Structures 16 (2018) 373–389
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Finite Element Modeling and Design of Single Angle Member Under Bi-axial Bending
T
⁎
A. Hussain, Yao-Peng Liu , Siu-Lai Chan Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Single angle Plastic and compact cross-sections Eccentric compression Bi-axial bending Plastic design
Practical single angle members are mostly subjected to eccentric forces due to the end connections to their legs. Plastic Class 1 and compact Class 2 angle cross-sections are allowed for using their plastic reserves. However, most of the interaction equations in the current design codes are linear whereas the nonlinear equations associated with plastic surface are provided in the literature. In this study, a new design method for single angle member to address the plastic failure as well as stability checks is proposed. The first yield moments at different critical points due to the interaction between the normal forces and the biaxial bending are provided. Moreover, the elastic flexural and flexural-torsional buckling loads for single angle member are calculated using a simplified equation rather than the complex design charts or series of equations in literature or design codes. New buckling curves, taking the Eurocode 3 design rules into account, are presented for single angle member subjected to eccentric load about both major and minor principal axes. A reliable and efficient finite element model is provided to verify the proposed equations and to investigate the different parameters affecting the buckling behavior. Finally, various initial geometric imperfections together with a residual stress distribution pattern are studied and provided in this paper.
1. Introduction Angle section members are widely used in the light-weight steel structures as web or bracing members because of their ease of connections. One of these structures is latticed steel transmission towers. Single angle members are often subjected to normal forces as a truss member connecting at both ends by one or two bolts in one of their two legs. The eccentricities at end connections are causes for bi-axial bending in angle members. Further, in transmission towers, the leg or bracing angle members are directly subjected to bi-axial bending through the transverse wind load. In the lower part of the tower, the axial force dominates due to high overturning moment, but in the upper part, the bending action may be critical because of high wind pressure. It means that in some cases the angle members have to be designed under the combination of axial force and bending moments. Furthermore, when the angle member is subjected to transverse loading in the direction of its geometrical axes which parallel to the legs, the geometry of the angle section causes bi-axial bending about its two principal axes. Different design codes [1–4] provide simplified methods for design of angle members, which mainly based on the modified effective slenderness ratio or on the effective area considering end eccentricities. ⁎
This method is applicable only to the case of welded or minimum two bolted connections, whereas in the case of single bolted connection, the interaction formula has been proposed to combine the effect of bi-axial bending. On the other hand, the over-simplified and conservative design approaches in different design codes are mostly based on the doubly symmetric sections. Thus, there are an expected underestimation and conservatism for the failure loads in case of monosymmetric and asymmetric sections for equal angles and unequal angles respectively. The second-order direct analysis is an alternative approach for angle trusses avoiding the error of estimating effective length and considering different non-linear effects [5–7]. However, in provided models available in literature, the residual stresses and initial imperfections have been considered in the P-Δ and P-δ moments accounting for geometric imperfections and residual stresses in the analysis, but torsional buckling effects are not considered in sufficient detail. Angle sections are usually designed using the theory of elasticity in which the elastic stress distribution at different critical points are calculated, and these values are then limited by the design strength, which equals to the yield stress divided by the material partial factor. Aydın and Doğan [8] provided a design procedure for single angle member subjected to bi-axial bending using dimensionless coefficient to
Corresponding author. E-mail addresses:
[email protected] (A. Hussain),
[email protected] (Y.-P. Liu),
[email protected] (S.-L. Chan).
https://doi.org/10.1016/j.istruc.2018.11.001 Received 27 July 2018; Received in revised form 6 October 2018; Accepted 4 November 2018 Available online 10 November 2018 2352-0124/ © 2018 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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estimate the first yield moment. Following the same procedure used by Aydın and Doğan [8] the first yield moment for angle section under the bi-axial bending and normal forces were predicted. Furthermore, angle sections, especially Class 1 (i.e. plastic) and Class 2 (i.e. compact) cross sections, have to be designed using the full plastic capacity of the angle sections. The interaction formulas for the plastic capacity of angle section under bi-axial bending have been proposed by some researchers [7,9,10]. The idealization of the angle section to single lines without geometrical thickness, under an assumption that small leg thickness when compared with leg widths, has been used by different researchers to simplify the calculations of the plastic surface of angle section. However, this idealization may lead to a significant error especially when the neutral axis intersects the leg at a small angle [9]. As a result, the interaction equation proposed by Charalampakis [9] was referenced in this study to estimate the plastic capacity for further stability checks. Buckling resistance of angle sections extensively attract many researchers. Residual stresses (RS) and initial geometric imperfections are the most significant factors affecting the buckling resistance, especially in the inelastic range. Different residual stress distribution patterns were provided in literature [11,12]. Most of the experimental investigations of the RS distribution patterns have been focused on the steel equal-leg angle, while the study on the unequal angle was insufficient. The different residual stress distribution patterns with different amplitudes were incorporated in the present study to achieve the well-recognized distribution pattern for residual stress to be comparable with the realistic distribution. The inelastic flexural and flexuraltorsional buckling of axially loaded angle members were investigated theoretically in literature. It has been found that the flexural buckling dominates in the most of struts except for single unequal angle members flexural-torsional buckling mode is the dominant failure mode [13]. Furthermore, lateral torsional buckling (LTB) of steel beams are extensively studied in literature. Trahair provided a number of related papers concerned with the theoretical investigation of the buckling behavior of steel angle beam [14–17]. Little research has been devoted to the investigation of the buckling behavior of the eccentric equal-leg and unequal-leg angle member using finite element approach in which the actual residual stress distribution pattern could be incorporated during the analysis. Earls [18,19] used the finite element techniques to study the behavior of equal leg single angle bent about only the geometric axis. Liu and Hui [20] provided a finite element model for equalleg single angle beam-column, and an interaction equation based on the AISC [1] had been proposed using the critical eccentricity scenario. However, the compression and flexural resistance were kept the same as in AISC [1]. The purpose of the current paper is to develop a reliable and efficient three-dimensional finite element (FE) model to investigate different failure modes and ultimate capacities for steel equal-leg and unequal-leg single angle member under bi-axial bending and normal forces. The finite element software ABAQUS [21] has been used to study the member behavior and the different buckling modes such as flexural (FB) and flexural-torsional buckling (FTB) for single angle strut under eccentric compression. Also, the lateral torsional buckling (LTB) of single angle beam under uniform bending moment, which represents the critical case for LTB, is considered. The stress-strain curves and different residual stress distribution patterns measured experimentally and mentioned in the literature are incorporated in this model using subroutine SIGINI in ABAQUS [21]. Moreover, the extensive parametric studies are provided. The cross-section configuration, steel grade, residual stresses and initial geometric imperfections are investigated. Finally, the results for this model are compared to different design formulations reported in different design codes such as EC3 [2] and AISC [1] to calibrate against code values and a new interaction equation based on the full plastic capacity of single angle section is presented.
2. Finite element model A three-dimensional finite element model was developed using the commercial software ABAQUS [21] to investigate the single angle member behavior under different load conditions and combinations. Various single angle cross-sections (i.e. equal and unequal angles) and a wide range of slenderness ratios were simulated in this model. The details of the current model including the elements description, material model, boundary conditions, loading applications, initial geometric imperfections and residual stress are summarized in the following sections. 2.1. The shell finite element model The actual geometry of the angle section is more precisely modeled using the solid finite elements. However, using the shell finite element model is more economical, regarding computational time, to represent the efficient configuration of the angle cross-sections with sufficiently accurate prediction [12]. Može et al. [12] reported that relative difference between the solid element model with actual geometry and the shell element model using uniform thickness is < 1%. Thus, the latter can predict accurate and consistent results for extensive parametric study. Furthermore, the shell finite elements are more convenient to assign the residual stresses values at different integration points, especially at the angle heal and leg edges allowing for different residual stresses distribution patterns. For this reason, the 4-node shell finite element with reduced integration (S4R) allowing for shear deformation in ABAQUS was used in the current model. For all angle sections studied in this paper, their breadth-to-thickness (b/t) ratios are > 5 and therefore they can be effectively simulated by shell element. The nominal thickness of the angle section was adopted by the thickness of the shell element, while the leg widths were described by the nominal angle widths minus the half of the leg thickness as shown in Fig. 1. The lengths of the angle members were varied to cover the slenderness ratios λv from 50 to 200 within the practical range. The angle width-to-thickness ratio was limited to avoid the local buckling failure before achieving the full plastic capacity of the cross-section. This paper aims to determine the full plastic capacity of the angle member subjected to an eccentric load. To allow for the eccentric point load application and to avoid possible bending and interference with the angle elements, a discrete rigid plate attached to the top nodes of the angle section was modeled as an end-loading plate. The material modeling was the elastic-plastic model with a modulus of elasticity (E) equal to 210 GPa and Poisson's ratio of 0.3. The strain hardening ratio of 0.01 as shown in Fig. 2 is used while fy is taken the characteristic yield strength. The steel grades S235, S275, S355 and S460 complied with EC3 have been studied in this paper. In addition, to fully control the finite element sizing and meshing the NODE, NGEN and NFILL functions have been used to generate the input file with the leg widths divided into 20 equal parts with further subdivisions to model the residual stress distribution pattern, while in the longitudinal direction the size of the element is equal to 1/20 of the leg width. Thus, the integration points were identified with ordered numbers and became well-defined to assign the initial residual stress values especially at the edges and the center of the angle width. The end-loading plate was meshed using the same sequence of the angle member and coinciding with the top angle nodes as shown in Fig. 3. 2.2. Boundary conditions and applied loads Due to the symmetry of the simply supported conditions and applied loads, only half length of the angle member was modeled, seen in Fig. 3. The reference point at top of the angle member was coupled with the top-edge nodes, and the displacements in the plane of the cross-section (Ux and Uy) as well as the rotation about the longitudinal axis (Rz) were constrained. Moreover, a rigid body constraint was adopted between 374
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Fig. 1. Typical cross-section details and shell elements.
the top loading plate and the corresponding reference point. Note that in reality the critical section can be any section close to the mid-span due to initial imperfections. Generally, the design codes use this idealized “simply supported column” model with codified imperfections such as the geometric imperfections and the residual stresses for practical design. Once the imperfections are incorporated into the idealized model, which will be introduced in next section, the actual conditions as well as the uncertainty can be taken into account indirectly and practically. The load cases in the current study were divided into three cases, i.e. concentric loading, uniform bending and eccentric loading. A concentrated force was applied to the top-reference point in case of concentrically and eccentrically loaded cases, where the eccentricity is controlled by coordinates of the loading point. However, in case of a uniform bending about one or two principal axes (i.e. u-axis and v-axis), two geometrical axes concentrated-moment components were applied
Fig. 2. Material modeling in the FE model.
Fig. 3. Meshing details, boundary conditions and load applications in the FE model. 375
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Fig. 4. First eigenvalue buckling mode shape: (a) flexural buckling; (b) flexural-torsional buckling.
It can be seen that the displacement contour is uniform along the member length for flexural buckling about weak v-axis while the flexural-torsional buckling is non-uniform. Note that the local imperfections were not included in this study for the Class 1 and Class 2 crosssections and only the global imperfections following the FB or FTB mode shape were considered. It was reported that the axially loaded equal angle members are generally dominated by FB while the unequal angle members are governed by FTB [13]. However, for most eccentrically loaded cases, FTB is the dominant buckling mode for both equal and unequal angles. In Section 5.3 of this paper, a simple equation for determination of the elastic critical buckling load which is related to the first buckling mode (FB or FTB) is proposed for the calculation of the non-dimensional slenderness ratio. The residual stress distribution patterns for single equal-leg angles were experimentally investigated, and the different patterns were evaluated in different studies [11,12]. The maximum amplitude of the
to the top-reference point which was coordinated at the center of gravity of the cross-section. It should be noted that the uniform bending case with these boundary conditions represents the most critical case in related to lateral torsional buckling. 2.3. Initial geometric imperfections and residual stresses The initial geometric imperfections and the residual stresses have a significant effect on the inelastic buckling behavior of the steel members. To model geometric imperfection, an eigenvalue buckling analysis was conducted first to obtain the first buckling mode which was scaled as initial imperfection mode before nonlinear analysis. For both FB and FTB, the maximum amplitude of the initial imperfection shape was suggested as 1/1000 of the member length or 3 mm, whichever is larger [22]. The first buckling modes for different load cases are shown in Fig. 4. 376
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Fig. 5. Residual stress distribution models: (a) Shi et al. [28]; (b) Može et al. [12] and (c) Može et al. [12].
residual stress (RS) did not generally exceed 25% of the yield stress. It should be noted that the RS only had a noticeable influence for the concentrically loaded member, while in case of an eccentrically loaded member, this influence did not generally exceed 5% of the single member capacity [23]. In the current study, the well-recognized 3points RS distribution pattern with peak value 0.2fy shown in Fig. 5(c) was incorporated into the proposed FE model. This pattern was considered as the more conservative distribution and provided in most of the specifications and similar documents. To adapt the initial stress field for the previously identified elements in the current model, a FORTRAN subroutine was written to be called by the ABAQUS before the analysis step, and a self-equilibrating analysis for residual stress was carried out in the first step before the loading step. Because the RS distribution pattern was evaluated in most cases for equal leg angle, an experimental investigation needs to be carried out to recognize the favorable pattern for unequal-leg angle, especially for the short leg length ratio equal to a half of the long leg.
Ne 3 {r22 − uo2 − vo2} − Ne 2 {(Pu + Pv + Pz ) r22 − Pv uo2 − Pu vo2} − Ne r22 {Pu Pv + Pv Pz + Pz Pu} − {Pu Pv Pz r22} = 0
(1)
in which the three reference buckling loads associated with the major flexural buckling load (Pu), minor flexural buckling load (Pv) and torsional buckling load (Pz) are calculated as follows.
Pu =
π 2EIu π 2EIv , Pv = , Pz = (GJ + π 2EIw / L2)/ r22 2 L L2
(2)
where EIu is the flexural rigidity about major axis; EIv is the flexural rigidity about minor axis; EIw is the warping rigidity; GJ is the Saint Venant torsional rigidity; uo and vo are the coordinates of the shear center with respect to the centroid; and r22 = uo2 + vo2 + (Iu + Iv)/A. For uniform bending about major axis case, the elastic critical buckling moment (Me) is given by 2
β Pv β Pv ⎤ ⎡ Me = ⎢± 1 + ⎛ u ⎞ + ⎛ u ⎞ ⎥ Mcr 2Mcr ⎠ 2Mcr ⎠ ⎝ ⎝ ⎣ ⎦ ⎜
2.4. Analysis procedure
⎟
⎜
⎟
(3)
in which,
The whole analysis procedure requires two finite element models. The first model was for eigenvalue buckling analysis and the second was for nonlinear analysis. The commercial software ABAQUS [21] is used in this study. The eigenvalue analysis was implemented firstly to verify the FE model and, as mentioned earlier, to initiate the first buckled mode to be used as the initial geometric imperfection. The first buckling load for each load case was recorded and denoted as Ne or Me depending on whether it was a concentric load case or a uniform bending case. The static nonlinear analysis allowing for material yielding was carried out after the eigenvalue buckling analysis. The modified-Riks method is adopted to solve the nonlinear problems. The limit point in the load-displacement curve was recorded as the ultimate capacities Nult or Mult depended on the loading cases.
Mcr =
π 2EIv ⎛ π 2EIw ⎞ GJ + 2 L ⎝ L2 ⎠ ⎜
⎟
(4)
where βu is the monosymmetric constant. For unequal angles, this constant is positive for short legs in compression and negative for long legs in compression; while it is zero for equal angles.
βu =
1 Iu
∫ v (u2 + v2) dA − 2vo A
(5)
Various angle cross-sections with different leg length ratios β (=b/ h) from 0.5 to 1.0, different slenderness ratios and different width-tothickness ratios, as shown in Table 1, were simulated by the proposed FE model and evaluated using linearized eigenvalue buckling analysis. Totally 720 models have been analyzed and the corresponding results were summarized in Table 1. The analytical solutions from Eqs. (1) and (3) were used for validation purpose. From Table 1, it can be seen that both the elastic buckling loads and the elastic buckling moments from FE eigenvalue buckling analysis agree well with the theoretical values obtained from Eqs. (1) and (3), respectively. Thus, the proposed FE model has sufficient accuracy to predict the buckling behavior of angle members with extensive configurations. For further validation of the FE model, the experimental tests reported by Liu [25,26] including concentrically and eccentrically loaded single equal and unequal angle members were studied here. Details of the specimen configuration, material properties, loading conditions and test results, as well as the corresponding FE results, were tabulated in Table 2. Note that due to the limited data for actual residual stresses and geometric imperfections of the tested members, the key parameters
3. Verification of the proposed finite element model To validate the FE model, the theoretical solutions for the elastic critical buckling loads referring to Trahair [24] were used to check the accuracy of FE buckling analysis results considering three different loading cases as: Case 1: Axial compression; Case 2: Uniform bending about major axis (short leg compression); and Case 3: Uniform bending about major axis (short leg tension). For axial compression case, the elastic buckling load (Ne) is the lowest root of the cubic equation below 377
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Table 1 Comparison between the FE results and the theoretical solutions. λ h/t Case 1, Case 2, and Case 3 β 0.5
Slenderness ratios Leg width to thickness ratios Loading type Leg length ratios (b/h) Ntheoritical/NFEM (Case 1) Mtheoritical/MFEM (Case 2 & 3)
Avg. COV Avg. COV
1.008 3.50% 1.08 10.21%
50, 60, 80, 100, 120, 140, 150, 160, 180, 200 8, 10, 12.5, 15 0.60
0.70
0.80
0.90
1.00
1.014 2.56% 1.09 9.58%
1.015 2.45% 1.06 8.20%
1.013 2.60% 1.01 7.30%
1.017 2.20% 1.02 8.50%
1.018 1.85% 1.016 4.51%
Note: Case 1 is axial compression; Case 2 is uniform bending about major axis (short leg compression), and Case 3 is uniform bending about major axis (short leg tension).
