Interaction Diagram - Tied Reinforced Concrete Column (Using CSA A23.3-94)
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 1 – Reinforced Concrete Column Cross-Section
Contents
1.1. Nominal axial resistance at zero eccentricity
1.2. Factored axial load resistance at zero eccentricity
1.3. Maximum factored axial load resistance
2. Bar Stress Near Tension Face of Member Equal to Zero, ( ε_{s} = f_{s} = 0 )
2.1. c, a, and strains in the reinforcement
2.2. Forces in the concrete and steel
3. Bar Stress Near Tension Face of Member Equal to 0.5 f_{y}, ( f_{s} = - 0.5 f_{y} )
3.1. c, a, and strains in the reinforcement
3.2. Forces in the concrete and steel
4. Bar Stress Near Tension Face of Member Equal to f_{y}, ( f_{s} = - f_{y} )
4.1. c, a, and strains in the reinforcement
4.2. Forces in the concrete and steel
5.1. c, a, and strains in the reinforcement
5.2. Forces in the concrete and steel
6.1. Strength under pure axial tension (P_{rt})
6.2. Corresponding Moment (M_{rt})
7. Column Interaction Diagram - spColumn Software
8. Summary and Comparison of Design Results
Code
Design of Concrete Structures (CSA A23.3-94)
Reference
Reinforced Concrete Mechanics and Design, 1^{st} Canadian Edition, 2000, James MacGregor and Fred Michael Bratlett, Prentice Hall Canada Inc.
Design Data
f_{c}’ = 35 MPa
f_{y} = 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, ( f_{s} = 0 )
Point 3: Bar stress near tension face of member equal to 0.5 f_{y} ( f_{s} = - 0.5 f_{y} )
Point 4: Bar stress near tension face of member equal to f_{y} ( f_{s} = - f_{y} )
Point 5: Pure bending
Point 6: Pure tension
Figure 2 – Control Points
CSA A23.3-94 (Equation 10-1)
Since this column is a tied column with steel strain in compression:
CSA A23.3-94 (Equation 10-10)
Where:
CSA A23.3-94 (8.4.2)
CSA A23.3-94 (8.4.3)
CSA A23.3-94 (Equation 10-9)
Figure 3 – Strains, Forces, and Moment Arms (ε_{t} = f_{s} = 0)
Strain ε_{s} 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 A23.3-94 (12.15 and 16)
Where c is distance from extreme compression fiber to neutral axis.
CSA A23.3-94 (10.0)
CSA A23.3-94 (10.1.7)
Where:
a = Depth of equivalent rectangular stress block CSA A23.3-94 (10.0)
CSA A23.3-94 (Equation 10-2)
CSA A23.3-94 (8.4.2)
CSA A23.3-94 (8.4.3)
CSA A23.3-94 (10.1.3)
CSA A23.3-94 (10.1.7)
The area of the reinforcement in this layer has been included in the area (ab) used to compute C_{c}. As a result, it is necessary to subtract α_{1}ϕ_{c}f_{c}’ from ϕ_{s}f_{s}’ before computing C_{rs}:
Figure 4 – Strains, Forces, and Moment Arms (f_{s} = - 0.5 f_{y})
CSA A23.3-94 (8.4.2)
CSA A23.3-94 (8.4.3)
CSA A23.3-94 (10.1.3)
Where c is depth of the neutral axis measured from the compression edge of the column section.
CSA A23.3-94 (10.0)
CSA A23.3-94 (10.1.7)
Where:
a = Depth of equivalent rectangular stress block CSA A23.3-94 (10.0)
CSA A23.3-94 (Equation 10-2)
CSA A23.3-94 (10.1.7)
The area of the reinforcement in this layer has been included in the area (ab) used to compute C_{c}. As a result, it is necessary to subtract α_{1}ϕ_{c}f_{c}’ from ϕ_{s}f_{s}’ before computing C_{rs}:
Figure 5 – Strains, Forces, and Moment Arms (f_{s} = - f_{y})
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.
CSA A23.3-94 (8.4.2)
CSA A23.3-94 (8.4.3)
CSA A23.3-94 (10.1.3)
Where c is depth of the neutral axis measured from the compression edge of the column section.
CSA A23.3-94 (10.0)
CSA A23.3-94 (10.1.7)
Where:
a = Depth of equivalent rectangular stress block CSA A23.3-94 (10.0)
CSA A23.3-94 (Equation 10-2)
CSA A23.3-94 (10.1.7)
The area of the reinforcement in this layer has been included in the area (ab) used to compute C_{c}. As a result, it is necessary to subtract α_{1}ϕ_{c}f_{c}’ from ϕ_{s}f_{s}’ before computing C_{rs}:
Figure 6 – Strains, Forces, and Moment Arms (Pure Moment)
Where c is depth of the neutral axis measured from the compression edge of the column section.
CSA A23.3-94 (10.0)
CSA A23.3-94 (10.1.7)
Where:
CSA A23.3-94 (Equation 10-2)
CSA A23.3-94 (10.1.3)
CSA A23.3-94 (8.4.2)
CSA A23.3-94 (8.4.3)
CSA A23.3-94 (10.1.7)
The area of the reinforcement in this layer has been included in the area (ab) used to compute C_{c}. As a result, it is necessary to subtract α_{1}ϕ_{c}f_{c}’ from ϕ_{s}f_{s}’ before computing C_{rs}:
The assumption that c = 78.55 mm is correct
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.
Since the section is symmetrical
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 A23.3-94. 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 8 – spColumn Model Editor (spSection)
Figure 9 – Column Section Interaction Diagram about the X-Axis (spColumn)
Table 1 - Comparison of Results |
||||||
Support |
P_{r}, kN |
M_{r}, kN.m |
||||
Hand |
Reference^{*} |
Hand |
Reference^{*} |
|||
Max compression |
4490 |
4490 |
4490 |
0 |
0 |
0 |
Allowable compression |
3592 |
3592 |
3592 |
--- |
--- |
--- |
f_{s} = 0.0 |
2945 |
2945 |
2945 |
229 |
229 |
229 |
f_{s} = 0.5 f_{y} |
2015 |
2015 |
2016 |
330 |
330 |
330 |
Balanced point |
1251 |
1253 |
1251 |
403 |
403 |
403 |
Pure bending |
0 |
0 |
0 |
285 |
285 |
285 |
Max tension |
1904 |
1904 |
1904 |
0 |
0 |
0 |
^{*} Reinforced Concrete Mechanic and Design, 1^{st} Canadian Edition, James MacGregor and Fred Bartlett – 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.
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 f_{s} = f_{y}).
Figure 10 – Strains, Forces, and Moment Arms (f_{s} = - f_{y} 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 M_{x} and M_{y} 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)