JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 4, pp. 859-868, Warsaw 2004 ELASTIC BUCKLING OF A POROUS BEAM Krzysztof Magnucki Piotr Stasiewicz Institute of Applied Mechanics, Poznań University of Technology e-mail: krzysztof.magnucki@put.poznan.pl; piotr.stasiewicz@put.poznan.pl Thework deals with the problem a straight beamof a rectangular cross- section pivoted at both ends and loaded with a lengthwise compressive force. The beam is made of an isotropic porous material. Its proper- ties vary through thickness of the beam. The modulus of elasticity is minimal on the beam axis and assumesmaximum values at its top and bottom surfaces.The principle of stationarity of the total potential ener- gy enables one to define a system of differential equations that govern the beam stability. The system is analytically solved, which leads to an explicit expression for the critical load of the compressed beam. Results of the solution are verified on an example beam by means of the Finite ElementMethod (COSMOS). Key words: buckling, porous beam, shear deformable beam 1. Displacements of a porous beam Amathematical description of composite structures obviously includesma- ny simplifying assumptions. Librescu and Hause (2000) provided a review of sandwich structures, paying attention to their stability problems. Vinson (1999) discussed sandwich structures made of isotropic and composite ma- terials. Kołakowski and Kowal-Michalska (1999) presented some problems of stability of thin-walled composite structures. The abovementionedworks pro- vide descriptions of displacements (strains) in cross-sections that are based on the linearEuler-Bernoulli hypothesis.Thus, the effect of shearingdue to trans- verse forces is omitted. A separate group includes three-layered structures, in the which shearing is taken into account. Lok andCheng (2000) characterized propertiesof structures,with special attentionpaid to themiddle layer, subject 860 K.Magnucki, P.Stasiewicz mainly to the shearing. Magnucki and Ostwald (2001) presented problems of stability andoptimal shapingof three-layered structures.Displacements occur- ring in cross-sections of such structures (Lok, Cheng andMagnucki, Ostwald) were described using the broken-line hypothesis. Romanów (1995) assumed a hyperbolic pattern of the normal stress distribution in the cross-section of a three-layered wall. The works of Lok, Cheng and Magnucki, Ostwald and Romanów took the shearing effect into account as well.Wielgosz andThomas (2002) discussed the results of an analytical solution, taking into consideration the shearing effect and experimental studies related to panel bending. Bart- Smith et al. (2001) presented the problem of bending of a sandwich structure with the middle layer made of a cellular metal. This work is concerned with an isotropic porous beam of a rectangular cross-section pivoted at both ends and loaded with a lengthwise compressive force. Mechanical properties of the material vary through thickness of the beam. Young’s modulus is minimal on the beam axis and assumes maximum values at its top and bottom surfaces. For such a case, the of Euler-Bernoulli or Timoshenko beam theories do not correctly determine displacements of the cross-section of the beam. Wang et al. (2000) discussed in details the effect of non-dilatational strain of middle layers on bending of beams and plates subject to various load cases. Fig. 1. Scheme of porous beam Aporousbeam(Fig.1) is a generalized sandwichbeam. Its outside surfaces (top and bottom) are smooth, without pores. The material