Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 1, pp. 41-53, Warsaw 2017 DOI: 10.15632/jtam-pl.55.1.41 BENDING AND BUCKLING OF A METAL SEVEN-LAYER BEAM WITH A LENGTHWISE CORRUGATED MAIN CORE – COMPARATIVE ANALYSIS WITH THE SANDWICH BEAM Ewa Magnucka-Blandzi Poznan University of Technology, Institute of Mathematics, Poznań, Poland e-mail: ewa.magnucka-blandzi@put.poznan.pl Marcin Rodak Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland e-mail: marcin.rodak@put.poznan.pl The subject of analytical investigations is a metal seven-layer beam, a plate band with a lengthwise trapezoidal corrugatedmain core and two crosswise trapezoidal corrugated cores of faces. The hypothesis of deformation of normal to the middle surface of the beam after bending is formulated. Equations of equilibrium are derived based on the theorem of mini- mum total potential energy.The equations are analytically solved.Three point bending and buckling for axially compressionof the simply supportedbeamare theoretically studied.The deflection and the critical axial force are determined for different values of the trapezoidal corrugationpitch of themain core.Moreover, an adequatemodel of the sandwichbeamwith steel foam core is formulated. The deflection and the critical axial force are determined for this sandwich beam.The results studied of the seven layer beamand the adequate sandwich beam are compared and presented in tables and figures. Keywords: layered plate-band, trapezoidal corrugated cores, deflection, critical load 1. Introduction The primary scientific description referring to the analysis and design of sandwich structures is the monograph by Allen (1969). A review of problems related to modelling and calculations of sandwich structures was presented by Noor et al. (1996), Vinson (2001) and Carrera and Bri- schetto (2009). A developed and analytical model of corrugated composite cores was described by Kazemahvazi and Zenkert (2009). The quasi-isotropic bending response of sandwich plates with bi-directionally corrugated cores was presented by Seong et al. (2010). The mathematical modelling of a rectangular sandwichplate under in plane compression is describedbyMagnucka- Blandzi (2011). The theoretical study of transverse shear modulus of elasticity for thin-walled corrugated cores of sandwich beams was presented by Magnucka-Blandzi andMagnucki (2014) and Lewinski et al. (2015). The problem of an equivalent plate model for corrugated-core san- dwich panels was presented by Cheon andKim (2015). The subject of thepaper is themetal seven-layer beam–aplate band.Thebeam is composed of a lengthwise trapezoidal corrugatedmain core, two inner flat sheets, two crosswise trapezoidal corrugated cores of the faces and two outer flat sheets. 