A Mathematical Model of a Five Layer Sandwich Beam

Transcription

1 Paper 4 A Mathematial Model o a Five ayer Sandwih Beam Civil-Comp Press 0 Proeedings o the Eleventh International Conerene on Computational Strutures Tehnology B.H.V. Topping (Editor) Civil-Comp Press Stirlingshire Sotland K. Magnuki M. Smyzynski and P. Jasion Institute o Applied Mehanis Poznan University o Tehnology Poland Astrat The sujet o this paper is an analysis o deletion o a ive layer sandwih eam. The mehanial and physial properties vary through the thikness o the eam and depend on the material o eah layer. Two aes o the eam are thin aluminium sheets and the ore is made o aluminium oam. Between the aes and the ore there are two thin inding glue layers. The main goal o the paper is to present a mathematial model o the ive layer eam and to ompare the results o the analyses otained analytially and numerially. Keywords: sandwih struture deletion metal oam mathematial modelling ive layer eam numerial model. Introdution Sandwih strutures with a metal oam ore are sujet o ontemporary studies. These strutures are haraterized y impat and heat resistane aousti and viration redution and easy assemly. Plantema [] and Allen [] desried the ases o the theory o sandwih strutures. Banhart [] delivered manuature haraterisation and appliation o ellular metals and metal oams. Noor et al. [4] and Vinson [5] presented strength and staility prolems o sandwih strutures. Wang et al. [] desried the strutural response o lamped sandwih eams with aluminium oam ore sujeted to impat loading. Grigolyuk and Chulkov [7] provided the irst hypothesis o ross setion deormations o sandwih strutures. Wang et al. [8] disussed the higher order hypotheses inluding shearing o eams and plates. He et al. [9] presented preise ending stress analysiso orrugated-ore sandwih panels. Carrera [0] ormulated the zig-zag hypotheses or multilayered plates. Iaarino et al. [] presented eet o a thin sot ore on the ending ehavior o a sandwih. Chakraarti et al. [] developed a new FE model ased on higher order zig-zag theory or the stati analysis o laminated sandwih eams with

2 a sot ore. Steeves et al. [] and Qin et al. [4] presented analytial models o ollapse mehanisms o sandwih eams under transverse ore. Rakow and Wass [5] presented mehanial properties o an aluminium oam under shear. Birman [] presented modeling and analysis o untionally graded materials. Magnuka- Blandzi and Magnuki [7] and Magnuki et al. [8] desried strength and ukling prolems o sandwih eams with a metal oam ore. Zenkert [9] presented strength o sandwih eams with deondings in the interae etween the ae and ore. A simply supported ive layer sandwih eam o the length and the width arries a onentrated ore F as shown in Figure. The ore F is loated in the middle o the eam. Figure. Sheme o the loaded eam. Analytial analysis The deormation o the lat ross setion o the ive layer eam is shown in Figure. Figure. Sheme o displaements the hypothesis or the eam. The ield o displaements is ormulated as ollows:. the upper ae ( ) ζ ( )

3 ζ dw d () ζ ( ) = t ζ u. the upper inding layer ( ) dw ( ) = t ζ ζ ( ) u ζ () d. the ore / ζ / dw u( ζ ) = tζ d () 4. the lower inding layer / ζ dw u t d 5. the lower ae / ζ / ( ζ ) = ζ ( ) ζ ( ( ) ( )) (4) dw u( ζ ) = t ζ d (5) = t / t = t / t ζ = z / t ( ) = u( ) / t ( ) = u / t. Strains o the layers o the eam are deined y the geometri relationship in the ollowing orm:. the upper ae ε = t ζ d d. the upper inding layer dw γ = 0 () d d dw ε = t ζ ζ d d d d γ = [ ] (7). the ore dw ε = tζ γ d 4. the lower inding layer d d d = (8) dw ε = t ζ ζ d d d d

4 5. the lower ae γ = [ ] (9) ε = t ζ d d dw γ = 0 (0) The physial relationships aording to Hooke s law or individual layers are σ = Eε τ = Gγ. () The ending moment o any ross setion o the eam dw M = σ da= t E E E A d () E E ( 4) E E ( ) d d = ( ) = ( ) ( ) = ( ) 4. 4 = The transverse ore o any ross setion o the eam E G = ( ν ) ( ) τ () A Q = da = t G G G G E = ( ν ).. Equations o equilirium The potential energy o the elasti strain o the eam is U ε = ( ε σ γ τ ) dv = t ( E E E )d (4) V 0 d w d w E = E t d d d d 4

5 E = E t d E d w d d d d 4 d w d d d ( ) ( ) G [ ] d d w = E 4G t d d. The work o the eternal load is d dw W = qwd F0 d 0 0 d. (5) The system o three partial dierential equations otained rom the priniple o stationary total potential energy δ ( U ε W ) = 0 ater integrating over the thikness o the eam and integrating y parts over the length o the eam takes the ollowing orm: δ w) 4 dw d t E E E E 4 E( 4) d d () d d w E E( ) = q F 0 d d δ ) dw d d E E ( 4) E E E d d d (7) G [ ] = 0 t dw d d ( ) ( ) δ ) E E E E E d d d G 4 G G = 0. t t (8) The irst equation () o the system is equivalent to the ending moment (). Thereore or urther analysis purpose the system o three Eqs. () (7) and (8) is applied.. Deletion o a eam The simply supported sandwih eam is loaded y ore F. The ending moment or this load ase is written in the orm M = F. Ater simply transormations o three equations () (7) and (8) two equations are otained 5

6 4 d d k 4 k = k0q d d (9) t Q ( ) d 0 = 0 t d t (0) B C D aa aa k = k = k0 = A= t A A A aa 0 aa a aa aa 0 G 0 G B = C = aa aa 0 0 a 0 at G D aa a aa aa = t = a 0 a aa aa aa aa a a G 0 = aa aa aa aa a a G a 0 = 4 G aa aa aa aa aa aa a a G = a = E E E aa aa aa aa a = E E ( 4 ) a = E E ( ) a = a a = E E a = E a = a a = a a = ( E E ). The two unknown untions are assumed orms o the irst word o the Fourier series π = os Q q π ( ) = os. () The equations (9) and (0) are approimately solved y means o the Bunov- Galerkin method. Ater simply transormations o this equations and equation () the deletion is otained t π F w = a a a t t 9t q 0 π π 0. () The eamples o deletions or the middle o the eam depending on Young modulus and thikness o the inding layers are shown in Tale. The parameters

7 o the eam are: t = mm H = 0 mm E = 500 MPa E = 00 MPa = 00 mm = 50 mm ν = ν = 0. F = kn. Tale. Deletions o a eam. Numerial analysis The numerial model FEM o the ive layer sandwih eam was uilt with use o D rik elements or the ore and two inding layers. The aes was modelled with the use o D shell elements. Between partiular layers the tie onditions have een imposed. Beause o the symmetry o a model only the quarter o the eam has een modelled. Figure shows a model and graphial visualisation o the distriution o the deletions o a eam. Figure. Numerial model o the eam. The stati analysis has een perormed and the deletions has een otained. The dimensions and the material properties were the same as in the analytial analysis. The omparison o the results otained in the analytial and numerial (FEM) analysis is shown in Figure 4. 7

8 Figure 4 The omparison o the results otained analytially and numerially The dierene etween results otained numerially and analytially is aout 0%. 4 Conlusions In this paper a mathematial model o a ive layer eam was presented. The aes are glued to the ore with thin inding layers. The glue is treated as a separate layer. The inluene o the inding layer thikness and properties on the deletion o the eam under ending was analysed. The results otained rom the FE analysis have een ompared with those given y the analytial model proposed in the paper. A poor agreement an e seen etween these two analyses (0%). Proaly it is eause o a too large approimation o the untions (). An inrease o degrees o reedom or these untions should redue the dierene to a ew perent. Aknowledgements The studies are supported y the Ministry o Sienes and Higher Eduation in Poland Grant No. DS-MK -88/0. Reerenes [] F.J. Plantema Sandwih onstrution New York ondon Sydney John Wiley&Sons 9. [] H.G. Allen Analysis and design o strutural sandwih panels Oord ondon Edinurgh New York Toronto Sydney Paris Braunshweig Pergamon Press 99. 8

9 [] J. Banhart Manuature haraterisation and appliation o ellular metals and metal oams Progress in Material Siene [4] A.K. Noor W.S. Burton C.W. Bert Computational models or sandwih panels and shells Applied Mehanis Reviews ASME [5] J. R. Vinson Sandwih strutures Applied Mehanis Reviews ASME [] Z. Wang. Jing J. Ning. Zhao The strutural response o lamped sandwih eams sujeted to impat loading Composite Strutures [7] E.I. Grigolyuk P.P. Chulkov Staility and virations o three layers shells Mosow Mashinostroene 97. [8] C.M. Wang J.N. Reddy K.H. ee Shear deormale eams and plates Amsterdam aussane New York Oord Shannon Singapore Tokyo Elsevier 000. [9]. He Y.-S. Cheng J. iu Preise ending stress analysis o orrugatedore honeyom-ore and X-ore sandwih panels Composite Strutures [0] E. Carrera Historial review o Zig-Zag theories or multilayered plates and shells Applied Mehanis Reviews [] P. Iaarino C. eone M. Durante G. Caprino A. amoglia Eet o a thin sot ore on the ending ehavior o a sandwih with thik CFRP aings Journal o Sandwih Strutures and Materials [] A. Chakraarti H.D. Chalak M.A. Iqal A.H. Sheikh A new FE model ased on higher order zigzag theory or the analysis o laminated sandwih eam with sot ore Composite Strutures [] C.A. Steeves N.A. Flek Collapse mehanisms o sandwih eams with omposite aes and a oam ore loaded in three-point ending. Part I: analytial models and minimum weight design International Journal o Mehanial Sienes 4(4) [4] Q.H. Qin T.J. Wang An analytial solution or the large deletions o a slender sandwih eam with a metalli oam ore under transverse loading y a lat punh Composite Strutures [5] J.F. Rakow A.M. Waas Size eets and the shear response o aluminium oam Mehanis o Materials 7() [] V. Birman.W. Byrd Modeling and analysis o untionally graded materials and strutures Applied Mehanis Reviews [7] E. Magnuka-Blandzi K. Magnuki Eetive design o a sandwih eam with a metal oam ore Thin-Walled Strutures [8] K. Magnuki P. Jasion W. Szy M. Smyzynski Strength and ukling o a sandwih eam with thin inding layers etween aes and a metal oam ore The 0 World Congress on Advanes in Strutural Engineering and Mehanis Seoul 8-85 Korea 0. [9] D. Zenkert Strength o sandwih eams with interae deondings Composite Strutures