Vibration and Shock Handbook 13

Vibration and Shock Handbook 13 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

Vibration and Shock Problems of Civil Engineering Structures

Seismicity and Ground Motions † Influence of Local Site Conditions † Response of Structures to Ground Motions † Dynamic Analysis † Earthquake Response Spectra † Design Philosophy and the Code Approach †

Analysis Options for Earthquake Effects †

Soil–Structure Interaction † Active and Passive Control Systems † Worked Examples

13.3 Dynamic Effects of Wind Loading on Structures 13-22

Introduction † Wind Speed † Design Structures for Wind Loading † Along and Across-Wind Loading †

Wind Tunnel Tests † Comfort Criteria: Human Response

to Building Motion † Dampers † Comparison with Earthquake Loading

13.4 Vibrations Due to Fluid–Structure Interaction 13-33

Added Mass and Inertial Coupling † Wave-Induced Vibration of Structure

13.5 Blast Loading and Blast Effects on Structures 13-34

Explosions and Blast Phenomenon † Explosive Air-Blast Loading † Gas Explosion Loading and Effect of Internal Explosions † Structural Response to Blast Loading †

Material Behaviors at High Strain Rate † Failure Modes

of Blast-Loaded Structures † Blast Wave–Structure Interaction † Effect of Ground Shocks † Technical Design Manuals for Blast-Resistant Design † Computer Programs for Blast and Shock Effects

13.6 Impact loading 13-47

Structural Impact between Two Bodies — Hard Impact and Soft Impact † Example — Aircraft Impact

13.7 Floor Vibration 13-51

Introduction † Types of Vibration † Natural Frequency

of Vibration † Vibration Caused by Walking † Design for Rhythmic Excitation † Example — Vibration Analysis of a Reinforced Concrete Floor

Summary

This chapter provides a concise guide to vibration theory, sources of dynamic loading and effects on structures,options for dynamic analyses, and methods of vibration control Section 13.1 gives an introduction to

13-1

different types of dynamic loads Section 13.2 covers the basic theory underlying earthquake engineering andseismic design In this section, seismic codes and standards are reviewed including American, British, and Europeanpractices Active and passive control systems for seismic mitigation are also discussed This section containsanalytical and design examples on seismic analysis and building response to earthquakes Section 13.3 introducesthe nature of wind loading, dynamic effects, and the basic principles of wind design This section includes formulae,charts, graphs, and tables on both static and dynamic approaches for designing structures to resist wind loads.Types of dampers to reduce vibration in tall buildings under wind loads are also introduced Section 13.4 gives abrief overview of vibration due to fluid–structure interaction Section 13.5 extensively covers the effects of explosion

on structures An explanation of the nature of explosions and the mechanism of blast waves in free air is given.This section also introduces different methods to estimate blast loads and structural response Section 13.6 dealswith the impact loading An analytical example of aircraft impact on a building is given Section 13.7 looks in detail

at the problems of floor vibration Charts and tables are given for designing floor slabs to avoid excessive vibrations

A comprehensive list of references is provided

13.1 Introduction

The different types of dynamic loading considered in this chapter include: earthquakes, wind, floorvibrations, blast effects, and impact- and wave-induced vibration The effects of these loadings ondifferent engineering structures are also discussed It is standard practice to use equivalent statichorizontal forces when designing buildings for earthquake and wind resistance This is the simplest way

of obtaining the dimensions of structural members Dynamic calculations may follow to check, andperhaps modify, the design However, vibrations caused by extreme loads such as blast and impact must

be assessed by methods of dynamical analysis or by experiment

Some examples of dynamic loading are shown in Figure 13.1 The first (a) is a record offluctuating wind velocity Corresponding fluctuating pressures will be applied to the structure Therandom nature of the loading is evident, and it is clear that statistical methods are required forestablishing an appropriate design loading The next figure (b) shows a typical earthquakeaccelerogram As shown, the maximum ground acceleration of the El-Centro earthquake was about0.33g The third figure (c) shows the characteristic shape of the air pressure impulse caused by a

Dynamic

−505

El Centro ground acceleration

Time (sec)a/g

Positivephase Negativephase

bomb blast The shapes of air-blast curves are usually quite similar, having an initial peak followed

by an almost linear decay and often followed by some suction The duration of the impulsiveloads and their amplitudes depend on many factors, for example, distance from blast and chargeweight

Vibration of structures is undesirable for a number of reasons, as follows:

1 Overstressing and collapse of structures

2 Cracking and other damage requiring repair

3 Damage to safety-related equipment

4 Impaired performance of equipment or delicate apparatus

5 Adverse human response

With modern forms of construction, it is feasible to design structures to resist the forces arisingfrom dynamic loadings such as major earthquakes The essential requirement is to prevent totalcollapse and consequent loss of life For economic reasons, however, it is the accepted practice toabsorb the earthquake energy by ductile deformation, therefore accepting that repair might berequired

Some forms of loading are quite well defined and may be quantified by observation or experiment.Many forms of loading are not at all well defined and require judgment on the part of the engineer.London’s Millennium Bridge, which is a 350-m pedestrian bridge, opened in June 2000 However, localauthorities shut it down after two days due to vibration problems Engineers found that the

“synchronous lateral excitation” caused the problem and fitted 91 dampers to reduce the excessivemovement In January 2001, a 2000-strong crowd marched across the bridge to check the performance ofthe structure before it was reopened to the public

Data on certain types of dynamic loading, such as earthquakes and wind, are readily available inmany design codes Other types of loading are less well covered, though much data may be available

in published research papers One of the aims of this chapter is to discuss the nature of the mostimportant types of dynamic loading and to direct the reader to relevant literature for furtherinformation

13.2 Earthquake-Induced Vibration of Structures

13.2.1 Seismicity and Ground Motions

The most common cause of earthquakes is thought to be the violent slipping of rock masses alongmajor geological fault lines in the Earth’s crust, or lithosphere These fault lines divide the global crustinto about 12 tectonic plates, which are rigid, relatively cool slabs about 100 km thick Tectonic platesfloat 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 thecontinuing deformation of the crustal structure eventually leads to stresses/strains which exceed thematerial strength A rupture will then initiate at some critical point along the fault line and will propagaterapidly through the highly stressed material at the plate boundary In some cases, the plate margins aremoving 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 theother and, depending on the density of the material, it may resurface in the form of volcanoes In boththese scenarios, there may be volcanoes and earthquakes at the plate boundaries, both being caused by thesame mechanism of movement in the Earth’s crust Another possibility is that the plate boundaries willslide sideways past each other, essentially retaining the local surface area of the plate It is believed thatapproximately 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

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

Earthquakes do occur at locations away from the plate boundaries Such events are known as plate earthquakes and they are much less frequent than interplate earthquakes They are also muchless 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 ofCalifornia, 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 EastCoast and Missouri in the interior However, because of the low population density at the time, thedamage 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 thatfrequently occur along the plate boundaries in California (It is thought that, because tectonic plates arenot homogeneous or isotropic, areas of local high stress are developed as the plate attempts to move as arigid body Accordingly, rupture within the plate, and the consequent release of energy, are believed togive rise to these intraplate events.)

intra-The point in the Earth’s crustal system where an earthquake is initiated (the point of rupture) is calledthe hypocenter or focus of the earthquake The point on the Earth’s surface directly above the focus iscalled the epicenter and the depth of the focus is the focal depth Earthquake-occurrence maps usuallyindicate the location of various epicenters of past earthquakes and these epicenters are located byseismological analysis of the effect of earthquake waves on strategically located receiving instrumentscalled seismometers

When an earthquake occurs, several types of seismic wave are radiated from the rupture The mostimportant of these are the body waves (primary (P) and secondary (S) waves) P waves are essentiallysound waves traveling through the Earth, causing particles to move in the direction of wave propagationwith 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 withparticle motion transverse to the direction of propagation S waves tend to travel at about 60% of thevelocity of P waves, so they always arrive at seismometers after the P waves The time lag between arrivalsoften provides seismologists with useful information about the distance of the epicenter fromthe recorder

The total strain energy released during an earthquake is known as the magnitude of the earthquakeand it is measured on the Richter scale It is defined quite simply as the amplitude of the recordedvibrations 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 notsufficient to indicate whether structural damage can be expected at a particular site The distance ofthe structure from the source has an equally important effect on the response of a structure, as dothe local ground conditions The local intensity of a particular earthquake is measured on thesubjective Modified Mercalli scale (Table 13.1) which ranges from 1 (barely felt) to 12 (totaldestruction) The Modified Mercalli scale is essentially a means by which damage may be assessedafter an earthquake In a given location, where there has been some experience of the damagingeffects of earthquakes, albeit only subjective and qualitative, regions of varying seismic risk may beidentified 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 diminishwith distance from the source, although anomalies due to local soil and geological conditions arequite common

Modem seismometers (or seismographs) are sophisticated instruments utilizing, in part,electromagnetic principles These instruments can provide digitized or graphical records of earth-quake-induced accelerations in both the horizontal and vertical directions at a particular site.Accelerometers provide records of earthquake accelerations and the records may be appropriatelyintegrated to provide velocity records and displacement records Peak accelerations, velocities, anddisplacements are all in turn significant for structures of differing stiffness (Figure 13.2)

13.2.2 Influence of Local Site Conditions

Local geological and soil conditions may have a significant influence on the amplitude and frequencycontent of ground motions These conditions affect the earthquake motions experienced (and hence thestructural response) in one, or more, of the following ways:

* Interaction between the bedrock earthquake motion and the soil column will modify the actualground accelerations input to the structure This manifests itself by an increase in the amplitude ofthe ground motion over and above that at the bedrock, and a filtering of the motion so that therange of frequencies present becomes narrow with the high-frequency components beingeliminated This condition particularly arises in areas where soft sediments and alluvial soil overlybedrock 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 underbase motions of lower amplitude

* The soil properties in the proximity of the structure contribute significantly to the effectivestiffness of the structural foundation This may be a significant parameter in determining theoverall structural response, especially for structures that would be characterized as stiff underother environmental loadings

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

evaluating the intensity of a likely earthquake It is usual to use this information, together with theregional 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 thedistribution 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 accelerationsgenerated by the foundation level(s) of the structure intercepting and being influenced by transientground motions Specifically, the product of the structural mass and the total acceleration produces theinertia loading experienced by the structure This is an expression of Newton’s Second Law It isimportant 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 absoluteacceleration is low This occurs because the ground acceleration and the acceleration of the buildingrelative to the ground tend to oppose one another In this case, the earthquake loading is approximatelyproportional 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 accelerationsthat are significant; whereas for structures that are largely horizontal in extent, the effect of the verticalaccelerations is dominant Moreover, if the plan distributions of mass and stiffness are dissimilar invertical structures, significant twisting motions may arise

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).

