Simplified Uniaxial Column Interaction Charts

This paper presents analytical method for generating the interaction diagrams for design of reinforced concrete (RC) columns. Due to the introduction of new classes in concrete compressive strength (fc ′ )with somewhat different parameters for the steel grades (fy), it has become necessary to develop new interaction diagrams. These proposed interaction diagrams take into consideration the different values of gamma (γ), concrete compressive strength (fc ′ ) and different steel reinforcement ratios (ρ). The interaction diagram of any desire level of gamma (γ) can be generated to find the required axial load capacity (Pc) and moment capacity (Mc) of the columns with different reinforcement ratios. This study also analyzed some numerical examples using the proposed interaction charts to find the values of Pc and Mc for the uniaxial columns and their results obtained are later compared with the computer software (SP-Column). The data obtained from Interaction charts showed a promising result as the values are quite close to the ones obtained from the computer software.

The strength of reinforced concrete columns is usually expressed using interaction diagrams [2] to relate the design axial load ∅ to the design bending moment ∅ . Figure 4 explains the control points for the column interaction curve (∅ − ∅ ). Each point on the curve represents one combination of design axial load ∅P n and design bending moment ∅ corresponding to a neutral-axis location. The interaction diagram is separated into a tension control region and a compression control region. The balanced condition occurs when the failure develops simultaneously in tension (i.e., steel yielding) and in compression (concrete crushing).  [3] In this study, the proposed expressions for generating the interaction diagram for RC column are discussed. These interaction diagrams will also take into consideration the different values of gamma ( ) , concrete compressive strength ( ′ ) and different steel reinforcement ratio (ρ). Numerical examples will also be analyzed using the interaction charts to find the values of Pc and Mc and their results will later be compared with the computer software (SP-Column) [9]. w w w . a j e r . o r g w w w . a j e r . o r g

II. INTERACTION CHARTS FORMULATION -ACI CODE DESIGN
The stress and strain distribution of a rectangular column section for the calculation of Pu and Mu is given in Figure 5, [10][11].  Figure 5, the moment about the midpoint of the section ( ) can be computed as; The and values for the plain concrete section are calculated as; w w w . a j e r . o r g w w w . a j e r . o r g Page 244

Tension Steel Section:
The Internal Tensile force Ts is computed as; = .
(5) where; = Area of tensile steel reinforcement = Computed steel stress in tensile steel The value of the internal moment is; = .
( − ′ ) (6) The and values for the tension steel section are calculated as; Substituting the value of 2 in the Eqn-8; where; fy = Yield stress of reinforcing steel ′ = Distance from extreme compression fiber to centroid of reinforcing steel

Compression Steel Section:
The Internal compressive force Cs is computed as; = .
′ ′ (10) where; ′ = Area of compression steel reinforcement ′ = Computed compressive stress in compression steel The value of the internal moment is; = .
′ ′ ( − ′ ) (11) The and values for the compression steel section are calculated as; Substituting the value of 3 in Eqn-13;

Construction of Interaction Chart:
The column axial load capacity Pc is summation of all internal forces = ∅ where; = − + (15) Therefore, where; = + + (18) Therefore Computing the values of and from the above equations (17) and (20).

Steps to find the Pc and Mc
The following steps need to be followed to compute the values of Pc and Mc for an economical design.
Step-1: Find the value of from the moments and the given cross-section. Step-2: Find the value of from the axial load and the given cross-section.
Step  The results obtained are also compared with the Finite Element software SP column and are shown in Table 2. The  columns are having different reinforcement ratios (ρ)with different values of gamma (γ). The input data for these columns are given in Table 3.

IV. RESULTS AND DISCUSSIONS
The results obtained from the Column interaction charts showed a safe and conservative column design strength when compared with the results obtained from the finite element software. The column (C5) with a higher reinforcement ratio ( =8%), also showed promising results with the difference of only 9 % with the finite element software. The bar charts in Figure 11 and 12 compares the values of Pc and Mc for the selected columns (C1 to C5) respectively.

A X I A L L O A D C A P A C I T Y ( P C )
Interaction Charts

V. CONCLUSION:
In the present work, an analytical model is derived for the hand computation of ( − ) interaction diagram of reinforced concrete column design. The charts with the different gamma values ( = 0.6, 0.7, 0.8 and 0.9), having different reinforcement ratios were formulated. Moreover, the charts for any desired value of gamma such as 6.5, 7.2 etc; can also be generated using the steps mentioned in this study to find the required values of Pc and Mc respectively. This study derives the interaction charts having the ′ = 30 MPa and fy = 415 MPa but is not limited to these parameters. The charts can also be updated based on the required concrete compressive strength ( ′ ) and steel yield stress ( ).
Several numerical examples of RC column design were analyzed by using the developed column interaction charts having different gamma values (γ) with different symmetrical reinforcement layout that have different reinforcement ratios ( ). The analytical results obtained from these interaction charts are compared with the finite element software (SP-Column). The results obtained are in close agreement with the finite element method. The average variation of analytically computed values to the finite element software was not more than 10% which shows relatively satisfactory results. Therefore, the developed interaction charts can help in finding the required Pc and Mc for the preliminary design of reinforced uniaxial concrete columns with symmetrical reinforcement layout.

M O M E N T C A P A C I T Y ( M C )
Interaction Charts