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7/23/2019 Column Interaction Curve (1)
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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
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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.
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To nd Stress for Steel(fs) and Concrete(fc)from Strain() al!es
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>
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To nd Stress for Steel(fs) and Concrete(fc)from Strain() al!es
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.
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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
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Limiting Cases de1ending on Ne!tral @xis
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
+orresponding max. stress
> 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)
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Limiting Cases de1ending on Ne!tral @xis
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
+orresponding max. stress
> 7.==* fc!
!u 2 3
D
?min >
7
?max
(7.7789)
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Limiting Cases de1ending on Ne!tral @xis
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
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Limiting Cases de1ending on Ne!tral @xis
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
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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
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4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON
=) 7or e
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4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON
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
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4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON
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
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4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON
=) 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!)
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4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON
=) 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
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4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON
&) 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
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4+OC8DU+8 @DO4T8D TO 7#ND MOM8NT C@4@C#TH O7 COLUMNS8CT#ON
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'