Serviceability-Based Design of Tall Buildings Subjected to Vortex Shedding
by
ARCHIVES
MASSACLUK[E TT! PITm TE
OF fECHNOLOLGY
Abram Wasef
JUL 022015
B.Eng. in Civil Engineering
McGill University, 2014
LIBRARIES
SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVIRONMENTAL
ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF ENGINEERING IN CIVIL AND ENVIRONMENTAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
2015 Abram Wasef. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part in any mediunr now known or
hereafter created.
Signature of Author:
Signature redacted
Department of Civil and Environmental Engineering
May 15, 2015
Certified by:
Signature redacted
Pierre Ghisbain
Lecturer in Civil and nvironmental Engineering
Thesis Supervisor
Certified by:
Signature redacted
/
J. Connor
IJerome
Professor of Civil and Environmental Engineering
Thesis Co-Advisor
Accepted by:
Signature redacted
f
Q eidi Nepf
Donald and Martha Harleman Professor of Civil and Environmental Engineering
Chair, Departmental Committee for Graduate Students
Serviceability-Based Design of Tall Buildings Subjected to Vortex Shedding
by
Abram Wasef
Submitted to the Department of Civil and Environmental Engineering on May 15, 2015
in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in
Civil and Environmental Engineering
ABSTRACT
With the increasing rate of population, there is an increase in demand for housing for
people and their families. Due to the limited amount of land space, one of the most viable
and feasible solutions is increase the number and height of residential and office
buildings leading to a requirement of having a special design for these tall buildings. Due
to the advancement of technology leading to an increase in the strength of materials used
in construction, these types of buildings can be built. This leads to lesser amounts of
materials used and resulting in lightweight structures that are flexible. As the height of
the buildings increases, these lightweight structures become more flexible making them
susceptible to excessive wind-induced motion. Although there are multiple factors that
govern serviceability in tall buildings, it has been deduced from the literature, that
acceleration is a very important factor, and that as the level of acceleration increases,
people become more uncomfortable. Moreover, across wind response caused mainly due
to vortex shedding becomes a very important phenomenon that needs to be dealt with,
and which also contributes a significant amount of acceleration on the building.
Acceleration due to vortex shedding is the focus of this thesis.
To determine a solution, information on factors affecting serviceability of tall buildings,
how increasing effects of these factors would affect occupants, and how current standards
and codes deal with serviceability requirements were obtained. Using this information, a
methodology similar to the Pacific Earthquake Engineering Research Center (PEER)
criteria was developed to determine the relationship between these different factors.
All of these factors were incorporated in different cost functions and combined together
to evaluate the serviceability of tall buildings over their lifetime from an economical
perspective. A flexible parametric approach was used to analyze how varying the level of
damping, stiffness and the negative effects due to wind-induced acceleration will affect
the cost of tall buildings. Moreover, a detailed example was presented to show how the
methodology works by analyzing the CAARC Building. Also, the analysis includes
varying the location by applying the methodology to three different states to determine
how stiffness and damping changed.
Thesis Supervisor: Pierre Ghisbain
Title: Lecturer of Civil and Environmental Engineering
Thesis Co-Advisor: Jerome J. Connor
Title: Professor of Civil and Environmental Engineering
ACKNOWLEDGEMENTS
I would like to thank my parents, and siblings for their support, trust and for being there
for me.
Moreover, I would like to thank Dr. Ghisbain for introducing me to the topic and
increasing my interest in performance-based design engineering and for always being
available to guide me throughout my thesis, and answer my questions and doubts.
In addition, I would like to also thank Prof. Connor for introducing me to motion-based
design and for being available to answer my questions.
TABLE OF CONTENTS
ABSTRACT.......................................................................................................................
3
ACKNOW LEDGEM ENTS ..........................................................................................
5
1.
2.
INTRODUCTION...............................................................................................
15
1.1
Need for Tall Buildings and a Methodology..................................................
15
1.2
Issues with Tall Buildings ...............................................................................
15
1.3
Problem Statement .............................................................................................
16
LITERATURE REVIEW ...................................................................................
17
2.1
Factors affecting occupant's perception...........................................................
17
2.2
Vortex Shedding.............................................................................................
19
2.3
Influence of different levels of acceleration....................................................
21
2.4
Factors affecting serviceability .....................................................................
23
2.4.1
Parameters affecting serviceability ........................................................
23
2.4.2
2.4.2 Comparing different parameters ...................................................
25
CURRENT PRACTICES....................................................................................
27
Current standards and codes...........................................................................
27
3.
3.1
3.1.1
North American Standards......................................................................
27
3.1.2
Asian Standards ........................................................................................
27
3.1.3
International Organization for Standardization (ISO) Standards............. 29
Comparing different codes and standards ......................................................
3.2
33
4.
CHALLENGES....................................................................................................
35
5.
PROPOSED SOLUTION AND METHODOLOGY........................................
37
5.1
Proposed Solution ..........................................................................................
37
5.2
M ethodology ...................................................................................................
39
6.
5.2.1
Period of the Structure ............................................................................
39
5.2.2
Modal Damping Ratio.............................................................................
40
5.2.3
Deflection of a building ..........................................................................
41
5.2.4
Acceleration of a building........................................................................
42
5.2.5
Parameters used to develop the cost functions ........................................
44
DETAILED EXAMPLE............................................
49
7.
6.1
CAARC Building: Structural Analysis ..........................................................
6.2
Relating wind speed with various parameters: Hazard Analysis .................... 53
6.3
C ost A nalysis....................................................................................................
57
6.4
Sensitivity Study ............................................................................................
60
6.4.1
Effects of damping....................................................................................
60
6.4.2
Effects of stiffness....................................................................................
64
6.4.3
Varying cost coefficient..........................................................................
68
6.4.4
Effect of the geographical location ..........................................................
69
APPLICATIONS .................................................................................................
7.1
Cost effectiveness of damping ........................................................................
73
73
7.1.1
Varying cost coefficient and damping ...................................................
73
7.1.2
Economical outcomes of increasing damping ........................................
75
7.2
8.
49
Cost effectiveness of stiffness ........................................................................
78
7.2.1
Varying cost coefficient and stiffness......................................................
78
7.2.2
Economical outcomes of increasing stiffness ..........................................
80
7.3
Comparing the cost-effectiveness of damping and stiffness...........................
83
7.4
Increasing damping and stiffness for different locations ................................
84
7.4.1
Economical analysis for Illinois...............................................................
84
7.4.2
Economical analysis for New York ........................................................
87
7.4.3
Economical analysis for Massachusetts.................................................
89
7.4.4
Evaluating different factors and locations ...............................................
91
CONCLUSIONS.................................................................................................
93
RIEFERIENCES ....................................................................
95
LIST OF FIGURES
Figure 2.1 - Wind Response Directions .......................................................................
19
Figure 2.2 - Effect on vortex shedding on response......................................................
20
Figure 2.3 - Annoyance Threshold Vibrations for Residences, Offices and Schools...... 24
Figure 2.4 - Relative effects on building motions of changing mass, stiffness, and
dam pin g.............................................................................................................................
26
Figure 3.1 - Probabilistic perception thresholds given in AIJ-GBV-2004....................
28
Figure 3.2 - Suggested satisfactory magnitudes of horizontal motion of buildings used for
general purposes (curve 1) and of off-shore fixed structures (curve 2)........................ 30
Figure 3.3 - Average (curve 2) and lower threshold (curve 1) of perception of horizontal
m otion by hum ans.............................................................................................................
30
Figure 3.4 - ISO 10137 acceleration limits. Curve 1 - maximum horizontal acceleration
for office buildings. Curve 2 - maximum horizontal acceleration for residential buildings
32
...........................................................................................................................................
Figure 5.1 - Flowchart presenting the modified PEER methodology ........................... 37
Figure 5.2 - Cost function presenting the negative effects of acceleration using the cost
coefficient, g . ....................................................................................................................
45
Figure 6.1 - Schematic diagram showing the dimension of the CAARC building..... 49
Figure 6.2 - The response of the CAARC building without increasing damping or
stiffn ess.............................................................................................................................
51
Figure 6.3 - Relationship between return period and wind speed.................................
55
Figure 6.4 - Relationship between exceedance rate density, r,(V), and wind speed, V.. 56
Figure 6.5 - Modifying the cost function for negative effects of acceleration, Ca. ......... 57
Figure 6.6 - Relating Ca* r,(V) and wind speed with unchanged stiffness and damping.
58
...........................................................................................................................................
Figure 6.7 - Relating CT and wind speed with unchanged stiffness and damping. ...... 58
Figure 6.8 - The response of the CAARC building increase in damping and unchanged
stiffn ess.............................................................................................................................
60
Figure 6.9 - Relating Ca and wind speed with increase in damping and unchanged
61
stiffn ess. ............................................................................................................................
Figure 6.10 - Relating Ca * r,(V) and wind speed with increase in damping and
62
unchanged stiffness........................................................................................................
Figure 6.11 - Relating CT and wind speed with increase in damping and unchanged
stiffn ess. ............................................................................................................................
63
Figure 6.12 - The response of the CAARC building with increase in stiffness and
unchanged damping. .........................................................................................................
64
Figure 6.13 - Relating Ca and wind speed with increase in stiffness and unchanged
d am pin g.............................................................................................................................
65
Figure 6.14 - Relating Ca * r,(V) and wind speed with increase in stiffness and
unchanged damping. .........................................................................................................
66
Figure 6.15 - Relating CT and wind speed with increase in stiffness and unchanged
d am pin g.............................................................................................................................
67
Figure 6.16 - Increase in cost coefficient with unchanged damping and stiffness, and
relating CTB and cost coefficient.................................................................................
68
Figure 6.17 - Relating exceedance rate density, r,(V), and wind speed for different states.
...........................................................................................................................................
69
Figure 6.18 - The exceedance rate density, r,(V), around the critical wind speed for
different states...................................................................................................................70
Figure 6.19 - Relating Ca * r,(V) and wind speed with unchanged stiffness and damping
for different states.............................................................................................................
70
Figure 6.20 - Relating CT and wind speed with unchanged stiffness and damping for
different states...................................................................................................................7
1
Figure 7.1 - Relating CTB and cost coefficient, p, with increase in damping and
unchanged stiffness........................................................................................................
73
Figure 7.2 - Relating CTB and increase in damping ratio with increase in cost coefficient
and unchanged stiffness. ................................................................................................
74
Figure 7.3 - Cost function for the dampers used in the CAARC Building. .................
75
Figure 7.4 - Increasing the cost coefficient to show the cost effectiveness of increasing
d amp ing.............................................................................................................................
76
Figure 7.5 - Varying the cost coefficient to evaluate the economic impact of increasing
the damping ratio..............................................................................................................
76
Figure 7.6 - Relating CTB and cost coefficient, p, with increase in stiffness and
unchanged dam ping. .........................................................................................................
78
Figure 7.7 - Relating CTB and increase in stiffness with increase in cost coefficient and
unchanged damping. .........................................................................................................
79
Figure 7.8 - Cost function for the increase in stiffness used in the CAARC Building.... 80
Figure 7.9 - Increasing the cost coefficient to measure the cost effectiveness of increasing
stiffn ess. ............................................................................................................................
81
Figure 7.10 - Varying the cost coefficient to evaluate the economic impact of increasing
stiffn ess. ............................................................................................................................
81
Figure 7.11 - Increasing the cost coefficient to show the cost effectiveness of increasing
dam ping for the state of Illinois.....................................................................................
84
Figure 7.12 - Varying the cost coefficient to evaluate the economic impact of increasing
dam ping for the state of Illinois.....................................................................................
85
Figure 7.13 - Increasing the cost coefficient to measure the cost effectiveness of
increasing stiffness for the state of Illinois. ..................................................................
85
Figure 7.14 - Varying the cost coefficient to evaluate the economic impact of increasing
stiffness for the state of Illinois......................................................................................
86
Figure 7.15 - Increasing the cost coefficient to measure the cost effectiveness of
increasing damping for the state of New York. ...........................................................
87
Figure 7.16 - Varying the cost coefficient to evaluate the economic impact of increasing
damping for the state of New York...............................................................................
87
Figure 7.17 - Increasing the cost coefficient to measure the cost effectiveness of
increasing stiffness the state of New York. ..................................................................
88
Figure 7.18 - Varying the cost coefficient to evaluate the economic impact of increasing
stiffness the state of N ew Y ork......................................................................................
88
Figure 7.19 - Increasing the cost coefficient to measure the cost effectiveness of
increasing damping for the state of Massachusetts. .....................................................
89
Figure 7.20 - Varying the cost coefficient to evaluate the economic impact of increasing
damping for the state of Massachusetts. ......................................................................
89
Figure 7.21 - Increasing the cost coefficient to measure the cost effectiveness of
increasing stiffness for the state of Massachusetts. ......................................................
90
Figure 7.22 - Varying the cost coefficient to evaluate the economic impact of increasing
stiffness for the state of Massachusetts..........................................................................
90
LIST OF TABLES
Table 1 - Human Perception Levels ............................................................................
21
Table 2 - Multiplication factors applied to the ISO 10137 base curve to provide maximum
rm s acceleration ................................................................................................................
32
Table 3 - Comparing different codes and guidelines to limit acceleration and ensure
occupant's com fort............................................................................................................
34
Table 4 - Values provided for the coefficients Ct and x...............................................
39
Table 5 - Structural parameters of the CAARC building.............................................
50
Table 6 - Wind speed limits used to determine the return period using data obtained from
th e N OA A .........................................................................................................................
53
Table 7 - Data from the NOAA used to determine the relationship between return period
and wind speed for the state of Florida..........................................................................
54
Table 8 - Determining the relationship between return period and wind speed in the last
6 4 years. ............................................................................................................................
55
Table 9 - Relating CTB and increase in damping with unchanged stiffness. ................. 63
Table 10 - Determining the relationship between the natural frequency of the building
and the percentage increase in stiffness........................................................................
64
Table 11 - Relating CTB and increase in stiffness with unchanged damping................ 67
Table 12 - Relating CTB and different locations with unchanged damping and stiffness...
...........................................................................................................................................
71
Table 13 - Comparing CTB for increasing values of damping and stiffness ................
83
1. INTRODUCTION
1.1
Need for Tall Buildings and a Methodology
Cohen (2003) has mentioned in his research that by 2050 the human population will
increase by 2 to 4 billion people, which is approximately a fifty percent increase to the
world's current population. As population increases, there is a high demand in
constructing tall buildings to provide space for people to live. This is has lead to
designing a relatively new trend of tall buildings for residential use, as tall buildings have
traditionally been mainly used as offices. This is evident in several examples such as the
432 Park Avenue Building in New York City, and the Princess Tower and 23 Marina
Building in Dubai, UAE, which are all taller than the Empire State Building. With the
growing number of tall buildings especially in North America and the Middle East, it is
important to ensure that occupants of these buildings feel comfortable when there is
wind-induced motion. With different codes and standards using different values and
limits to investigate this issue, there is a need to develop an evaluation process that is
universally applicable, and one that can be used by clients and structural engineers to
ensure occupant's comfort. Although there have been advancements in technology, there
still has not been a widely accepted international code or standard to ensure occupant's
comfort for tall buildings. A partial reason for the lack of development of such an
important standard is the subjectivity of the serviceability limits, and the complexity of
the issue as one of the influential factors affecting wind-induced response is the shape of
the building. Therefore, to find a solution to this issue, different factors and their effects
on people were analyzed.
1.2
Issues with Tall Buildings
After World War II, the average density of buildings was 18 pcf (288 kg/m 3) compared to
the last fifty years where the average density of buildings was approximately 9- 12 pcf
(144 to 192 kg/m3 ). This is due to advancements in construction materials that caused
these materials to be stronger. Hence lesser amounts of these materials were used leading
to lighter structures. Moreover, lighter facades, other non-structural members and lesser
partitions were used, which has lead tall buildings to be very light causing them to be
15
susceptible to wind-induced motion (Islam et al., 1990). Although these vibrations and
motions are not enough to cause structural damage, they still tend to cause discomfort to
occupants. Furthermore, although buildings are designed to limit the maximum lateral
story drift to minimize damage, this does not guarantee occupant's comfort (Chang et al.,
2009). Studies have recorded various events when wind-induced motions of buildings
have caused discomfort to occupants. Hansen et al. (1973) has conducted a building
survey on two buildings, and found out that in one building 36% of the building
occupants experienced motion sickness, whereas 47% of the occupants in the second
building experienced motion sickness. Moreover, Goto (1983) has also conducted a
survey after a typhoon, where more than 95% of the occupants above the 13th floor had
reported that they felt building motions. In addition, seventy-two percent of the occupants
had reported symptoms of motion sickness, which include headaches and feeling of
uneasiness that were more significant with occupants in higher floors than the lower ones
(Lamb et al., 2013).
1.3
Problem Statement
Wind loads, especially windstorms, occur more frequently than earthquakes, and may last
for several hours affecting a wide range of buildings. Moreover, in seismic regions,
designers tend to make the buildings flexible enough to overcome destructive effects of
earthquakes, which would cause wind loading to be the dominant load for lateral
resistance (Tallin & Ellingwood, 1984). As wind loadings are frequent, their induced
responses on tall buildings will be frequent as well. Although no structural damage would
occur as the lateral story drift is limited, vibrations and motions caused by wind may
cause discomfort to occupants. Therefore, a methodology needs to be developed that
includes an economical analysis, which would help designers and clients decide on the
acceptable level of motion to meet the serviceability limits of tall buildings.
16
2. LITERATURE REVIEW
Factors affecting occupant's perception
2.1
To understand how to limit serviceability, the factors affecting occupant's discomfort
from a physiological perspective needs to be understood. The affect of motion on the
biodynamical response of the human body can be quantified as:
R = KS"
(Eq. 1)
where
R is sensory greatness
S is the stimulus
n is the exponent
K is a constant
Therefore, to limit the affect of motion on the occupants, the stimuli they experience
should be limited, which would lead to an improvement in the comfort of occupants in
tall buildings (Bashor et al., 2005). The various stimuli that would cause occupants to
perceive motion and affect their comfort would depend on (Tallin & Ellingwood, 1984):
" the frequency of motion,
" the building's acceleration,
" the presence of visual or auditory cues,
* the duration of motion, and
" the occupant's activity.
The frequency of motion is important, as vibrations at frequencies less than 1 Hz are
problematic causing occupants to experience motion sickness. This is because the natural
frequency of tall buildings is small, and wind being a low frequency load can easily
match that natural frequency, making tall buildings more susceptible to resonance
(Ellingwood & Tallin, 1984). When this occurs, the acceleration of the tall building
amplifies, which could reach levels where the occupants feel uncomfortable.
17
Moreover, visual cues of shifting contents and moving horizons with respect to fixed
objects are significant in causing discomfort in occupants especially that occupants do not
expect buildings to move. As torsion enhances this phenomenon, it is an important factor
that needs to be maintained to low levels while designing the structure (Bashor et al.,
2005).
Furthermore, another important factor is the duration of motion. Larger amplitudes of
motion that dampen within a few cycles are more bearable than smaller amplitudes with
longer loading periods. Therefore, the rate of decay of motion is important when
evaluating the serviceability of tall buildings (Ellingwood & Tallin, 1984).
Also, the level of acceleration that occupants could tolerate would depend on the their
activity, as people experiencing excessive motion while undergoing physical activities,
such as exercising in the gym, would tolerate higher levels of acceleration than people in
residential or office buildings. In this thesis, the focus will be on two out of the five
factors, acceleration and frequency, as these parameters are easy to quantify and familiar
to engineers.
18
Vortex Shedding
2.2
As the height of buildings and wind speed increase, the across-wind response of these tall
buildings becomes more critical, and most of the time more important than the building's
along-wind response. In addition, the most common source of across-wind excitation is
vortex shedding.
Torsion
o Along-Wind
Cross-Wind
Wind Directon
Figure 2.1 - Wind Response Directions
Source: Mendis et al., 2007
Vortex shedding is a periodic loading that is caused by fluctuations in pressure on the
&
sides of the building perpendicular to the direction of the wind loading (Tamura
Kareem, 2013). The vortex shedding frequency can be calculated using the Strouhal
Number:
S = f(Eq.
where
S = Strouhal Number
D
=
width of the building
V = wind speed
fl= vortex shedding frequency
19
2)
A typical value for the Strouhal Number lies between 0.1 and 0.4, and this depends on the
cross-sectional shape of the building, Reynolds Number of the wind, surface roughness
and free stream turbulence.
Tall buildings experience excessive accelerations when the vortex shedding frequency
due to the wind load matches one of the natural frequencies of the building causing the
building to resonate. Using equation 2, the critical wind speed, Vcr, which is the wind
speed at which resonance would occur, can be calculated by equating the vortex shedding
frequency to the natural frequency of the building, and making the wind speed the subject
of the formula:
Vcr =
(Eq. 3)
s
where, Wn is the natural frequency of the building.
As shown in Figure 2.2, there are two parameters that would change the across-wind
acceleration of the building due to vortex shedding:
"
Vortex Shedding Force
*
Amplification factor due to resonance
4
Crosswind
Response
Vortex shedding
No vortex shedding
Wind velocity
Figure 2.2 - Effect on vortex shedding on response
Source: Irwin et al., 2008
20
2.3
Influence of different levels of acceleration
Irwin (1978), along with many researchers, has used acceleration as the parameter to
evaluate occupant's comfort in tall buildings confirming that it is the most suitable
parameter (Johann et al., 2015). According to Mendis, P., et al. (2007), Table 1 presents
how different levels of acceleration affect people. These results were obtained by taking
into account the various important physiological and psychological parameters affecting
occupant's perception to different levels of acceleration in tall buildings.