The peak ground acceleration is of importance in the response of stiff structures and peak grounddisplacements are of importance in the response of flexible structures, with peak ground velocity being ofimportance for structures of intermediate stiffness Stiff structures tend to move in unison with theground while flexible structures, such as high-rise buildings, experience the ground moving beneaththem, 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

sub-jected 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

mð€y þ €zÞ ¼ 2ky 2 c_y ð13:1ÞRearranging Equation 13.1, and replacing m; k;

and c by the system’s radial frequency v and

damping ratioj; gives

€y þ 2jv_y þ v2y ¼ 2€z ð13:2ÞGiven 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 givennatural period and damping ratio, and can also be used to derive maximum responses for constructing aresponse spectrum(Figure 13.6).Owing to the random nature of earthquake ground motion, numericalsolution 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

multiple-degree-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

€Yiþ 2jivi_Yiþv2

iYi¼ Li

Mi€z ð13:3ÞHere, Yiis 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

Li

Mi ¼

Xn j¼1fijmj

Xn j¼1

FIGURE 13.3 Single-DoF system.

FIGURE 13.4 Typical modes for multistory buildings.

In Equation 13.4,fij; describes the modal displacement of the jth mass in the ith mode The higher modesoften have very low values of ðLi=MiÞ; and their contribution can then be omitted In this way, thecomputational 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 anearthquake A plot of the maximum value of a response quantity as a function of the natural vibrationfrequency of the structure, or as a function of a quantity which is related to the frequency such as naturalperiod, constitutes the response spectrum for that quantity(see Chapter 17andChapter 31)

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

SE ¼ 12MS2vHence, the pseudo-relative velocity and acceleration spectra are defined as

13.2.5.2 Smoothed Design Spectra

Owing to the highly random nature of earthquake ground motions, the response spectrum for a realearthquake record contains many sharp peaks and troughs, especially for low levels of damping Thepeaks and troughs are determined by a number of uncertain factors, such as the precise location of theearthquake source, which are unlikely to be known precisely in advance Therefore, spectra for designpurposes are usually smoothed envelopes of spectra for a range of different earthquakes; indeed, one ofthe advantages of response spectrum analysis over time history analysis is that it can represent theenvelope response to a number of different possible earthquake sources from a single analysis, and is notdependent 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 areseveral times those at first yield, Dy; as represented inFigure 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,

time (sec)a/g

Dmax; to yield displacement, Dy) The reduction in acceleration response of the yielding system comparedwith the elastic one is period dependent; for structural periods greater than the predominant earthquakeperiods, the reduction is approximately 1=m; for very stiff systems there is no reduction, while atintermediate 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 theductility-modified spectrum used exactly like a normal elastic spectrum However, deflections derivedfrom this treatment must be multiplied bym to allow for the plastic deformation

It is now standard practice to analyze DoF systems in the same way That is, a yielding DoF system is treated as elastic, and an appropriate ductility-modified spectrum is substituted for anelastic one Acceleration and force responses are derived directly and deflections are multiplied bym:However, this procedure is not (contrary to the case for single-DoF systems) rigorously correct Although

multi-it gives satisfactory answers for regular structures, multi-it can be seriously in error for structures (such as thosewith weak stories) where the plasticity demand is not evenly distributed Nevertheless, most codes ofpractice allow the use of ductility-modified spectra for design, and give appropriate values for thereduction factors (called q; or behavior factors in Eurocode 8 and R factors in UBC) to apply to elasticresponse 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 thedilemma of being required to design for an event, the magnitude of which has only a small chance of

Stiffness, kMass, m

occurring during the life of the facility If the designer adopts conservative performance criteria for thefacility, 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 inthese 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 theseismic design The facility should be proportioned to resist such an earthquake withoutsignificant damage This “damageability” limit state should ensure safety, limited nonstructuraldamage, and the continued performance of facilities and services, particularly in those withimportant postearthquake functions The list includes hospitals, police, fire and civil defensefacilities, 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, theprotection of industrial complexes, in their own right, as well as the protection of individualitems of equipment in other buildings and facilities, is a necessary consequence of adoption ofthis limit state

2 The most severe, credible earthquake that may be expected to occur at the site is used to testsafety In this ultimate limit state, significant structural and nonstructural damage is expectedbut neither collapse nor loss of life should occur

The main strategy for preventing collapse has traditionally been provision of ductility This is theopposite quality to brittleness, and may be defined as the ability to sustain repeated excursionsbeyond the elastic limit without fracture Owing to the cyclic, imposed displacement nature ofearthquake 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 thanimposed loads

Achieving ductility is partly a matter of choosing the right structural system, and partly a matter ofdetailing In the former category comes the important concept of “capacity design,” as described byPaulay (1993) This involves ensuring a hierarchy of strengths within a structure to ensure that yieldingoccurs in ductile modes (such as flexure) rather than brittle modes (such as buckling or, for reinforcedconcrete, shear) There are other aspects of structural form which are important, particularly regularity inelevation (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 andPaulay, 1975)

