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Buckling of a uniformly compressed rectangular SSCF composite sandwich plate
Composite Structures 105 (2013) 108–115
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Buckling of a uniformly compressed rectangular SSCF composite sandwich plate A.V. Lopatin a, E.V. Morozov b,⇑ a b
Department of Aerospace Engineering, Siberian State Aerospace University, Krasnoyarsk, Russia School of Engineering & IT, The University of New South Wales, Canberra, Australia
a r t i c l e
i n f o
Article history: Available online 14 May 2013 Keywords: Buckling Composite sandwich plates Kantorovich procedure Generalised Galerkin method
a b s t r a c t Solution of the buckling problem for a uniformly compressed rectangular composite sandwich plate having two parallel edges simply supported, one edge clamped and the remaining edge free (the SSCF sandwich plate) is presented in the paper. A variational buckling equation is derived based on the Lagrange principle and first-order shear deformation theory (FSDT). The Kantorovich procedure is applied to reduce the original equation to a one-dimensional form. Subsequent application of the generalised Galerkin method leads to an analytical formula for the critical load which is verified by using finite-element analysis. Efficiency of the analytical formula derived is demonstrated for design cases in which constraints are imposed on the value of critical load. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Composite sandwich panels, being parts of various aerospace, marine and civil engineering structures, are often loaded by compressive and/or shear in-plane forces exerted on the panel edges. Although the sandwich panels usually have high bending stiffness due to their specific structure (thin facings separated by the lightweight core), this loading still can cause buckling when the loads reach their critical values. Hence, buckling analysis is an important part of the design of these structural elements. This explains significant interest of the research community to the development of relevant methods and techniques that allow design engineers to efficiently analyse and solve buckling problems. Results of numerous research investigations could be found in the monographs published by Allen [1], Plantema [2], Zenkert [3], Vinson [4], Sullins et al. [5], Volmir [6], Kollar and Springer [7], Aleksandrov et al. [8], and Grigoluk and Chulkov [9]. Growing application of advanced composite materials in the designs of sandwich panels boosted new research activities and developments in this field. The solution of the buckling problems formulated for sandwich plates is complicated by the higher order of differential equations modelling plates’ behaviour. Also, the selection of approximating functions is rather difficult for the plates with boundary conditions different from the simple support. In such cases, finite-element methods are normally employed for the solution of buckling problems for sandwich plates. Being efficient for direct buckling analyses, these methods could be overly computationally expensive when applied ⇑ Corresponding author. Tel.: +61 2 6268 9542; fax: +61 2 6268 8276. E-mail address: [email protected] (E.V. Morozov). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.04.036
to the related design problems. In the design cases, it would be advantageous to use so-called ‘‘fast solution’’ techniques based on analytical formulas that provide solutions for buckling problems with minimal computational effort. A development of such technique for the composite sandwich plate having two parallel edges simply supported, one edge clamped and the remaining edge free (the SSCF sandwich plate), and subjected to a uniformly distributed over the simply supported edges compressive load is of particular practical interest. This problem was solved for the first time by Timoshenko for the isotropic plate [10]. Liew et al. [11] and Xiang et al. [12] have found a critical load for the SSCF plate based on a model that takes into account transverse shear deformation. They have been using a joint analytical–numerical approach. Firstly, the Levy method has been applied to the buckling equations. Then, the resulting system of ordinary differential equations of the sixth order has been integrated using classical Euler’s method. The corresponding fundamental matrix and boundary conditions for the clamped and free edges have been formed numerically, and the value of critical load has been found as a result of iterative computational procedure. Note that to date there is no solution to the above problem formulated for the composite sandwich plate that would provide an analytical formula for the critical buckling load. This solution is proposed in the present work for the SSCF sandwich plate composed of two identical composite facings and orthotropic core. The variational governing equation based on the Lagrange principle has been used to solve the buckling problem. In this equation, the variation of total energy of compressed sandwich plate has been determined using the first-order shear deformation theory (FSDT) [13,14]. Firstly, the Kantorovich method [15] has been
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
applied to reduce the variational equation to a one-dimensional form. Conventional trigonometric functions have been employed to approximate variations of the kinematic variables between the simply supported edges of the plate. Then the generalised Galerkin method [16,17] has been used to solve the buckling problem. Variations of the kinematic variables between the clamped and free edges have been approximated by the functions obtained from the solution of a cantilever beam bending problem. The beam has been modelled using the theory that takes into account the transverse shear strains. The approximating functions satisfied kinematic boundary conditions for the clamped edge. The static boundary conditions for the free edge have been fulfilled approximately according to the generalised Galerkin method. Such an approach has yielded an analytical formula for the critical buckling load. This solution has been verified by the finite element analysis. Application of the formula to design problems has been illustrated by numerical examples. 2. Buckling equation and solution procedure Consider a rectangular sandwich a b plate composed of two identical composite facings and orthotropic core. The middle plane of the plate is referred to Cartesian coordinate frame (x, y) as shown in Fig. 1. The two parallel edges of the plate (x = 0, a) are simply supported, one edge (y = 0) is fully clamped and the remaining edge (y = b) is free. The plate is subjected to a compressive load N uniformly distributed over the edges x = 0, a. The plate is modelled using equations of the first order shear deformation theory (FSDT). The solution of buckling problem is based on variational Lagrange equation which, in the case under consideration, has the following form [18]: Z
a
0
þ
Z Z
ðMx dkx þ My dky þ Mxy dkxy þ Q x dwx þ Q y dwy Þdxdy
0
a
Z bh i N 0x xx dxx þ N0xy ðxy dxx þ xx dxy Þ þ N0y xy dxy dxdy ¼ 0
ð1Þ
where Mx, My, and Mxy are the bending and twisting moments; Qx, and Qy are the transverse shear forces, kx, ky and kxy are the curvature changes and twisting of the middle plane, respectively; wx and wy are transverse shear strains; xx and xy are the angles of rotation of the lines tangent to coordinates x and y; N0x ; N 0xy , and N0y are the membrane internal prebuckling forces caused by external loading. The moments and shear forces are related to the curvatures and shear strains by the following constitutive equations:
Q x ¼ Sx wx
@hx @x
ky ¼
@hy @y
kxy ¼
ð3Þ
It is assumed that the original plane stress state of the plate is uniform under given loading. Hence the forces N0y and N 0xy are zero and the force N 0x is determined by the external compressive load exerted on the edges x = 0, a, i.e.,
N0x ¼ N
N0y ¼ 0 N0xy ¼ 0
ð4Þ
The variational equation, Eq. (1) can be expressed in terms of unknown deflection, w and angles of rotation, hx, hy. Substituting Eqs. (2) and (3) into Eq. (1) and taking into account Eq. (4), the following equation can be derived:
@hx @hy @hx @hx @hy @hy d þ D21 d þ D12 þ D22 @x @y @x @x @y @y 0 0 @hx @hy @hx @hy @w @w d þ Sx h x þ d hx þ þD33 þ þ @x @x @y @x @y @x @w @w @w @w þSy hy þ d hy þ N dx dy ¼ 0 ð5Þ d @y @y @x @x Z
a
Z
b
D11
ð2Þ
Q y ¼ Sy wy
in which D11, D12, D21, D22, D33 (D12 = D21), and Sx, Sy are the bending and transverse shear stiffnesses of the plate, respectively.
ð6Þ
Substituting for Mx its expression in Eq. (2) and allowing for Eq. (3), these conditions are transformed to the following form:
w ¼ 0 hy ¼ 0 D11
@hx @hy þ D12 ¼0 @x @y
ð7Þ
This form allows the Kantorovich procedure to be implemented to reduce the dimension of the variational equation, Eq. (5). Variations of the deflection, w and angles of rotation, hx, hy along the axis x can be approximated by using trigonometric functions as follows:
wðx; yÞ ¼ wm ðyÞ sin km x hx ðx; yÞ ¼ hxm ðyÞ cos km x
M y ¼ D21 kx þ D22 ky
Mxy ¼ D33 kxy
@hx @hy þ @y @x @w @w wx ¼ hx þ wy ¼ hy þ @x @y @w @w xx ¼ xy ¼ @x @y kx ¼
w ¼ 0 hy ¼ 0 M x ¼ 0
0
Mx ¼ D11 kx þ D12 ky
Out-of-plane deformation of the plate is characterised by the deflection w, and the angles of rotations hx and hy of the element normal to the middle plane. These variables are related to the curvatures kx, ky, kxy, shear deformations wx, wy, and angles xx, xy as follows:
Boundary conditions at the simply supported edges of the plate x = 0 and x = a are given by
b
0
109
ð8Þ
hy ðx; yÞ ¼ hym ðyÞ sin km x where wm(y), hxm(y), and hym(y) are unknown functions of y, km = mp/a, m is the number of buckling half-waves in the x-direction. Clearly, these approximations satisfy the boundary conditions given by Eq. (7). Substituting Eq. (8) into Eq. (5) and integrating with respect to x, as per the Kantorovich procedure, yields the following one-dimensional functional: Z b
dhym Sx k2m wm þ km hxm dwm þ Sx km wm þ D11 k2m þ Sx hxm D12 km dhxm dy 0 dwm dhxm dwm dwm þ D33 km þ D33 k2m þ Sy hym dhym þ Sy þ hym d þ Sy dy dy dy dy dhxm dhxm dhym dhym þ km hym d þ D21 km hxm þ D22 d þD33 dy dy dy dy Nk2m wm dwm dy ¼ 0 ð9Þ Fig. 1. Uniformly compressed rectangular SSCF sandwich plate.
Varying this functional yields the following equations:
110
Z
b
0
Z
b
0
Z
b
0
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
Lm dwm dy þ ½Q ym dwm b0 ¼ 0 Lxm dhxm dy þ ½M xym dhxm b0 ¼ 0
ð10Þ
Lym dhym dy þ ½M ym dhym b0 ¼ 0
in which 2
Lm ¼ Sx k2m wm Sy
d wm dy
2
þ Sx km hxm Sy
dhym Nk2m wm dy
2 d hxm dhym Lxm ¼ Sx km wm þ D11 k2m þ Sx hxm D33 ðD12 þ D33 Þkm 2 dy dy 2 dwm dhxm d hym þ ðD21 þ D33 Þkm þ D33 k2m þ Sy hym D22 Lym ¼ Sy 2 dy dy dy ð11Þ
wm ðyÞ ¼ W m UðyÞ hxm ðyÞ ¼ T xm UðyÞ hym ðyÞ ¼ T ym VðyÞ
and
dwm þ hym dy dhxm þ km hym M xym ¼ D33 dy dhym M ym ¼ D21 km hxm þ D22 dy
ð12Þ
There are two approaches that could be applied to the solution of buckling problem specified by Eq. (10). The first approach is based on the traditional reduction of Eq. (10) to the following homogeneous buckling equations written for each buckling mode m
ð13Þ
and supplemented by the boundary conditions at y = 0
wm ¼ 0 hxm ¼ 0 hym ¼ 0
ð14Þ
and at y = b
Q ym ¼ 0 M xym ¼ 0 M ym ¼ 0
ð16Þ
where Wm, Txm, Tym are unknown coefficients, and U (y), V (y) are the approximating functions given by
Q ym ¼ Sy
Lm ¼ 0 Lxm ¼ 0 Lym ¼ 0
Based on the above, it follows that the use of this approach involves substantial algorithmic and computational efforts. Clearly, this hinders the application of this technique to the solution of design problems in which constraints are imposed on the value of critical buckling load. The second approach, adopted in this work, is based on the generalised Galerkin method. In this case, the governing equations are given by Eq. (10). According to this method, the functions wm(y), hxm(y), and hym(y) are replaced with their analytical approximations which would closely reflect the variations of the plate deflection and angles of rotation along the y-axis. In the case under consideration, these approximating functions are selected based on the solution of bending problem for a cantilever beam subjected to uniformly distributed load as shown in Fig. 2. The beam is modelled using a theory which takes into account the transverse shear deformation. As a result, the deflection and angles of rotation are presented as follows:
ð15Þ
Such a reduction is based on the arbitrariness of variations d wm, dhxm, and dhym in Eq. (10). It follows from Eq. (15) that the shear force Qy, and bending and twisting moments, My and Mxy are equal to zero at the free edge of the plate. Assume that Nm is the value of compressive load for which there exists a non-trivial solution of the homogeneous boundary problem specified by Eqs. (13)–(15). Note that the system of differential equations, Eq. (13) is of sixth order. Thus, the six roots of the corresponding characteristic polynomial would need to be found to solve the problem. Each root corresponds to a particular solution of the system of differential equations, Eq. (13). Clearly, the general solution is given by a linear combination of these particular solutions. In this case, functions wm(y), hxm(y), and hym(y) would include six constants of integration that could be found by using the boundary conditions given by Eqs. (14) and (15). This would lead to the system of algebraic equations of sixth order. Equating the determinant of this system to zero, the nonlinear algebraic equation with respect to the unknown force Nm could be derived. This equation should be solved numerically. Hence, in order to find Nm the two aforementioned numerical procedures should be applied. In the process of numerical solution for the nonlinear equation obtained from the fulfilment of boundary conditions, the six roots of the corresponding polynomial should be found numerically. It follows from Eq. (11) that the value of each of these roots depends on the force Nm. This force is calculated for m = 1, 2, 3, . . . and the critical buckling load corresponds to the minimal value of Nm.
y y y3 y2 y 4 2 þ 6 12cy 2 3 b b b b b y y2 y þ1 VðyÞ ¼ b 3b2 b
UðyÞ ¼
ð17Þ
The dimensionless parameter cy characterises a transverse shear compliance of the sandwich plate along the y-axis and is calculated as follows:
cy ¼
D22
ð18Þ
2
Sy b
It follows from Eqs. (16) and (17) that U(y = 0) = 0 and V(y = 0) = 0. So, the approximations selected as per Eqs. (16) and (17) satisfy kinematic boundary conditions, Eq. (14) at the clamped edge of the plate. However, these approximations do not fulfil the static boundary conditions, Eq. (15) at the free edge, i.e. Qym (y = b) – 0, Mxym (y = b) – 0, Mym(y = b) – 0. This is acceptable, since the application of the generalised Galerkin procedure does not require the exact fulfilment of static boundary conditions. As can be seen, the equations in Eq. (10) automatically provide an approximate satisfaction of these conditions at the free edge y = b. Note that the efficiency of applying the approximating functions U(y) and V(y) has been demonstrated for the solutions of free vibration problems for composite sandwich plates considered in articles [19,20]. Variations of the functions wm, hxm, and hym given by Eq. (16) have the form
dwm ¼ UdW m
dhxm ¼ UdT xm
dhym ¼ VdT ym
ð19Þ
Substituting Eqs. (16) and (19) into Eqs. (10)–(12), and taking into account the arbitrariness of the variations dWm, dTxm and dTym, the following governing equations of the generalised Galerkin method are derived:
Fig. 2. Cantilever beam subjected to uniformly distributed load
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Z
b
0
Z
b
0
Z
b
0
Lm Udy þ ½Q ym Ub0 ¼ 0 Lxm Udy þ ½M xym Ub0 ¼ 0
b Lym Vdy þ M ym V 0 ¼ 0
in which 2
Lym
d U
ð29Þ
!
dV W m þ Sx km UT xm Sy T ym Nk2m UW m 2 dy dy " # 2 d U dV T ym ¼ Sx km UW m þ D11 k2m þ Sx U D33 2 T xm ðD12 þ D33 Þkm dy dy " # 2 dU dU d V ¼ Sy W m þ ðD21 þ D33 Þkm T xm þ D33 k2m þ Sy V D22 2 T ym dy dy dy Sx k2m U Sy
Lm ¼ Lxm
ð20Þ
Taking these equations into account, the system of equations in Eq. (23) can be re-arranged to the following form:
Sx k2m I1 Sy I2 W m þ Sx km I1 T xm þ Sy J 3 T ym Nk2m I1 W m ¼ 0 h
i Sx km I1 W m þ D11 k2m þ Sx I1 D33 I2 T xm þ ðD12 I3 þ D33 J 3 Þkm T ym ¼ 0 h
i Sy J 3 W m þ ðD21 I3 þ D33 J 3 Þkm T xm þ D33 k2m þ Sy J 1 D22 J 2 T ym ¼ 0
ð21Þ
and
dU W m þ VT ym dy dU T xm þ km VT ym Mxym ¼ D33 dy dV T ym Mym ¼ D21 km UT xm þ D22 dy Q ym ¼ Sy
ð22Þ
where
I2 ¼
24 c2 35b
ð30Þ
and
c2 ¼ 15 þ 84cy þ 280c2y
ð31Þ
The equations in Eq. (29) can be transformed into a form with dimensionless coefficients. Substituting Eqs. (26) and (30) into the equations in Eq. (29), taking into account that km = mp/a, and mul2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tiplying each of them by 630ba = D11 D22 , the following equations are derived:
16a11 W m þ 16a12 F xm þ 108a13 F ym 16b11 gm W m ¼ 0 16a21 W m þ 16a22 F xm þ 18a23 F ym ¼ 0
ð32Þ
108a31 W m þ 18a32 F xm þ 9a33 F ym ¼ 0 in which
After integration in Eq. (20), the latter are transformed into the following homogeneous system of linear algebraic equations:
F xm ¼ aT xm
h
The coefficients in the system of equations given by Eq. (32) are calculated as follows:
i Sx k2m I1 Sy ðI2 PÞ W m þ Sx km I1 T xm
Sy ðI3 ZÞT ym Nk2m I1 W m ¼ 0 h
i Sx km I1 W m þ D11 k2m þ Sx I1 D33 ðI2 PÞ T xm
a11 ¼ p2m c1
ð23Þ a33 ¼ 5
þ ½D12 I3 D33 ðI3 ZÞkm T ym ¼ 0
in which
Z
b
U 2 dy I2 ¼
Z
0
J1 ¼
Z
2
b
U
0 b
2
V dy J 2 ¼
0
Z
dy I3 ¼
2
dy
V
d V dy
2
Z
b
U
0
2
b
0
d U
dy J 3 ¼
Z 0
b
dV dy dy
dU dy V dy
dU P¼ U dy
b 0
Z¼
½UVb0
b dV R¼ V dy 0
ð24Þ
ð25Þ
I1 ¼
ð26Þ
12 ð1 þ 4cy Þ R ¼ 0 Z ¼ 1 þ 4cy b
þ
1
!
þ
14
D11 Sx a2 sffiffiffiffiffiffiffiffi 2 D11 b a¼ 2 D22 a
cx ¼
ð34Þ
ð35Þ D12 b12 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22
D33 b33 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22
ð36Þ ð37Þ
The dimensionless parameter cx in Eq. (35) characterises the transverse shear compliance of the sandwich plate in the xz-plane (see Fig. 3). The parameter gm in the first equation of Eq. (32) is a dimensionless buckling coefficient corresponding to the buckling mode m and related to the external load as follows 2
Nb D11 D22
ð38Þ
Putting the determinant of the matrix of coefficients in Eq. (32) equal to zero, the following equation can be obtained
ð27Þ
Substituting Eq. (17) into Eq. (25), the latter can be transformed as follows:
P¼
p2m b33
gm ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
in which
c1 ¼ 91 þ 999cy þ 3024c2y c2 ¼ 5 þ 28cy þ 560c2y c3 ¼ 5 þ 56cy c3 ¼ 5 þ 14cy
pm ¼ mp
Substituting for U(y) and V(y) in Eq. (24) their expressions given by Eq. (17) and integrating, the following formulas are derived:
8b 12 c c I2 ¼ c I3 ¼ 3 315 1 35b 2 35 b 1 6 J1 ¼ J ¼ J ¼ c3 14 2 5b 3 35
a c2 1 þ 27 a ¼ p2m þ c a þ 27c2 b33 cx cy a 22 cx 1
where
and
ð33Þ
cy a a a c3 a12 ¼ a21 ¼ pm c1 a13 ¼ a31 ¼ cx cy a 3 Þ b11 ¼ p2m c1 a23 ¼ a32 ¼ pm ðb12 c3 þ 6b33 c
Sy J3 W m þ ½D21 ðJ3 ZÞ þ D33 J 3 km T xm h
i þ D33 k2m þ Sy J1 D22 ðJ2 RÞ T ym ¼ 0
I1 ¼
F ym ¼ bT ym
ð28Þ
8 > < 16a11 16b11 gm det 16a21 > : 108a31
16a12 16a22 18a32
9 108a13 > = ¼0 18a23 > ; 9a33
ð39Þ
Solving this equation for gm, taking into account that a12 = a21, a13 = a31, a23 = a32, the buckling coefficient can be calculated as follows:
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The transverse shear stiffness coefficients for the sandwich plate under consideration are given by
Sx ¼ Gxzt tfxz
Sy ¼ Gyzt tfyz
ð47Þ
where
2 h h fxz ¼ 1 þ 1þ t t 2 h h fyz ¼ 1 þ 1þ t t
Gxzt Gxzh Gyzt Gyzh
1 ð48Þ
1
Substituting for D11, D22 and Sx, Sy in Eqs. (18) and (35) their expressions given by Eqs. (43) and (47) yields Fig. 3. Structure of sandwich plate with composite facings and orthotropic core.
gm ¼
4a11 a22 a33 þ 108a12 a13 a23 9a11 a223 324a22 a213 4a33 a212 b11 4a22 a33 9a223 ð40Þ
Using this formula, the values g1, g2, . . . , gk can be calculated for m = 1, 2, 3, . . . , k for given k. Then the critical buckling coefficient can be found using the following condition:
gcr ¼ minðg1 ; g2 ; . . . ; gk Þ
ð41Þ
This coefficient depends on the dimensionless parameters cx, cy, a, b12 and b33 that contain all the information on the elastic and geometrical characteristics of the sandwich plate. Once the critical buckling coefficient is found, the corresponding critical force is calculated using Eq. (38) solved for N as
Ncr ¼ gcr
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22
ð42Þ
2
b
As indicated above, the parameters cx, cy, a, b12 and b33 are calculated in terms of the characteristics of sandwich plate. Consider the plate shown in Fig. 3. The coordinate z is measured from the middle surface. The total thickness of the facings is equal to t, whereas h is the thickness of the core. It is assumed, that the face sheets are made from an orthotropic material with the moduli of elasticity Ext, Eyt in x and y directions; shear moduli Gxyt, Gxzt and Gyzt in the planes xy, xz, and yz, respectively; and Poisson’s ratios mxyt and myxt. The similar respective characteristics of the core material are denoted as: Exh, Eyh, Gxyh, Gxzh, Gyzh, mxyh and myxh. The bending stiffness coefficients of the sandwich plate modelled by the firstorder shear deformation theory are calculated as follows [18]:
D11 D22
t3 ¼ Ext n 12 11 3 t ¼ Eyt n 12 22
D12 D33
t3 ¼ D21 ¼ Ext mxyt n 12 12 3 t ¼ Gxyt n 12 33
ð43Þ
3
n11
3
n12
Exh mxyh h ¼nþ Ext mxyt t3
3
n22 ¼ n þ
3
Eyh h Eyt t 3
n33 ¼ n þ
Gxyh h Gxyt t3
Ext Eyt Eyt ¼ 1 mxyt myxt 1 mxyt myxt Eyh Exh ¼ Eyh ¼ 1 mxyh myxh 1 mxyh myxh
ð44Þ
ð45Þ
Ext ¼ Exh
cy ¼
2 1 b Eyt n22 12 t Gyzt fyz
ð49Þ
Similarly, substituting Eq. (43) into Eq. (36), the parameters a, b12 and b33 are determined as follows
sffiffiffiffiffiffisffiffiffiffiffiffiffi 2 Ext n11 b a¼ 2 Eyt n22 a
Ext mxyt n12 b12 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi n 11 n22 Ext Eyt
Gxyt n33 b33 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi n 11 n22 Ext Eyt
ð50Þ
Taking into account the expressions for D11 and D22 as per Eq. (43), the formula for critical force, Eq. (42) can be presented in the following form:
Ncr ¼ gcr
qffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi Ext Eyt n11 n22 2 12b t3
ð51Þ
Using this equation, the critical buckling force can be calculated for the SSCF sandwich plate composed of the orthotropic facings and core that have the aforementioned elastic characteristics. The equations could be simplified for the plate with lightweight honeycomb core. In this case, it can be assumed that Exh = Eyh = Gxyh = 0. Subsequently, it follows from Eq. (44) that
n11 ¼ n12 ¼ n22 ¼ n33 ¼ n
ð52Þ
and the equations specifying parameters cx and cy are transformed as follows:
cx ¼
1 a 2 Ext n 12 t Gxzt fxz
cy ¼
2 1 b Eyt n 12 t Gyzt fyz
ð53Þ
Substituting Eq. (52) into Eq. (50) yields
sffiffiffiffiffiffi 2 Ext b a¼ Eyt a2
Ncr ¼ gcr
2
h h n¼1þ3 þ3 2 t t
1 a 2 Ext n11 12 t Gxzt fxz
Ext mxyt b12 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi Ext Eyt
Gxyt b33 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi Ext Eyt
ð54Þ
Taking into account Eq. (52), the formula for the critical buckling load, Eq. (51) takes the form
where
Exh h ¼nþ Ext t 3
cx ¼
ð46Þ
2 qffiffiffiffiffiffiffiffiffiffiffiffi tn b Ext Eyt 12 t
ð55Þ
The above equations, Eqs. (51) or (55) provide a ‘‘fast solution’’ to the buckling problem under consideration. The use of these formulae does not require complex computations and they can be efficiently used in the structural design process. 3. Numerical analysis The calculations of buckling load have been performed for composite sandwich plates with various honeycomb cores. The face sheets of the panels are made of carbon fibre reinforced plastic with the following properties: Ext = 54.55 GPa, Eyt = 54.55 GPa,
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Gxyt = 20.67 GPa, Gxzt = 3.78 GPa, Gyzt = 3.78 GPa, mxyt = 0.32, myxt = 0.32. In order to assess the applicability and limitations of the proposed solution, the panels with four different honeycomb cores have been considered: (a) 5052 Aluminium with the cell size of 3 mm, having transverse shear moduli Gxzh = 930 MPa, Gyzh = 372 MPa; (b) 3003 Aluminium with the cell size of 6 mm, having Gxzh = 440 MPa, Gyzh = 220 MPa; (c) 3003 Aluminium with the cell size of 19 mm, having Gxzh = 110 MPa, Gyzh = 55 MPa; and (d) HRH10 Nomex (Aramid) with the cell size of 3 mm, having Gxzh = 25 MPa, Gyzh = 17 MPa [21]. The plates have the overall thicknesses of the facings t = 0.001 m and the thicknesses of the core layer h = 0.01 m, 0.05 m, and 0.1 m. The in-plane dimensions of the plates are: b = 1 m, and a = 1, 3, and 5 m. Critical forces have been calculated using Eq. (51). The results of calculations are presented in Tables 1–4. It can be seen that, as expected, the value of Ncr is substantially increases with the increase in the core thickness. An increase in the length of plate a from 1 to 5 m leads to the reduction of critical force. It follows from the analysis that the buckling of the square plate (a = 1 m) occurs with the formation of one halfwave. For the plates with a = 3 and 5 m, the numbers of the buckle half-waves are equal to 2 and 3, respectively. Note, that the above analyses have been performed for k = 5 (see Eq. (41)). The results of calculations have been verified using a finite element analysis. The numerical solutions have been obtained using MSC Nastran [22]. A 20 20 mm 4-node finite element Laminate with the option suitable for the analysis of sandwich panels has been implemented to model the aforementioned plates. The finite-element models for 1 m 1 m, 1 m 3 m, and 1 m 5 m plates have been composed of 2500 (2601 nodes), 7500 (7701 nodes), and 12,500 (12,801 nodes) elements, respectively. The finite-element mesh for the square plate (a = b = 1 m) is shown in Fig. 4. The buckling mode shapes for the SSCF sandwich plates with a = 1, 3 and 5 m, and b = 1 m are shown in Fig. 5. The corresponding results of finite-element analyses are presented in Tables 1–4. It follows from the data given in these tables that the maximum differences between the values of critical loads Ncr calculated according to Eq. (51) and those obtained from the finite-element analysis
