MECHANICAL ANALYSIS OF REINFORCED PLATE STRUCTURES

Transcription

1 Budapet 9-30Ma 008 MECHANICA ANAYSIS OF REINFORCED PATE STRUCTURES Gábo M Vöö Depatment of Applied Mechanic Budapet Univeit of Technolog and Economic H-151 Budapet Hunga voo@mmbmehu Abtact: The pape peent the development of a new plate/hell tiffene element and the ubequent application in detemine buckling load and mode of diffeent tiffened panel The fomulation of the tiffene i baed on a geneal beam theo which include the containt toional waping effect and the econd ode tem of finite otation A pat of the validation of the method complete hell finite element anale wee made fo tiffened plate Kewod: Stiffene Fee vibation Buckling load Containt toion 1 INTRODUCTION Man mechanical engineeing tuctue conit of tiffened thin plate and hell pat to impove the tength/weight atio The buckling and vibation chaacteitic of tiffened plate and hell ubjected to initial o dead load ae of conideable impotance to mechanical and tuctual enginee Among the known olution technique the finite element method i cetainl the mot favouable It i a common featue of the finite element baed method that in ode to attain diplacement continuit a igid fictitiou link i applied to connect one node in the plate element to the beam node heaing the ame ection The igid link appoach neglect the outof-plane waping diplacement of the beam ection and thi i wh the uual fomulation oveetimate the tiffene toional igidit It i known that duing toion the beam ection doe not emain plain If thi toional waping i eticted b containt then the ate of toion will alo change along the beam ai The theo of containt toion wa developed b Vlaov [1] Invetigation of tand-alone beam tuctue poved that an appoimate o moe accuate modelling of the toional tiffne can conideabl modif the eult A the objective of thi pape i to tud the effect of containt toion the hea defomation of the tiffene i neglected and the fomulation i baed on the well-known Benoulli Vlaov theo Fo the finite element anali the tiffene element ha two node with even degee of feedom pe node In ode to maintain diplacement compatibilit between the beam and the tiffened element a pecial tanfomation i ued which include the coupling of toional and bending otation and the eccenticit of intenal foce between the tiffene and the plate/hell element BEAM EEMENT In thi wok the baic aumption ae a follow: the beam membe i taight and pimatic the co-ection i not defomed in it own plane but i ubjected to toional waping otation ae lage but tain ae mall the mateial i homogeneou iotopic and lineal elatic et u have a taight pimatic beam membe with an abita co-ection a it i hown in Fig 1 The co-odinate ae and z ae paallel to the pincipal ae maked a and The poition of the centoid C and hea cente S in the plane of each ection ae given b the elative co-odinate NC z NC and z The etenal load ae applied along point P located SP and z SP fom the hea cente 1 / 6

2 Budapet 9-30Ma 008 The vitual wok pinciple fo the beam tuctue ubjected to initial tee i epeed a δ Π=δ Π + Π + Π W = 0 (1) ( ) G Ge whee Π Π G Π Ge ae the linea elatic tain eneg the eneg change due to initial te eultant and the potential eneg due to eccentic initial nodal load epectivel and W i the wok of load incement on incemental diplacement The fit two tem of total potential (1) can be ewitten a: 1 Π = EAu EIw EIv EI ω GJ d α + α () 0 1 Π G = N( v w ) MW M1( v w vw ) M( v v ) M3( w w ) + + α + + α α + α α (3) 0 + ( Vw Vv ) α ( Vv + Vw )( u v w z ) d Diplacement paamete ae u v and w the tanlation in the and z diection of point S and α denote otation (twit) about the hea cente ai paallel to In Eq () E and G ae the Young and hea moduli epectivel The initial te eultant in Eq (3) ae: N the aial foce V and V the hea foce acting at the hea cente M 1 and M M 3 ae the twiting and bending moment with epect to hea cente S epectivel and M W i known a the Wagne effect: M (( ) ( ) ) W = + z σ da = N ip + Mβ Mβ (4) A X N 1 Z Y N i N z z SP z z NC N i z C 3 P S NC SP F z F Fig1 Beam element local tem and eccenticitie Alo ectional popetie ae defined a ϕ ϕ I = da I = da I ω = ϕ da J = I + I da A A A A I + I 1 1 ip = + + z β = ( ) da z ( ) da A I + β = I + (5) A A In thee equation the waping function φ and the S hea cente location ae the ame a in the cae of fee toion Fo thin-walled ection φ = - ω the ecto aea co-odinate The thid tem of Eq (1) i the incemental wok of initial etenal load Conideing conevative initial foce F F and F z acting at point P ( SP z SP ) a igned on Fig 1 of the i- th nodal ection the incemental wok of thee action i 1 ΠGe = F ( SP zsp ) F ( zsp SP ( )) Fz ( SP z SP ( )) β+ γ α+ βγ γ +α + βγ β +α (6) i The moe detailed deivation of Π and Π G Π Ge ma be efeed to Ref [] - [3] / 6

3 Budapet 9-30Ma Finite element model The vecto of even local diplacement and the element diplacement paamete ae defined a [ ] T T T i = u v w α β γ ϑ U i E = 1 A linea intepolation i adopted fo the aial diplacement and a cubic Hemitian function fo the lateal deflection and the twit Subtituting the hape function into Eq () and (3) and integating along the element length elementa matice can be defined a: T T δ Π = δuku E E δ Π G =δuk E GU E (7) The eplicit eactl integated (1414) element k linea tiffne and k G geometic tiffne matice can be found in Ref [3] 3 STIFFENER TRANSFORMATION Majoit of publication uing the finite element anali of tiffened panel whee the tiffene ae modelled uing beam element the beam node ae foced to undego the diplacement and otation pecibed b the coeponding plate/hell node In thi cae the containt condition i intoduced b conideing a igid fictiou link between the beam ection and the plate/hell node N on the common nomal In uch a model the diplacement and otation of a nodal point N with the co-odinate = - NC and = -z NC (ee Fig 1) in the plane of the co-ection will be a follow: u = u β NC +γ z NC u = v +α ( znc + z ) uz = w α ( NC + ) (8) Θ =α Θ =β Θ z =γ whee u u u z Θ Θ Θ z ae the nodal local diplacement and otation Thi tanfomation take the eccenticit into account but obvioul neglect the effect of toional waping 31 Continuit of otation t If a beam i connected to anothe component not onl in it co-ection but along a naow tipe on it F N i uface the tanfomation (8) i not ufficient to aue C R SN the equied diplacement continuit Duing toion while the co-ection tun aound point S b an angle α the oiginall taight connecting line coing point N S aume a pial hape The otation aiing thee i popotional to the ditance between point S and N Uing the notation of Fig the vecto of pial otation can be decibed a α Fig Joint line otation 0 d α Φ = R SN =ϑ ( RNC + R ) =ϑ NC + d znc + z (9) Supplementing bending otation in Eq (8) b thee pial otation component: u = u β NC +γ z NC u = v +α ( znc + z ) uz = w α ( NC + ) Θ =α Θ =β+ (NC + ) ϑ Θ z = γ+ (znc + z ) ϑ (10) which ield the modified tanfomation between the diplacement paamete 3 / 6

4 Budapet 9-30Ma Eccenticit of intenal foce The calculation of k Ge load tiffne mati of the tiffene element equie ome emak Initial intenal foce o contact foce between the tiffene and the plating along the contact line can be calculated fom the equilibium condition of initial tate Heeinafte the contact point hould be the node N and uing the notation a indicated on in Fig 1 the load eccenticitie if N = P ae: SP = SN = ( NC + ) zsp = zsn = ( znc + z ) Thee i a imple wa to calculate the tiffene load tiffne if the cubic element ae ued to define the initial te tate It follow fom the cubic hape function that the N nomal and V V hea foce ae contant along a taight beam element but diffeent fom element to element Thi intenal foce ditibution can be poduced b etenal foce acting at the two end node of an element: F1 = N F1 = V Fz1 = V F =+ N F =+ V Fz =+ V (11) With thee end load and eccenticitie in Eq (6) the additive tiffne due to off ai contact load acting along the joint line i epeed a 1 Π = ( α α )( V + Vz ) + ( β β ) Vz + ( γ γ ) V Ge 1 SN SN 1 SN 1 SN ( ) ( ) ( )( ) N Nz V z V αβ 1 1 αβ SN α1γ1 αγ SN β1γ1 βγ SN + SN 4 NUMERICA ANAYSIS AND DISCUSSIONS With the aembled tem matice the geneal equation fo buckling poblem i ( ) K + λ KG + KGe U= 0 whee λ i the citical load paamete The goal of the following numeical tud i to compae the adequac of two tiffene coupling tanfomation detailed in Section 3 Fit i the uual igid leve am coupling accoding to Eq (8) and the othe i the popoed tiffene coupling tanfomation including the intenal foce eccenticit in accodance to Eq (10) and (1) In the following thee will be called of BM (beam) and ST (tiffene) coupling epectivel In ode to veif the BM and ST eult of peent tud a COSMOS/M hell model wa emploed In that model the plate and the thin walled beam wa compoed of the ame fou node hell4t thick hell element A ectangula tiffened panel on Fig 3 conit of a flat plate with equall paced thin walled T-tiffene Becaue of the mmet in the tuctue onl a potion of the plate of width b with a T-tiffene wa modelled In the finite element model the otation about the longitudinal X ai and the lateal diplacement wee uppeed along the longitudinal edge to imulate the panel continuit and the X = 0 and X = end of the panel ae fied The mateial popetie ae: E = MPa ν = 03 In ode to model the wide ange of behaviou of the panel the plate dimenion and the beam hape unchanged the aea of tiffene wa caled in popotion to the web thickne The co ectional popetie fo the T-beam and the non-dimenional plate to beam aea atio paamete a the function of t w ae: 4 4 t = t b = 10t b = b A = 30t I = 1405t I = 85t f w f f w f w w w 4 / 6 ( ) 4 J = 10 t w I ω = 0 β = 16 t w z NC = - 135t w+ z = -7t w δ= A bt+bt f f w w tw A = bt = 80 p (1)

5 Budapet 9-30Ma 008 To invetigate the effect of BM and ST coupling method on the buckling load and mode the elatic buckling of the panel ubjected to longitudinal compeion i tudied in thi ection Thi kind of uniaial compeion can be poduced b an U 0 aial diplacement of the X = end of the panel Hee U 0 = 1 mm initial compeion coepond to E σ 0 = U 0 = MPa initial nomal te t = 4 mm z N Z X b w t w C z NC z =1800 mm b = 600 mm Y t f b f S Fig 3 Panel dimenion The buckling mode hape of the tiffened panel fo diffeent tiffene ection ae hown in Fig 4 The Fig 5 how the change of buckling mode and λ citical load paamete in tem of δ ize paamete If the tiffene ection i mall the buckling mode i the global (ometime called of Eule buckling) fleual mode and the eult i independent of the coupling (BM o ST) method Fo highe tiffene ection toional buckling mode called of tipping will occu pio to fleual buckling In contat with fleual mode thee i a ignificant diffeence in the buckling load of BM-t1 and ST-t1 coupling method and the ST coupling eult in a le igid model with malle citical load It i obeved fom the Table 1 that in cae of δ = 0 the ate of deceae i aound 30% A the tiffene tipping i a coupled lateal toional-bending motion the accuate modelling of toional popetie ae of geat impotance A detailed anali of the diffeent buckling mode of tiffened panel including the paametic anali of tipping can be found in ecent pape of Yuen et al [4] and Sheikh at all [5] a b c δ = 00; λ=0094 δ = 0; λ=035 δ = 09; λ=03813 Fig 4 Global (a) tipping (b) and plate buckling (c) mode With inceaing tiffene ize and igidit the diffeence between BM and ST eult vanih The unifom amptotic citical load in Fig 5 indicate the buckling of plate between the tiffene a it i hown in Fig 4 On Fig 5 quite atifacto ageement can be een fo the citical load paamete between the ST and COSMOS/M hell eult maked with black dot 5 / 6

6 Budapet 9-30Ma 008 δ / t w (mm) BM Eg (8) ST Eq (10 1) hell COSMOS/M 000 / / Tab 1 Buckling load paamete λ 05 l BM-t1 BM-ST-b1 ST-t1 Shell 0 d=a/ap Fig 5 Change of buckling mode and load paamete 5 CONCUSIONS In thi tud a detailed numeical evaluation ha been pefomed to pove the efficienc of the popoed tiffene plate/hell coupling method It wa hown that in all toion elated cae the popoed ST method lead to a le igid model The eult how good ageement with complete hell olution Thi fact indicate that the application ange can be etended Though futhe wok can be undetaken to pefom dnamic and buckling anali of eall cuved panel with tiffene the newl developed coupling method can be ueful fo futue eeache REFERENCES [1] Vlaov VZ Thin-walled elatic beam National Science Foundation Wahington1961 [] Kim MY Chang SP Pak HG Spatial potbuckling anali of nonmmetic thin-walled fame I: Theoetical conideation baed on emitangential popet J Engineeing Mechanic (ASCE) 001;17(8): [3] Vöö GM An impoved fomulation of pace tiffene Compute and Stuctue 007;85(7-8): [4] Yuen H Bozen C Jiulong S Tipping of thin-walled tiffene in the aiall compeed tiffened panel with lateal peue Thin-Walled Stuctue 000;371-6 [5] Seikh IA Elwi AE Gondin GY Stiffened teel plate unde uniaial compeion J Contuctional Steel Reeach 00; [6] Vöö GM Buckling and Vibation of Stiffened Plate Intenational Review of Mechanical Engineeing (IREME) 007;1(1): / 6