The overall view and the schematic view of our-point bending test arrangements are shown in Figs. (1 and 2). The beams were simply supported. The hinge support was simulated by roller support and slide support. The roller support was simulated by round bar which allows in-plane rotation and the slide support was simulated by slide block under the round bar which allows translation movement along the specimen. Two lipped channels were tested at the same time that bolted to T-shaped steel connective blocks. These steel connectors were located at the loading points and end supports. The sectional views at the loading points and end supports of the T-shaped steel blocks are detailed in Fig. (3). The four-point bending tests were loaded symmetrically at two points to the T-shaped steel blocks through a spread beam. A 100kN hydraulic jack testing machine was used to apply a downward force to the spread beam, as shown in Fig. (3). Round bar were also used at the two loading points. In this testing arrangement, pure in-plane bending of the specimens can be obtained between the two loading points without the presence of shear and axial forces, keep the two test members work together, and minimize the web crippling at the ends and load points. Near the 90% ultimate load, the static load was paused for half minute and loaded again. This allowed the stress relaxation with the occurrence of the plastics. A data acquisition system was used to record the displacement transducers and load readings during the tests. Displacement control was adopted to drive the hydraulic actuator a constant rate of 0.3 mm/min for the four-point bending tests. Three displacement transducers were positioned at the tension flange near the flange-web junctions of the two loading points and the mid-span point to measure the vertical deflection and curvature of the beams. One transducer was positioned near the compressive flange of web at mid-span to monitor the possible local buckling or distortional buckling.

Fig. (2). Sketch view of test set-up and specimens.

Fig. (3). View of the special steel connector.

Tension tests were conducted following the provisions of Metallic materials--Tensile testing--Part 1: Method of test at room temperature (GB/T228.1-2010) [13]. Six tensile coupons were cut from two ends of a test member including the web flat, the compression flange, and the tension flange. A 200kN capacity testing machine was used for applying load. The mean values of six coupon test results are summarized. The specimen yield stress, f_y, is 295MPa, and the steel elastic modulus, E, is assumed as 2.074x10⁵Mpa, and the specimens elongation is 32%.

Fig. (4). Specimen measurement nomenclature.

The dimensions and nomenclature for each specimen is presented in Fig. (4), and the measured value was recorded at 1/4 and 1/2 points in pure bending part, so there are three measurement locations for each specimen. The measured mean values for specimens are summarized in Table 1. The test specimens were labeled with characteristic information such that each beam can be easily identified. For example, the label “B600H160-1a”, where “B” means pure bending member, “600” indicates the pure bending length of specimen, “H160” refers that the nominal width of web is 160mm, If a test was repeated, then a symbol of “1” was added which means different repeated group. Finally, “a” indicates one lipped channel of a pair of channels tested at the same time. The inside bend radius of specimens is 2t, where t is the base thickness of members. The elastic theoretical local and distortional buckling moments (M_crl and M_crd) determined using CUFSM [14]. The yield moments (M_y), the plastic moments (M_p), and the nominal effective length (L) which is the distance between the loaded points for each specimen are reported in Table 1. The theoretical elastic buckling moments including local and/or distortional buckling of specimens are approximate to three times the yield moments which can ensure the occurrence of the elastic local or distortional buckling.

The general finite element non-linear analysis program ABAQUS was used to simulate the inelastic experimental behavior of the lipped channel bending members.

The back-to-back lipped channel sections were used in a 4-point bending test. Because of the special steel connector in the ends and the load points of specimens, the rotations of top flanges were prevented. So an idealized finite element model was selected to analysis the buckling failure mode and ultimate load carrying capacities of specimens. The analysis model is the constant moment region of the specimens, which can be considered as a pure bending test. At the two ends of the member, the warping keeps constraint and the sections keep from uniform rotation.

The material model adopts directly the results of the tensile tests. The ideal elastic-plastic curve is used based on the experimental steel elastic modular and yield stress for simplify finite element analysis. Residual stresses, residual strains and cold-work effects were not considered.

The ABAQUS S9R5 shell element was used to model bending members. The element aspect ratio was kept below 3:1 for lips, flange, web, and corners along the length direction. In the cross-section, 6, 18, 2, and 2 elements was used for the flange, web, lip, and corner, respectively. The ABAQUS solution control employed was the modified Riks method and the arc length method.

Imperfection sensitivity was considered in the finite element analysis. The value of the geometric imperfections adopted in analysis was L/750 according to Chinese cold-formed steel specification. The shape of the geometric imperfection was obtained from the first buckling mode shape of a finite element Eigen buckling analysis in ABAQUS. The finite element model for specimen is depicted in Fig. (10).

Fig. (10). Finite element model.

The deformed shapes for the typical specimens obtained from the FEM analysis are presented in Figs. (5c and 6b), which is agree to the test failure mode depicted in Figs. (5a, b and 6a). This observation suggests that the idealized model can well model the buckling mode of the bending members.

