Global and local buckling of sandwich circular and beam-rectangular plates with metal foam core

Thin-Walled Structures 61 (2012) 154–161

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Global and local buckling of sandwich circular and beam-rectangular plates with metal foam core P. Jasion a,n, E. Magnucka-Blandzi b, W. Szyc a, K. Magnucki a a b

´ University of Technology, ul. Piotrowo 3, 60-965 Poznan ´ , Poland Institute of Applied Mechanics, Poznan ´ University of Technology, ul. Piotrowo 3A, 60-965 Poznan ´ , Poland Institute of Mathematics, Poznan

a r t i c l e i n f o

a b s t r a c t

Available online 15 May 2012

The paper is devoted to the analytical, numerical and experimental studies of the global and local buckling–wrinkling of the face sheets of sandwich beams and sandwich circular plates. A mathematical model of displacements, which includes a shear effect, is presented. The governing differential equations of sandwich plates are derived. The equations are analytically solved and the critical loads are obtained. Finite element models of the plates are formulated and the critical loads and buckling modes are calculated. Moreover, experimental investigations are carried out for the family of sandwich beam-plates. The values of the critical load obtained by the analytical, numerical (FEM), and experimental methods are compared. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Sandwich structures Global and local buckling Critical loads

1. Introduction Global and local buckling problems of sandwich structures are investigated since the sixties of the twentieth century. When designing sandwich structures, strength and stability conditions are active constraints. Stability conditions include global and local buckling problems. The foundations of the theory of sandwich structures are described in the literature [1–3]. Stability of deformable systems is presented in the monograph [4]. Local buckling of sandwich structures in the form of wrinkling of the face treating like the beam on the elastic foundation, the core, is described in the monographs [3,5]. The theory of elastic foundation and stability problems of beams, plates and shells on elastic foundations is discussed in detail in the monograph [6]. Strength and stability problems of plates and shells are reviewed and described in the monographs [7,8]. The numerical simulation and analysis of the elastic–plastic beam-on-foundation is presented in the paper [9]. Theoretical and experimental investigations of faces wrinkling in sandwich structures are described in the papers [10–13]. Analytical models of collapse mechanisms of sandwich beams under transverse force are presented in the papers [14–16]. Mechanical properties of aluminium foam under shear are presented in the paper [17]. Elastic buckling problem of a sandwich rectangular plate loaded in plane is solved in the papers [18,19]. Strength and buckling problems of sandwich beams with metal foam core are described in the papers [20,21].

n

Corresponding author. Tel.: þ48 506325880. E-mail address: [email protected] (P. Jasion).

0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.04.013

Global buckling of simply supported and clamped circular plates on elastic foundation has been studied in [22], and with the use of Bessel function critical loads has been obtained. A geometrically nonlinear theory which describes both global and local (wrinkling) structural behaviour of sandwich plates is presented in [23]. Different aspects of wrinkling of the face layers as one of the typical failure modes in sandwich structures are discussed in [10], where finite element calculations and experimental studies are also performed to verify the analytical results. Cores of the sandwich structures are often assumed as an isotropic material with low compressive stiffness and transverse shear stiffness. The metal foam cores with different properties depending on the foam density are studied in [24–26]. Variability in elastic and plastic properties of metal foams is presented there. Critical loads and post-critical behaviour of elastic–plastic sandwich annular plates are calculated in [27]. The authors assumed the plate to be made of a compressive elastic–plastic material with kinematic–isotropic strain-hardening described by the theory of plastic flow equations. The axisymmetric buckled states of an annular sandwich plate subjected to uniform radial compression are studied in [28]. The results of analysis of the deflection of a circular plate, as a special case of a shell of revolution resting on a Winkler-type linear elastic foundation are presented in [29]. The problem of the axisymmetric dynamic stability of sandwich circular plates subjected to a periodic uniform radial loading is studied in [30] by the discrete layer annular finite element method. Three-dimensional analytical formulation of axially symmetrical isotropic sandwich circular plates is presented in [31], where the values of stresses, strains and deflection of the transverse loaded plate given by analytical and FE methods are compared. Buckling load of circular sandwich plates calculated by

P. Jasion et al. / Thin-Walled Structures 61 (2012) 154–161

nonlinear theory with the use of the spline collocation method is studied in [32]. Experimental investigation of sandwich plates with metallic foam cores under simulated blast loading is presented in [33], where the results of the experiments are compared with FE model calculations. The static stability problem of threelayered annular plates with a soft core is analysed in [34,35]. With the use of the finite difference method, critical load values are obtained from a system of differential equations describing plate deflections. The obtained results have been compared with the numerical FEM calculations. The dynamic stability problem of similar plates under uniform radial compression rapidly linearly increasing in time is studied in [36]. Shear forces appearing in the core of sandwich structures usually do not influence strength or buckling resistance of the structures. For that reason most analytical models known from literature do not take these forces into account. The goal of this paper is to elaborate the mathematical model of the sandwich structure which includes the shear effect and to investigate the influence of the core properties on the behaviour of the structure. The formulae describing the critical load corresponding to the global and local buckling are looked for. The model will be verifying by finite element method and experimental tests. Two simple sandwich structures widely explored in the literature have been chosen to investigate that is a beam (Fig. 1) and a circular plate (Fig. 2). The outer layers, the faces, of the sandwich structures under consideration are isotropic and of thickness tf. They are assumed to be made of aluminium. The inner layer, the core, is assumed to be made of aluminium foam and is also isotropic. The thickness of the core is denoted by tc. Material properties of the layers are: Young’s moduli Ef, Ec and Poisson’s ratios nf , nc for the faces and the core respectively. The beam of the length L and the width b carries a compressive axial force F0, the first load case; two transverse forces F1, the second load case (Fig. 1). The depth of the beam is H ¼ t c þ 2t f . The first load case is related to the global buckling, while the second one to the local buckling–wrinkling of the face of the beam under pure bending. The radius of an examined three-layered circular plate equals R. Two cases of loading are also considered (Fig. 2). The first one is devoted to a sandwich circular plate subjected to the compressive force with the intensity NR in its middle plane. In this case global buckling may occurs. The second one concerns a plate under pure bending. The upper face of a plate is compressed while the lower face is under tension (Fig. 2). This case is related to the local buckling. The results of the investigations presented below are based on papers [37,38].

tf H

F0

F1

F1

tc tf

x L1 L

Fig. 1. Scheme of the loaded sandwich beam.

155

2. Global buckling of a sandwich beam 2.1. Analytical solution The bending moment M b ðxÞ and the shear force VðxÞ of the sandwich beam, based on the paper [20], are in the following forms: " #  2   1 d w 1 dc0 3 Ec E 2Ef a1 þ  E a þ Mb ðxÞ ¼ btc , ð1Þ c 2 f 2 12 6 dx dx VðxÞ ¼ 2Gc btc c0 ðxÞ,

ð2Þ

where Ef, Ec and Gc ¼ Ec =½2ð1þ nc Þ, elastic moduli of the metal faces and the metal foam core; nc , Poisson ratio of the core; w(x), deflection; c0 ðxÞ, dimensionless displacement function; x1 ¼ t f =t c , 1 a1 ¼ 12 ð3 þ 6x1 þ4x21 Þx1 , a2 ¼ ð1 þx1 Þx1 , dimensionless parameters. The bending moment and the shear force of the buckled beam are Mb ðxÞ ¼ F 0 wðxÞ

and

VðxÞ ¼ F 0

dw : dx

ð3Þ

Substitution of the right hand sides of Eqs. (3) into Eqs. (1) and (2) gives the critical compressive axial force in the following form: F 0,CR ¼

p2 Ec bt3c f 01 12L2 f 02

,

ð4Þ

where f 01 ¼ 1þ 2ðEf =Ec Þa1 ,f 02 ¼ 1 þ 16 ðpðt c =LÞÞ2 ð1 þ nc Þð1 þ 6ðEf =Ec Þa2 Þ. The distribution of the compressive axial force in the particular layers is directly proportional to the longitudinal rigidities of the layers. Thus, stresses in the faces and the core for the critical force (4) are as follows:

