Flexural Buckling Analysis of Thin Walled Lipped Channel Cross Section Beams with Variable Geometry

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1 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 Fexura Bucking Anaysis of Thin Waed Lipped Channe Cross Section Beams with Variabe Geometry Sudhir Sastry Y B 1, Y Krishna, Anirudh Koduganti 3 1, Professor, Dept. of Aeronautica Engg, Institute of Aeronautica Engineering, Hyderabad, India, scientist G, Head, Structura Test Faciity, DRDL, Hyderabad, India, 3, Asst.Professor, Dept. of Aeronautica Engg, Institute of Aeronautica Engineering, Hyderabad, India, ABSTRACT-The structura components of an aircraft consist mainy of thin pates stiffened by arrangements of ribs and stringers. Thin waed structure is a structure whose thickness is sma compared to its other dimensions but which is capabe of resisting bending in addition to membrane forces. Which is basic part of an aircraft structure, Thin pates (or thin sections or thin waed structures) under reativey sma compressive oads are prone to bucke and so must be stiffened to prevent this. The determination of bucking oads for thin pates in isoation is reativey straight forward but when stiffened by ribs and stringers, the probem becomes compex and frequenty reies on an empirica soution. The bucking of the thin pates is a phenomenon which coud ead to destabiizing and faiure of the aircraft structure; in this paper it is considered C cross section with variabe geometry and ength. The critica bucking oads have been studied for severa combinations of the geometry parameters of the beam with the hep of ANSYS and drown the resut pots KEYWORDS: Thin waed beams, Channe sections, bucking anaysis, Finite eement anaysis I. LITERATURE SURVEY A great dea of attention has been focused on pates subjected to shear oading over the past decades. One main fact in design of such eements, which fas in the category of thin-waed structures, is their bucking behavior. Pate girders and recenty shear was are being widey used by structura engineers, as we as ship and aircraft designers. The roe of stiffeners is proved to be vita in design of such structures to minimize their weight and cost. iao-ting et a [1] presented an anaytica mode for predicting the atera torsiona bucking of thin waed channe section beams restrained by meta sheeting when subjected to an upift oad. And cacuated the critica oad from critica energy theory and showed that the critica bucking moment in the pure bending case is ess than haf of the critica moment, it is more effective to use the anti sag bars in the simpy supported beams than in the fixed beams, the coser the oading point to the centre the ower the critica oad. M.Ma et a [] deveoped energy method for anayzing the atera bucking behavior of the monosymmetric I beams subjected to distributed vertica oad, with fu aowance for distortion of web the method assumed that the fanges bucke as rigid the rectanguar section beams, but the web distorts as an eastic pate during bucking, it is shown that the disparity between the distortiona and cassica critica oad increases as h/ increases and that for short beams the cassica method seriousy over estimates the critica oad. B. W. Schafer [3] worked on cod-formed thin-waed open cross-section stee coumns and provided oca, distortiona, and fexura-torsiona bucking. Experimenta Copyright to IJIRSET

