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REPORT
ON
B.TECH PROJECT
ANALYSIS AND DESIGN OF REINFORCED CONCRETE
CHIMNEYS
BY
N. RAVI KIRAN
CE 97062
UNDER THE GUIDANCE OF
DR DEVDAS MENON
DEPARTMENT OF CIVIL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY, MADRAS
MAY 2001
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Certificate
This is to certify that the report titled ANALYSIS AND DESIGN OF
REINFORCED CONCRETE CHIMNEYS, submitted by Nagavarapu
Ravi Kiran, to the Indian Institute of Technology, Madras, in partial
fulfillment of the requirements for the award of the degree of Bachelor ofTechnology in Civil Engineering is a bona fide record of the work done by
him under the guidance of Prof. Devdas Menon during the academic year
2000-2001
Dr. Devdas Menon
Project Guide
Associate Professor
Dept. Of Civil EngineeringIIT Madras.
Dr. V.Kalyanaraman
Professor and Head
Dept. Of Civil EngineeringIIT Madras.
Department of Civil Engineering,
Indian Institute of Technology, Madras.
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Acknowledgements
I would like to thank my project guide Dr. Devdas Menon who has been
extremely patient with me during the last one year and without whose help and guidance
this project would not have been possible. I am very indebted to him.
I would like to place on record my thanks to all the faculty of IIT Madras who
have been extremely cooperative and helpful during my stay at the Institute.
I would also like to thank all my class mates, friends and wing mates who have
made my stay at this place a wonderful experience.
N. Ravi Kiran
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Abstract
The present thesis deals with the analysis and design aspects of Reinforced
Concrete Chimneys. The thesis reviews the various load effects that are incident upon tall
free standing structures such as a chimney and the methods for estimation of the same
using various codal provisions. Various loads are incident upon a chimney such as, wind
loads, seismic loads and temperature loads etc. The codal provisions for the evaluation of
the same have been studied and applied. Comparison has also been done between the
values obtained of these load effects using the procedures outlined by various codes.
The design strength of the chimney cross sections has also been estimated. Design
charts have also been prepared that can be used to ease the process of the design of the
chimney cross sections and the usage exemplified.
A typical chimney of 250m has been analyzed and designed using the processes
already outlined. Drawings have been prepared for the chimney. The foundation for the
chimney too has been designed.
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Contents
Title Pno
Chapter 1: Introduction 1
Chapter 2: Estimation of Wind Load Effects 2
2.1 Along Wind Effects 2
2.1.1 Basic Design wind speed 3
2.1.2 Wind Profile 3
2.1.3 Design Wind pressure 5
2.1.4 Force Resultants 6
2.1.5 Dynamic Effects and Gust factor 9
2.1.6 Analysis using STRAP 10
2.1.7 Expected Maximum Moments 11
2.2 Across Wind Effects 12
2.2.1 Vortex Shedding 13
2.2.2 Chimney modeling and estimation of shape factorand time period
14
2.2.3 Estimation of Moments 14
2.2.4 Variation of Moments with change in H/D ratio 20
2.2.5 Conclusions of the variational Analysis 21
2.3 Conclusions 24
Chapter 3: Estimation of Earthquake load Effects 25
3.1 Introduction 25
3.2 Estimation of loads 26
3.2.1 Design seismic coefficients 28
3.3 Calculations for a typical case 29
3.4 Conclusions 32
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Chapter 4: Estimation of Temperature load Effects 33
4.1 Introduction 33
4.2 Equations for evaluation of stresses 354.3 Conclusions 39
Chapter 5: Estimation of Design Resistance and Development of Design
Charts40
5.1 Introduction 40
5.2 Characteristic Stress-Strain Curve for Steel 41
5.3 Characteristic Stress-Strain Curve for Concrete 42
5.4 Calculation of Ultimate Moments 44
5.5 Interaction Curve 46
5.5.1 Family of Interaction Curves 48
5.5.2 Derivation of Equations used 50
5.6 Conclusions 52
Chapter 6: Design and detailing of Example Chimney 53
6.1 Introduction 53
6.2 Design of chimney 53
6.3 Design of foundation 57
6.4 Conclusions 60
Chapter 7: Summary and Conclusion 61
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Acknowledgements i
Abstract iiContents iii
List of figures vi
List of tables vii
List of important symbols viii
Appendix I
References XI
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List of figures
Figure Description
2.1 Wind Profile and Response
2.2 Moment Profiles (comparative)
2.3 Across wind Effect
2.4 Mode shapes
2.5 IS Simplified method & Figure
2.6 IS Random Response Method
2.7 IS Approximate Method Mode 1
2.8 IS Approximate Method Mode 2
2.9 IS Random Response Method Mode 1
2.10 IS Random Response Method Mode 2
3.1 Shear force due to seismic loads
3.2 Bending Moment due to seismic loads
4.1 Thermal Stresses
5.1 Stress-strain curve (steel)5.2 Stress-strain curve (concrete)
5.3 Chimney Cross-section
5.4 Stress and strain distributions
5.5 Strain profile variation
5.6 Interaction Curves
6.1 The Chimney
6.2 Sectional plan view Vertical reinforcement
6.3 Sectional elevation view Horizontal reinforcement
6.4 The foundation (representation)
6.5 Load and eccentricity
6.6 Actual loading pattern
6.7 The foundation and the connection
6.8 Design of staircase tread
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List of Tables
Table Description
2.1 Chimney Attributes
2.2 Results of dynamic analysis
2.3 Base Gust factors (comparative)
2.4 Base Moments (ACI)
2.5 ACI method (all moments)
4.1 Vertical Stresses
4.2 Hoop Stresses
5.1 Values of the interaction curve parameter
6.1 List of chimney parameters used
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List of important symbols used
Symbol Description
Vb , Vh Basic wind speed
Vz Wind profile
CD Drag Coefficient
wm Wind loading
G Gust factor
pz Pressure profile with height
g Peak factor, Acceleration due to gravity
Sn Strouhal number
oi Peak deflection due to vortex shedding in ith mode
Normalized mode shape
Fzoi Force due to vortex shedding
Mzoi Moment due to vortex shedding
Vcr Critical Velocity of flowui Normalized response
ui Actual response
h Seismic Coefficient
I Importance factor
Soil Coefficient
Sa Seismic acceleration
Es
Modulus of Elasticity (Steel)
Ec Modulus of Elasticity (Concrete)
Tx Temperature gradient
st Thermal stress in steel
ct Thermal stress in concrete
Coefficient of thermal expansion
k Location of neutral axis
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sml Limiting stress in steel
cul Limiting stress in concrete
s Partial safety factor for steelc Partial safety factor for concrete
r Radius of the chimney
t Thickness of the chimney shell
m Dimensionless quantity for moment
n Dimensionless quantity for normal force
Percentage of steel
fs
Stress-strain curve for steel
fpc Stress-strain curve for steel
Strain
0 Location of the neutral axis
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Chapter 1. Introduction
This project deals with the analysis and design of Reinforced Concrete (RC)
chimneys. Such chimneys (with heights up to 400m) are presently designed in conformity
with various codes of practice (IS 4998, ACI 307, CICIND etc.). The main loads to be
considered during the analysis of tall structures such as chimneys are wind loads,
temperature loads and seismic loads in addition to the dead loads. The design is done
using limit state concepts (which are yet to be incorporated into IS 4998).
The wind load effects are of two distinct types along-wind effects and across-
wind effects. While the along-wind loads deal with the effect of direct action of the wind
on the face of the chimney, the across-wind loads deals with the aerodynamic action of
the bluff body cross section of the chimney in a wind flow. The evaluation of along-wind
is straight forward, while the across-wind load estimation is more involved requiring
dynamic analysis. The loads are idealized as those on a acting on a cantilever, for the
purpose of evaluation of the load resultants on the chimney.
The seismic loads are another cause of natural loads on the chimney. These loads,
caused by earthquakes are generally dynamic in nature. However the codes provide forquasi-static methods for the evaluation of these loads. Codal provisions normally
recommend amplification of the normalized response of the chimney with a factor that
depends on the local soil conditions and the intensity of the earthquake.
The temperature load effects too are an important consideration in the analysis of
loads effects on chimneys taking into consideration the fact that the chimneys are used
for the venting of hot gasses. This develops a temperature gradient with respect the
ambient temperature outside and hence causes stresses in the reinforced concrete shell.
There is a considerable difference between the methods employed and the
assumptions made by the various codes. Hence the values predicted by the various codes
too vary a lot. A comparison has also been done between the values reported by the
various codes for the wind load effect analysis.
The design of the chimneys requires the estimation of the resistance of the tubular
cross section of the chimney. Also suitable design charts were constructed to serve as
design aids.
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Chapter 2. Estimation of Wind Load Effects
Wind forms the predominant source of loads, in tall freestanding structures like
chimneys. The effect of wind on these tall structures can be divided into two
components, known respectively as
x along-wind effect
x across-wind effect
Along-wind loads are caused by the drag component of the wind force on the
chimney, whereas the across-wind loads are caused by the corresponding lift
component. The former is accompanied by gust buffetting causing a dynamic response
in the direction of the mean flow, whereas the latter is associated with the phenomenon
of vortex shedding which causes the chimney to oscillate in a direction perpendicular
to the direction of wind flow. Estimation of wind effects therefore involves the
estimation of these two types of loads.
2.1 Along Wind Effects
Along-wind effect is due to the direct buffeting action, when the wind acts on the
face of a structure. For the purpose of estimation of these loads the chimney is modeled
as a cantilever, fixed to the ground. The wind is then modeled to act on the exposed face
of the chimney causing predominant moments in the chimney. Additional complications
arise from the fact that the wind does not generally blow at a fixed rate. Wind generally
blows as gusts. This requires that the corresponding loads, and hence the response be
taken as dynamic. True evaluation of the along-wind loads involves modeling the
concerned chimney as a bluff body having incident turbulent wind flow. However, the
mathematical rigor involved in such an analysis is not acceptable to practicing
engineers. Hence most codes use an equivalent static procedure known as the gust
factor method. This method is immensely popular and is currently specified in a number
of building codes including the IS (IS:4998) code. This process broadly involves the
determining of the wind pressure that acts on the chimney due to the bearing on the face
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of the chimney, a static wind load. This is then amplified using the gust factor to take
care of the dynamic effects.
This study involves the evaluation of the along-wind loads by using the methodsspecified in a number of codes like
x CICIND (Model Code for Concrete Chimneys, 1998)
x ACI 307-95
x IS 4998 (Part 1) : 1992
2.1.1 Basic Design Wind speed
One of the primary steps to finding the along-wind loads is to get the basic
design wind speed. The determination of the effective wind pressure is based on the
basic wind speed. The basic wind speed (Vb) is defined (by the CICIND code) as the
mean hourly wind speed at 10m above the ground level in open flat country without
having any obstructions. This means that the wind speed is measured at a height of 10m
above the ground at the location of the chimney and is averaged over an hour. The ACI
code suggests a wind speed averaged over a period of the order of 20min to 1hr. The IS
code however uses the basic wind speed based on peak gust velocity averaged over a
short time interval of about three seconds. The value of the basic wind speed must be
established by meteorological measurement. Normally though it is not necessary to
actually do the measurement for a particular region. The values as suggested from
published Wind Maps specified by the codes may be used. Basic wind speeds generally
have been worked out for a return period of 50 yrs.
It may me noted that the ACI follows the FPS system and therefore in the
following discussion the formulae by the code appear different from the SI system of the
other two codes.
2.1.2 Wind Profile
Wind flow is retarded by frictional contact with the earths surface. The effect of
this retardation is diffused by turbulence in wind flow across a region known as the
atmospheric boundary layer. The thickness of this boundary layer depends on the wind
speed, terrain roughness and angle of latitude. The rougher the terrain, the more
effective the retardation to the mean flow, and hence, greater is the gradient height. The
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effect of this gradient is the wind flow now assumes a profile that varies with height
from 0 at the surface to the maximum at the end of the atmospheric boundary layer.
The variation of mean wind speed with height Vz is generally described by thepower law.
(2.1)Vz = Vb (Z / Zo)
Where Vz is the profile with respect to height. Vb is the basic wind speed, Z is
the height above ground level, Zo) is a height of the boundary layer and is the terrain
factor. The values of the various factors are specified by the respective codes.
The CICIND code suggests the following code for the purpose of evaluation of
the wind speed profile.
(2.2)V(z) = Vb k(z) kt ki
Where:
V(z) is the hourly mean wind speed at level z
z is the height above ground level
Vb is the basic wind speed specified
k(z) is given by the equation
(2.3)k(z) = ks (z / 10)
ks scale factor, equal to 1.0 in open flat country
is the terrain factor
kt topographical factor
ki interference factor
The ACI code gives the following formula for obtaining the Wind profiles
V(z) = (1.47)0.78(80/VR)0.09 VR(z/33)
0.14(2.4)
Where VR is the basic wind speed. The equation also converts from the basic
wind speed in mph to ft/s as required for the calculations.
The IS:875 however does not give a wind profile but gives a wind velocity at any
height Vz.(2.5)
Vz = Vb k1 k2 k3
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Where Vz is the required wind speed, Vb is the basic wind speed. k1 is aprobability factor (risk), k2 is the terrain, height and structure size factor,k3 is a
topography factor. The values of these factors can be gauged from the Tables given in
the IS code.
2.1.3 Design Wind Pressure
The obtained wind velocities are assumed to act on the face of the chimney. The
corresponding pressure on the surface has to be evaluated next. This is done with the
help of the drag coefficient. This coefficient is defined in a number of ways in all the
codes. The main concept however is that the square of the velocity acting at any point is
to be multiplied by this coefficient to get the pressure acting at that point. The
coefficient takes into account factors like slenderness of the column, ribbed quality of
the surface, the effect of having a curved surface etc.
The wind pressure then is multiplied with the density of air and the exposed area
to get the actual static loads acting on the chimney.
The CICIND code calculates the loads with the following formula
wm(z) = 0.5 a v(z)2 CD d(z) (2.6)
Which is more than just the pressure calculation. However the term CD refers to
the coefficient that depends on the slenderness of the column. The value of this
coefficient depends on the h/d ratio and can be obtained from the code. It varies between
0.6 and 0.7 for change in the h/d ratio from 5 to 25. The term wm(z) is basically the
weight acting on the cantilever for which it has to be designed.
The Indian code converts the velocity profile into its corresponding pressure
profile with the help of the following formula
pz = 0.6 Vz2
(2.7)
The value of 0.6 is the drag coefficient specified.
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The ACI code suggests a very similar function, however specifying the
coefficient to be 0.0013 as opposed to 0.6, mainly to keep it consisting with the FPS
system used by the code.
6
Figure 2.1 Wind profile and Response
2.1.4 Force resultants
The pressure values obtained in the earlier case are then converted into the
corresponding force values. The chimney is idealized to be a vertical cantilever, fixed to
the ground. The load that acts can be takes as a continuous load acting on this cantilever.
The calculation of the force resultants of shear and moment are trivial.
In reality the base of the chimney is broad. Hence the shear resisting capacity of
the chimney is high. In fact shear also may manifest itself as moment due to the deep
beam effect. Hence the more important resultant to calculate here is the moment as
compared to either the shear or the axial force.
Moment
Wind
Profile
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The moment at any point on the cantilever can be calculated by integrating the
moment from the end to that point. Hence the functions given to calculate the moment
too are integrals.The CICIND code gives the following main formula for the purpose of
calculation of the gust factor moments in chimneys
h
mg zdzzwh
zGzw
0
2)(
)1(3)( (2.8)
where
G is the gust factor (will be looked into later)
h is the height of the top of the shell above the ground levelz is the height above the ground level
wm(z) is the mean hourly wind load per unit height at height z
The IS code gives two methods for the evaluation of along-wind loads on
chimneys, both of which are discussed below.
The IS simplified method
This method, as the name suggests, is a simple procedure to come up with the
load values for a given configuration. The formula suggested for this method is
Fz = pz.CD.dz (2.9)
Where the factor CD is to be taken as 0.8. This is actually a vast simplification of
the procedure outlined in the IS:875 which specifies the distribution of the value of the
drag coefficient around the periphery of the cylindrical shell. This method however does
not take into account the effect of the dynamic quality of the incident wind on the
chimney.
The second method given by the code is the random response method. The
equations for the same are given below and terms explained. The need and use of the
Gust factor however is discussed later.
