Earthquakes have wreaked destruction since oldest antiquity, and it is only in the last 50 years that our knowledge of earthquakes and of their impact on buildings has resulted in the design of earthquakeresistant structures. These are built with particularly strong lateral bracing systems capable of resisting the jerking forces of an earthquake. Even so, the number of quake victims is still high all over the world. When 27,000 people died in the Guatemala earthquake of 1967, we thought we had seen the worst, but when 242,000 people died in an earthquake later in the same year in the region north of Peking. The Earth's crust floats over a core of molten rock and some of its parts have a tendency to move with respect to one another. This movement creates stresses in the crust, which may break out along fractures called faults. The break occurs through a sudden sliding motion in the direction of the fault and jerks the buildings in the area. Since the dynamic impact forces due to this jerky motion are mostly horizontal, they can be resisted by the same kind of bracing used against wind.
Earthquakes have wreaked destruction since oldest antiquity, and it is only in the last 50 years that our knowledge of earthquakes and of their impact on buildings has resulted in the design of earthquakeresistant structures. These are built with particularly strong lateral bracing systems capable of resisting the jerking forces of an earthquake. Even so, the number of quake victims is still high all over the world. When 27,000 people died in the Guatemala earthquake of 1967, we thought we had seen the worst, but when 242,000 people died in an earthquake later in the same year in the region north of Peking. The Earth's crust floats over a core of molten rock and some of its parts have a tendency to move with respect to one another. This movement creates stresses in the crust, which may break out along fractures called faults. The break occurs through a sudden sliding motion in the direction of the fault and jerks the buildings in the area. Since the dynamic impact forces due to this jerky motion are mostly horizontal, they can be resisted by the same kind of bracing used against wind.
Earthquake strengths are evaluated on scales like the Richter scale, which measures the magnitude of the energy in the earthquake. For example, an earthquake measuring 4 or 5 on the Richter scale does little damage to wellbuilt buildings, while one measuring 8 or above collapses buildings and may cause many deaths. Not all parts of the earth are subjected to earthquakes, but there are two wide zones on the Earth's surface where the worst earthquakes take place. One follows a line through the Mediterranean, Asia Minor, the Himalayas, and the East Indies, and the other the western, northern, and eastern shores of the Pacific.
Earthquakes are catastrophic events that occur mostly at the boundaries of portions of the Earth's crust called tectonic plates. When movement occurs in these regions, along faults, waves are generated at the Earth's surface that can produce very destructive effects.
Aftershocks are smaller quakes that occur after all large earthquakes. They are usually most intense in size and number within the first week of the original quake. They can cause very significant reshaking of damaged structures, which makes earthquakeinduced disasters more hazardous. A number of moderate quakes (6+ magnitude) have had aftershocks that were very similar in size to the original quake. Aftershocks diminish in intensity and number with time. They generally follow a pattern of there being at least 1 large (within 1 Richter magnitude) aftershock, at least 10 lesser (within 2 Richter magnitude) aftershocks, 100 within 3, and so on. The Loma Prieta earthquake had many aftershocks, but the largest was only magnitude 5.0 with the original quake being magnitude 7.1.
Some of the most destructive effects caused by earthquake shaking are those that produce lateral loads in a structure. The input shaking causes the foundation of a building to oscillate back and forth in a more or less horizontal plane. The building mass has inertia and wants to remain where it is, and therefore, lateral forces are exerted on the mass in order to bring it along with the foundation. For analysis purposes, this dynamic action is simplified as a group of horizontal forces that are applied to the structure in proportion to its mass and to the height of the mass above the ground. In multistory buildings with floors of equal weight, the loading is further simplified as a group of loads, each being applied at a floor line and each being greater than the one below in a triangular distribution (see Figure 3.1). Seismically resistant structures are designed to resist these lateral forces through inelastic and primarily for economic reason and must, therefore, be detailed accordingly. These loads are often expressed in terms of a percent of gravity weight of the building and can vary from a few percent to near 50% of gravity weight. There are also vertical loads generated in a structure by earthquake shaking, but these forces rarely overload the verticalloadresisting system. However, earthquakeinduced vertical forces have caused damage to structures with high dead load compared to design live load. These vertical forces also increase the chance of collapse due to either increased or decreased compression forces in the columns. Increased compression overloads columns, while decreased compression reduces column bending strength.
Figure 3.1 Lateral loads due to earthquakes.
In earthquake engineering, we deal with random variables, and therefore, the design must be treated differently from the orthodox design. The orthodox viewpoint maintains that the objective of design is to prevent failure; it idealizes variables as deterministic. This simple approach is still valid, applied to design under only mild uncertainty. But when confronted with the effects of earthquakes, this orthodox viewpoint seems so over trustful as to be worthless. In dealing with earthquakes, we must contend with appreciable probabilities that failure will occur in the near future. Otherwise, all the wealth of this world would prove insufficient to fill our needs: the most modest structures would be fortresses. We must also face uncertainty on a large scale while designing engineering systems—whose pertinent properties are still debated to resist future earthquakes—about whose characteristics we know even less.
Although over the years, experience and research have diminished our uncertainties and concerns regarding the characteristics of earthquake motions and manifestations, it is unlikely, though, that there will be such a change in the nature of knowledge to relieve us of the necessity of dealing openly with random variables. In a way, earthquake engineering is a parody of other branches of engineering. Earthquake effects on structures systematically bring out the mistakes made in design and construction, even the minutest mistakes. Add to this the undeniable dynamic nature of disturbances, the importance of soil–structure interaction, and the extremely random nature of it all; in a manner of speaking, earthquake engineering is to the rest of the engineering disciplines what psychiatry is to other branches of medicine. This aspect of earthquake engineering makes it challenging and fascinating and gives it an educational value beyond its immediate objectives. If structural engineers are to acquire fruitful experience in a brief span of time, expose them to the concepts of earthquake engineering, even if their interest in earthquakeresistant design is indirect. Sooner or later, they will learn that the difficulties encountered in seismic design are technically intriguing and begin to exercise that nebulous trait called engineering judgment to make allowance for these unknown factors.
To understand the seismic behavior of buildings, it is helpful to study strongmotion seismograms (also called time histories). The familiar wiggly line graphic records shown in Figure 3.2 are not the actual motion of the ground but have been filtered in some way by both the recording instrument and by the agency providing the data. In most cases, however, for practical applications, the engineer need not be concerned about the difference.
Modern instruments capable of recording large motions strategically placed in structures provide information on the structural response. In this case, it is evident that there is amplification of both shortperiod and longperiod motions in the upper floors. This effect is reflected in seismic design by applying larger loads up the building height.
Figure 3.2 Strongmotion seismogram accelegram from El Centro earthquake, May 18, 1940 (NS Component).
Until the 1990s, seismic building codes used a single map of the United States that divided the country into numbered seismic zones (0, 1, 2, 3, 4) in which each zone was assigned a single acceleration value in %g, which was used to determine seismic loads on the structure.
Starting in the 1970s, new hazard maps began to be developed on a probabilistic basis. In the 1994 National Earthquake Hazards Reduction Program (NEHRP) Recommended Provisions, two maps of the United States were provided, showing effective peak acceleration coefficients and effective peak velocity–related coefficients by use of contour lines that designate regions of equal value. The ground motions were based on estimated probabilities of 10% of exceedance in various exposure times (50, 100, and 250 years).
The probabilistic analysis is typically represented in maps in the form of a percentage probability of exceedance in a specified number of years. For example, commonly used probabilities are a 10% probability of exceedance in 50 years (a return period of about 475 years) and 2% probability of exceedance in 50 years (a return period of about 2,500 years). These maps show ground motions that may be equaled but are not expected to be exceeded in the next 50 years: the odds that they will not be exceeded are 90% and 98%, respectively.
Seismic hazard probability maps are produced by the United States Geological Survey (USGS). The latest sets of USGS of maps provide a variety of maps for peak ground acceleration and spectral acceleration, with explanatory material, and are available on the USGS website.
The USGS map is a probabilistic representation of hazard for the contiguous United States. This shows the spectral acceleration in %g with a 2% probability of exceedance in 50 years: this degree of probability is the basis of the maps used in the building codes.
The return period of 1 in 2,500 years may seem very infrequent, but this is a statistical value, not a prediction, so some earthquakes will occur much sooner and some much later. The design dilemma is that if a more frequent earthquake—for example, the return period of 475 years—was used in the lower seismic regions, the difference between the high and lowprobability earthquakes is a ratio of between 2 and 5. Design for the highprobability earthquake would be largely ineffective when the lowprobability event occurred.
In practical terms, the building designer must assume that the large earthquake may occur at any time. Thus, use of the 2,500 return period earthquakes in the lower seismic regions ensures protection against rare earthquakes, such as the recurrence of the 1811–1812 earthquake sequence in New Madrid, Missouri, or the 1898 Charleston, South Carolina, earthquake. The selection of 2% in 50year likelihood as the maximum considered earthquake (MCE) ground motion is believed to result in acceptable levels of seismic safety for the nation.
The acceleration experienced by a building will vary depending on the period of the building, and in general, shortperiod buildings will experience more accelerations than longperiod buildings. The USGS maps recognize this phenomenon by providing acceleration values for periods of 0.2 s (short) and 1.0 s (long). These are referred to as spectral acceleration, and the values are approximately what are experienced by a building (as distinct from the peak acceleration that is experienced at the ground). The spectral acceleration is usually considerably more than the peak ground accelerations.
The USGS maps are based on MCE ground motion—the most severe earthquake considered in the US seismic standards. They are based on a 2% probability of occurrence in 50 years.
These USGS probability maps provide the basis for the maps used in building codes that provide design values for spectral acceleration used by structural engineers to calculate the seismic forces on a structure. These design value maps differ by use of an MCE for the regions. For most regions of the country, the MCE is defined as ground motion with a uniform likelihood of exceedance of 2% in 50 years (a return period of about 2,500 years) and is identical to the USGS probability maps. However, in regions of high seismicity, such as coastal California, the seismic hazard is typically controlled by largemagnitude events occurring on a limited number of welldefined fault systems. For these regions, rather than using the 2% in 50year likelihood, it is considered more appropriate to directly determine the MCE ground motions based on the characteristic earthquakes of those defined faults.
It is to be noted that the acceleration values shown on the maps are not used directly for design. Instead, they are reduced by twothirds of this value to determine the design earthquake (DE) and are the values used by engineers for design. The reason for this is that it is believed by engineers that the design provisions contain at least a margin of 1.5 against structural failure. MCE is inferred to provide collapse prevention level, while the actual design is done using the DE, which is 2/3 MCE for codelevel, lifesafety protection level. This belief is the result of the study of the performance of many types of buildings in earthquakes, mostly in California.
The building response to earthquake shaking occurs over the time of a few seconds. During this time, several types of seismic waves are combining to shake the building in ways that are different in detail for each earthquake. In addition, as the result of variations in fault slippage, differing rock through which the waves pass, and the different geological nature of each site, the resultant shaking at each site is different. The characteristics of each building are different, whether in size, configuration, material, structural system, age, or quality of construction: each of these characteristics affects the building response.
In spite of the complexity of the interactions between the building and the ground during the few seconds of shaking, there is broad understanding of how different building types will perform under different shaking conditions. This understanding comes mainly from extensive observation of buildings in earthquakes all over the world and to a lesser extent from analytical and experimental research.
Understanding the ground and building characteristics discussed in this chapter is essential to give designers a feel for how their building will react to shaking, which is necessary to guide the conceptual design of their building.
In this chapter, we
The seismic waves create internal forces within the building. Inertial forces are generated when an outside force tries to make the building move if it is at rest or change its rate or direction of motion if it is moving.
Figure 3.3 Analytical model for a freestanding water tower. (a) Water tower and (b) equivalent SDOF cantilever.
Consider a freestanding water tower shown in Figure 3.3a subjected to earthquake ground motions. A simplified analytical model for the tower may be represented by a cantilever column with a concentrated mass M at top, as shown in Figure 3.3b. When the base of the cantilever is subjected to sudden ground motion, the initial tendency for the water tower, that is, the mass M, is to stay put. The shifting of the ground is too rapid for the tower to keep up.
After a moment, the tower accelerates laterally to catch up with the movement of the ground. From Newton's second law of motion, we can surmise that the equivalent lateral force (ELF) F at the top is equal to the mass M of the tower and the acceleration at the base. Thus,
where
This part of Newton's law explains why light buildings, such as woodframe houses, tend to perform better in earthquakes than large heavy ones.
The acceleration, or the rate of change of the velocity of the seismic waves setting the building in motion, determines the percentage of the building mass or weight that must be dealt with as an equivalent horizontal force.
Acceleration is measured in terms of the acceleration due to gravity or g. One g is the rate of change of velocity of a freefalling body in space. This is an additive velocity of 32 ft/s. Thus, at the end of the first second, the velocity is 32 ft/s; a second later, it is 64 ft/s, and so on. When parachutists or bungee jumpers are in the free fall, they are experiencing an acceleration of 1g. While the rollercoaster riders reach as much as 4g. The aerobatic pilots are undergoing about 9g. The human body is very sensitive and can feel accelerations as small as 0.001g, such as when you shake hands with another person. A building in an earthquake experiences for a fraction of a second very high forces in one direction before they abruptly change direction. Poorly constructed buildings begin to suffer damage at about 10% g (or 0.1g). In a moderate earthquake, vibration may last for a few seconds, and accelerations may be approximately 0.2g. Short accelerations may, for a fraction of a second, exceed 1.0g. In the Northridge earthquake in 1994, a recording station in Tarzana, 5 miles from the epicenter, recorded 1.92g.
Acceleration is a key factor in determining the forces on a building, but a more significant measure is that of acceleration combined with duration, which takes into account the impact of earthquake forces over time. In general, a number of cycles of moderate acceleration, sustained over time, can be much more difficult for a building to withstand than a single much larger peak. Continued shaking weakens a building structure and reduces its resistance to earthquake damage.
The duration of strong motion, termed the bracketed duration, is measured above a certain threshold acceleration value, commonly taken as 0.05g, and is defined as the time between the first and last peaks of motion that exceeds this threshold value. In the San Fernando earthquake of 1971, the bracketed duration was only about 6 s. In both the Loma Prieta and the Northridge earthquakes, the strong motion lasted a little over 10 s yet caused much destruction. In the 1906 San Francisco earthquake, the severe shaking lasted 45 s, while in Alaska, in 1964, the severe motion lasted for over 3 min.
Two other measures of wave motion are directly related to acceleration and can be mathematically derived from it. Velocity, which is measured in inches or centimeters per second, refers to the rate of motion of the seismic waves as they travel through the earth. This is very fast. Typically the P wave travels at between 3 and 8 km/s or 7,000–18,000 mph. The S wave is slower, traveling at between 2 and 5 km/s or 4,000–11,000 mph.
Displacement refers to the distance that points on the ground are moved from their initial locations by the seismic waves. These distances, except immediately adjacent to or over the fault rupture, are quite small and are measured in inches or centimeters. For example, in the Northridge earthquake, parking structures at Burbank, about 18 miles (29 km) from the epicenter, recorded displacements at the roof of 1.6 in. (4.0 cm) at an acceleration of 0.47g. In the same earthquake, the Olive View hospital in Sylmar, about 7.5 miles (12 km) from the epicenter, recorded a roof displacement of 13.5 in. (34 cm) at an acceleration of 1.5g.
The velocity of motion on the ground caused by seismic waves is quite slow—huge quantities of earth and rock are being moved. The velocity varies from about 2 cm/s in a small earthquake to about 60 cm/s in a major shake. Thus, typical building motion is slow and the displacements are small, but because thousands of tons of steel and concrete are wrenched in all directions several times a second, building failure or severe damage is likely to occur.
In earthquakes, the values of ground displacement, velocity, and acceleration (DVA) vary a great deal in relation to the frequency of the wave motion. Highfrequency waves (higher than 10 Hz) tend to have high amplitudes of acceleration but small amplitudes of displacement, compared to lowfrequency waves, which have small accelerations and relatively large velocities and displacements.
Earthquake shaking is initiated by a fault slippage in the underlying rock. As the shaking propagates to the surface, it may be amplified, depending on the intensity of shaking, the nature of the rock, and the surface soil type and depth.
A layer of soft soil, measuring from a few feet to a hundred feet or so, may result in an amplification factor ranging from 1.5 to 6 over the rock shaking. This amplification is most pronounced at longer periods and may not be so significant at short periods. The amplification also tends to decrease as the level of shaking increases.
As a result, earthquake damage tends to be more severe in areas of soft ground. This characteristic became very clear when the effects of 1906 San Francisco earthquake were studied. Also, inspection of records from soft clay sites during the 1989 Loma Prieta earthquake indicated a maximum amplification of longperiod shaking of three to six times. In this earthquake, extensive damage was caused to buildings in San Francisco's Marina District, which was largely built on filled ground, some of it rubble deposited after the 1906 earthquake.
Because of the possibility of considerable shaking amplification related to the nature of the ground, seismic codes have some very specific requirements that relate to the characteristics of the site. These require the structure to be designed for higher force levels if it is located on poor soil. Specially designed foundations may also be necessary.
All buildings have a natural or fundamental period; this is the rate at which they will move back and forth if they are given a horizontal push (Figure 3.4). In fact, without pulling and pushing it back and forth, it is not possible to make an object vibrate at anything other than its natural period.
Another characteristic of importance in seismic design is the period of frequency of earthquake waves. Whether the waves are quick and abrupt or slow and rolling is particularly important for determining building seismic forces.
When a building's motion is started with a seismic push, to be effective, this shove must be as close as possible to the natural period of the building. When earthquake motion starts a building vibrating, it will tend to sway back and forth at its natural period.
Periods are the time in seconds (or fractions of a second) that is needed to complete one cycle of a seismic wave. Frequency is the inverse of this—the number of cycles that will occur in a second— and is measured in hertz. One hertz is one cycle per second.
Natural periods vary from about 0.05 s for a piece of equipment, such as a filing cabinet, to about 0.1 s for a onestory building. Period is the inverse of frequency, so the cabinet will vibrate at 1 divided by 0.05 = 20 cycles a second or 20 Hz.
A fourstory building will sway at about 0.5 s period, and taller buildings between about 10 and 20 stories will swing at periods of about 1–2 s.
A rule of thumb for preliminary design is that the building period equals the number of stories divided by 10; therefore, period is primarily a function of building height. The 60story Citicorp office building in New York has a measured period of 7 s; give it a push, and it wall sway slowly back and forth completing a cycle every 7 s. Other factors, such as the building's structural system, its construction materials, its content, and its geometric proportions, also affect the period, but height is the most important consideration (Figure 3.4).
Figure 3.4 Effect of building height on period.
The building's period may also be changed by earthquake damage, which has the effect of increasing the structure's period of vibration: the structure is softening. This may result in the structure's period approaching that of the ground and experiencing resonance, which may prove fatal to an already weakened structure.
In structural dynamics, resonance is defined as the tendency of the system to oscillate with larger amplitude at some frequencies than at others. It should be noted that frequency, typically denoted as ω, is the reciprocal of the period T. Thus,
At resonant frequencies, even small periodic driving forces can produce large amplitude oscillations.
A familiar example of resonance is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) will make the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo will result in smaller arcs. This is because the energy the swings absorb is maximized when the pushes are in phase with the swing oscillations, while some of the swing's energy is actually extracted by the opposing force of the pushes when they are not.
Avoiding resonance disasters is a major concern in every building, tower, and bridge construction project. As a countermeasure, shock mounts can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 730ton pendulum—a tuned mass damper—to cancel resonance. Buildings in seismic zones are often constructed to take into account the oscillating frequencies of expected ground motion.
It is of interest to note that resonance was first recognized by Galileo Galilei with his investigations of pendulums and musical strings beginning in 1602.
It is reassuring to know that in practice, exact resonance does not really occur, because buildings never completely respond as linear elastic systems. As distortions become large, the characteristics of the building change because of nonlinear response due to plastic deformation. Therefore, the concern as to whether building displacements become infinite is of course of academic interest only. Furthermore, the maximum amplitude is attained not after just a few cycles but many cycles of vibration. The important engineering conclusion is that, at or near resonance, the deflection of the building is likely to become very large and hence problematic.
When a vibrating building is given further pushes that are also at its natural period, its vibrations increase dramatically in response to even rather small pushes, and in fact, its accelerations may increase as much as four to five times.
Perhaps it is hard to imagine, but the ground obeys the same physical law and also vibrates at its natural period, if set in motion by an earthquake. The natural period of ground varies from about 0.4 to 2 s, depending on the nature of the ground. Hard ground or rock will experience shortperiod vibration. Very soft ground may have a period of up to 2 s, but unlike a structure, it cannot sustain longerperiod motions except under certain unusual conditions. Since this range is well within the range of common building periods, it is quite possible that the pushes that earthquake ground motion impacts to the building will be the natural period of the building. This may create resonance, causing the structure to encounter accelerations of perhaps 1g when the ground is only vibrating with accelerations of 0.2g. Because of this, buildings suffer the greatest damage from ground motion at a frequency close or equal to their own natural frequency.
Figure 3.5 Vibration modes.
The terrible destruction in Mexico City in the earthquake of 1985 was primarily the result of response amplification caused by coincidence of building and ground motion periods. Mexico City was some 250 miles from the earthquake focus, and the earthquake caused the soft ground in margins of the old lake bed under the downtown buildings to vibrate for over 90 s at its long natural period of around 2 s. This caused buildings that were between about 6 and 20 stories in height to resonate at a similar period, greatly increasing the accelerations within them. Taller buildings suffered little damage. This amplification in building vibration is very undesirable. The possibility of it happening can be reduced although not always, by trying to ensure that the building period will not coincide with that of the ground. Thus, on soft (longperiod) ground, it would be best to design a short, stiff (shortperiod) building.
Taller buildings also will undergo several modes of vibration so that the building will wiggle back and forth like a snake (Figure 3.5).
However, higher modes of vibration are generally less critical than the natural period, although they may be significant in a highrise building. For lowrise buildings, the natural period (which, for common structures, will always be relatively short) is the most significant. Note, however, that the lowperiod, lowtomidrise building is more likely to experience resonance from the more common shortperiod ground motion.
From the preceding discussion, it is evident that buildings with different periods (or frequency responses) will respond in widely differing ways to the same earthquake ground motion. Conversely, any given building will act differently during different earthquakes, so for design purposes, it is necessary to represent the building's range of response to ground motion of different frequency content. Such a representation is termed a site response spectrum. A site response spectrum is a graph that plots the maximum response values of DVA against period (and frequency).
Figure 3.6 shows a simplified version of a response spectrum. These spectra show, on the vertical ordinate, the DVA that may be expected at varying periods (the horizontal ordinate). Thus, the response spectrum illustrated shows a maximum acceleration response at a period of about 0.3 s—the fundamental period of a midrise building. This shows how building response varies with building period: as the periods lengthen, accelerations decrease and displacement increases. On the other hand, one or twostory buildings with short periods undergo higher accelerations but smaller displacements.
In general, a taller, more flexible building may be expected to experience proportionately lesser accelerations than a stiffer lowrise building. A glance at a response spectrum will show why this is so: as the period of the building lengthens (moving toward the right of the horizontal axis of the spectrum), the accelerations reduce. However, there's a tradeoff, in that the lower accelerations in taller buildings come at the expense of more displacement. This increased displacement may be such that the building may suffer considerable damage to its nonstructural components, such as ceilings and partitions, in even a modest earthquake.
Figure 3.6 Simplified response spectra for acceleration, velocity, and displacement.
The response spectrum enables the engineer to identify the resonant frequencies at which the building will undergo peak accelerations. Based on this knowledge, the building design might be adjusted to ensure that the building period does not coincide with the site period of maximum response. This is perhaps easier said than done for a building but is quite possible for a nonbuilding structure such as a flagpole shown in Figure 3.7.
Figure 3.7 Turning response of a flagpole: these changes shorten its period.
If a building is made to vibrate, the amplitude of the vibration will decay over time and eventually cease. Damping is a measure of this decay in amplitude, and it is principally due to internal friction and absorbed energy. The nature of the structure and its connections affects the damping; a heavy concrete structure will provide more damping than a light steel frame.
Architectural features such as partitions and exterior façade construction also contribute to the damping.
Damping is measured by reference to a theoretical damping level termed critical damping. This is the least amount of damping that will allow the structure to return to its original position without any continued vibration. For most structures, the amount of damping in the system will vary between 3% and 10% of critical. The higher values would apply to older buildings (such as offices and government buildings) that employed a structure of steel columns and beams encased in concrete together with some structural walls, which also had many heavy fixed partitions (often concrete block or hollow tiles), and would have high damping values. The lower values would apply to a modern office building with a steel moment frame, a light metal and glass exterior envelope, and open office layouts with a minimum of fixed partitions.
The main significance of damping is that accelerations generated by ground motion decrease rapidly as the damping value increases. The response spectra in Figure 3.8 show that the peak acceleration is about 3.2g for a damping value of 0%, 0.8g for a damping value of 5%, and 0.65g for a damping value of 10%.
Response spectra generally show acceleration values for 5% damping. A damping value of zero might be used in the design of a simple vibrator, such as a flagpole or a water tank supported on a single cantilever column. For typical structures, engineers generally use a value of 5% critical.
Damping used to be regarded as a fixed attribute of buildings, but in recent years, a number of devices have been produced that enable the engineer to increase the damping and reduce the building response. This greatly increases the designer's ability to provide a tuned response to the ground motion.
Figure 3.8 Response spectra for a number of damping values.
This is the property of certain materials to fail only after considerable inelastic deformation has taken place, meaning that the material does not return to its original shape after distortion. This deformation, or distortion, dissipates earthquake energy.
This is why it is much more difficult to break a metal spoon by bending it than one made of plastic. The metal spoon will remain intact, through distorted, after successive bending to and fro, while the plastic spoon will snap suddenly after a few bends. The metal is far more ductile than the plastic.
The plastic deformation of the metal absorbs energy and defers absolute failure of the structure. The material bends but does not break and so continues to resist forces and support loads, although with diminished effectiveness. The effect of earthquake motion on a building is rather like that of bending a spoon rapidly back and forth: the heavy structures are pushed back and forth in a similar way several times a second (depending on its period of vibration).
Brittle materials, such as unreinforced masonry or inadequately reinforced concrete, fail suddenly, with a minimum of prior distortion. However, the steel bars embedded in reinforced concrete with heavier and more closely spaced ties and special detailing of their placement can give this material considerable ductility equal to that of structural steel.
Ductility and reserve capacity are closely related: past the elastic limit (the point at which forces cause permanent deformation); ductile materials can take further loading before complete failure.
In addition, the member proportions, end conditions, and connection details will also affect ductility. Reserve capacity is the ability of a structure to resist overload and is dependent on the ductility of its individual members and connections. The only reason for not requiring ductility is to provide so much resistance that members would never be pushed beyond elastic limits.
Thus, buildings are designed in such a way that in the rare case when they are subjected to forces higher than those required by code, the materials and connections will distort but not snap. In so doing, they will safely absorb the energy of the earthquake vibrations, and the building, although distorted and possibly beyond repair, is at least still standing.
Earthquakes have long been feared as one of nature's most terrifying phenomena. Early in human history, the sudden shaking of the earth and the death and destruction that resulted were seen as mysterious and uncontrollable. We now understand the origin of earthquakes and know that they must be accepted as a natural environmental process. Scientific explanations, however, have not lessened the terrifying nature of the earthquake experience. Earthquakes continue to remind us that nature can, without warning, in a few seconds create a level of death and destruction that can only be equaled by the most extreme weapons of war.
This uncertainly, together with the terrifying sensation of earth movement, creates our fundamental fear of earthquakes. Beyond the threat to life is the possibility of the destruction of public and private property. Jobs, services, and business revenues can disappear instantly, and for many, homelessness can suddenly be very real. The aftermath of a great earthquake can endure for years or even decades.
Other types of phenomena sometimes accompany earthquakecaused ground shaking and are generally identified as geologic hazards. These are the following:
For all the aforementioned geologic hazards, the only truly effective defense is the application of good landuse practices that limit development in hazardprone locations. Seismic design and construction is aimed at reducing the consequences of earthquakecaused ground shaking, which is by far the main cause of damage and casualties.
Earthquakes in the United States are a national problem. Most people now know that earthquakes are not restricted to just a few areas in the United States, most notably California and Alaska. Structural engineers recognize that two of the greatest earthquakes known occurred not in California, but near New Madrid, MO, in 1811 and 1812. As can be seen on a map of earthquake probability in the United States, more than 40 of the 50 states are at risk from earthquakecaused damage, life loss, injuries, and economic impacts (see ASCE 710 for seismic maps). Certainly the likelihood of a damaging earthquake occurring west of the Rocky Mountains, and particularly in California, the states of Oregon and Washington, and Salt Lake City is much greater than it is in the East, Midwest, or South. However, the New Madrid, Missouri, and Charleston, South Carolina, regions are subject to the possibility of severe earthquakes, although with a lesser probability than the Western United States.
There are several common measures of earthquakes. Perhaps the most familiar is the Richter magnitude, devised by Professor Charles Richter of the California Institute of Technology in 1935. Richter scale is based on the maximum amplitude of certain seismic waves recorded on a standard seismograph at a distance of 100 km from the earthquake epicenter. Because the instruments are unlikely to be exactly 100 km from the source, Richter devised a method to allow for the diminishing of wave amplitude with increased distance. The Richter scale is logarithmic, and each unit of magnitude indicates a 10fold increase in wave amplitude. The energy increase represented by each unit of scale is approximately 31 times. The scale is open ended, but a magnitude of about 9.5 represents the largest possible earthquake.
Table 3.1 shows significant earthquakes (magnitude 6 or over) that occurred in 47 of the 50 US states between 1568 and 1989.
Records show that some seismic zones in the United States experience moderatetomajor earthquakes approximately every 50–70 years, while other areas have recurrence intervals for the same size earthquake of about 200–400 years. These frequencies of occurrence are simply statistical probabilities and one or several earthquakes could occur in a much shorter than average period. With current knowledge, there is no practical alternative to assume that a large earthquake is likely to occur at any time and to take appropriate action.
Moderate and even very large earthquakes are inevitable, although very infrequent, in areas of normally low seismicity. Consequently, in these regions, buildings are very seldom designed to deal with an earthquake threat; therefore, they are extremely vulnerable. In other places, however, the earthquake threat is quite familiar. Buildings in many areas of California and Alaska will be shaken by an earthquake perhaps two or three times a year, and some level of earthquakeresistant design has been accepted as a way of life since the early twentieth century.
Although, on a national basis, the areas where earthquakes are likely to occur and the potential size or magnitude of these earthquakes are well identified and scientists have a broad statistical knowledge of the likelihood of their occurrence, it is not yet possible to predict the nearterm occurrence of a damaging earthquake. Therefore, lacking useful predictions, it makes sense in any seismic region to take at least the minimum affordable prudent actions directed at saving lives. Because most lives are lost in earthquakes when buildings collapse, US seismic building code provisions focus on requiring that the minimum measures necessary to prevent building collapse are taken.
I, s 
1/2F(t), ft/s^{2} 
1000y, ft/s^{2} 
y, Eq. ft/s^{2} 
y (Δt)^{2}, ft 
y, ft 

0.0 
25 
0 
25.0 
0.0100 
0 
0.02 
30 
5.0 
25.0 
0.0100 
0.0050* 
0.04 
35 
20.0 
15.0 
0.0060 
0.0200 
0.06 
40 
41.0 
−1.0 
−0.0004 
0.0410 
0.08 
45 
61.6 
−16.6 
−0.0066 
0.0616 
0.10 
50 
75.6 
−25.6 
−0.0102 
0.0756 
0.12 
37.5 
79.4 
−41.9 
−0.0168 
0.0794 
0.14 
25 
66.4 
−41.4 
−0.0166 
0.0664 
0.16 
25 
36.8 
−11.8 
−0.0047 
0.0368 
0.18 
25 
2.5 
22.5 
0.0090 
0.0004 
0.20 
25 
−22.8 
47.8 
0.0191 
−0.0228 
0.22 
25 
−29.0 
54.0 
0.0216 
−0.0290 
0.24 
25 
−13.6 
38.6 
0.0154 
−0.0136 
0.26 
25 
17.2 
7.8 
0.0031 
0.0172 
0.28 
25 
51.1 
−26.1 
−0.0104 
0.0511 
0.30 
25 
74.6 
−49.6 
−0.0198 
0.0746 
0.32 
25 
78.3 
−53.3 
−0.0123 
0.0783 
0.34 
— 
— 
— 
— 
0.0607 
Until quite recently, the United States was divided into a number of seismic zones, which were shown on the maps in the model codes. Zones ranged from Zone 0 (indicating no seismicity) to Zones 1, 2A, 2B, 3, and 4. Zone 4 indicates the highest level of seismicity (see Figure 3.9). Each zone was allocated a factor, or coefficient, from 0.075 to 0.40; this value was a multiplier representing the acceleration value for which the building was to be designed. These values indicate a fourfold range in acceleration values between Zones 1 and 4. Within a zone, all buildings must be designed to the same acceleration value.
Current seismic standards, such as the ASCE 710, define site seismicity in a different way. The United States is still divided into zones by contour lines, but their areas are much smaller. Numerical values are also shown on the maps and also represent the acceleration value to be used for design, but they are calculated in a different way, and many more values are shown that reflect greater precision of knowledge. Also, acceleration values for both long and shortperiod buildings are shown in a separate series of maps. The simplicity of the old seismic zones is lost, but the design information is much more detailed.
Seismic performance in current codes is specified by selecting a maximum tolerable damage level for a given earthquakeshaking intensity. The shaking intensity can be specified probabilistically, derived by considering all future potential shaking at the site regardless of the causative fault, or deterministically, giving the expected shaking at the site for a given sized earthquake on a given fault.
Figure 3.9 1997 UBC seismic zone map of the United States. The map is based on 10% probability of exceedance in 50 years.
For some time, the earthquake shaking used by building codes for a new building has been described probabilistically, as shaking with a 10% chance of being exceeded in a 50year time period (50 years being judged as the average life of buildings). This can also be specified, similar to methods used with storms or floods, as the shaking with a return period of 475 years. (Actually, for ease of use, the return period is often rounded to 500 years, and since actual earthquake events are more understandable than probabilistic shaking, the most common term, although slightly inaccurate, is the 500year event.)
Nationally applicable building codes were therefore based on the level of shaking intensity expected at any site once every 500 years (on average). However, engineers in several areas of the country, most notably Salt Lake City, Utah; Charleston, South Carolina; and Memphis, Tennessee, felt that this standard was not providing sufficient safety in their regions because very rare, exceptionally large earthquakes could occur in those areas, producing shaking intensities several times that of the 500year event. Should such a rare earthquake occur, the building code design would not provide the same level of protection provided in areas of high seismicity, particularly California, because rare, exceptionally large shaking in California is estimated to be only marginally larger (about 1.5 times) than the 500year shaking. It was therefore decided to determine the national mapping parameters on a much longer return period—one that would capture the rare events in the regions at issue, and a 2.500year event was chosen (known as the MCE). Finally, it was judged unnecessary, and in fact undesirable, to significantly change seismic design practices in California, so the MCE was multiplied by 2/3 to make California design shaking levels about the same as before.
If the new shaking level—about 1.5 times the old—were multiplied by 2/3, the final design parameter would not change. However, in a region where the MCE is three times the previously used 500year event, the new parameter of 2/3 MCE would result in a shaking level twice that previously used—providing the soughtafter additional level of safety in those regions. Currently, national standards such as the ASCE define the level of shaking to be considered for evaluation of existing buildings to be 2/3 MCE, which, as previously explained, is about the same as the 500year event for much of California.
Probabilistic seismic hazard analysis (PSHA) provides an estimate of the likelihood of hazard from earthquakes based on geological and seismological studies. It is probabilistic in the sense that the analysis takes into consideration the uncertainties in the size and location of earthquakes and the resulting ground motions that could affect a particular site. Seismic hazard is commonly described as the probability of occurrence of some particular earthquake characteristic (such as peak ground acceleration). For statistical reasons, these probabilities cover a range of values, and because risk involves values being greater than expected, the word exceedance has been coined in their usage.
The effects of ground shaking on building response are well known and extensively documented. It is also acknowledged that severe ground shaking can significantly damage buildings designed in accordance with seismic codes and can even cause partial collapse of buildings with inadequate seismic resistance.
Seismic shaking is typically quantified using a parameter of motion, such as DVA. In current seismic codes, seismic design forces are defined in terms that relate to acceleration in the horizontal direction.
The earthquake groundshaking hazard for a given region or site can be determined in two ways: deterministically or probabilistically. A deterministic hazard assessment estimates the level of shaking, including the uncertainty in the assessment, at the building site for a selected earthquake scenario. Typically, the earthquake is selected as the maximummagnitude earthquake considered to be capable of occurring on an identified active earthquake fault; this maximummagnitude earthquake is termed a characteristic earthquake. A deterministic analysis is often made when there is a welldefined active fault for which there is a sufficiently high probability of a characteristic earthquake occurring during the life of the building. The known past occurrence of such an earthquake, or geologic evidence of the periodic occurrence of such earthquakes in the past, is often considered to be indicative of a high probability for a future repeat occurrence of the event.
Probabilistic hazard assessment expresses the level of ground shaking with a specific, low probability of being exceeded in a selected time period, for example, 10% probability of being exceeded in 50 years or 2% probability of being exceeded in 50 years, where 50 years is commonly chosen as the building design life. The seismic loading criteria in current US building codes define design force levels based on ground motions specified in probabilistic seismic hazard maps. Such maps show expected peak ground acceleration response at different building periods of vibration S_{s} and S_{1} defined elsewhere in this text.
These maps indicate that, although the level of earthquake activity is high in California, most parts of the United States are also exposed to a significant earthquake groundshaking hazard. In fact, large historic earthquakes in the United States have occurred outside California, in Missouri, Arkansas, South Carolina, Nevada, Idaho, Montana, Washington, Alaska, and Hawaii. Furthermore, current geologic studies have shown increasing evidence for large earthquake potential in areas that are popularly believed to be relatively quiet. Examples include the nowrecognized subduction zones in Oregon and Washington, the Wasatch fault zone in Utah, and the Wabash Valley seismic zone in Illinois and Indiana.
Decisions to mitigate seismic risk require a logical and consistent approach to evaluate the effects of future earthquakes on people and structures. One method to achieve this logic and consistency is the PSHA, which gives a probabilistic description (a frequency of exceedance) of earthquake characteristics such as ground motion amplitudes and fault displacement.
There is a distinction between earthquake damage and loss: damage refers to physical effects, such as the effect on structures. Loss is the associated monetary or social consequence.
Structural engineers can estimate damage, but estimating loss may involve considering additional factors like inflated costs of labor and materials after a major earthquake (which is called demand surge) or insurance deductibles. The destruction of a building's contents and the interruption of business are additional losses that might occur. Buildings suffer damage during an earthquake, but the owner (or perhaps the insurer) incurs the loss. Confusion occurs between these terms because both are often quantified as a percentage of the replacement value of the structure.
The term exposure time is often used in estimating natural hazards. It is used, for example, in defining the ground motion with a 2.475year return period as the motion “that will be exceeded with a 2% probability during an exposure time of 50 years, “with 50 years being the nominal lifetime of major civil structures.
A 10% probability of exceedance in 50 years corresponds to a 475year return period, and the question sometimes arises, “What is special about the 475year return period?” This period is derived by assuming a Poisson process for ground motion occurrences, wherein the probability of an event, P, is related to the annual frequency of exceedance of the ground motion Y and the exposure time t through
Rearranging this gives
Substituting a probability P = 0.1 and an exposure time t = 50 years gives y = 0.002107 per year, which is 1/475 years.
Whereby using PSHA, a geotechnical engineer determines and integrates contributions to the probability of exceedance of a ground motion level from all earthquake faults and magnitudes that could produce potentially damaging ground shaking at the site. Using the results from PSHA, ground motions can be readily obtained for any selected probability of exceedance and building design life.
For applications in performancebased design discussed in Chapter 14, both a probabilistic approach and a deterministic approach for the groundshaking hazard assessment may be used. Using a probabilistic approach, the seismic hazard can be integrated with the building resistance characteristics to estimate the probability of exceeding some level of damage during a time period of significance such as the anticipated building life. Using a deterministic approach, the probability of exceeding a specified damage level may be assessed for an earthquake likely to occur during the anticipated building life.
These are several performance descriptions that are currently available. However, it is believed that those given in ASCE 710 cover the full range of performance criteria typically used in seismic evaluation. These are the following:
For many decades, seismic building codes focused exclusively on the structure of the building, that is, the system of columns, beams, walls, and diaphragms that provides resistance against earthquake forces. Although this focus remains dominant for obvious reasons, experience in more recent earthquakes has shown that damage to nonstructural components is also of great concern. In most modern buildings, the nonstructural components account for 60%–80% of the value of the building. Most nonstructural components are fragile (compared to the building structure), easily damage, and costly to repair or replace.
The distinction between structural and nonstructural components and system is, in many instances, artificial. The engineer labels as nonstructural all those components that are not designed as part of the seismic lateralforceresisting system (LFRS). Nature, however, makes no such distinction and tests the whole building. Many nonstructural components may be called upon to resist forces even though not designed to do so.
The nonstructural components or system may modify the structural response in ways detrimental to the safety of the building. Examples are the placing of heavy nonstructural partitions in locations that result in severe torsion and stress concentration or the placement on nonstructural partitions between columns in such a way as to produce a shortcolumn condition, as described later in this text. This can lead to column failure, distortion, and further nonstructural damage. Failure of the fire protection system, because of damage to the sprinkler system, may leave the building vulnerable to postearthquake fires caused by electrical or gas system damage.
While distance does not always guarantee safety (San Francisco was approximately 60 miles from the focus of the Loma Prieta earthquake), in general, being a substantial distance from the earthquake will lessen the effects of the earthquake on the building and its nonstructural components. Nonstructural failures are commonly seen at greater distances than structural failures.
Historically, the model earthquake codes paid little attention to the vertical component of the shaking generated by earthquakes. As a rule of thumb, the maximum vertical ground motion is generally 60%–70% of the maximum horizontal ground motion. While it may be unnecessary to consider the vertical motions of the structure as a whole, this is often not the case with nonstructural design. The model codes have little reference to vertical acceleration design requirements for nonstructural components. The building, usually due to its configuration, can act as an amplifier for both horizontal and vertical motions. Therefore, even though the code most often does not require vertical design resistance, the designer must be cognizant of the implications of vertical motions during an earthquake and their potential effects.
It is quite evident for some time that seismic design of regular and prismatic buildings and their nonstructural components is at once easy to visualize and execute. The current vogue for complex shapes in architectural design has, however, increased the complexity of the nonstructural systems. This increase in complexity decreases our ability to visualize how systems and components will respond and interact.
Since many nonstructural issues involve the intermixing of several engineering disciplines, perhaps the best way to address the concern is to sit down with all the engineering professionals early and often to discuss earthquake performance objectives of the facility. This should help in visualizing potential interactions between building systems and components.
Nonstructural components may also, however, influence structural performance in response to ground shaking. Structural analysis assumes a bare structure. Nonstructural components that are attached to the structure and heavy contents, depending on their location, may introduce torsional forces. Characteristic examples of structural/nonstructural interaction are as follows:
Elements attached to the floors of buildings (e.g., mechanical equipment, ornamentation, piping, nonstructural partitions) respond to floor motion in much the same manner as the building responds to ground motion. However, the floor motion may vary substantially from the ground motion. The highfrequency components of the ground motion that correspond to the natural periods of vibrations of the building tend to be magnified. If the elements are rigid and are rigidly attached to the structure, the forces on the elements will be in the same proportion to the mass as the forces on the structure. But elements that are flexible and have periods of vibration close to any of the predominant modes of the building vibration will experience forces substantially greater than the forces on the structure.
When individual elements of the structure are analyzed separately, it becomes necessary to consider seismic effects differently than for lateralloadresisting systems including diaphragms. The reason for this is that certain elements that are attached to the structure respond dynamically to the motion of the structure rather than to the motion of the ground. Resonance between the structure and the attached elements may occur.
The ELF procedure simplifies the dynamic effects of earthquakes by using a static model. Historically, the procedure was used for the design of all structures, but the current codes restrict its application to small buildings of regular configuration and larger buildings of limited height constructed with flexible diaphragms that are not considered to be essential or hazardous to the public.
where
In an earthquake, buildings experience ground motions that cause high accelerations and proportionately large internal forces in the building structure for short durations. In the ELF procedure, static loads with a lesser magnitude than the actual earthquake forces are applied. This relies on the anticipated ability of structures to withstand larger forces for short periods of time and allows for a less conservative, more affordable seismic design. The seismic base shear V is specified as a given percentage of the building weight. The value is determined by combining factors representing properties of the structure, soil, and use of the building.
The tendency for the building to sway from side to side in response to ground motion produces greater accelerations in the upper parts of the building. This backandforth motion, called the fundamental mode, dominates the response of most regular building structures. To model this effect statically, the ELF procedure redistributes the load applied to the building floors by considering for their distance from the base of the building.
A linear dynamic analysis is useful for evaluating irregular or dynamically complex buildings. An irregular building is defined as having a distribution of mass or stiffness that is nonuniform and is often created in buildings that have complex space planning requirements or asymmetrical configurations. Dynamic complexity is common in flexible structural systems. Flexibility is greatly influenced by the selection of structural system and building height. Flexible buildings tend to have a significant response to higher mode shapes. Mode shapes are movement patterns that occur naturally in structures that have been set in motion by ground shaking. Schematics of the mode shapes of a fourstory building are shown in Figure 3.5.
Designers use linear dynamic analysis to determine the degree of influence each mode shape will have on a structure's performance. The importance of higher modes depends on the relationship between the fundamental mode of the structure and the dynamic groundshaking characteristics of the site. Designers express mode shape influence in terms of the percent of building mass assigned to a particular mode. If the building mass vibrates primarily in the first or fundamental mode, a static analysis is permitted by the code. Although linear dynamic analysis methods are becoming routine in engineering practice, they are more complicated because they require detailed information about ground motion. When linear dynamic analysis is used, the structure–ground shaking interaction is usually modeled using a response spectrum. The ASCE 710 includes a procedure for developing a design response spectrum, as explained elsewhere in this text.
An alternative and significantly more complex method for modeling ground shaking, called a timehistory analysis, examines modal response using actual ground motion data. The seismic standards, such as ASCE 710, require that timehistory analyses consider several different ground motion records to insure that the structure response is sufficiently representative to account for future unknown ground motion patterns.
Selecting a good structure requires engineering common sense. Common sense requires understanding the earthquake motion and its demands and understanding the structural behavior of the individual systems available. There are differences of scale between elastic and inelastic behavior and dynamic responses and seismic energy dissipation. Structural and architectural configurations (such as regular versus irregular forms) are also significant in the performance. The many variables often make it difficult to select an appropriate system. The ASCE 710 lists numerous structural systems, but it does not provide guidance in the selection of a system, and the many systems are not equal in performance.
Key performance issues are elastic behavior, inelastic behavior, and the related cyclic behavior resulting from pushing a structure back and forth. The cyclic behavior, often referred to as hysteretic behavior, should be stable, nondegrading, predictable, and capable of dissipating a large amount of seismic energy.
The ELF approach to seismic design requires diminishing an acceleration spectra plot by use of an R value. Elastic design is expressed by an R value referred to as seismic response modification coefficient. This is a simple but frequently questionable method. It does not consider performance, nonlinear cyclic behavior, or—most importantly—energy dissipation.
Inelastic design is a better indication of realistic lateral drift or deflection that results from actual earthquake motions. Nonlinear drift impacts structural and nonstructural behavior. For significant seismic energy dissipation, the drift should be large, but for favorable nonstructural behavior, this drift should be small. A building with a large but unstable structural drift is likely to collapse. A building with a limited or small structural drift generally will not dissipate significant seismic energy without substantial damage.
A good measure of seismic performance is stable cyclic hysteretic behavior. The plot of load versus deformation of a member, for motion in both directions, represents cyclic behavior (Figure 3.10). If the load curves are full, undiminished, without necking down, they represent a stable system that is ductile and has sufficient capacity to deliver a constant level of energy dissipation during the shaking imposed by an earthquake. Degrading cyclic systems may, however, be acceptable if they degrade slowly and in a predictable manner.
The aforementioned attributes show that a range of possibilities exist for selecting a structural system. The favorable systems will do the following:
The design reduction value R, discussed at length elsewhere in this work, does not necessarily correlate with performance. The R value was a consensus value developed for conventional elastic design. With the advent of performance design based on nonlinear evaluation, the R value serves only as a rough estimate of system behavior, but not a realistic estimate of performance.
We have learned from detailed investigations of past earthquake damages that problems occur because of
Figure 3.10 Hysteretic behavior: (a) curve representing large energy dissipation and (b) curve representing limited energy dissipation. (From Taranath, B.S., Structural Analysis and Design of Tall Buildings, CRC Press, Boca Raton, FL, 2011.)
Figure 3.11 Structural systems with excellent to acceptable seismic performance. (From FEMA454, Designing for Earthquakes: A Manual for Architects. Washington, DC: Federal Emergency Management Agency, 2006.)
Why, with all our accumulated knowledge, does all this failure continue? It is because buildings tend to be constructed essentially in the same manner, even after an earthquake. It takes a significant effort to change habits, styles, and techniques of construction.
Sometimes bad seismic ideas get passed on without too much investigation and modification. And finally as a guide to engineers are, shown in Figure 3.11, six schematic structures that have exhibited excellent to acceptable performance in previous earthquakes.
A building's structural system is directly related to its architectural configuration, which largely determines the size and location of structural elements such as walls, columns, horizontal beams, floors, and roof structure. In today's flamboyant architecture, it is more than likely that engineers are faced with seismic design of building structures that are at once unique and at the same time are of irregular configuration.
In the following sections, the effects of irregular configurations on seismic performance are explained using the two main conditions created by vertical and plan irregularities. A number of deviations from regular characteristics are identified as problematical from a seismic viewpoint. Four of these deviations are then discussed in more detail, and conceptual solutions are provided for reducing or eliminating the detrimental effects.
Seismic designers have the choice of three basic alternative types of vertical lateralloadresisting systems (VLLRSs) as illustrated in Figure 3.12.
These basic systems have a number of variations, mainly related to the structural materials used and the ways in which the members are connected. See Table 3.2 for a summary of seismic performance of structural systems in previous earthquakes.
Shown in Figures 3.13 and 3.14 are some examples of structural systems suitable for different site conditions and occupancy types.
Shear walls are designed to receive lateral forces from diaphragms and transmit them to the ground. The forces in these walls are predominately shear forces in which material fibers within the wall try to slide past one another. To be effective, shear walls must run from the top of the building to the foundation with no offsets and a minimum of openings.
Figure 3.12 The three basic vertical seismic systems. (From FEMA454, Designing for Earthquakes: A Manual for Architects. Washington, DC: Federal Emergency Management Agency, 2006.)
Structural System 
Earthquake Performance 
Specific Building Performance and Energy Absorption 
General Comments 

Reinforced Concrete Wall 
San Francisco, 1957 Alaska, 1964 Japan, 1966 Los Angeles, 1994 Variable to Poor 


Steel Brace 
San Francisco, 1906 Taft, 1952 Los Angeles, 1994 Variable 


Steel Moment Frame 
Los Angeles, 1971 Japan, 1978 Los Angeles, 1994 ? Good 


Concrete Shear Wall 
Caracas, 1965 Alaska, 1964 Los Angeles, 1971 Algeria, 1980 Variable 


Reinforced Concrete Ductile Moment Frame 
Los Angeles, 1971 ? Good 


Braced frames act in the same way as shear walls; however, they generally provide less resistance but better ductility depending on their detailed design. They provide more architectural design freedom than shear walls.
There are two general types of braced frames: conventional concentric and eccentric. In the concentric frame, the center lines of the bracing members meet the horizontal beam at a single point.
In the eccentricbraced frame, the braces are deliberately designed to meet the beam some distance apart from one another: the short piece of beam between the ends of the braces is called a link beam. The purpose of the link beam is to provide ductility to the system: under heavy seismic forces, the link beam will distort and dissipate the energy of the earthquake in a controlled way, thus protecting the remainder of the structure.
A momentresistant frame is the engineering term for a frame structure with no diagonal bracing in which the lateral forces are resisted primarily by bending in the beams and columns mobilized by strong joints between columns and beams. Momentresistant frames provide the most architectural design freedom.
Figure 3.13 Example structural systems for site conditions. (From FEMA389, Primer for Design Professionals: Communicating with Owners and Managers of New Buildings on Earthquake Risk. Washington, DC: Federal Emergency Management Agency, 2004.)
Figure 3.14 Structural systems for occupancy types. (From FEMA389, Primer for Design Professionals: Communicating with Owners and Managers of New Buildings on Earthquake Risk. Washington, DC: Federal Emergency Management Agency, 2004.)
These systems are, to some extent, alternatives, although designers oftentimes mix systems, using one type in one direction and another type in the other or combining them in a given direction. This must be done with care, however, mainly because the different systems are of varying stiffness (shearwall systems are much stiffer than momentresisting frame systems, and braced systems fall in between), and it is difficult to obtain balanced resistance when they are mixed. However, for highperformance structures, there is now increasing use of dual systems. Examples of effective mixed systems are the use of shearwall core together with a perimeter momentresistant frame or a perimeter steel moment frame with interior eccentricbraced frames. Another variation is the use of shear walls combined with a momentresistant frame in which the frames are designed to act as a failsafe backup case of shearwall failure.
The framing system must be chosen at an early stage in the design because the seismic system plays the major role in determining the seismic performance of the building and more importantly has considerable influence on the architecture of design. For example, if shear walls are chosen as the seismicforceresisting system, the building planning must be able to accept a pattern of permanent structural walls with limited openings that run uninterrupted through every floor from roof to foundation.
The term diaphragm is used to identify horizontal resistance members that transfer lateral forces between vertical resisting elements such as shear walls or frames. The diaphragm action is generally provided by the floor and roof systems of the building; sometimes, however, horizontal bracing systems independent of the roof or floor structure serve as diaphragms.
The diaphragm can be visualized as a wide horizontal beam with components at its edges, termed chords, designed to resist tension and compression: chords are similar to the flanges or a vertical beam (Figure 3.15).
A diaphragm that forms part of a resistant system may act either in a flexible or rigid manner, depending partly on its size (the area between enclosing resistance elements) and also on its material. The flexibility of the diaphragm, relative to that of the vertical lateralloadresisting (VLLR) elements such as shear walls whose forces it is transmitting, also has a major influence on the nature and magnitude of those forces. With flexible diaphragms made of steel decking without concrete, walls take loads according to tributary areas (if mass is evenly distributed). With rigid diaphragms (usually concrete slabs and steel deck with concrete topping), walls share the loads in proportion to their stiffness.
Perhaps, now it is as good a time as any to discuss behavior of diaphragms with particular emphasis on the following:
See Figure 3.16 for diaphragm design terminology.
Figure 3.15 Deep beam action of diaphragm.
Figure 3.16 Diaphragm design terminology: (a) plan, (b) shear force diagram, and (c) bending moment diagram. Note: Fpx diaphragm design force (see ASCE 710 Section 12.10.1.1).
Collectors, also called drag struts or ties, are diaphragm framing members that collect or drag diaphragm shear forces from laterally unsupported areas to vertical resisting elements (Figure 3.16).
Floors and roofs, more often than not, have to be penetrated by openings required for certain features such as staircases, elevators and duct shafts, skylights, and atria. The size and location of these penetrations are critical to the effectiveness of the diaphragm. The reason for this is not hard to see when the diaphragm is visualized as a beam. For example, it can be seen that openings cut in the tension flange of a beam will seriously weaken its loadcarrying capacity. In a verticalloadbearing situation, a penetration through a beam flange would occur in either a tensile or compressive region. In a lateralload system, the opening would be in a region of both tension and compression, since the seismic loading alternates rapidly in direction.
Earthquakeresistant design requires that components of the structure be connected or tied together in such a manner that they behave as a unit. Diaphragms are an important structural element in achieving this interconnection. Diaphragms are horizontally spanning members, analogous to deep beams that distribute the seismic loads from their origin to the vertically oriented lateralforceresisting frames (braced frames, moment frames, etc.). Diaphragms are commonly analyzed as simplespan or continuously spanning deep beams and hence is subject to shear, moment, and axial forces (for truss diaphragms and collectors) as well as the associated deformations. Figure 3.16 shows typical loading, shear and moment diagrams for the analysis and design of diaphragms.
In steel buildings, the floor or roof deck is usually designed as the shearresistant member (which is analogous to the web of a beam), and the beams or supplemental deck reinforcing at the boundaries of the diaphragms is designed as the flexuralresistant member or chord (which is analogous to the flanges of a beam).
Diaphragms are classified into one of three categories: rigid, flexible, or semirigid. Rigid diaphragms are those that possess the strength and stiffness to distribute the lateral forces to the lateralforceresisting frames in proportion to the relative stiffness of the individual frames, without significant deformation in the diaphragm. On the other hand, the distribution of the lateral forces through a flexible diaphragm is independent of the relative stiffness of the lateralforceresisting frames.
A semirigid diaphragm, as the name implies, distributes lateral forces in proportion to the stiffness of the diaphragm and the relative stiffness of the lateralforceresisting frames. Semirigid diaphragms are often analyzed using the analogy of a beam on elastic supports, where the beam represents the stiffness of the diaphragm and the elastic supports represent the stiffness of the lateralforceresisting frames.
Since many buildings have lateralforceresisting frames that are not uniformly spaced and continuous around the diaphragm boundaries, collector elements are utilized. Collector elements are tension and compression members that serve to deliver the diaphragm forces to the lateralforceresisting frames. A redistribution of collector forces can occur as yield mechanisms form in the lateralforceresisting frames.
The purpose of diaphragm as stated earlier is to distribute lateral forces to the elements of the VLLRS. In doing so, it
There are many types of materials and system for use as floor and roof diaphragms such as
In merits noting, however, that standing seam roof panels are not considered as diaphragms.
As stated previously, diaphragms are typically classified into three categories: rigid, flexible, and semirigid. A diaphragm is considered rigid if it exhibits the following characteristics:
Examples of rigid diaphragm are
A diaphragm is considered flexible if
As an example, an untopped steel deck, under certain conditions, may be considered as a flexible diaphragm.
A semirigid diaphragm, as the name implies, is one whose behavior is in between those of rigid and flexible diaphragms. It should be observed, however, that the classification is only for analytical purpose because diaphragms in building design are neither completely rigid nor completely flexible.
The behavior of a semirigid diaphragm may be considered analogous to that of a beam on elastic foundation. This is because deflection of VLLR elements and diaphragm under horizontal forces are of the same order of magnitude. Therefore, the design must account for relative rigidities of VLLR elements and diaphragm. Most seismic standards mandate that a diaphragm be considered flexible when the maximum lateral deformation of the diaphragm is more than two times the average story drift of the associated story. It is quite evident that in order to classify a diaphragm, one must compare the stiffness of the diaphragm to that of the VLLRS. The Steel Deck Institute (SDI), however, gives us the following guidelines for diaphragm classifications.
SDI diaphragm classifications
Shear Rigidity G′ 
Classification 
Deck Type 

6.67–14.3 
Flexible 
Bare steel deck 
14.3–100 
Semiflexible 
Bare steel deck 
100–1000 
Semirigid 
Concretefilled steel deck 
>1000 
Rigid 
Concretefilled steel deck 
Refer to SDI manual for definition of shear rigidity, G′.
It is perhaps selfevident that the inplane deflection of the diaphragm shall be limited to that which will permit the attached element to maintain its structural integrity under the applied lateral loads and continue to support selfweight and vertical load if applicable.
It should be noted that the generally accepted drift ratio of h/400 is for the inplane deflection experienced by the cladding and the outofplane drift limits are considerably less stringent (see Figure 3.17 for schematic explanation of this concept).
Another aspect equally important in diaphragm classification is the geometry (spantodepth ratio) of the diaphragm itself. For example, shown in Figure 3.18a is a diaphragm that has a spantodepth ratio of 62.5/50 = 1.25, which, for all practical purposes, would allow us to judge the diaphragm as rigid.
Next, let us consider the same diaphragm with the interior VLLR elements removed from the system as shown in Figure 3.18b. The diaphragm now has to span the entire width of 250 ft resulting in a spantowidth ratio = 250/50 = 5. This would generally require the diaphragm to be considered as flexible or semiflexible.
Figure 3.17 Outofplane drift. Note: The ratio h/400 limit is generally for the inplane deflections. The outofplane drift limits are generally less stringent.
Figure 3.18 Rigid and flexible diaphragms: (a) rigid diaphragm with a spantodepth ratio of 62.5/50 = 1.25 and (b) flexible diaphragm with a spantodepth ratio of 250/50 = 5.
To understand diaphragm behavior and design procedures, it is convenient to define certain terms unique to diaphragm design (see Figure 3.16).
Shown therein are the schematic bending moment and shear force diaphragms resulting from beam action of the diaphragm spanning between the VLLR elements. The term F_{px} is the diaphragm design force occurring at this level due to inertial force. To this, we must add any shear force resulting from discontinuity of VLLR element such as a shear wall, as shown in Figure 3.19.
Basically, there are two design approaches. In the first method, the entire diaphragm shear is transferred to the VLLRS without using collector beams or drag struts.
The shear transfer is assumed to occur only over the length of the VLLRS (see Figure 3.20).
In the second approach, the diaphragm shear is transferred to collector beams and then to the VLLRS (see Figure 3.21). Observe that this approach is required when there is no direct connection of the diaphragm to the VLLR element or when the length of VLLR element is insufficient to transfer the diaphragm shear. As an example of the former, shown in Figure 3.22 is a diaphragm that has an opening adjacent to the VLLRS. Because there is no direct connection between the diaphragm and beam BM2, there is no direct transfer of diaphragm shear force from the diaphragm to this beam. It does, however, carry the axial force V_{2} resulting from the shear accumulated by the beam BM1 (see Figure 3.22).
Figure 3.19 Transfer of VLLR element: (a) plan and (b) schematic elevation.
Figure 3.20 Diaphragm design procedures, approach 1: (a) partial plan of diaphragm and (b) elevation. Note: In this method, the entire diaphragm shear is transferred directly to the VLLRS. Connections 1 and 2 or beams 1 and 2 need not be designed for diaphragm forces.
Figure 3.21 Diaphragm design procedure, approach 2: (A) (a) partial plan of diaphragm; (b) elevation; (c) axial load in beam B1 due to diaphragm shear; (d) gravity moment Mu; and (e) axial load Ω0 PQE. (B) schematic diaphragm, shear transfer to VLLR without using collectors. Note: In this approach, diaphragm shear not directly transferred to the VLLR system is through collector beams.
Figure 3.22 Diaphragm with opening along drag beam, approach 2: (a) plan and (b) elevation. Note: (1) Beam BM2 does not collect diaphragm shear forces but must be designed to carry the axial force V2, (2) provide connection between diaphragm and collector beam BM1 to transfer diaphragm force, and design collector beam for this force. (3) Connections 1 and 2 must be designed to carry the axial force V2 through the beam–column connection.
Castinplace concrete diaphragm
The strength of castinplace concrete diaphragms discussed in Chapter 21 of ACI 31811 is governed by Equation 3.6:
and shall not exceed
Observe that
By definition, a rigid diaphragm has no relative inplane displacement of joints within the diaphragm. Beams that are part of VLLRSs with both end joints connected to a rigid diaphragm do not experience, in an analytical sense, any axial force. And none is reported in a typical computer analysis output. Therefore, any design that uses post processors without proper adjustment of axial loads in the beams would be wrong. How can we overcome this stumbling block?
One method, often attempted in practice, is to release the beam ends strategically from the rigid diaphragm so that axial force and deformation are captured in the analysis. It should, however, be kept in mind that release of excessive joints may create instability. To overcome this problem, it seems that we have no choice but to revert back to separate manual or computer analysis of each frame for proper determination of axial force in beams that are part of VLLRSs.
Observe that in braced frames, the building stiffness is likely to be overestimated by as much as 15% if axial deformation of beams is not considered. Keep in mind that computer analysis results do not tell the engineer if there is a problem in the analytical model.
A similar problem is likely to occur when modeling storydeep transfer trusses subjected to gravity and lateral loads (see Figure 3.23). If the diaphragms attached to the top and bottom chords of the truss are assumed rigid, then the computer thinks that the areas of chords are infinitely large.
Thus, A_{top chord} = A_{bot.chord} = ∞.
Figure 3.23 Top and bottom chords of storydeep truss attached to rigid diaphragms. Note: To capture correct axial forces in the truss top and bottom chords, release all but those at center of truss.
Since the moment of inertia of the truss (neglecting the flexibility of diagonals) is equal to
any analysis based on this assumption of rigid diaphragms is obviously incorrect.
One method of circumventing this problem is to release all but the center joint from the rigid diaphragm (see Figure 3.23).
The design of the diaphragm, at least in theory, is no more complicated than that of a deep beam. However, there are many reasons why you may have diaphragmrelated problems. Chief among them are as follows:
Schematics of diaphragm conditions that require particular attention are summarized in Figures 3.24 through 3.32.
Figure 3.24 Diaphragm design loads. Notes: (1) Floors and roofs used as diaphragms are designed for a lateral force of Fp acting in any direction. Generally, it is assumed that the inplane mass of a shear wall does not contribute to the diaphragm loading unless the shear wall is interrupted at the specific level. In case a shear wall does not extend below the floor level, both its horizontal and vertical loads must be distributed to the remaining walls with due considerations to major differences in rigidities. (2) The total shear Fp at any level will be distributed to the various vertical elements of the lateralforceresisting system in proportion to their rigidities considering the rigidity of the diaphragm.
Figure 3.25 Various types of diaphragm: (a) floor slab diaphragms, (b) roof deck diaphragm, and (c) truss diaphragm. Note: A diaphragm may be considered analogous to a plate girder laid in a horizontal plane or inclined in the roof plane where the floor or roof deck performs the function of the plate girder web, the joints or beams function as web stiffeners, and the peripheral beams or integral reinforcement functions as flanges.
For nearoptimum seismic performance, the following characteristics are desirable:
Figure 3.26 Relative effects of diaphragm stiffness. (a) Schematic plan, (b) rigid diaphragm, (c) semirigid diaphragm, and (d) flexible diaphragm.
In seismic terms, engineers refer to a building that has the desirable characteristics as a regular building. As the building characteristics deviate from this model, the building becomes increasingly irregular. It is these irregularities that affect the building's seismic performance.
Figure 3.27 Diaphragm with limited connection to VLLRS.
Figure 3.28 Isolate VLLR element with no direct connection to diaphragm: Note: Entire force for VLLR 2 must come through the collector beam. Beam and connections must be designed for these forces.
Figure 3.29 Narrow diaphragm with a VLLR element results in high shears.
Figure 3.30 Large opening in diaphragm results in high shear and/or bending stress in diaphragm.
Figure 3.31 Diaphragm with a reentrant corner.
Figure 3.32 Criteria for diaphragm classification. (a) Plan and (b) elevation.
Configuration irregularity is largely responsible for two undesirable conditions: stress concentrations and torsion. These conditions often occur concurrently.
Irregularities tend to create abrupt changes in strength or stiffness that may concentrate forces in an undesirable way resulting in stress concentration.
Stress concentration occurs when large forces are concentrated at one or a few elements of the building, such as a particular set of beams, columns, or walls. These few members may fail and, by a chain reaction, damage or even bring down the whole building.
Stress concentrations can be created by both horizontal and vertical stiffness irregularities. The shortcolumn phenomenon discussed elsewhere in this book is an example of stress concentration created by vertical dimensional irregularity in the building design. In plan, a configuration that is most likely to produce stress concentrations is the reentrant corner condition as exemplified in buildings with plan forms such as an L or a T. Other reentrant corner configurations are shown in Figure 3.43.
The vertical irregularity of the soft or weakstory types can produce dangerous stress concentrations along the plane of discontinuity. Soft and weak stories are discussed in more detail elsewhere in this book.
Configuration irregularities in plan may cause torsional forces to develop, which contribute a significant component of uncertainty to an analysis of building resistance, and are perhaps the most frequent cause of structural failure. Torsional forces are created in a building eccentricity between the center of mass and the center of resistance. This eccentricity originates either in the lack of symmetry in the arrangement of the perimeterresistant elements or in the plan configuration of the building, as in the reentrant corner forms discussed earlier.
Torsional moment is generated whenever the cg of the lateral forces fails to coincide with the center of rigidity (cr) of the vertical resisting elements, provided that the diaphragm is sufficiently rigid to transfer torsion. The magnitude of the torsional moment that is required to be distributed to the vertical resisting elements by a diaphragm is determined by the sum of the moments created by the physical eccentricity of the translational forces at the level of the diaphragm from the cr of the resisting elements (M_{T} = F_{p}e, where e = distance between cg and cr) and the accidental torsion of 5%. The accidental torsion is an arbitrary code requirement equivalent to the story shear acting with an eccentricity of not less than 5% of the maximum building dimension at that level. The torsional distribution by the more rigid diaphragms to the resisting elements is assumed to be in proportion to the stiffness of the elements and its distance from the cr. Negative torsional shears are neglected. Flexible diaphragms are not used for torsional distribution. Cantilever diaphragms on the other hand will distribute translational forces to vertical resisting elements, even if the diaphragm is flexible. In this case, the diaphragm and its chord act as a flexural beam on supports (vertical resisting elements) whose resistance is in the same direction as the forces.
Diaphragm deflections: A diaphragm is designed to provide such stiffness and strength so that walls and other vertical elements laterally supported by the diaphragm can safely sustain the stresses induced by the response to seismic motion. The total computed deflection (Δ_{d}) of diaphragms under seismic forces consists of the sum of two components. The first component is the flexural deflection (Δ_{f}) of the diaphragm, which is determined in the same manner as the deflection of beams. The assumption that flexural stresses on the diaphragm web are neglected is used except for reinforced concrete slabs. For such slabs, the proportional flexural stresses also may be assumed to be carried by the web. The second component is the web deflection (Δ_{w}) of the diaphragm. The specific nature of the web deflection will vary depending on the type of diaphragm. The deflection of the diaphragm under seismic forces is used as the criteria for the adequacy of the stiffness of a diaphragm.
Many of the configuration conditions that present seismic problems were identified by observers early in the twentieth century. However, the configuration problem was first defined for code purposes in the 1975 Commentary to the Structural Engineers Association of California (SEAOC) Recommended Lateral Force Requirements (commonly called the SEAOC Blue Book). In this commentary, over 20 specific types of irregular structures or framing systems were noted as examples of designs that should involve further analysis including dynamic consideration, rather than the use of the simple equivalent static force method in unmodified form. These irregularities vary in importance in their effect, and their influence also varies in degree, depending on which particular irregularity is present. Thus, while in an extreme form, the reentrant corner is a serious plan irregularity; in a lesser form, it may have little or no significance. The determination of the point at which a given irregularity becomes serious was left up to the judgment of the engineer.
Because of the belief that this approach was ineffective, in the 1988 codes, a list of six horizontal (plan) and six vertical (section and elevation) irregularities was provided that, with minor changes, is still in today's codes. This list also stipulated dimensional or other characteristics that established whether the irregularity was serious enough to require regulation and also provided the provisions that must be met in order to meet the code.
The seismic provisions of ASCE 710 provide descriptions of 10 irregularities—5 for horizontal and 5 for vertical irregularities as shown in Figures 3.33 and 3.34. Observe that the ASCE 710 provides only descriptions of these conditions; the diagrams are added in this text to illustrate each condition by showing how it would modify our optimized configuration and to also illustrate the failure pattern that is created by the irregularity.
Figure 3.33 Plan irregularities. (a) Torsional irregularity may cause localized damage and collapse in extreme cases. (b) Reentrant corners may cause load damage to diaphragm and attached elements. Collapse may occur in extreme conditions. (c) Diaphragm eccentricity and large cutouts may cause localized damage. (d) Nonparallel lateralloadresisting system may lead to torsional instability and localized damage. (e) Outofplane offsets of VLLRS may cause collapse mechanism in severe offsets.
Figure 3.34 Vertical irregularities. (a) Stiffness irregularity, soft story (common collapse mechanism, deaths, and much damage in 1994 Northridge earthquake). (b) Weight/mass irregularity (collapse mechanism in extreme instances). (c) Vertical geometric irregularity (localized structure failure). (d) Inplane irregularity in vertical LFRS (localized structural failure). (e) Capacity discontinuity, weak story (collapse mechanism).
For the most part, code provisions seek to discourage irregularity in design by imposing penalties, which are of three types:
It should be noted that the code provisions treat the symptom of irregularity rather than the cause. The irregularity is still allowed to exist; the hope is that the penalties will be sufficient to cause the designers to eliminate the irregularities. Increasing the design forces or improving the analysis to provide better information does not, in itself, solve the problem. The problem must be solved by design.
The codedefined irregularities serve as a checklist for ascertaining the possibility of configuration problems. Four of the more serious configuration conditions are described in more detail in the following sections along with some conceptual suggestions for their solution.
Four configuration conditions (two vertical and two in plan) that have the potential to seriously impact seismic performance are
The term weak story has commonly been applied to buildings whose groundlevel story is less stiff than those above. The building code distinguishes between soft and weak stories. Soft stories are less stiff, or more flexible, than the story above; weak stories have less strength. A soft or weak story at any height creates a problem, but since the cumulative loads are greatest toward the base of the building, a discontinuity between the first and second floor tends to result in the most serious condition.
The way in which severe stress concentration is caused at the top of the first floor is shown in the diagram sequence in Figure 3.35. Normal drift under earthquake forces that is distributed equally among the upper floors is shown in Figure 3.35a. With a soft story, almost all the drift occurs in the first floor, and stress concentrates at the secondfloor connections. This concentration overstresses the joints along the secondfloor line, leading to distortion or collapse (see Figure 3.35c).
Three typical conditions create a soft first story (Figure 3.36). The first condition (Figure 3.36a) is where the vertical structure between the first and second floor is significantly more flexible than that of the upper floors. (The seismic code provides numerical values to evaluate whether a softstory condition exists.) This discontinuity most commonly occurs in a frame structure in which the firstfloor height is significantly taller than those above, resulting in a large discrepancy in stiffness.
The second form of soft story (Figure 3.36b) is created by a common design concept in which some of the vertical framing elements do not continue to the foundation but rather are terminated at the second floor to increase the openness at ground level. This condition creates a discontinuous load path that results in an abrupt change in stiffness and strength at the plane of change.
Finally, the soft story may be created by an open first floor that supports heavy structural or nonstructural walls above (Figure 3.36c). This situation is most serious when the walls above are shear walls acting as major lateralforceresisting elements.
The best solution to the soft and weakstory problem is to avoid the discontinuity through architectural design. There may, however, be good programmatic reasons why the first floor should be more open or higher than the upper floors. In these cases, careful design must be employed to reduce the discontinuity. Some conceptual methods for doing this are shown in Figure 3.37.
Figure 3.35 Soft and weak stories: (a) building drift without soft story, (b) drift in soft story, and (c) collapse due to soft story.
Figure 3.36 Three types of soft first story: (a) flexible first floor, (b) discontinuity—indirect load path— and (c) heavy superstructure.
Figure 3.37 Design solutions for softstory condition: (a) softstory condition, (b) add columns, (c) add bracing, and (d) add external buttresses.
Not all buildings that show slender columns and high first floors are soft stories. For a soft story to exist, the flexible columns must be the main lateralforceresistant system.
When shear walls form the main lateralforceresistant elements of a structure, and there is no continuous load path through the walls from roof to foundation, the result can be serious, overstressing at the points of discontinuity. This discontinuous shear wall condition represents a special, but common, case of the softfirststory problem.
The discontinuous shear wall is a fundamental design contradiction: the purpose of the shear wall is to collect diaphragm loads at each floor and transmit them as directly and efficiently as possible to the foundation. To interrupt this load path is undesirable; to interrupt it at its base, where the shear forces are greatest, is a major error. Thus, the discontinuous shear wall that terminates at the second floor represents perhaps a worst case of the softfirstfloor condition. A discontinuity in vertical stiffness and strength leads to a concentration of stresses, and the story that must hold up all the rest of the stories in a building should be the last, rather than the first, element to be sacrificed.
Olive View hospital, which was severely damaged in the 1971 San Fernando, California, earthquake, represents an extreme form of the discontinuous shear wall problem. The general vertical configuration of the main building was a softtwostory layer of rigid frames on which was supported by a fourstory (five, counting penthouse), stiff, shearwallplusframe structure (Figures 3.38 through 3.40). The second floor extends out to form a large plaza. Sever damage occurred in the softstory portion. The upper stories moved as a unit and moved so much that the column at ground level could not accommodate such a high displacement between their bases and tops and hence failed. The largest amount by which a column was left permanently out of plumb was 2 ft 6 in. (Figure 3.40). The building did not collapse, but two occupants in intensive care and maintenance person working outside the building were killed.
Figure 3.38 Olive View Hospital, cross section. Note: The shear walls stop at first floor. (From FEMA454, Designing for Earthquakes: A Manual for Architects. Washington, DC: Federal Emergency Management Agency, 2006.)
Figure 3.39 Olive View Hospital, secondfloor plaza and the discontinuous shear wall. (From FEMA454, Designing for Earthquakes: A Manual for Architects. Washington, DC: Federal Emergency Management Agency, 2006.)
Figure 3.40 Olive View Hospital, San Fernando earthquake, 1971. Note the extreme deformation of the columns above the plaza level. (From FEMA454, Designing for Earthquakes: A Manual for Architects. Washington, DC: Federal Emergency Management Agency, 2006.)
The solution to the problem of the discontinuous shear walls is unequivocally to eliminate the condition. To do this may create architectural problems of planning or circulation or image. If this is so, it indicates that the decision to use shear walls as resistant elements was wrong from the inception of the design. If the decision is made to use shear walls, then their presence must be recognized from the beginning of schematic design, and their size and location made the subject of careful architectural and engineering coordination early.
This problem may occur in buildings whose configuration is geometrically regular and symmetrical, but nonetheless irregular for seismic design purposes.
A building's seismic behavior is strongly influenced by the nature of the perimeter design. If there is wide variation in strength and stiffness around the perimeter, the center of mass will not coincide with the center of resistance, and torsional forces will tend to cause the building to rotate around the center of resistance (see Figure 3.41).
A common instance of an unbalanced perimeter is that of openfront design in buildings, such as fire stations and motor maintenance shops in which it is necessary to provide large doors for the passage of vehicles. Stories, individually or as a group in a shopping mall, are often designed as boxes with three solid sides and an open glazed front.
Figure 3.41 Torsion due to wide variation in strength and stiffness around the building perimeter.
Figure 3.42 Some solutions to storefronttype unbalancedperimeterresistance conditions.
The solution to this problem is to reduce the possibility of torsion by balancing the resistance around the perimeter. The example shown is that of the store front. A number of alternative design strategies can be employed that could also be used for the other building type conditions noted (Figure 3.42).
The strategy is to design a frame structure of approximately equal strength and stiffness for the entire perimeter. The opaque portion of the perimeter can be constructed of nonstructural cladding, designed so that it does not affect the seismic performance of the frame. This can be done either by using lightweight cladding or by ensuring that heavy materials, such as concrete or masonry, are isolated from the frame.
A second approach would be to increase the stiffness of the open facades by adding sufficient shear walls, at or near the open face, designed to approach the resistance provided by the other walls. A third solution would be to use a strong momentresisting or braced frame at the open front, which approaches the solid wall in stiffness. The ability to do this will depend on the size of the facades. However, it should be noted that a long steel frame can never approach a long concrete wall in stiffness.
The possibility of torsion may be accepted, and the structure designed to have the capacity to resist it, through a combination of moment frames, braced frames, and shear walls.
The reentrant corner is the common characteristic of building forms that, in plan, assume the shape of L, T, H, etc., or a combination of these shapes (Figures 3.43 and 3.44).
Invariably these forms result in torsion due to eccentricity of center of mass and center of resistance shown schematically in Figure 3.45.
Figure 3.43 Reentrant corner plan forms.
Figure 3.44 Stress concentrations at reentrant corner.
Figure 3.45 Torsion due to eccentricity of center of mass and center of resistance.
There are two problems created by these shapes. The first is that they tend to produce differential motions between different wings of the building that, because of stiff elements that tend to be located in this region, result in local stress concentrations at the reentrant corner or notch.
The second problem of this form is torsion caused because the center of mass and the cr cannot geometrically coincide for all possible earthquake directions. The result is rotation. The resulting forces are very difficult to analyze and predict. Observe that the stress concentration at the notch and the torsional effects are interrelated. The magnitude of the forces and the severity of the problems will depend on
Reentrant corner plan forms are, however, the most useful set of building shapes for urban sites, particularly for residential apartments and hotels. This is because large plan areas may be accommodated in relatively compact form yet still provides a high percentage of perimeter rooms with access to air and light.
There are two basic alternative approaches to the problem of reentrant corner forms: structurally to separate the building into simpler shapes or to tie the building together more strongly with elements positioned to provide a more balanced resistance (Figure 3.46).
Once the decision is made to use separation joints, the structurally separated entities of a building must be fully capable of resisting vertical and lateral forces on their own, and their individual configuration must be balanced horizontally and vertically.
To design a separation joint, the maximum drift of the two units must be calculated. The worst case is when the two individual structures would lean toward each other simultaneously. Therefore, the dimension of the separation space must allow for the statistical sum of the building deflections.
Several considerations arise if it is decided to dispense with the separation joint and tie the building together. Collectors at the intersection can transfer forces across the intersection area, but only if the design allows for these beamlike members to extend straight across without interruption. Since the portion of the wing that typically distorts the most is the free end, it is desirable to place stiffening elements at that location.
The use of splayed rather than rightangle reentrant corners lessens the stress concentration at the notch (Figure 3.47). This is analogous to the way a rounded hole in a steel plate creates less stress concentration than a rectangular hole or the way a tapered beam is structurally more desirable than an abruptly notched one.
Figure 3.46 Solutions for the reentrant corner condition.
Figure 3.47 Relieving the stress on a reentrant corner by using a splay.
When flexible structures are subjected to lateral forces, the resulting horizontal displacements lead to additional overturning moments because the gravity load is also displaced. Thus, in the simple cantilever model of Figure 3.48a, the total base moment is
Therefore, in addition to the overturning moments produced by lateral force, V_{u}, the secondary moment P_{u}Δ must also be resisted. This moment increment in turn will produce additional lateral displacement, and hence Δ will increase further. In very flexible structures, instability, resulting in collapse, may occur. However, analyses of typical building frames have indicated that Pdelta effects are small when maximum interstory drift is less than 1%.
Figure 3.48 (a) P–⋄ effect, simple cantilever model, Mu = VuH + Pu⋄. (b) Pdelta effect: shearwallframe system. (c) Collapse due to Pdelta effect.
Although a building mass or weight, as part of the F = MA equation, determines the horizontal forces, there is, however, another way in which the building's weight may act under earthquake forces to overload the building and cause damage or even collapse.
Vertical members such as columns or walls may fail by buckling when the mass of the building exerts its gravity force on a member distorted or moved out of plumb by the lateral forces. This phenomenon as stated previously is the Pdelta effect, where P is the gravity force or weight and delta is the eccentricity or the extent to which the force is offset (see Figure 3.48b and c for schematics of Pdelta effects).
The geometrical proportions of the building also may have a great influence on whether the Pdelta effect will pose a problem. A tall, slender building is much more likely to be subject to overturning forces than a low, squat one. It should be noted, however, that if the lateral resistance is provided by shear walls of braced frames, it is their proportions that are significant rather than those of the building as a whole.
Figure 3.49 Why buildings generally fall down, not over: (a) filing cabinet and (b) building structure.
However, in earthquakes, buildings seldom overturn, because structures are not homogenous but are composed of many elements connected together. The earthquake forces will pull the components apart, and the building will fall down, not over. Strong homogeneous structures such as filing cabinets, however, will fall over (see Figure 3.49 for schematics).
Structures are commonly designed so that under sever shaking, the beams will fail before the columns. This reduces the possibility of complete collapse. The shortcolumn effect, discussed elsewhere in this text, is analogous to a weakcolumn, strongbeam condition, which is sometimes produced inadvertently when strong or stiff nonstructural members are inserted between columns.
Vertical setbacks can introduce discontinuities, particularly if columns or walls are offset at the plane of the setback. A horizontal plane of weakness can be created by the placement of windows or other openings that may lead to failure.
We typically accept higher risks of damage under seismic design loads than under other comparable extreme loads, such as maximum live load or wind loads. The reason being that the seismic forces generated during severe ground motions are too high to be resisted within the elastic range of material response. Common practice therefore is to design for forces that are a fraction of those corresponding to elastic response. We then expect the structures to survive strong earthquakes by large inelastic deformations and energy dissipation characteristics.
Although earthquake shaking causes damage to a structure, it is the gravity load that causes collapse. Redundancy for the load path and ductile behavior of critical members can prevent or reduce the extent of collapse. On the other hand, brittle behavior enhances possibility and increases extent of collapse.
With increased awareness that excessive strength is not essential or even necessarily desirable, the emphasis in seismic design has shifted from the resistance of large seismic forces to the evasion of these forces. Inelastic structural response has become an essential reality in structural design for earthquake forces. Inelastic deformations that provide ductility are considered essential for preventing building collapse while the structure is subjected to backandforth motions during severe ground shaking.
Seismic design encourages structural forms that are more likely to exhibit ductility than those that do not. Thus, for concrete structures, the shear strength provided in a member must exceed its actual flexural strength.
As stated earlier, one of the most common causes of building failures during earthquakes is the softstory mechanism that may develop when one level, typically the lowest, is weaker than upper levels. This condition results from a functional desire to open up lower levels for circulation. When subjected to strong ground motions, the columns develop high compression strains due to the combined effects of axial force and bending moment. Unless adequate, closely spaced, welldetailed transverse reinforcement is placed in the potential plastic hinge region, spalling of concrete followed by instability of the compression reinforcement will follow. It must be recognized that even with a weak beam/strong column design in which seismic energy dissipation is primarily in wellconfined beam plastic hinges, a column plastic hinge may still form at the base of the column, resulting in partial or total collapse.
While there is something new to be learned from each earthquake, it may be said that the majority of structural lessons have been learned. However, there is still plenty to learn about the unpredictable and unquantifiable effects of earthquakes.
Wellestablished techniques, used for design of structures for various static loads, including wind forces, cannot simply be extended and applied to conditions that arise during earthquakes. In earthquake design, it is imperative that we consider forces corresponding to the largest seismic displacement.
A source of major damage, particularly in columns, repeatedly observed in earthquakes, is the interference with the deformations of members by rigid nonstructural elements, such as infill walls. As Figure 3.50a and b shows, the top edge of a brick wall will reduce the effective length of one of the columns, thereby increasing its lateral stiffness. Since seismic forces are proportional to the stiffness, the braced column will attract larger horizontal shear forces than it would otherwise. The failure of such gravityloadcarrying members may lead to collapse of the entire building. Therefore, it is important to ensure that the inelastic column deformations can take place without interference from nonstructural construction.
Figure 3.50 Creation of inadvertent short columns: (a) partial floorheight panel infill and (b) failure pattern.
Earthquake damage or even failures can occur in buildings that have poorly detailed joints. The backandforth cycling of the structure during severe ground shaking may cause momentresistant joints to unravel. The gravity load can no longer be supported by these columns. Consequently, the structure is driven earthward until it stops on the ground or lower floors that have sufficient strength to stop the falling mass as shown schematically in Figure 3.51. The resulting collapse may be a pancaked group of slabs held apart by broken columns and building contents or a condition where columns are left standing, punched through the slabs.
These types of failures usually occur in taller structures (see Figure 3.52a, b, and c). The tension that is concentrated at the edges of a concrete frame or shear wall can produce very rapid loss of stability. In walls, if the reinforcing steel is inadequately proportioned or poorly embedded, it can fail in tension and result in rapid collapse of the wall by overturning. A more common condition occurs, when the tension causes the joints in a concrete moment frame to lose bending and shear strength. A rapid degradation of the structure can result in a partial or complete pancaking as in a beam/column failure.
Figure 3.51 Breaking up of poorly detailed joints may result in failure of columns.
Figure 3.52 Collapse patterns: (a) inadequate shear strength, (b) inadequate beam/column strength, and (c) tension compression failure due to overturning.
In this case, stability is lost for both the roof system and the wall. The vertical support of the roof is lost, as well as the horizontal outofplane support of the wall.
This can lead to loss of stability and/or progressive collapse in part of a structure as shown schematically in Figure 3.53. Observe that in most collapses, the driving force is the gravity load acting on a structure that has become unstable due to horizontal offset or insufficient vertical capacity. In addition, subsequent lateral loads from aftershocks can increase the offset, exaggerating the instability. The structure is often disorderly as it collapses. Some parts may remain supported by uncollapsed adjacent bays.
Schematically shown in Figure 3.54, this type of collapse can be partial or complete. It is usually caused when columns or walls, weakened by earthquake motion, are unable to support the heavy floors. Tall, moment frame structures may overturn, but more often, they collapse within their plan boundaries due to high gravity forces. Many partially collapsed concrete frame structures will contain parts of slabs and/or walls that are hanging off an uncollapsed area. This has been observed in corner buildings when only the streetfront bays collapse due to torsion effects.
Figure 3.53 Local column failure.
Figure 3.54 Heavy floor collapse. Major force is due to gravity loads due to inertial force at each floor. If column fails, heavy floors collapse.
Figure 3.55 Torsion effect: unsymmetrical placement of walls can lead to collapse.
Torsion may occur in frame structures when an infill wall is placed in between columns. These infilled bays become stiffer than other parts of the building and cause an eccentric condition that can lead to collapse (see Figure 3.55).
This occurs in buildings that are configured such that they have significantly less stiffness because much fewer or no walls are provided in the first story than in the stories above (see Figure 3.56). The collapse is often limited to the one story only, as the building becomes one story shorter.
This can occur when a midstory is configured with much different stiffness than the stories above and below. Examples are when a story has no walls and the ones above and below have significant walls, or when a story has stiff, short columns and the ones above and below have longer, more limber columns.
Damage due to pounding normally occurs when two adjacent buildings have floors that are at different elevations. The very stiff/strong edge of a floor in one building may cause damage to or even collapse of the adjacent building's column when they collide.
Figure 3.56 Softstory collapse: lower story that is weakened by too many openings becomes racked resulting in failure of firststory columns.
Earthquakes are catastrophic events that occur mostly at the boundaries of portions of the Earth's crust called tectonic plates. When movement occurs in these regions, along faults, waves are generated at the Earth's surface that can produce very destructive effects.
Aftershocks are smaller quakes that occur after all large earthquakes. They are usually most intense in size and number within the first week of the original quake. They can cause very significant reshaking of damaged structures, which makes earthquakeinduced disasters more hazardous. A number of moderate quakes (6+ magnitude) have had aftershocks that were very similar in size to the original quake. Aftershocks diminish in intensity and number with time.
They generally follow a pattern of there being at least one large (within one Richter magnitude) aftershock, at least 10 lesser (within two Richter magnitude) aftershocks, 100 within 3, and so on. The Loma Prieta earthquake had many aftershocks, but the largest was only magnitude 5.0 with the original quake being magnitude 7.1.
Some of the most destructive effects caused by earthquake shaking are those that produce lateral loads in a structure. The intense ground shaking causes the foundation of a building to oscillate back and forth in a more or less horizontal plane. The building mass has inertia and wants to remain where it is, and therefore, lateral forces are exerted on the mass in order to bring it along with the foundation. For analysis purposes, this dynamic action is simplified as a group of horizontal forces that are applied to the structure in proportion to its mass and to the height of the mass above the ground. In multistory buildings with floors of equal weight, the loading is further simplified as a group of loads, each being applied at a floor line and each being greater than the one below, thus resulting in a triangular distribution. Seismically resistant structures are designed to resist these lateral forces through inelastic action and must, therefore, be detailed accordingly. Earthquake loads are often expressed in terms of a percent of gravity weight of the building and can vary from a few percent to near 50% of gravity weight. There are also vertical loads generated in a structure by earthquake shaking, but these forces rarely overload the verticalloadresisting system. However, earthquakeinduced vertical forces have caused damage to structures with high dead load compared to design live load. These vertical forces also increase the chance of collapse due to either increased or decreased compression forces in the columns. Increased compression may overload columns while decreased compression reduces column bending strength (Taranath, Book 6, Chapter 3).
In earthquake engineering, we deal with random variables, and therefore, the design must be treated differently from the orthodox design. The orthodox viewpoint maintains that the objective of design is to prevent failure; it idealizes variables as deterministic. This simple approach is still valid when applied to design under only mild uncertainty. But when confronted with the effects of earthquakes, this orthodox viewpoint is no longer applicable. This is because in dealing with earthquakes, we must contend with appreciable probabilities that failure will occur in the near future. Otherwise, all the wealth of this world would prove insufficient to design and construct structures that are earthquake safe—the most modest structures would be fortresses. We must also face considerable uncertainty while designing engineering systems—whose pertinent properties are still debated to resist future earthquakes—about whose characteristics we know even less.
Although over the years, experience and research have diminished our uncertainties and concerns regarding the characteristics of earthquake motions and manifestations, it is unlikely, though, that there will be such a change in the nature of knowledge to relieve us of the necessity of dealing openly with random variables. In a way, earthquake engineering is a parody of other branches of engineering. Earthquake effects on structures systematically bring out the mistakes made in design and construction, even the minutest mistakes. Add to this the undeniable dynamic nature of disturbances, the importance of soil–structure interaction, and the extremely random nature of it all; in a manner of speaking, earthquake engineering is to the rest of the engineering disciplines what psychiatry is to other branches of medicine. This aspect of earthquake engineering makes it at once challenging and fascinating and gives it an educational value beyond its immediate objectives. If structural engineers are to acquire fruitful experience (Taranath, Book 6, Chapter 3) in a brief span of time, they should be exposed to the concepts of earthquake engineering, even if their interest in earthquakeresistant design is indirect. Sooner or later, they will learn that the difficulties encountered in seismic design are technically intriguing and begin to exercise that nebulous trait called engineering judgment to make allowance for these unknown factors.
Regardless of building type, size, or function, it is clear that the attempt to encourage or enforce the use of regular configurations is frequently not going to succeed; the architect's search for original forms is very powerful.
The seismic standards such as those in ASCE 710 are oriented toward everyday economical building and go a modest route of imposing limited penalties on the use of irregular configurations in the form of increased design forces and, for larger buildings, the use of more advanced analytical methods; both these measures translate into cost penalties that may be acceptable by the building owners. Only two irregularities are banned outright: extreme soft stories and extreme torsion in essential buildings in high seismic zones (i.e., buildings assigned to Seismic Design Category [SDC] E or F). This suggests a strategy that exploits the benefits of the ideal configuration but permits the architect to use irregular forms when they suit the design intentions.
Extreme irregularities may require extreme engineering solutions; these may be costly, but it is likely that a building with these conditions will be unusual and important enough to justify additional costs in materials, finishes, and systems.