2 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 and numerica studies indicated that post bucking strength in the distortiona mode is ess than in the oca mode. In pinended ipped channe and zed coumns, oca and Euer interaction is we estabished. A direct strength method is proposed for coumn design. The method uses separate coumn curves for oca bucking and distortiona bucking with the senderness and maximum capacity in each mode controed by consideration of Euer equation. Attard Mario et a [4] investigated atera-torsiona bucking behavior of open-section thin-waed beams based on a geometricay noninear formuation, which considers the effects of shear deformations, aso made Comparisons between the resuts based on fuy noninear anaysis and inearized bucking anaysis in order to iustrate the effects of pre-bucking deformations as we as the shear deformations on the bucking oad predictions. Ing. Antonin pistek,[5] anaytica method for imit oad capacity Cacuation Of thin waed aircraft structures focused on description and Comparison of different methods for imit oad Capacity cacuation of thin waed aircraft Structures considering a possibe forms of Bucking and faiures on noninear behavior of The structure under graduay increased Loading. Carine Louise Nisen, et a [6] found that the behavior of thin-waed stee sections, incuding oca bucking, distortiona bucking, goba bucking and shear bucking have been we understood and appropriate design methods existed. Foudi Mohria et a [7] derived anaytica soutions Based on a non-inear stabiity mode, for simpy supported beam-coumn eements with bi-symmetric I sections under combined bending and axia forces. Jaehong Lee et a [8] expained atera bucking of thin-waed composite beams with monosymmetric sections. A genera geometricay noninear mode for thin waed aminated composites with arbitrary open cross-section and genera aminate stacking sequences is given by using systematic variationa formuation based on the cassica amination theory. The oad capacity of cod-formed thin-waed beams is usuay restricted by their stabiity and post-bucking behaviour. Strength was considered by Cheng and Schafer [9], Trahair [10] Experimenta investigations, stress and dispacement distribution of cod-formed beams were shown by Paczos et a [11] Other exampes of papers directy connected with the subject of this work are Biegus et a [1] Magnucki et a [13] Magnucki et a [14] Paczos, Jeyaragan et a [15] Lawrence W. Rehfied et a [16] design methodoogy for bucking of thin-waed aminated composite beams shoed bucking by pure bending and bending- torsion couped modes can occur and that transition among modes are governed by structura parameters. Tomasz kubiak et a [17] presented anaysis of oca bucking of thin-waed beamcoumns, taking account Goba pre critica bending within the first order approximation. Marco et a [18] pubished a paper on bucking of thin-waed structures is presented using the 1D finite eement based refined beam theory formuation Deepak et a [19] cod formed c and z sections because of their easy connections but they suffer from certain bucking modes. here the structura behavior of channe ipped beams due to atera bucking and oad carrying capacity is evauated. Load Vs defection curves are potted in comparison with the experimenta resuts attained through FEA modeing of the software as part of resuts. Brad ford [0] atera-distortiona bucking of stee I section members presented how the usua types of bucking of stee members assumed in design are atera-torsiona and oca bucking modes. In atera-torsiona bucking, the cross-sections of the member transate and twist as rigid bodies. On the other hand, oca bucking is characterized by ocaized distortions of the cross-section over a short waveength in the absence of atera transation. II. INTRODUCTION A Thin pate or a thin waed structure is a structure whose thickness is sma compared with its other dimensions but which is capabe of resisting bending in addition to membrane forces. Such a pate forms a basic part of an aircraft structure, being, for exampe, the area of stressed skin bounded by adjacent stringers and frames in a fuseage. The structura components of an aircraft consist mainy of thin pates stiffened by arrangements of ribs and stringers. Thin pates (or thin sections or thin waed structures) under reativey sma compressive oads are prone to bucke and so must be stiffened to prevent this. The determination of bucking oads for thin pates in isoation is reativey straight forward but when stiffened by ribs and stringers, the probem becomes compex and frequenty reies on an empirica soution. The bucking of the thin pates is a phenomenon which coud ead to destabiizing and faiure of the aircraft, hence we study the bucking phenomenon on thin pates or thin waed structures with the hep of the finite eement anaysis software ANSYS. Copyright to IJIRSET

3 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 The first significant contribution to the theory of the bucking of coumns was made as eary as 1744 by Euer. [] His cassica approach is sti vaid, and ikey to remain so, for sender coumns possessing a variety of end restraints. Our initia discussion is therefore a presentation of the Euer theory for the sma eastic defection of perfect coumns. However, we investigate first the nature of bucking and the difference between theory and practice. It is common experience that if an increasing axia compressive oad is appied to a sender coumn there is a vaue of the oad at which the coumn wi suddeny bow or bucke in some un- predetermined direction. This oad is patenty the bucking oad of the coumn or something very cose to the bucking oad. Ceary this dispacement impies a degree of asymmetry in the pane of the bucke caused by geometrica and/or materia imperfections of the coumn and its oad. However, in our theoretica stipuation of a perfect coumn in which the oad is appied precisey aong the perfecty straight centroida axis, there is perfect symmetry so that, theoreticay, there can be no sudden bowing or bucking. We therefore require a precise definition of bucking oad which may be used in our anaysis of the perfect coumn. Assume that it is in the dispaced state of neutra equiibrium associated with bucking so that the compressive oad P has attained the critica vaue P CR. Simpe bending theory d v EI M so that the differentia equation of bending of the coumn is d v P EI CR or v 0 d v EI P CR v () (1) The we-known soution [] P CR Critica oad Other vaues of P CR corresponding to n=, 3,..., P CR v Acos z B sin z π 4 n EI EI 9, EI,... (5) By using the energy methods can aso be cacuated the critica oad. The tota potentia energy of the coumn in the neutra equiibrium of its bucked state is therefore EI d v PCR dv U V 0 0 (6) And is capabe, within the imits for which it is vaid and if suitabe vaues for the constant coefficients are chosen, of representing any continuous curve. We are therefore in a position to find P CR exacty. 4 EI 4 PCR U V n A 3 n n An 4 n1 4 n1 (7) In genera form n EI P CR (8) Copyright to IJIRSET (3) (4)

4 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 4EI EI P CR (9) Where I is moment of Inertia, E is Young s moduus of the materia; is the effective ength of the eement. C - Section thin waed beam fanges are behave simiar to pate bucking, the foowing equations are vaid for thin pate simpy supported aong edges. mx ny w= Amn sin sin m1 n1 a b (10) Aso, the tota potentia energy of the pate is a b w w w w w w U+V= D v N x dxdy x y x y x y x (11) The tota potentia energy of the pate has a stationary vaue in the neutra equiibrium of its bucked state i.e. Nx=Nx, CR k D N x, CR b (1) Where the pate bucking coefficient k is given by the minimum vaue of mb k a a mb (13) Where a is ength of the pate, b is width of the pate, m and n are the number of haf-waves in the x and y directions, The critica stress of the pate is given by the equation CR k E 1 1 v (14) Therefore the critica bucking oad(pcr) is = Critica stress (σ CR )/ Cross sectiona Area (A) (15) The oca faiure stress in ongitudinay stiffened panes was determined by Gerard using a sighty modified form Where g is number of cuts + Fanges f cy t b gtskt g A st E cy 1 m (16) III. MODELING Considered ipped channe section with ip dimension s Fange width b web height h and thickness t, c is shear center, o is centroid. Copyright to IJIRSET

5 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 Figure 1: Schematic diagram of the cross section of the ipped channe beam with different ip engths has been considered in the present work IV. CLASSIFICATION OF CHANNELS: Case1: represents a channe cross section beam with b = 50mm, h = 30mm, t = 0.8mm, and s = 0mm. Case : represents a Lipped channe cross section beam with b = 40mm, h = 30mm, t = 0.8mm, and s = 10mm. Case 3: represents a Lipped channe cross section beam with b = 35mm, h = 40mm, t = 0.8mm, and s = 10mm. Case 4: represents a Lipped channe cross section beam with b = 30mm, h = 50mm, t = 0.8mm, and s = 10mm. Case 5: represents a Lipped channe cross section beam with b = 5mm, h = 60mm, t = 0.8mm, and s =10mm. Case 6: represents a Lipped channe cross section beam with b = 30mm, h = 60mm, t = 0.8mm, and s = 5mm. V. ELASTIC BUCKLING OF THE BEAM Anaysis has been done using FEM the foowing parameters have been used: 7x10 3 N/mm =Young moduus (E); 0.3=Poisson s ratio (μ); 30 mm =depth of a beam (h); 50mm = Width of the fange; 150 mm=ength of a beam (L) for mode case The critica oad (P cr ) π E th (6b + h) 1 P = 4 I = (6b + h) Where I = Moment of Inertia. By using the above equation and by ansys, the critica oad obtained for thin waed channe section beam is x 10 3 N/mm and x 10 3 N/mm respectivey Copyright to IJIRSET

