Doina VERDEŞ
BASICS OF SEISMIC ENGINEERING
UTPRESS Cluj-Napoca, 2011
Editura U.T.PRESS Str. Observatorului nr. 34 C.P. 42, O.P. 2, 400775 Cluj-Napoca Tel.:0264-401999; Fax: 0264 - 430408 e-mail:
[email protected] http://www.utcluj.ro/editura Director: Consilier editorial:
Prof.dr.ing. Daniela Manea Ing. Călin D. Câmpean
Copyright © 2011 Editura U.T.PRESS Reproducerea integrală sau parţială a textului sau ilustraţiilor din această carte este posibilă numai cu acordul prealabil scris al editurii U.T.PRESS. Multiplicarea executata la Editura U.T.PRESS. ISBN 978-973-662-641-8 Bun de tipar: 25.05.2011 Tiraj: 100 exemplare
BASICS OF SEISMIC ENGINEERING By Doina Verdes
THE CONTENTS
CHAPTER 1 THE SEISMICITY OF THE TERRITORY 1.1 Introduction 1.2 Seismicity 1.3 The earthquake and the types of seismic waves 1.4 Measures of Earthquake Size 1.5 Record of the ground motion 1.6 Significant earthquakes produced in the world
CHAPTER 2 THE ANALYSIS OF SEISMIC RESPONSE OF SINGLE DEGREE OF FREEDOM SYSTEM 2.1 Modeling the buildings 2.2 The degrees of freedom 2.3 The Response Spectrum Analysis 1
2.4 The relative displacement response 2.5 The response spectrum and the pseudospectrum 2.6 Response to seismic loading: step-by-step methods 2.7 The Beta Newmark Methods 2.8. The seismic response of the SDOF nonlinear system using the step by step numerical integration 2.9 The energy balance procedure 2.10 Seismic response spectra of the SDOF inelastic systems
CHAPTER 3 ANALYSIS OF SEISMIC RESPONSE MULTIDEGREE OF FREEDOM SYSTEMS 3.1Vibration Frequencies and Mode Shapes 3.2 Earthquake Response Analysis by Mode Superposition 3.3 Response Spectrum Analysis for Multi-degree of Freedom Systems 3.4 Step-by-Step Integration
CHAPTER 4 METHODS OF SEISMIC ANALYSIS OF STRUCTURES 4.1 Introduction 4.2 Lateral force method of analysis Romanian Code P100/1-2006 4.3 Lateral force method of analysis - EC8 4.4 Time - history representation 4.5 Non-linear static (pushover) analysis
2
CHAPTER 5 EARTHQUAKE RESISTANT DESIGN 5.1 Introduction 5.2 Performance Based Engineering 5.3 Performance Requirements and Compliance Criteria 5.4 The guiding principles governing the conceptual design against seismic hazard
CHAPTER 6 INELASTIC DYNAMIC BEHAVIOR 6.1 Introduction 6.2 Global and local ductility condition 6.3 Ductility of reinforced concrete elements (local ductility) 6.4 Requirements for ductility of reinforced concrete frames 6.5 The damages of the reinforced concrete frames under seismic loads
CHAPTER 7 DESIGN CONCEPTS FOR EARTHQUAKE RESISTANT REINFORCED CONCRETE STRUCTURES 7.1 Energy dissipation capacity and ductility 7.2 Structural types 7.3 Design criteria at Ultimate Limit State (ULS) 7.4 The Global Ductility 7.5 Design criteria at Safety Limit State (SLS) 7.6 Structural types with stress concentration 7.7 The local effect of infill masonry 3
CHAPTER 8 NONSTRUCTURAL ELEMENTS 8.1 Defining nonstructural elements 8.2 Earthquake effects on buildings and nonstructural elements 8.3 Interstory displacement 8.4 The performances of nonstructural elements 8.5 Protection Strategies 8.6 Nonstructural design approaches for cladding 8.7 Prefabricated wall panels 8.8 Precast Concrete Cladding 8.9 Cladding which increase the seismic energy dissipation 8.10 Examples of damages
CHAPTER 9 THE STRUCTURAL CONTROL OF SEISMIC RESPONSE 9.1. Introduction 9.2. The control of structural response 9.3. Passive control system 9.4 The base isolation system 9.5 The energy dissipation systems 9.6 Advanced Technology Systems (9A) 9.7 Active structural Control (9B)
REFERENCES THE TEST ON SHAKE TABLE OF A HIGH BUILDING MODEL EQUIPPED WITH FRICTION DAMPERS 4
BASICS OF SEISMIC ENGINEERING
By Doina Verdes
CHAPTER I THE SEISMICITY OF THE TERRITORY
Contents
1.1 Introduction 1.2 Seismicity 1.3 The earthquake and the types of seismic waves 1.4 Measures of earthquake size 1.5 Record of the ground motion 1.6 Significant earthquakes produced in the world
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1.1 Introduction The detailed study of earthquakes and earthquake mechanisms lies in the province of seismology, but in his or her studies the earthquake engineer must take a different point of view than the seismologist Seismologists have focused their attention primarily on the global or long-range effects of earthquakes and therefore are concerned with very small amplitude ground motions which induce no significant structural responses. . Doina Verdes Basics of Seismic Engineering 2011
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Engineers, on the other hand, are concerned mainly with the local effects of large earthquakes, where the ground motions are intense enough to cause structural damage
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1.2
Seismicity.
The seismicity of a region determines the extent to which earthquake loadings may control the design of any structure planned for that location. The principal indicator of the degree of seismicity is the historical record of earthquakes that have occurred in the region. Because major earthquakes often have had disastrous consequences, they have been noted in chronicles dating back to the beginning of civilization. The earthquake occurrences are not distributed uniformly on the surface of the earth; instead they tend to be Concentrated along well-defined lines which are knownto be associated with the boundaries of “plates” of the earth’s crust. Doina Verdes Basics of Seismic Engineering 2011
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Fig. 1.1. Global distribution of seismicity* *http://geology.about.com Doina Verdes Basics of Seismic Engineering 2011
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Fig 1.2. Europe seismic map *
*http://geology.about.com Doina Verdes Basics of Seismic Engineering 2011
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Structure of the Earth
6370
Outer core (liquid)
0 24
Inner core (solid)
500 0 20 00 0 50
The earth consists of several discrete concentric layers: -the inner core, is a very dense solid thought to consist mainly of iron; -outer core is a layer of similar density, but thought to be a liquid because shear waves are not transmitted through it; - next is a solid thick envelope of lesser density around; - the core that is called the mantle, - the rather thin layer at the earth’s surface called the crust.
Crust Mantle
Fig. 1.3. Structure of the Earth
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the mantle is considered to consist of two distinct layers: the upper mantle together with the crust form a rigid layer called the lithosphere. Below that, the layer, called the asthenosphere, is thought to be partially molten rock consisting of solid particles incorporated within a liquid component. Although the asthenosphere represents only a small fraction of the total thickness of the mantle, it is because of its highly plastic character that the lithosphere does move as a single unit, however; instead it is divided into a pattern of plates of various sizes, and it is the relative movements along the plate boundaries that cause the earthquake occurrence patterns. *AFPS Brochure
Doina Verdes Basics of Seismic Engineering 2011
Fig. 1.4 The mantle is divided into a pattern of plates *
10
Earthquake Faults
From the study of geology, it has become apparent that the rock near the surface of the earth is not as rigid and motionless as it appears to be. There is ample evidence in many geological formations that the rock was subjected to extensive deformations at a time when it was buried at some depth. When such ruptures occurred, relative sliding motions were developed between the opposite side of the rupture surface creating what is called a geological fault. The orientation of the fault surface is characterized by its “strike”, the orientation from north of its line of intersection with the horizontal ground surface, and by its “dip”, the angle from horizontal of a line drawn on the fault surface perpendicular to this intersection line. Doina Verdes Basics of Seismic Engineering 2011
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Fig, 1.5. San Andreas fault, California
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Fig. 1.6. San Andreas fault, California [21] Doina Verdes Basics of Seismic Engineering 2011
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*BSSC California 2001
Fig 1.7 Types of fault slippage * Doina Verdes Basics of Seismic Engineering 2011
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1.3 The earthquake and the types of seismic waves
The important fact about any fault rupture is that the fracture occurs when the deformations and stresses in the rock reach the breaking strength of the material. Accordingly it is associated with a sudden release of strain energy which then is transmitted through the earth in the form of vibratory elastic waves radiating outward in all directions from the rupture point. These displacement waves passing any specified location on the earth constitute what is called an earthquake. The point on the fault surface where the rupture first began is called the earthquake focus, and the point on the ground surface directly above the focus is called the epicenter. Doina Verdes Basics of Seismic Engineering 2011
Fig. 1.8. The earthquake focus characteristics
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The types of seismic waves
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, and “S” waves, in which the material particles move in a direction perpendicular to the wave propagation path, thus inducing shear 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. 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.
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P-wave S-wave surface wave
1
2
3
Fig. 1.9 The time of seismic waves arrival
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The surface waves
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 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 the distance below the surface.
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Fig. 1.10 The types of seismic waves [21]
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Earthquake focus
Reflection at the surfaces
Mantle Core
Seismograph station Refraction at the core
Fig. 1. 11 The seismic waves travel into the earth
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1.4
Measures of Earthquake Size
The most important measure of size from a seismological point of view is the amount of strain energy released at the source, and this is indicated quantitatively as the magnitude. By definition, Richter magnitude is the (base 10) logarithm of the maximum amplitude, measured in micrometers (10-6 m) of the earthquake record obtained by Wood-Anderson seismograph, corrected to a distance of 100 Km. This magnitude rating has been related empirically to the amount of earthquake energy released E by the formula: log E = 11.8 + 1.5 M
in which M is the magnitude. 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. Doina Verdes Basics of Seismic Engineering 2011
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The magnitude of an earthquake by itself is not sufficient to indicate whether structural damage can be expected. This is a measure of the size of the earthquake at its source, but the distance of the structure from the source has an equally important effect on the amplitude of its response. The severity of the ground motions observed at any point is called the earthquake intensity; it diminishes generally with the distance from the source, although anomalies due to local geological conditions are not uncommon. The oldest measures of intensity are based on observations of the effects of the ground motions on natural and man-made objects. The standard measure of intensity for many years has been the Modified Mercalli (MM) scale. This is a 12-point scale ranging from I (not felt by anyone) to XII (total destruction). Results of earthquakeintensity observations are typically compiled in the form of isoseismal maps. Doina Verdes Basics of Seismic Engineering 2011
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Modified Mercalli (MM) Intensity Scale
I. No felt by people. VII. People are frightened; it is II. Felt only by a few persons at difficult to stand. Automobile drivers rest,especially on upper floors of notice the shaking. Hanging objects buildings. quiver. Furniture breaks. Weak chimneys break. Loose bricks, III. Felt indoors by many people. stones, tiles, cornices, unbraced Feels like the vibration of a light parapets, and architectural truck passing by. Hanging ornaments fall from buildings. objects swing. May not be Damage to masonry D. recognized as an earthquake. IV. Felt indoors by most people … and outdoors by a few. Feels like XI. Most masonry and wood the vibration of a heavy truck structures collapse. Some bridges passing by. Hanging objects destroyed. swing noticeably XII. Damage is total. Large rock V. Felt by most persons masses are displaced. Waves are indoors and outdoors; sleepers seen on the surface of the ground. awaken. Liquids disturbed, with Lines of sight and level are distorted. some spillage. Small objects Objects are thrown into the air. displaced or upset; VI. Felt by everyone. Many people are frightened, some run outdoors. People move unsteadily. Dishes, glassware, and some windows break. Doina Verdes Basics of Seismic Engineering 2011
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The seismic scale grades: MSK 1964; EMI; MM; JAPAN; RUSSIA
MSK 1964
I
II
III
EMI (PS69)
I
II
III
MERCALLI MODIFIED 1956
I
II
JAPAN
0
I
RUSSIA
I
II
IV
III
IV
V
IV
V
II III
V
III IV
VI VI VI IV
V
VI
VII VII
VIII VIII
VII
VIII
V VII
IX IX IX
X X X
XI XI
IX
XII
XI
VI VIII
XII
XII VII
X
XI
XII
maximum acceleration of the soil mouvement 0.002g 0.004g 0.008g 0.015g 0.020g 0.030g 0.130g 0.200g 0.300g 0.500g 1.000g
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18
Largest earthquake
10,000,000x10
Nuclear bomb 1964 Alaska earthquake 1906 San Francisco earthquake
18
1,000,000x10
Daily U.S. electrical energy consumption
1976 Guatemala earthquake
18
Energy (ergs)
1971 San Fernando earthquake 1983 Coalinga earthquake Atomic bomb 1,000x10 18
18
100x10
Se ism ic e ne rgy
1980 Italy earthquake
10,000x10
of ea rth qu ak es
18
100,000x10
1978 Santa Barbara earthauake
18
10 x 10
18
1 x 10
4
5
6
7
8
9
Richter magnitude
Fig. 1.12 Earthquakes: Magnitude/energy
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The three components of ground motion recorded by a strongmotion accelerograph provide a complete description of the earthquake which would act upon any structure at that site. However, the most important features of the record obtained in each component, from the standpoint of its effectiveness in producing structural response, are the amplitude, the frequency content, and the duration. The amplitude generally is characterized by the peak value of acceleration or sometimes by the number of acceleration peaks exceeding a specific level. The frequency content can be represented roughly by the number of zero crossings per second in the accelerogram and the duration by the length of time between the first and the last peaks exceeding a given threshold level. It is evident, however, that all these quantitative measures taken together provide only a very limited description of the ground motion and certainly do not quantify its damage-producing potential adequately Doina Verdes Basics of Seismic Engineering 2011
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Fig, 1.13 Seismoscop – Antic China
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Seismographs The motion of the ground is recorded during earthquakes by instruments known as seismographs. These instruments were first developed around 1890, so we have recordings of earthquakes only since that time. Today, there are hundreds of seismographs installed in the ground throughout the world, operating as part of a worldwide seismographic network for monitoring earthquakes and studying the physics of the earth.
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Seismograms
l
records of soil displacements produced by seismographs, called seismograms, are used in calculating the location and magnitude of an earthquake.
M L
Fig. 1.14 The principle of seismoscop Doina Verdes Basics of Seismic Engineering 2011
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1.5 Record of the ground motion
The motion of the ground at any point is three-dimensional, which means that the point moves in space and not merely in a plane or in a straight line. To completely record this motion, three seismometers must be built into each seismograph. These seismometers move in three perpendicular directions, two horizontal and one vertical, and generate three corresponding seismograms. Seismographs are designed to record small displacements caused by distant earthquakes and are used by seismologists interested in locating hypocenters, estimating magnitudes, and studying the mechanics of earthquakes – the kind of shaking that causes damage. To record this type of ground shaking requires a different type of instrument, one that measures ground acceleration instead of ground displacement. Such instruments are called accelerographs, and the mass-spring system is called accelerometer. Doina Verdes Basics of Seismic Engineering 2011
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Accelerogram-Accelerograph The record generated, known as an accelerogram, has the general appearance of a seismogram, but its mathematical characteristics are quite different. Acceleorgraphs do not have a continuous recording system, as seismographs do; instead, they are triggered by an earthquake and operate form batteries (because the power often is disrupted during an earthquake).
Fig,1.15 North-south component of horizontal ground acceleration recorded at El Centro, Califonia during the Imperial Valey Irrigation district of 18 May 1940
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Fig. 1.16 The accelerogram Vrancea March 1977 Doina Verdes Basics of Seismic Engineering 2011
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1.6 Significant earthquakes and tsunamis produced in the world
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Fig. 1.17 Annual number of earthquakes recorded in the 20th century * *according with the NEC/US GS Global Hypocenter Data Base Doina Verdes Basics of Seismic Engineering 2011
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Date 780 B.C.
373
Location
Magni tude
Deaths
Widespread destruction west of Xian
China; Shaanxi Province Greece
B.C.
1202 May 20
Middle East 30,000
1455 Dec.5 1531 Jan.26 1556 Jan.23
Italy Portugal; Lisbon China; Shaanxi Province
Remarks
40,000
Helice, on the Gulf of Corinth, was destroyed. Much of the city slid into the sea. Felt over an area of 800,000 square miles, including Egypt, Syria, Asia Minor, Sicily, Armenia, and Azerbai-jan. Variously reported as occuring in 1201 or 1202 with over a million deaths (which is highly improbable). Naples badly damaged.
30,000
8.0
830,000
Greatest natural disaster in history. Occured at night in the densely populated region around Xian. Thousands of landslides on the hillsides, which consists of soft rock. Many peasants living in caves were killed. Many villages destroyed and thousands of deaths when houses collapsed.
35
Fig. 1.18 View of an old tile fresco placed on a house wall from Sintra, Portugal, mentioning the 1731 earthquake.
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1626 July 30 1667 Nov. 1668 July 25 1688 July 5 1693 Jan.9 1703 Dec.3 0 1737 Oct.11 1755 Nov.1
Italy; Naples
70,000
Azerbaijan
80,000
1783 Feb.5 1868 Aug.1 3 1891 Oct.28
Italy; Calabria Chile and Peru Japan; Nobi Plain
7.9
1897 June 12
India; Assam
8.7
1906 Apr.18
U.S.A.; San Francisco
8.3
China; Shandong Province Turkey
8.5
15,000 Damage along Aegean coast. 60,000 Catania destroyed.
Sicily Japan; Tokyo region India; Calcutta Portugal; Lisbon
50,000 Widespread destruction throughout province.
8.2
5,200 Tsunami.
300,000
8.6
8.5
60,000 All Saints’ Day; many killed when churches collapsed and fire ravaged the city. Large tsunami killed many. 50,000 First earthquake to be investigated scientifically. 25,000 Large tsunami devasted Arica (now in Chile, but then in Peru). 7,300 Also known as Mino-Owari earthquake (Mino and Owari Provinces are now part of Gifu Prefec-ture). Many buildings destroyed. Large ground displacements. 1,500 Large fault scarp formed (vertical displacement 35 feet. Much building damage in Shillong. 700 San Andreas fault ruptured for 270 miles. Great fire burned much of the city.
37
1908 Dec28
7.5
1920 Dec.1 6 1923 Sept.1
China; Ningxia Province Japan; Tokyo
8.6
1931 Feb.3
New Zealand; Hawke Bay Romania; Vrancea district Japan; south of Shikoku Island Japan; Fukui Prefecture India; Assam (eastern) Algeria; El Asnam Mexico; Guerrero
7.8
1940 Nov. 10 1946 Dec. 21 1948 June 28 1950 Aug. 15 1954 Sept.9 1957 July 8
1968 Aug31
Iran (eastern); Khorasan
8.3
58,000 Messina destroyed.
200,000 Many landslides covered villages and towns. 99,300 Known as Kanto earth-quake. Major damage over a large area, including Tokyo and Yokohama. Great fire in Tokyo. Large tsunami inundated coastal regions. 225 Many buildings damaged in Napier.
7.4
1,000 Severe damage to buildings in Bucharest.
8.4
1,360 Known as the Nankai earthquake. Great tsunami.
7.3
5,400 Only known instance of a person being crushed in a ground fissure. 150 Damage in region along border with Tibet Landslides and floods. 1,240 El Asnam (then Orléansville) destroyed 68 Tall buildings damaged in Mexico City, 180 miles away.
8.7
6.8 7.9
7.3
12,100
About 60,000 people homeless.
38
1970 Peru; May31 Chimbote
7.8
1975 Feb.4
7.3
China; Liaoning Province; Haicheng 1976 China; July Hebei 28 Province; Tangshan 1977 Romania; Mar.4 Vrancea district 1979 Yugoslavia Apr.15 southern Montenegr o
7.8
7.2
7.0
67,000 Greatest earthquake disaster in the Western Hemisphere. About 800,000 people home-less. Huge landslide on Mt. Huascarán buried 18,000 people in Ranrahirca and Yungay. 1,300 Earthquake successfully predicted and population evacuated. Heavy damage, but many lives saved. 243,000 Major industrial city totally destroyed. Four aftershocks on same day with magnitudes 6.5, 6.0, 7.1, and 6.0. 1,570 Many buildings collapsed in Bucharest.
156 Near the Adriatic coast. Extensive damage.
39
17 January
1994 1995 26
Northdrige USA Kobe Japan Sumatra
9
Decem ber
2004 2009
Aquila Italy
Damages to buildings and bridges
M 6.8
6.3
6,500 deaths 240,000 Major damage, deaths The tsunami waves damaged the coast
308 Several buildings collapsed deaths 1500 injuried
12 January 2010
Haiti
7
316,000 250,000 residences and deaths 30,000 commercial buildings were severely damaged
11 March 2011
Tohoku Japan trench
9
Began on 9 March with a M 7.2, and continued with a further three earthquakes greater than M 6.0 on the same day, the major was on 11 march with 9M -explosion hit a petrochemical plant
-Major damage in the Fukushima nuclear plant -Four trains were missed along the coast
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Date 1755 Nov.1
Significant tsunamis produced in the world
1868 Apr.2
1883 Aug.27
Origin Lisbon, Portugal (off the coast, in the Atlantic Ocean); earthquake of magnitude 8.6 (60,000 deaths) Island of Hawaii (south slope of Mauna Loa); volcanic earthquake of magnitude 7.7 Island of Krakatoa (in the Sunda Strait, between Java and Sumatra); volcanic eruption (36,000 deaths)
1896 June 15
Japan (off the Sanriku coast); earthquake of magnitude 7.5 (27,000 deaths)
1923 Sept.1
Japan (Tokyo and vicinity); earthquake of magnitude 8.3 (99,300 deaths)
Remarks Several large waves washed ashore in Portugal, Spain, and Morocco. Major damage and many deaths in Lisbon from tsunamis Local tsunami destroyed many houses and killed 46 people Violent explosion of Krakatoa volcano. Great tsunami felt in harbors around the world. Tsunami caused much damage and loss of life on nearby islands. Numerous villages entirely destroyed by tsunami; maximum wave height 15 meters. Many lives lost by drowning. Known as the Kanto earthquake (epicenter in Kanto Plain), Major damage over a large area, including Tokyo and Yokohama; great fire in Tokyo. Tsunami in Sagami Bay struck the shore 5 minutes after the earthquake; maximum wave height 10 meters. Tsunami killed 160 people.
41
Date
Origin
Remarks
1946 Apr.1
Aleutian Islands (south of Unimak Island in the Aleutian trench); earthquake of magnitude 7.5 (173 deaths)
Major damage in Hilo, Hawaii (96 deaths). Minor damage in California (one death in Santa Cruz)
1956 July 9
Greece (Dodecanese Islands); earthquake of magnitude 7.8 (53 deaths)
Tsunami struck the coasts.
1960 May 22
Chile; Arauco Province (along the continental shelf, near the coast, south of Conception); earthquake of magnitude 8.5 (2,230 deats)
Major damage in Hilo (61 deaths), and Japan (120 deaths). Wave height 5 meters on Sanriku coast of Japan. Local tsunami in Chile.
1976
Philippine Islands (Moro Gulf); earthquake of magnitude 8.0 (6,500 deaths)
Major damage and many deaths from tsunami.
2004 December 26
Sumatra islands magnitude 8.0 About 47,000 more people died, from Thailand to Tanzania, when the tsunami struck without warning during the next few hours.
Major damage and 240,000 people died The worst part of it washed away whole cities in Indonesia, but every country on the shore of the Indian Ocean was also affected
2011 March 11
Tohoku earthquake was a massive earthquake with magnitude 9 Japan trench
-10m wave struck the port of Sendai, carrying ships, vehicles and other debris inland -The tsunami rolled across the Pacific at 800km/h - hitting Hawaii and the US West Coast
Aug.17
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1.7 Seismic hazard in Romania
The seismic hazard in Romania is due to contribution of two factors : (i) the major contribution of subcrustal seismic zone Vrancea (ii) others contributions due to the surface seismic zone contributions spread to country territory.
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Romanian earthquakes Data
1903
Ora (GMT) h:m:s
13 Septemrie
Lat.
°
N
08:02:7
Long.
E
°
H Adâncimea focarului, km
Catalogul RADU C, 1994
Catalogul MARZA, 1980
I
M
M
Mws
Mw
I
w1
26.6
>60
7
6.6
6.3
5.7
6.3
6.5
45.7
26.6
75
6
-
5.7
6.3
6.6
6
45.7 (45.5)
26.5
150(12
8
7.1
6.8
6.8
7.1
8
6.7
7
45.7 1904
6 Februarie
02:49:00
1908
6 Octombrie
21:39:8
5)
1912
25 Mai
18:01:7
45.7
27.2
80(90)
7
6.3
6.0
6.4
1934
29 Martie
20:06:51
45.8
26.5
90
7
6.6
6.3
6.3
1939
5 Septembrie
06:02:00
45.9
26.7
120
6
-
5.3
6.1
6.2
6
1940
22 Octombrie
06:37:00
45.8
26.4
122
7/8
6.8
6.5
6.2
6.5
7
1940
10 Noiembrie
01:39:07
45.8
26.7
9
7.7
7.4
7.4
7.7
9
1945
1 Septembrie
15:48:26
45.9
26.5
75
7/8
6.8
6,5
6.5
6.8
7.5
1945
9 Decembrie
06:08:45
45.7
26.8
80
7
6.3
6.0
6.2
6.5
7
1948
29 Mai
04:48:55
45.8
26.5
130
6/7
-
5.8
6.0
6.3
6.5
1977
4 Martie
19:22:15
45.3
26.30
109
8/9
7.5
7.2
7.2
7.4
9
45.5
26.47
133
8
7.2
7.0
-
7.1
-
45.8
26.90
91
8
7.0
6.7
-
6.9
-
45.8
26.89
79
7
6.4
6.1
-
6.4
-
140150*
6.6
8
4 1986
30 August
21:28:37 3
1990
30 Mai
10:40:06 2
1990
31Mai
00:17:49 3
Doina Verdes Basics of Seismic Engineering 2011
The design acceleration and seismic zones of Romanian territory
1. National territory is subdivided into seismic zones, depending on the local hazard. By definition, the hazard within each zone is assumed to be constant. 2.the hazard is described in terms of a single parameter, i.e. the value of the reference peak ground acceleration on rock or firm soil ag. 3. The reference peak ground acceleration, chosen by the National Authority for each seismic zone, corresponds to the reference return period chosen by the same authority. To this reference average return period for Romanian territory is call “the design soil acceleration” Doina Verdes Basics of Seismic Engineering 2011
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Fig. 1.19 Romanian seismic network
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The design acceleration, Conforming the Romanian Code P100/1-2006, for each zone of seismic hazard corresponds to an average return period of reference equal 100 years.
Fig. 1.20 Seismic zones of Romanian territory depending on soil design acceleration ag for seismic events with average return period (of magnitude) IMR = 100 years
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The control period and the design accelerations of some Romanian cities [22]
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The control period
The local soil conditions are described by values of control period TC of the response spectrum for the specific location. These values characterize synthetically the frequencies composition of the seismic movement. The control period represents the border between the zone of the maximum values in the spectrum of absolute accelerations and the zone of maximum values in the spectrum of relative velocity. TC is expressed in the seconds. The average interval of return earthquake magnitude IMR=100 years For the ultimate limit stage
Values of control periods TB , s TC , s TD , s
0,07 0,7 3
0,10 1,0 3
0,16 1,6 2
Fig. 1.21 Control periods for Romanian territory Doina Verdes Basics of Seismic Engineering 2011
49
Fig.1.22 The map of the Romanian territory with the zones on terms of TC for the horizontal components of the seismic movements due to earthquakes having the IMR=100 years.
Doina Verdes Basics of Seismic Engineering 2011
50
BASICS OF SEISMIC ENGINEERING
By Doina Verdes
CHAPTER 2 THE ANALYSIS OF SEISMIC RESPONSE OF SINGLE DEGREE OF FREEDOM SYSTEM
Doina Doina Verdes Verdes BASICS BASICS OFOF SEISMICAL SEISMICAL ENGINEERING ENGINEERING 2011 2011
2
Contents 2.1 Modeling the buildings 2.2 The degrees of freedom 2.3 The Response Spectrum Analysis 2.4 The relative displacement response 2.5 The response spectrum and the pseudospectrum 2.6 Response to seismic loading: step-by-step methods 2.7 The Newmark Beta Methods 2.8. The seismic response of the SDOF nonlinear system using the step by step numerical integration 2.9 The energy balance procedure 2.10 Seismic response spectra of the SDOF inelastic systems Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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2.1 Modeling the buildings
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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Dynamic models • •
•
• •
Dynamic model of the resistance structure It has to describe the behavior to seismic action. It has to represent adequately : - the general configuration – geometry, joints, material - the distribution of inertial characteristics: mass of the levels, inertia moments of the level mass - the stiffness and damping characteristics The model of building can contain the resistance system involved into vertical and lateral loads, connected trough slabs (horizontal diaphragms) The deformability model of the structure can involve also the beamcolumn connection and /or structural walls; the model can be done also by structural elements with nonstructural elements – ex: the partition walls, or panels which can significantly increase the stiffness of the framed structure. Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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• The behavior of the material of structural elements could be linear-elastic (a) or nonlinear (b)
a.
b. Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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The distribution of inertial characteristics: mass of the levels, inertia moments. m k, ξ
The model for a single span frame m k, ξ
The model for a frame multiple spans Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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Fn
F1
The model for a multilevel framed system
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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2.2 The degree of freedom
The degree of freedom (DOF) is by definition: the number of pendulum which block the movement of the mass. The methods to obtain the dynamic model are: - the concentrated mass; - the system with finite elements.
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How can be appreciate the degrees of freedom?
a. The case of an bridge
The horizontal translations of the mass of bridge’s deck The translations along the axis O-x and O-y => Two degrees of freedom
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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b. The case of one level building Simplified model: Three degrees of freedom due to horizontal translations and rotation on the vertical axis of the mass (concentrated at the roof level)
Important assumptions: The building has rigid foundation slab The movement of soil due to seismic excitation is synchronic
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c. The case of one level building subjected to foundation's rotation Results in one degree of freedom
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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Linear Elastic Calculus System FA(t)
FS(t)
c
k 1
1
y& (t )
y(t)
a.
b.
FS= Elastic force FA= Damping force K = Stiffness C = damping coefficient y(t)= displacement y& (t ) =velocity Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
Non-linear Calculus System Tangenta la curbă Tangenta la curbă
FA(t)
FS(t)
FA1 Fs1
∆FS Secanta la curbă
∆Fs
FA0
∆ y& (t )
Fs0 y(t) yo
∆y
y& 0
y1
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
y&1
y& (t )
14
Level of damping in different structures The damping varies with: the materials used, the form of the structure, the nature of the subsoil, and the nature of the vibration. Large-amplitudes post-elastic vibration is more heavily damped than small-amplitude vibration; Buildings with heavy shear walls and heavy cladding or partitions have greater damping than lightly clad skeletal structures.
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Damping coefficient in different structures Type of construction
Damping ν ξ percentage of critical
Steel frame, welded, with all walls of flexible construction
2
Steel frame, welded, with normal floors and cladding
5
Steel frame, bolted, with normal floors and cladding
10
Concrete frame, with all walls of flexible construction
5
Concrete frame, with stiff cladding and all internal walls flexible
7
Concrete frame, with concrete or masonry shear walls
10
Concrete and/or masonry shear wall buildings
10
Timber shear walls construction
15
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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2.3 The Response Spectrum Analysis
Response spectrum analysis is the dominant contemporary method for dynamic analysis of building structures under seismic loading.
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Typical SDOF system subjected to base seismically excitation unidirectional translation yg(t)
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The equilibrium of the forces based on D’Alembert low Fi (t ) + FD (t ) + Fe (t ) = 0
(1)
Fi (t)= the inertia force FD (t)= the damping force Fe (t)= the elastic force
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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Fi (t ) = m( &y&g + &y&)
(2)
FD (t ) = cy&
(3)
Fe (t ) = ky
(4)
m= the mass of system c= the viscous damping cœfficient k= the stifness
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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The equation of equilibrium becomes:
m&y&(t ) + cy& (t ) + ky (t ) = − m&y&g (t )
(5)
m&y&(t ) + cy& (t ) + ky (t ) = − FS (t )
(6)
The frequency equation
&y&(t ) + 2ωξy& (t ) + ω 2 y (t ) = − &y&g (t )
(7)
ω = k /m
ξ = c/2mω
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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The general solution of the seismic equilibrium equation is:
y(t ) = A exp −ξωt sin ωt +ϕ +
1 mω
t ∫0 − m&y&g (τ )sin ω D (t −τ )exp[−ξω (t −τ )]dτ D
(8)
The first term represents the free vibration of the system The second term represents the forced vibrations under seismic action. Neglecting the free vibrations contribution due to the quick damping of these the solution becomes:
y (t ) =
1 mω D
t
∫ − m&y& (τ )sin ω (t − τ )exp[− ξω(t − τ )]dτ 0
g
D
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
(9)
22
2.4 The relative displacement response
The relative displacement response of the frame to a single component of ground acceleration yg(t) may be expressed in the time domain by means of the Duhamel integral
y (t ) =
1 mω D
t
∫ − m&y& (τ )sin ω (t − τ )exp[− ξω(t − τ )]dτ 0
g
D
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
(9)
23
y(t) – the mass displacement ω D – the circular damped frequency ξ - the critically damper coefficient ξ = c/ccr c= the viscous damping coefficient ccr = critically damping coefficient ξ = 0.02 … 0.1 m= the mass
&y&g (τ ) = the ground acceleration at timeτ ω = k/m
(10)
ξ = c/2mω
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
(11)
24
When the difference between the damped and the undamped frequency is neglected, as is permissible for small damping ratios usually representative of real structures (say ξ < 0.10), and when it is noted that the negative sign has no real significance with regard to earthquake excitation, this equation can be reduced to:
y (t ) =
1
t
&y& (τ ) sin ω (t − τ ) exp[− ξω (t − τ )]dτ ∫ ω 0
g
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
(12)
25
North – south component of horizontal ground acceleration El Centro 1940 Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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2.5 The response spectrum and the pseudospectrum
The response spectrum used in seismical engineering are: - the velocity spectrum - the absolute acceleration spectrum - the displacement spectrum
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
• Taking the first time derivative of Eq.(12), one obtains the corresponding relative velocity time-history
t
y& (t ) = ∫ &y&g (τ ) cos ω (t − τ ) exp[− ξω (t − τ )]dτ 0
t
− ξ ∫ &y&g (τ )sin ω (t − τ ) exp[− ξω (t − τ )]dτ 0
(13)
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Further, substituting Eqs. (12) and (13) into the forced-vibration equation of motion, written in the form
&y&t (t) = -2ωξy& (t ) − ω 2 y(t) one obtains the total acceleration relation:
(
)
t
&y&t (t ) = ω 2ξ − 1 ∫ &y&g (τ )sin ω (t − τ ) exp[− ξω (t − τ )]dτ 2
0
t
− 2ωξ ∫ &y&g (τ ) cos ω (t − τ ) exp[− ξω (t − τ )]dτ 0
(14) Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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The absolute maximum values of the response given by Eqs. (12), (13), and (14) are called: - the spectral relative displacement, - the spectral relative velocity, and - the spectral absolute acceleration, These will be denoted herein as : Sd(ξ ,ω), Sv(ξ ,ω), Sa(ξ ,ω), respectively.
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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As will be shown subsequently, it is usually necessary to calculate only the so-called pseudo-velocity spectral response Spv(ξ ,ω) defined by t S pv (ξ , ω ) ≡ ∫ &y&g (τ ) sin ω (t − τ ) exp[− ξω (t − τ )]dτ 0 max
(15)
Now from Eq. (12), it is seen that
y (t ) =
1
t
ω ∫0
&y&g (τ )sin ω (t − τ ) exp[− ξω (t − τ )]dτ
S d (ξ , ω ) =
1
ω
S pv (ξ , ω )
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
(12)
(16) 31
and from Eqs. (13) and (15) that (for ξ = 0)
t S v (0, ω ) ≡ ∫ &y&g (τ ) cos ω (t − τ )dτ 0 max
t S pv (0, ω ) ≡ ∫ &y&g (τ )sin ω (t − τ )dτ 0 max
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
(17)
(18)
32
which are identical except for the trigonometric terms. It has been demonstrated by Hudson that Sv(0 ,ω) and Spv(0 ,ω) differ very little numerically, except in the case of very long period oscillators, i.e. very small values of ω. For damped systems, the difference between Sv and Spv is considerably larger and can differ by as much as 20 percent for ξ = 0.20. Also from Eq. (14) for ξ = 0 that
(
)
t
&y&t (t ) = ω 2ξ − 1 ∫ &y&g (τ )sin ω (t − τ ) exp[− ξω (t − τ )]dτ 2
0
t
− 2ωξ ∫ &y&g (τ ) cos ω (t − τ ) exp[− ξω (t − τ )]dτ 0
t S a (0, ω ) ≡ ω ∫ v&&g (t )sin ω (t − τ )dτ 0 max
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
(19)
33
• thus, from Eq. (19),
S a (0 , ω ) = ω S pv (0 , ω )
(19)
It can be shown that Eq. (19) is very nearly satisfied for damping values over the range 0 < ξ < 0.20; therefore, we are able to use the approximate relation
S a (ξ ,ω ) = ωS pv (ξ ,ω )
(20)
with little error being introduced. Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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• The entire quantity on the right hand side of Eq. (20) is called the pseudo-acceleration spectral response and it is denoted herein as Spa(ξ ,ω). This quantity is particularly significant since it is a measure of the maximum spring force developed in the oscillator
f s , max = kS d (ξ , ω ) = ω 2 mS d (ξ , ω ) = mS pa (ξ , ω )
(21)
• The other response spectra can be easily obtained there from using the relations
S d (ξ , ω ) =
1
ω
S pv (ξ , ω )
S pa (ξ , ω ) = ωS pv (ξ , ω ) Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
(22)
(23)
35
• As indicated above these response quantities depend not only on the ground motion time-history but also on the natural frequency and damping ratio of the oscillator. • Thus for any given earthquake accelerogram, by assuming discrete values of damping ratio and natural frequency, it is possible to calculate the corresponding discrete values of Spv(ξ ,ω) using Eq. (22) and to calculate corresponding values of Sd(ξ ,ω) and Spa(ξ ,ω) using Eqs. (22) and (23), respectively.
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• • • •
Graphs of the values for Spv(ξ ,ω), Sd(ξ ,ω), and Spa(ξ ,ω)
• plotted as functions of frequency (or functions of period T = 2π/ω) for discrete values of damping ratio are called • pseudo-velocity response spectra, • displacement response spectra, and • pseudo-acceleration response spectra, • respectively. If plotted in linear form, each type of spectra must be plotted separately similar to the set of Spv(ξ ,T) shown in Figure 2.3. for the El Centro, California, earthquake of May 18, 1940 (N S component).
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a) Ground acceleration (El Centro)
a
b) The deformation response of three SDF systems
c) deformation response spectrum
T=0,5 s
ξ =2%
T=1 s
ξ =2%
T=2 s
ξ =2%
b
c
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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However, due to the simple relationships existing among the three types of spectra as given by Eqs. (22) and (23) it is possible to present them all in a single plot. This may be accomplished by taking the log (base 10) of Eqs. (24) and (25) to obtain
log S d (ξ , ω ) = log S pv (ξ , ω ) − log ω (24)
log S pa (ξ , ω ) = log S pv (ξ , ω ) + log ω
Combined D-V-A RESPONSE SPECTRUM for El Centro 1940 ground motion
(25) Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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Combined D-V-A response spectrum for El Centro ground motion
Doina Verdes Doina Verdes BASICS OF SEISMICAL ENGINEERING BASICS OF SEISMICAL ENGINEERING 2011 2011
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From these relations, it is seen that when a plot is made with log Spv(ξ,ω) as the ordinate and logω as the abscissa, Eq. (24) is a straight line with slope of +45°for a constant value of logSd(ξ,ω) and Eq. (25) is a straight line with slope of – 45° for a constant value of logSpa(ξ,ω). Thus, a four-way log plot allows all three types of spectra to be illustrated on a single graph. When interpreting such plots, it is important to note the following limiting values:
[
]
lim S d (ξ , ω ) = y g (t ) max
ω →0
[
]
lim S pa (ξ , ω ) = &y&g (t ) max
ω →0
Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
(26)
(27)
41
These limiting conditions mean that all response spectrum curves on the four-way log plot, approach asymptotically the maximum ground displacement with increasing values of oscillator period (or decreasing values of frequency) and the maximum ground acceleration with decreasing values of oscillator period (or increasing values of frequency) for typical values of damping ratio, say ξ = 0.20.
