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On buckling loads for edge-loaded orthotropic plates including transverse shear
Composite Structures 65 (2004) 1–6 www.elsevier.com/locate/compstruct
On buckling loads for edge-loaded orthotropic plates including transverse shear Tomas Nordstrand
*
SCA Research, Box 716, 85121 Sundsvall, Sweden Available online 23 April 2004
Abstract Corrugated board usually exhibits low transverse shear stiffness, especially across the corrugations. In the present study the transverse shear is included in an analysis to predict the critical buckling load of an edge-loaded orthotropic linear elastic sandwich plate with all edges simply supported. In the analysis, effective (homogenised) properties of the corrugated core are used. Classical elastic buckling theory of orthotropic sandwich plates predicts that such plates have a finite buckling coefficient when the aspect ratio, i.e. the ratio between the height and width of the plate, becomes small. However, inclusion in the governing equilibrium equations of the additional moments, produced by the membrane stresses in the plate at large transverse shear deformations, gives a buckling coefficient which approaches infinity when the aspect ratio goes to zero. This improvement was first included in the buckling theory of helical springs by Harinx [Proc. Konjlike Nederland Akademie Wettenschappen, vol. 45, 1942, Amsterdam, Holland, pp. 533–539, 650–654] and later applied to orthotropic plates by Bert and Chang [J. Eng. Mech. Division, Proc. Am. Soc. Civil Engrs. 98 (EM6) (1972) 1499–1509]. Some inconsistencies in the latter analysis have been considered. The critical buckling load calculated with corrected analysis is compared with a predicted load obtained using finite element analysis of a corrugated board panel, and also with the critical buckling load obtained from panel compression tests. 2003 Elsevier Ltd. All rights reserved. Keywords: Critical buckling; Corrugated board; Orthotropic; Panel; Transverse shear
1. Introduction Corrugated board usually exhibits low transverse shear stiffness, especially across corrugations [1,2]. This will reduce the critical buckling load according to classical theory of orthotropic sandwich panels [3,4]. In this small-deflection theory it is customary to assume that the membrane forces are unchanged during plate deflection and equal to their initial values. However, due to the large transverse shear strains, the change in direction of the membrane forces over a small plate element cannot be disregarded. This gives additional moments that are introduced in the governing moment equilibrium equations of the panel. Such additional moments were first included in the buckling theory of helical springs by Harinx [5]. Later this was applied to shear deformable plates by Bert and Chang [6] although their work contains some inconsistencies that are corrected herein. Furthermore, in the corrected analysis the expression for *
Tel.: +46-60-193076; fax: +46-60-193804. E-mail address: [email protected] (T. Nordstrand).
0263-8223/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00154-5
the buckling coefficient is shown to reduce to the classical formulation of an orthotropic plate without shear deformation when the transverse shear stiffnesses become large. It is also shown that the buckling coefficient goes to infinity when the height–width ratio of the plate is decreased towards zero. In the following analysis the corrugated board panel is regarded as a laminated shear deformable orthotropic linear elastic plate [7]. Thus, effective (homogenised) properties of the corrugated core are used [8,9]. The papers in the facings are also regarded as orthotropic linear elastic materials [10,11]. The analysis was used to confirm predicted critical buckling load from a finite element analysis of a corrugated board panel modelled with eight-node multi-layered isoparametric shell elements [12–14]. Predicted critical buckling load is also compared to buckling loads obtained from compression tests of corrugated board panels [15]. 2. Analysis Fig. 1 shows an element of a corrugated board panel of thickness h. Core height is hc and wavelength of
2
T. Nordstrand / Composite Structures 65 (2004) 1–6
z tf
h hc t c
3
y,CD 2 1
λM
b
x,MD
Fig. 1. Schematic diagram of corrugated board.
corrugations is kM . Facing thickness is tf and thickness of core sheet is tc . The principal axes of elastic symmetry of the face sheets and the core are aligned with the Cartesian coordinate system xyz. The 2-axes of the corrugated medium is parallel with the y-axes. It is assumed that the facings and core sheet are thin compared to the total thickness of the panel and that the transverse shear strains are uniform in the core layer. Furthermore, the deflections and slopes are assumed to be small compared to the thickness of the plate. Transverse shear deformation of the plate is accommodated by assuming that cross-sections remains straight but not necessarily normal to the mid-plane of the plate during bending [6]. The membrane forces Nx , Ny , Nxy , transverse shear forces Qxz , Qyz , bending and twisting moments Mx , My , Mxy are acting on respective four sides of a plate element (see Fig. 2). The transverse shear strains (see Fig. 3) are determined by [4] cxz ¼ A55 Qxz
ð1Þ
cyz ¼ A44 Qyz
where A44 and A55 are the transverse shear stiffnesses [1,2]. The plate displacement w is then related to the applied moments as follows [4]: x
z dy Mxydy M ydx h
M xdy Nx dy
M xydx Nxydx Nxydy
Qxdy Nydx
oby obx D12 ox oy oby obx D22 My ¼ D12 ox oy 1 obx oby Mxy ¼ D66 þ 2 oy ox
Mx ¼ D11
Qydx
Fig. 2. Forces and moments acting on a plate element h dy dx.
ð2Þ
where bx ¼
ow cxz ; ox
by ¼
ow cyz oy
ð3Þ
and D11 , D22 , D12 and D66 are the bending and twisting stiffnesses defined according to Refs. [7–11]. Fig. 3 shows a cross-sectional view of the deformed plate element in the x–z plane. Considering that the plate is loaded in compression, the normal forces Nx , Ny and the shear force Nxy are much larger than the transverse shear forces Qyz and Qxz and have to be accounted for in the lateral equilibrium of the differential plate element. Subsequently, after algebraic manipulation the equation of equilibrium in the zdirection is obtained as, oby oQxz oQyz ob þ þ Nx x þ Ny ox oy ox oy obx oby þ 2Nxy þ ¼0 oy ox
dx
y
Fig. 3. Cross-section of a differential plate element.
ð4Þ
The transverse shear strains at the left and right crosssections in Fig. 3 reduce the slope slightly more of the right cross-section than the left cross-sections. This will rotate the normal forces so that their action will not be through the centre of the differential plate element. Consequently, the normal forces will generate additional moments. These moments are taken into account in the present theory. This is the basic difference between the present theory and the classical sandwich theory [4]. The derivation of moment equilibrium around an axis through the centre of the differential element and parallel with the y-axes in Fig. 3 is then as follows:
T. Nordstrand / Composite Structures 65 (2004) 1–6
oMx dx dy Mx dy þ Myx dx Mx þ ox oMyx oQxz dy dx Qxz þ dx dy dx Myx þ oy ox oNx oc dx dy dx cxz þ xz dx ¼ 0 þ Nx þ ox ox
only one-term solutions are used for w, cxz and cyz , respectively. w ¼ W sinðAxÞ sinðByÞ cxz ¼ Cxz cosðAxÞ sinðByÞ ð5Þ
If both sides in Eq. (5) are divided by dxdy and letting dx ! 0 and dy ! 0, Eq. (5) reduces to oMx oMxy Qxz þ Nx cxz ¼ 0 ox oy
ð6aÞ
Similarly, moment equilibrium around an axis through the centre of the differential element parallel with the x-axis yields oMy oMyz Qyz þ Ny cyz ¼ 0 oy ox
Mxy ¼ 0
cxz ¼ 0
and at x ¼ 0 and x ¼ a w ¼ 0 Mx ¼ 0 Mxy ¼ 0
cyz ¼ 0
According to Navier’s procedure [7], a solution of the three simultaneous differential equations that satisfy the boundary conditions above can be obtained by assuming that the out-of-plane displacement, w, and transverse shear strains cxz , cyz can be represented by double trigonometric series. However, in the present analysis y a
z
ð7Þ
cyz ¼ Cyz sinðAxÞ cosðByÞ where W , Cxz and Cyz are corresponding amplitudes. A and B are A¼
mp ; a
B¼
np b
ð8Þ
Integers m and n are number of buckles, i.e. m and n half sine waves, in the x and y directions. In the subsequent analysis it is convenient to define a number of parameters of the homogenised sandwich plate [4]:
ð6bÞ
Substitution of Eqs. (1)–(3) in equilibrium Eqs. (4) and (6) forms a system of three simultaneous differential equations in terms of the out-of-plane displacement w and the transverse shear strains cxz , cyz . It is assumed that the plate is simply supported along its edges, i.e. the edges of the panel are prevented from moving out-of-plane and are not rotationally restrained. The edges are also free to move in-plane and transverse shear strains are prevented by edge stiffeners. The edges of the panel, parallel to the x-axis, are compressed uniformly by a load of intensity, py , per unit length (see Fig. 4). Thus, boundary conditions at y ¼ 0 and y ¼ b w ¼ 0 My ¼ 0
3
k¼
na 2
;
f¼
mb D12 þ 2D66 g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; D11 D22
rffiffiffiffiffiffiffi D11 D66 ; w ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; D22 D11 D22 A55 k A44 k ; syz ¼ sxz ¼ py px
where k¼
a 2 p y;crit pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; np D11 D22
Pcrit;theor ¼ a py;crit
ð9Þ
py;crit , is the critical load intensity (load/unit length) to cause panel buckling. The total critical buckling load is Pcrit;theor . Substitution of the trigonometric expressions Eq. (7) in the differential equations (4), (6a) and (6b) leads to the following system of equations shown in matrix form 2 3 k A1 ðk þ syz Þ Bk sxz 6 7 k 6 1 þ kg 1 1 þ w þ k þ s 7 ðg wÞ yz 6f 7 A kf B 4 5 w Bk f þ k þ sxz kf þ g A1 ðg wÞ 2 3 2 3 W 0 6 7 6 7 ð10Þ 4 Cxz 5 ¼ 4 0 5 Cyz 0 Correct expressions of the elements in the stiffness matrix of Eq. (10) are given instead of those in the stiffness formulation [6]. Solution of Eq. (10) different than the trivial one, W ¼ Cxz ¼ Cyz ¼ 0 are possible when the determinant of the matrix vanishes. This criterion leads to a second order equation of k Pk 2 þ Qk þ R ¼ 0
p
b
y
where P ¼fþ
x
ð11Þ
w þ sxz k
ð12aÞ 2
Fig. 4. Schematic diagram of a simply supported panel in edgewise compression.
