2 LATERAL TORSIONAL BUCKLING OF AN IDEAL BEAM WITHOUT LATERAL RESTRAINTS

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1 0.55/tvsb Transactons of the VŠB Techncal Unversty of Ostrava No., 05, Vol. 5, Cvl Engneerng Seres paper #0 Ivan BALÁZS, Jndřch MELCHER STABILITY OF THIN-WALLED BEAMS WITH LATERAL CONTINUOUS RESTRAINT Abstract Metal beams of thn-walled cross-sectons have been wdely used n buldng ndustry as members of load-bearng structures. Ther resstance s usually lmted by lateral torsonal bucklng. It can be ncreased n case a beam s laterally supported by members of claddng or celng constructon. The paper deals wth possbltes of determnaton of crtcal load of thn-walled beams wth lateral contnuous restrant whch s crucal for beam bucklng resstance assessment. eywords Crtcal load, fnte dfference method, lateral torsonal bucklng, numercal analyss, stablty, thn-walled beam. INTRODUCTION Resstance of slender metal beams of thn-walled cross-sectons wth no lateral restrants s usually lmted by lateral torsonal bucklng. In case they are utled as purlns or grts, members of roof or wall claddng can be attached to them. It results n lateral restrant that prevents dsplacement of the beam cross-secton and therefore contrbutes to ts bucklng resstance. If t s correctly taken nto account, more economcal desgn of the cross-secton can be acheved. The paper focuses on possbltes of determnaton of crtcal load of thn-walled metal beams wth lateral contnuous restrant. A numercal method for crtcal load computaton s descrbed and results are compared wth results obtaned usng software based on fnte element method. LATERAL TORSIONAL BUCLING OF AN IDEAL BEAM WITHOUT LATERAL RESTRAINTS Lateral torsonal bucklng of an deal beam (wth no ntal mperfecton) wth no lateral restrants s charactered by deformaton of the member cross-secton consstng of two components lateral dsplacement of the cross-secton out of plane of bendng v and angle of rotaton φ []. For a beam of monosymmetrc cross-secton ths effect s llustrated n Fg. where q s vertcal load (n the XZ plane), C g ndcates cross-secton center of gravty, C s cross-secton shear center, a dstance of center of gravty and shear center and e dstance of the poston of load applcaton and center of gravty. Compressed part of the cross-secton tends to buckle out of plane of bendng. Lateral torsonal bucklng of an deal beam occurs when crtcal moment M cr (caused by crtcal load) s reached. The problem of stablty of an arbtrary thn-walled member of open crosssecton n bendng and compresson s defned by Vlasov n form of dfferental equatons []. Ing. Ivan Balás, Insttute of Metal and Tmber Structures, Faculty of Cvl Engneerng, Brno Unversty of Technology, Veveří 33/95, Brno, Cech Republc, phone: (+40) , e-mal: balas.@fce.vutbr.c. Prof. Ing. Jndřch Melcher, DrSc., Insttute of Metal and Tmber Structures, Faculty of Cvl Engneerng, Brno Unversty of Technology, Veveří 33/95, Brno, Cech Republc, phone: (+40) , e-mal: melcher.j@fce.vutbr.c. Unauthentcated

