# Essay: Generation Pu-Mu Interaction Diagram for Column

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ABSTRACT

In any structure column is a vertical member carries compressive loading. There is very few column structure which can carry load on its axis. So that due to loading some eccentricity generated in column effects the stress-strain curve of the section which ultimately affect the Interaction curve of the section. With various parameter, the interaction diagram of column section will change. In this paper various parameter are taken for development of the interaction diagram of column. Along with it the shape of column is also varies to obtain the graph. The parameter such as grade of concrete ,percentage of steel , Arrangement of reinforcement , variation of distance of neutral axis from outer most side, grade of steel ,shape of the section and ratio of d’/D(reinforcement cover). By the use of these parameters the interaction diagram for various section can be computed with the help of computer programming.

TABLE OF CONTENT
Title Page””””””””””””””’.. i
Certificate””””””””””””””’. ii
Compliance Certificate”””””””””””’… iii
Acknowledge””””””””””””’………… iV
Abstract”””””””””””””’………… V
Table of Content””..””””””””””’… Vii
List of Table ””..”””””””””””’.. Ix
List of Figure ..””””””””””””………… X
List of Abbreviation ””””””””””””. Xi
1 INTRODUCTION 1
1.1 General 1
1.2 Classification of Column according to Load 1
1.3 Pu – Mu Interaction curve 3
1.4 Need of Interaction curve 4
1.5 Methodology for development of Interaction curve 4
1.6 Objective of the Study 5
1.7 Scope of work 6
2 LITERATURE REVIEW 7
2.1 General 7
2.2 Literature Survey 7
3 FUNDAMENTAL OF INTERACTION CURVE 10
3.1 Assumption taken in design of column 10
3.2 Type of Column Failure 13
3.2.1 Balance Failure 13
3.2.2 Compression Failure: 14
3.2.3 Tension Failure 14
3.3 Introduction of Stress Block Parameter 16
3.3.1 NA INSIDE THE SECTION ( k < 1 , xu < D ) 16
3.3.2 NA OUTSIDE THE SECTION ( k > 1 , xu > D ) 17
3.4 Stress – Strain relationship for Steel reinforcement 19
4 DEVELOPMENT OF Pu -Mu INTERACTION CURVE 20
4.1 Interaction curve from SP 16 20
4.2 Method of generation of curve for Rectangular section 20
4.3 Interaction diagram for T section column 27
4.4 Working Example: 32
5 REFERENCES 60

LIST OF FIGURE

Figure 1 Dia. of Axial load , axial moment, Biaxial moment 2
Figure 2 Interaction diagram for general section 3
Figure 3 Stress Strain curve for concrete 11
Figure 4 Stress Strain relationship for steel 12
Figure 5 Modes of Failure of column section 15
Figure 6 Stress – Strain diagram for rectangle when NA inside the section 16
Figure 7 Stress – Strain diagram for rectangle when NA outside the section 17
Figure 8 Stress strain curve for steel 19
Figure 9 Stress Strain value for steel 19
Figure 10 T section when NA inside the flange 28
Figure 11 T section when NA inside the web 29
Figure 12 T section when NA outside the section 31
Figure 13 Diagram of T section with reinforcement details 33
Figure 14 Interaction diagram for T section 33

LIST OF ABBREVIATIONS

Xu ””””””””’. Depth of neutral axis
D ””””””””’. Depth of column section
e ””””””””… Eccentricity
Fck ””””””””.. Characteristic strength of concrete
Fy ””””””””. Characteristic strength of steel
Pu ””””””””.. Axial force ( load )
Mu ”””””””……… Uni axial Moment
IS ”””””””………
Pc ”””””””………
Ps ”””””””………
Muc ”””””””………
Mus ””””””””..
Leff ””””””””..
Indian Standard
Moment carrying capacity of concrete
Moment carrying capacity of steel
Effective length of column

INTRODUCTION

General

A Column is most important member in building/structure. So that designing of column is very important with respect to accuracy. Mostly columns are design as a compression member but some percentage of tension also develop in column section. Due to that design of column member with axial loading and bending is somewhat difficult and complex. So according to loading condition various eccentricity generated which leads to generate the Pu v/s Mu Interaction curve.
As compression members are subjected to axial compression force which causes buckling of member. Mostly compression member carry load along its length ( l ) in vertical direction. such members are column, pedestal , strut, etc. Strut is a member which can carry load along its any direction.
Classification of Column :

Classification according to Length
Column is classified according to its length ( based on slenderness ratio ) is given as…
Short column
Long column
According to IS:456-2000 the column section is classified as Leff/D ratio is less than 12 then it is known as short column and if the ratio is greater than 12 then column is known as long column or slender column.
Where, Leff = effective length of column
D = total depth of column
column are also classified corresponding to loading condition which is given as follows..
Concentric axial load ( fig 1)
Axial load and Uni – axial bending ( fig 2)
Axial Load and Biaxial bending (fig 3)

Figure 1 Dia. of Axial load , axial moment, Biaxial moment

Classifiction according to Shape :
Column is also divide in to different shape and section like square section , rectangular section column , circular column. These type of column is very common in practical use of design. It is easy to design as simple and symmetrical section. Other than these Ell( L) shaped column , T shaped column and Cruciform shaped column is also use for structure. But these type of column have two component as flange and web. So design for Axial and Biaxial loaded column is difficult in actual practice. The fig shows some section of the column below.

Figure 2 T , Cruciform , L shaped column

Pu – Mu Interaction curve
The different combination of Pu and Mu for each failure mode of given column section are determined and plotted , the resulting curve is known as Interaction Curve. The interaction diagram is very useful for designing and analysis of the section. A column interaction diagram is visual representation of combine load (axial and bending) that will cause the column to fail. where Pu is known as Ultimate Axial Load in kN and Mu stands as Ultimate Moment in kN.m in the section. The fig. shows general column typical interaction curve.

Figure 3 Interaction diagram for general section
The interaction diagram is useful to know the percentage of reinforcement in the section for given grade of concrete and steel. From the value of Pu and value of Mu eccentricity at each point can be calculated.
From the Interaction diagram failure zone of section can also find out according to three modes of failure and type of section can be found out as under reinforced , over reinforced and balance section. The above fig also shows Pu at zero eccentricity and moment at pure bending.

Need of Interaction curve

A reinforced concrete column is design with longitudinal reinforcement and lateral ties. The ultimate load carrying capacity and bending moment carrying capacity ( Pu and Mu) is dependent upon eccentricity of load before the column collapse.
Pu and Mu are to be determined for a column with specific percentage of longitudinal steel bars assuming different positions of the neutral axis. Finding out such pair of Pu & Mu is very time consuming process. Thus Interaction is very helpful to designers as it makes the design process very easy.

“””Methodology for development of Interaction curve

Generally Interaction curve is represent in the form of Pu/(Fck*Ag) v/s Mu/(Fck*Ag*D) . SP 16 has given different curve for rectangular section. This curve is mainly depended on P/Fck value where p is the percentage of reinforcement and fck is grade of concrete. One should have the give initial data like depth , width ect.. and designer can get percentage of r/f according to depth and other specification mention by designer. Along with it load and moment can also calculate from the curve as P/Fck is known to us.
Application of Interaction Curve

As mention above, Interaction diagram is the source to find out each value in column if one parameter is know. So it is applicable in every type of column whether it is short column or long column ( slender column ). This Interaction diagram is useful to check Biaxial moment for Biaxial loaded column and from that safe design can be applied.

Advantages And Limitation of Interaction Curve
Net search

Objective of the Study

The Interaction chart is help designer to design column with any given specification easily .

To generate graph for different shape of column like flanged section ( T section ).

It also gives the stress ‘ strain relationship for various value of depth of N.A. along with placement of reinforcement and curvature of axial moment.

To understand the failure pattern of column for various section of column from depth of neutral axis.

To understand the compression and tension failure pattern according to position of eccentricity.

Scope of work

To fulfill above objective, the scope of project is as follows..
The first step is to understand the basic fundamentals of Interaction curves. It includes fundamental study of strain-stress block of concrete, understanding the basic formula for axial force and moment produced in the column, considering the effect of location of neutral axis in the section , effect of location of the bars etc.

The computer programming of generation of Interaction diagram can be used for actual practice work without any complex calculation.

Pu-Mu Interaction diagram used to shows for various ratio of d’/D ( thickness of effective cover )

LITERATURE REVIEW

General

Literature review gives general concept and idea about specific work process. So that based on the literature review, research work can be done easily and particular field work can be done with different parameters and methods. So the problem of work can be understand easily.
Literature Survey

For above research work , most of the papers are studied and applied for the research work where it needed. Review of papers are given below.