Q ¼ ðsyz þ w kf 2gÞP þ
ðkf þ gÞ ðg wÞ k
2
ð12bÞ
4
2 w ðkf þ gÞ PH fþ H þ syz k k 1 syz P kf þ 2g þ kf
10 9
ð12cÞ
and 1 H ¼ 1 þ w kf þ 2g þ g2 kf
ð12dÞ
The non-trivial solution of Eq. (8) can thus be found when rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q Q2 R
ð13Þ k¼ 2P 4P 2 P where only positive values of k are valid since the buckling load must be compressive. The critical buckling load, Pcrit;theor , is given by Eq. (9), where n ¼ 1 and k is the smallest positive value given by Eq. (11). Using Eq. (12), the two ratios in Eq. (13) are: 3 2 2
2
Q syz 6 w kf 2g ðkf þ gÞ ðg wÞ 7 ¼ 41 þ þ 5 w fþ 2P syz 2 s s k 1þ k xz yz
sxz
ð14aÞ 2 2 R 1 ðkf þ gÞ 6 ¼ syz 4 kf þ 2g þ þ w fþ P kf sxz k 1 þ sxzk 3 1 þ w kf þ 2g þ kf1 g2 7 þ 5 w fþ k syz 1 þ sxz
8
With Shear Without Shear
7
Sandwich theory
6 5 4 3 2 0
0.5
1
1.5
2
Fig. 5. Buckling coefficient k, according to present theory, Eqs. (13) and (14), the theory for orthotropic plate without shear [4] and the classical sandwich theory [4].
pffiffiffiffiffiffiffiffiffiffiffi 1 1 1 þ v ¼ 1 þ v v2 þ 2 4 where 4 kf þ 2g þ kf1 v¼ syz
ð17Þ
ð18Þ
If Eqs. (17) and (18) substituted into Eq. (16), the expression on the right hand side is reduced to the buckling coefficient for an orthotropic plate [4] ð14bÞ k ¼ kf þ 2g þ
Attention is now turned to analysis of the limit case of infinite large transverse shear stiffnesses in order to show that the buckling coefficient k, determined by Eq. (13), for that limit is reduced to the buckling coefficient for orthotropic plates without shear deformation [7]. If the transverse shear stiffnesses A44 and A55 , corresponding to syz and sxz , approach infinity then it is evident from Eq. (14) that Q syz ! 2P 2
Buckling coefficient, k
R¼
T. Nordstrand / Composite Structures 65 (2004) 1–6
ð15aÞ
1 kf
ð19Þ
when syz goes to infinity. In Fig. 5 the buckling coefficient for a plate with and without transverse shear is plotted versus the plate width/height ratio. The material data used is typical for a common corrugated board grade and defined in Table 2. Notice that the buckling coefficient of the plate including transverse shear has no limit when the plate height/width becomes small as classical sandwich buckling theory predicts [3,4].
and R 1 ! syz kf þ 2g þ P kf Substitution of Eq. (15) into Eq. (13) gives vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ffi1 u 1 u 4 kf þ 2g þ t kf C syz B k ¼ @1 1 þ A syz 2
ð15bÞ
ð16Þ
The square root in Eq. (16) can be expanded according to the binomial series
3. Comparison with finite element analysis of a corrugated board panel In a finite element analysis of a simply supported corrugated board panel, with side lengths a ¼ b ¼ 400 mm, following eigenvalue analysis was made to obtain the critical buckling load [12]. A multiply eight node isoparametric shell element where first order transverse shear deformation is accounted for is used in the analysis. A quarter of the panel was modelled, due to symmetry, in a 6 6 element mesh. The side length ratio
T. Nordstrand / Composite Structures 65 (2004) 1–6
5
Table 1 Effective material properties of the layers in the panel Layer
Ex (GPa)
Ey (GPa)
Ez (GPa)
1 2 3
8.25 0.005 8.18
2.9 0.231 3.12
2.9 3.0 3.12
Gxy (GPa)
Gxz (GPa)
Gyz (GPa)
1 2 3
1.89 0.005 1.95
0.007 0.0035 0.007
0.070 0.035 0.070
mxy
mxz a
myz a
0.43 0.05 0.43
0.01 0.01 0.01
0.01 0.01 0.01
1 2 3 a
The Poisson’s ratios are assumed small because of the plane stress condition in the board. Fig. 6. Rig and corrugated board panel tested under compression.
