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Buckling and vibrations of metal sandwich beams with trapezoidal corrugated cores – the lengthwise corrugated main core

MARK

⁎

E. Magnucka-Blandzia, Z. Walczaka, P. Jasionb, , L. Wittenbecka a b

Poznan University of Technology, Institute of Mathematics, Poznan, Poland Poznan University of Technology, Institute of Applied Mechanics, Poznan, Poland

A R T I C L E I N F O

A B S T R A C T

Keywords: Mathematical modelling Sandwich structure Buckling Vibrations Stability Unstable regions

The paper is devoted to the stability of an orthotropic multi-layered beam. This beam is an untypical sandwich structure the faces of which consist of three layers. The original mathematical model of the beam is formulated taking into account diﬀerent properties of each layer. From the Hamilton's principle the system of equations of motion is derived which is the base for the analysis of buckling and vibration problems. As a result of the analysis the buckling load and natural frequencies of exemplary plates have been obtained. The results are compared with these given by the numerical solution realised with the use of the ﬁnite element method in the ANSYS and ABAQUS systems.

1. Introduction The theory of sandwich structures is developed from the middle of 20th century, that is evidenced by the papers published in journals and at conferences. Basic facts on sandwich structures, their generalizations and applications, can be found in, e.g., Libove, Hubka [1], Allen [2] and Ventsel, Krauthammer [3]. Besides monographs there are also many papers on this subject. Computational models for sandwich plates and shells, predictor-corrector procedures, and the sensitivity of the sandwich response to variations in the diﬀerent geometric and material parameters have been studied by Noor, Burton and Bert [4]. Carlsson, Nordstrand, Westerlind derived tension, shear, bending and twisting rigidities for sandwich structures with corrugated core. The paper [5] is devoted to the computation of the eﬀective properties of corrugated core sandwich panels. Talbi, Batt, Ayad, Guo [6] presented an analytical homogenization model for corrugated cardboard and its numerical implementation in a shell element. In the paper [7] by Cheng, Le, Lu the ﬁnite element method (FEM) is used to derive equivalent stiﬀness properties of sandwich structures with various types of cores. Similar results can be found in [8,9]. The present work is a continuation of the research on multi-layered structures with corrugated core conducted by the authors and coauthors. First works [10,11] were derived to classical sandwich structures. Strength and stability of aluminium beams with lengthwise and crosswise corrugated core have been analysed. To increase the ﬂexural stiﬀness of such beams the modiﬁcation has been introduced presented in works [12,15]. In structures presented here the faces are

⁎

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

http://dx.doi.org/10.1016/j.tws.2016.12.013 Received 29 September 2016; Accepted 19 December 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.

composed of two elements: inside one which is a corrugated plate and outside one in the form of a ﬂat sheet. The corrugations of the core and the faces are perpendicular to each other. The results of experiments on these ﬁve-layered beams presented in papers [16,17] showed that although the stiﬀness of the beam is much higher when compare to the stiﬀness of a three-layered beam, the connection between two corrugated plates can be the weak point of the structure. Further modiﬁcation has been made then by introducing a ﬂat sheet between two corrugated plates. This way a seven-layered beam has been obtained – a sandwich beam with three-layered faces, as can be seen in Fig. 1. The thin-walled beam presented in this paper is an innovatory orthotropic structure, not referred to in the literature. The main core is a lengthwise corrugated sheet. The two faces are three-layered structures the core of which, referred to as face core, is a crosswise corrugated sheet. The internal and external sheets of the faces are ﬂat. All layers of the beam are made of the same material which is isotropic and homogeneous. A characteristic feature of the beam consists in diﬀerentiation of shear eﬀects in particular layers, according to the core's corrugation direction. The deformation of the beam's cross section also depends on this direction. An original mathematical model of the structure will be formulated in the following sections. The model will include the hypothesis of deformation of the cross-section as well as rigidities of the layers in particular directions.

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E. Magnucka-Blandzi et al.

Fig. 1. Scheme of the beam with lengthwise corrugated core.

