Numerical investigation of the cold-formed I-beams bending strength with different web shapes

The wide use of cold-formed sections (CFS) in the field of steel constructions, favored by the multiple advantages they offer (lightness, ease of installation, etc.), has led us to reflect on a new process for manufacture of metal beams allowing the design of very large span hangars and a reduction in instability problems. This paper presents a study of the theoretical and numerical behavior of a large span CFS beam with different webs, a solid web, a triangular corrugated web, and a trapezoidal corrugated web. These beams are stressed by a concentrated bending load at mid-span. Numerical modeling was done using the finite element software ABAQUS. The results were validated with those theoretically found, based on the effective width method adopted in standard EN1993-1-3. The load capacity and failure modes of the beams were discussed. According to numerical and analytical analysis, corrugated web beams perform better than all other sections.


INTRODUCTION
tructural elements in steel construction are classified into two categories: hot-rolled sections and cold-formed sections. Cold-formed steel sections (CFS) are preferable to hot-rolled sections because of their greater versatility and because they are well suited to be economically constructed [1]. Cold formed steel is also known as light steel. In the field of civil engineering, cold-formed steel elements are generally used in industrial buildings as purlins, columns, trellises or structural members, storage supports, vehicle bodies, and various types of equipment [2]. In recent years, the use of CFS S structures has increased rapidly due to significant improvements in manufacturing technologies. CFS elements are made from steel sheets and are formed into different shapes, either by press bending of sheared sheets or coils or, more commonly, by rolling at room temperature [3,4]. Structural elements in CFS retain their position in the light construction sector. This is due to the important advantages of CFS: high strength/weight ratio, ease of transport, the possibility of using all conventional assembly methods, cost efficiency, very good quality control, dimensional stability, and flexibility in the manufacture of profiles by compared to hot-rolled sections [5,6]. In thin-walled or cold-formed steel sections, the width/thickness ratio of the plate elements is always important and the bending failure occurs by buckling and not by deformation, which limits its loading capacity [7]. Generally, cold formed steel sections such as U, Z, and I sections are effectively used as bending elements for purlins, wall grids, and roof slabs [8]. The strength of the elements used in the design is generally governed by buckling. They are commonly referred to as "light gauge sections" because their thickness is normally less than 12.0mm. However, more recent developments have made it possible to form cold sections up to 25mm thick, and open sections with a thickness of around 8mm are becoming common in building construction. The steel used for these sections can have a yield strength ranging from 250 MPa to 550 MPa. The elasticity value is increased from 15 to 30% [9]. In steel constructions, I sections are normally used as beams and columns. The current form of this beam is constructed from two parallel flanges a smooth web. One of the recent developments in construction technology is the use of the corrugated core instead of a solid core [10]. Corrugated web beams have been used for steel buildings and road bridges in Europe since the early 1960s and for road bridges in Europe and Japan since the 1980s [11]. In addition, the shear stresses which develop due to external loads are taken up by thin cores. If the web of an I-section is unstable (buckling), stiffeners can also be used to compensate for stability issues. To eliminate the use of stiffeners, corrugations in the web portion can be provided as a solution as this corrugated web provides greater lateral stiffness compared to flat webs [10,12]. CFS members are known as members subject to local, global buckling instabilities and strain, even at stress levels below the yield point. To minimize buckling instabilities by eliminating free edges, cross sections have been developed and subjected to extensive studies to find their varying structural behavior in the context of bending, shearing, and crushing of the web [13]. Corrugated web beams are advantageous for the construction industry, due to the maximum lateral stiffness of the beam. The corrugated core can be sinusoidal, triangular [14], trapezoidal and rectangular) [3,11]. A corrugated core beam is constructed with thin-walled corrugated cores and wing plates. The profiling of the webs avoids the rupture of the beam due to the loss of stability before the plastic limit load of the webs is reached (class 4). The main assumptions for corrugated steel plates are negligible bending capacity with adequate out-of-plane stiffness [8].
In the literature, many researchers have attempted to use corrugated plates in the webs of hot-rolled I-beams. This overcomes the drawbacks of conventional stiffened flat cores. Such as web instability due to bending stress and fatigue failure. Previous researchers studied I-beams with trapezoidal corrugation for hot-rolled sections. Abbas and al [11] treated a comparison between a beam with a trapezoidal web and the other a beam with a sinusoidal web, comparing with an I beam. The work confirms that corrugated webs have better shear stability and fatigue resistance compared to the standard flat web I-beam. Sumathi and al [1] showed a study of the behavior of a CFS with a solid web, a triangular corrugated web, and a trapezoidal corrugated web. This study involves the examination of theoretical and experimental investigations of serial specimens. Altogether three specimens were tested with a length of 1200 mm, they are tested under two points bending load with single support condition. Theoretical data used by Sumathi and al are calculated using the Indian standard code IS 801-1975 [1]. In our study, the numerical model was validated by the theoretical model, which is based on the calculation code Eurocode 3. It was found that the beam with a trapezoidal corrugation embedded web of 300 and 450 not only increases the load capacity but also the bearing capacity compared to the beam with a smooth web in the work of Divahar and al [8]. The use of cold corrugated webs makes it possible to increase the buckling rigidity of the beams where the folds will play the role of stiffeners. Thus, the use of these corrugated sheets will make it possible to lighten the beams, which will lead to a gain in weight (low thickness) and application to beams of great range.