between the corner element and mid element was 1:5. The finite element eigenvalue analysis is ð½K þ v½S ref Þfwg ¼ f0g
ð20Þ
where [K] is the global stiffness matrix of the finite element model, ½S ref is the ‘‘stress stiffness matrix’’, v is the factor used to multiply the loads which generate the stresses and fwg is the generalised displacement vector of the nodes [13]. The load fP g is also scaled by v and it alters the intensity of the membrane stresses but not the distribution of the stresses such that fP g ¼ vfP gref () ½S ¼ v½S ref
Table 2 Corrugated board data (transverse shear stiffness is measured) Basis weight (g/m2 ) Thickness (mm) Corrugated wavelength (mm) ECT neck down (kN/m) Bending stiffness (N m)
Transverse shear stiffness (kN/m)
D11 D22 D12 D66 A44 A55
556 4.02 7.26 10.14 14.6 5.43 2.71 3.34 39.2 5.6
ð21Þ
As v is increased, the overall stiffness of the plate, (½K þ ½S ), is reduced until a critical load fP gcr corresponding to the eigenvalue vcr is reached and the plate becomes unstable, i.e. detð½K þ ½S Þ goes to zero. The corrugated board analysed has 0.23 mm thick liners and a corrugated medium with wall thickness 0.25 mm and wavelength 7.26 mm. The height of the core layer is hc ¼ 3:65 mm (see Fig. 1). Using the material data in Table 1, the buckling load of the corrugated board panel was calculated to Pcr;fem ¼ 849 N. This value is in excellent agreement with the value obtained by the closed form solution Pcr;theor ¼ 846 N (see Eqs. (9) and (13)). Sandwich theory gives a critical buckling Pcr;sand ¼ 815 N and an orthotropic plate without shear Pcr;ortho ¼ 898 N.
4. Comparison with experiments Panels size 400 · 400 mm were cut from corrugated board and tested under compression in a rig that furnishes simply supported boundary conditions [15]. Panels were oriented with the cross direction (CD) in the direction of loading (see Fig. 6). Material data for the board is given in Table 2. The critical buckling load as estimated from the test results
by means of a non-linear regression analysis method [15] was 814 N. This value is consistent with the analytically predicted critical buckling load of 870 N using the present buckling analysis, i.e. an analysis of a plate including transverse shear deformation. In comparison, a plate without shear deformation is predicted to have a critical buckling load of 924 N, which is exactly the same result as obtained from the classical theory of orthotropic plates [7].
5. Conclusions An explicit equation for the buckling load of a simply supported orthotropic linear elastic plate in edgewise compression has been derived taking into account first order transverse shear deformation. There is major difference between present theory and classical sandwich theory in the additional moments that are introduced in the governing moment equilibrium equations of the panel, due to change in directions of the membrane forces over a small plate element that has large transverse shear strains. When the transverse shear stiffness goes to infinity the critical buckling load, predicted by the present theory, is shown to be reduced to the critical
6
T. Nordstrand / Composite Structures 65 (2004) 1–6
buckling load of an orthotropic plate without transverse shear deformation. Furthermore, the buckling coefficient does not have a limit in the present theory when the plate height/width becomes small, as classical sandwich buckling theory predicts. The present theory is approximate due to one-term approximations of the deflection wðx; yÞ and the transverse shear strains cxz ðx; yÞ and cyz ðx; yÞ. Verification by finite element analysis suggests that the present explicit equation for the buckling load is accurate, the deviation is typically less than 0.5%. However, the discrepancy is larger between present theoretical buckling load and the experimental buckling load of corrugated board panels. This may partly be due to the difficulties involved in evaluation of the buckling load from the experimental results [15] partly due to the non-linear material behaviour of paper.
References [1] Nordstrand TM, Allen HG, Carlsson LA. Transverse shear stiffness of structural core sandwich. Compos Struct 1994;27:317– 29. [2] Nordstrand T, Carlsson LA. Evaluation of transverse shear stiffness of structural core sandwich plates. Compos Struct 1997;37:145–53.
[3] Reissner E. The effect of transverse shear deformation on bending of elastic plates. J Appl Mech 1945;12:A69–77. [4] Plantema FJ. Sandwich construction. New York, NY: John Wiley & Sons, Inc.; 1966. [5] Harinx JA. On buckling and the lateral rigidity of helical compression springs. Proceedings of the Konjlike Nederland Akademie Wettenschappen 45. Amsterdam, Holland, 1942. p. 533–9, 650–4. [6] Bert CW, Chang S. Shear-flexible orthotropic plates loaded inplane. J Eng Mech Division, Proc Am Soc Civil Eng 1972;98(EM6):1499–509. [7] Jones RM. Mechanics of composite materials. New York: Hemisphere Publishing Corp.; 1975. [8] Carlsson LA, Nordstrand T, Westerlind B. On the elastic stiffnesses of corrugated core sandwich. J Sandwich Sturct Mater 2001;3:253–67. [9] Libove C, Hubka RE. Elastic constants for corrugated core sandwich plates. NACA, TN 2289, 1951. [10] Baum GA, Brennan DC, Habeger CC. Orthotropic elastic constants of paper. Tappi 1981;64:97–101. [11] Paetow R, G€ ottsching L. Poisson’s ratio of paper. Das Papier 1990;6:229–37. [12] Nordstrand T. Parametrical study of the post-buckling strength of structural core sandwich panels. Compos Struct 1995;30:441–51. [13] ANSYS User’s Manual, Swanson Analysis System Inc., vol. 1–2, 1989. [14] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finite elements. Int J Numer Meth Eng 1970;(2):419–51. [15] Nordstrand T. Analysis and testing of corrugated board panels into the post-buckling regime. Compos Struct 2004;63:189–99.