2. Analytical studies

In the above formulae tc1, tc2, ts (see Fig. 2) describes the thicknesses of particular layers. The indexes c1, c2 and s corresponds to the inner core, the core of the faces and ﬂat sheets, respectively. For the assumed hypothesis of deformation of the cross section – no shear eﬀect in the faces – the geometric relations, the strains, are

2.1. Mathematical model of the beam The classical approach to modelling of sandwich structures is to assume a broken line hypothesis as to the ﬁeld of displacements. For more than three layers the zig-zag hypotheses can be used formulated by Carrera [18]. In the proposed structure the stiﬀness of the core of the faces is considerably higher than the stiﬀness of the main core. For this reason it is assumed that the three-layered faces deforms according to Kirchhoﬀ-Love hypothesis and the shear eﬀect is present in the main core only. Consequently, the ﬁeld of displacements of the cross section of the beam takes the form shown in Fig. 2. Introducing dimensionless function describing displacements ψ (x ) = u1 (x )/ tc1 it can be expressed as follows:

•

the upper facing

u (x , z ) = − z

• •

+ 2ts + tc2 ) ≤ z ≤

1 t 2 c1

≤z≤

σx(s) = Eεx(s),

the lengthwise corrugated main core

1 − 2 tc1

σx(c1) = Ex(c1) εx(c1).

f 1 b1

t 01 , tc1

xf 1 =

bf 1 b 01

,

b 01 tc1

(2)

(indexes 01 and 02 corresponds to the main core and the ∼(c1) inner corrugated layer of facings, respectively); Gxz is dimensionless shear modulus described in details in [19]. E is the Young's modulus of the material of the beam. The Hamilton's principle can be written as follows:

(3)

δ

1 t 2 c1

⎡ dw ⎤ u (x , z ) = − z ⎢ − 2ψ ( x ) ⎥ ⎣ dx ⎦

τxz(c1) = Gxz(c1) γxz(c1),

a1

xb1 =

≤z≤

(4)

1 ulus in which s∼a1 = [(1 − x 01)2 + xb21 ( 2 − x f 1)2]1/2 , x 01 =

+ 2ts + tc2

dw + u1 (x ), dx

∂u , ∂x

(5) ( c 1) ∼ Stiﬀness moduli of the main core can be expressed as Ex(c1) = E x E and xb1x 3 ∼(c1) ∼(c1) Gxz(c1) = Gxz E where E x = 2(s∼ + x01x ) is dimensionless Young's mod-

(1) 1 t 2 c1

γxz(c2) = 0, εx(c1) =

Since the stiﬀness of each layer is diﬀerent, depending on the geometry, the physical relations, according to Hooke's law, have to be written in the form

1 − 2 tc1

dw − u1 (x ), dx

the lower facing

u (x , z ) = − z

1 −( 2 tc1

∂u ∂u , γxz(s) = 0, εx(c2) = , ∂x ∂x ∂u dw γxz(c1) = + = 2ψ (x ). ∂z dx εx(s) =

∫t

t2

[T − (Uε − W )] dt = 0

1

(6)

where 1 ∂w 2 1 T = 2 ∫ ∫ ∫ ρ ( ∂t ) dV – the kinetic energy, Uε = 2 ∫ ∫ ∫ (σx εx + τxz γxz ) dV V V – the elastic strain energy, L 2 ∂w 1 W = 2 ∫ F0 ( ∂x ) dx – the work of the load, t1, t2 – the initial and ﬁnal 0 times, ρ – the mass density of the beam, L – the length of the beam, F0 – the compressive force. Based on the above principle the equations of motion have been derived

⎧ ⎡ ∂ 3ψ ⎤ ∂ 2w ∂ 4w ∂ 2w ⎪ btc1 cρ ρs · + Ebtc31 ⎢2cww 4 − cwψ 3 ⎥ = −F0 2 ⎪ ⎣ ∂t 2 ∂x ∂x ⎦ ∂x ⎨ ψ (x ) ∼(c1) ∂ 2ψ ∂ 3w ⎪ ⎪ cwψ ∂x 3 − 2cψψ ∂x 2 + 4 t 2 Gxz = 0 ⎩ c1 where

Fig. 2. Scheme of deformation of beam's cross section.

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(7)

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E. Magnucka-Blandzi et al.