2 Fg. : Lateral torsonal bucklng of an deal beam After modfcaton for case of a member n bendng only the problem s defned by two homogenous dfferental equatons of fourth order () and () and approprate boundary condtons []. For a member smply supported n bendng as well as n torson boundary condtons (3) apply, for a fxed member boundary condtons (4) apply []. EI ( y ) v M 0, () EI GI b ( M ) q ( e a ) M v t y y 0, () v( 0) v( 0, v(0) v( 0, (0) ( 0, (0) ( 0, (3) v( 0) v( 0, v(0) v( 0, (0) ( 0, (0) ( 0, (4) where E s modulus of elastcty, G shear modulus of elastcty, I second moment of area, I ω warpng constant, I t torson constant, q vertcal load (n the XZ plane), M y bendng moment, v and φ unknown functons of deformaton and b s Wagner coeffcent. The expresson for ts determnaton can be found e.g. n []. Actual standard for desgn of steel structures [3] ndcates t as j. In the mathematcal pont of vew the crtcal load s gven as egenvalue problem of dfferental equatons () and () wth approprate boundary condtons. Derved expresson for crtcal moment of an solated beam (wth no lateral restrant) of at least monosymmetrc crosssecton loaded by a load that crosses ts shear center can be found n the standard [3]. 3 LATERAL TORSIONAL BUCLING OF AN IDEAL BEAM WITH LATERAL CONTINUOUS RESTRAINT Let us consder that there s lateral contnuous restrant located n a dstance of c from the cross-secton center of gravty (at the pont C lat ) that prevents lateral dsplacement of the crosssecton. Therefore rotaton of the cross-secton around fxed axs occurs []. The stuaton can be seen n Fg.. Fg. : Lateral torsonal bucklng of an deal beam wth lateral restrant Unauthentcated

3 The lateral restrant s consdered to be perfectly rgd. The beam span s L. An deal beam s consdered (wth no ntal mperfectons). 3. Crtcal load determnaton Snce lateral dsplacement of the cross-secton s prevented, the only unknown functon of deformaton s angle of rotaton φ and dfferental equatons () and () are modfed accordngly. It results n one homogenous dfferental equaton of order four (5) [] (n general wth nonconstant coeffcents) wth boundary condtons (6) for smply supported beam and (7) for fxed beam: EI EI ( c a ) GI ( c a b )( M ) q ( e c ) 0 t y, (5) ( 0) ( 0, (0) ( 0, (6) ( 0) ( 0, (0) ( 0, (7) where q s the magntude of load at bfurcaton of equlbrum (crtcal load). In the mathematcal pont of vew t s an egenvalue problem. In ths case t s the so called Sturm-Louvlle egenvalue problem of dfferental equaton of fourth order [4]. Soluton of ths complex problem s possble usng combnaton of selected numercal methods, e.g. fnte dfference method and nverse power method. Both methods are easly algorthmable. The procedure of the methods s descrbed. Equaton (5) s transformed to the form (8): A B CM y qd, (8) where A EI EI ( c a ), (9) B GI t, (0) C ( c a b ), () D ( c e ). () Equaton (8) wll be used as ntal for egenvalue problem soluton (crtcal load determnaton). The nterval <0; L> (beam span) s dvded nto N subntervals usng a step of h and N equdstant nodes x n such a way that 0 = x 0 < x < < x N- < x N = L, where x = h and h = / N (Fg. 3). Certan value of the functon φ s assgned to each node x. It arses from the boundary condtons (6) and (7) that φ 0 = φ(0) = 0 and φ N = φ( = 0. Let us ntroduce frst (3), second (4) and fourth (5) central dfference for frst, second and fourth dervatve approxmaton: f, (3) h f f, h (4) (5) Applcaton of the central dfferences to nodes x 0 and x N requres functonal values n fcttous nodes x = h and x N+ = L + h. It s possble to determne the functonal values φ and φ N+ usng boundary condtons. For smply supported beam applcaton of boundary condtons (6) results n functonal values φ = φ and φ N+ = φ N. For fxed beam applcaton of (7) results n functonal values φ = φ and φ N+ = φ N. Functonal values at nodes from x to x N are to be solved. h 3 Unauthentcated

4 Fg. 3: Dfference scheme For each node x, where =,, N equaton (6) s assembled puttng central dfferences (4) and (5) nto equaton (8) nstead of second and fourth dervatves: A B CM y qd, (6) Ths modfcaton leads to a system of N algebrac equatons that can be expressed as follows (7): N 5, N 3 N 4, N 3 N 3, N 3 N, N 3 N, N 3 N 4, N N 3, N N, N N, N N 3, N N 3, N N 3, N 3 N 3 N N q Dh Indvdual members of the system matrx are defned by followng expressons:,, A for =,, N 3, (8) 4 3 N 3 N N (7) A S for =,, N, (9), 4 4A S for =,, N, (0), S for =,, N. (), 6A For members, and N,N boundary condtons at nodes x 0 = 0 and x N = L have to be taken nto account. For a smply supported beam expresson () apples, for a fxed beam expresson (3) apples. S for ={; N }, (), 5A, 7A S for ={; N }. (3) Expresson S s defned as follows (4): S h ( CM y B). (4) System of algebrac equatons (7) can be expressed as follows (bold letters ndcate vectors): φ = q D h 4 φ, (5) where φ = φ φ φ 3 φ N-3 φ N- φ N- T. (6) It can be modfed to form (7): G φ = q φ, (7) where the matrx G s defned as expresson (8): 4 Unauthentcated

5 G. (8) 4 Dh For the egenvalue problem soluton (crtcal load q determnaton) equaton (7) s utled. For practcal purposes the most mportant s the mnmum postve egenvalue [5] (the lowest load that causes lateral torsonal bucklng of an deal beam). Approxmaton of the mnmum egenvalue and approprate egenvector (bucklng shape) of the matrx G can be computed e.g. usng nverse power method [6] (power method appled to the matrx G - ). The algorthm of the power method can be found e.g. n [6]. Another possblty s applcaton of the teratve QR algorthm [7] that gves all the egenvalues wth approprate egenvectors of the matrx G (complete egenvalue problem soluton). The algorthm s brefly outlned [7]: G G0 Q0R, 0 (9) Gk RkQk, k, (30) where matrces Q k and R k are products of the so called QR decomposton of the matrx G k. It s possble to decompose the matrx G k to matrces Q k and R k e.g. usng Gram-Schmdt orthonormalaton process [8]. Expresson (3) then apples: lm E, (3) G k k where E s a dagonal matrx wth egenvalues located on ts dagonal (other members are equal to ero). The egenvectors (bucklng shapes) are gven as columns of a matrx resultng from matrx Q k+ and Q k multplcaton. 3. Example of the crtcal load calculaton For the example of the crtcal load calculaton a steel beam of double symmetrc cross-secton accordng to Fg. 4 s chosen. There s lateral contnuous restrant located at a dstance of c = 50 mm from the center of gravty. Fg. 4: Cross-secton of the nvestgated beam The beam span s L = 5 m. There s vertcal unformly dstrbuted load of a magntude of kn/m actng at the top flange. Two types of support condtons accordng to boundary condtons (6) and (7) are consdered. The procedure explaned n secton 3. wth a step of h = 0,0 m (number of subntervals N = 50) s appled. The soluton leads to dfferental equaton (3) wth boundary condtons (6) or (7), respectvely:,3 4848,8 0,M y 0, 5q, (3) whch s transformed to a system of 49 algebrac equatons usng fnte dfference method and solved by nverse power method. It results n mnmum egenvalue of 4550 (smply supported beam) and 5 Unauthentcated

