International Journal of Polymer Science

Volume 2018, Article ID 9832894, 11 pages

https://doi.org/10.1155/2018/9832894

## Analysis and Design of Short FRP-Confined Concrete-Encased Arbitrarily Shaped Steel Columns under Biaxial Loading

^{1}School of Water Conservancy and Environment, Zhengzhou University, Zhengzhou, Henan Province, China^{2}Zhengzhou Metro Co. Ltd., Zhengzhou, Henan Province, China^{3}School of Civil and Transportation Engineering, Guangdong University of Technology, Guangzhou, Guangdong Province, China^{4}Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong^{5}College of Petroleum Engineering, China University of Petroleum, Beijing, China

Correspondence should be addressed to Bing Fu; nc.ude.tudg@gnibufec

Received 25 June 2018; Revised 26 September 2018; Accepted 30 September 2018; Published 25 November 2018

Academic Editor: Mehdi Salami-Kalajahi

Copyright © 2018 Wei He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The FRP-confined concrete-encased steel column is a new form of hybrid column, which integrates advantages of all the constituent materials. Its structural performance, including load carrying capacity, ductility, and corrosion resistance, has been demonstrated to be excellent by limited experimental investigation. Currently, no systematic procedure, particularly for that with reinforced structural steel of arbitrary shapes, has been proposed for the sectional analysis and design for such novel hybrid columns under biaxial loading. The present paper aims at filling this research gap by proposing an approach for the rapid section analysis and providing rationale basis for FRP-confined concrete-encased arbitrarily shaped steel columns. A robust iterative scheme has been used with a traditional so-called fiber element method. The presented numerical examples demonstrated the validity and accuracy of the proposed approach.

#### 1. Introduction

Lateral confinement leads to the compressed concrete under multiaxial compression and results in enhancements in both ductility and strength of the compressed concrete [1]. Such a feature is highly preferable for design and constructing columns in a region of high seismic risk, where adequate ductility of columns is necessary to ensure high moment redistribution capacity of structures and avoid collapse of structures due to the shaking from large earthquakes [2]. In conventional reinforced concrete structures, the lateral confinement to the compressed concrete is mainly provided by the transverse steel reinforcement in the form of either spirals or hoops. Concrete-filled steel tubular columns, which have been widely used in high-rise buildings, bridges, etc., are also utilizing the increased strength and deformability of confined concrete to achieve a high structural performance [3, 4]. In such a composite column, an outer steel tube is used to replace longitudinal and transverse steel reinforcements in conventional reinforced concrete columns and to provide continuous confinement to concrete infilled. However, such a concrete-filled steel tubular column, especially in a harsh environment, is susceptible to severe corrosion problem due to the direct exposure of the outer steel tube to ambient environment.

Fiber reinforced polymer (FRP) composites in the form of wraps or jackets have been widely used to serve as a confining device for seismic retrofit of existing RC columns [5, 6]. Its wide application is mainly attributed to the superior properties of FRP composites, such as high strength-to-weight ratio and excellent corrosion resistance. Recently, combining FRP composites with traditional construction materials (e.g., steel and concrete) has gained increasing research attention to form a hybrid column to achieve high structural performance by integrating advantages of the constituent materials. A successful example is the hybrid FRP-concrete double-skin tubular columns (DSTCs), which was proposed by Teng et al. [7] and then received a great deal of follow-on research (e.g., [8–13]). A DSTC consists of an FRP outer tube, a steel inner tube, and a layer of concrete sandwiched between them [7]. The FRP tube offers mechanical resistance primarily in the hoop direction to confine the concrete and to enhance the shear resistance of the member; the steel tube provides the main longitudinal reinforcement and prevents the concrete from inward spalling and the steel tube from outward local buckling deformations. Optimal use of FRP, steel, and concrete in the manner of DSTC therefore makes it an economical and corrosion-resistant column form.