section size L150 × 100 × 12, and under concentric and eccentric compression. Fig. 7 shows that initial imperfection has a significant effect for concentric compression when compared with the eccentric angle members. The relative variations, between the ultimate load capacities of imperfect to perfect members, are plotted in Fig. 7(a) and (b). In case of concentric members (i.e. e = 0.0), the maximum variation was recorded as 35% from the perfect column at slenderness ratio of 80 (i.e. λ′ = 1.05), while it did not exceed 21% in case of eccentric compression. Further, the buckling strength of the member reduces when the magnitude of the initial imperfections increases. It is clear that the variations increased with higher slenderness ratio and start decreasing again after slenderness ratio equal to 80, and then it becomes approximately constant in the elastic range (i.e. λv > 100). By plotting the deviation between the ultimate load of curve “b” calculated from the clause 6.3.1 in EC3 [2] and the perfect case in FEM, Fig. 7(a) shows that the capacity for member with imperfection of L/250 and free from residual stress provides a lower bound for the buckling curve “b” in the inelastic range, while members with L/380 cover the lower bound capacities in the elastic range. Thus, the limited magnitude of initial imperfection not < 3 mm recommended by different researchers and applied in the present study is appropriate. In addition, the value of initial imperfection of L/1000 or 3 mm with residual stress agrees well with the buckling curve “b”. On the other hand, Fig. 7(b) shows that the member with imperfection and eccentric moment provides higher capacity than the perfect column, especially in the inelastic range (i.e. λv < 80). It is probably due to the combination of the bending action and imperfection. Because the residual stresses with the geometric imperfections have a combined effect on the buckling strength of the column, the different residual stress distribution pattern shown in Fig. 5 were incorporated in the FEM considering the greater of initial geometric imperfection of L/ 1000 and 3 mm. The peak value coefficient (ξ) for different RS distribution varied from 0.2, 0.25 and 0.3. From Fig. 8, it is clearly seen that RS only had a noticeable influence for concentrically loaded members, while, in case of eccentrically loaded member, this influence did not exceed 6% on average the capacity of the single member, with the exception of higher peak coefficient (i.e. ξ = 0.3) with 3-points residual stress distribution pattern, where it almost reached 8.7% at slenderness ratio (λv) equal to 80. On the other hand, the concentrically loaded member shown in Fig. 8(a) indicates higher variations in the inelastic range (e.g., 12.5% for a 3-point RS with peak value of 0.3fy).
described in Section 2 were incorporated. From these results, the mean values of the ratios of the experimental results Pexp to the predicted FEM results PFEM are 1.04 with coefficient of variance (COV) equal to 6.39% for equal angle and 8.18% for unequal angle. On the other hand, for more credibility and reliability of the FEM, Fig. 6 shows the load versus deflection curves from the FE analysis using second-order nonlinear analysis against those from experimental tests by Liu and Chantel [25]. The discrepancies mainly come from the limited data of the actual geometric imperfection and residual stresses and therefore the codified imperfections were used for conservative design. In general, it could be concluded that FEM can predict the ultimate load capacity and the behavior of the single angle member in good agreement with the experimental tests. 4. Parametric study After verification, the proposed FE model is further used to study the different parameters affecting single angle member capacity. Liu and Hui [20] provided a parametric study for the single equal angle member, including slenderness parameter λ′ from 0.27 to 2.39, eccentricities with respect to major and minor axes (e = 0.5, 1.0, 1.5 and 2Uo), where Uo is the distance between the shear center and the centroid of the cross-section measured in the direction of the major axis (U) and the leg width-to-thickness ratios (h/t). In the present study, different parameters such as yield stress or design strength, initial geometric imperfections and the residual stress distribution patterns were investigated to evaluate the effect of these parameters on buckling behavior of single angle members and their section capacities. The parametric studies are also carried out on the unequal angle sections L150 × 100 × 12 to investigate the influences of the above parameters. These investigations on unequal angles were rarely reported in literature. The same material model mentioned in previous sections is incorporated into the present study. 4.1. Effects of geometric imperfections and residual stresses Different initial imperfections were provided in the literature. To investigate the sensitivity of the flexural-torsional buckling behavior of the angle member, different imperfections such as L/250, L/380, L500, L/1000, and L/1500 have been analyzed using the proposed FE model. The perfect members (i.e. without imperfections) were also studied for comparison. The angle member was made of steel grade S355 with Table 2 Comparison of numerical and experimental results. Source
Liu and Chantel [25] Liu and Hui [26]
No. of specs
26 28
λ = L/rv
Angle size
fy (MPa)
Loading condition
110-160 95-125-155
76 × 51 × 6.4 51 × 51 × 6.4
356 330–348
CL, EL (Major), and EL (Minor). CL, EL (Major), and EL (Minor).
Avg. of Pexp/PFEM
COV %
1.00 1.04
8.18 6.39
Note: CL = concentrically loaded member; EL (Major) = eccentrically loaded causing major-axis bending, EL (Minor) = eccentrically loaded causing minor-axis bending. 378
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Fig. 6. Comparison of numerical and experimental results (Liu and Chantel [25]).
Fig. 7. Relative deviation between FEM “perfect member” and different initial imperfections: (a) concentrically loaded; (b) eccentrically loaded.
By comparing the three incorporated RS distribution patterns shown in Fig. 5, it could be concluded that the 4-points RS distribution pattern has the least significant effect on the ultimate load capacities of the angle member, and in most cases gives higher load capacities than the case without considering the RS. However, the 3-points RS and the stepped RS show a good agreement with each other, which are considered the more conservative patterns. In addition, Fig. 8 shows that the increase of the peak value (ξ) decrease the buckling strength of the member. From the previous studies, and due to the fact that the effect of the initial imperfections and RS are less significant in case of eccentric members, the 3-point RS distribution pattern with peak value 0.2fy
together with initial imperfection L/1000 or 3 mm whichever is greater have been used in the following analysis. 4.2. Influence of steel grade In addition to the effect of the geometric imperfections and the residual stresses, the effect of different steel grades with different yield stress on the buckling strength of the concentrically and eccentrically loaded angle members has been investigated in this section. Fig. 9 shows the relationship between the ultimate load versus the axial shortening of the eccentrically loaded member, considering four steel 379
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Fig. 8. Relative deviation between FEM “without RS” and different RS distribution patterns: (a) concentrically loaded; (b) eccentrically loaded.
same procedures proposed by Aydın and Doğan [8] with the elastic stress distribution assumption are used to determine the most critical stress at angle section.
grades (S235, S275, S355 and S460), and the lengths of two members associated with slenderness ratios λv equal to 50 and 150. Fig. 9(a) provides the load-displacement curves for an axially loaded member, of which the ultimate load for the short column with λv equal to 50 and steel grade S275, is 661 kN, being 33.6% lower than the ultimate load of steel grade S460. However, in case of the slender column with λv equal to 150, the reduction decreases to only 7.6%. Moreover, for the eccentrically loaded member, the ultimate load for the short column with steel grade S275 is 424.2 kN, which is 38.4% lower than the load for steel grade S460 and this ratio also drops to 4.2% for the slender column as shown in Fig. 9(b). As a result, Fig. 10 concludes and agrees with the fact that the influences of changes of steel grades in higher slenderness ratios are negligible, this is due to the buckling controls for high slenderness members. Further, in lower slenderness regions, increasing the steel grades will enhance the member capacities up to approximately 88% for eccentrically loaded member and 73% for concentric members if the steel grade S460 is used in place of grade S235 at slenderness ratio λv of 50. Finally, it should be noticed that as the slenderness ratio increases, the effect of the eccentricities on the buckling strength decreases.
σ=±
N M M ± uv ± vu A Iu Iv
(6)
where, N, Mu and Mv are the applied design axial force and the applied bending moment about two principal axes; A, Iu and Iv are the crosssection area and the second moments of area about the principal u- and v-axes respectively; u and v are the coordinates of the critical points with respect to principal axes. The most critical stress should be calculated at the critical points marked as I to VI (seen in Fig. 11) and limited to the yield stress fy. The signs of the three terms have been taken as follows. The sign of the normal force is negative for compression and positive for tension while the sign of the two bending terms follows the coordinates of the critical points. As shown in Fig. 11, the positive coordinates for point (III) will lead to positive terms for the bending stresses as they are under tensile stress. Thus, the first yield moment about the major axis (Mu-el) considering the interaction between the normal force and the bi-axial bending is given by the following equations.
5. Proposed design method 5.1. First yield moment
Mu − el = m yu (h2t ) f y [1 − sign(σy ) n]
(7a)
Mv − el = αMu − el
(7b)
where, myu is the dimensionless coefficient given in Appendix B and obtained using the same method in reference [8] but with the consideration of normal force; h and t are the angle height and thickness respectively; fy is the yield stress; sign(σy) is the signum function for the yield stress which is equal to 1 and −1 for positive and negative yield stresses respectively (see Table B.1); n = N/NPL, NPL = Afy, and α is the ratio between minor to major principal axis bending changing between
Aydın and Doğan [8] proposed the first yield moment at the critical points of the single angle cross-section accounting for the combinations of the major and minor axis bending or the bi-axial bending case. However, the presence of the normal force has a significant effect on the stresses at these points as they mainly depend on the sign of the normal force as compression or tension force. From this perspective, the 380
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+1.0 and −1.0, which are considered as the most practical values, and follow the positive direction shown in Fig. 11. For other cross-sectional properties such as second moment of area Iu and Iv, shear center coordinates with respect to center of gravity of the cross-section in the two directions (i.e. geometrical and principal axes directions), and the principal axes inclination with respect to geometrical axes (α), Fig. 12(b) should be referred. Note that the values in Fig. 12(b) have been obtained by fitting the actual properties of the different angle cross-sections, which were calculated for a wide range of variables (i.e. β and h/t) to get the actual dimensions (h × βh × t). 5.2. Full plastic capacity To design the single angle member, especially Class 1 (plastic) and Class 2 (compact) cross-sections and to take the advantage of the plastic capacity of an angle section, the plastic surface of the angle section has to be defined. Some researchers idealize the cross-section to a single line without geometrical thickness to facilitate the calculation of the interaction formula of the plastic surface [7,8]. However, this idealization entailed a significant error especially when the plastic neutral axis intersects the leg at a small angle [9]. Thus, the interaction formula proposed by Charalampakis [9] has been implemented in the present studies for further stability checks to enhance the prediction of the single angle member capacity under eccentric compression. Fig. 12(a) plotted the full-plastic interaction curves for single angle cross-section under various limits of compression force (i.e. n = 0.0 to −0.9). Defining Mu-pL and Mv-pL as the full plastic capacity for the single angle section subjected to major and minor axis bending respectively, and applying the same method in reference [7] with different leg length ratios (β = 0.5 to 1.0), we have Table 3 as below. Regarding the cross-section classifications, Table 5.2 in EC3 [2] should be referred for the angle sections subjected to pure compression or the values of the flange outstands of hot-rolled I-section beams subjected to flexural bending. Trahair [14] provided an extensive study for the angle section classification using local buckling coefficients accounting for elastic and plastic stress distribution. Furthermore, the optimistic and conservative limits associated with plastic and compact cross-sections have been presented. Table 4 shows the plastic limit (λp) and compact limit (λc) for angle sections under major and minor axis bending. All cross-sections with different steel grades were verified using the procedures proposed in reference [14]. An example demonstrating the section classification will be detailed in Section 5.6.
Fig. 9. Effect of yield stress of steel members: (a) concentrically loaded; (b) eccentrically loaded about major axis (e = Uo).
5.3. Elastic buckling loads For further stability checks of single angle members, the elastic critical buckling loads including the flexural buckling (FB) and flexuraltorsional buckling (FTB) due to the axial compression, and the lateral torsional buckling (LTB) due to major axis bending need to be investigated. Trahair carried out theoretical investigations of the buckling behavior of steel angle beams [15–17]. Aydın and Doğan [8] provided an equation using dimensionless coefficient to predict the elastic critical buckling moment Me for LTB and reproduced as
Me = mb (Et 3)
(8)
where, mb is the dimensionless coefficient for beam buckling which can be obtained from Aydın and Doğan [8]. This coefficient is depended on the slenderness ratio λ (=Lt/h2) and the leg length ratio β. On the basis of theoretical Eq. (1), and using the FE results, the curve-fitting technique has been implemented to derive the following simplified equation to predict the elastic critical buckling force due to FB and FTB. Fig. 10. Effect of yield stress on buckling resistance for concentrically and eccentrically loaded members.
Ne = mc (Et 3/ h)
(9)
where, mc is the dimensionless coefficient from Eq. (10), and λ is the slenderness ratio and equals to Lt/h2. 381
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Fig. 11. Positive directions for the elastic stress distributions.
Fig. 12. Plastic design of angle section: (a) full plastic surface [9]; (b) section properties.
mc = (0.55β 2 + 0.35β − 0.16) λ(−0.167β
2
+ 0.2β − 2.035)
Table 4 Plastic and compact limits for cross-sections classifications [14].
(10)
A large number of FE results, which have been summarized in Table 1, were fitted to propose Eqs. (9) and (10). The parameters λ (=Lt/h2) and β (=b/h) have clear physical meanings. This equation is applicable for both equal and unequal angles undergoing FB and FTB. From this study, the elastic flexural and flexural-torsional buckling loads for single angle member can be calculated using a simplified equation rather than the complex equations as indicated in Eqs. (1) and (2). The proposed equation is also able to automatically find the critical buckling mode which may be FB or FTB for concentric loads. Fig. 13 shows that the proposed equation for predicting the elastic
Load condition
Major bending (u-axis) Minor bending (v-axis)
Plate slenderness ratio
λ=
h t
fy 250
Plastic limit λp
Compact limit λc
λp, u = 12 λp, v = 10
λc, u = 16 λc, v = 14
critical buckling load is in a good agreement with the FEM results and the theoretical Eq. (1) for both equal (e.g. L100 × 100 × 10) and unequal (e.g. L100 × 50 × 10) angles, especially at higher slenderness
Table 3 Plastic moment capacities about two principal axes. β = b/h = 0.5
0.6
0.7
0.8
0.9
1.0
γ1 γ2 Mu − pL
0.49 0.26 0.448(h2t) fy
0.59 0.209 0.487(h2t) fy
0.692 0.158 0.529(h2t) fy
0.794 0.106 0.579(h2t) fy
0.898 0.052 0.639(h2t) fy
0 1.0 0.707(h2t) fy
γ1 γ2 Mv, pL′
0.066 0.684 0.1249(h2t) fy
0.136 0.663 0.172(h2t) fy
0.216 0.634 0.221(h2t) fy
0.304 0.596 0.269(h2t) fy
0.399 0.55 0.315(h2t) fy
0.5 0.5 0.353(h2t) fy
Note: γ1 and γ2 are the ratios of the points of intersection of the plastic neutral axis with the legs; approximate equations for Mu-pL and Mv-pL are provided in Fig. 12. 382
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Fig. 13. Elastic critical buckling load for single angle column.
angle member under axial compression and bi-axial bending has been proposed. The proposed method is suitable for both Class 1 and Class 2 cross-sections accounting for the section plasticity and the member buckling such as FB and FTB. Charalampakis [9] firstly proposed a set of interaction equations for the determination of plastic resistance of angle sections under bi-axial bending and normal force. To extend the work of Charalampakis [9] allowing for more complex behaviors such as FB and FTB, the first step of our work is to simplify the interaction equations as a single equation rather than a set of equations. The original interaction equation is rewritten as
regions. However, there is an overestimation of the buckling load in the case of unequal angle with smaller leg length ratio (β = 0.5) and lower slenderness ratio as shown in Fig. 13(b). In addition to the elastic buckling load for an axially loaded single-angle column, and elastic buckling moment for a single-angle beam, the elastic buckling capacity for single angle beam-column under interactive flexural-torsional and flexural buckling for beams and columns has to be investigated. This interaction is not easy to interpret physically especially for asymmetric cross-sections [27]. For this reason, the simplified interaction Eq. (11) for elastic buckling capacity proposed by Trahair [24] is incorporated in the present study.
N Mu + =1 Ne (1 − N / Pu ) Me
|mu | ⎞ ω = n2 + μs m v + ρ ⎜⎛ ⎟ ⎝ 1 + μs n ⎠
(11)
=1 (13)
with,
in which N and Mu are the applied design axial force and moment, Ne and Me are the elastic critical loads from Eqs. (9) and (8) respectively, and Pu is the major flexural buckling load from Eq. (2). From the previous discussion, it could be concluded that the minimum load amplifier of the design loads to reach the elastic critical loads with regards to lateral or lateral torsional bucking (denoted as αcr) could be calculated as follows.
1 N Mu = + α cr Ne (1 − N / Pu ) Me
α
nμs ≥ 0: α = 2, ρ = 1 nμs < 0: α = 2 − |n|, ρ =
(1 − |n|)(1 + 3 |n|) (1 + |n|)2 − |n |
μs = sign(m v − 2n (1 − |n|) where ω is the section capacity factor. In Eq. (13), the sign of the second term μsmv can be considered by the signum function to cover the two parts of the interaction curve. Note that the sign (.) is the signum function with sign (x ≥ 0) = +1 and sign (x < 0) = −1. Thus, μs is equal to +1 for the upper part of the interaction curve (i.e. mv ≥ 2n (1 − |n|)), and μs = −1 for the lower part (i.e. mv < 2n (1 − |n|)). The second step was to calculate the reduction factor (χop) accounting for FB and FTB of single angle member. The reduction factor is mainly based on 2160 FEM results considering geometrical and material nonlinearities. The following parameters have been considered in our parametric study:
(12)
For evaluation of Eq. (12), two angle cross-sections (L100 × 100 × 10 and L150 × 100 × 12) with different slenderness ratios (λv) equal to 50, 60, 80, 100, 120, 140, 150, 160, 180, and 200 and under eccentric compression with respect to major u-axis have been investigated using the proposed FE model. Eccentricities (e) are also varied as 0.0, 0.5, 1.0, 1.5, 2 and 3Uo, where Uo is the distance between the shear center and the centroid of the cross-section in the direction of the major u-axis. The ultimate load “Nult” from FE analysis, the major axis bending Mult = Nult ∗ e, the elastic critical loads (Ne, Me and Pu) mentioned earlier have been incorporated in Eq. (12) to obtain the value of αcr. In addition, the ratio of the elastic loads to ultimate loads from FE analysis to get the minimum load amplifiers of the ultimate loads to obtain the elastic critical loads have been recorded as αcr, FEM. The mean value of the ratios of the predicted FEM results “αcr, FEM” to the numerical results “αcr-num” is 1.006 With the COV of 8.53% for unequal angle, whereas for the equal angle the mean value is 1.1 with COV equal to 9.2%, which means that the results of Eq. (12) are in a good agreement with the FEM results.
(1) Leg length ratios for unequal to equal angle members, β = 0.5, 0.6, 0.7, 0.8, 0.9 and 1; (2) Width to thickness ratio, h/t = 8, 10 and 12.5; (3) Different slenderness ratios, λv (=L/rv) = 50 to 200; (4) Two load cases, i.e. axial compression and eccentric compression causing major and/or minor axes bending with wide range of eccentricities (e = ± 0.5, ± 1.0, ± 1.5, ± 2 and ± 3Uo), where Uo is the distance between the shear center and the centroid of the crosssection measured in the direction of the major u-axis.
5.4. Proposal of a new design equation From the above, a new interaction equation is proposed and expressed as
Based on the previous sections, a new design method for single 383
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Table 5 Buckling curve parameters. Loading type
CL
αimp β λo Corresponding EC3 buckling curve
0.34 1 0.2 Curve b
EL(Minor)
EL(Major)
1 1 0.2 –
0.49 0.9 0.5 –
EL(Biaxial) 0.76 1 0.2 Curve d
Note: EL(Bi-axial) = eccentrically loaded causing bi-axial bending.
|mu | ⎞ ω = n2 + μs m v + ρ ⎜⎛ ⎟ ⎝ 1 + μs n ⎠
α
≤ χop
(14)
where, χop is the reduction factor to account for lateral and lateral torsional buckling, which can be calculated from Eq. (15) using the non-dimensionless slenderness ratio λop.
χop =
1 Φ+
Φ2 − βλop2
≤1
Fig. 15. Buckling curve for eccentric angle member and FEM results (minor axis bending).
Φ = 0.5[1 + αimp (λ op − λ 0) + βλop2] λ op =
αult α cr
the elastic critical buckling indicated as Ne and Me, as the flexuraltorsional buckling is an interaction between buckling of column and beam. On the other hand, for minor axis bending the buckling is of the inelastic type when αult ≤ αcr, as the buckling behavior is mostly due to the column buckling without consideration of the lateral-torsional buckling about major-axis bending. It could be concluded from Figs. 14 and 15 that the proposed equation represents a lower bound for the FEM results in case of major-axis bending and minor-axis bending. In summary, the design procedure with the proposed method is shown in Fig. 16.
(15)
where αult is the minimum load amplifier of the design loads to reach the characteristic resistance of the cross-section without considering lateral or lateral torsional bucking equal to 1/ω, αcr has been illustrated previously and could be calculated from Eq. (12), and by fitting the results of the numerical analysis the new values of the buckling curve parameters (i.e. αimp, β, and λo), which mainly depend on the load conditions such as concentric or eccentric members as shown in Table 5. Figs. 14 and 15 show the FEM results together with the proposed buckling curves in case of major and minor axis bending. For eccentrically loaded member about the major axis, the crosssection may achieve the full plastic capacity without reduction due to buckling if the global slenderness ratio (λop) is < 0.5 (i.e. αult/ αcr ≤ 0.25). Moreover, for αult ≤ 4αcr (i.e. λop ≤ 2), the buckling is in inelastic stress range, which means that the section capacity varies between the full plastic capacity and the first yield limit capacity. While, in case of the αult > 4αcr, the elastic critical buckling loads is of greater importance and the member capacity is mainly resulted from
5.5. Evaluation of the proposed equation and comparison with current design codes To evaluate the proposed equation, the experimental tests by Liu and Chantel [25] and Liu and Hui [26] were studied here. The axial resistance and the section capacity factor obtained from the proposed Eq. (14), EC3 [2] and AISC [1] will be summarized in Table 6 against the test results. Pexp and ωexp are the axial resistance and the section capacity factor from experimental tests respectively. PEC3 is calculated from the interaction equation for uniform members subjected to bending and axial compression, seen the clause 6.3.3 of EC3 [2]. PAISC is determined from the interaction equations in AISC [1] (i.e. Eq. (H.1-1) for singly symmetric members subjected to flexural and axial force and Eq. (H.2-1) for unsymmetric members). It should be noted that when using Eq. (14) to get Ppredicted the equality of ω and χop could be achieved by trials and errors. Moreover, in the interaction Eq. (13), the plastic capacity for major and minor axis bending (Mu-pL and Mv-pL) were limited to 1.5 times of the yield stress at the corresponding critical point as provided in AISC specifications. It could be clearly seen from Table 6 that the interaction equation provided in the EC3 gives more conservative values, probably due to the interaction factors in the EC3 equation based on the doubly symmetric section, which is considered a special case for asymmetric section. However, the proposed Eq. (14) achieves the good prediction for both Class 1 and Class 2 cross-sections.
5.6. Numerical example An unbraced member made of L150 × 100 × 12 unequal angle and simply supported over a span of 3 m is shown in Fig. 17. The yield strength fy is 355 MPa and Young's modulus E is 210 GPa. The member is subjected to four loading cases as below:
Fig. 14. Buckling curve for eccentric angle member and FEM results (major axis bending). 384
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Fig. 16. Design flow chart for single angle member under bi-axial bending.
stability of the member are summarized and listed in Tables 7 and 8. For elastic buckling, and according to Eq. (9), Ne = 303.72 kN, Pu = 1743.2 kN, and applying Eq. (8) with negative βu (long leg in compression) Me = 48.14 kN·m. It should be noted that the proposed Eq. (14) leads to safe design (i.e. ω < χop), while the interaction equations specified in EC3 and AISC (i.e. clause 6.33 and clause H. respectively) will probably lead to unsafe design. Furthermore, the ultimate load predicted by the proposed Eq. (14) agrees well with the FEM result (i.e. χop < χop[FEM]), seen in Table 8.
Table 6 Summary of comparison for test-to-predicted values from various methods. Average value
Liu and Hui [26] Liu and Chantel [25]
a) b) c) d)
Axial resistance: Pexp/Ppredicted
Section capacity: ωexp/ωpredicted
Eq. (14)
Eq. (14)
EC3
AISC
EC3
AISC
1.07
1.33
1.19
1.15
1.86
1.51
1.08
1.27
1.22
1.2
1.73
1.66
6. Conclusions
N = −245 kN (concentric); N = −130 kN, and eu = 130 mm (eccentric); N = −130 kN, and ev = −50 mm (eccentric); N = −130 kN, eu = 75 mm, and ev = −25 mm (eccentric).
The interaction between the normal force and bi-axial bending occurs because of unavoidable end connections and transverse wind loading in single angle member. The linear interaction equations provided in different design codes are generally conservative, especially for the plastic design of Class 1 and Class 2 angle cross-sections. However, the relationship between the axial force and the bending moments is nonlinear which can be clearly observed in Fig. 12. To take advantage of the plastic capacity of the single angle members and considering the buckling resistance, a new interaction equation is proposed in this paper based on extensive finite element (FE) analysis results to achieve a more economical design.
For the cross-section classification, h = 138 mm (the outstand leg), h
f
y λ= t = 13.7 . 250 For major axis bending, λc, u = 16 > 13.7 and therefore the section is compact. For minor axis bending, λc, v = 14 > 13.7 and therefore the section is compact. The verification of the cross-section capacity and the check of the
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Fig. 17. Numerical example – angle section L150 × 100 × 12. Table 7 Cross-section properties and plastic moment capacity. Load case
n (−ve)
Mu- ela (kN·m)
Mv-el (kN·m)
Wel(u)b (cm3)
Wel(v) (cm3)
Mu-pLc (kN·m)
Mv-pL (kN·m)
(a) (b)
0.241 0.128
– 23.0
– –
– –
– 38.5
– –
(c)
0.128
–
9.88
– 72.5 (II) –
–
16.8
(d)
0.128
−18.1
6.03
−31.6 (I) 27.3 (VI)
48.7
14.6
−97.4 (VI)
• •
Note: a First yield moment about major axis due to the interaction of normal force and bi-axial bending. b Elastic section modulus at the most critical point (seen in Fig. 17). c The plastic moment capacity from Fig. 12(b) and limited to 1.5 times the first yield moment (=1.5 × Wel × fy).
using dimensionless coefficients, seen in Appendix B. This equation considers the interaction between the normal force and the bi-axial bending. The proposed Eqs. (9) and (10) can predict the elastic flexural and flexural-torsional buckling loads for single angle member. This equation is applicable for both equal and unequal angles undergoing FB and FTB and able to automatically find the critical buckling mode which may be FB or FTB for concentric loads. The proposed interaction Eq. (14) is suitable for the design of both Class 1 and Class 2 cross-sections accounting for the section plasticity and the member buckling such as FB and FTB. This equation is applicable for different load cases compiled with different buckling curves as demonstrated in Table 5.
Nowadays, the second-order direct analysis method (DAM) has been increasingly used in stability design of many steel structures. The unique advantage of DAM is that it only needs section capacity check as the stability at both member and system levels has been reflected in the nonlinear analysis process. The paper provides a simple method to be incorporated into DAM for practical design of steel structures made of angle members such as transmission towers.
Firstly, a FE model was proposed to investigate the buckling behavior of the single angle members under normal force and bi-axial bending. The model was well verified and then used to study the effect of different parameters on the buckling resistance. The initial imperfections such as geometric imperfections and residual stresses have been explicitly incorporated in the proposed FE model. It is found that the initial imperfections will significantly affect the behaviors of single angle members. Also, the imperfections have less influence on the eccentrically loaded members than the concentrically loaded ones. Secondly, a new design equation concerning the elastic and inelastic buckling of single angle member is proposed and summarized in Fig. 16. In summary, the following conclusions can be drawn from this paper.
Acknowledgments The authors are grateful for financial support from the Research Grant Council of the Hong Kong SAR Government on the projects “Second-order Analysis of Shallow Dome Structures made of Tapering Members (PolyU 152047/17E)” and “Second-Order Analysis of Flexible Steel Cable Nets Supporting Debris (PolyU 152008/15E)”; from the Innovation and Technology Fund of the Hong Kong SAR Government for the project “Development of an Energy Absorbing Device for Flexible Rock-Fall Barriers (ITS/059/16FP)”; and from the Hong Kong Branch of the Chinese National Engineering Research Centre for Steel Construction of The Innovation and Technology Fund of the Hong Kong SAR Government for the project “Advanced Numerical Analyses for
• The proposed Eqs. (7a) and (7b) can be used for calculating the first yield (elastic) moments at critical points of the angle cross-section
Table 8 Verification of the cross-section and stability of the member. Load case
ρ
mu
mv
μs
ωa
αcrb
αult
λop
χopc
(a) (b) (c) (d)
– 0.9631 – 0.9631
– −0.43 – −0.20
– – 0.38 −0.22
– +1 +1 +1
0.24 0.28 0.40 0.14
1.24 1.23 2.30 1.54
4.15 3.57 2.50 6.99
1.820 1.698 1.040 2.130
0.247 0.282 0.402 0.159
χop[FEM] 0.26 0.41 0.41 0.29
Note: a Plastic capacity of single angle section according to Eq. (14). b Minimum load amplifier of the design loads to reach the elastic critical loads with regards to lateral or lateral torsional bucking according to Eq. (12). c The reduction factor to account for flexural and flexural-torsional buckling calculated from Eq. (15). 386
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Building Structures Using High Performance Steel Materials”.
Appendix A. Notation The following symbols are used in this paper.
E fy h/t βh u and v X and Y Uo and Vo Xo and Yo βu α Iu and Iv Wel Mu-el and Mv-el Mu-pl and Mv-pl NPL = PY Ne and Me N, Mu and Mv eu and ev n = N/NpL mu = Mu/Mu-pL mv = Mv/Mv-pL λv λ′
Modulus of elasticity. Yield stress. Angle leg height-to-thickness ratio. Short leg length. Major and minor principal axes of angle section respectively. Geometrical axes parallel to short and long leg respectively. Distances between the shear center and the centroid of the cross-section in the direction of the major and minor axis respectively. Distances between the shear center and the centroid of the cross-section in the direction of the geometrical axes. The monosymmetric constant for unequal leg angles. Inclination of principal axis, U, to geometrical axis, X Second moment of area about principal axes. Elastic section modulus. Elastic bending moments, major and minor principal axes = First yield moment for bi-axial bending. Full plastic bending moments, major and minor principal axes. Plastic resistance to normal force or yield load = Afy. Critical elastic buckling load and critical lateral torsional buckling moment. Applied normal force, major axis bending and minor axis bending. Eccentricities with respect to major and minor axes respectively. Ratio of normal force to full plastic resistance. Ratio of bending moment to plastic bending moment, major principal axis Ratio of bending moment to plastic bending moment, minor principal axis Slenderness ratio L/rv
λop
Global non-dimensional slenderness ratio
Non-dimensional slenderness ratio
Npl/ Ne αult / αcr
Appendix B. Dimensionless coefficients for first yield moment Table B.1 Dimensionless coefficient values myu for the calculation of the first yield moment Mu-el = myu (h2t) fy (1 − sign (σy) ∗ n) and Mv-el = α Mu-el. α = Mv/Mu
h/t = 6
8
10
12.5
15
20
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
β = b/h = 0.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 Mv-el
0.0554 0.0605 0.0668 0.0744 0.0841 0.0966 0.1135 0.1376 0.1746 0.2390 0.2869 0.1926 0.1450 0.1162 0.0970 0.0830 0.0714 0.0626 0.0558 0.0503 0.0458 0.0648
0.0578 0.0652 0.0748 0.0856 0.0940 0.1043 0.1170 0.1333 0.1548 0.1846 0.2149 0.2131 0.2113 0.2096 0.2079 0.1972 0.1512 0.1226 0.1031 0.0890 0.0782 0.0511
0.0604 0.0661 0.0729 0.0813 0.0919 0.1057 0.1244 0.1511 0.1924 0.2647 0.3130 0.2027 0.1499 0.1189 0.0985 0.0832 0.0717 0.0630 0.0561 0.0506 0.0461 0.0704
0.0589 0.0664 0.0762 0.0894 0.1021 0.1129 0.1261 0.1429 0.1649 0.1948 0.2267 0.2342 0.2423 0.2509 0.2601 0.2109 0.1623 0.1319 0.1111 0.0959 0.0844 0.0517
0.0647 0.0707 0.0780 0.0871 0.0984 0.1132 0.1332 0.1618 0.2060 0.2834 0.3301 0.2097 0.1537 0.1213 0.1002 0.0843 0.0727 0.0639 0.0570 0.0514 0.0469 0.0754
0.0604 0.0681 0.0783 0.0919 0.1084 0.1195 0.1330 0.1501 0.1721 0.2018 0.2343 0.2483 0.2641 0.2821 0.3026 0.2257 0.1737 0.1412 0.1189 0.1027 0.0904 0.0528
0.0689 0.0753 0.0831 0.0927 0.1048 0.1205 0.1417 0.1720 0.2188 0.3005 0.3446 0.2160 0.1573 0.1237 0.1019 0.0858 0.0740 0.0651 0.0581 0.0524 0.0478 0.0804
0.0621 0.0701 0.0806 0.0947 0.1143 0.1256 0.1394 0.1565 0.1786 0.2078 0.2407 0.2603 0.2834 0.3109 0.3444 0.2420 0.1861 0.1511 0.1272 0.1098 0.0966 0.0540
0.0722 0.0789 0.0871 0.0971 0.1097 0.1261 0.1482 0.1797 0.2284 0.3131 0.3548 0.2206 0.1601 0.1256 0.1033 0.0870 0.0751 0.0661 0.0590 0.0533 0.0486 0.0844
0.0635 0.0717 0.0825 0.0970 0.1178 0.1301 0.1440 0.1612 0.1832 0.2121 0.2451 0.2686 0.2970 0.3322 0.3664 0.2555 0.1961 0.1591 0.1339 0.1156 0.1016 0.0550
0.0769 0.0840 0.0927 0.1033 0.1166 0.1340 0.1573 0.1906 0.2417 0.3303 0.3679 0.2268 0.1639 0.1283 0.1053 0.0888 0.0767 0.0676 0.0604 0.0545 0.0497 0.0902
0.0655 0.0741 0.0853 0.1005 0.1222 0.1362 0.1503 0.1675 0.1893 0.2176 0.2507 0.2792 0.3150 0.3612 0.3971 0.2757 0.2111 0.1711 0.1438 0.1240 0.1090 0.0566
β = b/h = 0.6 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3
0.0715 0.0782 0.0863 0.0963 0.1088 0.1251 0.1472 0.1787
0.0829 0.0888 0.0957 0.1037 0.1132 0.1246 0.1385 0.1560
0.0783 0.0857 0.0946 0.1056 0.1194 0.1374 0.1619 0.1968
0.0864 0.0959 0.1031 0.1116 0.1215 0.1333 0.1477 0.1656
0.0837 0.0916 0.1011 0.1128 0.1276 0.1469 0.1730 0.2104
0.0895 0.1012 0.1086 0.1173 0.1275 0.1396 0.1543 0.1724
0.0888 0.0971 0.1072 0.1196 0.1353 0.1557 0.1834 0.2230
0.0925 0.1053 0.1136 0.1225 0.1329 0.1453 0.1601 0.1783
0.0926 0.1013 0.1118 0.1247 0.1411 0.1623 0.1911 0.2323
0.0949 0.1081 0.1173 0.1263 0.1369 0.1493 0.1643 0.1826
0.0979 0.1071 0.1182 0.1319 0.1491 0.1715 0.2018 0.2451
0.0982 0.1119 0.1222 0.1314 0.1422 0.1548 0.1698 0.1882
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Table B.1 (continued) α = Mv/Mu
h/t = 6
8
10
12.5
15
20
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 Mv-el
0.2273 0.3123 0.3052 0.2231 0.1758 0.1451 0.1235 0.1068 0.0927 0.0818 0.0733 0.0663 0.0606 0.0835
0.1784 0.2085 0.2303 0.2380 0.2462 0.2551 0.2645 0.2511 0.1931 0.1568 0.1320 0.1140 0.1003 0.0700
0.2511 0.3466 0.3311 0.2359 0.1833 0.1498 0.1267 0.1085 0.0943 0.0834 0.0748 0.0678 0.0619 0.0911
0.1884 0.2186 0.2439 0.2596 0.2776 0.2982 0.3221 0.2703 0.2084 0.1696 0.1430 0.1236 0.1088 0.0721
0.2685 0.3708 0.3479 0.2445 0.1885 0.1534 0.1293 0.1104 0.0961 0.0851 0.0763 0.0692 0.0633 0.0973
0.1953 0.2252 0.2525 0.2737 0.2988 0.3289 0.3659 0.2881 0.2222 0.1809 0.1525 0.1319 0.1161 0.0741
0.2844 0.3925 0.3621 0.2519 0.1931 0.1566 0.1317 0.1124 0.0980 0.0868 0.0779 0.0707 0.0647 0.1032
0.2012 0.2309 0.2597 0.2855 0.3171 0.3564 0.4068 0.3064 0.2363 0.1923 0.1621 0.1401 0.1233 0.0761
0.2961 0.4082 0.3720 0.2571 0.1965 0.1589 0.1335 0.1140 0.0994 0.0881 0.0791 0.0718 0.0657 0.1078
0.2054 0.2348 0.2647 0.2937 0.3298 0.3761 0.4375 0.3209 0.2473 0.2011 0.1695 0.1465 0.1289 0.0777
0.3120 0.4293 0.3847 0.2640 0.2009 0.1622 0.1360 0.1162 0.1014 0.0900 0.0809 0.0734 0.0672 0.1142
0.2109 0.2399 0.2710 0.3041 0.3464 0.4023 0.4798 0.3420 0.2632 0.2139 0.1802 0.1556 0.1370 0.0798
β = b/h = 0.7 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 Mv-el
0.0896 0.0982 0.1086 0.1215 0.1378 0.1591 0.1884 0.2307 0.2976 0.4192 0.3180 0.2484 0.2039 0.1729 0.1500 0.1306 0.1144 0.1018 0.0917 0.0834 0.0765 0.1026
0.0962 0.1029 0.1107 0.1197 0.1304 0.1431 0.1586 0.1778 0.2023 0.2346 0.2504 0.2664 0.2846 0.3055 0.3297 0.2888 0.2254 0.1848 0.1566 0.1359 0.1200 0.0924
0.0981 0.1075 0.1189 0.1331 0.1510 0.1746 0.2069 0.2539 0.3285 0.4651 0.3433 0.2630 0.2131 0.1791 0.1545 0.1336 0.1173 0.1045 0.0943 0.0859 0.0788 0.1118
0.1027 0.1098 0.1179 0.1274 0.1384 0.1516 0.1675 0.1871 0.2120 0.2445 0.2661 0.2898 0.3183 0.3529 0.3960 0.3111 0.2434 0.1999 0.1696 0.1473 0.1301 0.0961
0.1044 0.1145 0.1267 0.1417 0.1609 0.1861 0.2206 0.2707 0.3505 0.4969 0.3597 0.2724 0.2193 0.1835 0.1577 0.1362 0.1197 0.1068 0.0964 0.0879 0.0807 0.1190
0.1075 0.1148 0.1231 0.1328 0.1441 0.1575 0.1737 0.1935 0.2185 0.2510 0.2760 0.3049 0.3407 0.3859 0.4449 0.3300 0.2584 0.2123 0.1801 0.1564 0.1383 0.0991
0.1103 0.1209 0.1338 0.1498 0.1700 0.1966 0.2330 0.2861 0.3703 0.5250 0.3734 0.2805 0.2246 0.1872 0.1605 0.1387 0.1221 0.1090 0.0985 0.0898 0.0825 0.1257
0.1117 0.1192 0.1277 0.1376 0.1491 0.1627 0.1791 0.1991 0.2241 0.2564 0.2842 0.3175 0.3597 0.4148 0.4824 0.3486 0.2729 0.2243 0.1903 0.1653 0.1461 0.1020
0.1147 0.1257 0.1391 0.1557 0.1767 0.2044 0.2422 0.2973 0.3848 0.5453 0.3829 0.2860 0.2283 0.1899 0.1625 0.1405 0.1238 0.1107 0.1000 0.0913 0.0839 0.1307
0.1147 0.1223 0.1310 0.1410 0.1527 0.1664 0.1829 0.2030 0.2280 0.2601 0.2899 0.3262 0.3729 0.4353 0.5025 0.3630 0.2841 0.2334 0.1980 0.1720 0.1520 0.1041
0.1208 0.1324 0.1464 0.1639 0.1860 0.2150 0.2548 0.3126 0.4043 0.5723 0.3951 0.2933 0.2332 0.1935 0.1652 0.1431 0.1262 0.1129 0.1021 0.0932 0.0858 0.1377
0.1188 0.1266 0.1354 0.1456 0.1574 0.1713 0.1879 0.2081 0.2331 0.2650 0.2971 0.3373 0.3901 0.4624 0.5312 0.3834 0.2999 0.2463 0.2089 0.1814 0.1603 0.1070
β = b/h = 0.8 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 Mv-el
0.1100 0.1209 0.1343 0.1510 0.1724 0.2009 0.2407 0.3002 0.3988 0.4206 0.3295 0.2708 0.2299 0.1997 0.1762 0.1534 0.1358 0.1218 0.1104 0.1010 0.0930 0.1215
0.1077 0.1154 0.1241 0.1344 0.1465 0.1609 0.1786 0.2006 0.2287 0.2562 0.2780 0.3038 0.3348 0.3729 0.4110 0.3071 0.2451 0.2040 0.1746 0.1527 0.1356 0.1182
0.1200 0.1320 0.1466 0.1649 0.1884 0.2198 0.2636 0.3293 0.4386 0.4645 0.3543 0.2864 0.2403 0.2070 0.1806 0.1576 0.1398 0.1256 0.1141 0.1044 0.0963 0.1321
0.1139 0.1219 0.1310 0.1416 0.1541 0.1690 0.1871 0.2096 0.2381 0.2684 0.2961 0.3303 0.3733 0.4293 0.4423 0.3313 0.2649 0.2207 0.1891 0.1654 0.1470 0.1238
0.1274 0.1401 0.1557 0.1751 0.2001 0.2335 0.2801 0.3501 0.4668 0.4930 0.3702 0.2963 0.2471 0.2118 0.1841 0.1610 0.1430 0.1286 0.1169 0.1071 0.0988 0.1401
0.1183 0.1265 0.1358 0.1467 0.1594 0.1746 0.1930 0.2156 0.2443 0.2760 0.3076 0.3472 0.3987 0.4679 0.4674 0.3505 0.2804 0.2336 0.2002 0.1752 0.1557 0.1280
0.1341 0.1475 0.1639 0.1844 0.2108 0.2459 0.2951 0.3689 0.4920 0.5169 0.3834 0.3047 0.2528 0.2160 0.1873 0.1640 0.1459 0.1313 0.1194 0.1095 0.1011 0.1475
0.1222 0.1306 0.1401 0.1512 0.1641 0.1795 0.1981 0.2209 0.2497 0.2824 0.3171 0.3614 0.4201 0.5017 0.4916 0.3687 0.2950 0.2458 0.2107 0.1844 0.1639 0.1318
0.1391 0.1530 0.1700 0.1913 0.2186 0.2551 0.3061 0.3826 0.5102 0.5334 0.3925 0.3104 0.2568 0.2189 0.1897 0.1663 0.1480 0.1333 0.1213 0.1113 0.1028 0.1530
0.1250 0.1335 0.1431 0.1543 0.1674 0.1829 0.2016 0.2246 0.2534 0.2868 0.3235 0.3711 0.4350 0.5256 0.5099 0.3824 0.3060 0.2550 0.2186 0.1912 0.1700 0.1345
0.1459 0.1605 0.1784 0.2006 0.2293 0.2675 0.3210 0.4012 0.5349 0.5546 0.4041 0.3178 0.2619 0.2228 0.1929 0.1693 0.1508 0.1360 0.1238 0.1136 0.1050 0.1605
0.1287 0.1373 0.1472 0.1585 0.1718 0.1875 0.2063 0.2294 0.2583 0.2924 0.3318 0.3835 0.4543 0.5571 0.5354 0.4015 0.3212 0.2676 0.2294 0.2007 0.1784 0.1383
β = b/h = 0.9 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.1329 0.1468 0.1640 0.1858 0.2142 0.2529 0.3087 0.3961 0.5251 0.4159
0.1181 0.1267 0.1367 0.1484 0.1623 0.1791 0.1997 0.2258 0.2584 0.2843
0.1445 0.1597 0.1784 0.2021 0.2332 0.2754 0.3364 0.4321 0.5942 0.4551
0.1240 0.1329 0.1432 0.1553 0.1696 0.1868 0.2079 0.2343 0.2682 0.2986
0.1529 0.1690 0.1889 0.2140 0.2469 0.2917 0.3564 0.4579 0.6403 0.4802
0.1281 0.1372 0.1478 0.1601 0.1746 0.1921 0.2134 0.2401 0.2744 0.3077
0.1606 0.1775 0.1984 0.2248 0.2594 0.3064 0.3744 0.4812 0.6730 0.5012
0.1317 0.1410 0.1517 0.1643 0.1790 0.1967 0.2183 0.2451 0.2795 0.3151
0.1663 0.1838 0.2054 0.2327 0.2685 0.3172 0.3876 0.4981 0.6968 0.5157
0.1343 0.1437 0.1546 0.1672 0.1821 0.2000 0.2217 0.2486 0.2831 0.3202
0.1739 0.1922 0.2148 0.2434 0.2808 0.3318 0.4054 0.5210 0.7288 0.5342
0.1377 0.1473 0.1583 0.1712 0.1863 0.2043 0.2261 0.2533 0.2878 0.3268
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Table B.1 (continued) α = Mv/Mu
h/t = 6
8
10
12.5
15
20
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
σy (−ve)
σy (+ve)
0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 Mv-el
0.3444 0.2938 0.2562 0.2271 0.1991 0.1754 0.1568 0.1417 0.1293 0.1189 0.1100 0.1399
0.3159 0.3554 0.4062 0.4740 0.4036 0.3132 0.2559 0.2164 0.1874 0.1653 0.1478 0.1476
0.3687 0.3099 0.2673 0.2350 0.2051 0.1812 0.1622 0.1469 0.1342 0.1235 0.1144 0.1519
0.3368 0.3863 0.4527 0.5467 0.4360 0.3388 0.2770 0.2343 0.2030 0.1791 0.1602 0.1552
0.3842 0.3202 0.2744 0.2401 0.2095 0.1853 0.1662 0.1506 0.1377 0.1268 0.1175 0.1607
0.3500 0.4060 0.4832 0.5967 0.4606 0.3580 0.2928 0.2477 0.2146 0.1893 0.1694 0.1607
0.3970 0.3286 0.2804 0.2445 0.2133 0.1890 0.1697 0.1539 0.1408 0.1298 0.1204 0.1688
0.3610 0.4224 0.5090 0.6402 0.4833 0.3757 0.3073 0.2600 0.2253 0.1988 0.1778 0.1657
0.4058 0.3345 0.2845 0.2475 0.2161 0.1917 0.1722 0.1563 0.1431 0.1319 0.1224 0.1747
0.3684 0.4336 0.5268 0.6711 0.5002 0.3889 0.3181 0.2691 0.2332 0.2057 0.1840 0.1693
0.4170 0.3419 0.2898 0.2515 0.2198 0.1952 0.1755 0.1594 0.1461 0.1348 0.1251 0.1828
0.3779 0.4480 0.5499 0.7119 0.5233 0.4068 0.3328 0.2815 0.2439 0.2152 0.1925 0.1741
β = b/h = 1.0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 Mv-el
0.1582 0.1757 0.1977 0.2260 0.2636 0.3163 0.3954 0.5272 0.5101 0.4263 0.3661 0.3208 0.2855 0.2537 0.2224 0.1979 0.1784 0.1623 0.1489 0.1375 0.1278 0.1582
0.1278 0.1375 0.1489 0.1623 0.1784 0.1979 0.2224 0.2537 0.2855 0.3208 0.3661 0.4263 0.5101 0.5272 0.3954 0.3163 0.2636 0.2260 0.1977 0.1757 0.1582 0.1801
0.1714 0.1905 0.2143 0.2449 0.2857 0.3428 0.4285 0.5714 0.5689 0.4629 0.3902 0.3372 0.2969 0.2618 0.2301 0.2052 0.1853 0.1688 0.1550 0.1434 0.1333 0.1714
0.1333 0.1434 0.1550 0.1688 0.1853 0.2052 0.2301 0.2618 0.2969 0.3372 0.3902 0.4629 0.5689 0.5714 0.4285 0.3428 0.2857 0.2449 0.2143 0.1905 0.1714 0.1902
0.1810 0.2011 0.2263 0.2586 0.3017 0.3620 0.4525 0.6033 0.6074 0.4862 0.4053 0.3475 0.3041 0.2673 0.2354 0.2103 0.1901 0.1734 0.1594 0.1474 0.1372 0.1810
0.1372 0.1474 0.1594 0.1734 0.1901 0.2103 0.2354 0.2673 0.3041 0.3475 0.4053 0.4862 0.6074 0.6033 0.4525 0.3620 0.3017 0.2586 0.2263 0.2011 0.1810 0.1973
0.1897 0.2108 0.2371 0.2710 0.3162 0.3794 0.4743 0.6323 0.6401 0.5056 0.4178 0.3560 0.3101 0.2721 0.2400 0.2147 0.1942 0.1773 0.1631 0.1510 0.1406 0.1897
0.1406 0.1510 0.1631 0.1773 0.1942 0.2147 0.2400 0.2721 0.3101 0.3560 0.4178 0.5056 0.6401 0.6323 0.4743 0.3794 0.3162 0.2710 0.2371 0.2108 0.1897 0.2037
0.1961 0.2178 0.2451 0.2801 0.3268 0.3921 0.4901 0.6535 0.6627 0.5188 0.4263 0.3618 0.3142 0.2755 0.2433 0.2179 0.1972 0.1802 0.1658 0.1536 0.1430 0.1961
0.1430 0.1536 0.1658 0.1802 0.1972 0.2179 0.2433 0.2755 0.3142 0.3618 0.4263 0.5188 0.6627 0.6535 0.4901 0.3921 0.3268 0.2801 0.2451 0.2178 0.1961 0.2083
0.2047 0.2274 0.2558 0.2924 0.3411 0.4094 0.5117 0.6823 0.6919 0.5358 0.4372 0.3693 0.3196 0.2799 0.2476 0.2220 0.2012 0.1839 0.1694 0.1570 0.1463 0.2047
0.1463 0.1570 0.1694 0.1839 0.2012 0.2220 0.2476 0.2799 0.3196 0.3693 0.4372 0.5358 0.6919 0.6823 0.5117 0.4094 0.3411 0.2924 0.2558 0.2274 0.2047 0.2144
2002;128(11):1387–93. [15] Trahair N. Lateral buckling strengths of steel angle section beams. J Struct Eng 2003;129(6):784–91. [16] Trahair N. Biaxial bending of steel angle section beams. J Struct Eng 2004;130(4):554–61. [17] Trahair N. Buckling and torsion of steel unequal angle beams. J Struct Eng 2005;131(3):474–80. [18] Earls CJ. Single angle geometric axis flexure: II. Design recommendations. J Constr Steel Res 2001;57(6):625–48. [19] Earls CJ. Single angle geometric axis flexure I. Background and model verification. J Constr Steel Res 2001;57(6):603–23. [20] Liu Y, Hui L. Finite element study of steel single angle beam–columns. Eng Struct 2010;32(8):2087–95. [21] SIMULIA. Abaqus/CAE 6.14-1. 2016. [22] Bradford MA, Liu X. Flexural-torsional buckling of high-strength steel beams. J Constr Steel Res 2016;124:122–31. [23] Temple MC, Sakla SS. Single-angle compression members welded by one leg to a gusset plate. II. A parametric study and design equation. Can J Civ Eng 1998;25(3):585–94. [24] Trahair NS. Flexural-torsional buckling of structures. London, UK: E & FN Spon, an imprint of Chapman & Hall; 1993. [25] Liu Y, Chantel S. Experimental study of steel single unequal-leg angles under eccentric compression. J Constr Steel Res 2011;67(6):919–28. [26] Liu Y, Hui L. Behaviour of steel single angles subjected to eccentric axial loads. Can J Civ Eng 2010;37(6):887–96. [27] Trahair NS. Flexural-torsional buckling of structures (beam-columns). London, UK: E & FN Spon, an imprint of Chapman & Hall; 1993. p. 205–18. [28] Shi G, Liu Z, Ban H, Zhang Y, Shi Y, Wang Y. Tests and finite element analysis on the local buckling of 420 MPa steel equal angle columns under axial compression. Steel Compos Struct 2012;12(1):31–51.
References [1] AISC specification. Specification for structural steel buildings. Chicago (IL): American Institute of Steel Construction; 2016. [2] British Standards Institution (BSI). BS EN 1993-1-1: 2005 Eurocode 3: design of steel structures part 1-1: general rules and rules for building. London: BSI; 2005. [3] ASCE specification. Design of latticed steel transmission structures. American Society of Civil Engineers; 2000. [4] British Standards Institution (BSI). BS EN 1993-1-8: 2005 Eurocode 3: design of steel structures part 1-8: design of joints. London: BSI; 2005. [5] Chan S, Cho S. Second-order P-Δ-δ analysis and design of angle trusses allowing for imperfections and semi-rigid connections. Int J Adv Steel Constr 2005;1(1):157–72. [6] Chan S, Cho S. Second-order analysis and design of angle trusses part I: elastic analysis and design. Eng Struct 2008;30(3):616–25. [7] Cho S, Chan S. Second-order analysis and design of angle trusses, part II: plastic analysis and design. Eng Struct 2008;30(3):626–31. [8] Aydın R, Doğan M. Elastic, full plastic and lateral torsional buckling analysis of steel single-angle section beams subjected to biaxial bending. J Constr Steel Res 2007;63(1):13–23. [9] Charalampakis A. Full plastic capacity of equal angle sections under biaxial bending and normal force. Eng Struct 2011;33(6):2085–90. [10] Vayas I, Charalampakis A, Koumousis V. Inelastic resistance of angle sections subjected to biaxial bending and normal forces. Steel Constr 2009;2(2):138–46. [11] Ban H, Shi G, Shi Y, Wang Y. Residual stress tests of high-strength steel equal angles. J Struct Eng 2012;138(12):1446–54. [12] Može P, Cajot L-G, Sinur F, Rejec K, Beg D. Residual stress distribution of large steel equal leg angles. Eng Struct 2014;71:35–47. [13] Kitipornchai S, Lee H. Inelastic buckling of single-angle, tee and double-angle struts. J Constr Steel Res 1986;6(1):3–20. [14] Trahair N. Moment capacities of steel angle sections. J Struct Eng
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