CHAPTER

CHAPTER (3)
THEORITICAL STUDY

3-1- Introduction:

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3-2-Methodology:
Methodology that is used to achieve our main goal in the present study is dynamic analysis of three-dimensional model of foundation which is resting on different soil and carrying different number of floors by time history as a method of earthquake modeling, also with taking nonlinear behavior of element materials in consideration. Modeling and analysis to our problem will be done by finite element method as a numerical analysis method by using one of its application software like ANSYS. So that it is necessary to explain model element behavior, material properties of each element and dynamic loads properties.

3-3-Review of structural elements and dynamic loads relationship:
The structural system is consist of some elements that behave as a one unit to achieve stability against any collapse. These elements divide to some categories and each category has its role in stability of structure. These categories are represented in substructure as foundation, superstructure elements, soil that carries all structural system and loads affected on system. So we can focus on some factors that will be used in our research as the following:
i. Soil.
ii. Foundation.
iii. Superstructure.

3-3-1- Soil:
Soil is one of the major factors which has a great effect on analysis operation. In this section, we will care about some soil factors that we will be more interested in it in our study and analysis operation. These factors are as the following:
i- Soil types.
ii- Bearing capacity of soil.
iii- Settlement of soil under foundation.
iv- The contact pressure under foundation.

3-3-1-1- Soil types:
It is obviously shown that soil types have important effect on foundation analysis. Choosing type, area, depth and construction method of foundation is mainly supported on soil types. Also, soil types play main role in determination of structure cost. Soil divided into a lot of types depending on its grains, proportion of pores and compressibility. Soil is generally divided into three categories:
A) Coarse grained soil.
Coarse grained type is cohesion less soil. It is divided into gravel and sand. This type is excellent soil for structure as its small settlement that is called immediate settlement and for its compressibility resisting.

B) Fine grained soil.
Fine grained type is cohesive soil. It is divided into clay and silt. This type is a weak soil for structure as its large settlement that is caused by its compressibility. This type has a small grain size therefore Porosity and void ratio have a small value.
C) Organic soil.
Organic soil is a very bad type for structure as many reasons related to its components. It has a lot of problems due to it may decays leading to increasing in settlement or into incidence failure in soil under foundations.

3-3-1-2-Bearing capacity of the soil:
The soil which is carrying the foundation that transmits the loads produced by superstructure should be more flexible and strong enough so that the structure–foundation–soil system is safe and stable besides being serviceable without excessive settlements. Bearing capacity is the power of foundation soil to hold the forces from the superstructure without undergoing shear failure or excessive settlement. Foundation soil is that portion of ground which is subjected to additional stresses when foundation and superstructure are constructed on the ground. The following are a few important terminologies related to bearing capacity of soil. Ultimate bearing capacity is represented in Terzagi’s equation.

. Terzagi. 1943 3-1

qu = ultimate bearing capacity.
C = cohesion of soil.
q = surcharge.
= soil unit weight.
Nc, Nq and N? = Terzagi’s bearing capacity constants.
B = width of foundation in meter.
Depending on the stiffness of foundation soil and depth of foundation, the following are the modes of shear failure experienced by the foundation soil:

a) General shear failure:
The general shear failure is the most common mode. This type of failure is seen in dense and stiff soil. The following are some characteristics of general shear failure.
i- Continuous, well defined and distinct failure surface develops between the edge of footing and ground surface.
ii- Dense or stiff soil that undergoes low compressibility experiences this failure.
iii- Continuous bulging of shear mass adjacent to footing is visible.
iv- Failure is accompanied by tilting of footing.
v- Failure is sudden and catastrophic with pronounced peak in P – ? curve.
vi- The length of disturbance beyond the edge of footing is large.
vii- State of plastic equilibrium is reached initially at the footing edge and spreads gradually downwards and outwards.

b) Local shear failure:
This type of failure is seen in relatively loose and soft soil. The following are some characteristics of general shear failure.
i- A significant compression of soil below the footing and partial development of plastic equilibrium is observed.
ii- Failure is not sudden and there is no tilting of footing.
iii- Failure surface does not reach the ground surface and slight bulging of soil around the footing is observed.
iv- Failure surface is not well defined.
v- Failure is characterized by considerable settlement.
vi- Well defined peak is absent in P – ? curve.

c) Punching shear failure:
The punching shear failure occurs in very loose sands. This type of failure is seen in loose and soft soil and at deeper elevations. The following are some characteristics of general shear failure.
i- This type of failure occurs in a soil of very high compressibility.
ii- Failure pattern is not observed.
iii- Bulging of soil around the footing is absent.
iv- Failure is characterized by very large settlement.
v- Continuous settlement with no increase in P is observed in P – ? curve.