Detailing of the structure is also important to ensure ductility For concrete structures, this primarilyinvolves reinforcement detailing and in steel structures connection detailing The latter aspect has beenparticularly 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 incodes of practice, such as Eurocode 8 (ENV 1998, 1994-8) Part 1.3 and UBC/IBC (2000) Textbooksdiscussing these issues for concrete include Paulay and Priestley (1992), Booth (1994), Penelis andKappos (1996), FEMA 273/274, FEMA 356/357 “NEHRP guidelines for seismic rehabilitation ofbuildings,” FEMA 368/369 “NEHRP recommended provisions for seismic regulations for new buildingsand other structures,” and FEMA 306/307/308 “Evaluation and repair of earthquake damaged concreteand masonry buildings.” Textbooks covering failures of steel structures in recent earthquakes includeBurdekin (1996) and FEMA 350-354 (2000)

Another important aspect of detailing is to allow for the maximum inelastic deflections caused by thedesign earthquake Nonseismic-resisting elements of a structure such as cladding and infill walls must beable 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 Inaddition, adequate separation between adjacent structures must be provided Codes of practice (e.g.,Eurocode 8 and UBC) give guidance on suitable limits

13.2.6.1 Performance-Based Design

In recent years, seismic design codes throughout the world have been shifting toward the adoption ofperformance-based design philosophy The goal of a performance-based design procedure is to producestructures that have predictable seismic performance under multiple levels of earthquake intensity Inorder 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 designprocedure and have to be designed rationally The next generation of codes is expected to be based onperformance-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 arethe other vital components for ensuring good seismic performance This section provides a brieftheoretical review of the basis for seismic analysis, describing the main analytical techniques currentlyused by designers More details are given by Clough and Penzien (1993), a standard general text fordynamic 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 forearthquake loading on a structure In all methods, the dual damage criteria discussed above may beapplied 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 underabout 75 m in height A response spectrum analysis is required for taller buildings, because higher modeeffects may become important, and also for buildings with plan or elevational eccentricities becausetorsional effects or nonstandard mode shapes may be significant Codes of practice such as Eurocode 8and UBC (2000) specify the degree of eccentricity at which such analysis is required Unusual or veryimportant structures may require nonlinear time history analysis, and this may also be required wherethe inaccuracies implicit in the use of ductility-modified response spectrum analysis becomeunacceptable Displacement-based approach is a new method for seismic design, which is gainingpopularity 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 structurethrough a complete time history of earthquake ground motions The advantage of the method is that itcan give direct information on nonlinear response, the duration of response (and hence the number ofloading cycles), and the relative phasing of response between various parts The method involvessubjecting 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 aregenerally of most interest The analysis must be performed for a number of different earthquake timehistories 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 tohave 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 recordcomes from another country or continent Moreover, the method is expensive and time-consuming, sothat only for special structures can its use be justified If, in addition, inelastic response calculations areinvolved, another level of complexity (and uncertainty) is introduced Response then becomes

dependent, often heavily so, on the nonlinear models chosen and this is in addition to that inherent inchoosing 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 extensionpoint, so that the maximum force in the spring is given by

It is important to realize that the spectral acceleration, Sa; is an absolute value (the true acceleration ofthe 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 anmulti-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, Equation13.4 is modified to give the maximum base shear in the ith mode as

Fi¼

Xn j¼1fijmj

0

@

1A

2

Xn j¼1

f2mj

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 beneglected Since the sum of effective masses, meff;i; of all modes equals the total mass, a good test ofwhether the first r modes are sufficient to capture response adequately is

Xr i¼1

not occur simultaneously, and hence simple addition produces an overestimate of response A commonand usually adequate approximation is the square root of the sum of the squares (SRSS) rule, wherethe 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, andother methods, such as the complete quadratic combination (CQC) method, are available (Gupta, 1990).There are many commercially available computer programs which can perform response spectrumanalysis, 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 itsaccuracy upon response being dominated by one mode of vibration in each direction In the case ofbuildings, a quantity usually referred to as the “total base shear” is calculated from the product of theweight 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 soilconditions and their effect on structural responses Once the total base shear has been calculated, it isdistributed up the structure as a series of horizontal loads at each floor level and the structure is analyzedwith 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

RT but V #

2:5CaI

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

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

Cv; Ca¼ 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 returnperiod ¼ 0.075 to 0.4 in UBC

Nv¼ factor allowing for proximity to active faults ¼ 1.0 to 2.0 in UBC

ðV=WÞ represents the shape of a standard design response spectrum with a peak amplification on groundacceleration for 5% damping of 2.5, and a minimum value at long period to allow for the uncertainty inlong-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 torsionaleffects

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 ratherthan 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-baseddesign procedure together with a check on the displacement/drift limit, or by employing a directdisplacement-based procedure The idea of displacement-based design was introduced by Gulkanand Sozen (1974) They developed the concept of substitute structure to estimate the nonlinearstructural response through an equivalent elastic model, assuming a linear behavior and a viscous

damping equivalent to the nonlinear response This idea has been adopted recently by Priestley andKowalsky (2000) for a direct displacement design of single-DoF and multi-DoF reinforced concretestructures Another direct displacement-based design approach was proposed by Fajfar (2000) based onthe 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-purposespectrum, nonlinear elastic behavior of a structural system can be accounted for by either an equivalentelastic response spectrum or an inelastic response spectrum The former is associated with effectiveviscous dampingjeffand the latter is directly constructed based on the relation between reduction factorsand ductility Although the elastic acceleration design spectrum is available from codes, it is notappropriate to be a basis for the determination of the elastic displacement design spectrum because thedisplacement 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 theeffect of ground compliance on response Although this rigid base assumption may lead to anunderestimate of deflections, it is usually conservative as far as forces are concerned, because groundcompliance reduces stiffness and usually moves structural periods farther from resonance with theground motion However, this conservatism may not always apply, and Eurocode 8 Part 5 lists thefollowing 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/secAllowance 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 thesoil 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 dynamiccharacteristics 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 reducingenergy dissipation demand on primary structural members and minimizing possible structuraldamage