EFFECT
ACCELERATION
(m/sec 2
)
LEVEL
Humans cannot perceive motion
< 0.05
a) Sensitive people can perceive
motion;
b) hanging objects may move
slightly
a) Majority of people will perceive
motion;
b) level of motion may affect desk
work;
c) long - term exposure may produce
motion sickness
a) Desk work becomes difficult or
almost impossible;
b) ambulation still possible
a) People strongly perceive motion;
b) difficult to walk naturally;
c) standing people may lose balance.
Most people cannot tolerate motion
and are unable to walk naturally
People cannot walk or tolerate
motion.
Objects begin to fall and people may
be injured
0.05-0.1
0.1 -0.25
0.25 - 0.4
0.4-0.5
0.5-0.6
0.6-0.7
> 0.85
Table 1 - Human Perception Levels
Source: Mendis et al., 2007
21
Chang (1973) proposed different thresholds for acceleration using data from the aerospace
industry. These limits are:
*
Non-perceptible: a < 5 milli-g;
*
Perceptible: 5 milli-g < a < 10-15 milli-g;
*
Annoying: 10-15 milli-g < a < 50 milli-g;
*
Very Annoying: 50 milli-g < a < 150 milli-g;
"
Unbearable: 150 milli-g < a.
where 1 milli-g is equal to 1/1000th of acceleration due to gravity (Johann et al., 2015).
22
2.4
Factors affecting serviceability
2.4.1
Parameters affecting serviceability
To reduce a building's acceleration, the following factors can be modified:
" Damping
*
Mass
*
Stiffness
*
Shape of the building
Other than the fact that increasing damping, decreases acceleration, there is a distinct
advantage of damping. Increasing damping would decrease the duration of motion the
occupants are going to experience. This is evident from Figure 2.3, which shows that as
damping increases, the acceleration tolerated also increases. This is due to the fact that
people are willing to experience high levels of acceleration if it will only be for a short
period of time (Ellingwood & Tallin, 1984).
23
I
I
I
I
-
0.50
0.20 -
Transient
Idamping = 0,12)/
Transient
damping = 0.0)
.0
-
0.05
/
0.10-
/
Transient
IS 0.02
(damping = 0.03)
0.01-
1
Continuous
2
-
'rani
/
0.0050.005
S0.021
5
10
FREQUENCY (HzJ
20
Figure 2.3 - Annoyance Threshold Vibrations for Residences, Offices and Schools.
Source: Ellingwood and Tallin, 1984
The Scruton Number is a dimensionless factor used to evaluate the effect of vortex
shedding on a structure, and is proportional to the structure's damping and to the ratio
between the vibrating mass and the mass of the air displaced by the structure. Moreover,
an increase in the Scruton Number would represent a decrease in the effect of vortex
shedding. Therefore, if the Scruton Number is less than 30, then excessive vibrations due
to vortex shedding should be taken into consideration while designing the structure
(Baker et al., 2012).
Sc
where
me = effective mass per length
p = density of air
24
= 2me5
2
pD
(Eq. 4)
6
=
structural damping by the logarithmic decrement
6 = 2ni
=
(Eq. 5)
structural damping ratio
2.4.2
Comparing different parameters
The different parameters, damping, mass and stiffness, are compared from an analytical
and structural point of view to measure their effectiveness relatively to one another.
Substituting equation 5 in equation 4:
Sc
= 4nkme
2
PD
(Eq. 6)
From equation 6, it is shown that the damping ratio is proportional to the Scruton Number
by a factor of 1.
Substituting ( = 2cV-ik
in equation 6:
Sc =
41tk 1/ 22m3 /2
(E.7
(Eq. 7)
Equation 7 shows that the Scruton Number is proportional to the square root of stiffness.
The reason for such an effect is that an increase in stiffness would increase the separation
between the vortex shedding frequency and the natural frequency of the building, and this
decreases the chances of resonance to occur. Moreover, increasing the stiffness would
increase the natural frequency, and from equation 3 as the natural frequency increases the
critical wind speed also increases, which decreases the chance of resonance due to vortex
shedding to occur.
Furthermore, equation 7 shows that the Scruton Number is proportional to the mass by a
factor of 1.5. The reason for such an effect is the same as the increase in stiffness, as
increasing the mass increases the separation between the vortex shedding frequency and
the natural frequency of the building, and this decreases the chance of resonance.
25
Increasing stiffness is more beneficial than increasing mass, as it reduces deflection.
Moreover, increasing mass would be a disadvantage during earthquake events as the
effective force the building experiences is a proportional to the mass of the building and
the ground acceleration due to the earthquake. Furthermore, according to Irwin, et al.
(2008) and as shown on Figure 2.4, increasing damping is more effective than increasing
either mass or stiffness to decrease acceleration. In addition to being the most effective
parameter, the unique benefit of dampers is it provides engineers the flexibility in design.
Also, it helps in reducing the structural response to both wind and earthquake loads
(Irwin, 2009).
ADD
A,
3W0
MASS
1% Damping
ADD 30% STIFFNESS
INCREASE DAMPING
0
3% Damping
Design Variables
Figure 2.4 - Relative effects on building motions of changing mass, stiffness, and damping.
Source: Irwin et al., 2008
26
3. CURRENT PRACTICES
3.1
Current standards and codes
The standards and codes specify different limits on structural response to wind that differ
depending on the usage of the building, whether it is a residential building, office
building, hotel or retail.
3.1.1
North American Standards
The ASCE 7-05 (American Society of Civil Engineers - Minimum Design Loads for
Buildings and Other Structures) does not provide limits on wind-induced acceleration.
However, the peak acceleration limits used in practice in the US, which are based on a
10-year return period, are the following (Choi, 2009):
*
Residential = 10 - 15 milli-g
*
Hotel = 15 - 20 milli-g
*
Office
*
Retail
20 - 25 milli-g
=
25 milli-g < a
On the other hand, the National Building Code of Canada (2005) provides a limit on the
peak acceleration to ensure serviceability. The following limits are based on a 10-year
return period: 0.01g for residential buildings and 0.03g for office buildings (Kwok et al.,
2009).
3.1.2
Asian Standards
In the Chinese Building Code, there are different limits of acceleration not only for the
usage of the building but also if it was made of concrete or steel. According to the
"Technical specification for concrete structures rise building"(JGJ3-2010), concrete
buildings that exceed the height of 150 meters would have the following peak
acceleration limits based on a 10-year return period: 0.15 m/s2 for residential buildings
and 0.25 m/s 2 for office buildings. Moreover, according to the "Technical Specification
for Steel Tall Buildings" (JGJ99-98), tall steel buildings would have the following limits
based on a 10-year return period: 0.20 m/s 2 for residential buildings and 0.28 m/s2 for
public buildings.
27
The Architectural Institute of Japan (AIJ) has developed guidelines for the Evaluation of
Habitability to Building Vibration (AIJ-GBV-2004)
to provide limits for peak
acceleration in the form of multiple curves. Unlike the other standards previously
discussed, the acceleration limits provided are frequency dependent. These limits are
based on a one year period, and calculating the acceleration within ten minutes of the
maximum response of the building. The guidelines provide the curves shown in Figure
3.1. Each curve indicates the percentage of people who will perceive a particular motion.
For example, the H-10 curve represents that 10% of the people will perceive motion.
According to this standard, one of these curves will be selected either upon the request of
the owners or upon the requirements of the architect [16, 17].
'.4
0
20
10
*1
0
5
2
2
-1-
IH-10
2
1
0.1
0.2
0.5
1
2
5
Frequency (Hz)
Figure 3.1 - Probabilistic perception thresholds given in AIJ-GBV-2004
Source: Tamura et al., 2004
28
3.1.3
3.1.3.1
International Organization for Standardization (ISO) Standards
ISO 6897 (1984)
ISO 6897 (1984) is a standard that evaluates human comfort for occupants experiencing
horizontal vibrations between 0.063 to 1Hz. This standard provides four curves
presenting the root-mean-square (RMS) acceleration versus frequency. These curves are
based on ten minutes of the response of the structure experiencing wind load, and a return
period of five years. The curves are also based on the idea that the lower the vibration
frequency, the more comfortable the people will be (Lin et al., 2014).
The four curves are the following (ISO, 1984):
*
Figure 3.2, Curve 1: Evaluates the comfort of occupants of buildings that are
experiencing wind loadings.
*
Figure 3.2, Curve 2: Evaluates the comfort of occupants experiencing vibrations
on marine structures.
" Figure 3.3, Curve 1: Evaluates the average acceleration perceived by occupants of
tall buildings due to wind loads.
*
Figure 3.3, Curve 2: Represents the minimum acceleration perceived by
occupants of tall buildings due to wind loads.
29
1.00
-
-_ - -
E
--
-
-_-_
0.80
463
0.50
0.315
0.25Q20
f4
0.16-
0.1250.10
0,000
6
0.063
-
0.050
0.040
0.031 S
-- ------
0.025
0.020
0.016
-
0.0125
0.063 0.06 0.10 0.125
0.16 0.20
0.25 0.315 0.40
.50
0.63
z
0.80 1.00
Frequency, Hz
Figure 3.2 - Suggested satisfactory magnitudes of horizontal motion of buildings used for general
purposes (curve 1) and of off-shore fixed structures (curve 2)
Source: ISO 1984, 1984
0,0630.040
-
0.025
0,0200.016-
*
0
.0105
0.0060.00630.0050
0,0040D,003 IS --
0,002S0.00200.001 6-
),001 25
0.001 0 LL03 -0.06- 0.10
I
o125
Q%16n20
.
. O25 0.31S 040 0 SO 063 0 60 1
)
.
. ..,.0
Frequency, Hz
Figure 3.3 - Average (curve 2) and lower threshold (curve 1) of perception of horizontal motion by
humans
Source: ISO 1984, 1984
30
3.1.3.2
ISO 10137 (2007)
ISO 10137 (2007) is a frequency dependent standard that provides different peak
acceleration limits for a return period of one year to ensure occupant's comfort by
incorporating a group of important factors that affect occupant's comfort in buildings.
These factors include:
*
The surrounding environment whether it is peaceful or active, which is depends on the
usage of the structure whether it is residential, office building, etc.
*
Frequency of the vibration
*
Duration of the vibration
*
Time of the day during which the vibration has occurred, as vibrations at night are more
irritating than the day.
