Torsion Stiffness of Thin-walled Steel Beams with Web Holes

Transcription

1 Torson Stffness of Thn-walled Steel Beams wth Web Holes MARTN HORÁČEK, JNDŘCH MELCHER Department of Metal and Tmber Structures Brno Unversty of Technology, Faculty of Cvl Engneerng Veveří 331/95, 62 Brno CZECH REPUBLC Abstract: Ths paper summarzes the pure torson stffness examnaton of the perforated thn-walled steel beams generally used n the storage systems and expedent floorng structures. The crcular perforaton s beng placed n the web of the beam. The expermental verfcaton of the St. Venant torson parameter s a partal task to determne the ultmate and desgn strength of the beams wth respect to lateral torsonal bucklng. Frst the pure torson parameter s theoretcally calculated for the substtute cross-secton and then the examned beams are subjected to the laboratory nvestgaton. The resultng pure torson parameters obtaned from the test for dfferent beam lengths are approxmated by exponental curves and the formula descrbng the relatonshp between the pure torson stffness and the angle of rotaton s beng establshed. Key-Words: steel beam, torson stffness, web holes, torson moment, warpng, lateral torsonal bucklng. 1 ntroducton The St. Venant torson constant can be derved accordng to procedure for the castellated beams from the parameters of the full cross-secton and from the cross-secton weakened by hole [1]. Based on weghted average of those values the substtute St. Venant torson constant for the whole beam s beng establshed. The calculated pure torson parameters are verfed by tests executed on three dfferent lengths of beams (2, 3 and 4 meters). Totally 3 test were performed (1 test per each beam length). The experments weree evaluated and the results are compared wth the theoretcally calculated values for the substtute cross-secton. On the bass of the performed test results the formula for the expermental substtute St. Venant torson constant of the thn-walled beams wth the web holes has been derved. beams are not restran to lateral torsonal bucklng) or by the chpboards connected to the floor grders wth screws (lateral torsonal bucklng effect s elmnated). Fg. 1 The steel beam wth Sgma cross-secton 2 General The Sgma beams (as they are called n practce) are the mono-symmetrc profles loadedd n the plane perpendcular to the axs of symmetry. The beam heght s 26 mm, the flange wdth s 5 mm and the secton thckness s 2.5 mm; steel grade s S 355. The applcaton of the Sgma beams s n the bult-n floors n warehouses as the floor grders. n the structure the floor grders are bolted through the truss plate to the columns, whch support the floor. The floor structure s covered by the steel grds whch are not connected wth the floor grders (the Fg. 2 The floor structure arrangement SBN:

2 3 Theoretcal calculaton nsomuch as there s no verfed soluton of the pure torson constant calculaton for the beams weakened by holes, the procedure for the castellated beams was used. Frst accordng to Eq. (1) the torson stffness for the full cross-secton (Fg. 3 secton A-A) and for the weakened cross-secton (Fg.3 secton B-B) s calculated: 1 3 t = b t ( 1) 3 Fg. 3 Desgnated sectons on Sgma beam Then the pure torson stffness for the substtute cross-secton s establshed based on weghted average of the pure torson parameter for the full cross-secton and for the weakened cross-secton: T, sub, theor a = t, A + b a+ b t, B ( 2 ) where t,a, t,b are St. Venant torson parameters calculated for the sectons A-AA and B-B; a,b are dmensons accordng to Fg. 4. Fg. 5 The test equpment scheme Examned profles were vertcally hanged on the console. The top end of the specmen was nserted n between the couple of angle profles. The attachment of the top beam end to the console was realzed by means of pn (Ø 8mm) on whch the examned profle was hanged (Fg.6 and Fg. 7). Fg. 4 Web holes confguraton = =1856 mm T, sub, theor = 4 Expermental verfcaton Experment descrpton Expermental verfcaton of the pure torson constant was executed on the specal test equpment (Fg. 5) [2]. Fg. 6 The specmen hanged on the console SBN:

3 The bottom end of the beam s also nserted n between two angle profles whch are bolted to the loadng dsc (Fg. 8) Fg. 9 Load schemee smple couple The torson moment was realzed by the calbrated set of 1kg weghts hangng on steel cable attached to the rotatng loadng dsc (Fg. 11). Fg. 7 Actual detal of the top beam end attachment to the console Fg. 1 Two steel cables attached to the loadng dsc Fg. 8 The bottom beam end attached to the loadng dsc n relaton to the torson the specmen s supported as a cantlever beam. The top end of the specmen s fxed wth regard to torson, but the warpng s not restraned. The bottom end s free and t s loaded wth the torson moment (smple couple on arm of 6 mm Fg. 9). Durng the test the angle of bottom end rotaton s measured n dependence on the sze of torson moment. n the Fg 1 the bottom part of the test equpment wth specmen s dsplayed durng the testng. Fg. 11 Weghts hanged on the steel cable SBN:

4 4.2 Process of loadng Totally 3 tests were performed. Each specmen was loaded sx tmes (three tmes counter-clockwse and three tmes clockwse); thus overall 18 loadng cycles or 228 load steps has been realzed. The sze of the load was establshed for the shear stresses wthn elastc range. The load steps and load ranges for each beam lengths are lsted n the Table 1. Beam length Load step Load range L = 2m 1 kg 9 kg L = 3m 1 kg 7 kg L = 4m 1 kg 5 kg Table 1 Load steps and ranges The recorded processes of the loadng of the beams for each beam lengths are dsplayed n graphs 1, 2 and 3 as a dependence of the force F z [N] on the angle of rotaton φ z [ ] (rotaton of the bottom beam end related to the top end). F z [N] Graph 1 Record of loadng process - beams L = 2m F z [N] 1, 9, 8, 7, 6, 5, 4, 3, 2, 1,, 8, 7, 6, 5, 4, 3, 2, 1,,, 1, 2, 3, φ Z [ ], 5, 1, 15, 2, 25, 3, 35, 4, φ Z [ ] Graph 2 Record of loadng process - beams L = 3m F z [N] 6, 5, 4, 3, 2, 1,,, 5, 1, 15, 2, 25, 3, 35, 4, φ Z [ ] Graph 3 Record of loadng process - beams L = 4m The testng procedure shows the non-lnear progress. At the begnnng of loadng the nonlnear behavor s due to ntal test devce clearances. Durng the loadng advancement the nonlnearty s caused by excessve deformatons of the beam and related daphragm stresses. Based on these fndngs the pure torson stffness has been calculated separately for each load step. 4.3 Evaluaton prncple For each partal load step the dependence of smple couple magntude on the angle of rotaton s determned (the angle of rotaton s frst expressed n degrees and then n radans) by: 18 k ϕ Z, [ rad] FZ, = k ϕ Z, [ ] =. ( 3) π The ncrease of the smple couple magntude can be expressed as: FZ, = F F 1 ( 4), where F s magntude of smple couple actng by the -th load step, F 1 s magntude of smple couple actng by the prevous load step. The rotaton angle ncrease of lower beam end compared wth upper fxed beam end can be expressed as: ϕ = ϕ ϕ, ( 5) Z, 1 where ϕ s angle of rotaton correspondng to the -th load step, ϕ 1 s angle of rotaton correspondng to the prevous load step. The slope of the lne k ndcatng a lnear relatonshp between rotaton angle and the sze of a par of forces for the -th load step can be expressed as: SBN:

5 k F = ϕ F ϕ 1 1. ( 6) The calculaton of the pure torson stffness for each load step s based on the formula: M l ϕ Z, [ rad] = ( 7) G The expressed pure torson stffness from Eq. (7) for each partal load step s: M F, d l l Z t, = = ( 8) G ϕ rad] G ϕ [ rad] t, Z, [ Z, The torson moment n Eq. (8) can be replaced by smple couple actng on arm d and for the F Z, can be used expresson n Eq. (3); consequently: F, d l 18 k, [ rad] d l z ϕ Z t, = = ( 9) G ϕ π G ϕ [ rad] Z, Z, By adjustng the Eq. (9) we get the formula for the partal pure torson parameter calculaton for the -th load step: 18 k d l = ( 1 t, ) π G where k s the slope of the lne ndcatng a lnear relatonshp between rotaton angle and smple couple magntude, d s arm of smple couple [mm], l s theoretcal beam length [mm], G s shear modulus of steel [MPa]. 4.4 Results The resultant partal pure torson stffness calculated for ndvdual rotaton angles converted to degrees per meter of beam length are dsplayed separately for each beam length n graphs 4, 5 and 6. T [mm 4 ] , 2, 4, 6, 8, 1, 12, 14, Graph 4 Partal torson stffness - beams L = 2m T1-2 T2-2 T3-2 T4-2 T5-2 T6-2 T8-2 T9-2 T1-2 The plotted ponts n the graphs are nterspaced by usng the method of least squares wth an exponental curve representng the dependence of the pure torson stffness on the angle of rotaton relatve to the one meter of the beam length. T [mm 4 ] T [mm 4 ] Graph 5 Partal torson stffness - beams L = 3m Graph 6 Partal torson stffness - beams L = 4m The resultng formula for the pure torson stffness calculaton based on performed test s:,,exp = (,, 57 L) + 57 L e T sub, 2, 4, 6, 8, 1, 12, T sub theor, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1,,27 ϕ T1-3 T2-3 T3-3 T4-3 T5-3 T6-3 T7-3 T8-3 T9-3 T1-3 T1-4 T2-4 T3-4 T4-4 T5-4 T6-4 T7-4 T8-4 T9-4 T1-4 where,, s theoretcally calculated pure torson stffness for the substtute crosssecton [mm 4 ], s theoretcal beam length [m], s angle of rotaton converted to degrees per one meter of beam length [ /m]. The expermentally verfed substtute pure torson parameters for each beam lengths are plotted n the graph 7 dependng on angle of rotaton relatve to the one meter of beam length. The horzontal lne n that chart represents the theoretcally calculated pure SBN:

6 torson stffness for the substtute cross-secton. From the presented charts follows that the torson stffness for the small angle of rotaton s comparable wth the theoretcal value,,. n case of the larger rotaton angles the torson stffness s ncreasng more sgnfcantly for the longer tested beams. T [mm 4 ] t,sub,exp,l2 t,sub,exp,l3 t,sub,exp,l4 t,sub,theor, 2, 4, 6, 8, 1, 12, 14, 16, Graph 7 The exponental curves representng the expermentally verfed substtute pure torson stffness 5 Concluson On the bass of the performed experments t was found that the magntude of the St. Venant torson parameter vares dependng on the sze of the rotaton angle. Ths phenomenon s most lkely the result of excessve deformaton and assocated membrane stressed. For very small rotaton angles the pure torson stffness s comparable wth the theoretcally calculated stffness for the substtute cross-secton. For larger rotaton angles the stffness ncreases exponentally wth a dfferent trend for each beam lengths. For the longer specmens the ncrease s more sgnfcant. Acknowledgment: Ths paper has been elaborated wthn the support of the Projects of MSMT No. FAST-J-11-5/1174 and FAST-S-11-32/1252 and GACR reg. No. P15/12/314. References: [1] Horáček, M, Melcher, J, St. Venant Torson Stffness of Thn-walled Beams wth Holes, n Proceedngs of 6th European Conference on Steel and Composte Structures held n Budapest, Hungary, Vol. A, ECCS, 211, pp [2] Melcher, J, Skutečné působení členěných centrcky tlačených prutů (Real behavour of the centrc compresson lattce bars), Fnal report of research project P /3.c, VUT Brno, [3] ČSN EN Desgn of steel structures - Part 1-3: General rules Supplementary rules for cold formed thn gauge members and sheetng, Czech standards nsttute, Prague, 28. [4] Horáček, M, Melcher, J.: Expermental verfcaton of pure torson stffness of thnwalled beam wth holes, Report P12 of research centre CDEAS, project 1M579, VUT Brno, 21. [5] Horáček, M, Melcher, J.: Torson stffness of thn-walled steel beams, Report P28 of research centre CDEAS, project 1M579, VUT Brno, 211. SBN: