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13.2 Earthquake-Induced Vibration of Structures

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13.2.1 Seismicity and Ground Motions

The most common cause of earthquakes is thought to be the violent slipping of rock masses along

major geological fault lines in the Earth’s crust, or lithosphere. These fault lines divide the global crust

into about 12 tectonic plates, which are rigid, relatively cool slabs about 100 km thick. Tectonic plates

float on the molten mantle of the Earth and move relative to one another at the rate of 10 to

100 mm/year.

The basic mechanism causing earthquakes in the plate boundary regions appears to be that the

continuing deformation of the crustal structure eventually leads to stresses/strains which exceed the

material strength. A rupture will then initiate at some critical point along the fault line and will propagate

rapidly through the highly stressed material at the plate boundary. In some cases, the plate margins are

moving away from one another. In those cases, molten rock appears from deep in the Earth to fill the gap,

often manifesting itself as volcanoes. If the plates are pushing together, one plate tends to dive under the

other and, depending on the density of the material, it may resurface in the form of volcanoes. In both

these scenarios, there may be volcanoes and earthquakes at the plate boundaries, both being caused by the

same mechanism of movement in the Earth’s crust. Another possibility is that the plate boundaries will

slide sideways past each other, essentially retaining the local surface area of the plate. It is believed that

approximately three quarters of the world’s earthquakes are accounted for by this rubbing – sticking –

slipping mechanism, with ruptures occurring on faults on boundaries between tectonic plates.

Vibration and Shock Problems of Civil Engineering Structures 13-3

Earthquake occurrence maps tend to outline the plate boundaries. Such earthquakes are referred to as

interplate earthquakes.

Earthquakes do occur at locations away from the plate boundaries. Such events are known as intraplate

earthquakes and they are much less frequent than interplate earthquakes. They are also much

less predictable than events at the plate margins and they have been observed to be far more severe.

For example, the Eastern United States, which is located well away from the tectonic plate boundaries of

California, has recorded the largest earthquakes in the history of European settlement in the country.

These major intraplate earthquakes occurred in the middle of last century in South Carolina on the East

Coast and Missouri in the interior. However, because of the low population density at the time, the

damage caused was minimal. It is significant to note, however, that these intraplate earthquakes,

although very infrequent, were considerably larger than the moderately sized interplate earthquakes that

frequently occur along the plate boundaries in California. (It is thought that, because tectonic plates are

not homogeneous or isotropic, areas of local high stress are developed as the plate attempts to move as a

rigid body. Accordingly, rupture within the plate, and the consequent release of energy, are believed to

give rise to these intraplate events.)

The point in the Earth’s crustal system where an earthquake is initiated (the point of rupture) is called

the hypocenter or focus of the earthquake. The point on the Earth’s surface directly above the focus is

called the epicenter and the depth of the focus is the focal depth. Earthquake-occurrence maps usually

indicate the location of various epicenters of past earthquakes and these epicenters are located by

seismological analysis of the effect of earthquake waves on strategically located receiving instruments

called seismometers.

When an earthquake occurs, several types of seismic wave are radiated from the rupture. The most

important of these are the body waves (primary (P) and secondary (S) waves). P waves are essentially

sound waves traveling through the Earth, causing particles to move in the direction of wave propagation

with alternate expansions and compressions. They tend to travel through the Earth with velocities of up

to 8000 m/sec (up to 30 times faster than sound waves through air). S waves are shear waves with

particle motion transverse to the direction of propagation. S waves tend to travel at about 60% of the

velocity of P waves, so they always arrive at seismometers after the P waves. The time lag between arrivals

often provides seismologists with useful information about the distance of the epicenter from

the recorder.

The total strain energy released during an earthquake is known as the magnitude of the earthquake

and it is measured on the Richter scale. It is defined quite simply as the amplitude of the recorded

vibrations on a particular kind of seismometer located at a particular distance from the epicenter.

The magnitude of an earthquake by itself, which reflects the size of an earthquake at its source, is not

sufficient to indicate whether structural damage can be expected at a particular site. The distance of

the structure from the source has an equally important effect on the response of a structure, as do

the local ground conditions. The local intensity of a particular earthquake is measured on the

subjective Modified Mercalli scale (Table 13.1) which ranges from 1 (barely felt) to 12 (total

destruction). The Modified Mercalli scale is essentially a means by which damage may be assessed

after an earthquake. In a given location, where there has been some experience of the damaging

effects of earthquakes, albeit only subjective and qualitative, regions of varying seismic risk may be

identified. The Modified Mercalli scale is sometimes used to assist in the delineation of these regions.

A particular earthquake will be associated with a range of local intensities, which generally diminish

with distance from the source, although anomalies due to local soil and geological conditions are

quite common.

Modem seismometers (or seismographs) are sophisticated instruments utilizing, in part,

electromagnetic principles. These instruments can provide digitized or graphical records of earthquake-

induced accelerations in both the horizontal and vertical directions at a particular site.

Accelerometers provide records of earthquake accelerations and the records may be appropriately

integrated to provide velocity records and displacement records. Peak accelerations, velocities, and

displacements are all in turn significant for structures of differing stiffness (Figure 13.2).

13-4 Vibration and Shock Handbook

13.2.2 Influence of Local Site Conditions

Local geological and soil conditions may have a significant influence on the amplitude and frequency

content of ground motions. These conditions affect the earthquake motions experienced (and hence the

structural response) in one, or more, of the following ways:

* Interaction between the bedrock earthquake motion and the soil column will modify the actual

ground accelerations input to the structure. This manifests itself by an increase in the amplitude of

the ground motion over and above that at the bedrock, and a filtering of the motion so that the

range of frequencies present becomes narrow with the high-frequency components being

eliminated. This condition particularly arises in areas where soft sediments and alluvial soil overly

bedrock. The degree of amplification is dependent on the strength of shaking at the bedrock.

Because of nonlinear effects in the soil, the amplification ratio is less in strong shaking than under

base motions of lower amplitude.

* The soil properties in the proximity of the structure contribute significantly to the effective

stiffness of the structural foundation. This may be a significant parameter in determining the

overall structural response, especially for structures that would be characterized as stiff under

other environmental loadings.

