Study on Calculation of Bearing Capacity of Axially Loaded CFRP-Strengthened Cold-Formed Thin-Walled Lipped Channel Steel Columns

With the development of carbon fiber reinforced composites and the continuous improvement of the properties of bonding agents, scholars recommended using carbon fiber reinforced plastics (CFRP) to enhance cold-formed thin-walled C-shaped steel structures. It can provide a fast and effective way to strengthen and repair damaged steel structures. However, discussion on the bearing capacity calculation of cold-formed thin-walled C-section steel column strengthened by CFRP was limited. Also, the relevant influencing factors (the number of CFRP reinforcement layers), the orientation of CFRP (horizontal, vertical), and the location of CFRP reinforcement (web + flanges + lips, web + flanges, web, and flanges) were overlooked in calculating the bearing capacity of cold-formed thin-walled C-section steel column strengthened by CFRP. .en, the calculation result of the load capacity will be inaccurate. .is work, therefore, studied the effects of CFRP reinforcement layers, CFRP direction, and CFRP reinforcement position on the ultimate load of CFRP-strengthened cold-formed thin-walled C-section steel column. A threedimensional (3D) finite element model of cold-formed thin-walled steel strengthened by CFRP was established to discuss the bearing capacity under axial compression. Furthermore, a method for calculating the bearing capacity of the CFRP-strengthened cold-formed thin-walled C-section steel column was proposed based on the direct strength methods (DSM). .e results indicate that not only the slenderness ratio, section size, and length of members but also the number of CFRP reinforcement layers and orientation of CFRP have an impact on the calculation of bearing capacity. .e equation modified in this work has excellent accuracy and adaptability. Predicting the bearing capacity of reinforced members is necessary to give full play to the performance of CFRP accurately. .us, the methods proposed can provide a reference value for practical engineering.


