Example 1 of Flat &Column

8/16/2019 Example 1 of Flat &Column

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[Type the document title]

EXAMPLE 1

Design of punching shear reinforcement

A flat slab is supported by columns spaced on grid of L=6m. the proposed fleural reinforcement

is sho!n in the figure belo!. "ind suitable shear reinforcement.

#he nominal loads are$

2.1 Elastic and yield line analysis theory of slabs

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Under overload condition in a slab failing in flexure, the reinforcement will yield first in a region

of high moment. When that occurs, this portion of the slab act as plastic hinge, only able to resist

its hinging moment. When the load is increased further, the hinging rotates plastically, and the

moments due to additional loads are redistributed adjacent sections, causing them to yield as

shown in figure 1.9. The band in which yielding has occurred are referred to as yield lines anddivide the slab in to series of elastic plates. ventually, enough yield lines exist to form a plastic

mechanism in which the slab deform plastically without an increase in applied load.

!igure 1.9" #ield criteria

$n general design, moments and shear from an elastic analysis are compared to plastic member

strengths, using approximate load factor and strength%reduction factor. $n the yield%line method

for slabs, the loads re&uired to develop a plastic mechanism are compared directly to the plastic

resistance 'nominal strength( of the member. )oad factor and strength reduction factor can be

incorporated in to the procedure.

* yield%line analysis uses rigid plastic theory to compute the failure load corresponding to

given plastic moment resistance in various parts of the slab. $t does not give any information

about deflection or about the loads at which yielding first starts. The yield%line analysis theory is

used widely for slab design in the +candinavian countries. The yield%line concept is presented

here to aid in understanding of slab behavior between service load and failure.

#ield riteria

To limit deflection, floor slabs are generally considerably thic-er than re&uired for flexure, and

as a result, they seldom have reinforcement ration exceeding ./ to .0 times the balanced

reinforcement ratio defined in e&. 0.2. $n this stage of reinforcement ratios, the moment%

curvature response is essentially elastic%plastic with a plastic moment capacity conservatively

assumed to be e&ual to φ3n, the flexural strength of the section.

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$f yielding occurs along a line at an angle % to the reinforcement, as shown in fig. 1.9, the

bending and twisting moments are assumed to be redistributed uniformly along the yield line and

are the maximum values provided by the flexural capacities of reinforcement layers crossed by

yield line. $t is further assumed that there is no -in-ing of reinforcement as it crossed by the yield

line. $n fig.1.9, the reinforcement in and y directions provide moment capacities of mx and my per unit width. The bending moment, mb, and the twisting moment, mt , per unit length of the

yield line can be calculated from the moment e&uilibrium of element. $n this calculation, 4 will

be measured countercloc-wise from the axis5 the bending moments m & m y and mb will be

positive if they cause tension in the bottom of the slab5 and twisting moment, m t, will be positive

if the moment vector points away from the section as shown.

onsider the e&uilibrium of the elemnt in fig. 1.9 b"

mb L=m x ( L sinα )sinα +m y ( Lcosα ) cosα

This e&uation gives the bending moment mb

mb=m xsin2

α +m ycos2α 666666666666 '1.7(

The twisting moment mt is

mt =m x−m y

2sin2α 66666666666..'1.9(

These e&uation apply only for orthogonal reinforcement . $f mx 8 my, these two e&uation reduce

to mb8m 8 m y and mt 8 , regardless of the angle of yield. This is referred to as isotropic

reinforcement.

Location of axes and yield lines

When yield lines are formed, all further deformations are concentrated at the yield lines, and the

slab deflects as a series of stiff plates joined together by long hinge, as shown in fig. 1.1. The

pattern of deformation is controlled by axes that pass along line supports and over columns, as

shown in fig. 1.11 and by the yield line. ecause the individual plates rotate about the axes and

or yield, these and line must be straight. To satisfy compatibility of deformations at points suchas * and in fig. 1.1, yield line dividing two plates must intersect the intersection of the axes

about which those plates are rotating. !igure 1.11 shows the location of axes and yield lines in a

number of slabs subjected to uniform loads. The yield mechanism in fig. 1.1 and 1.11 are

referred to as 'inematically admissible mechanism because the displacement and rotations of

adjacent plate segments are compatible. $f more than one -inematically admissible mechanism

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;nce the general pattern of yielding and rotation has been established by applying the guidelines,

the specific location and orientation of the axes of rotation and failure load of the slab can be

established.

Basic Assumptions

The structure 'slab( is collapsing because of moment. lastic method of analysis

#) and strip method of analysis for slabs. >lastic hinge theory for framed structures.

ehavior of slab loaded failure in flexure

efore crac-ing, the slab acts as an elastic plate and the deformations, stresses and strains can be predicted from an elastic analysis.

*fter crac-ing elastic solution is still a good approximation provided that the reinforcementshave not yielded.

#ielding of reinforcement eventually starts in one or more regions of high moment andspreads through the slab as moments are redistributed from yielded regions to areas that arestill elastic. With further load, the regions of yielding, -nown as yield lines, divide the slab

into a series of trape?oidal or triangular elastic plates. The load corresponding to this stage of

behavior can be estimated using a #) analysis.

onsider a rectangular panel 'l(b)1( subjected to uniform load, which is increasing up to failure.

Upper and lower ound theorems

The lower bound theorem and the upper bound theorem, when applied to slabs, can be

stated as follows"

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Lower ound theorem! - $f, for a given external load, it is possible to find a distribution of

moments that satisfies e&uilibrium re&uirements, with the moment not exceeding the yield

moment at any location, and if the boundary conditions are satisfied, then the given load is a

lower bound of the true carrying capacity.

Upper ound theorem! - $f, for a small increment of displacement, the internal wor- done by

the slab, assuming that the moment at every plastic hinge is e&ual to the yield moment and that

boundary conditions are satisfied, is e&ual to the external wor- done by the given load for that

same small increment of displacement, then that load is an upper bound of the true carrying

capacity.

The yield line method of analysis for slabs is an upper bound method, and conse&uently the

failure load calculated for a slab with -nown flexural resistance may be higher than the truevalue. $f an incorrect set of yield lines is chosen, the analysis may be on the unsafe side.

"ules for yield lines

@uidelines for establishing axes of rotation and yield lines are summari?ed as follows"

1. #)s are straight lines bAc they represent the intersection of two planes.

. #)s represent axes of rotation.

/. The supported edges of the slabs will also establish axes of rotation. $f the edge is fixed,a Bve #) may form providing constant resistance to rotation. $f the edge is simply

supported the axis of rotation provides ?ero resistance.

0. * #) between two slab segments must pass through the point of intersection of the axesof rotation of the adjacent slab segment in order to satisfy compatibility of deformations.

2. The axis of rotation will pass along line supports and over columns.

Method of Solution

*fter a -inematically admissible yield mechanism has been selected, it is possible to compute the

values of m necessary to support a given set of loads or vice versa. The solution can be carriedout by the e&uilibrium method, in which the e&uilibrium e&uations are written for each plate

segment or by the virtual%wor- method, in which some parts of the slab is given a virtual

displacement and the resulting wor- is considered.

When the e&uilibrum method is used, considerable care must be ta-en to show all of the forces

acting on each element, including the twisting moments, especialy when several yield linesintersect or when yield lines intersect free edges. *t this location, off%setting vert ical nodal

forces wll be given the wrong sign or location, some building codes re&uire that yield line

calculation be done by the virtual wor- method.

Method of virtual work

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;nce the yield lines have been chosen, some point on the slab is given a virtual displacement, C,

as shown in figure 1.1 c below. The external wor- done by the loads when displaced this

amount is

¿

∬q δ d

xd

y

8 ∑ (W ∆c ) 666666666666666..'1.1(

Where" &8uniformly distributed load on the area

C8deflection of that element

W8total load on a plate element

∆c 8deflection of the centroid of that element

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The right hand side of e&uation 1.1 can be expressed as * times the diplaced volume for the slab

yield mechanism. The total external wor- done by rotating the yield lines is"

$nternal wor- 8 ∑ (mblθ) 6666666666666. '1.11(

Where" mb8bending moment per unit of yield line

l 8 length of the yield line

θ

8angle change at yield line corresponding to the virtual displacement, C

The total internal wor- done during the virtual displacement is the sum of the internal wor- done on each

separate yield line because, the #) are assumed to have formed before the virtual displacement.

The principle of virtual wor- states that, for conservation of energy,

external wor- 8 internal wor-

or

∑ (W ∆c )=∑ (mb lθ ) 6666666. '1.1(

The virtual wor- solution is the upper%bound solution5 that is, the load W is e&ual to or higher than the

true failure load. $f an incorrect sets of #)s is chosen, W will be too large for a given m, or conversely,

the value of m for a given W will be too small.

D*3>) 1. 'D%10. pp EF1(

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hapter

Plastic Moment Redistribution

+.1 ,ntroduction

+ome methods of elastic analysis are generally used to calculate the forces in concrete structures,

despite the fact that the structure does not behave elastically near its ultimate load. The

assumption of elastic behaviour is reasonably true for low levels5 but as a section approaches its

ultimate moment of resistance, plastic deformation will occur.


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+tress%strain diagram for


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!or example, for H redistribution

(/2'77.2.1

00.7.7. MPa f for

d

c' ≤=

−=⇒=δ

(/2'19.2.1

2F.7. MPa f for

d

c' >=

−=

$n moment redistribution usually it is the maximum support moments which are 'adjusted(

reduced so that economi?ing in reinforcing steel and also reducing congestion of bars at the

columns.


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% *ssume d8dMM and read

ρ

'correction factor( from table Ko. 1a. using = mA= mJ and

dAd.

%


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Where w is the load to cause the first plastic hinge5 thus the beam may carry a load of 1.//w

with redistribution.

!rom the design point of view, the elastic 3: can be obtained for the re&uired ultimate loading

in the ordinary way. +ome of these moments may then be reduced5 but this will necessitate

increasing others to maintain the static e&uilibrium of the structure. Usually it is the maximum

support moments which are reduced, so economi?ing in reinforcing steel and also congestion at

columns. The re&uirements for applying moment redistribution are"

1. &uilibrium between internal and external forces must be maintained, hence it

is necessary to reduce the span 3 and +! for the load case involved.

. The continuous beam or slabs are predominantly subjected to flexure./. The ratio of adjacent span be in the range of .2 to ..

0. The column design moments must not be reduced.

There are other restrictions on the amount of moment redistribution in order to ensure ductility of

the beams or slabs. This entails limitations on the grades of the reinforcing steel and of the area

of tensile reinforcement and hence the depth of neutral axis.

E*AM+LE 2#2 $ouly reinforced section with moment redistriution

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Assignment No-2

1. The ultimate load to be resisted by a beam is given as 0-KAm '!igure below(.a( ompute elastic moments,

b( :etermine the design moment assuming a H redistribution of bending moments at the

supports and mid%span,

c( >lot all the elastic 3:, 3oment curvature behaviour, free body diagram for

determining redistributed forces, combined plastic and elastic moments, and design

bending moment envelope and extent of reinforcements.

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!igure" eam and ultimate loading

2. onsider the continuous two span beam shown here in !ig. . *pply a 12H redistribution

of bending moments at the internal support and find the design bending moment and

shear force envelopes for the beam. The characterstic loads are self weight @- 82-KAm

and live load O- 81-KAm.

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Pitchspiral tie


Chapter ,

Columns! Comined Axial Load and Bending,#& ntroduction

* column is a vertical structural member subjected to axial compressive force, with or without

bending moment and transmitting the load to the ground through the foundation. The cross%

sectional dimensions of a column are generally considerably less than its height. 'h)0l D(.

/. Tied and +piral olumns 'lassification(

# Tied columns B main longitudinal reinforcement are held in position by separate ties spaced at

e&ual intervals by the ties along its length.

# +piral columns % 3ain bars are wrapped by closely spaced spiral

# omposite columns B consist of steel or cost iron structural member enclosed in concrete. Kominal main reinforcement positioned with ties or spirals are placed around the structural

member.

# $n filled columns % those having steel pipe filled with plain concrete or lightly reinforced

concrete.

a( Tied columns b( +piral columns c( omposite column $nfilled column

!igure /.1 lassification of column cross%sections

Tied and spiral columns are by far the most common than other types of columns. ;n the basis of slenderness ratio, columns may be further classified as short or long columns $f the moments induced by slenderness effects wea-en a column appreciably it is

referred to as a slender 'long( column otherwise it is a short column.

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The slenderness ratio Pλ

M of columns is defined as the ratio of the effective buc-ling length

' Lei( to the radius of gyration 'r i 8

( Ai

,i

, where Pi Prefers to the axes x and y of the crosssection.

λ8

Lei

ri=

Lei

√ I i

A i

6666666666666666666666. '/.1(

The effective length 'l ei( of a column is the distance between two consecutive points of contra

flexure or ?ero bending moments. $n other words it is the height of a theoretical column of

e&uivalent section but pined at both ends. This depends on the degree of fixity of column ends

which in turn depend on the relative stiffnesses of the columns and beam connected to either

ends of the column under consideration. !or this consider the following.

Le = L Le = .2 L Le = .3 L Le = +. L

"igure 0.+$ 4ehaior of columns

*ccording to +%, 1992, the effective buc-ling length, )e, of a column is given by"

a( Kon%sway mode

E.7.

0.≥

++

=m

me

L

L

α

α

666666666666666 '/.1(

b( +way mode

12.12.E

F.1('02.E

1

11 ≥+++++

=α α

α α α α

L

Le

66666666 '/.(

or

12.17.1 ≥+= me

L

Lα

!or theoretical model shown below" 41 and 4 may computed as

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55 1

1

111

1

1

α α α α α

+=

++

=++

= mcc

/ /

/ /

/ /

/ /

/ 1& / + B are column stiffness coefficients at the ends 1 and ' E,(L(.

/ c B is the stiffness coefficient ' E,(L( of the column being designed.

/ i5 B is the effective beam stiffness coefficient ' E,(L(

8 1. when opposite end restrained.

8 .2 Q Q Qfree to rotate

8 for a cantilever beam.

Limiting alues of slenderness ratio

1. The slenderness ratio of concrete column shall not exceed 10.

. second order effects in compression member need not be ta-en in to account in the

following cases"

a( !or sway frames, greater of λ ≤2 or λ ≤15

vd shall be used.

vd= N sd

f cd A c

b( !or non%sway frames, λ R 2 −25 ( M

1

M 2)

shall be used.

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Where" 31 and 3 are first order calculated moments, 3 being always positive and greater than

31 in magnitude and 31 is positive if member is bent in single curvature and negative if bent in

double curvature.

* frame may be classified as non%sway if its response to hori?ontal forces is sufficiently stiff

enough to neglect any additional internal forces or moments arising from hori?ontal

displacements of its nodes. !or a given load case a non%sway frame satisfies the criterion"

!here 6

6

cr

sd ,1.≤

666666666666666. '/./(

sd B total design vertical load

cr B the critical axial load, given by

e

e

cr L

E, 6

Π=

E, e = .+E c , c7E s , s )= .8E c , c '9 : concrete& ;-steel (

D*3>) /.1 +hort or slender column

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,#,"einforcement arrangement and minimum re.uirement

,#,#& Longitudinal reinforcement

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The area of longitudinal reinforcement shall not be less than 0.008 Ac nor more than 0.08 Ac.

The minimum number of longitudinal bars shall be 6 for circular columns and 4 for rectangular columns.

The minimum cross sectional si?e of rectangular column is 12mm and for circular,diameter 8 mm

The diameter of longitudinal bars shall not be less than 1mm.

,#,#2 Lateral reinforcement

The diameter of ties or spirals shall not be less than Fmm or one &uarter of diameter or longitudinal bars.

The center%to%center spacing of lateral reinforcement shall not exceed" %o 1J diameter of the longitudinal bars

o

b 'least dimension(o /mm

The pitch of spirals shall not exceed 1 mm.

,#,#, Axial compression

The ultimate capacity of an axially loaded short column can be computed from5

P du = f cd 1-? g @7 ? g A g f yd

8 A g >f cd


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,#/ nteraction $iagram

*n interaction diagram is a plot of axial load capacity of a column against the bending moment it

sustains. olumns, which are concentrically loaded, are rare and hence they are subjected to

bending moment, in addition, which decreases the axial load capacity. This may be due to

misalignment of load on the column as shown in fig. /./b or may result from the column

resisting a portion of the unbraced moment at the ends of the beam supported by column fig. /./

c. The distance e is referred to as eccentricity of the load. The two cases are the same because the

eccentric load > in fig././ b can be replaced by a load > acting along the centroidal axis, plus a

moment 38>e about the centroid. The load > and the moment 3 are calculated with respect to

geometric cenroidal axis because the moments and the forces obtained from structural analysis

normally are referred to this axis.

To illustrate conceptually the interaction between moments and axial load in a column, an

ideali?ed homogenous and linearly elastic column with a compressive strength, f cu, e&ual to itstensile strength, f tu, will be considered.

+uch a column would fail in compression when the maximum stress is reached, f cu give by"

P

f cu A+

M

f cu I =1,dividingboth sides by f cu

1max =+⇒= , f

My

A f

P f

cucu

cuσ

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '/.2(

ut P ma 8 f cu A and

y

, f M cu=max

!here" #$I % area and moment o& inertia o& the '(section respecti)ely

y % distance &rom the centroidal a'is to the most hi*hly compressedsur&ace +sur&ace #(# in ,* -.- a

P% a'ial load$ positi)e in compression

/%moment$ positi)e as sho!n in ,*. -.- c

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C

0

#

('max

99 M

M ('

max

9 M

M

('

max

9 P

P

('max

# P

P

(1.

(1.

1.

1.

0

C

#

.3

1.

('max

99A M

M

('max9

P

P

('max

# P

P

(.3 .3


!igure /./ )oad and moment on column

+ubstituting in '/.2(

1maxmax

=+ M

M

P

P

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '/.F(

&uation. '/.F( is -nown as an interaction e&uation because it shows the interaction of, or

relationship between P and M at failure.

$nteraction :iagrams

$nteraction diagram for an elastic column, Bf cuB = Bf tuS

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.3

.23

(.43 .43


a 3aterial with f tu = 0

5 3aterial with

cu

tu

f f =

!igure /. 0" $nteraction diagrams

nteraction diagrams for elastic columns 0f cu0≠

0f tu0#


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K . *dh

b

* s

* s 1

f c df s c

f s t

x . 7 x

3 u

> u

, s 8 * s 1 f s c

T s 8 * s f s t

, c 8 . 7 x f c d b

+ e c t i o n + t r a i n s + t r e s s e s $ n t e r n a l f o r c e s < e s u l t a n t f o r c e s

ε s 1

ε s

ε c u 8 . / 2

a( small axial force, so that ( x< 0.8h )

b( large axial force, so that ( x˃ 0.9h )

!igure /.2" alculation of >n and 3n for a given strain distribution

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!i"nificant oints in the interaction dia"ram

"igure 0.6$ ;train distributions corresponding to points on interaction diagram of areinforced concrete column

*ny combination of loading within the curve is a safe loading. *ny combination of loading outside the curve represents a failure combination. *ll combinations of P n and M n between points * will cause the concrete to fail in

compression before the bottom reinforcement, A s yields.

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*ll combinations of P n and M n between points ! will result in tensile yielding of A s before the concrete fails in compression.

When a member is subjected to combined axial compression P d and moment M d , it ismore convenient to replace the axial load and the moment with an e&uivalent P d , applied

at eccentricity ed .

D*3>) /. alculation of interaction diagram

,#1 $esign of columns according to EBCS-2 &1

$esign of solated columns

olumns may be considered as isolated column when they are isolated compression members

'such as individual isolated columns or columns with articulations in a non%sway structures( or

compression members which are integral parts of the of a structure but which are considered to

be isolated for design purpose 'such as slender bracing element considered as isolated columns,and columns with restrained ends in a non%sway structure(

+hort columns are usually designed using charts such as those developed in the previous section.+electing e&ual &uantities of tension and compression reinforcement may not be the most

economic solution but it has an important practical advantage.

1( Total eccentricity.

• The total eccentricity to be used for the design of columns of constant cross

section is given by

etot = ee7ea7e+ ,

where

ee % is e&uivalent first%order eccentricity of the design axial load.

ea % is additional eccentricity to account for geometric imperfection

mm L

e ea /

≥=

' Le % effective length mm(

e+ % is the second order eccentricity.

• !or first order eccentricity e e&ual at both ends of a column,

ee 8 e

• !or first order moments varying linearly along the length, the e&uivalent

eccentricity is the higher of the following two values"

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ee = .6e+ 7 .8e1 or

ee = .8e+

Where e1 and e+ are the first order eccentricities at the ends, e+ being positive

and greater in magnitude than e1.

• !or different eccentricities at the ends 'negative and positive( the critical end

section shall be chec-ed for first order moments.

etot = e+7ea

( +econd B order eccentricity

• !or non%sway frames, the second order eccentricity e+ of an isolated column may

be obtained as

(1

'1

1

r

L /

e e

=

1=

!

20−0.75

where Le is the effective buc-ling length of the column

E2.

1 −= λ

/

for 12 Iλ

I /2

/ 1=1. for L/2

r

1

is the curvature at the critical section.

/

1J(2

'1 −=

d /

r , where

d B is the column dimension in the buc-ling plane less the cover to the center of

the longitudinal reinforcement.

bal

d

M

M / =

M d % is the design moment at the critical section including second order effects.

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M bal B is the balanced moment capacity of the column.

The appropriate value of / + may be found iteratively ta-ing an initial valuecorresponding to 1st order actions.

$n the amplified sway moments method, the sway moments found by a first%order analysis shall be increased by multiplying them by the moment magnification factor"

cr

sd

s

6

6 −

=1

1δ

, where

sd B is the design value of the total vertical load.

cr B is the critical value for failure in a sway mode.

The second order eccentricity can be neglected when5• !or sway frames

=≤

ccd

sd

d

d A f

6 ν

ν

λ 12

2

• !or non%sway frames

1

22 M

M −


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* column is said to be uniaxial if it is loaded with a bending moment about one axis in addition

to axial force. !or the design of such a column interaction charts are prepared using non%

dimensional parameters,

µ υ and

, in which

bh f

6

cd

d =υ

,

bh f

M

cd

h= µ

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '/.F(

$n using these charts for design, the following procedures may be adopted"

@iven P d , M d 8 P d ed 8 P d etot

*ssume a cross section bAh *ssume dM and evaluate dMAh to chose appropriate chart Ko.

alculate ν and µ using 'J(

$f the coordinate 'ν

,

µ

( is within the boundary of the curve, the assumed cross

section is ade&uate5 otherwise, larger section should be tried.

The coordinate 'ν

,

µ

( gives the value of Pω

M

yd

cd st

f

bhf A

ω =

hec- that =

=≤c s

c s

pro. st A A A A A

7.7.

max,

min,

,

D*3>) /./ :esign of olumn using interaction diagram

/.E :esign of column for biaxial bending

There are situations in which axial compression is accompanied by simultaneous bending about both

principal axes of the section. This is the case in corner columns, interior or edge columns with

irregular column layout. For such columns, the determination of failure load is extremely laboriousand making manual computation difficult.

Consider the Rc column section shown under axial force p acting with eccentricities e x and e y, such

that ex = M yp, e y = Mxp from centroidal axes !Fig. ".#c$.

%n Fig. ".# a the section is sub&ected to bending about the y axis only with eccentricity e x. The

corresponding strength interaction cur'e is shown as Case !a$ !see Fig. ".#d$. (uch a cur'e can be

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established by the usual methods for uni)axial bending. (imilarly, in Fig. ".# b the section is

sub&ected to bending about the x axis only with eccentricity e y. The corresponding strength

interaction cur'e is shown as Case !b$ !see Fig. ".# d$. For case !c$, which combines x and y axis

bending, the orientation of the resultant eccentricity is defined by the angle λ*

n

ny

y

M

M

e

earctanarctan ==λ

+ending for this case is about an axis defined by the angle θ with respect to the x)axis. For other

'alues of λ, similar cur'es are obtained to define the failure surface for axial load plus bi)axial

bending.

ny combination of -u, Mux, and Muy falling out side the surface would represent failure. ote that

the failure surface can be described either by a set of cur'es defined by radial planes passing

through the -n axis or by a set of cur'es defined by hori/ontal plane intersections, each for aconstant -n, defining the load contours !see Fig. ".# d$.

Fig. ".# %nteraction diagram for compression plus bi)axial bending

Computation commences with the successi'e choice of neutral axis distance c for each 'alue of 0.

Then using the strain compatibility and stress)strain relationship, bar forces and the concrete

compressi'e resultant can be determined. Then - n, Mnx, and Mny !a point on the interaction surface$

can be determined using the e0uation of e0uilibrium !see below$.

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%'/.7(%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>>> ! stsccnh ++=⇒=∑

where*

concreteinforceresultant, f * > icic ∑=

steelncompressioinforceecompressivresultant, f *>sciscisc ∑=

steel in tensionforcetensileresultant, f *> stistist ∑=

∑ ∑ ∑ ∑++= %'/.9(%%%%%%%%%%%%%%%%%%%%%%%%%yf * yf * yf *3 stististiscisciscicicicinx

∑ ∑ ∑ ∑++= '/J( xf * xf * xf *3 stististisciscisciciciciny

(ince the determination of the neutral axis re0uires se'eral trials, the procedure using the abo'e

expressions is tedious. Thus, the following simple approximate methods are widely used.

a) Load contour method* %t is an approximation on load 'ersus moment interaction surface

!see Fig. ".#$. ccordingly, the general non)dimensional interaction e0uation of family of

load contours is gi'en by*

133

33

dyo

dy

dxo

dx =

+

nn α α

.1.12 and p

p1.FFE .FFE n

do

da ≤≤

+= α α n

where* Mdx = pd e y

Mdy = pd ex

Mdxo = Mdx when Mdy = 1 !design capacity under uni)axial bending about x$

Mdyo = Mdy when Mdx = 1 !design capacity under uni)axial bending about y$

b) Reciprocal method/Bresler’s equation* %t is an approximation of bowl shaped failure

surface by the following reciprocal load interaction e0uation.

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dodyodod p p p p

1111−+=

)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) !".21$

3here* -d = design !ultimate$ load capacity of the section with eccentricities edy and edx

-dxo = ultimate load capacity of the section for uni axial bending with edx only !edy = 1$

-dyo = ultimate laod capacity of the section for uni axial bending with edy only !edx = 1$

-do = concentric axial load capacity !edx = edy = 1$

4owe'er interaction charts prepared for biaxial bending can be used for actual design. The

procedure in'ol'es !considering the cross)section shown below$*

- (elect cross section dimensions h and b and also h5 and b5

- Calculateh

hN

andb

bN

!range of 'alues of 1.16, 1.2, 1.26, 7,

1.86 are a'ailable$

- Compute*

- ormal force

ratio*ccd A f

6 =υ

- Moment ratios*

h A f

M

ccd

h

h = µ

,

b A f

M

ccd

bb= µ

- (elect suitable chart which satisfyh

hN

andb

bN

ratio*

- 9nter the chart to obtain ω

- Compute yd

cd c

f

f Aω =tot*

- Check tot satisfies the maximum and minimum pro'isions

- :etermine the distribution of bars in accordance with the charts re0uirement

Circular columns

3hen load eccentricities are small, spirally reinforced columns show greater ductility !greater

toughness$ than tied columns; howe'er the difference fades out as the eccentricity is increased.

Interaction Diagram for Circular columns

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h

b b N b N

h N

h N

3 b

3 h


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The strain compatibility solution described in the preceding section can also be used to calculate

the points on an interaction diagram for circular columns

Consider the following circular cross section reinforced with < longitudinal bars

K . *

c r o s s % s e c t i o n s t r a i n s s t r e s s e s

ε s 1

ε s /

ε c u

ε s

f c df s c

f s t 1

x . 7 x

f s t

3 u

> u

, o m p r e s s i o n ? o n e < e s u l t a n t ! o r c e s

y

y

xx

Figure ". Calculation of -n and Mn for a gi'en strain distribution

Calculations can be carried out in the same way as in the pre'ious section except that for circular

columns the concrete compression /one sub&ect to the e0ui'alent rectangular stress distribution

has the shape of a segment of a circle !Fig. ".$.

To compute the compressi'e force and its moment about the centroid of the column, it is necessary

to be able to compute the area and centroid of the segment.

, e n t r o i d o f c o m p r e s s i o n ? o n e

y

' . 7 x % h A (

θ

φ

' h A % . 7 x (h

. 7 x

yθ

, e n t r o i d o f c o m p r e s s i o n ? o n e

Case 2 * 1.x ≤ h8, θ > ?1o Case 8 * 1.x @ h8, θ @ ?1o

−= −

A

7.Acos

1

h

hθ

−= −

A

A7.cos

1

h

h φ

$ θ % 16o ( φ

The area of the segment is*

−=

0

cossinh*

θ θ θ

The moment of this area about the centre of the column is*

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=

1

sinhy*

//

V θ

3here θ is expressed in radians.

The shape of interaction diagram of a circular column is affected by the number of bars and their

orientation relati'e to the direction of the neutral axis. Thus the moment capacity about axis x)x

!see abo'e$ is less than that about axis y)y.

(ince the designer has little control o'er the arrangement of bars in a circular column, the

interaction diagram should be computed using the least fa'orable bar orientation. +ut for circular

columns with more than bars, this problem 'anishes as the bar placement approaches a continuous

ring.

:esign or analysis of spirally reinforced columns is usually carried out by means of design aids.

Design of as per EBC !

I" #eneral

The internal forces and moments may generally be determined by elastic global analysis using either

first order theory or second order theory.

b$ First)order theory, using the initial geometry of the structure, may be used

in the following cases

on)sway frames

+raced frames

:esign methods which make indirect allowances for second)order

effects.

c$(econd)order theory, taking into account the influence of the deformation of the structure,

may be used in all cases.

II" Design of $on s%a& 'rames

%ndi'idual non)sway compression members shall be considered to be isolated elements and be

designed accordingly. !:esign re0uirements were listed in section ".6$

III" Design of %a& 'rames"

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The second order effects in the sway mode can be accounted using either of the following two

methods*

a$ (econd)order elastic global analysis* 3hen this analysis is used, the resulting forces and

moment may directly be used for member design.

b$ mplified (way Moments Method* %n this method, the sway moments found by a first)orderanalysis shall be increased by multiplying them by the moment magnification factor*

cr

sd

7

71

1

−=δ

where sd is the design 'alue of the total 'ertical load

cr is its critical 'alue for failure in a sway mode.

The amplified sway moments method shall not be used when the critical load ratio

23.7

7

cr

sd

>

(way moments are those associated with the hori/ontal translation of the top of story relati'e to

the bottom of that story. They arise from hori/ontal loading and may also arise from 'ertical

loading if either the structure or the loading is asymmetrical.

s an alternati'e to determiningcr

sd

7

7

direct, the following approximation may be used in beam and)

column type frames

89

7

7

7

cr

sd δ=

!see section 6.6$

where δ, A, 4 and are as defined before.

%n the presence of torsional eccentricity in any floor of a structure, unless more accurate methods

are used, the sway moments due to torsion should be increased by multiplying them by the larger

moment magnification factor δs, obtained for the two orthogonal directions of the lateral loads

acting on the structure.

Effect of Creep

Creep effects may be ignored if the increase in the first)order bending moments due to creep

deformation and longitudinal force does not exceed 21B.

The effect of creep can be accounted by*

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a$ For isolated columns in non)sway structures, creep may be allowed for by multiplying the

cur'ature for short)term loads! see the expression of cur'ature in second order

eccentricity$ by !2 βd$, where βd, is the ratio of dead load design moment to total design

moment, always taken as positi'e.

b$ For sway frames, the effecti'e column stiffness may be di'ided by !2 βd$, where βd, is as

defined abo'e.

Detailing Requirements

i(e The minimum lateral dimension of a column shall be at least 261 mm.

Longitudinal Reinforcement *see section +"+)

Lateral Reinforcement *see section +", also)

a$ Ties shall be arranged such that e'ery bar or group of bars placed in a corner and alternatelongitudinal bar shall ha'e lateral support pro'ided by the corner of a tie with an included

angle of not more than 2"61 and no bar shall be further than 261 mm clear on each side

along the tie from such a laterally supported bar! see Fig. $

b$ Dp to fi'e longitudinal bars in each corner may be secured against lateral buckling by means

of the main ties. The center)to)center distance between the outermost of these bars and

the corner bar shall not exceed 26 times the diameter of the tie !see Fig.$

smax = "61 mm

c$ (pirals or circular ties may be used for longitudinal bars located around the perimeter of a

circle. The pitch of spirals shall not exceed 211 mm.

!

a$ Measurement between laterally !b$ Re0uirements for main

supported column bars and intermediate ties

Fig. 6.2"

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ample Design charts ta-en from EBC !. part !

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Chapter /

Slender Columns

They are columns with high slenderness ratio and their strength may be significantly reduced

by lateral deflection.

3hen an unbalanced moment or as moment due to eccentric loading is applied to a column, the

member responds by bending as shown in Fig. E.2 below. %f the deflection at the centre of the

member is, d, then at the centre there is a force - and a total moment of M - δ. The second

order bending component, -δ, is due to the extra eccentricity of the axial load which results

from the deflection. %f the column is short δ is small and this second order moment is negligible.

%f on the other hand, the column is long and slender, δ is large and -δ must be calculated and

added to the applied moment M.

