Wind Effects on Structures: Modern Structural Design for Wind

Dedication

For Devra, SueYeun,

Zohar,Nitzan, Abigail, and Arin

Preface to the Fourth Edition

The quarter of a century that elapsed since the publication of the third edition of Wind Effects on Structures has seen a number of significant developments in micrometeorology, extreme wind climatology, aerodynamic pressure measurement technology, uncertainty quantification, the optimal integration of wind and structural engineering tasks, and the use of big data for determining and combining effectively multiple directionality‐dependent time series of wind effects of interest. Also, following a 2004 landmark report by Skidmore Owings and Merrill LLP on large differences between independent estimates of wind effects on the World Trade Center towers, it has increasingly been recognized that transparency and traceability are essential to the credibility of structural designs for wind. A main objective of the fourth edition of Wind Effects on Structures is to reflect these developments and their consequences from a design viewpoint. Progress in the developing Computational Wind Engineering field is also reflected in the book.

Modern pressure measurements by scanners, and the recording and use of aerodynamic pressure time series, have brought about a significant shift in the division of tasks between wind and structural engineers. In particular, the practice of splitting the dynamic analysis task between wind and structural engineers has become obsolete; performing dynamic analyses is henceforth a task assigned exclusively to the structural engineering analyst, as has long been the case in seismic design. This eliminates the unwieldy, time‐consuming back‐and‐forth between wind and structural engineers, which typically discourages the beneficial practice of iterative design. The book provides the full details of the wind and structural engineers' tasks in the design process, and up‐to‐date, user‐friendly software developed for practical use in structural design offices. In addition, new material in the book concerns the determination of wind load factors, or of design mean recurrence intervals of wind effects, determined by accounting for wind directionality.

The first author contributed Chapters 1–3; portions of Chapter 4; Chapters 5, 7, and 8; Sections 9.1 and 9.3; Chapters 10–12 and 15; portions of Chapter 17 and Part III; Part IV; and Appendices A, B, D, and E. The second author contributed Chapter 6; Section 9.2; and Section 23.5. The authors jointly contributed Chapters 13, 14, 16, and 18. They reviewed and are responsible for the entire book. Professor Robert H. Scanlan contributed parts of Chapter 4 and of Part III. Appendix F, authored by Skidmore Owings and Merrill LLP, is part of the National Institute of Standards and Technology World Trade Center investigation. Chapter 17 is based on a doctoral thesis by Dr. F. Habte supervised by the first author and Professor A. Gan Chowdhury. Dr. Sejun Park made major contributions to Chapters 14 and 18 and developed the attendant software. Appendix C is based on a paper by A. L. Pintar, D. Duthinh, and E. Simiu.

We wish to pay a warm tribute to the memory of Professor Robert H. Scanlan (1914–2001) and Dr. Richard D. Marshall (1934–2001), whose contributions to aeroelasticity and building aerodynamics have profoundly influenced these fields. The authors have learned much over the years from Dr. Nicholas Isyumov's work, an example of competence and integrity. We are grateful to Professor B. Blocken of the Eindhoven University of Technology and KU Leuven, Dr. A. Ricci of the Eindhoven University of Technology, and Dr. T. Nandi of the National Institute of Standards and Technology for their thorough and most helpful reviews of Chapter 6. We thank Professor D. Zuo of Texas Tech University for useful comments on cable‐stayed‐bridge cable vibrations. We are indebted to many other colleagues and institutions for their permission to reproduce materials included in the book.

The references to the authors' National Institute of Standards and Technology affiliation are for purposes of identification only. The book is not a U.S. Government publication, and the views expressed therein do not necessarily represent those of the U.S. Government or any of its agencies.

Rockville, Maryland

Emil Simiu

DongHun Yeo

Introduction

The design of buildings and structures for wind depends upon the wind environment, the aerodynamic effects induced by the wind environment in the structural system, the response of the structural system to those effects, and safety requirements based on uncertainty analyses and expressed in terms of wind load factors or design mean recurrence intervals of the response. For certain types of flexible structure (slender structures, suspended‐span bridges) aeroelastic effects must be considered in design.

1.1 The Wind Environment and Its Aerodynamic Effects

For structural design purposes the wind environment must be described: (i) in meteorological terms, by specifying the type or types of storm in the region of interest (e.g., large‐scale extratropical storms, hurricanes, thunderstorms, tornadoes); (ii) in micrometeorological terms (i.e., dependence of wind speeds upon averaging time, dependence of wind speeds and turbulent flow fluctuations on surface roughness and height above the surface); and in extreme wind climatological terms (directional extreme wind speed data at the structure's site, probabilistic modeling based on such data). Such descriptions are provided in Chapters 1–3, respectively.

