# In-Plane Interactive Buckling of High Strength Steel Welded Wide Web I-Section Beam-Columns

Authors: Pei Xin Liang; Bin Huang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

A numerical investigation on the overall-local interactive buckling performance of high strength steel (HSS) wide web I-section members was conducted using finite element (FE) models verified against experimental data. The effect of overall slenderness, the height-thickness ratio of the webhw/tw, and the width-thickness ratio of the flanges (bf/tf) on ultimate load was parametrically analyzed. The result demonstrated that (i) for the beam-columns with higher steel grade, the Pz-Uz curve presents a relatively obvious deformation plateau, meaning a certain degree of ductility. (ii) The ultimate load decreased essentially linearly resulting from higher hw/tw, and the trend was graphically consistent under various steel grades. (iii) bf/tf lower than 0.6 led to the failure mode of shearing buckling, and the corresponding ultimate load increased with an increase of bf/tf. In the case of bf/tf greater than 0.6, the ultimate load decreased as a result of the higher bf/tf. Based on the concept of the direct strength method, the local postbuckling strength prediction formula was fitted with stub beam-column FE results and examined with experiment data. The correlation formula of compression and bending moment was proposed to evaluate the resistance of the interactive buckling of the HSS wide web I-section underpinned by the FE results of beam-columns with intermediate length and experiment data.


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## Body

## 1. Introduction

High strength steel (HSS) members have been gradually employed in steel structures, such as buildings, bridge structures, transmission towers, and offshore structures.Compared with conventional mild steel, lower cross-section dimension and material consumption are entailed for the HSS members due to their higher yield strength. However, the HSS members with smaller section sizes are susceptible to buckling failure. Studies on the buckling performance of HSS welded I-section compression members have been conducted to serve the engineering practice. For the overall buckling performance, experimental studies of 460 MPa [1, 2], 690 MPa [3], 800 MPa [4], and 960 MPa [5, 6] I-section columns have been carried out to investigate the overall buckling reduction factor and the application of existing design standards. For the local buckling study, the experimental steel grades included 460 MPa [7] and 960 MPa [8], the achievement of which mainly focused on the ratio limit of width to thickness, section classification, and assessment of the resistance. The numerical investigations [9, 10] were operated in parallel with experiments, where the parametrical analysis was executed systematically, especially for the effect of geometric parameters, residual stresses, and geometric out-of-straightness detected in the experiments.In addition to the failure modes of local buckling and overall buckling, for the columns of intermediate length, overall buckling and local buckling may appear coupled, which is usually called overall-local interactive buckling (abbreviated as “interactive buckling”). The corresponding studies, including experimental studies [11], theoretical research [12, 13], and numerical simulation [14–16], are being implemented extensively.For the HSS I-section beam-columns, the study separately focused on the overall or local buckling performance [17–19], involving steel grades including 460 MPa [17], 690 MPa [18], and 800 MPa [19]. As for the interactive buckling performance, Gu and Chen [20] published a set of experimental data on wide web I-section beam-columns with a 238 MPa yield strength. Hasham and Rasmussen [21] reported experiment results of stubs and intermediate length beam-columns with yield strength up to 410 MPa and reassessed the application of various design codes.Based on a literature review of the buckling of beam-columns, it was found that the study of interactive buckling performance for HSS I-section beam-columns was relatively scarce, especially for the members with higher steel grades. In this study, the numerical simulation was carried out to explore the interactive buckling performance of wide web I-section beam-columns of steel grade 460 MPa, 690 MPa, 800 MPa, and 960 MPa.


## 2. Finite Element Model and Verification

### 2.1. Finite-Element Model Description

The specimens of the HSS wide web I-section in this study were marked as IH×B×tf×tw, the geometric notation of which is shown in Figure 1.Figure 1
Geometric notation for the I-section.The FE software ANSYS (Manual 15.0) was employed to carry out the numerical simulation. The nonlinear analysis capability of the software accounting for geometric and material nonlinearity was activated. Numerical specimens were modelled using Shell181 shell elements, which support the definition of laminated composite layers, various material directions, and a different number of integral points of each layer. Reasonable element division can not only meet the requirements of calculation accuracy but also ensure calculation efficiency. The number of elements in the width direction of the flange was 20. The number of elements in the height direction of the web and in the length direction of the models was determined according to the ratiohw/B and L/B (L was the length of the member), respectively. For the boundary conditions of the FE models, the nodes on the end profile were linked to the centroid of the end section via rigid body coupling to prevent local buckling. The boundary condition setting method was even more efficient for the beam-columns with a lower slenderness ratio of λl. Meanwhile, lateral supports were arranged on the flanges to prevent the buckling with respect to the minor axis as shown in Figure 2(a). The typical deformation of I-section beam-columns is shown in Figure 2(b).Figure 2
Typical finite element model: (a) boundary condition; (b) typical deformation.
(a)(b)


### 2.2. Material Properties

In accordance with the existing material test data [18, 19, 22, 23], the Quadruple-linear isotropic hardening material model was adopted as the strain-stress relationship of 460 MPa specimens. The trilinear isotropic hardening material model was adopted for the 690 MPa, 800 Mpa, and 960 MPa FE models. Owing to the randomness of material mechanical properties, the standard material parameters were used to ensure the universality of the investigation, including elastic modulus E0, yield strength fy, and tensile ultimate strength fu, which have been put into practice in the numerical analysis of HSS columns [10], as shown in Table 1.Table 1
Parameters for constitutive model of HSS.
GradeE (MPa)fy (MPa)fu (MPa)εst (%)εu (%)Material constitutive relationQ4602.06 × 1054605502.0014.0Q6902.06 × 105690770fy/E8.0Q8002.06 × 105800840fy/E7.0Q9602.06 × 105960980fy/E5.5


### 2.3. Imperfections

#### 2.3.1. Residual Stresses

Residual stresses are almost inevitable because of uneven input of temperature in the welding process, which results in the reduction of bending stiffness and buckling load of the beam-columns. In 2014, based on the test study of the Q460 and Q960 I-section columns, Ban et al. [23] summarized the relationship between the longitudinal residual stresses and the plate width-thickness ratio and proposed a simplified residual stress model for the HSS I-section, as shown in Figure 3 (denoted as the “Ban” model). One notable feature of the “Ban” model was that the effect of the plate depth-thickness ratio on the magnitude of the residual stresses was taken into account. Meanwhile, Ban [1] points out that the value of residual stresses has no significant variation as steel grades increase. The consistent conclusion was also illustrated in [24]. The longitudinal residual stresses of the mode “Ban” were uniformly applied to the five Gauss integral points along the thickness direction of the shell elements in the FE models.Figure 3
Residual stress pattern “Ban.”


#### 2.3.2. Geometric Imperfections

Because of the sensitivity of interactive buckling to geometric imperfections [25], overall geometric imperfection and local imperfections were all introduced into the FE models. By adjusting the thickness of plates, the linear buckling analysis was repeated twice to derive overall and local buckling modes, respectively, to update the FE models using the command “UPGEOM.” The overall imperfection profile was based on the overall buckling mode with an amplitude of L/1000. The local imperfection profile was derived from the local buckling mode with an amplitude of B/100 and hw/200 [26] for the flanges and the web, respectively.