3-3-1-3- Settlement of soil under foundation:
i- Settlement has a great danger on structure as it causes a lot of damage in structure and foundation also. Settlement has many reasons such as structure load is a very large and caused a big stress on soil.
ii- Soil under foundation failure.
iii- Under ground water table level is decreased.
The settlement consists of one or more than one of the following: (a)-Immediate Settlement, (b)-Consolidation settlement and (c)-secondary compression and creep settlement.

3-3-1-4- Contact pressure distributions under foundations:
The contact pressure distribution beneath foundation which produced by soil reaction is an important factor for foundation design. By means of contact pressure distribution, the internal forces can be calculated by using the structural theories. There are some methods to calculate the contact pressure distribution which include the condition of stability and the deformation in shape of soil and foundation. The rectilinear contact pressure method (conventional analysis), or the modulus of subgrade reaction method, and the method based on elastic theory show different contact pressure distribution, thus leading to very different foundation thicknesses and reinforcement figure. (3-1). The contact pressure distribution under foundation must be the real distribution as much as possible because any small change in contact pressure distribution will lead to a big error in moment calculation in all sections of foundation. And these methods is used to calculate the contact pressure under shallow foundation resting on cohesion soil or cohesion less soil while the foundation is under vertical loads, as in inhabited buildings, commercial buildings, tanks, and stores. This method is used if the soil consists of one horizontal layer or layers with regular characteristics. For an irregular soil, the behavior of the soil in vertical and horizontal directions must be taken into consideration. The distribution of contact pressure under foundation depends on the properties of soil and foundation.

Figure 3-1. Effect of contact pressure distribution on internal forces.
ECP (2001)

For foundation properties, contact pressure distribution depends on:
i- Foundation width (B)
ii- The relation between (L/B) and thickness, where L : Foundation length
iii-Elastic modulus of foundation material.
For soil properties, contact pressure distribution depends on:
i- Thickness of compressibility layer
ii- Compressibility index (Young’s modulus).
Another important factor for contact pressure distribution is the relative rigidity between soil and foundation.
ECP2001 3-2
Where:
Kr = relative rigidity between soil and foundation.
E = Modulus of elasticity of the foundation.
Es = Modulus of elasticity of the soil.
t = thickness of foundation.
B = width of foundation.

3-3-2- Foundation:
Foundation is the part that play an important role in structural stability. Foundation is the part that carry all loads of superstructure and transmits these loads to soil as stress with minimum value. There are many factors affecting foundation analysis. Some factors may be related to foundation itself as the following:
i- Foundation dimensions.
ii- Foundation material properties.

3-3-2–1- Foundation dimensions:
Foundation design mainly depend on soil bearing capacity. Stresses between soil and foundation are inversely proportional with foundation dimensions, also these stresses are inversely proportional with bearing capacity. This mean that when foundation area is more bigger stresses will be smaller so soil bearing capacity will be more resisting against any settlement. Also, Foundation thickness has a great effect whenever it is larger whenever foundation is more rigid and has no any deformation.

3-3-2–2- Foundation material properties:
From things that have a great effect in foundation design are its properties. Foundation components play an important role to make foundation more quality such as amount of cement, steel percentage and water content. Quality of foundation is one of the main reasons leading to safety of foundation design and soil structure interaction also as it has become more rigid.

3-3-3- Superstructure:
Superstructure has a great role in foundation analysis especially foundation that subjected to dynamic loads. Superstructure has many factors affected on analysis operation as the following:
i- The superstructure type.
ii- The superstructure loads acting on foundation.
iii- Superstructure material properties.

3-3-3-1-The superstructure type:
The superstructure type has an important role in estimating foundation types. Foundations are generally divided related to its depth from ground level into two categories shallow and deep foundations. Choosing foundation type is depending on superstructure type as the load acting on foundation is variable with this type. For example, Bridges are structures that subjected to big static and dynamic loads will be need to special foundations has a large dimension. Thus, we must consider superstructure type while design foundation.