sðfCRÞ ¼

F ðf0 Þ bt f

and

sðcÞ CR ¼

F ðcÞ 0 , bt c

ð5Þ

where F ðf0 Þ ¼ ðEf t f =ðEc t c þ 2Ef t f ÞÞF 0,CR , the force in the face; F 0ðcÞ ¼ ðEc t c =ðEc t c þ 2Ef t f ÞÞF 0,CR , the force in the core. 2.2. Numerical calculations—FEM analysis A discrete model of the sandwich beam consists of shell elements – the faces – and the solid elements – the core. Faces are offset from the core about half of theirs thickness. Bonding conditions are imposed between particular layers. Due to the symmetry of the problem a quarter of the beam has been modelled with proper boundary conditions on symmetry planes. The critical forces and stresses are calculated for the family of beams: L¼ 800 mm, b¼50 mm, tf ¼ 1 mm, tc ¼18 mm, Ef ¼65,600 MPa, nf ¼ 0:33, Ec ¼ /10; 50,100; 400,800; 1200S MPa, nc ¼ 0:3, sðfy Þ ¼ 112 MPa. An example of the global buckling mode is shown in Fig. 3. The results of the numerical calculations based on the analytical solution of Eqs. (4) and (5) and FEM are specified in Table 1. Good agreement can be seen between the results obtained with both methods—the discrepancies were below 6%. The values of stresses indicate that the global buckling has an elastic character. In Fig. 4 a strong influence of Ec on the value of

Fig. 2. The scheme of a sandwich circular plate under compression or under pure bending.

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Fig. 3. Global buckling mode of a compressed sandwich beam—a half of the beam is shown.

Table 1 Critical forces and stresses. Ec (MPa)

10 2.61

F ðAnalÞ 0,CR (kN) ðf -AnalÞ sCR (MPa) sðc-AnalÞ (MPa) CR

F ðFEMÞ 0,CR (kN) ðf -FEMÞ CR ðc-FEMÞ CR

s s

(MPa) (MPa)

26.1

50

100

6.10

400

7.34

60.6

8.73

72.4

82.8

800 9.14 82.4

0.01

0.05

0.11

0.50

1.00

2.68

6.18

7.39

8.75

9.15

26.8 0.004

61.9

72.9

0.05

82.3

0.11

0.51

82.5 1.01

1200 9.38 80.6 1.47 Fig. 5. Scheme of wrinkling of the face.

9.38 80.6 1.48

The scheme of wrinkling of the compressed face is shown in Fig. 5. Displacements of the core for the critical state are assumed in the following form: uðx,zÞ  0,wðx,zÞ ¼ w1  wðzÞ  sin

F0, CR [kN]

wðt c =2Þ ¼ 1

Analytical

and

wðt c =2Þ ¼ 0:

ð8Þ

Thus, the deflection of the face is

5

wðx,zÞ ¼ w1 sin

400

ð7Þ

where w1, parameter; m, natural number; w(z), unknown dimensionless function with the following boundary conditions:

FEM

10

10 100

mpx , L1

800

1200

mpx : L1

ð9Þ

The elastic strain energy of the core (t c =2 rz r t c =2)  Z tc =2 Z L1  Ec b 1nc 2 e2x þ 2nc ex ez þ e2z þ gxz dx dz, U ðcÞ e ¼ 2 2 2ð1nc Þ tc =2 0

EC [MPa]

Fig. 4. Comparison of the results of both methods.

the critical load is observed in the range up to 200 MPa. For higher values of Ec the critical load increase slowly and reach some asymptotic value. The global buckling for the family of sandwich beams is elastic in character. Differences in the values of the critical forces obtained from both methods are below 6%.

ð10Þ

where ex ¼ @u=@x  0, ez ¼ @w=@z, gxz ¼ @u=@z þ @w=@x ¼ @w=@x, strains. The elastic strain energy of the compressed face (z ¼ t c =2) U ðfe Þ ¼

1 3 E bt 24 f f

Z

!2 2 d w

L1

2

dx

0

dx ¼

  1 mp 4 3 Ef btf L1 w21 : 48 L1

ð11Þ

The work of the load-compression force (6) 3. Local buckling of a sandwich beam W¼

3.1. Analytical solution The sandwich beam under pure bending (the second load case, Fig. 1) loses stability by buckling–wrinkling of the compressed face. The compression force in the face, based on the paper [20] is Nðfc Þ ¼

where Db ¼ ing moment-load.

Z 0

L1



dw dx

2

dx ¼

  1 mp 2 ðf Þ N c L1 w21 : 4 L1

ð12Þ

The principle of stationary total potential energy ðf Þ dðU ðcÞ e þ U e WÞ ¼ 0,

ð13Þ

allowed to obtain the equation of equilibrium

Ef bt f ðt c þt f Þ M0 , 2Db 3 1 2 12 bt c ½2Ef x1 ð3þ 6x1 þ4x1 Þ þEc ,

1 ðf Þ N 2 c

ð6Þ

2

d wðzÞ 2

M0 ¼

1 2 F 1 ðLL1 Þ,

bend-

dz

2

k wðzÞ ¼ 0, 2

where k ¼ ðð1nc Þ=2Þðmp=L1 Þ2 .