Introduction
Currently, lightweight steel structures with cold-formed thin-walled steel frames become increasingly widely used in the field of structural engineering. Cold-formed thin-walled steel, wood, bamboo, and other materials can be combined into truss beams, columns, and other major bearing components. Also, it can be used for roof panels, floor slabs, and other minor components. Cold-formed thin-walled steel is a type of high-efficiency section steel, in which a thin steel plate is bent into various section shapes by rolling or stamping at room temperature. It can bear larger loads with less material by changing the section shape rather than increasing the section area [1]. While cold-formed thinwalled steel has been used efficiently, due to the improvement in the yield strength of the materials used, ultrathin plate and increasingly complex cross-section forms have led to problems with component stability. Generally, local buckling, distorted buckling, and overall buckling will occur on cold-formed open-section members. e interaction between two or three of buckling modes can also occur under certain conditions. In local buckling, the plate rotates around the intersection line of the plate, but the intersection line remains straight. In overall buckling, the whole cross-section rotates or moves laterally, but the cross-section shape does not change. In distorted buckling, the plate revolves around the intersection line of the plate, and some of the intersection lines of the plate no longer maintain straight lines. e cross-section shape and contour size also change. Under these circumstances, external strengthening using CFRP has been identified as a potential option to enhance the strength of existing columns, as this can provide a high strength-to-weight ratio and stiffness [2,3]. Furthermore, the durability and corrosion-resistant characteristics of CFRP will enhance the long-term structural performance of steel columns. e relevant research shows that the use of CFRP to strengthen the stability of steel structures is effective [4], and the use of CFRP to reinforce cold-formed steel lipped channel columns is also increasingly common.
Previous studies focused on the repair of naturally corroded I-girders. Girders were artificially notched in the lab to simulate damage, fatigue-damaged members. Also, the durability of the bond between steel and FRP materials can be used to investigate the occurrence of galvanic corrosion associated with combining steel and carbon fiber. Narmashiri et al. [5] conducted static load tests on 13 CFRPstrengthened beams and compared different reinforcement forms. It can be concluded that there were four failure modes in CFRP-strengthened beams, that is, transverse cracking of the carbon plate, interfacial peeling at the loading position, interlaminar failure of the carbon plate at the end, and interfacial peeling at the end. Yuan et al. [6] applied the basic theory of fracture mechanics to propose cohesive contact, cohesive element, and extended finite element computational models to simulate an experimental H-beam. e failure parameters of the model were determined by comparing the model results with the tests, and the cohesive contact model was used for numerical modeling and parameter analysis, and the cohesive contact model was used for numerical modeling and parameter analysis. Bambach et al. [7] investigated the effect of CFRP strengthening in short square hollow section (SHS) columns constructed of slender cross sections. ey achieved up to twice the axial compression capacity by a combination of transverse and longitudinal layers. Both transverse and longitudinal CFRP layers were used to reduce the membrane strains that develop in both directions with local buckling in the column. It can be found that slender section columns undergo larger buckling deformations, and, consequently, a greater increase in capacity can be achieved using CFRP. Imran et al. [8] developed a numerical simulation in ABAQUS in which CFRP and an adhesive were modeled using continuum and cohesive elements by employing Hashin and cohesive law criteria. ereby a new set of design equations based on the direct strength method (DSM) was proposed to determine the axial compression capacity of CFRP-strengthened SHS columns subject to local buckling. Fernando et al. [9] performed an experimental study aimed at clarifying the effects of adhesive properties on the failure mode and load-carrying capacity. Four different failure modes were observed in these tests, including adhesion failure, cohesion failure, combined adhesion and cohesion failure, and interlaminar failure of the CFRP plates. e tests also revealed that an adhesive with greater ultimate tensile strain leads to the higher load-carrying capacity of the strengthened rectangular hollow section (RHS) tube. Al-Saidy [10] focused on the behavior of steel composite beams that were intentionally damaged at their tension flange to simulate corrosion and then repaired with CFRP plates attached to the tension area side. A nonlinear analytical procedure was also developed to predict the behavior of the section/member in the elastic, inelastic, and ultimate states to facilitate the implementation of this strengthening technique. e test results showed a significant increase in the strength and stiffness of the repaired beams. us, the research on CFRP strengthening of steel structures has gradually matured. It also focused on three aspects, including enhancing axially compressed members (e.g., SHS, T, H, circular steel tube, and composite steel members) to improve the bearing capacity after strengthening, reinforcing damaged steel beams and nondestructive steel beams, and improving the fatigue resistance of steel members. e use of fiber-reinforced polymer (FRP) sheets and strips has been successfully demonstrated in retrofits of concrete structures. However, studies on using FRP for the retrofitting of steel structures were scarce. Most experiments and numerical simulations in the literature involved flexural members [4,[10][11][12][13][14][15]. Only a few considered compressive members [3,16,17]. Furthermore, research on particular section forms strengthened by CFRP, such as cold-formed thin-walled C-section steel, has rarely been performed. Also, there were few studies on calculating the bearing capacity of cold-formed thin-walled C-section steel strengthened with CFRP. e calculation accuracy of the proposed formulas was poor. Scholars compared the ultimate load of CFRPstrengthened cold-formed thin-walled steel under axial compression obtained with experimental, numerical, and direct strength methods. Results showed that the average values of the three ways were very similar. Nevertheless, the analysis revealed that the results of DSM were occasionally conservative, and the maximum error of the three methods was approximately 20%, which will lead to a failure to utilize the full performance of the material. ese experimental studies [3,7,16] have led to the development of design equations for predicting the axial compression capacity of CFRP-strengthened short steel tubular columns. Bambach et al. [7] developed a design approach in which the theoretical, critical elastic buckling stress of a CFRP-strengthened short SHS column was used to determine its axial compression capacity. e assumption of the model proposed was that the effects of both the transverse and longitudinal CFRP layers were the same in restricting the local buckling failure, contrary to the existing results that the effect of CFRP strengthening depended on the orientation of the CFRP layers [16,18]. erefore, this work aimed to improve upon the works of Kalavagunta et al. [19], thereby proposing a new set of design equations based on the DSM to predict the axial compression capacity of CFRP-strengthened cold-formed thinwalled C-section steel columns. Cold-formed thin-walled C-section steel is seldom used for axially compressed members. us, it is particularly important to investigate the changes in mechanical properties, the stability of the members, and the calculation formula for bearing capacity for this type of member under axially compressed conditions after the attachment of CFRP. Specifically, a 3D FE model was developed to verify the correctness of the numerical simulation and predict the ultimate bearing capacity of coldformed thin-walled C-section steel strengthened with CFRP based on the AISI-DSM. e number of CFRP reinforcement layers, CFRP orientation (horizontal, vertical, and oblique 45°), and CFRP reinforcement location (web + flanges + lips, web + flanges, web-only, and flanges only) were investigated in terms of previous research variables. Load-displacement curves were obtained, and the impact on the ultimate bearing capacity of the components was also analyzed. Besides, the calculation formulas for bearing capacity proposed in previous studies were modified and improved. e influence of the number of CFRP reinforcement layers and orientation of the CFRP were taken into account in the improved formulas to obtain more accurate bearing capacity results. Computed results and numerical simulation were compared in Section 4 to examine the validity and reliability of the proposed method. Overall, the investigation aims to provide engineers with a basic understanding of the performance of CFRP-reinforced cold-formed thin-walled steel. With better estimates of the utilization rate of materials, while ensuring the strength of structural components, the material can be saved, which can provide a reference for future engineering construction projects.

