Interaction Diagram - Tied Reinforced Concrete Column



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. Determine seven 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 1 – Reinforced Concrete Column Cross-Section


 

Contents

1.   Pure Compression. 3

1.1. Nominal axial compressive strength at zero eccentricity. 3

1.2. Factored axial compressive strength at zero eccentricity. 3

1.3. Maximum (allowable) factored axial compressive strength. 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. ϕPn and ϕMn. 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. ϕPn and ϕMn. 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. ϕPn and ϕMn. 9

5.   Bar Strain Near Tension Face of Member Equal to 0.005 in./in., ( εs = - 0.005 in./in.) 10

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

5.2. Forces in the concrete and steel 10

5.3. ϕPn and ϕMn. 11

6.   Pure Bending. 12

6.1. c, a, and strains in the reinforcement 12

6.2. Forces in the concrete and steel 12

6.3. ϕPn and ϕMn. 13

7.   Pure Tension. 14

7.1. Pnt  and ϕPnt 14

7.2. Mn  and ϕMn. 14

8.   Column Interaction Diagram - spColumn Software. 15

9.   Summary and Comparison of Design Results. 21

10. Conclusions & Observations. 22

 

                                                                                                                  

                                                                                                                  

                                                                                                                     


Code

Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14)

 

Reference

Reinforced Concrete Mechanics and Design, 6th Edition, 2011, James Wight and James MacGregor, Pearson

 

Design Data

fc’ = 5000 psi

fy = 60,000 psi

Cover = 2.5 in. to the center of the reinforcement

Column 16 in. x 16 in.

Top reinforcement = 4 #9

Bottom reinforcement = 4 #9

 

Solution

Use the traditional hand calculations approach to generate the interaction diagram for the concrete column section shown above by determining the following seven 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: Bar strain near tension face of member equal to 0.005

Point 6: Pure bending 

Point 7: Pure tension


Figure 2 – Control Points

 


 

1.       Pure Compression

1.1.     Nominal axial compressive strength at zero eccentricity

                                                                                                    ACI 318-14 (22.4.2.2)

1.2.     Factored axial compressive strength at zero eccentricity

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

                                                                                                                                  ACI 318-14 (Table 21.2.2)

                                                                                                                                            

1.3.     Maximum (allowable) factored axial compressive strength

                                                                ACI 318-14 (Table 22.4.2.1)


 

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.                                                                                                                                            ACI 318-14 (10.7.5.2.1 and 2)

 

2.1.     c, a, and strains in the reinforcement

 

Where c is the distance from the fiber of maximum compressive strain to the neutral axis.

                                                                                                                                                       ACI 318-14 (22.2.2.4.2)

                                                                                          ACI 318-14 (22.2.2.4.1)

Where:

a = Depth of equivalent rectangular stress block

[IMA1]         ACI 318-14 (Table 22.2.2.4.3)

                                                                                                                              ACI 318-14 (Table 21.2.2)

                                                                                                                                     ACI 318-14 (22.2.2.1)

2.2.     Forces in the concrete and steel

                                             ACI 318-14 (22.2.2.4.1)

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 0.85fc’ from fs’ before computing Cs:

2.3.     ϕPn and ϕMn


 

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

                                                                                                                              ACI 318-14 (Table 21.2.2)

                                                                                                                                     ACI 318-14 (22.2.2.1)

 

Where c is the distance from the fiber of maximum compressive strain to the neutral axis.

                                                                                                                                                       ACI 318-14 (22.2.2.4.2)

                                                                                          ACI 318-14 (22.2.2.4.1)

Where:

                    ACI 318-14 (Table 22.2.2.4.3)

3.2.     Forces in the concrete and steel

                                                ACI 318-14 (22.2.2.4.1)

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 0.85fc’ from fs’ before computing Cs:

3.3.     ϕPn and ϕMn


 

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. It also marks the start of the transition zone for ϕ for columns in which ϕ increases from 0.65 (or 0.75 for spiral columns) up to 0.90.

4.1.     c, a, and strains in the reinforcement

                                                                                                                              ACI 318-14 (Table 21.2.2)

                                                                                                                                     ACI 318-14 (22.2.2.1)

 

Where c is the distance from the fiber of maximum compressive strain to the neutral axis.

                                                                                                                                                       ACI 318-14 (22.2.2.4.2)

                                                                                           ACI 318-14 (22.2.2.4.1)

Where:

                     ACI 318-14 (Table 22.2.2.4.3)

4.2.     Forces in the concrete and steel

                                               ACI 318-14 (22.2.2.4.1)

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 0.85fc’ from fs’ before computing Cs:

4.3.     ϕPn and ϕMn


 

5.       Bar Strain Near Tension Face of Member Equal to 0.005 in./in., ( εs = - 0.005 in./in.)

Figure 6 – Strains, Forces, and Moment Arms (εs = - 0.005 in./in.)

This corresponds to the tension-controlled strain limit of 0.005. It is the strain at the tensile limit of the transition zone for ϕ, used to define a tension-controlled section.

5.1.     c, a, and strains in the reinforcement

                                                                                                                                ACI 318-14 (Table 21.2.2)

                                                                                                                                     ACI 318-14 (22.2.2.1)

 

Where c is the distance from the fiber of maximum compressive strain to the neutral axis.

                                                                                                                                                       ACI 318-14 (22.2.2.4.2)

                                                                                           ACI 318-14 (22.2.2.4.1)

Where:

                    ACI 318-14 (Table 22.2.2.4.3)

5.2.     Forces in the concrete and steel

                                               ACI 318-14 (22.2.2.4.1)

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 0.85fc’ from fs’ before computing Cs:

5.3.     ϕPn and ϕMn


 

6.       Pure Bending

Figure 7 – Strains, Forces, and Moment Arms (Pure Moment)

This corresponds to the case where the nominal axial load capacity, Pn, is equal to zero. Iterative procedure is used to determine the nominal moment capacity as follows:

6.1.     c, a, and strains in the reinforcement

Where c is the distance from the fiber of maximum compressive strain to the neutral axis.

                                                                                                                                                       ACI 318-14 (22.2.2.4.2)

                                                                                           ACI 318-14 (22.2.2.4.1)

Where:

                     ACI 318-14 (Table 22.2.2.4.3)

                                                                                                                                     ACI 318-14 (22.2.2.1)

                                                                                                                                ACI 318-14 (Table 21.2.2)

6.2.     Forces in the concrete and steel

                                                  ACI 318-14 (22.2.2.4.1)

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 0.85fc’ from fs’ before computing Cs:

6.3.     ϕPn and ϕMn

The assumption that c = 3.25 in. is correct


 

7.       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.

7.1.     Pnt  and ϕPnt

                                                                  ACI 318-14 (22.4.3.1)

                                                                                                                                    ACI 318-14 (Table 21.2.2)

7.2.     Mn  and ϕMn

Since the section is symmetrical

 


 

8.       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 318-14. 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 8 – Generating spColumn Model

 

Figure 9 – spColumn Model Editor (spSection)


 

Figure 10 – Column Section Interaction Diagram about the X-Axis (spColumn)


 

  


9.       Summary and Comparison of Design Results

 

Table 1 - Comparison of Results

Support

ϕPn, kip

ϕMn, kip.ft

Hand

Reference*

spColumn

Hand

Reference*

spColumn

Max compression

997

997

997

0

0

0

Allowable compression

798

798

798

---

---

---

fs = 0.0

622

622

622

170

170

170

fs = 0.5 fy

422

422

422

220

220

220

Balanced point

271

271

271

251

251

251

Tension control

175

175

175

288

288

288

Pure bending

0

0

0

214

214

214

Max tension

432

432

432

0

0

0

* Reinforced Concrete Mechanic and Design, 6th Edition, James Wight & MacGregor – Example 11-1

 

In all of the hand calculations and the reference used illustrated above, the results are in precise agreement with the automated exact results obtained from the spColumn program.


 

10.    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 11 – 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 12 – 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 13 – Nominal & Design Interaction Diagram in Two Directions (Biaxial) (spColumn)

 

 


-           [IMA1]Instead of x