rc chimneys

8/13/2019 Rc Chimneys

http://slidepdf.com/reader/full/rc-chimneys 1/21

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

f s 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

f cyk

Es

sml sy

sml = 0.070



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 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 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 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

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 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

The stress strain curve is parabolic rather than linear, even under the

short term loading under consideration. If f cu = 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 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 f cu = (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

pc s pc f C f 1 (5.1)

where

85.0

1.095.0max

N

N

C s

(5.2)

43



The curve is as shown in the following figure.

f cu 0.85 Cs f’ck

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.

44



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 N N N

usucu M M M (5.4)

0

Wind



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

Mu

f cu

Steel

Stresses

-f syk

f syk

(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 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



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

rt f

N n

ck ' (5.6)

t r f

M m

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

' (5.8)

Where

is the percentage of steel

f syk and f’ck are the strengths of steel and concrete.

48



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

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

49



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

f (5.9)

And so on.

The values of the terms utilized to arrive at the values are given below

f’ck f syk (f syk /f’ck )

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)ck s f C fcu '85.0

Where

Cs is the short term loading factor that varies linearly as explained earlier.

cul = 0.002



The stress-strain curve for concrete is given below

2

11

2ccc

cu pc

f f

(5.11)

The stress-strain curve for steel is given below

sy

sy sy s

fsyk

E

fs (5.12)

Where

s s

syk

sy E

f

(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

0

)()1(2 d f rt N pcc

(5.14)

0

)cos()()1(2 2d f t r M pcuc

(5.15)

0

)(2 d f rt N s s(5.16)

0

2 )cos()(2 d f t r M 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

51



0

)('

)1(2

d f f n pcck

c (5.18)

0

)cos()('

)1(2d f

f m pc

ck

c(5.19)

0

)('

2d f

f n s

ck

s(5.20)

0

)cos()('

2 d f f

m s

ck

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.

52



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

D b/t b

20-50 35

Mean, base thickness ratio tm/t b 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.

53



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 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 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 use20mm 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

56

650mm

Figure 6.2 – Sectional plan view – Vertical Reinforcement

Figure 6.3 – Sectional elevation view – horizontal Reinforcement

75mm

500mm

75mm

150mm

650mm



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

Z

M

A

W

Where the load is W, and the moment is M

The area A is given by

4

2 D

A

And the value of Z is

32

3 D

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

57

Figure 6.4 – The foundation (representation) 60m

19m - approx



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

58

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

22

2

16

3

1)ln(28baw

a

b

b

aW M

e P



Where

2

4

D

W w

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

59



60

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.

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.

24mm bars along with

vertical reinforcement

Top of foundation

12mm @ 150mm c/c

Bottom of foundation30mm @ 100mm c/c

8mm @ 220mm

310mm

1200mm 3-10mm300mm

rc chimneys

Documents