AXIAL DEFORMATION OF COLUMN IN TALL STRUCTURES

By Er.P.K.Mallick, Dy.Chief Engineer,Cuttack, Odisha.

pk_mallick1962@rediffmail.com

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Concrete members experience time-dependent behaviour caused by creep and

shrinkage .During the past, much research has been made in this area providing means

for good understanding of the effect of creep and shrinkage on concrete and processes

through which they evolve. Basically, two types of behaviour are distinct because of

creep and shrinkage:

1)Creep and shrinkage lead to increased deformations in plain concrete.

2)In reinforced concrete ,creep and shrinkage cause stress redistribution between the

compressive zone in concrete and steel reinforcement. The direction of stress transfer

in reinforced concrete column is normally from concrete section to reinforcement,

leading to an increase in steel stress and decrease in concrete stress with time.

A reinforced concrete column also undergoes axial shortening due to creep and

shrinkage and this phenomenon is known as time-dependent shortening of column.

With the increase in height of buildings, the importance of time-dependent shortening

of columns and shear walls become more critical owing to cumulative nature of such

shortening. It is known that column with varying percentage of reinforcement and

varying volume to surface ratio will undergo varying strains due to creep and shrinkage

under similar stresses.

In a multistoried building, adjacent columns may have different percentage of

reinforcement due to different tributary areas or different wind loads. As a result, the

differential elastic and inelastic shortening will produce moments in the connecting

beams or slabs and will cause load transfer to the element that shortens less. As

number of stories increase, the cumulative differential shortening also increases, and

the related effect become more severe. A common example is the case of a large,

heavily reinforced column attracting additional loads from adjacent shear wall which has

higher creep and shrinkage due to lower percentage of reinforcement and lower volume

to surface ratio. Significance differential shortening may also occur due to a time gap

between a slip-formed core and the columns. In this case the columns are subjected to

full amount of creep and shrinkage, while the core may have had the bulk of its

inelastic shortening occurring prior to casting of adjacent columns.

EFFECT OF TIME DEPENDENT SHORTENING OF COLUMNS- CASE HISTORIES

AND REAL ISSUES ASSOCIATED WITH CALCULATION OF AXIAL SHORTENING:

It is customary , at present, to neglect the effect on the frame of elastic and inelastic

shortening of columns and walls. For a low and intermediate height structures this may

be acceptable, however, neglecting the differential shortening in ultra-high-rise building

may lead to distress in the structure and in a non-structural elements of the building.

In a number of tall buildings in the United States built in the early sixties, structural

cracking and partition distress were observed as a result of differential creep between

shear walls and highly reinforced columns in close proximity to each other. Another

example of the reality of differential creep and shrinkage of vertical elements is of fifty

story building in Australia in which the measured differential shortening at the roof level

between the concrete core and peripheral column was 27.94mm after about four and

half years. Fortunately, no problems were experienced, the long span of about 11m

between the core and peripheral columns caused only small slab rotations. The elevator

rails had to be adjusted twice over the years to accommodate the shortening of

elevator shafts.

Building up to 30 stories with flexible slab systems ,such as flat plate slabs of average

spans or long span joint systems, are not adversely affected structurally by differential

shortening of supports. In those cases the knowledge of the total shortening is needed

to make allowance in architectural details to avoid further distress of partitions,

windows, cladding, and other nonstructural elements.

Differential shortening can be minimized by proportioning adjacent columns or walls to

have similar stress of the transformed section and similar percentage of reinforcement.

The volume to surface ratio has a lesser effect on differential shortening.

Although a large amount of research information is available on shrinkage and creep

strains, it is not directly applicable to columns of high-rise building. The available

shrinkage data must be modified since they are obtained on small standard prisms or

cylinders stored in controlled laboratory environment. The available creep research is

based on application of loads in one increment and such creep information , therefore,

is applicable to flexural elements of reinforced concrete and to elements of prestressed

concrete.

In the construction of high-rise building, however columns are loaded in as many

increments as there are stories above the level under consideration. If a 50 storied

building is constructed in 50weeks,then the first story columns receive 2% of their

design load every week during construction period. Incremental loading over a long

period of time makes considerable difference in this magnitude of creep too.

It has already been high- lighted that though we have lot of research data on creep and shrinkage, those are not directly applicable for prediction of inelastic shortening of column.

An effort has been made here for comprehensive review and comparison of method of prediction of inelastic shortening including that developed recently. The purpose of this review is to comment on theoretical validity and to compare them in terms of their efficiency .accuracy and practical value. The methods are considered in more or less, their chronological order of development. Considered roughly in their order of sophistication, the methods reviewed are: 1)Method developed by Mark Fintel & Fazlur .R.Khan. 2)Method developed by Mark Fintel , H.Iyenger & S.K.Ghosh. 3)Method developed by Raed M.Samra. Though the method developed by Raed M.Samra ,I call it as most recent ,but it in reality it was first published in the year 1995 in Journal of Structural Engineering. As far as I know there is no significant development after the work of Raed M. Samra. But if it has happened, I would like to be updated on that. METHOD DEVELOPED BY M.FINTEL & F.R.KHAN: Though it is long recognized fact that in reinforced concrete columns, creep result in gradual transfer of load from concrete to reinforcement, the procedure for prediction of the amount of creep and shrinkage strains was first outlined in the late 1969."Effect of

creep and shrinkage in tall structures -prediction of inelastic column shortening " was perhaps the first paper on this subject to be published in ACI Journal December 1969 issue and credit goes to M.Fintel and F.R.Khan for this publication. The procedure takes care:

a) Loading History of Columns.

b) Volume to Surface Ratio of Sections.

c)Effect of percentage of Reinforcement. For Structural Engineering practice, the specific creep has been considered. The specific creep Σc is defined as the ultimate creep strain per unit of sustained stress. Since creep decreases with age of concrete at load application, each subsequent incremental loading contributes a smaller specific creep to the final average specific of the column.

Determination of Specific Creep, Σc : There are two ways to determine the value of specific creep. It can be obtained by extrapolation from number of laboratory samples prepared in advance from actual mix to be used in structure. It is obvious that sufficient time for such tests must be allowed prior to start of construction, since the reliability of the prediction improves with length of time over which creep is actually measured. An alternative method to predict basic creep is from elastic modulus of elasticity. In the mentioned article a curve(we call fig-1) is shown which give the creep magnitude as related to initial modulus of elasticity for different load durations. For design purposes ,the 20 year creep can be regarded as the ultimate creep. Thus from the specified 28days strength, the basic specific creep for loading at 28days can be determined and then modified for construction time, member size and percentage of reinforcement. Effect of construction time on creep: To determine the effect of construction time on creep, this method takes the help of curve(we call fig-2) giving relationship between creep and age at loading. The total creep strain for an incrementally loaded column "N" stories below the roof will be

Σc = ΣN

i fci Σci

Where fci Σci are creep strains produced by the stress increment fci .Individual value for specific creep can be obtained from fig-1 or from the creep of a test specimen loaded at 28days and then modified for various age at loading using fig-2.

The procedure gives formula for weighted average of specified creep where load increments are unequal. Another formula is given for where load increments are equal. Then the procedure gives formula for total creep strain. The procedure explained above has been further simplified and a curve (we call fig-3) has been developed which gives relationship between "Time of Construction" and "Coefficient for incremental loading". The Coefficient for incremental loading plotted in figure-3 is used to convert the 28day creep into average specific creep for a column load with equal load increment at equal time intervals. In continuation to explanation of above method let us look into the rest of the issues associated with the method.