ISO 10137 includes a group of multipliers that are used depending on the usage of the
building. As shown in Table 2, the large range of multipliers is used to show that there is
a diverse range of acceleration limits that can be used for different occupants. Figure 3.4
shows the ISO 10137 curves where A is peak acceleration (m/s 2) and fo is the first natural
frequency in a structural direction of a building and in torsion (Hz). Moreover, it provides
two curves that represent the maximum acceleration limit of a residential and an office
building, where the limit for office buildings are 50% higher than that of residential
buildings. These maximum values are obtained using the highest value of the along wind
acceleration, across wind acceleration and torsion acceleration for the first-order
frequency of structures experiencing wind loading. Torsional acceleration can be
converted to an equivalent translational acceleration as follows:
r *A (t)
(Eq. 8)
where:
r represents the distance from the center of torsion to the point under consideration,
A (t) represents the angular acceleration of the torsional vibration [16, 20].
31
A
0,5
0,3
0,21
0,2
0,15
0,14
0,1
0,08
0,06
0,04
L
0,02 i i i||
0,06 0,1
0,2
0,3
0,5
2
1
2
3
5
fo
Figure 3.4 - ISO 10137 acceleration limits. Curve 1 - maximum horizontal acceleration for office
buildings. Curve 2 - maximum horizontal acceleration for residential buildings.
Source: Lin et al., 2014
These multipliers presented in Table 2 are applied to the base curves provided in Figure
3.4 to obtain the maximum acceleration limit that ensure occupants' comfort.
Time
Place
Critical work areas (eg some Day
hospital operating theatres, some Night
Multiplying factor to base curve
Impulsive vibration
Continuous vibration
& intermittent
with several
vibration
occurrences per day
1
I
1
1
precision laboratories, etc.)
Residential (eg
hospitals, etc)
flats,
homes, Day
Night
Offices (eg schools, offices)
Workshops
Any
Any
2 to 4
1.4
4
8
30 to 90
1.4 to 20
60 to 128
90 to 128
Table 2 -Multiplication factors applied to the IS010137 base curve to provide maximum RMS
acceleration
Source: King, 1999
32
3.2
Comparing different codes and standards
Table 3 shows a comparison between all the discussed codes and standards. This table
shows the difference between these guidelines, and therefore emphasizes the need to
develop an evaluation process that is universally applicable, and one that can be used by
clients and structural engineers to ensure occupant's comfort (Kwok et al., 2009).
The comparison is based on various factors, which include the return period, whether
RMS or peak acceleration was used and whether a certain limit is presented or will it
change depending on the frequency at which the building vibrates. It is shown that only
the ISO 6897 uses the RMS acceleration as the limit, whereas all the other guidelines use
the peak acceleration. Also, it is only the NBCC and Chinese Building Code that have a
certain limit on acceleration, whereas the acceleration limit for the other guidelines
changes depending on the frequency of the vibration. In addition, it is clear that there are
three different return periods used: 1 year, 5 years and 10 years. Return period is one of
the controlling factors to provide occupant's comfort, as the larger the return period the
fewer times occupants will experience acceleration beyond the limits proposed.
Therefore, using a return period of 10 years would mean that occupants would experience
larger acceleration than what is proposed not more than once every 10 years. Hence,
using a 10-year return period is a stricter limit than either 1 or 5-year return period, if the
same requirement on acceleration is considered (Lin et al., 2014).
But although the AIJ-GBV-2004 has a return period of one year compared to either the
Chinese building code and the NBCC that have a return period of 10 years, the
acceleration limits provided by the AIJ-GBV-2004 are lower than the limits provided by
the Chinese Building Code and the NBCC. This is because the AIJ-GBV-2004 provides
these limits based on people's perception of motion, while the limits provided by the
Chinese Building Code and the NBCC are based on limiting occupant's discomfort.
33
Chinese Building Code
NBCC
AIJ-GBV-
IS06897
IS010137
2004
(1984)
(2007)
10 years
10 years
10 years
1 year
5 years
1 year
Peak
Peak
Peak
Peak
RMS
Peak
acceleration
acceleration
acceleration
acceleration
acceleration
acceleration
Limits
0.15
0.20
0.10
Curve
Curve
Curve
(m/s2)
0.25
0.28
0.30
Load
return
period
Index
Curve
Table 3 - Comparing different codes and guidelines to limit acceleration and ensure occupant's
comfort
Source: Lin et al., 2014
34
4. CHALLENGES
Unlike ultimate limit states where the limits of structural members are well defined and
can be easily calculated, different people can react differently to serviceability limit
states. Moreover, defining these limits for acceptable levels of acceleration is a complex
task, as different levels of acceleration can affect people depending on multiple factors
such as the occupant's activity, and their physiological and psychological state. Hence,
there is no clear distinction between acceptable and unacceptable limits of acceleration
and that deciding on a certain limit to satisfy occupant's comfort is subjective (Lamb et
al., 2013)).
In addition to the multiple factors involving occupants, there are also multiple factors
related to the structure itself, and varying each of these factors in order to decrease
acceleration can cause negative outcomes such as increasing deflection or increasing in
the negative effect of earthquakes. This is evident in how increasing mass to minimize
vortex shedding can increase the force the building is experiencing during an earthquake.
Moreover, it has been mentioned that wind induced vibrations can change by changing
the building shape, which adds to the complexity of determining the acceleration of the
building. This is because it will be more difficult to calculate the different parameters
required to determine the acceleration of the building, which also means that errors are
prone to occur while calculating for these parameters (Huang et al., 2012).
35
5. PROPOSED SOLUTION AND METHODOLOGY
The proposed solution and methodology will include the addition of dampers and
increase in stiffness to solve the problem. There will be no change in mass, as adding
mass would worsen the situation during earthquake events (Irwin, 2009).
5.1
Proposed Solution
The Performance-Based Earthquake Engineering framework developed by the Pacific
Earthquake
Engineering Research Center (PEER) was adapted
to present the
Performance-Based Wind Engineering (PBWE) methodology (Ciampoli et. al., 2011).
Although Ciampoli et al. (2011) have already proposed a PBWE, this methodology was
further modified in this thesis into the criteria shown in Figure 5.1.
Hazard Analysis
Structural Analysis
I
Damage Analysis
Cost Analysis
Decision-making
Figure 5.1 - Flowchart presenting the modified PEER methodology
37
Hazard Analysis
In the hazard analysis, the wind speed is the main factor involved and this will depend on
the location of the tall building.
Structural Analysis
In the structural analysis, the wind speed determines the vortex shedding frequency and
force, which are used to calculate the acceleration of the building. Moreover, along with
the wind speed, the dimensions and structural properties of the building will affect:
" the natural frequency of the building
" the vortex shedding of the building
*
the Strouhal Number
Damage Analysis
As the focus of this thesis is regarding serviceability issues, factors that can be considered
as damage due to wind-induced acceleration are:
" People experiencing motion sickness
*
Affect on people's work and productivity
Cost Analysis
From the damage analysis, a cost analysis will be made using cost functions that will take
into account the cost of the negative effects of increasing acceleration. Moreover, there
will be two scenarios to compare the effects of increasing dampers and increasing
stiffness. These scenarios will include:
" Cost of increase in materials to increase stiffness
*
Cost of dampers and materials used to retrofit the structure to increase damping
38
5.2
Methodology
5.2.1
Period of the Structure
There are multiple ways to approximately calculate the fundamental period and the modal
damping ratio of a building structure. In this thesis two approaches will be presented, one
using the ASCE 7-10 code (Minimum Design Loads for Buildings and Other Structures,
2010) and another using the Japanese Damping Database (Tamura & Kareem, 2013).
ASCE 7-10 code
Approximately, the fundamental period is determined as the following:
Ta = Ct * h'
(Eq. 9)
where hn is the height of the structure, and Ct and x are coefficients that depend on the
structural system of the building and are provided in Table 4. These values are obtained
from Table 12.8-2 from the ASCE 7-10 code.
Structure Type
Moment-resisting frame systems in which the frames resist 100% of the
required seismic force and are not enclosed or adjoined by components
that are more rigid and will prevent the frames from deflecting where
subjected to seismic forces:
Ct
x
Steel moment-resisting frames
Concrete moment-resisting frames
Steel eccentrically braced frames
Steel buckling-restrained braced frames
All other structural systems
0.028 (0.0724)a
0.016 (0.0466)a
0.03 (0.0731 )a
0.03 (0.073 1)a
0.02 (0.0488)a
0.8
0.9
0.75
0.75
0.75
Table 4 -Values provided for the coefficients Ct and x
Source: Minimum Design Loads for Buildings and Other Structures, 2010
Japanese Damping Database (JDD)
Equations 10 to 15 have been provided by Tamura and Kareem (2013), and researchers
involved with the AIJ. These values where obtained from the database after analyzing
285 buildings and structures, which include reinforced concrete and steel buildings, and
other tall structures that are not buildings. The following equations were the result of the
39
analysis made, where the fundamental natural period is proportional to the building
height H (m) (Tamura & Kareem, 2013):
5.2.2
T, = 0.0 15H for reinforced concrete buildings
(Eq. 10)
T, = 0.020H for steel buildings
(Eq. 11)
Modal Damping Ratio
ASCE 7-05
Using the ASCE 7-10 Commentary, the suggested damping values are 1% for steel
buildings and 2% for concrete buildings.
JDD
Moreover, in the JDD, equations were developed to obtain the damping ratios for
reinforced concrete and steel buildings. Equations 11 and 12 are the natural frequency of
reinforced concrete (RC) and steel (S) buildings respectively, and equations 13 and 14 are
the fundamental damping ratios of reinforced concrete and steel buildings respectively
(Tamura & Kareem, 2013).
1
-
=
T,
1
1
-
0.O15H
=
0.020H
67
H
for reinforced concrete buildings
(Eq. 12)
for steel buildings
(Eq. 13)
H
= 0.014f1 + 470 !H - 0.0018 =
0+
H
470 HH- 0.0018
RC buildings (Eq. 14)
= 0.013f, + 4 00 H- + 0.0029 =
0.5+
400 -+
S buildings
0.0029
(Eq. 15)
Estimating the damping ratios mentioned using equations 13 and 14 are limited to a
particular range of height of the buildings, where the range is 10 to 100 m for reinforced
concrete buildings and 30 to 200 m for steel buildings. The height of the building used as
an example in section 6 is within the limits provided by the JDD.