N FEM do not exceed 4%, 5%, 5.8% and 10.1% for the plates having cr
Table 2 (kN/m) for the SSCF sandwich Critical loads Ncr (kN/m) (analytical solution) and N FEM cr plate with the honeycomb core having Gxzh = 440 MPa, Gyzh = 220 MPa (3003 Aluminium, cell size 6 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
a=5m
Ncr
N FEM cr
Ncr
N FEM cr
Ncr
N FEM cr
27.85 628.10 2414.29
27.91 618.06 2339.01
21.37 484.03 1870.48
20.99 469.48 1793.60
21.09 478.09 1848.87
20.52 459.73 1765.93
Table 3 Critical loads Ncr (kN/m) (analytical solution) and N FEM (kN/m) for the SSCF sandwich cr plate with the honeycomb core having Gxzh = 110 MPa, Gyzh = 55 MPa (3003 Aluminium, cell size 19 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
a=5m
Ncr
N FEM cr
Ncr
N FEM cr
Ncr
N FEM cr
27.38 576.86 2059.83
27.01 558.43 1954.31
21.07 451.22 1638.97
20.44 431.38 1552.05
20.81 446.59 1625.69
20.03 423.31 1536.63
Table 4 Critical loads Ncr (kN/m) (analytical solution) and N FEM (kN/m) for the SSCF sandwich cr plate with the honeycomb core having Gxzh = 25 MPa, Gyzh = 17 MPa (HRH10 Nomex (Aramid), cell size 3 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
Ncr
N FEM cr
25.74 440.63 1324.03
24.73 415.69 1221.43
a=5m
Ncr
N FEM cr
Ncr
N FEM cr
20.13 367.85 1164.22
19.08 344.11 1068.09
19.93 367.56 1161.85
18.78 342.91 1055.27
the aforementioned honeycomb cores (a–d), respectively. The highest discrepancy has been observed for the largest and thickest plate having very compliant core. These results show that the solution method proposed in this work provides the values of critical buckling loads with sufficient accuracy and can be efficiently applied to the solution of relevant design problems. 4. Examples of sandwich panel design Calculations of the buckling loads for the composite SSCF sandwich panels under consideration based on the formulas presented in this paper (i.e., Eqs. (51) or (55)) do not require any computationally expensive procedures. Therefore, the design problems involving buckling constraints can be efficiently solved with relative ease.
Ncr
N FEM cr
Ncr
N FEM cr
Consider a number of examples where thicknesses of the core, h and facings, t can be found for the sandwich plate compressed with e The elastic properties of the materials of facthe specified load N. ings and core are given as follows: Ext = Eyt = 54.55 GPa, Gxyt = 20.67 GPa, Gxzt = Gyzt = 3.78 GPa, mxyt = myxt = 0.32, Gxzh = 440 MPa, e the latter can be Gyzh = 220 MPa. Replacing Ncr in Eq. (55) with N, written as
21.41 489.31 1910.94
21.05 479.71 1846.32
21.13 483.07 1887.12
20.62 467.01 1814.28
2 qffiffiffiffiffiffiffiffiffiffiffiffi e g tn b N Ext Eyt ¼ 0 cr 12 t
Table 1 Critical loads Ncr (kN/m) (analytical solution) and NFEM (kN/m) for the SSCF sandwich cr plate with the honeycomb core having Gxzh = 930 MPa, Gyzh = 372 MPa (5052 Aluminium, cell size 3 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
Ncr
N FEM cr
27.93 637.06 2482.56
27.87 631.64 2432.13
Fig. 4. Finite-element mesh for the SSCF sandwich plate with a = b = 1 m.
a=5m
ð56Þ
114
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
Fig. 6. Contour curves of Ncr(t, h) for the SSCF sandwich plate with a = 2 m, b = 1 m.
Fig. 5. Buckling mode shapes of the SSCF sandwich plate: (a) a = 1 m; (b) a = 3 m; and (c) a = 5 m.
where gcr is determined by Eqs. (40), (41), taking into account Eqs. (34), (53), and (54). This nonlinear equation relates the e with the thicknesses of the core, h specified compressive load N and facings, t for given in-plane dimensions and elastic properties of the constituent materials. For example, consider the sandwich plate with a = 2 m, b = 1 m, and t = 0.001 m loaded with the force e ¼ 300kN=m. Calculations of gcr using Eqs. (40), (41) and Eqs. N (34), (53), (54) and subsequent solution of the equation, Eq. (56) yield: h = 38.68 mm. In another example, for a 1 1 m plate having the thickness of the core h = 0.02 m and subjected to compressive load e ¼ 200 kN=m, the application of the same procedure provides N t = 1.838 mm. In the third example, thicknesses of the facings and the core delivering the minimum mass of the sandwich panel subjected to the specified critical load are determined. The mass of the sandwich plate having the facings of equal thickness is calculated as follows:
M ¼ abðqt t þ qh hÞ
ð57Þ
where qt and qh are the densities of the facings and core material, respectively. Consider a sandwich plate with a = 2 m and b = 1 m. The values of critical load have been calculated using the aforementioned procedure for 0.5 mm 6 t 6 2 mm and 10 mm 6 h 6 80 mm. The function Ncr(t,h) can be presented by the contour curves shown e takes the in Fig. 6. Assume that the specified compressive load N following values: 100, 200, 300, 400, 500, and 600 kN/m. It follows e can be from Fig. 6 that for each of these values the condition N cr ¼ N fulfilled for various combinations of t and h that belong to a corresponding contour curve. In this example, equation, Eq. (57) defines the objective function whereas Eq. (56) can be treated as a constraint in the form of equality. The design domain for the variables t and h is determined by the respective ranges presented above. Using subroutines available in the library IMSL [23] the FORTRAN code has been developed to solve the optimisation problem. The results of optimisation are presented in Table 5 where the values
Table 5 Optimal parameters of the SSCF sandwich plate. e (kN/m) N
Mmin (kg)
t (mm)
h (mm)
KN
100 200 300 400 500 600
6.348 8.011 9.184 10.121 10.915 11.611
0.723 0.911 1.043 1.148 1.237 1.314
26.115 32.987 37.849 41.736 45.027 47.921
1605.8 2544.9 3329.8 4028.7 4669.6 5267.6
of t and h delivering the minimum mass Mmin for the specified vale are shown. These results can be used ues of the compressive load N to assess the weight efficiency of the SSCF sandwich plates under consideration. The following coefficient could be adopted as a measure of this efficiency:
KN ¼
e Nb M min g
ð58Þ
where g = 9.81 M/s2 is the gravitational acceleration. The corresponding values of this coefficient are given in Table 5. Analysis of e shows that the weight efficiency of the sandthe dependence K N ð NÞ wich plate increases with increase in the compressive load. In particular, the coefficient for the plate subjected to the highest load e ¼ 600 kN=mÞ is 3.28 times the value of the coefficient correðN sponding to the plate compressed with the lowest load e ¼ 100 kN=mÞ. ðN 5. Conclusions The buckling problem has been solved for the SSCF composite sandwich plate subjected to uniform in-plane compressive load. The problem has been solved using the Kantorovich procedure and generalised Galerkin method and the analytical formula providing the ‘‘fast solution’’ has been obtained. The formula has been successfully verified by using finite-element analysis. The efficiency of the solution developed in this work has been demonstrated for various examples of the design of sandwich plates for which the constraint is imposed on the value of critical compressive load. It has been shown that the maximum weight efficiency
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
of the sandwich plates under consideration can be achieved when the compressive load has the highest value. References [1] Allen HC. Analysis and design of structural sandwich panels. Oxford: Pergamon Press; 1969. [2] Plantema FJ. Sandwich construction. New York: Wiley & Sons, Inc.; 1966. [3] Zenkert D. An introduction to sandwich construction. London: Chameleon Press Ltd.; 1995. [4] Vinson JR. The behavior of sandwich structures of isotropic and composite materials. Technomic, Lancaster; 1999. [5] Sullins RT, Smith GW, Spier EE. Manual for structural stability analysis of sandwich plates and shells. NASA Contractor report CR-1457; 1969. [6] Volmir AS. Stability of deformable systems. Moscow: State Publishing House Nauka; 1967 [in Russian]. [7] Kollar LP, Springer GS. Mechanics of composite structures. Cambridge University Press; 2003. [8] Aleksandrov AY, Bryukker LE, Kurshin LM, Prusakov AP. Analysis of sandwich plates. Mashinostroenie; 1960 [in Russian]. [9] Grigoluk EI, Chulkov PP. Buckling and vibrations of sandwich shells. Mashinostroenie; 1973. in Russian. [10] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. New York: McGraw-Hill; 1961.