A soft or weak story should perhaps never be used: this does not mean that high stories or varied story heights cannot be used, but rather that appropriate structural measures be taken to ensure balanced resistance.
In looking at architectural design through a seismic filter, it appears that many useful and common architectural forms are in conflict with seismic design needs.
The ultimate solution to these conflicts depends on the architect and engineer working together on building design from the outset of the project and engaging in knowledgeable negotiation. It is unfair to expect the engineer to convince the architect of some of the conventional virtues of seismic design, such as simplicity, symmetry, and regularity.
Such discussions on building configurations are valid only for projects in which economy is the paramount objectives. When the architect and the client are looking for highstyle design, the forms will probably be irregular, unsymmetrical, and fragmented. The successful engineer will enjoy the challenges. New methods of analysis will help, but engineers must also continue to develop their own innate feeling on how buildings perform and be able to visualize the complex interaction of building elements that result from many influences, both functional and aesthetic.
In one sense, the static loading condition may be considered merely as a special form of dynamic loading. It is convenient, however, to analytically distinguish between the static and the dynamic components of the applied loading, to evaluate the response to each type of loading separately, and then to superpose the two response components to obtain their total effect. When treated thusly, the static and dynamic methods of analysis become fundamentally different in character.
The term dynamic may be defined simply as time varying; thus, a dynamic load is any load of which its magnitude, direction, and/or position varies with time. Similarly, the structural response to a dynamic load, that is, the resulting stresses and deflections, is also time varying or dynamic.
Even though seismic ground motions are highly oscillatory and irregular in character, the resulting loads may be considered as deterministic dynamic loads, as opposed to random dynamic loads. Therefore, the emphasis in earthquake engineering here is on development of methods of deterministic dynamic analysis, rather than on random or nondeterministic analysis that requires statistical information about dynamic response quantities.
Figure 3.57 Characteristics of dynamic loading: (a) simple harmonic, (b) impulsive, and (c) long duration. Note: denotes acceleration.
In general, structural response to any dynamic loading is expressed basically in terms of the displacements of the structure. Thus, a deterministic analysis leads directly to displacement time histories corresponding to the prescribed loading history; other related response quantities, such as stresses, strains, and internal forces, are usually obtained as a secondary phase of the analysis. On the other hand, a nondeterministic analysis provides only statistical information about the displacements resulting from the statistically defined loading; corresponding information on the related response quantities are then generated using independent nondeterministic analysis procedures.
From an analytical point of view, it is convenient to divide deterministic dynamic loadings into two distinct categories, periodic and nonperiodic. A periodic loading exhibits the same time rotating machinery variation successively for a large number of cycles as typified by the unbalanced rotating machinery.
Nonperiodic loadings, on the other hand, may be either shortduration impulsive loading or relatively longduration general loading. A blast or explosion is a typical source of impulsive load; for such shortduration loads, special simplified forms of dynamic analysis may be employed. On the other hand, a general, relatively longduration loading suck as might from an earthquake can be treated only by a completely general dynamic analysis procedure (see Figure 3.57 for characteristics of different types of dynamic loading described earlier).
A good number of loads that occur in buildings may be considered stationary requiring only static analysis. Although almost all loads except dead loads are transient, meaning they change with time, customarily in structural design, they are treated as static loads. For example, lateral loads due to transitory wind gusts that are often dynamic are usually treated as static loads. To realize acceptable results, the dynamic characteristics due to the sudden variation of wind velocity are taken into account by including a gust factor in the determination of wind loads. Therefore, the analysis reduces to a static case requiring but one unique solution.
Let us consider a building that, instead of being buffeted by wind, is subjected to ground motions due to an earthquake. The seismic shock causes the foundation of the building to oscillate back and forth, principally in a horizontal plane. The building would follow the movement of the ground without experiencing lateral loads if the ground oscillation took place very slowly over a long period of time. The building would merrily ride to the new displaced position as if no load was ever applied to it. On the other hand, when the ground moves suddenly as in an earthquake, the building mass, which has inertia, attempts to stay in its preearthquake position, thus resisting the accelerations of the structure. The resulting initial forces in a building with several floors may be visualized as a group of horizontal forces applied at various floors in proportion to the floor mass and its height above the ground.
These earthquake forces are considered dynamic, because they vary with time. Since the load is time varying, the response of the structure, including deflections, axial and shear forces, and bending moments, is also time dependent. Therefore, instead of a single solution, a separate solution is required to capture the response of the building at each instant of time for the entire duration of an earthquake. Because the resulting inertial forces are a function of building accelerations, which are themselves related to the inertial forces, it is necessary to formulate the dynamic problem in terms of differential equations.
A special feature of earthquake excitation of structures is that it is applied in the form of support motions that are neither harmonic nor periodic. Defining such excitations that vary arbitrarily with time is perhaps the most difficult and uncertain phase of predicting structural response to earthquakeinduced ground motions. However, when these motions have been established, the calculation of response of a given structure is a standard structural dynamics problem.
Simply stated, the dynamic load factor (DLF) is the ratio of the dynamic response such as deflection of an elastic system to the corresponding static deflection that results from the static application of a time–load function Ft. In a static problem, time variation of the load has no effect on the response since it is assumed that the load is applied in a gradual manner over a long period of time. In dynamic problems, however, the time it takes to apply the load has a large influence on the structural behavior and thus must be given consideration in the analysis. Shown in Figure 3.58 are some examples of time–load function, Ft, in which load variation is graphed with respect to time t. The graphs are for unit impulse force, step force, ramp or linearly increasing force, and step force with finite rise time.
Figure 3.58 Time–load functions: (a) unit impulse force, (b) stepforce (c–d) ramp or linearly increasing force, (e) triangular pulse force, and (f) halfcycle sine pulse force.
Figure 3.59 Dynamic response of cantilever. Note: DLF = 2 for suddenly applied point load.
Consider the cantilever beam shown in Figure 3.59 with an empty container of weight W_{D} at the free end. Ignoring the selfweight of the beam and using common notations, the deflection of the cantilever due W_{D} is given by
Let the container be filled with water of weight W_{w}, flowing gradually, say, drop by drop into the container. The additional deflection due to the weight of water W_{w} is
Instead of a dropbydrop loading, if all water is suddenly gushed into the container instantly, the cantilever deflection will no longer be the same as the static deflection Δ_{w}. We know, by intuition, it will be larger than Δ_{w} (and in our case, it can be shown that it is two times the static deflection Δ_{w}).
Hence, the DLF = 2 for this suddenly applied load case. Observe that in addition to experiencing a larger deflection due to dynamic load, the cantilever also springs up and then down and continues to do so about its static position. In other words, the cantilever, when subject to a sudden load, exhibits a dynamic response by vibrating typically in a sinusoidal manner. But for the effects of damping, it would continue to do so indefinitely as shown schematically in Figure 3.60a, as opposed to damped free vibrations shown in Figure 3.60b.
Earthquake ground motions are erratic; they are neither harmonic nor periodic and vary arbitrarily with time and last no more than a few seconds. The longest recorded earthquake lasted only about 90 s although it may have seemed like an eternity for those experiencing the trauma. Although ground motions are unrhythmical in an earthquake, we may, for analytical purposes, consider their effects as equivalent to the summation of the responses to appropriately scaled forces such as those shown in Figure 3.58.
Thus, the response spectrum of singledegreeoffreedom (SDOF) systems to ground acceleration, as explained presently in this chapter, can be plotted from the DLFs determined for a chosen set of time–load functions such as those shown in Figure 3.58.
It should be noted that the DLF also varies with time (see Figure 3.61). However, in structural engineering, we are interested in the maximum design values (such as displacements, stresses) that occur during an earthquake. Therefore, the maximum response, irrespective when it occurs during an earthquake, is of concern. It is worthwhile to note again that when we use response spectra to calculate design values for SDOF systems, it is not necessary to amplify the design values by DLF, because the computationally intensive dynamic analysis has been completed in generating the response spectra.
Figure 3.60 Vibrations of oneDOF systems: (a) undamped and (b) damped.
Figure 3.61 Maximum response of oneDOF undamped systems to rectangular and triangular pulse having zero rise time. Note: T = system period and td = pulse duration.
In our daytoday structural engineering practice, we are quite comfortable with the idea that if a member is overstressed, say in flexure, we simply increase the member size, and thus its section modulus, to decrease the flexural stresses. This concept is as old as engineering itself and has served us quite well in our design of structures subject to static loads. However, this thought of automatically reducing the stresses by increasing member sizes may not always hold together when designing structures subject to seismic ground motions.
In the case of earthquake excitation, the increase in member size resulting in higher stiffness shortens the natural vibration period that may have the effect of increasing the equivalent static force. Whether the corresponding stress decreases or increases by increasing the size of the member depends on the increase in section modules, S, and the increase in the equivalent static force that, in turn, depends on the response spectrum.
A 15 ft tall cantilever, an 8 in. nominal diameter standard steel pipe supporting a 40 kip weight at the top as shown in Figure 3.62. The properties of the column are
Figure 3.62 Cantilever columns with weight W at top.
Figure 3.63 Acceleration response spectrums.
This example (admittedly cooked up to make a point) would make you think twice before you add stiffness to seismic structural systems. The concept of designing with sufficient strength has been viewed traditionally as a key to more effectively control the behavior of buildings. However, in certain instances, the benefits associated with an increase in strength may be small or even have a negative impact. The doctoredup example given here is not intended to discount strength as an important design consideration but rather to allude to the possible negative impacts. Keep in mind the excessive strength of the yielding element will impose more demand on the brittle components along the lateral load path. A better design strategy would be to increase ductility; treat ductility as a wealthy person treats money—you cannot have too much of it.
Perhaps the reader will recall that in Chapter 2, it was stated that the load on a building due to a wind gust becomes a dynamic load if the rise time of the gust is considerably shorter than the fundamental period of the building. Conversely, if the rise time is the same as or higher than the building's period, it may be considered as a static load.
Figure 3.64 DLF for undamped SDOF system: (a) Tg/TB = 0.2, (b) Tg/TB = 2.5. Note: DLF = 2 for 1 s gust, and 1.0 for 12.5 s gust.
This characteristic that depends entirely on rise time of wind gust, T_{g}, is shown in Figure 3.64, T for two values of (T_{g}/T_{B}), the ratio of rise time T_{g} of the gust to the fundamental period, T_{B}, of the building. Observe that for (T_{g}/T_{B}) = 2.5 (which signifies a wind gust of relatively long rise time), DLF is nearly equal to 1.
To understand the significance of these figures, consider a tall building with a period, T_{B}, of 5 s (frequency of 0.2 H_{z}) subject to a constant wind pressure of 30 psf and then suddenly to a 1 s gust of 15 psf. A 1 s gust simply means that wind pressure has increased to 30 + 15 = 45 psf in a time period of 1 s. In other words, the 30 psf of constant pressure has increased to 45 psf, with the increase taking place in a short time interval of 1 s, and remains at 45 psf for the time duration of interest.
Referring to Figure 3.50a, for (T_{g}/T_{B}) = (1/5) = 0.20, the DLF is equal to 2.0 signifying that the dynamic effect of wind gust is as though it is an impulse load culminating with a maximum DLF equal to 2.0.
Next, consider the same increase in the wind load but assume that the increase occurs over a relatively long time interval of say, 12.5 s. As can be seen in Figure 3.64b, for the ratio $\left({T}_{g}/{T}_{B}\right)=\left(12.5/5\right)=2.5$
, the corresponding DLF is close to unity, signifying that the dynamic effect can, in fact, be ignored for this case.The concept of DLF presented earlier for wind gust is equally applicable to seismic design, except the load–time functions are referenced to support accelerations.
A structural dynamic problem differs from its static loading counterpart in two important respects. The first difference as implied by the definition is the timevarying nature of the dynamic problem. Because both loading and response vary with time, it is evident that a dynamic problem does not have a single solution, as static problem does; instead, the engineer must establish a succession of solutions corresponding to all times of interest in the response history. Thus, a dynamic analysis is clearly more complex and time consuming than a static analysis.
The second and more fundamental distinction between static and dynamic problems is illustrated in Figure 3.65. If a singlebay, singlestory frame is subjected to a static load F, as shown in Figure 3.65, the internal moments and shears in the columns and deflected shape of the frame depend only upon this load, and they can be computed by established principles of force equilibrium. On the other hand, if the F(t) is applied dynamically, as shown in Figure 3.66, the resulting displacements of the frame depend not only upon this load but also upon inertial forces, which oppose the accelerations producing them. Thus, the corresponding internal moments and shears in the columns must equilibrate not only the externally applied force F(t) but also the inertial forces resulting from the accelerations of the beam.
Inertial forces, which resist accelerations of the structure in this way, are the most important distinguishing characteristic of a structural dynamics problem. In general, if the inertial forces represent a significant portion of the total load equilibrated by the internal elastic forces of the structure, then the dynamic character of the problem must be accounted for in its solution. On the other hand, if the motions are so slow that the inertial forces are negligibly small, the analysis of response for any desired instant of time may be made by static structural analysis procedures even though the load and response may be time varying.
Figure 3.65 Portal frame subject to static loads. Note: F = ky, K = 24 EIc/h3.
Figure 3.66 Portal frame subject to earthquake ground motions.
The design of structures to perform satisfactorily under expected seismic conditions requires that realistic earthquake loadings be specified and that the structural components be proportioned to resist these and other combined loads within the limits of certain design requirements. In regions of high seismicity, earthquake loading is often critical among the types of loading that must be considered because a large earthquake will usually cause greater stresses and deformations in the various critical components of a structure than will all other loadings combined, yet the probability of such an earthquake occurring within the life of the structure is very low. In order to deal effectively with this combination of extreme loading and low probability of occurrence, a strategy based on the following dual criteria has usually been adopted:
Currently, the trend is to strengthen the second of these criteria for critical and expensive structures by calling for limited repairable damage, thus focusing not only on life safety but on protection of financial investment as well.
A special feature of earthquake excitation of structures, compared with most other forms of dynamic excitation, is that it is applied in the form of support motions rather than as external loads; thus, the effective seismic loadings must be established in terms of these motions. Defining the support motions is the most difficult and uncertain phase of the problem of predicting structural response to earthquakes. When these input motions have been established, however, the calculation of the corresponding stresses and deflections in any given structure is a standard problem in structural dynamics.
The earthquake excitation considered to act on a structure is the freefield ground motion at support points. Inherent in this treatment is the assumption that the same freefield ground motion acts simultaneously at all support points of the structure. This assumption is equivalent to considering the foundation soil or rock to be rigid. This hypothesis clearly is not consistent with the concept of earthquake waves propagating through the Earth's crust from the source of energy release; however, if the base dimensions of the structure are small relative to the predominant wavelengths, the assumption is acceptable.
When specifying input motions at the base of a structure, it should be recognized that the actual structure base motions during an earthquake may be significantly different from the corresponding freefield motions that would have occurred without the structure being present. This soil–structure interaction effect will be of slight importance if the foundation is relatively stiff and the structure is relatively flexible; in this case, the structure transmits little energy into the foundation and the freefield motions are adequate measures of the actual foundation displacements. On the other hand, if a stiff structure is supported on a deep, relatively soft layer of soil, considerable energy will be transferred from the structure to the soil and the base motions will differ from those experienced by the soil under freefield conditions. This soil–structure interaction mechanism is independent of, and in addition to, the effect of local soil conditions on the freefield motions.
Given
A moment frame with bay length L = 16 ft, height h = 12 ft, elastic modulus E = 29,000 ksi. The columns are W8 × 67 with I_{c} = 272 in.^{4} and are fixed at the base (see Figure 3.67a).
The beams are relatively heavy, W36 × 330 with I_{b} = 23,300 in.^{4}
Required
Assuming the beam is infinitely rigid as compared to the column, compute the maximum displacement and maximum base shear produced in this frame by an earthquake having the acceleration response spectrum shown in Figure 3.67b. Assume that the effective weight including the weight of the beam is equal to 596 kips.
Solution
The first step is to determine the vibration period of the structure. The circular frequency of the structure is given by
where
Figure 3.67 Example of portal frame subject to ground motions. (a) Portal frame, (b) acceleration spectrum, and (c) tripartite (DVA) spectrum.
Because the beam is considered rigid, that is, its flexural stiffness is large as compared to the column stiffness, the lateral stiffness, k, of the frame is
The mass
From Figure 3.67b, the spectral acceleration for this period is 0.61g. Hence, the maximum base shear
Shown in Figure 3.67 is a tripartite response spectrum of the same earthquake shown in Figure 3.63. The procedure for developing tripartite response spectrum is explained presently. Suffice here to note that the displacement for the example portal frame can be read directly from the tripartite response spectrum without performing the calculations.
We begin our study of dynamic equilibrium in structural dynamics with a simple singlestory portal frame shown in Figure 3.68, where the mass is distributed along the girder and only horizontal motions are considered. The supporting structure consisting of the beams and columns is assumed massless. The rigid frame is defined as a onedegree system in which only one type of motion is possible, or in other words, the position of the mass at any instant can be defined in terms of a single coordinate, which, in our case, is the horizontal displacement u. Thus, the mass can move in a horizontal direction only. As an example of dynamic analysis, let us determine the motion of this mass resulting from the application of a timevarying horizontal force.
Figure 3.68 Idealized SDOF system.
A convenient method of deriving the equation of motion is by the use of D'Alembert's principle of dynamic equilibrium. Having been trained to think in terms of equilibrium of forces, structural engineers may find this method particularly appealing. This principle states that with inertial forces included, a system is in dynamic equilibrium at each time instant. The inertial force, a force equal to the product of mass times its acceleration and acting in a direction opposite to the acceleration, is a fictitious force that allows us to treat structural dynamics in exactly the same manner as a problem in static equilibrium. From Figure 3.69, the equilibrium equation is
where
Throughout this text, $\dot{u}$
and ü will be used to the first and second derivatives of displacement u with respect to time t. In other words, $\dot{u}=\left(du/dt\right)$ is the velocity and $\ddot{u}=\left(d{u}^{2}/d{t}^{2}\right)$ is the acceleration.The rearranged and expanded form of Equation 3.13:
This is a differential equation of second order, which may be solved by using the principles of calculus. However, the solution of the differential equations will not be discussed herein since it is assumed that the reader is familiar with such procedures. The deliberate omission of mathematical techniques in this text enables us to concentrate on the physical phenomena involved and helps us to develop a physical feel and intuition for dynamic response, which is necessary for successful analysis of more complicated dynamic problems.
Figure 3.69 D'Alembert's principle of dynamic equilibrium: (a) portal frame subject to dynamic force Ft, (b) equivalent dynamic model, and (c) freebody diagram.
Consider, first, the elementary case where F_{t} equals zero, that is, there is no external dynamic excitation. Such a structure is said to be undergoing free vibration when it is disturbed from its static equilibrium position and then allowed to vibrate. Motion occurs only if the system is given an initial disturbance, such as an initial displacement u_{0}. Imagine that the mass m shown in Figure 3.70 is pulled horizontally by a rope and then suddenly released at t = 0. The resulting motion unaffected by any external force is the free vibration. The differential equation of motion for this case is given by
The solution for this equation in terms of displacement u is given by
A plot of displacement for initial displacement u_{0} is given in Figure 3.71. The parameter ω in the equation is called the natural frequency of the system and is given by the relation
As stated previously, D'Alembert's principle, named after its discoverer, the French mathematician Jean le Rond d'Alembert, permits the reduction of a problem in dynamics to one in statics. This is accomplished by intruding a fictitious force equal in magnitude to the product of the mass of the body and its acceleration and directed opposite to the acceleration. While D'Alembert's principle is merely another way of writing Newton's second law of motion, F = Ma, it has the advantage of changing a problem in engineering dynamics into a problem in statics.
Figure 3.70 Concept of free vibrations.
Figure 3.71 Displacement plots of free vibrations with damping.
The difficulties encountered with the design of structures to withstand earthquakeinduced ground motions of the base structure are technically intriguing. The major problem lies in the prediction of the character and intensity of the earthquakes to which a structure might be subjected to during its life. Another difficulty lies in the fact that a realistic analysis for earthquakes should account for inelastic behavior of the structure because very few structures could withstand a strong earthquake without some plastic deformation. In this section, attention will be restricted to a few basic concepts, which will provide a foundation for understanding seismic design principles.
Consider a portal frame subject to a displacement of the ground denoted by u_{g}. The total displacement u^{t} of the mass at each instant of time is given by
The rigid body component of the displacement of the structure produces no internal forces; only the relative motion u between the base and the mass produces elastic forces.
Thus, the relative displacement u(t) of the structure due to ground acceleration ü(t) will be identical to the displacement u(t) of the structure if its base were stationary and if it were subjected to an external force = mü_{g}(t) acting in a direction opposite to the ground motion. The ground acceleration can therefore be replaced by an effective earthquake force
This force is equal to mass times the ground acceleration acting opposite to its acceleration.
It was seen earlier that the deflection of the cantilever due to sudden application of the load is twice as much as the static deflection. If we knew this information, beforehand, we could have obtained the maximum dynamic response without going through the dynamic analysis procedure. However, the procedure, in a manner of speaking, is still a dynamic analysis, because it uses the vibration properties of the structure and the dynamic characteristics of ground motion. It is just that the engineers do not have to carry out any excruciating dynamic analysis. Somebody has done these in telling us that DLF = 2 for this particular load function.
Observe that this argument also holds well when we use acceleration values from a response spectrum. There is no need to multiply the acceleration value by a DLF because this has already been done prior to plotting the response spectrum.
The beamtocolumn stiffness ratio p indicates how much the system may be expected to behave as a frame. For p = 0, the beams impose no restraint on joint rotations behaving as if they were pin connected to the columns. The portal frame then behaves as if it consists of two independent cantilever columns fixed at the base. For p = ∞, the beam restrains completely the joint rotations, and the portal frame behaves as a shear beam with doublecurvature bending of the columns in each story. It is thus evident that an intermediate value of p represents a combination of both the cantilever bending mode of columns and the shear mode deformations due to beam bending. Observe that strong column–weak beam design is what we try to realize in seismic design of moment frames.
To understand the need for numerical integration in structural dynamics, perhaps it is instructive to recall the definition of differential equations and their application in engineering.
A differential equation in a broad sense is a mathematical formulation for the determination of an unknown variable that itself relates to the derivatives of the variable orders.
Take, for example, the use of differential equations in determining the velocity of a ball falling through the air. Consider only gravity and air resistance. The ball's acceleration toward the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance is proportional to the ball's velocity. Therefore, the ball's acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation.
Very few simple differential equations can be solved by using explicit formulas. If an explicit solution is not available, which usually is the case in seismic design, the solution is approximated using computers. In earthquake engineering, we call these methods timehistory or numerical methods.
Anyone who tries to explain the concept of numerical integration in less than 30 pages is asking for trouble. Here, we take the risk and go for relatively fewer pages by describing only the dynamic response of SDOF systems. The reader is referred to the vast body of literature that exists in several textbooks for further study.
Structural dynamics is too often taught as a course in advanced mathematics, making this approach unnecessarily difficult for practicing engineers. Many engineers may find the mathematical manipulation so intriguing that they fail to develop physical understanding so essential to good design. In this text, we avoid mathematical complexities, which, although are useful in understanding advanced dynamic topics, are not necessary for developing an insight into practical seismic designs.
Today, in the year 2013, the use of closedform solutions for analyzing practical dynamic problems, very rarely if ever, is undertaken. As in other fields, most often numerical analysis techniques are used to execute solutions to dynamic problems. Therefore, the author believes that summarizing a numerical analysis executed by hand develops a physical feel for dynamic behavior much more rapidly than does the closedform solution using differential equations.
Consider an SDOF system, such as the portal frame shown in Figure 3.72. In terms of dynamic behavior, the system is equivalent to a spring connected to a mass, a model most often illustrated in structural dynamics textbooks. Let us recognize at the outset that this simple system is not only just an ideational model but in fact represents practical singlebay, singlestory portal frames.
Figure 3.72 Portal frame subject to ground motions.
Figure 3.73 Equivalent load–time function due to base motions.
To demonstrate the numerical integration procedure, suppose that the base of the portal frame is subjected to a given set of ground accelerations. Let us assume further that the accelerations result in an equivalent dynamic load at the top, as shown in Figure 3.73 by the load–time function. To simplify the numerical solution, we will make two assumptions: first, the weights of the columns are negligible, and second, the girder is sufficiently rigid to prevent significant rotation at the tops of the columns. These assumptions are not necessary for demonstrating the analysis but will serve to simplify the problem and, in fact, are essentially correct for many actual frames of this type.
In the load–time function shown in Figure 3.73, the dynamic load is equal to 50 lbs at the time t = 0 s, rising linearly to 100 lbs at t = 0.10 s and dropping back to 50 lb at time t = 0.14 s. The purpose here is to determine the variation of the horizontal displacement of the beam, with time, starting with the system at rest t = 0 all the way to t = 0.14 s.
The example portal frame is an SDOF system defined as one in which only one type of motion is possible. Or, in other words, the position of the system at any instant can be defined in terms of a single coordinate. In our case, the mass m can move in a horizontal direction only.
Before considering a specific example, we shall discuss the process of numerical integration in general terms. This is a procedure by which the differential equation of motion is solved step by step, starting at zero time, when the displacement and velocity are presumably known. The time scale is divided into discrete intervals, and one progresses by successively extrapolating the displacement from one time station to the next.
There are many such methods available. One of the simpler versions presented by John M. Biggs is called the constantvelocity or lumpedimpulse procedure. The reader is referred to Biggs’ textbook for further explanation of this procedure.
To illustrate the numerical integration procedures, consider again the portal frame subjected to the load–time function shown in Figure 3.73. It is desired to determine the variation of displacement with time, starting with the system at rest at t = 0. Substituting into Equation 3.13 and being careful to keep the units consistent, the equation of motion is written as follows:
or
Thus, knowing the load and the displacement at any time enables one to compute acceleration at the time.
The next step is to select a time interval for the numerical integration. As mentioned earlier, this should not be greater than onetenth of the natural period of the system. The natural period T of a onedegree system is given by
This, in this example, is 0.198 s. The natural frequency f is the inverse of the natural period or 5.04 cps. The natural circular frequency ω is 2πf or 31.6 rad/s.
Onetenth of the natural period is approximately 0.02 s, and this value will be used in the computation. However, a second criterion must be considered when selecting the time interval: the interval should be small enough to represent properly the variation of load with time. In this example, it will be noted that the stations (i.e., Δ = 0.02) occur at the sudden breaks in the load function and, furthermore, that the interval is small enough so that points at this spacing accurately represent the load function (Figure 3.73). Therefore, the time interval selected is satisfactory.
The result of the complete calculation is plotted in Figure 3.74 where displacement versus time is shown. The ordinate also represents the time variation of the spring force if the displacements are multiplied by the spring constant k. To assist in the interpretation of the result, the hypothetical displacement corresponding to the static application of the load at any instant is also plotted. The maximum displacement, which occurs at 0.12 s, corresponds to a spring force of 159 lb, which is 1.59 times the maximum external load. Subsequent peaks are somewhat smaller and correspond to a spring force of 157 lbs. The latter peak value would remain constant indefinitely since we have not included damping in the present example. The time interval between successive positive (or negative) peaks is exactly equal to the natural period of the system. After the load becomes constant at 0.14 s, the spring force carries, in a sinusoidal fashion, the positive and negative peaks being equidistant above and below the value of the external load.
The parameters of the idealized system needed in performing the numerical integration are as follows:
The spring constant k is simply equal to the inverse of the deflection at the top of the frame due to a unit horizontal load, and the equation employed may be easily verified by simple elastic analysis. Having these parameters, we would then proceed to determine the dynamic displacements for the given load function. These displacements would be equal to the actual horizontal deflections at the top of the frame. The spring force computed for the ideal system would at all times be equal to the total shear in the two columns, and the maximum column bending moment would be given by 6Elu/h^{2}. Thus, the dynamic stresses at any time could be easily determined.
Figure 3.74 (a) Dynamic response of portal frame showing plot of displacement versus time. (b) Numerical integration, lumpedimpulse procedure.
The process of numerical integration, also referred to as timehistory analysis, is a method in which the differential equation of motion is solved step by step, starting at zero time, when the displacement and velocity are presumably known. The time duration of earthquake of interest is divided into discrete intervals, Δt, and we progress by successively extrapolating the displacement from one time station to the next.
There are many methods available for successive extrapolation of displacements, but only one of the more simple versions called the constantvelocity technique is discussed here.
Suppose that the curve shown in Figure 3.74b represents the displacement–time variation for a dynamic system. Suppose, further, that a successive numerical analysis for the determination of displacement is in progress with the displacements x_{t} at time station t and x_{t−1} at time station (t − 1) previously known. Our task is to determine the next displacement at station (t + 1) by extrapolation. This could be done by the following selfevident formula:
where
Substituting Equation 3.23 into Equation 3.22, the following recurrence formula is obtained:
The acceleration ${\ddot{x}}_{t}$
at time station t can be determined by the use of D'Alembert's principle of dynamic equilibrium:In keeping with the policy stated at the beginning of this section, we will not go further into explaining the process of numerical integration. Suffice it to note that a recurrence formula such as the one given in Equation 3.24 plays a pivotal role in timehistory analysis of dynamic systems. The reader is referred to standard textbooks on structural dynamics for an indepth discussion of numerical integration techniques.
In order to develop a physical feeling for dynamic behavior, consider what happens to our portal frame that has columns hinged to the ground, when the ground moves. Assume that you are sitting at the top of the beam and an earthquake occurs. The ground moves, but you do not, at least for the first small interval of time. You will, of course, move along with ground motions right after the first shock.
The initial differential displacement, d_{o}, between the mass (that is you) and the ground causes strain energy to be stored in the columns. The displacement may be, for analytical purposes, considered as though it is due to an equivalent force F_{o} applied at top.
F_{o} is given by
This force, F_{o}, which is in dynamic equilibrium, is given by Newton's second law of motion. Force equals mass times acceleration:
The acceleration you feel during the ride is given by
The inertial force imparted on the mass is given by F = k(d − x). Using subscripts to identify sequential increments, the acceleration, $\ddot{x}$
, experienced by the mass is ${\ddot{x}}_{i}=\left(K/M\right)\left({d}_{i}{x}_{i}\right)$ . Once we know the acceleration experienced by the mass, we can determine the corresponding values for the velocity and displacement. The velocity $\left(\dot{x}\right)$ and displacement (x) of the mass can be calculated through the use of integral calculus if the acceleration can be described in terms of a continuous function. If not, we have no choice but to use numerical techniques.Earthquake motions are erratic. It is not always possible to describe ground accelerations as continuous functions. Therefore, numerical integration is a more appropriate means of predicting successive velocities and displacements. The change in velocity experienced by the mass during a time interval of Δt is then determined by multiplying the average acceleration, which occurs during this time interval by the interval (Δt). The change in displacement is determined by multiplying the average velocity experienced by the mass during the time interval (Δt). Several numerical methods can be used to predict incremental changes in velocity and displacement.
It should be noted that given a small enough time interval, these methods will describe the characteristics of the dynamic response and maximum values sought in structural design, to a degree of accuracy consistent with our ability to predict ground motions.
In order to perform responsehistory analysis, it is necessary to have a digitized ground motion acceleration record. In linear responsehistory (LRH) analysis, the stiffness of the structure K is assumed to be independent of the prior displacement history. In nonlinear responsehistory (NRH) analysis, the structure's stiffness at a given instant of time t is dependent on the displacement history up to that point in time and varies to account for yielding, buckling, and other behaviors that may have occurred earlier in the structure's response.
Responsehistory analysis is useful because it allows solutions of the deflected shape and force state of the structure at each instant of time during an earthquake. Since each earthquake record has different characteristics, the results obtained from responsehistory analysis are valid only for the particular earthquake record analyzed. Therefore, when performing responsehistory analysis to determine forces and displacements for use in design, it is necessary to run a suite of analyses, each using different ground motion records as input. Present building codes require a minimum of three records. If three records are used, the maximum forces and displacements obtained from any of the analyses must be used for design purposes. If seven or more records are used, the code permits use of the mean forces and displacements obtained from the suite of analyses.
In design practice, LRH analysis is seldom used. This is because for design purposes, one is usually interested only in the maximum values of the response quantities (forces and displacements), and these quantities can more easily be approximated by an alternative form of analysis known as response spectrum analysis. However, NRH analysis is an essential part of the design of structures using seismic isolation or energy dissipation technologies. It is also routinely used in highend performancebased design approaches.
The word spectrum in seismic engineering conveys the idea that the response of buildings having a broad range of periods is summarized in a single graph. For a given earthquake motion and a percentage of critical damping, a typical response spectrum gives a plot of earthquakerelated responses such as acceleration, velocity, and deflection for a complete range, or spectrum, of building periods. An understanding of the concept of response spectrum is pivotal to performing seismic design.
A response spectrum (Figures 3.75 and 3.76) may be visualized as a graphical representation of the dynamic response of a series of progressively long cantilever pendulums with increasing natural periods subjected to a common lateral seismic motion of the base. Imagine that the fixed base of the cantilevers is moved rapidly back and forth in the horizontal direction, its motion corresponding to that occurring in a given earthquake. A plot of maximum dynamic response, such as accelerations versus the periods of the pendulums, gives us an acceleration response spectrum as shown in Figure 3.75, for the given earthquake motion. In this figure, the absolute value of the peak acceleration occurring during the excitation for each pendulum is represented by a point on the acceleration spectrum curve. As an example, an acceleration response spectrum from the 1940 EI Centro earthquake is illustrated in Figure 3.77. Using ground acceleration as an input, a family of response spectrum curves can be generated for various levels of damping, where higher values of damping result in lower spectral response.
To establish the concept of how a response spectrum is used to evaluate seismic lateral forces, consider two SDOF structures: (1) an elevated water tank supported on columns and (2) a revolving restaurant supported at the top of a tall concrete core (see Figure 3.78). We will neglect the mass of the columns supporting the tank and consider only the mass m_{1} of the tank in the dynamic analysis. Similarly, the mass m_{2} assigned to the restaurant is the only mass considered in the second structure. Given the simplified models, let us examine how we can calculate the lateral loads for both these structures resulting from an earthquake, for example, one that has the same ground motion characteristics as the 1940 El Centro earthquake shown in Figure 3.79. To evaluate the seismic lateral loads, we shall use the recorded ground acceleration for the first 30 s. Observe that the maximum acceleration recorded is 0.33g occurring about 2 s after the start of the record.
Figure 3.75 Graphical description of response spectrum.
Figure 3.76 (a and c) Concept of response spectrum (b and d) pendulums of varying heights representing progressively taller building.
Figure 3.77 Acceleration response spectrum: El Centro earthquake.
Figure 3.78 Examples of SDOF systems: (a) elevated water tank and (b) restaurant atop tall concrete core. Note: The acceleration for the water tank = 26.25 ft/s2 for T = 0.5 s and β = 0.05 and the acceleration for the water tank = 11.25 ft/s2 for T = 1.00 s and β = 0.10.
As a first step, the base of the two structures is analytically subjected to the same acceleration as the El Centro recorded acceleration. The purpose is to calculate the maximum dynamic response experienced by the two masses during the first 30 s of the earthquake. The maximum response such as DVA response of an SDOF system such as the two examples considered here may be obtained by considering the earthquake effects as a series of impulsive loads and then integrating the effect of individual impulses over the duration of the earthquake. This procedure, known as Duhamel integration method, requires considerable analytical effort. However, it is not necessary for us to carry out this procedure because maximum response for postulated earthquakes at a given site is already established. The spectral acceleration responses for the north–south component of the El Centro earthquake, shown in Figure 3.79, are one such example.
Figure 3.79 Recorded ground acceleration: El Centro earthquake.
To determine the seismic lateral loads, assume that the tank and restaurant structures weigh 720 and 2400 kips, with corresponding periods of vibration of 0.5 and 1 s, respectively. Since the response of a structure is strongly influenced by damping, it is necessary to estimate the damping factors for the two structures. Let us assume that the percentage of critical damping £ for the tank and restaurant are 5% and 10% of the critical damping, respectively. From Figure 3.77, the acceleration for the tank structure is 26.25 ft/s^{2}, and the horizontal force in kips would be equal to the mass at the top times the acceleration.
Thus, the horizontal force, F, for the water tank is equal to
Similarly, for the restaurant structure, the force at top is equal to
The two structures can then be designed by applying the seismic loads at the top and determining the associated forces, moments, and deflections. The lateral load, evaluated by multiplying the response spectrum acceleration by the effective mass of the system, is referred to as base shear, and its evaluation forms one of the major tasks in earthquake analysis.
In the example, SDOF structures were chosen to illustrate the concept of spectrum analysis. A multistory building, however, is not typically modeled as an SDOF system because it will have as many modes of vibration as its degrees of freedom (DOFs). For practical purposes, the distributed mass of a building is lumped at discrete levels to reduce the DOFs to a manageable number. In multistory buildings, the masses are typically lumped at each floor level.
Thus, in the 2D analysis of a building, the number of modes of vibration corresponds to the number of levels, with each mode having its own characteristic frequency. The actual motion of a building is a linear combination of its natural modes of vibration. During vibrations, the masses vibrate in phase with the displacements as measured from their initial positions, always having the same relationship to each other. Therefore, all masses participating in a given mode pass the equilibrium position at the same time and reach their extreme positions at the same instant.
Using certain simplifying assumptions, it can be shown that each mode of vibration behaves as an independent SDOF system with a characteristic frequency. This method, called the modal superposition method, consists of evaluating the total response of a building by statistically combining the response of a finite number of modes of vibration.
A building, in general, vibrates with as many mode shapes and corresponding periods as its DOFs. Each mode contributes to the base shear, and for elastic analysis, this contribution can be determined by multiplying a percentage of the total mass, called effective mass, by an acceleration corresponding to that modal period. The acceleration is typically read from the response spectrum modified for damping associated with the structural system. Therefore, the procedure for determining the contribution of the base shear for each mode of a multidegreeoffreedom (MDOF) structure is the same as that for determining the base shear for an SDOF structure, except that an effective mass is used instead of the total mass. The effective mass is a function of the lumped mass and deflection at each floor with the largest value for the fundamental mode, becoming progressively less for higher modes. The mode shape must therefore be known in order to compute the effective mass.
Because the actual deflected shape of a building consists of a linear combination of its modal shapes, higher modes of vibration also contribute, although to a lesser degree, to the structural responses. These can be taken into account through use of the concept of a participation factor. The base shear for each mode is determined as the summation of products of effective mass and spectral acceleration at each level. The force at each level for each mode is then determined by distributing the base shear in proportion to the product of the floor weight and displacement. The design values are then computed using modal combination methods, such as the complete quadratic combination (CQC) or the square root of the sum of the squares (SRSS), the preferred method being the former.
Response spectrum analysis is a means of using acceleration response spectra to determine the maximum forces and displacements in a structure that remains elastic when it responds to ground shaking. For SDOF structures, the maximum elastic structural displacement is given by
In this equation, T is the structure's period and S_{a} is the spectral acceleration obtained from the response spectrum plot at period T. The maximum force demand on the structure is given by
For MDOF structures, the response of the structure can be determined by combining the response quantities for a series of SDOF structures having the same period and mass as each of the structure's modes. For mode i, the maximum inertial force produced in the structure by the earthquake, which is also termed the modal base shear, V_{i}, is given by
In this equation, M_{i} is the modal mass for mode i and S_{a} is the spectral acceleration obtained from the response spectrum at natural period T_{i}.
The results of the analyses conducted for the various modes must be combined in order to obtain an estimate of the structure's actual behavior. Since it is unlikely that peak structural response in all modes will occur simultaneously, statistical combination rules are used to combine the modal results in a manner that more realistically assesses the probable combined effect of these modes. One such combination method takes the combined value as the SRSS of the peak response quantities in each mode.
When several modes have similar periods, the SRSS method does not adequately account for modal interaction. In this case, the CQC technique is used. Most structural analysis software used in design offices today provides the capability to perform these computations automatically.
For SDOF structures, the response spectrum analysis gives exact results, as long as the response spectrum that is used to represent the loading accurately represents the ground motion. However, the response spectra contained in building codes only approximate the ground motion from real earthquakes, and therefore, analysis using these spectra will be approximate. For MDF structures, response spectrum analysis is always approximate because the way that the peak displacements and forces from the various modes are combined does not accurately represent the way these quantities will actually combine in a real structure subjected to ground shaking. Although the results of response spectrum analysis are approximate, it is universally accepted as a basis for earthquakeresistant design, when properly performed.
An acceleration response spectrum as stated previously is a plot of the maximum acceleration that SDOF structures having different periods, T, would experience when subjected to a specific earthquake ground motion. This plot is constructed by performing responsehistory analyses for a series of structures, each having a different period, T, obtaining the maximum acceleration of each structure from the analysis, and plotting this as a function of T. Linear acceleration response spectra are most common and are obtained by performing LRH analysis.
Although the response spectra obtained from each earthquake record will be different, spectra obtained from earthquakes having similar magnitudes on sites with similar characteristics tend to have common characteristics. This has permitted the building codes to adopt standard response spectra that incorporate these characteristics and envelop spectra that would be anticipated at a building site during a DE. The response spectra contained in the building code are called smoothed design spectra because the peaks and valleys that are common in the spectrum obtained from any single record are averaged out to form smooth functional forms that generally envelope the real spectra.
Earthquake response spectrum gives engineers a practical means of characterizing ground motions and their effects on structures. Introduced in 1932, it is now a central concept in earthquake engineering that provides a convenient means to summarize the peak response of all possible linear SDOF systems to a particular ground motion. It also provides a practical approach to apply the knowledge of structural dynamics to the design of structures and development of lateral force requirements in building codes.
A plot of the peak value of response quantity as a function of the natural vibration period T_{n} of the system (or a related parameter such as circular frequency ω_{n} or cyclic frequency f_{n}) is called the response spectrum for that quantity. Each such plot is for SDOF systems having a fixed damping ratio ξ. Oftentimes, several such plots for different values of ξ are included to cover the range of damping values encountered in actual structures. Whether the peak response is plotted against f_{n} or T_{n} is a matter of personal preference. In this text, we use the latter because engineers are more comfortable in using natural period rather than natural frequency because the period of vibration is a more familiar concept and one that is intuitively appealing. Although a variety of response spectra can be defined depending on the chosen response quantity, it is almost always the acceleration response spectrum, a plot of pseudoacceleration, against the period T_{n} for a fixed damping, ξ, that is most often used in the practice of earthquake engineering.
A similar plot of displacement u is referred to as the deformation spectrum, while that of velocity $\dot{u}$
is called a velocity spectrum.It is worthwhile to note that only the deformation u(t) is needed to compute internal forces. Obviously, then, the deformation spectrum provides all the information necessary to compute the peak values of deformation and internal forces. The pseudovelocity and pseudoacceleration response spectra are important, however, because they are useful in studying the characteristics of response spectra, constructing design spectra, and relating structural dynamics results to building codes.
To explain the procedure for determining the deformation response spectrum, we start with the spectrum developed for El Centro ground motion, which has been studied extensively in textbooks.
The acceleration recorded for the first 30 s of ground motion is shown in Figure 3.80a. The corresponding deformations in the three SDOF systems of varying periods are presented in Figure 3.80b.
The peak deformations are
u_{0} = 2.67 in. for a system with natural period T_{n} = 0.5 s and damping ratio ξ = 2%
u_{0} = 5.97 in. for a system with T_{n} = 1 s and ξ = 2%
u_{0} = 7.47 in. for a system with T_{n} = 2 s and ξ = 2%
The u_{0} value so determined for each system provides one point on the deformation response spectrum. Repeating such computations for a range of values of T_{n} while keeping c constant at 2% provides the deformation response spectrum shown in Figure 3.81. The spectrum shown is for a single damping value, ξ = 2%. However, a complete response spectrum would include such spectrum curves for several values of damping.
The pseudovelocity response spectrum is a plot of V as a function of the natural vibration period T_{n} or natural vibration frequency f_{n} of the system. For a given ground motion, the peak pseudovelocity V for a system with natural period T_{n} can be determined from the following equation using the deformation ξ = 2% of the same system from the response spectrum of Figure 3.81:
Figure 3.80 (a) Ground acceleration; (b) deformation response of three SDOF systems with β = 2% and Tn = 0.5, 1, and 2 s; and (c) deformation response spectrum for β = 2%.
Figure 3.81 Response spectra (β = 2%) for El Centro ground motion: (a) deformation response spectrum, (b) pseudovelocity response spectrum, and (c) pseudoacceleration.
As an example, for a system with T_{n} = 0.5 s and ξ = 2%, D = 2.67 in.:
Similarly, for T_{n} = 1.0 s and ξ = 2%, D = 5.97 in.:
And for T_{n} = 2.0 s and the same damping ξ = 2%, D = 7.47 in.:
These three values of peak pseudovelocity V are identified in Figure 3.81b. Repeating such computations for a range of values of T_{n} while keeping ξ constant at 2% provides the pseudovelocity spectrum shown in Figure 3.81b. The prefix pseudo is used for V because it is not equal to the peak velocity, although it has the same units for velocity.
It has been stated many times in this chapter that the base shear is equal to the inertial force associated with the mass m undergoing acceleration A. This acceleration A is generally different from the peak acceleration of the system. It is for this reason that A is called the peak pseudoacceleration; the prefix pseudo is used to avoid possible confusion with the true peak acceleration, just as we did for velocity V. The pseudoacceleration response spectrum is a plot of acceleration A as a function of the natural vibration period T_{n} or natural vibration frequency f_{n} of the system. For a given ground motion, peak pseudoacceleration A for a system with natural period T_{n} and damping ratio ξ can be determined from the following equation using the peak deformation D of the system from the response spectrum:
As an example, for a system with T_{n} = 0.5 s and ξ = 2%, D = 2.67 in.:
where g = 386 in./s^{2}.
Similarly, for a system with T_{n} = 1 s and ξ = 2%, D = 5.97 in.:
And for a system with T_{n} = 2 s and the same damping ξ = 2%, D = 7.47 in.:
The three values T_{n} of pseudoacceleration, A, are shown in Figure 3.82. Repeating similar computations for a range of T_{n} values, while keeping β constant at 2%, yields the pseudoacceleration spectrum shown in Figure 3.81c.
It was shown in the previous section that each of the deformation, pseudovelocity, and pseudoacceleration response spectra for a given ground motion contains the same information, no more and no less. The three spectra are simply distinct ways of displaying the same information on structural response. With knowledge of one of the spectra, the other two can be derived by algebraic operations using the procedure given in the previous section.
If each of the spectra contains the same information, why do we need three spectra? There are two reasons. One is that each spectrum directly provides a physically meaningful quantity: The deformation spectrum provides the peak deformation of a system, the pseudovelocity spectrum gives the peak strain energy stored in the system during the earthquake, and pseudoacceleration spectrum yields directly the peak value of the equivalent static force and base shear. The second reason lies in the fact that the shape of the spectrum can be approximated more readily for design purposes with the aid of all three spectral quantities rather than any one of them alone. For this purpose, a combined plot showing all three of the spectral quantities is especially useful. This type of plot was developed for earthquake response spectra for the first time by A.S. Veletsos and N.M. Newmark in 1960.
Figure 3.82 Pseudoacceleration responses of SDOF systems to El Centro ground motion.
In an integrated DVA spectrum, the vertical and horizontal scales for V and T_{n} are standard logarithmic scales. The two scales for D and A sloping at +45° and −45°, respectively, to the T_{n}axis are also logarithmic scales but not identical to the vertical scale. The pairs of numerical data for V and T_{n} that were plotted in Figure 3.81 on linear scales are replotted in Figure 3.83 on logarithmic scales. For a given natural period T_{n}, the D and A values can be read from the diagonal scales. As an example, for T_{n} = 2 s in Figure 3.83, it gives D = 7.47 in. and A = 0.191g. The fourway plot is a compact presentation of the three—deformation, pseudovelocity, and pseudoacceleration—response spectra, for a single plot of this form replaces the three plots.
The benefit of the response spectrum in earthquake engineering may be recognized by the fact that spectra for virtually all ground motions strong enough to be of engineering interest are now computed and published soon after they are recorded. From these, we can get a reasonable idea of the kind of motion that is likely to occur in future earthquakes. It should be noted that for a given ground motion response spectrum, the peak value of deformation, pseudovelocity, and base shear in any linear SDOF can be readily read from the spectra without resorting to dynamic analyses. This is because the computationally intense dynamic analysis has been completed in generating the response spectrum.
Figure 3.83 Combined DVA response for El Centro ground motion, β = 2%.
Given these advantages, it makes good sense to have geotechnical engineers provide tripartite response spectrum rather than just an acceleration spectrum, when sitespecific studies are commissioned.
An example of tripartite response spectrum is shown in Figure 3.84. The spectrum is for maximum capable earthquake (MCE) of magnitude 8.5 occurring at San Andreas fault. The project site is in downtown Los Angeles located at a distance of 34 miles from San Andreas fault.
The response spectrum tells us that the forces experienced by buildings during an earthquake are not just a function of the quake but are also their dynamic response characteristics to the quake. The response primarily depends on the period of the building being studied. A great deal of singlemode information can be read directly from the response spectrum. Referring to Figure 3.85, the horizontal axis of the response spectrum expresses the period of the building being affected by the quake. The vertical axis shows the velocity attained by this building during the quake. The diagonal axis running up toward the lefthand corner reads the maximum accelerations to which the building is subjected. The axis at right angles to this will read the displacement of the building in relation to the support. Superimposed on these tripartite scales are the response curves for an assumed 5% damping. Now let us study how buildings with different periods respond to the earthquake described by these curves.
If the building is to be studied had a natural period of 1 s, we would start at the bottom of the chart at T = 1 s and reference vertically until we intersect the response curve. From this intersection, point A, we travel to the extreme right and read a velocity of 16 in./s. Following a displacement line diagonally down to the right, we find a displacement of 2.5 in. Similarly following the acceleration line down to the left, we see that the building will experience an acceleration of 0.25g. If we then move to the 2 s period, point B, in the same sequence, we find that we will have the same velocity of 16 in./s, a displacement of 4 in., and a maximum acceleration of 0.10g. If we then move to 4 s, point C, we see a velocity of 16 in./s, a displacement of 10 in., and an acceleration of 0.06g. If we run all out to 10 s, point D, we find a velocity of 7 in./s, a displacement of 10 in. the same as for point C, and an acceleration of 0.01g. Notice that the values vary widely, as started earlier, depending on the period of building exposed to this particular quake.
Figure 3.84 Tripartite response spectrum.
We now study the important properties of earthquake response spectra. For this purpose, we use once again an idealized response spectrum for El Centro ground motion shown in Figure 3.85. The damping, ξ, associated with the spectrum is 5%. The period T_{n} plotted on a logarithmic scale covers a wide range, T_{n} = 0.01–10 s.
Consider a system with a very short period, say 0.03 s. For this system, the pseudoacceleration A approaches the ground acceleration, while the displacement D is very small. There is a physical reasoning for this trend. For purposes of dynamic analysis, a veryshortperiod system is extremely stiff and may be considered essentially rigid. Such a system would move rigidly with the ground as if it is a part of the ground itself. Thus, its peak acceleration would be approximately equal to the ground acceleration as shown in Figure 3.86.
Figure 3.85 Velocity, displacement, and acceleration readout from response spectra.
Next, we examine a system with a very long period, say T_{n} = 10 s. The acceleration A, and thus the force in the structure, which is related to mA, would be small. Again, there is a physical reasoning for this trend: A verylongperiod system is extremely flexible. Therefore, the mass at top would remain stationary while the base moves with the ground below (see Figure 3.87).
Based on these two observations, and those in between the two periods (not examined here), it is logical to divide the spectrum into three period ranges: (1) the longperiod region to the right of point D, called the displacementsensitive region because structural response is most directly related to ground displacement; (2) the shortperiod region to the left of point C, called the accelerationsensitive region because structural response is most directly related to ground acceleration; and (3) the intermediateperiod region between points C and D, called the velocitysensitive region because structural response appears to be better related to ground velocity than to other ground motion parameters.
Figure 3.86 Idealized response spectrum for El Centro ground motion.
Figure 3.87 Schematic response of rigid and flexible systems: (a) rigid system, acceleration at top is nearly equal to the ground acceleration; (b) flexible system, structural response is most directly related to ground displacement.
The preceding discussion should be helpful in recognizing the usefulness of fourway logarithmic plot of the combined deformation, pseudovelocity, and pseudoacceleration response spectra. These observations would be difficult to discover from the three individual spectra.
We now turn our attention to damping that has significant influence on the earthquake response of buildings. It reduces the response of a structure, as expected. However, the reduction achieved with a given amount of damping is different in the three spectral regions. In the limit as T_{n} → ∞, damping does not affect the response because the structural mass stays still while the ground underneath moves. Among the three period regions, the effect of damping tends to be greatest in the velocitysensitive region of the spectrum. In this spectral region, the effect of damping depends on the ground motion characteristics. If the ground motion is harmonic over many cycles as it was in the Mexico City earthquake of 1985, the effect of damping would be especially large for systems near resonance.
The motion of structure and the associated forces could be reduced by increasing the effective damping of the structure. The addition of dampers achieves this goal without significantly changing the natural vibration period of the structure. Viscoelastic dampers have been used in many structures; for example, 10,000 dampers were installed throughout the height of each tower of the now nonexistent World Trade Center in New York City to reduce windinduced motion to within a comfortable range for the occupants. In recent years, there is a growing interest in developing dampers suitable for structures in earthquakeprone regions. Because the inherent damping in most structures is small, their earthquake response can be reduced significantly by the addition of dampers. These can be especially useful in improving the seismic safety of existing structures.
As stated previously, the response of an SDOF system at the extremes of the fundamental periods is intuitively obvious. For example, a system with a very short period, that is, a very stiff system, experiences peak acceleration approaching that of the ground. The distortion of the system, however, is negligible since the motion of the mass is the same as that of the ground. On the other hand, for systems with a very long period, that is, flexible buildings, the mass remains stationary while the ground moves beneath it. The relative displacement between the two is equal to the ground displacement. However, the acceleration of the mass is zero or nearly so. Thus, the force in the structure related to ma would also be small. This concept is illustrated in Figure 3.88.
The response spectra for a given earthquake are typically a jagged plot representing the peak response of SDOF systems. It is a unique plot representative of the particular ground motion. The design spectrum is, however, a smooth plot specifying the level of seismic force as a function of the systems’ period and damping ratio. This is the first difference. The second difference is the design spectrum is an envelope of two or more different elastic spectra that could affect ground motions at a given site. After determining the design spectrum for each of the postulated earthquakes, the design spectrum for the site is developed by enveloping the design spectra for all earthquakes considered for the site, as shown schematically in Figure 3.88a.
This is a welldevised method of determining peak response of a system directly from the response spectrum for the ground motion without having to carry a timehistory analysis. For SDOF systems, the method is quite accurate. For MDOF systems, in a manner of speaking, the result is not exact. However, the result is accepted in practice, as being accurate enough for seismic design applications.
In general, modal responses attain their peaks at different time instants. Therefore, approximations must be used in combining the peak responses determined from response spectrum analysis because no information is available as to when these peak modal values occur. Summing up the absolute values of the maximum values would certainly give us an upper bound solution but this is not used in building design practice because the results tend to be too conservative.
Figure 3.88 (a) Comparison of design and actual response spectra and (b) concept of effective modal masses and effective modal heights.
In practice, two other modal combination rules, the SRSS and CQC, both of which are based on random vibration theory, are popular. The latter is the preferred method applicable to a wider class of structures as it is also applicable to structures with notsowellseparated periods.
The response spectrum analysis is a procedure for dynamic analysis of a structure subjected to earthquake ground motions. But it reduces to a series of static analysis. For each mode considered, static analysis of the structure subjected to corresponding modal shears provides the peak modal static response. This procedure avoids the dynamic analysis of SDOF systems necessary for timehistory analysis. Thus, the response spectrum analysis may be considered as a pseudodynamic analysis procedure, because it indirectly uses the vibration properties of the structure and the dynamic characteristics of the ground motion. The intensive timehistory calculations are not required because in developing smooth design spectra, somebody has already done these by converting the dynamic loads into equivalent static loads. Conceptually equivalent loads may be thought of as static loads multiplied by their corresponding deflection load factors.
Response spectrum analysis is a procedure to compute the peak response of a structure during an earthquake directly from the earthquake design spectrum without the need for responsehistory analysis of the structure. The procedure is not an exact predictor of peak response, but it provides an estimate that is considered sufficiently accurate for structural design applications.
To assist us in developing an understanding of the response behavior of multistory buildings, a conceptual explanation of modal analysis is given in Figure 3.88b.
It should be obvious by now that the effective modal masses and effective modal heights for a given building depend on the characteristics of the response spectrum and the dynamic properties of the building itself. Therefore, the values shown for a sixstory example building in Figure 3.88b are for conceptual purposes only and are not based on any given design response spectrum or building properties.
The ability of structural elements to withstand deformations in the elastic range is called ductility. In a major earthquake, a structure will not remain elastic but will be forced into the inelastic range. Inelastic action absorbs significantly more energy from the system. Therefore, if a structure is properly detailed and constructed so that it can perform in a ductile manner (i.e., deform in the inelastic range), it can be designed for considerably smaller lateral forces. A good way to visualize the concept of ductility is perhaps to follow through the load–deformation characteristics of a member subject to an applied cyclic loading.
If dynamic loads in a seismic event deform a structure beyond the elastic range of the material, the resulting motion is called inelastic response. Such excursions beyond the elastic range are not usually permitted in normal operating conditions such as under gravity and wind loads. However, the behavior of and the resulting damage in structures subject to extreme load conditions are quite important in structural design. For example, a building subjected to blast or severe earthquake loading will probably be deformed inelastically, and therefore, it is of interest to evaluate its stability to assure that the building can sustain gravity loads without collapse.
Let us consider an idealized 2D steel frame shown in Figure 3.89 subjected to a lateral load P applied at the top. If the flexural rigidities of the columns are less than that of the beam and the load is increased infinitely, at some point in the loading history, socalled plastic hinges will form at the ends of the column. A plot of the load P against the displacement x is linear up to the value of P_{y1} (see the line labeled 1 in Figure 3.90), where yielding of the material begins. Subsequently, it curves (see the curve labeled 2), due to softening of columns at the base and top. Upon unloading, the material rebounds elastically, as indicated by part 3 of the plot in Figure 3.90.
If a reverse loading is then applied, the parts of Figure 3.90 labeled 4 and 5 result, and a subsequent unloading produces line 6. If the maximum positive and negative forces P_{m1} and −P_{m2} (the ordinates of points B and E on the diagram) are numerically equal, the hysteresis loop formed by cyclic loading is symmetric with respect to the origin.
The curved portions of Figure 3.90 are often replaced by straight lines approximating the true behavior. Figure 3.91 illustrates such a simplified load–displacement diagram, called a bilinear inelastic restoring force. It consists of two parallel lines (labeled 2 and 4 in Figure 3.91) for inelastic behavior and a family of parallel lines (of which those labeled 1 and 3 are representative) for elastic behavior. If the slopes of lines 2 and 4 are zero, as in Figure 3.92, the diagram represents an elastoplastic restoring force. That is, the plot of P against x consists of straightline segments, where the behavior is assumed to be either perfectly elastic or perfectly plastic. For example, let us reconsider the frame in Figure 3.89 and suppose that the load P increases to the value P_{m} (see point A in Figure 3.92). If the plastic hinges in Figure 3.92 are assumed to form instantaneously, the displacement increases without a corresponding increase of the load, as shown by the horizontal line from A to B in Figure 3.92. A decrease in the load causes a decrease of displacement in accordance with line (3) in Figure 3.93 and so on.
Figure 3.89 Idealized steel portal frame subject to load P at top.
Figure 3.90 Displacement plot of load P versus displacement x.
Figure 3.91 Plot of idealized displacement x versus load P: observe the curved portion is replaced by straight lines depicting bilinear inelastic behavior.
Figure 3.92 Elastoplastic hysteresis loop.
Figure 3.93 Rigidplastic hysteresis loop.
The hysteresis loop inherent in elastoplastic analyses represents a discretized form of structural damping. In this case, all of the dissipated energy is tacitly assumed to be absorbed at the plastic hinges. This dissipative mechanism is often referred to as elastoplastic damping, which represents a particular case of hysteretic damping.
It is interesting to note that if the slopes of inclined lines 1 and 2 are taken to be infinite (meaning that the deformation of the portal in its elastic range is assumed to be small in comparison to the plastic deformation), the hysteresis loops simplify to a rigidplastic dissipative mechanism as shown in Figure 3.93.
The importance of understanding the use of hysteresis loops in seismic design may be explained as follows:
Inelastic response occurs when the amplitude of earthquake shaking is strong enough to cause forces in a structure that exceed the strength of any of the structure's elements or connections. When this occurs, the structure may experience a variety of behaviors. If the elements that are strained beyond their elastic strength limit are brittle, they will tend to break and lose the ability to resist any further load. This type of behavior is typified by a steel tension member that is stretched such that the force in the brace exceeds the ultimate strength of its end connections or by an unreinforced concrete element that is strained beyond its cracking strength. If the element is ductile, it may exhibit plastic behavior, being able to maintain its yield strength as it is strained beyond its elastic limit. This type of behavior is typified by properly braced, compact section beams in moment frames; by the cores of bucklingrestrained braces; and by the shear links in eccentrically braced frames. Even elements that are ductile and capable of exhibiting significant postyielding deformation without failure will eventually break and lose loadcarrying capacity due to lowcycle fatigue if plastically strained over a number of cycles.
Modern structural analysis software provides the capability to analyze structures at deformation levels that exceed their elastic limit. In order to do this, these programs require input on the hysteretic (nonlinear force vs. deformation) properties of the deforming elements. Hence, knowledge of how these loops are generated is of importance in seismic design.
The essential background for practice in the field of earthquake engineering is knowledge about the earthquake itself. The detailed study of earthquakes and earthquake mechanisms lies in the province of seismology, but in their studies, the earthquake engineers must take a different point of view than the seismologist. Seismologists have focused their attention primarily on the global effects of earthquakes and are also concerned with smallamplitude ground motions that induce no significant structural response. 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. Nevertheless, even though the objectives of earthquake engineers differ from those of seismologist, there are many topics in seismology that are of immediate engineering interest.
The seismicity of a region determines the extent to which earthquake loadings may control the design of a structure planned for that location and the principal indicator of the degree of seismicity in 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 beginnings of civilization. In China, records have been kept that are thought to include every major destructive seismic event for a time span of nearly 3000 years.
More recently, beginning from the 1970s, earthquake occurrence information has been compiled from strong seismograph motion, which indicates the location and magnitude of all earthquakes. The most obvious conclusion to be drawn from these maps is that earthquake occurrences are not distributed uniformly over the surface of the Earth. Instead, they tend to be concentrated along welldefined lines, which are known to be associated with the boundaries of segments or plates of the Earth's crust.
Within the structure of the Earth, the mantle is considered to consist of two distinct layers. The upper mantle together with the crust forms a rigid layer called the lithosphere. Below that, a zone called the asthenosphere is thought to be partially molten rock consisting of solid particles incorporated within a liquid component. Because of its highly plastic character, the lithosphere can act as if it is floating on a liquid and thus can be subjected to large crustal deformations. The lithosphere does not 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. The detailed description of the motions of these plates is subjectedly called plate tectonics; development of understanding of this subject is one of the great advances of geology and seismology during the present century.
A special feature of earthquake excitation of structures, compared with most other forms of dynamic excitation, is that it is applied in the form of support motions rather than by external loads; thus, the effective seismic loadings must be established in terms of these motions. Defining the support motions is the most difficult and uncertain phase of the problem of predicating structural response to earthquakes. When these input motions have been established, however, the calculation of the corresponding stresses and deflections in any given structure is a standard problem of structural dynamics.
The most important aspect of an earthquake's ground motions is the effect they will have on structures, that is, the stresses and deformations or the amount of damage they would produce. This damage potential is, of course, least partly dependent on the size of the earthquake, and a number of measures of size are used for different purposes. 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, measured in micrometers (10^{−6}), of the earthquake record obtained by a Wood–Anderson seismograph, corrected to a distance of 100 km. This magnitude rating has been related empirically to the amount of earthquake energy released by the formula
in which M is the magnitude. By this formula, the energy increases by a factor of 12 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.
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 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 manmade objects. In the United States, the standard measure of intensity for many years has been the Modified Mercalli (MM) scale. This is a 12point scale ranging from I (not felt by anyone) to XII (total destruction). Results of earthquakeintensity observations are typically complied in the form of isoseismic maps. Although such subjective intensity ratings are very valuable in the absence of any instrumented records of an earthquake, deficiencies in providing criteria for the design of earthquakeresistant structures are obvious.
Basic information on the characteristics of earthquakes comes from strongmotionrecording accelerographs. Although their installations are radically increasing in the seismic region of the United States and various other parts of the world, the distribution of instruments is quite limited. Consequently, basic data concerning the influence of such factors as magnitude, distance, and local soil conditions of the characteristics of earthquake motions are still very scarce.
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 specified 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 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 damageproducing potential adequately.
Although structural design for seismic loading is primarily concerned with structural safety during major earthquakes, serviceability and the potential for economic loss are also of concern.
As such, seismic design requires an understanding of the structural behavior under large inelastic, cyclic deformations. Behavior under this loading is fundamentally different from wind or gravity loading. It requires a more detailed analysis and application of a number of stringent detailing requirements to assure acceptable seismic performance beyond the elastic range. Some structural damage can be expected when the building experiences design ground motions.
The seismic analysis and design of buildings has traditionally focused on reducing the risk of loss of life in the largest expected earthquake. Building codes base their provisions on the historic performance of buildings and their deficiencies and have developed provisions around lifesafety concerns, by focusing their attention to prevent collapse under the most intense earthquake expected at a site during the life of a structure. These provisions are based on the concept that the successful performance of buildings in areas of high seismicity depends on a combination of strength, ductility manifested in the details of construction, and the presence of a fully interconnected, balanced, and complete LFRS. In regions of low seismicity, the need for ductility reduces substantially. And in fact, strength may even substitute for a lack of ductility. Very brittle LFRSs can be excellent performers as long as they are never pushed beyond their elastic strength.
Seismic provisions specify criteria for the design and construction of new structures subjected to earthquake ground motions with three goals:
Some structural damage can be expected as a result of design ground motion because the codes allow inelastic energy dissipation in the structural system. For ground motions in excess of the design levels, the intent for the codes is for structures to have a low likelihood of collapse.
In most structures that are subjected to moderatetostrong earthquakes, economical earthquake resistance is achieved by allowing yielding to take place in some structural members at predetermined locations. It is generally impractical to design a structure to respond in the elastic range when subjected to the maximum expected earthquake ground motions. Therefore, in seismic design, yielding is permitted in predetermined structural members or locations, with the provision that the verticalloadcarrying capacity of the structure is maintained even after strong earthquakes. However, for certain types of structures such as nuclear facilities, yield cannot be tolerated, and as such, the design needs to be elastic.
Structures that contain facilities critical to postearthquake operations—such as hospitals, fire stations, power plants, and communication centers—must not only survive without collapse but must also remain operational during and after an earthquake. Therefore, in addition to life safety, damage control is an important design consideration for structures deemed vital to postearthquake functions.
An idea of the behavior of a building during an earthquake may be grasped by considering the simplified response shown in Figure 3.94. As the ground on which the building rests is displaced, the base of the building moves with it. However, the building above the base is reluctant to move with it because the inertia of the building mass resists motion and causes the building to distort. This distortion wave travels along the height of the structure and, with continued shaking of the base, causes the building to undergo a complex series of oscillations.
Although both wind and seismic forces are essentially dynamic, there is a fundamental difference in the manner in which they are induced in a structure. Wind loads, applied as external loads, are characteristically proportional to the exposed surface of a structure, while the earthquake forces are principally internal forces resulting from the distortion produced by the inertial resistance of the structure to earthquake motions. Whereas in wind design, one would feel greater assurance about the safety of a structure made up of heavy sections, in seismic design, this does not necessarily produce a safer design.
Figure 3.94 (a and b) Building behavior during earthquakes.
Earthquakes can cause structural collapse in several different ways. First, if the structure is not adequately connected and tied together, the motions induced in the structure by earthquake shaking can allow the building components to pull apart and, if one member is supported by another, to eventually collapse. This type of collapse is observed in bridges and other long structures that incorporate expansion joints.
Another way that earthquakes can cause structures to collapse is by overstressing gravityloadbearing elements such that they lose their loadcarrying capacity. As an example, if the overturning loads on the columns in a braced frame exceed the buckling capacity of the columns, these columns could buckle and lose their ability to continue to support the structure above.
The third way that earthquakes cause collapse is by inducing sufficient lateral displacement into a building to allow pΔ effects to induce lateral sideway collapse of the frame. Sideway collapse can occur in a single story or can involve multiple stories. Often, it is difficult to distinguish these collapses from the local failures of elements because the large displacements associated with sideway collapse can often trigger concurrent local collapse.
If the base of a structure is suddenly moved, as in a seismic event, the upper part of the structure will not respond instantaneously but will lag because of the inertial resistance and flexibility of the structure. The resulting stresses and distortions in the building are the same as if the base of the structure were to remain stationary while timevarying horizontal forces are applied to the upper part of the building. These forces, called inertial forces, are equal to the product of the mass of the structure times acceleration, that is, F = ma (the mass m is equal to weight divided by the acceleration of gravity, i.e., m = w/g). Because earthquake ground motion is 3D (one vertical and two horizontal), the structure, in general, deforms in a highly complex manner. Generally, the inertial forces generated by the horizontal components of ground motion require greater consideration for seismic design since adequate resistance to vertical seismic loads is usually provided by the member capacities required for gravityload design.
The behavior of a building during an earthquake is thus a vibration problem. The seismic motions of the ground do not damage a building by impact, as does a wrecker's ball, or by externally applied pressure such as wind but by internally generated inertial forces caused by vibration of the building mass. An increase in mass, in this context, has two undesirable effects on the earthquake design. First, it results in an increase in the force, and second, it can cause buckling or crushing of columns and walls when the mass pushes down on a member bent or moved out of plumb by the lateral forces. This effect is known as the pΔ effect, and the greater the vertical forces, the greater the movement due to pΔ. It is almost always the vertical load that causes buildings to collapse; in earthquakes, buildings very rarely fall over—they fall down.
In general, tall buildings respond to seismic motions quite differently than lowrise buildings as shown schematically in Figure 3.95. They are invariably more flexible than lowrise buildings and, in general, experience much lower accelerations. But a tall building subjected to ground motions for a prolonged period may experience much larger forces if its natural period is near that of the ground waves. Thus, the magnitude of lateral forces is not a function of the acceleration of the ground alone but is influenced to a great extent by the type of response of the structure itself and its foundation as well. This interrelationship of building behavior and seismic ground motion also depends on the building period as formulated in the socalled response spectrum, explained in this chapter.
Figure 3.95 Schematic magnitude of seismic force. Note: The magnitude of seismic force depends on the building mass M, ground acceleration a, and the dynamic response of the building itself.
An effective seismic design generally includes the following:
Earthquakeinduced motions, even when they are more violent than those induced by wind, evoke a totally different human response—first, because earthquakes occur much less frequently than windstorms, and second, because the duration of motion caused by an earthquake is generally short. People who experience earthquakes are grateful that they have survived the trauma and are less inclined to be critical of the building motion. Earthquakeinduced motions are, therefore, a safety rather than a human discomfort issue.
Lateral deflections that occur during earthquakes should be limited to prevent distress in structural members and architectural components. Nonloadbearing infills, external wall panels, and window glazing should be designed with sufficient clearance or with flexible supports to accommodate the anticipated movements.
Drift is generally defined as the lateral displacement of one floor relative to the floor below. Drift control is necessary to limit damage to interior partitions, elevator and stair enclosures, and glass and cladding systems. Stress or strength limitations in ductile materials do not always provide adequate drift control, especially for tall buildings with relatively flexible momentresisting frames.
Total building drift is the absolute displacement of any point relative to the base. Adjoining buildings or adjoining sections of the same building may not have identical modes of response and therefore may have a tendency to pound against one another. Building separations or joints must be provided to permit adjoining buildings to respond independently to earthquake ground motion.
Buildings are often built right up to property lines in order to make maximum use of space. Historically, buildings have been built as if the adjacent structures do not exist. As a result, the buildings may pound during an earthquake. Building pounding can alter the dynamic response of both buildings and impart additional inertial loads to them.
Buildings that are the same height and have matching floors are likely to exhibit similar dynamic behavior. If the buildings pound, floors will impact other floors, so damage usually will be limited to nonstructural components. When floors of adjacent buildings are at different elevations, the floors on one building will impact the columns of the adjacent building, causing structural damage. When buildings are of different heights, the shorter building may act as a buttress for the taller neighbor. The shorter building receives an unexpected load, while the taller building suffers from a major discontinuity that alters its dynamic response. Since neither is designed to weather such conditions, there is potential for extensive damage and possible collapse.
One of the basic goals in seismic design is to distribute yielding throughout the structure. Distributed yielding dissipates more energy and helps prevent the premature failure of any one element or group of elements. For example, in moment frames, it is desirable to have strong columns relative to the beams to help distribute the formation of plastic hinges in the beams throughout the building and prevent a story collapse mechanism.
A continuous load path, or preferably more than one path, with adequate strength and stiffness should be provided from the origin of the load to the final lateralloadresisting elements. The general path for load transfer is in reverse to the direction in which seismic loads are delivered to the structural elements. Thus, the path for load transfer is as follows: Inertial forces generated in an element, such as a segment of exterior curtain wall, are delivered through structural connections to a horizontal diaphragm (i.e., floor slab or roof); the diaphragms distribute these forces to vertical components such as moment frames and braced frames; and finally, the vertical LFRS transfers the forces into the foundations. While providing a continuous load path is an obvious requirement, examples of common flaws in load paths include a missing collector, or a discontinuous chord because of an opening in the floor diaphragm, or a connection that is inadequate to deliver diaphragm shear to LFRF.
The horizontal elements such as floor and roof slabs distribute lateral forces to the LFRS acting as horizontal diaphragms. In special situations, horizontal bracing may be required in the plane of diaphragms to transfer large shears from discontinuous walls or braces.
A complete load path is a basic requirement. There must be a complete gravity and LFRS that forms to a continuous load path between the foundation and all portions of the building. If there is a discontinuity in the load path, the building is unable to resist seismic forces regardless of the strength of the elements. Interconnecting the elements needed to complete the load path is necessary to achieve the required seismic performance. Examples of gaps in the load path in addition to those stated earlier would include a shear wall that does not extend to the foundation, a missing shear transfer connection between a diaphragm and vertical elements, a discontinuous chord at a diaphragm's notch, or a missing collector.
A good way to remember this important design strategy is to ask yourself the question, How does the inertial load get from here (meaning the point at which it originates) to there (meaning the shear base of the structure, typically the foundations)?
A building with an irregular configuration may be designed to meet all code requirements, but it will not perform as well as a building with a regular configuration. If the building has an odd shape that is not properly considered in the design, good details and construction are of a secondary value.
Two types of structural irregularities, vertical and plan irregularities, are typically defined in most seismic standards. These irregularities result in building responses significantly different from those assumed in the equivalent static force procedure and, to a lesser extent, from the dynamic analysis procedure. Although seismic provisions give certain recommendations for assessing the degree of irregularity and corresponding penalties and restrictions, it is important to understand that these recommendations are not an endorsement of their design; rather, the intent is to make the designer aware of the potential detrimental effects of irregularities.
Consider, for example, a reentrant corner, resulting from an irregularity characteristic of a building's plan. If the plan configuration has an inside corner, as shown in Figure 3.96, then it has a reentrant corner. It is, however, unavoidable in buildings of L, H, T, and X plan shapes.
Two problems related to seismic performance are created by these shapes: (1) differential lateral deformation modes between different wings of the building may result in a local stress concentration at the reentrant corner, and (2) torsion may result because the eccentricity between the center of rigidity and center of mass.
Figure 3.96 Reentrant corners in L, T, and Hshaped buildings (as a solution, add collector elements and/or stiffen end walls A, B, C, D, E, F, G, and J).
There are two alternative solutions to this problem: Tie the building together at lines of stress concentration and locate seismicforceresisting elements at the extremity of the wings to reduce torsion or separate the building into simple shapes. The width of the separation joint must allow for the estimated inelastic deflections for adjacent wings. The purpose of the separation is to allow adjoining portions of buildings to respond to earthquake ground motions independently without pounding on each other. If it is decided to dispense with the separation joints, collectors at the intersection must be added to transfer forces across the intersection areas. Since the free ends of the wings tend to distort most, it is beneficial to place seismicforceresisting members at these locations.
The seismic design of regular buildings is based on two concepts. First, the linearly varying lateral force distribution is a reasonable and conservative representation of the actual response distribution due to earthquake ground motions. Second, the cyclic inelastic deformation demands are reasonably uniform in all of the seismicforceresisting elements. However, when a structure has irregularities, these concepts may not be valid, requiring corrective factors and procedures to meet the design objectives.
The impact of irregular parameters in estimating seismic force levels, first introduced into the UBC in 1973, long remained a matter of engineering judgment. Beginning in 1988, however, some configuration parameters were quantified to establish the condition of irregularity. Additionally, specific analytical treatments and/or corrective measures have been mandated to address these flaws.
Typical building configuration deficiencies include an irregular geometry, a weakness in a story, a concentration of mass, or a discontinuity in the LFRS. Vertical irregularities are defined in terms of strength, stiffness, geometry, and mass. Although these are evaluated separately, they are related to one another and may occur simultaneously. For example, a building that has a tall first story can be irregular because of a soft story, a weak story, or both, depending on the stiffness and strength of this story relative to those above.
Those who have studied the performance of buildings in earthquakes generally agree that the building's form has a major influence on performance. This is because the shape and proportions of the building have a major effect on the distribution of earthquake forces as they work their way through the building. Geometric configuration, type of structural members, details of connections, and materials of construction all have a profound effect on the structural dynamic response of a building. When a building has irregular features, such as asymmetry in plan or vertical discontinuity, the assumptions used in developing seismic criteria for buildings with regular features may not apply. Therefore, it is best to avoid creating buildings with irregular features. For example, omitting exterior walls in the first story of a building to permit an open ground floor leaves the columns at the ground floor level as the only elements available to resist lateral forces, thus causing an abrupt change in rigidity at that level. When irregular features are unavoidable, special design considerations are required to account for the unusual dynamic characteristics and the load transfer and stress concentrations that occur at abrupt changes in structural resistance. Examples of plan and elevation irregularities are illustrated in Figures 3.97 and 3.98. Note that plan irregularities are also referred to as horizontal irregularities.
The ASCE 710 qualifies irregularity by defining geometrically or by use of dimensional ratios, the points at which a specific irregularity becomes an issue requiring remedial measures. These issues are discussed later in this chapter. It is worth noting that no corrective course is required for certain irregularities other than performing modal analysis for determining design seismic forces.
Irregularities are divided into two broad categories: (1) vertical and (2) plan irregularities. Vertical irregularities include soft or weak stories, large changes in mass from floor to floor, and discontinuities in the dimensions or inplane locations of lateralloadresisting elements. Buildings with plan irregularities include those that experience substantial torsion when subjected to seismic loads, or have reentrant corners and discontinuities in floor diaphragms or in the lateral force path, or have lateralloadresisting elements that are not parallel to each other or to the principal axes of the building.
Figure 3.97 Plan irregularities: (a) geometric irregularities, (b) irregularity due to massresistance eccentricity, and (c) irregularity due to discontinuity in diaphragm stiffness. Note: CR, center of resistance; CM, center of mass.
The intensity of ground motion reduces with the distance from the epicenter of the earthquake. The reduction, called attenuation, occurs at a faster rate for highfrequency (shortperiod) components than for lowerfrequency (longperiod) components. The cause of the change in attenuation rate is not understood, but its existence is certain. This is a significant factor in the design of tall buildings, because a tall building, although situated farther from a causative fault than a lowrise building, may experience greater seismic loads because longperiod components are not attenuated as fast as the shortperiod components. Therefore, the area influenced by ground shaking potentially damaging to, say, a 50story building is much greater than for a 1story building.
Figure 3.98 Elevation irregularities: (a) abrupt change in geometry, (b) large difference in floor masses, and (c) large difference in story stiffnesses.
As a building vibrates due to ground motion, its acceleration will be amplified if the fundamental period of the building coincides with that of the soil it rests upon. It is worth noting that natural periods of soil are typically in the range of 0.5–1.0 s. Therefore, it is entirely possible for a building and ground it rests upon to have the same fundamental period. This was the case for many 5 to 10story buildings that were damaged in the September 1985 earthquake in Mexico City.
Experience in several other earthquakes has confirmed that local soil conditions can have a significant effect on earthquake response. It is perhaps more challenging to picture, but the soil layers beneath a structure have a period of vibration T_{soil} similar to the period of vibration of a building T. Greater structural damage is likely to occur when the period of the underlying soil is close to the fundamental period of the structure. In these cases, a partial resonance effect may develop between the structure and the underlying soil. These conditions are addressed in ASCE 710 by classifying soil profiles, into Site Class A through F.
Ductility is the property exhibited by certain structural elements and structures composed of such elements that enable them to sustain loads when strained beyond their elastic limit. For structures that have welldefined yield and ultimate deformation capacities, ductility, μ, is defined by
In this equation, δ_{u} and δ_{y} are the displacements at which failure and yielding, respectively, initiate.
Ductility is an important parameter in computing seismic resistance. It enables structures that do not have adequate elastic strength to survive strong ground motions through inelastic response. The principal benefit of ductile response is that it makes it possible to place ductile elements at key locations in the seismicloadresisting system to protect other nonductile elements from being overstressed. This is a key strategy in design of structures for seismic resistance.
As stated earlier, in seismic design, structures are designed for forces much smaller than those the design ground motion would produce in a structure with completely linear elastic response. The reduction is possible for a number of reasons. As the structure begins to yield and deform inelastically, the effective period of response of the structure tends to lengthen, resulting in a reduction in strength demand. Furthermore, the inelastic action results in a significant amount of energy dissipation through hysteretic damping.
The energy dissipation resulting from hysteretic behavior can be measured as the area enclosed by the force–deformation curve of the structure as it experiences several cycles of excitation. Some structures have far more energy dissipation capacity than do others. The extent of energy dissipation capacity is largely dependent on the amount of stiffness and strength degradation of the structure as it experiences repeated cycles of inelastic deformation.
Let us consider the load–deformation curves for a beam–column assembly earlier in Figure 3.10. Hysteretic curve (a) in the figure is representative of the behavior of substructures that have been detailed for ductile behavior. The substructure can maintain nearly all of its strength and stiffness over a number of large cycles of inelastic deformation. The resulting force–deformation loops are quite wide and open, resulting in a large amount of energy dissipation capacity. Hysteretic curve (b) represents the behavior of a substructure that has not been detailed for ductile behavior. It rapidly loses stiffness under inelastic deformation and the resulting hysteretic loops are quite pinched. The energy dissipation capacity of such a substructure is much lower than that for the substructure (a). Hence, structural systems with large energy dissipation capacity are assigned higher R values, resulting in design for lower forces, than systems with relatively limited energy dissipation capacity.
In providing for ductility, it should be kept in mind that severe penalties are imposed by seismic provisions on structure with nonuniform ductility.
Some examples of nonuniform ductility due to vertical discontinuities are shown in Figure 3.99. Avoid them, if you can.
Redundancy is a fundamental characteristic for good seismic performance. It provides a building with a redundant system such that failure of a single connection or component does not adversely affect the entire lateral stability of the structure
Figure 3.99 Examples of nonuniform ductility in structural systems due to vertical discontinuities.
Buildings do not resonate with the purity of a tuning fork because they are damped; the extent of damping depends upon the construction materials, type of connections, and the influence of nonstructural elements on the stiffness characteristics of the building. Damping is measured as a percentage of critical damping. In a dynamic system, critical damping is defined as the minimum amount of damping necessary to prevent oscillation altogether. To visualize critical damping, imagine a tensioned string immersed in water. When the string is plucked, it oscillates about its rest position several times before stopping. If we replace water with a liquid of high viscosity, the string will oscillate but (Taranath, Book 6, Chapter 3) certainly not as many times as when it was in water. By progressively increasing the viscosity of the liquid, it is easy to visualize that a state can be reached where the string, once plucked, will return to its neutral position without ever crossing it. The minimum viscosity of the liquid that prevents the vibration of the string altogether can be considered equivalent to the critical damping.
The damping of structures is influenced by a number of external and internal sources. Chief among them are the following:
Figure 3.100 Concept of 100% g. A building subjected to 1g acceleration conceptually behaves as if it cantilevers horizontally from a vertical surface. Note: This is not a misprint!
For analytical purposes, it is a common practice to lump different sources of damping into a single viscous damping. For nonbaseisolated buildings, analyzed for codeprescribed loads, the damping ratios used in practice vary anywhere from 1% to 10% of critical. The lowend values are for wind, while those for the upper end are for seismic design. The damping ratio used in the analysis of seismic baseisolated buildings is rather large compared to values used for nonisolated buildings and varies from about 0.20 to 0.35 (20%–35% of critical damping).
Base isolation, discussed elsewhere in this book, consists of mounting a building on an isolation system to prevent horizontal seismic ground motions from entering the building. This strategy results in significant reductions in interstory drifts and floor accelerations, thereby protecting the building and its contents from earthquake damage.
A level of ground acceleration on the order 0.1g, where g is the acceleration due to gravity, is often sufficient to produce some damage to weak construction. An acceleration of 1.0g, or 100% of gravity, is analytically equivalent, in the static sense, to a building that cantilevers horizontally from a vertical surface as shown in Figure 3.100.
As stated previously, the process by which free vibration steadily diminishes in amplitude is called damping. In damping, the energy of the vibrating system is dissipated by various mechanisms, and often, more than one mechanism may be present at the same time. In simple laboratory models, most of the energy dissipation arises from the thermal effect of repeated elastic straining of the material and from the internal friction. In actual structures, however, many other mechanisms also contribute to the energy dissipation. In a concrete building, these include opening and closing of microcracks in concrete and friction between the structure itself and nonstructural elements such as partition walls. Invariably, it is impossible to identify or describe mathematically each of these energydissipating mechanisms in an actual building.
Therefore, the damping in actual structures is usually represented in a highly idealized manner. For many purposes, the actual damping in structures can be idealized by a linear viscous damper or dashpot. The damping coefficient is selected so that the vibrational energy that dissipates is equivalent to the energy dissipated in all the damping mechanisms. This idealization is called equivalent viscous damping.
Figure 3.101 shows a linear viscous damper subjected to a force f_{D}. The damping forces f_{D} are related to the velocity $\dot{u}$
across the linear viscous damper bywhere the constant c is the viscous damping coefficient; it has units of force × time/length.
Figure 3.101 Linear viscous damper. Damping is defined as a force that resists dynamic motion. A simple and yet realistic damping model for analysis purposes is to assume that the damping force, fD, is proportional to viscous friction of a fluid in a dash pot, and therefore, it is called viscous damping.
Unlike the stiffness of a structure, the damping coefficient cannot be calculated from the dimensions of the structure and the sizes of the structural elements. This is understandable because it is not feasible to identify all the mechanisms that dissipate vibrational energy of actual structures. Thus, vibration experiments on actual structures provide the data for evaluating the damping coefficient. These may be freevibration experiments that lead to measured rate at which motion decays in free vibration. The damping property may also be determined from forced vibration experiments.
The equivalent viscous damper is intended to model the energy dissipation at deformation amplitudes within the linear elastic limit of the overall structure. Over this range of deformations, the damping coefficient c determined from experiments may vary with the deformation amplitude. This nonlinearity of the damping property is usually not considered explicitly in dynamic analyses. It may be handled indirectly by selecting a value for the damping coefficient that is appropriate for the expected deformation amplitude, usually taken as the deformation associated with the linearly elastic limit of the structure. Additional energy is dissipated due to inelastic behavior of the structure at larger deformations. Under cyclic forces or deformations, this behavior implies formation of a force–deformation hysteresis loop as shown in Figure 3.102. The damping energy dissipated during one deformation cycle between deformation limits ±u_{0} is given by the area within the hysteresis loop abcda (Figure 3.102).
This energy dissipation is usually not modeled by a viscous damper, especially if the excitation is earthquake ground motion. Instead, the most common and direct approach to account for the energy dissipation through inelastic behavior is to recognize the inelastic relationship between resisting force and deformation. Such force–deformation relationships are obtained from experiments on structures or structural components at slow rates of deformation, thus excluding any energy dissipation arising from ratedependent effects.
Figure 3.102 Force–displacement hysteresis loop: the area inside of the loop is a measure of energy dissipation due to nonelastic behavior. Note: Ke = initial elastic stiffness, Kp = stiffness in the plastic range, Fy = stress at yield, Dy = deformation at yield.
Buildings are composed of vertical and horizontal structural members that resist lateral forces caused by wind and seismic. The primary purpose of a diaphragm, which consists of the roof and floors of a building, is to transfer the horizontal forces to the vertical elements resisting the lateral loads.
It is customary to consider a diaphragm analogous to a deep plate girder laid in a horizontal plane. In a composite floor system, the steel deck and concrete topping perform the function of the plate girder web. Just as the plate girder flanges carry flexural forces, so must the diaphragm elements located at the perimeter. These edge members are commonly referred to as chords using the analogy of a truss spanning between supports. The resulting forces in the chords are either tension or compression acting in a direction perpendicular to the direction of lateral loading width consideration. The diaphragm chord forces at any point along the diaphragm boundary member are equal to the diaphragm moment at that point divided by the depth of the diaphragm at that location. In the absence of such chord members to take the moment couple, the diaphragm must act as a deep plate resisting both bending and shear forces.
Drag force is the tension or compression force in the diaphragm boundary member parallel to the direction of loading under consideration. Where reentrant corner irregularities are present, drag struts (also called collectors) are required to prevent localized tearing of the diaphragm. The collector essentially collects the diaphragm shear and delivers it to the vertical elements resisting the lateral loads.
The stiffness of the diaphragm has an important effect on the proportionate distribution of lateral loads to various components of the lateral support system. This effect is displayed in Figure 3.103, which shows how the relative rigidities of diaphragm and lateral support elements influence load distribution. At one extreme is the rigid diaphragm that distributes the lateral loads to the supports in proportion to just their rigidities. At the other extreme is the flexible diaphragm, which delivers lateral loads based on tributary area. The diaphragm, which falls in between the two extremes, is the semirigid diaphragm in which the load distribution involves both aspects of behavior. All three types of diaphragm behavior require effective transfer of bending and shear forces in the plane of the diaphragm, necessitating careful design detailing of connections between diaphragm elements and the lateral support system.
Figure 3.103 Relative effects of diaphragm stiffness.
In steel buildings, floor diaphragms most commonly consist of composite steel deck with concrete topping. Of concern are the design of the diaphragm itself and the transfer of diaphragm forces into the lateralloadresisting system.
Depending upon the magnitude of lateral load to be transferred to the vertical system, the designer has two choices in detailing the force transfer: assigns the transfer to occur either along the entire width frame of the line or to a selected segment of the frame width. A combination of the two is, of course, quite logical.
As stated previously, tension and compression chord forces are developed at the perimeter of the diaphragm due to lateral loads. The topping concrete slab over steel deck can typically resist the compression chord forces. Tension chord forces can be resisted by spandrel steel beams, continuous steel closure plates, or by mild steel reinforcement placed within the concrete topping.
Earthquake loads at any given level of a building are distributed to the lateralloadresisting vertical elements through the floor and roof slabs. For analytical purpose, the diaphragms are assumed to behave as deep beams. The slab is the web of the deep beam carrying shear, and the perimeter spandrel or shear wall, if any, is the flange of the beam resisting bending (see Figure 3.104a and b). In the absence of perimeter members, the slab is analyzed as a plate subjected to inplate bending.
Figure 3.104 (a) Diaphragm action of floor or roof system. Note: VLLRS. (b) Schematic drag and chord for north–south seismic loads.
Figure 3.105 Diaphragm web failure due to large opening.
Three factors are important in diaphragm design:
Inappropriate location or largesize openings such as for stairs or elevator cores, atriums, and skylights create problems similar to those related to cutting of beam flanges and holes in the beam web of a beam adjacent to the flange. This reduces the ability of the diaphragm to transfer the chord forces and, if not designed properly, may cause rupture in the web (see Figure 3.105).
Seismic vulnerability may be reduced by increasing the performance of the building, thereby reducing the damage expected in earthquakes. There are two methods by which this is typically accomplished:
Reduce the response of the building to earthquake shaking. An earthquake generates inertial forces in a building that are a function of the structure's mass, stiffness, and damping, and of the acceleration and frequency of the earthquake motion. While the actual mass of the building (the weight of the structure, contents, and people) typically cannot be significantly altered, the effective mass can be changed by providing special devices, such as passive or active mass dampers, that can effectively reduce the overall mass that is accelerated by the earthquake. Stiffness can be altered by modifying the structural system. The building's fundamental period, which is an important parameter in determining building response, can be significantly increased (and resulting forces reduced) by providing seismic isolating devices at the building foundation.
The seismic design and construction requirements contained in US building codes have been developed over many years mostly by observing the way actual structures perform in earthquakes. Following each major earthquake that causes damage to modern engineered construction, engineers investigate the behavior of typical buildings. When these engineers observe that certain design and construction practices lead to unacceptable types of damage in buildings, they develop building code provisions to discourage the continued used of these practices in future design and construction. This process has been underway in the United States and worldwide for more than 100 years and, to be sure, will continue forever. This section summarizes some of the more significant lessons that have been learned from past earthquakes and how they are implemented in the US codes.
The great M7.9 San Francisco earthquake of April 18, 1906, remains one of the worst natural disasters to affect the United States and had great significance with regard to the development of US building codes.
Primary lessons learned from this earthquake included the importance of soil type on structural behavior and the benefits of having a complete verticalloadcarrying steel frame for seismic resistance.
Most of the urban center of San Francisco was constructed on land that was reclaimed from the surround San Francisco Bay. The fill soils used to reclaim this land were of mixed characteristics, consisting of debris from building construction, excavation spoils from building basements on the adjacent dry land, and even the rotting hulls of ships, abandoned in the bay.
Observers noted that buildings constructed on this made or infirm ground performed far worse than buildings that had been constructed on the natural ground that defined San Francisco prior to 1849. This observation was included in early building code provisions for seismic resistance, developed in the 1920s, which required higher design forces for buildings sited on infirm sites. As time progressed, memory of these effects faded, and the building codes of the 1940s no longer included this factor. It was to be rediscovered in the 1970s and 1980s and reinstituted into present building codes.
It was also observed that the only buildings in the commercial center of San Francisco that remained standing were those constructed with complete verticalloadcarrying steel frames and infill masonry walls. This observation led to the eventual requirement contained in building codes that tall buildings have complete verticalloadcarrying space frames and that buildings taller than 240 ft have momentresisting space frames as part of their seismicloadresisting system.
On March 10, 1933, a moderate M6.3 earthquake struck near Long Beach, California, causing extensive damage to many unreinforced masonry buildings. The massive damage to these buildings prompted California to adopt legislation prohibiting the further construction of unreinforced masonry buildings in the state and also empowering the Office of the State Architect to adopt occupancyspecific design, construction, and quality assurance requirements for the construction of schools. This legislation marked the beginning of the recognition that some buildings are more important than others and, therefore, should be designed and constructed with greater precautions to protect the safety of the public.
The 1940 Imperial Valley earthquake, with a magnitude of nearly M7.0, occurred in the California desert, east of San Diego. This earthquake affected few buildings in this sparsely populated region. However, the USGS did obtain a highquality, strong ground motion recording from this event. Much of the work performed by researchers in developing response spectra contained in building codes was based on these recorded motions.
The 1952 Kern County earthquake was a large event, with a magnitude of nearly M7.4. Located in the arid area east of Bakersfield, California, this earthquake also caused little building damage of significance, but it did result in extensive damage to oil field and refining facilities in the region. Following this earthquake, ASCE and SEAOC formed a joint committee to make recommendations for seismic design provisions in building codes. The recommendations of the committee eventually resulted in adoption of the building codes requirements to determine seismic design forces for building based on spectral response analysis concepts.
The M6.6 San Fernando earthquake of February 9, 1971, though not of great magnitude, was one of the most significant events with regard to its effect on building codes. Prior to the 1971 earthquake, the seismic provisions in building codes were largely limited to specification of minimum design lateral forces and contained few requirements related to structural detailing.
Because a largemagnitude earthquake had not affected California in nearly 20 years prior to this event, engineers felt confident that the building codes in effect at the time were capable of providing reliable protection of buildings in earthquakes. However, the San Fernando earthquake severely damaged many modern codeconforming buildings. Among the most famous of these damaged buildings was the Olive View hospital, a large multibuilding complex located near the epicenter of this earthquake. This complex consisted of a series of reinforced concrete frame buildings. Although the 1967 edition of the UBC contained provisions for ductile reinforced concrete frame design, these requirements were not mandatory and had not been included in the hospital's design. One of the buildings, which housed ambulances and emergency vehicles, collapsed. Stair towers separated and fell away from the main hospital building, and a singlestory mechanism formed in the columns at the first level of the main structure, resulting in large permanent drift and leaving the building unrepairable.
In response to this damage, a number of major revisions were introduced into the building codes in the following years. Perhaps the most important of these changes was the recognition of the importance of ductile detailing to seismic performance. The voluntary provisions for ductile concrete moment frames were made mandatory in regions of high seismicity, and similar provisions began to be introduced into the code for other structural systems, essentially resulting in the precedent for the special, intermediate, and ordinary classifications of seismicloadresisting systems contained in building codes today.
Another important change related to more formal consideration of building occupancy when determining the seismic design requirements. Following this earthquake, the concept of occupancy categories was introduced into the building code, with higher design forces required for the design of hospitals and other buildings deemed to be essential to the public safety.
Additionally, following the San Fernando event, engineers once again recognized that the types of soil present at a site had great significance with regard to the intensity and character of ground motions experienced by buildings. This resulted in the formal introduction of site classes into the determination of seismic design forces.
The M6.4 Imperial Valley earthquake of October 15, 1979, affected relatively few buildings due to the sparse population of the affected region, near the California–Mexico border. However, one building, the sixstory Imperial County Services Building, did experience noteworthy damage. This sixstory concrete shear wall building had an outofplane offset between the shear walls and frames above the first story and those below, in order to accommodate an arcade feature at the ground level. Overturning forces from the shear walls above the first story crushed the firststory columns immediately below the walls and frames.
Research into the behavior of this building led to the present code requirement to design columns beneath discontinuous walls and frames for the amplified forces that consider the overstrength of the structure above. Later, the building code applied this same requirement to other irregularities.
Prior to the great M8.1 Mexico City earthquake of September 19, 1985, there had been little record of any significant damage to steel frame buildings. However, in this earthquake, two highrise steel frame buildings, one 22story tall and other 16story tall, collapsed. These buildings were braced steel frame structures that utilized builtup box section columns. Investigation of damage suggested that overturning forces imposed on the columns resulted in local buckling of plate sections in the builtup box section columns, which then led to failure of the seam welds in the boxes.
This observation led to the design requirement to proportion the columns in steel seismicloadresisting systems with adequate strength to resist the maximum axial forces that can be delivered to the columns, considering the overstrength of the structural system supported above, whether or not the structural system is irregular.
The M5.9 Whittier Narrows earthquake of October 17, 1987, was a relatively modest event, both in size and effect. However, it did cause damage to some modern buildings prompting several code changes. One of the most significant of these changes was based on observation of damage sustained by the California Federal Savings company's data processing center. This braced steel frame building employed chevronpattern braces. The braces buckled in compression and the floor beams at the apex of the chevrons were bent downward, causing damage to the floor systems. Observations of this damage led to the introduction of provisions requiring design of beams at the apex of chevronpattern braces for the unbalanced forces that result following buckling of one of these braces.
The 1989 M7.1 Loma Prieta earthquake, occurring approximately 70 miles south of San Francisco, caused relatively little damage to modern structures designed to recent editions of the building code. However, it caused extensive damage to older buildings and structures.
The large array of ground motion obtained in this earthquake was key to the determination that the ground shaking in the region within the near field—that is, within a few kilometers of the zone of fault rupture—is not only stronger but also significantly different from the shaking experienced farther from the rupture zone. Recordings obtained in this event, together with the limited nearfault recordings available from other earthquakes (such as the 1971 San Fernando and 1992 Landers and Big Bear events), allowed seismologists to identify the pulselike characteristics of nearfield motions. However, it was only after the 1994 Northridge earthquake when these effects were again noted, and the building code was actually modified to account for these effects.
The M6.7 earthquake that struck Northridge, California, on January 17, 1994, was one of the most significant earthquakes of the past century with regard to the wealth of engineering data that were obtained and analyzed and subsequently implemented into the building codes. This is because, like the San Francisco Bay area in 1989, the affected region had many strong ground motion instruments present, but also, unlike the Loma Prieta earthquake, this event damaged many modern codeconforming buildings.
The Northridge earthquake provided valuable earthquake experience data on the performance of four types of structures: concrete tiltup buildings with wood roofs, precast concrete parking structures, braced steel frames, and momentresisting steel frames. As a result of the observations of damage that occurred in this earthquake, extensive revisions were made to the 1997 editions of the UBC, as well as the NEHRP provisions, which form the basis for seismic design requirements of SEI/ASCE 7.
With regard to braced steel frames, the observation of fractures in hollow structural section (HSS) braces, following buckling, resulted in significant restrictions on the permissible width–thickness ratios for braced elements. It also resulted in severe restrictions of the use of ordinary concentric braced frame (OCBF).
Perhaps the most significant lesions learned were associated with the unanticipated discovery of fractures in the welded joints of modern steel moment frames. These fractures were attributed to a variety of factors, including connection geometries that resulted in stress concentrations and high restraint, limiting ductility; wide variations in the yield and tensile strengths of the common ASTM A36 and A572 grades of material used in building construction at that time; the low toughness of weld filler metals commonly used in steel construction; and poor adherence to the requirements of the American Welding Society (AWS) welding code when making welded joints. Another important contributing factor to these unexpected failures was that the connection practices commonly in use prior to the earthquake had been validated years earlier by testing of specimens that were much lighter than those present in the damaged buildings. Over the years since the initial research, design practice had deviated away from the use of highly redundant frames with relatively small members, relatively few participating members, and use of very large sections. The earlier testing was not applicable to these heavier frames, but this was not understood until after the earthquake and discovery of the damage.
These observations resulted in a number of changes in the building codes and design practice as well as a major rewrite of AISC 341 and the introduction of the AWS D1.8 seismic supplement to AWS D1.1. Significant changes included the introduction of the new ASTM A992 grade of structural steel, with controlled yield strengths and yield to tensile ratios; requirements to demonstrate (through laboratory testing of fullscale specimens) that moment connections are capable of attaining minimum inelastic deformation demands; requirements to use weld filler metals with minimum rated notch toughness in seismicloadresisting systems; and requirements to remove weld backing bars and weld tabs from critical joints.
Another major feature introduced into the building codes following the Northridge earthquake was the requirement to quantify the redundancy inherent in a structural design to adjust the design seismic forces and permissible drifts for the structure, based on this redundancy. This was based on the observation that many modern structures that had been severely damaged in the earthquake were less redundant than earlier structures that performed better.
Current seismic building codes and accepted engineering practice do not assure that all structures designed and built in strict compliance with the codes will not be severely damaged in a moderatetostrong earthquake shaking. This basic purpose of a building code is to provide public safety. Codes are not concerned with high quality, serviceability, or limited repairable damage; the owner who is interested in these has to establish his or her own criteria. He or she may prescribe more stringent criteria so the owner can minimize property damage as well as protecting the occupants. The engineer may desire to apply more stringent requirements for any number of reasons, such as to lower insurance rates, to reduce expensive repair costs by lowering earthquake damage risk, to provide better quality for improved service and longerlife structures, or to reduce maintenance and operating costs. Performance can be improved at little, if any, increase in cost of proper structural concepts, and detailing techniques are selected.
The current ASCE 710 seismic standard is an advance in the art and standard that a building conforming to the seismic won't suffer damage in an earthquake; it is not allinclusive; there are areas where good judgment and experience of the earthquakeconscious engineer will have to prevail in order to produce a sound, safe, serviceable, and economical structure. It provides improved definitions and more specific guidance in several instances, but because of the great number of variables and complexity of the problem, the structural engineer should realize that the requirements of the code are minimum that emphasis must be placed on the structural concepts and detailing techniques rather than on refinement of stress calculations.
There are three basic steps in the development of adequate earthquakeresistant design: (1) architectural concept, (2) stress analysis, and (3) proper details. The seismic design begins with and is dominated by the architectural concept of the building. The materials of construction, the nonstructural component configuration (both horizontal and vertical), fenestration, and the nature of the structural elements all have a profound influence upon the success of the seismic design. Codeprescribed seismic coefficients are applicable only to a few simplified, stereotyped structures. Consequently, the structural engineer must visualize the response of the structure to earthquake ground motions and adapt the design to the distortions and stress paths, which will occur in the building. Assumptions used in the calculations are consistent with the asbuilt structure. Details for earthquakeresistant structures differ in basic concept from details for wind or vertical forces. Foremost among requirements vital to earthquakeresistant design of all types of buildings is the necessity of tying the various components together so that they act as a unit or the isolation of certain components so that the structure will respond as assumed in the calculations. Generally, these requirements cannot be stated in detail by a building code but have to be left to the understanding and judgment of the responsible engineer. By proper coordination in the development of basic concepts of the particular facility, the design team can secure adequate resistance, frequently without significant premium cost, just by better planning, simple mathematics, and the execution of better details.
Field observations made following several earthquake emphasized that improperly designed and constructed joints between vertical and horizontal resisting elements, as well as joints between precast and prestressed components, performed poorly under the abnormal earthquake intensity. Further, observations and computer dynamic analysis of a typical building show that a moderately strong earthquake will cause deflections and forces that are considerably greater than those that would result from application of the static seismic loads required by the code. Thus, during any reasonably severe earthquake, it can be expected that significant inelastic (nonlinear) deformations will be produced in a typical building. Hence, ductility that will increase the energyabsorbing capacity is essential if structures are to tolerate deformations beyond the range of elastic (linear) behavior.
Principal elements that influence damage to buildings are (1) intensity of the earthquake waves, (2) duration of earthquake motion, (3) response of the structure to ground motions, and (4) design of the building components and their connections. The code substitutes static design forces for a dynamic response of a structure to a complex earthquake ground motion. The code approach is to apply a horizontal force, in each direction alternately, to each principal mass, such as at each floor. Such static forces cause the structure to deflect in a rocking motion. However, in an actual earthquake with oscillating ground motion, the response of the structure may result in a floor moving in one direction, while at the same time, an adjacent floor may be moving in an opposite direction. When this happens, a building does not tilt just back and forth, but it deflects in a weaving, snakelike fashion. Using dynamic loading and a computer analysis, one can more nearly predict how a proposed building will act and deform under ground motions of a specific earthquake. It will be found that this response may sometimes cause deflections, joint rotations, and stresses in components quite different than that determined from the static loadings.
An earthquake causes random erratic vibratory ground motions at the base of a structure and the structure actively responds to these motions. Seismic design involves two distinct steps: (1) the prediction of ground motions at the base of the structure based on the seismicity of the site and (2) the selection of a structure that will deform without catastrophic collapse when responding to these ground motions.
Destructive earthquakes are all due to tectonic motions. According to the generally accepted elastic rebound theory, defaulting is the cause of earthquakes, not a consequence of them. The primary mechanism is the fracturing of the rock in a zone of large deformation along a fault. The consequent sudden displacement as the deformation and strain energy are released initiates vibrations in the Earth's crust (bedrock). Vibrations of such intensity as to cause significant damage to structures have been found only in the immediate environs (up to 150+ miles) of the fault movement. As the response of buildings depends on the characteristic of the ground motion, it is highly desirable that a quantitative description of the ground motion be available at the site of the building. Large amplitude vibrations near faults are too great to be recorded by ordinary seismographs, and special strongmotion seismographs have been developed, which directly record the three components of ground accelerations as functions of time. Unfortunately, there are absolutely no ground motion records available at the location of destructive damage or maximum ground motion for the truly major earthquakes such as Alaska (1964) or San Francisco (1906). Seismologists classify earthquakes according to the Richter scale of magnitude (M), which characterizes the total energy released during an earthquake. From the viewpoint of effect on a particular structure, the engineers are interested in the intensity of the vibrations of the Earth's surface at the site. Such intensity of vibrations is a function of (1) amount of energy released (M), (2) distance from the fault to the structure, and (3) the character and thickness of the foundation material. Seismological analysis allows prediction on a regional basis of the magnitude of future earthquakes. Mathematical theories have been developed for predicting the effect of distance from the fault movement for various soil conditions underlying a site, using assumed bedrock vibrations. But as mentioned previously, ground motion records of destructive earthquakes are not sufficiently available to substantiate these theories. Each new earthquake appears to differ from the previous one. Hence, there are too many unknowns to be able to predict with any firm degree of certainty (quantitatively) the timevarying foundation vibration input for some unknown future earthquake. On the other hand, these historical records indicate the general trend, give the best estimate available, and should be used where appropriate. Such appropriate analytical procedures provide the capability of obtaining a semiquantitative guide as well as a qualitative guide to site response during future earthquakes. Qualitatively, the following are apparent:
The loads or forces that a structure is called upon to sustain due to earthquake motions result directly from the distortions induced in the structure by the motion of the ground on which it rests. Base motion is characterized by displacements, velocities, and accelerations, which are erratic in direction, magnitude, duration, and sequence. Earthquake loads are essentially inertial forces related to the mass, stiffness, and damping characteristics of the structure. The problem is completely defined by these physical properties of the structural system and by the timevarying displacements applied at its foundation. Thus, the evaluation of these structural properties and the selection of an appropriate earthquake input are the most critical factors in the earthquake response analysis. Assuming that the structural and earthquake characteristics are known, the problem of the determination of the vibratory response of the structure can be formulated in theory and solved in very general terms, even for situations involving plastic (inelastic) deformations. In the formulation of any dynamic response analysis, it must be recognized that the structure generally will be subjected to static loadings (e.g., gravitational forces) in addition to the dynamic excitation. If the structure is linearly elastic, so that the principle of superposition is applicable, it is convenient to consider separately the static and dynamic loadings; then the total structural response is obtained by adding the static stresses and deflections to the results of the dynamic analysis. However, if the structure yields or is subject to some other nonlinear behavior during the dynamic loading, superposition is not valid. Since the mechanics and mathematics of dynamic analysis are given in standard textbooks, it will suffice here to point out that the application of such analysis has certain limitations. The structural properties and the earthquake input must be specified, but these properties are simplified and idealized for the model study as an assembly of masses interconnected by springs and damping elements. Since the mechanics of dynamic analysis requires that a separate solution must be obtained for each instant of time during the entire history of interest, computations using computers are necessary. The specific object of this theory is to predict the stresses and deflections that will be developed in any given structural system. Unlike the stiffness and mass properties of a structure, damping capacity cannot be calculated. Therefore, in formulating a mathematical model, it is important that some results on the damping capacity of structures similar to the one being modeled are available. Our knowledge of the response of buildings to earthquake motions is far from exact or complete. However, qualitatively, the following are apparent:
In this analysis, we substitute assumed equivalent design static lateral forces for the true response to ground motions. This approach attempts to recognize our limited recorded experience and the qualitative dynamic analysis of simplified structures. The ASCE formulas for base shear V may be rewritten in a compact form as follows:
where
The seismic response coefficient, C_{s},
The value of C_{s} computed need not exceed the following:
C_{s} shall not be less than
In addition, for strutures located where S_{1} is equal to or greater than 0.6g, C_{s} shall not be less than
These terms in these equations are defined in ASCE 710 (and elsewhere in this book) and need no further explanation except to indicate that the shear diagram corresponds roughly to the shear envelope obtainable from a dynamic analysis (for a uniform section and weight). Invariably, the design forces specified by the ASCE are smaller than the corresponding maximum values that would be indicated by an elastic–dynamic analysis, using moderateintensity input. Therefore, their application must be used with sound engineering judgment. The type of structure must be selected, detailed, and constructed to fit the intent of the ASCE and the particular site conditions. Even though the assumed equivalent static forces specified in the ASCE are only approximations, they are applicable to a few types of simple structures and under restricted conditions. The assignment of lateral load design coefficients to inappropriate type of structures or conditions could be disastrous in case of a strong earthquake. When unusual structures are involved, a dynamic analysis should be made to aid in making decisions. It is incumbent upon the architect, in cooperation with the structural engineer, to develop a concept that will avoid problems.
As previously indicated, the seismic forces specified are quite small relative to the actual forces expected at least once in the life cycle of any building. Therefore, they are meant to be used with certain limitations. These limitations and principles are important since they constitute the basic justification for designing with lateral earthquake forces of 3%–13% gravity (G) as compared with spectrum analysis requirements of over 50% G.
Accelerations derived from actual earthquakes are surprisingly high as compared to the ASCE forces used in ordinary design. Any design that merely comes up to the code is apt to be marginal. The manipulation of numbers in accordance with code requirements will certainly not assure adequate earthquake resistance in case of a major earthquake. Important basic assumptions are necessary before design enters the mathematical stage. Generally, these requirements cannot be stated in detail by a building code but must be left to the judgment of the responsible engineer. The most sensible approach is to design on some reasonable basis, to recognize the uncertain nature of the demands, and to provide all the reserve capacity that can be accomplished at little or no extra cost in initial construction or at only a slight sacrifice in architectural features. Of course, unusual risks should have additional design considerations and refinements. If potential elements of distress can be eliminated before calculations are made, the problems are much simpler. The first step is to develop an architectural concept that will avoid as many problems as feasible. The shape of the building (both horizontal and vertical configuration and fenestrations), the materials of construction, and detailing all have a profound effect on the response to lateral forces. Architects and engineers must realize the following:
The objective is to reduce project costs to a minimum without compromising its function, quality, or reliability. Final selection of materials and systems will be made with due considerations to cost of construction, architectural requirements, fire and other safety hazards, low maintenance and operating costs over the life cycle of the facility, and fund limitations. Of equal importance along with costs is the necessity to make certain that the most efficient systems, methods, and materials are employed and that the physical appearance is a credit to the design agency.
Usually, the major structural–architectural components of a building that mostly affect the cost of construction are the exterior walls, partitions, floor and roof decks, and the structural framing system. In some instances, the selection of a type of foundation may be a major factor in a cost study. Skillful planning, simple detailing, and arrangement of space to be compatible with repetitive construction all add greatly to reducing total building costs; however, such saving may not be easily measured. Including contractor's options will create competition and thereby lower costs. On the other hand, the use of exotic, unconventional, sophisticated, and/or proprietary materials and methods of construction is usually not well suited to open competitive bidding requirements, and the reliability of the earthquakeresistance performance may be difficult to determine.
Participation of all disciplines of the design team in the conceptual planning in a selection of basic construction materials, at an early stage, will pay big dividends in ensuring the most efficient design at lowest construction cost and minimize the total design effort. Procedures in the approach to develop a concept will vary depending upon the type of facility and the experience of the individuals on the design team.
The design of a structure in accordance with the seismic provisions will not ensure against earthquake damage, since the horizontal deformations from design loads are lower than can be expected during a major earthquake. However, a number of things can be done without increasing construction cost to limit damage, which otherwise would be expensive to repair following strong earthquake. An important factor to keep in mind is the nature and geometry of the building when it responds to earthquake motions. As a rough guide, it should be assumed that deflections (story drift) may be four times that resulting from the required lateral forces. A list of features, which can aid in avoiding excessive damage, follows:
For gravity loads, it has been a longstanding practice to design for strength and deflections within the elastic limits of the members. However, to control design within elastic behavior for likely horizontal seismic forces is impractical. Hence, designers need to resort to other techniques to meet acceptable performance. A number of structural features contributing to seismic resistance under earthquake conditions are enumerated and discussed as follows:
Earthquake forces are dynamic because they vary with time. Since the load is time varying, the response of the structure, including deflections, axial and shear forces, and bending moments, is also time dependent. Therefore, instead of a single solution, a separate solution is required to capture the response of the building at each instant of time for the entire duration of an earthquake. Because the resulting inertial forces are a function of building accelerations, which are themselves related to the inertial forces, it is necessary to formulate the dynamic problem in terms of differential equations.
Consider a portal frame, shown in Figure 3.106, consisting of an infinitely stiff beam supported by flexible columns that have negligible mass as compared to that of the beam. For horizontal motions, the structure can be visualized as a springsupported mass, as shown in Figure 3.107a, or as a weight W suspended from a spring, as shown in Figure 3.107b. Under the action of a force W, the spring will extend by a certain amount x. If the spring is very stiff, x is small, and vice versa. The extension x can be related to the stiffness of the spring k by the relation The spring constant or spring stiffness k denotes the load required to produce the unit extension of the spring. If W is measured in kip and the extension in inches, the spring stiffness will have a dimension of kip per inch. The weight W comes to rest after the spring has extended by the length x. Equation 3.47 expresses the familiar static equilibrium condition between the internal force in the spring and the externally applied force W.
Figure 3.106 Singlebay, singlestory postal frame.
Figure 3.107 Analytical models for SDOF stem: (a) model in horizontal position and (b) model in vertical position.
If a force is applied or removed suddenly, vibrations of the system are produced. Such vibrations, maintained by the elastic force in the spring along, are called free or natural vibrations. The weight moves up and down and therefore is subjected to an acceleration x given by the second derivative of displacement x, with respect to time t. At any instant t, there are three forces acting on the body: the dynamic force equal to the product of the body mass and its acceleration, the force W acting downward, and the force in the spring equal to W + Kx for the position of weight shown in Figure 3.108. These are in a state of dynamic equilibrium for an undamped system given by the relation
The preceding equation of motion is called Newton's law of motion and is governed by the equilibrium of inertial force that is a product of the mass W/g and acceleration x and the resisting forces that are a function of the stiffness of the spring.
Figure 3.108 Damped oscillator: (a) analytical model and (b) forces in equilibrium.
The principle of virtual work can be used as an alternative to derive Newton's law of motion. Although the method was first developed for static problems, it can be readily applied to dynamic problems by using D'Alembert's principle. The method establishes dynamic equilibrium by including inertial forces in the system.
The principle of virtual work can be stated as follows: For a system in equilibrium, the work done by all the forces during a virtual displacement is equal to 0. Consider a damped oscillator subjected to a timedependent force F_{t}, as shown in Figure 3.108. The freebody diagram of the oscillator subjected to various forces is shown in Figure 3.108b.
Let δ_{x} be the virtual displacement. The total work done by the system is zero and is given by
Since δx is arbitrarily selected,
This is differential equation of motion of the damped oscillator.
The equation of motion for an undamped system can also be obtained from the principle of conservation of energy. It states that if no external forces are acting on the system, and there is no dissipation of energy due to damping, then the total energy of the system must remain constant during motion, and consequently, its derivative with respect to time must be equal to zero.
Consider again the oscillator shown in Figure 3.108 without the damper. The two energies associated with this system are the kinetic energy of the mass and the potential energy of the spring.
The kinetic energy of the spring
where $\dot{x}$
is the instantaneous velocity of the mass.The force in the spring is kx; work done by the spring is kx δx. The potential energy is the work done by this force and is given by
The total energy in the system is a constant. Thus,
Differentiating with respect to x, we get
Since $\dot{x}$
cannot be zero for all values of t, we getThis differential equation has a solution of the form
where
Maximum kinetic energy is given by
Maximum potential energy is
Since T = V,
or
This is the natural frequency of the simple oscillator. This method, in which the natural frequency is obtained by equating maximum kinetic energy and maximum potential energy, is known as Rayleigh's method.
In these systems, the displacement configuration is determined by a finite number of displacement coordinates. The true response of multidegree system can be determined only by evaluating the inertial effects at each mass particle because structures are continuous systems with an infinite number of DOF's. Although analytical methods are available to describe the behavior of such systems, these are limited to structures with uniform material properties and regular geometry. The methods are complex, requiring the formulation of partial differential equations. However, the analysis is greatly simplified by replacing the entire displacement of the structure by a limited number of displacement components and assuming the entire mass of the structure is concentrated in a number of discrete points.
Figure 3.109 Multistory analytical model with lumped masses.
Consider a multistory building with n DOFs, as shown in Figure 3.109. The dynamic equilibrium equations for undamped free vibration can be written in the general form.
Writing the equations in matrix form
where
If the effect of damping is included, the equations of motion would be in the form
where
General methods of solutions of these equations are available but tend to be cumbersome. Therefore, in solving seismic problems, simplified methods are used; the problem first is solved by neglecting damping. The absence of precise data on damping does not usually justify a more rigorous treatment. Neglecting damping results in dropping the second term, and limiting the problem to free vibrations results in dropping the righthand side of Equation 3.61b. The resulting equations of motion will become identical to Equation 3.61a. During free vibrations, the motions of the system are simple harmonic, which means that the system oscillates about the stationary position in a sinusoidal manner; all masses follow the same harmonic function, having similar angular frequency, ω. Thus,
or in matrix notation,
where
Substituting for {x} and its second derivative $\left\{\ddot{\text{x}}\right\}$
in Equation 3.61a results in a set of algebraic expressions:Using a procedure known as Cramer's rule, the preceding expressions can be solved for determining the frequencies of vibrations and relative values of amplitudes of motion a_{11}, a_{12}, …, a_{n}. The rule states that nontrivial values of amplitudes exist only if the determinant of the coefficients of a is equal to 0 because the equations are homogeneous, meaning that the righthand side of the equation is 0. Setting the determinant of the equation equal to 0, we see that the values for all the stiffness coefficients k_{11}, k_{12}, etc., and the masses m_{1}, m_{2}, etc., are known, the determinant of the equation can be expanded, leading to an ω^{2} polynomial expression. The solution of the polynomial gives one real root for each mode vibration. Hence, for a system with n DOFs, n natural frequencies are obtained. The smallest of the values obtained is called the fundamental frequency and the corresponding mode, the fundamental or first mode.
In mathematical terms, the vibration problem is similar to those encountered in stability analyses: The determination of frequency of vibrations can be considered similar to the determination of critical loads, while the modes of vibration can be likened to the evaluation of buckling modes. Such types of problems are known as eigenvalue, or characteristic value, problems. The quantities ω^{2}, which are analogous to critical loads, are called eigenvalues, or characteristic values, and in a broad sense can be looked upon as unique properties of the structure similar to geometric properties such as area or moment of inertia of individual elements.
Unique values for characteristic shapes, on the other hand, cannot be determined because the substitution of ω^{2} for a particular mode into the dynamic equilibrium equation results in exactly n unknowns for the characteristic amplitudes x_{1}, …, x_{n} for that mode. However, it is possible to obtain relative values for all amplitudes in terms of any particular amplitude. We are, therefore, able to obtain the pattern or the shape of the vibrating mode, but not its absolute magnitude. The set of modal amplitudes that describe the vibrating pattern is called eigenvector or characteristic vector.
In this method, the equations of motions are transformed from a set of n simultaneous differential equations to a set of n independent equations by the use of normal coordinates. The equations are solved for the response of each mode, and the total response of the system is obtained by superposing individual solutions. Two concepts are necessary for the understanding of the modal superposition method: (1) the normal coordinates and (2) the property of orthogonality.
In a static analysis, it is common to represent structural displacements by a Cartesian system of coordinates. For example, in a planar system, coordinates x and y and rotation q are used to describe the position of a displaced structure with respect to its static position. If the structure is restrained to move only in the horizontal direction and if rotations are of no consequence, only one coordinate x is sufficient to describe the displacement. The displacements can also be identified by using any other independent system of coordinates. The only stipulation is that a sufficient number of coordinates are included to capture the deflected shape of the structure. These coordinates are commonly referred to as generalized coordinates and their number equals the number of DOFs of the system. In dynamic analysis, however, it is advantageous to use freevibration mode shapes known as normal modes to represent the displacements. While a mathematical description of normal modes and their properties may be intriguing, there is nothing complicated about their concept. Let us indulge in some analogies to bring home the idea. For example, normal modes may be considered as being similar to the primary colors red, blue, and yellow. None of these primary colors can be obtained as a combination of the others, but any secondary color such as green, pink, or orange can be created by combining the primary colors, each with a distinct proportion of the primary colors. The proportions of the primary colors can be looked upon as scale factors, while the primary colors themselves can be considered similar to normal modes. To further reinforce the concept of generalized coordinates, recall beam bending problems in which the deflection curve of beam is represented in the form of trigonometric series. Considering the case of a simply supported beam subjected to vertical loads, as shown in Figure 3.110, the deflection y, at any point, can be represented by the following series:
Geometrically, this means that the deflection curve can be obtained by superposing the simple sinusoidal curves shown in Figure 3.110.
Figure 3.110 Generalized displacement of a simply supported beam (a) loading, (b) fullsine curve, (c) halfsine curve, (d) onethirdsine curve, and (e) onefourthsine curve.
The first term in Equations 3.62 represents the fullsine curve, the second term, the halfsine, etc. The coefficients a_{1}, a_{2}, a_{3}, etc., represent the maximum ordinates of the curves, while the numbers 1, 2, 3, etc., the number of waves or mode shapes. By determining the coefficients a_{1}, a_{2}, a_{3}, etc., the series can represent the deflection curve to any desired degree of accuracy, depending on the number of terms considered in the series.
This force–displacement relationship is rarely used in static problems but is of great significance in structural dynamics. This is best explained with an example shown in Figure 3.111.
Consider a twostory, lumpedmass system subjected to free vibrations. The system's two modes of vibrations can be considered as elastic displacements due to two different loading conditions, as shown in Figure 3.111. We will use a theorem known as Betti's reciprocal theorem to demonstrate the derivation of orthogonality conditions. This theorem states that the work done by one set of loads on the deflections due to a second set of loads is equal to the work done by the second set of loads acting on the deflections due to the first. Using this theorem with reference to Figure 3.111, we get
This can be written in matrix form
or
If the two frequencies are not the same, that is, ω_{1} ≠ ω_{2}, we get
This condition is called the orthogonality condition, and the vibrating shapes, {x_{a}} and {x_{b}}, are said to be orthogonal with respect to the mass matrix [M]. By using a similar procedure, it can be shown that
Figure 3.111 Twostory lumpedmass system illustrating Betti's reciprocal theorem: (a) lumped model (b) forces during the first mode of vibration and (c) forces acting during second mode of vibration.
The vibrating shapes are therefore orthogonal with respect to the stiffness matrix as they are with respect to the mass matrix. In the general case of the structures with damping, it is necessary to make a further assumption in the modal analysis that the orthogonality condition also applies for the damping matrix. This is for mathematical convenience only and has no theoretical basis. Therefore, in addition to the two orthogonality conditions mentioned previously, a third orthogonality condition of the form
is used in the modal analysis.
To bring out the essentials of the normal mode method, it is convenient to consider the dynamic analysis of a twoDOF system. We will first analyze the system by a direct method and then show how the analysis can be simplified by the modal superposition method.
Consider a twostory dynamic model of a shear building shown in Figure 3.112a through c, subject to free vibrations. The masses m_{1} and m_{2} at levels 1 and 2 can be considered connected to each other and to the ground by two springs having stiffnesses k_{1} and k_{2}. The stiffness coefficients are mathematically equivalent to the forces required at levels 1 and 2 to produce unit horizontal displacements relative to each level.
It is assumed that the floors and, therefore, the masses m_{1} and m_{2} are restrained to move in the direction x and that there is no damping in the system. Using Newton's second law of motion, the equations of dynamic equilibrium for masses m_{1} and m_{2} are given by
Rearranging terms in these equations gives
Figure 3.112 Twostory shear building, free vibrations: (a) building with masses, (b) mathematical model, and (c) freebody diagram with masses.
The solutions for the displacements x_{1} and x_{2} can be assumed to be of the form
where
The substitution of Equations 3.72 and 3.73 into Equations 3.70 and 3.71 gives the following equations:
To obtain the solution for the nontrivial case of A and B ≠ 0, the determinant of the coefficients of A and B must be equal to 0. Thus,
The expansion of the determinant gives the relation
or
The solution of this quadratic equation yields two values for ω^{2} of the form
where
As mentioned previously, the two frequencies ω_{1} and ω_{2}, which can be considered intrinsic properties of the system, are uniquely determined.
The magnitudes of the amplitudes A and B cannot be determined uniquely but can be obtained in terms of ratios r_{1} = A_{1}/B_{1} and r_{2} = A_{2}/B_{2} corresponding to ${\omega}_{1}^{2}$
and ${\omega}_{2}^{2}$ , respectively. Thus,The ratios r_{1} and r_{2} are called the amplitude ratios and represent the shapes of the two natural modes of vibration of the system.
Substituting the larger angular frequency ω_{1} and the corresponding ratio r_{1} in Equations 3.72 and 3.73, we get
These expressions describe the first mode of vibration, also called the fundamental mode. Substituting the larger angular frequency ω_{2} and the corresponding ratio r_{2} in Equations 3.72 and 3.73, we get
The displacements x′_{1} and x″_{2} describe the second mode of vibration. The general displacement of the system is obtained by summing the modal displacements:
Thus, for systems having two DOFs, we are able to determine the frequencies and mode shapes without undue mathematical difficulties. Although the equations of motions for multidegree systems have similar mathematical form, solutions for modal amplitudes in terms of geometrical coordinates become unwieldy. The use of orthogonal properties of modes shapes makes this laborious process unnecessary. We will demonstrate how the analysis can be simplified by using the modal superposition method. Consider again the equations of motion for the idealized twostory building in the previous section. As previously stated, damping is neglected, but instead of free vibrations, we will consider the analysis of the system subject to timevarying force functions F_{1} and F_{2} at levels 1 and 2. The dynamic equilibrium for masses m_{1} and m_{2} is given by
These two equations are interdependent because they contain both the unknowns x_{1} and x_{2}. These can be solved simultaneously to get the response of the system, which was indeed the method used in the previous section to obtain the values for frequencies and mode shapes. The modal superposition method offers an alternate procedure for solving such problems. Instead of requiring the simultaneous solution of equations, we seek to transform the system of interdependent or coupled equations into a system of independent or uncoupled equations. Since the resulting equations contain only one unknown function of time, solutions are greatly simplified. Let us assume that the solution for the preceding dynamic equations is of the form
What we have done in the preceding equations is to express displacements x_{1} and x_{2} at levels 1 and 2 as a linear combination of properly scaled values of two independent modes. For example, a_{11} and a_{12}, which are the mode shapes at level 1, are combined linearly to give the displacement x_{1}; z_{1} and z_{2} can be looked upon as scaling functions. Substituting for x_{1} and x_{2} and their derivatives x_{1} and x_{2} in equilibrium (Equations 3.87 and 3.88), we get
We seek to uncouple Equations 3.87 and 3.88 by using the orthogonality conditions. Multiplying Equation 3.91 by a_{11} and Equation 3.92 by a_{21}, we get
Adding the preceding two equations, we get
Similarly, multiplying Equations 5.48 and 5.49 by a_{12} and a_{22} and adding, we obtain
Equations 3.91 and 3.92 are independent of each other and are the uncoupled form of the original system of coupled differential equations. These can be further written in a simplified form by making use of the following abbreviations:
Using these notations, each of Equations 3.94 and 3.95 takes the form similar to the equations of motion of an SDOF system:
The solution of these uncoupled differential equations can be found by any of the standard procedures given in textbooks on vibration analysis. In particular, Duhame's integral provides a general method of solving these equations irrespective of the complexity of the loading function. However, in seismic analysis, usually a response spectrum is used instead of a forcing function to obtain the maximum values of the response corresponding to each modal equation. The direct superposition of modal maximum would, however, give only an upper limit for the total system that, in many engineering problems, would be too conservative. To alleviate this problem, approximations based on probability considerations are generally employed. One method employs the socalled rootmeansquare (RMS) procedure, also called the SRSS method. As the name implies, a probable maximum value is obtained by evaluating the SRSS of the modal quantities. Although this method is simple and widely used, it is not always a conservative predictor of earthquake response because more combinations of modal quantities can occur, for example, when two modes have nearly the same natural period. In such cases, it is more appropriate to use the CQC procedure.
The aim of this section is to bring out the essentials of structural dynamics as related to seismic design of buildings. A certain amount of mathematical presentation has been unavoidable. Lest the reader lose the physical meaning of the various steps, it is worthwhile to summarize the essential features of dynamic analysis.
The dynamic analysis of buildings is performed by idealizing them as MDOF systems. The dead load of the building together with a percentage of live load (estimated to be present during an earthquake) is considered as lumped masses at each floor level. In a planar analysis, each mass has one DOF corresponding to lateral displacement in the direction under consideration, while in a 3D analysis, it has three DOF corresponding to two translational and one torsional displacements. Free vibrations of the buildings are evaluated, without including the effect of damping. The damping is taken into account by modifying the design response spectrum. The dynamic model representing a building has the number of mode shapes equal to the number of DOFs of the model. Mode shapes have the property of orthogonality, which means that no given mode shaped can be constructed as a combination of others, yet any deformation of the dynamic model can be described as a combination of its mode shapes, each multiplied by a scale factor. Each mode shape has a natural frequency of vibration. The mode shapes and frequencies are determined by solving for the eigenvalues. The total response of the building to a given response spectrum is obtained by statistically summing a predetermined number of modal responses. The number of modes required to adequately determine the design forces is a function of the dynamic characteristics of the building. Generally, for regular buildings, 6–10 modes in each direction are considered sufficient. Since each mass responds to earthquakes in more than one mode, it is necessary to evaluate effective modal mass values. These values indicate the percentage of the total mass that is mobilized in each mode. The acceleration experienced by each mass undergoing various modal deformations is determined from the response spectrum, which has been adjusted for damping. The product of the acceleration for a particular mode, multiplied by the effective modal mass for that mode, gives the static equivalent for forces at each discrete level. Since these forces do not reach their maximum values simultaneously, statistical methods such as SRSS or CQC are used for the combinations. The resulting forces are used as design static forces.
Given: A weight W when attached to the end of a rubber band produces in it a static elongation δ_{st} = 5 in. If the weight is raised until the tension in the band is zero and then released without initial velocity, what maximum elongation will be produced in the band due to this sudden application of load and with what frequency will the suspended weight W oscillate?
Solution: Taking the position of static equilibrium of the suspended weight as the origin and considering downward displacement as positive, we conclude that at the moment of release, the weight has the initial displacement
and that the initial velocity is 0. Hence, from the equation of simple harmonic motion, the displacement of weight W from the position of static equilibrium at any instant t is
When the angle pt = π, cos pt = −1 and the displacement y have its maximum positive value,
We conclude that the total elongation produced in the band by the sudden application of the load W is twice that produced by the same load when gradually applied. The frequency of vibration is given by
Now that we have learned the fundamentals of dynamic analysis in excruciating detail, perhaps it is instructive to scrutinize how dynamic analysis is performed internally in computer programs. This will enhance our understanding of the modal superposition techniques used in computer programs.
Two examples presented illustrate the modal analysis method. In the first part of each example, the analysis is performed to determine the base shear for each mode using given building characteristics and ground motion spectra. In the second part, the story forces, accelerations, and displacements are calculated for each mode and are combined statistically using the SRSS combination.
The base shear is determined from
where
The modal base shear participation factor, a_{m}, for the mth mode is determined from
The story modal participation, PF_{xm}, for the mth mode is determined from
where
The modal story lateral displacement, δ_{xm}, is determined from
where
Given: The example is illustrated in Figure 3.113.
Weights and masses
Figure 3.113 Threestory building example: dynamic analysis.
Figure 3.114 Threestory building; response spectrum.
Periods
Spectral acceleration: From the response spectrum of Figure 3.114, the spectral accelerations are
Required
Solution: The results of the modal analysis are shown in Tables 3.3 through 3.5. It should be noted that higher modes of response become increasingly important for taller or irregular buildings. For the regular threestory building, the first mode dominates the lateral response as shown in the comparison of the modal story shears and the SRSS story shears in Table 3.5. For example, if only the firstmode shears had been used for analysis, we would have obtained 89% of the SRSS shear at the roof, 99% at the third floor, and 95% at the second floor. While the secondmode shear at the roof is 50% of the firstmode shear, when combined on SRSS basis, the first mode accounts for 79% of the SRSS response, with 20% for the second mode and 0.6% for the third mode. These percentages are 91%, 8%, and 1% at the base. The effective modal weight factor, a_{m}, also shows the relative importance of each mode. In this example, with a_{1} = 0.804, a_{2} = 0.149, and a_{3} = 0.048, this indicate that 80.4% of the building mass participation is in the first mode, 14.9% in the second, and 4.8% in the third.
Given: See the sevenstory building illustration in Figure 3.115.
Weights and masses
Periods:
T_{1} = 0.880 s
T_{2} = 0.288 s
T_{3} = 0.164 s
Level 
PF_{xm} 
$\frac{{m}_{x}{\varphi}_{xm}}{{\displaystyle \sum {m}_{x}{\varphi}_{xm}}}$ 
F_{xm} (k) 
V_{xm} (k) 
ΔOTM_{xm} (ft · k) 
OTM_{xm} (ft · k) 
${a}_{xm}=\frac{{F}_{xm}}{{w}_{x}}$ 
δ_{xm} (in.) 
Δ_{xm} (in.) 

(a) Mode 1 

R 
1.346 
0.476 
63.2 
63.2 
772 
0 
0.337 
3.065 
1.182 
3 
0.829 
0.369 
48.9 
112.1 
1233 
772 
0.208 
1.892 
1.101 
2 
0.349 
0.155 
20.6 
132.7 
1416 
2005 
0.087 
0.791 
0.791 


1.000 



3421 



(b) Mode 2 

R 
−0.416 
−0.793 
−31.9 
−31.9 
−389 
0 
−0.171 
−0.212 
0.407 
3 
0.384 
0.923 
37.1 
5.2 
57 
−389 
−0.157 
0.195 
0.011 
2 
0.362 
0.870 
35.0 
40.2 
429 
−332 
−0.148 
0.184 
0.184 


1.000 



97 



(c) Mode 3 

R 
0.070 
0.420 
5.5 
5.5 
67 
0 
−0.029 
0.0094 
0.037 
3 
−0.212 
−1.599 
−20.8 
−15.3 
−168 
67 
−0.087 
−0.028 
0.066 
2 
0.289 
2.179 
28.3 
13.0 
139 
−101 
0.118 
0.038 
0.038 


1.000 



38 



(d) SRSS combination 

R 


71.0 
71.0 
867 
0 
0.379 
3.072 
1.251 
3 


64.8 
113.3 
1246 
867 
0.275 
1.893 
1.094 
2 


49.5 
139.3 
1486 
2035 
0.208 
0.812 
0.813 






3423 





Mode 1 
Mode 2 
Mode 3 


Level 
V_{SRSS} 
V_{1} 
V_{1}/V_{SRSS} 
(V_{1}/V_{SRSS})^{2} 
V_{2} 
(V_{2}/V_{SRSS})^{2} 
V_{1} 
(V_{3}/_{VSRSS})^{2} 
R 
71.0 
63.2 
0.89 
0.79 
−31.9 
0.202 
5.5 
0.006 
3 
119.3 
112.1 
0.989 
0.98 
5.2 
0.002 
−15.3 
0.018 
2 
139.3 
132.7 
0.953 
0.91 
40.2 
0.083 
13.0 
0.009 
Spectral accelerations: From the response spectrum of Figure 3.116a through c, the spectral accelerations are
S_{a1} = 0.276g
S_{a2} = 0.500g
S_{a3} = 0.500g
Observe that all three parts of Figure 3.116 contain the same information related to the acceleration response, S_{a}. Only the format is different. Figure 3.116a shows the building periods and spectral accelerations in a format similar to that in 1997 UBC and IBC03. Figure 3.116b is a tripartite response spectrum with additional values for displacements and velocities. Figure 3.116c shows the building periods and response accelerations in tabular format.
Figure 3.115 Sevenstory building example: dynamic analysis.
It should be noted that in the computer program used for the calculation of the eigenvalues, each mode is normalized for a value of ${\sum}_{g}^{w}{\Phi}_{2}=1.0$
. In some programs, Φ is normalized to 1.0 at the uppermost level.Required
Solution: From the modal analysis results shown in Table 3.6, the sum of the participation factors, PF_{xm}, and a_{m} adds up to 1.08 and 0.986, respectively. These values being close to 1.0 indicate that most of the modal participation is included in the three modes considered in the example. The story accelerations and the base shears are combined by the SRSS. The modal base shears are 2408, 632, and 200 kip for the first, second, and third modes, respectively. These are used in Figure 3.117 to determine story forces. The SRSS base shear is 2498 kip.
Story forces, accelerations, and displacements: Tables 3.6 through 3.9 are set up in a manner similar to the static design procedure described previously. In the static lateral procedure, Wh/ΣWh is used to distribute the force on the assumption of a straightline mode shape. In the dynamic analysis, the more representative WΦ/SWΦ distribution is used to distribute the forces. The story shears and overturning moments are determined in the same manner for each method. Modal story accelerations are determined by dividing the story force by the story weight. Modal story displacements are calculated from the accelerations and the period by using the following equations:
Figure 3.116 Response spectrum for threestory building example: (a) acceleration spectrum, (b) tripartite diagram, and (c) response spectra numerical representation.
Figure 3.117 Sevenstory building, modal analysis summary: (a) modal story forces, kip; (b) modal story shears, kip; (c) modal story overturning moments, kipft; (d) modal story accelerations, g; and (e) modal lateral displacement, inches.


Mode 1 
Mode 2 
Mode 3 
SRSS 


Level 
$\frac{w}{g}\left(\frac{k{s}^{2}}{\text{ft}}\right)$ 
ϕ_{1} 
$\frac{w}{g}{\varphi}_{1}$ 
$\frac{w}{g}{\varphi}_{1}^{2}$ 
a_{1}(g) 
ϕ_{2} 
$\frac{w}{g}{\varphi}_{2}$ 
$\frac{w}{g}{\varphi}_{2}^{2}$ 
a_{2}(g) 
ϕ_{3} 
$\frac{w}{g}{\varphi}_{3}$ 
$\frac{w}{g}{\varphi}_{3}^{2}$ 
a_{3}(g) 
a_{x}(g) 
Roof 
43.78 
0.0794 
3.48 
0.276 
0.362 
0.0747 
3.27 
0.744 
−0.235 
0.0684 
2.99 
0.205 
0.120 
0.448 
7 
45.34 
0.0745 
3.38 
0.252 
0.340 
0.0411 
1.86 
0.076 
−0.129 
−0.0040 
−0.18 
0.001 
−0.007 
0.364 
6 
45.34 
0.0666 
3.02 
0.201 
0.304 
−0.0042 
−0.19 
0.001 
0.013 
−0.0644 
−2.92 
0.188 
−0.113 
0.325 
5 
45.34 
0.0558 
2.53 
0.141 
0.254 
−0.0471 
−2.14 
0.101 
0.148 
−0.0630 
−2.86 
0.180 
−0.111 
0.314 
4 
45.34 
0.0425 
1.93 
0.082 
0.194 
−0.0718 
−3.26 
0.234 
0.226 
−0.0023 
−0.10 
0.000 
−0.004 
0.298 
3 
45.34 
0.0279 
1.27 
0.035 
0.127 
−0.0697 
−3.16 
0.220 
0.219 
0.0604 
2.74 
0.166 
0.106 
0.275 
2 
56.83 
0.0149 
0.85 
0.013 
0.068 
−0.0467 
−2.65 
0.124 
0.147 
0.0677 
3.85 
0.261 
0.119 
0.201 
1 
— 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 

Σ 
327.31 

16.46 
1.000 


−6.27 
1.000 


3.52 
1.001 


PF_{roof} 
Eq. (3.8) 
$\frac{16.46}{1000}\left(0.0794\right)=1.31$ 
$\frac{6.37}{1000}\left(0.0747\right)=0.47$ 
$\frac{3.52}{1.001}\left(0.0684\right)=0.24$ 
Σ = 1.08 

α 
Eq. (3.9) 
$\frac{{\left(16.46\right)}^{2}}{\left(327.31\right)\left(1.000\right)}=0.828$ 
$\frac{{\left(6.27\right)}^{2}}{\left(327.31\right)\left(1.000\right)}=0.120$ 
$\frac{{\left(3.52\right)}^{2}}{\left(927.31\right)\left(1.001\right)}=0.038$ 
Σ = 0.986 

T 

0.880 sec 
0.288 sec 
0.164 sec 


S_{a} 

0.276 g 
0.500 g 
0.500 g 


a_{roof} 
Eq. (3.10) 
(1.31)(0.276) = 0.362 g 
(−0.47)(0.500) = −0.235 g 
(0.24)(0.500) = 0.120 g 
0.448 

V 
Eq. (3.11) 
(0.828)(0.276)(10,539) = 2408 kips 
(0.12)(0.500)(10,539) = 632 kips 
(0.038)(0.500)(10,539) = 200 kips 
2498 kips (SRSS) 

V/W 

0.229 
0.060 
0.019 
0.237 

$W={\displaystyle \sum \left(\frac{w}{g}\right)\times g=327.31\times 32.2=10,539\text{kips}=\text{Buildingweight}\text{.}}$ A_{G} = 0.20 g Site PGA. β = 0.05 Damping factor. 
T_{1} = 0.880 s 


Modal Base Shear V_{1} = 2408 kips 

(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 


Story 
ϕ 
h ft 
Δh ft 
w kips 
$\frac{w\varphi}{{\displaystyle \sum w\varphi}}$ 
F kips (V_{1}) × (6) 
V kips ∑ (7) 
ΔOTM Kft (4)–(8) 
OTM Kft ∑ (9) 
Accel. g (7) ÷ (5) 
δ^{a} ft 
Δδ ft 
Roof 
0.0794 
65.7 
8.7 
1410 
0.211 
508 
508 
4420 
0 
0.360 
0.228 
0.014 
7 
0.7450 
57.0 
8.7 
1460 
0.205 
494 
1002 
8717 
4420 
0.338 
0.214 
0.022 
6 
0.0666 
48.3 
8.7 
1460 
0.184 
443 
1445 
12,572 
13,137 
0.303 
0.192 
0.031 
5 
0.0558 
59.6 
8.7 
1460 
0.154 
371 
1816 
15,799 
25,709 
0.254 
0.161 
0.039 
4 
0.0425 
30.9 
8.7 
1460 
0.117 
282 
2098 
10,253 
41,508 
0.193 
0.122 
0.042 
3 
0.0279 
22.2 
8.7 
1460 
0.077 
185 
2283 
19,862 
59,761 
0.127 
0.080 
0.057 
2 
0.0149 
13.5 
13.5 
1830 
0.052 
125 
2408 
32,508 
79,623 
0.068 
0.043 
0.043 
Grd. 
0 
0 

0 
0 
0 


112,131 
0 
0 





∑ 
1.000 
2408 

112,191 




T_{2} = 0.288 s 


Modal Base Shear V_{2} = 632 kips 

(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 


Story 
ϕ 
h ft 
Δh ft 
w kips 
$\frac{w\varphi}{{\displaystyle \sum w\varphi}}$ 
F kips (V_{2}) × (6) 
V kips ∑ (7) 
ΔOTM Kft (4) × (8) 
OTM Kft Z(9) 
Accel. g (7) ÷ (5) 
δ^{a} ft 
Δδ ft 
Roof 
0.0747 
65.7 
8.7 
1410 
0.522 
−330 
−330 
−2871 
0 
−0.234 
−0.016 
0.007 
7 
0.0411 
57.0 
8.7 
1460 
0.297 
−188 
−518 
−4507 
−2871 
−0.129 
−0.009 
0.010 
6 
−0.0042 
48.3 
8.7 
1460 
0.030 
19 
−499 
−4341 
−7378 
0.013 
0.001 
0.009 
5 
−0.0471 
39.6 
8.7 
1460 
0.341 
216 
−283 
−2462 
−11,719 
0.148 
0.010 
0.005 
4 
−0.0718 
30.9 
8.7 
1460 
0.520 
329 
46 
400 
−14,181 
0.225 
0.015 
0.000 
3 
−0.0697 
22.2 
8.7 
1460 
0.504 
319 
365 
3176 
−13,781 
0.219 
0.015 
0.005 
2 
−0.0467 
13.5 
13.5 
1830 
0.423 
267 
632 
8532 
−10,605 
0.146 
0.010 
0.010 
Grd. 
0 
0 






−2073 
0 
0 





∑ 
0.999 
632 

−2073 




T_{3} = 0.164 s 


Modal Base Shear V_{3} = 200 kips 

(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 


Story 
ϕ 
h ft 
Δh ft 
w kips 
$\frac{w\varphi}{{\displaystyle \sum w\varphi}}$ 
F kips (V_{3}) × (6) 
V kips ∑ (7) 
ΔOTM Kft (4) × (8) 
OTM Kft ∑ (9) 
Accel. g (7) ÷ (5) 
δ^{a} ft 
Δδ ft 
Roof 
0.0684 
65.7 
8.7 
1410 
0.849 
170 
170 
1479 
0 
0.121 
0.003 
0.003 
7 
−0.0040 
57.0 
8.7 
1460 
−0.051 
−10 
160 
1392 
1479 
−0.007 
0.000 
0.003 
6 
−0.0644 
48.3 
8.7 
1460 
−0.830 
−166 
−6 
−52 
2871 
−0.114 
−0.003 
0.000 
5 
−0.0630 
39.6 
8.7 
1460 
−0.813 
−163 
−169 
−1470 
2819 
−0.112 
−0.003 
0.003 
4 
−0.0023 
30.9 
8.7 
1460 
−0.028 
−6 
−175 
−1523 
1349 
−0.004 
0.000 
0.002 
3 
0.0604 
22.2 
8.7 
1460 
0.778 
156 
−19 
−165 
−174 
0.107 
0.002 
0.001 
2 
0.0677 
13.5 
13.5 
1830 
1.094 
219 
200 
2700 
−339 
0.120 
0.003 
0.003 
Grd. 
0 
0 






2361 
0 
0 





∑ 
0.999 
200 

2361 




Modal interstory drifts are calculated by taking the difference between the values of adjacent stories. The values shown in Tables 3.6 through 3.9 are summarized in Figure 3.117.
The fundamental period of vibration as determined from a computer analysis is 0.88 s. The periods of the second and third modes of vibration are 0.288 and 0.164 s, respectively. From Tables 3.7 through 3.9 using a response curve with 5% of critical damping (β = 0.05), it is determined that the second and thirdmode spectral accelerations are 0.276g. On the basis of mode shapes and modal participation factors, modal story forces, shears, overturning moments, accelerations, and displacements are determined.
Figure 3.117 shows story forces obtained by multiplying the story acceleration by the story mass. The shapes of story force curves are quite similar to the shapes of the acceleration curves because the building mass is essentially uniform.
Figure 3.117b shows story shears that are a summation of the modal story forces in Figure 3.117. The higher modes become less significant in relation to the first mode because the forces tend to cancel each other due to the reversal of direction. The SRSS values do not differ substantially from the firstmode values.
Figure 3.117c shows the building overturning moments. Again, the higher modes become somewhat less significant because of the reversal of force direction. The SRSS curve is essentially equal to the firstmode curve.
Figure 3.117d shows story accelerations. Observe that the second and third modes do play a significant role in the structure's maximum response. While the shape of an individual mode is the same for displacements and accelerations, accelerations are proportional to displacements divided by the squared value of the modal period, which accounts for the greater accelerations in the higher modes. The shape of the SRSS combination of the accelerations is substantially different from the shapes of any of the individual modes because it accounts for the predominance of the various modes at different story levels.
Figure 3.117e shows the modal displacements. Observe that the fundamental mode predominates, while the second and thirdmode displacements are relatively insignificant. The SRSS combination does not differ greatly from the fundamental mode. It should be noted, however, that for taller and irregular buildings, the influence of the higher modes will become larger.