thesis on tall building
TRANSCRIPT
Heaven’s Light is Our Guide
RAJSHAHI UNIVERSITY OF ENGINEERING & TECHNOLOGY
DRIFT AND DYNAMIC ANALYSIS ON
TALL BUILDING DUE TO WIND LOADS
Supervised By
Prepared By
DR. SHAIKH MD NIZAMUD-DOULAH Professor Department of Civil Engineering Rajshahi University of Engineering & Technology
G.M. JAKIRULLAH NOORUDDIN ROLL NO: 030061
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ACKNOWLEDGEMENT
The work presented here was carried out under the supervision of Prof. Dr. Shaikh
Md. Nizamud-Doulah, Professor, Department of Civil Engineering, Rajshahi University of
Engineering & Technology. The author wishes to express his deep gratitude to him for his
patient guidance and affectionate encouragement from the starting till the end of the thesis.
Without his inspiration, constant guidance and invaluable suggestions at all phases, the
work could hardly be materialized.
The author wishes to convey his thanks to the teachers, friends and well wishers,
who have helped me, suggested me with a view to accomplishing the project work.
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ABSTRACT
An analytical study has been performed for the effect of wind loading on tall
buildings. This paper mainly deals with the drift and dynamic analysis on tall buildings by
the action of wind loads.
In this study a differential equation is formed and solved to determine the drift of
the building. A program is also developed with the help Visual Basic language to analysis
the drift. In this theoretical study dynamic action of tall building due to along-wind and
cross-wind phenomena is discussed.
The analytical results are presented in tabular form and as well as in graphical
form. The variation of drift and along-wind acceleration and cross-wind acceleration with
respect to building height and building width is analyzed and compared.
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CONTENTS
ACKNOWLEDGEMENT i ABSTRACT ii LIST OF FIGURES iii LIST OF TABLES iii LIST OF SYMBOLS iv
CHAPTER-1 INTRODUCTION
1.1 Introduction 1 1.2 Objectives. 2
CHAPTER -2 NATURE OF WIND
2.1 Introduction 3 2.2 Types of Wind 3 2.3 Extreme Wind Condition 4 2.4 Characteristics of Wind 4 2.5 Variation of Wind Velocity with Height 5 2.6 Turbulent nature of wind 6 2.7 Vortex Shedding Phenomenon 9 2.8 Dynamic Nature of Wind 12
CHAPTER -3 ANALYSIS FOR DRIFT DUE TO WIND LOADS
3.1 Introduction. 14 3.2 Analysis for Drift. 14 3.3 Components of Drift 15 3.3.1 Story Drift due to Girder Flexure. 17 3.3.2 Story Drift due to Column Flexure. 19 3.3.3 Total Drift 19
3.4 Derivation of the Governing Differential Equations (Coupled-shear Wall Structure).
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3.4.1 Boundary Conditions 27 3.5 Derivation of the Governing Differential Equations
(Wall- frame Structure). 29
3.5.1Solution for Uniformly Distributed Loading. 31
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CHAPTER -4 DYNAMIC RESPONSES DUE TO WIND LOADS
4.1 Introduction 33 4.2 Sensitivity of Structures to Wind Forces 33 4.3 Dynamic Structural Responses Due to Wind Forces 34 4.4 Along-wind Response 36 4.4.1 Peak Along-Wind Accelerations 38 4.5 Cross-wind Response 38
CHAPTER -5 THEORATICAL ANALYSIS & RESULTS
5.1 Drift Analysis 45 5.2 Problems on Dynamic Action 51
CHAPTER -6 CONCLUSION
5.1 Conclusion 55 5.2 Recommendations 55
References 56
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LIST OF FIGURES:
Name of Figure Page No
1. Variation of wind velocity with height. 05 2. Variation of wind velocity with time. 07 3. Schematic representation of mean wind and gust velocity 08 4. Simplified two-dimensional flow of wind. 09 5. Vortex formation in the wake of a bluff object. 11 6. Vortex shedding phenomenon 11 7. Forced and deformations caused by external shear. 15 8. Forced and deformation caused by external moment. 15 9. Deflection of portal frame. a) Frame subjected to lateral loads b) typical story segment 16
10. Lateral deflection of typical story due to bending of columns 17 11. Lateral deflection of typical story due to bending of girders 17 12. Representing of coupled shear walls by continuous model 20 13. Internal forces in coupled shear walls 21 14. Relative displacement at line of contra-flexure. 23 15. Planar wall- frame structure; (b) Continuous analogy for wall- frame
structure; (c) Free body diagrams for wall and frame 30
16. Model Structure 33 17. Interfacing of the Software 34 18. Variation of roughness factor with building height 42 19. Variation of background turbulence factor with height and aspect ratio of
building 42
20. Variation of size reduction factor with reduced frequency and aspect ratio of building 43
21. Variation of gust energy ratio with inverse wavelength. 44 22. Variation of peak factor with average fluctuation rate 44 23. Variation of Drift of Different Buildings 48 24. Variation of Drift with Respect to Building Widths 50 25. Variation of Cross-wind Acceleration with Respect to the Width at Along- wind Direction 53
26. Variation of Cross-wind Acceleration with Respect to the width at Cross- wind Direction
54
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LIST OF TABLES:
Name of the Table Page no 1. Result of drift analysis on 10-storey building 45 2. Result of drift analysis on 15-storey building 46 3. Result of drift analysis on 20-storey building 47 4. For Drift analysis at different width of the building 49 5. Result of Along-wind Acceleration & Cross-wind Acceleration 52 6. Results for the cross –wind acceleration with increase in the
width of along-wind direction. 52
7. Results for the cross –wind acceleration with increase in the width of cross-wind direction.
53
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LIST OF SYMBOLS: Symbols used in this paper shall have the following meaning-
Vz= the mean wind speed at the height Z above the ground surface
Vg= gradient wind speed assumed constant above the boundary layer
Z= height above the ground
Zg= depth of boundary layer
a=power law coefficient
Gv =the gust factor
Vg =the gust speed
V=the mean wind speed
V = Shear force
h = Height of the storey
Ig = Moment if inertia of girder
Ic = Moment of inertia of Column
L = Clear distance between two column
? g = Deflection of girder
? c = Deflection of column
? s = Storey Drift
E = Modulus of Elasticity
g = Acceleration of gravity
y= Drift
M= Moment
w= Uniformly Distributed Load
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CHAPTER-1
INTRODUCTION
1.1 Introduction
Wind loads are of important, particularly in the design of large structures. The
wind velocity that should be considered in the design of structure depends upon the
geological location and the exposure of the structure.
Wind is a phenomenon of great complexity because of the many flow situations
arising from the interaction of wind with structures. Wind is composed of a multitude of
eddies of varying sizes and rotational 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 a time period
of the order of ten minutes or more tends to increase with height, while the gustiness
tends to decrease with height.
Some structures, particularly those that are tall or slender, respond dynamically to
the effects of wind. There are several different phenomena giving rise to dynamic
response of structures in wind. These include buffeting, vortex shedding, galloping and
flutter. Slender structures are likely to be sensitive to dynamic response in line with the
wind direction as a consequence of turbulence buffeting. Transverse or cross-wind
response is more likely to arise from vortex shedding or galloping but may also result
from excitation by turbulence buffeting. Flutter is a coupled motion, often being a
combination of bending and torsion, and can result in instability. For building structures
flutter and galloping are generally not an issue.
An important problem associated with wind induced motion of buildings is
concerned with human response to vibration and perception of motion. At this point it
will suffice to note that humans are surprisingly sensitive to vibration to the extent that
motions may feel uncomfortable even if they correspond to relatively low levels of stress
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and strain. Therefore, for most tall buildings serviceability considerations govern the
design and not strength issues.