is of continuous characteristics. The beam is porous inside, with the degree of porosity varying in the transverse direction, assuming the maximum value on the beam axis. A rectangular system of coordinates is introduced, with the x-axis oriented along the beam, and the y-axis in the beam-depth direction. Themoduli of elasticity are defined as follows E(y)= E1[1−e0 cos(πη)] G(y) = G1[1−e0 cos(πη)] (1.1) Elastic buckling of a porous beam 861 where e0 – coefficient of beam porosity, e0 =1−E0/E1 E0,E1 – Young’s moduli at y =0 and y =±h/2, respectively G0,G1 – shearmoduli (modulus of rigidity) for y =0 and y =±h/2, respectively Gj – relationship between the moduli of elasticity for j = 0,1, Gj = Ej/[2(1+ν)] ν – Poisson’s ratio (constant for the entire beam) η – dimensionless coordinate, η = y/h h – thickness of the beam. Fig. 2. Geometric model of broken-line hypothesis The field of displacements (geometric model) in the rectangular cross- section of the beam is shown in Fig.2. The cross-section, being initially a planar surface, becomes curved after the deformation. The surface perpendi- cularly intersects the top and bottom surfaces of the beam. The geometric model is similar to that obtained bymaking use of the broken-line hypothesis applied to three-layered structures. Such a definition of the displacement mo- del gives a basis for adopting a field of displacements in any cross-section in the following form u(x,y)=−h { η dv dx − 1 π [ψ1(x)sin(πη)+ψ2(x)sin(2πη)cos 2(πη)] } (1.2) v(x,y)= v(x,0)= v(x) 862 K.Magnucki, P.Stasiewicz where u(x,y) – longitudinal displacement along the x-axis v(x) – deflection (displacement along the y-axis) ψ1(x),ψ2(x) – dimensionless functions of displacements. The geometric relationships, i.e. components of the strain field are εx = ∂u ∂x =−h { η d2v dx2 − 1 π [dψ1 dx sin(πη)+ dψ2 dx sin(2πη)cos2(πη) ]} (1.3) γxy = ∂u ∂y + dv dx = ψ1(x)cos(πη)+ψ2(x)[cos(2πη)+cos(4πη)] where εx is the normal strain along the x-axis, and γxy – the angle of shear (shear strain). The physical relationships, according to Hooke’s law are σx = E(y)εx τxy = G(y)γxy (1.4) Moduli of elasticity (1.1) occurring here are variable and depend on the y- coordinate. 2. Equations of stability The field of displacements for in the thus defined problem includes three unknown functions: v(x), ψ1(x) and ψ2(x). Hence, three equations are neces- sary for a complete description of the problem. They may be formulated on the grounds of the principle of stationarity of the total potential energy of the compressed beam δ(Uε−W)= 0 (2.1) where Uε is the energy of elastic strain Uε = ht 2 L ∫ 0 1/2 ∫ −1/2 (σxεx+ τxyγxy) dxdη W is the work of the load (compressive force) W = F 2 L ∫ 0 (dv dx )2 dx Elastic buckling of a porous beam 863 and L – length of the beam, t – width of the rectangular cross-section of the beam. A system of three equations of stability for the porous beam under com- pression is formulated in the following form E1h 3t ( C1 d2v dx2 −C2 dψ1 dx −C3 dψ2 dx ) +Fv(x)= 0 C2 d3v dx3 −C4 d2ψ1 dx2 −C5 d2ψ2 dx2 + 1 2(1+ν) 1 h2 (C7ψ1+C8ψ2)= 0 (2.2) C3 d3v dx3 −C5 d2ψ1 dx2 −C6 d2ψ2 dx2 + 1 2(1+ν) 1 h2 (C8ψ1+C9ψ2)= 0 where C1 = 1 12 − π2−8 2π3 e0 C2 = 1 π2 (2 π − 1 4 e0 ) C3 = 1 π2 ( 3 16 − 32 75π e0 ) C4 = 1 π2 (1 2 − 2 3π e0 ) C5 = 1 π2 ( 8 15π − 1 8 e0 ) C6 = 1 π2 ( 5 32 − 128 315π e0 ) C7 = 1 2 − 4 3π e0 C8 = 8 15π − 1 4 e0 C9 =1− 832 315π e0 Moreover, appropriate boundary conditions are formulated (for x = 0 and x = L) [v′′δv′−v′′′δv] ∣ ∣ ∣ L 0 =0 [ψ′kδv ′ −ψ′′kδv] ∣ ∣ ∣ L 0 =0 (v′′δψk) ∣ ∣ ∣ L 0 =0 ψ′kδψk) ∣ ∣ ∣ L 0 =0 