The problems of calculating the load-bearing structures made of light steel thin-walled profiles

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1 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE-06 The problems of alulating the load-bearing strutures made of light steel thin-walled profiles Vera Ro,*, Oxana Izvekova, and Issa Togo Peter the Great St. Petersburg Poltehni Universit, 9 Politehniheskaa St., St. Petersburg, 955, Russia Abstrat. The artile presents the results of a stud of bearing apait of thin-walled old-formed steel beam of the guide profile. Suh profiles have a small thikness and omplex ross-setional shape. Bending deformation develops in the ross-setional plane under the influene of loads in beam. In addition, deformation of onstrained torsion and warping arise. These deformations influene the stress distribution at the points of the rosssetion of the beam and thereb determine its load-bearing apait. Introdution Toda in Russia buildings with load-bearing frame of thin-walled old-formed setions are widel used []. The have a number of tehnologial and operational advantages (lightness, high speed of eretion, et.) []. As pratie shows, often due to errors in the design of buildings of light steel thinwalled strutures (LSTS), frequent ases of emergen [3]. Sustaining the work of suh strutures is a major problem to the widespread introdution of LSTS in massive onstrution. Obviousl, the thin-walled strutures at the design stage requires a thorough analsis of their bearing apait. Not paing attention to the growing popularit of suh strutures and a large amount of researh on all LSTS stage of development, toda in Russia there are signifiant defiienies in the presriptive, methodologial and omputational databases of design of LSTS whih does not make it eas and lear to make their alulation. Theor of alulation based on the hpothesis of plane setions, are not appliable to the thin-walled bars, due to their small thikness and mismath the enters of gravit and bending. To solve engineering problems of alulation of elements of thin-walled strutures an distinguish two groups of was to alulate: based on the modeling of shells and rods. The first group of methods is assoiated with the representation of a thin-walled rod in the form of a shell and further numerial alulation, as a rule, using the finite element method, in the program omplexes SCAD, Lira, SOFiSTiK et. Suh methods of alulation are aurate enough, but ver time-onsuming in the engineering design ativities espeiall in terms of the omplex struture analsis. In the seond group of methods an be * Corresponding author: veraroa@gmail.om The Authors, published b EDP Sienes. This is an open aess artile distributed under the terms of the Creative Commons Attribution Liense 4.0 (

2 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE-06 distinguished analtial and numerial methods for alulation of thin rods assoiated with the introdution of additional "seventh" degree of freedom and the orresponding power fator bimoment [, 4, 5]. Stud of the bearing apait, deformabilit and manufaturing tehnolog of steel strutures of old-formed setions are devoted to [6-]. In [6,, ] are onsidered the possibilit of aounting the seventh degree of freedom in the reation of a thin-walled finite element. Statement of the researh problem Let us analze the degree of influene of omponent of warping on the stress-strain state of the thin-walled bar. We will lead the alulation for antilever balon beam with ross setion GP [] like for beam having strain of warping. Fig.. Design sheme.. Calulation of the strength of solid setion bent elements Fig.. Cross setion []. Geometri harateristis in alulation are taken the lower of two: b assortment [] and b alulation in the program TONUS (PC SCAD Offie) table.

3 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE-06 Table. Geometri harateristis of ross setion GP Parameter Value aording to STO[] Value aording to TONUS Units A Cross setional area m I t Moment of inertia at free torsion m 4 I w Setorial moment of inertia m 6 W u+ W v+ I u I v Y b Z b The maximum moment of resistane about an axis U The maximum moment of resistane about an axis V The maximum moment of inertia The mininmum moment of inertia Coordinate of the enter of the bend on the x axis Coordinate of the enter of gravit in the x diretion m m m m m m Flexural-torsional harateristis (it is a phsial onstant of ross setion and mathematiall representing a harateristi number of differential equation of equilibrium 4th order) [3] amount to: k GJ EJ t m Upon appliation of a uniforml distributed load and bringing it up to the maximum (destrutive) value due to a suffiientl high dutilit of the top shelf, whih will be signifiantl deformed, the profile loses its original geometri shape. Load, in most ases being a "stamping" that is more rigid than the setion b its nature, will not have tightl fit to the shelf. To indiret aounting for this geometri nonlinearit we will appl the load is not uniform along the shelf, but on the law of the triangle, whih is a fairl ommon method. Then, the resultant load vetor will pass through the enter of gravit of the load diagram, ling in the intersetion of the medians of the triangle, that is, at a distane (b/3) from the wall edge [4]. As an be seen from Fig. 3 full eentriit of load appliation will onsist of the eentriit, due to the mismath between the enter of gravit and the enter of the () 3

4 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE-06 bending ross setion of x, (equal Y b - oordinate of shear enter -axis) and the diret eentriit of appliation of the load equal: b 5.8 e x m () 3 3 Fig. 3. Sheme of axes and load appliation, points numbering of σ. Fig. 4. Diagram of setoral oordinates []. Diagram of setoral oordinates of setion is shown in Fig. 4. The maximum setoral oordinates are neessar for deteting the stress from bimoment; the omprise: h 5.34 ( x b) ( ) m (3) h x. 9 m (4) Aording to p SR [4], alulation of the strength of the beams should be arried out under the ation of moments in the two prinipal planes (and bimoment available) aording to the formula 4

5 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE-06 I xn M x R I M x R I xn B (5) R where x and the distane from the main axis to the point of the setion under onsideration; while the ation in the wall of the beam moment and shear fores. The bending moment in the referene setion: M = P t P t P 3 t 3 (6) M = ) = = = 8540 kg m (7) where t the distane from the end of beam to the point of load appliation, P applied load Assuming that the bending ours onl in one plane, taking M = 0 The formula for the bimoment in the ase of a antilever beam, aording to the tehnial literature []: B Pe sh( k( l t)) sh( kl) k h ( kl) (8) B = kg m (9) B = kg m (0) B 3 = kg m () Bimoment in the referene setion obtain as a superposition of moments of individual fores: B = B + B + B 3 () B = = kg m (3) The maximum normal stress σ, σ, σ 3, σ 4 ourring in the most tense ross-setion, loated in the middle of the span, will be different in the four harateristi points (,, 3 and 4, respetivel) of the ross-setion will be: M B (4) x x Wx I kg / m (5) Similarl, we alulate the value of the normal stress σ, σ 3, σ 4 : σ = kg/m (6) σ 3 = kg/m (7) σ 4 = 947. kg/m (8) Allowable stress: σ per =R γ = = 880 kg/m (9). The alulation using the finite element method (analsis of stress-strain state) Aording to p SR [4], values of stress in the wall of the beam must be heked aording to the following formula in the two prinipal planes of bending: 0.87 х у х у 3 ху (0) R 5

longitudinal axis of the beam; τ x shear stress in the wall.

6 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE-06 x () R where σ x normal stress in the median plane of the wall, parallel to the longitudinal axis of the beam; σ same, perpendiular to the longitudinal axis of the beam; τ x shear stress in the wall. Q S x () I t w Stress σ x and σ, taken in formula (0) with their signs and τ x, should be determined in the same point of the beam wall. For this, we use the finite element method in a stati statement and solve the problem in the software pakage SCAD Offie. Fig. 5. Normal stress σ у. Minimum value kg/m, Maximum value kg/m Fig. 6. Normal stress σ х. Minimum value kg/m, Maximum value kg/m 6

MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/0686007 IPICSE-06 Fig. 7. Shear stress τ ху. Minimum value 86.9 kg/m, Maximum value 56.

7 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE-06 Fig. 7. Shear stress τ ху. Minimum value 86.9 kg/m, Maximum value kg/m For this, we use the finite element method in a stati statement and solve the problem in the software pakage SCAD Offie (3) kg / m Conlusions Aording to the alulation results, we an onlude that the urvature of the ross setion of the thin-walled old-formed beams signifiantl affets their stress-strain state. Contribution of bimoment's omponent is an essential. In the analsis of the stress-strain state revealed that the strength ondition is not met, under a given load the beam of will lose the bearing apait. Referenes. N.I. Vatin, A.S. Sinel nikov, Constrution of Unique Buildings and Strutures, (0). V.A. Rbakov, Osnov stroitel'noj mehaniki legkih stal'nh tonkostennh konstrukij: uheb. Posobie (SPbGPU, SPb, 0) 3. M.R. Garifullin, N.I. Vatin, Constrution of Unique Buildings and Strutures 6 (), 3-57 (04) 4. V.A. Rbakov, O.S Gamaunova, Constrution of Unique Buildings and Strutures 7, 79-3 (03) 5. V.A Rbakov, Appliation of half-moving theor VI Slivkera for analsis of stressstrain state of thin-walled sstems (0) 6. E.L. Aruman, Ereting and Speial Works in Constrution 3, -7 (008) 7. V.V. Lalin, V.A Rbakov, A. Serge, Applied Mehanis and Materials , (04) 8. A.I. Alkhimenko, N.I. Vatin, V.A. Rbakov, The light steel onstrutions teholog (SPbGPU, SPb, 008) 7

8 MATEC Web of Conferenes 86, 007 (06) DOI: 0.05/ mateonf/ IPICSE V.A Rbakov, O.S Gamaunova, Stroprofil, 8-30 (008) 0. D.A. Trubina, D. Abdulaev, E. Pihugin, V.A. Rbakov, Applied Mehanis and Materials , (04). D.A. Trubina, D. Abdulaev, E. Pihugin, V.A. Rbakov, Applied Mehanis and Materials , (04). D.A. Trubina, E. Pihugin, V.A. Rbakov, M.R. Garifullin, O. Sokolova, Applied Mehanis and Materials 75-76, (05) 3. V.A. Rbakov, A. Panteleev, G. Sharbabhev, E. Epshtein, Applied Mehanis and Materials , (04) 4. V.A. Rbakov, A. Serge, Applied Mehanis and Materials 75-76, (05) 5. V.A. Rbakov, N. Molhanova, V. Laptev, A. Suslova, A. Sivokhin, MATEC Web of Conferenes 53, 007 (06) 6. N.I. Vatin, V.A. Rbakov, Stroprofil, (007) 7. A.R. Tusnin, O.A. Tusnina, Industrial and Civil Engineering 8, 6-65 (0) 8. A.O. Gordeeva, N.I Vatin, Engineering and Constrution Journal 3 (), (0) 9. V.V. Lalin, V.A. Rbakov, S.A Morozov, Engineering and Constrution Journal (7), (0) 0. V.V. Lalin, V.A. Rbakov, Engineering and Constrution Journal 8 (6), (0). A.R.Tusnin, O.A. Tusnina, Industrial and Civil Engineering 8, 6-65 (0). STO Standard Organization. Steel bent profile, for prodution of buildings and strutures on the LSTC tehnolog (008) 3. V.Z. Vlasov, Tonkostenne uprugie sterzhni (Strojizdat, Mosow, 940) 4. SP Steel strutures. Updated edition SNIP II-3-8(0) 8

Bending stress strain of bar exposed to bending moment

Bending stress strain of bar exposed to bending moment Elastiit and Plastiit Bending stress strain of ar eposed to ending moment Basi priniples and onditions of solution Calulation of ending (diret) stress Design of ar eposed to ending moment Comined stress