ANALYTICAL AND NUMERICAL ANALYSIS OF NON SYMMETRICAL ALL STEEL SANDWICH PANELS UNDER UNIFORM PRESSURE LOAD

Transcription

1 INTERNATIONAL DESIGN CONFERENCE - DESIGN 00 Durovnik, Ma 14-17, 00. ANALYTICAL AND NUMERICAL ANALYSIS OF NON SYMMETRICAL ALL STEEL SANDWICH PANELS UNDER UNIFORM PRESSURE LOAD Kujala Peni and Klanac Alan Kewords: Seel sandwic panel, laser welding, sear siffness 1. Inroducion Te demand for igger, faser and liger moving veicles, suc as sips and rains as increased e demand for efficien srucural arrangemens. Sandwic panels form one pe of efficien srucures enaling e use of seel, aluminium or composies for e consrucions. (Kujala e al., 1995). Developmen of laser welding ecniques enales e fas producion of ig quali and dimensionall accurae all seel sandwic panels. Helsinki Universi of Tecnolog/Sip Laoraor as developed during e las 7 ears a numer of various applicaions of laser welded seel sandwic panels. Te work carried ou includes e.g. developmen of design and producion meods wi exensive sreng esing for various pes of sandwic panels (as summarised Romanoff and Kujala, 00). Seel sandwic panels welded laser can offer % weig savings compared o e convenional seel srucures. In is paper analical analsis of all seel sandwic panels, suc as sown on Figure 1a, ave een performed in order o deermine e srucural eaviour under uniform laeral pressure load, especiall e sear siffness ransverse o e core siffeners are under consideraion. Sandwic consrucion as een analsed as a simpl suppored eam, Figure 1. Specific formulaions ave een developed and closed form soluion esalised. Wen analsing all seel sandwic consrucions under ending loads one as o ake ino accoun e influence of sear siffness S on a consrucion and no onl e ending siffness D. We will define ere e accurae sear siffness relaion. Anoer ing o consider wen analsing sandwic eams is ick faces effec a we will explain in e ex. Numerical finie elemen models ave een developed o verif e resuls of ere presened analical relaions. Models ave een creaed using FEM applicaion Aaqus. Beam numerical models ave een defined wi varing face plae icknesses. Figure 1a. I core all seel sandwic panel Figure 1. Sandwic panel modeled as a eam ANALYSIS TECHNOLOGIES 105

2 . Background Laser welded seel sandwic panels applicaions and design meods ave een under acive researc in USA and Europe during e las decade. Te US Nav as sudied e applicaions of laser welded corrugaed core sandwic since 1987 (Marsico e al., 1993). ). Tese applicaions include ulkeads and decks in accommodaion areas, deckouses, deck edge elevaor doors, angar a division doors and applicaions on SWATH srucures. In German Meer Werf as o designed and produced seel sandwic panels since 1994 (Roland & Mesckow, 1997). Te ave o produced and designed seel sandwic panels for example for sip applicaions suc as saircase landings, decks, ulkeads, ec. Te ave also made several sreng and faigue ess ogeer wi o analical and finie elemen analses for seel panels. MacGregor as developed a new kind of seel sandwic panel were e core is made ou of seel wires. Te also sell is produc as a Ro-Ro decks ec. (Macgregor, 000). In Finland e researc relaed o all seel sandwic panels was iniiaed in 1988 in e Sip Laoraor of Helsinki Universi of Tecnolog. Te researc was iniiaed suding e applicaion of sandwic panels in e sell srucures of an icereaker. Since en ere ave een several researc projecs in Finland regarding e manufacuring, design and design opimisaion of seel sandwic panels. Te opics, wic ave een invesigaed and relaed o design or design opimisaion of seel sandwic panels (Romanoff & Kujala, 00), see Figure for pical seel sandwic cross secions sudied. A e momen ere is also a European union funded projec called Advanced Composie Sandwic Seel Srucures (Sandwic, 001) were an exensive es program, ogeer wi e developmen of design formulaions and proope designs are carried ou. Te formulaions presened in asic exooks for sandwic srucures (e.g. Allen 1969, Zenker, 1995) concenrae mainl on e calculaion of e elasic response of sandwic panels in general. For seel sandwic panels ese ooks include lile informaion aou e special feaures of e seel sandwic panels. Te seel sandwic panels ave srongl ororopic eaviour due o e aving e core siffeners onl in one direcion. Tis ororopic eaviour and especiall e sear siffness ransverse o e core siffeners ave crucial imporance for developmen of e design meods for ese panels. Tis opic as een sudied earlier e.g. Fung e al. (1994) and Lok e al. (000) for smmeric seel sandwic panels. Here e analsis is exended o cover also e possiili of aving non-smmeric cross secion i.e. e op and oom plae ave differen icknesses. In addiion, e effec of e ick face plaing is included in e analsis as follows. I-core O-core Vf/V-core Figure. Tpical seel sandwic panel cross secions 3. Defining sear siffness S All seel sandwic panels wic ave a core siffeners in onl one direcion and are erefore srongl ororopic in sear, iniiae a quesion of deerminaion of sear siffness in e direcion perpendicular o e core. We assume a e face plaes are carring e sear forces in direcion, alongside wi e ending momens. Sear siffness is defined (1), from wic emerges e prolem of finding e sear srain γ z. Sear force T is divided on e par carried e op plae and on e par carried e oom plae. We also consider e influence of sear force wen i is applied a a disance. Tis causes orizonal force T p [Fung e al. 1994], were p = g. S T =, (1) γ z 106 ANALYSIS TECHNOLOGIES

3 Now, we can creae e analical model. Since i is smmerical aou siffener we consider jus alf of e segmen. Analical model on Figure 3 is a frame, simpl suppored in poins B and C o preven rigid od moion. We predic small displacemen in orizonal direcion since e face plaes are muc more rigid in compression/ension en in ending for approximael same ampliude of force. Due o smmer of e segmen and posiion of e verical forces we also predic small verical displacemens of e poins B and C. Figure 3. Half segmen Figure 4. Deformed alf segmen Le sear force T e of magniude 1. In Figure 3 e sear force as een divided ino wo pars, ( 1 z) and z o e carried e op and oom plae. Te value of z is a funcion of face plae icknesses and i will e defined laer. To calculae e oal deformaion of e segmen under inernal forces e displacemens of poins A and D mus e defined. Since e read of e frame model par in direcion x is larger an e leng p, see Figure 1, frame par is in plane srain so Young s modulus akes e value E 1 ν [Fung e al. 1994]. Wen calculaing e eam wic read of e par is smaller an one quarer of is leng we use e one dimensional Young s modulus E. Le us sae wo condiions a ave o e followed wen defining e sear srain: a) during e deformaion angles a poins B and C sa consan, wi e value of 90, A D ) verical displacemens of poins A, δ z, and D, δ z, are equal, a wa keeping e ickness of e panel consan. Sear srain γ z is separaed ino verical and orizonal par, γ v and γ eac defined roug values of displacemen of poin A and D [Lok 000]. From Figure 4 we can wrie: A δ z γ v = and γ p δ A + δ D =. () Verical and orizonal displacemens of poins A and D are defined for uni sear force load and susiued ino equaions (): 4 ( 1 ν ) ( 1 z) p p ( 3z) ( 1 ν ) γ + and = v = 3 3 E c Te value of facor z can e found from condiion (): z ( pc + 3 ) p ( + ) c p E γ (3) = (4) ANALYSIS TECHNOLOGIES 107

4 Sear srain for e plae S can now e deermined wi equaion (5) and for eam S wi read of wi equaion (6): S 1 = γ + γ v (5) S = S (6) 4. Closed form soluion for eam ending Le us consider now alf of a eam in Figrure 1 under e uniform line load, see Figure 5. Is sear siffness S is small compared o is ending siffness D so is srucural response will e largel influenced sear siffness. Tis is e reason w we use eam analsis o deermine e accurac of e sear siffness model. Beam is analsed wi e effec of ick faces o oain more accurae resuls in srucural response. Effec of ick face plaes of e sandwic panel assumes a e sear deformaions cause no onl e local core deformaions u also e local deformaions of e face plaes [Allen 1969]. Tese deformaions can e defined as ending of e faces aou eir own neural axes. Wen e sandwic ends as a wole e faces deform locall and gloall. In addiion local ending siffness descried is effec influences sear deformaions of e core. Sandwic eams wi inner face plaes are less influenced wile e ones wi icker are more. Teir face plaes ave significan local ending siffness D f and no longer eave as flexile memranes. Bending momens are: D = E ( + ) + A e + A e, D ( + ) 1 p p p p f =, (7) were A s are cross secion areas of face plaes and e s are disance eween e face plaes and a neural line. Full derivaion of e closed form soluion is omied ere [Allen 1969], jus final expression is given in (8). qx L x q D f x βl w = x 48D L S D 4θ 1 ( 1 cosα ), (8) = S D f 1 D were: α, D f L = α θ, β θ 1 cos =. (9) θ Te closed form soluion wi and wiou e ick plae effec (for e case 1 in Tale ) is sown in Figure 6 ogeer wi e FEM resuls as descried in furer deail in e nex secion. As can e seen, including e ick face effec decreases remarkal e deflecions of e eam and gives resuls close o e FEM calculaions. No ick face effec Wi ick face effec FEM Figure 5. Half of a sandwic eam Figure 6. Beam displacemen in mm 108 ANALYSIS TECHNOLOGIES