42 E.Magnucka-Blandzi,M. Rodak 2. Theoretical model of the seven-layer beam with the lengthwise corrugated main core The seven-layer simply supported beam of length L, width b, thicknesses of the main core tc1, facing cores tc2 and flat sheets ts is shown in Fig. 1. Fig. 1. Scheme of the seven-layer beamwith the lengthwise corrugatedmain core Thedirections of corrugations of themain core and the face cores are orthogonal. Trapezoidal corrugations of the main core and facing cores are shown in Fig. 2. The index i = 1 refers to the main core, while the index i =2 refers to the face cores. Total depth of the cores is tci and length of one pitch of the corrugation is b0i. Fig. 2. Scheme of trapezoidal corrugations of the main core (i =1) or face cores (i =2) Taking into account the layered structures of the beam, the hypothesis of the broken line (Fig. 3) is assumed. The plane cross-section before bending does not remain plane and normal after bending. The hypothesis for multi-layer structures was described in details by Carrera (2003), Magnucka-Blandzi (2012) andMagnucki et al. (2016). The displacements with consideration of this hypothesis are as follows: — the upper sandwich facing for−(0.5+2x1+x2)¬ ζ ¬−0.5 u(x,y,z) =−tc1 [ ζ dw dx +ψ(x) ] (2.1) — the main corrugated core for−0.5¬ ζ ¬ 0.5 u(x,z) =−tc1ζ [dw dx −2ψ(x) ] (2.2) Bending and buckling of a metal seven-layer beam... 43 Fig. 3. Scheme of the hypothesis of the seven-layer beam —the lower sandwich facing for 0.5¬ ζ ¬ 0.5+2x1+x2 u(x,y,z) =−tc1 [ ζ dw dx −ψ(x) ] (2.3) where x1 = ts/tc1, x2 = tc2/tc1 are dimensionless parameters, ζ = z/tc1 – dimensionless coordi- nate, ψ(x) = u1(x)/tc1 – dimensionless functions of displacements, u1(x) – displacement in the x direction and w(x) – deflection (Fig. 3). Thus, linear relations for the strains are as follows: — the main corrugated core εx =−tc1ζ (d2w dx2 −2 dψ dx ) γxz =2ψ(x) (2.4) — the upper/lower sandwich facings εx =−tc1 ( ζ d2w dx2 ± dψ dx ) γxz =0 (2.5) The sign “+” refers to the upper facing (u), and the sign “−” refers to the lower facing (l). Strains (2.4) and (2.5) and Hook’s law make a basis for the formulation of elastic strain energy of the seven-layer beam. 3. The equations of equilibrium of the seven-layer beam The elastic strain energy of the beam is a sum of the energy of particular layers U(beam)ε = U (c−1) ε +U (s−i) ε +U (c−2) ε +U (s−o) ε (3.1) 44 E.Magnucka-Blandzi,M. Rodak The addends are as follows: ➢ energy of the main corrugated core U(c−1)ε = 1 2 Esbtc1 L∫ 0 1 2∫ − 1 2 [Ẽ(c−1)x ε 2 x + G̃ (c−1) xz γ 2 xz] dζ dx (3.2) whereEs isYoung’smodulus, dimensionless longitudinal elasticmodulus of themain corrugated core is calculated based on the monograph of Ventsel and Krauthammer (2001) Ẽ(c−1)x = xb1 2(xf1xb1+ s̃a1) x301 (3.3) dimensionless shear elastic modulus of themain trapezoidal corrugated core based on the paper of Lewinski et al. (2015) G̃(c−1)xz = 1−x01 4(1−ν2)xb1fu (x01 s̃a1 )3 (3.4) and dimensionless parameters x01 = t01 tc1 xf1 = bf1 b01 xb1 = b01 tc1 s̃a1 = √ (1−x01)2+x 2 b1 (1 2 −xf1 )2 (3.5) Substituting expressions (2.4) for strains into expression (3.2) and after integration, the elastic energy of the main corrugated core is obtained in the following form U(c−1)ε = Esbt 3 c1 L∫ 0 { 1 24 Ẽ(c−1)x [(d2w dx2 )2 −4 d2w dx2 dψ dx +4 (dψ dx )2] +2G̃(c−1)xz (ψ(x) tc1 )2} dx (3.6) ➢ energy of the inner sheets U(s−i)ε = 1 2 Esbtc1 L∫ 0    − 1 2∫ − ( 1 2 +x1 ) ε2x,up dζ + 1 2 +x1∫ 1 2 ε2x,low dζ    dx (3.7) Substitution