H
zmzf zdzFH
z
H
gF
0
2
)1(3(2.10)
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8
8
Where Fzf is the wind load in N/m height due to the fluctuating component of the
wind at height z. The whole load is given by
(2.11)zfzmz FFF
The wind load due to the hourly mean wind component is given by
where pz gives the design pressure at hourly mean wind component and is
pbtained by the equation
zDzzm dCpF (2.12)
zVpz2
6.0 (2.13)
In the equation for the fluctuating component of the wind load the gust factor G
is used. The equations and the concept involved are discussed later.
The ACI code gives the following code for the purpose of calculation of the
along-wind load. This code too divides the load due to the wind into two parts the
mean load and the fluctuating component. The mean load is calculated by the formula
)()()()( zpzdzCzw dr (2.14)Where the value
Cdr= 0.65 for z < h-1.5d(h)
Cdr= 1 for z > h-1.5d(h)
And the value of the mean pressure has been given.
The fluctuating load component has been taken equal to
3
' )(0.3)('h
bMzGzw ww (2.15)
Where M is the base bending moment due to the constant load acting on the
chimney. It is basically an integral of the weight acting on the chimney multiplied with
the distance from the base. The Gust factor G is calculated by
> @86.0
47.0
1
)16(
)33(0.1130.0'
h
VTGw
(2.16)
For a preliminary design the Time period of oscillation can be calculated with
the help of an equation suggested by the code. However the code requires the time
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period to be calculated with the help of dynamic analysis for the final design. Analysis
here was done by modeling the chimney using a program STRAP.
2.1.5 Dynamic Effects and the Gust Factor
All along-wind loads that act on the chimney are not due to the static wing
bearing on the surface of the chimney alone. There is a significant change in the applied
load due to the inherent fluctuations in the strength of wind that acts on the chimney. It
is not possible of feasible to take the maximum load that can ever occur due to wind
loads and design the chimney for the same. At the same time it is very difficult to
quantify the dynamic effect of the load that is incident on the chimney. Such a process
would be very tedious and time consuming. So most of the codes make use of the gust
factor to account for this dynamic loading. To simplify the incident load due to the mean
wind is calculated and the result is amplified by means of a gust factor to take care of the
dynamic nature of the loading.
The gust factor is defined as the ratio of the expected maximum moment M0 to
the mean moment Mm0 at the base of the chimney. It is accordingly denoted as G0 and is
referred to as the base gust factor.
The CICIND code gives the following formula for the calculation of the Gust
factor.
]ES
BgiG 21 (2.17)
Where g is peak factor with
vTvTg
e
elog2
577.0log2 (2.18)
the turbulence intensity
hi 10log089.0311.0 (2.19)
88.063.0
2651
h
Bbackground turbulence(2.20)
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83.0
42.0
2
1
21.01
3301
123
hV
f
hV
f
E
b
benergy density
(2.21)spectrum
88.0
98.0
14.1
178.51
h
V
fS
b
size reduction factor (2.22)
damping is a fraction of the critical damping and is taken as 0.016. f1 is the
natural frequency in the first mode of vibration.
h is the height of the shell above the ground in m and Vb is the basic wind speed.
T is the sample period and v is effective cycling rate.
The equation for the Gust factor used by the ACI code is given earlier.
The IS code probably borrowed its gust factor equation from the CICIND code
as both the equations are remarkably similar. Only the names given to some of the
factors are different. The factors and the equations themselves are the same
A typical chimney of 250m was chosen to calculate the along-wind loads. The
dynamic analysis was done using a structural analysis program called STRAP.
2.1.6 Analysis using STRAP
For the purpose of analysis the chimney was modeled in STRAP. The chimney
was idealized into 32 components outside the ground and one component inside the
ground (to take care of fixity and the effect of the foundation), a total of 33 components.
The various components were taken to be cylindrical objects. Hence the chimney was
idealized as 33 hollow cylinders stacked upon each other.
The thickness of the components of the chimney were varied according the
thickness of the actual chimney at the middle of each section. A fixed joint was assumed
after 32 nodes.
For the purpose of dynamic analysis the weight data was calculated by the
program itself. This however was strictly not correct because there would be the
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additional weight of the lining inside the chimney. Hence a lining of a layer of bricks
was assumed and the weight calculated by the program was corrected with a factor to
account for the weight of the lining. The calculation of the factors was done with thehelp of a small program that actually calculated the volume ratios for the purpose.
The chimney itself was assumed to be of a standard dimensions and ratios as
given below.
Attribute Value
Height 250m
Height to Base Diameter 7
Top Diameter to Base Diameter 0.6Base Diameter to base thickness 35
Top thickness to base thickness 0.4675
Table 2.1 Chimney Attributes
The results of dynamic analysis of the modeled chimney are given below
Mode Time Period
1 0.23452 1.0266
3 2.4826
4 3.6286
5 4.4460
Table 2.2 Results of dynamic analysis
These values of time periods of oscillations and the corresponding frequencies
(1/Time Period) were used for the calculations of the Gust factor.
2.1.7 Expected maximum moments
The moments were calculated for the model chimney assumed earlier and the
results are shown in the graph below
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0
50
100
150
200
250
300
0 500 1000 1500 2000
CICIND ACI IS
Figure 2.2 Moment profiles (comparative)
As is visible, there is considerable difference in the expected maximum base
moments of the chimney using the three codal methods.
Additionally the base gust factors for the three methods are given below
Code Base Gust factor
IS 1.85
CICIND 1.85
ACI 1.993
Table 2.3 Base Gust factors (comparative)
2.2 Across Wind Effects
Recommendations for considering the across-wind loads have been included into
the codes only recently. In spite of considerable research the problem of accurately
predicting the across-wind response has to be fully resolved. Hence the CICIND code
does not take into account across-winds. For this study the codes used therefore were the
IS 4998(Part 1): 1992 and the ACI 307-95.
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A tall body like the chimney is essentially a bluff body as opposed to a
streamlines one. The streamlined body causes the oncoming wind flow to go smoothly
past it and hence is not exposed to any extra forces. On the other hand the bluff bodycauses the wind to separate from the body. This separated flow causes high negative
regions in the wake region behind the chimney. The wake region is a highly turbulent
region that give rise to high speed eddies called vortices. These discrete vortices are
shed alternately giving rise to lift forces that act in a direction perpendicular to the
incident wind direction.
13
Figure 2.3 Across wind effect
These lift forces cause the chimney to oscillate in a direction perpendicular to the
wind flow.
2.2.1 Vortex Shedding
The phenomena of alternately shedding the vortices formed in the wake region is
called vortex shedding. This is the phenomena that gives rise to the across-wind forces.
This phenomena was reported by Strouhal, who showed that shedding from a
circular cylinder in a laminar flow is describable in terms a non-dimensional number S n
called the Strouhal number.
CHIMNEY
velocityflowmean
cylinderofdiameterfrequencysheddingSn
__
___ u (2.23)
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The phenomena of vortex shedding and hence the across-wind loads depends on
a number of factors including wind velocity, taper factors etc., that are specified by the
codes. Codal estimation of the across-wind loads also involves the estimation of themode-shape of the chimney in various modes of vibration. This is obtained as follows.
2.2.2 Chimney Modeling and estimation of shape factor and time period
As discussed earlier dynamic analysis of the chimney was done using the
structural analysis program STRAP. A model chimney with the parameters shown
earlier was modeled and dynamic analysis performed on it. The required mode shapes
were obtained from the program itself.
The results from the analysis are given below with the normalized mode shapes
on the left and the corresponding frequencies of vibration on the right. It may be noted
that although four mode shapes have been assumed for the purpose of analysis, in reality
only the first two modes are actually active. This is because the wind velocity required
to make the chimney vibrate in higher mode shapes is very high.
Mode shapes 1 to 4
Frequencies:
Mode 1: 0.2345 hz
Mode 2: 1.0266 hz
Mode 3: 2.4826 hz
Mode 4: 3.6286 hz0
5
10
15
20
25
30
35
-1.2 -0.7 -0.2 0.3 0.8
Figure 2.4 Mode shapes
2.2.3 Estimation of Moments
The various codal methods for the purpose of estimation of along-wind loads are
as follows.