Numerical Modelling
A numerical investigation is performed using the ABAQUS CAE finite element (FE) software to simulate axial compression tests and the bearing capacity of CFRP-strengthened cold-formed steel lipped channel columns. Many factors, such as the element type, material properties, grid generation, boundary conditions, and solver setting, are considered in this work, to build an appropriate numerical model.

Initial Conditions.
e FE model is divided into three parts, including the cold-formed steel lipped channel columns, adhesive, and CFRP. Figure 1 shows the CFRPstrengthened cold-formed thin-walled short columns built by ABAQUS/CAE with a thickness of 0.1 mm. From the literature [20,21], the typical thickness of the adhesive is 0.1-0.3 mm. e adhesive is established in the cold-formed lipped channel steel and the first layer of CFRP during the modeling process. As interlayer debonding does not occur in CFRP-CFRP [8], bond failure can only be simulated between the cold-formed thin-walled lipped channel steel and the CFRP. e selection of appropriate element types plays an essential role in ensuring the accuracy of the simulation results. e cold-formed steel lipped channel columns are simulated using 4-node S4R shell elements with reduced integration and hourglass control. An 8-node 3D cohesive element (COH3D8) is used to model the adhesion layer to predict adhesive behavior and failure [20,[22][23][24]. e carbon fiber sheet is developed by 8-node quadrilateral inplane general-purpose continuum shell elements (SC8R) with reduced integration and hourglass control. Continuous shell elements can accurately predict the failure of composite materials and are thus more suitable for CFRP modeling [20,[22][23][24][25][26]. Material orientations are assigned to each CFRP layer by creating two different sections for longitudinal and transverse layers. Besides, the element deletion option is enabled for both cohesive and continuum elements with a maximum degradation of 95%. A mesh density of 10 mm is adopted for the short column, while a mesh density of 30 mm is used for the long column, and the interacting surfaces are connected by sharing the nodes of the base surface in the FE model.

Cold-Formed Steel.
e cold-formed steel lipped channel columns are built as an elastic-plastic material with strain hardening. e experimentally measured stress-strain curve along with a yield strength and elastic modulus of 550 MPa and 205 GPa, respectively, is used to develop the FE model [19]. e stress-strain curve of cold-formed steel is shown in Figure 2.

2.2.2.
Adhesive. MC-Dur 1280 epoxy resin (adhesive for CFRP strips and flat-bar steel for structural strengthening) is used to bond the carbon fabric to the cold-formed channel columns. e adhesive layer between the steel and the first CFRP layer is established by a coupled cohesive zone model based on traction separation law. e traction separation law considers traction and separation in three directions, including the normal direction and two shear directions. e normal direction and shear directions are denoted by t n , t s , and t t , respectively, while the corresponding separations are represented by δ n , δ s , and δ t , respectively. Consequently, the interface behavior of the adhesive from the initial reaction to failure can be expressed as follows: where k nn , k ss , and k tt are the elastic stiffness in the normal and shear directions, respectively: where E a is the elastic modulus and T 0 is the thickness of the adhesive [22]: where G a is the shear modulus of the adhesive. e stiffness of the adhesive should be large enough to provide reasonable Advances in Civil Engineering stiffness but not large enough to cause interface vibration [27]. k ss and k tt are determined by comparing the experimental results [28] (Table 1). e failure criterion used for damage is QUADS, expressed in (4). Damage initiation occurs when the damage function reaches a value of 1 based on this criterion: where σ max is the tensile strength and τ max is shear strength, which is determined from the following [22]: An energy-based linear softening approach is applied in ABAQUS. Benzeggagh-Kenane (BK) fracture energy-based mixed-mode law is used in this work [29]: where G n , G s , and G t represent the normal and shear directions, respectively. G I and G II are the corresponding maximum fracture energies, which cause failures in the normal and shear directions, respectively. e values of G I and G II are obtained from the literature [20]. e value of η is 1.55 [29]. Not all parameters used in the FE method are mentioned in [19]. e above data obtained from the material properties test is selected for the FE simulations, which is in agreement with the experimental data in [19]. e material properties of the adhesive layer are summarized in Table 1.

CFRP.
e CFRP composite is developed by a laminar-type elastic material. e damage of the composite is simulated using the Hashin failure criteria [30,31]. Owing to CFRP exhibits no visible plastic deformation in the initial stage of damage. e Hashin damage model can predict the destruction of these materials successfully.
e damage evolution begins once the damage criterion for a given failure mode is satisfied. e energy dissipation is equal to the critical fracture energy for a given failure mode. In the FE model, it is necessary to provide the fracture energy for the failure modes. e tensile strength of CFRP is 4830 MPa with a tensile modulus of 230 GPa, a weight of 300 g/m 2 , an ultimate elongation of 2%, and fiber thickness of 0.166 mm. e compressive strength of commonly used CFRP varies from 9% to 60% of the tensile strength [26,32]. Hence, the longitudinal compressive strength of the CFRP is assumed to be 20% of the tensile strength [8]. e transverse tensilecompressive and longitudinal-transverse shear strengths are supposed to be 10% of the tensile strength [25,33]. Poisson's ratio of the CFRP is 0.33 [34]. Finally, a viscosity coefficient of 0.0001 is used in the FE model [26]. e related material parameters of CFRP can be obtained from the material properties experiments, as summarized in Table 2. All the material properties are obtained from the technical data provided by the manufacturer.

Boundary Conditions and Analysis Procedure.
A hinge loads the thin-walled steel at both ends. UX, UY, and URZ are restrained at the loading end, while UX, UY, UZ, and URZ are restrained at the nonloading end. Loading is performed in the Z-direction. It is transformed into concentrated force loading. e axial compression load is incrementally applied to the top reference point. All the translational and rotational degrees of freedom are fixed at the top and bottom reference points ( Figure 3). An independent reference node is created at the center of the crosssection at each end of the column. All the peripheral nodes of the column are constrained to these reference nodes via beam-type multiple point constraints (MPC). e reference point is defined at the shear center of the cold-formed thinwalled steel. e shear center position is determined as specified in Chinese Standard (GB50018-2002)- approximate formula B.2.6 for inward edging channel steel [35]. e following equation is used to calculate the position of the shear center. e section diagram is shown in Figure 4: where Z 0 is the position of the shear center. We couple the reference nodes with cold-formed thinwalled steel sections. e adhesive layer is established between the cold-formed thin-walled steel and the first CFRP layer. Tie constraints are applied between the adhesive layer and the cold-formed thin-walled steel and between the adhesive layer and the CFRP.
Firstly, an elastic buckling analysis is conducted to determine the buckling load and buckling mode of the column. Next, the risk method is used to perform a nonlinear analysis to assess the load-displacement characteristics and the failure load of the column.
For cold-formed thin-walled channel steel, the initial defects include residual stress and geometric defects. To date, there have been few studies on the initial residual stress of cold-formed thin-walled channel steel. us, the influence of residual stress is neglected in this paper. For initial geometric imperfections, the overall initial bending and torsion deformation amplitude is L/1000 according to the technical code of cold-formed thin-walled steel structures (China Standard (GB50018-2002)) [35]. An eigenvalue buckling analysis is conducted to obtain the first-order buckling mode. en the buckling mode is multiplied by L/1000 to get the overall geometric defect of the specimen.

Results and Discussion
e results of CFRP-strengthened cold-formed steel-lipped channel section simulations are always affected by various factors. e ultimate bearing capacity calculated for three types of component sizes and CFRP reinforcement positions in cold-formed thin-walled steel (plain, full, and web) is first compared to experimental data from [19]. e FE models are also validated using the experimental results in [19]. Also, a parametric study is then conducted to investigate the effects of parameters such as the number of CFRP reinforcement layers, the orientation of the CFRP, and location of CFRP reinforcements.