6 47480 (fxed beam). In the physcal pont of vew these values represent crtcal loads (n N/m). The approprate crtcal moments are calculated usng (33) for smply supported beam and (34) for fxed beam: M M 8 cr qcr L, (33) cr qcr L, (34) For the smply supported beam the crtcal moment s equal to 454,8 knm and for the fxed beam 984,3 knm. For comparson: calculaton of the crtcal moment of a beam of the same crosssecton, span and boundary condtons but wth no lateral restrant usng procedure accordng to [3] results n M cr = 6, knm (smply supported beam) and M cr = 885,9 knm (fxed beam). In ths case the lateral restrant causes ncrease of crtcal moment of about 74 % (smply supported beam) and % (fxed beam). Applcaton of the QR algorthm results n mnmum egenvalue of 4660 (smply supported beam) and (fxed beam). Approprate crtcal moment determned usng (33) s 445,8 knm (smply supported beam) and usng (34) 963,3 knm (fxed beam). In Fg. 5 three lowest normaled egenvectors (bucklng shapes) of the matrx G obtaned usng QR algorthm are dsplayed (for clarty reasons hgher egenvectors are not dsplayed). In Fg. 6 graphcal dstrbuton of matrx G egenvalues as a result of the QR algorthm s shown. Fg. 5: Three lowest bucklng shapes Fg. 6: Egenvalues of the matrx G 3.3 Soluton usng fnte element method and comparson of results The example mentoned n the secton 3. s solved usng the fnte element method code ANSYS 4.0 [9]. The thckness of each part of the cross-secton s low n comparson wth other dmensons. For ths type of constructonal members the shell fnte elements are sutable. For the analyss the SHELL8 fnte element s utled. The beam cross-secton, poston of lateral restrant and load s n accordance wth the example solved n secton 3.. Two types of support condtons are consdered: smply supported beam and fxed beam. For the numercal analyss the fnte element edge length 0 mm s specfed. The detal of modeled beam ncludng lateral restrant mplementaton can be seen n Fg. 7 (fxed beam; warpng of support cross-secton prevented). In case of the smply supported beam the so called fork support condton apples. Accordng to [0] ths support condton s consdered for lateral torsonal bucklng analyss. It prevents lateral dsplacement at supports whle warpng s allowed. Frst order theory statc analyss and lnear bucklng analyss (egenvalue analyss) were performed. It results n mnmum postve egenvalue of 4660 (smply supported beam) and (fxed beam). Results of crtcal moments approxmatons obtaned by fnte dfference method wth subsequent nverse power method or QR algorthm, respectvely, and fnte element method usng the ANSYS 4.0 code are summared n Tab.. In Fg. 8 there s one of the results of the numercal analyss usng the ANSYS code frst postve egenvalue for the case of the fxed beam. 6 Unauthentcated

7 Fg. 7: Fnte element model Fg. 8: Frst postve bucklng shape Tab. : Comparson of results approxmatons of crtcal moments (n knm) Method smply supported Beam fxed Fnte dfference method + nverse power method 454,8 984,3 Fnte dfference method + QR algorthm 445,8 963,3 Fnte element method (ANSYS) 458, 904,9 3.4 Algorthmsaton of the numercal methods for crtcal load determnaton and parametrc studes Algorthms of the fnte dfference method, nverse power method and QR algorthm were programmed n the VBA language (programmng language for the MS Excel applcaton). After enterng of nput data (cross-secton characterstcs, beam span, poston of lateral restrant, poston of the load, step h of the nterval mesh and requred accuracy of teratve calculaton the code meshes the beam span (dscretaton of the problem), assembles matrces and vectors and teratvely calculates the mnmum egenvalue. Followng charts show some results obtaned by the code. In Fg. 9 there s relatonshp between egenvalue and se of the step h, Fg. 0 shows convergence of the teratve calculaton of the nverse power method to the mnmum egenvalue for dfferent beam span mesh. The convergence s very fast. The nput data s dentcal wth the data utled for the example n secton 3. for the case of the fxed beam. Fg. 9: Relatonshp between crtcal load and beam span mesh 7 Unauthentcated

8 Fg. 0: Convergence of teratve calculaton to frst postve egenvalue Fg. shows comparson of crtcal loads of a smply supported beam wth no lateral restrant (calculaton accordng to [3] and the ANSYS code) and analogous laterally restraned beam (calculaton usng fnte dfference method FDM and the ANSYS code) of varous spans. Fg. shows the comparson for a fxed beam (for clarty purposes spans from 5 m to 0 m only are dsplayed). Fg. : Comparson of methods of crtcal load determnaton smply supported beam Fg. : Comparson of methods of crtcal load determnaton fxed beam 8 Unauthentcated

