Study of the conditional flexibility effect of compressed chord on the overall stability of corrugated beams

. In this paper, the calculation of cut corrugated beams loaded with uniformly distributed load in the plane of the wall, with a wall height of 333 and 500 mm (54 sizes), chord width from 160 to 400 mm and chord thickness from 6 to 30 mm for general stability in accordance with Building Codes (BC) 294.1325800.2017 is made. A geometric criterion has been defined which defines the boundaries of the load-carrying capacity of Russian corrugated beams between the flat bending form stability and the strength of the normal section under the action of the bending moment in the plane of the wall.


Introduction
Welded I-beams with transversely corrugated walls (hereinafter referred to as corrugated beams) have been used as elements of steel frameworks, structures of buildings and constructions since the 60s of the last century. The effectiveness of corrugated beams has been shown in works [1][2][3][4][5][6][7][8][9][10]. At the same time, corrugated beams are used both with a sinusoidal wall profile as well as triangular, trapezoidal and other shapes [7][8].
In [11][12] a methodology for calculating the overall stability of corrugated beams has been proposed using clause. 8.4 BC16.13330.2017 "Steel structures". It is proposed to use the methodology for rolled I-beams (formula Zh.4). The torsional moment of inertia Jk is determined by the results of calculations in PC LIRA-SAPR.
In [13][14] the issues of stress-strain state of corrugated beams and general stability of corrugated beams under bending are considered.
In the Russian normative documentation, the calculation of beams with transverse corrugated wall (corrugated beams) are regulated by BC 294.1325800.2017 with Amendments 1, 2, 3 "Steel structures. Design Rules", where section 20.6 deals with the general provisions, dimensions and calculation of the load-bearing capacity of corrugated beams with sinusoidal and triangular corrugations.
Thus, paragraph 20.6.3.11 of BC 294.1325800.2017, with Amendments 1, 2, 3 defines the procedure for calculating the bending in the wall plane corrugated beams for stability, where instead of calculating the overall stability of the corrugated beam as such, the calculation for the stability of a single compressed chord as a centrally compressed rod according to BC 16.13330.2017 "Steel structures".
In this paper, we made the calculation of 54 cut corrugated beams from the domestic variety according to GOST 70571-2022 "Two-beam steel welded with a cross-corrugated wall for building structures. Range", loaded with uniformly distributed load in the plane of the wall, for general stability and strength under normal bending stresses in accordance with BC 294.1325800.2017. Based on the results of the calculation, conclusions were made as to which sections of the corrugated beam have the bearing capacity determined by the stability of the flat bending form, and for which sections the normal stress strength under the action of the bending moment.

Main Part
The paper considers 54 cut corrugated beams of domestic assortment according to GOST 70571-2022, loaded with uniformly distributed load q in the plane of the wall, of the span L = 6.0 m, with the fastening of the compressed chord from the plane of the wall with a pitch = 2.0 ( Fig. 1 For each compressed chord using Table 1, the critical force for loss of stability has been determined Ncr in the girder plane (in the zy plane according to Fig. 3 here Ry = 240 MPa -design resistance of steel C245 of the chord, γс = 1.0 is a duty factor, = * -compressed zone, -is the central compression resistance coefficient determined according to the formula (8), clause 7.
here is a length conversion factor, -calculated rod length in the xz plane, is the moment of inertia of the compressed chord about the y-axis, = 2.08 * 10 5 MPa -elastic modulus of the chord material.
In determining the notional flexibility of the compressed chord ̅̅̅̅̅ the calculated length The central compression girder was assumed to be equal to the girder stiffening step from the wall panel plane = 2.0 m, and a hinged attachment at the ends with a length conversion factor µу = 1.0 (Fig. 1, 3).
Moment of inertia was determined according to the formula for a rectangular crosssection: According to the critical force found the value of the critical transverse load qcr total loss of stability of the corrugated beam compression girder has been determined: The expression (4) is derived from the following relation in clause 20.6.3.4 of BC 294.1325800.2017 (Fig. 1, 2): . . here х . = * 2 /8 -maximum bending moment between the bracing points of the corrugated web, . . -is the distance between the centres of gravity of the girder chords. For the resulting critical load values qcr the value of the maximum normal bending stress of the beam is determined here -the moment of resistance of the corrugated beam in relation to the x-axis, determined according to the formula: here = ℎ + 2 * is a full height section, is a moment of inertia of the section relative to the axis x.
When designing corrugated beams for bending (clause 20.6.3 of BC 294.1325800.2017) it is assumed that the bending moments are taken up only by the chords, so the moment of inertia is calculated according to the formula: The results of the calculations are presented in Table 2. The interpretation of the calculation results in table 2 is as follows: if the maximum normal bending stresses in the beam elements m less than the assumed design resistance Ry = 240 MPa -The load-bearing capacity of the corrugated beam is limited by the stability of the flat bending form. Otherwise, it is limited by the bending moment strength. Table 1 shows that for corrugated beams with numbers 1-30, the loss of flat bending stability occurs before the normal-stress strength condition is violated, i.e. these beams must be calculated/checked for overall stability. For beams 30-54, the normal bending stress strength condition must be calculated/verified first.
In order to visualise the results of the calculation, a graph is plotted using the data in Table 2 ( ) from the reduced flexibility of the compressed chord between the bracing points ̅̅̅̅̅ (Fig.4). The graph in Fig. 4 shows that if the flexibility of the compressed chord is more than ̅̅̅̅̅ > 2.55 , the loss of flat bending stability occurs before the maximum bending stress of the beam reaches the design resistance Ry = 240 MPa.

Conclusion
For a sample of 54 cut corrugated beams of the Russian variety according to GOST 70571-2022, span L = 6.0 m, with an estimated length from the wall plane ℓef = 2.0 m with Ry = 240 MPa, bent by uniformly distributed transverse load in the wall plane, the critical load of loss of overall stability is determined as qcr, and the acting maximum normal bending stresses.
The boundary of conditional flexibility of the compressed chord between the bracing points is determined ̅̅̅̅̅ separating the areas where the load-bearing capacity is limited by the stability of the flat bending form and the bending normal stress strength in the beam: