PREPARATION OF INTERACTION DIAGRAM AS PER IS456:2000

To prepare the interaction diagram, four different cases shall be fulfilled.

a) When the neutral axis is at infinity, i.e. \(kD\to \infty \), pure axial load is applied on the column.

b) When the neutral axis is outside the cross-section of the column i.e. \(∞>kD \geq D\).

c) When the neutral axis is within the cross-section of the column i.e \( kD \leq D\).

d) When the column behaves like a steel beam.

a) WHEN THE NEUTRAL AXIS IS AT INFINITY \(kD\to \infty \), PURE AXIAL LOAD IS APPLIED ON THE COLUMN

The strain profile EF represents the condition of pure compression. The pure axial force of concrete and steel are given as.

curve

Fig. Stress-strain diagram of column section for pure bending

Pure compresive force of concrete

\(C_s=0.446f_{ck}bD\)

Pure compresive force of Steel

\(C_s=A_{st}(f_{sc}-0.446f_{ck})\)

Total compressive force at the section

\(P_u=C_s+C_s\)

Now

\(\frac{P_u}{bD}=\frac{(0.446f_{ck}bD+A_{st}(f_{sc}-0.446f_{ck}))}{bD} \)

And

\(\frac{M_u}{bD^2}=0 \)

b) WNEN THE NEUTRAL AXIS IS OUTSIDE THE CROSS-SECTION OF THE COLUMN i.e. \(∞>kD \geq D\).

The strain profile JK in above diagram represents this condition

\( C_c=C_1f_{ck}bD \)


\(C_1=0.446[1-(\frac{4}{21})(\frac{4}{7k-3})^2]\)


\( C_s = \sum_{i=1}^n A_{sci} (f_{si} - f_{ci}) \)


\(P_u=C_c+C_s\)


\(\frac{P_u}{f_{ck}bD}=C_1+\frac{\sum_{i=1}^n A_{sci} (f_{si} - f_{ci})}{f_{ck}bD} \)

And Moment is given by;

\(M_u=C_1f_{ck}bD(\frac{D}{2}-C_2D)+\sum_{i=1}^n A_{sci} (f_{si} - f_{ci})y_i \)

Where,

\(C_2=\frac{[\frac{1}{2}-\frac{8}{49}(\frac{4}{7k-3})^2]}{[1-\frac{4}{21}(\frac{4}{7k-3})^2]} \)

So,

\(\frac{M_u}{f_{ck}bD^2}=C_1(0.5-C_2)+\frac{\sum_{i=1}^n A_{sci} (f_{si} - f_{ci})y_i}{f_{ck}bD^2} \)

The values of \(C_1\) and \(C_2\) are given in the Table below.

Here,

\(k=\frac{x}{D}\)

Table: The coefficients \(C_1\) and \(C_2\)

\(K\) \(C_1\) \(C_2\)
1.00 0.361 0.416
1.05 0.374 0.432
1.10 0.384 0.443
1.20 0.399 0.458
1.30 0.409 0.468
1.40 0.417 0.475
1.50 0.422 0.480
2.00 0.435 0.491
2.50 0.440 0.495
3.00 0.442 0.497
4.00 0.444 0.499

Computation of compressive stress of Concrete

If Strain \(ε<0.002\) the stress \(f_c\) is given by:

\( f_c = 0.446 f_{ck} \left[ 2 \frac{\varepsilon}{0.002} - \left( \frac{\varepsilon}{0.002} \right)^2 \right] \)

Otherwise if

\( 0.002 \leq \varepsilon \leq 0.0035\)

stress is given by,

\( f_c = 0.446 f_{ck} \)

curve

Fig.: Stress-strain diagram of column when neutral axis lies outside the section.

c) WHEN THE NEUTRAL AXIS IS WITHIN THE CROSS-SECTION OF THE COLUMN i.e \( kD \leq D\).

Compressive force of Concrete

\( C_s=0.36kf_{ck}bD\)

Compressive force of Steel

\( C_s=\sum_{i=1}^n A_{sci}(f_{si}-f_{ci})\)

Total compressive force

\(\frac{P_u}{f_{ck}bD}=0.36k+\frac{\sum_{i=1}^n A_{sci}(f_{si}-f_{ci})}{f_{ck}bD} \)

And Moment Capacity is given by;

\(M_u=0.36f_{ck}kbD(0.5-0.42k)D+\sum_{i=1}^n A_{sci}(f_{si}-f_{ci}) \frac{y_i}{D} \)

And

\(\frac{M_u}{f_{ck}bD^2}=0.36(0.5-0.42k)+\sum_{i=1}^n\frac{A_{sci}(f_{si}-f_{ci})\frac{y_i}{D}}{f_{ck}bD^2} \)

curve

Fig.: Stress-strain diagram of column section when neutral axis lies within the section.

d) WHEN THE COLUMN BEHAVES LIKE A STEEL BEAM

This condition occurs when the column is subjected to pure moment only i.e., \(M_u=M_o\)

So,

\(\frac{P_u}{f_{ck}bD}=0 \)

And Moment Capacity is given by;

\(M_u=\sum_{i=1}^n 0.87f_yA_{sci}(\frac{y_i}{D}) \)

And

\(\frac{M_u}{f_{ck}bD^2}=\frac{\sum_{i=1}^n 0.87f_y A_{sci}(\frac{y_i}{D})}{f_{ck}bD^2}\)

By fulfilling these conditions, the interaction diagram can be prepared.