On buckling loads for edge-loaded orthotropic plates including transverse shear Tomas Nordstrand
*
SCA Research, Box 716, 85121 Sundsvall, Sweden Available online 23 April 2004
Abstract Corrugated board usually exhibits low transverse shear stiffness, especially across the corrugations. In the present study the transverse shear is included in an analysis to predict the critical buckling load of an edge-loaded orthotropic linear elastic sandwich plate with all edges simply supported. In the analysis, effective (homogenised) properties of the corrugated core are used. Classical elastic buckling theory of orthotropic sandwich plates predicts that such plates have a finite buckling coefficient when the aspect ratio, i.e. the ratio between the height and width of the plate, becomes small. However, inclusion in the governing equilibrium equations of the additional moments, produced by the membrane stresses in the plate at large transverse shear deformations, gives a buckling coefficient which approaches infinity when the aspect ratio goes to zero. This improvement was first included in the buckling theory of helical springs by Harinx [Proc. Konjlike Nederland Akademie Wettenschappen, vol. 45, 1942, Amsterdam, Holland, pp. 533–539, 650–654] and later applied to orthotropic plates by Bert and Chang [J. Eng. Mech. Division, Proc. Am. Soc. Civil Engrs. 98 (EM6) (1972) 1499–1509]. Some inconsistencies in the latter analysis have been considered. The critical buckling load calculated with corrected analysis is compared with a predicted load obtained using finite element analysis of a corrugated board panel, and also with the critical buckling load obtained from panel compression tests. 2003 Elsevier Ltd. All rights reserved. Keywords: Critical buckling; Corrugated board; Orthotropic; Panel; Transverse shear
1. Introduction Corrugated board usually exhibits low transverse shear stiffness, especially across corrugations [1,2]. This will reduce the critical buckling load according to classical theory of orthotropic sandwich panels [3,4]. In this small-deflection theory it is customary to assume that the membrane forces are unchanged during plate deflection and equal to their initial values. However, due to the large transverse shear strains, the change in direction of the membrane forces over a small plate element cannot be disregarded. This gives additional moments that are introduced in the governing moment equilibrium equations of the panel. Such additional moments were first included in the buckling theory of helical springs by Harinx [5]. Later this was applied to shear deformable plates by Bert and Chang [6] although their work contains some inconsistencies that are corrected herein. Furthermore, in the corrected analysis the expression for *
Tel.: +46-60-193076; fax: +46-60-193804. E-mail address: [email protected] (T. Nordstrand).
0263-8223/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00154-5
the buckling coefficient is shown to reduce to the classical formulation of an orthotropic plate without shear deformation when the transverse shear stiffnesses become large. It is also shown that the buckling coefficient goes to infinity when the height–width ratio of the plate is decreased towards zero. In the following analysis the corrugated board panel is regarded as a laminated shear deformable orthotropic linear elastic plate [7]. Thus, effective (homogenised) properties of the corrugated core are used [8,9]. The papers in the facings are also regarded as orthotropic linear elastic materials [10,11]. The analysis was used to confirm predicted critical buckling load from a finite element analysis of a corrugated board panel modelled with eight-node multi-layered isoparametric shell elements [12–14]. Predicted critical buckling load is also compared to buckling loads obtained from compression tests of corrugated board panels [15]. 2. Analysis Fig. 1 shows an element of a corrugated board panel of thickness h. Core height is hc and wavelength of
2
T. Nordstrand / Composite Structures 65 (2004) 1–6
z tf
h hc t c
3
y,CD 2 1
λM
b
x,MD
Fig. 1. Schematic diagram of corrugated board.
corrugations is kM . Facing thickness is tf and thickness of core sheet is tc . The principal axes of elastic symmetry of the face sheets and the core are aligned with the Cartesian coordinate system xyz. The 2-axes of the corrugated medium is parallel with the y-axes. It is assumed that the facings and core sheet are thin compared to the total thickness of the panel and that the transverse shear strains are uniform in the core layer. Furthermore, the deflections and slopes are assumed to be small compared to the thickness of the plate. Transverse shear deformation of the plate is accommodated by assuming that cross-sections remains straight but not necessarily normal to the mid-plane of the plate during bending [6]. The membrane forces Nx , Ny , Nxy , transverse shear forces Qxz , Qyz , bending and twisting moments Mx , My , Mxy are acting on respective four sides of a plate element (see Fig. 2). The transverse shear strains (see Fig. 3) are determined by [4] cxz ¼ A55 Qxz
ð1Þ
cyz ¼ A44 Qyz
where A44 and A55 are the transverse shear stiffnesses [1,2]. The plate displacement w is then related to the applied moments as follows [4]: x
z dy Mxydy M ydx h
M xdy Nx dy
M xydx Nxydx Nxydy
Qxdy Nydx
oby obx D12 ox oy oby obx D22 My ¼ D12 ox oy 1 obx oby Mxy ¼ D66 þ 2 oy ox
Mx ¼ D11
Qydx
Fig. 2. Forces and moments acting on a plate element h dy dx.