* The strength (and response) of the local soil under earthquake shaking may be critical to the

overall stability of the structure.

It is also important that information on relevant geological features, such as faulting, be assessed.

Geological information on suspected active faults near the site can assist in providing a basis for

TABLE 13.1 Modified Mercalli Intensity Scale

I. Not felt except by a very few under especially favorable circumstances

II. Felt only by a few persons at rest, especially on upper floors of buildings. Delicately suspended objects may swing

III. Felt quite noticeably indoors, especially on upper floors of buildings, but many people do not recognize it as an

earthquake. Standing motor cars may rock slightly. Vibration like passing truck. Duration estimated

IV. During the day felt indoors by many, outdoors by few. At night some awakened. Dishes, windows, and doors

disturbed; walls make creaking sound. Sensation like heavy truck striking building. Standing motorcars rock

noticeably

V. Felt by nearly everyone; many awakened. Some dishes, windows, etc., broken; a few instances of cracked plaster;

unstable objects overturned. Disturbance of trees, poles, and other tall objects sometimes noticed. Pendulum

clocks may stop

VI. Felt by all; many frightened and run outdoors. Some heavy furniture moved; a few instances of fallen plaster or

damaged chimneys. Damage slight

VII. Everybody runs outdoors. Damage negligible in buildings of good design and construction, slight to moderate

in well-built ordinary structures; considerable in poorly built or badly designed structures. Some chimneys

broken. Noticed by persons driving motor cars

VIII. Damage slight in specially designed structures; considerable in ordinary substantial buildings, with partial

collapse; great in poorly built structures. Panel walls thrown out of frame structures. Fall of chimneys, factory

stacks, columns, monuments, walls. Heavy furniture overturned. Sand and mud ejected in small amounts.

Changes in well water. Persons driving motorcars disturbed

IX. Damage considerable in specially designed structures; well-designed frame structures thrown out of plumb;

great in substantial buildings, with partial collapse. Buildings shifted off foundations. Ground cracked

conspicuously. Underground pipes broken

X. Some well-built wooden structures destroyed; most masonry and frame structures destroyed with foundations;

ground badly cracked. Rails bent. Landslides considerable from riverbanks and steep slopes. Shifted sand and

mud. Water splashed over banks

XI. Few, if any (masonry), structures remain standing. Bridges destroyed. Broad fissures in ground. Underground

pipelines completely out of service. Earth slumps and land slips in soft ground. Rails bent greatly

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

the air

Source: Data from Wood, H.O. and Neumann, Fr., Bull. Seis. Soc. Am., 21, 277 – 283, 1931.

Vibration and Shock Problems of Civil Engineering Structures 13-5

evaluating the intensity of a likely earthquake. It is usual to use this information, together with the

regional seismicity data, to determine the likely level of seismic activity.

13.2.3 Response of Structures to Ground Motions

The effect of ground motion on the various categories of structures is dictated almost entirely by the

distribution of mass and stiffness in the structure. It is important to appreciate that, in an earthquake,

loads are not applied to the structure. Rather, earthquake loading arises because of accelerations

generated by the foundation level(s) of the structure intercepting and being influenced by transient

ground motions. Specifically, the product of the structural mass and the total acceleration produces the

inertia loading experienced by the structure. This is an expression of Newton’s Second Law. It is

important to appreciate that the total acceleration is the absolute acceleration of the structure, namely,

the sum of the ground acceleration and that of the structure relative to the ground.

If the structure is stiff there is little, if any, additional acceleration relative to the ground motion and,

therefore, the earthquake loading experienced is essentially proportional to the building mass, that is,

Feq / M:

For structures that are flexible, for example, those in the high-rise or long-span category, the absolute

acceleration is low. This occurs because the ground acceleration and the acceleration of the building

relative to the ground tend to oppose one another. In this case, the earthquake loading is approximately

proportional to the square root of the mass, that is Feq / M0:5:

For structures in the cantilever category, which are essentially vertical, it is the horizontal accelerations

that are significant; whereas for structures that are largely horizontal in extent, the effect of the vertical

accelerations is dominant. Moreover, if the plan distributions of mass and stiffness are dissimilar in

vertical structures, significant twisting motions may arise.

−4

−3

−2

−1

0 10 15 20 25

−40

−30

−20

−10

0 10 15 20 25 30

−25

−20

−15

−10

−5

0 10 15 20 25 30

(A) Ground acceleration

(B) Ground velocity

Time (sec)

cm cm/sec a/g

FIGURE 13.2 El-Centro earthquake, north – south component. (A) Record of the ground acceleration; (B) ground

velocity, obtained by integration of (A); (C) ground displacement, obtained by integration of (B).

13-6 Vibration and Shock Handbook

The peak ground acceleration is of importance in the response of stiff structures and peak ground

displacements are of importance in the response of flexible structures, with peak ground velocity being of

importance for structures of intermediate stiffness. Stiff structures tend to move in unison with the

ground while flexible structures, such as high-rise buildings, experience the ground moving beneath

them, their upper floors tending to remain motionless.

13.2.4 Dynamic Analysis

13.2.4.1 Equations of Motion for Linear Single-Degree-of-Freedom Systems

Consider the linear single-degree-of-freedom

(single-DoF) system shown in Figure 13.3 subjected

to a time varying ground displacement, zðtÞ:

Let the relative displacement of the system to the

ground be, yðtÞ; y is then the extension of the

spring and dashpot. From the equation of motion,

it follows that

mðy€ þ z€Þ ¼ 2ky 2 cy_ ð13:1Þ

Rearranging Equation 13.1, and replacing m; k;

and c by the system’s radial frequency v and

damping ratio j; gives

y€ þ 2jvy_ þv2y ¼ 2z€ ð13:2Þ

Given a description of the input motion, zðtÞ; (for

example, from an accelerograph recording), the

solution of Equation 13.2 provides a complete time history of the response of a structure with a given

natural period and damping ratio, and can also be used to derive maximum responses for constructing a

response spectrum (Figure 13.6). Owing to the random nature of earthquake ground motion, numerical

solution techniques are needed for Equation 13.2, as described by Clough and Penzien (1993).

13.2.4.2 Equations of Motion for Linear Multiple-Degree-of-Freedom Systems

The dynamic response of many linear multipledegree-

of-freedom (multi-DoF) systems can be split

into decoupled natural modes of vibration

(Figure 13.4), each mode effectively representing a

single-DoF system. A modified form of Equation

13.2 then applies to each mode, which for mode i

becomes

Y€i þ 2jivi

Y_i þ ____________v2i

Yi ¼

z€ ð13:3Þ

Here, Yi is the generalized modal response in the ith

mode. ðLi=MiÞ is a participation factor, which

depends on the mode shape and mass distribution,

and describes the participation of the mode in

overall response to a particular direction of ground

motion. For a two-dimensional (2D) structure with

n lumped masses, responding in one horizontal direction

Mi ¼

j¼1

fijmj

j¼1

ijmj

ð13:4Þ

Stiffness, k Viscous

damper, c

Mass, m

Displacement

y (t)

Force

f (t)

FIGURE 13.3 Single-DoF system.

FIGURE 13.4 Typical modes for multistory buildings.

Vibration and Shock Problems of Civil Engineering Structures 13-7

In Equation 13.4, fij; describes the modal displacement of the jth mass in the ith mode. The higher modes

often have very low values of ðLi=MiÞ; and their contribution can then be omitted. In this way, the

computational effort is greatly reduced. In cases where only the first mode in each direction is significant

(often the case for low- to medium-rise building structures), equivalent static analysis may be sufficient,

as described later.

13.2.5 Earthquake Response Spectra

13.2.5.1 Elastic Response Spectra

For design purposes, it is generally sufficient to know only the maximum value of the response due to an

earthquake. A plot of the maximum value of a response quantity as a function of the natural vibration

frequency of the structure, or as a function of a quantity which is related to the frequency such as natural

period, constitutes the response spectrum for that quantity (see Chapter 17 and Chapter 31).

The peak relative displacement is usually called Sd and the peak strain energy of the oscillator is

SE ¼

KS2d

SE ¼

S2d

SE ¼

MS2

Hence, the pseudo-relative velocity and acceleration spectra are defined as

Svz ¼ vSd ð13:5Þ

Saz ¼ v2Sd ð13:6Þ

Figure 13.5 shows that a record of peak relative displacement response of an single-DoF oscillator can be

plotted for a given earthquake, given damping, and a range of periods, typical of structures.

The structure’s natural period (T or 1=n) is conventionally taken as the abscissa, and curves are drawn

for various levels of damping (Figure 13.6). It should be noted that the response spectrum gives no

information about the duration of response (and hence the number of damaging cycles) that the

structure experiences, which can have a very significant influence on the damage sustained.

13.2.5.2 Smoothed Design Spectra

Owing to the highly random nature of earthquake ground motions, the response spectrum for a real

earthquake record contains many sharp peaks and troughs, especially for low levels of damping. The

peaks and troughs are determined by a number of uncertain factors, such as the precise location of the

earthquake source, which are unlikely to be known precisely in advance. Therefore, spectra for design

purposes are usually smoothed envelopes of spectra for a range of different earthquakes; indeed, one of

the advantages of response spectrum analysis over time history analysis is that it can represent the

envelope response to a number of different possible earthquake sources from a single analysis, and is not

dependent on the precise characteristic of one particular ground motion record. Codes of practice such

as UBC (2000) and Eurocode 8 (ENV 1998, 1994-8) provide smoothed spectra for design purposes.

13.2.5.3 Ductility-Modified Response Spectrum Analysis

In a ductile structure, or subassemblage, the resistance, R; may be sustained at displacements that are

several times those at first yield, Dy ; as represented in Figure 13.7.

For yielding single-DoF systems, ductility-modified acceleration response spectra can be drawn,

representing the maximum acceleration response of a system as a function of its initial (elastic) period,

T; damping ratio, j; and displacement ductility ratio, m (m is the ratio of maximum displacement,

13-8 Vibration and Shock Handbook

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Period (sec)

Sd (m)

−0.5

−0.3

−0.1

0.1

0.3

0.5

0 10 20 30 40 50

Natural period (sec)

Deformation (m)

−0.5

−0.3

−0.1

0.1

0.3

0.5

0 10 20 30 40 50

Natural period (sec)

Deformation (m)

−0.5

−0.3

−0.1

0.1

0.3

0.5

0 10 20 30 40 50

Natural period (sec)

Deformation (m)

T = 0.5s

x = 2%

T = 1s

x = 2%

T = 2s

x = 2%

El Centro ground acceleration

−4

−3

−2

−1

0 5 10 15 20 25 30

time (sec)

a /g

Max deformation = 0.043

Max deformation = 0.074

Max deformation = 0.281

1 2 3

FIGURE 13.5 Compilation of (relative) displacement response spectra.

Vibration and Shock Problems of Civil Engineering Structures 13-9

Dmax; to yield displacement, Dy ). The reduction in acceleration response of the yielding system compared

with the elastic one is period dependent; for structural periods greater than the predominant earthquake

periods, the reduction is approximately 1=m; for very stiff systems there is no reduction, while at

intermediate periods a reduction factor between 1=m and 1 applies.

To derive peak accelerations and internal forces, the system can be treated as linear elastic and the

ductility-modified spectrum used exactly like a normal elastic spectrum. However, deflections derived

from this treatment must be multiplied by m to allow for the plastic deformation.

It is now standard practice to analyze multi-DoF systems in the same way. That is, a yielding multi-

DoF system is treated as elastic, and an appropriate ductility-modified spectrum is substituted for an

elastic one. Acceleration and force responses are derived directly and deflections are multiplied by m:

However, this procedure is not (contrary to the case for single-DoF systems) rigorously correct. Although

it gives satisfactory answers for regular structures, it can be seriously in error for structures (such as those

with weak stories) where the plasticity demand is not evenly distributed. Nevertheless, most codes of

practice allow the use of ductility-modified spectra for design, and give appropriate values for the

reduction factors (called q; or behavior factors in Eurocode 8 and R factors in UBC) to apply to elastic

response spectra.

13.2.6 Design Philosophy and the Code Approach

In areas of the world recognized as being prone to major earthquakes, the engineer is faced with the

dilemma of being required to design for an event, the magnitude of which has only a small chance of

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Undamped natural period of structure, s

10%

20%

Spectral acceleration

Peak ground acceleration

Peak spectral acceleration

Peak ground acceleration

Viscous damper, c

Stiffness, k

Mass, m

Damping ratio ξ =

Undamped natural period T =2π m

FIGURE 13.6 Acceleration response spectrum for El-Centro 1940 earthquake.

Δy Δmax Deflection

Resistance Yield

Peak deflection

Unloading

FIGURE 13.7 A simple bilinear elasto-plastic curve of response, representative of ductile performance.

13-10 Vibration and Shock Handbook

occurring during the life of the facility. If the designer adopts conservative performance criteria for the

facility, the client (often society) is faced with costs which may be out of proportion to the risks involved.

On the other hand, to ignore the possibility of a major earthquake could be construed as negligent in

these circumstances.

To overcome this problem, a dual design philosophy has been developed, by which procedure:

1. A moderate earthquake, such as may reasonably be expected at the site, is used as a basis for the

seismic design. The facility should be proportioned to resist such an earthquake without

significant damage. This “damageability” limit state should ensure safety, limited nonstructural

damage, and the continued performance of facilities and services, particularly in those with

important postearthquake functions. The list includes hospitals, police, fire and civil defense

facilities, water supply, telecommunications, electricity generation and distribution systems,

and so on. Almost as important is the maintenance of road and rail communications,

particularly for food distribution (including warehouses and their contents). Similarly, the

protection of industrial complexes, in their own right, as well as the protection of individual

items of equipment in other buildings and facilities, is a necessary consequence of adoption of

this limit state.

2. The most severe, credible earthquake that may be expected to occur at the site is used to test

safety. In this ultimate limit state, significant structural and nonstructural damage is expected

but neither collapse nor loss of life should occur.

The main strategy for preventing collapse has traditionally been provision of ductility. This is the

opposite quality to brittleness, and may be defined as the ability to sustain repeated excursions

beyond the elastic limit without fracture. Owing to the cyclic, imposed displacement nature of

earthquake loading, a ductile structure can absorb very large amounts of energy without collapse;

the designer must think in terms of designing for maximum imposed displacements, rather than

imposed loads.

Achieving ductility is partly a matter of choosing the right structural system, and partly a matter of

detailing. In the former category comes the important concept of “capacity design,” as described by

Paulay (1993). This involves ensuring a hierarchy of strengths within a structure to ensure that yielding

occurs in ductile modes (such as flexure) rather than brittle modes (such as buckling or, for reinforced

concrete, shear). There are other aspects of structural form which are important, particularly regularity in

elevation (to avoid “soft” or weak stories) and regularity in plan (to minimize torsional response).

These aspects are described in many textbooks and are quantified in some codes of practice (Park and

Paulay, 1975).

Detailing of the structure is also important to ensure ductility. For concrete structures, this primarily

involves reinforcement detailing and in steel structures connection detailing. The latter aspect has been

particularly recognized following the failure of H-welded connections in the Northridge earthquake of

1994 (Burdekin, 1996). The primary reliance is on empirical solutions to these problems, as described in

codes of practice, such as Eurocode 8 (ENV 1998, 1994-8) Part 1.3 and UBC/IBC (2000). Textbooks

discussing these issues for concrete include Paulay and Priestley (1992), Booth (1994), Penelis and

Kappos (1996), FEMA 273/274, FEMA 356/357 “NEHRP guidelines for seismic rehabilitation of

buildings,” FEMA 368/369 “NEHRP recommended provisions for seismic regulations for new buildings

and other structures,” and FEMA 306/307/308 “Evaluation and repair of earthquake damaged concrete

and masonry buildings.” Textbooks covering failures of steel structures in recent earthquakes include

Burdekin (1996) and FEMA 350-354 (2000).

Another important aspect of detailing is to allow for the maximum inelastic deflections caused by the

design earthquake. Nonseismic-resisting elements of a structure such as cladding and infill walls must be

able to accommodate these deflections safely, as must (crucially) the gravity load-bearing structure,

which still suffers the seismic displacements even when not contributing to seismic resistance. In

addition, adequate separation between adjacent structures must be provided. Codes of practice (e.g.,

Eurocode 8 and UBC) give guidance on suitable limits.

Vibration and Shock Problems of Civil Engineering Structures 13-11

13.2.6.1 Performance-Based Design

In recent years, seismic design codes throughout the world have been shifting toward the adoption of

performance-based design philosophy. The goal of a performance-based design procedure is to produce

structures that have predictable seismic performance under multiple levels of earthquake intensity. In

order to do so, it is important that the behavior of the structures is targeted in advance, both in the elastic

as well as the inelastic ranges of deformation. The four important parameters in seismic design, strength,

stiffness, ductility, and deformation, become the primary elements of a performance-based design

procedure and have to be designed rationally. The next generation of codes is expected to be based on

performance-based principles such as Asian Model Concrete Code (ACMC, 2001).

13.2.7 Analysis Options for Earthquake Effects

Analysis is only one part of the design process; conceptual design, detailing, and proper construction are

the other vital components for ensuring good seismic performance. This section provides a brief

theoretical review of the basis for seismic analysis, describing the main analytical techniques currently

used by designers. More details are given by Clough and Penzien (1993), a standard general text for

dynamic analysis, and Chopra (2001), which deals specifically with concerns for earthquake engineers.

Essentially, an earthquake engineer is faced with four possible methods of analysis/design for

earthquake loading on a structure. In all methods, the dual damage criteria discussed above may be

applied. The four analysis/design options available are:

* Dynamic time history analysis

* Response spectrum analysis

* Equivalent static approach (or force-based approach)

* Displacement-based approach

Equivalent static methods are usually adequate for conventional, regular building structures under

about 75 m in height. A response spectrum analysis is required for taller buildings, because higher mode

effects may become important, and also for buildings with plan or elevational eccentricities because

torsional effects or nonstandard mode shapes may be significant. Codes of practice such as Eurocode 8

and UBC (2000) specify the degree of eccentricity at which such analysis is required. Unusual or very

important structures may require nonlinear time history analysis, and this may also be required where

the inaccuracies implicit in the use of ductility-modified response spectrum analysis become

unacceptable. Displacement-based approach is a new method for seismic design, which is gaining

popularity. The above four analysis options are discussed next.

13.2.7.1 Dynamic Time History Analysis

The most rigorous form of dynamic analysis involves stepping a nonlinear model of the structure

through a complete time history of earthquake ground motions. The advantage of the method is that it

can give direct information on nonlinear response, the duration of response (and hence the number of

loading cycles), and the relative phasing of response between various parts. The method involves

subjecting an appropriate finite element computer model of the building, or structural system, to a given,

previously recorded, earthquake record and examining its response in real time. Response peaks are

generally of most interest. The analysis must be performed for a number of different earthquake time

histories to reduce dependence on the random characteristics of a particular record.

There are certain special circumstances where this procedure is useful but, for general seismic design, it

is of little value as the actual earthquake that the structure may have to resist cannot be guaranteed to

have sufficiently similar characteristics to the design earthquake. In particular, the intensity, duration,

and frequency content of the earthquake may be unsuitable especially if, as often happens, the record

comes from another country or continent. Moreover, the method is expensive and time-consuming, so

that only for special structures can its use be justified. If, in addition, inelastic response calculations are

involved, another level of complexity (and uncertainty) is introduced. Response then becomes

13-12 Vibration and Shock Handbook

dependent, often heavily so, on the nonlinear models chosen and this is in addition to that inherent in

choosing to use one particular record.

13.2.7.2 Response Spectrum Analysis

13.2.7.2.1 Response Spectrum Analysis of Single-Degree-of-Freedom Systems

With a knowledge of the natural period and damping of an single-DoF system, its peak (i.e., spectral)

acceleration, Sa; can be determined directly from an appropriate response spectrum (see Chapter 17).

In undamped systems, this peak response occurs when the equivalent spring is at its maximum extension

point, so that the maximum force in the spring is given by

F ¼ mSa ð13:7Þ

From Equation 13.7, the peak (i.e., spectral) displacement, Sx ; of the spring is given by

Sx ¼

k ¼ mSa

4p2m ¼

SaT2

4p2 ð13:8Þ

For structures with relatively small viscous damping, the same relationships are still approximately true,

because the maximum acceleration occurs when the velocity is low and hence the damping force (which

is velocity proportional) is also low. Therefore, Equation 13.7 and hence also Equation 13.8 are still very

good approximations for lightly damped systems.

Thus the two most important parameters of structural response — maximum force and

displacement — can be determined for a linear single-DoF system directly from the acceleration

response spectrum, provided only that the mass, natural period, and damping are known.

It is important to realize that the spectral acceleration, Sa; is an absolute value (the true acceleration of

the structure in space) whereas the spectral displacement, Sx ; is a relative value, measured in relation

to the ground, which itself is moving in the earthquake. This at first sight may seem confusing, until it

is remembered that the absolute acceleration of the mass is determined by the force on it (Equation 13.7),

which itself is determined by the relative compression of the spring with respect to the ground

(Equation 13.8).

13.2.7.2.2 Response Spectrum Analysis of Multi-Degree-of-Freedom Systems

By considering the response of each mode separately, a response spectrum analysis is also possible for an

multi-DoF system, if generalized modal quantities are used (compare Equation 13.2 and Equation 13.3).

For example, for a 2D structure with n lumped masses, responding in one horizontal direction, Equation

13.4 is modified to give the maximum base shear in the ith mode as

Fi ¼

j¼1

fijmj

j¼1

ijmj

Sa ¼ meff ;iSa ð13:9Þ

where Sa is the spectral acceleration corresponding to the damping and frequency of mode i:

Higher modes with low effective masses, meff ;i; may contribute little to response and can usually be

neglected. Since the sum of effective masses, meff ;i; of all modes equals the total mass, a good test of

whether the first r modes are sufficient to capture response adequately is

i¼1

meff ;i $ 0:9

i¼1

mi ¼ 0:9 ðtotal massÞ ð13:10Þ

A response spectrum analysis gives the maximum response of the structure for each mode of vibration

considered. Although it is rigorously correct to add the response in each mode at any time to obtain

the total response, the maximum responses in each mode, calculated from response spectrum analysis, do

Vibration and Shock Problems of Civil Engineering Structures 13-13

not occur simultaneously, and hence simple addition produces an overestimate of response. A common

and usually adequate approximation is the square root of the sum of the squares (SRSS) rule, where

the maximum total response is estimated as the SRSS combination of the individual modal responses.

However, this may not be conservative enough for closely spaced or high-frequency modes, and

other methods, such as the complete quadratic combination (CQC) method, are available (Gupta, 1990).

There are many commercially available computer programs which can perform response spectrum

analysis, and it is now regarded as a standard rather than a specialist technique.

13.2.7.3 Equivalent Static Analysis (Force-Based Approach)

This is the type of analysis presented in most contemporary codes of practice, and it is conditional for its

accuracy upon response being dominated by one mode of vibration in each direction. In the case of

buildings, a quantity usually referred to as the “total base shear” is calculated from the product of the

weight of the building and a coefficient. This coefficient takes into account the location and importance

of the structure, its ductility or energy absorption capacity, its dynamic characteristics, and the local soil

conditions and their effect on structural responses. Once the total base shear has been calculated, it is

distributed up the structure as a series of horizontal loads at each floor level and the structure is analyzed

with these equivalent horizontal loads applied.

The maximum lateral base shear is first calculated. Equation 13.11 gives the relevant formulae in UBC

(2000). Other current codes follow similar formats

V ¼

CvIW

but V #

2:5CaI

W and $ 0:11CaIW ð13:11Þ

In addition, V $ ð0:8ZNv I=RÞW (high seismicity, Zone 4 only), where:

V ¼ ultimate seismic base shear (force units, e.g., kN)

Cv ; Ca ¼ seismic coefficients, depending on the zone factor Z as given in UBC

I ¼ importance factor ¼ 1 to 1.25 in UBC

R ¼ reduction coefficient depending on the ductility of structure ¼ 2.8 to 8.5 in UBC

T ¼ first mode period of the building (sec)

W ¼ building weight (force units, e.g., kN)

Z ¼ zone factor expressed as the peak ground acceleration on rock (in gravity units) for a 475-year return

period ¼ 0.075 to 0.4 in UBC

Nv ¼ factor allowing for proximity to active faults ¼ 1.0 to 2.0 in UBC

ðV =W Þ represents the shape of a standard design response spectrum with a peak amplification on ground

acceleration for 5% damping of 2.5, and a minimum value at long period to allow for the uncertainty in

long-period motions and for proximity to active faults.

The base shear calculated by these methods is then applied to the structure as a set of horizontal forces,

with a vertical distribution based on the first mode shape of regular vertical cantilever structures.

Horizontal distribution follows the mass distribution, with some additional allowance for torsional

effects.

13.2.7.4 Displacement-Based Approach

In the development of performance-based earthquake engineering, which stresses the inelastic behavior

of structural system under severe earthquake ground motions (high seismic region), displacement rather

than force has been recognized as the most suitable and direct performance or damage indicator.

Deformation-controlled design can be achieved either by using the traditional force/strength-based

design procedure together with a check on the displacement/drift limit, or by employing a direct

displacement-based procedure. The idea of displacement-based design was introduced by Gulkan

and Sozen (1974). They developed the concept of substitute structure to estimate the nonlinear

structural response through an equivalent elastic model, assuming a linear behavior and a viscous

13-14 Vibration and Shock Handbook

damping equivalent to the nonlinear response. This idea has been adopted recently by Priestley and

Kowalsky (2000) for a direct displacement design of single-DoF and multi-DoF reinforced concrete

structures. Another direct displacement-based design approach was proposed by Fajfar (2000) based on

the capacity spectrum method (Chopra and Goel, 1999).

In all the above references, seismic demand is specified as either a displacement response spectrum

(D-T format) or an acceleration displacement response spectrum (ADRS format). For a general-purpose

spectrum, nonlinear elastic behavior of a structural system can be accounted for by either an equivalent

elastic response spectrum or an inelastic response spectrum. The former is associated with effective

viscous damping jeff and the latter is directly constructed based on the relation between reduction factors

and ductility. Although the elastic acceleration design spectrum is available from codes, it is not

appropriate to be a basis for the determination of the elastic displacement design spectrum because the

displacement increases with period even at longer periods.

13.2.8 Soil – Structure Interaction

Structural analyses usually assume that ground motions are applied via a rigid base, thus neglecting the

effect of ground compliance on response. Although this rigid base assumption may lead to an

underestimate of deflections, it is usually conservative as far as forces are concerned, because ground

compliance reduces stiffness and usually moves structural periods farther from resonance with the

ground motion. However, this conservatism may not always apply, and Eurocode 8 Part 5 lists the

following cases where soil – structure interaction (SSI) should be investigated:

1. Structures where P-Delta (second order) effects need to be considered

2. Structures with massive or deep-seated foundations, such as bridge piers, offshore caissons,

and silos

3. Slender, tall structures such as towers and chimneys

4. Structures supported on very soft soils with an average shear wave velocity less than 100 m/sec

Allowance for SSI effects is usually a specialist task. The simplest method is to present the soil flexibility

by discrete springs connected to the foundation. These require a knowledge of the shear stiffness of the

soil. Further information is given by Pappin (1991) and Wolf (1985, 1994).

13.2.9 Active and Passive Control Systems

Alternative strategies of designing for earthquake resistance involve modification of the dynamic

characteristics of structure to improve seismic response. The systems can be classified as either passive

or active. The basic role of these systems is to absorb a portion of the input energy, thereby reducing

energy dissipation demand on primary structural members and minimizing possible structural

damage.

The most common type of passive system involves lengthening the structure’s fundamental period of

vibration by mounting the superstructure on bearings with a low horizontal stiffness; this is known as

base or seismic isolation. Where this increases, the fundamental period above the predominant periods of

earthquake excitation, the acceleration (but not necessarily displacement) response is significantly

reduced. Usually, additional damping is provided in the seismic isolation bearing to control deflections.

The principle of seismic isolation is illustrated by Figure 13.8. The reduction in response, often of the

order of 50, has proved highly effective in recent earthquakes in reducing damage to both building

structure and building contents. UBC (2000) provides codified guidance for seismic isolation of

buildings while AASHTO (1991) and Eurocode 8 (ENV 1998, 1994-8) Part 2 treat bridge structures.

Seismic isolation has been incorporated in many hundreds of recent structures, particularly in bridges,

and also in buildings such as hospitals with contents that must remain functional after an earthquake.

It has also been used to improve the seismic resistance of existing structures. Another form of passive

Vibration and Shock Problems of Civil Engineering Structures 13-15

system is the provision of additional structural damping in the form of discrete viscous, frictional, or

hysteretic dampers.

Active systems modify the dynamic characteristics of a structure in real time during an earthquake, by

computer-controlled devices such as active mass dampers. Presently, very few buildings are actually

constructed in this way, but there has been a recent large international research effort (Casciati, 1996;

Kabori, 1996; Soong, 1996). Owing to their adaptability, active systems are less dependent for their

effectiveness on the precise nature of the input motion (a concern for passive systems, particularly where

they are very close to the earthquake source) but they must have a very high degree of reliability to ensure

they function during the crucial few seconds of an earthquake.

13.2.10 Worked Examples

Example 13.1 Seismic Analysis of a 30-Story Frame

A 30-story building has the effective stiffness, Ke ¼ 2:5 £ 103 kN/m, together with a mass per unit height

of m ¼ 30 tons=m:

0.5

1.5

2.5

3.5

4.5

0 0.5 1 1.5 2 2.5

Period (sec)

Spectral acceleration of base shear

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5

Period (sec)

Spectral displacement (m)

(a)

(b)

Period shift

Isolated Reduction in Force

period

x= 2%

x = 10%

x = 20%

x = 2%

x = 10%

x = 20%

Period shift

leads to

increase of

displacement

Period shift

Increased

damping leads

to decrease in

displacement

FIGURE 13.8 Effect of seismic isolation on forces and displacements for an earthquake with predominant period

around 0.5 sec. (a) Effect of period shift on design forces; (b) Effect of period shift and damping on relative

displacement between ground and structure.

13-16 Vibration and Shock Handbook

The mass of the building is uniform over its height of 120 m. An appropriate first mode shape for the

structure is the parabola, as shown in Figure 13.9.

For this frame, using the response spectrum for 2% damping (Figure 13.10), find:

1. The peak tip deflection

2. The peak base shear

3. The overturning moment

4. The peak interstory drift at the top of the frame

5. The peak acceleration at the top of the frame

The effective mass, Me or Mi; (see Equation 13.4)

Me ¼ f2m ¼

ðH

mðxÞ

􀀏 􀀐4

dx ¼

mðxÞ

􀀈 􀀈 􀀈 􀀈 􀀈

0 ¼

5 ¼

30 £ 120

5 ¼ 720 tons

This is the effective mass tributary to one frame.

The effective earthquake mass, Meq or Li (see Equation 13.4)

Meq ¼ fm ¼

ðH

mðxÞ

􀀏 􀀐2

dx ¼

mðxÞ

􀀈 􀀈 􀀈 􀀈 􀀈

0 ¼

3 ¼

30 £ 120

3 ¼ 1200 tons

The natural period is 3.37 sec, a long-period structure, given:

Keq ¼ 2:5 £ 103 kN/m

T ¼ 2pð720=2500Þ0:5

The participation factor is

PF ¼

Meq

Me ¼

1200

720 ¼ 1:667

(i) Peak tip deflection

From Figure 13.9

Sd ¼ 0:32 m ¼ 320 mm ¼ PF £ Sd

Dtip ¼ 1:667 £ 320 ¼ 533:44 mm

(ii) Peak base shear ¼ PF £ Fmax

V0 ¼ PFðDtipKeÞ ¼ 1:667ð0:53344 £ 2500Þ ¼ 2:22 MN

(iii) Peak overturning moment

Mot ¼ V0x􀀊

120 m

42 m

1st mode shape

Parabolic mode shape

Δ(x,t) = Δt

FIGURE 13.9 First mode shape of the 30-story frame.

Vibration and Shock Problems of Civil Engineering Structures 13-17

x􀀊 ¼

ðH

xmðxÞ

􀀏 􀀐2

ðH

mðxÞ

􀀏 􀀐2

ðH

mðxÞ

H2 dx

ðH

mðxÞ

H2 dx

x􀀊 ¼

4 ¼

3 £ 120

4 ¼ 90 m

Mot ¼ 2223:11 £ 90 ¼ 2:0008 £ 105 kN m

Equivalent static loading ¼ mðtÞv2 £ PF £ Sd

􀀏 􀀐2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Period (sec)

Sd (m)

0.2

0.4

0.6

0.8

1.2

0 2 4

Period (sec)

Sv (m/sec)

0.2

0.4

0.6

0.8

1.2

1.4

1.6

0 1 2 3

Period (sec)

Sa (g)

(a)

(b)

(c)

1 3

0 1 2 3 4

FIGURE 13.10 (a) Deformation (or displacement); (b) pseudo-velocity, and (c) pseudo-acceleration response

spectra. El-Centro ground motion. Damping ratio j ¼ 2%:

13-18 Vibration and Shock Handbook

v ¼

ffiffiffiffiffi

ffiffiffiffiffiffiffiffi

2500

720

¼ 1:86 rad=sec

Hence, equivalent static loading ¼ 30 £ 1.862 £ 1.667 £ 0.32 £ 1 ¼ 55.365 kN/m.

The equivalent static load, shear force, and bending moment diagrams of the frame are shown

in Figure 13.11.

(iv) Peak interstory drift ¼ 533:44 2

116

120

􀀏 􀀐2

£ 533:44 ¼ 34:97 mm

This will create significant constraints on component details such as partitions, windows, and panels

at the particular level. To avoid potential problems, the structure might have to be stiffened laterally.

(v) Peak acceleration ¼

mðxÞ ¼

55:365

30 ¼ 1:8455 m/sec2

This acceleration is 18.8% of gravity. It is important that the facade attachment, mechanical utilities,

or electrical utilities of the structure are appropriately designed according to the peak acceleration.

Example 13.2 Response of Buildings to an Earthquake

In the following example, a 52-story office/residential building is considered. The structure is founded on

a highly soft soil and located in UBC Zone 4 in the USA, which represents a relatively active seismic area.

The lateral load resisting system is a concrete core system with concrete moment frames for the perimeter.

According to UBC (2000), the structure needs to resist an equivalent horizontal seismic force of

79,113 kN, representing nearly 9.14% of the effective vertical load. Details of story weights, elevation, and

interstory height are given in Table 13.2.

Sample calculation

The base shear value is obtained based on UBC (2000) approach:

T ¼ CtH3=4

n ¼ 0:03 £ 682:43=4 ¼ 4:01 sec ðheight input must be in feetÞ

V ¼

CvIW

but V #

2:5CaI

W and V $

0:8ZNv I

W ðsee Equation 13:11Þ

V ¼

0:96 £ 1 £ 865:3 £ 103

3:5 £ 4:01 ¼ 59:25 MN

V $

0:8 £ 0:4 £ 1 £ 1 £ 863:5 £ 103

3:5

$ 79:113 MN

Hence, the lower limit applies, V ¼ 79:113 MN.

100 m

55.365 kN/m

Equivalent

static load

V0 = 2.22x103 kN

SFD

M0t = 2x105 kNm

BMD

FIGURE 13.11 Equivalent static load, shear force diagram, and bending moment diagram of the frame.

Vibration and Shock Problems of Civil Engineering Structures 13-19

TABLE 13.2 Calculation Details for Structure’s Response to an Earthquake

Height

(m)

Height

(ft)

(sec)

Ct Cv Ca Nv Na R I Z

208.00 682.41 4.01 0.03 0.96 0.36 1.00 1.00 3.50 1.00 0.40

Level Story

Height (m)

Height

(m)

Story Weight

(kN)

Wi Hi Force

(kN)

Shear

(kN)

Moment

(kN m)

Top 0 212 0 0 19,778 19,778 0

52 4.0 208.0 9,300.0 1,934,400 1,319 21,097 79,113

51 4.0 204.0 9,300.0 1,897,200 1,294 22,391 163,503

50 4.0 200.0 9,300.0 1,860,000 1,268 23,659 253,067

49 4.0 196.0 9,300.0 1,822,800 1,243 24,902 347,705

48 4.0 192.0 16,900.0 3,244,800 2,213 27,115 447,315

47 4.0 188.0 16,900.0 3,177,200 2,167 29,282 555,776

46 4.0 184.0 16,900.0 3,109,600 2,120 31,402 672,903

45 4.0 180.0 16,900.0 3,042,000 2,074 33,477 798,511

44 4.0 176.0 16,900.0 2,974,400 2,028 35,505 932,418

43 4.0 172.0 16,900.0 2,906,800 1,982 37,487 1,074,437

42 4.0 168.0 16,900.0 2,839,200 1,936 39,423 1,224,386

41 4.0 164.0 16,900.0 2,771,600 1,890 41,313 1,382,079

40 4.0 160.0 16,900.0 2,704,000 1,844 43,157 1,547,331

39 4.0 156.0 16,900.0 2,636,400 1,798 44,955 1,719,960

38 4.0 152.0 16,900.0 2,568,800 1,752 46,707 1,899,779

37 4.0 148.0 16,900.0 2,501,200 1,706 48,412 2,086,606

36 4.0 144.0 16,900.0 2,433,600 1,660 50,072 2,280,254

35 4.0 140.0 16,900.0 2,366,000 1,613 51,685 2,480,541

34 4.0 136.0 16,900.0 2,298,400 1,567 53,252 2,687,282

33 4.0 132.0 16,900.0 2,230,800 1,521 54,774 2,900,292

32 4.0 128.0 16,900.0 2,163,200 1,475 56,249 3,119,386

31 4.0 124.0 16,900.0 2,095,600 1,429 57,678 3,344,381

30 4.0 120.0 16,900.0 2,028,000 1,383 59,061 3,575,092

29 4.0 116.0 16,900.0 1,960,400 1,337 60,398 3,811,335

13-20 Vibration and Shock Handbook

28 4.0 112.0 16,900.0 1,892,800 1,291 61,688 4,052,926

27 4.0 108.0 16,900.0 1,825,200 1,245 62,933 4,299,679

26 4.0 104.0 16,900.0 1,757,600 1,199 64,131 4,551,410

25 4.0 100.0 16,900.0 1,690,000 1,152 65,284 4,807,936

24 4.0 96.0 16,900.0 1,622,400 1,106 66,390 5,069,072

23 4.0 92.0 16,900.0 1,554,800 1,060 67,450 5,334,633

22 4.0 88.0 16,900.0 1,487,200 1,014 68,465 5,604,435

21 4.0 84.0 16,900.0 1,419,600 968 69,433 5,878,293

20 4.0 80.0 16,900.0 1,352,000 922 70,355 6,156,024

19 4.0 76.0 16,900.0 1,284,400 876 71,230 6,437,442

18 4.0 72.0 16,900.0 1,216,800 830 72,060 6,722,364

17 4.0 68.0 16,900.0 1,149,200 784 72,844 7,010,605

16 4.0 64.0 16,900.0 1,081,600 738 73,581 7,301,981

15 4.0 60.0 16,900.0 1,014,000 691 74,273 7,596,306

14 4.0 56.0 16,900.0 946,400 645 74,918 7,893,398

13 4.0 52.0 16,900.0 878,800 599 75,518 8,193,071

12 4.0 48.0 16,900.0 811,200 553 76,071 8,495,141

11 4.0 44.0 16,900.0 743,600 507 76,578 8,799,424

10 4.0 40.0 16,900.0 676,000 461 77,039 9,105,735

9 4.0 36.0 16,900.0 608,400 415 77,454 9,413,890

8 4.0 32.0 16,900.0 540,800 369 77,822 9,723,705

7 4.0 28.0 16,900.0 473,200 323 78,145 10,034,994

6 4.0 24.0 16,900.0 405,600 277 78,422 10,347,575

5 4.0 20.0 16,900.0 338,000 230 78,652 10,661,261

4 4.0 16.0 16,900.0 270,400 184 78,837 10,975,870

3 4.0 12.0 16,900.0 202,800 138 78,975 11,291,216

2 4.0 8.0 16,900.0 135,200 92 79,067 11,607,116

1 4.0 4.0 16,900.0 67,600 46 79,113 11,923,384

GF 4.0 0.0 16,900.0 0 0 79,113 11,923,384

Sum 865,300.0 kN 87,012,000 kNm 79,113 kN Seismic

base shear

Seismic

overturning

moment

Vibration and Shock Problems of Civil Engineering Structures 13-21

Distribution of lateral forces

UBC (2000) specifies the load at the top to be

Ft ¼ 0:07TV ¼ 19:778 MN

Extracting calculation for level 48,

WiHi ¼ 16:9 £ 192 ¼ 3:245 £ 103 MN

Fx ¼ ðV 2 FtÞðW48H48Þ X52

WiHi

¼ ð79:113 2 19:778Þð3:245 £ 103Þ

87:012 £ 103 ¼ 2:21 MN

Shear force ¼ Shear49 þ F48 ¼ 24:9 þ 2:21 ¼ 27:1 MN

Moment ¼ Moment49 þ ðShear49 £ story heightÞ ¼ 347:7 þ ð24:9 £ 4Þ ¼ 447:3 MN m

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