k =1,2 (2.3) The system of differential equations (2.2) may be approximately solved with the use of Galerkin’s method. Hence, three unknown functions satisfying bo- undary conditions (2.3) are assumed in the following form v(x)= va sin ( nπ x L ) ψk(x)= ψak cos ( nπ x L ) k =1,2 where va, ψa1, ψa2 are parameters and n is a natural number. Substituting these functions into equations (2.2) and usingGalerkin’s me- thod yields a system of three homogeneous algebraic equations A(3×3)X =0 (2.4) 864 K.Magnucki, P.Stasiewicz where A=    a11−f a12 a13 a21 a22 a23 a31 a32 a33    X =    −va ψa1 ψa2    a11 = C1α 2 0 a12 = C2α0h a13 = C3α0h a21 = C2α 3 0 a22 = [ C4α 2 0+ C7 2(1+ν) ] h a23 = [ C5α 2 0+ C8 2(1+ν) ] h a31 = C3α 3 0 a32 = [ C5α 2 0+ C8 2(1+ν) ] h a33 = [ C6α 2 0+ C9 2(1+ν) ] h α0 = nπ h L and f is the dimensionless longitudal compressive force (0 < f) f = F E1ht The condition detA=0 (2.5) enables determination of the dimensionless force f. Limiting the considerations to the matrix A(2×2) and taking into account condition (2.5) yields f = a11− a12a21 a22 = C1α 2 0 [ 1−2(1+ν) C22α 2 0 C1(C7+2(1+ν)C4α 2 0) ] (2.6) and the dimensionless critical load fCR =min n f =π2 (h L )2 C1 [ 1−2(1+ν) C22 C1C0 ] (2.7) for n =1, where C0 =2(1+ν)C4+ C7 π2 λ2 and λ = L/h is the relative length of the beam. The critical force is FCR = π2E1h 3t L2 C1 [ 1−2(1+ν) C22 C1C0 ] (2.8) In a particular case of a beam made of an isotropic non-porous material, the elasticity coefficients do not depend on the coordinate y (e0 =0, C1 =1/12). The negligence of the transverse force effect (C2 =0) gives the classical Euler force. Apart from varying elasticity constants, the effect of shear strain on the critical force is also taken into consideration in expression (2.8). Elastic buckling of a porous beam 865 3. Numerical analysis of the stress state A family of beamsof the constant height h =100mmandwidth t =1mm was assumed. The beam lengths were: L =2000mm, 2500mmand 5000mm, the material constants: E1 = 2.05 · 10 5MPa, e0 = 0.99, ν = 0.3. Numerical analysis was carried out by means of the Finite Element Method – System COSMOS/M. The symmetry of the system enabled modeling of a half of the beam only by imposing suitable boundary conditions at one end for x = 0 (zero-displacement in the y-axis direction) and in themiddle cross-section for x = L/2 (zero-displacement in the x-axis direction). The beam was buckled only in the xy plane. Thematerial properties varying through thickness of the cross-section were discretized with 20 layers of constant properties. Particular layers were characterized by elasticity constants adopted according to (1.1) for points located in themiddle of each of the layers (Fig.3). For the purpose of strength analysis, the beam was subject to a transverse load of a constant intensity distributed at its whole length. Fig. 3. Discretization of material properties Fig. 4. Normal stress at cross-section: (a) theory, (b) FEM Figure 4 presents an example of normal stress distribution in points lo- cated in the middle cross-section of the beam. The theoretical distribution 866 K.Magnucki, P.Stasiewicz (Fig.5a) was determined from Hooke’s law (1.4) based on the adopted for- mulas of moduli of elasticity (1.1) and the broken-line hypothesis assumed for determination of displacements (1.2). The stress distribution patterns ob- tained analytically and numerically (FEM) are very similar, which seems to confirm the justness of the broken-line hypothesis. Fig. 5. Critical load as function of beam length 4. Numerical analysis of buckling The