Abstract

This paper presents a mathematical model for the analysis of reinforced concrete (RC) uniaxial and biaxial columns. This 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 with different loading conditions. In addition, the studied columns are subjected to both tension and compression failures. The detailed examples for both uniaxial and biaxial columns (one for each case) are also presented in this study. The studied columns are also analysed using computer software spColumn. The average variation of the mathematically computed values to the finite element software is not more than 10%, showing promising computational results.

1. 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. The moments can be converted to a load and eccentricities . The 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.

Figure 1 

Concentrically loaded columns.

(a)

(b)

(a)
(b)

Figure 2 

Eccentrically loaded column.

Figure 3 

Uniaxially and biaxially loaded column.

The strength of reinforced concrete columns is normally expressed using interaction diagrams to relate the design axial load to the design bending moment [4, 5]. Each control point on the column interaction curve represents one combination of design axial load, and design bending moment, , corresponding to a neutral-axis location (Figure 4) [6].

Figure 4 

Control points for column interaction curve .

Extensive studies have been carried out on the interaction diagrams (uniaxial and biaxial columns) of reinforced concrete (RC) rectangular columns [6–12]. 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. They 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. They 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. This 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 , channel, and T-shaped cross sections. In another study, Hsu [22] suggested a general equation for the analysis and design of reinforced concrete short and tied rectangular columns.

This study proposes a mathematical model to analyse and design the uniaxial and biaxial columns based on ACI building code of design [23]. This 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 . A simplified flow chart has also been developed to find the required column capacity using the proposed model approach. The previous research studies of the mathematical model approach are limited to columns having compression failure only. This study includes the numerical examples of columns using the proposed mathematical model approach for both compression and tension failure cases. This 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. These columns are analysed having different steel reinforcement ratios , different values of steel yield strength , concrete compressive strength , and different load capacity conditions. Moreover, the results obtained from this proposed model are compared with computer software spColumn 2016 [24].

2. Mathematical Model Formulation: ACI Code Design

The stress and strain distribution of a rectangular column section (uniaxial column) for the calculation of Pn and Mn is given in Figure 5.

Figure 5 

Calculation of Pn and Mn for a given strain distribution. (a) Section. (b) Strain. (c) Stress. (d) Internal forces. (e) Resultant forces.

The resultant force is equal to the summation of all internal forces:

Similarly, the resultant moment is equal to the summation of all internal moments:

The following steps revealed the calculation of the required internal forces and internal moments for a rectangular uniaxial RC column.

2.1. Plain Concrete Section

The Internal concrete compressive force is computed aswhere  = internal concrete compression force,  = compressive concrete strength,  = column width,  = depth of the compression stress block, , and  = distance from extreme compression fiber to the neutral axis.

Referring to Figure 5, the moment about the midpoint of the section can be computed aswhere  = column total depth, ,  = column effective depth , and  = distance from extreme compression fiber to centroid of top reinforcing steel.

2.2. Tension Steel Section

The internal tensile force is computed aswhere  = area of tensile steel reinforcement and  = yield stress of reinforcing steel.The internal moment is

2.3. Compression Steel Section

The internal compressive force is computed as [25]where  = area of compression steel reinforcement and (if the compression steel yields).

The internal moment is

3. Mathematical Model Analysis

The 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 :where , , , and (if the compression steel yields, then ).

3.1. Tension Failure Analysis

Tension failure will occur when the balanced eccentricity value is less than load eccentricity . Substituting the values of , and in equation (1) and solving for will be a second-degree equation [14]:where , , , and , .

Solve for to get

Substitute the value of in equation (3) to calculate and from equations (5) and (7) to compute and values. These obtained values are substituted in equation (1) for ,

The column axial load capacity and moment capacity can therefore be computed as(where is the column reduction factor having the value of 0.65).

3.2. Compression Failure Analysis

Compression failure will occur when the balanced eccentricity value is bigger than the load eccentricity (e). Substituting the values of , and in equation (1) and solving for (a) will be a cubic equation [14]:where , , , and ,where .

Once the values of are calculated, the value of 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 , 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 and moment capacity .

The following flow chart (Figure 6) can be followed to find the required capacity of the rectangular uniaxial column section.

Figure 6 

Flow chart of the mathematical model for the rectangular uniaxial rectangular column.

4. 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. These columns are having different reinforcement ratios in addition to different failure types, both tension and compression failures. The design input load data for these columns are illustrated in Table 1.

The 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. The results obtained are depicted in Table 2.

These 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]. These available methods are only available for the columns having the compression failure. There are no examples available for the columns with the tension failure cases. The results comparison is mentioned in Table 3.

Figure 7 

Column C-4.

4.1. Detailed Numerical Example for Column C-4 (400 × 400)

 Input Data: Figure 7  = 400 kN  = 100 kN·m  = 30 MPa  = 415 MPa  = 1000 mm²  = 1000 mm²  = 80 mm  = 0.65 Solution:(1)Finding the value of  = 250 mm(2) = 189.16 mm(3) = 160.788 mm(4) = 1.64 × 10³ kN(5) = 3.895 × 10² kN(6) = 4.15 × 10² kN(7) = 1.615 × 10³ kN(8) kN·m(9) (TENSION Failure)(10)Finding the value of (a) using the Quadratic Equation , where; ( = 370 mm)(11)Check if the tension steel has yielded;, steel yields, (12) kN·m(13) = 650.35 kN(14) = 162.56 kN·m

5. 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. These columns are also having different reinforcement ratios in addition to different failure types, that is, tension-tension, compression-compression, and tension-compression failures. The design input load data for these columns are illustrated in Table 4. The 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 and later for the eccentricity in the y-direction to obtain the required values of load capacities in x- and y-direction . These values are later used in Bresler’s formula [15] (equation (19)) to find the value of . Moreover, the Mcx and Mcy values can be found by using equations (21) and (22) accordingly:where  = maximum permissible column load,  = total area of steel, and .

Figure 8 

Biaxial column cross section.

The moments in the x- and y-direction can be found as

The 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. The results obtained are depicted in Table 5.

Figure 9 

Column CB-7 (X-X axis).

Figure 10 

Column CB-7(Y-Y axis).

5.1. Detailed Numerical Example for Column CB-7 (400 × 1200)

 Input Data: Figure 9 Pu = 1500 kN Mux = 300 kN·m Muy = 300 kN·m = 20 MPa fy = 300 MPa As = 3080 mm² = 3080 mm² = 60 mm  = 0.65 Solution: (Solving for the X-direction)(1)Finding the value of  = 200 mm(2) = 760 mm(3) = 706.8 mm(4) = 4.81 × 10³ kN(5) = 8.71 × 10² kN(6) = 9.24 × 10² kN(7) = 4,754 kN(8) kN·m(9) (COMPRESSION Failure)(10)Finding the value of (a) using the Cubic Equation , where; ( = 740 mm)(11)Check if the tension steel has yielded; mm , steel yields, (12) kN·m(13) = 4594 kN(14) = 918.79 kN·m (Solving for the Y-direction) (Figure 10)(1)Finding the value of  = 200 mm(2) = 226.67 mm(3) = 210.8 mm(4) = 4.3 × 103 kN(5) = 8.71 × 102 kN(6) = 9.24 × 102 kN(7) = 4.248 × 103 kN(8)   kN·m(9) (TENSION Failure)(10)Finding the value of (a) using the Quadratic Equation   , where; ( = 340 mm)   (11)Check if the tension steel has yielded;  mm  , steel yields, (12)    kN·m(13) = 2091 kN(14) = 418.12 kN·m

Finding the value of Pc using Bresler’s equation (19):where , .