Stabilisation of Beams by Trapezoidal Sheeting: Parametric Study

Transcription

1 Stabilisation o Beams by Trapezoidal Sheeting: Parametric Study IV BLÁZS, JIDŘICH ELCHER, RTI HORÁČEK Institute o etal and Timber Structures Br University o Techlogy, Faculty o Civil Engineering Veveří 331/95, Br CZECH REPUBLIC balazs.i@ce.vutbr.cz, melcher.j@ce.vutbr.cz, horacek.m1@ce.vutbr.cz bstract: - Planar members such as metal trapezoidal sheeting or sandwich panels are commonly used as roo or wall cladding or in ceiling construction. These members provide t only a covering unction, but also certain rotational spring stiness and shear stiness or beams supporting the sandwich panel or trapezoidal sheeting and thereby contribute to their higher buckling resistance. The beams are usually purlins, members o wall supporting substructure or ceiling beams. The goal o this paper is to quantiy the stiening eect o selected trapezoidal sheetings on beams in bending and compression by using provisions given in European standards, check these beams and compare the results with checks o identical beams with stabilisation. Key-Words: - Steel structures, purlin, trapezoidal sheeting, shear stiness, rotational spring stiness, stability 1 Introduction Planar members as trapezoidal sheetings or sandwich panels are widely used in civil engineering. They serve as roo or wall cladding or in constructions o ceilings. They also play a signiicant role in stabilisation o supporting beams in bending and compresion (e. g. purlins). The stabilisation behaviour is eected by shear stiness and rotational spring stiness o planar member. Those stinesses contribute to higher resistance o beam against losses o stability (lexural buckling and lateral-torsional buckling) and thereby they enable us to save material due to use beams with smaller cross-section area. It is actually a problem o bound deormation along the span o the beam (the delection and rotation can t develop reely) [1]. The actors having inluence on this behaviour are or example the way o transer o loading to the purlin cross-section, parameters o connection o the beam and the planar member, stiness o sheeting or sandwich panel and possible presence o bracing between the purlins. The constructional solution usually allows delection in the direction perpendicular to the plane o the planar member only, while the movement in the plane o the planar member is prevented [1]. rigorous theoretical solution is a complex problem and may be assisted by experiments. The presence o trapezoidal sheeting (roo or wall cladding) and its contribution to overall stiness and load bearing capacity o a structure is also advantageous or design o a structure. It is the case o the so-called stressed skin design []. There are some limiting conditions in the standard [] or this design, ia. the trapezoidal sheeting should be connected to the substructure at all our edges. There are researches dealing with behaviour o sheetings connected to the substructure at two edges only [3]. These problems and problem o stressed skin action in general is beyond the scope o this study. This paper ocuses on steel beams in bending and compression (purlins) stiened by steel trapezoidal sheeting astened at the upper lange o the beam. The beam is loaded by rmal orce Ed and uplit uniormly distributed load q Ed. The acting o uplit loading is selected in order the lower (ree) lange o the purlin to be in compression and prone to loss o stability. The upper lange, astened to trapezoidal sheeting is in tension. In real construction, the uplit loading may be caused by wind acting on a structure. Study considerations steel, simply supported Z-purlin with length o 6,0 m is considered. Six cross-sections o purlin are dealt with; they all are o cross-section corresponding to a scheme below (Fig. 1) with varying dimensions speciied in the Table 1. The inclination o the purlin cross-section is 10, which might be an usual inclination o roo. The grade o steel used or purlins is S355 (yielding stress o y = 355 Pa). ISB:

2 moment o area o the cross section about the mir axis. Shear stiness o trapezoidal sheeting is deined as ollows []: ( b ) s = roo, () h 3 S 1000 t + w Fig. 1: Z-purlin cross-section o. purlin h t b 1 Z15 15,00 3,40 76,00 Z03 03,00 3,40 76,00 3 Z9 9,00 3,40 76,00 4 Z54 54,00 3,40 76,00 5 Z ,00 3,40 76,00 6 Z ,00 3,40 76,00 where t is thickness o the sheeting, b roo width o roo, s spacing between purlins and h w proile depth o trapezoidal sheeting. The connection o the beam to the sheeting might be considered as partial restraint in torsion. It might be deined by rotational spring stiness C D. It is possible to substitute the rotational spring stiness C D with equivalent lateral spring o stiness K []. simpliied model o purlin is shown in Fig. 3. Table 1: Dimensions o the purlins There is a trapezoidal sheeting astened to the upper lange o purlin. The astening o the sheeting is carried out at each rib o the sheeting. The purlin is thereby restraint by rotational spring stiness and shear stiness o the sheeting. Several types o sheeting will be treated in order to determine their inluence on the stabilisation. scheme o sheeting can be seen in Fig.. The sheeting is in so called rmal (positive) position (narrower lange o sheeting is situated on the purlin). The material o sheeting is steel o grade o S30 (yielding stress y = 30 Pa). Fig. : Trapezoidal sheeting cross-section ccording to provision given in European standard [], the beam might be considered as restraint in the plane o sheeting, i equation (1) is ulilled: S π L π L 70 h EI w + GI t + EI z 0,5h, (1) where S is shear stiness provided by trapezoidal sheeting, h height o the purlin, L its length, I w warping constant, I t torsion constant, I z second Fig. 3: Simpliied model The determination o lateral spring stiness K is based on the procedure given in []. The procedure takes into account a distance between astener and web o the purlin, height o the purlin, width o lange o the purlin, thickness o the purlin, parameters o steel and rotational spring stiness C D. There is an assumption that the sheet-to-purlin asteners are positioned centrally on the lange o the purlin. The value o C D is inluenced by parameters o trapezoidal sheeting (eective second moment o area per unit width o the sheeting, spacing o the purlins, position o sheeting and its thickness, corrugation width), direction o loading (gravity or uplit) and width o purlin lange []. Having determined the above-mentioned characteristics o the system purlin sheeting and having checked the beam to be restraint in the plane o sheeting eq. (1), we can approach the purlin check. The purlin is loaded by rmal orce Ed, bending moment yed caused by uniormly distributed load q Ed and equivalent lateral loading o ree lange q h,ed caused by torsion and lateral bending, which results in bending moment z,ed. ISB:

3 Fig. 4: Superposition o stresses While determination o rmal orce Ed and bending moment yed is clear, or quantiying o equivalent lateral loading o ree lange o purlin q h,ed we have to use the above-mentioned lateral spring stiness K, geometrical parameters o purlin cross-section, span o the purlin, second moment o area o cross-section consisting o ull area o ree lange o purlin and interacting part o purlin web, which is given as 1/5 o web height. On the basis o q h,ed value, bending moment in ree lange z,ed can be quantiied. Detailed procedure o determination o z,ed is to be ound in []. scheme o the purlin with loading and o superposition o stresses can be seen in Fig. 4 []. 3 Results o parametric study It is possible to check maximum rmal stress in the purlin (it should t excess the yielding stress y ) and to check buckling resistance o the purlin, ie. take a possibility o a loss o stability into account. We will have a look at both cases with particular respect to stability. Following checks on beams with stabilisation are carried out and compared with checks o beams with stabilisation. The trapezoidal sheetings are selected rom rcelor ittal ironworks sortiment. 3.1 ormal stress Restrained lange o purlin []: σ y,ed Ed y max,ed = +, (3) We,y e ree lange o purlin []: σ y,ed Ed z,ed y max,ed = + +, (4) We,y e W z where σ max,ed is maximum value o rmal stress, W e,y and e are eective parameters o purlin cross-section, W z cross-section modulus o cross section consisting o ull area o ree lange and 1/5 o height o purlin web, y yielding stress (maximum allowable stress) o steel and saety actor. Eective parameters o cross-section were determined by using standard methods given in [4]. The check o purlin without stabilisation by trapezoidal sheeting is carried out by using ormula (5) taken rom []: Ed c,rd y,ed z,e d + + 1, (5) cy,rd cz,rd where c,rd is resistance o beam in compression, cy,rd and z,rd resistances o beam in bending about axes y and z. Comparison between both cases (with and without stabilisation) is done by ratio o maximum stress and yielding stress σ max,ed / y and can be seen in the Table or trapezoidal sheeting thickness 0,75 mm, mm and 1,50 mm (the other geometric parameters remain the same b R = 75 mm, h w = 101 mm; b R is the corrugation width, h w is the sheeting proile depth). There are more unavourable results rom eq. (3) and (4) selected and compared with result o purlin with stabilization taken rom eq. (5). In the case o the purlin stabilized by trapezoidal sheeting, the orces in the plane o the roo are transerred by the sheeting having suicient shear stiness S. In the case o purlin with stabilization, the beam has to transer all o the acting orces. In the Fig. 5 there is a graph with results or each purlin a or tree investigated thicknesses o sheeting. There are designations o purlins (numbers rom 1 to 6) on the horizontal axis and results o checks on the vertical axis o the graph. with stabilization with trap. sheeting thickness o. t=0,75 t= t=1,50-1 Z15 0,90 0,90 0,90 0,77 Z03 0,66 0,66 0,65 0,58 3 Z9 0,59 0,58 0,58 0,51 4 Z54 0,54 0,53 0,5 0,47 5 Z305 0,46 0,46 0,45 0,40 6 Z356 0,4 0,4 0,40 0,33 Table : Comparison o rmal stress checks ISB:

4 3,00 σed / y 0,90 0,80 0,70 0,60 0,40, t=0,75,t=,t=1,50 stability check,50,00 1,50,t=0,75,t=,t=1,50 0,30 0,0 0,10 o Fig. 5: Comparison o rmal stress checks 3. Buckling resistance The purlin is loaded by compression rmal orce and bending moment. These internal orces may cause a loss o stability o the beam. Losses o stability have bad inluence on slender beams resistance and must be checked careully [5]. The study conirms a crucial inluence o sheeting on stability o purlin. The buckling resistance o ree lange o stabilized beam is carried out by ollowing ormula (6) given in []: 1 χ LT W y,ed Ed z,ed y + +, (6) e,y e W where χ LT is reduction actor or lateral-torsional buckling (lexural buckling o the ree lange). Comparison o inal checks by using eq. (6) can be seen in the Table 3 and Fig. 6 or dierent thicknesses o sheeting. with stabilization trapezoidal sheeting thickness o. t=0,75 t= t=1,50-1 Z15 1,1 1,16 1,10,51 Z03 0,98 0,93 0,87 1,57 3 Z9 0,9 0,87 0,80 1,55 4 Z54 0,88 0,8 0,76 1,53 5 Z305 0,83 0,77 0,70 1,50 6 Z356 0,8 0,76 0,69 1,48 z 1 Table 3: Comparison o stability checks Fig. 6: Comparison o stability checks Following comparison is carried out or our dierent trapezoidal sheetings. Their parameters are described in the Table 4. The symbol b R is or corrugation width o the trapezoidal sheeting, h w is sheeting proile depth and t is or its thickness. The results or above-mentioned sheetings (named TS1 TS4) are to be seen in the Table 5. In the Table 5, there is also a result o check o purlin with stabilisation. In that case, the purlin is prone to lexural buckling and lateral-torsional buckling [5]. Both checks were carried out, in the Table 5 there are more unavourable inal results rom those two checks selected. For the calculation o lexural and lateral-torsional buckling o Z-purlin standard methods given in the standards were used. In the Fig. 7, a graphical representation o the results is to be seen. o. trap. sheeting b R h w t TS1 35/07 07,00 3,00 TS 100/75 75,00 10 TS3 135/ ,00 TS4 00/ Table 4: Trapezoidal sheetings parameters with stabilization by trap. sheeting purlin TS1 TS TS3 TS4-1 Z15 1,13 1,16 1,18 1,9,51 Z03 0,90 0,93 0,96 1,08,00 3 Z9 0,83 0,87 0,89 1,0 1,80 4 Z54 0,79 0,8 0,85 0,97 1,66 5 Z305 0,73 0,77 0,80 0,9 1,50 6 Z356 0,7 0,76 0,79 0,91 1,48 Table 5: Comparison o stability checks o. ISB:

5 stability check 3,00,50,00 1,50,TS1,TS tab.,ts3,ts4 o Fig. 7: Comparison o stability checks The last comparison shows a substantial inluence o presence o trapezoidal sheetings on stability o beam in compression and bending moment. This positive inluence is naturally expected. The Fig. 7 illustrates how strong the stabilisation eect might be. In the cases investigated in this study, the maximum value o the dierence between stabilised and ree purlin check is approximately 13%, the minimum value o this dierence is approximately 69%. Reerences: [1] elcher, J., Bending, torsion and stability o steel beams (Ohyb, kroucení a stabilita ocelových sníků), Br University o Techlogy, [] E , Eurocode 3: Design o steel structures Part 1-3: General rules Supplementary rules or cold-ormed members and sheeting, Czech Standards Institute, 008. [3] Seidel, F., Lindner, J., Stiening o beams in bending by trapezoidal sheeting connected with the substructure at two edges only (ussteiung von biegedrillknickgeährdeten Biegeträgern durch zweiseitig gelagerte Trapezproile), Stahlbau, o.11, 011, pp [4] E , Eurocode 3: Design o steel structures Part 1-5: Plated structural elements, Czech Standards Institute, 008. [5] Březina, V., Buckling resistance o metal bars and beams (Vzpěrná ússt kovových prutů a sníků), Czechoslovak cademy o Sciences, Conclusion Trapezoidal sheeting or similar planar members have a big inluence on the behavior o beams in compression and bending. It is good perceptible especially in stability analysis o such beams. The dierences between members stabilized by sheeting and members with stabilization might be decisive or design o the beams. n omission o the calculation o degree o stabilization can lead to unecomical design. 5 ckowledgement This paper was created within the rame o the projects FST-S , GCR P105/1/0314 and OPVK CZ.1.07/..00/ ISB: