Column Interaction Curve (1)

7/23/2019 Column Interaction Curve (1)

1/18

Design of COLUMNS with Uniaxial Moment

Analyse the given section whether it can carry given axial load

and moment( one axis at a time)

Design Parameters used as input for design are

i. Dimensions b and D of the rectangular cross-section,

ii. rades of concrete (fc!) and steel (fy)

iii. "ongitudinal steel reinforcing #ars $ %o. of #ars and its

distri#utionalong #& and D&. (%ot pt'fc! as per P *)iv. +over to longitudinal reinforcement d& (%ot d&'D as per P

*)

v. Axial "oad (Pu) for all the design cases

. ased on the given design input parameters, graph of two non-dimensional parameters , Pu'fc!/#/d (for axial load) v's0u'fc!/#/d1 (for moment) are produced for the given columnsection from stress #loc! parameters.

. 2or given axial load Pu and thus Pu'fc!/#/d, corresponding valueof 0u'fc!/#/d1 is found out #y interpolation. 2rom this value


2/18

To nd Stress for Steel(fs) and Concrete(fc)from Strain() al!es

+3%+4565

0aximum allowa#le strain in concrete in axial compression is7.771 while compression plus #ending is 7.7789.

train due to tension in concrete is :ero.(+oncrete neglected intension :one)

;dealised tress-strain curve for concrete shows that up to strainof 7.771, stress varies para#olically with the e ==*/? ( $ 197/?)/ fc! for? @ 7.771 (Sim1lied form!la for c!re)

> 7.==*/fc!for ? 7.771.


3/18


655"

o

2e197 B Stress is linearl. 1ro1ortional to strain !1

to .ield 1oint(i/e/ /$%f. 2 '3%/& m4a)/

Therefore for strain !1 to('3%/&5'x3&)2/3$%&6 stress islinear/

7or strain -e.ond /3$%&6 stress isconstant

i/e/ fs 2 x 8s for 9/3$%&

fs 2 /$%f. 2 '3%/& M4a for :

/3$%&

o CD ars (2e=9, 2e977, 2e*77)B

Stress is linearl. 1ro1ortional to strain!1 to stress al!e of (/$ f.d 2 /$ x/$% f.)

7rom /$f.d to 3/f.d6 stress ariation is

(7e'&)

7e=3&67e&67e>


4/18


e.g. 2or 2e=9, at 7.E9 fydF

#nelastic strain from gra1h 2 /3

8lastic strain 2 /$&x/$%x=3& 5 'x3&2 /3&?

Total strain 2 /3 < /3&? 2 /3>?

Ta-le 1roided in s1readsheet

Ta-le is 1roided in the s1readsheet with strains corres1onding tostresses ar.ing from /$f.d to 3/f.d for reference /

Gith the help of these stress-strain relations for concrete andsteel, for any value of strain (?), corresponding stress in

concrete (fc) and steel(fs) can #e found out.


5/18

Limiting Cases de1ending on Ne!tral @xis

Depth of neutral axis (xu

) is varied with respect to total depth (D)

through !u factor.

o +ase B Axial "oad with :ero eccentricity (no ending moment)

The entire cross;section is !nder direct com1ression with max/Strain of /'

Pu

0

ax strain of 7.771 (in red)

allowed

+orresponding max. stress

> 7.==* fc!

!u 2 ;nHnity (1.9x77

used in

D

Stress AlocB when B!

2

So!rce " *Design of +CCStr!ct!res, -. Dr/ Shah


6/18


o +ase 1B %eutral Axis "ying outside the section ( xu I D F !u I )

The tensile strains from -ending moments are less than the axialcom1ression strains/

The entire cross;section is !nder com1ression with max/ Strain of/?& allowed for concrete

Pu J

0u

0ax strain of 7.7789 (in red)

allowed


> 7.==* fc! train at 8D'K (as shown in

Hg) is always 7.771 when xuI D and therefore used asreference point for

calculations.

D

?min

?max(7.778

9)


7/18


o +ase 8B %eutral Axis "ying along the 5dge (xu > D)

The tensile strains from -ending moments are eE!al to the axialcom1ression strains in magnit!de/

The entire cross;section is !nder com1ression with max/ Strain of/?& and minim!m strain is F8+O

Pu J

0u

0ax strain of 7.7789 (in

red) allowed


> 7.==* fc!

!u 2 3

D

?min >

7

?max

(7.7789)


8/18


o +ase 8B %eutral Axis "ying within the section (xu @ D)

The tensile strains from -ending moments are more than the axialcom1ression strains in magnit!de/

The strain ariation across the section is do!-le triang!lar with max/Com1ressie Strain of /?& and tensile strain(negatie strain) at theother edge

Pu J

0u

0ax strain of 7.7789 (in red)allowed

+orresponding max. stress >

7.==* fc!

6ensile strain of concrete

ignored train and therefore stress in

steel #ars in tension side areta!en negative (as shown in

D

?max

(7.7789)

? > -ve


9/18


o +ase 8B %eutral Axis "ying within the section (xu @ D)

) alanced section point(!u > !u,max) B Aoth tensile andcom1ressie strains reach .ield

1) Pure Lexure point (Pu>7) and

8) Pure tensile axial load point (Mu N 7, Pu @ 7 , 0u > 7)

(f)

4t/ (c) " 4!re com1ression 1oint (B!2

)4t/ (d) " Com1ression < Aending (B!23 )

4t/ (-) " Aalanced Section 1oint4t/ (a) " 4!re 7lex!re 4oint4t(f) " 4!re tension 1ointGence if !

uis aried from

to innit.6 all the 1oints of4!;M! #nteraction C!recan -e 1lotted for giencross section


10/18

4+OC8DU+8 @DO4T8D TO 7#ND MOM8NTC@4@C#TH O7 COLUMN S8CT#ON

3) @ss!me diIerent al!es of ne!tral axis with res1ect to de1th of section/

i/e/ !u is aried from /3 (1!re tension 1oint) to '/&e33 2 innit. (1!recom1ression)

') De1ending if !u J 3 6 or !u 3 6 two form!las to calc!late strain al!e at

diIerent leels of steel -ars are !sed/

?) Kith the relationshi1s form!lated -etween strains and stresses -efore6 stressesfor concrete in com1ression and for each row or section of steel -ars are calc!lated/

!uIi2 /' xi5 (x!

?D5%)

!uO i2 /?& xi 5 x!

Khere i2 strain of steel at ithrow

xi2 distance of ithrow from ne!tral axis


11/18

4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON

=) 7or e


12/18


Case '" Khen ne!tral axis l.ing o!tside the section !u 3

4!c 2 @rea of stress

-locBP-

2 /==>(3;C?5>)Q

PfcB PD

2 C3PfcBPD

Khere C?form!la is as


13/18


Case ' (Contd)" Khen ne!tral axis l.ing o!tside the section !u 3

7or nding centroid from highl. com1ressed edge

Centroid from com1ressed

edge 2 C'PD

Khere C'form!la is as

shown in the 1ict!re

a-oe


14/18


=) c

c

c2 Moment of resistance oIered -. concrete in com1ression

2 4!cx leer arm

De1ending if !uJ 3 6 or !u 3 6 two form!las are !sed to calc!late leer arm/

!uI

Leer arm 2 (/&D C'D)

!uO

Leer arm 2 (/&D /=3>B!)


15/18


=) c

c

2 Total axial com1ressie resistance oIered -. steel at diIerent leels in thesection

(same for -oth cases of !u)

> PusJ Pus1J RR.. J Pusn

>Khere

i 2 serial n!m-er of the row of reinforcement

n 2 n!m-er of rows of steel -ars

2 cross;sectional area of steel in the ithrow

fsi 2 stress in steel in the ithrow ( Calc!lated from stress;strain relation)

(Com1ressiestress taBen as 1ositieand tensilestress taBen negatie)

fci 2 Com1ressie stress in concrete at the ithrow of reinforcement

2 @lge-raic s!m of the moments of resistance oIered -. steel at diIerent leels

(same for -oth cases of !u)

> / yi) ( where .i2 distance of the ithrow from the centroid ofthe section)

OC O O O C C O CO


16/18


&) 7or diIerent al!es of !ustarting from /3 ( almost Qero 2 1!re tension case)

till

'/&x333

(innit. 2 1!re com1ression case)6 al!es of 4! and M! are fo!nd o!tfor the gien section/

>) The interals taBen -etween these two extreme al!es are as follows/

%) 7iner interals are taBen from /? to /% to get closer to the -alanced section

1oint which !s!all. lies in this range for an. section/

$) 7rom 4! 6 we get al!es of 4!5fcBP-PD/

P P '

6rial !u

3 /3

' /&

? /3

= /3& /&

> /3

% /&

$ /%&

6rial ! u

R /3

3 /'

33 /?

3' /?&

3? /=

3= /=&

3& /&

3> /&&

6rial ! u

3% />

3$ />&

3R /%

' /%&'3 /$

'' /R

'? 3/

'= 3/3

6rial ! u

'& 3/'

'> 3/??

'% 3/&

'$ ''R '/&x3&

4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMN


17/18


3) Aoth these series are 1lotted as a c!re on a gra1h with increasing !u/

H;axis ;;; 4!5fcBP-PD

;axis ;;; M!5fcBP-PD'

Date post:	17-Feb-2018
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Column Interaction Curve (1)

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