ð14Þ

P. Jasion et al. / Thin-Walled Structures 61 (2012) 154–161

The solution of this equation with respect to the boundary conditions (8) is in the following form:    1 1 z  sinh C c , ð15Þ wðzÞ ¼ sinhðC c Þ 2 tc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C c ¼ ktc ¼ mp ð1nc Þ=2ðt c =L1 Þ. Substitution of this function into the principle (13) gives the critical force   b1 Þ þ b2 C 2c , Nðfc,CR ¼ min ð16Þ Cc C c tanhðC c Þ

157

properties of the aluminium alloy of the faces and the aluminium foam of the core are shown in Fig. 8. The material constants for the aluminium alloy: Ef ¼65,600 MPa, nf ¼ 0:33, sðfy Þ ¼ 112 MPa and the metal foam: Ec ¼1200 MPa, nc ¼ 0:3. The results of experimental tests are compared with numerical (FEM) calculations in Fig. 9. The results correspond to each other very well. In Fig. 10 the displacements sensors and the bended beam are shown. The local elastic–plastic buckling–wrinkling of the upper face was observed.

where b1 ¼ Ec bt c =2ð1 þ nc Þ, b2 ¼ ðEf bt f =6ð1nc ÞÞðt f =t c Þ2 , parameters. Taking into account (6), the critical bending moment M 0,CR ¼ 2

Db Þ bt ðt c þ t f ÞNðfc,CR : Ef f

ð17Þ

Table 2 Critical moments-loads and stresses. Ec (MPa)

3.2. Numerical calculations—FEM analysis Discrete model of the sandwich beam is equivalent to the one for global buckling (see Section 2.2). An example of the local buckling mode, wrinkling, is shown in Fig. 6. The critical loads are calculated for the family of beams: L¼800 mm, b ¼50 mm, sðfy Þ ¼ 112 MPa, tf ¼1 mm, tc ¼18 mm, Ef ¼ 65,600 MPa, nf ¼ 0:33, nc ¼ 0:3, Ec ¼ /10; 50,100; 400,800; 1200S MPa. The results of the numerical calculations based on the analytical solution of Eqs. (15) and (16) and FEM are specified in Table 2 and shown in Fig. 7. Here, differently than in the case of global buckling, the value of the critical load increase monotonically in the whole range of Ec which was considered. Again, good agreement can be seen between analytical and FEM analysis, especially for small values of Ec (discrepancies are less than 5%). The results given in Table 2 show that elastic buckling may appear for beams with flexible core for which Ec o 10 MPa. For higher values of Ec, including aluminium foam considered in the experiment—Ec ¼1200 MPa, an elastic–plastic buckling takes place. 3.3. Experimental investigations In the experimental investigation sandwich beams with a metal foam core were tested. The dimensions of the beams, according to Fig. 1, are as follows: L¼1100 mm, L1 ¼ 300 mm, b ¼ /50; 100S mm, tf ¼ 1 mm, t c ¼ /38; 48S mm. The mechanical

10

50

100

400

800

1200 2815.3

M ðAnalÞ 0,CR (Nm)

130.3

340.1

526.7

1314.6

2115.7

ðf -AnalÞ sCR (MPa)

137.0

356.9

551.7

1360.2

2154.4

2821.9

M ðFEMÞ 0,CR (Nm)

124.5

336.0

527.9

1353.3

2203.9

2951.6

sðfCR-FEMÞ (MPa)

123.0

331.3

519.4

1316.1

2110.3

2783.4

M0, CR [kNm] 3.0 FEM

2.5 2.0

Analytical

1.5 1.0 0.5 10 100200

400 500

800

1000 1200

EC [MPa]

Fig. 7. Comparison of the results of both methods.

Fig. 6. Local buckling mode of sandwich beam under bending—a half of the beam is shown.

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Fig. 8. Stress–strain diagram for the aluminium alloy (a) and the metal foam (b).

 for the core  12 r z r 12 erðcÞ ¼

u r

@u , @r

ðcÞ eðcÞ f ¼ , grz ¼

1 @u @w ¼ 2c0 ðrÞ: þ t c @z @r

Stresses in all layers of the plate according to the Hooke’s law are given in formula

 for the upper and lower face (i ¼1 and i¼2 respectively) sðfiÞ r ¼ Fig. 9. Comparison of the results of both methods.