The most common type of passive system involves lengthening the structure’s fundamental period ofvibration by mounting the superstructure on bearings with a low horizontal stiffness; this is known asbase or seismic isolation Where this increases, the fundamental period above the predominant periods ofearthquake excitation, the acceleration (but not necessarily displacement) response is significantlyreduced Usually, additional damping is provided in the seismic isolation bearing to control deflections.The principle of seismic isolation is illustrated byFigure 13.8 The reduction in response, often of theorder of 50, has proved highly effective in recent earthquakes in reducing damage to both buildingstructure and building contents UBC (2000) provides codified guidance for seismic isolation ofbuildings 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

system is the provision of additional structural damping in the form of discrete viscous, frictional, orhysteretic dampers.

Active systems modify the dynamic characteristics of a structure in real time during an earthquake, bycomputer-controlled devices such as active mass dampers Presently, very few buildings are actuallyconstructed 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 theireffectiveness on the precise nature of the input motion (a concern for passive systems, particularly wherethey are very close to the earthquake source) but they must have a very high degree of reliability to ensurethey 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, Ke¼ 2:5 £ 103kN/m, together with a mass per unit height

Period shift

Increaseddamping leads

to decrease indisplacement

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.

The mass of the building is uniform over its height of 120 m An appropriate first mode shape for thestructure 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, Meor Mi; (see Equation 13.4)

Me¼f2m ¼ðH

0 mðxÞ Hx 4dx ¼ mðxÞ5 Hx54

H

0¼ mH5 ¼ 30 £ 1205 ¼ 720 tonsThis is the effective mass tributary to one frame

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

Meq¼fm ¼ðH

0 mðxÞ Hx 2dx ¼ mðxÞ3 Hx32

H

0¼ mH3 ¼ 30 £ 1203 ¼ 1200 tonsThe natural period is 3.37 sec, a long-period structure, given:

From Figure 13.9

Sd¼ 0:32 m ¼ 320 mm ¼ PF £ Sd

Dtip¼ 1:667 £ 320 ¼ 533:44 mm(ii) Peak base shear ¼ PF £ Fmax

V0¼ PFðDtipKeÞ ¼ 1:667ð0:53344 £ 2500Þ ¼ 2:22 MN(iii) Peak overturning moment

2

00.050.10.150.20.250.30.350.40.45

Period (sec)

00.20.40.60.811.2

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 116120 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 f

mðxÞ ¼

55:365

30 ¼ 1:8455 m/sec2This 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 of79,113 kN, representing nearly 9.14% of the effective vertical load Details of story weights, elevation, andinterstory height are given inTable 13.2

Sample calculation

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

T ¼ CtHn3=4¼ 0:03 £ 682:43=4¼ 4:01 sec ðheight input must be in feetÞ

100 m

55.365 kN/m

Equivalentstatic load V0= 2.22x10SFD 3kN M0t= 2x10

5kNmBMDFIGURE 13.11 Equivalent static load, shear force diagram, and bending moment diagram of the frame.

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

Distribution of lateral forces

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

Ft¼ 0:07TV ¼ 19:778 MNExtracting calculation for level 48,

Wind is a phenomenon of great complexity because of the many flow situations arising from theinteraction of wind with structures Wind is composed of a multitude of eddies of varying sizes androtational characteristics carried along in a general stream of air moving relative to the Earth’s surface.These eddies give wind its gusty or turbulent character The gustiness of strong winds in the lower levels

of the atmosphere largely arises from interaction with surface features The average wind speed over atime period of the order of 10 min or more tends to increase with height, while the gustiness tends todecrease with height

A further consequence of turbulence is that dynamic loading on a structure depends on the size of theeddies Large eddies, whose dimensions are comparable with the structure, give rise to well-correlatedpressures as they envelop the structure On the other hand, small eddies result in pressures at variousparts of the structure being practically uncorrelated Eddies generated around a typical structure areshown in Figure 13.12

FIGURE 13.12 Generation of eddies (a) Elevation; (b) plan.

Some structures, particularly those that are tall or slender, respond dynamically to the wind The known structural collapse due to wind was the Tacoma Narrows Bridge which occurred in 1940 at a windspeed of only about 19 m/sec It failed after it had developed a joint torsional and flexural mode ofoscillation.

best-There are several different phenomena giving rise to dynamic response of structures in wind Theseinclude buffeting, vortex shedding, galloping, and flutter Slender structures are likely to be sensitive todynamic response in line with the wind direction as a consequence of turbulence buffeting Transverse orcrosswind response is more likely to arise from vortex shedding or galloping, but may be excited byturbulence buffeting also Flutter is a coupled motion, often a combination of bending and torsion, andcan result in instability

An important problem associated with the wind-induced motion of buildings is concerned with thehuman response to vibration At this point, it will suffice to note that humans are surprisingly sensitive tovibration, to the extent that motions may feel uncomfortable even if they correspond to relativelyunimportant stresses The next few sections give a brief introduction to the dynamic response ofstructures in wind More details can be found in wind engineering texts (e.g., Sachs, 1978; Holmes, 2001)

13.3.2 Wind Speed

At great heights above the surface of the Earth, where frictional effects are negligible, air movements aredriven by pressure gradients in the atmosphere, which in turn are the thermodynamic consequence ofvariable solar heating of the Earth This upper level wind speed is known as the gradient wind velocity.Different terrains can be categorized according to the roughness length Table 13.3 shows the differentcategories specified in the Australian/New Zealand wind code, AS/NZS 1170.2 (2002) Closer to thesurface, the wind speed is affected by frictional drag of the air over the terrain There is a boundary layerwithin which the wind speed varies from almost zero, at the surface, to the gradient wind speed at aheight known as the gradient height The thickness of this boundary layer, which may vary from 500 to

3000 m, depends on the type of terrain, as depicted inFigure 13.13 As can be seen, the gradient heightwithin a large city center is much higher than it is over the sea where the surface roughness is less

In practice, it has been found useful to start with a reference wind speed based on statistical analysis ofwind speed records obtained at meteorological stations throughout the country The definition of thereference wind speed varies from one country to another For example, in Australia/New Zealand, it is the3-sec gust wind speed at a height of 10 m above the ground assuming terrain category 2 Maps ofreference wind speeds applying to various countries are usually available

An engineering wind model for Australia has been developed by Melbourne (1992) from the Deavesand Harris (1978) model This model is based on extensive full-scale data and on the classic logarithmiclaw in which the mean velocity profile in strong winds applicable in noncyclonic regions (neutral stabilityconditions) is given by Equation 13.13

Vz< up0:4 loge

The numerical values are based on a mean gradient wind speed of 50 m/sec

TABLE 13.3 Terrain Category and Roughness Length ðz 0 Þ

Length ðz 0 Þ Exposed open terrain with few or no obstructions and water surfaces at serviceability wind speeds 0.002 Water surfaces, open terrain, grassland with few, well-scattered obstructions having heights

Terrain with numerous closely spaced obstructions 3 to 5 m high such as areas of suburban

Terrain with numerous large, high (10.0 to 30.0 m high) and closely spaced obstructions such as

For values of z , 30:0 m the z=zgvalues become insignificant and the above equation simplifies to

Vz< up0:4loge

z

where:

Vz¼ the design hourly mean wind speed at height z; in m/sec

up¼ the friction velocity

z ¼ the distance or height above ground, in m

zg¼ the gradient height in meters (the value ranges from 2700 to 4500 m), see Table 13.4 (derived by theauthors)

zg¼ 6 £ 10up24

As given inTable 13.3, there is an interaction between roughness length and terrain category, so it isnecessary to define a terrain category to find the design hourly wind speeds and gust wind speeds Thelink between hourly mean and gust wind speeds is as follows:

Elevation

FIGURE 13.13 Mean wind profiles for different terrains.

TABLE 13.4 Roughness Length, Friction Velocity, and Gradient Height

h ¼ 1:0 2 zz

g

!

ð13:17Þ

For design, the basic wind speed is classified into three different speeds as follows:

Vs¼ V20 yr¼ serviceability limit state design speed having an estimated probability of exceedance of

5% in any one year, for the serviceability limit states

Vp¼ V50 yr¼ permissible, or working, stress design wind speed and can be obtained directly from Vu

using the relation Vp¼ Vu=ð1:5Þ0:5

Vu ¼ V1000 yr¼ ultimate limit state design wind speed having an estimated probability of exceedance

of 5% in a lifetime of 50 years, for the ultimate limit statesUsing rigorous analysis incorporating probability distribution of wind speed and direction, basicdesign wind speeds for different directions and different return periods can be derived For example,AS/NZS 1170.2 provides a wind direction multiplier, which varies from 0.80 for wind from the east to 1.0for wind from the west, and having wind speeds up to a 2000-year return period

13.3.3 Design Structures for Wind Loading

The characteristics of wind pressures are a function of the characteristics of the approaching wind, thegeometry of the structure, and the geometry and proximity of the upwind structures The pressures arenot steady, but highly fluctuating, partly as a result of the gustiness of the wind, but also because of localvortex shedding at the edges of the structures themselves The fluctuating pressures result in fatiguedamage to structures, and in dynamic excitation, if the structure happens to be dynamically windsensitive The pressures are also not uniformly distributed over the surface of the structure, but varywith position

The complexities of wind loading should be kept in mind when designing a structure Because of themany uncertainties involved, the maximum wind loads experienced by a structure during its lifetime mayvary widely from those assumed in the design Thus, the failure or nonfailure of a structure in awindstorm cannot necessarily be taken as an indication of the nonconservativeness, or conservativeness,

of the wind-loading standard The standards do not apply to buildings or structures that are of unusualshape or location Wind loading governs the design of some types of structures, such as tall buildings andslender towers Experimental wind tunnel data may be used in place of the coefficients given in the codefor these structures

13.3.3.1 Types of Wind Design

Typically, for wind-sensitive structures, three basic wind effects need to be considered:

* Environmental wind studies — to study the wind effects on the surrounding environment caused

by erecting the structure (e.g., a tall building) This study is particularly important to assess theimpact of wind on pedestrians and motor vehicles and so on, which utilize the public domainwithin the vicinity of the proposed structure

* Wind loads for facade — to assess design wind pressures throughout the surface area of thestructure to design the cladding system Owing to the significant cost of typical facade systems inproportion to the overall cost of very tall buildings, engineers cannot afford the luxury ofconservatism in assessing design wind loads With due consideration to the complex buildingshapes and dynamic characteristics of the wind and building structure, even the most advancedwind codes generally cannot accurately assess design loads Wind tunnel tests to assess designloads for cladding are now a normal industry practice, with the aim of minimizing initial capitalcosts, and more significantly, to avoid the expensive maintenance costs associated withmalfunctions due to leakage and/or structural failure

* Wind loads for structure — to determine the design wind load so as to design the lateral resisting structural system of a structure and therefore satisfy the various design criteria.

load-13.3.3.2 Design Criteria

In terms of designing a structure for lateral wind loads, the following design criteria need to be satisfied:

* Stability against overturning, uplift, and/or sliding of the structure as a whole

* The strength of the structural components of the building, and stresses that must be withstoodwithout failure during the life of the structure

* Serviceability, for example for buildings, where interstory and overall deflections are withinacceptable limits The control of deflection and drift is imperative for tall buildings in order tolimit damage and cracking to nonstructural members such as the facade, internal partitions,and ceilings

As adopted by most international codes, to satisfy stability and strength limit state requirements, ultimatelimit state wind speed is used In many codes, such a speed has a 5% probability of being exceeded in a1-year period

An additional criterion that requires careful consideration in wind-sensitive structures such as tallbuildings is the control of accelerations when subjected to wind loads under serviceability conditions.Acceptability criteria for vibrations in buildings are frequently expressed in terms of acceleration limitsfor a 1- or 5-year return period wind speed, and are based on human tolerance to vibration discomfort inthe upper levels of buildings Wind response is relatively sensitive to both mass and stiffness, and responseaccelerations can be reduced by increasing either or both of these parameters However, this is in conflictwith earthquake design optimization where loads are minimized in buildings by reducing both the massand stiffness Increasing the damping results in a reduction in both the wind and earthquake responses.The detailed procedure described in wind codes is subdivided into static analysis and dynamic analysismethods The static approach is based on a quasi-steady assumption It assumes that the building is afixed rigid body in the wind The static method is not appropriate for tall or slender structures orstructures susceptible to vibration in the wind In practice, static analysis is normally appropriate forstructures up to 50 m in height The subsequently described dynamic method is for exceptionally tall,slender, or vibration-prone buildings The codes not only provide some detailed design guidance withrespect to dynamic response, but also state specifically that a dynamic analysis must be undertaken todetermine overall forces on any structure with both a height (or length) to breadth ratio greater than five,and a first mode frequency less than one

Wind-loading codes may give the impression that wind forces are relatively constant with time Inreality, wind forces vary significantly over short time intervals, with large amplitude fluctuations at high-frequency intervals The magnitude and frequency of the fluctuations is dependent on many factorsassociated with the turbulence of the wind and local gusting effects caused by the structure andsurrounding environment

To simplify this complex wind characteristic, most international codes have adopted a simplifiedapproach by utilizing a quasi-steady assumption This approach simply uses a single value equivalent,static wind pressure, to represent the maximum peak pressure the structure would experience

13.3.3.3 Static Analysis

This method assumes the quasi-steady approximation It approximates the peak pressures on thebuilding surfaces by the product of gust dynamic wind pressure and the mean pressure coefficients Themean pressure coefficients are measured in a wind-tunnel or full-scale tests and are given by pbar=qzðbarÞ:The implied assumption is that the pressures on the building surface (external and internal) faithfullyfollow the variations in upwind velocity Thus, it is assumed that a peak value of wind speed isaccompanied by a peak value of pressure or load on the structure The quasi-steady model has beenfound to be fairly good for small structures

In static analysis, gust wind speed, Vz; is used to calculate the forces, pressures, and moments on thestructure.

The main advantages and disadvantages of the quasi-steady/peak gust format can be summarized asfollows:

* Advantages:

* Simplicity

* Continuity with previous practice

* Pressure coefficients should need little adjustment for different upwind terrain types

* Existing meteorological data on wind gusts are used directly

The dynamic wind pressure at height z is given by

qz¼ 0:6V2

where

Vz¼ the design gust wind speed at height z; in meters per second ¼ VMðz;catÞMzMtMi

V ¼ the basic wind speed

The multiplying factors ðMÞ take into account the type of terrain ðMtÞ; height above ground level ðMzÞ;topography, and the importance of the structure ðMiÞ: The above derivation essentially forms the basis ofmost international codes

The mean base overturning moment Mbaris determined by summing the moments resulting from thenet effect of the mean pressure and leeward sides of the structure given by

Fz¼Xcp;eqzAz

or for structures with discrete elements:

where

Fz¼ the hourly mean net horizontal force acting on a structure at height z

Cp;e¼ the pressure coefficients for both windward and leeward surfaces

Az¼ the area of a structure or a part of a structure, at height z; in square meters

