Deflection analysis of welded steel I‐girders with corrugated webs based on first yield

Sinusoidal corrugated profile webs have been popularly used in steel structural designs to replace the flat webs in conventional welded beams, while there are better performances in corrugated web beams (CWBs) regarding more stability and less material used to against beam failures caused by buckling. Previous studies have provided that CWBs enabled numerous favourable benefits to be recognised as alternatives to the traditional weld beams in designing structures. Furthermore, as CWBs are proposed as the major load‐carrying elements, the maximum deflection in the elastic range is one of the important beam properties that should be precisely estimated and calculated. To find an appropriate method in computing the maximum deflection of CWBs based on the first yield for civil communities in Australia, proposed equations based on other standards will be employed to calculate the theoretical results for the comparisons with simulation‐based results. While applying the linear analysis simulations provided by SAP 2000, ultimate limit state design theory has also been used with requirements stated by AS 4100. In this study, the results in theoretical calculations and numerical simulations have been compared to conclude that the highly defined equations by ASTM [37] and Sause et al. [38] could precisely estimate the maximum deflections of CWBs based on the first yield in conjunction with requirements and limitations in Australian standards, which could be adequate for the structural design calculations in Australian design fields.


Introduction and background
Corrugated web beams (CWBs) and girders have been widely utilised as the main load-carrying components in the existing and new-developing structures [1,2]. Previous studies state that the nature of corrugated webs can thin the plates and stiffeners, avoid the out-plane buckling failures and provide stronger internal capacities [3][4][5].
The shapes of the corrugation, such as sinusoidal, trapezoidal and rectangular shapes, have been validated to have more favourite properties, including stronger stiffness, higher ductility ratio and more energy dissipation capacity than the normal beams [6][7][8][9]. As a result, structures assembly, including continuous girders and rigid frames from CWBs, has been utilised in many major composite structures [10][11][12]. Zevallos et al. [10] and Wang et al. [12] also mentioned that the first bridge built with corrugated web girders and concrete decks was located in Cognac Bridge in France.
Based on previous studies, CWBs are actively analysed and studied for different structural behaviour [13]. Kim et al. [14] pointed out that beams with corrugated webs had better ductility and serviceability compared to conventional beams. They provided a series of experimental results which showed the loading-resistance capacity of CWBs was significantly higher than the conventional steel beams. Lin et al. [15] stated that the stability against shear yielding buckling of the beams was improved by the corrugated nature after comparing the experimental and numerical results, where the profile of the corrugated webs could act as a kind of horizontal-direction stiffener, which can dramatically increase the total beam stiffness and stability. Moreover, because of the improved stability to the lateral torsional buckling, the CWBs can resist higher shear forces than the traditional beams [16]. According to Farzampour [4], this alternative option of beams was modelled in finite element (FE) simulation as the main lateral load resisting elements in a beam-column system to investigate the shear capacity, in which the author found that the experimental results were higher than the results of the flat web beams, which could be considered as a conservative option to replace the ordinary beams in the building design. According to Park et al. [17], the weight of CWBs could be reduced by 30 % to 40 % comparing with the traditional flat plate girders because the significant amount of reinforcing stiffeners was replaced by the corrugated webs. They also stated that the corrugated members could still provide high material efficiency and better shear resistance with less material used. Similarly, by using the concept of the beam optimisation to the CWBs to thin the web thickness and reduce the beam height, the CWBs could achieve the material saving up to 20 % while maintaining the same level of capacity in both the shear and flexural behaviours [18]. Lin et al. [18] mentioned that using the shorter section of CWBs to replace the selected ordinary beams in a proposed building design could satisfy the needs of the economic design because the lighter beam features could assist designers to save the raw material cost and the total saved beam height could be utilised to add more floors to that building proposal.
Several studies [28][29][30][31][32][33][34] performed different types of FE simulations and deflection limit tests to validate the experimental data against the theoretical results. Due to the imprecise determination of the shape factor k s and assumptions of pure bending or shear acting to the structures, the deformations of CWBs were accidently underestimated or overestimated with inaccurate coefficient estimations [35]. They also indicated that some researchers assumed varied live-to-dead load ratios in the functional settings, which would affect the calculation processes to gain precise results. Ultimately, the precise estimation to the deflection of CWBs is unable to be achieved by the above-mentioned researchers. The CWBs have been popularly used in some main structures of the civil engineering projects in Australia. However, there is no direct approach to analysis the beam deflection of CWBs, in which the deformation limit of CWBs is one of the main requirements of the safety design. As a result, a comprehensive approach to accurately calculate the deflection of CWBs is highly required in order to utilise CWBs in Australian engineering design fields, where the steel design standard AS 4100 [36] in Australia does not include a section listed any methods to calculate the beam deformations.

Theoretical background
In this study, a reasonable approach to estimate the deflection of CWBs will be developed, as the main objective, for Australian civil engineering communities, by considering the equations stated by the supplementary guidelines ASTM [37]. Based on the research performed by Sause et al. [38], the conservative method to calculate the deflection of CWBs with the trapezoidal shape will be selected as a component of the equations to calculate the deflection of sinusoidal shape web beams because the equations from Sause et al. [38] could acceptably estimate the deformation of CWBs [35]. Thus, this study will combine the approaches from ASTM [37] and Sause et al. [38] to establish the purposed equations to generate the theoretical results, where Eq. (8) introduced by ASTM [37] will be used as the formula to precisely represent the beam total stiffness. Eqs. (6) and (7) from Sause et al. [38] will be considered as the starting point to calculate the bending stiffness and the shear stiffness respectively of the CWBs.
where E is the Young's modulus, G is the shear modulus of elasticity equal to 80 × 10 3 MPa, L is the corrugation CWBs are the basic elements in the structural design which play critical roles to resist and transfer the loads to the foundation by the beam-column system [19][20][21][22]. Hence, the deflection of CWBs should be clearly investigated, which is essential for the deflection limit check. A functional method called active deflection control to analyse the deflection of the CWBs has been mainly focused on some studies [23]. Wang et al. [24] conducted investigations to study the displacement of beams with corrugated webs by using the stiffness factors. The deflection of the I-girder with corrugated webs can be determined by D = P/k 0 . k 0 is the initial stiffness in Eq. (3), k s is the shear stiffness in Eq. (2), and k b is the bending stiffness in Eq. (1), respectively. (1) where E is the Young's modulus; I is the moment of inertia about the major axis; A w represents the sectional area of the corrugated steel web; L represents the span length of the simply supported I-girder; G is the shear modulus; and h indicates the shape coefficient of CWBs. After comparing the theoretical results and simulation-based results, the significant differences in the comparison result indicated that these highly structured formulas had underestimated the deflection of CWBs [24].
Leblouba and Tabsh [25] examined the displacement analysis of CWBs by considering the reliability criterion stated in specifications in AISC-360-16 [26], which is the Specification for Structural Steel Buildings by the American Institute of Steel Construction and CSA-S16-14 [27] being the Design of Steel Structures by the Canada Standard Association. They used Eqs. (4) and (5) to compute the deflection appeared in the mid-span of the CWBs. (4) where G is the shear modulus of the steel material and a is the shear span of the beam; t w is the web thickness and h w indicates the web depth. They demonstrated 49 shear performance tests for four steel I-girders with corrugated webs, where there was not a good agreement between results from the shear tests and the theoretical results. The main reason of the overestimated deflection was that the author assumed the shear load to be uniformly distributed along the web.

Numerical simulation assumptions
There will be several assumptions to properly complete the FE model simulations in SAP 2000, and a typical model with meshed elements of CWBs shown in Fig. 2, in which the uniformly distributed load (UDL) will be applied on the top flanges of CWBs. Accordingly, the distribution load labelled in each meshed area is displayed as green colour by the SAP 2000 software to present the area load shown in Fig. 2. The 3, 5 and 7 m of the beam length will be selected, which were popularly adopted in main beam structures in the Australia buildings, and the supported conditions were selected as pin and roller supports. Loads applying on the compression flanges will be precisely represented by the constant pressures, while the top flange was fully retrained to resist lateral torsional buckling for each CWB. There were two 10-mm-thick stiffener plates applied on both sides of the beam to prevent the twisting along the beam. The pin and roller supports were simulated with node applying along the middle depth of the stiffener. However, modelling the same stiffener nodes restricted at only one end in the longitudinal direction can eliminate rigid body deformation along the beam axis. EF simulation standard shell elements supported by SAP 2000 can mesh all elements of the beam.