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Combined D-V-A RESPONSE SPECTRUM for El Centro 1940 with different damping coefficient values
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In fact, these response spectra show directly the extent to which real SDOF structures (with specific values of damping ratio and natural period) respond to the input ground motion. The only limitation in their application is that the response must be linear elastic because linear response is inherent in the Duhamel integral. Therefore, such response spectra cannot accurately represent the extent of damage to be expected from a given earthquake excitation, as damage involves inelastic (nonlinear) deformations. Nevertheless, the maximum amount of elastic deformation produced by an earthquake is a very meaningful indication of ground motion intensity.
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Moreover, such response spectra indicate the maximum deformations for all structures having periods within the range for which they were evaluated; hence, the integral of a single response spectrum over an appropriate period range can be used as an effective measure of ground motion intensity. Housner originally introduced such a measure of ground motion intensity when he suggested defining the integral of the pseudo-velocity response spectrum over the period range 0.1 < T < 2.5 sec as the spectrum intensity:
SI (ξ ) ≡ ∫
2.5
0.1
S pv (ξ , T )dT
(28)
As indicated, this integral can be evaluated for any desired damping ratio; however, Housner recommended using ξ = 0.20.
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Usually, it is assumed that the shapes of the design spectra are the same for both the design and maximum probable earthquakes but than they differ in intensity as measured by peak ground acceleration. Thus, it has been common practice to first normalize the intensity of these design spectra to the 1 g peak acceleration level so that Eq. (27) becomes:
lim S pa (ξ ,ω ) = 1 g
ω →0
(29)
and then later to scale them down to the appropriate peak acceleration levels representing the design and maximum probable earthquakes. Once the shapes of these common normalized spectra have been developed, taking into consideration local soil conditions, appropriate scaling factors are applied representing the intensity levels of the peak free-field surface ground accelerations (PGA) produced by the design and maximum probable earthquakes. Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
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BASICS OF SEISMIC ENGINEERING
By Doina Verdes
CHAPTER 2 THE ANALYSIS OF SEISMIC RESPONSE OF SINGLE DEGREE OF FREEDOM SYSTEM
Doina Doina Verdes Verdes BASICS BASICS OFOF SEISMICAL SEISMICAL ENGINEERING ENGINEERING 2011 2011
2
Contents 2.1 Modeling the buildings 2.2 The degrees of freedom 2.3 The Response Spectrum Analysis 2.4 The relative displacement response 2.5 The response spectrum and the pseudospectrum 2.6 Response to seismic loading: step-by-step methods 2.7 The Beta Newmark Methods 2.8. The seismic response of the SDOF nonlinear system using the step by step numerical integration 2.9 The energy balance procedure 2.10 Seismic response spectra of the SDOF inelastic systems Doina Verdes Basics of Seismic Engineering 2011
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2.6 Response to seismic loading: step-by-step methods
Doina Verdes Basics of Seismic Engineering 2011
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The step-by step procedure • A severe earthquake will induce inelastic deformation in a code-designed structure. The step-by step procedure is suited to analysis of nonlinear response in earthquake engineering. • There are many different step-by-step methods, but in all of them the loading and the response history are divided into a sequence of time intervals or ‘steps’. The response during each step then is calculated from the initial conditions (displacement and velocity) existing at the beginning of the step and from the history of loading during the step. Doina Verdes Basics of Seismic Engineering 2011
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The response for each step • Thus the response for each step is an independent analysis problem, and there is no need to combine response contribution within the step. Nonlinear behavior may be considered easily by this approach merely by assuming that the structural properties remain constant during each step, and causing them to change in accordance with any specified form of behavior from one step to the next; hence the nonlinear analysis actually is a sequence of linear analyses of a changing system. • Any desired degree of refinement in the nonlinear behavior may be achieved in this procedure by making the time steps’ short enough; also it can be applied to any type of nonlinearity, including changes of mass, and damping properties as well as the more common nonlinearities due to changes of stiffness.
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Step-by-step methods The simplest step-by-step method for analysis the SDOF system is based on the exact solution of the equation of motion for response of a linear structure to a loading that varies linearly during a discrete time interval. The loading history is divided into time intervals, usually defined by significant changes of shape in the actual loading history; between this points, it is assumed that the slope of the load curve remains constant. The other step-by-step methods employ numerical procedures to approximately satisfy the equation of motion during each time step using numerical differentiation or numerical integration. The general numerical approach to step-by step dynamic response analysis makes use of integration to step forward from the initial to the final conditions for each time step. The essential concept is represented by the following equations: Doina Verdes Basics of Seismic Engineering 2011
7
y&1 = y& 0 + ∫
h
y1 = y0 + ∫
h
0
0
y& (τ )dτ
(1)
y& (τ )dτ
(2)
which express the final velocity and displacement in terms of the initial values of these quantities plus an integral expression. The change of velocity depends on the integral of the acceleration history, and the change of displacement depends on the corresponding velocity integral. In order to carry out this type of analysis, it is necessary first to assume how the acceleration varies during the time step; this acceleration assumption controls the variation of the velocity as well and thus makes it possible to step forward to the next time step.
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2.7 The Newmark Beta Methods A more general step-by-step formulation was proposed by Newmark, which includes the preceding method as a special case, but also may be applied in several other versions. In the Newmark formulation, the basic integration equation [Eqs. (1,2)] for the final velocity and displacement are expressed as follows:
y& 1 = y& 0 + (1 − γ )h&y&0 + γh&y&1
(3)
1 2 & y1 = y 0 + h y 0 + − β h &y&0 + β h 2 &y&1 2
(4)
h=time step h = ti+1 – ti
(5) Doina Verdes Basics of Seismic Engineering 2011
9
• It is evident in Eq. (3) that the factor γ provides a linearity varying weighting between the influence of the initial and the final accelerations on the change of velocity; the factor β similarly provides for weighting the contributions of these initial and final accelerations to the change of displacement. • From study of the performance of this formulation, it was noted that the factor γ controlled the amount of artificial damping induced by this step-by-step procedure; there is no artificial damping if γ = 1/2, so it is recommended that this value be use for standard SDOF analyses.
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The constant variation of acceleration during the incremental h time
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The variation of acceleration during the incremental h time interval c. β= 1/6 e. β= 1/8
Doina Verdes Basics of Seismic Engineering 2011
12
These results also may be derived by assuming that the acceleration varies linearly during the time step between the initial and final values of ÿ and ÿ1, thus the Newmark β = 1/6 method is also known as the linear acceleration method. The linear acceleration method is only conditionally stable referring the incremental value of time step: h=t –t i+1
&y&s (t )
i
Conditions for step time
h p 3/π T
&y&si +1
&y&si
h p 0.55 T ti
ti+1
(6)
t h
Doina Verdes Basics of Seismic Engineering 2011
13
Coeficient β= 1/6 (γ = 1/2)
• β = 1/6 ( γ = 1/2),
&y&si &y&si −1
h/T ≤ √3/π = 0.55
i-1
Doina Verdes Basics of Seismic Engineering 2011
h∆h
h∆h
i
&y&si +1
i+1
14
Linear variation of acceleration during time interval “h”
β = 1/6 (for γ = ½,)
h y& 1 = y& 0 + ( &y&0 + &y&1 ) 2
h2 h2 &y&0 + &y&1 y1 = y0 + y& 0 h + 3 6
Doina Verdes Basics of Seismic Engineering 2011
(7)
(8)
(9)
15
Step 1
m&y&1 (t ) + c(t ) y&1 (t ) + k (t ) y1 (t ) = m&y&s1 (t )
h y&1 = y& 0 + ( &y&0 + &y&1 ) 2 h2 h2 y1 = y0 + y& 0 h + &y&0 + &y&1 3 6
Doina Verdes Basics of Seismic Engineering 2011
(10) (11)
(12)
16
STEP 1 • Initial moment: ground acceleration is =0 and the response in accelerations, velocity and displacement
y& 0 = 0 &y&0 = 0 y0 = 0 h y&1 = ( &y&1 ) 2
(13)
2
h &y&1 y1 = 6
(14)
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The displacement and velocity increments using eq. 13 and 14 h h2 &y&1 = −m&y&1s m&y&1 + c &y&1 + k 2 6 &y&1 = −m&y&1s
y&1 = −m&y&1s
1 h h2 m+c +k 2 6
1 h ⋅ h h2 2 m+c +k 2 6
1 h2 y1 = −m&y&1s ⋅ h h2 6 m+c +k 2 6 Doina Verdes Basics of Seismic Engineering 2011
(15)
(16)
(17)
(18)
18
Summary of the Linear Acceleration Procedure For any given time increment, the above described explicit linear acceleration analysis procedure consists of the following operations which must carried out consecutively in the order given: Using the initial velocity and displacement values y& o and y0, which are known either from the values at the end of the preceding time increment or as initial conditions of the response at time t = 0, and the specified properties of the system; (1) Determine the displacement and velocity increments using Eqs. (13 and 14); (2) Finally, evaluate the velocity and displacement at the end of the time increment. Doina Verdes Basics of Seismic Engineering 2011
19
Linear systems can also be treated by this same procedure, which becomes simplified due to the physical properties remaining constant over their entire time-histories of response. • As with any numerical-integration procedure the accuracy of this step-by-step method will depend on the length of the time increment h. • The factors which must be considered in the selection of this interval: the complexity of the nonlinear damping and stiffness properties, and the period T of vibration of the structure. The time increment must be short enough to permit the reliable representation of all these factors, the last one being associated with the free-vibration behavior of the system. Doina Verdes Basics of Seismic Engineering 2011
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2.8 The seismic response of the SDOF nonlinear system using the step by step numerical integration We have to know: • - the behavior of the material done by the diagram (the model can be elastic-linear or nonlinear) • - the digitalised accelerogram The equation of equilibrium at the time step t1
m&y&1 (t ) + c(t ) y&1 (t ) + k (t ) y1 (t ) = m&y&s1 (t )
(1)
c(t) – the damping coefficient k(t) – the stiffness
The coefficients c(t) and k(t) are variable time depending Doina Verdes Basics of Seismic Engineering 2011
21
The calculus model for the non-elastically behavior of the material
a. The symmetrical elastic-plastic model
b. The asymmetrical elastic-plastic model
c. The bilinear elastic-plastic model Doina Verdes Basics of Seismic Engineering 2011
22
The nonlinear system
FS(t)
FA(t)
Tangent
Tangent
FA1 Fs1
∆ FA
∆Fs
Secant
FA0
Fs0 y(t) yo
∆y
y1
b. The damping
a. The stiffness
Doina Verdes Basics of Seismic Engineering 2011
23
&y&s (t )
&y&si +1
&y&si
ti
ti+1
t ∆h
The digitalized accelerograme
Doina Verdes Basics of Seismic Engineering 2011
24
We assume that the acceleration varies linearly during the time step between the initial and final values of ÿ0 and ÿ1, thus the Newmark β = 1/6 method is also known as the linear acceleration method. The linear acceleration method is only conditionally stable referring the incremental value of time step
&y&s (t )
h = ti+1 – ti Conditions for step time to
&y&si+1
h p 3/π T
&y&si
ti
ti+1
t h
Doina Verdes Basics of Seismic Engineering 2011
h p 0.55 T
(2)
25
The incremental form of the seismic equation During the incremental time step h the system behavior is elastically
∆FI + ∆FD + ∆FS = ∆F ef
(3)
∆FI (t ) = FI (t + h ) − FI (t ) = m∆&y&(t )
(4)
∆FD (t ) = FD (t + h ) − FD (t ) = c∆y& (t )
(5)
∆FS (t ) = FS (t + h ) − FS (t ) = k∆y (t )
(6)
∆Fef (t ) = Fef (t + h ) − Fef (t ) = m∆&y&s (t )
(7)
m∆&y&(t ) + c∆y& (t ) + k∆y (t ) = − m∆&y&s (t )
(8)
The equation can be solved using the β Newmark integration method Doina Verdes Basics of Seismic Engineering 2011
26
2.9 The energy balance procedure • Is based on the comparison of two energies which are found on the structure during the earthquake: • The input energy into structure by the earthquake • The energy dissipated or stored by the structure The equation of energy balance is useful if it can be computed in each step of integration • Assumption: the induced energy is computed for an elastic linear system mS v Ei = 2
2
(9)
• Sv is the pseudo-velocity spectrum
Doina Verdes Basics of Seismic Engineering 2011
27
The energy balance equation EI = EE +EH = (EES + EK )+ (EHξ + EHµ) • • • • • • •
(10)
EI = Input Energy EE = Elastic energy of the system EH = Energy due to deformations EES= Energy elastic strains EK = Kinetic energy EHξ = Energy dissipated by the damping EHµ= Energy dissipated by the plastic deformation
Doina Verdes Basics of Seismic Engineering 2011
28
All types of energy are computed in the moment of structure collapse The collapse may be produced by: The fatigue at a reduced number of cycles; By reaching the maximum deformation of the structural elements; By the overturning effect due to the large lateral displacements.
Doina Verdes Basics of Seismic Engineering 2011
29
The energetic procedure based on the ultimate displacements ECAP=Ep+EH
(11)
F FE
Elastic behavior Elastic-plastic behavior
Fy - the seismic design force
Fy
∆∆y C ∆Ue
ECAP =
∆u
∆
1 FC ∆ C + FC (∆U − ∆ C ) = FC ∆ C (ρ D − 0,5) 2 Doina Verdes Basics of Seismic Engineering 2011
(12)
30
2.10 Seismic response spectra of the inelastic systems The spectrum one obtains from elastical spectrum by ductility factors. These can be computed using two proceedings : i)
ii)
The spectral displacement of the nonlinear system is equal with those of a linear system; The energia of the nonlinear system is equal with the energy of the linear elastically system.