Effect of Member size on Creep:

Creep is less sensitive to member size than shrinkage since only the drying creep component of total creep is affected by size and shape of members,where as basic creep is independent of size and shape. It appears from a laboratory investigation that drying creep has its effect only during the initial three months.Beyond 100days,the rate of creep is equal to basic creep. Shrinkage Strains-Adjusted For Column Size:

Shrinkage of concrete is caused by evaporation of moisture from the surface. Similar to creep,the rate of shrinkage is high at early ages,decreasing with increase of age,until the curve becomes asymptotic to final value of shrinkage.Since evaporation occurs only from the surface of members the volume to surface ratio of a member has a pronounced effect on the amount of its shrinkage. The amount of shrinkage decreases as the size of specimen increases. Much of the shrinkage data available in the literature is obtained on 27.9 cm long prisms of a 7.6* 7.6 cm section. Obviously, such data cannot be applied to usual size columns without considering side effect. The relationship between the magnitude of shrinkage and the volume-to-surface ratio has been plotted in a curve (we call fig-4). The size coefficient for shrinkage shown in fig-4 is used to convert shrinkage data obtained in 6inch cylinders to any other size columns. Effect of relative humidity on shrinkage:

The shrinkage specimen should be stored under conditions similar to those for actual structures. If this is not possible, the shrinkage results of a specimen not stored under field humidity conditions of structure must be modified to account for humidity conditions of structure. The curve developed by C.L.Freyermuth showing relative humidity percentage and shrinkage humidity correction factor must be used in this regard.

Progress of Creep and Shrinkage with Time:

Both creep and shrinkage have similarity regarding the rate of progress with respect to time. A curve (we call fig-5) is developed to show ratio of creep or shrinkage at anytime to final value at time infinity. This curve can be used to extrapolate the ultimate creep and shrinkage values from laboratory testing of certain duration time. Effect of Reinforcement on creep and shrinkage:

Long term test has shown that on columns with low percentage of reinforcement the stress in steel increased until yielding while in highly reinforced columns after entire load had been transferred to steel ,further shrinkage actually caused some tensile stresses in the concrete. It should be noted that despite the redistribution of load between concrete and steel, the ultimate steel capacity of the columns remains unchanged. The total creep and shrinkage strains of a non-reinforced column are

Σ = fc Σc + Σs

where fc =Initial elastic stress in the concrete. Σc = ultimate specific creep strain of plain concrete ΣS = Ultimate shrinkage strain of plain concrete. A curve (we call fig-5) has been developed to determine residual creep and shrinkage strains of reinforced column from the total creep and shrinkage strain of identical column without reinforcement for various percentage of reinforcement,varying specific creep and modulus of elasticity of concrete. THEREFORE,THE TOTAL STRAIN IN COLUMN DUE TO CREEP AND SHRINKAGE IS SUM TOTAL OF STRAINS CALCULATED DUE TO VARIOUS FACTORS AFFECTING SHRINKAGE AND CREEP.

LIMITATION OF METHOD DEVELOPED BY M.FINTEL & F.R.KHAN:

This solution for creep and shrinkage of columns were prepared during late sixties

based on state of art in that era. Hence the limitation in the procedure is apparent.

a) Effect of relative humidity on creep is not considered.

b) Effect of water/cement ratio on creep is not considered.

c) Creep also depends on fine aggregate to total aggregate ratio. This effect is not

taken into account in the analysis.

d) Similarly percent of air content in concrete has pronounced effect on creep and this

method is silent on this aspect.

Further, the method to predict basic creep (without testing) from elastic modulus of

elasticity is based on results of limited tests on normal weight concrete conducted at

Bureau of Reclamation in Denver.

METHOD DEVELOPED BY M.FINTEL,H.IYENGER AND S.K.GHOSH:

This method is an extension of method-1. However, the procedure has been

computerized to ease the burden of meticulous arithmetical calculation and extensive

book keeping of data.

The developed computerized procedure is applicable to concrete, steel and composite

structure and consider separately the elastic and creep component due to gravity loads

and also shrinkage shortening.

Since structural effects result from differential distortions caused by column shortening

after slab has been installed, the procedure separates the shortening of supports that

occur after slab installations.