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The IS code, gives two methods for the estimation of across-wind loads. These
are called respectively the simplified method and the random response method. The
amplitude of the vortex excited oscillation is to be calculated by the equation.
sin
L
H
zi
H
ziz
oiKS
C
dz
dzd
2
0
2
0
4SI
I
K u
-
(2.24)
Where oi is the peak tip deflection due to vortex shedding in the ith mode of
vibration in m, CL = 0.16, H is the height in meters, Ksi is the damping parameter for the
ith more of vibration, Sn
strouhal number = 0.2 and zi
is the normalized mode shape.
Calculations of oscillation calculated using this formula are acceptable till 4
percent of the effective diameter. For values more that this the resultant is amplified
using a given formula.
Once this value is obtained the sectional shear force Fzoi and Bending moment
Mzoi at any height zo for the ith mode of vibration, as obtained as follows.
H
zo
ziziozoidzmfF IKS 21
24 (2.25)
H
zo
ziziozoidzmfF IKS 21
24 (2.26)
Where fi is the natural frequency in the ith mode of vibration and mz is the mass
per unit length of the chimney at section z in kg/m.
The mass damping factor Ksi required for the earlier equation is calculated using
the formula
2
2
d
mK sei
is V
G (2.27)
mei is the equivalent mass per unit length in kg/m in the ith mode of vibration, s
= 2, and = 0.016 (structural damping factor), is the mass density of air taken as 1.2
kg/m3 and d is the effective diameter taken as average diameter over the top 1/3 height
of the chimney in m.
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The equivalent mass per unit length in the ith mode of vibration can be calculated
using the formula given below. It is basically dependant on the amount of mass that is
available given the mode shape.
H
zi
H
ziz
ei
dz
dzm
m
0
2
0
2
I
I
(2.28)
The oscillation is caused by the wind. The mode in which the chimney vibrates is
decided by the wind speed. Higher modes need a higher wind speed for excitation.
Hence it is possible to know the wind velocities that causes shedding in the i th mode. It is
done with the help of the following equation.
n
criS
dfV 1 (2.29)
Since higher wind speeds are required to excite higher modes of vibration, it is
not necessary to consider all the modes of vibration for the purpose of design. All modes
which can be excited up to wind speeds of 10 percent above the maximum expected at
the height of the effective diameter shall be considered for subsequent analysis. If the
critical winds for any mode of vibration, exceeds the limits specified earlier, the code
allows the assumption that the problem of vortex excited resonance will not be a design
criteria for that and higher modes. In these cases across-wind analysis may not be
required.
The across-wind analysis using the random response method is also specified by
the code. The relevant expressions are given for chimneys of two types those with
little or no taper and those with significant taper. Taper is defined as
H
ddtaper
topav )(2 (2.30)
When the value of the taper is less than 1 in 50 (or 2 percent) the chimney is said
to have little taper.
For chimneys with little or no taper, the expression to calculate the modal
response at critical wind speed as given in equation 2.24 earlier
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ei
azi
ein
L
oi
m
dkdz
H
m
Ld
S
HdC
22
2
22
1
1
)2(225.1
VEI
SV
S
I
K
-
u
(2.31)
Where the RMS lift coefficient is taken as 0.12, correlation length in diameters is
taken as 1.0 and the aerodynamic damping coefficient is taken as 0.5.
Chimneys that are significantly tapered have the following equation
H
ei
aziziei
zeizeL
oi
m
dkdmS
tLHdC
0
2222
14
2
2
VEIS
SIIVK (2.32)
Where zei is the height in m at which a given expression is maximum in the ith
mode of vibration. The term in the expression is the power law exponent which was
discussed earlier with respect to the wind profiles. The value of this depends on the
Terrain Category and varies from 0.10 to 0.34.
The critical wind speed for exciting the mode of vibration is determined by the
equation.
n
ize
criS
dfV
1 (2.33)
Calculations begin by first taking zei =H and progressively decreasing till a
maximum in oi is observed. Also if the required velocity for excitation in any mode is
greater than the maximum velocity, the chimney will not be assumed to experience
much across-wind loads in that and higher modes. If this applies to the first mode of
vibration itself then the chimney has negligible across-wind loads.
The ACI code considers the across-wind loads due to vortex shedding for in the
design of chimneys when the critical velocity is between 0.5 and 1.3 Vzcr. Across-wind
loads are not considered outside this range.
Te critical velocity is calculated using the function.
t
cr
S
ufdV
)( (2.34)
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Where the St is the Strouhal number and is calculated using
)(log206.0333.025.0
udhS et (2.35)
d(u) is the mean outside diameter of the upper 1/3 of the chimney in feet, and h is
the height above the ground level.
The peak base moment at the critical velocity if determined by the equation.
x
x
E
p
as
crLSa
Cud
h
LShudV
aCS
g
GM
)(
2
4)(
2
22
EESU
(2.36)
Ma is evaluated over a range of wind speeds in the specified range of 0.5 to 1.3
Vcr to determine the maximum response. For values of velocity greater than Vcr the
value of Ma is multiplied with
4.1
1
)(
)(
3
44.1
-
cr
cr
zV
zVV(2.37)
The values of the various terms are given in the code including the peak factor,
mode shape factor and specific gravity of air.
The code also gives a formula for the calculation of the time period in the second
mode of vibration, although the final design needs a dynamic analysis. The values
obtained from the STRAP program were used in this calculations.
The results of the analysis are given below
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0
50
100
150
200
250
300
-400 -200 0 200 400 600 800
Mode 1 Mode 2
Figure 2.5 IS Simplified method & Figure 2.6 IS Random Response Method
0
50
100
150
200
250
300
-1000 -500 0 500 1000 1500
Mode 1 Mode 2
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The first graph refers to the result of the IS simplified method, whereas the
second graph refers to the IS Random response method.
As can be seen from the graph the moments in the first mode of vibration arevery similar for both the methods of calculation, whereas the moments for the second
mode of vibration vary a lot. The moments obtained from the Random response method
are almost double that obtained using the simplified method. In fact the Random
response method given higher moments for the second mode of vibration and lower
moments for the first mode of vibration, as compared to the simplified method.
The base moments as calculated using the ACI method are given below
(All values MNm) Across-wind Along-wind Gust Factor Max Moment
Mode 1 125.46 432.98 1.8854 825.922
Mode 2 98.86 432.98 1.592 696.56
Table 2.4 Base Moments (ACI)
It is seen that the values obtained using the ACI method are very small as
compared to the IS method. This is especially true of the across-wind loads.
2.2. Variation of moments with change in H by D ratioAn analysis was done to find the change in across-wind loads with change in
Height to Base diameter ratio.
For the purpose of the Analysis, Chimneys with the following parameters were
used
Height : 250 m
Height to Base diameter Ratio : 7, 9, 11, 12, 13, 15, 17
Top diameter to Base diameter Ratio : 0.6Base diameter to Base thickness Ratio : 35
Top thickness to Base thickness Ratio : 0.4675
The following methods were employed for the same
1. IS Approximate Method
2. IS Random Response Method
3. ACI 95 Method (Also CICIND approved)
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Estimation of Free Vibration parameters like the mode-shapes the free
frequency and the Weight data for the calculations were calculated by modeling thechimney in STRAP. The modeling was done with the chimney broken down into 32
elements. Vibration Analysis was done for modes 1 to 5 but only the first two were
required for the purpose of Moment calculations.
2.2.5 Conclusions from the variational analysis
x The Across-Wind Moments were inversely proportional to the H by D Ratio.The Moments consistently increased with fall in the H/D Ratio for all methods of
estimation.
x The Approximate method of the IS code gave consistently higher moments as
compared to the Random Response Method for vibrations in the first mode.
x The Approximate method of the IS code gave consistently lower moments as
compared to the Random Response Method for vibrations in the second mode.
x The IS method gave higher moments in the second mode of vibration as
compared to the first mode in both its methods.
x The ACI method gave very small values as compared to the IS methods for the
base moment in all cases
x Anomalously the moments in the second mode were lower in the ACI method as
compared to those in the first mode.
All relevant Data can be found in the subsequent pages. It may be noted that the
higher moment curves correspond to lower H/D ratio.
