Design charts for channel shaped reinforced concrete short columns subjected to axial compressive load and uniaxial bending

Eight design charts are presented for reinforced concrete short C-columns subjected to axial compressive load plus uniaxial bending. For design these charts can be used for determining t he re quired c olumn dimensions a nd amount o f steel, wh ile fo r a nalysis these charts can be used for estimating the loaded column capacity. Four examples are given to e xplain the use of design charts for both d esign and analysis, two of which are design examples while the other two are analysis. It has been shown by these examples that t he n ew proposed c harts a re very simple to use in structural applications.


Introduction
Arbitrarily shaped reinforced concrete members subjected to uniaxial or biaxial bending with axial compression are frequently used in multistory tall buildings and bridge piers.
In the last decades some methods have been presented for the ultimate strength analysis of various concrete sections, such as L-, T-and channel-shaped, under symmetrical bending or combined biaxial bending and axial compression (1)(2)(3)(4)(5) . These methods compute the ultimate flexural capacity of section. For design purposes they require trial and error procedures.
The present research also aims at obtaining direct relationships between the compression load and the uniaxial bending capacities which can be used as ready design charts for short C-columns.

Research significance
This search deals with reinforced concrete C-shaped cross sections commonly are used as columns and enclosures of the elevator shafts.
The principal aims of this work is to present a method for analysing tied short columns under the combined action of axial compressive load and uniaxial bending that is simple in concept and can be beneficially used in providing easy way to deal with charts for the design of such columns.

Description of the procedure
For columns subjected to uniaxial bending, the neutral axis (N.A) always remains parallel to the axis about which the moment is being applied. Since the position of the neutral axis depends on the value of the eccentricity (e), therefore the variation of the neutral axis position may in general leads to the two possible cases of compression zone shown in figure (1).

Estimating concrete compression force
Depending on an equivalent rectangular compression block for concrete, defined by ACI-318M-05 (6) the compressive force of the concrete is C c =area of compression zone * Based on the chosen value of ultimate usable strain at extreme concrete compression fibre (ε cu ) which is equal to 0.003 based on the ACI-318M-05 Code (6) and the linear strain distribution across the depth of the cross section; (figure 2 ), a correlation between the strain (ε si ) in any arbitrary reinforcing bar and the depth of the compression zone (c) can be obtained. Let ( d si ) denotes the distance from the extreme compression fibre to the centroid of any arbitrary reinforcing bar.
Referring to figure (2), the strain in any steel bar can therefore be obtained since steel can be idealized as elasticperfectly plastic material with maximum value of stress ( f y ), therefore the stress in any steel bar is simply where f si = stress in the reinforcement of the layer i, E s = modulus of elasticity of the steel reinforcement, and f y = specified yield strength of the reinforcement.

Equilibrium criteria
For a given eccentricity (e), the value of the compressive load (P) can be estimated from the following simple equilibrium equation The associated uniaxial bending moment (M) can also be estimated by summing up the moment of the resulting forces on the cross section around the plastic centroid (PC) of the section, M = C c * its lever arm to the PC + ∑ C si * its lever arm to the PC + ∑ T si * its lever arm to the PC … (5) where C si and T si represent the compressive and tensile force in the i th reinforcing bar respectively, figure (2). The subscript (i) refers to the reinforcing steel layer position.

Program description
The computer program is developed in Microsoft Quick-Basic Version 4.5. It is capable of producing points that describe the axial load versus moment interaction diagram for any short C-column under uniaxial bending.
Input data for program include: the material and section properties, and the area and coordinates of each longitudinal bar. The output of the program consists of a series of data points ( P and M values ) that could be used in drawing the interaction diagram for the column.
The program assumed a linear variation of strain over the depth of the section. Strain hardening of steel, tensile strength of concrete, and slenderness effects are ignored. In addition, the Design charts for channel shaped reinforced concrete short columns subjected to axial compressive load and uniaxial bending output does not include the capacity reduction factor (φ ).

Flowchart
In order to simplify the analysis procedure of columns, flowchart is presented which demonstrate the steps that are followed for the analysis of channel short columns subjected to axial compression load plus uniaxial bending. The flowchart is used as step by step guides for manual computations.

Design charts
In order to make direct use of the present method in the design of reinforced concrete C-columns subjected to axial load and uniaxial bending moment, charts are constructed in a manner analogous to those given in text books for the case of rectangular or circular columns with uniaxial bending. Charts 1 through 8 have been prepared for the case of uniaxial bending of Ccolumns. These charts are designed to cover a wide range of the cross sectional parameters. Figure (3) shows a cross section of a typical reinforced concrete C-columns. For convenience, these design charts are presented in this study which cover the following cases

Conclusions
The analysis and design of reinforced concrete C-sections subjected to axial compression and uniaxial bending are tedious and time consuming because 1. In the analysis, a trial and error procedure is required to find the depth of the neutral axis satisfying the equilibrium conditions. 2. In the design process, a trial and error procedure is required to find the steel ratio ( t ρ ) satisfying the strength requirements. While the simplicity of the present approach enabled the construction of new design charts can be used directly in design.