6 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 (a) (b) (c) (d) (e) (f) Figure : (a) channe cross section beam with b = 50mm, h = 30mm, t = 0.8mm, and s = 0mm.(b) Lipped channe cross section beam with b = 40mm,h = 30mm,t = 0.8mm, and s = 10mm, (c) Lipped channe cross section beam with b = 35mm,h = 40mm,t = 0.8mm, and s = 10mm.(d) Lipped channe cross section beam with b = 30mm,h = 50mm,t = 0.8mm, and s = 10mm,.(e) Lipped channe cross section beam with b = 5mm,h = 60mm,t = 0.8mm, and s = 10mm,.(f) Lipped channe cross section beam with b = 30mm,h = 60mm,t = 0.8mm, and s = 5mm, (a) (b) (c) (d) (e) (f) Figure 3: Von mises stresses of (a) channe cross section beam with b = 50mm, h = 30mm, t = 0.8mm, and s = 0mm.(b) Lipped channe cross section beam with b = 40mm,h = 30mm,t = 0.8mm, and s = 10mm,.(c) Lipped channe cross section beam with b = 35mm,h = 40mm,t = 0.8mm, and s = 10mm.(d) Lipped channe cross section beam with b = 30mm,h = 50mm,t = 0.8mm, and s = 10mm,.(e) Lipped channe cross section beam with b = 5mm,h = 60mm,t = 0.8mm, and s = 10mm,.(f) Lipped channe cross section beam with b = 30mm,h = 60mm,t = 0.8mm, and s = 5mm. Y FREQ=-.8E+09 Y Y Y Y (b) (c) (d) (e) (f) Y (a) Copyright to IJIRSET

7 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 Figure 4: First bucking mode shapes of (a) channe cross section beam with b = 50mm, h = 30mm, t = 0.8mm, and s = 0mm.(b) Lipped channe cross section beam with b = 40mm,h = 30mm,t = 0.8mm, and s = 10mm, (c) Lipped channe cross section beam with b = 35mm, h = 40mm, t = 0.8mm, and s = 10mm. (d) Lipped channe cross section beam with b = 30mm, h = 50mm, t = 0.8mm, and s = 10mm, (e) Lipped channe cross section beam with b = 5mm, h = 60mm, t = 0.8mm, and s = 10mm, (f) Lipped channe cross section beam with b = 30mm, h = 60mm, t = 0.8mm, and s = 5mm. VI. RESULT AND DISCUSSION The graphs for different C sections indicating the reationship between modes and critica bucking are given beow: Length of the channe mm: Bucking Anaysis: Combined graph for a cases of C section 6.00E+08 Critica Bucking.00E E C section case1 case case3 case4 case5 case6-6.00e+08 Mode Figure 5: Bucking Anaysis of channe section beams (case 1 to case 6), the graph critica bucking v s mode shapes, case 5 observed that the more critica bucking vaues in first mode than other cases and case more critica bucking vaues at fifth mode for the channe ength L = 150mm Length -300 mm: Bucking Anaysis: Combined graph for a cases of C sections: Copyright to IJIRSET

8 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June E+08 Critica Bucking 3.00E+08.00E E E C section case1 case case3 case4 case5 -.00E+08 case6-3.00e+08 Mode Figure 6: Bucking Anaysis of channe section beams (case 1 to case 6), the graph critica bucking v s mode shapes, case 1 observed that the more critica bucking vaues in first mode than other cases and case 5 and case more critica bucking vaues at fifth mode of the channe ength L = 300mm Length mm: Bucking Anaysis: Combined graph for a cases of C sections: 3.00E+08 Critica Bucking.00E E E E E C section case1 case case3 case4 case5 case6-5.00e+08 Mode Copyright to IJIRSET

9 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 Figure 7: Bucking Anaysis of channe section beams (case 1 to case 6), the graph critica bucking v s mode shapes, case 4 observed that the more critica bucking vaues at first mode than other cases and case 5 more critica bucking vaues at fifth mode of the channe ength L = 450mm Length4-600mm: Bucking Anaysis: Combined graph for a cases of C sections: Critica Bucking.00E E C section case1 case case3 case4 case5 - Mode case6 Figure 8: Bucking Anaysis of channe section beams (case 1 to case 6), the graph critica bucking v s mode shapes, case 1 observed that the more critica bucking vaues in first mode than other cases and case 5 and case 6 more critica bucking vaues at fifth mode of the channe ength L = 600mm Length5-750mm: Bucking Anaysis: Combined graph for a cases of C sections: 6.00E+08 Critica Bucking.00E E E Mode C section case1 case case3 case4 case5 case6 Figure 9: Bucking Anaysis of channe sections beams (case 1 to case 6), the graph critica bucking v s mode shapes, case 5 observed that the more critica bucking vaues in first mode than other cases and case more critica bucking vaues at fifth mode of the channe ength L = 750mm Copyright to IJIRSET

10 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 L/ H rat io CASE 1 CASE CASE 3 CASE 4 CASE 5 CASE 6 L/ Critica L/H Critica L/H Critica L/H Critica L/H Critica H Bucking ratio Bucking ratio Bucking ratio Bucking ratio Bucking rati o Critica Bucking E E E E E E E E E E E E E E E E E E E E E E E E E+08 Critica Moment 3.00E+08.00E E L/H CASE 1 CASE CASE 3 CASE 4 CASE 5 CASE 6 Figure 10: Length by Height ratio v s Critica Bucking for channe section beams (case 1 to case 6) The figures from 5 to 9 are combined graph for a cases of different Channe and ipped channe sections indicating the reationship between modes and critica bucking are observed, the engths are increased in five cases from 150mm to 750mm with the increment of 150 mm. In figure 5,7and 9, the fifth case ipped channe critica bucking vaues are more. Lipped channe cross section beam represents with b = 5mm, h = 60mm, t = 0.8mm, and s =10mm, in this case h = 60mm, and s =10mm, both are vertica dimensions so that moment of inertia is more and it is taking more oad. Out of a the channes case 5 is preferabe. From figure 10, the behavior of M cr with ambda, the curve is going down with increase of ambda because ength ( L) is inversey propositiona to M CR therefore if M CR increase λ decrease. Where λ = L / H. In genera aso if ength of the beam increases stabiity wi decrease according to Euers Equation that is Critica oad P CR = π EI / VII. CONCLUSIONS Thin-waed C sections with different cross-sections and different engths of extrusion have been considered for bucking anaysis in this paper. It is observed that in bucking anaysis, at ength L = 150mm, at the first mode, the vaue of critica bucking is higher for case 5 whereas at the fifth mode, case is giving higher vaue of critica bucking. At ength L = 300mm, at the first mode, case 1 is giving higher vaue of critica bucking at the fifth mode, case 5 and case are giving higher vaues of critica bucking. At ength L = 450mm, at the first mode, case 4 is giving higher vaue of critica bucking Copyright to IJIRSET