of expressions (2.5) for the strains with regard to the upper/lower facings and after integration provides U(s−i)ε = Esbt 3 c1 L∫ 0 [ 1 12 x1(3+6x1+4x 2 1) (d2w dx2 )2 −x1(1+x1) d2w dx2 dψ dx +x1 (dψ dx )2] dx (3.8) ➢ energy of the corrugated cores of the facings U(c−2)ε = 1 2 Es b b02 L∫ 0 [ ∫ A (c−2) TR ε2x,up dA (c−2) TR + ∫ A (c−2) TR ε2x,low dA (c−2) TR ] (3.9) Bending and buckling of a metal seven-layer beam... 45 where the area of the trapezoid A (c−2) TR =2t 2 c2x02(xf2xb2+ s̃a2) (3.10) and dimensionless parameters x02 = t02 tc2 xf2 = bf2 b02 xb2 = b02 tc2 s̃a2 = √ (1−x02)2+x 2 b2 (1 2 −xf2 )2 (3.11) Substituting expressions (2.5) for strains into expression (3.9) and after integration, the elastic energy of the corrugated cores of facings is obtained in the following form U(c−2)ε = Esbt 3 c1 x2x02 xb2 L∫ 0 [ C(c−2)ww (d2w dx2 )2 −C (c−2) wψ d2w dx2 dψ dx +C (c−2) ψψ (dψ dx )2] dx (3.12) where dimensionless parameters are as follows C(c−2)ww = 1 2 [1 3 x22(1−x02) 2(3xf2xb2+ s̃a2)+(1+2x1+x2) 2(xf2xb2+ s̃a2) ] C (c−2) wψ =2(1+2x1+x2)(xf2xb2+ s̃a2) C (c−2) ψψ =2(xf2xb2+ s̃a2) ➢ energy of the outer sheets U(s−o)ε = 1 2 Esbtc1 L∫ 0    − ( 1 2 +x1+x2 ) ∫ − ( 1 2 +2x1+x2 ) ε2x,up dζ + 1 2 +2x1+x2∫ 1 2 +x1+x2 ε2x,low dζ    dx (3.13) Substitution of expressions (2.5) for the strains with regard to the upper/lower facings and after integration gives U(s−o)ε = Esbt 3 c1 L∫ 0 [ C(s−o)ww (d2w dx2 )2 −x1(1+3x1+2x2) d2w dx2 dψ dx +x1 (dψ dx )2] dx (3.14) where the dimensionless parameter C (s−o) ww =(1/12)x1[28x 2 1+3(1+2x2)(1+6x1+2x2)]. Therefore, the elastic strain energy of the inner and outer sheets is as follows U(s)ε = U (s−i) ε +U (s−o) ε = Esbt 3 c1 L∫ 0 [ C(s)ww (d2w dx2 )2 −C (s) wψ d2w dx2 dψ dx +2x1 (dψ dx )2] dx (3.15) where dimensionless parameters C(s)ww = 1 6 x1[16x 2 1+6x1(2+3x2)+3(1+2x2+2x 2 2)] C (s) wψ =2x1(1+2x1+x2) Thus, the elastic strain energy of the seven-layer beam (6) is in the following form U(beam)ε = Esbt 3 c1 L∫ 0 [1 2 Cww (d2w dx2 )2 −Cwψ d2w dx2 dψ dx + 1 2 Cψψ (dψ dx )2 +2G̃(c−1)xz (ψ(x) tc1 )2] dx (3.16) 46 E.Magnucka-Blandzi,M. Rodak where dimensionless parameters Cww = 1 12 Ẽ(c−1)x +2 x2x02 xb2 C(c−2)ww +2C (s) ww Cwψ = 1 6 Ẽ(c−1)x + x2x02 xb2 C (c−2) wψ +C (s) wψ Cwψ = 1 3 Ẽ(c−1)x +2 x2x02 xb2 C (c−2) wψ +2C (s) wψ The work of the load W = L∫ 0 [ qw(x)+ 1 2 F0 (dw dx )2] dx (3.17) where q is the intensity of the transverse load, F0 – axial compressive force of the beam. The system of the equations of equilibrium – two ordinary differential equations derived based on the theorem ofminimumpotential energy δ(U (beam) ε −W)= 0, is in the following form Cww d4w dx4 −Cwψ d3ψ dx3 = 1 Ebt3c1 ( q−F0 d2w dx2 ) Cwψ d3w dx3 −Cψψ d2ψ dx2 +4G̃(c−1)xz ψ(x) t2c1 =0 (3.18) The bendingmoment of the seven-layer beam Mb(x)= ∫ A zσx dA =−Esbt 3 c1 ( Cww d2w dx2 −Cwψ dψ dx ) (3.19) Integration is analogical as in the case of the elastic strain energy, from which the following equation is obtained Cww d2w dx2 −Cwψ dψ dx =− Mb(x) Esbt 3 c1 (3.20) Equations (3.18)1 and (3.20) are equivalent, therefore, bending and buckling analysis of the seven-layer beam is based on the system of two differential equations (3.18)2 and (3.20). 