Interaction Diagram - Tied Reinforced Concrete Column (Using CSA A23.3-14)


Interaction Diagram - Tied Reinforced Concrete Column

Develop an interaction diagram for the square tied concrete column shown in the figure below about the x-axis using CSA A23.3-14 provisions. Determine six control points on the interaction diagram and compare the calculated values in the Reference and with exact values from the complete interaction diagram generated by spColumn engineering software program from StructurePoint.

 

Figure 1Reinforced Concrete Column Cross-Section


 

Contents

1.   Pure Compression. 3

1.1. Nominal axial resistance at zero eccentricity. 3

1.2. Factored axial load resistance at zero eccentricity. 3

1.3. Maximum factored axial load resistance. 3

2.   Bar Stress Near Tension Face of Member Equal to Zero, ( εs = fs = 0 ) 4

2.1. c, a, and strains in the reinforcement 4

2.2. Forces in the concrete and steel 4

2.3. Pr and Mr 5

3.   Bar Stress Near Tension Face of Member Equal to 0.5 fy, ( fs = - 0.5 fy ) 6

3.1. c, a, and strains in the reinforcement 6

3.2. Forces in the concrete and steel 6

3.3. Pr and Mr 7

4.   Bar Stress Near Tension Face of Member Equal to fy, ( fs = - fy ) 8

4.1. c, a, and strains in the reinforcement 8

4.2. Forces in the concrete and steel 9

4.3. Pr and Mr 9

5.   Pure Bending. 10

5.1. c, a, and strains in the reinforcement 10

5.2. Forces in the concrete and steel 10

5.3. Pr and Mr 11

6.   Pure Tension. 12

6.1. Strength under pure axial tension (Prt) 12

6.2. Corresponding Moment (Mrt) 12

7.   Column Interaction Diagram - spColumn Software. 13

8.   Summary and Comparison of Design Results. 19

9.   Conclusions & Observations. 20

 

                                                                                                                  

                                                                                                                  

                                                                                                                     


Code

Design of Concrete Structures (CSA A23.3-14)

 

Reference

Reinforced Concrete Mechanics and Design, 1st Canadian Edition, 2000, James MacGregor and Fred Michael Bratlett, Prentice Hall Canada Inc.

 

Design Data

fc = 35 MPa

fy = 400 MPa

Cover = 55 mm to the center of the reinforcement

Column 400 mm x 400 mm

Top reinforcement = 4 No. 30

Bottom reinforcement = 4 No. 30

 

Solution

Use the traditional hand calculations approach to generate the interaction diagram for the concrete column section shown above by determining the following six control points:

Point 1: Pure compression

Point 2: Bar stress near tension face of member equal to zero, ( fs = 0 )

Point 3: Bar stress near tension face of member equal to 0.5 fy  ( fs = - 0.5 fy )

Point 4: Bar stress near tension face of member equal to fy  ( fs = - fy )

Point 5: Pure bending 

Point 6: Pure tension


Figure 2 – Control Points

 


 

1.       Pure Compression

1.1.     Nominal axial resistance at zero eccentricity

                                                                                                                                              

                                                                          CSA A23.3-14 (Equation 10.1)

1.2.     Factored axial load resistance at zero eccentricity

Since this column is a tied column with steel strain in compression:

                                                                              CSA A23.3-14 (Equation 10.11)

Where:

                                                                                                                                        CSA A23.3-14 (8.4.2)

                                                                                                                                   CSA A23.3-14 (8.4.3(a))

1.3.     Maximum factored axial load resistance

                                                                          CSA A23.3-14 (Equation 10.9)


2.       Bar Stress Near Tension Face of Member Equal to Zero, ( εs = fs = 0 )

Figure 3 – Strains, Forces, and Moment Arms (εt = fs = 0)

 

Strain εs is zero in the extreme layer of tension steel. This case is considered when calculating an interaction diagram because it marks the change from compression lap splices being allowed on all longitudinal bars, to the more severe requirement of tensile lap splices.                                                                                                                                             CSA A23.3-14 (12.15 and 16)

2.1.     c, a, and strains in the reinforcement

 

Where c is depth of the neutral axis measured from the compression edge of the column section.