40
5.2.3
Deflection of a building
Periodic wind loading can be expressed as:
P = peiwt
The response of the structure can then be given as:
U = uei(ft-d)
(Eq. 16)
The amplitude of the response with respect to one of the loading can be shown to be:
2
2
U [(k- fl m) + (nc)
(Eq. 17)
2
(
u =
Rearranging the above formula gives:
u =
[(1i- p 2 ) 2 +(2 p Q)22]E
(Eq . 18)
p =
-
where
k
Simplifying equation 18:
u =EH,
k
(Eq. 19)
where
H
[(1-
1
[j p 2 ) 2 + (pg)
41
2
]
(Eq. 20)
5.2.4
Acceleration of a building
Acceleration of the structure can be given as:
a
Substituting u =
k
(Eq. 21)
= uf2
H 1 in equation 21:
a = EH
(Eq. 22)
f2
Substituting k = W m in equation 22:
a =
Substituting
bg22
G)2 =
p2
(Eq. 23)
2H,
in equation 23:
(Eq. 24)
a = Pp2 H,
Substituting Hjp 2 = H 2 in equation 24:
(Eq. 25)
where
p2
H2
=
2 2
[(1- p ) + (2pk) 2 ]
(Eq. 26)
[
2
(Eq. 27)
Therefore,
a =
M j[(1- p2)2+ (2pk)
Now, the affect of vortex shedding needs to be incorporated into the acceleration
equation.
p
42
sv
-,sv
in p
f
-
Substituting the vortex shedding frequency equation, 2 =
(Eq. 28)
Substituting p = S t in equation 27:
2
(SV
a =PDomS
(Eq. 29)
22
Simplifying equation 29 to:
1
where 3 =
(Eq. 30)
-
a =
Simplifying equation 30 to:
(Eq. 31)
a a=-H
= LHs
where Hs is the amplification factor:
H 5 =-
-
19
2
0
(Eq. 32)
with p the magnitude of the vortex shedding force:
p =
2
pCLV DH
2
(Eq. 33)
where
*
p is the density of air
*
V is the wind speed
"
D is the building dimension perpendicular to the direction of wind
*
H is the height of the building
*
CL is the lift coefficient, which depends on the shape of the building and the flow of fluid
around the building.
43
5.2.5
Parameters used to develop the cost functions
Cost function for damping
Inherent damping ratio (%): a
Maximum damping ratio (%): qj
Percentage of the cost of the building that must be added to reach the maximum damping
ratio (%): A
Damping ratio actually required (%):
Cost of the space acquired by the dampers that could have been used to gain profit: CR
Total cost of damping as a fraction of the cost of the building: CD
CD
-a+CR
(Eq. 34)
A square root relationship is established between the cost of the dampers and the
damping ratio. Such a relationship is appropriate, as the cost required to increase
damping is large with small damping ratios, and decreases with large damping ratios.
With increase in damping, the structure is retrofitted, and the cost to retrofit decreases
every time the damping ratio of the building is increased, which also justifies the square
root relationship.
Cost of damping as a percentage of the cost of the building, CDB,
CDB = CD * 100
44
(Eq. 35)
Cost function for negative effects of acceleration
Acceleration less than 0.05 m/s 2 is imperceptible, and therefore its cost will be 0. A linear
relationship was made to evaluate the cost of increasing acceleration.
Ca
1
a (m/s 2
)
0.05
Figure 5.2 - Cost function presenting the negative effects of acceleration using the cost coefficient, p.
The cost function, Ca, shown in Figure 5.2 is presented in the form of an equation:
0,
C
a < 0.05 m/s 2
(20a - 1), 0.05 M/s
a
2
> a
Substituting equations 31, 32 and 33 in equation 36:
0,
Ca=
Ca
19
(2 0 p
m
[(p2-1)2+
2
(2pg) I
a < 0.05 m/s 2
- 1), 0.05 m/s2 > a
(Eq. 37)
t is a measure of how clients and structural engineers value occupant's comfort.
Moreover, p is a cost coefficient that is expressed as a percentage of the cost of the
building when acceleration of the building is 1M/s 2 . Hence, an increase in p would mean
prioritizing occupant's comfort by magnifying the cost of occupant's discomfort.
The cost of the negative effects of acceleration as a percentage of the cost of the building,
CaB, is:
CaB = Ca * 100
45
(Eq. 38)
Moreover, the cost of the negative effects of acceleration expressed as a fraction of the
cost of the building experiencing wind speeds up to a certain magnitude for one year, CT,
is:
CT
=
fv
Ca * r,(V) dV
(Eq. 39)
Total cost of the negative effects of acceleration as a fraction of the cost of the building
for one year is CTB, and the percentage of the total cost of the negative effects of
acceleration on the building for one year is C%TB-
CTB
is the value of CT integrating over
wind speeds up to 200 m/s. Keeping the wind speed limit to 200 m/s is a fair
approximation, as the additional cost due to wind speeds exceeding this value is
insignificant due to their low occurrence probability.
CTBI
C0
a *rET(V) dV
(Eq. 40)
200
C%TB
=
f20 0 Ca * rT(V) dV * 100
(Eq. 41)
Cost function for stiffness
Assuming:
only rectangular sections are used
Stiffness of a frame structure, k = I)
where f = 12 for fixed-fixed columns and f= 3 for pinned-pinned columns
x = Percentage increase in stiffness
knew = (1 + x)kinitial
fE I
3
fEl
new = (1+ x) ()initial
'new =
(1 + X)Iinitial
bh 3
bhW
(2)new
=
(1
)initial
(h 3 )new = (1 + x)(h 3 )initial
46
hnew = hinitiaiVf1i+U
(Eq. 42)
Assuming the lengths or heights of the sections are constant, equation 42 is changed to:
Vnew = Vinitial
1+ X
Cnew = Cinitialivl+ X
Therefore the cost of adding stiffness as a fraction of the cost of the building, Cs, is
Cs = VY +x
(Eq. 43)
where I = moment of inertia, E = modulus of elasticity, b = smaller side of the section,
h = larger side of the section, Vne, = volume of the new section, Vinitial = volume of the
current section, Cnew:= cost of the new section, Cinitial = cost of the current section.
The cost of the increase in stiffness as a percentage of the cost of the building, CsB, is:
CSB
=
CS * 100
(Eq. 44)
Total serviceability cost
There are two scenarios that will be analyzed separately, one for increase in damping and
the other for increase in stiffness.
Total serviceability cost as a fraction of the cost of the building for one year by increasing
damping, Ctotal,D, and the percentage of the cost of the building required to meet
serviceability of the building for one year by increasing damping, Ctotal,%D, are:
Ctotal,D
Ctotal,%D
CTB + CD
Ctotal,D * 100
(Eq. 45)
(Eq. 46)
Total serviceability cost as a fraction of the cost of the building for one year by increasing
stiffness, Ctotai,s, and percentage of the cost of the building required to meet
serviceability of the building for one year by increasing stiffness, Ctotal,%s, are:
Ctotai,s = CTB + CS
Ctotal,%S = Ctotal,s * 100
47
(Eq. 47)
(Eq. 48)
The total serviceability cost as a percentage of the cost of the building for N years by
increasing damping:
Ctotal,ND
where r =
[
*
too
+ [CR
*
100] + CTB)
*
(-)rN
(Eq. 49)
and is the discount rate.
Total additional cost as a percentage of the cost of the building required to meet
serviceability of the building for N number of years by increasing stiffness:
Ctotal,NS =
CSB + [C%TB
48
* ((Eq.
50)
6. DETAILED EXAMPLE
The analyses presented in this section were based on the following assumptions:
*
The calculations were made by taking the building as a single-degree-of-freedom
(SDOF).
"
The structure analyzed is a rectangular cross-section building with a flat top, no parapets
and no geometric irregularities.
In this section, a detailed example is provided to show how the methodology works using
wind data for the state of Florida. The same building was analyzed in three different
states, Illinois, New York and Massachusetts, to evaluate the affect of different
geographic locations and impact of different patterns of wind events on the acceleration
of the building.
6.1
CAARC Building: Structural Analysis
The CAARC Building has been used as an example by several researchers, and its
structural properties have been studied carefully using wind tunnel testing. A schematic
diagram of the building is shown in Figure 6.1, and the values presented in Table 5 were
provided by Cui and Caracoglia (2015).
Wind
h
B
Figure 6.1 - Schematic diagram showing the dimension of the CAARC building.
Source: Cui and Caracoghia, 2015
49
Value
30.5 m
45.7 m
183 m
223224 kg/m
1%
0.2 Hz
(z/h)Y: y = 1
0.116
0.287
Quantity
B
D
H
m(z)
nox, noy
< x(z)
S
CL
Table 5 - Structural parameters of the CAARC building
Source: Cui and Caracoglia, 2015
The structural parameters in Table 5 represent the following:
*
m(z) is the mass per vertical length of the building
*
k is the inherent damping ratio
*
nox is the fundamental natural frequency along the x direction
"
noy is the fundamental natural frequency along the y direction
S<px (z) is the fundamental mode shape
*
S is the Strouhal Number
"
CL is the lift coefficient
The information presented on the CAARC Building was used to carry out the structural
analysis, and find the acceleration of the building for different wind speeds.
Using equation 33, the vortex shedding force is:
2
PCL V DH
p=
2
where
" p is the density of air = 1.25 kg/m 3
*
D is the dimension perpendicular to the direction of wind = 45.7 m
*
H is the height of the building
*
CL is the lift coefficient = 0.287
*
V is the wind speed
=
183 m
50
1.25*0.287*45.7*183*V
2
2
= 1500V 2
Using equation 30, the values provided in Table 5 and the vortex shedding force, the
relationship between wind speed and the acceleration of a building is:
150OV
a
1
p
a =
M[(p2 _ 1)2 +
(2 Pk2]
Dw
45.7*0.2
78.8
5V
0.116*V
V
2
1
223224.2
((7.
2
)
-p-) -
(Eq. 51)
.o
2
+
2*-V-*0.01)
2
12
10 4
IA
0
(Ul
6
M
4
0
0
20
40
60
100
80
120
140
160
180
200
Wind speed, V (m/s)
Figure 6.2 - The response of the CAARC building without increasing damping or stiffness.
51
Figure 6.2 was obtained using equation 51. As it is shown in Figure 6.2, there are two
factors that would increase the acceleration of the building in relation with increase in
wind speed:
" Resonance in a short range of wind velocities where the vortex shedding frequency is
equal to the natural frequency of the building.
*
Increase in the vortex shedding force at high wind velocities
Also, it is evident in Figure 6.2, that each of these factors govern in a particular range of
wind speeds. Resonance occurs at the critical wind speed and dominates in evaluating the
acceleration of the building. But as the wind speed exceeds the critical wind speed, the
resonance effect decreases and the vortex shedding force dominates, making it the
governing factor in evaluating the acceleration of the building.
52
6.2
Relating wind speed with various parameters: Hazard Analysis
The data required to determine the relationship between the return period and wind speed
were obtained from National Oceanic and Atmospheric Administration (NOAA) for the
last 64 years from 1950 to 2014 for the state of Florida.
The limits for the different wind speeds used to compute the return period were
determined using the Fujita-Pearson Tornado Damage Scale as shown in Table 6.
