Verification of lira-sapr calculation of i-beams with two symmetry axes for general stability in transverse bending in the plane with maximum stiffness

. The paper considers verification of calculation of split I-beams with two symmetry axes on the general Eulerian stability at transverse bending in the plane of maximum stiffness in the software package (PC) LIRA-SAD. Calculations for various loads are given. Numerical calculation results in PC LIRA-SAP are compared with calculation by analytical formulas of S.P. Timoshenko and V.Z. Vlasov. Conclusions about calculation accuracy in PC LIRA-SAD in comparison with analytical formulas analysis are made.


Introduction
When calculating building structures in modern software packages using the finite element method (LIRA CAD, SCAD office, etc.) the verification of calculations is an important task. One of the easiest ways of calculation verification is comparison of results obtained by numerical simulation and by analytical expressions.
In "Verification report on LIRA-SAD software" (pp. 78-68, [3]) considered the question of verification of calculation of rods for general stability at transverse bending. Here the authors consider the static calculation of a rod model that does not take into account the sectional moment of inertia of the section, i.e. the stability calculation is performed without taking into account the deplanation of the sections (no support ribs, no constriction torsion at the loss of stability). This means that the LIRA CAD software inadequately reflects the performance of thin-walled rods under torsion that accompanies the loss of beam stability.
It is obvious that the rod model does not fully correspond to the real operation of an Ibeam at the stage of stability loss.
In this paper an attempt is made to calculate the overall stability of I-beams in the PC LIRA-SAPR when modelling the flanges, wall, support ribs by shell finite elements which is not considered in the verification report [3].
Thus, the question of calculation of real I-beams on the general stability in PC "LIRA-SAPR" remains open.
For the first time the question of general stability of I-beams at transverse bending is considered in the work of academician S.P. Timoshenko "Stability of elastic systems" [1]. In this paper the author obtained formulas for determining critical loads on a rod element subjected to compression and bending in two planes. As a special case, [1] gives an expression for the calculation of general stability of a sectioned I-beam, the ends of which are subjected to bending moments in the plane of maximum rigidity.
On the calculation of thin-walled rods and, in particular, I-beams was dealt with by Professor V.Z. Vlasov. In his paper "Thin-walled elastic rods" [2] he developed a bending and torsional model of thin-walled rods, and showed the effect of support and intermediate ribs on the torsional stiffness of the rods. Also, in [2] general conditions of stability of thinwalled rods in transverse bending are considered and expressions for determining the critical force applied in the middle of the span and the critical uniformly distributed load in the plane of maximum stiffness for a sectional I-beam with two symmetry axes are given.
In [8 -10] authors conducted researches to clarify methods of calculation of I-beams for general stability in transverse bending in the plane of maximum stiffness.
In this paper comparison of results of calculation of sectional I-beams with two symmetry axes on general stability by analytical formulas of S.P. Timoshenko, Z.V. Vlasov and with numerical simulation in PC LIRA-SAPR taking into account deplanation of sections and constrained torsion is made.

Main Part
Critical loads of the first form of I-beam stability loss from wall plane according to analytical formulas of S.P. Timoshenko, V.Z. Vlasov and PC LIRA-SAPR for sectional scheme without bracing in the span (calculated span l ef ) with a cross-section having two axes of symmetry ( fig. 1).
Calculations have been carried out for the following loads: -Multidirectional bending moments applied at the ends of the beam ( Fig.1(a)); -concentrated load in the middle of the span ( Fig.1 b)); -uniformly distributed load ( Fig. 1(c)).  Sectoral moment of inertia I w and moment of inertia of the section in pure torsion I t have been determined in the SCAD office software package when modelling the section according to Fig. 2.
The values of moments of inertia of the investigated beam are presented in Table 1. It is shown in [2] that the presence of support ribs slightly increases the overall stability of the beam at transverse bending in the wall plane.
At the same time, it is impossible to carry out a calculation in PC LIRA-SAPR for the overall stability of a beam without support ribs, because in the absence of support ribs the local loss of wall stability in the area of support sections occurs under a smaller load than the loss of overall stability of the beam. Therefore, the minimum thickness of support ribs was selected to ensure the stability of the girder wall with manifestation of the first form of general stability loss.
As a result of the simulation, it was not possible to completely eliminate the supporting effect of the support ribs on the overall stability of the beam. From this it follows logically that the critical load determined in the LIRA CAD software must be somewhat greater than the values determined from the analytical expressions. Figure 3 shows the calculated beam models in LIRA CAD for the three considered loading types.  Table 2: Table 2. Calculation results for beams for overall stability in LIRA CAD Diagram of Fig. 1 (a) (bending moments at girder ends), critical load М cr , tcf*m Further, for three types of loading according to Fig. 1, critical loads were calculated using analytical formulas of S.P. Timoshenko and V.Z. Vlasov.
At differently directed bending moments applied at beam ends ( Fig.1 (a)) critical moment calculation М 1cr is made according to the formula given by S.P. Timoshenko (p. here l ef -girder span without bracing in the span, Еmodulus of elasticity of beam material, С = GI t -torsional stiffness, C 1 = EI w -deplanation rigidity. With a concentrated load in the middle of the span ( Fig. 1(b)), the analytical expression for the critical load is Р cr is given in V.Z. Vlasov's paper on p. 378 [2] (formula 4.8): here -coefficient determined according to  For a better approximation from the data in Table 3, the dependence is plotted ( 2 ) (Fig. 4) and the value of the coefficient k = 26.1. For a split beam with a uniformly distributed load (Fig. 1 (c)) acting in the wall plane, the analytical expression for the critical value of (ql) cr , is given in V.Z. Vlasov's paper on p. 379 [2]: here k is the coefficient determined according to Table 4 depending on the parameter m 2 . Using the data in Table 4, a graph is plotted ( 2 ) (Fig. 5) and with the previously calculated 2 = 7.392 (3), the coefficient value k = 43.5 is determined for the formula (4). The results of the calculations for the three loading options are shown in the Table 5. PC LIRA-SAD with sufficient accuracy for engineering calculations calculates critical loads of general stability loss of I-beams at bending in the plane of the wall when modelling beams by shell finite elements taking into account constraint by torsion due to the presence of support ribs.

Conclusion
According to the results of the study, the difference between numerical and analytical solutions for the considered I-beam loading schemes does not exceed 3%. In all calculated cases, the critical load determined numerically in LIRA-SAP is greater than the values determined analytically.
The difference is caused by the fact that the cross ribs in the support sections of the beams contribute to the preservation of the flat bending shape of the beam. At the same time, the analytical formulas developed by S.P. Timoshenko and Z.V. Vlasov for the rod model do not take into account the influence of support ribs on the overall stability of beams.
As shown in S.P. Timoshenko [1], [4] and V.Z. Vlasov [2], [5], the critical load depends on the static moment of inertia of the section relative to the axis y I y , section moment of inertia in pure torsion J t and sectorial moment of inertia of the section J w . However, in the presence of supporting ribs, pure torsion is not possible, since in the supporting sections the flanges of the beam cannot freely rotate with respect to the vertical axis. Due to the cramped torsion of the beam, the critical load on the beam in transverse bending, determined numerically, is somewhat higher than the analytical formulas.
Thus, the accuracy of beams calculation for general stability in LIRA-SAPR corresponds to the calculation according to the known analytical expressions with the accuracy acceptable in engineering calculations, and the calculation error not exceeding 3% is explained by the influence of supporting ribs.