4. Global buckling of a circular plate

In the analytical description the classical broken line hypothesis of deformation of a field of displacements is applied (Fig. 11). The analysis of global buckling behaviour of a sandwich circular plate is limited to an elastic range only. The plate is loaded like shown in Fig. 2 (the left hand side picture). The longitudinal displacements are assumed as follows:

 for the upper face ð12 þx1 Þ r z r  12 

 dw þ c0 ðrÞ , uðr, zÞ ¼ t c z dr

rzr    dw 2c0 ðrÞ , uðr, zÞ ¼ t c z dr

 for the core

 12

1 2

sðfiÞ f ¼

Ef ðeðfiÞ þ nf eðfiÞ r Þ, 1n2f f

 for the core srðcÞ ¼

4.1. Analytical description and solution

Ef ðeðfiÞ þ nf eðfiÞ f Þ, 1n2f r

Ec ðeðcÞ þ nf eðcÞ f Þ, 1n2c r

sðcÞ f ¼

Ec ðeðcÞ þ nf erðcÞ Þ, 1n2c f

ðcÞ trz ¼ Gc gðcÞ rz ,

where Gc ¼ Ec =½2ð1 þ nc Þ is a shear modulus of the core. The elastic strain energy is defined as follows: ðf 2Þ U e ¼ U ðfe 1Þ þ U ðcÞ e þU e ,

where U ðfe 1Þ , an elastic strain energy of the upper face Z R Z 1=2 Ef t c ðe2 þ 2nf er ef þ e2f Þr dz dr, U ðfe 1Þ ¼ p 2 1nf 0 ð1=2 þ x1 Þ r U ðcÞ e , an elastic strain energy of the core  Z R Z 1=2  Ec t c 1 e2r þ 2nc er ef þ e2f þ ð1nc Þg2rz r dz dr, U ðcÞ e ¼ p 1n2 2 1=2 c 0 U ðfe 2Þ , an elastic strain energy of the lower face Z R Z 1=2 þ x1 Ef t c ðe2r þ 2nf er ef þ e2f Þr dz dr: U ðfe 2Þ ¼ p 1n2f 0 1=2 The work of a load is

 for the lower face r z r þ x1   1 2

uðr, zÞ ¼ t c z

dw c0 ðrÞ , dr

1 2

W ¼ pN R

Z

R 0

 2 dw r dr, dr

where N R , an intensity of the load (Fig. 2). Based on the principle of stationary total potential energy where x1 ¼ t f =t c , dimensionless parameter; c0 ¼ u1 ðrÞ=t c , dimensionless functions shaping displacements; w(r), function of deflection. In particular case, if c0 ðrÞ  0 then the field of displacements is the Kirchhoff–Love’s hypothesis. Strains are defined as follows:

 for the upper and lower face (i¼1 and i ¼2 respectively) eðfiÞ r ¼

@u , @r

u r

ðfiÞ eðfiÞ f ¼ , grz ¼

1 @u @w ¼ 0, þ t c @z @r

dðU e WÞ ¼ 0, the following system of two partial differential equations has been obtained: 8         > Ec t 3c d d 1 d dw d dw > > r  r A A r c ðrÞ ¼ N > R 1 2 0 2 < 1n dr dr r dr dr dr dr c    , 3   > E t d 1 d dw c c > > þ 4Gc c0 ðrÞ ¼ 0 > : 1n2 dr r  dr r A2 dr A3 c0 ðrÞ c ð18Þ

P. Jasion et al. / Thin-Walled Structures 61 (2012) 154–161

159

Fig. 10. View of the test stand and wrinkling of the upper face.

Table 3 The values of an intensity of the critical load obtained from analytical and FEM calculation. Ec (MPa)

10

50

100

400

800

1200

N ðAnalÞ R,CR (N/mm)

66.44

255.72

399.86

699.11

808.13

860.14

N ðFEMÞ R,CR

68.86

259.56

404.58

704.32

811.36

861.42

(N/mm)

Fig. 11. Geometric model of displacements.