                                                                                                                                                              CSA A23.3-14 (3.2)

                                                                                            CSA A23.3-14 (10.1.7)

Where:

a = Depth of equivalent rectangular stress block                                                                         CSA A23.3-14 (3.2)

                                CSA A23.3-14 (Equation 10.2)

                                                                                                                                        CSA A23.3-14 (8.4.2)

                                                                                                                                   CSA A23.3-14 (8.4.3(a))

                                                                                                                                  CSA A23.3-14 (10.1.3)

2.2.     Forces in the concrete and steel

                                      CSA A23.3-14 (10.1.7)

The area of the reinforcement in this layer has been included in the area (ab) used to compute Cc. As a result, it is necessary to subtract α1ϕcfc from ϕsfs before computing Crs:

2.3.     Pr and Mr

 


 

3.       Bar Stress Near Tension Face of Member Equal to 0.5 fy, ( fs = - 0.5 fy )

Figure 4 – Strains, Forces, and Moment Arms (fs = - 0.5 fy)

3.1.     c, a, and strains in the reinforcement

                                                                                                                                        CSA A23.3-14 (8.4.2)

                                                                                                                                   CSA A23.3-14 (8.4.3(a))

                                                                                                                                  CSA A23.3-14 (10.1.3)

Where c is depth of the neutral axis measured from the compression edge of the column section.

                                                                                                                                                              CSA A23.3-14 (3.2)

                                                                                            CSA A23.3-14 (10.1.7)

Where:

a = Depth of equivalent rectangular stress block                                                                         CSA A23.3-14 (3.2)

                                CSA A23.3-14 (Equation 10.2)

3.2.     Forces in the concrete and steel

                                     CSA A23.3-14 (10.1.7)

The area of the reinforcement in this layer has been included in the area (ab) used to compute Cc. As a result, it is necessary to subtract α1ϕcfc from ϕsfs before computing Crs:

3.3.     Pr and Mr


 

4.       Bar Stress Near Tension Face of Member Equal to fy, ( fs = - fy )

Figure 5 – Strains, Forces, and Moment Arms (fs = - fy)

This strain distribution is called the balanced failure case and the compression-controlled strain limit. It marks the change from compression failures originating by crushing of the compression surface of the section, to tension failures initiated by yield of longitudinal reinforcement.

4.1.     c, a, and strains in the reinforcement

                                                                                                                                        CSA A23.3-14 (8.4.2)

                                                                                                                                   CSA A23.3-14 (8.4.3(a))

                                                                                                                                  CSA A23.3-14 (10.1.3)

Where c is depth of the neutral axis measured from the compression edge of the column section.

                                                                                                                                                              CSA A23.3-14 (3.2)

                                                                                            CSA A23.3-14 (10.1.7)

Where:

a = Depth of equivalent rectangular stress block                                                                         CSA A23.3-14 (3.2)

                                CSA A23.3-14 (Equation 10.2)

4.2.     Forces in the concrete and steel

                                     CSA A23.3-14 (10.1.7)

The area of the reinforcement in this layer has been included in the area (ab) used to compute Cc. As a result, it is necessary to subtract α1ϕcfc from ϕsfs before computing Crs:

4.3.     Pr and Mr


 

5.       Pure Bending

Figure 6 – Strains, Forces, and Moment Arms (Pure Moment)

This corresponds to the case where the factored axial load resistance, Pr, is equal to zero. Iterative procedure is used to determine the factored moment resistance as follows:

5.1.     c, a, and strains in the reinforcement

Where c is depth of the neutral axis measured from the compression edge of the column section.

