Stress Analysis of Steel Beams Made of Sigma Cross-Section

This paper presents a stress analysis of elements made of a steel cold-formed sigma cross-section, uniformly loaded in a plane parallel to the web and not passing through the shear centre. Such an application of a load very often occurs in engineering practice and corresponds to the application of a load to the upper fl ange of the crosssection. It usually result in an additional torsional moment. In this paper, special attention is paid to normal stresses from the bi-moment, and shear stresses from restrained and free torsion. The contribution of these stresses to the section utilization was evaluated on the example of a sigma cross-section with diff erent thicknesses of the wall. Furthermore, the paper also included the stresses analysis concerning diff erent load locations at the upper fl ange. All numerical calculations were made using analytical approach based on Vlasov beam theory.


INTRODUCTION
Thin-walled bars, due to the high slenderness of their walls, are sensitive to the loss of stability phenomenon, which is described as a sudden change in shape (deformation) of a structural element under critical load. There are distinguished three basic forms of loss of stability [1,2], namely global local and distorsional. Global beam instability was developed by Vlasov [3] as a well-known Vlasov beam theory, which is based on previous work of Timoshenko [4] and dedicated to the thin-walled steel elements with the open cross-section. The basis to calculate local buckling comes from the theory of plates from [5] and the experimental corrections done by Winter for the preparation of the fi rst edition of the American Iron and Steel Institute Specifi cation for the Design of Cold-Formed Steel Structural Members [6]. For practical reasons, the concepts of the eff ective width and eff ective thickness were proposed and introduced in Eurocode [7]. A description of the thin-walled element theory is also described in the works [8,9]. Despite of cold-formed steel elements bearing capacity procedures are widely known in engineering practice, there is still a place for further investigations for the cross-sections with one axis of symmetry, such as sigma cross-sections.
As it can be observed, the list of publications referring to sigma cross-sections analysis is very limited. Although, in recent years there has been an increased interest in solutions related to thinwalled, cold-formed steel structures. The latest study by [10] was conducted to describe the behaviour of compressed thin-walled steel columns with a sigma cross-section. The authors of [11] investigated the buckling behaviour of cold-formed steel sigma beam-column members. In their study, the analyses indicate that the failure modes are mainly depending on the stress distribution of the cross-section. The buckling analyses of cold-formed steel sigma cross-sections in purlin-sheeting systems subjected to uniformly distributed uplift load were conducted in [12]. Axially loaded cold-formed sigma profi les were investigated in order to defi ne their local and distortional buckling behaviour [13].