Computer utilization is particularly significant because consideration of shrinkage and

creep requires extensive computation and summations as every story -high column

segment in a multistoried building is loaded as many increments as there are stories

above and for each loading increment of each column segment has now new time

dependent properties ,modulus of elasticity, creep coefficients ,shrinkage coefficients,

changing column sizes i.e. volume to surface ratio and varying reinforcement ratios.

Since the method is similar to method-1, this does not necessitate a detailed discussion.

METHOD DEVELOPED BY RAED M.SAMRA

This method is due to Prof.Raed M. Samra and was published in Journal of Structural

Engineering in March 1995 issue. This is an improvement over his previous study of

creep model which requires the use of an iterative procedure for the solution of the

creep strain and creep stress under sustained load, published in Journal of American

Concrete Institute (1998) under the title “Creep model for Reinforced Concrete

Columns”. The new approach has a great advantage from practical point of view since

the results for axially loaded column are identical to those obtained by using the

iterative procedure, but can derived from a direct calculation. The best advantage of

the current procedure is that it requires little input, including creep data and section

and material properties ,which makes it useful in most case of commonly encountered

design problems. The brief outline of the procedure is as follows:

To evaluate the change in stress and deformation arising in reinforced concrete section

under sustained loads, the procedure has been evolved based on the algebraic

constitutive law to describe the creep deformation of the concrete. The law is expressed

as follows:

Єct = fci /Ec (1+Øt ) - ∆ ft /Ec (1+ 0.8 Øt )

Where ,

Єct = concrete strain at time “t”.

fci = Initial concrete stress.

∆ ft = stress decrement in concrete at time “t”.

Øt = Creep coefficient at time “t”.

Ec = Modulus of elasticity of concrete.

The law is particular case of well-known age-adjusted effective modulus method.

For axially loaded columns, the requirements of strain compatibility, equilibrium and

stress-strain relationship apply at any time under load. These requirements are written

at time t > to, a long time after load application.

From the strain compatibility and equilibrium the stress in steel ,the stress in steel can

be expressed by the formula:

Fst = n Fci {(1- 0.8 ) *Øt + Fct/Fci (1+0.8 Øt)}

Hence strain in steel due to creep= Fst/ Єct

The above equation completely solves the problem in closed form. A comparison

between results of tests conducted by Troxell etal. on concrete columns and results

predicted by the analytical procedure presented in the Prof.Samra’s paper shows a good

correlation between measured and computed value.

Although this paper only deals with creep, the most cases of practical interest it is

important to superpose the results of a shrinkage model proposed by Park and Pauley

with the outcome of the Samra’s creep model, because the final stresses and strains

under the combined effect of creep and shrinkage are normally of interest.

Since the equilibrium and compatibility are satisfied in the individual models, they will

be also satisfied in the final superposed model. The shrinkage model is simple to use

and is based on sound theoretical basis in which it is assumed that the restraint of

shrinkage concrete by reinforcement will induce tensile stresses in concrete

accompanied by compressive stress in steel. Since the shrinkage can occur even in the

absence of any external load, the requirement of equilibrium would dictate that total

force induced in concrete is equal and opposite of the total force in steel. Simple

mathematical derivation yields that the stress in steel, Fs, at any time equal to

Fs=Fc*Ac/As Where Fs=Steel compressive stress due to shrinkage.

Fc =X/Y

X=єsh

Y= (1+Øt)/Ec +( Ac/As*Es) Analysis of Method developed by Prof. Samra.

Basically this method consists of two parts.

1) Prediction of creep coefficient and ultimate shrinkage.

2) Calculation of axial deformation.

The beauty of Prof. Samra method is that the prediction of creep coefficient and

ultimate shrinkage has been separated from the main problem and therefore latest

codal practices in the field of creep and shrinkage can be accounted for where as in

earlier two methods (Method developed by M.Fintel and F.R.Khan and subsequent

method by M.Fintel,H.Iyenger & S.K.Ghosh) do not take the help of latest developments

in the field of Creep and Shrinkage and still depend on research data of late sixties.