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[11] Liew KM, Xiang X, Kitipornchai SS. Analytical buckling solutions for Mindlin plates involving free edges. Int J Mech Sci 1996;38(10):1127–38. [12] Xiang X, Liew KM, Kitipornchai SS. Exact buckling solutions for composite laminates: proper free edge conditions under in-plane loadings. Acta Mech 1996;117:115–28. [13] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Trans ASME, J Appl Mech 1945;12(2):69–77. [14] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Trans ASME, J Appl Mech 1951;18:8–31. [15] Kantorovich LV, Krylov VI. Approximate methods of higher analysis. New York: John Wiley & Sons; 1958. [16] Pratusevich YA. Variational methods in structural mechanics. MoscowLeningrad: OGIZ; 1948 [in Russian]. [17] Pilkey WD, Wunderlich W. Mechanics of structures. Variational and computational methods. CRC Press; 1994. [18] Vasiliev VV. Mechanics of composite structures. Taylor & Francis; 1993. [19] Morozov EV, Lopatin AV. Fundamental frequency of the CCCF composite sandwich plate. Compos Struct 2010;92:2747–57. [20] Lopatin AV, Morozov EV. Fundamental frequency and design of the CFCF composite sandwich plate. Compos Struct 2011;93:983–91. [21] HEXCEL. Honeycomb sandwich design technology. Technical report AGU 075; 1998. [22] MSC Nastran. Quick reference guide. [23] IMSL. Math/library users manual. IMSL Inc., 2500 CityWest Boulevard, Houston TX 77042.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Buckling of a uniformly compressed rectangular SSCF composite sandwich plate A.V. Lopatin a, E.V. Morozov b,⇑ a b
Department of Aerospace Engineering, Siberian State Aerospace University, Krasnoyarsk, Russia School of Engineering & IT, The University of New South Wales, Canberra, Australia
a r t i c l e
i n f o
Article history: Available online 14 May 2013 Keywords: Buckling Composite sandwich plates Kantorovich procedure Generalised Galerkin method
a b s t r a c t Solution of the buckling problem for a uniformly compressed rectangular composite sandwich plate having two parallel edges simply supported, one edge clamped and the remaining edge free (the SSCF sandwich plate) is presented in the paper. A variational buckling equation is derived based on the Lagrange principle and first-order shear deformation theory (FSDT). The Kantorovich procedure is applied to reduce the original equation to a one-dimensional form. Subsequent application of the generalised Galerkin method leads to an analytical formula for the critical load which is verified by using finite-element analysis. Efficiency of the analytical formula derived is demonstrated for design cases in which constraints are imposed on the value of critical load. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Composite sandwich panels, being parts of various aerospace, marine and civil engineering structures, are often loaded by compressive and/or shear in-plane forces exerted on the panel edges. Although the sandwich panels usually have high bending stiffness due to their specific structure (thin facings separated by the lightweight core), this loading still can cause buckling when the loads reach their critical values. Hence, buckling analysis is an important part of the design of these structural elements. This explains significant interest of the research community to the development of relevant methods and techniques that allow design engineers to efficiently analyse and solve buckling problems. Results of numerous research investigations could be found in the monographs published by Allen [1], Plantema [2], Zenkert [3], Vinson [4], Sullins et al. [5], Volmir [6], Kollar and Springer [7], Aleksandrov et al. [8], and Grigoluk and Chulkov [9]. Growing application of advanced composite materials in the designs of sandwich panels boosted new research activities and developments in this field. The solution of the buckling problems formulated for sandwich plates is complicated by the higher order of differential equations modelling plates’ behaviour. Also, the selection of approximating functions is rather difficult for the plates with boundary conditions different from the simple support. In such cases, finite-element methods are normally employed for the solution of buckling problems for sandwich plates. Being efficient for direct buckling analyses, these methods could be overly computationally expensive when applied ⇑ Corresponding author. Tel.: +61 2 6268 9542; fax: +61 2 6268 8276. E-mail address: [email protected] (E.V. Morozov). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.04.036
to the related design problems. In the design cases, it would be advantageous to use so-called ‘‘fast solution’’ techniques based on analytical formulas that provide solutions for buckling problems with minimal computational effort. A development of such technique for the composite sandwich plate having two parallel edges simply supported, one edge clamped and the remaining edge free (the SSCF sandwich plate), and subjected to a uniformly distributed over the simply supported edges compressive load is of particular practical interest. This problem was solved for the first time by Timoshenko for the isotropic plate [10]. Liew et al. [11] and Xiang et al. [12] have found a critical load for the SSCF plate based on a model that takes into account transverse shear deformation. They have been using a joint analytical–numerical approach. Firstly, the Levy method has been applied to the buckling equations. Then, the resulting system of ordinary differential equations of the sixth order has been integrated using classical Euler’s method. The corresponding fundamental matrix and boundary conditions for the clamped and free edges have been formed numerically, and the value of critical load has been found as a result of iterative computational procedure. Note that to date there is no solution to the above problem formulated for the composite sandwich plate that would provide an analytical formula for the critical buckling load. This solution is proposed in the present work for the SSCF sandwich plate composed of two identical composite facings and orthotropic core. The variational governing equation based on the Lagrange principle has been used to solve the buckling problem. In this equation, the variation of total energy of compressed sandwich plate has been determined using the first-order shear deformation theory (FSDT) [13,14]. Firstly, the Kantorovich method [15] has been
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
applied to reduce the variational equation to a one-dimensional form. Conventional trigonometric functions have been employed to approximate variations of the kinematic variables between the simply supported edges of the plate. Then the generalised Galerkin method [16,17] has been used to solve the buckling problem. Variations of the kinematic variables between the clamped and free edges have been approximated by the functions obtained from the solution of a cantilever beam bending problem. The beam has been modelled using the theory that takes into account the transverse shear strains. The approximating functions satisfied kinematic boundary conditions for the clamped edge. The static boundary conditions for the free edge have been fulfilled approximately according to the generalised Galerkin method. Such an approach has yielded an analytical formula for the critical buckling load. This solution has been verified by the finite element analysis. Application of the formula to design problems has been illustrated by numerical examples. 2. Buckling equation and solution procedure Consider a rectangular sandwich a b plate composed of two identical composite facings and orthotropic core. The middle plane of the plate is referred to Cartesian coordinate frame (x, y) as shown in Fig. 1. The two parallel edges of the plate (x = 0, a) are simply supported, one edge (y = 0) is fully clamped and the remaining edge (y = b) is free. The plate is subjected to a compressive load N uniformly distributed over the edges x = 0, a. The plate is modelled using equations of the first order shear deformation theory (FSDT). The solution of buckling problem is based on variational Lagrange equation which, in the case under consideration, has the following form [18]: Z
a
0
þ
Z Z
ðMx dkx þ My dky þ Mxy dkxy þ Q x dwx þ Q y dwy Þdxdy
0
a
Z bh i N 0x xx dxx þ N0xy ðxy dxx þ xx dxy Þ þ N0y xy dxy dxdy ¼ 0
ð1Þ
where Mx, My, and Mxy are the bending and twisting moments; Qx, and Qy are the transverse shear forces, kx, ky and kxy are the curvature changes and twisting of the middle plane, respectively; wx and wy are transverse shear strains; xx and xy are the angles of rotation of the lines tangent to coordinates x and y; N0x ; N 0xy , and N0y are the membrane internal prebuckling forces caused by external loading. The moments and shear forces are related to the curvatures and shear strains by the following constitutive equations:
Q x ¼ Sx wx
@hx @x
ky ¼
@hy @y
kxy ¼
ð3Þ
It is assumed that the original plane stress state of the plate is uniform under given loading. Hence the forces N0y and N 0xy are zero and the force N 0x is determined by the external compressive load exerted on the edges x = 0, a, i.e.,
N0x ¼ N
N0y ¼ 0 N0xy ¼ 0
ð4Þ
The variational equation, Eq. (1) can be expressed in terms of unknown deflection, w and angles of rotation, hx, hy. Substituting Eqs. (2) and (3) into Eq. (1) and taking into account Eq. (4), the following equation can be derived:
@hx @hy @hx @hx @hy @hy d þ D21 d þ D12 þ D22 @x @y @x @x @y @y 0 0 @hx @hy @hx @hy @w @w d þ Sx h x þ d hx þ þD33 þ þ @x @x @y @x @y @x @w @w @w @w þSy hy þ d hy þ N dx dy ¼ 0 ð5Þ d @y @y @x @x Z
a
Z
b
D11
ð2Þ
Q y ¼ Sy wy
in which D11, D12, D21, D22, D33 (D12 = D21), and Sx, Sy are the bending and transverse shear stiffnesses of the plate, respectively.
