PREPARATION OF INTERACTION DIAGRAM AS PER BS 8110: 1997

Input parameters.

The following conditions shall be fulfilled to prepare the interaction diagram.

i) When neutral axis lies in the section.

Computation of strains

Strain of compression bars

And

And so on........depending upon the number of rows of rebar where \(\varepsilon_{cu} = 0.0035\)

Strain at tensile reinforcement

Computation of Stresses

At compression rebar

If \(f_{sc1}\geq 0.87f_y\), then

Else

And

If \(f_{sc2}\geq 0.87f_y\), then

Else

And so on........depending upon the number of rows of rebar.

At tension rebar

If \(f_{st}\geq 0.87f_y \) then

Else

Computation of forces

Of Concrete:

Of Compression rebar

And

And so on……depending on the number of rows of rebar

Of tension rebars

Resultant forces in concrete

Calculation of moment capacities (Taking moment at center of the section).

Of Compression rebar

And

Of Concrete

Where \(s=0.9x\)

Of Tension rebar

ii) When neutral axis lies outside the section

Computation of Strains

Row-1

Row-2

And so on......

Note: all the rebars are on compression side.

Computation of Stresses

Row-1

But not greater than \(0.87f_y\).

Row-2

But not greater than \(0.87f_y\).

And so on......

Compression forces

Row-1

Row-2

And so on......

Compressive stress of Concrete

Total force at the section

Moment at the center of the section

And so on............

Total Moment

The moment due to rows of reinforcement below half of the section shall be deducted.

Description of interaction diagram

a) Pure bending

This point represents that of a beam in bending. Note that the presence of a small axial force will generally increase the moment capacity of a beam

b) Balanced point

This is the point where the concrete reaches its ultimate strain at the same time the tension reinforcement yields. For combinations of N and M that will fall below the balance point, the failure mode is ductile with the reinforcement yielding before the concrete fails in compression. Otherwise, failure mode is brittle; the concrete crushes without yielding of the tension reinforcement. Unfortunately, the failure mode in a column can’t be controlled by reinforcement quantities as it can be in beams.

c) Pure axial compression

At this point, the column is subjected to an axial force with only M=0. The capacity of the section is equal to Nuz. Note that the tension reinforcement in compression for this case.

d) Zero strain in the tension reinforcement

Moving from point (b) to point (c) the neutral axis increases from xbal to infinity as N increases. The strain in the tension reinforcement changes from yielding in tension to yielding in compression, passing through zero at point (d). Moving from points (d) to (c) the neutral axis will fall outside the section and the strain distribution will eventually change from triangular to uniform.

Between points (b) and (c) an increase in axial load N will lead to a smaller moment capacity M at failure. Conversely, below the balance point an increase in N increases will increase the moment capacity of the section.

e) Yielding of the compression reinforcement

As the axial force N increases and the neutral axis x increases, the strain in compression reinforcement will often change from elastic to yielding. This will clearly be influenced by the strength of the reinforcement and its position within the section. This point will typically correspond to a change in slope of interaction diagram as shown at point (e).

DESIGN EXAMPLE

INPUTS

CALCULATION

Effective length coefficient \(beta=0.75\)

Effective length \(L_e=\beta L=0.75×3.0=2.25m \)

Check for long or short column, for braced column

So, the column is short

Ratio of effective depth to overall depth

Minimum Eccentricity

Check for dominancy of the moments in the axes.

In width

Moment per width

In depth

Moment per width

Check for dominancy

Since \(\frac{M_y}{h'}>\frac{M_x}{b'}\) , \(M_y\) dominance over \(M_x\).

Multiplying factor \(\beta\) shall be calculated using following table

So, \(\beta = 0.853\)

Design moment

\(Since,\frac{M_y}{b'}>\frac{M_x}{h'}\), then \(M'_y=M_y+\beta\frac{b'}{h'}M_x \)

Preparation of Interaction diagram

A. In Width (X-Chart)

Reinforcement distribution (Equally distributed in all side) in rows and columns

Spacing of neutral axis from top

Start of NA, \(x_1=0.05\)

Condition-1: When neutral axis lies in section

For example \(x=0.35m\)

Strain:

Stress:

If \(\varepsilon\leq\frac{0.87f_y}{E_s}\) then

Else

Calculation of Strain and stress at each rows of the rebar

Negative sign indicates the tension reinforcement

Calculation of compressive and tensile forces

Compressive force of concrete

Compressive/Tensile force of steel

CALCULATION TABLE

Bending moments

Similar process shall be carried out different positions of neutral axis. In case of neutral axis lies outside of the sections, the summation of forces and moments are carried out by following formula

And so on....

Total moment

A similar process shall be followed in another axis as well. After doing detailed calculations, following interaction diagram has been achieved.

Check for linear approximation

Moment at x-axis for \(N=500kN\), \(M_{ux}=320.55kNm\)

Similarly, Moment at y-axis for \(N=500kN\), \(M_{uy}=320.55kNm\)

Result: Assumption of 3% of reinforcement is okay for this section.