1.2 Objectives of Thesis
The main objectives of the thesis are summarized below:
1 To observe the drift analysis on high-rise structure.
2. To observe the along-wind acceleration on high-rise structure.
3. To observe the cross-wind acceleration on high-rise structure.
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CHAPTER-2
NATURE OF WIND
2.1 Introduction
Windy weather poses a variety of problems in new skyscrapers, causing concern
for building owners and engineers alike. The forces exerted by winds on buildings
increase dramatically with the increase in building heights. The velocity of wind
increases with height, and the pressure increase as the square of the velocity of wind.
Wind is the term used for air in motion and is usually applied to the natural
horizontal motion of the atmosphere. Motion in a vertical or near vertical direction is
called a current. Winds are produced by difference in atmospheric pressure, which are
primarily attributable to differences in temperature. These temperature differences are
caused largely by unequal distribution of heat from the sun, together with the difference
in thermal properties of land and ocean surfaces. When temperatures of adjacent regions
become unequal, the warmer and lighter air tends to rise and flow over the colder,
heavier air. Winds initiated in this way are usually greatly modified by the rotation of
earth. Movement of air near the surface of the earth is three-dimensional nature, with a
horizontal motion which is much greater than the vertical motion.
2.2 Types of Wind
Of the several types of wind that encompass the earth’s surface, winds which are
of interest in the design of tall buildings can be classified into three major types: the
prevailing winds, seasonal winds, and local winds.
1. The prevailing winds: Surface air moving from the horse latitudes toward the
low pressure equatorial belt constitutes the prevailing winds on trade winds.
2. The seasonal winds: The air over the land is warmer in summer and cooler in
winter than the air adjacent to oceans during the same seasons.
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3. The local winds: Corresponding with the seasonal variation in temperature and
pressure over land and water, daily changes occur which have a similar but local
effect. Similar daily changes in temperature occur over irregular terrain and cause
mountain and valley breezes.
2.3 Extreme Wind Condition
Extreme winds such as thunderstorms, hurricanes, tornadoes, and typhoons,
impose loads on structures that are many times more than those normally assumed in
their design.
2.3.1 Thunderstorms: Thunderstorms are one of the most familiar features of
temperature summer weather, characterized by long hot spells punctuated by release of
torrential rain. The essential conditions for the development of thunderstorm are warm,
moist air in the lower atmosphere and cold, dense air at higher altitudes. Wind speeds of
20 to 70 mph (9 to 31 m/s) are typically reaches in a thunderstorm and are often
accompanied with swirling wind action exerting high suction forces on roofing and
cladding elements.
2.3.2 Hurricanes: Hurricanes originate in the tropical regions of the Atlantic
Ocean or Caribbean Sea. They travel north, northwest, or northeast from their point of
origin and usually cause heavy rainfall. In a fully developed hurricane, winds reach
speeds up to 70 to 80 mph (31 to 36 m/s), and in sever hurricanes can attain velocities as
high as 200mph (90m/s).
2.3.3 Tornadoes: Tornadoes develop within severe thunderstorms and sometimes
hurricanes and consists of a rotating column of air usually accompanied by a funnel-
shaped downward extension of a dense cloud having vortex of several hundred feet,
typically 200 to 800 ft (61 to 144 m) in diameter whirling destructively at speeds up to
300 mph (134 m/s).
2.4 Characteristics of Wind
Wind is a phenomenon of great complexity because of the many flow situations
arising from the interaction of wind with structures. However, in wind engineering
simplifications are made to arrive at meaningful predictions of wind behavior by
characterizing the flow states into the following distinguishing features:
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Fig 2.1 Variation of wind velocity with height.
1. Variation of wind velocity with height
2. Turbulent nature of wind
3. Vortex shedding phenomenon
4. Dynamic nature of wind structure interaction
2.5 Variation of Wind Velocity with Height
At the interface between a moving fluid and solid surface, viscosity manifests
itself in the creation of shear forces aligned opposite to the direction of fluid motion. A
similar effect occurs between the surface of the earth and the atmosphere. Viscosity
reduces the air velocity adjacent to the earth’s surface to almost zero. A retarding effect
occurs in the layers near the ground, and these inner layers in turn successively slow
down the outer layers. The slowing down is less at each layer and eventually becomes
negligibly small. It is evident that the velocity increase which takes place along a vertical
line must be continuous from zero on the surface to a maximum at some distance away.
The height at which the velocity ceases to increase is called the gradient height, and the
corresponding velocity, the gradient velocity. The shape and size of the curve depends
less on the velocity of the air than the type and predominance of the turbulent and
random eddying motions in the wind, which in turn are effected by the type of terrain
over which the wind is blowing (Fig: 3.1). This important characteristic of variation of
wind velocity with height is a fairly well understood phenomenon and is reflected in
higher design pressures given at higher elevations in most building codes.
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The variation of velocity with height can be considered as a gradual retardation of
the wind nearer the ground as a result of surface friction. At heights of approximately
1200 ft (366 m) from the ground, the wind speed is virtually unaffected by surface
friction and its movement is solely dependent prevailing seasonal and local wind effects.
The height through which the wind speed is affected by the topography at ground level is
called the atmospheric boundary layer. The wind speed profile within this layer is in the
domain of turbulent flow. The variation of wind speed in this layer can be
mathematically predicted from a logarithmic equation. However, in engineering practice
wind profile in the atmospheric boundary layer is well represented by the so called
power law expression of the form:
Vz = Vg (Z/Zg)a …………………(2.1)
Where Vz= the mean wind speed at the height Z above the ground surface
Vg= gradient wind speed assumed constant above the boundary layer
Z= height above the ground
Zg= depth of boundary layer
a=power law coefficient
Therefore the mean wind speed at gradient height and the value of exponent a,
the wind speeds are easily calculated by using Eq. (2.1). The exponent a and the depth of
boundary layer Zg varies with terrain roughness. The value of a ranges from a law of
0.14 for open country to about 0.5 for built-up urban areas, signifying that wind speed
reaches its maximum value over a longer height in an urban terrain than in open country.
The pressure and suction on a tall building generated by wind are a function of the wind
speed, and therefore they increase with the building height.
2.6 Turbulent nature of wind
The motion of wind is turbulent. Any motion of air at speeds greater than 2 to 3
mph (0.9 to 1.3 m/s) is turbulent, causing particles of air to move in all directions.
The variation of wind velocity with height describes only one aspect of wind in
the boundary layer. Superimposed on the mean wind speed is the turbulence or gustiness
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of wind, which produces deviations in the wind speed above or below the mean,
depending upon whether there is a gust or lull in the wind action. Turbulence is created
as a result of shearing velocity gradient in viscous fluid. The layers of wind slide
relatively to one another because wind near a solid boundary has a near-zero velocity,
whereas the adjacent layers have a definite velocity giving rise to gradient distribution.
Flow of air near the earth’s surface changes in speed and direction because of the
obstacles which introduce random vertical and horizontal components at right angles to
the main direction of flow. Turbulence thus generated generally influences the wind flow
not only in the immediate vicinity, but it may persist downward from projections as
much as a hundred their height. These gusts have a random distribution over a wide
range of frequencies and amplitudes, both in time and space. Shown in Fig. 2.2 is a
schematic representation of wind speed as measured by a typical anemometer, which
clearly shows the unsteady nature of wind.
The scale and intensity of turbulence can be linked to the size and rotating speed
of the eddies or vortices that make up the turbulence. It is generally found that the size of
the flow affects the size of the turbulence within it. Thus, the flow of a large mass of air
has a larger overall turbulence than a corresponding flow of air mass of air. Because of
the randomness of its variation, the properties of wind are studied statistically. A
statistical property is the mean or the average.
For structural engineering purposes, the characteristics of the natural wind
in the atmosphere near the earth’s surface can be considered as being made up of a mean
velocity whose value increases with height in some way and on which are superimposed
Fig 2.2 Variation of wind velocity with time.
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turbulent fluctuations as velocity components along the wind direction. Figure 2.3
schematically represents the fluctuation of mean and gust velocity along the height of the
building.
Rapid bursts in the velocity of wind are called gusts. Tall buildings are sensitive
to gusts that last about one second. Therefore, the fastest mile wind is inadequate for
design of tall buildings. One must use the gust speed rather than the mean wind speed
can be obtained by multiplying the mean wind speed by a gust factor Gv.
Thus
Vg = Gv V
Where
Gv =the gust factor
Vg =the gust speed
V=the mean wind speed
Not all buildings are equally sensitive to gusts. In general, the more flexible a
structure is the more sensitive it is to gusts. The only accurate way to determine the gust
factor is to conduct a wind tunnel test.
Fig 2.3 Schematic representation of mean wind and gust velocity
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2.7 Vortex Shedding Phenomenon
The flow of wind can be considered to be two-dimensional, as shown in Fig. 3.4
Along-wind ( Normal wind): The along-wind loading or response of a
building due to buffeting by 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 a
fluctuating component due to wind speed variations from the mean. The fluctuating
wind is a random mixture of gusts or eddies of various sizes with the larger eddies
occurring less often (i.e. with a lower average frequency) than for the smaller eddies.
The natural frequency of vibration of most structures is sufficiently higher than the
component of the fluctuating load effect imposed by the larger eddies. i.e. the average
frequency with which large gusts occur is usually much less than any of the structure's
natural frequencies 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 be treated in a similar way as that due to the
mean wind. The smaller eddies, however, because they occur more often, may induce
the structure to vibrate at or near one (or more) of the structure's natural frequencies of
Fig 2.4 Simplified two-dimensional flow of wind.
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vibration. This in turn induces a magnified dynamic load effect in the structure which
can be significant.
The separation of wind loading into mean and fluctuating components is the basis
of the so-called "gust- factor" approach, which is treated in many de-sign codes. The
mean load component is evaluated from the mean wind speed using pressure and load
coefficients. The fluctuating 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).
The dynamic response of buildings in the along-wind direction can be predicted
with reasonable accuracy by the gust factor approach, provided the wind flow is not
significantly affected by the presence of neighboring tall buildings or surrounding
terrain.
Cross-wind (transverse wind): There are many examples of slender
structures that are susceptible to dynamic motion perpendicular to the direction of the
wind. Tall chimneys, street lighting standards, towers and cables frequently exhibit this
form of oscillation which can be very significant especially if the structural damping is
small. Crosswind excitation of modern tall buildings and structures can be divided into
three mechanisms (AS/NZ1170.2, 2002) and their higher time derivatives, which are
described as follows:
(a) Vortex Shedding. 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 from the surface of the
structure, rather than follow the body contour (Fig. 4). For a particular structure,
the shed vortices have a dominant periodicity that is defined by the Strouhal
number. Hence, the structure is subjected to a periodic cross 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 the critical
velocity effect. The asymmetric pressure distribution, created by the vortices
around the cross section, results in an alternating transverse force as these
vortices are shed. If the structure is flexible, oscillation will occur transverse to
the wind and the conditions for resonance would exist if the vortex shedding
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Fig 2.6 Vortex shedding phenomenon
frequency coincides with the natural frequency of the structure. This situation can
give rise to very large oscillations and possibly failure.
Consider a cylindrical shaped building subjected to a smooth wind flow. The
originally parallel stream lines are displaced on either side of the cylinder, and these
results in spiral vortices being periodically from the sides of the cylinder into the
downstream flow of wind which is called the wake. At low speeds the vortices are shed
symmetrically in pairs one from each side. These vortices can be thought of as imaginary
projections attached to the cylinder that increase the drag force on the cylinder. When the
vortices are shed, i.e., break away from the surface of the cylinder, an impulse is applied
to the cylinder in the transverse direction. This phenomenon of alternating shedding of
vortices for rectangular tall building is shown in Fig-2.6.
Fig 2.5 Vortex formation in the wake of a bluff object.
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(b) The incident turbulence mechanism. The ‘incident turbulence’
mechanism refers to the situation where the turbulence properties of the natural
wind give rise to changing wind speeds and directions that directly induce
varying lift and drag forces and pitching moments on a structure over a wide
band of frequencies. The ability of incident turbulence to produce significant
contributions to crosswind response depends very much on the ability to generate
a crosswind (lift) force on the structure as a function of longitudinal wind speed
and angle of attack. In general, this means sections with a high lift curve slope or
pitching moment curve slope, such as a streamline bridge deck section or flat
deck roof, are possib le candidates for this effect.
(c) Higher derivatives of crosswind displacement: There are three
commonly recognized displacement dependent excitations, i.e., ‘galloping’,
‘flutter’ and ‘lock- in’, all of which are also dependent on the effects of turbulence
in as much as turbulence affects the wake development and, hence, the
aerodynamic derivatives. Many formulae are available to calculate these effects
(Holmes, 2001) recently computational fluid dynamics techniques (Tamura,
1999) have also been used to evaluate these effects.
2.8 Dynamic Nature of Wind
When wind hits a blunt body in its path, it transfers some of its energy to
the body. The measure of amount of energy transferred is called the gust response
factor. The gust response factor is dependent on the roughness of the terrain and
the height of the ground. A tall, slender, and flexible structure could have a
significant dynamic response to wind because of buffeting. This dynamic
amplification of response would depend on how the gust frequency correlates
with the natural frequency of structure and also on the size of the gust in the
relation to the building size. Unlike the mean flow of wind, which can be
considered as static, wind loads associated with gustiness or turbulence change
rapidly and abruptly, creating effects much larger than if the same loads were
applied gradually.
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Under the wind pressure, the building bend slightly and its top moves. It
first moves in the direction of wind, and then starts oscillating back and forth. Its
top goes through its neutral position, then moves in the opposite direction, and
continues oscillating back and forth until it eventually stops. The action of a wind
gust depends not only on how long it takes to reach its maximum value and
decrease again, but on the period of the building on which it acts. If the wind gust
reaches its maximum value and vanishes in a much shorter than the period of the
building, its affects are dynamic. The gusts can be considered as static loads if the
wind loads increases and vanishes in a time much longer than the period for the
building.
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CHAPTER-3
ANALYSIS FOR DRIFT DUE TO WIND LOAD
3.1 Introduction
As building heights increase, the forces of nature begin to dominate the structural
system and take on importance in the overall building system. The analyses and design
of tall building are affected by lateral loads, particularly drift or sway caused by such
loads. Drift or sway is the magnitude of the lateral displacement at the top of the building
relative to its base.
3.2 Analysis for Drift
When the initial sizes of the frame members have been selected, an approximate
check on the horizontal drift of the structure can be made. The drift in a non-slender rigid
frame is mainly caused by racking (Fig.-3.1). This racking may be considered as
comprising two components: the first is due to rotation of the joints, as allowed by the
double bending of the girders (Fig.-3.5), while the second is caused by double bending of
the columns (Fig.-3.4). If a rigid frame is slender, a contribution to drift caused by the
overall bending of the frame, resulting from axial deformations of the columns, may be
significant (Fig.-3.2). If the frame has a height width ratio less than 4:1, the contribution
of overall bending to the total drift at the top of the structure is usually less than 10% of
that due to racking.
The following method of calculation for drift allows the separate determination
of the components attributable to beam bending, column bending, and overall cantilever
action.
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Fig 3.1 Forced and deformations caused by external shear.
Fig 3.2 Forced and deformation
caused by external moment.
3.3 Components of Drift
It is assumed that the drift analysis that points of contra-flexure occur in frame at he mid
story level of the columns and the mid span of the girders. This is a reasonable
assumption for high-rise frames for all stories except near the top and bottom.
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Fig 3.3 Deflection of portal frame. a) Frame subjected to lateral loads b) typical
story segment
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3.3.1 Story Drift due to Girder Flexure. Consider a story-height segment of a frame at
floor level ‘i’ consisting of a line of girders and half story-height columns above and
below each joint (Fig.-3.3). To isolate the effect of girder bending, assume the columns
are flexurally rigid.
The average rotation of the joints can be expressed approximately as
Total moment carried by the joints
? i-g =
Total rotational stiffness of the joints
Fig 3.5 Lateral Deflection of Typical storey due to bending of Columns
Fig 3.4 Lateral deflection of typical story due to bending of columns.