4. Deflection of the seven-layer beam under three-point bending Three-point bending of the seven-layer beam of length L is shown in Fig. 4. Fig. 4. Scheme of the three-point bending of the beam The system of two differential equations (3.18)2 and (3.20) is reduced to one differential equation in the following form d2ψ dx2 − ( k tc1 )2 ψ(x)=−Cq Q(x) Esbt 3 c1 (4.1) Bending and buckling of a metal seven-layer beam... 47 where Q(x)= dMb/dx is the shear force, k, Cq – dimensionless parameters k =2 √√√√ CwwG̃ (c−1) xz CwwCψψ −C 2 wψ Cq = Cwψ CwwCψψ −C 2 wψ The general solution to equation (4.1) is in the form ψ(x) = C1 sinh ( k x tc1 ) +C2cosh ( k x tc1 ) +ψp(x) (4.2) where C1,C2 are integration constants, ψp(x) – particular solution. The shear force in thehalf beam(Fig. 4) isQ(x)= F1/2, for 0¬x ¬ L/2, then theparticular solution ψp = Cwψ 8CwwG̃ (c−1) xz F1 Esbtc1 (4.3) Taking into account theboundaryconditions for thehalf beam(dψ/dx)|x=0 =0andψ(L/2)= 0, the integration constants C1 = 0 and C2 = −cosh −1[kL/(2tc1)]ψ0 are determined, hence, the function of displacement (4.3) is in the following form ψ(x) = ( 1− cosh kx tc1 cosh kL 2tc1 ) ψp (4.4) Substituting this function, and the bending moment Mb(x) = F1x/2, for 0 ¬ x ¬ L/2 to equation (3.20), one obtains w(x) = C4+C3x+ Cwψ Cww ( x− tc1 k sinh kx tc1 cosh kL 2tc1 ) ψp − F1 12CwwEsbt 3 c1 x3 (4.5) Taking intoaccount theboundaryconditions for thehalf beam w(0)= 0and (dw/dx)|x=L/2 =0, the integration constants C3 =F1L 2/(16CwwEsbt 3 c1) and C4 =0 are determined. Themaximum deflection – the deflection for the middle of the beam is w(7−lay)max = w (L 2 ) = [ 1+3 ( 1− 2tc1 kL tanh kL 2tc1 ) C2wψ CwwG̃ (c−1) xz (tc1 L )2] F1 48CwwEsb ( L tc1 )3 (4.6) 5. Critical load of the seven-layer beam subjected to axial compression The axial compression of the simply supported seven-layer beam is shown in Fig. 5. Fig. 5. Scheme of the simply supported seven-layer beamwith the axial force F0 The system of two differential equations (3.18)2 and (3.20) is reduced to one differential equation in the following form (CwwCψψ −C 2 wψ) d4w dx4 − 4 t2c1 G̃(c−1)xz Cww d2w dx2 = [ 4 t2c1 G̃(c−1)xz Mb(x)−Cψψ d2Mb dx2 ] 1 Esbt 3 c1 (5.1) where the bendingmoment Mb(x)= F0w(x) (Fig. 5). 48 E.Magnucka-Blandzi,M. Rodak Differential equation (5.1)with oneunknown functionw(x) is approximately solved assuming this function in the form w(x) = wa sin πx L (5.2) where wa is the parameter of the function, L – length of the beam. Substituting this function into the equation (5.1) the critical force is obtained F (7−lay) 0,CR = ( Cww − C2wψ α1 )π2Esbt3c1 L2 (5.3) where α1 = Cψψ + ( 2L πtc1 )2 G̃(c−1)xz 6. Equivalent sandwich beam Comparative analysis is carried out for the classical sandwich beam (Fig. 6) equivalent to the seven-layer beam (Fig. 1). This classical sandwich beam consists of two steel faces of thickness tf = ts and the steel foam core of thickness tc = tc1+2(ts+tc2). Its sizes andmass are identical to the seven-layer beam. Fig. 6. Scheme of the sandwich (three-layer) beam equivalent to the