11 ISSN: (An ISO 397: 007 Certified Organization) Vo. 3, Issue 6, June 014 whereas at the fifth mode, 5 is giving higher vaue of critica bucking. At ength L = 600mm, at the first mode, case 1 is giving higher vaue of critica bucking whereas at the fifth mode, case 5 and case 6 are giving higher vaues of critica bucking. At ength L = 750mm, at the first mode, case 5 is giving higher vaue of critica bucking whereas at the fifth mode, case is giving higher vaue of critica bucking. VIII. REFERENCES 1) iao-ting Chu, Roger Kette, Long-yuan Li. "Latera-torsion bucking anaysis of partia-ateray restrained thin-waed channe-section beams. Journa of Constructiona Stee Research 60 (004) ) M. Ma & O. Hughes. Latera Distortiona Bucking of Monosymmetric I-Beams under Distributed Vertica Load. Thin-Waed Structures Vo. 8, No., pp , ) B. W. Schafer. Loca, Distortiona, and Euer Bucking of Thin-Waed Coumns. Journa of Structura Engineering, Vo. 18, No. 3, March 1, 00. 4) R. Emre Erkmen, Mario M. Attard. Latera torsiona bucking anaysis of thin-waed beams incuding shear and pre-bucking deformation effects. Internationa Journa of Mechanica Sciences - INT J MECH SCI 01/011; 53(10): ) Prof. Ing. Antonin pistek. Anaytica Method for Limit Load Capacity Cacuation of Thin Waed Aircraft Structures 8 th Internationa Congress of the Aeronautica Sciences. 6) Carine Louise Nisen, Md Azree Othuman Mydin and Mahyuddin Rami 01" Performance of ightweight thin-waed stee sections: theoretica and mathematica considerations." Advances in Appied Science Research, 01, 3 (5): ) Foudi Mohria, Cherif Bouzerirab, Miche Potier-Ferryc of France in the year 008" Latera bucking of thin-waed beam-coumn eements under combined axia and bending oads." Thin-Waed Structures 46 (008) ) Jaehong Lee. Latera bucking anaysis of thin-waed aminated composite beams with monosymmetric sections Engineering Structures 01/006. 9) Cheng Yu, Benjamin W. Schafer. Simuation of cod-formed stee beams in oca and distortiona bucking with appications to the direct strength method. Journa of Constructiona Stee Research 63 (007) ) N. S. Trahair. Bucking Anaysis Design of Stee Frames. Journa of Constructiona Stee Research, 65(7), ) K. Magnucki, P. Paczos. Theoretica shape optimization of cod-formed thin-waed channe beams with drop fanges in pure bending. Journa of Constructiona Stee Research 65 (009) ) A. Biegus, D. Czepiżak. Experimenta investigations on combined resistance of corrugated sheets with strengthened cross-sections under bending and concentrated oad. Thin-Waed Structures (Impact Factor: 1.3). 01/008; 46(3): ) K. Magnucki, M. Rodak, J. Lewinski. Optimization of mono and anti-symmetrica I-sections of cod-formed thin-waed beams. Thin-Waed Structures 44 (006) ) P. Paczos, P. Wasiewicz. Experimenta investigations of bucking of ipped, cod-formed thin-waed beams with I-section. Thin-Waed Structures 47 (009) ) S. Jeyaragan and M. Mahendran. Experimenta Investigation of the New Buit-up Litestee Beams. Fifth Internationa Conference on Thin-Waed Structures Brisbane, Austraia, ) Lawrence W. Rehfied and Urich Mueer. Design Methodoogy for Bucking of Thin-Waed Laminated Composite Beams. ICCM - 1 Europe ) Tomasz Kubiak. Interactive Bucking in Thin-waed Beam-coumns with Widthwise Varying Orthotropy. Journa of Theoretica and Appied Mechanics 44, 1, pp , Warsaw ) Syed Muhammad Ibrahim, Erasmo Carrera, Marco Petroo, Enrico appino 01. "Bucking of thin-waed beams by a refined theory." J hejiang Univ-Sci A (App Phys & Eng) 01 13(10): ) M.S.Deepak, R.Kandasamy, Dr R.Thenmozhi. Investigation on Latera Torsiona Bucking Performance of Cod-formed Stee C-Channe Sections. Internationa Journa of Emerging Trends in Engineering and Deveopment Issue, Vo. 4 (May-01). 0) M.A. Bradford. Latera-Distortiona bucking of stee I-Section members Journa of Constructiona Stee Research 3, ) Sudhir Sastry YB, Y Krishna, Pattabhi R. Budarapu Parametric studies on bucking of thin waed channe beams Manuscript submitted to Journa of Computationa Materias Science, Esevier Editoria System ) T. H. G. Megson, Aircraft-Structures-for-Engineering-Students-Fourth-Edition 3) Sudhir Sastry Y B, Y Krishna, Pattabhi R. Budarapu, Anirudh Koduganti. Fexura Bucking Anaysis of Thin Waed T Cross Section Beams with Variabe Geometry ", Vo. 3 - Issue 3 (March - 014), Internationa Journa of Engineering Research & Technoogy, ISSN: ) Sudhir Sastry Y B, Sindhura.G, Sarwade A. G. Mode Anaysis of Composite Fuseage ike Structures with Cutouts ", Vo. - Issue 11 (November - 013), Internationa Journa of Engineering Research & Technoogy, ISSN: ) Sudhir Sastry Y B, Bhargavi Rachana I, Durga Rao K. Stress Anaysis of Heicopter Composite Bade Using Finite Eement Anaysis ", Vo. - Issue 1 (December - 013), Internationa Journa of Engineering Research & Technoogy, ISSN: Copyright to IJIRSET