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The total moment = Vi hi/2 + V(i+1) h (i+1)/2 ………………. (3.1)
And the total rotational stiffness =
6E [Ig1/L1 + (Ig1/L1 +Ig2/L2) + (Ig2/L2 +Ig3/L3) + Ig3/L3] ……… (3.2)
From Eqs. (3.1) to (3.2)
Vi hi/2 + V(i+1) h (i+1)/2 ? i-g = ………………………………………….. (3.3) 24E S (Ig /L)
A similar expression may be obtained for the average joint rotation in the floor i-
1 below, but with subscripts (i+1) replaced by i, and I by (i-1).
Referring to Fig.-3.5, the drift in story i due to the joint rotations is
? i-g = hi/2 (?i-1 + ? i) ……………………………………….………………… (3.4)
that is Vi-1 hi-1 + Vi hi Vi hi + V (i+1) h (i+1)
? i-g=hi/2 [ + ] 24E S (Ig /L)i-1 24E S (Ig /L)i
………………………… (3.5)
Assuming that the girders in floors i-1 and i are the same, the story heights are the
same, and the average of Vi+1 and Vi-1 is equal to Vi.
Vi hi
2 ? i-g = ………………………………………. (3.6)
12E S (Ig /L)
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3.3.2 Story Drift due to Column Flexure. Referring to Fig. – 3.4, in which the drift due
to bending of the columns is isolated by assuming the girders are rigid, the drift of the
structure in story I is
Vi hi3
? i-c = ……………………………………. (3.7) 12E S Ici
Vi hi2
? i-c= ………………………………. (3.8)
12E S (Ici /h)
3.3.3 Total Drift:
The total frame shear deflection is given by ? s
Vihi2 hi 1
? s = ? c + ? g = { + } ……………… (3.9) 12 (SEI) col (SEI/L) beams
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3.4 Derivation of the Governing Differential Equations (Coupled- Shear Wall
Structure)
Consider the plane coupled-wall structure sown in Fig.-3.6a subjected to
distributed lateral loading if intensity w per unit height. A general form of loading is
used to illustrate the derivation of the governing differential equation, before solutions
are derived for common standard design load cases.
The basic assumptions made in the analysis are as follows:
1. The properties of the walls and connecting beams do not change over the height,
and the storey heights are constant.
2. Plane sections before bending remain plain after bending for all structural
members.
3. The discreet set of connecting beams, each of flexural rigidity EIb, may be
replaced by an equivalent continuous connecting medium of flexural rigidity
Fig 3.6 Representing of Coupled Shear Walls by Continuous Model
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EIb/h per unit height, where h is the story height (Fig.-3.6b). Strictly speaking, for
this analogy to be correct, the inertia of the top beam should be half of the other
beams.
4. The walls deflected equally horizontally, as a result of the high in-plane rigidity
of surrounding floor slabs and the axial stiffness of connecting beams. It follows
that the slopes of the wall are every where equal along the height, and thus, using
a straightforward application of the slope-deflection equations, it may be shown
that the connecting beams, and hence the equivalent connecting medium, deform
with a point of contra-flexure at mid span. It also follows from this assumption
that the curvatures of the walls are equal throughout the height, and so the
bending moment in each wall will be proportional to its flexural rigidity.
5. The discreet set of axial forces, shear forces, and bending moments in the
connecting beams may then be replaced by equivalent continuous distributions of
intensity n, q, and m, respectively, per unit height.
Fig 3.7 Internal Forces in Coupled Shear Walls
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In particular, if the connecting medium is assumed cut along the vertical line of contra-
flexure, the only forces acting there are a shear flow of intensity q (z) per unit height and
an axial force of intensity n (z) per unit height, as in Fig- The axial force N in each wall
at any level z will then be equal to the integral of the shear flow in the connecting
medium above that level, that is,
Consider now the condition of vertical compatibility along the cut line of contra-
flexure of Fig.-3.7 Relative vertical displacement will occur at the cut ends of the
cantilevered laminas due to the following four basic actions. [In the derivation, positive
relative displacements are taken to mean that the end of the left-hand lamina (1) moves
downward relative to the end of the right hand lamina (2).]
1. Rotations of the wall cross-sections due to bending (Fig.-3.8a). Under the action
of a bending moment, the wall will deflect, and cross-sections will rotate as
shown if Fig.-3.8a Two forms of bending action occur; first, the free bending of
the wall due to the applied external moments and second, the reverse bending.
The relative vertical displacement d1 is given by (Fig.-3.8a)
d1= (b/2 + d1) dy/dz + (b/2 + d2)dy/dz = l dy/dz
where dy/dz is the slope of the centroidal axes of the walls at level z due to the
combined bending actions.
2. Bending and shearing deformations of the connecting beams under the action of
the shear flow (Fig.-3.8b). Consider a small element of the connecting medium of
depth dz, which may be assumed cantilevered from the inner edge of the wall.
H N=? q dz
z or, on differentiating
q= - dN/dz
23
The flexural rigidity of this small lamina is (EIb/h) dz, and the cantilever is subjected to a
tip shear force of q dz.
Fig 3.8 Relative Displacement at Line of Contra-flexure.
24
Due to bending only, the relative displacement d2 is given by
Where b is the clear span of the beams.
The effects of shearing deformations in the connecting beams may be readily
included by replacing the true flexural rigidity EIb by an equivalent flexural rigidity
EIc,where
Ic= Ib/ (1+r)
And r=12EIb?/b2GA
In which GA is the shearing rigidity and ? is the cross-sectional shape factor for
shear equal to 1.2 in the case of rectangular sections. The correction is necessary only in
the case of connecting beams with a span-to-depth ratio less than about 5.
The evaluation of d2 has assumed that the connecting beam is rigidity connected
to the wall, and just ignores the effects of local elastic deformations at the beam wall
junction that will increase the flexibility of the lamina. Both elasticity and finite element
studies have shown that the additional flexibility that arises may be included by the
simple expedient of extending the beam length by a further quarter beam depth into the
wall at each end. The length b in Eq (3.8) should thus be taken as the true length b+1/2
beam depth.
The axial forces N,
q dz qb3h d2 = -2 (b/2) = - ……………………….. (3.10) 3 (EIb/h)dz 12 EIb
qb3h dN ? 2 = + ………………………………..……… (3.11) 12 EIb dz
25
3. Axial deformations of the walls under the actions of the axial forces N (Fig-3.8c).
The action of the shear forces in the connecting beams will be to induce tensile
forces in the windward wall 1 and compressive forces in the leeward wall2.
consequently, the relative displacement, d3 at level z will be
z
? 3 =- (1/E) (1/A1+1/A2) ? N dz ………………………………….………….. (3.12)
0
Where A1 and A2 are the cross-sectional areas of walls 1 and 2, respectively.
4. Any vertical or rotational relative displacements at the base (Fig-3.8d). Vertical
or rotational deformations of the base may be occur as a result of displacements
of the foundations (proportional to the modulus of sub-grade reaction, for
example) or as a result of the flexibility of the supporting substructure, such
foundation displacements will induce rigid body movements of the superstructure
above, and will give rise to displacements are constant over the height as shown
in Fig-3.8d.
Assuming relative displacements (dv ) and rotations ( d? ) occur in the same senses as
the axial forces and moments, the relative vertical displacement d4 is
? 4 = - dv + l dd = db (say) ……………………………………………………………. (3.13)
In the original deflected structure there can be no relative vertical displacement
on the line of contra-flexure of the connecting beams. Consequently, the condition of
vertical compatibility at this position is
? 1 + d2 + d3 + d4 =0
26
Or, using the appropriate expressions for each.
The last term will be zero in the common case of a rigid base.
On considering both the free bending due to the externally applied moment M
and the reverse bending due to the shears and axial forces in the connecting medium
(Fig-3.7), the moment-curvature relationships for the two walls are, at any level.
