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The Collapse Analysis of the Lateral-Torsional Buckling of I-Shaped Stepped Steel Beams

ABSTRACT The use of a non-prismatic member such as a stepped beam as a design method has the ability to function as a tool for steel beams optimization. A cover plate is partially welded on the upper and lower flange of the member at the maximum bending moment location to increase its flexural strength and, under critical load, flexural members bend about its strong axis, displace to the lateral direction, and twist coincidentally through a phenomenon known as the Lateral-Torsional Buckling (LTB). There is, however, no equations in the AISC 360-16 specification to calculate the critical moment of a stepped beam (Mst). Therefore, this research focuses on developing Mst for a simply supported stepped beam which deforms on its shear center under static-transverse loading through the use of a collapse analysis and the behavior of the beam. The results showed the welded cover plates consequently increased the LTB resistance of the prismatic I-shaped steel beam from 9.8% to 202% while the critical moment increased more significantly with an increment in the ratio of the cover plate length to the unbraced length (α). The cover plate thickness was observed to have dominantly affected only a large α ratio while the post-buckling characteristic of large α showed a sudden collapse phenomenon. Furthermore, the LTB modification factor was generated in this study due to the initial geometrical imperfection from the first mode of Eigen shape with maximum amplitude Lb/2000 (Cb1) and stepped beam shape (Cst) which were required to estimate the critical moment of a stepped beam based on the AISC equation for a prismatic beam.


INTRODUCTION
Steel has always been a useful material for building constructions compared to other materials due to some of its advantages such as the high strength-to-weight ratio required in long-span bridges, tall buildings, and poorsituated steel structures to provide high strength for them despite their small weights. There is, however, the need for additional steel to stiffen these structures against buckling caused by slenderness and which reduces their economic value (McCormac and Csernak, 2012).
Non-prismatic member is one of the substantial methods to provide more efficient steel design optimization and it involves partially fabricating web-tapered beams and welding cover plates. For instance, the web-tapered beam has a geometric deeper web section on the maximum bending moment segment and a shallow depth is required on the minimum. Moreover, this design involves the allocation of more strength and stiffness using minimum weight (Kaehler et al., 2011) with the ratio of bending moment to moment capacity becoming larger.
Welded cover plates are also applicable to the flanges section using the design which serves the same purpose as those web-tapered. Therefore, the beam which undergoes a sudden crosssectional change in a particular segment is commonly known as the stepped beam, and its failure and geometry in real practices which indicates the maximum bending moment on mid-span are presented in Figure 1.  (Yossef, 2015) b) Geometry of a stepped beam.
Structural engineers prefer using stepped beams to web-tapered beams due to the lesser complexity in their fabrication as well as the Inexpensive cost of production (Park and Kang, 2004). This has led to an extensive understanding of the stepped beam by researchers as observed in the study of LTB simply supported stepped beam by generating the elastic buckling analysis by Trahair and Kitipornchai (1971). Moreover, Park and Kang (2004) proposed the critical moment of doubly and singly stepped beams located at the end supports with or without continuous lateral topflange bracing. Surla and Park (2015;2014) proved the theoretical findings through the experimental study of lateral-torsional buckling strength of monosymmetric stepped I-beams. Recently, Alolod and Park (2018) examined the inelastic buckling capacity of the stepped beam at mid-span under a uniform moment. Meanwhile, AISC (2016) has considered a critical moment equation for prismatic flexural and webtapered members but none has been found for the critical moment of a stepped beam (Mst) which functions as a base to determine the flexural strength.
This study aimed to extend the findings of a previous study on stepped beam which was used to generate elastic buckling analysis to form the critical moment formula without considering the initial geometrical imperfection and residual stress effect. Therefore, the inelastic and elastic critical moment of simply supported stepped beam (Mst) located at mid-span under concentrated and uniformly distributed load was developed using Finite Element Analysis (FEA) as well as the collapse behavior of the beam. In addition, the research also examined the doubly-symmetric I-shaped beam with welded cover plates on its flanges and this is different from the investigation on stepped beam caused by the modification of the flange thickness conducted by other researchers.
The development of the modification factor for the LTB critical moment due to the effect of the cover plate added on upper and lower flange (Cst) and the initial geometrical imperfection (Cb1) through collapse analysis makes it possible to nearly approximate the critical moment of a stepped beam (Mst) to the value for the prismatic beam before the cover plates were added. Cst Is fundamentally defined as the ratio of the critical moment of the stepped beam (Mst) to the critical moment of the prismatic I-shaped member (Mcr). Therefore, the Mst is obtainable using the following Equation (1). AISC specification considered the nominal flexural strength of compact doubly-symmetric I-shaped steel beams and channels. Moreover, the plastic moment (Mp) and Lateral-Torsional Buckling (LTB) are the limit state for compact flexural members depending on their undergoing stresses and unbraced length (Lb). The relationship between critical stress (Fcr) and Lb is, therefore, presented in the following Figure 2. The flexural members experience inelastic LTB when Lp < Lb< Lr as the critical moment is shown in Equation (3).