Fd¼ the hourly mean drag force acting on discrete elements

Cd¼ the drag force coefficient for an element of the structure

13.3.4 Along and Across-Wind Loading

Not only is the wind approaching a building a complex phenomenon, but the flow patterngenerated around a building is complicated by the distortion of the mean flow, the flow

separation, the vortex formation, and the wake

development Large wind pressure fluctuations

due to these effects occur on the surface of a

building As a result, large aerodynamic loads

are imposed on the structural system and

intense localized fluctuating forces act on the

facade of such structures Under the collective

influence of these fluctuating forces, a building

vibrates in rectilinear and torsional modes, as

illustrated in Figure 13.14 The amplitude of

such oscillations is dependant on the nature of

aerodynamic forces and the dynamic

character-istics of the building

13.3.4.1 Along-Wind Loading

The along-wind loading or response of a building

due to the gusting wind can be assumed to consist

of a mean component due to the action of the mean wind speed (e.g., the mean hourly wind speed), and afluctuating component due to wind speed variations from the mean The fluctuating wind is a randommixture of gusts or eddies of various sizes, with the larger eddies occurring less often (i.e., with a loweraverage frequency) than smaller eddies The natural frequency of vibration of most structures issufficiently higher than the component of the fluctuating load effect imposed by the larger eddies That is,the average frequency with which large gusts occur is usually much less than any of the structure’s naturalfrequencies of vibration and so they do not force the structure to respond dynamically The loading due

to those larger gusts (which are sometimes referred to as “background turbulence”) can therefore betreated in similar way to that due to the mean wind speed The smaller eddies, however, because theyoccur more often, may induce the structure to vibrate at or near one of the structure’s natural frequencies

of vibration This in turn induces a magnified dynamic load effect in the structure which can besignificant

The separation of wind loading into mean and fluctuating components is the basis of theso-called “gust factor” approach, which is the basis of many design codes The mean loadcomponent is evaluated from the mean wind speed using pressure and load coefficients Thefluctuating loads are determined separately by a method which makes an allowance for the intensity

of turbulence at the site, size reduction effects, and dynamic amplification (Davenport, 1967;Vickery, 1971)

The dynamic response of buildings in the along-wind direction can be predicted with reasonableaccuracy by the gust factor approach, provided the wind flow is not significantly affected by the presence

of neighboring tall buildings or surrounding terrain

13.3.4.2 Across-Wind Loading

There are many examples of slender structures that are susceptible to dynamic motionperpendicular to the direction of the wind Tall chimneys, street lighting standards, towers, andcables frequently exhibit this form of oscillation, which can be very significant, especially if thestructural damping is small Crosswind excitation of modern tall buildings and structures can bedivided into three mechanisms (AS/NZS 1170.2, 2002) These and higher time derivatives aredescribed as follows:

1 The most common source of crosswind excitation is that associated with “vortex shedding.”Tall buildings are bluff (as opposed to streamlined) bodies that cause the flow to separate fromthe surface of the structure, rather than follow the body contour (Figure 13.15) For a

Along-windTorsion

FIGURE 13.14 Wind response directions.

particular structure, the shed vortices have

a dominant periodicity that is defined by

the Strouhal number Hence, the structure

is subjected to a periodic pressure loading,

which results in an alternating crosswind

force If the natural frequency of the

structure coincides with the shedding

frequency of the vortices, large amplitude

displacement response may occur, and this

is often referred to as critical velocity effect The asymmetric pressure distribution created

by the vortices around the cross section results in an alternating transverse force as they areshed If the structure is flexible, oscillation will occur transverse to the wind, and the conditionsfor resonance would exist if the vortex shedding frequency coincided with the naturalfrequency of the structure This situation could give rise to very large oscillations andpossibly failure

In practice, vertical structures are exposed to a turbulent wind in which both the windspeed and the turbulence level vary with height, so that excitation due to vortex shedding iseffectively broadband Therefore, the term “wake excitation” is used to include all forms ofexcitation associated with the wake and not just those associated with the critical wind velocity

2 The “incident turbulence” mechanism refers to the situation where the turbulence properties ofthe natural wind give rise to changing wind speeds and directions that directly induce varyinglift and drag forces and pitching moments on the structure over a wide band of frequencies.The ability of incident turbulence to produce significant contributions to crosswind responsedepends very much on the ability to generate a crosswind (lift) force on the structure as afunction of longitudinal wind speed and angle of attack In general, this means that sectionswith a high lift curve slope or pitching moment curve slope, such as a streamlined bridge decksection or a flat deck roof, are possible candidates for this effect

3 Higher derivatives of crosswind displacement: there are three commonly recognizeddisplacement-dependent excitations (i.e., “galloping,” “flutter,” and “lock-in”), all of whichare also dependent on the effects of turbulence (turbulence affects the wake development, andhence, the aerodynamic derivatives) Many formulae are available to calculate these effects(Holmes, 2001) Recently, computational fluid dynamics techniques have also been used(Tamura, 1999) to evaluate these effects

13.3.5 Wind Tunnel Tests

There are many situations in which analytical methods cannot be used to estimate certain types of windloads and the associated structural response For example, when the aerodynamic shape of the building israther uncommon, or the building is very flexible so that its motion affects the aerodynamic forces acting

on the building In such situations, more accurate estimates of wind effects on buildings are obtainedthrough aeroelastic model tests in a boundary-layer wind tunnel

Wind tunnel tests currently being conducted on buildings and other structures can be divided into twotypes The first is concerned with the determination of wind-loading effects to enable the design of awind-resistant structure The second is concerned with the flow fields induced around the structure, such

as its effects on pedestrian comfort and safety at ground level or air intake concentration levels of exhaustpollutants

Wind tunnel studies involve blowing wind on the subject building model and its surrounding atvarious angles relative to the building orientation, representing the wind directions This is typicallyachieved by placing the complete model on a rotating platform within the wind tunnel Once testing is

FIGURE 13.15 Vortex formation in the wake of a bluff object.