seven-layer beam Themass of the metal foam core of this sandwich beam (three-layer beam) m(3−lay)c = [1+2(x1+x2)]tc1bLρc (6.1) where ρc is the mass density of the metal foam core. However, mass of the material (steel with mass density ρs) located between the two outer sheets of the seven-layer beam (Fig. 1) is a sum of the mass of particular layers m(7−lay)c = m (c−1) c +2m (s−i) c +2m (c−2) c (6.2) where the mass of the main corrugated core m(c−1)c = A (c−1) TR b01 bLρs (6.3) Substituting the expression for the area of the trapezoid A (c−1) TR = 2t 2 c1x01(xf1xb1 + s̃a1) to the above expression with consideration of the dimensionless parameters (3.5) one obtains m(c−1)c =2x01 ( xf1+ s̃a1 xb1 ) tc1bLρs (6.4) Bending and buckling of a metal seven-layer beam... 49 and m(s−i)c = tsbLρs = x1tc1bLρs m (c−2) c = A (c−2) TR b02 bLρs (6.5) wherem (s−i) c is themass of the inner sheets,m (c−2) c –mass of the corrugated cores of the facings. Substituting the expression for the area of trapezoid (3.10) with dimensionless parameters (3.11), one obtains m(c−2)c =2x2x02 ( xf2+ s̃a2 xb2 ) tc1bLρs (6.6) Thus, mass (6.2) is in the following form m(7−lay)c =2 [ x01 ( xf1+ s̃a1 xb1 ) +x1+x2x02 ( xf2+ s̃a2 xb2 )] tc1bLρs (6.7) Then, from the equivalence condition m (3−lay) c = m (7−lay) c (Eqs. (6.1) and (6.7)) of these two beams, the proportion of mass densities of the metal foam core to steel is obtained ρ̃c = ρc ρs = [ x01 ( xf1+ s̃a1 xb1 ) +x1+x2x02 ( xf2+ s̃a2 xb2 )] 2 1+2(x1+x2) (6.8) Taking into account the experimental results related to themechanical properties ofmetal foams presented in details by Ashby et al. (2000), Smith et al. (2012) and Szyniszewski et al. (2014), the relationship for Young’s moduli and mass densities of the metal foams and the reference material (steel) is as follows Ẽc = Ec Es = 3 4 (ρc ρs )2 (6.9) where Ec and Es are Young’s moduli of the metal foam and the steel. 7. Bending and buckling of the equivalent sandwich beam The hypothesis of deformation of the plane cross-section after bending of the sandwich (three- -layer) beam is assumed as the broken line (Fig. 7). The detailed description of this hypo- thesis and derivation of the equations of equilibrium for the sandwich beam was presented by Magnucka-Blandzi (2012). The displacements with consideration of this hypothesis are as follows: — the upper/lower facing for−(0.5+x0)¬ ζ ¬−0.5 and 0.5¬ ζ ¬ 0.5+x0 u(x,z) =−tc [ ζ dw dx ±ψ0(x) ] (7.1) — the metal foam core for−0.5¬ ζ ¬ 0.5 u(x,z) =−tcζ [dw dx −2ψ0(x) ] (7.2) where x0 = tf/tc is the dimensionless parameter, ζ = z/tc – dimensionless coordinate, ψ0(x) = uf(x)/tc – dimensionless functions of displacements, uf(x) – displacement in the x di- rection and w(x) – deflection (Fig. 7). 