ð6Þ
Substituting for Mx its expression in Eq. (2) and allowing for Eq. (3), these conditions are transformed to the following form:
w ¼ 0 hy ¼ 0 D11
@hx @hy þ D12 ¼0 @x @y
ð7Þ
This form allows the Kantorovich procedure to be implemented to reduce the dimension of the variational equation, Eq. (5). Variations of the deflection, w and angles of rotation, hx, hy along the axis x can be approximated by using trigonometric functions as follows:
wðx; yÞ ¼ wm ðyÞ sin km x hx ðx; yÞ ¼ hxm ðyÞ cos km x
M y ¼ D21 kx þ D22 ky
Mxy ¼ D33 kxy
@hx @hy þ @y @x @w @w wx ¼ hx þ wy ¼ hy þ @x @y @w @w xx ¼ xy ¼ @x @y kx ¼
w ¼ 0 hy ¼ 0 M x ¼ 0
0
Mx ¼ D11 kx þ D12 ky
Out-of-plane deformation of the plate is characterised by the deflection w, and the angles of rotations hx and hy of the element normal to the middle plane. These variables are related to the curvatures kx, ky, kxy, shear deformations wx, wy, and angles xx, xy as follows:
Boundary conditions at the simply supported edges of the plate x = 0 and x = a are given by
b
0
109
ð8Þ
hy ðx; yÞ ¼ hym ðyÞ sin km x where wm(y), hxm(y), and hym(y) are unknown functions of y, km = mp/a, m is the number of buckling half-waves in the x-direction. Clearly, these approximations satisfy the boundary conditions given by Eq. (7). Substituting Eq. (8) into Eq. (5) and integrating with respect to x, as per the Kantorovich procedure, yields the following one-dimensional functional: Z b
dhym Sx k2m wm þ km hxm dwm þ Sx km wm þ D11 k2m þ Sx hxm D12 km dhxm dy 0 dwm dhxm dwm dwm þ D33 km þ D33 k2m þ Sy hym dhym þ Sy þ hym d þ Sy dy dy dy dy dhxm dhxm dhym dhym þ km hym d þ D21 km hxm þ D22 d þD33 dy dy dy dy Nk2m wm dwm dy ¼ 0 ð9Þ Fig. 1. Uniformly compressed rectangular SSCF sandwich plate.
Varying this functional yields the following equations:
110
Z
b
0
Z
b
0
Z
b
0
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
Lm dwm dy þ ½Q ym dwm b0 ¼ 0 Lxm dhxm dy þ ½M xym dhxm b0 ¼ 0
ð10Þ
Lym dhym dy þ ½M ym dhym b0 ¼ 0
in which 2
Lm ¼ Sx k2m wm Sy
d wm dy
2
þ Sx km hxm Sy
dhym Nk2m wm dy
2 d hxm dhym Lxm ¼ Sx km wm þ D11 k2m þ Sx hxm D33 ðD12 þ D33 Þkm 2 dy dy 2 dwm dhxm d hym þ ðD21 þ D33 Þkm þ D33 k2m þ Sy hym D22 Lym ¼ Sy 2 dy dy dy ð11Þ
wm ðyÞ ¼ W m UðyÞ hxm ðyÞ ¼ T xm UðyÞ hym ðyÞ ¼ T ym VðyÞ
and
dwm þ hym dy dhxm þ km hym M xym ¼ D33 dy dhym M ym ¼ D21 km hxm þ D22 dy
ð12Þ
There are two approaches that could be applied to the solution of buckling problem specified by Eq. (10). The first approach is based on the traditional reduction of Eq. (10) to the following homogeneous buckling equations written for each buckling mode m
ð13Þ
and supplemented by the boundary conditions at y = 0
wm ¼ 0 hxm ¼ 0 hym ¼ 0
ð14Þ
and at y = b
Q ym ¼ 0 M xym ¼ 0 M ym ¼ 0
ð16Þ
where Wm, Txm, Tym are unknown coefficients, and U (y), V (y) are the approximating functions given by
Q ym ¼ Sy
Lm ¼ 0 Lxm ¼ 0 Lym ¼ 0
Based on the above, it follows that the use of this approach involves substantial algorithmic and computational efforts. Clearly, this hinders the application of this technique to the solution of design problems in which constraints are imposed on the value of critical buckling load. The second approach, adopted in this work, is based on the generalised Galerkin method. In this case, the governing equations are given by Eq. (10). According to this method, the functions wm(y), hxm(y), and hym(y) are replaced with their analytical approximations which would closely reflect the variations of the plate deflection and angles of rotation along the y-axis. In the case under consideration, these approximating functions are selected based on the solution of bending problem for a cantilever beam subjected to uniformly distributed load as shown in Fig. 2. The beam is modelled using a theory which takes into account the transverse shear deformation. As a result, the deflection and angles of rotation are presented as follows:
ð15Þ
Such a reduction is based on the arbitrariness of variations d wm, dhxm, and dhym in Eq. (10). It follows from Eq. (15) that the shear force Qy, and bending and twisting moments, My and Mxy are equal to zero at the free edge of the plate. Assume that Nm is the value of compressive load for which there exists a non-trivial solution of the homogeneous boundary problem specified by Eqs. (13)–(15). Note that the system of differential equations, Eq. (13) is of sixth order. Thus, the six roots of the corresponding characteristic polynomial would need to be found to solve the problem. Each root corresponds to a particular solution of the system of differential equations, Eq. (13). Clearly, the general solution is given by a linear combination of these particular solutions. In this case, functions wm(y), hxm(y), and hym(y) would include six constants of integration that could be found by using the boundary conditions given by Eqs. (14) and (15). This would lead to the system of algebraic equations of sixth order. Equating the determinant of this system to zero, the nonlinear algebraic equation with respect to the unknown force Nm could be derived. This equation should be solved numerically. Hence, in order to find Nm the two aforementioned numerical procedures should be applied. In the process of numerical solution for the nonlinear equation obtained from the fulfilment of boundary conditions, the six roots of the corresponding polynomial should be found numerically. It follows from Eq. (11) that the value of each of these roots depends on the force Nm. This force is calculated for m = 1, 2, 3, . . . and the critical buckling load corresponds to the minimal value of Nm.
y y y3 y2 y 4 2 þ 6 12cy 2 3 b b b b b y y2 y þ1 VðyÞ ¼ b 3b2 b
UðyÞ ¼
ð17Þ
The dimensionless parameter cy characterises a transverse shear compliance of the sandwich plate along the y-axis and is calculated as follows:
cy ¼
D22
ð18Þ
2
Sy b
It follows from Eqs. (16) and (17) that U(y = 0) = 0 and V(y = 0) = 0. So, the approximations selected as per Eqs. (16) and (17) satisfy kinematic boundary conditions, Eq. (14) at the clamped edge of the plate. However, these approximations do not fulfil the static boundary conditions, Eq. (15) at the free edge, i.e. Qym (y = b) – 0, Mxym (y = b) – 0, Mym(y = b) – 0. This is acceptable, since the application of the generalised Galerkin procedure does not require the exact fulfilment of static boundary conditions. As can be seen, the equations in Eq. (10) automatically provide an approximate satisfaction of these conditions at the free edge y = b. Note that the efficiency of applying the approximating functions U(y) and V(y) has been demonstrated for the solutions of free vibration problems for composite sandwich plates considered in articles [19,20]. Variations of the functions wm, hxm, and hym given by Eq. (16) have the form
dwm ¼ UdW m
dhxm ¼ UdT xm
dhym ¼ VdT ym
ð19Þ
Substituting Eqs. (16) and (19) into Eqs. (10)–(12), and taking into account the arbitrariness of the variations dWm, dTxm and dTym, the following governing equations of the generalised Galerkin method are derived:
Fig. 2. Cantilever beam subjected to uniformly distributed load
111
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
Z
b
0
Z
b
0
Z
b
0
Lm Udy þ ½Q ym Ub0 ¼ 0 Lxm Udy þ ½M xym Ub0 ¼ 0
b Lym Vdy þ M ym V 0 ¼ 0
in which 2
Lym
d U
ð29Þ
!
dV W m þ Sx km UT xm Sy T ym Nk2m UW m 2 dy dy " # 2 d U dV T ym ¼ Sx km UW m þ D11 k2m þ Sx U D33 2 T xm ðD12 þ D33 Þkm dy dy " # 2 dU dU d V ¼ Sy W m þ ðD21 þ D33 Þkm T xm þ D33 k2m þ Sy V D22 2 T ym dy dy dy Sx k2m U Sy
Lm ¼ Lxm
ð20Þ
Taking these equations into account, the system of equations in Eq. (23) can be re-arranged to the following form:
Sx k2m I1 Sy I2 W m þ Sx km I1 T xm þ Sy J 3 T ym Nk2m I1 W m ¼ 0 h
i Sx km I1 W m þ D11 k2m þ Sx I1 D33 I2 T xm þ ðD12 I3 þ D33 J 3 Þkm T ym ¼ 0 h
i Sy J 3 W m þ ðD21 I3 þ D33 J 3 Þkm T xm þ D33 k2m þ Sy J 1 D22 J 2 T ym ¼ 0
ð21Þ
and
dU W m þ VT ym dy dU T xm þ km VT ym Mxym ¼ D33 dy dV T ym Mym ¼ D21 km UT xm þ D22 dy Q ym ¼ Sy
ð22Þ
where
I2 ¼
24 c2 35b
ð30Þ
and
c2 ¼ 15 þ 84cy þ 280c2y
ð31Þ
The equations in Eq. (29) can be transformed into a form with dimensionless coefficients. Substituting Eqs. (26) and (30) into the equations in Eq. (29), taking into account that km = mp/a, and mul2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tiplying each of them by 630ba = D11 D22 , the following equations are derived:
16a11 W m þ 16a12 F xm þ 108a13 F ym 16b11 gm W m ¼ 0 16a21 W m þ 16a22 F xm þ 18a23 F ym ¼ 0
ð32Þ
108a31 W m þ 18a32 F xm þ 9a33 F ym ¼ 0 in which
After integration in Eq. (20), the latter are transformed into the following homogeneous system of linear algebraic equations:
F xm ¼ aT xm
h
The coefficients in the system of equations given by Eq. (32) are calculated as follows:
i Sx k2m I1 Sy ðI2 PÞ W m þ Sx km I1 T xm
Sy ðI3 ZÞT ym Nk2m I1 W m ¼ 0 h
i Sx km I1 W m þ D11 k2m þ Sx I1 D33 ðI2 PÞ T xm
a11 ¼ p2m c1
ð23Þ a33 ¼ 5
þ ½D12 I3 D33 ðI3 ZÞkm T ym ¼ 0
in which
Z
b
U 2 dy I2 ¼
Z
0
J1 ¼
Z
2
b
U
0 b
2
V dy J 2 ¼
0
Z
dy I3 ¼
2
dy
V
d V dy
2
Z
b
U
0
2
b
0
d U
dy J 3 ¼
Z 0
b
dV dy dy
dU dy V dy
dU P¼ U dy
b 0
Z¼
½UVb0
b dV R¼ V dy 0
ð24Þ
ð25Þ
I1 ¼
ð26Þ
12 ð1 þ 4cy Þ R ¼ 0 Z ¼ 1 þ 4cy b
þ
1
!
þ
14
D11 Sx a2 sffiffiffiffiffiffiffiffi 2 D11 b a¼ 2 D22 a
cx ¼
ð34Þ
ð35Þ D12 b12 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22
D33 b33 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22
ð36Þ ð37Þ
The dimensionless parameter cx in Eq. (35) characterises the transverse shear compliance of the sandwich plate in the xz-plane (see Fig. 3). The parameter gm in the first equation of Eq. (32) is a dimensionless buckling coefficient corresponding to the buckling mode m and related to the external load as follows 2
Nb D11 D22
ð38Þ
Putting the determinant of the matrix of coefficients in Eq. (32) equal to zero, the following equation can be obtained
ð27Þ
Substituting Eq. (17) into Eq. (25), the latter can be transformed as follows:
P¼
p2m b33
gm ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
in which
c1 ¼ 91 þ 999cy þ 3024c2y c2 ¼ 5 þ 28cy þ 560c2y c3 ¼ 5 þ 56cy c3 ¼ 5 þ 14cy
pm ¼ mp
Substituting for U(y) and V(y) in Eq. (24) their expressions given by Eq. (17) and integrating, the following formulas are derived:
8b 12 c c I2 ¼ c I3 ¼ 3 315 1 35b 2 35 b 1 6 J1 ¼ J ¼ J ¼ c3 14 2 5b 3 35
a c2 1 þ 27 a ¼ p2m þ c a þ 27c2 b33 cx cy a 22 cx 1
where
and
ð33Þ
cy a a a c3 a12 ¼ a21 ¼ pm c1 a13 ¼ a31 ¼ cx cy a 3 Þ b11 ¼ p2m c1 a23 ¼ a32 ¼ pm ðb12 c3 þ 6b33 c
Sy J3 W m þ ½D21 ðJ3 ZÞ þ D33 J 3 km T xm h
i þ D33 k2m þ Sy J1 D22 ðJ2 RÞ T ym ¼ 0
I1 ¼
F ym ¼ bT ym
ð28Þ
8 > < 16a11 16b11 gm det 16a21 > : 108a31
16a12 16a22 18a32
9 108a13 > = ¼0 18a23 > ; 9a33
ð39Þ
Solving this equation for gm, taking into account that a12 = a21, a13 = a31, a23 = a32, the buckling coefficient can be calculated as follows:
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A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
The transverse shear stiffness coefficients for the sandwich plate under consideration are given by
Sx ¼ Gxzt tfxz
Sy ¼ Gyzt tfyz
ð47Þ
where
2 h h fxz ¼ 1 þ 1þ t t 2 h h fyz ¼ 1 þ 1þ t t
Gxzt Gxzh Gyzt Gyzh
1 ð48Þ
1
Substituting for D11, D22 and Sx, Sy in Eqs. (18) and (35) their expressions given by Eqs. (43) and (47) yields Fig. 3. Structure of sandwich plate with composite facings and orthotropic core.
gm ¼
4a11 a22 a33 þ 108a12 a13 a23 9a11 a223 324a22 a213 4a33 a212 b11 4a22 a33 9a223 ð40Þ
Using this formula, the values g1, g2, . . . , gk can be calculated for m = 1, 2, 3, . . . , k for given k. Then the critical buckling coefficient can be found using the following condition:
gcr ¼ minðg1 ; g2 ; . . . ; gk Þ
ð41Þ
This coefficient depends on the dimensionless parameters cx, cy, a, b12 and b33 that contain all the information on the elastic and geometrical characteristics of the sandwich plate. Once the critical buckling coefficient is found, the corresponding critical force is calculated using Eq. (38) solved for N as
Ncr ¼ gcr
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22
ð42Þ
2
b
As indicated above, the parameters cx, cy, a, b12 and b33 are calculated in terms of the characteristics of sandwich plate. Consider the plate shown in Fig. 3. The coordinate z is measured from the middle surface. The total thickness of the facings is equal to t, whereas h is the thickness of the core. It is assumed, that the face sheets are made from an orthotropic material with the moduli of elasticity Ext, Eyt in x and y directions; shear moduli Gxyt, Gxzt and Gyzt in the planes xy, xz, and yz, respectively; and Poisson’s ratios mxyt and myxt. The similar respective characteristics of the core material are denoted as: Exh, Eyh, Gxyh, Gxzh, Gyzh, mxyh and myxh. The bending stiffness coefficients of the sandwich plate modelled by the firstorder shear deformation theory are calculated as follows [18]:
D11 D22
t3 ¼ Ext n 12 11 3 t ¼ Eyt n 12 22
D12 D33
t3 ¼ D21 ¼ Ext mxyt n 12 12 3 t ¼ Gxyt n 12 33
ð43Þ
3
n11
3
n12
Exh mxyh h ¼nþ Ext mxyt t3
3
n22 ¼ n þ
3
Eyh h Eyt t 3
n33 ¼ n þ
Gxyh h Gxyt t3
Ext Eyt Eyt ¼ 1 mxyt myxt 1 mxyt myxt Eyh Exh ¼ Eyh ¼ 1 mxyh myxh 1 mxyh myxh
ð44Þ
ð45Þ
Ext ¼ Exh
cy ¼
2 1 b Eyt n22 12 t Gyzt fyz
ð49Þ
Similarly, substituting Eq. (43) into Eq. (36), the parameters a, b12 and b33 are determined as follows
sffiffiffiffiffiffisffiffiffiffiffiffiffi 2 Ext n11 b a¼ 2 Eyt n22 a
Ext mxyt n12 b12 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi n 11 n22 Ext Eyt
Gxyt n33 b33 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi n 11 n22 Ext Eyt
ð50Þ
Taking into account the expressions for D11 and D22 as per Eq. (43), the formula for critical force, Eq. (42) can be presented in the following form:
Ncr ¼ gcr
qffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi Ext Eyt n11 n22 2 12b t3
ð51Þ
Using this equation, the critical buckling force can be calculated for the SSCF sandwich plate composed of the orthotropic facings and core that have the aforementioned elastic characteristics. The equations could be simplified for the plate with lightweight honeycomb core. In this case, it can be assumed that Exh = Eyh = Gxyh = 0. Subsequently, it follows from Eq. (44) that
n11 ¼ n12 ¼ n22 ¼ n33 ¼ n
ð52Þ
and the equations specifying parameters cx and cy are transformed as follows:
cx ¼
1 a 2 Ext n 12 t Gxzt fxz
cy ¼
2 1 b Eyt n 12 t Gyzt fyz
ð53Þ
Substituting Eq. (52) into Eq. (50) yields
sffiffiffiffiffiffi 2 Ext b a¼ Eyt a2
Ncr ¼ gcr
2
h h n¼1þ3 þ3 2 t t
1 a 2 Ext n11 12 t Gxzt fxz
Ext mxyt b12 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi Ext Eyt
Gxyt b33 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi Ext Eyt
ð54Þ
Taking into account Eq. (52), the formula for the critical buckling load, Eq. (51) takes the form
where
Exh h ¼nþ Ext t 3
cx ¼
ð46Þ
2 qffiffiffiffiffiffiffiffiffiffiffiffi tn b Ext Eyt 12 t
ð55Þ
The above equations, Eqs. (51) or (55) provide a ‘‘fast solution’’ to the buckling problem under consideration. The use of these formulae does not require complex computations and they can be efficiently used in the structural design process. 3. Numerical analysis The calculations of buckling load have been performed for composite sandwich plates with various honeycomb cores. The face sheets of the panels are made of carbon fibre reinforced plastic with the following properties: Ext = 54.55 GPa, Eyt = 54.55 GPa,
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A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
Gxyt = 20.67 GPa, Gxzt = 3.78 GPa, Gyzt = 3.78 GPa, mxyt = 0.32, myxt = 0.32. In order to assess the applicability and limitations of the proposed solution, the panels with four different honeycomb cores have been considered: (a) 5052 Aluminium with the cell size of 3 mm, having transverse shear moduli Gxzh = 930 MPa, Gyzh = 372 MPa; (b) 3003 Aluminium with the cell size of 6 mm, having Gxzh = 440 MPa, Gyzh = 220 MPa; (c) 3003 Aluminium with the cell size of 19 mm, having Gxzh = 110 MPa, Gyzh = 55 MPa; and (d) HRH10 Nomex (Aramid) with the cell size of 3 mm, having Gxzh = 25 MPa, Gyzh = 17 MPa [21]. The plates have the overall thicknesses of the facings t = 0.001 m and the thicknesses of the core layer h = 0.01 m, 0.05 m, and 0.1 m. The in-plane dimensions of the plates are: b = 1 m, and a = 1, 3, and 5 m. Critical forces have been calculated using Eq. (51). The results of calculations are presented in Tables 1–4. It can be seen that, as expected, the value of Ncr is substantially increases with the increase in the core thickness. An increase in the length of plate a from 1 to 5 m leads to the reduction of critical force. It follows from the analysis that the buckling of the square plate (a = 1 m) occurs with the formation of one halfwave. For the plates with a = 3 and 5 m, the numbers of the buckle half-waves are equal to 2 and 3, respectively. Note, that the above analyses have been performed for k = 5 (see Eq. (41)). The results of calculations have been verified using a finite element analysis. The numerical solutions have been obtained using MSC Nastran [22]. A 20 20 mm 4-node finite element Laminate with the option suitable for the analysis of sandwich panels has been implemented to model the aforementioned plates. The finite-element models for 1 m 1 m, 1 m 3 m, and 1 m 5 m plates have been composed of 2500 (2601 nodes), 7500 (7701 nodes), and 12,500 (12,801 nodes) elements, respectively. The finite-element mesh for the square plate (a = b = 1 m) is shown in Fig. 4. The buckling mode shapes for the SSCF sandwich plates with a = 1, 3 and 5 m, and b = 1 m are shown in Fig. 5. The corresponding results of finite-element analyses are presented in Tables 1–4. It follows from the data given in these tables that the maximum differences between the values of critical loads Ncr calculated according to Eq. (51) and those obtained from the finite-element analysis
N FEM do not exceed 4%, 5%, 5.8% and 10.1% for the plates having cr
Table 2 (kN/m) for the SSCF sandwich Critical loads Ncr (kN/m) (analytical solution) and N FEM cr plate with the honeycomb core having Gxzh = 440 MPa, Gyzh = 220 MPa (3003 Aluminium, cell size 6 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
a=5m
Ncr
N FEM cr
Ncr
N FEM cr
Ncr
N FEM cr
27.85 628.10 2414.29
27.91 618.06 2339.01
21.37 484.03 1870.48
20.99 469.48 1793.60
21.09 478.09 1848.87
20.52 459.73 1765.93
Table 3 Critical loads Ncr (kN/m) (analytical solution) and N FEM (kN/m) for the SSCF sandwich cr plate with the honeycomb core having Gxzh = 110 MPa, Gyzh = 55 MPa (3003 Aluminium, cell size 19 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
a=5m
Ncr
N FEM cr
Ncr
N FEM cr
Ncr
N FEM cr
27.38 576.86 2059.83
27.01 558.43 1954.31
21.07 451.22 1638.97
20.44 431.38 1552.05
20.81 446.59 1625.69
20.03 423.31 1536.63
Table 4 Critical loads Ncr (kN/m) (analytical solution) and N FEM (kN/m) for the SSCF sandwich cr plate with the honeycomb core having Gxzh = 25 MPa, Gyzh = 17 MPa (HRH10 Nomex (Aramid), cell size 3 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
Ncr
N FEM cr
25.74 440.63 1324.03
24.73 415.69 1221.43
a=5m
Ncr
N FEM cr
Ncr
N FEM cr
20.13 367.85 1164.22
19.08 344.11 1068.09
19.93 367.56 1161.85
18.78 342.91 1055.27
the aforementioned honeycomb cores (a–d), respectively. The highest discrepancy has been observed for the largest and thickest plate having very compliant core. These results show that the solution method proposed in this work provides the values of critical buckling loads with sufficient accuracy and can be efficiently applied to the solution of relevant design problems. 4. Examples of sandwich panel design Calculations of the buckling loads for the composite SSCF sandwich panels under consideration based on the formulas presented in this paper (i.e., Eqs. (51) or (55)) do not require any computationally expensive procedures. Therefore, the design problems involving buckling constraints can be efficiently solved with relative ease.
Ncr
N FEM cr
Ncr
N FEM cr
Consider a number of examples where thicknesses of the core, h and facings, t can be found for the sandwich plate compressed with e The elastic properties of the materials of facthe specified load N. ings and core are given as follows: Ext = Eyt = 54.55 GPa, Gxyt = 20.67 GPa, Gxzt = Gyzt = 3.78 GPa, mxyt = myxt = 0.32, Gxzh = 440 MPa, e the latter can be Gyzh = 220 MPa. Replacing Ncr in Eq. (55) with N, written as
21.41 489.31 1910.94
21.05 479.71 1846.32
21.13 483.07 1887.12
20.62 467.01 1814.28
2 qffiffiffiffiffiffiffiffiffiffiffiffi e g tn b N Ext Eyt ¼ 0 cr 12 t
Table 1 Critical loads Ncr (kN/m) (analytical solution) and NFEM (kN/m) for the SSCF sandwich cr plate with the honeycomb core having Gxzh = 930 MPa, Gyzh = 372 MPa (5052 Aluminium, cell size 3 mm, [21]). h (m)
0.01 0.05 0.10
a=1m
a=3m
Ncr
N FEM cr
27.93 637.06 2482.56
27.87 631.64 2432.13
Fig. 4. Finite-element mesh for the SSCF sandwich plate with a = b = 1 m.
a=5m
ð56Þ
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A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
Fig. 6. Contour curves of Ncr(t, h) for the SSCF sandwich plate with a = 2 m, b = 1 m.
Fig. 5. Buckling mode shapes of the SSCF sandwich plate: (a) a = 1 m; (b) a = 3 m; and (c) a = 5 m.
where gcr is determined by Eqs. (40), (41), taking into account Eqs. (34), (53), and (54). This nonlinear equation relates the e with the thicknesses of the core, h specified compressive load N and facings, t for given in-plane dimensions and elastic properties of the constituent materials. For example, consider the sandwich plate with a = 2 m, b = 1 m, and t = 0.001 m loaded with the force e ¼ 300kN=m. Calculations of gcr using Eqs. (40), (41) and Eqs. N (34), (53), (54) and subsequent solution of the equation, Eq. (56) yield: h = 38.68 mm. In another example, for a 1 1 m plate having the thickness of the core h = 0.02 m and subjected to compressive load e ¼ 200 kN=m, the application of the same procedure provides N t = 1.838 mm. In the third example, thicknesses of the facings and the core delivering the minimum mass of the sandwich panel subjected to the specified critical load are determined. The mass of the sandwich plate having the facings of equal thickness is calculated as follows:
M ¼ abðqt t þ qh hÞ
ð57Þ
where qt and qh are the densities of the facings and core material, respectively. Consider a sandwich plate with a = 2 m and b = 1 m. The values of critical load have been calculated using the aforementioned procedure for 0.5 mm 6 t 6 2 mm and 10 mm 6 h 6 80 mm. The function Ncr(t,h) can be presented by the contour curves shown e takes the in Fig. 6. Assume that the specified compressive load N following values: 100, 200, 300, 400, 500, and 600 kN/m. It follows e can be from Fig. 6 that for each of these values the condition N cr ¼ N fulfilled for various combinations of t and h that belong to a corresponding contour curve. In this example, equation, Eq. (57) defines the objective function whereas Eq. (56) can be treated as a constraint in the form of equality. The design domain for the variables t and h is determined by the respective ranges presented above. Using subroutines available in the library IMSL [23] the FORTRAN code has been developed to solve the optimisation problem. The results of optimisation are presented in Table 5 where the values
Table 5 Optimal parameters of the SSCF sandwich plate. e (kN/m) N
Mmin (kg)
t (mm)
h (mm)
KN
100 200 300 400 500 600
6.348 8.011 9.184 10.121 10.915 11.611
0.723 0.911 1.043 1.148 1.237 1.314
26.115 32.987 37.849 41.736 45.027 47.921
1605.8 2544.9 3329.8 4028.7 4669.6 5267.6
of t and h delivering the minimum mass Mmin for the specified vale are shown. These results can be used ues of the compressive load N to assess the weight efficiency of the SSCF sandwich plates under consideration. The following coefficient could be adopted as a measure of this efficiency:
KN ¼
e Nb M min g
ð58Þ
where g = 9.81 M/s2 is the gravitational acceleration. The corresponding values of this coefficient are given in Table 5. Analysis of e shows that the weight efficiency of the sandthe dependence K N ð NÞ wich plate increases with increase in the compressive load. In particular, the coefficient for the plate subjected to the highest load e ¼ 600 kN=mÞ is 3.28 times the value of the coefficient correðN sponding to the plate compressed with the lowest load e ¼ 100 kN=mÞ. ðN 5. Conclusions The buckling problem has been solved for the SSCF composite sandwich plate subjected to uniform in-plane compressive load. The problem has been solved using the Kantorovich procedure and generalised Galerkin method and the analytical formula providing the ‘‘fast solution’’ has been obtained. The formula has been successfully verified by using finite-element analysis. The efficiency of the solution developed in this work has been demonstrated for various examples of the design of sandwich plates for which the constraint is imposed on the value of critical compressive load. It has been shown that the maximum weight efficiency
A.V. Lopatin, E.V. Morozov / Composite Structures 105 (2013) 108–115
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