Design and analysis parameters
Some researchers [4,7,15] have mentioned that the corrugated feature can improve the bending and shear strength of the CWBs with less deflection. To start with the load determination, firstly, the UDL w* is set to be the length, a is equal to a quarter of the corrugation length, I is the second moment of inertia about the major axis, α is the corrugation angle, b is the ratio of a and L, k b is the bending stiffness, and k s is the shear stiffness.
After gathering the results of the beam stiffness k 0 for each CWB, the deflection d will be calculated from where P is the maximum load that each CWB can safely resist. As a result, the above-mentioned highly structured equations will be used to estimate the maximum deflection of CWBs, which will be utilised to compare with the results from FE simulations via strictly implementing the requirements and properties of steel beams stated in Australian Standards. Hot Rolled and Structural Steel Products property tables [39] will be used to set the necessary beam properties and features in SAP 2000 in order to accurately perform the FE simulations. This software has been used by several researchers [40][41][42][43][44] to investigate mechanical behaviour of structures. Moreover, all other essential information including factors of load combination, factors for buckling checks, and modification factors of required loads will be extracted from AS/NZS 1170.1 [45].

Methodology
Lin et al. [18] stated that the CWBs with a corrugated length of 400 mm and the corrugated angle of 30° performed favourable mechanical capacities comparing with other corrugated sizes, and this selected size will be used to replace the flat web in the conventional beams. A CWB numerical model sample has been illustrated in Fig. 1, where it presents the geometry dimensions of the simulated model in this study. In addition, the cross-sectional geometry for the fundamental dimensions is selected as the same values cited from the Hot Rolled and Structural Steel Products property tables [39].
Based on Fig. 1, the factor of h w indicating the web depths of the CWBs and the factor of t w indicating the web thickness will be used to estimate the deflection in Eq. (7). Due to the remodelling of the web geometries, the elastic buckling investigation should be performed to avoid buckling failure on the new webs. Moreover, as required by AS/NZS 1170.1 [45], all buckling factors should be greater than 1.0 while applying design load combinations to the analysis, in which the buckling factors being less than 1.0 will result in web buckling, including lateral torsional buckling, flexural torsional buckling, global buckling and local buckling [46]. SAP 2000 FE analysis software [46] has been used to comprehensively investigate the maximum deflections for CWBs based on the first yield with favourable functions for analysis, where the deflection capacities gained from numerical investigations will be compared with the results obtained from the theoretical results by Eqs. (6), (7) and (8). Lastly, the viability of the highly structured equations could be tested while strictly applying requirements stat- Fig. 1 Typical illustration of a CWB same FE simulation approach with shell elements utilised in this study was selected to carry out the investigations on the required beam to obtain the essential data for the comparison in the next step. Finally, the results generated from the same FE simulation technique utilised in the CWB models in this research have been validated against the load-deflection curve published by Martins et al. [48] in Fig. 3. Fig. 3 has clearly illustrated that the curve of the load-deformation gained from parametric experiments of FE models in SAP 2000 consistently fits with the practical results demonstrated by Martins et al. [48]. Lastly, the numerical model developed in this research can be utilised as an adequate model to illustrate the mechanical and physical properties of steel I-girder for corrugated webs. The credibility of the developed FE model in SAP 2000 has been verified throughout the consistent agreement between numerical analysed results and the practical results tested by Martins et al. [48]. As a result, the FE model provides a high degree of accuracy for predicting different behaviours of CWBs.

Results for 5 m CWBs
In this research, based on Hot Rolled and Structural Steel Products property table [39], five different welded beams from each category are selected to estimate the deformation based on the first yield, in which the traditional flat webs of selected beams will be reformed with corrugated webs to form new-designated CWBs.  (7) to calculate the bending stiffness k b and shear stiffness k s . The bending stiffness factor k b is quotient by the multiplication of corrugated length L, Young's modulus E, and secondary moment area I over highest load that could be safely resisted by each CWB while reaching the maximum allowable stress on the first yield. According to AS 4100 [36], the maximum allowable shear stress is 0.6 f y and that for bending stress is 0.9 f y . A trial-and-error process will be conducted to determine the maximum UDL w*, where each trial-and-error process will be paused if the flexural or shear stress limitation is approached. After that, the maximum UDL and the maximum deformation will be recorded.
Based on AS/NZS 1170.1 [45], the primarily designed load input to the FE simulation in SAP 2000 is calculated by load combination of 1.2DL + 1.5LL, where the LL is live load and DL is the dead load applying on the beams. According to suggestions practised by the Australian civil engineering communities, the equation DL = 3LL could be used to properly represented the relationship between the live load and the dead load. Hence, the load relationships of DL = w*/1.7 and LL = w*/5.1 can be utilised as the inputs to SAP 2000. To clearly present the conducted methodology in this research, a flow chart is demonstrated in Fig. 3. In this research, two different structural steel grades of 300PLUS-300 and 300PLUS-280 are selected to model the CWBs, which have the same mass density of 7849 kg/m 3 .

Validation of the developed numerical model
Previous studies [47,48] provided the numerical and experimental models to verify different performances for CWBs. To confirm the credibility of the FE simulation models performed by SAP 2000 in this study, the experimental results collected from this model had been contrasted to the practical results conducted by Martins et al. [48]. They comprehensively demonstrated parametric experiments on three full-scale composite connections for CWBs, including the PSS 600 × 150 × 12.5 × 2.0 for specimens 1 and 2 and PSS 600 × 150 × 8/12.5 × 2.0 for specimen 3. Since the similar features performing on the beam to investigate the corrugated beam models, the experimental results for specimen 1 were chosen to verify the functional numerical results in this study. Then, the X. Zhang, H. Far, X. Lin: Deflection analysis of welded steel I-girders with corrugated webs based on first yield ARTICLE the special relationship between total corrugated length L and quarter of the length a. The stiffness k s is computed with the individual beam features, including corrugated length L, the height of beam h w , the thickness of beam t w , the shear module of elasticity G, and a quarter of corrugated length a. When these two factors are derived, the initial stiffness k 0 would be computed to estimate the beam's deflection. Besides, P is the applied unit force calculated by distribution load w. Lastly, the deflection would be generated throughout by the initial stiffness using the equation D = P/k 0 illustrating in Tab. 3.

Effects of length variation
To further explore the applicability of the above-mentioned equations [37,38] to calculate the maximum deflection based on the first yield, the analyses to other popular beam lengths including 3 m and 7 m are performed. Similar to the investigations to 5 m CWBs, the buckling analysis should be performed to investigate the web conditions against the buckling. Tab. 4 clearly shows that the smallest buckling factors for 3 m CWB are much higher than the minimum value of 1. Therefore, the conclusion could be made that the replacements in the webs could safely resist different types of web buckling. The inputs to SAP 2000 and the corresponding results are shown in Tab. 5 by practical simulation. In this length series analysis, the applied UDL of 3 m length would re- Following the procedures mentioned in Section 5.1, the values of bending stiffness k b , shear stiffness k s , and bending stiffness factor k b are computed, where the maximum elastic deflections for 3 m CWBs would be generated throughout by the initial stiffness using the equation D = P/k 0 illustrating in Tab. 7. By using the same process as the above-mentioned sections, Tab. 4 depicts the smallest buckling factor of 7 m CWBs, which are also much higher than the minimum factor of 1. Thus, the 7 m CWBs could be safely used because the webs remain in safe conditions to resist against the web bucking failures. Similarly, bending stiffness k b , shear stiffness k s , and bending stiffness factor k b and the corresponding deflections are summarised in Tab. 7.
Many researchers [15,24,49] have reported that there were many benefits in using the CWBs, including its material efficiency, better strengths, stronger stability, and less deformation mentioned, where these favourable features could attract more engineering designers specialising in various design regions. In this study, the highly consistent results in the comparisons indicate that the numerical methods by ASTM [37] and Sause et al. [38] could appropriately estimate the maximum deflections in in SAP 2000. The trial-and-error process in finding the maximum UDL will be stopped when the flexural or shear stress approaches the corresponding stress limits, which is 0.9* f y for the flexural stress check and 0.6* f y for shear stress check. Comparing these results through these two methods, the differences are found between 1 % and 3.5 %, in which this range is considered as reasonable and acceptable results in the requirements of the conservative safety design. By reviewing the result comparisons between formula-based approach and results in FE simulations, the high degree of consistency suggests that it is conservatively safe to conclude that the high-structured equations from ASTM [37] and Sause et al. [38], which has been suggested by He et al. [35] to calculate the deflections of CWBs based on Chinese standards, could properly predict the maximum deflection within an acceptable accuracy based on the first yield for CWBs. Overall, the proposed numerical method combined with ASTM [37] and Sause et al. [38] can be easily adopted with high level of accuracy.
the elastic range for the sinusoidal CWBs, and this numerical approach could also be properly combined with requirement and limitations to comply with Australian standard, in which the highly structured equations are beneficial to calculate the deflection limitations during the structural designs for Australian engineers with its high credibility and applicability (Tab. 8).

Conclusion
In this parametric study, FE analysis has been conducted generate results in order to compare them with the theoretical calculations from the relevant equations so as to find proper equations in estimating the maximum deflection based on the first yielding for steel I-girders and beams with corrugated webs. Ultimate limit state design theory has been utilised to cover these studies containing numerical analysis with conjunction of AS 4100 [36] and the FE simulation with material and geometric linearity