Analysis and Design of RC Chimney 120m
Short Description
Analysis and Design of RC Chimney 120m...
Description
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
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 of Technology 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 Engineering
Dr. V.Kalyanaraman Professor and Head Dept. Of Civil Engineering IIT Madras.
IIT Madras.
Department of Civil Engineering, Indian Institute of Technology, Madras.
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
i
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.
ii
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 factor and 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
iii
Chapter 4: Estimation of Temperature load Effects
33
4.1 Introduction
33
4.2 Equations for evaluation of stresses
35
4.3 Conclusions
39
Chapter 5: Estimation of Design Resistance and Development of Design Charts
40
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
iv
Acknowledgements
i
Abstract
ii
Contents
iii
List of figures
vi
List of tables
vii
List of important symbols
viii
Appendix
I
References
XI
v
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
vi
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
vii
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 flow
ui
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
viii
%sml
Limiting stress in steel
%cul
Limiting stress in concrete
&s
Partial safety factor for steel
&c
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
ix
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 acrosswind 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 for quasi-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.
1
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 ! along-wind effect ! 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
2
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 methods specified in a number of codes like ! CICIND (Model Code for Concrete Chimneys, 1998) ! ACI 307-95 ! 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 earth’s 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 3
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 the power 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. V(z) = Vb k(z) kt ki
(2.2)
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 k(z) = ks (z / 10)
(2.3)
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. Vz = Vb k1 k2 k3 4
(2.5)
Where Vz is the required wind speed, Vb is the basic wind speed. k1 is a probability 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 The value of 0.6 is the drag coefficient specified.
5
(2.7)
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.
Moment
Wind Profile
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.
6
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 3(G # 1) z "0 wm ( z ) zdz h2 h
wg ( z ) $
(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 level z 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.
3( g # 1) z Fzf $ Fzm zdz H 2 H "0 H
7
(2.10)
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)
Fz $ Fzm % Fzf
The wind load due to the hourly mean wind component is given by
Fzm $ p z C D d z (2.12) where pz gives the design pressure at hourly mean wind component and is pbtained by the equation 2
p z $ 0.6V z
(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
w( z ) $ C dr ( z )d ( z ) p( z )
(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
w' ( z ) $
3.0 zG w' M w (b) h3
(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
&
'
11.0 T1V (33) Gw' $ 0.30 % (h % 16) 0.86
0.47
(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
8
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. G $ 1 % 2 gi B %
ES (
(2.17)
Where g is peak factor with g $ 2 log e vT % the turbulence intensity
background turbulence
0.577 2 log e vT
i $ 0.311 # 0.089 log10 h
. 4 h 1 0.63 + B $ ,1 % 2 / ) -, 3 265 0 *)
9
(2.18)
(2.19)
#0.88
(2.20)
energy density
E$
spectrum
size reduction factor
4 f 1 12322 1 //h 0.21 3 Vb 0 . 4 330 f 1 2 + 1 // h 0.42 ) ,1 % 22 ,- 3 Vb 0 )*
0.83
1.14 4 1 4 1 f 1 S $ 21 % 5.7822 // h 0.98 / 2 / 3 Vb 0 3 0
#0.88
(2.21)
(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
10
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 the help 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
0.6
Base 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
0.2345
2
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
11
300 250 200 150 100 50 0 0
500
1000 CICIND
1500 ACI
2000
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.
12
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 body causes 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.
CHIMNEY
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 Sn called the Strouhal number. Sn $
shedding _ frequency 5 diameter _ of _ cylinder mean _ flow _ velocity
13
(2.23)
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 the mode-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.
Frequencies:
35 30 25 20
Mode 1: 0.2345 hz
15 10
Mode 2: 1.0266 hz
5
Mode 3: 2.4826 hz
0 -1.2
-0.7
-0.2
Mode 4: 3.6286 hz 0.3
0.8
Mode shapes 1 to 4
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.
14
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. ?H < 9 " d z7 zi dz 9 CL 9 9 8 oi $ > 0H ;5 9 7 2 zi dz 9 46S 2 n K si 9 "0 9 = :
(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
Fzoi $ 46 2 f1 8 o i " m z7 zi dz 2
(2.25)
zo
H
Fzoi $ 46 2 f1 8 o i " m z7 zi dz 2
(2.26)
zo
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
Ksi $
2mei A s @d 2
(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.
15
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
"m 7
2
zi
z
mei $
dz
0 H
"7
2
zi
dz
(2.28)
0
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 ith mode. It is done with the help of the following equation.
Vcri $
f1d Sn
(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 taper $
2(d av # d top )
(2.30)
H
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
16
8 oi $
? 6L < @d 2 > ; 1.25C L d7H 1 = 2( % 2) : 5 mei 6 2S 2n k @d 1 7 2 zi dz B # a " H mei
(2.31)
2
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 @ C L d 4 ze7 zei7H 1
8 oi $
H
26 S mei " 7 2
2
2
zi
d zi
0
6L 2t
B # k a @d 2 mei
(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.
Vcri $
f1 d ze i Sn
(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. Vcr $
fd (u ) St
17
(2.34)
Where the St is the Strouhal number and is calculated using 4 h 1 / S t $ 0.2522 0.333 % 0.206 log e d (u ) /0 3
(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. Ma $
G Ea 2 S S CL V cr d (u )h 2 g 2
6 2L Sp 4CB s % B a D . h + , d (u ) % C E ) *
(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 ?9 4 .V # V ( z cr ) + 1.4 # , ); 3 - V ( z cr ) * 9: 1.4 9=
(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
18
300 250 200 150 100 50 0 -400
-200
0
200 Mode 1
400
600
800
Mode 2
Figure 2.5 – IS Simplified method & Figure 2.6 – IS Random Response Method 300 250 200 150 100 50 0 -1000
-500
0
500
Mode 1
19
Mode 2
1000
1500
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 are very 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)
Mode 1 Mode 2
Across-wind 125.46 98.86
Along-wind 432.98 432.98
Gust Factor 1.8854 1.592
Max Moment 825.922 696.56
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 ratio An 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
:
0.6
Base 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)
20
Estimation of Free Vibration parameters like the mode-shapes the free frequency and the Weight data for the calculations were calculated by modeling the chimney 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 ! 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. ! The Approximate method of the IS code gave consistently higher moments as compared to the Random Response Method for vibrations in the first mode. ! The Approximate method of the IS code gave consistently lower moments as compared to the Random Response Method for vibrations in the second mode. ! The IS method gave higher moments in the second mode of vibration as compared to the first mode in both its methods. ! The ACI method gave very small values as compared to the IS methods for the base moment in all cases ! 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.
21
35 30 25 20 15 10 5 0 0
2000
4000
6000
Figure 2.7 – IS Approximate Method – Mode 1
35 30 25 20 15 10 5 0 -10000
-5000
0
Figure 2.8 IS Approximate Method – Mode 2
22
5000
10000
35 30 25 20 15 10 5 0 0
1000
2000
3000
4000
5000
20000
30000
Figure 2.9 IS Random Response Method – Mode 1
35 30 25 20 15 10 5 0 -20000
-10000
0
10000
Figure 2.10 IS Random Response Method – Mode 2
23
H/d Mode 1 Mode 2
7 641.15 411.482
9 340.566 225.483
11 204.425 142.764
12 144.783 107.867
13 114.783 87.404
15 77.271 59.786
17 55.244 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.
24
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 earth’s 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.
25
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 of a 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
26
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 shapes by 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
ui
ui N i
(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. 2
pi Ti a s (Ti ) 4! 2
Ni
(3.2)
The as is the response function described earlier. The value of pi is obtained from h
" u ( z)m( z )dz i
pi
0 h
"u
2 i
(3.3)
( z )m( z )dz
0
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.
27
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 from strong 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 soilfoundation 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.
28
#h
$I# 0
(3.4)
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.
#h
$IF0
Sa g
(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 A
%
! 2 2 d out ' d in 4 29
&
(3.6)
A = 45.0 m2 The moment of inertia at the base is calculated by
%
! 4 4 d out ' d in 64
I
&
(3.7)
The value of I = 2742.5 m4 Radius of gyration r is given by
I A
r
(3.8)
r = 7.806 Hence the slenderness ration l/r is given by l r
32.02
(3.9)
The coefficient CT
CT
57.822
(3.10)
!DmeanTH t (
(3.11)
Weight of the chimney Wt
Weight = 17495583 kg The period of vibration is now given by T
CT
Wt h ' EAg
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)
30
(3.12)
F0 = 0.25 (assuming the chimney to be in the zone IV) Shear force and Bending moments The design shear force at a distance of X’ from the top is given by V
.5 4 X ' 1 2 4 X ' 12 + CV # hWt , 2 / ' 2 / ) ,- 3 3 h' 0 3 3 h' 0 )*
(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.
300 250 200 150 100 50 0 0
500
1000
1500
kN
Figure 3.1 – Shear force due to seismic loads The bending moment can be calculated using the formula
M
1 4+ . 2 X ' 4 1 4 X'1 ) , # hWt h 0.62 / ' 0.42 / , 3 h' 0 3 h' 0 ) *
31
(3.14)
Again the value of X’ is varied and the expression evaluated. The resultant graph is given below.
300 250 200 150 100 50 0 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.
32
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 f’’CTV refers to the concrete stresses and the value f’’STV refers to the stress in steel. f ' 'CTV ! " te c Tx E c
(4.1)
f ' ' STV ! " te (c $ 1 # % 2 ) Tx nE c
(4.2)
And
Where the terms are explained below
33
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 concrete c is given by the equation
c ! $ *n'% 1 # 1( #
)*n'% 1 # 1(&2 # 2 *n)% 2 # % 1 (1 $ % 2 )&
(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 0 . + td ci Ti $ To . + Tx ! td ci d ci + Cc d c . 1 .K #C d # K d + c c o co , / i
(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 f STV ! " ie c' Tx E c
34
(4.6)
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 f ' ' CTC ! " ie c' Tx E c and the same for steel is f ' ' STC ! " ie (% 2 ' c' ) Tx E s
(4.7) (4.8)
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
1 1 ct kt c ! t s1 st ! pt c1 st 2
(4.9)
Which gives on solving for the stress in steel
7 at $ kt c 71 4 1 st ! 55 ct 22 ! m1 ct 55 c 6 2p 3 6 kt c
35
4 'a $ k ( 22 ! m1 ct k 3
(4.10)
The following figures are a representation of the case
atc tc
Air Gap
Lining
Temperature Gradient To ktc st ct
Net Strain in Steel (Tension)
T! - e T!
e
Net Strain in Concrete (Compression)
Figure 4.1 – Thermal Stresses The expressions for stress in steel in turn give the following equation for the value of k2 k 2 ! 2 pm'a $ k (
(4.11)
k ! $ mp # 2mpa # p 2 m 2
(4.12)
Wherein the value of k is
36
Rise in temperature in reinforcement is Free expansion of steel is
(1 $ a)T
(4.13)
(1 $ a )"T
(4.14)
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
1 st ! Es)'1 $ k ("T $ '1 $ a ("T &
(4.15)
1 st ! Es"T (a $ k )
(4.16)
or
similarly stress in concrete is given by
1 ct ! Ec"kT
(4.17)
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 k’tc = 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
37
The equations for the calculation of the stresses are given as
1 ' s ! m1 ' c
'a $ k '( k'
(4.18)
k ' ! 2 pma # p 2 m 2 $ pm
(4.20)
)1 ' s # m1 'c & ! E s"Ta
(4.21)
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
38
#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
1 ' s ! 18.31 ' c
(4.22)
1 ' s #111 ' c ! 168.9
(4.23)
And 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.
39
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. !
Stress-strain relationship of concrete in compression
!
Limiting compressive strain in concrete
40
!
Limiting tensile strain in steel
!
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 sml = 0.070
fcyk
Es
s sy
sml
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
41
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 Hognestad’s 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 value of the strain
cu
cul
corresponds to the maximum
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 of
cul
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 shortterm 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
42
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 the overall 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 !
The stress strain curve is parabolic rather than linear, even under the short term loading under consideration.
!
If fcu = 0.85 f’ck is assumed then it is reasonable to assume an increase of approximately 10% for relatively short time loading.
!
The value of the ultimate compressive strength
cul
corresponding to
this peak may be assumed to be approximately 0.002 for both shortterm and long-term loading. On the basis of the above discussion the following curve is assumed as the stressstrain 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 f’ck) 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 f pc1 " C s f pc
(5.1)
( N % '0.95 ) 0.1 $ N max # & Cs " 0.85
(5.2)
where
43
The curve is as shown in the following figure.
fcu
0.85 Cs f’ck
c1 cul
0.002 Figure 5.2 – Stress-strain curve (Concrete) 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.
44
N " Nc * Ns M u " M uc * M us
(5.3) (5.4)
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.
Neutral Axis
!0
! Wind
Figure 5.3 – Chimney Cross section The distribution of strains and the corresponding stresses are given in the below
45
Neutral Axis Strains cu
= 0.002
Concrete Stresses fcu Steel Stresses -fsyk fsyk 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
+"
Mu N
46
(5.5)
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 within the 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. !=90
! = maximum Neutral axis location not possible
The maximum compressive strain in steel
!=0 Figure 5.5 – Strain profile variation
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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 on the 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 n"
N f ' ck rt
(5.6)
m"
Mu f ' ck r 2 t
(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
,
f syk f ' ck
Where # is the percentage of steel fsyk and f’ck are the strengths of steel and concrete.
48
(5.8)
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 that particular 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 9 8 7
2.075 8.3 15.56 20.75 25.94 31.125
6 n
5 4 3 2 1 0 0
1
2 m
Figure 5.6 – Interaction curves
49
3
4
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
,
f syk f ' ck
" 2.075
(5.9)
And so on. The values of the terms utilized to arrive at the values are given below
#
f’ck
fsyk
#(fsyk/f’ck)
0.2
40
415
2.075
0.8
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. fcu " 0.85C s f ' ck Where Cs is the short term loading factor that varies linearly as explained earlier. cul
= 0.002
50
(5.10)
The stress-strain curve for concrete is given below
f pc
2 f cu (- 3 + 0 3 + 0 %" '21 . ) 1 . $ 4 c - 12 + c1 ./ 12 + c1 ./ & #
(5.11)
The stress-strain curve for steel is given below ( E s + 6 )+ sy 7 + 7 + sy 3+ 0 fs " ' 1 . 6 + sy 5 + fsyk 1+ . 2 / &
(5.12)
Where
+ sy "
f syk
4 s Es
(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 9
N c " 2rt (1 ) , ) 8 f pc (+ )d:
(5.14)
:0
9
M uc " 2r 2 t (1 ) , ) 8 f pc (+ ) cos(: )d:
(5.15)
:0
9
N s " 2rt, 8 f s (+ )d:
(5.16)
0
9
M us " 2r t, 8 f s (+ ) cos(: )d: 2
(5.17)
0
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
51
9
2(1 ) , ) nc " f pc (+ )d: f ' ck :80
(5.18)
9
2(1 ) , ) mc " f pc (+ ) cos(: )d: f ' ck :80 ns "
ms "
2, f ' ck
(5.19)
9
2, f ' ck
8f
(+ )d:
(5.20)
(+ ) cos(: )d:
(5.21)
s
0
9
8f
s
0
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 20-50
35
Db/tb 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
54
Parameter = 13.83 (percentage of steel) With 1% steel, using the curve with a parametric value of 15.56 – the cross section 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 steel also needs to be changed with increase in height. Since increasing the spacing alone is not a good option, keeping in mind the requirements of
55
temperature gradient and for preventing the surface cracks. Hence increase in spacing needs to be coupled along with decrease in diameter. Increase spacing to 180mm after a height of 100m. from a height of 150m use 20mm diameter steel bars at the initial spacing of 150mm. From 200m onwards use 20mm bars at 180mm center to center. The following is a sketch of the reinforcement details at the base of the chimney. 150mm
75mm
500mm
650mm
Figure 6.2 – Sectional plan view – Vertical Reinforcement 75mm
150mm
650mm Figure 6.3 – Sectional elevation view – horizontal Reinforcement
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6.3 Design of foundation Load effects Deal load = 175 MN Wind load = 1553 MNm Earthquake load = 800 MNm (Not Critical) For no tension design we consider W A
M Z
Where the load is W, and the moment is M The area A is given by A !
D2 4
!
D3 32
And the value of Z is Z Substituting and calculating Diameter = 70.99m Adopt a smaller diameter and not go for a no-tension approach. Adopt a diameter of 60m A rough diagram of the chimney base is shown here
19m - approx
Figure 6.4 – The foundation (representation)
57
60m
Apportioning the chimney base The stress distribution on the base of the chimney is calculated by trial and error method. The maximum permissible stress of the soil is assumed to be 300 kN/mm2 (assuming a rock strata) The force diagram looks like e
P
Figure 6.5 – Load and eccentricity The value of the eccentricity is 9.1m With a liftoff are segment of 4 meters the resistance capacities are F = 189476 kN P = 39.18 from the left of the diagram. Hence the eccentricity of 9.1 meters (total of 30+9.1 = 39.1m) is taken care of. The loads are calculated keeping in mind the distribution of loads that occur on the foundation as shown
Figure 6.6 – Actual loading pattern Calculating the maximum moment from the cantilevered part of the chimney M
2 W * a 0b- ' 3 2 2 (2 ln( ) 1 1 $ . + % $ w"a $ b # 8! )( b a 16 / , &%
58
Where w
4W !D 2
Moment acting at the base = 11 MNm Required deff = 1.795m Assume 75mm cover Use a total depth of 2m under the shell of the chimney Taper the thickness of the foundation to 600mm at the end of the chimney Reinforcement Although the footing is circular, the reinforcement is provided orthogonal. Calculating the steel requirement Pt/100 = 3.5% Provide 30mm bars at 100mm center to center The moment acting on the other direction is due to the landfill and is not very high. Hence providing a nominal reinforcement on the sloping side of the foundation. Provide 12mm bars at 150 centers. Also take a local thickening of the shell to take care of transfer of loads. Hence the final detailing of the foundation and the base connection is given below
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24mm bars along with vertical reinforcement Top of foundation 12mm @ 150mm c/c
Bottom of foundation 30mm @ 100mm c/c Figure 6.7 – The foundation and the connection Design of Staircase A staircase of independent tread slabs can be constructed inside or outside the chimney for the purpose of maintenance and cleaning etc. The cross section of such a stair is given below. The end of the stair is embedded into the shell. 8mm @ 220mm 310mm
300mm
1200mm
3-10mm
Figure 6.8 – Design of staircase tread
6.4 - Conclusion The example chimney has been designed and the reinforcement requirements and other details worked out. The foundation for the chimney has been designed too. The interaction curves developed were used in the process of design.
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7. Summary and Conclusion In the preceding chapters, various aspects involved in the analysis and design of Reinforced Concrete Chimneys were looked into. The loads that are to be considered during the analysis phase of the chimney were taken into account. Of these loads the most important were found to be the wind loads. Although there were also earthquake loads and seismic loads these were not found to be critical for design. The wind loads were of two types – along-winds and across-winds, depending on the type of response. The reasons for both these types of loads were studied. Various codal methods were employed for the purpose of evaluation of these wind loads. During the calculation for calculation of the along-wing loads calculations were performed to evaluate the wind profiles, the pressures of the incident wind, and the resultant loads and load resultants on the chimney. The dynamic effects were adjusted with the calculation of the gust factor. The reasons for the occurrence of across-wind loads were identified. The concept of bluff body flow and vortex shedding were discussed. The chimney was also modeled using STRAP for the purpose of estimating the dynamic effects. It was found during a comparative evaluation of the various codes for the calculation of along-wind loads, to have a considerable difference in the reported value for similar chimney and wind conditions. The values of the gust factors too were found o be different. The across wind loads were also estimated using the provisions in various codes. The two methods suggested by the IS code too were studied. Since the random response method, suggested by the code is more realistic in evaluation of the across-wind loads, it was felt that the simplified method could be done away with completely. A variational analysis was also done, calculating the moments for different height to base diameter ratios. The conclusions from this analysis were listed. The occurrence, identification and estimation of loads from seismic action were also studied. Various terms relating to seismic action were also introduced. Estimation of loads was done for an example chimney by estimating the design coefficient for a particular case.
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The two methods suggested by the IS code were studied. Calculation was however done using the response spectrum method. The moment profile was calculated and plotted. It was found that the loads due to seismic action were almost half that from the wind velocities, and hence would not be a major consideration in design, given their rare occurrences. Temperature loads due to a temperature gradient across the chimney shell were noted and expressions to evaluate the same were derived. The resultant stresses for a case were calculated. It was found that they are normally taken care of by good detailing unless the temperature differences were very high. The design resistance of a cross-section was estimated. The normal force on a cross-section was related to the moment carrying capacity of the section and interaction curves were drawn for the same. Using the loads obtained earlier and the design principles above a typical chimney was apportioned and detailed. The foundation (raft) to take care of the loads for the chimney was also detailed. A design of a staircase tread was also given. Also all cases for design were not considered. The chimney was assumed to have no openings and designed as such.
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Appendix Listing –1 Across wind moments (ACI method) #include #include #include #include #define SQ(a) ((a)*(a)) void main() { float Vel(float, float); float Dia(float); // This program calculates the Moments due to Across Wind effects // Using the ACI method // Data assumed for the chimney const float ht = 250; // meters const float bdia = 20.8333; // meters const float tdia = 12.5; // meters const float bthk = 0.595238; // meters const float tthk = 0.278274; // meters // Data required as input for each mode shape - File input float freq; // time period in that mode shape - to calculate freq later on float mass56; // mass value at 5/6 the height int modeno; // Variables char fname[20]; // The input file name // Get the file name clrscr(); cout > fname; ifstream fin(fname,ios::in); fin >> modeno; fin >> freq; fin >> mass56; mass56 = mass56 * 32 / (250 * 100); // weight per unit meter kN/m mass56 = mass56 * 68.53; // lb/ft conversion fin.close(); float Vzcr, Vcr; // The control wind velocities to be calculated float strouhal; strouhal = 0.25 * (0.333 + 0.206*log(820.25/45.569) ); Vzcr = 40*pow((5*250)/(6*10),.14); Vcr = freq * 13.89 / strouhal; // Convert to ft/sec Vzcr = 3.281 * Vzcr; Vcr = 3.281 * Vcr;
I
// Some variables float term1, term2, term3; // The three terms the eqn is divided into float i, cl0, fb, cl; float V; float modeshp = modeno==1?0.57:0.18; // Calculations i = 1/(log((820.25*5/6)/0.018) ); cl0 = -0.243 + 5.648*i - 18.182*i*i; fb = -0.089 + 0.337*log(820.25/45.569); if (fb > 1.0) fb = 1.0; if (fb < 0.2) fb = 0.2; cl = cl0 * fb; term1 = (4.0/32.2)*modeshp*cl*(0.075/2)*SQ(Vcr)*45.569*SQ(820.25); //cout
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