50 E.Magnucka-Blandzi,M. Rodak Fig. 7. Scheme of the hypothesis of the sandwich (three-layer) beam Continuation of the procedure similar to the one applied to the seven-layer beam gives a system of two differential equations of equilibrium for the classical sandwich beam presented by Magnucka (2012) in the following form Bww d2w dx2 −Bwψ dψ0 dx =− Mb(x) Esbt3c Bwψ d3w dx3 −Bψψ d2ψ0 dx2 +4G̃c ψ0(x) t2c =0 (7.3) where dimensionless parameters Bww =2C2f + 1 12 Ẽc Bwψ = C1f + 1 6 Ẽc Bψψ =2x0+ 1 3 Ẽc C1f =(1+x0)x0 C2f = 1 12 (3+6x0+4x 2 0)x0 andmoduli Ẽc = Ec Es G̃c = Ẽc 2(1+νc) This system of equations is analogical to the one of the seven-layer beam, (3.20) and (3.18)2. Then, themaximumdeflection and the critical force of the sandwich equivalent beam are as follows w(3−lay)max = w (L 2 ) = [ 1+3 ( 1− 2tc k0L tanh k0L 2tc ) B2wψ BwwG̃c (tc L )2] F1 48BwwEsb (L tc )3 (7.4) and F (3−lay) 0,CR = ( Bww − B2wψ α0 )π2Esbt3c L2 (7.5) Bending and buckling of a metal seven-layer beam... 51 where α0 = Bψψ + (2L πtc )2 G̃c 8. Illustrative detailed analysis for selected beams Adetailed analysis for an examplary steel seven-layer beamand the equivalent sandwichbeam is carried out for the following test data: L =1620mm, b =240mm, ts =0.8mm, tc1 =32.0mm, t01 = 0.8mm, bf1 = 10.0mm, b01 = [32.4,36.0,40.5,45.0]mm, tc2 = 16.0mm, t02 = 0.8mm, bf2 = 8.0mm, b02 = 40.0mm and material-steel constants Es = 2 · 10 5MPa, ν = 0.3, ρs =7850kgm −3. Moreover, tf = ts =0.8mm and tc = tc1+2(ts + tc2)= 65.6mm. The values of maximum deflections (4.6) and critical forces (5.3) of the seven-layer beam are specified in Table 1. The values of maximum deflections (7.4) and critical forces (7.5) of the sandwich (three-layer) beam are specified in Table 2. Table 1.Maximum deflections and critical forces of the seven-layer beam b01 [mm] 32.4 36.0 40.5 45.0 w (7−lay) max [mm] 3.49 3.18 2.98 2.88 F (7−lay) 0,CR [kN] 490.1 535.8 568.5 587.3 Table 2.Maximum deflections and critical forces of the sandwich beam b01 [mm] 32.4 36.0 40.5 45.0 ρ̃c Eq. (6.8) 0.0892374 0.0863605 0.0835631 0.0814007 Ẽc Eq. (6.9) 0.005972 0.005594 0.005237 0.004970 w (3−lay) max [mm] 5.13 5.16 5.21 5.24 F (3−lay) 0,CR [kN] 328.1 325.6 323.2 321.3 Moreover, the values ofmaximumdeflections and critical forces of the seven-layer beam and the equivalent sandwich beam are presented in Figs. 8 and 9. Fig. 8. Maximum deflections of the two beams 52 E.Magnucka-Blandzi,M. Rodak Fig. 9. Critical forces of the two beams 9. Conclusions The analytical modelling of the seven-layer beamwith a lengthwise trapezoidal corrugatedmain core and two crosswise trapezoidal corrugated cores of faces leads to the conclusions: • hypotheses of the flat cross-sections deformations of these two beams as the broken line are analogous, • equations of equilibrium of these two beams are similar, • proportion of the maximum deflections of these two beams for the studied family of the beams is w (3−lay) max /w (7−lay) max =1.47-1.82, • proportion of the critical force of these two beams for the studied family of the beams is F (7−lay) 0,CR /F (3−lay) 0,CR =1.49-1.83, • stiffness of the seven-layer beam is decidedly greater than that of the equivalent classical sandwich (three-layer) beam. Acknowledgements The project was funded by theNational Science Centre allocated on the basis of the decision number DEC-2013/09/B/ST8/00170. References 1. Allen H.G., 1969,Analysis and Design of Structural Sandwich Panels, PergamonPress, Oxford, London, Edinburgh, NewYork, Toronto, Sydney, Paris, Braunschweig 2. Ashby M.F., Evans A., Fleck N.A., Gibson L.J., Hutchinson J.W., Wadley H.N.G., 2000,Metal Foams, A Design Guide, Butterworth-Heinemann, An Imprint of Elsevier 3. CarreraE., 2003,Historical reviewofZig-Zag theories formulti-layeredplates and shells,Applied Mechanics Reviews, 56, 3, 287-308 4. Carrera E., Brischetto S., 2009, A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates,Applied Mechanics Reviews, 62, 1, 010803 5. Cheon Y.-J., Kim H.-G., 2015, An equivalent plate model for corrugated-core sandwich panels, Journal of Mechanical Science and Technology, 29, 3, 1217-1223 6. Kazemahvazi S., Zenkert D., 2009, Corrugated all-composite sandwich structures. Part 1:Mo- deling,Composite Science and Technology, 69, 7/8, 913-919 Bending and buckling of a metal seven-layer beam... 53 7. Lewinski J., Magnucka-Blandzi E., Szyc W., 2015, Determination of shear modulus of ela- sticity for thin-walled trapezoidal corrugated cores of seven-layer sandwich plates, Engineering Transactions, 63, 4, 421-437 8. Magnucka-Blandzi E., 2011, Mathematical modelling of a rectangular sandwich plate with a metal foam core, Journal of Theoretical and Applied Mechanics, 49, 2, 439-455 9. Magnucka-Blandzi E., 2012, Displacement models of sandwich structures (in Polish), [In:] Strength and Stability Sandwich Beams and Plates with Aluminium Foam Cores, K. Magnucki, W. Szyc (Eds.), Pub. House of PoznanUniversity of Technology, Poznan, 109-120 10. Magnucka-Blandzi E., Magnucki K., 2014, Transverse shear modulus of elasticity for thin- walled corrugated cores of sandwich beams, Theoretical study, Journal of Theoretical and Applied Mechanics, 52, 4, 971-980 11. Magnucka-Blandzi E.,MagnuckiK.,WittenbeckL., 2015,Mathematicalmodelling of she- aring effect for sandwich beamswith sinusoidal corrugated cores,Applied Mathematical Modelling, 39, 2796-2808 12. MagnuckiK.,Magnucka-Blandzi E.,WittenbeckL., 2016,Elastic bending andbuckling of a steel composite beamwith corrugatedmain core and sandwich faces –Theoretical study,Applied Mathematical Modelling, 40, 1276-1286 13. Noor A.K., Burton W.S., Bert C.W., 1996, Computational models for sandwich panels and shells,Applied Mechanics Reviews, 49, 3, 155-199 14. Seong D.Y., Jung C.G., Yang D.Y., Moon K.J., Ahn D.G., 2010, Quasi-isotropic bending responses ofmetallic sandwich plates with bi-directionally corrugated cores,Materials and Design, 31, 6, 2804-2812 15. Smith B.H., Szyniszewski S., Hajjar J.F., Schafer B.W., Arwade S.R., 2012, Steel fo- am for structures: A review of applications, manufacturing and material properties, Journal of Constructional Steel Research, 71, 1-10 16. Szyniszewski S., SmithB.H., Hajjar J.F., SchaferB.W., Arwade S.R., 2014,Themecha- nical properties and modelling of a sintered hollow sphere steel foam, Materials and Design, 54, 1083-1094 17. VentselE.,KrauthammerT., 2001,ThinPlates andShells, Theory, Analysis andApplications, Marcel Dekker, NewYork, Basel 18. Vinson JR., 2001, Sandwich structures,Applied Mechanics Reviews, 54, 3, 201-214 Manuscript received February 11, 2016; accepted for print May 5, 2016