F Fe
Fy=F Fc=F p pl
Doina Verdes Basics of Seismic Engineering 2011
∆∆yc
∆u (∆e max)
∆
31
i) The spectral desplacement of the nonlinear system is equal with those of a linear system The displacements in the ultimate stage are: ∆e max= ∆u F
Fy Fe Fc =
=
Fe
ρd
∆y
Fe
∆u =
mS a
ρd
Fy=F Fcp=Fpl (13)
∆∆yc
∆u (∆e max)
∆
Sa – Elastic acceleration spectrum. ρd – desplacement ductility factor Doina Verdes Basics of Seismic Engineering 2011
32
The energy of the nonlinear system is equal with the energy of the linear elastical system F
1 1 ∆ C FC + (∆ u − ∆ e ) FC = ∆ e Fe 2 2 Fc =
1 2ρ d − 1
F e=
mS a 2ρ d − 1
(14)
Fe
Fcp=Fpl Fy=F
The spectral response for the elasticplastic systems one obtains by dividing elastic spectrum to the ductility factor ρ d or by the equation
∆∆yc
∆u (∆e max)
2ρ d − 1
Doina Verdes Basics of Seismic Engineering 2011
33
∆
Newmark inelastic Spectrum (for pseudo acceleration)*
The Newmark-Hall spectrum may be converted into an “inelastic design response spectrum” by making the appropriate adjustments. To determine strength demands, the spectrum is divided by ductility in the higher period (equal displacement) realm but is divided by (2µ - 1) in the short period *Source: FEMA Instructional Material (equal energy) Complementing FEMA 451 Doina Verdes Basics of Seismic Engineering 2011
34
Elastic-plastic response spectrum for El Centro 1940 with 5% damping coeficient and ductilities 1; 1.5; 2; 4; 8.
Doina Verdes Basics of Seismic Engineering 2011
35
BASICS OF SEISMIC ENGINEERING
By Doina Verdes
CHAPTER 3 ANALYSIS OF SEISMIC RESPONSE MULTIDEGREE OF FREEDOM SYSTEMS
Contents
3.1Vibration Frequencies and Mode Shapes
3.2 Earthquake Response Analysis by Mode Superposition
3.3 Response Spectrum Analysis for Multi-degree of Freedom Systems
3.4 Step-by-Step Integration
Doina Verdes Basics of Seismic Engineering 2011
3
3.1 Introduction • In the dynamic analysis of most structures it is necessary to assume that the mass is distributed in more than one discrete lump. For most buildings the mass is assumed to be concentrated at the floor levels and to be subjected to lateral displacement only. • To illustrate the corresponding multi-degree-of-freedom analysis, consider a three story-building (Figure 3.1.). Each story mass represents one degree-of-freedom each with an equation of dynamic equilibrium.
Doina Verdes Basics of Seismic Engineering 2011
4
mc
Axis of reference
mb
ma
yc(t)
uc,1
yb(t)
ub,1
&y&g
Hypothesis - the mass is assumed to be concentrated at the floor levels - the mass is assumed to be subjected to lateral displacement only (the building base is very rigid and the ground movement is assumed to be synchronically, in the same phase)
ya(t)
ua,1
Mode 1
uc,2 uc,3
ub,3
ub,2
ua,2
Mode 2
ua,3
Mode 3
Shapes of vibration due to mode 1 to 3
Each mass has 2 DOF Due to two Horizontal Translations and rotation Doina Verdes Basics of Seismic Engineering 2011
5
The equations of dynamic equilibrium
FI a + FDa + FS a = Fa (t )
[1]
FI b + FDb + FSb = Fb (t )
[2]
FI c + FDc + FSc = Fc (t )
[3]
Doina Verdes Basics of Seismic Engineering 2011
6
The inertia forces in equation (1) are:
FI a = ma ⋅ u&&a
[4]
FI b = mb ⋅ u&&b
[5]
FI c = mc ⋅ u&&c
[6]
Doina Verdes Basics of Seismic Engineering 2011
7
The inertia forces in matrix form: F1a ma F1b = 0 F 0 1c
0 mb 0
0 0 mc
u&&a u&&b u&& c
[7]
or more generally:
FI = M ⋅ &y&
[8]
FI is the inertia force vector, M is the mass matrix and &y& is the acceleration vector. Doina Verdes Basics of Seismic Engineering 2011
8
• It should be noted that the mass matrix is of diagonal form for a lumped sum-system, giving no coupling between the masses. • In more generalized shape co-ordinate systems, coupling generally exists between the coordinates, complicating the solution. This is a prime reason for using the lumped-mass method.
Doina Verdes Basics of Seismic Engineering 2011
9
The elastic forces in equation (1) depend on the displacement and using stiffness influence coefficients they may be expressed:
FS a = k aa u a + k ab u b + k ac u c FSb = k ba u a + k bb u b + k bc u c F = k u + k u + k u ca a cb b cc c Sc
Doina Verdes Basics of Seismic Engineering 2011
[9]
10
In matrix form FSa k aa FSb = k ba F k Sc ca
k ab k bb k cb
k ac k bc k cc
u a ub u c
[10]
or more generally:
FS = k ⋅ u
[11]
F S is the elastic force vector, k is the stiffness matrix and u is the displacement vector The stiffness matrix k generally exhibits coupling and will be best handled by a standard computerized matrix analysis. Doina Verdes Basics of Seismic Engineering 2011
11
By analogy with the expression (9), (10) and (11) the damping forces may be expressed
FD = c ⋅ y&
[12]
F D is the damping force vector, c is the damping matrix and y& o is the velocity vector. In general it is not practicable to evaluate c and damping is usually expressed in terms of damping coefficients. Doina Verdes Basics of Seismic Engineering 2011
12
Using the Eqs. (8), (11) and (12) the equation of dynamic equilibrium (1) may be written generally as:
FI + FD + FS = F (t )
[13]
which is equivalent to
Mu&& + cu& + ku = − mu&&g (t )
Doina Verdes Basics of Seismic Engineering 2011
[14]
13
3.2 Vibration Frequencies and Mode Shapes • The dynamic response of a structure is dependent upon the frequency (or period T) and the displaced shape • The first step in the analysis of a MDOF system is to find its free vibration frequencies and mode shapes. In free vibration there is no external force and damping is taken as zero.
Doina Verdes Basics of Seismic Engineering 2011
14
• The equation of motion (14) becomes:
Mu&& + ku = 0
[15]
Making the necessary steps of calcullus on obtains:
kuˆ − ω 2 Muˆ = 0
[16]
the eigenvalue equation and is readily solved for ω by standard computer programs
Doina Verdes Basics of Seismic Engineering 2011
15
• An important simplification can be made in equations of motion because of the fact that each mode has an independent equation of exactly equivalent form to that for a single degree of freedom system. Because of orthogonality properties of mode shapes, Eq. (14) can be written T φ 2 n F (t ) & & & Yn + 2ξ nω nYn + ω n Yn = T φ n Mφ n
Yn is a generalized displacement in mode n leading to the actual displacement and ønT is the row mode vector corresponding to the column vector øn.
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16
Earthquake Response Analysis by Mode Superposition • The dynamic analysis of a multi-degree-of-freedom system can be simplified to the solution of Eq. (14) for each mode, and the total response is then obtained by superposing the modal effects. • In terms of excitation by earthquake ground motion üg(t) Eq. (15) becomes:
Doina Verdes Basics of Seismic Engineering 2011
17
Y&&n + 2ξω nY&n + ω n2Yn =
Ln u&&g (t ) T φ n Mφ n
[16]
The response of the n–th mode at any time demands the solution of Eq for Yn(t). where Yn is a generalized displacement in mode n leading to the actual displacement and T is the row mode vector corresponding to the Φn column vector øn.
Doina Verdes Basics of Seismic Engineering 2011
18
This may be done by evaluating the Duhamel integral:
Ln 1 Yn (t ) = T ⋅ φ n Mφ n ω n
∫
t
0
u&&g (σ )e −ξω n (t −σ ) dσ
Doina Verdes Basics of Seismic Engineering 2011
[17]
19
• This displacement of floor (or mass) i at t is then obtained by superimposing the response of all modes evaluated at this time t: N
u i = ∑ φ inYn (t )
[18]
n =1
where øin is the relative amplitude of displacement of mass i in mode n. • It should be noted that in structures with many degrees of freedom most of the vibration energy is absorbed in the lower modes, and it is normally sufficiently accurate to superimpose the effects of only the first few modes. Doina Verdes Basics of Seismic Engineering 2011
20
The earthquake forces • The earthquake forces in the structure may then be expressed in terms of the effective accelerations 2 & & Yn eff (t ) = ω n Yn (t )
[19]
from which the acceleration at any floor i is
&u&in eff (t ) = ω n2φ inYn (t )
[20]
and the earthquake force at any floor “i” is
[
]
qin (t ) = mi ω n2φinY&&n (t ) Doina Verdes Basics of Seismic Engineering 2011
[21] 21
Superimposing all the modal contributions, the earthquake forces in the total structure may be expressed in matrix form as:
[
u a max ≈ u a2,1 max + u a2,2 max + u a2,3 max
1 2
]
[22]
the entire history of displacement and force response can be defined for any multi-degree of freedom system, having first determined the modal response amplitudes.
Doina Verdes Basics of Seismic Engineering 2011
22
R (t)
First mode Time u 1max
Second mode Time
u 2max
Nth mode
Time u n max
Superimposing all the modal contributions
Doina Verdes Basics of Seismic Engineering 2011
23
3.3 Response Spectrum Analysis for Multidegree of Freedom Systems • As with single degree-of-freedom structures considerable simplification of the analysis is achieved if only the maximum response to each mode is considered rather than the whole response history. • If the maximum value Yn max of the Duhamel equation (17) is calculated, the distribution maximum displacement in that mode is:
Doina Verdes Basics of Seismic Engineering 2011
24
u n max = φ nYn max
Ln S vn = φn T ⋅ φ n Mφ n ω n
[23]
and the distribution of maximum earthquake forces in that mode is: 2 n n max
q n max = Mφ nω Y
Ln = Mφ n T ⋅ S an φ n Mω n
[24]
Where Svn is the spectral velocity for mode n; San is the spectral acceleration for mode n. Eqs. (23) and (24) enable the maximum response in each mode to be determined Doina Verdes Basics of Seismic Engineering 2011
25
• As the modal maxima do not necessarily occur at the same time, not necessarily have the same sign, they cannot be combined to give the precise total maximum response. The best that can be done in a response spectrum analysis is to combine the modal responses on a probability basis. Various approximate formula for superposition are used, the most common being the Square Root of Sum of Squares (SRSS) procedure. As an example the maximum deflection at the top of a three-story structure (three masses) would be:
[
u a max ≈ u a2,1 max + u a2,2 max + u a2,3 max
Doina Verdes Basics of Seismic Engineering 2011
1 2
]
[25]
26
Exemple of a three stories frame Response Spectrum Analysis [21]
Doina Verdes Basics of Seismic Engineering 2011
27
Solutions for System in Undamped Free Vibration Mode Shapes for Idealized 3-Story Frame
Doina Verdes Basics of Seismic Engineering 2011
28
Concept of Linear Combination of Mode Shapes (Transformation of Coordinates)
U=ФY
Doina Verdes Basics of Seismic Engineering 2011
29
Orthogonality conditions
The orthogonality condition is an extremely important concept as it allows for the full uncoupling of the equations of motion. The damping matrix (which is not involved in eigenvalue calculations) will be diagonalized as shown only under certain conditions. In general, C will be diagonalized if it satisfies the Caughey criterion: CM-1K = KM-1C Doina Verdes Basics of Seismic Engineering 2011
30
Development of uncoupled Equations of motions
Doina Verdes Basics of Seismic Engineering 2011
31
The explicit form
Doina Verdes Basics of Seismic Engineering 2011
32
Modal Damping Matrix • For structures without added dampers, the development of an explicit damping matrix, C, is not possible because discrete dampers are not attached to the dynamic DOF. However, some mathematical entity is required to represent natural damping. • An arbitrary damping matrix cannot be used because there would be no guarantee that the matrix would be diagonalized by the mode shapes. • The two types of damping shown herein allow for the uncoupling of the equations.
