Mathematical Model for Analysis of Uniaxial and Biaxial Reinforced Concrete Columns

,is paper presents amathematical model for the analysis of reinforced concrete (RC) uniaxial and biaxial columns.,is proposed model is a quick and faster approach for the analysis and design of reinforced concrete rectangular columns without going through the interaction charts procedure as well as other iterative methods for the computation of required axial load capacity (Pc) and moment capacity (Mc). A simplified flow chart has also been developed to find the required column capacity using this mathematical model. Eight uniaxial columns (C-1 to C-8) and seven biaxial columns (CB-1 to CB-7) are analysed in this study. Each column is analysed having different steel reinforcement ratios (ρ) with different loading conditions. In addition, the studied columns are subjected to both tension and compression failures.,e detailed examples for both uniaxial and biaxial columns (one for each case) are also presented in this study. ,e studied columns are also analysed using computer software spColumn. ,e average variation of the mathematically computed values to the finite element software is not more than 10%, showing promising computational results.


Introduction
Columns are the vertical compression members, which transmit loads from the upper floors to the lower levels and to the soil through the foundations [1]. Based on the position of the load on the cross section, columns are classified as concentrically loaded (Figure 1) or eccentrically loaded columns ( Figure 2). Eccentrically loaded columns are subjected to moments, in addition to axial force. e moments can be converted to a load P and eccentricities e X and e Y . e moments can be uniaxial, as in the case when two adjacent panels are not similarly loaded, such as columns A and B in Figure 3 [2]. A column is considered as biaxially loaded when the bending occurs about the x-and y-axis, such as in the case of corner column C in Figure 3. In a recent study [3], Al-Ansari and Afzal also presented an analytical model for generating interaction diagram charts for biaxial columns. e strength of reinforced concrete columns is normally expressed using interaction diagrams to relate the design axial load 2∅P n to the design bending moment ∅M n [4,5]. Each control point on the column interaction curve (∅P n − ∅M n ) represents one combination of design axial load, ∅P n and design bending moment, ∅M n , corresponding to a neutral-axis location ( Figure 4) [6].
Extensive studies have been carried out on the interaction diagrams (uniaxial and biaxial columns) of reinforced concrete (RC) rectangular columns [6][7][8][9][10][11][12]. Several studies have also been performed on providing numerical approaches for the analysis and design of reinforced concrete columns. Furlong et al. [13] provided an overview of the analysis and design of reinforced concrete columns subjected to biaxial bending. ey reviewed several methods of analysis that use traditional design methods and compared their results with the obtained data from physical tests of normal strength concrete columns subjected to short-term axial loads and biaxial bending's. ey concluded that the elliptic load contour equation [14] and the reciprocal equation [15] are the simplest to use, as they do not require complicated calculations.
Chen et al. [16] proposed an iterative numerical method for rapid section analysis and design of short concrete composite columns subjected to biaxial bending. Wang and Hsu [17] proposed the numerical method approach for the determination of load-moment curvature relationship for short and slender columns.
is numerical method approach is also applicable for columns, made of different materials, and shows good agreement with the different experimental results obtained in their study.
Whitney [18] and Hsu et al. [19] provided major research studies on numerical method approaches. Whitney suggested an approximate equation to estimate the nominal compressive strength of columns subjected to compression failure. Hsu in different research projects [10,17,20,21] also presented the results of experimental and analytical studies on the strength and deformation of biaxially loaded short and tied columns with L− , channel, and T-shaped cross sections. In another study, Hsu [22] suggested a general   Advances in Civil Engineering equation for the analysis and design of reinforced concrete short and tied rectangular columns. is study proposes a mathematical model to analyse and design the uniaxial and biaxial columns based on ACI building code of design [23]. is model is a quick and easy approach for analysing and designing the reinforced rectangular columns without going through the interaction charts for the computation of the required axial load capacity (Pc) and moment capacity (Mcx, Mcy). A simplified flow chart has also been developed to find the required column capacity using the proposed model approach.
e previous research studies of the mathematical model approach are limited to columns having compression failure only.
is study includes the numerical examples of columns using the proposed mathematical model approach for both compression and tension failure cases.
is relatively new approach will also be useful to the undergraduate and graduate students as well as researchers to calculate the required column capacities using this approach in their research-related activities.
Numerical examples for the selected reinforced concrete columns (uniaxial and biaxial columns) are also illustrated to check the adequacy of this proposed model. Eight uniaxial columns (C-1 to C-8) and seven biaxial columns (CB-1 to CB-7) are analysed in this study. ese columns are analysed having different steel reinforcement ratios (ρ), different values of steel yield strength (fy), concrete compressive strength (f c ′ ) , and different load capacity conditions. Moreover, the results obtained from this proposed model are compared with computer software spColumn 2016 [24].