THEORETICAL METHOD
n order to have an economic effect of the use of bent profiles in constructions, it is necessary to study new optimal cross-sectional shapes, new calculation methods, new construction, and assembly technologies. This reason led us to think about a new process for manufacturing large span metal beams (12 m) of form I allowing the design of hangars. These beams are made by a welded assembly of plates and cold folded sheets.

I
The dimensional characteristics are shown in Fig. 1 and the verification of the geometric proportions are presented in the following formulas [15]: Regarding the minimum thickness to be used, Eurocode 3 part 1-3 proposes the following values [15]: To ensure sufficient rigidity and to prevent buckling of the edge stiffener, the dimensions of the edge stiffener should be between the following values [15]:     The cross sections of the profiles can be classified according to their ability to reach their limit state. This classification is influenced by the ability of sections to plasticize and the influence of instabilities. The Eurocode defines four types of classes. Gatheeshgar and al [13] describe the calculation of the effective properties of section I with flanged edges in the upper and lower flange in more detail. The properties of effective cross-sections in class 4 are based on the effective widths of the compression parts. To understand the phenomenon of instability of cold-formed elements, we have recourse to calculation codes, the most widely used, namely the European code (Eurocode), in the design philosophy of this type of profile is based on the principle of the effective width. This principle currently forms the basis of most of the calculation codes for cold-formed thin sections.
CFS profiles are placed in class 4 because of their thicknesses and their susceptibility to local instability. In this class, the bending moment or compressive strength of a cross section must be determined with explicit consideration of the effects of local buckling [15] (Fig. 5).   [16].
The general iterative procedure should be applied to calculate the effective properties of the flange and the compressed edges (plane element with edge stiffener). The calculation is carried out in three steps: The first step is to obtain an initial effective cross section for the stiffener using the effective widths of the flanges which are determined by considering that the compressed flanges are doubly supported as well as the stiffener provides full support (   K ). The determination of the effective width of the compressed flanges takes into consideration the stress ratio:   1 (uniform compression) and the buckling coefficient   4 b are based on the following formulas [15]: The second step is the use of the initial effective cross section to determine the reduction coefficient  d , taking into account the effects of continuous elastic retention i.e the effective parts of the edge stiffener behave as a member fully supported by elastic springs of rigidity K along its central axis (Figs. 6a and 6b). The critical elastic buckling stress  , cr s of the edge stiffener is  cr, s 2 KEIs σ As (16) with: K is the stiffness of the elastic support per unit length. For the upper edge stiffener; Is is the effective moment of inertia and As is the effective area of the edge stiffeners. The thickness reduction coefficient  d for the edge stiffener is presented in Fig. 6c. The reduced slenderness  d is given by the following formula [15,17]: In the last step, we can determine the area of the effective cross section (Fig. 7) and the effective inertia and modulus of resistance of the effective section: we repeat the first step by calculating the effective width with the reduced compressive stress  , cr s . The iteration is optional in Eurocode 3 with [17]:

Bending check
In a single span simple beam, failure occurs when the value of the bending moment (Msd) exceeds the resistance moment of the cross section, the magnitude of which depends on the shape of the profile, the strength of the material, and the classification of the section. In cases where the shear force exerted on the cross-section can be considered small enough that its effect on the design resistance moment can be neglected (EC3 sets a shear force value of 50% of the plastic design resistance to shear). the bending resistance of the cross-section must satisfy the following conditions [15]: The resistance: with: C,Rd M is the design value of the bending resistance with respect to a principal axis of the section; ed M is the bending moment at the ultimate limit state [15].
Shearing effort: The plastic shear strength V pl.Rd shall be determined by [15]: with: . pl Rd V : Design value of the plastic shear strength.
 : Inclination angle of the web relative to the flanges.
Regarding Eurocode 3, the hot-rolled sections calculation procedure is used for cold-formed sections, except that this requires the determination of the effective section, which is not easy. To enhance the analytical results, the numerical study has been performed.

FINITE ELEMENT MODELING
he finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. The basic concept of finite element analysis is that a structure is divided into a finite number of elements having finite dimensions and reducing the structure having infinite degrees of freedom to finite degrees of freedom [18]. For finite element analysis, the advanced ABAQUS software was used for our numerical simulation [2,[19][20][21][22]. The goal of our project is the design of beam models on two supports subjected to bending. The models studied are I-beams with a span of 12 m in cold formed steel with different web sections ie. be a solid core and a corrugated core (trapezoidal or triangular) 3 mm thick. The ripple angle for the trapezoidal model is 55° and 45° in the triangular model as shown in Tab. 1. The cross sections of the three beams are 150 × 300 × 3 mm 3 . The beams were stressed by a concentrated load in the T middle (three-point bending) with increasing static loading in order to develop a more efficient element. The beams are considered to be made of high yield strength steel for cold forming (S355 according to EN10149-2). During the production process of cold formed elements, the initial mechanical properties of steel are often changed. The shaping operation is usually accompanied by an increase in the elastic limit yb f and the tensile strength u f . Tab. 2 summarizes the mechanical properties of the beam with a single web and the beams with corrugated webs (triangular and trapezoidal). When the forming force is applied to the sheet, the sheet will deform and plasticize to the desired shape reaching certain stress. It will represent the new elastic limit if we recharge immediately. On the other hand, if we recharge after a certain time, the elastic limit will be more important [23]. In reality, the increase in tensile strength fu is much smaller than that in elastic limit yb f so the shape of the stress-strain curve of the steel will change and be like that shown in Fig. 8. In this case, the elastic limit y f is determined for a strain equal to 0.002, Young's modulus, Poisson's ratio are  5 2.1 10 . MPa and 0.3 respectively. In the numerical model, the manufacturing process is considered taking into account the mechanical properties of materials determined after the production process due to residual stresses.

Geometry and mesh
The mesh used represents a good compromise between the calculation time and the precision of the results. For the nonlinear mechanical analysis, the beams are modeled as plate elements assembled together, then for the mesh, it refers to the library of ABAQUS and according to a case study proposed by it. In our study; elements of the S8R type are used, including S8R: A thick shell with 8 nodes doubly curved, reduced integration. This brick element can be used effectively in geometric and material nonlinear analysis which have been taken into account, including plasticity, contact, large displacements, and fracture. An example of finite element mesh of modeled beams is presented in Fig. 9.

