Optimization of the thin-walled rod with an open profile

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1 (1) DOI: 1.151/ matecconf/181 IPICSE-1 Optimization of te tin-walled rod wit an open profile Vladimir Andreev 1,* Elena Barmenkova 1, 1 Moccow State University of Civil Engineering, Yaroslavskoye s., Moscow 1911, Russia Peoples' Friendsip University of Russia, Mikluko-Maklaya str., Moscow , Russia Abstract. Te purpose of tis study is to develop optimization of te cross-section of te inomogeneous tin-walled rod wit an open profile compared to te traditionally used omogeneous. Te idea of te developed metod of optimization is to determine te law of variation of te elasticity modulus, at wic te stress across te crosssection "aligned" and approacing to a constant value. Tus, it is possible to use all material of rod, wic leads to optimization of its structure, i.e. an increase efficiency of rod. 1 Introduction Due to te widespread of tin-walled structures in te construction practice, wic caracterized by reduced material consumption and weigt, we substantiate te optimization metod of suc structures. A number of works on optimization of tinwalled open profile rod provides te variation of different parameters subject to te conditions of te problem of optimal design (for example, conditions of strengt) [1-5]. In tis paper we consider optimization metod, wic based on solving te inverse problem of te teory of elasticity of inomogeneous bodies, te essence of wic is to determine te law of canging te modulus of elasticity on te rod s eigt for wic stress state will be given. Strengt analyses Below are te results of te strengt calculations of te roof run. Te design sceme is presented in te form of simply-supported at its ends tin-walled rod (Fig. 1). As te profile of te rod is considered te steel cannel, because it is te most profitable for te roof runs, te most desirable structurally and in terms of te convenience of attacing it to a belt truss (Fig. ). Te initial data: Lengt of te roof run: l = cm. Roof pitc: = 5. Design load on te run: q = kg/m. Geometric caracteristics of te profile: 1P by GOST 84-89: * Corresponding autor: asv@mgsu.ru Te Autors, publised by EDP Sciences. Tis is an open access article distributed under te terms of te Creative Commons Attribution License 4. (ttp://creativecommons.org/licenses/by/4./).

2 (1) DOI: 1.151/ matecconf/181 IPICSE-1 = 1 cm, b =.4cm, d =.5 cm, t =.84cm, x = 1.97 cm, Wx = 9.8 cm, Wymin= 1.4 cm, I x = 75 cm 4, I y = 7.8 cm 4. Fig.1. Design sceme. Fig.. Cannel cross section. Caracteristics of te material steel: R y = 4MPa, Е = MPa, G = MPa. Te roof run, resting on an inclined plane and being under te influence of vertical loads, sould be calculated on te combined effect of bending wit torsion. So it is necessary to carry out te calculation of te strengt of elements under te action of moments (M x, M y ) in te two principal planes and te presence of bimoment (B ω ) (paragrap 8..1 SP ). Bending moments in te two principal planes: qр l qр l М x = sin p = 117. khcm ; М y= cos p= 145 khcm. 8 8 To determine te bimoment calculate te sectorial geometric section properties. Sectorial coordinate: 1 t d 1 = b =4. cm. Coordinate of te torsion center: y x = I I.4cm, x 1 t d 5 Here Iy= b t 185 cm sector-linear moment of inertia (static moment). Te values of te main sectorial coordinate in te caracteristic points of te cross section (Fig. ): lower te leftmost point of te profile 1 t R=- x cm. lower rigtmost point of te profile 1 t t d S=- x b 8.17 cm. Sectorial moment of inertia:

3 (1) DOI: 1.151/ matecconf/181 IPICSE-1 d t b R I=t S R R Sd R 975 cm. Moment of inertia at free torsion considering factor for cannel β = 1.1: 1 4 Id = bt d.58 cm ; Eccentricity (Fig. ): b e= x tgpcosp 4.9 cm. Bimoment taking into account te design sceme: kl qр ec 1 B=.71 khcm, kl k c G I were k= d,1 flexural-torsional caracteristics. E I Figure sows te following diagrams of normal stresses: ( М x), ( М y) diagrams of bending in te two principal planes, - sectorial stresses diagram, - diagram of total stresses. Fig.. Te diagrams of normal stresses ( М x ), ( М x ),,, [MPa]. Tus, for te most strenuous point te value of total stress is: max =195.1 MPa Te analysis of strengt: max MPa R y c MPa. Strengt is provided. As can be seen from te results, for run cannel profile account torsion gives reduction of normal stresses tat is actually bimoment downloads profile, and it will be a reserve of strengt. Tus, te strengt analysis sowed te strengt of te rod is provided taking into account te unloading of profile of due to sectorial stress.