LTB modification factor (Cb)
In most cases, Equation (2) usually produces a conservative result due to the fact that most beams are not subjected to uniform moment and most supports are not simple (Galambos and Surovek, 2008 Where Mmax is the absolute value of maximum moment in the unbraced segment, MA is the absolute value of moment at the quarter-point of the unbraced segment, MB is the absolute value of moment at the centerline of the unbraced segment, and MC is the absolute value of moment at the three-quarter point of the unbraced segment.

ANALYSIS METHOD
Two methods were used for analysis in this study with the first being the bifurcation theory which is applicable only in an idealized beam with perfectly straight geometry, fully linear elastic material, and with the applied load resultant passing through the shear center. This theory leads to an Eigen problem which when solved produces Eigenvalue with the minimum value used as the elastic buckling load. The theory is also known as the linearized buckling due to the assumption of a small displacement assumption and this further makes the differential equation linear.
The second method is the collapse analysis procedure, and this involves an incremental increase in the applied load while the displacement is monitored until the beam completely collapses. Therefore, the output produces a load-displacement curve which Is further used to analyze the maximum applied load. Meanwhile, a beam with an initial geometrical imperfection from the bifurcation theory exhibits instability deformation and this means the maximum load on the curve as Indicated in Figure 3 is equal to the maximum buckling load. The collapse analysis was preferred in this study due to its similarity with real conditions as observed with most of the beams mostly having an initial imperfection due to fabrication as well as the inelastic and plastic behavior they experience after reaching the yield stress. Moreover, the use of large-displacement kinematics makes the differential equation nonlinear and this is suitable for analyzing inelastic buckling.

FINITE ELEMENT MODELING
The ADINA program was used as the non-linear analysis software and all the models used a 3D shell element with 4 nodes. The boundary conditions applied at both ends which are supported vertically using a pin-roller including full restraint on the torsional rotation and free warping.

Rigid Link
The stepped beam consisted of a prismatic Ishaped beam with the cover plates perfectly attached to the weld line. Meanwhile, the rigid link connections in FEM performed as the weld line on the periphery of the cover plate and the gap between the cover and flange plates was found to be 0.

Steel material
The steel material has a true stress-strain curve with a yield stress (Fy) value of 250 MPa and also assumed to be plastic-multilinear, isotropic, homogenous, and linear. Meanwhile, every fabricated steel has residual stress before being subjected to any load due to the fabrication process. Moreover, the existing internal stress increases throughout the section up to the attainment of the critical stress during loading. The maximum residual stress of hot-rolled steel was valued at 0.3 x Fy with a triangular linear pattern and Van der Aa (2015) has found the difference between the parabolic and triangular patterns of residual stress to be negligible. Therefore, a linear-triangular pattern was used in the FEM analysis as illustrated in According to Salmon et al. (2008), the initial outof-straightness for L/1500 amplitude has been calculated in the flexural column strength curves. Meanwhile, Dharma and Suryoatmono (2019) applied the initial out-of-straightness of L/1500 value for the non-prismatic column to the node where the maximum scale of deformation pattern of the first mode of Eigen is located based on recommendation while the other nodes follow the geometrical imperfection shape. In this research, the FEM accommodates the initial imperfection of the first buckling mode of Eigen obtained from the linear buckling analysis and this shape has been reported to be the most suitable for LTB compared to others due to some certain reasons ( Van der Aa, 2015). This theory has also been supported by Boissonnade and Somja (2012) in their journal.
The Lb/2000 amplitude, which is smaller, was applied to a certain node where the maximum amplitude of the first mode of Eigen is located and it produced the insignificant reduction effect of the moment capacity of the stepped beam due to the incomplete consideration of any effect of the Initial geometrical imperfection by the theoretical solution of buckling. Therefore, the Cst equation was used accurately to approximate the moment of the stepped beam (Mst) from the theoretical solution (Mcr) but the reduction effect due to the imperfection was considered. Figure 5 illustrates the initial geometrical imperfection on FEM.

Loading
The concentrated and the uniformly distributed loads were assigned to the models on the shear center as shown In Figure 6. The concentrated load requires a stiffener on mid-span to apply each nodal load throughout the stiffener for distribution to avoid the nodal stress concentration. Moreover, the stiffener also functions as a solidifier for the unstiffened element against local buckling.