3-3-3-2-The superstructure loads acting on foundation:
There are multi load forms that acting on superstructure such as wind loads, earthquake loads and superstructure own weight. Superstructure transform these loads to foundation and therefore to soil finally. Superstructure loads that are transformed to foundation control in value of stress on soil so that foundation area, dimensions and safety against bearing capacity of soil depend on the value of these loads. Thus, we must take these loads effect in consideration.
3-3-3-3- Superstructure material properties:
Superstructure material properties such as concrete ultimate strength, steel ultimate stress and any addition components improve structure quality increase structure resisting against any deformations. So whenever the superstructure will be more rigid and resistance against deformation, the stress between soil and foundation reduces as a result. So, superstructure material properties is an important factor in foundation analysis.

Dynamic loads were the ghost that impended engineers for a long time and this for many reasons. Non-uniform value, suddenly happened and unknown direction of dynamic loads are the most important factors that made dynamic loads as a big danger on structure and had to be taken in consideration during design. So it is important to know the nature of dynamic loads and how it happen to understand the behavior of foundation under these loads. So it is important to explain some points as the following:
ii. Difference between dynamic and static loads.
iv. Causes of earthquakes.
v. Types of earthquake waves.
vi. Measurement of earthquakes.
vii. Effects of earthquake.
viii. Methods used for earthquake analysis.
ix. Time-History method ECPLF 2012.

3-3-4-1- Dynamic load types Rajasekaran, S. 2009:

Figure 3-2 Different types of dynamic loads: (a) simple harmonic; (b) non-harmonic (periodic); (c) non-periodic (short duration); (d) non-periodic (long duration). Rajasekaran, S. (2009).

3-3-4-2- Difference between dynamic and static loads Rajasekaran, S. 2009:
Dynamic and static loads have different effect on structure so the behavior of structure changes for each case. Static load is stable and has a linear relation with time. But dynamic load is antithesis not stable and has a variable relation with time. A dynamic load is different from a static load from two aspects. The first and most obvious difference is the time-varying load and the time-varying response also. This needs analysis over a specific interval of time. Hence dynamic analysis is complex and computationally extensive and expensive compared with static analysis. The other difference between dynamic and static problems is the major occurrence of inertia forces when the loading is dynamically applied. Consider a water tank as shown in Figure. (3-3) subjected to load F at the top.

Figure 3-3 Water tank subjected to static and dynamic loads: (a) Static load;
(b) Dynamic load. Rajasekaran, S. (2009).

The resulting deflection, shear force and bending moment can be calculated on the basis of static structural analysis principles as shown in Figure. (3-3a). On the other hand, if the time-varying load F(t) is applied at the top, the structure is set to motion or vibration and experiences accelerations. According to Newton’s second law, inertia force is proportional to acceleration. Inertia forces are proportional to the mass and they develop in the structure that resists these accelerations. Depending on the contribution made by inertia force to shear and bending moment will determine whether dynamic analysis is warranted as shown in figure. (3-3b). The most famous type of dynamic loads is earthquake loads and this type which we will talk about its effect on our life.

Earthquakes are the Earth’s natural means of releasing stress. Earthquakes are the most famous type of dynamic loads and they in the fact are the motion or trembling of the ground produced by sudden displacement of rock in the earth’s crust, volcanism, landslides, rock bursts, and man-made explosions. Of these, naturally occurring tectonic-related earthquakes are the largest and most important. These are caused by the fracture and sliding of rock along faults within the Earth’s crust. A fault is a zone of the earth’s crust within which the two sides have moved – faults may be hundreds of miles long, from one to over one hundred miles deep, and are sometimes not readily apparent on the ground surface. Earthquakes initiate a number of phenomena or agents, termed seismic hazards, which can cause significant damage to the built environment – these include fault rupture, vibratory ground motion, inundation and various kinds of permanent ground failure, fire, or hazardous materials release. In a particular earthquake event, any particular hazard can dominate, and historically each has caused major damage and great loss of life in particular earthquakes. For most earthquakes, shaking is the dominant and most widespread agent of damage. Shaking near the actual earthquake rupture lasts only during the time when the fault ruptures, a process that takes seconds or at most a few minutes. The seismic waves generated by the rupture propagate long after the movement on the fault has stopped. Typically, earthquake ground motions are powerful enough to cause damage only in the near field – in a few instances, long period motions have caused significant damage at great distances, to selected lightly damped structures. Earthquake cause a lot of damages to structure and these due to many reasons. These reasons may be because of soil or structure itself.