4 (1) DOI: 1.151/ matecconf/181 IPICSE-1 Stress state optimization metod To optimize tin-walled open profile rods on torsion wit bending will use te iterative metod. As te initial solution, we take te solution for a omogeneous material, represented in te above paragrap. According to te presented stress diagram (Fig. ) for a omogeneous rod, obviously, for equal-stress profile it is necessary to reduce te modulus of elasticity sould be reduce as compared wit a omogeneous material on te lower and upper fibers and increase at te neutral layer. As a first approximation we consider te stress state of te rod, wen te modulus of elasticity varies linearly: (1) E1 E E ( y) E1 y, were E 1 = MPa (te elastic modulus at y =), E =.11 5 MPa (te elastic modulus at y = /). Dependence of Young's modulus of te coordinate y is searced in te range ( /, ), tis is caused by te symmetry of te cross section and presented in point strengt analysis, according to wic te lower-left point of te profile is te most strenuous. (1) Normal stresses can be calculated by te formula: (1) (1) Mx M y B ( y) E ( y) y x ( y). EIx 1 EI y EI 1 1 Here t (1) (1) 7 EIx de ( y) y dy be ( y) y dy.51 kn cm 1, t t x d b x (1) (1) EI y E ( y) x dxdy E ( y) x dxdy.41 kn cm 1 x x t te reduced bending stiffness, d t b (1) P (1) EI d E ( y) y dy 1 S R R S E ( y) dy t t kncm te sectorial sti ffnes s, On te next step of iteration process using condition const we define te function E () (y): () Е ( y). Mx M y B y x ( y) EIx 1 EI y EI 1 1 Tus, we ave obtained te second distribution of te elastic modulus on eigt of cross-section. Distribution E () (y) represented in te Table 1. 4

5 (1) DOI: 1.151/ matecconf/181 IPICSE-1 Now we can define te stresses = () (y) by using formula for E(y) = E () (y) () () Mx M y B ( y) E ( y) y x ( y) EIx EI y EI Stiffness are increased in comparison wit te values on te first step of iteration process: t () () 7 EIx de ( y) y dy be ( y) y dy.1 kn cm, t t x d b x EI y E ( y) x dxdy E ( y) x dxdy 4.1 kn cm x x t t d b (1) P (1) EI de ( y) y dy S R RS E ( y) dy t t kn cm. Te function E () (y) is given by: () Е ( y). Mx M y B y x ( y) EIx EI y EI Te expression for () (y) takes te following form: () ( y) E () () () M ( y) x x y EI EI M y y x B ( y) EI Te calculation results are sown in Table 1. According results, it can be argued tat in te considered problem will be enoug execute tree approximations. y, m Table 1. Te values of te modulus of elasticity Е (i) and te total stresses (i) for te stages of te iterative process. Е () () Е (1) (y) (1) Е () (y) () Е () (y) () MPa

6 (1) DOI: 1.151/ matecconf/181 IPICSE-1 As seen from te results, taking into account inomogeneity of te material leads to a substantial redistribution of te stresses in te structure. Tis redistribution as quantitative and qualitative caracter. In te case of a omogeneous material stresses in te extreme fibers differ more tan by. times, wence it follows tat te profile is generally underloaded. In te inomogeneous rod for te case E(y) = E () (y) stresses are constant over te cross section, tat allows using all material of rod, wic generally increases its efficiency. To determine te effect of an optimized model of an inomogeneous structure, we (om) (in) introduce te efficiency ratio of work of inomogeneous beam: тах / тах [7- (om) 11]. In formula тах и (in) тах const are respectively te maximum stresses in te omogeneous and inomogeneous rods. In te above case, tis factor is: () () тах / тах 1. (see Table 1). Te question naturally arises ow to create a beam wit a continuous cange of its stiffness on eigt cross-section. We can offer a way to create a piecewise omogeneous structure, as was done in [7-11]. Tus, we come to te problem of multilayered structures. Mecanical caracteristics of eac layer of te multilayer structure are assigned from te solution of te inverse problem of determining te dependence of E(y). In tis case, naturally, te coefficient for multilayer-rod will be smaller tan for te beam wit continuous inomogeneity, but wit an increasing number of layers it will be closer to te above value, wic will give economic effect. 4 Conclusions Te obtained solution of problem for te cannel profile at combined effect of bending wit torsion gives justification to use te developed optimization metod for tinwalled rod of various profiles. Te used metod allows to improve te structure of tin-walled rod wit te open profile under bending wit torsion. Te developed metod of optimization leads to a significant reduction of stresses in inomogeneous rod compared to te traditionally used omogeneous and te stresses over te cross section are approacing to a constant value. Tus, it is possible to use all material of rod, wic leads to optimization of its structure, i.e. an increase efficiency of rod. Te reduction in stresses in comparison wit te omogeneous rod can increase te loads, reduce te size of te rod, so it will allow to receive some economic effect. Acknowledgment Tis work was financially supported by te Ministry of Russian Education (state task #14/14). References 1. D.V. Byckov, Structural Mecanics of tin-walled structures, (19). G.I. Grebenyuk, A.A. Gavrilov, E.V. Yankov, Proceedings of te universities. Building 7, -11 (1). T.L. Dmitrieva, Vestnik of Irkutsk State Tecnical University 5, (1)

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