EFFECTIVE BUCKLING LENGTH OF COLUMNS IN SWAY FRAMEWORKS: COMPARISONS

Transcription

1 IV EFFETIVE BUING ENGTH OF OUMN IN WAY FRAMEWOR: OMARION Ojectives In the present context, two different pproches re eployed to deterine the vlue the effective uckling length eff n c of colun n c out the locl xis (X The first is the siplified pproch to B, while the second is ore ccurte ethod This chpter nswers two questions; (i when does the vlue of the effective uckling length fctor deterined y the B pproch differ fro the effective uckling length fctor clculted y the direct ethod? (ii wht re resons for the different results of eff when pplying the different pproches? The elstic stility of ulti y single storey freworks peritted to swy re investigted Here, coprisons hve een de etween nd where n c refers to the colun er under considertion

2 Effective Buckling ength of oluns in wy Freworks: oprisons The developed FORTRAN progr for the stility nlysis hs een used nd verifiction of results hs een perfored using ANY oprisons etween nd The stility nlysis of structurl steel freworks is sed on the following ssuptions: The frework teril is perfectly elstic Buckling out plne of the frework is prevented The effect of the xil defortion of ers produced y second order forces is neglected The effect of the second order vritions in the xil forces of frework ers due to sideswy is neglected when deterining the stility futions The deflections of coluns nd es re sll, thus the reltionship etween the ending oent nd the curvture cn e pproxitely expressed y the second order differentil eqution discussed in ection Exple : ingle y single storey frework onsider the rectngulr fixed se frework ABD shown in Figure which is peritted to swy lterlly The frework is sujected to two verticl lods t it corners (B nd nd horizontl force α where α equls The coluns of the frework hve different types of cross sections The stility futions of left hnd side (H colun, colun, re different fro those of right hnd side (RH colun, colun When the verticl lods rech their criticl vlues cr, ny sll sideswy cn tke plce This results in two opposite horizontl forces (H nd H when using the finite eleent nlysis The differee etween (H nd H is however

3 Effective Buckling ength of oluns in wy Freworks: oprisons neglected when using the direct ethod The opertions of swy nd rottions re then uilt up for every er of the frework seprtely s shown in Figure c corresponding to the distorted configurtion ssued in Figure of the frework ie t the criticl lod there re no externl oents set up t the corners to keep the frework in the distorted position, the su of oents M M, BA B n H BA D α B A D H H ( oding pttern ( Distorted configurtion n H n H O H O H (c Opertions of rottion nd swy Figure ingle y single storey frework

4 Effective Buckling ength of oluns in wy Freworks: oprisons Dividing this eqution y : H n nd ( B D M M, H n Dividing this eqution y : H n ( ie the sideswys t joints B nd re equl, the su of the corresponding coponents of pure sher nd no sher swy of the two coluns cn e expressed s AB D, ( ( H H, ( ( H ( Eliinting the three unknowns nd, H fro (, ( nd (, the generl for ( of the criticl uckling lod is otined: ( ( det n n (

5 Effective Buckling ength of oluns in wy Freworks: oprisons The solution of this for is otined following the flow chrt given in Figure y the tril nd error ethod, where the vriles re the height of the coluns, the spn of the e nd the properties of the cross section of ech er of the frework These properties cn e tken fro teel onstruction Institute ( To verify the results otined fro eqution (, the elstic criticl lod of two single y single storey freworks, hving the se diensions of, nd the sections descried in Tle, hve een copred to the results otined using ANY, where E kn/ Tle Diensions nd cross sections of single y single storey frworks se study H colun RH colun Be Frework U U UB Frework U U UB The ANY dt file is prepred in sequee of three steps: preprossesing, solution nd postprocessing In the preprocessing stge, nodes nd ers of the frework re defined, where ech er is divided into severl eleents to get dequte ccurcy Four different divisions into eleents for ech er re studied: se : five eleents for ech er, se : ten eleents for ech er, se : twenty eleents for ech er nd se : one hunderd eleents for ech er In the solution stge, the type of nlysis required s well s the end conditions nd loding ptteren re defined Finlly, ANY retric Design nguge (AD is used in the postprocessing stge to otin the results

6 Effective Buckling ength of oluns in wy Freworks: oprisons Tle descries the coprisons etween the criticl lod ANY, nd tht DM cr y using the direct ethod FE cr, otined using Tle ingle y single storey frework: verifiction of otined results se study riticl lods FE cr (kn se se se se DM cr (kn Frework Frework To copre the effective length fctors nd, the preters, nd cross section of the e re fixed while the cross sections of coluns re vried, thus, two sets of preters re considered: et :, nd the cross section of the e is UB, et :, nd the cross section of the e is UB The vlue of is coputed y n c ( ρ where ρ F, F is the xil force t the criticl uckling lod n c E, n c DM cr nd E, π EI oprisons etween nd re listed in Tles nd for the frework hving preter sets nd respectively Fro these tles, It cn e oserved tht when the sections of coluns re the se, the vlues of equl to their corresponding vlues of