Stress Analysis of Steel Beams Made of Sigma Cross-Section
Maciej Adam Dybizbański 1* , Katarzyna Rzeszut 1 , Aleksandra Szczepańska 1 1 Faculty of Civil and Transport Engineering, Poznań University of Technology, ul. Marii Skłodowskiej-Curie 5, 60-965 Poznań, Poland An analytical approach was devised for calculating the bending-torsion coupled random response of thin-walled beams with monosymmetrical cross-sections in [14]. The authors of [15] investigated the Goldenvejzer solution for the system of governing differential equations of stability of centrically loaded members with rigid open cross-sections. Based on a combination of the Vlasov assumption and the Kirchhoff assumption of plate/shell theory, an analytical formulation for the torsional warping function of a thin-walled open-section beam is proposed in [16]. In [17] the authors investigated a simple thin-walled beam carrying a uniformly distributed transverse load. The post-buckling analysis of thin-walled elements with open sections was investigated in [18,19]. Finite element analysis of thin-walled open section beam structures was presented in [20]. Furthermore, other research in the field of thin-walled elements includes work presented in [21,22,23,24]. However, due to many difficult-to-solve problems, that still have not been discovered despite many years of research, engineers' usage of thin-walled monosymmetric members is significantly limited. Studies mostly concern topics related to buckling analysis, however, there is a gap in the knowledge of stress analysis of sigma crosssections. In engineering practice, it is common to neglect the free and restrained torsion components, however, in this paper, the emphasis is placed on analysing the contribution of both free and restrained torsion to the stress block. For these reasons, in this paper, the stress analysis of a sigma cross-section is carried out. For bending beams of the sigma section, loaded in a plane parallel to the web and not passing through the shear centre, normal stresses from torsional warping σ ω occur in addition to the normal stresses from bending σ x . The normal stresses are accompanied by shear stresses from torsional warping τ ω , uniformly distributed over the wall thickness, which are associated by shear stresses from free torsion τ t .
In the Vlasov beam theory of restrained torsion, it is assumed that the relationships derived for free torsion are valid. The angle of rotation is defined as: where: M T -torsional moment, G -Kirchoff's modulus, I T -torsional moment of inertia.
In the case of a profile, which consists of many rectangular parts, the torsional moment of inertia is roughly equal to the sum of torsional moments of individual walls: where: δ -thickness of a wall, -width of each wall into which the profile can be separated, s -experimental coefficient, which depends on the shape of a cross-section.
The experimental coefficient is based on results obtained from a laboratory tests by Föppl [25], who computed this parameter for torsion of a narrow rectangular cross-section.
Shear stresses resulting from free torsion, constant in all sections, are defined by formula: The longitudinal displacement in the coordinate system (x, y, z) of the member section is proportional to the sectional coordinate: The axis of rotation during torsion with constrained warping passes through the shear center. During deformation, all the remaining fibers, except those on the axis of rotation, are curved. Since the warping due to varying torsion angles is not constant, elongations ε ω = du/dx and normal stresses from warping torsion σ ω = Eε ω arise in the direction of the longitudinal axis of the member. Under the influence of variable normal stresses from warping torsion ε ω , shear stresses from warping torsion τ ω are evenly distributed over the thickness of the walls.
The lack of freedom of deplaning induces a normal stresses from warping torsion equal to: where: E -Young's modulus, φ -section torsion angle. Additionally: The normal stresses due to bi-moment are calculated from the formula: The shear stresses from warping torsion are calculated according to the following formula: where: B ω -bi-moment, M ω -warping-torsional moment, S ω -sectional static moment, I ω -sectional moment of inertia, t -wall thickness, ω -sectorial coordinate.
The warping-torsional moment is described as: Bi-moment is defined as pairs of moments of same amplitude and opposite direction: The formula for calculating normal stresses due to normal forces with bending and torsion for a member is: where: σ x N -normal stresses from compression/ tension, σ x My -normal stresses from bending in respect to "y" axis, σ x Mz -normal stresses from bending in respect to "z" axis, σ x Bω -normal stresses from torsional warping.
The formula (11) can also be presented as: 12) where: N -normal force, A -area of the crosssection, M y -bending moment in respect to "y" axis, I y -moment of inertia in respect to "y" axis, z -distance from crosssection's centre of gravity in respect to "y" axis, M z -bending moment in respect to "z" axis, I z -moment of inertia in respect to "z" axis, y -distance from cross-section's centre of gravity in respect to "z" axis, B ω -bi-moment, I ω -a sectional moment of inertia, ω -sectorial coordinate.
The first three components of the equation (12) correspond to the beam theory regarding compression in combination with bending. The last component contains the bi-moment B ω , the sectional moment of inertia I ω , and the sectorial coordinate ω.
Formula (12) is a generalization of eccentric compression, where the last component defines the normal stresses due to warping. These stresses are distributed in the cross-section according to the concept of the sectoral surfaces.
The formula for calculating shear stresses due to shear forces with bending and torsion for a member with an open profile is: where: τ Vz -shear stresses from forces in "z" axis direction, τ Vy -shear stresses from forces in "y" axis direction, τ Mω -shear stresses from warping-torsional moment, τ Mt -shear stresses from torsional moment.
The formula (13) can also be presented as: where: V z -shear force in "z" axis direction, S ymoment of stability in respect to "y" axis, I y -moment of inertia in respect to "y" axis, V y -shear force in "y" axis direction, S z -moment of stability in respect to "z" axis, I z -moment of inertia in respect to "z" axis, M ω -warping-torsional moment, S ω -sectional static moment, I ω -a sectional moment of inertia, M T -torsional moment, I T -a torsional moment of inertia, t -wall thickness.

PROBLEM FORMULATION
The purpose of the research conducted in this paper was to determine the stress state when considering a load located at the upper flange and not passing through the shear centre. This approach is a departure from the well-known Vlasov beam theory, which assumes that the load is located in a plane passing through the shear centre. The estimation of the contribution of normal and shear stresses from free and restrained torsion to the section was carried out for cold-formed beams made of sigma cross-sections Σ200×2.00, Σ200×2.50, and Σ200×3.00. The calculations were conducted for members with the static scheme of a simply supported beam. The beam span was assumed to be 4 m. A uniform load of 2.00 kN/m along the entire length was considered. The load was applied to the upper flange of the beam in three different plane locations. The first case refers to the situation, where linear load passes through the middle of the fl ange width. The second one passes through the quarter of the fl ange width, and the third passes near the web of the beam. The last one is recommended by EC3 1-3 for gravitational load case, due to high fl exural sensibility of the upper fl ange. Henceforth, an additional torsional moment was considered. The structural elements were made of S350 steel grade. Several geometrical characteristics have been determined following the sigma profi le catalogue [26]. Figure  1 presents the geometrical and Figure 2 the load cross-sections schemes of analysed problem.

STRESS ANALYSIS IN CONTEXT OF VLASOV THEORY
All numerical calculations were carried out for a computational model in accordance with the currently relevant Eurocode, characterised by fl at walls with sharp edges. The stress values were calculated for ten points located in the mid span cross-section of the beam, shown in Figure 2.

Consideration of additional torsion
Linear uniformly distributed load was located in the distance as shown in Figure 3. The value of was assumed as: Linear uniformly distributed load caused the additional torsional moment: To determine the bi-moment and the torsion angle the following formula was used: where: Fig. 1. Geometrical cross-section scheme The functions of the bi-moment B ω and the torsional moment m s will be calculated after solving the diff erential equation for the torsion angle ψ. When the factor ϑl is suffi ciently small, this equation can be simplifi ed considerably by assuming that GI S = 0. The multiplier GI_S is neglected if: • for double-sided fork support (ψ = ψ'' = 0), ϑl < 0.75. • for full restraint on both sides (ψ = ψ' = 0), ϑl < 1.50. • for one end free and the other fully restrained, ϑl < 0.50.
The boundary conditions in the considered case for the free-supported beam (ψ(0) = ψ(l) = 0, ψ''(0) = ψ''(l) = 0) correspond to the fork support conditions. Therefore, the coeffi cient ϑl was analysed in terms of fork support: Due to the support conditions, the diff erential equation for the torsional angle takes the form: The boundary conditions in the considered case (ψ(0) = ψ(l) = 0, ψ''(0) = ψ''(l) = 0) correspond to those of a simply supported beam (w(0) = w(l) = 0, w''(0) = w''(l) = 0). Due to this analogy the diagrams of B ω and M ω , shown in Figure 4, can be created. With the analogy of a simply supported beam, the formula for the value of the bimoment at mid-span of the beam can be determined: Furthermore, the formula for warping-torsional moment near the beam supports may be specifi ed as:

Research program
The process of the analysis was started by the determination of geometrical characteristics of sigma sections considering cross-sections with sharp edges according to [7]. Subsequently, the values of bi-moment, as well as normal and shear stresses, were concluded. Finally, the comparison  analysis was carried out. Figure 5 shows the fl owchart of the analysis process.

Numerical calculations
The results of numerical calculations were collected in Tables 1-9. Figure 6 presents the diagrams of shear stresses from shear forces and warping torsion, as well as normal stresses from bending and warping torsion for the one selected case of considered several examples namely Σ200×2.00 cross-section for load in the middle of the fl ange. The shear stresses from shear forces and warping torsion, as well as normal stresses from bending and warping torsion for the one selected case of considered several examples namely Σ200x2.00 cross-section for load near the web are presented in Figure 7.          Table 9. Stress values of the Σ200x3.00 section -load near the web (in point 7) Section BP/S200x3.00, Span L = 4.00 m, Load q = 2.00 kN/m, Location b 3 = 0 For the same thickness, it can be observed that the normal stresses from warping torsion are roughly 50% higher when the load is positioned in the middle of the flange than when the load is located in the quarter of the flange. However, for the same thickness, the difference is approximately 10-15 times greater when the load is located in the middle of the flange and near the web. Furthermore, the normal stresses from warping torsion for mid-span cross-sections reached approximately 6-10% of the bending stresses for cases with load near the web. A value of approximately 89-93% was observed for cases with load in the middle of the flange. Finally, for cases with load in the flange quarter, the value ranged between 40 and 43%.
In the case of shear stresses, the stresses force from different load placement lwas up to about 30 times greater than the shear stresses from warping torsion, depending on the thickness of the wall and load location.

FREE AND WARPING TORSION STRESSES
From an engineering and scientifi c point of view, it seems very interesting to determine the eff ect of stresses resulting from additional torsion, including warping torsion in the load capacity of the cross-section. Hence the estimation of the contribution of shear stresses determined for warping torsion and shear forces were expressed in the form of the parameter κ: Figure 8 shows the parameter change for each point on mid-span sigma cross-sections describing the contribution of shear stresses from warping torsion and shear forces concerning diff erent load locations for Σ200x2.00. Similar values may be noted for Σ200×2.50 and Σ200×3.00.
It should be noted that for the same thickness the load location heavily infl uences the parameter. Furthermore, the greatest diff erence in parameter κ can be seen on points l and 8, both located on the opposite corner of the cross-section, at the end of the fl ange.
A similar analysis for load in the middle of the fl ange was carried out. Figure 9 depicts the variation of the parameter concerning diff erent wall thicknesses of Σ200. For the load located at the centre of the flange, the parameter is not much different, which proves that in this case, the section thickness has little effect on the shear stresses from warping torsion and shear forces. Figure  10 depicts the κ parameter values for each point on mid-span sigma cross-sections, demonstrating how different wall thickness of S200 affects warping torsion and shear forces for load located near the web.
The parameter for the load near the end of the fl ange varies considerably with wall thickness. The greatest diff erences in contribution between shear stress from warping torsion and shear force from load location can be observed for the thickest wall.
Overall, it can be concluded that the infl uence of shear stresses from warping torsion τ ω is theoretically not so signifi cant, however it should not be neglected in the case of the sigma cross-section, as it diff ers with diff erent load location. Moreover, it is especially critical for elements with thicker walls and the load located close to the web.
In next step, in order to formulate the relation between the normal stresses caused by warping and bending the parameter was formulated: A normal stresses ratio is represented in Figure 11 for points located on mid-span sigma cross-section for Σ200×2.00 with respect to the diff erent load location.
For the load located in the middle of the fl ange, the normal stresses from warping torsion were up to approximately 220% of the stresses from bending, as it can be observed in Figure 11. Furthermore, it was found that for load location near the web, the normal stresses  from warping torsion are close to 0 and may even change their direction.
As it is seen in Figure 12, the normal stresses from warping torsion were up to about 220% of stress from bending for the load located in the middle of the fl ange. Additionally, no signifi cant difference between for each thickness can be noted.
For the load location near the web, the normal stresses from warping torsion were up to about 21% of stresses from bending, as shown in Figure  13. Moreover, the results of this analysis reveal that in comparison the impact of normal stresses from warping torsion σ ω is greater than the impact of shear stresses caused by warping torsion τ ω , especially in the case when load is placed near the web.

CONCLUSIONS
The article presents an analysis of the stress state referring to the theory of thin-walled members, considering the normal and shear stresses from bending and torsion for three diff erent sigma sections: Σ200×2.00, Σ200×2.50, and 200×3.00. Furthermore, a stress analysis of the diff erent load locations at the upper fl ange was performed. Such a load application, as typical engineering practice, caused the additional torsional moment which should not be neglected. As an outcome, special attention was focused, in the paper, on bearing capacity of normal and shear stresses from warping torsion. It is worthy to note, that this analysis provides important insight into the complex relationship between normal stresses and shear stresses induced by warping. Moreover, it can be used to develop an accurate bearing capacity approach for the sigma cross-section. Based on the conducted examples, it was determined that the change in the location of the external load at the upper fl ange results in the reasonably large increase in stresses caused by warping. Based on the coeffi cient, developed in order to defi ne the contribution of the shear stresses caused by warping torsion and shear forces, it was found that the warping shear stresses increased for the load location in the middle of the fl ange. Simultaneously, based on the coeffi cient, introduced respectively to defi ne the contribution of the normal stresses caused by warping and bending, one can noticed, the similar phenomenon.
Additionally, in the paper, the infl uence of the wall thickness on the distribution of stresses was investigated. It can be noted, that for the thicker wall the increase in warping, as well for the normal and shear stresses, is observed. Therefore, it should be pointed out, that the eff ect of warping defi nitely has to be taken into account in the case of the sigma cross-section. Thus neglecting the shear and normal stresses caused by warping can lead to the reasonably large mistakes. Moreover, the thickness of the wall can play signifi cant infl uence, especially when the load is located close to the web.