H
EI1 d2y/dz2 = M1= M – (b/2 + d1) ? q dz – Ma …………………….…………. (3.15)
z
H
EI2 d2y/dz2 = M2= M – (b/2 + d2) ? q dz – Ma …………………….…………. (3.16)
z
where Ma is the moment caused by the axial forces in the connecting beams.
The addition of Eqs (3.15) and (3.16) yields the overall moment-curvature
relationship for the coupled walls,
H
E(I1 + I2) d2y/dz2 = l ? q dz = M - lN ………………………………...………….(3.17)
z
Differentiating Eq (3.14) with respect to z and combining with Eq (3.17) to
eliminate the curvature d2y/dz2 gives
d2N / dz2 – (ka)2 N = - a2 /l M ………………………..(3.18)
qb3h dN z
l dy/dz + - 1/E (1/A1 + 1/A2) ? N dz + db ………..(3.14) 12 EIb dz 0
27
This is the governing equation for coupled wall expressed in terms of the axial
force N.
The parameters in the equation are defined as
a2 = 12Icl2/b3hI, k2 = 1 + AI/A1A2l2
and I = I1 + I2, A= A1 + A2
As usual, the left-hand side of Eq (3.18) describes the inherent physical
properties of the structure, and the right-hand side involves the form of applied loading.
Alternately, eliminating the axial force N from Eqs (3.14) and (3.17) gives
d4y/dz4 – (ka)2d2y/dz2 = I/EI (d2M/dz2 – (ka)2 (k2-1)/k2 M) ………………………(3.19)
This is the governing equation for coupled walls expressed in terms of the lateral
deflection y.
The general solution of Eq (3.19) is
y = C1+ C2z +C3 cosh kaz +C4 sinh kaz – (1/EI (ka) 2) [1/D2 + 1/ (ka)2+
D2/(ka)4 + D4/(ka)6 + ….] [d2M/dz2 – (ka)2 (k2-1)/k2 M] ……..…………………(3.20)
where C1 to C4 are constants to be determined from the boundary conditions expressed in
terms of the variable y,
and M= w(H-z)2/2.
3.4.1Boundary Conditions
By considering conditions of compatibility and equilibrium at the top and bottom
of the structure, appropriate boundary conditions may be derived for a range of base
conditions.
28
For example, for a structure that is free at the top and rigidly built in at the base,
the four boundary conditions for Eq (3.20) will be
At z = 0, y = 0 …………….. (3.21)
dy/dz = 0 ………….. (3.22)
The second boundary condition at the top may be readily be derived by
substituting for N and its first derivative dN/dz from Eq (3.17) into the compatibility Eq
(3.14), and making use of Eq (3.22). The boundary conditions are then
At z = H, d2y/dz2 = 0 ……………………………………………………(3.23)
H
d3y/dz3- (ka)2dy/dz = 1/EI [dM/dz – a2(k2 - 1) ? M dz ] ……………….(3.24)
0
From these boundary conditions the constants are
C1 = w/ EI (ka)4 k2 cosh kaH + wH sinh kaH/EI (ka)3 k2 cosh kaH
C2 = wh3/6EI
C3 = - w/EI k2 cosh kaH [1/(ka)4 + Hsinh kaH/(ka)3]
C4 = wH/EI k2 (ka)3
Put these values in Eq (3.20) and simplifying
y = wH4/EI [1/24 {(1- z/H)4 + 4z/H - 1} + 1/k2 {1/2(kaH)2 [2z/H – (z/H)2
]– 1/24 [(1-z/H)4 + 4z/H -1] – 1/(kaH)4 cosh kaH [1 + kaH sinh kaH – cosh kaH – kaH
sinh ka (H-z)]}]
29
3.5 Derivation of the Governing Differential Equation (Wall-Frame
Structure)
The planer wall- frame in Fig-3.9a may be taken to represent either a structure
with walls and frames interacting in the same plane, or one with walls and frames in
parallel planes. Since, in a no-twisting structure, parallel walls and frames translate
identically, they may be simulated by a planar linked model.
The analytical solution requires the structure to be presented by a uniform
continuous model (Fig-3.9b), with all components deflecting identically. The following
assumptions are adopted to achieve this:
1. The properties of the wall and the frame members do not change over the height.
2. The wall may be represented by a flexural cantilever, that is, one which deform is
bending only.
3. The frame may be represented by a continuous shear cantilever, which deforms in
shear only. This implies that the frame deflects only by reverse bending of the
columns and girders, and that the columns are axially rigid.
4. The connecting members may be represented by a horizontally rigid connecting
medium that transmit horizontal forces only and that causes the flexural and shear
cantilevers to deflect identically.
Considering the wall and frame separately, as in Fig-3.9c, w and q are
respectively, the distributed external loading and the distributed internal interactive
force, whose intensities vary with height. QH is a horizontal concentrated force that,
as will be demonstrated later, acts between the top of the wall and the frame.
The differential equation for shear in the flexural member is
H
- EI d3y/dz3 = ? [w (z) – q (z) dz - QH] …………………………………….… (3.25)
z
And, for shear in the shear cantilever is
H
(GA) dy/dz = ? q (z) dz + QH …………………………………………………. (3.26)
z
30
Fi
g 3.
9 (
a) P
lana
r wal
l-fr
ame
stru
ctur
e; (b
) Con
tinuo
us a
nalo
gy fo
r wal
l-fr
ame
stru
ctur
e; (c
) Fre
e bo
dy d
iagr
ams
for w
all a
nd
fram
e.
31
In which the parameter (GA) represents the story height average shear rigidity of the
frame, as though it were a shear member with an effective shear area A and a shear
modulus G. note that G is not the shear modulus of the frame material nor is A the
area of its members.
Differentiating and summing Eqs (3.25) and (3.26) gives
EI d4y/dz4 – (GA) d2y/dz2 = w (z) …………………. (3.27)
Or, d4y/dz4 – a2 d2y/dz2 = w (z)/EI …………...… (3.28)
In which a2 = (GA)/EI .………………………………… (3.29)
Equation (3.28) is the characteristic differential equation for the deflection of a
wall- frame.
3.5.1 Solution for Uniformly Distributed Loading
The solution of Eq. (3.27) for uniformly distributed external loading w can be written as
y (z) = C1 + C2z + C3 cosh az + C4 sinh az – wz2/2EIa 2 ………………..(3.30)
The boundary conditions for the solution of constants C1 to C4 are
1. fixity at the base
y (0) = dy/dz (0) = 0 …………..(3.31)
2. zero moment at the top of the flexural cantilever
Mb (H) = EI d2y/dz2 = 0 ……………… (3.32)
And
32
3. zero resultant shear at the top of the structure
EI d3y/dz3 (H) – (GA) dy/dz (H) = 0 …………….. (3.33)
Equations (3.31), (3.32) and (3.33) are used to determine C1 to C4 to give the
deflection equation:
C1 = - w/EIa4 [(aH sinh aH + 1)/ cosh aH]
C2 = wH/EIa2
C3 = w/EIa4 [(aH sinh aH + 1)/ cosh aH]
C4 = - awH/ EIa4
Put these values of constants in Eq (3.30) and simplifying
y (z) = wH4/EI {1/(aH)4 [(aH sinh aH + 1) (cosh az - 1)/ cosh aH –
aH sinh a z + (aH)2 [ z/H – ½ (z/H)2]]}
33
A typical system of plane coupled shear walls is shown in Fig-3.10. In the Fig
the total height of the structure is H, AB and CD is the wall-1 and wall-2 respectively,
BC is the connecting beam between two walls
Fig 3.10 Model Structure
34
The interfacing of the program to analysis the drift of the structure is shown in
Fig-3.11. This program is applicable for the model structure of coupled shear walls
shown in Fig-3.10. In the program it is assumed that, the beam dimension is constant
(2.5m) and also the level height is constant (3m). Others parameters are considered as
changeable. The result shows the drift of five storeys because in the program assumed
that the width of the wall is constant upto five storey. This program is applicable for any
height of coupled shear wall.
Fig 3.11 Interfacing of the Software
35
CHAPTER NO-4
DYNAMIC RESPONSE DUE TO WIND LOADING
4.1 Introduction
Dynamic motions refers to those caused by time dependent dynamic forces,
notably seismic accelerations, short period wind loads, blasts, and machinery vibrations,
the first two usually being of the greatest concern. If the building is exceptionally slender
or tall, or if it is located in extremely sever exposure conditions, the effective wind
loading on the building may be increased by dynamic interaction between the motion of
the building and the gusting of the wind.
Dynamic wind pressures produce sinusoidal or narrow-band random vibration
motions of the building, which will generally oscillate in both along wind and cross-wind
directions, and possibly rotate about a vertical axis. The magnitude of the displacement
components will depend on the velocity distribution and direction of wind, and on the
shape, mass, and stiffness properties of the structure. In certain cases, the effect of cross-
wind motions of the structure may be greater than those due to along cross-wind
motions.
4.2 Sensitivity of Structures to Wind Forces
The principal structural characteristics that affect the decision to make a dynamic
design analysis are the natural frequencies of the first few normal modes of vibration and
the effective size of the building. When a structure is small, the whole building is loaded
by gusts so that the full range of frequencies from both boundary layer turbulence will be
encountered. On the other hand, when the building is relatively large or tall, the smaller
gusts will not act simultaneously on all the parts, and will tend to offset each other’s
effects, so that only the lower frequencies are significant.
If the structure is stiff, the first few natural frequencies will be relatively high, and
there will be little energy in the spectrum of atmospheric turbulence available to excite
resonance. The structure will thus tend to follow any fluctuating wind forces without
36
appreciable amplification or attenuation. The dynamic deflections will not be significant,
and the main design parameter to be considered is the maximum loading to which the
structure will be subjected during its lifetime. Such a structure is termed as ‘static’, and it
may be analyzed under the action of equivalent wind forces.
If the structure is flexible, the first few natural frequencies will be relatively low,
and the response will depend on the frequency of the fluctuating wind forces. At
frequencies below the first natural frequency, the structure will tend to follow closely the
fluctuating force actions. The dynamic response attenuates at frequencies above the
natural frequency, but will be amplified at frequencies at or near the natural frequency;
consequently the dynamic deflections may be appreciably greater than the static values.
The lateral deflection of the structure then an important design parameter, and the
structure is classified as ‘dynamic’. In such structure, the dynamic stresses must also be
determined by design process. Furthermore, the accelerations induced in dynamic
structures may be important with regard to the comfort of occupants of building and
must be considered.
When a structure is very flexible, its oscillations may interact with the
aerodynamic forces to produce various kinds of instability, such as vortex-capture
resonance, galloping oscillations, divergence, and flutter. In this exceptional case, the
potential for disaster is so great that the designed must be changed or the aerodynamic
effects modified to ensure that this form of unstable behavior cannot occur.
4.3 Dynamic Structural Responses Due to Wind Forces
The prediction of the structural response involves two stages: (1) the prediction
of the occurrence of various mean wind speeds and their associated directions, and (2)
given the occurrence of the wind, the prediction of the maximum dynamic response of
the structure. The former requires an assessment of the wind climate of the region,
adjusted to take account of the local topography of the site, and of the local wind
characteristics (mean velocity profile and turbulent of structure). The steady pressures
and forces due to the mean wind, and the fluctuating pressures on the exterior, may then
37
be determined. The properties of the mean wind can be conveniently expressed only in
statistical terms.
Although the design of cladding may be strongly influenced by local pressures,
the response of the building as a whole depends on the integrate values over the different
faces of the building.
The exciting forces on a structure due to wind actions tend to be random in
amplitude and spread over a wide range of frequencies. The structures response to
dominated by the actions of its resonant response to wind energy available in the narrow
bands close to the natural frequencies of the structure. The major part of the exciting
energy will generally be frequencies much lower than the fundamental natural frequency,
and amount of energy decreases with increasing frequency. Consequently, for design
purposes, it is usually necessary to consider the structure’s response only in the
fundamental modes; the contribution from higher modes is rarely significant.
The fluctuations in the response of a structure can be consider as those associated
with the mean wind speed, and those associated with the turbulence of the wind, which
are predominantly dynamic in character. Consequently, it is convenient to describe wind
speeds, forces, deflections, etc. in terms of an hourly mean value together with the
average maximum fluctuation likely to occur in an hour. When these are added, the sum
can be used as an average hourly maximum, or peck response, to define equivalent static
design data.
The peak value can be expressed statistically in terms of the number of
standard deviations by which exceeds the mean value. For design purposes, the
conventional practice is to define the peak value of the variable, x (max) say, by the
relationship
x (max) = x + gp s ………………(4.1)
where x(max), x, and s are the peak, mean, and standard deviations, respectively, of the
variable x concerned, referred to a record period of one hour, and gp is the ‘peak’ factor.
38
When considering the response of a tall building of wind actions, both along-
wind and cross-wind motions must be considered. These arise from different forcing
mechanism, the former being due primarily to buffeting effects caused by turbulence,
while the latter is due to primarily to alternate-side vortex shedding. The cross-wind
response may be of particular importance with regard to the comfort of the occupants.
4.4 Along-wind Response
The pioneering work of Davenport and Vickery has shown that the along-wind
response of most structures is due almost entirely to the action of the incident turbulence
of the longitudinal component of the wind velocity, superimposed on a mean
displacement due to the mean drag. The resulting analytical methods, using spectral and
spatial correlation considerations to predict the structural response have been developed
to such a level that they are now employed in modern design wind Codes. The work has
led to the development of the gust factor method for the prediction of the building
response.
The gust factor method is based on the assumption that the fundamental mode of
vibration of a structure has an approximately linear mode shape. In essence, the aim of
the method is to determine a gust factor G that relates the peak to mean response in terms
of an equivalent static design, or load effect Q, such that
Design value, Q (max) = G Q ……………… (4.2)
Where, Q defines the mean value of quantity concerned.
For example, if the mean pressures acting on the face of a tall building are
summed to give the mean base overturning moment M, the design dynamic base
overturning moment M (max) will be obtained by multiplying M by gust factor G.
M (max) = GM ………………… (4.3)
The gust factor can be regarded as a rela tion ship between the wind gusts and the
magnification due to the structural properties. As such, it will depend on properties on
39
the structure (height H, and breadth height ratio W/H), fundamental natural frequency
n0, and critical damping ratio ß, the mean design wind speed V, and the particular
location of the building (i.e., whether it is sited in the centre of the city, in suburbs or
wooded areas, or in flat open country).
It may be shown that the gust factor G may be expressed as
G= 1+ gp r (B+R) ½ ………………… (4.4)
In Eq.(4.4) gp is a peak factor that accounts for the time history of the excitation
and is determined from the duration time T over which the mean velocity is averaged and
fundamental frequency of vibration n0; in practice, T is taken as 3600 sec (1 hour), r is a
roughness factor, which depends on the background turbulence or gust energy, which
depends on the height and aspect ratio of the building; and R is the excitation by the
turbulence resonant with the structure, which depends on the size effect S, the gust
energy ratio at the natural frequency of the structure, F, and the critical damping ratio
,that is,
R=SF/ ß …………………… (4.5)
The size reduction factor S depends on the aspect ratio W/H, the natural
frequency n0, and the mean wind velocity at the top of the structure, Vw, as shown in
Fig…… The gust energy ratio F is the function of the inverse wavelength, n0, / VH, as
shown in Fig….
If resonant effects are small, then R will be small compared to the background
turbulence B, and vise-versa.
The peak factor gp in Eq. (4.4) gives the no of standard deviations by which the
peak load effect is expected to exceed the mean load effect, and shown in Fig. as a
function of average fluctuation rate ? given by
?= n0 / (1+B/R) ½ ……………………(4.6)
40
In the above formulas, the variables VH, n0, and ß must relate to the along-wind
direction.
Substitution of the known values of gp, r, B, and R into Eq. (4.4) then produces
the desired value of gust factor.
Once the gust factor G has been determined, the peak dynamic forces and
displacements may be determined by multiplying the values due to the mean wind
loading by G.
4.4.1 Peak Along-Wind Accelerations
The peak along-wind accelerations are the most important criterion for the
comfort of the building’s occupants. The maximum acceleration aD in the along-wind
direction may be estimated from the expression
aD= 4? 2n02gpr R½ (?/G)………………..(4.7)
where, ? = the maximum wind- induced deflection at the top of the building in the along-
wind direction (m).
The natural frequency n0 and ß damping ratio must be again in the along-wind
direction. The other symbols have been defined previously in connection with Eq. (4.4).
4.5 Cross-wind Response
The cross-wind excitation of tall building is due predominantly to vortex
shedding. However, generalized empirical methods of predicting the response have been
difficult to drive, even assuming that the motions are due entirely to wake excitation,
because of the effects of building geometry and density, structural damping, turbulence,
operating reduced frequency range, and interference from upstream buildings. The last
effect can alter significantly the cross-wind motions. Consequently, as yet, the most
accurate method of determining the cross-wind structural response has been from tests
on an aero-elastic model in a wind tunnel.
41
The work of Saunders, Melbourne, and Kwok using the results of empirical wind
tunnel data, has led to an approximate analysis that can take into account the most
important variables concerned. The technique employed to calculate he response due to
wake excitation is to solve the equation of motion for a lightly damped structure in
modal from with the forcing function mode generalized in spectral format.
Although it is generally found that the maximum lateral wind loading and
deflection are in the along-wind direction, the maximum acceleration of the building,
which is particularly important for human comfort, mat often occur in the cross-wind
direction. Across –wind acceleration are likely to exceed along-wind accelerations if the
building is slender about both axis, such that the geometric ratio (WD) ½ / H is less than
one-third, where D is the along-wind plan dimension.
Based on a wide of turbulent boundary layer wind tunnel studies, a tentative
formula is given in the NBCC for the peak acceleration aW at the top of the building,
namely,
aW = n02gp [WD ½ (ar / ?g ½ ) ……………….. (4.8)
where, ?= average density of the building (kg/m3)
ar =78.5*10-3 [VH/ n0WD½ ]3.3 (pa)
g= acceleration due to gravity (m/sec2)
Because of relative sensitivities of the expressions in Eq. (4.7) and (4.8) to the
natural frequencies, it is recommended that the latter be determined using fairly rigorous
analytical method s, and that approximate formulas be used with caution.
42
Height of Structure H (m)
Fig 4.1 Variation of roughness factor with building height
Rou
ghne
ss F
acto
r, r
City Centre
Suburbs, wooded area
Flat open country
Height (m)
Fig 4.2 Variation of background turbulence factor with height and aspect ratio of building
Bac
kgro
und
Tur
bule
nce
Fac
tor
B
43
Size
red
uctio
n fa
ctor
S
Reduced Frequency H/VH.n0
Fig 4.3 Variation of size reduction factor with reduced frequency and aspect ratio of building
44
Inverse wavelength n0/VH (waves/m)
Fig 4.4 Variation of gust energy ratio with inverse wavelength
Gus
t ene
rgy
ratio
F
Average fluctuation rate v (c/s) Fig 4.5 Variation of peak factor with average fluctuation rate
Peak
fact
or g
p
45
CHAPTER-5
THEORATICAL ANALYSIS AND RESULTS
5.1 DRIFT ANALYSIS
Drift is the lateral displacement or sway. Drift is one of the major key factor to
design of a tall building. Drift is mainly depends on the wind loads and height and width
of the structure. The equation of drift for coupled shear wall is shown in below:
Drift,y = wH4/EI [1/24 {(1- z/H)4 + 4z/H - 1} + 1/k2 {1/2(kaH)2 [2z/H –
(z/H)2 ]– 1/24 [(1-z/H)4 + 4z/H -1] – 1/(kaH)4 cosh kaH [1 + kaH sinh kaH – cosh kaH
– kaH sinh ka (H-z)]}] By using this equation and with the help of the software the drift of 10-storey, 15-storey
and 20-storey buildings analyzed in Table-5.1, Table-5.2 and Table-5.3 respectively. The
uniformly distributed load is considered as 1.5 ksf.
TABLE 5.1 For result of drift analysis on 10-storey building
Level Length of Wall-1
(ft)
Length of Wall-2
(ft)
Width of Wall
(ft) Drift (ft)
10 16 23 1 .018
09 16 23 1 .0175
08 16 23 1 .0168
07 16 23 1 .0157
06 16 23 1 .0143
05 16 23 1 .0126
04 16 23 1 .01063
03 16 23 1 .00836
02 16 23 1 .005823
01 16 23 1 .00303
46
TABLE 5.2 For result of drift analysis on 15-storey building
Level Length of Wall-1 (ft)
Length of Wall-2 (ft)
Width of Wall (ft)
Drift (ft)
15 16 23 1 .0475
14 16 23 1 .046
13 16 23 1 .044
12 16 23 1 .043
11 16 23 1 .0408
10 16 23 1 .038
09 16 23 1 .0356
08 16 23 1 .0325
07 16 23 1 .0292
06 16 23 1 .026
05 16 23 1 .0218
04 16 23 1 .01782
03 16 23 1 .0136
02 16 23 1 .00921
01 16 23 1 .00466
47
TABLE 5.3 For result of drift analysis on 20-storey building
Level Length of Wall-1
(ft)
Length of Wall-2
(ft)
Width of
Wall (ft) Drift (ft)
20 16 23 1 .1022
19 16 23 1 .09949
18 16 23 1 .0964
17 16 23 1 .0934
16 16 23 1 .0894
15 16 23 1 .0854
14 16 23 1 .0811
13 16 23 1 .0765
12 16 23 1 .0716
11 16 23 1 .0666
10 16 23 1 .06121
09 16 23 1 .0557
08 16 23 1 .0499
07 16 23 1 .044
06 16 23 1 .0379
05 16 23 1 .0317
04 16 23 1 .025
03 16 23 1 .0191
02 16 23 1 .01269
01 16 23 1 .0063
48
The variation of drift of different buildings is shown in Fig-5.1. In this graph X-
axis represents the storey level and Y-axis represents the drift in ft. The drift at different
storey level of 10-storey, 15-storey and 20-storey is plotted and from these three curves,
the drift is increased with increase in the height of the structure. Again the drift at the
same level increases with increase in the total he ight of the structure. The drift of 8th
(say) level is 0.0168 ft, 0.0325 ft and 0.0499 ft on the structure height of 10-storey, 15-
storey and 20-storey respectively.
0
0.02
0.04
0.06
0.08
0.1
0.12
0 5 10 15 20 25
Storey Level
Dri
ft (f
t) 20-storey
15-storey10-storey
Fig 5.1 Variation of Drift of Different Buildings
49
The drift of different width of the building is analyzed in Table-5.4. Here the drift
of a 20-storey building at different width of 30ft, 35ft, 40ft, 45ft and 50ft is analyzed.
The uniformly distributed load and width of wall is considered as 1.5ksf and 1ft
respectively.
TABLE 5.4 For Drift analysis at different width of the building
Storey Level
Length of Wall (w1=12,w2=18)
Length of Wall (w1=14,w2=21)
Length of Wall (w1=16,w2=23)
Length of Wall (w1=18,w2=27)
Length of Wall (w1=20,w2=30)
1 .015 .00893 .006322 .0037 .0025 2 .0297 .0178 .01269 .0075 .0052 3 .044 .02664 .0191 .011 .00793 4 .0582 .03534 .0254 .0154 .0108 5 .072 .044 .0317 .0194 .014 6 .085 .0522 .0379 0234 .0166 7 .098 .0604 .044 .0274 .01962 8 .11 .068 .0499 .0313 .0226 9 .1215 .076 .0557 .0352 .026 10 .1325 .0831 .0612 .039 .0285 11 .1430 .0900 .0666 .0427 .0313 12 .1530 .0970 .0720 .0462 .0340 13 .1620 .1030 .0765 .0497 .0370 14 .1700 .1086 .0810 .0530 .0395 15 .1780 .1140 .0854 .0560 .0420 16 .1840 .1190 .0894 .0590 .0444 17 .1910 .1230 .0930 .0620 .0470 18 .1960 .1270 .0960 .0650 .0490 19 .2010 .1300 .0995 .0670 .0510 20 .2004 .1343 .1022 .0693 .0530
50
The variation of drift with respect to different building widths is shown in Fig-
5.2. This Fig indicates when the building width is decreased gradually the drift is
increased dramatically.