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0
5
10
15
20
25
30
35
0 2000 4000 6000
Figure 2.7 IS Approximate Method Mode 1
0
5
10
15
20
25
30
35
-10000 -5000 0 5000 10000
Figure 2.8 IS Approximate Method Mode 2
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0
5
10
15
20
25
30
35
0 1000 2000 3000 4000 5000
Figure 2.9 IS Random Response Method Mode 1
0
5
10
15
20
25
30
35
-20000 -10000 0 10000 20000 30000
Figure 2.10 IS Random Response Method Mode 2
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H/d 7 9 11 12 13 15 17
Mode 1 641.15 340.566 204.425 144.783 114.783 77.271 55.244
Mode 2 411.482 225.483 142.764 107.867 87.404 59.786 42.523
Table 2.5 ACI Method (all modes)
Conclusion
The wind loads form the major sources for moments on Tall free standing
structures like chimneys. We have looked at the two kinds on wind-loads that act on
chimneys and also have presented the calculations for a standard chimney.
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Chapter 3. Estimation of Earthquake Load Effects
3.1 Introduction
Seismic action on chimneys forms an additional source of natural loads on the
chimney.
Seismic action or the earthquake is a short and strong upheaval of the ground.
This naturally is the cause for loads on any structure. Any structure under seismic loading
is subjected to cyclical loading for a short period of time.
An earthquake is described by its intensity and it epicenter.
The intensity of and earthquake at a place is a measure of the degree of shaking
caused during the earthquake and thus characterizes the effect of the earthquake. Most of
the study of earthquakes up to the beginning of the twentieth century dealt with the
effects of earthquakes and to quantitatively describe these effects a number of intensity
scales were introduced. Initially there was the Rossi-Forel scale that had ten divisions. In
1888 Mercalli proposed a scale with 12 subdivisions to permit a clear distinction in
shocks of extreme intensity. After a number of changes the Modified Mercalli scale or
simply the MM scale is generally used by engineers today. Another revision made in
1956 to the MM scale by Richter is also in use.
The focus is the source for the propagation of seismic waves. It is also called the
hypocenter. The depth of the focus from the surface of the earth directly above is referred
to as the focal depth. The point on the earths surface directly above the focus is known
as the epicenter.
The structure experiences cyclic loading during the process of seismic action.
This causes energy to build up in the system leading to its collapse. The friction with air,friction between particles that constitute the structure, friction at junctions of structural
elements, yielding of the structural material and other processes of energy dissipation
depress the amplitude of motion of a vibrating structure and the vibrations die out in
course of time. When such internal and or external friction fully dissipates the energy of
the structural system during its motion from a displaced position to its initial position of
rest, inhibiting oscillations of the structure, the structure is said to be critically damped.
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Thus the damping beyond which motion will not be oscillatory is called critical
damping.
The effect of energy dissipation in reducing successive amplitude of vibrations ofa structure from the position of static equilibrium is called damping and is expressed as a
percentage of critical damping.
There are other terms that are important with respect to seismic analysis. During
earthquakes there occurs a sate in saturated cohesion less soil where in the effective shear
strength is reduced to a negligible value, for all engineering purposes. Un this condition
the soil tends to behave like a fluid mass.
A system is said to be vibrating in its normal mode or principal mode when all its
masses attain maximum values of displacement simultaneously and they also pass
through the equilibrium positions simultaneously. When a system is vibrating in its
normal mode, the amplitude of the masses at any particular time expressed as a ratio of
the amplitude of one of the masses is known as the mode shape coefficient.
During an earthquake ground vibrated (moves) in all directions. The horizontal
component of the ground motion is generally more intense than that of the vertical
components during string earthquakes. The ground motion is generally random in nature
and generally the random peaks of various directions may not occur simultaneously.
Hence for design purposes, at one time, it is assumed that only the horizontal component
acts in any one direction. All structures are designed to withstand their own weight. This
could be deemed as though a vertical acceleration of 1g is applied to the various masses
of the system. Since the design vertical forces proposed in the codes are small as
compared to the acceleration of 1 gravity, the same emphasis has not been given to the
vertical forces as compared to the horizontal forces. However for structures where
stability is a criterion it may become necessary to take into account these vertical forces.
3.2 Estimation of loads
The seismic action is described by means of a standardized acceleration response
spectrum. The CICIND code suggests a general response spectra. The response spectra is
a relation between the maximum effective peak ground acceleration at the location of the
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chimney. This is in relation with the natural time period of the structure and the soil type
existing at the site.
The movement of the chimney is found by calculating the first few mode shapesby modal analysis of the chimney. The result of such a modal analysis will yield the
values for the deflection, the shear force and the moment.
The modal analysis can determine the functions of the deflection, shear and the
moment only up to a constant factor. Thus if the mode shape calculated is known, then a
constant times the mode shape too is a possible solution.
Hence the actual value of the shear force or the bending moment is found by
multiplying the normalized response with a scaling factor.
Hence if u is the value of the normalized mode shape then the true mode shape is
given by
iii Nuu (3.1)
Where they refer to the ith mode of vibration, and Ni is the scaling factor. The
scaling factor is determined by the following equation.
)(4 2
2
isii Ta
TpNi
S (3.2)
The as is the response function described earlier. The value of pi is obtained from
h
i
h
i
i
dzzmzu
dzzmzu
p
0
2
0
)()(
)()(
(3.3)
The code also assumes the vertical movements to result in a value of resultants
that are 0.3 times the horizontal forces.
The ACI code also assumes the vertical component to be negligible with respect
to the horizontal one. The code also suggests the spectral values for the values of
maximum ground acceleration.
The following calculations are based on the IS code. The code used is the
IS:1893-1975.
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Since the earthquakes occur without any warning, it is very necessary to avoid
construction practices that cause sudden failure or brittle failure. The current philosophy
relies heavily on the action of members to absorb all the vibrational energy resulting fromstrong ground motion by designing the member to behave in a ductile manner. In this
manner even if an earthquake occurs that is stronger than that which has been foreseen,
total collapse of the building can be avoided.
Earthquake resistant designs are generally performed by pseudo-static analysis,
the earthquake loads on the foundations are considered as static loads and hence capable
of producing settlement as dead loads. Therefore as the footings are generally designed
for equal stresses under them, the footings for exterior columns will have to be made
wider. Permissible increase in safe bearing pressure will have to depend in the soil-
foundation system. Where small settlements are likely to occur larger increase can be
allowed and vice versa.
3.2.1 Design seismic coefficients for different zones
The force attracted by any structure during an earthquake is dynamic in nature
and is a function of the ground motion and the properties of the structure itself. the
dominant effect is equivalent to a horizontal force varying over the height of the
structure. Therefore the assumption of a uniform force to be applied along one axis at a
time is an oversimplification which can be justified for reasons of saving effort in
dynamic analysis. However a large number of structures designed on this basis have
withstood earthquake shocks in the past. This is a justification of a uniform seismic
coefficient in seismic design. In the code, therefore, it is considered adequate to provide
uniform seismic coefficients to ordinary structures.
The IS code suggests two methods for the purpose of evaluation of the earthquake
loads. This is similar to the two methods suggested for the calculation of across-wind
loads. Both methods calculate the design value of the horizontal coefficient.
Seismic coefficient method
The value of the horizontal seismic design coefficient shall be calculated using the
following expression.
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2929
(3.4)0DED Ih
Where is a coefficient depending on the soil type. This value varies between 1.0 and
1.5.
I is the importance factor.
0 is the basic horizontal seismic coefficient.
The response spectrum method
The response acceleration is first obtained for the natural time period and
damping of the structure and the design value of horizontal seismic coefficient is
computed using the following expression.
g
SIF ah 0ED (3.5)
Here
F0 is a seismic zone factor.
Sa/g is the average acceleration coefficient depending on the natural period and
damping of the structure.
3.3 Calculations for a typical case
The calculation of the earthquake load for a typical chimney is given below. The
assumptions made are also specified.
The weight data for the case has been taken from the STRAP model of the
chimney.
Period of vibration
Diameter of the base = 22.72 m
Base Thickness = 0.649m
Inner diameter at the base is 21.422m
Area of cross section at the base is
(3.6)
22
4 inoutddA
S
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A = 45.0 m2
The moment of inertia at the base is calculated by
4464
inout ddI S (3.7)
The value of I = 2742.5 m4
Radius of gyration r is given by
A
Ir (3.8)
r = 7.806
Hence the slenderness ration l/r is given by
02.32r
l(3.9)
The coefficient CT
(3.10)822.57TC
Weight of the chimney
(3.11)US tmeanTHDWt
Weight = 17495583 kg
The period of vibration is now given by
EAg
hWCT tT
' (3.12)
Substituting the values the value of T = 125.6
Design seismic coefficient
Using the Response Spectrum method and the equation **
ah = 0.03975
the value assumed are
= 1.0 (assuming a hard/medium soils)
I = 1.0 (importance factor)
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F0 = 0.25 (assuming the chimney to be in the zone IV)
Shear force and Bending momentsThe design shear force at a distance of X from the top is given by
2
'
'
3
2
'
'
3
5
h
X
h
XWCV thVD
(3.13)
Where the value of CV has been found to be 0.2 for the very large time period
obtained. Varying the value of X from 0 to 250 the profile of the shear force has been
calculated.
0
50
100
150
200
250
300
0 500 1000 1500
kN
Figure 3.1 Shear force due to seismic loads
The bending moment can be calculated using the formula
42
1
'
'4.0
'
'6.0
h
X
h
XhWM thD (3.14)
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Again the value of X is varied and the expression evaluated. The resultant graph
is given below.
0
50
100
150
200
250
300
0 200 400 600 800 1000
MNm
Figure 3.2 Bending Moment due to seismic loads
As can be seen from the graph, the maximum moment at the base of the chimney
is about 800 MNm.
3.4 Conclusions
The reasons and assumptions involved in the evaluation of earthquake loads have
been studied. The codal provisions for the calculation of the same have been understood.
A sample calculation has been done to calculate the shear force and bending moment
caused due to earthquake loading on chimneys. The loads in this case have been found to
be significantly lower that those obtained in the wind analysis. Hence earthquake loads
do not normally form the main loads to be considered for design.
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3333
Chapter 4. Estimation of Temperature Load Effects
4.1 Introduction
In walls of Reinforced Concrete chimney, stresses are developed due to the
temperature difference between the inner and the outer surface of the walls. This
temperature difference from inside to outside tends to expand the inner surface relative to
the outer one. Due to the monolithic action of the entire wall, differential expansion is not
possible and hence equal expansion takes place so that the shell is compressed on its
inside surface and pulled on its outside surface. As a whole there is an average increase in
length of the chimney due to the temperature gradient.
Various codes given different methods for the evaluation of the resultant
temperature stresses. The CICIND code does not explicitly give equations for the
evaluation of these stresses. Instead it asks the designer to account for them assuming the
shell wall to be a straight Reinforced Concrete wall.
The ACI code gives a code that is shortly discussed. There is another method that
is discussed by the book Advanced Reinforced Concrete Design by Dr. N.Krishna Raju
which will be used to calculate the stresses on a typical cross section.
The temperature stresses are of two different types. The stresses that occur in the
vertical part of the cross section and the stresses that occur in the horizontal part of the
cross section. Also calculations must be performed for the steel on the inside face as well
as the outside face of the chimney.
The ACI code gives the following equations to calculate the maximum vertical
stresses occurring in steel at the inside of the chimney due to temperature difference.
Note that fCTV refers to the concrete stresses and the value fSTV refers to the stress insteel.
cxteCTV ETcf xxxD'' (4.1)
And
cxteSTV nETcf xxx )1('' 2JD (4.2)
Where the terms are explained below
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3434
@
te = thermal coefficient of expansion of the concrete and of the reinforcing steel,
to be taken as 0.0000065 per deg F
Ec = modulus of elasticity of concretec is given by the equation
> @ > )1(211 2122
11 JJJUJUJU nnnc (4.3)
= ratio of the total area of the vertical outside face reinforcement to total area of
concrete chimney shell at the section under consideration
1 = ratio of the inside face vertical reinforcement area to the outside face vertical
reinforcement area.
2 = ratio of distance between inner surface of chimney shell and center line of
outer face vertical reinforcement to total shell thickness
n = Es/Ec (4.4)
Tx is the temperature gradient across the shell.
The code gives a number of formulas for the calculation of this gradient
depending on the type of shell. The shell type could be any of unlined chimneys, lined
chimneys with insulation completely filling the space between the lining and the shell,
lined chimneys with unventilated air space between the lining and the shell or lined
chimneys with ventilated space between the lining and the shell.
The equation for the unlined case is given
coo
ci
cc
ci
i
oi
cc
ci
dK
d
dC
td
K
TT
dC
tdTx
1
(4.5)
Where the factors are dependant on the cross section under consideration.
The terms Ko and Ki are the coefficients for transfer of heat. These can be
obtained from curves given by the code.
The maximum stress in the vertical steel fstv occurring at the outside face of the
chimney shell due to the temperature gradient can be computed using
cxieSTV ETcf xxx 'D (4.6)
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3535
An additional kind of temperature stress that is taken into account by the ACI
code is the circumferential temperature stress. The equation for the evaluation of the
same is
(4.7)cxieCTC ETcf xxx ''' D
and the same for steel is
(4.8)sxieSTC ETcf xxx )''('' 2JD
4.2 Equations for evaluation of stresses
The following is a derivation of the equations for the temperature stresses.
Assume that
To is the temperature difference between inside and outside with a linear
temperature gradient.
is the coefficient of expansion of steel and concrete.
e is the strain difference in temperature
m is the modular ratio
ts
is the area of reinforcement per unit width
tc is the area of concrete per unit width
ct is the stress in concrete due to temperature
st is the stress in steel due to temperature
p is (ts/tc)
k is the neutral axis depth constant.
Referring to the figures below and considering the force equilibrium we have
stcstscct pttkt VVV 2
1(4.9)
Which gives on solving for the stress in steel
k
kam
kt
ktatm
pct
c
ccct
ctst
VV
VV
2(4.10)
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The following figures are a representation of the case
36
atc
tcAir Gap
Figure 4.1 Thermal Stresses
The expressions for stress in steel in turn give the following equation for the value
of k2
Wherein the value of k is
Lining
Temperature
Gradient To
ktc
stct
Net Strain in
Steel
(Tension) T- e
Te Net Strain in
Concrete
(Compression)
(4.11) kapmk 22
(4.12)222 mpmpampk
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3737
Rise in temperature in reinforcement is
(4.13)Ta)1(
Free expansion of steel is(4.14)Ta D)1(
The tensile stress due to the difference between that due to strain e and due to
temperature rise (1-a)T
Hence the stress in steel is
(4.15) > @TaTkEsst DDV 11
or
(4.16))( kaTEsst DV
similarly stress in concrete is given by
(4.17)kTEcct DV
Stresses in horizontal reinforcement
At high temperatures, the inner surface of the chimney is prevented from
expansion and therefore gets compressed. The outer surface will expand more than the
natural expansion and will be in tension. Due to temperature stresses, generally the hoop
tries to expand and consequently tensile stresses develop in the hoop reinforcement.
Using the above figures and the following notation
ktc = position of the neutral axis
c = compressive strength in concrete
s = compressive strength in steel
As = area of hoop reinforcement per unit height
As = cross sectional area of steel
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3838
The equations for the calculation of the stresses are given as
'
'''kkam cs VV (4.18)
(4.20)pmmppmak 222'
(4.21)> @ TaEm scs DVV ''
Knowing the value of k the stresses can be calculated.
Sample calculations
The following is a sample calculation for a simple 4000mm concrete Reinforced
Concrete wall. The derivation does not take into account the curvature of the shell
directly. Hence they can also be applied to any wall. Also the assumed thickness of the
wall is quite typical of the chimneys looked into so far.
tc = 4000
assuming a steel cover of about 100 mm
atc = 3900
hence a = 3900/4000 = 0.975
assuming a 1% steel reinforcement
p = 0.01
assume a temperature difference of 75oC
other values are
= 11*10-6 /oC
m = 11
Es = 210000
Ec = 19090.9
Calculating the value of k using equation 4.20
k = 0.366025
Hence the vertical stresses are calculated to be
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c 5.765 N/mm2
s 105.5 N/mm2
Table 4.1 Vertical Stresses
The hoop stresses are calculated by solving the following equations
cs '3.18' VV (4.22)
And
(4.23)9.168'11' cs VV
Wherein the solutions are
c 5.764 N/mm2
s 105.49 N/mm2
Table 4.2 Hoop Stresses
4.3 Conclusions
The cause for the occurrence of thermal stresses in chimney shells were studied.Equations describing the phenomena were derived and stress resultants related. The
thermal stresses in the cross section were calculated.