ð2Þ
where bx ¼
ow cxz ; ox
by ¼
ow cyz oy
ð3Þ
and D11 , D22 , D12 and D66 are the bending and twisting stiffnesses defined according to Refs. [7–11]. Fig. 3 shows a cross-sectional view of the deformed plate element in the x–z plane. Considering that the plate is loaded in compression, the normal forces Nx , Ny and the shear force Nxy are much larger than the transverse shear forces Qyz and Qxz and have to be accounted for in the lateral equilibrium of the differential plate element. Subsequently, after algebraic manipulation the equation of equilibrium in the zdirection is obtained as, oby oQxz oQyz ob þ þ Nx x þ Ny ox oy ox oy obx oby þ 2Nxy þ ¼0 oy ox
dx
y
Fig. 3. Cross-section of a differential plate element.
ð4Þ
The transverse shear strains at the left and right crosssections in Fig. 3 reduce the slope slightly more of the right cross-section than the left cross-sections. This will rotate the normal forces so that their action will not be through the centre of the differential plate element. Consequently, the normal forces will generate additional moments. These moments are taken into account in the present theory. This is the basic difference between the present theory and the classical sandwich theory [4]. The derivation of moment equilibrium around an axis through the centre of the differential element and parallel with the y-axes in Fig. 3 is then as follows:
T. Nordstrand / Composite Structures 65 (2004) 1–6
oMx dx dy Mx dy þ Myx dx Mx þ ox oMyx oQxz dy dx Qxz þ dx dy dx Myx þ oy ox oNx oc dx dy dx cxz þ xz dx ¼ 0 þ Nx þ ox ox
only one-term solutions are used for w, cxz and cyz , respectively. w ¼ W sinðAxÞ sinðByÞ cxz ¼ Cxz cosðAxÞ sinðByÞ ð5Þ
If both sides in Eq. (5) are divided by dxdy and letting dx ! 0 and dy ! 0, Eq. (5) reduces to oMx oMxy Qxz þ Nx cxz ¼ 0 ox oy
ð6aÞ
Similarly, moment equilibrium around an axis through the centre of the differential element parallel with the x-axis yields oMy oMyz Qyz þ Ny cyz ¼ 0 oy ox
Mxy ¼ 0
cxz ¼ 0
and at x ¼ 0 and x ¼ a w ¼ 0 Mx ¼ 0 Mxy ¼ 0
cyz ¼ 0
According to Navier’s procedure [7], a solution of the three simultaneous differential equations that satisfy the boundary conditions above can be obtained by assuming that the out-of-plane displacement, w, and transverse shear strains cxz , cyz can be represented by double trigonometric series. However, in the present analysis y a
z
ð7Þ
cyz ¼ Cyz sinðAxÞ cosðByÞ where W , Cxz and Cyz are corresponding amplitudes. A and B are A¼
mp ; a
B¼
np b
ð8Þ
Integers m and n are number of buckles, i.e. m and n half sine waves, in the x and y directions. In the subsequent analysis it is convenient to define a number of parameters of the homogenised sandwich plate [4]:
ð6bÞ
Substitution of Eqs. (1)–(3) in equilibrium Eqs. (4) and (6) forms a system of three simultaneous differential equations in terms of the out-of-plane displacement w and the transverse shear strains cxz , cyz . It is assumed that the plate is simply supported along its edges, i.e. the edges of the panel are prevented from moving out-of-plane and are not rotationally restrained. The edges are also free to move in-plane and transverse shear strains are prevented by edge stiffeners. The edges of the panel, parallel to the x-axis, are compressed uniformly by a load of intensity, py , per unit length (see Fig. 4). Thus, boundary conditions at y ¼ 0 and y ¼ b w ¼ 0 My ¼ 0
3
k¼
na 2
;
f¼
mb D12 þ 2D66 g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; D11 D22
rffiffiffiffiffiffiffi D11 D66 ; w ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; D22 D11 D22 A55 k A44 k ; syz ¼ sxz ¼ py px
where k¼
a 2 p y;crit pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; np D11 D22
Pcrit;theor ¼ a py;crit
ð9Þ
py;crit , is the critical load intensity (load/unit length) to cause panel buckling. The total critical buckling load is Pcrit;theor . Substitution of the trigonometric expressions Eq. (7) in the differential equations (4), (6a) and (6b) leads to the following system of equations shown in matrix form 2 3 k A1 ðk þ syz Þ Bk sxz 6 7 k 6 1 þ kg 1 1 þ w þ k þ s 7 ðg wÞ yz 6f 7 A kf B 4 5 w Bk f þ k þ sxz kf þ g A1 ðg wÞ 2 3 2 3 W 0 6 7 6 7 ð10Þ 4 Cxz 5 ¼ 4 0 5 Cyz 0 Correct expressions of the elements in the stiffness matrix of Eq. (10) are given instead of those in the stiffness formulation [6]. Solution of Eq. (10) different than the trivial one, W ¼ Cxz ¼ Cyz ¼ 0 are possible when the determinant of the matrix vanishes. This criterion leads to a second order equation of k Pk 2 þ Qk þ R ¼ 0
p
b
y
where P ¼fþ
x
ð11Þ
w þ sxz k
ð12aÞ 2
Fig. 4. Schematic diagram of a simply supported panel in edgewise compression.