3-3-4-4- Causes of earthquakes Nelson, A, S. 2013:
Earthquakes occur when energy stored in elastically strained rocks is suddenly released. This release of energy causes intense ground shaking in the area near the source of the earthquake and sends waves of elastic energy, called seismic waves, throughout the Earth. Earthquakes can be generated by bomb blasts, volcanic eruptions, and sudden slippage along faults. Earthquakes are definitely a geologic hazard for those living in earthquake prone areas, but the seismic waves generated by earthquakes are invaluable for studying the interior of the Earth. The causes of earthquakes are generally divided into:
i- Faults.
ii- Tectonic causes.
iii- Volcanic causes.
iv- Surface causes.

3-3-4-4-1- Faults:
A fault is a zone of the earth’s crust within which the two sides have moved. Faults are the physical expression of the boundaries between adjacent tectonic plates and thus may be hundreds of miles long. Generally, the longer a fault the larger the earthquake it can generate. Beyond the main tectonic plates, there are many smaller sub plates, ”platelets,” and simple blocks of crust that occasionally move and shift due to the ”jostling” of their neighbors and the major plates. The existence of these many sub plates means that smaller but still damaging earthquakes are possible almost anywhere, although often with less likelihood. Faults are typically classified according to their sense of motion. Scientists use the angle of the fault with respect to surface (known as the dip) and the direction of slip along the fault to classify faults. Faults in general are divided into three categories:
i- Normal fault.
ii- Thrust (reverse) fault.
iii- A left-lateral Strike-slip fault.
iv- A right-lateral Strike-slip fault.

A) Normal fault:
This type is a dip-slip fault in which the rock or block above the fault move downward relative to the rock or block below as in figure (3-4).

Figure 3-4 normal fault type. Nelson, A, S. (2013)

B) Thrust (reverse) fault:
This type is a dip-slip fault in which the upper rock or block, above the fault plane, move up and over the rock or block below as in figure (3-5).

Figure 3-5 thrust fault type. Nelson, A, S. (2013)

C) A left-lateral strike-slip fault:
In this type the displacement of the far block is to the left when viewed from either side as in figure (3-6).

Figure 3-6 left lateral strike-slip fault type. Nelson, A, S. (2013)

D) A right-lateral strike-slip fault:
In this type the displacement of the far block is to the right when viewed from either side as in figure (3-7).

Figure 3-7 right lateral strike-slip fault type. Nelson, A, S. (2013)

3-3-4-4-2- Tectonic causes:
The tectonic plate theory is the main cause of earthquakes. A theory of global tectonics in which the lithosphere is divided into a number of plates that act like rigid bodies and that interact with one another at their boundaries causing earthquakes. So, in this type earthquakes occur due to displacement in underground layer of soil when rocks in the Earth’s crust break resulting of geological forces created by movement of tectonic plates.

3-3-4-4-3- Volcanic causes:
Earthquakes related to volcanic activity may produce hazards which include ground cracks, ground deformation, and damage to manmade structures. There are two general categories of earthquakes that can occur at a volcano:
i- Volcano-tectonic earthquakes.
ii- Long period earthquakes.

A) Volcano-tectonic earthquakes
In volcano-tectonic type earthquakes produced by stress changes in solid rock due to the injection or withdrawal of magma are called volcano-tectonic earthquakes (Chouet, 1996). These earthquakes can cause land to subside and can produce large ground cracks. These earthquakes can occur as rock is moving to fill in spaces where magma is no longer present. Volcano-tectonic earthquakes don’t indicate that the volcano will be erupting but can occur at any time.

B) Long period earthquakes
In long period type earthquakes produced by the injection of magma into surrounding rock. These earthquakes are a result of pressure changes during the unsteady transport of the magma. When magma injection is sustained a lot of earthquakes are produced (Chouet, 1996). This type of activity indicates that a volcano is about to erupt. Scientists use seismographs to record the signal from these earthquakes. This signal is known as volcanic tremor. People living near an erupting volcano are very aware of volcanic earthquakes. Their houses will shake and windows rattle from the numerous earthquakes that occur each day before and during a volcanic eruption. To prevent damage from being done, structures should be built according to earthquake standards, building foundations should be constructed on firm ground and not unconsolidated material which may amplify earthquake intensity, and buildings should be constructed on stable slopes in areas of low hazard potential.

3-3-4-4-4- Surface causes:
There are some causes for earthquakes occur on ground surface. These causes may be from man-made or natural such as, great explosions, landslides, slip on steep coasts, dashing of sea waves, railway trains, heavy trucks and some large engineering projects cause minor tremors. These loads may be have small values but by repeating its value will be large enough to make earthquakes.
3-3-4-5- Types of earthquake waves:
During earthquake some waves are associated with it. These waves are created when stress is released as energy in earthquakes. These waves are divided into three categories as shown in figure (3-8):
i- P-waves
ii- S-waves
iii- Surface waves

A) P-waves
The P wave, or primary wave, is the fastest of the three waves and the first detected by seismographs. They are able to move through both liquid and solid rocks. P waves, like sound waves, are compressional waves, which mean that they compress and expand matter as they move through it.

B) S-waves
S waves, or secondary waves, are the waves directly following the P waves. As they move, S waves shear, or cut the rock they travel through. S waves cannot travel through liquid because, while liquid can be compressed, it can’t shear. S waves are the more dangerous type of waves because they are larger than P waves and produce vertical and horizontal motion in the ground surface.

C) Surface waves
The third type of wave, and the slowest, is the surface wave. These waves move close to or on the outside surface of the ground. There are two types of surface waves: Love waves, that move like S waves but only horizontally, and Rayleigh waves, that move both horizontally and vertically in a vertical plane pointed in the direction of travel.

Figure 3-8 earthquake waves types. Nelson, A, S. (2013)

3-3-4-6- Measurement of earthquakes Chen, W. F. and Lui, E. M. 2005:
Scientists are very interested in studying earthquakes and its effects on our life. To study these effects they studied the seismic waves that are accompanied with earthquakes. Seismic waves are the vibrations from earthquakes that travel through the Earth; they are recorded on instruments called seismographs. Seismographs record a zig-zag trace that shows the varying amplitude of ground oscillations beneath the instrument. Sensitive seismographs, which greatly magnify these ground motions, can detect strong earthquakes from sources anywhere in the world. The time, locations, and magnitude of an earthquake can be determined from the data recorded by seismograph stations. The earthquakes measurement methods are divided into:

i- Magnitudes.
ii- Intensity.
iii- Time history.
iv- Elastic Response Spectra.
v- Inelastic Response Spectra.

3-3-4-6-1- Magnitudes:
The magnitude of an earthquake is an estimate of the energy released by it. The amount of this energy has formed the basis for measuring the earthquake event. The scale that divides the size of earthquakes into categories called magnitudes such as Richter scale as in figure (3-9). Richter was the first to define magnitude of the earthquake, as:
ML = Log A – Log Ao Richter 1935 3-3

Figure 3-9. Relationship between moment magnitude and various magnitude scales Campbell, K.W. (1985).

Where ML is the local magnitude (which Richter only defined for Southern California), A is the maximum trace amplitude in micrometers recorded on a standard Wood–Anderson short-period torsion seismometer, 3 at a site 100 km from the epicenter, and log A0 is a standard value as a function of distance for instruments located at distances other than 100 km and less than 600 km. Subsequently, a number of other magnitudes have been defined, the most important of which are surface wave magnitude MS, body wave magnitude mb, and moment magnitude MW. Due to the fact that ML was only locally defined for California (i.e., for events within about 600 km of the observing stations), Surface wave magnitude MS was defined analogously to ML, using teleseismic observations of surface waves of 20 s period. Magnitude, which is defined on the basis of the amplitude of ground displacements, can be related to the total energy in the expanding wave front generated by an earthquake, and thus to the total energy release — an empirical relation by Richter is:

Log?0 ES = 11.8 + 1.5 Ms Richter 1935 3-4

Where ES is the total energy. Subsequently, due to the observation that deep-focus earthquakes commonly do not register measurable surface waves with periods near 20 s, a body wave magnitude mb was defined (Gutenberg and Richter 1954), which can be related to MS (Darragh et al. 1994):

mb = 2.5 + 0.63 Ms Richter 1935 3-5

Body wave magnitudes are more commonly used in eastern North America, due to the deeper earthquakes there. A number of other magnitude scales have been developed, most of which tend to saturate — that is, asymptote to an upper bound due to larger earthquakes radiating significant amounts of energy at periods longer than used for determining the magnitude (e.g., for MS, defined by measuring 20 s surface waves, saturation occurs at about MS>7.5). More recently, seismic moment has been employed to define a moment magnitude MW (Hanks and Kanamori 1979; also denoted as boldface M), which is finding increased and widespread use:

Log Mo = 1.5 Mw + 16.0 Richter 1935 3-6

Where seismic moment M0 (dyne cm) is defined as (Lomnitz 1974)

Mo = µAu? Richter 1935 3-7

Where µ is the material shear modulus, A is the area of fault plane rupture, and u? is the mean relative displacement between the two sides of the fault (the averaged fault slip). Comparatively, MW and MS are numerically almost identical up to magnitude 7.5. Figure (3-9) indicates the relationship between moment magnitude and various magnitude scales.