Title – 1 : ” An Analytical Investigation on Behavior of Cross ( + ) Shaped RC Columns Using Pu-Mu Interaction Diagrams”

Shanmukh Shetty , R.M Subrahmanya, B.G. Naresh Kumar and H.S Narahimhan
In this paper the section of column is taken as flanged shaped cross (+) and by varying some parameters the interaction cure is developed. The reacherch work is mainly based on IS 456-2000 and SP 16 codes. In the research work different grade of concrete is use ( up to M40) and compare result with each other. They also used Etabs for further comparison of result analysis. With the same grade of steel and same diameter of steel , the interaction curve is generated manually and with the help of Etabs result analysis is compared. This can help designer to design column as it is very long and time consuming process. Again accuracy is very important so that in this paper Etabs result also checked.

Title – 2 : ” Study on the Behavior of Rectangular column with Unequally Spaced Longitudinal Reinforcement.”
Preekanth Lloyd Dsouza, Subrahmanya R.M , B.G Naresh Kumar

The present code of analysis and design is not allow for designing a column with unequal spacing of reinforcement. So in this paper authors have done formation of unequal spacing of reinforcement for same type of reinforcement and grade of steel and concrete. Only position is changed in the section and Interaction curve is created.

With the use of rectangular column Ultimate load and Ultimate moment is determine for two different case. The stress – strain behavior of column is different for equal and unequally spaced reinforcement. These leads the column to resist more moment .

Title – 3 :”COMPUTER AIDED ANALYSIS OF REINFORCED CONCRETE COLUMNS SUBJECTED TO AXIAL COMPRESSION AND BENDING-I L-SHAPED SECTIONS.”

As L section column is very familiar in column construction the authors develop some basic fundamental equations for calculation of Pu and Mu which is used in this paper. With the use of basic stress and strain relationship, axial load and Bending moment is generated for L section column . parameters like B’/B , B1/B , fy etc. is changing for development of interaction diagram. They have taken four different cases with respect to eccentricity. Number of curve is created for both axis but only for one pattern of reinforcement.

Title – 4 : ” BIAXIAL INTERACTION DIAGRAM FOR SHORT COLUMN OF ANY CROSS SECTION ”
J.A. Rodriguez & J. Dario Aristizabal-Ochoa

This paper mainly deals with Interaction diagram for biaxial bending moment. The section taken for work is hollow rectangular , Rectangular , L section etc. This research work follows mathematical formula for developing curve. The graph is develop for Pu-Mux and Pu-Muy for rectangular section. With the use of basic integration and Quasi – Newton formula non linear equation is generated and got the value of load and moment.

Title – 5 : ” PRACTICAL METHOD FOR ANALYSIS AND DESIGN OF SLENDER REINFORCED CONCRETE COLUMNS SUBJECTED TO BIAXIAL BENDING AND AXIAL LOADS .”
T . Bouzid and K. Demagh

The main aim of this research paper is to establish relationship between axial load and biaxial bending moment for slender reinforced concrete column. The experimental approach is performed in this research area. Comparison between short and long column with respect to axial loading is shown and experimental result is also mention. With the use of ACI 318-99 design provision for slender column is taken and moment magnification method is used for calculating additional moment in slender column.
Both square and L section specimens are tested for practic purpose and also for analytical purpose.

Title – 6 : ” COMPUTER-AIDED ANALYSIS OF REINFORCED CONCRETE COLUMNS SUBJECTED TO AXIAL COMPRESSION AND BENDING. PART II: T-SHAPED SECTIONS”

This paper presents the writer’s experience in investigating numerically the strength behaviour of short T-shaped reinforced concrete (RC) columns in order to provide design aids for structural engineers. This paper deals with the definitions and materials that are used in the limit state design of reinforced concrete structures alone with it paper also represent the stress and strain relationship of T section column and a computer program is generated from it. The parameters are as usual for any design of column ( rectangular ) and different interaction diagram is generated for both major axis and minor axis of column. Number of pattern is used for section design in this paper.

Title – 7 : ‘Possible use of T section columns in RC frame’
Saubhagya k Panigrahi & Ashok k Sahoo

The paper present the comparative study of rectangle section and T section with same reinforcement and same cross section area of section. As per Indian Standard code, both theoretical and experimental result is calculated and then compared with each other. with 1.5 percentage of steel and 300mm*100mm flange and 100mm*100mm web section is calculated for T column for the experimental work. Whereas 200 * 200 mm section of square is taken for research work process. The aim of experiment was to check strength , deflection pattern , failure pattern cracks formation of particular section.

Title – 8 : ‘L-SHAPED COLUMN DESIGN FOR BIAXIAL ECCENTRICITY ‘
L.N. Ramamurthy and T. A. Hafeez Khan

This paper present the load contours in L section for equivalent column and number of example is given for different section type. for different inclination angle biaxial moment is calculated with simple formula. Analytical and design both type of problem is solved in this paper.
Summary :

The summarized explanation is given below for the above research work of development of interaction curve.
From the study of research we can say to development of Interaction diagram basic fundamental stress and strain of steel and column is to be analyzed first.

With the change of section , the value of loading and moment is changing. so if we want to provide L section then we have to take capacity of that particular section. we cannot convert L section to square or rectangular.

The difference in axial load and bending moment is very large for regular shape and arbitrary shapes as flanged and web have different dimension.

Ratio of depth and cover is also very important as the interaction curve is derived from this.

As the grade of steel and concrete is changes, the values of Pu – Mu changes

Position of reinforcement is very important in calculation purpose as stress is calculated with respect to NA.
FUNDAMENTAL OF INTERACTION CURVE

Assumption Regarding Design of Column

The assumption for limit state of collapse due to compression is given in IS:456-2000 , cl. 39.1
The plane section normal to the axis of column before deformation remain plane after deformation that means the strain at any point is proportional to its distance from the Neutral Axis.

Figure 4 Stress Strain curve for concrete

The acceptable stress-strain curve of concrete is assumed to be parabolic.

The relationship between compressive stress distribution in concrete and strain in concrete may be assumed to be rectangular, trapezoidal, parabola and any other shape according to N.A. condition. For the design purpose the compressive strength of concrete is assume to be 0.67 time fck. But the design strength of concrete is taken as 0.446 fck.

The maximum compressive strain in concrete for axial compression is taken as 0.002.

Figure 5 Stress Strain relationship for steel

The maximum compressive strain in concrete at the outer most compression fiber is taken as 0.0035 in bending when N.A. lies within the section.

The tensile strength of concrete is ignored.

The maximum strain in the tension reinforcement in the section at failure shall not be less than fy/(1.15 Es) + 0.002, where fy is the characteristic strength of steel and Es = modulus of elasticity of steel.

“The maximum compressive strain at the highly compressed extreme fibre in concrete subjected to axial compression and bending and when there is no tension on the section shall be 0.0035 minus 0.75 times the strain at the least compressed extreme fibre”.

Type of Column Failure

Failure of column can be classified as three type of mode which are as follows…
Balance Failure

with further increment of eccentricity the depth of N.A. further decreases. But the tensile strain and tensile stress also increase. In this zone compressive force and tensile force is equal so that it is called Balance section. At maximum value of eccentricity, tension steel and concrete in extreme compression fiber both attain its limiting value simultaneously.
At the balance failure the value of N.A. is called ‘xu’_bal and corresponding value of eccentricity is called e_b. As per IS 456-2000 the limiting value of strain at tension fiber is given as below.
”=(0.87*fy)/Es+0.002
where fy = grade of steels
Es = modules of elasticity of steel = 2’*10’^5 MPa
for mild steel, ”=(0.87 * fy)/Es
Compression Failure:

If the eccentricity is relatively small , all the column and reinforcement are in compression. The compression behaviors will be predominant at the failure. The failure of column will be known as compression failure.
In this case , the neutral axis lies outside the section and also in some case of xu , NA lies inside the section till the balanced failure occurs .Fig shows the compression failure of column for rectangular section. In terms of eccentricity when e < e_b( normal eccentricity less than balance eccentricity) and k > 1 then compression failure occurs. here k is ratio of depth of NA ( xu ) and total depth of section .
In above interaction curve fig. the compression zone is clearly mention.
Tension Failure

As the eccentricity increase, yielding of steel is increases that leads to failure which is tension failure of the section. As steel is ductile material it will continue to increase the strain with the increase in eccentricity.
In this type of failure, steel undergoes large ductility but concrete cannot take large deformation as concrete is brittle material. so that the failure is occurs due concrete.
Here normal eccentricity is greater than balance eccentricity ( e > e_b )
fingers for all the three mode of failure is shown below in stress block parameters.

Figure 6 Modes of Failure of column section

fig. (a) = when load at center of column
fig. (b) = when NA is outside the section ( compression failure)
fig. (c) = when NA is at end of the section ( k=1 , balance failure )
fig. ( d) = when NA is inside the section ( tension failure)
Introduction of Stress Block Parameter

NA INSIDE THE SECTION ( k < 1 , xu < D )

Figure 7 Stress – Strain diagram for rectangle when NA inside the section

The section stress diagram for concrete and steel are shown in fig. The stress block diagram for reinforced concrete column is same as RCC beam as given in IS 456-2000. When N.A. is inside the section the maximum strain at compression fiber is 0.0035.
Thus Ultimate load is written as,
Pu = 0.36*f_ck*b*x_u+ ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si
Mu = 0.36*f_ck*b*x_u*D*(0.5-0.416*xu/D )+ ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_(si )*xi
NA OUTSIDE THE SECTION ( k > 1 , xu > D )

Figure 8 Stress – Strain diagram for rectangle when NA outside the section

The stress is uniformly 0. 446 fck for a distance of ( 3*D/7) from the highly compressed edge because the strain is more than 0.002 and thereafter the stress diagram is parabolic.
When xu grater than total depth of section at that time section feels compression failure. In that case k > 1 and many stress and strain value have to consider from the given IS and SP 16 codes which is shown here.
k_u Area of stress block(C1) Distance of centroid from highly compression edge(C2)
1.00 0.361*fck*D 0.416*D
1.05 0.374*fck*D 0.432*D
1.1 0.384*fck*D 0.443*D
1.2 0.399*fck*D 0.458*D
1.3 0.409*fck*D 0.468*D
1.4 0.417*fck*D 0.475*D
1.5 0.422*fck*D 0.480*D
2.0 0.435*fck*D 0.491*D
2.5 0.440*fck*D 0.495*D
3.0 0.442*fck*D 0.497*D
4.0 0.444*fck*D 0.499*D

In general Area of stress block ,
= 0.446*fck*D*(1-4/21*((‘4/(7k-3))’^2 ) )

so the ultimate load when N.A. is outside the section is given as..
Pu = C1*f_ck*b*D+ ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si
Mu = C1*f_ck*b*D*(D/2-C2*D)+ ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_(si )*xi
where C1 = coefficient of area of stress block
C2 = distance of centroid from compression fiber

Stress – Strain relationship for Steel reinforcement

Stress can be calculated from the value of strain in steel bars.

Figure 9 Stress strain curve for steel
here , for fe 415 and fe 500 the strain and stress value change which is given in SP 16
This value is dependent on value of strain at particular distance from N.A.
Figure 10 Stress Strain value for steel
DEVELOPMENT OF Pu -Mu INTERACTION CURVE

Interaction curve from SP 16

The interaction diagram of two section , rectangular section and circular section is given by SP 16. In this code many parameters are taken for the design of column. Design charts are prepared by taking grade of steel as Fe 250 , Fe 415 and Fe 500. along with grade of concrete is M20. for ratio of d’/D the values taken is 0.05 , 0.1, 0.15 and 0.2 and the percentage of steel there is one parameter which gives the steel amount is p . With the change in p/Fck value different graph is generated for different value of reinforcement and for the variation of place of reinforcement. For example in SP16 Chart 27 is given for Two side reinforcement and chart 39 is given for four side reinforcement.
Method of generation of curve for Rectangular section

As mention in above section, the N.A. is classified in three zone
For the case of Neutral Axial lies outside the section.

Capacity of concrete to carry axial load is..

Pc = C1*Fck*b*xu

where Fck = Characteristic compressive strength of concrete in N/’mm’^2
C1 = coefficient area of stress block
b = width of the section

Capacity of steel to carry the axial load

Ps = ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si

For calculating fsi and fci we have to calculate the strain in steel for each row for the section. for that calculation basic formula for stress and strain limit of reinforced concrete section is used which is given as,

fsi= stress in i^th row of reinforcement compression being positive and
tension being negative

This value is taken from the above fig. The value of stress in concrete is calculated by following formula,

fci= 446*fck* ”s*(1-(250*”s) ) if ”s<0.002

fci=0.446*fck if ”s>0.002

fci = stress in concrete at the level of i row of reinforcement .