Fig. E.2

lenderness Ratio

The significance of -δ !i.e. whether a column is short or slender$ is defined by a slenderness ratio.

%n 9+C( 8, the slenderness ratio is defined as follows*

a$ For isolated columns, the slenderness ratio is defined by*

i

9e=λ

where* Ae is the effecti'e buckling length

i is the minimum radius of gyration. The radius of gyration is e0ual to

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A , i =

where* % is the second moment of area of the section

is cross sectional area

b$ For multistory sway frames comprising rectangular sub frames, the following expression

may be used to calculate the slenderness ratio of the columns in the same story.

L /

A

l

1=λ

where* is the sum of the cross)sectional areas of all the columns of the story

l is the total lateral stiffness of the columns of the story !story rigidity$, with

modulus of elasticity taken as unity

A is the story height

Limits of lenderness

The slenderness ratio of concrete columns shall not exceed 2E1

(econd order moment in a column can be ignored if

a$ For sway frames, the greater of

23≤λ

d υ λ

12≤

whereccd sd d A f 6 =υ

b$ For non)sway frames

2

1

/

/233−≤λ

3here M2 and M8 are the first)order !calculated$ moments at the ends, M 8 being always positi'e

and greater in magnitude than M2, and M2 being positi'e if member is bent in single cur'ature and

negati'e if bent in double cur'ature

Effectie Length of Columns

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9ffecti'e buckling length is the length between points of inflection of columns and it is the length

which is effecti'e against buckling. The greater the effecti'e length, the more likely the column is

to be buckle.

The effecti'e length of the column, Ae, can be determined from Fig. 6., alignment charts !see Fig.

6.?$, or using approximate e0uations.

a$ Figure is used when the support conditions of the column can be closely represented by those

shown in the figure below.

Fig. 6. 9ffecti'e length factors for centrally loaded columns with 'arious ideali/ed conditions

b$ The alignment chart !see Fig. 6.?$ is used for members that are parts of a framework. The

effect of end restrained is 0uantified by the two end restrain factors G2 and G8

bbcm

col col cm

L , E L , E or A

A(' 1∑∑=

β α α

3here 9cm is modulus of elasticity of concrete

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Acol is column height

Ab is span of the beam

%col, %b are moment of inertia of the column and beam respecti'ely

β is factor taking in to account the condition of restraint of the beam at the opposite end

β = 2.1 opposite end elastically or rigidly restrained

β = 1.6 opposite end free to rotate

β = 1 for cantile'er beam

$ote that if the end of the column is fixed, the theoretical 'alue of G is 1, but an G 'alue of 2 is

recommended for use. Hn the other hand, if the end of the member is pinned, the theoretical 'alueof G is infinity, but an G 'alue of 21 is recommended for use. The rational behind the foregoing

recommendations is that no support in reality can be truly fixed or pinned.

Fig. 6.? lignment Chartsomograph for effecti'e length of columns in continuous frames

c$ The following approximate e0uations can be used pro'ided that the 'alues of G 2 and G8 don5t

exceed 21 !see 9+C( 8$.

!a$ on)sway mode

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4.6.

:.

9

9

m

me ≥+α+α

=

!b$ %n (way mode

13.13.4

;.1+:3.499

21

2121e ≥α+α+ αα+α+α+=

Hr Conser'ati'ely,

1.16.19

9m

e ≥α+=

3here G2 and G8 are as defined abo'e and Gm is defined as*

1 α α α

+=m

$ote that for flats slab construction, an e0ui'alent beam shall be taken as ha'ing the width and

thickness of the slab forming the column strip.

lender columns bent about the ma0or a1is

slender column bent about the ma&or axis may be treated as bi)axially loaded with initial

eccentricity ea acting about the minor axis

Bia1ial Bending of Columns

a) mall Ratios of Relatie Eccentricit&

Columns of rectangular cross)section which are sub&ected to biaxial bending may be checked

separately for uni)axial bending in each respecti'e direction pro'ided the relati'e eccentricities

are such that k≤

1.8; where k denotes the ratio of the smaller relati'e eccentricity to the

larger relati'e eccentricity.

The relati'e eccentricity, for a gi'en direction, is defined as the ratio of the total eccentricity,

allowing for initial eccentricity and second)order effects in that direction, to the column width

in the same direction.

b) 2ppro1imate 3ethod

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Columns of rectangular cross)section which are sub&ected to biaxial bending may be checked

separately for uni)axial bending in each respecti'e.

%f the abo'e condition is not satisfied, the following approximate method of calculation can be

used, in the absence of more accurate methods.

For this approximate method, one)fourth of the total reinforcement must either be distributed

along each face of the column or at each corner. The column shall be designed for uni)axial

bending with the following e0ui'alent uni)axial eccentricity of load, e e0 along the axis parallel to

the larger relati'e eccentricity*

( )α+=


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Documents

Example 1 of Flat &Column