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Interaction Diagrams

The design of RC columns is more difficult than the design of

RC beams. In practice the longitudinal steel in an RC column isusually chosen with the aid of an interaction diagram. An

interaction diagram is a graphical summary of the ultimate

bending capacity of a range of RC columns with different

dimensions and areas of longitudinal reinforcement. Thus RC

column design is via RC analysis.

Consider the column section shown above. This section is on the

point of failure since the strain in the most compressive fibre is

0.0035. The axial load which the column is resisting is given by

F f A f Ault c s s s s= + +' '

(assume compressive stresses positive)

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In a similar manner the moment which the section resists can be

expressed as,

Mult Fc

hx fs As

hd fsAs d

h

=

+

2 0 4 2 2. ' ' '

both these quantities the axial capacity and moment capacity are

functions of the position of the neutral axis. These equations are

the basis of any interaction diagram.

Dimension-less Interaction Diagrams

In practice it is cumbersome to construct an interaction diagram

for each column one designs. Therefore it is customary to draw.

or use pre-drawn, dimension-less interaction diagrams.

In the following section we first consider a concrete section

without reinforcement. In this case

Nf

b x fconcck

ck= =0 85

150 8 0 453

.

.. . bx

By altering the equation for the axial capacity a dimension-less

variable related to the axial capacity can be derived.

concconc

ck

N

f bh

x

h

= =

0453.

This expression describes the axial capacity in terms of one

variablex

h

.

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In a similar manner an expression for the moment capacity of an

unreinforced concrete section

Mf

b xh x

f bxhx

h

conc ck

ck

=

=

0 85

15 08 2

0 8

2

0 227 1 08

.

. ..

. .

can be developed to give a dimension-less quantity which

describes the moment capacity in terms ofx

h

.

concconc

ck

M

f bh

x

h

x

h= =

2

0 227 1 0 8. .

The dimension-less formulae for axial capacity are altered if the

column is reinforced on the tension (less compressed) side.

sS

ck

s s

ck

s s

ck

N

f bh

A f

f bh

A

bh

f

f= = =

ss

ck

s

M

f bh

d

h= =

2

1

2

The change in axial load and moment capacity are dependent onthe stress in the steel and thus on the strain in the steel,

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( )

( )0 00350 0035

0 0035

..

.

x x d

x d

x

x

h

d

h

x

h

ss

s

=

=

=

and hence the change in the capacities are dependent on

x

h

.

Similarly the compression steel (steel on the most compressed

side of the section) also alters the axial capacity

sS

ck

s s

ck

s s

ck

N

f bh

A f

f bh

A

bh

f

f'

' ' ' '= = =

'

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and the moment capacity

s sck

s

M

f bh

d

h' ' '

'

= =

2

1

2

as before the strain in the compression steel is a function of the

depth to the neutral axis and hence the variablex

h

.

( ) ( )0 0035 0 0035

0 0035

.'

. '

.

'

'

'

x x dx d

x

x

h

d

h

x

h

ss

s

= =

=

Thus the overall axial capacity,

N

f bhf

x

h

d

h

d

h

A

bh

A

bhckc s s

s s= + + =

'

', ,', ,

and the overall moment capacity,

M

f bhf

x

h

d

h

d

h

A

bh

A

bhckc s s

s s

2= + + =

'

', ,', ,

are both functions offx

h

d

h

d

h

A

bh

A

bhs s, ,

', , '

.

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Interaction Diagram Procedure

Step 1

Calculated

h

d

h

A

bh

A

bhs,

', & 's . These are all constants once the areas

of steel and the dimensions of the section and the concrete cover

are defined.

Step 2

Choose a value forxh

Step 3

Using this value ofx

hcalculate the strain in the top and bottom

reinforcement, s '& s , using the formulae

s

x

h

d

h

x

h

=

0 0035. & s

x

h

d

h

x

h

' .

'

=

0 0035

and thus the stresses in the top and bottom reinforcement,

.f fs s' &

Step 4Use these values for in the formulae forfs '& fs &

=

+

+

0453. ' '

x

h

A

bh

f

f

A

bh

f

fs s

ck

s s

ck

and

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=

+

0 227 1 0 8

1

2

1

2. .

' ' 'x

h

x

h

d

h

A

bh

f

f

d

h

A

bh

f

fs s

ck

s s

ck

Step 5

Repeat steps 2-4 to generate a series of failure points on the

& diagram. Join adjacent points to generate a continuous

curve which indicates combinations of& at failure

Example

d

h

d

h

A

bh

A

bhs s

= = = =

= =

= =

255

300085

45

300015

314 3

300 3000 02 2%

.'

.

.'