For calculating the value of strain at different distance of the reinforcement can be done by the given method,

”= (0.0035*xi)/((xu-(3*D/7)))

where xi = distance of position of reinforcement from NA
xu = depth of NA
D = total depth of section
fck = characteristic strength of concrete ( N/mm2)

Pu = Pc + Ps

Pu=C1*Fck*b*xu+’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si

where,
Pu = Ultimate load carrying capacity of section
Pc = load carrying capacity of concrete
Ps = load carrying capacity of steel

Capacity of concrete to carry Bending Moment is..

Muc = C1*Fck*b*xu*(CG-(C2*xu))

where
Fck = Characteristic compressive strength of concrete in N/’mm’^2
C1 = coefficient area of stress block
C2 = centroidal distance of stress block from compressed edge
CG = center of gravity from highly compressed edge
b = width of the section

Capacity of steel to carry the Bending Moment

Mus = ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si*(lever arm distance)

Where,

fsi= stress in i^th row of reinforcement compression being positive and
tension being negative

fci = stress in concrete at the level of i row of reinforcement .

For calculating the value of strain at different distance of the reinforcement can be done by the given method,

”= (0.0035*xi)/((xu-(3*D/7)))

xi = distance of position of reinforcement from NA
xu = depth of NA
D = total depth of section
fck = characteristic strength of concrete ( N/mm2)
Lever arm distance = (CG – xi )

Total load carrying capacity of Bending Moment

Mu = Muc + Mus

Mu=C1*Fck*b*xu*(CG-(C2*xu)) +’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si*(lever arm distance)

where,
Mu = Ultimate moment carrying capacity of section
Muc = Bending moment carrying capacity of concrete
Mus = Bending moment carrying capacity of steel

For the case of Neutral Axial lies inside the section.

Capacity of concrete to carry axial load of column section

Pc = 0.361*Fck*b*xu
where,
Fck = Characteristic compressive strength of concrete in N/’mm’^2
xu = depth of neutral axis
b = width of the section

Capacity of steel to carry the axial load of column section

Ps = ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si

fsi= stress in i^th row of reinforcement compression being positive and
tension being negative

This value is taken from the above fig. The value of stress in concrete is calculated by following formula,

fci= 446*fck* ”s*(1-(250*”s) ) if ”s<0.002

fci=0.446*fck if ”s>0.002

fci = stress in concrete at the level of i row of reinforcement .
For calculating the value of strain at different distance of the reinforcement can be done by the given method,

”= (0.0035*xi)/((xu))

where xi = distance of position of reinforcement from NA
xu = depth of NA
fck = characteristic strength of concrete ( N/mm2)

Pu = Pc + Ps

Pu=0.361*Fck*b*xu+’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si

where,
Pu = Ultimate load carrying capacity of section
Pc = load carrying capacity of concrete
Ps = load carrying capacity of steel
Capacity of concrete to carry Bending moment of column section

Muc = 0.361*Fck*b*xu*(CG-(0.416*xu))

where,
Fck = Characteristic compressive strength of concrete in N/’mm’^2
xu = depth of neutral axis
b = width of the section
CG = center of gravity from compressed edge
Capacity of steel to carry the Bending Moment of column section

Mus = ‘_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si*Yi

fsi= stress in i^th row of reinforcement compression being positive and
tension being negative
fci = stress in concrete at the level of i row of reinforcement .

For calculating the value of strain at different distance of the reinforcement can be done by the given method,

”= (0.0035*xi)/((xu))

where,
xi = distance of position of reinforcement from NA
xu = depth of NA
fck = characteristic strength of concrete ( N/mm2)
Yi = lever arm distance
= (CG – xi )

Total Moment carrying capacity of column section,

Mu = Muc + Mus

Mu=0.361*Fck*b*xu*(CG-(0.416*xu))+’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si*Yi
where,
Mu = Ultimate Bending moment carrying capacity of section
Muc = Bending moment carrying capacity of concrete
Mus = Bending moment carrying capacity of steel

For the case of Neutral axis lies at infinite distance ( at zero eccentricity )

Pu = 0.446 * fck * Ac + fs* Asc

where,
Ac = area of concrete
fs = 0.79*fy for Fe 415
fs = 0.75 * fy for Fe 500
Asc = area of compressive steel

Interaction diagram for T section column

T section columns are very useful most in long span structure like auditorium, drawing hall, etc. due to flange section load carrying capacity more than regular shape column. In this work short T shape column is consider for the development of interaction curve.
The calculation of axial load and bending moment is almost same as rectangular shape column. That is why in practice most of the column are design as rectangular column. but analytically the result are different. Here in this chapter basic mathematical equations are develop for the calculation of P – M Interaction curve.