It is mentioned in the article of Prof. Samra that accuracy of proposed method has been

checked and found to be comparable with actual measurement taken on water tower

place and lake point tower of U.S.A.

The beauty of Prof.Samra’s method is that it separates the issue of the prediction of

creep coefficient and ultimate shrinkage from the main problem of calculation of axial

shortening. Therefore, one can use the codal provision of his own country or the output

of latest research findings to calculate creep coefficient and ultimate shrinkage. This

flexibility is not available other two methods discussed (Method developed by M.Fintel &

F.R.Khan and the Method developed by M.Fintel, H.Iyenger & S.K.Ghosh.)

For solving axial deformation problem by Prof.Samra’s method, I prefer to use the code of ACI 209R-92(Reapproved in 1997)-PREDICTION OF CREEP, SHRINKAGE AND TEMPERATURE EFFECTS IN CONCRETE STRUCTURES for calculation of creep coefficient and ultimate shrinkage. Hence let me explain the provisions of ACI 209R-92(Reapproved in 1997). CALCULATION OF CREEP COEFFICIENT BY ACI 209R – 92 (REAPPROVED IN 1997) METHOD

This code expresses the creep coefficient Ø (t, t o ) as a function of time

Ø (t, t o) = X/Y (Ø∞ (t o ))

X= (t- t o ) 0.6

Y= 10 + (t- t o )0.6

Where creep coefficient is the ratio of specific creep C (t, t o ) at age ‘t’ due to a unit

stress applied at the age ‘t’ to a unit stress applied at the age ‘t o’ , where age ‘t o ‘ is

measured in days.

Since the initial elastic strain under a unit stress is equal to the reciprocal of the

modulus of elastically Ec (t0)

Ø (t, t o ) = C (t, t o ) x Ec(t0)

(t, t o ) is the time since application of load and Ø∞ (t, t o ) is the ultimate

creep coefficient, which is given by

Ø∞ (t, t o ) = 2.35 K1 K2 K3 K4 K5 K6

For age at application of load greater that 7 days for moist curing, or greater that 1 to 3

days for steam curing, the coefficient ‘K1’ is estimated from:

For moist curing:

K1 = 1.25 t o -0.118

For steam curing:

K1 = 1.13 t o -0.095

The coefficient “k2” is dependent upon the relative humidity ‘h’ (percent)

K2 = 1.27 – 0.006h

For h ≥ 40

The coefficient ‘K3’ allows for member size in terms of volume/surface ratio, V/s which

is defined as the ratio of the cross sectional area to the perimeter exposed to

drying. For values of V/s smaller that 37.5mm, K3 is given below.

Value to Surface Ratio (mm) Coefficient (K3)

12.5 1.3

19.0 1.17

25.0 1.11

31.0 1.04

37.5 1.0

When V/s is between 37.5 and 95mm, K3 is given by:

For (t-to) ≤ 1 year:

K3 = 1.14 – 0.00364 v/s

For (t-to) > 1 year:

K3 = 1.1 – 0.00268 v/s

When v/s ≥ 95mm

K3 = 2/3 [1+1.13e-0.0212(v/s) ]

The coefficients to allow for composition of concrete are K4, K5.

Coefficient K4 is given by :

K4 = 0.82 + 0.00264S

Where S = slump of fresh concrete.

Coefficient K5 depends on the fine aggregate/total aggregate ratio, Af/A, in percent and

is given by :

K5 = 0.88 + 0.0024(AF/A)

Coefficient K6 depends on the air content ‘a’ (percent)

K6 = 0.46 + 0.09a ≥1

The elastic strain plus creep deformation under a unit stress is termed the creep

function Ø, which is given by:

Ø (t, t o ) = [1/Ec(t0)]* [1 + Ø (t, t o )]

Where Ec (t0) is related to the compressive strength of test of cylinders. If the strength

at age ‘t0’ is not known, it can be found from the following relation

Fcy (t0) = (t0/(X+Yt0)) * (fcy28)