The ultimate bending capacities (M_ab) obtained from the FEM are compared with the test ultimate capacities (M_t) as shown in Table 2. Meanwhile, the bending capacities of every specimen calculated using Chinese cold-formed steel specification are shown in Table 2, Where M_CI and M_C are the bending capacity predicted using Chinese cold-formed steel specification with considering the interaction of the elements and without considering the interaction of the elements, M_ey is the yield strength of bending members considering effective width. The mean value of the ratio of experimental-to-FEM ultimate bending capacities is 1.04 and the coefficient of variation is 0.0347 for all bending specimens. The comparison of ultimate capacity demonstrates that the ultimate bending capacity obtained from the FEM is close to the experimental ultimate capacity and the idealized model can also analyze the ultimate bending capacity well.

To provide a stress distribution of the bending member under the different loading stages including the yield moment, peak moment, and peak displacement, the moment–rotation response of cold-formed steel beam analyzed by Abaqus is depicted in Fig. (11). The bending moment applied to the specimen B900H140-2b is 4.96 kN.m, 5.55 kN.m and 3.8kN.m for each of the three cases, respectively. Fig. (12) shows the Misses stress and stress along longitudinal direction for the specimen B900H140-2b in three different stages.

Fig. (11). Moment-rotation curve of the specimen B900H140-2b analyzed by Abaqus.

Fig. (12). Moment-rotation curve of the specimen B900H140-2b analyzed by Abaqus.

At the yield moment (Fig. 12a, b), the Misses stress and longitudinal stress of the flange reach the yield stress while the stress of the web is less than the yield stress. So the moment of member attains the yield moment. At the peak moment (Fig. 12c, d), the Misses stress and longitudinal stress of the flange reach the yield stress and the stress of the partially compressive web reaches the yield stress. This phenomenon shows that the inelastic reserve of the partially compressive web is worked and the ultimate moment is higher than the yield moment. Finally at the peak displacement (Fig. 12e, f), the Misses stress and longitudinal stress of the flange and the partially compressive web reach the yield stress and the maximum stress in the web is concentrate at the a small portion of member, which shows that the plastic hinge of beam occurs and the moment of member decreases increasingly.

The strength and stability capacity should be calculated in Chinese cold-formed steel specification for bending members.

The strength of the bending member, M, is

(8)

Where W_en is the moment of the effective net section of the bending member.

The stability capacity of the bending member, M_e, is

(9)

Where W_e is the moment of the effective section of the bending member φ_x, and is the global stability coefficient of the bending member.

The relationship between the global stability coefficient (φ_x) and the global relative slenderness ratio is depicted in Fig. (13). The maximum value of the global stability coefficient equals 1 which means that the bending capacity of members should be less than or equal the yield strength (M_y) of the bending members.

Fig. (13). Comparison between the proposed design curve and the test values.

The effective width of elements can be calculated using following expression when the effective moment of section (W_e) need to be determined.

(10)

where b is the width of the element; t are the thickness of the element; b_e is the effective width of the element; b_c is the compressive width of the element; α is a coefficient, and α=1.15-0.15ψ; ψ is an uneven coefficient of the compression stress distribution (for pure bending members, ψ=-1); ρ is a calculating coefficient, ; k is the buckling stability coefficient of the element; k₁ is the interaction buckling coefficient of elements.

There are two existing methods to consider the inelastic reserve capacity in North American cold-formed steel specification (draft), which includes the Element-based method and DSM. The Element-based method is aligned with the EWM and only using the compression strain factor (C_y) varies from 1 to 3 to take into account the plasticity of compressive flange and partly compressive web. The compression strain factor can be predicted by using the following equation:

(11)

Where and w is the width of flange. The Element-based method can be also used to calculate the interaction of local buckling and yielding and global buckling.

The nominal strength considering inelastic flexural reserve capacity for DSM can be calculated as follows for global, local, and distortional buckling, respectively.

The expression of the inelastic global buckling capacity, M_nℓ, is:

For M_cre > 2.78M_y

(12)

Where M_cre is the critical elastic lateral-torsional buckling moment.

The inelastic local buckling capacity, M_nℓ is:

For λ_l ≤ 0.776

(13)

Where , M_ne is the global buckling capacity defined as expression (12) and M_crl is the critical elastic local buckling moment.

The inelastic distortional buckling capacity, M_nd, is:

For λ_d ≤ 0.673

(14)

Where M_crd is the critical elastic distortional buckling moment.

The inelastic bending capacity for the bending member is the minimum value calculated using expression (12), (13), and (14).

The bending capacities of every specimen calculated using Chinese cold-formed steel specification, Element-based method, and DSM in North America cold-formed steel specification (draft) considering inelastic reserve capacity are provided in Tables 2 and 3, respectively, including ratios of test-to-calculated capacities using various design methods for each specimen, Where M_NDSM is the ultimate capacity predicted using direct strength method (the minimum value of local buckling and distortional buckling capacity), and M_NEWM is the bending capacity calculated using Element-based method. The mean value of ratios of test results to calculated bending capacities M_CI and M_C obtained using Chinese specification is 1.1251 and 1.1188, respectively. The comparison results indicate that the current Chinese cold-formed steel specification is too conservative to predict the inelastic bending capacity of bending members.

The mean value of ratios of test results to calculated bending capacities using Element-based method considering inelastic reserve capacity in North America specification (draft) is 1.0334 and the coefficient of variation is 0.0443 for bending members. Meanwhile, the mean value of ratios of test results to calculated bending capacities using DSM considering inelastic reserve capacity in North America specification (draft) is 1.0885 and the coefficient of variation is 0.0367 for bending members. The comparison demonstrates that Element-based method and DSM in North America cold-formed steel specification (draft) well takes into account the increase on capacity because of inelastic reserve strength of the bending member. The DSM is more conservative and accurate than Element-based method.

In international cold-formed steel specifications, the compression strain factor (C_y) is usually used to consider the increase of moment because of inelastic reserve capacity, where the compression strain factor is the ratio of the ultimate strain to the yield strain and can be determined using expression (11). The partial plasticity shall be considered for the web of bending members in order to utilize the inelastic reserve capacity. Expression (11) means that the plastic height of web equals one third of the height of overall web when the flange is full effective. So the plastic height of web proposed for Chinese cold-formed steel specification can be determined using expression (15) based on expression (11) and EWM of Chinese cold-formed steel specification when the bending strength are calculated using formula (8).

(15)

The plastic strength of the lip of bending member is negligible in order to simplify calculation when the inelastic strength is calculated.

As shown in expression (16), the increasing of the global stability coefficient can increase the stability bending capacity. So the suggested method for stability capacity considering the inelastic reserve capacity can be put forward via increasing the global stability coefficient in inelastic stage. The global stability coefficient is increased at 1.15 and 1.05 when the global relative slenderness ratio equals 0.25 and 0.7, respectively. The revised curve for the global stability coefficient is depicted in Fig. (13).

The proposed global stability coefficient, φ_x, is

(16)

Where φ_bx is the elastic global stability coefficient.

Fig. (13) compares the current Chinese design curve and the proposed design curve considering inelastic capacity based on EWM with the experimental results, in which effective section yield moment was used to determine the ratio of test-to-predicted results. Meanwhile, the test results performed by Maduliat, Bambach, and Zhao [11] are illustrated in the same figure. The comparison demonstrates that the proposed design method to predict the ultimate bending capacity is better than the existing Chinese method and less conservative.

The bending capacities of each specimen calculated using proposed method are provided in Table 3. M_CIP and M_CP are the bending capacity calculated using proposed method with considering the interaction of the elements and without considering the interaction of the elements, respectively. The mean value of ratios of test results to predicted results M_CIP and M_CP using the proposed method is 1.0412 and 1.0354 and the coefficient of variation is 0.0293 and 0.0298, respectively.

Meanwhile, the mean value of ratios of test results conducted by Maduliat, Bambach, and Zhao [11] to predicted results M_CIP and M_CP is 1.057 and 1.0699 and the coefficient of variation is 0.0742 and 0.0728, respectively. While the mean value of ratios of test results [11] to calculated results M_CI and M_C is 1.1568 and 1.1709 and the coefficient of variation is 0.0818 and 0.0804, respectively. The comparison indicates that the proposed method is less conservative. Additionally, the proposed method is slightly less conservative than DSM considering inelastic reserve capacity in draft of North America cold-formed steel specification.

Reliability analysis was performed in order to evaluate the appropriateness of the proposed effective width method to calculate the inelastic buckling load-carrying capacities for the cold-formed steel channel sections subjected to bending. The second-order moment method is used to estimate the design reliability of the bending member in this study. When the calculated reliability index (β) is larger than the target reliability index(β₀), the proposed design method is regarded to be probabilistically safe. The target reliability index for cold-formed steel structural members is 3.2 according to the Chinese cold-formed steel specification.

The resistance of structural members is predicted by the following equation:

(17)

Where R_K is the characteristics value of the resistance of structural members; K_M is the uncertainty of the material strength of structural members; K_F is the uncertainty of the geometric characteristics of structural members; K_P is the uncertainty of the calculation method.

Statistical parameters of the uncertainties of different kinds of external load in China can be obtained according to the existing research reference [15].

The load combinations of 1.2DL+1.4LL+0.84W as specified in the Chinese load specification were adopted in the calculation, where DL is the dead load, LL is the live load, and W is the wind load. The resistant partial coefficient for steel Q235 is taken as 1.087.The live load to dead load ratio used in calculation is 0.25, 0.5, 1, and 2. Based on the given resistance partial coefficient and calculation, the reliability indexes of the bending members are 4.1611, 4.2075, 4.1615, and 4.0207 for different load ratio and the mean value for reliability indexes is 4.1377 which is larger than the target reliability index(3.2). It is shown that the modified formulae of effective width method are accurate and reliable for bending members. Hence, proposed effective width method formulae (Eqs. (15)–(16)) are recommended for the design of cold-formed steel channel sections.