507 33 Powerpoint-slides Ch14 DRCS

8/12/2019 507 33 Powerpoint-slides Ch14 DRCS

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Oxford University Press 2013. All rights reserved.

Design of Reinforced

Concrete Structures

N. Subramanian


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Chapter 14

Design of Design ofColumns with Moments


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Introduction

Axially loaded columns are rare in actual practice. Most of the

columns are subjected to bending moments, about one or both the axes

of cross section, in addition to direct compressive loads.

The bending action may produce tensile forces over a part of the cross

section depending on the magnitude of the axial compressive force as

well as the bending moment.

Despite the presence of tensile stresses, columns are generally

referred to as compression members or beam-columns, as the

compressive forces or stresses dominate their behaviour.


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Such compression members include columns rigidly connected to

beams, columns of multi-storeyed buildings, portal frames, columns

supporting crane loads in industrial buildings, and arches.

In multi-storeyed buildings, the edge columns are usually subjected to

uniaxial bending and the corner columns are subjected to biaxial

bending.

Even the internal columns may be subjected to bending if there are

lateral loads or when the adjoining spacing of columns are different on

either side of the column.

Introduction


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Design of Column with Axial Load and

Uniaxial BendingAssumptions Made in Limit States Design for Columns

The failure of concrete is governed by the maximum strain criteria.

For members under concentric load, the ultimate compressive strain

in concrete is taken uniformly as 0.002 across the section.

The ultimate strain in concrete at the outermost compression fibre for

bending is taken as 0.0035.


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Assumptions Made in Limit States Design

for Columns

When the neutral axis lies along one edge of the section (see Fig.

14.1), the strain varies from 0.0035 at the highly stressed compressed

edge to zero at the opposite edge.

As shown in Fig. 14.1, the strain distribution lines for these two cases

intersect each other at a depth of 3D/7 (Point F in Fig. 14.1) from the

highly compressed edge.

This point F is assumed to act as a fulcrum for the strain distribution

line when the neutral axis falls outside the section, as shown in Fig. 14.1.


7/99 Oxford University Press 2013. All rights reserved.

Failure Strain in Concrete under

Compressive Load and Moment

Fig. 14.1 Failure strain in concrete under compressive load and moment



Design Stress-strain Curve for

Concrete The compressive strength of concrete in the

structure is assumed to be 0.67 times the

characteristic strength of concrete.

In addition, a partial factor of safety equal to 1.5 isapplied to the strength of concrete. Thus, the design

strength of concrete is taken as 0.67fck/1.5 = 0.447

fck.

The equation of the parabolic part of the curve is

taken as



Design Stress-strain Curve for Steel

Reinforcements The partial factor of safety for the strength of steel

reinforcement is taken in IS 456 as 1.15, and hence

the design strength is fy/1.15 or 0.87fy.

The non-linear stress-strain curve beyond a stress of0.8 x 0.87fy= 0.696 fyfor HYSD bars may be obtained

using (with taken as strain x 1000)



Derivation of Basic EquationsThe ultimate load carrying capacity of a uniaxially eccentrically loaded

column depends on the following parameters:

1. The size of the column

2. The disposition of reinforcements

3. The stressstrain curves of the materials used

4. The yield limits of the materials

5. The eccentricity of the load



Derivation of Basic Equations

The relation between the axial force, P, and moment, M, in a

symmetrically reinforced rectangular column section is derived by

considering different positions of neutral axis.

Three cases are considered as follows:

Case 1Eccentricity eemin(see Fig. 14.2)

Wherefscis the compressive stress in steel correspondingto a strain of 0.002 (equals 0.79fyfor Fe 415 grade steeland 0.746fyfor Fe 500 steel). The second term within

parenthesis is usually neglected for convenience



Stress and Strain Diagram

Fig. 14.2 Stress and strain diagrams for e emin

(a) Section (b) Strain (c) Stress


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Case 2Neutral axis lies outside section (see Fig. 14.3)





Fig. 14.3 StressStrain diagram when neutral axis is outside the section (a) Column section

(b) Strain diagram (c) Concrete stress diagram (d) Steel forces


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Case 3Neutral axis lies inside section (see Fig. 14.4)



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Fig. 14.4 StressStrain diagram when neutral axis is inside the section (a) Column section

(b) Strain diagram (c) Concrete stress diagram (d) Steel forces

S ti ith A t i R i f t


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Sections with Asymmetric Reinforcement

and Plastic Centroid

Usually, reinforced concrete (RC) columns are symmetrically

reinforced about the axis of bending.

However, in certain situations, such as columns of portal frames or

arches where the eccentricity is large, asymmetric reinforcement is

provided with more rods on the tension side, as shown in Fig. 14.5.

Pl ti C t id f A t i ll


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Plastic Centroid of an Asymmetrically

Reinforced Column

Fig. 14.5 Plastic centroid of an asymmetrically reinforced column

S ti ith A t i R i f t


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Sections with Asymmetric Reinforcement

and Plastic Centroid

For an asymmetrically reinforced column, the resultant load must pass

through the plastic centroid to produce uniform strain at failure. The

plastic centroid represents the location of the resultant force produced

by the steel and concrete (see Eqn. 14.22 of the book)

For symmetrical sections, the plastic centroid coincides with the

centroid of the column cross section. The moments are taken about the

plastic centroid.


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Analysis of Circular Columns

The determination of the ultimate strength of circular columns is

based on the same principles as in the case of rectangular or square

columns.

However, in this case, the geometry of the compression zone and the

circular arrangement of steel bars pose complications.


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The circular column is replaced by an equivalent rectangular column.

The area of the equivalent column is made equal to the area of the

actual circular column, and its depth in the direction of bending is taken

as 0.8 times the outside diameter of the real column (see Fig 14.6).

The values of Pnand Mnare calculated as for rectangular columns.


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Fig. 14.6 Replacing circular column with an equivalent rectangular

column (a) Actual circular column (b) Equivalent rectangular column


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The stress block parameters for rectangular sections are notapplicable to circular sections.

The extreme fibre strain for circular section may be taken as 0.0035,

even though the failure strain in compression for circular sections may

be less than that of rectangular sections.


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While developing the interaction curves of SP 16, the circular sectionwas divided into strips and the forces on each of these strips were

summed up for determining the total forces and moments due to

stresses in concrete.

To compute the compressive force and its moment about the centroid

of the column, we need to compute the area and centroid of the

segment (see Eqn. 14.23 and Eqn. 14.24 of the book).


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Fig. 14.7 Circular column under direct load and moments (a) Section (b) Strains (c) Stresses

(d) Compression zones


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Interaction Curves

Designers often use spreadsheets, computer programs, or computer-generated interaction curves or tables for column design.

Fig. 14.8 shows a curve that is drawn for a column as the load changes

from one of pure axial load through varying combinations of axial loadsand moments to a pure bending case.

Interaction curves are useful for studying the strength of columns with

various proportions of loads and moments.

Any combination of loading that falls inside the curve is generally

satisfactory, whereas any combination falling outside the curve is not

satisfactory and may represent failure.


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Interaction Curves

Fig. 14.8 Column interaction curve


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The points and regions on the interaction curve that are important are

as follows:1. Point Apure axial load: This point corresponds to a strain

distribution that represents uniform axial compression without

moment, sometimes referred to as pure axial load.

2. Point Bzero tension, onset of cracking The strain distribution at

this point corresponds to the axial force and moment on the onset

of the crushing of the concrete, when the strain in the concrete at

the least compressed edge is zero and the concrete begins to crack.Since the tensile strength of concrete is ignored in strength

calculations, failure loads below point B in the interaction curve

represent cases where the section is partially cracked.

Interaction Curves


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3. Region ACcompression-controlled failure: The columns withaxial load capacity Pnand moment capacity Mnthat fall in this region

of interaction curve initially fail due to the crushing of concrete in

the compression face, before the yielding of tensile steel. Hence,

they are called compression-controlled columns.

4. Point Cbalanced failure: This point is called the balanced failure

point and represents the balanced loading case, where theoreticallyboth the crushing of the concrete in the compression face and the

yielding of reinforcement in the tension face develop

simultaneously.

Interaction Curves


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5. Point Dtension-controlled limit: This point denotes the ductilefailure of column, where the tensile strain in the extreme layer of

the tension steel is sufficiently large, that is, equal to or great than

about 2.5 times the yield strain in steel.

6. Region CDtransition region: The columns that fall in the region

CD are termed transition region columns.

7. Point Epure bending: This point represents the bending strengthof the member, that is, when it is subjected to moment alone with

zero axial loads.

Interaction Curves


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Design AidsThe non-dimensional interaction curves given in SP 16:1980 consider

the following three types of symmetrically reinforced columns:

1. Rectangular columns with reinforcement on two sidesCharts 27

to 38: The two sides refer to the sides parallel to the axis of bending.

There are no interior rows of bars, and each outer row has an areaof 0.5As and includes four-bar reinforcement.

2. Rectangular columns with reinforcement on four sidesCharts 39

to 50: These charts have been prepared for a section with 20 barsequally distributed on all four sides, but they can be used for any

number of bars greater than eight.


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Fig. 14.9 Typical interaction diagram for rectangular columns (Chart 32 of SP 16)


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Design Aids-Circular Columns

3. Circular columnsCharts 51 to 62: These charts have beenprepared for a section with eight bars, but can be used for any

section having more than six bars.

These charts have been prepared for three grades of steel (Fe

250, Fe 415, and Fe 500) and for four values of cover ratio d/D (0.05,0.10, 0.15, and 0.20) for each of the three types of columns.

The dotted lines in these charts (see Figs 14.9 and 14.10) indicatethe stress in the bars nearest to the tension face of the column.


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Fig. 14.10 Interaction diagram for circular columns (Chart 56 of SP 16)


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Design AidsThe linefst= 0 indicates that the neutral axis lies along the outermost

row of reinforcement.

For points lying above this line on the chart, all the bars in the section

will be in compression.

The line forfst=fydindicates that the outermost tension reinforcement

will reach the design yield strength.

For points lying below this line on the chart, all outermost tension

reinforcement will undergo inelastic deformation, whereas successive

inner rows may reach the stress offyd.


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Design Procedure

1. Determine the axial load and bending moments acting on thecolumn for different load cases. From this, determine the maximum

axial force and bending moment that has to be supported by the

column. Calculate the factored load and factored bending moment.

2. Select trial cross-sectional dimensions based on experience, grade of

concrete, minimum permissible column size and minimum size and

cover based on fire resistance and environment exposure

requirements.

3. Check for minimum eccentricity


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Design Procedure

4. Choose rebar size and size of cover based on exposure condition.Calculate d/D.

5. Calculatepuand mu.

6. Depending on the values ofpu, mu, d/D, and grade of steel, select

the corresponding chart from SP 16. If there is no exact match for

the calculated d/D value, the values from two charts should be

taken and interpolated.Be sure that the column diagram shown at the upper right side

of the interaction curve matches with the column being considered.


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Design Procedure7. Calculate the steel areaAs. Check whether the calculated area of

reinforcement is within the bounds of the specified code, that is,above 0.8 per cent and below 34 per cent; revise the section if

necessary and repeat the calculations.

8. Design ties as per Clause 26.5.3.2 and detail the reinforcementstaking into consideration Clause 26.5.3.1 of IS 456.

For eccentricity ratios, e/D, less than about 0.1, a spiral circular

column is more efficient in terms of load capacity.

For e/Dratios greater than 0.2, a rectangular column with bars

in the faces farthest from the axis of bending is economical.


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Splicing of Reinforcement

In non-seismic zones, the longitudinal bars of columns are spliced justabove each floor using indirect splices (lap splicing), or by direct splicing

(welded splices or mechanical splices).

In seismic zones, the lap splices should be located only in the mid-height of the column.

The design interaction curves contain dotted lines (see Figs 14.9 and

14.10), which indicate various tensile stresses occurring in thereinforcement closest to the tension face of the column.


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Fig. 14.11 Required lap splice length

f


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Transverse ReinforcementDiamond- and octagonal-shaped ties facilitate

to keep the centre of the column open and freeof cross-ties, resulting in easy placing and

vibration of concrete.

Welded reinforcement grids (WRG) will

improve the constructability and speed ofconstruction but significantly lower ductility in

columns than conventional reinforcement.

Specially fabricated welded wire reinforcement

cages incorporating the longitudinal bars and

ties can also be used.

Bigger column sizes will also result in better

reinforcement detailing.


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Fig. 14.12 Typical arrangements of column ties (a) Square columns (b) Rectangular columns

l f l


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Typical Arrangements of Column Ties

Fig. 14.13 Typical arrangements of column ties (c) Large square column (d) L-shaped columns

(e) I-shaped column

C S d


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Case Study

Design of Columns with Axial Load


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Design of Columns with Axial Loadand Biaxial Bending

Many columns are subjected to biaxial bending, that is, bending aboutboth axes. The most commons ones are the corner columns in buildings,

where beams frame into the corner column in two perpendicular

directions and transfer their end moments into the column.

Similar loading may also occur in the interior columns if the column

layout is irregular, at the columns supporting heavy spandrel beams, and

at the bridge piers.

In addition, beams supporting helical or freestanding stairs, or

oscillating and rotating machinery, have to resist biaxial bending with or

without axial load.



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One of the methods for the design of members subjected tocombined axial load and biaxial bending is based on the conditions of

equilibrium with a suitably chosen inclined neutral axis.

A symmetrically reinforced concrete column section subjected tobiaxial bending is shown in Fig. 14.14.

If the column has more than four bars, the extra steel forces should

also be considered. The use of rectangular stress block will simplify thecalculations.




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Fig. 14.14 Biaxial bending of symmetrically reinforced concrete columns



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The analysis involves a triangular or trapezoidal area of compressedconcrete, as well as a neutral axis that is not usually perpendicular to

the direction of eccentricity; it is inclined with an angle depending on

the moment values as well as the section properties.

For a given cross section and reinforcement, by varying the inclination

of the neutral axis, a series of interaction diagrams can be drawn.

From Fig. 14.15, it is seen that the complete set of diagrams for allangles will result in an interaction surface, which is the failure surface

for the given section.




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Fig. 14.15 Three-dimensional interaction surface for an RC column with biaxial bending

M th d f S iti


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Methods of SuperpositionSome simple methods of superposition have been developed, which

reduce the inclined bending to bending about major axis of the section,thus allowing the use of interaction diagrams developed for uniaxial

bending.

One such method involves the following steps:a) Determine the requiredAsin the x-direction considering Puand

Mux.

b) Determine the requiredAsin the y-direction considering Puand

Muy.c) Determine the total required area of steel by adding the two

areas obtained in steps (a) and (b).

M th d f S iti


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This method has no theoretical basis and may lead to unsafe designs

because the full strength of concrete is considered twice in the design.

However, this method can be conveniently used in the design of long

L-, T-, and +-shaped columns as the overlapping area in thex- and y-

directions will be small.

Methods of Superposition

M th d f E i l t U i i l E t i it


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Methods of Equivalent Uniaxial Eccentricity

In these methods, the biaxial eccentricities are replaced by an

equivalent uniaxial eccentricity and the column is designed for uniaxialbending and axial load.

This procedure is limited in application to columns with doubly

symmetric cross sections having the ratio of longer to shorter dimensionbetween 0.5 and 2 and reinforced with equal reinforcement on all the

four faces.

Clause 3.8.4.5 of the UK code BS 8110-Part 1: 1997 suggests anapproximate method for symmetrically reinforced rectangular sections.

It suggests that the two moments acting on the column can be reduced

to a single moment about a given axis (see Fig. 14.17).

M th d f E i l t U i i l E t i it


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Methods of Equivalent Uniaxial Eccentricity

Fig. 14.16 Definition of terms for biaxially loaded columns

C l d Bi i l B di


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Column under Biaxial Bending

Fig. 14.17 Column under biaxial bending as per BS 8110

BS 8110-Part 1: 1997 suggests that the

two momentsMxandMyacting onthe column can be reduced to asingle moment about a given axis byusing the following:

Methods Based on Approximations for


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Shape of Interaction Surface

1. Breslers Reciprocal Load Method (ACI 318):

This method is more suitable for analysis than for design and,

hence, is often used to check designs.

The capacity predicted by this method is in reasonable

agreement with theoretical as well as experimental results.

The results obtained by Breslers reciprocal formula are not

realistic for columns with high-end restraints or columns that are

very slender.

M th d B d A i ti f


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2. BreslersLoad Contour Method (IS 456):

A plot of interaction curves for different

values of Pu/Pnzare shown in Fig. 14.17.Any combination of biaxial moments falling

inside these curves for the given value of

Pu

/Pnz

is considered safe.


Shape of Interaction Surface

Interaction Curves for Biaxial Moments


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Interaction Curves for Biaxial Moments

Fig. 14.17 Interaction curves for biaxial moments for

different values of Pu/P

nz

Design Procedure for Columns with Biaxial


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es g ocedu e o Co u s t a aMoments

The procedure for the design of columns subjected to factored axial loadPuand biaxial moments Muxand Muyconsists of the following steps:

1. Assume cross-sectional dimensions and the area of steel and its

distribution.

2. Compute concentric load capacity Pnzand Pu/Pnz. Chart 63 of SP

16 can also be used to evaluate the value of Pnz.

3. Determine the uniaxial capacities Mnxand Mnyof the section

combined with the given axial load Puwith the use of interactioncurves for axial load and uniaxial moment.

4. Determine the adequacy of the column section using Fig. 14.17.

Design Aids


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Design Aids

The design of columns subjected to factored axial load Puand biaxialmoments Muxand Muyis iterative.

Sinha and his associates have developed interaction curves for typical

reinforcement distribution in rectangular and square columns for axialload, biaxial moments, effective cover to reinforcement, and area of

steel.

These curves can be used directly to determine the area of steel,without any trial and error process.

Design of L T and + Columns


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Design of L, T, and + ColumnsL-, T-, and cross (+)-shaped columns are often used at outside and re-

entrant building corners for architectural purposes.

Interaction curves of L, T, and + columns subjected to axial load and

biaxial bending are available.

Charts for the design of hollow rectangular sections and of circular

ring-shaped columns are available.

A number of computer programs for biaxial bending, like PCAColumn

(current version 4.10), developed by Portland Cement Association,

Illinois, are also commercially available.

Slender Columns


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Slender Columns

A slender column may be defined as a column that has significantreduction in its axial load capacity due to moments resulting from lateral

deflections of the column (see Fig. 14.19).

Slender concrete columns may fail by buckling in the elastic or

inelastic stress state or they may fail when the compressive strain in the

concrete reaches its limit of 0.0035. The former is classified as instabilityfailure and the latter as material failure.

Slender Columns


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Slender Columns

Fig. 14.19 Examples of slender columns (a) 50 m tall column for runway in Portugal

(b) Slender columns in a building in Chicago

Definition of Slender Columns


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Definition of Slender ColumnsA compression member is considered slender when either of the

slenderness factors Lex/Dand Ley/Bis greater than 12, where Lexand Leyare the effective lengths with respect to the major and minor axis,

respectively, and Band Dare the width and depth of the column.

In addition, the following limits are recommended:1. Columns with both ends restrained: Unsupported length should

not exceed 60 times the least lateral dimension of a column.

2. Columns with one end unrestrained: Unsupported lengthshould not exceed 100 B2D, where Bis the width and Dis the

depth of column measured in the plane under consideration.


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Fig. 14.20 Single and double curvature bending in braced frames (a) Braced (non-

sway) frame (b) Single curvature bending (c) Double curvature bending

Behaviour of Slender Columns


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The behaviour of a slender column shown in Fig. 14.20(a) underincreasing load is illustrated by the PM interaction diagram of Fig.

14.20(c). It also illustrates the different types of failure.

Slender column behaviour for particular loading and end conditioncan be illustrated by the use of slender column interaction diagrams

(see Fig. 14.21).

The three most significant variables affecting the strength andbehaviour of slender columns have been identified as the

slenderness ratio , the end eccentricity ratio , and the ratio of end

eccentricities (see Fig. 14.22).



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Fig. 14.20 Behaviour of slender columns (a) Column with eccentric loads (b) Free body

diagram (c) PMinteraction diagram



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Fig. 14.21 Construction of slender column interaction diagrams (a) Slender

column behaviour (b) Slender column PM interaction diagrams



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Fig. 14.22 Effect of curvature on interaction diagrams for slender hinged columns

Factors Affecting Behaviour of Slender


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gColumns

1. The ratio of unsupported length to section depth, the end

eccentricity ratio, and the ratio and signs of end eccentricitiesthe

effects of these variables are strongly interrelated.

2. The degree of rotational restraintstiffer beams at the ends of

columns provide greater column strength.

Factors Affecting Behaviour of Slender


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gColumns

3. The degree of lateral restrainta braced column is significantlystronger than a column unbraced against end displacements.

4. The amount of steel reinforcement and the strength of concreteanincrease in thep/fck ratio provides increased stability.

5. The duration of loadingcreep of concrete during sustained loading

increases the concrete deflections and decreases the strength of

slender columns.


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Exact Method Based on Non-linear


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The sections may be proportioned to resist these actions without anymodification, as the effect of column slenderness has been considered

in the determination of member forces and moments.

The main factors to be included in the second-order analysis are theP and Pdmoments due to the lateral deflections of the columns in

the structure.

Clause 39.1 of IS 456 and clause 10.10.3 of ACI 318 recommend thistype of second-order analysis. However, these rational methods are not

usually used in design offices as they are time consuming and may be

expensive.

Second-order Analysis

Exact Method Based on Non-linear


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These geometric non-linear effects are typically distinguishedbetween Pdeffects, associated with deformations along the members,

measured relative to the member chord, and P effects, measured

between member ends and commonly associated with storey drifts in

buildings.

In buildings subjected to earthquakes, P effects are much more of a

concern than Pdeffects, and provided that members conform to theslenderness limits for special systems in high seismic regions, Pdeffects

do not generally need to be modelled in non-linear seismic analysis.

Second-order Analysis

Moment Magnifier Method


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Moment Magnifier MethodIn this approximate moment magnifier method (Clause 10.10.5 of

ACI), moments computed from the first-order analysis are multiplied bya moment magnifier to account for the second-order effects.

The moment magnifier is a function of the factored axial load and the

critical buckling load for the column.

Using this method, non-sway and sway frames are treated separately.

Sway and non-sway frames can be identified based on two criteria:increase in column end moments from second-order effects not

exceeding five per cent of the first-order end moments or the stability

index is less than 0.04.

Columns in Non-sway or Braced Frames


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Columns in Non sway or Braced Frames

Braced column is not subjected to side sway, and hence, there is nosignificant relative lateral displacement between the top and bottom

ends of the column.

Normally, the ends of a braced column will be partially restrainedagainst rotation by the connecting beams.

For each load combination, the factored moments at the top and

bottom of the column are calculated using first-order frame analysis.

Columns in Non-sway or Braced


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Columns in Non sway or Braced

Frames The magnified moment, Mc(for each load combination), is

found by multiplying the larger factored end moment, M2 :

with

EIfor cracked section may be determined fromequations given in ACI code(see Eqns. 10.42 of thebook).

Values of Cmfor Different End


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Moment Cases

Fig. 14.23 Values of Cmfor different end moment cases

The correction factor for equivalent uniform momentdiagram is

M1/M2is takenas +ve if the

column is bentin singlecurvature and -

ve if bent in

doublecurvature

Columns in Sway or Unbraced Frames


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Unbraced frame is subjected to side sway, and hence, there will besignificant displacement between the top and bottom ends of the

column.

Such a sway is possible in asymmetric frames or in frames subjected tolateral loads. A simple frame subjected to side sway is shown in Fig.

14.24(a).

The additional moments at the ends of the column caused by theaction of the vertical load acting on the deflected configuration of the

unbraced column is called lateral drift effect (see Fig. 14.24b).


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Fig. 14.24 Unbraced columnlateral drift effect (a) Sway frame (b) Deflected shape of column

(c) Moments



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In unbraced frames, the action of primary moments generally result indouble curvature, as shown in Fig. 14.24(b).

Moreover, the moments at the unbraced column ends will be the

maximum; it is due to the primary moments being enhanced by thelateral drift effect (see Fig. 14.24c).

For each load combination, the factored non-sway moments and the

factored sway moments are calculated at the top and bottom of thecolumn using first-order elastic frame analysis. The magnified sway

moments are added to the unmagnified non-sway moments at each end

of the column.


l i b d


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The magnified moments at each end of the column (M1andM2) are calculated as:


The ACI code gives two alternate methods to calculate s. Inthe first method, it is taken as

If scalculated by the above equation exceeds 1.5, take it as

Additional Moment Method


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Additional Moment MethodThis additional moment method has been adopted in IS 456.

The moment at the failure section of the column can also be taken as

equal to the sum of the applied moment Mand a complementary or

additional moment Maequal to load times the complementary

eccentricity. This complementary moment represents the momentinduced by the column deflections.

The column is designed for the axial load Puand the moment (Me+

Ma).

In this method, the deflectionof the column is computed from the

curvature diagram as shown in Fig. 14.25.



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Fig. 14.25 Deflection of beam-column based on curvature (a) Deflected shape

of slender column at ultimate load (b) Moment diagram (c) Idealized Mrelationship (d) Curvature diagram

Additi l M t M th d


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As per clause 39.7.1 of IS 456, the additionalmoments Maxand Mayshould be calculated as


If the failure is not a balanced one, a moment

reduction factor, kis specified in clause 39.7.1.1

Additi l M t M th d


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The value of Pbmay be evaluated forrectangular and circular sections as below

(Table 60 of SP 16):


For the values of k1and k2see Tables 14.5 and

14.6 of the Book.

Reduction Factor Method


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Reduction Factor MethodThe reduction factor method (Clause B-3.3 of IS 456) implies that the

same eccentricity is maintained in both the slender and analogous shortcolumns.

This is contrary to the actual behaviour of slender columns, where the

reduction in load carrying capacity is caused by the increasedeccentricity due to secondary deflection moments.

This is a severe shortcoming in the case of unbraced fames, since the

magnitude of the secondary moments is extremely important.

Moreover, owing to practical considerations, many important

variables are neglected to keep the formula simple.

Slender Columns Bent about Both Axes


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When slender columns are subjected to significant bending about

both the axes, additional moments have to be calculated for bothdirections of bending.

These additional moments are combined with the initial moments

found from the first-order analysis to obtain the design moments in theprincipal directions.

However, the minimum eccentricity is to be assumed to act only about

one axis at a time.

With these moments, the columns may be designed for biaxial

bending using the design charts given in SP 16.

Design Procedure for Slender


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g

Columns

The various steps involved are:

1. Assume initial sizes based on experience; If

the slenderness factor is greater than 12 or

more, about any of the axes, the column hasto be designed as a slender column about

that axis. If it is slender about both axes, the

additional moments about both the axesshould be considered.



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The following steps are done in XX axis.

1. From first order analysis determine the end

moments Mu1and Mu2.

2. Determine the moments caused by

accidental eccentricity, Mmin

3. Choose Mux1as the larger of Mu1and Mmin

and Mux2as the larger of Mu2and Mmin



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4. Calculate the additional moment, Madd= kxMaxusing Eqs. 14.50 and 14.51. Pnzand Pbcan

be determined using an assumed area of

longitudinal reinforcement of about 2.5% to3%. Chart 63 of SP 16 may be used to find Pnz,

Table 60 of SP 16 may be used to calculate Pbx,

and using the value of Pbx/Pnz, the value of kxcan be determined using chart 65 of SP 16.



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5. Calculate

6. Determine the value of design momentsMdxbyadding the additional momentMaddwith Mux

7.Calculate Pu/fckBD and using appropriateinteraction diagram of SP 16, determine Mnx, for theassumed area of steel



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8. If the column is slender in YY axis, repeatsteps 2 to 8 for YY axis also.

9. Check the following Equation

10. Change reinforcement or size and repeat the abovecalculation, if the left-hand side of equation results invalues higher than 1.0 or much lower than 1.0.

Earthquake Considerations


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For columns situated in earthquake zones, the axial loads and bendingmoment(s) acting on them can be found by a suitable first-order or

second-order analysis; the design is similar to that in non-seismic zones.

For bond limitations of beam bars passing through interior beam-column joints to be satisfied, the depth of the column needs to be up to

30 times the diameter of beam bars.

Smaller column sizes relative to that of beams will result in a strongbeamweak column system, leading to catastrophic storey (or side

sway) collapse mechanisms.



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To meet the strong columnweak beam requirement, the sum of thenominal flexural strengths of the columns framing into a joint must be at

least 1.1 times the sum of the nominal flexural strengths of the beams

framing into the joint (NZS Code suggests 1.4 times).

It is required to include the developed slab reinforcement within the

effective flange width as beam flexural tension reinforcement whencomputing beam strength. This check must be verified independently

for sway in both directions and in each of the two principal framing

directions.



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The minimum number of bars in seismic columns should be eight insquare and rectangular columns and six in the case of circular columns.

Longitudinal bars should not be farther apart than 200 mm centre to

centre or one-third of the cross-sectional dimension in the directionconsidered in the case of rectangular columns, or one-third the

diameter in the case of circular columns.

Bundled bars grouped in the four corners of a column are undesirablein seismic zones.



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Splicing of column bars should be provided only in the middle half of acolumn and not near its top or bottom ends, where plastic hinges are

likely to form.

Moreover, only up to 50 per cent of the vertical bars in the columnare to be lapped at a section in any storey. Furthermore, when laps are

provided, ties must be provided along the length of the lap at a spacing

not more than 150 mm.

Mechanical couplers should be used where the reinforcement ratio is

greater than three per cent.



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Welded splices should never be located in potential plastic hingeregions. It is also necessary to anchor the column rods in the

foundations and provide special confining reinforcements in footings.

L-, T-, or +-shaped columns should not be used in earthquake zones asthey may crack at the re-entrant corners and fail subsequently without

reaching their ultimate capacities.

Similarly, slender columns should not be used in seismic-dominatedductile frames.

Thank You!


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Thank You!

507 33 Powerpoint-slides Ch14 DRCS

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