The description of the wind flows' micrometeorological features is needed for three main reasons. First, those features directly affect the structure's aerodynamic and dynamic response. For example, the fact that wind speeds increase with height above the surface means that wind loads are larger at higher elevations than near the ground. Second, turbulent flow fluctuations strongly influence aerodynamic pressures, and produce in flexible structures fluctuating motions that may be amplified by resonance effects. Third, micrometeorological considerations are required to transform measured or simulated wind speed data at meteorological stations or other reference sites into wind speed data at the site of interest.

Micrometeorological features are explicitly considered by the structural designer if wind pressures or forces acting on the structure are determined by formulas specified in code provisions. However, for designs based on wind‐tunnel testing this is no longer the case. Rather, the structural designer makes use of records of non‐dimensional aerodynamic pressure data and of measured or simulated directional extreme wind speeds at the site of interest, in the development of which micrometeorological features were taken into account by the wind engineer and are implicit in those records. However, the integrity of the design process requires that the relevant micrometeorological features on which those records are based be fully documented and accounted for.

To perform a design based on aerodynamic data obtained in wind‐tunnel tests (or in numerical simulations) the structural engineer needs the following three products:

Time series of pressures at large numbers of taps, non‐dimensionalized with respect to the wind tunnel (or numerical simulation) mean wind speed at the reference height (commonly the elevation of the building roof) (Chapters 4–6).

Matrices of directional mean wind speeds at the site of interest, at the prototype reference height.

Estimates of uncertainties in items (1) and (2) (Chapter 7).

These products, and the supporting documentation consistent with Building Information Modeling (BIM) requirements to allow effective scrutiny, must be delivered by the wind engineering laboratory to the structural engineer in charge of the design. The wind engineer's involvement in the structural design process ends once those products are delivered. The design is then fully controlled by the structural engineer. In particular, as was noted in the Preface, dynamic analyses need no longer be performed partly by the structural engineer and partly by the wind engineer, but are performed solely, and more effectively, by the structural engineer. This eliminates unwieldy, time‐consuming back‐and‐forth between the wind engineering laboratory and the structural design office, which typically discourages the beneficial practice of iterative design. Chapters 1–7 constitute Part I of the book.

1.2 Structural Response to Aerodynamic Excitation

The structural designer uses software that transforms the wind engineering data into applied aerodynamic loads. This transformation entails simple weighted summations performed automatically by using a software subroutine. Given a preliminary design, the structural engineer performs the requisite dynamic analyses to obtain the inertial forces produced by the applied aerodynamic loads. The effective wind loads (i.e., the sums of applied aerodynamic and inertial loads) are then used to calculate demand‐to‐capacity indexes (DCIs), inter‐story drift, and building accelerations with specified mean recurrence intervals. This is achieved by accounting rigorously and transparently for (i) directionality effects, (ii) combinations of gravity effects and wind effects along the principal axes of the structure and in torsion, and (iii) combinations of weighted bending moments and axial forces inherent in DCI expressions. Typically, to yield a satisfactory design (e.g., one in which the DCIs are not significantly different from unity), successive iterations are required. All iterations use the same applied aerodynamic loads but different structural members sizes. Part II of the book presents details on of the operations just described, software for performing them, and examples of its use supported by a detailed user's manual and a tutorial. Also included in Part II is a critique of the high‐frequency force balance technique, commonly used in wind engineering laboratories before the development of multi‐channel pressure scanners, material on wind‐induced discomfort in and around buildings, tuned mass dampers, and requisite wind load factors and design mean recurrence intervals of wind effects.

Part III presents fundamentals and applications related to aeroelastic phenomena: vortex‐induced vibrations, galloping, torsional divergence, flutter, and aeroelastic response of slender towers, chimneys and suspended‐span bridges. Part IV contains material on trussed frameworks and plate girders, offshore structures, tensile membrane structures, tornado wind and atmospheric pressure change effects, and tornado‐ and hurricane‐borne missile speeds.

Appendices A–E present elements of probability and statistics, elements of the theory of random processes, the description of a modern peaks‐over‐threshold procedure that yields estimates of stationary time series peaks and confidence bounds for those estimates, elements of structural dynamics based on a frequency‐domain approach still used in suspended‐span bridge applications, and elements of structural reliability that provide an engineering perspective on the extent to which the theory is, or is not, useful in practice. The final Appendix F is a highly instructive Skidmore Owings and Merrill report on the estimation of the World Trade Center towers response to wind loads.

Part I

Atmospheric Flows, ExtremeWind Speeds, Bluff Body Aerodynamics

1

Atmospheric Circulations

Wind, or the motion of air with respect to the surface of the Earth, is fundamentally caused by variable solar heating of the Earth's atmosphere. It is initiated, in a more immediate sense, by differences of pressure between points of equal elevation. Such differences may be brought about by thermodynamic and mechanical phenomena that occur in the atmosphere both in time and space.

The energy required for the occurrence of these phenomena is provided by the sun in the form of radiated heat. While the sun is the original source, the source of energy most directly influential upon the atmosphere is the surface of the Earth. Indeed, the atmosphere is to a large extent transparent to the solar radiation incident upon the Earth, much in the same way as the glass roof of a greenhouse. That portion of the solar radiation that is not reflected or scattered back into space may therefore be assumed to be absorbed entirely by the Earth. The Earth, upon being heated, will emit energy in the form of terrestrial radiation, the characteristic wavelengths of which are long (in the order of 10 μ) compared to those of heat radiated by the sun. The atmosphere, which is largely transparent to solar but not to terrestrial radiation, absorbs the heat radiated by the Earth and re‐emits some of it toward the ground.

1.1 Atmospheric Thermodynamics

1.1.1 Temperature of the Atmosphere

To illustrate the role of the temperature distribution in the atmosphere in the production of winds, a simplified version of model circulation will be presented. In this model the vertical variation of air temperature, of the humidity of the air, of the rotation of the Earth, and of friction are ignored, and the surface of the Earth is assumed to be uniform and smooth.

The axis of rotation of the Earth is inclined at approximately 66° 30′ to the plane of its orbit around the sun. Therefore, the average annual intensity of solar radiation and, consequently, the intensity of terrestrial radiation, is higher in the equatorial than in the polar regions. To explain the circulation pattern as a result of this temperature difference, Humphreys [1] proposed the following ideal experiment (Figure 1.1).

Figure 1.1 Circulation pattern due to temperature difference between two columns of fluid.

Source: From Ref. [1]. Copyright 1929, 1940 by W. J. Humphreys.

Assume that the tanks A and B are filled with fluid of uniform temperature up to level a, and that tubes 1 and 2 are closed. If the temperature of the fluid in A is raised while the temperature in B is maintained constant, the fluid in A will expand and reach the level b. The expansion entails no change in the total weight of the fluid contained in A. The pressure at c therefore remains unchanged, and if tube 2 were opened, there would be no flow between A and B. If tube 1 is opened, however, fluid will flow from A to B, on account of the difference of head (b – a). Consequently, at level c the pressure in A will decrease, while the pressure in B will increase. Upon opening tube 2, fluid will now flow through it from B to A. The circulation thus developed will continue as long as the temperature difference between A and B is maintained.

If tanks A and B are replaced conceptually by the column of air above the equator and above the pole, in the absence of other effects an atmospheric circulation will develop that could be represented as in Figure 1.2. In reality, the circulation of the atmosphere is vastly complicated by the factors neglected in this model. The effect of these factors will be discussed later in this chapter.

Figure 1.2 Simplified model of atmospheric circulation.

The temperature of the atmosphere is determined by the following processes:

Solar and terrestrial radiation, as discussed previously

Radiation in the atmosphere

Compression or expansion of the air

Molecular and eddy conduction

Evaporation and condensation of water vapor.

1.1.2 Radiation in the Atmosphere

As a conceptual aid, consider the action of the following model. The heat radiated by the surface of the Earth is absorbed by the layer of air immediately above the ground (or the surface of the ocean) and reradiated by this layer in two parts, one going downward and one going upward. The latter is absorbed by the next higher layer of air and again reradiated downward and upward. The transport of heat through radiation in the atmosphere, according to this conceptual model, is represented in Figure 1.3.

Figure 1.3 Transport of heat through radiation in the atmosphere.

1.1.3 Compression and Expansion. Atmospheric Stratification

Atmospheric pressure is produced by the weight of the overlying air. A small mass (or particle) of dry air moving vertically thus experiences a change of pressure to which there corresponds a change of temperature in accordance with the Poisson dry adiabatic equation

1.1 equation

A familiar example of the effect of pressure on the temperature is the heating of compressed air in tire pump.

If, in the atmosphere, the vertical motion of an air particle is sufficiently rapid, the heat exchange of that parcel with its environment may be considered to be negligible, that is, the process being considered is adiabatic. It then follows from Poisson's equation that since ascending air experiences a pressure decrease, its temperature will also decrease. The temperature drop of adiabatically ascending dry air is known as the dry adiabatic lapse rate and is approximately 1°C/100 m in the Earth's atmosphere.

Consider a small mass of dry air at position 1 (Figure 1.4). Its elevation and temperature are denoted by h1 and T1, respectively. If the particle moves vertically upward sufficiently rapidly, its temperature change will effectively be adiabatic, regardless of the lapse rate (temperature variation with height above ground) prevailing in the atmosphere. At position 2, while the temperature of the ambient air is T2, the temperature of the element of air mass is = T1 – (h2 – h1) γa, where γa is the adiabatic lapse rate. Since the pressure of the element and of the ambient air will be the same, it follows from the equation of state that to the difference − T2 there corresponds a difference of density between the element of air and the ambient air. This generates a buoyancy force that, if T2 < , acts upwards and thus moves the element farther away from its initial position (superadiabatic lapse rate, as in Figure 1.4), or, if T2 > , acts downwards, thus tending to return the particle to its initial position. The stratification of the atmosphere is said to be unstable in the first case and stable in the second. If T2 = , that is, if the lapse rate prevailing in the atmosphere is adiabatic, the stratification is said to be neutral. A simple example of the stable stratification of fluids is provided by a layer of water underlying a layer of oil, while the opposite (unstable) case would have the water above the oil.

Figure 1.4 Lapse rates.

1.1.4 Molecular and Eddy Conduction

Molecular conduction is a diffusion process that effects a transfer of heat. It is achieved through the motion of individual molecules and is negligible in atmospheric processes. Eddy heat conduction involves the transfer of heat by actual movement of air in which heat is stored.

1.1.5 Condensation of Water Vapor

In the case of unsaturated moist air, as an element of air ascends and its temperature decreases, at an elevation where the temperature is sufficiently low condensation will occur and heat of condensation will be released. This is equal to the heat originally required to change the phase of water from liquid to vapor, that is, the latent heat of vaporization stored in the vapor. The temperature drop in the saturated adiabatically ascending element is therefore slower than for dry air or moist unsaturated air.

1.2 Atmospheric Hydrodynamics

The motion of an elementary air mass is determined by forces that include a vertical buoyancy force. Depending upon the temperature difference between the air mass and the ambient air, the buoyancy force acts upwards (causing an updraft), downwards, or is zero. These three cases correspond to unstable, stable, or neutral atmospheric stratification, respectively. It is shown in Section 2.3.3 that, depending upon the absence or a presence of a stably stratified air layer above the top of the atmospheric boundary layer, called capping inversion, neutrally stratified flows can be classified into truly and conventionally neutral flows.

The horizontal motion of air is determined by the following forces:

The horizontal pressure gradient force per unit of mass, which is due to the spatial variation of the horizontal pressures. This force is normal to the lines of constant pressure, called isobars, that is, it is directed from high‐pressure to low‐pressure regions (Figure 1.5). Let the unit vector normal to the isobars be denoted by n, and consider an elemental volume of air with dimensions dn, dy, dz, where the coordinates n, y, z are mutually orthogonal. The net force per unit mass exerted by the horizontal pressure gradient along the direction of the vector n is

1.2 equation

where p denotes the pressure, and ρ is the air density.

The deviating force due to the Earth's rotation. If defined with respect to an absolute frame of reference, the motion of a particle not subjected to the action of an external force will follow a straight line. To an observer on the rotating Earth, however, the path described by the particle will appear curved. The deviation of the particle with respect to a straight line fixed with respect to the rotating Earth may be attributed to an apparent force, the Coriolis force

1.3 equation

where m is the mass of the particle, f = 2ω sin ϕ is the Coriolis parameter, ω = 0.7292 × 10−4 s−1 is the angular velocity vector of the Earth, ϕ is the angle of latitude, and v is the velocity vector of the particle referenced to a coordinate system fixed with respect to the Earth. The force Fc is normal to the direction of the particle's motion, and is directed according to the vector multiplication rule.

The friction force. The surface of the Earth exerts upon the moving air a horizontal drag force that retards the flow. This force decreases with height and becomes negligible above a height δ known as gradient height. The atmospheric layer between the Earth's surface and the gradient height is called the atmospheric boundary layer (see Chapter 2). The wind velocity speed at height δ is called the gradient velocity,¹ and the atmosphere above this height is called the free atmosphere (Figure 1.6).

Figure 1.5 Direction of pressure gradient force.

Figure 1.6 The atmospheric boundary layer.

In the free atmosphere an elementary mass of air will initially move in the direction of the pressure gradient force – the driving force for the air motion − in a direction normal to the isobar. The Coriolis force will be normal to that incipient motion, that is, it will be tangent to the isobar. The resultant of these two forces, and the consequent motion of the particle, will no longer be normal to the isobar, so the Coriolis force, which is perpendicular to the particle motion, will change direction, and will therefore no longer be directed along the isobar. The change in the direction of motion will continue until the particle will move steadily along the isobar, at which point the Coriolis force will be in equilibrium with the pressure gradient force, as shown in Figure 1.7.

Figure 1.7 Frictionless wind balance in geostrophic flow.

Within the atmospheric boundary layer the direction of the friction force, denoted by S, coincides with the direction of motion of the particle. During the particle's steady motion the resultant of the mutually orthogonal Coriolis and friction forces will balance the pressure gradient force, that is, will be normal to the isobars, meaning that the friction force – and therefore the motion of the particle − will cross the isobars (Figure 1.8). Since the friction force, which retards the wind flow and vanishes at the gradient height, decreases as the height above the surface increases, the velocity increases with height (Figure 1.6). The Coriolis force, which is proportional to the velocity, also increases with height. The combined effect of the Coriolis and friction forces causes the angle between the isobars and the direction of motion within the ABL, shown as α0 in Figures 1.8 and 1.9, to increase from zero at the gradient height to its largest value at the Earth's surface. The wind velocity in the boundary layer can therefore be represented by a spiral, as in Figure 1.9. Under certain simplifying assumption regarding the effective flow viscosity the spiral is called the Ekman spiral (see Section 2.3.1).

Figure 1.8 Balance of forces in the atmospheric boundary layer.

Figure 1.9 Wind velocity spiral in the atmospheric boundary layer.

If the isobars are curved, the horizontal pressure gradient force as well as the centrifugal force associated with the motion on a curved path will act on the elementary mass of air in the direction normal to the isobars, and the resultant steady wind will again flow along the isobars. Its velocity results from the relations

1.4 equation

where r is the radius of curvature of the air trajectory. If the mass of air is in the Northern Hemisphere, the positive or the negative sign is used according as the circulation is cyclonic (around a center of low pressure) or anticyclonic (around a center of high pressure).

1.3 Windstorms

1.3.1 Large‐Scale Storms

Large‐scale wind flow fields of interest in structural engineering may be divided into two main types of storm: extratropical (synoptic) storms, and tropical cyclones. Synoptic storms occur at and above mid‐latitudes. Because their vortex structure is less well defined than in tropical storms, their winds are loosely called straight winds.

Tropical cyclones, known as typhoons in the Far East, and cyclones in Australia and the Indian Ocean, generally originate between 5° and 20° latitudes. Hurricanes are defined as tropical cyclones with sustained surface wind speeds of 74 mph or larger. Tropical cyclones are translating vortices with diameters of hundreds of miles and counterclockwise (clockwise) rotation in the Northern (Southern) hemisphere. Their translation speeds vary from about 3–30 mph. As in a stirred coffee cup, the column of fluid is lower at the center than at the edges. The difference between edge and center atmospheric pressures is called pressure defect. Rotational speeds increase as the pressure defect increases, and as the radius of maximum wind speeds, which varies from 5 to 60 miles, decreases.

The structure and flow pattern of a typical tropical cyclone is shown in Figure 1.10. The eye of the storm (Region I) is a roughly circular, relatively dry core of calm or light winds surrounded by the eye wall. Region II contains the storm's most powerful winds. Far enough from the eye, winds in Region V, which decrease in intensity as the distance from the center increases, are parallel to the surface. Where Regions V and II intersect the wind speed has a strong updraft component that alters the mean wind speed profile and is currently not accounted for in structural engineering practice. The source of energy that drives the storm winds is the warm water at the ocean surface. As the storm makes landfall and continues its path over land, its energy is depleted and its wind speeds gradually decrease. Figure 1.11 shows a satellite image of Hurricane Irma. In the United States hurricanes are classified in accordance with the Saffir–Simpson scale (Table 1.1).²

Figure 1.10 Structure of a hurricane.

Figure 1.11 Satellite view of hurricane Irma.

Source: National Oceanic and Atmospheric Administration photo.

Table 1.1 Saffir‐Simpson scale and corresponding wind speedsa.

aFor the definition of 1‐minute and 3‐second wind speeds see Section 2.1. Official speeds are in mph.

1.3.2 Local Storms

Foehn winds (called chinook winds in the Rocky Mountains area) develop downwind of mountain ridges. Cooling of air as it is pushed upwards on the windward side of a mountain ridge causes condensation and precipitation. The dry air flowing past the crest warms as it is forced to descend, and is highly turbulent (Figure 1.12). A similar type of wind is the bora, which occurs downwind of a plateau separated by a steep slope from a warm plain.

Figure 1.12 Foehn wind.

Jet effect winds are produced by features such as gorges.

Thunderstorms occur as heavy rain drops, due to condensation of water vapor contained in ascending warm, moist air, drag down the air through which they fall, causing a downdraft that spreads on the earth's surface (Figure 1.13). The edge of the spreading cool air is the gust front. If the wind behind the gust front is strong, it is called a downburst. Notable features of downbursts are the typical difference between the profiles of their peak gusts near the ground and those of large‐scale storms, and the differences among the time histories of various thunderstorms [3] (Figure 1.14). According to [5], the maximum winds (i.e., design level winds) rarely occur at the locations where profiles differ markedly from the logarithmic law.

Figure 1.13 Section through a thunderstorm in the mature stage.