5 5. Numerical models and comparison of resuls Te numerical models are creaed o enale e deerminaion of e accurac of e analical model of sear siffness and also o verif e relaion (8). Using e smmer of oundar condiions of a eam onl one alf as een modelled on wic mesing of 83 eig nodal quadraic sell elemens as een applied, see Figure 7. Figure 7. Elemen mesing applied o descrie e eam We calculae e maximal displacemen for e eam wi e caracerisics given in Tale. Uniform pressure load q = 1000 Pa as een applied over op plae surface. For e analical eam calculaions i as een ransformed ino a line load of q = 50 N/m. We presen ere 5 pical cases o compare e analical and numerical analsis. Tale 1. Caracerisics of eam numerical models Maerial prop. Beam prop. Ticknesses E = 06.8 Gpa l = 000 mm = 1 5 ν = 0.9 = 50 mm c = 4 = 100 mm = 3 p = 100 mm g = 00 mm Tale sows e liss of maximal deflecions acquired e analical and FEM calculaions and e sear siffness calculaed wi e analical formula for e analsed models. Also e margins of error of analical model o e FEM are presened. As can e seen e maximum error is onl.3 %. Tale. Maximal displacemens of e calculaed eams Tick[mm] Siff.[Nm] Max. deflecion [mm] Error c S Analic FEM [%] ,43 1,139 0,740 1, ,78 8,69 8,490, ,71 3,64 3,74 -, ,31 3,6 3,87-0, ,65,489,438,05 6. Conclusions Analical calculaions sown in e paper resul in small margin of error, so we can e sure a e ave een developed correcl o saisf e need of sandwic panel design. If we know a e ending siffnesses D and D f are, in manner we ave descried i, considered as an accurae values, ANALYSIS TECHNOLOGIES 109

6 we can conclude a e analical model for e deerminaion of sear siffness is accurae since e eam deflecion is dependen onl on ese ree variales. Sear siffness S was derived for e eam and e plae wi e same assumpions and in e same general manner, so en in fuure analsis of sandwic plaes relaions presened ere could e considered correc. We can also confirm a e closed form soluion for deerminaion of deflecions is accurae and a e assumpions made wen analsing ick faces effec are correc. In fuure ese formulaions will e applied o opimise e sandwic panel configuraions for various applicaions onoard sips suc as movale car decks. Acknowledgemens Tis repor is ased mainl on e resuls oained in one Finnis researc projec and one European Union funded researc projec. Te naional projec as een carried ou under e following ecnolog programs: Kenno Lig Srucures. Tis projec as een funded TEKES (Tecnolog Developmen Cenre) and several Finnis indusrial parners. Te EU-funded researc projec is called: Advanced Composie Sandwic Seel Srucures (Sandwic, p://sandwic.alpor.com/ ). Tis financial suppor is ere graefull acknowledged. References Allen, H.G., "Analsis and Design of Srucural Sandwic Panels", Pergamon Press Oxford UK, Fung, T.C., Tan, K.H., Lok, T.S., "Elasic Consans for Z Core Sandwic Panels", Journal of Srucural Engineering, Vol. 10, 1994, pp Kujala, P., Mesä, A. & Nallikari, M., All Seel Sandwic Panels for Sip Applicaions, Sipard 000: Spinoff Projec, Helsinki Universi of Tecnolog, Sip Laoraor, Oaniemi, 1995, M-196. Lok, T.S., Ceng, Q.H., "Elasic Siffness Properies and Beaviour of Truss Core Sandwic Panel", Journal of Srucural Engineering, Vol. 16, 000, pp MacGregor, COREX Panels Increase Revenue-earning Capaci, MacGregor News, No. 143, 000. Marsico, T.A. e al., Laser Welding of Ligweig Srucural Seel Panels, Proceedings of e Laser Maerials Processing Conference, ICALEO 93, Orlando. Roland, F., & Mesckow, B., Laser Sandwic Panels for Sipuilding and Srucural Seel Engineering, Meer Werf, Papenurg, Romanoff, J, Kujala, P., Formulaions for e Sreng Analsis of All Seel Sandwic Panels, Helsinki Universi of Tecnolog, Sip Laoraor, Oaniemi, 00, M-66. Sandwic, p://sandwic.alpor.com/, Zenker, D., An Inroducion o Sandwic Consrucion, Cameleon Press LTD, London, Peni Kujala, Dr.Tec, Senior Researc Scienis Helsinki Universi of Tecnolog Sip Laoraor P.O.Box HUT Finland Tel Fax ANALYSIS TECHNOLOGIES