critical loads determined on the grounds of the analytical solution to equation (2.8) for a family of beams are specified in Table 1. Moreover, the critical loads are determined bymeans of FEM. The subspace Iteration algo- rithmwas applied. Values of the loads are shown in Table 1. The comparison of the solutions obtained with both methods shows that the error does not exceed 4 percent for the beam of the length L =5000mm (Fig.5). Table 1.Values of critical loads L [m] FCR [N] Eq. (2.8) FEM 2.0 25808 24915 2.5 16795 16346 5.0 4295 4166 Elastic buckling of a porous beam 867 5. Conclusions The above proposal of analytical description of the field of strains in a beam properties varying through thickness is a generalization of an approach to multi-layered composite beams. The linear Euler-Bernoulli hypothesis for beams subject to bending makes a particular case of the description. The general solution to three equations of stability enable one define a simple formula for the critical load of the beam. The critical loads obtained from the analytical and numerical (FEM) solutions are similar, with themaximum difference not exceeding 4 percent. The work was presented at the 10th Symposium of Structure Stability in 2003 (Zakopane). References 1. Bart-Smith H., Hutchinson J.W., Evans A.G., 2001, Measurement and analysis of the structural performance of cellular metal sandwich construction, International Journal of Mechanical Sciences, 43, 1945-1963 2. Hutchinson J.R., 2001, Shear coefficients for Timoshenko beam theory, Jo- urnal of Applied Mechanics, 68, 87-92 3. Kołakowski Z., Kowal-Michalska K. (edit.), 1999, Selected Problems of Instabilities in Composite Structures, Publishers of Technical University of Lodz, Łódź 4. Librescu L., Hause T., 2000, Recent developments in the modeling and be- havior of advanced sandwich constructions: a survey, Composites Structure, 1-17 5. Lok T.-S., Cheng Q.-H., 2000, Elastic stiffness properties and behavior of truss-core sandwich panel, Journal of Structural Engineering, 5, 552-559 6. Magnucki K., Ostwald M. (edit.), 2001, Stateczność i optymalizacja kon- strukcji trójwarstwowych,Wyd. InstytutuTechnologiiEksploatacjiwRadomiu, Poznań-ZielonaGóra 7. Romanów F., 1995, Wytrzymałość konstrukcji warstwowych, Wyd. Wyższej Szkoły Inżynierskiej, ZielonaGóra 8. VinsonJ.R., 1999,TheBehavior of Sandwich Structuresof Isotropic andCom- posite Materials, Technomic Publishing Company, Inc. Lancaster, Basel 9. WangC.M.,ReddyJ.N.,LeeK.H., 2000,ShearDeformableBeams andPla- tes, ElsevierScience,Amsterdam-Lausanne-NewYork-Oxford-Singapore-Tokyo 868 K.Magnucki, P.Stasiewicz 10. Wielgosz C., Thomas J.-C., 2002, Deflections of inflatable fabric panels at high pressure,Thin-Walled Structures, 40, 523-536 Wyboczenie sprężyste belki porowatej Streszczenie Przedmiotem pracy jest prosta belka o przekroju prostokątnym, podparta prze- gubowo na obu końcach, obciążona wzdłużną siłą ściskającą. Belka wykonana jest zmateriału izotropowegoporowatego.Właściwości tegomateriału są zmienne nawy- sokości belki. Na osi belki moduł sprężystości jest najmniejszy, natomiast na po- wierzchniach górnej i dolnej największy. Z zasady stacjonarności całkowitej energii potencjalnej wyznaczono układ równań różniczkowych stateczności belki. Układ ten rozwiązanoanalitycznie i zapisanowpostaci zamkniętej wyrażenie na obciążenie kry- tyczne ściskanej belki.Wyniki tego rozwiązania zweryfikowanodla przykładowejbelki za pomocąmetody elementów skończonych (SystemCOSMOS). Manuscript received January 6, 2004; accepted for print March 30, 2004