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Chapter 5. Estimation of Design Resistance and
development of Interaction Curves
5.1 Introduction
This chapter deals with the calculation of the ultimate moment of resistance of the
Reinforced Concrete tubular section of the tower. There are many methods prescribed in
the codes for the purpose of estimation of the ultimate loads. These methods differ
primarily with regard to the model used to represent the stress strain curve of concrete in
compression.
The ultimate moment capacity of the tubular Reinforced Concrete section depends
on the normal compressive load that acts at that point. The interaction of this normal
force with the ultimate moment, corresponds particularly to the location of the neutral
axis which generally falls within the section for the high eccentricities in loading usually
encountered under extreme wind speeds.
The following are some of the assumptions commonly adopted for the purpose of
estimation.
1. Place sections remain plane after bending. This means that a linear strain
distribution is assumed at the cross section.
2. Extreme fibre stresses are computed at the center line of the concrete shell.
The mean radius is representative of all stresses.
3. The vertical reinforcing steel is replaced by an equivalent thin steel shell,
located at the mean radius.
4. The stress-strain relationship of steel is assumed to be elasto-plastic, and is
assumed to be identical in tension and compression.
5. Tensile stresses in concrete are ignored. The section is assumed to be fully
cracked in the tension region of the neutral axis.
In addition, the following are some requirements before the calculations can be
done.
x Stress-strain relationship of concrete in compression
x Limiting compressive strain in concrete
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x Limiting tensile strain in steel
x Modulus of elasticity of steel
The differences in the various codal methods are basically caused due to
dissimilarities in the above assumptions.
This paper calculates the design resistance using the standard stress-strain curve
for steel and a proposed stress-strain curve for concrete. This curve was proposed by Dr.
Devdas Menon in his Ph.D. thesis.
5.2 Characteristic Stress-Strain Curve for Steel
The stress-strain curve for steel is more or less standard and is used by all the
codal provisions. It is an idealized elasto-plastic relationship. The values to be assumed
are the Es (modulus of elasticity for steel) and the sml (limiting tensile strain in steel).
A diagrammatical representation of the Steel stress-strain curve is given below
fs Es = 200000MPa
41
Figure 5.1 Stress-strain curve (steel)
As has been indicted the value of
Es = 200,000 N/mm2
sml = 0.07 (as initially proposed by the ACI code)
The value for the limiting tensile strain is assumed for some codes to be a very
conservative 0.05. This is probably to take care of the excessive cracking in concrete on
the tension side. This however is not strictly called for at ultimate loads, in the limit state
s
fcyk
Es
smlsy
sml = 0.070
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of collapse, since the crack control is checked for separately as part of the serviceability
requirements.
5.3 Characteristic stress-strain curve for concrete
Various codes give various stress-strain codes for concrete.
The ACI code for example employs the Hognestads curve, originally proposed
for eccentrically loaded columns. The curve has two parts. The first is a parabolic curve
and the second is a straight line that continues from the end of the parabolic curve that
represents the downward trend of the curve. It assumes a limiting strain under direct
compression of 0.002 and an ultimate strain in flexure of 0.003.On the other hand, the CICIND has a very elaborate curve. It is a parabolic-linear
curve that distinguishes between the effects of dynamic, short-term loading and static
long-term loading.
The curve that is used for the purpose of estimation of resistance and for the
purpose of generation of the interaction curves is a new curve. This curve has been
proposed taking into account the effect of tubular geometry and the effect of short-term
wind loading.
The limiting compressive strain in concrete cul corresponds to the maximum
value of the strain cu at the middle of the concrete shell thickness at the extremity of
compression. Since the shell is extremely thin in comparison to its very large diameter,
the distribution of stress across the thickness of the shell is almost uniform. The behavior
of thin walled chimneys is very different from the behavior of solid Reinforced Concrete
sections which can accommodate a large strain variation across the cross section.
Hence the value ofcul should not be as large as 0.003 as suggested by the codes.
Rather it must be restricted to a value usually specified under conditions of uniform
compression, that is cul = 0.002.
The CICIND code proposition of distinctively accounting for the dynamic short-
term loading effect of wind merits consideration. However the premises on which the
curve is based are questionable. It is, for example, observed that the wind loads are
extremely short-lasting, while the meteorological practice is to compile hourly mean
wind speeds. The values for the code are taken from practical tests where the loading was
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done by reversed cyclic bending. However since the dynamic nature of wind consists of
random velocity fluctuations about a mean, rather than complete change of direction in
short periods. Since the mean response to wind loading is fairly substantial and theoverall response is quasi-static in nature, the behavior is better approximated by
monotonic loading rather that reversed cyclic loading; the duration of the loading to be
considered is approximately 2 to 5 hours.
On the basis of the results of a large number of tests on eccentrically loaded
concrete cylinders under varying load conditions the following conclusions can be drawn
x The stress strain curve is parabolic rather than linear, even under the
short term loading under consideration.x If fcu = 0.85 fck is assumed then it is reasonable to assume an increase
of approximately 10% for relatively short time loading.
x The value of the ultimate compressive strength cul corresponding to
this peak may be assumed to be approximately 0.002 for both short-
term and long-term loading.
On the basis of the above discussion the following curve is assumed as the stress-
strain curve for concrete under compression. It employs a simple parabolic curve with a
limiting ultimate limiting strain of 0.002 and a value of fcu = (0.85 fck) CS. Here the term
CS is called the short term loading factor, having a value that depends on the normal
compression on the tower section; it is assumed to vary linearly between a maximum
value of (0.95/0.85) for normal load = 0 and to unity when the value of normal load is
maximum that is under pure compression.
The formula for the curve is given below
pcspc fCf 1 (5.1)
where
85.0
1.095.0max
-
N
N
Cs
(5.2)
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The curve is as shown in the following figure.
fcu 0.85 Cs fck
c1
cul
Figure 5.2 Stress-strain curve (Concrete)0.002
Design Stress-Strain Curve
The characteristic stress-strain curve refers to the actual characteristic values of
the stress-strain values. These are multiplied by the partial safety factors to get the design
curves. The values of the partial safety factors assumed are as follows
s = 1.15
c = 1.50
these design curves are used to calculate the design ultimate moment carrying
capacity of the Reinforced Concrete tubular section.
The codes also specify either the design or the characteristic curves. The CICIND
code for example specifies the design curves along with the characteristic curves whereas
the ACI method specifies the design curve which is to be multiplied with a resistance
factor of 0.8. The code does not recommend any design stress-strain curves.
5.4 Calculation of Ultimate moments
The ultimate moment carrying capacity Mu of tubular section, corresponding to
any given normal compression N is determined by solving the following equilibrium
equations.
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45
(5.3)
where Nc and Ns are the resultant normal forces obtained from the concrete and
steel stress blocks respectively. Muc and Mus denote the respective moments of the
concrete and steel blocks about the centerline.
The following diagram is a representation of the various components involved in
the estimation of the design interaction curves.
Figure 5.3 Chimney Cross section
The distribution of strains and the corresponding stresses are given in the below
Neutral Axis
sc NNN
usucu MMM (5.4)
0
Wind
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Neutral Axis
Strains
46
Figure 5.4 Stress and strain distributions
These diagrams are merely depictive. They do not show the actual values.
As can be seen from the diagrams, for a neutral axis there exists a strain
distribution. This strain distribution is linear because of the assumption we had made in
the starting of the chapter. This in turn determines the stresses in the concrete and steel
block. The summation of these stresses gives rise to the resistive strength of the
chimneys.
5.5 Interaction Curve
The interaction curve is a complete graphical representation of the design strength
of a Reinforced Concrete chimney. Each point on the curve corresponds to the design
strength values of N and Mu. That is to say that if the load of N were to be applied to the
Reinforced Concrete chimney with an increasing eccentricity then the value of the
eccentricity where this line would intersect with the interaction curve is given by
cu = 0.002
Concrete
Stresses
N
MuH
fcu
Steel
Stresses
-fsyk
fsyk
(5.5)
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The interaction curve is the failure envelope. Any point inside the curve is safe.
That is any combination of moment and compressive strength where the point lies withinthe curve will not cause failure of the Reinforced Concrete chimney.
In reality the loading is not done in this manner. Given values the moment and the
compressive stress, it should be possible to check whether the chimney cross section is
safe.
The magnitude of N determines the neutral axis. This location is specified by the
angle 0 in the equation and the diagram given above. On location of the neutral axis the
strain distribution is known. This can then be used to solve for the value of N and the
ultimate moment Mu. It is therefore obvious that the solution to the above set of equations
can be found as a closed form solution. This is because the location of the neutral axis is
required for the calculation of the normal force N, while the value of N is itself required
for the location of the neural axis.
For the purpose of developing the interaction curves the location the neutral axis
was assumed and the values of the normal force and the moment were calculated. The
neutral axis was then changed to calculate a new set of N and Mu. This was repeated to
get the interaction curves of N Vs Mu.
Not all locations of the neutral axes are realistically feasible, as will be seen in the
following discussion.
The following diagram depicts the variation of the strain profile with change in
the location of the neutral axis.
47
Figure 5.5 Strain profile variation
The maximum
compressive
strain in steel=0
=90 = maximum
Neutral axislocation not
possible
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As the angle that locates the neutral axis changes from 0 the location of the
neutral and hence the participation of steel in taking the load varies. This continues as
more and more participation of steel in tension occurs and the net compressive force onthe chimney reduces. At a particular value of the value of steel in tension effectively
nullifies the effect of the compression of the concrete block. Any increase in the value of
is not possible because it follows that the chimney in overall tension, which is not
possible.
Although the interaction curve is plotted between the value of N and Mu, in the
interest of greater flexibility, the interaction curve is rendered non dimensional by use of
the following relations
rtf
Nn
ck' (5.6)
trf
Mm
ck
u
2' (5.7)
Where r is the value of the radius of the section in consideration of the Reinforced
Concrete chimney, and t is the thickness of the section.
5.5.1 Family of interaction curves
Since we are using the non dimensional parameters m and n, the curves are no
longer applicable to one chimney alone. It is possible to plot a family of curves that vary
with respect to one parameter. Once the parameter value is known, it is possible to
calculate the corresponding value for any new chimney and then reuse these curves for
that particular chimney.
The parameter that was used for the purpose of generating a family of curves was
ck
syk
f
f
'U (5.8)
Where
is the percentage of steel
fsykand fckare the strengths of steel and concrete.
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A program was written in C++ that was used to that calculate the values of pairs
of values of n and m. The iteration was done by varying the value of the angle of the
neutral axis in incremental steps of 1 degree. Then the strain distribution for thatparticular neutral axis was evaluated. The total force contributed by the concrete and steel
sections was evaluated by integration. Then the value obtained was non-dimensionalised
using the factors as appropriate. This was continued till the value of the total normal force
evaluated to zero, signaling that the limit of the neutral axis was achieved. The program
listing is given in the appendix.
The interaction curve is given below.
Interaction Curves
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4
m
n
2.075
8.3
15.56
20.75
25.94
31.125
Figure 5.6 Interaction curves
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The family of curves is from the parametric variation of the term given earlier.
The values of the parameter for each of the curve is given in the chart. The curves have to
be read left to right. That is the first curve on the left refers to the value
075.2'
ck
syk
f
fU (5.9)
And so on.
The values of the terms utilized to arrive at the values are given below
fck fsyk (fsyk/fck)
0.2 40 415 2.0750.8 40 415 8.3
1.5 40 415 15.5625
1.5 30 415 20.75
1.5 24 415 25.94
1.5 20 415 31.125
Table 5.1 Values of the interaction curve parameter
From the table the ranges assumed for the values are also visible. The percentage
of steel is assumed from 0.2% to 1.5% which is the normal range. The value of f ck too is
assumed to be varying from 20 to 40, that is use of concrete of grades M20 to M40 has
been assumed.
The usage of these curves for the estimation of strength is shown in the chapter
Design and detailing of Example Chimney.
5.5.2 Derivation of equations used
The derivation of the equations for the calculation is given below.
50
(5.10)cks fCfcu '85.0
Where
Cs is the short term loading factor that varies linearly as explained earlier.
cul = 0.002
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The stress-strain curve for concrete is given below
-
2
11
2ccc
cupc
ff
H
H
H
H
J
(5.11)
The stress-strain curve for steel is given below
-
o
ddo
HH
H
H
HHHH
sy
sysys
fsyk
E
fs (5.12)
Where
ss
syk
syE
f
JH (5.13)
Let Nc and Ns refer to the compressive forces in the concrete and steel blocks
respectively. Similarly Mc and Ms refer to the moments in the two blocks. Then the
integration equations are
S
D
DHU
0
)()1(2 dfrtN pcc(5.14)
S
D
DDHU
0
)cos()()1(2 2 dftrM pcuc (5.15)
S
DHU
0
)(2 dfrtN ss (5.16)
S
DDHU
0
2 )cos()(2 dftrM sus (5.17)
But it is not necessary to calculate the value of the whole normal force or the
moment. It is only required to calculate the value of the non dimensional parameters.
Using the relations given in equation 5.6 and 5.7 we have
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S
D
DH
U
0
)('
)1(2
dffn pcckc (5.18)
S
D
DDHU
0
)cos()('
)1(2df
fm pc
ck
c(5.19)
S
DHU
0
)('
2df
fn s
ck
s(5.20)
S
DDHU
0
)cos()('
2 dff
m sck
s(5.21)
Note that 0 is the parameter for varying the location of the neutral axis.
These four equations form the basis for the calculation of the interaction curves
shown above.
5.6 Conclusions
The stress-strain curves of the steel and the special curve for concrete were
formed and justified. The ultimate strength equation was formulated. The interaction
curve between moment and compressive force was calculated and plotted. The necessary
equations for the same were also derived and listed.
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6. Designing and Detailing of example chimney
6.1 Introduction
In the earlier chapters about analysis of the various loads that are incident on a
chimney, a number of calculations have been performed on some typical chimneys.
Those results will be brought together towards the design of a sample chimney.
Then the detailing of such a chimney is also shown.
In addition the last part of the chapter deals with the design of the footing for the
chimney.
6.2 Design of a chimney
The following table gives the list of the various parameters of a chimney and their
typical values.
Name of parameter Practical range Typical value
Slenderness ratio h/Do 7-17 11
Taper ratio Dt/Do 0.3-1.0 0.6
Base diameter to thickness ratio
Db/tb
20-50 35
Mean, base thickness ratio tm/tb 0.3-0.8 0.55
Top mean thickness ratio tt/tm 0.7-1.0 0.85
Table 6.1 Chimney parameters
These values determine the section of the chimney which is given below with the
dimensions of the various parameters.
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13.6 m
Top Thickness
0.3035 m
250 m
Base Thickness
0.65 m
22.72 m
Figure 6.1 The chimney
Checking the viability of the cross section
Taking the values of the forces as follows, which have been calculated in the
earlier chapters. It may be noted that this calculation is for the worst case of the wind
load.
Moment = 1552.8 MNm
Axial force = 175 MN
Calculating the values of m and n to be used in the design charts, assuming
M30 concrete.
m = 3.089
n = 3.955
The parameter value for use in the design charts without the value of the steel
comes to
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Parameter = 13.83 (percentage of steel)
With 1% steel, using the curve with a parametric value of 15.56 the crosssection is safe.
The design
Fe415 steel
M30 concrete
Steel = 1%
Since the loads and other effects are totally reversible, the steel must be applied
equally on both faces of the chimney shell. Hence each face has 0.5 percent of the steel.
The detailing is done as follows and the figure is given later.
Using bars of 25mm diameter
Area of a meter length (circumferential) of the chimney = 6500mm2
Area of reinforcing bar = 490.9 mm2
Number of bars = 6.6
Spacing between the bars = 150 mm
Provide a cover of 75 mm on either face
This is the scheme to be followed for both the horizontal and the vertical
reinforcement at the base of the chimney. The changes to be done are given below
Curtailment
Since the chimney tapers with height, the area of concrete available decreases
along with the reinforcement requirement. Hence the vertical steel needs to be curtailed
in stages. The following scheme may be followed for the same.
Curtail 1 out of every 6 six bars at about a height of 120m. Curtail a second bar
out of the original six (now five) at a height of 200m.
The horizontal s
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