Loading and boundary conditions
The tested beams are simply supported, the boundary conditions result in a blocking of displacements at the level of the first support where one has (    0 x y z U U U ). On the other hand, the other support there is a blocking of displacements and rotation which ensures (    0 x y x U U R ) (Fig. 10). The model taken is that respecting an isostatic beam where the model symmetry has not been exploited. The external loading is bending and presented by a concentrated load applied to the middle of the upper flange (three-point bending) and increasing in intensity, which varies up to the load  10 P kN (Fig.  11). The case taken is that of the applied charge seen that our case is a main element of resistance.  The loading adopted for the tested beams.

RESULTS AND DISCUSSION
orrugated web beams are advantageous for the construction industry, due to the maximum lateral stiffness. The main objective of this study is to analyze the behavior of beams with corrugated webs (trapezoidal or triangular) and a flat web, by an analytical approach and a theoretical approach. It is shown that for a load of 6 kN, the corrugated web beams (trapezoidal and triangular) exhibited a decrease in displacement y U of around 36% and 17% respectively, compared to the single web beam. Besides the beam I shows a concentration of displacements in the area of application of the load with  105 max U m m (central area) as shown in Fig. 12a. The triangular web beam presented a significant lateral displacement compared to the other beams (simple and trapezoidal), this lateral displacement results in instability. On the other hand, we notice a reduction in displacement x U . of about 21% for the beam model with trapezoidal web, compared with the solid web beam. The beam with a trapezoidal corrugated web has a maximum load capacity compared to the smooth web [13]. A cold formed steel beam with a corrugated web has a lower deflection than steel beams with a smooth web [18] (Fig. 12b). For a load of 6 kN, it can be seen that the trapezoidal corrugated web beam and the I-beam presented a very remarkable rigidity from a deformation reduction point of view. A 15% reduction in deformation compared to the normal web beam. The central area is the most stressed (the area of load application) as shown in Fig. 13. This type of deformation is of

Normal web
Trapezoidal web Triangular web C importance for long beams with low lateral and torsional stiffness. Similar behavior from a deformation point of view for the simple model and the triangular model. A coincidence of curves except for the load application area has been observed.  For the different beams studied, we measured the state of stress deformation in the center point of the lower flange. By describing the constitutive law of the three studied beams, the three models have a nonlinear behavior. In the linear mode, at the beginning of the loading, the beams presented an elastic behavior whose strains are proportional to the applied forces (    2% elas ). Plasticization of the material is obtained for higher values. Plasticity results in the dissipation of energy during deformation, mechanical energy is transformed and leading to the irreversibility of the behavior of the material, this mechanism also reflects the ductility of the material, which allows the beam to undergo elongation significant before breaking up as shown in fig. 14. All three beams get plasticized before the elastic limit (355 MPa) is reached, confirming the theory of efficient section and class 4 [15]. The single-core model exhibited large deformations. Failure occurs in the cold formed solid core beam by distortional lateral buckling and local buckling. The triangular core beam showed an increase in yield strength compared to other models. The stress concentration occurs near the center point. Failure of the corrugated beam is accomplished by lateral torsional buckling. Fig. 15 shows the state (load -horizontal displacement Ux) in the center point of the lower flange and it was found In the range of 0 to 7 kN, the beams with wavy (triangular and trapezoidal) have shown small displacement which translates into a remarkable linearity. Above 7 kN, the beam with a trapezoidal web core has a lower lateral displacement of the order of 44% compared to the beam with a single web. Models (beam with single and triangular webs) have a risk of torsional lateral buckling [12]. The triangular soul beam undergoes significant lateral displacement compared to other beams.   Fig. 16 shows the failure mode of the single web beam modeled with the ABAQUS software which was chosen because of its high performance and precision [19][20][21][22]24]. It is noted that the beam I has undergone a local buckling in the upper flange at the level of the zone of application of the load in addition to a distortional lateral buckling. This last mode occurs because

Load (kN) Horizontal deflection Ux ( mm)
Normal web Trapezoidal web Triangular web the beam is slender and is characterized by the fibre, which moves laterally, and the transverse load exerts torsion (low lateral stiffness and low torsional stiffness). The excessive compression in the compressed part of the beam causes this part to buckle, which causes it to come out of its main plane of bending while the tensioned part remains in its plane. It is observed that the ripple angle increases when the load capacity increases (Fig. 15). The rupture of the core is eliminated thanks to the presence of corrugations in the core [25]. The trapezoidal corrugated blade has better stiffness in bending and shearing out of the plane [26]. Fig. 17 presents the failure mode of the beam with the trapezoidal web. It showed local buckling in the load application area. The beam is subjected to compressive forces, which is why it tends to be veiled locally. This mode only involves the out-of-plane flexion of the top sole without deformation of the flanged junction lines, that is to say just one rotation and no deformation (Fig. 17). It is shown from the rupture mode that the lateral buckling resistance of the web increases with the trapezoidal corrugation web. The results are in the same trend as those of Divahar and Joanna [26].  Fig. 18 presents the failure mode of the beam with the triangular web. The triangular core beam breaks by lateral torsional deformation due to the large gap in the web section, this finding has the same tendency as those of Preethi et al [12]. The stress concentration occurs near the center point. Failure of the corrugated beam is accomplished by torsional lateral buckling (Fig. 18). From the point of view (load -vertical displacement Uy) at the level of the central point of the lower flange, during the increase of the applied load, it is noted that the beams with corrugated webs (trapezoidal and triangular) presented a maximum capacity i.e for a load of 7 kN, a reduction of displacement of order 84% and 31% for the model with trapezoidal and triangular undulation respectively compared to the model of the simple beam (Fig. 19).
The results of the numerical study are shown in Tab. 3, which shows the maximum deflections, maximum stresses, major moments, and the ultimate failure mode of beams with a single web and a corrugated web [18].

Vertical deflection Uy ( mm)
Normal web Trapezoidal web Triangular web For a load of 6 kN, Fig. 18 shows a very good correlation between numerical results and those proposed by Eurocode 3 part 1-3, a difference less than 13%. The numerical results confirm that the triangular corrugated web beam has a maximum moment capacity compared to all models, which is 25% higher than the ordinary web. The trapezoidal core configuration has greater strength compared to the triangular and ordinary core configuration. Eurocode 3 gives a more unfavorable state of moment compared to other models because Eurocode is based on experimentation, which is why some failures can be found. Therefore, the corrugated steel beam is economical in all respects [27]. The beam whose web is trapezoidal corrugated has greater moment resistance and greater rigidity compared to the single-web beam. This displacement ductility can influence the increase in stiffness of the corrugated web [26].

CONCLUSIONS
he numerical study and the theoretical approach on section I beams with different web shapes allowed us to draw the following conclusions: • Tried metal beams are made of relatively thin parts and with significant slenderness. They are particularly sensitive to the phenomena of instability. The single web beam exhibited a significant lateral displacement compared to the corrugated web beams (triangular and trapezoidal) and a local buckling, this lateral displacement results in instability i.e a distortional lateral buckling. • The beam with trapezoidal web showed a failure in the upper flange of the central zone which is defined by local buckling. • The beam with the triangular web exhibited a significant torsional lateral displacement in the central zone.
• Analysis shows that an I-beam with a corrugated web will twist out of the plane at the same time as it deforms in the plane under the action of loads in the plane. • By increasing the load, and in the midpoint of the bottom flange. It can be seen that the beams with corrugated webs (trapezoidal and triangular) showed a reduction in displacement y U of the order of 84% and 31% respectively.
• The trapezoidal corrugated web presented a maximum capacity compared to the beam model with regular and triangular web: -A decrease of 21% in x U displacement point of view. -A 36% decrease in y U displacement point of view.
• Corrugated web beams are advantageous for the construction industry due to their maximum lateral stiffness.
• Corrugation improves the bending capacity and the beam stiffness, especially for trapezoidal web beams.
• The load capacity is higher for beams with corrugated webs compared to beams with a normal web.
• The European Eurocode 3 regulation is effective for dedicating to the sizing and design of thin cold formed beams.