Figure 1.14 Time histories of eight thunderstorm events.

Source: Reprinted from Ref. [3], with permission from Elsevier.

Microbursts were defined by Fujita [4] as slow‐rotating small‐diameter columns of descending air which, upon reaching the ground, burst out violently (Figure 1.15). A number of fatal aircraft accidents have been caused by microbursts. According to [5], because of the higher frequency and large individual area of a microburst, probabilities of structural damage by microbursts with 50–100 mph wind speeds could be much higher than those of tornadoes.

Figure 1.15 Andrews Air Force Base microburst on 1 August 1 1983. Its 149.7 mph peak speed was the highest recorded in a microburst in the U.S [4].

Tornadoes are small vortex‐like storms, and can contain winds in excess of 100 m s−1 (Figure 1.16) [6, 7].

Figure 1.16 Tornado funnel (

Source: National Oceanic and Atmospheric Administration photo).

For unvented or partially unvented structures, the difference between atmospheric pressure at the tornado periphery and the tornado center (i.e., the pressure defect) typical of cyclostrophic storms is a significant design factor. For such structures, the difference between the larger atmospheric pressure that persists inside the structure and the lower atmospheric pressure acting on the structure during the tornado passage results in large, potentially destructive net pressures that must be accounted for in design (see Chapter 27).

The National Weather Service and the U.S. Nuclear Regulatory Commission are currently classifying tornado intensities in accordance with the Enhanced Fujita Scale (EF‐scale), agreed upon in a forum organized by Texas Tech University in 2001. The EF‐scale, shown in Table 1.2, replaced the original Fujita scale following a consensus opinion that the latter overestimated tornado wind speeds (see, e.g., [8]). The EF scale is based on the highest 3‐second wind speed estimated to have occurred during the tornado's life, and is shown in Table 1.2.

Table 1.2 Tornado enhanced Fujita Scale.

As noted in [9], no tornado has been assigned an intensity of EF6 or greater, and there is some question whether an EF6 or greater tornado would be identified if it did occur. For tornadoes that occur in areas containing no objects capable of resisting events with intensity EF0 (e.g., in a corn field), no intensity estimate is possible. An additional difficulty is that intensity estimates depend upon quality of construction. Since there are no measurements of tornado speeds at heights above ground comparable to typical building heights, it is necessary to rely on largely subjective estimates, based primarily on observations of damage.

For additional material on tornadoes, see Sections 3.4 and 5.3, and Chapters 27 and 28.

References

1 Humphreys, W.J. (1940). Physics of the Air. New York: McGraw‐Hill.

2 ASCE, Minimum design loads for buildings and other structures (ASCE/SEI 7‐16), in ASCE Standard ASCE/SEI 7‐16, Reston, VA: American Society of Civil Engineers, 2016.

3 Lombardo, F.T., Smith, D.A., Schroeder, J.L., and Mehta, K.C. (2014). Journal of Wind Engineering and Industrial Aerodynamics 125: 121–132. http://dx.doi.org/10.1016/j.jweia.2013.12.004.

4 Fujita, T.T. (1990). Downbursts: meteorological features and wind field characteristics. Journal of Wind Engineering and Industrial Aerodynamics 36: 75–86.

5 Schroeder, J. L., Personal communication, Nov. 21, 2016.

6 Lewellen, D.C., Lewellen, W.S., and Xia, J. (2000). The influence of a local swirl ratio on tornado intensification near the surface. Journal of the Atmospheric Sciences 57: 527–544.

7 Hashemi Tari, P., Gurka, R., and Hangan, H. (2010). Experimental investigation of tornado‐like vortex dynamics with swirl ratio: the mean and turbulent flow fields. Journal of Wind Engineering and Industrial Aerodynamics 98: 936–944.

8 Phan, L. T. and Simiu, E., Tornado aftermath: Questioning the tools, Civil Engineering, December 1998, 0885‐7024‐/98‐0012‐002A. https://www.nist.gov/wind.

9 Ramsdell, J. V. Jr. and Rishel, J. P., Tornado Climatology of the Contiguous United States, A. J. Buslik, Project Manager, NUREG/CR‐4461, Rev. 2, PNNL‐15112, Rev. 1, Pacific Northwest National Laboratory, 2007.

Notes

1For straight winds (i.e., winds whose isobars are approximately straight), the term geostrophic is substituted in the meteorological literature for gradient.

2See Commentary, ASCE 7‐16 Standard [2].

2

The Atmospheric Boundary Layer

As indicated in Chapter 1, the Earth's surface exerts on the moving air a horizontal drag force whose effect is to retard the flow. This effect is diffused by turbulent mixing throughout a region called the atmospheric boundary layer (ABL). In strong winds the depth of the ABL ranges from a few hundred meters to a few kilometers, depending upon wind speed, roughness of terrain, angle of latitude, and the degree to which the stratification of the free flow (i.e., the flow above the ABL) is stable. Within the ABL the mean wind speed varies as a function of elevation.

This chapter is devoted to studying aspects of ABL flow of interest from a structural engineering viewpoint. Section 2.1 is concerned with the dependence of the wind speed on averaging time. Section 2.2 presents the equations of mean motion in the ABL. Sections 2.3 and 2.4 pertain to horizontally homogeneous flows over flat uniform surfaces, and contain, respectively, theoretical as well as empirical results on the dependence of wind speeds on height above the Earth's surface, and the structure of atmospheric turbulence. Section 2.5 concerns horizontally non‐homogeneous flows (i.e., flows affected by changes of surface roughness or by topographic features, and flows in tropical storms and thunderstorms). Since the structural engineer is concerned primarily with the effect of strong winds, it will be assumed that the ABL flow is neutrally stratified. Indeed, in strong winds turbulent transport dominates the heat convection by far, so that thorough turbulent mixing tends to produce neutral stratification, just as in a shallow layer of incompressible fluid mixing tends to produce an isothermal state. In flows of interest in structural engineering, a layer of stably stratified flow, called the capping inversion, is present above the ABL and significantly affects the ABL's height.

2.1 Wind Speeds and Averaging Times

If the flow were laminar wind speeds would be the same for all averaging times. However, owing to turbulent fluctuations, such as those recorded in Figure 2.1, the definition of wind speeds depends on averaging time.

Figure 2.1 Wind speed record.

The peak 3‐second gust speed is the peak of a storm's speeds averaged over 3 seconds. In 1995 it was adopted in the ASCE Standard as a measure of wind speeds. Similarly, the peak 5‐second gust speed is the largest speed averaged over 5 seconds. The 5‐second speed is reported by the National Weather Service ASOS (Automated Service Observing System), and is about 2% less than the 3‐second speed. The 28‐mph peak of Figure 2.1 is, approximately, a 3‐second speed.

The hourly wind speed is the speed averaged over 1 hour. It is commonly used as a reference wind speed in wind tunnel simulations. Hence the need to estimate the hourly speed corresponding to a 3‐second (or a 1‐minute, or a 10‐minute) speed specified for design purposes or recorded at weather stations. In Figure 2.1 the statistical features of the record do not vary significantly (i.e., the record may be viewed as statistically stationary, see Appendix B) over an interval of almost two hours; the hourly wind speed is about 18.5 mph, or about 1/1.52 times the peak 3‐second gust.

Sustained wind speeds, defined as wind speeds averaged over intervals in the order of 1 min, are used in both engineering and meteorological practice. The fastest 1‐minute wind speed or, for short, the 1‐minute speed, is the storm's largest 1‐minute average wind speed. The fastest‐mile wind speed Uf is the storm's largest speed in mph averaged over a time interval tf = 3600/Uf. For example, a 60 mph fastest‐mile wind speed is averaged over a 60‐second time interval.

Ten‐minute wind speeds are wind speeds averaged over 10 min, and are used in World Meteorological Organization (WMO) practice as well as in some standards and codes.

The ratio between the peak gust speed and the mean wind speed is called the gust factor. Expressions for the relation between wind speeds with different averaging times are provided in Section 2.3.7 as functions of parameters defined subsequently in this chapter.

2.2 Equations of Mean Motion in the ABL

The motion of the atmosphere is governed by the fundamental equations of continuum mechanics, which include the equation of continuity – a consequence of the principle of mass conservation, – and the equations of balance of momenta, that is, the Navier–Stokes equations (see also Chapters 4 and 06). These equations must be supplemented by phenomenological relations, that is, empirical relations that describe the specific response to external effects of the medium being considered. (For example, in the case of a linearly elastic material the phenomenological relations consist of the so‐called Hooke's law.)

If the equations of continuity and the equations of balance of momenta are averaged with respect to time, and if terms that can be shown to be negligible are dropped, the following equations describing the mean motion in the boundary layer of the atmosphere are obtained:

2.1

2.2

2.3 equation

2.4 equation

where U, V, and W are the mean velocity components along the axes x, y, and z of a Cartesian system of coordinates whose z‐axis is vertical; p, ρ, f, and g are the mean pressure, the air density, the Coriolis parameter, and the acceleration of gravity, respectively; and τu, τv are shear stresses in the x and y directions, respectively. The x‐axis is selected, for convenience, to coincide with the direction of the shear stress at the surface, denoted by τ0 (Figure 2.2).

Figure 2.2 Coordinate axes.

It can be seen, by differentiating Eq. (2.3) with respect to x or y, that the vertical variation of the horizontal pressure gradient depends upon the horizontal density gradient. For the purposes of this text it will be sufficient to consider only flows in which the horizontal density gradient is negligible. The horizontal pressure gradient is then invariant with height and thus has, throughout the boundary layer, the same magnitude as at the boundary layer's top:

2.5 equation

where Vgr is the gradient velocity, r is the radius of curvature of the isobars, and n is the direction of the gradient wind (see Eq. [1.4]).

The geostrophic approximation corresponds to the case where the curvature of the isobars can be neglected. The gradient velocity is then called the geostrophic velocity and is denoted by G. Eq. (2.5) then becomes

(2.6a,b)2.6a,b equation

where Ug and Vg are the components of the geostrophic velocity G along the x‐ and y‐axes.

The boundary conditions for Eqs. (2.1)–(2.4) may be stated as follows: at the ground surface the velocity vanishes, while at the top of the ABL the shear stresses vanish and the wind flows with the gradient velocity Vgr. In addition, an interaction between the ABL and the capping inversion occurs (see Section 2.3.3).

2.3 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces

It may be assumed that in large‐scale non‐tropical storms, within a flat site of uniform surface roughness with sufficiently long fetch, a region exists over which the flow is horizontally homogeneous. The existence of horizontally homogeneous atmospheric flows is supported by observations and distinguishes ABLs from two‐dimensional boundary layers such as occur along flat plates. In the latter case the flow in the boundary layer is decelerated by the horizontal stresses, so that the boundary‐layer thickness grows as shown in Figure 2.3 [1]. In atmospheric boundary layers. In atmospheric boundary layers, however, the horizontal pressure gradient – which below the free atmosphere is only partly balanced by the Coriolis force (Figure 1.8) – re‐energizes the flow and counteracts the boundary‐layer growth. Horizontal homogeneity of the flow is thus maintained [2].

Figure 2.3 Growth of a two‐dimensional boundary layer along a flat plate.

Under equilibrium conditions, in horizontally homogeneous flow Eqs. (2.1) and (2.2), in which Eq. (2.6a,b) are used, become

(2.7a,b)2.7a,b equation

The Ekman spiral was the first attempt to describe the ABL in mathematical terms, and is presented in Section 2.3.1 for the sake of its historical interest. In the 1960s and 1970s a major advance was achieved in the field of boundary‐layer meteorology, based on an asymptotic approach. As shown in Section 2.3.2, the asymptotic approach yields the unphysical result that the mean speed component V vanishes throughout the boundary layer's depth, except at its top, where it has the value Vg. In addition, the 1960s and 1970s work did not consider the important effect of the capping inversion on the ABL height. Section 2.3.3 introduces the contemporary classification of neutrally stratified ABLs as functions of the Brunt‐Väisäla frequency. The latter characterizes the interaction between the ABL and the capping inversion, and provides expressions for the height of the ABL that account for that interaction. Section 2.3.4 presents the logarithmic description of the mean wind speed within the lower layer of the ABL, called the surface layer, as well as estimates of the surface layer's depth. Section 2.3.5 presents the power law representation of the wind speed profile which, though obsolete, is still being used in some codes and standards, including the ASCE 7‐16 Standard [3]. Section 2.3.6 discusses the relation between characteristics of the ABL flows in different surface roughness regimes. Section 2.3.7 provides details on the relation between wind speeds with different averaging times.

2.3.1 The Ekman Spiral

The Ekman spiral model is obtained if it is assumed in Eq. (2.7a,b) that the shear stresses are proportional to a fictitious constant K, called eddy viscosity, such that

(2.8a,b)2.8a,b equation

Equations (2.7) and (2.8) then become a system of differential equations with constant coefficients. With the boundary conditions U = V = 0 for height above the surface z = 0, and U = Ug, V = Vg for z = ∞, the solution of the system is

(2.9a,b)2.9a,b

where . Equations (2.9a,b), which describe the Ekman spiral, are represented schematically in Figure 1.9. Observations are in sharp disagreement with these equations. For example, while according to Eq. (2.9a,b) the angle α0 between the surface stress τ0 and the geostrophic wind direction is 45°, observations indicate that this angle may range approximately between approximately 5° and 30° (see Section 2.3.3). The cause of the discrepancies is the assumption, mathematically convenient but physically incorrect, that the eddy viscosity is independent of height.

2.3.2 Neutrally Stratified ABL: Asymptotic Approach

A vast literature is available on the numerical solution of the equations of motion of the fluid. A different type of approach, based on similarity and asymptotic considerations, was developed in [2]. The starting point of the asymptotic approach is the division of neutral boundary layers into two regions, a surface layer and an outer layer. In the surface layer the shear stress τ0 induced by the boundary‐layer flow at the Earth's surface must depend upon the flow velocity at a distance z from the surface, the roughness length z0 that characterizes