Deflection and strength of a sandwich beam with thin binding layers between faces and a core

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1 Arh. Meh., 65, 4, pp , Warszawa 2013 Defletion and strength of a sandwih beam with thin binding layers between faes and a ore K. MAGNUCKI, M. SMYCZYŃSKI, P. JASION Institute of Applied Mehanis Poznań University of Tehnology Jana Pawła II Poznań, Poland mikolaj.smyzynski@put.poznan.pl The subjet of the paper is the analysis of defletion of a five-layer sandwih beam under bending. The mehanial and physial properties vary through the thikness of the beam and depend on the material of eah layer, whih are: the metal fae, binding material (the glue) and metal foam ore. The main aim of the paper is to present the analytial model of the five-layer beam and to ompare the results of bending analysis obtained theoretially and numerially. In the paper, a mathematial model of the field of displaements, whih inludes a shear effet and a bending moment, is presented. The system of partial differential equations of equilibrium for the five-layer sandwih beam is derived on the basis of the priniple of stationary total potential energy. The equations are analytially solved and the formula desribing the defletion of the beam is obtained. The influene of the thikness and mehanial properties of the binding layer on the defletion of the beam under bending is analysed. The omparison of the results obtained in the analytial and numerial (FEM) analysis is shown in graphs and figures. Key words: sandwih struture, metal foam, multi-layered beam, bending. Copyright 2013 by IPPT PAN 1. Introdution Sandwih strutures with a metal foam ore are the subjet of present-day studies. These strutures are haraterized by the high impat and heat resistane, good aousti absorption, vibration s redution and easy assembly. The bases of the theory of sandwih strutures were desribed by Plantema 1] and Allen 2]. Vinson 3] provided a general harateristi of sandwih strutures pointing out theirs advantages. A great number of publiations onerning sandwih beam, plates and shells are ited. Banhart 4] desribed in detail the proesses of manufaturing of different types of ommerial metal foams. The author also presented the ways the destrutive and non-destrutive tests, in whih the ellular materials an be haraterized. The fields of engineering in whih the ellular metals an be used are disussed. Noor et al. 5] in their review work onsider different models of sandwih strutures. The researhes on-

2 302 K. Magnuki, M. Smyzyński, P. Jasion erning different types of problems, like vibrations, bukling or thermal stress are presented. Grigolyuk and Chulkov 6] provided the first hypothesis of the ross-setion deformation of sandwih strutures. Wang et al. 7] disussed the higher-order hypotheses whih inlude shearing of beams and plates. Carrera 8] formulated the zigzag hypotheses for multilayered plates. Iaarino et al. 9] desribed the effet of a thin soft ore on the bending behavior of a sandwih beams. Chakrabarti et al. 10] developed a new FE model based on higher-order zigzag theory for the stati analysis of laminated sandwih beams with a soft ore. Steeves et al. 11] and Qin et al. 12] presented analytial models of ollapse mehanisms of sandwih beams under transverse fore. Rakow and Wass 13] desribed the mehanial properties of aluminium foam under shear. Birman 14] presented modeling and analysis of funtionally graded materials. Magnuka-Blandzi and Magnuki 15] and Magnuki et al. 16] desribed strength and bukling problems of sandwih beams with a metal foam ore and effetive design of these strutures. Zenkert 17] investigated debondings in foam ore sandwih beams assuming that raks in the interfae between the fae and ore are present. This paper is devoted to the strength analysis of a simply-supported sandwih beam. The goal is to elaborate a mathematial model of the beam in whih the binding layer will be treated as a separate layer. This way the influene of the mehanial properties and the thikness of the glue an be investigated what is usually omitted when sandwih strutures are analyzed. For simpliity reasons, a lassial broken line hypothesis has been assumed to desribe the deformation of the ross-setion of the beam. Suh an approah allowed toobtain a formula with whih the defletion of the five-layer beam an be determined. The beam onsists of five layers: the upper and lower fae, the ore and the thin binding layers between the faes and the ore. The sheme of the beam is shown in Fig. 1. The beam has the length L, the width b and the depth H. The thikness of partiular layers that is the faes, the ore and the binding layers are denoted by t f, t and t b, respetively. The load has the form of a onentrated fore F loated in the mid-length of the beam. Fig. 1. Sheme of the loaded beam.

3 2. Analytial analysis Defletion and strength of a sandwih beam The deformation of the flat ross-setion of the five-layer beam is shown in Fig. 2. Fig. 2. Sheme of displaements the hypothesis for the beam. The field of displaements is formulated as follows: 1. The upper fae (1/2 + x 1 + x 2 ) ζ (1/2 + x 1 ) (2.1) u(x,ζ) = t ζ dw ] + ψ 1(x). 2. The upper binding layer (1/2 + x 1 ) ζ 1/2 (2.2) u(x,ζ) = t ζ dw + ψ 2(x) 1 ( ζ + 1 ) ] (ψ 1 (x) ψ 2 (x)). x The ore 1/2 ζ 1/2 (2.3) u(x,ζ) = t ζ ] dw 2ψ 2 (x). 4. The lower binding layer 1/2 ζ 1/2 + x 1 (2.4) u(x,ζ) = t ζ dw ψ 2 (x) 1 x 1 ( ζ The lower fae 1/2 + x 1 ζ 1/2 + x 1 + x 2 (2.5) u(x,ζ) = t ζ dw ] ψ 1 (x), ) ] (ψ 1 (x) ψ 2 (x)).

4 304 K. Magnuki, M. Smyzyński, P. Jasion where x 1 = t b /t, x 2 = t f /t, ζ = z/t, ψ 1 (x) = u 1 (x)/t, ψ 2 (x) = u 2 (x)/t. Strains of the layers of the beam are defined by the geometri relationships in the following form: 1. The upper fae (2.6) ε x = t ζ d2 w 2 + dψ ] 1 (x), γ xz = 0. (2.7) 2. The upper binding layer 2. The ore ε x = t ζ d2 w 2 + ψ 2 (x) 1 x 1 γ xz = 1 x 1 ψ 1 (x) ψ 2 (x)]. (2.8) ε x = t ζ (2.9) 4. The lower binding layer ( ζ + 1 ) ( dψ1 (x) dψ )] 2 (x), 2 d 2 ] w 2 2dψ 2 (x), γ xz = 2ψ 2 (x). ε x = t ζ d2 w 2 ψ 2 (x) 1 x 1 γ xz = 1 x 1 ψ 1 (x) ψ 2 (x)]. ( ζ 1 ) ( dψ1 (x) dψ )] 2 (x), 2 5. The lower fae (2.10) ε x = t ζ d2 w 2 dψ ] 1 (x), γ xz = 0. The physial relationships for individual layers aording to Hooke s law are (2.11) σ x = Eε x, τ xz = Gγ xz. The bending moment of any ross-setion of the beam is (2.12) M b (x) = A σ x zda = bt 3 x ] 1 E f 1f + E b 6 (3 + 4x dψ1 1) {( 2E f 2f + 2E b 2b + 1 ) d 2 12 E w E E bx 1 (3 + 2x 1 ) ] dψ2 },

5 Defletion and strength of a sandwih beam where 1b = x 1 (1 + x 1 ), 2b = 1 12 x 1 ( 3 + 6x1 + 4x 2 1), 1f = x 2 (1 + 2x 1 + x 2 ), 2f = 1 12 x 2 12x1 (1 + x 1 + x 2 ) x 2 + 4x 2 ] 2, and E f, E, E b are: Young s modulus of the faes (E f ), Young s modulus of the ore (E ), Young s modulus of the glue layers (E b ). The transverse fore of any ross-setion of the beam is (2.13) Q(x) = τ xz da = 2bt G b ψ 1 (x) + (G G b ) ψ 2 (x)], A where G = E 2(1 + ν ), G b = E b 2(1 + ν b ) Equations of equilibrium The potential energy of the elasti strain of the beam is (2.14) U ε = 1 2 V (ε x σ x + γ xz τ xz )dv = 1 2 bt (f Ef + f Eb + f E ), L 0 where ( d f Ef = 2E f t 2 2 w 2f 2 ( d f Eb = 2E b t 2 2 w 2b ) 2 1f d 2 ( ) w dψ 2 ] x dψ1 2, ) dψ 1 x dψ 2 x 1 + ( 1b d 2 w 2x dψ 1 x )( dψ 2 dψ1 x ( dψ1 4x 1 (1 + 2x 1) dψ ) 2 ] 2 (1 + 2x 1) dψ 2 ) f E = 1 12 E t x 1 G b ψ 1 (x) ψ 2 (x)] 2, d 2 ] 2 w 2 2dψ 2 + 4G ψ 2 2(x).

6 306 K. Magnuki, M. Smyzyński, P. Jasion The work of the external load is (2.15) W = L 0 qw. The system of three differential equations obtained from the priniple of stationary total potential energy δ(u ε W) = 0 has the following form: (2.16) δw) bt 3 (2.17) δψ 1 ) (2.18) δψ 2 ) {( 2E f 2f + 2E b 2b + 1 ) d 4 12 E w 4 E f 1f + 16 ( )] d 3 E ψ 1 bx x } 6 E + E b x 1 (3 + 2x 1 )] d3 ψ 2 3 E f 1f + 1 ] d 3 ( 6 E w bx 1 (3 + 4x 1 ) 3 2 E f x ) d 2 3 E ψ 1 bx 1 2 = q F 0 d 2 w 2, 1 3 E d 2 ψ 2 bx G b x 1 t 2 ψ 1 (x) ψ 2 (x)] = 0, 1 6 E + E b x 1 (3 + 2x 1 )] d3 w E d 2 ψ 1 bx (E + 2E b x 1 ) d2 ψ 2 2 2G b x 1 t 2 ψ 1 (x) + 1 t 2 (4G + 2x1 G b ) ψ 2 (x) = 0. The first equation (2.16) of the system is equivalent to the bending moment (2.12). Therefore, for further analysis purpose, the system of three equations that is Eqs. (2.12), (2.17) and (2.18) is used Defletion of a beam The simply-supported sandwih beam is loaded with the fore F. The bending moment for this load ase is written in the form M b (x) = 0.5Fx. After simple transformations Eqs. (2.12), (2.17) and (2.18) an be redued to two equations of the following form: (2.19) ( ) a 2 d 2 ψ 1 12 a 11 a (a 12a 13 a 11 a 23 ) d2 ψ 2 2 2G b F + a 11 x 1 t 2 ψ 1 (x) ψ 2 (x)] = a 12 2bt 3,

7 Defletion and strength of a sandwih beam (2.20) (a 12 a 31 a 11 a 32 ) d2 ψ (a 13a 31 a 11 a 33 ) d2 ψ 2 2 where: G F + 4a 11 t 2 ψ 2 (x) = a 31 2bt 3, a 11 = 2E f 2f + 2E b 2b E, a 12 = E f 1f E bx 1 (3 + 4x 1 ), a 13 = 1 6 E + E b x 1 (3 + 2x 1 )], a 22 = 2(E f x E bx 1 ), a 23 = 1 3 E bx 1, a 31 = E f 1f + E b x 1 (1 + x 1 ) E, a 33 = 1 3 (E + 3E b x 1 ). The system of equations is approximately solved by means of the Bubnov Galerkin method. The three unknown funtions, ψ 1, ψ 2 and w, are assumed in the form of Fourier series (2.21) ψ 1 (x) = ψ 11 os πx L + ψ 13 os 3πx L + + ψ 1k os kπx L, ψ 2 (x) = ψ 21 os πx L + ψ 23 os 3πx L + + ψ 2k os kπx L, w(x) = w 1 sin πx L + w 3 sin 3πx L + + w k sin kπx L, k = 1,3,5,.... As a result of the orthogonalization proess the formula desribing the maximum defletion of the five-layer beam is obtained in the following form: (2.22) w = 2FL3 π 4 D, where: k 2 a 11 D = 1 bt 3, α k = a 12b 22 a 31b 12, + a k 2 12 α k + a 13 β k b 11 b 22 b 12 b 21 β k = a 31b 11 a 12 b 21 b 11 b 22 b 12 b 21, b 11 = k 2 11 a 11 2 π 2 G b b 12 = k a 11 2 π 2 G b x 1 b 22 = k 2 22 a 11 4 π 2G ( ) L 2, b 21 = k 2 21, t ( L t ) 2, 11 = a 2 12 a 11 a 22, x 1 ( ) L 2, 12 = a 12 a 13 a 11 a 23, 21 = a 12 a 31 a 11 a 32, 22 = a 13 a 31 a 11 a 33. t

8 308 K. Magnuki, M. Smyzyński, P. Jasion Example alulations have been performed for the beam with the following dimensions: L = 100 mm, H = 20 mm, b = 50 mm, t f = 1 mm. The material properties were: E f = MPa, E = 1200 MPa and ν = ν b = 0.3. Different values of E b and t b have been onsidered aording to Table 1, in whih the values of defletion determined with the use of equation (2.22) are presented. The beam was loaded with the fore F = 1 kn. Table 1. Defletions of a beam w for k = 3. E b MPa] t b mm] Numerial analysis The finite element model of the beam has been built using the ABAQUS ode. For modelling of the ore and of the binding layers 3D brik elements with eight nodes have been used. The faes of the beam have been modelled with the use of four-node thin shell elements. The tie onditions have been applied between the layers. Beause of the symmetry of the problem only a quarter of the beam has been used with proper boundary onditions in the symmetry planes. To obtain the boundary onditions orresponding to the ones assumed in the analytial model all layers have been joined with a rigid plate at the edge of the beam. Similar solution has been applied in the mid-length of the beam. Here, the rigid plate distributes the applied fore equally to all layers whih prevents from loal deformations. The FE model of the beam as well as an example of deformation is shown in Fig. 3. The stati analysis has been performed in whih the defletion in the midlength of the beam has been measured. The dimensions of the beam and the material properties were the same as in the analytial alulations. The omparison of the results obtained analytially and numerially (FEM) is shown in Fig. 4. The differene between results obtained numerially and analytially for k = 1 is about 10 15%. However, the differene is muh less (2 2.5%) for k = 3.

9 Defletion and strength of a sandwih beam Fig. 3. Numerial model of the beam. Fig. 4. The omparison of the results obtained analytially and numerially. 4. Conlusions In the paper, a mathematial model of a five-layer beam was presented. The faes were glued to the ore with thin binding layers. The glue was treated as a separate layer. The influene of the thikness and the material properties of the binding layer on the defletion of the beam under bending was analyzed. The results obtained from the FEM analysis have been ompared with those given by the analytial model proposed in the paper. A good agreement an be seen between these two approahes the disrepany was 2 2.5% at the most. From the results given in Table 1 it an be observed that for high values of E b the thikness of the binding layers does not influene onsiderably the defletion of the beam. Similarly, the thinner the binding layer the smaller its influene on the stiffness of the beam.

10 310 K. Magnuki, M. Smyzyński, P. Jasion Aknowledgements The studies were supported by the Ministry of Sienes and Higher Eduation in Poland Grant No. DS-MK /2011. Referenes 1. F.J. Plantema, Sandwih onstrution, John Wiley&Sons: New York, London, Sydney, H.G. Allen, Analysis and design of strutural sandwih panels, Pergamon Press: Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunshweig, J.R. Vinson, Sandwih strutures, Applied Mehanis Reviews, ASME, 54, 3, , J. Banhart, Manufature, haraterisation and appliation of ellular metals and metal foams, Progress in Material Siene, 46, , A.K. Noor, W.S. Burton, C. W. Bert, Computational models for sandwih panels and shells, Applied Mehanis Reviews, ASME, 49, 3, , E.I. Grigolyuk, P.P. Chulkov, Stability and vibrations of three layers shells, Mosow, Mashinostroene (in Russian), C.M. Wang, J.N. Reddy, K.H. Lee, Shear deformable beams and plates, Elsevier: Amsterdam, Laussane, New York, Oxford, Shannon, Singapore, Tokyo, E. Carrera, Historial review of Zig-Zag theories for multilayered plates and shells, Applied Mehanis Reviews, 56, 3, , P. Iaarino, C. Leone, M. Durante, G. Caprino, A. Lamboglia, Effet of a thin soft ore on the bending behavior of a sandwih with thik CFRP faings, Journal of Sandwih Strutures and Materials, 13, 2, , A. Chakrabarti, H.D. Chalak, Mohd. Ashraf Iqbal, Abdul Hamid Sheikh, A new FE model based on higher order zigzag theory for the analysis of laminated sandwih beam with soft ore, Composite Strutures, 93, 2, , C.A. Steeves, N.A. Flek, Collapse mehanisms of sandwih beams with omposite faes and a foam ore, loaded in three-point bending. Part I: analytial models and minimum weight design, International Journal of Mehanial Sienes, 46, 4, , Q.H. Qin, T.J. Wang, An analytial solution for the large defletions of a slender sandwih beam with a metalli foam ore under transverse loading by a flat punh, Composite Strutures, 88, 4, , J.F. Rakow, A.M. Waas, Size effets and the shear response of aluminium foam, Mehanis of Materials, 37, 1, 69 82, V. Birman, L.W. Byrd, Modeling and analysis of funtionally graded materials and strutures, Applied Mehanis Reviews, 60, 5, , E. Magnuka-Blandzi, K. Magnuki, Effetive design of a sandwih beam with a metal foam ore, Thin-Walled Strutures, 45, 4, , 2007.

11 Defletion and strength of a sandwih beam K. Magnuki, P. Jasion, W. Szy, M. Smyzynski, Strength and bukling of a sandwih beam with thin binding layers between faes and a metal foam ore, The 2011 World Congress on Advanes in Strutural Engineering and Mehanis, Seoul, Korea 2011, , D. Zenkert, Strength of sandwih beams with interfae debondings, Composite Strutures, 17, 4, , Reeived Deember 4, 2012; revised version May 3, 2013.