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Semi-analytical stability analysis of doubly-curved orthotropic shallow panels — considering the effects of boundary conditions
International Journal of Non-Linear Mechanics 37 (2002) 659–667
Semi-analytical stability analysis of doubly-curved orthotropic shallow panels — considering the e(ects of boundary conditions D.H. van Campena; ∗ , V.P. Bouwmanb , G.Q. Zhangb , J. Zhanga , B.J.W. ter Weemeb a Department
of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands b Center For Industrial Technology=Philips, P.O. Box 218, 5600 MD Eindhoven, Netherlands
Abstract This paper focuses on the development of the partitioned solution method (PSM) for analyzing the stability behavior of doubly-curved shallow orthotropic panels under external pressure, covering both the buckling and postbuckling responses. Adjacent equilibrium method (AEM) is used to verify the developed PSM method and the associated stability results. The equilibrium and compatibility equations are derived using Donnell-type thin shell theory, with the Airy stress function and the out-of-plane displacement as unknowns. Based on AEM and PSM, both an eigenvalue problem and non-linear algebraic equations are obtained which are used as the basis for the stability criteria, respectively. Results obtained from those two methods are presented and compared with each other for a few arbitrary sets of system parameters, wherein no postbuckling solutions are presented with AEM. The in>uence of the boundary conditions on the stability behavior is also investigated using the PSM. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bifurcation; Buckling; Stability; Shell
1. Introduction Doubly-curved shallow orthotropic panels under external pressure are frequently encountered in engineering practice and their stability behavior can be one of the important design speci@cations. Consequently, the development for stability analysis methodology and the associated simulation tools have drawn extensive attention of scientists [1– 6]. Among them, some semi- analytical analysis methods are developed based on the adjacent equilibrium ∗
Corresponding author.
method (AEM). The nature of this method is that the solutions at the neutral equilibrium position are perturbated and the criterion of instability is then the existence of such perturbations. AEM will lead to an eigenvalue problem that can be used to calculate the bifurcation buckling load [2]. In addition, the asymptotic postbuckling analysis method established by Koiter [7] yields information about the initial postbuckling path. In order to obtain more information about the buckling and specially the postbuckling path of thin-walled structures, FEM is frequently used. The general applicability and >exibility of the @nite element method (FEM) make it a very strong tool
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Nomenclature AEM FEM PSM a; b; t Ex ; E y Gxy
xy ; yx Mx ; My ; Mxy Nx ; Ny ; Nxy u; v; w
adjacent equilibrium method @nite element method partitioned solution method length, width and thickness of panel elasticity moduli in length and width directions shear modulus Poisson’s ratios bending and twisting moment resultants normal and shearing stress resultants middle surface displacement components
x ; y ; xy ; x ; panel strain components y ; xy ’ airy stress function principal curvatures in length and kx ; ky width directions unknown coeKcients for function wnm w q uniform load
in development and engineering. FEM is specially useful for analyzing complex structures, with for example spatial dependent material and thickness properties, cut-outs and no-regular boundary conditions, for which semi-analytical methods are not robust enough. However, there are several obvious reasons to study the stability behavior of doubly-curved orthotropic panels using semi-analytical approaches. First, it is very time consuming to obtain stability solutions by FEM. To identify the buckling solution alone, already many simulations have to be carried out. Secondly, it is diKcult (if not impossible) to obtain multi-equilibrium solutions inherited in certain structures by using FEM, such as multiple or clustered bifurcation. Some numerical algorithms for the application of FEM have been developed such as the path following procedure, which is generally constructed by the arc-length scheme [8–18].
Even with the help of those specially developed algorithms, the reliability of the stability solutions obtained by FEM can still be very sensitive to the elements and the analysis parameters. Thirdly, it is believed that semi-analytical solutions may provide more physical insight than the FEM results. Hence, due to the complexity of the non-linear stability behavior, it is desirable to have a semi-analytical method and tool that can be used to provide a reliable and eKcient estimation of the characteristic stability behavior and to verify the FEM results. To obtain more information on the postbuckling behavior of doubly-curved shallow panels, a semi-analytical method, partitioned solution method (PSM) [19], was proposed. The major advantages of the PSM are that the stability solutions are not limited to the initial postbuckling region, and the clustered bifurcation problem can be analyzed. In this paper, stability predictions obtained from both PSM and AEM are presented. Based on PSM, the in>uence of combinations of boundary conditions on the stability behavior of doubly-curved orthotropic panels is also investigated. Notice that although only results of perfect panels are described in this paper, results of panels with general described con@gurations using PSM method are also available, which are under experimental veri@cations. The e(ect of geometric imperfections can be taken into account by the PSM method in several di(erent ways. One of them is to use the expression of the real panel con@guration including imperfections in the derivation of the stability governing equations. Another possibility is to carry out a geometric parameter sensitivity study for the e(ect of geometric imperfections. 2. Fundamental equations Fig. 1 shows the doubly-curved shallow panel subjected to uniform external pressure. The Donnell-type thin shell theory and assumptions associated with doubly-curved shallow panels are used in this investigation. The strain–displacement relations are x = u; x + kx w + 12 (w; x )2 ;
x = − w; xx ;
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the following equilibrium and compatibility equations are obtained, respectively: D1 w; xxxx + D2 w; yyyy − D3 w; xxyy = − ’; yy (kx − w; xx ) − ’; xx (ky − w; yy ) − 2’; xy w; xy − q;
(5)
D4 ’; xxxx + D5 ’; yyyy + D6 ’; xxyy = (w; xy )2 − w; xx w; yy + kx w; yy + ky w; xx ; where Ex t 3 D1 = ; 12B
Fig. 1. Doubly-curved panel.
y = v; y + ky w + 12 (w; y )2 ; xy = u; y + v; x + w; x w; y ;
(1)
where (:); x means @(:)=@x. Taking for the constitutive equations Hooke’s law for orthotropic materials, the stress–strain relationship can be expressed as Ex t Ex t 3 (x + xy y ); Mx = (x + xy y ); Nx = B 12B Ey t Ex Ny = y + xy x ; B Ey Ey t 3 My = 12B
Ex y + xy x ; Ey
Gxy t 3 xy ; (2) 12 where B = 1 − 2xy Ex =Ey . Using the stationary potential energy criterion, the non-linear equilibrium equations become Nxy = Gxy txy ;
Mxy =
Nx; x + Nxy; y = 0; Nxy; x + Ny; y = 0; Mx; xx + 2Mxy; xy + My; yy − kx Nx − ky Ny − q + Nx w; xx + 2Nxy w; xy + Ny w; yy = 0:
(3)
Combining the above equations with an Airy stress function ’ de@ned by Nx = ’; yy ;
Ny = ’; xx ;
Nxy = − ’; xy
Ey t 3 ; 12B
Gxy t 3 Ex ; D2 − Ey 3 1 1 D5 = ; D6 = − 2 xy D4 : Ex t Gxy t
D3 = − xy D1 − xy
y = − w; yy ; xy = − 2w; xy ;
D2 =
(6)
(4)
D4 =
1 ; Ey t
The left-hand side of the equilibrium equation can be de@ned as a linear operator LD (w). Then {sin(n x) sin(m y) with = =a, = =b and n; m = 1; 2; 3; : : : + ∞} is one of the sets of its eigenfunctions. The solution of the equilibrium equation can therefore be projected on the space spanned by the following eigenfuncions that are complete in that space: wnm sin(n x) sin(m y): (7) w(x; y) = − To complete the description of the model of the doubly-curved orthotropic panel, the following four types of boundary conditions are investigated: (1) Simply supported on both sides w = N1 = M1 = 0 on x = 0; a, w = N2 = M2 = 0 on y = 0; b. (2) Hinged on both sides w = M1 = 0 on x = 0; a, w = M2 = 0 on y = 0; b. (3) Hinged on x = 0, a and simply supported on y = 0; b w = M1 = 0 on x = 0; a; w = N2 = M2 = 0 on y = 0; b. (4) Simply supported on x = 0; a and hinged on y = 0; b w = N1 = M1 = 0 on x = 0; a, w = M2 = 0 on y = 0; b.
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In the following, the stability solutions of the doubly-curved shallow orthotropic panels will be developed following both AEM and PSM, using the derived governing equations and boundary conditions. 3. Adjacent equilibrium method (AEM) Prior to buckling, the panel is assumed to be in a primary equilibrium state (w0 ; ’0 ). Near the point of buckling an adjacent equilibrium state (w1 ; ’1 ) is assumed to exist, which is suKciently close to the primary equilibrium state such that w1 and ’1 are in@nitesimal quantities. On account of this, the equations for the solution of the prebuckling state are supplemented with equations governing the buckling problem. Substituting w = w0 + w1 ;
’ = ’0 + ’1
(8)
into the equilibrium and compatibility equations and omitting the quadratic terms, equations governing the prebuckling state and buckling state can be obtained. Prebuckling state: D1 w;0xxxx + D2 w;0yyyy − D3 w;0xxyy = − ’0; yy nkx − ’0; xx ky − q;
(9)
D4 ’0; xxxx + D5 ’0; yyyy + D6 ’0; xxyy = kx w;0yy + ky w;0xx : (10) Buckling state: D1 w;1xxxx + D2 w;1yyyy − D3 w;1xxyy =−
’1; yy kx
−
+ ’1; yy w;0xx
’1; xx ky +
+
+
(11)
D4 ’1; xxxx + D5 ’1; yyyy + D6 ’1; xxyy = kx w;1yy + ky w;1xx + 2w;0xy w;1xy − w;0xx w;1yy − w;0yy w;1xx :
(13)
Consequently, the stress function satisfying the hinged boundary conditions, can be obtained directly by resolving the compatibility equation after substituting the expression for w0 : 0 ’0 = D70 w11 sin( x) sin( y) + 12 px0 y2 + 12 py0 x2 ;
(14)
where ky 2 + k x 2 : D5 4 + D 4 4 + D 6 2 2 Because of the hinged boundary conditions, the average distance between two opposite boundaries is @xed. The terms px0 and py0 can then be obtained with the following equations: 1 b a 0 u d x dy = 0; Qu = b 0 0 ;x 1 a b 0 v dy d x = 0: (15) Qv = a 0 0 ;y D70 =
0 ; px0 = D80 w11
’1; xx w;0yy
− 2’0; xy ; w;1xy − 2’1; xy w;0xy ;
0 sin( x) sin( y): w0 = − w11
Resolving these equations after substituting the linear strain–displacement relations in them yields:
’0; yy w;1xx
’0; xx w;1yy
are not presented here and only the equations for the system with hinged boundary conditions are developed further. This method however, can be extended to cases with other boundary conditions like for example those speci@ed in the previous section, or even, by changing the set of eigenfunctions in Eq. (7) to cover di(erent boundary conditions. The @rst mode pair of the eigenfunction found in the previous section is used to approach the prebuckling state, which is close to the membrane state. Accordingly, the out-of-plane displacement of the prebuckling equations can be expressed as
(12)
Notice that for the sake of simplicity, the governing equations for the initial postbuckling state
0 ; py0 = D90 w11
where C 1 C2 − C3 C4 D80 = ; C4 C6 − C2 C5
(16) D90 =
C 3 C5 − C1 C6 ; C4 C6 − C2 C5
yx 4 1 4 4 + D70 + kx ; Ey t Ex t 1 C2 = ab ; Ey t
C1 = −D70
D.H. van Campen et al. / International Journal of Non-Linear Mechanics 37 (2002) 659–667
C3 = −D70
xy 4 1 4 4 + D70 + ky ; Ey t Ex t
C4 = −ab
yx ; Ey t
1 ; C5 = ab Ex t
compatibility equations yield two governing equations for the fundamental path and two governing equations for the bifurcated path: Fundamental path D1 w;0xxxx + D2 w;0yyyy − D3 w;0xxyy
xy C6 = − ab : Ex t
Finally, the governing equation of the prebuckling state can be obtained by Galerkin procedure as follows: 4 0 0 q = 0; (17) w11 − D10 where 0 D10
4 (D0 kx + D90 ky ): 8
1 DR wR 1 = 0:
(19) 1
The existence of perturbation wR depends on ma1 1 trix DR , i.e. det(DR ) = 0. Notice that the relation 1 det(DR ) = 0 is a linear function of load q and thus can be considered as an eigenvalue problem. The smallest load q is de@ned as the bifurcation buckling load. 4. Partitioned solution method (PSM)
0
’ = ’0 + ’1 : 0
D4 ’0; xxxx + D5 ’0; yyyy + D6 ’0; xxyy = kx w;0yy + ky w;0xx + (w;0xy )2 − w;0xx w;0yy :
(22)
0
+ ’1; yy w;0xx + ’0; xx w;1yy + ’1; xx w;0yy − 2’0; xy w;1xy − 2’1; xy w;0xy − q1 ;
(23)
D4 ’1; xxxx + D5 ’1; yyyy + D6 ’1; xxyy = kx w;1yy + ky w;1xx + (w;1xy )2 − w;1xx w;1yy + 2w;0xy w;1xy − w;0xx w;1yy − w;0yy w;1xx :
(24)
In this case, the governing equations are developed further for systems with simply supported boundary conditions. The @rst mode pair of the eigenfunction found previously is used to approach the fundamental path. Again the stress function satisfying the simply supported boundary conditions can be obtained by resolving the compatibility equation after substituting the expression for w0 : 0 0 2 0 0 2 ’0 = D11 (w11 ) cos(2 x) + D12 (w11 ) cos(2 y)
The solutions of the equilibrium and compatibility equations can be partitioned into two parts: w = w0 + w1 ;
(21)
− ’1; xx (ky − w;1yy ) − 2’1; xy w;1xy + ’0; yy w;1xx
This yields the following solution for bifurcation buckling:
q = q0 + q1 ;
− 2’0; xy w;0xy − q0 ;
D1 w;1xxxx + D2 w;1yyyy − D3 w;1xxyy = − ’1; yy (kx − w;1xx )
With the same procedure and the known prebuckling solution, one can solve the buckling equations with the following assumption for the deformations: 1 wnm sin(n x) sin(m y): (18) w1 (x; y) = −
1 1 D10 wnm = 0 or
= − ’0; yy (kx − w;0xx ) − ’0; xx (ky − w;0yy )
Bifurcated path (postbuckling)
ab = (D1 4 + D2 4 − D3 2 2 + D70 kx 2 4 + D70 ky 2 ) −
663
(20)
With q as the bifurcation load, w ; ’ the fundamental solution at this bifurcation point and w1 ; ’1 the solution on the bifurcated path (postbuckling). Substituting these relations in the equilibrium and
0 0 + D13 w11 sin( x) sin( y);
where 2 ; 32D5 2
0 = D11
; 32D4 2
0 = D13
kx 2 + ky 2 : D5 4 + D 4 4 + D 6 2 2
0 D12 =
(25)
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The governing equation of the fundamental path can thus be obtained as 0 0 3 0 0 2 0 0 (w11 ) + D15 (w11 ) + D16 w11 − D14
4 0 q = 0;
(26)
where 0 D14
2 2 ab 0 0 (D11 + D12 = ); 2
0 =− D15
0 D16 =
8 0 0 0 2 2 (2D12 kx 2 + 2D11 ky 2 + D13 ); 3
ab (D1 4 + D2 4 − D3 2 2 4 0 0 + D13 kx 2 + D13 ky 2 ):
Under the assumption that the interaction between modes can be ignored, the out-of-plane displacement at the bifurcated path can be obtained as 1 sin(n x) sin(m y): w1 = − wnm
(27)
With this assumption and the known solution for the fundamental path, the stress function at the bifurcated path can be obtained by resolving the compatibility equation 1 1 2 1 1 2 (wnm ) cos(2n x) + D12 (wnm ) cos(2m y) ’1 = D11 1 1 wnm sin(n x) sin(m y) + D13 1 1 + D1i cos( i x) cos( i y)wnm ;
where n = 1 and m = 1. Finally, the stability equation governing the bifurcated path can be obtained as 1 1 3 1 1 2 (wnm ) + D15 (wnm ) D14
4 q1 = 0: n m
1 1 → G(wnm ); wnm 1 =− wnm
1 4 D14 D1 1 3 1 2 (wnm ) − 15 (wnm ) + q1 ; 1 1 1 D16 D16 n m D16 (30)
1 where D16 = 0: Let wR 1nm be a @xed point of the point mapping, then the following equation can be used to determine the bifurcation load q0 leading to the saddle-node bifurcation on the bifurcated path: D1 D1 1 DG 1 wR nm = − 3 14 (wR 1nm )2 − 2 15 wR nm = 1: #= 1 1 1 Dwnm D16 D16 (31)
The in>uence of the fundamental solution at the bifurcation load on the bifurcated solution is in1 1 and D16 . At the saddle-node cluded in the terms D15 bifurcation, there exist two points that coincide with each other. The necessary condition for saddle-node bifurcation is therefore 1 2 1 D14 D15 − 12 = 0: (32) $=4 1 1 D16 D16 Because of the non-linearity of this equation, it needs an iterative algorithm in combination with the equations governing the fundamental path to approach the solution. The limit load is de@ned as the load where the saddle-node bifurcation will occur on the @rst bifurcated branch (postbuckled).
(28)
i=4;5;6;7
1 1 + D16 wnm −
following form:
(29)
Snap-through buckling will occur along this new branch and the limit load is the load at which a saddle-node bifurcation occurs. To analyze the stability of the equilibrium positions at this postbuckled branch, a point mapping is constructed. The bifurcation equation is therefore rewritten in the
5. Numerical results In the following, some stability analysis results are presented. For a few arbitrary sets of system parameters, the results of AEM and PSM are compared with each other. The in>uences of the boundary conditions on both the buckling and postbuckling solutions are investigated using PSM. 5.1. Comparison between AEM and PSM Both AEM and PSM are used to analyze the buckling behavior of doubly-curved orthotropic shallow panels, described by the parameters in Table 1.
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665
Table 1 System parameters A B C D E
Ex (N=m2 )
Ey (N=m2 )
Gxy (N=m2 )
xy
kx
ky
a (m)
b (m)
3e10
3e10
1:154e10 1e10 1e10 1:154e10 1:154e10
0.3 0.1 0.1 0.3 0.3
0.1375 0.2 0.3 0.15 0.3
0.1375 0.3 0.4 0.25 0.2
0.2 0.2 0.4 0.2 0.2
0.2 0.2 0.3 0.5 0.2
2e10 2e10 3e10 3e10
4e10 4e10 3e10 3e10
Table 2 Results from AEM and PSM AEM=PSM
Bifurcation load (N=m2 )
Buckling load at the postbuckled branch (N=m2 )
Mode (x; y)
A B C D E
31.922=25.143 76.706=57.860 139.95=134.96 50.593=47.351 88.836=73.651
..=25.678 ..=61.179 ..=137.09 ..=47.830 ..=77.755
(1; 2)=(1; 3) (1; 3)=(1; 3) (1; 6)=(1; 5) (1; 8)=(1; 7) (4; 1)=(3; 1)
Table 3 System parameters Ex (N=m2 )
Ey (N=m2 )
Gxy (N=m2 )
xy
kx
ky
a (m)
b (m)
t (m)
3e10
3e10
1:154e10
0.3
0.25
0.5
0.6
0.34
0.00022
Notice that the thickness is the same for all cases (t = 0:00022 m) and that for some cases the material properties are isotropic. With AEM the bifurcation buckling load and the corresponding mode are obtained, while with PSM the bifurcation load and the corresponding modes, the limit load at the postbuckled branch are obtained. The results are shown in Table 2. As can be seen in Table 2, the bifurcation buckling results obtained from both methods are quite comparable, and the trends in bifurcation buckling loads are similar. The di(erences between the absolute values of the buckling loads and the corresponding buckling modes are caused by di(erences in the prebuckling states of both methods. The buckling loads obtained by following the postbuckled branches are a little higher. 5.2. In:uence of boundary conditions The PSM is used to analyze the in>uence of the boundary conditions on the stability behavior of the
doubly-curved shallow isotropic panel. The arbitrary system parameters are given in Table 3. With these system parameters four calculations are performed with various boundary conditions: Case 1: All edges hinged. Case 2: All edges simply supported. Case 3: Simply supported on the short and hinged on the long edges. Case 4: Hinged on the short and simply supported on the long edges. The results of these calculations are shown in Table 4 and for the @rst case, the response of the center of the panel is depicted in Fig. 2. As can be seen from Table 4, the boundary conditions have signi@cant in>uence on both the buckling loads and modes. As could be expected, the boundary conditions on the longer edges are dominant in the results due to the smaller distance between these edges. The lower buckling loads for cases with hinged boundary conditions are caused by prebuckling compression. In Fig. 2, the stable prebuckling
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Table 4 Results for various boundary conditions Case
Bifurcation load (N=m2 )
Buckling load at postbuckled branch (N=m2 )
Mode (x; y)
1 2 3 4
157.43 187.70 143.92 181.06
158.79 187.71 144.98 181.07
(1; 7) (16; 1) (1; 7) (16; 1)
tial postbuckling path. It is found that the buckling results obtained from both methods are comparable. Di(erences between absolute values of the buckling loads and modes are caused by the di(erences in the assumed prebuckling states. What is more, the PSM method can give stability predictions at the postbuckled branch. In addition, it is found that the in>uence of boundary conditions on both the buckling and postbuckling solutions can be significant. Hinged boundary conditions result in lower buckling loads due to prebuckling compression. Fig. 2. Load-displacement diagram for Case 1.
path with the out-of-plane displacement described by the @rst mode pair of the eigenfunctions, becomes unstable when it is intersected by the secondary equilibrium path. Along this path, with the out-of-plane displacement described by the bifurcation buckling mode, the limit load is found to be slightly higher than the bifurcation buckling load. 6. Conclusions In order to investigate the stability behavior (both buckling and postbuckling) of doubly-curved shallow orthotropic panels under external pressure, the PSM is further developed and presented in this paper. This method can be used to obtain both the bifurcation buckling and postbuckling solutions that are not con@ned in the initial postbuckling region. For some arbitrary sets of system parameters, the results of the PSM are compared with those obtained from the AEM that can only obtain the bifurcation buckling solutions and information of the ini-
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D.H. van Campen et al. / International Journal of Non-Linear Mechanics 37 (2002) 659–667 [12] J. Shi, Computing critical points and secondary paths in non-linear stability analysis by the @nite element method, Comput. Structures 58 (1996) 203–220. [13] A. Eriksson, C. Pacoste, A. Zdunek, Numerical analysis of complex instability behaviors using incremental-iterative strategies, Comput. Methods Appl. Mech. Eng. 136 (1996) 59–92. [14] J.G. Teng, Y.F. Lou, Post-collapse bifurcation analysis of shells of revolution by the accumulated arc-length method, Internat. J. Numerical Methods Eng. 40 (1997) 2369–2383. [15] H.B. Hellweg, M.A. Chris@eld, A new arc-length method for handling sharp snapback, Comput. Structures 66 (1998) 705–709.
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Semi-analytical stability analysis of doubly-curved orthotropic shallow panels — considering the e(ects of boundary conditions D.H. van Campena; ∗ , V.P. Bouwmanb , G.Q. Zhangb , J. Zhanga , B.J.W. ter Weemeb a Department
of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands b Center For Industrial Technology=Philips, P.O. Box 218, 5600 MD Eindhoven, Netherlands
Abstract This paper focuses on the development of the partitioned solution method (PSM) for analyzing the stability behavior of doubly-curved shallow orthotropic panels under external pressure, covering both the buckling and postbuckling responses. Adjacent equilibrium method (AEM) is used to verify the developed PSM method and the associated stability results. The equilibrium and compatibility equations are derived using Donnell-type thin shell theory, with the Airy stress function and the out-of-plane displacement as unknowns. Based on AEM and PSM, both an eigenvalue problem and non-linear algebraic equations are obtained which are used as the basis for the stability criteria, respectively. Results obtained from those two methods are presented and compared with each other for a few arbitrary sets of system parameters, wherein no postbuckling solutions are presented with AEM. The in>uence of the boundary conditions on the stability behavior is also investigated using the PSM. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bifurcation; Buckling; Stability; Shell
1. Introduction Doubly-curved shallow orthotropic panels under external pressure are frequently encountered in engineering practice and their stability behavior can be one of the important design speci@cations. Consequently, the development for stability analysis methodology and the associated simulation tools have drawn extensive attention of scientists [1– 6]. Among them, some semi- analytical analysis methods are developed based on the adjacent equilibrium ∗
Corresponding author.
method (AEM). The nature of this method is that the solutions at the neutral equilibrium position are perturbated and the criterion of instability is then the existence of such perturbations. AEM will lead to an eigenvalue problem that can be used to calculate the bifurcation buckling load [2]. In addition, the asymptotic postbuckling analysis method established by Koiter [7] yields information about the initial postbuckling path. In order to obtain more information about the buckling and specially the postbuckling path of thin-walled structures, FEM is frequently used. The general applicability and >exibility of the @nite element method (FEM) make it a very strong tool
0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 9 0 - 7
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Nomenclature AEM FEM PSM a; b; t Ex ; E y Gxy
xy ; yx Mx ; My ; Mxy Nx ; Ny ; Nxy u; v; w
adjacent equilibrium method @nite element method partitioned solution method length, width and thickness of panel elasticity moduli in length and width directions shear modulus Poisson’s ratios bending and twisting moment resultants normal and shearing stress resultants middle surface displacement components
x ; y ; xy ; x ; panel strain components y ; xy ’ airy stress function principal curvatures in length and kx ; ky width directions unknown coeKcients for function wnm w q uniform load
in development and engineering. FEM is specially useful for analyzing complex structures, with for example spatial dependent material and thickness properties, cut-outs and no-regular boundary conditions, for which semi-analytical methods are not robust enough. However, there are several obvious reasons to study the stability behavior of doubly-curved orthotropic panels using semi-analytical approaches. First, it is very time consuming to obtain stability solutions by FEM. To identify the buckling solution alone, already many simulations have to be carried out. Secondly, it is diKcult (if not impossible) to obtain multi-equilibrium solutions inherited in certain structures by using FEM, such as multiple or clustered bifurcation. Some numerical algorithms for the application of FEM have been developed such as the path following procedure, which is generally constructed by the arc-length scheme [8–18].
Even with the help of those specially developed algorithms, the reliability of the stability solutions obtained by FEM can still be very sensitive to the elements and the analysis parameters. Thirdly, it is believed that semi-analytical solutions may provide more physical insight than the FEM results. Hence, due to the complexity of the non-linear stability behavior, it is desirable to have a semi-analytical method and tool that can be used to provide a reliable and eKcient estimation of the characteristic stability behavior and to verify the FEM results. To obtain more information on the postbuckling behavior of doubly-curved shallow panels, a semi-analytical method, partitioned solution method (PSM) [19], was proposed. The major advantages of the PSM are that the stability solutions are not limited to the initial postbuckling region, and the clustered bifurcation problem can be analyzed. In this paper, stability predictions obtained from both PSM and AEM are presented. Based on PSM, the in>uence of combinations of boundary conditions on the stability behavior of doubly-curved orthotropic panels is also investigated. Notice that although only results of perfect panels are described in this paper, results of panels with general described con@gurations using PSM method are also available, which are under experimental veri@cations. The e(ect of geometric imperfections can be taken into account by the PSM method in several di(erent ways. One of them is to use the expression of the real panel con@guration including imperfections in the derivation of the stability governing equations. Another possibility is to carry out a geometric parameter sensitivity study for the e(ect of geometric imperfections. 2. Fundamental equations Fig. 1 shows the doubly-curved shallow panel subjected to uniform external pressure. The Donnell-type thin shell theory and assumptions associated with doubly-curved shallow panels are used in this investigation. The strain–displacement relations are x = u; x + kx w + 12 (w; x )2 ;
x = − w; xx ;
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the following equilibrium and compatibility equations are obtained, respectively: D1 w; xxxx + D2 w; yyyy − D3 w; xxyy = − ’; yy (kx − w; xx ) − ’; xx (ky − w; yy ) − 2’; xy w; xy − q;
(5)
D4 ’; xxxx + D5 ’; yyyy + D6 ’; xxyy = (w; xy )2 − w; xx w; yy + kx w; yy + ky w; xx ; where Ex t 3 D1 = ; 12B
Fig. 1. Doubly-curved panel.
y = v; y + ky w + 12 (w; y )2 ; xy = u; y + v; x + w; x w; y ;
(1)
where (:); x means @(:)=@x. Taking for the constitutive equations Hooke’s law for orthotropic materials, the stress–strain relationship can be expressed as Ex t Ex t 3 (x + xy y ); Mx = (x + xy y ); Nx = B 12B Ey t Ex Ny = y + xy x ; B Ey Ey t 3 My = 12B
Ex y + xy x ; Ey
Gxy t 3 xy ; (2) 12 where B = 1 − 2xy Ex =Ey . Using the stationary potential energy criterion, the non-linear equilibrium equations become Nxy = Gxy txy ;
Mxy =
Nx; x + Nxy; y = 0; Nxy; x + Ny; y = 0; Mx; xx + 2Mxy; xy + My; yy − kx Nx − ky Ny − q + Nx w; xx + 2Nxy w; xy + Ny w; yy = 0:
(3)
Combining the above equations with an Airy stress function ’ de@ned by Nx = ’; yy ;
Ny = ’; xx ;
Nxy = − ’; xy
Ey t 3 ; 12B
Gxy t 3 Ex ; D2 − Ey 3 1 1 D5 = ; D6 = − 2 xy D4 : Ex t Gxy t
D3 = − xy D1 − xy
y = − w; yy ; xy = − 2w; xy ;
D2 =
(6)
(4)
D4 =
1 ; Ey t
The left-hand side of the equilibrium equation can be de@ned as a linear operator LD (w). Then {sin(n x) sin(m y) with = =a, = =b and n; m = 1; 2; 3; : : : + ∞} is one of the sets of its eigenfunctions. The solution of the equilibrium equation can therefore be projected on the space spanned by the following eigenfuncions that are complete in that space: wnm sin(n x) sin(m y): (7) w(x; y) = − To complete the description of the model of the doubly-curved orthotropic panel, the following four types of boundary conditions are investigated: (1) Simply supported on both sides w = N1 = M1 = 0 on x = 0; a, w = N2 = M2 = 0 on y = 0; b. (2) Hinged on both sides w = M1 = 0 on x = 0; a, w = M2 = 0 on y = 0; b. (3) Hinged on x = 0, a and simply supported on y = 0; b w = M1 = 0 on x = 0; a; w = N2 = M2 = 0 on y = 0; b. (4) Simply supported on x = 0; a and hinged on y = 0; b w = N1 = M1 = 0 on x = 0; a, w = M2 = 0 on y = 0; b.
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In the following, the stability solutions of the doubly-curved shallow orthotropic panels will be developed following both AEM and PSM, using the derived governing equations and boundary conditions. 3. Adjacent equilibrium method (AEM) Prior to buckling, the panel is assumed to be in a primary equilibrium state (w0 ; ’0 ). Near the point of buckling an adjacent equilibrium state (w1 ; ’1 ) is assumed to exist, which is suKciently close to the primary equilibrium state such that w1 and ’1 are in@nitesimal quantities. On account of this, the equations for the solution of the prebuckling state are supplemented with equations governing the buckling problem. Substituting w = w0 + w1 ;
’ = ’0 + ’1
(8)
into the equilibrium and compatibility equations and omitting the quadratic terms, equations governing the prebuckling state and buckling state can be obtained. Prebuckling state: D1 w;0xxxx + D2 w;0yyyy − D3 w;0xxyy = − ’0; yy nkx − ’0; xx ky − q;
(9)
D4 ’0; xxxx + D5 ’0; yyyy + D6 ’0; xxyy = kx w;0yy + ky w;0xx : (10) Buckling state: D1 w;1xxxx + D2 w;1yyyy − D3 w;1xxyy =−
’1; yy kx
−
+ ’1; yy w;0xx
’1; xx ky +
+
+
(11)
D4 ’1; xxxx + D5 ’1; yyyy + D6 ’1; xxyy = kx w;1yy + ky w;1xx + 2w;0xy w;1xy − w;0xx w;1yy − w;0yy w;1xx :
(13)
Consequently, the stress function satisfying the hinged boundary conditions, can be obtained directly by resolving the compatibility equation after substituting the expression for w0 : 0 ’0 = D70 w11 sin( x) sin( y) + 12 px0 y2 + 12 py0 x2 ;
(14)
where ky 2 + k x 2 : D5 4 + D 4 4 + D 6 2 2 Because of the hinged boundary conditions, the average distance between two opposite boundaries is @xed. The terms px0 and py0 can then be obtained with the following equations: 1 b a 0 u d x dy = 0; Qu = b 0 0 ;x 1 a b 0 v dy d x = 0: (15) Qv = a 0 0 ;y D70 =
0 ; px0 = D80 w11
’1; xx w;0yy
− 2’0; xy ; w;1xy − 2’1; xy w;0xy ;
0 sin( x) sin( y): w0 = − w11
Resolving these equations after substituting the linear strain–displacement relations in them yields:
’0; yy w;1xx
’0; xx w;1yy
are not presented here and only the equations for the system with hinged boundary conditions are developed further. This method however, can be extended to cases with other boundary conditions like for example those speci@ed in the previous section, or even, by changing the set of eigenfunctions in Eq. (7) to cover di(erent boundary conditions. The @rst mode pair of the eigenfunction found in the previous section is used to approach the prebuckling state, which is close to the membrane state. Accordingly, the out-of-plane displacement of the prebuckling equations can be expressed as
(12)
Notice that for the sake of simplicity, the governing equations for the initial postbuckling state
0 ; py0 = D90 w11
where C 1 C2 − C3 C4 D80 = ; C4 C6 − C2 C5
(16) D90 =
C 3 C5 − C1 C6 ; C4 C6 − C2 C5
yx 4 1 4 4 + D70 + kx ; Ey t Ex t 1 C2 = ab ; Ey t
C1 = −D70
D.H. van Campen et al. / International Journal of Non-Linear Mechanics 37 (2002) 659–667
C3 = −D70
xy 4 1 4 4 + D70 + ky ; Ey t Ex t
C4 = −ab
yx ; Ey t
1 ; C5 = ab Ex t
compatibility equations yield two governing equations for the fundamental path and two governing equations for the bifurcated path: Fundamental path D1 w;0xxxx + D2 w;0yyyy − D3 w;0xxyy
xy C6 = − ab : Ex t
Finally, the governing equation of the prebuckling state can be obtained by Galerkin procedure as follows: 4 0 0 q = 0; (17) w11 − D10 where 0 D10
4 (D0 kx + D90 ky ): 8
1 DR wR 1 = 0:
(19) 1
The existence of perturbation wR depends on ma1 1 trix DR , i.e. det(DR ) = 0. Notice that the relation 1 det(DR ) = 0 is a linear function of load q and thus can be considered as an eigenvalue problem. The smallest load q is de@ned as the bifurcation buckling load. 4. Partitioned solution method (PSM)
0
’ = ’0 + ’1 : 0
D4 ’0; xxxx + D5 ’0; yyyy + D6 ’0; xxyy = kx w;0yy + ky w;0xx + (w;0xy )2 − w;0xx w;0yy :
(22)
0
+ ’1; yy w;0xx + ’0; xx w;1yy + ’1; xx w;0yy − 2’0; xy w;1xy − 2’1; xy w;0xy − q1 ;
(23)
D4 ’1; xxxx + D5 ’1; yyyy + D6 ’1; xxyy = kx w;1yy + ky w;1xx + (w;1xy )2 − w;1xx w;1yy + 2w;0xy w;1xy − w;0xx w;1yy − w;0yy w;1xx :
(24)
In this case, the governing equations are developed further for systems with simply supported boundary conditions. The @rst mode pair of the eigenfunction found previously is used to approach the fundamental path. Again the stress function satisfying the simply supported boundary conditions can be obtained by resolving the compatibility equation after substituting the expression for w0 : 0 0 2 0 0 2 ’0 = D11 (w11 ) cos(2 x) + D12 (w11 ) cos(2 y)
The solutions of the equilibrium and compatibility equations can be partitioned into two parts: w = w0 + w1 ;
(21)
− ’1; xx (ky − w;1yy ) − 2’1; xy w;1xy + ’0; yy w;1xx
This yields the following solution for bifurcation buckling:
q = q0 + q1 ;
− 2’0; xy w;0xy − q0 ;
D1 w;1xxxx + D2 w;1yyyy − D3 w;1xxyy = − ’1; yy (kx − w;1xx )
With the same procedure and the known prebuckling solution, one can solve the buckling equations with the following assumption for the deformations: 1 wnm sin(n x) sin(m y): (18) w1 (x; y) = −
1 1 D10 wnm = 0 or
= − ’0; yy (kx − w;0xx ) − ’0; xx (ky − w;0yy )
Bifurcated path (postbuckling)
ab = (D1 4 + D2 4 − D3 2 2 + D70 kx 2 4 + D70 ky 2 ) −
663
(20)
With q as the bifurcation load, w ; ’ the fundamental solution at this bifurcation point and w1 ; ’1 the solution on the bifurcated path (postbuckling). Substituting these relations in the equilibrium and
0 0 + D13 w11 sin( x) sin( y);
where 2 ; 32D5 2
0 = D11
; 32D4 2
0 = D13
kx 2 + ky 2 : D5 4 + D 4 4 + D 6 2 2
0 D12 =
(25)
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The governing equation of the fundamental path can thus be obtained as 0 0 3 0 0 2 0 0 (w11 ) + D15 (w11 ) + D16 w11 − D14
4 0 q = 0;
(26)
where 0 D14
2 2 ab 0 0 (D11 + D12 = ); 2
0 =− D15
0 D16 =
8 0 0 0 2 2 (2D12 kx 2 + 2D11 ky 2 + D13 ); 3
ab (D1 4 + D2 4 − D3 2 2 4 0 0 + D13 kx 2 + D13 ky 2 ):
Under the assumption that the interaction between modes can be ignored, the out-of-plane displacement at the bifurcated path can be obtained as 1 sin(n x) sin(m y): w1 = − wnm
(27)
With this assumption and the known solution for the fundamental path, the stress function at the bifurcated path can be obtained by resolving the compatibility equation 1 1 2 1 1 2 (wnm ) cos(2n x) + D12 (wnm ) cos(2m y) ’1 = D11 1 1 wnm sin(n x) sin(m y) + D13 1 1 + D1i cos( i x) cos( i y)wnm ;
where n = 1 and m = 1. Finally, the stability equation governing the bifurcated path can be obtained as 1 1 3 1 1 2 (wnm ) + D15 (wnm ) D14
4 q1 = 0: n m
1 1 → G(wnm ); wnm 1 =− wnm
1 4 D14 D1 1 3 1 2 (wnm ) − 15 (wnm ) + q1 ; 1 1 1 D16 D16 n m D16 (30)
1 where D16 = 0: Let wR 1nm be a @xed point of the point mapping, then the following equation can be used to determine the bifurcation load q0 leading to the saddle-node bifurcation on the bifurcated path: D1 D1 1 DG 1 wR nm = − 3 14 (wR 1nm )2 − 2 15 wR nm = 1: #= 1 1 1 Dwnm D16 D16 (31)
The in>uence of the fundamental solution at the bifurcation load on the bifurcated solution is in1 1 and D16 . At the saddle-node cluded in the terms D15 bifurcation, there exist two points that coincide with each other. The necessary condition for saddle-node bifurcation is therefore 1 2 1 D14 D15 − 12 = 0: (32) $=4 1 1 D16 D16 Because of the non-linearity of this equation, it needs an iterative algorithm in combination with the equations governing the fundamental path to approach the solution. The limit load is de@ned as the load where the saddle-node bifurcation will occur on the @rst bifurcated branch (postbuckled).