1 where A1 ¼ 2c2 a1 þc1 , A2 ¼ c2 a2 þ2c1 , A3 ¼ 2ðc2 x1 þ 2c1 Þ, c1 ¼ 12 , c2 ¼ Ef =Ec  ð1n2c Þ=ð1n2f Þ. The unknown functions w and c0 are assumed as follows:   r 2 r 3  r 2  r þ b3 , c0 ðrÞ ¼ ca0 þ b4 , ð19Þ wðrÞ ¼ wa 1þ b2 R R R R

where b3 ¼ 2ðð2a1 c2 ð1 þ nf Þ þc1 ð1 þ nc ÞÞ=ð2a1 c2 ð4þ nf Þ þ c1 ð4 þ nc ÞÞÞ, b4 ¼ ða2 c2 ð1 þ nf Þ þ 2c1 ð1 þ nc ÞÞ=ða2 c2 ð2 þ nf Þ þ 2c1 ð2 þ nc ÞÞ, b2 ¼ 1b3 . The functions, as defined in Eq. (19), satisfy boundary conditions  dw ¼ 0, Mr 9r ¼ R ¼ 0, wðRÞ ¼ 0, dr r ¼ 0 where Mr , a bending moment "  # 2 Ec t 3c d w dc0 1 dw ~ ~ A 1 þ A 2 c0 , þ A1 2 þ A2 Mr ¼ r dr dr 1n2c dr A~ 1 ¼ 2nf c2 a1 þ nc c1 , A~ 2 ¼ nf c2 a2 þ 2nc c1 . Substituting the assumed functions (19) into the system (18) and applying Galerkin’s method the following homogeneous system of algebraic equations has been derived: " # " w #   a A1 K A2 0 R  ¼ , ð20Þ ca0 A2 A3 2b6 b7 0

Fig. 12. A global buckling mode of a sandwich circular plate.

2 2 ð1n2c Þððb3 4b3 þ 10Þ=ð8b3 Þb3 Þ  N R R2 =Ec t3c , b6 ¼ ð1nc Þ where K ¼ 15 2 2 ðR=t c Þ , b7 ¼ ð6b4 þ 15b4 þ 10Þ=15ð2b4 þ3Þb4 . A homogeneous system of algebraic equations (20) has a nontrivial solution if a determinant of coefficient matrix is equal to zero, that means    A1 K  A2   det ð21Þ  ¼ 0:  A2 A3 2b6 b7 

Therefore from the above equation an intensity of the critical load has been calculated NðAnalyticalÞ R,CR

15 ð8b3 Þb3 A22 A1   ¼ A3 2b6 b7 2ð1n2c Þ b23 4b3 þ 10

!

Ec t 3c R2

:

ð22Þ

The values of an intensity of the critical load for exemplary family of plates obtained from Eq. (22) are given in Table 3 and compared with the results obtained from FEM.

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P. Jasion et al. / Thin-Walled Structures 61 (2012) 154–161

illustrated in Fig. 15. The values of the critical load increase rapidly in the whole range of Ec.

4.2. Numerical calculations—FEM analysis Numerical calculations have been conducted with the use of finite element method. The discrete model of a sandwich circular plate is similar the one of beam (see Section 2.2). An example of global bucking mode obtained from FEM is shown in Fig. 12. The values of an intensity of the critical load have been analytically and numerically calculated for the family of plates with the radius R¼250 mm, the thickness of each face tf ¼1 mm, the thickness of the core tc ¼18 mm, Young’s modulus of the faces Ef ¼65,600 MPa, Poisson’s ratio nf ¼ 0:33 and nc ¼ 0:3 for faces and the core respectively. These results are presented in Table 3 for different values of Young’s modulus Ec of the core. The obtained results have been compared with the analytical one and illustrated in Fig. 13. The biggest difference between critical loads given by analytical and FEM calculations does not exceed 4% and occurs for Ec ¼10 MPa. The difference is smaller if the value of Young’s modulus in the core (Ec) is bigger. The smallest one is less than 0.2% for Ec ¼1200 MPa.

5. Local buckling of a circular plate Local buckling has only been studied numerically. Numerical calculations have been conducted with the use of finite element method. An example of local bucking mode obtained from FEM is shown in Fig. 14. The discrete model of a sandwich circular plate is the same as the one in global buckling analysis. The plate is simply supported and the lower edge and loaded according to Fig. 2 (the right hand side picture). The values of an intensity of the critical load have been numerically calculated for the same family of plates as previously (Section 4.2—example for global buckling). The results are presented in Table 4 for different values of Young’s modulus Ec of the core. The obtained results are