Q ¼ ðsyz þ w kf 2gÞP þ
ðkf þ gÞ ðg wÞ k
2
ð12bÞ
4
2 w ðkf þ gÞ PH fþ H þ syz k k 1 syz P kf þ 2g þ kf
10 9
ð12cÞ
and 1 H ¼ 1 þ w kf þ 2g þ g2 kf
ð12dÞ
The non-trivial solution of Eq. (8) can thus be found when rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q Q2 R
ð13Þ k¼ 2P 4P 2 P where only positive values of k are valid since the buckling load must be compressive. The critical buckling load, Pcrit;theor , is given by Eq. (9), where n ¼ 1 and k is the smallest positive value given by Eq. (11). Using Eq. (12), the two ratios in Eq. (13) are: 3 2 2
2
Q syz 6 w kf 2g ðkf þ gÞ ðg wÞ 7 ¼ 41 þ þ 5 w fþ 2P syz 2 s s k 1þ k xz yz
sxz
ð14aÞ 2 2 R 1 ðkf þ gÞ 6 ¼ syz 4 kf þ 2g þ þ w fþ P kf sxz k 1 þ sxzk 3 1 þ w kf þ 2g þ kf1 g2 7 þ 5 w fþ k syz 1 þ sxz
8
With Shear Without Shear
7
Sandwich theory
6 5 4 3 2 0
0.5
1
1.5
2
Fig. 5. Buckling coefficient k, according to present theory, Eqs. (13) and (14), the theory for orthotropic plate without shear [4] and the classical sandwich theory [4].
pffiffiffiffiffiffiffiffiffiffiffi 1 1 1 þ v ¼ 1 þ v v2 þ 2 4 where 4 kf þ 2g þ kf1 v¼ syz
ð17Þ
ð18Þ
If Eqs. (17) and (18) substituted into Eq. (16), the expression on the right hand side is reduced to the buckling coefficient for an orthotropic plate [4] ð14bÞ k ¼ kf þ 2g þ
Attention is now turned to analysis of the limit case of infinite large transverse shear stiffnesses in order to show that the buckling coefficient k, determined by Eq. (13), for that limit is reduced to the buckling coefficient for orthotropic plates without shear deformation [7]. If the transverse shear stiffnesses A44 and A55 , corresponding to syz and sxz , approach infinity then it is evident from Eq. (14) that Q syz ! 2P 2
Buckling coefficient, k
R¼
T. Nordstrand / Composite Structures 65 (2004) 1–6
ð15aÞ
1 kf
ð19Þ
when syz goes to infinity. In Fig. 5 the buckling coefficient for a plate with and without transverse shear is plotted versus the plate width/height ratio. The material data used is typical for a common corrugated board grade and defined in Table 2. Notice that the buckling coefficient of the plate including transverse shear has no limit when the plate height/width becomes small as classical sandwich buckling theory predicts [3,4].
and R 1 ! syz kf þ 2g þ P kf Substitution of Eq. (15) into Eq. (13) gives vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ffi1 u 1 u 4 kf þ 2g þ t kf C syz B k ¼ @1 1 þ A syz 2
ð15bÞ
ð16Þ
The square root in Eq. (16) can be expanded according to the binomial series
3. Comparison with finite element analysis of a corrugated board panel In a finite element analysis of a simply supported corrugated board panel, with side lengths a ¼ b ¼ 400 mm, following eigenvalue analysis was made to obtain the critical buckling load [12]. A multiply eight node isoparametric shell element where first order transverse shear deformation is accounted for is used in the analysis. A quarter of the panel was modelled, due to symmetry, in a 6 6 element mesh. The side length ratio
T. Nordstrand / Composite Structures 65 (2004) 1–6
5
Table 1 Effective material properties of the layers in the panel Layer
Ex (GPa)
Ey (GPa)
Ez (GPa)
1 2 3
8.25 0.005 8.18
2.9 0.231 3.12
2.9 3.0 3.12
Gxy (GPa)
Gxz (GPa)
Gyz (GPa)
1 2 3
1.89 0.005 1.95
0.007 0.0035 0.007
0.070 0.035 0.070
mxy
mxz a
myz a
0.43 0.05 0.43
0.01 0.01 0.01
0.01 0.01 0.01
1 2 3 a
The Poisson’s ratios are assumed small because of the plane stress condition in the board. Fig. 6. Rig and corrugated board panel tested under compression.