0
0.05
0.1
0.15
0.2
0.25
0 5 10 15 20 25
Storey Level
Dri
ft (f
t)
w1=12,w2=18
w1=14,w2=21w1=16,w2=23w1=18,w2=27w1=20,w2=30
Fig 5.2 Variation of Drift with Respect to Building Widths
51
5.2 PROBLEMS ON DYNAMIC ACTION
To illustrate the calculations involved in the estimation of the peak wind load effects,
Considered of tall building in the centre of the city.
Height H= 70m
Breadth B= Depth D=15m
Estimated fundamental natural frequency ?0 = .2 Hz
Estimated critical damping ratio, ß = .015
Mean wind speed at top of the building, VH =30m/s
Estimated maximum deflection at top of the building, ? = 0.36m
Estimated average building density, =175 kg/m3
1) GUST FACTOR:
From Fig-4.1 Roughness factor, r = 0.445
Aspect ratio W/H, = 15/70 = 0.2143
From Fig-4.2, background turbulence factor, B =0.94285
Reduced frequency, ?0.H/VH = (0.2*70/30)=0.47
From Fig-4.3 size reduction factors, S= 0.235
Inverse wavelength, ?0/VH = .2/30 = 0.0066
From Fig-4.4 gust energy ratio, F= 0.26
Resonant turbulence factor, R = SF/ß =0.235*0.26/0.015 = 4.07 > B (0.94285)
Therefore, the resonant turbulence excitation is greater than the background
turbulence excitation.
Average fluctuation rate, v = ?0 / (1+B/R) ½ = 0.2/(1+0.94285/4.07)½ = 0.18
From Fig-4.5 peak factor, gp = 3.75
Gust factor, G= 1+gp.r. (B+R) ½ =1+3.75*0.445*(0.94285+4.07)½ = 4.74
That is, the peak dynamic forces and displacement are obtained by multiplying all
static values due to the wind loading by 4.74.
2. Along-wind Acceleration
aD= 4? 2n02gpr R1/2 (?/G)
= 4? 2 *0.22*3.75 * 0.445*4.07½ (.36/4.74)
=0.4038 m/sec2 (4.12% of g)
52
3. Cross-wind Acceleration
ar =78.5*10-3 [VH/ (n0W) ]3..3
= 78.5*10-3 [30/(0.2 * 15)]3..3
=156.63 Pa
aW = n02gp [WD]½ (ar / ?g ß½ )
= 0.22*3.75 * [15*15]½ *(156.63 / 175*9.81* 0.015½ )
=1.676 m/sec2 (17.10% of g)
The along-wind and cross –wind acceleration has found in Table-5.5, Table-5.6,
and Table-5.7. The along-wind acceleration and cross-wind acceleration of one sample
of 70m storey has calculated above and other building of 80m, 90m and 100m in height
is calculated in same procedure and recorded in Table-5.5. In Table-5.6 and Table-5.7
the along-wind acceleration and cross-wind acceleration of a typical building in 90m
height has found with change in the width of along-wind direction and cross-wind
direction respectively.
TABLE 5.5 Result of Along-wind Acceleration & Cross-wind Acceleration
Along wind Acceleration (m/sec2)
Cross wind Acceleration (m/sec2)
Height (m)
[WD]½ /H
aD % of g aW % of g 70 .2143 0.404 4.120 1.676 17.10
80 .1875 0.421 4.292 1.685 17.18 90 .167 0.428 4.363 1.694 17.27 100 .150 0.433 4.414 1.70 17.33
TABLE 5.6 Results for the cross –wind acceleration with increase at width of the along-wind direction. Height
(m) [WD]½
/H Width at
along wind
Direction, D (m)
Width at cross wind Direction,
W (m)
Along-wind Acceleration, aD (m/sec2)
Cross-wind Acceleration, aW (m/sec2)
% of g
.149 12 1.515 15.44
.167 15 1.694 17.27
.183 18 1.856 18.92
.192 20 1.956 19.94
90
.215 25
15
0.167
2.052 20.92
53
TABLE 5.7 Results for the cross –wind acceleration with increase at width of the cross-wind direction.
Height (m)
[WD]½
/H Width at
along wind
Direction, D (m)
Width at cross wind Direction,
W (m)
Along-wind Acceleration, aD (m/sec2)
Cross-wind Acceleration, aW (m/sec2)
% of g
.149 12 .398 3.131 31.92
.167 15 .394 1.676 17.08
.183 18 .387 1.01 10.3
.192 20 .374 .750 7.65
90
.215
15
25 .36 .401 4.09
The variation of cross-wind acceleration with respect to the width of along –wind
direction is shown in Fig-5.3. This graph indicates that the cross-wind acceleration is
increased with the increase in the width of the along-wind direction. Because the cross-
wind acceleration depends on the ar and both the width of along-wind and cross-wind
direction. Though the width of cross-wind direction is remain constant the cross-wind
acceleration increases with the increase in the width of the along-wind direction.
0
0.5
1
1.5
2
2.5
10 12 14 16 18 20 22 24 26
Width (m)
Cro
ss-w
ind
Acc
eler
atio
n (m
/s2)
Fig 5.3 Variation of Cross-wind Acceleration with Respect to the Width at Along-wind Direction
54
The variation of cross-wind acceleration with respect to the width of the cross-
wind direction is shown in Fig-5.4. This graph indicates that the cross-wind acceleration
is decreased with the increase in the width of the cross-wind direction. Because when the
width of the cross-wind direction increases ar is decreased abruptly, so the cross-wind
acceleration is decreased with increase in the width of the cross-wind direction.
0
0.5
1
1.5
2
2.5
3
3.5
10 12 14 16 18 20 22 24 26 28 30
Width (m)
Cro
ss-w
ind
Acc
eler
atio
n (m
/s2)
Fig 5.4 Variation of Cross-wind Acceleration with Respect to the width at Cross-wind Direction
55
CHAPTER- 6
CONCLUSION
6.1 CONCLUSION
A theoretical investigation has been made to study the drift, along-wind and
cross-wind behavior of tall building due to wind loads. The following conclusions can be
drawn from this theoretical study:
1. The drift of the structure increases with increase in the height of the structure.
The drift of 10-storey, 15-storey and 20-storey is increased due to increase in the
height of the structure.
2. The drift at the same level increases with increase in the total height of the
structure.
3. The drift of the structure decreases with the increase in the width of the structure.
4. The cross-wind acceleration increases with increase in the total height of the
structure.
5. The cross-wind acceleration increases with the increase in the width of the
structure at the along-wind direction.
6. The cross-wind acceleration decreases with the increase in the width of the
structure at the cross-wind direction.
6.2 RECOMMENDATIONS
In this theoretical study only wind loads is considered. The following should be
kept in mind for fur ther study:
1. The drift and dynamic response due to seismic load should be considered.
2. The differential equation is only formed for coupled shear wall and wall- frame
structure, this equation should be formed for all type of structural system.
3. The program is only applicable for coupled shear wall; it should be modified for
all type of tall structure.
56
References
1. Stafford Smith Bryan & Alex Coul, “Tall Building Structure: Analysis and
Design”. John Wiley & Sons, INC.
2. Taranath B.S. (1988), “Structural Analysis and Design of Tall Buildings”.
McGraw-Hill Book Company.