Doina Verdes Basics of Seismic Engineering 2011
33
Rayleigh proportional Damping
Doina Verdes Basics of Seismic Engineering 2011
34
Response Spectrum Method
Doina Verdes Basics of Seismic Engineering 2011
35
Doina Verdes Basics of Seismic Engineering 2011
• As the modal maxima do not necessarily occur at the same time, not necessarily have the same sign, they cannot be combined to give the precise total maximum response. The best that can be done in a response spectrum analysis is to combine the modal responses on a probability basis. Various approximate formula for superposition are used, the most common being the Square Root of Sum of Squares (SRSS) procedure. As an example the maximum deflection at the top of a three-story structure (three masses) would be:
[
u a max ≈ u a2,1 max + u a2,2 max + u a2,3 max
Doina Verdes Basics of Seismic Engineering 2011
1 2
]
[25]
37
3.4 Step-by-Step Integration Generally the response history is divided into very short time increments, during each of which the structure is assumed to be linearly elastic. Between each interval the properties of the structure are modified to match the current state of deformation. Therefore, the nonlinear response is obtained as a sequence of linear responses of successively differing system. In each time increment the following computation are made:
Doina Verdes Basics of Seismic Engineering 2011
38
•
• •
• •
The stiffness of the structure for that increment is computed, based on the state of displacement existing at the beginning of the increment. Changes of displacement are computed assuming the accelerations to vary linearly during the interval. These changes of displacement are added to the displacement state of the beginning of the interval to give the displacement at the end of the interval. Stresses appropriate to the total displacement are computed. In the above procedure the equations of motion must be integrated in their original form during each time increment. For this purpose Eq. (14) may be written:
M∆u&& + c(t )∆u& + k (t )∆u = ∆F (t ) Doina Verdes Basics of Seismic Engineering 2011
[26] 39
∆FI + ∆FD + ∆FS = ∆F ef
Tangenta la curba
FS(t)
Fs1 Secanta la curba
?Fs
∆FI (t ) = FI (t + h ) − FI (t ) = m∆&y&(t )
Fs0
∆FD (t ) = FD (t + h ) − FD (t ) = c∆y& (t )
y(t) yo
?
y
y1
∆FS
∆FS (t ) = FS (t + h ) − FS (t ) = k∆y (t )
&y&s (t )
∆Fef (t ) = Fef (t + h ) − Fef (t ) = m∆&y&s (t )
∆y y1
&y&si +1
&y&si
ti
ti+1
t
m∆&y&(t ) + c∆y& (t ) + k∆y (t ) = −m∆&y&s (t )
∆h
Doina Verdes Basics of Seismic Engineering 2011
40
• In order to avoid instability in the response calculated by these equations the length of the time step must be limited by the condition 1 h≤ TN 1.8
(6)
&y&g &y&gi +1
&y&gi
ti
ti+1
t
h
where TN is the vibration period of the highest mode (i.e., the shortest period) associated with the system eigenproblem. Doina Verdes BASICS OF SEISMICAL ENGINEERING 2011
41
• where • k(t) is the stiffness matrix for the time increment beginning at the time t, • ∆u is the change in displacement during the interval. • The determination of k for each increment is the most demanding part of the analysis, as all the individual member stiffness must be found each time or their current state of deformation. • The integration may be obtained applying the procedure ß Newmark.
Doina Verdes Basics of Seismic Engineering 2011
42
Modal Analysis Equivalent Lateral Force Procedure Empirical period of vibration • Smoothed response spectrum • Compute total base shear,, as if SDOF • Distribute T along height assuming “regular” geometry • Compute displacements and member forces using standard procedures Doina Verdes Basics of Seismic Engineering 2011
43
BASICS OF SEISMIC ENGINEERING
By Doina Verdes
CHAPTER 4. METHODS OF SEISMIC ANALYSIS OF STRUCTURES
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Contents • 4.1 Introduction • 4.2 Lateral force method of analysis Romanian Code P100/1-2006 • 4.3 Lateral force method of analysis- EC8 • 4.4 Time - history representation • 4.5 Non-linear static (pushover) analysis
Doina Verdes Basics of Seismic Engineering 2011
4.1 Introduction The many methods for determining seismic forces in structures fall into two distinct categories: • Equivalent static force analysis; • Dynamic analysis. The three main techniques currently used for dynamic analysis are: Direct integration of the equation of motion by stepby-step procedures; Normal mode analysis; Response spectrum techniques. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• a) the “lateral force method of analysis” for common buildings • b) the “modal response spectrum analysis", which is applicable to all types of buildings. As alternative to a linear method, a non-linear methods may also be used, such as: • c) non-linear static (pushover) analysis; • d) non-linear time history (dynamic) analysis
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The Equivalent Lateral Force Procedure • Empirical computation of vibration period • Smoothed response spectrum • Compute total base shear seismic force • Distribute the base shear seismic force along height assuming “regular” geometry • Compute displacements and member forces using standard procedures Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
5.2 Lateral force method of analysis Code P100/1-2006 procedure
• The design acceleration for each zone of seismic hazard corresponds to an average return period of reference equal 100 years. • The zonation of soil design acceleration ag of Romanian territory for seismic events with average return period of magnitude is noted: IMR = 100 years Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The zonation of Romanian territory depending on soil design acceration ag for seismic events with average return period (of magnitude) IMR = 100 years
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The control period and the ag for Romanian territory (part of the table [22])
Basic representation of the seismic action • The earthquake motion at a given point of the surface is generally represented by an elastic ground acceleration response spectrum, henceforth called “elastic response spectrum”. • The horizontal seismic action is described by two orthogonal components considered as independent and represented by the same response spectrum.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Shape of horizontal elastic response spectrum of accelerations for Vrancea sources a), b), c) and Banat d)
a)
TC = 0.7s
c)
TC = 1.6s
TC = 1.0 s
b)
d)
TC = 0.7s
Design spectrum for non-linear analysis • The capacity of structural systems to resist seismic actions in the non-linear range generally permits their design for forces smaller than those corresponding to a linear elastic response.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
FA(t)
FS(t)
c
k 1
1
y(t)
Linear elastic behavior
y& (t )
Stiffness
Nonlinear elastic behavior Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Damping
Base shear force • The seismic base shear force Fb, for each horizontal direction in which the building is analysed, is determined as follows: (4.1) Fb = γI Sd (T1) m λ where: • Sd (T1) ordinate of the design spectrum at period T1; • T1 fundamental period of vibration of the building for lateral motion in the direction considered; • m total mass of the building, above the foundation or above the top of a rigid basement, • λ correction factor, the value of which is equal to: • λ = 0,85 if T1 < 2 TC and the building has more than two storeys, or λ = 1,0 otherwise •
γI the importance factor
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The deformed shape for the 1st mode: a. Computed by methods of structural dynamics b. approximated by horizontal displacements increasing linearly along the height of the building
Fn
sn
sn Fi
si
si zn zi
F1
s1
s1 z1
a. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
b.
The fundamental period of vibration period T1 • For the determination of the fundamental period of vibration period T1 of the building, expressions based on methods of structural dynamics (e.g. by Rayleigh method) may be used. • Alternatively, the estimation of T1 (in s) may be made by the following expression:
T1 = 2 u
(4.2)
• where: • u - lateral elastic displacement of the top of the building, in m, due to the gravity loads applied in the horizontal direction. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Determination of the fundamental vibration periods T1 • For the determination of the fundamental vibration periods T1 of both planar models of the building, expressions based on methods of structural dynamics (e.g. by Rayleigh method) may be used for buildings with heights up to 40 m the value of T1 may be approximated by the following expression: T1 = Ct ⋅ H 3/ 4
(4.3)
Where: • T1 - fundamental period of building, in s, • C t is function of the structure type • 0,050 for all other structures • 0,075 for moment resistant space concrete frames and for eccentric braced • 0,085 for moment resistant space steel frames • H height of the building, in m. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Design spectrum • design spectrum for the accelerations Sd(T) is an:
Inelastic response spectrum •Which can be obtained with the equation : β0
S d (t ) = a g [1 +
q
−1
TN
T]
(4.4)
• For the horizontal components of the seismic action the design spectrum, Sd(T), • is defined by the following expressions [EC8]: Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Case “a”
0 p T ≤ TB
β0 −1 q S d (T ) = a g 1 + ⋅T TB
T > TB β (T ) S d (T ) = a g q
(4.5)
Case “b”
T the vibration period ag soil design acceleration q behavior factor Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
(4.6)
ß(T) elastic response spectrum; T vibration period of a linear single-degree-of-freedom system; ag design ground acceleration on type A ground (ag); TB, TC limits of the constant spectral acceleration branch; TD value defining the beginning of the constant displacement response range of the spectrum; ß0 amplification factor of maximum horizontal acceleration of the soil by the structure;
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Values of control periods for Romanian territory Table 4.1 T h e a v e ra g e in te rv a l o f re tu rn e a rth q u a k e m a g n itu d e IM R = 1 0 0 y e a rs F o r th e u ltim a te lim it s ta g e
V a lu e s o f c o n tro l p e rio d s T B, s T C, s TD, s
0 ,0 7 0 ,7 3
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
0 ,1 0 1 ,0 3
0 ,1 6 1 ,6 2
The behaviour factor q •
The behaviour factor q is an approximation of the ratio of the seismic forces, that the structure would experience if its response was completely elastic with 5% viscous damping, to the minimum seismic forces that may be used in design - with a conventional elastic response model - still ensuring a satisfactory response of the structure.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Behaviour factors for horizontal seismic action Nr.crt
DCM Sistem strctural Clădiri cu un nivel
1.
Cadre
Table 4.2
DCH
EC8
P100-1/2006
EC8
P100-1/2006
3,30
4,025
4,95
5,75
Clădiri cu mai multe niveluri şi cu o singură deschidere
4,00; 5,00
3,60
4,375
5,40
6,25
Clădiri cu mai multe niveluri şi cu mai multe deschideri
2.
Dual
Structuri cu cadre preponderente
P100- 92 (1/Ψ) 5,00; 6,66
4,00; 5,00
3,90
4,725
5,85
6,75
3,90
4,025; 4,375; 4,725;
5,85
5,75; 6,25; 6,75;
-
Structuri cu pereţi preponderenţi -
Structuri cu doi pereţi în fiecare direcţie 3.
Pereţi Structuri cu mai mulţi pereţi
3,60
4,375
5,40
3
3
3 3
3 3
4,00
3
3
4,00
6,25 4,00
4,00 4,00
4,00
Structuri cu pereţi cuplaţi Flexibil la torsiune(nucleu) 4. 5.
Pendul inversat
4,00
3,60 2 2 1,5 1,5
4,375 2 2 2 2
5,40 3 3 3 3
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
6,25 3 3 3 3
2,86
23
Nr.crt
Sistem strctural
4
DCM P1001/2006 2,5; 4
4
2,5; 4
4
4
4 4 4 2
4 4 4 2
2 4
EC8
DCH EC8
Structuri parter 1.
Cadre necontravântuite
2.
Cadre contravântuite centric
2,50; 5,00; 5,50
6,00; 6,50. 4 4 2,5
6,00; 6,50 4 4 2,5
2 4
2,5
2,5
2,00; 2,50 5,00
3.
Cadre contravântuite excentric
4
4
6,00
6,00
4.
Pendul inversat Structuri cu nuclee sau pereţi de beton
2 2 2 2
2 2 2 2
6,00 3 3
6,00 3 3
4
4
4
4
4,8
-
4
-
-
4
-
5. Cadre necontrav. asociate cu cadre contravântuite în X şi alternante 6.
4,8
Cadre duale Cadre necontrav. asociate cu cadre contravântuite excentric
P10092 (1/Ψ) 2,94; 3,46; 5,00; 5,88
5,50
Structuri etajate Contravântuiri cu diagonale întinse Contravântuiri cu diagonale in V
P1001/2006 2,5;
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
6,00
5,88 4,00; 5,00
1,54; 2,00 2,00; 2,20; 4,00; 5,00 2,00; 2,20; 4,00; 5,00
24
Distribution of the horizontal seismic forces Fb = γI Sd (T1) m λ
(4.7)
• The fundamental mode shapes in the horizontal directions of analysis of the building may be calculated using methods of structural dynamics or • may be approximated by horizontal displacements increasing linearly along the height of the building.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The deformed shape for the 1st mode
Fn
sn
sn Fi
si
si zn zi
F1
s1
s1 z1
a. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
b.
The seismic action effects shall be determined by applying, to the two planar models, horizontal forces Fi to all storeys.
Fi = Fb ⋅
mi ⋅ si n
∑m ⋅s i
(4.8)
i
i =1
where: Fi horizontal force acting on storey i; Fb seismic base shear according to expression (4.1 ); si, sj displacements of masses mi, mj in the fundamental mode shape; mi, mj storey masses Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• When the fundamental mode shape is approximated by horizontal displacements increasing linearly along the height, the horizontal forces Fi are given by:
Fi = Fb ⋅
mi ⋅ zi
(4.9)
n
∑m ⋅ z i
i
i =1
zi, zj heights of the masses; mi, mj above the level of application of the seismic action (foundation or top of a rigid basement).
Fi
The horizontal forces Fi shall be distributed to the lateral load resisting system assuming rigid floors. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
i
Torsional effects if lateral stiffness and mass are symmetrically distributed in plan • If the lateral stiffness and mass are symmetrically distributed in plan and unless the accidental eccentricity is taken into account . • Whenever a spatial model is used for the analysis, the accidental torsion effects referred may be determined as the envelope of the effects resulting from the application of static loadings, consisting of sets of torsion moments Mai about the vertical axis of each storey i: Mai = eai Fbi (4.10) e=0,05Li Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
(4.11)
Mx torsional moment applied at storey i about its vertical axis; e 1x – the accidental eccentricity on o-x axis e 1y – the accidental eccentricity on o-y axis CM – the center of mass Fbx – the seismic force on o-x direction Fby – the seismic force on o-y direction Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Reason for Consideration of Accidental Torsion [22]
Fk,n
Fk,n – the seismic level force at k level , in “n” th mode of vibration Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The case of “natural” eccentricity
Mtx=Tbx e ix Mty=Tby e iy e ix ,e iy = the “natural” eccentricity
e 0ix ,e 0iy = the distance between the center of masse and center of rigidity at level “i” e 1ix ,e 1iy = the accidental eccentricity Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
(4.12) (4.13)
(4.14) (4.15)
The distribution of seismic force to structural vertical elements (4.16)
(4.17)
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Ground conditions The construction site and the nature of the supporting ground should normally be free from risks of: • ground rupture, • slope stability and • permanent settlements caused by liquefaction or densification in the event of an earthquake.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
5.3 Lateral force method of analysis- EC8 • This type of analysis may be applied to buildings whose response is not significantly affected by contributions from higher modes of vibration. • These requirements are deemed to be satisfied in buildings which fulfil the two following conditions: a) they have fundamental periods of vibration T1 in the two main directions smaller than the following values where TC is given in Codes’ Tables; b) they meet the criteria for regularity in elevation
T 1≤ 2 s T1 ≤ 4TC Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
(4.18) (4.19)
Base shear force • The seismic base shear force Fb, for each horizontal direction in which the building is analysed, is determined as follows: (4.20) Fb = γI Sd (T1) m λ where: • Sd (T1) ordinate of the design spectrum at period T1; • T1 fundamental period of vibration of the building for lateral motion in the direction considered; • m total mass of the building, above the foundation or above the top of a rigid basement, • λ correction factor, the value of which is equal to: • λ = 0,85 if T1 < 2 TC and the building has more than two storeys, or λ = 1,0 otherwise •
γI the importance factor
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The design spectrum •
For the horizontal components of the seismic action the design spectrum, Sd(T), is defined by the following expressions: (4.21) (4.22)
Where: Sd(T) ordinate of the design spectrum, (4.23) q behaviour factor, β lower bound factor for the spectrum Values of the parameters S, T B, (4.24) T C, and T D are given in following tables
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Elastic response spectrum, Type 1
Values of the parameters describing the Type 1 elastic response spectrum
Elastic response spectrum, Type 2
Values of the parameters describing the Type 2 elastic response spectrum
Classification of subsoil classes EC8
• • • • • • • • •
Where: Se (T) ordinate of the elastic response spectrum, T vibration period of a linear single degree of freedom system, ag design ground acceleration (ag = agR γI), k modification factor to account for special regional situations, TB, TC limits of the constant spectral acceleration branch, TD value defining the beginning of the constant displacement response range of the spectrum, S soil parameter, ξ damping correction factor with reference value ξ =1 for 5% viscous damping Factor λ accounts for the fact that in buildings with at least three storeys and translation degrees of freedom in each horizontal direction, the effective modal mass of the 1st (fundamental) mode is smaller – on average by 15% - than the total building mass. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Design spectrum for elastic analysis The capacity of structural systems to resist seismic actions in the non-linear range generally permits their design for forces smaller than those corresponding to a linear elastic response. To avoid explicit inelastic structural analysis in design, the capacity of the structure to dissipate energy, through mainly ductile behaviour of its elements and/or other mechanisms, is taken into account by performing an elastic analysis based on a response spectrum reduced with respect to the elastic one, henceforth called ''design spectrum'', This reduction is accomplished by introducing the behaviour factor q. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The behaviour factor q •
The behaviour factor q is an approximation of the ratio of the seismic forces, that the structure would experience if its response was completely elastic with 5% viscous damping, to the minimum seismic forces that may be used in design - with a conventional elastic response model - still ensuring a satisfactory response of the structure.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• The value of the behaviour factor q, which also accounts for the influence of the viscous damping being different from 5%, are given for the various materials and structural systems and according to the relevant ductility classes in the various Parts of EN 1998. • The value of the behaviour factor q may be different in different horizontal directions of the structure, although the ductility classification must be the same in all directions.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
q the factor of structure behavior; the values are standard function of structure type and the capacity of energy dissipation Example the EC8 formula for reinforced concrete buildings where: q 0 basic value of the behavior factor dependent on the type of the structural system k w factor reflecting the prevailing failure mode in structural systems
(4.25)
Basic value of q 0 of behavior factor for systems regular in elevation
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• The reference method for determining the seismic effects is the modal response spectrum analysis, using a linear-elastic model of the structure and the design spectrum. • Depending on the structural characteristics of the building one of the following two types of linearelastic analysis may be used:
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Horizontal elastic response spectrum (1) For the horizontal components of the seismic action, the elastic response spectrum ß(T) is defined by the following expressions for damping correction factor for 5% viscous damping (2) If for special cases a viscous damping ratio different from 5% is to be used, this value will be given in the relevant Part of EN 1998.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Case “a”
T ≤ TB ( β 0 − 1) β (T ) = 1 + T TB
(4.26)
Case “b”
TB p T ≤ TC
β (T ) = β 0
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
(4.27)
Case “c”
TC p T ≤ TD TC β (T ) = β 0 T
(4.27)
Case “d”
T f TD TCTD β (T ) = β 0 2 T
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
(4.28)
Importance categories and importance factors Buildings are generally classified into 4 importance categories, which depend on the size of the building, on its value and importance for the public safety and on the possibility of casualties in case of collapse
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Table 4.3 Importance category I
II
Buildings
Buildings whose integrity during earthquakes is of vital importance for civil protection, e.g. hospitals, fire stations, power plants, etc. Buildings whose seismic resistance is of importance in view of the consequences associated with a collapse, e.g. schools, assembly halls,cultural institutions etc.
III IV
Ordinary buildings, not belonging to the other categories Buildings of minor importance for public safety, e.g. agricultural buildings, etc. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Seismic zones • For the purpose of EN 1998, national territories shall be subdivided by the • National Authorities into seismic zones, depending on the local hazard. By definition, • the hazard within each zone is assumed to be constant. • (2) For most of the applications of EN 1998, the hazard is described in terms of a single parameter, i.e. the value of the reference peak ground acceleration on rock or firm soil agR.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• Additional parameters required for specific types of structures are given in the relevant Parts of EN 1998. • The reference peak ground acceleration, chosen by the National Authorities for each seismic zone, corresponds to the reference return period chosen by National Authorities.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
5.4 Time - history representation • The seismic motion may also be represented in terms of ground acceleration time-histories and related quantities (velocity and displacement). • When a spatial model is required, the seismic motion shall consist of three simultaneously acting accelerograms. The same accelerogram may not be used simultaneously along both horizontal directions. • The description of the seismic motion may be made by using artificial accelerograms and recorded or simulated accelerograms.
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Non-linear methods • The mathematical model used for elastic analysis shall be extended to include the strength of structural elements and their post-elastic behaviour. • As a minimum, bilinear force – deformation envelopes should be used at the element level. In reinforced concrete and masonry buildings, the elastic stiffness of a bilinear force-deformation relation should correspond to cracked sections.
Bilinear force – deformation relation of the element
Zero post-yield stiffness may be assumed, If strength degradation is expected
In ductile elements, expected to exhibit post-yield excursions during the response, the elastic stiffness of a bilinear relation should be the secant stiffness to the yield-point. Trilinear envelopes, which take into account pre-crack and post-crack stiffnesses, are allowed.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
5.5 Non-linear static (pushover) analysis Pushover analysis is a non-linear static analysis under constant gravity loads and monotonically increasing horizontal loads. It may be applied to verify the structural performance of newly designed and of existing buildings for the following purposes: a) to verify or revise the overstrength ratio values αu/α1; b) to estimate expected plastic mechanisms and the distribution of damage; c) to assess the structural performance of existing or retrofitted buildings; Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• Buildings not complying with the regularity criteria shall be analysed using a spatial model. • For buildings complying with the regularity the analysis may be performed using two planar models, one for each main horizontal direction. • For low-rise masonry buildings, in which structural walls are dominated by shear, each storey may be analysed independently.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Lateral loads The vertical distributions of lateral loads which should be applied are at least two : − “uniform” pattern, based on lateral forces that are proportional to mass regardless of elevation (uniform response acceleration) - “modal” pattern, proportional to lateral forces consistent with the lateral force distribution determined in elastic analysis Lateral loads shall be applied at the location of the masses in the model. The torsion due to accidental eccentricity shall be considered. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Plastic mechanism
Determination of the idealized elasto - perfectly plastic force – displacement relationship.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Capacity curve
The relation between base shear force and the control displacement (the “capacity curve”) should be determined by pushover analysis for values of the control displacement ranging between zero and the value corresponding to 150% of the target displacement. The control displacement may be taken at the centre of mass at the roof of the building.
Overstrength factor
When the overstrength (αu/α1) should be determined by pushover analysis, the lower value of overstrength factor obtained for the two lateral load distributions should be used. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Plastic mechanism The plastic mechanism shall be determined for both lateral load distributions. The plastic mechanisms should comply with the mechanisms on which the behaviour factor q used in the design is based.
Target displacement
Target displacement is defined as the seismic demand in terms of the displacement of an equivalent single-degree-of-freedom system in the seismic design situation. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
BASICS OF SEISMIC ENGINEERING
By Doina Verdes
CHAPTER 5
EARTHQUAKE RESISTANT DESIGN
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Contents
5.1 Introduction 5.2 Performance Based Engineering 5.3 Performance Requirements and Compliance Criteria 5.4 The guiding principles governing the conceptual design against seismic hazard
Doina Verdes Basics of Seismic Engineering 2011
3
5.1 Introduction • The basic principle of any design is that the product should meet the owner’s requirements, which may be reduced to the criteria: • Function; • Cost; • Reliability.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Reliability • While the terms function and cost are simple in principle, reliability concerns various technical factors relating to serviceability and safety. • As the above three criteria are interrelated, and because of the normal constraints on cost, compromises with function and reliability generally have to be made
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• The term reliability is used here in its normal language qualitative sense and in its technical sense, where it is a quantitative measure of performance stated in terms of probabilities (of failure or survival). • The required reliability is achieved if enough of the elements of the design behave satisfactorily under the design earthquake. The elements that may be required to behave in agreed ways during earthquakes include structure, architectural elements, equipment, and contents.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Up to the mid-1980s it was common practice to design normal structures or equipment to meet two criteria: (1) in moderate, frequent earthquakes the structure or equipment should be undamaged; (2) in strong, rare earthquakes the structure or equipment could be damaged but should not collapse.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• The main intention of the second of these criteria was to save human lives, while the definition of the terms “strong”, “rare”, “moderate”, and “frequent” have varied from place to place, and have tended to be rather imprecise because of the uncertainties in the state-of-the-art. • Indeed, design has generally only been carried out explicitly for criterion (2), the assumption being made that, in so doing, it could be deemed that criterion (1) would automatically be satisfied.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
In our days the Seismic requirements provide minimum standards for use in building design to maintain public safety in an extreme earthquake. • Seismic requirements do not necessarily limit damage, maintain function, or provide for easy repair. • Design forces are based on the assumption that a significant amount of inelastic behavior will take place in the structure during a design earthquake.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
For reasons of economy and affordability, the design forces are much lower than those that would be required if the structure were to remain elastic. • In contrast, wind-resistant structures are designed to remain elastic under factored forces. • Specified code requirements are intended to provide for the necessary inelastic seismic behavior. • The buildings survival in large earthquakes depends directly on the ability of their resistance systems to dissipate hysteretic energy while undergoing (relatively) large inelastic deformations. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
5.2 Performance Based Engineering
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Selection of performance design objectives The three phases of the design process of the entire building system, i.e., - conceptual overall design; - preliminary numerical design; - final design and detailing. The acceptability checks of the designs arrived at in the above three phases. Quality assurance during construction (NOT in the last point).
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
PERFORMANCE BASED ENGINEERING CHECK SUITABILITY OF THE SITE DISCUSS WITH CLIENT THE PERFORMANCE LEVEL AND SELECT THE MINIMUM PERFORMANCE DESIGN OBJECTIVES
SITE SUITABILITY ANALYSIS (USE MICROZONATION MAP
• USE PERFORMANCE MATRIX • SERVICEABILITY UNDER MINOR EARTHQUAKES • FUNCTIONALITY UNDER MODERATE EARTHQUAKES • STRUCTURAL STABILITY UNDER EXTREME EARTHQUAKES
CONDUCT CONCEPTUAL OVERAL DESIGN, SELECTING CONFIGRATION STRUCTURAL LAYOUT, STRUCTURAL SYSTEM, STRUCTURAL MATERIALS AND NONSTRUCTURAL COMPONENTS
NO
ACCEPTABILITY CHECKS OF CONCEPTUAL OVERAL DESIGN
USE GUIDELINES
USE PEER REVIEW
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
NUMERICAL PRELIMINARY DESIGN
NO
•USE LINEAR AND NONLINEAR STATIC PUSHOVER DINAMIC TIME HISTORY ANALYSIS METHODS •USE PEER REVIEW
ACCEPTABILIT Y CHECKS OF PRELIMINARY DESIGN YES FINAL DESIGN AND DETAILING
NO
DESIGN TO COMPLY SIMULTANOUSLY WITH AT LEAST TWO LIMIT STATES (Ultimate limit states, Serviceability limit states)
ACCEPTABILITY CHECKS OF FINAL DESIGN AND DETAILING
•USE LINEAR AND NONLINEAR -STATIC PUSHOVER AND -DINAMYC TIME HISTORY ANALYSIS METHODS •EXPERIMENTAL DATA AND •INDEPENDENT REVIEW •USE LINEAR AND NONLINEAR -STATIC PUSHOVER AND -DINAMYC TIME HISTORY ANALYSIS METHODS •EXPERIMENTAL DATA AND •INDEPENDENT REVIEW
YES
QUALITY ASSURANCE DURING CONSTRUCTION MONITORING, MAINTENANCE AND FUNCTION Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Site suitability analysis of the selected site (Ground conditions)
The construction site and the nature of the supporting ground should normally be free from risks of: •ground rupture, •slope stability and •permanent settlements caused by liquefaction or densification in the event of an earthquake.
The collapse of a bridge placed on the seismic fault during the earthquake Taiwan 1999
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
1989 Earthquake in Loma Prieta, California, Bridge failure.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Site suitability analysis of the selected site
Romanian Territory the design acceleration and Control period TC of the soil
Elastic response spectra for horizontal components of soil movement (Romanian Territory ) TC = 0.7s
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
5.3 Performance Requirements and Compliance Criteria i) Selection of performance design objectives SEAOC Vision 2000, 1999
ii) Conforming Eurocode 8 iii) Conforming P100/2006
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Seismic performance design matrix (SEAOC Vision 2000, 1999)
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Building Performance Levels and Ranges*
Source: FEMA Instructional Material Complementing FEMA 451
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Total costs for different performance design objectives
Conforming SEAOC Vision 2000, 1999 Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Quality assurance during construction • Maintenance (modification and repairs) • Monitoring of occupancy (function) • Evaluation of seismic vulnerability of existing buildings • Seismic upgrading of existing hazardous buildings • Massive education and information dissemination programs
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Performance requirements and compliance criteria Conforming: EUROCODE 8 and P100/2006
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Fundamental requirements Structures in seismic regions shall be designed and constructed in such a way, that the following requirements are met, each with an adequate degree of reliability: No collapse requirement Damage limitation requirement
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
a. Requirement No collapse : The structure shall be designed and constructed to withstand the seismic action without local or global collapse, thus retaining its structural integrity and a residual load bearing capacity after the seismic events. The reference seismic action is associated with a reference probability of excedance in 50 years and a reference return period.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
IZMIT Earthquake, 1999 Turkey
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
b. Requirement: Damage limitation The structure shall be designed and constructed to withstand a seismic action having a larger probability of occurrence than the seismic action used for the verification of the “no collapse requirement”, without the occurrence of damage and the associated limitations of use (the costs of which would be disproportionately high in comparison with the costs of the structure itself).