Scale
Wind speed (mph)
FO
<73
Fl
73-112
F2
113-157
F3
158-206
F4
207-260
F5
261-318
Table 6 - Wind speed limits used to determine the return period using data obtained from the NOAA
53
Wind speed
(mph)
Number of
tornadoes
Number of
thunderstorms
<73
1553
5979
73-112
816
102
113-157
327
158-206
207-260
E
1553 +5979
=7532
7532
8819- 7532=
1287
816+102=
7532+918
8819-8450=
918
= 8450
369
1
327+1=
328
8450+328
= 8778
8819-8778=
41
37
0
37+0=37
8815
8819-8815=4
4
0
4+0= 4
8815+4=
8819
8819 -8819 = 0
8778 + 37
=
Total number of events for each range
Q = Cumulative number of events for each range
E = Total number of events - Cumulative number of events for each range
=
Table 7 - Data from the NOAA used to determine the relationship between return period and wind
speed for the state of Florida.
A = 2014 - 1950 = 64 years, the range of time that was be used to determine the return
period, as it is the time period for the data obtained.
The middle value of each range of the wind speeds shown in Table 7 were chosen as the
points that will be used find the return periods, which was used to find the relationship
between the wind speed and the return period, as shown in Table 8.
54
Return period,
Wind speed
Wind speed
(mph)
(m/s)
e
TR (years)
42
1287
64
-0.0497
1287=
135
61
369
182
82
41
234
105
4
93
93
=
- = 0.173
64
-=
1.56
61
-4= 16
Table 8 - Determining the relationship between return period and wind speed in the last 64 years.
The values in Table 8 were used to draw the graph in Figure 6.3 to obtain the equation
that relates the return period to the wind speed.
18
16
14
12
10
0
8
CL
*0.
6
4-:
4
2
0
0
20
60
40
80
100
120
Wind Speed, V (m/s)
Figure 6.3 - Relationship between return period and wind speed.
Therefore, the equation relating the return period, TR, and wind speed, V, is:
TR
=
0.0008e0.0926V
55
(Eq. 52)
The exceedance rate function, R(V), gives the average number of events of magnitude
exceeding wind speed, V, over a period
T:
T
RT(V) =
O.OO
T
0.
92 6 V
(Eq. 53)
The exceedance rate density function, rT(V), can be used to aggregate the consequences
of successive events. This is used in our analysis to find the consequences of wind speed
over the lifetime of a building.
dRT(V) - 115.75te(-0 0.
rT(V) -
T is
dV
92 6 V)
the expected lifetime of the building. In this example,
T
(Eq. 54)
is 50 years making the
exceedance rate density function equal to:
_dR
rT(V)
-
(V)
= 5787.5e
dV
dV
(-0.0926V)
(Eq. 55)
Figure 6.4 is the graph expressing equation 55.
120
100
L-
80
At
60
CU
40 1U
1.X
20
0
0
20
60
40
80
100
120
Wind Speed, V (m/s)
Figure 6.4 - Relationship between exceedance rate density, r, (V), and wind speed, V.
56
6.3
Cost Analysis
The results in the following graphs are obtained by keeping the cost coefficient, R,
presented in equation 37 at a constant value of 0.1%.
0.014
0.01
-
0.012
0.008
0.006
Modified
Cost
Function
0.004
-Actual
Cost
Function
0.002
0
0
100
50
150
200
Wind speed, V (m/s)
Figure 6.5 - Modifying the cost function for negative effects of acceleration, Ca-
In Figure 6.5, the cost function has been modified so that the maximum cost for negative
effects of acceleration, Ca, which occurs at the critical wind speed, is the same for all
wind speeds exceeding the critical wind speed. The reason for this modification is that
the speed of wind does not increase immediately. Therefore, a recorded value of wind
speed exceeding the critical wind speed value would have had to increase gradually
reaching the critical value causing the building to resonaate. Then it would exceed the
critical value to reach the recorded value.
57
0.001
0.0009
0.0008
- _______
0.0007
I-
0.0006
I-.
0.0005
*
1~
(U
K
0.0004
0.0003
0.0002
0
0
20
40
60
100
80
120
140
160
180
200
Wind speed, V (m/s)
Figure 6.6 - Relating Ca * rT(V) and wind speed with unchanged stiffness and damping.
0.04
0.035
0.03
0.025
s
0.02
0.015
0.01
-
0.005
0
0
20
40
60
100
80
120
140
160
180
200
Wind speed, V (m/s)
Figure 6.7 - Relating CT and wind speed with unchanged stiffness and damping.
58
Figure 6.6 was obtained by multiplying the values obtained from Figures 6.4 and 6.5, and
the area under the curve represents the total cost of the negative effects of acceleration on
the occupants of the building. Moreover, Figure 6.6 shows that the critical wind speed
has a significant impact on the cost of the negative effects of acceleration on the
occupant's comfort. Figure 6.7 shows the cost of the negative effects of acceleration
expressed as a fraction of the cost of the building experiencing wind speeds up to a
certain magnitude for one year. The values for the wind speed were stopped at 200m/s, as
the increase in cost for wind speeds exceeding 200m/s are insignificant.
59
6.4
Sensitivity Study
6.4.1
Effects of damping
The results in the following graphs are obtained by keeping the cost coefficient parameter
presented in equation 34 constant at a value of 0.1%.
12
10
8
E
--
-
4---
-
--
-
---
-
= 1%
k= 10%
.
= 20%
2
01
0
20
40
60
80
100
120
140
160
180
200
Wind speed, V (m/s)
Figure 6.8 - The response of the CAARC building increase in damping and unchanged stiffness.
In Figure 6.8, as the damping ratio increases, the amplification of the vortex shedding due
to the resonance effect decreases. This is evident in Figure 6.8, when the damping ratio is
20%, the curve is almost linear showing that the vortex shedding force is always the
governing factor. Moreover, it is also clear from Figure 6.8, that damping has a great
effect in reducing the amplification factor HS (equation 32), but a minimal effect in
reducing the vortex shedding force p (equation 33).
60
0.014
0.012
0.01
0.008
-
=1%
-
= 2%
--
0.006
=5%
--
=
10%
---
=
20%
0.004
0.002
0
0
50
100
150
200
Wind speed, V (m/s)
Figure 6.9 - Relating Ca and wind speed with increase in damping and unchanged stiffness.
The modification made in Figure 6.5 was made in Figure 6.9 presenting different
damping ratios. But these modifications only need to be made at damping ratios less than
5% since the cost functions for damping ratios exceeding 5% do not decrease after the
critical wind speed.
In addition, in Figure 6.9, as the damping ratio increases, the cost function becomes
almost linear, which is evident in the curve representing the damping ratio of 20%, which
is the same effect seen in Figure 6.8.
61
0.001
0.0009
0.00080.0007
-
--
0.0006
0-.
1.
= 1%
0
0.0004
10%
=
0.0003-
20%
0.0002
0.0001
0
0
50
100
150
200
Wind speed, V (m/s)
Figure 6.10 - Relating Ca
*
r,(V) and wind speed with increase in damping and unchanged stiffness.
The area under the curves in Figure 6.10 represents the total cost of the negative effects
of acceleration on the building. It is clear that as the damping ratios increases, the area
under the curves decreases, which represents a decrease in the cost of the negative effects
of acceleration on the building. This effect is shown in Figure 6.11, which shows the cost
due to negative effects of acceleration on the building for wind speeds up 200 m/s.
Moreover, as the damping ratio increases, the curves in Figure 6.10 shift from having a
peak and a significant value at the critical wind speed to the area under the curves
spanning mostly over lower wind speeds than the critical value.
62
0.03
-
0.035
-
A A4
0.025
(in)
k(=2%
----
(
5%
=
10%
-
0.015
1%
=
---
0.02
--
(=20%
0.01
0.005
0
0
100
50
150
200
Wind speed, V (m/s)
Figure 6.11 - Relating CT and wind speed with increase in damping and unchanged stiffness.
(%)
1
2
5
10
20
Table 9 - Relating
CTB
CTB
0.0376
0.0252
0.0157
0.0106
0.00595
and increase in damping with unchanged stiffness.
CTB is the CT value at 200 m/s, which is the total cost of the negative effects of
acceleration on the occupants as a fraction of the cost of the building. These values are
obtained from Figure 6.11 and are tabulated in Table 9. This table shows that as damping
increases, CTB decreases, which shows how affective and economically beneficial
damping could be. The cost of additional damping has not been included yet with these
results.
63
6.4.2
Effects of stiffness
The values used to analyze the affect of increasing stiffness on the acceleration of the
building were determined by calculating the stiffness required to increase the natural
frequency of the building by 0.01Hz. The values for the increase in stiffness are presented
in Table 10.
Increase in stiffness (%)
0
10
k
1.63*106
1.80*106
o (Hz)
0.20
0.21
0.22
1.98*106
21
0.23
0.24
2.16*10"
2.35*106
32
44
0.25
2.55*106
56
Table 10 - Determining the relationship between the natural frequency of the building and the
percentage increase in stiffness.
The results in the following graphs are obtained by keeping the cost coefficient, [t,
presented in equation 37 at a constant value of 0.1%.
25
20
IA
E
-il
15
'U
-
k = 0%
-
k = 10%
k= 21%
-
-
0
-k
= 32%
Ir
M
10
= 44%
-k
-k
0
0
50
100
150
=
56%
200
Wind speed, V (m/s)
Figure 6.12 - The response of the CAARC building with increase in stiffness and unchanged
damping.
64
In Figure 6.12, there are two effects that are clearly depicted when the stiffness is
increased. Increasing stiffness, increases the natural frequency of the building, therefore
for resonance to occur, the frequency for the vortex shedding force also has to increase.
This is achieved by increasing the wind speed, and therefore the critical wind speed at
which resonance
occurs increases. Moreover, increasing stiffness, increases the
maximum acceleration a building could experience due to resonance. Therefore, unlike
damping, stiffness does not decrease the acceleration of the building at the critical wind
speed but increases it.
0.02
0.018
-
-
-
--
-
-
0.014
-
-
-
0.016
0.012
--S
0.01
-
-
---
-
-
0.008 -
k=0%
k = 10%
k =21%
____
k =32%
0.006
---
0.004
-
k=44%
-
k=56%
0.002
0
0
20
40
60
80
100
120
140
160
180
200
Wind speed, V (m/s)
Figure 6.13 - Relating Ca and wind speed with increase in stiffness and unchanged damping.
The modification made for Ca shown in Figure 6.5 was made in Figure 6.13 with
different stiffness values. Unlike damping, these modifications are made for all the
stiffness values, as the cost due to maximum acceleration at critical wind speed is always
greater than the acceleration due to the vortex shedding force. Hence, the amplified
acceleration at critical wind speed will always govern the cost due to negative effects of
acceleration on the occupants of the building.
65
0.001
0.0009
0.0008
-
0.0007
0.0006
(U
LI
-
k = 10%
-
k = 21%
-
k = 32%
0.0005
*
1~
k = 0%
0.0004
k = 44%
0.0003
k = 56%
0.0002
0.0001
""" """" ~~~~ ~
----
,
0
50
0
100
150
200
Wind speed, V (m/s)
*
Figure 6.14 - Relating Ca
r (V) and wind speed with increase in stiffness and unchanged damping.
The area under the curves in Figure 6.14 would represent the cost of the negative effects
of acceleration on the building. It is clear that as stiffness increases, the area under the
curves also decreases, which represents a decrease in the cost of the negative effects of
acceleration on the building. The main reason for this effect is increasing stiffness causes
an increase in critical wind speed, causing the probability of that wind speed to occur to
be minimized leading to a decrease in CT as shown in figure 6.15. The cost due to
negative effects of acceleration on the building was shown in Figure 6.15 for wind speeds
up to 200 m/s, as an increase in cost for wind speeds exceeding 200 m/s is very small and
insignificant. Moreover, unlike increase in damping, the critical wind speed would
always have an impact on the CT value, as it shown in Figure 6.14 that the area under
Ca * r,(V) curve peaks at the critical wind speed when stiffness is increased. This is
because increasing stiffness does not reduce the acceleration's amplification factor.
66
004
0.03
--- ----
-
--
k = 0%
0.025
k = 10%
I---
I-
0.02
F
-- -k=21%
-
-
--
-
0.015
k = 32%
k = 44%
k = 56%
0.005
0
0
20
40
60
80
100
120
140
160
180
200
Wind speed, V (m/s)
Figure 6.15 - Relating CT and wind speed with increase in stiffness and unchanged damping.
Increase in stiffness (%)
0
CTB
0.0376
5
10
16
21
27
32
38
44
50
56
0.0339
0.0306
0.0276
0.0250
0.0225
0.0204
0.0185
0.0168
0.0152
0.0138
Table 11 - Relating CTB and increase in stiffness with unchanged damping.
acceleration on
CTB is the CT value at 200 m/s, which is the cost of the negative effects of
the occupants as a fraction of the cost of the building. These values are obtained from
Figure 6.15 and are tabulated in Table 11. This table shows that as stiffness increases,
stiffness could
CTB decreases, which shows how affective and economically beneficial
be. The cost of additional stiffness has not been included yet with these results.
67
6.4.3
Varying cost coefficient
0.8
0.7
0.6
0.4
-----------
0.3
-I-
0.1
0
0
0.2
0.4
0.6
0.8
1.2
1
1.4
1.6
1.8
2
Cost coefficient, p (%)
Figure 6.16 - Increase in cost coefficient with unchanged damping and stiffness, and relating
cost coefficient.
CTB
and
Figure 6.16 shows that as the cost coefficient increases, the CTB also increases, which is
what is expected. This shows that the analysis is moving towards the right direction, and
as decision makers value occupant's comfort, which is expressed by increasing t, the
cost due to negative effects of acceleration, CTB, also increases.
68
6.4.4
Effect of the geographical location
Different locations will be used to study the affect of how the relationship between the
return period and the wind speed could greatly impact the results obtained. Four different
states were analyzed to show how increasing the different factors, damping and stiffness,
could affect a building from an economically perspective.
Figure 6.17 shows the exceedance rate density versus the wind speed curves. The
exceedance rate density as discussed previously is obtained from the relationship between
the return period and wind speed. From the figure, it is clear that the exceedance rate
density of Florida is the highest for low wind speeds, followed by Illinois, New York and
then Massachusetts. It has been calculated from the structural analysis section that the
critical wind speed is about 79 m/s, therefore the exceedance rate density around that
wind speed is the most important. Figure 6.18 shows the exceedance rate density around
the critical wind speed region and it clear that Illinois has the highest exceedance rate
density, followed by Florida, New York and then Massachusetts.
120
80
-
100
60
-Florida
lIlnois
New York
a,40
-
-
20
Massachusetts
0
0
10
20
40
30
50
60
Wind speed, V (m/s)
Figure 6.17 - Relating exceedance rate density, r,(V), and wind speed for different states.
69
0.4
j.
0.35
0.3
0.25
;A-
0.2
-
Florida
0.15
-
New York
"0.
llnois
U,
Massachusetts
x
III
0.1
0.05
0
72
70
74
78
76
80
82
84
86
88
90
Wind speed, V (m/s)
Figure 6.18 - The exceedance rate density, r,(V), around the critical wind speed for different states.
The following results are obtained by keeping the cost coefficient, p., constant at 0.1%.
A
02
0.002
0.0015
0.001
Florida
-
Illinois
-
New York
-
Massachusetts
-
U
-
0.0005 4
0
0
--
| | -50
100
is
150
200
Wind speed, V (m/s)
Figure 6.19 - Relating Ca * r,(V) and wind speed with unchanged stiffness and damping for different
states.
70
0.12
0.1
-----
-
---- -- - ------------------
-
------
0.08
006
0.04
-
Florida
-
Illinois
-
New York
-
Massachusetts
0.02
0
150
100
50
0
200
Wind speed, V (m/s)
Figure 6.20 - Relating CT and wind speed with unchanged stiffness and damping for different states.
Table 12 - Relating
CTB
Site/Location
CTB
Florida
0.0376
Illinois
0.0990
New York
0.0171
Massachusetts
0.00839
and different locations with unchanged damping and stiffness.
As expected, Illinois had the highest cost due to the negative effects of acceleration on
the occupants of the building followed by Florida, New York and then Massachusetts,
which is evident in Figures 6.19, 6.20 and Table 12. This shows that the exceedance rate
density, which is dependent on the location of the building, can play a significant role in
increasing or decreasing the CTB-
71
7. APPLICATIONS
7.1
Cost effectiveness of damping
7.1.1
Varying cost coefficient and damping
Figures 7.1 and 7.2 show the effect of increasing the damping ratio from an economic
perspective by varying the cost coefficient parameter presented in equation 37. But these
figures do not include the cost of damping. From these two figures, it is evident that
increasing the damping ratio is always beneficial and that it would decrease the cost of
negative effects of acceleration on the occupants.
0.8
0.7
--
+
0.6
-
-
0.5
-=
1%
0.41 -__-k__=_2%
0.3
-
=5%
-
k= 20%
= 10%
--
0.2-Mw
0.1
OP
0
0.5
1
1.5
2
Cost coefficient, p (%)
Figure 7.1 - Relating CTu and cost coefficient, p, with increase in damping and unchanged stiffness.
73
0.8
--
0.7
0.6
0.5
=
0.5P
CO
0.4
-
0.3-----------
0.1%
p= 0.5%
--
p= 1%
--
p=2%
p
-- =1.5%
-
0.2
0 4
0
2
4
6
8
12
10
14
16
18
20
Damping ratio, k,(%)
Figure 7.2 - Relating CTB and increase in damping ratio with increase in cost coefficient and
unchanged stiffness.
74
7.1.2
Economical outcomes of increasing damping
In this example, viscoelastic dampers will be combined with structural braces, and hence
it will not take any space that could be rented or used to gain profit. Therefore, the graph
presented in Figure 7.3 represents the cost function of damping when the value of CR is 0.
0.06
T
0.05
0.04
0
0.03
0.02
0.01
~1
0
0
0.02
0.04
0.06
0.1
0.08
0.12
0.14
0.16
0.18
0.2
Damping ratio, , (%)
Figure 7.3 - Cost function for the dampers used in the CAARC Building.
Decision makers can use Figures 7.4 and 7.5, which are graphs that incorporate the cost
of increasing damping with the cost of the negative effects acceleration. These graphs are
two different ways of presenting the same information to help decision makers see the
economic impact of increasing damping by varying the cost coefficient.
75
0.8
0.7
0.6
0.5 4-
=1%
0
0.4
=2%
-.......
=
0.3
10%
=20%
0.2
0.1 41
0
0
0.2
0.4
0.6
0.8
1.2
1
1.4
1.6
1.8
2
Cost coefficient, g (%)
Figure 7.4 - Increasing the cost coefficient to show the cost effectiveness of increasing damping.
0.8
0.7
0.6
0.5
p= 0.1%
-
+-0
0.4
p=0.5%
-
--p=1%
--
0.3
p=1.5%
--p=2%
0.2
0.1
-
0
0
2
4
6
8
12
10
14
16
18
20
Damping ratio, k (%)
Figure 7.5 - Varying the cost coefficient to evaluate the economic impact of increasing the damping
ratio.
76
Figure 7.4 shows that all the curves representing different damping ratios intersect at one
point. Cost coefficients to the left of that point would represent that increasing damping
ratios is an ineffective and uneconomical solution to solve the serviceability issue. This is
evident in figure 7.5, where the curve representing the cost coefficient equal to 0.1%
shows that an increase in damping ratio increases the additional total cost due to negative
effects of acceleration,
Ctotal,D.
This shows that the more decision-makers value
occupant's comfort, which is expressed by increasing the cost coefficient, the more costeffective the increase in damping will be. Also, this means that if occupant's comfort is
not valued enough, then the cost of adding dampers to increase the damping ratio could
govern the total cost function,
Ctotal,D
causing the increase in damping to become an
uneconomical option.
77
7.2
Cost effectiveness of stiffness
7.2.1
Varying cost coefficient and stiffness
Figures 7.6 and 7.7 show the effects of increasing stiffness from an economic perspective
by varying the cost coefficient, y, but they do not include the cost of increasing stiffness.
From these two figures, it is evident that increasing stiffness is always beneficial and that
it would decrease the cost of negative effects of acceleration on the building.
0.8
0.7
0.6
0.5
0.4
0.3
-
k = 0%
-
k = 10%
-
k = 21%
-
k = 32%
k = 44%
k=56%
0.2
-
0.1
0 -i0
I
0.2
0.4
0.6
0.8
1.2
1
1.4
1.6
1.8
2
Cost coefficient, pL (%)
Figure 7.6 - Relating
CTB
and cost coefficient, pt, with increase in stiffness and unchanged damping.
78
0.8
0.7
0.6
-
0.5
p =0.1%
CC--.4
- ----
-
-
-
0.3
= 0.5%
p= 1.5%
0.2
-V
- --
2%
=
0.1
0
10
20
30
40
50
60
Increase in stiffness, k (%)
Figure 7.7 - Relating CTB and increase in stiffness with increase in cost coefficient and unchanged
damping.
79
7.2.2
Economical outcomes of increasing stiffness
Figure 7.8 represents the cost of increasing stiffness, which is obtained using equation 43
and values from Table 10.
0.180
0.160
0.140
0.120
0.100
0.080
zlo-
0.060
0.040
0.020
0.000
0
10
30
20
40
50
60
Increase in stiffness, k (%)
Figure 7.8 - Cost function for the increase in stiffness used in the CAARC Building.