Mathematical Model Formulation: ACI
Code Design e stress and strain distribution of a rectangular column section (uniaxial column) for the calculation of Pn and Mn is given in Figure 5. e resultant force P N is equal to the summation of all internal forces: Balanced failure and compression controlled limit Compression ϕP n , ϕM n Axial load,

Advances in Civil Engineering
Similarly, the resultant moment M N is equal to the summation of all internal moments: (2) e following steps revealed the calculation of the required internal forces and internal moments for a rectangular uniaxial RC column.

Plain Concrete Section.
e Internal concrete compressive force (C conc ) is computed as where C conc � internal concrete compression force, f c ′ � compressive concrete strength, b � column width, a � depth of the compression stress block, β � 0.85 − 0.008(f c ′ − 30) ≥ 0.65, and c � distance from extreme compression fiber to the neutral axis.
Referring to Figure 5, the moment about the midpoint of the section (M conc ) can be computed as where h � column total depth, d ″ � ((h/2) − d ′ ), d � column effective depth (h − d ′ ), and d ′ � distance from extreme compression fiber to centroid of top reinforcing steel.

Tension Steel Section. e internal tensile force T s is computed as
where A s � area of tensile steel reinforcement and f y � yield stress of reinforcing steel. e internal moment M T is

Compression Steel
Section. e internal compressive force C s is computed as [25] where A s ′ � area of compression steel reinforcement and

Mathematical Model Analysis
e following steps should be revealed to calculate the design axial load and moment capacity of the required rectangular RC column section. Columns may be subjected to tension failure or compression failure depends on the balanced eccentricity value (e b ):

Advances in Civil Engineering
where 3.1. Tension Failure Analysis. Tension failure will occur when the balanced eccentricity value (e b ) is less than load eccentricity (e). Substituting the values of C c , C s , and T s in equation (1) and solving for (a) will be a second-degree equation [14]: (1)), Substitute the value of a in equation (3) to calculate C c and from equations (5) and (7) to compute T s and C s values.
ese obtained values are substituted in equation (1) for e column axial load capacity and moment capacity can therefore be computed as (where ∅ is the column reduction factor having the value of 0.65).

Compression Failure Analysis.
Compression failure will occur when the balanced eccentricity value (e b ) is bigger than the load eccentricity (e). Substituting the values of C c , C s , and T s in equation (1) and solving for (a) will be a cubic equation [14]: Once the values of A, B, C, and D are calculated, the value of a can be determined by the trial method or directly by using MATLAB or any scientific calculator. Moreover, the cubic equation can also be solved using different numerical methods, for example, Newton Raphson Method. After getting the required value of (a), similar equations from (14) to (18) (as mentioned in the Tension Failure Analysis) should be used to get the required value of column axial load capacity (P c ) and moment capacity (M c ).
e following flow chart ( Figure 6) can be followed to find the required capacity of the rectangular uniaxial column section.

Numerical Examples for Uniaxial Columns
Eight reinforced rectangular columns (C-1 to C-8) having different column sizes are analysed using the numerical method approach. ese columns are having different reinforcement ratios (ρ) in addition to different failure types, both tension and compression failures. e design input load data for these columns are illustrated in Table 1.
e above eight columns C1 to C8 are analysed using the mathematical model approach to find the required values of axial load capacity, Pc, and moment capacity, Mc. Moreover, these values are also compared with the computer software spColumn. e results obtained are depicted in Table 2.
ese above columns are also analysed with different available methods, Whitney's 1 st approximation method [18], Whitney's second approximation method [18], and the method provided by HSU [19]. ese available methods are only available for the columns having the compression failure. ere are no examples available for the columns with the tension failure cases. e results comparison is mentioned in Table 3.

Advances in Civil Engineering
(10) Finding the value of (a) using the Quadratic Equation

Numerical Examples for Biaxial Columns
Seven reinforced biaxial rectangular columns (CB-1 to CB-7) having different column sizes are also analysed using the proposed model. ese columns are also having different reinforcement ratios (ρ) in addition to different failure types, that is, tension-tension, compression-compression, and tension-compression failures. e design input load data for these columns are illustrated in Table 4. e column cross section subjected to biaxial bending is shown in Figure 8. A similar flow chart has to be adopted (as discussed in the uniaxial column sections), once for the case of eccentricity in the x-direction (ex) and later for the eccentricity in the y-direction (ey) to obtain the required values of load capacities in x-and y-direction (∅P x , ∅P y ).
ese values are later used in Bresler's formula [15] (equation (19)) to find the value of Pc. Moreover, the Mcx and Mcy values can be found by using equations (21) and (22) accordingly: where ∅P N max � maximum permissible column load, Ast � total area of steel, and Ag � (Gross area of cross section) − (sectional area of concrete member member).
e moments in the x-and y-direction can be found as M cy � P C × e ux . (22) e above seven columns (CB-1 to CB-7) are analysed with mathematical model approach to find the required values of axial load capacity Pc, using reciprocal formula. Moreover, the values of Pc are also compared with the computer software spColumn. e results obtained are depicted in Table 5.

Validation of the Mathematical Model
In order to validate the proposed mathematical model approach, the model is validated with the existing experimental results of columns subjected to uniaxial and biaxial loadings. e experimental results data has been extracted from the test results provided by HSU [22]. Two uniaxial columns as provided by Bresler (B-1 and B-2) and one biaxial column as provided by Anderson and Lee (SC-4) are selected from the research article [22] to compare the results with the mathematical model. Table 6 illustrates the experimental testing data provided by HSU. e data and the results are provided in imperial units (Kips-ft) units. erefore, they are converted to metric units accordingly to compare the values with our results. Table 7 provides the experimental test results as well as the validation of the test data with the proposed mathematical model. e column capacity (Pc) results obtained from the experimental data are quite close to the mathematical model results, showing satisfactory computational results.

Results and Discussions
e results obtained from the mathematical model approach for both uniaxial and biaxial columns showed a safe and conservative column design method. e results of eight uniaxial column sections (C-1 to C-8) using the proposed model are also compared with different available mathematical models, provided by Whitney's 1 st approximation method, Whitney's second approximation method, and the method provided by HSU. Columns C1 to C-4 were subjected to tension failure, whereas columns C-5 to C-8 were the compression failure cases. e other three mathematical studies (Whitney's 1 st approximation, Whitney's second    ese studied columns (C-1 to C-8) are also analysed using the computer software spColumn and the comparison results for axial load capacities (Pc) and moment capacities (Mc) are displayed in bar charts (Figures 11 and 12).
For the biaxial columns (CB-1 to CB-7), the axial load capacity results for mathematical model approach using Bresler's formula and the computer software spColumn are displayed in the bar chart ( Figure 13). e values of Pc obtained using the mathematical model are quite close to the computer software results, showing relatively satisfactory computational results.

Conclusion
In this study, the mathematical model is presented to analyse and design the uniaxial and biaxial columns without going through the column interaction charts to find the required axial load capacities and moment capacities. A simplified flow chart has also been developed to solve the required column section following the mathematical model steps.
Eight (RC) uniaxial columns (C-1 to C-8) and seven (RC) biaxial columns (CB-1 to CB-7) are analysed in this study. ese columns are analysed having different steel reinforcement ratios (ρ), different values of steel yield strength (fy), concrete compressive strength (f c ′ ), and different load capacity conditions. Moreover, the studied columns are subjected to both tension and compression failures.
For the uniaxial columns, the proposed mathematical model results are also compared with the different available numerical approaches done by Whitney's 1 st approximation, Whitney's 2 nd approximation, and the method provided by HSU. All of these three methods were formulated based on the case of compression failure only. ese uniaxial columns are also analysed using the computer software spColumn.
e results obtained showed that this proposed mathematical approach showed good agreement with the computer software spColumn showing relatively satisfactory results. e studied biaxial columns are subjected to different failure conditions, that is, tension-tension failure, compression-compression failure, and tension-compression failure. Bresler's formula was used to find the required capacity (Pc) after finding the (Px)and(Py) from the mathematical model approach.
e biaxial columns were also analysed with the computer software. e average variation of the mathematically computed values for biaxial columns to the finite element software was not more than 10%. Moreover, the results obtained for the columns subjected to tension failure are quite close with the computer software spColumn. Moreover, this mathematical model has also been validated with the existing experimental results conducted by HSU.
In short, this newly proposed mathematical model is a good and quick approach to analyse the reinforced concrete uniaxial and biaxial columns. is model can also help the students and the academic researchers to find the column capacities without going through the column interaction charts and other long iterative approaches.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.