The effect of columns twisting on the behavior of steel frame of industrial building

. The bracing system plays a very important role in the steel frame of an industrial building. The braces ensure the fixation of structures and their required effective lengths, give the frame overall spatial stability and geometric invariability. The scheme of bracing system significantly affects the behavior of the frame and the distribution of forces in its elements. The influence of the design solution of vertical bracing systems along the columns on the stress-strain state of the frame is studied in the article. It is shown that when using solid I-columns with the large web height (over 500 mm), to prevent their twisting and excessive displacements, it is necessary to fix both flanges of the columns in the longitudinal direction. That is why, the use of a single-plane vertical bracing system for the columns with the large web height is not recommended. For this, either two-plane braces can be used (allowing to reduce the twisting angle of the columns by 95%), or various options for fixing the flanges of the columns to struts or other structures (provide a reduction in the twisting angle of up to 88%).


Introduction
The bracing system as part of the steel frame of a multi-storey industrial building plays an important role in ensuring its spatial stiffness and stability, and also affects the distribution of forces in the frame elements.
According to the current version of the Russian building code for the design of steel structures (SP 16.13330.2017"Steel structures") in buildings with two-branch columns, if the distance between the branches of the columns is more than 500 mm, vertical braces must be placed in the plane of each of the branches.
During the inspection of the existing frames the twisting of the solid I-section columns with a large web height (exceeding 500 mm) is often observed if single-plane vertical braces are attached to the I-columns in the middle of their web.So, in this case the thinwalled columns should be analyzed with account of restrained torsion [1].
Column twisting negatively affects the behavior of the frame and reduces its spatial stiffness.Torsional deformations of columns may cause additional off-design horizontal effects on the beams attached to them.This is especially true for beams, which are the crossbars of the frame and ensure its spatial work in the transverse direction.The behavior of the beams is significantly affected by the design solution of the beam-to-column joint to [2][3][4][5][6][7], and the deformability of the columns.Due to the twisting of the columns, the beams experience bending moments in the horizontal plane that accompanied by restrained torsion.
In addition to the torsion of columns, their increased deformability in the longitudinal direction is observed when using single-plane bracing system due to its low stiffness.
As a result, under the action of uneven loads on the frame, which is often typical for the frames of multi-storey industrial buildings and whatnots, adjacent rows of columns may have different displacements of large magnitude in the longitudinal direction, due to which transverse horizontal forces are transmitted to the transverse beams attached to these columns.
The work of I-beams on horizontal and torsional actions is very inefficient.In order to avoid this, it is possible to eliminate the twisting of columns by some constructive methods, ensuring fixation of their flanges in the longitudinal direction.With this aim, it is advisable to use two-plane braces installed in the planes of the I-column flanges.If it is impossible to implement such a solution, for example, for situation of reinforcement of the existing frame, it is possible to use local ties that attach the flanges of the columns to the one-plane struts located along the columns rows.
The article presents the results of theoretical studies using the finite element method of the influence of various design solutions of a bracing system on the deformations of columns and on the stress-strain state of the beams attached to them.
Various studies [8][9][10][11][12][13][14][15][16][17] show how the bracing system configuration affects the stiffness and displacement of the steel multi-storey frames.However, the review of the literature has shown, that there is no information in the world literature about the results of studies of the influence of the bracing system design solution on the torsion of columns.In this regard, the study of this issue is relevant and important.

Materials and Methods
The existing frame of the converter compartment of the converter shop No. 2 of the Novolipetsk Iron and Steel Works (NLMK) is considered in the article [18].
The converter compartment has a complex shape in plan with dimensions of 340x132 m.Its steel frame is multi-span and very heterogeneous in height which is vary from 30 min the zones of one-story bucket and scrap spans up to 83 m -multi-storey stacks of wet gas cleaning and a cooler boiler.
The spatial design scheme of the frame of the converter compartment was made using the SCAD Office 21.1 software package.To model the main load-bearing structures, bem finite elements with 6 degrees of freedom in the node were used.
The investigated zone of the frame is the area of the stack of bulk materials in the G-J axes, which is a multi-storey structure with solid I-section columns with the height of 1000 mm and floor and roof beams attached to them (at level about 64 m).According to the project and survey data, single-plane vertical braces are attached to the middle of the column web.
The calculation using bar finite elements showed overloading of the roof beams in the G-D axes according to the criterion of strength from the action of bending moment in the horizontal plane, which occurs in the beams even under the action of only a vertical load.
Roof beams are welded hinged I-beams with the web -1226x12 and flanges -300x16 made of steel 14G2.Roof beams are attached on the side to the columns on the D row, and on the G row supported by columns from above.
At the same time, according to the results of the calculation, it was found that the roof beams experience twisting, and the calculation using beam finite elements with the 6 degrees of freedom at the node does not allow taking into account the effect of restrained torsion.As a result, additional normal sectorial stresses in thin-walled I-beams of the roof are left out.Calculation of thin-walled beams, subjected to restrained torsion, must be performed with the account of the 7th degree of freedom in the node -warping [19].
To refine the calculation results and study the influence of the structural solution of the bracing system on the forces and displacements in the frame structures, the roof beams and columns were modeled using plate finite elements.
The following cases of the structural solution of the vertical bracing system along the columns were considered: single-plane braces (Fig. 1, a), two-plane braces (Fig. 1, b); single-plane braces in combination with the fixing of the column flanges to the struts with the use of ties made of angles (Fig. 1, c).The calculation takes into account the inclusion in the work of the roof shear disk, made of steel panels 3 m wide, welded at four corners to the roof beams [20].The thickness of the panel sheet is 4 mm, the sheet is reinforced by ribs made of angles.In the finite element model, the roof panels are modeled by plate finite elements.According to the structural solution, the shear disk of roof panels has a gap in the exhaust shaft area near the G row, as well as in the gas cleaning zones in the axes 13-14, 17-18 and 21-22.To compensate for this and ensure the necessary shear stiffness of the roof, horizontal bracing trusses made of angles are provided.Fragment of the finite element model of the converter compartment in the level of roof in the axes G-J is shown on Fig. 2.

Results
As the results of the finite element analysis showed, due to the inclusion in the spatial work of the frame, the I-beams of the roof experience various types of influences, including bending in the horizontal plane and restrained torsion.
Total normal stress at the i-th point of the beam cross-section taking into account restrained torsion can determined with the use of Vlasov torsion theory [21] by the equation: where N -axial force; M y , M z -bending moments about the axis Y and Z, relatively; Bbimoment; A -area of beam cross-section; I y , I z -moments of inertia of beam cross-section about the axis Y and Z, relatively; I ω -sectorial moment of inertia of beam cross-section; z i , y i -linear coordinates of the i-th point of beam cross-section; ω i -sectorial coordinate of the i-th point of beam cross-section.
In the result of the calculation, the total normal stresses in the plate finite elements were obtained.In order to determine the values of the forces (N, M y , M z , B) acting in the cross section of the roof beams, the following method is used.
A system of linear equations is compiled based on formula (1) to determine the normal stresses at the 4 most loaded points of the cross section (marking of points -see Fig. 3): In the system of equations ( 2), the values of the forces N, M y , M z , B are unknowns in the equations, and the stress values in the each point of cross section (σ 1 , σ 2 , σ 3 , σ 4 ) are taken from the results of the calculation.

Fig. 3. Marking of points of the roof beam cross-section
The forces obtained as a result of solving the system of equations (2) for the most loaded span cross section of the beam are given in Table .1.Besides the maximum values of the horizontal deflection and the twisting angle of the beam in the span are shown in Table 1 To analyze the influence of various loads, there are considered loadings of the frame with a uniformly distributed vertical load (snow load), a longitudinal horizontal load acting in the longitudinal direction (wind load) and a vertical crane load.The values of displacements and angles of rotation of the columns at the level of the roof beams are shown in the Table 2.

Conclusions
Solid I-section columns with a web height of more than 500 mm, fixed with single-plane braces attached to the middle of the column section web, experience torsion during work as part of the frame.Because of this, there are negative phenomena associated with large uneven displacements of the flanges of the columns.For this reason, beams attached to columns, when included in the spatial work of the frame, experience additional bending in the horizontal plane.Prevention of column twisting allows to increase the stiffness of the frame in the longitudinal direction and reduce the stresses in the beams.
Calculations have shown that the most effective structural solution is the use of a system of two-plane braces and struts attached to the column flanges.
In this case, when a vertical uniformly distributed load (for example, snow load, dead load, etc.) acts on the roof, the decrease in the longitudinal displacement of the top of the columns at the level of the roof beams is up to 90%, and the twisting angle of the columns is up to 95%.The horizontal deflection of the roof beams is reduced by 23%, and their twisting angle is reduced by almost 38%.In this case, the reduction of the maximum total normal stress in the cross section of the beam is up to 5%.
If it is impossible to install two-plane braces to fix the columns from twisting, it is possible to use ties that fasten the flanges to the struts.
Such a solution, under the action of a uniform vertical load on the roof, makes it possible to achieve a reduction in the twisting angle and longitudinal displacement of the top of the columns at the level of the roof beams up to 88%.In this case, the horizontal deflection of the roof beams is reduced by 14%, and their twisting angle -by almost 25%.The reduction of the maximum normal stress in the cross section of the beam is up to 3%.
The influence of the structural solution of the bracing system along the columns on the stress-strain state of the frame under the action of other types of loads, in particular, the longitudinal wind load and the vertical crane load, is less significant and represented in Table 1 and Table 2 respectively.
Thus, it can be concluded that for solid I-section columns with a web height of more than 500 mm, the use of a single-plane vertical bracing system is not recommended.To ensure the efficient work of the frame, both flanges of the columns should be fixed against displacement, for which it is recommended to use two-plane braces and struts.In addition, it is allowed to carry out local fastening of the column flanges to struts or other elements, which also gives a sufficient restrained effect.

Discussion
The horizontal shear stiffness of the roof affects the stress-strain state of the frame and its spatial work.
The shear stiffness of the roof is provided by a system of horizontal braces and, under certain conditions, by enclosing structures that act as shear diaphragms [20,[22][23].
The structural solution of the enclosing structures, their attachment joints to the supporting structures and to each other, and the layout of their location affect the value of the shear stiffness of roof disk.
In the frame of the converter compartment under consideration, the horizontal shear stiffness of the roof is largely determined by the configuration and continuity of the shear disk of enclosing metal steel panels.
The study of the influence of the layout of the roof steel panels on the behavior of the frame and the stress-strain state of the roof beams is a task for future research.

Fig. 1 .
Fig. 1.Finite-element model of beam-to-column joint on the row D: a -single-plane braces; b -twoplane braces; c -single plane braces combined with fixation of column flanges to struts by ties

Fig. 2 .
Fig. 2. Fragment of the finite element model of the converter compartment in the level of roof in the axes G-J (roof panels modeled by plate finite elements)

Fig. 4 .
Fig. 4. Deformed shape of the beam-to-column joint (on contour view the linear displacements along X-axis are shown): a -single-plane braces; b -two-plane braces; c -single plane braces combined with fixation of column flanges to struts by ties

Table 1 .
Maximum forces, stresses and displacements in the span cross section of the roof beam

Table 2 .
Maximum displacements of the column at the level of beam gravity centre