Comparison of Experimental Data and Numerical
Simulation. A total of three components (C7510, C7512, and C10010) are simulated based on the details listed in Table 3. e section sizes of the three series of members are shown in Figure 5. A cold-formed steel-lipped channel section with a yield stress of 550 MPa and modulus of elasticity of 205 GPa is used for all steel sections. e crosssectional dimensions of specimens C7510, C7512, and C10010 with lengths of 500 mm, 600 mm, and 700 mm, respectively, are considered in the modeling. e basic dimensions and properties of the three different components are summarized in Table 3. Figure 6 shows the experimental model of cold-formed thin-walled steel bonded with CFRP from [19]. Also, it gives that the surface of the steel needs to be cleaned to remove dust and other contaminants. e carbon fiber sheets are fabricated following the standard size. e epoxy adhesive is smeared uniformly on the surface of the components and the CFRP sheets. e side of the component to adhere to the CFRP is exposed to the top and gradually pressed along the direction of the fiber. ree types of specimens are prepared for testing, including plain cold-formed steel-lipped channel sections, a cold-formed steel section with a CFRPstrengthened web only and, next, a cold-formed steel section with full CFRP strengthening ( Figure 6). Note that the basic bond strength of the epoxy resin should reach at least 15 d when CFRP is used to reinforce cold-formed thin-walled steel members.
ree CFRP reinforcement types are adopted for the members in Table 1 for the numerical simulation to verify the FE model, which is the same as those used in the experiments: unreinforced reinforced webs and adequately reinforced. A comparison between the numerical simulation results and the experimental data is shown in Figure 7.
From Figure 7, the comparison between the experimental data in reference and the simulated results obtained with the FE model exhibit a slight error of approximately 5%. e reason is that it is difficult to control the axial loading in the experiment. Also, the shear center of the cold-formed thin-walled steel is hard to determine accurately. us, a slight difference in the introduction of the defect coefficient may also result in the error in the final data. In general, the numerical simulation results are in good agreement with the experimental data from [19]. In this work, the cooperative deformation of the steel plate and CFRP is mainly considered, but the failure mode between CFRP and steel interfaces is not found. Consequently, it can be considered that the bond performance of adhesive is excellent. Besides, there is no separation between CFRP and steel members in the process of ultimate load research. erefore, it can be concluded that the ultimate compressive strength increases between 6% and 12% after strengthening the webs with  CFRP. e average increase is 10%. After strengthening the webs with CFRP, the ultimate compressive strength increases by 11% to 18%. From all cases, the ultimate bearing capacity of the member will increase its stiffness and resistance to varying degrees depending on the location of the CFRP reinforcement.

Numerical Model Parameter Analysis.
In the modeling, the component size parameters are selected as follows: b w � 125 mm, b f � 102 mm, b l � 14 mm, t � 1 mm (short column), andt � 1.5 mm (long column). e column CFRP strengthening can always increase the critical buckling load, but it does not affect the length range corresponding to local, distorted, and global buckling [34]. Accordingly, the component lengths are chosen as L � 600 mm (short column) and L � 2200 mm (long column) to analyze the influence of CFRP on the bearing capacity of members under failure modes of different lengths. For analyzing the effect of the fiber orientation on the load-carrying capacity of the components, the variable fiber orientation is set to 90°(transverse, T) and 0°( longitudinal, L); the fiber direction diagram is shown in Figure 8 Table 4.     Advances in Civil Engineering e rule proposed by the works of Hashin and Rotem is adopted to simplify the method for introducing initial defects [30]. For long columns, 1.98t is used as the defect coefficient, and 1.55t is used for short columns.
(2) From Figure 9, the ultimate load-bearing capacity of the member is not significantly increased compared with that of the bare column when the web position of the short column is strengthened by transverse fiber. e reason is that the size of the flanges of the member is longer and is thus prone to large deformations. After strengthening the web, deformation at the flanges can occur more easily, which leads to a nonobvious increase in the bearing capacity. Nonetheless, three types of reinforcement methods (S-WF-T, S-F-T, and S-WFL-T) provide clear improvement in the ultimate bearing capacity, leading to ultimate load increases of 8.1%, 7.2%, and 10.2%, respectively.
Accordingly, considering the range of increase in the ultimate load of the members, S-WF-T is a more efficient reinforcement technique for a CFRP layer on short columns. Nonetheless, S-WFL-T is the desperately advantageous reinforcement method if the bearing capacity of members needs to be maximally increased. From Figures 9(a) and 9(b), the reinforcement effect is better for short columns when the components are reinforced with transversely oriented fibers. Besides, the CFRP reinforcement delays the brittle phenomenon of a sudden decrease in the axial stiffness of the members. It can also be seen that, for short columns, the reinforcement effect is better when the components are reinforced with transversely oriented fibers. e CFRP reinforcement delays the brittle phenomenon of a sudden decrease in the axial stiffness of the members.  Figure 9(c) shows that the bearing capacity of the members strengthened through the web bonded with CFRP is similar to that of short column S-W-T. Hence, it is not recommended to enhance long columns with longitudinal fibers. (4) Figure 9(d) gives that the ultimate load increases by 3.6% when the long column is strengthened with transversely oriented fibers, and both the web and the flanges are strengthened. Reinforcing only the flanges or the whole member results in improvements in the ultimate load of 5.3% and 7%, respectively.
It can be seen that the L-WFL-T has significant enhancement effect for long columns considering the amount of increase in the ultimate bearing capacity. Yet the reinforcement effect for long columns strengthened with one layer of CFRP is not equal to that for short columns. As a consequence, multistory reinforcement can be considered to enhance the ultimate bearing capacity of the members when an actual project requires long members. In summary, the cold-formed thin-walled steel reinforced with a layer of CFRP can provide a better reinforcing effect, especially for short columns. A detailed description of the image fluctuation trend is presented in Table 5.