Method for calculation

CASE 1 : NA INSIDE THE FLANGE SECTION

Figure 11 T section when NA inside the flange
xu < Df
Load carrying capacity of concrete of T section column,
Pc=(0.361*Fck*xu-((1/3)*0.446*fck*(4/7)*xu) )*D
Load carrying capacity of steel in T section column
Ps =’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si
Moment carrying capacity of concrete in T section column,
Muc=Pc*(CG-0.416*xu )
Moment carrying capacity of steel in T section,
Mus=’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si*Yi

CASE 2 : NA INSIDE THE WEB SECTION

Figure 12 T section when NA inside the web

Df>(3/7)*xu
Load carrying capacity of concrete in T section column

Pc=(0.361*Fck*bw*xu+(0.446*fck*(bf-bw)*yf) )
Load carrying capacity of steel in T section column
Ps =’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si
Moment carrying capacity of concrete in T section column

Muc=(0.361*Fck*bw*xu*(CG-0.416*xu )+(0.446*fck*(bf-bw)*yf) )*(CG-yf/2) )
Moment carrying capacity of steel in T section column
Mus=’_(i=1)^n”( f_si ‘ – f_(ci ))* A_si*Yi

Df<(3/7)*xu

Load carrying capacity of concrete in T section column

Pc=(0.361*Fck*bw*xu+(0.446*fck*(bf-bw)*Df) )
Load carrying capacity of steel in T section column

Ps =’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si

Moment carrying capacity of concrete in T section column
Muc=(0.361*Fck*bw*xu*(CG-0.416*xu )+(0.446*fck*(bf-bw)*Df) )*(CG-Df/2) )
Moment carrying capacity of steel in T section column
Mus=’_(i=1)^n”( f_si ‘ – f_(ci ))* A_si *Yi

CASE 3 : NA OUTSIDE THE SECTION

Figure 13 T section when NA outside the section

Load carrying capacity of steel in T section column
Pc=(0.446*fck*Df*bf)+(0.446*fck*((3*D/7)-Df)*bw)+(((0.446*fck*(4*D/7) )- ( (g/3)*(4*D/7) ) )*bw)
Load carrying capacity of steel in T section column
Ps =’_(i=1)^n”( f_(si-) ‘ f_(ci ))* A_si
Load carrying capacity of steel in T section column
Muc=Pc*(CG-Y)

Load carrying capacity of steel in T section column
Mus=’_(i=1)^n”( f_si ‘ – f_(ci ))* A_si*Yi
Where,
Pc = Load carrying capacity of concrete in column
Ps = Load carrying capacity of steel in column
Muc = Moment carrying capacity of concrete in column
Mus = Moment carrying capacity of steel in column
fck = Characteristic strength of concrete N/mm2
Xu = Depth of neutral axis
D = Depth of section
Df = Depth of flange
bf = Width of flange
bw = Width of web

Working Example:
A T section column is given in fig is take to generate Interaction diagram with M20 and Fe 415. Take d’/D is 0.05 and with un equal reinforcement in flange section and web section. Take equal diameter of bars.
Depth of Flange = 200 mm width of web = 300 mm
Depth of web = 800 mm Diameter of Bar = 20 mm
Width of Flange = 600 mm

Figure 14 Diagram of T section with reinforcement details

Figure 15 Interaction diagram for T section

Result and Conclusion

Result :

In this present work the section of column is taken as L shaped and T shaped for different cases. The study of this section is explain in above chapter and from that values Interaction curve is developed.
This graph are classified with respect to different parameters like , d’/D ratio , p/ fck ratio , different position of reinforcement , Grade of steel ,Grade of concrete etc. The graph is generated for Axial Load and Uni axial moment.
T Section :

T section with Un – equal number of reinforcement

The diagram shows the T shaped section with arrangement of reinforcement.

