Reliability assessment of reinforced concrete rectangular columns subjected to biaxial bending using the load contour method
Introduction
Various external loads applied to columns generally induce biaxial bending. Recent reliability-based design specifications and standards [1], [2], [3], [4], [5], [6], [7] provide design criteria for the design of reinforced concrete (RC) columns subjected to axial force and biaxial bending. The load contour method and the reciprocal load method proposed by Bresler [8] are widely employed to define the strengths of RC columns subjected to biaxial bending. The reciprocal load method describes the relation between the ultimate axial strength of an RC column and the eccentricity of axial force. The load contour defines the failure surface of an RC column subjected to biaxial bending with a family of curves using a surface exponent and P-M interaction diagrams for uniaxial bending (PMIDU) with respect to two principal axes of a cross-section.
The reciprocal load method has a limitation in terms of general applicability because it is unable to estimate the strength of an RC column subjected to biaxial bending induced by lateral load independent of axial force. Moreover, it is generally known that the reciprocal load method may yield erroneous results for columns subjected to strong bending [9], [10], [11], [12], and that the load contour method is applicable to a wider range of load effects than the reciprocal load method. This is why some design specifications [4], [5], [6], [7] adopt only the load contour method while other specifications [1], [2], [3] define the strength of an RC column by both methods depending on the magnitude of axial force.
Some succeeding studies on the load contour method have been reported [13], [14], [15], [16], [17]. Pannell [13] and Parme et al. [14] proposed approximated values of the surface exponent for the load contour method by graphical representations. Hsu [15], [16] formulated a simplified version of the load contour method with a fixed surface exponent. Bonet et al. [17] presented analytical expressions of the surface exponent in terms of a reinforcement ratio. Although such studies have been reported, the load contour method proposed by Bresler [8] is commonly adopted in recent reliability-based design codes [1], [2], [3], [4], [5], [6], [7] to define the strength of RC columns under biaxial bending.
Various approaches for assessing the reliability of RC columns subjected to uniaxial bending are available [18], [19], [20], [21]. However, the reliability levels of RC columns subjected to biaxial bending has been rarely reported except for one conference paper [22] that showed a simplified approach based on the reciprocal load method. The reliability levels of RC columns need to be evaluated accurately to determine a proper resistance factor and a target reliability index for various limit states used in reliability-based code calibration.
This paper applies the advanced first-order second-moment reliability method (AFOSM) [23] to the estimation of the reliability levels of RC columns subjected to biaxial bending for equivalent static design, in which dynamic load effect can be idealized by equivalent static load effect. The failure surface defined in the load contour method acts as the limit state function for the reliability analysis. The load parameters, the eccentricities of axial forces, the geometric and material properties of RC columns are selected as random variables. The Hasofer-Lind-Rackwitz-Fiessler (HL-RF) algorithm with the gradient projection method [24] is utilized to solve the minimization problem for the AFOSM.
Although the strain hardening of rebars and the traverse reinforcement increase the strength of an RC column, the statistical characteristics of their effects on the strength of an RC column are not available at the present time, and thus those two effects are not considered in this study. Consequently, the proposed scheme underestimates the strength of an RC column, and yields the lower bound of the reliability index. Since the microscopic behaviors of an RC column depending on ingredients in the concrete cannot be taken into account explicitly due to the macroscopic approach adopted in this study, it is assumed that they are approximately represented through the compressive strength and stress-strain relationship of concrete in a macroscopic sense.
Since the failure surface of an RC column subjected to biaxial bending is constructed with the PMIDU with respect to each principal axis of a cross-section, reliability analyses of biaxially loaded RC columns heavily rely on those for uniaxial bending. Kim et al. [21] recently proposed a general approach for constructing the PMIDU using the cubic spline interpolation for the reliability analysis. They also presented detailed expressions for the sensitivity calculations that are required in the HL-RF algorithm. This study adopts their approach to form the PMIDU and to evaluate the sensitivities of the failure surface to the random variables.
The reliability indices are calculated for a rectangular section and a square section as numerical examples. The surface exponent of each section is determined using the values presented by Bresler [8]. In the first example on the rectangular section, the load effects induced by a dead, live and lateral load are considered for different angles of attack of the lateral load to the column ranging from 0 to 360°. The effect of the surface exponent and the angle of attack on the results of the reliability analysis is discussed in detail. The uncertainty of the eccentricity of axial force in each principal axis is taken into account in the second example on the square section. The results indicate that the proposed approach yields stable and reasonable solutions for both examples.
Section snippets
Failure surface for biaxial bending
The load contour method proposed by Bresler [8] is adopted to evaluate the failure surface of an RC column subjected to combined axial and biaxial bending in a 3-dimensional axial force-biaxial moment space (P, , ). The failure surface defined by the load contour method is schematically illustrated in Fig. 1. The principal axes of a cross-section are denoted as the y and z axis, and and are bending moments about the axes specified in the subscripts, respectively. The load contour for
AFOSM and sensitivity
The strength parameters of an RC column, the load parameters and eccentricities of the axial force are considered to be random variables. All random variables are assumed to be statistically independent of each other in this study. For the compactness of the forthcoming derivations, the random variables are written in one vector, . In the case where all random variables are normally distributed and statistically independent of each other, the reliability index, β, and the
Applications and verification
The reliability indices are evaluated for two examples by the approach presented in the previous sections. In the first example, a rectangular section subjected to a lateral load is analyzed for three different surface exponents. The second example is adopted to investigate the effect of the eccentricity of axial force on the reliability assessment for a square cross-section. The reference values of the surface exponent for the examples are evaluated by extrapolating or interpolating the values
Summary and conclusions
The reliability indices and MPFPs of RC columns under biaxial bending conditions are calculated by combining the load contour method and the AFOSM. The strength parameters, the load parameters and the eccentricities of axial force are included in the random variables. The failure surface defined by the load contour method is utilized as the limit state function, and constructed by the PMIDU for an RC column with respect to each principal axis. The HL-RF algorithm with the gradient projection
Acknowledgements
This research was supported by a grant (17SCIP-B119964-02-000000) from the Ministry of Land, Transport and Maritime Affairs of the Korean government through the Korea Bridge Design & Engineering Research Center of Seoul National University.
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