Where ‘fcy28’ is the strength at 28 days and ‘X’ and ‘Y’ are given below in table:

Type of cement Curing condition Constant

X Y

Ordinary Portland Moist 4 0.85

Cement

Steam 1 0.95

Rapid hardening

Portland cement Moist 2.3 0.92

Stream

Steam 0.7 0.98

Comments by Dr.N.S who has replied that

The paper New Analysis for Creep Behavior in Concrete Columns by R.M. Samra, Journal of Structural Engineering, Vol. 121, No. 3, March 1995, pp. 399-407, referred by Er Mallick contains an example. The abstract of the paper is given below: This paper presents a new rational approach for the evaluation of the effects of creep on reinforced-concrete axially loaded columns at sustained service stresses. The analysis involves a straightforward computation based on a closed form procedure and the assumption of linear elastic materials for both concrete and steel. The analysis may be easily extended to cover the case of reinforcement at yield. The results of the proposed approach may be superposed with those from a shrinkage model presented by Park and Paulay in 1975, and the overall behavior of column axial shortening and stress transfer from concrete to steel may be described using the combined approach. The process involved is very convenient to use from an engineering view point since it requires few input parameters, which are easy to estimate or measure experimentally, such as the modulus of elasticity of concrete and the creep coefficient. The results of the theoretical approach correlate well with experimental tests conducted on specimens in the laboratory and with deformations of columns measured in the Water Tower Place and Lake Point Tower in Chicago. As I already informed Dr Taranath discusses these effects in his book- These effects should be considered when the no. of stories exceed 30. Taranath discusses about steel columns, the same can be applied to concrete also, which creeps more than steel. Prof. Samra's method is more refined for RCC. CALCULATION OF ULTIMATE SHRINKAGE BY ACI 209-R-92(REAPPROVED IN 1997) METHOD

According to AC I 209.R – 92, Shrinkage Sh (t,τo) at time t(days), measured from the start of drying at τo (days) is expressed as follows: For moist curing Sh (t, τo ) =(( t- τo)/(35 + (t – τo))) Sh∞ For steam curing Sh (t, τo) = ((t- τo)/(55 + (t- τo))) Sh∞ Where Sh∞= Ultimate shrinkage and Sh∞ = 780 x 10-6 K1 K2 K3 K4 K5 K6 K7 For curing times different from seven days for moist cured concrete, the age coefficient K1 is given below: Period of moist curing shrinkage coefficient (K’) 1 1.2 3 1.1 7 1.0 14 0.93 28 0.86 90 0.75 and for steam curing with a period of 1 to 3 days K1 = 1 The humidity coefficient K2 ’ is K2 = 1.4 – (0.01)h, where 40 ≤ h ≤ 80 K2 = 3.0 – 0.3h, where 80 ≤ h ≤ 100 Where h = Relative humidity (Percent) Coefficient K3 allows for the size of the member in terms of the Volume / surface ratio V/S.

For values of the V/S <37> 50) Where Af/A = Fine aggregate / total time aggregate ratio by mass K6 = 0.75 + 0.00061τ, Where τ = Cement content (Kg/m3 ) K7 = 0.95 + 0.008 A, Where A = Air content (Percent) CALCULATION OF ULTIMATE SHRINKAGE BY ACI 209-R-92(REAPPROVED IN 1997) METHOD According to AC I 209.R – 92, Shrinkage Sh (t,τo) at time t(days), measured from the start of drying at τo (days) is expressed as follows: For moist curing Sh (t, τo ) =(( t- τo)/(35 + (t – τo))) Sh∞ For steam curing Sh (t, τo) = ((t- τo)/(55 + (t- τo))) Sh∞ Where Sh∞= Ultimate shrinkage and Sh∞ = 780 x 10-6 K1 K2 K3 K4 K5 K6 K7 For curing times different from seven days for moist cured concrete, the age coefficient K1 is given below: Period of moist curing shrinkage coefficient (K’) 1 1.2 3 1.1 7 1.0 14 0.93 28 0.86 90 0.75 and for steam curing with a period of 1 to 3 days K1 = 1

The humidity coefficient K2 ’ is K2 = 1.4 – (0.01)h, where 40 ≤ h ≤ 80 K2 = 3.0 – 0.3h, where 80 ≤ h ≤ 100 Where h = Relative humidity (Percent) Coefficient K3 allows for the size of the member in terms of the Volume / surface ratio V/S. For values of the (V/S)<37> 50) Where Af/A = Fine aggregate / total time aggregate ratio by mass K6 = 0.75 + 0.00061τ, Where τ = Cement content (Kg/m3 ) K7 = 0.95 + 0.008 A, Where A = Air content (Percent)

2-11-12 NUMERICAL EXAMPLE BASED ON METHOD DEVELOVED BY PROF.SAMRA.

Assumed an inside column 50 stories below the roof. Floor to floor height is 3.5mm. The size of column is 750*1500mm. The column is reinforced with 4% of reinforcement. n=Es/Ec is taken as 8. Es=2*105Mpa. The column is subjected to load of 16, 8000 Newton per floor. Every fifth floor the reinforcement and size of column changes. The details of column sizes and reinforcement are as follows- SIZE REINFORCEMENT Ground to 5th floor 750x1500 4% 5TH to 10th floor 650x1400 3.5% 10th to 15th floor 550x1300 3% 15th to 20th floor 450x1200 3% 20th to 25th floor 450x1100 3% 25th to 30th floor 450x1000 3% 30th to 35th floor 450x9000 3% 35th to 40th floor 450x800 3% 40th t o 45th floor 450x700 2.5% 45th to 50th floor 450x600 2%

a) Planned construction two floors per week b) Concrete is moist cured. c) Relative humidity 40%. d) Slump of concrete 34%. e) Fine contents 34%. f) Air contents 5% g) Cement content 356kg/m3 h) Age at loading 28 days. SOLUTION STEP-1: Calculation of creep Coefficient. Since planned construction is quite fast i.e., two floors per week, the effect of incremental loading can be neglected. Ø∞ (t,to) = 2.35 K1 K2 K3 K4 K5 K6 K1 = 1.25 t0

-0.118 = 1.25(28-0.118 )= 0.843 K2 = 1.27 – 0.006(40) = 1.03 v/s = (750x1500)/(2(750+1500)) = 250 k3 =2/3 [1+1/13 e-0.0212(v/s) ] = 2/3 [1+1.13 e -0212(250) ] = 0.67 K4 = 0.82+0.00264 (100)] = 1.084 K5 = 0.88+0.0024(34) = 0.9616 K6 = 0.46+0.09(5) = 0.91<1 Hence k6 = 1 Ø∞(t,to) = 2.35 (0.843) (1.03) (0.67) (1.084) (0.9616) (1) = 1.425= Øt Step-2: Calculation of Ultimate Shrinkage Ultimate shrinkage, Sh∞ = 780X10 –6 K1 K2 K3 K4 K5 K6 K7 K1 = 1 (assumed 7 days curing) K2= 1.4-(0.01)(40)= 1 K3 = 1.2 e -0.00743(v/s) = 1.2 e-0.00743(v/s) =0.3678 K4 = 0.89+0.00264(100) =1.154 K5 = 0.30+0.014(34)=0.776 K6 = 0.75+0.00061 (356) =0.967 K7 = 0.95+0.008(5) = 0.99 Sh∞ = 780x10 -6(1) (1) (0.3678) (1.154) (0.776) (0.967) (0.99) = 2.459x10 -4 = єsh APPLICATION OF PROF SAMRA’S EQUATION.

As = 4/100X750X1500 = 45000mm2 Steel ratio with respect to gross section=0.04= Sr n=8 fci = P/ (Ac+nAs) = P/ (Ag + (n-1)As ) = 50X16, 8000/1440000 = 5.8333N/mm2 fct = fci(((1-Sr)+n(1-0.2 Øt)Sr)/((1-Sr)+n(1+0.8 Øt)Sr)) By substituting fci =5.8333N/mm2 n=8 Øt=1.425 Sr=0.4 We get fct =4.216 N/mm2 fst=nfci((0.2)Øt+ ((fct/ fci)(1+0.8 Øt))) Substituting all the values ,we get fst=85.4778 N/mm2 Hence deformation due to creep= fst/ES =85.4778/(2*105)=4.27 *(1/ 10000) mm Stress in steel due to shrinkage = єsh/ ( (1+ Øt)/Ec + (Ac/AsEs)) =27.69 N/mm2 Strain due to shrinkage=27.69/(2*105)=1.3845 *(1/10000) mm. Hence deformation for 3.5m high column= (1.3845+4.27)(1/10000) (3500) =1.979 mm Hence height deformation of column 50 storied below the roof = 1.979 mm. Going by same procedure axial deformation of each storied column i.e. column 50 storied below the roof to 1st storied below the roof can be calculated. Summation of deformation of each segments of column is equal to total axial deformation of the

column.

COMPUTER PROGRAM

From the numerical example based on Prof.Samra's method,it is evident that the procedure involves lot of arithmetic computation and highly repetitive as for each column segment the procedure has to be repeated. Back in 1997, I had developed computer program to calculate inelastic axial shortening based on Prof.Samra's method. The computer program was developed using Q basic language. The basic outlines of the program are as follows:

a) The program calculates the deformation of column due to creep and shrinkage for any number of storied buildings. b)The program is in interactive mode and asks information to user, one by one such as

no of stories, story height, relative humidity etc. Hence it can be operated by any person who understands basic engineering terminology. c) The program calculates creep coefficient and ultimate shrinkage by methods outlined in ACI 209R.Then the program uses the calculated creep coefficient and ultimate shrinkage in Prof.Samra's equations. The deformation for each segment of column is calculated and then cumulative inelastic axial deformation is calculated. d)The program assumes that floor to floor height of column and load coming per floor remain constant and the size and reinforcement of column change every fifth floor.This has been done to keep the inputs to a minimum but the program can be easily modified to suit a particular problem. e) The validity of the program has been checked by solving various problems and comparing computer output with manual calculation. References:

a) Dissertation for award of M.Tech submitted at School of Continuing and Distance Education Jawaharlal Nehru Technological University ,Hyderabad by P.K.Mallick under the guidance of (Late) Prof I.M.Reddy. b) Ghali.A and Favre.R "Concrete Structures: Stresses and Deformation" Chapman and Hall,London. c) Neville.A.M and Brooks.J.J (1944) "Concrete Technology" Longman Singapore Publishers. d) Smolira.M "Analysis of tall buildings by Force-Displacement Method" McGraw Hill,London. e) Park.R and Paulay.T(1975) "Reinforced Concrete Structures" John Wiley and Sons,New York. f) ACI 209R-92 (Reapproved in 1997) Prediction of Creep ,Shrinkage and Temperature effects in Concrete Structures. g) Fintel.M and Khan.R (1969)-Effect of column creep and shrinkage in tall structures-Prediction of inelastic column shortening.ACI Journal Proceedings,V.66,no-12,Dec 1969. h) Samra.R.M.(1995) "New Analysis for Creep behaviour in Concrete Columns" Journal of Structural Engineering.March 1995.

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Comment by Dr.N.S : Over a period of time I have lost the original article of Prof Samra and my memory does not help me too. I need a confirmation from you. In the worked out example, has Prof Samra assumed the creep coefficient and ultimate shrinkage or he derived those by method outlined in ACI 209 R ? I had solved the same problem of Prof Samra's article by using method outlined in ACI 209 R for creep coefficient and ultimate shrinkage. If Prof Samra has used ACI 209 R for creep coefficient and ultimate shrinkage and has not assumed the data, then I will not discuss the problem as anybody can see the worked out example in Prof Samra's article . In case he has assumed the data, then I will discuss the problem. Please confirm.

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