6. Conclusions In the paper the mathematical models of sandwich structures – the beam and the circular plater – are presented. They include the shearing forces acting in the core which is not typical for models known from literature. This makes possible to investigate the influence of the material properties of the core on the buckling load. The model of the beam allows to obtain the formula for the global buckling load when the beam is axially compressed and the formula for the local buckling–wrinkling load for the beam under pure bending. The model of circular plate is limited to global stability only and also gives the formula for the critical load. The local buckling of the circular plate is analysed with the use of finite element method only. Numerical calculations on a family of beams and plates have been performed not only using analytical formulae obtained in the paper but also with the use of FE method. Additionally for beams, experimental tests have been carried out. The values of critical loads obtained from each method correspond to each other very well. The discrepancies are smaller than 6%. Table 4 The values of an intensity of the critical load obtained from analytical calculation. Ec (MPa)

10

50

100

400

800

1200

N ðFEMÞ R,CR (N/mm)

130.84

339.75

519.94

1241.90

1940.00

2525.40

NR, CR [kN] 3000

NR, CR [kN] 1000

2500

800 600

FEM

2000

Analytical

1500

FEM

1000

400

500

200 10 50 100

400

800

1200 EC [MPa]

Fig. 13. A comparison of the analytical and FEM results.

10 50 100

400

800

1200 EC [MPa]

Fig. 15. A relation between the intensity of critical load and Young’s modulus in the core.

Fig. 14. A local buckling mode of a sandwich circular plate.

P. Jasion et al. / Thin-Walled Structures 61 (2012) 154–161

As to the axially compressed sandwich beams a strong influence of the shear forces on the buckling load is observed (Fig. 4). For very soft cores (Ec o 100 MPa) the value of critical load rises rapidly after which it reaches an asymptotic value. In all cases the global buckling had an elastic character. According to Table 2 the elastic local buckling may appear for beams with flexible core only (Ec o10 MPa). For higher values of Ec the elastic–plastic buckling occurs. It was also observed in experimental tests, in which beams with aluminium metal foam (Ec ¼1200 MPa) were tested. Differently than in the global buckling case here the value of the buckling load was rising monotonically in the whole range of Ec considered. Similar conclusions as to the influence of shearing force on the buckling load may be drawn for the circular plates. Also in this case the values of the critical load grow faster for the local buckling than for the global one. The results presented above indicate that shearing forces should not be ignored when designing sandwich structures. It is especially significant in cases when the Young modulus of the core is of small value (Ec o 100 MPa). Acknowledgments These studies are conducted with the support of the Ministry of Science and Higher Education in Poland—Grant no. 0807/B/ T02/2010/38. References [1] Libove C, Butdorf SB. A general small-deflection theory for flat sandwich plates. NACA TN 1526; 1948. [2] Reissner E. Finite deflections of sandwich plates. Journal of the Aeronautical Science 1948;15(7):435–40. [3] Plantema FJ. Sandwich construction: the bending and buckling of sandwich beams, plates and shells. New York: John Wiley and Sons; 1966. [4] Volmir AS. Stability of deformable systems. Moscow: Izd. Nauka Fiz-Mat-Lit; 1967 [in Russian]. [5] Allen HG. Analysis and design of structural sandwich panels. London: Pergamon Press; 1969. [6] Vlasov VZ, Leontev NN. Beams, plates and shells on elastic foundation. Moscow: Gosud. Izd. Fiz-Mat-Lit; 1960 [in Russian]. [7] Woz´niak C., editor. Mechanics of elastic plates and shells, mechanika techniczna, vol. VIII. Warszawa: Wyd. Naukowe PWN; 2001 [in Polish]. [8] Ventsel E, Krauthammer T. Thin plates and shells. Theory, analysis, and applications. New York/Basel: Marcel Dekker, Inc.; 2001. [9] Chen XW, Yu TX. Elastic–plastic beam-on-foundation under quasi-static loading. International Journal of Mechanical Sciences 2000;42:2261–81. [10] Stiftinger MA, Rammerstorfer FG. Face layer wrinkling in sandwich shells—theoretical and experimental investigations. Thin-Walled Structures 1997;29(1–4):113–27. [11] Hadi BK. Wrinkling of sandwich column: comparison between finite element analysis and analytical solutions. Composite Structures 2001;53:477–82. [12] Le´otoing L, Drapier S, Vautrin A. First applications of a novel unified model for global and local buckling of sandwich columns. European Journal of Mechanics: A/Solids 2002;21:683–701. [13] Koissin V, Shipsha A, Skvortsov V. Effect of physical nonlinearity on local buckling in sandwich beams. Journal of Sandwich Structures and Materials 2010;12(7):477–94. [14] Kesler O, Gibson LJ. Size effects in metallic foam core sandwich beams. Materials Science and Engineering A 2002;326:228–34.

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[15] Steeves CA, Fleck NA. Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part I: analytical models and minimum weight design. International Journal of Mechanical Sciences 2004;46:561–83. [16] Qin QH, Wang TJ. An analytical solution for the large deflections of a slender sandwich beam with a metallic foam core under transverse loading by a flat punch. Composite Structures 2009;88:509–18. [17] Rakow JF, Waas AM. Size effects and the shear response of aluminum foam. Mechanics of Materials 2005;37:69–82. [18] Magnucka-Blandzi E, Malinowski M, Magnucki K. Elastic buckling of a sandwich rectangular plate simply-supported by three edges and one edge free. In: Zingoni A, editor. Proceedings of the 3rd international conference on structural engineering, mechanics and computations, Cape Town, South Africa, SEMC 2007. Recent developments in structural engineering, mechanics and computations. Rotterdam: Mill-press Science Publishers; 2007. p. 355–6 [CD 940–945]. [19] Magnucka-Blandzi E. Mathematical modelling of a rectangular sandwich plate with a metal foam core. Journal of Theoretical and Applied Mechanics 2011;49(2):439–55. [20] Magnucka-Blandzi E, Magnucki K. Effective design of a sandwich beam with a metal foam core. Thin-Walled Structures 2007;45:432–8. [21] Magnucka-Blandzi E. Non-linear hypotheses of deformation of flat cross sections of elastic sandwich beam. Proceedings in Applied Mathematics and Mechanics 2007;9:703–4. [22] Kline LV, Hancock JO. Buckling of circular plate on elastic foundation. Journal of Engineering for Industry 1965;8:323–4. ¨ [23] Kuhhorn A, Schoop H. A nonlinear theory for sandwich shells including the wrinkling phenomenon. Archive of Applied Mechanics 1992;62:413–27. [24] Koza E, Leonowicz M, Wojciechowski S, Simancik F. Compressive strength of aluminium foams. Materials Letters 2003;58:132–5. [25] Ramamurty U, Paul A. Variability in mechanical properties of a metal foam. Acta Materialia 2004;52:869–76. [26] Rakow JF, Waas AM. Size effects in metal foam cores for sandwich structures. AIAA Journal 2004;42(7):1331–7. [27] Radwan´ska M, Waszczyszyn Z. Buckling and post-critical deflections of an elastic–plastic sandwich annular plate. Bulletin de l’Academie Polonaise des Sciences 1975;XXIII(4):165–71. [28] Lu-wu H, Chang-jun C. The buckled states of annular sandwich plates. Applied Mathematics and Mechanics 1992;13(7):623–8. [29] Luo YF, Teng JG. Stability analysis of shells of revolution on nonlinear elastic foundations. Computers and Structures 1998;69:499–511. [30] Wang HJ, Chen LW. Axisymmetric dynamic stability of sandwich circular plates. Composite Structures 2003;59:99–107. [31] Luo JZ, Liu TG, Zhang T. Three-dimensional linear analysis for composite axially symmetrical circular plates. International Journal of Solids and Structures 2004;41:3689–706. [32] Chao-sheng H, Shou-kai Z, Feng L. Cubic spline solutions of axisymmetrical nonlinear bending and buckling of circular sandwich plates. Applied Mathematics and Mechanics 2005;26(1):131–8. [33] Radford DD, McShane GJ, Deshpande VS, Fleck NA. The response of clamped sandwich plates with metallic foam cores to simulated blast loading. International Journal of Solids and Structures 2006;43:2243–59. [34] Pawlus D. Solution to the static stability problem of three-layered annular plates with a soft core. Journal of Theoretical and Applied Mechanics 2006;44(2):299–322. [35] Pawlus D. Approach to evaluation of critical static loads of annular threelayered plates with various core thickness. Journal of Theoretical and Applied Mechanics 2008;46(1):85–107. [36] Pawlus D. Solution to the problem of axisymmetric and asymmetric dynamic instability of three-layered annular plates. Thin-Walled Structures 2011;49(5):660–8. [37] Jasion P, Magnucka-Blandzi E, Szyc W, Wasilewicz P, Magnucki K. Global and local buckling of a sandwich beam-rectangular plate with metal foam core. In: The 6th int. conference on thin-walled structures. Timisoara, Romania; 2011. p. 707–4. [38] Jasion P, Magnucka-Blandzi E, Szyc W, Magnucki K. Global and local buckling of a sandwich circular plate with metal foam core. In: The 6th int. conference on thin-walled structures. Timisoara, Romania; 2011. p. 699–6.