Figure 3-10. Typical earthquake accelerograms courtesy Darragh, R.B., Huang, M.J. and Shakal, A.F. (1994).
3-3-4-6-2- Intensity:
Intensity is another way to measure the strength of an earthquake by using the Mercalli scale. Invented by Mercalli, G. (1902), this scale uses the observations of the people who experienced the earthquake to estimate its intensity. The Mercalli scale isn’t considered as scientific as the Richter scale, though. Some witnesses of the earthquake might exaggerate just how bad things were during the earthquake and you may not find two witnesses who agree on what happened; everybody will say something different. The amount of damage caused by the earthquake may not accurately record how strong it was either. In general, seismic intensity is a metric of the effect, or the strength, of an earthquake hazard at a specific location. While the term can be generally applied to engineering measures. Numerous intensity scales developed in pre-instrumental times, the most common in use today are the modified Mercalli (MMI) Wood and Neumann (1931) and its successor the European Macro-seismic Scale EMS-98 (1998), and Japan Meteorological Agency (JMA) Kanai (1983) scales. Modified Mercalli Intensity (MMI) is a subjective scale defining the level of shaking at specific sites on a scale of I to XII. (MMI is expressed in Roman numerals, to connote its approximate nature.) For example, moderate shaking that causes few instances of fallen plaster or cracks in chimneys constitutes MMI VI. It is difficult to find a reliable relationship between magnitude, which is a description of the earthquake’s total energy level, and intensity, which is a subjective description of the level of shaking of the earthquake at specific sites, because shaking severity can vary with building type, design and construction practices, soil type, and distance from the event Table 3-1.

TABLE 3-1 Modified Mercalli Intensity Scale of 1931 Wood and Neumann (1931)

I Not felt except by a very few under especially favorable circumstances
II
Felt only by a few persons at rest, especially on upper floors of buildings. Delicately suspended objects may swing
III

Felt quite noticeably indoors, especially on upper floors of buildings, but many people do not recognize it as an earthquake. Standing motor cars may rock slightly. Vibration like passing track. Duration estimated
IV

During the day felt indoors by many, outdoors by few. At night some awakened. Dishes, windows, and doors disturbed; walls make creaking sound. Sensation like heavy truck striking building. Standing motorcars rock noticeably
V

Felt by nearly everyone; many awakened. Some dishes, windows, etc., broken; a few instances of cracked plaster; unstable objects overturned. Disturbance of trees, poles, and other tall objects sometimes noticed. Pendulum clocks may
stop
VI
Felt by all; many frightened and run outdoors. Some heavy furniture moved; a few instances of fallen plaster or damaged chimneys. Damage slight
VII

Everybody runs outdoors. Damage negligible in buildings of good design and construction slight to moderate in well-built ordinary structures; considerable in poorly built or badly designed structures. Some chimneys broken. Noticed by
persons driving motor cars
VIII

Damage slight in specially designed structures; considerable in ordinary substantial buildings, with partial collapse; great in poorly built structures. Panel walls thrown out of frame structures. Fall of chimneys, factory stacks, columns, monuments, walls. Heavy furniture overturned. Sand and mud ejected in small amounts. Changes in well water. Persons driving motor cars disturbed.
IX

Damage considerable in specially designed structures; well-designed frame structures thrown out of plumb; great in substantial buildings, with partial collapse. Buildings shifted off foundations. Ground cracked conspicuously. Underground pipes broken
X

Some well-built wooden structures destroyed; most masonry and framestructures destroyed with foundations; ground badly cracked. Rails bent. Landslides considerable from river banks and steep slopes. Shifted sand and mud. Water splashed over banks
XI

Few, if any (masonry), structures remain standing. Bridges destroyed. Broad fissures in ground. Underground pipelines completely out of service. Earth slumps and land slips in soft ground. Rails bent greatly
XII

Damage total. Waves seen on ground surfaces. Lines of sight and level distorted. Objects thrown upward into the air

Note that MMI X is the maximum considered physically possible due to ”mere” shaking, and that MMI XI and XII are considered due more to permanent ground deformations and other geologic effects than to shaking.

3-3-4-6-3- Time history:
Time history is one of the earthquakes measurement methods that was not exist until a few years ago. Since the start of using sensitive strong motion seismometers, the actual ground motion recording has become possible. Typically, for many years the acceleration have been recorded by ground motion records, these records are termed accelerograms as in figure (3-10), in analog form on photographic film and, more recently, digitally. Analog records required considerable effort for correction due to instrumental drift, before they could be used. The time history is the sequence of values of any time-varying quantity (such as a ground motion measurement) measured at a set of fixed times. Also termed time series. So that, time histories theoretically contain complete information about the motion at the instrumental location. Time histories (i.e., the earthquake motion at the site) can differ dramatically in duration, frequency content, and amplitude. The maximum amplitude of recorded acceleration is termed the peak ground acceleration, PGA (also termed the ZPA, or zero period acceleration), peak ground velocity (PGV) and peak ground displacement (PGD) are the maximum respective amplitudes of velocity and displacement.

3-3-4-6-4- Elastic Response Spectra:
A response spectrum is simply a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series of oscillators of varying natural frequency that are forced into motion by the same base vibration or shock. The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation. One such use is in assessing the peak response of buildings to earthquakes. Response spectra can also be used in assessing the response of linear systems with multiple modes of oscillation (multi-degree of freedom systems), although they are only accurate for low levels of damping. Modal analysis is performed to identify the modes, and the response in that mode can be picked from the response spectrum. The science of strong ground motion may use some values from the ground response spectrum (calculated from recordings of surface ground motion from seismographs) for correlation with seismic damage. If the input used in calculating a response spectrum is steady-state periodic, then the steady-state result is recorded. If a single degree-of-freedom (SDOF) mass is subjected to a time history of ground (i.e., base) motion similar to that shown in Figure (3-10), the mass or elastic structural response can be readily calculated as a function of time, generating a structural response time history, as shown in Figure (3-11) for several oscillators with differing natural periods.

Figure 3-11. Computation of deformation (or displacement) response spectrum Chopra, A.K. (1981).
3-3-4-6-5- Inelastic Response Spectra:
The method used for computing the inelastic response spectra follows the methods used for computing an elastic response spectrum with the addition of an iterative approach to achieve the target ductility factor and provision to allow for the hysteretic and strength degradation behaviour of the system. Although the structures are designed to remain elastic under ground motion but they behave inelasticity. So, it was important to study its inelastic response. The inelastical behaviour can be defined by the ductility factor, µ:
µ = um / uy 3-8
Where (um) is the actual displacement of the mass under actual ground motions and (uy) is the displacement at yield.

3-3-4-7- Effects of earthquake:
The most terrifying thing about earthquakes is that they occur without warning. This is what makes it so difficult to protect people and property from destruction. Most earthquakes last less than a minute, but they can bring down entire cities and kill thousands in a matter of a few moments. The tremors during an earthquake can make buildings collapse. They can twist railway tracks, destroy bridges, open up cracks in the ground and damage dams. They can start up fires and cause floods and landslides. The collapse of buildings is usually the cause of death and injuries, though floods and fires (caused by earthquakes) have also been known to cause great human suffering. The structural damage is the most dangerous effect of earthquake due to the horrible disasters resulting about it. One of the most important earthquake effects is:
i- Structural damages.

3-3-4-7-1- Structural damages Chen, W. F. and Lui, E. M. 2005:
Structural damage is one of the most important consequences of the earthquake. People can be trapped in collapsed buildings. This is the type of damage that leads to the worst casualties. The worst thing to do in a quake is to rush out into the street during the quake. The danger from being hit by falling glass and debris is many times greater in front of the building than inside. Table 3-2 shows categories of damage, these categories guide engineers for predicting how structure will behave during earthquakes. Therefore, structures must to be designed to have sufficient ductility to survive an earthquake. This means that elements will yield and deform but they will be strong in shear and continue to support their load during and after the earthquake.

TABLE 3-2 Categories of Structural Damage (Chen, W. F. and Lui, E. M. 2005)
Damage state Functionality Repairs required Expected outage

None (preyield) (1) No loss None None

Minor/slight (2) Slight loss Inspect, adjust, patch

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