column interaction

The Column Design Section in Strunet contains two main parts: Charts to develop strength interaction diagrams for any given section, and ready -made Column Interaction Diagrams, for quick design of a given column. Concrete column is one of the most interes ting members in concrete structural design application. A structural design of a concrete column is quite complicated procedures. Evaluation, however, of a given column section and reinforcement is straightforward process. This is due to the fact that pure axial compression is rarely the case in column analysis. Some value of moment is always there due to end restraint, or accidental eccentricity due to out of alignment. ACI established the minimum eccentricity on a concrete column, regardless of the struct ural analysis proposed for the column, which is defined as the maximum axial compression load that a column can be designed for. Column Design Charts in Bullets: One of the demanding aspects in concrete column design is to define the controlling points on strength interaction diagram. The column strength interaction diagram is a curve plot of points; where each point has two ordinates. The first ordinate is bending moment strength and the second is the corresponding axial force. Both ordinates are linked with eccentricity. The shape of the curve, or the strength interaction diagram, can be defined by finding the ordinates of major seven points. Each point has specific requirement, as established by the code, and thus evaluating the requirement of this poin t will result of calculating the ordinates. The points and their respective requirements are as follows:

• Point 1: Pure compression. • Point 2: Maximum compression load permitted by code at zero

eccentricity. • Point 3: Maximum moment strength at the maximum a xial compression

permitted by code. • Point 4: Compression and moment at zero strain in the tension side

reinforcement. • Pont 5: Compression and moment at 50% strain in the tension side

reinforcement. • Point 6: Compression and moment at balanced conditions. • Point 7: Pure tension.

STRUNETCONCRETE DESIGN AIDS

Introduction to Concrete Column Design Flow Charts

Strunet.com: Concrete Column Design V1.01 -

a = depth of equivalent rectangular stress block, in. ab = depth of equivalent rectangular stress block at balanced

condition, in. Ag = gross area of column, in2. As = area of reinforcement at tension side, in2. A’s = area of reinforcement at compres sion side, in2. A’st = Total area of reinforcement in column cross section, in 2. b = column width dimension, in. c = distance from extreme compression fiber to neutral axis, in. cb = distance from extreme compression fiber to neutral axis at

balanced condition, in. Cc = compression force in equivalent concrete block. Cs = compression force in tension-side reinforcement, if any. C’s = compression force in compression-side reinforcement. d = distance from extreme compression fiber to centroid of tensi on-

side reinforcement d’ = distance from extreme compression fiber to centroid of

compression-side reinforcement e = eccentricity, in. eb = eccentricity at balanced condition, in. Es = modulus of elasticity of reinforcement, psi. f’c = specified compressive strength of concrete, psi. fy = specified tensile strength of reinforcement, psi. fs = stress in tension-side reinforcement at strain εs , ksi. f’s = stress in compression-side reinforcement at strain ε 's , ksi. h = overall column depth, in. Mb = nominal bending moment at balanced condition. Mn = nominal bending moment at any point. Po = nominal axial load strength at zero eccentricity. Pb = nominal axial force at balanced condition. Plim = limit of nominal axial load value at which low or h igh axial

compression can be defined in accordance with ACI 9.3.2.2. Pn = nominal axial load strength at any point. T = tension force in tension-side reinforcement. β? = factor as defined by ACI 10.2.7.3. εs = strain in tension-side reinforcement at calculated stress fs ε 's = strain in compression-side reinforcement at calculated stress f’s εy = yield strain of reinforcement. φ = strength reduction factor



Notations for Concrete Column Design Flow Charts

7

3

4

5

6

increasing φ

consistent φ

comp.control

tensioncontrol

balanced

pure tension

max. axial comp.

Axia

l Com

pres

sion

,φP n

Bending Moment, φMn

2

1

εs=0.0

εs=0.5εy



Main Points of Column Interaction Diagram

Point 1: axial compression at zero moment.Point 2: maximum permissible axial compression at zero eccentricit.Point 3: maximum moment strength at maximum permissible axial compression.Point 4: axial compression and moment strength at zero strain.Point 5: axial compression and moment strength at 50% strain.Point 6: axial compression and moment strength at balanced conditions.Point 7: moment strength at zero axial force.

( )0 85o c g st y stP . f A A f A′= − +st g c yA ,A ,f ,f′

Spiral?

0 85n oP (max) . P=0 80n oP (max) . P=

0 75.φ =0 70.φ =

0 64n oP (max) . Pφ =0 56n oP (max) . Pφ =

Finding Point 1

Finding Point 2

ACI 10.3.5.1

ACI 9.3.2.2ACI 9.3.2.2



Point 1: Axial Compression at Zero MomentPoint 2: Maximum Permissible Axial Compression at Zero Eccentricity

finding φ

Axial Tension andAxial Tension with

Flexure

0 90.φ =

Axial Compressionand Axial

Compression withFlexure

Spiral

0 75.φ =0 70.φ =

High Values of AxialCompression Low Values of Axial

Compression

60 0 70 Symmetric reinf.

y

s

f ksih d d . h

• ≤

′• − − ≤

•

0 150 9 n

L

. P.P

φ = −

0 10 c gL

. f AP

φ′

=

L nP P>

Spiral?

0 75.φ =0 70.φ =

0 133L c gP . f A′=

nP

Spiral?Spiral?

min(0 133 , )L c g bP . f A P′=min(0 143 , )L c g bP . f A P′= 0 20 9 n

L

. P.P

φ = −

ACI 9.3.2.2

0 143L c gP . f A′=

0 150 9 n

L

. P.P

φ = −0 20 9 n

L

. P.P

φ = −



Strength Reduction Factor

4000 cf psi′ ≤

1 0 85.β =1

40000 85 0 05 0 651000

cf. . .β′ − = − ≥

strain stress

Cc

C's0.003

c

b

ha

d'

εs

ε's

Cs

ds

A's

As

d

( )10 85c cC . f c bβ′=

( )0 003s

c d .c

ε−

=

( )0 85s s y cC A' f . f′ ′= −

( )0 85s s s cC A f . f ′= −

s s sf Eε=

( )0 0030 85s s s c

c d .C A E . f

c−

′= −

n c s sP C C C′= + +

( ) ( ) ( )1

0 0030 85 0 85 0 85n c s y c s s c

c d .P . f c b A' f . f A E . f

cβ

− ′ ′ ′= + − + −

of point 2nP

finding distance c

Result: c

( ) ( ) ( )0 5 0 5 0 5n c s s sM . C h a C . h d C . h d′ ′= − + − − −

1a cβ=

Compute: Cc , C's from above Eqs.

ACI 10.2.7.3



Point 3: Maximum Moment Strengthat Maximum Permissible Compression

s 0 0.ε =

c d=

1a cβ=

min( )s s s yf E ,fε′ ′=

( )0 85s s s cC A f . f′ ′ ′ ′= −

n c sP C C′= +

n

n

MeP

=

2 2 2n c sh a hM C C d ′ ′= − + −

0 85c cC . f ab′=

0 003sc d .c

ε′− ′ =

See finding φ&n nM Pφ φ

Finding Point 4

See Finding β1 in point 3

T=0.0

d'

strain stress

Cc

C's

0.003

0.0

c=d

b

h

aε's

As

A's



Point 4: Axial Compression and Moment at Zero Strain


( )0 85s s s cC A f . f′ ′ ′ ′= −

n c sP C C T′= + −

n

n

MeP

=

2 2 2 2n c sh a h hM C C d T d ′ ′= − + − + −


Finding Point 5

yy

s

fE

ε =

0 5s y.ε ε=

0 003sc d .c

ε′− ′ =

10 003

s

dc

.ε

= +

1a cβ= See Finding β1 in point 3

0 85c cC . f ab′=

( )0 5s yT A . f=

d'

T

strain stress

Cc

C's0.003

c

b

h

aε's

εs=0.5εy

As

A's



Point 5: Axial Compression andMoment Strength at 50% Strain

Finding Point 6

sy

ys

fE

ε ε= =

0 003 0 0030 003b

y

.c .. ε

= +

1b ba cβ=

( )0 85s s s cC A f . f′ ′ ′= −

y sT f A=

b c sP C C T= + −

2 2 2 2b

b c sah h hM C C d T d ′= − + − + −

bb

b

MeP

=

0 003b ss

b

c d .c

ε ′−′ =

0 85c c bC . f a b′=




d'

T

strain stress

Cc

C's0.003

c

b

h

aε's

εy=fy /Es

As

A's



Point 6: Axial Compression and MomentStrength at Balanced Conditions

Finding Point 7

See finding φ

0 85c cC . f ab′=

y sT f A=

0 85s y

c

A fa

. f b=

′

1

acβ

=

0 003 0 003s y. d .c

ε ε = − >

0 85c c bC . f a b′=

y sT f A=

2 2 2n ch a hM C T d = − + −

0 9.φ =

nMφ


d'

T

strain stress

Cc0.003

c

b

h

a

ε's~0.0

εs>εy

As

A's



Point 7: Moment Strength at Zero Axial Force

column interaction

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