.01
0.001
Disp
lacem
ent, i
n.
Problems with Current Spectrum:
It is for a single earthquake; otherearthquakes will have differentCharacteristics.
For a given earthquake,small variations in structural frequency (period) can producesignificantly different results.
-
SDOF Dynamics 3 - 90Instructional Material Complementing FEMA 451, Design Examples
0.1
1
10
100
0.01 0.1 1 10Period, Seconds
Pseu
do V
eloc
ity, I
n/Se
c
0% Damping5% Damping10% Damping20* Damping
1.0
10.0
0.1
0.01
Acceleration, g
0.001
10.0
0.10
1.0
0.01
0.001
Disp
lacem
ent, i
n.
For a given earthquake,small variations in structural frequency (period) can producesignificantly different results.
1940 El Centro, 0.35 g, N-S
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SDOF Dynamics 3 - 91Instructional Material Complementing FEMA 451, Design Examples
0.1
1.0
10.0
100.0
0.01 0.10 1.00 10.00Period, seconds
Pseu
so V
eloc
ity, i
n/se
c
El CentroLoma PrietaNorth RidgeSan FernandoAverage
5% Damped Spectra for Four California EarthquakesScaled to 0.40 g (PGA)
Different earthquakeswill have different spectra.
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SDOF Dynamics 3 - 92Instructional Material Complementing FEMA 451, Design Examples
Smoothed Elastic Response Spectra(Elastic DESIGN Response Spectra)
Newmark-Hall spectrum ASCE 7 spectrum
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SDOF Dynamics 3 - 93Instructional Material Complementing FEMA 451, Design Examples
0.1
1
10
100
0.01 0.1 1 10 100Period (sec)
Disp
lace
men
t (in
)
0% Damping5% Damping10% Damping
Newmark-Hall Elastic Spectrum
Observations
gu&&max
gu&max
gumax
gvv &&&& max
gvv max
at short T
at long T
0v
0v&&
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SDOF Dynamics 3 - 94Instructional Material Complementing FEMA 451, Design Examples
Very Stiff Structure (T < 0.01 sec)
Total accelerationZero
Ground accelerationRelative displacement
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SDOF Dynamics 3 - 95Instructional Material Complementing FEMA 451, Design Examples
Very Flexible Structure (T > 10 sec)
Relative displacement Total acceleration
Ground displacement Zero
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SDOF Dynamics 3 - 96Instructional Material Complementing FEMA 451, Design Examples
0.1
1
10
100
0.01 0.1 1 10Period, Seconds
Pse