### 2.4. Verification of FE Models

The results of FE models were verified against the existing experimental data, as summarized in Table2. The boundary conditions of all specimens were arranged according to the corresponding literature. For the case of residual stresses provided, the original residual stress mode was followed, otherwise the pattern “Ban” [23] was adopted. The geometric imperfection amplitude of the experiment specimens was taken as bf/100 for the flanges, and hw/200 for the web. Taken together, the average value of the ratio between the FEM results and the experiment data was 0.967 and the standard deviation was 5.45%, which proved that the FE models adopted in this study were reasonable and reliable. Moreover, the load-lateral deflection curves of specimen EH1P, EH2P, EH3P, and EH4P in [19], obtained from tests and FE models, are compared in Figure 4. Notably, the experiment data of Ma et al. [19] in Table 2 was on the overall buckling study due to the scarcity of the experiment data of interactive buckling study of steel grades higher than 460 MPa. In general, the FE models adopted in this study were feasible and reliable.Table 2
Compared of FEM and experiment specimens.
SpecimenLexNExp.NExp./NFEMfy(mm)(mm)(kN)(MPa)I 190 × 120 × 5×2.5(Gu et al., 1991)580022596.041.05238.6I 190 × 120 × 5×2.5580090165.620.99238.6I 220 × 120 × 5×2.5580076207.761.05238.6I 220 × 120 × 5×2.55800206121.031.08238.6I 250 × 120 × 5×2.5580090203.840.94238.6I 250 × 120 × 5×2.55800218121.521.12238.6I 280 × 120 × 5×2.55800233129.360.89238.6I 280 × 120 × 5×2.5 t580078231.280.96238.6I 310 × 120 × 5×2.55800230147.000.99238.6I 364.2 × 104.7 × 4.90×7.85 (Hasham and Rasmussen, 1998)1395.009631.02360I 365.0 × 105.0 × 4.82×7.951396.0258960.94360I 364.5 × 104.8 × 4.81×7.751395.0507600.92360I 364.8 × 104.8 × 4.75×7.761395.01256240.97360I 364.5 × 104.9 × 4.77×7.851394.02504640.91360I 363.2 × 104.8 × 4.85×7.821396.05003210.97360I 365.2 × 105.0 × 4.85×7.891397.07502100.98360I 365.2 × 105.2 × 4.86×7.891398.01000173.6(kN·m)0.96360I 259.5 × 174.9 × 4.89×4.831494.208520.96410I 258.8 × 175.1 × 4.83×4.821494.0257170.91410I 259.2 × 175.8 × 4.89×4.831494.0506021.02410I 260.0 × 175.1 × 4.83×4.881494.01254620.95410I 259.2 × 174.4 × 4.96×4.881495.02503340.98410I 259.8 × 174.1 × 4.95×4.881492.55002250.97410I 260.2 × 175.2 × 4.88×4.901494.57501241.01410I 259.5 × 175.5 × 4.81×4.701494.01000106.7(kN·m)1.04410I 140.0 × 119.6 × 9.90×5.83(T.Y. Ma, et al., 2017)1992.0101.53281.06756I 141.2 × 119.8 × 9.91×5.852790.196.12501.02756I 170.0 × 149.3 × 9.92×5.851993.3103.85271.06756I 170.0 × 149.7 × 9.92×5.852790.399.74180.95756I 231.8 × 201.5 × 15.98×9.921993.398.216981.04756I 231.7 × 200.7 × 15.97×9.952791.6102.413760.94756I 284.2 × 250.1 × 15.97×9.921990.5100.126621.06756I 280.2 × 249.9 × 15.93×9.932790.198.622761.02756Average0.994Standard deviation5.82%Note. 173.6 (kN·m) and 106.7(kN·m) was magnitude of the bending moment, which the test specimen was only subjected to.Figure 4
Comparison of test and FE model load-lateral deflection curves: (a) EH1P [19]; (b) EH2P [19]; (c) EH3P [19]; and (d) EH4P [19].
(a)(b)(c)(d)


## 2.1. Finite-Element Model Description

The specimens of the HSS wide web I-section in this study were marked as IH×B×tf×tw, the geometric notation of which is shown in Figure 1.Figure 1
Geometric notation for the I-section.The FE software ANSYS (Manual 15.0) was employed to carry out the numerical simulation. The nonlinear analysis capability of the software accounting for geometric and material nonlinearity was activated. Numerical specimens were modelled using Shell181 shell elements, which support the definition of laminated composite layers, various material directions, and a different number of integral points of each layer. Reasonable element division can not only meet the requirements of calculation accuracy but also ensure calculation efficiency. The number of elements in the width direction of the flange was 20. The number of elements in the height direction of the web and in the length direction of the models was determined according to the ratiohw/B and L/B (L was the length of the member), respectively. For the boundary conditions of the FE models, the nodes on the end profile were linked to the centroid of the end section via rigid body coupling to prevent local buckling. The boundary condition setting method was even more efficient for the beam-columns with a lower slenderness ratio of λl. Meanwhile, lateral supports were arranged on the flanges to prevent the buckling with respect to the minor axis as shown in Figure 2(a). The typical deformation of I-section beam-columns is shown in Figure 2(b).Figure 2
Typical finite element model: (a) boundary condition; (b) typical deformation.
(a)(b)


## 2.2. Material Properties

In accordance with the existing material test data [18, 19, 22, 23], the Quadruple-linear isotropic hardening material model was adopted as the strain-stress relationship of 460 MPa specimens. The trilinear isotropic hardening material model was adopted for the 690 MPa, 800 Mpa, and 960 MPa FE models. Owing to the randomness of material mechanical properties, the standard material parameters were used to ensure the universality of the investigation, including elastic modulus E0, yield strength fy, and tensile ultimate strength fu, which have been put into practice in the numerical analysis of HSS columns [10], as shown in Table 1.Table 1
Parameters for constitutive model of HSS.
GradeE (MPa)fy (MPa)fu (MPa)εst (%)εu (%)Material constitutive relationQ4602.06 × 1054605502.0014.0Q6902.06 × 105690770fy/E8.0Q8002.06 × 105800840fy/E7.0Q9602.06 × 105960980fy/E5.5


## 2.3. Imperfections

### 2.3.1. Residual Stresses

Residual stresses are almost inevitable because of uneven input of temperature in the welding process, which results in the reduction of bending stiffness and buckling load of the beam-columns. In 2014, based on the test study of the Q460 and Q960 I-section columns, Ban et al. [23] summarized the relationship between the longitudinal residual stresses and the plate width-thickness ratio and proposed a simplified residual stress model for the HSS I-section, as shown in Figure 3 (denoted as the “Ban” model). One notable feature of the “Ban” model was that the effect of the plate depth-thickness ratio on the magnitude of the residual stresses was taken into account. Meanwhile, Ban [1] points out that the value of residual stresses has no significant variation as steel grades increase. The consistent conclusion was also illustrated in [24]. The longitudinal residual stresses of the mode “Ban” were uniformly applied to the five Gauss integral points along the thickness direction of the shell elements in the FE models.Figure 3
Residual stress pattern “Ban.”


### 2.3.2. Geometric Imperfections

Because of the sensitivity of interactive buckling to geometric imperfections [25], overall geometric imperfection and local imperfections were all introduced into the FE models. By adjusting the thickness of plates, the linear buckling analysis was repeated twice to derive overall and local buckling modes, respectively, to update the FE models using the command “UPGEOM.” The overall imperfection profile was based on the overall buckling mode with an amplitude of L/1000. The local imperfection profile was derived from the local buckling mode with an amplitude of B/100 and hw/200 [26] for the flanges and the web, respectively.


## 2.3.1. Residual Stresses

Residual stresses are almost inevitable because of uneven input of temperature in the welding process, which results in the reduction of bending stiffness and buckling load of the beam-columns. In 2014, based on the test study of the Q460 and Q960 I-section columns, Ban et al. [23] summarized the relationship between the longitudinal residual stresses and the plate width-thickness ratio and proposed a simplified residual stress model for the HSS I-section, as shown in Figure 3 (denoted as the “Ban” model). One notable feature of the “Ban” model was that the effect of the plate depth-thickness ratio on the magnitude of the residual stresses was taken into account. Meanwhile, Ban [1] points out that the value of residual stresses has no significant variation as steel grades increase. The consistent conclusion was also illustrated in [24]. The longitudinal residual stresses of the mode “Ban” were uniformly applied to the five Gauss integral points along the thickness direction of the shell elements in the FE models.Figure 3
Residual stress pattern “Ban.”


## 2.3.2. Geometric Imperfections

Because of the sensitivity of interactive buckling to geometric imperfections [25], overall geometric imperfection and local imperfections were all introduced into the FE models. By adjusting the thickness of plates, the linear buckling analysis was repeated twice to derive overall and local buckling modes, respectively, to update the FE models using the command “UPGEOM.” The overall imperfection profile was based on the overall buckling mode with an amplitude of L/1000. The local imperfection profile was derived from the local buckling mode with an amplitude of B/100 and hw/200 [26] for the flanges and the web, respectively.


## 2.4. Verification of FE Models

The results of FE models were verified against the existing experimental data, as summarized in Table2. The boundary conditions of all specimens were arranged according to the corresponding literature. For the case of residual stresses provided, the original residual stress mode was followed, otherwise the pattern “Ban” [23] was adopted. The geometric imperfection amplitude of the experiment specimens was taken as bf/100 for the flanges, and hw/200 for the web. Taken together, the average value of the ratio between the FEM results and the experiment data was 0.967 and the standard deviation was 5.45%, which proved that the FE models adopted in this study were reasonable and reliable. Moreover, the load-lateral deflection curves of specimen EH1P, EH2P, EH3P, and EH4P in [19], obtained from tests and FE models, are compared in Figure 4. Notably, the experiment data of Ma et al. [19] in Table 2 was on the overall buckling study due to the scarcity of the experiment data of interactive buckling study of steel grades higher than 460 MPa. In general, the FE models adopted in this study were feasible and reliable.Table 2
Compared of FEM and experiment specimens.
SpecimenLexNExp.NExp./NFEMfy(mm)(mm)(kN)(MPa)I 190 × 120 × 5×2.5(Gu et al., 1991)580022596.041.05238.6I 190 × 120 × 5×2.5580090165.620.99238.6I 220 × 120 × 5×2.5580076207.761.05238.6I 220 × 120 × 5×2.55800206121.031.08238.6I 250 × 120 × 5×2.5580090203.840.94238.6I 250 × 120 × 5×2.55800218121.521.12238.6I 280 × 120 × 5×2.55800233129.360.89238.6I 280 × 120 × 5×2.5 t580078231.280.96238.6I 310 × 120 × 5×2.55800230147.000.99238.6I 364.2 × 104.7 × 4.90×7.85 (Hasham and Rasmussen, 1998)1395.009631.02360I 365.0 × 105.0 × 4.82×7.951396.0258960.94360I 364.5 × 104.8 × 4.81×7.751395.0507600.92360I 364.8 × 104.8 × 4.75×7.761395.01256240.97360I 364.5 × 104.9 × 4.77×7.851394.02504640.91360I 363.2 × 104.8 × 4.85×7.821396.05003210.97360I 365.2 × 105.0 × 4.85×7.891397.07502100.98360I 365.2 × 105.2 × 4.86×7.891398.01000173.6(kN·m)0.96360I 259.5 × 174.9 × 4.89×4.831494.208520.96410I 258.8 × 175.1 × 4.83×4.821494.0257170.91410I 259.2 × 175.8 × 4.89×4.831494.0506021.02410I 260.0 × 175.1 × 4.83×4.881494.01254620.95410I 259.2 × 174.4 × 4.96×4.881495.02503340.98410I 259.8 × 174.1 × 4.95×4.881492.55002250.97410I 260.2 × 175.2 × 4.88×4.901494.57501241.01410I 259.5 × 175.5 × 4.81×4.701494.01000106.7(kN·m)1.04410I 140.0 × 119.6 × 9.90×5.83(T.Y. Ma, et al., 2017)1992.0101.53281.06756I 141.2 × 119.8 × 9.91×5.852790.196.12501.02756I 170.0 × 149.3 × 9.92×5.851993.3103.85271.06756I 170.0 × 149.7 × 9.92×5.852790.399.74180.95756I 231.8 × 201.5 × 15.98×9.921993.398.216981.04756I 231.7 × 200.7 × 15.97×9.952791.6102.413760.94756I 284.2 × 250.1 × 15.97×9.921990.5100.126621.06756I 280.2 × 249.9 × 15.93×9.932790.198.622761.02756Average0.994Standard deviation5.82%Note. 173.6 (kN·m) and 106.7(kN·m) was magnitude of the bending moment, which the test specimen was only subjected to.Figure 4
Comparison of test and FE model load-lateral deflection curves: (a) EH1P [19]; (b) EH2P [19]; (c) EH3P [19]; and (d) EH4P [19].
(a)(b)(c)(d)


## 3. Parametric Study

### 3.1. Parameter Selection

The FE models of HSS wide web I-section beam-columns were designated in accordance with the design code GB 50017-2017 [27]. For the I-section beam-columns, the slenderness limit of the flange was mainly dependent on steel grades, and that of the web varies over the stress gradient in addition, as expressed in equations (1) and (2), respectively. Meanwhile, it can be found that the limitation of the plate slenderness was more stringent and resulted from the higher steel grade, from the perspective of the utilization rate of materials(1)bftf≤15235fy,(2)hwtw≤45+25α01.66235fy.Also,α0 is given by(3)α0=σmax−σminσmax,where σmax is the maximum compressive stress of web edge, and σmin is the stress corresponding to the other edge.The parameters of numerical specimens were designated as follows: (i) the slendernessλl of FE models were 20, 30, 40, 50, 60, and 80; (ii) the eccentricity ratio or relative eccentricity ε = e/(W/A) were 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 3.0, and 4.0, where e was the eccentricity, W was the section modulus, and A was the gross cross-sectional area); and (iii) bf/tf was less than the limit in accordance to equation (1), and hw/tw were 50, 60, 70, and 80, as shown in Table 3.Table 3
Parameters of FE models.
ParametersValuesSteel strengthfy (MPa)460, 690, 800, and 960λl20, 30, 40, 50, 60Ε0.2, 0.4, 0.8, 1.0, 1.4, 2.0, 3.0, and 4.0bf/tf5.0–9.0hw/tw50, 60, 70, and 80Using the verified FE models, a large amount numeral simulations were implemented to investigate the interactive buckling performance of HSS welded I-section beam-columns, including the property of load deformation curve, the effect of plate width-thickness ratio, relative eccentricity ratio, and the stress distribution et al.


### 3.2. Load-Displacement Curve

In this section, the equilibrium state evolution process of HSS wide web I-section beam-columns was investigated, by means of the analysis of the curve of axial compressive forcePz versus compressive deformation Uz. Taking the beam-column of cross-section I 250 × 80 × 5×4 and ε = 0.4 as an example, the curves of Pz versus Uz are plotted in Figures 5(a)–5(d). For the case of stub beam-columns (λl = 20), the ultimate capacity was mainly limited to the local buckling resistance. From the trend of the curves (λl = 20), it can be found that the ultimate load decreased sharply after reaching the peak. The phenomenon was more notable for the beam-columns with higher steel yield strength, which was observed from the comparison with Figures 5(a)–5(d). As the slenderness increased, the ultimate capacity and axial stiffness all decreased pronouncedly. Meanwhile, unlike the cases of stub beam-columns, the more distinguished deformation plateau of the Pz–Uz curve was observed as steel grade increased, which meant the failure mode showed a certain degree of ductility resulting from high steel grades.Figure 5
Axial compressive forcePz versus compressive deformation Uz curves: (a) the load–displacement of Q460; (b) the load–displacement of Q690; (c) the load–displacement of Q800; and (d) the load–displacement of Q960.
(a)(b)(c)(d)


### 3.3. Effect of Slenderness Ratio

The dimensionless axial ultimate load (Pu/Afy) versus overall slenderness λl curves are shown in Figure 6, where Pu was the ultimate load, A was the area of the cross-section, and fy was the yield strength. The Pu/Afy decreased as λl increased, the decreasing level of which was gradually weakened resulted from higher ε. For example, when ε = 0.4, the ultimate load of λl = 60 was about 0.73 of that of λl = 30. When ε = 2.0, the ratio increased to 0.76, as shown in Figure 6(a). The main reason was that, because of higher ε, the tensile zone of the cross section was elongated, which was propitious to rebate the second-order effect of axial load, whereas the ratio was reduced to 0.72, 0.71, and 0.70 with Q690, Q800, and Q960, respectively. This meant the effect of slenderness was amplified for higher steel grades. Since the ultimate load depended on the buckling resistance, the ratio of Pu/Afy became lower for beam-columns of higher steel grades.Figure 6
Axial nondimensional ultimate compressive load (Pu/Afy) versus overall slenderness λl under different &epsi; curves: (a) Q460; (b) Q690; (c) Q800, and (d) Q960.
(a)(b)(c)(d)


### 3.4. Effect of Height-to-Thickness Ratio

ThePu/Afy versus web height-to-thickness ratio hw/tw curves are shown in Figure 7 to reflect the effect of hw/tw on the ultimate load. It was found that the Pu/Afy decreased gradually resulted from larger hw/tw. The curve of Pu/Afy versus hw/tw were graphically straight lines under various ε. As expressed in (4), larger hw/tw is more likely to trigger the local buckling of the web, which further resulted in the additional load eccentricity of the cross-section and the reduction of the cross section bearing capacity. No significant disparity was inspected from the comparison of the models under various steel grades (Figures 7(a)–7(d)), because the local buckling happened at the elastic stage.(4)σcr=k121−ν2×π2Eb/t2,where k is the buckling factor; ν is the Poisson ratio; b and t are the width and thickness of the plates, respectively.Figure 7
Pu/Afy versus hw/tw under different &epsi; curves: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


### 3.5. Effect of Width-to-Thickness Ratio

For the wide web I-section beam-columns, the relative compact flanges are conducive to enhancing boundary constraint stiffness and the buckling load of the web, but for the wide web I-section, the above expectation was not always observed from the curves of ultimate loadPu/Afy versus bf/tf, as shown in Figure 8. For the case of bf/tf lower than 6.0, the failure mode of the web was mainly presented as shear buckling, the ultimate load corresponding to which increased resulted from a relatively strong constraint as shown in Figure 8. When bf/tf was higher than 6.0, the failure mode of the web presented as bending buckling, the ultimate load corresponding to which was decreased resulted from higher bf/tf. The variation trend of different steel grade did not present significant distinction.Figure 8
Pu/Afy versus bf/tf under different &epsi; curves: under different &epsi; curves: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


### 3.6. Stress Distribution of Stub Beam-Columns

The effect of eccentricityε on stress distribution of the web under ultimate load is shown in Figures 9(a)–9(d), where it can be observed that: Under axial compression (ε = 0), the stress was compressive and distributed symmetrically; it was obviously smaller in the middle of the web due to local buckling. Higher eccentricity ε meant a wider tension zone and a smaller compression zone, which was conducive avoiding the local buckling. With higher ε, tensile stresses gradually increased and the neutral axis moved towards the centroid of the section. The width of the buckled zone became narrower, and compressive stresses at the edge of the webs were gradually closer to the yield strength. Meanwhile the stress distribution on the cross-section was incrementally close to a straight line, which was more obvious for higher steel grade under higher ε, as shown in Figures 9(a)–9(d).Figure 9
Stress distribution of web of stubs corresponding to ultimate capacity under different&epsi;: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


## 3.1. Parameter Selection

The FE models of HSS wide web I-section beam-columns were designated in accordance with the design code GB 50017-2017 [27]. For the I-section beam-columns, the slenderness limit of the flange was mainly dependent on steel grades, and that of the web varies over the stress gradient in addition, as expressed in equations (1) and (2), respectively. Meanwhile, it can be found that the limitation of the plate slenderness was more stringent and resulted from the higher steel grade, from the perspective of the utilization rate of materials(1)bftf≤15235fy,(2)hwtw≤45+25α01.66235fy.Also,α0 is given by(3)α0=σmax−σminσmax,where σmax is the maximum compressive stress of web edge, and σmin is the stress corresponding to the other edge.The parameters of numerical specimens were designated as follows: (i) the slendernessλl of FE models were 20, 30, 40, 50, 60, and 80; (ii) the eccentricity ratio or relative eccentricity ε = e/(W/A) were 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 3.0, and 4.0, where e was the eccentricity, W was the section modulus, and A was the gross cross-sectional area); and (iii) bf/tf was less than the limit in accordance to equation (1), and hw/tw were 50, 60, 70, and 80, as shown in Table 3.Table 3
Parameters of FE models.
ParametersValuesSteel strengthfy (MPa)460, 690, 800, and 960λl20, 30, 40, 50, 60Ε0.2, 0.4, 0.8, 1.0, 1.4, 2.0, 3.0, and 4.0bf/tf5.0–9.0hw/tw50, 60, 70, and 80Using the verified FE models, a large amount numeral simulations were implemented to investigate the interactive buckling performance of HSS welded I-section beam-columns, including the property of load deformation curve, the effect of plate width-thickness ratio, relative eccentricity ratio, and the stress distribution et al.


## 3.2. Load-Displacement Curve

In this section, the equilibrium state evolution process of HSS wide web I-section beam-columns was investigated, by means of the analysis of the curve of axial compressive forcePz versus compressive deformation Uz. Taking the beam-column of cross-section I 250 × 80 × 5×4 and ε = 0.4 as an example, the curves of Pz versus Uz are plotted in Figures 5(a)–5(d). For the case of stub beam-columns (λl = 20), the ultimate capacity was mainly limited to the local buckling resistance. From the trend of the curves (λl = 20), it can be found that the ultimate load decreased sharply after reaching the peak. The phenomenon was more notable for the beam-columns with higher steel yield strength, which was observed from the comparison with Figures 5(a)–5(d). As the slenderness increased, the ultimate capacity and axial stiffness all decreased pronouncedly. Meanwhile, unlike the cases of stub beam-columns, the more distinguished deformation plateau of the Pz–Uz curve was observed as steel grade increased, which meant the failure mode showed a certain degree of ductility resulting from high steel grades.Figure 5
Axial compressive forcePz versus compressive deformation Uz curves: (a) the load–displacement of Q460; (b) the load–displacement of Q690; (c) the load–displacement of Q800; and (d) the load–displacement of Q960.
(a)(b)(c)(d)


## 3.3. Effect of Slenderness Ratio

The dimensionless axial ultimate load (Pu/Afy) versus overall slenderness λl curves are shown in Figure 6, where Pu was the ultimate load, A was the area of the cross-section, and fy was the yield strength. The Pu/Afy decreased as λl increased, the decreasing level of which was gradually weakened resulted from higher ε. For example, when ε = 0.4, the ultimate load of λl = 60 was about 0.73 of that of λl = 30. When ε = 2.0, the ratio increased to 0.76, as shown in Figure 6(a). The main reason was that, because of higher ε, the tensile zone of the cross section was elongated, which was propitious to rebate the second-order effect of axial load, whereas the ratio was reduced to 0.72, 0.71, and 0.70 with Q690, Q800, and Q960, respectively. This meant the effect of slenderness was amplified for higher steel grades. Since the ultimate load depended on the buckling resistance, the ratio of Pu/Afy became lower for beam-columns of higher steel grades.Figure 6
Axial nondimensional ultimate compressive load (Pu/Afy) versus overall slenderness λl under different &epsi; curves: (a) Q460; (b) Q690; (c) Q800, and (d) Q960.
(a)(b)(c)(d)


## 3.4. Effect of Height-to-Thickness Ratio

ThePu/Afy versus web height-to-thickness ratio hw/tw curves are shown in Figure 7 to reflect the effect of hw/tw on the ultimate load. It was found that the Pu/Afy decreased gradually resulted from larger hw/tw. The curve of Pu/Afy versus hw/tw were graphically straight lines under various ε. As expressed in (4), larger hw/tw is more likely to trigger the local buckling of the web, which further resulted in the additional load eccentricity of the cross-section and the reduction of the cross section bearing capacity. No significant disparity was inspected from the comparison of the models under various steel grades (Figures 7(a)–7(d)), because the local buckling happened at the elastic stage.(4)σcr=k121−ν2×π2Eb/t2,where k is the buckling factor; ν is the Poisson ratio; b and t are the width and thickness of the plates, respectively.Figure 7
Pu/Afy versus hw/tw under different &epsi; curves: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


## 3.5. Effect of Width-to-Thickness Ratio

For the wide web I-section beam-columns, the relative compact flanges are conducive to enhancing boundary constraint stiffness and the buckling load of the web, but for the wide web I-section, the above expectation was not always observed from the curves of ultimate loadPu/Afy versus bf/tf, as shown in Figure 8. For the case of bf/tf lower than 6.0, the failure mode of the web was mainly presented as shear buckling, the ultimate load corresponding to which increased resulted from a relatively strong constraint as shown in Figure 8. When bf/tf was higher than 6.0, the failure mode of the web presented as bending buckling, the ultimate load corresponding to which was decreased resulted from higher bf/tf. The variation trend of different steel grade did not present significant distinction.Figure 8
Pu/Afy versus bf/tf under different &epsi; curves: under different &epsi; curves: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


## 3.6. Stress Distribution of Stub Beam-Columns

The effect of eccentricityε on stress distribution of the web under ultimate load is shown in Figures 9(a)–9(d), where it can be observed that: Under axial compression (ε = 0), the stress was compressive and distributed symmetrically; it was obviously smaller in the middle of the web due to local buckling. Higher eccentricity ε meant a wider tension zone and a smaller compression zone, which was conducive avoiding the local buckling. With higher ε, tensile stresses gradually increased and the neutral axis moved towards the centroid of the section. The width of the buckled zone became narrower, and compressive stresses at the edge of the webs were gradually closer to the yield strength. Meanwhile the stress distribution on the cross-section was incrementally close to a straight line, which was more obvious for higher steel grade under higher ε, as shown in Figures 9(a)–9(d).Figure 9
Stress distribution of web of stubs corresponding to ultimate capacity under different&epsi;: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


## 4. Post-Buckling Strength of Stub Beam-Columns

Practical design methods to estimate the effect of local buckling are mainly divided into two categories: the effective width method (EWM) and the direct strength method (DSM). The former deems that the plate within a certain width can still bear normal load after local buckling; equivalent width (be) is used to calibrate the ultimate capacity. For the latter, based on the gross area properties, a continuous function is used to reduce the overall buckling ultimate capacity to consider the adverse effects of the plate local buckling.According to the EWM specification provided in GB50017-2017 [27], for beam-columns of ordinary steel grade, the equivalent width is determined by the stress gradient and the support condition of plates. The geometric characteristics of the section should be reevaluated, including area, moment of inertia, etc. Besides, the additional load eccentricity caused by the deviation of the centroid of the equivalent section needs to be considered in the bending moment calculation.Similar provisions of EWM can be found in other specifications, such as AS4100. [28] The main difference is the expression of the plate buckling coefficient kσ used in the dimensionless slenderness λn,p of the web, which is described as equations (5) and (6) in GB50017-2017 [27].(5)kσ=162−α0+2−α02+0.112α02,(6)λn,p=hw/tw28.1kσ1235/fy.Because of the geometric characteristics of the gross section adopted in the DSM, this approach may simplify the calculation process under certain conditions. In 2010, Rossi et al. [29] derived a formula using DSM to determine the bearing capacity of cold-formed thin-walled members, which was expressed as the following equation:(7)σcl=1−λ¯pλlimfufy−1+1,λ¯p≤λlim,1λ¯pma−bλ¯pmfy,λ¯p>λlim,where λ¯p=fy/σcrl. σcrlis the critical local buckling elastic stress of component.Notably,σcrl is the local buckling stress of a component, not the individual stresses of the flange or web, which is often derived with software, such as CUFSM [30], based on the concept of the finite strip method. In this study, the buckling load of the was taken as the approximate value of σcrl, using λn,p ((6)instead of λ¯p in the equation (7).The FE results of stub beam-columns of 460 MPa, 690 MPa, 800 MPa, and 960 MPa were used to establish the DSM assessment formula of the local postbuckling strength. The effective length (Leff) of all the FE models of stub beam-columns was designated to be 1.2hw. The web width-thickness ratio of the FE models was from 60 to 100, and the eccentricity was from 0.2 to 1.4. In terms of edge fiber yield criterion, the material strength correlation equation of axial compressive load and bending moment for stub beam-columns can be expressed as(8)PueAfy+MexWexfy=1.If part of the cross-section was allowed to enter the plastic stage, (8) was expressed as(9)PueAfy+MexγxWexfy=1,where γx is the plastic development coefficient of the cross-section. In GB20017-2017, it is recommended to be 1.05 for the I-section of bending moment around a strong axial.Considering the effect of local buckling of the web, the proportion of the load on the web is lower than that determined according to the ratio ofAw/A and Iw/I. The local postbuckling strength was reflected by the strength reduction factor ρ defined as (11) derived from the following equation:(10)PuAρfy+MuxγxWeρfy=1,(11)ρ=PuAfy+MuγxWefy,where Nu and Mu were derived via the longitude stress σz integration about the cross section and expressed as(12)Nu=∫σzdy,Mu=∫σzydy.The FE data of stub beam-columns of 460 MPa, 690 MPa, 800 MPa, and 960 MPa about plate slendernessλn,p and ρ are present in Figure 10. Based on the FEM results, the relation expression of λn,p against ρ was fitted as the following equation:(13)ρ=1λn,p0.3841.354−0.315λn,p0.384,λn,plim>1.151,1.000,λn,plim≤1.151.Figure 10
The curve ofλn,p versus strength reduction coefficient ρ.The reliability of the proposed formula (13) was further verified against test data in [21]. The average (AVE) and standard deviation (SD) of the ratio of them (experiment data [21]/proposed formula) were 1.05 and 4.52%, respectively.


## 5. Interactive Buckling Performance of HSS Beam-Columns

### 5.1. Axial Load and Bending Moment Correlation of Beam-Columns

At present, the most widely used method for describing buckling performance is essentially based on the material strength correlation formula, simultaneously considering the second order effect of the axial loads. According to the edge fiber yield criterion, considering the second order effect of axial compression, the overall buckling behavior can be expressed as(14)PPy+M+Pe0My1−P/PE=1,where P and M are the end loads of beam-columns; e0 is the initial imperfection including geometric imperfection and residual stresses;1/1−P/PE is the moment amplification factor considering second-order effect resulted from axial compression; PE=π2EI/leff2; Py=Afy; My=Wfy; W is the elastic resistance moment.SupposeM = 0 corresponding to the case of axial compression, the ultimate load Pcr was expressed as(15)PcrPy+Pcre0My1−Pcr/PE=1,where overall buckling factor φ=Pcr/Py; e0 can be expressed as(16)e0My=1−PcrPy1−PcrPE1Pcr,by substituting (16) into (14), then(17)PφPy+MMy1−φP/PE=1.For the interactive buckling resistance, the strength reduction factorρ using DSM ((13)was introduced into (17) to clarify the effect of the local buckling on the material strength. Meanwhile considering plastic development of cross section with γx, (17) was deduced as(18)PuφAρfy+MuγxWeρfy1−φP/PE=1.Based on equation (18), a more general form was assumed as(19)PuφAρfyξP+MuγxWeρfy1−φP/PEξM=1,where ξp and ξM are both form parameters fitted based on the results of existing experiment data and FE models.


### 5.2. Overall Bending Buckling Factor

In (19), it can be found that, in addition to ξp and ξM, the overall bending bucking reduction factor φ of HSS beam-columns was also a key factor in determining the ultimate load. Overall buckling reduction factors (column curves) in different codes, including GB50017-2017 [27], EN1993-1-1 [31], and ASCE360-05 [32], were summarized as follows.


#### 5.2.1. EN1993-1-1

Five column curves are provided as the overall buckling reduction factors, distinguished according to the cross-section properties in EN1993, as expressed in the following equation(20)χ=1.0,λc≤0.2,χ=1ψ+ψ2−λc2,λc>0.2,where(21)ψ=0.51+αλc−0.2+λc2,(22)λc=λπfyE,and α are 0.13, 0.21, 0.34, 0.49, and 0.76 for curves a0, a, b, c, and d, respectively.


#### 5.2.2. AISC360-05

The critical stress of axial compression component in AISC360-05 is(23)fcr=0.658fy/fefy,λ≤4.71Efy,0.877fe,λ>4.71Efy,where(24)fe=π2Eλ2;λ=leffi.By substitutingλcexpressed as equation (22) into equation (23), then, it evolved into(25)φ=fcrfy=0.658λc2,λc≤1.5,0.877λc2,λc>1.5.


#### 5.2.3. GB50017-2017

Four curves are used as the overall buckling factors in GB50017-20017 Chinese steel design code, expressed as(26)φ=1−α1λ¯2,λ¯≤0.215,φ=12λ¯2α2+α3λ¯+λ¯2−α2+α3λ¯+λ¯22−4λ¯21−α1λ¯2,λ¯>0.215,where λ¯=λ/πfy/E and α1, α2, α3 are listed in Table 4.Table 4
α1, α2, α3 for overall buckling factor in GB50017-2017.
Section typeα1α2α3A0.410.9860.152B0.650.9650.300cλ¯ ≤ 1.050.730.9060.595λ¯ > 1.051.2160.302dλ¯ ≤ 1.051.350.8680.915λ¯ > 1.051.3750.432For the overall slenderness in the range of 50 to 100 commonly used in practical engineering:(i)
Values of curve a in GB50017 are slightly higher than that of curve a in EN1993-1-1; values of curveb are close of the two; values of curve c and d in GB50017 were lower than that in EN1993-1-1. For the HSS I-section beam-columns, the curve b in EN1993-1-1 was selected as the overall buckling factor φ of HSS I-section beam-columns according to the study of Huang [15].(ii)
The overall bending buckling reduction factor of AISC360-05 for I-section columns are between that of curvea and curve b in EN1993-1-1.


### 5.3. Axial Load and Bending Moment Correlation Curves

The correlation curve of dimensionless axial loadPu/Py and bending moment Mu/My is shown in Figure 11. For the models of lower λl, Pu/Py and Mu/My were essentially linear. As the overall slenderness increased, the correlation curves slightly convexed downward, which was more pronounced for the models of high steel grade. Considering the curves’ description with dimensionless ratio, taking the effect of steel grade into account, it meant the effect of local buckling was more significant on the beam-columns resulting from high yield strength.Figure 11
Relative curves ofPu/Py versus Mu/My: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


### 5.4. Design Method for Overall-Local Interactive Buckling of High Strength Steel Beam-Columns

Based on the discussion in Section5.3, it was found that the correlation of axial load and bending moment was closely dependent on overall slenderness. Present dimensionless overall slenderness λ¯ = 0.30, 0.60, 1.00, and 1.30 as examples, the fitting curves of P/φAρfy and M/Wρfy1−φP/PE are shown in Figure 12, where ξp and ξm were the form parameters of equation (19).Figure 12
Relative curves ofP/φAρfyversus M/Wρfy1−φP/PE: (a) λ¯=0.3, (b) λ¯=0.6, (c) λ¯=1.0, and (d) λ¯=1.3.
(a)(b)(c)(d)In the same approach,ξp and ξM were derived, as shown in Table 5 and Figure 13, the fitting formulas of which about λ¯ were expressed as equations (27) and (28), rrespectively.Table 5
ξp, ξM as the form parameters fitting result.
λ¯0.300.450.600.750.900.370.550.740.920.40ξm0.3850.3300.2560.1970.1560.3350.2660.2000.1410.306ξp1.0881.0261.0651.0350.9551.1311.1291.1081.0751.152λ¯0.600.790.991.190.430.650.871.091.300.60ξm0.2440.1790.1210.0730.2690.2070.1460.0920.0580.244ξp1.1341.1091.0751.0641.2381.2211.1881.1591.0971.134(27)ξm=0.538−0.640λ¯+0.277λ¯2−0.055λ¯3,(28)ξp=0.973+0.787λ¯−1.200λ¯2+0.518λ¯3.Figure 13
ξp, ξM as the form parameter fitting curve.In summary, the formula for the ultimate load of interactive buckling prediction were generalized as follows:(29)PuφAρfyξP+MuγxWeρfy1−φP/PEξM=1,where(30)ρ=1λn,p0.3841.354−0.315λn,p0.384,λn,p>1.151,1.000,λn,p≤1.151,ξM=0.538−0.640λ¯+0.277λ¯2−0.055λ¯3,ξp=0.973+0.787λ¯−1.200λ¯2+0.518,λ¯3λn,p=hw/tw28.1kσ1235/fy,λ¯=λπfyE.The comparison against experiment data is shown in Figure14; the AVE of the error between the prediction formula and test data was 3.21% and the SD was 6.24%, which proved the reliability of the proposed formula.Figure 14
Comparison between the proposed formula and existing test result.


## 5.1. Axial Load and Bending Moment Correlation of Beam-Columns

At present, the most widely used method for describing buckling performance is essentially based on the material strength correlation formula, simultaneously considering the second order effect of the axial loads. According to the edge fiber yield criterion, considering the second order effect of axial compression, the overall buckling behavior can be expressed as(14)PPy+M+Pe0My1−P/PE=1,where P and M are the end loads of beam-columns; e0 is the initial imperfection including geometric imperfection and residual stresses;1/1−P/PE is the moment amplification factor considering second-order effect resulted from axial compression; PE=π2EI/leff2; Py=Afy; My=Wfy; W is the elastic resistance moment.SupposeM = 0 corresponding to the case of axial compression, the ultimate load Pcr was expressed as(15)PcrPy+Pcre0My1−Pcr/PE=1,where overall buckling factor φ=Pcr/Py; e0 can be expressed as(16)e0My=1−PcrPy1−PcrPE1Pcr,by substituting (16) into (14), then(17)PφPy+MMy1−φP/PE=1.For the interactive buckling resistance, the strength reduction factorρ using DSM ((13)was introduced into (17) to clarify the effect of the local buckling on the material strength. Meanwhile considering plastic development of cross section with γx, (17) was deduced as(18)PuφAρfy+MuγxWeρfy1−φP/PE=1.Based on equation (18), a more general form was assumed as(19)PuφAρfyξP+MuγxWeρfy1−φP/PEξM=1,where ξp and ξM are both form parameters fitted based on the results of existing experiment data and FE models.


## 5.2. Overall Bending Buckling Factor

In (19), it can be found that, in addition to ξp and ξM, the overall bending bucking reduction factor φ of HSS beam-columns was also a key factor in determining the ultimate load. Overall buckling reduction factors (column curves) in different codes, including GB50017-2017 [27], EN1993-1-1 [31], and ASCE360-05 [32], were summarized as follows.


### 5.2.1. EN1993-1-1

Five column curves are provided as the overall buckling reduction factors, distinguished according to the cross-section properties in EN1993, as expressed in the following equation(20)χ=1.0,λc≤0.2,χ=1ψ+ψ2−λc2,λc>0.2,where(21)ψ=0.51+αλc−0.2+λc2,(22)λc=λπfyE,and α are 0.13, 0.21, 0.34, 0.49, and 0.76 for curves a0, a, b, c, and d, respectively.


### 5.2.2. AISC360-05

The critical stress of axial compression component in AISC360-05 is(23)fcr=0.658fy/fefy,λ≤4.71Efy,0.877fe,λ>4.71Efy,where(24)fe=π2Eλ2;λ=leffi.By substitutingλcexpressed as equation (22) into equation (23), then, it evolved into(25)φ=fcrfy=0.658λc2,λc≤1.5,0.877λc2,λc>1.5.


### 5.2.3. GB50017-2017

Four curves are used as the overall buckling factors in GB50017-20017 Chinese steel design code, expressed as(26)φ=1−α1λ¯2,λ¯≤0.215,φ=12λ¯2α2+α3λ¯+λ¯2−α2+α3λ¯+λ¯22−4λ¯21−α1λ¯2,λ¯>0.215,where λ¯=λ/πfy/E and α1, α2, α3 are listed in Table 4.Table 4
α1, α2, α3 for overall buckling factor in GB50017-2017.
Section typeα1α2α3A0.410.9860.152B0.650.9650.300cλ¯ ≤ 1.050.730.9060.595λ¯ > 1.051.2160.302dλ¯ ≤ 1.051.350.8680.915λ¯ > 1.051.3750.432For the overall slenderness in the range of 50 to 100 commonly used in practical engineering:(i)
Values of curve a in GB50017 are slightly higher than that of curve a in EN1993-1-1; values of curveb are close of the two; values of curve c and d in GB50017 were lower than that in EN1993-1-1. For the HSS I-section beam-columns, the curve b in EN1993-1-1 was selected as the overall buckling factor φ of HSS I-section beam-columns according to the study of Huang [15].(ii)
The overall bending buckling reduction factor of AISC360-05 for I-section columns are between that of curvea and curve b in EN1993-1-1.


## 5.2.1. EN1993-1-1

Five column curves are provided as the overall buckling reduction factors, distinguished according to the cross-section properties in EN1993, as expressed in the following equation(20)χ=1.0,λc≤0.2,χ=1ψ+ψ2−λc2,λc>0.2,where(21)ψ=0.51+αλc−0.2+λc2,(22)λc=λπfyE,and α are 0.13, 0.21, 0.34, 0.49, and 0.76 for curves a0, a, b, c, and d, respectively.


## 5.2.2. AISC360-05

The critical stress of axial compression component in AISC360-05 is(23)fcr=0.658fy/fefy,λ≤4.71Efy,0.877fe,λ>4.71Efy,where(24)fe=π2Eλ2;λ=leffi.By substitutingλcexpressed as equation (22) into equation (23), then, it evolved into(25)φ=fcrfy=0.658λc2,λc≤1.5,0.877λc2,λc>1.5.


## 5.2.3. GB50017-2017

Four curves are used as the overall buckling factors in GB50017-20017 Chinese steel design code, expressed as(26)φ=1−α1λ¯2,λ¯≤0.215,φ=12λ¯2α2+α3λ¯+λ¯2−α2+α3λ¯+λ¯22−4λ¯21−α1λ¯2,λ¯>0.215,where λ¯=λ/πfy/E and α1, α2, α3 are listed in Table 4.Table 4
α1, α2, α3 for overall buckling factor in GB50017-2017.
Section typeα1α2α3A0.410.9860.152B0.650.9650.300cλ¯ ≤ 1.050.730.9060.595λ¯ > 1.051.2160.302dλ¯ ≤ 1.051.350.8680.915λ¯ > 1.051.3750.432For the overall slenderness in the range of 50 to 100 commonly used in practical engineering:(i)
Values of curve a in GB50017 are slightly higher than that of curve a in EN1993-1-1; values of curveb are close of the two; values of curve c and d in GB50017 were lower than that in EN1993-1-1. For the HSS I-section beam-columns, the curve b in EN1993-1-1 was selected as the overall buckling factor φ of HSS I-section beam-columns according to the study of Huang [15].(ii)
The overall bending buckling reduction factor of AISC360-05 for I-section columns are between that of curvea and curve b in EN1993-1-1.


## 5.3. Axial Load and Bending Moment Correlation Curves

The correlation curve of dimensionless axial loadPu/Py and bending moment Mu/My is shown in Figure 11. For the models of lower λl, Pu/Py and Mu/My were essentially linear. As the overall slenderness increased, the correlation curves slightly convexed downward, which was more pronounced for the models of high steel grade. Considering the curves’ description with dimensionless ratio, taking the effect of steel grade into account, it meant the effect of local buckling was more significant on the beam-columns resulting from high yield strength.Figure 11
Relative curves ofPu/Py versus Mu/My: (a) Q460; (b) Q690; (c) Q800; and (d) Q960.
(a)(b)(c)(d)


## 5.4. Design Method for Overall-Local Interactive Buckling of High Strength Steel Beam-Columns

Based on the discussion in Section5.3, it was found that the correlation of axial load and bending moment was closely dependent on overall slenderness. Present dimensionless overall slenderness λ¯ = 0.30, 0.60, 1.00, and 1.30 as examples, the fitting curves of P/φAρfy and M/Wρfy1−φP/PE are shown in Figure 12, where ξp and ξm were the form parameters of equation (19).Figure 12
Relative curves ofP/φAρfyversus M/Wρfy1−φP/PE: (a) λ¯=0.3, (b) λ¯=0.6, (c) λ¯=1.0, and (d) λ¯=1.3.
(a)(b)(c)(d)In the same approach,ξp and ξM were derived, as shown in Table 5 and Figure 13, the fitting formulas of which about λ¯ were expressed as equations (27) and (28), rrespectively.Table 5
ξp, ξM as the form parameters fitting result.
λ¯0.300.450.600.750.900.370.550.740.920.40ξm0.3850.3300.2560.1970.1560.3350.2660.2000.1410.306ξp1.0881.0261.0651.0350.9551.1311.1291.1081.0751.152λ¯0.600.790.991.190.430.650.871.091.300.60ξm0.2440.1790.1210.0730.2690.2070.1460.0920.0580.244ξp1.1341.1091.0751.0641.2381.2211.1881.1591.0971.134(27)ξm=0.538−0.640λ¯+0.277λ¯2−0.055λ¯3,(28)ξp=0.973+0.787λ¯−1.200λ¯2+0.518λ¯3.Figure 13
ξp, ξM as the form parameter fitting curve.In summary, the formula for the ultimate load of interactive buckling prediction were generalized as follows:(29)PuφAρfyξP+MuγxWeρfy1−φP/PEξM=1,where(30)ρ=1λn,p0.3841.354−0.315λn,p0.384,λn,p>1.151,1.000,λn,p≤1.151,ξM=0.538−0.640λ¯+0.277λ¯2−0.055λ¯3,ξp=0.973+0.787λ¯−1.200λ¯2+0.518,λ¯3λn,p=hw/tw28.1kσ1235/fy,λ¯=λπfyE.The comparison against experiment data is shown in Figure14; the AVE of the error between the prediction formula and test data was 3.21% and the SD was 6.24%, which proved the reliability of the proposed formula.Figure 14
Comparison between the proposed formula and existing test result.


## 6. Conclusion

Using FE models verified against existing test data, the parametrical analysis was carried out. The property of load-displacement curves of various slenderness, the effect of overall slenderness,bf/tf of the flanges, and hw/tw of the web were investigated. The establishment of the formulas for the resistance of the HSS wide web I section beam-columns were divided into two main steps: Firstly, based on the study of stub columns, the local postbuckling strength, i.e., the strength considering the effect of the web local buckling, was clarified using DSM. After the effect of local buckling on material strength was separated from the interactive buckling at the first step, the axial compression-bending moment-relevant formula was established based on the results of beam-columns with intermediate slenderness. In summary, the following conclusions were made:(1)
The ultimate capacity and axial stiffness of the beam-columns decreased resulted from higher slenderness. As the slenderness increased, the deformation plateau of the load-displacement curve was more distinguished, which was more pronounced with the beam-columns of higher steel grades.(2)
The dimensionless ultimate loadPu/Afy showed a linear decline as hw/tw increasing. The underlying mechanism was that the web with higher hw/tw was prone to being buckled, which further resulted in the additional eccentricity and impairment of the cross-section, and there is no obvious difference in the influence of hw/tw on beam-columns with various steel grades.(3)
Whenbf/tf was lower than 0.6, the ultimate load improved resulted from higher bf/tf,, and bf/tf greater than 0.6, the ultimate load decreased with the increase of bf/tf.(4)
Based on the concept of DSM, the prediction formula for local postbuckling strength of stub beam-columns was fitted using FE results, the validity of which was verified against experiment data, and the prediction formula was further used to separate the local buckling effect from the comprehensive effect of interactive buckling on the material strength.(5)
Continuing the result of the sub-column study, the estimation formulas for interactive buckling strength were proposed using the experiment data and FE results of beam-columns of intermediate length, where the curveb of the overall bending buckling reduction factor in EN1993-1-1 was adopted.In this work, the verified FE models combined with the existing test data were used to investigate the interactive buckling performance of HSS wide web I-section beam-columns. Due to the inconsistency of steel smelting processes and steel member processing technology in different regions, the mechanical properties, geometric imperfections, and residual stresses of members are not completely consistent. Therefore, more experimental research, numerical calculation, and engineering tests are needed.



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*Source: 7219975-2022-07-22.xml*