Decision makers can use Figures 7.9 and 7.10, which are graphs that incorporate the cost
of increasing stiffness with the cost of negative effects acceleration, to help them see the
economic impact of increasing stiffness by varying the cost coefficient. These graphs are
two different ways of presenting the same information to help decision makers see the
economic impact of increasing stiffness by varying the cost coefficient.
80
0.80
-
0.70
0.60
4w
0
-
-
-
-
0.40
--
-
0.50
-
--
-----
-
0.30
k =0%
---
k= 21%
-
k = 32%
0
k = 44%
k= 56%
0.10
- --
--
--
0.00
2
1.5
1
0.5
0
Cost coefficient, p (%)
Figure 7.9 - Increasing the cost coefficient to measure the cost effectiveness of increasing stiffness.
0.70
-
-
t
0.60
-
--
0.50
-
-
-
-
-
-
0.80
- --------- --
---
p =0.1%
-
0.40
--
--
-
Ip
=0.5%
1%
p = 1.5%
.-
p9=2%
0.20
+-
0.10
0.00
0
10
20
30
40
50
60
Increase in stiffness, k (%)
Figure 7.10 - Varying the cost coefficient to evaluate the economic impact of increasing stiffness.
81
It is shown in Figure 7.9 that all the curves, which represent different stiffness values,
intersect at one point, and that cost coefficients at the left of that point would represent
that increasing stiffness is an ineffective and uneconomical way to solve the
serviceability issue. This is evident in Figure 7.10, where the curves representing the cost
coefficients 0.1% and 0.5%, show that an increase in stiffness increases the additional
total cost due to negative effects of acceleration, Ctotai,s. This shows that the more
decision-makers value occupant's comfort, which is expressed by increasing the cost
coefficient, the more effective the increase in stiffness will be. Also, this means that if
occupant's comfort is not valued enough, then the cost of increasing stiffness could
govern the cost function, CtotaL,s causing increase in stiffness to be an uneconomical
solution.
Therefore, increasing stiffness is uneconomical for cost coefficients less than 0.5%,
whereas increasing damping is uneconomical for cost coefficients less than 0.1%. This
means that in this example damping is more economical for a wider range of cost
coefficients than that of stiffness.
82
7.3
Comparing the cost-effectiveness of damping and stiffness
Increase in stiffness (%)
CTB
0 (Original value)
5
10
16
21
27
32
38
44
50
56
0.0376
0.0339
0.0306
0.0276
0.0250
0.0225
0.0204
0.0185
0.0168
0.0152
0.0138
( (%)
C1B
1 (Original value)
0.0376
2
0.0252
5
0.0157
10
0.0106
20
0.00595
Table 13 - Comparing CTB for increasing values of damping and stiffness.
From Table 13, it is evident that increasing damping is very affective on decreasing the
acceleration of the building. This is because the CTB of 2% damping is close to the CTB
Of
21% increase in stiffness, and 5% damping is more cost-effective than 44% increase in
stiffness, and its value is close to the CTB of 50% increase in stiffness.
83
7.4
Increasing damping and stiffness for different locations
Along with Florida, the methodology was used for another three states, Illinois, New
York and Massachusetts, following the steps from 6.1 to 7.2.2, which produced the
results as shown below.
7.4.1
Economical analysis for Illinois
2.5
2
--
1.5
= 1%
k=2%
-
1
k=5%
=
-
-
k
10%
20%
0.5-
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Cost coefficient, p (%)
Figure 7.11 - Increasing the cost coefficient to show the cost effectiveness of increasing damping for
the state of Illinois.
84
2.5
2
1.5
0.1%
0,--
=0.5%
-
-p
1
- --
- -
- - - -
-
--
-
=1%
-
--
-
=
1.5%
=2%
0.5
0
2
0
6
4
12
10
8
16
14
18
20
Damping ratio, g(%)
Figure 7.12 - Varying the cost coefficient to evaluate the economic impact of increasing damping for
the state of Illinois.
2.50
1.50
-
2.00
k =0%
f
1.00
-
k = 10%
-
k = 21%
-
k =32%
k = 44%
k = 56%
--
-
0.50
0.00
0
0.5
1.5
1
2
Cost coefficient, p (%)
Figure 7.13 - Increasing the cost coefficient to measure the cost effectiveness of increasing stiffness
for the state of Illinois.
85
2.50
2.00
-
1.50
p =0.1%
--
1= 1%
-
0.50
-
0.00
0
10
20
30
40
50
60
Increase in stiffness, k (%)
Figure 7.14 - Varying the cost coefficient to evaluate the economic impact of increasing stiffness for
the state of Illinois.
86
7.4.2
Economical analysis for New York
0.4
0.3-
-
0.25
1%
-
0
ImJI
5%
-
-
-
-
0.15
-0%
--
(=20%
-
0.1
2%
=
---
0.05
--
0
0
0.2
0.4
0.8
0.6
1.2
1
1.4
1.6
1.8
2
Cost coefficient, g (%)
Figure 7.15 - Increasing the cost coefficient to measure the cost effectiveness of increasing damping
for the state of New York.
0.4
0.35-
0.3
0.25--
0.%
p =0.1%
-
0.2
-
=1%
1.5%
0.1s
-p=2%
0.1
0.05
0
-
--
---
0
2
4
6
8
10
12
14
16
18
20
Damping ratio, k(%)
Figure 7.16 - Varying the cost coefficient to evaluate the economic impact of increasing damping for
the state of New York.
87
0.40
0.35
0.30
--
-
k =0%
-
0.25
-
-k
.2-
= 10%
k = 21%
0.15
-
----
-
k = 32%
-
-
k = 44%
k = 56%
0.05
0.00
2
1.5
1
0.5
0
Cost coefficient, A(%)
Figure 7.17 - Increasing the cost coefficient to measure the cost effectiveness of increasing stiffness
the state of New York.
0.40
0.35
0.15
.0.1%
0.20
020-
-
=
0.10
0.5%
-- p=1%
0.15
-
- -
-
- -
-..-
p
--..
1.5%
--p=2%
--
-
0.10-0.05
0.00
0
10
20
30
40
50
60
Increase in stiffness, k (%)
Figure 7.18 - Varying the cost coefficient to evaluate the economic impact of increasing stiffness the
state of New York.
88
7.4.3
Economical analysis for Massachusetts
0.18
0.16
0.14
0.12
0
0.1
- =2%
= 5%
-
0.08
---
0.06
=
10%
=
20%
0.04
0.02
0
0
0.5
1
1.5
2
Cost coefficient, V (%)
Figure 7.19 - Increasing the cost coefficient to measure the cost effectiveness of increasing damping
for the state of Massachusetts.
0.18
0.16
0.14
0.12 I--
(U
0
_______
=
-p
0.1
0.08
0.1%
--
p= 0.5%
---
p= 1%
= 1.5%
----
0.06
p = 2%
0.04
0.02
0
0
5
10
15
20
Damping ratio, k(%)
Figure 7.20 - Varying the cost coefficient to evaluate the economic impact of increasing damping for
the state of Massachusetts.
89
.
0.30
0.25
0.20
WL
0.15
0%
-
k=
-
k = 10%
-
k = 21%
-
k = 32%
k = 44%
0.10 -t-
k = 56%
0.05
0.00
0
2
1.5
1
0.5
Cost coefficient, V (%)
-
Figure 7.21
Increasing the cost coefficient to measure the cost effectiveness of increasing stiffness
for the state of Massachusetts.
0.30
0.25
0.20
-4a
0.15
p = 0.1%
---
= 0.5%
---
= 1%
p=1.5%
0.10
--p=2%
0.05
I
0.00
0
10
20
30
40
-50
60
Increase in stiffness, k (%)
-
Figure 7.22
Varying the cost coefficient to evaluate the economic impact of increasing stiffness for
the state of Massachusetts.
90
7.4.4
Evaluating different factors and locations
The CAARC Building was analyzed in four states, where the stiffness and damping of
the building were changed to evaluate their effects on the building's performance from an
economic perspective. The four states can be divided into two categories: windy regions
and non-windy regions. Florida and Illinois are categorized as windy regions, whereas
New York and Massachusetts are categorized as non-windy regions.
It is clear from Figures 7.17, 7.18, 7.21 and 7.22 that increasing stiffness is an
uneconomical option for the non-windy regions, as it has caused an increase in cost. But
this is not the case for windy regions, as it is clear from Figures 7.6, 7.7, 7.13 and 7.14
that increasing stiffness decreases the cost making it an economical option.
In the case of changing damping, the situation is different. For the windy regions, as the
cost coefficient increases, increasing damping is cost-effective and economically
beneficial. But a general case cannot be made for the non-windy regions as the optimum
damping ratio is highly dependent on the cost coefficient, t, which means that the graphs
need to be carefully analyzed.
91
8. CONCLUSIONS
The methodology proposed is a modification of the PEER criteria to evaluate the
serviceability of vortex shedding on tall buildings. This methodology is divided into four
components: hazard analysis, structural analysis, damage analysis and cost analysis.
These four parts are used to help decision makers decide on various factors to provide an
economic solution.
From a structural point of view, the methodology has successfully captured the effect of
vortex shedding on the structures and how this effect can amplify the acceleration of a
building, causing occupants to feel uncomfortable. Moreover, it has also succeeded in
determining the effects of different factors such as stiffness and damping, where an
increase in either factor can change the structure in different ways to minimize
occupant's discomfort. Increasing damping would reduce the amplification due to
resonance, which shows that damping does not eliminate resonance but minimizes its
effect. On the other hand, increasing stiffness increases the critical wind speed at which
resonance occurs, but it also increases the value of acceleration at the critical wind speed
due to resonance. Hence, the main advantage in increasing stiffness is increasing the
critical wind speed, therefore minimizing the chance of resonance to occur.
To decide on the most suitable factors, an economical analysis was made to evaluate the
negative effects of increasing acceleration as a measure of occupant's discomfort, and to
analyze the increase of adding damping and stiffness. This was accomplished by
incorporating their affects on acceleration and the costs needed to achieve the necessary
increase in damping and stiffness. By taking CAARC Building as an example in
presenting how the method works, and choosing Florida as the location of the building, it
was found out that an increase in stiffness and damping are justified as long as there is an
increase in the cost coefficient.
93
Moreover, the location of the CAARC Building was changed to three other states,
Illinois, New York and Massachusetts, to analyze how varying wind loads and their
return periods can have a large impact on the analysis. The four states were divided into
two categories, windy and non-windy regions, where Florida and Illinois were
categorized as windy regions, and New York and Massachusetts were categorized as nonwindy regions. Varying stiffness and damping for windy and non-windy regions had
different results. For windy regions, increasing stiffness and damping are economically
beneficial as the cost coefficient increases, but for non-windy regions, finding the
optimum damping from an economic perspective is highly dependent on the cost
coefficient used, and increase in stiffness is an uneconomical option unless the cost
coefficient is extremely large to justify the increase in stiffness.
94
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