Another successful example that integrates advantages of all the constituent materials into a hybrid column is the FRP-confined concrete-encased steel composite columns (FCSCs), which was first proposed by Liu et al. [14] for retrofit of existing steel columns. In their study, five notched steel columns to simulate the corroded section were encased using FRP-confined concrete to enhance its load carrying capacity, and tests results demonstrated the feasibility of such a retrofit technique. Karimi and his coauthors [4, 15, 16] then introduced the FCSCs for the new construction of a column and conducted a systematic experimental study on the compressive behavior of both short and slender FCSCs. Recently, Yu et al. [17] presented a combined experimental and theoretical study on the behavior of FCSCs under concentric and eccentric compression and revealed that two flanges of H-shaped steel could provide additional confinement to infilled concrete thus enhancing the ductility and load carrying of the composite columns. In order to further enhance the confinement efficiency for FRP-confined concrete-encased steel composite square columns and inspired by Yu et al. [17], Huang and his coauthors [18] innovatively proposed a new form of FCSCs, in which a cross-shaped steel section was used to replace H-shaped (or I-shaped) steel. Their test results demonstrated that despite concrete infill in square section, it was effectively confined by both outer tube and cross-shaped steel and justified the rationales of the proposed new form of composite columns. In some cases, such as corner columns of buildings, irregular cross section or regular cross section placed asymmetrically is often encountered [19]. Despite the fact that the structural performance of FCSCs has been demonstrated by a number of concentric of eccentric compression tests, no approach has been proposed for a rapid section analysis and design of FCSCs with arbitrarily shaped steel and under biaxial loading.

Against the above background, the present paper proposes an approach for the rapid section analysis and provides rationale basis for FRP-confined concrete-encased arbitrarily shaped steel columns. The robust iterative scheme proposed by Chen et al. [19] has been adopted with a traditional so-called fiber element method, where all the constitutive materials were treated as fiber elements avoiding the integration for the stress block of the FRP-confined concrete. The presented numerical examples demonstrated the validity and accuracy of the proposed approach.

#### 2. Methodology

A proper approach is necessary for robust convergence of the exact location of the neutral axis. The iterative procedure proposed by Chen et al. [19] has been adopted herein, in which the iterative quasi-Newton procedure is employed within the Regula-Falsi numerical scheme for the solution of equilibrium equations. In addition, the use of the plastic centroidal axes of the cross section as the reference axes of loading guarantees the convergence of solution in the iterative process. All the constituent materials, including FRP-confined concrete under the combination of axial compression and bending, are treated as fiber elements to avoid the definition of stress block for the FRP-confined concrete, which is difficult to analytically determine. Appropriate constitutive models were used to simulate the mechanical properties of the constituent materials. For instance, strain gradient along the section has been taken into account in the constitutive model of the confined concrete, which is simulated by using a so-called variable confinement stress-strain model proposed in the Chinese code (GB50608-2010).

##### 2.1. Basic Assumptions

The sectional analyses and design in the present paper were conducted based on the following basic assumptions: (1)Section plane remains plane after loading; this assumption ensures that the strain at any point of the cross section is proportional to its distance from the neutral axis(2)Failure limit state is only defined by the attainment of the strain of the extreme compression fiber; this means no failure mode of steel/steel bar rupture has been considered(3)Tensile strength of confined concrete is neglected(4)No contribution of FRP tubes to compressive strength of composite columns has been taken directly into account in the section analysis

##### 2.2. Definition of the Reference-Loading Axes

If the usual definition of reference-loading axes (i.e., geometric centroid as its origin) is used, it is possible that the origin of the loading may fall outside the interaction curve, especially when the axial load is close to the axial load capacity of columns with irregular structural steel under concentric compression [19]. To overcome such divergence difficulty, Chen et al. [19] accepted the plastic centroid of the cross section of columns as the origin of the reference-loading axes. Such definition of the origin of the reference-loading axes ensures , which is the inclination of resultant bending moment resistance and is equal to arctan and increases monotonically from 0 to 2*π* with increase of , orientation of neutral axis from 0 to 2*π*. In this way, the existence and uniqueness of and convergence are guaranteed.

For any arbitrary cross section, the definition of the plastic centroid (i.e., the origin of loading-reference axes in this study) was given in (1) and (2): in which , , and are areas of concrete, shaped steel, and steel rebars, respectively; , , and are the safety of confined concrete, shaped steel, and steel rebars, respectively; and , , and are the respective characteristic strength of confined concrete, shaped steel, and steel rebars and set to be the ultimate strength of confined concrete, yielding strength of shaped steel and steel rebars, respectively.

##### 2.3. Fiber Element Method Adopted

In the present paper, all the structural components in the composite columns, including the confined concrete, structural steel, and steel bars, have been treated as fiber elements to calculate the stress resultants. This approach avoids the difficulty of the determination of the stress block for confined concrete under bending with/without axial compression. Cross section of any shape has been first meshed to determine the stress resultants of each component. The total resultants of each component can be obtained by a summation over all the fiber elements given by (3), (4), and (5): where , , and are the stress resultants of axial compression, moment over -axis, and moment over -axis, respectively; , , and are the numbers of concrete fibers, structural steel fibers, and reinforcing bar fibers, respectively; , , and are the stresses of each concrete fiber, structural steel fiber, and reinforcing bar fiber, respectively; and , , and are areas of respective concrete fiber, structural steel fiber, and reinforcing bar fiber, respectively.

##### 2.4. Modelling of Confined Concrete

The behavior of concrete confined by FRP jacketing has been extensively investigated, and a number of strength models have been developed (e.g., [5, 20–22]). All the strength models, which give the stress-strain relationship explicitly, can be used in the proposed approach; however, a simple and generally accurate model is more preferable for such a design purpose. A stress-strain model of confined concrete in the Chinese code (GB50608-2010), which reflects the effect of strain gradient along the section, has therefore been adopted in the present section analysis. In this model, the slope of the second portion of the stress-strain curve (i.e., ) is defined as a function of the load eccentricity and equal to be that of the confined model in Lam and Teng [5] for the concrete under the concentric compression (i.e., a zero load eccentricity) and to be zero for the concrete under pure bending (i.e., an infinite load eccentricity).

As shown in Figure 1, the strain-stress relationship of confined subjected to pure bending is formulated by (6) and (7), while that under eccentric compression is given in (8) and (9).
in which and are the axial stress and axial strain, respectively; is the design ultimate compressive strain of the concrete subjected to pure bending; is the elastic modulus of unconfined concrete; and is the design compressive strength of unconfined concrete.
in which is the design ultimate compressive strain of the concrete in sections subjected to eccentric compression and is the slope of the second linear portion of the stress-strain curve. Related parameters above are determined by using the following equations:
in which is the slope of the second portion of the stress-strain curves for the FRP-confined concrete under axial compression; is the ultimate strength of the confined concrete under concentric compression; *d* is the diameter of the concrete core; is the load eccentricity; and is the design ultimate compressive strain of the concrete subjected to pure bending. Such a parameter should be determined from tests and is defined as the smaller value of the design ultimate strains obtained from the axial compression tests on concrete-filled FRP tubes and hollow FRP tubes, respectively. Due to lack of test data, the present manuscript assumes be equal to the corresponding ultimate strain of FRP-confined concrete under concentric compression. Such an assumption cannot lead to a significant loss in accuracy and will not adversely affect the main purpose of the present manuscript, which is to validate the proposed approach for the rapid section analysis of FRP-confined concrete-encased arbitrarily shaped steel columns. , the ultimate strain of FRP-confined concrete under concentric compression, can be determined following [5] as follows:
in which and are the ultimate strains of the confined concrete under concentric compression and unconfined concrete, respectively; is the ultimate hoop strain of the FRP reinforcement; and is the actual maximum confining pressure given in the following:
where is elastic modulus of the FRP composite.