(28)
i=4;5;6;7
1 1 + D16 wnm −
following form:
(29)
Snap-through buckling will occur along this new branch and the limit load is the load at which a saddle-node bifurcation occurs. To analyze the stability of the equilibrium positions at this postbuckled branch, a point mapping is constructed. The bifurcation equation is therefore rewritten in the
5. Numerical results In the following, some stability analysis results are presented. For a few arbitrary sets of system parameters, the results of AEM and PSM are compared with each other. The in>uences of the boundary conditions on both the buckling and postbuckling solutions are investigated using PSM. 5.1. Comparison between AEM and PSM Both AEM and PSM are used to analyze the buckling behavior of doubly-curved orthotropic shallow panels, described by the parameters in Table 1.
D.H. van Campen et al. / International Journal of Non-Linear Mechanics 37 (2002) 659–667
665
Table 1 System parameters A B C D E
Ex (N=m2 )
Ey (N=m2 )
Gxy (N=m2 )
xy
kx
ky
a (m)
b (m)
3e10
3e10
1:154e10 1e10 1e10 1:154e10 1:154e10
0.3 0.1 0.1 0.3 0.3
0.1375 0.2 0.3 0.15 0.3
0.1375 0.3 0.4 0.25 0.2
0.2 0.2 0.4 0.2 0.2
0.2 0.2 0.3 0.5 0.2
2e10 2e10 3e10 3e10
4e10 4e10 3e10 3e10
Table 2 Results from AEM and PSM AEM=PSM
Bifurcation load (N=m2 )
Buckling load at the postbuckled branch (N=m2 )
Mode (x; y)
A B C D E
31.922=25.143 76.706=57.860 139.95=134.96 50.593=47.351 88.836=73.651
..=25.678 ..=61.179 ..=137.09 ..=47.830 ..=77.755
(1; 2)=(1; 3) (1; 3)=(1; 3) (1; 6)=(1; 5) (1; 8)=(1; 7) (4; 1)=(3; 1)
Table 3 System parameters Ex (N=m2 )
Ey (N=m2 )
Gxy (N=m2 )
xy
kx
ky
a (m)
b (m)
t (m)
3e10
3e10
1:154e10
0.3
0.25
0.5
0.6
0.34
0.00022
Notice that the thickness is the same for all cases (t = 0:00022 m) and that for some cases the material properties are isotropic. With AEM the bifurcation buckling load and the corresponding mode are obtained, while with PSM the bifurcation load and the corresponding modes, the limit load at the postbuckled branch are obtained. The results are shown in Table 2. As can be seen in Table 2, the bifurcation buckling results obtained from both methods are quite comparable, and the trends in bifurcation buckling loads are similar. The di(erences between the absolute values of the buckling loads and the corresponding buckling modes are caused by di(erences in the prebuckling states of both methods. The buckling loads obtained by following the postbuckled branches are a little higher. 5.2. In:uence of boundary conditions The PSM is used to analyze the in>uence of the boundary conditions on the stability behavior of the
doubly-curved shallow isotropic panel. The arbitrary system parameters are given in Table 3. With these system parameters four calculations are performed with various boundary conditions: Case 1: All edges hinged. Case 2: All edges simply supported. Case 3: Simply supported on the short and hinged on the long edges. Case 4: Hinged on the short and simply supported on the long edges. The results of these calculations are shown in Table 4 and for the @rst case, the response of the center of the panel is depicted in Fig. 2. As can be seen from Table 4, the boundary conditions have signi@cant in>uence on both the buckling loads and modes. As could be expected, the boundary conditions on the longer edges are dominant in the results due to the smaller distance between these edges. The lower buckling loads for cases with hinged boundary conditions are caused by prebuckling compression. In Fig. 2, the stable prebuckling
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D.H. van Campen et al. / International Journal of Non-Linear Mechanics 37 (2002) 659–667
Table 4 Results for various boundary conditions Case
Bifurcation load (N=m2 )
Buckling load at postbuckled branch (N=m2 )
Mode (x; y)
1 2 3 4
157.43 187.70 143.92 181.06
158.79 187.71 144.98 181.07
(1; 7) (16; 1) (1; 7) (16; 1)
tial postbuckling path. It is found that the buckling results obtained from both methods are comparable. Di(erences between absolute values of the buckling loads and modes are caused by the di(erences in the assumed prebuckling states. What is more, the PSM method can give stability predictions at the postbuckled branch. In addition, it is found that the in>uence of boundary conditions on both the buckling and postbuckling solutions can be significant. Hinged boundary conditions result in lower buckling loads due to prebuckling compression. Fig. 2. Load-displacement diagram for Case 1.
path with the out-of-plane displacement described by the @rst mode pair of the eigenfunctions, becomes unstable when it is intersected by the secondary equilibrium path. Along this path, with the out-of-plane displacement described by the bifurcation buckling mode, the limit load is found to be slightly higher than the bifurcation buckling load. 6. Conclusions In order to investigate the stability behavior (both buckling and postbuckling) of doubly-curved shallow orthotropic panels under external pressure, the PSM is further developed and presented in this paper. This method can be used to obtain both the bifurcation buckling and postbuckling solutions that are not con@ned in the initial postbuckling region. For some arbitrary sets of system parameters, the results of the PSM are compared with those obtained from the AEM that can only obtain the bifurcation buckling solutions and information of the ini-
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