Preliminary Verification
The models were supposed to be verified before the parametric study and both the elastic critical moment of finite element analysis and Equation (2) or (3) from verification models presented in Table 1 were considered to verify the finite element models. Meanwhile, Table 1 shows the results based on linearized buckling from several verification models.
The verification results showed the deviation ranged from -1.67% to 3.96%. This small amount of numerical error indicates the accuracy and validity of the model with all its assumptions as defined in Chapter 4 and this means it should be continuously applied in this study for further analysis. Moreover, the first buckling mode shape based on the linearized buckling results is shown in Figure 7. The WF 400X200 finite element models were tested in relation to the collapse analysis verification and results are presented in the following Table 2. * Ay is Lb/5000 initial geometrical imperfection amplitude All the models have the maximum initial geometrical imperfection Lb/5000 and the values obtained were relatively small to Indicate the insignificant effect of imperfection on the critical moment. Moreover, the verification through collapse analysis showed the models are accurate and reliable with a maximum deviation of 2.28%. The results are similar to linearized buckling results and the theoretical solution (equation 2), therefore, WF 250X125 verification through collapse analysis was not conducted due to the possibility of making reference to the linearized buckling results. Meanwhile, the second verification was conducted on several models for the rigid link and the findings are presented in Table 3.

Parametric Study
A previous study by Park and Kang (2004) showed the effect of adding cover plates to the critical moment of a stepped beam depends on Its unbraced segment length (Lb), cover plate thickness (tc), and its width (bc). This study only discusses the effect of length and thickness with both written in ratios such that parameter α represents the ratio of the cover plate length to the beam length and β Indicates the ratio of the cover plate width to flange width while the total thickness of the profile flange thickness was notated as γ. The illustration of the whole parameters Is shown In Figure 1.
The ratio of the unbraced length to section height (Lb/D) is a prismatic beam dimensionless parameter in the analytical model while the cover plate parameters for each Lb/D value including 25, 21.25, 17.5, 15, 12.5, and 10 in WF 400 x 200 are presented in Table 5. Meanwhile, those for other values such as 32, 28, 24, 18, and 16 are presented in Table 6.  The results on the effects of the parameters are visualized into the data distribution graphs with Figure 10 depicting the Mst Vs Lb curves obtained from the collapse analysis of the stepped beam under concentrated load on the shear center.

Stepped beam
The collapse shapes in Figures 9 show the maximum lateral displacement of 0.167, 0.333, 0,700, and 1,00 α ratios occurred in the mid-span and this is in accordance with the theoretical LTB deformed shape. Therefore, there Is no particular concern in the discussion of the stepped beam collapse shape. Moreover, Figure 11 shows the load vs. lateral displacement (P-δ curve) for several α values. Figure 11. The relationship between load (P) and lateral displacement (δ) for different α ratios.
The graph clearly shows a greater value of α leads to a higher value for the critical load (Pcr) and this means an increase in the α ratio has the ability to create a sudden collapse of the beam. This is evident from the slope of the curve after the critical load is reached and a beam was observed to have deformed significantly while the current load decreased suddenly. This is a post-buckling characteristic of the sudden collapse phenomenon but a smaller α ratio has a slight slope after the buckling point.
The effect of γ for each group of α is indicated in the distribution of data vertically connected by lines under both concentrated and distributed loads as in Figure 10. The influence of γ for α ≤ 0.333 was found to be very insignificant with respect to Mst but more influence was observed at larger α. This means α is strongly dominant in the stepped beam design due to the ability of Its ratio to determine the effect of the cover plate thickness on the section properties.
The torsion and warping constant are crosssectional properties used in determining the torsional stiffness of a beam. The addition of the cover plate along the flange significantly increased the critical moment of the LTB due to the Increment In the torsional stiffness. However, the length of the cover plate in a portion of the span length has an enormous effect on the intended stiffness.

Coefficient analysis
The rearrangement of the linear relationship in Equation (1) made the modification factor for the critical moment of the prismatic beam due to the addition of the partial cover plate on the upper and lower flange is shown in Equation (5).  The Cst equations were proposed reasonably according to the accuracy of each regression model and summarized in Table 8. Moreover, all the finite element models contain the initial geometrical imperfections following the first buckling mode shape with Lb/2000 amplitude. This, of course, influences the load-bearing capacity of the stepped beam due to the occurrence of excessive stress before the steel material reaches its yield stress. Therefore, the LTB modification factor coefficient due to the initial geometrical was required and the equation between the critical moment including the initial geometrical imperfection of Lb/2000 (M0cr) and the Idealized one (Equations 2&3) is presented as Equation (6). Figure 15 shows the proposed equation of Cb1 vs dimensionless unit (Lb/Lr) with the initial geometrical imperfection coefficient expected to be limited by 1.

SUMMARY
The research showed several conclusions concerning the LTB of the stepped beam and the major summary is the possibility of determining the critical moment of the stepped beam (Mst) using the Equation (7).
The γ ratio or cover plate thickness only has a significant effect on the critical moment of stepped beams (Mst) in the greater α ratio (α > 0.333). Therefore, it is possible to obtain the optimum cover plate thickness for α ≤ 0.333 from the minimum thickness of the compactness requirements of AISC 360-16. In addition, the inelastic post-buckling characteristic showed a sudden collapse for α > 0.700, and the rigid link connection was found to be applicable as a welding line in FEM. However, the unconnected nodes between the cover plate and flange do not affect the critical moment even though it experiences a torsional moment.