For the above section total 12 no. of graph is generated for different criteria.
Graph : 1
d’/D = 0.05 , M20 , Fe 415

Graph : 2
d’/D = 0.05 , M 20 , Fe 500

Graph : 3
d’/D = 0.05 , M 25 , Fe 415

Graph : 4
d’/D = 0.05 , M25 ,Fe 500

Graph : 5
d’/D = 0.05 , M30 ,Fe 415

Graph : 6
d’/D = 0.05 , M30 ,Fe 500

Graph : 7
d’/D = 0.1 , M20 ,Fe 415

Graph : 8
d’/D = 0.1 , M20 ,Fe 500

Graph : 9
d’/D = 0.1 , M25 ,Fe 415

Graph : 10
d’/D = 0.1 , M25 ,Fe 500

Graph : 11
d’/D = 0.1 , M30 ,Fe 415

Graph : 12
d’/D = 0.1 , M30 ,Fe 500

T section with equal number of reinforcement

Graph : 1
d’/D = 0.05 , M20 ,Fe 415

Graph : 2
d’/D = 0.05 , M20 ,Fe 500

Graph : 3
d’/D = 0.05 , M25 ,Fe 415

Graph : 4
d’/D = 0.05 , M25 ,Fe 500

Graph : 5
d’/D = 0.05 , M30 ,Fe 415

Graph : 6
d’/D = 0.05 , M30 ,Fe 500

Graph : 7
d’/D = 0.1 , M20 ,Fe 415

Graph : 8
d’/D = 0.1 , M20 ,Fe 500

Graph : 9
d’/D = 0.1 , M25 ,Fe 415

Graph : 10
d’/D = 0.1 , M25 ,Fe 500

Graph : 11
d’/D = 0.1 , M30 ,Fe 415

Graph : 12
d’/D = 0.1 , M30 ,Fe 500

L Section :
L Section with equal number of reinforcement

Graph : 1
d’/D = 0.05 , M20 ,Fe 415

Graph : 2
d’/D = 0.05 , M20 ,Fe 500

Graph : 3
d’/D = 0.05 , M25 ,Fe 415

Graph : 4
d’/D = 0.05 , M25 ,Fe 500

Graph : 5
d’/D = 0.05 , M30 ,Fe 415

Graph : 6
d’/D = 0.05 , M30 ,Fe 500

L Section with equal number of reinforcement and Un-equal diameter of reinforcement

Graph : 1
d’/D = 0.05 , M20 ,Fe 415

Graph : 2
d’/D = 0.05 , M20 ,Fe 500

Graph : 3
d’/D = 0.05 , M25 ,Fe 415

Graph : 4
d’/D = 0.05 , M25 ,Fe 500

Graph : 5
d’/D = 0.05 , M30 ,Fe 415

Graph : 6
d’/D = 0.05 , M30 ,Fe 500

REFERENCES

RESEARCH PAPER :
1. Resistance of reinforced concrete columns subjected to axial force and bending -BY Marek Lechman
2. Biaxial Intersection Diagram for Short RC columns of any cross section – BY J.A. Rodriguez & J. Dario Aristizabal-Ochoa
3. Study on the Behaviour of Rectangular Column with Unequally Spaced Longitudinal Reinforcement – BY Preekanth Lloyd Dsouza,Subrahmanya R.M & B.G Naresh Kumar
4. M- Pu diagrams for reinforced , Partially and Fully prestress concrete section under Biaxial bending and Axial Load.- BY J.A.Rodriguez-Gutierrez and J.Dario Aristizabal- Ochoa
5. Effective flexural stiffness of slender reinforced concrete columns under axial forces and biaxial bending – BY -J.L. Bonet , M.L. Romero and P.F. Miguel
6.Recangular stress block for high strength concrete – BY Togay Ozbakkaloglu and Murat saatcioglu
7.Experimental study on the seismic response of braced reinforced concrete frame with irregular column -BY ‘ Xiao Jianzhuang , Li Jie and Chen Jun
8.Arbitarily Shaped reinforced concrete member subjected to biaxial bending and axial load BY ‘ C.Dundar and B.Sahin
9. L shaped column design for Biaxial Eccentricity BY ‘L.N. Ramamurthy and T.A.Hafeez Khan

BOOKS REFFERENCE:
Limit state theory & design of reinforced concrete By V. L. Shah &
Dr. S. R. Karve

RCC Designs as per IS 456-2000 By Dr. B. C. PUNMIA , ASKHOK KUMAR JAIN,ARUN KUMAR JAIN

Reinforced concrete Vol 1(Based on IS 456-2000) By Dr. H. J. Shah

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