between the corner element and mid element was 1:5. The finite element eigenvalue analysis is ð½K þ v½S ref Þfwg ¼ f0g
ð20Þ
where [K] is the global stiffness matrix of the finite element model, ½S ref is the ‘‘stress stiffness matrix’’, v is the factor used to multiply the loads which generate the stresses and fwg is the generalised displacement vector of the nodes [13]. The load fP g is also scaled by v and it alters the intensity of the membrane stresses but not the distribution of the stresses such that fP g ¼ vfP gref () ½S ¼ v½S ref
Table 2 Corrugated board data (transverse shear stiffness is measured) Basis weight (g/m2 ) Thickness (mm) Corrugated wavelength (mm) ECT neck down (kN/m) Bending stiffness (N m)
Transverse shear stiffness (kN/m)
D11 D22 D12 D66 A44 A55
556 4.02 7.26 10.14 14.6 5.43 2.71 3.34 39.2 5.6
ð21Þ
As v is increased, the overall stiffness of the plate, (½K þ ½S ), is reduced until a critical load fP gcr corresponding to the eigenvalue vcr is reached and the plate becomes unstable, i.e. detð½K þ ½S Þ goes to zero. The corrugated board analysed has 0.23 mm thick liners and a corrugated medium with wall thickness 0.25 mm and wavelength 7.26 mm. The height of the core layer is hc ¼ 3:65 mm (see Fig. 1). Using the material data in Table 1, the buckling load of the corrugated board panel was calculated to Pcr;fem ¼ 849 N. This value is in excellent agreement with the value obtained by the closed form solution Pcr;theor ¼ 846 N (see Eqs. (9) and (13)). Sandwich theory gives a critical buckling Pcr;sand ¼ 815 N and an orthotropic plate without shear Pcr;ortho ¼ 898 N.
4. Comparison with experiments Panels size 400 · 400 mm were cut from corrugated board and tested under compression in a rig that furnishes simply supported boundary conditions [15]. Panels were oriented with the cross direction (CD) in the direction of loading (see Fig. 6). Material data for the board is given in Table 2. The critical buckling load as estimated from the test results
by means of a non-linear regression analysis method [15] was 814 N. This value is consistent with the analytically predicted critical buckling load of 870 N using the present buckling analysis, i.e. an analysis of a plate including transverse shear deformation. In comparison, a plate without shear deformation is predicted to have a critical buckling load of 924 N, which is exactly the same result as obtained from the classical theory of orthotropic plates [7].
5. Conclusions An explicit equation for the buckling load of a simply supported orthotropic linear elastic plate in edgewise compression has been derived taking into account first order transverse shear deformation. There is major difference between present theory and classical sandwich theory in the additional moments that are introduced in the governing moment equilibrium equations of the panel, due to change in directions of the membrane forces over a small plate element that has large transverse shear strains. When the transverse shear stiffness goes to infinity the critical buckling load, predicted by the present theory, is shown to be reduced to the critical
6
T. Nordstrand / Composite Structures 65 (2004) 1–6
buckling load of an orthotropic plate without transverse shear deformation. Furthermore, the buckling coefficient does not have a limit in the present theory when the plate height/width becomes small, as classical sandwich buckling theory predicts. The present theory is approximate due to one-term approximations of the deflection wðx; yÞ and the transverse shear strains cxz ðx; yÞ and cyz ðx; yÞ. Verification by finite element analysis suggests that the present explicit equation for the buckling load is accurate, the deviation is typically less than 0.5%. However, the discrepancy is larger between present theoretical buckling load and the experimental buckling load of corrugated board panels. This may partly be due to the difficulties involved in evaluation of the buckling load from the experimental results [15] partly due to the non-linear material behaviour of paper.
References [1] Nordstrand TM, Allen HG, Carlsson LA. Transverse shear stiffness of structural core sandwich. Compos Struct 1994;27:317– 29. [2] Nordstrand T, Carlsson LA. Evaluation of transverse shear stiffness of structural core sandwich plates. Compos Struct 1997;37:145–53.
[3] Reissner E. The effect of transverse shear deformation on bending of elastic plates. J Appl Mech 1945;12:A69–77. [4] Plantema FJ. Sandwich construction. New York, NY: John Wiley & Sons, Inc.; 1966. [5] Harinx JA. On buckling and the lateral rigidity of helical compression springs. Proceedings of the Konjlike Nederland Akademie Wettenschappen 45. Amsterdam, Holland, 1942. p. 533–9, 650–4. [6] Bert CW, Chang S. Shear-flexible orthotropic plates loaded inplane. J Eng Mech Division, Proc Am Soc Civil Eng 1972;98(EM6):1499–509. [7] Jones RM. Mechanics of composite materials. New York: Hemisphere Publishing Corp.; 1975. [8] Carlsson LA, Nordstrand T, Westerlind B. On the elastic stiffnesses of corrugated core sandwich. J Sandwich Sturct Mater 2001;3:253–67. [9] Libove C, Hubka RE. Elastic constants for corrugated core sandwich plates. NACA, TN 2289, 1951. [10] Baum GA, Brennan DC, Habeger CC. Orthotropic elastic constants of paper. Tappi 1981;64:97–101. [11] Paetow R, G€ ottsching L. Poisson’s ratio of paper. Das Papier 1990;6:229–37. [12] Nordstrand T. Parametrical study of the post-buckling strength of structural core sandwich panels. Compos Struct 1995;30:441–51. [13] ANSYS User’s Manual, Swanson Analysis System Inc., vol. 1–2, 1989. [14] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finite elements. Int J Numer Meth Eng 1970;(2):419–51. [15] Nordstrand T. Analysis and testing of corrugated board panels into the post-buckling regime. Compos Struct 2004;63:189–99.