7 Tle ingle y single storey fre: coprisons etween nd when using preter set H colun RH colun se study ross section ρ Direct ethod B Direct ethod B X, X, ross section ρ X, X, U U U U U U U U U U U U U U U U U U U U

8 Tle ingle y single storey fre: coprisons etween nd when using preter set H colun RH colun se study ross section ρ Direct ethod B Direct ethod B X, X, ross section ρ X, X, U U U U U U U U U U U U U U U U U U U U

9 Effective Buckling ength of oluns in wy Freworks: oprisons Exple : Two y single storey frework In this section, the two y single storey frework shown in Figure is studied Following the se procedure descried in ection, equtions ( cn e deduced, where the distorted configurtion s well s the opertions of rottions of the frework re given in Figure nd c respectively: B M, H n, ( M, H n, ( E M, H H n, ( AB EF, ( ( ( H H ( nd EF D H H ( ( ( (

10 Effective Buckling ength of oluns in wy Freworks: oprisons α B E A D F ( oding pttern BA D EF H H H H ( Distorted configurtion n H n H n ( H H O H O H O ( H H (c Opertions of rottion nd swy Figure Two y single storey frework

11 Effective Buckling ength of oluns in wy Freworks: oprisons Eliinting the unknowns,,, H nd H fro ( (, the criticl uckling lod ( is derived: det yetric ( where, n,,, n,, n,,, ( (, nd ( ( ( This eqution ( is used to otin DM cr when diensions of the frework re, The cross sections of ers, given in Tle, re ssued Tle Two y single storey frework: ssued cross sections Meeer Outer coluns Inner olun Bes ross section U U UB The finite eleent odel of the frework ws uilt s illustrted in ection, then FE cr ws otined to verify the results Tle shows the coprison etween FE cr nd DM cr ssuing E kn/

12 Effective Buckling ength of oluns in wy Freworks: oprisons Tle Two y single storey frework: verifiction of otined results riticl lods FE cr (kn se se se se DM cr (kn Using ( of the criticl uckling lod, the effective length fctors of coluns of freworks ( cses re coputed ssuing, nd the cross section of e is UB Then, the effective length fctors nd for ech er re copred s given in Tle It is lso ssued tht the two outer coluns, coluns nd, hve the se type of cross section Fro this tle, it cn e deduced tht the vlue of the effective length fctor cn e either greter or sller thn the vlue of the effective length fctor nd this depends on the section properties of the frework ers, ie the ending stiffness of the ers

13 Tle Two y single storey fre: coprisons etween nd Outer coluns Inner colun se study ross section ρ ρ Direct ethod B Direct ethod B X, X, ross section ρ X, X, U U U U U U U U U U U U U U U U where X, X,, X, X, nd

14 Effective Buckling ength of oluns in wy Freworks: oprisons Exple : Three y single storey frework The effective length fctors nd of the coluns of the three y single storey frework shown in Figure re copred following the se procedure descried in ection The generl eqution of the elstic criticl lod is otined: det yetric ( where,,,,, n, n, n, n, -, (, ( ( nd ( ( ( ( The unknowns of this for re,,, H, H nd H,

15 α B E G ( oding pttern A D F J BA D EF GJ ( Distorted configurtion H H H H H H (c Opertions of rottion n H n H n H n ( H H H nd swy O H O H O H O ( H H H Figure Three y single storey frework

16 Effective Buckling ength of oluns in wy Freworks: oprisons To verify the results when using the developed progr, FE cr nd DM cr, the results re copred ssuing diensions of the frework, the cross sections of its ers re s given in Tle nd Tle Three y single storey frework: used cross sections Meeer Outer coluns Inner oluns Bes ross section U U UB The finite eleent odel of the frework ws uilt s deonstrted in ection The otined results re then copred s shown in Tle ssuing E kn/ Tle Three y single storey frework: verifiction of otined results riticl lods FE cr (kn se se se se DM cr (kn Applying ( of the criticl uckling lod, cses (Tle re investigted In this investigtion, the effective length fctors nd re otined considering different types of cross sections of coluns The frework diensions, nd the cross sections of es of UB re ssued It is lso ssued tht outer coluns, coluns nd, hve the se cross section s well s inner coluns, colun nd

17 Tle Three y single storey fre: coprisons etween nd Outer coluns Inner coluns se study ross section ρ ρ Direct ethod B Direct ethod B X, X, ross section ρ ρ X, X, U U U U U U U U U U U U U U U U where X, X,, X, X,, X, X,, X, X, nd

18 Effective Buckling ength of oluns in wy Freworks: oprisons Exple : Four y single storey frework The next exple to investigte is the four y single storey frework shown in Figure Following the se procedure descried in ection, the generl eqution ( of the criticl uckling lod is otined: det yetric ( where,,,,,, n, n, n, n, n, -, (, ( (, ( ( nd ( ( n

19 α B E G U ( oding pttern A D F J V BA D EF GJ UV ( Distorted configurtion H H H H H H H H (c Opertions of rottion nd swy n n H n H H n H O H O H O H O H n ( H H H H O ( H H H H Figure Four y single storey frework

20 Effective Buckling ength of oluns in wy Freworks: oprisons ( ( The unknowns of this for re,,,, H, H, H nd H, To verify the results otined fro (, the frework is odelled using ANY nd stility nlysis is crried out where diensions of the frework re, nd cross sections of ers re ssued s given in Tle Tle Four y single storey frework: used cross sections Meeer oluns Bes ross section U UB The finite eleent odel ws uilt s deonstrted in ection The otined results re then copred s shown in Tle ssuing E kn/ Tle Four y single storey frework: verifiction of otined results riticl lods FE cr (kn se se se se DM cr (kn Eight cses (Tle re studied where the frework diensions of, s well s the es sections of UB re ssued It is lso ssued tht the two outer coluns, coluns nd, hve the se section s well s the inner coluns, colun, nd

21 Tle Four y single storey fre: coprisons etween nd Outer coluns Inner coluns se study ross section ρ ρ Direct ethod B Direct ethod B X, X, ross section ρ ρ X, ρ X, U U U U U U U U U U U U U U U U where X, X,, X, X, X,, X, X,, X, X, X, nd

22 Effective Buckling ength of oluns in wy Freworks: oprisons Exple : Five y single storey frework Following the procedure descried in ection, the criticl uckling lod eqution ( of the five y single storey frework shown in Figure ws otined: det ( where,,,,,, -, n, n, n, n, n, n, ( (, ( -, ( -, yetric

23 α B E G U W ( oding pttern A D F J V Y BA D EF GJ UV WY ( Distorted configurtion H H H H H H H H H H (c Opertions of rottion nd swy n H O H n H O H n H O H n H O H n H O H n ( H H H H H O ( H H H H H Figure Five y single storey frework

24 Effective Buckling ength of oluns in wy Freworks: oprisons, ( (, ( ( nd ( ( ( ( The unknowns of this for re,,,,, H, H, H, H nd H, To vlidte the results otined fro (, the frework is odelled using ANY nd stility nlysis is crried out considering diensions of the frework of, nd cross sections of ers s given in Tle Tle Five y single storey frework: used cross sections Meeer oluns Bes ross section U UB Following the se technique illustrted in ection for uilding the frework odel (Appendix B, the otined results FE cr nd DM cr re copred s given in Tle ssuing E kn/

25 Effective Buckling ength of oluns in wy Freworks: oprisons Tle Five y single storey frework: verifiction of otined results riticl lods FE cr (kn se se se se DM cr (kn Using (, cses (Tle re investigted to copre the effective fctors nd where diensions of the frework re, nd cross section of es of UB re ssued It is lso ssued tht the two outer coluns, coluns nd, hve the se of cross section s well s the inner coluns, coluns,, nd Fro this tle, it cn e extrted tht the vlue of the effective length fctor the vlue of the effective length fctor cn e either greter or less thn nd this depends on the section of the frework ers, ie the ending stiffness of the ers

26 Tle Five y single storey fre: coprisons etween nd Outer coluns Inner coluns se study ross section ρ ρ Direct ethod B Direct ethod B X, X, ross section ρ ρ ρ ρ X, U U U U U U U U U U U U U U U U X, where X, X,, X, X, X, X,, X, X,, X, X, X, X, nd

27 Effective Buckling ength of oluns in wy Freworks: oprisons oluding rerks In this chpter, the stility coept of steelwork hs een investigted The criticl uckling lod equtions of five ulti y single storey freworks were otined using the direct ethod of nlysis To verify the results otined using these equtions, the finite eleent pckge, ANY, ws used Fro the results presented in this chpter, the oservtions cn e surized s follows: It cn e oserved tht there is differee etween the effective length fctors deterined y the B pproch nd the effective uckling length fctor clculted y the direct ethod This differee vries ccording to the geoetric diensions nd section properties of ech er of the frework under considertion This differee is otined ecuse the siplified ethod dpted y Wood ( nd presented in B inly depend on two ssuptions: involving the no sher stility fution only, while the generl cse of swy is superposition of no sher swy nd pure sher swy s discussed in hpter the considertion of liited frework which contins the colun under considertion plus ll ers, in the frework in t either end This ens, the effect of other ers not iluded in the liited frework were neglected When the single y single storey frework is syetric nd syetriclly loded, the vlue ρ for oth coluns ecoes identicl ecuse the effect of n c the pure sher stility futions is very sll onsequently, the vlues of nd re lost equl This indictes tht when the

28 Effective Buckling ength of oluns in wy Freworks: oprisons ssuptions de y Wood ( re et, the siplified pproch gives the se results s those y ore ccurte pproch The coprison shows tht the effective length fctor cn e either greter or sller thn nd this depends on the geoetric diensions nd section properties of ech er of the frework under considertion Four questions now rise First, wht is the xiu differee tht could occur etween the effective length fctor when using the code pproch nd the ore ccurte pproch? econd, wht is the position of the colun er where the xiu differee is otined? Third, wht re the cross sections of the frework ers t the xiu differee position? Fourth, how does the ethod of the deterintion of the effective length fctor influee the optiu design? To nswer these questions, ppliction of optiiztion techniques is needed onsequently, the i of next chpter is to introduce n lgorith, tune its preters, suggest new odifictions nd finlly test the lgorith