(s ) cww = cww +

cψψ = 2x1 + (c2) cψψ

x 2 x 02 (c2) c xb2 ww x 2 x 02 (c2) c xb2 ψψ

= 2(x f 2 xb2

+

1 ∼(c1) E , 24 x

(s ) cwψ = cwψ +

1 ∼(c1) + 6 Ex ,

x 2 x 02 (c2) c xb2 wψ

1 ∼(c1) + 6 Ex ,

2.3. Vibrations of the beam

(c2) cwψ = 2(1 + 2x1 + x2 )(x f 2 xb2 + s∼a2 ),

+ s∼a2 ),

(s ) cwψ

The system (7) has been reduced to one equation of motion. The following formulas of three unknown functions in the equations of motion (7) have been assumed

= 2x1 (1 + 2x1 + x2 ),

2 1/2 1 s∼a2 = [(1 − x 02 )2 + xb22 ( 2 − x f 2 ) ] , cρ = ρ∼c1 + 2x2 ρ∼c2 + 4x1, x x ρ∼c1 = 2 x01 (x f 1 xb1 + s∼a1), ρ∼c2 = 2 x02 (x f 2 xb2 + s∼a2 ), b1

x2 =

x1 =

bf 2 b 02

,

ts , tc1

x 02 =

t 02 , tc2

ψ (x, t ) = ψa (t )cos

πx , L

F0 (t ) = Fc + Fa cos(Θt ),

xb2 =

b 02 , tc2

(13) where Fc – the average value of the load, Fa – the amplitude of the load, Θ – the frequency of the load. Than the Mathieu's equation has been obtained

ρs − mass density of the sheet,

d 2wa + Ω 2 [1 − 2μ cos(Θt )] wa (t ) = 0, dt 2

b − width of the beam. (c2) cww

πx , L

b2

tc2 , tc1

xf 2 =

w (x, t ) = wa (t )sin

⎤ ⎛ 1⎡ 1 ⎞ = ⎢x 22 (1 − x 02 )2 ⎜x f 2 xb2 + s∼a2⎟ + (1 + 2x1 + x2 )2 (x f 2 xb2 + s∼a2 ) ⎥ ⎝ 2⎣ 3 ⎠ ⎦ (s ) , cww =

(14)

where

⎛ F ⎞ Ω 2 = ω 2 ⎜1 − c ⎟ , ⎝ F0CR ⎠

1 x1 [16x12 + 6x1 (2 + 3x2 ) + 3(1 + 2x2 + 2x 22 )]. 6

μ=

1 Fa . 2 F0CR − Fc

(15)

The Eqs. (7) are the basis for further analysis of the stability and vibrations of the beam proposed in the present work.

For this the expression for the angular frequency of the beam is as follows

2.2. Buckling of the beam

⎛ π ⎞2 Etc21 ω=⎜ ⎟ cFCR . ⎝ L ⎠ cρ ρs

Stability analysis has been carried out for a simply supported beam compressed by the axial force F0 (Fig. 3). Solving the buckling problem the following system of equilibrium equations have been taken into account, based on the system (7)

⎧ dψ d 2w −F0 = ⎪ 2cww 2 − cwψ ⎪ dx dx Ebtc31 ⎨ d 2ψ ψ ∼(c1) d 3w ⎪ ⎪ cwψ dx 3 − 2cψψ dx 2 + 4 t 2 Gxz = 0. ⎩ c1

The natural frequency in Hertz [Hz] is given by the formula

f=

πx ψ (x ) = ψa cos , L

•

(8)

w (L ) = 0,

⎛L⎞ ψ ⎜ ⎟ = 0. ⎝2⎠

•

(18)

the second unstable region

1 2 μ . 3

(19)

Below there are results of buckling and vibrations analyse performed on an exemplary family of beam. The results obtained with the use of the analytical formulae are compared with the results given by the ﬁnite element method.

(10) 3. Numerical studies – exemplary calculations 3.1. FE model of the plate

π 2Ebtc31 cFCR , L2

(11)

A family of beams of the width b=200 mm and the length L = {1104, 1288, 1472, 1656, 1840, 2024, 2208, 2392} mm has been considered in the analyses. The total thickness of the beam equals t=67 mm. The thickness of the sheets of all layers is equal to ts=0.8 mm. Dimensions of the corrugations are given in Fig. 4. The beam has been considered as a thin-walled structure. For all seven layers linear shell elements have been used to prepare the ﬁnite element model (FE model). Number of ﬁnite elements has been chosen to obtain two elements on the shorter base of the trapezoid. Particular layers of the beam have been connected by applying bonding conditions – the layers can not separate or move relative to each other during the analysis. Details of the FE model as well as deformation of the plate are shown in Fig. 5. The model is simply supported at both ends and the support is applied to the edges of all layers. The force in the buckling analysis is applied to the internal sheets of the faces which are stiﬀer than the external sheets. Since the symmetry plane can be deﬁned in the midlength of the plate only half of the structure has been modelled and

where

cFCR = 2cww −

the ﬁrst unstable region

Ω 1 − 2μ 2 < Θ < Ω 1 +

(9)

After substituting functions (9) into the system (8) the following expression for the critical load has been derived

F0CR =

(17)

2Ω 1 − μ < Θ < 2Ω 1 + μ ,

where wa – amplitude of deﬂection, ψa – amplitude of dimensionless function. The functions (9) identically satisfy the following boundary conditions

w (0) = 0,

ω . 2π

Then two unstable regions, according to the monograph [20] have been determined

The formulas of two unknown functions in the system of equations of statics (8) have been assumed as follows

πx w (x ) = wa sin , L

(16)

2 cwψ

⎛ 2L ⎞2 ∼(c1) 2cψψ + ⎜ ⎟ Gxz ⎝ πtc1 ⎠

. (12)

Fig. 3. Scheme of the simply supported beam.

80

Thin-Walled Structures 112 (2017) 78–82

E. Magnucka-Blandzi et al.

Fig. 4. Dimensions of the beam: a) crosswise direction; b: lengthwise direction.

Fig. 5. FE model of the beam: a) ﬁnite element mesh; b) deformation (x200).

Fig. 6. Results of analyses: a) buckling analysis; b) vibration analysis.

frequencies and the modes of vibration have been determined with the use of modal analysis. For each beam only the ﬁrst mode has been taken into account which had the form of one longitudinal half-wave. The values of natural frequencies obtained numerically and analytically Eq. (17) are presented in Fig. 6b. A good agreement between results obtained with all approaches can be seen for all analysed models.

analysed. The mechanical properties of the material used in analyses correspond to steel: Young's modulus E = 2·10 5 MPa , Poisson's ratio ν = 0.3 and the density ρ = 7800 kg/m3

3.2. Results of buckling and vibration analysis Linear buckling analysis has been performed to determine the buckling load and the corresponding buckling shape. Only the ﬁrst buckling mode is considered which is assumed to be a global one – one longitudinal half-wave should appear in the longitudinal direction. Such shape function has been chosen in the analytical solution (9). However, due to complex geometry of the plate a local buckling may also appear. The results obtained with both methods, analytical and numerical, are shown in Fig. 6a. The length of the beam is shown on the horizontal axis of the plot whereas the buckling load is presented on the vertical axis. In the plot the solid line represents the analytical solution according to Eq. (11), dashed line with diamonds – numerical results form ABAQUS and dashed line with crosses – numerical results from ANSYS. All three lines overlap each other for the beams longer than 1200 mm. The shortest plate buckles locally and the buckling mode has the form of a short waves located on the external sheet of the beam. The natural

4. Summary and conclusions Based on the results presented in the previous sections the properties of the seven layer beam can be easily determined. From the plot presented in Fig. 6 it is seen that the stiﬀness of the plate increase when its length decreases. It manifest itself in an increase of the value of both the critical load and natural frequency. The buckling mode and the mode of vibration is the same and has the form of one longitudinal halfwave. However, for beams shorter than 1200 mm, a local phenomenon prevails – external sheets may wrinkle. This type of buckling may also appear in longer beams if geometrical imperfections are introduced into the model which were not taken into account in this investigation. The small discrepancy between the analytical and numerical results suggests that the analytical model presented in the present paper is formulated correctly. The biggest diﬀerence in buckling load reaches 81

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E. Magnucka-Blandzi et al.

element modeling of corrugated cardboard, Compos. Struct. 88 (2009) 280–289. [7] Q.H. Cheng, H.P. Lee, C. Lu, A numerical analysis approach for evaluating elastic constants of sandwich structures with various core, Compos. Struct. 74 (2006) 226–236. [8] M.E. Biancolini, Evaluation of equivalent stiﬀness properties of corrugated board, Compos. Struct. 69 (2005) 322–328. [9] D.Y. Seong, C.G. Jung, D.Y. Yand, K.J. Moon, D.G. Ahn, Quasi-isotropic bending responses of metallic sandwich plates with bi-directionally corrugated cores, Mater. Des. 31 (2010) 2804–2812. [10] K. Magnucki, M. Kruś, P. Kuligowski, L. Wittenbeck, Strength of sandwich beams with corrugated core under pure bending, in: Proceedings of the 2011 World Congress on Advances in Structural Engineering and Mechanics, ASEM'11, Seoul Korea, CD, 2011, pp. 321–330. [11] K. Magnucki, P. Jasion, M. Kruś, P. Kuligowski, L. Wittenbeck, Strength and buckling of sandwich beams with corrugated core, J. Theor. Appl. Mech. 51 (2013) 15–24. [12] K. Magnucki, P. Kuligowski, M. Kruś, E. Magnucka-Blandzi, Bending and buckling of orthotropic sandwich beams aith three-layer faces, Coupled instabilities in metal structures, in: Proceedings of the 6th International Conference on Coupled Instabilities in Metal Structures held in Glasgow, Scotland, 2012, pp. 587–594. [15] L. Wittenbeck, M. Grygorowicz, P. Paczos, Numerical analysis of sandwich beam with corrugated core under three-point bending, AIP Conf. Proc. 1648 (2015) 800007-1–800007-3. [16] M. Grygorowicz, P. Paczos, L. Wittenbeck, P. Wasilewicz, Experimental three-point bending of sandwich beam with corrugated core, AIP Conf. Proc. 1648 (2015) 800002-1–800002-4. [17] P. Paczos, P. Wasilewicz, E. Magnucka-Blandzi, Experimental and numerical investigations of ﬁve-layered trapezoidal beams, Compos. Struct. 145 (2016) 129–141. [18] E. Carrera, Historical review of Zig-Zag theories for multi-layered plates and shells, Appl. Mech. Rev. 56 (2003) 287–308. [19] J. Lewinski, E. Magnucka-Blandzi, W. Szyc, Shear modulus of elasticity for thinwalled trapezoidal corrugated cores of seven-layer sandwich plates, Eng. Trans. 63 (4) (2015) 421–437. [20] M. Życzkowski, Technical Mechanics, Strength of Construction Elements, Publishing PWN, Warszawa (in Polish), 1988.

0.4% for ABAQUS and 3.4% for ANSYS; in natural frequencies the diﬀerences equals 0.4% and 5.8%, respectively. The structure presented in this paper seems to be a good alternative for classical sandwich structures. It is made of one type of material only but its properties can be controlled in many ways and in a wide range by changing the thickness of the sheets, changing the thickness of both cores and by changing the dimensions of corrugations of both cores. The analytical model of the beam thanks to the closed-form solution can be easily used by engineers when design this type of structures. A number of diﬀerent parameters can be taken into account and theirs inﬂuence on the behaviour of the plate can be investigated. The model can also serve as a base for a further optimisation process. Acknowledgements The project was funded by the National Science Centre allocated on the basis of the decision number DEC-2013/09/B/ST8/00170. References [1] C. Libove, R.E. Hubka, Elastic constants for corrugated-core sandwich plates, Technical Note 2289, Washington: NACA, 1951. [2] H.G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, London, Edinburgh, New York, Sydney, Paris, Braunschweig, 1969. [3] E. Ventsel, T. Krauthammer, Thin plates and shells. Theory, analysis and applications, Marcel Dekker, Inc, Basel, New York, 2001. [4] A.K. Noor, W.S. Burton, C.W. Bert, Computational models for sandwich panels and shells, Appl. Mech. Rev. 49 (1996) 155–199. [5] N. Buannic, P. Cartraud, T. Quesnel, Homogenization of corrugated core sandwich panels, Compos. Struct. 59 (2003) 299–312. [6] N. Talbi, A. Batt, R. Ayad, Y.Q. Guo, An analytical homogenization model for ﬁnite

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