Reinforced Concrete Example

Figure 1 shows the design and nominal interaction curves given by ACI. We consider the

cantilever column in Figure 2 with cross-section shown in Figure 3. The appropriate ACI

interaction curves are shown in Figure 4. We pick of the curve corresponding to the

reinforcement ratio ρg = 0.034 of the cross-section considered. The corresponding nominal

interaction curve is shown in Figure 5. The piecewise linear approximation and its extension to

the tension side are shown in Figure 6. An interaction diagram obtained from a detailed cross-

section analysis could be used instead. The procedure for analysis with LARSA is at this point

the same as for the previous example that used the steel cross-section. One point of note is that

the area and moments of inertia for the cross-section are the transformed values, i.e., the steel

areas are multiplied by the ratio of the steel elastic modulus to the concrete elastic modulus as

shown in Figure 8. As before, the analysis is performed by first applying an axial compression

and then applying lateral support displacement incrementally. A typical lateral force-

displacement plot from LARSA is shown in Figure 10. These curves for different values of acial

force are over-plotted in

Figure 1. Design and Nominal Interaction Curves (taken from Leet and Bernal, 1997)

Figure 2. Cantilever Column Example

Axial force, P

Lateral Displacement, ∆

8 ft

Figure 3. Example cross-section (taken from Leet and Bernal, 1997)

Figure 4. ACI Design Interaction curves (taken from Leet and Bernal, 1997)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

M/Agh (ksi)

P/A

g (

ksi)

Design Interaction Curve

Nominal Interaction Curve

Figure 5. Interaction Curves for Example Cross-section

h = 15 in

γ h = 9 in

12 in

4 #11 bars

Ast = 6.25 in2

Ag = 180 in2

ρg = 0.034

f’c = 3 ksi

fy = 60 ksi

Figure 6. Interaction curve input to LARSA

Figure 7. Material Properties

'57,000c

f

Figure 8. Section Properties

Figure 9. Interaction Diagram Input

Large numbers to

ignore these effects

Transformed Area

Not Used

Transformed Icr

To input interaction curve

To input family of M-φ curves

Lines from above

interaction curve

Mirror image about

axial force axis

Constrain y Moment

Figure 10. Typical Force-Displacement plot from LARSA

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Laterl Displacement (in)

Late

ral

Fo

rce (

kip

s)

Axial Force = 0 kips

Axial force = 200 kips

Axial force = 400 kips

AXial force = 600 kips

Figure 11. Lateral force-displacement plots for different axial force levels