9 There are certan notceable dfferences among values of crtcal loads for small spans (low slendernesses) of the beams. These dfferences can be apparently explaned by numercal fnte element analyss that takes nto account also effects of local bucklng of thn walls of the crosssectons. These local stablty effects act together wth global bucklng of a beam and nfluence the resultng values of crtcal loads. For greater spans the global stablty completely prevals and dfference between methods s not sgnfcant. The system matrx of the fnte dfference method was derved from the dfferental equatons of global stablty of a thn-walled beam (provded the crosssecton does not dstort). It follows that mentoned local effects are not ncluded. 4 CONCLUSIONS The paper focuses on lateral torsonal bucklng of thn-walled beams wth lateral contnuous restrant. The lateral restrant ncreases bucklng resstance of members; ts nfluence on crtcal moment was quantfed usng example of a beam of double symmetrc cross-secton. The results were compared wth results of an analogous beam wth no lateral restrant. For the crtcal load determnaton selected numercal methods were utled. A code for computered calculaton n the VBA language was created. It mples from the performed parametrc studes that the nfluence of lateral restrant on crtcal load s sgnfcant. If t s correctly taken nto account t mght result n more economcal cross-secton desgn that postvely nfluences materal consumpton. Constructonal desgn mght be therefore more effectve. The approxmatons of crtcal loads were compared wth results of numercal analyses performed usng a fnte element code. Certan dfferences between results for smaller beam spans are explaned by local stablty effects that arse from the fnte element method analyss. Descrbed numercal algorthms (fnte dfference method, nverse power method) are relatvely easy to algorthme and allow to create applcatons for egenvalue problems soluton. ACNOWLEDGMENT The paper has been supported by the projects of the specfc research program of the Brno Unversty of Technology No FAST-J and the Cech Scence Foundaton No P05//034. REFERENCES [] BŘEZINA, V. Bucklng Resstance of Metal Bars and Beams (Vpěrná únosnost kovových prutů a nosníků). Prague: State Publshng of Techncal Lterature, pp. [] VLASOV, V. Z. Thn-Walled Elastc Bars (Tenkostěnné pružné pruty). Prague: Cechoslovak Academy of Scences Publshng, pp. [3] ČSN EN Eurocode 3: Desgn of Steel Structures Part -: General Rules and Rules for Buldngs (Eurokód 3: Navrhování ocelových konstrukcí Část -: Obecná pravdla a pravdla pro poemní stavby). Prague: Cech Standard Insttute, pp. [4] RATTANA, A. & BÖCMANN, C. Matrx methods for computng egenvalues of Sturm- Louvlle problems of order four. Journal of Computatonal and Appled Mathematcs. Volume 49, pp , ISSN (Onlne) , DOI: 0.06/j.cam , 03. [5] INDMANN, R. & LAUMANN, J. Determnaton of egenvalues and modal shapes for members and frames (Ermttlung von Egenwerten und Egenformen für Stäbe und Stabwerke). Stahlbau. Volume 73, Issue, pp. 6 36, ISSN , 004. [6] DALÍ, J. Numercal methods (Numercké metody). Brno: CERM Academc Publshng, pp. ISBN [7] VONDRÁ, V. & POSPÍŠIL, V. Numercal methods I (Numercké metody I). [onlne] Unauthentcated

10 [8] BJORC, A. Numercs of Gram-Schmdt orthogonalaton. Lnear Algebra and ts Applcatons. Volume 97-98, pp , ISSN (Onlne) , DOI: 0.06/ (94) , 994. [9] ANSYS Academc Research, Release 4.0. [0] SEDLACE, G. & NAUMES, J. Excerpt from the Background Document to EN 993--: Flexural bucklng and lateral bucklng on a common bass: Stablty assessments accordng to Eurocode 3. Aachen: Insttut und Lehrstuhl für Stahlbau und Lechtmetallbau, pp. Revewers: Doc. Ing. Martn Psotný, PhD., Department of Structural Mechancs, Faculty of Cvl Engneerng, Slovak Unversty of Technology n Bratslava, Slovak Republc. Prof. Ing. Josef Včan, CSc., Department of Structures and Brdges, Faculty of Cvl Engneerng, Unversty of Žlna, Slovak Republc. 0 Unauthentcated