                                                                                                                                                              CSA A23.3-14 (3.2)

                                                                                         CSA A23.3-14 (10.1.7)

Where:

                                CSA A23.3-14 (Equation 10.2)

                                                                                                                                  CSA A23.3-14 (10.1.3)

                                                                                                                                        CSA A23.3-14 (8.4.2)

                                                                                                                                   CSA A23.3-14 (8.4.3(a))

5.2.     Forces in the concrete and steel

                                         CSA A23.3-14 (10.1.7)

The area of the reinforcement in this layer has been included in the area (ab) used to compute Cc. As a result, it is necessary to subtract α1ϕcfc from ϕsfs before computing Crs:

5.3.     Pr and Mr

The assumption that c = 78.55 mm is correct

 


 

6.       Pure Tension

The final loading case to be considered is concentric axial tension. The strength under pure axial tension is computed by assuming that the section is completely cracked through and subjected to a uniform strain greater than or equal to the yield strain in tension. The strength under such a loading is equal to the yield strength of the reinforcement in tension.

6.1.     Strength under pure axial tension (Prt)

                                                                              

6.2.     Corresponding Moment (Mrt)

Since the section is symmetrical

 


 

7.       Column Interaction Diagram - spColumn Software

 

spColumn program performs the analysis of the reinforced concrete section conforming to the provisions of the Strength Design Method and Unified Design Provisions with all conditions of strength satisfying the applicable conditions of equilibrium and strain compatibility. For this column section, we ran in investigation mode with control points using the CSA A23.3-14. In lieu of using program shortcuts, spSection (Figure 9) was used to place the reinforcement and define the cover to illustrate handling of irregular shapes and unusual bar arrangement.

 

Figure 7 – Generating spColumn Model

 

Figure 8spColumn Model Editor (spSection)


 

Figure 9 – Column Section Interaction Diagram about the X-Axis (spColumn)


 



8.       Summary and Comparison of Design Results

 

Table 1 - Comparison of Results

Support

Pr, kN

Mr, kN.m

Hand

Reference* **

spColumn

Hand

Reference* **

spColumn

Max compression

4705

4490

4705

0

0

0

Allowable compression

3764

3592

3764

---

---

---

fs = 0.0

3111

2945

3111

236

229

236

fs = 0.5 fy

2144

2015

2144

340

330

340

Balanced point

1355

1253

1355

414

403

414

Pure bending

0

0

0

286

285

286

Max tension

1904

1904

1904

0

0

0

*    Reinforced Concrete Mechanic and Design, 1st Canadian Edition, James MacGregor and Fred Bartlett – Example 11-1

**  The reference used CSA A23.3-94 where the resistance factor for concrete (ϕc) is 0.60. The hand calculation and spColumn used CSA A23.3-14 where the resistance factor for concrete (ϕc) is 0.65. (Check Column Interaction Diagram Using CSA A23.3-94 Example)


 

9.       Conclusions & Observations

 

The analysis of the reinforced concrete section performed by spColumn conforms to the provisions of the Strength Design Method and Unified Design Provisions with all conditions of strength satisfying the applicable conditions of equilibrium and strain compatibility.

 

In the calculation shown above a P-M interaction diagram was generated with moments about the X-Axis (Uniaxial bending). Since the reinforcement in the section is not symmetrical, a different P-M interaction diagram is needed for the other orthogonal direction about the Y-Axis (See the following Figure for the case where fs = fy).

 

Figure 10 – Strains, Forces, and Moment Arms (fs = - fy Moments About x- and y-axis)


 

When running about the Y-Axis, we have 2 bars in 4 layers instead of 4 bars in just 2 layers (about X-Axis) resulting in a completely different interaction diagram as shown in the following Figure.

Figure 11 – Comparison of Column Interaction Diagrams about X-Axis and Y-Axis (spColumn)

 

Further differences in the interaction diagram in both directions can result if the column cross section geometry is irregular.

 

In most building design calculations, such as the examples shown for flat plate or flat slab concrete floor systems, all building columns are subjected to Mx and My due to lateral forces and unbalanced moments from both directions of analysis. This requires an evaluation of the column P-M interaction diagram in two directions simultaneously (biaxial bending).

 

StucturePoint’s spColumn program can also evaluate column sections in biaxial mode to produce the results shown in the following Figure for the column section in this example.

 

Figure 12 – Nominal & Design Interaction Diagram in Two Directions (Biaxial) (spColumn)