Influence of CFRP Reinforcement Layer Number on Ultimate Bearing
Capacity. Two layers of CFRP are considered to investigate the effect of the number of CFRP reinforcement layers on the ultimate load-bearing capacity. e fiber orientation of the two layers of CFRP is divided into three modes (2T, 2L, and 1T1L). In the 2T mode, the fiber orientation of the first and second layers of CFRP is 90°. In this study, the ultimate load of the three different fiber orientations column is compared with a single-ply column and a bare column. By comparing the data in Section 3.3, the influence of the orientation and position variables of the two-layer CFRP columns on the bearing capacity is roughly consistent with the law of single-layer CFRP columns. According to the analysis, the transverse fiber reinforcement method has a more significant impact on the bearing capacity performance if ignoring long columns and short columns. Hence, only the components that have a considerable effect on the bearing capacity are drawn in Figure 11. Tables 6 and 7 show a detailed description of the image fluctuation trend.
From Figure 11(a) and Tables 6 and 7, it is interesting to note that, compared to the bare column, the gain for the S-W-1T1L columns with two layers of CFRP is minimal. It demonstrates that strengthening only the web cannot provide sound reinforcement. It is not recommended to use it in practical projects. e other reinforcement methods, S-F-1T1L, S-WF-1T1L, S-WFL-1T1L, S-F-2L, S-WF-2L, and S-WFL-2L, show improvements of 10.2%, 10.7%, 17.4%, 6.4%, 7%, and 17%, respectively, over the bare column. S-F-2L, S-WF-2L, and S-WFL-2L increase by 2.5%, 2.58%, and 11.3%, compared to S-F-L, S-WF-L, and S-WFL-L, respectively. S-F-1T1L, S-WF-1T1L, and S-WFL-1T1L lead to   Advances in Civil Engineering ultimate load increases of 4.1%, 4%, and 0.43% compared to S-F-2L, S-WF-2L, and S-WFL-2L, respectively. e above comparison of the bearing capacities indicates that the web + flange + lips (WFL) reinforcement of two layers of CFRP (1T1L) can provide the most significant improvement in the bearing load, while S-WFL-2L reinforcement can also play a similar role; the percentage increase is similar to that of 1T1L reinforcement.
Furthermore, the results can reveal that, for the short column, placing the fibers at 90°and gluing both layers of CFRP to the web provide almost no improvement in the bearing capacity compared to the bare column, while reinforcement as S-F-2T, S-WF-2T, and S-WFL-2T provides increases of 12.8%, 13.2%, and 15%, respectively, compared to S-N for the short column. Compared to S-F-T, S-WF-T, and S-WFL-T reinforcements, S-F-2T, S-WF-2T, and S-WFL-2T provide increases of 6%, 5.5%, and 5%, respectively. e ultimate bearing capacity with S-WFL-1T1L is 2.9% higher than that of S-WFL-2T. As a result, for the short column, the most useful CFRP strengthening is obtained by gluing two layers (1T1L) to the whole column. e values presented in Figure 11(b) and Tables 6 and 7 for a fiber orientation of 0°show that, compared to L-N, the L-F-1T1L, L-WF-1T1L, L-WFL-1T1L, L-F-2L, and L-WFL-2L, reinforcements exhibited increases of 5.5%, 2.2%, 8.5%, 2%, and 4%, respectively. Additional strengthening of the lips (L-WFL-1T1L) renders this increase slightly larger: 8.5%. It is preferable to orient one fiber layer at 0°and the other at 90°to increase the ultimate load, which leads to gains of up to 7% compared to L-WFL-L. Gluing one CFRP layer longitudinally and the other transversely to the flanges, both the web and flanges, or around the whole column, provide increases in the ultimate load of 3.6%, 3%, and 4.7% compared to L-F-2L, L-WF-2L, and L-WFL-2L, respectively. erefore, there is no apparent difference between the effect of bonding one layer or two layers of CFRP on the web or the web and flange when the orientation of the fiber is 0°. Generally, L-WFL-1T1L provides the best reinforcement effect. Figure 11(b) and Tables 6 and 7 show that when the flanges or both web and flanges are strengthened with two CFRP layers oriented transversely, the ultimate load increases by 8.3% and 7.2%, respectively, compared to the bare column. Additional strengthening of the lips (L-WFL-2T) renders this increase slightly more substantial by 12.8%. Interestingly, the ultimate bearing capacity with L-WF-1T1L is lower than with L-WF-T. Compared to L-F-T, L-WF-T, and L-WFL-T, reinforcement with L-F-2T, L-WF-2T, and L-WFL-2T exhibited increases of 3.1%, 3.7%, and 6.7%, respectively. Compared to L-F-1T1L, L-WF-1T1L, and L-WFL-1T1L, the improvements with L-F-2T, L-WF-2T, and L-WFL-2T are 2.9%, 5.1%, and 4.8%, respectively. As a result, observation of (3) and (4) makes it possible to   conclude that attaching two CFRP layers to the whole column provides the most significant reinforcement for long columns.
After the above analysis, it can be concluded that the range of increase in the reinforcement capacity for long columns is not as obvious as that for short columns. Hence, in practical engineering, to increase the ultimate load significantly, multi-ply reinforcement should be adopted for long columns. In conclusion, the fiber orientation, fiber reinforcement location, and the number of reinforcement layers all have a significant effect on the ultimate bearing capacity of members.

Evaluation of Ultimate Bearing Capacity
From the discussion in the last section, different reinforcement positions, orientations, and the number of layers of CFRP influence the ultimate bearing capacity of members. Hence, the orientation of the CFRP and the number of layers of CFRP should be considered in the calculation of the bearing capacity of members. e prediction equation given in [19] is modified based on the AISI-DSM. Also, an appropriately modified modulus design equation is proposed to predict the bearing capacity of CFRP-strengthened coldformed steel lipped channel columns.

Previous Studies.
e assumptions adopted in the design of the bearing capacity of axially compressed cold-formed thin-walled steel members strengthened by CFRP in [19] are as follows: (1) ere is an adequate bond between the steel and CFRP.
(2) Slenderness of the composite plate is considered.
(3) CFRP yield stress is ignored for the composite section. (4) e composite area is considered: where t t is thickness of composite section; t cf is CFRP thickness; t s is steel plate thickness; E s is elastic modulus of steel; E cf is elastic modulus of CFRP; and E cfrp is elastic modulus of the CFRP with steel. From the work of Sreedhar Kalavagunta, the cross section thickness of the composite material is composed of the thickness of the CFRP and steel plate. e thickness of the adhesive layer is not considered, as in (8). e modulus of elasticity is calculated by considering the ratio of the modulus of CFRP to that of cold-formed thin-walled steel, as in (9). ey apply the modified modulus of elasticity and properties of composite materials to the AISI-DSM to predict the bearing capacity of cold-formed thin-walled steel strengthened by CFRP.
rough the above analysis, it can be concluded that the number of CFRP-strengthened layers and the orientation of CFRP will have a significant impact on the modulus, which further affects the bearing capacity of CFRP-strengthened members. Yet the influence of the number and direction of CFRP reinforcement layers is not taken into account in (8) and (9). erefore, they have been revised to obtain a new modified modulus calculation formula.

Modified Design Provisions.
e FE simulation results show that the fiber orientation (vertical, transverse) and the number of reinforcement layers affect the increase in bearing capacity of reinforced members.
us, it is necessary to introduce these factors into the equation for the bearing capacity. A proportional factor, ξ, is introduced to consider the effect of the fiber orientation on the bearing capacity. N T (number of transverse layers) and N L (number of  12 Advances in Civil Engineering longitudinal layers) are introduced to reflect the effect of the number of reinforcement layers on the bearing capacity: E � E s t s + E CE t cf t s + t cf . (13) Assuming that the thickness of all transverse and longitudinal CFRP layers is equal, the equivalent stiffness method is used to calculate the CFRP modulus of elasticity. A proportional coefficient is introduced to calculate the CFRP modulus of elasticity. e proportional factor (ξ) is used to transform the effect of transverse layers into the equivalent for longitudinal layers. e value of ξ depends on the type of CFRP used. In this study, we tried to adopt the proportional factor proposed by experiments of Bambach [7]. Still, this proportional factor does not apply to 1T and 1L CFRP configurations. At present, there have been few studies on this topic. e selection of appropriate scale factors for different types of CFRP urgently needs to be investigated. So ξ depends on the mechanical properties of the CFRP composite. e value is assigned 0.8 according to the detailed FE analysis results of this study [8]. e orientation of the CFRP fibers varies with the number of reinforcement layers when using multilayer CFRP to reinforce cold-formed thin-walled steel. Meanwhile, the equivalent elastic modulus (E CE ) derived from (11) is adopted, in which E 1c represents the modulus of elasticity of CFRP in the tensile direction, which can be measured through tensile tests of the material or calculated using (12). In (12), ] denotes the volume ratio of fibers, E C and E a represent the elastic modulus of the CFRP and the adhesive layer, respectively, and t T is the total thickness of the composite section in terms of the CFRP composite layer thickness (t c ).

Direct Strength Method (DSM) Equations.
e DSM uses member elastic buckling solutions based on gross crosssectional properties instead of the effective width to determine the member strength. e DSM is an alternative design method, allowing computation of the compression capacities of sections, provided the elastic buckling load (P cr ) and yield load (P y ) of the section are known. e DSM is included in AS/NZS 4600 as well as Appendix 1 of the AISI S100 Standard [36][37][38].
where λ c � ������ P y /P cre , in which P y � A g f y and A s and A c are the gross section areas of steel and the CFRP, respectively; A g � A s + A c (E 1c /E s ) is the gross section area, where E s denotes the elastic modulus of the steel bars. P cre is the minimum critical elastic column buckling load in flexural, torsional, or flexural-torsional buckling [39][40][41].
(3) Distorted buckling (P nd ) e nominal axial strength (resistance), P nd , for distorted buckling should be calculated as follows: (a) For λ d ≤ 0.561, (b) For λ d ≻ 0.561, e column ultimate load is as follows: Tables 8-11, respectively, list the ultimate load values obtained using the above design method along with the ultimate load P u , obtained by the numerical calculation. In addition, the tables also show the ratio of the estimated limit load P u,ap to the numerical simulation value P u . Finally, the safety and accuracy of the ultimate load are assessed by the proposed new formula. e short and long column ultimate loads obtained with the proposed new method are quite accurate (2% and 5% mean errors, respectively). Moreover, the values also exhibit a rather low scatter. From the previous tables, it can be concluded that the results calculated with the revised formula are in good agreement with those obtained with the numerical simulation, which verifies the correctness of the revised formula.

Conclusions
(1) For a short column member with a fiber orientation of 0°, the comparatively effective of CFRP strengthening is associated with gluing the CFRP longitudinally to the web, flanges, and lips (S-WFL-L). e S-WFL-T is the best reinforcement method for a CFRP layer on short columns when the member is strengthened with transversely oriented fibers. Compared with the increase of the bearing capacity of the two short column reinforcement methods, S-WFL-T is also the best reinforcement method for the short column. Besides, the reinforcement effect is better when the components are reinforced with transversely oriented fibers compared with longitudinally. (2) ere is no apparent difference between the effect of bonding one layer of CFRP on the web for the long column when the orientation of the fiber is 0°. e effect of CFRP-strengthening the whole column with gluing the one-layer CFRP transversely is more significant for long columns. e cold-formed thinwalled steel reinforced with a layer of CFRP can provide a better reinforcing effect, especially for short columns. (3) Compared to the bare column, the gain for the S-W-1T1L columns with two layers of CFRP is minimal,    which demonstrates that strengthening only the web cannot provide sound reinforcement. It is not recommended to use this reinforcement method in practical projects for a short column. e relatively efficacious way to improve the ultimate bearing capacity of short columns is to apply a full section reinforcement with one layer of transverse CFRP and one layer of horizontal CFRP. (4) ere is no apparent difference between the effect of bonding two layers of CFRP on the web or the web and flange for the long column when the orientation of the fiber is 0°. A more useful approach to improve the ultimate bearing capacity of long columns is to use a full section reinforcement with two layers of transverse CFRP. Still, the reinforcement effect is not visible compared with those short columns. For this reason, multiplies strengthening can be considered to enhance the ultimate bearing capacity of members when the members are long in practical engineering. (5) e short and long column ultimate loads obtained with the proposed new method are entirely accurate (2% and 5% mean errors, respectively). Moreover, the values also exhibit a rather low scatter. So, the modified formula can accurately estimate the ultimate bearing capacity of the cold-bending thinwalled C-section steel column reinforced by fiber.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.