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The Codes Target reliabilities for the “no collapse requirement” and for the “damage limitation requirement” are established by the National Authorities for different types of buildings or civil engineering works on the basis of the consequences of failure. Reliability differentiation is implemented by classifying structures into different importance categories.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Importance classes for buildings cf EC8
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Compliance Criteria In order to satisfy the fundamental requirements the following limit states shall be checked : - Ultimate limit states are those associated with collapse or with other forms of structural failure which may endanger the safety of people. - Serviceability limit states are those associated with damage occurrence, corresponding to states beyond which specified service requirements are no longer met.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The structural system shall be verified as having the resistance and ductility. The resistance and ductility to be assigned to the structure are related to the extent to which its nonlinear response is to be exploited.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
If the building • configuration is symmetrical or quasi-symmetrical, • a symmetrical structural layout, well distributed in-plan, is an obvious solution for the achievement of uniformity. • The use of evenly distributed structural elements increases redundancy and allows a more favourable redistribution of action effects and widespread energy dissipation across the entire structure.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Criteria for regularity in elevation All lateral load resisting systems, like cores, structural walls or frames, run without interruption from their foundations to the top of the building or, if setbacks at different heights are present, to the top of the relevant zone of the building. Both the lateral stiffness and the mass of the individual storeys remain constant or reduce gradually, without abrupt changes, from the base to the top. In framed buildings the ratio of the actual storey resistance to the resistance required by the analysis should not vary disproportionately between adjacent storeys. Within this context the special aspects of masonry infilled frames have to be treated. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Criteria for structural regularity Building structures for the purpose of seismic design, are distinguished as regular and non-regular. This distinction has implications on the following aspects of the seismic design: − the structural model, which can be either a simplified planar or a spatial one, − the method of analysis, which can be either a simplified response spectrum analysis (lateral force procedure) or a multi-modal one, − the value of the behaviour factor q, which can be decreased depending on the type of non-regularity in elevation, i.e.: geometric non-regularity (exceeding the limits ), non-regular distribution of over strength in elevation (exceeding the limits). Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
With respect to the lateral stiffness and mass distribution, the building structure is approximately symmetrical in plan with respect to two orthogonal axes. The plan configuration is compact, i.e., at each floor is delimited by a polygonal convex line. If in plan set-backs (re-entrant corners or edge recesses) exist, regularity in plan may still be considered satisfied provided that these set-backs do not affect the floor in-plan stiffness and that, for each set-back, the area between the outline of the floor and a convex polygonal line enveloping the floor does not exceed 6 % of the floor area.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The in-plane stiffness of the floors is sufficiently large in comparison with the lateral stiffness of the vertical structural elements, so that the deformation of the floor has a small effect on the distribution of the forces among the vertical structural elements. In this respect, the L, C, H, I, X plane shapes should be carefully examined, notably as concerns the stiffness of lateral branches, which should be comparable to that of the central part, in order to satisfy the rigid diaphragm condition. The application of this paragraph should be considered for the global behaviour of the building. The slenderness η=Lx/Ly of the building in plan is not higher than 4.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
A simplified definition, for the classification of structural regularity in plan and for the approximate analysis of torsional effects, is possible if the two following conditions are satisfied: All lateral load resisting systems, like cores, structural walls or frames, run without interruption from the foundations to the top of the building. The deflected shapes of the individual systems under horizontal loads are not very different. This condition may be considered satisfied in case of frame systems and wall systems. In general, this condition is not satisfied in dual systems. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The foundation elements and the foundation-soil interaction It shall be verified that both the foundation elements and the foundation-soil are able to resist the action effects resulting from the response of the superstructure without substantial permanent deformations.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Modeling Procedures for Embedded Structures* The actual soil-foundation structure system is excited by a wave field that is incoherent both vertically and horizontally and which may include waves arriving at various angles of incidence. These complexities of the ground motions cause foundation motions to deviate from free-field motions. This complex ground excitation acts on stiff, but non-rigid, foundation walls and the base slab, which in turn interact with a flexible and nonlinear soil medium having a significant potential for energy dissipation. Finally, the structural system is connected to the base slab, and possibly basement asWITH well. *INPUT GROUNDto MOTIONS FOR TALLwalls BUILDINGS SUBTERRANEAN LEVELS Authors: Jonathan P. Stewart and Salih Tileylioglu Civil & Environmental Engineering Department, UCLA Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
There are two classical methods for modeling the problem soil – foundationstructure. The first is a direct approach, - a computational model of the full structure, foundation, and soil system is set up and excited by a complex and incoherent wave field. This problem is difficult to solve from a computational standpoint, and hence the direct approach is rarely used in practice.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
In the second approach (referred to as the substructure approach), the complex soil-foundation-structure interaction problem is divided into three steps: Kinematic interaction, Foundation - soil flexibility and damping, Foundation flexibility and damping.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Substructure approach to solution of soil-foundation-structure interaction using rigid foundation or flexible foundation assumption
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
θ g = the foundation rotation u s = the foundation translation a. Rigid foundation
b. Structure with foundation flexibility - flexibility and damping)
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Overturning and sliding stability • The structure as a whole shall be checked to be stable under the design seismic action. Both overturning and sliding stability shall be considered.
Influence of second order effects In the analysis the possible influence of second order effects on the values of the action effects shall be taken into account Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
5.4 The guiding principles governing the conceptual design against seismic hazard
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The guiding principles governing the conceptual design against seismic hazard are:
− uniformity, symmetry and redundancy - structural simplicity − bi-directional resistance and stiffness, − torsional resistance and stiffness, − diaphragmatic behaviour at storey level, − adequate foundation Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
The form in plan recommended in seismic design
a.
b.
c.
d.
e.
f.
• Uniformity is characterised by an even distribution of the structural elements which, if fulfilled in-plan, allows short and direct transmission of the inertia forces created in the distributed masses of the building. If necessary, uniformity may be realised by subdividing the entire building by seismic joints into dynamically independent units, provided that these joints are designed against pounding of the individual units.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Uniformity in the development of the structure along the height of the building is also important, since it tends to eliminate the occurrence of sensitive zones where concentrations of stress or large ductility demands might prematurely cause collapse. If the building configuration is symmetrical or quasisymmetrical, a symmetrical structural layout, well distributed in-plan, is an obvious solution for the achievement of uniformity. The use of evenly distributed structural elements increases redundancy and allows a more favourable redistribution of action effects and widespread energy dissipation across the entire structure. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Symmetry • In seismic area it has to be searched building shapes as simplest and symmetric as possible, in plan as much as in elevation. Many of the successful realizations aesthetic • Symmetry is desirable for much the same reasons. It is worth pointing out that symmetry is important in both directions in plan and in elevation as well. Lack of symmetry produces torsion effects which are sometimes difficult to asses and can be very destructive.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• The introduction of deep reentrant angles into the facades of buildings introduces complexities into the analysis which makes them potentially less reliable than simple forms. Buildings of H-, L-, T-, and Yshape in plan have often been severely damaged in earthquakes. • External lifts and stairwells provide similar dangers, and should be used with the appropriate attention to analysis and design. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
a. e.
f.
b.
c.
g.
d.
h.
Seismic joint condition Buildings shall be protected from earthquake-induced pounding with adjacent structures or between structurally independent units of the same building. If the floor elevations of the building or independent unit under design are the same as those of the adjacent building or unit, the above referred distance may be reduced by a factor of 0,7 (EC8).
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
This is deemed to be satisfied if the distance from the boundary line to the potential points of impact is not less than the maximum horizontal displacement of the adjacent parts according to expression.
∆= ∆ 1+ ∆ 2+20 mm
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Building separation to avoid pounding
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Length in plan Structures which are long in plan naturally experience greater variation in ground movement and soil conditions over their length than short ones. These variations may be due to out- of-phase effects or to differences in geological conditions, which are likely to be most pronounced along bridges where depth to bedrock may change from zero to very large. The effects on structure will differ greatly, depending on whether the foundation structure is continuous, or a series of isolated footings, and whether the superstructure is continuous or not.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
• Continuous foundations may reduce the horizontal response of the superstructure at the expense of push-pull forces in the foundation itself. Such effects should be allowed for in design, either by designing for the stressed induced in the structure or by permitting the differential movements to occur by incorporating movement gaps.
Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
Shape in elevation
h>4L1
L1
L2
a.
b.
Height/width ratios in excess of about 4 lead to increasingly uneconomical structures and require dynamic analysis for proper evaluation of seismic responses. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
h>4L1
those with sudden changes in width should be avoided in strong earthquakes areas.
h AS2
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The influence of the reinforcing ratio from tensioned zone M
Φu ρΦ = Φy Φ u1 ρ Φ1 = Φ y1
M
M Mu1; Mu2 My1 My2
Φu2 ρΦ = Φ y2 Φ Φ y2 Φ y1
Φu1 Φu2
Φ y1 f Φ y 2
ρ Φ1 p ρ Φ 2 The increasing of the reinforcement ratio of the transversal tensioned reinforcement, do not lead to increasing of ductility. Doina Verdes BASICS OF SEISMIC ENGINEERING 2011
ρФ = CURVATURE DUCTILITY COEFICIENT
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The influence of the reinforcing ratio of transversal reinforcement
G1
G2
A S2
A S3 Craking of concrete covering layer
M
h A S1
Mu1
h
My1
A S1
Mu2
b
b Φ Φy1 Φy2
AS2 to see how dampers work in the structure
The final model
Images of the first model: construction and testing 10th -12th of February 2011 SAN-DIEGO CA, USA
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Results for 5% damping for integration to artificial accelerogram GM3 (UCDavis)
Predictions for the structure were made using numerical analysis as follows: •
Computation of the seismic response of the structure using SAP2000 for 5% damping
•
Computation of the seismic response of the structure using SAP2000 for 15% damping Max displacement =5.57cm = 2.19in 10th -12th of February 2011 SAN-DIEGO CA, USA
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Max velocity =3.36m/s =11.02ft/s 10th -12th of February 2011 SAN-DIEGO CA, USA
Max acc =2.02m/s^2 = 6.62ft/s^2 32
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Results with 15% damping for intergration to artificial accelerogram GM3 (UCDavis)
Performances of the structure according to the rules of the competition • Annual Income : 735,000 $/year • Annual Initial Building Cost : 322,000 $/year • Annual Seismic Cost : 50,900 $/year
Max displacement =3.74cm = 1.47in
Max velocity =3.36m/s = 7.87ft/s
Max acc =2.02m/s^2 = 4.36ft/s^2
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The accelerograms
Accelerogram El Centro, 18 Mai 1940
Accelerogram Northridge, 1994
The model before the test on shake table at Seismic Design Contest
Artificial accelerogram UCDavis 35
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9. Conclusions after the test The model was subjected to three accelerograms - behavior of model was very good at all three accelerograms ; - the model bars were not damaged ; - the friction dampers have worked very well allowing the deformation of the structural elements and dissipating energy. The collapse of the model arrived after the test with sinus wave having the frequency equal fundamental frequency of the model. 10th -12th of February 2011 SAN-DIEGO CA, USA
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10. The award ceremony
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The 8th Seismic Design Competition, 2011 winner teams
The prize plaque Egor Popov Award for Structural Innovation for the model “Valahia tower” made by the team from Technical University of Cluj-Napoca, Romania
The top three teams: Oregon State University California Polytechnic State University, San Luis Obispo California Polytechnic State University, San Luis Obispo Charles Richter Award for the Spirit of the Competition: UC Davis Honorable Mention Nominees: Penn State University, Universiti Teknologi Malaysia Egor Popov Award for Structural Innovation: Technical University Cluj-Napoca, Romania Honorable Mention Nominee: UC Davis Fazlur Khan Award for Architectural Design: San Jose State University Honorable Mention Nominees: Brigham Young University, California Polytechnic University, Pomona
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The annoncement of Karthik Ramanathan vice president of SLC, of the award Egor Popov 10th -12th of February 2011 SAN-DIEGO CA, USA
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Romanian delegation together with colleagues from American universities Romanian delegation together with Nima Tafazolli, co president of SLC
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11. Models Presented by the Participants Universities
Romanian delegation together with colleagues from University of Technologi , Malaysia 10th -12th of February 2011 SAN-DIEGO CA, USA
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Oregon State University
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California Polytechnic State University, San Luis Obispo University of Illinois Urbana Champaign 10th -12th of February 2011 SAN-DIEGO CA, USA
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UC Davis
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Purdue University
California State University, Los Angeles 10th -12th of February 2011 SAN-DIEGO CA, USA
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Roger Williams University Roger Williams University
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Brigham Young University
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UC Irvine
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University of Massachusetts Amherst 10th -12th of February 2011 SAN-DIEGO CA, USA
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12. The tour in San Diego city
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Many thanks to the generous sponsors of the TUCN team !
Many thanks to the generous sponsors of the 2011 SDC!
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